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- [1] arXiv:2608.19228 [pdf, html, other]
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Title: Assembly Theory and the Smallest Grammar ProblemSubjects: Information Theory (cs.IT); Formal Languages and Automata Theory (cs.FL)
Assembly theory (AT) quantifies complexity through the assembly index (ASI) -- a metric proven equivalent to the size of the smallest straight-line program (SLP). While this equivalence links AT to data compression -- a field shaped by decades of research -- the practical efficacy of specific algorithms as ASI approximators remains largely unexplored. This study provides an empirical evaluation of eight compression algorithms, spanning grammar-based and dictionary schemes (CAs) across a dataset of 408 strings -- comprising 368 synthetic strings (including max-complexity strings and strings with varying levels of symbol distribution balance, quantified by Shannon entropy) and 40 natural biological sequences (genomic and proteomic). We introduce Re-Pair T-NDR, a branch-and-bound tie-resolving variant of Re-Pair, providing a tighter upper bound on the ASI than the other CAs researched in this study. Increasing alphabet size extends the regime in which CAs closely track the ASI, delaying a ``divergence point'', where a CA deviates from the optimal assembly path. The results establish compression algorithms as practical tools for ASI estimation.
- [2] arXiv:2608.19233 [pdf, html, other]
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Title: Microlocal analysis of non-linear artifacts in cone beam CTComments: 31 pages, 4 figuresSubjects: General Mathematics (math.GM)
We present a novel microlocal analysis of beam hardening artifacts arising in cone-beam X-ray CT, where the set of X-ray sources is restricted to a 1D smooth curve $ \gamma\subset \mathbb{R}^3$. We show that, when the CT data is modeled in the standard way using the Beer Lambert law, the exponential term in the model creates singularities in the data that are not present in the linear X-ray transform. We assume that the attenuation coefficient $\mu$ (the reconstruction target) has a jump discontinuity across a surface $\mathcal{S}\subset\mathbb{R}^3$, and is smooth otherwise. We prove that these additional singularities in the data occur when the X-ray beam is tangent to $\mathcal{S}$ at two points simultaneously. To investigate how the singularities in the data propagate to the reconstruction space, we apply Filtered Back Projection (FBP) type reconstruction. We prove that the artifacts due to beam hardening are locally of conormal type, and lie on a 2-D surface which is the union of all double tangent rays which intersect $\gamma$. The artifacts are notably weaker than the reconstructed jumps of $\mu$ (i.e., the desired singularities), and we quantify this using the order of the corresponding conormal distributions. While our primary theory applies to the regions of $\mathcal{S}$ that are smooth, we also extend our theory to non-smooth $\mathcal{S}$ with "ridges." These arise where $\mathcal{S}$ is locally the intersection of two smooth surface patches meeting transversely along a curve (e.g., the edge of a cuboid). In addition, we present simulated reconstructions of metal objects in circular cone-beam CT to validate our theory.
- [3] arXiv:2608.19240 [pdf, html, other]
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Title: A Nonmonotone Real-Rootedness Set for Symmetric Imaginary ShiftsComments: 7 pages, 1 table. AI Research Artifact with material AI use and author accountability disclosed in the paper. Corresponds to Zenodo v0.1.3, DOI: https://doi.org/10.5281/zenodo.21922629Subjects: Complex Variables (math.CV)
For a real polynomial $F$ and $\omega\geq 0$, set $A_\omega(z)=(F(z+i\omega)+F(z-i\omega))/2$ and $\Omega_F=\{\omega\geq0:A_\omega\text{ has only real zeros}\}$. We present an explicit rational even polynomial of degree eight for which $6/25$ and $12/25$ belong to $\Omega_F$, while $3/10$ does not. Exact Sturm certificates give respectively eight, four, and eight distinct real zeros. Consequently $\Omega_F$ is neither an interval nor an up-set. All zeros of $F$ lie in the strip $|\operatorname{Im}z|\leq11/25$, and the classical strip-contraction theorem gives the eventual tail $[11/25,\infty)\subset\Omega_F$. We also include a direct elementary proof of that tail and a standard-library exact verifier.
- [4] arXiv:2608.19250 [pdf, html, other]
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Title: A fully discrete LBRFD-IPDG method for linear fourth-order parabolic equationsComments: 28 pages, 10 figuresSubjects: Numerical Analysis (math.NA)
We propose a fully discrete method for linear fourth-order parabolic equations with Dirichlet boundary conditions, combining an implicit LBRFD multistep scheme in time with a mixed interior penalty discontinuous Galerkin (IPDG) method in space. The temporal discretization employs equispaced linear barycentric rational interpolants and incorporates a startup procedure. To facilitate the spatial discretization, the original problem is reformulated through an auxiliary variable. For certain parameter pairs $(n,d)$, the LBRFD method is shown to be $A(\alpha)$-stable and to possess a wider stability angle than the corresponding BDF$p$ method of the same order. Stability and a priori error estimates are established via a $G$-energy technique and the discrete Grönwall lemma. The theoretical analysis yields a total $L^2$ error estimate of order $h^{k-1}+\tau^p$, where $p=d$ if $n-d$ is even and $p=d+1$ if $n-d$ is odd. The reduced spatial convergence rate is attributed to boundary contributions on $\partial\Omega$. Despite this theoretical prediction, numerical experiments confirm the stability and demonstrate optimal convergence of order $h^{k+1}+\tau^p$.
- [5] arXiv:2608.19251 [pdf, html, other]
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Title: The Second Edge Theorem: The Asymptotic Collapse of Sample-Dependent Information Geometry to the Canonical Flat Canvas of Conventional Statistics in Large Sample LimitsSubjects: General Mathematics (math.GM)
This paper establishes the global proof of the Second Edge Theorem: as sample size tends to infinity, sample-dependent information-geometric manifolds---formed by the parameter space, sample-scaled Fisher metric, and dual alpha-connections---undergo metric-topological collapse onto the flat tangent space of Conventional Statistics at the true parameter. We first prove a universal tensor valence scaling law, under which tensor fields of valence one through four degenerate at rates determined by sample size. Score fluctuations stabilize, the Fisher metric freezes to its true-parameter value, affine connections dissolve, and Riemann curvature is annihilated. We then bridge geometric collapse with statistical decision theory, showing that Cheeger--Gromov flattening and Le Cam risk condensation are dual projections of the same asymptotic phase transition. A Fisher-compatible Ehresmann connection extends these results to over-parameterized and singular models, yielding horizontal leaf-space collapse and uniform local asymptotic normality. Unifying the First and Second Edge Theorems yields a nested dual-edge hierarchy: Conventional Statistics is the boundary of Information Geometry, which is itself the boundary of Statistical Mechanics and Geometry. Thus, Conventional Statistics is not a heuristic approximation but the unique zero-curvature thermodynamic attractor of regular parametric information manifolds. This redefines modern statistics as a dynamic non-equilibrium field theory of finite-sample fluctuations, phase transitions, and gauge-invariant interactions.
- [6] arXiv:2608.19253 [pdf, html, other]
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Title: Simplex--center configurations in dense subsets of Euclidean spaces and the integer latticeSubjects: Metric Geometry (math.MG); Combinatorics (math.CO); Number Theory (math.NT)
We obtain density Ramsey theorems for configurations consisting of the vertices of a simplex $\Delta_o$ together with their barycenter. We prove that any subset $A\subseteq\mathbb{R}^n$ of positive upper density contains an isometric copy of all sufficiently large dilates of $\Delta_o$ together with its barycenter. As this configuration is non-spherical such results are not possible with respect to the quadratic Euclidean metric, we consider general metrics $\rho$ defined by a positive-definite, homogeneous forms of even degree at least four. We prove the analogous result in the discrete setting, for subsets $A$ of the integer lattice $\mathbb{Z}^n$, under some natural and necessary congruence restrictions on the scales $\lambda$ at which the set $A$ can contain an isometric copy of the simplex.
- [7] arXiv:2608.19255 [pdf, html, other]
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Title: Totally positive field extensions and the pythagorean indexJournal-ref: Journal of Algebra and Its Applications, Vol. 23, No. 02, 2450034 (2024)Subjects: Number Theory (math.NT)
For a formally real field $F$, we study totally positive field extensions $K$ over $F$. We show that, if $K /F$ is Galois and totally positive then
so is the corresponding extension of their pythagorean closures $K_{\rm py}$ over $F_{\rm py}$. We also study the behaviour of weak isotropy and weak hyperbolicity of central simple algebras with an orthogonal involution over totally positive field extensions. We use some of these results to
prove new cases for which a conjecture due to Becher holds. - [8] arXiv:2608.19256 [pdf, html, other]
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Title: An Algebraic Approach to the Fundamental Theorem of AlgebraJournal-ref: Rendiconti del Circolo Matematico di Palermo, Vol 73, 3211 - 3215 (2024)Subjects: Number Theory (math.NT)
In this paper, we investigate the algebraic counterpart of the Fundamental Theorem of Algebra. We explore the concept of real closed fields and quadratic forms. We show, by means of Galois theory, that $F(\sqrt{-1})$ is algebraically closed if $F$ is real-closed. Lastly, we explain the algebraic closure of $\mathbb R(\sqrt{-1})=\mathbb C$ by demonstrating the real-closeness of $\mathbb R$.
- [9] arXiv:2608.19262 [pdf, html, other]
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Title: Many Representations as Sums of Three Prime CubesSubjects: Number Theory (math.NT)
Let $F_k(n)$ be the number of unordered representations \[
n=p_1^k+p_2^k+\cdots +p_k^k \] by primes, with repetitions allowed. Erdős stated that $\limsup_{n\to\infty} F_3(n)=\infty$, but his proof appears not to have been published. A complete unconditional proof is given. The principal input is the classical Hecke equidistribution theorem for the CM Fermat cubic; the rest of the argument uses standard estimates for primes in arithmetic progressions and elementary counting. The argument used for \(k=3\) does not extend to the case \(k=4\). Nevertheless, by applying the Green--Tao--Ziegler theorem to the linear forms arising from an admissible binary quartic identity, \(\limsup_{n\to\infty}F_4(n)\ge2\) is shown. - [10] arXiv:2608.19268 [pdf, other]
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Title: From the Half-Order Recurrence to General Fractional-Order Differentiation on Monomials: Functional Continuation and Operator CompositionComments: 92 pages. Continuation of arXiv:2607.19482Subjects: General Mathematics (math.GM)
This work develops a general-order fractional differentiation operator on monomials from the discrete half-order coefficient structure established in Part I. Starting from the one-step transfer law of the integer coefficient family, the discrete coefficient relation is continued to a real argument and a canonical Gamma-functional form is selected under the stated normalization and regularity requirements. The operator order is then extended beyond one half by dividing the first derivative into equal operator steps, leading from orders 1/m and k/m to general rational and real orders. The resulting coefficient is independently verified through the addition law for operator orders on the monomial system, subject to the requirement that all intermediate expressions in the composition are defined. On the corresponding parameter domain, the construction yields the standard Gamma-ratio monomial formula for fractional differentiation. For nonnegative integer orders it reduces to ordinary repeated differentiation, while for positive non-integer orders it agrees at the monomial level with the left-sided Riemann-Liouville formula with lower limit 0. Its relation to the Caputo operator is also discussed, including the distinction for constants and low-degree polynomial terms. The aim is not to introduce a new classical fractional derivative, but to provide a constructive algebraic-operator route from a discrete coefficient law to the general fractional-order monomial formula, clarifying the roles of functional continuation, Gamma normalization, and operator composition.
- [11] arXiv:2608.19270 [pdf, html, other]
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Title: Product formulas for multivariate and matrix-variate confluent hypergeometric functionsSubjects: Classical Analysis and ODEs (math.CA)
In this note, we revisit one of Erdélyi's product formulas for the univariate confluent hypergeometric function ${}_{1}F_{1}$ and extend it to the multivariate confluent hypergeometric functions $\Phi_2^{(k)}$, $\Psi_2^{(k)}$, as well as to a certain matrix-variate confluent hypergeometric function. A useful connection related to fractional calculus is also given.
- [12] arXiv:2608.19271 [pdf, html, other]
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Title: Information Geometry (IG) Lives at Edge or Boundary of SMG (statistically meaningful geometry): - the First Edge Theorem and ApplicationsSubjects: General Mathematics (math.GM)
Statistically Meaningful Geometry (SMG) is a differential-geometric and information-theoretic framework that lifts over-parameterized models into infinite-dimensional non-parametric Orlicz statistical fiber bundles with an Ehresmann connection, decoupling unobservable vertical gauge noise from horizontal statistically verifiable directions. We prove the First Edge Theorem: Amari's information geometry (IG) and conventional statistics (CS) are not autonomous statistical universes but degenerate boundary layers of the larger gauge-active SMG space. Taking the structural identifiability radius $R\to\infty$ breaks gauge symmetry, collapses vertical fibers, and forces the total space onto the IG manifold; subsequent local asymptotic normality as $N\to\infty$ flattens the remaining curvature, yielding \[ \mathrm{SMG}\xrightarrow{\;R\to\infty\;}\mathrm{IG}\xrightarrow{\;N\to\infty\;}\mathrm{CS}. \] The same fiber-bundle machinery transforms model non-identifiability from a singular collapse into a structured gauge space. Applications include resolving the deep-learning generalization paradox, constructing gauge-invariant gradient descent and holonomy-matched preference alignment for generative AI, and solving weak identification in structural econometrics via intrinsic horizontal geodesic search.
- [13] arXiv:2608.19276 [pdf, html, other]
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Title: ${}_5F_4$ evaluations and a family of $π^2+\log^2$ identitiesComments: 13 pagesSubjects: Number Theory (math.NT); Combinatorics (math.CO)
We evaluate the series $\sum_{n\ge1} z^n\big/\!\big(n^2\binom{4n}{n}\big)$, equal to $-\tfrac{z}{4}\,{}_5F_4\!\left(1,1,1,\tfrac43,\tfrac53;\tfrac54,\tfrac32,\tfrac74,2;\tfrac{27z}{256}\right)$, in closed form at an infinite family of algebraic points indexed by a rational angle $\theta=j\pi/N$. Each value equals $c\,\pi^2$ plus a universal rational quadratic form in three logarithms, with $c=-\tfrac13\left(1-\tfrac{2j}{N}\right)^2$. This is the quartic-base case reached but not evaluated by D'Aurizio and Di Trani. The proof is self-contained: an exact integer factor relating two weights, followed by Landen's identity, reduces the integral to a sum of squared logarithms.
- [14] arXiv:2608.19286 [pdf, html, other]
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Title: Minimum attaining operators on reducing subspaces: Spectral structure and densityComments: Submitted to a journal. Comments and suggestions are welcomeSubjects: Functional Analysis (math.FA); Operator Algebras (math.OA)
In this article, we introduce and investigate a new subclass $\mathcal{M}_r(H)$ of minimum attaining operators on a separable Hilbert space $H$. This class contains the absolutely minimum attaining operators and is properly contained in the class of minimum attaining operators. We establish several structural and spectral characterizations of operators in $\mathcal{M}_r(H)$. In particular, we characterize positive operators in $\mathcal{M}_r(H)$ in terms of their spectral representations. We prove that $\mathcal{M}_r(H)$ is dense in $\mathcal{B}(H)$ in the operator norm and, moreover, that the operators in $\mathcal{M}_r(H)$ having a nontrivial invariant half-space are also dense in $\mathcal{B}(H)$ in the operator norm. We further obtain a representation theorem for normal operators in $\mathcal{M}_r(H)$ and establish additional structural properties of this class.
- [15] arXiv:2608.19289 [pdf, html, other]
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Title: Existence of periodic solutions for Hamiltonian inclusion systems using Clarke dualitySubjects: Dynamical Systems (math.DS)
We prove the existence of periodic solutions for Hamiltonian differential inclusions under growth conditions involving a G-function.
- [16] arXiv:2608.19294 [pdf, html, other]
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Title: On Generalized Total Colourings of Planar GraphsComments: 12 pages, 7 color figuresSubjects: Combinatorics (math.CO)
In this paper we study generalised total colourings of graphs where the colour classes formed by vertices and edges, respectively, induce forests, while incident edges/vertices receive distinct colours. In [M. Borowiecki and I. Broere, Hamiltonicity and Generalised Total Colourings of Planar Graphs, Discussiones Mathematicae Graph Theory 36 (2016) 243--257] it was conjectured that for planar graphs, four colours suffice for this type of colouring. We confirm this conjecture for two infinite families of planar graphs.
- [17] arXiv:2608.19301 [pdf, html, other]
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Title: Disproof of the Yau--Tian--Donaldson conjectureComments: 79 pages. Main result of this paper obtained by generative AI. Including an appendix joint with Bin Dong and Guoxiong Gao as a detailed report on the use of generative AI in this paperSubjects: Differential Geometry (math.DG); Algebraic Geometry (math.AG); Complex Variables (math.CV)
We construct a polarized smooth projective fivefold and prove that it is K-polystable but does not admit a constant scalar curvature Kähler metric. This disproves the Yau--Tian--Donaldson conjecture for constant scalar curvature metrics.
The main result of this paper was obtained using generative AI, particularly GPT-5.6-sol, Fable 5, and the Danus system. A detailed report on the use of generative AI in this paper is enclosed in the appendix, joint with Bin Dong and Guoxiong Gao. - [18] arXiv:2608.19340 [pdf, html, other]
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Title: The Liquid Drop Model with a Yukawa Potential: Existence of Minimizers and Sharp Stability of the BallComments: 30 pages, comments welcome!Subjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph); Optimization and Control (math.OC)
We study a long-range perturbation of the perimeter functional by a nonlocal repulsive term defined through the Yukawa kernel, minimized under a volume constraint. Because the kernel decays exponentially, the competition between surface tension and repulsion is governed by two independent quantities, the screening rate and the volume. Specifically we prove that (i) above an explicit critical screening rate, minimizers exist at every volume, whereas earlier work required the volume to be large; (ii) below an explicit volume threshold minimizers exist for every screening rate; (iii) at small volume, and uniformly in the screening rate, the ball is the unique minimizer up to translation; and (iv) there exists a sharp volume threshold, depending on the screening rate, for the ball to be a stable volume-constrained critical point. The threshold is given in closed form for every screening rate and reduces to the known unscreened value when $\alpha=0$. Our proofs overcome new technical changes due to the lack of homogeneity in the Yukawa kernel.
- [19] arXiv:2608.19345 [pdf, html, other]
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Title: Counting thresholds for perfect matchings in hypergraphsComments: 18 pagesSubjects: Combinatorics (math.CO)
In a $k$-uniform hypergraph, the minimum $d$-degree for some $0\le d\le k-1$ is the minimum number of edges containing any given $d$-set of vertices. An extension of the classical Dirac theorem guarantees that whenever the minimum $d$-degree of a $k$-uniform $n$-vertex hypergraph, $k\mid n$, is larger than a certain Dirac threshold, it contains at least one perfect matching. Moreover, it has been known for some time, due to Kwan, Safavi, and Wang, that for $d\ge k/2$ such hypergraphs contain not only one, but ``many'' perfect matchings, that is, at least as many as are expected in a random hypergraph with the same edge density. However, it has also been known that such a result could not be hoped for in general, as it already fails for $(d,k)=(1,3)$.
In this paper we introduce new notions of the \emph{counting thresholds} and \emph{approximate counting thresholds}, above which a hypergraph is guaranteed to have at least this many perfect matchings. We show that these thresholds are well-defined and nontrivial for all $d,k,n$, that they are asymptotically related, and finally, we derive improved upper bounds by reducing to cases with smaller $d$ and $k$. - [20] arXiv:2608.19349 [pdf, html, other]
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Title: Connective constants of Grigorchuk graphsSubjects: Combinatorics (math.CO); Mathematical Physics (math-ph); Group Theory (math.GR)
The connective constant $\mu(G)$ of a graph $G$ is the exponential growth rate of the number of self-avoiding walks starting at a given vertex. We prove upper and lower bounds for the connective constants of Cayley graphs $G_\omega$ of a general Grigorchuk group encoded by a sequence $\omega\in\{0,1,2\}^{\Bbb N}$. In particular, $\mu(G_\omega) > \phi$ for any such Cayley graph (subject to a simple condition on $\omega$), where $\phi:= \frac12(1+\sqrt 5)$ is the golden mean. This extends earlier work of the author and Zhongyang Li in "Cubic graphs and the golden mean'', Discrete Math. 343 (2020), article 111638, where it was conjectured that $\mu(G)\ge\phi$ for all infinite, vertex-transitive, cubic graphs. The current work includes an analysis of the proportions of appearances of given label-sequences in the orbital Schreier graphs of general Grigorchuk groups.
- [21] arXiv:2608.19354 [pdf, html, other]
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Title: On the Finiteness of Isolated $j$-invariants for $X_1(N)$Comments: 15 pages. Comments welcome!Subjects: Number Theory (math.NT)
Characterizing isolated points on the modular curve $X_1(N)$ is a key obstruction to classifying all points of a fixed degree. These points do not lie in infinite parameterized families, making them difficult to obtain through geometric constructions. In this paper, we focus on the collection of "isolated $j$-invariants" for $X_1(N)$, which are the values obtained by mapping isolated points to the $j$-line. Prior work of the author in collaboration with Ejder, Liu, Odumodu, and Viray asks whether there are only finitely many isolated $j$-invariants lying in extensions of bounded degree. Here, we explore how this question relates to other uniformity problems in the field and give new finiteness results for isolated $j$-invariants in $\mathbb{Q}$. As an application, we show similar methods give sharpened polynomial bounds on torsion for non-CM elliptic curves having rational $j$-invariant.
- [22] arXiv:2608.19356 [pdf, html, other]
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Title: Topological Effects on Bubbling in the Critical Dirichlet Problem on Hyperbolic Domains: $3\leq N \leq5$Comments: 102 pages, comments are welcomeSubjects: Analysis of PDEs (math.AP)
Let \(\Omega\Subset\mathbb H^N\), \(N\in\{3,4,5\}\), be a bounded connected \(C^2\) domain. We prove that the pure critical Dirichlet problem
\[
-\Delta_{\mathbb H}u=u^{\frac{N+2}{N-2}}
\quad\text{in }\Omega,
\qquad
u=0
\quad\text{on }\partial\Omega
\] admits a positive solution whenever \(H_d(\Omega;\mathbb F_2)\neq0\) for some \(1\le d\le N-1\). This gives a hyperbolic Bahri-Coron theorem for \(3\le N\le5\) under \(C^2\) boundary regularity. Under conformal reduction, the hyperbolic geometry produces a positive potential and leads to dimension-dependent bubbling mechanisms. In dimension three, the required energy drop follows from the balance between diagonal corrections and pair interactions at fixed large multiplicity. In dimensions four and five, it is obtained at the matched scale through a normalized defect estimate and an all-pairs source-transfer bound. A unified Thom-barycenter construction converts these analytic estimates into the topological contradiction. Thus nontrivial domain topology forces existence despite critical loss of compactness and the additional geometric potential. - [23] arXiv:2608.19358 [pdf, html, other]
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Title: An optimal Poincaré inequality for the complex Ginibre log-gasComments: 12 pages, 2 graphicsSubjects: Probability (math.PR); Mathematical Physics (math-ph); Functional Analysis (math.FA)
We establish an optimal Poincaré inequality for real-valued symmetric observables of the complex Ginibre log-gas. Equality is attained by the real and imaginary parts of the center-of-mass observable. Equivalently, we determine the exact spectral gap of the associated overdamped Langevin dynamics, for real symmetric observables. The Hessian of the energy of this log-gas is unbounded below, so standard convexity arguments do not directly apply. Our proof instead combines a Vandermonde transform, a holomorphic projection, and a complex Gaussian d-bar spectral-gap estimate, corresponding to the constant-curvature case of the Hörmander-Berndtsson estimate.
- [24] arXiv:2608.19360 [pdf, html, other]
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Title: Hodge Coercivity and Global Dynamics in Two-Field Edge-Cochain Systems with MHD-Type CancellationSubjects: Dynamical Systems (math.DS); Social and Information Networks (cs.SI)
A finite-dimensional two-field system for divergence-free edge cochains is introduced. Its MHD-type designation refers only to a quadratic exchange pattern and exact total-energy cancellation; it is not a physical MHD discretization. A general cancellation class is separated from a corrected explicit realization: the anticommutator $D(a)J+JD(a)$ is skew-symmetric for diagonal $D(a)$ and skew-symmetric $J$, and its projected bilinear map has the required trilinear antisymmetry. The central result is a Hodge coercivity criterion: the full divergence-free space admits the Poincaré-type estimate needed for dissipativity if and only if its harmonic $1$-cochain space is trivial. Under this condition, global existence, an exact energy identity, an absorbing ball, and a compact global attractor follow. When harmonic modes are present, a harmonic-decoupled interaction class yields invariant harmonic affine fibres and fibre-wise attractors. Deterministic disk, annular, and two-hole examples illustrate the spectral criterion, energy law, and distinction between general harmonic exchange and harmonic-fibre invariance.
- [25] arXiv:2608.19368 [pdf, html, other]
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Title: A High-Order Rank-Adaptive Implicit Algorithm for Solving High Dimensional Diffusion Equations using the Hierarchical Tucker DecompositionComments: This work was done while PBP was an undergraduate at Swarthmore College. Project Advisor: Dr. Joseph Nakao (Swarthmore College)Subjects: Numerical Analysis (math.NA)
This paper presents a high-order rank-adaptive implicit integrator for the tensor solution of high-dimensional diffusion equations. We extend the 3D version of this method from the Tucker decomposition to higher dimensions using the hierarchical Tucker (HT) decomposition, since the storage complexity for the Tucker decomposition increases exponentially with the number of dimensions $d>3$. The HT format avoids this issue by decomposing the solution according to a binary tree consisting of bases for each dimension and core tensors which connect the bases. Spectral methods are considered for spatial discretization, and diagonally implicit Runge-Kutta methods are considered for time discretization. At each stage of the Runge-Kutta method, the bases computed at the previous stages are augmented to predict the upcoming basis and construct projection subspaces. By projecting onto these enriched subspaces, the bases and cores can be updated in a sequential manner going up the tree from leaf-to-root. Unlike the 3D Tucker method which has a single core tensor, the HT method also updates the intermediate core tensors. Numerical experiments demonstrate that the method observes high-order accuracy, and test how well the integrator captures the solution rank for various sets of time-dependent diffusion coefficients.
- [26] arXiv:2608.19374 [pdf, html, other]
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Title: Constructing solvable groups whose character degree graphs generalize the bowtieComments: 10 pages, 10 figures, comments welcomeSubjects: Group Theory (math.GR); Combinatorics (math.CO)
We present here a generalized construction of a finite solvable group whose prime character degree graph has the shape and structure of the bowtie graph. As with the original bowtie, the graphs obtained by this generalized construction, under certain restrictions, cannot be realized by the usual method of taking direct products of smaller graphs. Within the condition of $n=1$, we show how this recovers the original bowtie graph, which has five vertices. We also provide examples and explicit choices of primes which generate graphs with more vertices.
- [27] arXiv:2608.19384 [pdf, html, other]
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Title: A high-order, meshless, Lagrangian--Eulerian RBF-FD method for advection--diffusion--reaction on moving manifoldsComments: 27 pages + 12 page appendix, 21 figuresSubjects: Numerical Analysis (math.NA)
We present a high-order radial basis function-generated finite difference (RBF-FD) method for partial differential equations on moving manifolds $\mathcal M(t)\subset\mathbb R^3$ of co-dimension one. Our method builds on the tangent-plane formulation of surface RBF-FD and combines Lagrangian and Eulerian treatments: the manifold and material derivative are evolved in a Lagrangian fashion, while the remaining surface differential operators are reconstructed on the instantaneous point cloud and stabilized, when necessary, by a quasi-analytical hyperviscosity formulation. Lagrangian marker drift is handled by adaptive rearrangement using a compact global parametric model of the moving surface, which also supplies accurate normals and geometry-based quadrature. After rearrangement, we reconstruct the multistep history by backward semi-Lagrangian tracing and interpolation before resuming the Lagrangian time discretization. We exploit temporal coherence through three update strategies: defect correction for the local RBF-FD weights, a curvature-based update of the hyperviscosity coefficients between spectral recomputations, and a global defect-correction iteration that reuses an incomplete LU (ILU) factorization before preconditioned generalized minimal residual (GMRES) iterations. Finally, we enforce the prescribed global mass balance through a scalar projection based on the evolving surface quadrature. Numerical experiments demonstrate high-order convergence, conservation to roundoff in source-free problems, stable long-time integration, and substantial savings from the proposed update strategies.
- [28] arXiv:2608.19386 [pdf, html, other]
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Title: Weyl forms in parabolic contact geometriesComments: 31 pages, 1 figureSubjects: Differential Geometry (math.DG)
In a parabolic contact Cartan geometry a choice of contact form determines a Weyl form, an important piece of data for understanding the space from the perspective of Cartan geometry. The Weyl form decomposes into three components: a soldering form, Weyl connection and Rho tensor. An explicit description of the soldering form is well-known. We express the Rho tensor in terms of the Weyl connection in all torsion-free normal parabolic contact geometries. This includes CR, integrable Legendrian contact, and contact projective (vanishing contact torsion) geometries in addition to all flat geometries of contact type. Parabolic contact geometries with a specified contact form have a second canonical connection, the Tanaka--Webster connection, which is easier to define explicitly than the Weyl connection. In [7] \v Cap and Slovák give a simple recipe for expressing the Weyl connection in terms of the Tanaka--Webster connection. We apply this recipe to every parabolic contact geometry, giving a formula for the Weyl connection in terms of the Tanaka--Webster connection in each case. As an application of some of these results we derive formulas for the harmonic curvature components in integrable Legendrian contact geometry, which provide a complete obstruction to local isomorphism to the model space $\mathrm{Flag}_{1,n+1}(\R^{n+2})$, recovering a result in [16].
- [29] arXiv:2608.19393 [pdf, other]
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Title: Outer Contact BilliardsComments: 30 pages, 13 figuresSubjects: Dynamical Systems (math.DS); Symplectic Geometry (math.SG)
We introduce outer contact billiards as an odd dimensional counterpart to outer symplectic billiards. Until now, outer billiards have only been considered in even dimensional symplectic vector spaces. By projectivizing outer symplectic billiards, we obtain outer contact billiards, where the affine midpoint condition descends to its projective analog, namely harmonic conjugation.
We prove that outer contact billiards generate contactomorphisms. For quadratic surfaces in $\mathbb{RP}^3$, we show that the correspondence is completely integrable: its domain is foliated by invariant quadrics and, on each leaf, the dynamics is determined by the iteration of an explicit linear transformation. We also establish two rigidity results for periodic trajectories: outer contact billiards admit no 3-periodic orbits and, among quadratic tables, only one admits 4-periodic orbits. - [30] arXiv:2608.19396 [pdf, html, other]
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Title: Penalisation of Two-Dimensional Brownian MotionSubjects: Probability (math.PR)
We study a penalisation problem for two-dimensional Brownian motion. Starting from the Wiener measure, we consider a family of probability measures obtained by weighting paths by a nonnegative functional $F_t$ depending on $t \geq 0$, $F_t$ being measurable with respect to the $\sigma$-algebra generated by the path up to time $t$. Under suitable assumptions on the penalisation process, we establish the weak convergence of these measures when $t \rightarrow \infty$. The limiting law is identified explicitly in terms of a $\sigma-$finite measure $\mathbf{W}^{(2)}$, which admits a path decomposition involving the last hitting time of a circle. This decomposition plays a central role in the analysis and yields a martingale representation of the limiting measure. where ordering and local time techniques are no longer available. The proofs rely on Laplace transform methods and Tauberian theorems, which replace excursion-theoretic tools and allow a precise identification of the limiting measure and its structural properties.
- [31] arXiv:2608.19399 [pdf, html, other]
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Title: On the holonomy of Lie algebroidsSubjects: Differential Geometry (math.DG)
We introduce a holonomy groupoid for Lie algebroids. This construction generalizes both the holonomy groupoid of a foliation and the adjoint representation of a Lie algebra. We prove that the resulting groupoid is longitudinally smooth and compute its Lie algebroid. When the algebroid of the holonomy groupoid coincides with the given Lie algebroid, the holonomy groupoid is the canonical terminal integration: every source-connected integration admits a unique morphism into it inducing the identity infinitesimally. To construct the groupoid structure on the holonomy groupoid, we utilize the notion of the "flow product" of time-dependent sections of a Lie algebroid. This provides an alternative way to define the groupoid structure for the Weinstein groupoid.
- [32] arXiv:2608.19410 [pdf, html, other]
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Title: Induced Subgraphs of Order Seven and Their Frequencies in $srg(n,k,1,2)$Subjects: Combinatorics (math.CO)
In this paper, we examine the structure of strongly regular graphs with parameters $\lambda = 1$ and $\mu = 2$. In particular, we provide a complete classification of induced subgraphs of order seven and determine their relative frequencies. These findings contribute to a finer understanding of the local structure of such graphs and may be useful in related combinatorial and algebraic investigations.
- [33] arXiv:2608.19411 [pdf, html, other]
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Title: High-corank torsion in homotopy of unitary groups via topological modular forms and higher real $K$-theoriesComments: 13 pages, comments welcome!Subjects: Algebraic Topology (math.AT)
Work of Toda identifies groups of metastable vector bundles on even-dimensional spheres with stable homotopy groups of certain stunted projective spectra. Using Weiss' unitary calculus, the second-named author generalized this identification to show that metastable, stably trivial vector bundles on even cell complexes can naturally be identified with stable homotopy classes of maps into a shifted stunted projective spectrum. Thus, certain classical questions about vector bundles (or homotopy of unitary groups) can be rephrased as stable computations. In this note, we show that certain generalized cohomology theories arising in chromatic and equivariant homotopy theory can be used to deduce the existence of non-trivial, stably trivial vector bundles on spheres and complex projective spaces.
- [34] arXiv:2608.19414 [pdf, html, other]
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Title: The Non-Cancelling-Intersections Conjecture Fails for Left-Linear TreesSubjects: Combinatorics (math.CO); Discrete Mathematics (cs.DM)
First formulated by Amarilli, Monet, and Suciu (arXiv:2401.16210, 2024), the Non-Cancelling Intersections (NCI) conjecture is an open problem in combinatorics stating that any set union can be constructively built from its algebraically non-cancelling intersections using only disjoint unions and subset complements. In the same paper, two orthogonal possible strengthenings are proposed: using only left-linear trees, and using non-trivial intersections only positively or only negatively depending on the sign of their Möbius value. Here we show that using only left-linear trees, the conjecture is false (independent of the other strengthening). Our argument is non-constructive. We prove the existence of a counterexample, though it is of immense size.
- [35] arXiv:2608.19424 [pdf, html, other]
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Title: A rock-paper-scissors Mandelbrot setSubjects: Dynamical Systems (math.DS); Mathematical Software (cs.MS); Complex Variables (math.CV); Rings and Algebras (math.RA)
The titular object of this paper is an analogue of the Mandelbrot set over the 3-dimensional real algebra whose multiplication is the bilinear extension of the rock-paper-scissors operation. In order to study the dynamics of the mappings $x\mapsto x^2+c$ in this setting, a notion of holomorphy for functions on general finite-dimensional real algebras is introduced. Under a mild assumption it is shown that for such algebras holomorphy is always equivalent to solving a finite system of linear first-order PDEs generalizing the Cauchy-Riemann equations. Code is provided for generating animations of these fractals and for exploring them in a video game format.
- [36] arXiv:2608.19426 [pdf, html, other]
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Title: Implicit-adjoint finite-volume topology optimization of two-dimensional conjugate heat transferSubjects: Numerical Analysis (math.NA)
Designing compact, high-efficiency thermal architectures requires resolving the competing demands of solid conduction, fluid convection, and flow resistance within highly constrained physical envelopes. Density-based topology optimization provides a systematic framework for synthesizing these coupled layouts, yet the reproducibility and numerical stability of the resulting designs depend critically on the underlying discrete solvers and adjoint sensitivity mechanics. In this work, we present a transparent, self-contained two-dimensional finite-volume formulation on a staggered Marker-and-Cell grid for conjugate heat transfer governed by design-dependent energy transport coupled to Stokes--Brinkman or Darcy flow at fixed solid volume. To prevent spurious artificial thermal sources in porous, weakly compressible Brinkman domains, the discrete advection operator is constructed to satisfy the identity $\mathbf{u}\cdot\nabla T=\nabla\cdot(\mathbf{u}T)-T(\nabla\cdot\mathbf{u})$ cellwise, ensuring that uniform temperature fields remain exact discrete nullspaces even under inexact continuity satisfaction. Reverse-mode derivatives are evaluated via the implicit function theorem rather than unrolled iterative loops, yielding exact discrete adjoints with bounded memory requirements. The discrete operators are systematically validated through the method of manufactured solutions and directional Taylor remainder tests. Four representative thermofluid design benchmarks are optimized using a projected-gradient scheme with $\beta$-continuation, wherein candidate iterates are accepted and published only upon satisfying rigorous, predeclared gates on residual convergence, mass conservation, volume feasibility, and numerical finiteness. The resulting formulation provides an inspectable, deterministic reference stack for verifiable conjugate thermofluidic topology optimization.
- [37] arXiv:2608.19434 [pdf, html, other]
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Title: Heat--Airy Transport of Differential Operators and Moving BoundariesSubjects: Analysis of PDEs (math.AP)
We study the transport of polynomial differential operators under the two-parameter commuting evolution \[ P_{t,s} = \exp\left( \frac{t}{2}D^2-\frac{s}{3}D^3 \right), \qquad D=\frac{d}{dx}. \] Conjugation of the position operator gives \[ P_{t,s}xP_{t,s}^{-1} = x+tD-sD^2, \] which leads to a recursive normal-ordering expansion and to a family of Heat--Airy polynomials arising as its derivative-free coefficients.
Our main purpose is to study the interaction of this transport with moving absorbing boundaries. For functions satisfying \[ v_t=\frac12v_{xx}, \qquad v_s=-\frac13v_{xxx}, \qquad v(t,s,f(t,s))=0, \] the boundary condition generates a hierarchy of relations among the spatial jets of \(v\). Combined with the restrictions of transported differential equations and their spatial derivatives, these identities produce compatibility equations for \(f\).
We prove a general restriction principle showing that, at every spatial-jet level, the compatibility equations of the classical Heat problem are obtained by restricting the corresponding Heat--Airy hierarchy to the slice \(s=0\). For second-order operators, this framework yields a finite characteristic system and nonlinear boundary equations, together with explicit solvable examples. For a third-order linear-potential operator, the additional Airy-flow identity permits a stronger sequential recovery: the first Heat--Airy compatibility equation determines \(f_s(t,0)\), and the next spatial-jet equation then reproduces the nonlinear compatibility condition obtained in the pure Heat theory from a \(4\times4\) characteristic determinant. Thus the additional commuting flow reorganizes the moving-boundary compatibility problem into a hierarchy while retaining the classical Heat theory as a distinguished restriction. - [38] arXiv:2608.19435 [pdf, html, other]
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Title: A New Impossibility Region for the $5\times5$ Symmetric Nonnegative Inverse Eigenvalue ProblemComments: 45 pages, 1 figureSubjects: Rings and Algebras (math.RA); Spectral Theory (math.SP)
We present a new impossibility region for the $5\times5$ symmetric nonnegative inverse eigenvalue problem. The region lies in the low-trace regime and, to the best of our knowledge, has not been identified previously. The proof uses a suitable diagonal shift to transform the problem to a critical high-trace boundary and then reduces a complementary commuting matrix to a weighted five-cycle. The characteristic polynomial and spectral identities of this five-cycle provide the main tools for deriving the resulting contradiction.
- [39] arXiv:2608.19441 [pdf, html, other]
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Title: Pearl necklace knots with fewer vertices than an equivalent FCC lattice knotComments: 4 pages, 2 figures, tables in appendix, four ancillary filesSubjects: Geometric Topology (math.GT)
The pearl necklace number, $N_P$, of a knot is the smallest number of unit spheres required to construct a knot if each sphere is tangent to two neighbors and no sphere overlaps with another. It has been speculated that the pearl necklace number is equal to the minimum number of lattice sites required to embed a knot on a face centered cubic (FCC) lattice $N_L$, implying that a trefoil knot cannot be constructed from fewer than 15 spheres. A large language model (LLM) was prompted to find configurations of knot for which $N_P<N_L$, and this manuscript describes its findings, attempts at validating them, and their implications. No 14-vertex example was found for the trefoil knot, but examples with $N_P=N_L-1$ were found for all knots from 5 to 7 crossings as well as $8_{19}$, and $10_{124}$. One example, $8_1$, could be constructed with $N_P=N_L-2$.
- [40] arXiv:2608.19451 [pdf, html, other]
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Title: Factorization and Atomic Decomposition in Hardy-Orlicz Spaces on the Upper Half-Plane with Applications to Hankel OperatorsSubjects: Classical Analysis and ODEs (math.CA); Functional Analysis (math.FA)
In this work, we establish a strong factorization for Hardy-Orlicz spaces on the upper half-plane. We show that the product of two functions belonging respectively to Hardy-Orlicz spaces $H^{\Phi_{1}}$ and $H^{\Phi_{2}}$ lies in a third space $H^{\Phi_{3}}$, and every holomorphic function in $H^{\Phi_{3}}$ admits such a decomposition. We then provide an atomic decomposition for certain Hardy-Orlicz spaces, which allows us to describe the topological dual of these spaces when the associated function is concave. Finally, these results are applied to the study of the continuity of the Hankel operators in this setting.
- [41] arXiv:2608.19452 [pdf, html, other]
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Title: A problem for isolated singularities of surfacesComments: 2026 Preprint 1.7 (441), Dipartimento di Matematica, Sezione di Geometria e Algebra, Universita di Pisa, April 1989Subjects: Differential Geometry (math.DG); Complex Variables (math.CV)
We study the extension of the conformal structure of a Riemann surface, obtained as a submanifold of $\mathbf R^n$, across an isolated singular point, under hypotheses that are metric rather than analytic. The main result (1989) is that an isolated singularity is $\textbf{conformally point-like}$ (conformal to a punctured disc) whenever it is $\textbf{$M$-regular}$: the pair (surface minus the point, the point) satisfies a Whitney condition, and the length of the spherical slice $S(0,r)\cap E$ decreases at most linearly in $r$, a condition described as ``metric decay''. This is proved via a modulus-of-rings (extremal-length) argument, generalizing the classical planar technique to submanifolds of $\mathbf R^n$. Two classes of examples are treated: surfaces of revolution generated by a single curve, and subanalytic surfaces, for which $M$-regularity is verified directly from Hironaka's structure theory, giving as a corollary that every isolated singularity of an orientable subanalytic surface is conformally point-like. A further original result is a $C^\infty$ counterexample showing that a strict Whitney condition alone does $\textbf{not}$ imply the linear length bound: the two hypotheses in the definition of $M$-regularity are independent.
- [42] arXiv:2608.19455 [pdf, html, other]
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Title: Connected Counterexamples to the Henning--Yeo Conjecture on Identifying Vertex CoversComments: 12 pages, 2 figures. Submitted to The Electronic Journal of CombinatoricsSubjects: Combinatorics (math.CO)
Henning and Yeo conjectured an upper bound on the identifying vertex cover number of a graph in terms of its order, size, and maximum degree. We disprove the conjectured inequality with a two-parameter family $H_{t,r}$ of connected diameter-two graphs. After clearing denominators, the right-hand side minus the left-hand side is exactly $-(t-1)(r-1)$; hence a connected counterexample exists for every maximum degree at least four. Chaining copies through low-degree vertices preserves the maximum degree and allows the packing number to be determined exactly. At maximum degree five, this gives counterexamples of arbitrarily large order with additive gap $1/13$. For every fixed maximum degree $\Delta\ge6$, suitable chains have unbounded additive violation. Thus neither rounding nor a fixed additive correction repairs the conjecture. The supremal normalized additive gap at maximum degree $\Delta$ is $\Theta(1/\Delta)$. An exhaustive check of all graphs of order at most seven shows that the eight-vertex example $H_{2,2}$ has minimum possible order.
- [43] arXiv:2608.19457 [pdf, html, other]
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Title: Equidistribution and thermodynamics at infinityComments: 45 pages, no figuresSubjects: Dynamical Systems (math.DS)
We prove level-2 large deviation upper bounds for potentials on countable Markov shifts and for suspension semi-flows over countable Markov shifts. For strongly positive recurrent potentials, we establish equidistribution of weighted empirical measures toward the corresponding equilibrium state. We then apply these results to interval maps, obtaining, in particular, equidistribution and large-deviation estimates for measures supported on boundary points. For the Gauss map, this yields equidistribution results on rational numbers, including a theorem of David and Shapira.
- [44] arXiv:2608.19461 [pdf, html, other]
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Title: A minimality criterion for surfaces of general type with applications to product-quotient surfacesSubjects: Algebraic Geometry (math.AG)
We present a numerical criterion to establish the minimality of surfaces of general type. This approach is then adapted to the case of product-quotient surfaces, where the criterion determines whether the (-1)-curves are constrained to lie inside a fibre of one of the two natural fibrations.
- [45] arXiv:2608.19466 [pdf, html, other]
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Title: Classification of Collisions of Twisted Foulkes Character PolynomialsComments: 43 pagesSubjects: Combinatorics (math.CO)
The twisted Foulkes character polynomial is an algebraically defined polynomial attached to an integer partition. We determine precisely how much combinatorial information this polynomial encodes by completely classifying all pairs of partitions that give rise to the same polynomial. Our main result shows that equality of twisted Foulkes character polynomials admits a purely combinatorial characterization in terms of two explicit local operations on partitions.
- [46] arXiv:2608.19467 [pdf, html, other]
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Title: The minimum of the graph likelihoodComments: 19 pagesSubjects: Combinatorics (math.CO)
The likelihood of a finite simple undirected graph $G$ on $n$ vertices is the probability that the uniform sequential attachment process, which at each step joins a new vertex to a uniformly random subset of uniformly random size of the vertices already present, outputs a graph isomorphic to $G$. Dervovic, Mocherla and Severini conjectured that the likelihood is minimised by the balanced complete bipartite graph. We prove that, among complete bipartite graphs of a given order, the balanced one uniquely minimises the likelihood. Exact computation shows that it also minimises over all graphs for every order from $6$ through $14$, and that the first counterexample occurs at $n=15$. The blow-up of the five cycle by independent sets of size three, equivalently the circulant on fifteen vertices with connection set $\{1,4,6\}$, has likelihood $0.20128\ldots$ times that of $K_{7,8}$, and it is again triangle-free. We show that the failure is not sporadic by proving that the likelihood of the balanced complete bipartite graph is $2^{-(1/2-1/(8\ln 2)+o(1))n^2}$, whereas the minimum over all graphs of order $n$ is $2^{-(1/2+o(1))n^2}$, so the conjectured minimiser exceeds the minimum by a factor exponential in $n^2$. We also determine the Shannon entropy of the process to leading order, namely $n^2/(4\ln 2)$ bits, which shows that the conjectured minimiser is in fact more likely than a typical output of the process. The proofs rest on a vertex deletion recurrence which evaluates the likelihood in time $O(n\,2^n)$ and which closes on the blow-ups of any fixed base graph.
- [47] arXiv:2608.19468 [pdf, html, other]
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Title: Large Finite Point Sets Have 4 Collinear Points or a 6-CliqueComments: 7 pagesSubjects: Combinatorics (math.CO); Discrete Mathematics (cs.DM)
We prove that every finite point set of size at least $10^{11055931}$ has four collinear points or six points that pairwise see each other. This resolves the first open case of the big-line-big-clique conjecture of Kára, Pór, and Wood.
- [48] arXiv:2608.19470 [pdf, html, other]
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Title: Polar and torus manifoldsSubjects: Differential Geometry (math.DG)
We show that quasitoric manifolds can be equipped with an invariant metric for which the torus action is polar. More generally, we show that an infinitesimally polar action of a torus admits an invariant Riemannian metric making it polar if and only if the real orbifold Euler class of its quotient vanishes. Because the orbit spaces of quasitoric manifolds are contractible simple polytopes, this cohomological obstruction trivially vanishes.
- [49] arXiv:2608.19478 [pdf, html, other]
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Title: Products of nonconjugate maximal subgroupsSubjects: Group Theory (math.GR)
We prove that a finite group $G$ is solvable whenever $MN=G$ for every pair of nonconjugate maximal subgroups $M,N<G$. Equivalently, every finite nonsolvable group has two nonconjugate maximal subgroups whose setwise product is proper. This gives a negative answer to Problem 10.34 of the Kourovka Notebook. As an application, we also answer an open question raised by Guo.
- [50] arXiv:2608.19479 [pdf, html, other]
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Title: Solutions to a One-Dimensional Combustion-Type Free Boundary Problem via Maximal RegularityComments: 26 pagesSubjects: Analysis of PDEs (math.AP)
We study a one-dimensional free boundary problem arising in combustion theory, where the motion of the interface is governed by a prescribed Neumann boundary flux and a zero Dirichlet boundary condition. We treat both the half-line case and the bounded interval case. For both settings, we employ maximal $L^p$-$L^q$ regularity as our main analytical tool. In the half-line case, the solutions need not decay at infinity, even though the spatial derivatives belong to $L^q(\mathbb{R}_+)$. To handle the evolution law of the free boundary, we introduce a derivative formulation that avoids second-order boundary traces. By combining maximal $L^p$-$L^q$ regularity and Schauder estimates, we establish the local-in-time existence, uniqueness, and regularity of solutions, as well as the evolution law of the free boundary.
- [51] arXiv:2608.19481 [pdf, html, other]
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Title: Geometrically multiplicative non-archimedean normsComments: 15 pages, first version, comments are welcomeSubjects: Algebraic Geometry (math.AG)
Universally (or geometrically) multiplicative norms on Banach algebras over complete non-archimedean fields were used by Berkovich in his works on non-archimedean geometry, and later they were studied in some detail by Poineau. In this paper, we perform a more thorough study of the question when multiplicativity, spectrality and spectral multiplicativity of norms on algebras over real valued fields are preserved by ground field extensions. We obtain precise criteria quite analogous to the classical theory of geometric irreducibility and reducedness. In particular, we generalize the results of Poineau in a few aspects.
- [52] arXiv:2608.19483 [pdf, html, other]
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Title: Global Well-Posedness near Rayleigh-Jeans Equilibria for the Cubic NLS Wave Kinetic EquationComments: 45 pagesSubjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph)
We study the dynamics of the kinetic wave equation associated to the three dimensional Schrödinger equation close to Rayleigh-Jeans equilibria. We first prove that the linearised operator generates a semigroup of contractions in $L^2((0,\infty);\sqrt \omega \dd\omega )$. Considering the family of nonsingular Rayleigh-Jeans spectra, we prove that the linearised operator possesses a spectral gap, despite the non-compactness of the integral collisional operator, and thus obtain an exponential relaxation for the linear semigroup. We then prove bilinear and trilinear estimates in the relevant norm for the nonlinear terms and deduce global well-posedness and exponential relaxation for sufficiently small relative perturbations of Rayleigh-Jeans equilibria. To our knowledge, this is the first global strong well-posedness and asymptotic stability result near a nonzero thermodynamic equilibrium for the full spatially homogeneous four-wave kinetic equation associated with the cubic nonlinear Schrödinger equation.
- [53] arXiv:2608.19484 [pdf, html, other]
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Title: Deciding if a shadow resolves into a given link: linear-time algorithmsSubjects: Geometric Topology (math.GT)
A {\em shadow} (or {\em projection}) is obtained from a link diagram by ignoring the over/under information at each crossing. Given a fixed link $L$ we investigate the complexity of deciding whether an input shadow $S$ can be {\em resolved} into $L$, that is, whether we can assign over/under information to its crossings to obtain a diagram of a link isotopic to $L$. We show that if $L\in\{3_1,4_1,5_1,5_2,6_2,L2a1, L4a1, L5a1, L6n1\}$ then there exists a linear-time algorithm that decides whether an input shadow $S$ resolves into $L$.
- [54] arXiv:2608.19486 [pdf, html, other]
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Title: Beyond linear subspaces: Nonlinear moment matching meets quadratic manifoldsSubjects: Optimization and Control (math.OC); Numerical Analysis (math.NA)
Quadratic manifold-based model order reduction offers a viable pathway to circumvent the limitations of linear subspaces for linear control systems characterized by slow Kolmogorov $n$-width decay. However, a system-theoretic framework for constructing such quadratic approximations remains absent from the literature. This paper presents a system-agnostic, optimization-free framework for the direct construction of quadratic projection matrices. We prove that the synthesized reduced-order model matches the nonlinear moments of the full-order system and preserves its exact center manifold mapping, thereby ensuring asymptotic tracking of steady-state outputs under specific input classes. Numerical results on transport-dominated benchmark problems, namely, the one-dimensional damped wave and advection equations, show that the proposed framework achieves high-fidelity trajectory reconstruction within a significantly reduced-dimensional state space, yielding substantial online computational savings.
- [55] arXiv:2608.19496 [pdf, html, other]
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Title: A Method for Preserving Geometric Meaning in the Calculus of VariationsSubjects: Differential Geometry (math.DG)
We show that the Calculus of Moving Surfaces offers distinct advantages over the Euler-Lagrange equation for optimization problems originating in Geometry. An extension of Tensor Calculus, it provides tools for analyzing invariant representations rather than specific coordinate representations of relevant quantities, which allows us to avoid the myriad of hardships associated with the use of coordinates from untenable complexity of analytical expressions to virtual impossibility of recovering the geometric interpretation of the final result. As an illustration, we solve the brachistochrone problem and give a geometric characterization for the desired curve in terms of its curvature.
- [56] arXiv:2608.19499 [pdf, html, other]
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Title: Resolving the generalized hyperbolicity conjecture for shadowingComments: 16 pagesSubjects: Functional Analysis (math.FA); Dynamical Systems (math.DS)
It is known that generalized hyperbolicity implies the shadowing property for invertible bounded linear operators on a Banach space. Whether the converse holds has been a central open problem in linear dynamics and has been conjectured to have a positive answer. We show that this conjecture fails on general Banach spaces by constructing a counterexample, whereas it holds on separable Hilbert spaces. The distinction is explained by the gap that may occur on Banach spaces between surjectivity and right invertibility, a gap that disappears on Hilbert spaces. The main ingredients of the proofs are a recent spectral characterization of shadowing in terms of the surjective spectrum and a new characterization of generalized hyperbolicity in terms of right resolvent functions near the unit circle.
- [57] arXiv:2608.19507 [pdf, html, other]
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Title: On the number of solutions of systems of diagonal equations through diagonal GP-graphs: the general and the Hermitian-form casesComments: 45 pagesSubjects: Combinatorics (math.CO); Number Theory (math.NT)
For any $m, s \in \mathbb{N}$, we study the number $N_{m\times s,q}(\kappa, \beta)$ of solutions $(x_1,\ldots,x_s) \in (\mathbb{F}_q)^s$ of the monic system of diagonal equations
$$ X_{1}^{k_i} + \cdots + X_{s}^{k_i}= \beta_i, \qquad (1\le i \le m), $$ with $\kappa=(k_1,\ldots,k_m) \in \mathbb{N}^m$ and $\beta=(\beta_1,\ldots,\beta_m) \in (\mathbb{F}_q)^m$. We show that this number can be obtained in terms of some data of \textit{diagonal} GP-graphs $\Gamma(\kappa,q)$. This is a new family of graphs that we introduce here, i.e. Cayley graphs of the form
$$ \Gamma(\kappa,q) = Cay(\mathbb{F}_{q}^{m}, R_{\kappa}) \quad \text{where} \quad
R_{\kappa} = \{ (x^{k_1},\ldots,x^{k_m}) : x \in \mathbb{F}_{q}^*\}, $$ with $\kappa=(k_1,\ldots,k_m)\in \mathbb{N}^{m}$. In particular, we give three different expressions for $N_{m\times s,q}(\kappa, \beta)$: one in terms of walks, another in terms of adjacency matrices of $\Gamma(\kappa,q)$ and the last one in terms of the spectrum of $\Gamma(\kappa,q)$. Finally, we explicitly derive combinatorial formulas for the number of solutions $N_{m}(s,q) = N_{m\times s,q}(\kappa_\ell, 0)$ of monic homogeneous systems of diagonal equations of the form $$ X_1^{q^{\ell_i}+1} + \cdots + X_s^{q^{\ell_i}+1} = 0 \qquad (1\le i \le m),$$ with $\kappa_\ell=(\ell_1,\ldots,\ell_m)=(1,3,\ldots,2m-1)$ and $m\ge 2$, via the known spectrum of Hermitian-form graphs, which can be viewed as diagonal GP-graphs. For any $m,s \in \mathbb{N}$, we give general summation and recursive formulas for $N_m(s,q) \in \mathbb{Z}[q]$. For the small cases $N_{1}(s,q)$, $N_{2}(s,q)$ and $N_{m}(s,q)$, with $1\le s \le 5$, we give explicit expressions. - [58] arXiv:2608.19513 [pdf, html, other]
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Title: The Normal Procrustes Problem: A Riemannian Optimization ApproachComments: 41 pages, 10 figures. Code available at this https URLSubjects: Numerical Analysis (math.NA); Optimization and Control (math.OC)
For given $m \times n$ data matrices $X, Y$, we investigate the Normal Procrustes Problem---the least squares optimization problem that aims to minimize $\|AX-Y\|_F^2$, where $A$ is constrained to be a normal $m \times m$ matrix. As far as the author of this article is aware, no other method that attempts to solve the Normal Procrustes Problem exists in the literature; we thus propose what is, to our knowledge, the first such method. We, furthermore, adapt our approach to address the Real Normal Procrustes Problem, where $A$ must be real. In our treatment of these problems, we first reduce our complex and real objective functions to be purely optimizable over the Riemannian manifolds of the unitary and real orthogonal matrices, respectively. This reduction enables us to apply techniques in Riemannian manifold optimization to approximate solutions to both. The Closest Normal Matrix and Real Closest Normal Matrix Problems are both special cases of their respective Procrustes Problems and have been previously studied in the literature. Our approach thus recovers a novel Riemannian optimization method for approximating solutions to both these problems. We further numerically test the performance of our method across all such problems (including against previously developed algorithms on the Closest Normal Matrix Problems) and obtain competitive residuals and favorable scaling in wall-clock time.
