id
stringlengths 64
64
| published
stringlengths 19
25
| title
stringlengths 7
262
| description
stringlengths 6
54.4k
| link
stringlengths 31
227
| category
stringclasses 6
values | image
stringlengths 3
247
|
|---|---|---|---|---|---|---|
ee3efb0992368a5a8786cca15fb756251135781f43c2a75ee04941c9590e110c
|
2026-01-13T00:00:00-05:00
|
Lagrange multiplier expressions for matrix polynomial optimization and tight relaxations
|
arXiv:2506.12579v2 Announce Type: replace Abstract: This paper studies matrix constrained polynomial optimization. We investigate how to get explicit expressions for Lagrange multiplier matrices from the first order optimality conditions. The existence of these expressions can be shown under the nondegeneracy condition. Using Lagrange multiplier matrix expressions, we propose a strengthened Moment-SOS hierarchy for solving matrix polynomial optimization. Under some general assumptions, we show that this strengthened hierarchy is tight, or equivalently, it has finite convergence. We also study how to detect tightness and how to extract optimizers. Numerical experiments are provided to show the efficiency of the strengthened hierarchy.
|
https://arxiv.org/abs/2506.12579
|
Academic Papers
|
svg
|
4c6c26c66c8796190f1ef691edb155856265c6e8bbe3e807a06a7fcfa8e44cf5
|
2026-01-13T00:00:00-05:00
|
Generalized Frobenius Manifold Structures on the Orbit Spaces of Affine Weyl Groups I
|
arXiv:2506.13656v3 Announce Type: replace Abstract: We present an approach to construct a class of generalized Frobenius manifold structures on the orbit spaces of affine Weyl groups, and prove that their monodromy groups are parabolic subgroups of the associated affine Weyl groups.
|
https://arxiv.org/abs/2506.13656
|
Academic Papers
|
svg
|
9c7b44da7393e0029885e5bc44151e4ff4bf8124730b970b85350c9053d6659e
|
2026-01-13T00:00:00-05:00
|
Automorphism groups and linearizability of rational Fano conic bundle threefolds
|
arXiv:2506.15042v2 Announce Type: replace Abstract: We generalize the equivariant intermediate Jacobian torsor obstruction over $\mathbb{C}$ to algebraically closed fields of characteristic zero. It is an obstruction to the (projective) linearizability problem of finite group actions on threefolds. In addition, we calculate automorphism groups of general smooth Fano threefolds of No. 2.18. As an application, we prove that a general smooth Fano threefold X of No. 2.18 is linearizable for its automorphism group.
|
https://arxiv.org/abs/2506.15042
|
Academic Papers
|
svg
|
7d962ae2ae1de83e96aa4f7aef019a4f068fb32025e663d17622fdda6eff09b7
|
2026-01-13T00:00:00-05:00
|
Transporting a Dirac mass in a mean field planning problem
|
arXiv:2506.16041v2 Announce Type: replace Abstract: We study a mean field planning problem in which the initial density is a Dirac mass. We show that there exists a unique solution which converges to a self-similar profile as time tends to $0$. We proceed by studying a continuous rescaling of the solution, and characterizing its behavior near the initial time through an appropriate Lyapunov functional.
|
https://arxiv.org/abs/2506.16041
|
Academic Papers
|
svg
|
63a5dd7c2a35be3a16844e3793e958b727ac0044d098474bffaba1580e6e1ff5
|
2026-01-13T00:00:00-05:00
|
Deep random difference method for high-dimensional quasilinear parabolic partial differential equations
|
arXiv:2506.20308v2 Announce Type: replace Abstract: Solving high-dimensional parabolic partial differential equations (PDEs) with deep learning methods is often computationally and memory intensive, primarily due to the need for automatic differentiation (AD) to compute large Hessian matrices in the PDE. In this work, we propose a deep random difference method (DRDM) that addresses these issues by approximating the convection-diffusion operator using only first-order differences and the solution by deep neural networks, thus avoiding Hessian and other derivative computations. The DRDM is implemented within a Galerkin framework to reduce sampling variance, and the solution space is explored using stochastic differential equations (SDEs) to capture the dynamics of the convection-diffusion operator. The approach is then extended to solve Hamilton-Jacobi-Bellman (HJB) equations, which recovers existing martingale deep learning methods for PDEs [{\it SIAM J. Sci. Comput.}, 47 (2025), pp. C795-C819], without using stochastic calculus. The proposed method offers two main advantages: it avoids the need to compute derivatives in PDEs and enables parallel computation of the loss function in both time and space. Moreover, a rigorous error estimate is proven for the quasi-linear parabolic equation, showing first-order accuracy in $h$, the time step used in the discretization of the SDE paths by the Euler-Maruyama scheme. Numerical experiments demonstrate that the method can efficiently and accurately solve quasilinear parabolic PDEs and HJB equations in dimensions up to $10^5$ and $10^4$, respectively.
|
https://arxiv.org/abs/2506.20308
|
Academic Papers
|
svg
|
96de19c4a9efa1a90c0578670acf1beb5b72ef2098443cba1d7ac59cdaba8c4e
|
2026-01-13T00:00:00-05:00
|
On a minimal free resolution of the residue field over a local ring of codepth 3 of class $T$
|
arXiv:2506.21505v2 Announce Type: replace Abstract: Let $R$ be any noetherian local ring with residue field $k$, and $A$ the homology of the Koszul complex on a minimal set of generators of the maximal ideal of $R$. In this paper, we show that a minimal free resolution of $k$ over $R$ can be obtained from a graded minimal free resolution of $k$ over $A$. More precisely, this is done by the iterated mapping cone construction, introduced by the authors in a previous work, using specific choices of ingredients. As applications, using this general perspective, we exhibit a minimal free resolution of $k$ over a complete intersection ring of any codepth, and explicitly construct a minimal free resolution of $k$ over a noetherian local ring of codepth 3 of class $T$ in terms of Koszul blocks.
|
https://arxiv.org/abs/2506.21505
|
Academic Papers
|
svg
|
daf73ddffa04c182faacd504738d94b5e5c359e073a3220e633a9b9aad8db191
|
2026-01-13T00:00:00-05:00
|
The Dedekind-Hasse Criterion in Quaternion Algebras
|
arXiv:2506.22651v2 Announce Type: replace Abstract: We show that a criterion for an integral domain to be a principal ideal domain (PID), due to Dedekind and Hasse, can also be applied in quaternion orders, and that it can be used to build a finite algorithm to determine if a given order is a principal left (or right) ideal domain. Using this algorithm, we give an alternative proof that the maximal orders of discriminant 7 and 13, which are non-Euclidean, are PIDs. We also provide a completely arithmetic proof of a result of Gordon Pall that shows that, in an order that is a PID, an element of whose norm is divisible by an integer $m$ always has a left and a right divisor with norm $m$. This easily yields the existence and uniqueness (up to associates) of factorizations of a quaternion modeled on a factorization of its norm.
|
https://arxiv.org/abs/2506.22651
|
Academic Papers
|
svg
|
677f61dda199c8c309ae2e37799f2d328d1a73d6a90583acae7189dbbf261a2f
|
2026-01-13T00:00:00-05:00
|
An infinite family of pairs of distinct quartic Galois CM-fields with the same discriminant and regulator
|
arXiv:2506.23541v2 Announce Type: replace Abstract: We construct an infinite family of pairs of distinct imaginary biquadratic fields and pairs of distinct imaginary cyclic quartic fields with the same discriminant and regulator. We also construct an infinite family of imaginary biquadratic fields and imaginary cyclic quartic fields with the same regulator. Moreover, we give examples of a pair of distinct imaginary biquadratic fields and a pair of distinct imaginary cyclic quartic fields with the same discriminant, regulator and class number.
|
https://arxiv.org/abs/2506.23541
|
Academic Papers
|
svg
|
57b5c6eee927be1b74359124858a665946b62243643d6755533497dd277061e2
|
2026-01-13T00:00:00-05:00
|
Existence and Incompressible Limit of Weak and Classical Solutions to the Cauchy Problem for Compressible Navier-Stokes Equations with Large Bulk Viscosity and Large Initial Data
|
arXiv:2507.02497v2 Announce Type: replace Abstract: This paper investigates the Cauchy problem for the barotropic compressible Navier-Stokes equations in $\mathbb{R}^2$ with the constant state as far field, which could be vacuum or non-vacuum. Under the assumption of a sufficiently large bulk viscosity coefficient, we establish the global existence of weak, strong, and classical solutions for the non-vacuum far-field case, and long-time existence for the vacuum far-field case. It should be mentioned that this result is obtained without any restrictions on the size of the initial data. Moreover, we demonstrate that the solutions of the compressible Navier-Stokes equations converge to solutions of the inhomogeneous incompressible Navier-Stokes equations, as the bulk viscosity coefficient tends to infinity. The incompressible limit of the weak solutions holds even without requiring the initial velocity to be divergence-free.
|
https://arxiv.org/abs/2507.02497
|
Academic Papers
|
svg
|
18ac072d0b69839f2f5d35f173a3084bc802a8e9388e9322cbe2d1cb6574091e
|
2026-01-13T00:00:00-05:00
|
Skew Laurent Series and General Cyclic Convolutional Codes
|
arXiv:2507.05022v2 Announce Type: replace Abstract: Convolutional codes were originally conceived as vector subspaces of a finite-dimensional vector space over a field of Laurent series having a polynomial basis. Piret and Roos modeled cyclic structures on them by adding a module structure over a finite-dimensional algebra skewed by an algebra automorphism. These cyclic convolutional codes turn out to be equivalent to some right ideals of a skew polynomial ring built from the automorphism. When a skew derivation is considered, serious difficulties arise in defining such a skewed module structure on Laurent series. We discuss some solutions to this problem which involve a purely algebraic treatment of the left skew Laurent series built from a left skew derivation of a general coefficient ring, when possible.
|
https://arxiv.org/abs/2507.05022
|
Academic Papers
|
svg
|
35d2b8eec4abbd8d1c00de668c0adad91f67c4a0d6036a5c17b3ad684a6003bc
|
2026-01-13T00:00:00-05:00
|
Covering Complete Geometric Graphs by Monotone Paths
|
arXiv:2507.10840v2 Announce Type: replace Abstract: Given a set $A$ of $n$ points (vertices) in general position in the plane, the \emph{complete geometric graph} $K_n[A]$ consists of all $\binom{n}{2}$ segments (edges) between the elements of $A$. It is known that the edge set of every complete geometric graph on $n$ vertices can be partitioned into $O(n^{3/2})$ crossing-free paths (or matchings). We strengthen this result under various additional assumptions on the point set. In particular, we prove that for a set $A$ of $n$ \emph{randomly} selected points, uniformly distributed in $[0,1]^2$, with probability tending to $1$ as $n\rightarrow\infty$, the edge set of $K_n[A]$ can be covered by $O(n\log n)$ crossing-free paths and by $O(n\sqrt{\log n})$ crossing-free matchings. On the other hand, we construct $n$-element point sets such that covering the edge set of $K_n[A]$ requires a quadratic number of monotone paths.
|
https://arxiv.org/abs/2507.10840
|
Academic Papers
|
svg
|
717df43a9b14a9f308389bc7f4ef666d955a2903fd54cfe267de0ed50cea51a0
|
2026-01-13T00:00:00-05:00
|
Hierarchical Secure Aggregation with Heterogeneous Security Constraints and Arbitrary User Collusion
|
arXiv:2507.14768v2 Announce Type: replace Abstract: In hierarchical secure aggregation (HSA), a server communicates with clustered users through an intermediate layer of relays to compute the sum of users' inputs under two security requirements -- server security and relay security. Server security requires that the server learns nothing beyond the desired sum even when colluding with a subset of users, while relay security requires that each relay remains oblivious to the users' inputs under collusion. Existing work on HSA enforces homogeneous security where \tit{all} inputs must be protected against \tit{any} subset of potential colluding users with sizes up to a predefined threshold. Such a \homo formulation cannot capture scenarios with \tit{\het} \secty \reqs where \diff users may demand various levels of protection. In this paper, we study hierarchical secure aggregation (HSA) with heterogeneous security requirements and arbitrary user collusion. Specifically, we consider scenarios where the inputs of certain groups of users must remain information-theoretically secure against inference by the server or any relay, even if the server or any relay colludes with an arbitrary subset of other users. Under server security, the server learns nothing about these protected inputs beyond the prescribed aggregate sum, despite any such collusion. Under relay security, each relay similarly obtains no information about the protected inputs under the same collusion model. We characterize the optimal communication rates achievable across all layers for all parameter regimes. Furthermore, we study the minimum source keys required at the users to ensure security. For this source key requirement, we provide tight characterizations in two broad regimes determined by the security and collusion constraints, and establish a general information-theoretic lower bound together with a bounded-gap achievable scheme for the remaining regime.
|
https://arxiv.org/abs/2507.14768
|
Academic Papers
|
svg
|
0ff882ec9a029418e3d3c65ca4ab45edc4adfed063ea553f309e2f59ef52e616
|
2026-01-13T00:00:00-05:00
|
A result on spanning trees with bounded total excess
|
arXiv:2507.15139v2 Announce Type: replace Abstract: Let $G$ be a connected graph and $T$ a spanning tree of $G$. Let $\rho(G)$ denote the adjacency spectral radius of $G$. The $k$-excess of a vertex $v$ in $T$ is defined as $\max\{0,d_T(v)-k\}$. The total $k$-excess $\mbox{te}(T,k)$ is defined by $\mbox{te}(T,k)=\sum\limits_{v\in V(T)}{\max\{0,d_T(v)-k\}}$. A tree $T$ is said to be a $k$-tree if $d_T(v)\leq k$ for any $v\in V(T)$, that is to say, the maximum degree of a $k$-tree is at most $k$. In fact, $T$ is a spanning $k$-tree if and only if $\mbox{te}(T,k)=0$. This paper studies a generalization of spanning $k$-trees using a concept called total $k$-excess and proposes a lower bound for $\rho(G)$ in a connected graph $G$ to ensure that $G$ contains a spanning tree $T$ with $\mbox{te}(T,k)\leq b$, where $k$ and $b$ are two nonnegative integers with $k\geq\max\{5,b+3\}$ and $(b,k)\neq(2,5)$.
