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@@ -0,0 +1,1692 @@
+Anyon statistics through conductance measurements of time-domain interferometry
+Noam Schiller1, Yotam Shapira2, Ady Stern1, and Yuval Oreg1
+1Department of Condensed Matter Physics
+2Department of Physics of Complex Systems
+Weizmann Institute of Science, Rehovot 7610001, Israel
+We propose a method to extract the mutual exchange statistics of the anyonic excitations of
+a general Abelian fractional quantum Hall state, by comparing the tunneling characteristics of a
+quantum point contact in two different experimental conditions. In the first, the tunneling current
+between two edges at different chemical potentials is measured. In the second, one of these edges is
+strongly diluted by an earlier point contact. We describe the case of the dilute beam in terms of a
+time-domain interferometer between the anyons flowing along the edge and quasiparticle-quasihole
+excitations created at the tunneling quantum point contact. In both cases, temperature is kept
+large, such that the measured current is given to linear response. Remarkably, our proposal does
+not require the measurement of current correlations, and allows us to carefully separate effects of
+the fractional charge and statistics from effects of intra- and inter-edge interactions.
+Introduction.— It has been almost four decades since
+the initial proposal that the elementary quasiparticles
+of fractional quantum Hall (FQH) systems obey anyonic
+statistics [1]. Despite the apparent maturity of the field,
+the pursuit to definitively observe the physical quanti-
+ties and quantum numbers characterizing anyons [2, 3] is
+constantly being reinvigorated [4–20]. In particular, early
+2020 saw two major experimental steps forward: the ob-
+servation of anyonic braiding in a Fabry-Perot interfer-
+ometer [21], and demonstration of a so-called “anyon col-
+lider” [22, 23] using cross-correlation measurements.
+Here we show that anyonic statistics can be inferred di-
+rectly from conductance measurements, without requir-
+ing current correlation measurements or explicitly build-
+ing an interferometer. The configuration we propose to
+obtain this result consists of a quantum point contact
+(QPC) between two edges of a general Abelian FQH state
+which are driven out of equilibrium. The edges may be
+driven off-equilibrium by one of three methods: inject-
+ing a single quasiparticle into one of the edges; injecting
+a Poissonian, dilute beam of quasiparticles into one of
+the edges; and placing a finite bias voltage between the
+edges.
+Our proposed setup, shown in Fig. 1(a), allows a
+smooth transition between the dilute Poissonian beam
+and a full beam at finite bias voltage.
+This is ob-
+tained by tuning a second, injection QPC from fully open
+(a differential conductance, Ginj ≡ dIinj/dV , satisfying
+Ginj/σxy → 0) to fully closed (Ginj/σxy → 1). We hence-
+forth refer to these as the dilute and full limits, respec-
+tively.
+We propose sweeping Ginj through this range, and
+measuring the ratio I/Iinj, where I is the measured cur-
+rent after the tunneling QPC, and Iinj is the injected inci-
+dent current, as defined in Fig. 1(a). Comparing the val-
+ues at the dilute and full limits cancels out non-universal
+constants, yielding the relation,
+� I(T)
+Iinj(T)
+�
+dilute
+=
+νe2
+2πe∗
+1e∗
+2
+sin 2θ12
+� I(T)
+Iinj(T)
+�
+full
++ Gdirect
+Ginj
+. (1)
+Here, e∗
+1 is the tunneling quasiparticle charge, e∗
+2 the
+injected quasiparticle charge, δ1 is the tunneling quasi-
+particle scaling dimension, θ12 is the mutual statistics
+phase between the injected and tunneling quasiparticles,
+T is temperature, and Gdirect is a residual conductance
+corresponding to direct tunneling [24–26] through both
+QPCs.
+The full crossover between these two limits is
+shown schematically in Fig. 1(b).
+The mechanism leading to this result is a time-domain
+interferometer at the tunneling QPC which is created by
+the dilute incident beam. The interference is between two
+processes, in which a quasiparticle-quasihole excitation
+occurs at the tunneling QPC either before or after the
+arrival of an injected quasiparticle (see Fig. 2). A similar
+physical picture has been shown in Refs. [25, 27, 28]. We
+further find that this interference is sensitive to the mu-
+tual statistics phase between the injected and the tunnel-
+ing quasiparticles, θ12. We emphasize that these quasi-
+particles are not necessarily of the same type, although
+they must be supported by the same FQH liquid.
+Since our focus is the interference of two amplitudes
+which differ from one another by the orderings of events,
+the key point of our analysis is the identification of the
+phase differences between the two orderings.
+We find
+phase differences that are determined by the quasiparti-
+cle charge e∗, which is a fraction of the electron charge
+for non-integer values of ν [4–6]; the scaling dimension δ,
+which defines the zero-temperature time correlations of
+the quasiparticle via the relation ⟨ψ†(τ)ψ(0)⟩ ∼ τ −2δ
+[29–32]; and the exchange statistics phase θ, which for
+anyons take special values beyond the fermionic π and
+the bosonic 2π [1–3].
+We are interested here in isolating the effect of θ from
+the other two effects.
+In particular, we would like to
+separate it from the effect of δ.
+For non-interacting
+edges, in which all the modes propagate in the same di-
+rection, 2πδ = θ; however, in general δ is affected by
+non-universal factors, such as intra-edge and inter-edge
+interactions, 1/f noise or neutral modes [33–38]. This in
+stark contrast to the charge, exchange statistics phase,
+or filling factor, which are universal.
+We separate the effect of θ from that of δ by tuning
+arXiv:2301.00021v1  [cond-mat.mes-hall]  30 Dec 2022
+
+2
+(a)
+𝜈
+𝐼
+𝑉
+𝑒1
+∗, 𝛿1
+𝑒2
+∗, 𝛿2
+Injection 
+QPC
+Tunneling 
+QPC
+𝐼inj
+𝑢
+𝑑
+𝑎
+𝐷𝑎
+𝑆𝑢
+𝐷𝑢
+𝑆𝑑
+(b)
+200
+400
+600
+800
+1000
+0.30
+0.35
+0.40
+0.45
+400
+800
+0.1
+0.2
+FIG. 1. (a) Two counter-propagating edge modes (u/d) of
+a fractional quantum Hall droplet at filling factor ν are con-
+nected by a quantum point contact, through which quasipar-
+ticles of charge e∗
+1 and scaling dimension δ1 can tunnel. Cur-
+rent is measured at the lower edge’s drain, denoted by I. A
+current of Iinj is injected into the upper edge via a second, in-
+jection QPC, e.g. from a third auxiliary edge mode (a). The
+injection QPC is placed at a bias voltage of V , and allows
+tunneling of quasiparticles of charge e∗
+2 and scaling dimension
+δ2. All other sources and drains are grounded. (b) The ratio
+between I/Iinj in the dilute case and I/Iinj in the full case,
+as a function of temperature, for ν = e∗
+1/e = e∗
+2/e = 1/3,
+and for different scaling dimensions δ1. For the dilute case,
+we Iinj = 10pA, and assume kBT ≪ eV for all relevant tem-
+peratures, such that the contribution from Gdirect to Eq. (1)
+is negligible. In the full case, we use V = 10µV . Both cases
+use ξ = 72mK, τc = 10−13s. When the dilute case satisfies
+ℏIinj/e ≪ kBT ≪ eV ≪ ℏ/τc, and the full case satisfies
+ℏIinj/e = νeV/2π ≪ kBT ≪ ℏ/τc, the ratio approaches an
+asymptote that does not depend on scaling dimension, allow-
+ing extraction of the mutual statistics θ12. Inset: I/Iinj for the
+dilute and full cases as a function of temperature for δ1 = 1/6,
+the canonical value for a Laughlin 1/3 state.
+the system to a regime where δ only affects observables
+through a non-universal prefactor, which then cancels out
+in the ratio of currents given in Eq. (1). We arrive at this
+regime by employing a careful ordering of the various
+energy scales in the system, such that ℏIinj/e ≪ kBT
+throughout the entire crossover of Ginj.
+This ensures
+that the current I is given to linear response in Iinj. We
+present an analytic expression generalizing Eq. (1) out-
+side of this regime in App A, Eq. (A5).
+While in the full limit the edge that enters the tunnel-
+ing QPC is in equilibrium at chemical potential V , at the
+dilute limit we need the injection QPC to reflect only a
+small fraction of the impinging electrons, such that the
+resulting injection current is Poissonian and rare. Said
+differently, the injected current in this limit must satisfy
+Iinj ≪ σxyV . Furthermore, the beam must still be dilute
+when arriving at the tunneling QPC. As such, the dis-
+tance between the two QPCs must be sufficiently small
+that no equilibration or dephasing occurs along the way.
+Finally, we assume that tuning the injection QPC does
+not affect the transparency of the tunneling QPC, to en-
+sure that all non-universal constants are cancelled when
+examining the ratio of the two limits. [39]
+Easy extraction of θ12 requires Gdirect to be sub-
+dominant (see Eq. (1)). Quantitatively, this is the case
+if both kBT ≪ eV and 4δ1 < 2 are satisfied. These con-
+straints result from the direct tunneling process being
+dominated by short time scales. Naive theories describ-
+ing quasiparticles may satisfy this condition even if the
+aforementioned non-universal effects change the scaling
+dimension quite significantly. For example, theory gives
+δ = 1/2m for Laughlin quasiparticles.
+Edge theory.— We now define the system’s Hamilto-
+nian and derive the current. As shown by Wen, the edge
+theory of a general Abelian FQH state can be described
+by n-boson fields, φ(x, t) ≡ (φ1, φ2, · · · φn)T [2]. These
+define the theory in conjunction with a charge vector, q,
+which determines the electric charge carried by each bo-
+son field, and the so called K-matrix, which determines
+the commutation relations between the boson fields,
+[φi(x), ∂x′φj(x′)] = i2π(K−1)ijδ(x − x′).
+(2)
+The filling factor is then given by ν = qT K−1q, and the
+charge density is given by ρ = − 1
+2πq · ∂xφ. In terms of
+these fields, the Hamiltonian of a single FQH edge mode
+is given by
+Hedge = 1
+4π
+n
+�
+i,j=1
+ˆ
+dx∂xφiVij∂xφj,
+(3)
+where ˆV is a positive definite matrix describing the ve-
+locities of the modes and intra-edge interactions. These
+edges support quasiparticles of the form ψl ∼ eil·φ, where
+l is a vector of integers. The charge of these quasiparti-
+cles is then given by e∗
+l = qT K−1l.
+The configuration of Fig. 1(a) involves two edges, u
+and d, tunnel-coupled by a QPC. This is described by
+two copies of the Hamiltonian Hedge, time reversed with
+regard to one another, as well as a tunneling term, HT ,
+which we treat as a perturbation.
+Assuming only one
+type of quasiparticle, denoted by the vector l1 and car-
+rying charge e∗
+1, tunnels between the edges, this is given
+
+3
+by
+HT = ξ
+�
+ˆA + ˆA†�
+; ˆA(t) ≡ ei(l1·φ(u)(0,t)−l1·φ(d)(0,t)). (4)
+Here, ξ is a small tunneling amplitude, which we assume
+to be real, and φ(u) (φ(d)) are the bosonic field operators
+on the upper (lower) edge. We project the auxiliary edge
+a out of the Hamiltonian, as it is only used to “initialize”
+the state of the edge u.
+The current that tunnels from the upper edge to
+the lower edge is then given by the operator, ˆIT (t) =
+iξe∗
+1
+�
+ˆA†(t) − ˆA(t)
+�
+.
+Since the lower edge is grounded,
+we henceforth identify I = ⟨ˆIT ⟩. Expanding to leading
+order in ξ, the current is given by
+I(t) = e∗
+1ξ2
+ˆ t
+−∞
+dt′ ��
+ˆA†(t), ˆA(t′)
+�
++
+�
+ˆA†(t′), ˆA(t)
+��
+.
+(5)
+Here, [·, ·] denotes commutation, and expectation values
+are calculated with respect to the Hamiltonian in the
+absence of tunneling.
+Deviation from Equilibrium.— It is clear from Eq. (5)
+that one needs to derive correlation functions such as
+⟨ ˆA†(t) ˆA(t′)⟩. In equilibrium, at temperature T, the sys-
+tem is particle-hole symmetric, and the correlation func-
+tions are given by [2, 40]
+⟨ ˆA†(t) ˆA(t′)⟩0 = ⟨ ˆA(t) ˆA†(t′)⟩0
+(6)
+=
+�
+πTτc
+sinh (πT |t − t′|)
+�4δ1
+e−i2πδ1sgn(t−t′),
+where δ1 is the scaling dimension of the quasiparticle l1,
+and τc > 0 is a short time cutoff.
+Two main features are carried over from Eq. (6) to the
+correlation functions out of equilibrium - the exponen-
+tial decay at time difference larger than ℏ/T, and the
+phase e2πiδ1 associated with an interchange of the time
+arguments.
+We now consider two non-equibrium cases. In the first
+we introduce a constant bias voltage V ≡ Vu − Vd be-
+tween the edges. In the setup of Fig. 1(a), this corre-
+sponds to a fully closed injection QPC, i.e. Iinj = σxyV .
+The introduction of the voltages can be formally ab-
+sorbed into the boson fields by use of a simple gauge
+transformation, which maps φ(u/d)(x, t) �→ φ(u/d)(x, t)+
+K−1qVu/d (t ∓ x/v) /ℏ.
+This accordingly modifies the
+correlation functions by a phase factor
+⟨ ˆA†(t) ˆA(t′)⟩full = ⟨ ˆA†(t) ˆA(t′)⟩0ei
+e∗
+1 V
+ℏ
+(t−t′),
+⟨ ˆA(t) ˆA†(t′)⟩full = ⟨ ˆA(t) ˆA†(t′)⟩0e−i
+e∗
+1 V
+ℏ
+(t−t′).
+(7)
+In the second non-equilibrium driving, we consider in-
+jecting a single quasiparticle, denoted by the vector l2,
+into the upper edge at the location xinj < 0 and at time
+tinj. This is shown schematically in Fig. 2(a). In view
+of the commutation relations (2), the application of the
+quasiparticle creation operator e−il2·φ(u)(xinj,tinj) on the
+edge creates a soliton in each of the boson fields,
+φ(u)(x, tinj) �→ φ(u)(x, tinj) − 2πK−1l2Θ (x − xinj) . (8)
+We assume here the injection happens instantaneously.
+This assumption will be relaxed to find the subleading
+term of Eq. (1).
+The fields at general times can then be obtained using
+the equations of motion dictated by the Hamiltonian in
+Eq. (3). If all modes are chiral with the same velocity v,
+this amounts to replacing x−xinj → x−xinj −v (t − tinj).
+The soliton thus arrives at the QPC, x = 0, at time
+t0 ≡ tinj − xinj/v.
+The c-number shift in the bosonic field of Eq. (8) leads
+to a phase shift in the correlator Eq. (6). We see directly
+from the definition of the operator ˆA in Eq. (4) that
+⟨ ˆA†(t) ˆA(t′)⟩qp = ⟨ ˆA†(t) ˆA(t′)⟩0e2πil1K−1l2[Θ(t−t0)−Θ(t′−t0)],
+⟨ ˆA(t) ˆA†(t′)⟩qp = ⟨ ˆA(t) ˆA†(t′)⟩0e−2πil1K−1l2[Θ(t−t0)−Θ(t′−t0)].
+(9)
+The phase we obtain is the standard definition of
+mutual braiding statistics between two quasiparticles,
+θ12 ≡ πl1K−1l2 [2]. The expression in Eq. (9) shows
+that the product gains a phase of e2iθ12sgn(t−t′) if the
+arrival time t0 is between the times t′ and t, and a triv-
+ial phase of 1 otherwise. We emphasize how naturally
+this result came from the underlying theory: the only as-
+sumptions necessary to obtain this are the commutation
+relations, (2), and the existence of quasiparticles in the
+edge’s excitation spectrum.
+This result holds for different boson modes with differ-
+ent velocities if all solitons arrive at the tunneling QPC
+more or less concurrently, avoiding dephasing. This is
+the case if |xinj|/∆v ≪ ℏ/T, where ∆v is the velocity
+difference between the fastest and the slowest modes.
+Time-domain interferometry.— The appearance of the
+phase, θ12, can be understood as time-domain interfer-
+ometry of the two distinct ±e∗
+1 quasiparticle-quasihole
+excitations, before and after the injected e∗
+2 quasiparticle
+arrives at the QPC. A similar physical picture has been
+shown in Ref. [25, 27, 28].
+To show this we consider the configuration of a single
+injected particle, as described in Fig. 2(a). In this case
+the non-equilibrium correlation function takes the form,
+⟨ ˆA†(t) ˆA(t′)⟩qp = ⟨ψl2(t0) ˆA†(t) ˆA(t′)ψ†
+l2(t0)⟩0,
+(10)
+i.e., the expectation value is calculated with respect to
+the state resulting from exciting the ground state |0⟩ with
+a single quasiparticle. Here we omit the position variable
+from the quasiparticle injection operator ψ†
+l2(t0), and as-
+sume it arrives at the tunneling QPC x = 0 at time t0.
+The current in Eq. (5) is then given by integration
+over multiple terms of the form in Eq. (10). We define
+|t, t0⟩− ≡ ˆA(t)ψ†
+l2(t0) |0⟩ and |t, t0⟩+ ≡ ˆA†(t)ψ†
+l2(t0) |0⟩.
+
+4
+𝜈
+𝐼
+𝑉
+−𝑒1
+∗
+𝑒1
+∗
+𝑒2
+∗
+(a)
+(b)
+I Injection
+Time
+I Injection
+II Arrival
+III Pair
+Time
+I Injection
+III Pair
+II Arrival
+III Pair
+II Arrival
+FIG. 2. Time-domain interferometry. (a) I A quasiparticle
+is injected from the sourced, left edge, through the injection
+QPC, and into the upper edge. II The injected quasiparti-
+cle, by virtue of its chiral motion along the edge, arrives at
+the tunneling QPC. III A quasiparticle-quasihole pair is cre-
+ated at the tunneling QPC. (b) The two processes by which
+charge carriers may ultimately arrive at the drain. The in-
+jected quasiparticle arrives at the tunneling QPC either before
+(upper panel) or after (lower panel) the creation quasiparticle-
+quasihole pair. These two processes interfere, with a relative
+phase dictated by the mutual statistics phase, ei2θ12.
+Eq. (5) can now be re-written as
+I ∝ −
+ˆ t
+−∞
+dt′ �
+b=±
+b
+�� |t, t0⟩b + |t′, t0⟩b
+��2.
+(11)
+The expression above involves two interference terms.
+The term with b = − is an interference between cre-
+ation of −e∗
+1 quasiholes on the upper edge at the QPC at
+times t and t′. The two interfering processes are shown
+schematically in Fig. 2(b).
+As shown in the first row
+of Eq. (9), these two processes are distinguished by a
+non-trivial phase of ei2θ12 if the arrival time t0 is in be-
+tween the quasiholes’ creation times, t′ < t0 < t. Com-
+bined with the equilibrium correlation function Eq. (6),
+one finds that this interference gives a term proportional
+to cos (2θ12 − 2πδ). Using similar arguments, the term
+with b = + in Eq. (11), gives an interference term pro-
+portional to cos (2θ12 + 2πδ).
+The total contribution
+from the two terms in Eq. (11) is thus proportional to
+sin (2θ12) sin (2πδ) [41].
+This interference happens entirely in the time domain,
+and along only one edge. It is however crucial that this
+edge be part of a two-dimensional bulk. This is important
+both because the second edge is required to absorb the
+leftover quasiparticle or quasihole resulting from the pair
+creation at the QPC, and because the injected quasiparti-
+cle must be created within a bulk FQH droplet. Further-
+more, the bulk is intimately related to the edge through
+bulk-edge correspondence. This dictates that the statisti-
+cal phase contributing to time-domain interference along
+a single edge, which our setup measures, is the same as
+the phase obtained from spatial exchange.
+It is easy to generalize this to injection of multiple
+quasiparticles: as long as all injected quasiparticles are
+mutually independent, each injected quasiparticle con-
+tributes a phase of e2iθ12 if and only if the arrival time
+at the point contact was between t′ and t. If we assume
+this is a Poissonian process, with a quasiparticle injection
+rate of Iinj/e∗
+2, we obtain for t > 0
+⟨ ˆA†(t) ˆA(0)⟩dilute
+⟨ ˆA†(t) ˆA(0)⟩0
+=
+∞
+�
+n=0
+(tIinj/e∗
+2)ne−tIinj/e∗
+2
+n!
+e2inθ12
+= e−tIinj/e∗
+2(1−e2iθ12).
+(12)
+This is precisely the result given in Refs. [23, 25] for injec-
+tion along a single edge. Adding injected quasiparticles
+to the lower edge and generalizing for t < 0 are straight-
+forward using the same arguments.
+Currents.— The effect of driving the system out of
+equilibrium is completely encapsulated in the correlation
+functions obtained above.
+These can then be used to
+derive any observable of interest, such as charge or heat
+currents in any of the system’s drains, or their respective
+auto- and cross-correlations.
+For concreteness, we present the explicit results of such
+a calculation for the charge current at the lower drain,
+denoted as I in Fig. 1. We show that a simple cohort
+of current measurements is sufficient to obtain the mu-
+tual statistics θ12, without requiring correlation measure-
+ments.
+We focus on the regime where the temperature is large
+compared to the injected current ℏIinj/ekBT.
+For the
+full limit, this assumption guarantees linear response in
+the voltage and in the injected current, which in this
+limit is Iinj = σxyV . For the dilute limit, the exponen-
+tial suppression of the equilibrium correlation function at
+times larger than ℏ/T, guarantees that the exponent in
+Eq. (12) may be expanded to first order in Iinj. Conse-
+quently,
+⟨ ˆA†(t) ˆA(t′)⟩full/dilute
+⟨ ˆA†(t) ˆA(t′)⟩0
+≈ 1 + iωf/d (t − t′) ,
+(13)
+where the frequencies ωf/d are given by
+ωf = e∗
+1V
+ℏ
+= e∗
+1
+ℏ
+Iinj
+σxy
+;
+ωd = iIinj
+e∗
+2
+�
+1 − e2iθ12�
+.
+(14)
+The zeroth order term corresponds to the equilibrium
+state and does not contribute to the current. The ratio
+of the two first order contributions is Eq. (1).
+Explicit calculation of the resulting current in Eq. (5),
+given in App. A, finds that
+Ifull/dilute = 2πe∗
+1(ξτc)2(2πTτc)4δ1−2B (2δ1, 2δ1) Re
+�
+ωf/d
+�
+,
+(15)
+where B(x, y) is the Euler Beta function. It is thus imme-
+diately apparent that by focusing on the ratio between
+the full and dilute beams, all dependence on δ1, T and ξ
+drops out. Examining the ratio I/Iinj, and noting that
+σxyℏ/e∗
+1e∗
+2 = νe2/2πe∗
+1e∗
+2 we thus obtain Eq. (1).
+
+5
+For general temperatures, the current can no longer be
+treated as a linear response to the drive of the full or di-
+lute beams. We hence obtain the typical power laws char-
+acterizing tunneling in Luttinger liquids [2, 34, 42, 43].
+Comparing measurements of the full and dilute limits at
+the low temperature limit T ≪ e∗V, Iinj can still give a
+quantity related to the mutual statistics θ12, but will ex-
+plicitly depend on the value of δ1. We present general
+expressions for the current in this case in App. A.
+For a fermionic θ12 = π, Eq. (15) gives no current at all
+for a dilute electron beam. However, Landauer-Buttiker-
+Imry scattering theory [44] tells us the current is given
+by the product of the transparencies of the two QPCs
+along the electron’s path, regardless of whether they are
+close to full transmission or full reflection. This requires
+accounting for the direct tunneling term in Eq. (1), which
+now becomes the leading contribution.
+We do this by accounting for the finite width of the
+soliton. This leads to the required, Landauer-Buttiker-
+Imry consistent result of Idilute = 4π2τ 2
+c ξ2Iinj. The phys-
+ical intuition behind the requirement of a finite soliton
+width is that tunneling without time-domain interferom-
+etry, dubbed the direct tunneling process in [24, 25], is
+dominated by short times. Performing these calculations
+explicitly in App. B, we show that the ratio between
+the first term in Eq. (1) and Gdirect is ∝ (Tτs)4δ1−2,
+where τs is the soliton width. It has been shown [24, 25]
+that τ −1
+s
+∝ max{eV, kBT}; as such, to ensure Gdirect is
+sub-dominant, the dilute limit must be measured when
+kBT ≪ eV and 4δ1 < 2.
+Several contemporary experimental setups use the
+equivalent of non-interacting fermionic formulae to rea-
+sonable success [45], corresponding to the limiting value
+of 2δ1 = 1. In this case, the second term of Eq. (1) is
+a numerical coefficient of order one, which may depend
+solely on e∗, δ1 and θ12.
+For non-interacting fermions,
+this coefficient is easily found by comparing to known
+Landauer-Buttiker-Imry scattering theory [44], but it is
+straightforward to generalize. We discuss this coefficient
+further in App. B.
+Discussion.— We propose a simple method to extract
+anyonic exchange statistics.
+Our system consists only
+of a single quantum Hall droplet with two QPCs, which
+effectively create a time-domain interferometer, as can
+be identified from current measurements. We thus avoid
+both current correlation (or noise) measurements, and
+the need for a real space interferometer, making the iden-
+tification of the exchange statistics much more accessible
+than existing experiments. All time-domain interferom-
+etry is between pairs of an injected quasiparticle and a
+tunneling quasiparticle, and occurs at the same edge, as
+previously proposed in Ref. [25].
+Both the exchange statistics θ11 of the tunneling quasi-
+particle, and θ22 of the injection quasiparticle, do not
+appear in our derivation. Rather, it is the two particles’
+mutual statistics, θ12 that affect the modified correlation
+functions, and hence, the physical observables. Likewise,
+the scaling dimension and electric charge which directly
+effect observables are only those of the tunneling quasi-
+particle, δ1 and e∗
+1 (properties of the injected quasiparti-
+cles may implicitly enter through the injection rate).
+Only in the case where the injected and tunneling
+quasiparticles are identical, l1 = l2, do we obtain ex-
+change statistics for a single quasiparticle type. We re-
+mark that this is indeed the case in the experiment of
+Ref. [22], where all quasiparticles are Laughlin e∗ = e/3
+anyons, and subsequent recreations for the ν = 1/3 and
+ν = 2/5 cases [26, 46, 47].
+Interestingly, a recent ex-
+periment employing a similar setup, where the injected
+quasiparticle was a e/3 anyon and the tunneling quasi-
+particle was an electron, observed Andreev-like reflection
+[48]. This is consistent with a mutual statistics phase of
+θ12 = π, for which Eq. (1) gives no time-domain interfer-
+ometry signal.
+Acknowledgements.— We thank Tomer Alkalay, Moty
+Heiblum, Changki Hong, June-Young Lee and H.-S.
+Sim for insightful discussions and comments on the
+manuscript. This work was partially supported by grants
+from the ERC under the European Union’s Horizon 2020
+research and innovation programme (grant agreements
+LEGOTOP No. 788715 and HQMAT No. 817799), the
+DFG (CRC/Transregio 183, EI 519/7-1), the BSF and
+NSF (2018643), the ISF Quantum Science and Technol-
+ogy (2074/19). N.S. was supported by the Clore Scholars
+Programme.
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+Appendix A: Finite temperature current from time-domain interferometry
+Here derive explicit expressions for the tunneling current I at finite temperature T.
+This section neglects the
+contribution Gdirect (see Eq. (1), which is discussed in App. B. We begin with the expression for the current in
+Eq. (5). Writing this explicitly,
+I = e∗
+1ξ2
+ˆ t
+−∞
+dt′
+� �
+ˆA†(t) ˆA(t′)
+�
+−
+�
+ˆA(t′) ˆA†(t)
+�
++
+�
+ˆA†(t′) ˆA(t)
+�
+−
+�
+ˆA(t) ˆA†(t′)
+� �
+.
+(A1)
+In the case where the edges are not driven out of equilibrium, we plug the equilibrium correlation functions Eq. (6),
+and obtain I = 0, as expected. A similar expression can be written for the symmetrized current fluctuations,
+��
+δ ˆIT (t), δ ˆIT (t′)
+��
+= (e∗
+1)2ξ2
+� �
+ˆA†(t) ˆA(t′)
+�
++
+�
+ˆA(t′) ˆA†(t)
+�
++
+�
+ˆA†(t′) ˆA(t)
+�
++
+�
+ˆA(t) ˆA†(t′)
+� �
+,
+(A2)
+where we define δ ˆIT ≡ δ ˆIT − ⟨δ ˆIT ⟩. We do not focus on current fluctuations in this work, but note that our methods
+reproduce the known results of Refs. [23, 25].
+We now want to obtain the current for each of the three methods of driving the two edges out of equilibrium. Each
+of these leads to a corresponding multiplicative factor to the correlation functions. A finite bias voltage V , used for the
+“full” beam, gives the correlation functions of Eq. (7); injection of a single quasiparticle gives the correlation functions
+of Eq. (9); and a dilute, Poissonian beam of quasiparticles gives the correlation functions of Eq. (12). Plugging in
+these appropriate correlation functions gives after minor algebra and changes of variables
+Ifull = 2ie∗
+1ξ2
+ˆ ∞
+0
+d˜t sin
+�e∗
+1V
+ℏ
+˜t
+�� �
+πTτc
+i sinh
+�
+πT
+�˜t − iτc
+��
+�4δ1
+−
+�
+πTτc
+i sinh
+�
+πT
+�
+−˜t − iτc
+��
+�4δ1�
+,
+(A3a)
+Idilute = 2ie∗
+1ξ2
+ˆ ∞
+0
+d˜t
+sin
+�
+Iinj
+e∗
+2 ˜t sin 2θ12
+�
+exp
+�
+Iinj
+e∗
+2 ˜t (1 − cos 2θ12)
+�
+� �
+πTτc
+i sinh
+�
+πT
+�˜t − iτc
+��
+�4δ1
+−
+�
+πTτc
+i sinh
+�
+πT
+�
+−˜t − iτc
+��
+�4δ1�
+,
+(A3b)
+Iqp = 2ie∗
+1ξ2
+ˆ t
+−∞
+dt′ sin (2θ12 [Θ (t − t0) − Θ (t′ − t0)])
+� �
+πTτc
+i sinh (πT (t − t′ − iτc))
+�4δ1
+−
+�
+πTτc
+i sinh (πT (t′ − t − iτc))
+�4δ1�
+.
+(A3c)
+We proceed using the identity, correct at the limit τc → 0,
+i
+� �
+πTτc
+i sinh
+�
+πT
+�˜t − iτc
+��
+�4δ1
+−
+�
+πTτc
+i sinh
+�
+πT
+�
+−˜t − iτc
+��
+�4δ1�
+=
+�
+�
+�
+�
+�
+−2πτ 2
+c ∂˜tδ(˜t)
+2δ1 = 1
+�
+πT τc
+sinh(πT |˜t|)
+�4δ1
+2 sin (2πδ1)sgn(˜t)
+2δ1 ̸= 1
+, (A4)
+where δ(t) is the Dirac delta function. This identity is necessary to treat the case of δ1 = 1, which otherwise may lead
+to divergent integrals.
+
+2
+Standard manipulations of these integrals then give results in terms of the Euler Beta function and the incomplete
+Beta function, B (x; a, b) ≡
+´ x
+0 ta−1(1 − t)b−1dt, B (a, b) ≡ B (1; a, b). We thus obtain the general results
+Ifull = 2e∗
+1ξ2(2πT)4δ1−1τ 4δ1
+c
+sinh
+�e∗
+1V
+2T
+�
+B
+�
+2δ1 + i e∗
+1V
+2πT , 2δ1 − i e∗
+1V
+2πT
+�
+(A5a)
+Idilute = −
+2π
+cos (2πδ1) Γ (4δ1)e∗
+1ξ2(2πT)4δ1−1τ 4δ1
+c
+Im
+�
+�
+Γ
+�
+Iinj
+e∗
+2
+1−cos(2θ12)+i sin(2θ12)
+2πT
++ 2δ1
+�
+Γ
+�
+Iinj
+e∗
+2
+1−cos(2θ12)+i sin(2θ12)
+2πT
++ 1 − 2δ1
+�
+�
+�
+(A5b)
+Iqp = 4e∗
+1ξ2(2πT)4δ1−1τ 4δ1
+c
+sin (2θ12) sin (2πδ) B
+�
+e−2πT (t−t0); 1 + 2δ1, 1 − 4δ1
+�
+.
+(A5c)
+Here, Γ(a) is the Euler Gamma function, satisfying B(a, b) = Γ(a)Γ(b)
+Γ(a+b) , and Im[. . . ] denotes the imaginary part.
+The high temperature and zero temperature limits of the full beam and dilute beam are then immediately repro-
+ducible. For e∗V, ℏIinj/e∗
+2 ≪ kBT, one expands to leading order in e∗V/T and Iinj/e∗
+2T, one obtains
+Ifull ≈ 2πe∗
+1(ξτc)2(2πTτc)4δ1−2B (2δ1, 2δ1) e∗
+1V
+ℏ ,
+(A6a)
+Idilute ≈ 2πe∗
+1(ξτc)2(2πTτc)4δ1−2B (2δ1, 2δ1) Iinj
+e∗
+2
+sin (2θ12) .
+(A6b)
+We thus see that the mutual statistics are immediately extractable from the dilute case. While this expression does
+depend on the non-universal ξ and δ, as well as the temperature, these are all encoded in a prefactor which appears
+in the full case as well. We can hence lose this unwanted prefactor by examining the ratio between the two cases.
+For T ≪ e∗V, Iinj/e∗
+2, we use the identities
+lim
+x→∞ Γ (x + a) = Γ (x) xa,
+lim
+x→∞ sinh(πx)B (a + ix, a − ix) =
+π
+Γ(2a)x2a−1,
+to obtain
+Ifull ≈ 2πe∗
+1ξ2τ 4δ1
+c
+Γ (4δ1)
+�e∗
+1V
+ℏ
+�4δ1−1
+,
+(A7a)
+Idilute ≈ −
+2πe∗
+1ξ2τ 4δ1
+c
+cos (2πδ1)Γ (4δ1)
+�Iinj
+e∗
+2
+�4δ1−1
+Im
+��
+1 − cos (2θ12) + i sin (2θ12)
+�4δ1−1�
+.
+(A7b)
+By tuning 2δ1 → 1, we again obtain an expression from which the mutual statistics are easily extractable, with an
+identical non-universal prefactor appearing in both the full and dilute cases. However, once the scaling dimension is
+tuned to this critical value, the contribution from time-domain interferometry no longer dominates the direct tunneling
+process, as can be seen from the calculation of Gdirect in App. B.
+We note that for temperatures larger than the source voltage, one has to account for injection of both quasiparticles
+and quasiholes through the injection QPC. This can be done by modifying the Poissonian correlation function in
+Eq. (12) according to
+⟨ ˆA†(t) ˆA(0)⟩dilute
+⟨ ˆA†(t) ˆA(0)⟩0
+= e−tIinj/e∗
+2(1−e2iθ12)
+→ e−tIqp
+inj/e∗
+2(1−e2iθ12)e−tIqh
+inj/e∗
+2(1−e−2iθ12),
+(A8)
+where Iqp
+inj is the injection rate of quasiparticles, and Iqh
+inj is the injection rate of quasiholes. This is a similar expression
+to the three QPC setup considered in [23] and [25]. Performing the same algebra as in this section, and identifying
+Iinj ≡ Iqp
+inj − Iqh
+inj, one then reproduces Eq. (A6) for the high temperature limit.
+Finally, it is instructive to consider the current due to the injection of a single quasiparticle at time t0, which
+was obtained in Eq. (A5c). In this case we must examine the explicit temperature dependence, as tunneling of a
+single quasiparticle may be relevant, and we lack any other energy scale to serve as a cutoff for the RG flow of the
+process. This current exhibits a power-law decay for t − t0 ≪ 1/πT, consistent with the orthogonality catastrophe
+that characterizes injection into Luttinger liquid edges. For 2δ1 = 1, this results in ⟨ˆIT ⟩qp ∝ δ (t − t0). This gives
+some intuition as to what makes the 2δ1 = 1 case so unique - the QPC just scatters the incident particle with some
+probability, without inducing any long-time correlations, resulting in the direct tunneling process.
+
+3
+Appendix B: Finite soliton width: restoring Landauer-Buttiker-Imry for electrons and subleading corrections
+The results of App. A are seemingly inconsistent with the known non-interacting electron limits. Indeed, inserting
+e∗
+1 = e∗
+2 = e, 2δ1 = 2δ2 = 1 and θ12 = π into these results would indicate that the dilute electron beam gives no
+current at all. This is in direct contrast with the intuition of Landauer-Buttiker-Imry scattering theory, which would
+indicate that the current should be given by the product of the transparencies of the two QPCs along the electron’s
+path, regardless of whether they are close to full transmission or full reflection.
+The culprit of this result is a peculiarity of soliton physics. The boson field φ is compact under φ �→ φ + 2π. As
+such, a soliton of height 2πK−1l2 would appear to leave the boson field completely unperturbed if K−1l2 is an integer.
+This corresponds precisely to electron injection operators [2]. As such, our soliton description is ill-equipped to treat
+electrons without modifications.
+We solve this issue by introducing a finite width to the soliton, τs. To fully recreate the known non-interacting
+result, it is crucial to maintain an order of limits such that the soliton width is larger than the short-time cutoff, τc.
+We note that we still take care to ensure that τs < 1/T, (Iinj/e∗
+2)−1, i.e. the solitons are still narrow compared to
+the larger time scales in the problem. Previous works [24, 25], performing a full Keldysh calculation, have shown the
+soliton width (refered to in the cited papers as the temporal width) is given by the voltage, h/e∗V , if eV > kBT,
+and by the inverse temperature ℏ/kBT if eV ≲ kBT; as such, the dilute limit must be measured in the regime
+Iinj/e∗
+2 ≪ kBT ≪ eV .
+Formally, this means that injecting a quasiparticle into the upper edge at the location x0 and time t0 transforms
+the boson field according to
+φ(u)(x, t0) �→ φ(u)(x, t0) − 2πK−1l2
+� 1
+π tan−1
+�x − x0
+τs
+�
+− 1
+2
+�
+.
+Accordingly, the correlation functions of Eq. (9) are now replaced with
+⟨ ˆA†(t) ˆA(t′)⟩qp = ⟨ ˆA†(t) ˆA(t′)⟩0 exp
+�
+2iθ12
+π
+�
+tan−1
+�t − t0
+τs
+�
+− tan−1
+�t′ − t0
+τs
+���
+,
+⟨ ˆA(t) ˆA†(t′)⟩qp = ⟨ ˆA(t) ˆA†(t′)⟩0 exp
+�
+−2iθ12
+π
+�
+tan−1
+�t − t0
+τs
+�
+− tan−1
+�t′ − t0
+τs
+���
+.
+(B1)
+One indeed sees that at the limit τc → 0, one reproduces the immediate soliton results from the main text.
+To find the correlation function in the presence of a dilute, Poissonian beam of injected quasiparticles, we now
+must sum over the number of injected quasiparticles, in a manner similar to Eq. (12). However, this is now trickier,
+for two reasons. First, the accumulated phase explicitly depends on the time of the injected quasiparticle. Second,
+injected quasiparticles outside of the window [0, t] can still affect the correlation function, due to the long tails of the
+finite-width solitons.
+So generalizing the methods that lead to Eq. (12), the correlation function now changes to define
+⟨ ˆA†(t) ˆA(0)⟩fw
+⟨ ˆA†(t) ˆA(0)⟩0
+=
+�
+n
+�
+(t + 2cτc) Iinj
+e∗
+2
+�n
+e
+−(t+2cτc)
+Iinj
+e∗
+2
+n!
+�ˆ t+cτc
+−cτc
+dt0P (Particle injected at t0) e2i θ12
+π [tan−1(
+t−t0
+τs )−tan−1(
+0−t0
+τs )]
+�n
+.
+(B2)
+Here c is some unitless cutoff, chosen such that injected quasiparticles affect the correlation function only if they are
+injected in the window [−cτc, t + cτc], which we will eventually take to be infinite. The probability of injection at a
+particular time t0 is given by
+P (Particle injected at t0) =
+Iinj/e∗
+2e−Iinjt0/e∗
+2
+´ t+cτc
+−cτc dt0Iinj/e∗
+2e−Iinjt0/e∗
+2 .
+(B3)
+Performing this sum, and re-defining this integration with unitless variables, we find that the new correlation function
+is given in integral form by
+⟨ ˆA†(t) ˆA(0)⟩fw
+⟨ ˆA†(t) ˆA(0)⟩0
+= exp
+�
+− (t + 2cτc) Iinj
+e∗
+2
+�
+1 − Iθ12
+�Iinj
+e∗
+2
+τs, t
+2τs
+���
+,
+Iθ12 (a, b) ≡
+a
+2 sinh (a(b + c))
+ˆ b+c
+−b−c
+dxe−axe2i θ12
+π [tan−1(x+b)−tan−1(x−b)].
+(B4)
+
+4
+By plugging this new correlation function into the expression for the current in Eq. (5), one now finds
+Idilute = 2ie∗
+1ξ2
+ˆ ∞
+0
+d˜t
+sin
+�
+(˜t + 2cτc) Iinj
+e∗
+2 Im
+�
+Iθ12
+�
+Iinj
+e∗
+2 τs,
+t
+2τs
+���
+exp
+�
+(˜t + 2cτc) Iinj
+e∗
+2 Re
+�
+1 − Iθ12
+�
+Iinj
+e∗
+2 τs,
+t
+2τs
+���
+×
+� �
+πTτc
+i sinh
+�
+πT
+�˜t − iτc
+��
+�4δ1
+−
+�
+πTτc
+i sinh
+�
+πT
+�
+−˜t − iτc
+��
+�4δ1�
+.
+(B5)
+Careful re-application of the limit τc → 0 indeed replicates our previous result in Eq. (12).
+For general θ12, the integral Iθ12 (a, b) is difficult to solve analytically. In the main text, this is circumvented by
+taking the limit τc → 0, allowing use of Eq. (A4), in conjunction with replacing (˜t+2cτc) Iinj
+e∗
+2 Iθ12
+�
+Iinj
+e∗
+2 τs,
+t
+2τs
+�
+→ −i˜tωd.
+However, as noted previously, fermionic exchange statistics corresponding to values of θ12 that are integer multiples
+of π lead to ωd = 0, and hence give a vanishing current. As such, Eq. (B5) must be calculated in full while retaining
+a finite τc.
+To simplify these expressions, we assume that
+�
+Iinj
+e∗
+2
+�
+is significantly larger than any other time scale in the system.
+This makes sense from a physical perspective as well, as it corresponds to the assumption that injection is sufficiently
+rare such that solitons do not overlap. In this case, one can assume the probability of injection which appears in
+Eqs. (B2),(B3) is approximately uniform, i.e. P (Particle injected at t0) ≈ 1/(t + 2cτc). One can now safely take the
+limit c → ∞ without artificial divergences, giving the simpler result,
+⟨ ˆA†(t) ˆA(0)⟩fw
+⟨ ˆA†(t) ˆA(0)⟩0
+= exp
+�Iinj
+e∗
+2
+ˆ ∞
+−∞
+dt0
+�
+e2i θ12
+π [tan−1(
+t−t0
+τs )−tan−1(
+0−t0
+τs )] − 1
+��
+.
+(B6)
+Since we undertook this endeavor with the explicit goal of finding the correct result for non-interacting electrons,
+we wish to find this integral for 2δ1 = 1, θ12 = π, and e∗
+1 = e∗
+2 = e. This value of θ12 allows one to significantly
+simplify Eq. (B6) using trignometric identities; plugging the resulting correlation function in Eq. (A1), we obtain
+Iθ12=π
+dilute = 2ie∗
+1ξ2
+ˆ ∞
+0
+d˜t
+sin
+�
+Iinj
+e∗
+2
+2π˜t(2τs)2
+˜t2+(2τs)2
+�
+exp
+�
+Iinj
+e∗
+2
+2π˜t2(2τs)
+˜t2+(2τs)2
+�
+� �
+πTτc
+i sinh
+�
+πT
+�˜t − iτc
+��
+�4δ1
+−
+�
+πTτc
+i sinh
+�
+πT
+�
+−˜t − iτc
+��
+�4δ1�
+.
+(B7)
+As can be seen in Eq. (A4), the expression in the curled brackets is approximately zero for ˜t > τc. We can thus
+approximate the total integral as the contribution from short times, ˜t ≤ τc ≪ 1/πT. To leading order, this will be
+given by
+Iθ12=π,2δ1=1
+dilute
+≈ 2ie∗
+1ξ2τ 2
+c
+ˆ ∞
+0
+d˜tIinj
+e∗
+2
+2π˜t(2τs)2
+˜t2 + (2τs)2
+� �
+1
+i˜t + τc
+�2
+−
+�
+1
+−i˜t + τc
+�2�
+=
+(2τs)2
+(2τs + τc)2 4π2ξ2τ 2
+c Iinj.
+(B8)
+Now taking the limit τc ≪ τs, we compare to the electron case in, say, Eq. (15) or Eq. (A5). We find that the
+result we expect for non-interacting electrons is indeed 4π2ξ2τ 2
+c Iinj. This is consistent with - the current is linear in
+the injected current, and in the transparency of the tunneling QPC (which is given by ξ2τ 2
+c ).
+For general values of θ12 and δ1 this integral is more difficult to solve analytically. However, it is possible to re-write
+Eq. (B6) as
+⟨ ˆA†(t) ˆA(0)⟩fw
+⟨ ˆA†(t) ˆA(0)⟩0
+= exp Iinj
+e∗
+2
+�
+sin (2θ12) t + fθ12 (t, τc))
+�
+,
+(B9)
+fθ12 (t, τc)) ∝
+�
+�
+�
+�
+�
+t
+t ≲ τs
+τs
+t ≫ τs, θ12 ̸= π
+(τs)2/t
+t ≫ τs, θ12 = π.
+(B10)
+Plugging this into the general expression for the current, and expanding to linear response in Iinj
+e∗
+2 we find
+
+5
+Idilute = 2ie∗
+1ξ2 Iinj
+e∗
+2
+ˆ ∞
+0
+d˜t
+�
+sin (2θ12) t + fθ12 (t, τc))
+�� �
+πThwτc
+i sinh
+�
+πT
+�˜t − iτc
+��
+�4δ1
+−
+�
+πTτc
+i sinh
+�
+πT
+�
+−˜t − iτc
+��
+�4δ1�
+.
+(B11)
+The term proportional to sin (2θ12), as discussed at length above, is the main interest of this paper. This is calculated in
+Eq. (A6). We see there that the time scales in the system contribute a leading term of the form ∝ (ξτc)2(Tτc)4δ1−2Iinj.
+The term proportional to fθ12 (t, τc)) contains several contributions: at short times (˜t ∼ τc), we obtain a con-
+tribution of order (ξτc)2; at long times (˜t ∼ 1/πT) we obtain a contribution of order (ξτc)2(τs/τc) (Tτc)4δ1−1 for
+θ12 ̸= π and (ξτc)2(τs/τc) (Tτc)4δ1 for θ12 = π; and at intermediate times (˜t ∼ τs) we obtain contributions of order
+(ξτc)2(τs/τc)1−4δ1 and (ξτc)2(τs/τc)2−4δ1.
+We compare these contributions to the coefficients of Eq. (15) or Eq. (A6), which give the time-domain interferom-
+etry process, which is of order (ξτc)2 (Tτc)4δ1−2. Utilizing τc ≪ τs ≪ 1/πT, we see that the long time contribution
+is always subdominant, but the short time dominates for 2δ1 ≥ 1 - consistent with both Eq. (B8) and the known
+electron result. This is consistent with our physical intuition: direct tunneling dominates short times, which give
+the main contribution for 2δ1 ≥ 1, whereas time-domain interferometry dominates long times, which give the main
+contribution for 2δ1 < 1.
+Finally, if we indeed assume 2δ1 < 1, the intermediate time contribution dominates the entire direct process. In
+this case, the ratio between the time-domain interferometry process and the direct process is given by ∝ (Tτs)4δ1−2.
+This again confirms that we must have a soliton width smaller than the inverse temperature to ensure time-domain
+interferometry
+This method is also what we use to calculate the current for an almost full beam, i.e. σxy − Ginj ≪ 1. Since
+in this case, the beam can be treated as a conjoined full beam of fractional quasiparticles with a dilute beam of
+e∗ = e holes, we have 2θ12 = 2πn regardless of the tunneling quasiparticles. Defining the injection rate of holes as
+Iholes
+inj
+≡ σxyV − Iinj, we combine the full beam correlation function of Eq. (7) and the regularized Poissonian hole
+injection to obtain described in this section
+I|Ginj−σxy|≪1 = 2ie∗
+1ξ2
+ˆ ∞
+0
+d˜t
+sin
+�
+e∗
+1V
+ℏ ˜t −
+Iholes
+inj
+e
+2π˜t(2τs)2
+˜t2+(2τs)2
+�
+exp
+� Iholes
+inj
+e
+2π˜t2(2τs)
+˜t2+(2τs)2
+�
+×
+� �
+πTτc
+i sinh
+�
+πT
+�˜t − iτc
+��
+�4δ1
+−
+�
+πTτc
+i sinh
+�
+πT
+�
+−˜t − iτc
+��
+�4δ1�
+.
+(B12)
+In the relevant limits, the same methods as previously mention allow us to approximate the exponent in the
+denominator as 1, and to expand the sine in the numerator. We thus have the sum of two linear responses, one in in
+e∗
+1V
+ℏ
+and one in −
+Iholes
+inj
+e
+. Taking, as in the Landauer-Buttiker-Imry case, the limit τs ≫ τc, i.e. a soliton width that is
+larger than the short time cutoff, this can be re-written as
+I|Ginj−σxy|≪1 ≈ 2ie∗
+1ξ2
+ˆ ∞
+0
+d˜t
+�
+e∗
+1V
+ℏ
+− 2π Iholes
+inj
+e
+�
+˜t
+� �
+πTτc
+i sinh
+�
+πT
+�˜t − iτc
+��
+�4δ1
+−
+�
+πTτc
+i sinh
+�
+πT
+�
+−˜t − iτc
+��
+�4δ1�
+.
+(B13)
+Identifying
+�
+e∗
+1V
+ℏ
+− 2π
+Iholes
+inj
+e
+�
+= 2π
+e
+�
+σxyV − Iholes
+inj
+�
+≡ 2π
+e Iinj, we see that this is precisely the same integral that we
+had in Eq. (A3a) for the full beam case, with the replacement σxyV → Iinj. We note that we used here σxy = ee∗/h,
+which is correct only for Laughlin edge states, ν = 1/m; this is valid as Laughlin edges are the outer level of heirarchal
+FQH fluids, and thus are the states of interest for nearly full closed QPCs.
+

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@@ -0,0 +1,1082 @@
+ 
+ 
+Breakdown-Limited Endurance in HZO FeFETs: Mechanism and 
+Improvement Under Bipolar Stress 
+Kasidit Toprasertpong1*, Mitsuru Takenaka1, Shinichi Takagi1  
+1 Department of Electrical Engineering and Information Systems, the University of Tokyo, Tokyo, 
+Japan 
+* Correspondence:  
+Kasidit Toprasertpong 
+toprasertpong@mosfet.t.u-tokyo.ac.jp 
+Keywords: Ferroelectrics, MOSFET, reliability, oxide breakdown, substrate hole current. 
+Abstract 
+Breakdown is one of main failure mechanisms that limit write endurance of ferroelectric devices 
+using hafnium oxide-based ferroelectric materials. In this study, we investigate the gate current and 
+breakdown characteristics of Hf0.5Zr0.5O2/Si ferroelectric field-effect transistors (FeFETs) by using 
+carrier separation measurements to analyze electron and hole leakage currents during time-dependent 
+dielectric breakdown (TDDB) tests. Rapidly increasing substrate hole currents and stress-induced 
+leakage current (SILC)-like electron currents can be observed before the breakdown of the 
+ferroelectric gate insulator of FeFETs. This apparent degradation under voltage stress is recovered 
+and the time-to-breakdown is significantly improved by interrupting the TDDB test with gate voltage 
+pulses with the opposite polarity, suggesting that defect redistribution, rather than defect generation, 
+is responsible for the trigger of hard breakdown.  
+ 
+1 
+Introduction 
+HfO2-based ferroelectric thin films have been actively employed in recent electron device research 
+thanks to their CMOS compatibility, established know-how on the fabrication process, and high 
+scalability of thickness to 10 nm or lower (Böscke et al., 2011a; Müller et al., 2012; Park et al., 2015; 
+Migita et al., 2018b; Kim et al., 2018; Tan et al., 2021; Toprasertpong et al., 2022a; Schroeder et al., 
+2022). Ferroelectric field-effect transistors (FeFETs) with HfO2-based ferroelectric thin films as gate 
+insulators have received considerable attention, not only because of the maturity of the HfO2 
+deposition technology in the advanced transistor process, but also because of their low energy 
+consumption, high speed, and satisfactory retention during their operation. HfO2-based FeFETs have 
+been investigated as promising devices for low-power nonvolatile memory (Böscke et al., 2011b; 
+Trentzsch et al., 2016; Dünkel et al., 2017; Florent et al., 2018a; Müller et al., 2021) and non-von 
+Neumann computing applications (Jerry et al., 2018; Dutta et al., 2022; Matsui et al., 2021; Luo et 
+al., 2022; Toprasertpong et al., 2022b).  
+Despite their excellent properties, one of the most crucial issues to be dealt with towards the practical 
+use of HfO2-based FeFETs is the write endurance. There are two major mechanisms that have been 
+reported to determine the write endurance of FeFETs: the memory window narrowing and gate 
+dielectric breakdown. The memory window narrowing refers to a phenomenon where a separation of 
+
+Breakdown-Limited Endurance in HZO FeFETs: Mechanism and Improvement Under Bipolar Stress 
+ 
+2 
+the threshold voltages of the two states (high and low threshold voltage states) becomes gradually 
+smaller and eventually becomes zero after certain operating cycles. The polarization states are no 
+longer able to be read out through threshold voltages and FeFETs lose a capability as memory 
+devices. On the other hand, gate dielectric breakdown refers to a situation where the gate insulator 
+experiences hard breakdown under a certain amount of electrical stress. Hard breakdown makes gate 
+insulators conductive, electrically connects the gate and channel, and causes FeFETs to lose their 
+function as field-effect transistors.  
+Memory window narrowing and gate dielectric breakdown originate from different physics and occur 
+almost independently; therefore, the write endurance of FeFETs, i.e. a number of write operations 
+before failure, is determined by the mechanism that leads to earlier failure. The dominant mechanism 
+depends on the device property and the operation scheme of each specific device and application. 
+Write endurance of state-of-the-art FeFETs is typically dominated by the memory window narrowing 
+(Böscke et al., 2011b; Yurchuk et al., 2016; Trentzsch et al., 2016; Dünkel et al., 2017; Florent et al., 
+2018a; Gong et al., 2018) because of the presence of large density of trapped charges in the vicinity 
+of the interfacial layer (IL) between HfO2 and Si (Toprasertpong et al., 2019; Toprasertpong et al., 
+2020a), while there are only a few reports showing that endurance of FeFETs is limited by gate 
+dielectric breakdown (Ni et al., 2018; Peng et al., 2021). That is, the FeFET operation so far usually 
+reaches failure because of memory window narrowing before gate dielectric breakdown occurs; thus, 
+there is still a poor understanding of the gate dielectric breakdown mechanism in HfO2-based 
+FeFETs. On the other hand, a lot of effort has been put on the material and device-structure 
+engineering such that there have already been some reports in recent years demonstrating FeFET 
+memory devices with remarkably suppressed memory window narrowing (Sharma et al., 2020; Yan 
+et al., 2020; Tan et al., 2021; Liao et al., 2022). In such devices with suppressed memory window 
+narrowing, gate dielectric breakdown may become a dominant mechanism that limits write endurance 
+and play a crucial role in device reliability. Furthermore, there are some applications of FeFETs using 
+new-concept computing that are insensitive to memory window narrowing, such as reservoir 
+computing (Nako et al., 2022). In such applications, gate dielectric breakdown will be a dominant 
+endurance-limiting mechanism. Therefore, gaining an understanding of the mechanism of gate 
+dielectric breakdown is important to improve the overall write endurance characteristics of HfO2-
+based FeFETs.  
+In this study, we investigate the breakdown characteristics and the stress-induced degradation 
+behavior as well as the underlying physical mechanism in Hf0.5Zr0.5O2 (HZO)/IL/Si FeFETs. The 
+carrier separation measurement and interrupted stress for time-dependent dielectric breakdown 
+(TDDB) evaluation are employed to analyze the physical mechanism underlying gate dielectric 
+breakdown. 
+2 
+Sample Preparation 
+The process flow is shown in Figure 1A. We fabricated n-channel non-ferroelectric FETs (called here 
+as nonferro-FET) with a paraelectric HfO2 gate insulator and FeFETs with a ferroelectric HZO gate 
+insulator on p-type Si substrates with a moderate doping concentration of 4×1015 cm-3. After the 
+source and drain (S/D) regions were doped by phosphorus ion implantation and annealed to activate 
+dopants, the Si substrates were cleaned by hydrochloric-peroxide mixture (HPM)-last cleaning 
+process to grow a high-quality SiO2 IL (Toprasertpong et al., 2020). For FeFETs, 10-nm-thick 
+ferroelectric HZO was deposited by atomic layer deposition (ALD) using at using 
+tetrakis(ethylmethylamino)hafnium (TEMAH), tetrakis(ethylmethylamino)zirconium (TEMAZ), and 
+H2O at 300°C. For nonferro-FETs, 10-nm-thick HfO2 was deposited in a similar way but without 
+
+ 
+ Breakdown-Limited Endurance in HZO FeFETs: Mechanism and Improvement Under Bipolar Stress 
+ 
+ 
+3 
+TEMAZ. TiN was deposited as gate metal by sputtering and Al:Si was deposited as S/D contacts by 
+thermal evaporation. Samples were annealed at 400°C for 30 s in a N2 atmosphere to crystalized the 
+ferroelectric phase in FeFETs. The nonferro-FETs were also annealed at the same condition. Except 
+the ALD step, both samples were processed simultaneously in the same chamber to ensure the same 
+device condition. Figures. 1B and 1C show transmission electron microscopic (TEM) images of the 
+gate stacks of a nonferro-FET and a FeFET, respectively, indicating that HZO was crystallized 
+whereas HfO2 remained amorphous. The IL thickness was similar in the both samples. 
+3 
+Results and Discussion 
+3.1 
+Band diagram and breakdown position 
+Before we discuss the experimental results of the leakage and breakdown behaviors, we examine the 
+band diagram of the HZO (10 nm)/IL (0.7 nm)/Si gate stack and the possible gate leakage path. 
+Figure 2A depicts an example of an ideal band diagram of the HZO/IL/Si gate stack at 3 V when 
+HZO has ferroelectric polarization of 10 µC/cm2. Due to high ferroelectric polarization, most 
+literature considers a band diagram with a strong electric field across the IL, which significantly pulls 
+down the band position HZO, as shown in Figure 2A (Müller et al., 2016; Yurchuk et al., 2016; Gong 
+et al., 2018; Peng et al., 2021; Mulaosmanovic et al., 2021). In such a case, the breakdown of the IL 
+is supposed to determine the gate dielectric breakdown of FeFETs. However, it has been reported that 
+a large density of trapped charges near the HZO/IL interface electrically screens the polarization and 
+suppresses the electric field across the IL (Toprasertpong et al., 2019; Toprasertpong et al., 2022c). 
+Figure 2B depicts the band diagram with ferroelectric polarization of 10 µC/cm2 and 90% (Ichihara 
+et al., 2020) of induced electrons are trapped at the HZO/IL interface. It can be seen that the band of 
+HZO is not at such a low energy position. This fact indicates that electrons have to tunnel through a 
+thick HZO layer and thus the breakdown of HZO is necessary to describe the gate breakdown failure 
+of FeFETs.  
+3.2 
+Device characteristics  
+The I-Vg characteristics of the nonferro-FET and FeFET are shown in Figures 3A and 3B, 
+respectively, for gate current Ig, drain current Id, source current Is, and substrate current Isub. A gate 
+length L is 10 µm and a gate width W is 100 µm. As expected, the nonferro-FET exhibits the Id-Vg 
+characteristics with clockwise hysteresis, which is a feature of electron trapping during Vg scans. On 
+the other hand, the FeFET exhibits counterclockwise hysteresis, which is a feature of ferroelectricity, 
+with a memory window of approximately 1.8 V. Comparison of the I-Vg characteristics of the 
+nonferro-FET and FeFET indicates interesting features on Ig and Isub. Gate current Ig in the HZO 
+FeFET is much larger by several orders of magnitude than in nonferro-FETs having HfO2 with a 
+similar physical thickness. This can be understood from the fact that the poly-crystallinity and a lot of 
+defects such as oxygen vacancies in HZO can promote the gate leakage current, as shown in Figure 
+3C. It is also found that the substrate current Isub in the FeFET rapidly increases by four orders of 
+magnitude in a narrow range of Vg = 3.6 V to 4.0 V during the forward Vg scan, which is in the same 
+range that Ig also increases rapidly by two orders of magnitude. This finding suggests that a study of 
+the behavior of Isub would be helpful in understanding the behavior of the gate leakage and gate 
+dielectric degradation. The nonferro-FET in Figure 3A does not exhibit this Isub behavior. 
+3.3 
+Carrier Separation Measurements 
+
+Breakdown-Limited Endurance in HZO FeFETs: Mechanism and Improvement Under Bipolar Stress 
+ 
+4 
+Carrier separation measurements (Eitan et al., 1983; Weinberg et al., 1985) were carried out to 
+analyze the behavior of gate leakage and gate dielectric degradation. The electrical measurement tool 
+(Keysight B1500A with high-resolution source/monitor unit modules) was connected with FETs in a 
+way shown in Figure 4A, where Vd = Vs = Vsub = 0. The current detected at the S/D terminal 
+corresponds to the electron component of gate current, denoted by Ie, while the current detected at the 
+substrate corresponds to the hole component, denoted by Ih. When Vg is larger than the threshold 
+voltage, Ie corresponds to the tunneling current of inversion electrons from the Si substrate to the 
+gate, whereas Ih corresponds to the sum of the tunneling current of valance-band electrons in the Si 
+substrate to the gate (Weinberg et al., 1985; Schuegraf et al., 1994b; Shanware et al., 1999) and the 
+tunneling back current of holes from the gate to the Si substrate (Schuegraf et al., 1994a; Schuegraf et 
+al., 1994b; Kobayashi et al., 1995), as illustrated in Figure 4B.  