- [59] arXiv:2608.19516 [pdf, html, other]
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Title: Higher order logarithms of Bessel operators and an extension problemSubjects: Analysis of PDEs (math.AP)
We consider the Bessel operator defined by \[ B_\lambda =-\frac{d^2}{dx^2}+\frac{\lambda^2-1/4}{x^2}, \] on $(0,\infty)$, with $\lambda>-1$. We study the fractional power $B_\lambda^s$, $s\in (-1,1)$, $s\neq 0$, and the logarithm $\log^kB_\lambda $, $k\in \mathbb N$, of $B_\lambda$. We obtain pointwise representations of these operators and asymptotic Taylor expansions of the operators $B_\lambda^s$ in terms of logarithmic operators $\log^kB_\lambda $. We also obtain $\log B_\lambda$ as the solution of an extension problem.
- [60] arXiv:2608.19517 [pdf, html, other]
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Title: Height and distance-squared functions on implicit plane curvesSubjects: Differential Geometry (math.DG)
We introduce families of Lagrange functions associated with height and distance-squared functions on implicit plane curves. We show that degeneracies of these families characterize geometric properties of the curves, such as inflections and vertices, and give geometric interpretations of the corresponding Lagrange multipliers in terms of contact with lines and circles.
- [61] arXiv:2608.19525 [pdf, html, other]
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Title: Quantitative bounds for sets lacking polynomial progressions with shifted prime differenceComments: 60 pagesSubjects: Number Theory (math.NT); Combinatorics (math.CO)
We prove quantitative polynomial Szemerédi-type theorems involving polynomial progressions with shift parameter restricted to the set of shifted primes $\mathbb{P}-1$. The types of configurations covered are distinct degree progressions and progressions involving integer multiples of a fixed polynomial.
For nonlinear configurations of length at least three, these results provide the first quantitative versions of such theorems. In the linear case, our results improve on work by the last two authors. Our density bounds are strongest in the case of distinct degree polynomials, where they give polylogarithmic bounds, of the same shape as recent bounds by Shao and Wang with integer shifts.
The proofs combine recent quantitative results for polynomial configurations in the integers with quantitative Gowers uniformity bounds of the primes. For multiples of a fixed polynomial, we adapt a comparison argument of Altman and Sawhney to obtain uniformity over the polynomial families produced by the $W$-trick. For distinct degree progressions, we establish a comparison between prime-weighted and unweighted polynomial counts that is uniform throughout the density increment argument and accounts for a possible Siegel zero. - [62] arXiv:2608.19530 [pdf, other]
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Title: On the Value Function of Infinite-Horizon Optimal Control of Piecewise Affine SystemsComments: Submitted to IEEE Transactions on Automatic ControlSubjects: Optimization and Control (math.OC); Systems and Control (eess.SY)
In this paper, we study the structure of the value function in constrained infinite-time optimal control (CITOC) problems of piecewise affine (PWA) systems, with $\ell_1$ or $\ell_\infty$ stage cost. Existing works, such as [1], establish that the resulting value function is PWA in the state. However, existing results do not analyze whether the value function is a proper PWA function, i.e., with a finite number of affine pieces over compact sets, or whether the number of pieces can be infinite. We show that the latter case is indeed possible by means of an explicit example, which is also instrumental in establishing rigorous and easily verifiable sufficient conditions that ensure that the resulting value function is a proper PWA function. Our theoretical findings complement well-known results, e.g., the linear-quadratic case, and serve as support for recent learning-based control schemes for PWA systems. Throughout the paper, the proposed results are illustrated by means of a numerical example.
- [63] arXiv:2608.19531 [pdf, html, other]
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Title: Sharp Bilinear Decoupling under Shell-Type Restrictions and ApplicationsComments: 39 pages, 7 figuresSubjects: Analysis of PDEs (math.AP)
We establish a sharp bilinear decoupling inequality under shell-type Fourier support restrictions. The optimal coefficient exhibits a transition at $N_1=N_2^2$ and is strictly smaller than the corresponding thick-annulus coefficient of Fan--Staffilani--Wang--Wilson [FSWW18]. The proof combines a lossless $L^2$ cap--plate decomposition, a one-direction linear lifting argument, localized shell-type linear decoupling, and a complementary localization of the second factor.
For applications, we also derive a corresponding bilinear Strichartz estimate, toral eigenfunction estimates, mixed additive-energy bounds on lattice spheres, and a separated nonlinear smoothing result for the periodic Zakharov system. - [64] arXiv:2608.19542 [pdf, html, other]
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Title: The number of limit cycles of piecewise linear Liénard systemsSubjects: Dynamical Systems (math.DS)
For the planar Liénard differential system $\dot{x}=F(x)-y$, $\dot{y}=x$, where $F(x)$ is a piecewise linear function, Tonnelier (SIAM J. Appl. Math., 2002) conjectured that the maximum number of limit cycles of the system is $n$ when $F(x)$ has $n$ fold points and no jump points, and $2n$ when $F(x)$ has $n$ jump points and no fold points. This conjecture was confirmed by Llibre et al. (J. Nonlinear Sci., 2015) (resp. Chen et al. (J. London Math. Soc., 2026a)) when $F(x)$ has one fold point and no jump points (resp. two fold points). More recently, Chen et al. (J. London Math. Soc., 2026b) proved that the conjecture is correct when $F(x)$ has no fold points and one jump point. All other cases remain open.
Here we verify that the lower bound for the maximum number of limit cycles of the system can be $n$ when $F(x)$ has only $n$ fold points, and $2n$ when $F(x)$ has only $n$ jump points, thereby confirming the lower bound part of Tonnelier's conjecture. Moreover, when $F(x)$ has $m$ jump points and $n-m$ fold points, $0\le m\le n$, we also show that the system can have $n+m=(n-m)+2m$ limit cycles. In addition, a complete classification of the {dynamics} near infinity for this class of systems is provided. - [65] arXiv:2608.19548 [pdf, html, other]
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Title: The lattice of abstract elementary classes of modulesComments: 30 pagesSubjects: Logic (math.LO); Rings and Algebras (math.RA)
Let $R$ be a ring. We organize the abstract elementary classes whose underlying class is the class of all $R$-modules and whose strong submodel relation lies between the submodule and direct summand relations into a lattice $\mathscr{L}_{R}$, ordered by reverse inclusion. We establish the basic lattice-theoretic properties of $\mathscr{L}_{R}$ and investigate its two natural sublattices, below and above purity. Below purity, we isolate relations defined by first-order pp-formulas for which amalgamation, tameness, and stability hold. Above purity, we introduce relations defined by infinitary pp-formulas and prove a broad stability result. Specializing to abelian groups, we show that the lattice $\mathscr{L}_{\mathbb{Z}}$ has the following properties: it has a strong submodel relation that is not positive syntactic, it contains an uncountable antichain and a strictly increasing proper-class-sized chain, and it has a broad region above purity where amalgamation fails.
- [66] arXiv:2608.19550 [pdf, html, other]
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Title: Weak arcs and applications to the DNA-based storage access problemSubjects: Combinatorics (math.CO); Information Theory (cs.IT)
Weak arcs are point sets in PG$(n-1,q)$ meeting every general hyperplane (those are the hyperplanes not going through one of the points given by the standard basis vectors) in at most $n-1$ points. In this paper, we study weak arcs together with balanced variants which are contained on the sides of the fundamental simplex. We give an upper bound on the size of weak arcs, characterise the largest balanced quasi-arcs in the plane and construct large balanced quasi-arcs in PG$(3, q)$. We then use these configurations to build point sets for the random-access problem in DNA-based storage. The constructions are explicit, work over small fields, and attain recovery expectations matching the best known asymptotic bounds.
- [67] arXiv:2608.19559 [pdf, html, other]
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Title: The minimum number of Dehn $\mathbb Z$-colors of any nonsplittable $\mathbb Z$-colorable link is threeSubjects: Geometric Topology (math.GT); Combinatorics (math.CO)
Our previous papers [8, 9] are the first and second to discuss minimum numbers of ``region'' colors, while minimum numbers of arc colors such as Fox colors are well-studied. As the third installment, in this paper, we investigate the minimum number of Dehn $\mathbb{Z}$-colors. In particular, we show that the minimum number of Dehn $\mathbb{Z}$-colors of a nonsplittable $\mathbb{Z}$-colorable link is three.
- [68] arXiv:2608.19560 [pdf, html, other]
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Title: Computing transient probabilities in Markovian queues with balking conditional on a fixed number of joined customersSubjects: Probability (math.PR)
We analyze the transient behavior of a Markovian queue with balking, conditional on exactly $K$ customers joining the system in a finite time interval $[0, T]$. A key quantity of interest is the cumulative number of balking customers, for which we consider the probability generating function (PGF) and derive an equation it satisfies. This approach circumvents the computational burden arising from dealing with high-dimensional Markovian system, which arises when attempting to directly compute the joint distribution of the cumulative number of balking customers together with the total number of joining customers and the number of customers in the system. To compute state transitions in $(t, T]$ efficiently, we examine the conditional state probability at time $T$ given the system state at time $T - u$, and show that it satisfies a linear differential equation in $u$. For piecewise-constant arrival rates, we develop a numerical procedure for evaluating the moment of the total number of balking customers, jointly with the cumulative number of joining customers and the number of customers in the system, under the condition that $K$ customers joined. We also present numerical examples that highlight counterintuitive behaviors arising from this conditioning, along with explanations of the underlying mechanisms.
- [69] arXiv:2608.19563 [pdf, html, other]
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Title: An exact structural identity and its normal form consequences in a modified Leslie--Gower modelComments: 16 pages, 1 figureSubjects: Dynamical Systems (math.DS)
Zhao and Zhao (2026) established a codimension-four Bogdanov--Takens singularity in a modified Leslie--Gower predator--prey model through a recursive normal form computation. We show that the nonlinear coefficients governing the restriction of the reduced system to the singularity's distinguished eigendirection are not independent. Instead, a single exact rational identity, already present in the transformed equations, generates the entire family of pure Taylor coefficients and proves that their simultaneous vanishing is an exact all orders property rather than a finite-order coincidence. We further show that this structural identity propagates through the planar Bogdanov--Takens normal form algorithm of Kuznetsov (2005). Although the pure coefficients vanish identically, the corresponding normal form coefficients are generated by mixed quadratic interactions. This leads to an explicit characterization of the degenerate Bogdanov--Takens locus and reduces the search for higher-order Bogdanov--Takens degeneracies to a single explicit algebraic condition.
- [70] arXiv:2608.19565 [pdf, html, other]
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Title: A gap theorem for metric solitons and its applicationsComments: 27 pagesSubjects: Differential Geometry (math.DG)
In this paper, we prove a gap theorem for $\mathbb{F}$-limit metric solitons with respect to the asymptotic volume ratio (AVR): if the AVR of a metric soliton is sufficiently close to 1, then the metric soliton is Euclidean; this is a metric-soliton counterpart of Wang-Wang. Our result can be applied to Ricci flows to derive a gap theorem and an $\varepsilon$-regularity theorem: (1) an ancient Ricci flow with a type-I scalar curvature bound and AVR close enough to 1 must be the static Euclidean space, (2) a Ricci flow with locally type-I scalar curvature bound and local volume ratio close enough to 1 must be regular enough locally (in the sense that its curvature radius cannot be too small).
- [71] arXiv:2608.19570 [pdf, html, other]
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Title: Non-equivalence of pro-$p$-Iwahori invariantsComments: 8 pages. Comments welcome!Subjects: Representation Theory (math.RT); Number Theory (math.NT)
Suppose $F$ is a nonarchimedean local field whose residue field is a proper extension of $\mathbb{F}_p$ with $p > 3$. Generalizing results of Ghate--Le--Sheth, we show that any split, connected, reductive group $G$ over $F$ which is not a torus admits a smooth, irreducible, non-admissible mod $p$ representation. We use this to show that the functor of pro-$p$-Iwahori invariants does not induce an equivalence between the category of smooth mod $p$ $G$-representations generated by their pro-$p$-Iwahori invariant vectors and modules over the pro-$p$-Iwahori--Hecke algebra, contrary to what happens for $\textrm{GL}_2(\mathbb{Q}_p)$.
- [72] arXiv:2608.19571 [pdf, html, other]
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Title: Gradient regularity and potential estimates for fractional drift--diffusion equations in the critical and subcritical rangesSubjects: Analysis of PDEs (math.AP)
We establish scale-invariant interior $C^{1,\alpha}$ estimates for bounded viscosity solutions of $(-\Delta)^su+b\cdot\nabla u=f$ for $s\in[1/2,1)$ with locally Hölder $b$ and $f$. The critical case uses Silvestre's parabolic theorem; the subcritical case uses Schauder estimates and interpolation. Applying this viscosity estimate to drifted Green sections, for finite Radon data above the critical order we obtain sharp solution and gradient potentials of orders $2s$ and $2s-1$, together with weak-*--to--strong local $W^{1,1}$ stability; hence the Green-potential SOLA is approximation-independent. For the normalized whole-space kernels, we identify the classical second-order limits as $s\uparrow1$, including the logarithmic kernel in dimension two; at the critical order, the whole-space gradient becomes a zero-order singular integral. For zero-exterior problems with compactly supported drift, we also prove $u/d^s\in C^{s-\varepsilon}(\overline\Omega)$ and identify the obstruction when the drift reaches the boundary.
- [73] arXiv:2608.19572 [pdf, html, other]
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Title: Disuccessors, Gröbner-Shirshov bases and free L-dendriform algebrasComments: 28 pagesSubjects: Rings and Algebras (math.RA)
In this paper, we prove respectively that the disuccessor operations on the associative operad $\as$, the Lie operad $\lie$, and the pre-Lie operad $\prelie$ preserve Gröbner-Shirshov bases. This structural preservation enables the transfer of known bases to more complex operads. As a consequence, we introduce new methods for constructing Gröbner-Shirshov bases for the $\dend$ and $\prelie$ operads, and explicitly construct a Gröbner-Shirshov basis for the free L-dendriform algebra. This provides a conceptual and computationally efficient resolution of Madariaga's problem and offers new insights into the combinatorial structure of L-dendriform algebras.
- [74] arXiv:2608.19576 [pdf, html, other]
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Title: Convex order preservation for graphon mean-field systemsSubjects: Probability (math.PR); Optimization and Control (math.OC)
In this paper, we study the convex order preservation for the graphon mean-field systems on R^d. First, we establish the uniform-in-time Euler approximations for the graphon mean-field systems of interest, which extends previous results of Bayraktar and Wu(SPA, 2022) on graphon particle systems and is necessary for our main theorem. Building on this together with appropriate conditions on the graphon functions, we then proceed to prove the marginal convex order preservation. Second, by employing the weighted $L^2$-norm, we are able to further establish an infinite-horizon trajectory-level convergence result, which is key to the functional convex ordering as in Liu and Pages(AAP, 2023). Finally, we apply our results to the study of value function comparison of graphon mean-field games(MFGs).
- [75] arXiv:2608.19577 [pdf, html, other]
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Title: Quadratic generation of ideals defining nonsigular toric 3-foldsComments: 12 pages, 1 figureSubjects: Algebraic Geometry (math.AG); Combinatorics (math.CO)
Let $X$ be a projective line bundle over a nonsingular toric surface which is a blowup the projective plane along at most 4 invariant points, or a blowup the product of the projective lines along at most 4 invariant points. Let $L$ be an ample line bundle on $X$. Then $L$ defines a projectively normal embedding to big projective space. We show the ideal of this embedded $X$ is generated by elements of degree two.
- [76] arXiv:2608.19586 [pdf, html, other]
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Title: Strict Concavity of the Torsion Function for the Restricted Half-Laplacian in Bounded Convex DomainsSubjects: Analysis of PDEs (math.AP)
Let $D\subset\mathbb{R}^n$, $n\ge2$, be a bounded convex domain, and let $u_D$ be the torsion function for the restricted half-Laplacian. We prove that $D^2u_D$ is negative definite at every point of $D$. The argument is based on the reflected harmonic extension in a slit domain. Quantitative Schauder estimates in slit domains yield parameter-uniform estimates for the first and second derivatives of the edge remainder; a Schur-complement calculation then determines the inertia of the extended Hessian near the slit edge. Superharmonicity of the logarithmic Hessian determinant and the Gleason--Wolff zero-set theorem exclude interior degeneracy. A method of continuity starting from the unit ball proves the result for smooth uniformly convex domains, and an exhaustion argument treats arbitrary bounded convex domains.
- [77] arXiv:2608.19594 [pdf, html, other]
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Title: Mesoscopic Rectangular Spikes under Subspace Local Laws: Outlier Values and Singular SubspacesSubjects: Probability (math.PR)
A rectangular data matrix is often modelled as noise plus a signal of low rank. When that rank is fixed the picture is classical: a signal direction must exceed a critical strength before it produces a singular value outside the noise bulk, and above the threshold the singular vectors of the observation retain a definite, computable fraction of the planted direction. We ask what survives when the number of signal directions grows with the dimension.
Everything here rests on one hypothesis, which we call a subspace local law: seen from inside the signal subspace, the resolvent of the noise should look like a scalar. We show that this hypothesis alone locates the outliers and counts them, and, in a holomorphic form, determines the outlier singular subspaces as well. The conclusions are stated through spectral projectors rather than individual singular vectors, so they remain meaningful when spike strengths collide or come closer together than the error of the approximation, which at growing rank they must. We then verify the hypothesis in four genuinely different settings, from deterministic noise viewed through randomly oriented signal directions to independent entries with fixed deterministic ones, and for Marchenko--Pastur noise we compute every constant explicitly. - [78] arXiv:2608.19600 [pdf, html, other]
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Title: Uniform estimates for complex Monge-Ampère equations: big cohomology classesComments: Comments are welcomeSubjects: Differential Geometry (math.DG); Complex Variables (math.CV)
We prove uniform a priori estimates for solutions to degenerate complex Monge--Ampère equations in big cohomology classes, using both auxiliary-function technique developed by Guo, Phong and Tong [On $L^\infty$-estimates for complex Monge-Ampère equations, Ann. of Math. (2) 198 (2023), no.1, 393-418], and quasi-psh envelope approach developed by Guedj and Lu [Quasi-plurisubharmonic envelopes 1: Uniform estimates on Kähler manifolds, J. Eur. Math. Soc. (JEMS) 27 (2025), no. 3, 1185-1208.]. As an application, we apply our method to prove the Moser-Trudinger and Brezis-Merle-type inequalities for complex Monge-Ampère equations.
- [79] arXiv:2608.19603 [pdf, html, other]
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Title: An adaptive and conservative low-rank IMEX solver for the hybrid ion Vlasov-Fokker-Planck and fluid electron systemSubjects: Numerical Analysis (math.NA)
We present a macroscopically conservative, rank-adaptive method for solving a hybrid Vlasov-Fokker-Planck (VFP) equation in which the ions are treated kinetically, and the electrons are treated as a fluid. Solving this system poses several coupled computational challenges: the curse of dimensionality, conservation of macroscopic quantities, stiffness arising from the multiscale nature of the model, and structure preservation. To address these challenges, we combine several established methods, each targeting a specific difficulty, into a unified framework for solving this nonlinear system. Specifically, we employ the recent Reduced Augmentation Implicit Low-rank (RAIL) method for rank adaptivity in velocity space to combat the curse of dimensionality, high-order implicit-explicit time stepping to handle the multiscale stiffness, and the Local Macroscopic Conservative (LoMaC) procedure for conservative truncation of the solution. The RAIL and LoMaC methods are extended from Cartesian to cylindrical coordinates. The resulting framework accommodates implicit rank-adaptive time integration while remaining conservative and leveraging structure-preserving discretizations. We verify the scheme on a suite of test problems and simulate a standing shock to demonstrate the importance of conserving the macroscopic quantities.
- [80] arXiv:2608.19608 [pdf, html, other]
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Title: Tau-Rho Equality and Other Dependence Measures of a Subclass of Factorizable CopulasSubjects: Statistics Theory (math.ST)
Kendall's tau and Spearman's rho, two widely used dependence measures in statistics and risk management, are often treated as interchangeable, yet can disagree sharply: Schreyer et al.~(2017) established the exact region of attainable $(\tau,\rho)$ pairs. We study the complementary question of equality, namely, identifying nontrivial families of copulas $C$ satisfying $\tau_C=\rho_C$. We prove that this equality holds for every factorizable copula of the form $C_{e,\alpha}\ast C_{\beta,e}$, where $\alpha$ and $\beta$ are piecewise linear monotonic surjections (PLMS). For these PLMS-generated copulas, we study further dependence measures, including Chatterjee's rank correlation coefficient and tail dependence coefficients, revealing some useful algebraic formulas and unexpected phenomena. In particular, Chatterjee's coefficient can exhibit extreme asymmetry.
- [81] arXiv:2608.19609 [pdf, html, other]
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Title: The integer point enumerator of one irrational translate of P is a complete invariantComments: 12 pagesSubjects: Combinatorics (math.CO)
For a full-dimensional rational polytope $P\subset\mathbb{R}^d$ and a real dilation parameter $t>0$, the integer point enumerator is defined by $L_{P}(t):= |tP\cap\mathbb{Z}^d|$. We determine exactly which translation vectors $\mathbf y=(y_1,\ldots,y_d)\in\mathbb{R}^d$ have the property that the single translated counting function $t\longmapsto L_{P+\mathbf y}(t)$, with $t\in\mathbb{Q}_{>0}$, uniquely determines $P$ among all full-dimensional rational polytopes in $\mathbb{R}^d$. The necessary and sufficient condition is that $1,y_1,\ldots,y_d$ be linearly independent over $\mathbb{Q}$. In particular, we may use the explicit algebraic vector $\mathbf y^* := (2^{1/(d+1)},2^{2/(d+1)},\ldots,2^{d/(d+1)})$ in every dimension $d$. The sufficiency proof recovers the primitive facet inequalities from isolated discontinuities of the counting function, while necessity follows from an affine-unimodular obstruction.
- [82] arXiv:2608.19612 [pdf, html, other]
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Title: Arithmetic partial differential operators on ramified extensions of $\ZZ_p$Subjects: Number Theory (math.NT)
The notion of $p$-derivation as introduced by Buium has a rich history of applications in arithmetic geometry. Working over $\ZZ_p$, Buium-Ralph-Simanca showed that arithmetic differential operators built from these determine $p$-adic analytic functions and vice versa. Notably, over $\ZZ_p$, there is only one $p$-derivation. In this article, we consider the same question for ramified extensions $A_\pi$ with uniformizer $\pi$. This has the effect of passing from ordinary to partial differential operators, since such extensions can enjoy multiple $\pi$-derivations. We introduce a notion of analytic functions which are naturally determined by these operators in a way analogous to that in the unramified case, but with a significantly richer structure. We show that under mild conditions, analytic functions and partial differential operators determine each other in this setting.
- [83] arXiv:2608.19615 [pdf, html, other]
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Title: Evo-GTransNet for Parabolic PDEs: A Fixed-Feature Galerkin Method of Lines with Quadrature-Mass OrthonormalizationSubjects: Numerical Analysis (math.NA)
In this paper, we develop an evolutionary generalized transferable neural network (Evo-GTransNet) solver for parabolic partial differential equations, formulated as a retained-space fixed-feature Galerkin method of lines. A GTransNet provides the prescribed spatial dictionary, while only the retained output coefficients evolve, thereby avoiding nonlinear training during time integration. To address severe mass-matrix ill-conditioning, we apply a quadrature-weighted truncated singular value decomposition (SVD) to select the numerically resolved trial space, followed by a separate rescaling that makes its basis orthonormal with respect to the assembly-quadrature mass inner product. Rank truncation modifies the approximation space, whereas the subsequent orthonormalization changes only its coordinate representation and preserves the retained discrete functions in exact arithmetic. The resulting semidiscrete coefficient system has an identity mass matrix, and we establish a semidiscrete energy law for symmetric linear parabolic problems. With the implicit midpoint scheme for time discretization, we further prove the contractivity of the method and derive a conditional fully discrete error estimate in which the error is controlled by the retained-space approximation error and the consistency defects. Numerical experiments demonstrate second-order temporal convergence using repeated feature samples together with separate assembly and validation quadratures, and quantify the accuracy of the retained space in the presence of severe raw-mass ill-conditioning. For the high-frequency and multiscale benchmark problems considered here, GTransNet achieves the smallest mean validation errors among the tested fixed-feature dictionaries at the same nominal output dimension.
- [84] arXiv:2608.19617 [pdf, html, other]
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Title: Adaptive Time Windows for Discrete Adjoint Topology Optimization of Unsteady FlowsComments: 36 pages, 19 figuresSubjects: Optimization and Control (math.OC)
Rather than prescribing an evaluation interval a priori, the proposed framework characterizes each evolving unsteady flow using a sequence of time windows. A consecutive-window convergence criterion is introduced to automatically identify a representative time window within its fully developed stage. The objective evaluation and discrete adjoint analysis are then carried out consistently over the identified representative time window. The framework is implemented using a regularized lattice Boltzmann method-based large-eddy simulation (LBM-LES) solver together with a partial bounce-back fluid-solid model. The consecutive-window convergence criterion is first validated using the backward-facing step flow. Cylinder-flow applications are then employed to investigate the influence of different flow regimes on the proposed framework. The wake-flow recovery problem verifies its effectiveness for unsteady topology optimization, while U-bend optimization further demonstrates its capability to identify and reorganize complex vortical structures.
- [85] arXiv:2608.19618 [pdf, html, other]
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Title: Borelness of Moduli Spaces of Metrics Implies SeparabilitySubjects: General Topology (math.GN); Metric Geometry (math.MG)
Let X be a metrizable space, and let Met(X) denote the space of metrics compatible with the topology of X, regarded as a subspace of the space of continuous pseudometrics with the supremum-metric topology. We first prove that if Z is a discrete space of cardinality aleph-one, then Met(Z) is not Borel. As a consequence, if Met(X) is Borel, then X is separable. Combined with a theorem of Koshino, our result yields that the space of bounded compatible metrics on a metrizable space X is completely metrizable if and only if X is sigma-compact. We also establish non-Archimedean analogues for spaces of ultrametrics.
- [86] arXiv:2608.19619 [pdf, html, other]
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Title: Moore--Read construction and explicit monodromies of Laughlin states on Riemann surfacesComments: 24 pagesSubjects: Mathematical Physics (math-ph); Strongly Correlated Electrons (cond-mat.str-el); High Energy Physics - Theory (hep-th)
We revisit the Moore--Read construction of the $\nu=1/k$ Laughlin states on compact Riemann surfaces, deriving their conformal blocks from $U(1)_k$ Chern--Simons theory through the CS/WZW correspondence. We compute their explicit monodromies under quasi-hole transport and Aharonov--Bohm flux insertion, and conjecture that these coincide with the corresponding adiabatic holonomies. Under this identification, we recover classical results on Laughlin states including the flux-averaged Hall conductance $1/k$ via a vector bundle of Laughlin states over $Jac(\Sigma)$. We also relate our construction to the recently developed algebro-geometric approach to higher genus Laughlin states.
- [87] arXiv:2608.19623 [pdf, other]
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Title: Palette Sparsification for General Uniform HypergraphsComments: 9 pagesSubjects: Combinatorics (math.CO); Discrete Mathematics (cs.DM); Data Structures and Algorithms (cs.DS)
We prove a palette sparsification theorem for general $r$-uniform hypergraphs. For all sufficiently large $n$, every $r\ge 3$, and every $\alpha\ge 7.1$, we show that an $n$-vertex $r$-uniform hypergraph of maximum degree $\Delta$ is w.h.p. colorable from independently sampled lists of size $O(\sqrt{\log n})$ drawn from an ambient palette of size $\lceil \alpha \Delta^{1/(r-1)}\rceil$. The $\sqrt{\log n}$ dependence is asymptotically tight.
- [88] arXiv:2608.19627 [pdf, html, other]
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Title: Uniformly Rotating Vortex Patches with Arbitrarily Many Genuine HolesComments: 39 pages, 1 figureSubjects: Analysis of PDEs (math.AP)
For every prescribed integer $N\ge2$, we construct uniformly rotating unit-vorticity vortex patches $\omega=\mathbf{1}_D$ for the planar Euler equation such that $D$ is connected and $\mathbb{R}^2\backslash D$ has exactly $N$ bounded connected components. These components are genuine zero-vorticity holes, rather than opposite-sign vortex patches or regions carrying a second nonzero vorticity level. The angular velocities lie in the rigidity-compatible interval $(0,1/2)$, and the domains converge in measure to the Rankine disk as the holes collapse.
The construction starts from a fixed co-rotating polygonal configuration of $N$ unit-vorticity vortex patches and removes a shrinking spatial copy of that configuration from the Rankine disk. An exact complement identity solves all inner-boundary equations before the outer circle is perturbed. The remaining defect is generated by the $N$-th exterior multipole and has size $\varepsilon^{N+2}$. The resulting outer correction feeds back into the normalized inner problem at size $\varepsilon^{2N}$. Separate renormalization of the outer and inner equations produces a limiting affine system with a lower-triangular derivative. Its diagonal blocks are the nonresonant Rankine operator and the angular-velocity-augmented linearization of the fixed seed configuration. We also determine the first corrections to the outer boundary, the hole boundaries, and the angular velocity. - [89] arXiv:2608.19640 [pdf, html, other]
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Title: Convex cocompact right-angled Gromov-Thurston polyhedraComments: 9 pages, 1 figure. Comments welcomeSubjects: Geometric Topology (math.GT); Group Theory (math.GR)
Lee and Marquis exhibited convex cocompact hyperbolic reflection groups in dimension 5 whose limit sets are homeomorphic to the 3-sphere, but none of whose finite-index subgroups can be realized as 4-dimensional real hyperbolic lattices. Using different methods, we furnish right-angled examples.
- [90] arXiv:2608.19641 [pdf, html, other]
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Title: A sharp eigenvalue theorem for mixed elliptic problems under mixed boundary conditionsSubjects: Analysis of PDEs (math.AP)
In this paper, we study a class of eigenvalue problems involving both local and nonlocal operators, namely the classical Laplacian and the fractional Laplacian, under mixed boundary conditions. More precisely, we consider the problem \begin{equation}\label{1} \left\{ \begin{aligned} \mathcal{L}u &= \lambda f(u), \quad u>0 &&\text{in }\Omega,\\ u&=0 &&\text{in }U^c,\\ \mathcal{N}_s(u)&=0 &&\text{in }\mathcal{N},\\ \frac{\partial u}{\partial\nu}&=0 &&\text{on }\partial\Omega\cap\overline{\mathcal{N}}, \end{aligned} \right. \tag{$P_\lambda$} \end{equation} where \( U=\Omega\cup\mathcal{N}\cup \bigl(\partial\Omega\cap\overline{\mathcal{N}}\bigr), \) \(\Omega\subseteq\mathbb{R}^n\) is a bounded open set with smooth boundary, \(\lambda>0\) is a real parameter, \(f\) is continuous function with \(f(0)=0\), and \[ \mathcal{L}=-\Delta+(-\Delta)^s, \qquad s\in(0,1). \] We establish a characterization theorem for the existence of positive weak solutions to problem (P_{\lambda}). Motivated by the classical elliptic framework developed by Molica Bisci and Rădulescu \cite{MolicaBisciRadulescu2017}, we establish a corresponding characterization result for mixed local-nonlocal operators under mixed boundary conditions.
- [91] arXiv:2608.19660 [pdf, html, other]
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Title: Norm of the generalized Hilbert operator on Hardy spacesSubjects: Complex Variables (math.CV)
We study the generalized Hilbert operator \[ \mathcal{H}_b f(z)=\int_0^1 f(t)\,\frac{(1-t)^b}{(1-tz)^{b+1}}\,dt, \qquad b>0, \] acting on the Hardy spaces $H^p$ for $1\leq p\leq \infty$. We establish the precise operator norm \[ \|\mathcal{H}_b\|_{H^p\to H^p}=B\!\left(\frac1p,b+1-\frac1p\right) \] for every $1<p<\infty$ and, by continuous extension to $b=0$, recover the classical norm $\pi/\sin(\pi/p)$. We also prove that $\mathcal{H}_b$ is bounded on $H^1$ for every $b>0$, in contrast with the classical Hilbert operator, and we obtain the sharp restricted norm estimate \[ \|\mathcal{H}_b\|_{H^1_0\to H^1}=B(1,b). \] We also determine the exact norm \[ \|\mathcal{H}_b\|_{H^\infty\to \mathcal B}=\frac{1}{b+1}+2. \]
- [92] arXiv:2608.19663 [pdf, html, other]
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Title: Topological degree and existence of nontrivial normalized solutions for mass-critical Gross-Pitaevskii systemsSubjects: Analysis of PDEs (math.AP)
In this paper, we study the existence of nontrivial solutions for the following Gross-Pitaevskii system involving mass-critical exponent: \[ \left\{ \begin{array}{ll} -\Delta u_{1}+V_1(x)u_{1}=a_{1}u_{1}^3+\beta u_{1}u_{2}^2+\mu u_{1}& \hbox{ in }\Omega,\\ -\Delta u_{2}+V_2(x)u_{2}=a_{2}u_{2}^3+\beta u_{2}u_{1}^2+\mu u_{2}&\hbox{ in }\Omega,
u_{1},u_{2}\ge 0 &\hbox{ in }\Omega,
u_1=u_2=0 &\hbox{ on }\partial\Omega, \end{array}\right. \] with the constraint \[ \int_\Omega (u_1^2+u_2^2)=1, \] where $\Omega$ is an unbounded smooth domain in $\mathbb{R}^2$, $a_1, a_2, \beta$ are positive parameters, $V_i$ are trapping potentials, and $\mu\in\mathbb{R}$ is an unknown Lagrange multiplier. We derive the existence of solutions by computing the Leray-Schauder degree for the parameters $a_1,a_2, \beta$, which are away from some critical values. The system may have semi-trivial solutions of the form $(u_1, 0)$ or $(0, u_2)$. Our novelty is that we provide mechanisms ensuring that the solutions we find are nontrivial, i.e., $u_1>0$ and $u_2>0$. - [93] arXiv:2608.19676 [pdf, html, other]
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Title: Nonexistence of Frame Measures for Separated Uniform Consecutive-Digit Bernoulli ConvolutionsComments: 14 pagesSubjects: Functional Analysis (math.FA)
We investigate the existence problem of frame measures for uniform consecutive-digit Bernoulli convolutions under the separated scaling condition. Given parameters $N\geq 2$ and $0<\rho<1/N$, we prove that if $\rho^{-m}=B$ for some integers $m\geq 1$ and $B\geq 2$, with $N\nmid B$, then the associated $N$-Bernoulli convolution $\mu_{\rho,N}$ admits no frame measure. The proof introduces a new cyclic mask-quotient obstruction adapted to the multi-character structure of the consecutive-digit framework.
- [94] arXiv:2608.19686 [pdf, html, other]
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Title: Clique number and triangle densities in $C_4$-free graphsComments: 25 pages, 3 figuresSubjects: Combinatorics (math.CO); Commutative Algebra (math.AC)
For a $C_4$-free graph $G$ on $n$ vertices --- one with no induced cycle on four vertices --- we study the two-sided extremal problem for the triangle density $\tau$: How large and how small can $\tau$ be for given edge density $\varepsilon$ and clique-number density $\kappa = \omega(G)/n$?
We give lower and upper bounds for $\tau$ in terms of $\kappa$ and $\varepsilon$. The two bounds sandwich $\tau$, and their compatibility forces a lower bound for $\kappa$ in terms of $\varepsilon$. When the clique complex of $G$ is $2$-Leray over a field $\Bbbk$, the resulting bound on the clique-number density lies between the previous best $C_4$-free bound and the sharp chordal bound. It improves on the former {\it for every} $\varepsilon \in (0,1)$.
The lower bound is elementary. The upper bound is homological, obtained by passing to the Stanley--Reisner ring of the clique complex. When the complex is $2$-Leray, its Betti table has at most two linear strands. The two first entries in the first strand encode edge and triangle densities, and the strong structural form of a Boij--Söderberg decomposition constrains what these entries can be, yielding the upper bound.
For $2$-Leray graphs with no holes in the range $[4,g]$ we give a conjecturally sharp bound. We further ask questions concerning the triangle bound for any $C_4$-free graph. - [95] arXiv:2608.19688 [pdf, html, other]
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Title: Learning Deterministic and Stochastic Forced Hamiltonian SystemsSubjects: Numerical Analysis (math.NA); Machine Learning (cs.LG); Symplectic Geometry (math.SG)
We develop a geometric framework for learning deterministic and stochastic forced Hamiltonian systems with neural networks. Motivated by the Lagrange-d'Alembert principle and the theory of variational integrators, we introduce the notion of a Lagrange-d'Alembert map and establish a $C^r$ convergence theorem for first-order one-step methods. Building on these results, we propose Generalized Forced Hamiltonian Neural Networks (GFHNNs), a class of structure-preserving neural networks obtained by concatenating Lagrange-d'Alembert-Euler maps, and prove a universal approximation theorem for this architecture. We further extend the framework to parameter-dependent systems, leading to Parametric Generalized Forced Hamiltonian Neural Networks (PGFHNNs). By interpreting the multiple Stratonovich integrals appearing in the Stratonovich-Taylor expansion as parameters, the same framework can be applied to stochastic forced Hamiltonian systems whenever information about the underlying Wiener process is available. Our numerical experiments demonstrate that the proposed geometric architectures provide significantly improved long-time stability and accuracy compared to non-geometric residual neural networks, while requiring substantially less training data to achieve a comparable level of performance.
- [96] arXiv:2608.19697 [pdf, html, other]
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Title: Open cover specification property and ChaosSubjects: Dynamical Systems (math.DS)
This paper investigates the open cover specification property and its role in the study of topological dynamical systems. We show that this property is a topological invariant and a natural extension of the classical topological specification property from uniform spaces to general topological spaces, and that it is preserved under infinite products with compositions. We further prove that on regular spaces, the strong open cover specification property implies Devaney chaos, and that on locally compact, first countable Hausdorff spaces, it guarantees both distributional chaos and Li Yorke chaos. These results demonstrate the significance of the open cover specification property in linking topological structure with dynamical complexity.
- [97] arXiv:2608.19700 [pdf, html, other]
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Title: Calibrated submanifolds, adiabatic limit, and gradient graphsComments: 31 pagesSubjects: Differential Geometry (math.DG)
We study the compactness question for certain special Lagrangians in semiflat SYZ fibrations (resp. associative submanifolds in Donaldson's proposal of collapsing coassociative K3 fibrations), and give some criterion for when gradient graphs emerge from the adiabatic limit. This gives a partial converse to the Donaldson-Scaduto proposal.
- [98] arXiv:2608.19702 [pdf, html, other]
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Title: Exact hierarchical algorithms for accelerating particle--mesh coupling in sparse-grid particle-in-cell methodsSubjects: Numerical Analysis (math.NA)
In this paper, we propose two hierarchical algorithms for charge deposition and electric-field interpolation that apply to both the sparse-grid combination technique (SGCT-PIC) and hierarchical sparse-grid (HSG-PIC) particle-in-cell methods. The two algorithms are inspired by the fast multipole method (FMM) and exploit clusters of particles associated with a directed acyclic graph (DAG) of particle-populated boxes to reduce the number of particle--mesh interactions. The particle--mesh interactions are governed by piecewise-polynomial kernels, so that the associated multipole expansions are exact, requiring neither truncation nor approximation, and are valid in both near- and far-field regions, thereby eliminating the need for multipole-to-local translations. The arithmetic complexity of the charge deposition and field interpolation steps is reduced from $Ø(p^d n^{d-1}N)$ to $Ø(p^d(N+M))$, where $M=2^{dn}$ denotes the number of full-grid mesh nodes and is typically no larger than the particle population in the considered regime, $M\lesssim N$. Numerical experiments in two-dimensional configurations demonstrate charge-deposition speedups of $8.2\times$--$66.9\times$ for SGCT-PIC and $3.1\times$--$18.8\times$ for HSG-PIC, and field-interpolation speedups of $4.1\times$--$62.6\times$ and $4.2\times$--$13.7\times$, respectively, depending on the particle-per-cell ratio, while preserving the exact particle--mesh interactions. The speedups increase with the particle-per-cell ratio, reflecting the reduced dependence of the hierarchical algorithms on the number of particles and their increasing advantage for large particle populations.
- [99] arXiv:2608.19704 [pdf, html, other]
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Title: Liouville theorem for a class of p-Laplace type equations on manifoldsSubjects: Analysis of PDEs (math.AP)
We study a class of $p$-Laplace equations $$\Delta_p u-\lambda u^{p-1}+ u^{q-1}=0$$ on a closed $n$-dimensional Riemannian manifold $(M,g)$ with $\operatorname{Ric}\geqslant(n-1)g$. For $1<p<2$, $p<q<p^*$, and $0<\lambda<S_{p,q}^{-1}$, where $$S_{p,q}=\frac{q-p}{2}(\frac p n)^{\frac p 2}\Big(\frac{(p^*-1)^2(2-p)}{(p_*-1)(q-1)(p^*-q)}\Big)^{\frac{2-p}{2}},$$ with $p_*=\frac{(n-1)p}{n-p}$ and $p^*=\frac{np}{n-p}$, we prove that the constant $\lambda^{\frac{1}{q-p}}$ is the unique positive solution of the equation. In contrast, for $p>2$ and $p<q<p^*$, the uniqueness fails for every $\lambda>0$; aside from the constant solution, the equation admits a positive nonconstant solution. This answers Véron's problem raised in \cite{Ver92}.
- [100] arXiv:2608.19706 [pdf, html, other]
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Title: Forced Shadows of an Obstructed Hyperbolic Kac-Moody DenominatorComments: 29 pages. Also deposited at Zenodo, doi:https://doi.org/10.5281/zenodo.21973291Subjects: Number Theory (math.NT)
The four orders of the quaternion algebra B6 carry four reflective wall data on lattices of signature (3,2); three integrate to Borcherds denominators and one is obstructed. The failed denominator survives as a weakly harmonic Maass form, and we prove its shadow is a Hecke eigenform on the line of the newform 6.4.a.a, with zero twist component. The mechanism is invariance selection: the obstruction functional is invariant under the discriminant isometry group, whose invariants in S_{5/2} are one-dimensional; the same mechanism, verified at quaternion discriminants 10 and 22, places the shadows there on 10.4.a.a and 22.4.a.c. On the weight-1/2 layer we prove a determination theorem: the canonical form exists unconditionally and uniquely precisely when the obstruction space vanishes, and combining the finiteness bound of Bruinier-Ehlen-Freitag with a finite computation this happens exactly for D in {6, 10, 22}, the genus-zero compact Shimura curves, whose maximal-order ternaries are reflective with integral Weyl chambers of ranks 3, 4, 4. Parity confines this layer to odd channels; on the obstructed orientation the section layer is obstructed outside an explicit 40-element locus of orientations, giving a double shadow, CM (36.2.a.a) at weight 3/2 and newform at weight 5/2, while the deck-symmetric directions instead carry a unique canonical weight-1/2 form. The defect invariant satisfies ||Xi||^2 <v_+,v_+> = 144 exactly and equals L(f,2)/(48 pi^2 <f,f>) to 31 digits. On the section layer the Petersson geometry is rigid: the Gram matrix of S_{3/2} is a single transcendental multiple of an exact rational form, and that transcendental is identified, to 40 digits, as 3 Gamma(1/3)^3 / (2^{7/3} pi^2). The weight-3/2 shadow norms lie in the Chowla-Selberg ring from which the weight-5/2 norm is excluded.
- [101] arXiv:2608.19711 [pdf, html, other]
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Title: High-Multiplicity Flexible Job Shops: From Exact Recurrent Fluid Attainment to Structure-Guided Finite-Horizon SchedulingSubjects: Optimization and Control (math.OC)
High-multiplicity flexible job shops involve many copies of a small set of job types that must be scheduled on alternative machines. Fluid relaxations provide scalable workload lower bounds, but their fractional machine allocations do not define feasible schedules for individual jobs. We show that, after a suitable finite scaling, an optimal fluid allocation can be realized exactly by a feasible repeating discrete schedule. The construction scales the fluid allocation to integer operation counts, places the resulting operations in nonoverlapping machine intervals, repeats this arrangement, and links operations across repetitions into individual jobs without moving any interval, thereby enforcing job precedence while preserving machine feasibility. For growing finite instances with the same job-type composition, even with a fixed number of extra jobs, the gap between the optimal makespan and the fluid lower bound remains bounded by a constant; hence the relative gap vanishes as the instance grows. Guided by this repeated structure, we develop type-based cyclic template replay (TCTR), which searches job-type templates using an optimization model whose size does not grow with the number of job copies and replays the selected template on the full instance. On 676 multiplicity-expanded public flexible-job-shop instances, TCTR is feasible in every case and achieves a 1.54% mean gap to the fluid lower bound, compared with 5.32% for a job-indexed adaptive large-neighborhood search and 6.18% for a hybrid genetic algorithm.
- [102] arXiv:2608.19717 [pdf, html, other]
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Title: Well-Posedness for Cauchy Problems with Singular Time-Measurable Pseudo-Differential Operators in Quasi-decreasing Weighted $\mathrm{L}_2$-SpacesComments: 39 pagesSubjects: Analysis of PDEs (math.AP)
This study examines Cauchy problems governed by highly singular, time-measurable pseudo-differential operators (singular measurable families of Fourier multipliers). We show that the symbols of these operators can exhibit arbitrary blow-up behavior. In particular, we prove the existence and uniqueness of solutions even when the symbols grow super-exponentially in time and frequency. As a concrete application, we solve evolutionary equations driven by fractional Laplacians of any negative order. Additionally, we establish unique strong solutions under the sole condition that the symbol is locally integrable in frequency, even in the presence of severe blow-up at the initial time.
- [103] arXiv:2608.19732 [pdf, html, other]
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Title: Improved quantitative stability for the critical Hardy inequalityComments: 19 pagesSubjects: Analysis of PDEs (math.AP)
We establish a quantitative stability estimate for the critical Hardy inequality on bounded domains containing the origin. Our result improves the existing quantitative stability estimate by reducing the exponent in the distance function from $N^{2}$ to $N$, replacing the Lorentz-Zygmund framework with the Luxemburg norm of the critical exponential Orlicz space $\operatorname{Exp}L^{\frac{N}{N-1}}(\Omega)$, and avoiding any cut-off modification of the virtual extremizers. As a consequence, we also obtain an improved quantitative stability estimate for the critical Hardy inequality with the logarithmic weight considered by Cianchi and Ferone, where the exponent is likewise reduced from $N^{2}$ to $N$ and the distance is measured directly from the virtual extremizer without truncation. The proof is completely rearrangement-free and relies on a critical Hardy inequality with a remainder term, scale-invariant Sobolev inequalities, and a refined dyadic summation argument. These ingredients yield stronger quantitative stability estimates for both forms of the critical Hardy inequality.
- [104] arXiv:2608.19733 [pdf, html, other]
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Title: Real polynomials with given multiplicities of real roots: Complete conjectural description of homologyComments: 10 pages, no figuresSubjects: Algebraic Topology (math.AT)
Following \cite {KSW} we continue the study the cellular complexes formed by polynomials of a given degree having a given sequence of multiplicities of real roots.
A computer-assisted calculation disproves the earlier conjecture that homology of one point compactification of the closure of such cell is concentrated in at most one degree. Namely, for $\omega=(3,1,1,3)$ in degree $d=18$, the reduced homology is $\ZZ^2$ in degree $7$. The obstruction is already visible in a signed cell count, whose value is $-2$.
We relate that count to the rational signed weight enumerator $F_\omega(t)=\sum_{\eta\preceq\omega}(-1)^{\elln(\eta)}t^{|\eta|}$. For the counterexample, $F_\omega(t)=-t^{14}/(1+t^2)^2$, which gives an exact linear formula for the Euler characteristic and forces the total rational Betti number to be unbounded.
We prove a number of results and formulate a complete conjecture describing the above homology. - [105] arXiv:2608.19736 [pdf, html, other]
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Title: Tri-Hybrid Beamforming for T-RIS-Enabled Base StationSubjects: Information Theory (cs.IT)
Transmissive reconfigurable intelligent surfaces (T-RISs) integrated into the transmitter provide a viable realization of tri-hybrid multiple-input multiple-output (MIMO), where spatial processing is distributed across the digital, analog radio-frequency (RF), and electromagnetic (EM) domains. However, unlike conventional RIS-assisted links, a transmitter-native T-RIS directly participates in radiation formation, making T-RIS front-end modeling and weighted sum-rate (WSR)-oriented joint precoding challenging under practical hardware constraints. This paper develops a unified modeling and precoding framework for transmitter-native T-RIS tri-hybrid multi-user (MU) downlink transmission. Specifically, starting from a continuous-field description, the received field is characterized by the interaction among the feed array, the programmable T-RIS aperture, and the user-side propagation, leading to a cascaded baseband input-output model for MU precoding. Furthermore, the same representation is instantiated in the Fresnel, Fraunhofer, and mixed-field regimes, so that near-field focusing and far-field angular steering can be handled within one front-end model. Additionally, a WSR maximization problem is formulated over the digital precoder, analog network, and T-RIS coefficients under power and quantization constraints. A two-level solver is then developed by coupling outer weighted minimum mean-square error (WMMSE) updates with WMMSE-induced aperture-field shaping and hardware projection. Simulations validate the modeling accuracy and convergence, and show that the proposed full tri-hybrid design improves WSR over baselines while suppressing mixed-field cross-regime leakage.
- [106] arXiv:2608.19742 [pdf, html, other]
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Title: Ribet Points, geometric divisibility sequence and order of reductions on semiabelian varietiesComments: 38 pages, comments very welcome!Subjects: Number Theory (math.NT); Algebraic Geometry (math.AG)
Silverman conjectured that the geometric divisibility sequence attached to a Zariski-dense point on an irreducible commutative algebraic group of dimension at least two, with no unipotent part, returns to its initial value infinitely often. We construct, for the firsst time, unconditional examples of this phenomenon on geometrically nonsplit semiabelian varieties over number fields. More precisely, let $A/K$ be a positive-dimensional abelian variety over number field $K$, let $$ 1\longrightarrow\mathbf G_m\xrightarrow{\iota}G_q\xrightarrow{\pi}A
\longrightarrow0 $$ be the extension represented by $q\in A^\vee(K)$, and let $R_\beta(q)\in G_q(K)$ be the normalized Ribet point associated with a homomorphism $\beta:A^\vee\to A$. Assume that $\delta$ is an isogeny and that the cyclic subgroup generated by $\delta q$ is Zariski dense in $A$. Let $t\in\mathbf G_m(K)$ be a torsion point. Then $G_q$ is geometrically nonsplit and $P=R_\beta(q)+\iota(t)$ has Zariski-dense cyclic orbit. Put $\delta=\beta-\widehat{\beta}$. If $e_\delta$ denotes the exponent of $\ker\delta$ and $h=\operatorname{ord}(t)$, define $$ N_{\delta,t} := \prod_{\ell} \ell^{ \max\left\{ 0, \left\lceil \frac{v_\ell(h)-2v_\ell(e_\delta)}{2} \right\rceil \right\}}. $$ If $N_{\delta,t}>1$, then $N_{\delta,t}\mid d_v(P)$ for all but finitely many finite places $v$, where $d_v(P)$ denotes the order of the reduction of $P$. Consequently, there exists a squarefree integer $Q>1$ such that $$ (n,Q)=1 \quad\Longrightarrow\quad \mathfrak d_{\mathcal N}(nP) = \mathfrak d_{\mathcal N}(P), $$ where $\mathfrak d_{\mathcal N}$ denotes the full denominator ideal on the Néron lft-model $\mathcal N$. - [107] arXiv:2608.19744 [pdf, html, other]
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Title: Conditioned Brownian motion and local equivalence of path ensemblesComments: 34 pagesSubjects: Probability (math.PR); Mathematical Physics (math-ph)
We study Brownian motion in R^d conditioned so that the time average of a continuous confining potential remains below a fixed level. On every fixed initial time interval, we prove that the conditioned process converges in total variation to the ground-state diffusion associated with a suitable Schrödinger operator. We also obtain sharp asymptotics for the probability of the conditioning event, including bounded perturbations of the constraint. The proof is based on a local limit theorem for the corresponding Feynman-Kac measures. Our results extend the previously known one-dimensional quadratic case to arbitrary finite dimension and a broad class of confining potentials, therefore resolving a conjecture of Aurzada, Lifshits and Schickentanz. The presented approach also works when Brownian motion is replaced by suitable reversible Markov processes, including multidimensional Ornstein-Uhlenbeck processes, CIR processes and continuous-time Markov chains.
- [108] arXiv:2608.19747 [pdf, html, other]
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Title: Dangling points in area-minimizing Meissner polyhedraSubjects: Optimization and Control (math.OC); Metric Geometry (math.MG)
Meissner polyhedra are constant-width bodies obtained from extremal finite sets of unit diameter. Such a generating set may contain dangling points, namely points having exactly two diametric neighbors. This article studies whether these points can play an essential role in surface-area minimization. Given an extremal set we show that the smallest surface area among the Meissner polyhedra based on it cannot increase by deleting a dangling point. Adding a dangling point cannot decrease the smallest achievable surface area. This reduces the search for area-minimizing Meissner polyhedra to generating sets without dangling points.
- [109] arXiv:2608.19753 [pdf, html, other]
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Title: Uniform large deviation principles for the stochastic heat equation over unbounded sets of initial dataSubjects: Probability (math.PR); Analysis of PDEs (math.AP)
We study small-noise large deviations for the stochastic heat equation (SHE) on the torus with unbounded, multiplicative space-time white noise. We establish uniform large deviation principles (ULDPs) over unbounded sets of initial data, alongside novel well-posedness and regularity results. The ULDPs are obtained for both continuous-in-space and $L^p$ initial data. For these two classes, the ULDP holds uniformly over $L^q$-bounded subsets, with $q<\infty$ and $q<p$ allowed respectively, so these sets are highly unbounded in the state space. The admissible range of $q$ is dictated by the growth of the noise coefficient in the SHE. Crucially, our methods allow us to reach down to $q=1$. This yields ULDPs over $L^1$-bounded subsets, enabling the study of exit times in physical systems with mass conservation.
- [110] arXiv:2608.19754 [pdf, html, other]
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Title: A Controllability Gramain Shaping with LMI Constraints under Bures--Wasserstein DistanceComments: 8 pages, 4 figures. Accepted for presentation at the 65th IEEE Conference on Decision and Control (CDC 2026), Honolulu, HI, USA,December 15-18, 2026. This is the initially submitted versionSubjects: Optimization and Control (math.OC); Systems and Control (eess.SY)
This paper proposes a controller design method for shaping the controllability Gramian into a desired form to design the effect from exogenous inputs to the system state. Using the Bures--Wasserstein distance, we formulate the shaping problem as the minimization of the distance between the system Gramian and a desired Gramian, and the objective function is shown to be strictly convex on the set of symmetric positive definite matrices. In addition, by deriving a semidefinite programming formulation via a linear matrix inequality (LMI), computational efficiency is improved and additional LMI constraints can be incorporated. When the exogenous input is modeled as Gaussian white noise, the proposed framework is closely related to $H_2$ control, which can be interpreted as a special case of optimal transport. Numerical examples demonstrate anisotropic controllability design for a guidance robot and verify the ability to impose additional directional constraints through LMIs. The numerical examples also confirm that the proposed method approaches $H_2$ control as the desired Gramian tends to zero.