|
https://arxiv.org/abs/2507.15139
|
Academic Papers
|
svg
|
0298c46d849780b96aa0840d3d686fa89c9d197fbc13516333ab104c72b42687
|
2026-01-13T00:00:00-05:00
|
Riesz representers for the rest of us
|
arXiv:2507.19413v3 Announce Type: replace Abstract: The application of semiparametric efficient estimators, particularly those that leverage machine learning, is rapidly expanding within epidemiology and causal inference. This literature is increasingly invoking the Riesz representation theorem and Riesz regression. This paper aims to introduce the Riesz representation theorem to an epidemiologic audience, explaining what it is and why it's useful, using step-by-step worked examples.
|
https://arxiv.org/abs/2507.19413
|
Academic Papers
|
svg
|
0877e07ff9b92c664391659923d85b470fbdf6b6e663fdaf0e4d25ba9860ceed
|
2026-01-13T00:00:00-05:00
|
Time-complexity of sampling from a multimodal distribution using sequential Monte Carlo
|
arXiv:2508.02763v2 Announce Type: replace Abstract: We study a sequential Monte Carlo algorithm to sample from the Gibbs measure with a non-convex energy function at a low temperature. We use the practical and popular geometric annealing schedule, and use a Langevin diffusion at each temperature level. The Langevin diffusion only needs to run for a time that is long enough to ensure local mixing within energy valleys, which is much shorter than the time required for global mixing. Our main result shows convergence of Monte Carlo estimators with time complexity that, approximately, scales like the fourth power of the inverse temperature, and the square of the inverse allowed error. We also study this algorithm in an illustrative model scenario where more explicit estimates can be given.
|
https://arxiv.org/abs/2508.02763
|
Academic Papers
|
svg
|
527508042d4353351c7aec85468e8e1b9f7a5e1a8f70deb1a209bf38870bbf67
|
2026-01-13T00:00:00-05:00
|
Extendability of $1$-decomposable complexes
|
arXiv:2508.04555v3 Announce Type: replace Abstract: A well-known conjecture of Simon (1994) states that any pure $d$-dimensional shellable complex on $n$ vertices can be extended to $\Delta_{n-1}^{(d)}$, the $d$-skeleton of the $(n-1)$-dimensional simplex, by attaching one facet at a time while maintaining shellability. The notion of $k$-decomposability for simplicial complexes, which generalizes shellability, was introduced by Provan and Billera (1980). Coleman, Dochtermann, Geist, and Oh (2022) showed that any pure $d$-dimensional $0$-decomposable complex on $n$ vertices can similarly be extended to $\Delta_{n-1}^{(d)}$, attaching one facet at a time while preserving $0$-decomposability. In this paper, we investigate the analogous question for $1$-decomposable complexes. We prove a slightly relaxed version: any pure $d$-dimensional $1$-decomposable complex on $n$ vertices can be extended to $\Delta_{n + d - 3}^{(d)}$, attaching one facet at a time while maintaining $1$-decomposability.
|
https://arxiv.org/abs/2508.04555
|
Academic Papers
|
svg
|
b2e5319b61bcb9511238294f662e5ea9cf18c6b2ac59d7ebaf3eb9035ffac01a
|
2026-01-13T00:00:00-05:00
|
The modified diagonal cycles of Hypergeometric curves
|
arXiv:2508.06008v2 Announce Type: replace Abstract: For each $N\geq 2$, Asakura and Otsubo have recently introduced a smooth family of algebraic curves $\{X_{N,\lambda}\}_{\lambda \in \mathbb{P}^1\setminus \{0, 1, \infty\}}$ in characteristic 0 that is closely related to hypergeometric functions and the Fermat curve of degree $N$. In this paper, we study the Gross-Kudla-Schoen modified diagonal 1-cycles of these curves. We prove that if $p \ge 3$ is a prime, then for every $\lambda$ the Griffiths Abel-Jacobi image of the modified diagonal cycle of $X_{p,\lambda}$ is nontrivial for every cuspidal choice of a base point. On the other hand, we show that the modified diagonal cycle and hence the Ceresa cycle of $X_{3,\lambda}$ is torsion in the Chow group for every $\lambda$ and every choice of a base point.
|
https://arxiv.org/abs/2508.06008
|
Academic Papers
|
svg
|
98a82862bedc77348e7d263b4137255dfb536dd5bcb4f9f1b3fc90605c6171dc
|
2026-01-13T00:00:00-05:00
|
Connected components of Berkovich fixed locus: Potential good reduction
|
arXiv:2508.07156v2 Announce Type: replace Abstract: Let $\mathbbm{P}^{1,an}$ be the Berkovich projective line over a complete, algebraically closed, non-Archimedean field. Let $\phi$ be a degree $\geq 2$ rational map with potential good reduction, acting on $\mathbbm{P}^{1,an}$. In this article, we study the topology of the fixed locus of $\phi$. we show that the reduction of $\phi$ at its type~II totally ramified fixed point dictates the topological structure of the fixed locus of $\phi$. We give an easily verifiable equivalent criterion for the fixed locus of $\phi$ to be connected as well as an equivalent criterion for the fixed locus of $\phi$ to be finite. Moreover, we provide a sharp upper bound for the number of connected components of the fixed locus of a rational map with potential good reduction.
|
https://arxiv.org/abs/2508.07156
|
Academic Papers
|
svg
|
86c3a0040e9313190ddee79b4a1b66d5313ba1298b381d227cfac1c390137fa8
|
2026-01-13T00:00:00-05:00
|
A Generalized Crystalline Equivalence Principle
|
arXiv:2508.10978v3 Announce Type: replace Abstract: We prove a general version of the crystalline equivalence principle which gives an equivalence of categories between a category of TQFTs defined on a generic space with $G$-symmetry, and a category of TQFTs with internal symmetry. We give a definition and classification of anomalies associated to TQFTs in the presence of spatial symmetry, which we then generalize to a definition of an anomaly for a categorical symmetry.
|
https://arxiv.org/abs/2508.10978
|
Academic Papers
|
svg
|
c8072f045f6b8b26c23c370af95554283365b5384ed7e578d3a0402335457844
|
2026-01-13T00:00:00-05:00
|
Reciprocity for GL(2) L-functions twisted by Dirichlet characters
|
arXiv:2508.12401v2 Announce Type: replace Abstract: A formula connecting a moment of L-functions and a dual moment in a way that interchanges the roles of certain key parameters on both sides is known as a reciprocity relation. We establish a reciprocity relation for a first moment of GL(2) L-functions twisted by Dirichlet characters. This extends, via a new and simple argument, some results of Bettin, Drappeau, and Nordentoft.
|
https://arxiv.org/abs/2508.12401
|
Academic Papers
|
svg
|
4c745990fc41630b1d8d639a91232dc0e4743029e1c351feb1e8aab5d402a90b
|
2026-01-13T00:00:00-05:00
|
Stabilized automorphism groups and full groups of odometers
|
arXiv:2508.20005v2 Announce Type: replace Abstract: In this article, we show that the stabilized automorphism group of free exact odometers arising from actions of finitely generated residually finite groups coincides with the topological full group of the odometer acting on itself by right multiplication. We then prove that two free exact odometers have isomorphic stabilized automorphism groups if and only if they have isomorphic clopen subgroups of the same index. As a consequence, continuous orbit equivalence implies isomorphic stabilized automorphism groups, while for free $\mathbb{Z}^d$-odometers, isomorphic stabilized automorphism groups imply orbit equivalence. In general, neither continuous orbit equivalence nor orbit equivalence is equivalent to having isomorphic stabilized automorphism groups.
|
https://arxiv.org/abs/2508.20005
|
Academic Papers
|
svg
|
99a7bfc343f1ad9d7f98b64792edb1227d34f6f320e55b50ba0f69cde6a73736
|
2026-01-13T00:00:00-05:00
|
The linear minimal 4-chart with three crossings
|
arXiv:2509.04114v2 Announce Type: replace Abstract: Charts are oriented labeled graphs in a disk. Any simple surface braid (2-dimensional braid) can be described by using a chart. Also, a chart represents an oriented closed surface embedded in 4-space. In this paper, we investigate embedded surfaces in 4-space by using charts. Let $\Gamma$ be a chart, and we denote by $Cross(\Gamma)$ the set of all the crossings of $\Gamma$, and we denote by $\Gamma_m$ the union of all the edges of label $m$. For a 4-chart $\Gamma$, if the closure of each connected component of the set $(\Gamma_1\cup \Gamma_3)-Cross(\Gamma)$ is acyclic, then $\Gamma$ is said to be {\it linear}. In this paper, we shall show that any linear minimal $4$-chart with three crossings is lor-equivalent (Label-Orientation-Reflection equivalent) to the chart describing a $2$-twist spun trefoil knot by omitting free edges and hoops.
|
https://arxiv.org/abs/2509.04114
|
Academic Papers
|
svg
|
8b2c8e77f451862d92db72d40753209b8370a3471c5a180aa27560211e936237
|
2026-01-13T00:00:00-05:00
|
Unitary equivalence in Generalized Uncertainty Principle theories
|
arXiv:2509.05477v2 Announce Type: replace Abstract: We analyze the issue of unitary equivalence within Generalized Uncertainty Principle (GUP) theories in the one-dimensional case. For a deformed Heisenberg algebra, its representation in terms of Hilbert space and conjugate operators is not uniquely determined, raising the question of whether different realizations of the same algebra are equivalent and describe the same physics. After proposing a definition of a quantum GUP theory, we establish conditions for unitary equivalence. Using this framework, we rigorously prove that two commonly used representations are unitarily equivalent, specifying the conditions under which this equivalence holds. We demonstrate this equivalence explicitly by providing a unitary map and showing how both GUP formulations yield the same physical results in two examples: the quantum harmonic oscillator and a free-falling particle. Finally, we discuss a case in which equivalence fails, suggesting that a generalization of the Stone-von Neumann theorem may not be possible within the GUP framework under our definition of unitary equivalence.
|
https://arxiv.org/abs/2509.05477
|
Academic Papers
|
svg
|
e51caec39948373a2ef0b3a152c92dc6f4614d181eebee91d2c77681c45edd62
|
2026-01-13T00:00:00-05:00
|
Robust Confidence Intervals for a Binomial Proportion: Local Optimality and Adaptivity
|
arXiv:2509.05568v2 Announce Type: replace Abstract: This paper revisits the classical problem of interval estimation of a binomial proportion under Huber contamination. Our main result derives the rate of optimal interval length when the contamination proportion is unknown under a local minimax framework, where the performance of an interval is evaluated at each point in the parameter space. By comparing the rate with the optimal length of a confidence interval that is allowed to use the knowledge of contamination proportion, we characterize the exact adaptation cost due to the ignorance of data quality. Our construction of the confidence interval to achieve local length optimality builds on robust hypothesis testing with a new monotonization step, which guarantees valid coverage, boundary-respecting intervals, and an efficient algorithm for computing the endpoints. The general strategy of interval construction can be applied beyond the binomial setting, and leads to optimal interval estimation for Poisson data with contamination as well. We also investigate a closely related Erd\H{o}s--R\'{e}nyi model with node contamination. Though its optimal rate of parameter estimation agrees with that of the binomial setting, we show that adaptation to unknown contamination proportion is provably impossible for interval estimation in that setting.
|
https://arxiv.org/abs/2509.05568
|
Academic Papers
|
svg
|
27800406464f34d46af7347037c4f666ff833a46955363a7743e90d07dbc96b1
|
2026-01-13T00:00:00-05:00
|
Convergence analysis for the Barrett--Garcke--Nurnberg method of transport type for evolving curves
|
arXiv:2509.07834v2 Announce Type: replace Abstract: In this paper, we propose a Barrett-Garcke-Nurnberg (BGN) method for evolving geometries under general flows and present the corresponding convergence analysis. Specifically, we examine the scenario where a closed curve evolves according to a prescribed background velocity field. Unlike mean curvature flow and surface diffusion, where the evolution velocities inherently exhibit parabolicity, this case is dominated by transport which poses a significant difficulty in establishing convergence proofs. To address the challenges imposed by this transport-dominant nature, we derive several discrete energy estimates of the transport type on discretized polynomial surfaces within the framework of the projection error. The use of the projection error is indispensable as it provides crucial additional stability through its orthogonality structure. We prove that the proposed method converges sub-optimally in the L2 norm, and this is the first convergence proof for a fully discrete numerical method solving the evolution of curves driven by general flows.
|
https://arxiv.org/abs/2509.07834
|
Academic Papers
|
svg
|
0e059185625dc2369cf96aa2ab54f82e2b24f3f1ece3bac16b9ede0c7a816ac4
|
2026-01-13T00:00:00-05:00
|
Finite time blow-up and global solutions for the viscoelastic wave equation with combined power-type nonlinearities
|
arXiv:2509.08462v3 Announce Type: replace Abstract: The main objective of this manuscript is to investigate the global behavior of the solutions to the viscoelastic wave equation with a linear memory term of Boltzmann type, and a nonlinear damping modeling friction, as well as a supercritical source term which is a combined power-type nonlinearities. The global existence of the solutions is obtained provided that the energy sink dominates the energy source in an appropriate sense. In more general scenarios, we prove the global existence of the solutions if the initial history value $u_0$ is taken from a subset of a suitable potential well. Based on global existence results, the energy decay rate is derived which depends on the relaxation kernel as well as the growth rate of the damping term. In addition, we study blow-up of solutions when the source is stronger than dissipation.
|
https://arxiv.org/abs/2509.08462
|
Academic Papers
|
svg
|
107c53188dbd84e153531aa86e6d773aa8a96c090101c31fdf55b7ced4e71bd1
|
2026-01-13T00:00:00-05:00
|
Rank of the family of elliptic curves $y^2 = x^3- 5px$
|
arXiv:2509.09169v3 Announce Type: replace Abstract: This article considers the family of elliptic curves given by $E_{p}: y^2=x^3-5px$ and certain conditions on an odd prime $p$. More specifically, we have shown that if $p \equiv 7, 23 \pmod {40}$, then the rank of $E_{p}$ is zero for both $ \mathbb{Q} $ and $ \mathbb{Q}(i) $. Furthermore, if the prime $ p $ is of the form $ 40k_1 + 3 $ or $ 40k_2 + 27$, where $k_1, k_2 \in \mathbb{Z}$ such that $(5k_1+1)$ or $(5k_2 +4)$ are perfect squares, then the given family of elliptic curves has rank one over $\mathbb{Q}$ and rank two over $\mathbb{Q}(i)$. Moreover, if the prime $ p $ is of the form $ 40k_3 + 11 $ or $ 40k_4 + 19$ where $k_3 ~\text{and}~ k_4 \in \mathbb{Z}$ such that $(160k_3+49)$ or $(160k_4 + 81) $ are perfect squares, then the given family of elliptic curves has rank at least one over $\mathbb{Q}$ and rank at least two over $\mathbb{Q}(i)$.
