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<|MaskedSetence|> <|MaskedSetence|> It shows that the operator Op(a)Op𝑎\text{Op}(a)Op ( italic_a ) belongs to the ∗*∗-algebra C∞(AdU)superscript𝐶Ad𝑈C^{\infty}(\text{Ad}\,U)italic_C start_POSTSUPERSCRIPT ∞ end_POSTSUPERSCRIPT ( Ad italic_U ) of smooth elements for the representation AdUAd𝑈\text{Ad}\,UAd italic... | **A**:
h∈ℝℎℝh\in\mathbb{R}italic_h ∈ blackboard_R, 1⩽k⩽n1𝑘𝑛1\leqslant k\leqslant n1 ⩽ italic_k ⩽ italic_n, combined with the estimate (3.23), give (3.22) in the case |α|+|β|=1𝛼𝛽1|\alpha|+|\beta|=1| italic_α | + | italic_β | = 1.
**B**: The equality for general α,β∈ℕn𝛼𝛽superscriptℕ𝑛\alpha,\beta\in\mathbb{N}^{n}... | ABC | ABC | ABC | ABC | Selection 4 |
<|MaskedSetence|> Organization
This paper is organized as follows. In Section 2 we fix notations and present some results in commutative algebra that are needed in the paper. In Section 3 we give the nonsingularity conditions which guarantee that (1.3) holds for some d𝑑ditalic_d.
In Section 4 we establish the effect... | **A**: In appendix we present some deferred proofs..
**B**: In Section 7 we conclude and provide some perspectives on future research.
**C**:
1.4.
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<|MaskedSetence|> Thus it is natural to define the relative class of a given tropical disc to be that of the holomorphic disc which intersects the toric divisors in accordance with unbounded edges of the tropical disc. Throughout, we will frequently use the terms such as relative classes, symplectic areas, Maslov indi... | **A**: On the other hand, the intersection patterns of a holomorphic disc with toric divisors completely determine its topological type, and in particular its symplectic area and Maslov index.
**B**: Namely, we fix a set of generic points in ℝ2superscriptℝ2\mathbb{R}^{2}blackboard_R start_POSTSUPERSCRIPT 2 end_POSTSUP... | ABC | ABC | ABC | CAB | Selection 3 |
<|MaskedSetence|> Botnan et al. <|MaskedSetence|> <|MaskedSetence|> Asashiba et al. provided a criterion for determining whether or not a given multiparameter persistence module is interval decomposable without having to explicitly compute indecomposable decompositions
[1].
Dey and Xin proposed an efficient algorith... | **A**: introduced notions of signed barcode and rank decomposition for encoding the rank invariant of multiparameter persistence modules as a linear combination of rank invariants of indicator modules [10].
**B**: In their paper, Möbius inversion was utilized for computing the rank decomposition, characterizing the ge... | CAB | BAC | CAB | CAB | Selection 3 |
Organization
The rest of the paper is organized as follows. <|MaskedSetence|> <|MaskedSetence|> In Section 4, we show how to choose this sum of squares so that (P1), (P3) are satisfied and finish the proof of Theorem 3 and Theorem 4. In Section 5, we extend our proof technique to the upper bounds to obtain Theorem... | **A**: In Section 2, we introduce some notations and cover the necessary preliminaries on orthogonal (Gegenbauer) polynomials.
**B**: Finally, we give a proof of some technical statements in Appendix A..
**C**: In Section 3, we present closed form expressions of the Christoffel-Darboux kernel and use them to obtain k... | CAB | ACB | ACB | ACB | Selection 2 |
Remark \the\thmnumbering.
It was claimed in [GK, Example 8.15] that the canonical metric on any line bundle of an abelian variety is locally the tensor product of a smooth and a formal metric. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> Therefore, it remains an open question whether canonical metrics ar... | **A**: However, this was done under the incorrect assumption that canonical tropicalization maps of abelian varieties are smooth.
**B**: The latter means that locally there is a smooth tropicalization map such that the first Chern current is induced by a (1,1)11(1,1)( 1 , 1 )-form with piecewise smooth coefficients on... | CAB | CBA | CBA | CBA | Selection 2 |
The above theorem leaves us with an intriguing question for harmonic maps with a higher degree. <|MaskedSetence|> There we have shown that similar quantitative stability for higher degree ones is only true in the local sense. <|MaskedSetence|> More precisely, given a harmonic map (or a compact set of harmonic maps),... | **A**: We have addressed a similar question for half-harmonic maps in [9].
**B**: In this paper, we shall prove that a similar phenomenon happens here.
**C**: The bound in general will depend on the given harmonic map (or the compact set).
To that end, let us introduce some notations..
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In particular, real nilpotent Lie algebras with an ad-invariant metric are classified up to dimension 10101010 in [17]. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> Lie algebras with a nice basis, called nice Lie algebras in this paper, possess a strong algebraic structure, which can be encoded in a specia... | **A**: Nice bases are mostly studied in the context of Einstein Riemannian metrics on solvmanifolds.
**B**: In [6], we showed that the irreducible Lie algebras appearing in this classification admit a unique ad-invariant metric (up to sign); as a step in the proof, we proved that all of them admit a nice basis.
**C**... | ABC | BAC | BAC | BAC | Selection 2 |
<|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> In contrast, the Brownian transport map allows us to prove estimates on the Lipschitz constant of the transport map in expectation, which is what is needed to make the connection with the Kannan–Lovász–Simonovits conjecture; cf. Theorem 1.4. (We remark however th... | **A**:
(b) The second way to apply the Hessian estimate is to use it within the context of the heat flow transport map of Kim and Milman [37].
**B**: This approach avoids the issues mentioned in part (a).
**C**: On the other hand, the usage of this transport map is only suitable if we want to prove pointwise estima... | BAC | ABC | ABC | ABC | Selection 4 |
<|MaskedSetence|> <|MaskedSetence|> This result was extended to the subanalytic category by A. <|MaskedSetence|> The author gives in [31] a construction of locally bi-Lipschitz trivial stratifications for definable sets in polynomially bounded o-minimal structures (expanding ℝℝ\mathbb{R}blackboard_R) for which the l... | **A**: Parusiński [22], to polynomially bounded o-minimal structures expanding ℝℝ\mathbb{R}blackboard_R in [21], and to polynomially bounded o-minimal structures expanding an arbitrary real closed field in [15].
**B**:
T.
**C**: Mostowski [20, Proposition 1.21.21.21.2] proved that every complex analytic set admits a... | BCA | BCA | BCA | BCA | Selection 2 |
<|MaskedSetence|> Previous results on proper permutations
Proper permutations were first introduced in [BHY20]. We highlight that our definition differs slightly from the original definition. The original definition of properness was motivated by the study of Levi-spherical Schubert varieties in GLn/B𝐺subscript𝐿𝑛... | **A**:
1.3.
**B**: This updated definition is considerably more natural in the general type setting.
.
**C**: To study Levi-spherical Schubert varieties in G/B𝐺𝐵G/Bitalic_G / italic_B, for G𝐺Gitalic_G a simple Lie group, requires a definition of properness that corresponds to Levi-spherical Schubert varieties in ... | BAC | ACB | ACB | ACB | Selection 3 |
<|MaskedSetence|> Ref. [16] showed that space-time block coding (STBC) with single polarization outperforms STBC with dual polarization in Rayleigh and Ricean fading channels. <|MaskedSetence|> It is noteworthy that the extent of benefit from dual-polarized antennas depends on the associated schemes to exploit the ch... | **A**: A MIMO system with dual-polarized antenna elements can have lower spatial diversity but higher spatial multiplexing gain than a conventional MIMO system with single-polarized antennas, particularly, in Ricean fading channels with high K𝐾Kitalic_K-factor [17].
**B**: Various channel sounding campaigns and chann... | ACB | CAB | CAB | CAB | Selection 4 |
Several other results address the non-existence of a second bound state for a Schrödinger equation in one dimension, and they could potentially be applicable here. For instance, Calogero [14] established various results of the kind.
One well-known result by Calogero requires the monotonicity of the potential function, ... | **A**: See, for instance, [46].
**B**: While any of these results could be useful here, the choice of [37] (or [5]) is made because it is analytically simpler and has an interesting physical interpretation as the distance to a piece-wise constant shock.