+The results of the carrier separation measurements are shown in Figures 4C and 4D for the HfO2 
+nonferro-FET and HZO FeFET, respectively. In these measurements, Vg of pristine samples was 
+scanned from 0 V to the positive voltage where breakdown occurs. It can be seen that tunneling of 
+inversion electrons is the main contribution of Ig for both the nonferro-FETs and FeFET. Ih is found 
+to be under detection limit in a low Vg regime, but it rapidly increases at Vg close to the breakdown 
+voltage. The breakdown voltage VBD of the nonferro-FET is approximately 5.2 V, whereas the FeFET 
+reaches hard breakdown much earlier at approximately VBD = 4.1 V. Earlier breakdown is contributed 
+to more defects in HZO than those in HfO2, in agreement with larger gate current shown in Figures 
+3A and 3B. Hard breakdown of the nonferro-FET occurs at comparatively low Ih, whereas Ih of HZO 
+FeFET keeps noisy until very high level of Ih. After breakdown, the electrical properties of the gate 
+insulators of both the devices become ohmic and dominated by electron current, as shown in Figures 
+4E and 4F.  
+The band alignments are shown in Figures 4G-4I. At small Vg, it is clear from the band alignment 
+that electrons in the conduction band of Si can easily tunnel to the gate. At Vg in the mid-range, both 
+electrons in the valence band and holes generated at the gate can tunnel more easily, resulting in 
+increasing Ih. At large Vg, an electric field across HZO is so large that hole tunneling back can reach 
+the valence band of HZO, resulting in large Ih. Increasing hole tunneling back consequently causes 
+breakdown in the gate insulator, as the hole tunneling back is known to be the main cause of damage 
+in the gate insulator (Schuegraf et al., 1994a; Schuegraf et al., 1994b; Takayanagi et al., 2001).  
+Results of repeated measurements of Ie and Ih in a Vg scan range of -2 V to 4 V are shown in Figure 5. 
+It is interesting that rapidly increasing Ih and Ie at Vg > 3.5 V in the FeFET, together with noisy 
+signals before breakdown, are recovered during the Vg backward scan, resulting in repeatable Ih-Vg 
+and Ie-Vg characteristics. These results imply that, although rapidly increasing Ih is an indication that 
+breakdown is going to be triggered, the permanent degradation still does not occur yet in this 
+condition and occurs when Ih increases in a step-wise manner, which can be observed in Figure 4D at 
+Vg = 4.1 V.  
+The analysis above suggests that Ih is a convenient indicator for determining appropriate operating 
+range of Vg. Figure 6 shows the I-Vg characteristics of the FeFET when Vg was kept below 3.5 V. In 
+this Vg range, the ferroelectric hysteresis can still be achieved with a satisfactory memory window of 
+1.7 V while Ih is suppressed to under the detection limit. Note that Isub at negative Vg is due to gate-
+induced drain leakage (GIDL), which is unrelated to gate leakage currents. Although Ih does not 
+necessarily imply to device degradation as discussed in Figure 5, hole tunneling back is flowing and 
+leads to a higher probability that breakdown is triggered; therefore, the operating condition with high 
+Ih should be avoided. The reliability of FeFETs operating in this way is notably improved and we 
+cannot observe breakdown under electrical stress for a practically long time (> 105 s).  
+
+ 
+ Breakdown-Limited Endurance in HZO FeFETs: Mechanism and Improvement Under Bipolar Stress 
+ 
+ 
+5 
+3.4 
+Time-dependent dielectric breakdown: Constant voltage stress and interrupted test 
+TDDB tests with a carrier separation setup were carried out to gain more insights into the breakdown 
+behavior of FeFETs. Ie(t) and Ih(t) under constant voltage stress (CVS) as a function of stress time t 
+are shown in Figures 7A and 7B for nonferro-FETs and FeFETs, respectively. Both Ie(t) and Ih(t) of 
+the FeFET increase with time, which is in the opposite direction of Ie(t) of nonferro-FETs in the early 
+stage. Note that Ih of nonferro-FETs is so low that cannot be measured until breakdown, indicating 
+that there is less hole tunneling back in nonferro-FETs. We call the behavior of FeFETs having Ie(t) 
+increasing with time as a SILC-like behavior, as stress-induced leakage current (SILC) refers to a 
+phenomenon that a leakage current increases with electrical stress. This SILC-like behavior of Ie(t) of 
+FeFETs can be fitted with a power-law function to be 
+e ∝
+I
+t , independent of Vg stress, as displayed 
+in Figures 7C. Increasing gate current over time becomes positive feedback to the damage in the gate 
+insulator, leading to breakdown when Ie is raised to the order of A/cm2. The Ie and Ih levels that 
+trigger breakdown are almost independent of the stress voltage Vg.  
+Time-to-breakdown tBD under CVS are summarized in Figures 7D and 7E for nonferro-FETs and 
+FeFETs, respectively. Not only the breakdown at lower Vg than nonferro-FETs but also tBD more 
+sensitive to Vg can be observed for FeFETs, with tBD of approximately 103 s at Vg = 3.75 V reduced to 
+approximately 10-1 s at Vg = 4.2 V. The results of charge-to-breakdown QBD for electrons Qe = 
+e( )
+∫ I t dt  and holes Qe = 
+h( )
+∫ I
+t dt  are summarized in Figures 7F and 7G for non-ferro FETs and 
+FeFETs, respectively. An obvious difference in the QBD-Vg properties in FeFETs and nonferro-FETs 
+can be observed. While the total electron fluence Qe of nonferro-FETs at which the breakdown of 
+HfO2 gate insulators occurs has only a weak dependence on stress voltage (note that Qh could not be 
+extracted as Ih was too low), the total electron Qe and hole fluences Qh at which FeFETs reach 
+breakdown vary in a wide range, implying that the total fluence is not a factor that is responsible for 
+the trigger of breakdown of HZO insulators in FeFETs. Figure 7H shows the ratio of Qe/Qh at 
+different stress voltages. It is interesting that the electron-to-hole ratio of QBD of FeFETs is almost 
+constant independent of stress voltage. This behavior is remarkably different from conventional 
+SiO2-gate MOSFETs, where the hole fluence Qh triggers gate dielectric breakdown and the Qe/Qh 
+ratio is not a constant (Chen et al., 1986; Schuegraf et al., 1994a). This finding indicates that the gate 
+dielectric breakdown mechanism in FeFETs should be different from SiO2-gate MOSFETs. We 
+could not compare with nonferro-FETs as Qh was below the detection limit, so further investigation 
+of the Qe/Qh ratio in nonferro-FETs is needed to specify whether or not the constant Qe/Qh ratio is a 
+unique feature of FeFETs. Further studies of what physical parameters trigger the breakdown of HZO 
+insulators in FeFETs would provide a clearer understanding of the interaction between the leakage 
+current and gate dielectric breakdown event in FeFETs. 
+We have observed from Figure 7B that gate leakage increases with stress time, as similar to a SILC-
+like behavior. Here, we investigate the device behavior during the increase of gate leakage current. 
+Figures 8A and 8C show the I-Vg characteristics before and after a CVS at 4 V for 10 s shown in 
+Figure 8B. Although Ie(t) and Ih(t) increase by approximately 100 times during the 10-s CVS test, it 
+is found that an only small change of the I-Vg characteristics can be observed after stress. This 
+implies that increases of Ie(t) and Ih(t) in FeFETs are not similar to typical SILC, where increasing 
+current cannot be easily recovered: increasing currents in FeFETs can be recovered after releasing the 
+stress.  
+
+Breakdown-Limited Endurance in HZO FeFETs: Mechanism and Improvement Under Bipolar Stress 
+ 
+6 
+This peculiar behavior of the gate leakage current is further investigated by applying interrupt pulses 
+during TDDB tests. Figure 9A displays a voltage waveform when TDDB tests stressed at Vg = 4 V 
+were interrupted by Vg = 0 V for 1 s every stress time of ts. Figures 9B,C show Ie(t) and Ih(t) for each 
+stress cycle when ts = 10 s (cycles of 4 V for 10 s and 0 V for 1 s). Ie(t) and Ih(t) increase cycle by 
+cycle regardless of interrupts by 0 V, implying that electrical stress keeps accumulated. Figure 9D 
+summarizes the time-to-breakdown tBD (excluding interrupt time at 0 V). tBD independent of interrupt 
+frequency indicates that the interrupts at 0 V have no significant effect on tBD. On the other hand, 
+interrupting with negative voltage of Vg = -4 V is different. Figure 9E displays a voltage waveform 
+when interrupted by Vg = -4 V for 1 s every stress time ts. Figures 9F,G illustrate that the SILC-like 
+gate leakage current is recovered after interrupted with Vg = -4 V for 1 s: increasing Ie(t) and Ih(t) are 
+recovered back almost to Ie(t = 0) and Ih(t = 0), respectively, in every cycle. Note that only the 
+current at the first cycle was slightly different because the polarization state of pristine devices is 
+different. This is in agreement with the repeatable Ig-Vg and Isub-Vg in Figure 5. Due to the recovery 
+of SILC-like behavior, applying negative voltage interruption in this way helps extend the time-to-
+breakdown tBD by more than an order of magnitude, as summarized in Figure 9H. 
+3.5 
+Mechanism under voltage stress 
+The behavior of stress recovery by negative interrupt pulses can be found as well in HfO2 nonferro-
+FETs, as shown in Figures 10A,B. These facts suggest that although the leakage current and 
+breakdown voltage of HfO2 nonferro-FETs and HZO FeFETs are different in detail due to 
+differences in crystallinity or defect density, the fundamental mechanisms of the breakdown and 
+recovery behavior should be generally similar in HfO2-based materials, for instance, same type of 
+defect generation. 
+Considering the above findings, we propose the mechanism under high Vg stress, shown in Figure 11. 
+Typically, SILC as well as noisy gate leakage current (PBD; progressive breakdown) under electrical 
+stress before hard breakdown are attributed to the generation of defects such as oxygen vacancies 
+(Olivo et al., 1988; Rofan et al., 1991; DiMaria et al., 1995; Degraeve et al, 1995). On the other hand, 
+the recovery and repeatable behavior of apparently degraded gate leakage currents observed in 
+FeFETs suggests that the defect redistribution should be the main contribution of apparently 
+degraded characteristics rather than the generation of new defects. These defects are redistributed 
+again after applying an opposite voltage pulse, recovered to the condition close to the initial one 
+before stress. This model is supported by the fact that oxygen vacancies can move during the voltage 
+cycling (Pešić et al., 2016; Florent et al., 2018b). However, if the stress is large enough for defects to 
+move to the condition that triggers hard breakdown, suddenly increasing current generates a huge 
+density of defects, which forms a permanent conduction path and results in the failure of the device. 
+Then, the recovery is no longer available for devices that reach the breakdown condition.  
+Such a memory operation that the polarization states are frequently switched in a bipolar manner can 
+help extend the device lifetime in terms of breakdown failure. In other words, not only the 
+improvement in the material aspect but also choosing an appropriate memory operation is important 
+for the reliability of FeFETs. Whereas bipolar operation is favorable to improving the breakdown-
+limited endurance, the memory-window-limited endurance has been reported to have the opposite 
+behavior: memory window narrowing is degraded in a bipolar operation faster than in a unipolar 
+operation (Yurchuk et al, 2014). These findings address that the ideal writing operation on the 
+aspects of breakdown and MW narrowing are different. Thus, the endurance tests for evaluating the 
+real lifetime should be carefully designed. Conventional endurance tests of FeFETs using bipolar 
+
+ 
+ Breakdown-Limited Endurance in HZO FeFETs: Mechanism and Improvement Under Bipolar Stress 
+ 
+ 
+7 
+stress evaluates only one aspect of device endurance, resulting in underestimation of gate dielectric 
+breakdown and overestimation of MW narrowing. 
+4 
+Conclusion 
+We investigated the behavior of stress-induced degradation and gate dielectric breakdown in FeFETs 
+with ferroelectric HZO as gate dielectrics on Si substrates. It was observed that gate dielectric 
+breakdown in FeFETs is dominated by the breakdown in the HZO layer, not in the IL. Increasing 
+gate and substrate hole currents under stress, due to the defect movement in HZO, were observed 
+before gate dielectric breakdown occurs. These increasing currents are not a permanent phenomenon: 
+temporary degradation is recovered by applying opposite voltage because of defect redistribution. We 
+found that continuous electrical stress with the same polarity leads to easier hard breakdown, whereas 
+bipolar stress frequently recovers the device distribution and help extend the time-to-breakdown. 
+Because bipolar stress suppresses the breakdown-limited endurance while accelerates the memory 
+window-limited endurance, accurate endurance tests should be carried out to correctly evaluate the 
+endurance characteristics of FeFETs in practical memory operations. 
+5 
+Conflict of Interest 
+The authors declare that the research was conducted in the absence of any commercial or financial 
+relationships that could be construed as a potential conflict of interest. 
+6 
+Author Contributions 
+K.T. and S.T. conceived and proposed the main concepts. K.T. fabricated devices and characterized 
+the electrical properties. K.T., M.T. and S.T analyzed the data and contributed to the in-depth 
+discussion. K.T. and S.T. wrote the manuscript. All authors contributed to the discussions regarding 
+the manuscript. 
+7 
+Funding 
+This paper is based on results obtained from a project, JPNP16007, commissioned by New Energy 
+and Industrial Technology Development Organization (NEDO) as well as JST CREST Grant Number 
+JPMJCR20C3 by the Japan Science and Technology Agency (JST).  
+8 
+Data Availability Statement 
+The raw data supporting the conclusion of this article will be made available by the authors, without 
+undue reservation. 
+9 
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+Low Operating voltage, improved breakdown tolerance, and high endurance in Hf0.5Zr0.5O2 
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+density, and excellent reliability,” in Proc. 2020 International Electron Device Meeting (IEDM), 75-
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+“Origin of the endurance degradation in the novel HfO2-based 1T ferroelectric nonvolatile 
+memories,” in Proc. 2014 IEEE International Reliability Physics Symposium (IRPS), 2E.5.1-2E.5.5. 
+doi: 10.1109/IRPS.2014.6860603  
+Yurchuk, E., Müller, J., Muller, S., Paul, J., Pesic, M., Bentum, R.v., et al. (2016). Charge-trapping 
+phenomena in HfO2-based FeFET-type nonvolatile memories. IEEE Trans. Electron Devices 63, 
+3501-3507. doi: 10.1109/TED.2016.2588439 
+ 
+ 
+
+Breakdown-Limited Endurance in HZO FeFETs: Mechanism and Improvement Under Bipolar Stress 
+ 
+12 
+ 
+FIGURE 1 | (A) Fabrication process flow. TEM images of (B) HfO2 nonferro-FET and (C) HZO 
+FeFET. HZO was crystallized whereas HfO2 remained amorphous. 
+ 
+FIGURE 2 | Schematic band diagram of HZO/IL/Si gate stack (A) when there is no interface charge 
+trapping and (B) when there is a large amount of interface charge trapping, where 90% of induced 
+electrons are trapped. Here, the Si band was scaled in the depth direction by 1:100 ratio to make the 
+band bending clear. 
+ 
+FIGURE 3 | Characteristics of Ig, Id, Is, and Isub for (A) HfO2 nonferro-FET and (B) HZO FeFET 
+with L/W = 10/100 µm. Ig of a HZO FeFET is around 103 times higher than that of a HfO2 nonferro-
+FET. Steeply increasing substrate current Isub can be found in FeFETs. (C) Leakage current path in 
+ferroelectric HZO gate insulator. 
+
+A
+nonferro-FET
+FeFET
+TiN
+TiN
+HfO2
+710 nm
+HZO
+B
+c
+D
+nonferro-FET
+FeFET
+S
+Si
+Si
+D
+TiN
+TiN
+Preparationof gate insulator
+O Grow IL by HPM
+OALD300℃
+10 nm
+HfO2
+HZO
+(TEMAHf + H,O) x 135 cycles
+for 10-nm HfO2
+(TEMAHf + H,O +
+0.7 nm
+IL
+IL
+TEMAZr + H,O) x 71 cycles
+Si
+Si
+for 10-nm Hfo.5Zro.5O2 (HZO)
+O TiN by sputtering
+5 nm
+5 nm
+O Anneal at 400℃, 30 s (N2)A
+IDEAL FeFET
+B
+ACTUAL FeFET
+without interface trap
+with large interface trap
+Si
+Si
+TiN
+TiN
+Free
+O
+electrons
+Trapped
+HZO
+electrons
+HZO
+IL
+IL
+ILBreakdown
+>FerroelectricBreakdownA
+nonferro-FET
+B
+FeFET
+c
+10~3
+N!I
+103
+Current (A)
+10
+5
+E
+10~5
+.
+HZO
+10'
+Is
+107
+-Id
+10-9
+Si
+Isub
+o Defects
+Tunneling
+10-13
+.2
+2
+0
+2
+3
+4
+-1
+0
+2
+3
+4
+- Trap-assisted tunneling
+-1
+Vg (M)
+Vg (v) 
+ Breakdown-Limited Endurance in HZO FeFETs: Mechanism and Improvement Under Bipolar Stress 
+ 
+ 
+13 
+ 
+FIGURE 4 | (A) Schematic of carrier separation measurement for analyzing gate current. (B) Current 
+components in gate current. Inversion electron tunneling flows through S/D, while tunneling of 
+valence-band electrons and generated holes appears as substrate current. Electron-component (blue 
+lines), hole-component (red lines), and total (circle symbols) gate currents of (C) nonferro-FET and 
+(D) FeFET when Vg was scanned from 0 V until the breakdown point. The electron component 
+dominates the gate current while the hole component rapidly increases near the breakdown voltage. 
+Gate currents after breakdown for (E) nonferro-FET and (F) FeFET, showing ohmic characteristics. 
+Band diagrams and expected gate current components at (G) low Vg, (H) Vg < VBD, and (I) Vg > VBD. 
+ 
+ 
+FIGURE 5 | Repeatedly measured electron and hole components of Ig in the FeFET. Repeatable 
+current implies that it is not a behavior of permanent trap generation. 
+ 
+FIGURE 6 | Characteristics of Ig, Id, Is, and Isub for HZO FeFET with L/W = 10/100 µm when the Vg 
+ranged is limited below 3.5 V. No substrate current Isub is observed at positive Vg. 
+
+c
+nonferro-FET
+D
+(i)
+FeFET
+Carrierseparation
+A
+B
+0
+102
+102
+Symbols:
+(A/cm²)
+100
+100
+Total I。
+(A/cm²
+G
+Ie
+10-2
+Symbols: Total I.
+10-2
+Ve
+① (ii)
+ 10-4
+ 10-4
+n
+10-6
+10-6
+(i) Inversion electron tunneling
+(ii)Valence-bandelectron tunneling
+10~8
+10-8
+I.
+0
+1
+2
+3
+4
+6
+0
+2
+3
+4
+5
+6
+(ili)Tunnelingbackofgeneratedholes
+Vg (V)
+Vg (M)
+E
+F
+120
+120
+100
+100
+G
+H
+(A/cm²)
+21
+80
+/cm
+80
+60
+Si
+60
+A
+Si
+40
+F
+10
+40
+Si
+TiN
+TiN
+Ih
+TiN
+20
+20
+00
+1
+2
+3
+4
+1
+2
+4
+Vg (V)
+Vg(V)
+V.<3V
+3V<V.<4V
+V.>4V100
+(A/cm²)
+lh
+1st V.
+scan
+-Ih 2nd V.
+ scan
+107
+-lh.
+ scan
+106
+ scan
+Ih
+108
+0
+1
+2
+3
+4
+Vg (V)10
+Current (A)
+10
+10
+10
+.9
+Isub
+10°
+10
+13
+2
+-1
+0
+2
+3
+4
+Vg (V)Breakdown-Limited Endurance in HZO FeFETs: Mechanism and Improvement Under Bipolar Stress 
+ 
+14 
+ 
+ 
+FIGURE 7 | TDDB results with carrier separation for (A) HfO2 nonferro-FET at Vg = 4.7 V and (B) 
+HZO FeFET at Vg = 3.9 V with L/W = 100/100 µm. A SILC-like behavior, with current increasing 
+with stress time, can be observed in FeFETs. (C) TDDB of FeFET at different stress voltage Vg. 
+Electron current increases with time by approximately 
+e ∝
+I
+t . The electron and hole current levels 
+at breakdown have weak dependency on Vg. Time-to-breakdown of (D) nonferro-FET and (E) FeFET 
+under constant voltage stress. The FeFET has a stronger dependence on Vg. Charge-to-breakdown 
+QBD for (F) nonferro-FET and (G) FeFET under constant voltage stress. QBD in the FeFET strongly 
+depends on stressing voltage, whereas QBD in the nonferro-FET is almost constant. (H) Qe/Qh ratio at 
+breakdown condition for FeFET. 
+ 
+ 
+FIGURE 8 | (A) I-Vg characteristics of FeFET before CVS. (B) Electron and hole components of 
+gate leakage current under CVS at Vg = 4 V for 10 s. (C) I-Vg characteristics of FeFET after CVS. 
+Although gate current increases during CVS, it has a negligible effect on I-Vg characteristics. 
+ 
+
+A
+B
+c
+102
+I.@BD = 2~5 A/cm2
+Current (A/cm²)
+10°
+10°
+Ie
+Ih
+3.8 V
+—Ih 3.9 V
+Current (
+10-2
+T。
+Ih 4.0 V
+10-4
+In
+-Ih 4.1 v
+104
+= 4.7 V
+I。
+Ih 4.15 V
+10-6FV
+10°
+100
+101
+102
+10°
+10°
+101
+102
+101
+100
+101
+10
+103
+Time (s)
+Time (s)
+D
+F
+H
+Time-to-breakdown (s)
+104
+104
+103
+103
+Qe
+103
+(C/cm²)
+10
+102
+10
+Qe
+Qh
+10°
+101
+10%
+..
+0
+?
+101
+10°
+00
+10
+107
+.01
+10°
+4.5
+4.75
+5
+5.253.5
+3.75
+4
+4.25
+4.5
+4.5
+4.75
+5.25
+3.5
+3.75
+4
+4.25
+4.5
+3.5
+4
+4.5
+Vg(v)
+Vg(v)
+V(v)
+Vg(V)
+Vg (V)A
+B
+c
+10°
+10-3
+Id
+E
+Id
+Isub
+Isub
+Current (A)
+10
+10°
+Current (A)
+10
+10
+10
+10
+10°
+T
+10
+10-11
+10-11
+10
+CVS at 4 V, 10 s
+10-13
+E
+10
+13
+-2
+-1
+0
+2
+3
+4
+10-1
+100
+101
+2
+-1
+0
+2
+3
+4
+Vg (v)
+Time (s)
+Vg (v) 
+ Breakdown-Limited Endurance in HZO FeFETs: Mechanism and Improvement Under Bipolar Stress 
+ 
+ 
+15 
+ 
+FIGURE 9 | (A) Applied voltage scheme with repeating stress of 4 V for time ts and 0 V for 1 s. 
+(B,C) Electron and hole components of gate leakage current at each 4-V stress cycle for ts = 10 s 
+when current is plotted in (B) log scale and (C) linear scale. Between each stress cycle, tests were 
+interrupted by 0 V for 1 s. (D) Total stress time (excluding 0 V interruption duration) before 
+breakdown for different time ts of 4-V stress. (E) Applied voltage scheme when the interrupted 
+voltage is -4 V for 1 s. (F,G) Electron and hole components of gate leakage current at each 4-V stress 
+cycle for ts = 10 s, which were interrupted at -4 V for 1 s between cycles, when current is plotted in 
+(F) log scale and (G) linear scale. (H) Total stress time (excluding -4 V interruption duration) before 
+breakdown for different time ts of 4-V stress. 
+ 
+ 
+ 
+ 
+FIGURE 10 | (A) Applied voltage scheme with repeating stress of 4.7 V for time ts and -4.7 V for 1 
+s. (B) Total stress time before breakdown of HfO2 nonferro-FETs. CVS indicates experiments 
+without recovery pulses. 
+
+le
+I
+1st stress cycle
+2nd stress cycle
+Te
+Ih
+3rd stress cycle
+Te
+Ih
+4th stress cycle
+B 
+104
+c
+A
+ (A/cm²)
+Time-to-breakdown
+103
+V
+100
+4 V
+3
+10
+oV
+Current (
+1 s 
+1 s
+1 s
+10-4
+00
+106
+101
+0
+10-1
+10
+101
+10-1
+100
+10
+0
+10
+20
+30
+CVS
+Time (s)
+Time (s)
+Stress time per cycle ts (s)
+F
+102
+G
+H
+E
+Time-to-breakdown
+103
+4
+上
+A
+100
+g.
+4 V
+3
+[Φ]
+102
+2
+Q0
+-4 V
+1 s 
+1s1s
+106
+101
+10-1
+100
+101
+10-
+100
+101
+0
+1020
+30
+CVS
+Stress time per cycle ts (s)
+Time (s)
+Time (s)B
+55
+TiN
+S
+10
+HfO2
+A
+S
+Si
+D
+10
+4.71
+OXO
+-4.7 V
+1 s
+1 s
+1S
+10
+0
+20
+40
+60
+CVS
+Stress time per cycle ts (s)Breakdown-Limited Endurance in HZO FeFETs: Mechanism and Improvement Under Bipolar Stress 
+ 
+16 
+ 
+FIGURE 11 | Mechanism under electrical stress. The SILC-like behavior is attributed to the 
+redistribution of defects rather than permanent defect generation as recovery is observed. Too much 
+stress will trigger breakdown. 
+
+Voltage stress
+More voltage stress
+TiN
+TiN
+TiN
+81
+HZO
+OZH
+OZH
+Si
+Si
+Si
+o Defects
+Breakdown
+Opposite voltage pulse

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3tFQT4oBgHgl3EQf3jaF/content/tmp_files/2301.13428v1.pdf.txt

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+Contrast and Clustering: Learning Neighborhood Pair Representation for
+Source-free Domain Adaptation
+Yuqi Chen1 , Xiangbin Zhu1 and Yonggang Li2 and Yingjian Li1 and Yuanwang
+Wei2 and Haojie Fang 3
+1Zhejiang Normal University
+2Jiaxing University
+3Zhejiang Sci-Tech University
+mochizuki@zjnu.edu.cn, zhuxb@zjnu.cn, liyonggang@zjxu.edu.cn, liyingjian li@163.com,
+yuanwang wei@zjxu.edu.cn, 18868901971@163.com
+Abstract
+Domain adaptation has attracted a great deal of at-
+tention in the machine learning community, but it
+requires access to source data, which often raises
+concerns about data privacy. We are thus motivated
+to address these issues and propose a simple yet ef-
+ficient method. This work treats domain adaptation
+as an unsupervised clustering problem and trains
+the target model without access to the source data.
+Specifically, we propose a loss function called con-
+trast and clustering (CaC), where a positive pair term
+pulls neighbors belonging to the same class together
+in the feature space to form clusters, while a neg-
+ative pair term pushes samples of different classes
+apart. In addition, extended neighbors are taken into
+account by querying the nearest neighbor indexes in
+the memory bank to mine for more valuable nega-
+tive pairs. Extensive experiments on three common
+benchmarks, VisDA, Office-Home and Office-31,
+demonstrate that our method achieves state-of-the-
+art performance. The code will be made publicly
+available at https://github.com/yukilulu/CaC.
+1
+Introduction
+The appetite for massive labeled training data has been success-
+fully addressed in unsupervised learning. Significant degra-
+dation will occur if the data distributions in the source and
+target domains are very different, which is formally denoted
+as domain/distribution shift. To tackle the generalization of
+the model to unseen domains, domain adaptation (DA) meth-
+ods [Huang et al., 2022; Lin et al., 2022] based on cotrain-
+ing of source and target data are conceptually simple, i.e.,
+transferring learned knowledge from the source domain to
+the target domain. However, with increasing concern about
+data privacy and data transfer bottlenecks of large datasets, it
+is extremely unrealistic to require the coexistence of source
+and target data. In this privacy-preserving scenario, previ-
+ous unsupervised DA methods could not be deployed, and
+thus, source-free domain adaptation (SFDA) has emerged
+over time. The purpose of SFDA is to obtain high perfor-
+mance in an unlabeled target domain, where the source data
+are not available during the target adaptation process. Exist-
+ing SFDA methods [Roy et al., 2022; Hou and Zheng, 2021;
+Qiu et al., 2021] try to better learn domain invariant/variant
+representations; however, these methods either require an
+auxiliary network[Li et al., 2020; Xia et al., 2021], or
+complex extra data processing is used[Lee et al., 2022;
+Kundu et al., 2022]. Other methods [Liang et al., 2021;
+Wang et al., 2022] are negatively affected by noisy labels
+may predict incorrect target pseudolabels.
+The above observation motivates us to tackle the data shift
+issue in SFDA. There are two obstacles: one is the unlabeled
+target data, and the other is that the source data cannot be
+obtained directly, relying only on the pretrained source model.
+Based on the fact that classes are shared between the source
+and target domains under closed set DA[You et al., 2019;
+Kundu et al., 2020], it is reasonable to assume that the pre-
+trained source model can learn the class representation of the
+target data. Therefore, even if the source and target data are
+shifted in the feature space, the features extracted by the source
+model on the target data can form rough clusters through in-
+trinsic class representation information (e.g., husky should
+never be classified as parrot), where the softmax output of
+similar features should be highly consistent.
+To achieve without the need for specialized source train-
+ing or changing the model structure, we expect to use the
+knowledge learned from the source model for self-supervised
+learning on unlabeled target data. Inspired by recent con-
+trastive learning[He et al., 2020; Oord et al., 2018] (which,
+as the name implies, learns feature representations by com-
+paring positive and negative samples), it is shown that the
+data itself provide supervision for network learning. Unlike
+previous methods that only add samples or change the defini-
+tion of positive pairs, we first define two probability functions,
+the probability of having the same class as their positive and
+negative samples. Then, our method, named Contrast and
+Clustering(CaC), is obtained by taking the negative logarithm
+of these two probability functions. As illustrated in the Figure
+1, we define the nearest neighbor samples as positive pairs and
+the neighbors of other samples as negative pairs to achieve
+contrastive clustering with more sample pairs. Simultaneously,
+considering that harder negative pairs[Kalantidis et al., 2020;
+Mitrovic et al., 2020] facilitate better and faster learning, we
+introduce extended neighbors to exclude similar samples in the
+arXiv:2301.13428v1  [cs.CV]  31 Jan 2023
+
+1
+2
+1
+1
+0
+1
+1
+2
+2
+2
+Expanded 
+neighbor
+Nearest 
+neighbor
+Before Adaptation
+After Adaptation
+...
+...
+...
+...
+...
+...
+...
+Feature bank Output bank Neighbor bank
+1
+z
+2
+z
+N
+z
+1
+y
+2
+y
+N
+y
+1
+1i
+1
+ki
+2
+1i
+2
+ki
+N
+1i
+N
+ki
+...
+Misclassified samples
+Decision Boundary
+Target samples
+1
+1
+1
+2
+2
+0
+1
+1
+Figure 1: An overview of the proposed method. CaC learns features of the unlabeled target data from pretrained source network, and then these
+prototype features are used for unsupervised learning clustering.
+negative pool to extract more valuable negative pairs. These
+extended neighbors are computationally costless because they
+are obtained by querying the nearest neighbor index in the
+memory bank. In practice, negative pair term is not always
+effective. We find that the contrastive self-supervised per-
+formance degrades on the class-imbalanced dataset, and we
+address this problem by recursively decoupling positive and
+negative samples[Yeh et al., 2022] during training. The exper-
+imental results show that the proposed method is sufficiently
+effective on three source-free domain adaptation benchmarks,
+and it outperforms recent state-of-the-art methods on the chal-
+lenging dataset VisDA.
+The primary contributions of this work are as follows:
+1. We propose a contrastive clustering loss function for
+SFDA, which uses nearest neighbors to learn more infor-
+mation for intraclass compactness and interclass separa-
+tion in a self-supervised manner.
+2. Extended neighbors are taken into account to mine more
+valuable negative pairs, and these extended neighbors are
+obtained by querying the nearest neighbor indexes in the
+memory bank without incurring additional computational
+cost.
+3. The experimental results on three datasets demonstrate
+the effectiveness and state-of-the-art of our method.
+2
+Related Work
+Domain Adaptation
+Domain adaptation attempts to learn
+a powerful classifier from the source domain to the target do-
+main. To generate similar feature distributions from different
+domain data, the early adversarial adaptation method [Ganin
+et al., 2016] combines domain adaptation with a two-player
+game similar to generative adversarial networks. CDAN[Long
+et al., 2018] extends the conditional adversarial mechanism
+to enable discriminative and transferable domain adaptation.
+SRDC[Tang et al., 2020] directly reveals intrinsic target dis-
+crimination by discriminative clustering of target data. CaCo
+[Huang et al., 2022] introduces contrastive learning, encourag-
+ing networks to learn representations with different categories
+but different domains. However, the source data are not di-
+rectly available in practice due to privacy issues, making these
+methods unapplicable.
+Source-Free Domain Adaptation
+The abovementioned
+normal domain adaptation methods need to access source do-
+main data during target adaptation. Recently, many methods
+have emerged to tackle source-free domain adaptation(SFDA),
+which has no way of accessing source data. SHOT[Liang et
+al., 2020] tunes the source classifier to encourage interclass
+feature clustering by maximizing mutual information and pseu-
+dolabeling. 3C-GAN[Li et al., 2020] is based on conditional
+GAN to provide supervised adaptation by regularizing the
+source domain information gradually. NRC[Yang et al., 2021]
+proposes neighborhood clustering, which performs predictive
+consistency among local neighborhoods. CPGA[Qiu et al.,
+2021] proposes a contrastive prototype generation strategy to
+generate feature prototypes for each class. U-SFAN[Roy et
+al., 2022] accounts for uncertainty by placing priors on the
+parameters of the source model. DIPE[Wang et al., 2022] cap-
+tures such domain-invariant parameters in the source model to
+generate domain-invariant representations.
+Contrastive Learning
+Contrast learning is a type of self-
+supervised learning, in which knowledge is learned by con-
+structing pair samples: similar data (positive samples) and
+dissimilar data (negative samples). There are various ways to
+construct positive and negative sample pairs. [Ye et al., 2019;
+Chen et al., 2020] construct pair samples from the current mini-
+batch, where the enhanced samples are positive pair. [Tian et
+al., 2020] treat data from different views of the same scene as
+positive pairs and data from different scenes as negative pairs.
+NNCLR[Dwibedi et al., 2021] uses data augmentation and
+its nearest neighbors in the memory bank as positive sample
+pairs. DCL[Yeh et al., 2022] removes positive sample pairs
+from the denominator in contrast loss to achieve positive and
+negative sample decoupling.
+
+3
+Method
+3.1
+Problem Definition
+Given the model Ms trained on the labeled source domain
+Ds = {xsi, ysi}M
+i=1 and the unlabeled target domain Dt =
+{xti}N
+i=1, assume the feature space Xs = Xt, label space
+Ys = Yt, but marginal probability Ps (xs) ̸= Pt (xt) with
+conditional probability Q (ys | xs) ̸= Q (yt | xs). The target
+domain and source domain have the same C classes in this
+paper (known as the closed-set problem). Our method splits
+the model Ms into two parts: a feature extractor f, and a
+classifier C = f(x)T W + b. Therefore, the output of the
+model is denoted as z(x) = δ(C(f(x))), where δ is denoted
+as the softmax function.
+The goal of source-free domain adaptation is to learn a
+feature extractor f and a classifier C to predict the labels
+yt ∈ Yt for xt in the target domain Dt. The first obstacle is
+that the source data are not accessible, and the second is the
+tremendous discrepancy between these two domains.
+3.2
+InfoNCE Revisit
+InfoNCE is a loss function widely used for contrastive learning.
+It defines the augmented data of each sample i as its positive
+sample i+, and B negative sample i−. This loss function is as
+follows:
+LInfoNCE =
+N
+�
+i=1
+log
+ez(i)T z(i+)/τ
+�B
+b=1 ez(i)T z(i−)/τ + ez(i)T z(i+)/τ
+(1)
+where z(i), z(i+) is called the positive pair and z(i), z(i−)
+is called the negative pair. τ ∈ R+ is a scalar temperature
+parameter.
+3.3
+Motivation
+Similar samples (e.g., husky and alaskan malamute) should
+have similar predictions, while dissimilar samples (e.g., husky
+and parrot) should have different predictions. Unlike the pre-
+vious methods that regard an augmented sample as a positive
+pair, we set the neighbor samples (the top-k similar samples
+in the feature embedding) as positive pairs. This allows us to
+perform clustering directly on the data without any generative
+techniques, and it allows us to consider a greater number of
+positive pairs. The samples in the k-nearest neighbor set (mea-
+sured by cosine similarity or Euclidean distance) K are chosen
+as the positive pairs of the instance xi, and the other samples
+not in this set are chosen as the negative pairs. Based on this
+setting, InfoNCE could expand to contain multiple positive
+pairs, and the loss function L can be defined as follows:
+L = −
+N
+�
+i=1
+�
+j∈K
+log
+ez(i)T z(j)
+�
+b̸=i∪b̸=j ez(i)T z(b) + ez(i)T z(j)
+(2)
+Intuitively, similar samples (e.g., husky and alaskan mala-
+mute) belong to different categories. However, the above loss
+function simply extends InfoNCE to multiple positive pairs,
+encouraging the network to classify samples with high simi-
+larity into the same category and samples with low similarity
+into different categories. When two categories have similar
+Algorithm 1 Learning Nearest Pair Representations for SFDA
+Input: Source-pretrained model Ms, unlabeled target data
+Dt.
+1: Build three memory banks to store all the target features
+(F) and predictions (P) and the indexes of neighbors (N).
+2: Store the values of the feature bank
+3: while Adaptation do
+4:
+Sample a mini-batch T from Dt and update memory
+banks P and F.
+5:
+For each feature in T , find its K-nearest neighbors
+topK(z(i)) and update memory bank N.
+6:
+Retrieve and expand the neighbors from memory bank
+N to generate Wsim
+7:
+Compute the loss function LCaC
+8:
+Back-propagate with the loss function and update the
+network parameters
+9: end while
+10: return solution
+features, the samples are likely to be misclassified. To avoid
+incorrectly pulling closer to neighbors of different categories,
+the network needs to encourage neighbors of the same cate-
+gory to be closer together and neighbors of different categories
+to be more distant.
+3.4
+Contrast and Clustering
+Assuming that the target feature of the source pretrained
+feature extractor can form clusters[Liang et al., 2020; Wang
+et al., 2022], we exploit this intrinsic ability of the pretrained
+model to perform SFDA by considering neighborhood infor-
+mation. The probability of xi belonging to class j in C-class
+classification is:
+P(Y = j|X = i) =
+exp(z(ij))
+�C−1
+c=0 exp(z(ic))
+(3)
+where z(ik) = f(xi)T wk and it can be interpreted as the
+probability that instance xi belongs to class k.
+Now, we consider the following conditions. Given data x, its
+k-nearest neighbor set is K (the method for finding the nearest
+k-neighbor set K for each sample is described in detail in
+Sec.3.5), and the set B denotes the other samples in the mini-
+batch. Intuitively, x and the neighbor set K should belong
+to the same category, meaning that their one-hot outputs are
+highly consistent, so the outputs of x and its k-nearest neighbor
+set K should be more similar to those of the k-nearest neighbor
+set of the other data in the current batch Bk.
+Therefore, we define two likelihood functions P same
+i,j
+: the
+probability that xi has the same category as its neighbors, and
+P dis
+i,j : the probability that xi has the same category as the
+neighbors of the other data in the current mini-batch.
+P same
+i,j
+=
+�
+j∈K
+ez(i)T z(j)
+�
+q̸=i ez(i)T z(q)
+(4)
+P dis
+i,j =
+�
+j∈Bk
+ez(i)T z(j)
+�
+q̸=i ez(i)T z(q)
+(5)
+
+where Bk denotes the corresponding k-nearest neighbors of B.
+We then propose to achieve target feature clustering by mini-
+mizing the following negative logarithmic objective function,
+denoted as CaC:
+LCaC = − 1
+N
+N
+�
+i=1
+log P same
+i,j
+P dis
+i,j
+= 1
+N
+N
+�
+i=1
+(
+�
+j∈Bk
+z(i)T z(j)
+�
+��
+�
+neg:negative pairs
+−
+�
+j∈K
+z(i)T z(j)
+�
+��
+�
+pos:positive pairs
+)
+(6)
+3.5
+Finding the Nearest Neighbors
+To retrieve the nearest neighbors for batch training, we build
+three memory banks: F ∈ RN×Dim stores all target features,
+P ∈ RN×C stores the corresponding prediction scores, and
+N ∈ RN×K stores the corresponding nearest data. For two
+memory banks, F and P, which are initialized to all target
+features and their predictions, only the features and their pre-
+dictions computed in each small batch are used to update these
+two repositories, as in a previous study [Liang et al., 2021;
+Yang et al., 2021].
+Our work differs from previous work in that we store the in-
+dexes of the nearest neighbor samples to facilitate finding the
+extended neighborhoods and then use these extended neigh-
+borhoods to generate the weights Wsim. This step works
+efficiently because the memory bank N is initialized to empty
+and is only updated after the sample similarity is computed
+in each mini-batch, meaning that as features are fed into the
+network, their corresponding nearest neighbor samples are
+updated in N. Note that updating the nearest neighbor bank
+stores only the indexes1 of the corresponding nearest neigh-
+bors, without any additional computation.
+For each feature f(i), its nearest neighbors, denoted as
+topK(f(i)), are the topK with the highest similarity to the
+memory bank F and are used to compute the positive pairs in
+Eq.(6). The similarity between the two samples is at a maxi-
+mum when the two softmax outputs have the same prediction
+class and are close to a one-hot vector. For the negative pair
+term in Eq.(6), since other samples in the min-batch may come
+from the same category as f(i), we claim that these similar
+samples should be excluded in the corresponding B, because
+these similar samples play a relatively unimportant part in
+the negative pair term. To find these similar features, rather
+than using a larger K to find more neighbors, we utilize the
+expanded neighbors of each feature, i.e., the nearest neigh-
+bors of each feature and the nearest neighbors of these nearest
+neighbors. The feature j is regarded as a similar feature of
+i if f(j) ∈ topK(topK(z(i))). For each feature in the cur
+mini-batch, the weight Wsim ∈ RN×N is used to exclude
+those similar features. Taking the i-th feature as an example,
+if the j-th feature is its similar feature, then the j-th column of
+the i-th row in Wsim is 0 and the other positions are 1. Finally,
+1Each sample is given a particular index, which is the same in the
+dataset and the memory banks.
+the objective function is denoted as:
+LCaC = 1
+N
+N
+�
+i=1
+(
+�
+j∈Bk
+Wsim · z(i)T z(j) −
+�
+j∈K
+z(i)T z(j))
+(7)
+These two terms interact to achieve self-supervision of the
+features, where positive pairs are expected to improve the
+consistency of the one-hot outputs while negative pairs are
+expected to improve the diversity of the outputs. The weights
+Wsim are used to mine more valuable negative pairs. Our
+algorithm is illustrated in Algorithm 1.
+4
+Experiments
+Datasets.
+We conduct the experiments on three benchmark
+datasets: VisDA is a more challenging dataset, with 12-
+class synthetic-to-real object recognition tasks. Its source
+domain consists of 152k synthetic images while the target
+domain contains 55k real object images. Office-Home con-
+tains 4 domains(Art, Clipart, Real World, Product) with 65
+classes and a total of 15,500 images. Office-31 contains 3
+domains(Amazon, Webcam, DSLR) with 31 classes and 4652
+images.
+Evaluation.
+The column SF in the tables denotes source-
+free setting. For VisDA, we show accuracy for all classes and
+average over those classes (Avg in the tables). For Office-31
+and Office-Home, we show the results of each task and the
+average accuracy over all tasks (Avg in the tables).
+Baselines.
+We compare CaC with three types of baselines:
+(1) source-only: ResNet[He et al., 2016]; (2) unsupervised
+domain adaptation with source data: DANN[Ganin et al.,
+2016], CDAN[Long et al., 2018], SRDC[Tang et al., 2020],
+CaCo[Huang et al., 2022]; and (3) source-free unsupervised
+domain adaptation: SHOT[Liang et al., 2020], 3C-GAN[Li et
+al., 2020], NRC[Yang et al., 2021], CPGA[Qiu et al., 2021],
+U-SFAN+[Roy et al., 2022], DIPE[Wang et al., 2022].
+Implementation details.
+To ensure fair comparison with
+related methods, we use the same network architecture as
+SHOT and adopt SGD with momentum 0.9 and batch size of
+64 for all datasets. Specifically, we adopt the backbone of
+ResNet50 for Office-Home and Office31, and ResNet101 for
+VisDA. The learning rate for Office-Home and Office31 is set
+to 1e-3 for all layers, except for the last two newly added fc
+layers, where we apply 1e-2. Learning rates are set 10 times
+smaller for VisDA. We train 15 epochs for VisDA, 40 epochs
+for Office-Home and 100 epochs for Office-31.
+4.1
+Comparison with State-of-the-Art Methods
+In this section, we compare our proposed CaC with state-of-
+the-art methods on three DA benchmarks. In Table 1, Table3
+and Table 4, the top part shows the results for the DA methods
+with access to source data during adaptation. The bottom
+shows the results for the SFDA methods. The best results are
+bolded, and the second-best results are underlined.
+Specifically, CaC outperforms other SOTA methods on the
+more challenging dataset VisDA, achieving the best results in
+various categories and ultimately obtaining excellent results,
+
+Method
+SF
+VisDA
+plane bicycle
+bus
+car
+horse
+knife mcycl person plant sktbrd train
+truck
+Avg
+ResNet-101
+55.1
+53.3
+61.9
+59.1
+80.6
+17.9
+79.7
+31.2
+81.0
+26.5
+73.5
+8.5
+52.4
+DANN
+81.9
+77.7
+82.8
+44.3
+81.2
+29.5
+65.1
+28.6
+51.9
+54.6
+82.8
+7.8
+57.4
+CDAN
+85.2
+66.9
+83.0
+50.8
+84.2
+74.9
+88.1
+74.5
+83.4
+76.0
+81.9
+38.0
+73.9
+CaCo
+90.4
+80.7
+78.8
+57.0
+88.9
+87.0
+81.3
+79.4
+88.7
+88.1
+86.8
+63.9
+80.9
+SHOT
+✓
+94.3
+88.5
+80.1
+57.3
+93.1
+94.9
+80.7
+80.3
+91.5
+89.1
+86.3
+58.2
+82.9
+3C-GAN
+✓
+94.8
+73.4
+68.8
+74.8
+93.1
+95.4
+88.6
+84.7
+89.1
+84.7
+83.5
+48.1
+81.6
+NRC
+✓
+96.8
+91.3
+82.4
+62.4
+96.2
+95.9
+86.1
+80.6
+94.8
+94.1
+90.4
+59.7
+85.9
+CPGA
+✓
+94.8
+83.6
+79.7
+65.1
+92.5
+94.7
+90.1
+82.4
+88.8
+88.0
+88.9
+60.1
+84.1
+DIPE
+✓
+95.2
+87.6
+78.8
+55.9
+93.9
+95.0
+84.1
+81.7
+92.1
+88.9
+85.4
+58.0
+83.1
+U-SFAN+
+✓
+94.9
+87.4
+78.0
+56.4
+93.8
+95.1
+80.5
+79.9
+90.1
+90.1
+85.3
+60.4
+82.7
+CaC(Ours)
+✓
+96.9
+91.0
+83.3
+72.3
+96.9
+96.1
+90.7
+81.6
+95.1
+92.9
+92.0
+63.2
+87.7
+Table 1: Accuracies (%) on VisDA(Synthesis → Real) for ResNet101-based methods.
+SHOT
+LCaC
+Wsim
+Avg
+✓
+82.9
+✓
+87.15
+✓
+✓
+87.7
+Table 2: Accuracy comparison with different components on VisDA
+as shown in Table 1. For Office-Home, the proposed CaC ob-
+tains competitive results compared with other SFDA methods,
+as shown in Table 3. Note that our method is superior in the
+tasks A→C, A→R, C→R and P→C. In addition, CaC obtains
+similar results to the SOTA in Office-31, as shown in Table 4.
+CaC slightly underperforms DIPE(requires a special parame-
+ter update strategy and combines five objective functions) and
+CPGA(epoch set to 400) on Office-Home and Office-31, but
+CaC achieves the best results on VisDA, far surpassing these
+SOTA methods. The main result is that VisDA provides suffi-
+cient data for learning positive and negative pairs so that CaC
+can learn better domain-invariant representations to achieve
+clustering of same-class samples. Moreover, CaC is able to
+outperform recent methods with source data (e.g., CaCo and
+SRDC), which demonstrates the superiority of our proposed
+method.
+4.2
+Analyzing and Ablating
+Ablation study on the proposed LCaC and weight Wsim
+To investigate the loss of adaptation, we show the quantitative
+results of the model optimized by different losses. As shown
+in Table 2, our proposed comparison and clustering loss LCaC
+achieves better results on VisDA than SHOT due to the pseudo-
+labeling that may give high confidence values for incorrect
+samples. Such results validate that neighbor pairs can give
+the network excellent self-supervised clustering ability. More-
+over, we obtained the best performance when extracting more
+valuable negative sample pairs by using the weights Wsim
+generated from the nearest neighbors and extended neighbors.
+Number of neighbors K
+For the number of neighbors K used for feature clustering
+in Eq.(7), we show in Table 5 that our method is robust to
+the choice of K. From Eq.(7), we can see that K is correlated
+with the sample size of the dataset, requiring a larger value
+on the larger VisDA dataset and a relatively smaller value for
+Office-Home. Additionally, the smallest dataset Office-31 is
+not sensitive to the size of K. As can be summarized from the
+results, larger K values consider more pairs, which is better for
+learning a robust bound. However, setting too large a K value
+may also include samples from other categories, bringing more
+noisy samples, which leads to performance degradation.
+Figure 2: Accuracy of datasets.
+(a) 65 classes
+(b) 12 classes
+Figure 3: The proportion of classes on Office-Home and VisDA.
+
+85
+ww
+80
+Accuracy(%)
+75
+Dataset
+70
+Office-Home
+VisDA(with decay)
+VisDA(without decay)
+65
+0
+50
+100
+150
+ItervalOffice-Home
+100
+80
+Number
+60
+40
+20
+0
+ClassVisDA
+Class
+10000
+1:plane
+2:bicycle
+3:bus
+4:car
+5:horse
+8000
+6:knife
+7:mcycl
+8:person
+9.plant
+10:sktbrd
+Number
+11:train
+6000
+12:truck
+4000
+2000
+123456789101112
+ClassMethod
+SF
+Office-Home
+A→C A→P A→R C→A
+C→P
+C→R
+P→A
+P→C
+P→R
+R→A R→C
+R→P
+Avg
+ResNet-50
+34.9
+50.0
+58.0
+37.4
+41.9
+46.2
+38.5
+31.2
+60.4
+53.9
+41.2
+59.9
+46.1
+DANN
+45.6
+59.3
+70.1
+47.0
+58.5
+60.9
+46.1
+43.7
+68.5
+63.2
+51.8
+76.8
+57.6
+CDAN
+50.7
+70.6
+76.0
+57.6
+70.0
+70.0
+57.4
+50.9
+77.3
+70.9
+56.7
+81.6
+65.8
+SRDC
+52.3
+76.3
+81.0
+69.5
+76.2
+78.0
+68.7
+53.8
+81.7
+76.3
+57.1
+85.0
+71.3
+SHOT
+✓
+57.1
+78.1
+81.5
+68.0
+78.2
+78.1
+67.4
+54.9
+82.2
+73.3
+58.8
+84.3
+71.8
+NRC
+✓
+57.7
+80.3
+82.0
+68.1
+79.8
+78.6
+65.3
+56.4
+83.0
+71.0
+58.6
+85.6
+72.2
+DIPE
+✓
+56.5
+79.2
+80.7
+70.1
+79.8
+78.8
+67.9
+55.1
+83.5
+74.1
+59.3
+84.8
+72.5
+U-SFAN+
+✓
+57.8
+77.8
+81.6
+67.9
+77.3
+79.2
+67.2
+54.7
+81.2
+73.3
+60.3
+83.9
+71.9
+CaC(Ours)
+✓
+59.0
+79.5
+82.0
+67.6
+79.2
+79.5
+66.7
+56.5
+81.3
+74.2
+58.3
+84.7
+72.4
+Table 3: Accuracies (%) on Office-Home for ResNet50-based methods.
+Method
+SF
+Office-31
+A→D A→W D→A D→W W→A W→D
+Avg
+ResNet-50
+68.9
+68.4
+62.5
+96.7
+60.7
+99.3
+76.1
+DANN
+82.0
+96.9
+99.1
+79.7
+68.2
+67.4
+82.2
+CDAN
+92.9
+94.1
+71.0
+98.6
+69.3
+100.0
+87.7
+SRDC
+95.8
+95.7
+76.7
+99.2
+77.1
+100.0
+90.8
+CaCo
+89.7
+98.4
+100.0
+91.7
+73.1
+72.8
+87.6
+SHOT
+✓
+94.0
+90.1
+74.7
+98.4
+74.3
+99.9
+88.6
+3C-GAN
+✓
+92.7
+93.7
+98.5
+99.8
+75.3
+77.8
+89.6
+NRC
+✓
+96.0
+90.8
+75.3
+99.0
+75.0
+100.0
+89.4
+CPGA
+✓
+94.4
+94.1
+98.4
+99.8
+76.0
+76.6
+89.9
+DIPE
+✓
+96.6
+93.1
+75.5
+98.4
+77.2
+99.6
+90.1
+U-SFAN+
+✓
+94.2
+92.8
+74.6
+98.0
+74.4
+99.0
+88.8
+CaC(Ours)
+✓
+95.2
+94.0
+74.7
+99.1
+76.5
+99.8
+89.9
+Table 4: Accuracies (%) on Office-31 for ResNet50-based methods.
+Dataset
+K
+Avg
+Office-Home
+1
+69.79
+3
+72.16
+4
+71.08
+5
+70.22
+Office-31
+1
+88.47
+3
+72.16
+4
+89.87
+5
+89.36
+VisDA
+4
+85.94
+5
+87.22
+6
+87.12
+8
+85.71
+Table 5: Comparison in three datasets using
+different values of K.
+Office-Home
+β
+Avg
+0
+72.16
+0.25
+72.07
+0.5
+72.00
+0.75
+71.92
+1
+71.77
+2
+71.26
+Office31
+β
+Avg
+0
+89.73
+0.25
+89.79
+0.5
+89.78
+0.75
+89.77
+1
+89.80
+2
+89.87
+VisDA
+β
+Avg
+1
+82.66
+5
+85.74
+10
+87.05
+15
+87.44
+18
+87.65
+20
+87.2
+Table 6: Comparison in three datasets using different values of β.
+Interesting impact of negative pairs
+We find that CaC can maintain the accuracy improvement
+on Office-Home, but degrades at a later stage on VisDA, as
+shown in the green curve in Figure 2. We first consider the
+class comparison of these two datasets. As shown in Figure
+3, VisDA suffers from a worse class imbalance problem than
+Office-Home, which has a smaller number of classes and, even
+worse, a large gap in class proportions. Taking the fourth class
+of VisDA:car as an example, it is clear that a large proportion
+of the samples in a mini-batch belong to the car class, and
+the contrast loss treats the other samples in the mini-batch as
+potentially negative pairs. Ultimately, the network separates
+these samples that belong to the same class, resulting in a
+decrease in accuracy. As shown in Figure 4, the method is
+much less accurate for classes with large quantities (car, mcycl,
+and sktbrd) than other classes.
+Dataset
+Runtime(s/epoch)
+Avg
+SHOT
+485
+82.9
+CaC(Ours)
+471
+87.7
+30% for memory bank
+466
+87.5
+Table 7: Runtime analysis of SHOT and our methods. 30% denote
+the percentage of target features stored in the memory bank.
+Observing that sampling negative examples with truly dif-
+ferent labels improved performance in [Chuang et al., 2020],
+we utilized extended nearest neighbors to find more valu-
+able negative samples; however, the contribution of the neg-
+ative pair term to the loss may still be significant. There-
+fore, we introduce a factor α = (
+max iter
+max iter+iter)β to con-
+trol the impact of negative pairs.
+As the epoch time in-
+creases, the impact of the negative pairs will be reduced, where
+max iter = batch size × epoch and β controls the rate of
+decrease; the larger its value is, the faster the negative pair
+impact decreases, as shown in Table 6. The comparison results
+with and without decay are shown in Figure 2. After introduc-
+ing the decay factor, the accuracy can be steadily improved on
+VisDA.
+Runtime analysis
+We compare the runtime in one epoch with SHOT in Table
+7. For SHOT, the pseudo-label is computed by clustering
+in each iteration. In contrast, our nearest neighbor is a dot
+product operation on the features, and the nearest neighbor of
+
+Figure 4: Accuracy of each class on VisDA.
+the current sample is stored in the memory bank each time.
+Even though the weights Wsim need to use extended near-
+est neighbors, no additional computational consumption is
+required because these extended nearest neighbors can be di-
+rectly retrieved from the bank. Therefore, our method can
+improve the performance with a relatively small amount of
+computation. Additionally, we reduce the size of the reposi-
+tory, which does not incur significant performance loss and
+maintains competitive results.
+5
+Conclusion
+In this paper, we propose a source-free domain adaptation
+method with a self-supervised loss function that encourages
+one-hot output consistency with nearest neighbors to cluster
+of similar samples. Our method differs from previous methods
+in that we build an indexed memory bank for nearest neighbor
+samples to facilitate the retrieval of expanded neighbors, which
+are used to mine more valuable negative pairs without increas-
+ing the computational cost. Extensive experiments verify the
+importance of nearest neighbors and the impact of negative
+pairs, as well as proving that the proposed method outperforms
+other state-of-the-art source-free domain adaptation methods
+on several benchmarks.
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+

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+Mixed-state Entanglement for AdS Born-Infeld Theory
+Peng Liu 1,∗ Zhe Yang 1,† Chao Niu 1,‡ Cheng-Yong Zhang 1,§ and Jian-Pin Wu 2,3¶
+1 Department of Physics and Siyuan Laboratory,
+Jinan University, Guangzhou 510632, China
+2 Center for Gravitation and Cosmology,
+College of Physical Science and Technology,
+Yangzhou University, Yangzhou 225009, China
+3 School of Aeronautics and Astronautics,
+Shanghai Jiao Tong University, Shanghai 200240, China
+Abstract
+We study the mixed-state entanglement for AdS Born-Infeld (BI) theory.
+We calculate the
+mixed-state entanglement and investigate the relationship between it and the system parameters.
+We find that the holographic entanglement entropy (HEE) and mutual information (MI) exhibit
+monotonically increasing and decreasing behavior with BI factor b. However, the entanglement
+wedge cross-section (EWCS) exhibits a very rich set of phenomena about system parameters.
+EWCS always increases with b when b is small and then monotonically decreases with b. These
+behaviors suggest that increasing the BI factor, which is essentially enhancing the coupling be-
+tween the background geometry and the transport properties can always enhance the EWCS. The
+coupling between the entanglement and the transport behaviors has also been studied in condensed
+matter theories and is important to construct a stable quantum circuit. We also provide analytical
+understanding of the above phenomenon.
+∗Electronic address: phylp@email.jnu.edu.cn
+†Electronic address: yzar55@stu2021.jnu.edu.cn
+‡Electronic address: niuchaophy@gmail.com
+§Electronic address: zhangcy@email.jnu.edu.cn
+¶jianpinwu@yzu.edu.cn, corresponding author
+1
+arXiv:2301.04854v1  [hep-th]  12 Jan 2023
+
+Contents
+I. Introduction
+2
+II. Holographic Born-Infeld Theory And Information-Related Quantities
+4
+A. The AdS Born-Infeld Model
+4
+B. Holographic information-related quantities
+7
+C. Computations of holographic geometric quantities
+8
+1. The minimum surface
+9
+2. The EWCS
+10
+III. The Holographic Entanglement Entropy And The Holographic Mutual
+Information
+11
+IV. The Holographic Entanglement Wedge Cross-Section
+14
+V. Discussion
+18
+Acknowledgments
+19
+References
+19
+I.
+INTRODUCTION
+Quantum entanglement is the most distinguishing characteristic between quantum and
+classical systems. Holographic gravity, condensed matter theory, quantum information, and
+other areas have recently overlapped with each other on quantum entanglement. Numerous
+quantum entanglement measurements have been discovered to be capable of diagnosing the
+quantum phase transition of strong correlation systems and the topological quantum phase
+transitions, as well as playing a key role in the emergence of spacetime [1–8].
+There are numerous types of quantum entanglement measurements, including entangle-
+ment entropy (EE), mutual information (MI), R´enyi entanglement entropy, and negativity.
+Among these quantum entanglement measurements, EE is commonly considered a useful
+measure of pure state entanglement. However, EE is not applicable to measure the more
+common mixed-state entanglement. To measure mixed-state entanglement, numerous new
+2
+
+entanglement measurements, such as the entanglement of purification, non-negativity, and
+the entanglement of formation, have been proposed [9, 10]. On the other hand, calculating
+the mixed-state entanglement measures is extremely difficult.
+Gauge/gravity duality is a powerful tool for analyzing strongly correlated systems because
+it connects entanglement-related physical quantities to geometric objects in dual gravity sys-
+tems. In the dual gravity system, the holographic entanglement entropy (HEE) connects
+the EE of a subregion on the boundary with the area of the minimum surface [5]. HEE
+has been demonstrated to be able to detect quantum phase transitions and thermodynamic
+phase transitions [11–15]. Recently, the R´enyi entropy was proposed to be proportional to
+the minimal area of cosmic branes [16]. Moreover, the butterfly effect that reflects the dy-
+namic properties of quantum systems, has been extensively studied in holographic theories
+[17–26]. In addition, holographic duality of quantum complexity, a new information-related
+quantity from the EE, was also proposed [27–33]. More recently, the EWCS was associated
+with the area of the minimum cross-section of the entanglement wedge [34, 35]. The geomet-
+ric prescription of EWCS provides a novel and powerful tool for studying the mixed-state
+entanglement in holographic theories.
+Among all the models in holographic theories, the Born-Infeld (BI) model is a special
+class of models for nonlinear electromagnetic field theories. It was first proposed to eliminate
+the divergent self-energy of Maxwell theory. Later, it was found that the BI theory can be
+naturally derived from the string theory under the low-energy approximation. The BI model
+under the holographic theories can be dual to the quantum chromodynamics (QCD) systems
+[36, 37], and some condensed matter systems with novel transport behaviors, such as the
+quantum liquid [38], the Mott-insulator [39], and the novel magneto-resistance phenomenon
+[40, 41], which is consistent with the experimental phenomenon in [42, 43]. Various prop-
+erties of the BI model, such as its thermodynamic properties, transport properties [44], the
+complexity [45], have been extensively investigated. However, the question of how exactly
+the BI factor b, which embodies the nonlinearity of this nonlinear electromagnetic field the-
+ory, affects the properties of the system, especially the mixed-state entanglement properties,
+remains to be answered.
