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43c6ce2c245a83b9219643a8565c99550b215697e816a4f803a195744e257e20
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2026-01-13T00:00:00-05:00
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Relative tensor products and Koszul duality in monoidal oo-categories
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arXiv:2210.11861v3 Announce Type: replace Abstract: This semi-expository work covers central aspects of the theory of relative tensor products as developed in Higher Algebra, as well as their application to Koszul duality for algebras in monoidal oo-categories. Part of our goal is to expand on the rather condensed account of loc. cit. Along the way, we generalize various aspects of the theory. For instance, given a monoidal oo-category Cc, an oo-category Mm which is left-tensored over Cc, and an algebra A in Cc, we construct an action of A-A-bimodules N in Cc on left A-modules M in Mm by an "external relative tensor product" N \otimes_A M. (Up until now, even the special ("internal") case Cc = Mm appears to have escaped the literature. As an application, we generalize the Koszul duality of loc. cit. to include modules. Our straightforward approach requires that we at this point assume certain compatibilities between tensor products and limits; these assumptions have recently been shown to be unnecessary in work by Brantner, Campos and Nuiten (arXiv:2104.03870).
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https://arxiv.org/abs/2210.11861
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2c626976c99ee92eab1f21ae5ef987949c0c1d73ee58c6853aae8b2f7520d8fa
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2026-01-13T00:00:00-05:00
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Joint Binary-Continuous Fractional Programming: Solution Methods and Applications
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arXiv:2211.02152v3 Announce Type: replace Abstract: In this paper, we investigate a class of non-convex sum-of-ratios programs relevant to decision-making in key areas such as product assortment and pricing, and facility location and cost planning. These optimization problems, characterized by both continuous and binary decision variables, are highly non-convex and challenging to solve. To the best of our knowledge, no existing methods can efficiently solve these problems to near-optimality with arbitrary precision. To address this challenge, we propose an innovative approach based on logarithmic transformations and piecewise linear approximation (PWLA) to approximate the nonlinear fractional program as a mixed-integer convex program with arbitrary precision, which can be efficiently solved using cutting plane (CP) or Branch-and-Cut (B&C) procedures. Our method offers several advantages: it allows for a shared set of binary variables to approximate nonlinear terms and employs an optimal set of breakpoints to approximate other non-convex terms in the reformulation, resulting in an approximate model that is minimal in size. Furthermore, we provide a theoretical analysis of the approximation errors associated with the solutions derived from the approximated problem. We demonstrate the applicability of our approach to constrained competitive joint facility location and cost optimization, as well as constrained product assortment and pricing problems. Extensive experiments on instances of varying sizes, comparing our method with several alternatives, including general-purpose solvers and more direct PWLA-based approximations, show that our approach consistently achieves superior performance across all baselines, particularly in large-scale instances.
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https://arxiv.org/abs/2211.02152
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46c383a9d670e3f21f75cdab459beed55ab7de072ee6f4d19abdc752e8ffc4c7
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2026-01-13T00:00:00-05:00
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Partition regularity of infinite parallelepiped sets
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arXiv:2212.06887v2 Announce Type: replace Abstract: A proper infinite parallelepiped (IP) set in a semigroup is an infinite set consisting of a sequence $\myseq{a}$ and its finite sums, or a superset of such a set. Hindman's theorem asserts that the proper IP sets of natural numbers are partition regular: for each finite coloring of a proper IP set of natural numbers there is a monochromatic proper IP subset. Furstenberg generalized this question to arbitrary semigroups, in which the analogous result does not hold in general. We provide a complete classification of the semigroups for which the proper IP sets are partition regular, and show that this property is equivalent to other fundamental notions of additive Ramsey theory.
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https://arxiv.org/abs/2212.06887
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7cb9f810edbcce1b45ca977249cf98660301fef5287cfcfa93711a8a71d898a7
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2026-01-13T00:00:00-05:00
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Cliques and independent subgroups of the Birkhoff polytope graph
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arXiv:2212.12655v2 Announce Type: replace Abstract: The Birkhoff polytope $\Omega_n$ is the polytope of doubly stochastic matrices of order $n$. The Birkhoff polytope graph $G(\Omega_n)$ is the skeleton of $\Omega_n$; it is the Cayley graph whose vertex set consists of the elements of the symmetric group ${\rm Sym}(n)$ of degree $n$, where two permutations are adjacent if one equals the product of the other with a cycle. We study the combinatorial structure of this graph, focusing on its maximal and maximum cliques and on its independent subgroups (subgroups of ${\rm Sym}(n)$ whose elements are pairwise nonadjacent in the graph). We obtain maximal subgroups of $G(\Omega_n)$ and establish both a lower bound and an upper bound for its clique number. Especially, we prove that if $K$ is a subset of ${\rm Sym}(n)$ consisting of 3-cycle permutations such that $\delta_1^{-1}\delta_2$ is a single cycle for all $\delta_1,\delta_2\in K$, then the maximum size of $K$ is $\lfloor (n-1)^2/4\rfloor$, which can be viewed as an Erd\H{o}s-Ko-Rado-type theorem for ${\rm Sym}(n)$.
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https://arxiv.org/abs/2212.12655
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7e5779169a681c115f0c39c3999bfd5c38e98e41c6196d6d99865901adfbe433
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2026-01-13T00:00:00-05:00
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A factorization of metric spaces
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arXiv:2212.13409v3 Announce Type: replace Abstract: We first prove that for every metrizable space $X$, for every closed subset $F$ whose complement is zero-dimensional, the space $X$ can be embedded into a product space of the closed subset $F$ and a metrizable zero-dimensional space as a closed subset. Using this theorem, we next show the existence of extensors of metrics and ultrametrics, which preserve properties of metrics such as the completeness, the properness, being an ultrametrics, its fractal dimensions, and large scale structures. This result contains some of the author's extension theorems of ultrametrics.
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https://arxiv.org/abs/2212.13409
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1cf8ca49b7dd5307587785172f0513e21bcbc0cf5c299d26cd0bf8e4fbce75a6
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2026-01-13T00:00:00-05:00
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Topology and convergence on the space of measure-valued functions
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arXiv:2301.04361v3 Announce Type: replace Abstract: In these notes, uniform convergence on compacta is studied on the space of functions taking values in the set of finite Borel measures. Related limit theorems, including L\'evy's continuity theorem and functional limit theorems for (classical and non-commutative) additive processes, are also described. N.B.: the contents of this manuscript have been incorporated into another manuscript (arXiv:2412.18742).
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https://arxiv.org/abs/2301.04361
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35c0c47a50cbb0b396fbdd4e82518b2d3bf85d1ee3ae8de663f54c35ca0df5bc
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2026-01-13T00:00:00-05:00
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Heat kernel-based p-energy norms on metric measure spaces
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arXiv:2303.10414v4 Announce Type: replace Abstract: We investigate heat kernel-based and other $p$-energy norms (1<\infty) on bounded and unbounded metric measure spaces, in particular, on nested fractals and their blowups. With the weak-monotonicity properties for these norms, we generalise the celebrated Bourgain-Brezis-Mironescu (BBM) type characterization for p\neq2. When there admits a heat kernel satisfying the two-sided estimates, we establish the equivalence of various $p$-energy norms and weak-monotonicity properties, and show that these weak-monotonicity properties hold when p=2 (in the case of Dirichlet form). Our paper's key results concern the equivalence and verification of various weak-monotonicity properties on fractals. Consequently, many classical results on p-energy norms hold on nested fractals and their blowups, including the BBM type characterization and Gagliardo-Nirenberg inequality.
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https://arxiv.org/abs/2303.10414
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ed81989a6b4b7bc53b999ab6b7c179fe0541e644e923802c8b61a3c91436a5bf
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2026-01-13T00:00:00-05:00
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Generative Modeling via Hierarchical Tensor Sketching
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arXiv:2304.05305v2 Announce Type: replace Abstract: We propose a hierarchical tensor-network approach for approximating high-dimensional probability density via empirical distribution. This leverages randomized singular value decomposition (SVD) techniques and involves solving linear equations for tensor cores in this tensor network. The complexity of the resulting algorithm scales linearly in the dimension of the high-dimensional density. An analysis of estimation error demonstrates the effectiveness of this method through several numerical experiments.
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https://arxiv.org/abs/2304.05305
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0a7cff32077ec21aa0b49d5f92cd7570f444922bbff51dba1a247808e172ac08
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2026-01-13T00:00:00-05:00
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Faster List Decoding of AG Codes
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arXiv:2304.07083v2 Announce Type: replace Abstract: In this article, we present a fast algorithm performing an instance of the Guruswami-Sudan list decoder for algebraic geometry codes. We show that any such code can be decoded in $\tilde{O}(s^2\ell^{\omega-1}\mu^{\omega-1}(n+g) + \ell^\omega \mu^\omega)$ operations in the underlying finite field, where $n$ is the code length, $g$ is the genus of the function field used to construct the code, $s$ is the multiplicity parameter, $\ell$ is the designed list size and $\mu$ is the smallest positive element in the Weierstrass semigroup of some chosen place.
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https://arxiv.org/abs/2304.07083
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e8c8ccf3c7cb0c30720792419cc11ffba149e9cb1cc768a6ad3edefe0891610c
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2026-01-13T00:00:00-05:00
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Nonorientable genus embedding of nearly complete bipartite graphs
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arXiv:2305.10008v2 Announce Type: replace Abstract: The nearly complete bipartite graph $G(m,n,k)$ is obtained by removing $k$ independent edges from the complete bipartite graph $K_{m,n}$. In this paper, we prove that for any nearly complete bipartite graph $G(m,n,k)$ with $m, n\geq 3$, and $(m,n,k)\notin\{(5,4,4)$, $(4,5,4)$, $(5,5,5)\}$, there exists a nonorientable genus embedding $\Pi$ satisfying $\tilde{\gamma}(\Pi)=\max\{\lceil \big((m-2)(n-2)-k\big)/2\rceil, 1\}$. This embedding can be constructed by starting from an embedding of some $G(p,q,h)$ with $h\leq 6$ and $p,q\leq 7$, and then iteratively adding multiple copies of $G(2,2,2)$, $G(2,0,0)$ and $G(0,2,0)$. As a consequence, the previously unresolved nonorientable genus $\tilde{\gamma}(G(n+1,n,n))$ for even $n$ and $\tilde{\gamma}(G(n,n,n))$ for arbitrary $n$ are now determined.
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https://arxiv.org/abs/2305.10008
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5860d39e973a4e490cf418b1ff0b3846477be828aa7e94883a5bd39115cd286d
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2026-01-13T00:00:00-05:00
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Smooth projective surfaces with bounded cohomology property
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arXiv:2306.07830v4 Announce Type: replace Abstract: In this paper, we first prove that every Mori dream surface $X$ satisfies the bounded cohomology property (BCP for short). Namely, there exists a constant $c_X>0$ such that $h^1(\mathcal O_X(C))\le c_Xh^0(\mathcal O_X(C))$ for every curve $C$ on $X$. We then prove that there is a positive constant $m(Y)$ such that $l_C:=(K_Y\cdot C)(C^2)^{-1}\le m(Y)$ for every ample curve $C$ on a geometrically ruled surface $Y$ over a curve of genus $g$, and $Y$ satisfies the BCP if $g\le1$.
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https://arxiv.org/abs/2306.07830
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e6312771641671849cd36f4363c61f196f619a57a14443a6ca6de06287567068
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2026-01-13T00:00:00-05:00
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Commutation of Smyth and Hoare Power Constructions in Well-filtered Dcpos
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arXiv:2311.14261v3 Announce Type: replace Abstract: Prior work [11] established a commutativity result for the Hoare power construction and a modified version of the Smyth power construction consisting of strongly compact sets, which is defined for Us-admitting dcpos, where Us-admissability is well-filteredness with compact sets replaced by strongly compact sets. In this paper, we consider the Hoare power construction H and the Smyth power construction Q on the category WF of well-filtered dcpos with Scott-continuous maps. Actually, the functors H and Q can be extended to monads. We prove that H and Q commute, that is, HQ(L) is isomorphic to QH(L) for a well-filtered dcpo L, if and only if L satisfies a property similar to consonance that we call (KC) and the Scott topology coincides with the upper Vietoris topology on Q(L). We also investigate the Eilenberg-Moore category of the monad composed by H and Q under a distributive law on WF and characterize it to be a subcategory of the category Frm, which is composed of all frames and all frame homomorphisms.
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https://arxiv.org/abs/2311.14261
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67f4fdd70d67c8bf9988ee90cf8c01adf105d0c7c0bad0f1667463891d021464
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2026-01-13T00:00:00-05:00
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M\"obius function and primes: an identity factory with applications
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arXiv:2312.05138v3 Announce Type: replace Abstract: We investigate the sums $\sum_{n\le X, (n,q)=1}\frac{\mu(n)}{n^s}\log^k\left(\frac{X}{n}\right)$, where $k\in\{0,1\}$, $s\in\mathbb{C}$, $\Re s>0$. Our goal is to obtain explicit asymptotic estimations for these quantities. To achieve this, we develop a broad framework of identities that we use to derive several applications. Building on similar principles, we also provide an appendix establishing the inequality $\sum_{n\le X}\Lambda(n)/n\le \log X$, valid for any $X\geq 1$.