- [111] arXiv:2608.19773 [pdf, html, other]
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Title: Non-persistence of equality between chromatic polynomials and list-color functionsSubjects: Combinatorics (math.CO)
For any graph $G$, let $P(G,k)$ and $P_{\ell}(G,k)$ denote the chromatic polynomial and the list-color function of $G$, respectively. It remains an open problem whether, for every graph $G$ and integer $k$, the equality $P(G,k)=P_{\ell}(G,k)>0$ implies that $P(G,k+1)=P_{\ell}(G,k+1)$ also holds. In this paper, we answer this question in the negative. For every integer $k\ge 3$, we construct an infinite family of graphs $G$ such that $P(G,k)=P_{\ell}(G,k)>0$ while $P(G,k+1)>P_{\ell}(G,k+1)$. Moreover, using this infinite family of graphs as attachment gadgets, we further show that any graph $H$ with $P(H,k)=P_{\ell}(H,k)>0$ can be developed into an infinite family of graphs $H'$ with $P(H',k)=P_{\ell}(H',k)>0$ and $P(H',k+1)>P_{\ell}(H',k+1)$.
- [112] arXiv:2608.19780 [pdf, html, other]
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Title: Real-rooted flow polynomials have only integer rootsSubjects: Combinatorics (math.CO)
In this article, we show that for any bridgeless graph $G$, if its flow polynomial $F(G,x)$ has real zeros only, then $G$ is the dual of a chordal plane graph and each zero of $F(G,x)$ is an integer in the set $\{1,2,3\}$.
- [113] arXiv:2608.19782 [pdf, html, other]
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Title: The irrationality measure of arctan1/2 is at most 8.585166Subjects: Number Theory (math.NT)
This paper establishes a new upper bound for the irrationality measure of arctan(1/2), namely 8.585166... Following the strategy and proof techniques of Zudilin and Zeilberger (who improved the bound for pi), we use a family of complex contour integrals with symmetric integrands. The integrality and divisibility of the partial-fraction coefficients are derived via p-adic valuations and a specially defined prime set P_n. Combined with saddle-point asymptotics, the growth rates of the integral sequence and its rational/logarithmic components yield the desired irrationality measure.
- [114] arXiv:2608.19785 [pdf, html, other]
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Title: Learning piecewise-smooth dynamical systemsSubjects: Dynamical Systems (math.DS); Machine Learning (cs.LG); Numerical Analysis (math.NA)
Discovering dynamical systems from trajectory data is a central problem in applied mathematics and engineering. Whilst recent advances in machine learning have led to strong progress in data-driven system identification, much less attention has been given to systems with discontinuous dynamics. These systems are nevertheless highly relevant in applications, including climate dynamics and mechanical systems with friction. In this work, we consider the problem of identifying piecewise-smooth dynamical systems directly from trajectory data. Compared with the smooth setting, this requires recovering the governing equations and detecting the switching hyperplanes that separate different dynamical regimes and characterising their behaviour, such as sliding motion. We present a modular framework for discovering such systems by first estimating switching hyperplanes from data and then learning smooth dynamics within each region using geometry-constrained neural networks. The geometry-learning phase is studied from a statistical perspective, analysing the identifiability of the discontinuities and the robustness of the procedure. We also introduce a novel neural network architecture with a prescribed discontinuity set, and provide a theoretical analysis of its approximation properties. The approach is tested on low-dimensional benchmark problems, including dry-friction oscillators and the PP04 climate model for the ice ages.
- [115] arXiv:2608.19791 [pdf, html, other]
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Title: Modules with few Jordan blocks for rank $1$ groups of Lie type and related groupsSubjects: Group Theory (math.GR); Representation Theory (math.RT)
Suppose that $p$ is a prime and $X$ is a finite group with a strongly $p$-embedded subgroup, for example a rank $1$ group of Lie type in characteristic $p$. Let $m$ denote the $p$-rank of $X$ and assume that $m \ge 2$. We say that a faithful $\FF_p X$-module is $k$-active if some element of order $p$ in $X$ acts with exactly $k$ non-trivial Jordan blocks. In this paper, we determine the non-trivial composition factors of the faithful $\FF_pX$-modules which are $k$-active for some $k\leq m$.
- [116] arXiv:2608.19797 [pdf, html, other]
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Title: Optimal Sobolev Approximation by Deterministic and Random Shallow Sigmoidal NetworksSubjects: Numerical Analysis (math.NA)
Shallow networks with prescribed or randomly sampled hidden parameters are widely used as numerical trial spaces, yet their optimal Sobolev approximation power with standard smooth sigmoidal activations in general dimension remains unresolved. We establish the corresponding optimal rates for a class of smooth sigmoidal activations with Schwartz-class derivative decay, including $\tanh$, the logistic sigmoid, and the error function $erf$. We first construct deterministic direction--offset dictionaries with $M$ features such that every $u\in H^k(\Omega)$ can be approximated with error of order $M^{-(k-m)/d}$ in $H^m(\Omega)$ for all $0\le m\le k$. This rate is optimal in the sense of Kolmogorov widths for Sobolev balls. We further prove that dictionaries obtained by independent parameter sampling from any prescribed density bounded away from zero attain the same approximation exponent with high probability, up to logarithmic oversampling. The analysis develops a sigmoidal ridge representation and combines it with deterministic or probabilistic quadrature in direction--offset space while retaining polynomial control of the output coefficients. Numerical experiments across a broad range of dimensions, target regularities, and Sobolev error norms recover the predicted algebraic rates for both deterministic and random feature dictionaries.
- [117] arXiv:2608.19798 [pdf, html, other]
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Title: The subregular and submaximal $p$-cellsComments: 33 pagesSubjects: Representation Theory (math.RT); Combinatorics (math.CO)
Cells for the canonical and $p$-canonical bases organise the representation theory and geometry of Hecke categories. We determine the relevant $p$-canonical basis elements and the resulting $p$-cell structure for the subregular and submaximal cells in all classical types.
- [118] arXiv:2608.19806 [pdf, html, other]
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Title: Boundary-Weighted Fourier Inequalities for Convex DomainsComments: 34 pagesSubjects: Classical Analysis and ODEs (math.CA); Functional Analysis (math.FA)
We consider the natural family of Fourier inequalities for the Paley--Wiener space $\mathrm{PW}^q(\Omega)$, consisting of $L^q$-functions with Fourier support in a convex set $\Omega \subset \mathbb{R}^n$, $n \geq 2$, free of affine lines. Namely,
\[
\int_{\Omega}\dfrac{|\hat{f}(x)|^p}{\omega_{\Omega}^d(x)}dx\leq C\|f\|_{L^q}^p,\quad f\in \mathrm{PW}^q(\Omega).
\]
Here $\hat{f}$ is the Fourier transform of $f$, $1 \leq p, q < \infty$, $d \in \mathbb{R}$, and $\omega_\Omega$ is the frequency multiplier weight associated with the Paley--Wiener space of $\Omega$,
\[
\omega_{\Omega}(x)=m(\Omega\cap (2x-\Omega)), \qquad x \in \Omega.
\]
For an arbitrary polyhedron $P$, we completely characterize the triples $(p,q,d)$ which yield valid Fourier inequalities. For a ball $B$, we characterize the valid triples when $p \geq 2$. When $p < 2$, the situation is different for the ball, and natural critical inequalities fail. However, we show that the spherical restriction conjecture implies a family of subcritical Fourier inequalities for the ball, which in turn imply the Kakeya conjecture (in its Minkowski-form). Finally, we link our family of Fourier inequalities to the theory of truncated Hankel operators acting on the Paley--Wiener space of $\Omega$. - [119] arXiv:2608.19815 [pdf, html, other]
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Title: Trustworthy Decisions in Reliability Set Estimation under Insufficient Model InformationComments: 30 pages,3 figuresSubjects: Statistics Theory (math.ST); Methodology (stat.ME)
Reliability set estimation identifies input regions where a response probability exceeds a target level, bridging estimation and safety-critical decisions. Practitioners typically start with a working model, an imperfect approximation of the true response surface. Relying on this imperfect model may incur decision risk, potentially certifying unsafe regions as safe. We develop a unified framework that turns such a working model into a trustworthy decision rule. First, a modeling-then-calibration procedure decouples estimation from decision. Since the true set is unobservable, we introduce an asymmetric, observable surrogate loss and use a separate calibration set to select a bias-correcting threshold, reducing decision risk and achieving $O_P(1/n)$ volume convergence. Second, we leverage conformal risk control with the surrogate loss to control false inclusion risk, which is the most safety-critical error, at a pre-specified level regardless of working model quality. Together, these calibration procedures show that a separate calibration set is necessary for risk control. Third, an adaptive design concentrates observations on the reliability set and its boundary, improving model quality where errors most affect decisions while controlling budget elsewhere. Numerical studies show not only more accurate set estimates but also calibrated finite-sample risk control that classical plug-in methods lack.
- [120] arXiv:2608.19829 [pdf, html, other]
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Title: A Note on Topological Hochschild Homology Relative to $\Sphere_{W(k)}[x_0,x_1,\ldots,x_n]$Comments: 29 pagesSubjects: Algebraic Topology (math.AT)
We explain the relation between the relative topological Hochschild homology $\THH(R/\Sphere_{W(k)}[x_0,\ldots,x_n])$ and the Nygaard completed Frobenius twisted relative prismatic cohomology $\widehat{\Prism}^{(1)}_{R/W(k)[x_0,\ldots,x_n]^\wedge}$, where $W(k)[x_0,x_1,\ldots,x_n]\rightarrow R$ is relatively quasiregular semiperfectoid. As an application, for $R=\Z_p[x]/(px)$, we compute $\pi_*\THH(R)^\wedge_p$ by descent along $\THH(R)^\wedge_p\rightarrow \THH(R/\Sphere_p[z,x])$, where $R=\Z_p[x]/(px)$ is regarded as an $\Einfty$-$\Sphere_p[z,x]$-algebra through $\Sphere_p[z,x]\xrightarrow{z\mapsto p,x\mapsto x}\Z_p[x]/(px)$.
- [121] arXiv:2608.19830 [pdf, html, other]
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Title: Categorification in Representation TheoryComments: survey article for Proceedings from ICRA 2024Subjects: Representation Theory (math.RT); Category Theory (math.CT)
In this survey article, we give an introduction to the relatively young subject of $2$-representation theory, which studies categorifications of important players in classical representation theory. In particular, we provide a streamlined exposition of the main results leading to the classification of simple $2$-representations for Soergel bimodules associated to finite Weyl groups in characteristic $0$, obtained in [M. Mackaay, V. Mazorchuk, V. Miemietz, D. Tubbenhauer and X. Zhang, Simple transitive $2$-representations of Soergel bimodules for finite Coxeter types. Proc. London Math. Soc. 126 (2023)].
- [122] arXiv:2608.19832 [pdf, html, other]
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Title: Sharp Convex Concentration for Symmetric Random Tensors with Subgaussian CoordinatesSubjects: Probability (math.PR)
Let $X=(X_1,\ldots,X_n)$ have independent coordinates with mean zero, variance one, and $\|X_i\|_{\psi_2}\le K$, and let $H_d=(\mathbb R^n)^{\otimes_2 d}$. Let $L>0$ and let $f:H_d\to\mathbb R$ be convex and $L$-Lipschitz. We prove that, for $0\le t\le c_KLn^{d/2}$, \[
\textsf{P}\left\{
\left\lvert f(X^{\otimes d})-\textsf{E}f(X^{\otimes d})\right\rvert >t
\right\}
\le C\exp\left[-c_K\mathcal I_{n,d}\left(
\frac{t}{L n^{(d-1)/2}}
\right)\right], \] where \[
\mathcal I_{n,d}(s)=
\min\left\{
\frac{s^2}{d^2},
\frac{s^2}{d\log(e+nd/s^2)}
\right\},\qquad s>0,
\qquad \mathcal I_{n,d}(0)=0. \] The first rate is forced by changes in $\|X\|$. The second comes from changes of $X$ when its norm is nearly fixed. The proof constructs one coupling that controls both the coordinatewise conditional displacement and the mean squared Euclidean distance, and combines these bounds with a second-order estimate for $x\mapsto x^{\otimes d}$. The rate is minimax sharp, scale by scale, even when the subgaussian norms are bounded by an absolute constant. For bounded coordinates the logarithm in the second rate disappears. - [123] arXiv:2608.19835 [pdf, html, other]
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Title: On symmetric systems of transport equationsSubjects: Analysis of PDEs (math.AP)
We study a symmetric system of transport equations with solenoidal coefficients. This system reduces to an evolutionary equation with a skew-symmetric spatial operator in the real Hilbert space of square-integrable vector-functions, and by general results we claim that there always exists a generalized solution of the Cauchy problem. Uniqueness of this solution is equivalent to skew-adjointness of the spatial operator. We demonstrate that in the case of locally Lipschitz coefficients satisfying a linear growth condition the spatial transport operator is indeed skew-adjoint. For scalar transport equation this result remains true under the weaker DiPerna-Lions conditions.
- [124] arXiv:2608.19839 [pdf, html, other]
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Title: Sharp Dimension Bounds for Spline Spaces over T-meshes with Highest Order of SmoothnessSubjects: Numerical Analysis (math.NA)
The dimension of a polynomial spline space of bi-degree $(d_1,d_2)$ over a T-mesh $\mathscr{T}$ with the highest order of smoothness $(d_1-1,d_2-1)$ depends on both mesh topology and geometric configurations. Under the assumption that the T-connected components of the T-mesh $\mathscr{T}$ contain no vanishable T $l$-edges, we develop explicit upper and lower bounds of the dimension of the polynomial spline space. By introducing a decoupling technique within the completely non-diagonalizable component (CNDC) of the T-mesh $\mathscr{T}$, we separate tightly coupled multi-vertex constraints and transform global conformality conditions into localized linear equations along each interior large edge. Based on the decoupling technique, a new dimension formula of the polynomial spline space is then presented, and from which sharp upper and lower bounds of the dimension are obtained. The bounds are sharp in the sense that different geometric realizations of T-meshes with the same topology can attain the lower and upper bounds for the dimension of the polynomial spline space. We further prove that the new formula is consistent with Mourrain's homological dimension formula, and a sharper lower bound is obtained by our method.
- [125] arXiv:2608.19844 [pdf, html, other]
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Title: Circles determined by planar point setsComments: 27 pages, 5 figures; ancillary files contain exact verification code and Lean 4 formalizationsSubjects: Combinatorics (math.CO)
For $n\geq 4$, let $c(n)$ be the minimum number of distinct circles containing at least three points of an $n$-point set in the Euclidean plane, where the set is neither collinear nor concyclic. Put[F(n)=1+\binom{n-1}{2}-\left\lfloor\frac{n-1}{2}\right\rfloor.]We determine $c(n)$ for every $n\geq 4$: it equals $F(n)$ apart from three exceptional orders. We also solve the variant in which no three points are collinear; that variant has a single exceptional order. The proofs and exact finite verifications were developed through a collaboration between human researchers and artificial-intelligence systems.
- [126] arXiv:2608.19847 [pdf, html, other]
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Title: A Fixed-Penalty Linearized Augmented Lagrangian Method with Classical Multiplier UpdatesComments: 26 pages, 2 figures, 3 tablesSubjects: Optimization and Control (math.OC)
Augmented Lagrangian methods are effective for nonlinear equality-constrained optimization, but solving their nonlinear primal subproblems can be expensive. For smooth nonconvex problems with deterministic or stochastic objectives, we propose a nonlinear-residual linearized augmented Lagrangian method (NR-LALM) that replaces this subproblem by a regularized Gauss-Newton-type step while retaining the classical multiplier update based on the nonlinear constraint residual. The resulting step is computed from one symmetric positive-definite linear system, but the mismatch between the linearized primal model and the nonlinear-residual update produces a quadratic constraint-linearization error in the multiplier identity. We show that this error can be controlled under local regularity; multiplier boundedness and trajectory localization are derived rather than assumed. With fixed, accuracy-independent parameters, deterministic NR-LALM finds an $\varepsilon$-approximate Karush-Kuhn-Tucker (KKT) pair in $O(\varepsilon^{-2})$ iterations and first-order oracle evaluations. For stochastic objectives, a projected stochastic path-integrated differential estimator with safeguarded restarts requires, in expectation, $O(\varepsilon^{-3})$ stochastic-gradient evaluations and $O(\varepsilon^{-2})$ constraint and Jacobian evaluations. Compactness and a Kurdyka-Lojasiewicz condition further yield finite-length convergence of the deterministic primal-dual sequence. An optional minimum-norm second-order correction reduces the constraint-linearization error from second to fourth order without changing the complexity orders. All theoretical results are formalized in Lean 4. Numerical experiments confirm the predicted error orders and show favorable performance on high-dimensional deterministic and stochastic problems.
- [127] arXiv:2608.19850 [pdf, html, other]
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Title: Resilience in Trustworthy Wireless SystemsComments: 7 pages, 6 figures, submitted to IEEE journal for possible publicationsSubjects: Information Theory (cs.IT); Signal Processing (eess.SP)
Resilience has emerged as a fundamental capability for future wireless systems operating in dynamic and uncertain environments. Although resilience has attracted growing attention across academia, industry, and standardization, its conceptual scope, enabling mechanisms, and realization techniques remain fragmented. This paper presents a systematic framework of resilience in wireless systems from a trustworthiness perspective. We first formalize the concept of resilience and distinguish it from related uncertainty-aware terminologies, including reliability, robustness, adaptability, survivability, and recoverability. We then establish a hierarchical framework that organizes resilience into capability dimensions and enabling mechanisms, and that quantifies resilience through different technical aspects. We further use physical links and unmanned aerial vehicle networks as representative wireless scenarios to demonstrate how resilience can be systematically realized through the joint design of architectures, operations, and algorithms. Finally, we examine the fundamental trade-offs in resilience engineering: Improving resilience generally incurs costs in resource efficiency, nominal performance, information acquisition, implementation complexity, and latency.
- [128] arXiv:2608.19851 [pdf, html, other]
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Title: Weighted Perimeters and pth Moments of Inertia of Convex Curves and SurfacesSubjects: Analysis of PDEs (math.AP); Optimization and Control (math.OC)
We study a shape optimization problem among convex bodies in $\mathbb{R}^n$ that minimize or maximize weighted perimeters of the form $\int_{\partial\Omega} \phi(|x|) \, \mathrm{d} H^{n-1}(x)$ under a standard perimeter constraint. We prove the existence of extremals for general weight functions in any dimension. In dimension two, we prove that the degenerate needle configuration $(-a,a)\times \{0\} \subset \mathbb{R}^2$ is the optimizer for a wide family of weights, including $|x|^p$ for $p \in (0,2]$ and $|x|^{-\alpha}$ for $\alpha \in (0,1)$, among convex curves satisfying a symmetry assumption.
- [129] arXiv:2608.19852 [pdf, html, other]
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Title: Signed list edge coloring in graphs of bounded treewidthSubjects: Combinatorics (math.CO)
Vizing conjectured that the list edge chromatic number of any graph with maximum degree $\Delta$ is at most $\Delta + 1$. This conjecture has been confirmed for several important classes of graphs, in particular, Lang proved that it holds for all graphs of treewidth $3$. In this paper, we introduce the list edge coloring of signed graphs, a framework that generalizes both classical list edge coloring and the signed edge coloring introduced by Behr. We extend Lang's result by proving the signed analogue of Vizing's conjecture for all signed graphs of treewidth $3$, as well as for signed graphs of treewidth $4$ with maximum degree $\Delta \ge 10$.
- [130] arXiv:2608.19862 [pdf, html, other]
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Title: To Control or not to ControlSubjects: Optimization and Control (math.OC)
We introduce a model with control vacations instead of standard queueing control systems with permanent control. The researched model is an M/M/1 queue with temporary periods of service rate control with two available service rates. After a control period of exponentially distributed length, a control vacation is initiated during which a fixed service rate $\mu$ is used. The start of the next control period needs to be scheduled directly at a certain cost. We will use the Markov Decision Process from Kanavetas et al. arXiv:2605.31573 to find a sufficient condition that ensures that the average expected cost can be reduced w.r.t. the model that only uses the fixed service rate. Under this condition we will use properties of this related process to construct a cost reducing policy. The process with control vacations under specific policies induces a renewal reward process. We use a Tauberian theorem to relate the average expected cost of this renewal reward process to a vanishing discount method and analytically determine a lower bound of the average expected cost reduction. Finally, we study the actual attained average cost reduction for these policies through simulation.
- [131] arXiv:2608.19869 [pdf, html, other]
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Title: Sub-optimality of Marton's Inner Bound for the Two-Receiver Broadcast ChannelSubjects: Information Theory (cs.IT); Discrete Mathematics (cs.DM)
Marton's inner bound, the best-known achievable region for a general discrete memoryless broadcast channel, was proposed by Katalin Marton in 1979, and whether it always achieves the capacity region has remained open since then. In this paper, we establish its strict sub-optimality: we show that the capacity region of some discrete memoryless broadcast channels can be strictly larger than Marton's inner bound.
- [132] arXiv:2608.19870 [pdf, html, other]
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Title: Hammock localization via Segal animaeComments: 41 pages, comments welcome!Subjects: Category Theory (math.CT); Algebraic Topology (math.AT)
We give a short, conceptual account of Dwyer--Kan's hammock localization in the setting of $\infty$-categories. Starting from Mazel-Gee's formula for localization via the relative Rezk nerve, we show that its Segalification can be described explicitly as a Segal anima of zig-zags. We also compute its mapping animae and recover and generalize Dwyer--Kan's hammock formula. As an application, we show that, when a relative $\infty$-category supports fractions, the mapping animae of localization admit a simple description. This gives a unifying treatment for several formulas of this type in the existing literature.
- [133] arXiv:2608.19872 [pdf, html, other]
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Title: New upper bounds on covering codes K_q(n,R) for alphabets of size six and sevenComments: 6 pages. Ancillary files contain the nine codes and a standalone verifierSubjects: Combinatorics (math.CO); Information Theory (cs.IT)
We present improved upper bounds for nine entries of the standard tables of bounds on K_q(n,R), the minimum cardinality of a q-ary code of length n with covering radius R, for q in {6,7}: K_6(7,3)<=232, K_6(8,3)<=1045, K_6(8,4)<=167, K_6(9,4)<=703, K_6(9,5)<=123, K_6(10,4)<=2951, K_6(10,5)<=610, K_7(8,4)<=329, and K_7(9,4)<=1743. The previous best bounds, recorded in Keri's tables (last updated 2011), all arose from general constructions (direct sums and related product rules) rather than from explicit search; to our knowledge these are the first improvements to any upper bound on K_q(n,R) with q>=5 since 2011. The new bounds were found by focused local search seeded with the construction-based incumbents. All nine codes are given explicitly in the ancillary files, together with a standalone verifier; each code was checked by four independent exhaustive verification methods.
- [134] arXiv:2608.19874 [pdf, html, other]
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Title: Combining Concurrent and Historical Functional Linear RegressionSubjects: Statistics Theory (math.ST); Methodology (stat.ME)
We study a function-on-function linear regression model in which the response at time $t$ depends on both the past trajectory of a predictor and its concurrent value. The model combines an $L^2$-historical effect with a point-evaluation effect, and these two coefficient functions are not automatically identifiable. We characterize the resulting non-identifiability and show that the concurrent and historical effects are separately identifiable whenever the covariance eigenfunctions of the covariance operator of the predictor are not pointwise square-summable. This mild and novel condition prevents the concurrent point evaluation from being represented by an $L^2$-historical effect.
Building on an orthogonalized representation of the predictor process, we propose a smoothing-spline estimator for both coefficient functions and establish consistency rates. The rates reveal an interesting trade-off between path regularity and eigenvalue decay: smoother predictor trajectories lead to faster convergence of the historical-effect estimator, whereas rougher trajectories lead to faster convergence of the concurrent-effect estimator. Simulation studies and two real-data applications demonstrate the practical importance of disentangling concurrent and historical effects. - [135] arXiv:2608.19876 [pdf, html, other]
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Title: Partizan Serial NimSubjects: Combinatorics (math.CO)
A combinatorial game is a two-player game without hidden information or chance elements. The main object of combinatorial game theory is to determine the outcome (i.e., which player has a winning strategy) of a given position in combinatorial games. NIM is a well-known and fundamental ruleset in combinatorial game theory. This paper proposes a novel partizan variant of NIM called PARTIZAN-SERIAL-NIM, defined as follows: there are $n$ piles of stones indexed by $1, 2, \ldots, n$; the two players have permutations $\mathbf{\sigma}^L$ and $\mathbf{\sigma}^R$ of $(1, 2, \ldots, n)$, respectively; a move is to remove any positive number of stones from the non-empty pile with the minimum value in the player's permutation; the player who cannot make a move loses. This ruleset is a generalization of SERIAL-NIM and PARTIZAN-END-NIM. We give an algorithm to compute the outcome of a given position in PARTIZAN-SERIAL-NIM in $O(n^2)$ time, provided that each arithmetic and comparison operation is performed in $O(1)$ time. Also, for the case where all non-empty piles have the same number $m$ of stones, we prove that the outcome does not depend on $m$ for $m \geq 2$ and present an algorithm to compute the outcome in $O(n)$ time. Further, we prove that the atomic weight of every position in PARTIZAN-SERIAL-NIM is an integer.
- [136] arXiv:2608.19886 [pdf, html, other]
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Title: Gap spectra and densities of slow Fibonacci walksComments: 12 pagesSubjects: Number Theory (math.NT); Combinatorics (math.CO)
Let $F_1=F_2=1$ and $F_{t+2}=F_{t+1}+F_t$ for $t\geq1$. For every $n\geq2$, there are unique integers $a,b,t$ such that $n=aF_t+bF_{t-1}$ with $t\geq2$ and $1\leq a\leq b\leq F_t$. The Fibonacci walk with initial pair $(b,a)$ reaches $n$ as late as possible, and the term following $n$ in this walk is $\lfloor\phi n\rfloor$ when $t$ is even and $\lceil\phi n\rceil$ when $t$ is odd, where $\phi=(1+\sqrt5)/2$. Let $D=\{d_1<d_2<\cdots\}$ and $U=\{u_1<u_2<\cdots\}$ be the sets corresponding to even and odd $t$, respectively. For $\ell,m\geq1$, define $D_\ell=\{d_{k+\ell}-d_k:k\geq1\}$, $U_\ell=\{u_{k+\ell}-u_k:k\geq1\}$, $D_\ell(m)=\{d_k:d_{k+\ell}-d_k=m\}$ and $U_\ell(m)=\{u_k:u_{k+\ell}-u_k=m\}$. Chung, Graham and Spiro conjectured that $D_\ell=U_\ell$ for all $\ell$, and asked for the densities of $D_\ell(m)$ and $U_\ell(m)$, especially when $\ell=1$. In this paper, we determine the third and fourth order gap spectra, and show that the conjecture holds for $\ell=3$ but fails for $\ell=4$. We also answer their density question by characterizing when $D_\ell(m)$ and $U_\ell(m)$ have natural densities and proving that their logarithmic densities always exist and are equal. For $\ell=1$, we give the exact logarithmic densities.
- [137] arXiv:2608.19904 [pdf, other]
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Title: Meet obstructions and saturation for the constant window convolution on graded posetsComments: 66 pagesSubjects: Algebraic Topology (math.AT); Combinatorics (math.CO); Category Theory (math.CT)
Let $\mathsf{P}$ be a finite graded poset and $\Delta_a^{\mathsf{P}}$ the height-$a$ thickening of its diagonal. We study the \emph{window convolution} $C_a=q_{1\sharp}(k_{\Delta_a^{\mathsf{P}}}\otimes^{\mathbf L}q_2^\ast(-))$ on $\mathrm{Shv}(\mathsf{P};k)$. An interleaving distance needs the left derived $\mathbb{L}C_a$ to compose as a flow, $\mathbb{L}C_a\mathbb{L}C_b\simeq\mathbb{L}C_{a+b}$; the total meet functor $\Phi$ gives rise to the canonical comparison. Finality is sufficient, and necessary where the finality defect of $\Phi$ is essential; where $\Phi$ is total at a minimal apex with unit windows, it is the failure of a length-two interval to have a single interior element. The flow fails at every branching length-two interval, and with it on the face poset of every finite regular cell complex of dimension $\ge2$. It survives on tame posets, where $\mathrm{id}\Rightarrow\mathbb{L}C_a$ gives a canonical extended interleaving pseudometric on $\operatorname{D^{b}}(\mathrm{Shv}(\mathsf{P};k))$; in the saturation cases computed here it takes no finite value above the length of $\mathsf{P}$, and is finite if and only if the derived colimits agree.
- [138] arXiv:2608.19905 [pdf, html, other]
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Title: Ramanujan Cayley Graphs with Normal Connection Sets in Ratio-One Frobenius GroupsComments: 8 pagesSubjects: Combinatorics (math.CO)
Let $G=N\rtimes H$ be a finite Frobenius group with $|N|=q$ and $|H|=q-1$. We classify all Ramanujan Cayley graphs of $G$ whose connection sets are normal, in the sense of being unions of conjugacy classes. The group-theoretic input is a simple blow-up phenomenon: every such Cayley graph is either $Y[\overline{K_q}]$ or $Y[K_q]$ for a connected regular Cayley graph $Y$ on the complement $H$. We first prove a graph-theoretic result classifying all Ramanujan graphs of these two forms when $Y$ is an arbitrary connected regular graph on $q-1$ vertices. The proof combines the classical characterization of regular graphs with least eigenvalue greater than $-2$ with a second-moment identity in the bipartite case. Translating the resulting five graph types back to $G$ yields a complete classification for all ratio-one Frobenius groups, and in particular for $\operatorname{AGL}(1,q)$ over every finite field.
- [139] arXiv:2608.19908 [pdf, html, other]
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Title: A Layered Simplex Architecture for Large AlphabetsComments: 30 pages, 7 figuresSubjects: Information Theory (cs.IT); Machine Learning (cs.LG); Machine Learning (stat.ML)
Probability estimation over large alphabets under log loss is a well-studied problem, with celebrated methods such as the Good-Turing estimator. We introduce and study a new Bayesian estimator with four notable properties. First, its construction is exceptionally simple: multiply independent uniform draws from the probability simplex coordinate-wise and renormalize. Depth is the only structural parameter, and averaging over depths eliminates the need to tune it. Second, the regret of the resulting mixture, the excess code length it pays relative to a code that knows the source, admits an explicit and efficiently computable expression. Third, despite its simplicity and lack of tuned constants, the estimator is competitive across a diverse set of synthetic and real-text benchmarks with substantially more specialized methods, including Good-Turing. Fourth, the tractability of its regret allows us to identify scaling laws in data, alphabet size, and depth. For Zipf targets with exponent above one, the regret has a simple reading as long as the sample reveals only a small fraction of the alphabet. It closely matches the description length of the set of discovered symbols, at one bit of code per bit of description, plus a further cost per symbol. The data exponent is therefore the rate at which new symbols are discovered.
- [140] arXiv:2608.19909 [pdf, other]
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Title: On the Local boundedness and higher integrability for the subcritical doubly nonlinear parabolic systemsSubjects: Analysis of PDEs (math.AP)
We consider the inhomogeneous doubly nonlinear parabolic systems of the form \begin{equation*}\partial_t (|u|^{q-1}u)-\operatorname{div}(|Du|^{p-2}Du)=\operatorname{div}(|F|^{p-2}F)\end{equation*} in a bounded space-time cylinder $\Omega_T=\Omega\times(0,T)\subset \mathbb{R}^{N+1}$. We study the local regularity properties for weak solutions in the subcritical range $p\leq\frac{N(q+1)}{N+q+1}$ and $0<q<\frac{N+2}{N-2}$. Under an extra integrability assumption $|u|\in L_{\loc}^{\rr}(\Omega_T)$, we establish a quantitative bound for $|u|$. Here, the exponent $\rr$ satisfies $\mathrm{\lambda}_{\rr}=N(p-q-1)+p\rr>0$. In addition, we prove local higher integrability of $|Du|$ in the range $p=\frac{N(q+1)}{N+q+1}$ and $\frac{N+p}{N-p}<q<\frac{N+2}{N-2}$, provided that $|u|\in L_{\loc}^{\rr}(\Omega_T)$ holds.
- [141] arXiv:2608.19913 [pdf, html, other]
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Title: On the Numerical Range of Linear Relations in Banach SpacesSubjects: Functional Analysis (math.FA)
This paper is devoted to the study of the numerical range of linear relations in Banach spaces. We present a new definition adapted to the multivalued nature of linear relations and analyze its main properties. We establish spectral inclusion results showing that the spectrum is contained in the union of the closure of the numerical range of a linear relation and the numerical range of its Banach adjoint, together with resolvent estimates related to the distance to the numerical range. As an application, we derive spectral enclosures for operator pencils by associating them with suitable linear relations and introduce corresponding numerical ranges for operator pencils in Banach spaces. We show that this approach may provide sharper information than classical numerical ranges of operator pencils. Furthermore, we use the numerical abscissas to show that the Banach space numerical range can yield strict exponential decay for semigroups that cannot be obtained from the corresponding Hilbert space numerical range.
- [142] arXiv:2608.19915 [pdf, html, other]
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Title: Efficient flatness-based computation of trajectories for vehicles with many trailersSubjects: Optimization and Control (math.OC)
We address the practical computation of feedforward steering inputs for a car with n trailers via differential flatness. The standard route (iterated symbolic differentiation of the flat output) scales prohibitively with n: even with computer-algebra software the expression sizes grow super-exponentially and become intractable beyond n approx. 5. We propose an algorithm that combines four ingredients (angular intermediate variables, factorisation of the recursion in submaps, rescaled higher-order product/quotient rules, and a composition rule based on truncated formal power series instead of Faa di Bruno's formula). We benchmark the proposed method against (i) direct symbolic differentiation in SymPy and (ii) Faa di Bruno's formula evaluated via Bell polynomials. Both reference methods hit a clear computational wall: SymPy reaches a 20 s timeout at derivative order r = 14, while the Bell-polynomial recursion exceeds a 60 s timeout beyond r = 24. The proposed approach has O(r3) arithmetic complexity in the derivative order and remains under one millisecond up to r = 40, with numerical agreement to machine precision in the regime where the reference methods succeed. As an end-to-end illustration, a 20-trailer parking manoeuvre is computed and animated.
- [143] arXiv:2608.19916 [pdf, html, other]
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Title: How many cherry-picking sequences are needed to reduce all subtrees of a phylogenetic tree?Comments: 21 pages, 7 figuresSubjects: Combinatorics (math.CO); Discrete Mathematics (cs.DM)
Phylogenetic networks are graphs that represent the evolutionary history of species. Recently, the class of orchard phylogenetic networks, which can be reduced by so-called cherry-picking sequences, has gained attention for its computational and biological aspects. In this paper, we study a fundamental question on orchards and their cherry-picking sequences by considering the CoveringNumber problem: given an orchard network $N$, how many cherry-picking sequences are needed to reduce all subnetworks of $N$? We initiate this study by considering the problem for trees. We then show that the covering number can be computed for binary trees recursively using a similar but more fine-grained notion of survival covering number. We also give a recursive formula for the survival covering number of non-binary trees. However, computing the covering number for non-binary trees appears to be considerably more challenging. For this case, we show that the covering number of star trees (whose root is adjacent to all leaves) is equivalent to the so-called SubsetConnectivity problem, which we introduce in this paper. Finally, we show that if there is no restriction on the sequence length, a single sequence of minimum length $\binom{n}{2}$ suffices to reduce all subtrees of a tree on $n$ leaves.
- [144] arXiv:2608.19921 [pdf, html, other]
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Title: Recollements of derived categories from $n$-term big tilting complexesComments: Constructive comments are highly appreciatedSubjects: Representation Theory (math.RT); Rings and Algebras (math.RA)
Let $A$ be a ring and let $\bf T$ be an $n$-term big tilting complex over $A$, represented by a bounded complex $\cpx{P}$ of projective $A$-modules. Set $B=\End_{\D{A}}(\cpx{P}), \Lambda:=\dotEnd_A(\cpx{P})$ and $\Delta:=\tau_{\leq 0}\Lambda$. The associated complex of $A$-$B$-bimodule $\cpx{T}=\cpx{P}\otimesL_{\Delta}B$ is given. We construct an extension-closed exact subcategory $\mathscr E$ of $B\Modcat$ and prove that the derived category $\D{B}$ admits a recollement by $\D{\mathscr E}$ and $\D{A}$. The proof passes through a dg double-centralizer description of a projective model of $\cpx{T}$ and an exact realisation theorem identifying $\D{\mathscr E}$ with the kernel of the derived tensor functor. We further show that $\mathscr E$ is $d$-symmetric for every $d$ not smaller than the amplitude of a perfect right $B$-model of $\cpx{T}$. Moreover, $\mathscr E$ is abelian if and only if the kernel is stable under the standard $t$-structure, equivalently, the recollement is induced by a homological ring epimorphism. In right $B$-amplitude at most one, the kernel is therefore abelian, and our construction specializes to the recollement arising from universal localisation in the two-term case. Thus the classical two-term picture extends to arbitrary finite-term big tilting complexes, with exact categories replacing abelian kernels in higher amplitude. Finally, we construct genuinely $n$-term non-compact big tilting complexes for every $n\geq2$, including an explicit three-term example whose left-hand term is determined by an algebraic Calkin-type quotient.
- [145] arXiv:2608.19923 [pdf, html, other]
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Title: On the Generalized Rational Exponents ConjectureComments: 11 pagesSubjects: Combinatorics (math.CO)
For fixed graphs $H$ and $F$, let $\ex(n,H,F)$ denote the maximum number of copies of $H$ in an $n$-vertex $F$-free graph. In this note, we prove the generalized rational exponents conjecture, posed by Gerbner and Palmer, showing that for every rational number $\alpha\ge1$, there exist fixed graphs $H_\alpha$ and $F_\alpha$ such that \[ \ex(n,H_\alpha,F_\alpha)=\Theta(n^\alpha). \] Furthermore, the counting graph $H_\alpha$ can always be chosen connected with diameter at most $3$. Our argument hinges on a localization--compression--shift framework, which transforms the Bukh--Conlon finite family construction for edges into a generalized Turán problem setting with a single forbidden graph.
- [146] arXiv:2608.19925 [pdf, html, other]
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Title: Dissipative eigenvalue problems for a singular quantum Dirac systemSubjects: Spectral Theory (math.SP)
In this paper, dissipative singular $q$-Dirac operators are examined in Hilbert space $\mathcal{L}_{q}^{2}(q^{\mathbb{Z}};\mathbb{C}^{2})$, where they arise as extensions of a minimal symmetric operator in the limit-point case. We develop a selfadjoint dilation and get its incoming and outgoing spectral representations, enabling the explicit computation of the scattering matrix associated with the dilation. Furthermore, we construct a functional model for the dissipative operator and express its characteristic function in terms of the Titchmarsh-Weyl function of the corresponding selfadjoint operator. Finally, we establish results concerning the completeness of the system of eigenvectors and associated vectors for these dissipative Dirac operators.
- [147] arXiv:2608.19931 [pdf, html, other]
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Title: Energy rigidity and weak-strong uniqueness for the 2D anisotropic Navier-Stokes equationsSubjects: Analysis of PDEs (math.AP)
In two dimensions, we show that dissipation in one spatial direction is sufficient to enforce the energy equality for every weak solution at the natural energy level. In particular, neither anomalous energy loss nor creation can occur. The main difficulty is that the missing directional regularity prevents the usual self-testing argument. We overcome this obstruction through two observations: The pressure is square-integrable by a directional Riesz-transform estimate, and the less regular component is still a renormalized solution. As an application of energy rigidity, we derive a weak-strong uniqueness principle.
- [148] arXiv:2608.19947 [pdf, html, other]
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Title: Enumeration of plane hypermaps with a mixed boundary IComments: 28 pages, 11 figuresSubjects: Combinatorics (math.CO); Mathematical Physics (math-ph); Probability (math.PR)
Plane hypermaps are plane maps endowed with a proper coloration of their inner faces in black or white. We consider the problem of enumerating plane hypermaps with prescribed face degrees and a $k$-alternating boundary condition: by this we mean the colors of inner faces incident to the outer face alternates at most $2k$ times when turning around the hypermap. The present paper deals with the cases $k=1,2$, the general case being left to the forthcoming part II. Our approach relies on the so-called slice decomposition and uses crucially the notion of accessibility, which exploits the canonical orientation of hypermaps and the marking variable $t$ associated with vertices, to enumerate pointed hypermaps by decomposing them according to the set of all vertices that can access to the marked vertex. This process enables us to express the generating functions of hypermaps with mixed boundaries in terms of the generating functions of hypermap slices and to recover, in a purely combinatorial way, some formulas previously obtained through algebraic methods.
- [149] arXiv:2608.19949 [pdf, html, other]
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Title: Classification of global solutions to the singular equation $-Δu=f(X)\cdot u^{-γ}$ in a Lipschitz epigraphical coneSubjects: Analysis of PDEs (math.AP)
We construct and classify all global solutions to the singular equation $$-\Delta u=f(X)\cdot u^{-\gamma}$$ supported in a general Lipschitz epigraphical cone, where $f(X)$ is a locally Dini continuous function with $0<\lambda\leq f(X)\leq\Lambda$. The existence and non-existence of a global solution is solely determined by the exponent $\gamma$ of the equation and the ``frequency" of the cone.
Moreover, in order to classify all global solutions, we introduce several new methods. First, we use the local data to estimate the global growth rate, which in turn establishes the boundedness of the ``asymptotic slope" of the global solution. Second, by establishing a nonlinear variant of Kemper's boundary Harnack principle, we classify all global solutions through an ``oscillation reduction" argument on the ``asymptotic slope". - [150] arXiv:2608.19951 [pdf, html, other]
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Title: Decompositions of good involutions on quandlesComments: 27 pagesSubjects: Geometric Topology (math.GT)
We study the behavior of good involutions under two fundamental constructions of quandles: interaction-free unions and direct products. In particular, we show that the set of good involutions on a quandle decomposes naturally into the set of involutions on its maximal trivial component and the set of good involutions on the complement. Moreover, we construct an example of a connected noninvolutory symmetric quandle with multiple good involutions, which serves as a counterexample to a conjecture by Ta.
- [151] arXiv:2608.19958 [pdf, html, other]
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Title: The Symmetry and Linear Stability of Convex 1+5 Coorbital Central Configurations with Homogeneous PotentialComments: 14 pages, 1 figures, 28 conferencesSubjects: Dynamical Systems (math.DS)
For the planar Newtonian 1+N-body problem when the N masses tend to zero, the corresponding relative equilibria become coorbital around the dominant mass. In this work, we focus on convex central configurations in the planar 1+N coorbital problem. For the 1+5 coorbital problem with the homogeneous potential, we prove that any convex coorbital central configuration with symmetric masses must have an axis of symmetry. Furthermore, under explicit restrictions on the angular variables in a homogeneous potential, we prove the linear stability of both convex symmetric 1+5 and convex 1+N coorbital central configurations.
- [152] arXiv:2608.19961 [pdf, html, other]
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Title: Fractional revival on oriented Cayley and semi-Cayley graphs over abelian groupsSubjects: Combinatorics (math.CO)
Fractional revival (FR), a generalization of perfect state transfer (PST), is a significant phenomenon in quantum state transfer that allows quantum information to be transmitted between two qubits with a certain probability. The existence of FR has been extensively studied on many classes of graphs. However, oriented graphs have not yet been investigated. In this paper, we investigate the existence of proper FR on oriented graphs. We first establish necessary and sufficient conditions for oriented graphs to admit proper FR between strongly cospectral vertices. Furthermore, we prove that oriented Cayley graphs over abelian groups do not admit proper FR, and we subsequently characterize the conditions under which oriented semi-Cayley graphs over abelian groups admit proper FR.
- [153] arXiv:2608.19962 [pdf, html, other]
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Title: Spectral minimal partitions of combinatorial graphsSubjects: Spectral Theory (math.SP)
This paper investigates spectral minimal partitions for weighted graphs, thus extending the extensive class of results that are currently available on domains and, to a lesser extent, manifolds and metric graphs. We provide a rigorous framework for analyzing graph Laplacians under Dirichlet, Neumann, and boundaryless energy formulations; a central focus of the study is establishing existence theorems for minimal partitions. While existence is straightforward for finite connected graphs due to the finiteness of the class of admissible partitions, infinite graphs require advanced topological and functional-analytic machinery. Specifically, we introduce the notion of canonical compactifiability, which relates to compact embeddings and uniform Poincaré-type constants for Neumann and boundaryless energies; and an appropriate notion of subgraph convergence. In this way, we can relax the spectral minimal problem on infinite graphs by reducing it to the study of finite graphs; and can, thus, guarantee that optimal spectral energies are actually attained by appropriate partitions even in non-compact settings.
- [154] arXiv:2608.19970 [pdf, html, other]
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Title: Explicit birational models of marked elliptic surfaces of relative degree twoComments: 36 pages, 2 figures, 5 tables. Comments are welcome!Subjects: Algebraic Geometry (math.AG)
We introduce explicit construction methods for two distinct birational models of elliptic surfaces equipped with a bisection --- or, more generally, a relative polarisation of degree two --- and double fibres. For Halphen surfaces of index two and Enriques surfaces, we illustrate how these give simple descriptions inside a toric variety.
- [155] arXiv:2608.19972 [pdf, html, other]
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Title: Exceptional sets for compositions involving Euler's function,the divisor-sum function and Dedekind's functionSubjects: Number Theory (math.NT)
Let \(\psi(n)\), \(\phi(n)\), and \(\sigma(n)\) denote Dedekind's arithmetic function, Euler's totient function, and the sum-of-divisors function, respectively. We study exceptional sets associated with the compositions \(\phi(\psi(n))\), \(\phi(\sigma(n))\), \(\psi(\psi(n))\), and \(\psi(\sigma(n))\). For every fixed \(c>0\), we obtain quantitative upper bounds for the numbers of integers \(n\le x\) satisfying \(\phi(\psi(n))\ge cn\) and \(\phi(\sigma(n))\ge cn\), thereby refining density results of Sándor and Dixit and Bhattacharjee, respectively. We further establish quantitative density-zero estimates for the sets of integers \(n\le x\) for which \(\psi(\psi(n))\le cn\) or \(\psi(\sigma(n))\le cn\). More generally, we allow the fixed thresholds in the first two problems to be replaced by thresholds involving non-decreasing functions subject to mild growth conditions.
- [156] arXiv:2608.19976 [pdf, html, other]
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Title: Random valuationsComments: 33 pagesSubjects: Probability (math.PR); Metric Geometry (math.MG)
A valuation is a finitely additive function on the family of compact convex sets in $\mathbb{R}^d$. We study non-negative infinitely divisible random valuations, with particular emphasis on monotone, $\sigma$-continuous models with independent increments along nested families. After separating the deterministic part, we show that the Lévy measure of such a valuation is generated by pairs $(F,r)$, where $F$ is a non-empty closed convex set and $r>0$, with each pair contributing $r\mathbf{1}_{F\cap K=\emptyset}$. This yields a Poisson representation and an equivalent formulation through a pure-jump completely random measure on the space of closed convex sets. For stationary valuations, we derive a cylinder-Grassmannian representation of the Lévy measure. In the stationary isotropic case, we obtain a McMullen-type decomposition, at the level of one-dimensional distributions, into independent components stable under dilation of the argument.
- [157] arXiv:2608.19978 [pdf, html, other]
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Title: Generalized Hamming weights of codes arising from complete intersectionComments: 5 pagesSubjects: Commutative Algebra (math.AC); Information Theory (cs.IT)
We provide a positive answer to a conjecture proposed by Tohǎneanu and Van Tuyl regarding the minimum distance of codes whose underlying set of points is a reduced complete intersection. Despite the technical nature of the conjecture, we show that it follows directly from a not-well-known refinement of the classical Bézout bound for overdetermined polynomial systems. For completeness, this paper presents a self-contained proof of this refined bound. Furthermore, we show that using the same approach, it is possible to obtain a bound on the generalized Hamming weights of such a code and, more generally, to control the minimum distance of the codes obtained by evaluating forms of degree $d$ on the points of a zero-dimensional complete intersection.
- [158] arXiv:2608.19984 [pdf, html, other]
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Title: Internal numerical semigroupsComments: 23 pages for main paper + 16 pages of appendix with dataSubjects: Group Theory (math.GR); Combinatorics (math.CO)
In this paper the tree structure of numerical semigroups is studied. An internal numerical semigroup is a semigroup located in an internal node of the tree. Analogous a leaf numerical semigroup is placed in a leaf node. Internal semigroups with fixed multiplicity, Frobenius number or genus are studied by providing algorithms to construct all of them. Several conjectures are established, for example, in each case (fixed multiplicity, fixed Frobenius number and fixed genus respectively), the results suggest that there are always more internal than leaf numerical semigroups. Finally, numerical semigroups with fixed multiplicity and Frobenius number simultaneously are investigated. In this case with two invariants fixed, moreover closed formulas to count the number of internal and leaf semigroups are provided for some values of multiplicity and Frobenius number.
- [159] arXiv:2608.19986 [pdf, html, other]
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Title: Weakly stable solutions of Serrin's problemComments: All comments welcomeSubjects: Analysis of PDEs (math.AP)
We prove that weakly stable solutions of Serrin's problem are compact and therefore round balls.
- [160] arXiv:2608.19989 [pdf, other]
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Title: On facet gaps of order and chain polytopesComments: 6 pagesSubjects: Combinatorics (math.CO)
We discuss three questions from a recent paper of Bhandari, Cunningham, Morrell, Oh and Smith. We obtain an exact local formula for the facet gap between the order and the chain polytope of a finite poset. This gives a classification of the case $\gap(P)=2$ in terms of star elements. For marked chain--order polytopes, the same local weights determine the facet differences between all admissible decompositions.
- [161] arXiv:2608.19991 [pdf, html, other]
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Title: Mean curvature bounds for the obstacle problemSubjects: Analysis of PDEs (math.AP)
We prove optimal $(n-1)$-semiconvexity estimates for solutions to the classical obstacle problem, possibly with a smooth right-hand side. We deduce that the mean curvature of the free boundary is universally bounded above in all dimensions. Existing examples show that full curvature bounds fail in dimensions three and higher. Noticeably, these results are novel even for a constant right-hand side in the $n=2$ case.
- [162] arXiv:2608.19996 [pdf, html, other]
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Title: Hamiltonian Floer Theory for Quantum Electrodynamics up to First Order in $\hbar$Subjects: Symplectic Geometry (math.SG); Mathematical Physics (math-ph); Probability (math.PR)
We use Hamiltonian Floer theory to prove a cuplength result about the existence of periodic solutions of particle-field systems with a Gaussian random field. As a concrete model we study stochastic electrodynamics, which approximates quantum electrodynamics up to first order in $\hbar$, and is even exact in the case of low-order Hamiltonians or when the particles are treated classically.
- [163] arXiv:2608.19997 [pdf, html, other]
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Title: Generalized inverses of strictly monotone transformationsSubjects: Classical Analysis and ODEs (math.CA); Functional Analysis (math.FA)
Let $ C \subseteq \mathbb{R}^n $ be a convex set. The mapping $ f : C \longrightarrow \mathbb{R}^n $ is strictly increasing, if $ \langle f(x)-f(y), x-y \rangle > 0 $ for all distinct elements $ x,y \in C $. Applying classical theorems of finite dimensional convex geometry and convex analysis, we show that the inverse functions has a unique extension $ f^{(-1)} : \mathrm{conv} (f(C)) \longrightarrow C $ such that $ f^{(-1)} $ is monotone, continuous and it acts as a left-inverse of $ f $. As an application we introduce the concept of vector-valued weighted quasi-arithmetic means and discuss their equality problem.
- [164] arXiv:2608.19999 [pdf, other]
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Title: On a visco-elastic Mullins-Sekerka SystemSubjects: Analysis of PDEs (math.AP)
We introduce a novel visco-elastic Mullins-Sekerka system with a prescribed constant contact angle at the boundary. The system is derived as an $H^{-1}$-$H^1$-type gradient flow of an energy consisting of the perimeter together with capillary, elastic, and second-gradient contributions. Building on the framework of Hensel and Stinson (Arch. Ration. Mech. Anal. 248, 2024), we introduce a measure-valued solution concept featuring a sharp De Giorgi-type energy-dissipation inequality. Moreover, we establish existence of solutions via an implicit time discretization scheme, and prove existence of $BV$ solutions under an energy-conservation hypothesis.
- [165] arXiv:2608.20001 [pdf, html, other]
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Title: Bayesian inference and retrodiction for faithful states on von Neumann algebrasComments: Comments are welcome! 30 pages + appendix + bibliographySubjects: Operator Algebras (math.OA); Category Theory (math.CT); Quantum Physics (quant-ph)
Retrodiction is the act of inferring a cause from its effects, the most common example of which is Bayesian inference. Retrodiction can be defined by its structural process-theoretic properties, which are mathematically captured by category theory. This categorical definition of retrodiction has recently been shown to potentially isolate the Petz recovery map as a unique universal candidate for quantum Bayesian inference. This paper extends these results to the infinite-dimensional setting on von Neumann algebras. In the process, we provide a pedagogical review of the Petz recovery map in infinite dimensions and its relation to the more commonly used expression in the finite-dimensional setting. We formalize the open question as to whether these categorical axioms for retrodiction do in fact uniquely characterize the Petz recovery map. If such a characterization holds, this would show that Bayesian inversion and the Petz recovery map are structural necessities and not simply useful algorithms for classical and quantum inference.
- [166] arXiv:2608.20002 [pdf, html, other]
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Title: Forgotten charactersComments: 13 pagesSubjects: Combinatorics (math.CO)
A partial permutation of $[n] := \{1,\dots,n\}$ is a bijection $g: I \to J$ between two subsets $I,J \subseteq [n]$. Given a partial permutation $g$ of $[n]$, let $a_g \in \mathbb{C}[\mathfrak{S}_n]$ be the group algebra sum of those permutations $w \in \mathfrak{S}_n$ which extend $g$. Informally, a partial permutation $g$ is obtained by forgetting some data in a genuine permutation. The forgotten symmetric functions are the least-studied of the six `standard' bases for the ring of symmetric functions. We show that forgotten symmetric functions arise naturally in class function evaluations on partial permutations.
- [167] arXiv:2608.20004 [pdf, html, other]
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Title: Sets of nice recurrence are partition regularComments: 6 pagesSubjects: Dynamical Systems (math.DS); Combinatorics (math.CO)
A set of positive integers $R$ is called a set of nice recurrence if for any measure preserving system $(X,\mu,T)$, for all measurable $A\subseteq X$, and each $\varepsilon>0$, there exists $n\in R$ such that $\mu(A\cap T^{-n}A)\geqslant \mu(A)^2 - \varepsilon$. Answering a long-standing question of Bergelson, we show that sets of nice recurrence have the following Ramsey property: any finite colouring of a set of nice recurrence admits a monochromatic set of nice recurrence.
- [168] arXiv:2608.20007 [pdf, html, other]
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Title: Real Interference Alignment for Active IRS-Aided Systems: A Rate-Profile Learning-Based ApproachSubjects: Information Theory (cs.IT)
With additional spatial degrees of freedom provided by the active intelligent reflecting surface (IRS), interference alignment (IA) can be achieved at low cost. In this letter, we propose a real IA scheme for an active IRS-aided system. The proposed scheme only requires the IRS to know the instantaneous channel coefficients under the assumption of blocked direct links. To maximize the achievable sum rate subject to individual minimum rate requirements and transmission power constraints, we propose a rate-profile learning-based algorithm. The algorithm uses offline-trained achievable rate profiles to decouple the original problem into multiple feasibility subproblems, which are then solved by generalized eigenvalue decomposition. Simulation results demonstrate that our proposed algorithm outperforms the conventional weighted minimum mean square error algorithm, while requiring significantly less program execution time.