+This paper focuses on the effect of the BI factor on two measures of mixed-state en-
+tanglement - MI and EWCS. When b → 0, the background geometry is AdS-Schwarzschild
+solution, and the entanglement property of the system is decoupled from the transport prop-
+3
+
+erty of the system; while for non-zero b, the transport behaviors can affect the entanglement
+property. Therefore, we interpret b as the degree of correlation between the entanglement
+and transport properties of the metric when b increases from zero. Remind also that the
+coupling between the transport properties and the entanglement is also an important topic
+in condensed matter field theory, and is crucial for the construction of a stable quantum cir-
+cuit [46–48]. For b → ∞, the system goes to the AdS-RN black brane system with a linear
+Maxwell field. Therefore, the range b ∈ (0, ∞) represents the process that the Maxwell field
+turns on and converges to a linear Maxwell field case. Our main goal is to explore how BI
+factor b affects the MI and EWCS.
+We organize this paper as follows: we introduce the holographic BI model in Sec. II A,
+entanglement measures (HEE, MI, EWCS) and their holographic duality in Sec. II B. We
+discuss the properties of HEE, MI (III) and EWCS (IV) systematically. Finally, we summa-
+rize in Sec. V.
+II.
+HOLOGRAPHIC BORN-INFELD THEORY AND INFORMATION-RELATED
+QUANTITIES
+First, we review the holographic BI model. Following that, we review the concepts of the
+HEE, MI, and EWCS with their holographic dual. Then, we elaborate upon our algorithms
+proposed to calculate minimum surfaces and minimum cross-sections.
+A.
+The AdS Born-Infeld Model
+The action of the 4-dimensional holographic BI model is,
+S =
+�
+d4x√−g
+�
+R − 3Λ
+16πG +
+b2
+4πG
+�
+1 −
+�
+1 + 2F
+b2
+��
+.
+(1)
+The parameter b is the BI factor, and Λ = − 3
+l2 with l the AdS radius. The solution of the
+BI theory is,
+ds2 = −f(r)dt2 +
+1
+f(r)dr2 + r2hijdxidxj,
+(2)
+with
+f(r) = r2
+l2 − 2M
+r
++
+4Q22F1
+�
+1
+4, 1
+2; 5
+4; − Q2
+b2r4
+�
+3r2
++ 2b2r2
+3
+�
+1 −
+�
+Q2
+b2r4 + 1
+�
+,
+(3)
+4
+
+Q is the electric charge and M is the mass of the black brane. For l2 < 0 and l2 > 0 the
+system is asymptotically dS and AdS, respectively. Here, we fix l2 = 1 for concreteness.
+For k = 1, 0, −1 the hij denotes a sphere, a Ricci flat surface, and a hyperbolic surface,
+respectively. Here, we focus on the planar case, i.e., k = 0.
+At the horizon r = rh we have f(rh) = 0, and hence we arrive at the ADM mass
+M =
+4l2Q2 2F1
+�
+1
+4, 1
+2; 5
+4; − Q2
+b2r4
+h
+�
+− 2b2l2r4
+h
+�
+Q2
+b2r4
+h + 1 + 2b2l2r4
+h + 3r4
+h
+6l2rh
+.
+(4)
+The Hawking temperature is,
+T = rh
+4π
+�
+3 − 2b2
+��
+Q2
+b2r4
+h
++ 1 − 1
+��
+.
+(5)
+The planar case is always thermodynamically stable [49]. Therefore, in this BI black brane
+system, there is no thermodynamic phase transition.
+The system is invariant under the rescaling,
+(t, 1/r, x, y) → α(t, 1/r, x, y), Q → Q/α2, T → T/α, rh → αrh.
+Other parameters such as b, β are all dimensionless. Therefore, we can fix rh = 1. Here,
+we adopt √Q as the scaling unit, consequently, we need to divide physical quantity with
+scaling dimension [n] by Qn/2.
+For numerical convenience, we transform r into z ≡ rh/r such that the semi-infinite
+domain r ∈ (rh, ∞) becomes a finite domain z ∈ [0, 1]. Then the metric becomes,
+ds2 = 1
+z2
+�
+−hdt2 + r2
+hdz2
+h
++ r2
+hdx2 + r2
+hdy2
+�
+,
+(6)
+with
+h(z) ≡4
+3Q2z3
+�
+z 2F1
+�1
+4, 1
+2; 5
+4; −Q2z4
+b2
+�
+− 2F1
+�1
+4, 1
+2; 5
+4; −Q2
+b2
+��
+− 2
+3b2
+�
+z3
+�
+1 −
+�
+Q2
+b2 + 1
+�
++
+�
+Q2z4
+b2
++ 1 − 1
+�
+− z3 + 1.
+(7)
+And the dimensionless Hawking temperature becomes,
+T =
+b2
+�
+2 − 2
+�
+Q2
+b2 + 1
+�
++ 3
+4π√Q
+.
+(8)
+5
+
+FIG. 1: The contour plot of the Hawking temperature in the plane (b, rh), where the temperature
+is only positive in the shaded region.
+From the dimensionless Hawking temperature (8) we can find that,
+lim
+Q→0 T → ∞,
+lim
+Q→
+√
+12b2+9
+2b
+T → 0.
+(9)
+Also, we can find that,
+∂QT = 2b
+�
+b −
+�
+b2 + Q2
+�
+− 3
+�
+Q2
+b2 + 1 − 2Q2 < 0.
+(10)
+Therefore, the quantity Q is restricted to the range [0,
+√
+12b2+9
+2b
+] and that the temperature
+T decreases as Q increases. This system is described by three variables (T, b , rh), with
+only two of them being independent. We have also observed that for any given value of
+b, the temperature T always increases with increasing rh, thus, the value of rh is uniquely
+determined by a given temperature T. This can be seen in the Fig. 1. Therefore, we can
+simplify the system to a two-parameter system (b, T).
+When the parameter b → ∞, the background solution of our system converges to the
+AdS-RN solution, and when b → 0, it becomes the AdS-Schwarzschild solution. When b is
+zero, there is an electromagnetic field present, but the background solution is still the AdS-
+Schwarzschild solution. This means that the entanglement-related physical quantities are
+not affected by the conductivity of the system. However, as b increases, the electromagnetic
+fields starts to affect the background solution, and thus has an impact on the entanglement
+6
+
+5
+4 
+0.99
+0.88
+0.77
+3
+0.66
+0.55
+0.44
+2
+0.33
+0.22
+0.11
+0
+1
+0
+0
+1
+2
+3
+4
+5
+bstructure of the system. Therefore, we refer to increasing b from zero to infinity as the
+process of turning on the coupling between the background and the conductivity, and finally
+resulting in an AdS-RN system.
+It is worth noting that the relationship between conductivity and entanglement-related
+quantities is of great importance in condensed matter theories. Recent experiments have
+shown that entanglement between quantum dots can persist despite the influence of surface
+plasmon polariton (SPPs) transmission [47, 48, 50]. These findings are crucial for the devel-
+opment of stable quantum circuits. Additionally, it has been found that at specific values
+of the inter-dot distance d or detuning δ, the two-quantum-dot system can be in a highly
+entangled state and be separate from the transmission of SPPs [46]. However, when d or δ
+deviate from these values, the entanglement of quantum dots becomes highly correlated with
+the transmission of SPPs. This suggests that decoupling of entanglement and transport can
+exist in real physical systems and can be characterized by certain parameters.
+Next, we will focus on how the entanglement-related physical quantities change as we
+vary the parameter b.
+B.
+Holographic information-related quantities
+Entanglement is a fundamental and intriguing aspect of quantum mechanics. One way
+to quantify entanglement is through entanglement entropy (EE), which measures the degree
+of entanglement between a subset of a system and the rest of the system. Specifically, the
+entanglement entropy SA between subsets A and B of a system A ∪ B is defined as the von
+Neumann entropy in terms of the reduced density matrix ρA.
+SA(|ψ⟩) = −Tr [ρA log ρA] ,
+ρA = TrB (|ψ⟩⟨ψ|) .
+(11)
+It is easy to find that SA = SB for pure states [51]. Holographic duality theory relates the
+holographic entanglement entropy (HEE) to the area of the minimum surface in dual gravity
+systems [5] (see the left plot of Fig. 2).
+EE is often used to measure the degree of entanglement in pure states, but it is not as
+effective in measuring mixed state entanglement. For example, even when subsystems A and
+B are not entangled, they can still have non-zero EE in a system composed of direct product
+of the density matrices of ρA and ρB. This is because EE takes into account both quantum
+7
+
+entanglement and classical correlation, so it does not always provide a accurate measure
+of the entanglement. As a result, other measures for mixed-state entanglement have been
+proposed in the literature [9, 10]. The most direct mixed-state entanglement measure is MI.
+For the subsystem A ∪ C separated by B, the mutual information (MI) is defined as:
+I (A, B) := S (A) + S (B) − S (A ∪ B) ,
+(12)
+This measures the mixed-state entanglement between A and B. It can be easily verified
+that I (A, B) = 0 when ρAB = ρA ⊗ ρB, therefore MI have the property that direct product
+states have zero entanglement. However, MI is not a perfect measure of mixed-state entan-
+glement, as it is closely related to EE, and it’s properties are sometimes dominated by EE
+or thermal entropy in certain situations. This indicates that other measures of mixed-state
+entanglement should be used.
+The entanglement wedge cross-section (EWCS) has been associated with the duality of
+certain mixed-state entanglement measures, such as entanglement of purification, logarith-
+mic negativity, and reflect entropy [53–55]. Takayanagi proposed that EWCS EW (ρAB) is
+the area of the minimum cross-section ΣAB in connected entanglement wedge [34], i.e. (see
+the right plot in Fig. 2),
+EW (ρAB) = min
+ΣAB
+�Area (ΣAB)
+4GN
+�
+.
+(13)
+It is important to note that if the entanglement wedge is disconnected, meaning the minimum
+cross-section does not exist, the EWCS will be zero, it corresponds to cases with vanish-
+ing MI. Additionally, the EWCS also satisfies some important inequalities as its quantum
+information counterparts [34, 56]
+Next, we present the algorithm for obtaining the minimum surfaces and EWCS.
+C.
+Computations of holographic geometric quantities
+We examine the EWCS of an infinite strip with a homogeneous background for numerical
+simplicity. For a generic homogeneous background
+ds2 = gttdt2 + gzzdz2 + gxxdx2 + gyydy2,
+(14)
+where z = 0 represents the boundary of the asymptotic AdS spacetime. The left plot in
+Fig. 3 is a visual representation of the minimum surface for an infinite strip along the y-
+8
+
+x
+y
+z
+x
+y
+z
+FIG. 2: The left plot: The minimum surface for a given width w. The right plot: The minimum
+cross-section (green surface) of the entanglement wedge.
+axis. Since the background is homogeneous, all metric components gµν only depend on the
+coordinate z.
+1.
+The minimum surface
+The minimum surface near the AdS boundary is perpendicular to the boundary, making
+the spatial direction x an unsuitable parameter for finding the minimum surface. Ref. [58]
+adopted the angle θ with tan θ = z/x, as the parameter for the minimum surface (see Fig.
+3). Using this method, we can parametrize a surface as (x(θ), z(θ)) with area A given by
+A = 2
+� π/2
+0
+�
+x′(θ)2gxxgyy + z′(θ)2gyygzzdθ.
+(15)
+The resultant equations of motion read,
+x′(θ)z′(θ)2
+� g′
+xx
+2gxx
++ g′
+yy
+gyy
+− g′
+zz
+2gzz
+�
++ x′(θ)3 �
+gyyg′
+xx + gxxg′
+yy
+�
+2gxxgzz
++ x′′(θ)z′(θ) − x′(θ)z′′(θ) = 0,
+z(θ) − tan(θ)x(θ) = 0.
+(16)
+where g′
+## ≡ g′
+##(z). The boundary conditions are,
+z(0) = 0,
+x(0) = w,
+z′(π/2) = 0,
+x(π/2) = 0,
+(17)
+where w is the width of the strip.
+9
+
+2.
+The EWCS
+Given a biparty subsystem with minimum surfaces C1(θ1), C2(θ2), we solve the minimum
+surface Cp1,p2 connecting p1 ∈ C1 and p2 ∈ C2. We parametrize Cp1,p2 with z, then the area
+of Cp1,p2 reads,
+A =
+�
+Cp1,p2
+�
+gxxgyyx′(z)2 + gxxgzzdz.
+(18)
+The resultant equation of motion becomes,
+x′(z)3
+� gxxg′
+yy
+2gyygzz
++ g′
+xx
+2gzz
+�
++ x′(z)
+�g′
+xx
+gxx
++ g′
+yy
+2gyy
+− g′
+zz
+2gzz
+�
++ x′′(z) = 0,
+(19)
+with boundary conditions,
+x(z(θ1)) = x(θ1),
+x(z(θ2)) = x(θ2).
+(20)
+To obtain the EWCS, we need to locate the global minimum of the minimum surfaces
+connecting C1(θ1), C2(θ2), i.e., the minimum cross-section.
+Finding the minimum cross-section is a challenging task as it involves searching through
+a two-dimensional parameter space (θ1, θ2).
+However, it can be noted that the globally
+minimum cross-section must be perpendicular to the minimum surfaces at the point of
+intersection. This observation serves as a local constraint, which can greatly speed up the
+search process. We demonstrate the methods of solving the EWCS in Fig. 3. For numerical
+stability, it is better to implement the perpendicular conditions with normalized vectors as,
+Q1(θ1, θ2) ≡
+gab
+� ∂
+∂z
+�a �
+∂
+∂θ1
+�b
+�
+gcd
+� ∂
+∂z
+�c � ∂
+∂z
+�d
+�
+gmn
+�
+∂
+∂θ1
+�m �
+∂
+∂θ1
+�n
+��������
+p1
+= 0,
+Q2(θ1, θ2) ≡
+gab
+� ∂
+∂z
+�a �
+∂
+∂θ2
+�b
+�
+gcd
+� ∂
+∂z
+�c � ∂
+∂z
+�d
+�
+gmn
+�
+∂
+∂θ2
+�m �
+∂
+∂θ2
+�n
+��������
+p2
+= 0.
+(21)
+Note that Q1 and Q2 are both functions of the θ1 and θ2. Now, the search of the EWCS is
+equivalent to finding the minimum surface ending at (θ1, θ2) where (21) is satisfied.
+To determine the correct EWCS, we first select an initial seed (θ1, θ2) and use the Newton
+iterative method to obtain feedback (δθ1, δθ2). By repeating this process, we can find the
+minimum cross-section, which is the EWCS. It is crucial to carefully choose the initial
+10
+
+-1.5
+-1.0
+-0.5
+0.5
+1.0
+1.5
+x
+0.2
+0.4
+0.6
+0.8
+1.0
+z
+p1
+p2
+θ1
+θ2
+C1(θ1)
+C2(θ2)
+FIG. 3:
+The demonstration of the EWCS. The p1 and p2 are the intersection points of the
+minimum surface connecting those two minimum surfaces. The solid blue curve (parametrized
+with θ1) and solid orange curve (parametrized with θ2) are minimum surfaces.
+The thick red
+curve is the minimum surface connecting p1 and p2. The blue arrows at the p1 and p2 are the
+tangent vector
+� ∂
+∂z
+�a���
+p1 and
+� ∂
+∂z
+�a���
+p2 along the Cp1,p2, while the purple arrows are the tangent
+vectors
+�
+∂
+∂θ1
+�a���
+p1 and
+�
+∂
+∂θ2
+�a���
+p2 along C1, C2, respectively. The dark dashed horizontal line is the
+horizon.
+values of (θ1, , θ2) for the iterations to converge. The numerical reliability is ensured by
+the convergence of results when using different initial values or increasing the density of
+discretization. For more technical details, refer to reference [57].
+Using the techniques outlined above, we will now examine mixed-state entanglement
+measures for the BI model. Additionally, we will examine the correlation between the BI
+factor b and information-related quantities.
+III.
+THE HOLOGRAPHIC ENTANGLEMENT ENTROPY AND THE HOLO-
+GRAPHIC MUTUAL INFORMATION
+We begin by examining the relationship between HEE, system parameters b and T. As
+shown in Fig. 4, HEE, represented by S, increases monotonically with both b and T, but
+their rate of increase is different. Initially, S increases slowly with T and its growth rate
+with T becomes more pronounced as T increases. On the other hand, S increases quickly
+with b at first and then slows down as b decreases. Next, we explain the behavior of S with
+11
+
+b=0.0001000 b=0.008791
+b=0.01765
+b=0.02868
+b=0.03912
+b=0.05289
+0.05
+0.10
+0.15
+0.20
+0.25
+T
+-1.80
+-1.75
+-1.70
+-1.65
+-1.60
+S
+w=0.8
+T=0.00100 T=0.09967 T=0.1389
+T=0.1614
+T=0.1856
+T=0.2110
+0.1
+0.2
+0.3
+0.4
+b
+-1.80
+-1.75
+-1.70
+-1.65
+-1.60
+-1.55
+S
+w=0.8
+FIG. 4: HEE vs T and b at width w = 0.8, respectively.
+b and T, respectively.
+When the horizon radius of the black brane increases, the minimum surface tends to be
+closer to the horizon of the black brane, which makes the thermodynamic entropy dominate
+the behavior of the HEE. Therefore, the growth of HEE with T as well as b, can be un-
+derstood from the relation between rh and T or b. According to (5) we can deduce that rh
+increases with increasing temperature and b, this can be seen by taking the derivative of rh
+with respect to T and b. The results are,
+∂Trh =
+r4
+h
+�
+Q2
+b2r4
+h + 1
+r4
+h
+�
+2b2
+��
+Q2
+b2r4
+h + 1 − 1
+�
++ 3
+�
+Q2
+b2r4
+h + 1
+�
++ 2Q2,
+∂brh =
+2rh
+�
+Q2 − 2b2r4
+h
+��
+Q2
+b2r4
+h + 1 − 1
+��
+b
+�
+r4
+h
+�
+2b2
+��
+Q2
+b2r4
+h + 1 − 1
+�
++ 3
+�
+Q2
+b2r4
+h + 1
+�
++ 2Q2
+�.
+(22)
+From the above equation, it is clear that ∂Trh is always positive, indicating that rh increases
+as T increases. However, ∂brh can be positive or negative, depending on the specific pa-
+rameter range. Further examination shows that ∂brh is always greater than zero when rh is
+relatively large. This means that rh increases with b when rh is large, or when the minimum
+surface is closer to the horizon of the black brane.
+When b is relatively large, the system is approximately the AdS-RN system. The ar-
+gument presented in [59] can be applied to prove that ∂TS > 0. Furthermore, for small
+subregions, it can be inferred from the equations in [59] that ∂TS is close to 0, which ex-
+plains the flat behavior of S along T for small temperatures.
+After studying HEE, we proceed to investigate the behavior of MI with T and b. In the BI
+model, the configurations for MI and EWCS are subsystems composed of a and b separated
+12
+
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.2
+0.4
+0.6
+0.8
+1.0
+b
+T
+I(3, 0.10, 2)
+5.85
+6.63
+7.41
+8.19
+8.97
+9.75
+10.53
+11.31
+12.09
+12.87
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+0.2
+0.4
+0.6
+0.8
+1.0
+b
+T
+I(3, 0.25, 2)
+0.42
+0.84
+1.26
+1.68
+2.10
+2.52
+2.94
+3.36
+3.78
+4.20
+FIG. 5:
+MI as a function of b and T for different configurations.
+b=0.0001000 b=0.04554
+b=0.09392
+b=0.1591
+b=0.2526
+b=0.4000
+0.1
+0.2
+0.3
+0.4
+0.5
+T
+0.5
+1.0
+1.5
+I
+(a, p, c) = (0.5, 0.2, 0.35)
+T=0.00100 T=0.1182 T=0.1614
+T=0.1856
+T=0.2110 T=0.2372
+0.1
+0.2
+0.3
+0.4
+b
+1.30
+1.35
+1.40
+1.45
+1.50
+1.55
+I
+(a, p, c) = (0.5, 0.2, 0.35)
+FIG. 6: MI as a function of b and T for different configurations.
+by region p. As seen in Fig. 5, MI decreases with increasing temperature and b. This is in
+contrast to the behavior of HEE. Moreover, it is worth noting that MI can decrease to zero,
+which is an indication of a disentanglement phase transition. We have also plotted the MI
+for smaller configurations (see Fig. 6), and the qualitative phenomena remain the same.
+As the subsystem c and the separation p change, the system undergoes a disentangling
+phase transition, at which point the entanglement of two subsystems a and c vanishes. The
+critical value of subsystem cc and separation pc are shown in Fig. 7. The left plot of Fig. 7
+shows that the critical value of subsystem cc increases with b and T; however, the right plot
+of Fig. 7 shows that the critical value of the separation pc decreases with b and T. This is
+as expected since increasing the temperature or b will tends to destroy the entanglement,
+13
+
+0.0
+0.1
+0.2
+0.3
+0.4
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+b
+T
+cc (a=0.6, p=0.3)
+0.4234
+0.4408
+0.4582
+0.4756
+0.4930
+0.5104
+0.5278
+0.5452
+0.5626
+0.5800
+0.0
+0.1
+0.2
+0.3
+0.4
+0.0
+0.1
+0.2
+0.3
+0.4
+0.5
+b
+T
+pc (a=0.6, c=0.3)
+0.1988
+0.2030
+0.2072
+0.2114
+0.2156
+0.2198
+0.2240
+0.2282
+0.2324
+0.2366
+FIG. 7: Critical configurations of cc and pc.
+resulting in a larger subregion cc or a smaller separation pc.
+Next, we explore the mixed-state entanglement through the EWCS.
+IV.
+THE HOLOGRAPHIC ENTANGLEMENT WEDGE CROSS-SECTION
+In Fig.
+8, we present the minimum surfaces and the corresponding minimum cross-
+sections. It can be observed that the minimum surface is flatter when the temperature is
+lower. This is due to the fact that the coordinate z is related to the horizon radius rh, and at
+lower temperatures, a small rh will rescale z to zrh, resulting in a flatter minimum surface.
+This makes it challenging to obtain precise enough solutions for the minimum surface since
+the ���at case is more singular in the θ coordinate. To overcome this issue, we redefine the
+angle as z = ηx tan(θ), where η is a number related to the temperature. Only with this
+technique, we can achieve precise enough solutions.
+We show the EWCS vs b in Fig. 9, from which we can find that the EWCS can show very
+delicate behaviors. The EWCS increases with b at first in a very narrow range of b, however,
+it starts to decrease with b when b is relatively large and monotonically decreases with b.
+This is in sharp contrast to the behavior of the HEE and MI, that only shows monotonical
+behaviors (see Fig. 4 and Fig. 5). In addition, the EW changes slower with b than that with
+T. The typical change is of order 10−4 and 10−3, respectively. This delicate behavior can
+be captured precisely because the precision of our numerical methods can be up to 10−7.
+Notice that the background is an AdS-Schwarzschild solution when b is 0, meanwhile, its
+14
+
+-����
+-����
+����
+����
+�
+����
+����
+����
+����
+����
+����
+�
+�=������
+�=������
+�=������
+�=������
+�=������
+FIG. 8: The illustration of EWCS. At the same configuration (a, p, c) = (0.1, 0.05, 0.06925) we
+see that the minimum surface becomes flatter when decreasing the temperature. Meanwhile, the
+minimum cross-section always ends at the point near the tops of the inner minimum surface, while
+ends at the point away from the tops of the outer minimum surface.
+0.0
+0.1
+0.2
+0.3
+0.4b
+18.360
+18.365
+18.370
+18.375
+Ew
+T=0.1565 T=0.2492 T=0.2537
+T=0.2654 T=0.2758 T=0.2971
+0.05
+0.10
+0.15
+0.20b
+-0.02
+0.02
+0.04
+∂bEw
+FIG. 9: EWCS vs T. This plot is obtained at (a, p, c) = (0.1, 0.05, 0.06925). When b is relatively
+large, the EW converges to certain fixed values. For T = 0.2971 it can first increase, and later
+decreases, and after that increases with b. Therefore, for very small b the EWCS increases with b,
+irrespective of the values of the T and the configurations.
+electromagnetic field is non-zero. At this point, the charge transport behavior of the system
+is significantly different from that of the genuine AdS-Schwarzschild system.
+Moreover,
+since its geometry is still AdS-Schwarzschild, the entanglement-related geometric quantities
+will be decoupled from the charge transport.
+As b gradually increases, the background
+geometry will receive back reactions from the Maxwell field. At this time, the entanglement-
+related geometry starts to couple with the charge transports. Therefore, b can play a role
+in measuring the relationship between entanglement and transport when b is small. As we
+15
+
+T=0.1211
+T=0.1364
+T=0.1461
+T=0.1569
+T=0.1687
+0.0
+0.1
+0.2
+0.3
+0.4b
+4.864
+4.866
+4.868
+4.870
+4.872
+4.874
+4.876
+Ew
+FIG. 10: The EWCS vs b for a larger configuration (a, p, c) = (0.5, 0.2, 0.3875).
+have pointed out, the EWCS increases with b when b is very small, i.e., when the coupling
+has just occurred. And when b increases further, the EWCS gradually shows a decreasing
+behavior. Notice that simpler geometric quantities such as HEE, and MI only show a very flat
+monotonic behavior. This indicates that EWCS, as a mixed-state entanglement, captures
+very different properties from HEE and MI.
+To understand the above behavior more clearly, we implement the following analytical
+treatments. For small values of b, we can expand the expression of the EW (18) integral
+with respect to b as,
+EW =
+�
+Σ
+�
+� 1
+z2
+�
+dx2 +
+dz2
+(1 − z3) + bdz2 �
+Γ
+� 1
+4
+�
++ 8Γ
+� 5
+4
+�� �
+dz2 + dx2 (1 − z3)
+�−1/2
+2
+�
+3πT (1 − z3) (z2 + z + 1) Γ
+� 1
+4
+�
++ O(b2)
+�
+� ,
+(23)
+where the second term shows us that
+dEW
+db
+> 0 for small values of b. This explains the
+ubiquitous existence of the monotonically increasing behavior of EW vs b for small values of
+b. From the holographic dual picture, it means that when the Maxwell field starts to turn
+on from the BI case, the EW is increased. However, when further increasing b we find that
+EW reaches local maximums and starts to decrease. When b is large, it can be expected that
+the background system approaches the AdS-RN, a fixed background geometry. Therefore,
+the EW will starts to converge to some fixed value.
+Next, we show the EWCS in larger configurations in Fig. 10. As seen in Fig. 10, the
+non-monotonicity of EW with b becomes more pronounced as the width of the configuration
+increases. This means that the non-monotonicity exists over a wider interval. The reason
+for this is that when the width is relatively small, the minimum surface and the minimal
+16
+
+b=0.1355
+b=0.1753
+b=0.2492
+b=0.2758
+b=0.3084
+0.20
+0.25
+0.30
+0.35
+0.40T
+18.32
+18.33
+18.34
+18.35
+18.36
+18.37
+18.38
+Ew
+b=0.0100
+b=0.0744
+b=0.1389
+b=0.2033
+b=0.2678
+0.20
+0.21
+0.22
+0.23
+0.24
+0.25
+0.26
+0.27T
+4.81
+4.82
+4.83
+4.84
+4.85
+4.86
+Ew
+FIG. 11: EWCS vs T. The left plot is obtained at (a, p, c) = (0.1, 0.05, 0.06925); while the right
+plot is obtained for a larger configuration (a, p, c) = (0.5, 0.2, 0.3875).
+cross-section are only slightly different from the properties of AdS. However, as the width
+increases, they deviate more significantly from AdS.
+Next, we examine the behavior of EWCS with temperature. When the configuration is
+relatively small in BI systems, EWCS decreases monotonically with temperature, as shown
+in the left plot of Fig. 11. It is worth noting that the non-monotonic behavior of EWCS at
+extremely small temperatures has been studied in [59] for AdS-RN systems. Additionally,
+we illustrate the behavior of EWCS with temperature for larger configurations in the right
+plot of Fig. 11, which also shows that EWCS decreases monotonically with temperature.
+Although the monotonic decreasing behaviors are similar, the EWCS curves for small con-
+figurations differ from those for large configurations. By comparing the two plots in Fig.
+11, crossovers of the EWCS curves with temperature can be observed in the larger configu-
+ration, which reflects the non-monotonic behavior of EWCS with b. These findings suggest
+that the behavior of EWCS is generally monotonically decreasing with temperature, and
+this behavior is consistent with that of MI.
+In order to more clearly demonstrate the relationship between the EWCS and variables
+b and T, a contour plot of EWCS as a function of b and T is presented in Figure 12. This
+plot illustrates the non-monotonic nature of EWCS with respect to b and the monotonic
+decrease of EWCS as T increases.
+17
+
+FIG. 12: The EWCS vs b for a larger configuration (a, p, c) = (0.5, 0.2, 0.3875).
+V.
+DISCUSSION
+In this paper, we study the behavior of HEE, MI, and the mixed-state entanglement
+measure EWCS in the BI model. Our results shows that HEE increases monotonically with
+both b and T, while MI decreases monotonically with both b and T.
+Interestingly, the
+behavior of EWCS with respect to b shows a non-monotonic trend. Specifically, when b is
+small, EWCS increases with b, but it begins to decrease as b increases further. In contrast,
+EWCS exhibits a consistent monotonically decreasing trend with T.
+Moreover, we provide analytical explanations for the non-monotonic behavior of EWCS
+with respect to b. Note that when b is small, b serves as a measure of the coupling between
+the entanglement-related quantities and the charge transport of the system. Based on this
+observation, we conjecture that increasing the coupling between the entanglement-related
+quantities and the transport properties can enhance the EWCS of the system. This cou-
+pling between transport behaviors and entanglement is also a topic of significant interest in
+condensed matter theory, as seen in previous studies on nanowires [46], plasmonics [47, 50],
+and plasmons [48].
+18
+
+Ew at (a,p,c)=(0.5,0.2,0.3875)
+0.40
+0.35
+5.066
+4.998
+0.30
+4.930
+4.862
+T
+4.794
+0.25
+4.726
+4.658
+4.590
+0.20
+0.15 E
+0.0
+0.1
+0.2
+0.3
+0.4
+bNext, we point out the issues that deserve further investigation. To begin, we can exam-
+ine other BI-like theories, such as the BI theory with massive gravity, the BI theory with
+Axions, and so on, to see if the non-monotonic behavior observed in this paper is general.
+Furthermore, we can examine the effect of more general nonlinear EM field theories on the
+entanglement-related physical quantities of the system, such as the more general nonlinear
+EM fields [39, 60]. We are working on these directions.
+Acknowledgments
+Peng Liu would like to thank Yun-Ha Zha for her kind encouragement during this work.
+Zhe Yang would like to express appreciation to Feng-Ying Deng. This work is supported by
+the Natural Science Foundation of China under Grant No. 11805083, 11905083, 12005077,
+12147209, the Science and Technology Planning Project of Guangzhou (202201010655) and
+Guangdong Basic and Applied Basic Research Foundation (2021A1515012374). J.-P.W. is
+also supported by Top Talent Support Program from Yangzhou University.
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+The persistent homology of genealogical networks
+Zachary M. Boyda,∗, Nick Callorb, Taylor Gledhillc, Abigail Jenkinsd, Robert Snellmane, Benjamin Webbf,
+Raelynn Wonnacottg
+aDepartment of Mathematics, Brigham Young University, Provo, UT 84602, USA, zach boyd@byu.edu
+bDepartment of Mathematics, Brigham Young University, Provo, UT 84602, USA, n.b.callor@gmail.com
+cDepartment of Mathematics, Brigham Young University, Provo, UT 84602, USA, gledhilltaylor2@gmail.com
+dDepartment of Mathematics, Brigham Young University, Provo, UT 84602, USA, jenkins.abby@gmail.com
+eDepartment of Mathematics, Brigham Young University, Provo, UT 84602, USA, snellman@mathematics.byu.edu
+fDepartment of Mathematics, Brigham Young University, Provo, UT 84602, USA, bwebb@mathematics.byu.edu
+gDepartment of Mathematics, Brigham Young University, Provo, UT 84602, USA, raelynnwo@gmail.com
+Abstract
+Genealogical networks (i.e. family trees) are of growing interest, with the largest known data sets now
+including well over one billion individuals. Interest in family history also supports an 8.5 billion dollar
+industry whose size is projected to double within 7 years [FutureWise report HC-1137]. Yet little mathemat-
+ical attention has been paid to the complex network properties of genealogical networks, especially at large
+scales.
+The structure of genealogical networks is of particular interest due to the practice of forming unions, e.g.
+marriages, that are typically well outside one’s immediate family. In most other networks, including other
+social networks, no equivalent restriction exists on the distance at which relationships form. To study the
+effect this has on genealogical networks we use persistent homology to identify and compare the structure
+of 101 genealogical and 31 other social networks. Specifically, we introduce the notion of a network’s
+persistence curve, which encodes the network’s set of persistence intervals. We find that the persistence
+curves of genealogical networks have a distinct structure when compared to other social networks. This
+difference in structure also extends to subnetworks of genealogical and social networks suggesting that, even
+with incomplete data, persistent homology can be used to meaningfully analyze genealogical networks. Here
+we also describe how concepts from genealogical networks, such as common ancestor cycles, are represented
+using persistent homology. We expect that persistent homology tools will become increasingly important in
+genealogical exploration as popular interest in ancestry research continues to expand.
+Keywords: persistent homology, genealogical networks, social networks, persistence curves, bottleneck
+distance
+1. Introduction
+The study of genealogical networks, that is networks relating parents with children and spouses with
+each other through successive generations is of rapidly growing interest, both because of genealogy’s pop-
+ular appeal and its applications in genetics [1], sociology [2], population sciences [3], and economics [4].
+Growing data availability of rich, temporally resolved data is also driving interest in genealogy. For example,
+∗Corresponding Author
+arXiv:2301.11965v1  [q-bio.MN]  27 Jan 2023
+
+FamilySearch has constructed a human family tree with over 1.40 billion individuals, based on 2.21 billion
+sources, including 4.78 billion images (https://www.familysearch.org/en/newsroom/company-facts). Popu-
+larization of DNA testing services and increasing availability of audio sources, geographic tags, occupation
+metadata, and migration records combine to make genealogical networks some of the largest, most richly
+featured, geospatially embedded temporal networks in existence. Examples of relevant academic studies
+include methods for automatically constructing networks from documents [5, 6], analyzing marriage pat-
+terns [4], structured population modeling, branching processes [7], and biconnected components [2, 8]. Of
+particular interest to us are works that study distance to recent common ancestors, both theoretically and via
+simulation (e.g. [9, 3]). A growing body of literature also uses genealogical networks for genetic inference,
+as in [1].
+Related to these genealogical endeavors, a major goal of network science is to describe the structure
+of such real-world networks. In this paper, we consider persistent homology as a tool to both analyze and
+explore the structure of genealogical networks. Persistent homology, roughly speaking, is a method of
+representing voids or gaps in the structure of a network, that distinguishes how significant these voids are to
+the overall network structure. Persistent homology can be used to compare these voids across two networks
+without requiring a correspondence between the individual vertices or edges, or even requiring the networks
+to be the same size. The basic idea involves “filling in” the network with simplices (points, edges, triangles,
+tetrahedra, etc.) and keeping track of how the network changes as we do so (see Section 3 for details).
+Some similar applications of persistent homology in the study of networks include [26], [28], [30], [27].
+The collaboration networks studied in [26] are similar to the social networks that we use for comparison
+in this paper, though our focus is primarily on distinguishing these from genealogical networks. Both [28]
+and [30] apply persistent homology techniques to general randomized networks of various forms. It is also
+possible to vary the technique for generating a topological object from a network, as in [27] where three
+methods are compared. We also recommend [29] and [36] as good overviews of the general methods of
+applying persistent homology.
+For this paper, our method of constructing a topological representative for each network follows the same
+general pattern as the work cited above. However, we also acknowledge the wide variety of alternatives for
+encoding such information. [32] and [33] encode their information as point-clouds rather than graphs. A
+higher-dimensional version of persistent homology is presented in [34], which may permit the inclusion
+of time-varying networks. Finally, the formulation in [35] may allow for better analysis of corrupted or
+too-large datasets.
+We also wish to bring attention to four particular applications that demonstrate the versatility of persistent
+homology. In each of these applications, persistent homology has been used to identify structural voids in
+data and then to associate these voids to recognizable features in the underlying networks. It is the latter
+use that we wish to emphasize. Robins et al. [10] have shown that voids found using persistent homology
+correspond to percolating spheres in a porous material. In [11], structural voids arise when several groups
+of neurons are strongly connected sequentially, but out-of-sequence pairs are only weakly connected. In
+these neurological networks, persistent homology provides a way to identify and classify these different
+sequences as well as quantify the strength of these connections. The application in [31] provides a method
+for extending traditional genetic analysis tools to a parameterized family of datasets by constructing an
+appropriate topological object. Lastly, [12] shows that structural voids or gaps can also represent much
+more abstract concepts. In this case persistent voids are shown to correspond to the atonality in music
+compositions.
+Intuitively, the voids or gaps in genealogical networks should be quite different when compared with
+2
+
+other networks, such as social networks, since unions1 (such as marriages) in genealogical networks typically
+form at specific distances, rather than through other mechanisms e.g. triadic closure. That is, distances
+between individuals who form unions are typically not too small or too large (see Section 2). In contrast,
+in other social networks, new connections can form at any distance but are often quite small [13]. This
+difference in network growth between genealogical and other social networks causes differences in network
+topology that are reflected in the network’s persistent homology. Thus persistent homology is a useful
+descriptive tool for exploring and modeling the structure of genealogical networks.
+Here, we propose a new method for representing persistent homology, which we call a persistence curve
+(see Section 4). The persistence curves of many genealogical networks are very similar to each other,
+and importantly the persistence curves of subsets of genealogical networks, that is, sampled genealogical
+networks, are also similar to the persistence curves of unsampled genealogical networks (see Section 6).
+To give our study of genealogical networks context we also study the persistent homology of social net-
+works. We find that the same result holds for the social networks we consider, in that the persistence curves
+of social networks show a common pattern and the persistence curves for social and sampled social networks
+are similar (see Section 6). We confirm our analysis using another tool for comparing persistent homologies,
+the bottleneck distance, which is also capable of detecting and differentiating the distinct homology patterns
+between genealogical and other social networks.
+In summary, we make the following contributions:
+• Introduce the notion of a persistence curve and introduce the use of this together with the bottleneck
+distance as a tool for the analysis of general networks.
+• Report the distinct persistent homology structure of genealogical networks using both persistence
+curves and the bottleneck distance.
+• Link this structure to genealogically relevant concepts.
+• Similarly, report the distinct persistence homology structure of social networks and compare this to
+the structure of genealogical networks.
+• Report evidence that persistent homology methods work well even in the presence of incomplete data.
+This is particularly relevant given that genealogical data is often, if not necessarily, incomplete.
+Throughout the paper, examples from family networks are contrasted with other social networks to highlight
+the unique features of genealogical networks from a persistent homology point of view.
+The paper is organized as follows. In Section 2 we describe both genealogical and social networks. In
+Section 3 we define the persistent homology of a network and introduce the notion of persistence curves. In
+Section 4 we define the bottleneck distance and show how both this distance and persistence curves can be
+used to compare networks. In Section 5 we describe the genealogical and social data sets we use in our study
+and give our experimental results in Section 6. Section 6 also includes a discussion of how certain structural
+features of social and genealogical networks are represented using persistent homology. In Section 7 we
+summarize our results and conclude with a discussion regarding the use of persistent homology as a tool for
+analyzing general network structure and recovering network features. Throughout we give examples of each
+of the concepts we introduce.
+1In order to be inclusive of various relevant relationships in this paper, we use the word “union” to describe not only legal marriages
+and common law marriages but also some others, including any relationship that produced children.
+3
+
+2. Background: Genealogical and Social Networks
+We represent genealogical networks with a graph G = (V, E), where V = {1, 2, . . . , n} are the individ-
+uals within the network, and E are the (genealogical) relationships. These relationships consist of both
+parent-child edges and spouse (or more generally union) edges. For the sake of simplicity, these edges are
+considered to be undirected. We note that the structure of a genealogical network is often thought of as
+Out[1858]=
+Tikopia Genealogical Network
+Residence Hall Social Network
+Figure 1: Left: The largest connected component of the Tikopia genealogical network consisting of 288 individuals from the island of
+Tikopia in Polynesia from the year 1930 to 2008, is shown [14]. Parent-child edges are shown in blue and union edges are shown in
+red. Right: The largest connected component of the Residence Hall social network consisting of 217 individuals and their friendships
+from the Australian National University campus is shown [15].
+being “tree-like”, since genealogical networks are often constructed from an individual, their parents, their
+grandparents, and so on, ignoring union edges. The result is a tree, i.e. a connected acyclic graph, if we
+create only a few generations of the family. However, full genealogical networks are not trees due to the
+presence, for example, of triangles consisting of two parents and a child (with the two parent-child edges
+and one union edge). Because of the frequency of such cycles and the fact that they are the smallest possible
+cycles, we refer to them as trivial cycles. The other typical familial cycle, or cycle found within a family
+consisting of two parents and some number of children, is a cycle of length four consisting of two parents
+and two children.
+Although familial cycles are ubiquitous in genealogical networks, they are not the only cycles that can
+form. Going far enough through an individual’s ancestors, it is often possible to find a nearest common
+ancestor, i.e., a common ancestor of one’s father and mother. If such an ancestor exists (and it usually does
+exist), then the genealogical network has a nontrivial cycle. We refer to this as a common ancestor cycle,
+which consists of only parent-child edges. Other nontrivial cycles are possible in genealogical networks via
+unions. For instance, a “double cousins” relationship occurs when two siblings from one family form unions
+with two siblings from another family. The result is a union cycle, or a cycle that contains only union edges
+and the parent-child edges connecting siblings. In genealogical networks, union and parent-child edges can
+combine in any number of ways to create complex non-tree structures (see Figure 1 left).
+A feature that is particular to genealogical networks is that union edges typically form at specific dis-
+tances within these networks. Here the distance d(i, j) between i and j is the shortest path distance between
+these individuals if such a path exists. Otherwise, it is infinite. In a genealogical network we refer to the
+distance between two individuals before they form a union as the couple’s distance to union. For cultural,
+genetic, and other reasons these distance are typically not small, i.e. usually larger than four. Consequently,
+4
+
+0
+5
+10
+15
+20
+25
+30
+0.00
+0.05
+0.10
+0.15
+Distance to Union
+Fraction of Unions at Distance
+Figure 2: The histogram representing the finite “distance to union” distances is shown where data is collected from 104 genealogical
+networks from kinsources.net. The height of each bar represents the fraction of unions that form at a specific distance.
+genealogical networks do not typically have small nonfamilial cycles and often have large extended cycles.
+This is illustrated in Figure 2 where distance to union data is collected from 104 publicly available genealog-
+ical networks given in Table 2 in the Appendix. Here familial cycles are omitted and the height of each bar
+represents the fraction of unions that form at a specific distance. Noticeably, few unions form at distances
+less than five with the large majority of distance falling between 5 and 10.
+The observation that genealogical networks have large extended cycles is illustrated in Figure 3. Shown
+left in orange is the distribution of cycle lengths of the San Marino genealogical network, a network of the
+population of the Republic of San Marino from the 15th to the end of the 19th century [14]. In this network,
+which consists of 28,586 individuals, there are 7,146 familial cycles of length three and 8,636 familial cycles
+of length four. These are omitted in the figure so we can observe the lengths of the cycles forming a basis
+of nonfamilial cycles in the network. For the sake of contrast, in blue is the distribution of cycle lengths
+in a basis of the cycles found in the Deezer Europe social network, consisting of 28,281 individuals. Here,
+similar to genealogical networks, a social network is represented by a graph G = (V, E) where the vertices V
+also represent individuals. The difference is that in a social network the edges represent some type of social
+interaction(s). The Deezer network is an online music streaming platform whose social network represents
+individuals in Europe who use the platform where edges represent mutual user-follower relationships.
+Noticeably, the San Marino network has relatively few nonfamilial basis cycles under length ten but
+quite a few cycles with lengths greater than thirty. In contrast, the Deezer social network has a much tighter
+distribution of basis cycles ranging from roughly five to fifteen in length.
+To understand the extent to which these cycle distributions are related to the local structure of the associ-
+ated networks we compare these to the cycle distribution of the associated configuration models of these two
+networks, respectively. The configuration model is a model for generating random networks with a given
+degree sequence [16]. Taking the degree sequences from both the San Marino genealogical and Deezer so-
+cial network, we create ten versions of these networks each with the same degree sequences. The result of
+averaging the basis cycle length distributions of these versions of the San Marino and Deezer networks is
+shown in Figure 3 (center and right in red and green, respectively). While the cycle distribution for the San
+Marino network is quite different from what the configuration model produces, the Deezer social network
+is quite similar to the distribution predicted by its configuration model. This suggests that much of the cy-
+cle structure in the Deezer social network is dominated by local interactions, whereas the cycles in the San
+Marino genealogical network are affected by nonlocal mechanisms that form the network. This includes,
+5
+
+Out[51]=
+0
+10
+20
+30
+40
+0.00
+0.05
+0.10
+0.15
+0.20
+0
+10
+20
+30
+40
+0.00
+0.05
+0.10
+0.15
+0.20
+0
+5
+10
+15
+0.00
+0.05
+0.10
+0.15
+0.20
+0.25
+0.30
+0.35
+SM and DE Cycle Lengths
+SM Configuration Model
+DE Configuration Model
+Figure 3: Left: Shown in orange is the distribution of the lengths of the cycles forming a basis of the nonfamilial cycle lengths in the
+San Marino (SM) genealogical network. The analogous distribution of cycle lengths is shown in blue for all cycles in the Deezer Europe
+(DE) social network. Center: Shown in orange is again the basis cycle length distribution of the San Marino genealogical network. In
+red is the distribution of the basis cycle lengths averaged over ten realizations of the (loopy, multi-edged) configuration model on the
+San Marino network. Since the configuration model generates graphs with the same degree distribution as the SM network, this panel
+indicates that SM’s longer cycles do not arise simply from the degree distribution. Right: Shown in blue is again the basis cycle length
+distribution of the Deezer social network. In green is the distribution of the basis cycle lengths averaged over ten realizations of the
+configuration model on the Deezer social network. For this social network, the cycle length distribution can be mostly explained by the
+degree distribution alone.
+presumably, the nonlocal distance to union phenomena described above.
+The relations we see in Figure 3 between the cycle length distribution for the San Marino genealogical
+network and the Deezer social network are typical of the genealogical and social networks we consider in
+Section 5. This suggests that cycle length distribution is a feature that can be used to distinguish genealog-
+ical from social networks. Specifically, when we consider two networks with a similar number of cycles,
+genealogical networks have a much wider distribution of cycle lengths than social networks. However, the
+method used to calculate the cycle length distribution in Figure 3 does not provide any further insight into
+this phenomenon. This limitation motivates us to apply tools from persistent homology which provides ways
+to describe and measure the relation between any two network cycles. The additional structure that can be
+obtained by these methods allow us to further distinguish the structure of genealogical and social networks
+(see Section 6.1) and to relate the structural differences demonstrated in Figure 3 to mechanisms that produce
+genealogical and social networks, respectively (see Section 6.3).
+3. Persistent Homology of Networks
+Persistent homology provides a method for studying cycles in a network. For the purposes of this paper,
+a brief explanation of persistent homology will be given from the context of simplicial homology. For a
+more in-depth treatment of simplicial homology, see Chapter 2.1 of [17]. For those readers who are either
+familiar with the basics of persistent homology or who wish to skip the following technical discussion it is
+possible to proceed to Section 5 where we discuss the social and genealogical networks we analyze.
+For a network given by a graph G = (V, E) we define the distance matrix D(G) = [di j] to have entries
+di j = d(i, j), which is the length of the shortest path between individual i and j. For each value δ that
+appears in the distance matrix D(G), we form a simplicial complex Gδ as follows. The set of 0-simplices
+is equivalent to the set of vertices of G, where each 0-simplex is identified with a single vertex. Since the
+distinction between 0-simplices and vertices is purely formal, we will use the terms 0-simplex and vertex
+interchangeably, and the 0-simplices will be indexed the same way as the vertices. The set of 1-simplices Eδ
+corresponds to the set of edges {i, j} such that d(i, j) ≤ δ, where the edge {i, j} is identified with the 1-simplex
+6
+
+formed by i and j. Again the distinction here is unnecessary for our present discussion, so we will use the
+same notation for 1-simplices and edges. However, the simplicial complex Gδ may also contain objects that
+do not have equivalent representatives in the graph G, namely the n-simplices for n ≥ 2. For each integer
+n ≥ 2, the set of n-simplices in Gδ consists of all n-simplices [a0
+a1
+. . .
+an] such that d(ai, aj) ≤ δ for
+0 ≤ i < j ≤ n. That is, Gδ includes an n-simplex σ if each vertex listed in σ is within δ of every vertex listed
+in σ.
+In order to simplify our remaining definitions, we extend our definition of Gδ to include all non-negative
+integers. For i ≥ 0, let δi be the greatest entry of D(G) such that δi ≤ i. Let Gi = Gδi. This definition together
+with our construction of Gδ ensures the following three important properties are true for all Gi.
+1. For i < j, Gi is a subcomplex of G j, i.e. every simplex of Gi is a simplex of G j.
+2. For i ≥ 1, there exists a subcomplex of Gi that can be identified with the original graph G.
+3. Since G is finite, let M = maxij d(i, j), then, for all i ≥ M, Gi = GM.
+(a) G0
+(b) G1 = G
+(c) G2
+(d) G3
+Figure 4: The hexagonal network G = G1 in Example 3.1 is filled in as i increases from 0 to 3. This produces the simplicial complexes
+G0,G1,G2,G3 shown left to right.
+Example 3.1. (Hexagonal Network) Consider the hexagonal network G = (V, E) with six vertices, forming
+a single cycle, shown in Figure 4(b). This network has the distance matrix
+D(G) =
+�������������������������
+0
+1
+2
+3
+2
+1
+1
+0
+1
+2
+3
+2
+2
+1
+0
+1
+2
+3
+3
+2
+1
+0
+1
+2
+2
+3
+2
+1
+0
+1
+1
+2
+3
+2
+1
+0
+�������������������������
+.
+For the values i = 0, 1, 2, 3, we form four simplicial complexes, G0, G1, G2, and G3 where we let Gi =
+(Vi, Ei). For i = 0, E0 is empty. Thus, G0 consists of six vertices. For i = 1 the set E1 contains the six
+edges that form the network’s single cycle, so G1 = G. This graph has no trivial cycles (i.e., triangles), so
+G1 contains no simplices of dimension greater than 1 (i.e., no n-simplices for n > 1). For i = 2 the set E2
+gains six additional edges. We also now have eight trivial cycles. Each of these cycles is the boundary of
+a 2-simplex, so G2 contains these eight 2-simplices as well. However, no subset of these 2-simplices forms
+the boundary of a 3-simplex, so G2 has no simplices of dimension greater than 2. For i = 3 the set E3
+contains all possible edges between the vertices of G, so all possible trivial cycles are present. Additionally,
+all possible 2-simplices, and hence all possible n-simplices, are also present in G3. In particular, G3 is a
+6-simplex with its boundary. Since M = 3 is the largest value we see in the distance matrix, then Gi = G3
+for i ∈ Z, i > 3.
+7
+
+The persistent homology of the network G measures how the homology of Gi changes as i increases. If
+certain features can be identified across multiple values of i, we say they persist. Intuitively, features that
+arise from the actual network structure should persist for many values of i, while features that arise because
+of measurement error, ‘noise’, should only appear sporadically. The Stability Theorem (the Main Theorem
+of [18]) states that if the error in measuring a network is bounded by some constant C, then the persistent
+homology of the true network and the persistent homology of the noisy network will differ by at most C. We
+will make this statement more precise in Section 4.1.
+Here we give a formal definition of persistent homology in terms of simplicial homology, which we will
+immediately follow this with equivalent definitions in the context of networks. We use Hp(Gi) to denote the
+dimension-p simplicial homology of the simplicial complex Gi with coefficients in Z2, as Hp(X) is a vector
+space of Z2.
+Definition 1. (pth Persistent Homology) For a graph G, and integers i, j with 0 ≤ i ≤ j, let the function
+φi, j : Hp(Gi) → Hp(G j) be the linear map induced by the inclusion Gi → G j. The pth persistent homology
+of G, PHp(G) is the pair ({Hp(Gi)}i≥0, {φi,j}0≤i<j).
+Our analysis in Sections 4-6 only requires the first few dimensions of persistent homology to distinguish
+the genealogical and social networks we consider. In order to better understand what persistent homology
+calculates, in what follows we will provide equivalent definitions for PH0, PH1, and PH2 using network
+concepts. We also illustrate how these definitions apply to the hexagonal network in Figure 4(b). (See
+Examples 3.3, 3.4, and 3.5 for PH0, PH1, and PH2; respectively.)
+Definition 2. (Births and Deaths) Let G = (V, E) be a network with simplicial complexes G0,G1,G2, · · · .
+The pth persistent homology of G provides maps φi, j between the pth homology of Gi and the pth homology
+of G j. Suppose that basis elements have been chosen for each Hp(Gi) so that if α is a basis element of
+Hp(Gi), then φi,j(α) is either trivial in Hp(G j) or a basis element of Hp(G j). The birth of a basis element
+α ∈ Hp(G j) is the minimum index i such that α = φi, j(ˆα) for some basis element ˆα ∈ Gi. The death of α is
+the minimum index k such that φj,k(α) is trivial.
+Remark 3.2. Those already familiar with persistent homology will find that the preceding definition is
+somewhat nonstandard, although it is equivalent to the standard definition. We have taken this approach
+to reduce the notation burden on non-specialist readers. We have done similarly with some of the other
+persistent homology definitions.
+We will demonstrate how to choose such representatives for H0, H1, and H2 in the following definitions.
+Given such representatives, though, the maps φi,j and φ j,k are simply the maps on homology induced by
+the inclusion maps Gi ⊂ G j ⊂ Gk. That is, if a represents α ∈ Hp(Gi), then a also represents φi, j(α).
+The Fundamental Theorem of Persistent Homology ensures that we can choose a single representative that
+corresponds to α ∈ Hp(G j), ˆα ∈ Hp(Gi), and φj,k(α) ∈ Hp(Gk). The birth of α is then just the first Gi in which
+the representative exists, and the death of α is the first Gk in which the representative is null-homotopic i.e.,
+homotopic to a trivial cycle.
+Definition 3. (Representing Persistent Homology: Dimension 0) Let G = (V, E) be a network with vertices
+V = {1, 2, . . . , n} which form k connected components. Then H0(G0) � Zn
+2, so we can identify the basis
+for H0(G0) with the set of all n vertices. Likewise, we may choose k vertices, one from each connected
+component, to represent the basis for H0(Gi) � Zk
+2 for i ≥ 1. Thus, we will refer to the vertices of G as
+representatives of PH0(G). (In fact, PH0(G) is a vector space whose basis elements are equivalence classes
+of formal sums of 0-simplices.)
+8
+
+Example 3.3. We now consider PH0(G) for the hexagonal network G in Figure 4, with G0, G1, G2, and G3
+in the same figure. Recall that G has six distinct vertices forming one connected component. If we take any
+numbering of the vertices, V = {1, 2, 3, 4, 5, 6}, then H0(G0) � Z6
+2, which is equivalent to the vector space
+over Z2 with basis V. For i > 0, H0(Gi) � Z2, which is equivalent to the vector space over Z2 with basis
+{1}. For any v ∈ V, since i = 0 is the first time we see v, we call this the birth of v. At i = 1, since we have
+removed all vertices except 1 from the basis, we say this is the death of those five 0-simplices. Since 1 will
+always be in the basis for Gi, the death of 1 is said to be ∞.
+Definition 4. (Representing Persistent Homology: Dimension 1) Let G = (V, E) be a network with one
+connected component. For each i ≥ 0, we can identify the basis of H1(Gi) with a set Ci of cycles in Gi. The
+Fundamental Theorem of Persistent Homology allows us to choose these cycles so that if σ is a cycle in Ci,
+then exactly one of the following is true for any integer j ≥ 0:
+1. σ does not exist in G j, in which case j < i,
+2. σ is trivial or null-homotopic in G j, in which case i < j,
+3. σ is a cycle in C j.
+Thus, we will refer to the cycles in �
+i≥0 Ci as the representatives of PH1(G). (Again, PH1(G) is actually
+much larger than this. These are actually representatives of equivalence classes that form a basis for PH1(G)
+as a vector space.)
+We note that C0 is always empty, since there are no edges in G0. Furthermore, rank(H1(Gi)) = |Ci| for
+all i ≥ 0. Because of the construction of the Gi all representatives of PH1(G) will be present in G1. One
+can think of the representatives of PH1(G) as representing “large” cycles. More specifically, if a cycle σ is
+contained in �
+s≤i≤t Ci, then it must have a diameter of at least t and at least one pair of consecutive vertices
+distance s apart.
+Example 3.4. We now consider PH1(G) for the hexagonal network G in Figure 4(b). In both Figure 4(a)
+and 4(b) we see that G0 has no cycles, G1 has exactly one cycle, and that the cycle in G1 is non-trivial.
+In Figures 5(a) and 5(b), we have indicated some of the cycles in G2, namely the cycles 1,2,3,1; 3,4,5,3;
+1,5,6,1; and 1,3,5,1 in Figure 5(a) and the cycle 1,2,3,5,1 in Figure 5(b). In fact, Figure 5(c) shows us that
+G2 is an octahedron and therefore every cycle in G2 is either trivial or null-homotopic. Finally, G3 contains
+even more cycles than G2, such as 1,3,6,1; but these are all null-homotopic since G3 also contains every
+possible 2-simplex for six vertices. Therefore, PH1(G) has only one representative, the cycle 1,2,3,4,5,6,1;
+which appears in G1, so we say that t = 1 is the birth of the cycle. The cycle is null-homotopic in G2, so
+t = 2 is the death of the cycle.
+We now turn our attention to PH2(G), but in order to represent PH2(G) we need to introduce some new
+structure for the induced graphs. A triangle [a
+b
+c] in Gi is a set of three vertices, a, b, and c, that form a
+trivial cycle in Gi. That is, the edges {a, b}, {b, c}, and {a, c} are all present in Gi. A closed surface in Gi is a
+set of distinct triangles so that for each [a
+b
+c] in the set there is exactly one other triangle [a
+b
+d] also
+in the set. A closed surface in Gi is trivial if the corresponding set of 2-simplices is null-homotopic in Gi.
+That is, the closed surface is “filled in” by some collection of 3-simplices in Gi. For example, the octahedron
+in Figure 5(c) is a non-trivial closed surface in G2 because there are no 3-simplices in G2. In G3, however,
+we add edges between vertices at distance 3. In turn, we gain several 3-simplices, including [1
+2
+3
+6],
+[1
+3
+5
+6], [3
+4
+5
+6], and [2
+3
+4
+6]. Figure 5(d) shows three of these 3-simplices to demon-
+strate how the closed surface from G2 is filled in by all four.
+9
+
+1
+2
+3
+4
+5
+6
+1
+2
+3
+4
+5
+6
+1
+2
+3
+4
+5
+6
+1
+2
+3
+4
+5
+6
+(a) G2 trivial cycles
+(b) G2 null-homotopic cycle
+(c) G2 sphere
+(d) G3 select 3-simplices
+Figure 5: A visual depiction of simplices and cycles present in G2. Left: Four trivial cycles filled by individual 2-simplices: [1
+2
+3],
+[3
+4
+5], [1
+5
+6], and [1
+3
+5]. Center Left: A non-trivial, but null-homotopic cycle, 1, 2, 3, 5, 1 filled in by two 2-simplices
+[1
+2
+3] and [1
+3
+5]. Center Right: All eight 2-simplices represented as the faces of a regular octahedron. Right: The closed
+surface of G2 is filled in by four 3-simplices [1
+2
+3
+6], [1
+3
+5
+6](notshown), [3
+4
+5
+6], [2
+3
+4
+6].
+Definition 5. (Representing Persistent Homology: Dimension 2) Let G = (V, E) be a network with one
+connected component. For each i ≥ 0, we can identify the basis for H2(Gi) with a set S i of non-trivial closed
+surfaces in Gi. The Fundamental Theorem of Persistent Homology allows us to choose these representatives
+so that if σ is a closed surface in S i, then exactly one of the following is true for any integer j ≥ 0
+1. σ does not exist in G j, in which case j < i,
+2. σ is trivial in G j, in which case i < j,
+3. σ is a cycle in S j.
+Thus we will refer to the closed surfaces in �
+i≥0 S i as the representatives of PH2(G).
+The geometric intuition for PH2(G) is similar to that of PH1(G) in identifying large ‘voids’ in G. If
+σ ∈ �
+s≤i≤t S i, then σ is a closed surface with diameter at least t. The value of s is harder to describe, but is
+related to the density of vertices.
+Example 3.5. We now consider PH2(G) for the hexagonal graph G in Example 3.1. Recall from Example
+3.4 that G0 and G1 have no trivial cycles, and therefore contain no closed surfaces. We can see in Figure
+5 that G2 has exactly one closed surface and it must be non-trivial, since there are no 3-simplices. Finally,
+G3 has many closed surfaces, but because it contains every possible 3-simplex on six vertices, these are all
+trivial. Therefore, PH2(G) has only one representative, the octahedral closed surface in G2. This surface
+first appears in G2, so t = 2 is its birth, and the surface is filled by a solid in G3, so t = 3 is its death.
+Definition 6. (Persistence Intervals) Recall that the birth of a representative σ ∈ PHp(G) (vertex, cycle,
+or closed surface) of the persistent homology of a network G is the smallest integer i so that σ ∈ Gi, and
+the death of σ is the largest integer j so that σ ∈ G j−1 and σ is trivial in Gk for k ≥ j, if such an integer
+exists. The persistence interval for σ is [a, b), where a and b are the birth and death of σ, respectively.
+This represents the set of all parameter values i for which the equivalence class corresponding to σ is a
+non-trivial element of Hp(Gi). The persistence of σ is b − a.
+Example 3.6. We now finish our consideration of the persistent homology of G from Figure 4(b). Recall
+from Example 3.3 that PH0(G) has six representatives. These all have birth t = 0. Five of these have a death
+of t = 1, and one of these has a death of ∞. Therefore the persistence intervals for PH0(G) are [0, 1) × 5 and
+[0, ∞) × 1.
+10
+
+From Example 3.4, we know PH1(G) has one representative, with birth t = 1 and death t = 2. Therefore
+the corresponding persistence interval is [1, 2). Note that the diameter of the cycle is 3 and every pair of
+consecutive vertices is distance 1 apart. This follows the idea mentioned earlier that the representatives of
+PH1(G) indicate ‘large’ cycles. Specifically, the diameter of σ is at least the death of σ, and the birth of σ
+is the maximum distance between consecutive vertices.
+From Example 3.5, PH2(G) has one representative, with birth t = 2 and death t = 3. Therefore, the
+persistence interval for that element is [2, 3). Note that the diameter of the corresponding set of vertices is 3
+in G. This also follows the idea mentioned earlier that PH2(G) identifies large ‘voids’ in G. Specifically, the
+death of σ is a lower bound on the diameter of σ.
+Given the representatives chosen in Definitions 3, 4, 5, and 6, we have the following three observations
+regarding the persistent homology of a finite, undirected, unweighted graph G:
+(i) If G has n vertices, then PH0(G) will have exactly n persistence intervals, with exactly one [0, ∞) interval
+for each connected component and the rest will be [0, 1) intervals.