.
**C**: Nevertheless, such a Calogero bound will play a signi... | CAB | BAC | CAB | CAB | Selection 1 |
In particular, 𝒫0=:𝒫\mathcal{P}_{0}=:\mathcal{P}caligraphic_P start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = : caligraphic_P, the well known Carathéodory class. <|MaskedSetence|> Further, let 𝒮𝒮\mathcal{S}caligraphic_S be a subclass of 𝒜1subscript𝒜1\mathcal{A}_{1}caligraphic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ... | **A**: Suppose 𝒜psubscript𝒜𝑝\mathcal{A}_{p}caligraphic_A start_POSTSUBSCRIPT italic_p end_POSTSUBSCRIPT be the subclass of ℋ[0,p]ℋ0𝑝\mathcal{H}[0,p]caligraphic_H [ 0 , italic_p ] consisting of functions normalized by f(p)(0)=1.superscript𝑓𝑝01f^{(p)}(0)=1.italic_f start_POSTSUPERSCRIPT ( italic_p ) end_POSTSUPER... | ACB | ACB | ACB | BAC | Selection 3 |
<|MaskedSetence|> In Section 2 we start with the definition of a standard set, along with some basic definitions and properties on the set of monomials in a polynomial ring. <|MaskedSetence|> <|MaskedSetence|> In Section 4 we use strongly Lech-independence to analyze the colength of powers of ideals and derive inequ... | **A**: In Section 3 we define strongly Lech-independence and expansion property and prove some equivalent conditions.
**B**: The paper is organized in the following way.
**C**: There are also some examples showing the relation between strongly Lech-independence and other notions.
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We can reverse the procedure to construct an arbitrary meander from prime ones.
Unfortunately, there are meanders that can be constructed in different ways. For instance, consider a direct snake of order (3,0)30(3,0)( 3 , 0 ). <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> For this purpose, we need to introdu... | **A**: That is, if we consider only the sequences of insertions in which no direct snake is inserted into another snake (neither direct nor inverse), each meander can be constructed in a unique way.
It is convenient to consider another way of constructing meanders.
**B**: In fact, the insertion of a direct snake int... | CBA | BCA | CBA | CBA | Selection 4 |
In the present paper, we examine absorbing random walks on graphs in which different nodes can have different absorption rates, inducing an “effective” network structure that is reflected only partially by the edge weights of a network. Many notions of network community structure arise from the analysis of random walk... | **A**: In our adaptation, we apply InfoMap to absorption-scaled graphs, which account for absorption by scaling the edge weights of a network [16].
**B**: There is intense interest in understanding how community structure and node characteristics combine to influence contagions on networks [24, 34, 37].
We develop c... | CAB | BCA | BCA | BCA | Selection 2 |
<|MaskedSetence|> Organization of the paper
In Section 2, we include the definitions of adjacency matrices of hypergraphs. <|MaskedSetence|> <|MaskedSetence|> The proof for the correctness of our algorithms for Theorem 1.7 and Corollary 1.9 are given in Section 5. The proof of Theorem 1.6, as well as the proofs of ... | **A**: The concentration results for the adjacency matrices are provided in Section 3.
**B**: The algorithms for partial recovery are presented in Section 4.
**C**:
1.4.
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A major difference between the models of adaptive walks/flights and adaptive dynamics is that the former assume a fitness landscape that is random but fixed in time, while in the latter case the fitness landscape is dynamic and depends on the current resident traits. As mentioned before, the notion of local fitness max... | **A**: Moreover, if equal competition between all traits is assumed in the adaptive dynamics model, the fitness landscape can again be regarded as fixed.
**B**: Chapter 3 is devoted to the discussion of a number of examples that highlight different aspects of the complicated limiting dynamics in an easy set up.
**C**... | ABC | ABC | ABC | ABC | Selection 4 |
<|MaskedSetence|> <|MaskedSetence|> ℒ≥n(X)subscriptℒabsent𝑛𝑋\mathcal{L}_{\geq n}(X)caligraphic_L start_POSTSUBSCRIPT ≥ italic_n end_POSTSUBSCRIPT ( italic_X )) denote the set of words of length n𝑛nitalic_n
(resp. <|MaskedSetence|> We also let ℒ(x)ℒ𝑥\mathcal{L}(x)caligraphic_L ( italic_x ) denote the set
of fac... | **A**: ≥nabsent𝑛\geq n≥ italic_n) in ℒ(X)ℒ𝑋\mathcal{L}(X)caligraphic_L ( italic_X ).
**B**: set of all factors of the sequences x∈X𝑥𝑋x\in Xitalic_x ∈ italic_X.
**C**: We let
ℒn(X)subscriptℒ𝑛𝑋\mathcal{L}_{n}(X)caligraphic_L start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ( italic_X ) (resp.
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Environmental fluctuations have emerged as a significant factor in the study of diseases, particularly in the context of the coronavirus. Consequently, it becomes crucial to investigate the impact of random disturbances on epidemic models. Disease spread is inherently stochastic, and the introduction of stochastic noi... | **A**: Our objective is to derive the basic reproduction number, a critical determinant governing the extinction or persistence of the disease (infection).
**B**: The reproduction number represents the average number of secondary infections produced by a single infected individual in a susceptible population, making i... | ABC | ABC | CBA | ABC | Selection 4 |
<|MaskedSetence|> <|MaskedSetence|> The first step computes the cluster mean. The second step minimizes the within-cluster distance. Just like k𝑘kitalic_k-means clustering, the two-step optimization is then iterated until convergence. <|MaskedSetence|> | **A**: If μ𝜇\muitalic_μ is given and fixed, the identification of clusters C𝐶Citalic_C can be done easily by assigning each network to the closest mean.
**B**: Thus the topological clustering algorithm can be written as the two-step optimization similar to the expectation maximization (EM) algorithm often used in va... | ABC | ABC | ABC | CBA | Selection 1 |
Next, we present a test case to illustrate the performance of the proposed approach under disturbance. We consider a scenario where an adversary injects a cyberattack in the form of a disturbance to the battery module to induce overdischarge. The disturbance is injected at 700s700𝑠700s700 italic_s as current drain fr... | **A**: 4.
Figure 3: State-of-Charge (SOC) of the battery module under nominal condition and under disturbance through battery overdischarge.
.
**B**: It can be seen that the disturbance was initiated around 700s700𝑠700s700 italic_s, and consequently, after 1098s1098𝑠1098s1098 italic_s the modules goes into the o... | CAB | CBA | CBA | CBA | Selection 2 |
Problems involving the optimization of pathways are critical for a number of applications across different fields. The general problem stemming from optimizing collision-free paths for robots in the realm of numerical analysis is studied in [1]. In graph theory, Dijkstra’s algorithm is used to look at the shortest path... | **A**: With this goal in mind, we will define new net substructures, and using them identify where these shortest paths can lie as well as the lengths of these paths (for any two given points).
.
**B**: The same type of problem relating to robotics and motion planning is explored by using sequence trees in [3].
**C**... | BCA | BCA | BCA | BCA | Selection 3 |
<|MaskedSetence|> Influence of spatial correlations has been investigated in other contexts such as localization of directed polymers with space-time noise [31], Brownian motion in correlated Poisson potential [32, 33, 39] as well as Anderson localization, both by mathematicians [26, 27, 34] and physicists [2, 14, 46]... | **A**: Those are related to the heavy tails of the gap distribution.
**B**:
Since the obstacles are drawn according to a renewal process, our work provides an exemple of localization in a correlated disordered environment.
**C**: Influence of heavy tails in localization phenomena has been considered in the context... | BCA | BAC | BAC | BAC | Selection 3 |
<|MaskedSetence|> See, e.g. <|MaskedSetence|> In particular, the Faber-Krahn inequality [12, 17, 18] states that the (Lebesgue) principal frequency of a domain K𝐾Kitalic_K of a fixed Lebesgue measure is minimized when K𝐾Kitalic_K is an Euclidean ball. <|MaskedSetence|> [22]) states that, conversely, the torsional ... | **A**: The result of Saint-Venant (see e.g.