- [169] arXiv:2608.20008 [pdf, html, other]
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Title: An arithmetic approach to parabolic multiplicity in complex dynamicsComments: 26 pagesSubjects: Dynamical Systems (math.DS)
When $\omega$ is a primitive $n$-th root of unity, the quadratic polynomial $F(z) = \omega z (1 -z)$ and the entire map $F(z) = \omega z \mathrm{e}^{-z}$ both have a parabolic fixed point at $0$. Their parabolic multiplicity is equal to $1$, that is, $F^{\circ n}(z) = z \bigl( 1 +c z^n +\mathcal{O}(z^{n+1}) \bigr)$ with $c \neq 0$. The classical proof of this fact is transcendental. We present an arithmetic proof which may be extracted from [Towards global models near homoclinic tangencies of dissipative diffeomorphisms; H. Broer, C. Simó, J.C. Tatjer] in the transcendental case and requires working in $\mathbb{Z}/(n -1) \mathbb{Z}$, and which is new in the polynomial case and requires working in the $p$-adic field $\mathbb{Q}_p$ for a suitable prime $p$ such that the order of $2$ in $(\mathbb{Z}/p \mathbb{Z})^\times$ is exactly $n$.
- [170] arXiv:2608.20010 [pdf, html, other]
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Title: Graphs with connectivity $3/4 - \varepsilon$ are globally synchronizingComments: 17 pagesSubjects: Combinatorics (math.CO); Dynamical Systems (math.DS)
We study synchronization in the Kuramoto model on finite graphs. We prove that there is an absolute constant $\eta>0$ such that every finite simple graph $G$ on $n$ vertices with minimum degree at least $(3/4-\eta)n$ has no local minima of the Kuramoto energy other than the fully synchronized states. This strictly improves the previous $3/4$ upper bound and refutes a conjecture of Bandeira, Kireeva, Maillard, and Rödder.
- [171] arXiv:2608.20012 [pdf, html, other]
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Title: Proof of Lichiardopol's conjecture on disjoint directed cycles of distinct lengthsSubjects: Combinatorics (math.CO)
There is a fascinating array of interrelated questions studying which structures can be guaranteed in digraphs of large minimum out-degree. These often have intriguingly simple statements, yet seem surprisingly difficult to approach. A well-known example is Lichiardopol's conjecture (2014), stating that there exists a function $g:\mathbb{N}\rightarrow \mathbb{N}$ such that every digraph with minimum out-degree at least $g(k)$ contains $k$ vertex-disjoint directed cycles of distinct lengths.
In this paper, building on earlier work of the second author, we confirm this conjecture in full generality. We also generalise this result to a weighted setting. Our proof uses and combines many ingredients from structural digraph theory such as butterfly minors, directed tangles, a directed analogue of the Tangle-Wall Theorem due to Robertson and Seymour as well as a local variant of the Directed Flat Wall Theorem due to Giannopoulou, Kawarabayashi, Kreutzer and Kwon. These techniques, which are somewhat atypical in the study of minimum degree conditions, may be of independent interest and may find further applications. - [172] arXiv:2608.20023 [pdf, html, other]
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Title: A Variational Characterization of Positive Scalar Curvature Kähler metricsComments: 45 pages, all comments are welcomeSubjects: Differential Geometry (math.DG)
We introduce the prescribed scalar curvature measure equation on a compact Kähler manifold. For a Kähler class of positive total scalar curvature, we prove that the following are equivalent: the existence of a positive scalar curvature Kähler metric, solvability of this equation for every admissible measure, $d_1$-coercivity of the associated functionals, and uniform geodesic stability along finite-energy $d_1$-geodesic rays. As a consequence, in each fixed Kähler class, the space of positive scalar curvature Kähler metrics is either empty or contractible. We further prove that every Kähler class on a positive-dimensional compact smooth toric Kähler manifold contains a torus-invariant metric of positive scalar curvature. Therefore, for every Kähler class, the prescribed scalar curvature measure equation admits a smooth solution for every admissible measure, unique modulo constants.
- [173] arXiv:2608.20028 [pdf, html, other]
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Title: Simultaneous approximation to pairs of real numbersSubjects: Number Theory (math.NT)
Let $B_{m}$ and $D_{n}$ be the denominators of the $m$th and $n$th convergent of the real numbers $\alpha$ and $\beta$, respectively. We introduce an abnormal method in the theory of simultaneous Diophantine approximation to the pair $\alpha,\beta$. Namely, the question of finding simultaneous approximation is turned into study of small solutions of a linear Diophantine equation $xB_{m} + yB_{m+1} = zD_{n} + vD_{n+1}$. The Thue-Siegel's lemma guarantees the existence of a non-zero integer vector $(x,y,z,v)$ in such a way that $|x|,|y|,|z|,|v|$ are bounded above by $\big( B_{m} + B_{m+1} + D_{n} + D_{n+1} \big)^{1/3}$. Thereby we can construct an integer $q:=xB_{m} + yB_{m+1} = zD_{n} + vD_{n+1}\ge 1$, a common denominator, which by the theory of continued fractions gives simultaneous approximations to $\alpha$ and $\beta$. We give also a variety of explicit constructions without the Thue-Siegel's lemma. Let $\|\alpha\|:=\underset{k\in\mathbb{Z}}\min\{|\alpha-k|\}$. For a class of numbers, including particular equivalent numbers $\alpha$ and $\beta$, we show there exist a real number $\kappa=\kappa(\alpha,\beta)>1/2$ and infinitely many explicitly constructible positive integers $q$ such that $\|q\alpha\| \le \frac{1}{q^{\kappa}}$ and $\|q\beta\| \le \frac{1}{q^{\kappa}}$. As the result improves Dirichlet's theorem on simultaneous approximation it also confirms the classical Littlewood conjecture for such a pair $\alpha,\beta$. In addition, we present general criteria for the classical Littlewood conjecture as well as for its $p$-adic counterpart.
- [174] arXiv:2608.20029 [pdf, html, other]
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Title: Universal torsors over quartic del Pezzo surfaces and stable rationalityComments: 24 pagesSubjects: Algebraic Geometry (math.AG)
Let $S$ be a smooth quartic del Pezzo surface over a field $k$, of characteristic zero. We prove that a universal torsor $\mathcal T$ over $S$ is $k$-rational, provided $\mathcal T$ has $k$-points. As an application, we obtain examples of stably rational smooth cubic hypersurfaces over $\mathbb Q$ in every dimension greater than 2.
- [175] arXiv:2608.20030 [pdf, html, other]
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Title: Positive scalar curvature Kähler metrics and shifted $K$-stabilityComments: 17 pages, all comments are welcomeSubjects: Differential Geometry (math.DG); Algebraic Geometry (math.AG)
In this paper, we show that the quantitative data underlying $K$-stability, originally developed in the study of canonical metrics, also detect the existence of positive scalar curvature Kähler metrics in a fixed class.
- [176] arXiv:2608.20034 [pdf, html, other]
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Title: A Comprehensive p-VEM Framework for Advanced Variable Stiffness Plates with Arbitrary ShapesComments: 55 pages, 24 figures, 6 tablesSubjects: Numerical Analysis (math.NA)
This paper presents a comprehensive, high-order (p-version) Virtual Element Method (VEM) framework for the structural analysis of innovative variable stiffness plates. VEM is particularly suited for complex configurations due to its ability to handle arbitrary polygonal meshes, including curved edges. However, its mathematical formulation may hinder its spread in the engineering community. This work illustrates a formulation with an accessible implementation using well-known FEM notation and integrating at the same time a set of new advanced capabilities. Specifically, both standard stabilized and advanced self-stabilized strategies are adopted. To further improve the robustness of VEM in the presence of variable coefficients, polynomial projections taking into account the coefficients are employed. This approach is referred to as Variable Coefficients-VEM approach (VC-VEM). This unified framework is applied to linear static, free-vibration, and buckling analyses, and validated against analytical solutions and numerical benchmarks. In particular, plates with cutouts and problems featuring high-gradient solutions are investigated, demonstrating that the proposed comprehensive approach provides a flexible and ready-to-implement tool for advanced structural design.
- [177] arXiv:2608.20036 [pdf, html, other]
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Title: Broadcast Domination Number is at Most Twice the Multipacking NumberSubjects: Combinatorics (math.CO); Discrete Mathematics (cs.DM)
For a graph $ G = (V, E) $ with a vertex set $ V $ and an edge set $ E $, a function
$ f : V \rightarrow \{0, 1, 2, . . . , diam(G)\} $ is called a \emph{broadcast} on $ G $. For each
vertex $ u \in V $, if there exists a vertex $ v $ in $ G $ (possibly, $ u = v $) such that $ f (v) > 0 $ and
$ d(u, v) \leq f (v) $, then $ f $ is called a dominating broadcast on $ G $. The cost of the dominating broadcast $f$ is the quantity $ \sum_{v\in V}f(v) $. The minimum cost of a
dominating broadcast is the broadcast domination number of $G$, denoted by $ \gamma_{b}(G) $.
A multipacking is a set $ M \subseteq V $ in a
graph $ G = (V, E) $ such that for every vertex $ v \in V $ and for every integer $ r \geq 1 $, the
ball of radius $ r $ around $ v $ contains at most $ r $ vertices of $ M $, that is, there are at most
$ r $ vertices in $ M $ at a distance at most $ r $ from $ v $ in $ G $. The
multipacking number of $ G $ is the maximum cardinality of a multipacking of $ G $ and
is denoted by $ mp(G) $.
It is known that $mp(G)\leq\gamma_b(G)$. In 2014, Hartnell and Mynhardt proved that $\gamma_b(G)\leq 3mp(G)-2$ whenever $mp(G)\geq2$. In 2019, Beaudou, Brewster, and Foucaud improved this bound to $\gamma_b(G)\leq 2 mp(G)+3$ and conjectured that $\gamma_b(G)\leq 2 mp(G)$. We solve their conjecture by proving that $\gamma_b(G)\leq 2 mp(G)$ for every graph $G$. Our proof is constructive and yields a polynomial-time $2$-approximation algorithm for Maximum Multipacking problem which improves the earlier approximation factor $2+o(1)$. - [178] arXiv:2608.20037 [pdf, html, other]
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Title: Inverse Hessian Curvature Flow in Minkowski Space II: The Dirichlet problem at infinitySubjects: Differential Geometry (math.DG)
This paper studies self-shrinkers and the long-time behavior of the inverse $\sigma_k$ curvature flow for noncompact entire spacelike strictly convex hypersurfaces in Minkowski space. In contrast to the co-compact setting, where the corresponding self-shrinker is rigid, the noncompact problem admits a rich family of self-shrinkers determined by their asymptotic data. More precisely, we formulate the self-shrinker equation as a fully nonlinear Dirichlet problem on hyperbolic space with prescribed data on its ideal boundary, and prove that every continuous negative boundary value determines a unique entire spacelike strictly convex self-shrinker. We further study the inverse $\sigma_k$ curvature flow starting from an entire spacelike strictly convex hypersurface satisfying a subsolution condition at infinity and a uniform positive lower bound for its $\sigma_k$ curvature. We prove global existence of the flow and show that the normalized flow converges locally smoothly to the unique self-shrinker with the same prescribed boundary value at infinity.
- [179] arXiv:2608.20040 [pdf, html, other]
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Title: Rigidity of shrinking gradient ricci soliton with constant scalar curvatureSubjects: Differential Geometry (math.DG)
Let $(M^n, g, f)$ be a complete shrinking gradient Ricci soliton with constant scalar curvature. Under the assumptions that
(i) $Ric \geq \frac{\nabla_{\nabla f}Ric}{f}$ on $M\setminus D$, where $D$ is a compact set over $M$;
(ii) $(M^n, g, f)$ smoothly converges to $\mathbb{R}^2 \times \mathbb{S}^{n-2}$,
we conclude that $(M^n, g, f)$ is isometric to $\mathbb{R}^2 \times \mathbb{S}^{n-2}$. Notably, condition \textup{(i)} is weaker than the radial flatness condition in \cite{Petersen-Wylie2}. - [180] arXiv:2608.20045 [pdf, html, other]
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Title: A dichotomy for the number of vertex-critical ($P_5$, $H$)-free graphs when $H$ is bipartiteSubjects: Combinatorics (math.CO)
A graph $G$ is $k$-vertex-critical if $\chi(G)=k$, but $\chi(H)<k$ for every induced subgraph $H$ of $G$. A graph $G$ is $(H_1,H_2,\dots,H_m)$-free if does not contain $H_i$ as an induced subgraph for any $i\in\{1,2,\dots,m\}$.We provide the following dichotomy that for bipartite graphs $H$ and any fixed integer $k\ge 5$ , there are only finitely many $k$-vertex-critical $(P_5,H)$-free graphs if and only if $H$ is $2P_2$-free. This leads us to pose the problem about determining for which graphs $H$ with $\chi(H)\ge 3$ there are infinitely many $k$-vertex-critical $(P_5,H)$-free graphs for all $k\ge 5$. Toward this problem, we show that there only finitely many $k$-vertex-critical $(P_5, K_{s,t}+e)$-free graphs for all $k,s,t\ge 1$, where $K_{s,t}+e$ is a complete bipartite graph plus a single edge. On the other hand, we show that there are infinitely many $k$-vertex-critical $(P_5,\operatorname{net},\operatorname{co-net},\overline{C_5},\overline{C_6},\dots\overline{C_{k-1}})$-free graphs for all $k\ge 5$. We also show that there are only finitely many $k$-vertex-critical $(P_4+\ell P_1,\overline{L(K_{2,n})})$-free graphs for all $\ell,n\ge 0$, providing the largest known subfamily of $(P_4+\ell P_1)$-free graphs to satisfy this property.
Our results, together with known results, imply the existence of new polynomial-time certifying algorithms to determine the $k$-colourability of many subfamilies of $P_5$-free and $(P_4+\ell P_1)$-free graphs for fixed $k\ge 5$. Our proof techniques apply a powerful theorem of Chudnovsky, Kim, Oum, and Seymour (2016) on prime graphs that we expect to be of interest and have further applications to bounding the number of $k$-vertex-critical graphs in other hereditary families of graphs. - [181] arXiv:2608.20048 [pdf, html, other]
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Title: The Prescribed-Vertex Semidegree Threshold for Directed $3q$-Cycles in Oriented GraphsSubjects: Combinatorics (math.CO)
For every $q\ge2$, we prove that every oriented graph $G$ on $n\ge45q-8$ vertices whose minimum semidegree satisfies \[
\delta^0(G)\ge \left\lceil\frac n3\right\rceil \] contains a directed cycle of length $3q$ through every vertex. The semidegree bound is sharp. This closes the one-unit gap left by the prescribed-vertex theorem of Kelly, Kühn and Osthus when $3\mid n$. We also prove that if an oriented graph $H$ has order $N$, minimum semidegree $d\ge3$, and $7d\ge2N+3$, then every ordered pair of distinct vertices is joined by a path of length three, four, or five. The constant $+3$ is best possible. As a consequence, the order hypothesis $n\ge10^{10}\ell$ in the general prescribed-vertex theorem of Kelly, Kühn and Osthus can be replaced by $n\ge15\ell-60$ for $\ell\ge7$. - [182] arXiv:2608.20051 [pdf, html, other]
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Title: A Canonical m-Atomic Decomposition of Bipartite Graphs via a Grid ModelComments: 60 pages. Structural decomposition of bipartite graphs via a grid model. Introduces the characteristic m of a brick, determines it for disconnected excessive bricks, shows that m-excessiveness implies m-extendability with the same m, extending Plummer's notion to unbalanced graphs, and classifies bipartite graphs into eleven structural classesSubjects: Combinatorics (math.CO); Data Structures and Algorithms (cs.DS)
We study finite, connected, simple bipartite graphs in a grid model, in which a graph is drawn as a rectangular array and its structure is read off from empty subrectangles, called holes. In this model we attach to every brick a numerical invariant, its characteristic m, the difference between the number of rows and the largest proper independent set. A brick is excessive if m > 0.
Our main results concern this invariant. We determine the characteristic of a disconnected excessive brick from those of its components, showing that m = min_i min{m_i, imb(W_i)} while the imbalance is additive; and we prove that an m-excessive brick is m-extendable, that is, every matching of size m extends to a maximum matching. Since Plummer's notion of n-extendability is defined only for graphs carrying a perfect matching, and our proof nowhere uses balance, the characteristic extends that notion canonically to unbalanced bipartite graphs. Using the characteristic we partition bipartite graphs into eleven structural classes.
The underlying decomposition into atomic blocks is the classical decomposition into elementary components, and the description of the maximum proper independent sets by ideals of the block poset is likewise classical; the paper states precisely which results are classical and are not claimed here. What the grid model adds is a single geometric framework in which holes, characteristics and the block triangular form are read off from one picture. - [183] arXiv:2608.20058 [pdf, html, other]
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Title: Nearly tight framelet systems from nested Marcinkiewicz--Zygmund measures on compact Riemannian manifoldsComments: 31 pages, 7 pagesSubjects: Numerical Analysis (math.NA)
Tight framelet systems on manifolds provide exact energy preservation and one-pass reconstruction, but their standard semi-discretization relies on polynomial-exact quadrature rules, which are difficult to reconcile with scattered and progressively refined data. We develop nested Marcinkiewicz--Zygmund (MZ) measures on compact Riemannian manifolds as a quantitative alternative. Using dyadic quasi-uniform nested point sets and local partition weights, we establish a sequence of deterministic MZ inequalities for diffusion polynomial spaces. The result has two complementary forms: a fixed-tolerance version, in which the polynomial bandwidth grows dyadically with the level, and a fixed-bandwidth refinement version, in which the MZ tolerance improves as more nested nodes are added. When these measures are used to discretize continuous tight framelets, the MZ tolerance transfers directly to the deviation of the frame bounds from one. Thus quadrature exactness is relaxed with a controlled loss of tightness, and near-tightness improves along the same nested hierarchy. For the fully discrete setting, we develop filter-bank analysis and synthesis procedures, characterize one-pass and canonical reconstruction, and show that the MZ tolerance also controls the conditioning of the frame-operator equation. Fast implementation of filter-bank transforms is also presented. Numerical experiments on the sphere and the flat torus illustrate the resulting multiscale decompositions and reconstruction behavior.
- [184] arXiv:2608.20060 [pdf, html, other]
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Title: Stability of admissible solutions for coexisting phase transitions for one-dimensional compressible van der Waals fluidsComments: 29 pages, 3 figuresSubjects: Analysis of PDEs (math.AP)
In this paper, we investigate the dynamic stability of certain steady-state solutions to the periodic boundary value problem for compressible isentropic Navier-Stokes system under the van der Waals equation of state in one space dimension. These steady-state solutions correspond to the admissible solutions describing two-phase coexisting phase transitions, where the integral average of the specific volume belongs to the Maxwell region. We first construct a semi-discrete staggered grid difference scheme to prove the local existence of solutions to the periodic problem, without imposing the standard stability hypothesis \(p_v<0\). Then, by virtue of rigorous piecewise a priori estimates, we demonstrate that the periodic boundary value problem for van der Waals fluids possesses a global solution existing for all time, and this solution converges uniformly to the admissible steady state as time tends to infinity. This result firmly establishes the nonlinear stability of the admissible phase-transition solutions under general small initial disturbances.
- [185] arXiv:2608.20064 [pdf, html, other]
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Title: Classification of Deza graphs from anisotropic association schemes of quadricsSubjects: Combinatorics (math.CO)
Let $Q^\varepsilon(3,q)$, where $\varepsilon\in\{+,-\}$ and $q>3$ is odd, be a non-degenerate hyperbolic or elliptic quadric of $PG(3,q)$. Fix one of the two quadratic classes of anisotropic points. Since the line joining two distinct points of this class is tangent, secant, or external to the quadric, one obtains a $3$-class association scheme. We classify all non-trivial unions of its relations which define Deza graphs. In addition to the previously known tangency family, exactly four exceptional strictly Deza graphs occur, with parameters $(360,135,54,45)$, $(369,108,36,27)$, $(65,34,18,15)$ and $(168,111,75,70)$. We determine their spectra and Deza children and give geometric or group-theoretic descriptions of all four exceptional graphs.
- [186] arXiv:2608.20066 [pdf, html, other]
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Title: Resolution of Singularities in Positive Characteristic: Frobenius-Hasse Towers and Exceptional-History DescentComments: 798 pages in nine parts. Program page is available at this https URL Comments are welcomeSubjects: General Mathematics (math.GM)
Let k be a perfect field of characteristic p>0. We introduce an object-level construction for canonical strong embedded resolution and principalization over k. The construction program replaces monotonicity of pointwise numerical invariants by a well-founded history of addressed comparison factors. Starting from the differential-integral saturation of a marked Rees algebra, we construct total-Hasse activity packets, filtered coefficient cubes, semilinear Frobenius-Hasse sources, and literal transform data for ordinary permissible blowups. A six-row defect calculus routes local problems to certified surface, toroidal-monomial, binomial, and additive-type procedures, after which clean centre portfolios are serialized and descended on a global nerve.
The central structure is a global replacement certificate transporting successor addresses, paid quotients, typed traces, displayed parents, reopening data, and terminal truth across macroblocks. We prove that a complete state equipped with this certificate admits a strict multiset replacement in a single dependent well-founded order; hence the iteration terminates and the exhausted state reconstructs a regular strict transform having normal crossings with the ordered boundary. We further formulate an object-level realization of the certificate through rigid generation, cross-generation no-reset, complete wild-capacity control, a centre-or-typed-exit alternative, structured cofibres, displayed-parent allocation, and literal terminal truth. The program page is also available at website this https URL - [187] arXiv:2608.20068 [pdf, html, other]
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Title: Flexibility for the Three-Dimensional Navier-Stokes Equations via Moving Hill VorticesSubjects: Analysis of PDEs (math.AP)
We construct weak solutions of the three-dimensional incompressible Navier--Stokes equations on the torus. The convex-integration scheme is based on the moving-dipole construction of Bruè, Colombo, and Kumar~\cite{BrueColomboKumar2024}. For the explicit exponent $\bar p=\frac65+5\times10^{-5},$ and for any two mean-zero, divergence-free vector fields in $L^2(\mathbb T^3)$, we construct a weak solution whose traces at times $0$ and $1$ approximate the prescribed fields arbitrarily well and which satisfies
\[
u\in C([0,1];L^2(\mathbb T^3)),
\qquad
\nabla u\in C([0,1];L^{\bar p}(\mathbb T^3)).
\] Exploiting the time-locality of the iteration, we also obtain exact nonuniqueness for a dense set of initial data in \(L^2_\sigma(\mathbb T^3)\).The principal perturbations are localized, rescaled copies of Hill's spherical vortex. The Hill scaling preserves both the kinetic-energy scale and the \(L^{6/5}\)-scale of the velocity gradient. The construction uses localization of the potential exterior flow, long-orbit averaging of moving vortex cores, an auxiliary source correction, and a temporal corrector compatible with uniform-in-time Sobolev control. - [188] arXiv:2608.20070 [pdf, html, other]
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Title: Calderón-Zygmund estimates for parabolic systems with $p$-growth and non-divergence dataComments: 25 pagesSubjects: Analysis of PDEs (math.AP)
We obtain Calderón--Zygmund estimates for weak solutions to nonlinear parabolic systems with polynomial $p$-growth and the right-hand side in non-divergence form. Our approach does not require differentiability of the vector field with respect to the gradient variable and provides a unified treatment of both the singular and degenerate regimes. In particular, we establish local higher-integrability estimates for the gradient of the solution under appropriate dependency on the integrability of the right-hand side.
- [189] arXiv:2608.20072 [pdf, html, other]
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Title: Scattering for the focusing $H^{1/2}$-critical nonlinear Schrödinger equation with large dataSubjects: Analysis of PDEs (math.AP)
In this article we prove that the solution to the focusing $H^{1/2}$-critical nonlinear Schrödinger equation in dimension $d\geq 5$ scatters outside finitely many arbitrarily small spacetime cones. We will discuss the relationship between Theorem~\ref{thm:finite-bad-directions} and the resolution of solitons. There are two key ingredients in the proof: the localized below-ground-state scattering in Theorem~\ref{thm:cone-scattering}, and the characterization in Theorem~\ref{thm:measure-convergence} of the asymptotic behaviour of the residual part (i.e., after subtracting the scattering part).
As a byproduct, we also establish an upgrading machinery: if the solution is below the ground state (resp. small) along a time sequence in a spacetime cone, then it is also below ground state (resp. small) uniformly for large time in a shrinked spacetime cone. We expect this to be useful in future works using concentration compactness and our spacetime cone approach. - [190] arXiv:2608.20073 [pdf, html, other]
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Title: Dual Control: On Exploration-Exploitation in Linear SystemsComments: To be published in Annual Review of Control, Robotics, and Autonomous Systems Vol. 10 (2027)Subjects: Optimization and Control (math.OC); Systems and Control (eess.SY)
The term "dual control" refers to the dual objective of simultaneously balancing exploration and exploitation. Problems of this kind have been studied for nearly a century. This paper is devoted to theory and methodology relevant for optimal control of linear time-invariant systems whose parameters are initially unknown and must be learned by active probing. We review the main ideas underlying four major research directions: Multi-armed bandits, self-tuning regulators, regret rate minimizing controllers, and minimax optimal dual controllers. The first three have a long history and rich literature, whereas the fourth provides a promising framework for robust dual control.
- [191] arXiv:2608.20078 [pdf, html, other]
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Title: Domain derivative and shape reconstruction for an inverse backscattering problem for the wave equationSubjects: Numerical Analysis (math.NA)
We consider an inverse backscattering problem for time-dependent acoustic waves with a compactly supported penetrable scattering obstacle in unbounded three-dimensional free space. Assuming that dynamic backscattering far field data of scattered waves corresponding to a few time-dependent incident plane waves are available, the goal is to reconstruct the shape of the scattering obstacle. We establish Fréchet differentiability of the time-dependent far field pattern with respect to the shape of the scattering obstacle and give a characterization of the associated temporal domain derivative in terms of its Laplace transform. This characterization is then utilized in an efficient implementation of a regularized Gauß-Newton method for the inverse backscattering problem using convolution quadrature and a boundary element method. Numerical examples demonstrate potentials and limitations of the algorithm.
- [192] arXiv:2608.20082 [pdf, html, other]
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Title: Completely isometric subspaces of noncommutative $\mathrm{L}^p$-spaces and contractive projectionsComments: 22 pagesSubjects: Functional Analysis (math.FA); Operator Algebras (math.OA)
We investigate the relation between the complete isometry class of subspaces of noncommutative $\mathrm{L}^p$-spaces and their contractive complementability, where $1 < p < \infty$ with $p \not= 2$. We show that if $P \colon \mathrm{L}^p(\mathcal{M}) \to \mathrm{L}^p(\mathcal{M})$ is a positive contractive projection whose range is completely isometric to another noncommutative $\mathrm{L}^p$-space, then $P$ is necessarily completely positive. This provides a converse to the main result of [ArR24] and the positivity assumption on $P$ is essential. We further establish a rectangular analogue of this result. More precisely, we prove that every closed subspace of a noncommutative $\mathrm{L}^p$-space which is completely isometric to a rectangular noncommutative $\mathrm{L}^p$-space of the form $e\mathrm{L}^p(\mathcal{N})(1-e)$ is the range of a contractively decomposable projection. Combined with the known converse implication, this yields a characterization of the ranges of contractively decomposable projections as precisely the subspaces completely isometric to rectangular $\mathrm{L}^p$-spaces associated with $\mathrm{W}^*$-ternary rings of operators.
- [193] arXiv:2608.20089 [pdf, html, other]
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Title: Backstepping-Guided Reinforcement Learning for Wide-Range Saint-Venant Canal RegulationSubjects: Analysis of PDEs (math.AP); Systems and Control (eess.SY)
Backstepping control provides local stability guarantees for nonlinear Saint-Venant systems, but its regulation performance may degrade when the system operates far from the nominal equilibrium. This letter proposes a backstepping-guided soft actor-critic (SAC) controller framework that incorporates model-based control knowledge into reinforcement learning (RL). The nominal backstepping control law is first learned by deep operator network (DeepONet) and embedded into the actor and critic networks as prior informed feature representations. The learned prior is further combined with the SAC policy to generate the final control input, while a transfer-learning strategy preserves the useful backstepping knowledge during adaptation to the nonlinear dynamics. Simulation results on the Sambre River model demonstrate that the proposed method improves learning efficiency and maintains effective regulation over larger initial deviations than backstepping control.
- [194] arXiv:2608.20094 [pdf, html, other]
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Title: Nonlinear Controllability and the Propagation of Local Information: From the Kalman Family to Lie Brackets, Rotation Groups, and Reachable SubgroupsComments: No original claim in this paper. It is a pedagogical exposition giving important links to the literature, espcially with Yamabe Theorem on arcwise connected subgroupsSubjects: Optimization and Control (math.OC)
This article develops a self-contained Lie-theoretic route from linear controllability to nonlinear controllability on matrix Lie groups. The organizing question is whether local algebraic information can be propagated into statements about reachable sets. The linear case supplies the model: the matrix exponential, Cayley-Hamilton theorem, trajectory formula and Kalman family B,AB,...,An-1B reduce controllability to finite-dimensional linear algebra. In the nonlinear setting, Lie brackets replace matrix powers, but the associated Lie algebra may be infinite dimensional and the finite-dimensional Lie correspondence can fail. Particular emphasis is placed on rotation groups. Skew-symmetric matrices, their exponentials, commutators, one-parameter subgroups, and the Baker-Campbell-Hausdorff formula provide a concrete laboratory for seeing how infinitesimal directions propagate on SO(n). A numerical SO(3) simulation illustrates the geometry of controlled rotations, while explicit SO(4) and SO(7) commutator sequences make the propagation of a localized control direction visible. The article then develops the attainable-subgroup argument for right-invariant systems, gives a detailed proof sequence behind the Yamabe step, and works through solvable, nilpotent and ideal examples for upper-triangular matrix algebras. The finite-dimensional mechanism is finally contrasted with Sussmann's local controllability theory for general nonlinear systems.
- [195] arXiv:2608.20098 [pdf, html, other]
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Title: Superconvergence and aliasing saturation in Sloan iteration for spherical integral equationsComments: 18 pages, 2 figuresSubjects: Numerical Analysis (math.NA)
Sloan iteration raises the convergence order of Galerkin and degenerate-kernel approximations to second-kind integral equations. After quadrature discretization, these become a discrete Galerkin method and a product-integration Nyström method, respectively. How much of this improvement survives quadrature discretization? For zonal integral equations on the sphere, we give a sharp answer by expressing the Sloan error identity in terms of spherical harmonics. In the absence of quadrature, Sloan iteration fully exploits the smoothing of the integral operator. Quadrature can destroy this gain by aliasing unresolved information into low-frequency modes, where further iteration no longer improves the asymptotic rate. This yields a unified tail--aliasing description of these four methods. For positive-weight polynomially exact quadrature and multipliers of exact algebraic order, we derive sharp two-sided worst-case bounds that separate the spectral-tail and aliasing contributions. Their balance yields a depth-dependent recovery--saturation threshold: sufficient overintegration recovers the quadrature-free rate, while below the threshold aliasing determines the sharp order. We also extend the analysis beyond polynomial exactness using Marcinkiewicz--Zygmund stability and Gram-corrected least squares. Numerical experiments illustrate both regimes.
- [196] arXiv:2608.20101 [pdf, html, other]
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Title: Largest bulk gap of the complex Ginibre ensembleComments: 25 pagesSubjects: Probability (math.PR)
Let $M_n(B)$ be the largest distance from an eigenvalue of an $n\times n$ complex Ginibre matrix, with entries of variance $1/n$, lying in a fixed bulk set $B$ compactly contained in the unit disk and of planar area $|B|$, to its nearest other eigenvalue. Lopatto and Otto proved that $ \sqrt{n} M_n(B)/(4\log n)^{1/4} \to 1$ in probability. Here we prove that $\beta_n^{3/4}\bigl(\sqrt n\,M_n(B)-\beta_n^{1/4}\bigr)$ converges in distribution to a Gumbel random variable, and we determine $\beta_n$ explicitly.
- [197] arXiv:2608.20103 [pdf, html, other]
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Title: Implicit BDF2 dual time-stepping positivity-preserving entropy-stable schemes for unsteady compressible viscous flowsSubjects: Numerical Analysis (math.NA)
This paper presents a rigorous extension of the explicit, high-order, positivity-preserving, and entropy-stable spectral collocation schemes developed in Upperman 2023 and Yamaleev 2023 for the 3D compressible Navier-Stokes equations to a time-implicit formulation. The time derivative terms are discretized by using the second-order implicit backward difference formula (BDF2) that is well suited for solving time-variable viscous flows at high Reynolds numbers. The nonlinear system of discrete equations resulting from the BDF2 discretization at each physical timestep is solved using a dual time-stepping (DTS) technique. The BDF2 DTS scheme is entropy-stable and positivity-preserving in the pseudotime and provides unconditional stability properties in the physical time. Numerical results demonstrate the efficiency and accuracy of the positivity-preserving BDF2 DTS scheme as compared with its explicit counterpart are presented for supersonic flows with strong shock waves and contact discontinuities.
- [198] arXiv:2608.20105 [pdf, html, other]
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Title: An AI-discovered smooth random fast dynamo on $\mathbb{T}^3$Comments: 12 pages; AI discovered; human writtenSubjects: Analysis of PDEs (math.AP); Probability (math.PR)
We construct a random, time-dependent divergence-free velocity field on $\mathbb{T}^3$---refreshing iid on finite time blocks and obeying deterministic $C^\infty_{t,x}$ bounds---that exhibits fast dynamo behavior. That is, for every fixed, sufficiently small resistivity, the almost sure exponential growth rate of the magnetic field solving the associated linear resistive induction equation is at least $1/2$; the exceptional null set may depend on the resistivity. We in fact get a time-uniform lower bound---with a random prefactor obeying a uniform-in-$\kappa$ inverse moment bound. The argument relies on a particular algebraic structure in Fourier space of the induction equation solution operator that allows us to propagate expected growth of the logarithmic size of three specially chosen Fourier modes. This allows us to reduce to a simple recursion, avoiding the complicated infinite-dimensional dynamics typical to the dynamo problem. The central proof idea was generated autonomously by ChatGPT 5.6 Sol Ultra; the manuscript was written (and verified) by the author.
- [199] arXiv:2608.20121 [pdf, html, other]
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Title: Worst-Case Probability Bounds for Finite-Horizon Safety under Moment UncertaintySubjects: Optimization and Control (math.OC)
This paper addresses the problem of estimating upper bounds on the probability that a dynamical system will enter an undesirable region at some point within a finite time horizon. The primary source of uncertainty lies in the system's initial state, for which only a finite set of moments is known or within a prescribed interval. To tackle this problem, we formulate a measure-based program and propose its relaxation using the moment-sum-of-squares (moment-SOS) framework. The corresponding dual problem is introduced as a functional program, which is subsequently strengthened into a sum-of-squares (SOS) program. Notably, this dual formulation bears a structural resemblance to classical barrier function techniques for certifying system safety, with the key distinction that it yields a probabilistic certificate. The effectiveness of the proposed approach is demonstrated through multiple case studies, including a case involving an object in orbit.
- [200] arXiv:2608.20125 [pdf, html, other]
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Title: Modularity theorems for Eisenstein congruences in prime-power levelComments: 20 pagesSubjects: Number Theory (math.NT)
Let $p, N \geq 5$ be primes such that $N \equiv 1 \bmod p$. We prove modularity theorems at levels $N$ and $N^2$, showing that suitable Eisenstein localizations of the weight-$2$ $p$-adic Hecke algebra at these levels are isomorphic to certain natural quotients of a universal pseudodeformation ring. This universal ring parametrizes pseudorepresentations that are residually Eisenstein, unramified outside $Np$ and finite-flat at $p$, and satisfy appropriate conditions at $N$ depending on the level.
- [201] arXiv:2608.20126 [pdf, html, other]
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Title: RMWorld: Task-Aware Radio World Models with Value-of-Information Guided Multi-Trial Learning for Multi-UAV Communication ControlSubjects: Information Theory (cs.IT)
Reliable multi-UAV communication control depends on predicting which aerial links will serve traffic before measurements are available. Radio world models (radio WMs) make such planning tractable, but their errors are nonuniform: a globally accurate model may still fail along high-demand corridors or association boundaries where rate errors reverse control decisions. This mismatch creates a learning challenge. Link queries must reduce decision-relevant channel uncertainty, while counterfactual trials must be filtered so that biased rollouts do not corrupt the policy. Existing acquisition and model-based control treat these budgets separately, valuing uncertainty, coverage, or optimistic return rather than risk reduction. We present RMWorld, a task-aware radio-WM framework that couples value-of-information channel calibration with credibility-diversity multi-trial selection. A biased propagation formula is corrected by a Bayesian residual, and each link is valued by its exact one-label reduction in locally linearized task-integrated posterior rate variance. Counterfactual branches are selected by a task-gated log-determinant objective, followed by conflict projection and fixed-batch validation. We derive the variance-reduction identity, prove posterior task-risk equivalence and the submodular greedy guarantee, and establish a scoped first-order non-interference result. Across 100 paired 3GPP trials RMWorld reaches 0.949~bit/s/Hz task-weighted RMSE, and across 30 severe-load DeepMIMO trials it reduces median backlog by 0.967 versus Ensemble UCB at 37.5\% more offline rollouts.
- [202] arXiv:2608.20130 [pdf, html, other]
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Title: Variation of Iwasawa Invariants for Ordinary RepresentationsComments: 14 pagesSubjects: Number Theory (math.NT)
Let $K$ be a number field and $p$ be an odd prime. Greenberg introduced a natural topology on the space of all $\mathbb{Z}_p$-extensions of $K$ and established several boundedness results for the classical Iwasawa invariants. We extend this framework to compare the Iwasawa invariants of Selmer groups attached to an ordinary $p$-adic representation across $\mathbb{Z}_p$-extensions lying in a Greenberg neighbourhood in the sense of Greenberg. We also establish analogous results for the fine Selmer groups. Finally, in a neighbourhood of the cyclotomic $\mathbb{Z}_p$-extension, we provide evidence for the expected connection between the characteristic ideal of the Selmer group and the conjectural $p$-adic $L$-function introduced by Disegni.
- [203] arXiv:2608.20143 [pdf, html, other]
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Title: The entropy spectrum of hyperbolic surfacesComments: 62 pages, 14 figuresSubjects: Geometric Topology (math.GT); Differential Geometry (math.DG); Dynamical Systems (math.DS)
This article introduces and studies the entropy spectrum of a hyperbolic surface, that is the set of entropies of its subsurfaces. The main results are that the entropy spectrum is a reverse well-ordered multiset, with finite multiplicities, and that there is a quantifiable gap around the value $1$. This gap comes from a counting result on the number of non-filling geodesics which in turn comes from explicit estimates on the number of curves on surfaces with boundary in terms of geometric data. The geometric data includes lengths of boundary geodesics and so-called boundary width, which measures maximal distance to the boundary but can also be interpreted in terms of the topology of the surface and the systole.
- [204] arXiv:2608.20150 [pdf, html, other]
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Title: The nested derivative criterion for determining elementary solutions to certain non-linear PDEs in one-dimensional space-timeSubjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph)
In this article, we have studied a particular criterion to establish if a particular class of nonlinear PDEs $\mathcal{O}_t u=\mathcal{O}_x u $ in unidimensional space-time $(x,t)$ admits elementary solutions respect to the spatial coordinate $x$, through the nested derivative method. Where a nested derivative is a generalization of the usual partial derivative of a order higher than one and, in terms of differentials operators, is a generalization of the Laguerre operator. This method integrated with Liouville's theorem and Galois theory e method by integrate by parts establish if solution of nested derivative equations admits elementary solutions respect with $x$. These solutions could help to study better evolutions of possible physical systems in space-time.
- [205] arXiv:2608.20152 [pdf, html, other]
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Title: The Generalized Random Access Problem for Linear CodesSubjects: Information Theory (cs.IT); Combinatorics (math.CO)
Random access is a central requirement in DNA-based storage systems: one would like to recover selected information symbols without sequencing the whole encoded object. A recent combinatorial model associates to a generator matrix $G\in F_q^{k\times n}$ the random variable $\tau_i(G)$, measuring the number of sampled columns needed to recover the information vector $e_i$. We study the cardinality-based extremal and finite-geometric aspects of simultaneous multi-symbol recovery. For a nonempty set $I\subseteq[k]$, let $\tau_I(G)$ denote the number of random column samples needed until all vectors $e_i$, $i\in I$, lie in the span of the observed columns. This variable interpolates between the singleton random access problem and the full-recovery problem underlying coverage depth. For each $m$, we introduce uniform worst-case and average parameters over all requested sets $I$ with $|I|=m$. Using the known subset-counting formula for $E[\tau_I(G)]$, we establish general upper and lower bounds for these parameters. In particular, the lower bounds are expressed through order statistics of the singleton recovery variables and specialize to the known singleton bounds when $m=1$. For systematic MDS encoders, we record an equivalent form of the known multi-symbol expectation formula and derive monotonicity and asymptotic consequences. For simplex encoders in arbitrary dimension, we obtain closed formulae in terms of Gaussian binomial coefficients; the full-recovery endpoint agrees with the known coverage-depth formula for simplex codes. Finally, in dimension three we study balanced quasi-arcs and compare their values with the simplex and MDS benchmarks.
- [206] arXiv:2608.20158 [pdf, html, other]
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Title: On Cap-Decorated Floer Persistence modulesComments: 45 pages; Comments are very welcomeSubjects: Symplectic Geometry (math.SG)
For a closed symplectically aspherical manifold, we introduce cap-decorated Floer persistence by refining the action-filtered Hamiltonian Floer persistence module with layers defined by iterated actions of homogeneous cohomology ideals. We prove functoriality and stability of these layers, and construct a universal cap-profile pseudodistance bounded above by Hofer distance. As an application, we obtain new invariants of Hamiltonian conjugacy classes, detecting symplectic mapping-class displacement phenomena invisible to ordinary Floer barcodes and spectral invariants. In particular, for a genus-two surface we construct Hamiltonian diffeomorphisms with identical ordinary Floer data but positive separation in the Hamiltonian-conjugacy quotient.
- [207] arXiv:2608.20159 [pdf, html, other]
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Title: Numerical Study of a Surface Growth Model with Singular NoiseComments: 28 pages, 4 figuresSubjects: Numerical Analysis (math.NA)
We study a stochastic model for epitaxial thin-film growth driven by spatially rough additive noise in a regime where the noise is singular and regularization via truncation in Fourier space is used to give a meaning to the solution, leading to a vanishing nonlinearity in the limit. In order to study this phenomenon numerically, the nonlinearity is discretized by a spectral Galerkin projection, while time integration is performed with an exponential Euler scheme. For roughness stronger than space-time white noise, we derive strong error estimates in $L^p(\Omega;C([0,T];\mathcal H^1))$ that display explicitly the interaction between the spatial cut-off, the time step, and the decay of the nonlinear current.
We also quantify the growth of the truncated stochastic convolution and the corresponding vanishing rate of the nonlinearity. Numerical experiments illustrate the transition from persistent hill formation to noise-dominated dynamics as the roughness parameter increases. - [208] arXiv:2608.20165 [pdf, html, other]
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Title: Orbit equivalence and total weak mixing of free group actionsComments: 6 pagesSubjects: Dynamical Systems (math.DS); Group Theory (math.GR); Logic (math.LO)
We prove that the orbit equivalence class of every free ergodic probability-measure-preserving (pmp) action of a free group contains a totally weak mixing action. Equivalently, every ergodic treeable pmp equivalence relation of cost $n\in\mathbf{N}\cup\{\infty\}$ is generated by a free totally weak mixing action of $\mathbf{F}_n$. This answers a question of Miller and Tserunyan.
The proof goes by considering a Polish space of edge slidings along a fixed mixing transformation and proving that for every $w\not=e\in\mathbf{F}_n$ the set of edge slidings that produce an action with $w$ weakly mixing forms a comeager set. - [209] arXiv:2608.20171 [pdf, html, other]
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Title: Koszul cohomology of Čech cohomology modulesComments: Comments are welcome!Subjects: Commutative Algebra (math.AC)
Let $R$ be a commutative Noetherian ring, let $\mathbf{x}=x_1,\ldots,x_n$ be an $R$-regular sequence, and let $\mathbf{y}=y_1,\ldots,y_m$ be a sequence of elements of $R$. Put $I=(\mathbf y)$. Let $\mathcal S$ be a Serre subcategory of the category of $R$-modules. We consider the double complex obtained from the Koszul co-complex with respect to $\mathbf x$ and the Čech complex with respect to $\mathbf y$. Using the two spectral sequences associated with this double complex, we prove that \[ Ext_R^i(R/(\mathbf x),H_I^j(R))\in\mathcal S \quad\text{for all }i,j\in \mathbb N_0 \] implies \[ H_I^j(R/(\mathbf x))\in\mathcal S \quad\text{for all }j\in\mathbb N_0. \]
- [210] arXiv:2608.20176 [pdf, html, other]
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Title: The exact Spearman rho-footrule region via optimal transport with applications to finite rankings, mixability, and Chatterjee's rank correlationComments: 46 pages, 7 figuresSubjects: Statistics Theory (math.ST)
We solve the open problem of determining the maximal value of Spearman's rho when Spearman's footrule is prescribed, thereby completing the exact attainable region of these two quantities. To prove this result, we reformulate the underlying copula optimization problem as an optimal transport problem with a linear moment constraint and construct the unique optimal coupling through a matching feasible dual potential and its contact set. Equivalently, this coupling minimizes the variance of $|U-V|$ among all couplings of $U,V\sim\mathcal{U}(0,1)$ with prescribed mean $\mathbb{E}|U-V|$. Our main result admits several applications: First, in the context of finite rankings, we obtain an improved Cauchy--Schwarz inequality between Spearman's footrule distance and the associated quadratic rank difference. Second, in the framework of generalized mixability, we characterize the attainable constant values of $|U'+V'|$, for $U',V'\sim\mathcal{U}(-1/2,1/2)$, and determine the minimal quadratic deviation for a given mean. Third, we derive explicit bounds relating Chatterjee's rank correlation $\xi(X,Y)$, which can detect complex functional dependence of $Y$ on $X$, to the copula correlation ratio--a rank-based fraction of explained variance--by exploiting their conditional i.i.d. representations in terms of Spearman's footrule and Spearman's rho.
- [211] arXiv:2608.20177 [pdf, html, other]
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Title: Hofer-Zehnder capacity as a geodesic selectorSubjects: Symplectic Geometry (math.SG)
We compute the Hofer-Zehnder capacity of the unit disk cotangent bundle of every ellipsoid in $\mathbb R^3$. The capacity is determined by the smaller of two distinguished quantities in the geodesic length spectrum: twice the systole and the length of the shortest simple closed geodesic of Morse index 3. For the lower bound, we use Riemannian billiards on a suitable cut of the ellipsoid. For the upper bounds, we develop two complementary methods. The first combines an argument by Hofer-Viterbo with neck-stretching and yields, more generally, an upper bound for positively curved Riemannian two-spheres in terms of closed geodesics of prescribed index. The second uses the pair-of-pants product in symplectic homology and the Viterbo isomorphism to bound the Hofer-Zehnder capacity of any disk cotangent bundles of Riemannian two-spheres by twice the diastole; for positive curvature, the diastole agrees with the systole.
- [212] arXiv:2608.20179 [pdf, html, other]
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Title: Dynamic Portfolio Optimization under CVaR ConstraintsSubjects: Optimization and Control (math.OC); Portfolio Management (q-fin.PM); Risk Management (q-fin.RM)
We study continuous-time dynamic portfolio optimization under a Conditional Value-at-Risk (CVaR) constraint on the investor's terminal loss. For a general class of convex trading objectives, we exploit the auxiliary-threshold representation of CVaR to establish the existence of an optimal strategy and strong duality without requiring market completeness. These results motivate a dual-based nested bisection--golden-search algorithm over the threshold and Lagrangian multiplier, where the inner iterations reduce to standard unconstrained stochastic control problems. We prove that the resulting strategies converge to the optimal control as the number of iterations tends to infinity. Numerical experiments recover the Merton policy when the risk constraint is nonbinding. When the constraint is binding, the optimal strategy becomes state dependent: the investor reduces risky exposure following adverse outcomes but preserves, and near maturity may increase, exposure following favorable outcomes. Thus, a terminal CVaR constraint produces an asymmetric reallocation across states rather than uniform de-risking. Nontraded endowment risk amplifies the conservative adjustment, whereas price impact lowers desired positions and adjustment speeds.
- [213] arXiv:2608.20188 [pdf, html, other]
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Title: A note on interior curvature estimates for strictly convex solutions to the equation of prescribed curvature quotientComments: to appearSubjects: Analysis of PDEs (math.AP); Differential Geometry (math.DG)
In this note, we prove a priori interior curvature bounds for strictly convex solutions to elliptic Weingarten curvature quotient equations. The proof does not employ the advanced methods involving integral estimates or compactness arguments. Instead, it relies on a concavity inequality for the equation operator and a novel choice of the auxiliary function to carry out an elementary maximum-principle argument. Interior curvature estimates for strictly convex solutions to the special Lagrangian curvature equation in low dimensions also follow as a consequence.
- [214] arXiv:2608.20189 [pdf, html, other]
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Title: On the closed solution of a problem coupling fluid infiltration with a hydration reactionComments: 8 pages, 1 figureSubjects: Analysis of PDEs (math.AP)
We present a one-dimensional model for water infiltration coupled with a hydration reaction, relevant to coupled transport and chemical processes in the Earth's subsurface. In this model the sharp interface separating the saturated and dry regions evolves over time, leading to a moving free boundary problem. Consistent with recently presented numerical treatments in the literature, this solution indicates an interesting dynamic for the free boundary. At early time, for a given domain porosity, the infiltration front advances with a specific square-root-in-time behavior. At later time, depending on the consumption of the hydration reaction, the front advance can exhibit square-root, linear, or exponential forms. The presented closed solution provides an analytical tool that can be used to quantify important behavior in coupled transport and reaction systems.
- [215] arXiv:2608.20190 [pdf, html, other]
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Title: Small circumference in regular sublinear expandersComments: 11 pages, 2 figuresSubjects: Combinatorics (math.CO)
Sublinear expansion is weak enough to be extracted from arbitrary graphs while retaining nearly all of their average degree, yet it has proved strong enough to force global structures in many sparse extremal problems. Letzter, Methuku and Sudakov [JLMS 2026] developed methods yielding nearly Hamilton cycles in sufficiently dense regular sublinear expanders, and Montgomery [ICM 2026] subsequently conjectured that, every sufficiently large (but constant) degree $d$-regular sublinear expander is Hamiltonian. We disprove this conjecture in a strong form by constructing $n$-vertex $d$-regular sublinear expanders with degree $d=\left(\frac12+o(1)\right)\log^2 n$, which does not even has a cycle covering a positive fraction of its vertices.
The construction blows up one side of a biregular Ramanujan graph into almost-complete blocks while keeping the other side independent. The Ramanujan incidence graph certifies expansion for arbitrary mixtures of partial blocks and separator vertices, whereas the independent side forms a sparse vertex separator that prevents a cycle from visiting enough blocks. The construction also explains why $\log^2 n$ is the natural degree scale for this obstruction. - [216] arXiv:2608.20191 [pdf, html, other]
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Title: Spectrum of the refined Diophantine exponentSubjects: Combinatorics (math.CO); Formal Languages and Automata Theory (cs.FL); Dynamical Systems (math.DS); Number Theory (math.NT)
The refined Diophantine exponent, recently introduced by the author, is a quantity that measures the periodicity of an infinite word. In this article, we study this exponent from combinatorial and topological viewpoints. First, we show that, over a ternary alphabet, the spectrum of the refined Diophantine exponent is $[1,\infty]$. Second, we show that this exponent has topological properties similar to those of the set of Liouville numbers. Finally, we provide concrete examples with the Champernowne, Rudin--Shapiro, and Thue--Morse words, words coming from coding a rotation by intervals, and bracket words.
- [217] arXiv:2608.20200 [pdf, html, other]
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Title: A note on 4-forms in 8-dimensionsComments: 14 pages, no figuresSubjects: Differential Geometry (math.DG)
We present a note on two related questions on 4-forms in 8-dimensions. In particular, we clarify that the Cayley form is not unique in determining a metric via the formula of Karigiannis. Additionally, we discuss and prove part of a conjecture of Salamon and Walpuski.
- [218] arXiv:2608.20203 [pdf, html, other]
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Title: The arithmetic de Rham stackComments: 82 pages. Comments welcome!Subjects: Algebraic Geometry (math.AG); Number Theory (math.NT)
We define and study the arithmetic de Rham stack $X^{\operatorname{arith}}$ of a scheme $X$ over a field of characteristic $p$, and analyze its relation with related stacks such as the Hyodo--Kato stack. We show that $X^{\operatorname{arith}}$ gives a stack-theoretic approach to rigid cohomology and its coefficients, known as overconvergent isocrystals and arithmetic $D$-modules, which avoids the long-standing problem of frame-choosing in earlier approaches. We finally give some arithmetic applications of our formalism, such as a proof of Berthelot's conjecture on the preservation of overconvergent isocrystals by smooth proper pushforward.
- [219] arXiv:2608.20205 [pdf, html, other]
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Title: Carleson measures, Volterra integral operators and multipliers for $Q_K$ spacesSubjects: Complex Variables (math.CV)
We characterize the positive Borel measures $\mu$ on the unit disc $\mathbb{D}$ for which the Möbius-invariant space $Q_K$ embeds continuously or compactly into $L^2(\mu)$. The characterization is given in terms of a discrete dyadic capacity $C^{(b)}_{K,\mathcal{R}}(\mu)$ built from a polar dyadic resolution of $\mathbb{D}$, and of an equivalent capacity $D^{(b)}_{K,\mathcal{R}}(\mu)$ expressed as a semidefinite program. The equivalence of the two capacities is established through conic duality and the complex Grothendieck inequality. As an application, we characterize the boundedness and compactness of the Volterra integral operator $T_g$ on $Q_K$, bridging and completely resolving the gap between the sufficient and necessary conditions established by Li and Wulan (2010). We also obtain a complete non-testing characterization of the pointwise multipliers $\mathcal{M}(Q_K)$ on $Q_K$, thereby answering an open problem posed in the survey of Bao and Wulan (2021).
- [220] arXiv:2608.20207 [pdf, html, other]
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Title: Direct vs. Indirect Data-Driven Control: Case-study of Switching Systems StabilityComments: Preprint version. Submitted to AutomaticaSubjects: Optimization and Control (math.OC)
A central methodological question in data-driven control is whether to adopt a direct or indirect approach. Direct methods infer a controller or certificate directly from data, while indirect methods first identify a system model and then apply model-based control techniques. Recent developments of the direct method have led to finite-sample guarantees for the data-driven stability analysis of switched linear systems under various settings. However, for the indirect method, such guarantees remain largely elusive. In this paper, we provide a novel framework for the stability analysis of switched linear systems from noisy state measurements, using the indirect approach and quadratic Lyapunov analysis. Our framework comes with finite-sample guarantees on the convergence rate of the system. For that, we combine generalization bounds from machine learning and system identification with sensitivity analysis from quadratic Lyapunov analysis. To enable comparison, we also extend existing direct data-driven methods to handle measurement noise beyond the bounded noise case currently available in the literature. Finally, we compare the two approaches through numerical experiments, revealing that under moderate-to-high noise levels the indirect approach yields tighter probabilistic guarantees as well as greater robustness to noise and outliers than the direct approach
- [221] arXiv:2608.20215 [pdf, html, other]
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Title: On the proof of Bray's conjectureSubjects: Differential Geometry (math.DG)
Let $(M^n,g)$ be a connected, closed, smooth Riemannian manifold with dimension $n\geq 3$. There exists a positive constant $\varepsilon_n<1$ such that, if Ricci curvature $\operatorname{Ric}_g\geq \varepsilon_n(n-1)g$ and the scalar curvature $R_g\geq n(n-1)$, then the volume $V_g(M^n)$ is less than or equal to the volume of standard $n$-sphere. This confirms a conjecture by Bray in 1997.