+(ii) In dimension 1, PH1(G) describes the number and sizes of the non-trivial cycles in the original network.
+The persistence intervals will all be of the form [1, b) for some integer b > 1. The value of b is related to
+the diameter of the corresponding cycle. In the networks we have studied, we note that a persistence interval
+[1, b) in PH1(G) corresponds to a simple cycle with between 3b − 2 and 3b vertices, inclusive.
+(iii) In dimension 2, the voids we detect in PH2(G) tell us about the nontrivial intersections of cycles. Such
+intersections are hard to visualize but, roughly speaking, a representative in PH2(G) can only form if several
+large cycles intersect each other pairwise.
+4. Comparing Networks using Persistent Homology
+In this section we demonstrate how methods based on persistent homology can be used to compare
+different networks. The two methods we introduce in this paper are based on using (a) the bottleneck
+distance and (b) the persistence curves of a given set of networks. Both (a) and (b) rely on first computing
+persistence intervals then analyzing the differences in these intervals.
+The two networks we consider throughout this section to demonstrate these methods are the Tikopia ge-
+nealogical network from Figure 1 (left) and the hexagonal network from Figure 4. The persistence intervals
+for these networks are given in Table 1, respectively.
+Dimension
+Interval Type and Persistence
+Tikopia
+Hexagon
+Dimension 0
+[0, ∞) × 8, [0, 1) × 286
+[0, ∞) × 1, [0, 1) × 1
+Dimension 1
+[1, 2) × 16, [1, 3) × 19, [1, 4) × 5, [1, 5) × 3,
+[1, 6) × 2, [1, 7) × 1
+[1, 2) × 1
+Dimension 2
+[2, 3) × 4, [3, 4) × 11, [4, 5) × 12, [5, 6) × 4,
+[6, 7) × 5, [7, 8) × 1, [8, 9) × 1
+[2, 3) × 1
+Table 1: The persistence intervals of the Tikopia genealogical network and the hexagon network are shown. Here the notation [a, b) × k
+indicates that the network has k persistence intervals [a, b). The corresponding persistence diagrams are shown in Figure 6 and the
+corresponding persistence curve for the Tikopia network is shown in Figure 7.
+11
+
+4.1. Persistence Diagrams and Bottleneck Distance
+One common way to represent persistence intervals is to plot them as points in R × (R ∪ {∞}), which
+is typically referred to as a persistence diagram. While this method of visualizing a network’s persistent
+homology does not indicate how often a given persistence interval occurs, it does provide information on
+what kind of persistence intervals occur for a given network.
+Definition 7. (Persistence Diagrams) Let PHp(G) be the pth persistent homology of a network G. The
+persistence diagram for PHp(G) is a multiset of points in R × (R ∪ {∞}) defined as follows.
+• For each σ ∈ PHp(G) with persistence interval [a, b), we include one copy of the point (a, b).
+• For each c ∈ R, we include infinitely many copies of the point (c, c).
+Note that we include the points (a, a) to represent features in G that are considered trivial in PHp(G),
+such as cycles consisting of exactly three vertices. This inclusion is necessary for us to define a meaningful
+metric on the space of persistence diagrams. The metric we use here is called the bottleneck distance.
+Definition 8. (Bottleneck Distance) Let S 1 and S 2 be persistence diagrams for two graphs G and H, re-
+spectively. Let η range over the set of bijections from S 1 to S 2. Then the bottleneck distance between S 1 and
+S 2 is
+dB(S 1, S 2) = inf
+η sup
+x∈S 1
+∥x − η(x)∥∞.
+The Fundamental Theorem of Persistent Homology (introduced in [19], explained well in [36] and [29])
+ensures that if two graphs are isomorphic, the corresponding persistence diagrams will be equal, and thus the
+bottleneck distance will be 0. However, it is possible for non-isomorphic graphs to have identical persistence
+diagrams.
+Example 4.1. (Bottleneck Distance Between the Tikopia and Hexagonal Networks) Notice that the per-
+sistence intervals for the Tikopia genealogical network (see Table 1) include, as a subset, the persistence
+intervals from the hexagonal network we considered in Example 3.6. We can form a bijection between the
+persistence diagrams of the Tikopia and hexagonal network by identifying the non-trivial intervals from the
+hexagonal network with those of the Tikopia network. We then map any additional intervals from the Tikopia
+network of the form [a, b) to the trivial interval [ a+b
+2 , a+b
+2 ). (The perceptive reader may notice that this is not
+clearly a bijection, but there is a standard technique from set theory for modifying it to be bijective.)
+This mapping is shown in Figure 6 (right). Here, [1, 7) is mapped to [4, 4). As this pair of points is
+further apart than any other pair in this bijection, the bottleneck distance for the two networks is at most
+three, since we take an infimum over all possible bijections. Conversely, there is no interval in the hexagonal
+persistence diagram that is closer to [1, 7) than 3, so the bottleneck distance is at least three. Thus, the
+bottleneck distance for these two persistence diagrams is exactly 3.
+Suppose that two networks, each of which is connected, admit isometric embeddings in Rn. The Stability
+Theorem [18] guarantees that if the Hausdorff distance between the embeddings is δ, then the bottleneck
+distance for the corresponding persistence diagrams is at most δ. For example, if the PH1 persistence
+diagrams differ by δ, then any attempt to pair up cycles in the networks must include at least one pair of
+cycles for any isometric embedding that are δ apart in that embedding. In Section 6.1 we apply this idea to
+a large collection of genealogical and social networks.
+12
+
+Hexagonal Network PD
+Tikopia Network PD
+Bottleneck Bijection
+Figure 6: Left: The persistence diagram of the hexagonal network in Figure 4(b) is shown. Center: The persistence diagram of
+the Tikopia genealogical network in Figure 1 (left) is shown. Right: A bottleneck bijection between the persistence intervals of the
+hexagonal and Tikopia family network is shown. Orange lines show which points are matched to points of the form (a, a) where a ∈ R.
+4.2. Persistence Curves
+For the network data we consider, persistence diagrams obfuscate a key difference that we consider
+important: the number of persistence intervals. For a simple example of this, consider networks of the form
+V = {1, 2, . . . , n} with edges of the form {i, i + 1} for 1 ≤ i < n. For n ≥ 2, any network of this type will have
+persistence intervals [0, 1) × (n − 1) and [0, ∞) × 1. However, when plotting the persistence diagram we will
+only ‘see’ two points: (0, 1) and (0, ∞).
+To address this limitation, we introduce the notion of a persistence curve as a new way to visualize
+the persistent homology of a network (see Definition 9). The difference between the persistence curve and
+the persistence diagram of a network is that the persistence curve also includes the number of intervals of
+a particular type. To create a persistence curve we first compute a network’s persistence intervals, then
+sort the intervals of a given dimension by their persistence into a bar graph. For instance, in dimension
+1 the Tikopia genealogical network has thirteen [1, 2) intervals, nineteen [1, 3) intervals, etc. which are
+sequentially stacked as shown in Figure 7 (left) to create what we will call a barcode. To create the associated
+persistence curve we connect the endpoints of each subsequent bar as shown in Figure 7 (right).
+In dimension-one, the birth times of our intervals will all start at 1, as the networks we consider are
+unweighted, undirected, and connected. This means that in this dimension the resulting bar graph is also a
+plot of the death times for each interval. For higher-dimensions, which have varied birth times, we also plot
+the lengths of the intervals but for simplicity we start at 1 as in dimension-one.
+A formal definition of a network’s persistence curves is the following.
+Definition 9. (Persistence Curves) Let G = (V, E) be a network with nonempty vertex and edge sets. Let
+{[a j, bj)} be the set of all persistence intervals for each σj ∈ PHn(G) where j ∈ N. For all n ∈ N the
+persistence curve PHn(G) is the linear interpolation of the set of points {(bj − (aj − 1), j)} where b j−1 −
+(aj−1 − 1) ≤ bj − (aj − 1).
+Visualizing persistence intervals as a curve allows us to compare the persistent homology of different
+networks in a similar fashion to persistence diagrams while retaining different information. In particular, we
+can see how many intervals there are of a given persistence, whereas the persistence diagram only indicates
+the presence of such an interval. In what follows we will typically plot the persistence curves of multiple
+networks on the same axes to indicate what differences exist in the persistent homology of different networks
+(cf. Section 6).
+13
+
+6-Cycle Persistence Diagram
+Tikopia Persistence Diagram
+Superimposed Persistence Diagram
+1
+8-
+8
+8
+6
+4
+eath
+: 
+(a, a)
+a, a)
+2
+(e 'e)
+Dimension 0
+Dimension 0
+Dimension 1
+Dimension 1
+Matched Points
+Dimension 2
+Dimension 2
+Tikopia Network
+0
+2
+4
+6
+8
+0
+2
+4
+6
+8
+0
+2
+4
+6
+8
+1
+Birth
+Birth
+BirthTikopia Network Barcode
+Tikopia Network Persistence Curve
+Figure 7: Left: The barcode of the Tikopia genealogical network in dimension 1 is shown. The individual bars are formed from the
+persistence intervals given in Table 1. Right: The associated persistence curve for the Tikopia network in Figure 1 is shown.
+5. Data
+The data we consider in this paper is of two types; genealogical network data and other social network
+data. The genealogical networks we consider are drawn from ninety-seven genealogical networks found
+in[14], which range in size from n = 17 to 5, 016 individuals. The social network data we use is taken from
+twenty-seven different social networks obtained from [20, 21, 22, 23]. These range in size from n = 16 to
+2, 539 individuals. (See Table 2 in the Appendix for a full description of this data set.)
+Although many larger genealogical and social network data sets are available we are limited by both the
+temporal and spacial complexity of the algorithm used to compute persistence intervals. The program we
+used, called Ripser (from the python package Ripser) [24], has a computational and spacial complexity of
+O((n + m)3) where n is the number of individuals and m is the number of edges in a network. The number
+n+m is the number of simplicies in the network. In the genealogical networks we consider there are between
+n + m = 41 to 15, 735 simplicies and in the social networks we consider between n + m = 41 to 19, 056
+simplices.
+To understand how a network’s persistence intervals are effected by the completeness or incompleteness
+of data we also consider subnetworks sampled from a few, much larger, genealogical and social networks.
+These sampled networks are created by randomly selecting an individual with a single neighbor, i.e. a
+vertex of degree 1, then performing a breadth-first-search starting with this individual to find the η closest
+individuals in the network to this individual. Because of the spatial and computational limitations of Ripser
+we choose 600 ≤ η ≤ 3, 000 to ensure we can compute the persistence intervals of these sampled networks.
+In total we sampled from four different genealogical networks and four different social networks. These are
+the Advogat, LastFM Asia, Deezer HU and Deezer RO social networks and the genealogical networks 96–99
+shown in Table 2, respectively. We sampled from each of these networks five times each to create a total of
+20 sampled genealogical networks and 20 sampled social networks. The reason we begin our breadth-first
+search with a vertex of degree 1 is to ensure that our sampled networks have vertices both on the boundary
+and the interior of the original network we sampled to better mimic the structure of the original genealogical
+and social networks.
+Apart from the (i) genealogical and social networks we consider and (ii) sampled versions of these
+networks, we also consider what we refer to as (iii) atypical genealogical networks. There are a number of
+14
+
+FamilyDimension1
+40
+30
+Interval Index
+20
+10
+0
+1
+2
+m
+4
+-5
+6
+1
+7
+Intervals (Birth and Death Time)FamilyDimension1
+40
+Interval Index
+20
+10
+2
+3
+4
+5
+6
+7
+Intervals(BirthandDeathTime)genealogical networks that appear to be created with no attempt to represent all or even a fraction of the
+familial relationships. For example, the US Presidents network, cited as Atyp. Gen. Network 2 in Table 2,
+follows the shortest genealogical path between presidents leaving out extraneous relationships. We consider
+a number these atypical genealogical networks, which form a contrast to the more standard genealogical
+networks we consider especially in terms of their peristent homology. A description of each of the (i)
+genealogical, social, (ii) sampled genealogical, sampled social, and (iii) atypical genealogical networks we
+consider is given at the end of the Appendix.
+Figure 8: PCA projections of the bottleneck distances between networks are shown. Left: The bottleneck distance between each of
+the twenty sampled genealogical and sampled social networks is shown. Center: The bottleneck distances are shown between the
+genealogical, social, and atypical genealogical networks we consider. Right: The bottleneck distances in the center panel are shown for
+only the genealogical and social networks we consider.
+6. Results
+Here we compare genealogical and other social networks using the (a) bottleneck distance and the (b)
+persistence curves defined in Section 4 (see Definitions 8 and 9, respectively). For those who have skipped
+Sections 3 and 4, the bottleneck distance gives us a distance between two networks based on the differences
+in their persistent homology. Persistence curves give us a way of visualizing this difference but in greater
+detail (cf. Figure 7).
+6.1. Network Comparison using Bottleneck Distance
+Here we compute the bottleneck distance between every pair from the social and genealogical networks
+we consider. To visualize these results we use principal component analysis to identify the two components
+that account for the most variance and then plot this data in R2 (see Figure 8).
+From each part of Figure 8 we can see that genealogical networks are generally separated from social net-
+works and form clusters that are easily distinguished. For the sampled networks (shown left), we can easily
+separate genealogical and social networks, and we can identify at least two distinct subclasses of genealogi-
+cal networks. However, the bottleneck distance does an inferior job separating the non-sampled genealogical
+and social networks (shown center and right). The exception are the atypical genealogical networks, whose
+persistence intervals differ significantly enough from all of the other networks to be distinguishable as a third
+class of networks (shown center).
+15
+
+Sampled Genealogical and Social Networks
+Genealogical, Social, and atypical Network
+Genealogical and Social Networks
+Genealogical
+24
+Genealogical
+Genealogical
+Social
+Social
+8
+Social
+Atypical
+15
+6
+-2
+-
+1
+21
+31
+t
+-
+15
+2Figure 9: Comparison of persistence curves for full networks vs sampled networks, grouped by dimension and type of network. Upper
+Row: Sampling social networks typically stretches the persistence curve in only one axis without affecting the other axis. Lower Row:
+Sampling genealogical networks typically shrink the persistence curve in both axes. Overall the average slope for social networks tends
+to increase when sampled, while genealogical networks experience a decrease in average slope.
+6.2. Comparison of Genealogical and Social Networks using Persistence Curves
+Persistence curves give us a new alternative way of comparing networks. The advantage of using these
+curves compared to the bottleneck distance is that these curves give us a more detailed picture of how
+the number of persistence intervals varies from network to network. This allows us to better differentiate
+the structure of genealogical networks from social networks as well as observe the structure common to
+genealogical networks and those common to social networks, respectively.
+In Figure 9 the persistence curves for the unsampled genealogical and unsampled social networks are
+shown in blue and red, respectively. The atypical genealogical networks are shown in green. The social
+networks have persistence curves that are quite vertical in both dimension 1 and dimension 2. For dimension
+1, this indicates that most cycles in a social network are close to being trivial; either because they have a
+relatively small circumference or because they can be decomposed into a union of cycles with small circum-
+ferences. In particular, most of the social networks have a maximum death time of three (see Definition 2),
+which corresponds to having a basis of cycles whose maximal circumference is at most nine. In other words,
+any cycle of circumference ten or more decomposes as the union of smaller cycles. For dimension 2, the
+steepness of the persistence curves indicate the presence of many distinct, yet similar, paths between certain
+pairs of vertices.
+In contrast, the genealogical networks have persistence curves that have a much more horizontal profile
+16
+
+Social vs. Sampled Sccial Networks: Dimension 1
+Social vs. Sampled Sccial Networks: Dimension 2
+O-
+Social
+50D0 
+Social
+Sampled Social
+Sampled Social
+ofIntervals
+3100
+31D0
+2400 
+aqnn
+2400
+8DO
+0 -
+0 
+2D
+25
+3.D
+3.5
+4.D
+2D
+25
+3.D
+3.5
+4.D
+Genealogical vs. Sampled Genealaogical Networks: Dimensian 1
+Genealogical vs. Sampled Genealagical Networks: Dimensian 2
+160.0
+Atypical
+3500
+Atypical
+1400
+Genealogical
+310
+Genealogical
+Number of Intervaba
+1200
+Sampled Genealogical
+of Intervals
+Sampled Genealogical
+2500
+24D0
+aqnn
+1500
+1400
+240
+500
+0 -
+0-
+25
+5.D
+SL
+14.0
+12.5
+15.0
+17.5
+25
+5.D
+14.0
+12.5
+15.0
+17.5indicating that most cycles are quite long and there are fewer ‘alternate paths’ between pairs of vertices.
+In the extreme, the atypical genealogical networks are nearly flat in dimension 1, which reflects the fact
+that these atypical networks were intentionally constructed to have very few cycles. In dimension 2, the
+atypical networks show a similar slope to most of the typical genealogical networks, but the size of the
+alternative paths in these networks are much larger. This is likely due to the high number of individuals
+who were added only to link distant individuals, e.g. presidents. In a typical genealogical network, the
+additional relationships between such individuals would allow large cycles to decompose but in the atypical
+genealogical networks this in not the case.
+Figure 10: Upper Row: Comparison of persistence curves for full networks by type. Lower Row: Comparison of persistence curves for
+sampled networks by type, excluding atypical genealogical networks. In each dimension, the average slope for genealogical networks
+is typically lower than the average slope for a social network. The atypical genealogical networks have the lowest average slope and
+much greater total length. The behavior for average slopes is more pronounced for sampled networks than for full networks.
+In Figure 10, we see the persistence curves for the sampled genealogical and sampled social networks
+shown in blue and red, respectively. The atypical genealogical networks are shown in green. Again the social
+networks have persistence curves that are quite vertical in both dimensions, although these curves are not as
+tall as in the case of unsampled social networks. This indicates that as a social network is sampled it retains a
+similar proportion of close-to-trivial cycles, but may lose many of the alternative paths between vertices that
+appear in dimension 2. By contrast, for genealogical networks the persistence curves indicate the complete
+loss of very large cycles in conjunction with a proportional loss of close-to-trivial cycles. In dimension
+2, genealogical networks experience a more severe loss of alternative paths than the social networks. As a
+result, though sampling shrinks the scale of the persistence curves for social and genealogical networks, they
+remain visually distinct.
+17
+
+Genealogical vs. Sccial: Dimension 1
+Genealogical vs. Sccial Networks: Dimension 2
+O
+Atypical
+5000
+Atypical
+Genealogical
+Genealogical
+Social
+Number of Intervals
+Social
+40
+3100
+2400
+24D0
+0
+25
+5.D
+7.5
+14.0
+12.5
+15.0
+17.5
+25
+5.D
+7.5
+14.0
+012.515.017.5
+Sampled Geneakogical vs. Sampled Social Networks: Dimensian 1
+Sampled Geneakogical vs. Sampled Social Networks: Dimensian 2
+24D0
+Sampled Genealogical
+24D0
+Sampled Genealogical
+Sampled Social
+Sampled Social
+of Intervals
+of Intervaba
+1500
+1500
+1400
+1400
+500
+0
+2D
+25
+3.0
+3.5
+4.0
+4.5
+5.D
+2
+3
+6As in the bottleneck distance plots, genealogical and social networks appear to cluster together in that
+they have similar types of persistence curve. In fact, this is true whether or not the networks are sampled or
+unsampled. This suggests that even with incomplete data social network and genealogical networks have a
+distinguishable persistent homology, at least at the scales we consider.
+It is worth mentioning that, while the bottleneck distance plots show us to an extent how different ge-
+nealogical and social networks are the persistence curves show us what are differences are. The distance
+plots in Figure 8 do have the advantage of simplicity, however, and could presumably be used to more
+quickly identify differences in networks that are not as apparent as those we find between genealogical and
+social networks.
+6.3. Connections
+It is also possible to use persistent homology to study properties of a network, such as the number of
+connected components, the typical size of cycles, or even “missing links” in the data. For genealogical
+and social networks, we can convert these mathematical concepts into more familiar ideas such as family
+groups or common ancestors. This also allows us to make conjectures about the persistent homology for
+such networks by converting standard assumptions about families or social networks into the language of
+persistence.
+In dimension 0, the number of connected components determines the number of [0, ∞) intervals, and the
+total number of distinct vertices is the number of [0, ∞) intervals plus the number of [0, 1) intervals. In the
+context of a genealogical network, each connected component represents a family group that is not related
+to the other family groups by any known connection. Thus, if a given family network is indeed a single
+“family” of relatives, there should be exactly one [0, ∞) interval. In our Tikopia example we have eight
+[0, ∞) intervals each of which correspond to exactly one connected component of this genealogical network.
+(Note that Figure 1 (left) shows only the largest of these components). In this example, most of the the other
+‘family groups’ are actually individuals with no relation edges in the network.
+In social networks, the connected components create what could be referred to as friend groups. Unlike
+genealogical networks, there are usually few restrictions on which edges form in a social network. As
+such, we do not have a conjecture about the number of [0, ∞) intervals in this setting in general. However,
+sampling any network as described in Section 5 will result in a new network with a single [0, ∞) interval.
+Moving to dimension 1, persistence intervals in this dimension describe the way that each connected
+component is internally structured. In sufficiently large genealogical networks, we will see three kinds of
+features that we call common ancestors, union cycles, and hybrid cycles. A common ancestor cycle occurs
+when two descendants of an individual form a union or have a child together. We use the term union cycle to
+refer to situations where a cycle is formed through union edges and edges connecting two siblings. The final
+type of cycle of note, the hybrid cycles, are those formed by any other combination of parent-child edges
+and union edges, which includes everything that is not a strict common ancestor or union cycle. These three
+types of cycles are illustrated in Figure 11, where marriage edges are indicated by red edges and parent-
+child edges are indicated by blue edges. We show a common ancestor in Figure 11(a). Figure 11(b) is an
+example of a union cycle in which two siblings in one family form unions with two siblings in another,
+where only a single parent in each family is shown. In Figure 11(c) we give an example of a θ-cycle, which
+is the union of a common ancestor cycle and two overlapping hybrid cycles. This example comes from
+siblings of one family marrying cousins from another family. These cycles can be any length theoretically,
+but cultural norms affect the typical size and number of each type of cycle differently. Recording practices
+and incomplete data also limit whether these cycles appear in a given dataset. Thus having a description
+of these cycles together with an understanding of the culture may help identify errors in the recorded data.
+18
+
+Conversely, understanding the distribution of cycles in high fidelity datasets can help identify the underlying
+cultural norms and help extrapolate where individuals are missing in incomplete data sets.
+(a) Common Ancestor Cycle
+(b) Union Cycle
+(c) θ-Cycle
+Figure 11: Left: A common ancestor cycle. The top most vertex is a common ancestor of the lowest vertex. The horizontal red line is a
+marriage, all other lines are parent-children edges. Center: A union cycle, specifically the double cousin situation described in Section
+2. The left-most and right-most vertices are parents of their neighboring vertices. The two horizontal red lines are marriage edges.
+Right: A θ-cycle formed by a common ancestor cycle with two overlapping hybrid cycles.
+Since many cultures avoid marrying close relatives, common ancestor cycles tend to have a fairly large
+circumference. In the Tikopia network (see Figure 1) we see persistence intervals with death values as high
+as 7 corresponding to cycles with a circumference of at least 21 individuals, which appear to be common
+ancestor cycles. This partially explains why persistence curves are so flat: there are relatively few minimal
+common ancestor cycles in a network, but they have very high persistence. More precisely, if the distance to
+union (the total number of individuals in a common ancestor cycle) is n, then the persistence of that cycle is
+⌊n/3⌋. However, the representatives of persistent homology only include a basis for these cycles, instead of
+including every possible distinct cycle. In particular, a large common ancestor cycle will decompose into the
+union of two hybrid cycles if the hybrid cycles are each shorter than the common ancestor cycle, as shown
+in Figure 11(c). Persistent homology will reflect the size of the two smaller cycles instead of the larger
+common ancestor cycle. We note that it is possible to identify the actual cycles chosen for our basis, but the
+software we used does not provide that information and size of the networks prohibits us from identifying
+the cycles manually.
+In social networks, we see that highly persistence cycles are quite rare. In order to have a cycle of
+persistence 3, for instance, we need a loop with circumference 9 or higher with no shorter paths between any
+two vertices in the loops. It may be that phenomena like the small-world effect or, more colloquially, six-
+degrees of freedom limit the maximal persistence of social networks. We see this reflected in our example
+data sets with a maximum persistence of 3 for all but one of the social networks.
+7. Conclusion
+In this paper, we explore the persistent homology structure of genealogical networks, motivated by the
+observation that family links tend to form in a fixed range of intermediate distances, which makes genealog-
+ical networks homologically distinct from most other social networks. We also introduce the notion of a
+persistence curve, which can be used to summarize and compare the persistent homology structure of any
+19
+
+network. We also relate specific genealogical structures, such as the common ancestor cycle, to homology
+objects.
+We find that, in the presence of incomplete data homology analysis is still genealogically useful. We
+note missing data due to recording practices and incomplete data (a ubiquitous feature of real genealogical
+networks), limits the kind of cycles that appear in a given dataset. Thus having a description of these cycles
+together with an understanding of the culture may help identify errors in the recorded data. Conversely,
+understanding the distribution of cycles in high fidelity datasets can help identify the underlying cultural
+norms and help extrapolate where individuals are missing in incomplete data sets.
+There are several interesting directions in which this work could be expanded. For example, our work
+has made it clear that there is a real need to analyze the persistent homology of large networks, with at least
+tens of thousands of nodes, since family formation generally takes place at these scales. The Ripser library
+we relied on was not able to reach these scales. Additionally, we are very interested in creating random
+graph models which reflect the actual homology of human family networks—a first attempt at this by our
+group has been fairly successful at the scale of hundreds of nodes [25]. More broadly, there is a need to
+model the ground truth human family network. All the extant data sources represent biased, limited, and
+noisy subnetworks, while the true interest of the genealogical community is in the ground truth network.
+Tools for signal denoising, image inpainting, and graph extrapolation, for example, could be useful in this
+context. Finally, an important aspect of genealogical networks is the relationship between various support-
+ing documents/metadata and the links that are discoverable through them. For example, one can consider
+optimal document collection strategies with a limited budget or document collection that is fair in terms of
+capturing minority information, which is often underrepresented.
+8. Declarations
+8.1. Availability of data and materials
+Links to the datasets generated and/or analysed during the current study can be found in Table 2. Code to
+replicate and extend this work can be found at https://github.com/AbigailJ32/The-persistent-homology-of-
+genealogical-networks.
+8.2. Competing interests
+The authors declare that they have no competing interests.
+8.3. Funding
+ZB, BW, and AJ, were supported by a BYU CPMS CHIRP grant. ZB was additionally supported by NFS
+award #2137511 and Army Research Office grant #W911NF-18-1-0244, and the James S. McDonnell Foun-
+dation 21st Century Science Initiative—Complex Systems Scholar Award grant #2200203. BW was addi-
+tionally supported by the Simons Foundation grant #714015. The views and conclusions contained in this
+document are those of the authors and should not be interpreted as representing the official policies, either
+expressed or implied, of the Army Research Office or the U.S. Government. The U.S. Government is autho-
+rized to reproduce and distribute reprints for Government purposes notwithstanding any copyright notation
+herein.
+8.4. Authors’ contributions
+Designed the experiments: ZB, NC, BW, RW. Performed the experiments: RF, RW. Wrote the paper: ZB,
+NC, TG, AJ, RS, BW, RW. All authors read and approved the final manuscript.
+20
+
+8.5. Acknowledgements
+We acknowledge helpful conversations with Joseph Price and the FamilySearch Engineering Research team.
+We also acknowledge Kolton Baldwin for helping to improve our code and simulations.
+9. Appendix
+Here we indicate both the genealogical and social networks used in our persistent homology computa-
+tions (see Section 6). We distinguish the datasets by network type: Friendship/Acquaintance, Social Media,
+Collaboration/Business, Disease Transmission, Information Sharing, Genealogical, and Atypical Genealog-
+ical networks. We also provide the network name, number of vertices and edges in the network, and a
+citation where the network can be found. Also, a special thanks to Kolton Baldwin for help with numerical
+simulations on this paper.
+Table 2: Social and Genealogical Network Data Sets.
+Network Data
+Network Type & Name
+Vertices
+Edges
+Citation
+Social Networks
+Friendship &
+Aquaintance
+Dolphins
+62
+159
+http://www-personal.umich.edu/∼mejn/netdata/
+Zachary Karate Club
+34
+78
+http://vlado.fmf.uni-lj.si/pub/networks/data/ucinet/ucidata.htm#zachary
+Residence Hall
+217
+2672
+http://konect.cc/networks/moreno oz/
+Highland Tribes
+16
+58
+http://konect.cc/networks/ucidata-gama/
+Seventh Graders
+29
+376
+http://konect.cc/networks/moreno seventh/
+Physicians
+241
+1098
+http://konect.cc/networks/moreno innovation/
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+23
+
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+Comparing Ordering Strategies For Process
+Discovery Using Synthesis Rules
+Tsung-Hao Huang and Wil M. P. van der Aalst
+Process and Data Science (PADS), RWTH Aachen University, Aachen, Germany
+{tsunghao.huang, wvdaalst}@pads.rwth-aachen.de
+Abstract. Process discovery aims to learn process models from ob-
+served behaviors, i.e., event logs, in the information systems. The dis-
+covered models serve as the starting point for process mining techniques
+that are used to address performance and compliance problems. Com-
+pared to the state-of-the-art Inductive Miner, the algorithm applying
+synthesis rules from the free-choice net theory discovers process mod-
+els with more flexible (non-block) structures while ensuring the same
+desirable soundness and free-choiceness properties. Moreover, recent de-
+velopment in this line of work shows that the discovered models have
+compatible quality. Following the synthesis rules, the algorithm incre-
+mentally modifies an existing process model by adding the activities in
+the event log one at a time. As the applications of rules are highly depen-
+dent on the existing model structure, the model quality and computation
+time are significantly influenced by the order of adding activities. In this
+paper, we investigate the effect of different ordering strategies on the dis-
+covered models (w.r.t. fitness and precision) and the computation time
+using real-life event data. The results show that the proposed ordering
+strategy can improve the quality of the resulting process models while
+requiring less time compared to the ordering strategy solely based on the
+frequency of activities.
+Keywords: Process discovery · Synthesis rules · Ordering strategy.
+1
+Introduction
+Process mining, a discipline bridging the gap between process science and data
+science [2], offers techniques and tools to analyze event data, i.e., event logs, gen-
+erated during the process execution. The analysis generated by process mining
+techniques provides valuable data-driven insights for the stakeholders.
+Process discovery is one of the three main research fields in process mining
+among conformance checking and process enhancement. Process discovery tech-
+niques aim to learn end-to-end process models from the event data. With the
+discovered models, knowledge workers can apply other process mining techniques
+to generate further insights for optimization.
+While various algorithms have been proposed, only a few ensure desirable
+properties such as soundness and free-choiceness. On the one hand, the sound-
+ness property guarantees that (1) it is always possible to finish the process (2)
+arXiv:2301.02182v1  [cs.DB]  4 Jan 2023
+
+2
+T. Huang and W. M. P. van der Aalst
+a process can be properly completed (3) no inexecutable transitions exist in the
+model [1]. On the other hand, the free-choice property separates the choice and
+synchronization constructs of a process model (Petri net). Such property is de-
+sirable as it allows easy conversions from the discovered model to widely-used
+notations such as BPMN [3]. Moreover, free-choice nets are supported by an
+abundance of analysis techniques developed from the theory [5].
+State-of-the-art techniques, such as the Inductive Miner (IM) [9] family, dis-
+cover process models guaranteed to be sound and free-choice. IM can provide
+such guarantees by exploiting its internal process representation - the process
+tree. However, such representation can also be a double-edged sword. Due to
+the representational bias, the discovered models by IM are doomed to be block-
+structured, i.e., the model must compose of parts that have a single entry and
+exit [9]. This implies that only a subset of sound free-choice workflow nets can
+be discovered by IM.
+To provide a more flexible process representation while keeping the same
+guarantees, we proposed a novel discovery algorithm, the so-called Synthesis
+Miner in [8]. The Synthesis Miner utilizes the synthesis rules from the free-
+choice net theory [5]. Activities in the event log are gradually added to a model
+under construction using predefined patterns. Following the rules ensures that
+the discovered process models are always sound and free-choice. Moreover, it is
+shown that the discovered models have compatible quality compared to the ones
+from Inductive Miner. Nevertheless, the possible applications of synthesis rules
+are highly dependent on the existing model structure. Different orders of adding
+activities can result in different models. Therefore, an open research question
+is the influence of the order in which the activities are added to an existing
+model on the final process model quality. In this paper, we address the research
+question by comparing the ordering strategies for the Synthesis Miner and taking
+a deeper look into the impacts of the activity adding order to the model quality
+and computation time. The experiment using four publicly available real-life
+event logs shows that advanced ordering strategies can significantly improve the
+model quality and the computation time.
+The remainder of the paper is structured as follows. Related work is presented
+in Sect.2. We introduce the necessary notations and concepts used throughout
+the paper in Sect. 3. Then, the proposed ordering strategies are introduced in
+Sect. 4. The evaluation using publicly available real-life event logs is presented
+in 5. Finally, Sect. 6 concludes this paper.
+2
+Related Work
+For a general introduction to process mining, we refer to [2]. Additionally, a re-
+view and benchmark of the recent development in process discovery can be found
+in [4]. In this paper, we focus on process discovery techniques that incrementally
+modify a model under construction to derive the final process.
+Incremental process mining allows users to learn a process model from event
+logs by gradually integrating different traces into an existing model [14]. As the
+ordering strategy has a significant impact on the model quality, a study [13] is
+
+Comparing Ordering Strategies For Process Discovery Using Synthesis Rules
+3
+conducted to investigate the interplay. Nevertheless, it is the trace that is added
+to the algorithm iteratively rather than the activity. Therefore, it is less relevant
+to this paper.
+Dixit et al. [6] were among the first to use synthesis rules from free-choice
+net theory [5] to discover process models. Inspired by [6], [8] introduces the
+Synthesis Miner that automates the discovery by introducing predefined patterns
+and a search space pruning mechanism. Both [6] and [8] introduce a few ordering
+strategies for their approaches. However, the choice of ordering is left to the user
+as an input parameter. The impact of the ordering strategies on the model
+quality and computation time is not thoroughly investigated. Furthermore, the
+interplay between the ordering strategies and the search space pruning has not
+been explained. Last but not least, a comparison between different ordering
+strategies is needed. In this paper, we aim to address the open research question
+and provide users with a rule of thumb.
+3
+Preliminaries
+In this section, we introduce the necessary concepts and notations that are used
+throughout the paper.
+For an arbitrary set A, we denote the set of all possible sequences as A∗
+and the set of all multi-sets over A as B(A). Given σ1, σ2 ∈ A∗, σ1 · σ2 de-
+notes the concatenation of the two sequences. Let A be a set and X ⊆ A be
+a subset of A. For σ ∈ A∗ and a ∈ A, we define ↾X∈ A∗→X∗ as a projec-
+tion function recursively with ⟨⟩↾X = ⟨⟩, (⟨a⟩ · σ)↾X = ⟨a⟩ · σ↾X if a ∈ X and
+(⟨a⟩ · σ)↾X = σ↾X if a /∈ X. For example, ⟨x, y, x⟩↾{x,z} = ⟨x, x⟩. The pro-
+jection function can also be applied to a multi-set of sequences. For example,
+[⟨x, y, x⟩4, ⟨x, y⟩2, ⟨y, x, z⟩6]↾{y,z} = [⟨y⟩6, ⟨y, z⟩6]. We denote UA as the universe
+of activity labels.
+Definition 1 (Trace & Log). A trace σ ∈ U∗
+A is a sequence of activity labels.
+A log is a multi-set of traces, i.e., L ∈ B(U∗
+A).
+Definition 2 (Log Properties [8]). Let L ∈ B(U∗
+A) and a, b ∈ UA be two
+activity labels. We define the following log properties:
+– #(a, L) = Σσ∈L|{i ∈ {1, 2, ..., |σ|}|σ(i) = a}| is the times a occurred in L.
+– #(a, b, L) = Σσ∈L|{i ∈ {1, 2, ..., |σ| − 1}|σ(i) = a ∧ σ(i + 1) = b}| is the
+number of direct successions from a to b in L.
+– caus(a, b, L) =
+�
+#(a,b,L)−#(b,a,L)
+#(a,b,L)+#(b,a,L)+1
+if a ̸= b
+#(a,b,L)
+#(a,b,L)+1
+if a = b is the strength of causal rela-
+tion (a, b).
+– Apre
+c
+(a, L) = {apre ∈ UA|caus(apre, a, L) ≥ c} is the set of a’s preceding
+activities, determined by threshold c.
+– Afol
+c
+(a, L) = {afol ∈ UA|caus(a, afol, L) ≥ c} is the set of a’s following
+activities, determined by threshold c.
+
+4
+T. Huang and W. M. P. van der Aalst
+Definition 3 (Petri Net). Let N = (P, T, F, l) be a Petri net, where P is the
+set of places, T is the set of transitions, P ∩ T = ∅. F ⊆ (P × T) ∪ (T × P) is
+the set of arcs, and l ∈ T → UA ∪ {τ} is a labeling function that assigns activity
+labels to transitions. A transition t ∈ T is invisible (or silent) if l(t) = τ.
+Definition 4 (Path & Elementary Path). A path of a Petri net N = (P, T, F)
+is a non-empty sequence of nodes ρ = ⟨x1, x2, ..., xn⟩ such that (xi, xi+1) ∈ F
+for 1 ≤ i < n. ρ is an elementary path if xi ̸= xj for 1 ≤ i < j ≤ n. For
+X, X′ ∈ P ∪ T, elemPaths(X, X′, N) ⊆ (P ∪ T)∗ is the set of all elementary
+paths from some x ∈ X to some x′ ∈ X′.
+Definition 5 (Workflow Net (WF-net) [1]). Let N = (P, T, F, l) be a Petri
+net. W = (P, T, F, l, i, o, ⊤, ⊥) is a WF-net iff (1) it has a dedicated source
+place i ∈ P: •i = ∅ and a dedicated sink place o ∈ P: o• = ∅ (2) ⊤ ∈ T:
+•⊤ = {i} ∧ i• = {⊤} and ⊥ ∈ T: ⊥• = {o} ∧ •o = {⊥} (3) every node x is on
+some path from i to o, i.e., ∀x∈P ∪T (i, x) ∈ F ∗ ∧ (x, o) ∈ F ∗, where F ∗ is the
+reflexive transitive closure of F.
+Definition 6 (Activity Order). Let L ∈ B(U∗
+A) and A = �
+σ∈L{a ∈ σ}.
+γ ∈ A∗ is an activity order for L if {a ∈ γ} = A and |γ| = |A|.
+Synthesis Miner: Process Discovery Using Synthesis Rules In previous
+work [8], we introduced the Synthesis Miner that guarantees to discover sound
+and free-choice workflow nets by applying the synthesis rules defined in [5] with
+an additional dual abstraction rule [8].
+Given a workflow net W, the abstraction rule (ψA) allows to add a place
+p and a transition t between a set of transitions R ⊆ T and a set of places
+S ⊆ P if they are fully connected, i.e., (R × S ⊆ F) ∧ (R × S ̸= ∅). The linear
+transition/place rule (ψT /ψP ) allows to add a transition t/place p if it is linearly
+dependent on the other transitions/places in the corresponding incidence matrix.
+The dual abstraction rule (ψD) can add a transition t and a place p between a
+set of places S and a set of transitions R if (S × R ⊆ F) ∧ (S × R ̸= ∅). All
+four rules1 preserve sound and free-choice properties [5,8]. Fig. 1 shows a few
+examples of rules applications.
+Given a log L, the Synthesis Miner first determines an activity order γ. Then,
+the iteration is initiated. In iteration i (where 1 ≤ i ≤ |γ|), activity γ(i) is added
+to an existing net2 from the i − 1 iteration. The procedure for every iteration
+is as follows: (1) use heuristics from the projected log Li = L↾{γ(1),γ(2),...γ(i)}
+to find the most likely position for the to-be-added activity γ(i) on the existing
+WF-net (Wi), (2) apply predefined patterns (derived from synthesis rules) to get
+the set of candidate nets, and (3) select the best net (w.r.t. fitness and precision)
+from the set of candidates for the next iteration.
+1 For the formal definitions of the rules, we refer to [5,8].
+2 The existing net in the first iteration is initiated by the initial net, as shown in the
+example for the abstraction rule in Fig.1.
+
+Comparing Ordering Strategies For Process Discovery Using Synthesis Rules
+5
+𝑖
+𝑝1
+𝑜
+⊥
+⊤
+𝑖
+𝑝1
+𝑜
+⊥
+⊤
+a
+𝑝2
+𝑡1
+𝑖
+𝑝1
+𝑜
+⊥
+⊤
+a
+𝑝2
+𝑡1
+𝑖
+𝑝1
+𝑜
+⊥
+⊤
+a
+𝑝2
+𝑡1
+𝑡2
+𝑖
+𝑝1
+𝑜
+⊥
+⊤
+a
+𝑝2
+𝑡1
+𝑡2
+⊤
+⊥
+𝑖
+𝑝1
+𝑜
+a
+𝑝2
+𝑡1
+𝑡2
+𝑝3
+⊤
+⊥
+𝑖
+𝑝1
+𝑜
+a
+𝑝2
+𝑡1
+𝑡2
+𝑝3
+⊤
+⊥
+𝑖
+𝑝1
+𝑜
+a
+𝑝2
+𝑡1
+𝑡2
+𝑝3
+b
+𝑝4
+𝑡3
+⊤
+⊥
+𝑡1
+𝑡2
+𝑖
+-1
+0
+0
+0
+𝑝1
+0
+-1
+1
+1
+𝑝2
+1
+0
+-1
+-1
+𝑜
+0
+1
+0
+0
+𝑝3
+1
+-1
+0
+0
+⊤
+⊥
+𝑡1
+𝑡2
+𝑖
+-1
+0
+0
+0
+𝑝1
+0
+-1
+1
+1
+𝑝2
+1
+0
+-1
+-1
+𝑜
+0
+1
+0
+0
+linear dependent transition rule 𝜓𝑇
+abstraction Rule 𝜓𝐴
+linear dependent place rule 𝜓𝑃
+dual abstraction rule 𝜓𝐷
+R
+S
+S
+R
+initial net
+Fig. 1: Some examples of the synthesis rules applications. ψA allows to add p2 and t1
+by R = {⊤} and S = {p1}. t2 is added by ψT as it is linearly dependent on t1. p3 is
+added by ψP as it is a linear combination of p1 and p2. ψD allows to add t3 and p4
+with S = {p1, p3} and R = {⊥}.
+As step (1) is directly affected by the ordering strategy, we formally define3
+how the search space is limited to only a subset of the nodes on a workflow net
+using log heuristics.
+Definition 7 (Reduced Search Space). Let a ∈ U∗
+A be an activity, L∈B(U∗
+A)
+be a log, W = (P, T, F, l, i, o, ⊤, ⊥) be a WF-net, and 0 ≤ c ≤ 1. T pre is the
+set of transitions labeled by the preceding activities of a in log L. T pre = {t ∈
+T|l(t) ∈ Apre
+c
+(a, L)} if Apre
+c
+(a, L) ̸= ∅, otherwise T pre = {⊤}. T fol is the set of
+transitions labeled by the following activities of a in log L.T fol = {t ∈ T|l(t) ∈
+Afol
+c
+(a, L)} if Afol
+c
+(a, L) ̸= ∅, otherwise T fol = {⊥}. The reduced search space
+is reduce(a, L, W, c) = {x ∈ ρ|ρ ∈ elemPath(T pre, T fol, W)}.
+The function reduce first finds the preceding and following activities and the
+corresponding sets of labeled transitions for the to-be-added activity γ(i). Then,
+it returns the set of nodes, denoted as Vi, that are on the path between the pre-
+ceding and following transitions. Vi is used to confine the application of synthesis
+rules. To be more precise, the set of transitions R and the set of places S used
+as the preconditions for applying rules ψA and ψD need to be a subset of Vi,
+i.e., S ⊆ V ∧ R ⊆ V . As for rule ψT /ψP , the new transition/place (t′/p′) cannot
+have arcs connected to any node other than Vi. This step helps us to limit the
+search space to the most likely nodes on a workflow net to add activity γ(i).
+Fig. 2 shows an example for reducing the search space.
+3 As the formal definitions of steps (2) and (3) are out of scope, we refer to [8].
+
+6
+T. Huang and W. M. P. van der Aalst
+⊤
+⊥
+𝑖
+𝑝2
+𝑜
+x
+𝑝1
+𝑡1
+z
+𝑝3
+𝑡2
+𝐿3 = [ 𝑥, 𝑦, 𝑧 66, 𝑥, 𝑧 66]
+𝑇𝑝𝑟𝑒 = 𝑡1 , 𝑇𝑓𝑜��� = {𝑡2}
+(a) W2, the existing net from the last iteration
+⊤
+⊥
+𝑖
+𝑝2
+𝑜
+x
+𝑝1
+𝑡1
+z
+𝑝3
+𝑡2
+y
+𝑡3
+𝑡4
+𝑝4
+(b) W3, the net after adding y
+Fig. 2: An example showing how the search space is reduced. Consider the log L3 =
+[⟨x, y, z⟩66, ⟨x, z⟩66]. y is the activity which we want to add to the net W2. Using c = 0.9,
+we get T pre = {t1} and T fol = {t2}. Therefore, the function reduce would return the
+set of nodes between t1 and t2, which means V3 = {t1, p2, t2} as highlighted by the
+green dashed line in (a). The application of synthesis rules would then only consider
+these three nodes. Finally, the best net is selected as W3 from the candidates and is
+visualized in (b).
+4
+Ordering Strategies
+In this section, we introduce different ordering strategies. To illustrate the order-
+ing strategy, consider the following log Ls = [⟨b, c, d, e, f, g⟩, ⟨b, e, c, d, f, g⟩, ⟨b, e, c,
+f, g, d⟩, ⟨b, e, c, f, d, g⟩, ⟨b, c, e, d, f, g⟩, ⟨b, c, e, f, g, d⟩, ⟨b, c, e, f, d, g⟩, ⟨e, b, c, d, f, g⟩,
+⟨e, b, c, f, g, d⟩, ⟨e, b, c, f, d, g⟩].
+b
+10
+c
+10
+d
+10
+e
+10
+f
+10
+g
+10
+7
+3
+3
+3
+3
+4
+3
+1
+1
+3
+3
+3
+7
+3
+3
+7
+3
+3
+7
+Fig. 3: The DFG for log Ls.
+The corresponding directly follows
+graph (DFG) is shown in Fig. 3.
+The
+first
+ordering
+strategy
+is
+frequency-based and it is relatively
+straightforward. The activities are
+simply ordered by their frequency in
+the log.
+Definition 8 (Frequency-Based Ordering). Let L∈B(U∗
+A). Frequency-based
+ordering function is orderfreq(L) = γ such that γ is an activity order and
+∀1≤i<j≤|γ|#(γ(i), L) ≥ #(γ(j), L).
+If activities have the same frequency, we order them alphabetically. Using the
+example log Ls for illustration, the order would be orderfreq(Ls) = ⟨b, c, d, e, f, g⟩.
+The other ordering strategies are more involved as they consider not only the
+frequency of activities but also the connections between them. Before introducing
+the other ordering strategies, we first define a helper function that ranks the
+directly-follow activities based on the strength of connections.
+Definition 9 (Directly-Follow Activities Sorting). Let L∈B(U∗
+A) and a∈UA.
+A={b ∈ UA|#(a, b, L)>0} is the set of activities directly-follow a in L at least
+once and σ ∈ A∗. Directly-follow activities sorting is sortDFA(a, L) = σ such
+that {b ∈ σ} = A and |σ| = |A| and ∀1≤i<j≤|σ| #(a, σ(i), L) ≥ #(a, σ(j), L).
+For example, sortDFA(b, Ls) = ⟨c, e⟩. This is because activities c and e have
+incoming arcs from b and the strength #(b, c, Ls) ≥ #(b, e, Ls). With the func-
+tion for sorting directly-follow activities defined, we are now ready to define the
+Breadth-First-Search-Based ordering strategy in Algo. 1.
+
+Comparing Ordering Strategies For Process Discovery Using Synthesis Rules
+7
+Algorithm 1: Breadth-First-Search-Based Ordering, orderBFS
+Input
+: A log L ∈ B(U∗
+A)
+Output : An activity order γ for L
+A ← �
+σ∈L{a ∈ σ} ;
+// the set of activities in L
+As ← {σ(1) | σ ∈ L ∧ σ ̸= ⟨⟩} ;
+// the set of start activities in L
+σ ← orderfreq(L)↾As ;
+// the sequence of start activities ordered by frequency
+i ← 1;
+while |σ| ̸= |A| :
+A′ ← A \ {a ∈ σ} ;
+// the set of activities that are not in σ
+σ′ ← sortDFA(σ(i), L)↾A′ ; // sort σ(i)’s following activities & project on A′
+σ ← σ · σ′ ;
+// update σ
+i ← i + 1;
+γ ← σ;
+return γ;
+BFS-based ordering strategy starts by building a sequence of start activities
+in a log and iteratively append the sequence of directly-follow activities using
+the function in Def. 9. Applying the function to the example log Ls, we get
+orderBFS(Ls) = ⟨b, e⟩ · ⟨c⟩ · ⟨f, d⟩ · ⟨⟩ · ⟨g⟩ = ⟨b, e, c, f, d, g⟩. σ is initiated with
+⟨b, e⟩. Then, in iteration i, σ is appended by the sequence of σ(i)’s directly-
+follow activities sorted by sortDFA(σ(i), Ls) with the set of activities already in
+σ filtered out. The loop continues until σ includes every activity in the log. As
+its name suggests, the ordering prioritizes the exploration of the directly-follow
+activities.
+Next, we introduce another ordering strategy in Algo. 2 that is Depth-First-
+Search-based. While also considering the connection between the activities as
+BFS-based ordering strategy, DFS-based ordering prioritizes depth over breadth.
+That is, the directly-follow activities are not explored thoroughly until activities
+with higher depth have been explored. Applying DFS-based ordering to log Ls,
+we get orderDFS(Ls) = ⟨b, c, f, g, d, e⟩.
+Note that although we define the BFS- and DFS-based ordering strategies to
+start from the start activities, one can also initiate the exploration from another
+direction, i.e., from the end activities and subsequently explore the directly-
+precede activities for ordering. Using Ls as an example, if starting from the set
+of end activities, we would get ⟨g, f, c, b, e, d⟩ with DFS-based ordering on log Ls
+and ⟨g, d, f, c, e, b⟩ with BFS-based ordering.
+To explain how the progression of the process discovery influenced by the dif-
+ferent ordering strategies, Fig. 4 shows all the intermediate nets when applying
+Synthesis Miner to log Ls using the three different ordering strategies. DFS-
+based ordering tends to build the process from start to end at the beginning
+before adding the activities in the parallel/choice branches. On the contrary,
+BFS-based ordering prioritizes the construction of local control flows. For ex-
+ample, the difference is observable from iteration 1 to 2. While all the ordering
+strategies produce the same net in iteration 1, BFS-based ordering suggests to
+add the concurrent activity e for b in iteration 2 and DFS-based ordering adds
+
+8
+T. Huang and W. M. P. van der Aalst
+Algorithm 2: Depth-First-Search-Based Ordering orderDFS
+Input
+: A log L ∈ B(U∗
+A)
+Output : An activity order γ for L
+A ← �
+σ∈L{a ∈ σ} ;
+// the set of activities in L
+As ← {σ(1) | σ ∈ L ∧ |σ| ̸= 0} ;
+// the set of start activities in L
+σs ← orderfreq(L)↾As ;
+// the sequence of start activities ordered by frequency
+σ ← ⟨σs(1)⟩ ;
+// initiate the sequence with the most frequent start activity
+σs ← σs↾{As\{σs(1)}} ;
+// update σs to be the stack
+while |σ| ̸= |A| :
+A′ ← A \ {a ∈ σ} ;
+// set of activities that are not in σ
+σf ← sortDFA(σ(|σ|), L)↾A′ ;
+// sort σ(|σ|)’s following activities
+if |σf| = 0 :
+σ ← σ · ⟨σs(1)⟩ ;
+// append the 1st element from the stack σs to σ
+else :
+σ ← σ · ⟨σf(1)⟩ ;
+// append the 1st element from σf to σ
+σs ← (σf↾A\{a∈σ∨a∈σs}) · (σs↾A\{a∈σ}) ;
+// update the stack σs
+γ ← σ;
+return γ;
+b
+b
+b
+e
+f
+b
+c
+e
+c
+b
+e
+d
+b
+c
+e
+c
+b
+f
+b
+c
+g
+b
+c
+f
+g
+b
+c
+f
+d
+e
+𝛾 = 𝑜𝑟𝑑𝑒𝑟𝑓𝑟𝑒𝑞(𝐿𝑠) = 〈𝑏, 𝑐, 𝑑, 𝑒, 𝑓, 𝑔〉
+𝛾 = 𝑜𝑟𝑑𝑒𝑟𝐵𝐹𝑆(𝐿𝑠) = 〈𝑏, 𝑒, 𝑐, 𝑓, 𝑑, 𝑔〉
+𝛾 = 𝑜𝑟𝑑𝑒𝑟𝐷𝐹𝑆(𝐿𝑠) = 〈𝑏, 𝑐, 𝑓, 𝑔, 𝑑, 𝑒〉
+𝑖 = 1
+𝑖 = 2
+𝑖 = 3
+𝑖 = 4
+𝑖 = 5
+𝑖 = 6
+g
+b
+c
+f
+d
+g
+b
+c
+f
+e
+d
+b
+c
+b
+d
+b
+c
+f
+b
+c
+e
+d
+Fitness: 1
+Precision: 1
+Fitness: 1
+Precision: 1
+Fitness: 1
+Precision: 1
+Fitness: 1
+Precision: 1
+Fitness: 1
+Precision: 0.94
+Fitness: 1
+Precision: 1
+Fitness: 1
+Precision: 1
+Fitness: 1
+Precision: 1
+Fitness: 1
+Precision: 1
+Fitness: 1
+Precision: 1
+Fitness: 1
+Precision: 1
+Fitness: 1
+Precision: 1
+Fitness: 1
+Precision: 1
+Fitness: 1
+Precision: 1
+Fitness: 1
+Precision: 1
+Fitness: 1
+Precision: 1
+f
+b
+c
+e
+d
+Fitness: 1
+Precision: 0.95
+Fig. 4: A comparison of different ordering strategies for log Ls. Each column repre-
+sents an ordering strategy and each row corresponds to the intermediate workflow net
+in iteration i after adding γ(i). The green dashed lines highlight the nodes representing
+the reduced search space. The metrics fitness and precision are measured using the cor-
+responding projected log Li = L↾{γ(1),γ(2),...γ(i)}. Note that the final model discovered
+by the BFS- and DFS-based ordering strategies are the same in this example.
+the directly-follow activity c of b first. The frequency ordering doesn’t seem to
+have clear patterns for the discovery.
+We expect that the choice of ordering can significantly influence the computa-
+tion time of discovery. The main difference stems from the time required to check
+
+Comparing Ordering Strategies For Process Discovery Using Synthesis Rules
+9
+the feasibility of the linear dependency rules. As the WF-net grows, it becomes
+more expensive (w.r.t. time) to check if a candidate place/transition is linear
+dependent. Thus, it is preferable to limit the search space as small as possible,
+especially in the later iterations. Recall that the reduced search space (Def. 7) is
+a set of nodes confining the application of synthesis rules. The green dashed lines
+in Fig. 4 highlight the reduced search space Vi in iteration i. As shown in Fig. 4,
+generally, BFS-based ordering can keep the search space smaller than the other
+strategies because it prioritizes the connected activities. In contrast, the search
+space of DFS-based ordering is more likely to be large in the later iterations. As
+the parallel/alternative activities are added later, the preceding and following
+activities of the to-be-added activity γ(i) is highly likely to be spread across
+the existing net. Together with the effect of search space reduction, it results
+in a relatively large search space, which indicates more nodes to be considered.
+Examples can be seen in iterations 4 and 5 for the DFS-based ordering in Fig. 4.
+Although it is assumed that BFS-based ordering would have relatively lower
+computation time, search space reduction might introduce trade-offs between the
+optimal solution and time. In the following section, we aim to investigate the
+impact of the ordering strategy on both model quality and the time to discover
+the process model in the experiment.
+5
+Evaluation
+In this section, we present the experiment used to evaluate the ordering strategies
+including the setup and a discussion of the result4.
+5.1
+Experimental Setup
+For the experiment, we use four publicly available real-life event logs [7,10,11,12].
+The logs are filtered to focus on the mainstream behaviors (at least 95% of the
+traces) where the most frequent trace variants are used. For the BPI2017 log [7],
+we split it into three logs using the activity prefix (A, W, O). This results in six
+logs in total.
+For every event log, we apply different ordering strategies for the Synthesis
+Miner [8] with default values for the other parameters. For the BFS- and DFS-
+based ordering strategies, we apply the ordering from both directions (start and
+end activities). Therefore, we evaluate five ordering strategies. To measure the
+effect of ordering strategies on search space pruning, we keep track of the ratio
+of reduced search space. This is evaluated by
+|Vi|
+|Pi∪Ti|−2, where Vi is the set of
+reduced nodes, Pi and Ti are the set of places and transitions in the existing WF-
+net Wi. The −2 in the denominator is there to exclude the two places (source
+and sink) that can never be connected by new nodes by Def. 5. Using Fig. 4 as
+an example, the value of
+|V3|
+|P3∪T3|−2 for the frequency ordering strategy would be
+9
+11−2 = 1 in iteration 3. This indicates that all the possible nodes are considered
+for the application of synthesis rules to add the next activity. Furthermore, we
+evaluate the final model in terms of fitness, precision, and F1 score (the harmonic
+mean of fitness and precision).
+4 https://github.com/tsunghao-huang/synthesisRulesMiner
+
+10
+T. Huang and W. M. P. van der Aalst
+5.2
+Results and Discussion
+Search Space Reduction and Computation Time Fig. 5 shows the result
+of the comparison among the five ordering strategies regarding their effects on
+the search space reduction. The value in the y-axis
+|Vi|
+|Pi∪Ti|−2 is the average
+across six event logs. As indicated, the metric keeps track of the reduced search
+space ratio for adding the next activity, which indicates the number of possible
+synthesis rule applications. In general, we can observe from the figure that the
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+Number of activities added (i)
+0.1
+0.2
+0.3
+0.4
+0.5
+0.6
+0.7
+0.8
+0.9
+1.0
+Ratio of nodes considered 
+|Vi|
+|Pi
+Ti|
+2
+freq
+bfs_start
+bfs_end
+dfs_start
+dfs_end
+(a) Average ratio of reduced search space
+2
+3
+4
+5
+6
+7
+8
+9
+10
+11
+Number of activities added(i)
+0
+200
+400
+600
+800
+1000
+1200
+1400
+Average time(sec) to add an activity
+freq
+bfs_start
+bfs_end
+dfs_start
+dfs_end
+(b) Average time to add an activity
+Fig. 5: Comparisons of ordering strategies on the effects of search space reduction as
+well as the computation time for each step. Note that it is preferable to have a lower
+value for
+|Vi|
+|Pi∪Ti|−2.
+ordering strategies behaved as expected. As shown in Fig. 5a, in the later stage
+of the discovery (i ≥ 8), the BFS-ordering strategies (bfs_start, bfs_end) keep
+the ratio of reduced search space at a low level while the value for frequency and
+DFS-based ordering strategies show that they are more likely to include a large
+portion of the nodes in the search space.
+Fig. 5b shows the average time to add an activity to the existing WF-net for
+each step of six logs. Comparing the two figures, one can see the effect of search
+space reduction on the computation time. As shown in Fig. 5b, the bfs_end
+strategy keeps the average computation time for each step at a fairly low level.
+This is also the case for the bfs_start strategy despite the two peaks when adding
+the 7th and 10th activity. The two peaks in the 7th and 10th steps are especially
+severe for the dfs_end strategy. Both took more than 10 minutes to add a single
+activity to the existing model. Also, the longest duration to add an activity also
+happens in the 11th step of the dfs_start strategy.
+In short, due to its interplay with the search space reduction, the BFS-based
+ordering strategies have significant advantage in terms of computation time.
+Model Quality Table 15 shows the result of the model quality using the five
+different ordering strategies. As expected, we observe that the BFS-based or-
+dering strategies have the lowest computation time in all six event logs. This
+5 To provide a reference to the state of the art, we also present the results from IMf
+(marked by gray color). The best model generated by IMf (w.r.t. F1 score) is selected
+from a set of nets using five different values ([0.1, 0.2, 0.3, 0.4, 0.5]) for the filter.
+
+Comparing Ordering Strategies For Process Discovery Using Synthesis Rules
+11
+Table 1: Quality of the models discovered by different ordering strategies.
+Log
+Ordering Strategy & IMf Fitness Precision
+F1
+time(sec)
+frequency
+0.971
+0.947
+0.958
+685
+BFS_start
+0.973
+1.000
+0.986
+893
+BFS_end
+0.990
+0.935
+0.961
+334
+DFS_start
+0.963
+0.868
+0.913
+1850
+DFS_end
+0.999
+0.986
+0.993
+1248
+BPI2017A
+IMf(0.2)
+0.999
+0.936
+0.967
+10
+frequency
+0.993
+0.962
+0.978
+537
+BFS_start
+0.985
+0.963
+0.974
+165
+BFS_end
+0.989
+1.000
+0.995
+231
+DFS_start
+0.996
+1.000
+0.998
+498
+DFS_end
+0.993
+0.962
+0.978
+360
+BPI2017O
+IMf(0.2)
+0.997
+0.907
+0.950
+7
+frequency
+0.993
+0.726
+0.838
+3617
+BFS_start
+0.974
+0.864
+0.914
+1626
+BFS_end
+0.993
+0.888
+0.936
+579
+DFS_start
+0.974
+0.864
+0.914
+1732
+DFS_end
+0.993
+0.901
+0.944
+5397
+BPI2017W
+IMf(0.2)
+0.923
+0.897
+0.910
+14
+frequency
+0.974
+0.984
+0.978
+51
+BFS_start
+0.974
+0.984
+0.978
+52
+BFS_end
+0.983
+0.976
+0.979
+43
+DFS_start
+0.974
+0.984
+0.978
+49
+DFS_end
+0.989
+0.963
+0.976
+64
+helpdesk
+IMf(0.2)
+0.967
+0.950
+0.958
+1
+frequency
+0.945
+0.810
+0.879
+509
+BFS_start
+0.931
+0.922
+0.936
+314
+BFS_end
+0.988
+0.935
+0.961
+383
+DFS_start
+0.931
+0.970
+0.961
+2154
+DFS_end
+0.943
+0.883
+0.920
+2359
+hospital
+billing
+IMf(0.2)
+0.982
+0.906
+0.943
+45
+frequency
+0.967
+0.930
+0.945
+274
+BFS_start
+0.967
+0.930
+0.945
+202
+BFS_end
+0.972
+0.720
+0.825
+388
+DFS_start
+0.991
+0.933
+0.960
+366
+DFS_end
+0.942
+0.858
+0.903
+443
+traffic
+IMf(0.4)
+0.904
+0.720
+0.801
+28
+corresponds to the findings in the previous section. Moreover, despite the search
+space being considerably reduced, the models discovered using BFS-ordering
+strategies have the highest F1 score in two out of the six logs.