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https://arxiv.org/abs/2312.05138
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2d4954b93b30ce3a1ddae53a56bbb943bb461356895bc235a0919b8e7f537373
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2026-01-13T00:00:00-05:00
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The FBSDE approach to sine-Gordon up to $6\pi$
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arXiv:2401.13648v2 Announce Type: replace Abstract: We develop a stochastic analysis of the sine-Gordon Euclidean quantum field $(\cos (\beta \varphi))_2$ on the full space up to the second threshold, i.e. for $\beta^2 < 6 \pi$. The basis of our method is a forward-backward stochastic differential equation (FBSDE) for a decomposition $(X_t)_{t \geqslant 0}$ of the interacting Euclidean field $X_{\infty}$ along a scale parameter $t \geqslant 0$. This FBSDE describes the optimiser of the stochastic control representation of the Euclidean QFT introduced by Barashkov and one of the authors. We show that the FBSDE provides a description of the interacting field without cut-offs and that it can be used effectively to study the sine-Gordon measure to obtain results about large deviations, integrability, decay of correlations for local observables, singularity with respect to the free field, Osterwalder-Schrader axioms and other properties.
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https://arxiv.org/abs/2401.13648
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bb26997b6e31858e6b9c700293617172f8b34508e3a3a506fc9cef56ddb2decc
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2026-01-13T00:00:00-05:00
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Kubota-type formulas and supports of mixed measures
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arXiv:2401.16371v2 Announce Type: replace Abstract: Kubota's integral formula expresses the intrinsic volumes of a convex body as averages over its projections onto linear subspaces. In this work, we introduce a new class of Kubota-type formulas for mixed area measures adapted to rotations around a fixed axis, which encode a crucial disintegration property. Our construction is motivated by applications to valuations on convex functions. In the latter framework, we obtain corresponding statements for (conjugate) mixed Monge-Amp\`ere measures. As a by-product, we characterize supports of mixed area and mixed Monge-Amp\`ere measures, thereby confirming a special case of a conjecture by Schneider.
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https://arxiv.org/abs/2401.16371
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7b1aa55549abee8679cdfcabdc37605836b402bf6434323c6436eaf916d8f652
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2026-01-13T00:00:00-05:00
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Kinetic shock profiles for the Landau equation
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arXiv:2402.01581v2 Announce Type: replace Abstract: The physical quantities in a gas should vary continuously across a shock. However, the physics inherent in the compressible Euler equations is insufficient to describe the width or structure of the shock. We demonstrate the existence of weak shock profiles to the kinetic Landau equation, that is, traveling wave solutions with Maxwellian asymptotic states whose hydrodynamic quantities satisfy the Rankine-Hugoniot conditions. These solutions serve to capture the structure of weak shocks at the kinetic level. Previous works considered only the Boltzmann equation with hard sphere and angular cut-off potentials.
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https://arxiv.org/abs/2402.01581
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31d191e022c6a39b1dab69a5a5a03ce912fe5b4252ea134ce8fc76f1f956f8e8
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2026-01-13T00:00:00-05:00
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On smooth adic spaces over $\mathbb{B}_{\mathrm{dR}}^+$ and sheafified $p$-adic Riemann--Hilbert correspondence
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arXiv:2403.01363v3 Announce Type: replace Abstract: Let $C$ be a completely algebraic closed non-archimedean field over $\mathbb{Q}_p$ and $\alpha,r$ be two positive integers. Denote by $B_\alpha$ the ring $\mathbb{B}_{\mathrm{dR}}^+(C)/(\ker\theta)^\alpha$. This paper first constructs a sheafified $p$-adic Riemann--Hilbert correspondence. Specifically, we construct a canonical sheaf isomorphism on $X_{\mathrm{\acute{e}t}}$, \[ R^1\nu_*\big( \mathrm{GL}_r(\mathbb{B}_{\mathrm{dR}}^+/(\ker\theta)^\alpha) \big) \cong \mathrm{MIC}_{r}(X)\{-1\}, \] where the first term is identified with the sheaf of isomorphism classes of $v$-vector bundles with coefficients in $\mathbb{B}_{\mathrm{dR}}^+/(\ker\theta)^{\alpha}$, and the second term is defined as the sheaf of isomorphism classes of integrable connections of rank $r$. We then define the moduli space of integrable connections on $X$ and the moduli space of $v$-vector bundles on $X$ with coefficients in $\mathbb{B}_{\mathrm{dR}}^+/(\ker\theta)^{\alpha}$, and prove that they are small $v$-stacks in the sense of Scholze. These constructions generalize Heuer's work on $p$-adic Simpson correspondence.
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https://arxiv.org/abs/2403.01363
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43b464117bd357603a53c46bef06f74d249bf7a48a9130d3c2d0048b904685d5
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2026-01-13T00:00:00-05:00
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The classification of homomorphism homogeneous oriented graphs
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arXiv:2403.04393v2 Announce Type: replace Abstract: The modern theory of homogeneous structures begins with the work of Roland Fra\"iss\'e. The theory developed in the last seventy years is placed in the border area between combinatorics, model theory, algebra, and analysis. We turn our attention to its combinatorial pillar, namely, the work on the classification of structures for given homogeneity types, and focus onto the homomorphism homogeneous ones, introduced in 2006 by Cameron and Ne\v{s}et\v{r}il. An oriented graph is called homomorphism homogeneous if every homomorphism between finite induced subgraphs extends to an endomorphism. In this paper we present a complete classification of the countable homomorphism homogeneous oriented graphs.
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https://arxiv.org/abs/2403.04393
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b94c961c44bd3f15393ce92ac329521d3bb92919dd310b25226d1a6fed0af96c
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2026-01-13T00:00:00-05:00
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Directional testing for one-way MANOVA in divergent dimensions
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arXiv:2403.07679v4 Announce Type: replace Abstract: Testing the equality of mean vectors across $g$ different groups plays an important role in many scientific fields. In regular frameworks, likelihood-based statistics under the normality assumption offer a general solution to this task. However, the accuracy of standard asymptotic results is not reliable when the dimension $p$ of the data is large relative to the sample size $n_i$ of each group. We propose here an exact directional test for the equality of $g$ normal mean vectors with identical unknown covariance matrix in a high dimensional setting, provided that $\sum_{i=1}^g n_i \ge p+g+1$. In the case of two groups ($g=2$), the directional test coincides with the Hotelling's $T^2$ test. In the more general situation where the $g$ independent groups may have different unknown covariance matrices, although exactness does not hold, simulation studies show that the directional test is more accurate than most commonly used likelihood{-}based solutions, at least in a moderate dimensional setting in which $p=O(n_i^\tau)$, $\tau \in (0,1)$. Robustness of the directional approach and its competitors under deviation from the assumption of multivariate normality is also numerically investigated. Our proposal is here applied to data on blood characteristics of male athletes and to microarray data storing gene expressions in patients with breast tumors.
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https://arxiv.org/abs/2403.07679
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6e48d2947cebf6121ca3f454f54d741ee0250759d7954f64cc7cd38b96c1ab70
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2026-01-13T00:00:00-05:00
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Compactly supported $\mathbb{A}^1$-Euler characteristics of symmetric powers of cellular varieties
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arXiv:2404.08486v2 Announce Type: replace Abstract: The compactly supported $\mathbb{A}^1$-Euler characteristic, introduced by Hoyois and later refined by Levine and others, is an anologue in motivic homotopy theory of the classical Euler characteristic of complex topological manifolds. It is an invariant on the Grothendieck ring of varieties $\mathrm{K}_0(\mathrm{Var}_k)$ taking values in the Grothendieck-Witt ring $\mathrm{GW}(k)$ of the base field $k$. The former ring has a natural power structure induced by symmetric powers of varieties. In a recent preprint, Pajwani and P\'al construct a power structure on $\mathrm{GW}(k)$ and show that the compactly supported $\mathbb{A}^1$-Euler characteristic respects these two power structures for $0$-dimensional varieties, or equivalently \'etale $k$-algebras. In this paper, we define the class $\mathrm{Sym}_k$ of symmetrisable varieties to be those varieties for which the compactly supported $\mathbb{A}^1$-Euler characteristic respects the power structures and study the algebraic properties of $\mathrm{K}_0(\mathrm{Sym}_k)$. We show that it includes all cellular varieties, and even linear varieties as introduced by Totaro. Moreover, we show that it includes non-linear varieties such as elliptic curves. As an application of our main result, we compute the compactly supported $\mathbb{A}^1$-Euler characteristics of symmetric powers of Grassmannians and certain del Pezzo surfaces.
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https://arxiv.org/abs/2404.08486
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4bc485b2952618f788d1e12a69da0d36f44d7645d656d7a1043a0bac530260a5
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2026-01-13T00:00:00-05:00
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Global existence of a strong solution to the initial value problem for the Nernst-Planck-Navier-Stokes system in high space dimensions
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arXiv:2404.16433v5 Announce Type: replace Abstract: We study the existence of a strong solution to the initial value problem for the Nernst-Planck-Navier-Stokes (NPNS) system in $\mathbb{R}^N, N\geq 3$. The system describes the electrodiffusion of ions in a viscous Newtonian fluid. A strong solution is obtained in any dimension of space without constraints on the number of species or the size of the given data.
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https://arxiv.org/abs/2404.16433
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88db1e46e5304a482dc15de1c0ecdf35752c527cf1a2d4e1826a331c7eefeeb7
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2026-01-13T00:00:00-05:00
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Parametric set-theoretic Yang-Baxter equation: p-racks, solutions & quantum algebras
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arXiv:2405.04088v3 Announce Type: replace Abstract: The theory of the parametric set-theoretic Yang-Baxter equation is established from a purely algebraic point of view. The first step towards this objective is the introduction of certain generalizations of the familiar shelves and racks called parametric (p)-shelves and racks. These objects satisfy a parametric self-distributivity condition and lead to solutions of the Yang-Baxter equation. Novel, non-reversible solutions are obtained from p-shelf/rack solutions by a suitable parametric twist, whereas all reversible set-theoretic solutions are reduced to the identity map via a parametric twist. The universal algebras associated to both p-rack and generic parametric, set-theoretic solutions are next presented and the corresponding universal R-matrices are derived. The admissible universal Drinfel'd twist is constructed allowing the derivation of the general set-theoretic universal R-matrix. By introducing the concept of a parametric coproduct we prove the existence of a parametric co-associativity. We show that the parametric coproduct is an algebra homomorphism and the universal R-matrices satisfy intertwining relations with the algebra coproducts
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https://arxiv.org/abs/2405.04088
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8083a9b6764f41adf0f2da448e0c2429f583bb59299989e621f21a9dd132c065
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2026-01-13T00:00:00-05:00
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Universal-existential theories of fields
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arXiv:2405.12771v4 Announce Type: replace Abstract: We study various universal-existential fragments of first-order theories of fields, in particular of function fields and of equicharacteristic henselian valued fields. For example we discuss to what extent the theory of a field k determines the universal-existential theories of the rational function field over k and of the field of Laurent series over k, and we find various many-one reductions between such fragments.
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https://arxiv.org/abs/2405.12771
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8c020ddd08bc3e65b43a404fc5e98c91bca5a3f0e402cc4d5ffcd701b9a3e1c6
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2026-01-13T00:00:00-05:00
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The hierarchies of identities and closed products for multiple complexes
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arXiv:2405.16271v3 Announce Type: replace Abstract: We consider infinite $\Z_\Z$-index complexes $\mathcal C$ of spaces with elements depending on a number of parameters, complete with respect to a linear associative regular inseparable multilinear product. The existence of nets of vanishing ideals of orders of and powers of differentials is assumed for subspaces of $\mathcal C$-spaces. In the polynomial case of orders and powers of the differentials, we derive the hierarchies of differential identities and closed multiple products. We prove that a set of maximal orders and powers for differentials, differential conditions, together with coherence conditions on indices of a complex $\mathcal C$ elements generate families of multi-graded differential algebras.
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https://arxiv.org/abs/2405.16271
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ec1a40ba215d26165f6fe230f106da57af55c84fc3ea680812ff4196e56645b9
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2026-01-13T00:00:00-05:00
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Bounded geometry for PCF-special subvarieties
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arXiv:2405.17343v3 Announce Type: replace Abstract: For each integer $d\geq 2$, let $M_d$ denote the moduli space of maps $f: \mathbb{P}^1\to \mathbb{P}^1$ of degree $d$. We study the geometric configurations of subsets of postcritically finite (or PCF) maps in $M_d$. A complex-algebraic subvariety $Y \subset M_d$ is said to be PCF-special if it contains a Zariski-dense set of PCF maps. Here we prove that there are only finitely many positive-dimensional irreducible PCF-special subvarieties in $M_d$ with degree $\leq D$. In addition, there exist constants $N = N(D,d)$ and $B = B(D,d)$ so that for any complex algebraic subvariety $X \subset M_d$ of degree $\leq D$, the Zariski closure $\overline{X\cap\mathrm{PCF}}~$ has at most $N$ irreducible components, each with degree $\leq B$. We also prove generalizations of these results for points with small critical height in $M_d(\bar{\mathbb{Q}})$.