**B**:
In the case when μ𝜇\muitalic_μ is the Lebesgue measure and L=Δ,𝐿ΔL=\Delta,italic_L = roman_Δ , these quantities have been studied extensively, and are intimately tied with the subject of isoperimetric inequalities.
**C**: Kawohl [13], Pólya and Szegö [22],
Burcha... | BCA | BCA | CAB | BCA | Selection 1 |
<|MaskedSetence|> In the subsequent part, we present a synthesis scheme to allocate excitation and measurement signals for identifiability of an acyclic network model set. <|MaskedSetence|> In [6], all the vertices in a directed network are measured, and the allocation of excitations is based on decomposing the under... | **A**: In this work, we extend the work in [6] to the partial excitation and measurement case.
**B**: [16] for the definition.
**C**: This will lead to alternative and more compact identifiability conditions.
To this end, we resort to a graph covering approach as introduced in [6] for general (cyclic) dynamic netwo... | BCA | BCA | BCA | BAC | Selection 2 |
<|MaskedSetence|> Section 2 introduces the necessary background of differential privacy and presents the estimation of the noisy bi-degree sequence based on the moment equation, and obtains unified asymptotic properties for the differentially private estimation as the number of nodes goes to infinity. Section 3 applie... | **A**: Some further discussion is given in Section 5.
**B**:
The rest of this article is organized as follows.
**C**: Section 4 carries out the simulations under the Probit model and a real data analysis.
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<|MaskedSetence|> <|MaskedSetence|> This group is called the special Euclidean group and is denoted by SE(2)𝑆𝐸2SE(2)italic_S italic_E ( 2 ). In many applications, the congruence with respect to other groups is considered. <|MaskedSetence|> If a light source can be considered to be infinitely far away (like a sun... | **A**: For example, two shadows cast by the same object onto two different planes by blocking the rays of light emitted from a lamp are related by a projective transformation.
**B**: However, since a reflection changes the orientation of an object, a group of orientation-preserving rigid motions, consisting of rotatio... | CBA | BCA | CBA | CBA | Selection 1 |
A substantial review of variants of coordinate descent algorithms can be found in [4, Section 6.5.1]. The cyclic selection of coordinates is normally assumed to ensure convergence of the algorithm. <|MaskedSetence|> <|MaskedSetence|> Obviously, this is not guaranteed for each instance of the algorithm. <|MaskedSet... | **A**: Particularly, it is shown in [31] that randomization leads to faster convergence in terms of expectation.
**B**: On the other hand, the use of an irregular order is then considered by researchers to accelerate convergence.
**C**: The Gauss-Southwell method leads to faster convergence at the cost of extra comp... | BAC | CBA | BAC | BAC | Selection 1 |
<|MaskedSetence|> R. Graham and J. M. <|MaskedSetence|> Biquard [6].
Similar results, yet unpublished, has been obtained simultaneously by J. <|MaskedSetence|> | **A**: Lee [19, 23].
The general case of asymptotically symmetric metrics have been covered by O.
**B**: Roth in his Ph.D thesis for the complex hyperbolic case [25].
.
**C**: They all admit a sphere at infinity, which is endowed with a particular geometric structure closely related to the Riemannian metric of these ... | CAB | CAB | CAB | CAB | Selection 3 |
<|MaskedSetence|> <|MaskedSetence|> We start with a slight reorganization and minor refinement of ideas introduced by Lempert and Szőke in [LS-2014], and then we introduce definitions that are already suggested or implied in [LS-2014]. For example, in [LS-2014] the introduction of a ∂¯¯\bar{\partial}over¯ start_ARG ∂... | **A**: In fact, [B-2009] considers subfields in the case of trivial families of complex manifolds..
**B**:
2.
**C**: Abstract Theory of Berndtsson-Lempert-Szőke Fields
In this section we discuss the general geometric structure with which ℋℋ{\mathscr{H}}script_H and ℒℒ{\mathscr{L}}script_L are eventually endowed.
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<|MaskedSetence|> In the next Section we formulate main results of the paper. In Section 2 we set up the notation. Section 3 contains all complete proofs, for instance, in §3.1 we have considered an important special case — a level computation for exterior squares of elementary groups. <|MaskedSetence|> <|MaskedSete... | **A**: Finally, a level reduction for exterior powers is proved in §3.5.
Acknowledgment.
**B**: In §3.2–§3.4 we develop a technique for an arbitrary general exterior power.
**C**: The present paper is organized as follows.
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<|MaskedSetence|> Outline of the paper
In section 2 we recall some basic facts of T-varieties and the combinatorial framework to describe them. In particular, we recall the definitions of p-divisors and of divisorial fans. In Section 3 we restrict to the complexity one case and we describe the invariant subvarieties ... | **A**: 1.1.
**B**: Then we recall the classification of T-invariant Cartier divisors in terms of Cartier divisorial support functions, following the work in [AIPSV] and [PS].
.
**C**: It turns out that these invariant subvarieties generate the pseudoeffective cone of the T-variety and we give a list of its generator... | ACB | ACB | ABC | ACB | Selection 4 |
Note that this discrepancy between Bayesian and frequentist measures differs considerably from the situation in standard statistical inference with non-adaptive samples. For a fixed-sample problem, the Bernstein–von Mises theorem describes the asymptotic equivalence of Bayesian and frequentist inference. However, in a... | **A**: We demonstrate several instances in which it is feasible to perform exact analyses of dynamic programming regardless of the need to project the evolution of the posterior over extended future periods.
**B**: Bayesian algorithms are robust up to a polynomially small underestimation, whereas frequentist algorithm... | BCA | CBA | CBA | CBA | Selection 4 |
As well as the results of Čulík [6], Guy [10] and Roman [19] already discussed, there has been some more recent work that determines exact values for Zarankiewicz numbers. Goddard, Henning and Oellermann [9] and Collins, Riasanovsky, Wallace and Radziszowski [4] determined exact values for various small parameters. Da... | **A**: Section 4 is devoted to the cases where ⌊Aλk(m,n)⌋subscriptsuperscript𝐴𝑘𝜆𝑚𝑛\lfloor A^{k}_{\lambda}(m,n)\rfloor⌊ italic_A start_POSTSUPERSCRIPT italic_k end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_λ end_POSTSUBSCRIPT ( italic_m , italic_n ) ⌋ is achieved, while Section 5 is devoted to the cases where ⌊Bλ... | BAC | BAC | BAC | CAB | Selection 3 |
2}-1}^{\prime}lk ( italic_p , italic_M ) = italic_P ( italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_p start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , … , italic_p start_POSTSUBSCRIPT italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT , italic_x , italic_q , italic_y , italic_p start_POSTSUBSCRIPT ... | **A**: Here, x,y,pi,pj′,qk𝑥𝑦subscript𝑝𝑖superscriptsubscript𝑝𝑗′subscript𝑞𝑘x,y,p_{i},p_{j}^{\prime},q_{k}italic_x , italic_y , italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , italic_p start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ′ end_POSTSUPERSCRIPT , italic_q start_POSTSUBSCRIP... | CAB | ACB | CAB | CAB | Selection 1 |
In this section we equip an S𝑆Sitalic_S-adic shift with an invariant measure and study the existence of measurable eigenvalues. <|MaskedSetence|> In particular
we state results from [BKMS13], [Men91] and [CDHM03] which are couched in terms of Bratteli diagrams, or cutting-and-stacking transformations, but which we re... | **A**: The combination of strong primitivity and finitary ensures that our systems are of exact finite rank,
a condition which has been extensively studied in the literature [Men91, Bos92], and one which is useful in the proof of Theorem 5.7.
.
**B**: For this, Theorem 2.7 is particularly useful, as, with the conditi... | ACB | BCA | BCA | BCA | Selection 3 |
<|MaskedSetence|> Because with the Steiner system there is an association between pilot sequences and user IDs, observing a certain pilot sequence in a given slot automatically indicates which user is active and where to look for its remaining replicas. <|MaskedSetence|> One possibility is to look for the correlation... | **A**: This is not the case when Random selection scheme is used so additional procedures might be required.
**B**:
Another caveat is that, clearly, the receiver must know where each replica of each user is located in order to perform combining through MRC.