+As for the DFS-based ordering strategies, they have an apparent disadvantage
+for computation time but get the highest F1 score in the other four event logs.
+The result matches our assumption as search space reduction introduces a trade-
+off between the optimal solution and time. Lastly, the frequency ordering strategy
+has no significant advantage in model quality and computation time. The results
+show that the ordering strategies that take the connections between activities
+into consideration can improve the Synthesis Miner than the frequency-based
+ordering strategy.
+6
+Conclusion
+In this paper, we introduced five ordering strategies for the process discovery al-
+gorithm using synthesis rules [8]. We investigated the impact of ordering strate-
+gies on model quality and computation time. The results show that compared
+to the ordering strategy solely based on the frequency of activities, the proposed
+ordering strategies considered the connection between activities (Breadth-First-
+
+12
+T. Huang and W. M. P. van der Aalst
+Search-based and Depth-First-Search-based) have superior performance w.r.t.
+time and model quality respectively. It is shown in the result that the introduced
+BFS-based ordering strategies can speed up the computation. Nevertheless, the
+overall discovery time of the Synthesis Miner is still not comparable to the state
+of the art despite being able to discover models with better quality. Therefore,
+for future work, we plan to speed up the Synthesis Miner by further exploiting
+the log heuristics and investigating more sophisticated ordering strategies. An-
+other direction for improvement is the ability to cope with infrequent behaviors
+as we use the most frequent trace variants to capture the mainstream process. It
+would be valuable to introduce a filtering mechanism to the Synthesis Miner so
+that it can directly work on the original log without depending on pre-filtering
+the log.
+Acknowledgements. We thank the Alexander von Humboldt (AvH) Stiftung
+for supporting our research.
+References
+1. van der Aalst, W.M.P.: The application of Petri nets to workflow management. J.
+Circuits Syst. Comput. 8(1), 21–66 (1998)
+2. van der Aalst, W.M.P.: Process Mining - Data Science in Action, Second Edition.
+Springer (2016)
+3. van der Aalst, W.M.P.: Using free-choice nets for process mining and business
+process management. In: FedCSIS 2021. vol. 25, pp. 9–15 (2021)
+4. Augusto, A., Conforti, R., Dumas, M., Rosa, M.L., Maggi, F.M., Marrella, A.,
+Mecella, M., Soo, A.: Automated discovery of process models from event logs:
+Review and benchmark. IEEE Trans. Knowl. Data Eng. 31(4), 686–705 (2019)
+5. Desel, J., Esparza, J.: Free Choice Petri Nets. No. 40, Cambridge university press
+(1995)
+6. Dixit, P.M., Buijs, J.C.A.M., van der Aalst, W.M.P.: Prodigy : Human-in-the-loop
+process discovery. In: RCIS 2018. pp. 1–12. IEEE (2018)
+7. van
+Dongen,
+B.:
+BPI
+Challenge
+2017
+(2017).
+https://doi.org/10.4121/uuid:5f3067df-f10b-45da-b98b-86ae4c7a310b
+8. Huang, T., van der Aalst, W.M.P.: Discovering sound free-choice workflow nets
+with non-block structures. In: EDOC 2022. vol. 13585, pp. 200–216. Springer
+(2022). https://doi.org/10.1007/978-3-031-17604-3_12
+9. Leemans, S.J.J., Fahland, D., van der Aalst, W.M.P.: Scalable process discovery
+and conformance checking. Softw. Syst. Model. 17(2), 599–631 (2018)
+10. de Leoni, M.M., Mannhardt, F.: Road Traffic Fine Management Process (2015).
+https://doi.org/10.4121/uuid:270fd440-1057-4fb9-89a9-b699b47990f5
+11. Mannhardt,
+F.:
+Hospital
+Billing
+-
+Event
+Log
+(2017).
+https://doi.org/10.4121/uuid:76c46b83-c930-4798-a1c9-4be94dfeb741
+12. Polato, M.: Dataset belonging to the help desk log of an Italian Company (2017).
+https://doi.org/10.4121/uuid:0c60edf1-6f83-4e75-9367-4c63b3e9d5bb
+13. Schuster, D., Domnitsch, E., van Zelst, S.J., van der Aalst, W.M.P.: A generic trace
+ordering framework for incremental process discovery. In: IDA 2022. vol. 13205, pp.
+264–277. Springer (2022)
+14. Schuster, D., van Zelst, S.J., van der Aalst, W.M.P.: Incremental discovery of
+hierarchical process models. In: RCIS 2020. vol. 385, pp. 417–433. Springer (2020)
+

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+Journal of Machine Learning Research 23 (2023) 1-24
+Submitted 1/23; Revised -; Published -
+Uncertainty Estimation based on Geometric Separation
+Gabriella Chouraqui
+CHOURAGA@POST.BGU.AC.IL
+Department of Computer Science
+Ben-Gurion University
+Israel
+Liron Cohen
+CLIRON@BGU.AC.IL
+Department of Computer Science
+Ben-Gurion University
+Israel
+Gil Einziger
+GILEIN@BGU.AC.IL
+Department of Computer Science
+Ben-Gurion University
+Israel
+Liel Leman
+LEMAN@POST.BGU.AC.IL
+Department of Computer Science
+Ben-Gurion University
+Israel
+Editor:
+Abstract
+In machine learning, accurately predicting the probability that a specific input is correct is crucial
+for risk management. This process, known as uncertainty (or confidence) estimation, is particularly
+important in mission-critical applications such as autonomous driving. In this work, we put for-
+ward a novel geometric-based approach for improving uncertainty estimations in machine learning
+models. Our approach involves using the geometric distance of the current input from existing
+training inputs as a signal for estimating uncertainty, and then calibrating this signal using standard
+post-hoc techniques. We demonstrate that our method leads to more accurate uncertainty estima-
+tions than recently proposed approaches through extensive evaluation on a variety of datasets and
+models. Additionally, we optimize our approach so that it can be implemented on large datasets in
+near real-time applications, making it suitable for time-sensitive scenarios.
+Keywords: uncertainty estimation, geometric separation, calibration, confidence evaluation.
+1. Introduction
+Machine learning models, such as neural networks, random forests, and gradient boosted trees, are
+widely used in various fields, including computer vision and transportation, and are transforming the
+field of computer science Niculescu-Mizil and Caruana (2006); Zhang and Haghani (2015). How-
+ever, the probabilistic nature of classifications made by these models means that misclassifications
+are inevitable. As a result, estimating the uncertainty for a particular input is a crucial challenge in
+machine learning. In fact, many machine learning models have some built-in measure of confidence
+that is often provided to the user for risk management purposes. The field of uncertainty calibration
+©2023 Gabriella Chouraqui et al.
+License: CC-BY 4.0, see https://creativecommons.org/licenses/by/4.0/. Attribution requirements are provided at
+http://jmlr.org/papers/v23/-.html.
+arXiv:2301.04452v1  [cs.LG]  11 Jan 2023
+
+CHOURAQUI ET AL.
+aims to improve the accuracy of the confidence estimates made by machine learning models Guo
+et al. (2017a).
+Confidence evaluation, or the model’s prediction of its success rate on a specific input, is a
+crucial aspect of mission-critical machine learning applications, as it provides a realistic estimate
+of the probability of success for a classification and enables informed decisions about the current
+situation. Even a highly accurate model may encounter an unexpected situation, which can be com-
+municated to the user through confidence estimation. For example, consider an autonomous vehicle
+using a model to identify and classify traffic signs. The model is very accurate, and in most cases,
+its classifications are correct with high confidence. However, one day, it encounters a traffic sign
+that is obscured, e.g., by heavy vegetation. In this case, the model’s classification is likely to be
+incorrect. Estimating confidence, or uncertainty, is a crucial tool for assessing unavoidable risks,
+allowing system designers to address these risks more effectively and potentially avoid unexpected
+and catastrophic consequences. For example, our autonomous vehicle may reduce its speed and ac-
+tivate additional sensors until it reaches higher confidence. Therefore, all popular machine learning
+models have mechanisms for determining confidence that can be calibrated to maximize the qual-
+ity of confidence estimates Niculescu-Mizil and Caruana (2005); Guo et al. (2017b); Kumar et al.
+(2019), and there is ongoing research to calibrate models more effectively and enable more reliable
+applications Leistner et al. (2009); Sun et al. (2007).
+Existing calibration methods can be divided into two categories: post-hoc methods that perform
+a transformation that maps the raw outputs of classifiers to their expected probabilities Kull et al.
+(2019); Guo et al. (2017a); Gupta and Ramdas (2021), and ad-hoc methods that adapt the training
+process to produce better calibrated models Thulasidasan et al. (2019); Hendrycks et al. (2019a).
+Post-hoc calibration methods are easier to apply because they do not change the model and do not
+require retraining. However, ad-hoc methods may lead to better model training in the first place and
+more reliable models. With the success of both approaches, recent research has focused on using
+ensemble methods whose estimates are a weighted average of multiple calibration methods Ashukha
+et al. (2020); Ma et al. (2021); Zhang et al. (2020); Pakdaman and Cooper (2016); Naeini et al.
+(2015). Another recent line of work attempts to further refine the uncertainty estimations by refining
+the grouping of confidence estimations, e.g., Perez-Lebel et al. (2022); Hebert-Johnson et al. (2018).
+In principle, post-hoc calibration can be viewed as cleaning up a signal, namely the model’s
+original confidence estimate. Interestingly, if we follow this logic, it is clear that the maximal
+attainable benefit lies in the quality of the signal. To see this, consider a model that plots the same
+confidence for all inputs. In this case, the best result that can be achieved is to set that confidence
+to the model’s average accuracy over all inputs. Therefore, finding better signals to calibrate is a
+promising direction for research.
+In this work, we introduce a novel approach for improving uncertainty estimates in machine
+learning models using geometry. We first provide an algorithm for calculating the maximal geo-
+metric separation of an input. However, calculating the geometric separation of an input requires
+evaluating the whole space of training inputs, making it a computationally expensive method that
+is not always feasible. Therefore, we suggest multiple methods to accelerate the process, including
+a lightweight approximation called fast-separation and several data reduction methods that shorten
+the geometric calculation.
+We demonstrate that using our geometric-based method, combined with a standard calibration
+method, leads to more accurate confidence estimations than calibrating the model’s original signal
+across different models and datasets. Even more, our approach yields better estimation even when
+2
+
+UNCERTAINTY ESTIMATION BASED ON GEOMETRIC SEPARATION
+compared to state-of-the-art calibration methods Kumar et al. (2019); Gupta and Ramdas (2021);
+Guo et al. (2017a); Zhang et al. (2020); Naeini et al. (2015); Kull et al. (2017). Additionally, we
+show that our approach can be implemented in near real-time on a variety of datasets through the
+use of multiple levels of approximation and optimization. This is particularly useful for practical
+applications that require rapid decision-making, such as autonomous driving. The entire code is
+available at our Github Leman et al. (2022).
+2. Related Work
+As mentioned above, uncertainty calibration is about estimating the model’s success probability of
+classifying a given example. Post-hoc calibration methods apply some transformation to the model’s
+confidence (without changing the model) such transformations include Beta calibration (Beta) Kull
+et al. (2017), Platt scaling (Platt) Platt (1999), Temperature Scaling (TS) Guo et al. (2017a); Kull
+et al. (2019), Ensemble Temperature Scaling (ETS) Zhang et al. (2020), and cubic spline Gupta
+and Ramdas (2021). In brief, these methods are limited by the best learnable mapping between the
+model’s confidence estimations, and the actual confidence. That is, post-hoc calibration map each
+confidence value to another calibrated value whereas our method introduces a new signal that can
+be calibrated just like the model’s original signal. Another work that uses a geometric distance in
+this context is Dalitz (2009). There, the confidence score is computed directly from the geometric
+distance, while we first fit a function on a subset of the data to learn the specific behavior of the
+dataset and model. Moreover, the work in Dalitz (2009) only applies to the k-nearest neighbor
+model, while our method is applicable to all models.
+The recently proposed Scaling Binning Calibrator (SBC) of Kumar et al. (2019) uses a fitting
+function on the confidence values, divides the inputs into bins of equal size, and outputs the func-
+tion’s average in each bin. Histogram Binning (HB) Gupta and Ramdas (2021) uses a similar idea
+but divides the inputs into uniform-mass (rather than equal-size) bins. Interestingly, while most
+post-hoc calibration methods are model agnostic, recent methods have begun to look at a neural
+network non-probabilistic output called logits (before applying softmax) Guo et al. (2017b); Ding
+et al. (2020); Wenger et al. (2019). Thus, some new post-hoc calibration methods apply only to
+neural networks.
+Ensemble methods are similar to post-hoc calibration methods as they do not change the model,
+but they consider multiple signals to determine the model’s confidence Ashukha et al. (2020); Ma
+et al. (2021). For example, Bayesian Binning into Quantiles (BBQ) Naeini et al. (2015) is an exten-
+sion of HB that uses multiple histogram binning models with different bin numbers, and partitions
+then outputs scores according to Bayesian averaging. The same methodology of Bayesian averaging
+is applied in Ensemble of Near Isotonic Regression Pakdaman and Cooper (2016), but instead of
+histogram binning, they use nearly isotonic regression models.
+Ad-hoc calibration is about training models in new manners aimed to yield better uncertainty es-
+timations. Important techniques in this category include mixup training Thulasidasan et al. (2019),
+pre-training Hendrycks et al. (2019a), label-smoothing M¨uller et al. (2019), data augmentation
+Ashukha et al. (2020), self-supervised learning Hendrycks et al. (2019b), Bayesian approxima-
+tion (MC-dropout) Gal and Ghahramani (2016); Gal et al. (2017), Deep Ensemble (DE) Laksh-
+minarayanan et al. (2017), Snapshot Ensemble Huang et al. (2017a), Fast Geometric Ensembling
+(FGE) Garipov et al. (2018), and SWA-Gaussian (SWAG) Maddox et al. (2019). A notable approach
+is to use geometric distances in the loss function while training the model Xing et al. (2020). The
+3
+
+CHOURAQUI ET AL.
+authors work with a representation space that maximizes intra-class distances, minimizes inter-class
+distances, and uses the distances to estimate the confidence. Ad-hoc calibration is perhaps the best
+approach in public as it tackles the core of models’ calibration directly. However, because it offers
+specific training methods, it is of less use to large and already trained models, and the impact of each
+workshop is limited to a specific model type (e.g., DNNs in Garipov et al. (2018)). In comparison,
+post-hoc and ensemble methods (and our own method) often work for numerous models.
+Our geometric method is largely inspired by the approach of robustness proving in machine
+learning models. In this field, formal methods are used to prove that specific inputs are robust to
+small adversarial perturbations. That is, we formally prove that all images in a certain geometric
+radius around a specific train-set image receive the same classification Narodytska et al. (2018);
+Katz et al. (2017); Huang et al. (2017b); Gehr et al. (2018); Ehlers (2017); Einziger et al. (2019).
+These works rely on formal methods produced in an offline manner and thus apply only to training
+set inputs (known apriori). Whereas confidence estimation reasons about the current input. How-
+ever, the underlying intuition, i.e., that geometrically similar inputs should be classified in the same
+manner is also common to our work.
+Indeed, our work shows that geometric properties of the inputs can help us quantify the uncer-
+tainty in certain inputs and that, in general, inputs that are less geometrically separated and are ’on
+the edge’ between multiple classifications are more error-prone than inputs that are highly separated
+from other classes. Thus our work reinforces the intuition behind applying formal methods to prove
+robustness and supports the intuition that more robust training models would be more dependable.
+3. Geometric Separation
+In this section, we define a geometric separation measure that reasons about the distance of a
+given input from other inputs with different classifications. Our end goal is to use this measure
+to provide confidence estimations. Formally, a model receives a data input, x, and outputs the pair
+⟨C(x), conf (x)⟩, where C(x) is the model’s classification of x and conf (x) reflects the probability
+that the classification is correct. We estimate the environment around x where inputs are closer to
+inputs of certain classifications over the others. Our work assumes that the inputs are normalized,
+and thus these distances carry the same significance between the different inputs.
+In Section 3.1, we define geometric separation and provide an algorithm to calculate it. Our
+evaluation shows that geometric separation produces a valuable signal that improves confidence
+estimations. However, calculating geometric separation is too cumbersome for real-time systems,
+so we suggest a lightweight approximation in Section 3.2. Finally, Section 3.3 explains how we
+use the geometric signal to derive conf (x). That is, mapping a real number corresponding to the
+geometric separation to a number in [0, 1] corresponding to the confidence ratio.
+3.1 Separation Measure
+We look at the displacement of x compared to nearby data inputs within the training set. Intuitively,
+when x is close to other inputs in C(x) (i.e., inputs with the same classification as x) and is far
+from inputs with other classifications, then the model is correct with a high probability, implying
+that conf (x) should be high. On the other hand, when there are training inputs with a different
+classification close to x, we estimate that C(x) is more likely to be incorrect.
+Below we provide definitions that allow us to formalize this intuitive account. In what follows,
+we consider a model M to consist of a machine learning model (e.g., a gradient boosted tree or a
+4
+
+UNCERTAINTY ESTIMATION BASED ON GEOMETRIC SEPARATION
+ 
+ 
+Figure 1: Geometric representation of safe and dangerous inputs, maximal zones, and separation
+values. The various classifications are illustrated via different shapes, and the safe (danger) zones
+of x (y) are illustrated via green (red) circles.
+neural network), along with a labeled train set, Tr, used to generate the model. We use an implicit
+notion of distance and denote by d(x, y) the distance between inputs x and y, and by D(x, A) the
+distance between the input x and the set A (i.e., the minimal distance between x and the inputs in
+A).
+Definition 1 (Safe and Dangerous inputs). Let M be a model. For an input x in the sample space
+we define:
+FM(x) := {x′ ∈ Tr : C(x′) = C(x)}.
+We denote by F M(x) the set Tr \ FM(x). An input x ∈ X is labeled as safe if D(x, FM(x)) <
+D(x, F M(x)), and it is labeled as dangerous otherwise.
+Definition 2 (Zones). Let x be a safe (dangerous) input. A zone for x, denoted zx, is such that for
+any input y, if d(x, y) < zx, then D(y, FM(x)) < D(y, F M(x)) (D(y, FM(x)) ≥ D(y, F M(x))).
+For each x we denote the maximal such zone by Z(x).
+In other words, a zone of a safe (dangerous) input x is a radius around x such that all inputs in
+this ball are closer to an input in FM(x) (F M(x)) than to any input in F M(x) (FM(x)). Z(x) is
+the maximal zone attainable of x.
+Figure 1 provides a geometric illustration of the safe and danger zones of a given input and of
+the separation values. For illustration purposes, the figure uses the L2 norm with two dimensions,
+whereas our data usually includes many more dimensions. For example, a 30×30 traffic sign image
+will have 900 dimensions. In the figure, the shapes represent the classification of training set inputs.
+In yellow, we see a new input (x on the left-hand-side and y on the right-hand-side) which the model
+classifies as a triangle. x is a safe input because it is closer to other triangles in the training set than it
+is to the squares. The green highlighted ball represents its maximal zone. The input y is dangerous
+because the closest training set input is a square. The red highlighted ball represents its maximal
+zone which dually represents how far we need to distance ourselves from y so that inputs classified
+as triangles may become closer than other inputs.
+Definition 3 (Separation). The separation of a data input x with respect to the model M is Z(x)
+when x is a safe input, and −1 · Z(x) when x is a dangerous input.
+5
+
+CHOURAQUI ET AL.
+That is, the separation of x encapsulates the maximal zone for x (provided by the absolute
+value) together with an indication of whether the input is safe or dangerous (provided by the sign).
+The separation of x depends only on the classification of x by the model and the train set. This
+is because our definition partitions the inputs in Tr into two sets: one with C(x), FM(x), and one
+with all other classifications, F M(x). These sets vary between models only when they disagree on
+the classification of x. Note that x’s for which the distance from FM(x) equals the distance from
+F M(x) are considered dangerous inputs, and their separation measure will be zero.
+As mentioned, Definition 2 and Definition 3 use an implicit notion of distance and can accept
+any distance metric (e.g., L1, L2 or L∞). However, throughout this work, we use L2 as it is a stan-
+dard measure for safety features in adversarial machine learning Moosavi-Dezfooli et al. (2017), in
+addition to it being easy to illustrate and intuitive to understand. Moreso, as our work targets real-
+time confidence estimations using L2 allows us to leverage standard and well-optimized libraries.
+Accordingly, all our definitions and calculations assume the L2 metrics (Euclidean distances). Nev-
+ertheless, Section 4.2.1 shows that other metrics are also feasible.
+Next, we provide a formula for calculating the separation of a given input x within the L2
+distance metric.
+Definition 4. Given a model M and an input x, define:
+S
+M(x) =
+min
+x′′∈F M(x)
+max
+x′∈FM(x)
+d2(x, x′′) − d2(x, x′)
+2d(x′, x′′)
+Lemma 1. Let x, x′, x′′ ∈ Rn be inputs such that d(x, x′) < d(x, x′′). The maximal distance
+M(x, x′, x′′) for which if y ∈ Rn such that d(x, y) < M(x, x′, x′′), then d(y, x′) < d(y, x′′) is
+d2(x, x′′) − d2(x, x′)
+2d(x′, x′′)
+.
+Proof Since any three points in space define a plane we focus on the plane defined by these three
+points.
+Figure 2: Illustration of the proof of Lemma 1
+Figure 2 demonstrates a geometric positioning of the points and the main constructions in the
+proof. The perpendicular bisector to the line between x′ and x′′ divides the plane into two parts: one
+in which all the points are closer to x′′ than to x′ (the lower part in the figure) and one in which all
+the points are closer to x′ than to x′′ (the upper part in the figure). Our goal is thus to establish the
+distance between x and the lower part of the plane. Hence, M(x, x′, x′′) amounts to the distance
+6
+
+UNCERTAINTY ESTIMATION BASED ON GEOMETRIC SEPARATION
+from x to the perpendicular bisector to the line between x′ and x′′. Using trigonometric calculations,
+it is straightforward to verify that indeed
+M(x, x′, x′′) = d2(x, x′′) − d2(x, x′)
+2d(x′, x′′)
+.
+Proposition 1. S
+M(x) is the separation of x with respect to the model M (in Definition 3).
+Proof Let x be a safe input, and y be an input such that:
+d(x, y) <
+min
+x′��∈F M(x)
+max
+x′∈FM(x)
+d2(x, x′′) − d2(x, x′)
+2d(x′, x′′)
+.
+We first show that y is closer to FM(x) than to F M(x). Let z′′ ∈ F M(x), it suffices to show that
+there exist z′ ∈ FM(x) such that d(y, z′) < d(y, z′′). Notice that:
+d(x, y) <
+max
+x′∈FM(x)
+d2(x, z′′) − d2(x, x′)
+2d(x′, z′′)
+.
+Therefore, there exist a z′ ∈ FM(x) for which:
+d(x, y) < d2(x, z′′) − d2(x, z′)
+2d(z′, z′′)
+Thus, since x is a safe input, using Lemma 1, we conclude that d(y, z′) < d(y, z′′). The proof
+follows similar arguments for dangerous inputs, taking the distances as −S
+M and flipping the in-
+equalities.
+To show the maximality, observe that the intersection point marked by w in Figure 2, which is
+at distance S
+M(x) from x, can be easily shown to be of equal distances from FM(x) and F M(x).
+While separation provides the maximal zone, it is expensive to calculate. As can be seen in Def-
+inition 4, to estimate the separation of one specific input, we go over many triplets of inputs. The
+exact amount is unbounded and depends on the dataset. Thus, separation is infeasible to compute
+in near real-time. Therefore, when time or computation resources are limited, we require a differ-
+ent and computationally simpler notion. Accordingly, the following section provides an efficient
+approximation of the separation measure.
+3.2 Fast-Separation Approximation
+We approximate the separation of a given input using only its distance from FM(x) and its distance
+from F M(x). This simplification allows us to calculate a zone for any given input, which is not
+necessarily the maximal one. The reliance on these two distances enables a faster calculation since
+we do not perform an exhaustive search over many triplets of inputs. In particular, we do not
+consider the geometric positioning of the inputs that determine the distance from these sets.
+7
+
+CHOURAQUI ET AL.
+4
+3
+(a) SM(x) = S
+M(x) = 0.5
+3
+1
+(b) 0.5 = SM(x) ̸= S
+M(x) = 3.5
+Figure 3: Geometric representation of the induced zones of SM and S
+M for different input align-
+ments. SM is represented by blue arrows and S
+M by green arrows.
+Definition 5 (Fast-Separation). Given a model M, the fast-separation of an input x, denoted
+SM(x), is defined as:
+SM(x) = D(x, F M(x)) − D(x, FM(x))
+2
+Notice that just as is the case for separation, if x is a safe input, its fast-separation value will be
+strictly positive and non-positive otherwise.
+Figure 3 illustrates the notion of fast-separation. In particular, it exemplifies why it only provides
+an approximation of the more accurate separation measure. It encapsulates a zone that is less than
+or equal to that of separation. Sub-figure (a) demonstrates a case in which SM(x) = S
+M(x), while
+sub-figure (b) presents a case where S
+M(x) is considerably larger than SM(x).
+The separation measure defined as the maximal safe zone is applicable to all norms. However,
+the explicit formula S
+M, given in Definition 4 is only applicable in L2. The following proposition
+demonstrates that fast separation, SM, calculates a zone that is always contained in the maximal
+zone for any distance metric. Thus, it approximates the geometric separation for all metrics as the
+proof only requires the triangle inequality.
+Proposition 2. For any metric ℓ, and for any input x, SM(x) (calculated with respect to ℓ) is a
+zone of x. That is, |SM(x)| ≤ Z(x). Furthermore, SM(x) has the same sign as the separation of
+x.
+Proof Let x be a safe input, we show that SM(x) is a zone of x.
+Let y be a point such that
+d(x, y) < SM(x) = D(x, F M(x)) − D(x, FM(x))
+2
+.
+We show that D(y, FM(x)) < D(y, F M(x)). Take z′ ∈ FM(x) and z′′, w ∈ F M(x) such that
+d(x, z′) = D(x, FM(x)), d(x, z′′) = D(x, F M(x)), and d(y, w) = D(y, F M(x)). Using the
+triangle inequality we get:
+D(y, FM(x)) ≤ d(y, z′) ≤ d(x, z′) + d(x, y)
+< d(x, z′) + d(x, z′′) − d(x, z′)
+2
+= d(x, z′′) + d(x, z′)
+2
+= d(x, z′′) − d(x, z′′) − d(x, z′)
+2
+< d(x, z′′) − d(x, y)
+≤ d(x, w) − d(x, y) ≤ d(y, w) = D(y, F M(x))
+For dangerous inputs, the proof follows similar arguments, switching FM(x) and F M(x).
+8
+
+UNCERTAINTY ESTIMATION BASED ON GEOMETRIC SEPARATION
+For the sign of SM(x) it is easy to see that for a safe (dangerous) input, SM(x) will be positive
+(negative) and therefore has the same sign as the separation.
+Proposition 2 shows that SM(x) induces a zone that is always smaller than the maximal zone
+for any distance metric. In the case of L2, we have a formula for calculating the maximal zone
+(S
+M(x)), and the following proposition provides an approximation bound.
+Proposition 3. The following holds for any point x:
+|S
+M(x) − SM(x)| ≤ D(x, FM(x)) + D(x, F M(x))
+2
+.
+Proof We here prove the bound for safe inputs x, the proof for dangerous inputs is similar. Let x
+be a safe input. By definition:
+|S
+M(x) − SM(x)| = S
+M(x) − SM(x) =
+=
+min
+x′′∈F M(x)
+max
+x′∈FM(x)
+d2(x, x′′) − d2(x, x′)
+2d(x′, x′′)
+− D(x, F M(x)) − D(x, FM(x))
+2
+Let z′′ ∈ F M(x) be an input such that d(x, z′′) = D(x, F M(x)), and let z′ ∈ FM(x) be a input
+for which the maximum on the expression above is obtained. Then, we have:
+|S
+M(x) − SM(x)|
+≤
+max
+x′∈FM(x)
+d2(x, z′′) − d2(x, x′)
+2d(x′, z′′)
+− d(x, z′′) − D(x, FM(x))
+2
+(1)
+=d2(x, z′′) − d2(x, z′)
+2d(z′, z′′)
+− d(x, z′′) − D(x, FM(x))
+2
+(2)
+≤d(x, z′′) + d(x, z′)
+2
+− d(x, z′′) − D(x, FM(x))
+2
+(3)
+=d(x, z′) + D(x, FM(x))
+2
+(4)
+≤D(x, FM(x)) + D(x, F M(x))
+2
+(5)
+The first inequality (Equation (1)) holds due to the definition of the minimum function. The second
+inequality (Equation (3)) is due to the triangle inequality. The last inequality (Equation (5)) holds
+because, since x is a safe input, the maximal distance between x and z′ can’t be greater than the
+distance from x to F M(x).
+Notice that the above bound is tight, in the sense that there exists an example witnessing the
+exact bound, as shown in Figure 4 below.
+3.3 Calibration of the Geometric Separation
+In this section, we use the geometric notions of SM(x) and S
+M(x) to derive confidence estimations
+(conf (x)). Notice that conf (x) ∈ (0, 1) while the geometric notions are in (−∞, +∞). Next, we
+explain how to translate between the two.
+9
+
+CHOURAQUI ET AL.
+Figure 4: Example of a input x with |S
+M(x) − SM(x)| = D(x,F M(x))+D(x,FM(x))
+2
+For each value of SM(x) (S
+M(x)), we need to match a confidence value. To do so, we split the
+data into a Validation set, Vs, which is disjoint from the train and test sets. Such a methodology is
+commonly used in post-hoc calibration methods Guo et al. (2017b); Platt (1999); Kull et al. (2017);
+Mozafari et al. (2018); Tomani et al. (2022); Zhang et al. (2020); Gupta and Ramdas (2021); Kumar
+et al. (2019). We then measure the accuracy for inputs with similar SM(x) (or S
+M(x)) on Vs.
+At this point, we have pairs (y, z) where y is a geometric separation value, and z is the desired
+confidence value (as measured by the accuracy on Vs). The next step is to find a low-dimensionality
+function that maximizes accuracy.
+Hence, we perform a fitting between SM (or S
+M) values and the ratios of correct classifications
+(on Vs) for each unique value. E.g., if for SM value of 10 we see that 90% of the points are classified
+correctly, then we’ll add the pair ⟨10, 0.9⟩ to the fitting function. Intuitively, we expect very low
+confidence values for highly negative distances and approach 100% confidence when the distances
+are large and positive.
+4. Experimental Results
+In this section, we evaluate the effectiveness of our geometric approach. First, we explain the
+evaluation methodology in Section 4.1, including the datasets and models. Then we continue our
+experiment results step by step by gradually explaining the tradeoffs and design decisions we take
+throughout this work.
+4.1 Methodology
+4.1.1 DATASETS
+Our evaluation uses the following standard datasets:
+• Modified National Institute of Standards and Technology database (MNIST) LeCun and Cortes
+(2010). A dataset that consists of hand-written images designed for training various image
+processing systems. It includes 70,000 28×28 grayscale images belonging to one of ten labels.
+• Fashion MNIST (Fashion) Xiao et al. (2017). A dataset comprising of 28×28 grayscale images
+of 70,000 fashion products from 10 categories.
+• German Traffic Signs Recognition Benchmark (GTSRB) Houben et al. (2013). A large image
+set of traffic signs for the single-image, multi-class classification problem. It consists of
+50,000 RGB images of traffic signs, belonging to 43 classes.
+10
+
+UNCERTAINTY ESTIMATION BASED ON GEOMETRIC SEPARATION
+• American Sign Language (SignLang) Techperson (2017). A database of hand gestures repre-
+senting a multi-class problem with 24 classes of letters. It consists of 30,000 28×28 grayscale
+images.
+• Canadian Institute for Advanced Research (CIFAR10) Krizhevsky et al. (2009). A dataset
+containing 32x32 RGB images of 60,000 objects from 10 classes.
+For each dataset, we randomly partitioned the data into three subsets: train set Tr (60%), validation
+set Vs (20%) and test set Ts (20%). As is standard practice, we used normalized datasets (e.g.,
+the same image size for all images), see Leman et al. (2022) for details. The train set is used to
+calculate (fast-)separation and train the model. The validation set is used to evaluate the confidence
+estimation associated with each (fast-)separation value. These values, in turn, are used to fit an
+isotonic function. Finally, the test set is used to evaluate the confidence on new inputs that were not
+present in the train and validation sets.
+4.1.2 MODELS
+In our evaluation, we use the following popular machine learning models: Random Forest (RF)
+Breiman (2001), Gradient Boosting Decision Trees (GB) Mason et al. (1999), and Convolutional
+Neural Network (CNN) Gu et al. (2018). We chose these models because they are different: RF
+and GB are tree-based, while CNN is a neural network. For RF and GB, we configured the hy-
+perparameters (e.g., the maximal depth of trees) by cross-validation on the train set via the random
+search technique Bergstra and Yoshua (2012). For CNN, we used the configuration suggested by
+practitioners. Our specific configurations as well as the accuracy scores of each of the models are
+detailed in Leman et al. (2022).
+4.1.3 EVALUATION ALGORITHMS
+To evaluate our method, we compare our (fast-)separation-based confidence estimation to the fol-
+lowing methods: the built-in isotonic regression calibration implemented by Sklearn library, Iso
+Zadrozny and Elkan (2002); the built-in Platt scaling calibration method implemented by Sklearn
+library, Platt Platt (1999); the scaling-binning calibrator, SBC Kumar et al. (2019) implemented
+by the same authors repository; the histogram-binning, HB Gupta and Ramdas (2021) implemented
+by the same authors repository; the beta calibrator, Beta Kull et al. (2017) implemented by K¨uppers
+et al. (2020); the bayesian binning into quantiles calibrator, BBQ Naeini et al. (2015) implemented
+by K¨uppers et al. (2020); the temperature scaling calibrator, TS Guo et al. (2017a) implemented
+by Kerrigan et al. (2021); and the ensemble temperature scaling calibrator, ETS Zhang et al. (2020)
+implemented by Kerrigan et al. (2021). Notice that TS and ETS are calibration methods for neural
+networks thus we only apply those to CNNs.
+Each method receives the same baseline model as an input yielding a slightly different cali-
+brated model. Note that our method is evaluated against the uncalibrated model as our method does
+not affect the model. Moreover, it allows us to compare our method against different calibration
+methods, as shown in Table 2.
+To evaluate the confidence predictions, we use the Expected Calibration Error (ECE), which is
+a standard method to evaluate confidence calibration of a model Xing et al. (2020); Krishnan and
+Tickoo (2020). Concretely, the predictions sample of size n are partitioned into M equally spaced
+bins (Bm)m≤M, and ECE measures the difference between the sample accuracy in the mth bin and
+11
+
+CHOURAQUI ET AL.
+Dataset
+Model
+L1
+L2
+L∞
+CNN
+0.17 ±0.03
+0.18 ±0.05
+0.08 ±0.03
+GB
+0.28 ±0.09
+0.36 ±0.07
+0.37 ±0.06
+MNIST
+RF
+0.37 ±0.07
+0.39 ±0.06
+0.37 ±0.05
+CNN
+0.38 ±0.14
+0.42 ±0.15
+0.36 ±0.16
+GB
+1.08 ±0.19
+0.65 ±0.11
+0.41 ±0.09
+GTSRB
+RF
+0.54 ±0.19
+0.37 ±0.04
+0.32 ±0.05
+CNN
+0.05 ±0.03
+0.05 ±0.04
+0.05 ±0.03
+GB
+0.00 ±0.00
+0.08 ±0.03
+0.17 ±0.02
+SignLang
+RF
+0.00 ±0.00
+0.08 ±0.02
+0.14 ±0.03
+CNN
+0.74 ±0.07
+0.79 ±0.15
+0.55 ±0.12
+GB
+0.64 ±0.21
+0.73 ±0.13
+0.85 ±0.17
+Fashion
+RF
+0.68 ±0.13
+0.74 ±0.16
+0.83 ±0.14
+CNN
+1.20 ±0.19
+1.20 ±0.30
+3.03 ±0.79
+GB
+1.59 ±0.51
+1.25 ±0.21
+1.17 ±0.24
+CIFAR10
+RF
+1.08 ±0.45
+1.15 ±0.24
+1.30 ±0.18
+Table 1: ECE(%) measures with 95% confidence intervals comparing the results of the fast-
+separation-based method using L1, L2 and L∞ norms.
+the the average confidence in it Naeini et al. (2015). Formally, ECE is calculated by the following
+formula:
+ECE =
+M
+�
+m=1
+|Bm|
+n
+|acc (Bm) − conf (Bm)|
+where: acc (Bm) =
+1
+|Bm| · |{x ∈ Bm : C(x) is correct}|, and conf (Bm) =
+1
+|Bm|
+�
+x∈Bm conf (x).
+4.2 Empirical Study
+4.2.1 DISTANCE METRICS
+As mentioned in Section 3.1, the notion of geometric separation is applicable to any norm. In fact, as
+shown in Proposition 2, the fast-separation approximation provides a zone under any norm. Thus,
+we have evaluated the ECE obtained from fast-separation under different norms. The results are
+given in Table 2. As can be observed, the ECE is low regardless of the selection of norm indicating
+the attractiveness of the geometric signal. However, while some norms are more accurate for some
+datasets, there is no universally superior norm. Thus, the following experiments focus on the L2
+norm from the reasons specified in Section 3.1.
+4.2.2 FITTING FUNCTION
+As mentioned in Section 3.3, for our fitting function we can use any existing calibration function.
+Post-hoc calibration methods based on fitting functions typically use either a logistic (Sigmoid) or
+an isotonic regression Zadrozny and Elkan (2002). Isotonic regression fits a non-decreasing free-
+form line to a sequence of observations. In comparison, Sigmoid is a continuous step function. We
+used both fitting functions on our fast-separation values and obtained similar accuracy. We opt here
+to present the isotonic regression as it provides the best empirical results, as motivated by Figure 5.
+12
+
+UNCERTAINTY ESTIMATION BASED ON GEOMETRIC SEPARATION
+−200
+0
+200
+400
+600
+Fast Separation Score SM
+0.0
+0.2
+0.4
+0.6
+0.8
+1.0
+Accuracy on Validation Set
+Less than 100 samples
+More than 100 samples
+Sigmoid fitting
+Isotonic regression
+Figure 5: An illustration of the inputs to the fitting function (blue diamonds and red dots), and the
+functions fitted by Sigmoid (black line) and isotonic regression (green line). The inputs are for the
+MNIST dataset and the Random Forest model.
+Figure 5 illustrates an example of the success ratio of the Random Forest model for MNIST
+inputs with varying values of SM scores (similar behavior was observed for the various models
+and datasets). We clustered inputs with a similar score together (into 50 bins overall) as each
+classification is correct or not, and we are looking for the average. The black line represents the
+Sigmoid function, and the green line represents the isotonic regression. As can be observed, both
+regressions are nearly identical on all the points with positive SM values. We eventually chose
+isotonic regression because it better fitted the few points with negative SM values. Interestingly,
+these points were consistently a poor fit for the Sigmoid regression rendering it slightly less accurate
+on average. Also, observe that the transition is around the value 0, indicating that the distinction
+between safe and dangerous points is meaningful in confidence evaluation.
+4.3 Confidence Evaluation
+This section presents the experimental results of the confidence estimation.
+4.3.1 ESTIMATING CONFIDENCE
+Table 2 presents the main experimental results of our work. The table summarizes ECEs for our
+method (with bin size 15). Each entry in the table describes the ECE and the 95% confidence
+interval. We highlight the most accurate method for each experiment in bold. In this experiment,
+we perform one hundred random splits of the data into train, validation, and test sets for each
+model and dataset. We then measure the ECE of the confidence estimation for all test set inputs,
+average the result and take the 95% confidence intervals. 1 First, observe that SM and S
+M yield
+very similar ECEs, and that the differences between them are usually statistically insignificant.
+1. For Plat and Iso we used the standard SKlearn implementation. However, we used Pytorch for CNN models, and
+Pytorch does not have Plat and Iso, so we implemented them for our Pytorch-based CNN models.
+13
+
+CHOURAQUI ET AL.
+Table 2: ECE(%) measures with 95% confidence intervals when varying the calibration method,
+model, and dataset.
+Dataset Model
+SM
+S
+M
+Iso
+Platt
+SBC
+HB
+BBQ
+Beta
+TS
+ETS
+CNN
+0.15±.01 0.15±0.01
+0.17±0.01
+0.52±0.04
+8.91±0.16
+0.32±0.02 0.22±0.01
+0.64±0.02
+0.20±0.01 0.20±0.01
+RF
+0.35±.02 0.36±0.02
+0.92±0.03
+1.49±0.02
+3.92±0.11
+0.46±0.02 1.13±0.03
+0.37±0.02
+-
+-
+GB
+0.34±.02 0.34±0.02
+1.74±0.03
+1.97±0.03
+8.46±0.07
+0.45±0.02 0.65±0.03
+0.47±0.02
+-
+-
+MNIST
+CNN
+0.37±.04 0.37±0.04
+0.38±0.04
+2.83±0.53
+29.01±0.49 1.22±0.18 1.08±0.21
+1.98±0.25
+0.90±0.11 0.77±0.09
+RF
+0.37±.02 0.38±0.02
+2.55±0.04
+4.19±0.03
+13.99±0.11 0.85±0.05 3.08±0.04
+0.56±0.03
+-
+-
+GB
+0.61±.03 0.63±0.03 10.04±0.07 19.63±0.83 31.25±0.12 1.42±0.05 9.28±0.11
+5.36±0.10
+-
+-
+GTSRB
+CNN
+0.09±.05 0.10±0.06
+0.09±0.05
+0.12±0.07
+17.77±0.21 1.24±1.03 1.24±1.03
+1.24±1.04
+0.11±0.01 0.12±0.01
+RF
+0.08±.01 0.08±0.01
+0.46±0.02
+1.76±0.02
+17.34±0.18 0.16±0.02 0.86±0.02
+0.29±0.01
+-
+-
+GB
+0.07±.01 0.07±0.01
+4.01±0.06
+5.93±0.06
+31.01±0.08 0.46±0.03 0.78±0.05
+0.70±0.03
+-
+-
+SignLang
+CNN
+0.75±.03 0.75±0.04
+0.71±0.03
+6.60±0.72
+7.36±0.20
+1.10±0.05 2.18±0.15
+9.15±0.10
+0.82±0.04 0.89±0.04
+RF
+0.78±.04 0.82±0.04
+1.03±0.05
+3.75±0.04
+3.52±0.10
+1.07±0.05 1.23±0.05
+0.83±0.03
+-
+-
+GB
+0.79±.04 0.79±0.04
+3.82±0.06
+5.01±0.65
+3.90±0.12
+1.01±0.04 1.41±0.05
+0.97±0.05
+-
+-
+Fashion
+CNN
+1.12±.07 1.16±0.07
+1.28±0.06
+6.05±0.21
+3.57±0.10
+4.10±0.10 5.31±0.24 24.76±0.21 3.68±0.08 3.45±0.12
+RF
+1.37±.07 1.38±0.07
+3.33±0.07
+4.54±0.09
+3.01±0.09
+2.27±0.09 3.84±0.08
+1.65±0.06
+-
+-
+GB
+1.41±.07 1.38±0.06
+7.70±0.08
+8.58±0.09
+2.63±0.09
+2.40±0.10 1.51±0.07
+2.94±0.08
+-
+-
+CIFAR10
+Thus, we conclude that SM is a good approximation of S
+M despite being considerably simpler
+to compute. The next interesting comparison is between SM and Iso. We use the same fitting
+function (isotonic regression) in both cases, but Iso performs the calibration on the model’s natural
+uncertainty estimation, and SM performs the calibration on geometric distances. Our SM almost
+consistently improves the confidence estimations across the board compared to Iso, Platt, SBC,
+HB, TS, ETS, Beta, and BBQ. Specifically, we derive improvements up to 99% in almost all
+tested models and datasets. Such results demonstrate the potential of geometric signals to improve
+the effectiveness of uncertainty estimation.
+Table 3 describes the improvement of our fast-separation-based method over recently proposed
+posthoc calibration techniques. The improvement is calculated using the ratio of the difference be-
+tween our ECE and the competitor’s ECE. Observe that our method always improves the alternatives
+except for CNNs on the Fashion dataset, where it loses by 5%. Such results position our geometric
+method as a competitive approach for confidence estimation. However, note that our fast-separtion
+can be used alongside the existing methods.
+4.3.2 TABULAR DATA
+Fast-separation was designed for image data sets since they appear to be most governed by geom-
+etry, that is, different images will likely be geometrically separable. Nonetheless, as a controlled
+14
+
+UNCERTAINTY ESTIMATION BASED ON GEOMETRIC SEPARATION
+Table 3: Relative improvement percentage of ECE of SM over other calibration methods.
+Dataset
+Model
+Iso
+Platt
+SBC
+HB
+BBQ
+Beta
+TS
+ETS
+CNN
+11.8%
+71.2%
+98.3%
+53.1%
+31.8%
+76.6%
+25.0%
+25.0%
+RF
+62.0%
+76.5%
+91.1%
+23.9%
+69.0%
+5.4%
+-
+-
+GB
+80.5%
+82.7%
+96.0%
+24.4%
+47.7%
+27.7%
+-
+-
+MNIST
+CNN
+2.6%
+86.9%
+98.7%
+69.7%
+65.7%
+81.3%
+58.9%
+51.9%
+RF
+85.5%
+91.2%
+97.4%
+56.5%
+88.0%
+33.9%
+-
+-
+GB
+93.9%
+96.9%
+98.0%
+57.0%
+93.4%
+88.6%
+-
+-
+GTSRB
+CNN
+0.0%
+25.0%
+99.5%
+92.7%
+92.7%
+92.7%
+18.2%
+25.0%
+RF
+82.6%
+95.5%
+99.5%
+50.0%
+90.7%
+72.4%
+-
+-
+GB
+98.3%
+98.8%
+99.8%
+84.8%
+91.0%
+90.0%
+-
+-
+SignLang
+CNN
+-5.6%
+88.6%
+89.8%
+31.8%
+65.6%
+91.8%
+8.5%
+15.7%
+RF
+24.3%
+79.2%
+77.8%
+27.1%
+36.6%
+6.0%
+-
+-
+GB
+79.3%
+84.2%
+79.7%
+21.8%
+44.0%
+18.6%
+-
+-
+Fashion
+CNN
+12.5%
+81.5%
+68.6%
+72.7%
+78.9%
+95.5%
+69.6%
+67.5%
+RF
+58.9%
+69.8%
+54.5%
+39.6%
+64.3%
+17.0%
+-
+-
+GB
+81.7%
+83.6%
+46.4%
+41.2%
+6.6%
+52.0%
+-
+-
+CIFAR10
+experiment, we also tested our method on non-visual tabular data. Here, we have no apriori intuition
+that the geometric signal is feasible. We used two datasets: Red wine quality Cortez et al. (2009),
+which contains a total of twelve variables and 1,599 observations and six classes, and airline passen-
+ger satisfaction Klein (2019), which contains a total of twenty-five variables, 129,880 observations,
+and two classes.
+In most experiments, we saw a small improvement of ranging between 1% to 77% in accuracy.
+Thus, we conclude that our method achieves good results on tabular data as well. However, the
+improvement was not uniform and there were a few cases where Iso was superior to our own. Thus,
+the geometric signal may also be useful for non-visual data but further investigations are required
+to adapt the method to various datasets.
+5. Optimizing Performance
+As shown in the previous section, the fast-separation approximation yields competitive confidence
+estimations promptly for small and medium-sized datasets. Nonetheless, our approach may still
+be too slow to handle large datasets due to the need to calculate geometric notions on the entire
+training set. To address this bottleneck, we explore the impact of several standard methods for
+dimensionality reduction on the quality of our approach for confidence estimation.
+15
+
+CHOURAQUI ET AL.
+5.1 Handling Large Datasets
+Large datasets are datasets with a large number of images or with large images with many pixels.
+In such cases, each calculation of fast separation requires potentially going over many comparisons
+that slow down the process. Here, we explore ways to either reduce the image size, or to reduce
+the number of images.2 The following list reviews various known techniques for reducing the
+dimensionality of the data, the first four reduce the number of pixels, and the last two reduce the
+number of images in the set used to calculate geometric distances.
+In order to have a fair comparison, we define the reduction parameter t to indicate the amount
+of data reduced in each method. In each method, a reduction parameter t implies that we reduce
+the dataset size by a factor of t2. E.g., in the Pooling technique, we can reduce 2x2 images into a
+pixel reducing the image size, while K-means would reduce the number of images, and both would
+reduce it by a factor of four so that the total number of pixels in the set is the same for each reduction
+parameter value for all the methods.
+Pooling Mosteller (1948) is an operation that calculates a function for patches of a feature map
+and uses it to create a down-sampled (pooled) feature map. For example, if one wants a 2-pool of
+an image, one reduces its size by 2x2, and every square of 2x2 is then represented as the output of
+the function on the squared elements. Some broadly used functions for pooling are average (pool)
+and maximum (maxpool).
+Principal Component Analysis (PCA) F.R.S. (1901) linearly transforms the data into a new
+coordinate system where most of the variation in the data can be described with fewer dimensions
+than the initial data. For reduction parameter 2 we reduce each image to a new smaller image with
+a reduction factor of four in the number of pixels.
+Resizing using a Bilinear Interpolation (RBI) Smith (1981) is a generalization of single di-
+mension linear interpolation. RBI performs linear interpolation in one direction and then again in
+the other direction. Resizing using a Bilinear Interpolation is common in computer vision applica-
+tions that are based on convolutional neural networks. For a reduction parameter 2 we resize the
+image to a new image in which both the length and width are two times smaller, ending with an
+image four times smaller than the original one.
+Sampling random pixels (Randpix) reduces the number of pixels in the metadata by a random
+sample. Notice that this approach can be viewed as the baseline for other pixel-reducing techniques.
+We chose the number of pixels sampled to be the original pixel number divided by the squared
+reduction parameter.
+K-means MacQueen (1967) clustering is a vector quantization method aiming to partition n
+observations into k clusters in which each observation belongs to the cluster with the nearest mean
+(cluster centroid). K-means clustering minimizes variances in the clusters (squared Euclidean dis-
+tances). Here, we set k to be the reduced dimension of the compressed dataset. E.g., if the original
+dataset had 10,000 images, and we set k = 1, 000, we get a reduction factor of x10 from 10, 000
+dimensions to 1, 000. When using this method we first find the centroids of the dataset, and then
+use these as the metadata for calculating geometric separation.
+Sampling the training set (Randset) reduces the number of inputs in the training set, by pick-
+ing a random sample. We chose the sample size to be the dataset size divided by the squared
+reduction parameter.
+2. One can also directly manipulate the searching algorithm to improve the calculation complexity of the nearest neigh-
+bor by, e.g., randomization or special data structures. We leave exploring this option for future work.
+16
+
+UNCERTAINTY ESTIMATION BASED ON GEOMETRIC SEPARATION
+pool
+maxpool
+RBI
+PCA
+Randpix
+Randset
+Kmeans
+Reduction Method
+0
+100
+200
+300
+400
+500
+600
+700
+Predictions per second
+(a) MNIST
+pool
+maxpool
+RBI
+PCA
+Randpix
+Randset
+Kmeans
+Reduction Method
+0
+100
+200
+300
+400
+500
+600
+700
+Predictions per second
+(b) Fashion
+pool
+maxpool
+RBI
+PCA
+Randpix
+Randset
+Kmeans
+Reduction Method
+0
+100
+200
+300
+400
+Predictions per second
+(c) GTSRB
+pool
+maxpool
+RBI
+PCA
+Randpix
+Randset
+Kmeans
+Reduction Method
+0
+100
+200
+300
+400
+500
+600
+700
+Predictions per second
+None
+2
+3
+4
+pool
+maxpool
+RBI
+PCA
+Randpix
+Randset
+Kmeans
+Reduction Method
+0
+50
+100
+150
+200
+250
+300
+Predictions per second
+(d) CIFAR10
+pool
+maxpool
+RBI
+PCA
+Randpix
+Randset
+Kmeans
+Reduction Method
+0
+200
+400
+600
+800
+1000
+Predictions per second
+(e) SignLanguage
+Figure 6: Time comparison between various reduction methods on all datasets with Random Forest
+model. The error bar show the 95% confidence interval over 10 shuffles. The colors denote the
+various reduction parameters.
+5.2 Experimental Results
+To evaluate the effect of these reductions on our algorithm, we apply the reduction to the whole
+dataset and then calculate the fast-separation values on the reduced dataset. Note that the models
+are trained on the original dataset, so accuracy is not affected. For RGB images, we further changed
+the color to grayscale images, which reduced the size of images by a factor of 3 while keeping the
+image as close as possible to the original one. The experiments were executed on a desktop PC with
+Intel(R) 16 Cores(TM) i7-10700 CPU @ 2.90GHz, and 16GB RAM.
+Figures 6 and 7 show a comparison of the speed and accuracy of the various methods in the Ran-
+dom Forest model. As shown in Figure 6, all data optimizations increase the number of predictions
+per second, and we can readily reach several hundred estimations per second which is a sufficient
+speedup for our needs. All methods show almost the same speedup on the algorithm for each hy-
+perparameter value, except for k-means which sometimes has a better speedup and RBI which has
+a slightly lower speedup. Since there is little variability in the experiments, the confidence intervals
+are barely visible.
+Our experiments show that the time performance is not affected by the model. Thus, Figure 6
+presents only the results for the Random forest model, i.e., the average number of predictions per
+second. Observe that the number of predictions per seconds is the same for all other models.
+Importantly, this improvement in runtime does not come at a meaningful cost for the confidence
+estimation, as shown in Figure 7. While the error in the calibration estimation slightly changes
+across different methods and reduction parameters, the changes seem insignificant. Specifically, in
+17
+
+CHOURAQUI ET AL.
+pool
+maxpool
+RBI
+PCA
+Randpix
+Randset
+Kmeans
+Reduction Method
+0.000
+0.001
+0.002
+0.003
+0.004
+0.005
+0.006
+ECE
+(a) MNIST
+pool
+maxpool
+RBI
+PCA
+Randpix
+Randset
+Kmeans
+Reduction Method
+0.0000
+0.0025
+0.0050
+0.0075
+0.0100
+0.0125
+0.0150
+0.0175
+0.0200
+ECE
+(b) Fashion
+pool
+maxpool
+RBI
+PCA
+Randpix
+Randset
+Kmeans
+Reduction Method
+0.000
+0.001
+0.002
+0.003
+0.004
+0.005
+ECE
+(c) GTSRB
+MNIST
+Fashion
+SignLang
+GTSRB
+CIFAR-10
+Datasets
+0.000
+0.002
+0.004
+0.006
+0.008
+0.010
+0.012
+ECE
+None
+2
+3
+4
+pool
+maxpool
+RBI
+PCA
+Randpix
+Randset
+Kmeans
+Reduction Method
+0.000
+0.002
+0.004
+0.006
+0.008
+0.010
+0.012
+0.014
+0.016
+ECE
+(d) CIFAR10
+pool
+maxpool
+RBI
+PCA
+Randpix
+Randset
+Kmeans
+Reduction Method
+0.0000
+0.0005
+0.0010
+0.0015
+0.0020
+0.0025
+ECE
+(e) SignLanguage
+Figure 7: ECE comparison between various reduction methods on all datasets with Random forest
+model. The error bar show the 95% confidence interval over 10 shuffles. The black line shows the
+ECE score of the method without any reduction. The colors denote the various reduction parameters.
+MNIST
+Fashion
+SignLang
+GTSRB
+CIFAR-10
+Datasets
+0.000
+0.002
+0.004
+0.006
+0.008
+0.010
+0.012
+0.014
+0.016
+ECE
+None
+2
+3
+4
+(a) GB
+MNIST
+Fashion
+SignLang
+GTSRB
+CIFAR-10
+Datasets
+0.000
+0.002
+0.004
+0.006
+0.008
+0.010
+0.012
+0.014
+0.016
+ECE
+None
+2
+3
+4
+(b) RF
+MNIST
+Fashion
+SignLang
+GTSRB
+CIFAR-10
+Datasets
+0.000
+0.002
+0.004
+0.006
+0.008
+0.010
+0.012
+0.014
+ECE
+None
+2
+3
+4
+(c) CNN
+Figure 8: ECE measures with 95% confidence intervals on 10 shuffles for various datasets on several
+models after applying max-pooling with different reduction parameters.
+the SignLanguage dataset we observe an increase in the ECE, which we believe is due to the fact
+that the original dataset is already quite small, rendering the datasize optimization pointless. Our
+experiments also show similar behavior across different models. For example, Figure 8 shows that
+all three models obtain similar errors with maxpooling with different parameters. Moreover, our
+results outperform most state-of-the-art algorithms even with a 4-pool.
+When using our method one needs to ship besides the model and the fitting function also the
+dataset itself, since the calculation of the fast-separation requires the calculation of distances to all
+images in the training set. This may imply memory overhead, which can be critical when using
+big datasets. Using a reduction of the dataset allows us to reduce the training set size needed to
+18
+
+UNCERTAINTY ESTIMATION BASED ON GEOMETRIC SEPARATION
+be shipped. As we have shown, the reduced dataset still obtains improved results, thus freeing
+memory usage of the algorithm. Users can also predetermine the trade-off they would like between
+throughput or memory and ECE and adjust the reduction parameters accordingly.
+6. Conclusion
+Our work introduces geometric separation-based algorithms for confidence estimation in machine
+learning models. Specifically, we measure a geometric separation score and use the specific model
+to translate each score value into a confidence value using a standard post-hoc calibration method.
+Thus, inputs close to training set examples of the same class receive higher confidence than those
+close to examples with a different classification. Thus, our algorithms depend on the specific model
+but as a black box resulting in methods that work for all machine learning models.
+Our evaluation shows that geometric separation improves confidence estimations in visual work-
+loads. However, calculating geometric separation is computationally complex and time intensive.
+Thus, we suggest multiple approximation techniques to speed up the process and bring it to prac-
+ticality. Our extensive evaluation shows that such approximations retain most of the benefits of
+geometric separations and drastically improve confidence estimation along with supporting many
+calculations per second, enabling real-time applications. For example, we can process live camera
+feeds at multiple hundreds of calculations per second.
+Our work is unique because it extracts a new external signal to derive confidence estimations.
+Thus, we can leverage the existing post-hoc calibration techniques to calibrate our signal and meet
+various optimization criteria. We showed that the same calibration method (Isotonic regression)
+yields a lower ECE when performed on the geometric signal rather than on the model’s original
+signal. The achieved accuracy improves on a diverse set of recently proposed calibration meth-
+ods. Notably, our approach reduces the error in confidence estimations by up to 99% compared to
+alternative methods (depending on the specific dataset and model).
+Looking into the future, we plan to address the dependence of this work on normalized inputs
+and tackle datasets with variable-sized images. In such datasets, the geometric distances may also
+depend on the resolution and alignment of the object. As a director, we plan to use the CNN middle
+layer latent space as the feeding vector (rather than the original images) in the geometric separation
+calculation. In any case, the ability to derive fast approximations of geometric separation would be
+a valuable tool in future research.
+References
+A. Ashukha, A. Lyzhov, D. Molchanov, and D. Vetrov. Pitfalls of In-Domain Uncertainty Estimation
+and Ensembling in Deep Learning. In International Conference on Learning Representations,
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+J. Bergstra and B. Yoshua. Random search for hyper-parameter optimization. Journal of machine
+learning research, 13(2), 2012.
+L. Breiman. Random Forests. Machine Learning, 45(1):5–32, 2001.
+P. Cortez, A. Cerdeira, F. Almeida, T. Matos, and J. Reis. Modeling wine preferences by data
+mining from physicochemical properties. 2009. https://archive.ics.uci.edu/ml/
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+Z. Ding, X. Han, P. Liu, and M. Niethammer. Local temperature scaling for probability calibration.
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+els. In The Thirty-Third Conference on Artificial Intelligence,, pages 2446–2453. AAAI, 2019.
+K. Pearson F.R.S. Liii. on lines and planes of closest fit to systems of points in space. The London,
+Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 2(11):559–572, 1901.
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+arXiv:2301.01963v1  [math.RA]  5 Jan 2023
+BOGOMOLOV MULTIPLIER OF LIE ALGEBRAS
+PRADEEP K. RAI
+Abstract. In the work of Rostami et al., the Bogomolov multiplier of a Lie
+algebra L over a field Ω is defined as a particular factor of a subalgebra of the
+exterior product L ∧ L. If L is finite dimensional, we identify this object as
+a certain subgroup of the second cohomology group H2(L, Ω) by deriving a
+Hopf-Type formula. As an application, we affirmatively answer two questions
+posed by Kunyavski˘ı regarding the invariance of the Bogomolov multiplier
+under isoclinism of Lie algebras and the existence of a family of Lie algebras
+with Bogomolov multipliers of unbounded dimension.
+1. Introduction
+The Bogomolov multiplier of a finite group G is a cohomological invariant of G
+that has been studied in connection with Noether’s problem, which asks whether
+the fixed subfield k(G) of the function field k(xg : g ∈ G) is purely transcendental
+over an algebraically closed field k of characteristic zero. The Bogomolov multiplier
+has played a key role in the discovery of counter examples to Noether’s problem over
+the complex numbers C. Saltman [7] was the first to provide such counter examples,
+showing that if the unramified cohomology group H2
+nr(k(G), C) is nonzero, then G
+has a negative solution to Noether’s problem over C. Bogomolov later showed that
+H2
+nr(k(G), C) is naturally isomorphic to the subgroup B0(G) of the second coho-
+mology group H2(G, C) consisting of classes that vanish when restricted to the
+abelian subgroups of G [2]. This has led to the discovery of numerous other counter
+examples to Noether’s problem. Kunyavski˘ı later gave the name “Bogomolov mul-
+tiplier” to B0(G) [3], and Moravec provided a homological description of B0(G) as
+a quotient of H2(G, C) [5]. Following Moravec’s construction, Rostami et. al [6]
+extended the notion of Bogomolov multiplier to Lie algebras. For the convenience
+of the reader we define it here.
+Let L be a Lie algebra over Ω. The exterior square of L is defined to be the
+Lie algebra L∧L generated by the symbols m ∧ n, where m, n ∈ L, subject to the
+following relations:
+(i) α(m ∧ n) = αm ∧ n = m ∧ αn,
+(ii) (m + m′) ∧ n = m ∧ n + m′ ∧ n,
+(iii) m ∧ (n + n′) = m ∧ n + m ∧ n′,
+(iv) [m, m′] ∧ n = m ∧ [m′, n] − m′ ∧ [m, n],
+(v) m ∧ [n, n′] = [n′, m] ∧ n − [n, m] ∧ n′,
+(vi) [(m ∧ n), (m′ ∧ n′)] = −[n, m] ∧ [m′, n′],
+(vii) m ∧ n = 0 whenever m = n,
+2010 Mathematics Subject Classification. 17B56, 14E08.