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https://arxiv.org/abs/2405.17343
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43acf18af09ae5197f6efb9e2a5712c9b3023d6a94b7359bcbd74783d4b9b396
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2026-01-13T00:00:00-05:00
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Fast Numerical Approximation of Linear, Second-Order Hyperbolic Problems Using Model Order Reduction and the Laplace Transform
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arXiv:2405.19896v2 Announce Type: replace Abstract: We extend our previous work [F. Henr'iquez and J. S. Hesthaven, arXiv:2403.02847 (2024)] to the linear, second-order wave equation in bounded domains. This technique uses two widely known mathematical tools to construct a fast and efficient method for the solution of linear, time-dependent problems: the Laplace transform (LT) and the model-order reduction (MOR) techniques, hence the name LT-MOR method. The application of the Laplace transform yields a time-independent problem parametrically depending on the Laplace variable. Following the two-phase paradigm of the reduced basis method, first in an offline stage we sample the Laplace parameter, compute the high-fidelity solution, and then resort to a Proper Orthogonal Decomposition (POD) to extract a basis of reduced dimension. Then, in an online phase, we project the time-dependent problem onto this basis and proceed to solve the evolution problem using any suitable time-stepping method. We prove exponential convergence of the reduced solution computed by the proposed method toward the high-fidelity one as the dimension of the reduced space increases. Finally, we present numerical experiments illustrating the performance of the method in terms of accuracy and, in particular, speed-up when compared to the full-order model.
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https://arxiv.org/abs/2405.19896
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49862b9a2234d97893729b566cc95fd31f1686a82c4fa04bf4dc05db3d7e41c7
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2026-01-13T00:00:00-05:00
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Topological Stability and Latschev-type Reconstruction Theorems for Spaces of Curvature Bounded Above
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arXiv:2406.04259v4 Announce Type: replace Abstract: We consider the problem of homotopy-type reconstruction of compact subsets $X\subset\R^N$ that have the Alexandrov curvature bounded above ($\leq$ $\kappa$) in the intrinsic length metric. The reconstructed spaces are in the form of Vietoris--Rips complexes computed from a compact sample $S$, Hausdorff--close to the unknown shape $X$. Instead of the Euclidean metric on the sample, our reconstruction technique leverages a path-based metric to compute these complexes. As naturally emerging in the framework of reconstruction, we also study the Gromov--Hausdorff topological stability and finiteness problem for general compact for subspaces of curvature bounded above. Our techniques provide novel sampling conditions as an alternative to the existing and commonly used techniques using weak feature size and $\mu$--reach. To the best of our knowledge, this is the first work that establishes homotopy-type reconstruction guarantees for spaces with vanishing reach and $\mu$--reach, a regime not covered by existing sampling conditions.
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https://arxiv.org/abs/2406.04259
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74659e7c2a4e81eff67ad2127322f1953bc48815001b62d315bafb02f857586e
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2026-01-13T00:00:00-05:00
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Homogeneous G2 and Sasakian instantons on the Stiefel 7-manifold
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arXiv:2406.06753v2 Announce Type: replace Abstract: We study homogeneous instantons on the seven dimensional Stiefel manifold V in the context of $G_2$ and Sasakian geometry. According to the reductive decomposition of V we provide an explicit description of all invariant $G_2$ and Sasakian structures. In particular, we characterise the invariant $G_2$- structures inducing a Sasakian metric, among which the well known nearly parallel $G_2$-structure (Sasaki- Einstein) is included. As a consequence, we classify the invariant connections on homogeneous principal bundles over V with gauge group U(1) and SO(3), satisfying either the $G_2$ or the Sasakian instanton condition. In addition, we study infinitesimal deformations of $G_2$--instantons on coclosed $G_2$--manifolds using a spinorial approach. By means of a Weitzenb\"ock--type formula with torsion, we obtain curvature obstructions to the existence of non--trivial infinitesimal deformations and prove rigidity results for certain homogeneous $G_2$--instantons
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https://arxiv.org/abs/2406.06753
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135249dc12b2ca9ff04390ba362bbd0d4c056699b26600f39b46a3eef38ccfd1
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2026-01-13T00:00:00-05:00
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Slalom numbers
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arXiv:2406.19901v3 Announce Type: replace Abstract: The paper is an extensive and systematic study of cardinal invariants we call slalom numbers, describing the combinatorics of sequences of sets of natural numbers. Our general approach, based on relational systems, covers many such cardinal characteristics, including localization and anti-localization cardinals. We show that most of the slalom numbers are connected to topological selection principles, in particular, we obtain the representation of the uniformity of meager and the cofinality of measure. Considering instances of slalom numbers parametrized by ideals on natural numbers, we focus on monotonicity properties with respect to ideal orderings and computational formulas for the disjoint sum of ideals. Hence, we get such formulas for several pseudo-intersection numbers as well as for the bounding and dominating numbers parametrized with ideals. Based on the effect of adding a Cohen real, we get many consistent constellations of different values of slalom numbers.
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https://arxiv.org/abs/2406.19901
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8a605ac7d63e30cc1b77445c8a38cb3318888d26a05d89f2e4f10e119de1a390
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2026-01-13T00:00:00-05:00
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The geometry of preperiodic points in families of maps on $\mathbb{P}^N$
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arXiv:2407.10894v2 Announce Type: replace Abstract: We study the dynamics of algebraic families of maps on $\mathbb{P}^N$, over the field $\mathbb{C}$ of complex numbers, and the geometry of their preperiodic points. The goal of this note is to formulate a conjectural characterization of the subvarieties of $S \times\mathbb{P}^N$ containing a Zariski-dense set of preperiodic points, where the parameter space $S$ is a quasiprojective complex algebraic variety; the characterization is given in terms of the non-vanishing of a power of the invariant Green current associated to the family of maps. This conjectural characterization is inspired by and generalizes the Relative Manin-Mumford Conjecture for families of abelian varieties, recently proved by Gao and Habegger, and it includes as special cases the Manin-Mumford Conjecture (theorem of Raynaud) and the Dynamical Manin-Mumford Conjecture (posed by Ghioca, Tucker, and Zhang). We provide examples where the equivalence is known to hold, and we show that several recent results can be viewed as special cases. Finally, we give the proof of one implication in the conjectural characterization.
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https://arxiv.org/abs/2407.10894
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70680b8d87e1b515c8b94eb982626f3e0e8e06bfb3a7f767cb7c5e29cde06023
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2026-01-13T00:00:00-05:00
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Obstructions to homotopy invariance of loop coproduct via parametrised fixed-point theory
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arXiv:2407.13662v2 Announce Type: replace Abstract: Given $f: M \to N$ a homotopy equivalence of compact manifolds with boundary, we use a construction of Geoghegan and Nicas to define its Reidemeister trace $[T] \in \pi_1^{st}(\mathcal{L} N, N)$. We realize the Goresky-Hingston coproduct as a map of spectra, and show that the failure of $f$ to entwine the spectral coproducts can be characterized by Chas-Sullivan multiplication with $[T]$. In particular, when $f$ is a simple homotopy equivalence, the spectral coproducts of $M$ and $N$ agree.
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https://arxiv.org/abs/2407.13662
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9b4f8dbe4d791bfb4cc11dd1a82583791c8de23ca26a90ced4506559806a5d08
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2026-01-13T00:00:00-05:00
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Universal Approximation of Dynamical Systems by Semi-Autonomous Neural ODEs and Applications
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arXiv:2407.17092v3 Announce Type: replace Abstract: In this paper, we introduce semi-autonomous neural ordinary differential equations (SA-NODEs), a variation of the vanilla NODEs, employing fewer parameters. We investigate the universal approximation properties of SA-NODEs for dynamical systems from both a theoretical and a numerical perspective. Within the assumption of a finite-time horizon, under general hypotheses we establish an asymptotic approximation result, demonstrating that the error vanishes as the number of parameters goes to infinity. Under additional regularity assumptions, we further specify this convergence rate in relation to the number of parameters, utilizing quantitative approximation results in the Barron space. Based on the previous result, we prove an approximation rate for transport equations by their neural counterparts. Our numerical experiments validate the effectiveness of SA-NODEs in capturing the dynamics of various ODE systems and transport equations. Additionally, we compare SA-NODEs with vanilla NODEs, highlighting the superior performance and reduced complexity of our approach.
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https://arxiv.org/abs/2407.17092
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6159236d5d00127bf7ef15cea2d0835bf1750304331a420112fe68c5b2fa7431
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2026-01-13T00:00:00-05:00
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The flux norm, Bohr-Sommerfeld Quantization Rules and the scattering problem for h $\Psi$DO's on the real line
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arXiv:2407.18893v3 Announce Type: replace Abstract: We revisit the well known Bohr-Sommerfeld quantization rule (BS) of order 2 for a self-adjoint 1-D h-Pseudo-differential operator within the algebraic and microlocal framework of Helffer and Sjoestrand; BS holds precisely when Gram matrix consisting of scalar products of some WKB solutions with respect to the ``flux norm'' (or microlocal Wronskian) is not invertible. We simplify somewhat our previous proof [A. Ifa H. Louati and M. Rouleux. Bohr-Sommerfeld Quantization Rules Revisited: the Method of Positive Commutators. J. Math. Sci. Univ. Tokyo, 25(2):2018] by working in spatial representation only, as in complex WKB theory for Schroedinger operator. We consider also the scattering problem.
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https://arxiv.org/abs/2407.18893
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080903a7cc0a0bd1765ba8881081fd0eca1568570b0d3a88b02bd99bca87e865
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2026-01-13T00:00:00-05:00
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Deterministic approximate counting of colorings with fewer than $2\Delta$ colors via absence of zeros
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arXiv:2408.04727v3 Announce Type: replace Abstract: Let $\Delta,q\geq 3$ be integers. We prove that there exists $\eta\geq 0.002$ such that if $q\geq (2-\eta)\Delta$, then there exists an open set $\mathcal{U}\subset \mathbb{C}$ that contains the interval $[0,1]$ such that for each $w\in \mathcal{U}$ and any graph $G=(V,E)$ of maximum degree at most $\Delta$, the partition function of the anti-ferromagnetic $q$-state Potts model evaluated at $w$ does not vanish. This provides a (modest) improvement on a result of Liu, Sinclair, and Srivastava, and breaks the $q=2\Delta$-barrier for this problem. As a direct consequence we obtain via Barvinok's interpolation method a deterministic polynomial time algorithm to approximate the number of proper $q$-colorings of graphs of maximum degree at most $\Delta$, provided $q\geq (2-\eta)\Delta$.
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https://arxiv.org/abs/2408.04727
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4e4a3ee7245586bd5e89358ebe11203cb802c35bbcbfe09ebbd3ebc838d04f19
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2026-01-13T00:00:00-05:00
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A multi-objective mixed integer linear programming model for supply chain planning of 3D printing
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arXiv:2408.05213v4 Announce Type: replace Abstract: 3D printing is considered the future of production systems and one of the physical elements of the Fourth Industrial Revolution. 3D printing will significantly impact the product lifecycle, considering cost, energy consumption, and carbon dioxide emissions, leading to the creation of sustainable production systems. Given the importance of these production systems and their effects on the quality of life for future generations, it is expected that 3D printing will soon become one of the global industry's fundamental needs. Although three decades have passed since the emergence of 3D printers, there has not yet been much research on production planning and mass production using these devices. Therefore, we aimed to identify the existing gaps in the planning of 3D printers and to propose a model for planning and scheduling these devices. In this research, several parts with different heights, areas, and volumes have been considered for allocation on identical 3D printers for various tasks. To solve this problem, a multi-objective mixed integer linear programming model has been proposed to minimize the earliness and tardiness of parts production, considering their order delivery times, and maximizing machine utilization. Additionally, a method has been proposed for the placement of parts in 3D printers, leading to the selection of the best edge as the height. Using a numerical example, we have plotted the Pareto curve obtained from solving the model using the epsilon constraint method for several parts and analyzed the impact of the method for selecting the best edge as the height, with and without considering it. Additionally, a comprehensive sensitivity and scenario analysis has been conducted to validate the results.
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https://arxiv.org/abs/2408.05213
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2e568ebca808e9297e85460b82b8ef944558035237869dc866fe105967011056
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2026-01-13T00:00:00-05:00
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Topology of total cut complexes and cut complexes of grid graphs
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arXiv:2408.07646v3 Announce Type: replace Abstract: Inspired by the work of Fr{\"o}berg (1990) and Eagon and Reiner (1998), Bayer et al. recently introduced two new graph complexes: total cut complexes and cut complexes. In this article, we investigate these complexes specifically for (rectangular) grid graphs, focusing on $2 \times n$ and $3 \times n$ cases. We extend and refine the work of Bayer et al., proving and strengthening several of their conjectures, thereby enhancing the understanding of these graph complexes' topological and combinatorial properties.
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https://arxiv.org/abs/2408.07646
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2b0d2388092ff5f1cd35bc60c960a7a5fff049a0deaae370cb5a589d4b300358
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2026-01-13T00:00:00-05:00
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Rigidity results for non-K\"ahler Calabi-Yau geometries on threefolds
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arXiv:2408.09648v5 Announce Type: replace Abstract: We derive a canonical symmetry reduction associated to a compact non-K\"ahler Bismut-Hermitian-Einstein manifold. In real dimension $6$, the transverse geometry is conformally K\"ahler, and we give a complete description in terms of a single scalar PDE for the underlying K\"ahler structure. In the case when the soliton potential is constant, we show that that the Bott-Chern number $h^{1,1}_{BC} \geq 2$, and that equality holds if and only if the metric is Bismut-flat, and hence a quotient of either $\SU(2) \times \mathbb R \times \mathbb C$ or $\SU(2) \times \SU(2)$.