|
https://arxiv.org/abs/2509.09169
|
Academic Papers
|
svg
|
330def60830d4a799d562a9abfc51a7e599bfab2a31b87c47c25f15369688e9d
|
2026-01-13T00:00:00-05:00
|
Long-time behavior of a nonlocal Cahn-Hilliard equation with nonlocal dynamic boundary condition and singular potentials
|
arXiv:2509.10304v2 Announce Type: replace Abstract: We investigate the long-time behavior of a nonlocal Cahn-Hilliard equation in a bounded domain $\Omega\subset\mathbb{R}^d$ $(d\in\{2,3\})$, subject to a kinetic rate-dependent nonlocal dynamic boundary condition. The kinetic rate $1/L$, with $L\in[0,+\infty)$, distinguishes different types of bulk-surface interactions. For general singular potentials, including the physically relevant logarithmic potential, we establish the existence of a global attractor $\mathcal{A}_m^L$ in a suitable complete metric space for any $L\in[0,+\infty)$. Moreover, we verify that the global attractor $\mathcal{A}_m^0$ is stable with respect to perturbations $\mathcal{A}_m^L$ for small $L>0$. When $L\in(0,+\infty)$, based on the strict separation property of global weak solutions, we further prove the existence of exponential attractors via a short-trajectory type technique, which also implies that the global attractor has finite fractal dimension. Finally, for this case, we show that every global weak solution converges to a single equilibrium in $\mathcal{L}^\infty$ as time goes to infinity, using a generalized {\L}ojasiewicz-Simon inequality and an Alikakos-Moser type iteration.
|
https://arxiv.org/abs/2509.10304
|
Academic Papers
|
svg
|
35c990fb23e81a59cf65e6b6734a98e49396d72d65913db9fc50ea37e8fbd9d2
|
2026-01-13T00:00:00-05:00
|
Decomposition of the Tschirnhausen module for coverings on decomposable $\mathbb{P}^1$-bundles
|
arXiv:2509.11227v3 Announce Type: replace Abstract: In this note, we show that for a smooth algebraic variety $Y$ and a smooth $m$-section $X$ of the $\mathbb{P}^1$-bundle \[ f : \mathbb{P}(\mathcal{O}_Y \oplus \mathcal{O}_Y(E)) \longrightarrow Y, \] where $E$ is an effective divisor on $Y$ satisfying $H^1(Y, \mathcal{O}_Y(kE)) = 0$ for all $k = 1, \ldots, m-1$, the Tschirnhausen module of the induced covering $ f|_X : X \longrightarrow Y $ is completely decomposable. We then apply it to coverings of curves arising in such a way.
|
https://arxiv.org/abs/2509.11227
|
Academic Papers
|
svg
|
3e4391cf27d7706d58a885de3babed3e5daad2f6c6b13ae77756e71830288cb0
|
2026-01-13T00:00:00-05:00
|
Spotlight inversion by orthogonal projections
|
arXiv:2509.15512v2 Announce Type: replace Abstract: Many computational problems involve solving a linear system of equations, although only a subset of the entries of the solution are needed. In inverse problems, where the goal is to estimate unknown parameters from indirect noisy observations, it is not uncommon that the forward model linking the observed variables to the unknowns depends on variables that are not of primary interest, often referred to as nuisance parameters. In this article, we consider linear problems, and propose a novel projection technique to eliminate, or at least mitigate, the contribution of the nuisance parameters in the model. We refer to this approach as spotlight inversion, as it allows to focus on only the portion of primary interest of the unknown parameter vector, leaving the uninteresting part in the shadow. The viability of the approach is illustrated with two computed examples, one where it works as model reduction for a finite element approximation of an elliptic PDE, the other amounting to local fanbeam X-ray tomography, spotlighting the region of interest that is part of the full target.
|
https://arxiv.org/abs/2509.15512
|
Academic Papers
|
svg
|
6ce61d31c67b5dbb229d2769e698e6bd335d86290645ce6718b4d7c2fb31c809
|
2026-01-13T00:00:00-05:00
|
Quantum Howe duality and Schur duality of type AIII
|
arXiv:2509.18982v2 Announce Type: replace Abstract: We establish a new connection between the iHowe duality of type AIII established by Luo-Xu and the iSchur duality established by Bao-Wang. We show that iweight $\overline{\rho}$ space in the iHowe duality is naturally isomorphic to the tensor space in the iSchur duality. Under this isomorphism, we show that the relative braid group action on this iweight space coincides with the action of the type B Hecke algebra in the iSchur duality. As a consequence, we derive from multiplicity-free decompositions that the iweight $\overline{\rho}$ spaces of irreducible modules over iquantum groups are irreducible modules over the type B Hecke algebra. Meanwhile, in the iHowe duality, we identify the relative braid group action from one side with the action of $K$-matrices and $R$-matrices from the other side.
|
https://arxiv.org/abs/2509.18982
|
Academic Papers
|
svg
|
c8e293e328ed74bec3e8d14fdf4206afdf094693426691555b2a5663672dbae3
|
2026-01-13T00:00:00-05:00
|
Higher-order Sobolev and Rellich inequalities via iterated Talenti's principle
|
arXiv:2509.19198v2 Announce Type: replace Abstract: In this paper we establish higher-order Sobolev and Rellich-type inequalities on non-compact Riemannian manifolds supporting an isoperimetric inequality. We highlight two notable settings: manifolds with non-negative Ricci curvature and having Euclidean volume growth (supporting Brendle's isoperimetric inequality) and manifolds with non-positive sectional curvature (satisfying the Cartan-Hadamard conjecture or supporting Croke's isoperimetric inequality). Our proofs rely on various symmetrization techniques, the key ingredient is an iterated Talenti's comparison principle. The non-iterated version is analogous to the main result of Chen and Li [J. Geom. Anal., 2023].
|
https://arxiv.org/abs/2509.19198
|
Academic Papers
|
svg
|
b5b967576202ec8fab99c340c38463615dc9e168552229fbba31b989cd595730
|
2026-01-13T00:00:00-05:00
|
ALNS for Tugboat Scheduling in Inland Waterway
|
arXiv:2509.19718v2 Announce Type: replace Abstract: This paper focuses on the barges shipping problem, also known as the tugboats scheduling problem, within the context of a scenario where a single tugboat has the capacity to tow multiple barges and conduct multiple trips in a drop-and-pull mode during a daily work shift. The problem is mathematically formalized as mixed-integer programming models. To tackle real-world-sized problem instances, an adaptive large neighborhood search (ALNS) algorithm integrated with a decoding mathematical model is proposed. When applied to large-scale instances, the ALNS algorithm showcases performance superiority over the strengthened mathematical model.
|
https://arxiv.org/abs/2509.19718
|
Academic Papers
|
svg
|
46431be3d783f9b29b5c6e92ddeff485629d94f9d8bf396360e3f960193fe258
|
2026-01-13T00:00:00-05:00
|
Souriau-Fisher metric and Completely integrable system on Lie groups SO(2) and SO(3)
|
arXiv:2509.20910v2 Announce Type: replace Abstract: We study the generalize Fisher metric on SO(2) and SO(3) via the thermodynamics Lie group theories of Souriau. Then we give the effect of 2-cocycle on the integrability of gradient systems due to the Fisher metric and Souriau-Fisher metric. In addition, we show how the cocycle can locally modify the Fisher metric on a coadjoint orbit, in explicit terms of brackets and central extensions on the Lie groups SO(2) and SO(3).
|
https://arxiv.org/abs/2509.20910
|
Academic Papers
|
svg
|
d9981c83e0f1c7ef39df573ba001f99998dc9c4339141f02e642776cc80b0f5e
|
2026-01-13T00:00:00-05:00
|
Optimal Control of a Navier-Stokes-Cahn-Hilliard System for Membrane-fluid Interaction
|
arXiv:2509.22069v2 Announce Type: replace Abstract: We consider an optimal control problem for a two-dimensional Navier-Stokes-Cahn-Hilliard system arising in the modeling of fluid-membrane interaction. The fluid dynamics is governed by the incompressible Navier-Stokes equations, which are nonlinearly coupled with a sixth-order Cahn-Hilliard type equation representing the deformation of a flexible membrane through a phase-field variable. Building on the previously established existence and uniqueness of global strong solutions for the coupled system, we introduce an external forcing term acting on the fluid as the control variable. Then we seek to minimize a tracking-type cost functional, demonstrating the existence of an optimal control and deriving the associated first-order necessary optimality conditions. A key issue is to establish sufficient regularity for solutions of the adjoint system, which is crucial for the rigorous derivation of optimality conditions in the fluid dynamic setting.
|
https://arxiv.org/abs/2509.22069
|
Academic Papers
|
svg
|
8cd92cfe8a90d7e10364bc11ef56c8e2ca642269c1e5543199d1753a78800b3d
|
2026-01-13T00:00:00-05:00
|
Advances in the Shannon Capacity of Graphs
|
arXiv:2509.24600v2 Announce Type: replace Abstract: We derive exact values and new bounds for the Shannon capacity of two families of graphs: the $q$-Kneser graphs and the tadpole graphs. We also construct a countably infinite family of connected graphs whose Shannon capacity is not attained by the independence number of any finite strong power. Building on recent work of Schrijver, we establish sufficient conditions under which the Shannon capacity of a polynomial in graphs, formed via disjoint unions and strong products, equals the corresponding polynomial of the individual capacities, thereby reducing the evaluation of such capacities to that of their components. Finally, we prove an inequality relating the Shannon capacities of the strong product of graphs and their disjoint union, which yields streamlined proofs of several known bounds. In addition to contributing to the computation of the Shannon capacity of graphs, this paper is intended to serve as an accessible entry point to those wishing to work in this area.
|
https://arxiv.org/abs/2509.24600
|
Academic Papers
|
svg
|
9732d2f77a6945aae0d6a4f8c9a72a9c0cb9c14694094ae09fa13a5c185e7b9d
|
2026-01-13T00:00:00-05:00
|
The Euler characteristic of $\ell$-adic local systems on $\mathcal{A}_n$
|
arXiv:2510.00656v2 Announce Type: replace Abstract: We study the Euler characteristic of $\ell$-adic local systems on the moduli stack $\mathcal{A}_n$ of principally polarized abelian varieties of dimension $n$ associated to algebraic representations of $\mathbf{GSp}_{2n}$, as virtual representations of the absolute Galois group of $\mathbb{Q}$ and the unramified Hecke algebra of $\mathbf{GSp}_{2n}$. To this end we take the last steps of the Ihara-Langlands-Kottwitz method to compute the intersection cohomology of minimal compactifications of Siegel modular varieties in level one, following work of Kottwitz and Morel, proving an unconditional reformulation of Kottwitz' conjecture in this case. This entails proving the existence of $\mathrm{GSpin}$-valued Galois representations associated to certain level one automorphic representations for $\mathbf{PGSp}_{2n}$ and $\mathbf{SO}_{4n}$. As a consequence we prove the existence of $\mathrm{GSpin}$-valued Galois representations associated to level one Siegel eigenforms, a higher genus analogue of theorems of Deligne (genus one) and Weissauer (genus two). Using Morel's work and Franke's spectral sequence we derive explicit formulas expressing the Euler characteristic of compactly supported cohomology of automorphic $\ell$-adic local systems on Siegel modular varieties in terms of intersection cohomology. Specializing to genus three and level one, we prove an explicit conjectural formula of Bergstr\"om, Faber and van der Geer for the compactly supported Euler characteristic in terms of spin Galois representations associated to level one Siegel cusp forms. Specializing to trivial local systems we give explicit formulas for the number of points of $\mathcal{A}_n$ over finite fields for all $n \leq 7$.
|
https://arxiv.org/abs/2510.00656
|
Academic Papers
|
svg
|
c6d02c1bd68af6c1dee1765e5d61d8e26c6203b23741e2d788195d8b22656bdd
|
2026-01-13T00:00:00-05:00
|
Agile Tradespace Exploration for Space Rendezvous Mission Design via Transformers
|
arXiv:2510.03544v2 Announce Type: replace Abstract: Spacecraft rendezvous enables on-orbit servicing, debris removal, and crewed docking, forming the foundation for a scalable space economy. Designing such missions requires rapid exploration of the tradespace between control cost and flight time across multiple candidate targets. However, multi-objective optimization in this setting is challenging, as the underlying constraints are often nonconvex, and mission designers must balance accuracy (e.g., solving the full problem) with efficiency (e.g., convex relaxations), slowing iteration and limiting design agility. To address these challenges, this paper proposes an AI-powered framework that enables agile and generalized rendezvous mission design. Given the orbital information of the target spacecraft, boundary conditions of the servicer, and a range of flight times, a transformer model generates a set of near-Pareto optimal trajectories across varying flight times in a single parallelized inference step, thereby enabling rapid mission trade studies. The model is further extended to accommodate variable flight times and perturbed orbital dynamics, supporting realistic multi-objective trade-offs. Validation on chance-constrained rendezvous problems in Earth orbits with passive safety constraints demonstrates that the model generalizes across both flight times and dynamics, consistently providing high-quality initial guesses that converge to superior solutions in fewer iterations. Moreover, the framework efficiently approximates the Pareto front, achieving runtimes comparable to convex relaxation by exploiting parallelized inference. Together, these results position the proposed framework as a practical surrogate for nonconvex trajectory generation and mark an important step toward AI-driven trajectory design for accelerating preliminary mission planning in real-world rendezvous applications.