**C**: Furthermore, it entails exhaustive search and, hence... | BAC | BAC | BCA | BAC | Selection 4 |
<|MaskedSetence|> In the course of proving Theorem 6.8 we will prove a bundle construction-type theorem, Theorem 3.15, which gives an equivalence between a category of plotwise defined cocycles on X𝑋Xitalic_X and the category of diffeological principal G𝐺Gitalic_G-bundles over X𝑋Xitalic_X. This result may be of ind... | **A**: Furthermore, higher topos theory already has definitions for higher principal bundles (called bundle gerbes) and connections on such objects inherently built into it.
**B**:
Thus ∞\infty∞-stack cohomology of diffeological spaces also encompasses nonabelian cohomology.
**C**: In particular we believe that whil... | BCA | CBA | BCA | BCA | Selection 1 |
<|MaskedSetence|> <|MaskedSetence|> Then it follows from Lemma 2.12 that there exists a holomorphic function f~jsubscript~𝑓𝑗\tilde{f}_{j}over~ start_ARG italic_f end_ARG start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT on {Ψ1<0}∩DjsubscriptΨ10subscript𝐷𝑗\{\Psi_{1}<0\}\cap D_{j}{ roman_Ψ start_POSTSUBSCRIPT 1 end_PO... | **A**: holds for any j∈ℤ>0𝑗subscriptℤabsent0j\in\mathbb{Z}_{>0}italic_j ∈ blackboard_Z start_POSTSUBSCRIPT > 0 end_POSTSUBSCRIPT and r1∈(0,1]subscript𝑟101r_{1}\in(0,1]italic_r start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∈ ( 0 , 1 ].
Without loss of generality, assume that there exists r0∈(0,1]subscript𝑟001r_{0}\in(0,1]i... | ACB | ACB | ACB | ABC | Selection 2 |
<|MaskedSetence|> If p≡7(mod8)𝑝annotated7pmod8p\equiv 7\pmod{8}italic_p ≡ 7 start_MODIFIER ( roman_mod start_ARG 8 end_ARG ) end_MODIFIER, there is only one down 2222-isogeny from E1subscript𝐸1E_{1}italic_E start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT. If E1subscript𝐸1E_{1}italic_E start_POSTSUBSCRIPT 1 end_POSTSUBSCRIP... | **A**: Moreover, each form corresponds to one of {E2,i}i∈{1,2,3}subscriptsubscript𝐸2𝑖𝑖123\{E_{2,i}\}_{i\in\{1,2,3\}}{ italic_E start_POSTSUBSCRIPT 2 , italic_i end_POSTSUBSCRIPT } start_POSTSUBSCRIPT italic_i ∈ { 1 , 2 , 3 } end_POSTSUBSCRIPT.
**B**:
In fact, the curve E1subscript𝐸1E_{1}italic_E start_POSTSUBSCRI... | BAC | BAC | ABC | BAC | Selection 4 |
Biological sciences are the field where SSPs have been most investigated over the past three decades, raising several challenges in both methods and applications. <|MaskedSetence|> [2019], which shows how large scale genomic data provide a fertile ground for SSPs. Although sequencing technologies have advanced the und... | **A**: This is testified by the work of Deng et al.
**B**: Deng et al.
**C**: Due to the impossibility of sequencing DNA libraries up to complete saturation, it is common to make use of the observed samples, typically collected under suitable budget constraints, to infer the molecular heterogeneity of additional unob... | ACB | CAB | ACB | ACB | Selection 3 |
<|MaskedSetence|> Its long intervals terminate at 8.58.58.58.5 and start from 10.510.510.510.5. <|MaskedSetence|> <|MaskedSetence|> It also contains a Red join gadget in [1.5,4.5]1.54.5[1.5,4.5][ 1.5 , 4.5 ]. Its long intervals terminate at 1.51.51.51.5 and start from 4.54.54.54.5.
. | **A**:
The fifth buffer contains the second part of the second switch gadget in [8.5,11.5]8.511.5[8.5,11.5][ 8.5 , 11.5 ].
**B**: Its long intervals terminate at {12,15.5}1215.5\{12,15.5\}{ 12 , 15.5 } and start from {17.5,21}17.521\{17.5,21\}{ 17.5 , 21 }.
**C**: It also contains the third switch gadget in [12,21]1... | ACB | ACB | BCA | ACB | Selection 1 |
<|MaskedSetence|> There are several topological complications to be addressed, but the outcome is relatively satisfactory: the invariants are well-defined and finite in many interesting cases and interrelate to shed a light on the structure theory of certain natural classes of topological groups.
In Section 2 we int... | **A**: A small final epilogue in Section 5 is devoted to finiteness of the asymptotic dimension of totally disconnected, locally compact, second countable groups..
**B**:
Here instead of finitely generated discrete groups one has to work with compactly generated locally compact groups (usually second countable, to ma... | BCA | BCA | BCA | BCA | Selection 4 |
<|MaskedSetence|> Our notion relies on [Ken92] notion of bifilteredness, together with a convenient notion of bicompact objects (Definition 3.1.1) enjoying the analogous property of compact objects against bifiltered bicolimits; we define finitely bi-accessible categories as those having bifiltered bicolimits and an e... | **A**: We then prove that categories of flat pseudofunctors are bi-accessible (Corollary 4.2.6) and bipresentable (Theorem 4.3.6) if their domain admits finite weighted bilimits, the latter result being part of a categorification of the well known Gabriel-Ulmer duality established in Section 5.3.
**B**: To fix this, w... | BAC | BAC | BAC | BCA | Selection 2 |
Proof.
Let (V1,V2,V3,V4)subscript𝑉1subscript𝑉2subscript𝑉3subscript𝑉4(V_{1},V_{2},V_{3},V_{4})( italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_V start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_V start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , italic_V start_POSTSUBSCRIPT 4 end_POSTSUBSCRIPT ) be an ℳ𝒢𝒫ℳ�... | **A**: If u∈V1𝑢subscript𝑉1u\in V_{1}italic_u ∈ italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, then by Condition 1 of Definition 5 there are no further vertices in V1∪V3subscript𝑉1subscript𝑉3V_{1}\cup V_{3}italic_V start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ∪ italic_V start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT.
**B**:... | BAC | ACB | BAC | BAC | Selection 4 |
Our approach to formulating risk-averse MDPs is grounded in the understanding that law-invariant convex risk measures can be interpreted as functionals defined on the space of probabilities over ℝℝ\mathbb{R}blackboard_R. This perspective has been effectively employed in various contexts to further the development of r... | **A**: This, in turn, allows for greater flexibility in risk-averse modeling.
**B**: [61, 1, 26, 32]).
**C**: Moreover, it seamlessly integrates latent costs and random actions through the concept of regular conditional distributions.
| BCA | ACB | BCA | BCA | Selection 3 |
We note that the junta approximation results of [10] and [13], as well as the results for sets from [33] and [12] came together with stability results that characterized the approximate structure of families that are close to extremal. <|MaskedSetence|> <|MaskedSetence|> And, of course, this opens the way to explorin... | **A**: Our general results in Section 3 and 4 allow for similar results in a sparse setting that have no analogues via the junta method.
**B**: We show the following result that answers a question of Narayanan.
.
**C**: Our approach also gives such structural results, moreover, it allows finding structure in families... | CAB | CAB | CAB | CAB | Selection 2 |
<|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> Traveling along the curve beginning at the base point in the direction of the orientation. An ordered pair (a,b)𝑎𝑏(a,b)( italic_a , italic_b ) of two crossings a𝑎aitalic_a and b𝑏bitalic_b are called parallel if and only if smoothing two crossings along the or... | **A**: Figure 1.
**B**: The label L𝐿Litalic_L (resp. R𝑅Ritalic_R) indicates “left” (resp. “right”).
Let C𝐶Citalic_C be a curve with a base point.
**C**: Smoothing.
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There are additional complications in real-world implementations of Elo. <|MaskedSetence|> <|MaskedSetence|> Sometimes rating updates are batched. That is, one accumulates their pot winnings and losses over several games, and updates their rating once at the end. <|MaskedSetence|> All these details and even more c... | **A**: For legibility and practicality, fractional and negative rating points are avoided by scaling and shifting points up and rounding to the nearest integer, and by imposing an artificial floor on possible ratings (by gifting a player points if they would otherwise dip below the floor).