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https://arxiv.org/abs/2408.09648
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c4956a4b6e41e18013270450236d8ca38d22bc31d04acd90f17aaa26b56967ef
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2026-01-13T00:00:00-05:00
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Turk's head knots and links: a survey
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arXiv:2409.20106v3 Announce Type: replace Abstract: We collect and discuss various results on an important family of knots and links called Turk's head knots and links $Th (p,q)$. In the mathematical literature, they also appear under different names such as rosette knots and links or weaving knots and links. Unless being the unknot or the alternating torus links $T(2,q)$, the Turk's head links $Th (p,q)$ are all known to be alternating, fibered, hyperbolic, invertible, non-split, periodic, and prime. The Turk's head links $Th (p,q)$ are also both positive and negative amphichiral if $p$ is chosen to be odd. Moreover, we highlight and present several more results, focusing on Turk's head knots $Th (3,q)$. We finally list several open problems and conjectures for Turk's head knots and links. We conclude with a short appendix on torus knots and links, which might be of independent interest.
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https://arxiv.org/abs/2409.20106
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df01e60022c488fb8d830aa3d9f6b634fd5fba3b2076c32a511caba8894476ae
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2026-01-13T00:00:00-05:00
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Improved performance guarantees for Tukey's median
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arXiv:2410.00219v3 Announce Type: replace Abstract: Is there a natural way to order data in dimension greater than one? The approach based on the notion of data depth, often associated with John Tukey, is among the most popular. Tukey's depth has found applications in robust statistics, graph theory, and the study of elections and social choice. We present improved performance guarantees for empirical Tukey's median, a deepest point associated with a given sample, when the data-generating distribution is elliptically symmetric and possibly anisotropic. Some of our results remain valid in the wider class of affine equivariant estimators. As a corollary of our bounds, we show that the typical diameter of the set of all empirical Tukey's medians scales like $o(n^{-1/2})$ where $n$ is the sample size. Moreover, when the data follow the bivariate normal distribution, we prove that with high probability, the diameter is of order $O(n^{-3/4}\log^{1/2}(n))$. On the technical side, we show how affine equivariance can be leveraged to improve concentration bounds; moreover, we develop sharp strong approximation results for empirical processes indexed by halfspaces that could be of independent interest.
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https://arxiv.org/abs/2410.00219
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0f08363017dcef94f0186b552bb1d06d0f789f1f050925403988c319312f4728
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2026-01-13T00:00:00-05:00
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Constrained Consensus-Based Optimization and Numerical Heuristics for the Few Particle Regime
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arXiv:2410.10361v2 Announce Type: replace Abstract: Consensus-based optimization (CBO) is a versatile multi-particle optimization method for performing nonconvex and nonsmooth global optimizations in high dimensions. Proofs of global convergence in probability have been achieved for a broad class of objective functions in unconstrained optimizations. In this work we adapt the algorithm for solving constrained optimizations on compact and unbounded domains with boundary by leveraging emerging reflective boundary conditions. In particular, we close a relevant gap in the literature by providing a global convergence proof for the many-particle regime comprehensive of convergence rates. On the one hand, for the sake of minimizing running cost, it is desirable to keep the number of particles small. On the other hand, reducing the number of particles implies a diminished capability of exploration of the algorithm. Hence numerical heuristics are needed to ensure convergence of CBO in the few-particle regime. In this work, we also significantly improve the convergence and complexity of CBO by utilizing an adaptive region control mechanism and by choosing geometry-specific random noise. In particular, by combining a hierarchical noise structure with a multigrid finite element method, we are able to compute global minimizers for a constrained $p$-Allen-Cahn problem with obstacles, a very challenging variational problem.
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https://arxiv.org/abs/2410.10361
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fc6e1c2df9f76170415eba423105065b49e726248909f5eb9284e7bd06ba4927
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2026-01-13T00:00:00-05:00
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Semi-boolean and Yosida $\ell$-groups, Martinez and Yosida frames, and the $G+B$ construction
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arXiv:2410.11930v2 Announce Type: replace Abstract: The class of semi-boolean $\ell$-groups was introduced in 1968 by A. Bigard. These are the $\ell$-groups $G$ in which the principal convex $\ell$-subgroup $G(a)$ generated by any $a \in G$ is equal to the polar $a^{\perp \perp}$. Examples include all hyperarchimedean $\ell$-groups and all existentially closed abelian $\ell$-groups. Ordered by inclusion, the set of convex $\ell$-subgroups of a semi-boolean $\ell$-group is a \Mart frame (an algebraic frame with FIP in which every element is a $d$-element). Related are the Yosida $\ell$-groups, i.e., the $\ell$-groups whose frame of convex $\ell$-subgroups is a Yosida frame (an algebraic frame with FIP in which every compact element is a meet of maximal elements). Applying results on \Mart frames and Yosida frames, we obtain new characterizations of the semi-boolean and Yosida $\ell$-groups, show that the former constitute a radical class and the latter do not, and present new examples with special properties. To build some of our examples, we introduce the $G+B$ construction for $\ell$-groups, an adaptation of the $A+B$ construction from commutative algebra.
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https://arxiv.org/abs/2410.11930
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ac011d40159f24c6bf51517eadc001fbdc375fe2c0d03039f16ed2f3011cccc5
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2026-01-13T00:00:00-05:00
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An iterative construction of complete K\"ahler--Einstein metrics
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arXiv:2410.12599v3 Announce Type: replace Abstract: We extend Tsuji's iterative construction of complete K\"ahler--Einstein metrics with negative scalar curvature to noncompact K\"ahler manifolds with bounded geometry, using Berndtsson's method from the compact setting. Consequently, given a holomorphic surjective map $p:X\to Y$, where $X$ is a weakly pseudoconvex K\"ahler manifold and $Y$ is a complex manifold, and where the smooth fibers admit K\"ahler--Einstein metrics with negative scalar curvature and bounded geometry, we show that the fiberwise K\"ahler--Einstein metrics induce a semipositively curved metric on the relative canonical bundle $K_{X/Y}$. Moreover, our approach also applies to the plurisubharmonic variation of cusp K\"ahler--Einstein metrics.
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https://arxiv.org/abs/2410.12599
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c36dcd7ceca2ced442866d76ecaad3007dd40e773d28416f6911d672c42437d6
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2026-01-13T00:00:00-05:00
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Hypoellipticity and Higher Order Gaussian Bounds
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arXiv:2410.14456v2 Announce Type: replace Abstract: Let $(\mathfrak{M},\rho,\mu)$ be a metric measure space satisfying a doubling condition, $p_0\in (1,\infty)$, and $T(t):L^{p_0}(\mathfrak{M},\mu)\rightarrow L^{p_0}(\mathfrak{M},\mu)$, $t\geq 0$, a strongly continuous semi-group. We provide sufficient conditions under which $T(t)$ is given by integration against an integral kernel satisfying higher-order Gaussian bounds of the form \[ \left| K_t(x,y) \right| \leq C \exp\left( -c \left( \frac{\rho(x,y)^{2\kappa}}{t} \right)^{\frac{1}{2\kappa-1}} \right) \mu\left( B_\rho\left(x,\rho(x,y)+t^{1/2\kappa}\right) \right)^{-1}, \] where $B_\rho$ denotes the metric ball. We also provide conditions for similar bounds on ``derivatives'' of $K_t(x,y)$ and our results are localizable. If $A$ is the generator of $T(t)$ the main hypothesis is that $\partial_t -A$ and $\partial_t-A^{*}$ satisfy a hypoelliptic estimate at every scale, uniformly in the scale. We present applications to subelliptic PDEs.
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https://arxiv.org/abs/2410.14456
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2bf4b6a2cd31385f0aa393b42848ca5772668f6e4c2b9e869e027a7956b71d33
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2026-01-13T00:00:00-05:00
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$s$-almost $t$-intersecting families for finite sets
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arXiv:2410.20185v4 Announce Type: replace Abstract: A family $\mathcal{F}$ of $k$-subsets of an $n$-set is called $s$-almost $t$-intersecting if each member is $t$-disjoint with at most $s$ members. In this paper, we prove that, if $\left|\mathcal{F}\right|$ is maximum, then $\mathcal{F}$ consists of all $k$-subsets containing a fixed $t$-subset. Consequently, it is natural to consider the maximum-sized $\mathcal{F}$ with $\left|\bigcap_{F\in\mathcal{F}} F\right|<t$. The famous Hilton-Milner theorem settles the case where $\mathcal{F}$ is $t$-intersecting. We characterize the remaining case completely.
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https://arxiv.org/abs/2410.20185
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10454dc61796f9d53733adeb0a745249299fca33962a59bc57ec4fe2c97402e1
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2026-01-13T00:00:00-05:00
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Effectively Leveraging Momentum Terms in Stochastic Line Search Frameworks for Fast Optimization of Finite-Sum Problems
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arXiv:2411.07102v3 Announce Type: replace Abstract: In this work, we address unconstrained finite-sum optimization problems, with particular focus on instances originating in large scale deep learning scenarios. Our main interest lies in the exploration of the relationship between recent line search approaches for stochastic optimization in the overparametrized regime and momentum directions. First, we point out that combining these two elements with computational benefits is not straightforward. To this aim, we propose a solution based on mini-batch persistency. We then introduce an algorithmic framework that exploits a mix of data persistency, conjugate-gradient type rules for the definition of the momentum parameter and stochastic line searches. The resulting algorithm provably possesses convergence properties under suitable assumptions and is empirically shown to outperform other popular methods from the literature, obtaining state-of-the-art results in both convex and nonconvex large scale training problems.
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https://arxiv.org/abs/2411.07102
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b90025f03251950b5b663f56262e82de33a680e6f3077f89a34f03b7d2d16eef
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2026-01-13T00:00:00-05:00
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A Hodge-Tate decomposition with rigid analytic coefficients
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arXiv:2411.07366v2 Announce Type: replace Abstract: Let $X$ be a smooth proper rigid analytic space over a complete algebraically closed field extension $K$ of $\mathbb{Q}_p$. We establish a Hodge--Tate decomposition for $X$ with $G$-coefficients, where $G$ is any commutative locally $p$-divisible rigid group. This generalizes the Hodge--Tate decomposition of Faltings and Scholze, which is the case $G=\mathbb{G}_a$. For this, we introduce geometric analogs of the Hodge--Tate spectral sequence with general locally $p$-divisible coefficients. We prove that these spectral sequences degenerate at $E_2$. Our results apply more generally to a class of smooth families of commutative adic groups over $X$ and in the relative setting of smooth proper morphisms $X\rightarrow S$ of smooth rigid spaces. We deduce applications to analytic Brauer groups and the geometric $p$-adic Simpson correspondence.
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https://arxiv.org/abs/2411.07366
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c693e54cab3992e4d6a66f31f55381dfcbb730ce51cba2b513cafc5e175b471e
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2026-01-13T00:00:00-05:00
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The transcendence degree of the reals over certain set-theoretical subfields
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arXiv:2412.00616v2 Announce Type: replace Abstract: It is a well-known result that, after adding one Cohen real, the transcendence degree of the reals over the ground-model reals is continuum. We extend this result for a set $X$ of finitely many Cohen reals, by showing that, in the forcing extension, the transcendence degree of the reals over a combination of the reals in the extension given by each proper subset of $X$ is also maximal. This answers a question of Kanovei and Schindler.
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https://arxiv.org/abs/2412.00616
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35375708f872d77ab113ec8217efd0cd8d8d93539f46bcce75d13b603d5e0505
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2026-01-13T00:00:00-05:00
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Adelic C*-correspondences and parabolic induction
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arXiv:2412.02379v2 Announce Type: replace Abstract: In analogy with the construction of representations of adelic groups as restricted products of representations of local groups, we study restricted tensor products of Hilbert C*-modules and of C*-correspondences. The construction produces global C*-correspondences from compatible collections of local C*-correspondences. When applied to the collection of C*-correspondences capturing local parabolic induction, the construction produces a global C*-correspondence that captures adelic parabolic induction.
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https://arxiv.org/abs/2412.02379
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f63bb328e6eb7385e29b97c98004d708dcda37fbcf2669bd58ccfccd01096b6f
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2026-01-13T00:00:00-05:00
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Convergence of a discrete selection-mutation model with exponentially decaying mutation kernel to a Hamilton-Jacobi equation
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arXiv:2412.06657v2 Announce Type: replace Abstract: In this paper we derive a constrained Hamilton-Jacobi equation with obstacle from a discrete non-linear integro-differential model of population dynamics, with exponentially decaying mutation kernel. The exponential decay of the kernel leads to a modification of the classical Hamilton-Jacobi equation obtained previously from continuous models in \cite{BMP}. We consider a population composed of individuals characterized by a quantitative trait, subject to selection, mutation and competition. In a regime of small mutations, small spatial discretization step and large time we prove that the WKB transformation of the density converges to a viscosity solution of a constrained Hamilton-Jacobi equation with obstacle.
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https://arxiv.org/abs/2412.06657
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882908b29d00c22f1fe185f452e94ed7f3a53d6ec05db275e7fdc71080945f28
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2026-01-13T00:00:00-05:00
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When entropy meets Tur\'an: new proofs and hypergraph Tur\'an results
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arXiv:2412.08075v3 Announce Type: replace Abstract: In this paper, we provide a new proof of a density version of Tur\'an's theorem. We also rephrase both the theorem and the proof using entropy. With the entropic formulation, we show that some naturally defined entropic quantity is closely connected to other common quantities such as Lagrangian and spectral radius. In addition, we also determine the Tur\'an density for a new family of hypergraphs, which we call tents. Our result can be seen as a new generalization of Mubayi's result on the extended cliques.