|
https://arxiv.org/abs/2510.03544
|
Academic Papers
|
svg
|
f042ed95a4dc4a53832e096fa2ee74a28235b3bbed7cde245c6bc23c9e460ab2
|
2026-01-13T00:00:00-05:00
|
The $\mathbb{A}^1$-Euler characteristic of symmetric powers
|
arXiv:2510.06922v2 Announce Type: replace Abstract: The $\mathbb{A}^1$-Euler characteristic is a refinement in algebraic geometry of the classical topological Euler characteristic, which can be constructed using motivic homotopy theory. This invariant is a quadratic form rather than an integer, which carries a lot of information, but is difficult to compute in practice. In this survey, we discuss a conjectural way for computing the $\mathbb{A}^1$-Euler characteristic of the symmetric powers of a variety in terms of the $\mathbb{A}^1$-Euler characteristic of the variety itself formulated using the theory of power structures. We discuss evidence towards the conjecture so far, techniques to approach it, and applications.
|
https://arxiv.org/abs/2510.06922
|
Academic Papers
|
svg
|
fd31859667576421e72efb2757527789b8ffa48683feda5294616f2cfb29d87e
|
2026-01-13T00:00:00-05:00
|
A quantum N-dimer model
|
arXiv:2510.07543v2 Announce Type: replace Abstract: We study a quantum version of the $n$-dimer model from statistical mechanics, based on the formalism from quantum topology developed by Reshetikhin and Turaev (the latter which, in particular, can be used to construct the Jones polynomial of a knot in $\mathbb{R}^3$). We apply this machinery to construct an isotopy invariant polynomial for knotted bipartite ribbon graphs in $\mathbb{R}^3$, giving, in the planar setting, a quantum $n$-dimer partition function. As one application, we compute the expected number of loops in the (classical) double dimer model for planar bipartite graphs.
|
https://arxiv.org/abs/2510.07543
|
Academic Papers
|
svg
|
74bd5fc4a7683f84672ba4d2b9cd83423832186b76de4a8d2c3c66c80395fdaf
|
2026-01-13T00:00:00-05:00
|
Metaplectic time-frequency representations
|
arXiv:2510.09322v3 Announce Type: replace Abstract: Time-frequency representations stemmed in 1932 with the introduction of the Wigner distribution. For most of the 20th century, research in this area primarily focused on defining joint probability distributions for position and momentum in quantum mechanics. Applications to electrical engineering were soon established with the seminal works of Gabor and the researchers at Bell Labs. In 2012, Bai, Li and Cheng used for the first time metaplectic operators, defined in the middle of 20th century by Van Hove, to generalize the Wigner distribution and unify effectively the most used time-frequency representations under a common framework. This work serves as a comprehensive up-to-date survey on time-frequency representations defined by means of metaplectic operators, with particular emphasis on the recent contributions by Cordero and Rodino, who exploited metaplectic operators to their limits to generalize the Wigner distributions. Their idea provides a fruitful framework where properties of time-frequency representations can be explained naturally by the structure of the symplectic group.
|
https://arxiv.org/abs/2510.09322
|
Academic Papers
|
svg
|
04465aaff4ee60ccaa446a8382da9552c46356e1f4a2d42e270f9e25858ed233
|
2026-01-13T00:00:00-05:00
|
Elliptic Harnack inequalities for mixed local and nonlocal $p$-energy form on metric measure spaces
|
arXiv:2510.12404v2 Announce Type: replace Abstract: In the context of metric measure spaces, we introduce an axiomatic formulation of mixed local and nonlocal $p$-energy forms. Within this framework, we use the Poincar\'{e} inequality, the cutoff Sobolev inequality, and mild assumptions on the jump measure to establish the weak and strong elliptic Harnack inequalities for such mixed forms. Our approach is based on the De Giorgi--Nash--Moser method and extends the corresponding results for Dirichlet forms without the killing part, as well as for mixed energy forms on Euclidean spaces.
|
https://arxiv.org/abs/2510.12404
|
Academic Papers
|
svg
|
d69667e0020b92125fbaffa3370b3801f7e602f9008e7583b8eb0f5f16115a84
|
2026-01-13T00:00:00-05:00
|
Approximation by elements of finite spectra for C* Algebras of higher real rank
|
arXiv:2510.17271v3 Announce Type: replace Abstract: In this article, we extend a well known result about real rank zero C* Algebras to higher real rank C* Algebras. The main technique used here is similar to the method in which we approximate continuous functions using projections. What we reach at the end, is similar to the fact that the self-adjoint elements of a real rank zero C* Algebra can be approximated by elements of finite spectrum. We achieve the result for the diagonal of the self-adjoint elements of A^2, where A is a real rank one C* Algebra.
|
https://arxiv.org/abs/2510.17271
|
Academic Papers
|
svg
|
18ca9ffeeeb21015774b35c88652dd643da6a27c04241aa27c31f2572e165564
|
2026-01-13T00:00:00-05:00
|
Twisted commutativity and conjugacy ratio in groups
|
arXiv:2510.18980v2 Announce Type: replace Abstract: In this paper we introduce and study the degree of twisted commutativity and the twisted conjugacy ratio of a finitely generated group $G$. The degree of twisted commutativity $\mathrm{tdc}_X(\varphi, G)$ generalises the degree of commutativity of $G$, by measuring the density of pairs of elements with trivial twisted commutators in the ball of radius $n$ of $G$, as $n \rightarrow \infty$, where the twisting is done with respect to an endomorphism $\varphi$ of $G$. We compute $\mathrm{tdc}_X(\varphi, G)$ for several classes of groups, including virtually abelian groups, groups of subexponential growth, and free groups. We then study the twisted conjugacy ratio $\mathrm{tcr}_{X}(\varphi, G)$, which is the limit at infinity of the quotient of the twisted conjugacy and standard growth functions. We compute $\mathrm{tcr}_{X}(\varphi, G)$ for virtually abelian groups, and give examples of groups of exponential growth such that $\mathrm{tcr}_{X}(\varphi, G) = 0$.
|
https://arxiv.org/abs/2510.18980
|
Academic Papers
|
svg
|
1625aaa0e6ef8240d9bebd4c93c3f13781d1056813c2d23ea9778d0baededc50
|
2026-01-13T00:00:00-05:00
|
On $2n+4$ normals conjecture for convex polytopes in $\mathbb{R}^n$
|
arXiv:2510.22695v2 Announce Type: replace Abstract: We prove that for $n>3$ each generic simple polytope in $\mathbb{R}^n$ contains a point with at least $2n+4$ emanating normals to the boundary. This result is a piecewise-linear counterpart of a long-standing problem about normals to smooth convex bodies.
|
https://arxiv.org/abs/2510.22695
|
Academic Papers
|
svg
|
4e2e4ed6d33db8f8da4a4d4eeb617295f8ad52a0b15bc105f7f0196864f00b85
|
2026-01-13T00:00:00-05:00
|
Monomial algebras and G_a^n-equivariant embeddings into toric varieties
|
arXiv:2510.22820v2 Announce Type: replace Abstract: An induced additive action on a projective variety X in P^n is a regular action of the group G_a^n on X with an open orbit that can be extended to a regular action on P^n. Such actions are described with pairs (A, U), where A is a local algebra and U is a generating subspace lying in the maximal ideal. Such pairs are called S-pairs. We study additive actions on projective toric varieties and on Hirzebruch surfaces in particular. We prove that for any linearly normal toric variety with an additive action normalized by a torus, the corresponding S-pair consists of a monomial algebra and a subspace generated by variables. Also, we describe S-pairs for normalized and non-normalized induced additive actions on Hirzebruch surfaces.
|
https://arxiv.org/abs/2510.22820
|
Academic Papers
|
svg
|
b33373e8edb5038c73afb6f6c3a7ebf1d32cbd984c4afa7b792cc883ef31dcb5
|
2026-01-13T00:00:00-05:00
|
Injective envelopes of $C^*$-algebras as maximal rigid multiplier covers
|
arXiv:2510.24441v3 Announce Type: replace Abstract: We present a non-commutative analogue of B{\l}aszczyk's elegant two-step construction of the Gleason cover: first maximise irreducibility, then compactify. The key insight is that extremal disconnectedness (respectively, injectivity) emerges naturally from a maximality principle rather than being imposed by hand. We work with $A$-multiplier covers: non-unital $C^*$-algebras $E$ equipped with a faithful, non-degenerate $*$-homomorphism $\iota:A\to M(E)$. Covers are ordered by $A$-preserving ucp maps between multiplier algebras in the forward direction. A cover is rigid if every ucp endomorphism of $M(E)$ fixing $A$ pointwise is the identity. We prove that if $(E_{\max},\iota_{\max})$ is maximal among rigid covers, then the inclusion $A\hookrightarrow M(E_{\max})$ is rigid and essential; consequently, \[ I(A)\ \cong\ M(E_{\max}) \] canonically over $A$ by Hamana's characterisation. In the commutative case $A=C(X)$, we recover B{\l}aszczyk's picture: for a maximal regular refinement $(X,T^\ast)$ with irreducible identity, $G(X)=\beta(X,T^\ast)$ and $I(C(X))\cong C(G(X))\cong M(C_0(U))$ for any dense cozero set $U\subseteq G(X)$ (in particular, for any dense open $F_\sigma$ set, e.g., an increasing union of clopen sets). Thus the paradigm \emph{maximise first, then compactify} provides a unified conceptual framework on both sides of Gelfand duality.
|
https://arxiv.org/abs/2510.24441
|
Academic Papers
|
svg
|
504cd1f2af9388ba681fa49c791e7e2186f6dda20d39a3f96126fe4112191e77
|
2026-01-13T00:00:00-05:00
|
Modular Periodicity of Random Initialized Recurrences
|
arXiv:2510.24882v4 Announce Type: replace Abstract: Classical studies of the Fibonacci sequence focus on its periodicity modulo $m$ (the Pisano periods) with canonical initialization. We investigate instead the complete periodic structure arising from all $m^2$ possible initializations in $\mathbb{Z}/m\mathbb{Z}$. We discover perfect mirror symmetry between the Fibonacci recurrence $a_n = a_{n-1} + a_{n-2}$ and its parity transform $a_n = - a_{n-1} + a_{n-2}$ and observe fractal self-similarity in the extension from prime to prime power moduli. Additionally, we classify prime moduli based on their quadratic reciprocity and demonstrate that periodic sequences exhibit weight preservation under modular extension. Furthermore, we define a minima distribution $P(n)$ governed by Lucas ratios, which satisfies the symmetric relation $P(n)=P(1-n)$. For cyclotomic recurrences, we propose explicit counting functions for the number of distinct periods with connections to necklace enumeration. These findings imply potential connections to Viswanath's random recurrence, modular forms and L-functions.
|
https://arxiv.org/abs/2510.24882
|
Academic Papers
|
svg
|
e0b828225db132d56219f6913862e18dd79f658e9a244bbab6700e315969c28f
|
2026-01-13T00:00:00-05:00
|
The product measures of cross $t$-intersecting families
|
arXiv:2510.26642v2 Announce Type: replace Abstract: We investigate the product measures of intersection problems in extremal combinatorics. Invoking a recent result of He--Li--Wu--Zhang, we prove that for any $ n \geq t \geq 3$ and $ p_1, p_2 \in (0, \frac{1}{t+1})$, if $ \mathcal{F}_1, \mathcal{F}_2 \subseteq 2^{[n]}$ are cross $ t$-intersecting families, then $\mu_{p_1}(\mathcal{F}_1)\mu_{p_2}(\mathcal{F}_2)\le (p_1p_2)^t$. Secondly, we study the intersection problems for integer sequences by proving that if $\mathcal{H}_1, \mathcal{H}_2 \subseteq [m]^{n}$ are cross $t$-intersecting with $ m > t+1$, then $|\mathcal{H}_1|| \mathcal{H}_2|\leq (m^{n-t})^2$. These results confirm two classical conjectures of Tokushige. As an application, we strengthen a recent theorem of Frankl--Kupavskii, generalizing the well-known IU-Theorem. Finally, we show that if $ p \geq \frac{1}{2}$ and $ \mathcal{F}_1, \mathcal{F}_2 \subseteq 2^{[n]}$ are cross $t$-intersecting families, then $\min \left\{\mu_{p}(\mathcal{F}_1),\mu_{p}(\mathcal{F}_2)\right\} \leq \mu_{p}(\mathcal{K}(n,t))$, where $\mathcal{K}(n,t)$ denotes the Katona family. This recovers an old result of Ahlswede--Katona.
|
https://arxiv.org/abs/2510.26642
|
Academic Papers
|
svg
|
99659768f667b7d4e765064409682f0305c1354e06efdf0a3227d1048c908f10
|
2026-01-13T00:00:00-05:00
|
Forget BIT, It is All about TOKEN: Towards Semantic Information Theory for LLMs
|
arXiv:2511.01202v2 Announce Type: replace Abstract: Large language models (LLMs) have demonstrated remarkable capabilities in numerous real-world applications. While the vast majority of research conducted from an experimental perspective is progressing rapidly, it demands substantial computational power, data, and other resources. Therefore, how to open the black-box of LLMs from a theoretical standpoint has become a critical challenge. This paper takes the theory of rate-distortion function, directed information, and Granger causality as its starting point to investigate the information-theoretic principles behind LLMs, leading to the development of semantic information theory for LLMs, where the fundamental unit is token, rather than bits that lacks any semantic meaning. By defining the probabilistic model of LLMs, we discuss structure-agnostic information-theoretic measures, such as the directed rate-distortion function in pre-training, the directed rate-reward function in post-training, and the semantic information flow in inference phase. This paper also delves deeply into the theory of token-level semantic embedding and the information-theoretically optimal vectorization method. Thereafter, we propose a general definition of autoregression LLM, where the Transformer architecture and its performance such as ELBO, generalization error bound, memory capacity, and semantic information measures can be derived theoretically. Other architectures, such as Mamba/Mamba2 and LLaDA, are also discussed in our framework. Consequently, this paper provides a theoretical framework for understanding LLMs from the perspective of semantic information theory, which also offers the necessary theoretical tools for further in-depth research.
|
https://arxiv.org/abs/2511.01202
|
Academic Papers
|
svg
|
a75891eba5bb50126d4662670fa4f17fa33dfe5e0ff6b211693c4ef24fe4bdd4
|
2026-01-13T00:00:00-05:00
|
Lectures on local theta correspondence
|
arXiv:2511.07849v2 Announce Type: replace Abstract: This set of lecture notes on local theta correspondence is the written version of a mini-course the author gave in March of 2025 for the program ``Representation Theory and Noncommutative Geometry" at the Institut Henri Poincar\'e, Paris. The emphasis is on the Archimedean theory, which concerns representations of classical Lie groups. Section 1 is about the basic theory, including Howe's Duality Theorem, and the conservation relations. Section 2 highlights the invariant-theoretic nature of local theta correspondence via the proof of the conservation relations. Sections 3 and 4 explain how two fundamental invariants of representations behave under local theta correspondence. The final section discusses applications to unitary representation theory.
