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**A**: We also mention related works
[33, 13, 53, 12, 46, 54] on the non-normalizability (and other issues)**B**: See, in particular, Lemma 3.4 and the proof of (3.42)**C**: blowup profile, such that the Wick-ordered L2superscript𝐿2L^{2}italic_L start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT-cutoff does not exclude this blowup profile for any cutoff size K>0𝐾0K>0italic_K > 0
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**A**: coarsenings of 𝒩𝒩\mathcal{N}caligraphic_N**B**: In particular, 𝒩cic={ℝd}subscript𝒩𝑐𝑖𝑐superscriptℝ𝑑\mathcal{N}_{cic}=\{\mathbb{R}^{d}\}caligraphic_N start_POSTSUBSCRIPT italic_c italic_i italic_c end_POSTSUBSCRIPT = { blackboard_R start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT } is
possible even if 𝒩𝒩\mathcal{N}caligraphic_N has inscribable coarsenings**C**: It is clear that
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**A**: Lastly we will like to talk about applications of the Silverman-Tate height inequality. The inequality was used originally in [Sil83] to compare generic Neŕon-Tate heights to the fiberwise ones via a limiting process**B**: We prove similar partial results in the case where the base-space is a higher dimensional projective space and not only one dimensional curves. As mentioned earlier, this inequality is used together with other sophisticated methods to prove Uniformity in Mordell-Lang in the recent paper [dimitrov2020uniformity]. Another application of the Silverman Tate inequality is in the proof of the Geometric Bogomolov Conjecture for the function field of a curve defined over ℚ¯¯ℚ\overline{\mathbb{Q}}over¯ start_ARG blackboard_Q end_ARG in the paper [gao2019heights] by Gao and Habegger.
**C**: This relation was in turn used to show a specialisation theorem as in Theorem C of [Sil83]
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**A**: (2016); Chen et al. (2016); Dalalyan (2017); Chen et al. (2017); Raginsky et al. (2017); Brosse et al. (2018); Xu et al**B**: See, e.g., Welling and Teh (2011); Chen et al. (2014); Ma et al. (2015); Chen et al. (2015); Dubey et al. (2016); Vollmer et al**C**: (2018); Cheng and Bartlett (2018); Chatterji et al. (2018); Wibisono (2018); Bernton (2018); Dalalyan and Karagulyan (2019); Baker et al. (2019); Ma et al. (2019a, b); Mou et al. (2019); Vempala and Wibisono (2019); Salim et al. (2019); Durmus et al. (2019); Wibisono (2019) and the references therein.
Among these works,
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**A**: Furthermore, the Stokes phenomenon was used to construct a Drinfeld twist killing the KZ associator. The proof in this section is mainly motivated by [64].
**B**: The system is equivalent to the two variables Knizhnik–Zamolodchikov (KZ) equations with irregular singularities [22]**C**: Under the assumption that the eigenvalues of huℎ𝑢huitalic_h italic_u are purely imaginary, the canonical solutions of the KZ equation with prescribed asymptotics was first studied in a formal setting by Toledano Laredo [64]
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**A**:
The main results of this work holds also for analytic groups over non-archimedean local fields**B**: However since the Nevo–Zimmer factor theorem [NZ02, NZ99] is written for real Lie groups, we decided to restrict to that case as well. We remark that the proof of the Nevo–Zimmer theorem also applies with minor changes to the non-archimedean setup.**C**: The same proofs can be carried out in that generality with minor adaptations
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**A**: We record the argument in this section for the reader’s convenience**B**: First we need some standard estimates.**C**:
While the paper of Magee–Naud [48] does not explicitly state Proposition 3.6, it can be derived from Proposition 3.7 together with some additional estimates provided in their work
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**A**:
In practice, not all nodes that enter the network have the same degree, and thus it would be interesting to extend our result to the case of a random initial degree distribution**B**: Promising results on this model have been obtained in [7, 8]**C**: Moreover, in this paper we assume that the parameters of the model are known, but in many practical situations one is given a realization of the graph and the task is estimating the unknown parameters, see [10, 20, 21]. If we consider a more general class of preferential attachment graphs, for which a model-free approach is used and therefore the exact distribution of the graph is not known (see for instance [11]), we expect that the techniques presented in this paper could be used to derive central limit theorem for all the degree counts. This is an interesting open problem.
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**A**: Their method covers the Euclidean case and the algorithm has O(1/N)𝑂1𝑁O(1/N)italic_O ( 1 / italic_N ) convergence rate.
Our paper proposes an algorithm based on adding Lagrangian multipliers to consensus constraints, which is analogical to [61], but our method works in a general proximal smooth setup and achieves O(1/N)𝑂1𝑁O(1/N)italic_O ( 1 / italic_N ) convergence rate**B**: Paper [61] introduced an Extra-gradient algorithm for distributed multi-block SPP with affine constraints**C**: Moreover, it has an enhanced dependence on the condition number of the network.
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**A**: We start with a simple lemma.**B**:
Now we will focus on the regularity of the ℒℒ{\mathcal{L}}caligraphic_L-Hodge moduli space**C**: We will switch to use the ℒℒ{\mathcal{L}}caligraphic_L-connections interpretation of the points of the moduli space in our study
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**A**: In [5] a unified perspective of the problem is presented. The authors show that the MCB problem is different in nature for each class. For example in [10] a remarkable reduction is constructed to prove that the MCB problem is NP-hard for the strictly fundamental class, while in [11] a polynomial time algorithm is given to solve the problem for the undirected class. Some applications of the MCB problem are described in [5, 11, 10, 12].**B**: This problem was formulated by Stepanec [7] and Zykov [8] for general graphs and by Hubicka and Syslo [9] in the strictly fundamental class context. In more concrete terms this problem is equivalent to finding the cycle basis with the sparsest cycle matrix**C**:
The length of a cycle is its number of edges. The minimum cycle basis (MCB) problem is the problem of finding a cycle basis such that the sum of the lengths (or edge weights) of its cycles is minimum
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**A**: The second subsection will handle the case where X𝑋Xitalic_X is the increment of a centered continuous Gaussian process with nonnegatively correlated stationary increments.
**B**: This section will be broken up into two subsections**C**: The first subsection will handle the case when X𝑋Xitalic_X is a centered continuous Gaussian process with nonnegatively correlated increments
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**A**: This may be viewed as an average form of GRH for Dirichlet L𝐿Litalic_L-functions**B**: As part of his proof [6], Bombieri proved a strong form of the zero density estimate in Theorem 1.1 for Dirichlet L𝐿Litalic_L-functions**C**: We call any θ𝜃\thetaitalic_θ for which (2.7) holds a level of distribution for the primes. Elliott and Halberstam conjectured that any fixed θ<1𝜃1\theta<1italic_θ < 1 is a level of distribution for the primes.