The total size of the pot K𝐾... | ABC | ABC | CBA | ABC | Selection 4 |
This has been studied in [wu] under some restrictions. The results were then generalized using the same approach in [parsur]. <|MaskedSetence|> <|MaskedSetence|> In a [guhsur], the authors generalize the main Theorem 7.9 in some special cases.
For a detailed account on the notions of this section, we refer the rea... | **A**: In both cases, the authors show a Hasse principle for homogeneous varieties under unitary groups.
**B**: Assume, moreover, that k𝑘kitalic_k is a local field and chark≠2.char𝑘2{\mathrm{char}\ k\neq 2}.roman_char italic_k ≠ 2 ..
**C**: We give here a brief summary of the interpretation of these results in our... | ACB | ACB | ACB | BCA | Selection 1 |
<|MaskedSetence|> <|MaskedSetence|> However none of these works comes with an explicit convergence rate. The work [stoltz2018longtime] takes a
very nice approach, producing a perturbative expansion in ϵitalic-ϵ\epsilonitalic_ϵ of the invariant measure of the slow-fast system, and therefore proving convergence of the ... | **A**: The only other UiT results we are aware of are those in [ilyin1998global] (and references therein), for deterministic systems; for stochastic dynamics, aside from the work [barr2020fast], we are only aware of [cheng2023second], which deals with Stochastic Partial Differential equations, and of [stoltz2018longtim... | CAB | BCA | CAB | CAB | Selection 4 |
Recall that R¯¯𝑅\overline{R}over¯ start_ARG italic_R end_ARG is finitely generated as an R𝑅Ritalic_R-module, and also it is noetherian and local, because R𝑅Ritalic_R is complete (for example R¯=k[[x4,y4,x3y,xy3,x2y2]]¯𝑅𝑘delimited-[]superscript𝑥4superscript𝑦4superscript𝑥3𝑦𝑥superscript𝑦3superscript𝑥2supe... | **A**: By
Serre’s characterization of normality, see [24, Theorem 23.8], we know R¯¯𝑅\overline{R}over¯ start_ARG italic_R end_ARG satisfies Serre’s condition (S2)subscript𝑆2(S_{2})( italic_S start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) (for the definition, see [24, page 183]).
**B**: Now, let a,b𝑎𝑏a,bitalic_a , itali... | BAC | BAC | BAC | CAB | Selection 2 |
2. Basic definitions and main theorem
In this section, we state our main theorem. <|MaskedSetence|> <|MaskedSetence|> To an n×n𝑛𝑛n\times nitalic_n × italic_n matrix A,𝐴A,italic_A , we can associate a weighted digraph D(A),𝐷𝐴D(A),italic_D ( italic_A ) , with vertex set [n]delimited-[]𝑛[n][ italic_n ] and for e... | **A**: See [4] for details.
**B**: The sign of a linear subdigraph γ𝛾\gammaitalic_γ is (−1)n+c(γ),superscript1𝑛𝑐𝛾(-1)^{n+c(\gamma)},( - 1 ) start_POSTSUPERSCRIPT italic_n + italic_c ( italic_γ ) end_POSTSUPERSCRIPT , where n𝑛nitalic_n is the order of the matrix A.𝐴A.italic_A .
**C**: Before that, let us briefl... | CAB | CAB | CAB | CAB | Selection 1 |
Now gXng−1=−Xn⇒gXn=−Xng⇒g2,1=0andg1,1=−g2,2𝑔subscript𝑋𝑛superscript𝑔1subscript𝑋𝑛⇒𝑔subscript𝑋𝑛subscript𝑋𝑛𝑔⇒subscript𝑔210andsubscript𝑔11subscript𝑔22gX_{n}g^{-1}=-X_{n}\Rightarrow gX_{n}=-X_{n}g\Rightarrow g_{2,1}=0\ \hbox{and}%
\ g_{1,1}=-g_{2,2}italic_g italic_X start_POSTSUBSCRIPT italic_n end_POST... | **A**: But gXsg−1=−Xs⇒g1,1𝐢=−𝐢g1,1𝑔subscript𝑋𝑠superscript𝑔1subscript𝑋𝑠⇒subscript𝑔11𝐢𝐢subscript𝑔11gX_{s}g^{-1}=-X_{s}\Rightarrow g_{1,1}\mathbf{i}=-\mathbf{i}g_{1,1}italic_g italic_X start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT italic_g start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT = - italic_X start_P... | CAB | CAB | CAB | CAB | Selection 3 |
<|MaskedSetence|> <|MaskedSetence|> Another possibility is stacking of infinitely many copies of a certain code to create a code with higher dimension. <|MaskedSetence|> In this section we discuss stacking and coarse-graining (in particular how they affect invariants of a code), but compactifications are postponed t... | **A**: For example, one may “compatify” some (even all) spatial directions, i.e. replace ΛΛ\Lambdaroman_Λ by a quotient group.
**B**: Finally, one has coarse-graining, which does not change the code, but forgets about some of its translation symmetry.
**C**: 4 Operations on Pauli stabilizer codes
One Pauli stabilize... | CAB | CAB | ABC | CAB | Selection 2 |
<|MaskedSetence|> Theorem 1. <|MaskedSetence|> Lemma 2.1). <|MaskedSetence|> For experts familiar with the ways ReLU neural network functions can degenerate, Lemma 3.12 should provide a clear explanation for why most ReLU neural network functions are not PL Morse.
. | **A**: Indeed, one should view the analogues of critical cells (both Morse and non-Morse) in the PL category as an appropriate subset of the flat cells.
**B**:
Unsurprisingly, the key to understanding how the topology of sublevel sets changes as one varies the threshold are the PL analogues of points where the gradie... | BCA | BCA | ABC | BCA | Selection 4 |
<|MaskedSetence|> <|MaskedSetence|> We are also grateful to D. Kubrak, S. Mondal, S. Petrov, and A. Prikhodko for many useful conversations. We thank A. Langer for several useful comments on an earlier draft. <|MaskedSetence|> Finally, we would like to thank S. Naprienko for creating the website Thuses.com, without ... | **A**: In particular, we thank him for the suggestion to use the results from [BMS19] to attack Theorem 1.1.1 and for explaining the relevant background.
**B**: We thank the referee for many useful suggestions which greatly improved this paper.
**C**:
We are very grateful to B. Bhatt for many fruitful discussions.
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We note here that the model above does not include a latent period, however since the results in this paper relate to the final outcome of an SIR epidemic they are insensitive to quite general assumptions concerning a latent period (see e.g. <|MaskedSetence|> Moreover, the results of the present paper (suitably mod... | **A**: In particular, all results in this paper apply without change to corresponding SEIR epidemics.
**B**: Adjusting our results to allow for
a fixed number of initial infectives spread in some specified way through the communities is also straightforward, but it is rather lengthy to describe.
**C**: Ball, 1986).
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<|MaskedSetence|> <|MaskedSetence|> Seq2seq strategy was proposed in [16], where the PINN learns to predict the solution at each time step, instead of all times. <|MaskedSetence|> We take the prediction at t=𝑡absentt=italic_t =d𝕋𝕋\mathbb{T}blackboard_T by using the model of the first sequence and use this as the ... | **A**: Note that the only data available of the first sequence is from the PDE itself, i.e., just the initial condition.
**B**: In complex cases, this can be more difficult to learn.
**C**:
(3.5)
The original PINN approach trains the NN model to predict the entire space-time at once.
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We experiment with Voronoi diagrams in which the cells in the center of the diagram tend to be smaller. A point is sampled by first choosing a random cell and then choosing a uniform point on its boundary. <|MaskedSetence|> <|MaskedSetence|> Further details of the data generation process may be found in Appendix B.
... | **A**: We further inject additive noise.
**B**: The sample points are shown in Figure 9, the persistence diagrams are shown in Figure 11, and the significant loops found by oracle and subsample bootstrapping are shown in Figure 10.
.