- [222] arXiv:2608.20216 [pdf, html, other]
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Title: Sharp Summability of Nevanlinna Defects for Finite-Lower-Order Holomorphic CurvesSubjects: Complex Variables (math.CV)
For a countable family of hyperplanes $H_j\subset \mathbb{P}^m$, $j\in\mathbb{N}$, in general position and a linearly nondegenerate holomorphic curve $f\colon \mathbb{C}\to \mathbb{P}^m$ of finite lower order, we prove that the Nevanlinna defects $\delta_f(H_j)$ satisfy
$$
\sum_{j=1}^{\infty}\delta_f(H_j)^{1/3}<\infty.
$$
This resolves a long-standing open problem in Nevanlinna theory and extends Weitsman's celebrated scalar endpoint theorem (the case $m=1$) as well as Krutin's results for exponents strictly greater than $1/3$. The same uniform finite-family estimate yields the corresponding endpoint theorem for divisors cut out on a projective variety by ambient hypersurfaces of uniformly bounded degree, assuming that the divisors are in general position with respect to the variety and that the curve is not contained in the support of any divisor. - [223] arXiv:2608.20217 [pdf, html, other]
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Title: Schäffer's matrix inequality: the exact asymptotic constantSubjects: Functional Analysis (math.FA); Spectral Theory (math.SP)
Let $S_n$ denote the smallest constant such that \[ |\det T|\|T^{-1}\| \leq S_n \|T\|^{n-1} \] for every invertible operator $T$ on every $n$-dimensional complex Banach space. In Hilbert space the optimal constant is $1$. For arbitrary Banach spaces, J. J. Schäffer proved in 1970 that \[S_n\leq \sqrt{en}. \] Subsequent work showed that $S_n$ grows like $\sqrt n$, but the sharp asymptotic constant has remained open for more than five decades. We resolve this problem by proving \[ \lim_{n\to\infty}\frac{S_n}{\sqrt n}=\sqrt e. \] Thus Schäffer's upper bound is asymptotically sharp, including its constant. Our proof is constructive, providing explicit Banach-space norms through duality and explicit matrices through the theory of model operators. At the analytic core of the argument, an extremal formulation of Schäffer's problem in the Wiener algebra reduces the matching asymptotic lower bound for $S_n$ to uniformly controlling the Taylor coefficients of products $QB_n$, where $B_n$ is a finite Blaschke product of degree $n$ and $Q$ is a polynomial factor. We optimize simultaneously the zero distribution of $B_n$ and the choice of $Q$. The resulting zeros follow a logarithmic asymptotic distribution, and a sharp uniform asymptotic analysis of the Taylor coefficients of $QB_n$ yields the constant $\sqrt e$. The corresponding model operators then yield matrices with these spectra that asymptotically attain Schäffer's bound.
- [224] arXiv:2608.20223 [pdf, html, other]
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Title: Self-Normalizing Denominators in Rational Causal EstimationSubjects: Statistics Theory (math.ST)
Rational causal estimators in linear structural equation models take the form of one covariance polynomial divided by another, and a small denominator is commonly interpreted as weak identification. We show that, under Gaussian sampling, some denominators cannot enter this regime at first order. Their sampling variation is exactly proportional to their magnitude, so the standardized denominator is constant in every sample. Products of powers of nested covariance minors have this property in every dimension and admit an exact Wishart pivot. The converse is complete in dimension two. In dimension three, one mixed family remains open, while a factor-and-rank criterion classifies all denominators with linear or quadratic determinant-free factors and covers instrumental-variable, front-door and proximal formulas. For linear front-door adjustment, Wald inference remains asymptotically valid even as the mediator residual variance vanishes at an arbitrary rate, provided the treatment--mediator coefficient is nonzero. In simulations, proximal Wald coverage fell as a naive treatment--proxy diagnostic strengthened, while front-door coverage stayed nominal, and right-heart-catheterization data distinguished naive from denominator-relevant diagnostics.
- [225] arXiv:2608.20225 [pdf, html, other]
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Title: The Local Langlands Correspondence for Middle Supercuspidal Representations of $p$-adic $\text{GL}(2n)$Comments: 26 pagesSubjects: Representation Theory (math.RT); Number Theory (math.NT)
Let $\text{F}$ be a non-archimedean local field of characteristic zero with residual characteristic $p$. In this paper we give an explicit description of the local Langlands correspondence for middle supercuspidal representations of $\text{GL}(2n,\text{F})$, under the tameness condition $p\nmid 2n$, in terms of the maximal simple types that define them. We achieve this by explicitly computing and comparing the gamma factors on the automorphic and Galois sides of the local Langlands correspondence. The computation on the automorphic side requires neither the tameness condition nor the characteristic zero condition.
- [226] arXiv:2608.20234 [pdf, html, other]
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Title: Algorithms, Complexity, and Entropy of the Bernard-Letac Fair-Sampling ConstructionComments: 34 pages, 1 figureSubjects: Information Theory (cs.IT); Probability (math.PR)
Bernard and Letac (1971) introduced a method for uniform random sampling among m outcomes from an unknown biased source of independent and identically distributed symbols. The process terminates when the multinomial coefficient of the cumulative symbol counts equals zero modulo m. This study extends the computational and information-theoretic analysis of their construction by presenting five algorithms with formal correctness guarantees and comprehensive complexity analyses. For prime m = p, the Bernard-Letac framework is analyzed in greater detail. The Rényi entropies of the source yield an exact product formula for the expected number of draws. A first-order approximation consistently overestimates this value, and the entropy lower bound is never attained. As p approaches 1, the expected cost converges to a constant greater than 1, determined by the entire source distribution. Furthermore, a seven-state automaton computes the mod-2 first-passage kernel of the binary walk, reducing the fair assignment cost from quadratic to nearly linear.
- [227] arXiv:2608.20241 [pdf, html, other]
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Title: Directional Subdifferentials of the Value Function in Asplund SpacesSubjects: Optimization and Control (math.OC)
Directional subdifferentials of the value function provide a quantitative measure of optimal value response to perturbations. While existing results are largely limited to finite-dimensional settings, this paper develops a comprehensive variational framework in Asplund spaces. We establish essential directional calculus rules, extending directional nonsmooth analysis to infinite dimensions. To address the lack of compactness of bounded sets in infinite-dimensional spaces, we introduce a new directional condition, under which we derive upper estimates for directional limiting and singular subdifferentials of the value function. These results provide a refined analytical foundation for sensitivity analysis in infinite-dimensional hierarchical systems.
- [228] arXiv:2608.20242 [pdf, html, other]
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Title: Quantitative bounds for regular $3$-wise intersecting familiesComments: 8 pages, comments welcome!Subjects: Combinatorics (math.CO)
Frankston, Kahn and Narayanan proved that every regular increasing $3$-wise intersecting family of subsets of $[n]$ has cardinality $o(2^n)$ using Friedgut's junta theorem. We give a short quantitative proof using elementary tools from the analysis of Boolean functions and entropy. More precisely, if $\mathcal{A}\subseteq\mathcal{P}_n$ is a nonempty $3$-wise intersecting family that is both regular and increasing, then $$ \log\frac{2^n}{|\mathcal{A}|}\ge \frac{n}{2}\left(\frac{|\mathcal{A}|}{2^n-|\mathcal{A}|}\right)^2, $$ and consequently $|\mathcal{A}|\le 2^n\sqrt{W(n)/n}$, where $W$ is the principal Lambert function defined by $W(x)e^{W(x)}=x$ for $x\ge0$. We also give a purely Fourier-analytic proof of the weaker estimate $$ |\mathcal{A}|\le \frac{2^n}{1+n^{1/3}}. $$
- [229] arXiv:2608.20248 [pdf, html, other]
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Title: Intersecting families of permutations with a fixed number of cyclesSubjects: Combinatorics (math.CO)
Let $\mathrm{Sym(n,k)}$ denote the set of permutations on $\{1,2,\ldots,n\}$ with exactly $k$ cycles. A family $\mathcal{F}\subset\mathrm{Sym}(n,k)$ is said to be intersecting if $\sigma^{-1}\tau$ has a fixed point for all $\sigma,\tau\in\mathcal{F}$. In this paper, we investigate the size and structure of maximum-sized intersecting families of permutations in $\mathrm{Sym}(n,k)$. In the regime $k\leq n^{0.25}$, we show that every maximum-sized intersecting family is a star, meaning it consists of all permutations in $\mathrm{Sym}(n,k)$ that agree at a given point in $[n]$. We establish this result by proving a stronger stability result that bounds the maximum possible size of a non-centred intersecting family. Specifically, in the regime $k\leq n^{0.25}$, the size of any non-centred intersecting family is at most $\left(2/3+o(1)\right)$ times the maximum possible size of a star. In the tighter polylogarithmic regime $k\leq (\ln n)^{d}$, we improve this bound to $\left(1-1/e+o(1)\right)$ times the maximum possible size of a star; we show that this bound is asymptotically sharp. Thus, we establish both an Erdős--Ko--Rado theorem and its corresponding stability version for $\mathrm{Sym}(n,k)$.
- [230] arXiv:2608.20249 [pdf, html, other]
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Title: Boundary layers and vanishing diffusivity in run-and-tumble modelsComments: 57 pages, 3 figuresSubjects: Analysis of PDEs (math.AP)
A notable feature of confined active matter systems is the tendency for motile particles to accumulate near solid boundaries. In various linear models with no-flux boundary conditions, this accumulation is realized through the development of sharp boundary layers at small particle diffusivity $\kappa$. In this paper, we present the first rigorous investigation of nonlinear boundary layers in the context of confined active matter. Specifically, we consider a family of 1D run-and-tumble models with nonlinear advection and tumbling on the half-line $\mathbb{R}_+$. We rigorously prove the vanishing diffusivity limit with quantitative convergence rates. In the limiting system, the boundary mass enters as a new variable which solves a nonlinear ODE, coupled to the PDE through a dynamic boundary condition. Interestingly, the nonlinearity on the boundary at $\kappa = 0$ cannot be obtained without reference to the boundary layer analysis at $\kappa \ll 1$. Numerically, these models exhibit rich behavior, including phase transition and hysteresis in the boundary layer.
- [231] arXiv:2608.20252 [pdf, html, other]
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Title: Notes on Hydrodynamic Limits and Related TopicsSubjects: Probability (math.PR); Mathematical Physics (math-ph)
In these lecture notes, we discuss various `hydrodynamic LLN' and `CLT' scaling limits, among others, in types of stochastic interacting particle systems, connecting `microscopic' behaviors to continuum laws. Via `short stories', the aim is to present some of the `basics' for students and those entering the field, as a complement to books such as Kipnis-Landim 1999, Komorowski-Landim-Olla 2012, Liggett 1985, Liggett 1999. To be concrete, attention is restricted to a few `mass conservative' systems on discrete spaces, namely exclusion and zero-range processes, that have proved robust in the study of different phenomena.
After preliminaries, we discuss the `entropy' and `relative entropy' methods to prove hydrodynamic limits of the bulk mass in finite volume, as well as other items such as construction of systems in infinite volume and the structure of their invariant measures, and scaling limits of local functionals, such as occupation times of sites and the motion of a tagged particle. In the last part, we also discuss equilibrium fluctuations of the bulk mass when the process starts from an invariant measure. - [232] arXiv:2608.20259 [pdf, html, other]
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Title: Operational Foundations for Quaternionic and Octonionic Quantum Models: Exact Quaternionic Channels and Error Correction, Para-Linear Operators, and Categorical Closure Boundaries Beyond AssociativityComments: 25 pages; no figuresSubjects: Quantum Algebra (math.QA)
A scalar field alone does not determine a quantum theory: states, effects, processes, symmetries, composition, and discard are equally structural. Realification illustrates this point: an orthogonal complex structure J^2 = -I selects the physical real operators and the balanced composite. Quaternionic quantum mechanics has an analogous exact representation on a doubled complex space selected by an antiunitary symplectic structure Theta^2 = -I. Within that sector, a Choi fixed-point condition characterizes when a complex channel admits quaternionic Kraus operators, while exact correction of a finite-dimensional right-quaternionic code is characterized by compression coefficients in the real center and admits an explicit recovery. For octonions, nonassociativity precludes a unique continuation; para-linear, categorical, sectorial, Jordan, Clifford-envelope, and Moufang models are compared by the operational structures they retain and the additional data required to define composites, channels, or recovery. Across these cases, an exact ambient representation does not erase the complex or symplectic structure that selects the physical theory.
- [233] arXiv:2608.20264 [pdf, html, other]
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Title: Systole Increasing Deformations to Maximal Translation SurfacesComments: 23 pages, 15 figuresSubjects: Geometric Topology (math.GT)
A unit-area translation surface is called \emph{maximal} if it maximizes the length of the shortest saddle connection among all surfaces in the same stratum. We investigate whether a non-maximal translation surface can be continuously deformed into a maximal one while the systole increases strictly monotonically. Although such a deformation does not exist in general due to the existence of local but not global maxima of the systole function, we prove that it exists for square-tiled surfaces and translation surfaces obtained from regular hexagons and regular octagons. In each case, we explicitly construct a continuous deformation to a maximal translation surface along which the systole is strictly increasing. Finally, the preceding construction yields another family of translation surfaces admitting continuous systole-increasing deformations to maximal surfaces.
- [234] arXiv:2608.20265 [pdf, html, other]
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Title: Milnor's cartography problemComments: 8 pages, 7 figuresSubjects: Differential Geometry (math.DG); Metric Geometry (math.MG)
We solve John Milnor's problem: among all convex regions of a given area on the sphere, the round disk requires the greatest distortion in cartographic projections onto the plane.
- [235] arXiv:2608.20266 [pdf, html, other]
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Title: Necessary conditions for deterministic and stochastic maximal regularityComments: 31 pagesSubjects: Functional Analysis (math.FA); Analysis of PDEs (math.AP); Probability (math.PR)
We study the role of Banach space geometry in deterministic and stochastic maximal regularity. We first construct an example showing that the UMD assumption in Weis' characterisation of maximal $L^p$-regularity in terms of $R$-sectoriality cannot be omitted. Combining this construction with an equivalence between stochastic maximal regularity and deterministic maximal regularity on the $2$-concavification of the underlying space, we obtain an operator on a UMD Banach function space of type $2$ that has a bounded $H^\infty$-calculus of angle zero, but fails stochastic maximal $L^p$-regularity (SMR$_p$) for every $p\in[2,\infty)$.
Motivated by this example, we study the Banach space geometry hypothesis underlying SMR$_p$ more closely. This is an $R$-boundedness condition $(S_p)$ for stochastic convolution operators. For UMD spaces $X$ of type $2$, we show that this condition is not only sufficient, but also necessary for two canonical test operators: a diagonal multiplier on a Rademacher space and, for $q>2$, the Laplacian on $L^q(\mathbb R^d;X)$. Finally, we prove that its interval-kernel and exponential-kernel formulations are equivalent and that, at the endpoint $p=2$, condition $(S_2)$ holds if and only if $X$ is isomorphic to a Hilbert space. - [236] arXiv:2608.20267 [pdf, html, other]
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Title: Uniform weak type $(1,1)$ bounds for Riesz transforms on stratified Lie groupsSubjects: Classical Analysis and ODEs (math.CA)
Let $G$ be a stratified Lie group and $\mathcal L$ its sub-Laplacian. We prove that the full horizontal Riesz transform $\nabla_{H} \mathcal L^{-1/2}$ is of weak type $(1,1)$ on real-valued functions, with constant at most $2$. In particular, the constant is independent of the horizontal dimension, the homogeneous dimension, the step, and the underlying group structure of $G$. Our result provides a noncommutative generalization of the dimension-free Euclidean theorem of Ouyang, Spector, and Stockdale arXiv:2608.18068 [math.CA], with the same universal constant. Our proof constitutes a fractional obstacle problem formulated through the heat semigroup and the associated Dirichlet form, adapted to the stratified structure. This extends the method of Ouyang, Spector, and Stockdale arXiv:2608.18068 [math.CA] without using the Fourier analytic ingredients.
- [237] arXiv:2608.20270 [pdf, html, other]
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Title: Two results on asphericitySubjects: Geometric Topology (math.GT); Group Theory (math.GR)
We construct an explicit finite chain of non-aspherical group presentation complexes $K(\mathcal X)\subset K(\mathcal Y)\subset K(\mathcal Z)$ in which $K(\mathcal Y)$ is Cockcroft, both inclusion-induced maps on $\pi_2$ are zero, and the pair $(K(\mathcal Y),K(\mathcal X))$ has the identity property. This answers a question of Hitchman. The construction is given directly from a short balanced presentation of the binary icosahedral group. As the second result, we construct an integral Lie ring presentation in which a subpresentation has a nontrivial identity among relations, whereas the presentation itself has none.
- [238] arXiv:2608.20272 [pdf, html, other]
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Title: Optimal regularity of stable harmonic maps to spheresComments: 18 pages. Comments are welcome!Subjects: Analysis of PDEs (math.AP); Differential Geometry (math.DG)
In this paper, we show that the codimension of the singular set of a stable stationary harmonic map to a round $k$-sphere is at least $k+1$ when $k$ is between 3 and 6, and is at least 7 when $k$ is at least 7. The result is sharp in the sense that there exist energy minimizing 0-homogeneous maps in the aforementioned critical dimensions. We also establish non-trivial index bounds for the non-constant harmonic maps from $n$-spheres to $k$-spheres, provided that $n$ is less than $k$.
- [239] arXiv:2608.20279 [pdf, html, other]
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Title: Robustness of random-walk Metropolis for steep potentialsComments: 11 pages, no figuresSubjects: Probability (math.PR); Statistics Theory (math.ST); Computation (stat.CO)
In Markov chain Monte Carlo sampling, light-tailed target distributions present something of a poisoned chalice: their light tails offer good confinement, and tend to imply good mixing properties for natural continuous-time dynamics, but the steepness of their tail decay means that they often fall outside of the scope of modern quantitative convergence theory. For usual gradient-based samplers, this reflects a genuine instability issue, whereby Metropolis acceptance rates can degrade badly. In this work, we study the gradient-free random-walk Metropolis sampler, and show that for a wide range of light-tailed targets, the acceptance probability remains stable for reasonable choices of proposal variance, from which effective and favourable mixing time estimates can be deduced. The analysis relies on a simple relationship between the first and second derivatives of the log-density of the target distribution.
- [240] arXiv:2608.20287 [pdf, html, other]
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Title: The Honeycomb Framework for Code BoundsSubjects: Information Theory (cs.IT); Discrete Mathematics (cs.DM)
We introduce the honeycomb hierarchy, a representation-theoretic framework that gives new asymptotic upper bounds on $R_2(\delta)$. Its first level is the two-row hyperoctahedral representation graph associated with type $S^{(n-k,k)}$. Retaining every two-row irreducible and every coordinate box-transfer channel, together with a moving-projection theorem, yields an explicit four-parameter exponent $\kappa_{\mathrm{HC}}$. The earlier whole-cube exponent $\kappa_H$ is a boundary restriction of this optimization, whereas the fully optimized second MRRW exponent $M_2$ is an exact symmetric slice.
The prior best curve is the combined $\kappa_{\mathrm{bin}}=\min\{\kappa_{\mathrm{CW}},\kappa_H\}$, which uses a constant-weight branch $\kappa_{\mathrm{CW}}$. Replacing only the whole-cube branch by the honeycomb bound gives $\kappa_{\mathrm{best}}=\min\{\kappa_{\mathrm{CW}}, \kappa_{\mathrm{HC}}\}$. We prove, on $0<\delta<1/2$, \[
R_2(\delta)\le \kappa_{\mathrm{best}}(\delta)
\le \kappa_{\mathrm{bin}}(\delta)
\le R_{\mathrm{2MQC}}(\delta)<M_2(\delta),\\[-1mm]
\kappa_{\mathrm{best}}(\delta)
\le \min\{\kappa_{\mathrm{CW}}(\delta),
\kappa_{\mathrm{bal}}(\delta)\}
<R_{\mathrm{2MQC}}(\delta),
\qquad
\kappa_H(\delta)=R_{\mathrm{MQC}}(\delta). \]
The hierarchy has two further directions. Increasing the representation depth replaces scalar by matrix-valued transfers on the hive. Increasing the anchor depth localizes it in a stable-set hierarchy. The resulting bounds are monotone in both directions and eventually recover $A_2(n,d)$. A complementary Horn--channel hierarchy gives matrix optimizations whose $2\times2$ level is $\kappa_{\mathrm{HC}}$ and whose $3\times3$ level is a stronger bound. Already at low levels, they can be used to improve the strongest previous general bounds, while the honeycomb framework provides a route towards tighter bounds. - [241] arXiv:2608.20289 [pdf, html, other]
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Title: Spectral Viterbo isomorphism: complex-oriented versus framedComments: 30 pagesSubjects: Symplectic Geometry (math.SG); Algebraic Topology (math.AT)
The Viterbo isomorphism relates the symplectic cohomology of a cotangent bundle to the homology of the free loop space of its base. We lift this to a relation of modules over (1) the complex bordism spectrum MU and (2) the sphere spectrum $\mathbb{S}$. In particular, by a result of Porcelli and the present author [BP26], it is not the case that, in general, the $\mathbb{S}$-level statement recovers the MU-level statement after base-change -- even in the case the base is spin.
- [242] arXiv:2608.20292 [pdf, html, other]
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Title: A new lower bound for the growth rate of Av(1324)Comments: 50 pages, 2 tablesSubjects: Combinatorics (math.CO)
The growth rate of Av(1324) is the last unknown Stanley-Wilf limit of a length-four pattern. The best rigorous lower bound has been 10.271012 since Bevan, Brignall, Elvey Price and Pantone obtained it in 2020; we raise it to 10.617. Their scheme relaxes an interleaving rule in one direction only. Relaxing it in both is valid, and the Harris inequality then bounds the resulting count below by the product of its two marginals. We remove that inequality, the last one the scheme contains: both neighbours of a connecting cell are placed against one and the same sequence of skew components, so their joint count is a single transfer operator on the square of one cell's state space, and the matrix the Catalan series is applied to is unipotent, so the series terminates and the count is exact. Its rate is concave in the strip profile, which reduces the minimisation to finitely many vertices, and the vertices the aggregating weight does not reach are classified. Two further ingredients enter: an algebraic tilt of the domino ensemble off the leaf and empty-strip densities at which their construction holds it, and the k-leaf strip densities in closed form, which they could not obtain even for k = 1.
- [243] arXiv:2608.20301 [pdf, html, other]
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Title: A two-point phase recovering with spherical wave referenceSubjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph)
We consider a reference wave, a radiation solution, and the sum of these solutions (total solution) for the Helmholtz equation in an exterior region. We give two-point formulas for approximate phase recovering of the radiation solution from the intensity of the total solution for the case of spherical reference wave. We show that these formulas can be used, in particular, for approximate phase recovering from holographic data on a single plane. By these formulas, we continue previous studies with plane wave reference.
- [244] arXiv:2608.20303 [pdf, html, other]
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Title: Kahn--Lovász-type inequalities for graph factorsComments: 27 pagesSubjects: Combinatorics (math.CO)
The Kahn--Lovász theorem gives a sharp upper bound on the number of perfect matchings in a graph in terms of its degree sequence, extending the classical Brégman--Minc inequality for bipartite graphs. In this paper, we establish an asymptotically sharp extension of the Kahn--Lovász theorem to $F$-factors for every Hamiltonian graph $F$. As a consequence, we asymptotically determine the maximum number of $F$-factors in an $n$-vertex $m$-edge graph, yielding an $F$-factor analogue of Kruskal--Katona-type theorems.
We also prove a multigraph analogue of the Kahn--Lovász theorem. Combining this with our results for Hamiltonian graphs, we obtain an asymptotically sharp Kruskal--Katona-type bound for a further class of connected graphs $F$, including those containing two vertex-disjoint cycles of equal length whose union spans $V(F)$. - [245] arXiv:2608.20309 [pdf, html, other]
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Title: The Additive Arithmetic of Linear OrdersSubjects: Logic (math.LO)
We present a systematic development of the arithmetic of the class of linear orders under the ordered sum $(LO, +)$ and prove a number of new results. Our approach is based on a Euclidean algorithm for pairs of linear orders that almost additively commute.
Among our results:
(i.) We generalize and give unified proofs of the main classical theorems for $(LO, +)$, including Lindenbaum's division theorem for $(LO, +)$ and a representation theorem for additively commuting pairs of linear orders due to Aronszajn.
(ii.) We solve the following problem, posed by Tarski in 1956: is it true that for every pair of linear orders $A, B$ and quadruple of natural numbers $n, m, k, l \geq 1$, if $nA + mB \cong kB + lA$ then $A + B \cong B + A$? Tarski and Chang showed the answer is yes for certain choices of the coefficients $n, m, k, l$. We show the answer is yes in general.
(iii.) We prove the following characterization of the additively commuting pairs in $LO$: $A + B \cong B + A$ if and only if $\omega A$ embeds initially in $\omega B$ and $\omega^* A$ embeds finally in $\omega^* B$, or vice versa. We show this can be viewed as a correctly revised version of a refuted conjecture of Tarski.
(iv.) We characterize the commutative semigroups $(S, \oplus)$ that can be represented in $(LO, +)$ and show in particular they are all naturally totally ordered commutative semigroups in the sense of Clifford. - [246] arXiv:2608.20313 [pdf, html, other]
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Title: Complete Symbols of Equivariant Pseudodifferential Operators on Noncompact Symmetric SpacesSubjects: Representation Theory (math.RT); Analysis of PDEs (math.AP); Operator Algebras (math.OA)
We study $G$-equivariant Hörmander pseudodifferential operators on a noncompact symmetric space $G/K$. We define a notion of a complete symbol function, called the Harish-Chandra symbol function, on the spherical tempered dual of $G$, for operators that satisfy a rapid off-diagonal decay condition on their Schwartz kernels, and we characterize the class of such operators in terms of their Harish-Chandra symbol function.
- [247] arXiv:2608.20321 [pdf, html, other]
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Title: Large Sample Properties of Higher Order Markov ModelsSubjects: Statistics Theory (math.ST); Probability (math.PR)
We study large-sample properties of higher-order Markov chains on a finite alphabet $\Sigma$ when the order $m_n$ is allowed to grow with the sequence length $n$. By embedding the process into a first-order chain on $\Sigma^{m_n}$ and exploiting return-time decompositions, we establish a central limit theorem for additive functionals $\sum_{t}\! g_n(Y_t^{(n)})$ under natural ergodicity and sparsity conditions. The normalization involves the stationary return time to a suitably chosen state and accommodates triangular arrays with $m_n\!\to\!\infty$ and $m_n/n\!\to\!0$. We further illustrate the assumptions in a binary variable length Markov chain (VLMC), deriving explicit lower bounds on stationary masses that yield a concrete growth regime (e.g., $m_n\log m_n/n \to 0$) ensuring the CLT. These results provide asymptotic foundations for inference in sparse/partitioned higher-order models; including VLMCs and sparse Markov models (SMMs) where the effective dimensionality grows with the sample size.
- [248] arXiv:2608.20337 [pdf, html, other]
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Title: Information on trajectories: martingales and random timesSubjects: Probability (math.PR); Information Theory (cs.IT); Machine Learning (cs.LG); Statistics Theory (math.ST)
Accounting for information flow on the path space of trajectories of a nonnegative martingale yields exact variational identities for it, even at arbitrary random times. This recovers the widely used classical concentration inequalities, from Ville to PAC-Bayes, and measures what each one discards. The tail a bound controls is itself a relative entropy, resolved by the chain rule into per-step conditional divergences. The discarded slack has an exact form in each of three geometries: a Gibbs tilt for the Azuma-Hoeffding and PAC-Bayes bounds, the crossing itself for Ville's and for pooled tests, and a dominating certificate for the $L^p$ maximal bound. That certificate's optional-stopping deficit resolves per step into Bregman divergences of the running maximum. On a path-time space, the same identity gains one factor that prices anticipation: an arbitrary random time carries an e-process ``peeking penalty.'' The partition function can be read as a coalescent--a prefix-sharing probability of independent copies--and geometric mixtures of test martingales gain a pooling benefit for multi-model safe testing.
New submissions (showing 248 of 248 entries)
- [249] arXiv:2608.19221 (cross-list from q-fin.PR) [pdf, html, other]
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Title: Filtering Credit Risk with Stochastic DiscontinuitiesSubjects: Pricing of Securities (q-fin.PR); Probability (math.PR); Mathematical Finance (q-fin.MF)
We develop a structural credit-risk model under incomplete information in which investors observe firm value only indirectly through noisy market signals and scheduled corporate disclosures. While disclosure dates are known in advance, their informational content is random, leading to stochastic discontinuities in the observation process. We derive the Kushner-Stratonovich equation for structural credit-risk models with endogenous default by applying the nonlinear filtering framework with predictable jumps. We then study the valuation and local risk-minimization hedging for default-sensitive securities under partial information. The interaction between predictable disclosure events and endogenous default produces discrete adjustments in the conditional default compensator, leading to announcement-driven distortions in credit spreads and hedge ratios that are absent from classical diffusion-based and inaccessible-jump models. Numerical experiments illustrate how scheduled disclosures affect filtered default probabilities, Credit Default Swaps (CDS) spreads, and hedging strategies, generating characteristic pre-announcement dynamics in credit spreads.
- [250] arXiv:2608.19273 (cross-list from cs.MS) [pdf, html, other]
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Title: DSLHyPE-a DSL kernel language for the Exascale Hyperbolic PDE Engine ExaHyPESubjects: Mathematical Software (cs.MS); Programming Languages (cs.PL); General Relativity and Quantum Cosmology (gr-qc); Numerical Analysis (math.NA)
We introduce a bilingual domain-specific language (DSL) for modelling compute kernels within a generic solver for hyperbolic partial differential equations (PDEs). Users express PDE terms, i.e.~the underlying physics, in a familiar native language such as C or C++, while the numerical scheme is specified in a Python-embedded DSL, DSLHyPE. DSLHyPE's compiler lowers the Python description to MLIR and introduces a translation pass that integrates it with native code likewise mapped to MLIR. Our approach keeps the numerical representation and the physics implementation separate for as long as possible, while delegating optimization to the compiler through existing MLIR optimization passes. This separation of concerns benefits researchers developing numerical schemes on top of existing PDE implementations or with applications involving nonlinear systems whose PDE terms must solve PDEs themselves. We demonstrate the feasibility of the approach using a gravitational-wave solver and a matter-evolution solver on x86 processors and H200 GPUs.
- [251] arXiv:2608.19275 (cross-list from stat.ME) [pdf, html, other]
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Title: A Self-Exciting Model of Eddy Formation at SubmesoscaleSubjects: Methodology (stat.ME); Probability (math.PR)
Ocean eddies are highly dynamic structures marked by frequent splitting and merging events. They exhibit complex spatio-temporal clustering that traditional Poisson models fail to capture. In this paper, a novel spatio-temporal Hawkes process is introduced to model self-exciting eddy fields on the basis of high-frequency ocean flow data. We formulate a triggering kernel that couples the self-excitation intensity with spatial deformation caused by the strain rate magnitude. To characterize the asymptotic behavior of the eddy field, we derive a Volterra integral equation that governs the expected eddy intensity over time and space. We develop an Expectation-Maximization (EM) algorithm that treats the unobserved parent-child relationships as latent branching structures for parameter estimation. We further extend this framework to accommodate a time-varying, non-homogeneous background intensity, modifying both the EM updates and the analytical Volterra solution accordingly. Finally, we propose a two-stage simulation framework utilizing a cluster representation algorithm. The simulated empirical paths of eddy formation are compared together with numerical solution of the Volterra equation for mean rate as validation of the Hawkes model.
- [252] arXiv:2608.19309 (cross-list from hep-th) [pdf, html, other]
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Title: 3D Einstein action from 6D Kodaira-Spencer gravityComments: 47 pages, 5 appendicesSubjects: High Energy Physics - Theory (hep-th); General Relativity and Quantum Cosmology (gr-qc); Mathematical Physics (math-ph)
In view of embedding 3D gravity into topological string theory, we show that the dimensional reduction of 6D Kodaira-Spencer gravity on $\text{AdS}_3\times S^3$, i.e. the low-energy description of twisted holography, contains a subsector that coincides with the action of chiral 3D gravity on $\text{AdS}_3$. Furthermore, we show that the full 3D Einstein action is obtained as the reduction of two independent complex-conjugate copies of 6D Kodaira-Spencer gravity. We perform the dimensional reduction explicitly at the level of the classical actions, extending our previous work that related the equations of motion of the two theories. Our reduction procedure is based on a novel rewriting of the non-local 6D Kodaira-Spencer action as a local 6D holomorphic Chern-Simons action for the gauge group $\mathrm{SL}(2,\mathbb{C})$, valid within a subsector of complex structure deformations that we identify. Our results provide a necessary step toward embedding the Euclidean path integral of 3D gravity into topological string theory.
- [253] arXiv:2608.19314 (cross-list from quant-ph) [pdf, html, other]
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Title: Proof of the hiding conjecture for Gaussian boson sampling with an arbitrary number of squeezed input modesComments: 25 pages, 1 figureSubjects: Quantum Physics (quant-ph); Computational Complexity (cs.CC); Mathematical Physics (math-ph)
Gaussian boson sampling (GBS) is a sampling task proposed to demonstrate quantum advantage. We consider Gaussian boson sampling on $M$ optical modes, with $K$ equally squeezed input modes and $N$ observed photon counts. We complete the proof of the hiding conjecture for Gaussian boson sampling with an arbitrary number of squeezers $K$, which is a part of the argument for classical hardness of GBS. In particular, we show that for any $K$ and $N=o(\sqrt{K})$, the symmetric product $MK^{-1/2}U_{NK}U_{NK}^T$, for $U_{NK}$ the top left $N\times K$ submatrix of an $M\times M$ Haar random unitary $U$, is close in total variation distance to both an $N\times N$ symmetric complex Gaussian matrix $\mathbf G$ with independent entries, and the symmetric product $GG^T/\sqrt{K}$ for $G$ an $N\times K$ matrix of iid standard complex Gaussians. We show however that the density-based instance generating method of [Aaronson and Arkhipov, Theory Comput. 9, 143 (2013), Lemma 5.8] used to efficiently implement a hiding procedure fails for Gaussian boson sampling with $K=cM$ if $c<1/2$. Instead we use approximate instance generating to implement the hiding for the usual classical hardness reduction.
- [254] arXiv:2608.19315 (cross-list from quant-ph) [pdf, html, other]
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Title: Thermal Throttling of Quantum State TransferComments: 18 pages, 3 figuresSubjects: Quantum Physics (quant-ph); Statistical Mechanics (cond-mat.stat-mech); Strongly Correlated Electrons (cond-mat.str-el); Mathematical Physics (math-ph); Atomic Physics (physics.atom-ph)
Quantum state transfer on qubit lattices is a crucial step in a panoply of quantum information processing tasks. Understanding its fundamental limits in the presence of practical imperfections remains a pressing open question. In particular, it is unclear how thermal noise in the intermediate ancilla sites impedes transfer. In this work, we derive tight lower bounds on the necessary growth of commutator norms to achieve approximate state transfer. Specializing to 1D power-law systems with thermal ancilla states, we demonstrate that state transfer runtimes depend sensitively on the scaling of temperature with system size, improving logarithmic bounds to algebraic ones. Our work extends previous results on exact state transfer to the qualitatively different and more practically relevant regime of approximate state transfer.
- [255] arXiv:2608.19324 (cross-list from hep-th) [pdf, html, other]
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Title: Black Holes, the Bethe Ansatz, and Elliptic Calogero--Moser SystemsComments: v1: 14 pages, 1 appendix, 10 tables, 4 figures, 1 ancillary Mathematica file with plots and numerical solutions for B2, G2 and D4Subjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Exactly Solvable and Integrable Systems (nlin.SI)
We define a map from solutions of the Bethe Ansatz equations (BAEs) of four-dimensional $\mathcal{N}=4$ super-Yang--Mills with arbitrary semisimple gauge algebra $\mathfrak{g}$ to extrema and poles of the potential of the untwisted elliptic Calogero--Moser system of type $\mathfrak{g}$. We conjecture the map to be a bijection on the preimage of the Calogero--Moser extrema, and show that it intertwines the symmetries of the two systems, both the gauge ones (torus, Weyl and center invariance) and a $\mathrm{PSL}(2,\mathbb{Z})$ action, so that solutions on both sides organize into orbits, each BAE orbit mapping onto a single orbit of Calogero--Moser extrema or poles. That the system produced is the untwisted one has a consequence: the conjectured correspondence between BAE solutions and vacua of the $\mathcal{N}=1^\ast$ deformation of $\mathcal{N}=4$ on $\mathbb{R}^{3,1}$, which are extrema of the twisted system, cannot extend to non-simply-laced $\mathfrak{g}$. It also fails within the simply-laced cases, though not for $\mathfrak{su}(N)$: we exhibit an $\mathfrak{so}(8)$ solution that flows to a pole of the Calogero--Moser potential rather than to an extremum, and so has no $\mathcal{N}=1^\ast$ counterpart. We illustrate the map in detail for every rank-two $\mathfrak{g}$, classical and exceptional alike, and use these cases as evidence for the conjecture.
- [256] arXiv:2608.19369 (cross-list from cs.CL) [pdf, html, other]
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Title: Linguistic Holonomy and Statistical Watermarks: Inner Geometry of Meaning-Preserving TransformationsSubjects: Computation and Language (cs.CL); Cryptography and Security (cs.CR); Differential Geometry (math.DG)
Statistical watermarks for language models live in the freedom of the signifier: they choose among tokens that are nearly equivalent in meaning, and they are therefore eroded by exactly those transformations which move the form of a text while leaving its content in place. The literature measures such transformations by their endpoint, through the semantic similarity between the original and the rewritten text. We show that the endpoint is the wrong statistic. Adapting the formalism of linguistic loops, we prove that the invariant of a chain of meaning-preserving transformations factorises canonically into an endpoint part and a holonomy in the stabiliser of the initial state, the second of which the semantic deficit cannot see; the loop rotation is parallel transport on the unit sphere of the embedding space, so that the analogy with the Wilson loop becomes a theorem rather than a figure of speech. On the side of the detector we prove an exact identity: the residual statistic is proportional to the number of positions whose seeding window survived intact, from which the decay law $\rho^{h+1}$ follows as the independent-edit corollary. The identity has a disconcerting consequence, which we confirm to three decimal places: at one and the same retention rate the surviving signal may be one half of the original, one quarter of it, or exactly nothing, according only to where the edits fall.
- [257] arXiv:2608.19370 (cross-list from quant-ph) [pdf, html, other]
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Title: Rethinking Quantum CircuitsComments: 75 pages, 26 figures (incl. cover figure), 10 tables; based on a four-lecture series delivered by the author during the Niels Bohr Quantum Summer School in Odense, DK, 10-13 August 2026Subjects: Quantum Physics (quant-ph); Mesoscale and Nanoscale Physics (cond-mat.mes-hall); Information Theory (cs.IT); Mathematical Physics (math-ph); Category Theory (math.CT)
These notes develop four interconnected ways of reading a quantum circuit. A circuit for us begins as an operational composition of gates; then, it becomes a diagram whose local equalities may be used as calculations; next, it becomes a protected process once errors, syndromes, and logical degrees of freedom are separated; and finally, it becomes geometric when its connectivity, topology, and boundary data are treated as physical design parameters. The development begins at the level of bits and qubits before appealing to Deutsch's and Grover's algorithms as basic examples of quantum circuits. With the basics in hand, we interpret quantum circuits diagramatically, leading us to compact closed string diagrams and the ZX-calculus. After that, we consider how to correct quantum circuits by introducing the Knill--Laflamme condition, homological surface codes, and related concepts with a view towards thinking of these as operations on diagrams. The lectures eventually arrive at the properties of hyperbolic quantum codes and the prospect of physical superconducting circuits emulating the negatively-curved lattices needed to support those codes. These mathematical ideas and physical experiments, taken together, represent one way to impart a geometric layer onto quantum circuits. By the very end, we bring the ideas nearly full circle by assessing the extent to which these device physics experiments operationalize the basic ZX diagrams encountered much earlier in the story. While the later material reports on original research, and while the discussion becomes increasingly mathematical as the sections progress, no prior knowledge of quantum information, quantum computing, or quantum error correction is actually assumed.
- [258] arXiv:2608.19394 (cross-list from q-fin.CP) [pdf, html, other]
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Title: Deep-MKV-TS: Path-Dependent McKean--Vlasov Control for Financial Time Series GenerationSubjects: Computational Finance (q-fin.CP); Computational Engineering, Finance, and Science (cs.CE); Machine Learning (cs.LG); Optimization and Control (math.OC)
We introduce Deep-MKV-TS, a path-dependent McKean-Vlasov framework for financial scenario generation. The stochastic dynamics are chosen by matching selected path and volatility features of generated scenarios to those observed in the data. Starting from an interpretable reference model, Deep-MKV-TS preserves the reference drift and adjusts its volatility, while a regularization penalty limits unnecessary departures from the calibrated dynamics. We solve the resulting control problem using a neural, sample-based implementation of the stochastic maximum principle.
We validate the method against an exactly computable oracle. On Heston and Heston-mixture models, Deep-MKV-TS substantially reduces path-dependent and volatility-related deficiencies of the reference model. In delayed-volatility experiments, the correction remains effective as the forecasting horizon increases, while direct training becomes less reliable. On held-out intraday equity-index futures, the corrected model improves conditional forecasts relative to the reference and reaches a level of performance comparable to flexible generative and historical baselines. The resulting scenarios also support greater exposure than the reference under a fixed drawdown-risk target. These results show that path-dependent McKean-Vlasov control can enrich an interpretable reference model without replacing it. - [259] arXiv:2608.19423 (cross-list from stat.ME) [pdf, html, other]
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Title: Shape-Preserving Covariate Adjustment via Empirical Likelihood in Randomized ExperimentSubjects: Methodology (stat.ME); Statistics Theory (math.ST)
Covariate adjustment improves estimation efficiency in randomized experiments, but standard calibration and augmentation methods, when applied to distribution or survival functions, do not preserve monotonicity---a fundamental property of the estimand. We propose using empirical likelihood with covariate-balancing constraints to construct a covariate-adjusted empirical measure for each treatment arm. Estimators of a broad class of distributional functionals, including cumulative distribution functions, survival functions, quantiles, and restricted mean survival times, are then derived as plug-in functionals of this measure, automatically inheriting proper shape constraints. We establish asymptotic normality with an explicit, guaranteed efficiency gain over unadjusted estimators. The asymptotic distributions are invariant to the randomization scheme, providing a unified inference procedure under simple randomization and all commonly used covariate-adaptive designs satisfying a mild balancing condition. This unified construction, adjusting the empirical measure once and deriving all estimators from it, offers a principled reconciliation of covariate adjustment with shape preservation. Simulations and an application to the SURPASS-4 trial confirm the theoretical gains.
- [260] arXiv:2608.19440 (cross-list from physics.soc-ph) [pdf, html, other]
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Title: Second-Order Continuum Model for Disordered TrafficComments: 44 pages, submitted to Transp. Res. Part BSubjects: Physics and Society (physics.soc-ph); Numerical Analysis (math.NA)
In this study, a two-dimensional second-order macroscopic traffic flow model is proposed to capture the complex dynamics of disordered traffic by explicitly incorporating coupled longitudinal and lateral interactions. The framework extends the classical continuity equation to two spatial dimensions and introduces acceleration equations consisting of self-driven, interaction-induced, and road-boundary-related components. The macroscopic longitudinal acceleration is formulated based on the Full Velocity Difference Model (FVDM), while lateral dynamics are governed by interaction principles consistent with the Optimal Velocity Model (OVM). The explicit inclusion of road-boundary effects enables the representation of vehicle confinement and space-sharing behavior that are essential in lane-free and weakly lane-disciplined traffic systems. The model is examined through a series of numerical experiments in which longitudinal and lateral dynamics are analysed individually as well as simultaneously within a coupled two-dimensional setting under a range of initial conditions, including transitions between free-flow, medium congestion, and heavy congestion, and different lateral density configurations. The simulations demonstrate the model's ability to reproduce key traffic features such as shockwave propagation, lateral dispersion, vehicle rearrangement, and stable density evolution across the road width. The numerical scheme, based on upwind and Lax-Friedrichs discretisations, ensures stable solutions of the coupled partial differential equations. Overall, the proposed framework provides a robust macroscopic description of disordered traffic and offers a consistent basis for analysing two-dimensional vehicular flow dynamics.
- [261] arXiv:2608.19448 (cross-list from quant-ph) [pdf, html, other]
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Title: Efficient Classical Simulation of Weakly Interacting Fermion DynamicsSubjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph); Numerical Analysis (math.NA)
We consider the task of simulating the real-time dynamics of weakly interacting fermionic systems. In particular, we focus on computing the expectation value of a local observable $A$ at time $t$. By analyzing the convergence of the perturbative expansion in the interaction strength $\lambda$ for the Heisenberg-picture observable, we propose a polynomial-time algorithm for estimating this expectation value in the weakly interacting regime $\lambda |t|^{2D+1}=\mathcal{O}(1)$, when the Hamiltonian is geometrically local on a $D$-dimensional lattice. Importantly, this condition is independent of the system size. If the goal is instead to approximate the time-evolved observable in normalized Frobenius norm, we extend the convergence regime to $\lambda |t|=\mathcal{O}(1)$ with quasi-polynomial runtime. When the non-interacting part exhibits Anderson localization, our polynomial-time algorithm can be extended up to $\lambda |t|=\mathcal{O}(1)$, modulo polylogarithmic factors. Our algorithm brings together ideas from continuous-time QMC, diagrammatic QMC, and Majorana Propagation, but with a new Heisenberg-picture operator-growth analysis that makes the sampling complexity rigorously controllable. This leads to provably efficient classical algorithms in regimes where the interaction is weak enough that the sampling variance remains bounded independently of system size. Together, these results identify broad regimes in which weak interactions, locality, and localization can be leveraged to make real-time fermionic dynamics classically tractable.
- [262] arXiv:2608.19459 (cross-list from quant-ph) [pdf, html, other]
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Title: Is the Quantum-Entangled Universe a Small World?Comments: 16 pages, 3 figures, in preparation for ChaosSubjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph)
Partial entanglement may provide enough connections in the universe to satisfy the definition of a small world. To investigate this possibility, we define a network of particles on a given space-like hyper-surface with a long-range link between any two particles that are connected by a chain of exchanged particles involving less than some small maximum number of interactions. Considering the mean free paths of particles in different regions of space and the resulting probability distributions of entanglement connections vs. distance, we find evidence of small-world or random network structure on all but the smallest scales - corresponding to stars and planets - on which other types of connections would need to be added to complete the small world picture.
- [263] arXiv:2608.19464 (cross-list from physics.optics) [pdf, html, other]
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Title: Which pulse maximizes resonant nonlinear conversion?Subjects: Optics (physics.optics); Mathematical Physics (math-ph); Applied Physics (physics.app-ph)
At fixed pulse energy, which drive waveform extracts the most $n$th-order nonlinear conversion from a resonator ($n=2$ for second harmonic)? A short pulse couples poorly to a narrow resonance, a long one dilutes its energy, and no linear rule fixes the compromise. We solve the problem exactly for a single mode of amplitude decay rate $\kappa$. Eliminating the drive turns fixed incident energy into a constraint on the stored field alone, and the optimization becomes a sharp Gagliardo--Nirenberg inequality whose extremal is the ground-state soliton of the nonlinear Schrödinger equation. The optimal stored field is $\mathrm{sech}^{1/(n-1)}[(n-1)\kappa t]$, sustained by an asymmetric input that rises as $e^{\kappa t}$ and falls as $e^{-(2n-1)\kappa t}$; the largest converted energy follows in closed form. A rising exponential, the time-reversal recipe, retains at most $79.0\%$ of the bound at $n=2$ and $2/e$ at large $n$; a two-rate pulse retains above $97\%$. Critical coupling generalizes to $n$-fold overcoupling, with optimal input coupling $n$ times the intrinsic loss rate. The bound applies from microrings to superconducting circuits and caps the per-pulse brightness of broadband photon-pair sources.
- [264] arXiv:2608.19471 (cross-list from cs.AR) [pdf, other]
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Title: Exact Multistate Reliability and Upgrade Design for Heterogeneous HBM Systems via Threshold-Pruned BATSubjects: Hardware Architecture (cs.AR); Performance (cs.PF); Optimization and Control (math.OC)
High Bandwidth Memory (HBM) systems can exhibit partial service rather than only full service or complete isolation: a controller-visible service unit may deliver full, reduced, or zero bandwidth because of sub-channel isolation, lane remapping, or protection overhead. This paper develops an exact multistate reliability framework in which each service unit carries an arbitrary finite set of bandwidth states and reliability is the probability that aggregate delivered bandwidth meets a demand. The binary k-out-of-n model is recovered as a special case, while closed-form binary-mapping error relations quantify mean-bandwidth distortion and provide a screening test for the simpler abstraction. For exact single-threshold evaluation a threshold-pruned multistate Binary-Addition-Tree (TP-mBAT) algorithm is proposed. It is deliberately regime-specific: fixed-grid dynamic programming is preferable on a compact common grid, where a 16-unit commensurate control required 273 pruned-DP updates versus 362,506 TP-mBAT node visits. On a reproducible 14-unit incommensurate benchmark, TP-mBAT is compared against a dynamic program carrying the same threshold rules, so that no baseline is weakened. Both expand the same state space, 6,862 nodes against 6,861 updates, and the separation lies entirely in retained state: 17 traversal entries against 412,121 probability states at central demand, reducing measured peak storage from 25.02 MB to 1,152 B. An exact probability-transfer sensitivity identifies when moving mass from a degraded state to a higher-bandwidth state changes system success, and a reserved-unit floor model admits a third exact pruning rule that is vacuous without such floors. A latent package-state mixture captures shared stress, where ignoring dependence overstates reliability by 8.73 percentage points.
- [265] arXiv:2608.19474 (cross-list from econ.TH) [pdf, html, other]
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Title: Monotone Allocations without Single-Crossing: When to Bunch and When to JumpComments: 85 pages, 10 figures. Replication code: this https URLSubjects: Theoretical Economics (econ.TH); Optimization and Control (math.OC)
A principal screens an agent whose technology has a minimum efficient scale, so the Spence-Mirrlees condition fails along a monotone dividing curve: the locus at which every type values marginal output equally. For the class in which this curve and the relaxed solution are both strictly monotone, the optimal contract obeys a trichotomy, governed by how the two meet: a jump is impossible when they never meet, unavoidable across a flat dividing curve, a choice across a strictly increasing one. The optimum is found, not conjectured: each solution is certified as globally optimal among all implementable allocations, deterministic or random, by dualizing the family of binding constraints through an explicit weight; the certificates require neither linear primitives nor any restriction on the shape of the contract. Under mild regularity the class comprises exactly forty configurations; each is mapped to its forced shape, solved in closed form, and certified.
- [266] arXiv:2608.19489 (cross-list from quant-ph) [pdf, html, other]
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Title: Fast Algorithms for Stoquastic Spin SystemsComments: 30 pages, 0 figuresSubjects: Quantum Physics (quant-ph); Computational Complexity (cs.CC); Data Structures and Algorithms (cs.DS); Combinatorics (math.CO)
We establish a general framework for developing fast sampling and counting algorithms for stoquastic spin systems at high temperature. Our framework is based on a rapidly mixing Markov chain for polymer models and a subcritical percolation process for sampling individual polymers. We apply our framework to obtain fast algorithms for approximating the partition function and sampling from the thermal distribution of (1) general stoquastic spin systems, (2) ferromagnetic Heisenberg models, and (3) antiferromagnetic Heisenberg models on bipartite graphs. For the Heisenberg models, we obtain an improved bound on the inverse temperature by using their respective cycle and loop representations.
- [267] arXiv:2608.19491 (cross-list from cs.LG) [pdf, html, other]
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Title: DeltaMomentum: A Key-Value based Anisotropic Momentum Update via Delta RuleSubjects: Machine Learning (cs.LG); Computation and Language (cs.CL); Optimization and Control (math.OC); Machine Learning (stat.ML)
Most modern optimizers form their momentum as an exponential moving average (EMA) of past gradients, forgetting every direction at one fixed rate. However, the inputs a deep network sees during training can be highly anisotropic, with a few directions queried frequently while most are seen rarely. Recent methods address this anisotropy by wrapping extra processing around this buffer, leaving the momentum update itself unchanged. We propose DeltaMomentum, which builds direction-awareness into the momentum update rule. The main observation is that the gradient of a linear layer splits into an input that acts as a key and an output-side error that acts as a value. Exploiting the key-value structure, DeltaMomentum updates the momentum buffer by the canonical delta rule, so each direction is forgotten at a rate set by how often it appears. We prove that it is a valid momentum, that it applies the input-side curvature correction without matrix inversion, and that it clears stale directions faster than EMA under both a fixed and a drifting optimum. It is a drop-in replacement for the momentum buffer of any optimizer, its coefficient transfers across widths under $\mu$P, and its extra compute stays between $22.2\%$ and $25.0\%$ of a gated-MLP block's linear cost with no persistent memory. In FineWeb-Edu pretraining, AdamW with DeltaMomentum (DeltaAdamW) reaches AdamW's validation loss in up to $46.39 \pm 4.32\%$ fewer steps at 67M and $22.12 \pm 0.80\%$ at 370M over three seeds, and the gain persists at 1B on a Chinchilla-optimal budget. A Muon baseline tuned under the same protocol sits above DeltaAdamW at both language-model scales, and the gain holds for SGD, ResNet-18, and ViT-Tiny on CIFAR-10. Training-time diagnostics confirm the predicted mechanism, better gradient tracking and healthier input directions.
- [268] arXiv:2608.19519 (cross-list from quant-ph) [pdf, html, other]
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Title: Beyond Integrability Preserving Renormalization-Group Protocol in Non-Hermitian Hamiltonians with Time-Dependent Interaction StrengthsSubjects: Quantum Physics (quant-ph); Strongly Correlated Electrons (cond-mat.str-el); Mathematical Physics (math-ph)
It is well established that in time-dependent quantum systems, integrability preserving time-dependent interaction strengths are identical to the renormalization group (RG) trajectories of the corresponding static model when time `$t$' in the driven model is identified with the logarithm of the cutoff `$\log\Lambda$' of the static model. We refer to this integrability preserving driving as the RG protocol. In this work we extend the class of time-dependent integrable models to include non-Hermitian quantum models with time-dependent interaction strengths. Using the recently formulated generalized Bethe ansatz framework [P. R. Pasnoori, Phys. Rev. B 112, L060409 (2025)], we show that the constraints imposed by integrability are more general: The interaction strengths of the static model that flow in the RG follow the respective RG trajectories in the corresponding time-dependent model as described above. In addition, the interaction strengths of the static model that are RG invariant can either be constant or have a specific time-dependence in the corresponding time-dependent model which is constrained by integrability. Thus we establish that in the context of time-dependent non-Hermitian systems, the set of integrability preserving time-dependent strengths is larger than the set corresponding to the RG protocol.
- [269] arXiv:2608.19539 (cross-list from hep-th) [pdf, html, other]
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Title: Higher Chern--Simons Theory in $2n+2$ Dimensions for Balanced 2-term $L_\infty$-AlgebrasSubjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
We construct a semistrict higher Chern--Simons (HCS) gauge theory in $2n+2$ dimensions associated with balanced 2-term $L_\infty$-algebras. Starting from the homotopy Maurer--Cartan theory, we first introduce 2-term $L_\infty$-algebra gauge theory, and show that there is a four-dimensional HCS construction. Then we extend invariant bilinear pairings to invariant multilinear forms of the appropriate degree, and define a $(2n+3)$-dimensional higher Pontryagin--Chern form, which is closed and invariant under infinitesimal gauge transformations. Its transgression yields an explicit $(2n+2)$-dimensional HCS form. We further establish a higher Chern--Weil theorem that generates higher transgression forms, and prove that the HCS theory is a distinguished instance of the higher transgression gauge theory. Finally, we apply the extended Cartan homotopy formula in this semistrict setting, and show that it is a common origin of both the higher Chern--Weil theorem and the associated triangle equation.
- [270] arXiv:2608.19579 (cross-list from cs.AI) [pdf, html, other]
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Title: Enforcing LLM Safety through DMD-based Classification of Prompt-Response Embedding DynamicsSubjects: Artificial Intelligence (cs.AI); Dynamical Systems (math.DS)
Large Language Models (LLMs) are increasingly deployed in high-stakes applications, yet their tendency to generate toxic, harmful, or policy-violating content poses significant risks. Detecting these unsafe outputs efficiently in a black-box manner remains an open challenge. In this paper, we extend a recently proposed dynamical systems framework designed for hallucination detection to LLM safety classification. By projecting both prompts and responses into high-dimensional embedding spaces and fitting separate Koopman-based predictive models for safe and unsafe regimes, we classify new outputs using a new differential residual score that compares prediction errors of the safe and unsafe regimes. A key contribution is the incorporation of the prompt and response embedding dynamics, yielding fitted Koopman operators that capture crucial interaction patterns. We evaluate our black-box method across three safety benchmarks using three embedding models. Our results show that incorporating prompt embeddings yields consistent improvements, particularly for interaction-dependent violations when paired with causal decoders (e.g., in Llama-3), while response-only violations benefit more from dense semantic embedding representations. These findings opens the door for using dynamical systems to analyze AI systems rather than the dominant paradigm of using AI to model dynamical systems.