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**A**: However, it is interesting that the proof hinges on Property (T) for suitable automorphism groups, a theme related with the circle of ideas at the origin of [PW13]. Regarding Property (T), we recall the definitions here; for full details see [BdlHV08].**B**: As we already observed, this does not imply an analog of the NFO conjecture in the realm of free groups**C**:
In this section we will show that there are extensions of free (and free abelian) groups that have no virtual excessive homology
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**A**:
The proof of Theorem 2.1 is quite involved and builds on the method of constrained chain maps developed in [18, 35] to study intersection patterns via homological minors [37]**B**: A major part of this paper, all of Sections 3 and 4, is devoted to adapt it to handle the k𝑘kitalic_k-partite structure of colorful intersection patterns.**C**: This technique, which we briefly outline here, was specifically designed for complete intersection patterns
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**A**: The normalization of transfer factors for non-strongly regular elements is needed in the analysis of the trace formula for the cohomology of Shimura varieties**B**: In particular, the results of this paper are used in work of the author that uses the cohomology of Shimura varieties to deduce new formulas for the cohomology of Rapoport-Zink spaces ([Ber21]) and related work of the author and K.H**C**: Nguyen proving the Kottwitz conjecture on the cohomology of Rapoport-Zink spaces for odd unramified unitary similitude groups ([BN23]).
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**A**: Again repeating the Frobenious map φ𝔖(aijp)subscript𝜑𝔖superscriptsubscript𝑎𝑖𝑗𝑝\varphi_{\mathfrak{S}}(a_{ij}^{p})italic_φ start_POSTSUBSCRIPT fraktur_S end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ) lands into 𝔫p2superscript𝔫superscript𝑝2\mathfrak{n}^{p^{2}}fraktur_n start_POSTSUPERSCRIPT italic_p start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_POSTSUPERSCRIPT, and so on.**B**:
In the 1st case, when the constant term of aijsubscript𝑎𝑖𝑗a_{ij}italic_a start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT is zero, the Frobenious map φ𝔖(aij)subscript𝜑𝔖subscript𝑎𝑖𝑗\varphi_{\mathfrak{S}}(a_{ij})italic_φ start_POSTSUBSCRIPT fraktur_S end_POSTSUBSCRIPT ( italic_a start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ) lands into 𝔫psuperscript𝔫𝑝\mathfrak{n}^{p}fraktur_n start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT**C**: This happens because if u∈𝔫𝑢𝔫u\in\mathfrak{n}italic_u ∈ fraktur_n then up∈𝔫psuperscript𝑢𝑝superscript𝔫𝑝u^{p}\in\mathfrak{n}^{p}italic_u start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT ∈ fraktur_n start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT i.e., all power series divisible by upsuperscript𝑢𝑝u^{p}italic_u start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT are in 𝔫psuperscript𝔫𝑝\mathfrak{n}^{p}fraktur_n start_POSTSUPERSCRIPT italic_p end_POSTSUPERSCRIPT
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**A**: We give a brief introduction of the existence and uniqueness of ground state solutions**B**: Uniqueness was obtained in dimensions N≥3𝑁3N\geq 3italic_N ≥ 3 by Yanagida [36] (see**C**: The existence of the ground state was obtained by Genoud-Stuart [19, 24] in dimensions N≥2𝑁2N\geq 2italic_N ≥ 2, and
by Genoud [20] in dimension N=1𝑁1N=1italic_N = 1
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**A**: The authors are grateful to Rafe Mazzeo, Richard Melrose and Michael Singer for helpful discussions, as well as to an anonymous referee for useful suggestions**B**: This paper is also based in part on work supported by NSF under grant DMS-1440140 while CK was in residence at the Mathematical Sciences Research Institute in Berkeley, California during the Fall 2019. FR was supported by NSERC and a Canada research chair.
**C**: CK was supported by NSF Grant number DMS-1811995
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**A**: For two vertices u𝑢uitalic_u and v𝑣vitalic_v of ΓΓ\Gammaroman_Γ, a path from u𝑢uitalic_u to v𝑣vitalic_v is a sequence of vertices (u0,u1,…,uk)subscript𝑢0subscript𝑢1…subscript𝑢𝑘(u_{0},u_{1},\dots,u_{k})( italic_u start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_u start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) of ΓΓ\Gammaroman_Γ such that u0=usubscript𝑢0𝑢u_{0}=uitalic_u start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = italic_u, uk=vsubscript𝑢𝑘𝑣u_{k}=vitalic_u start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_v and each
step ui→ui+1→subscript𝑢𝑖subscript𝑢𝑖1u_{i}\to u_{i+1}italic_u start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT → italic_u start_POSTSUBSCRIPT italic_i + 1 end_POSTSUBSCRIPT is a directed edge in ΓΓ\Gammaroman_Γ**B**: Let ΓΓ\Gammaroman_Γ be a weighted lattice graph**C**: For a path p𝑝pitalic_p,
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**A**: starting with 00**B**: Since n=1𝑛1n=1italic_n = 1 is trivial, we only consider
n⩾2𝑛2n\geqslant 2italic_n ⩾ 2**C**: While U2subscript𝑈2U_{2}italic_U start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT is an involution, Unsubscript𝑈𝑛U_{n}italic_U start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT has order 4444
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**A**: One of the key features of the Boltzmann equation (or more generally, kinetic equations) is their relation to the fluid equations. Generally speaking, there is an important dimensionless number for the Boltzmann equation—the Knudsen number ε>0𝜀0\varepsilon>0italic_ε > 0, which is the ratio of the mean free path to the macroscopic length scale, as mentioned above. The Knudsen number measures how fluid-like of the Boltzmann equation is**B**: In other words, the solutions to the Boltzmann equation can converge to compressible/incompressible Navier-Stokes and Euler equations. For the formal derivations of these fluid equations, see [2]. Rigorously justifying these limiting process in different contexts of solutions (renormalized, or classical, or mild solutions) is an active research field in the last four decades.