|
https://arxiv.org/abs/2511.07849
|
Academic Papers
|
svg
|
a4438fb124215566d3be924556e6e72210f27fe744c3d4d0fb508a445979ee54
|
2026-01-13T00:00:00-05:00
|
Conditional stability in determining source terms of semilinear parabolic partial differential equations
|
arXiv:2511.08460v3 Announce Type: replace Abstract: We study an inverse source problem for a semilinear parabolic equation in a bounded domain, where the nonlinearity depends on the unknown function and its gradient through a quadratic reaction term and a Burgers-type convection term. From partial boundary observation of the time derivative and its spatial gradient on an open portion of the boundary, together with an interior snapshot of the solution at a fixed time, we aim to recover an unknown spatial source factor. The analysis combines (i) a paradifferential paralinearization in Besov spaces on a short time window, which converts the nonlinear model into a linear parabolic equation with small bounded coefficients, and (ii) a Carleman estimate for the time-differentiated equation, yielding conditional Holder stability. The approach extends the cut-off free Carleman method for linear inverse source problems to a nonlinear setting while keeping the observation geometry unchanged.
|
https://arxiv.org/abs/2511.08460
|
Academic Papers
|
svg
|
1cee9108ccf87466cc5ac152be7ee316741d4b0c71bd3fcec754ad9599ebbc01
|
2026-01-13T00:00:00-05:00
|
Instantaneous Type I blow-up and non-uniqueness of smooth solutions of the Navier-Stokes equations
|
arXiv:2511.09556v2 Announce Type: replace Abstract: For any smooth, divergence-free initial data, we construct a solution of the Navier--Stokes equations that exhibits Type~I blow-up of the $L^\infty$ norm at time $T_*>0$, while remaining smooth in space and time on $\mathbb T^d\times([0,T]\setminus\{T_*\})$. An instantaneous injection of energy from infinite wavenumber initiates a bifurcation from the classical solution, producing an infinite family of spatially smooth solutions with the same data and thereby violating uniqueness of the Cauchy problem. A key ingredient is the first known construction of a complete inverse energy cascade realized by a classical Navier--Stokes flow, which transfers energy from infinitely high to low frequencies. The result holds in all dimensions $d\geq2$.
|
https://arxiv.org/abs/2511.09556
|
Academic Papers
|
svg
|
4c2095c86df09b2978a4709f5fb8cff9092727e98050bf62bcbd45a5817bd85f
|
2026-01-13T00:00:00-05:00
|
Surface homeomorphisms with big rotation set
|
arXiv:2511.15220v2 Announce Type: replace Abstract: This article consists in applications of [arXiv:2511.14232] in the case of homemomorphisms of higher genus surfaces whose homological rotation set is big enough -- a class of dynamics that is open. We first prove a structure theorem for the rotation set of such homeomorphisms: it is a finite union of convex sets, we get an optimal bound for the number of such pieces. This bound can be improved in the case of transitive (in this case the rotation set is convex) and non-wandering dynamics (and for such homeomorphisms we get the existence of a family of invariant essential open sets). We also get boundedness of deviations for homeomorphisms with big rotation set and some consequences of it, including a answer to Boyland's conjecture in our framework.
|
https://arxiv.org/abs/2511.15220
|
Academic Papers
|
svg
|
70fc1ca8e210aacf0ad494df388f447924a254984529cdca8d353766f6ddcd13
|
2026-01-13T00:00:00-05:00
|
A Stochastic Approach to the Definition of the Path Integral Measure
|
arXiv:2511.15772v3 Announce Type: replace Abstract: We to define a Path Integral in Lorentzian time by restricting the relevant domain of integration on $C([0,1],M)$ over a Riemannian configuration manifold $(M,g)$ and considering the dynamics of a particle evolving between to fixed endpoints with a referential non-degenerate classical trajectory, formulating a framework around a quadratic Lagrangian. Through fibration, we reduce the infinite-dimensional space under consideration to an $L^2$-isometric flux spaces in which we consider a stochastic process associated to a Gaussian measure. The Path Integral is subsequently defined as an expectation value with respect to the Gaussian measure, allowing us to rigorously formulate the former as a functional integral. We prove mathematical correspondence between the Stochastic Path Integral and the Euclidean Path Integral theory formulated rigorously under the Feynman-Kac theorem.
|
https://arxiv.org/abs/2511.15772
|
Academic Papers
|
svg
|
e7e420d6910f242eb14c13715c6f0c04966248b76bb2b0c88e9d663159374cc1
|
2026-01-13T00:00:00-05:00
|
Optimal local central limit theorems on Wiener chaos
|
arXiv:2511.21496v2 Announce Type: replace Abstract: This paper investigates a local central limit theorem for a normalized sequence of random variables belonging to a fixed order Wiener chaos and converging to the standard normal distribution. We prove, without imposing any additional conditions, that the optimal rate of convergence of their density functions to the standard normal density in the Sobolev space $W^{k,r}(\mathbb{R})$, for every $k \in \mathbb{N} \cup \{0\}$ and $r \in [1,\infty]$, is determined by the maximum of the absolute values of their third and fourth cumulants. We also obtain exact asymptotics for this convergence under an additional assumption. Our approach is based on Malliavin--Stein techniques combined with tools from the theory of generalized functionals in Malliavin calculus.
|
https://arxiv.org/abs/2511.21496
|
Academic Papers
|
svg
|
61ec9d7c45eaab0c56b0df925ef666932084b7e5be9108582a0c5604d81609b0
|
2026-01-13T00:00:00-05:00
|
Logistic elliptic and parabolic problem for the fractional $p$-Laplacian
|
arXiv:2511.23272v2 Announce Type: replace Abstract: In this paper we prove existence, uniqueness of weak solutions of the following nonlocal nonlinear logistic equation \begin{equation*} \begin{cases} (-\Delta)_p^s u_\lambda=\lambda u_\lambda^q - b(x)u_\lambda^r \quad \text{in} \;\Omega,\\ u_\lambda=0 \quad \text{in} \; ( \mathbb{R}^d \backslash \Omega), \\ u_\lambda>0 \text{ in} \; \Omega. \end{cases}\ \end{equation*} We also prove behavior of $u_\lambda$ with respect to $\lambda,$ underlining the effect of the nonlocal operator. We then study the associated parabolic problem, proving local and global existence, uniqueness and global behavior such as stabilization, finite time extinction and blow up.
|
https://arxiv.org/abs/2511.23272
|
Academic Papers
|
svg
|
e180319abba3fce969749347e278cfe66adec0e2d9d8e925046c267a28ce939d
|
2026-01-13T00:00:00-05:00
|
Nonsmooth bifurcations in families of one-dimensional piecewise-linear quasiperiodically forced maps
|
arXiv:2512.04234v2 Announce Type: replace Abstract: We study nonsmooth bifurcations of four types of families of one-dimensional quasiperiodically forced maps of the form $F_i(x,\theta) = (f_i(x,\theta), \theta+\omega)$ for $i=1,\dots,4$, where $x$ is real, $\theta\in\mathbb{T}$ is an angle, $\omega$ is an irrational frequency, and $f_i(x,\theta)$ is a real piecewise linear map with respect to $x$. The first two types of families $f_i$ have a symmetry with respect to $x$, and the other two could be viewed as quasiperiodically forced piecewise-linear versions of saddle-node and period-doubling bifurcations. The four types of families depend on two real parameters, $a\in\mathbb{R}$ and $b\in\mathbb{R}$. Under certain assumptions for $a$, we prove the existence of a continuous map $b^*(a)$ where for $b=b^*(a)$ there exists a nonsmooth bifurcation for these types of systems. In particular we prove that for $b=b^*(a)$ we have a strange nonchaotic attractor. It is worth to mention that the four families are piecewise-linear versions of smooth families which seem to have nonsmooth bifurcations. Moreover, as far as we know, we give the first example of a family with a nonsmooth period-doubling bifurcation.
|
https://arxiv.org/abs/2512.04234
|
Academic Papers
|
svg
|
5ac1a9f7b18dc9df0991716bcbd83c2adaa45b7f92e27c9df5ed8252b0b6e070
|
2026-01-13T00:00:00-05:00
|
A poset representation for stable contracts in a two-sided market generated by integer choice functions
|
arXiv:2512.05942v2 Announce Type: replace Abstract: Generalizing a variety of earlier problems on stable contracts in two-sided markets, Alkan and Gale introduced in 2003 a general stability model on a bipartite graph $G=(V,E)$ in which the vertices are interpreted as ``agents'', and the edges as possible ``contract'' between pairs of ``agents''. The edges are endowed with nonnegative capacities $b$ giving upper bounds on ``contract intensities'', and the preferencies of each ``agent'' $v\in V$ depend on a \emph{choice function} (CFs) that acts on the set of ``contracts'' involving $v$, obeying three well motivated axioms of \it{consistence}, \it{substitutability} and \it{cardinal monotonicity}. In their model, the capacities and choice functions can take reals or discrete values and, extending well-known earlier results on particular cases, they proved that systems of \it{stable} contracts always exist and, moreover, their set $\cal S$ constitutes a distributive lattice under a natural comparison relation $\prec$. In this paper, we study Alkan--Gale's model when all capacities and choice functions take integer values. We characterize the set of rotations -- augmenting cycles linking neighboring stable assignments in the lattice $(\cal S,\prec)$, explain how to construct the rotations efficiently, and devise a weighted poset in which the lattice of closed functions is isomorphic to $(\cal S,\prec)$, thus obtaining an explicit representation for the latter. We show that in general the size of the poset is at most $b^{\rm max}|E|$, where $b^{\rm max}$ is the maximal capacity, and the poset can be constructed in pseudo polynomial time. Then we explain that by imposing an additional condition on CFs, the size of the poset becomes polynomial in $|V|$, and the total time reduces to a polynomial in $|V|,\log b^{\rm max}$.
|
https://arxiv.org/abs/2512.05942
|
Academic Papers
|
svg
|
52c0138266121a9cdbf4f203a082cd4a27278bba9e2e63d8229cd4f014383325
|
2026-01-13T00:00:00-05:00
|
Spoke topological Hochschild homology
|
arXiv:2512.11338v2 Announce Type: replace Abstract: Fix primes $p$ and $\ell$, and let $C_p$ be the cyclic group of order $p$. We compute the $C_p$-equivariant spoke topological Hochschild homology of $\underline{\mathbb{F}}_{\ell}$ and prove it exhibits a form of B\"okstedt periodicity. Here spoke topological Hochschild homology is a variant of topological Hochschild homology where one replaces the circle in the construction with the unreduced suspension of $C_p$. As an application, we use this result to give a new proof of the Segal conjecture for the cyclic group of order an odd prime $p$.
|
https://arxiv.org/abs/2512.11338
|
Academic Papers
|
svg
|
44e33259219807e9bc6aeaeb64c7a01ce6d91162fe553fa0f4e27f6e58f12d62
|
2026-01-13T00:00:00-05:00
|
Decomposition theorems for unmatchable pairs in groups and field extensions
|
arXiv:2512.12942v2 Announce Type: replace Abstract: A theory of matchings for finite subsets of an abelian group, introduced in connection with a conjecture of Wakeford on canonical forms for symmetric tensors, has since been extended to the setting of field extensions and to that of matroids. Earlier approaches have produced numerous criteria for matchability and unmatchability, but have offered little structural insight. In this paper, we develop parallel structure theorems which characterize unmatchable pairs in both abelian groups and field extensions. Our framework reveals analogous obstructions to matchability: nearly periodic decompositions of sets in the group setting correspond to decompositions of subspaces involving translates of a subfield in the linear setting. This perspective not only recovers previously known results through short proofs, but also leads to new matching criteria and guarantees the existence of nontrivial unmatchable pairs.
|
https://arxiv.org/abs/2512.12942
|
Academic Papers
|
svg
|
122ff14fa196f21907642d0beac4c8dc1c35dca0d1e4ed115c1d3756c836a93d
|
2026-01-13T00:00:00-05:00
|
Ensemble Parameter Estimation for the Lumped Parameter Linear Superposition (LPLSP) Framework: A Rapid Approach to Reduced-Order Modeling for Transient Thermal Systems
|
arXiv:2512.14467v2 Announce Type: replace Abstract: This work introduces an ensemble parameter estimation framework that enables the Lumped Parameter Linear Superposition (LPLSP) method to generate reduced order thermal models from a single transient dataset. Unlike earlier implementations that relied on multiple parametric simulations to excite each heat source independently, the proposed approach simultaneously identifies all model coefficients using fully transient excitations. Two estimation strategies namely rank-reduction and two-stage decomposition are developed to further reduce computational cost and improve scalability for larger systems. The proposed strategies yield ROMs with mean temperature-prediction errors within 5% of CFD simulations while reducing model-development times to O(10^0 s)-O(10^1 s). Once constructed, the ROM evaluates new transient operating conditions in O(10^0 s), enabling rapid thermal analysis and enabling automated generation of digital twins for both simulated and physical systems.
|
https://arxiv.org/abs/2512.14467
|
Academic Papers
|
svg
|
7cc919124fca9db0fc0c91857074c104fae0437f6353e69d6c5631d20b296d55
|
2026-01-13T00:00:00-05:00
|
Using the Jones Polynomial to Prove Infinite Families of Knots Satisfy the Cosmetic Surgery Conjecture
|
arXiv:2512.20511v2 Announce Type: replace Abstract: This paper computes the Jones polynomial and the invariants obstructing cosmetic surgery which are derived from it for two infinite families of knots, proving they satisfy the Purely Cosmetic Surgery Conjecture. Both the method of computation and the method for generating families of knots extend.
|
https://arxiv.org/abs/2512.20511
|
Academic Papers
|
svg
|
0a18b93f501ca45c9da46c34071ae1f3e070e56bd5582979ae6fe42dd32e9e6b
|
2026-01-13T00:00:00-05:00
|
Relation between generalized and ordinary cluster algebras
|
arXiv:2512.21062v2 Announce Type: replace Abstract: Recently, Ramos and Whiting showed that any generalized cluster algebra of geometric type is isomorphic to a quotient of a subalgebra of a certain cluster algebra. Based on their idea and method, we show that the same property holds for any generalized cluster algebra with $y$-variables in an arbitrary semifield. We also present the relations between the $C$-matrices, the $G$-matrices, and the $F$-polynomials of a generalized cluster pattern and those of the corresponding composite cluster pattern.