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https://arxiv.org/abs/2412.08075
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88aad512408e33ebfe32f13c0f8f93fc8271baf3f9416f1ce45eaefb086b0a2b
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2026-01-13T00:00:00-05:00
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A Detailed Analysis on Sharpened Singular Adams-Type Inequalities
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arXiv:2412.11176v3 Announce Type: replace Abstract: We establish a sharp Adams-type inequality in higher-order function spaces with singular weights on $\mathbb{R}^n$. A sharp singular concentration-compactness principle, improving Lions' result, is also proved. The study distinguishes between critical and subcritical sharp singular Adams-type inequalities and shows their equivalence. Furthermore, we analyze the asymptotic behavior of the associated bounds and relate the suprema of the critical and subcritical cases. A new compact embedding, crucial to our analysis, is also derived. Moreover, as an application of these results, by employing the mountain pass theorem, we study the existence of nontrivial solutions to a class of nonhomogeneous quasilinear elliptic equations involving the $(p,\frac{n}{2})$-biharmonic operator with singular exponential growth.
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https://arxiv.org/abs/2412.11176
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9e809a4efeb551b76edeeeb9201d83d5ef33977333c8d5ef8376d60b00494349
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2026-01-13T00:00:00-05:00
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Defining extended TQFTs via handle attachments
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arXiv:2412.14649v2 Announce Type: replace Abstract: We give a finite presentation of the cobordism symmetric monoidal bicategory of (smooth, oriented) closed manifolds, cobordisms and cobordisms with corners as an extension of the bicategory of closed manifolds, cobordisms and diffeomorphisms. The generators are the standard handle attachments, and the relations are handle cancellations and invariance under reversing the orientation of the attaching spheres. In other words, given a categorified TQFT and 2-morphisms associated to the standard handles satisfying our relations, we construct a once extended TQFT.
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https://arxiv.org/abs/2412.14649
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ad156f6fc20df38791859e30f75899191e8a12fb9f4aabd259fad83742518041
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2026-01-13T00:00:00-05:00
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Dimension reduction for path signatures
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arXiv:2412.14723v2 Announce Type: replace Abstract: This paper focuses on the mathematical framework for reducing the complexity of models using path signatures. The structure of these signatures, which can be interpreted as collections of iterated integrals along paths, is discussed and their applications in areas such as stochastic differential equations (SDEs) and financial modeling are pointed out. In particular, exploiting the rough paths view, solutions of SDEs continuously depend on the lift of the driver. Such continuous mappings can be approximated using (truncated) signatures, which are solutions of high-dimensional linear systems. In order to lower the complexity of these models, this paper presents methods for reducing the order of high-dimensional truncated signature models while retaining essential characteristics. The derivation of reduced models and the universal approximation property of (truncated) signatures are treated in detail. Numerical examples, including applications to the (rough) Bergomi model in financial markets, illustrate the proposed reduction techniques and highlight their effectiveness.
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https://arxiv.org/abs/2412.14723
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13cc35a0ecd912222a73656196ba5b584d08122bb185f5ff746e4b00d8934164
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2026-01-13T00:00:00-05:00
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Fluctuations in Various Regimes of Non-Hermiticity and a Holographic Principle
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arXiv:2412.15854v3 Announce Type: replace Abstract: The variance of the number of particles in a set is an important quantity in understanding the statistics of non-interacting fermionic systems in low dimensions. An exact map of their ground state in a harmonic trap in one and two dimensions to the classical Gaussian unitary and complex Ginibre ensemble, respectively, allows to determine the counting statistics at finite and infinite system size. We will establish two new results in this setup. First, we uncover an interpolating central limit theorem between known results in one and two dimensions, for linear statistics of the elliptic Ginibre ensemble. We find an entire range of interpolating weak non-Hermiticity limits, given by a two-parameter family for the mesoscopic scaling regime. Second, we considerably generalize the proportionality between the number variance and the entanglement entropy between Fermions in a set $A$ and its complement in two dimensions. Previously known only for rotationally invariant sets and external potentials, we prove a holographic principle for general non-rotationally invariant sets and random normal matrices. It states that both number variance and entanglement entropy are proportional to the circumference of $A$.
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https://arxiv.org/abs/2412.15854
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6376a03a296c69ad5b3f21133c762defa1303a839ba2633d100ab0b292a9bb2e
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2026-01-13T00:00:00-05:00
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Additive processes on the real line and Loewner chains
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arXiv:2412.18742v2 Announce Type: replace Abstract: This paper investigates additive processes with respect to several different independences in non-commutative probability in terms of the convolution hemigroups of the distributions of the increments of the processes. In particular, we focus on the relation of monotone convolution hemigroups and chordal Loewner chains, a special kind of family of conformal mappings. Generalizing the celebrated Loewner differential equation, we introduce the concept of ``generator'' for a class of Loewner chains. The locally uniform convergence of Loewner chains is then equivalent to a suitable convergence of generators. Using generators, we define homeomorphisms between the aforementioned class of chordal Loewner chains, the set of monotone convolution hemigroups with finite second moment, and the set of classical convolution hemigroups on the real line with finite second moment. Moreover, we define similar homeomorphisms between classical, free, and boolean convolution hemigroups on the real line, but without any assumptions on the moments.
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https://arxiv.org/abs/2412.18742
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497ffd8e28763ab7579b50ba35e205411d4be01508aa2f2f0d262fd80727ad6e
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2026-01-13T00:00:00-05:00
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On reconstructing Morse functions with prescribed level sets on $3$-dimensional manifolds and a necessary and sufficient condition for the reconstruction
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arXiv:2412.20626v3 Announce Type: replace Abstract: We discuss a necessary and sufficient condition for reconstruction of Morse functions with prescribed (regular) level sets on $3$-dimensional manifolds. The present work strengthens a previous result of the author where only sufficient conditions are studied. Our new work is also regarded as a kind of addenda.
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https://arxiv.org/abs/2412.20626
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e725d9b54cc6e91560782f04b4836370291df53a29b8d0f3c4475f282453ae4d
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2026-01-13T00:00:00-05:00
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Scalar behavior for a complex multi-soliton arising in blow-up for a semilinear wave equation
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arXiv:2501.04505v2 Announce Type: replace Abstract: This paper deals with blow-up for the complex-valued semilinear wave equation with power nonlinearity in dimension 1. Up to a rotation of the solution in the complex plane, we show that near a characteristic blow-up point, the solution behaves exactly as in the real-valued case. Namely, up to a rotation in the complex plane, the solution decomposes into a sum of a finite number of decoupled solitons with alternate signs. The main novelty of our proof is a resolution of a complex-valued first order Toda system governing the evolution of the positions and the phases of the solitons.
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https://arxiv.org/abs/2501.04505
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0fedb712f693d017603ef72b429865a656d4501e4b25ff20d273c9b0c1068597
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2026-01-13T00:00:00-05:00
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Uniformizable foliated projective structures along singular foliations
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arXiv:2501.04626v2 Announce Type: replace Abstract: We consider holomorphic foliations by curves on compact complex manifolds, for which we investigate the existence of projective structures along the leaves varying holomorphically (foliated projective structures), that satisfy particular uniformizability properties. Our results show that the singularities of the foliation impose severe restrictions for the existence of such structures. A foliated projective structure separates the singularities of a foliation into parabolic and non-parabolic ones. For a strongly uniformizable foliated projective structure on a compact K\"ahler manifold, the existence of a single non-degenerate, non-parabolic singularity implies that the foliation is completely integrable. We establish an index theorem that imposes strong cohomological restrictions on the foliations having only non-degenerate singularities that support foliated projective structures making all of them parabolic. As an application of our results, we prove that, on a projective space of any dimension, a foliation by curves of degree at least two, with only non-degenerate singularities, does not admit a strongly uniformizable foliated projective structure.
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https://arxiv.org/abs/2501.04626
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040e8911def2dc5d20453f344a97f3878825a01480bc355bff917d87268a90e4
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Maximality of the futures of points in globally hyperbolic maximal conformally flat spacetimes
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arXiv:2501.05856v2 Announce Type: replace Abstract: Let M be a globally hyperbolic conformally spacetime. We prove that the indecomposable past/future sets (abbrev. IPs/IFs) -in the sense of Penrose, Kronheimer and Geroch -of the universal cover of M are domains of injectivity of the developing map. This relies on the central observation that diamonds are domains of injectivity of the developing map. Using this, we provide a new proof of a result of completeness by C. Rossi, which notably simplifies the original arguments. Furthermore, we establish that if, in addition, M is maximal, the IPs/IFs are maximal as globally hyperbolic conformally flat spacetimes. More precisely, we show that they are conformally equivalent to regular domains of Minkowski spacetime as defined by F. Bonsante.
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https://arxiv.org/abs/2501.05856
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d12bbda9f12fab7521a8a4198f5e4a994814fd401be5b71922dc00a29813f713
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2026-01-13T00:00:00-05:00
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Fourier Extension Based on Weighted Generalized Inverse
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arXiv:2501.16096v4 Announce Type: replace Abstract: This paper introduces a weighted generalized inverse framework for Fourier extensions, designed to suppress spurious oscillations in the extended region while maintaining high approximation accuracy on the original interval. By formulating the Fourier extension problem as a compact operator equation, we propose a weighted best-approximation solution that incorporates a priori smoothness information through suitable weight operators on the Fourier coefficients. This leads to a regularization scheme based on the generalized truncated singular value decomposition (GTSVD). Under algebraic and exponential smoothness assumptions, convergence analysis demonstrates optimal $L^2$ accuracy and improved stability for derivatives. Compared with classical Fourier extension using standard TSVD, the proposed method effectively controls high-frequency components and yields smoother extensions. A practical discretization using uniform sampling is developed, along with an adaptive design of weight functions. Numerical experiments confirm that the method significantly improves derivative approximations and reduces oscillations in the extended domain without compromising accuracy on the original interval.
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https://arxiv.org/abs/2501.16096
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a2abfd1fddcc15d6b6a1070a4bd47fab2e104ee5167e6a91fdcd99159f2cac24
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2026-01-13T00:00:00-05:00
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Shrinking vs. expanding: the evolution of spatial support in degenerate Keller-Segel systems
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arXiv:2501.19119v2 Announce Type: replace Abstract: We consider radially symmetric solutions of the degenerate Keller-Segel system \begin{align*} \begin{cases} \partial_t u=\nabla\cdot (u^{m-1}\nabla u - u\nabla v),\\ 0=\Delta v -\mu +u,\quad\mu =\frac{1}{|\Omega|}\int_\Omega u, \end{cases} \end{align*} in balls $\Omega\subset\mathbb R^n$, $n\ge 1$, where $m>1$ is arbitrary. Our main result states that the initial evolution of the positivity set of $u$ is essentially determined by the shape of the (nonnegative, radially symmetric, H\"older continuous) initial data $u_0$ near the boundary of its support $\overline{B_{r_1}(0)}\subsetneq\Omega$: It shrinks for sufficiently flat and expands for sufficiently steep $u_0$. More precisely, there exists an explicit constant $A_{\mathrm{crit}} \in (0, \infty)$ (depending only on $m, n, R, r_1$ and $\int_\Omega u_0$) such that if \begin{align*} u_0(x)\le A(r_1-|x|)^\frac{1}{m-1} \qquad \text{for all $|x|\in(r_0, r_1)$ and some $r_0\in(0,r_1)$ and $A0$ and $\zeta>0$ such that $\sup\{\, |x| \mid x \in \operatorname{supp} u(\cdot, t)\,\}\le r_1 -\zeta t$ for all $t\in(0, T)$, while if \begin{align*} u_0(x)\ge A(r_1-|x|)^\frac{1}{m-1} \qquad \text{for all $|x|\in(r_0, r_1)$ and some $r_0 \in (0, r_1)$ and $A>A_{\mathrm{crit}}$}, \end{align*} then we can find $T>0$ and $\zeta>0$ such that $\sup\{\, |x| \mid x \in \operatorname{supp} u(\cdot, t)\,\}\ge r_1 +\zeta t$ for all $t\in(0, T)$.
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https://arxiv.org/abs/2501.19119
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10b6189d944eb137daae6f7d077a885c74e6d2366dac703c71fb4b5f1994b5b1
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2026-01-13T00:00:00-05:00
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Density-valued symplectic forms from a multisymplectic viewpoint
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arXiv:2502.02224v4 Announce Type: replace Abstract: We give an intrinsic characterization of multisymplectic manifolds that have the linear type of density-valued symplectic forms in each tangent space, prove Darboux-type theorems for these forms, and investigate their symmetries.
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https://arxiv.org/abs/2502.02224
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754ad6ed5d20753b06d4ee1c673a2b852d3bb936c9b0301ce46bfedac7e4f7ac
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Nondeterministic Behaviours in Double Categorical Systems Theory
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arXiv:2502.02517v2 Announce Type: replace Abstract: In this paper, we build double theories capturing the idea of nondeterministic behaviors and trajectories. Following Libkind and Myers' Double Operadic Theory of Systems, we construct monoidal semi double categories of interfaces, along with what we call semimodules of systems, in the case of Moore machines, working with Markov categories with conditionals to handle nondeterminism. We use conditional products in these Markov categories to define trajectories in a compositional way, and represent nondeterministic systems using Markov maps; channels between systems are assumed to be deterministic.