**C**: The smaller the Knudsen number is, the more fluid-like the Boltzmann equation is. In this sense, we call the limiting process of ε→0→𝜀0\varepsilon\rightarrow 0italic_ε → 0 as hydrodynamic limit, or fluid limit. For the different physical scalings, the Boltzmann equation can formally converge to different fluid equations, both compressible and incompressible, or both viscous and inviscid
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**A**: Including the energy as an additional datum may seem a bit superfluous at first sight, but it is indispensable in order to obtain the semigroup property for every time. Otherwise one would only obtain a statement for a.e. time as in [5, 18]. We refer to Remark 3.10 and to [28, Section 6] for a further discussion of this issue.**B**:
Inspired by [5, 6, 7], we overcome this issue by including an auxiliary variable, i.e. the energy E𝐸Eitalic_E, as a part of the solution. Accordingly, a weak solution in our context actually consists of a triple [u,p,E]𝑢𝑝𝐸[u,p,E][ italic_u , italic_p , italic_E ] of the fluid velocity and the pressure together with the energy**C**: In addition, the Navier-Stokes system (1.1), (1.2) needs to be supplemented by a suitable form of an energy inequality. As usual, see for e.g. [26, Section 4.1.2], the pressure can be recovered from the velocity so we do not specify it in our results or in the definition of a solution, see Definition 2.5
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**A**: Following Const ,
we adopt a compact set K⊂S𝐾𝑆K\subset Sitalic_K ⊂ italic_S required by Lemma 2.1.**B**: For the unforced case a=0𝑎0a=0italic_a = 0, similar result has been presented in (Lin, , pp**C**: 1824-1825) originated from (Const, , Proof of Theorem 1.4)
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**A**: In the literature, additional methods for testing the independence of two multidimensional random variables have emerged, including those based on the L1subscript𝐿1L_{1}italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT-norm between the distribution of the vector and the product of the distributions of its components (Gretton and Györfi (2010)), on ranks of distances (Heller, Heller and Gorfine, 2013), on nearest neighbor methods (Berrett and Samworth, 2019) and on applying distance covariance to center-outward ranks and signs (Shi, Drton and Han, 2022a). In Jin and Matteson (2018), three distinct measures of mutual dependency are proposed**B**: Due to the “curse of dimensionality,” these tests may fail to detect the non-linear dependence when the random vectors are of high dimension. To address this issue, Qiu, Xu and Zhu (2023) proposed a general framework for testing the dependence of two random vectors by randomly selecting two subspaces composed of vector components.
Zhang, Gao and Ng (2023) proposed a new class of independence measures based on the maximum mean discrepancy in Reproducing Kernel Hilbert Space**C**: One of the approaches extends the concept of distance covariance from pairwise dependency to mutual dependence, and the remaining two measures are derived by summing the squared distance covariances. Finally,
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**A**:
The plan of the paper is as follows**B**: In section three we will use a modification of the power series method of [Br] to prove Theorem 1.4. In section four we will use integral representation of hℎhitalic_h and construction of appropriate auxilliary functions to prove the asymptotic behaviour of the solution of (1.11). In section five we will prove Theorem 1.8 and Theorem 1.9.**C**: In section two we will use fixed point technique to prove the existence of unique solution of (1.8)
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**A**: These situations naturally happen if we consider stchastic differential equations on orthonormal frame bundles on general complete Riemannian manifolds. Lévy processes on Riemannian manifolds constructed in [App95] are typical examples. **B**:
This result is a generalization of the result of [PE]**C**: In [PE] this kind of a result was shown only in the case where the jumps of semimartingales can be uniquely connected by a minimal geodesic, but our result includes some cases where this assumption is not satisfied
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**A**: For this reason we always use the notation V𝑉Vitalic_V instead of L𝐿Litalic_L**B**: We only do this to simplify the introduction below to ordinal definability and minimal extensions, and provide a more concrete context for the reader who is less familiar with the subject.**C**: This can be done as all the equivalence relations we talk about are in L𝐿Litalic_L, and the questions of Borel reducibility and strong ergodicity between them are absolute.
This restriction to L𝐿Litalic_L is not necessary, and will never in fact be used
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**A**: Thanks to Lemma 2.3 we have that 𝒜𝒜\mathcal{A}caligraphic_A is a dissipative operator**B**: From Lemma 2.4 we showed that the operator 𝒜𝒜\mathcal{A}caligraphic_A is maximal and Lemma 2.5 gives that the operator ℬℬ\mathcal{B}caligraphic_B is Lipschitz and continuous. Finally, Thanks to the nonlinear Hille-Yosida theorem ([8]), we deduce that the Cauchy problem**C**:
Now, let us summarize the previous results
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**A**: The same argument holds for Bogdan, which allows keeping the basic structure of trades quite simple in most situations. In an economy of scale, Aki and Bogdan can transact over long distances by relying on a network of mutual bonds of trusts defining a banking system.
**B**: In a legally standard environment, all parties in a trade can enforce exchange on the spot: if Aki pays Bogdan, but Bogdan refuses to give Aki apples, Aki can go to the police, refuse to leave the shop until a refund or the goods are received, or hit Bogdan on the head, depending on the jurisdiction in which they are**C**: Now, an environment for a transaction is ‘standard’ if trades happen in-person and in a situation where at least some basic form of law enforcement exists
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**A**: See [48] for a rapid overview and references, [69, Section 5.3] for a ‘monotone’ version closely related to that used here, and**B**:
A standard criterion for non-vanishing of Floer cohomology for a Lagrangian torus is the existence of a critical point of an appropriate potential (usually, a potential defined from the curved A∞subscript𝐴A_{\infty}italic_A start_POSTSUBSCRIPT ∞ end_POSTSUBSCRIPT-structure on a space of weak bounding cochains, or a disc potential in the sense introduced previously, cf**C**: Remark 4.1)
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**A**: The paper is organized as follows**B**: In sections 2 and 3, we stated the analogs of Ramanujan’s Master Theorem for the k𝑘kitalic_k-gamma function and p𝑝pitalic_p-k𝑘kitalic_k gamma function along with their examples**C**: In section 4, we established certain extensions of Ramanujan’s Master Theorem along with their corollaries and applications to calculate some definite integral. Next, in section 5, we have established a double integral identity of Ramanujan’s Master Theorem along with its examples and corollaries.
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**A**: The idea of the proof is very similar to the one in Jaggi [2013]. In a nutshell, as the primal progress per iteration is directly related to the step size times the Frank-Wolfe gap, we know that the Frank-Wolfe gap cannot remain indefinitely above a given value, as otherwise we would obtain a large amount of primal progress, which would make the primal gap become negative. This is formalized in Theorem 2.6.**B**: More specifically, the minimum of the Frank-Wolfe gap over the run of the algorithm converges at a rate of 𝒪(1/t)𝒪1𝑡\mathcal{O}(1/t)caligraphic_O ( 1 / italic_t )**C**:
Furthermore, with this simple step size we can also prove a convergence rate for the Frank-Wolfe gap, as shown in Theorem 2.6
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**A**: In fact, not much modification will be needed.
**B**: In the previous section, we showed how to construct solutions and prove sharp decay in the case of polynomially decaying boundary data on hypersurfaces of constant r=R𝑟𝑅r=Ritalic_r = italic_R**C**: We now outline how to generalise to spherically symmetric hypersurfaces on which r𝑟ritalic_r is allowed to vary
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**A**: Moving into the 1980s, mathematicians delved into the representation type of triangular matrix algebras over different classes of algebras, for example, see [S] for Nakayama algebras, [HM] for self-injective algebras, [L2] for radical square zero algebras, etc. The most recent progress in this field can be attributed to Leszczyn´´n\acute{\text{n}}over´ start_ARG n end_ARGski and Skowron´´n\acute{\text{n}}over´ start_ARG n end_ARGski, as clear in their series of papers [L1, LS1, LS2]**B**: These papers provide a complete description of non-wild tensor product algebras. However, it is still open to distinguish representation-finite cases and tame cases from non-wild cases.**C**:
The representation type of tensor product algebras has been studied in various contexts. In the 1970s, Bondarenko and Drozd [BD] considered the representation type of finite groups, while Auslander and Reiten [AR] dedicated their effort to the representation type of triangular matrix rings
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**A**: CPP enjoys large flexibility in both the compression method and the network topology**B**: We show CPP achieves linear convergence rate under strongly convex and smooth objective functions.