**C**: This results in a higher sampling density on boundaries of smaller cells.
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Our work is related to a line of recent work on the sample efficiency of reinforcement learning for POMDPs. In detail, Azizzadenesheli et al. (2016); Guo et al. <|MaskedSetence|> (2021) establish sample complexity guarantees for searching the optimal policy in POMDPs whose models are identifiable and can be estimated... | **A**: Their proposed algorithm can attain the optimal policy without estimating the exact model, but an observable component (Jaeger, 2000; Hsu et al., 2012), which is the same for our algorithm design, while only applies to tabular POMDPs.
In a broader context of reinforcement learning with partial observability, o... | CBA | BCA | CBA | CBA | Selection 4 |
<|MaskedSetence|> Since 𝒟𝒟{\mathscr{D}}script_D is a convex neighbourhood of the origin, it follows that tH∈𝒟¯𝑡𝐻¯𝒟tH\in{\overline{\mathscr{D}}}italic_t italic_H ∈ over¯ start_ARG script_D end_ARG is minimal for 0≤t≤10𝑡10\leq t\leq 10 ≤ italic_t ≤ 1. For t=1𝑡1t=1italic_t = 1 we have H∈∂𝒟𝐻𝒟H\in\partial{\math... | **A**: By using that Ad(K)Ad𝐾\mathrm{Ad}(K)roman_Ad ( italic_K ) conjugates every X∈𝔰𝑋𝔰X\in\mathfrak{s}italic_X ∈ fraktur_s to some H∈𝔱𝐻𝔱H\in\mathfrak{t}italic_H ∈ fraktur_t, it follows from Proposition 6.2 that the cut locus of S𝑆Sitalic_S is contained in the adjoint orbit of the cut locus of Tp𝑇𝑝Tpitalic_... | CBA | CBA | CBA | CBA | Selection 3 |
<|MaskedSetence|> It was defined by Björner, Korte and Lovász, and they give two equivalent definitions (greedoid, , Theorem 6.1). One of them is topological, and uses the (abstract) dual complex of the greedoid. <|MaskedSetence|> We will give two proofs for our main theorem, each based on these two definitions. <|M... | **A**: Hence we repeat here both definitions.
.
**B**: Another definition uses activities.
**C**:
One important invariant of a greedoid is the greedoid polynomial.
| CBA | CBA | CBA | BCA | Selection 3 |
Among the available approaches, the concept of control invariant set is one of the most exploited historically, since it ensures the existence of some feedback law able to steer the closed-loop trajectories of the uncertain system within a prescribed state set 25, 6, 8, 37. This is traditionally achieved by associating... | **A**: As a common drawback affecting both the implementations, however, fixing the input values at the vertices may result in poor control performance for the stabilization task.
A more sophisticated control method coincides with the selection-based policy.
**B**: These methods can therefore require significant memor... | BAC | BAC | CBA | BAC | Selection 1 |
The classical Keller-Segel model assumes that the density diffusion is not affected by the nonlocal behaviour of the organisms. <|MaskedSetence|> Then, the trajectories of the population of organisms are better described by the so called Lévy flights than Brownian motion (see [8, 9]). Lévy flights behaviour has been ... | **A**: This last consideration has been point out relevant in the analysis of the propagation of chaos for some aggregation-diffusion models [11].
**B**: However, in many situations found in nature, organisms develop alternative search strategies, particularly when chemoattractants, food, or other targets are sparse o... | ACB | BCA | BCA | BCA | Selection 2 |
Section 3 consists of the technical results on (relative) cyclic (co)homology of Lie algebras that we shall need in the sequel. In Subsection 3.1 we outline stable and unimodular stable AYD modules over Lie algebras, followed by the cyclic homology of a Lie algebra with coefficients in a stable AYD module in Subsection... | **A**: We recall in Subsection 4.1 the Hopf-cyclic homology of the coordinate algebra of functions on an algebraic group with coefficients in an SAYD module.
**B**: Then, in Subsection 4.2 we prove our first van Est isomorphisms between the Hopf-cyclic (co)homology of the coordinate algebra of functions on an algebrai... | ACB | ABC | ACB | ACB | Selection 3 |
This is due to the fact that the argument of our paper [17] is based on the Dirichlet-to-Neumann (DN) machinery. There is a possibility to modify the approach by invoking Neumann-to-Dirichlet maps instead, which would have two advantages: one could consider all rates of vertex volume decay in (iii), and certain geometr... | **A**: The alternative strategy will be followed up elsewhere, both in the present context and in the setting of [17].
The above results of course imply the Hausdorff spectral convergence, at the same time yielding a sharp estimate on its rate.
**B**: Moreover, in contrast to [56], our approach allows one to conside... | ABC | BCA | ABC | ABC | Selection 1 |
We have benefited from discussions and correspondence with J. Kollár, M. Mustaţă, and M. Popa on higher du Bois singularities while preparing [FL22]. The second author had several related discussions with M. Kerr and M. <|MaskedSetence|> We also thank J. de Jong, M. Saito and C. <|MaskedSetence|> <|MaskedSetence|> ... | **A**: Schnell for some further comments related to this paper.
**B**: After circulating an earlier version of this paper, M.
**C**: Saito while preparing previous joint work.
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<|MaskedSetence|> First, we consider a protocol that requires significance in two independent trials—we call this the standard protocol. <|MaskedSetence|> Second, we consider the case where only one trial is required, but at the 0.010.010.010.01 level for a two-sided test. <|MaskedSetence|> Lastly, we consider a pro... | **A**: The probability that a placebo drug is approved with this protocol is 0.0252=0.000625superscript0.02520.0006250.025^{2}=0.0006250.025 start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = 0.000625.
**B**:
To facilitate formal analysis, we analyze three simplified statistical protocols that track the FDA evaluatory pr... | BAC | BCA | BAC | BAC | Selection 4 |
The structure of the paper is as follows. In Section 2 we recall some basic definitions about rings and modules, and prove that the notion of rank is well defined. <|MaskedSetence|> In Section 4 we obtain several results of screw theory by using the formalism of 𝔻𝔻\mathbb{D}blackboard_D-module geometry. In Section ... | **A**: Again, they are obtained by using the formalism of 𝔻𝔻\mathbb{D}blackboard_D-module geometry.
**B**: Vectors.
**C**: In Section 3 we show how the Euclidean space emerges out of 𝔻𝔻\mathbb{D}blackboard_D-module geometry, in particular, Theorem 3 is proved.
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Theorem 1 predicts that in certain situations FNO has a systematic bias. Namely, if one trains FNO on an insufficiently fine grid, activation functions introduce distortions that FNO will learn to mitigate. When the grid is sufficiently refined, aliasing errors disappear, but since FNO was trained to mitigate them, it... | **A**: However, it is possible to decrease aliasing error even when FNO is used for interpolation.
**B**: Indeed, the right graph in Figure 2a shows that for the integration operator, which performs smoothing, aliasing leads to an approximately two-fold error increase, while for the differentiation operator the increa... | BAC | CAB | BAC | BAC | Selection 4 |
where K(f)(g)=∑n∈ℕ⟨f(n),g(n)⟩𝐾𝑓𝑔subscript𝑛ℕ𝑓𝑛𝑔𝑛K(f)(g)=\sum_{n\in\mathbb{N}}\left<f(n),g(n)\right>italic_K ( italic_f ) ( italic_g ) = ∑ start_POSTSUBSCRIPT italic_n ∈ blackboard_N end_POSTSUBSCRIPT ⟨ italic_f ( italic_n ) , italic_g ( italic_n ) ⟩ for each f∈ℓ1(ℕ,X**)𝑓superscriptℓ1ℕsuperscript𝑋absentf\... | **A**: Let P=K∘i*:ℓ1(ℕ,X)**→ℓ1(ℕ,X)**:𝑃𝐾superscript𝑖→superscriptℓ1superscriptℕ𝑋absentsuperscriptℓ1superscriptℕ𝑋absentP=K\circ i^{*}:\ell^{1}(\mathbb{N},X)^{**}\to\ell^{1}(\mathbb{N},X)^{**}italic_P = italic_K ∘ italic_i start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT : roman_ℓ start_POSTSUPERSCRIPT 1 end_POSTSUPERSC... | CBA | ABC | ABC | ABC | Selection 3 |
<|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> What makes the monopole-dimer model less physical is that configurations have a signed weight and they cannot be interpreted as energies anymore. On the other hand, the partition function here can be expressed as a determinant. Moreover, it is a perfect square fo... | **A**:
In another direction, a signed version of the monomer-dimer model called the monopole-dimer model has been
introduced [Ayy15] for planar graphs.