+Key words and phrases. Bogomolov multiplier, Lie algebras, Second cohomology group.
+1
+
+2
+P. K. RAI
+for all α ∈ Ω, m, m′, n, n′ ∈ L.
+It is easy to see that K : L × L �→ [L, L] given by (m, n) �→ [m, n] induces a
+homomorphism ¯K : L∧L �→ [L, L], such that ¯K(m ∧ n) = [m, n], for all m, n ∈ L.
+It is known that the kernel of ¯K is isomorphic to the Schur Multiplier H2(L, Ω)
+(defined below). We denote it by M(L). Define M0(L) to be the group ⟨m ∧ n |
+m, n ∈ L, [m, n] = 0⟩. The factor group
+M(L)
+M0(L) is defined to be the Bogomolov
+multiplier B(L) of the Lie algebra L.
+In this article, we define a cohomological object B0(L) for a finite dimensional
+Lie algebra L over a field Ω, and show that it is isomorphic to the Bogomolov
+multiplier B(L). Before that, we recall the definition of the Schur multiplier of a
+finite dimensional Lie algebra. Let L be a finite dimensional Lie algebra and A be
+a trivial L-module. A map f : L × L �→ A said to be a 2-cocycle if it is bilinear,
+alternating and satisfies
+f([x1, x2], x3) + f([x2, x3], x1) + f([x3, x1], x2) = 0.
+And f is said to be a 2-coboundry if there exists a linear σ : L �→ A such that
+f(x1, x2) = −σ([x1, x2]).
+The sets of 2-cocycles and 2-coboundries are denoted by Z2(L, A) and B2(L, A),
+respectively and form abelian groups with respect to usual addition. The group
+Z2(L, A)/B2(L, A) is said to be the second cohomology group with coefficients in
+A, and is denoted by H2(L, A).
+Schur multiplier of the Lie algebra L is defined as the abelian Lie algebra
+H2(L, Ω), considering Ω as a central L-module. We are now ready to define B0(L).
+Definition. For a finite dimensional Lie algebra L over Ω, we define B0(L) as
+follows:
+B0(L) = {f ∈ H2(L, Ω) | f(x1, x2) = 0 whenever [x1, x2] = 0}.
+Batten [1, Theorem 3.6] established the following Hopf Formula for the Schur
+multiplier of the Lie algebra L:
+H2(L, Ω) ∼= F ′ ∩ R
+[F, R] ,
+where 1 �→ R �→ F �→ L �→ 1 is a free a presentation of L and F ′ is the derived
+subalgebra of F.
+Let K(L) denote the set {[x, y] | x, y ∈ L}. In the following theorem we derive
+a Hopf-type formula for B0(L).
+Theorem 1.1. Let L be a finite dimensional Lie algebra with a free presentation
+L ∼= F/R. Then B0(L) ∼=
+F ′∩R
+⟨K(F )∩R⟩.
+The following corollary follows from [6, Proposition 4.1] and the Theorem 1.1.
+Corollary 1.2. Let L be a finite dimensional Lie algebra. Then B(L) ∼= B0(L).
+As an application we answer a couple of questions of Kunyavski˘ı [4]. He asked
+the following questions for finite dimensional Lie algebras L:
+Question 1.3. [4, Question 7.1] Can the dimension of B(L) be as large as possible?
+
+BOGOMOLOV MULTIPLIER OF LIE ALGEBRAS
+3
+Question 1.4. [4, Question 7.2] Is B(L) invariant under isoclinism of Lie algebras?
+Two Lie algebras L and K are said to be isoclinic if there exist isomorphisms α :
+L/Z(L) �→ K/Z(K) and β : L′ �→ K′ such that the following diagram commutes.
+L
+Z(L) ×
+L
+Z(L)
+φ
+−−−−→ L′
+α×α
+�
+β
+�
+K
+Z(K) ×
+K
+Z(K)
+θ
+−−−−→ K′
+where
+φ
+�
+l1 + Z(L), l2 + Z(L)
+�
+= [l1, l2] for l1, l2 ∈ L
+and
+θ
+�
+k1 + Z(K), k2 + Z(K)
+�
+= [k1, k2] for k1, k2 ∈ K.
+The pair (α, β) is called an isoclinism between L and K.
+In the following theorems we give an affirmative answer to Questions 1.3 and
+1.4.
+Theorem 1.5. Let n ≥ 1 be a natural number. There exists a finite dimensional
+nilpotent Lie algebra L of nilpotency class 2 such that dimension of B(L) is greater
+than or equal to n.
+Theorem 1.6. Let L and M be two isoclinic finite dimensional Lie Algebras over
+the field Ω. Then B(L) ∼= B(M).
+2. Hopf-Type Formula
+Consider a finite dimensional Lie algebra L over a field Ω, a central ideal H of
+L, and a trivial L-module A. The restriction map Res : Hom(L, A) → Hom(H, A)
+is defined as follows: for a homomorphism f : L → A, Res(f) is the restriction of f
+to H. There is also an inflation map Inf : Hom(L/H, A) → Hom(L, A) defined by
+sending a homomorphism α ∈ Hom(L/H, A) to the homomorphism α′ ∈ Hom(L, A)
+defined by α′(x, y) = α(β(x), β(y)) for all x, y ∈ L, where β : L → L/H is
+the natural group homomorphism. Another inflation map Inf : H2(L/H, A) →
+H2(L, A) is defined similarly by sending [α] ∈ H2(L/H, A) to [α′] ∈ H2(L, A)
+where α′(x, y) = α(β(x), β(y)) for all x, y ∈ L. Next, we define a transgression map
+Tra : Hom(H, A) → H2(L/H, A) as follows: for a fixed section µ of β, define a map
+f : L/H ×L/H → L by f(x, y) = [µ(x), µ(y)]−µ([x, y]) for all x, y ∈ L/H. Given a
+homomorphism χ ∈ Hom(H, A), we can verify that χf ∈ Z2(L/H, A) and that the
+cohomology class of χf does not depend on the choice of µ. The transgression map
+Tra is then defined as the map that sends a homomorphism χ to the cohomology
+class of χf
+We are now ready to quote some results required for our subsequent investiga-
+tions. The following 5-term exact sequence was established by P. Batten in her
+Ph.D. Thesis [1, Theorem 3.1]
+Theorem 2.1. Let L be a Lie algebra, H be central ideal of L, 1 �→ H �→ L �→
+L/H �→ 1 be the natural exact sequence and A be a trivial L-module. Then the
+
+4
+P. K. RAI
+induced sequence
+(2.1)
+1 −→ Hom(L/H, A)
+Inf
+−−→ Hom(L, A)
+Res
+−−→ Hom(H, A)
+Tra
+−−→ H2(L/H, A)
+Inf
+−−→ H2(L, A)
+is exact.
+Let L be a lie algebra and T be a subset of L. By ⟨T ⟩ we denote the subspace
+of L generated by T and by HomT (L, A) we denote the set of those homomor-
+phisms which maps T to 0. The following lemma is instrumental in our subsequent
+investigations.
+Lemma 2.2. Let L be a finite dimensional Lie algebra over the field Ω and H be a
+central ideal of L. Then Tra(λ) ∈ B0(L/H) if, and only if λ ∈ HomT (H, Ω) where
+T = ⟨K(L) ∩ H⟩.
+Proof. Let µ : L/H → L be a section such that µ(0) = 0 and let Tra be defined
+using µ as Tra(λ) = [λf], where
+f(x, y) = [µ(x), µ(y)] − µ([x, y]) ∀ x, y ∈ L/H.
+Let λ ∈ HomT (H, Ω) and x, y ∈ L/H be such that [x, y] = 0. Then λf(x, y) =
+λ([µ(x), µ(y)]). But [µ(x), µ(y)] ∈ T. Therefore λf(x, y) = 0.
+This proves that
+[λf] ∈ B0(L/H). Thus Tra(λ) ∈ B0(L/H).
+Conversly, suppose that Tra(λ) ∈
+B0(L/H). Let l1, l2 ∈ L such that [l1, l2] ∈ H. Notice that [l1, l2] = [µ(l1+H), µ(l2+
+H)] because H is central. Also µ([l1 + H, l2 + H]) = µ([l1, l2] + H) = 0 because
+[l1, l2] ∈ H and µ(0) = 0. Therefore,
+λ([l1, l2] = λ([µ(l1 + H), µ(l2 + H)]) − λµ([l1 + H, l2 + H]) = λf(l1 + H, l2 + H).
+Since Tra(λ) = [λf] ∈ B0(L/H) and [l1 + H, l2 + H] = 0 in L/H we have that
+λf(l1 + H, l2 + H) = 0. Thus λ([l1, l2]) = 0 whenever [l1, l2] ∈ H. This proves that
+λ ∈ HomT (H, Ω).
+□
+Theorem 2.3. Let L be a finite dimensional Lie algebra over the field Ω, H be
+central ideal of L, and T = ⟨K(L) ∩ H⟩. Then the induced sequence
+(2.2)
+1 −→ Hom(L/H, Ω)
+Inf
+−−→ Hom(L, Ω)
+Res
+−−→ HomT (H, Ω)
+tra
+−−→ B0(L/H)
+inff
+−−→ B0(L)
+is exact, where tra and inff are the restrictions of Tra and Inf.
+Proof. The theorem follows from the exactness of the Sequence 2.1, Lemma 2.2 and
+the following straight forward observations:
+(i) Res(Hom(L, Ω) ≤ HomT (H, Ω).
+(ii) Inf(B0(L/H) ≤ B0(L).
+□
+The next theorem follows from Lemma 2.2 and [1, Corollary 3.7].
+Theorem 2.4. Let L be a finite dimensional Lie algebra, L∗ be its cover with
+A ≤ L∗ satisfying the following three conditions
+(1) A ≤ Z(L∗) ∩ [L∗, L∗];
+
+BOGOMOLOV MULTIPLIER OF LIE ALGEBRAS
+5
+(2) A ∼= M(L);
+(3) L ∼= L∗/A,
+and T = ⟨K(L∗) ∩ A⟩. Then the map tra : HomT (A, Ω) �→ B0(L) is bijective.
+Corollary 2.5. Let L be a finite dimensional Lie algebra and L∗ be its cover with
+A ≤ L∗ satisfying the following three conditions
+(1) A ≤ Z(L∗) ∩ [L∗, L∗];
+(2) A ∼= M(L);
+(3) L ∼= L∗/A.
+Then B0(L) ∼=
+A
+⟨K(L∗)∩A⟩.
+Proof. Let T = ⟨K(L∗)∩A⟩. Since A is an abelian lie algebra, homomorphisms are
+nothing but linear transformations. It follows that HomT (A, Ω) ∼= Hom
+�
+A
+T , Ω
+�
+∼=
+A
+T . The result follows from Theorem 2.4.
+□
+Theorem 2.6. Let H be a central ideal in L and T = ⟨K(L)∩H⟩. Then
+L′∩H
+⟨K(L)∩H⟩
+is isomorphic to the image of the map tra : HomT (H, Ω) �→ B0(L/H). In particular,
+L′∩H
+⟨K(L)∩H⟩
+∼= B0(L/H) if the map tra is surjective.
+Proof. By Theorem 2.3, we have the following exact sequence
+Hom(L, Ω)
+Res
+−−→ HomT (H, Ω)
+tra
+−−→ B0(G/H).
+It follows that
+HomT (H,Ω)
+Res(Hom(L,Ω)) is isomorphic to the image of tra. Thus, to prove the
+theorem, we only need to show that
+L′ ∩ H
+⟨K(L) ∩ H⟩
+∼=
+HomT (H, Ω)
+Res(Hom(L, Ω)).
+Since H is abelian it follows that the natural restriction map Res1 : HomT (H, Ω) →
+HomT (L′ ∩ Z, Ω) is surjetive. Therefore
+HomT (H, Ω)
+ker Res1
+∼= HomT (L′ ∩ H, Ω).
+If we consider the natural restriction map Res2 : Hom(H, Ω) �→ Hom(L′ ∩ H, Ω),
+it is straight forward to note that ker Res1 = ker Res2 . Let J be the subset of
+Hom(H, Ω) consisting of all χ which can be extended to a homomorphism L → Ω.
+Invoking the proof of [1, Theorem 3.2] we have J = ker Res2 . But it is obvious that
+J = Res(Hom(L, Ω)). Hence, it follows that
+HomT (H, Ω)
+Res(Hom(L, Ω))
+∼= HomT (L′ ∩ H, Ω).
+But
+HomT (L′ ∩ H, Ω) ∼= Hom
+�L′ ∩ H
+T
+, Ω
+�
+∼= L′ ∩ H
+T
+because L′ ∩ H is abelian. This completes the proof.
+□
+Proof of Theorem 1.1: Let ¯R =
+R
+[F,R] and ¯F =
+F
+[F,R].
+Then ¯R is a cen-
+tral ideal of ¯F and L ∼=
+¯
+F
+¯
+R.
+By [1, Lemma 3.4] the transgression map Tra :
+Hom( ¯R, Ω) �→ H2(L, Ω) is surjective.
+Therefore by Lemma 2.2 the map tra :
+Hom⟨K( ¯
+F )∩ ¯R⟩( ¯R, Ω) �→ B0(L) is also surjective. It therefore follows, from Theorem
+
+6
+P. K. RAI
+2.6, that B0(L) ∼=
+¯
+R∩[ ¯
+F, ¯
+F ]
+⟨K( ¯
+F )∩ ¯R⟩. But ¯R ∩ [ ¯F , ¯F] = F ′∩R
+[F,R] and ⟨K( ¯F) ∩ ¯R⟩ = ⟨K(F )∩R⟩
+[F,R]
+.
+Hence B0(L) ∼=
+F ′∩R
+⟨K(F )∩R⟩.
+3. Applications
+The proof of the following proposition is exactly the same as the proof of [6,
+Proposition 4.3].
+Theorem 3.1. Let L be a Lie algebra with a free presentation L ∼= F/R, and M
+be an ideal of L, such that T = ker(F �→ L/M). Then the sequence
+0 �→ R ∩ ⟨K(F) ∩ T ⟩
+⟨K(F) ∩ R⟩
+�→ B0(L) �→ B0(L/M) �→
+M ∩ L′
+⟨K(L) ∩ M⟩ �→ 0,
+is exact.
+Definition. A Lie algebra L is called generalized Heisenberg of rank n if L′ = Z(L)
+and dim L′ = n.
+A freest generalized Heisenberg Lie algebra is a d-generated (minimally generated
+by d elements) generalized Heisenberg Lie algebra of rank 1
+2d(d−1) for some d ≥ 2.
+We shall denote it by Ld. Notice that Ld has the following presentation:
+⟨x1, . . . , xd, yij | [xi, xj] = yij, 1 ≤ i < j ≤ d, class 2⟩,
+and dim Ld = 1
+2d(d + 1).
+Theorem 3.2. Let Ld be the freest generalized Heisenberg Lie algebra of rank
+1
+2d(d − 1). Then B0(Ld) = 0.
+Proof. Let f : Ld × Ld �→ Ω be a 2-cocycle such that ¯f ∈ B0(Ld).
+Let B =
+{x1, . . . , xd, [xi, xj] | 1 ≤ i < j ≤ d}. Define a map µ : B �→ Ω as follows:
+µ(xi) = 0 for 1 ≤ i ≤ d,
+µ([xi, xj]) = −f(xi, xj) for 1 ≤ i < j ≤ d.
+Since B is basis for Ld we can extend this map linearly to Ld and call it σ so that
+σ is a linear transformation. Let x, y ∈ Ld. Since B is a basis we can write x and
+y as
+x =
+d
+�
+i=1
+αixi +
+�
+1≤i<j≤d
+αij[xi, xj],
+y =
+d
+�
+k=1
+βkxk +
+�
+1≤k<l≤d
+βkl[xk, xl],
+for some αi, βi, αij, βij ∈ Ω. Now using the bilinearity of f and the fact that
+f(a, b) = 0 whenever [a, b] = 0, we get that
+f(x, y) =
+d
+�
+i=1
+d
+�
+k=1
+αiβkf(xi, xk).
+
+BOGOMOLOV MULTIPLIER OF LIE ALGEBRAS
+7
+Also, using the bilinearity of Lie bracket, linearity of σ and the fact that f(l1, [l2, l3]) =
+0 for all l1, l2, l3 ∈ Ld, we get that
+σ([x, y]) = −
+d
+�
+i=1
+d
+�
+k=1
+αiβkf(xi, xk).
+Therefore f(x, y) = −σ([x, y]). Thus f is a coboundry and ¯f = 0. This completes
+the proof.
+□
+Rostami et al. [6] proved that the Bogomolov multiplier of a Heisenberg Lie
+algebra is trivial. We prove this fact as a Corollary of Theorem 3.2.
+A Heisenberg Lie algebra of dimension 2n + 1, n > 0, is given by the following
+presentation:
+⟨x1, . . . , x2n, v | [x2i−1, x2i] = v, [xj, xk] = 0, 1 ≤ i ≤ n, (j, k) ̸= (2i − 1, 2i)⟩.
+Corollary 3.3. Let H2n+1 be the Heisenberg Lie algebra of dimension 2n+1. Then
+B0(H2n+1) = 0.
+Proof. Let L2n be the generalized Heisenberg Lie algebra of rank n(2n − 1) and M
+be its ideal generated by [x2r−1, x2r] − [x2s−1, x2s], 1 ≤ r < s ≤ n and [xt, xu], 1 ≤
+t < u ≤ 2n, (t, u) ̸= (2i − 1, 2i) for any i ≤ n. Then it is easy to see that L2n/M is
+isomorphic to H2n+1. Since B0(L2n) = 0 by Theorem 3.2, it follows from Theorem
+3.1 that
+B0(L2n/M) ∼=
+M ∩ L′
+2n
+⟨K(L2n) ∩ M⟩.
+Since for 1 ≤ r < s ≤ n,
+[x2r−1 − x2s−1, x2r + x2s] = [x2r−1, x2r] − [x2s−1, x2s] + [x2r, x2s−1] + [x2r−1, x2s],
+it follows that M∩L′
+2n = ⟨K(L2n)∩M⟩. Thus B0(L2n/M) = 0 so that B0(H2n+1) =
+0.
+□
+We now proceed to prove Theorem 1.5.
+Proof of Theorem 1.5: Let d be a natural number greater than 4n and Ld be
+the d-generated freest generalized Heisenberg Lie algebra generated by x1, x2, . . . xd.
+Let M be the ideal generated by [x1, x2]+[x3, x4], [x5, x6]+[x7, x8], . . . [x4n−3, x4n−2]+
+[x4n−1, x4n]. Since B0(Ld) = 0, it follows from Theorem 3.1 that
+dim B0(Ld/M) = dim
+L′
+d ∩ M
+⟨K(Ld) ∩ M⟩.
+Next we prove that K(Ld) ∩ M = {0}. For this let l ∈ K(Ld) ∩ M. Since l ∈ M
+there exists γ′
+ks such that
+l =
+n−1
+�
+k=0
+γk+1
+�
+[x4k+1, x4k+2] + [x4k+3, x4k+4]
+�
+.
+
+8
+P. K. RAI
+Also there exist αi’s and βj’s such that
+l =
+�
+d
+�
+i=1
+αixi,
+d
+�
+j=1
+βjxj
+�
+because l ∈ K(Ld). Hence
+n−1
+�
+k=0
+γk+1
+�
+[x4k+1, x4k+2] + [x4k+3, x4k+4]
+�
+=
+d−1
+�
+i=1
+d
+�
+j=i+1
+(αiβj − αjβi)[xi, xj].
+It follows that
+(3.1)
+αiβj − αjβi = 0 when (i, j) ̸= (2k + 1, 2k + 2) for any k = 0, 1, . . . n − 1,
+and
+γk+1 = α4k+1β4k+2 − α4k+2β4k+1 = α4k+3β4k+4 − α4k+4β4k+3
+for any k = 0, 1, . . . n − 1. From Equation 3.1 we have
+(3.2)
+α4k+1β4k+3 − α4k+3β4k+1 = 0
+(3.3)
+α4k+1β4k+4 − α4k+4β4k+1 = 0
+(3.4)
+α4k+2β4k+3 − α4k+3β4k+2 = 0
+(3.5)
+α4k+2β4k+4 − α4k+4β4k+2 = 0,
+for k = 0, 1, . . .n − 1.
+Suppose α4k+1 = β4k+1 = 0. Then γk+1 = 0. Assume then that α4k+1 = 0 but
+β4k+1 ̸= 0. From Equations 3.2 and 3.3, α4k+3 = α4k+1 = 0. As a result, γk+1 = 0
+in this case as well. Thus we have shown that if α4k+1 = 0, then γk+1 = 0. Similarly,
+if either of α4k+2, α4k+3, α4k+4, β4k+1, β4k+2, β4k+3, β4k+4 is zero, then γk+1 = 0.
+Hence, we can now assume that neither of α4k+i, β4k+i is zero for i = 1, 2, 3, 4.
+From Equations 3.2 and 3.3 we can deduce that β4k+3/β4k+4 = α4k+3/α4k+4.
+Hence γk+1 = 0 so that l = 0. It follows that K(Ld) ∩ M = {0}.
+Also, M ≤ L′
+d.
+Hence dim B0(Ld/M) = dim(M) = n. By Corollary 1.2
+dim B(Ld/M) = n. Taking L to be Ld/M completes the proof.
+Proof of Theorem 1.6: Let (θ, φ) be an isoclinism between L and M, i.e., θ :
+L
+Z(L) �→
+M
+Z(M) and φ : γ2(L) �→ γ2(M) be isomorphisms and whenever θ(liZ(L)) =
+miZ(M) for i = 1, 2, we have φ([l1, l2]) = [m1, m2]. Let ¯f ∈ B0(L) where f :
+L × L �→ Ω be a cocycle. Define cf : M × M �→ Ω by cf(m1, m2) = f(l1, l2), where
+l1 and l2 are given by θ−1(mi + Z(M)) = li + Z(L) for i = 1, 2. The rest of the
+proof follows from the following lemmas:
+Lemma 3.4. Let cf be the map defined above. Then
+(i) cf is well defined.
+(ii) cf is a 2 cocycle.
+(iii) cf ∈ B0(M).
+
+BOGOMOLOV MULTIPLIER OF LIE ALGEBRAS
+9
+Proof. Since f is bilinear and f(k, l) = 0 whenever [k, l] = 0.
+It follows that
+f(l1 + z1, l2 + z2) = f(l1, l2) for every z1, z2 ∈ Z(L). This shows that cf is well-
+defined.
+To see that cf is a 2-cocycle, let m1, m2, m3 ∈ M and let θ−1(mi + Z(M)) =
+li + Z(L) for i = 1, 2, 3. It is obvious that θ−1(m1 + m2 + Z(M)) = l1 + l2 + Z(L).
+Therefore cf(m1 + m2, m3) = f(l1 + l2, l3) which equals f(l1, l3) + f(l2, l3) that
+is equal to cf(m1, m3) + cf(m2, m3). Similarly cf(m1, m2 + m3) = cf(m1, m2) +
+cf(m1, m3). Thus cf is bilinear. Also, it is easy to see that cf is alternating because
+f is alternating. Next, For i, j, k ∈ {1, 2, 3} note that cf([mi, mj], mk) = f([li, lj], lk)
+because
+θ−1�
+[mi, mj] + Z(M)
+�
+= θ−1��
+mi + Z(M), mj + Z(M)
+��
+=
+�
+θ−1�
+mi + Z(M)
+�
+, θ−1�
+mj + Z(M)
+��
+=
+�
+li + Z(L), lj + Z(L)
+�
+= [li, lj] + Z(L).
+t follows that cf is a 2-cocycle, since f is a 2-cocycle.
+To see that cf ∈ B0(M), suppose that [m1, m2] = 0.
+But then [l1, l2] = 0
+because φ([l1, l2]) = [m1, m2]. Since f ∈ B0(L) it follows that f(l1, l2) = 0. Hence
+cf(m1, m2) = 0.
+Lemma 3.5. The map η : B0(L) �→ B0(M) defined by η(f) = cf is an isomor-
+phism.
+Proof. We begin by ensuring that the map is well-defined. To verify this consider
+σ : L × L �→ Ω to be a coboundary. Then
+cf+σ(m1, m2) = (f + σ)(l1, l2) = f(l1, l2) + σ(l1, l2) = cf(m1, m2) + cσ(m1, m2).
+Thus we have, cf+σ = cf +cσ. Notice that cσ is a coboundary because σ is cobound-
+ary. Therefore cf = cf+σ, i.e., η(f) = η(f + σ). This proves that η is well-defined.
+In a similar fashion one can see that cf1+f2 = cf1 + cf2 and cαf1 = αcf1 for each
+α ∈ Ω and each cocycles f1, f2 from L × L to Ω. So that η(f1 + f2) = η(f1) + η(f2)
+and η(αf1) = αη(f1). Thus η is a linear map.
+Finally, in order to see that η is a bijection, we define another map χ : B0(M) �→
+B0(L) in the same way as η is defined from B0(L) to B0(M). Then it is easy to see
+that ηχ and χη both are identity maps and thus η is a bijection. This completes
+the proof.
+□
+□
+References
+[1] P. Batten, Covers and multipliers of Lie algebras, Dissertation, North Carolina State Uni-
+versity, 1993. 2, 3, 4, 5
+[2] F. A. Bogomolov, The Brauer group of quotient spaces of linear representations, Izv. Akad.
+Nauk SSSR, Ser. Mat. 51 (1987), no. 3, article no. 688. 1
+[3] B. Kunyavski˘ı, The Bogomolov multiplier of finite simple groups, Cohomological and geo-
+metric approaches to rationality problems, 209–217, Progr. Math., 282, Birkh¨auser Boston,
+Inc., Boston, MA, 2010. 1
+
+10
+P. K. RAI
+[4] B. Kunyavski˘ı, Some New Parallels Between Groups and Lie Algebras, or What Can Be
+Simpler than Multiplication Table?, EMS Newsl. 118 (2020), 5–13. 2, 3
+[5] P. Moravec, Unramified Brauer groups of finite and infinite groups, Am. J. Math. 134 (2012),
+no. 6, 1679-1704. 1
+[6] Z. A. Rostami, M. Parvizi, P. Niroomand, The Bogomolov multiplier of Lie algebras, Hacet.
+J. Math. Stat. 49 (2020), 1190- 1205. 1, 2, 6, 7
+[7] D. J. Saltman, Noether’s problem over an algebraically closed field, Invent. Math. 77 (1984),
+no. 1, 71-84. 1
+(Pradeep K. Rai) Mahindra University, Hyderabad, Telangana,, India
+Email address: raipradeepiitb@gmail.com
+

CtA0T4oBgHgl3EQfAf94/content/tmp_files/load_file.txt

@@ -0,0 +1,391 @@
+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf,len=390
+page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='01963v1  [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='RA]  5 Jan 2023  BOGOMOLOV MULTIPLIER OF LIE ALGEBRAS  PRADEEP K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' RAI  Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' In the work of Rostami et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=', the Bogomolov multiplier of a Lie  algebra L over a field Ω is defined as a particular factor of a subalgebra of the  exterior product L ∧ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' If L is finite dimensional, we identify this object as  a certain subgroup of the second cohomology group H2(L, Ω) by deriving a  Hopf-Type formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' As an application, we affirmatively answer two questions  posed by Kunyavski˘ı regarding the invariance of the Bogomolov multiplier  under isoclinism of Lie algebras and the existence of a family of Lie algebras  with Bogomolov multipliers of unbounded dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Introduction  The Bogomolov multiplier of a finite group G is a cohomological invariant of G  that has been studied in connection with Noether’s problem, which asks whether  the fixed subfield k(G) of the function field k(xg : g ∈ G) is purely transcendental  over an algebraically closed field k of characteristic zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' The Bogomolov multiplier  has played a key role in the discovery of counter examples to Noether’s problem over  the complex numbers C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Saltman [7] was the first to provide such counter examples,  showing that if the unramified cohomology group H2  nr(k(G), C) is nonzero, then G  has a negative solution to Noether’s problem over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Bogomolov later showed that  H2  nr(k(G), C) is naturally isomorphic to the subgroup B0(G) of the second coho-  mology group H2(G, C) consisting of classes that vanish when restricted to the  abelian subgroups of G [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' This has led to the discovery of numerous other counter  examples to Noether’s problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Kunyavski˘ı later gave the name “Bogomolov mul-  tiplier” to B0(G) [3], and Moravec provided a homological description of B0(G) as  a quotient of H2(G, C) [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Following Moravec’s construction, Rostami et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' al [6]  extended the notion of Bogomolov multiplier to Lie algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' For the convenience  of the reader we define it here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  Let L be a Lie algebra over Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' The exterior square of L is defined to be the  Lie algebra L∧L generated by the symbols m ∧ n, where m, n ∈ L, subject to the  following relations:  (i) α(m ∧ n) = αm ∧ n = m ∧ αn,  (ii) (m + m′) ∧ n = m ∧ n + m′ ∧ n,  (iii) m ∧ (n + n′) = m ∧ n + m ∧ n′,  (iv) [m, m′] ∧ n = m ∧ [m′, n] − m′ ∧ [m, n],  (v) m ∧ [n, n���] = [n′, m] ∧ n − [n, m] ∧ n′,  (vi) [(m ∧ n), (m′ ∧ n′)] = −[n, m] ∧ [m′, n′],  (vii) m ∧ n = 0 whenever m = n,  2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' 17B56, 14E08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Bogomolov multiplier, Lie algebras, Second cohomology group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  1  2  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' RAI  for all α ∈ Ω, m, m′, n, n′ ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  It is easy to see that K : L × L �→ [L, L] given by (m, n) �→ [m, n] induces a  homomorphism ¯K : L∧L �→ [L, L], such that ¯K(m ∧ n) = [m, n], for all m, n ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  It is known that the kernel of ¯K is isomorphic to the Schur Multiplier H2(L, Ω)  (defined below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' We denote it by M(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Define M0(L) to be the group ⟨m ∧ n |  m, n ∈ L, [m, n] = 0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' The factor group  M(L)  M0(L) is defined to be the Bogomolov  multiplier B(L) of the Lie algebra L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  In this article, we define a cohomological object B0(L) for a finite dimensional  Lie algebra L over a field Ω, and show that it is isomorphic to the Bogomolov  multiplier B(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Before that, we recall the definition of the Schur multiplier of a  finite dimensional Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Let L be a finite dimensional Lie algebra and A be  a trivial L-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' A map f : L × L �→ A said to be a 2-cocycle if it is bilinear,  alternating and satisfies  f([x1, x2], x3) + f([x2, x3], x1) + f([x3, x1], x2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  And f is said to be a 2-coboundry if there exists a linear σ : L �→ A such that  f(x1, x2) = −σ([x1, x2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  The sets of 2-cocycles and 2-coboundries are denoted by Z2(L, A) and B2(L, A),  respectively and form abelian groups with respect to usual addition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' The group  Z2(L, A)/B2(L, A) is said to be the second cohomology group with coefficients in  A, and is denoted by H2(L, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  Schur multiplier of the Lie algebra L is defined as the abelian Lie algebra  H2(L, Ω), considering Ω as a central L-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' We are now ready to define B0(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' For a finite dimensional Lie algebra L over Ω, we define B0(L) as  follows:  B0(L) = {f ∈ H2(L, Ω) | f(x1, x2) = 0 whenever [x1, x2] = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  Batten [1, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='6] established the following Hopf Formula for the Schur  multiplier of the Lie algebra L:  H2(L, Ω) ∼= F ′ ∩ R  [F, R] ,  where 1 �→ R �→ F �→ L �→ 1 is a free a presentation of L and F ′ is the derived  subalgebra of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  Let K(L) denote the set {[x, y] | x, y ∈ L}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' In the following theorem we derive  a Hopf-type formula for B0(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Let L be a finite dimensional Lie algebra with a free presentation  L ∼= F/R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Then B0(L) ∼=  F ′∩R  ⟨K(F )∩R⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  The following corollary follows from [6, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='1] and the Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Let L be a finite dimensional Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Then B(L) ∼= B0(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  As an application we answer a couple of questions of Kunyavski˘ı [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' He asked  the following questions for finite dimensional Lie algebras L:  Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' [4, Question 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='1] Can the dimension of B(L) be as large as possible?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  BOGOMOLOV MULTIPLIER OF LIE ALGEBRAS  3  Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' [4, Question 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='2] Is B(L) invariant under isoclinism of Lie algebras?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  Two Lie algebras L and K are said to be isoclinic if there exist isomorphisms α :  L/Z(L) �→ K/Z(K) and β : L′ �→ K′ such that the following diagram commutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  L  Z(L) ×  L  Z(L)  φ  −−−−→ L′  α×α  \uf8e6\uf8e6�  β  \uf8e6\uf8e6�  K  Z(K) ×  K  Z(K)  θ  −−−−→ K′  where  φ  �  l1 + Z(L), l2 + Z(L)  �  = [l1, l2] for l1, l2 ∈ L  and  θ  �  k1 + Z(K), k2 + Z(K)  �  = [k1, k2] for k1, k2 ∈ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  The pair (α, β) is called an isoclinism between L and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  In the following theorems we give an affirmative answer to Questions 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='3 and  1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Let n ≥ 1 be a natural number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' There exists a finite dimensional  nilpotent Lie algebra L of nilpotency class 2 such that dimension of B(L) is greater  than or equal to n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Let L and M be two isoclinic finite dimensional Lie Algebras over  the field Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Then B(L) ∼= B(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Hopf-Type Formula  Consider a finite dimensional Lie algebra L over a field Ω, a central ideal H of  L, and a trivial L-module A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' The restriction map Res : Hom(L, A) → Hom(H, A)  is defined as follows: for a homomorphism f : L → A, Res(f) is the restriction of f  to H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' There is also an inflation map Inf : Hom(L/H, A) → Hom(L, A) defined by  sending a homomorphism α ∈ Hom(L/H, A) to the homomorphism α′ ∈ Hom(L, A)  defined by α′(x, y) = α(β(x), β(y)) for all x, y ∈ L, where β : L → L/H is  the natural group homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Another inflation map Inf : H2(L/H, A) →  H2(L, A) is defined similarly by sending [α] ∈ H2(L/H, A) to [α′] ∈ H2(L, A)  where α′(x, y) = α(β(x), β(y)) for all x, y ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Next, we define a transgression map  Tra : Hom(H, A) → H2(L/H, A) as follows: for a fixed section µ of β, define a map  f : L/H ×L/H → L by f(x, y) = [µ(x), µ(y)]−µ([x, y]) for all x, y ∈ L/H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Given a  homomorphism χ ∈ Hom(H, A), we can verify that χf ∈ Z2(L/H, A) and that the  cohomology class of χf does not depend on the choice of µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' The transgression map  Tra is then defined as the map that sends a homomorphism χ to the cohomology  class of χf  We are now ready to quote some results required for our subsequent investiga-  tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' The following 5-term exact sequence was established by P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Batten in her  Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Thesis [1, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='1]  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Let L be a Lie algebra, H be central ideal of L, 1 �→ H �→ L �→  L/H �→ 1 be the natural exact sequence and A be a trivial L-module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Then the  4  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' RAI  induced sequence  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='1)  1 −→ Hom(L/H, A)  Inf  −−→ Hom(L, A)  Res  −−→ Hom(H, A)  Tra  −−→ H2(L/H, A)  Inf  −−→ H2(L, A)  is exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  Let L be a lie algebra and T be a subset of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' By ⟨T ⟩ we denote the subspace  of L generated by T and by HomT (L, A) we denote the set of those homomor-  phisms which maps T to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' The following lemma is instrumental in our subsequent  investigations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Let L be a finite dimensional Lie algebra over the field Ω and H be a  central ideal of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Then Tra(λ) ∈ B0(L/H) if, and only if λ ∈ HomT (H, Ω) where  T = ⟨K(L) ∩ H⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Let µ : L/H → L be a section such that µ(0) = 0 and let Tra be defined  using µ as Tra(λ) = [λf], where  f(x, y) = [µ(x), µ(y)] − µ([x, y]) ∀ x, y ∈ L/H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  Let λ ∈ HomT (H, Ω) and x, y ∈ L/H be such that [x, y] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Then λf(x, y) =  λ([µ(x), µ(y)]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' But [µ(x), µ(y)] ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Therefore λf(x, y) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  This proves that  [λf] ∈ B0(L/H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Thus Tra(λ) ∈ B0(L/H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  Conversly, suppose that Tra(λ) ∈  B0(L/H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Let l1, l2 ∈ L such that [l1, l2] ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Notice that [l1, l2] = [µ(l1+H), µ(l2+  H)] because H is central.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Also µ([l1 + H, l2 + H]) = µ([l1, l2] + H) = 0 because  [l1, l2] ∈ H and µ(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Therefore,  λ([l1, l2] = λ([µ(l1 + H), µ(l2 + H)]) − λµ([l1 + H, l2 + H]) = λf(l1 + H, l2 + H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  Since Tra(λ) = [λf] ∈ B0(L/H) and [l1 + H, l2 + H] = 0 in L/H we have that  λf(l1 + H, l2 + H) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Thus λ([l1, l2]) = 0 whenever [l1, l2] ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' This proves that  λ ∈ HomT (H, Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  □  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Let L be a finite dimensional Lie algebra over the field Ω, H be  central ideal of L, and T = ⟨K(L) ∩ H⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Then the induced sequence  (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='2)  1 −→ Hom(L/H, Ω)  Inf  −−→ Hom(L, Ω)  Res  −−→ HomT (H, Ω)  tra  −−→ B0(L/H)  inff  −−→ B0(L)  is exact, where tra and inff are the restrictions of Tra and Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' The theorem follows from the exactness of the Sequence 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='1, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='2 and  the following straight forward observations:  (i) Res(Hom(L, Ω) ≤ HomT (H, Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  (ii) Inf(B0(L/H) ≤ B0(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  □  The next theorem follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='2 and [1, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Let L be a finite dimensional Lie algebra, L∗ be its cover with  A ≤ L∗ satisfying the following three conditions  (1) A ≤ Z(L∗) ∩ [L∗, L∗];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  BOGOMOLOV MULTIPLIER OF LIE ALGEBRAS  5  (2) A ∼= M(L);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  (3) L ∼= L∗/A,  and T = ⟨K(L∗) ∩ A⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Then the map tra : HomT (A, Ω) �→ B0(L) is bijective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Let L be a finite dimensional Lie algebra and L∗ be its cover with  A ≤ L∗ satisfying the following three conditions  (1) A ≤ Z(L∗) ∩ [L∗, L∗];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  (2) A ∼= M(L);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  (3) L ∼= L∗/A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  Then B0(L) ∼=  A  ⟨K(L∗)∩A⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Let T = ⟨K(L∗)∩A⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Since A is an abelian lie algebra, homomorphisms are  nothing but linear transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' It follows that HomT (A, Ω) ∼= Hom  �  A  T , Ω  �  ∼=  A  T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' The result follows from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  □  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Let H be a central ideal in L and T = ⟨K(L)∩H⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Then  L′∩H  ⟨K(L)∩H⟩  is isomorphic to the image of the map tra : HomT (H, Ω) �→ B0(L/H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' In particular,  L′∩H  ⟨K(L)∩H⟩  ∼= B0(L/H) if the map tra is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='3, we have the following exact sequence  Hom(L, Ω)  Res  −−→ HomT (H, Ω)  tra  −−→ B0(G/H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  It follows that  HomT (H,Ω)  Res(Hom(L,Ω)) is isomorphic to the image of tra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Thus, to prove the  theorem, we only need to show that  L′ ∩ H  ⟨K(L) ∩ H⟩  ∼=  HomT (H, Ω)  Res(Hom(L, Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  Since H is abelian it follows that the natural restriction map Res1 : HomT (H, Ω) →  HomT (L′ ∩ Z, Ω) is surjetive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Therefore  HomT (H, Ω)  ker Res1  ∼= HomT (L′ ∩ H, Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  If we consider the natural restriction map Res2 : Hom(H, Ω) �→ Hom(L′ ∩ H, Ω),  it is straight forward to note that ker Res1 = ker Res2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Let J be the subset of  Hom(H, Ω) consisting of all χ which can be extended to a homomorphism L → Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  Invoking the proof of [1, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='2] we have J = ker Res2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' But it is obvious that  J = Res(Hom(L, Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Hence, it follows that  HomT (H, Ω)  Res(Hom(L, Ω))  ∼= HomT (L′ ∩ H, Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  But  HomT (L′ ∩ H, Ω) ∼= Hom  �L′ ∩ H  T  , Ω  �  ∼= L′ ∩ H  T  because L′ ∩ H is abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  □  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='1: Let ¯R =  R  [F,R] and ¯F =  F  [F,R].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  Then ¯R is a cen-  tral ideal of ¯F and L ∼=  ¯  F  ¯  R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  By [1, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='4] the transgression map Tra :  Hom( ¯R, Ω) �→ H2(L, Ω) is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  Therefore by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='2 the map tra :  Hom⟨K( ¯  F )∩ ¯R⟩( ¯R, Ω) �→ B0(L) is also surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' It therefore follows, from Theorem  6  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' RAI  2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='6, that B0(L) ∼=  ¯  R∩[ ¯  F, ¯  F ]  ⟨K( ¯  F )∩ ¯R⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' But ¯R ∩ [ ¯F , ¯F] = F ′∩R  [F,R] and ⟨K( ¯F) ∩ ¯R⟩ = ⟨K(F )∩R⟩  [F,R]  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  Hence B0(L) ∼=  F ′∩R  ⟨K(F )∩R⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Applications  The proof of the following proposition is exactly the same as the proof of [6,  Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Let L be a Lie algebra with a free presentation L ∼= F/R, and M  be an ideal of L, such that T = ker(F �→ L/M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Then the sequence  0 �→ R ∩ ⟨K(F) ∩ T ⟩  ⟨K(F) ∩ R⟩  �→ B0(L) �→ B0(L/M) �→  M ∩ L′  ⟨K(L) ∩ M⟩ �→ 0,  is exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  Definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' A Lie algebra L is called generalized Heisenberg of rank n if L′ = Z(L)  and dim L′ = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  A freest generalized Heisenberg Lie algebra is a d-generated (minimally generated  by d elements) generalized Heisenberg Lie algebra of rank 1  2d(d−1) for some d ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  We shall denote it by Ld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Notice that Ld has the following presentation:  ⟨x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' , xd, yij | [xi, xj] = yij, 1 ≤ i < j ≤ d, class 2⟩,  and dim Ld = 1  2d(d + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Let Ld be the freest generalized Heisenberg Lie algebra of rank  1  2d(d − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Then B0(Ld) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Let f : Ld × Ld �→ Ω be a 2-cocycle such that ¯f ∈ B0(Ld).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  Let B =  {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' , xd, [xi, xj] | 1 ≤ i < j ≤ d}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Define a map µ : B �→ Ω as follows:  µ(xi) = 0 for 1 ≤ i ≤ d,  µ([xi, xj]) = −f(xi, xj) for 1 ≤ i < j ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  Since B is basis for Ld we can extend this map linearly to Ld and call it σ so that  σ is a linear transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Let x, y ∈ Ld.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Since B is a basis we can write x and  y as  x =  d  �  i=1  αixi +  �  1≤i<j≤d  αij[xi, xj],  y =  d  �  k=1  βkxk +  �  1≤k<l≤d  βkl[xk, xl],  for some αi, βi, αij, βij ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Now using the bilinearity of f and the fact that  f(a, b) = 0 whenever [a, b] = 0, we get that  f(x, y) =  d  �  i=1  d  �  k=1  αiβkf(xi, xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  BOGOMOLOV MULTIPLIER OF LIE ALGEBRAS  7  Also, using the bilinearity of Lie bracket, linearity of σ and the fact that f(l1, [l2, l3]) =  0 for all l1, l2, l3 ∈ Ld, we get that  σ([x, y]) = −  d  �  i=1  d  �  k=1  αiβkf(xi, xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  Therefore f(x, y) = −σ([x, y]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Thus f is a coboundry and ¯f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' This completes  the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  □  Rostami et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' [6] proved that the Bogomolov multiplier of a Heisenberg Lie  algebra is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' We prove this fact as a Corollary of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  A Heisenberg Lie algebra of dimension 2n + 1, n > 0, is given by the following  presentation:  ⟨x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' , x2n, v | [x2i−1, x2i] = v, [xj, xk] = 0, 1 ≤ i ≤ n, (j, k) ̸= (2i − 1, 2i)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Let H2n+1 be the Heisenberg Lie algebra of dimension 2n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Then  B0(H2n+1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Let L2n be the generalized Heisenberg Lie algebra of rank n(2n − 1) and M  be its ideal generated by [x2r−1, x2r] − [x2s−1, x2s], 1 ≤ r < s ≤ n and [xt, xu], 1 ≤  t < u ≤ 2n, (t, u) ̸= (2i − 1, 2i) for any i ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Then it is easy to see that L2n/M is  isomorphic to H2n+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Since B0(L2n) = 0 by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='2, it follows from Theorem  3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='1 that  B0(L2n/M) ∼=  M ∩ L′  2n  ⟨K(L2n) ∩ M⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  Since for 1 ≤ r < s ≤ n,  [x2r−1 − x2s−1, x2r + x2s] = [x2r−1, x2r] − [x2s−1, x2s] + [x2r, x2s−1] + [x2r−1, x2s],  it follows that M∩L′  2n = ⟨K(L2n)∩M⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Thus B0(L2n/M) = 0 so that B0(H2n+1) =  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  □  We now proceed to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='5: Let d be a natural number greater than 4n and Ld be  the d-generated freest generalized Heisenberg Lie algebra generated by x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' xd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  Let M be the ideal generated by [x1, x2]+[x3, x4], [x5, x6]+[x7, x8], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' [x4n−3, x4n−2]+  [x4n−1, x4n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Since B0(Ld) = 0, it follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='1 that  dim B0(Ld/M) = dim  L′  d ∩ M  ⟨K(Ld) ∩ M⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  Next we prove that K(Ld) ∩ M = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' For this let l ∈ K(Ld) ∩ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Since l ∈ M  there exists γ′  ks such that  l =  n−1  �  k=0  γk+1  �  [x4k+1, x4k+2] + [x4k+3, x4k+4]  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  8  P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' RAI  Also there exist αi’s and βj’s such that  l =  �  d  �  i=1  αixi,  d  �  j=1  βjxj  �  because l ∈ K(Ld).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Hence  n−1  �  k=0  γk+1  �  [x4k+1, x4k+2] + [x4k+3, x4k+4]  �  =  d−1  �  i=1  d  �  j=i+1  (αiβj − αjβi)[xi, xj].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  It follows that  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='1)  αiβj − αjβi = 0 when (i, j) ̸= (2k + 1, 2k + 2) for any k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' n − 1,  and  γk+1 = α4k+1β4k+2 − α4k+2β4k+1 = α4k+3β4k+4 − α4k+4β4k+3  for any k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' From Equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='1 we have  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='2)  α4k+1β4k+3 − α4k+3β4k+1 = 0  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='3)  α4k+1β4k+4 − α4k+4β4k+1 = 0  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='4)  α4k+2β4k+3 − α4k+3β4k+2 = 0  (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='5)  α4k+2β4k+4 − α4k+4β4k+2 = 0,  for k = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  Suppose α4k+1 = β4k+1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Then γk+1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Assume then that α4k+1 = 0 but  β4k+1 ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' From Equations 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='3, α4k+3 = α4k+1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' As a result, γk+1 = 0  in this case as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Thus we have shown that if α4k+1 = 0, then γk+1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Similarly,  if either of α4k+2, α4k+3, α4k+4, β4k+1, β4k+2, β4k+3, β4k+4 is zero, then γk+1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  Hence, we can now assume that neither of α4k+i, β4k+i is zero for i = 1, 2, 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  From Equations 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='3 we can deduce that β4k+3/β4k+4 = α4k+3/α4k+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  Hence γk+1 = 0 so that l = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' It follows that K(Ld) ∩ M = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  Also, M ≤ L′  d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  Hence dim B0(Ld/M) = dim(M) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' By Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='2  dim B(Ld/M) = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Taking L to be Ld/M completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='6: Let (θ, φ) be an isoclinism between L and M, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=', θ :  L  Z(L) �→  M  Z(M) and φ : γ2(L) �→ γ2(M) be isomorphisms and whenever θ(liZ(L)) =  miZ(M) for i = 1, 2, we have φ([l1, l2]) = [m1, m2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Let ¯f ∈ B0(L) where f :  L × L �→ Ω be a cocycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Define cf : M × M �→ Ω by cf(m1, m2) = f(l1, l2), where  l1 and l2 are given by θ−1(mi + Z(M)) = li + Z(L) for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' The rest of the  proof follows from the following lemmas:  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Let cf be the map defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Then  (i) cf is well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  (ii) cf is a 2 cocycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  (iii) cf ∈ B0(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  BOGOMOLOV MULTIPLIER OF LIE ALGEBRAS  9  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Since f is bilinear and f(k, l) = 0 whenever [k, l] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  It follows that  f(l1 + z1, l2 + z2) = f(l1, l2) for every z1, z2 ∈ Z(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' This shows that cf is well-  defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  To see that cf is a 2-cocycle, let m1, m2, m3 ∈ M and let θ−1(mi + Z(M)) =  li + Z(L) for i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' It is obvious that θ−1(m1 + m2 + Z(M)) = l1 + l2 + Z(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  Therefore cf(m1 + m2, m3) = f(l1 + l2, l3) which equals f(l1, l3) + f(l2, l3) that  is equal to cf(m1, m3) + cf(m2, m3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Similarly cf(m1, m2 + m3) = cf(m1, m2) +  cf(m1, m3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Thus cf is bilinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Also, it is easy to see that cf is alternating because  f is alternating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Next, For i, j, k ∈ {1, 2, 3} note that cf([mi, mj], mk) = f([li, lj], lk)  because  θ−1�  [mi, mj] + Z(M)  �  = θ−1��  mi + Z(M), mj + Z(M)  ��  =  �  θ−1�  mi + Z(M)  �  , θ−1�  mj + Z(M)  ��  =  �  li + Z(L), lj + Z(L)  �  = [li, lj] + Z(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  t follows that cf is a 2-cocycle, since f is a 2-cocycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  To see that cf ∈ B0(M), suppose that [m1, m2] = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  But then [l1, l2] = 0  because φ([l1, l2]) = [m1, m2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Since f ∈ B0(L) it follows that f(l1, l2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Hence  cf(m1, m2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' The map η : B0(L) �→ B0(M) defined by η(f) = cf is an isomor-  phism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' We begin by ensuring that the map is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' To verify this consider  σ : L × L �→ Ω to be a coboundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Then  cf+σ(m1, m2) = (f + σ)(l1, l2) = f(l1, l2) + σ(l1, l2) = cf(m1, m2) + cσ(m1, m2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  Thus we have, cf+σ = cf +cσ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Notice that cσ is a coboundary because σ is cobound-  ary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Therefore cf = cf+σ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=', η(f) = η(f + σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' This proves that η is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  In a similar fashion one can see that cf1+f2 = cf1 + cf2 and cαf1 = αcf1 for each  α ∈ Ω and each cocycles f1, f2 from L × L to Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' So that η(f1 + f2) = η(f1) + η(f2)  and η(αf1) = αη(f1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Thus η is a linear map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  Finally, in order to see that η is a bijection, we define another map χ : B0(M) �→  B0(L) in the same way as η is defined from B0(L) to B0(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Then it is easy to see  that ηχ and χη both are identity maps and thus η is a bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' This completes  the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  □  □  References  [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
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+page_content=' 2, 3, 4, 5  [2] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
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+page_content=' Akad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='  Nauk SSSR, Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
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+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=', 282, Birkh¨auser Boston,  Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=', Boston, MA, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
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+page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
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+page_content=' 1  [6] Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
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+page_content=' Niroomand, The Bogomolov multiplier of Lie algebras, Hacet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
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+page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Saltman, Noether’s problem over an algebraically closed field, Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' 77 (1984),  no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' 1, 71-84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' 1  (Pradeep K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content=' Rai) Mahindra University, Hyderabad, Telangana,, India  Email address: raipradeepiitb@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}
+page_content='com' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CtA0T4oBgHgl3EQfAf94/content/2301.01963v1.pdf'}

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+arXiv:2301.04466v1  [hep-th]  7 Jan 2023
+Clifford odd and even objects in even and odd dimensional
+spaces describing internal spaces of fermion and boson fields
+Norma Susana Mankoˇc Borˇstnik
+Department of Physics, University of Ljubljana
+SI-1000 Ljubljana, Slovenia,
+norma.mankoc@fmf.uni-lj.si
+January 12, 2023
+Abstract
+In a long series of works, it has been demonstrated, that the spin-charge-family theory offers
+the explanation for all in the standard model assumed properties of the second quantized fermion
+and boson fields, offering several predictions as well as explanations for several of the observed
+phenomena. The theory assumes a simple starting action in even dimensional spaces with d ≥
+(13 + 1) with massless fermions interacting with gravity only. The internal spaces of fermion and
+boson fields are described by the Clifford odd and even objects, respectively. This contribution
+discusses the properties of the fermion and boson fields in odd dimensional spaces, d = (2n + 1),
+with the internal spaces of fermion and boson fields described again by the Clifford odd and even
+objects, respectively, pointing out that their properties differ essentially from the properties in even
+dimensional spaces, resembling the ghost needed when looking for final solutions with Feynman
+diagrams.
+Keywords: Second quantization of fermion and boson fields with Clifford algebra; beyond the standard
+model; Kaluza-Klein-like theories in higher dimensional spaces; Clifford algebra in odd dimensional
+spaces; ghosts in quantum field theories
+1
+introduction
+30 years ago, I recognized that there are two kinds of Clifford algebra objects, γa’s and ˜γa’s [1, 2, 3],
+originating in the Grassmann algebra. The Clifford and the Grassmann algebras can be used to describe
+the internal space of fermions in even dimensional spaces: The superposition of odd products of either
+γa’s or ˜γa’s, anti-commute, fulfilling on the vacuum states the anti-commutation relations [11] of the
+second quantization postulates for fermion fields [4, 5]. The superposition of odd products of either γa’s
+or ˜γa’s, appear in irreducible representations [8].
+Only one kind of fermions has been observed so far, appearing in several families. If we use one
+of the two kinds of the Clifford algebra objects, say γa’s, to describe the internal space of fermions,
+and the second kind of the Clifford algebra objects, ˜γa’s, to describe the family quantum numbers of
+each of the irreducible representation determined by γa’s, we are left with one kind of fermions [16, 19],
+Sect.(3.2.3) of [8].
+1
+
+In any even dimensional space there are 2
+d
+2 −1 of the Clifford odd “basis vectors”, appearing in 2
+d
+2 −1
+families. They are the superposition of odd products γa’s. All the members of any family are orthogonal
+to all the members of the same and all the other families. Their Hermitian conjugated partners appear
+in a separate group, again with 2
+d
+2 −1 members in 2
+d
+2−1 families.
+The Clifford odd “basis vectors” have in even dimensional spaces only left or only right handedness,
+depending on the definition (Γ = �d
+a(√ηaaγa) · (i)
+d
+2).
+In any even dimensional space there are two groups of 2
+d
+2 −1× 2
+d
+2 −1 of the Clifford even “basis vectors”,
+which are the superposition of even products γa’s. The family quantum number has no meaning for
+the Clifford even “basis vectors”. The members of one group are orthogonal to the members of another
+group. The members of any of the two groups of the Clifford even “basis vectors” have their Hermitian
+conjugated partners within the same group [12, 9].
+The superposition of even products of γa’s (or ˜γa’s), commute, fulfilling the commutation rela-
+tions [11] of the second quantization postulates for boson fields [4, 5, 6].
+The Clifford even “basis vectors” have properties of the gauge fields of the corresponding Clifford
+odd “basis vectors”, what becomes transparent after the algebraic multiplication, ∗A, of the Clifford
+even “basis vectors” on the Clifford odd “basis vectors” and opposite, as well as of the Clifford even
+“basis vectors” among themselves [12, 9].
+Algebraic multiplication is distributive and associative.
+The properties of the Clifford odd and the Clifford even “basis vectors” in even dimensional spaces
+is shortly overviewed in Sect. 2.1, showing that the Clifford odd “basis vectors”, applying on the
+appropriate vacuum states, manifest the postulates of the second quantized fermion fields, while the
+Clifford even “basis vectors” manifest the postulates for their gauge fields, the second quantized boson
+fields.
+The properties of the fermion and boson fields in odd dimensional spaces differ drastically from
+the properties of the fermion and boson fields in even dimensional spaces: The Clifford odd “basis
+vectors” do not manifest the properties of the second quantized fermion fields in even dimensional
+spaces. Although anti-commuting, they instead manifest properties of the Clifford even “basis vectors”
+in even dimensional spaces. And the Clifford even “basis vectors” do not manifest the properties of the
+second quantized boson fields in even dimensional spaces. Although commuting, they instead manifest
+properties of the Clifford odd “basis vectors” in even dimensional spaces.
+In addition, since the operator of handedness has in odd dimensional spaces the Clifford odd char-
+acter (Γ = �d
+a(√ηaaγa) · (i)
+d−1
+2 ), it transforms the Clifford odd “basis vectors” into the Clifford even
+“basis vectors” [17].
+The eigenstates of the operator of handedness are in odd dimensional spaces
+correspondingly the superposition of the Clifford odd and the Clifford even “basis vectors”.
+The properties of the Clifford odd and the Clifford even ”basis vectors” in odd dimensional spaces
+are discussed in Sect. 2.2.
+In d = (13 + 1) dimensional space the Clifford odd “basis vectors”, if analysed from the point of
+view of the subgroups of the standard model groups, offer the description of the internal spaces of all
+the so far observed quarks and leptons and antiquarks and antileptons as assumed by the standard
+model before the electroweak phase transition, including in addition the right handed neutrinos and left
+handed antineutrions. Quarks and antiquarks and leptons and antileptons appear as sixty-four (64)
+members in two times four families.
+The corresponding Clifford even “basis vectors” offer the description of the internal spaces of the
+corresponding vector and scalar gauge fields [8, 12, 9, 7].
+The spin-charge-family theory, describing the internal spaces of fermion and boson fields by using
+the Clifford odd and even algebras in d = (13+1)-dimensional space, offers not only the explanation for
+the postulates of the second quantized fermion and boson fields, and the explanation for all the standard
+model assumptions, but also for several observed phenomena, making several predictions [8]. The theory
+is built on the simple starting starting action in which fermion interacts with the gravitational fields
+2
+
+only
+A
+=
+�
+ddx E 1
+2 ( ¯ψ γap0aψ) + h.c. +
+�
+ddx E (α R + ˜α ˜R) ,
+p0a
+=
+f α
+ap0α + 1
+2E {pα, Ef α
+a}− ,
+p0α
+=
+pα − 1
+2Sabωabα − 1
+2
+˜Sab˜ωabα ,
+R
+=
+1
+2 {f α[af βb] (ωabα,β − ωcaα ωc
+bβ)} + h.c. ,
+˜R
+=
+1
+2 {f α[af βb] (˜ωabα,β − ˜ωcaα ˜ωc
+bβ)} + h.c. .
+(1)
+Here 1 f α[af βb] = f αaf βb − f αbf βa.
+I demonstrate in this paper that in odd dimensional spaces the Clifford odd and the Clifford even
+objects have drastically different properties than in even dimensional spaces, offering the explanation
+for postulated ghost fields appearing in several theories for taking care of the singular contributions in
+evaluating Feynman graphs.
+In Sect. 2 appropriate definition of the eigenstates of the Cartan subalgebra members are presented
+for even dimensional spaces, and extended to odd dimensional spaces.
+In Subsect. 2.1 the internal spaces described by the Clifford odd and the Clifford even ”basis vectors”
+for fermion and boson fields in even dimensional spaces are presented.
+In Subsect. 2.2 the internal spaces of fermion and boson fields in odd dimensional spaces are pre-
+sented.
+In Sect. 3, the internal spaces for fermion and boson fields in even and odd dimensional spaces for
+simple cases are discussed: In Subsect. 3.1 for the choices d = (1 + 1), d = (3 + 1) and in Subsect. 3.2
+for d = (2 + 1) and d = (4 + 1).
+In Refs. [13, 14, 15] from 20 years ago the authors discuss the question of q time and d − q dimen-
+sions in odd and even dimensional spaces for any q. Using the requirements that the inner product
+of two fermions is unitary and invariant under Lorentz transformations the authors conclude that odd
+dimensional spaces are not appropriate due to the existence of fermions of both handedness and cor-
+respondingly not mass protected. The recognition of this paper might further clarify the “effective”
+choice of Nature for one time and three space dimensions.
+In Sect. 4, the main idea of this note is overviewed.
+In App.A, some helpful relations of the Clifford algebra can be found.
+2
+Eigenstates of Cartan subalgebra members of Lorentz alge-
+bra for Clifford odd and Clifford even “basis vectors”
+In this section, the properties of the two kinds of Clifford algebra objects, γa’s and ˜γa’s, are shortly
+repeated following several papers [1, 2, 16, 11, 12, 7, 9, 10], in particular the reference ([8], and the
+references therein).
+1f αa are inverted vielbeins to eaα with the properties eaαf αb = δab, eaαf βa = δβ
+α, E = det(eaα).
+Latin indices
+a, b, .., m, n, .., s, t, .. denote a tangent space (a flat index), while Greek indices α, β, .., µ, ν, ..σ, τ, .. denote an Einstein index
+(a curved index). Letters from the beginning of both the alphabets indicate a general index (a, b, c, .. and α, β, γ, .. ), from
+the middle of both the alphabets the observed dimensions 0, 1, 2, 3 (m, n, .. and µ, ν, ..), indexes from the bottom of the al-
+phabets indicate the compactified dimensions (s, t, .. and σ, τ, ..). We assume the signature ηab = diag{1, −1, −1, · · · , −1}.
+3
+
+The two kinds of Clifford algebra objects, γa and ˜γa, each offering 2d superposition of products of
+either γa or ˜γa, fulfil the relation [1, 18, 19]
+{γa, γb}+
+=
+2ηab = {˜γa, ˜γb}+ ,
+{γa, ˜γb}+
+=
+0 ,
+(a, b) = (0, 1, 2, 3, 5, · · · , d) ,
+(γa)†
+=
+ηaa γa ,
+(˜γa)† = ηaa ˜γa .
+(2)
+Each of these two kinds of the Clifford algebra objects could be used to describe the internal spaces of
+fermion and boson fields.
+We can reduce the two possibilities to only one by deciding to describe the internal spaces of fermion
+and boson fields with the superposition of the Clifford odd (for fermion fields) and the Clifford even (for
+boson fields) products of γa’s, while using ˜γa’s to equip the irreducible representations of the Lorentz
+group in the internal space of fermions with the family quantum numbers by assuming
+{˜γaB
+=
+(−)B i Bγa} |ψoc > ,
+(3)
+with (−)B = −1, if B is (a function of) an odd product of γa’s, otherwise (−)B = 1 [19], |ψoc > is
+defined in Eq. (33). It is proven in [8] (App.I, Statement 3, 3.a, 3.b) that all the relations of Eq. (2)
+remain valid also after the assumption of Eq. (31).