**C**: In this paper, we consider decentralized optimization over general directed networks and propose a novel Compressed Push-Pull method (CPP) that combines Push-Pull/𝒜ℬ𝒜ℬ\mathcal{A}\mathcal{B}caligraphic_A caligraphic_B with a general class of unbiased compression operators
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**A**: We make a detailed comparison with them in Appendix C. Due to the fact that we consider a personalized setting, we can have a significant gain in communications. For example, when λ=0𝜆0\lambda=0italic_λ = 0 or small enough in (1) the importance of local models increases and we may communicate less frequently.
We now outline the main contribution of our work as follows (please refer also Table 1 for an overview of the results):**B**: In the literature, there are works on general (non-personalized) SPPs**C**: To the best of our knowledge, this paper is the first to consider decentralized personalized federated saddle point problems, propose optimal algorithms and derives the computational and communication lower bounds for this setting
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**A**: I thank Nguyen Viet Dang, Colin Guillarmou and Gabriel Rivière for fruitful discussions**B**: This project has received funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement No. 725967).
**C**: I also thank the anonymous referees for interesting suggestions that helped improve the quality of this manuscript
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**A**: In §3 we fix some quiver notation, and discuss Theorem 1.1**B**: §4 has the dual purpose of collecting all of the category theory that we will need in the paper, and establishing formality in general 2CY categories.**C**:
In §2 we collect some facts from the theory of mixed Hodge structures and mixed Hodge modules, as well as describing the extension of the theory to a certain class of stacks including all of the stacks mentioned in §1.1
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**A**: The idea is similar to what Frohn has done in the proof of [frohn2004accp, Lemma 1].
**B**: A quotient module of an ACCC module is not necessarily ACCC**C**: However, in the next lemma we show that certain chains of cyclic submodules in a quotient module of an ACCC module stabilize
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**A**: (7.15)] it was for singular values.
Second, in [11, Section 7] it was sufficient to consider singular values close to zero hence**B**: First, the DBM in this paper is for eigenvalues (see (4.7) below) while in [11, Eq**C**: (which itself is based upon [28]) with two minor differences
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**A**: In 1950, H**B**: Busemann and A.D**C**: Aleksandrov generalized the concept of geodesic metric spaces based on the concept of manifolds with a non-positive sectional curvature. Gromov suggested the notation CAT(0)CAT0\mathrm{CAT}(0)roman_CAT ( 0 ) for a non-positive curvature geodesic metric space. The letters C, A and T in CAT(0)CAT0\mathrm{CAT}(0)roman_CAT ( 0 ) stand for Cartan, Aleksandrov and Toponogov, respectively.
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**A**: In view of the (3.3),**B**: italic_k end_POSTSUBSCRIPT consisting
of the points (u1,…,um−2,cm−1,cm)subscript𝑢1…subscript𝑢𝑚2subscript𝑐𝑚1subscript𝑐𝑚(u_{1},\dots,u_{m-2},c_{m-1},c_{m})( italic_u start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_u start_POSTSUBSCRIPT italic_m - 2 end_POSTSUBSCRIPT , italic_c start_POSTSUBSCRIPT italic_m - 1 end_POSTSUBSCRIPT , italic_c start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT )**C**: Since (c1,…,cm)∈VN,ksubscript𝑐1…subscript𝑐𝑚subscript𝑉𝑁𝑘(c_{1},\dots,c_{m})\in V_{N,k}( italic_c start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , … , italic_c start_POSTSUBSCRIPT italic_m end_POSTSUBSCRIPT ) ∈ italic_V start_POSTSUBSCRIPT italic_N , italic_k end_POSTSUBSCRIPT, one obtains a sub-variety of the VN.ksubscript𝑉formulae-sequence𝑁𝑘V_{N.k}italic_V start_POSTSUBSCRIPT italic_N
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**A**: In fact, Chartrand, Gould, and Kapoor show in [Chartrand] that there exist HTNH graphs of any order n≥9𝑛9n\geq 9italic_n ≥ 9**B**: Hu and Zhan extended this in [HU21] to find families of regular HTNH graphs. In all of these papers, the authors construct graphs by starting with some base graph and building new graphs (with larger vertex sets) inductively.**C**:
It is natural to explore the reverse implications; more generally, one might ask ‘how many hamiltonian paths can a non-hamiltonian graph contain?’ Every cycle graph is hamiltonian but not hamiltonian-connected, and the Petersen graph is (perhaps unsurprisingly) an example of a homogeneously traceable non-hamiltonian (HTNH) graph
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**A**: We emphasize that this result is global**B**: The idea of the proofs in the following lemmas are parallel with those in MR2061575 ; MR2142598 .**C**:
We begin by showing reduction in the value of the objective function of Algorithm 1 when the backtracking line search (Armijo-rule) is satisfied
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**A**: In Section 11, Theorem 11.1, we prove Conjecture 9.1 in the form (9.4) for a semisimple adjoint F𝐹Fitalic_F-split group 𝐆𝐆{\mathbf{G}}bold_G**B**:
In [Re02], Reeder proves that this elliptic correspondence holds in the case of irreducible representations with Iwahori-fixed vectors of a split adjoint group**C**: In Section 11.4, we explain how this result could be extended to arbitrary isogenies and, in particular, in Corollary 11.13 we prove it in the Iwahori-fixed case for an arbitrary F𝐹Fitalic_F-split group 𝐆𝐆{\mathbf{G}}bold_G.
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**A**: It allows us to pass from the gap of compression on points to the relative distance to the origin**B**: Proposition 3.1 offers us an extremely useful identity**C**: It tells us that points under compression with a large gap must be far away from the origin than points with a relatively smaller gap under compression. That is to say, the inequality
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**A**: [VSCC93, Section II.5].**B**: We give a short application just to illustrate such idea, and refer to more general related analysis in the context of symmetric Markovian semigroups to e.g**C**:
As an immediate application of the Nash inequality, one can compute the decay rate for the heat equation for the sub-Laplacian, following Nash’s argument
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**A**:
We will need one lemma concerning infinitary derivations**B**: We take more care than is typical, carrying out the proof in 𝖠𝖢𝖠0subscript𝖠𝖢𝖠0\mathsf{ACA}_{0}sansserif_ACA start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT using the formalization of proofs given in Definition 2.6.**C**: Given the standard definition of infinitary proof calculi, one can establish the following lemma easily by transfinite induction
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**A**: But if we apply this procedure inductively we obtain the Gieseker-Harder-Narasimhan filtration.**B**: [HLHJ21]**C**: It should be pointed out that the filtration
f𝑓fitalic_f which maximizes ν𝜈\nuitalic_ν does not have the property that the graded pieces of the filtration are semistable
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**A**:
By Theorem 1.1, one can recover the classification of 3-dimensional complete gradient Yamabe solitons**B**: In fact, by non-existence theorem of compact gradient Yamabe solitons (cf**C**: [12]), (1) of Theorem 1.1 cannot happen. We will consider the case (3) of Theorem 1.1.