**B**: Configurations of the monopole-dimer model can be thought of as superpositions of two monomer-dimer configurations having monomers (called monopoles there) at... | ABC | BAC | ABC | ABC | Selection 4 |
After completing the proofs of Theorems 1.3 and 1.4, we were informed that Fujino has also proved these results see [Fuj22b]. We note that Fujino’s approach is based on [BCHM10] whereas our approach is inspired by [CL10]. <|MaskedSetence|> In Subsection 2.4 we prove two important results, namely Theorem 2.29 and 2.36.... | **A**: Part 2 of the article is devoted to proving finite generation as in [CL10].
**B**: We prove Theorem 1.3 and 1.4 in Section 4 of this part.
**C**: Another possible approach can be found in [Pau12], which is particularly suited to the analytic context.
This article is organized in the following manner: In Par... | CAB | CAB | CAB | BCA | Selection 2 |
Here and throughout the paper we use L2(D)superscript𝐿2𝐷L^{2}(D)italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( italic_D ) and H1(D)superscript𝐻1𝐷H^{1}(D)italic_H start_POSTSUPERSCRIPT 1 end_POSTSUPERSCRIPT ( italic_D ) to denote the standard Lebesgue and Sobolev spaces of functions on D𝐷Ditalic_D. Moreo... | **A**: ∥ start_POSTSUBSCRIPT 1 , italic_D end_POSTSUBSCRIPT,∥.∥−1,D\|.\|_{-1,D}∥ .
**B**: , .
**C**: The same symbols are used for inner product and norms
in the vector-valued versions of the spaces.
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<|MaskedSetence|> This exploration is carried out on a different nonlinear model, highlighting our approach’s flexibility. <|MaskedSetence|> q=1𝑞1q=1italic_q = 1. <|MaskedSetence|> A data-set is simulated according to a logistic growth model such as:
. | **A**: The other parameters are assumed to be shared among individuals and they are estimated jointly.
**B**: For ease of presentation, we consider variable selection in a one-dimensional setting, i.e.
**C**:
5.2.1 Model
Now, the performance of SAEMVS is studied under a variety of scenarios.
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To avoid the difficulties in analysis in the metric level and due to the interests from mirror symmetry, there are replacements for the original proposal from algebraic geometry and symplectic geometry. On the algebraic side, Kontsevich–Soibelman [2006-Kontsevich-affine-structures-and-non-archimedean-analytic-spaces] a... | **A**: We refer the readers to [Gross]*Section 7 and [CL]*Section 1 and references therein.
.
**B**: However, it is still unclear how the two approaches are related to the original conjecture explicitly, again due to the lacking of the existence of genuine SYZ fibrations.
**C**: In this paper, we will study the comp... | BCA | BCA | BCA | BCA | Selection 3 |
<|MaskedSetence|> It is not hard to see that the L𝐿Litalic_L-subdifferential is L𝐿Litalic_L-monotone [20, Proposition 1.9]. <|MaskedSetence|> Several authors have investigated conditions under which it is [10, 11, 18, 19, 17, among others]. Here we show that under Assumption 1 the result is also true. <|MaskedSete... | **A**: Our proof generalises the proof by [13].
.
**B**:
Our next result concerns the maximal monotonicity of the abstract subdifferential operator.
**C**: However, [4, example 3.1] showed that it is not generally maximal L𝐿Litalic_L-monotone.
| BCA | ABC | BCA | BCA | Selection 4 |
Related Work
Operations of AMoD systems are relevant to several mathematical problems, including dispatching, routing, scheduling, and rebalancing of service fleets. <|MaskedSetence|> Within the context of AMoD systems, fleet operations have been examined primarily using queuing-theoretic models [13] [14], simulatio... | **A**: More recently, AMoD fleet operations have been extended to a decentralized manner.
**B**: Examples include the VRP and its variants [4, 5, 6][8], where distance-minimized routes are to be planned to satisfy a set of transportation requests while considering different constraints.
**C**: [16] addressed the opti... | BCA | BCA | BCA | BAC | Selection 3 |
<|MaskedSetence|> <|MaskedSetence|> The focus of his undergraduate and master studies was statistics and discretization techniques and reduction order models for partial differential equations. He obtained his PhD in Applied Mathematics at Emory University in Atlanta (GA) in May 2018. The main topic of his doctorate ... | **A**: Massimiliano (Max) Lupo Pasini obtained his Bachelor of Science and Master of Science in Mathematical Engineering at the Politecnico di Milano in Milan, Italy.
**B**: He is currently the lead of the Artificial Intelligence for Scientific Discovery thrust of the ORNL Artificial Intelligence Initiative..
**C**: ... | CAB | CAB | BCA | CAB | Selection 2 |
<|MaskedSetence|> <|MaskedSetence|> This is formulated precisely in Proposition 3.8(3). Using this result we then proceed to proving an analogue of Lemma 2.9. <|MaskedSetence|> The strategy of proof is similar to that carried out in [11, §6.2], using results on 1-h-minimal fields, but working in mixed characteristic... | **A**: Section 3.2 is dedicated to the second main property that we need: local linearity of definable functions in K/𝒪𝐾𝒪K/{\mathcal{O}}italic_K / caligraphic_O, culminating in Corollary 3.32.
**B**:
We prove two geometric properties that play a crucial role in the sequel.
**C**: The first, comprising the heart o... | BCA | BCA | BAC | BCA | Selection 4 |
<|MaskedSetence|> More specifically, in these settings we derive novel upper and lower bounds for the limiting acceptance probabilities that clarify in which contexts the abc posterior is still well–defined for n𝑛nitalic_n large enough. When this is the case, it is further possible to obtain informative supersets for... | **A**: The former advantage is illustrated within Section 4 through a specific focus on mmd with routinely–implemented bounded and unbounded kernels, whereas the latter is clarified in Section 6, where we extend the theory from Section 3 to non–i.i.d.
**B**: This yields an improved understanding of the factors that go... | CBA | CBA | ABC | CBA | Selection 4 |
The simulation results are summarized in Fig. 4. <|MaskedSetence|> <|MaskedSetence|> To capture the dynamics of the evolution of the thin diffuse interface, the mesh size should be much smaller than ϵitalic-ϵ\epsilonitalic_ϵ, the width of the diffuse interface. Traditional numerical methods often use an adaptive or ... | **A**: In contrast, a neural network-based numerical scheme has a mesh-free feature.
**B**: Numerical simulation of phase-field type models is often challenging.
**C**: It clearly shows that our method can achieve comparable results with the FEM.
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<|MaskedSetence|> Outline
In Section 2, we provide basic preliminaries on Deligne’s Hodge theory and the perverse Leray filtration. Section 3 introduces the notion of a hybrid LG model, the primary focus of this paper, and examines its topological properties. Section 4 explores the Hodge theory of hybrid LG models, i... | **A**: In the Appendix, we review the theory of cohomological mixed Hodge complexes, which will be employed in Section 4.
**B**: Section 5 contains speculative discussions and proposes a relative version of homological mirror symmetry for the extended Fano/LG correspondence.
**C**: 1.5.
| BCA | CBA | CBA | CBA | Selection 3 |
<|MaskedSetence|> Hladký, Král, and Norine [13] confirmed a conjecture of Baker [2] relating the ranks of a divisor on a graph and on a tropical curve, and provided a purely combinatorial algorithm for computing the rank of a divisor on a tropical curve, which can be considered simply as a metric graph, see [18, 12]. ... | **A**:
Previous work
From the positive side, Luo [16] introduced the notion of rank-determining sets of metric graphs, and verified the existence of finite rank-determining sets constructively.