- [271] arXiv:2608.19584 (cross-list from cs.LG) [pdf, html, other]
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Title: Kähler landscapes for complex neural network descents and guarantees including a search and destroy of the Calabi-Yau manifoldComments: First versionSubjects: Machine Learning (cs.LG); Differential Geometry (math.DG); Machine Learning (stat.ML)
We study landscapes for complex-parameterized networks. Our approach is motivated with an information-theoretic manifold perspective of the parameter and via classical optimization guarantees although of complex geometric variety such as through Dolbeault asymptotics. The descent path admits a Kähler information metric under a cross-entropy via the Wirtinger Hessian on the log-likelihood potential. We restrict attention to a descent update rule with natural gradient descent via a differentiated loss scaled by the inverse metric, so the descent path remains in the holomorphic tangent bundle. We emphasize Calabi-Yau information manifolds which profane theoretical guarantees via an ill-curvature-conditioned landscape. Under a Calabi-Yau metric, specifically in a non-compact setting with a global potential so defined geometrically rather than invoking the topological requirements of the Calabi conjecture, a wedged nowhere-vanishing holomorphic form is the top exterior product of the Kähler form up to constants, yielding a constant determinant condition. Under a fixed determinant, a metric almost low rank up to an eigenvalue tolerance implies a blow-up effect. Moreover, it has been discovered that negative curvature subverts the loss landscape, specifically sectional curvature, so we expand on this and draw interconnections to negative-definite Ricci curvature. Our arguments primarily exist in a geometric analytic modality, although we establish roots in deep learning theory such as through asymptotics at initialization and connections through failure modes of neural network guarantees under vanishing and negative Ricci curvature.
- [272] arXiv:2608.19593 (cross-list from cs.RO) [pdf, html, other]
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Title: The Verification Gap in Networked Physical AI: A Post-Semantic Communication FrameworkComments: 9 pages, 3 figures, 3 tablesSubjects: Robotics (cs.RO); Information Theory (cs.IT); Networking and Internet Architecture (cs.NI)
A task-effective proposal is not yet a justified physical action. In networked Physical AI, a proposal may be understood while valid, timely, proposal-bound evidence or the authority required to finalize an action remains unavailable. We call this mismatch the verification gap and introduce a Post-Semantic Communication Framework for the systems interface between proposal formation and physical execution. The framework begins with application-declared evidence requirements, represents qualifying observations as evidence records, validates supporting and conflicting records through one path, and separates evidence sufficiency from authorized finalization and a downstream runtime gate. It further distinguishes evidence transfer, which can enlarge the record set reachable by a finalizer, from evidence coordination, which can suppress transmission around records already held at the finalization endpoint. Finite-state framework checks verify that the evaluator implements the declared distinctions consistently. Under the declared model, the controlled communication study exposes a finalizer-dependent asymmetry: sender-finalized Feedback uses evidence transfer to expand evidence reachability throughout the feasible plotted region, whereas receiver-finalized Feedback uses coordination to suppress redundant payload until loss, latency, freshness, and deadline costs shift selection to One-way. Finally, an episode-level reporting schema defines common denominators for future measured Physical-AI studies.
- [273] arXiv:2608.19605 (cross-list from cond-mat.stat-mech) [pdf, html, other]
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Title: Exact partition function of arithmetic Ising modelComments: 6 pages, 3 figuresSubjects: Statistical Mechanics (cond-mat.stat-mech); Strongly Correlated Electrons (cond-mat.str-el); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Combinatorics (math.CO)
We present a compact formula for the exact partition function of the $d$-dimensional arithmetic Ising model (AIM). For a $2\times2$ system, we express it analytically using the $q$-Hurwitz-Lerch zeta function and derive explicit forms for the free energy and entropy. Additionally, we find that the entropy increases at high temperatures, supporting the presence of entropic order.
- [274] arXiv:2608.19634 (cross-list from stat.ME) [pdf, html, other]
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Title: Curvature-Calibrated Quasi-Bayesian Updating for Moment-Restricted ModelsSubjects: Methodology (stat.ME); Econometrics (econ.EM); Statistics Theory (math.ST)
Moment restrictions provide a flexible basis for quasi-Bayesian inference when a full likelihood is unavailable, but the weighting matrix in a quadratic moment criterion determines both the relative importance of the moments and the information scale of posterior updating. We propose curvature-calibrated quasi-Bayesian updating, which uses the inverse of the covariance (or long-run covariance) of the moment conditions evaluated at a self-consistent quasi-posterior center. The resulting fixed-point procedure alternates between covariance estimation and simulation from a fixed-weight quasi-posterior, thereby avoiding parameter-dependent weighting during each simulation run. Under a Bernstein-von Mises condition for the fixed-weight quasi-posterior at the efficient population weight, we show that the calibration map is locally contractive, that its fixed point is consistent at the standard parametric rate, and that the Gaussian approximation continues to hold under the calibrated data-dependent weight, with covariance given by the inverse Godambe information matrix. Under a uniform fourth-moment condition, the scaled quasi-posterior covariance converges to the same matrix, so quasi-posterior and repeated-sampling uncertainty agree to first order. Simulations show improved covariance calibration and interval coverage after a few updates. An application to longitudinal binary-response data illustrates the method with within-subject dependence and overidentified residual moments.
- [275] arXiv:2608.19658 (cross-list from cs.LG) [pdf, html, other]
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Title: Rationally Enriched Chebyshev Trunk Bases for DeepONet Surrogates of High Péclet Entrance TransportComments: 39 pages, 9 figuresSubjects: Machine Learning (cs.LG); Numerical Analysis (math.NA)
This study demonstrates a rationally enriched Chebyshev (REC) trunk for deep operator network (DeepONet) surrogate models of singularly perturbed and high-Péclet transport problems whose solution profiles are characterized by thin localized boundary or wall layers. The REC trunk combines Chebyshev polynomial dictionary elements with rational dictionary elements constructed using the adaptive Antoulas-Anderson (AAA) algorithm. Over five independent training runs, the resulting REC-trunk DeepONet is evaluated against a vanilla DeepONet and a Chebyshev-trunk DeepONet whose prescribed dictionary consists only of Chebyshev polynomials across three problems whose singular perturbation parameters are diffusion-to-advection ratios: a singularly perturbed scalar boundary-value problem (BVP), the thermal entrance problem with a prescribed wall temperature, and the concentration entrance problem with an absorbing wall. Across the held-out test profiles, the REC-trunk DeepONet improves over the vanilla DeepONet and remains comparable to the Chebyshev-trunk DeepONet in predicting the scalar profile, with its clearest advantage over the Chebyshev-trunk DeepONet appearing when the perturbation parameter lies between $1.00\times10^{-4}$ and $1.78\times10^{-4}$, where it reduces the profile-error metrics by up to $19.5\,\%$ relative to the Chebyshev-trunk DeepONet. In predicting the wall-normal temperature and concentration profiles, the REC-trunk DeepONet reduces the profile-error metrics by up to $60.2\,\%$ and $32.2\,\%$ relative to the vanilla and Chebyshev-trunk DeepONets, respectively, while suppressing artificial near-wall oscillations as the Péclet or mass-transfer Péclet number ranges from $10^{2}$ to $10^{4}$.
- [276] arXiv:2608.19668 (cross-list from quant-ph) [pdf, html, other]
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Title: A charge selection rule fixes what a squeezed-light reservoir computer can compute and affordSubjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph)
Reading an optical quantum reservoir costs repetitions growing super-exponentially with feature order: what it can afford is set by its detector, not its optics. For reservoirs encoding data in a parametric pump's phase, one conservation law fixes what is readable and what it costs. Pairwise photon exchange conserves an integer phase charge: across an ensemble of input masks, order-D readout reaches exactly the assemblies of at most D unit-charge kernels; degree-one homodyne readout is universal for fading-memory functionals along weak-squeezing families, at shot cost polynomial in accuracy; and at fixed squeezing no finite degree reaches every sector, at a computable distance. Nonlinearity sits in the optics, not the detector, whose per-shot variance is fixed at every order. In a hardware-faithful digital twin-no device was built-the rule is visible: on an open RF corpus a displacement-encoded control at identical photon number loses 17.7 accuracy points, as the charge algebra predicts.
- [277] arXiv:2608.19679 (cross-list from quant-ph) [pdf, html, other]
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Title: Geometric phase of open paths and a geodesic-selection rule at a level degeneracyComments: 12 pages, 1 figureSubjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph)
When the control field of a qubit, a polarization state, or a spin-$\tfrac12$ system is swept through a level degeneracy, its direction traces an open curve on the Bloch sphere whose endpoints are antipodal, and the geodesic rule for the open-path geometric phase becomes ambiguous: infinitely many geodesics close the path, and different closures enclose different solid angles. We resolve this ambiguity in closed form. A coordinate-free monopole connection defines the open-path solid angle $\Omega[C]$ intrinsically, and displacing the degeneracy by $\epsilon\uhat$ closes the path with enclosed solid angle $\Omega(\epsilon\uhat)=\Omega[C]+2\alpha+O(\epsilon)$, where $\alpha$ is the azimuth of the transverse part of $\uhat$ measured from the principal normal of the control curve at the crossing. The identity between geometric phase and enclosed solid angle therefore holds for exactly one closing geodesic---the great circle in the osculating plane ($\alpha=0$)---supplied by the curvature at the degeneracy. Berry's $\pi$ invariant under reversal of the displacement and the values $\pm\pi/2$ under a reflection symmetry follow as corollaries, and the pure-state limit of the finite-temperature Uhlmann phase selects the osculating-plane closure automatically, turning the heuristic closing rules of the open-path literature into a computable prescription.
- [278] arXiv:2608.19685 (cross-list from quant-ph) [pdf, html, other]
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Title: Tight Entropy Contraction of Generalized Quantum DepolarizationComments: 24 pages, comments welcomeSubjects: Quantum Physics (quant-ph); Functional Analysis (math.FA); Operator Algebras (math.OA)
We establish upper and lower bounds for relative entropy contraction of generalized quantum depolarizing channels and semigroups. Our bound provides tight first order asymptotic of the contraction rate in terms of the dimension constant. One side estimate are based on sharp reverse ratio and convexity of relative entropy of two states, which can be derived from the recently introduced Hockey-Stick quantum $f$-divergence, and also independently, Bogoliubov--Kubo--Mori quantum Fisher information metric. The other side follows from the existence of index achieving pure state with respect to a general conditional expectation. Our results extend to the complete entropy contraction rate, tensor stable estimates for product dynamics. Examples include quantum depolarization, dephasing, and compact group symmetrization, with consequences for the decay rate of coherence and asymmetry.
- [279] arXiv:2608.19716 (cross-list from stat.AP) [pdf, html, other]
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Title: A Bayesian Time-Varying SEIARD Model for State-Level COVID-19 Transmission and Mortality in the United StatesSubjects: Applications (stat.AP); Probability (math.PR); Computation (stat.CO); Methodology (stat.ME)
We conduct a retrospective analysis of COVID-19 transmission dynamics across U.S. states using a modified population-based Susceptible-Exposed-Infectious-Asymptomatic-Recovered-Deceased (SEIARD) compartmental model. The proposed framework introduces time-varying transmission, reporting, and mortality rates to capture temporal variations in public behavior and policy interventions during the pandemic. In particular, the transmission rate is modeled as a function of population mobility (derived from Google Mobility Reports), with a residual time-decay term capturing the net effect of unobserved factors such as behavioral adaptation and control measures, while reporting is linked to nationwide testing strategies. We employ a Bayesian approach to integrate multiple data sources and quantify uncertainties in model parameters. The model explicitly distinguishes between symptomatic and asymptomatic infectious individuals and links the latent epidemic states to observable quantities, including reported cases and deaths, through a dynamic reporting function. This retrospective modeling framework provides insights into state-level epidemic trajectories and supports data-driven decision-making for optimal allocation of healthcare resources and evaluation of public health interventions during future pandemics. We further apply a clustering analysis to the posterior parameter estimates to identify groups of U.S. states exhibiting similar epidemiological characteristics, revealing substantial regional heterogeneity in transmission intensity, reproduction dynamics, and mortality burden.
- [280] arXiv:2608.19761 (cross-list from quant-ph) [pdf, html, other]
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Title: Krylov Tomography and Finite-Uncertainty Certification of Exceptional-Point DynamicsSubjects: Quantum Physics (quant-ph); Mesoscale and Nanoscale Physics (cond-mat.mes-hall); Mathematical Physics (math-ph)
Exceptional points (EPs) can produce striking responses, but locating one does not reveal how much of its dynamics an excitation accesses, which responses a detector distinguishes, or whether a missing signal is absent or undetected. We introduce Krylov tomography, a preparation- and measurement-aware framework that uses time-resolved data and models with stated uncertainty limits to answer these questions, while also bounding offset-induced departures from exact-EP behavior. As an explicit illustration of the general framework, we present finite-precision simulations of a red-sideband optomechanical model, whose second-moment coherence sector contains a third-order EP, certifying preparation-sensitive access to two and three directions. Krylov tomography thus bridges EP structure and finite-precision measurements, providing a general framework for non-Hermitian systems.
- [281] arXiv:2608.19762 (cross-list from cs.LG) [pdf, html, other]
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Title: Finite-Horizon Input-Output Dynamics of Minibatch Perturbations in AdamWSubjects: Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Optimization and Control (math.OC); Machine Learning (stat.ML)
A minibatch can influence training beyond the update at which it is observed because AdamW stores past gradient information in its optimizer states. We study this delayed effect through paired trajectories that differ only in one gradient update and share the same subsequent training sequence. We formulate AdamW as a finite-horizon input--state--output (ISO) system whose state contains the model parameters and first- and second-moment estimates. Linearizing the joint dynamics yields a signed response operator that maps a localized gradient perturbation to its future loss effects, revealing how optimizer memory shapes their magnitude, timing, and sign. We further derive an exact multistep error decomposition and establish first-order finite-horizon accuracy under local smoothness and controlled activation switching. Experiments validate the response mechanism and optimizer-state effects, while repeated-future analyses reveal substantial prospective structure in delayed influence that can be partially recovered from ISO approximations. Code is available at this https URL.
- [282] arXiv:2608.19775 (cross-list from hep-th) [pdf, html, other]
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Title: Supergroup Gauged Linear Sigma Models and their Physical MathematicsComments: 117 pp. + appendixSubjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Algebraic Geometry (math.AG)
We construct 2d $\mathcal{N}=(2,2)$ gauged linear sigma models with $\mathrm{U}(1|1)^N$ supergauge group possibly with superpotential. Despite being nonunitary, one can still study their space of supersymmetric states and explore their applications to mathematics. In particular, we find a relation between a nonlinear sigma model on a Calabi-Yau complete intersection of hypersurfaces in a super-Grassmannian and a supergauged Landau-Ginzburg orbifold, which can reduce to a regular Calabi-Yau/Landau-Ginzburg correspondence for complete intersections. This defines a super-Grassmannian/supergroup generalization of the correspondence proved by Clader [1] and Zhao [2]. Similarly, we find a relation between a nonlinear sigma model on a Calabi-Yau hypersurface in a product of super-Grassmannians and a hybrid NLSM/supergauged Landau-Ginzburg orbifold, which can reduce to a regular hybrid Calabi-Yau/Landau-Ginzburg correspondence for hypersurfaces in product space. This defines a super-Grassmannian/supergroup generalization of the correspondence proved by Fan-Jarvis-Ruan [3]. We also find that Calabi-Yau supervector bundles over a super-Grassmannian can undergo a physically related mild topology change which is reducible to a regular Atiyah-type flop transition. This defines a super-Grassmannian generalization of a birational equivalence of Calabi-Yau vector bundles in mathematics. Similarly, we find that a Calabi-Yau complete intersection of quadrics in a super-Grassmannian can also undergo a physically related topology change which is reducible to a regular conifold transition. This defines a super-Grassmannian generalization of a homological projective duality for Calabi-Yau quadrics by Kuznetsov-Perry [4] in mathematics.
- [283] arXiv:2608.19793 (cross-list from cond-mat.mtrl-sci) [pdf, html, other]
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Title: Constitutive modelling of open-porous neo-Hookean solidsSubjects: Materials Science (cond-mat.mtrl-sci); Soft Condensed Matter (cond-mat.soft); Mathematical Physics (math-ph); Classical Physics (physics.class-ph)
Open-porous materials exhibit pronounced compressibility, nonlinear densification, and power-law scaling of stiffness with density. In this work, we propose a thermodynamically consistent compressible neo-Hookean constitutive model for open-porous solids in which porosity serves as the primary governing variable. The strain-energy density is formulated to couple distortional elasticity of the solid skeleton with a volumetric response governed by deformation-induced porosity evolution, including a bounded representation of pore collapse. The formulation introduces a minimal set of parameters, namely the initial porosity, intrinsic skeleton moduli, and a scalar parameter controlling the onset of densification. A key feature of the model is a modified volumetric term in which the response is normalised by the current porosity, ensuring a physically consistent transition from a porous to a densified state without artificial stiffening. In the small-strain limit, the model recovers classical linear elasticity with effective moduli that may be chosen either from homogenisation bounds, such as the Hashin-Shtrikman estimates, or from Gibson-Ashby-type power-law scaling to capture topology-dependent behaviour. At finite strains, the formulation captures the characteristic nonlinear stiffening and convex stress-stretch response associated with progressive pore collapse. The proposed framework thus provides a compact, flexible, and extensible constitutive description that unifies effective-medium consistency with experimentally observed scaling behaviour, and is well suited for finite element implementation and multiscale modelling of highly compressible open-porous materials. The model is finally validated against available experimental data.
- [284] arXiv:2608.19814 (cross-list from physics.chem-ph) [pdf, html, other]
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Title: Rare Fluctuations from Normally Hyperbolic Invariant ManifoldsComments: 37 pages, 2 figuresSubjects: Chemical Physics (physics.chem-ph); Dynamical Systems (math.DS)
Freidlin--Wentzell theory converts weak-noise large deviations into a Hamiltonian variational problem. We study how a $k$-dimensional normally hyperbolic invariant manifold (NHIM) $N$ of the deterministic dynamics appears in this Hamiltonian system. Its zero-momentum copy $N_0=N\times\{0\}$ is invariant, but the Hamiltonian dynamics has $2k$ center directions near $N_0$: $k$ tangent to $N$ and $k$ conjugate covector directions. We construct the resulting local symplectic geometry and show that fluctuation extremals approaching $N_0$ backward in time at the strong normal rate form an $n$-dimensional exact Lagrangian invariant manifold carrying a single-valued action. A counterexample shows that a corresponding zero-energy section need not be normally hyperbolic within $H_{\mathrm{FW}}^{-1}(0)$. We then consider reaction dynamics. If a parameter moves a deterministic trajectory toward a codimension-one reactivity boundary, the minimum Freidlin--Wentzell action required to reach the boundary is quadratic in the distance from the threshold parameter. Its coefficient is determined by the relative motion of trajectory and boundary and by how effectively the available noise acts transversely. This deterministic boundary is distinct from a noise-dependent stochastic transition state or a committor surface. In a solvent--solute model, varying solvent mass moves the phase-space reactivity boundary while leaving the potential-energy surface fixed, changing the rare-event cost without changing the potential-energy barrier.
- [285] arXiv:2608.19818 (cross-list from cs.MS) [pdf, other]
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Title: Eigensolvers for polynomial roots and tensor decompositionEnrica Barrilli (AROMATH), Bernard Mourrain (AROMATH)Subjects: Mathematical Software (cs.MS); Symbolic Computation (cs.SC); Commutative Algebra (math.AC); Algebraic Geometry (math.AG)
Computing eigenvalues and eigenvectors is at the heart of the solution of many non-linear problems. For instance, finding the roots of polynomial systems reduces to computing joint eigenvectors of operators of multiplication. Similarly, tensor decomposition can be performed via the joint diagonalization of submatrices of the Catalecticant of the tensor. We describe and illustrate symbolic-numeric methods for computing the solutions of these algebraic problems from the computation of joint eigenvectors of commuting operators, and for analysing their multiplicity structure, as well as their implementation in the package this http URL.
- [286] arXiv:2608.19831 (cross-list from cs.AI) [pdf, html, other]
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Title: Causal Reasoning with Bipartite Graphical Causal ModelsComments: Figure 2 was added w.r.t. to the published version at Uncertainty in Artificial Intelligence (UAI) 2026Subjects: Artificial Intelligence (cs.AI); Probability (math.PR)
Causal Bayesian networks (CBNs) and structural causal models (SCMs) are the dominant frameworks for graphical causal reasoning, but they cannot adequately represent all real-world causal systems. In particular, systems at equilibrium---where feedback mechanisms create cyclic causal dependencies---can exhibit causal semantics that are fundamentally incompatible with these frameworks: different interventions that enforce the same variable value may have different effects, rendering the standard ``perfect intervention'' do($X = x$) ambiguous. We propose bipartite graphical causal models (BGCMs), in which the structure of a system of equations is encoded by a bipartite graph with variable and equation nodes. In this framework, a hard intervention do($f_j : X_v = \xi_v$) specifies which equation is replaced, which variable is targeted, and at what value---resolving the ambiguity of the standard notion. We demonstrate, through a detailed case study of a physical system, that this representation naturally corresponds to distinct real-world interventions. We formulate a Markov property in terms of a new graphical separation criterion (B-separation) that exploits the functional determinism inherent in the equations, and we extend it to settings with non-random inputs. We show how this gives rise to a do-calculus for reasoning about domain invariances. BGCMs strictly generalize CBNs and SCMs while retaining the ability to perform graphical causal reasoning.
- [287] arXiv:2608.19840 (cross-list from eess.SP) [pdf, html, other]
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Title: Applying the Spectral Method for Modeling Linear Filters: Bessel, Papoulis, and Legendre FiltersSubjects: Signal Processing (eess.SP); Systems and Control (eess.SY); Numerical Analysis (math.NA)
This paper proposes a new technique for computer simulation of linear filters. It allows simulating continuous-time linear filters based on the spectral method for analyzing linear control systems. Applying the spectral method implies that the input and output signals are represented by ordered sets of expansion coefficients in a chosen basis, and the filter itself is specified by a two-dimensional nonstationary transfer function. The described technique is tested on Bessel, Papoulis, and Legendre filters of various orders. For each filter, the corresponding two-dimensional nonstationary transfer function, i.e., the matrix of the linear transformation relating expansion coefficients of the input and output signals, is obtained.
- [288] arXiv:2608.19979 (cross-list from physics.comp-ph) [pdf, html, other]
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Title: Resonant state expansion for acoustic resonators. Part I. Eigenvalue problemEgor Domoratskii, Vladimir Igoshin, Nikolay Solodovchenko, Mingzhao Song, Yong Li, Mihail Petrov, Andrey BogdanovComments: 12 pages, 5 figures, Supplemental MaterialSubjects: Computational Physics (physics.comp-ph); Mathematical Physics (math-ph); Optics (physics.optics)
Resonant-state expansion (RSE) is a powerful modal framework for the perturbative analysis of open resonant systems, providing direct access to complex eigenfrequencies and eigenmodes. While RSE is well developed in electromagnetism, a comparably systematic formulation for acoustics remains less established. Here, we develop a general Green-function-based formalism for acoustic RSE and illustrate it for a class of two-dimensional acoustic resonators. Using the resonant states of an analytically solvable cylindrical reference system as a basis, we derive explicit perturbation matrix elements for uniform, radial, and sectoral variations of density and compressibility, representing homogeneous tuning, graded profiles, and symmetry-induced modal coupling. The resulting complex eigenfrequencies and eigenmodes are validated against exact analytical solutions and finite-element simulations, showing excellent quantitative agreement. The framework provides a systematic and physically transparent approach for analyzing perturbed open acoustic resonators and establishes a basis for resonant-state methods in acoustic metamaterials and non-Hermitian acoustics.
- [289] arXiv:2608.19992 (cross-list from physics.comp-ph) [pdf, html, other]
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Title: Resonant state expansion for acoustic resonators. Part II. Scattering problemEgor Domoratskii, Vladimir Igoshin, Nikolay Solodovchenko, Mingzhao Song, Yong Li, Mihail Petrov, Andrey BogdanovComments: 9 pages, 6 figures, Supplemental MaterialSubjects: Computational Physics (physics.comp-ph); Mathematical Physics (math-ph); Optics (physics.optics)
We develop a resonant-state expansion formulation for acoustic scattering by individual resonators. The scattered pressure and particle-velocity fields are expanded over the resonant states of the system, with excitation amplitudes determined by overlap integrals between the incident field and the resonant states over the resonator volume. Using the acoustic energy flux, we derive expressions for the extinction, scattering, and absorption cross-sections and show that the extinction spectrum can be resolved into contributions from individual resonant states. The formulation is first validated for a homogeneous two-dimensional cylinder, where it reproduces the analytical Mie-theory solution. We then consider a sectorally perturbed cylinder with coupled azimuthal modes and demonstrate agreement with finite-element simulations. Finally, we combine the eigenvalue and scattering formulations for a material-programmed hard-wall annular metaatom and reproduce its scattering spectra and near fields. The developed framework provides a physically transparent modal approach to acoustic scattering by open resonators with reduced symmetry and spatially structured material parameters.
- [290] arXiv:2608.20003 (cross-list from cond-mat.soft) [pdf, html, other]
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Title: Role of topology in scaling laws for studying mechanics in open-porous solids: Moving beyond classical Gibson-Ashby scalingSubjects: Soft Condensed Matter (cond-mat.soft); Materials Science (cond-mat.mtrl-sci); Mathematical Physics (math-ph)
The elastic modulus of porous materials is commonly described using power-law scaling relations with relative density, where the scaling exponent is often interpreted in terms of the underlying deformation mechanism. However, in highly disordered porous networks, changes in density are generally accompanied by changes in network topology, which can substantially modify the apparent scaling behavior. In this paper, we propose a topology-informed framework that separates the intrinsic mechanical contribution from the effects of network structure. Three representative topological descriptors are considered: the mean coordination number, the fraction of the load-bearing backbone, and the tortuosity of the load paths. For each case, the corresponding density-dependent contribution to the apparent modulus-scaling exponent is derived and analyzed. The results show that variations in connectivity, mechanical participation of the solid phase, and load-path efficiency can all lead to apparent scaling exponents exceeding the intrinsic exponent associated with the local deformation mechanism. These effects are particularly pronounced at low relative densities, where network topology evolves most strongly. The framework provides a physically interpretable basis for understanding anomalous modulus-density scaling in disordered porous materials and highlights the need to consider topology explicitly alongside relative density.
- [291] arXiv:2608.20017 (cross-list from nlin.CD) [pdf, html, other]
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Title: Understanding the superiority of multi-model ensemble forecasts through reservoir computingDaniel Estevez Moya, Francesco Martinuzzi, Edmilson Roque dos Santos, Erick Alejandro Madrigal Solis, Ernesto Estevez Rams, Holger KantzSubjects: Chaotic Dynamics (nlin.CD); Dynamical Systems (math.DS)
Weather forecasting and climate projection frequently use multi-model ensembles (MMEs) to improve short-term forecasts by averaging across models. However, this practice is often not well justified or validated. Using reservoir computing (RC) as a computationally efficient alternative to large-scale physical models, we assess the validity of the MME approach for chaotic time series. By training multiple randomly constructed RCs on the same dataset, we create a multi-model ensemble in which each model has its own unique error. These model errors lead to very different forecasting performances, with forecast error distributions that exhibit heavy tails. The arithmetic mean across forecasts from multiple models for the same target is usually closer to the ground truth than most individual forecasts, and further improvement is achieved by weighted arithmetic means where the weights are constructed based on each model's test-set performance. We show that iterated forecasts over many time steps deviate from the ground truth along the unstable manifold of the target point, in both directions, so that, if forecast errors were independent and had zero mean, the arithmetic mean forecast should approach the true target like $1/\sqrt{\nens}$ where $\nens$ is the size of the multi-model ensemble. We observe deviations from this behavior, which we attribute to the tails of the error distribution of random RCs.
- [292] arXiv:2608.20043 (cross-list from eess.SY) [pdf, other]
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Title: Wave-Based Bilateral Teleoperation between Nonlinear Manipulators with Direct Contact Force FeedbackComments: 65th IEEE Conference on Decision and Control (CDC), Honolulu, HI, USA, Dec. 2026Subjects: Systems and Control (eess.SY); Robotics (cs.RO); Dynamical Systems (math.DS); Optimization and Control (math.OC)
We study bilateral teleoperation between nonlinear, multi-DOF robotic manipulators in the presence of constant communication delays. Unlike classical wave-transformation architectures that transmit a coordinating force, we consider the case where the environmental force is reflected to the master side to enhance teleoperation transparency. Since direct contact force feedback might destabilize the closed-loop system, we first develop a passivity-shortage characterization for the Euler--Lagrange remote system using a linear matrix inequality (LMI) approach. An upper strictly passive communication law is then employed to compensate for the computed passivity shortage so that the closed-loop stability under delays as well as position and force synchronization are preserved under appropriate conditions. Simulations with nonlinear 2-DOF robotic manipulators in different settings illustrate our approach.
- [293] arXiv:2608.20151 (cross-list from quant-ph) [pdf, html, other]
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Title: A Zoology of Quantum Turing PatternsComments: A related work is available at arXiv:2607.26331. Code is publicly available in the author's GitHub repository. this https URLSubjects: Quantum Physics (quant-ph); Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph); Adaptation and Self-Organizing Systems (nlin.AO); Pattern Formation and Solitons (nlin.PS)
We explore quantum Turing pattern zoology, where the same Lindblad equation supports a morphology atlas of stripes, spots, holes, labyrinths, and defects. The stable stripe species provides a quantitatively controlled case in which morphology and Gaussian witness loss separate parametrically. In particular, visible Turing stripes can remain after two Gaussian witness margins associated with the same $k_*$ mode cross zero in a completely positive Lindblad lattice. The witness thresholds on the exact shell fall as $\mathcal{N}^{-1}$. The stripe nematic threshold tends to a nonzero value at fixed lattice size, time window, and morphology criterion. The ratio of the morphology threshold to either witness threshold therefore grows with $\mathcal{N}$. Imaging and momentum-resolved covariance measurements probe these sectors separately.
- [294] arXiv:2608.20183 (cross-list from cs.LG) [pdf, html, other]
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Title: Exact Algebraic Computation of Learning Coefficients for Two-Dimensional Singular ModelsGrégoire Sergeant-Perthuis (1), Elias Tsigaridas (2), Jules Tsukahara (2) ((1) CQSB, Sorbonne Université, (2) Ouragan Team, INRIA)Subjects: Machine Learning (cs.LG); Symbolic Computation (cs.SC); Algebraic Geometry (math.AG); Machine Learning (stat.ML)
Classical information criteria such as the Bayesian Information Criterion (BIC) rely on regularity assumptions that break down for singular models, leading to incorrect model selection in settings such as deep learning. The Widely Applicable Bayesian Information Criterion (WBIC) relies on local learning coefficients $\lambda$, which in the analytic case coincides with local Real Log Canonical Thresholds (RLCT) of the Kullback-Leibler divergence of the model, to capture correct marginal likelihood asymptotics. Exact computation of the learning coefficients has been limited to special cases, and only sampling-based estimation methods are generally applicable. We present the first deterministic algorithm that computes local RLCTs exactly for any two-dimensional model whose Kullback-Leibler distance is contact equivalent to a polynomial, derive a bound on its complexity, and demonstrate its effectiveness for a broad class of models, with applications including polynomial neural networks. Beyond providing ground truth to calibrate sampling-based estimators, exact computation reveals algebraic structure in learning coefficients that sampling cannot and out-speeds it in the shallow regime.
- [295] arXiv:2608.20209 (cross-list from quant-ph) [pdf, html, other]
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Title: Rigorous existence and location of quantum phase transitions in lattice Hamiltonian systemsComments: 16 pages, 4 figuresSubjects: Quantum Physics (quant-ph); Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph)
We extend the analysis of the class of quantum phase transitions (QPTs) that can be interpreted as condensations in state space, first introduced in [M. Ostilli and C. Presilla, J. Phys. A 54, 055005 (2021)], by generalizing the arguments of [M. Ostilli and C. Presilla, Phys. Rev. Lett. 127, 040601 (2021)] to prove the existence and determine the location (via simple bounds) of QPTs in general one-parameter lattice Hamiltonians. Unlike our original formulation, this extension also encompasses second-order QPTs, for which we provide the explicit example of the transverse-field Ising model. Our analysis suggests that, under conditions typically satisfied in physical contexts, any QPT taking place in lattice systems can be interpreted as a condensation in state space.
- [296] arXiv:2608.20273 (cross-list from quant-ph) [pdf, html, other]
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Title: Analytical Solutions of the Generalized Klein-Gordon Oscillator in Som-Raychaudhuri Space-Time via the Extended Nikiforov-Uvarov MethodSubjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph)
In this study, we present an analytical approach based on the extended Nikiforov-Uvarov method to solve the generalized Klein-Gordon oscillator in the presence of a uniform magnetic field within the Som-Raychaudhuri space-time. Exact eigenstate solutions are obtained for two distinct potential models, namely the linear potential and the Cornell potential. The corresponding energy eigenvalues are determined, and the associated eigenfunctions are analyzed and illustrated graphically.
- [297] arXiv:2608.20286 (cross-list from stat.ME) [pdf, html, other]
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Title: Fast high-dimensional mean testing via logistic regressionSubjects: Methodology (stat.ME); Statistics Theory (math.ST)
We propose computationally efficient tests for equality of mean vectors of two or more high-dimensional populations. Central to our approach is an equivalence between equality of means and a zero population logistic regression parameter. We establish this equivalence for independently distributed observations without imposing common distributional assumptions across populations. Our procedure uses logistic Lasso to screen informative variables and an unpenalized logistic refit for inference in the reduced dimension, yielding asymptotically correct size and consistency. For a specified two-sample Gaussian submodel and sparse discriminative class, the test also attains the minimax separation rate. The framework extends to multiple populations through multi-class logistic regression. Simulations demonstrate accurate size control, strong power, and favorable computational scaling compared with existing tests under unbalanced designs and variance heterogeneity. Applications to gene-expression data with more than twenty-two thousand variables illustrate the practical scalability of the proposed procedures.
- [298] arXiv:2608.20295 (cross-list from cs.LG) [pdf, html, other]
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Title: Physical-Support Confidence Sets for Highly Coherent DictionariesSubjects: Machine Learning (cs.LG); Signal Processing (eess.SP); Statistics Theory (math.ST)
Sparse pursuit after dictionary learning can yield a precise atom support even when its physical interpretation is not justified by the calibration data, especially for highly coherent dictionaries where alternative calibration-compatible dictionaries may assign different physical meanings to the same selected support. We develop resolution-aware physical-support inference that jointly accounts for uncertainty in the learned dictionary and in the representation of a deployment signal. Our cross-dictionary confidence correspondence retains calibration-compatible dictionaries and deployment-compatible sparse representations, then projects the surviving explanations onto physical-support space. For local coherent-atom classes with separation scale s, once the deployment data resolve the coherent-block explanation and its atom support, the minimax physical resolution from N calibration signals satisfies $\delta_{\mathrm{opt}}(N,s)\asymp\min\{s,\frac{1}{\sqrt{N}s^2}\}$, with relative resolution governed by the orientation-information scale $Ns^6$. Deployment replication improves physical localization only when orientation changes cannot be absorbed by adjusting the active coefficients. For computation, we introduce active endpoint bracketing (AEB), an adaptive finite-bank procedure that evaluates only candidates that can still affect the physical report and otherwise safely coarsens or abstains. Finite-bank experiments, including a four-region synthetic application, show that a point-valued plug-in selector can be physically overprecise, whereas AEB avoids unsupported refinement with fewer candidate evaluations.
- [299] arXiv:2608.20298 (cross-list from eess.SY) [pdf, html, other]
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Title: Zero-Sum Power Factor GamesSubjects: Systems and Control (eess.SY); Optimization and Control (math.OC)
Variable active power injections arising from device behavior or compromised dispatch complicate voltage regulation in electric power networks with distributed energy resources (DERs). An operator can limit the resulting voltage deviations by remotely selecting DER reactive power parameters before observing the active power injections. IEEE Standard 1547-2018 specifies constant power factor as one such control mode, coupling each device's reactive power to its realized active power. Using a linear voltage model, we formulate the operator's decision as a robust minimax problem in which the operator minimizes the largest feasible aggregate voltage deviation. We solve this problem by expressing the power factor decisions through continuous reactive to active power ratios and exactly decomposing the payoff according to the signs of the voltage deviations. When every feasible voltage residual remains on its initial side of nominal, the resulting ratios cancel each injection's contribution and yield a closed form minimax strategy. We identify realistic DER ratings for which this strategy applies and quantify the regulation capacity lost under restricted power factor ranges. Numerical tests check the cancellation computation, solve the complete minimax problem directly at a representative DER rating, and compare the linear voltage predictions with nonlinear AC power flow.
- [300] arXiv:2608.20300 (cross-list from eess.SY) [pdf, html, other]
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Title: Taming the Tilt: A Unified Pilot Control Concept for Transformational eVTOL AircraftComments: Accepted for publication in "Aerospace Systems", 31 pages, 16 figuresSubjects: Systems and Control (eess.SY); Optimization and Control (math.OC)
Transformational electric vertical take-off and landing (eVTOL) vehicles have gained significant attention over the past decade due to their efficient wing-borne cruise capabilities and reduced reliance on ground-based infrastructure. However, control system design for these vehicles remains challenging, as they must operate across multiple flight phases, each with distinct dominant dynamics. If left unaddressed, this complexity would significantly increase pilot workload, thus motivating the development of pilot control systems for multi-phase flight operations. The Simplified Vehicle Operations concept presents a promising strategy for reducing pilot workload. This study presents the design and implementation of a novel pilot control concept for eVTOL aircraft, validated through a tandem tilt-wing aircraft simulation on a full-motion simulator equipped with an active, force-feedback side stick. The system provides pilots with tactile feedback during specific flight phases, supporting intuitive control. The proposed approach enables seamless transitions and multi-phase flight maneuvers by leveraging the available degrees of freedom. Furthermore, an optimal-control-based methodology is proposed as a metric to evaluate command-filter-induced performance penalties and inceptor activities. The results show that the proposed command filter does not significantly increase the mission duration compared to the closed-loop system, while the active side stick helps reduce inceptor activity.
- [301] arXiv:2608.20330 (cross-list from cond-mat.str-el) [pdf, html, other]
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Title: Torus Berry Data Determine All-Genus Abelian Topological OrdersSubjects: Strongly Correlated Electrons (cond-mat.str-el); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
We show that for Abelian Chern-Simons topological orders, torus Berry matrices determine the all-genus extended TQFT. We identify the topological part of the projective Berry holonomy under metric deformations with the mapping-class-group representation of the Abelian Chern-Simons TQFT and prove that the normalized torus data reconstruct its finite quadratic module $(G,q)$. Recent work showed that $(G,q)$ classifies the extended theory up to symmetric monoidal natural isomorphism, then genus-one data determine the all-genus theory without choosing a $K$-matrix presentation. We also prove that, for normalized character row errors $\delta<21.96$%, nearest-row decoding recovers the Abelian fusion algebra independently of the number of anyons. The result applies to Abelian fractional quantum Hall and spin-liquid phases described by even-lattice Chern-Simons theories.
Cross submissions (showing 53 of 53 entries)
- [302] arXiv:1308.4108 (replaced) [pdf, other]
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Title: Limits of Boolean Functions on F_p^nComments: The paper is withdrawn due to a gap in the proof of Lemma 4.2, which affects the main theoremSubjects: Combinatorics (math.CO); Functional Analysis (math.FA)
We study sequences of functions of the form F_p^n -> {0,1} for varying n, and define a notion of convergence based on the induced distributions from restricting the functions to a random affine subspace. Using a decomposition theorem and a recently proven equi-distribution theorem from higher order Fourier analysis, we prove that the limits of such convergent sequences can be represented by certain measurable functions. We are also able to show that every such limit object arises as the limit of some sequence of functions. These results are in the spirit of similar results which have been developed for limits of graph sequences. A more general, albeit substantially more sophisticated, limit object was recently constructed by Szegedy in [Sze10].
- [303] arXiv:2107.07575 (replaced) [pdf, html, other]
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Title: Optimal tests of the composite null hypothesis arising in mediation analysisComments: 73 pages, 12 figuresSubjects: Statistics Theory (math.ST); Methodology (stat.ME)
The indirect effect of an exposure on an outcome through an intermediate variable can be identified by a product of two regression coefficients under certain causal and regression modeling assumptions. In this context, the null hypothesis of no indirect effect is a composite null hypothesis, as the null holds if either regression coefficient is zero. A consequence is that traditional hypothesis tests are severely underpowered near the origin (i.e., when both coefficients are small with respect to standard errors). We propose hypothesis tests that (i) preserve level alpha type 1 error, (ii) meaningfully improve power when both true underlying effects are small relative to sample size, and (iii) preserve power when at least one is not. One approach gives a closed-form test that is minimax optimal with respect to local power over the alternative parameter space. Another uses sparse linear programming to produce an approximately optimal test for a Bayes risk criterion. We discuss adaptations for performing large-scale hypothesis testing as well as modifications that yield improved interpretability. We provide an R package that implements our proposed methodology.
- [304] arXiv:2208.11255 (replaced) [pdf, html, other]
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Title: The Green's function of the parabolic Anderson model and the continuum directed polymerComments: To appear in Stochastics and Partial Differential Equations: Analysis and Computations. arXiv version differs slightly in formatting from the published version. 67 pages, 3 figuresSubjects: Probability (math.PR)
We build a regular version of the field $Z_{\beta}(t,x|s,y)$ which describes the Green's function, or fundamental solution, of the parabolic Anderson model (PAM) with white noise forcing on $\mathbb{R}^{1+1}$: $\partial_t Z_{\beta}(t,x | s,y) =$ $\frac{1}{2}\partial_{xx} Z_{\beta}(t,x|s,y) + \beta Z_{\beta}(t,x | s,y)W(t,x)$, $Z_{\beta}(s,x | s,y) = \delta(x-y)$ for all $-\infty < s \leq t < \infty$, all $x,y \in \mathbb{R}$, and all $\beta \in \mathbb{R}$ simultaneously. Through the superposition principle, our construction gives a pointwise coupling of all solutions to the PAM with initial or terminal conditions satisfying sharp growth assumptions, for all initial and terminal times. Using this coupling, we show that the PAM with a (sub-)exponentially growing initial condition admits conserved quantities given by the limits $\displaystyle \lim_{x\to \pm\infty} x^{-1}\log Z_{\beta}(t,x)$, in addition to proving many new basic properties of solutions to the PAM with general initial conditions. These properties are then connected to the existence, regularity, and continuity of the quenched continuum polymer measures. Through the polymer connection, we also show that the kernel $(x,y) \mapsto Z_{\beta}(t,x | s,y)$ is strictly totally positive for all $t>s$ and $\beta\in \mathbb{R}$.
- [305] arXiv:2212.09391 (replaced) [pdf, html, other]
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Title: Bounds for matrices of inclusion probabilities in rejective samplingSubjects: Probability (math.PR); Statistics Theory (math.ST)
Motivated by a desire to model n-sparse signals more realistically, we study matrices of inclusion probabilities for conditional Poisson or rejective sampling in the non-asymptotic regime. In particular, we provide bounds in the semi-definite ordering for matrices that collect first and second order inclusion probabilities as their diagonal and off-diagonal entries respectively, and operator norm bounds of their Hadamard products with other matrices. Finally, we demonstrate the usefulness on these bounds, which only involve first order inclusion probabilities, in a toy application from dictionary learning.
- [306] arXiv:2304.00924 (replaced) [pdf, html, other]
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Title: Limit theorems for random Motzkin paths near boundaryComments: Corrected misprint in Theorem 1.5 and some other typos. This is an expanded version of the paper with additional detailsJournal-ref: Bernoulli 2024, Vol. 30, No. 3, 2185-2206Subjects: Probability (math.PR)
We consider Motzkin paths of length $L$, not fixed at zero at both end points, with constant weights on the edges and general weights on the end points. We investigate, as the length $L$ tends to infinity, the limit behaviors of (a) boundary measures induced by the weights on both end points and (b) the segments of the sampled Motzkin path viewed as a process starting from each of the two end points, referred to as boundary processes.
Our first result concerns the case when the induced boundary measures have finite first moments. Our second result concerns when the boundary measure on the right end point is a generalized geometric measure with parameter $\rho_1\ge 1$, so that this is an infinite measure and yet it induces a probability measure for random Motzkin path when $\rho_1$ is not too large. The two cases under investigation reveal a phase transition. In particular, we show that the limit left boundary processes in the two cases have the same transition probabilities as random walks conditioned to stay non-negative. - [307] arXiv:2304.01937 (replaced) [pdf, html, other]
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Title: Analysis and systematic discretization of a Fokker-Planck equation with Lorentz forceComments: Corresponding code can be found at this https URLSubjects: Numerical Analysis (math.NA)
The propagation of charged particles through a scattering medium in the presence of a magnetic field can be described by a Fokker-Planck equation with Lorentz force. This model is studied both, from a theoretical and a numerical point of view. A particular trace estimate is derived for the relevant function spaces to clarify the meaning of boundary values. Existence of a weak solution is then proven by the Rothe method. In the second step of our investigations, a fully practical discretization scheme is proposed based on an implicit Euler method for the energy variable and a spherical-harmonics finite-element discretization with respect to the remaining variables. A complete error analysis of the resulting scheme is given and numerical test are presented to illustrate the theoretical results and the performance of the proposed method.
- [308] arXiv:2401.01129 (replaced) [pdf, html, other]
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Title: Variational Lifting and Optimal Gauges on Riemannian Homogeneous SpacesComments: 22 pages, 2 figuresSubjects: Optimization and Control (math.OC); Systems and Control (eess.SY); Differential Geometry (math.DG); Dynamical Systems (math.DS)
Building on standard Euler-Poincaré reduction on Lie groups, we develop a variational lifting framework for mechanical systems on Riemannian homogeneous spaces $H=G/K$. A mechanical action on $H$ is lifted to $G$ through a functional whose kinetic energy depends only on the horizontal component of the velocity, and we prove that its critical points project precisely onto those of the original action.
The lifted functional possesses a natural gauge invariance under $H^1$ curves with values in the isotropy subgroup $K$. Consequently, every lifted critical curve decomposes into a smooth horizontal representative and an arbitrary vertical gauge, and the associated Euler-Poincaré equations with symmetry breaking are obtained in reduced form. We then introduce a second variational problem that selects a distinguished lift by minimizing the vertical kinetic energy.
The optimal gauge is a length-minimizing geodesic on $K$, yielding an explicit decomposition of the total energy into projected and vertical contributions and a geometric interpretation in terms of holonomy for closed projected curves. The framework is illustrated first for pure quantum states on $\mathbb{CP}^{n}\cong SU(n+1)/S(U(1)\times U(n))$, where the gauge freedom corresponds to the isotropy of unitary lifts, and then for $S^2\cong SO(3)/SO(2)$ through an optimal camera-orientation problem for an axisymmetric satellite subject to an undesirable pointing region. - [309] arXiv:2405.12761 (replaced) [pdf, other]
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Title: Critical conditions of nonlinearities in the John problemComments: 22 pagesSubjects: Analysis of PDEs (math.AP)
In this paper, we determine the sharp criteria for the nonlinearities in the John problem $\partial_{t}^2 u-\Delta_{\R^3}u= F(u)$ so that the problem admits global solutions for small initial data with compact support. The criteria is of Dini type. For the situation in which the criteria is not satisfied, we show that the solution will blow up in finite time for some small initial data, together with an almost sharp estimates of the lifespan. For the proof, we mainly rely on the precise pointwise estimates of the solution, based on the fundamental solution, John's iteration argument as well as an adaption of the slicing method.
- [310] arXiv:2407.14778 (replaced) [pdf, html, other]
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Title: Minimax estimation of functionals in sparse vector model with correlated observationsJournal-ref: Electronic Journal of Statistics 20.2 (2026): 3770-3806Subjects: Statistics Theory (math.ST)
We consider the observations of an unknown $s$-sparse vector ${\boldsymbol \theta}$ corrupted by Gaussian noise with zero mean and unknown covariance matrix ${\boldsymbol \Sigma}$. We propose minimax optimal methods of estimating the $\ell_2$ norm of ${\boldsymbol \theta}$ and testing the hypothesis $H_0: {\boldsymbol \theta}=0$ against sparse alternatives when only partial information about ${\boldsymbol \Sigma}$ is available, such as an upper bound on its Frobenius norm and the values of its diagonal entries to within an unknown scaling factor. We show that the minimax rates of the estimation and testing are leveraged not by the dimension of the problem but by the value of the Frobenius norm of ${\boldsymbol \Sigma}$.
- [311] arXiv:2407.17972 (replaced) [pdf, html, other]
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Title: Strong Embeddings of 3-Connected Cubic Planar Graphs on Surfaces of non-negative Euler CharacteristicSubjects: Combinatorics (math.CO)
Whitney proved that 3-connected planar graphs admit a unique embedding on the sphere. In contrast, Enami investigated embeddings of 3-connected cubic planar graphs on non-spherical surfaces with non-negative Euler characteristic. He established that such an embedding exists if and only if the dual graph contains a particular subgraph. Here, strong embeddings are investigated motivated by the cycle double cover conjecture and the relation to triangulated surfaces. We provide a complete characterization of strong embeddings on the projective plane, the torus, and the Klein bottle in terms of a distinguished subset of Enami's subgraphs. This characterization not only deepens the structural understanding of graph embeddings on non-spherical surfaces, but also establishes a robust foundation for computing cycle double covers. As a direct consequence, we derive explicit criteria that determine when a graph does not admit a strong embedding on these surfaces-offering new tools for both theoretical analysis and algorithmic applications.
- [312] arXiv:2409.01346 (replaced) [pdf, html, other]
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Title: Multifractal spectrum of branching random walks on free groupsComments: 53 pages, 1 figure. Fixed several mistakes and filled some gaps. Comments are welcomeSubjects: Probability (math.PR)
Consider a symmetric branching random walk on a free group $\mathbb{F}$ in the transient regime $1<r\leq R$, where $r$ is the mean offspring number and $R$ is the reciprocal of the spectral radius of the underlying random walk. The limit set $\Lambda_r$--consisting of all ends of $\mathbb{F}$ to which the BRW's particle trajectories converge--is a proper random subset of the boundary $\partial \mathbb{F}$. Hueter and Lalley (2000) determined the Hausdorff dimension of $\Lambda_r$, and proved that $\dim_{\mathrm{H}} \Lambda_r \leq \frac{1}{2} \dim_{\mathrm{H}} \partial \mathbb{F}$ with equality possible only when $r = R$.
We further extend this study by conducting a multifractal analysis of the limit set $\Lambda_r$. We obtain the Hausdorff dimensions of the sub-fractals $\Lambda_r(\alpha) \subset \Lambda_r$ which consist of all ends of $\mathbb{F}$ approached by particle trajectories escaping at the rate $\alpha \in [0,1]$. Notably, there exists a unique $\alpha(r) \in [0,1]$ such that \begin{equation} \dim_{\mathrm{H}} \Lambda_r = \dim_{\mathrm{H}} \Lambda_r( \alpha(r) ). \end{equation} Moreover, the maximizing speed exhibits a phase transition: $\alpha(r)>0$ for $1<r<R$, whereas $\alpha(R)=0$. - [313] arXiv:2409.07948 (replaced) [pdf, html, other]
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Title: Quickest Change Detection Using Mismatched CUSUMComments: Extended version of extended abstract for the Allerton Conference on Communication, Control, and Computing, September 2024Subjects: Statistics Theory (math.ST); Information Theory (cs.IT)
Quickest change detection concerns estimation of an unknown change time \(\tau_a\) from a sequence of partial observations \(\{Y_k:k\ge 0\}\). We consider stopping rules of CUSUM form, \[
\mathcal{X}_{n+1}
=
\max\{0,\mathcal{X}_n+F(Y_{n+1})\},
\quad
\tau_s=\min\{n\ge 0:\mathcal{X}_n\ge \textrm{H}\}, \] where the function \(F\) and threshold \(\textrm{H}\) are design parameters.
The observations and change time are modeled jointly through a hidden Markov model, and \( F\) is selected from a prescribed function class \(\mathcal{G}\) to minimize the weighted criterion \[
\textsf{E}\bigl[
(\tau_s-\tau_a)_+
+
\kappa(\tau_s-\tau_a)_-
\bigr]. \]
When \(\mathcal{G}\) is a linear function class, the optimizer
\(F^*\) is characterized by a convex program, whose dual yields extensions of classical likelihood-ratio constructions. This conclusion is based on analysis that is asymptotic in the regime \(\kappa\to\infty\). We show that the hidden Markov model admits an asymptotically equivalent conditionally independent approximation of the type commonly used in the quickest change detection literature. We then develop the design and asymptotic theory for a substantially broader class of conditionally independent models, so that the resulting conclusions are not tied to the particular POMDP reduction.
Combining renewal theory and large deviations for reflected random walks, we obtain for each $F\in\mathcal{G}$ asymptotically accurate approximations of the optimal threshold and average cost, with error vanishing as \(\kappa\to\infty\). It is found in numerical experiments that the resulting approximations are accurate for moderate values of \(\kappa\). - [314] arXiv:2410.15955 (replaced) [pdf, html, other]
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Title: The mutual arrangement of Wright-Fisher diffusion path measures and its impact on parameter estimationSubjects: Statistics Theory (math.ST); Probability (math.PR); Populations and Evolution (q-bio.PE)
The Wright-Fisher diffusion is a fundamentally important model of evolution encompassing genetic drift, mutation, and natural selection. Suppose you want to infer the parameters associated with these processes from an observed sample path. Then to write down the likelihood one first needs to know the mutual arrangement of two path measures under different parametrizations; that is, whether they are absolutely continuous, equivalent, singular, and so on. In this paper we give a complete answer to this question by finding the separating times for the diffusion - the stopping time before which one measure is absolutely continuous with respect to the other and after which the pair is mutually singular. In one dimension this extends a classical result of Dawson on the local equivalence between neutral and non-neutral Wright-Fisher diffusion measures. Along the way we also develop new zero-one type laws for the diffusion on its approach to, and emergence from, the boundary. As an application we derive an explicit expression for the joint maximum likelihood estimator of the mutation and selection parameters and show that its convergence properties are closely related to the separating time.
- [315] arXiv:2411.06483 (replaced) [pdf, html, other]
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Title: Quantitative bounds for bounded solutions to the Navier-Stokes equations in endpoint critical Besov spacesSubjects: Analysis of PDEs (math.AP)
Building on Tao's quantitative regularity theory and triple-logarithmic blow-up estimate in $L^3$ in \cite{Tao_20}, we consider classical solutions $(u,P)$ of the three-dimensional incompressible Navier--Stokes equations on $[0,T]\times\mathbb{R}^3$. For $3<p<\infty$, under simultaneous uniform control of the two scaling-critical quantities $\|u\|_{L_T^\infty(\dot B_{p,\infty}^{-1+\frac{3}{p}})}$ and $\||D|^{-1+\frac{3}{p}}u\|_{L_T^\infty(L^p)}$, we obtain explicit quantitative estimates for all spatial derivatives of $u$. As a consequence, we derive a mixed blow-up criterion coupling a double exponential of the critical Besov norm with the $L^p$ norm of $|D|^{-1+\frac{3}{p}}u$, which forces quantified growth of at least one of these two critical quantities near any finite blow-up time. The low regularity and lack of dyadic summability in the endpoint Besov space are handled through a finite iterative decomposition that successively improves spatial integrability and produces an energy-class remainder, together with refined nonlinear energy estimates. The nonlocal signed quantity $|D|^{-1+\frac{3}{p}}u$ is treated by localized mean-zero vector tests and almost orthogonality across geometrically separated concentration scales.