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**A**: Secondly, we compare the proposed Algorithm 4 with AdaMirr algorithm, which was recently proposed in [1]**B**: In this section, in order to demonstrate the performance of the proposed Algorithms, we firstly consider some numerical experiments concerning the Intersection of Ellipsoids Problem (IEP)**C**: We also, consider some numerical experiments concerning the Support Vector Machine (SVM) [8, 13].
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**A**: [10] for a survey), where the colors of any two adjacent vertices have to differ by at least k𝑘kitalic_k and the colors of any two vertices within distance 2222 have to be distinct.
**B**: This description draws a comparison e.g**C**: to L(k,1)𝐿𝑘1L(k,1)italic_L ( italic_k , 1 )-labeling problem (see e.g
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**A**: Let α+β=γ𝛼𝛽𝛾{\alpha+\beta}=\gammaitalic_α + italic_β = italic_γ be a partition of γ𝛾\gammaitalic_γ**B**: There exists a canonical embedding
**C**: The above definition coincides with the Induction functor given in [KL09, Section 3.1] by [KL09, Propositional 3.3]
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Selection 2
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**A**: The reflection principle has been one of the key elements in various lattice path models providing explicit enumerative formulas. It was generalized by Gessel and Zeilberger [GZ] to lattice walks on Weyl chambers, which are the regions preserved under the actions of Weyl reflection groups. The above mentioned case corresponds to counting paths in Weyl chamber of type A1subscript𝐴1A_{1}italic_A start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT.
**B**: A similar method was introduced by André [Andre] in 19th century to solve the two candidate ballot problem by counting unfavorable records and subtracting them from the total number of records. The term ’reflection principle’ was attributed to André in the books of M. Feller [F] and J**C**: L. Doob [D]. For the detailed history of the reflection principle, see [R]
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**A**: We present several logical consequences of independence logic and demonstrate how they can be interpreted in the context of empirical and hidden-variable teams**B**: Semantic proofs are due to [1],
**C**: In many cases we can derive the logical consequence relation from the axioms of Definition 2.5
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**A**:
and ξμsuperscript𝜉𝜇\xi^{\mu}italic_ξ start_POSTSUPERSCRIPT italic_μ end_POSTSUPERSCRIPT are given by the Killing solution (4.2)**B**: This algebra coincides with that of the linear wave equation when the infinite-dimensional subalgebra is factored out.**C**: The corresponding symmetry algebra has a basis given by (5.5)
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Selection 4
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**A**: In particular, properties of projective spaces as well as the Segre variety appeared to be crucial**B**:
The question of existence of product vectors in a subspace of the tensor product of two Hilbert spaces can be formulated in terms of algebraic geometry**C**: For more information the reader may consult [14]. The following Proposition stated in [15] is a special case of basic theorem given in [14].
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Selection 2
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**A**: It is possible that one can resolve these issues and introduce surfaces into the free product situation - using something like the improved relative train tracks of [3], especially when the underlying group is free, and the given automorphism is relatively irreducible with respect to a free factor system - which one can think of as the main case of interest for the free product theory**B**: We would reiterate here that this process has not formally been done, and we feel that the subtleties and differences between free groups and free products require some caution in simply hand-waving through techniques which may not be entirely valid.
**C**: Taking that route would simplify our Section 6 a little, which is a generalisation of similar results in [2] and [10], but we have chosen not to do that since it seems to us that fully taking into account the subtleties of relative surfaces would add length and complexity in a different way
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Selection 3
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**A**: However, this happens outside of the variety which interests us,**B**: intersection Dy∩{∑Dzi+∑Dwi.}D_{y}\cap\{\sum D_{z_{i}}+\sum D_{w_{i}}.\}italic_D start_POSTSUBSCRIPT italic_y end_POSTSUBSCRIPT ∩ { ∑ italic_D start_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT + ∑ italic_D start_POSTSUBSCRIPT italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT **C**: }), where Dzisubscript𝐷subscript𝑧𝑖D_{z_{i}}italic_D start_POSTSUBSCRIPT italic_z start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT and Dwisubscript𝐷subscript𝑤𝑖D_{w_{i}}italic_D start_POSTSUBSCRIPT italic_w start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT are the
divisors that appear below
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Selection 2
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**A**: Note that τ≠2,α+1,Δ𝜏2𝛼1Δ\tau\neq 2,\alpha+1,\Deltaitalic_τ ≠ 2 , italic_α + 1 , roman_Δ**B**: Case 1: u∼s1similar-to𝑢subscript𝑠1u\sim s_{1}italic_u ∼ italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and φ(us1)𝜑𝑢subscript𝑠1\varphi(us_{1})italic_φ ( italic_u italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) is not a ΔΔ\Deltaroman_Δ-inducing color.
Let φ(us1)=τ𝜑𝑢subscript𝑠1𝜏\varphi(us_{1})=\tauitalic_φ ( italic_u italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = italic_τ**C**: We first show that us1𝑢subscript𝑠1us_{1}italic_u italic_s start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT can not be 1 under any (L,φ)𝐿𝜑(L,\varphi)( italic_L , italic_φ )-stable coloring.
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**A**: I was inspired by Han-Zhang [9]**B**: To state our results we first introduce some notations.**C**:
In this short note we obtain some local and global upper bounds for the Hessian of a positive solution to the conjugate heat equation coupled with the Ricci flow
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**A**: In Section 3, we provide an example of a Postnikov presentation for unbounded chain complexes**B**: It provides inductive methods to compute homotopy limits of chain complexes. In Section 4, we recall some of the results in [Pér20b] and show that comodules also admit an efficient Postnikov presentation. Finally, we apply our methods to prove our main result on Dold-Kan correspondence for comodules and A𝐴Aitalic_A-theory in Section 5.