|
https://arxiv.org/abs/2512.21062
|
Academic Papers
|
svg
|
f2872660dd84938eb729ed35a2d33874ec0f28b6be102acff4352b59bb52aee9
|
2026-01-13T00:00:00-05:00
|
On Rayleigh scattering in the massless Nelson model
|
arXiv:2512.21307v2 Announce Type: replace Abstract: Asymptotic completeness of Rayleigh scattering in models of atoms and molecules of non-relativistic QED is expected, but for a proof we still lack sufficient control on the number of emitted soft photons. So far, this obstacle has only been overcome for the spin-boson model. In a general class of models asymptotic completeness holds provided the expectation value of the photon number $N$ remains bounded uniformly in time. This has been established by Faupin and Sigal. We review and simplify their work, and, more importantly, we replace the bound on $N$ by a weaker assumption on the distribution of $N$ that is both necessary and sufficient for asymptotic completeness.
|
https://arxiv.org/abs/2512.21307
|
Academic Papers
|
svg
|
b82087d526fea92f8873c1d88f1cdffe0fcf6b63a76aad106ee076f5b31a1433
|
2026-01-13T00:00:00-05:00
|
Data-Free Asymptotics-Informed Operator Networks for Singularly Perturbed PDEs
|
arXiv:2512.22006v2 Announce Type: replace Abstract: Recent advances in machine learning (ML) have opened new possibilities for solving partial differential equations (PDEs), yet robust performance in challenging regimes remains limited. In particular, singularly perturbed differential equations exhibit sharp boundary or interior layers with rapid transitions, where standard ML surrogates often fail without extensive resolution. Generating training data for such problems is also costly, as accurate reference solutions typically require massive adaptive mesh refinement. In this work, we propose eFEONet, an enriched Finite Element Operator Network tailored to singularly perturbed problems. Guided by classical singular perturbation theory, eFEONet augments the operator-learning framework with specialized enrichment basis functions that encode the asymptotic structure of layer solutions. This design enables accurate approximation of sharp transitions without relying on large datasets, and can operate with minimal supervision-or even in a data-free manner under appropriate settings. We further provide a rigorous convergence analysis of the proposed method and demonstrate its effectiveness through extensive experiments on representative problems featuring both boundary and interior layers.
|
https://arxiv.org/abs/2512.22006
|
Academic Papers
|
svg
|
a485ed917d3d1989390a0abee97e57e2564d5838a7e4bea213eee61c5f97d239
|
2026-01-13T00:00:00-05:00
|
Generalized binomial edge ideals are Cartwright-Sturmfels
|
arXiv:2512.22012v2 Announce Type: replace Abstract: Binomial edge ideals associated to a simple graph G were introduced by Herzog and collaborators and, independently, by Ohtani. They became an ``instant classic" in combinatorial commutative algebra with more than 100 papers devoted to their investigation over the past 15 years. They exhibit many striking properties, including being radical and, moreover, Cartwright-Sturmfels. Using the fact that binomial edge ideals can be seen as ideals of 2-minors of a matrix of variables with two rows, generalized binomial edge ideals of 2-minors of matrices of m rows were introduced by Rauh and proved to be radical. The goal of this paper is to prove that generalized binomial edge ideals are Cartwright-Sturmfels. On the way we provide results on ideal constructions preserving the Cartwright-Sturmfels property. We also give examples and counterexamples to the Cartwright-Sturmfels property for higher minors.
|
https://arxiv.org/abs/2512.22012
|
Academic Papers
|
svg
|
681c9052ed5e1d12966b7ea2a5d18c32bd5eada10abcecd867f7687b1290a2d1
|
2026-01-13T00:00:00-05:00
|
Infinitesimal moments in free and c-free probability and Motzkin paths
|
arXiv:2512.22700v3 Announce Type: replace Abstract: Infinitesimal moments associated with infinitesimal freeness and infinitesimal conditional freeness are studied. For free random variables, we consider continuous deformations of moment functionals associated with Motzkin paths $w$, which provide a decomposition of their moments, and we compute their derivatives at zero. We show that the first-order derivative of each functional vanishes unless the path has exactly one local maximum. Geometrically, this means that $w$ is a pyramid path, which is consistent with the characteristic formula for alternating moments of infinitesimally free centered random variables. In this framework, infinitesimal Boolean independence is also obtained and it corresponds to flat paths. A similar approach is developed for infinitesimal conditional freeness, for which we show that the only moment functionals that have a non-zero first-order derivative are associated with concatenations of a pyramid path and a flat path. This charaterization leads to a Leibniz-type definition of infinitesimal conditional freeness at the level of moments.
|
https://arxiv.org/abs/2512.22700
|
Academic Papers
|
svg
|
6a55294ecc21e795f64be92a2408d56bf5879ddbdf1141409dbccc3ce98dc6ba
|
2026-01-13T00:00:00-05:00
|
Small-time global controllability of a class of bilinear fourth-order parabolic equations
|
arXiv:2512.23339v2 Announce Type: replace Abstract: In this work, we investigate the small-time global controllability properties of a class of fourth-order nonlinear parabolic equations driven by a bilinear control posed on the one-dimensional torus. The controls depend only on time and act through a prescribed family of spatial profiles. Our first result establishes the small-time global approximate controllability of the system using three scalar controls, between states that share the same sign. This property is obtained by adapting the geometric control approach to the fourth-order setting, using a finite family of frequency-localized controls. We then study the small-time global exact controllability to non-zero constant states for the concerned system. This second result is achieved by analyzing the null controllability of an appropriate linearized fourth-order system and by deducing the controllability of the nonlinear bilinear model through a fixed-point argument together with the small-time global approximate control property.
|
https://arxiv.org/abs/2512.23339
|
Academic Papers
|
svg
|
9d816fdc3b1729077f1e3575d2aa9e273e2896010b901d5612255ec3125be7a6
|
2026-01-13T00:00:00-05:00
|
Completeness and reflexivity type properties of $B_1(X)$
|
arXiv:2601.00733v2 Announce Type: replace Abstract: For a Tychonoff space $X$, $B_1(X)$ denotes the space of all Baire-one functions on $X$ endowed with the pointwise topology. We prove that the following assertions are equivalent: (1) $B_1(X)$ is a (semi-)Montel space, (2) $B_1(X)$ is a (semi-)reflexive space, (3) $B_1(X)$ is a (quasi-)complete space, (4) $B_1(X)=\mathbb{R}^X$, (5) $X$ is a $Q_f$-space. It is proved that $B_1(X)$ is sequentially complete iff $B_1(X)$ is locally complete iff $X$ is a $CZ$-space. In the case when $K$ is a compact space, we show that $B_1(K)$ is locally complete iff $K$ is scattered. We thoroughly study the case when $X$ is a separable metrizable space. Numerous distinguished examples are given.
|
https://arxiv.org/abs/2601.00733
|
Academic Papers
|
svg
|
8970800dfc47168fc0e7aed38a01d63a6b2d3034e3652856321c1f5d542e1926
|
2026-01-13T00:00:00-05:00
|
Characterizations of harmonic quasiregular mappings in function spaces
|
arXiv:2601.01017v2 Announce Type: replace Abstract: We study conjugate-type phenomena for complex-valued harmonic quasiregular mappings in the unit disk across three function space families: $Q(n,p,\alpha)$, $F(p,q,s)$, and the non-derivative $M(p,q,s)$. For a harmonic $K$-quasiregular mapping $f=u+iv$, we first show that if the real part $u$ belongs to $Q_h(1,p,\alpha)$ (with $\alpha>-1$ and $\alpha+1<\alpha+2$), the imaginary part $v$ lies in the same space with a $K$-dependent quantitative bound. An analogous stability result is established for the harmonic $F$-scale, with sharp $K$-dependence. These results are extended to harmonic $(K, K')$-quasiregular mappings, yielding explicit estimates with an additional inhomogeneous term involving $K'$. Finally, for normalized harmonic quasiconformal mappings, %$f\in\mathcal S_H(K)$, we derive membership criteria in the harmonic $M$- and $F$-scales, and obtain corresponding conclusions for their natural derivatives, with parameter ranges governed by the order $\alpha_K$ of the family of harmonic quasiconformal mappings.
|
https://arxiv.org/abs/2601.01017
|
Academic Papers
|
svg
|
4b0b7fa5026f9a4a29c999375a59373c259b02971ad37bcaa23cdf8bcbd885bf
|
2026-01-13T00:00:00-05:00
|
Notes on Poisson deformations of symplectic varieties
|
arXiv:2601.01131v2 Announce Type: replace Abstract: In this article we give corrections and addendum to the article ``Flops and Poisson deformations of symplectic varieties, Publ. Res. Inst. Math. Sci. {\bf 44} (2008) 259 - 314''.
|
https://arxiv.org/abs/2601.01131
|
Academic Papers
|
svg
|
3e7f1c196c50220f46b304fe5ab44cfdab3ad57cbf3afcd5a90cb2744eda89c9
|
2026-01-13T00:00:00-05:00
|
Stochastic Control Methods for Optimization
|
arXiv:2601.01248v2 Announce Type: replace Abstract: In this work, we investigate a stochastic control framework for global optimization over both Euclidean spaces and the Wasserstein space of probability measures, where the objective function may be non-convex and/or non-differentiable. In the Euclidean setting, the original minimization problem is approximated by a family of regularized stochastic control problems; using dynamic programming, we analyze the associated Hamilton--Jacobi--Bellman equations and obtain tractable representations via the Cole--Hopf transformation and the Feynman--Kac formula. For optimization over probability measures, we formulate a regularized mean-field control problem characterized by a master equation, and further approximate it by controlled $N$-particle systems. We establish that, as the regularization parameter tends to zero (and as the particle number tends to infinity for the optimization over probability measures), the value of the control problem converges to the global minimum of the original objective. Building on the resulting probabilistic representations, Monte Carlo-based numerical schemes are proposed and numerical experiments are reported to illustrate the effectiveness of the methods and to support the theoretical convergence rates.
|
https://arxiv.org/abs/2601.01248
|
Academic Papers
|
svg
|
7e4a610073f9813394f83b25ada7375a9575ab42fd30ea302b6f957f68c9b430
|
2026-01-13T00:00:00-05:00
|
On Hahn-Banach smoothness of $L_1$-preduals and related $w^*-w$ point of continuity of unit balls of dual spaces
|
arXiv:2601.01567v2 Announce Type: replace Abstract: In this article, we intend to study possible $(U)$-embeddings of a Banach space $X$ into $X^{**}$. The canonical embedding of $X$ in $X^{**}$ which possesses $(U)$-embedding is of particular interest and such spaces are known as Hahn-Banach smooth spaces. Separable $L_1$-preduals are characterized which are Hahn-Banach smooth. It is derived that, when $S$ is a compact convex set where each point in $ext(S)$ is a limit point of $ext(S)$ then no subspace of $A(S)$ retains property-$(wU)$ in $A(S)^{**}$. Moreover, if $X$ is an $L_1$-predual where $I:(B_{X^*},w^*)\rightarrow (B_{X^*},w)$ is continuous on $ext (B_{X^*})$ then $X$ is Hahn-Banach smooth, is observed. This means that not all finitely supported elements in $B_{\ell_1}$ can be points of continuity of $I:(B_{\ell_1},w^*(c))\rightarrow (B_{\ell_1},w)$, which is incorrectly stated in \cite{DMR}. Throughout this article this fact is established in a few ways. It is shown that if $L_1(\mu)$ possesses a predual that is weakly Hahn-Banach smooth, then $\mu$ must have a specific characteristic.
|
https://arxiv.org/abs/2601.01567
|
Academic Papers
|
svg
|
e4681d30d4f4f4d907972a8233a575516bc20bcd4c559acc3f62385a7a59a4eb
|
2026-01-13T00:00:00-05:00
|
On a Smoothed Dirichlet Divisor Problem
|
arXiv:2601.01905v2 Announce Type: replace Abstract: Hardy showed that $\sum_{n \ioe x}\tau(n)-x(\log x +2\gamma -1)$ is not $o(x^{1/4})$. In this article, we prove that $\sum_{n \ioe x}\tau(n)(1-\frac{x}{n})-xP(\log x)=\frac{1}{4}+O \left( \frac{\log x}{x^{1/4}} \right)$, where $P$ is a polynomial of degree 2. As a corollary, this estimate enables us to settle a conjecture surmised by Berkane, Bordell\`{e}s, and Ramar\'{e} dealing with the positivity of an integral of the error term in the Dirichlet divisor problem. All results are entirely explicit and allow us to study the proximity between the remainder of the Dirichlet divisor problem and its logarithmic version.
|
https://arxiv.org/abs/2601.01905
|
Academic Papers
|
svg
|
53c77d4148ef3fb8925daf022f54bf3865a667b8787fd96949a74f9c96e81f28
|
2026-01-13T00:00:00-05:00
|
On the sizes of the maximal prime powers divisors of factorials
|
arXiv:2601.03414v2 Announce Type: replace Abstract: Let p be any prime, and $p^(\nu_p(n!))$ the maximal power of $p$ dividing $n!$. It is proved that there exists a positive integer $n_0$, which depends only on $p$, such that $q^(\nu_q(n!)) p$. For twin primes $p$ and $q = p + 2$ it is proved that the minimal $n_0$ satisfying $q^(\nu_q(n!)) < p^(\nu_p(n!))$ for all $n \ge n_0$ is given by $n_0 = (p^2+p)/2$.