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https://arxiv.org/abs/2502.02517
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eef61d8b28830588704287e85f411049d3eb100dd109f1d2b871e8c32417eec2
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2026-01-13T00:00:00-05:00
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High-dimensional stochastic finite volumes using the tensor train format
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arXiv:2502.04868v3 Announce Type: replace Abstract: A method for the uncertainty quantification of nonlinear hyperbolic conservation laws with many uncertain parameters is presented. The method combines stochastic finite volume methods and tensor trains in a novel way: the dimensions of physical space and time are kept as full tensors, while all stochastic dimensions are compressed together into a tensor train. The resulting hybrid format has one tensor train for each spatial cell and each time step. The MUSCL scheme is adapted to the proposed hybrid format, and its feasibility is demonstrated through several test cases. For the scalar Burgers' equation, we conduct a convergence study and compare the results with those obtained using the full tensor train format with three stochastic parameters. The equation is then solved for an increasing number of stochastic dimensions.For systems of conservation laws, we focus on the Euler equations. A parameter study and a comparison with the full tensor train format are carried out for the Sod shock tube problem. As a more complex application, we investigate the Shu-Osher problem, which involves intricate wave interactions. The presented method opens new avenues for integrating uncertainty quantification with established numerical schemes for hyperbolic conservation laws.
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https://arxiv.org/abs/2502.04868
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67e57705c639494fa311bce9baa888bb9a51c59b6563998dc422854b964e430d
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2026-01-13T00:00:00-05:00
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Fixed-strength spherical designs
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arXiv:2502.06002v3 Announce Type: replace Abstract: A spherical $t$-design is a finite subset $X$ of the unit sphere such that every polynomial of degree at most $t$ has the same average over $X$ as it does over the entire sphere. Determining the minimum possible size of spherical designs, especially in a fixed dimension as $t \to \infty$, has been an important research topic for several decades. This paper presents results on the complementary asymptotic regime, where $t$ is fixed and the dimension tends to infinity. The main results in this paper are (1) a construction of smaller spherical designs via an explicit connection to Gaussian designs and (2) the exact order of magnitude of minimal-size signed $t$-designs, which is significantly smaller than predicted by a typical degrees-of-freedom heuristic. We also establish a method to ``project'' spherical designs between dimensions, prove a variety of results on approximate designs, and construct new $t$-wise independent subsets of $\{1,2,\dots,q\}^d$ which may be of independent interest. To achieve these results, we combine techniques from algebra, geometry, probability, representation theory, and optimization.
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https://arxiv.org/abs/2502.06002
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3d23fcf88d1f98591bd091e2ec5242e365bf57e02ca57a132d9abe1c43194247
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2026-01-13T00:00:00-05:00
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High-order BUG dynamical low-rank integrators based on explicit Runge--Kutta methods
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arXiv:2502.07040v5 Announce Type: replace Abstract: In this work, we introduce high-order Basis-Update & Galerkin (BUG) integrators based on explicit Runge-Kutta methods for large-scale matrix differential equations. These dynamical low-rank integrators extend the BUG integrator to arbitrary explicit Runge-Kutta schemes by performing a BUG step at each stage of the method. The resulting Runge-Kutta BUG (RK-BUG) integrators are robust with respect to small singular values, fully forward in time, and high-order accurate, while enabling conservation and rank adaptivity. We prove that RK-BUG integrators retain the order of convergence of the underlying Runge-Kutta method until the error reaches a plateau corresponding to the low-rank truncation error, which vanishes as the rank becomes full. This theoretical analysis is supported by several numerical experiments. The results demonstrate the high-order convergence of the RK-BUG integrator and its superior accuracy compared to other existing dynamical low-rank integrators.
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https://arxiv.org/abs/2502.07040
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58c51aedccad021473ee5b6d1356bf7cda293ea2003aa51bf94a5c4e1ddc2390
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2026-01-13T00:00:00-05:00
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Convex cocompact groups in real hyperbolic spaces with limit set a Pontryagin sphere
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arXiv:2502.09470v3 Announce Type: replace Abstract: We exhibit two examples of convex cocompact subgroups of the isometry groups of real hyperbolic spaces with limit set a Pontryagin sphere: one generated by $50$ reflections of $\mathbb{H}^4$, and the other by a rotation of order $21$ and a reflection of $\mathbb{H}^6$. For each of them, we also locate convex cocompact subgroups with limit set a Menger curve.
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https://arxiv.org/abs/2502.09470
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01d8f34a501611ae46314d5d2ee0688f1a26296d4641dd4905790aa1ad14b733
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2026-01-13T00:00:00-05:00
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On the global stability and large time behavior of solutions of the Boussinesq equations
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arXiv:2502.16226v3 Announce Type: replace Abstract: We study the two dimensional viscous Boussinesq equations, which model stratified flows in a circular domain under the influence of a general gravitational potential $f$. First, we show that the Boussinesq equations admit steady-state solutions only in the form of hydrostatic equilibria, $(\mathbf{u},\rho,p) = (0, \rho_s, p_s)$, where the pressure gradient satisfies $\nabla p_s = -\rho_s \nabla f$. Moreover, the relation between $\rho_s$ and $f$ is constrained by $(\partial_y \rho_s, -\partial_x \rho_s) \cdot (\partial_x f, \partial_y f) = 0$, which allows us to write $\nabla \rho_s = h(x,y) \nabla f$ for some scalar function $h(x,y)$. Second, we prove that any hydrostatic equilibrium $(0, \rho_s, p_s)$ is linearly unstable if $h(x_0, y_0) > 0$ at some point $(x, y) = (x_0, y_0)$. This instability coincides with the classical Rayleigh--Taylor instability. Third, by employing a series of regularity estimates, we reveal that although the presence of the Rayleigh--Taylor instability makes perturbations around the unstable equilibrium grow exponentially in time, the system ultimately converges to a state of hydrostatic equilibrium. The analysis is carried out for perturbations about an arbitrary hydrostatic equilibrium, covering both stable and unstable configurations. Finally, we derive a necessary and sufficient condition on the initial density perturbation under which the density converges to a profile of the form $-\gamma f + \beta$ with constants $\gamma, \beta > 0$. This result underscores the system's inherent tendency to settle into a hydrostatic state, even in the presence of Rayleigh--Taylor instability.
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https://arxiv.org/abs/2502.16226
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40bd0588ca49ee8ca8d5a3d83a07ab49ecde41cc76952cf1560c5a269abf5816
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2026-01-13T00:00:00-05:00
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Novel Constructions for Computation and Communication Trade-offs in Private Coded Distributed Computing
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arXiv:2502.17195v2 Announce Type: replace Abstract: Distributed computing enables scalable machine learning by distributing tasks across multiple nodes, but ensuring privacy in such systems remains a challenge. This paper introduces a novel private coded distributed computing model that integrates privacy constraints to keep task assignments hidden. By leveraging placement delivery arrays (PDAs), we design an extended PDA framework to characterize achievable computation and communication loads under privacy constraints. By constructing two classes of extended PDAs, we explore the trade-offs between computation and communication, showing that although privacy increases communication overhead, it can be significantly alleviated through optimized PDA-based coded strategies.
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https://arxiv.org/abs/2502.17195
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020bbd93280827af79e7d3c7a11ca5d45207d51e9f5651a62dad037d9eb7b7aa
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2026-01-13T00:00:00-05:00
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The multi-level friendship paradox for sparse random graphs
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arXiv:2502.17724v2 Announce Type: replace Abstract: In Hazra, den Hollander and Parvaneh (2025) we analysed the friendship paradox for sparse random graphs. For four classes of random graphs we characterised the empirical distribution of the friendship biases between vertices and their neighbours at distance $1$, proving convergence as $n\to\infty$ to a limiting distribution, with $n$ the number of vertices, and identifying moments and tail exponents of the limiting distribution. In the present paper we look at the multi-level friendship bias between vertices and their neighbours at distance $k \in \mathbb{N}$ obtained via a $k$-step exploration according to a backtracking or a non-backtracking random walk. We identify the limit of empirical distribution of the multi-level friendship biases as $n\to\infty$ and/or $k\to\infty$. We show that for non-backtracking exploration the two limits commute for a large class of sparse random graphs, including those that locally converge to a rooted Galton-Watson tree. In particular, we show that the same limit arises when $k$ depends on $n$, i.e., $k=k_n$, provided $\lim_{n\to\infty} k_n = \infty$ under some mild conditions. We exhibit cases where the two limits do not commute and show the relevance of the mixing time of the exploration.
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https://arxiv.org/abs/2502.17724
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3f8344fcdbfba9852a4723c9d4fc49a0cb185485cd1451425fbb3b8be6f73c9e
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2026-01-13T00:00:00-05:00
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Mass conservation, positivity and energy identical-relation preserving scheme for the Navier-Stokes equations with variable density
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arXiv:2503.00405v3 Announce Type: replace Abstract: In this paper, we consider a mass conservation, positivity and energy identical-relation preserving scheme for the Navier-Stokes equations with variable density. Utilizing the square transformation, we first ensure the positivity of the numerical fluid density, which is form-invariant and regardless of the discrete scheme. Then, by proposing a new recovery technique to eliminate the numerical dissipation of the energy and to balance the loss of the mass when approximating the reformation form, we preserve the original energy identical-relation and mass conservation of the proposed scheme. To the best of our knowledge, this is the first work that can preserve the original energy identical-relation for the Navier-Stokes equations with variable density. Moreover, the error estimates of the considered scheme are derived. Finally, we show some numerical examples to verify the correctness and efficiency.
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https://arxiv.org/abs/2503.00405
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33e9086ce2fc8fcab6919cd73b7f2e7a208e01af53e415020a9b4bcd3ed8a172
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2026-01-13T00:00:00-05:00
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On the Mordell-Weil rank and $2$-Selmer group of a family of elliptic curves
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arXiv:2503.04561v2 Announce Type: replace Abstract: We consider the parametric family of elliptic curves over $\mathbb{Q}$ of the form $E_{m} : y^{2} = x(x - n_{1})(x - n_{2}) + t^{2}$, where $n_{1}$, $n_{2}$ and $t$ are particular polynomial expressions in an integral variable $m$. In this paper, we investigate the torsion group $E_{m}(\mathbb{Q})_{\rm{tors}}$, a lower bound for the Mordell-Weil rank $r({E_{m}})$ and the $2$-Selmer group ${\rm{Sel}}_{2}(E_{m})$ under certain conditions on $m$. This extends the previous works done in this direction, which are mostly concerned with the Mordell-Weil ranks of various parametric families of elliptic curves.
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https://arxiv.org/abs/2503.04561
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4ace4b48e059eb39a58134b384e7863d5a273ede19e73e9016ca22c2305c08f1
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2026-01-13T00:00:00-05:00
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On Approximate Representation of Fractional Brownian Motion
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arXiv:2503.04575v3 Announce Type: replace Abstract: This paper considers the orthogonal expansion of the fractional Brownian motion relative to the Legendre polynomials. Such an expansion has not only theoretical but also practical interest, since it can be applied to approximate and simulate the fractional Brownian motion in continuous time. The relations for the mean square approximation error are presented, and a comparison with the previously obtained result is carried out.
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https://arxiv.org/abs/2503.04575
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ddc3ebcfedbe79393465f77734e0f59d5f526738875044eddf8425f21b9554da
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2026-01-13T00:00:00-05:00
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Algebraic and Combinatorial Stability of Independence Polynomials in Iterated Strong Products of Cycles
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arXiv:2503.07910v2 Announce Type: replace Abstract: This paper investigates the independence polynomials arising from iterated strong products of cycle graphs, examining their algebraic symmetries and combinatorial structures. Leveraging modular arithmetic and Galois theory, we establish precise conditions under which these polynomials factor over finite fields, highlighting modular collapses based on prime and composite cycle lengths. We demonstrate that while real-rootedness depends on cycle parity, the combinatorial structure ensures universal log-concavity and unimodality of coefficients. A toggling argument provides a combinatorial proof of unimodality, complementing algebraic methods and offering insights into polynomial stability. These findings bridge combinatorial and algebraic perspectives, contributing to graph-theoretic frameworks with implications in statistical mechanics and information theory.
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https://arxiv.org/abs/2503.07910
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abec732f7fc4d336617396fe42eb6744d35bcb8429062b05ef733bcec3698c93
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2026-01-13T00:00:00-05:00
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The LZ78 Source
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arXiv:2503.10574v3 Announce Type: replace Abstract: We study a family of processes generated according to sequential probability assignments induced by the LZ78 universal compressor. We characterize entropic and distributional properties such as their entropy and relative entropy rates, finite-state compressibility and log loss of their realizations, and the empirical distributions that they induce. Though not quite stationary, these sources are "almost stationary and ergodic;" similar to stationary and ergodic processes, they satisfy a Shannon-McMillan-Breiman-type property: the normalized log probability of their realizations converges almost surely to their entropy rate. Further, they are locally "almost i.i.d." in the sense that the finite-dimensional empirical distributions of their realizations converge almost surely to a deterministic i.i.d. law. However, unlike stationary ergodic sources, the finite-state compressibility of their realizations is almost surely strictly larger than their entropy rate by a "Jensen gap". We present simulations demonstrating the theoretical results. These sources allow to gauge the performance of sequential probability models, both classical and deep learning-based, on non-Markovian non-stationary data. As such, we apply realizations of the LZ78 source to the study of in-context learning in transformer models.