**C**: In Section 2, we introduce all the dual notions and methods of [SS03]: fibrantly generated model categories, Postnikov presentations of a model category, left-induced model categories, fibrant-compatible monoidal structure on model categories and weak opmonoidal Quillen equivalences between them
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**A**:
For β𝛽\betaitalic_β-log gases loop equations have proved to be a very efficient tool in the study of global fluctuations, see [BG24, BG13, Joh98, KS10, Shc13] and the references therein**B**: In the discrete setup loop equations have been used to study global fluctuations in [BGG17] and edge fluctuations in [GH19] for discrete β𝛽\betaitalic_β-ensembles. It is our strong hope that the multi-level loop equations we derive in the present paper can also be used to study the global fluctuations of continuous and discrete β𝛽\betaitalic_β-corners processes.**C**: Since their introduction loop equations have also been used to prove local universality for random matrices [BEY14, BFG15]
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**A**: Usually, D𝐷Ditalic_D is assumed to be a symmetric and semi-positive definite**B**: In this paper, we only make some assumptions
that can guarantee the invertibility of 𝒜𝒜\mathcal{A}caligraphic_A. We assume that A𝐴Aitalic_A and the Schur complement Schur(𝒜)Schur𝒜\mbox{Schur}(\mathcal{A})Schur ( caligraphic_A )**C**: KKT system or saddle point system
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**A**: Kilmer2008third introduced a novel form of tensor multiplication that enables the representation of a third-order tensor as a product of other third-order tensors**B**: The multiplication of tensors, a fundamental and crucial operation analogous to matrix multiplication, has garnered considerable attention across various scientific disciplines.
In 2008, Kilmer et al**C**: This development stemmed from their endeavor to extend the matrix singular value decomposition to the realm of tensors. For the purpose of clarification and distinction from other tensor product operations Golub2013matrix , this specific type of multiplication is referred to as tensor-tensor multiplication.
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Selection 4
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**A**: The concept of BEC was first introduced by Elias in 1955 InfThe **B**: Together with the binary symmetric channel (BSC), they are frequently used in coding theory and information theory because they are among the simplest channel models, and many problems in communication theory can be reduced to problems in a BEC. Here we consider more generally a q𝑞qitalic_q-ary erasure channel in which a q𝑞qitalic_q-ary symbol is either received correctly, or totally erased with probability ε𝜀\varepsilonitalic_ε.
**C**: In a binary erasure channel (BEC), a binary symbol is either received correctly or totally erased with probability ε𝜀\varepsilonitalic_ε
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Selection 1
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**A**: Their collaboration, which pioneered the use of approximation theory in complex dynamics, took place in the fall of 1983 in Kharkiv**B**:
This work follows in the footsteps of two world-leading Ukrainian mathematicians, Alex Eremenko and Misha Lyubich**C**: At that time, Alex Eremenko was based at the Institute of Low Temperature Physics and Engineering, and it was there that he formulated what is now known as Eremenko’s conjecture. The city of Kharkiv has been devastated during the ongoing invasion of
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**A**:
In this subsection we would like to prove an analogue of the results in [5] in positive characteristic**B**: For the case of characteristic zero, we also refer the reader to [11] for more details**C**: First let us consider logarithmic connections for the reductive group G𝐺Gitalic_G.
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**A**: The authors thank AIM for providing a supportive environment**B**:
The project was begun during a SQuaRE at the American Institute of Mathematics**C**: The authors also thank the anonymous referees for their careful reading and helpful comments which improved the paper enormously.
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Selection 1
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**A**: Since xn−1∈Psubscript𝑥𝑛1𝑃x_{n-1}\in Pitalic_x start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT ∈ italic_P, we have ωℱ↾P(xn−1)≥εsubscript𝜔↾ℱ𝑃subscript𝑥𝑛1𝜀\omega_{\mathcal{F}\restriction P}(x_{n-1})\geq\varepsilonitalic_ω start_POSTSUBSCRIPT caligraphic_F ↾ italic_P end_POSTSUBSCRIPT ( italic_x start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT ) ≥ italic_ε. Thus, knowing that Un−1subscript𝑈𝑛1U_{n-1}italic_U start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT is an open neighbourhood of xn−1subscript𝑥𝑛1x_{n-1}italic_x start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT, Player I can choose xn∈Un−1subscript𝑥𝑛subscript𝑈𝑛1x_{n}\in U_{n-1}italic_x start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∈ italic_U start_POSTSUBSCRIPT italic_n - 1 end_POSTSUBSCRIPT and fn∈ℱsubscript𝑓𝑛ℱf_{n}\in\mathcal{F}italic_f start_POSTSUBSCRIPT italic_n end_POSTSUBSCRIPT ∈ caligraphic_F such that**B**: Initially, Player I picks x0∈Psubscript𝑥0𝑃x_{0}\in Pitalic_x start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ∈ italic_P.
Let n>0𝑛0n>0italic_n > 0**C**: a perfect set P⊆X𝑃𝑋P\subseteq Xitalic_P ⊆ italic_X and ε>0𝜀0\varepsilon>0italic_ε > 0 such that ωℱ↾P(x)≥εsubscript𝜔↾ℱ𝑃𝑥𝜀\omega_{\mathcal{F}\restriction P}(x)\geq\varepsilonitalic_ω start_POSTSUBSCRIPT caligraphic_F ↾ italic_P end_POSTSUBSCRIPT ( italic_x ) ≥ italic_ε for each x∈P𝑥𝑃x\in Pitalic_x ∈ italic_P
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**A**: Now add a fifth color for the last two edges.**B**:
This process, illustrated with k=3𝑘3k=3italic_k = 3 for the graph C(13,{1,3})𝐶1313C(13,\{1,3\})italic_C ( 13 , { 1 , 3 } ) in Fig**C**: 4, may be viewed as taking the red coloring and rotating it three more times, switching colors for each rotation
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**A**: Our result on perpetuities is given in Section 3, it might be of some interest on its own. There, we also provide a more detailed discussion of the relevant literature concerning perpetuities. The proof of the main theorem is given in Section 4.
**B**: In Section 2 we recall the required notions on branching processes in a varying environment and establish the representation of the survival probability**C**: The paper is organized as follows
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**A**: In this section we define and prove some basic facts, many of which are well-known, about rational Cherednik algebras and their spherical subalgebras.
We emphasise that our notation will be the same as that of the companion paper [BLNS] and so for many standard concepts we will refer the reader to that paper rather than repeating the definitions here**B**: A survey on Cherednik algebras can also be found in [Be] and our notation is (usually) consistent with that given there**C**: Henceforth, the base field will be ℂℂ\mathbb{C}blackboard_C.
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Selection 1
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**A**: Then part (1) implies that f2sD∈Lsuperscriptsubscript𝑓2𝑠𝐷𝐿f_{2}^{s}D\in Litalic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT italic_D ∈ italic_L**B**:
(2) Suppose that L∋fsD=f1sf2sDcontains𝐿superscript𝑓𝑠𝐷superscriptsubscript𝑓1𝑠superscriptsubscript𝑓2𝑠𝐷L\ni f^{s}D=f_{1}^{s}f_{2}^{s}Ditalic_L ∋ italic_f start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT italic_D = italic_f start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT italic_f start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_s end_POSTSUPERSCRIPT italic_D for some D∈A𝐷𝐴D\in Aitalic_D ∈ italic_A and s≥1𝑠1s\geq 1italic_s ≥ 1**C**: By the analogue of part (1) for
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Selection 2
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**A**: It is also possible to lift a graded ring to the Witt vectors (simply lift the coefficients of the equations involved in the defining ideal)**B**: The proof of Theorem 1.3 contains the following principle: There is a good criterion to determine whether a given proper variety over a perfect field of characteristic p>0𝑝0p>0italic_p > 0 admits a flat lifting to the ring of Witt vectors**C**: However, it is hard to check if the lifting can be taken to be flat.