|
https://arxiv.org/abs/2601.03414
|
Academic Papers
|
svg
|
66f166cf7ae36827dee25ddf04edc40a74ced5324ebf6c39f4db7eca8ec51acd
|
2026-01-13T00:00:00-05:00
|
Mean-field limits for interacting particle systems on general adaptive dynamical networks
|
arXiv:2601.03742v2 Announce Type: replace Abstract: We study the large-population limit of interacting particle systems evolving on adaptive dynamical networks, motivated in particular by models of opinion dynamics. In such systems, agents interact through weighted graphs whose structure evolves over time in a coupled manner with the agents' states, leading to non-exchangeable dynamics. In the dense-graph regime, we show that the asymptotic behavior is described by a Vlasov-type equation posed on an extended phase space that includes both the agents' states and identities and the evolving interaction weights. We establish this limiting equation through two complementary approaches. The first follows the mean-field methodology in the spirit of Sznitman [28]. In this framework, we impose the additional assumption that the weight dynamics is independent of one of the agent's states, an assumption that remains well motivated from a modeling perspective and allows for a direct derivation of the mean-field limit. The second approach is based on the graph limit framework and is formulated in a deterministic setting. This perspective makes it possible to remove the aforementioned restriction on the weight dynamics and to handle more general interaction structures. Our analysis includes wellposedness and stability results for the limiting Vlasov-type equation, as well as quantitative estimates ensuring the propagation of independence. We further clarify the relationship between the continuum (graph limit) formulation and the mean-field limit, thereby providing a unified description of the asymptotic dynamics of interacting particle systems on adaptive dynamical networks.
|
https://arxiv.org/abs/2601.03742
|
Academic Papers
|
svg
|
2b6247397b77f7adc82fec8c2448fcf8dc661c8fa7d2664e4398bcb0fa5c537c
|
2026-01-13T00:00:00-05:00
|
Notes on Ecalle's and Brown's solutions to the double shuffle relations modulo products
|
arXiv:2601.03828v2 Announce Type: replace Abstract: We investigate relationships between polar/polynomial solutions to the double shuffle relations modulo products, which were independently introduced by Brown and Ecalle.
|
https://arxiv.org/abs/2601.03828
|
Academic Papers
|
svg
|
910ce1d0713d7e7b3cfa1fd621ca8cd67382dd942b3aa557a1b3e6bc2cf44ff7
|
2026-01-13T00:00:00-05:00
|
The Feldman-H\'ajek Dichotomy for Countable Gaussian Mixtures and their Asymptotic Separability in High Dimensions
|
arXiv:2601.03911v2 Announce Type: replace Abstract: This paper establishes the theoretical foundations for the asymptotic separability of Gaussian Mixture Models (GMMs) in high dimensions by extending the classical Feldman-H\'ajek theorem. We first prove that a countable mixture of Gaussian measures is a well-defined probability measure. Our primary result, the Gaussian Mixture Dichotomy Theorem, demonstrates that the mutual singularity of individual Gaussian components is a sufficient condition for the mutual singularity of the resulting mixtures. We provide a rigorous proof and further discuss the ``Mixed Case,'' where the presence of even a single equivalent pair of components leads to partial absolute continuity via the Lebesgue decomposition, thereby defining the theoretical limits of perfect classification in infinite-dimensional spaces.
|
https://arxiv.org/abs/2601.03911
|
Academic Papers
|
svg
|
712ba2c96da427cb5b72d6a6b35511fe57ed762845a974feeaa77b6792370cc8
|
2026-01-13T00:00:00-05:00
|
Non-Existence of Linear-Quartic Factorization for the Second Cuboid Quintic
|
arXiv:2601.04241v2 Announce Type: replace Abstract: Let $Q_{p,q}(t)\in\mathbb{Z}[t]$ be Sharipov's even monic degree-$10$ second cuboid polynomial depending on coprime integers $p\neq q>0$. Writing $Q_{p,q}(t)$ as a quintic in $t^{2}$ produces an associated monic quintic polynomial. After the weighted normalization $r=p/q$ and $s=r^{2}$ we obtain a one-parameter family $P_s(x)\in\mathbb{Q}[x]$ such that \[ Q_{p,q}(t)=q^{20}\,P_s\!\left(\frac{t^{2}}{q^{4}}\right)\qquad\text{with}\qquad s=\left(\frac{p}{q}\right)^{2}. \] We show that for every rational $s>0$ with $s\neq 1$ the equation $P_s(x)=0$ has no rational solutions. Equivalently, $P_s$ admits no $1+4$ factorization over $\mathbb{Q}$. The proof uses an explicit quotient by the inversion involution $(s,y)\mapsto(1/s,1/y)$ and reduces the rational-root problem for $P_s$ to rational points on the fixed genus-$2$ hyperelliptic curve \[ C:\quad w^2=t^5+21t^4+26t^3+10t^2+5t+1=(t+1)(t^4+20t^3+6t^2+4t+1). \] Using Magma and Chabauty's method on the Jacobian of $C$, we compute $C(\mathbb{Q})$ exactly and conclude that the only parameter value producing a rational root is the excluded case $s=1$ (equivalently $p=q$). As a consequence, for coprime $p\neq q>0$ the polynomial $Q_{p,q}(t)$ has no rational roots (hence no linear factor over $\mathbb{Q}$, and in particular no linear factor over $\mathbb{Z}$).
|
https://arxiv.org/abs/2601.04241
|
Academic Papers
|
svg
|
8c95866b326d0db453ed8a64dfaa3362fab394cc93921faa50ff5906142f84d5
|
2026-01-13T00:00:00-05:00
|
Sobre los teoremas de Shafarevich y Siegel
|
arXiv:2601.04284v2 Announce Type: replace Abstract: Presentaremos una nueva demostraci\'on del teorema de Shafarevich sobre finitud de curvas el\'ipticas con buena reducci\'on fuera de un conjunto finito de primos dado. Esto da un nuevo punto de entrada a teoremas fundamentales de finitud diofantina tales como el teorema de Siegel sobre la ecuaci\'on $S$-unidad. Nuestro argumento est\'a libre de aproximaci\'on diofantina o teor\'ia de trascendencia, y se acerca m\'as a las ideas de Faltings en su demostraci\'on de la conjetura de Mordell. -- We present a new proof of Shafarevich's theorem on finiteness of elliptic curves with good reduction outside a given finite set of primes. This gives a new entry point to fundamental diophantine finiteness theorems such as Siegel's theorem on the $S$-unit equation. Our proof is free from diophantine approximation or transcendence theory, and it is closer to the ideas of Faltings in his proof of Mordell's conjecture .
|
https://arxiv.org/abs/2601.04284
|
Academic Papers
|
svg
|
482ddadf0114032269930d778119539239ec085b2267007cd6426a10411d51be
|
2026-01-13T00:00:00-05:00
|
Infinitesimal Variations of Hodge Structure for Singular Curves I
|
arXiv:2601.04410v2 Announce Type: replace Abstract: We study the infinitesimal variation of Hodge structure associated with families of reduced algebraic curves with singularities. The analysis applies to curves beyond the nodal case and is not restricted to plane curves, encompassing curves lying on smooth projective surfaces as well as families with more general isolated singularities. Using deformation-theoretic and residue-theoretic methods, we describe how the infinitesimal period map decomposes into local contributions supported at singular points, together with global constraints arising from the geometry of the normalization. While nodal singularities give rise to nontrivial rank-one contributions, other singularities may contribute only through higher-order local data or may be invisible at the infinitesimal level. As a consequence, we obtain sharp criteria for maximal infinitesimal variation in terms of numerical invariants of the curve, notably when the number of nodes satisfies the inequality $\delta \ge g$, where $g$ denotes the genus of the normalization. We extend these results to curves on very general surfaces in projective three-space, showing that maximal variation persists on Picard-rank-one surfaces but fails for sufficiently large genus in the presence of higher ADE singularities. These results extend classical maximality phenomena in infinitesimal Hodge theory to a broader singular and geometric setting.
|
https://arxiv.org/abs/2601.04410
|
Academic Papers
|
svg
|
cf7bf1a3955aaf0c12b8462c6164e788e5ee528f81c3b06cc774dbaffca7e519
|
2026-01-13T00:00:00-05:00
|
Automorphic vector-forms using the Cohn-Elkies magic functions
|
arXiv:2601.04704v2 Announce Type: replace Abstract: In this study, we introduce the theory of what we call Hecke vector-forms. A Hecke vector-form can be viewed as a vector function representation of some quasiautomorphic form that transforms like an automorphic form on an arbitrarily chosen Hecke triangle group. In other words, because quasiautomorphic forms have complicated transformation behavior when compared with automorphic forms, the construction of a Hecke vector-form is to retrieve a transformation behavior analogous to the simpler, automorphic case. In this way, a Hecke vector-form can be viewed as the vector function analogue of an automorphic form. Since our work is for any quasi-automorphic form over an arbitrary Hecke triangle group, we briefly review the construction of such groups. Furthermore, we review the derivation of the hauptmodul, the automorphic forms, and the normalized quasiautomorphic form of weight 2 for any Hecke triangle group. We then proceed to the theory of Hecke vector-forms and establish the desired transformation behavior with respect to the generators of the associated group. A proof of this fact is strictly elementary, relying on fine properties of binomial coefficients. Lastly, we relate the vector-forms to Hecke automorphic linear differential equations, which are analogues of the frequently researched modular linear differential equations. Our results include Hecke vector-forms of the classical quasimodular forms as the simplest case.
|
https://arxiv.org/abs/2601.04704
|
Academic Papers
|
svg
|
93d9fe3c63aec265d13ffc535bfd5f91480f07f5de0bfe3e7b20dcf8b9121481
|
2026-01-13T00:00:00-05:00
|
Spectral Properties of $ C_{0}$-Semigroups of Positive Operators on C$^*$-Algebras
|
arXiv:2601.05025v2 Announce Type: replace Abstract: Let $ (T(t))_{t\geq0} $ be a positive $C_{0}$-semigroup with generator $A$ on a C$^*$-algebra or on the predual of a W$^*$-algebra. Then the growth bound $\omega_{0}$ equals $s(A)$. If the spectrum of $A$ is not empty, then $s(A)$, the spectral bound of $A$, is a spectral value.
|
https://arxiv.org/abs/2601.05025
|
Academic Papers
|
svg
|
bef51e97f3c382e2ef3f3eaabe1c444def25bb837c88b4a677aac2f9c942665a
|
2026-01-13T00:00:00-05:00
|
Information-Theoretic Limits on Exact Subgraph Alignment Problem
|
arXiv:2601.05173v2 Announce Type: replace Abstract: The graph alignment problem aims to identify the vertex correspondence between two correlated graphs. Most existing studies focus on the scenario in which the two graphs share the same vertex set. However, in many real-world applications, such as computer vision, social network analysis, and bioinformatics, the task often involves locating a small graph pattern within a larger graph. Existing graph alignment algorithms and analysis cannot directly address these scenarios because they are not designed to identify the specific subset of vertices where the small graph pattern resides within the larger graph. Motivated by this limitation, we introduce the subgraph alignment problem, which seeks to recover both the vertex set and/or the vertex correspondence of a small graph pattern embedded in a larger graph. In the special case where the small graph pattern is an induced subgraph of the larger graph and both the vertex set and correspondence are to be recovered, the problem reduces to the subgraph isomorphism problem, which is NP-complete in the worst case. In this paper, we formally formulate the subgraph alignment problem by proposing the Erdos-Renyi subgraph pair model together with some appropriate recovery criterion. We then establish almost-tight information-theoretic results for the subgraph alignment problem and present some novel approaches for the analysis.
|
https://arxiv.org/abs/2601.05173
|
Academic Papers
|
svg
|
83332478e4502e5929ed33d41c2ada7d8d0d6d700f67c4516775d53655864727
|
2026-01-13T00:00:00-05:00
|
Markovian Compression: Looking to the Past Helps Accelerate the Future
|
arXiv:2601.05398v2 Announce Type: replace Abstract: This paper deals with distributed optimization problems that use compressed communication to achieve efficient performance and mitigate communication bottleneck. We propose a family of compression schemes in which operators transform vectors fed to their input according to a Markov chain, i.e. the stochasticity of the compressors depends on previous iterations. The compressors are implemented in the vanilla Quantized Stochastic Gradient Descent algorithm (QSGD), and, to further improve the efficiency and convergence rate, in the momentum accelerated QSGD. We provide convergence results for our algorithms with Markovian compressors, the analysis covers non-convex, Polyak-Lojasiewicz, and strongly convex cases. To demonstrate the applicability of our approach to distributed data-parallel optimization problems, we conduct experiments on the CIFAR-10 and GLUE datasets with the Resnet-18 and DeBERTaV3 models. Practical results show the superiority of methods that use our compressor design over existing schemes.
|
https://arxiv.org/abs/2601.05398
|
Academic Papers
|
svg
|
73bed4702029bfc2c052d4fcb7282cda122911f24d3334fc35b40b882f3d4b19
|
2026-01-13T00:00:00-05:00
|
The B\'enabou-Roubaud theorem via string diagrams
|
arXiv:2601.05691v2 Announce Type: replace Abstract: We give a complete proof of the B\'enabou-Roubaud monadic descent theorem using the graphical calculus of string diagrams. Our proof links the monadic and Grothendieck's original viewpoint on descent via an internal-category-based characterization of the category of descent data, equivalent to the one of Janelidze and Tholen.
|
https://arxiv.org/abs/2601.05691
|
Academic Papers
|
svg
|
9c93fe150be74d1a58475a4ef30fe23e0ec4d04abea2fb2c5432226cbbf47d39
|
2026-01-13T00:00:00-05:00
|
Approximating Persistent Homology for Large Datasets
|
arXiv:2204.09155v3 Announce Type: replace-cross Abstract: Persistent homology is an important methodology in topological data analysis which adapts theory from algebraic topology to data settings. Computing persistent homology produces persistence diagrams, which have been successfully used in diverse domains. Despite its widespread use, persistent homology is simply impossible to compute when a dataset is very large. We study a statistical approach to the problem of computing persistent homology for massive datasets using a multiple subsampling framework and extend it to three summaries of persistent homology: H\"{o}lder continuous vectorizations of persistence diagrams; the alternative representation as persistence measures; and standard persistence diagrams. Specifically, we derive finite sample convergence rates for empirical means for persistent homology and practical guidance on interpreting and tuning parameters. We validate our approach through extensive experiments on both synthetic and real-world data. We demonstrate the performance of multiple subsampling in a permutation test to analyze the topological structure of Poincar\'{e} embeddings of large lexical databases.