- [316] arXiv:2411.06636 (replaced) [pdf, html, other]
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Title: The internal languages of univalent categoriesSubjects: Category Theory (math.CT); Logic in Computer Science (cs.LO)
Internal language theorems are fundamental in categorical logic, since they express an equivalence between syntax and semantics. One such theorem was proven by Clairambault and Dybjer, who corrected the result originally by Seely. More specifically, they constructed a biequivalence between the bicategory of locally Cartesian closed categories and the bicategory of democratic categories with families with extensional identity types, $\Sigma$-types, and $\Pi$-types. This theorem expresses that the internal language of locally Cartesian closed categories is extensional Martin-Löf type theory with dependent sums and products. In this paper, we study the theorem by Clairambault and Dybjer for univalent categories, and we extend it to various classes of toposes, among which are $\Pi$-pretoposes, elementary toposes, and elementary toposes with a universe. The results in this paper have been formalized using the proof assistant Rocq and the UniMath library.
- [317] arXiv:2411.11290 (replaced) [pdf, html, other]
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Title: Chebyshev's method for exponential mapsComments: 28 pages, 10 figuresSubjects: Dynamical Systems (math.DS)
It is proved that the Chebyshev's method applied to an entire function $f$ is a rational map if and only if $f(z) = p(z) e^{q(z)}$, for some polynomials $p$ and $q$. These are referred to as rational Chebyshev maps, and their fixed points are discussed in this article. It is seen that $\infty$ is a parabolic fixed point with multiplicity one bigger than the degree of $q$. Considering $q(z)=p(z)^n+c$, where $p$ is a linear polynomial, $n \in \mathbb{N}$ and $c$ is a non-zero constant, we show that the Chebyshev's method applied to $ pe^q$ is affine conjugate to that applied to $z e^{z^n}$. We denote this by $C_n$. All the finite extraneous fixed points of $C_n$ are shown to be repelling. The Julia set $\mathcal{J}(C_n)$ of $C_n$ is found to be preserved under rotations of order $n$ about the origin. For each $n$, the immediate basin of $0$ is proved to be simply connected. For all $n \leq 16$, we prove that $\mathcal{J}(C_n)$ is connected. For $n$ even, the non-existence of Herman ring and Siegel disk of $C_n$ is proved. Under some additional hypothesis, the same is also proved for odd $n$. The Newton's method applied to $ze^{z^n}$ is found to be conjugate to a polynomial, and its dynamics is also completely determined.
- [318] arXiv:2501.02358 (replaced) [pdf, html, other]
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Title: Chebyshev systems and Sturm oscillation theory for discrete polynomialsComments: 31 pagesSubjects: Classical Analysis and ODEs (math.CA)
We prove an analogue of Chebyshev's alternation theorem for linearly independent discrete functions $\Phi_n=\{\varphi_k\}_{k=1}^n$ on the interval $[0,q]_{\scriptscriptstyle\mathbb{Z}}=[0,q]\cap \mathbb{Z}$. In particular, we establish that the polynomial of best uniform approximation of a discrete function admits a Chebyshev alternance set of length $n+1$ if and only if $\Phi_n$ is a Chebyshev $T_{\scriptscriptstyle\mathbb{Z}}$-system. We also obtain a discrete version of Sturm's oscillation theorem, according to which the number of discrete zeros of the polynomial $\sum_{k=m}^{n}a_k\varphi_k$ is no less than $m-1$ and no more than $n-1$. This implies that $\Phi_n$ is a $T_{\scriptscriptstyle\mathbb{Z}}$-system and a discrete Sturm-Hurwitz spectral gap theorem is valid. As applications, we study the orthogonal polynomials with removed largest zeros. We~establish the monotonicity property of coefficients in the Fourier expansions of such polynomials, thereby strengthening the results of H.~Cohn and A.~Kumar. We apply this to solve a Yudin-type extremal problem for polynomials with spectral gap.
- [319] arXiv:2502.05180 (replaced) [pdf, html, other]
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Title: Proof of Soland's conjecture on the efficient solutions to a multicriteria optimization problemSubjects: Optimization and Control (math.OC)
This note proves Richard M. Soland's conjecture that given an efficient solution to a multicriteria optimization problem, there need not exist a continuous, strictly increasing and strictly concave criterion space function that attains its maximum at the vector of criteria values achieved by that solution. We work out an important implication of this result for multicriteria decision making.
- [320] arXiv:2502.10598 (replaced) [pdf, html, other]
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Title: Finite symmetric algebras in tensor categories and Verlinde categories of algebraic groupsComments: 27 pages. v3: correction in Lemma 3.2.5 compared to published versionJournal-ref: Adv. Math. 483 (2025) 110677Subjects: Representation Theory (math.RT); Category Theory (math.CT)
We investigate objects in symmetric tensor categories that have simultaneously finite symmetric and finite exterior algebra. This forces the characteristic of the base field to be $p>0$, and the maximal degree of non-vanishing symmetric and exterior powers to add up to a multiple of $p$. We give a complete classification of objects in tensor categories for which this sum equals $p$. All resulting tensor categories are Verlinde categories of reductive groups and we fill in some gaps in the literature on these categories.
- [321] arXiv:2502.19052 (replaced) [pdf, html, other]
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Title: Algorithmic approaches to avoiding bad local minima in nonconvex inconsistent feasibilityComments: 33 pages, 7 figures, 28 referencesSubjects: Optimization and Control (math.OC); Numerical Analysis (math.NA)
We report on the use of algorithms to avoid or move away from ``bad'' local minima in nonconvex optimization. Our study is phenomenological and empirical, focusing on the performance of cyclic projections, the cyclic relaxed Douglas-Rachford algorithm, and relaxed Douglas-Rachford splitting on the product space for orbital tomographic imaging from angle-resolved photon emission spectroscopy (ARPES) measurements, with both synthetic and laboratory data. Cyclic projections and Douglas-Rachford on the product space are both well-known methods, but cyclic relaxed Douglas-Rachford was only recently fully characterized in a companion paper to the present study. Only one other study of note has investigated the performance of all three of these algorithms for inconsistent nonconvex feasibility. We show that the relaxed Douglas-Rachford algorithm on the product space, while exhibiting very poor convergence rates, can be used to filter out bad local minima from all cyclic algorithms. Our numerical experiments lead to the following recommendation: run cyclic projections to find some fixed point, and from this fixed point run relaxed Douglas-Rachford algorithm on the product space with as large a relaxation parameter as is numerically stable in order to escape poor local minima. This advice runs counter to the current practice for phase retrieval, where a Douglas-Rachford-type algorithm is run for several iterations, and then cyclic projections is used to ``clean up'' the images.
- [322] arXiv:2503.05606 (replaced) [pdf, html, other]
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Title: Control analysis and synthesis for general control-affine systemsComments: The last version corresponds to the peer-reviewed article published in the SIAM Journal on Control and OptimizationSubjects: Optimization and Control (math.OC); Classical Analysis and ODEs (math.CA); Dynamical Systems (math.DS)
We study controllability and constructive synthesis for control-affine systems. We introduce trajectory-dependent Gramian maps that extend the linear time-varying Gramian and yield explicit fixed-point synthesis maps. On feasible coercivity classes (uniform eigenvalue lower bounds), the Gramian map is Lipschitz, and under a comparison estimate criterion, synthesis iterates exhibit decay, and the Banach fixed-point theorem gives a unique fixed point that steers the system and satisfies an energy identity. When, in addition, an orthogonality condition holds, this fixed point coincides with the unique global minimum-energy control on the feasible set; if the coercivity bound holds uniformly for all bounded controls, the same conclusion holds on the full bounded-control space. We provide structural conditions on the input matrix that ensure the nonemptiness of the feasible class (and, in fully actuated regimes, equality with the full space) and sufficient conditions for underactuated systems via bounded-amplitude reference controls. Case studies on Hopfield network dynamics illustrate refined estimates that enlarge reachable targets. A trajectory-freezing and compactness step extends the synthesis to general nonlinear control-affine systems. The results yield verifiable controllability criteria with explicit, numerically implementable controllers.
- [323] arXiv:2503.16093 (replaced) [pdf, html, other]
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Title: A Cheeger-type inequality for the drift Laplacian with Wentzell-type boundary conditionJournal-ref: Calculus of Variations and PDE, Vol. 65, Article No. 167, 2026Subjects: Probability (math.PR); Analysis of PDEs (math.AP); Differential Geometry (math.DG); Spectral Theory (math.SP)
We prove lower bounds for the first non-trivial eigenvalue of the drift Laplacian on manifolds with Wentzell-type boundary condition in terms of some Cheeger-type constants for bulk-boundary interactions. Our results are in the spirit of Cheeger's classical inequality.
- [324] arXiv:2503.19686 (replaced) [pdf, html, other]
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Title: Distinct differences of singular moduliComments: 25 pagesSubjects: Number Theory (math.NT); Algebraic Geometry (math.AG)
Let $E_1, E_2 / \mathbb{C}$ be non-isomorphic elliptic curves with complex multiplication. We prove that the pair $(E_1, E_2)$ is characterised, up to isomorphism, by the difference $j(E_1) - j(E_2)$ of the respective $j$-invariants. In other words, we show that if $x_1, x_2, x_3, x_4$ are singular moduli such that $x_1 - x_2 = x_3 - x_4$, then either $(x_1, x_2) = (x_3, x_4)$ or $(x_1, x_3) = (x_2, x_4)$.
- [325] arXiv:2504.06104 (replaced) [pdf, html, other]
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Title: Well-posedness of Heat Equations with Nonlinearities of Arbitrarily Rapid GrowthComments: Accepted version for Proc. London Math. SocSubjects: Analysis of PDEs (math.AP)
We address local- and global-in-time well-posedness of the Cauchy problem for nonlinear heat equations without imposing growth rate restrictions on the nonlinearity a priori. Our results constitute a non-trivial expansion of the classical $L^q$-theory for nonlinearities dominated by polynomial growth and the exponential-Orlicz space theory for nonlinearities of exponential growth, to one dealing with nonlinearities of arbitrarily large growth rate. A key ingredient is a new smoothing estimate for the action of the heat semigroup between two arbitrary Orlicz spaces, and in particular into $L^{\infty}$. For nonlinearities growing at least exponentially we are able to identify explicitly a critical space for local well-posedness and for small initial data global well-posedness.
- [326] arXiv:2504.10089 (replaced) [pdf, html, other]
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Title: Convergence Analysis of a Stochastic Interacting Particle-Field Algorithm for 3D Parabolic-Parabolic Keller-Segel SystemsSubjects: Numerical Analysis (math.NA)
Chemotaxis models describe the movement of organisms in response to chemical gradients. In this paper, we present a stochastic interacting particle-field algorithm with a random batch approximation (SIPF-$r$) for the three-dimensional (3D) parabolic-parabolic Keller-Segel (KS) system, also referred to as the fully parabolic KS system. The SIPF-$r$ method approximates the KS system by coupling particle-based representations of the density with a smooth field variable computed using spectral methods. By incorporating the random batch method (RBM), we bypass the mean-field limit and significantly reduce computational complexity. Under mild assumptions on the regularity of the original KS system and the boundedness of numerical approximations, we prove that the empirical measure of the SIPF-$r$ particle system converges, with high probability, to the exact measure of the limiting McKean-Vlasov process in the $1$-Wasserstein distance. Finally, we present numerical experiments to validate the theoretical convergence rates, and demonstrate the performance and robustness of the SIPF-$r$ method as a diagnostic tool for intense focusing and potential finite-time singularity in 3D, subject to critical initial mass thresholds in the system.
- [327] arXiv:2504.13064 (replaced) [pdf, html, other]
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Title: Minimal isometric immersions of flat n-tori into spheresComments: 31 pages. This is a corrected arXiv version of the paper submitted to the journal in January 2026. The embeddedness of the non-homogeneous examples was already addressed in that submitted version, concerning the embedding question discussed in Grigoriy Yakovlev's recent paper arXiv:2608.14688. Comments are welcomeSubjects: Differential Geometry (math.DG)
In 1985, Bryant established that a flat $2$-torus admits a minimal isometric immersion into some round sphere if and only if a certain rationality condition is satisfied. We show that when $n\geq 3$, the rationality criterion is no longer a necessary, but a sufficient condition for a flat $n$-torus to admit minimal isometric immersions into spheres. We also derive an upper bound for the algebraic irrationality degree of such immersions. When $n=3$, this bound is sharp and explicit embedded examples are provided respectively for each possible degree. Moreover, by constructing a family of non-homogeneous minimal flat $3$-tori, we show that minimal isometric immersions (embeddings) of flat $n$-tori are not necessarily homogeneous when $n \geq 3$. In addition, we establish a deformation theorem that every flat $n$-torus admitting a minimal isometric spherical immersion can be isometrically, minimally and homogeneously immersed into a sphere of dimension at most $n^2+n-1$.
- [328] arXiv:2504.19663 (replaced) [pdf, html, other]
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Title: The Boussinesq equation on the half-lineComments: 35 pages, 5 figuresSubjects: Analysis of PDEs (math.AP)
We study the initial-boundary value problem for the Boussinesq equation on the half-line. Assuming that the solution exists, we prove that it can be recovered from its initial-boundary values via the solution of a $3\times 3$ Riemann-Hilbert problem. The contour consists of $18$ arcs on the unit circle, $18$ segments and $18$ half-lines, and the associated jump matrices involve $9$ reflection coefficients.
- [329] arXiv:2505.04166 (replaced) [pdf, html, other]
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Title: An Equidistribution Result for Differences Associated to Square Pyramidal Numbers IIComments: 12 pagesJournal-ref: Dong, A., Saettone, K., Song, K., Zaharescu, A. On differences of square pyramidal numbers II. Aequat. Math. 100, 64 (2026)Subjects: Number Theory (math.NT)
This paper presents some new results concerned with uniform distribution properties associated with the sequence $(a_n)_{n\in\mathbb{N}}$, which is defined as the distance from the $n$-th square pyramidal number to the closest square. We also extend the results to arithmetic progressions.
- [330] arXiv:2505.09765 (replaced) [pdf, html, other]
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Title: Subspace correction as a general framework for convex optimization algorithmsComments: 49 pages, 1 figuresSubjects: Optimization and Control (math.OC); Numerical Analysis (math.NA)
This paper shows that subspace correction methods provide a common algorithmic and theoretical foundation for several classes of convex optimization algorithms, including operator splitting, alternating projection, and multiplier methods. The underlying principle is to decompose a problem into smaller subproblems and combine their solutions, a strategy that appears throughout iterative algorithms. The main tool is an iterate-level formalism, which we call dualization, that abstracts classical primal--dual correspondences and relates a subspace correction method for a dual problem to an algorithm for the corresponding primal problem via a primal--dual consistency relation. At the algorithmic level, dualizing successive subspace correction yields the Peaceman--Rachford and Douglas--Rachford splitting methods; the von Neumann and Dykstra alternating projection algorithms arise as special cases. Dualizing parallel subspace correction yields a parallel splitting method. Multiplier methods, including the alternating direction method of multipliers (ADMM), are connected to subspace correction through the same mechanism together with equivalent block formulations. In particular, a multi-block ADMM-type algorithm is obtained by dualizing a splitting method derived from successive subspace correction. At the theoretical level, the primal--dual consistency relation transfers convergence estimates from subspace correction to the derived algorithms under suitable assumptions. Thus, these operator splitting, alternating projection, and multiplier algorithms can be derived from subspace correction at both the algorithmic and theoretical levels.
- [331] arXiv:2505.10666 (replaced) [pdf, html, other]
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Title: Quantitative Carleson's conjecture for Ahlfors regular domainsComments: 30 pages, 3 figures. V2: typos and small errors corrected; minor adjustments in the proof Proposition 3.1. Accepted for publication in Analysis & PDESubjects: Classical Analysis and ODEs (math.CA); Analysis of PDEs (math.AP)
In this article, we prove a quantitative version of Carleson's $\varepsilon^2$ conjecture in higher dimension: we characterise those Ahlfors-David regular domains in $\mathbb{R}^{n+1}$ for which the Carleson's coefficients satisfy the so-called strong geometric lemma.
- [332] arXiv:2505.13064 (replaced) [pdf, html, other]
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Title: Symmetric Lyapunov Subcenter Manifolds for Periodic Regulation of Mechanical SystemsComments: 21 pages, 27 figures, submitted to AutomaticaSubjects: Dynamical Systems (math.DS); Robotics (cs.RO)
Multi-body mechanical systems have rich internal dynamics, whose solutions can be exploited as energy-efficient control targets. Yet, solutions non-trivially depend on system parameters, obscuring feasible properties for use as target trajectories. For periodic regulation tasks in robotics applications, we investigate properties of nonlinear oscillations collected in Lyapunov subcenter manifolds (LSMs) of conservative mechanical systems (CMs). Using a time-symmetry of CMs, it is shown that mild non-resonance conditions guarantee that LSMs exclusively consist of oscillations between two points of zero velocity. The existence of a unique generator is proven, which is a connected, 1D manifold that collects these points of zero velocity for a given LSM. Furthermore, it is shown that an additional spatial symmetry provides LSMs with yet stronger properties of Rosenberg manifolds. Here all oscillations pass through a unique equilibrium configuration, which can be favorable for control applications. These theoretical results are numerically confirmed on two mechanical systems: a double pendulum and a 5-link pendulum.
- [333] arXiv:2505.20005 (replaced) [pdf, html, other]
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Title: Existence of penalised likelihood estimates and posterior propriety of separable prior distributions for Gaussian precision matricesComments: 31 pagesSubjects: Statistics Theory (math.ST); Methodology (stat.ME)
Penalised likelihoods are often used for sparse estimation of a Gaussian precision matrix. In high dimensional settings where the matrix dimension is larger than the sample size, the sample covariance matrix $S$ is not of full rank and the maximum likelihood estimate of the precision matrix does not exist. An additional advantage of some penalised likelihood estimates, for example the graphical lasso, is that it can exist even in such high dimensional settings. This paper gives a thorough analysis of the existence of penalised likelihood estimates for positive semidefinite $S$. Specific tail conditions are provided on the diagonal and off-diagonal penalty functions that ensure existence of the estimate. This is also extended to the Bayesian setting where conditions on separable prior distributions are provided that ensure the resulting posterior distribution is proper.
- [334] arXiv:2506.19989 (replaced) [pdf, html, other]
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Title: An Ergodic Spectral Decomposition Theorem for Singular Star FlowsComments: 114 pages, 2 figuresSubjects: Dynamical Systems (math.DS)
For Axiom A diffeomorphisms and flows, Smale's Spectral Decomposition Theorem asserts that the non-wandering set decomposes into finitely many isolated hyperbolic basic sets, each given by a homoclinic class. For singular star flows, which may be viewed as "Axiom A flows with singularities", the corresponding spectral decomposition remains open and is known as the Spectral Decomposition Conjecture.
We provide a positive answer to an ergodic formulation of this conjecture: $C^1$-open and dense among singular star flows with positive topological entropy, there is a unique measure of maximal entropy. More generally, we prove the uniqueness of equilibrium states for Hölder continuous potentials under a mild and natural pressure gap condition. We further establish that $C^1$-open and dense star flows are almost expansive and that the topological pressure of continuous potentials varies continuously with respect to the vector field in the $C^1$ topology.
Our approach combines ergodic and geometric arguments adapted to the multi-singular setting. In this context, classical hyperbolic tools such as uniform local product structure or invariant splittings on the tangent bundle are no longer available. To overcome this, we develop new mechanisms to control the geometry of orbit segments and to produce transversal intersections on large subsets uniformly detected by good invariant measures. These ingredients allow us to extend classical arguments to the multi-singular setting through structural properties of equilibrium states combined with refined shadowing and specification at the level of invariant measures. - [335] arXiv:2507.06754 (replaced) [pdf, html, other]
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Title: Exact counts of elliptic curves of bounded height over $\mathbb F_q(t)$ in characteristics $2$ and $3$Comments: 32 pages. Refined the introduction and added the corrected exact formulas for the weighted count to the main theorem. Comments welcomeSubjects: Number Theory (math.NT); Algebraic Geometry (math.AG); K-Theory and Homology (math.KT)
Let $p\in\{2,3\}$, let $q=p^r$ with $r\geq1$, and put $K=\mathbb F_q(t)$. We determine the exact weighted and unweighted counts of $K$-isomorphism classes of elliptic curves of bounded Faltings height, equivalently of bounded minimal-discriminant degree. Writing $B$ for the discriminant height bound, the leading term in each count is of order $B^{5/6}$ and has the same coefficient in every characteristic, whereas the lower-order terms depend substantially on $p$ and on the arithmetic of the constant field. In the unweighted count, characteristic two produces a term of order $B^{5/12}$ and, when $r$ is even, a term of order $B^{1/4}$, neither of which occurs for $p>3$. Some characteristic-specific lower-order coefficients are negative.
These terms reflect two small-characteristic phenomena. First, the nonsmooth locus of generalized Weierstrass equations contains quasi-elliptic-type strata that are not detected by the rational-singular-section argument in de Jong's count: the unique geometric singular point of the generic fiber may be defined only after a nontrivial purely inseparable extension of $K$. Second, on the $j=0$ locus, the geometric origin-preserving automorphism groups are nonabelian and contain wild elements, while twisting affects which automorphisms descend to $K$. Consequently, the passage from weighted to unweighted counts requires a marked-inertia calculation involving conjugacy classes and centralizer weights.
Our formulas show that nonsmooth-locus corrections, extra-automorphism loci, and the removal of minimality defects collectively account for all lower-order terms. Together with the characteristic-greater-than-three formulas of Bejleri-Park-Satriano, this completes the exact weighted and unweighted bounded-height enumerations over $\mathbb F_q(t)$ in every characteristic. - [336] arXiv:2507.11958 (replaced) [pdf, html, other]
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Title: Interacting Hosts with Microbiome Exchange: An Extension of Metacommunity Theory for Discrete Interactions between HostsComments: 57 pages; revised versionSubjects: Dynamical Systems (math.DS); Statistical Mechanics (cond-mat.stat-mech); Social and Information Networks (cs.SI); Adaptation and Self-Organizing Systems (nlin.AO); Populations and Evolution (q-bio.PE)
Microbiomes, which are collections of interacting microbes in an environment, often substantially impact the environmental patches or living hosts that they occupy. In microbiome models, it is important to consider both the local dynamics within an environment and exchanges of microbiomes between environments. One way to incorporate these and other interactions across multiple scales is to employ metacommunity theory. Metacommunity models commonly assume continuous microbiome dispersal between the environments in which local microbiome dynamics occur. Under this assumption, a single parameter between each pair of environments controls the dispersal rate between those environments. This metacommunity framework is well-suited to abiotic environmental patches, but it fails to capture an essential aspect of the microbiomes of living hosts. Living hosts generally do not interact continuously with each other. Instead, living hosts interact with each other in discrete time intervals. In this paper, we develop a modeling framework that encodes such discrete interactions and uses two parameters to separately control the interaction frequencies between hosts and the amount of microbiome exchange during each interaction. We derive analytical approximations of models in our framework in three parameter regions and prove that they are accurate in these regions. We compare these approximations to numerical simulations for an illustrative model, and we demonstrate that both parameters in our modeling framework are necessary to determine microbiome dynamics. Key features of the dynamics, such as microbiome convergence across hosts, depend sensitively on the interplay between interaction frequency and strength.
- [337] arXiv:2507.13049 (replaced) [pdf, html, other]
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Title: A Note on Pliability and the Openness of the Multiexponential Map in Carnot GroupsJournal-ref: SIGMA 22 (2026), 078, 8 pagesSubjects: Metric Geometry (math.MG); Optimization and Control (math.OC)
In recent years, several notions of non-rigidity of horizontal vectors in Carnot groups have been proposed, motivated, in particular, by the characterization of monotone sets and Whitney extension properties. In this note we compare some of these notions.
- [338] arXiv:2508.03585 (replaced) [pdf, html, other]
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Title: Resolvent estimates for a function of a linear operatorComments: The last section has been added. 18 pagesSubjects: Functional Analysis (math.FA); Complex Variables (math.CV)
Let $T$ be a bounded linear operator on a Banach space and $f$ an analytic function, defined on the spectrum of $T$. We study the relations between the rate of growth of the resolvent of $T$ and that of $f(T)$. We also discuss whether the property of unconditional basisness of eigenspaces or root spaces of $f(T)$ implies the corresponding property for $T$, and related issues.
- [339] arXiv:2508.05120 (replaced) [pdf, html, other]
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Title: Turaev-Viro invariant from the modular double of $\mathrm {U}_{q}\mathfrak{sl}(2;\mathbb R)$Comments: 111 pages, 26 figures. We add the proof of the absolute convergence of the invariant for all $b$ in $(0,1)$, where previously we only had the convergence for sufficiently small $b.$Subjects: Geometric Topology (math.GT); Mathematical Physics (math-ph); Probability (math.PR); Quantum Algebra (math.QA)
We define a family of Turaev-Viro type invariants of hyperbolic $3$-manifolds with totally geodesic boundary from the $6j$-symbols of the modular double of $\mathrm U_{q}\mathfrak{sl}(2;\mathbb R)$, and prove that these invariants decay exponentially with the rate the hyperbolic volume of the manifolds and with the $1$-loop term the adjoint twisted Reidemeister torsion of the double of the manifolds.
- [340] arXiv:2508.09319 (replaced) [pdf, html, other]
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Title: Transcendence Meets Normality: Construction of Transcendentally Normal NumbersComments: 37 pages, simplified and corrected proof, added referencesSubjects: Number Theory (math.NT); Probability (math.PR)
In this work, we study real numbers $x$ for which $p(x)$ is (absolutely) normal for every non-constant integer-valued polynomial $p$. We call such numbers transcendentally normal. We prove that almost every real number is transcendentally normal and provide an explicit construction of such a number, based on Sierpinski's covering method and novel ideas involving the so-called stretch function. In the next step, we transform this construction into an algorithm that computes the digits of a t-normal number recursively in all integer bases. Moreover, we extend our covering approach to construct and compute LIL-normal numbers whose discrepancies are of the order predicted by the law of the iterated logarithm. We also take the opportunity to discuss several interesting open problems.
- [341] arXiv:2508.13759 (replaced) [pdf, html, other]
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Title: A Boundary Characterization of Turaev-Viro TQFTsJournal-ref: SIGMA 22 (2026), 079, 69 pagesSubjects: Quantum Algebra (math.QA); Geometric Topology (math.GT)
We consider three-dimensional topological field theories on manifolds with boundary defects and identify explicit boundary locality conditions. We show that these conditions imply a state sum construction of the given TQFT. As a consistency check, we prove that Turaev-Viro state sum models obey the boundary locality conditions. Recent progress [Faria Martins J., Meusburger C., Adv. Math. 494 (2026), 110923, 102 pages, arXiv:2410.18049] in the description of defects in Dijkgraaf-Witten theories enables us to show that these theories likewise satisfy boundary locality. This directly implies that Dijkgraaf-Witten TQFTs with boundary defects admit a state sum description.
- [342] arXiv:2509.05054 (replaced) [pdf, html, other]
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Title: Non-residually finite $\tilde{C}_2$-latticesComments: 55 pages, 10 figures; v2: include classification of ~A_2-latticesSubjects: Group Theory (math.GR)
We provide the first known examples of non-residually finite lattices on irreducible buildings. They contain the first known simple CAT(0)-groups with property (T), and the first known CAT(0)-groups that are not quasi-isometric to a direct product.
We also classify type-preserving vertex-regular lattices on buildings of type $\tilde{A}_2$ and thickness three, and discover an arithmetic example that is not commensurable to previously studied lattices. - [343] arXiv:2509.06050 (replaced) [pdf, html, other]
- [344] arXiv:2509.09633 (replaced) [pdf, html, other]
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Title: Orthogonal Latin Squares of Order Ten with Two Relations: A SAT InvestigationComments: A minor improvement has been made to Proposition 6, so results differ slightly from the published version in Discrete Mathematics, Algorithms and ApplicationsSubjects: Combinatorics (math.CO); Discrete Mathematics (cs.DM)
A $k$-net($n$) is a combinatorial design equivalent to $k-2$ mutually orthogonal Latin squares of order $n$. A relation in a net is a linear dependency over $\mathbb{F}_2$ in the incidence matrix of the net. A computational enumeration of all orthogonal pairs of Latin squares of order 10 whose corresponding nets have at least two nontrivial relations was achieved by Delisle in 2010 and verified by an independent search of Myrvold. In this paper, we confirm the correctness of their exhaustive enumerations with a satisfiability (SAT) solver approach instead of using custom-written backtracking code. Performing the enumeration using a SAT solver has at least three advantages. First, it reduces the amount of trust necessary, as SAT solvers produce independently-verifiable certificates that their enumerations are complete. These certificates can be checked by formal proof verifiers that are relatively simple pieces of software, and therefore easier to trust. Second, it is typically more straightforward and less error-prone to use a SAT solver over writing search code. Third, it can be more efficient to use a SAT-based approach, as SAT solvers are highly optimized pieces of software incorporating backtracking-with-learning for improving the efficiency of the backtracking search. For example, the SAT solver completely enumerates all orthogonal pairs of Latin squares of order ten with two nontrivial relations in under 2 hours on a desktop machine, while Delisle's 2010 search used 11,700 CPU hours. Although computer hardware was slower in 2010, this alone cannot explain the improvement in the efficiency of our SAT-based search.
- [345] arXiv:2509.12407 (replaced) [pdf, html, other]
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Title: Spectra of random graphs with discrete scale invarianceSubjects: Spectral Theory (math.SP); Disordered Systems and Neural Networks (cond-mat.dis-nn); Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph); Probability (math.PR)
Random graphs defined by an occurrence probability that is invariant under node aggregation have been identified recently in the context of network renormalization. The invariance property requires that edges are drawn with a specific probability that, in the annealed case, depends on a necessarily infinite-mean node fitness. The diverging mean determines many properties that are uncommon in models with independent edges, but at the same time widespread in real-world networks. Here we focus on the leading eigenvalues and eigenvectors of the adjacency matrix of the model, where the n nodes are assigned a Pareto($\alpha$)-distributed fitness with 0 < $\alpha$ < 1. We find that the leading eigenvalues are all of order square root of n, alternate in sign and are located at the intersection between the real axis and a logarithmic spiral in the complex plane, which we characterize analytically in terms of the Gamma function. We also calculate the associated eigenvectors, finding that they display complexvalued scaling exponents and log-periodicity, which are signatures of discrete scale invariance. In contrast with the typical finite-rank behaviour of random graphs with finite-mean variables, we find that a growing number of the leading eigenvalues emerges from the bulk, whose edge extends up to order square root of n and therefore reaches the same scale as that of the structural eigenvalues.
- [346] arXiv:2509.22036 (replaced) [pdf, html, other]
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Title: On the differentiability of the local time of the ($1+β$)-stable super-Brownian motionSubjects: Probability (math.PR)
We consider the local times of $(1+\beta)$-stable $d$-dimensional super-Brownian motion with $0<\beta<1$. Mytnik and Perkins (2003) proved that the local time, denoted by $L(t, x)$, is jointly continuous for $d=1$, whereas it is locally unbounded in $x$ for $d\geq 2$ where it exists. This paper strengthens the results of Mytnik and Perkins for $d=1$ by showing that when $X_0$ is atomless, $L(t,\cdot)$ is differentiable at every fixed deterministic point almost surely and is differentiable Lebesgue-a.e. However, with probability one, the local time is almost surely not differentiable at every spatial point simultaneously.
- [347] arXiv:2509.22386 (replaced) [pdf, html, other]
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Title: An upper bound for the size of the ideal class monoidComments: 21 pages, revisedSubjects: Number Theory (math.NT)
The ideal class monoid for an order $R$ in a finite field extension $E/F$ of a number field, denoted by $\overline{\mathrm{Cl}}(R)$, is a fundamental object to study in number theory which has useful applications in algebraic geometry and topology. In this paper, we describe an upper bound for $\#\overline{\mathrm{Cl}}(R)$, in terms of the class number of $E$ and (local) orbital integrals for $\mathfrak{gl}_n$. We also describe an upper bound for the class number of $E$ in terms of the Minkowski bound.
When $[E:F]\leq 3$ or when $R$ is a Bass order, we refine our upper bound, using a known formula for local orbital integrals in the authors' previous work. In particular, if $R=\mathbb{Z}[x]/(x^3-mx^2+(m-1)x-1)$ with $m\in \mathbb{Z}$ which arises in a study of Cappell-Shaneson homotopy 4-spheres in topology, then we further refine our upper bound in terms of the discriminants of $R$ and $E$, which is $\frac{2}{3^5} \Delta_R^{\frac{1}{2}}\cdot \Delta_E^{\frac{3}{2}}$, when $\Delta_E>3075$. - [348] arXiv:2510.06524 (replaced) [pdf, html, other]
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Title: Stable Central Limit Theorems for Discrete-Time Lag Martingale Difference Arrays: Applications to Dynamic Causal InferenceSubjects: Statistics Theory (math.ST); Probability (math.PR)
Recent work in dynamic causal inference introduced a class of discrete-time stochastic processes that generalize martingale difference sequences and arrays as follows: the random variates in each sequence have expectation zero given certain lagged filtrations but not given the natural filtration. We formalize this class of stochastic processes and prove stable central limit theorems (CLTs) via martingale-coboundary decomposition, leveraging the classical martingale CLT. We develop a variety of sufficient conditions, including conditions under which the limiting variance has a simple form that depends on variances and covariances of neighboring variates. We demonstrate the application of these results to inference for time-averaged treatment effects in switchback designs and present a simulation study supporting their validity. The CLTs enable various extensions to existing methodology for design-based approaches to dynamic causal inference, including time-lagged effects, random limiting variances, cross-unit dependence, and vector-valued estimands.
- [349] arXiv:2510.12223 (replaced) [pdf, html, other]
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Title: Dual Truncated Hankel Operators: Characterizations and PropertiesComments: 20 pages; To appear in the Journal of Operator TheorySubjects: Functional Analysis (math.FA)
We introduce the notion of the Dual Truncated Hankel Operator (DTHO) and provide several operator equation characterizations using the dual compressed shift operator. These characterizations are similar to classical results concerning Hankel operators and align with recent findings related to Truncated Hankel Operators (THO) \cite{GM}. Additionally, our work addresses comprehensive solutions to various operator equations encountered in studying THO and the classical Hankel operator. We have also established some fundamental operator-theoretic properties of DTHO that apply to general symbols and symbols under specific conditions.
- [350] arXiv:2510.12230 (replaced) [pdf, html, other]
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Title: Near Invariance of The Dual Compressed ShiftComments: 13 pagesJournal-ref: Results in Mathematics, 2026Subjects: Functional Analysis (math.FA)
We present the notion of the nearly dual compressed shift-invariant subspaces of the orthogonal complement of the model space and obtain their structure using Hitt's algorithm \cite{DH}.
- [351] arXiv:2510.15321 (replaced) [pdf, html, other]
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Title: Cantor's Non-Equinumerosity Theorems, InductivelyComments: 6 pagesSubjects: Logic (math.LO)
In the first, pre-college level part, we present a proof by mathematical induction for Cantor's theorem on the uncountability of the infinite binary strings and the real numbers. In the second, undergraduate-level part, we prove Cantor's powerset theorem by using transfinite induction. There, we will need the axiom of choice and the concept of ordinals. Some of these proofs by (mathematical and transfinite) induction seem to be new; comparisons will be made with older results.
- [352] arXiv:2510.22466 (replaced) [pdf, html, other]
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Title: On the generalized $m$-Kropina metricsComments: 25 pages, The results become more rigorous, some physical examples and physical meaning of the geometric objects have been added, all results of Cheng-Shen Finslerian field equation for positive-definite generalized m-Kropina metrics have been deleted as this equation could not be applied for our singular metricsSubjects: Differential Geometry (math.DG); General Relativity and Quantum Cosmology (gr-qc)
Generalized $m$-Kropina metrics appear naturally as a spacetime geometry compatible with Lorentz symmetry breaking, leading to useful applications in modified gravity and cosmology. We prove that a generalized $m$-Kropina metric $F$ is an almost rational Finsler metric. Thereby, we study the rationality of its Finslerian geometric objects in the directional variable $y$. For example, its geodesic spray coefficients are rational in $y$. Consequently, we prove that if $F$ is an Einstein metric with $m \notin \mathbb{Z}$, then it is Ricci-flat. Moreover, for $m \in 2 \mathbb{Z}$, the arithmetic nature of $m$ imposes strong rigidity constraints: if $F$ has isotropic mean Berwald curvature, or has relatively isotropic Landsberg curvature, or has almost vanishing $\mathbf{H}$-curvature, then $F$ is weakly Berwaldian, or $F$ is Landsbergian, or $\mathbf{H}=0$, respectively. Furthermore, we show that if $F$ has almost isotropic flag curvature ($m \in 2 \mathbb{Z}$ and $n \geq 3$), then the flag curvature is constant. We, hence, deduce under what conditions a generalized $m$-Kropina metric $F$ becomes an exact solution to "Pfeifer and Wohlfarth's vacuum field equation". Finally, we provide several four-dimensional examples arising in modified gravity and cosmology.
- [353] arXiv:2511.00168 (replaced) [pdf, html, other]
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Title: A Tight SDP Relaxation for the Cubic-Quartic Regularization ProblemComments: 37 pagesSubjects: Optimization and Control (math.OC)
This paper studies how to compute global minimizers of the cubic-quartic regularization (CQR) problem \[ \min_{s \in \mathbb{R}^n} \quad f_0+g^Ts+\frac{1}{2}s^THs+\frac{\beta}{6}\| s \|^3+ \frac{\sigma}{4} \| s\|^4, \] where $f_0$ is a constant, $g$ is an $n$-dimensional vector, $H$ is an $n$-by-$n$ symmetric matrix, and $\| s \|$ denotes the Euclidean norm of $s$. The parameter $\sigma$ is nonnegative while $\beta$ can have any sign. The CQR problem arises as a critical subproblem for getting efficient regularization methods for solving unconstrained nonlinear optimization. Its properties are recently well studied by Cartis and Zhu {\it [cubic-quartic regularization models for solving polynomial subproblems in third-order tensor methods, Math. Program, 2025]}. We propose a structured semidefinite programming (SDP) relaxation method for solving the CQR problem globally. The SDP relaxation has only three symmetric positive semidefinite matrix variables of sizes $(n+1)$-by-$(n+1)$, $3$-by-$3$ and $2$-by-$2$ respectively. We show that our SDP relaxation is tight if and only if $\| s^* \| ( \beta + 3 \sigma \| s^* \|) \ge 0$ holds for a global minimizer $s^*$. When $s^* \ne 0$, this aligns with the sufficient global optimality condition $\beta + 3 \sigma \| s^* \| \ge 0$ given by Cartis and Zhu. In particular, if either $\beta \ge 0$ or $H$ has a nonpositive eigenvalue, then the SDP relaxation is shown to be tight. Second, we show that all nonzero global minimizers have the same Euclidean norm for the tight case. Third, we give an algorithm to detect tightness and to obtain the set of all global minimizers. Numerical experiments demonstrate that our SDP relaxation method is both effective and computationally efficient. This paper gives a polynomial time algorithm for solving the CQR problem globally, under the sufficient global optimality condition.
- [354] arXiv:2511.00753 (replaced) [pdf, html, other]
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Title: $F$-intersection flatness of dagger and Berkovich Tate algebrasComments: 15 pages, comments welcomeSubjects: Commutative Algebra (math.AC); Algebraic Geometry (math.AG); Number Theory (math.NT)
We show, using the techniques developed in arXiv:2504.06444 and arXiv:2305.11139, that dagger algebras and Tate algebras in the sense of Berkovich in prime characteristic $p > 0$ have intersection flat Frobenius. Equivalently, if $S$ is such a ring, then $S^{1/p}$ is a flat and Mittag-Leffler $S$-module. As a consequence, we deduce that any ideal-adic completion of a reduced ring that is essentially of finite type over a dagger algebra or a Berkovich Tate algebra in prime characteristic has big test elements from tight closure theory.
- [355] arXiv:2511.04314 (replaced) [pdf, html, other]
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Title: There is no universal separable Banach algebraComments: 5 pp., accepted for publication in Journal of Functional AnalysisSubjects: Functional Analysis (math.FA); Operator Algebras (math.OA)
We prove that no separable Banach algebra is universal for homomorphic embeddings of all separable Banach algebras, whether embeddings are merely bounded or required to be contractive. The same holds in the commutative category.
The proof is short and direct. For each closed subspace $X$ of $c_0$ we build a separable commutative Banach algebra $A_X$ whose multiplication records the canonical pairing between $X^*$ and $X$. Any injective homomorphism of $A_X$ into a Banach algebra $B$ forces the identity operator on $X^*$ to factor through $B$. Thus a universal separable Banach algebra would be complementably universal for the family of duals of subspaces of $c_0$, contrary to a theorem of Johnson and Szankowski. - [356] arXiv:2511.06539 (replaced) [pdf, html, other]
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Title: Balanced Domination in Convex Polytopes, Trees, and Grid GraphsSubjects: Combinatorics (math.CO)
This paper addresses two open questions posed in [27] regarding the balanced domination number in graphs. We show that three new classes of graphs, those of convex polytopes A_n, D_n, and Rn'', are d-balanced. Further, we provide a characterization of d-balancedness for rooted trees with two levels of descendants and prove that each full binary tree is d-balanced. Several results for caterpillar graphs are established. Moreover, we determine and prove the exact balanced domination number for grid graphs. Finally, we conclude by providing several open problems of interest.
- [357] arXiv:2511.16295 (replaced) [pdf, other]
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Title: Genus two embedded minimal surfaces in $\mathbb{S}^3$ with dihedral symmetryComments: Withdrawn by the authors because Lemma 4.2 contains a gap: the moduli space of the relevant R_2-symmetric right-angled geodesic hexagons has an additional parameter omitted from the stated classification. This affects the reduction of the closing problem, so the main theorem is not proved by the present manuscript. A corrected version is in preparationSubjects: Differential Geometry (math.DG)
We prove that the Lawson surface $\xi_{2,1}$ is the unique closed embedded minimal surface of genus $2$ in $\mathbb{S}^3$ whose isometry group contains the dihedral group $D_4$ generated by two reflections across orthogonal totally geodesic two-spheres and a half-turn about a great circle. This weakens the full-symmetry hypotheses in previous characterizations of Lawson surfaces and leads to a substantially different geometric problem.
- [358] arXiv:2511.19503 (replaced) [pdf, html, other]
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Title: On Some Generalisations of Gauss SequencesSubjects: Number Theory (math.NT)
In this paper, we introduce integer sequences satisfying new congruence properties inspired by the Euler and Gauss congruences, which we call Euler--Gauss sequences. Noting that every Gauss sequence is an Euler--Gauss sequence, we compare them with certain generalisations of Gauss sequences and provide several counterexamples. Unlike Gauss sequences, Euler--Gauss sequences include sequences based on distinct prime factors, such as the Smallest Prime Factor and Greatest Prime Factor sequences (suitably defined at $1$). Moreover, we show that the prime-divisor subclass of Gauss sequences, given by $\sum_{p\mid n} pg_p$ for an integer sequence $(g_n)$, admits a natural extension to Euler--Gauss sequences of the form $\sum_{p\mid n} pf_p(\operatorname{rad}(n))$, where, for each prime $p$, $f_p$ is an integer-valued function and $\operatorname{rad}(n)$ denotes the square-free kernel of $n$. Further, we obtain $q$-analogs of the Euler--Gauss sequences, fill gaps in the literature on $q$-Gauss sequences, and conjecture a divisibility criterion for $q$-Euler--Gauss sequences, which we have verified computationally. We also show that not only do our $q$-Euler--Gauss sequences satisfy the Cyclic Sieving Phenomenon (CSP) exhibited by the $q$-Gauss sequences, but we also derive a new CSP condition for the SPF and GPF sequences, not hitherto known in the literature.
- [359] arXiv:2511.21144 (replaced) [pdf, html, other]
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Title: On the order-diameter ratio of girth-diameter cagesComments: 23 pagesSubjects: Combinatorics (math.CO); Discrete Mathematics (cs.DM)
For integers $k,g,d$, a $(k;g,d)$-cage (or simply girth-diameter cage) is a smallest $k$-regular graph of girth $g$ and diameter $d$ (if it exists). The order of a $(k;g,d)$-cage is denoted by $n(k;g,d)$. We determine asymptotic lower and upper bounds for the ratio between the order and the diameter of girth-diameter cages as the diameter goes to infinity. We also prove that this ratio can be computed in constant time for fixed $k$ and $g$.
We theoretically determine the exact values $n(3;g,d)$, and count the number of corresponding girth-diameter cages, for $g \in \{4,5\}$. Moreover, we design and implement an exhaustive graph generation algorithm and use it to determine the exact order of several open cases and obtain -- often exhaustive -- sets of the corresponding girth-diameter cages. The largest case we generated and settled with our algorithm is a $(3;7,35)$-cage of order 136. - [360] arXiv:2512.07554 (replaced) [pdf, html, other]
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Title: A simple proof of exponential decay for the near-critical planar Ising modelComments: 9 pages, 2 figuresSubjects: Probability (math.PR); Mathematical Physics (math-ph)
For the Ising model defined on $a\mathbb{Z}^2$ at critical temperature with external field $a^{15/8}h$, we give a simple and elementary proof that its truncated two-point function decays exponentially. The proof combines the high temperature expansion, random-cluster and random current representations. A new input in the proof is that, in the near-critical sourceless single current measure, there are many loops formed by a path on $a\mathbb{Z}^2$ with diameter of order $1$, together with two external edges that connect the path's endpoints to the ghost.
- [361] arXiv:2512.08375 (replaced) [pdf, html, other]
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Title: A Characterization of Functional Affine Surface AreasSubjects: Metric Geometry (math.MG); Functional Analysis (math.FA)
A characterization of valuations on the space of convex Lipschitz functions whose domain is a polytope in $\mathbb{R}^n$ is obtained. It is shown that every upper semicontinuous, equi-affine and dually epi-translation invariant valuation can be written as a linear combination of a constant term, the volume of the domain, and a functional affine surface area. In addition, dual statements for finite-valued convex functions are established.
- [362] arXiv:2512.11785 (replaced) [pdf, html, other]
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Title: Universal entrywise eigenvector fluctuations in delocalized spiked matrix models and asymptotics of rounded spectral algorithmsComments: 64 pages, 7 figures. v2: Results expanded to include analysis of multi-spectral and multi-frequency algorithms, as well as other small correctionsSubjects: Probability (math.PR); Data Structures and Algorithms (cs.DS); Statistics Theory (math.ST)
We consider the distribution of the top eigenvector $\widehat{v}$ of a spiked matrix model of the form $H = \theta vv^* + W$, in the supercritical regime where $H$ has an outlier eigenvalue of comparable magnitude to $\|W\|$. We show that, if $v$ is sufficiently delocalized, then the distribution of the individual entries of the projector $\widehat{v}\widehat{v}^*$ (not, we emphasize, merely the inner product $|\langle \widehat{v}, v\rangle|^2$) is universal over a large class of generalized Wigner matrices $W$ having independent entries, depending only on the first two moments of the distributions of the entries of $W$. This complements the observation of Capitaine and Donati-Martin (2021) that these distributions are not universal when $v$ is instead sufficiently localized. Further, for $W$ having entrywise variances close to constant and thus resembling a Wigner matrix, we show by comparing to $W$ drawn from the Gaussian orthogonal or unitary ensembles that averages of entrywise functions of $\widehat{v}\widehat{v}^*$ behave as they would if $\widehat{v}$ had Gaussian fluctuations around a suitable multiple of $v$. We also establish such results for several possibly dependent spiked matrices, showing that, if such matrices are entrywise uncorrelated, then their leading eigenvectors behave as they would with independent Gaussian fluctuations. We apply these results to spectral algorithms with rounding procedures for synchronization problems over the cyclic and circle groups, obtaining the first precise asymptotic error rates for such algorithms. Using our analysis of multiple spiked matrices, we also show that multi-frequency spectral algorithms using estimates from several matrices often have asymptotic error rate superior to that of naive spectral algorithms using just one matrix.
- [363] arXiv:2512.16599 (replaced) [pdf, html, other]
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Title: The $s$-chromatic Ramsey number for starsSubjects: Combinatorics (math.CO)
In 1977, Chung, Chung and Liu generalized the definition of the Ramsey number. They introduced the $s$-chromatic Ramsey number as follows. Let $1\leq s< t$ be integers and let $A_{1}, A_{2}, \dots, A_{c}$ be subsets with size $s$ of $[t]$, where $c= {t\choose s}$. For given graphs $G_{1}, G_{2}, \dots, G_{c}$, the {\it $s$-chromatic Ramsey number} $r^{s, t}(G_{1}, G_{2}, \dots, G_{c})$, is the minimum positive integer $N$ such that every $t$-coloring of $E(K_{N})$ yields a copy of $G_{i}$ whose edges are colored by colors in the color set $A_{i}$ for some $i\in [c]$. The {\it star-critical $s$-chromatic Ramsey number} $r_{*}^{s, t}(G_{1}, G_{2}, \dots, G_{c})$, is the minimum integer $\ell$ such that every $t$-coloring of the edges in $K_{N}- E(K_{1, N- 1- \ell})$ yields a copy of $G_{i}$ whose edges are colored by colors in the color set $A_{i}$ for some $i\in [c]$, where $N= r^{s, t}(G_{1}, G_{2}, \dots, G_{c})$. If $G_{1}= G_{2}= \dots= G_{c}= G$, then we simplify them to $r^{s, t}(G)$ (also called the {\it weakened Ramsey number}) and $r^{s, t}_{*}(G)$, respectively. In this paper, we determine all the values of $r^{s, t}(K_{1, m})$ and $r_{*}^{s, t}(K_{1, m})$, and part of the value of $r^{s, t}(K_{1, m_{1}}, K_{1, m_{2}}, \dots, K_{1, m_{c}})$.
- [364] arXiv:2512.16825 (replaced) [pdf, html, other]
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Title: Quivers with quantum Yang-Baxter equation and Hecke condition: Deformation of face algebrasSubjects: Quantum Algebra (math.QA); Rings and Algebras (math.RA); Representation Theory (math.RT)
In this paper, we initiate the study of quivers carrying quantum Yang--Baxter and Hecke structure. Calling a quiver $Q$ to be a solution of the QYBE when the adjacency matrix of its Kronecker square is a solution, we show that this holds precisely when the adjacency matrix $A$ satisfies $A^2 = \mu A$ for a scalar $\mu$. We determine exactly when Kulish's rank-one construction yields a Hecke $R$-matrix that satisfies both the braided QYBE and the Hecke condition. We deform Hayashi's face algebra by the resulting RTT relations, and prove that the quantum matrix algebra $\mathcal{O}_q(M_n)$ is isomorphic as a bialgebra to the Hecke-deformed face algebra of the rose quiver with $n$ petals, and that for a disjoint union of $m$ such roses the deformation is a genuine quantum groupoid with $m$-dimensional base, isomorphic as an algebra to $m^2$ copies of $\mathcal{O}_q(M_n)$.
- [365] arXiv:2512.23339 (replaced) [pdf, html, other]
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Title: Small-time global controllability of a class of bilinear fourth-order parabolic equationsComments: 28 pages, Comments are welcomeSubjects: Optimization and Control (math.OC); Analysis of PDEs (math.AP)
In this work, we address the small-time global controllability properties of a class of fourth-order nonlinear parabolic equations driven by bilinear controls 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 model through a fixed-point argument together with the small-time global approximate control property. To the best of our knowledge, this work provides the first contribution toward the study of controllability properties of fourth-order parabolic equations by means of bilinear controls.
- [366] arXiv:2512.23414 (replaced) [pdf, html, other]
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Title: On the existence of the KMS spectral gap in Gaussian quantum Markov semigroupsComments: Revised version with a new blockwise rank criterion for the KMS spectral gap, revised proofs and examples, and removal of some auxiliary material. 23 pagesSubjects: Functional Analysis (math.FA); Quantum Physics (quant-ph)
In arXiv:2405.04947, it was shown that a Gaussian quantum Markov semigroup on the $d$-mode bosonic Fock space with a unique faithful normal invariant state has a positive GNS spectral gap if and only if the matrix $[U \overline{V}]$, formed from the coefficients of the Kraus operators, has $2d$ linearly independent columns. In this paper, we establish the corresponding criterion for the KMS spectral gap. After standardizing the invariant Gaussian state and grouping the modes according to their inverse temperatures, let $U_n$ and $V_n$ denote the submatrices of $U$ and $V$, respectively, corresponding to the $n$-th equal-temperature block. We prove that $\ker U_n=\ker V_n$ for every $n$, and that the KMS spectral gap is positive if and only if this common kernel is trivial for every equal-temperature block. Thus, whereas the GNS spectral gap requires a global maximal-rank condition on the Kraus coefficients, the KMS spectral gap requires maximal column rank only within each equal-temperature block.
- [367] arXiv:2601.00972 (replaced) [pdf, html, other]
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Title: Improved decoding algorithms for surface codes under independent bit-flip and phase-flip errorsComments: Updated discussion of open problems to reflect subsequent results on the complexity and approximability of minimum-weight decodingSubjects: Information Theory (cs.IT); Quantum Physics (quant-ph)
We study exact decoding for the toric code and for planar and rotated surface codes under the standard independent \(X/Z\) noise model, focusing on Separate Minimum Weight (SMW) decoding and Separate Most Likely Coset (SMLC) decoding. For the SMW decoding problem, we show that an \(O(n^{3/2}\log n)\)-time decoder is achievable for surface and toric codes, improving over the \(O(n^{3}\log n)\) worst-case time of the standard approach based on complete decoding graphs. Our approach is based on a local reduction of SMW decoding to the minimum weight perfect matching problem using Fisher gadgets, which preserves planarity for planar and rotated surface codes and genus~\(1\) for the toric code. This reduction enables the use of Lipton--Tarjan planar separator methods and implies that SMW decoding lies in \(\mathrm{NC}\). For SMLC decoding, we show that the planar surface code admits an exact decoder with \(O(n^{3/2})\) algebraic complexity and that the problem lies in \(\mathrm{NC}\), improving over the \(O(n^{2})\) algebraic complexity of Bravyi \emph{et al.} Our approach proceeds via a dual-cycle formulation of coset probabilities and an explicit reduction to planar Pfaffian evaluation using Fisher--Kasteleyn--Temperley constructions. The same complexity measures apply to SMLC decoding of the rotated surface code. For the toric code, we obtain an exact polynomial-time SMLC decoder with \(O(n^{3})\) algebraic complexity. In addition, while the SMLC formulation is motivated by connections to statistical mechanics, we provide a purely algebraic derivation of the underlying duality based on MacWilliams duality and Fourier analysis. Finally, we discuss extensions of the framework to the depolarizing noise model and identify resulting open problems.
- [368] arXiv:2601.02024 (replaced) [pdf, html, other]
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Title: Prescribed Chern scalar curvature on complete noncompact Hermitian manifolds with nonpositive curvaturesComments: 20 pagesSubjects: Differential Geometry (math.DG)
In this paper, we investigate the problem of prescribing Chern scalar curvatures on complete noncompact Hermitian manifolds with nonpositive curvatures, and establish some existence results. In particular, we obtain some sufficient conditions for the existence of a constant negative Chern scalar curvature metric in the conformal class.
- [369] arXiv:2601.09314 (replaced) [pdf, html, other]
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Title: Tail behavior of Markov-modulated generalized Ornstein-Uhlenbeck processesSubjects: Probability (math.PR)
We study the tail behavior of Markov-modulated generalized Ornstein-Uhlenbeck processes -- that is, solutions to Langevin-type stochastic differential equations driven by a background continuous-time Markov chain. To this end, we consider a sequence of Markov modulated random affine functions $ \Psi_{n} : \mathbb{R} \to \mathbb{R} $, $ n \in \mathbb{N} $, and the associated iterated function system defined recursively by $ X_0^x := x $ and $ X_{n}^x := \Psi_{n-1}(X_{n-1}^x) $ for $ x \in \mathbb{R} $, $n \in \mathbb{N}$. We analyze the tail behavior of the stationary distribution of such a Markov chain using tools from Markov renewal theory. Our approach extends Goldie's implicit renewal theory~\cite{Goldie:91} and can be seen as an adaptation of Kesten's work on products of random matrices~\cite{Kesten:73} to the one-dimensional setting of random affine function systems. These results have applications in diverse areas of applied probability, including queueing theory, econometrics, mathematical finance, and population dynamics.