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**A**: In this section we report some progress of this direction**B**: an integral of an algebraic function over a semialgebraic set. Afterwards, we prove that when there is at most one pair of dominant complex roots, it is decidable whether the density is rational, in which case we can compute it exactly.
**C**: We begin by showing that when there are no non-trivial multiplicative relations among the roots, density is a period as defined by Kontsevich and Zagier [KZ01], i.e
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**A**:
I would like to thank my advisor Jeffrey Diller for his encouragement, helpful discussions, and expert advice on the exposition of this document**B**: I thank Claudiu Raicu for his excellent question asking whether the Minimal Stabilisation Algorithm is truly the ‘minimal’ method**C**: I also thank the diligent referee, especially for the suggestion to expand the scope of this paper to the concept of eventual algebraic stability.
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Selection 1
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**A**: As addressed in Remark 3.8, the computability of α𝛼\alphaitalic_α is not required for listing the instances of Axiom scheme 4 either**B**:
It is obvious that Axiom 2 and Axiom scheme 3 are recursively enumerable even if α𝛼\alphaitalic_α is not computable**C**: But, as mentioned in part (2) of Remark 2.2 and also in Remark 4.5, the computability of α𝛼\alphaitalic_α is needed and in fact suffices for Axiom schemes 1, 5, and 6 to be recursively enumerable.
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Selection 4
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**A**: In our main applications, ℬℬ\mathcal{B}caligraphic_B will not be merely a ∗*∗-algebra but rather a Fréchet ∗*∗-algebra, that is, a ∗*∗-algebra which is also a metrizable and complete locally convex topological vector space such that both its multiplication and its involution are continuous.111We recall that a separately continuous bilinear map from a Fréchet space to an arbitrary locally convex space is automatically (jointly) continuous [13, Theorem 1, p. 357], [13, p. 214]**B**: In this case, there are important results [4, Theorems 3.3 & 3.4] which will guarantee the validity of the hypotheses of Proposition 2.1, Theorem 2.3 and Theorem 2.5.
**C**: A particularly interesting situation appears when ℬℬ\mathcal{B}caligraphic_B is a Fréchet ∗*∗-algebra whose topology can be defined by a differential seminorm, as originally introduced by B. Blackadar and J. Cuntz [5] and later modified by S.J. Bhatt, A. Inoue and H. Ogi [4, Definition 3.1]
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Selection 1
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**A**: In §6 we introduce a refinement of Irie’s argument which, instead of a Weyl law, uses bounds on “spectral gaps” coming from ball packings in symplectic cobordisms, to detect the creation of periodic orbits. This method in fact leads to stronger, quantitative closing lemmas as in §1.2 above.
**B**: A Weyl law is really much stronger than necessary to detect the creation of periodic orbits**C**: Indeed, a Weyl law implies that during a suitable perturbation, infinitely many spectral invariants change, with certain asymptotics; but to detect the creation of a periodic orbit, it suffices to show that a single spectral invariant changes by any nonzero amount
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Selection 4
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**A**: Centering large amounts of data is often impractical due to limited communication bandwidth and data privacy concerns [33].**B**: For example, the data used for modern machine learning tasks are getting increasingly large, and they are usually collected or stored in a distributed fashion by a number of data centers, servers, mobile devices, etc**C**:
Decentralized optimization problems arise naturally in many real-world applications
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Selection 3
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**A**: That adjoint situation is comonadic.
This fact not only reveals the coalgebraic nature of equality, but provides a universal construction yielding elementary doctrines from primary ones.**B**: the 2-categories of primary doctrines and that of elementary ones that is, primary doctrines with equality**C**: It shows an adjoint situation between 𝐏𝐃𝐏𝐃\mathbf{PD}bold_PD and 𝐄𝐃𝐄𝐃\mathbf{ED}bold_ED, i.e
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**A**: We also note that the condition r≥1𝑟1r\geq 1italic_r ≥ 1 in Theorem 1.1 is not vacuous since finite nilpotent groups have abelian rank 00. In fact, a nilpotent group N𝑁Nitalic_N has abelian rank r≥1𝑟1r\geq 1italic_r ≥ 1 if and only if |N|=∞𝑁|N|=\infty| italic_N | = ∞.**B**: The MHS on G/T𝐺𝑇G/Titalic_G / italic_T is the natural one coming from the full flag variety G/B𝐺𝐵G/Bitalic_G / italic_B, where B⊂G𝐵𝐺B\subset Gitalic_B ⊂ italic_G is a Borel subgroup**C**:
In Theorem 1.1, W𝑊Witalic_W acts on (G/T)×Tr𝐺𝑇superscript𝑇𝑟(G/T)\times T^{r}( italic_G / italic_T ) × italic_T start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT via the standard action on the homogeneous space G/T𝐺𝑇G/Titalic_G / italic_T and by simultaneous conjugation on Trsuperscript𝑇𝑟T^{r}italic_T start_POSTSUPERSCRIPT italic_r end_POSTSUPERSCRIPT
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**A**: Those which are covariant, in a suitable way, form a symmetric Banach *{}^{*}start_FLOATSUPERSCRIPT * end_FLOATSUPERSCRIPT-algebra**B**:
In connection with a given Fell bundle, in Section 3.6 we introduce a class of kernel-sections that are convolution dominated with respect to a weight**C**: To get a better result, for a larger algebra, we define an enlarged Fell bundle, canonically associated to the initial one.
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**A**:
This paper is organized as follows**B**: In Section 3 we give the nonsingularity conditions which guarantee that (1.3) holds for some d𝑑ditalic_d.**C**: In Section 2 we fix notations and present some results in commutative algebra that are needed in the paper
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**A**: This generalizes the same fact which was previously proved for Meyniel graphs [22] (a class which contains chordal graphs, HHD-free graphs, Gallai graphs, parity graphs, distance-hereditary graphs…) and line graphs of bipartite graphs [3]**B**: Our proof is a generalization of the proof of the latter result by Bonamy, Groenland, Muller, Narboni, Pekárek and**C**:
We now prove our main result, that there are no ugly perfect graphs
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**A**: We start by giving a Donsker-type theorem for the k𝑘kitalic_k-NN estimator of the conditional cumulative distribution function**B**: We thereafter introduce a certain local-linear estimator based on the k𝑘kitalic_k-NN measure and we discuss ways to estimate its variance.