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https://arxiv.org/abs/2503.10574
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25ec4a6b161ba05e76977a24ed428d976afc958ea9dc805dad3cf2c7e9707fb4
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2026-01-13T00:00:00-05:00
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Two statistical problems for multivariate mixture distributions
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arXiv:2503.12147v4 Announce Type: replace Abstract: We address two important statistical problems: that of estimating for mixtures of multivariate normal distributions and mixtures of $t$-distributions based of univariate projections, and that of measuring the agreement between two different random partitions. The results are based on an earlier work of the authors, where it was shown that mixtures of multivariate Gaussian or $t$-distributions can be distinguished by projecting them onto a certain predetermined finite set of lines, the number of lines depending only on the total number of distributions involved and on the ambient dimension. We also compare our proposal with robust versions of the expectation-maximization method EM. In each case, we present algorithms for effecting the task, and compare them with existing methods by carrying out some simulati
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https://arxiv.org/abs/2503.12147
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18ba9cb79060b1707935d1291fee2c16b9881e5a83034cb4294830755ae911b9
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2026-01-13T00:00:00-05:00
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Strassen's LIL and a Phase transition for the capacity of the random walk under diameter constraints
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arXiv:2503.12728v2 Announce Type: replace Abstract: We discuss the relationship between the capacity and the geometry for the range of the random walk for $d=3$. In particular, we consider how efficiently the random walk moves or what shape it forms in order to maximize its capacity. In one of our main results, we show a functional law for the capacity of the random walk. In addition, we find that there is a phase transition for the asymptotics of the capacity of the random walk when we condition the diameter of the random walk.
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https://arxiv.org/abs/2503.12728
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f3b9212be68b28d4d3611151ffa54ab88e8b2d915ee42cb268ed94c3f5c41abc
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2026-01-13T00:00:00-05:00
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Disproving two conjectures on the Hamiltonicity of Venn diagrams
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arXiv:2503.18554v4 Announce Type: replace Abstract: In 1984, Winkler conjectured that every simple Venn diagram with $n$ curves can be extended to a simple Venn diagram with $n+1$ curves. His conjecture is equivalent to the statement that the dual graph of any simple Venn diagram has a Hamilton cycle. In this work, we construct counterexamples to Winkler's conjecture for all $n\geq 6$. As part of this proof, we computed all 3.430.404 simple Venn diagrams with $n=6$ curves (even their number was not previously known), among which we found 72 counterexamples. We also construct monotone Venn diagrams, i.e., diagrams that can be drawn with $n$ convex curves, and are not extendable, for all $n\geq 7$. Furthermore, we also disprove another conjecture about the Hamiltonicity of the (primal) graph of a Venn diagram. Specifically, while working on Winkler's conjecture, Pruesse and Ruskey proved that this graph has a Hamilton cycle for every simple Venn diagram with $n$ curves, and conjectured that this also holds for non-simple diagrams. We construct counterexamples to this conjecture for all $n\geq 4$.
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https://arxiv.org/abs/2503.18554
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58fa1f9693f2806c5313f659243d20e19477c7c6e308042fe4d56c2d8d433906
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2026-01-13T00:00:00-05:00
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Calibration Bands for Mean Estimates within the Exponential Dispersion Family
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arXiv:2503.18896v2 Announce Type: replace Abstract: A statistical model is said to be calibrated if the resulting mean estimates perfectly match the true means of the underlying responses. Aiming for calibration is often not achievable in practice as one has to deal with finite samples of noisy observations. A weaker notion of calibration is auto-calibration. An auto-calibrated model satisfies that the expected value of the responses for a given mean estimate matches this estimate. Testing for autocalibration has only been considered recently in the literature and we propose a new approach based on calibration bands. Calibration bands denote a set of lower and upper bounds such that the probability that the true means lie simultaneously inside those bounds exceeds some given confidence level. Such bands were constructed by Yang-Barber (2019) for sub-Gaussian distributions. Dimitriadis et al. (2023) then introduced narrower bands for the Bernoulli distribution. We use the same idea in order to extend the construction to the entire exponential dispersion family that contains for example the binomial, Poisson, negative binomial, gamma and normal distributions. Moreover, we show that the obtained calibration bands allow us to construct various tests for calibration and auto-calibration, respectively. As the construction of the bands does not rely on asymptotic results, we emphasize that our tests can be used for any sample size.
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https://arxiv.org/abs/2503.18896
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b6489355af3da5661b1741fc6c1a09f90cd74069d2f3c66aec2fa67c8173e0fd
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2026-01-13T00:00:00-05:00
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Convergent Power Series for Anharmonic Chain with Periodic Forcing
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arXiv:2503.23527v5 Announce Type: replace Abstract: We study the propagation of energy in one-dimensional anharmonic chains subject to a periodic, localized forcing. For the purely harmonic case, forcing frequencies outside the linear spectrum produce exponentially localized responses, preventing equi-distribution of energy per degree of freedom. We extend this result to anharmonic perturbations with bounded second derivatives and boundary dissipation, proving that for small perturbations and non-resonant forcing, the dynamics converges to a periodic stationary state with energy exponentially localized uniformly in the system size. The perturbed periodic state is described by a convergent power type expansion in the strength of the anharmonicity. This excludes chaoticity induced by anharmonicity, independently of the size of the system. Our perturbative scheme can also be applied in higher dimensions.
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https://arxiv.org/abs/2503.23527
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68dc0fa01143d94f9008bb81b03d2f3a41c5db4a2a5efbebdef4d498779f8e6a
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2026-01-13T00:00:00-05:00
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A Comparison of Parametric Dynamic Mode Decomposition Algorithms for Thermal-Hydraulics Applications
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arXiv:2503.24205v2 Announce Type: replace Abstract: In recent years, algorithms aiming at learning models from available data have become quite popular due to two factors: 1) the significant developments in Artificial Intelligence techniques and 2) the availability of large amounts of data. Nevertheless, this topic has already been addressed by methodologies belonging to the Reduced Order Modelling framework, of which perhaps the most famous equation-free technique is Dynamic Mode Decomposition. This algorithm aims to learn the best linear model that represents the physical phenomena described by a time series dataset: its output is a best state operator of the underlying dynamical system that can be used, in principle, to advance the original dataset in time even beyond its span. However, in its standard formulation, this technique cannot deal with parametric time series, meaning that a different linear model has to be derived for each parameter realization. Research on this is ongoing, and some versions of a parametric Dynamic Mode Decomposition already exist. This work contributes to this research field by comparing the different algorithms presently deployed and assessing their advantages and shortcomings compared to each other. To this aim, three different thermal-hydraulics problems are considered: two benchmark 'flow over cylinder' test cases at diverse Reynolds numbers, whose datasets are, respectively, obtained with the FEniCS finite element solver and retrieved from the CFDbench dataset, and the DYNASTY experimental facility operating at Politecnico di Milano, which studies the natural circulation established by internally heated fluids for Generation IV nuclear applications, whose dataset was generated using the RELAP5 nodal solver.
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https://arxiv.org/abs/2503.24205
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dab72c06934f3eb70872facf9bdfc4cacbbd28d1e865e5b78dce7159c3b0614f
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2026-01-13T00:00:00-05:00
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Uniform convergence to the equilibrium of the homogeneous Boltzmann-Fermi-Dirac Equation with moderately soft potential
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arXiv:2504.01341v2 Announce Type: replace Abstract: We concern the long-time behavior of mild solutions to the spatially homogeneous Boltzmann--Fermi--Dirac equation with moderately soft potential. Based on the well-posedness results in [X-G. Lu, J. Stat. Phys., 105, (2001), 353-388], we prove that the mild solution decays algebraically to the Fermi--Dirac statistics with an explicit rate. Under the framework of the level set analysis by De Giorgi, we derive an $L^\infty$ estimate which is uniform with respect to the quantum parameter $\varepsilon$. All quantitative estimates are independent of $\varepsilon$, which implies that they also hold in the classical limit, i.e., the Boltzmann equation.
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https://arxiv.org/abs/2504.01341
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c29d5f560aa338cfcdb7b1aaf832ece4a9f3b29e2b2a8d5b5fec25175c5b51f6
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2026-01-13T00:00:00-05:00
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Certified Model Order Reduction for parametric Hermitian eigenproblems
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arXiv:2504.02672v3 Announce Type: replace Abstract: This article deals with the efficient and certified numerical approximation of the smallest eigenvalue and the associated eigenspace of a large-scale parametric Hermitian matrix. For this aim, we rely on projection-based model order reduction (MOR), i.e., we approximate the large-scale problem by projecting it onto a suitable subspace and reducing it to one of a much smaller dimension. Such a subspace is constructed by means of weak greedy-type strategies. After detailing the connections with the reduced basis method for source problems, we introduce a novel error estimate for the approximation error related to the eigenspace associated with the smallest eigenvalue. Since the difference between the second smallest and the smallest eigenvalue, the so-called spectral gap, is crucial for the reliability of the error estimate, we propose efficiently computable upper and lower bounds for higher eigenvalues and for the spectral gap, which enable the assembly of a subspace for the MOR approximation of the spectral gap. Based on that, a second subspace is then generated for the MOR approximation of the eigenspace associated with the smallest eigenvalue. We also provide efficiently computable conditions to ensure that the multiplicity of the smallest eigenvalue is fully captured in the reduced space. This work is motivated by a specific application: the repeated identifications of the states with minimal energy, the so-called ground states, of parametric quantum spin system models.
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https://arxiv.org/abs/2504.02672
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4041388f876a5036979990056d11f8deb5d127ea6571f43809ddb63e552f0178
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2026-01-13T00:00:00-05:00
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Hybrid Nitsche method for distributed computing
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arXiv:2504.05036v2 Announce Type: replace Abstract: We extend a distributed finite element method built upon model order reduction to arbitrary polynomial degree using a hybrid Nitsche scheme. The new method considerably simplifies the transformation of the finite element system to the reduced basis for large problems. We prove that the error of the reduced Nitsche solution converges optimally with respect to the approximation order of the finite element spaces and linearly with respect to the dimension reduction parameter. Numerical tests with nontrivial tetrahedral meshes using second-degree polynomial bases support the theoretical results.
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https://arxiv.org/abs/2504.05036
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babf77cbb06c02a11f0146932455550d1547177a341f7a023af19ce001f94fee
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2026-01-13T00:00:00-05:00
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Instability of the Standing Pulse in Skew-Gradient Systems and Its Application to FitzHugh-Nagumo Type Systems
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arXiv:2504.11211v2 Announce Type: replace Abstract: Classical results from Sturm-Liouville theory establish that the Morse index of a one-dimensional Sturm-Liouville operator defined on $\mathbb{R}$ is equal to the number of its associated conjugate points. Recent advancements by Beck et al.~\cite{BCJLM18} have extended these results to higher-dimensional Sturm-Liouville operators on $\mathbb{R}$, utilizing the Maslov index to characterize the spectral stability of nonlinear waves in multi-component systems. In this paper, we extend this framework further to non-self-adjoint settings by investigating skew-gradient reaction-diffusion systems. By utilizing the Maslov index and spectral flow, we derive an instability criterion for standing pulses. This approach bridges the gap between variational methods and the stability index in systems where the standard self-adjoint structure is absent. As a primary application, we apply our results to FitzHugh-Nagumo type systems, where the reaction terms for both the activator and inhibitor exhibit intrinsic nonlinearities. This provides a robust topological method to account for the influence of nonlinear inhibition on pulse stability in the non-self-adjoint regime.
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https://arxiv.org/abs/2504.11211
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46a6b5d04fb2f9abb6390df0cc54d1912f38e841734a3d7df8f57cebc2f6395f
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2026-01-13T00:00:00-05:00
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Theoretical and computational investigations of superposed interacting affine and more complex processes
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arXiv:2504.13712v3 Announce Type: replace Abstract: We theoretically and computationally investigate long-memory processes based on the Markovian lifts of affine jump-diffusion processes. A nominal superposition process consisting of an infinite number of interacting affine processes is considered, along with its finite-dimensional version and associated generalized Riccati equations. We propose a splitting scheme suited to the Markovian lifts where jump and diffusion parts are dealt with separately based on recently developed exact discretization methods. We examine the computational performance of the scheme through comparisons with the analytical results. We also numerically investigate a more complex model arising in the environmental sciences and some extended cases in which superposed processes belong to a class of nonlinear processes that generalize affine processes.
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https://arxiv.org/abs/2504.13712
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0bdb6fdb27d66a604a7e1a18d88014acce99b2f7b0efd3bea58b2b902891bf08
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2026-01-13T00:00:00-05:00
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Asymptotically well-calibrated Bayesian $p$-value using the Kolmogorov-Smirnov statistic
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arXiv:2504.14077v3 Announce Type: replace Abstract: The posterior predictive $p$-value (ppp) is widely used in Bayesian model evaluation. However, due to double use of the data, the ppp may not be a valid $p$-value even in large samples: The asymptotic null distribution of the ppp can be non-uniform unless the underlying test statistic satisfies certain well-calibration conditions. Such conditions have been studied in the literature for asymptotically normal test statistics. We extend this line of work by establishing well-calibration conditions for test statistics that are not necessarily asymptotically normal. In particular, we show that Kolmogorov-Smirnov (KS)-type test statistics satisfy these conditions, such that their ppps are asymptotically well-calibrated Bayesian $p$-values. KS-type statistics are versatile, omnibus, and sensitive to model misspecifications. They apply to i.i.d. real-valued data, as well as non-identically distributed observations under regression models. Numerical experiments demonstrate that such $p$-values are well behaved in finite samples and can effectively detect a wide range of alternative models.
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https://arxiv.org/abs/2504.14077
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52945deaa318babe0834e071de34a189e74f7d8d32fb7583e41d528cfbd4541a
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2026-01-13T00:00:00-05:00
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Jacobson identities for post-Lie algebras in positive characteristic
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arXiv:2504.14540v2 Announce Type: replace Abstract: Let $p$ be a prime number. Given a restricted Lie algebra over a field of characteristic $p$ and a post-Lie operation over it, we prove the Jacobson identities for a $p$-structure built from the Lie bracket and the post-Lie operation, called sub-adjacent $p$-structure. Furthermore, we give sufficient conditions for the sub-adjacent Lie algebra to be restricted if equipped with this sub-adjacent $p$-structure. This construction is ''axiomatized'' by introducing the notion of restricted post-Lie algebras, and we work out several examples.