+The “basis vectors” describing internal spaces of fermion and boson fields are chosen to be eigenstates
+of all the Cartan subalgebra members. There are d
+2 commuting operators of the Lorentz algebra in the
+even dimensional spaces, Eq. (28), and d−1
+2
+in odd dimensional spaces, Eq. (29).
+If Sab, a ̸= b, (or ˜Sab or Sab = Sab + ˜Sab) are members of the Cartan subalgebra group of the
+Lorentz algebra in the internal space of fermion and boson fields, then it is not difficult to find the
+eigenstate of each of the members just by taking into account relations of Eq. (2: Sab 1
+2(γa + ηaa
+ik γb) =
+k
+2
+1
+2(γa+ ηaa
+ik γb) and Sab 1
+2(1+ i
+kγaγb) = k
+2
+1
+2(1+ i
+kγaγb), with k2 = ηaaηbb. The first eigenstate is nilpotent,
+( 1
+2(γa + ηaa
+ik γb))2 = 0 and the second eigenstate is projector ( 1
+2(1 + i
+kγaγb))2 = 1
+2(1 + i
+kγaγb).
+Let us introduce the graphic notation, following Ref. [9, 18, 19].
+ab
+(k):
+=
+1
+2(γa + ηaa
+ik γb) ,
+ab
+[k]:= 1
+2(1 + i
+kγaγb) ,
+ab
+˜
+(k):
+=
+1
+2(˜γa + ηaa
+ik ˜γb) ,
+ab
+˜[k]: 1
+2(1 + i
+k ˜γa˜γb) ,
+(
+ab
+(k))†
+=
+ab
+(−k) ,
+(
+ab
+(k))2 = 0 ,
+(
+ab
+[k])† =
+ab
+[k] ,
+(
+ab
+[k])2 =
+ab
+[k] .
+(4)
+After taking into account Eq. (2) the relations follow
+γa
+ab
+(k)
+=
+ηaa
+ab
+[−k],
+γb
+ab
+(k)= −ik
+ab
+[−k],
+γa
+ab
+[k]=
+ab
+(−k),
+γb
+ab
+[k]= −ikηaa
+ab
+(−k) ,
+˜γa
+ab
+(k)
+=
+−iηaa
+ab
+[k],
+˜γb
+ab
+(k)= −k
+ab
+[k],
+˜γa
+ab
+[k]=
+i
+ab
+(k),
+˜γb
+ab
+[k]= −kηaa
+ab
+(k) ,
+(5)
+More relations can be found in App. A.
+2.1
+Properties of Clifford odd and Clifford even “basis vectors” in even
+dimensional spaces
+In each even dimensional space there are 2
+d
+2 −1 members of the Clifford odd “basis vectors” appearing
+2
+d
+2−1 families, and the same number of 2
+d
+2 −1 their Hermitian conjugated partners appearing in 2
+d
+2 −1
+families.
+4
+
+There are two orthogonal groups of the Clifford even “basis vectors”. The members of each group
+have their Hermitian conjugated partners within the same group.
+Clifford odd “basis vectors”
+We find the Clifford odd “basis vectors”, describing the internal space of fermion fields, as products
+of odd numbers of nilpotents and the rest of projectors, if each nilpotent and each projector is the
+eigenstate of one of the Cartan subalgebra members.
+Let us call the Clifford odd ”basis vectors” ˆbm†
+f , if this is the mth member of the family f.
+Let us choose the first member ˆb1†
+1 , if d = 2(2n + 1), as the product of nilpotents only.
+d = 2(2n + 1) ,
+ˆb1†
+1 =
+03
+(+i)
+12
+(+)
+56
+(+) · · ·
+d−1 d
+(+) ,
+ˆb2†
+1 =
+03
+[−i]
+12
+[−]
+56
+(+) · · ·
+d−1 d
+(+) ,
+· · ·
+ˆb2
+d
+2 −1†
+1
+=
+03
+[−i]
+12
+[−]
+56
+(+) . . .
+d−3 d−2
+[−]
+d−1 d
+[−] ,
+· · · .
+(6)
+In the case that d = 4n, n = 1, 2, .., the first member must have one projector.
+d = 4n ,
+ˆb1†
+1 =
+03
+(+i)
+12
+(+)
+56
+(+) · · ·
+d−1 d
+[+] ,
+· · · .
+(7)
+All the rest members of the same family, 2
+d
+2 −1 − 1, follow by the application of all possible Sab on ˆb1†
+1 ,
+while all the rest 2
+d
+2 −1 − 1 families follow by the application of all possible ˜Sab on all the members of
+the starting family.
+The Hermitian conjugated partners (ˆbm†
+f )† of the “basis vectors” ˆbm†
+f
+follow from these 2
+d
+2 −1 × 2
+d
+2 −1
+“basis vectors” by replacing each nilpotent
+ab
+(k) with
+ab
+(−k).
+Choosing the vacuum state equal to
+|ψoc >=
+2
+d
+2 −1
+�
+f=1
+ˆbm
+f ∗Aˆbm†
+f
+| 1 > ,
+(8)
+for one of the members m, anyone, of the odd irreducible representation f, with | 1 >, which is the
+vacuum without any structure — the identity — it follows that ˆbm
+f |ψoc >= 0.
+Each Clifford odd “basis vector” carries the family quantum number, and so does its Hermitian
+conjugated partner. One correspondingly finds that the “basis vectors” and their Hermitian conjugated
+partners fulfil the postulates for the second quantized fermion fields.
+ˆbm
+f ∗A|ψoc >
+=
+0. |ψoc > ,
+ˆbm†
+f
+∗A|ψoc >
+=
+|ψm
+f > ,
+{ˆbm
+f ,ˆbm′
+f‘ }∗A+|ψoc >
+=
+0. |ψoc > ,
+{ˆbm†
+f ,ˆbm′†
+f‘ }∗A+|ψoc >
+=
+0. |ψoc > ,
+{ˆbm
+f ,ˆbm′†
+f‘ }∗A+|ψoc >
+=
+δmm′
+ff‘ |ψoc > ,
+(9)
+5
+
+where ∗A represents the algebraic multiplication of ˆbm†
+f
+and ˆbm′
+f′ among themselves and with the vacuum
+state |ψoc > of Eq.(8). Eq. (9) follows by taking into account Eq. (2).
+These “basis vectors” are not yet the representatives of the creation and annihilation operators:
+They must be tensor, ∗T, products of the “basis vectors” and the basis in ordinary momentum or
+coordinate space [8] 2.
+Clifford even “basis vectors”
+We can find the Clifford even “basis vectors” describing the internal space of the boson fields as
+products of even numbers of nilpotents and the rest of projectors if each nilpotent and each projector
+is the eigenstate of one of the Cartan subalgebra members.
+Let us call the Clifford even “basis vectors” iAm†
+f , i = I, II. There are namely two groups of Clifford
+even basis vectors”. Each group has 2
+d
+2 −1 × 2
+d
+2 −1 members.
+Let us choose the starting Clifford even “basis vector”, i=IA1†
+1 , to be the product of projectors
+ab
+[k],
+with k = i for S03, and k = 1 for the rest 2
+d
+2−1 − 1 members of the Cartan subalgebra.
+I ˆ
+A1†
+1 =
+03
+[+i]
+12
+[+] · · ·
+d−1 d
+[+] .
+(10)
+The starting Clifford even “basis vector” of the second group i=IIA1†
+1 can again be the product of pro-
+jectors only, but in this case with
+03
+[−i] instead of
+03
+[+i] and for all the rest 2
+d
+2 −1−1 members of the Cartan
+subalgebra with k = +1. (This starting member can not be obtained from IA1†
+1 by the application of
+Sab’s or ˜Sab’s, since these operators always change the eigenvalues of two Cartan subalgebra members.)
+II ˆ
+A1†
+1 =
+03
+[−i]
+12
+[+] · · ·
+d−1 d
+[+] .
+(11)
+The rest of the members of each group follow from the starting member by the application of either
+Sab’s or ˜Sab’s.
+Since S01 transforms
+03
+[+i]
+12
+[+] into
+03
+(−i)
+12
+(−1), while ˜S01 transforms
+03
+[+i]
+12
+[+] into
+03
+(+i)
+ab
+(+), we immedi-
+ately see that the Clifford even “basis vector” have the Hermitian conjugated partners within the same
+group of 2
+d
+2 −1 × 2
+d
+2 −1 members.
+Clifford even “basis vectors” applying on Clifford odd “basis vectors.
+Let us apply IA1†
+1 , which is made of projectors
+ab
+[k] only, with k = i for S03, and k = 1 for the rest
+members of the Cartan subalgebra, on ˆb1†
+1 , which is the product of nilpotents only, with eigenvalue of
+S03 equal k = i and of the rest of Cartan subalgebra members equal to k = 1.
+Taking into account Eqs. (34, 35) one sees that this application, IA1†
+1 ∗A ˆb1†
+1 , leaves ˆb1†
+1 unchanged.
+When applying IA2†
+1 , with the first two projectors transformed into two nilpotents,
+03
+(−i)
+12
+(−1), and all
+the rest remain the same, we see that this application transforms ˆb1†
+1 into ˆb2†
+1 (=
+03
+[−i]
+12
+[−1]
+56
+(+)
+78
+(+) .... (all
+the rest remains the same). The application of IA2†
+1 on ˆb1†
+1 obviously changes the eigenvalues of S03 and
+of S12 of ˆb1†
+1 for integer values, −i and −1, respectively.
+2In even dimensional spaces with d = 4n, one proceeds as we did in d = 2(2n + 1) dimensional case after taking
+into account the requirement that the odd number of nilpotents forms the anti-commuting “basis vectors” describing the
+internal space of fermions: The starting “basis vector” ˆb1†
+1
+must have one projector, while all the rest are nilpotents.
+Sab’s then generate all the members of one family, while ˜Sab’s generate all the families. The “basis vectors” and their
+Hermitian conjugated partners fulfil on the vacuum state, Eq. (33), the anti-commuting postulates of Eq. (9).
+6
+
+We conclude: The algebraic application, ∗A, of the Clifford even ”basis vectors” on the Clifford odd
+”basis vectors”, describing the internal space of fermion fields, change their eigenvalues of the Cartan
+subalgebra members for 0 or for integer values, ±i, or ±1, leading to
+I ˆ
+Am†
+f‘ ∗A ˆbm′†
+f
+→
+�
+ˆbm†
+f
+,
+or zero .
+(12)
+Clifford even “basis vectors” applying on Clifford even “basis vectors”
+It is not difficult to see, by taking into account Eqs. (34, 35), that the algebraic applications of
+IAf†
+1 ∗A IIAm′†
+f‘
+= 0 = IIAm′†
+f‘
+∗A IAm†
+f , for all (m, m′, f, f‘).
+The algebraic application, ∗A, of iAm†
+f ∗A iAm′†
+f‘
+within each of the two groups give in general non zero
+contribution, demonstrating the properties of the internal spaces of the gauge fields to the corresponding
+fermion fields, the internal space of which are described by the Clifford odd “basis vectors”.
+In each of the two groups, there are 2
+d
+2 −1 members, which are products of projectors only. They are
+self adjoint and have the eigenvalues of all the Cartan subalgebra members equal zero: Sab = Sab + ˜Sab.
+All the rest iAm†
+f
+(there are 2
+d
+2 −1 × (2
+d
+2 −1 − 1) members) appear in pairs; Hermitian conjugated to
+each other. Their mutual algebraic products form one of 2
+d
+2 −1 self-adjoint members.
+The algebraic multiplication of the Clifford even “basis vectors” on the Clifford even “basis vectors”
+lead to
+i ˆ
+Am†
+f
+∗A
+i ˆ
+Am′†
+f‘
+→
+�
+i ˆ
+Am†
+f‘ ,
+or zero . i = (I, II) .
+(13)
+The reader can find in Ref. [7, 9] the Clifford odd and the Clifford even ”basis vectors” in the case
+that the dimension of the space is d = (5 + 1), describing the internal space of fermion and boson fields,
+respectively, illustrated by figures.
+2.2
+Properties of the Clifford odd and Clifford even ”basis vectors” in odd
+dimensional spaces
+In this Subsect. 2.2 the Clifford odd and Clifford even “basis vectors” in odd dimensional spaces [12, 9]
+are discussed.
+While in even dimensional spaces the Clifford odd “basis vectors” fulfil the postulates for the second
+quantized fermion fields, Eq. (9), and Clifford even ”basis vectors” have all the properties of the internal
+spaces of their corresponding gauge fields, Eqs. (12, 13), the Clifford odd and even ”basis vectors”
+have in odd dimensional spaces unusual properties resembling properties of the internal spaces of the
+Faddeev-Popov ghosts, as we shall see in what follows.
+Looking in d = (2n+1)dimensional cases, n = 1, 2, . . . , for the Clifford odd and Clifford even “basis
+vectors” in 2n-dimensional part of space we find half of the “basis vectors” with properties presented
+in Eqs. (6, 7, 10). In Eqs. (14, 15) they are presented on the left hand side.
+The rest of the “basis vectors” follow applying S0 2n+1 on the obtained half of the Clifford odd and
+the Clifford even “basis vectors”. Since S0 2n+1 are Clifford even operators; they do not change oddness
+or evenness of the “basis vectors”.
+One finds for the Clifford odd “basis vectors” correspondingly the additional 2
+d−1
+2 −1 members, ap-
+pearing in 2
+d−1
+2 −1 families and the same number of their Hermitian conjugated partners on the right
+7
+
+hand side of Eq. (14).
+d =
+2(2n + 1) + 1
+ˆb1†
+1 =
+03
+(+i)
+12
+(+)
+56
+(+) · · ·
+d−2 d−1
+(+)
+,
+ˆb1†
+2
+d−1
+2
+−1+1 =
+03
+[−i]
+12
+(+)
+56
+(+) · · ·
+d−2 d−1
+(+)
+γd ,
+ˆb2†
+1 =
+03
+[−i]
+12
+[−]
+56
+(+) · · ·
+d−2 d−1
+(+)
+,
+ˆb2†
+2
+d−1
+2
+−1+1 =
+03
+(+i)
+12
+[−]
+56
+(+) · · ·
+d−2 d−1
+(+)
+γd ,
+· · ·
+· · ·
+ˆb2
+d−1
+2
+−1†
+1
+=
+03
+[−i]
+12
+[−]
+56
+(+) . . .
+d−2 d−1
+[−]
+,
+ˆb2
+d−1
+2
+−1†
+2d−12−1+1 =
+03
+(+i)
+12
+[−]
+56
+(+) . . .
+d−2 d−1
+[−]
+γd ,
+· · ·
+· · · .
+(14)
+The right handed half of “basis vectors” follows from the left handed “basis vectors” or from their
+Hermitian conjugated partners by the application of S0d on the left handed part. The application of
+˜S0d on the left handed part of the “basis vectors” generates the whole set of 2d−2 members of the Clifford
+odd ”basis vectors” from the right hand side 3.
+When applying on the Clifford even “basis vectors” appearing on the left hand side of Eq. (15)
+the operators S0 2n+1 the additional two groups of 2
+d−1
+2 −1× 2
+d−1
+2 −1 “basis vectors” follow, presented in
+Eq. (15) on the right hand side.
+The 2d−2 Clifford odd objects presented on the right hand side of Eq. (14), and for the special
+cases of Eqs. (23, 25), although they are the superposition of the Clifford odd products of γa’s, do not
+manifest properties of “basis vectors” and their Hermitian conjugated partners, presented on the left
+hand side of Eq. (14), and for the special cases of Eqs. ( 23, 25).
+The eigenstates appearing on the right hand side of Eq. (14) can be divided into two groups which
+are orthogonal to each other, having their Hermitian conjugated partners within the same group or are
+self adjoint. Although they are Clifford odd objects they resemble the properties of the Clifford even
+partners of the “basis vectors”, appearing on the left hand side of Eq. (15).
+Let us see the application of the operators S0d and ˜S0d on the Clifford even “basis vectors” on the
+even dimensional part of the d = 2(2n + 1) + 1 space. The Clifford even ��basis vectors” must have an
+even number of nilpotents, which means that in d = 2(2n + 1), we must have at least one projector.
+To obtain all the Clifford even “basis vectors” we must apply on these starting Clifford even “basis
+vectors”, presented in Eq. (15) on the left hand side, the operators S0d and ˜S0d.
+d =
+2(2n + 1) + 1
+IA1†
+1 =
+03
+(+i)
+12
+(+)
+56
+(+) · · ·
+d−2 d−1
+[+]
+,
+IA1†
+2d−12−1+1 =
+03
+[−i]
+12
+(+)
+56
+(+) · · ·
+d−2 d−1
+[+]
+γd ,
+IA2†
+1 =
+03
+[−i]
+12
+[−]
+56
+(+) · · ·
+d−2 d−1
+[+]
+,
+IA2†
+2d−12−1+1 =
+03
+(+i)
+12
+[−]
+56
+(+) · · ·
+d−2 d−1
+[+]
+γd ,
+· · ·
+· · ·
+IA2
+d−1
+2
+−1†
+1
+=
+03
+[−i]
+12
+[−]
+56
+[−] . . .
+d−2 d−1
+[+]
+,
+IA2
+d−1
+2
+−1†
+2d−12−1+1 =
+03
+(+i)
+12
+[−]
+56
+[−] . . .
+d−2 d−1
+[+]
+γd ,
+· · ·
+· · · .
+(15)
+The right hand side of Eq. (15), and for the special cases of the Clifford even part of Eqs. ( 23,
+25), are the Cliffdord even “basis vectors” as there are their left handed partners. But they resemble
+properties of the left handed “basis vectors”; presented in Eq. (14), and for the special cases of the
+Clifford odd part of Eqs. ( 23, 25). These Clifford even objects can be arranged into 2
+d−1
+2 −1 members
+3The application of S0d and ˜S0d on the left hand side part of the Hermitian conjugated group to the Clifford odd
+”basis vectors” generate the same 2d−2 Clifford odd “basis vectors” as the S0 d and ˜S0 d when applying on the left hand
+side “basis vectors”. Correspondingly we now have twice 2d−2 Clifford odd eigenstates of the d−1
+2
+Cartan subalgebra
+members.
+8
+
+in 2
+d−1
+2 −1 families of “basis vectors” and into a separate group of their Hermitian conjugated partners.
+However, they are the Clifford even “basis vectors”.
+Let us point out that the Lorentz transformations in internal spaces of fermion and boson fields
+transform the left hand sides of Eq. ((14) and of Eq. ((15) into the corresponding right hand sides and
+opposite.
+If we apply algebraically the Clifford even “basis vectors” appearing on the right hand side of Eq. (15)
+on the Clifford odd “basis vectors” appearing on the right hand side of Eq. (14), we end up with the
+Clifford odd “basis vector” appearing on the left hand side of Eq. (14), or on one of their Hermitian
+conjugated partners. Or we obtain zero.
+If we apply algebraically the Clifford even “basis vectors” appearing on the right hand side of Eq. (15)
+on the Clifford odd “basis vectors” appearing on the left hand side of Eq. (14), we end up with the
+Clifford odd “basis vectors” appearing on the right hand side of Eq. (14).
+In the next section, we discuss concrete cases to make discussions more transparent.
+Let us conclude this section with what we have learned:
+a. In d = 2n + 1 dimensional spaces, n = 1, 2, . . . , there are two kinds of the Clifford odd “basis
+vectors”:
+a.i. The “basis vectors” are products of an odd number of nilpotents and the rest of the projectors.
+These “basis vectors” appear in 2
+d−1
+2 −1 families, each family has 2
+d−1
+2 −1 members. They anti-commute,
+fulfilling together with their Hermitian conjugated partners the postulates for the second quantized
+fermion fields. Their Hermitian conjugated partners appear in a separate group.
+a.ii. Applying on these Clifford odd “basis vectors” the operators S0d and ˜S0d the additional two times
+2
+d−1
+2 −1× 2
+d−1
+2 −1 of the Clifford odd “basis vectors” follow. These Clifford odd “basis vectors” resemble
+the properties of the Clifford even “basis vectors” from the case b.i. presented below; They form two
+orthogonal groups. The members of each group have their Hermitian conjugated partners within the
+same group, or they are self-adjoint.
+b. In d = 2n + 1 dimensional spaces, n = 1, 2, . . . , there are two kinds of the Clifford even “basis
+vectors”:
+b.i. The “basis vectors” are products of even number of nilpotents and the rest of the projectors. These
+“basis vectors” appear in two orthogonal groups with 2
+d−1
+2 −1×2
+d−1
+2 −1 members. Each group have their
+Hermitian conjugated members within their own group, or they are self-adjoint. They commute, fulfill-
+ing the postulates for the second quantized boson fields, the gauge fields of the corresponding fermion
+fields of the case a.i..
+b.ii. Applying on these “basis vectors” the operators S0d and ˜S0d the additional two times 2
+d−1
+2 −1×
+2
+d−1
+2 −1 Clifford even “basis vectors” follow. These Clifford even “basis vectors” resemble the properties
+of the Clifford odd “basis vectors” of the case a.i.; They form two groups with 2
+d−1
+2 −1 members in
+each of the 2
+d−1
+2 −1 families. Their Hermitian conjugated partners appear in a separate group. But they
+commute.
+c.i. When Clifford even “basis vectors” of the kind b.i. algebraically apply on the Clifford odd
+“basis vectors” of the kind a.i. they transfer to the Clifford odd “basis vectors” the integer values of
+the Cartan subalgebra members (±i, ±1 or 0) or they give zero.
+c.ii. When Clifford even basis vectors” of the kind b.ii. algebraically apply on the Clifford odd “basis
+vectors” of the kind a.ii. they transfer to the Clifford odd “basis vectors” the integer values of the
+Cartan subalgebra members, (±i, ±1 or 0) or they give zero as in the case c.i..
+d.i. While the Clifford odd “basis vectors” in even dimensional spaces have well-defined handedness,
+since the operator of handedness is the Clifford even operator, Eq. (26), the eigenvectors of the operator
+9
+
+of handedness in odd dimensional spaces are the superposition of the “basis vectors” of the kind a.i.
+and of the kind a.ii..
+3
+“Basis vectors” in even, d = 2n for n = 1, 2, and odd, d = 2n+1
+for n = 1, 2, dimensional spaces
+The internal spaces for fermion and boson fields in even and odd dimensional spaces for simple cases
+are discussed: In Subsect. 3.1 for the choices d = (1+1), d = (3+1) and in Subsect. 3.2 for d = (0+1),
+d = (2 + 1) and d = (4 + 1). This section is meant as an illustration of Sect. 2.
+In Refs. [7, 9, 10, 8, 12, 11] the reader can find the definition of the “basis vectors” as the eigenstates
+of the Cartan subalgebra of the Lorentz algebra in internal spaces of fermion and boson fields. “Basis
+vectors” are written as superposition of the Clifford odd (for fermions) and the Clifford even (for bosons)
+products of γa’s. “Basis vectors” for fermions have either left or right handedness, Γd (the handedness
+is defined in Eq. (26)), and appear in families (the family quantum numbers are determined by ˜γa’s,
+with ˜Sab =
+i
+4{˜γa, ˜γb}−). The Clifford odd “basis vectors” have their Hermitian conjugated partners
+in a separate group. “Basis vectors” for bosons have no families and have their Hermitian conjugated
+partners within the same group, Sect. 2.
+The “basis vectors” in odd dimensional spaces differ in properties from the “basis vectors” in even
+dimensional spaces, as we have concluded in the previous Sect. 2.
+Half of the Clifford odd “basis vectors” have properties as in even dimensional spaces 4.
+The
+remaining half of the Clifford odd “basis vectors” gain properties of the Clifford even “basis vectors”.
+Half of the Clifford even “basis vectors” have properties as in even dimensional spaces. The remaining
+half of the Clifford even “basis vectors” gain properties of the Clifford odd “basis vectors”. Since the
+operator of handedness is is the Clifford odd object (it is the product of odd number of γa’s), only the
+superposition of the Clifford odd and the Clifford even “basis vectors” have a definite handedness 5.
+3.1
+“Basis vectors” in even dimensional spaces: d = (1 + 1), (3 + 1)
+To illustrate the differences in properties of the internal spaces of fermion and boson fields in even and
+odd dimensional spaces, simple cases are discussed. The definition of nilpotents and projectors and the
+relations among them can be found in Eq. (4) and App. A.
+d = (1 + 1)
+There are 4 (2d=2) “eigenvectors” of the Cartan subalgebra members, Eq. (28), S01 and S01 of the
+Lorentz algebra Sab and Sab = S01 + ˜S01 (Sab = i
+4{γa, γb}− ˜Sab = i
+4{˜γa, ˜γb}−), representing one Clifford
+odd “basis vector” ˆb1†
+1 =
+01
+(+i) (m=1), appearing in one family (f=1) and correspondingly one Hermitian
+conjugated partner ˆb1
+1 =
+01
+(−i) 6 and two Clifford even “basis vector” IA1†
+1 =
+01
+[+i] and IIA1†
+1 =
+01
+[−i], both
+self-adjoint.
+4The same choice of the Cartan subalgebra members is made in the case d = (2n + 1) and in the case of d = 2n. The
+Lorentz transformations in the internal space of fermion and boson fields transform in Eqs. (14, 15) the left hand sides
+into the right hand sides and opposite.
+5Correspondingly the eigenvectors of the Cartan subalgebra members have both handednesses, Γ(2n+1) = ±1.
+6It is our choice which one,
+01
+(+i) or
+01
+(−i), we choose as the “basis vector” ˆb1†
+1 , and which one is its Hermitian conjugated
+partner. The choice of the “basis vectors” determines the vacuum state |ψoc >, Eq. (8). For ˆb1†
+1 =
+01
+(+i), the vacuum state
+is |ψoc >=
+01
+[−i] (due to the requirement that ˆb1†
+1 |ψoc > is nonzero, while ˆb1
+1|ψoc > is zero), which is the Clifford even object.
+10
+
+Correspondingly we have, after using Eqs. (2, 32), two Clifford odd and two Clifford even eigenvectors
+of the Cartan subalgebra members
+Clifford odd
+ˆb1†
+1
+=
+01
+(+i) ,
+ˆb1
+1 =
+01
+(−i) ,
+Clifford even
+IA1†
+1
+=
+01
+[+i] ,
+IIA1†
+1 =
+01
+[−i] .
+(16)
+The two Clifford odd “basis vectors” are Hermitian conjugated to each other. The choice is made that
+ˆb1†
+1 is the “basis vector”, the second Clifford odd object is its Hermitian conjugated partner. Defining
+the handedness as Γ(1+1) = γ0γ1, Eq. (26), it follows, using Eq. (30), that Γ(1+1) ˆb1†
+1 = ˆb1†
+1 . ˆb1†
+1 is the
+right handed “basis vector” 7.
+Each of the two Clifford even “basis vectors” is self adjoint ((I,IIA1†
+1 )† = I,IIA1†
+1 ).
+Let us notice, taking into account Eqs. (30, 34), that
+{ˆb1
+1(≡
+01
+(−i)) ∗A ˆb1†
+1 (≡
+01
+(+i))}|ψoc >
+=
+IIA1†
+1 (≡
+01
+[−i])|ψoc >= |ψoc > ,
+{ˆb1†
+1 (≡
+01
+(+i)) ∗A ˆb1
+1(≡
+01
+(−i))}|ψoc >
+=
+0 ,
+IA1†
+1 (≡
+01
+[+i]) ∗A ˆb1†
+1 (≡
+01
+(+i))|ψoc >
+= ˆb1†
+1 (≡
+01
+(+i))|ψoc > ,
+IA1†
+1 (≡
+01
+[+i])ˆb1
+1(≡
+01
+(−i))|ψoc >
+=
+0 ,
+IA1†
+1 ∗A
+IIA1†
+1
+=
+0 = IIA1†
+1 ∗A
+IA1†
+1 .
+(17)
+The case with d = (3 + 1) is more informative:
+d = (3 + 1)
+In d = (3 + 1) there are 16 (2d=4) “eigenvectors” of the Cartan subalgebra members (S03, S12) and
+(S03, S12) of the Lorentz algebras Sab and Sab , Eq. (28).
+Half of them are the Clifford odd “basis vectors”, appearing in two families 2
+4
+2−1, f = (1, 2)),
+each with two (2
+4
+2−1, m = (1, 2)), members, ˆbm†
+f , and 2
+4
+2 −1× 2
+4
+2 −1 Hermitian conjugated partners ˆbm
+f
+appearing in a separate group (not reachable by Sab).
+There are 2
+4
+2 −1 × 2
+4
+2−1 Clifford even ”basis vectors”, the members of the group IAm†
+f , which are
+Hermitian conjugated to each other or are self adjoint, all reachable by Sab from any starting ”basis
+vector” IA1†
+1 . And there is another group of 2
+4
+2 −1 × 2
+4
+2−1 Clifford even ”basis vectors”, they are the
+members of IIAm†
+f , again either Hermitian conjugated to each other or are self adjoint. All are reachable
+from the starting vector IIA1†
+1 by the application of Sab.
+Choosing the right handed Clifford odd “basis vectors” as
+f = 1
+f = 2
+˜S03 = i
+2, ˜S12 = −1
+2
+˜S03 = − i
+2, ˜S12 = 1
+2
+S03
+S12
+ˆb1†
+1 =
+03
+(+i)
+12
+[+]
+ˆb1†
+2 =
+03
+[+i]
+12
+(+)
+i
+2
+1
+2
+ˆb2†
+1 =
+03
+[−i]
+12
+(−)
+ˆb2†
+2 =
+03
+(−i)
+12
+[−]
+− i
+2
+−1
+2 ,
+(18)
+7We could choose left handed “basis vectors” if choosing ˆb1†
+1 =
+01
+(−i), but the choice of handedness would remain only
+one.
+11
+
+we find for their Hermitian conjugated partners
+S03 = − i
+2, S12 = 1
+2
+S03 = i
+2, S12 = −1
+2
+˜S03
+˜S12
+ˆb1
+1 =
+03
+(−i)
+12
+[+]
+ˆb1
+2 =
+03
+[+i]
+12
+(−)
+− i
+2
+−1
+2
+ˆb2
+1 =
+03
+[−i]
+12
+(+)
+ˆb2
+2 =
+03
+(+i)
+12
+[−]
+i
+2
+1
+2 .
+(19)
+The vacuum state on which the Clifford odd ”basis vectors apply is equal to: |ψoc >=
+1
+√
+2(
+03
+[−i]
+12
+[+]
++
+03
+[+i]
+12
+[+]) 8.
+Let us recognize that all the Clifford odd ”basis vectors” are orthogonal:
+ˆbm†
+f
+∗A ˆbm′†
+f′
+= 0.
+Let us present 2
+4
+2−1 × 2
+4
+2−1 Clifford even ”basis vectors”, the members of the group IAm†
+f , which are
+Hermitian conjugated to each other or are self adjoint 9
+S03
+S12
+S03
+S12
+IA1†
+1 =
+03
+[+i]
+12
+[+]
+0
+0
+, IA1†
+2 =
+03
+(+i)
+12
+(+)
+i
+1
+IA2†
+1 =
+03
+(−i)
+12
+(−)
+−i
+−1
+, IA2†
+2 =
+03
+[−i]
+12
+[−]
+0
+0 ,
+(20)
+and 2
+4
+2−1 × 2
+4
+2 −1 Clifford even ”basis vectors”, the members of the group IIAm†
+f , m = (1, 2), f = (1, 2),
+which are again Hermitian conjugated to each other or are self adjoint
+S03
+S12
+S03
+S12
+IIA1†
+1 =
+03
+[+i]
+12
+[−]
+0
+0
+, IIA1†
+2 =
+03
+(+i)
+12
+(−)
+i
+−1
+IIA2†
+1 =
+03
+(−i)
+12
+(+)
+−i
+1
+, IIA2†
+2 =
+03
+[−i]
+12
+[+]
+0
+0 .
+(21)
+The Clifford even “basis vectors” have no families. The two groups which are not reachable by Sab are
+orthogonal.
+IAm†
+f
+∗A
+IIAm′†
+f‘
+= 0,
+for any (m, m′, f, f‘) .
+(22)
+Even dimensional spaces have the properties of the fermion and boson second quantized fields. The
+reader can find discussions about d = (5 + 1)- dimensional case in [9, 8] and the references therein.
+3.2
+“Basis vectors” in odd dimensional spaces with d = (2 + 1), (4 + 1)
+Half of the Clifford odd and Clifford even Clifford objects in 2n + 1-dimensional cases can be found by
+treating the Clifford odd “basis vectors” and their Hermitian conjugated partners and the Clifford even
+“basis vectors” in 2(2n + 1) (or 4n) dimensional part of space. The properties of these “basis vectors”
+are presented in Eqs. (6, 7, 10, 11).
+The rest of the “basis vectors” follow by the application of S0d on the “basis vectors” determining
+the internal space of fermion and boson fields in 2(2n + 1) (or 4n) dimensional part of space. Since S0d
+are the Clifford even operators, they do not change oddness or evenness of the “basis vectors” or their
+8The case SO(1, 1) can be viewed as a subspace of the case SO(3, 1), recognizing the “basis vectors”
+03
+(+i)
+12
+[+] and
+03
+(−)
+12
+[−] with
+03
+(+i) and
+03
+(−i), respectively, as carrying two different handedness in d = (1 + 1), but each of them carries a
+different “charge” S12. In the whole internal space, all the Clifford odd “basis vectors” have only one handedness.
+9Let be repeated that Sab = Sab + ˜Sab [9].
+12
+
+Hermitian conjugated partners. But they do change their properties:
+i.
+In even dimensional subspace, 2(2n + 1) of d = 2(2n + 1) + 1) (or 4n of d = 4n + 1) the
+Clifford odd “basis vectors”, ˆbm†
+f , have 2
+d−1
+2 −1 members, m, in 2
+d−1
+2 −1 families, f, and their Hermitian
+conjugated partners appear in a separate group of 2
+d−1
+2 −1 members in 2
+d−1
+2 −1 families. The Clifford even
+“basis vectors” appear in two mutually orthogonal groups, each with 2
+d−1
+2 −1× 2
+d−1
+2 −1 members.
+ii.
+The second part of “basis vectors” and their Hermitian conjugated partners, obtained from the
+first part by the application of S0d with the same number of either the Clifford odd or of the Clifford
+even objects as the first part, manifest:
+The Clifford odd “basis vectors” appear in two mutually orthogonal groups, each with 2
+d−1
+2 −1× 2
+d−1
+2 −1
+members, self adjoint or with the Hermitian conjugated partners within the same group. The Clifford
+even “basis vectors” appear in 2
+d−1
+2 −1 members, m, in 2
+d−1
+2 −1 families, f, and their Hermitian conju-
+gated partners appear in a separate group of 2
+d−1
+2 −1 members in 2
+d−1
+2 −1 families.
+iii. While ˆbm†
+f
+have in even dimensional spaces one handedness only (either right or left, depending
+on the definition of handedness), in odd dimensional spaces, the operator of handedness is a Clifford odd
+object — the product of an odd number of γa’s, Eq. (26), (still commuting with Sab) — transforming
+the Clifford odd “basis vectors” into Clifford even “basis vectors” and opposite. Correspondingly are
+the eigenvectors of the operator of handedness the superposition of the Clifford odd and the Clifford
+even basis vectors”, offering in odd dimensional spaces the right and left handed eigenvectors of the
+operator of handedness.
+Let us illustrate the above mentioned properties of the “basis vectors” in odd dimensional spaces,
+starting with the simplest case:
+d=(2+1)
+In d = (2 + 1) there are 8 (2d=3) “eigenvectors” of the Cartan subalgebra members (S01) and (S01)
+of the Lorentz algebras Sab and Sab , Eq. (29).
+Half of the Clifford odd and Clifford even “basis vectors” and their Hermitian conjugated partners
+can be taken from Eq. (16), the rest half are obtained by the application of S02 or ˜S02 on the first half.
+One obtains
+d =
+2 + 1
+Clifford odd
+ˆb1†
+1 =
+01
+(+i) ,
+ˆb1†
+2 =
+01
+[−i] γ2 ,
+ˆb1
+1 =
+01
+(−i) ,
+ˆb1
+2 =
+01
+[+i] γ2 ,
+Clifford even
+IA1†
+1 =
+01
+[+i] ,
+IA1†
+2 =
+01
+(−i) γ2 ,
+IIA1†
+1 =
+01
+[−i] ,
+IIA1†
+2 =
+01
+(+i) γ2 .
+(23)
+One clearly sees that the left hand side of the Clifford odd “basiss vectors” and the right hand side of
+the Clifford even “basis vectors”, although the first are the Clifford odd objects and the later Clifford
+even objects, have similar properties.
+Like:
+ˆb1
+1 ∗A ˆb1†
+1 = IA1†
+2 ∗A
+IIA1†
+2 =
+01
+(−i)
+01
+(+i)=
+01
+[−i]= |ψoc > .
+13
+
+And the right hand side of the Clifford odd “basis vectors” contains two self adjoint orthogonal
+“basis vectors” as the left hand side of the two Clifford even “basis vectors” does.
+Let us find the eigenvectors of the operator of handedness Γ(2+1) = iγ0γ1γ2. Since it is the Clifford
+odd object, its eigenvectors are the superposition of Clifford odd and Clifford even “basis vectors”.
+Γ(2+1){
+01
+[−i] ±i
+01
+[−i] γ2} = ∓{
+01
+[−i] ±i
+01
+[−i] γ2} ,
+Γ(2+1){
+01
+(+i) ±i
+01
+(+i) γ2} = ∓{
+01
+(+i) ±i
+01
+(+i) γ2} ,
+Γ(2+1){
+01
+[+i] ±i
+01
+[+i] γ2} = ±{
+01
+[+i] ±i
+01
+[+i] γ2} ,
+Γ(2+1){
+01
+(−i) γ2 ± i
+01
+(−i)} = ±{
+01
+(−i) γ2 ± i
+01
+(−i)} .
+(24)
+d=(4+1)
+In d = (4 + 1) there are 32 (2d=5) “eigenvectors” of the Cartan subalgebra members (S03, S12) and
+(S03, S12) of the Lorentz algebras Sab and Sab, Eq. (29).
+Half of the Clifford odd and Clifford even “basis vectors” and their Hermitian conjugated partners
+can be taken from Eqs. (18, 19, 20, 21), the rest half follows by the application of S05 or ˜S05 on the
+first half.
+d =
+4 + 1
+Clifford odd
+ˆb1†
+1 =
+03
+(+i)
+12
+[+] , ˆb1†
+2 =
+03
+[+i]
+12
+(+) ,
+ˆb1†
+3 =
+03
+[−i]
+12
+[+i] γ5 , ˆb1†
+4 =
+03
+(−i)
+12
+(+) γ5 ,
+ˆb2†
+1 =
+03
+[−i]
+12
+(−) , ˆb2†
+2 =
+03
+(−i)
+12
+[−] ,
+ˆb2†
+3 =
+03
+(+i)
+12
+(−) γ5 , ˆb2†
+4 =
+03
+[+i]
+12
+[−] γ5 ,
+ˆb1
+1 =
+03
+(−i)
+12
+[+] , ˆb1
+2 =
+03
+[+i]
+12
+(−) ,
+ˆb1
+3 =
+03
+[+i]
+12
+[+] γ5 , ˆb1
+4 =
+03
+(−i)
+12
+(−) γ5 ,
+ˆb2
+1 =
+03
+[−i]
+12
+(+) , ˆb2
+2 =
+03
+(+i)
+12
+[−] ,
+ˆb2
+3 =
+03
+(+i)
+12
+(+) γ5 , ˆb2
+4 =
+03
+[−i]
+12
+[−] γ5 ,
+Clifford even
+IA1†
+1 =
+03
+[+i]
+12
+[+] ,
+IA1†
+2 =
+03
+(+i)
+12
+(+) ,
+IA1
+3 =
+03
+(−i)
+12
+[+] γ5 ,
+IA1
+4 =
+03
+[−i]
+12
+(+) γ5 ,
+IA2†
+1 =
+03
+(−i)
+12
+(−i) ,
+IA2†
+2 =
+03
+[−i]
+12
+[−] ,
+IA2
+3 =
+03
+[+i]
+12
+(−) γ5 ,
+IA2
+4 =
+03
+(+i)
+12
+[−] γ5 ,
+IIA1†
+1 =
+03
+[−i]
+12
+[+] ,
+IIA1†
+2 =
+03
+(−i)
+12
+(+) ,
+IIA1†
+3 =
+03
+(+i)
+12
+[+] γ5 ,
+IIA1†
+4 =
+03
+[+i]
+12
+(+) γ5 ,
+IIA2†
+1 =
+03
+(+i)
+12
+(−) ,
+IIA2†
+2 =
+03
+[+i]
+12
+[−] ,
+IIA2†
+3 =
+03
+[−i]
+12
+(−) γ5 ,
+IIA2†
+4 =
+03
+(−i)
+12
+[−] γ5 .
+(25)
+One notices that the right hand side of the Clifford odd “basis vectors” appear in two mutually
+orthogonal groups, each one with either self-adjoint members or with the Hermitian conjugated partners
+within the same group.
+The members of one group
+ˆb1†
+3 =
+03
+[−i]
+12
+[+i] γ5 , ˆb1†
+4 =
+03
+(−i)
+12
+(+) γ5 , ˆb2†
+3 =
+03
+(+i)
+12
+(−) γ5 , ˆb2†
+4 =
+03
+[+i]
+12
+[−] γ5
+have the properties, except the commutativity (they are namely, the Clifford odd objects), as the group
+of Clifford even objects
+IIA1†
+1 =
+03
+[−i]
+12
+[+] , IIA1†
+2 =
+03
+(−i)
+12
+(+) , IIA2†
+1 =
+03
+(+i)
+12
+(−) , IIA2†
+2 =
+03
+[+i]
+12
+[−] .
+14
+
+The comparable properties also have the Clifford odd members of the group
+ˆb1
+3 =
+03
+[+i]
+12
+[+] γ5 , ˆb1
+4 =
+03
+(−i)
+12
+(−) γ5 , ˆb2
+3 =
+03
+(+i)
+12
+(+) γ5 , ˆb2
+4 =
+03
+[−i]
+12
+[−] γ5 ,
+and the Clifford even members of the group
+IA1†
+1 =
+03
+[+i]
+12
+[+] , IA1†
+2 =
+03
+(+i)
+12
+(+) , IA2†
+1 =
+03
+(−i)
+12
+(−i) , IA2†
+2 =
+03
+[−i]
+12
+[−] .
+The members of both groups have Hermitian conjugated partners within the same group or are self-
+adjoint.
+On the other side, the members of the Clifford even group
+IIA1†
+3 =
+03
+(+i)
+12
+[+] γ5 , IIA1†
+4 =
+03
+[+i]
+12
+(+) γ5 , IIA2†
+3 =
+03
+[−i]
+12
+(−) γ5 , IIA2†
+4 =
+03
+(−i)
+12
+[−] γ5 ,
+have their Hermitian conjugated partners in a separate group
+IA1
+3 =
+03
+(−i)
+12
+[+] γ5
+IA1
+4 =
+03
+[+i]
+12
+(−) γ5 , IA2
+3 =
+03
+[−i]
+12
+(+) γ5 , IA2
+4 =
+03
+(+i)
+12
+[−] γ5 ,
+just like the Clifford odd objects on the left hand side
+ˆb1†
+1 =
+03
+(+i)
+12
+[+] , ˆb1†
+2 =
+03
+[+i]
+12
+(+) , ˆb2†
+1 =
+03
+[−i]
+12
+(−) , ˆb2†
+2 =
+03
+(−i)
+12
+[−] ,
+which have their Hermitian conjugated partners in a separate group
+ˆb1
+1 =
+03
+(−i)
+12
+[+] , ˆb1
+2 =
+03
+[+i]
+12
+(−) , ˆb2
+1 =
+03
+[−i]
+12
+(+) , ˆb2
+2 =
+03
+(+i)
+12
+[−] .
+The “basis vectors” of the right hand side keep oddness if they are partners of the Clifford odd
+“basis vectors” on left hand side, but demonstrate properties of the Clifford even objects on the left
+hand side.
+The “basis vectors” of the right hand side keep evenness if they are partners of the Clifford even
+“basis vectors” on the left hand side, but demonstrate properties of the Clifford odd objects on the left
+hand side.
+After algebraically application of, for example, IIA1†
+3 (=
+03
+(+i)
+12
+[+] γ5 on ˆb1†
+4 =
+03
+(−i)
+12
+(+) γ5 we are left
+with ˆb1†
+2 =
+03
+[+i]
+12
+(+).
+The eigenvectors of the operator of handedness in d = (4 + 1), Γ(4+1) = γ0γ1γ2γ3γ5, are the su-
+perposition of the Clifford odd and Clifford even “basis vectors”, as for example: Γ(4+1)(ˆb1†
+1 [=
+03
+(+i)
+12
+[+]
+] ± IIA1†
+3 [=
+03
+(+i)
+12
+[+] γ5]) = ∓((ˆb1†
+1 ± IIA1†
+3 ).
+We can conclude that neither Clifford odd nor Clifford even “basis vectors”, have in odd dimensional
+spaces the properties which they do demonstrate in even dimensional spaces: The properties which
+empower the Clifford odd “basis vectors” to describe the internal space of fermion fields and the Clifford
+even “basis vectors” to describe the internal space of the corresponding gauge fields: After enlarging the
+“basis vectors” in a tensor product, ∗T, with the basis in ordinary space [9], the corresponding creation
+and annihilation operators manifest the properties required by the postulates for the second quantized
+either fermion or boson fields, respectively.
+In odd dimensional spaces, half of the Clifford odd “basis vectors” demonstrate properties of the
+Clifford even “basis vectors” and half of the Clifford even “basis vectors” demonstrate properties of the
+Clifford odd “basis vectors”. Arbitrary Lorentz transformations transform the left hand sides into the
+right sides and vice versa.
+These are properties of the internal spaces of the ghost scalar fields, used in the quantum field theory
+to make contributions of the Feynman diagrams finite.
+15
+
+4
+Discussion
+This article discusses the properties of the internal spaces of fermion and boson fields in even and odd
+dimensional spaces, if the internal spaces are described by the Clifford odd and even “basis vectors”,
+which are the superposition of odd or even products of operators γa’s. “Basis vectors” are arranged
+into algebraic products of nilpotents and projectors, which are eigenvectors of the Cartan subalgebra
+of the Lorentz algebra Sab in the internal space of fermion and bosons fields.
+The Clifford odd “basis vectors”, which are products of an odd number of nilpotents and the rest
+of projectors, offer in even dimensional spaces the description of the internal space of fermion fields.
+Each irreducible representation of the Lorentz algebra is equipped with the family quantum number
+determined by the second kind of the Clifford operators ˜γa’s. The Clifford odd “basis vectors” anti-
+commute. Their Hermitian conjugated partners appear in a different group. In a tensor product with
+the basis in ordinary space, the “basis vectors” and their Hermitian conjugated partners form the
+creation and annihilation operators which, applied on the vacuum state or on the Hilbert space ([8]
+and the references therein), fulfil the anti-commutation relations postulated for the second quantized
+fermion fields, offering therefore the explanation for the postulates.
+In d = 2(2n + 1), n ≥ 7, the creation and annihilation operators, applying on the vacuum state, or
+the Hilbert space, offer the description of all the properties of the observed quarks and leptons and
+antiquarks and antileptons ([8] and the references therein) 10.
+The Clifford even “basis vectors”, which are products of an even number of nilpotents and the rest of
+projectors offer in even dimensional spaces the description of the internal space of boson fields, the gauge
+fields of the corresponding fermion fields, described by the Clifford odd “basis vectors”. The Clifford
+even “basis vectors” commute. They do not appear in families and have their Hermitian conjugated
+partners in the same group or are self-adjoint. In a tensor product with the basis in ordinary space, the
+Clifford even “basis vectors” form the creation and annihilation operators, which fulfil the commutation
+relations postulated for the second quantized boson fields. In d = 2(2n + 1), n ≥ 7, these creation and
+annihilation operators offer the description of all the properties of the observed gauge fields as well as
+of Higgs’s scalar field, explaining also the Yukawa couplings.
+This way of describing the internal space of boson fields with the Clifford even “basis vectors”,
+although very promising, needs further studies to understand what new it can bring into understanding
+of the second quantization of fermion and boson fields. In particular, it must be understood what new,
+if anything, does bring the replacement in a simple starting action in d = 2(2n + 1), n ≥ 7, Eq. (1), of
+vielbeins, f aα, and the two kinds of the spin connection fields, ωabα (the gauge fields of Sab) and ˜ωabα
+(the gauge fields of ˜Sab) in the covariant derivative p0α
+p0α = pα − 1
+2Sabωabα − 1
+2
+˜Sab˜ωabα ,
+with
+p0α = pα −
+�
+mf
+I ˆ
+Am†
+f
+ICm
+fα −
+�
+mf
+II ˆ
+Am†
+f
+ICm
+fα .
+The relations among I ˆ
+Am†
+f
+ICm
+fα and ωabα, and II ˆ
+Am†
+f
+IICm
+fα and ˜ωabα, not discussed directly in this
+article [9], need additional study.
+Not only that the description of the internal spaces of the fermion and boson fields with the Clifford
+odd and Clifford even “basis vectors” in even dimensional spaces offers an explanation for the second
+quantized postulates for fermion and boson fields, for all the assumptions of the standard model, and
+for several so far observed phenomena, making several predictions, also the description of the internal
+spaces of the fermion and boson fields in odd dimensional spaces seems meaningful for an explanation
+10Quarks and leptons and antiquarks and antileptons appear in the same irreducible representation
+16
+
+for the ghosts, postulated by Fadeev and Popov [20]. introduced into gauge quantum field theories to
+take care of the consistency of the path integral formulation of the quantum field theory.
+Let us repeat what we have learned in this paper, Subsect. 2.2, Subsect. 3.2, about properties of the
+Clifford even and the Clifford odd objects in odd dimensional spaces:
+Neither Clifford odd nor Clifford even “basis vectors” have in odd dimensional spaces the properties
+which they do demonstrate in even dimensional spaces, the properties which empower the Clifford odd
+“basis vectors” to describe the internal space of fermion fields and the Clifford even “basis vectors” to
+describe the internal space of the corresponding gauge fields.
+In odd dimensional spaces, namely, half of the Clifford odd ”basis vectors”, although anticommuting,
+demonstrate properties of the Clifford even “basis vectors” in even dimensional spaces and half of the
+Clifford even “basis vectors”, although commuting, demonstrate properties of the Clifford odd “basis
+vectors” in even dimensional spaces. These “basis vectors” obviously resemble properties of the internal
+spaces of the ghost scalar fields, used in the quantum field theory to make contributions of the Feynman
+diagrams finite 11. These are properties of the internal spaces of the ghost scalar fields used in the
+quantum field theory to make contributions of the Feynman diagrams finite.
+Also, properties of the Clifford odd and the Clifford even ”basis vectors” in odd dimensional spaces
+need further study.
+A
+Some useful formulas
+This appendix contains helpful relations needed in this paper. For more detailed explanations, and for
+proofs, the reader is kindly asked to read [8] and the references therein.
+The operator of handedness Γd is for fermions determined as follows.
+Γ(d) =
+�
+a
+(√ηaaγa) ·
+�
+(i)
+d
+2 ,
+for d even ,
+(i)
+d−1
+2 ,
+for d odd,
+(26)
+The Clifford objects γa’s and ˜γa’s fulfil the relations
+{γa, γb}+
+=
+2ηab = {˜γa, ˜γb}+ ,
+{γa, ˜γb}+
+=
+0 ,
+(a, b) = (0, 1, 2, 3, 5, · · · , d) ,
+(γa)†
+=
+ηaa γa ,
+(˜γa)† = ηaa ˜γa .
+(27)
+In the paper the signature ηaa = diag(1, −1, −1, . . . , −1) is used.
+The choice of the Cartan subalgebra members is made for d even
+S03, S12, S56, · · · , Sd−1 d ,
+S03, S12, S56, · · · , Sd−1 d ,
+˜S03, ˜S12, ˜S56, · · · , ˜Sd−1 d ,
+Sab = Sab + ˜Sab ,
+(28)
+and for d odd
+S03, S12, S56, · · · , Sd−2 d−1 ,
+S03, S12, S56, · · · , Sd−2 d−1 ,
+˜S03, ˜S12, ˜S56, · · · , ˜Sd−2 d−1 ,
+Sab = Sab + ˜Sab .
+(29)
+11Arbitrary Lorentz transformations in odd dimensional spaces transform the left hand sides of Eqs. (14, 15, 23, 25)
+into the right sides and vice versa.
+17
+
+Nilpotents and projectors are defined as follows [1, 18, 19]
+ab
+(k):
+=
+1
+2(γa + ηaa
+ik γb) ,
+ab
+[k]:= 1
+2(1 + i
+kγaγb) ,
+(30)
+with k2 = ηaaηbb.
+One finds, taking Eq. (2) into account, and assuming
+{˜γaB
+=
+(−)B i Bγa} |ψoc > ,
+(31)
+with (−)B = −1, if B is (a function of) an odd products of γa’s, otherwise (−)B = 1 [19], |ψoc > is
+defined in Eq. (33), the eigenvalues of the Cartan subalgebra operators
+Sab
+ab
+(k)= k
+2
+ab
+(k) ,
+˜Sab
+ab
+(k)= k
+2
+ab
+(k) ,
+Sab
+ab
+[k]= k
+2
+ab
+[k] ,
+˜Sab
+ab
+[k]= −k
+2
+ab
+[k] .
+(32)
+The vacuum state for the Clifford odd ”basis vectors”, |ψoc >, is defined as
+|ψoc >=
+2
+d
+2 −1
+�
+f=1
+ˆbm
+f ∗Aˆbm†
+f
+| 1 > .
+(33)
+Taking into account Eq. (2) it follows
+γa
+ab
+(k)
+=
+ηaa
+ab
+[−k],
+γb
+ab
+(k)= −ik
+ab
+[−k],
+γa ab
+[k]=
+ab
+(−k),
+γb ab
+[k]= −ikηaa
+ab
+(−k) ,
+˜γa
+ab
+(k)
+=
+−iηaa ab
+[k],
+˜γb
+ab
+(k)= −k
+ab
+[k],
+˜γa
+ab
+[k]=
+i
+ab
+(k),
+˜γb
+ab
+[k]= −kηaa
+ab
+(k) ,
+ab
+(k)
+†
+=
+ηaa
+ab
+(−k) ,
+(
+ab
+(k))2 = 0 ,
+ab
+(k)
+ab
+(−k)= ηaa ab
+[k] ,
+ab
+[k]
+†
+=
+ab
+[k] ,
+(
+ab
+[k])2 =
+ab
+[k] ,
+ab
+[k]
+ab
+[−k]= 0 ,
+ab
+(k)
+ab
+[k]
+=
+0 ,
+ab
+[k]
+ab
+(k)=
+ab
+(k) ,
+ab
+(k)
+ab
+[−k]=
+ab
+(k) ,
+ab
+[k]
+ab
+(−k)= 0 ,
+ab
+˜
+(k)
+†
+=
+ηaa
+ab
+˜
+(−k) ,
+(
+ab
+˜
+(k))2 = 0 ,
+ab
+˜
+(k)
+ab
+˜
+(−k)= ηaa
+ab
+˜
+[k] ,
+ab
+˜
+[k]
+†
+=
+ab
+˜
+[k] ,
+(
+ab
+˜
+[k])2 =
+ab
+˜[k] ,
+ab
+˜
+[k]
+ab
+˜
+[−k]= 0 ,
+ab
+˜
+(k)
+ab
+˜[k]
+=
+0 ,
+ab
+˜
+[k]
+ab
+˜
+(k)=
+ab
+˜
+(k) ,
+ab
+˜
+(k)
+ab
+˜
+[−k]=
+ab
+˜
+(k) ,
+ab
+˜
+[k]
+ab
+˜
+(−k)= 0 .
+(34)
+One can further find
+Sac
+ab
+(k)
+cd
+(k)
+=
+− i
+2ηaaηcc
+ab
+[−k]
+cd
+[−k] ,
+Sac ab
+[k]
+cd
+[k]= i
+2
+ab
+(−k)
+cd
+(−k) ,
+Sac
+ab
+(k)
+cd
+[k]
+=
+− i
+2ηaa
+ab
+[−k]
+cd
+(−k) ,
+Sac ab
+[k]
+cd
+(k)= i
+2ηcc
+ab
+(−k)
+cd
+[−k] .
+(35)
+B
+Acknowledgment
+The author thanks Department of Physics, FMF, University of Ljubljana, Society of Mathematicians,
+Physicists and Astronomers of Slovenia, for supporting the research on the spin-charge-family theory by
+offering the room and computer facilities and Matjaˇz Breskvar of Beyond Semiconductor for donations,
+in particular for the annual workshops entitled ”What comes beyond the standard models”, and N.B.
+Nielsen, L. Bonora and M. Blagojevic for fruitful discussions which have just started on this topic and
+might hopefully continue.
+18
+
+References
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+of Math. Phys. 34 (1993) 3731-3745.
+[2] N. Mankoˇc Borˇstnik, ”Spin connection as a superpartner of a vielbein”, Phys. Lett. B 292 (1992)
+25-29.
+[3] N. Mankoˇc Borˇstnik, ”Unification of spin and charges in Grassmann space?”, hep-th 9408002,
+IJS.TP.94/22, Mod. Phys. Lett.A (10) No.7 (1995) 587-595.
+[4] P.A.M. Dirac Proc. Roy. Soc. (London), A 117 (1928) 610.
+[5] H.A. Bethe, R.W. Jackiw, ”Intermediate quantum mechanics”, New York : W.A. Benjamin, 1968.
+[6] S. Weinberg, ”The quantum theory of fields”, Cambridge, Cambridge University Press, 2015.
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+and boson fields, respectively, open new insight into next step beyond standard model”, contribution
+in this proceedings .
+[8] N. S. Mankoˇc Borˇstnik, H. B. Nielsen, ”How does Clifford algebra show the way to the
+second quantized fermions with unified spins,
+charges and families,
+and with vector and
+scalar gauge fields beyond the standard model”, Progress in Particle and Nuclear Physics,
+http://doi.org/10.1016.j.ppnp.2021.103890 .
+[9] N. S. Mankoˇc Borˇstnik, ”How Clifford algebra can help understand second quantization of fermion
+and boson fields”, [arXiv: 2210.06256. physics.gen-ph].
+[10] N. S. Mankoˇc Borˇstnik, ”Clifford odd and even objects offer description of internal space of
+fermions and bosons, respectively, opening new insight into the second quantization of fields”, The
+13th Bienal Conference on Classical and Quantum Relativistic Dynamics of Particles and Fields
+IARD 2022, Prague, 6 − 9 June [http://arxiv.org/abs/2210.07004].
+[11] N.S. Mankoˇc Borˇstnik, H.B.F. Nielsen, ”Understanding the second quantization of fermions in
+Clifford and in Grassmann space”, New way of second quantization of fermions — Part I and Part
+II, in this proceedings [arXiv:2007.03517, arXiv:2007.03516].
+[12] N. S. Mankoˇc Borˇstnik, ”How do Clifford algebras show the way to the second quantized fermions
+with unified spins, charges and families, and to the corresponding second quantized vector and scalar
+gauge field ”, Proceedings to the 24rd Workshop ”What comes beyond the standard models”, 5 - 11
+of July, 2021, Ed. N.S. Mankoˇc Borˇstnik, H.B. Nielsen, D. Lukman, A. Kleppe, DMFA Zaloˇzniˇstvo,
+Ljubljana, December 2021, [arXiv:2112.04378] .
+[13] N.S. Mankoˇc Borˇstnik, H.B. Nielsen, “Why odd space and odd time dimensions in even dimensional
+spaces?” Phys. Lett. B 486 (2000)314-321.
+[14] N.S.Mankoˇc Borˇstnik, H.B.Nielsen, ”Why Nature has made a choice of one time and three space
+coordinates?”, [hep-ph/0108269], J. Phys. A:Math. Gen. 35 (2002) 10563-10571.
+19
+
+[15] N.S. Mankoˇc Borˇstnik, H.B. Nielsen, D. Lukman, ”Unitary representations, noncompact groups
+SO(q, d-q) and more than one time”, Proceedings to the 5th International Workshop ”What Comes
+Beyond the Standard Model”, 13 -23 of July, 2002, VolumeII, Ed. Norma Mankoˇc Borˇstnik, Hol-
+ger Bech Nielsen, Colin Froggatt, Dragan Lukman, DMFA Zaloˇzniˇstvo, Ljubljana December 2002,
+hep-ph/0301029.
+[16] N.S. Mankoˇc Borˇstnik N S, ”The spin-charge-family theory is explaining the origin of families, of
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+Proceedings to the 25rd Workshop ”What comes beyond the standard models”, 6 - 12 of July, 2022,
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+[19] N.S. Mankoˇc Borˇstnik, H.B.F. Nielsen, “How to generate families of spinors”, J. of Math. Phys.
+44 4817 (2003) [arXiv:hep-th/0303224].
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+(1): 29. Bibcode:1967PhLB...25...29F. doi:10.1016/0370-2693(67)90067-6.
+20
+

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+Prompt-Based Editing for Text Style Transfer
+Guoqing Luo, Yu Tong Han, Lili Mou∗, Mauajama Firdaus
+Dept. Computing Science, Alberta Machine Intelligence Institute (Amii), University of Alberta
+∗Canada CIFAR AI Chair, Amii
+{gluo, yhan22}@ualberta.ca
+{doublepower.mou, mauzama.03}@gmail.com
+ABSTRACT
+Prompting approaches have been recently explored in text style transfer, where a textual prompt is
+used to query a pretrained language model to generate style-transferred texts word by word in an
+autoregressive manner. However, such a generation process is less controllable and early prediction
+errors may affect future word predictions. In this paper, we present a prompt-based editing approach
+for text style transfer. Specifically, we prompt a pretrained language model for style classification
+and use the classification probability to compute a style score. Then, we perform discrete search
+with word-level editing to maximize a comprehensive scoring function for the style-transfer task.
+In this way, we transform a prompt-based generation problem into a classification one, which is a
+training-free process and more controllable than the autoregressive generation of sentences. In our
+experiments, we performed both automatic and human evaluation on three style-transfer benchmark
+datasets, and show that our approach largely outperforms the state-of-the-art systems that have 20
+times more parameters. Additional empirical analyses further demonstrate the effectiveness of our
+approach.
+1
+Introduction
+Text style transfer aims to automatically rewrite a sentence by changing it from one style to another (McDonald and
+Pustejovsky, 1985), such as transferring the positive-sentiment sentence “He loves eating sandwiches” into a negative
+one “He hates eating sandwiches”. During the transfer, the style of the sentence must be changed, whereas the overall
+content should be preserved. Text style transfer has wide real-world applications, such as personalized response
+generation (Yang et al., 2017; Zheng et al., 2021), text debiasing (Nogueira dos Santos et al., 2018; Ma et al., 2020),
+text simplification (Woodsend and Lapata, 2011; Kumar et al., 2020), and stylistic headline generation (Jin et al., 2020;
+Zhan et al., 2022).