**C**: In this section, two illustrative applications of Theorem 3 are considered
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**A**: Namely, the tropical counterpart of the Maslov index two discs is shown to be exactly the broken lines in [GHK], which confirms the general expectation mentioned at the beginning**B**: We will prove that W𝑊Witalic_W can be calculated tropically, again by establishing the tropical/holomorphic correspondence for the Maslov index two discs**C**: For instance, wall-crossing in the holomorphic setting is caused by the bubbling of a Maslov index two disc into a union of Maslov index two and zero discs, and bending of a broken line precisely reflects this phenomenon. Hence, our method provides an explicit (algorithmic) count of Maslov index two discs, which would be very difficult if directly looking at holomorphic discs themselves. The following is a consequence of Theorem 5.8 and Lemma 5.11:
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**A**:
McCleary and Patel utilized the Möbius inversion formula for establishing a functorial pipeline to summarize simplicial filtrations over finite lattices into persistence diagrams [46]. Botnan et al**B**: introduced notions of signed barcode and rank decomposition for encoding the rank invariant of multiparameter persistence modules as a linear combination of rank invariants of indicator modules [10]. In their paper, Möbius inversion was utilized for computing the rank decomposition, characterizing the generalized persistence diagram in terms of rank decompositions**C**: Asashiba et al. provided a criterion for determining whether or not a given multiparameter persistence module is interval decomposable without having to explicitly compute indecomposable decompositions
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**A**: We are grateful to Giorgio Ottaviani for a helpful comment on Example 6 and to Luca Chiantini and Jarek Buczyński for related discussions as well**B**: The first author also wishes to express his gratitude to Hajime Kaji and Hiromichi Takagi for useful discussions.
**C**: The authors would like to give thank to Korea Institute for Advanced Study (KIAS) for inviting them and giving a chance to start the project
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**A**: In Section 5, we extend our proof technique to the upper bounds to obtain Theorem 5 and Theorem 6. Finally, we give a proof of some technical statements in Appendix A.**B**:
The rest of the paper is organized as follows. In Section 2, we introduce some notations and cover the necessary preliminaries on orthogonal (Gegenbauer) polynomials**C**: In Section 3, we present closed form expressions of the Christoffel-Darboux kernel and use them to obtain kernels whose associated operators satisfy (P2) and whose eigenvalues are given by the coefficients of a univariate sum of squares in an appropriate basis of orthogonal polynomials. In Section 4, we show how to choose this sum of squares so that (P1), (P3) are satisfied and finish the proof of Theorem 3 and Theorem 4
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**A**: Here, we wish to emphasize that a definable family of sets (resp**B**: mappings) is not only a family of definable sets (resp**C**: mappings): all the fibers must glue together into a definable set (resp. mapping). This confers to such families many uniform finiteness properties that are essential for our purpose (see [5, 18]).
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**A**: For non-Archimedean curves, they were investigated in Thuillier’s thesis [Thuillier].**B**: This is an important tool used
in Section 7, where we will explain the notion of harmonic functions**C**: In Section 6, we recall Temkin’s reduction of germs, then we define the residue of a piecewise -linear function at a point x∈X𝑥𝑋x\in Xitalic_x ∈ italic_X as a canonical invertible sheaf on the reduction of the germ at x𝑥xitalic_x, following [CD12, Section 6]
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**A**:
As mentioned in the introduction, gammoids are the main motivation behind this work, however they are not very well understood**B**: For a deeper understanding of gammoids, we refer the reader to [1]**C**: In this section we define gammoids and positroids, and show that positroids are actually gammoids. For that we are going to need the following definition.
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Selection 2
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**A**: We need to expand the integrand to the third order to prove the result**B**: Instead of diving into massive computations, we defer the proof of it to the next section and continue the main thread of our construction. Let
**C**: The proof of this lemma needs some involved integration
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**A**: Nice bases are mostly studied in the context of Einstein Riemannian metrics on solvmanifolds. They were first introduced in [19], and received their (now widely accepted) name in [23]**B**:
In particular, real nilpotent Lie algebras with an ad-invariant metric are classified up to dimension 10101010 in [17]. In [6], we showed that the irreducible Lie algebras appearing in this classification admit a unique ad-invariant metric (up to sign); as a step in the proof, we proved that all of them admit a nice basis**C**: Lie algebras with a nice basis, called nice Lie algebras in this paper, possess a strong algebraic structure, which can be encoded in a special kind of directed graph. Nice nilpotent Lie algebras have been classified up to dimension 6666 in [13, 20] and up to dimension 9999 in [7] (see also [16] for the particular case where the Nikolayevsky derivation is simple and the root matrix is surjective); these classifications show that in small dimension most, but not all, nilpotent Lie algebras are nice.
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Selection 3
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**A**: Section 4 focuses on the setting where the target measure is log-concave with compact convex support and shows that, in this setting, we can bound the moments of the derivative of the Brownian transport map; the main result is Theorem 4.2. In addition, section 4 contains a short explanation of the connection between stochastic localization and the Föllmer process. In section 5 we use the almost-sure contraction established in Theorem 3.1 to prove new functional inequalities**B**: In addition, section 5 contains our results on Stein kernels and their applications to central limit theorems. In section 6 we set up the preliminaries necessary for the study of contraction properties of transport maps on the Wiener space itself. In section 7 we show that causal optimal transport maps are not Cameron-Martin contractions even when the target measure is κ𝜅\kappaitalic_κ-log-concave, for any κ𝜅\kappaitalic_κ. Finally, section 8 is devoted to optimal transport on the Wiener space.
**C**: Section 2 contains the preliminaries necessary for this work including the definition of the Brownian transport map based on the Föllmer process. Section 3 contains the construction of the Föllmer process and the analysis of its properties which then leads to the almost-sure contraction properties of the Brownian transport map. The main results in this section are contained in Theorem 3.1
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**A**: The author gives in [31] a construction of locally bi-Lipschitz trivial stratifications for definable sets in polynomially bounded o-minimal structures (expanding ℝℝ\mathbb{R}blackboard_R) for which the local trivializations are in addition definable. The stratifications constructed in [22, 21, 31] can be required to be compatible with finitely many definable sets, and therefore, to refine a given stratification. Hence, the theorem below follows from [22, Theorem 1.4], [21, Theorem 2.62.62.62.6], or [31, Corollary 1.6.81.6.81.6.81.6.8].
**B**: T. Mostowski [20, Proposition 1.21.21.21.2] proved that every complex analytic set admits a locally bi-Lipschitz trivial stratification**C**: This result was extended to the subanalytic category by A. Parusiński [22], to polynomially bounded o-minimal structures expanding ℝℝ\mathbb{R}blackboard_R in [21], and to polynomially bounded o-minimal structures expanding an arbitrary real closed field in [15]
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**A**: Working with arbitrary signature forces us to restrict to the real analytic case**B**: This indicates that the harmful structure corresponding to this signature always extends to an Einstein Lorentzian manifold. Determining whether the spinor can be extended to obtain a Killing spinor will be the object of future work.
**C**: However, we note that the results of [15] indicate that smooth Riemannian hypersurfaces inside a Lorentzian Einstein manifold have a characterization similar to [22]
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Selection 4
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