**B**: For multigraphs with a constant number of vertices, Manjunath [17] gave a polynomial time algorithm that computes t... | ACB | ABC | ABC | ABC | Selection 2 |
<|MaskedSetence|> This problem has been studied by L. <|MaskedSetence|> <|MaskedSetence|> Kriventsov in [5], where the authors have proved the existence of a solution u𝑢uitalic_u for the problem and the regularity of its jump set. Another similar problem, in a non-linear context, has been deepened by D. Bucur and A... | **A**:
with v∈SBV(ℝn)𝑣SBVsuperscriptℝ𝑛v\in\operatorname{SBV}(\mathbb{R}^{n})italic_v ∈ roman_SBV ( blackboard_R start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT ) and v=1𝑣1v=1italic_v = 1 in ΩΩ\Omegaroman_Ω.
**B**: A.
**C**: Caffarelli and D.
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<|MaskedSetence|> <|MaskedSetence|> However, in the case that K𝐾Kitalic_K is of three-genus one and is algebraically slice, the invariant is determined by the Levine-Tristram signature functions of certain knots formed as simple closed curves on a genus one Seifert surface. <|MaskedSetence|> The paper [MR3096507] p... | **A**: Computing σ1τ(Bq,χ0)subscript𝜎1𝜏subscript𝐵𝑞superscript𝜒0\sigma_{1}\tau(B_{q},\chi^{0})italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_τ ( italic_B start_POSTSUBSCRIPT italic_q end_POSTSUBSCRIPT , italic_χ start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT ).
In general, there are few methods available f... | ACB | BAC | BAC | BAC | Selection 4 |
Now we prove (a). Suppose that n0=dk0minursubscript𝑛0superscriptsubscript𝑑superscriptsubscript𝑘0urn_{0}=d_{k_{0}^{\min}}^{\mathrm{ur}}italic_n start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_d start_POSTSUBSCRIPT italic_k start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_min end_POSTSUPERSCRIPT en... | **A**: the index i=n0𝑖subscript𝑛0i=n_{0}italic_i = italic_n start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT would be part of a segment (n1,n2)subscript𝑛1subscript𝑛2(n_{1},n_{2})( italic_n start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_n start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT ) with n1<n0<n2subscript𝑛1subscript𝑛0subscr... | ABC | BAC | ABC | ABC | Selection 4 |
Our pseudo-stabilization technique discussed in Appendix A is essential to the scalability of the JFP method and thus also for its application to practical computational problems. We emphasize that high-precision computations are required only for the computation of fractional integration matrices and not for the solut... | **A**: We conclude the paper with a summary and a discussion of topics for future work.
**B**: In addition, we do not require any of the inputs to the FIE or FDE (e.g., variable coefficients, the function on the right-hand side of the equation, boundary conditions, etc.) to be computable to high-precision accuracy.
... | BCA | BCA | BCA | BCA | Selection 3 |
On the other hand, processes with Moran type interactions are natural models for finite populations with either variety-increasing or variety-reducing effects such as genetic drift, genetic mutations and natural selection. First introduced by Moran [57], the Moran model describes the evolution of N𝑁Nitalic_N genes suc... | **A**: Thus, the independence between particles is lost.
We refer the reader to [33] and references therein for an overview of this model and of its extensions.
**B**: This type of resampling has since been employed in a range of particle system models to numerically solve Feynman-Kac models, see [15, 25, 26, 66] and ... | ABC | BAC | ABC | ABC | Selection 1 |
However, the study of linear convergence rates of DYS has been limited. Bredies, Chenchene, Lorenz, and Naldi studied the convergence of DYS under a wider range of stepsizes [4] and Aragón-Artacho and Torregrosa-Belén also proved the convergence of the DYS iteration over a wider range of stepsizes [1]. Dao and Phan ana... | **A**: Ryu, Taylor, Bergeling, and Giselsson [39] and Wang, Fazlyab, Chen, and Preciado [45] formulated the problem of computing tight contraction factors as SDPs using the performance estimation problem (PEP) and integral quadratic constraint (IQC) approaches, but these are numerical results, not analytic convergence ... | ACB | BAC | ACB | ACB | Selection 3 |
We introduce basic notation and briefly recall required properties of Bregman divergences in Section 2. <|MaskedSetence|> <|MaskedSetence|> In particular, information geometry is employed in a way that indicates how other convex programs could be handled in the same way, in principle. While we indicate multilevel ex... | **A**: The core part of this paper, Section 4, generalizes this scheme to a Riemannian setting.
**B**: Numerical experiments are reported in Section 5 for a range of problem instances and compared to a recent state-of-the-art method [HRX21].
**C**: Section 3 summarizes the basic scheme of two-level optimization in Eu... | ABC | CAB | CAB | CAB | Selection 3 |
Given the above result, it may seem that, similarly to the case of monotone networks with ReLU activations, the class of monotone networks with threshold activations is too limited, in the sense that it cannot approximate any monotone function with a constant depth (allowing the depth to scale with the dimension was... | **A**: Thereafter, a simple argument shows that monotone networks of bounded depth are universal approximators of monotone functions.
**B**: Our first main result supports the contrary to this belief.
**C**: One reason for such a belief is that, for non-monotone networks, depth 2222 suffices to ensure universality.
| BCA | CBA | CBA | CBA | Selection 3 |
The evolution of the estimation error {‖-Θtθoptt‖} is shown in Fig. 12 for each value of b listed in (102). Only ξct in (103) was used as the probing signal for these experiments. We see in Fig. 12 that tracking performance deteriorates as the rate of change of {θoptt} increases. When >t/T2, the signal {θoptt} is nearl... | **A**: That is, a second order filter is preferable when the rate of change of {θoptt} is small..
**B**: We see in Fig. 13 that the filtered estimates {Θ1Ft} yield lower objective values than {Θ2Ft} for ≤t/T2.
**C**: Again, the selected probing signal was ξct and the signal {θoptt} had period T0 with =b3 for both c... | BCA | CBA | CBA | CBA | Selection 4 |
<|MaskedSetence|> GKAD21220144, 2021GXNSFFA196004 and GKAD23026237, the NNSF of China Grant Nos. 12001478, 12101143 and 12371312, the China Postdoctoral Science Foundation funded project No. 2022M721560, the Startup Project of Doctor Scientific Research of Yulin Normal University No. G2023ZK13, and the European Union’... | **A**: It is also supported by the project cooperation between Guangxi Normal University and Yulin Normal University.
**B**: The second author is supported by the grant from the National Program for Research of the National Association of Technical Universities - GNAC ARUT 2023, ID: 220235047, ”Sisteme cuplate de ineg... | CAB | CAB | CAB | CAB | Selection 4 |
In the classical case of matroids independent sets need three axioms. For q𝑞qitalic_q-matroids, the straightforward q𝑞qitalic_q-analogue of these three axioms are not strong enough to get a q𝑞qitalic_q-matroid with a semimodular rank function. Therefore, a fourth axiom was added (see [10]).
This unexpected fourth ... | **A**: The second one is an alternative for the third axiom that is still a q𝑞qitalic_q-analogue of the classical case, but that obliterates the need for the fourth axiom.
**B**: The first one is a direct cryptomorphism between independent sets and circuits that was not shown before.
**C**: The first one is to remov... | CAB | CAB | CAB | BAC | Selection 2 |
<|MaskedSetence|> It is structured as follows. <|MaskedSetence|> <|MaskedSetence|> DG in the QMC framework is presented in Section 5 and the corresponding parametric regularity analysis is considered in Section 6. Numerical results, which confirm our theoretical findings, are given in Section 7, while a short conclu... | **A**: Section 3 describes randomly shifted lattice rules, and Section 4 gives a brief overview over the analysis of conforming FE methods.
**B**: Notations and preliminaries are introduced in Section 2.
**C**: However, since discontinuous Galerkin methods are non-conforming, we cannot directly apply the existing QMC... | CBA | CBA | CBA | ACB | Selection 3 |
Structure of the paper. In Section 2, we first recall fractal structures and self-affine structures. Then we estimate the lower bound of the Hausdorff dimension of the associated fractal set. <|MaskedSetence|> <|MaskedSetence|> <|MaskedSetence|> | **A**: In Section 3, we generalize the lattice point counting results in [LSST20, Section 3] to higher dimensional cases.
**B**: In Section 4, we construct a fractal set contained in the set of weighted singular vectors and prove Theorem 1.1 by estimating the Hausdorff dimension of the fractal set.
**C**:
.
| CBA | ABC | ABC | ABC | Selection 4 |
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