+Early work on text style transfer mainly falls into three categories: 1) Parallel supervision with labelled source–target
+sentence pairs in a sequence-to-sequence manner (Zhu et al., 2010; Rao and Tetreault, 2018; Zhang et al., 2020), 2) Non-
+parallel supervision with style labels only, such as learning latent representations of style and content separately (Shen
+et al., 2017; John et al., 2019; Goyal et al., 2021), and 3) Unsupervised generative methods, such as constructing
+non-parallel training data for learning (Lample et al., 2018b; Luo et al., 2019; Krishna et al., 2020).
+Very recently, prompting methods have been explored in text style transfer (Reif et al., 2022; Suzgun et al., 2022), as
+large-scale pretrained language models (PLMs) enable us to perform various natural language generation tasks in a
+zero-shot (Wei et al., 2022a; Sanh et al., 2022) or exemplar-based manner (Brown et al., 2020; Schick and Schütze,
+2021a). In this paper, we also follow the prompt-based setting. This does not require any training samples or labels, but
+directly performs inference with PLMs; thus, it is more challenging than the above three settings.
+Previous work uses a prompt (e.g., a piece of text “Rewrite the text to be positive:”) to query a PLM, which will then
+generate a style-transferred sentence in an autoregressive manner (Reif et al., 2022; Suzgun et al., 2022). However,
+such autoregressive generation is less controllable, as words are generated one after another by the PLM. It has the
+error accumulation problem where early prediction errors of the PLM will affect its future predictions, leading to less
+satisfactory performance in general.
+To this end, we propose a prompt-based editing approach to unsupervised style transfer. We prompt a PLM for style
+classification and use the classification probability to compute a style score. Then, we perform steepest-ascent hill
+climbing (SAHC) (Russell and Norvig, 2010) algorithm for discrete search with word-level editing (such as replacement,
+insertion, and deletion) to maximize a heuristically defined scoring function for style transfer. In this way, we transform
+arXiv:2301.11997v1  [cs.CL]  27 Jan 2023
+
+PLM
+  :
+}
+is
+The  sentiment  of  the  text 
+ 
+{
+Discrete Search
+ Rewrite   the   sentence   to   be   more   positive  
+bland
+is
+taco
+beef
+the
+Input
+:
+a.  Prompt-Based Generation
+PLM
+Candidate 
+tasty
+is
+taco
+beef
+the
+b.  Prompt-Based Editing
+bland
+taco
+beef
+is
+the
+Input
+  the    beef             is
+tasty
+taco
+Classification: negative/positive
+Figure 1: a) Prompt-based generation: previous work (Reif et al., 2022) uses a prompt to query a PLM, which generates
+a style-transferred sentence in an autoregressive manner. b) Our prompt-based editing approach involves one-word
+classification (e.g., positive or negative in sentiment transfer).
+a prompt-based generation problem into a classification problem, which involves only a style-word prediction and is
+generally believed to be easier than multiple-word predictions for sentence generation. Our approach is a training-free
+process and does not suffer from the error accumulation problem, because it performs word edits scattered throughout
+the entire sentence, rather than generating a sentence word by word. Further, we are able to combine the style score
+with other scoring functions such as fluency and semantic similarity, so that our generation process is more controllable.
+We use Eleuther AI’s GPT-J-6B (an off-the-shelf PLM)1 and conduct both automatic and human evaluations on three
+style-transfer benchmark datasets. Results show that our prompt-based editing approach largely outperforms the
+state-of-the-art prompting systems that have 20 times more parameters. Further empirical analysis verifies that our
+approach can achieve a balance between style transfer strength and content preservation, showing the effectiveness of
+our approach.
+2
+Related Work
+Prompting. Prompting methods use a piece of text to query a PLM to provide desired outputs (Liu et al., 2021). The
+simplest prompting method, perhaps, is zero-shot prompting (Wei et al., 2022a; Sanh et al., 2022; Suzgun et al., 2022),
+which directly prompts a PLM to perform a natural language processing task (see Figure 1a), but may result in returning
+less well-formatted or logical sentences (Reif et al., 2022). Another prompting method is few-shot prompting (Brown
+et al., 2020; Schick and Schütze, 2021a,b; Wei et al., 2022b); it requires several task-specific exemplars for the PLMs,
+but is able to achieve higher performance than zero-shot prompting, and thus is more widely applied in natural language
+processing tasks (Schick and Schütze, 2021a; Brown et al., 2020; Wei et al., 2022b).
+Prompting methods were initially applied to natural language classification tasks (Schick and Schütze, 2021a,b; Min
+et al., 2022), where PLMs are asked to predict the masked word given a piece of text containing the token “[MASK]”, and
+the predicted word is then projected to a label by a pre-defined verbalizer. With the emergence of various PLMs (Devlin
+et al., 2019; Radford et al., 2019; Brown et al., 2020; Raffel et al., 2020), prompting methods have recently been widely
+applied to natural language generation tasks (Liu et al., 2021), such as text style transfer (Reif et al., 2022; Suzgun et al.,
+2022), machine translation (Radford et al., 2019; Brown et al., 2020; Raffel et al., 2020), and generative commonsense
+reasoning (Wei et al., 2022a,b).
+Text style transfer. Traditional approaches to style-transfer generation can be accomplished by supervised methods
+with parallel training data (Xu et al., 2012; Zhang et al., 2015; Rao and Tetreault, 2018). However, obtaining parallel
+data is labor-intensive and time-consuming, which remains a significant challenge for this task.
+To mitigate the need for parallel data, one line of research focuses on non-parallel supervision, where it trains the model
+on a non-parallel but style-labelled corpus (Shen et al., 2017; Bao et al., 2019; Goyal et al., 2021). John et al. (2019)
+train an autoencoder of disentangled representation of content and style. Goyal et al. (2021) train multiple language
+models as discriminators for each of the target styles given the content representation. However, explicit separation of
+content and style is not always possible, because style can only be conveyed holistically for some sentences.
+1https://github.com/kingoflolz/mesh-transformer-jax
+2
+
+Another line of research is devoted to unsupervised generative methods, which constructs non-parallel training data for
+pretraining the model (Lample et al., 2018b; Li et al., 2018; Krishna et al., 2020; Riley et al., 2021). Luo et al. (2019)
+generate non-parallel training data via back-translation (Lample et al., 2018a) and apply policy gradient training to
+learn one-step mappings between the corpora of source and target styles. Reid and Zhong (2021) first train an attentive
+style classifier to perform synthesis of source-target style pairs, which are then used to train a Levenshtein editor and
+perform multi-span edits. However, these unsupervised generative methods require a complicated training process,
+which is not efficient. In addition, poor-quality data synthesis would possibly lead to low performance in general.
+Recently, researchers have developed several prompt-based approaches that generate style-transferred texts in a zero-
+shot (Suzgun et al., 2022) or exemplar-based manner (Reif et al., 2022). Such methods do not require a learning process
+or any training labels. Reif et al. (2022) use large PLMs to understand instructions inside a prompt to generate sentences
+with different styles. Suzgun et al. (2022) apply mutiple prompts to PLMs and then use a re-ranking mechanism to
+choose the candidate sentence with the highest quality.
+Our approach follows the prompt-based setting and directly performs style-transfer text generation without any training
+procedure. However, unlike other work, we transform the generation task into a classification task and perform discrete
+search, which is more controllable than autoregressive sentence generation.
+3
+Approach
+Given an input sentence x = (x1, · · · , xm), our goal is to generate a sentence y = (y1, · · · , yn) that transfers the style
+of x. Figure 1b depicts the framework of our prompt-based editing approach, where we propose to prompt a pretrained
+language model (PLM) to predict the style of a candidate sentence. Then, we perform discrete search and iteratively
+edit the candidate sentence to maximize a scoring objective that involves the PLM’s classification probability. Finally,
+the highest-scored candidate is taken as the style-transferred sentence.
+3.1
+Prompt-Based Classifier
+In previous work, researchers directly prompt a PLM to obtain style-transferred sentences (Figure 1a) (Reif et al., 2022).
+However, this could be especially challenging, as the PLM has to generate the sentence in a zero-shot or exemplar-based
+manner; such a process is autoregressive and less controllable.
+To address this, we propose to transform prompt-based generation into prompt-based classification. We query a PLM to
+obtain a style score, which involves only a one-step prediction and is much simpler than generating the whole sentence.
+Given a candidate sentence [y], we intuitively design the prompt as
+promptcls(y) ≡ The [t] of the text { [y] } is :
+(1)
+where [t] is the style-transfer task, i.e., “sentiment” or “formality” in our experiments, and “{” and “}” are text boundary
+markers (Reif et al., 2022). Notice that we have not performed prompt engineering, which is beyond the scope of this
+paper. Instead, our focus is to develop a prompt-based editing approach for text style transfer.
+Based on the above prompt, we perform next-word prediction to obtain a style probability.
+Specifically, the
+PLM computes the conditional probability of the next word w in the vocabulary given the prompt, denoted by
+PPLM(w | promptcls(y)).
+We denote si as the representative word of the ith style. This is simply chosen to be the most intuitive style word,
+namely, positive and negative for sentiment transfer and formal and informal for formality transfer. In general, the
+predicted probabilities of the two styles are PPLM(s1 | promptcls(y)) and PPLM(s2 | promptcls(y)).
+To compute the style score, we consider the ratio of the two styles. Suppose a sentence in style s1 is to be transferred to
+s2, we design the style score as:
+fsty(y) = PPLM(s2 | promptcls(y))
+PPLM(s1 | promptcls(y))
+(2)
+Such a ratio measures the candidate’s relative affiliation with different styles.2 It is more robust than the predicted
+target-style probability PPLM(s2| promptcls(y)), which could be affected by the data sample per se.
+2While our datasets only consider the transfer between two styles, our approach can be extended to multiple styles in a one-vs-one
+or one-vs-all manner.
+3
+
+Algorithm 1 Prompt-Based Editing
+1: Input: Original sentence x, iterative steps T
+2: y(0) = x
+3: for t ∈ {1, . . . , T} do
+4:
+Enumerate all edit positions and operations
+5:
+Obtain the highest-scored candidate y∗ by Eqn. (3)
+6:
+if fsty(y∗) > 1
+▷ PLM believes style transferred
+7:
+or y∗ = y(t−1)
+▷ Local optimum found
+8:
+then: return y∗
+9:
+else: y(t) = y∗
+10: return y(T )
+3.2
+Search Objective
+We apply an edit-based search for unsupervised style transfer. This follows the recent development of search-based text
+generation (Li et al., 2020; Kumar et al., 2020; Jolly et al., 2022; Liu et al., 2022; Mou, 2022), where local edits (e.g.,
+word changes) are performed to maximize a heuristically defined objective function. Specifically, our objective function
+involves three aspects:
+f(y; x) = fsty(y) · fflu(y) · fsem(y, x)
+(3)
+where the style scorer fsty is designed in §3.1; fflu and fsem are fluency and semantic scorers, mostly adopted from
+previous work and explained below.
+Language fluency. A language fluency scorer provides an approximation of how grammatically correct a candidate
+sentence y is. We follow Li et al. (2020) and use GPT2 (Radford et al., 2019) to obtain the fluency score of the candidate
+y by the geometric mean of predicted probabilities:
+fflu(y) =
+�
+�
+� t�
+i=1
+PGPT2(yi|y<i)
+� 1
+t �
+�
+α
+(4)
+where α is a hyperparameter balancing fflu with other scoring functions (Section 3.2)3.
+Semantic similarity. The semantic similarity scorer evaluates how an output y captures the semantics of an input x. In
+our work, we adopt word- and sentence-level semantic similarities as in Li et al. (2020).
+A word-level scorer focuses on keyword information, where the keywords in the input sentence x are extracted by the
+Rake system (Rose et al., 2010). Then, the RoBERTa model (Liu et al., 2019) is adopted to compute the contextualized
+representation, denoted by RBT(w, s), for a word w in some sentence s. The word-level semantic score is defined as
+the lowest similarity among all the keywords, given by
+fword(y, x) =
+min
+k∈keyword(x) max
+yi∈y cos(RBT(k, x), RBT(yi, y))
+(5)
+A sentence-level scorer computes the cosine similarity of two sentence vectors as
+fsent(y, x) = cos(y, x) =
+y⊤x
+||y|| · ||x||
+(6)
+where the sentence vectors y and x are also encoded by RoBERTa.
+Finally, the semantic similarity score is computed as the product of word- and sentence-level scores:
+fsem(y, x) = fword(y, x)β · fsent(y, x)γ
+(7)
+where β and γ are the weighting hyperparameters.
+3.3
+Discrete Search Algorithm
+We perform style-transfer generation by discrete local search using editing operations, such as word insertion, deletion,
+and replacement, following previous work (Miao et al., 2019; Li et al., 2020). However, we propose to use steepest-
+ascent hill climbing (SAHC) (Russell and Norvig, 2010) as our search algorithm.
+3Notice that a weighting hyperparameter is not needed for the style scorer fsty because the relative weight of different scorers are
+given in fflu, and fsem.
+4
+
+Our observation is that the average edit distance is 2.9 steps for sentiment transfer and 4.7 steps for formality transfer
+between the input sentences and reference outputs. Therefore, we set the maximum number of edit steps to 5 to maintain
+their resemblance. This, unfortunately, makes previous search algorithms—such as simulated annealing (SA) (Liu et al.,
+2020) and first-choice hill climbing (FCHC) (Schumann et al., 2020)—ineffective, as they cannot fully make use the
+limited search steps.
+In our work, we use the SAHC algorithm: at a search step t, SAHC enumerates every editing position and performs every
+editing operation (namely, word deletion, replacement, and insertion).4 Then it selects the highest-scored candidate
+sentence y(t) if the score f(y(t), x) is higher than f(y(t−1), x) before it reaches the maximum edit steps. Otherwise,
+SAHC terminates and takes the candidate y(t−1) as the style-transferred output. In this way, our SAHC greedily finds
+the best edit for every search step and is more powerful than SA and FCHC.
+Moreover, we design an additional stopping criterion such that the search terminates when the prompted PLM predicts
+that the source style has changed into the target one even if it has not reached the maximum edit steps. This not only
+improves time efficiency but also encourages content preservation.
+Our approach is summarized in Algorithm 1.
+4
+Experiments
+In this section, we will present an empirical evaluation of our proposed prompt-based editing approach. We will first
+introduce our datasets and setups. Then, we will show our main results, followed by detailed analyses.
+4.1
+Datasets
+We evaluated our approach on two standard style-transfer tasks: sentiment and formality.
+We used Yelp reviews (YELP) (Zhang et al., 2015) and Amazon reviews (AMAZON) (He and McAuley, 2016) for
+sentiment transfer. These two datasets are widely used in previous work (Li et al., 2018; Luo et al., 2019; John et al.,
+2019; Reif et al., 2022; Suzgun et al., 2022). YELP contains restaurant and other business reviews and was first used for
+text classification in Zhang et al. (2015). AMAZON contains product reviews obtained from the Amazon website. Both
+YELP and AMAZON datasets contain 500 positive and 500 negative sentences in the test set.
+In addition, we used Grammarly’s Yahoo Answers Formality Corpus (GYAFC) (Rao and Tetreault, 2018) for formality
+transfer. GYAFC consists of sentences that were extracted from a question-answering forum (Yahoo Answers). We
+chose the “Family & Relationships” domain following Suzgun et al. (2022). The test set contains 500 formal and 500
+informal sentences.
+4.2
+Implementation Details
+We used Eleuther AI’s off-the-shelf GPT-J-6B as the prompt-based classifier for the style score. We used a non-finetuned
+pretrained language model RoBERTa-Large (Liu et al., 2019) to encode the sentences (Section 3.2), and to predict top-k
+words as candidate edits (Section 3.3). We set k = 50 for all the sentiment and formality transfer datasets.
+For the weighting hyperparameters α, β, and γ of the search objective f(y) in Eqn. (3), they are 1
+4, 1
+6, and 1
+6 for both
+YELP and AMAZON datasets, and 1
+4, 3
+8, and 3
+8 for GYAFC dataset. This shows that the style scorer is the most important
+among all the scorers.
+We developed our proposed approach with Python 3.7 and Pytorch 1.11.0. The experiments were conducted on NVIDIA
+A100 SXM4 GPUs.
+4.3
+Evaluation Metrics
+We adopted the following automatic evaluation metrics:
+• Style transfer accuracy. This measures whether a generated output is correctly transferred. Following the
+practice in Reif et al. (2022) and Lai et al. (2021), we used a finetuned RoBERTa-Large (SiEBERT) (Hartmann
+et al., 2022) for sentiment classification, and finetuned a RoBERTa-Large (Liu et al., 2019) for formality
+classification.
+4For replacement and insertion, we follow Li et al. (2020) and choose top-k candidate words predicted by RoBERTa due to
+efficiency concerns.
+5
+
+Setting
+Method
+Model
+#Para
+YELP
+AMAZON
+(B)
+ACC%
+BLEU
+GM
+HM
+ACC%
+BLEU
+GM
+HM
+zero-shot
+Vanilla
+LLM
+128
+69.7∗
+28.6∗
+44.6
+40.6
+-
+-
+-
+-
+LLM-dialog
+128
+59.1∗
+17.6∗
+32.3
+27.1
+-
+-
+-
+-
+P&R†
+GPT-J-6B
+6
+68.6∗
+19.8∗
+35.2
+30.1
+57.1
+21.7
+35.2
+31.4
+Ours
+GPT-J-6B
+6
+73.0∗
+40.1∗
+54.1
+51.7
+72.7
+28.6
+45.6
+41.0
+few-shot
+Distant
+exemplars
+GPT-J-6B
+6
+52.8∗
+35.8∗
+43.5
+42.7
+51.0
+27.1
+37.2
+35.4
+GPT-3 babbage
+6.7
+57.8∗
+29.3∗
+41.2
+38.9
+53.3
+19.4
+32.2
+28.5
+GPT-3 curie
+13
+53.0∗
+48.3∗
+50.6
+50.5
+72.2
+22.9
+40.7
+34.8
+LLM
+128
+79.6∗
+16.1∗
+35.8
+26.8
+-
+-
+-
+-
+LLM-dialog
+128
+90.6∗
+10.4∗
+30.7
+18.7
+-
+-
+-
+-
+GPT-3 danvinci
+175
+74.1∗
+43.8∗
+57.0
+55.1
+87.3
+28.3
+49.7
+42.7
+P&R†
+GPT-J-6B
+6
+75.0∗
+42.5∗
+56.5
+54.3
+66.8
+20.5
+37.0
+31.4
+Ours
+GPT-J-6B
+6
+74.5∗
+48.9∗
+60.3
+59.0
+78.5
+37.1
+54.0
+50.4
+Table 1: Results on YELP and AMAZON test sets. #Para: Number of parameters. GM and HM: Geometric mean and
+harmonic mean of ACC% and BLEU. †We replicated Prompt & Rerank (Suzgun et al., 2022) by their released code, as
+the settings in Suzgun et al. (2022) are incompatible with other previous work. ∗Quoted from (Reif et al., 2022). Other
+results are given by our experiments. The performance of LLM and LLM-dialog is not available for AMAZON because
+these PLMs are not public.
+Method
+Model
+#Para (B)
+ACC%
+BLEU
+GM
+HM
+Distant exemplars
+GPT-J-6B
+6
+39.4
+33.1
+36.1
+36.0
+GPT-3 babbage
+6.7
+41.7
+28.8
+34.7
+34.1
+P&R
+GPT-J-6B
+6
+44.4
+32.9
+38.2
+37.8
+Ours
+GPT-J-6B
+6
+44.4
+33.4
+38.5
+38.1
+Table 2: Four-shot performance on the GYAFC dataset, considering both directions of informal ↔ formal.
+• BLEU. The BLEU score measures the semantic similarity between generated outputs and human-written
+references. Following Luo et al. (2019) and Reif et al. (2022), we used multi-bleu.perl to obtain the
+BLEU-4 score.
+• Geometric mean and harmonic mean. They are the average of the above-mentioned metrics, evaluating the
+overall performance of text style transfer. Again, this follows the standard practice in previous work (Luo
+et al., 2019; Li et al., 2020).
+We also performed human evaluation on selected style-transfer systems, detailed in Subsection 4.6.
+4.4
+Baselines
+Since our approach is based on prompting and does not require a training process, we compared our approach with the
+following state-of-the-art prompting systems:
+• Vanilla prompting. This baseline method prompts a PLM with “Here is some text: { [x] }. Here is a
+rewrite of the text, which is more [s]: {” where [x] is the input and [s] is the style word, to directly obtain a
+style-transferred sentence, shown in Figure 1a. No exemplars are used here.
+• Distant-exemplar prompting. We adopted the approach in Reif et al. (2022), which queries a large PLM
+(such as the LLM, LLM-dialog, and 175B-parameter GPT-35) with several style-transfer samples in a few-
+shot manner. However, their exemplars have a different target style from the test cases, and thus we call it
+distant-exemplar prompting.
+5We use the same prompt provided by Reif et al. (2022) to obtain results on the off-the-shelf GPT-3 babbage and GPT-J-6B for
+the YELP, AMAZON, and GYAFC datasets.
+6
+
+Dataset
+Method
+Style
+Content
+Fluency
+Average
+YELP
+Prompt & Rerank
+3.64
+3.55
+3.04
+3.41
+Our approach
+3.76
+4.24
+3.13
+3.71
+AMAZON
+Prompt & Rerank
+3.46
+3.52
+3.38
+3.45
+Our approach
+3.67
+3.97
+3.62
+3.75
+Table 3: Human evaluation on the sentiment transfer datasets. We show human ratings of style transfer strength (Style),
+content preservation (Content), and fluency. We also compute the average score of these metrics.
+• Prompt & Rerank. Suzgun et al. (2022) propose a method that generates multiple candidate outputs from
+different manually designed prompts; then, they rerank the outputs by a heuristically defined scoring function.
+It should be mentioned that the paper (Suzgun et al., 2022) adopts a setting that is non-compatible with
+prior work; specifically, they report different directions of sentiment transfer separately, while excluding
+informal-to-formal transfer in the formality experiment. Therefore, we replicated their work under the standard
+settings (Luo et al., 2019; Reif et al., 2022).
+To the best of our knowledge, Reif et al. (2022) and Suzgun et al. (2022) are the only prior studies of prompting
+methods on text style transfer.
+4.5
+Main Results
+Table 1 shows the performance of different prompting systems on the YELP and AMAZON datasets. Compared with the
+recently proposed prompting system, Prompt & Rerank (Suzgun et al., 2022), our approach outperforms by over 14 and
+3 points for GM, and 15 and 5 points for HM in the zero- and few-shot settings, respectively, averaged across the two
+datasets. Further, compared with the state-of-the-art system that uses 175B-parameters GPT-3 with distant exemplars
+(i.e., style-transfer exemplars containing source texts and outputs written in non-target styles), our approach yields
+higher GM and HM scores by more than 3, and 5 points, respectively, also averaged across the two datasets. This is a
+compelling result, as our approach yields a better balance between content preservation and style transfer strength while
+using a 20x smaller PLM.
+Table 2 shows the results of different prompting systems on the GYAFC dataset, where both informal-to-formal and
+formal-to-informal directions are considered (Luo et al., 2019; Reif et al., 2022). For a fair comparison with previous
+prompting systems, we followed Suzgun et al. (2022) and conducted experiments in a four-shot setting. As seen, our
+approach outperforms previous approaches in GM and HM scores, which is consistent with the results in Table 1. It is
+noticed that our approach achieves less improvement on the GYAFC than YELP and AMAZON datasets, as formality
+transfer is more challenging than sentiment transfer.
+4.6
+Detailed Analyses
+In this subsection, we conduct in-depth analyses to assess the effectiveness of our prompt-based editing approach. Due
+to limited time and resources, we chose the sentiment transfer datasets (YELP and AMAZON) as our testbed.
+Human Evaluation. We conducted human evaluation via pairwise comparison of system outputs to further confirm
+the superiority of our approach. Specifically, we randomly selected 100 outputs from the recently proposed Prompt-and-
+Rerank (P&R) system (Suzgun et al., 2022) and our approach based on the same GPT-J-6B model. Following Luo et al.
+(2019) and Krishna et al. (2020), we asked three human annotators, who were instructed to rate each sentence based on
+a 1–5 Likert scale (Stent et al., 2005) in terms of style transfer strength, content preservation, and fluency (Briakou et al.,
+2021). Our annotations were strictly blind; the samples from the two prompting approaches were randomly shuffled
+and the annotators did not know which approach generated the sample.
+We measured the inter-rater agreement by Fleiss’ Kappa score [1971] for the Likert scale ratings. They are 0.37, 0.42,
+and 0.39 for style transfer strength, content preservation, and fluency, respectively, and these scores are considered fair
+correlation (Fleiss, 1971).
+Table 3 presents the results of human evaluation. We observe that our prompt-based editing approach outperforms
+P&R in all three aspects, particularly in terms of content preservation. This is because with the proposed stopping
+criterion and discrete search, we avoid unnecessary edits and preserve the original content. Our approach also achieves
+a higher average score, which is consistent with the automatic evaluation results in Table 1, further demonstrating the
+effectiveness of our approach.
+7
+
+Dataset
+Model
+ACC% BLEU
+GM
+HM
+PPL
+YELP
+Full model
+73.0
+40.1
+54.1
+51.7
+122.7
+w/o style
+17.9
+25.1
+21.2
+33.9
+29.3
+w/o semantic
+74.0
+39.0
+53.7
+51.1
+124.0
+w/o fluency
+81.3
+39.3
+56.5
+53.0
+223.6
+w/o stop criterion 78.3
+25.2
+44.4
+38.1
+192.4
+AMAZON
+Full model
+72.7
+28.6
+45.6
+41.0
+137.2
+w/o style
+33.6
+20.2
+26.1
+25.3
+31.5
+w/o semantic
+71.1
+28.1
+44.7
+40.3
+116.3
+w/o fluency
+78.0
+28.6
+47.2
+41.8
+229.9
+w/o stop criterion 79.9
+19.3
+39.3
+31.1
+176.3
+Table 4: Ablation study on the sentiment transfer datasets in the zero-shot setting. PPL: Perplexity (the smaller, the
+better). In the “w/o style” setting, the model mainly optimizes toward fflu, so it achieves an extraordinarily low PPL;
+however, its style is usually not transferred, shown by extraordinarily low ACC%. Therefore, this is not a meaningful
+style-transfer setting, and is grayed out.
+Dataset
+Algorithm
+ACC%
+BLEU
+GM
+HM
+YELP
+SAHC
+73.0
+40.1
+54.1
+51.7
+FCHC
+67.2
+31.8
+46.2
+43.1
+SA
+66.0
+28.7
+43.5
+40.0
+AMAZON
+SAHC
+72.7
+28.6
+45.6
+41.0
+FCHC
+64.1
+24.8
+39.8
+35.7
+SA
+63.2
+23.7
+38.7
+34.4
+Table 5: Results of different search algorithms on the sentiment transfer datasets.
+Ablation Study. To evaluate the contribution of key components in our model, we conducted an ablation study of
+different scoring functions and our proposed stopping criterion.
+Table 4 shows that all the scorers play a role in our approach, and that the prompt-based style scorer is the most
+important one. This makes sense, as it is the only signal of the style. Without the style scorer, we would not be able to
+perform meaningful style transfer. Moreover, we find that the fluency scorer slightly hurts style accuracy and BLEU
+scores, which are the standard metrics in Luo et al. (2019). However, it significantly improves language model fluency
+(i.e., lower perplexity), which roughly estimates the fluency of text (John et al., 2019). Therefore, we deem the fluency
+scorer fflu essential to our text style transfer model.
+In addition, our approach involves a stopping criterion that terminates the search process if the PLM believes the style
+is successfully transferred. As seen from the last row of Table 4, more edit steps (w/o stop criterion) improve the style
+accuracy but drastically hurt BLEU scores. This shows that our stopping criterion is able to seek a balance between
+style transfer accuracy and content preservation.
+Discrete Search Algorithms. Our steepest-ascent hill climbing (SAHC) algorithm enumerates candidate edits, includ-
+ing word deletion, insertion, and replacement (where top-50 candidate words are considered for efficiency concerns).
+Then, SAHC selects the best one for the next round of editing, shown in Algorithm 1.
+We compare our SAHC with two stochastic optimization algorithms, first-choice hill climbing (FCHC) (Schumann
+et al., 2020) and simulated annealing (SA) (Liu et al., 2020), which are used in previous search-based text generation.
+Both FCHC and SA perform stochastic local changes to obtain a candidate sentence. If the proposed sentence is better
+than the current one, the algorithms will accept the new candidate. Otherwise, FCHC will keep the current candidate,
+while SA may still accept the candidate with a small probability.
+From Table 5, we observe that our SAHC algorithm significantly outperforms FCHC and SA in both style-transfer
+accuracy and the BLEU score. This is likely due to the limited number of edit steps, requiring that the algorithm should
+make an effective edit at every search step. The results confirm that SAHC is more suited than other discrete search
+algorithms in our scenario.
+8
+
+YELP
+Negative −→ Positive
+Positive −→ Negative
+Source
+so far i’m not really impressed
+their lunch special is a great value
+P&R
+The text is good now
+but their lunch is a great value
+Ours
+so far i’m really impressed
+their lunch special is not a great value
+AMAZON
+Negative −→ Positive
+Positive −→ Negative
+Source
+i like neutrogena products as a rule,
+for my purpose this is the perfect item.
+so this was a disappointment.
+P&R
+i like neutrogena products, so this was
+for my purpose this is the perfect item. So this text has
+a disappointment.
+two different purposes: to be a text and to be a rewrite...
+Ours
+overall i like neutrogena products as a
+but for my purpose this is not the perfect item.
+rule, so this was a success.
+GYAFC
+Informal −→ Formal
+Formal −→ Informal
+Source
+think about what good it brought about.
+i’m unsure concerning what i should do.
+P&R
+think about what good it will bring about ...
+i’m not certain about what to do next...
+Ours
+please think about what all the good news
+yeah lol really ... i’m unsure concerning what i ’ll do.
+has brought about.
+Table 6: Example outputs on the YELP, AMAZON, and GYAFC datasets. Improperly generated words are italicized.
+Case Study. We show in Table 6 several example outputs by P&R and our approach for YELP, AMAZON, and GYAFC
+datasets. We observe that the previous approach, which performs autoregressive generation, generates less controllable
+and satisfactory sentences. For example, given the source input “for my purpose this is the perfect item” in the
+positive-to-negative sentiment transfer of the AMAZON dataset, P&R generates an unrelated sentence starting with “So
+this text has”, leading to the subsequent improper word predictions “a text and to be a rewrite”.
+Our prompt-based editing approach, however, transfers the sentiment of a source sentence from positive to negative by
+inserting the words “but” and “not”, while maintaining other semantic content. This shows that our approach is able to
+generate more sensible and controllable sentences.
+In addition, we find that our approach is able to convert the style of source inputs with multiple edits. For example,
+given the source sentence “i’m unsure concerning what i should do” in formal-to-informal transfer, our approach inserts
+multiple words (“yeah”, “lol”, “really”, “...”) at the beginning and replaces “should” with “’ll” at the end, and the
+sentence is transferred to an informal one. By allowing iterative edits and examining all possible positions and editing
+operations, multiple word edits can be scattered throughout the sentence and result in a gradual transfer of style.
+5
+Conclusion
+In this paper, we propose a novel prompt-based editing approach to text style transfer that turns a prompt-based
+generation problem into a classification one. It is a training-free process and is more controllable than the autoregressive
+generation. Our experiments on sentiment and formality transfer benchmark datasets show that the proposed approach
+significantly outperforms the state-of-the-art prompting systems that has 20 times more parameters. Additional analyses
+highlight the balance between style transfer strength and content preservation, demonstrating the effectiveness of our
+approach.
+Limitation and Future Work. Our paper introduces the advantages of transforming a generation problem into a
+classification one in text style transfer, but it comes with the trade-off of requiring multiple rounds of overhead for
+search efficiency. Nevertheless, our algorithm can be implemented in a highly parallel manner when evaluating different
+candidates, and we only need five iterations. Therefore, the efficiency of our SAHC is already much higher than other
+search algorithms (such as SA) which requires several hundred search steps (Liu et al., 2020). Further, the efficiency
+can be improved by learning from the search results (Li et al., 2020), i.e., fine-tuning a PLM based on our outputs. In
+this way, our approach can be more computationally efficient.
+Another limitation is the need for manually designed prompts, which is inevitable in zero-shot prompting. Our current
+work adopts the most intuitive prompt and has not performed prompt engineering. In the future, we would like to
+investigate prompt tuning (Schick and Schütze, 2021b; Li and Liang, 2021; Wei et al., 2022a) to mitigate the reliance
+on designing prompts.
+9
+
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+

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+arXiv:2301.12927v1  [math.CV]  30 Jan 2023
+UNIVALENT FUNCTIONS WITH NON-NEGATIVE COEFFICIENTS
+INVOLVING CLAUSEN’S HYPERGEOMETRIC FUNCTION
+K. CHANDRASEKRAN, G. MURUGUSUNDARAMOORTHY, AND D. J. PRABHAKARAN
+Abstract. In this work, we derived the necessary and sufficient conditions on param-
+eters for 3F2(a,b,c
+b+1,c+1; z) Hypergeometric Function to be in the classes M∗(λ, α) and
+N ∗(λ, α) and information regarding the image of function 3F2(a,b,c
+b+1,c+1; z) belonging to
+Rτ(A, B) by applying the convolution operator in open unit disc D = {z : |z| < 1}.
+1. Introduction
+Let D = {z ∈ C : |z| < 1} be the open unit disc in the complex plane C. Let H denote
+the class of all analytic functions in D. Let A denote the family of analytic functions f
+of the form
+f(z) = z +
+∞
+�
+n=2
+an zn, z ∈ D
+(1)
+with f(0) = 0 and f ′(0) = 1 in the open unit disc D. Which is the subclass of H and
+Let, S ⊂ A,
+i.e. S denotes the class of all normalised functions that are analytic and
+univalent in open unit disc D. For the function f is given by (1) in A and g ∈ A with
+g(z) = z +
+∞
+�
+n=2
+bn zn, the convolution product of f and g is defined by
+(f ∗ g)(z) = z +
+∞
+�
+n=2
+an bn zn, z ∈ D.
+Note that the convolution product is called Hadamard Product. For more details refer [9]
+Definition 1.1. The subclass V of A consisting of functions of the form
+f(z) = z +
+∞
+�
+n=2
+an zn, z ∈ D, with an ≥ 0, n ∈ N, n ≥ 2.
+In [14], Uralegaddi et al. introduced the following two classes which are stated as:
+Definition 1.2. [14] The class M(α) of starlike functions of order α, with 1 < α ≤ 4
+3,
+defined by
+M(α) =
+�
+f ∈ A : ℜ
+�zf ′(z)
+f(z)
+�
+< α, z ∈ D
+�
+2000 Mathematics Subject Classification. 30C45, 33C20.
+Key words and phrases. Generalized Hypergeometric Series, Univalent Functions, Starlike Functions,
+Convex Functions and Alexander Integral Operator.
+Final Version as on 30-01-2023.
+1
+
+Definition 1.3. [14] The class N (α) of convex functions of order α, with 1 < α ≤ 4
+3,
+defined by
+N (α) =
+�
+f ∈ A : ℜ
+�
+1 + zf ′′(z)
+f ′(z)
+�
+< α, z ∈ D
+�
+= {f ∈ A : zf ′(z) ∈ M(α)}
+In this paper, we considere the two subclasses M(λ, α) and N (λ, α) of to discuss some
+inclusion properties based on Clausen’s Hypergeometric Function. These two subclasses
+was introduced by Bulboaca and Murugusundaramoorthy [2]. which are stated as follows:
+Definition 1.4. [2] For some α
+�
+1 < α ≤ 4
+3
+�
+and λ (0 ≤ λ < 1), the functions of the form
+(1) be in the subclass M(λ, α) of S is
+M(λ, α)
+=
+�
+f ∈ A : ℜ
+�
+zf ′(z)
+(1 − λ)f(z) + λz f ′(z)
+�
+< α, z ∈ D
+�
+Definition 1.5. [2] For some α
+�
+1 < α ≤ 4
+3
+�
+and λ (0 ≤ λ < 1), the functions of the form
+(1) be in the subclass N (λ, α) of S is
+N (λ, α)
+=
+�
+f ∈ A : ℜ
+� f ′(z) + zf ′′(z)
+f ′(z) + λz f ′′(z)
+�
+< α, z ∈ D
+�
+Also, let M∗(λ, α) ≡ M(λ, α) ∩ V and N ∗(λ, α) ≡ N (λ, α) ∩ V.
+Definition 1.6. [8] A function f ∈ A is said to be in the class Rτ(A, B), with τ ∈ C\{0}
+and −1 ≤ B ≤ A ≤ 1, if it satisfies the inequality
+����
+f ′(z) − 1
+(A − B)τ − B[f ′(z) − 1]
+���� < 1, z ∈ D
+Dixit and Pal [8] introduced the Class Rτ(A, B). Which is stated as in the definition
+1.6. If we substitute τ = 1, A = β and B = −β, (0 < β ≤ 1) in the definition 1.6, then
+we obtain the class of functions f ∈ A satisfying the inequality
+����
+f ′(z) − 1
+f ′(z) + 1
+���� < β, z ∈ D
+which was studied by Padmanabhan [12] and others subsequently.
+Definition 1.7. [1] The 3F2(a, b, c; d, e; z) hypergeometric series is defined as
+3F2(a, b, c; d, e; z) =
+∞
+�
+n=0
+(a)n(b)n(c)n
+(d)n(e)n(1)n
+zn, a, b, c, d, e ∈ C,
+(2)
+provided d, e ̸= 0, −1, −2, −3 · · · , which is an analytic function in open unit disc D.
+We consider the linear operator Ia,b,c
+b+1,c+1(f) : A → A defined by convolution product
+Ia,b,c
+b+1,c+1(f)(z) = z 3F2(a,b,c
+b+1,c+1; z) ∗ f(z) = z +
+∞
+�
+n=2
+An zn
+(3)
+where A1 = 1 and for n > 1,
+An
+=
+(a)n−1(b)n−1(c)n−1
+(b + 1)n−1(c + 1)n−1(1)n−1
+an.
+(4)
+2
+
+Motivated by the results in connections between various subclasses of analytic univalent
+functions, by using hypergeometric functions [3, 4, 5, 6, 7, 14], and Poisson distributions
+[2], we obtain the necessary and sufficient conditions on parameters for 3F2(a,b,c
+b+1,c+1; z)
+hypergeometric series to be in the classes M∗(λ, α) and N ∗(λ, α) and information regard-
+ing the image of functions 3F2(a,b,c
+b+1,c+1; z) hypergeometric series belonging to Rτ(A, B) by
+applying the Hadamard product.
+2. Main Results and Proofs
+First, we recall the following results to prove our main theorems.
+Lemma 5. [11] For some α (1 < α ≤
+4
+3) and λ (0 ≤ λ < 1), and if f ∈ V, then
+f ∈ M∗(λ, α) if and only if
+∞
+�
+n=2
+[n − (1 + nλ − λ)��]an
+≤
+α − 1.
+(6)
+Lemma 7. [11] For some α (1 < α ≤
+4
+3) and λ (0 ≤ λ < 1), and if f ∈ V, then
+f ∈ N ∗(λ, α) if and only if
+∞
+�
+n=2
+n [n − (1 + nλ − λ)α]an
+≤
+α − 1.
+(8)
+The following result is due to Miller and Paris [10] & Shpot and Srivastava [13].
+Theorem 9. For a, b, c > 0, c ̸= b and a < min(1, b + 1, c + 1),
+3F2
+�
+a,b,c
+b+1,c+1; 1
+�
+=
+bc
+c − bΓ(1 − a)
+�
+Γ(b)
+Γ(1 − a + b) −
+Γ(c)
+Γ(1 − a + c)
+�
+.
+(10)
+Now, we state the following lemma due to Chandrasekran and Prabhakaran [4] which
+is useful to prove our main results.
+Lemma 11. [4] Let a, b, c > 0. Then we have the following:
+(1) For b, c > a − 1, we have
+∞
+�
+n=0
+(n + 1)(a)n (b)n (c)n
+(b + 1)n (c + 1)n (1)n
+=
+bc Γ(1 − a)
+c − b
+� (1 − b)Γ(b)
+Γ(1 − a + b) − (1 − c)Γ(c)
+Γ(1 − a + c)
+�
+.
+(2) For b, c > a − 1, we have
+∞
+�
+n=0
+(n + 1)2(a)n (b)n (c)n
+(b + 1)n (c + 1)n (1)n
+=
+bc Γ(1 − a)
+c − b
+� (1 − b)2Γ(b)
+Γ(1 − a + b) − (1 − c)2Γ(c)
+Γ(1 − a + c)
+�
+.
+(3) For b, c > a − 1, we have
+∞
+�
+n=0
+(n + 1)3(a)n (b)n (c)n
+(b + 1)n (c + 1)n (1)n
+=
+bc Γ(1 − a)
+c − b
+� (1 − b)3Γ(b)
+Γ(1 − a + b) − (1 − c)3Γ(c)
+Γ(1 − a + c)
+�
+.
+3
+
+(4) For a ̸= 1, b ̸= 1, and c ̸= 1 with b, c > max{0, a − 1}, we have
+∞
+�
+n=0
+(a)n (b)n (c)n
+(b + 1)n (c + 1)n (1)(n+1)
+=
+bc
+(a − 1)(b − 1)(c − 1)
+×
+�Γ(2 − a)
+c − b
+� (c − 1)Γ(b)
+Γ(1 − a + b) − (b − 1)Γ(c)
+Γ(1 − a + c)
+�
+− 1
+�
+.
+Theorem 12. Let a ∈ C\{0}, b, c > 0, c ̸= b and |a| < min{1, b+ 1, c + 1}. A sufficient
+condition for the function z 3F2
+�
+a,b,c
+b+1,c+1; z
+�
+to belong to the class M∗(λ, α), 1 < α ≤ 4
+3
+and 0 ≤ λ < 1 is that
+((1 − α) − b(1 − αλ)) Γ(b)
+Γ(1 − |a| + b)
+≤ ((1 − α) − c(1 − αλ))) Γ(c)
+Γ(1 − |a| + c)
+(13)
+Proof. Let f(z) = z 3F2
+�
+a,b,c
+b+1,c+1; z
+�
+, then, by Lemma 5, it is enough to show that
+T1(α, λ)
+=
+∞
+�
+n=2
+[n − (1 + nλ − λ)α] |An| ≤ α − 1
+Using the fact |(a)n| ≤ (|a|)n, one can get
+T1(α, λ)
+=
+∞
+�
+n=2
+[n(1 − αλ) − α(1 − λ)]
+�
+(|a|)n−1 (b)n−1 (c)n−1
+(b + 1)n−1 (c + 1)n−1 (1)n−1
+�
+=
+(1 − αλ)
+∞
+�
+n=2
+n
+�
+(|a|)n−1 (b)n−1 (c)n−1
+(b + 1)n−1 (c + 1)n−1 (1)n−1
+�
+−α (1 − λ)
+∞
+�
+n=2
+�
+(|a|)n−1 (b)n−1 (c)n−1
+(b + 1)n−1 (c + 1)n−1 (1)n−1
+�
+=
+(1 − αλ)
+∞
+�
+n=0
+�(n + 1) (|a|)n (b)n (c)n
+(b + 1)n (c + 1)n (1)n
+�
+− (1 − αλ)
+−α (1 − λ)
+∞
+�
+n=0
+�
+(|a|)n (b)n (c)n
+(b + 1)n (c + 1)n (1)n
+�
++ α (1 − λ)
+Using the result (1) of Lemma 11 and the formula (10) in above mentioned equation, we
+derived that
+=
+(1 − αλ) bc Γ(1 − |a|)
+c − b
+� (1 − b)Γ(b)
+Γ(1 − |a| + b) −
+(1 − c)Γ(c)
+Γ(1 − |a| + c)
+�
+−α (1 − λ) bcΓ(1 − |a|)
+c − b
+�
+Γ(b)
+Γ(1 − |a| + b) −
+Γ(c)
+Γ(1 − |a| + c)
+�
++ α − 1
+=
+bc Γ(1 − |a|)
+c − b
+�(1 − b)(1 − αλ) Γ(b)
+Γ(1 − |a| + b)
+− (1 − c)(1 − αλ) Γ(c)
+Γ(1 − |a| + c)
+−α (1 − λ) Γ(b)
+Γ(1 − |a| + b) + α (1 − λ) Γ(c)
+Γ(1 − |a| + c)
+�
++ α − 1
+4
+
+=
+bc Γ(1 − |a|)
+c − b
+�(1 − α) − b(1 − αλ)) Γ(b)
+Γ(1 − |a| + b)
+− ((1 − α) − c(1 − αλ)) Γ(c)
+Γ(1 − |a| + c)
+�
++ α − 1
+The above expression is bounded above by α − 1 if and only if the equation (13) holds,
+which completes proof.
+□
+Theorem 14. Let a ∈ C\{0}, b, c > 0, c ̸= b and |a| < min{1, b+ 1, c + 1}. A sufficient
+condition for the function z 3F2
+�
+a,b,c
+b+1,c+1; z
+�
+to belong to the class N ∗(λ, α), 1 < α ≤ 4
+3
+and 0 ≤ λ < 1 is that
+(b − 1)(b(1 − αλ) − (1 − α))Γ(b)
+Γ(1 − |a| + b)
+≤
+(c − 1) (c(1 − αλ) − (1 − α))Γ(c)
+Γ(1 − |a| + c)
+(15)
+Proof. Let f(z) = z 3F2
+�
+a,b,c
+b+1,c+1; z
+�
+, then, by the Lemma 7, it is enough to show that
+T2(α, λ)
+=
+∞
+�
+n=2
+n [n − (1 + nλ − λ)α] |An| ≤ α − 1
+Using the fact |(a)n| ≤ (|a|)n, one can get
+T2(α, λ)
+=
+∞
+�
+n=2
+n [n(1 − αλ) − α(1 − λ)]
+�
+(|a|)n−1 (b)n−1 (c)n−1
+(b + 1)n−1 (c + 1)n−1 (1)n−1
+�
+=
+∞
+�
+n=2
+[n2 (1 − αλ) − α(1 − λ) n]
+�
+(|a|)n−1 (b)n−1 (c)n−1
+(b + 1)n−1 (c + 1)n−1 (1)n−1
+�
+Replace n = (n − 1) + 1 and n2 = (n − 1)(n − 2) + 3(n − 1) + 1 in above, we find that
+T2(α, λ)
+=
+∞
+�
+n=2
+[((n − 1)(n − 2) + 3(n − 1) + 1)]
+�(1 − αλ) (|a|)n−1 (b)n−1 (c)n−1
+(b + 1)n−1 (c + 1)n−1 (1)n−1
+�
+−
+∞
+�
+n=2
+[α(1 − λ) ((n − 1) + 1)]
+�
+(|a|)n−1 (b)n−1 (c)n−1
+(b + 1)n−1 (c + 1)n−1 (1)n−1
+�
+=
+(1 − αλ)
+∞
+�
+n=2
+�(n − 1)(n − 2) (|a|)n−1 (b)n−1 (c)n−1
+(b + 1)n−1 (c + 1)n−1 (1)n−1
+�
++(3 − 2α λ − α)
+∞
+�
+n=2
+�(n − 1) (|a|)n−1 (b)n−1 (c)n−1
+(b + 1)n−1 (c + 1)n���1 (1)n−1
+�
++(1 − α)
+∞
+�
+n=2
+�
+(|a|)n−1 (b)n−1 (c)n−1
+(b + 1)n−1 (c + 1)n−1 (1)n−1
+�
+=
+(1 − αλ)
+∞
+�
+n=3
+�
+(|a|)n−1 (b)n−1 (c)n−1
+(b + 1)n−1 (c + 1)n−1 (1)n−3
+�
+5
+
++(3 − 2α λ − α)
+∞
+�
+n=2
+�
+(|a|)n−1 (b)n−1 (c)n−1
+(b + 1)n−1 (c + 1)n−1 (1)n−2
+�
++(1 − α)
+∞
+�
+n=2
+�
+(|a|)n−1 (b)n−1 (c)n−1
+(b + 1)n−1 (c + 1)n−1 (1)n−1
+�
+=
+(1 − αλ)
+∞
+�
+n=0
+�
+(|a|)n+2 (b)n+2 (c)n+2
+(b + 1)n+2 (c + 1)n+2 (1)n
+�
++(3 − 2α λ − α)
+∞
+�
+n=0
+�
+(|a|)n+1 (b)n+1 (c)n+1
+(b + 1)n+1 (c + 1)n+1 (1)n
+�
++(1 − α)
+∞
+�
+n=1
+�
+(|a|)n (b)n (c)n
+(b + 1)n (c + 1)n (1)n
+�
+=
+(1 − αλ)
+� |a|(|a| + 1)b(b + 1)c(c + 1)
+(b + 1)(b + 2)(c + 1)(c + 2)
+� ∞
+�
+n=0
+�(|a| + 2)n (b + 2)n (c + 2)n
+(b + 3)n (c + 3)n (1)n
+�
++(3 − 2α λ − α)
+�
+abc
+(b + 1)(c + 1)
+�
+∞
+�
+n=0
+�
+(|a|)n+1 (b)n+1 (c)n+1
+(b + 1)n+1 (c + 1)n+1 (1)n
+�
++(1 − α)
+∞
+�
+n=0
+�
+(|a|)n (b)n (c)n
+(b + 1)n (c + 1)n (1)n
+�
+− (1 − α)
+Using the formula (10) in above mentioned equation, we find that
+=
+(1 − αλ)
+� |a|(|a| + 1)b(b + 1)c(c + 1)
+(b + 1)(b + 2)(c + 1)(c + 2)
+� �(b + 2)(c + 2)Γ(1 − (a + 2))
+(c + 2) − (b + 2)
+�
+×
+�
+Γ(b + 2)
+1 − (|a| + 2) + (b + 2) −
+Γ(c + 2)
+1 − (|a| + 2) + (c + 2)
+�
++(3 − 2α λ − α)
+�
+|a|bc
+(b + 1)(c + 1)
+� �(b + 1)(c + 1)Γ(1 − (|a| + 1))
+(c + 1) − (b + 1)
+�
+×
+�
+Γ(b + 1)
+1 − (|a| + 1) + (b + 1) −
+Γ(c + 1)
+1 − (|a| + 1) + (c + 1)
+�
++(1 − α) bcΓ(1 − |a|)
+c − b
+�
+Γ(b)
+Γ(1 − |a| + b) −
+Γ(c)
+Γ(1 − |a| + c)
+�
+− (1 − α)
+=
+(1 − αλ)
+�bc (−|a|)(−(|a| + 1)) Γ(1 − (|a| + 2))
+c − b
+� �(b + 1) b Γ(b)
+1 − |a| + b
+− (c + 1) c Γ(c)
+1 − |a| + c
+�
+−(3 − 2α λ − α)
+�bc(−|a|)Γ(1 − (|a| + 1))
+c − b
+� �
+b Γ(b)
+1 − |a| + b −
+c Γ(c)
+1 − |a| + c
+�
++(1 − α) bcΓ(1 − |a|)
+c − b
+�
+Γ(b)
+Γ(1 − |a| + b) −
+Γ(c)
+Γ(1 − |a| + c)
+�
+− (1 − α)
+6
+
+Using Γ(1 − a) = −aΓ(−a), the aforesaid equation reduces to
+=
+�bc Γ(1 − |a|)
+c − b
+�
+×
+�(b − 1)(b(1 − αλ) − (1 − α))Γ(b)
+Γ(1 − |a| + b)
+− (c − 1) (c(1 − αλ) − (1 − α))Γ(c)
+Γ(1 − |a| + c)
+�
++ α − 1
+The above expression is bounded above by α − 1 if and only if the equation (15) holds,
+which completes proof.
+□
+Lemma 16. [8] If f ∈ Rτ(A, B) is of the form (1), then
+|an|
+≤
+(A − B)|τ|
+n , n ∈ N ∖ {1}.
+(17)
+The result is sharp.
+Using the Lemma 16, we prove the following results:
+Theorem 18. Let a ∈ C\{0}, b, c > 0, c ̸= b and |a| < min{1, b + 1, c + 1} and
+f ∈ Rτ(A, B) ∩ V. Then Ia,b,c
+b+1,c+1(f)(z) ∈ N ∗(α, λ) if
+�bc Γ(1 − |a|)
+c − b
+�(1 − α) − b(1 − αλ)) Γ(b)
+Γ(1 − |a| + b)
+− ((1 − α) − c(1 − αλ)) Γ(c)
+Γ(1 − |a| + c)
+��
+×
+�
+(A − B) |τ|
+(1 − (A − B) |τ|)
+�
+≤
+α − 1.
+(19)
+Proof. Let f be of the form (1) belong to the class Rτ(A, B) ∩ V. Because of Lemma 7,
+it is enough to show that
+∞
+�
+n=2
+n [n(1 − αλ) − α(1 − λ)]
+�
+(|a|)n−1 (b)n−1 (c)n−1
+(b + 1)n−1 (c + 1)n−1 (1)n−1
+�
+|an| ≤ α − 1
+since f ∈ Rτ(A, B) ∩ V, then by Lemma 16, we have
+|an| ≤ (A − B)|τ|
+n , n ∈ N ∖ {1}.
+Letting
+T3(α, λ)
+=
+∞
+�
+n=2
+n [n(1 − αλ) − α(1 − λ)]
+�
+(|a|)n−1 (b)n−1 (c)n−1
+(b + 1)n−1 (c + 1)n−1 (1)n−1
+�
+|an|
+we derived that
+T3(α, λ)
+=
+(A − B) |τ|
+∞
+�
+n=2
+[n(1 − αλ) − α(1 − λ)]
+�
+(|a|)n−1 (b)n−1 (c)n−1
+(b + 1)n−1 (c + 1)n−1 (1)n−1
+�
+=
+(A − B) |τ|
+�
+(1 − αλ)
+∞
+�
+n=2
+n
+�
+(|a|)n−1 (b)n−1 (c)n−1
+(b + 1)n−1 (c + 1)n−1 (1)n−1
+�
+−α (1 − λ)
+∞
+�
+n=2
+�
+(|a|)n−1 (b)n−1 (c)n−1
+(b + 1)n−1 (c + 1)n−1 (1)n−1
+� �
+7
+
+=
+(A − B) |τ|
+�
+(1 − αλ)
+∞
+�
+n=0
+�(n + 1) (|a|)n (b)n (c)n
+(b + 1)n (c + 1)n (1)n
+�
+− (1 − αλ)
+−α (1 − λ)
+∞
+�
+n=0
+�
+(|a|)n (b)n (c)n
+(b + 1)n (c + 1)n (1)n
+�
++ α (1 − λ)
+�
+Using the result (1) of Lemma 11 and the formula (10) in above mentioned equation, we
+derived that
+=
+(A − B) |τ|
+�
+(1 − αλ) bc Γ(1 − |a|)
+c − b
+� (1 − b)Γ(b)
+Γ(1 − |a| + b) −
+(1 − c)Γ(c)
+Γ(1 − |a| + c)
+�
+−α (1 − λ) bcΓ(1 − |a|)
+c − b
+�
+Γ(b)
+Γ(1 − |a| + b) −
+Γ(c)
+Γ(1 − |a| + c)
+�
++ α − 1
+�
+=
+(A − B) |τ|
+�bc Γ(1 − |a|)
+c − b
+�(1 − b)(1 − αλ) Γ(b)
+Γ(1 − |a| + b)
+− (1 − c)(1 − αλ) Γ(c)
+Γ(1 − |a| + c)
+−α (1 − λ) Γ(b)
+Γ(1 − |a| + b) + α (1 − λ) Γ(c)
+Γ(1 − |a| + c)
+�
++ α − 1
+�
+=
+(A − B) |τ|
+�bc Γ(1 − |a|)
+c − b
+�(1 − α) − b(1 − αλ)) Γ(b)
+Γ(1 − |a| + b)
+− ((1 − α) − c(1 − αλ)) Γ(c)
+Γ(1 − |a| + c)
+�
++α − 1
+�
+The above expression is bounded above by α − 1 if and only if the equation (19) holds,
+which completes proof.
+□
+Theorem 20. Let a ∈ C\{0}, b, c > 0, c ̸= b and |a| < min{1, b + 1, c + 1} and
+f ∈ Rτ(A, B) ∩ V. Then Ia,b,c
+b+1,c+1(f)(z) ∈ M∗(α, λ) if
+�(1 − αλ) bc Γ(1 − |a|)
+c − b
+�
+Γ(b)
+Γ(1 − |a| + b) −
+Γ(c)
+Γ(1 − |a| + c)
+�
+−
+�
+α (1 − λ) bc
+(|a| − 1)(b − 1)(c − 1)
+� �Γ(2 − |a|)
+c − b
+� (c − 1)Γ(b)
+Γ(1 − |a| + b) −
+(b − 1)Γ(c)
+Γ(1 − |a| + c)
+�
+− 1
+� �
+×
+�
+(A − B) |τ|
+(1 − (A − B) |τ|)
+�
+≤ α − 1.
+(21)
+Proof. Let f be of the form (1) belong to the class Rτ(A, B) ∩ V. Because of Lemma 5,
+it is enough to show that
+∞
+�
+n=2
+[n(1 − αλ) − α(1 − λ)]
+�
+(|a|)n−1 (b)n−1 (c)n−1
+(b + 1)n−1 (c + 1)n−1 (1)n−1
+�
+|an| ≤ α − 1
+since f ∈ Rτ(A, B) ∩ V, then by Lemma 16 the inequality (17) holds. Letting
+T4(α, λ)
+=
+∞
+�
+n=2
+[n(1 − αλ) − α(1 − λ)]
+�
+(|a|)n−1 (b)n−1 (c)n−1
+(b + 1)n−1 (c + 1)n−1 (1)n−1
+�
+|an|
+8
+
+We get
+T4(α, λ)
+=
+(A − B) |τ|
+∞
+�
+n=2
+1
+n [n(1 − αλ) − α(1 − λ)]
+�
+(|a|)n−1 (b)n−1 (c)n−1
+(b + 1)n−1 (c + 1)n−1 (1)n−1
+�
+=
+(A − B) |τ|
+�
+(1 − αλ)
+∞
+�
+n=2
+�
+(|a|)n−1 (b)n−1 (c)n−1
+(b + 1)n−1 (c + 1)n−1 (1)n−1
+�
+−α (1 − λ)
+∞
+�
+n=2
+1
+n
+�
+(|a|)n−1 (b)n−1 (c)n−1
+(b + 1)n−1 (c + 1)n−1 (1)n−1
+� �
+=
+(A − B) |τ|
+�
+(1 − αλ)
+∞
+�
+n=0
+�
+(|a|)n (b)n (c)n
+(b + 1)n (c + 1)n (1)n
+�
+− (1 − αλ)
+−α (1 − λ)
+∞
+�
+n=0
+�
+(|a|)n (b)n (c)n
+(b + 1)n (c + 1)n (1)n+1
+�
++ α (1 − λ)
+�
+Using the formula (10) and the result (4) of Lemma 11 in above mentioned equation, we
+have
+=
+(A − B) |τ|
+�
+(1 − αλ) bc Γ(1 − |a|)
+c − b
+�
+Γ(b)
+Γ(1 − a + b) −
+Γ(c)
+Γ(1 − |a| + c)
+�
+−
+�
+α (1 − λ) bc
+(|a| − 1)(b − 1)(c − 1)
+� �Γ(2 − |a|)
+c − b
+� (c − 1)Γ(b)
+Γ(1 − |a| + b) −
+(b − 1)Γ(c)
+Γ(1 − |a| + c)
+�
+− 1
+�
++α − 1
+�
+The above expression is bounded above by α − 1 if and only if the equation (21) holds,
+which completes proof.
+□
+References
+[1] G.E.Andrews, R.Askey and R.Roy 1999, Special functions, Encyclopedia of Mathematics and its
+Applications, 71, Cambridge University Press, Cambridge.
+[2] T.Bulboaca and G.Murugusundaramoorthy, (2020), Univalent functions with positive coefficients
+involving Pascal distribution series, Commun. Korean Math. Soc. 35, no. 3, pp. 867–877.
+[3] K.Chandrasekran and D.J.Prabhakaran, Geometric Properties of Generalized Hypergeometric
+Functions and Stable Functions, Ph.D. Thesis, May 2022.
+[4] K.Chandrasekran and D.J.Prabhakaran, Geometric Properties of Clausen’s Hypergeometric Func-
+tions, Preprint.
+[5] K.Chandrasekran and D.J.Prabhakaran, Hohlov Type Integral Operator involving Clausen’s Hy-
+pergeometric Functions, Preprint.
+[6] K.Chandrasekran and D.J.Prabhakaran, Univalence, Starlikeness and Convexity properties of
+4F3(a1, a2, a3, a4
+b1, b2, b3
+; z) Hypergeometric Functions using convolution technique, Preprint.
+[7] K.Chandrasekran and D.J.Prabhakaran, Convolutions with Generalized Hypergeometric Functions,
+Preprint.
+[8] K. K. Dixit and S. K. Pal, On a class of univalent functions related to complex order, Indian J.
+Pure Appl. Math. 26 (1995), no. 9, 889–896.
+[9] A. W. Goodman, Univalent functions, Vol.I and Vol.II, Tampa Florida Mariner Publishing Com-
+pany, (1983).
+9
+
+[10] A. R. Miller and R. B. Paris, Clausen’s series 3F2(1) with integral parameter differences and trans-
+formations of the hypergeometric function 2F2(x), Integral Transforms Spec. Funct. 23 (2012),
+no. 1, 21–33.
+[11] G. Murugusundaramoorthy, Univalent functions with positive coefficients involving Poisson distri-
+bution series, Honam Math. J. 40 (2018), no. 3, 529–538.
+[12] K. S. Padmanabhan, On a certain class of functions whose derivatives have a positive real part in
+the unit disc, Ann. Polon. Math. 23 (1970/71), 73–81.
+[13] M. A. Shpot and H. M. Srivastava, The Clausenian hypergeometric function 3F2 with unit argument
+and negative integral parameter differences, Appl. Math. Comput. 259 (2015), 819–827.
+[14] B. A. Uralegaddi, M. D. Ganigi and S. M. Sarangi, (1994), Univalent functions with positive
+coefficients, Tamkang J. Math. 25, no. 3, pp. 225–230.
+K. Chandrasekran, Research Scholar, Department of Mathematics, MIT Campus, Anna
+University, Chennai 600 044, India
+Email address: kchandru2014@gmail.com
+G. Murugusundaramoorthy, School of Advanced Sciences, Vellore Institute of Tech-
+nology, Vellore-632014, India
+Email address: gmsmoorthy@yahoo.com
+D. J. Prabhakaran, Department of Mathematics, MIT Campus, Anna University, Chen-
+nai 600 044, India
+Email address: asirprabha@gmail.com
+10
+