|
https://arxiv.org/abs/2204.09155
|
Academic Papers
|
svg
|
c8043f7a240679359f453591766481787e9935626b9097ea47d8bf308fbd3db3
|
2026-01-13T00:00:00-05:00
|
Field Theory Equivalences as Spans of $L_\infty$-algebras
|
arXiv:2305.05473v3 Announce Type: replace-cross Abstract: Semi-classically equivalent field theories are related by a quasi-isomorphism between their underlying $L_\infty$-algebras, but such a quasi-isomorphism is not necessarily a homotopy transfer. We demonstrate that all quasi-isomorphisms can be lifted to spans of $L_\infty$-algebras in which the quasi-isomorphic $L_\infty$-algebras are obtained from a correspondence $L_\infty$-algebra by a homotopy transfer. Our construction is very useful: homotopy transfer is computationally tractable, and physically, it amounts to integrating out fields in a Feynman diagram expansion. Spans of $L_\infty$-algebras allow for a clean definition of quasi-isomorphisms of cyclic $L_\infty$-algebras. Furthermore, they appear naturally in many contexts within physics. As examples, we first consider scalar field theory with interaction vertices blown up in different ways. We then show that (non-Abelian) T-duality can be seen as a span of $L_\infty$-algebras, and we provide full details in the case of the principal chiral model. We also present the relevant span of $L_\infty$-algebras for the Penrose-Ward transform in the context of self-dual Yang-Mills theory and Bogomolny monopoles.
|
https://arxiv.org/abs/2305.05473
|
Academic Papers
|
svg
|
5710fc49b4bd55762d8fc66e819ee1d2953c2d13e1f8e441a305e012c8633d4a
|
2026-01-13T00:00:00-05:00
|
Asymptotic Error Analysis of Multilevel Stochastic Approximations for the Value-at-Risk and Expected Shortfall
|
arXiv:2311.15333v4 Announce Type: replace-cross Abstract: Cr\'epey, Frikha, and Louzi (2023) introduced a nested stochastic approximation algorithm and its multilevel acceleration to compute the value-at-risk and expected shortfall of a random financial loss. We hereby establish central limit theorems for the renormalized estimation errors associated with both algorithms as well as their averaged versions. Our findings are substantiated through a numerical example.
|
https://arxiv.org/abs/2311.15333
|
Academic Papers
|
svg
|
15f6fc6f8f7ef027a8585d7417d39cdc81751f68d417a03629b90d80b4475dbe
|
2026-01-13T00:00:00-05:00
|
Learning Operators with Stochastic Gradient Descent in General Hilbert Spaces
|
arXiv:2402.04691v4 Announce Type: replace-cross Abstract: This study investigates leveraging stochastic gradient descent (SGD) to learn operators between general Hilbert spaces. We propose weak and strong regularity conditions for the target operator to depict its intrinsic structure and complexity. Under these conditions, we establish upper bounds for convergence rates of the SGD algorithm and conduct a minimax lower bound analysis, further illustrating that our convergence analysis and regularity conditions quantitatively characterize the tractability of solving operator learning problems using the SGD algorithm. It is crucial to highlight that our convergence analysis is still valid for nonlinear operator learning. We show that the SGD estimator will converge to the best linear approximation of the nonlinear target operator. Moreover, applying our analysis to operator learning problems based on vector-valued and real-valued reproducing kernel Hilbert spaces yields new convergence results, thereby refining the conclusions of existing literature.
|
https://arxiv.org/abs/2402.04691
|
Academic Papers
|
svg
|
0b8537b985e86821720df994f9ffda1b12fb97dd08c70b313e0b7dc871664bad
|
2026-01-13T00:00:00-05:00
|
$\texttt{skwdro}$: a library for Wasserstein distributionally robust machine learning
|
arXiv:2410.21231v2 Announce Type: replace-cross Abstract: We present skwdro, a Python library for training robust machine learning models. The library is based on distributionally robust optimization using Wasserstein distances, popular in optimal transport and machine learnings. The goal of the library is to make the training of robust models easier for a wide audience by proposing a wrapper for PyTorch modules, enabling model loss' robustification with minimal code changes. It comes along with scikit-learn compatible estimators for some popular objectives. The core of the implementation relies on an entropic smoothing of the original robust objective, in order to ensure maximal model flexibility. The library is available at https://github.com/iutzeler/skwdro and the documentation at https://skwdro.readthedocs.io.
|
https://arxiv.org/abs/2410.21231
|
Academic Papers
|
svg
|
1f462b6c4868f382cbd84f9c78d398d378f496f06e9721159cf90dd6dd2bac34
|
2026-01-13T00:00:00-05:00
|
Schmidt Decomposition of Multipartite States
|
arXiv:2411.02473v4 Announce Type: replace-cross Abstract: Quantum states can be written in infinitely many ways depending on the choices of basis. Schmidt decomposition of a quantum state has a lot of properties useful in the study of entanglement. All bipartite states admit Schmidt decomposition, but this does not extend to multipartite systems. We obtain necessary and sufficient conditions for the existence of Schmidt decompositions of multipartite states. Moreover, we provide an efficient algorithm to obtain the decomposition for a Schmidt decomposable multipartite state.
|
https://arxiv.org/abs/2411.02473
|
Academic Papers
|
svg
|
682cd0ec8445eaac82b904e7c0dca8dfa91f012c3017f7156812d4c88eb82591
|
2026-01-13T00:00:00-05:00
|
Point processes with event time uncertainty
|
arXiv:2411.02694v2 Announce Type: replace-cross Abstract: Point processes are widely used statistical models for continuous-time discrete event data, such as medical records, crime reports, and social network interactions, to capture the influence of historical events on future occurrences. In many applications, however, event times are not observed exactly, motivating the need to incorporate time uncertainty into point process modeling. In this work, we introduce a framework for modeling time-uncertain self-exciting point processes, known as Hawkes processes, possibly defined over a network. We begin by formulating the model in continuous time under assumptions motivated by real-world scenarios. By imposing a time grid, we obtain a discrete-time model that facilitates inference and enables computation via first-order optimization methods such as gradient descent and variational inequality (VI). We establish a parameter recovery guarantee for VI inference with an $O(1/k)$ convergence rate using $k$ steps. Our framework accommodates non-stationary processes by representing the influence kernel as a matrix (or tensor on a network), while also encompassing stationary processes, such as the classical Hawkes process, as a special case. Empirically, we demonstrate that the proposed approach outperforms existing baselines on both simulated and real-world datasets, including the sepsis-associated derangement prediction challenge and the Atlanta Police Crime Dataset.
|
https://arxiv.org/abs/2411.02694
|
Academic Papers
|
svg
|
795788bebfa78924fd00dead91afb97d51d26127debb1bd70e95e13bb93d265d
|
2026-01-13T00:00:00-05:00
|
Color symmetry and ferromagnetism in Potts spin glass
|
arXiv:2412.20607v2 Announce Type: replace-cross Abstract: We consider the Potts spin glass with additional ferromagnetic interaction parametrized by $t$. It has long been observed that the Potts color symmetry breaking for the spin glass order parameter is closely related to the ferromagnetic phase transition. To clarify this, we identify a single critical value $t_\mathrm{c}$, which marks the onset of both color symmetry breaking and the transition to ferromagnetism.
|
https://arxiv.org/abs/2412.20607
|
Academic Papers
|
svg
|
54d20088c80156be1b133f1020d47bcca0ccea8545f4f7741a2f889a6b22ae49
|
2026-01-13T00:00:00-05:00
|
Coverage and Spectral Efficiency of NOMA-Enabled LEO Satellite Networks with Ordering Schemes
|
arXiv:2501.05946v2 Announce Type: replace-cross Abstract: This paper investigates an analytical model for low-earth orbit (LEO) multi-satellite downlink non-orthogonal multiple access (NOMA) networks. The satellites transmit data to multiple NOMA user terminals (UTs), each employing successive interference cancellation (SIC) for decoding. Two ordering schemes are adopted for NOMA-enabled LEO satellite networks, i.e., mean signal power (MSP)-based ordering and instantaneous signal-to-inter-satellite-interference-plus-noise ratio (ISINR)-based ordering. For each ordering scheme, we derive the analytical expression for the coverage probability of each typical UT. Moreover, we discuss how coverage is influenced by SIC, main-lobe gain, and tradeoffs between the number of satellites and their altitudes. Additionally, two user fairness-based power allocation (PA) schemes are considered, and PA coefficients with the optimal number of UTs that maximize their sum spectral efficiency (SE) are studied. Simulation results show that there exists a maximum effective signal-to-inter-satellite-interference-plus-noise ratio (SINR) threshold for each PA scheme that ensures the operation of NOMA in LEO satellite networks, and NOMA provides performance gains only when the target SINR is below a certain threshold. Compared with orthogonal multiple access (OMA), NOMA increases UTs' sum SE by as much as 35%. Furthermore, for most SINR thresholds, the sum SE increases with the number of UTs to the highest value, whilst the maximum sum SE is obtained when there are two UTs.
|
https://arxiv.org/abs/2501.05946
|
Academic Papers
|
svg
|
09f4a84676f10cf17d5fa084e459c1472d13668d4667a2969d84d02dca815bf1
|
2026-01-13T00:00:00-05:00
|
Ordinary Least Squares as an Attention Mechanism
|
arXiv:2504.09663v2 Announce Type: replace-cross Abstract: I show that ordinary least squares (OLS) predictions can be rewritten as the output of a restricted attention module, akin to those forming the backbone of large language models. This connection offers an alternative perspective on attention beyond the conventional information retrieval framework, making it more accessible to researchers and analysts with a background in traditional statistics. It falls into place when OLS is framed as a similarity-based method in a transformed regressor space, distinct from the standard view based on partial correlations. In fact, the OLS solution can be recast as the outcome of an alternative problem: minimizing squared prediction errors by optimizing the embedding space in which training and test vectors are compared via inner products. Rather than estimating coefficients directly, we equivalently learn optimal encoding and decoding operations for predictors. From this vantage point, OLS maps naturally onto the query-key-value structure of attention mechanisms. Building on this foundation, I discuss key elements of Transformer-style attention and draw connections to classic ideas from time series econometrics.
|
https://arxiv.org/abs/2504.09663
|
Academic Papers
|
svg
|
5810134f6d6f347eb502bcf65263f8e832863fac956687f312ab755de63dd013
|
2026-01-13T00:00:00-05:00
|
SAD Neural Networks: Divergent Gradient Flows and Asymptotic Optimality via o-minimal Structures
|
arXiv:2505.09572v3 Announce Type: replace-cross Abstract: We study gradient flows for loss landscapes of fully connected feedforward neural networks with commonly used continuously differentiable activation functions such as the logistic, hyperbolic tangent, softplus or GELU function. We prove that the gradient flow either converges to a critical point or diverges to infinity while the loss converges to an asymptotic critical value. Moreover, we prove the existence of a threshold $\varepsilon>0$ such that the loss value of any gradient flow initialized at most $\varepsilon$ above the optimal level converges to it. For polynomial target functions and sufficiently big architecture and data set, we prove that the optimal loss value is zero and can only be realized asymptotically. From this setting, we deduce our main result that any gradient flow with sufficiently good initialization diverges to infinity. Our proof heavily relies on the geometry of o-minimal structures. We confirm these theoretical findings with numerical experiments and extend our investigation to more realistic scenarios, where we observe an analogous behavior.
|
https://arxiv.org/abs/2505.09572
|
Academic Papers
|
svg
|
c45aadf927752a58e4a4f62ba580709bea1589fbbd5e393eebf2d62381a69d96
|
2026-01-13T00:00:00-05:00
|
Efficient Online Random Sampling via Randomness Recycling
|
arXiv:2505.18879v3 Announce Type: replace-cross Abstract: This article studies the fundamental problem of using i.i.d. coin tosses from an entropy source to efficiently generate random variables $X_i \sim P_i$ $(i \ge 1)$, where $(P_1, P_2, \dots)$ is a random sequence of rational discrete probability distributions subject to an \textit{arbitrary} stochastic process. Our method achieves an amortized expected entropy cost within $\varepsilon > 0$ bits of the information-theoretically optimal Shannon lower bound using $O(\log(1/\varepsilon))$ space. This result holds both pointwise in terms of the Shannon information content conditioned on $X_i$ and $P_i$, and in expectation to obtain a rate of $\mathbb{E}[H(P_1) + \dots + H(P_n)]/n + \varepsilon$ bits per sample as $n \to \infty$ (where $H$ is the Shannon entropy). The combination of space, time, and entropy properties of our method improves upon the Knuth and Yao (1976) entropy-optimal algorithm and Han and Hoshi (1997) interval algorithm for online sampling, which require unbounded space. It also uses exponentially less space than the more specialized methods of Kozen and Soloviev (2022) and Shao and Wang (2025) that generate i.i.d. samples from a fixed distribution. Our online sampling algorithm rests on a powerful algorithmic technique called \textit{randomness recycling}, which reuses a fraction of the random information consumed by a probabilistic algorithm to reduce its amortized entropy cost. On the practical side, we develop randomness recycling techniques to accelerate a variety of prominent sampling algorithms. We show that randomness recycling enables state-of-the-art runtime performance on the Fisher-Yates shuffle when using a cryptographically secure pseudorandom number generator, and that it reduces the entropy cost of discrete Gaussian sampling. Accompanying the manuscript is a performant software library in the C programming language.
|
https://arxiv.org/abs/2505.18879
|
Academic Papers
|
svg
|
693732402b0b62bf424bd141f41f48ae5e8bd7b3130cc3d1ceed29cde4d59ef2
|
2026-01-13T00:00:00-05:00
|
Conformable Scaling and Critical Dynamics: A Unified Framework for Phase Transitions
|
arXiv:2507.11782v2 Announce Type: replace-cross Abstract: We investigate the application of conformable derivatives to model critical phenomena near continuous phase transitions. By incorporating a deformation parameter into the differential structure, we derive unified expressions for thermodynamic observables such as heat capacity, magnetization, susceptibility, and coherence length, each exhibiting power-law behavior near the critical temperature. The conformable derivative framework naturally embeds scale invariance and critical slowing down into the dynamics without resorting to fully nonlocal fractional calculus. Modified Ginzburg-Landau equations are constructed to model superconducting transitions, leading to analytical expressions for the order parameter and London penetration depth. Experimental data from niobium confirm the model's applicability, showing excellent fits and capturing asymmetric scaling behavior around Tc. This work offers a bridge between classical mean-field theory and generalized scaling frameworks, with implications for both theoretical modeling and experimental analysis.
|
https://arxiv.org/abs/2507.11782
|
Academic Papers
|
svg
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.