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https://arxiv.org/abs/2504.14540
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5925179939e3a76de69194164773305330ec802242bba8b4056005458e2cf236
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2026-01-13T00:00:00-05:00
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Projection Coefficients Estimation in Continuous-Variable Quantum Circuits
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arXiv:2504.16246v3 Announce Type: replace Abstract: In this work, we propose a continuous-variable quantum algorithm to compute the projection coefficients of a holomorphic function in the Segal--Bargmann space by leveraging its isometric correspondence with single-mode quantum states. Using CV quantum circuits, we prepare the state $\ket{f}$ associated with $f(z)$ and extract the coefficients $c_n = \braket{n}{f}$ via photon-number-resolved detection, enhanced by interferometric phase referencing to recover full complex amplitudes. This enables direct quantum estimation and visualization of the coefficient sequence -- offering a scalable, measurement-based alternative to classical numerical integration for functional analysis and non-Gaussian state characterization.
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https://arxiv.org/abs/2504.16246
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46de2ec9c967ac3bf31dc7685055df6c48eb6acb8e122e9e8220976fa8e0a07f
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2026-01-13T00:00:00-05:00
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Secret Sharing in the Rank Metric
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arXiv:2504.18294v2 Announce Type: replace Abstract: The connection between secret sharing and matroid theory is well established. In this paper, we generalize the concepts of secret sharing and matroid ports to $q$-polymatroids. Specifically, we introduce the notion of an access structure on a vector space, and consider properties related to duality, minors, and the relationship to $q$-polymatroids. Finally, we show how rank-metric codes give rise to secret sharing schemes within this framework.
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https://arxiv.org/abs/2504.18294
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68e953952dd34f49a1288b97695818b9301973d8d6869a0c185884a234a638b3
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2026-01-13T00:00:00-05:00
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Efficient approximations of matrix multiplication using truncated decompositions
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arXiv:2504.19308v3 Announce Type: replace Abstract: We exploit the truncated singular value decomposition and the recently proposed circulant decomposition for an efficient first-order approximation of the multiplication of large dense matrices. A decomposition of each matrix into a sum of a sparse matrix with relatively few dominant entries and a dense residue can also use the above approach, and we present methods for multiplication using a Fourier decomposition and a cycle decomposition-based sparsifications. The proposed methods scale as $\mathcal{O}(n^2 \log n)$ in arithmetic operations for $n \times n$ matrices for usable tolerances in relative error $\sim$ 1\%. Note that different decompositions for the two matrices $A$ and $B$ in the product $AB$ are also possible in this approach, using efficient a priori evaluations for suitability, to improve further on the error tolerances demonstrated here.
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https://arxiv.org/abs/2504.19308
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18e72c533d73b2b2dd7114c53bb7ac72894eba502f14ef6139b0d4226de6b2bb
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2026-01-13T00:00:00-05:00
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Asymptotics of zeta determinants of Laplacians on large degree abelian covers
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arXiv:2505.01586v2 Announce Type: replace Abstract: Let $(M,g)$ be some smooth, closed, compact Riemannian manifold and $(M_N\mapsto M)_N$ be an increasing sequence of large degree cyclic covers of $M$ that converges when $N\rightarrow +\infty$, in a suitable sense, to some limit $\mathbb{Z}^p$ cover $M_\infty$ over $M$. Motivated by recent works on zeta determinants on random surfaces and some natural questions in Euclidean quantum field theory, we show the convergence of the sequence $ \frac{\log\det_\zeta(\Delta_{N})}{\text{Vol}(M_N)} $ when $N\rightarrow +\infty$ where $\Delta_N$ is the Laplace-Beltrami operator on $M_N$. We also generalize our results to the case of twisted Laplacians coming from certain flat unitary vector bundles over $M$.
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https://arxiv.org/abs/2505.01586
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0621f67fa6c2953239f168ba6eb7f9d716dc38cdaa3e9978fc483b71ba685f51
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2026-01-13T00:00:00-05:00
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The raspberries in three dimensions with at most two sizes of berry
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arXiv:2505.03408v2 Announce Type: replace Abstract: In three dimensional Euclidean space, a raspberry is defined to be an arrangement of spheres with pairwise disjoint interiors, where all spheres are tangent to a central unit sphere and such that the contact graph of the non-central spheres triangulates the central sphere. We discuss the relevance of these structures in related work. We present a catalog of all configurations of radii that permit the formation of raspberries that have at most two sizes of non-central spheres. Throughout, we discuss the construction of this catalog.
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https://arxiv.org/abs/2505.03408
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266e1bbee4c6e98717e306dc4d6cfc0fcfa92e5ed05c367146998ff6fdafa499
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2026-01-13T00:00:00-05:00
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m-Accretive Extensions of Friedrichs Operators
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arXiv:2505.03657v2 Announce Type: replace Abstract: The introduction of abstract Friedrichs operators in 2007-an operator-theoretic framework for studying classical Friedrichs operators has led to significant developments in the field, including results on well-posedness, multiplicity, and classification. More recently, the von Neumann extension theory has been explored in this context, along with connections between abstract Friedrichs operators and skew-symmetric operators. In this work, we show that all m-accretive extensions of abstract Friedrichs operators correspond precisely to those satisfying (V)-boundary conditions. We also establish a connection between the m-accretive extensions of abstract Friedrichs operators and their skew-symmetric components. Additionally, the three equivalent formulations of boundary conditions are unified within a single interpretive framework. To conclude, we discuss a constructive relation between (V)- and (M)-boundary conditions and examine the multiplicity of the associated M-operators. We demonstrate our results on two examples, namely, the first order ordinary differential equation on an interval, with various boundary conditions, and the second-order elliptic partial differential equation with Dirichlet boundary conditions.
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https://arxiv.org/abs/2505.03657
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c7c120bcc62aafacdb20bc768939d2c79d900be86a35a660c81dc5906baef827
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2026-01-13T00:00:00-05:00
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On some critical Riemannian metrics and Thorpe-type conditions
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arXiv:2505.06990v3 Announce Type: replace Abstract: We study critical metrics of higher-order curvature functionals on compact Riemannian $n$-manifolds $(M,g)$. For an integer $k$ with $2 \leq 2k \leq n$, let $R^k$ denote the $k$-th exterior power of the Riemann curvature tensor. We investigate the Riemannian functionals \[H_{2k}(g)=\int_M \operatorname{tr}(R^k)\,\mathrm{dvol}_g\quad\text{and}\quad G_{2k}(g)=\int_M \|R^k\|^2\,\mathrm{dvol}_g,\] which generalize the Hilbert--Einstein functional and the total squared norm curvature, obtained for $k=1$ respectively. Using the formalism of double forms, we develop a systematic variational framework yielding compact first variation formulas for these functionals. Two key lemmas streamline the variational computations. A central technical ingredient is a generalization of the classical Lanczos identity to symmetric double forms of arbitrary even degree, providing explicit algebraic relations between the tensors $\cc^{2k-1}(R^k \circ R^k)$ and $\cc^{4k-1}(R^{2k})$. As a main geometric application, we introduce $(2k)$-Thorpe and $(2k)$-anti-Thorpe metrics, defined by self-duality and anti-self-duality conditions on $g^{r-2k}R^k$ in even dimensions $n=2r$. In the critical dimension $n=4k$, these metrics are absolute minimizers of $G_{2k}$, with the minimum determined by the Euler characteristic. For $n>4k$, they satisfy a harmonicity property leading to rigidity results under suitable curvature positivity assumptions. We further establish equivalences among variational criticality conditions. For hyper-$(2k)$-Einstein metrics, characterized by $\cc R^k=\lambda g^{2k-1}$, being critical for $G_{2k}$ is equivalent to being $(4k)$-Einstein and to being weakly $(2k)$-Einstein. In the locally conformally flat setting, we classify all $4$-Thorpe metrics, showing that they are either space forms or Riemannian products $\mathbb{S}^r(c) \times \mathbb{H}^r(-c)$.
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https://arxiv.org/abs/2505.06990
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a261d3bbaba9fbfec6638d8084ccaafd97d2cd2472e93f09839b95e7aed14a90
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2026-01-13T00:00:00-05:00
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A 2-torsion invariant of 2-knots
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arXiv:2505.13699v2 Announce Type: replace Abstract: In this paper we describe what should perhaps be called a `type-2' Vassiliev invariant of knots S^2 -> S^4. We give a formula for an invariant of 2-knots, taking values in Z_2 that can be computed in terms of the double-point diagram of the knot. The double-point diagram is a collection of curves and diffeomorphisms of curves, in the domain S^2, that describe the crossing data with respect to a projection, analogous to a chord diagram for a projection of a classical knot S^1 -> S^3. Our formula turns the computation of the invariant into a planar geometry problem. More generally, we describe a numerical invariant of families of knots S^j -> S^n, for all n >= j+2 and j >= 1. In the co-dimension two case n=j+2 the invariant is an isotopy invariant, and either takes values in Z or Z_2 depending on a parity issue.
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https://arxiv.org/abs/2505.13699
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74dee641a76369ef7c704985068bbfe2ccaf7cfd8d7d7e3501c31629bd5e188e
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2026-01-13T00:00:00-05:00
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A new measure of dependence: Integrated $R^2$
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arXiv:2505.18146v4 Announce Type: replace Abstract: We introduce a novel measure of dependence that captures the extent to which a random variable $Y$ is determined by a random vector $X$. The measure equals zero precisely when $Y$ and $X$ are independent, and it attains one exactly when $Y$ is almost surely a measurable function of $X$. We further extend this framework to define a measure of conditional dependence between $Y$ and $X$ given $Z$. We propose a simple and interpretable estimator with computational complexity comparable to classical correlation coefficients, including those of Pearson, Spearman, and Chatterjee. Leveraging this dependence measure, we develop a tuning-free, model-agnostic variable selection procedure and establish its consistency under appropriate sparsity conditions. Extensive experiments on synthetic and real datasets highlight the strong empirical performance of our methodology and demonstrate substantial gains over existing approaches.
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https://arxiv.org/abs/2505.18146
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45ee8012e4d179506e30f49a9493f38e8e6a385d4e6974598ba44b70252c116f
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2026-01-13T00:00:00-05:00
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Second Order Properties of Thinned Counts in Finite Birth--Death Processes
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arXiv:2505.20174v2 Announce Type: replace Abstract: The paper studies the counting process arising as a subset of births and deaths in a birth--death process on a finite state space. Whenever a birth or death occurs, the process is incremented or not depending on the outcome of an independent Bernoulli experiment whose probability is a state-dependent function of the birth and death and also depends on whether it is a birth or death that has occurred. We establish a formula for the asymptotic variance rate of this process, also presented as the ratio of the asymptotic variance and the asymptotic mean. Several examples including queueing models illustrate the scope of applicability of the results. An analogous formula for the countably infinite state space is conjectured and tested.
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https://arxiv.org/abs/2505.20174
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aec1dd0dd083438b168ec4049e25381b787ad78d1a134070b8b6484f67fa674e
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2026-01-13T00:00:00-05:00
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Definable ranks
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arXiv:2506.00443v4 Announce Type: replace Abstract: We introduce the notion of the definable rank of an ordered field, ordered abelian group and ordered set, respectively. We study the relation between the definable rank of an ordered field and the definable rank of the value group of its natural valuation. Similarly, we compare the definable rank of an ordered abelian group to that of its value set with respect to the natural valuation. We describe the definable rank on the group-level by characterizing the definable convex subgroups. We also give a detailed comparison of field- and group-level, in particular for ordered fields with henselian natural valuation. We investigate definability of final segments in ordered sets and introduce definable condensation as a tool for further study.
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https://arxiv.org/abs/2506.00443
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3673d7bdd6a95bec59429bdafeeb83e4b4e3c05b7d257e839fa3796089910a7b
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2026-01-13T00:00:00-05:00
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R-PINN: Recovery-type a-posteriori estimator enhanced adaptive PINN
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arXiv:2506.10243v2 Announce Type: replace Abstract: In recent years, with the advancements in machine learning and neural networks, algorithms using physics-informed neural networks (PINNs) to solve PDEs have gained widespread applications. While these algorithms are well-suited for a wide range of equations, they often exhibit suboptimal performance when applied to equations with large local gradients, resulting in substantial localized errors. To address this issue, this paper proposes an adaptive PINN algorithm designed to improve accuracy in such cases. The core idea of the algorithm is to adaptively adjust the distribution of collocation points based on the recovery-type a-posterior error of the current numerical solution, enabling a better approximation of the true solution. This approach is inspired by the adaptive finite element method. By combining the recovery-type a-posteriori estimator, a gradient-recovery estimator commonly used in the adaptive finite element method (FEM) with PINNs, we introduce the Recovery-type a-posteriori estimator enhanced adaptive PINN (R-PINN) and compare its performance with a typical adaptive PINN algorithm, FI-PINN. Our results demonstrate that R-PINN achieves faster convergence with fewer adaptive points and significantly outperforms in the cases with multiple regions of large errors than FI-PINN. Notably, our method is a hybrid numerical approach for solving partial differential equations, integrating adaptive FEM with PINNs.
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https://arxiv.org/abs/2506.10243
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