- [370] arXiv:2601.11422 (replaced) [pdf, html, other]
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Title: Stein's method for the matrix normal distributionComments: 25 pages, 0 figuresSubjects: Statistics Theory (math.ST); Probability (math.PR)
This work presents the first systematic development of Stein's method for matrix distributions. We establish the basic essential ingredients of Stein's method for matrix normal approximation: we derive an extended-generator-based Stein identity from a matrix Ornstein-Uhlenbeck diffusion with two-sided scales, provide an explicit semigroup representation for the solution of the Stein equation, and obtain regularity estimates for the solution. The new methodology is demonstrated in three examples: (i) smooth Wasserstein distance bounds to quantify the matrix central limit theorem (a didactic example), (ii) a Wasserstein distance bound for the matrix normal approximation of the centered matrix $T$ distribution, and (iii) a Stein's method-of-moments approach to estimating the row and column covariance factors of the matrix normal, yielding a flexible class of weighted flip-flop Stein estimators that generalize Dutilleul's classical flip-flop algorithm and naturally accommodate row/column importance weights, systematic missingness, and projection onto structured covariance families. The latter two examples are intrinsically matrix-valued and cannot be treated using naive vectorization.
- [371] arXiv:2601.12017 (replaced) [pdf, html, other]
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Title: Deformation rigidity of some simple affine VOAsComments: Fill some gaps and corrected some mistakes and typos in the previous version. 10 Pages. Comments are welcomedSubjects: Quantum Algebra (math.QA); High Energy Physics - Theory (hep-th)
In this paper, we prove that simple affine vertex operator algebras with positive integral levels admit only trivial first-order deformations. Therefore, the deformation rigidity conjecture of strongly rational vertex operator algebras holds for these cases. We also show that the same holds simple affine vertex operator algebra of $\mathfrak{sl}_2$ at the non-integral admissible level $-4/3$. Therefore, neither $C_2$-cofiniteness nor rationality is a necessary condition for deformation rigidity of VOAs. We conjecture that the same should hold for every simple affine VOA that does not coincide with the corresponding universal affine VOA.
- [372] arXiv:2601.15767 (replaced) [pdf, html, other]
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Title: Recursive Flow: A Generative Framework for MIMO Channel EstimationZehua Jiang, Fenghao Zhu, Chongwen Huang, Richeng Jin, Zhaohui Yang, Xiaoming Chen, Zhaoyang Zhang, Mérouane DebbahSubjects: Information Theory (cs.IT)
Channel estimation is a fundamental challenge in massive multiple-input multiple-output systems, where estimation accuracy governs the spectral efficiency and link reliability. In this work, we introduce Recursive Flow (RC-Flow), a novel solver that leverages pre-trained flow matching priors to robustly recover channel state information from noisy, under-determined measurements. Different from conventional open-loop generative models, our approach establishes a closed-loop refinement framework via a serial restart mechanism and anchored trajectory rectification. By synergizing flow-consistent prior directions with data-fidelity proximal projections, the proposed RC-Flow achieves robust channel reconstruction and delivers state-of-the-art performance across diverse noise levels, particularly in noise-dominated scenarios. The framework is further augmented by an adaptive dual-scheduling strategy, offering flexible management of the trade-off between convergence speed and reconstruction accuracy. Theoretically, we analyze the Jacobian spectral radius of the recursive operator to prove its global asymptotic stability. Numerical results demonstrate that RC-Flow reduces inference latency by two orders of magnitude while achieving a 2.7 dB performance gain in low signal-to-noise ratio regimes compared to the score-based baseline.
- [373] arXiv:2602.02938 (replaced) [pdf, html, other]
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Title: A Correspondence between Billiards and GeodesicsComments: 26 pages, 4 figuresJournal-ref: Daniele Giannetto 2026 Nonlinearity 39 085015Subjects: Differential Geometry (math.DG)
From a geometric viewpoint, billiard trajectories and geodesics are related by mutual approximation results. In one direction, it is known that every geodesic curve on the boundary of a smooth convex body can be approximated by a sequence of billiard trajectories inside of it. We establish the other direction by proving that, for Riemannian billiard tables (under mild assumptions), there are families of fold-type surfaces such that every sequence of geodesic segments on these surfaces has a subsequence that converges to a billiard trajectory in the table. In particular, this is true for convex Euclidean tables. We also describe a more general class of tables for which this result holds and present explicit non-convex examples.
- [374] arXiv:2602.03299 (replaced) [pdf, html, other]
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Title: On Poincaré-Sobolev level involving fractional GJMS operators on hyperbolic spaceSubjects: Analysis of PDEs (math.AP)
This paper is devoted to a qualitative analysis of the Poincaré--Sobolev level associated with the fractional GJMS operators \(\mathcal{P}_s\) \(\bigl(s\in(0,\tfrac n2)\setminus\mathbb N\bigr)\) on the hyperbolic space \(\mathbb H^n\). In contrast to the integer-order case, when \(s\notin\mathbb N\) the operator \(\mathcal{P}_s\) does not enjoy the conformal covariance that allows one, in the upper half-space or ball model, to relate it to the Euclidean fractional Laplacian \((-\Delta)^s\); this link is crucial for importing Euclidean theory. We therefore introduce \(\widetilde{\mathcal{P}}_s\) (\(s>0\)), which is conformally related to \((-\Delta)^s\). Our purpose in the paper is to analyze the monotonicity, attainability, and strict-gap regions of the Poincaré--Sobolev levels associated with \(\mathcal{P}_s\) and \(\widetilde{\mathcal{P}}_s\). First, we reinterpret the Brezis--Nirenberg problem through the lens of Poincaré--Sobolev levels, connecting earlier results for the Euclidean Laplacian and for operators \(\mathcal{P}_k\) on \(\mathbb H^n\) with integer \(k\in(0,\tfrac n2)\). We then establish new, explicit lower bounds for the Hardy term in fractional Hardy--Sobolev--Maz'ya inequalities involving both \(\mathcal{P}_s\) and \(\widetilde{\mathcal{P}}_s\). By applying the concentration--compactness principle together with a detailed analysis of the strict-gap regions for the Poincaré--Sobolev levels, we prove the existence of solutions to the Brezis--Nirenberg problem on \(\mathbb H^n\) for both operators. Finally, combining the Hardy lower bounds with criteria for attainability, we obtain a complete characterization of the Poincaré--Sobolev levels \(H_{n,s}\) and \(\widetilde H_{n,s}\).
- [375] arXiv:2602.03717 (replaced) [pdf, html, other]
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Title: Curious crossing-critical edgesSubjects: Combinatorics (math.CO)
Motivated by Kuratowski's theorem, a Kuratowski subgraph of a graph is a subgraph that is a subdivided $K_5$ or a subdivided $K_{3,3}$. An edge is crossing-critical if the crossing number decreases after removing the edge. In this note, we present the following examples: a graph with an edge that is crossed in every optimal drawing of the graph, but the edge is not in any Kuratowski subgraph of the graph; a graph with an edge that is in every Kuratowski subgraph but is not crossed in any optimal drawing of the graph; and a graph with a crossing-critical edge that is not present in any Kuratowski subgraph and is not crossed in any optimal drawing of the graph. Fáry's theorem implies that the Kuratowski subgraphs are the only obstructions to a graph having a crossing-free drawing with all edges drawn as straight lines. The three example graphs given also hold if we restrict drawings to only have straight line edges, and thus also apply to the rectilinear crossing number.
- [376] arXiv:2602.12163 (replaced) [pdf, html, other]
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Title: NLS with exponential nonlinearity on compact surfacesComments: 35 pagesSubjects: Analysis of PDEs (math.AP)
In this paper, we establish a probabilistic global theory in $H^1$ for the NLS with a Moser-Trudinger nonlinearity posed on compact surfaces. This equation is known to be the two dimensional counterpart to the classical energy-critical Schrödinger equations \cite{CollianderIbrahimMajdoubMasmoudi2009}. The authors of \cite{CollianderIbrahimMajdoubMasmoudi2009} also identified a trichotomy around the criticality of the equation based on the size of the total energy. In particular, for supercritical regimes (large energy), the equation is known to exhibit instabilities : the (uniform) continuity of the flow fails to hold. Large data distributional non unique probabilistic solutions have been obtained in \cite{CasterasMonsaingeon2024}. The setting of \cite{CasterasMonsaingeon2024} does not handle the uniqueness issue for the $H^1$-data and therefore could not define a flow for this regularity. Our main focus here is to build a single probabilistic framework that provides both existence, uniqueness, and continuity with respect to the initial data in $H^1$. Our uniqueness and continuity are based on the so-called Yudowich argument \cite{Judovic1963}, and the probabilistic estimates are derived through the IID limit procedure \cite{Sy2019}. Beyond the difficulties related to the borderline nature of the context, the major challenge resides in the need to satisfy two features that tend to play against each other : obtaining both continuity property of the flow and large data in the support of the reference measure. This made the design of the dissipation operator inherent in the method, as well as the analysis of the resulting quantities, particularly difficult. Regarding the supercritical regime, we show that a modified energy, with regularity similar to the original total energy, admits values as high as desired, suggesting that the constructed set of data contains supercritical ones.
- [377] arXiv:2602.12678 (replaced) [pdf, html, other]
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Title: Soft Bitopological Groups via Soft ElementsComments: 15 pages, 0 figures. Substantially revised version: the selector topology is corrected to the box topology; the pairwise Hausdorff condition and the compactness and homomorphism arguments are corrected; conjugate soft paratopological-group results and finite/infinite parameter counterexamples are added; references are updatedSubjects: General Topology (math.GN)
Let $F$ be a soft group over a parameter set $A$. We equip the selector group $\mathrm{SE}(F)=\prod_{t\in A}F(t)$ with the topology generated by componentwise open boxes. This construction replaces a sectionwise rule that does not define a topology on arbitrary selector subsets. It depends only on the component topologies and may therefore lose correlations between parameters. A soft bitopological group is then a soft group with two selector topologies, each making $\mathrm{SE}(F)$ a topological group. We also study a soft paratopological group together with its inverse topology. In this conjugate pair, inversion interchanges the two selector topologies, and equality of the two is equivalent to continuity of inversion. We prove componentwise separation criteria, finite-parameter compactness and $\omega$-boundedness results, and infinite-parameter counterexamples caused by the box topology. We also characterize when a soft union creates mixed selectors outside the two original selector sets.
- [378] arXiv:2602.13663 (replaced) [pdf, html, other]
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Title: Cantor's Powerset Theorem, Graph-TheoreticallyComments: 2 pages (an extended version will appear in "the Mathematical Gazette" under the title "Graph-Theoretic Proofs of Cantor's Powerset Theorem")Subjects: History and Overview (math.HO); Logic (math.LO)
We study Cantor's powerset theorem from a graph-theoretic perspective, consider some alternative proofs to Cantor's original, and provide a new proof.
- [379] arXiv:2602.23207 (replaced) [pdf, html, other]
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Title: A Study of the Extreme Points in the Unit Ball of $JT$Comments: 16 pagesSubjects: Functional Analysis (math.FA)
In this note, we investigate the extreme points of the unit ball of the James Tree space ($JT$). We relate the geometric structure of $JT$ to the classical James space $J$ and provide partial characterizations of extremality based on the concept of separated vectors. We provide a complete characterization for positive vectors and establish the equal sums property for positive extreme points.
- [380] arXiv:2603.13025 (replaced) [pdf, html, other]
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Title: Maximal and minimal displacement of supercritical branching random walks on free products of groupsComments: minor changes and corrections suggested by the refereesSubjects: Probability (math.PR)
We prove that the maximal and minimal displacement of branching random walks with mean offspring number $\rho>1$ on free products of finite groups grows linearly almost surely. More precisely, we establish that the linear speed for the maximal (respectively minimal) displacement is given by the largest (respectively smallest) intersection point of the large deviation rate function of the underlying random walk with the horizontal line at height $\log\rho$. The proof is based on constructing an associated multitype branching process which consists of particles that travel fast enough, and distinguishing the types via the suffix of the particles locations.
- [381] arXiv:2603.26209 (replaced) [pdf, html, other]
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Title: Lieb-Robinson bounds for Bose-Hubbard Hamiltonians: A review with a simplified proofComments: 32 pages, 1 figure. final version to appear in PAFA special issue dedicated to Barry Simon on the occasion of his 80th birthdaySubjects: Mathematical Physics (math-ph); Analysis of PDEs (math.AP); Quantum Physics (quant-ph)
We review recent progress on state-dependent Lieb--Robinson bounds for Bose--Hubbard Hamiltonians. In particular, Kuwahara, Vu, and Saito established that, for general bounded-density initial states, the Lieb--Robinson velocity is bounded by $t^{d-1}$ for large times, where $d$ denotes the lattice dimension. We present a shorter proof of the weaker, but still polynomial velocity bound $t^{d+\epsilon}$.
- [382] arXiv:2603.28007 (replaced) [pdf, html, other]
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Title: Legendrian and Lagrangian higher torsionComments: minor revisionsSubjects: Symplectic Geometry (math.SG)
Let $M$ be a closed manifold. We introduce a family of Legendrian isotopy invariants for Legendrians in $J^1M$, which we collectively call Legendrian higher torsion. Given a choice of a class $\mathcal{F}$ of fibre bundles over $M$, equipped with suitable unitary local systems, the Legendrian higher torsion of a Legendrian $\Lambda \subset J^1M$ is the subset of $H^*(M;\mathbf{R})$ consisting of higher Reidemeister torsion cohomology classes of fibre bundles $W$ over $M$ in the class $\mathcal{F}$ such that $\Lambda$ admits a generating function on a stabilization of $W$. For the class of tube bundles in the sense of Waldhausen we call the invariant tube torsion. We show that the tube torsion of a nearby Lagrangian $L \subset T^*M$ is well-defined when the stable Gauss map $L \to U/O$ is trivial (for example when $L$ is a nearby Lagrangian homotopy sphere) and it consists of a union of cosets of a normalized version of the Pontryagin character. We also identify a distinguished coset, invariant under Hamiltonian isotopy of $L$, which we call nearby Lagrangian torsion. We do not know whether nearby Lagrangians must have trivial tube torsion, as would follow from the nearby Lagrangian conjecture. However, we show that there exist Legendrians $\Lambda \subset J^1M$ with nontrivial tube torsion whose projection $\Lambda \to M$ is homotopic to a diffeomorphism.
- [383] arXiv:2603.29978 (replaced) [pdf, html, other]
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Title: The van der Waerden Simplicial Complex and its Lefschetz PropertiesComments: 22 pages; revised version corrects some errors and rearranges some of the materialSubjects: Commutative Algebra (math.AC)
The van der Waerden simplicial complex, denoted ${\tt vdw}(n,k)$, is the simpicial complex whose facets correspond to the arithmetic progressions of length $k$ in the set $\{1,\ldots,n\}$. We study the Lefschetz properties of the Artinian ring $A({\tt vdw}(n,k)) = K[x_1,\ldots,x_n]/(I_{{\tt vdw}(n,k)} + \langle x_1^2,\ldots,x_n^2\rangle)$ where $I_{{\tt vdw}(n,k)}$ is the associated Stanley--Reisner ideal. If $k=1,2$ or $n-1$, the ring $A({\tt vdw}(n,k))$ will have the Weak Lefschetz Property for all $n > k$. When $k=3$, we classify the rings $A({\tt vdw}(n,3))$ that have the Weak Lefschetz Property when the characteristic is zero. We conjecture that $A({\tt vdw}(n,k))$ fails to have the Weak Lefschetz Property if $n \gg k \geq 3$ and $k$ odd. We also classify when ${\tt vdw}(n,k)$ is a pseudo-manifold, which allows us to show that $A({\tt vdw}(n,k))$ satisfies the Weak Lefschetz Property in some degrees by using a result of Dao and Nair.
- [384] arXiv:2604.01199 (replaced) [pdf, html, other]
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Title: A high-order, structure preserving scheme for the stochastic Galerkin shallow water equations -- unification and two-dimensional extensionSubjects: Numerical Analysis (math.NA)
Recently, two independent research efforts have been made to study the stochastic Galerkin formulation of the shallow water equations. Bender and Öffner developed entropy-conservative discontinuous Galerkin (DG) methods to solve the stochastic shallow water equations in a stochastic Galerkin framework using Roe variable transformation, while Dai, Epshteyn and collaborators proposed second-order, energy-stable and well-balanced schemes for the same class of problems with a specific projection step used inside the Galerkin projection together with high-order quadrature rules and a time-step restriction. In this paper, we provide a comprehensive comparison of the two methodologies mentioned, focusing on their theoretical properties and practical implementation aspects. We highlight shared foundational concepts and key differences of both approaches, with a particular focus on the selection of basis functions in the stochastic domain. As a highlight, we show that under specific conditions, the two formulations align, offering a unified framework that connects these distinct approaches. From our theoretical findings, we extend the development of high-order entropy conservative DG methods for the one-dimensional stochastic Galerkin shallow equations to two space dimensions; constructing entropy conservative two-point fluxes via primitive variables instead of entropy variables and applying it in our high-order DG setting. In numerical simulations, we verify and support our theoretical findings of a well-balanced and entropy-stable DG scheme which can be used to solve geophyiscal fluid flows with uncertainty.
- [385] arXiv:2604.06423 (replaced) [pdf, html, other]
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Title: The Chambolle-Pock method converges weakly with $0 < θ\le 1/2$ and $τσ\|L\|^{2} < 4θ(2-θ)/(1 - 2θ+ 9θ^{2} - 4θ^{3})$Subjects: Optimization and Control (math.OC)
The Chambolle-Pock method, also known as the primal-dual hybrid gradient method, is a standard first-order algorithm for convex-concave saddle-point problems and composite convex optimization. We establish weak sequential convergence of its primal-dual iterates in real Hilbert spaces for every $0<\theta\leq 1/2$ whenever $\tau\sigma\|L\|^{2}<4\theta(2-\theta)/(1-2\theta+9\theta^{2}-4\theta^{3})$. This extends the weak-convergence theory to a previously unexplored range of extrapolation parameters.
- [386] arXiv:2604.08773 (replaced) [pdf, html, other]
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Title: Neutral representations in dimension $\leq 3$ and fields of moduliSubjects: Algebraic Geometry (math.AG)
A representation $V$ of an algebraic group $G$ induces a vector bundle $[V/G] \to BG$. The representation $V$ of $G$ is neutral if, for every twisted form $\mathcal{V} \to \mathcal{G}$ of $[V/G] \to BG$ over a field $k$, we have $\mathcal{G}(k) \neq \emptyset$.
Twisted forms of representations arise in many ways, for instance as cohomology of families of varieties on residual gerbes of moduli spaces, and from quotient singularities. Moreover, every Tannakian category is the category of vector bundles on some gerbe. Because of this, studying neutral representations yields numerous applications, especially to problems about fields of moduli.
The present article has three main results. First, we completely classify neutral, faithful representations of finite groups in dimension $\leq 3$. Second, we give a very general, computation-friendly result for proving that representations of finite abelian groups are neutral, in arbitrary dimensions. Third, we develop the abstract concept of the normalizer $\mathcal{G} \to \mathcal{N} \to \mathcal{H}$ of a morphism of gerbes $\mathcal{G} \to \mathcal{H}$ on an arbitrary site (twisted representations correspond to morphisms of gerbes $\mathcal{G} \to B\mathrm{GL}_{n}$), and show that the normalizer $\mathcal{N}$ only depends on the geometric type of $\mathcal{G} \to \mathcal{H}$. - [387] arXiv:2604.09113 (replaced) [pdf, html, other]
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Title: A ROM-based BDDC solver for unfitted p-FEM level-set-based two-dimensional lattice structuresComments: 44 pages, 21 figures, 5 algorithmsSubjects: Numerical Analysis (math.NA)
We present a domain decomposition method for the fast simulation of large two-dimensional lattice structures described by level set functions. The method does not rely on homogenization or multiscale techniques, and therefore avoids their underlying assumptions such as scale separation and periodicity. Individual cells are defined through level set functions and mapped into physical space using arbitrary order mappings, which allows the creation of complex graded designs with varying geometries and topologies. The discretization is based on unfitted p-FEM, where each cell is approximated by a single high order element. This choice naturally handles the implicit geometric description and provides high accuracy with a moderate number of degrees of freedom. The solver is built on the Balanced Domain Decomposition by Constraints method, where each cell corresponds to one subdomain. To accelerate the assembly of the cell stiffness matrices, we combine a fast assembly technique that separates the contributions of the geometric mapping from the trimmed domain with a reduced order model based on the matrix discrete empirical interpolation method. The ROM surrogate is trained offline and can be reused for any geometric mapping, restricting the expensive quadrature on cut elements to the training stage. A stabilization term is introduced to ensure the scalability of the solver when using the ROM approximation, at the cost of a small and controllable error. We validate the method through a series of numerical experiments and demonstrate its performance on a 2D problem with more than 17,000 cells of varying geometry, which is solved in approximately 30 seconds on a standard laptop. The number of solver iterations grows only mildly as the number of subdomains increases, provided the ratio between subdomain and mesh sizes is kept constant, consistent with the scalability properties of BDDC methods.
- [388] arXiv:2604.23011 (replaced) [pdf, html, other]
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Title: Theoretical analysis of the effect of the kinetic energy operator on double heterostructure spectraSubjects: Mathematical Physics (math-ph)
We report an analytical and numerical study of the effect of kinetic energy operator (KEO) ambiguity on the energy spectra of double heterostructures when the mass of the charge carriers, subjected to a potential, depends on position. The spectra are calculated using two complementary techniques. In the first case, which is not always possible, the energy spectrum satisfies a transcendental equation, which is obtained through an analytical process. While in the second, which can be used practically in any double heterostructure (DH), the energy spectra correspond to the poles of the reflection coefficient of the heterostructure. The spectra thus obtained complement those already reported in a previous reference, since we now include the spectra of another KEO. Finally, we show how these methods allow us to critically analyze some statements that have been made about the relevance of some kinetic energy operators in some simpler heterostructures.
- [389] arXiv:2605.00784 (replaced) [pdf, html, other]
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Title: The structure of gauge invariant Gaussian quantum operations on finite Fermion systemsComments: This revision is the version accepted for publication in Advanced Nonlinear StudiesSubjects: Functional Analysis (math.FA); Quantum Physics (quant-ph)
Let ${\mathcal H}_1$ be a finite dimensional complex Hilbert space. Let $\psi\mapsto Z(\psi)$ be a canonical anti-commutation relations (CAR) field over ${\mathcal H}_1$ acting irreducibly on a Hilbert space ${\mathord{\mathscr K}}$. The $*$-algebra ${\mathscr A}_{{\mathcal H}_1}$ generated by the $Z(\psi)$, $\psi\in {\mathcal H}_1$, is simply all operators on ${\mathscr K}$. However, the CAR field endows ${\mathscr A}_{{\mathcal H}_1}$ with additional structure, and we are concerned with quantum operations acting in harmony with this structure. In particular, there is a "gauge" automorphism group generated by "second quantizing'' $\psi \mapsto e^{it}\psi$. The fixed point algebra of the gauge group, ${\mathscr G}_{{\mathcal H}_1}$, is a sub-algebra of ${\mathscr A}_{{\mathcal H}_1}$ studied by Araki and Wyss. It contains the density matrices of an important class of states, the {\em gauge invariant Gaussian states}, ${\mathfrak S}_{GIG}$.
Our focus is on semigroups $\{e^{t{\mathscr L}}\}_{t\geq 0}$ of quantum operations on ${\mathscr A}_{{\mathcal H}_1}$ that map ${\mathfrak S}_{GIG}$ into itself. Each $e^{t{\mathscr L}}$ is one-to-one, and our first main result is a structure theorem for such quantum operations on ${\mathscr G}_{{\mathcal H}_1}$ that map ${\mathfrak S}_{GIG}$ into itself.
We apply this to study semigroups of quantum operations on ${\mathscr G}_{{\mathcal H}_1}$ that map ${\mathfrak S}_{GIG}$ into itself. Our second main result is a structure theorem showing that they are parameterized by pairs $(G,A)$ where $G$ is a contraction semigroup generator on ${\mathcal H}_1$, and $0 \leq A \leq -G -G^*$. We then show that each of these semigroups has a natural extension to the full CAR algebra ${\mathscr A}_{{\mathcal H}_1}$. Further results are obtained under further assumptions on the pair $(G,A)$. - [390] arXiv:2605.03867 (replaced) [pdf, other]
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Title: Geometric Perspective on Concentration Phenomena in Frame TheoryComments: V2: Improved survey of related literatureSubjects: Functional Analysis (math.FA); Probability (math.PR)
Parseval and equal-norm frames play a fundamental role in frame theory and signal processing. It is known that a random frame, with unit vectors drawn independently from the uniform distribution on the sphere, will be nearly Parseval with high probability; asymptotic results go back at least to Goyal, Vetterli, and Thao and a non-asymptotic error bound was proved more recently by Kwok, Lau, and Ramachandran. In this work, we prove a dual result, which shows that random Parseval frames, with respect to the Haar measure, are nearly equal-norm with high probability. Our proofs are geometric in nature, and rely on general measure concentration principles in Riemannian manifolds. Using these techniques, we also give a novel probabilistic upper bound for the Paulsen problem.
- [391] arXiv:2605.06090 (replaced) [pdf, html, other]
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Title: A Sugawara--Legendre mechanism for the four-point Heisenberg algebraComments: 24 pagesSubjects: Representation Theory (math.RT)
This paper provides a representation-theoretic explanation for the recent discovery that families of orthogonal polynomials arise in the centers of universal central extensions of superelliptic affine Lie algebras. Working in the Verma modules of the Heisenberg subalgebra, we compute the canonical contravariant form in closed form on a distinguished family of weight vectors, and show that this family is the Legendre family, forced by the curve itself. The classical squared norms follow by a rescaling available under a positivity condition on the central charge, and the module is irreducible exactly when that charge is non-zero. The main result identifies a Sugawara element carried, by an explicit intertwiner, to the classical Legendre differential operator. The results are proved for the genus-zero, four-point case.
- [392] arXiv:2605.16004 (replaced) [pdf, html, other]
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Title: A proof of Esterle's conjecture on negative powers of Hilbert-space contractionsComments: 8 pagesSubjects: Functional Analysis (math.FA); Complex Variables (math.CV)
We establish the following result, confirming a conjecture of Jean Esterle. For each closed subset $E$ of the unit circle of Lebesgue measure zero, there exists a positive sequence $u_n\to\infty$ with the following property: if $T$ is a contraction on a Hilbert space such that $\sigma(T)\subset E$ and $\|T^{-n}\|=O(u_n)$ as $n\to\infty$, then $T$ is a unitary operator.
A key tool used in the proof is a result generalizing the well-known fact that closed subsets $E$ of the real axis of Lebesgue measure zero are removable for bounded holomorphic functions. We show that such sets remain removable even for certain unbounded holomorphic functions of moderate growth near $E$, where the notion of `moderate' depends on $E$. - [393] arXiv:2605.22706 (replaced) [pdf, html, other]
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Title: On the cohomological classification of vector bundles on smooth real affine surfaces and threefoldsComments: 19 pages. v2: added some material, reworked introduction and structureSubjects: Algebraic Geometry (math.AG); Commutative Algebra (math.AC); Algebraic Topology (math.AT); K-Theory and Homology (math.KT)
We study the cohomological classification of vector bundles on smooth real affine surfaces and threefolds. We show that, as was observed in joint work in A. Asok and J. Fasel and with S. Banerjee and J. Fasel, under suitable cohomological assumptions on the real locus of such varieties, this classification mirrors the one obtained on algebraically closed base fields by Mohan Kumar and Murthy and by Asok and Fasel. Using an argument due to Fasel, we also give an efficient proof of a theorem of Kucharz characterising the triples of algebraic cycles that can be realised as the Chern classes of a rank $3$ bundle on a smooth real affine threefold. We further answer the questions left open by Kucharz; to our knowledge, we give the first instance of a projective module over a smooth affine $\mathbb{R}$-algebra of dimension $3$ with trivial Chern classes which is not stably free.
- [394] arXiv:2605.22725 (replaced) [pdf, html, other]
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Title: Geometric fields, ranks, and generic derivationsComments: 23 pages, this version contains a faster proof of Theorem 2.5Subjects: Logic (math.LO)
In this note, we show various minimality results for a geometric theory of fields $T$: $T$ is stable if and only if it is strongly minimal, $T$ is simple if and only if it has SU-rank 1, and $T$ is rosy if and only if $T$ is surgical. Combining the first equivalence with an earlier result of Hrushovski, we deduce that algebraically bounded stable fields are precisely expansions of algebraically closed fields by constants.
We then consider algebraically bounded and o-minimal expansions of fields with generic derivations. We show that if $\mathbb{M}$ is a simple algebraically bounded structure and $\Delta$ is a generic tuple of derivations on $\mathbb{M}$, then $(\mathbb{M};\Delta)$ is supersimple if and only if the derivations commute. Similarly, if $\mathbb{M}$ is an o-minimal structure and $\Delta$ is a generic tuple of $T$-derivations on $\mathbb{M}$, then $(\mathbb{M};\Delta)$ is superrosy if and only if the derivations commute. We obtain explicit bounds on ranks using the Kolchin polynomial. - [395] arXiv:2606.01011 (replaced) [pdf, html, other]
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Title: Semiparametric Efficiency of Residual Correlation Testing under Gaussian Additive Noise ModelsSubjects: Statistics Theory (math.ST); Methodology (stat.ME)
This paper studies conditional independence testing under the Gaussian additive noise model (GANM), where two variables are modeled as nonlinear functions of covariates with independent bivariate Gaussian regression errors. Under this framework, conditional independence can be characterized by the correlation coefficient of the regression errors, which motivates a test based on the Pearson correlation coefficient computed from the fitted residuals. Despite its simple form, the asymptotic behavior and statistical efficiency of the resulting test have not been well understood. In this paper, we develop the semiparametric efficiency theory under GANM and show, surprisingly, that the efficient estimator coincides exactly with the ordinary residual Pearson correlation estimator. We further establish the asymptotic properties of the proposed test and develop the corresponding inference procedure. Simulation studies demonstrate that the proposed method achieves near-oracle efficiency and competitive empirical power while maintaining valid Type I error control. We further apply the proposed test to conditional dependence analysis of U.S. stock returns.
- [396] arXiv:2606.01554 (replaced) [pdf, html, other]
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Title: Fast Near-Optimal Estimation over Symmetric Norm BallsComments: fixed a bunch of typosSubjects: Statistics Theory (math.ST)
This short note proposes a polynomial-time algorithm for near-optimal Euclidean estimation of a signal constrained to lie in the unit ball of a symmetric norm, where the symmetry is with respect to a known basis and the norm is accessible through an evaluation oracle. We further extend the method to a random-design, moderate-dimensional linear regression setting, where the regression parameter is likewise assumed to belong to a constraint set defined by a symmetric norm.
- [397] arXiv:2606.02811 (replaced) [pdf, html, other]
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Title: Navier-Stokes Equations in Complex SpaceComments: Section 5 revisedSubjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph)
We prove global in time regularity of solutions of the Navier-Stokes equations defined in the complex space.
- [398] arXiv:2606.07885 (replaced) [pdf, html, other]
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Title: Weighted Recursions for the Smallest Parts FunctionComments: 8 pages, incorporate anonymous referee feedbackSubjects: Number Theory (math.NT)
We establish new polynomial-weighted recursions for Andrews' smallest parts function. Our results use the generating series for the spt function, a harmonic Maass form of weight 3/2, paired with the Dedekind eta function via the Rankin-Cohen bracket. In weights larger than 2, we find a nontrivial quasimodular component, which we determine for the relevant weights. We apply the holomorphic projection operator and the vanishing of cusp form spaces of small enough weight to obtain our results.
- [399] arXiv:2606.14336 (replaced) [pdf, html, other]
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Title: Ricci solitons from the perspectives of energy functionComments: 21 pages; The manuscript is modifiedSubjects: Differential Geometry (math.DG)
This article explores to what extent the geometry of gradient Ricci solitons extends to non-gradient Ricci solitons. The primary tool is the energy function $E$ of the soliton. We study consequences of various bounds on $E$. Under mild assumptions on the scalar curvature, we prove a weighted $L^1$-Liouville type theorem for both the usual Laplacian and the drifted Laplacian $\Delta_V$ associated to soliton vector field $V$, the former of which implies that Ricci solitons with bounded energy function have at most one nonparabolic end. Finally, we show that the measure $e^{-E} \mathrm{d} \mathrm{Vol}_{g}$ is finite for complete shrinking Ricci solitons, partially generalizing a result of Aaron Naber. As a consequence, non-gradient shrinking Ricci solitons also have finite fundamental groups, as in the gradient case.
- [400] arXiv:2606.14660 (replaced) [pdf, html, other]
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Title: Hilbert's basis theorem for Poisson Ore extensionsComments: 8 pages; added Remark 2 and Lemma 3 on localization, simplifying the Laurent case of Proposition 5 and the Laurent cases of Theorems 1 and 6, with some additional minor corrections and clarificationsSubjects: Rings and Algebras (math.RA); Commutative Algebra (math.AC); Quantum Algebra (math.QA)
We prove an analogue of Hilbert's basis theorem for Poisson Ore extensions and Poisson Laurent Ore extensions. We also obtain corresponding results for iterated Poisson Ore extensions and iterated Poisson Laurent Ore extensions associated to commuting Poisson-pairs. Finally, we give examples of Poisson Ore extensions that are Poisson-Noetherian without being Noetherian as ordinary algebras.
- [401] arXiv:2606.18407 (replaced) [pdf, html, other]
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Title: Determination of separable perturbations of an unbounded potential in the two-dimensional Schrödinger equationSubjects: Analysis of PDEs (math.AP)
We establish uniqueness and stability results for a class of perturbations of an unbounded potential in the two-dimensional Schrödinger equation, from the corresponding Dirichlet-to-Neumann map. We assume that the difference between the potentials has a separated product structure. Our proof relies on a specific Carleman inequality.
- [402] arXiv:2606.23398 (replaced) [pdf, html, other]
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Title: A first-exit proof of Cusick's sum-of-digits conjectureComments: Added and clarified references to prior results; no change to the main results or proofsSubjects: Number Theory (math.NT); Combinatorics (math.CO)
We prove Cusick's conjecture on the binary sum-of-digits function. More precisely, for every integer \(t\ge 1\) we show that \[ c_t:=\lim_{N\to\infty}\frac{1}{N} \#\{0\le n<N:\ s_2(n+t)\ge s_2(n)\}>\frac{1}{2}, \] and in fact obtain the explicit bound \[ c_t\ge \frac{1}{2}+2^{-2s_2(t)-1}, \] where \(s_2(m)\) denotes the number of ones in the binary expansion of \(m\). The proof is based on an exact deconvolution which replaces the distribution of \(s_2(n+t)-s_2(n)\) by a finite stopped random-walk law. The required bias is then proved through first-exit medians for principal subsequence ideals.
- [403] arXiv:2606.24555 (replaced) [pdf, html, other]
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Title: Nonexistence of finite-time blow-up for the equivariant harmonic map heat flow from $B^2$ to $S^2$Comments: Result was improved from D $\geq 3$ to $D \geq 2$. Several mistakes in the proof were also fixedSubjects: Analysis of PDEs (math.AP)
We consider $D$-equivariant solutions to the harmonic map heat flow from $B^2$ to $S^2$ under general time-dependent smooth boundary data and prove that there is no finite-time blow-up when $D \geq 2$.
- [404] arXiv:2606.24782 (replaced) [pdf, html, other]
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Title: A new perspective in linear Cauchy Elasticity: variational minimum principles for statics, dynamics, and heterogeneous materialsComments: This preprint was suggested as a submissiion for math.AP by arxiv - there is no separate category on arxiv for theoretical Solid MechanicsSubjects: Analysis of PDEs (math.AP); Optimization and Control (math.OC); Classical Physics (physics.class-ph)
A variational minimum principle for linear elastodynamics of a possibly heterogeneous material without a stored energy function is developed. It involves a change of variables to dual fields, and results in a degenerate elliptic Euler-Lagrange system, even when the primal elastodynamics is hyperbolic. Uniqueness assertions for the dual dynamic and static problems and implications of the degenerate ellipticity are sketched. Some implications pertaining to heterogeneous materials and ones with indefinite elastic moduli are discussed.
- [405] arXiv:2606.27063 (replaced) [pdf, html, other]
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Title: Brick infinite algebras admit infinitely many non-$τ$-rigid bricksComments: 11 pages. Added Example 4.1 for v2Subjects: Representation Theory (math.RT)
For finite dimensional algebras over algebraically closed fields, we settle a question previously known only for certain families of algebras. More specifically, motivated by some foundational interactions between bricks and $\tau$-rigid modules, we show that a given algebra is brick infinite if and only if it admits infinitely many bricks which are not $\tau$-rigid. This proves the $\tau$-analogue of an open conjecture asserting that if (almost) all bricks over an algebra $A$ are rigid, then $A$ should be brick-finite. In retrospect, we strengthen some recent contributions to the study of a series of challenging open problems related to the $2$nd brick-Brauer-Thrall conjecture. Moreover, motivated by some of our arguments, we pose the question whether there exists any algebra that admits an infinite semibrick consisting of Ext-orthogonal rigid bricks. In connection with this, we present an algebra $A$ of rank $n$ that admits a semibrick of cardinality strictly greater than $n$ consisting of Ext-orthogonal rigid bricks.
- [406] arXiv:2606.29615 (replaced) [pdf, html, other]
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Title: A Characterization of the Cumulants as Continuous Moment-Based StatisticsComments: 9 pagesSubjects: Probability (math.PR)
Cumulants are classical statistics associated with a random variable, defined as polynomial functions of its moments and distinguished by their additivity under convolution of distributions. Statistic is the name given to a function of a random variable, and a moment-based statistic is one that depends only on the moments $(\E[X^n])_{n\in \N}$. We prove a converse: any statistic depending continuously on finitely many moments and additive for independent sums must be a linear combination of cumulants.
The proof uses an algebraic reformulation of the problem via the Hurwitz product and a linearizing change of coordinates. This result also follows from the more general theorem of Mattner \cite{mattner}, but our approach is elementary and self-contained. - [407] arXiv:2606.30416 (replaced) [pdf, html, other]
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Title: Perfect closure detects injective dimensionSubjects: Commutative Algebra (math.AC)
Let $R$ be a noetherian local ring of prime characteristic $p$, and let $R^\infty$ denote its perfect closure. We prove that a finitely generated R-module $N$ has finite injective dimension if and only if $\operatorname{Ext}_R^i(R^\infty, N) = 0$ for all $i > 0$. As a Gorenstein counterpart, we show that finiteness of the Gorenstein injective (or projective) dimension of $R^\infty$ forces $R$ to be Gorenstein, and we relate this to a non-noetherian Cohen factorization of $R \to R^\infty$. Applications include preservation of Gorensteinness under weakly etale extensions, structural results on F-coherent and weakly F-nilpotent rings, along with the ascent of Frobenius closure of a parameter ideal to its powers.
- [408] arXiv:2607.01985 (replaced) [pdf, html, other]
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Title: A nine-line counterexample to a conjecture on the minimal degree of Jacobian relationsComments: 7 pages, version 1.1Subjects: Algebraic Geometry (math.AG); Commutative Algebra (math.AC); Combinatorics (math.CO)
We construct two arrangements of nine lines in the complex projective plane with isomorphic intersection lattices but with different minimal degrees of Jacobian relations. The common weak combinatorics is \[ (n_2,n_3,n_4)=(9,7,1), \] so the example is not the classical Ziegler-Yuzvinsky pair, whose weak combinatorics is $(n_{2},n_{3}) = (18,6)$. For the two defining equations $f$ and $g$ we prove \[ {\rm mdr}(f)=4,\qquad {\rm mdr}(g)=5. \] Since the degree is $d=9$, the first equality gives ${\rm mdr}(f)<d/2$. Hence the pair gives a counterexample to the Generalized Terao Conjecture.
- [409] arXiv:2607.04859 (replaced) [pdf, html, other]
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Title: Euclidean $\vee$-systems and real PK arrangementsComments: 30 pages. Added classification of rank-$3$ irreducible $\vee$-systems whose vectors have equal lengthSubjects: Differential Geometry (math.DG); Mathematical Physics (math-ph); Combinatorics (math.CO)
We establish a correspondence between two structures arising in the geometry of hyperplane arrangements: Euclidean $\vee$-systems and real polyhedral Kähler (PK) arrangements. We prove that every irreducible Euclidean $\vee$-system determines a real PK arrangement, and conversely that every real PK arrangement arises this way. As a consequence, we show that, up to equivalence, there are exactly three irreducible rank-three Euclidean $\vee$-systems whose vectors have equal length; their arrangements are the mirrors of the reflection groups of the regular tetrahedron, cube, and icosahedron. The correspondence also yields a description of the moduli space of Euclidean $\vee$-systems in a fixed projective class: it is homeomorphic to the relative interior of a polytope. We also give a direct proof that the hyperplane arrangement associated with a Euclidean $\vee$-system is simplicial. Among the currently known simplicial line arrangements, we identify precisely those that arise from $\vee$-systems. As a consequence, we prove that the Schreiber--Veselov catalog is complete for irreducible rank-three Euclidean $\vee$-systems with at most $27$ vectors.
- [410] arXiv:2607.04964 (replaced) [pdf, html, other]
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Title: Squarefree Powers of Edge Ideals under Graph Joins and ConesSubjects: Commutative Algebra (math.AC); Combinatorics (math.CO)
For $q\ge 1$, the $q$-th squarefree power $I(G)^{[q]}$ of the edge ideal of a graph $G$ is generated by the squarefree monomials supported on $q$-matchings of $G$; it is the Stanley--Reisner ideal of the complex $\Delta_q(G)=\{F\subseteq V(G):\nu(G[F])<q\}$, where $\nu$ denotes matching number. We prove a general formula for the matching number of an arbitrary graph join, \[ \nu(G\ast H) = \min\Big(\nu(G)+|V(H)|,\ \ \nu(H)+|V(G)|,\ \ \Big\lfloor\tfrac{|V(G)|+|V(H)|}{2}\Big\rfloor\Big), \] via the Tutte--Berge formula, and use it to decompose $\Delta_q(G\ast H)$ for arbitrary graphs $G,H$. Specializing to the wheel graph $\mathcal{W}_n = \mathcal{C}_n\ast\mathcal K_1$, we determine the Krull dimension and height of $R/I(\mathcal{W}_n)^{[q]}$ exactly for all $n\ge 3$, $1\le q\le\lfloor n/2\rfloor$, and -- combining our matching-number computations with a recent Tutte-type Cohen-Macaulayness criterion of Ficarra and Moradi -- prove that at the \emph{top} squarefree power $q=\nu(\mathcal{W}_n)=\lceil n/2\rceil$, the ideal $I(\mathcal{W}_n)^{[\nu(\mathcal{W}_n)]}$ is literally the squarefree Veronese ideal, so that $R/I(\mathcal{W}_n)^{[\nu(\mathcal{W}_n)]}$ is Cohen-Macaulay with \[ {\rm dim} = {\rm depth} = {\rm reg}\big(R/I(\mathcal{W}_n)^{[\nu(\mathcal{W}_n)]}\big) = 2\Big\lceil\frac n2\Big\rceil-1. \] This resolves all four classical invariants at the top power, and confirms there the pattern depth$(R/I(\mathcal{W}_n)^{[q]}) = 2q-1$ that our computational data (now extended to $n\le13$, every valid $q$) suggests holds throughout. We prove a general depth formula for squarefree powers of cone graphs, via a Betti-splitting exact sequence, that reduces this pattern to two more tractable statements about the underlying cycle alone; both are verified computationally in every case checked but left open in general.
- [411] arXiv:2607.11547 (replaced) [pdf, html, other]
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Title: On The Eigenvalue Rigidity of the Laguerre Unitary EnsembleComments: 40 pages,9 figuresSubjects: Mathematical Physics (math-ph); Probability (math.PR)
In this paper, we establish an optimal global rigidity estimate for the eigenvalues of the Laguerre unitary ensemble. Using the central limit theorem, we first construct a random measure via the eigenvalue counting function and then prove its convergence to a Gaussian multiplicative chaos measure, which yields the desired rigidity result. To prove this convergence, we apply a sufficient condition due to Claeys et al.[7] and carry out an asymptotic analysis of the corresponding exponential moments.
- [412] arXiv:2607.12183 (replaced) [pdf, html, other]
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Title: A Lefschetz type homomorphism for coincidence of several mapsSubjects: Algebraic Topology (math.AT)
Given $p$-maps $f_1, \cdots, f_p : X \to M,$ $p \geq 2,$ from an arbitrary topological space to an orientable closed connected $m$-manifold, in this paper we define a graded homomorphism $\Lambda_{f_1 \cdots f_p}: H(X) \to H(M^{p-1})$ of degree $-m(p-1)$ called by Lefschetz homomorphism. If the Lefschetz homomorphism is nontrivial then there is a point $x \in X$ such that $f_1(x) = \cdots = f_p(x).$ The Lefschetz homomorphism $\Lambda_{f_1 \cdots f_p}$ can be represented as a Knill-like trace.
- [413] arXiv:2607.14087 (replaced) [pdf, html, other]
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Title: Stochastic Domination of Gaussian Maxima: A Resolution of the Weak Simplex ConjectureComments: 41 pagesSubjects: Probability (math.PR); Information Theory (cs.IT); Metric Geometry (math.MG)
Let $R$ be an $m\times m$ correlation matrix satisfying $R-\mathbf{1}\mathbf{1}^{\mathsf T}/m\succeq0$, let $X\sim\mathcal{N}(0,R)$, and let $Z_1,\ldots,Z_m$ be independent standard Gaussian random variables. We prove $\max_i X_i\leq_{\mathrm{st}}\max_i Z_i$, with equality in distribution if and only if $R=I_m$. We use this comparison to resolve the Weak Simplex Conjecture: among $d+1$ equiprobable equal-energy signals in $\mathbb{R}^d$ transmitted over an additive white Gaussian noise channel, the regular simplex is the unique maximizer of the average probability of correct maximum-likelihood decoding at every signal-to-noise ratio. The same comparison proves the Simplex Mean Width Conjecture and gives the exact finite-energy performance of deterministic no-feedback AWGN codes with equiprobable messages, no restriction on the number of channel uses, and a maximal per-codeword energy constraint.
The proof uses a Gaussian product inequality for log-concave functions whose first moments with respect to standard Gaussian measure vanish. A variational argument chooses one exponential tilt and one truncation endpoint in each coordinate so that this product inequality applies and a Gaussian change of measure returns all coordinates to the prescribed common threshold. A strict form of the product inequality also shows that, unless $R=I_m$, $\mathbb{P}\{X\leq c\mathbf{1}\}>\Phi(c)^m$ for every finite $c$, and hence gives the distributional equality statement. A Lean formalization is available at this https URL. - [414] arXiv:2607.19001 (replaced) [pdf, html, other]
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Title: Minkowski geometry of finite Hurwitz continued fractionsSubjects: Classical Analysis and ODEs (math.CA); Number Theory (math.NT)
We study the Minkowski geometry of finite-level sets of Gaussian rationals defined by the lengths of their Hurwitz continued fraction expansions. For each $m\geq 1$, let $H_m$ be the set of points in the fundamental square whose Hurwitz continued fraction expansions have length exactly $m$. We also introduce the relaxed recursive sets defined by $G_0=\{0\}$ and $$G_m=\Big\{\frac{1}{u+v}: u \in\mathbb{Z}[i],\ v\in G_{m-1},\ |u+v|>1 \Big\}.$$ We prove that for every $m\geq 1$, $$\dim_{\rm M} H_m=\dim_{\rm M} G_m=1.$$ We further determine the critical one-dimensional Minkowski content of these sets. We have ${\mathcal M}^1(H_1)={\mathcal M}^1(G_1)=4\pi\log(1+\sqrt{2})$, whereas ${\mathcal M}^1(H_m)={\mathcal M}^1(G_m)=\infty$ for every $m\geq 2$.
- [415] arXiv:2607.19068 (replaced) [pdf, html, other]
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Title: Secure OFDM-IM ISAC With Artificial-Noise-Aided Index DeceptionComments: 5 pages, 4 figures, VTC 2027Subjects: Information Theory (cs.IT); Emerging Technologies (cs.ET)
We propose an artificial noise (AN)-aided, secure OFDM with index modulation (OFDM-IM) framework for integrated sensing and communication (ISAC). Consider a security-critical scenario, where the sensing target is also a potential eavesdropper. Instead of suppressing the signal power toward the target, the proposed method injects AN into inactive OFDM-IM subcarriers. The precoder spatially nulls the AN at the legitimate receiver, while projecting active-subcarrier energy levels toward the target to deceive its index detector. Since the monostatic ISAC transmitter knows the AN waveform, the same inactive subcarriers also contribute to sensing. We formulate this scenario as an optimization problem that maximizes transmit power toward the target while providing secure communication with a legitimate receiver. Numerical results show that the proposed scheme improves targeted transmit power compared to classical OFDM.
- [416] arXiv:2607.19554 (replaced) [pdf, html, other]
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Title: On the Ramsey number of a graph obtained by attaching a pendant to a path with length 1 modulo 4Comments: Several typos have been correctedSubjects: Combinatorics (math.CO)
In this paper, for odd $n$, we consider the graph $T_n$ with vertex set $\{x_1,....,x_n,x_{n+1}\}$ and edge set $\{x_{1}x_{2},x_{2}x_{3},...,x_{n-1}x_{n}\} \cup \{x_{\frac{n+1}{2}}x_{n+1}\}$ and prove that the Ramsey number $R(T_n,T_n)$ is equal to $\frac{3n+1}{2}$ for $n \in [1]_{4}$.
- [417] arXiv:2607.23585 (replaced) [pdf, html, other]
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Title: The one-period Kyle model has one equilibriumComments: 26 pages. v4: Stylistic changes, typographic fixes, no change in theorem or proofs. Title shorter to fit journal styleSubjects: Probability (math.PR); Econometrics (econ.EM); Theoretical Economics (econ.TH); Functional Analysis (math.FA)
Let $V$ and $U$ be independent standard normal random variables. For each Borel-measurable function $\phi\colon\mathbb{R}\to\mathbb{R}$, let $P_\phi\colon\mathbb{R}\to\mathbb{R}$ be a Borel version of the inverse regression $y\mapsto\mathbb{E}[V\mid\phi(V)+U=y]$. We prove that $\phi(v)\in\operatorname*{arg\,max}_{x\in\mathbb{R}} \mathbb{E}[(v-P_\phi(x+U))x]$ for every $v\in\mathbb{R}$ if and only if $\phi=\operatorname{id}_{\mathbb{R}}$.
The rigidity result implies that the one-period Gaussian Kyle (1985) insider-trading model has a unique Borel-measurable equilibrium strategy, namely Kyle's affine strategy. Building on preliminary results of McLennan, Monteiro, and Tourky (2017), the proof establishes that, at equilibrium, the total expected loss of noise traders attains a sharp universal upper bound and that any strategy attaining this bound must be affine almost everywhere. - [418] arXiv:2607.24666 (replaced) [pdf, html, other]
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Title: Global $W^{2,p}$ Regularity in Optimal TransportSubjects: Analysis of PDEs (math.AP)
In this paper we establish global $W^{2,p}$ estimates for the convex potentials of quadratic optimal transport between bounded convex domains with continuous positive densities. All the assumptions are optimal. The main new ideas include a blow-up analysis that allows for lower-dimensional collapse of the limiting source measure and reduces the limiting problem to a transport problem on its affine hull, and a good-bad scale decomposition in which rigidity controls the good scales while a counting argument shows that the proportion of bad scales tends to zero.
- [419] arXiv:2607.26271 (replaced) [pdf, html, other]
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Title: Sharp bounds for the fractional chromatic number of high-girth $d$-degenerate graphsComments: 18 pages plus referencesSubjects: Combinatorics (math.CO); Discrete Mathematics (cs.DM)
Martinsson and Steiner recently proved that the fractional chromatic number of any $d$-degenerate triangle-free graph $G$ satisfies $\chi_f(G) = O\left(\frac{d}{\log d}\right)$. They further conjectured a sharp leading constant $1 + o(1)$. In this paper, we confirm their upper bound conjecture for graphs having girth at least $5$. Our proof is constructive: it gives an efficient randomized algorithm that, with high probability, computes a fractional coloring of weight at most $(1 + o(1))\frac{d}{\log d}$ in such graphs.
Furthermore, we establish their conjectured lower bound in a stronger form: for any constant $g \ge 4$, there exist $d$-degenerate graphs having girth at least $g$ with $\chi_f(G) \ge (1 - o(1))\frac{d}{\log d}$. This lower bound is achieved by analyzing a random graph based on the uniform attachment model. Notably, our results reveal that this model lacks the typical computational complexity barriers found in Erdős-Rényi graphs, where there is a conjectured factor-$2$ algorithmic gap for this problem. - [420] arXiv:2608.00020 (replaced) [pdf, html, other]
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Title: Quaternionic Kittaneh Numerical Radius InequalitiesComments: 7 Pages, 0 Figures, Quaternionic Crouzeix Problem addedSubjects: Functional Analysis (math.FA)
We show that for certain classes of matrices over quaternions, a more restrictive version of breakthrough numerical radius inequalities obtained by Kittaneh [\textit{{Stud. Math.}, 2005}] hold.
- [421] arXiv:2608.00197 (replaced) [pdf, other]
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Title: Inertial manifolds for the nonlocal parabolic problemSubjects: Analysis of PDEs (math.AP); Dynamical Systems (math.DS); Functional Analysis (math.FA)
This paper provides an abstract framework for studying inertial manifolds associated with a class of nonlocal parabolic problems. In particular, by suitably modifying the nonlocal term outside the absorbing ball and changing the scale of time, we derive a corresponding spectral gap condition. As applications, we establish the existence of inertial manifolds for two classes of two-dimensional modified nonlocal parabolic equations on a square domain, whose diffusion coefficients depend on the $L^2$-norm of the solution and of its gradient, respectively.
- [422] arXiv:2608.00781 (replaced) [pdf, html, other]
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Title: Bessel-Like Multiple Orthogonal Polynomials of Mixed TypeSubjects: Classical Analysis and ODEs (math.CA); Mathematical Physics (math-ph)
This article constructs a Bessel-like family of mixed-type multiple orthogonal polynomials for a $q\times p$ matrix weight on the unit circle. Unlike a rank-one product weight, this matrix has generic rank $\min\{q,p\}$ outside a finite subset of the circle. Its reciprocal-Gamma moments recover the multiple Bessel system when $q=1$ and the Bessel-like system of Wolfs when $p=1$. The same matrix is obtained as a scaled Markov-Stieltjes limit of a rank-one Jacobi-like system, although the interval measures themselves have no finite limit. For balanced near-diagonal indices, explicit formulas are obtained for the mixed $A$ and $B$ polynomial vectors. Their orthogonality and weak normality are proved, and componentwise strong normality is characterized. The components have terminating generalized hypergeometric representations; the $B$ components also admit finite Kampé de Fériet representations and a matrix Rodrigues-type formula. In the one-row reduction, the bivariate representation becomes a generalized hypergeometric polynomial governed by a reflected type-II multiple Hahn polynomial. Finite Gamma-Pochhammer formulas give the near-diagonal and step-line recurrence coefficients. The corresponding banded recurrence matrix has a bidiagonal Christoffel factorization: the lower factors are evaluated from transformed polynomial vectors, while the upper factors are expressed through finite tau determinants. When $q=1$, every Christoffel step remains within the multiple Bessel family, and Gamma-Vandermonde determinants yield the complete factorization.
- [423] arXiv:2608.00788 (replaced) [pdf, html, other]
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Title: Positive Bidiagonal Factorizations for Banded Markov ProcessesSubjects: Classical Analysis and ODEs (math.CA); Mathematical Physics (math-ph); Probability (math.PR); Spectral Theory (math.SP)
An ordered positive bidiagonal factorization (PBF) is used to develop a spectral and probabilistic theory for Markov transition matrices of arbitrary finite bandwidth. The factors determine two families of mixed-type multiple orthogonal polynomials, an entrywise positive $q\times p$ matrix of measures, and a sequence of elementary death-or-stay and birth-or-stay transitions. This yields Karlin-McGregor formulas for transition probabilities, Green kernels, resolvents, potentials, and first-passage transforms without reversibility or block symmetrizability. Rational stochastic PBFs are characterized by ordered finite-urn experiments. Cyclic reorderings give Darboux intertwinings, while a factor-resolved continued fraction gives the first-return law. Grouping states produces a finite-phase quasi-birth-and-death process and a matrix continued fraction for return time and phase. For bounded continuous-time generators, uniformization preserves the spectral data. For unbounded rates, a conservative generator whose shifted leading truncations all admit scalar PBFs must be tridiagonal; nevertheless, $Q=V(T-I)$ preserves arbitrary fixed bandwidth and separates the embedded chain from the holding rates. The theory is explicit for mixed Piñeiro and Jacobi-like systems. For Piñeiro, the exact PBF region and larger structural-band positivity regions are obtained. Jacobi-like beta-convolution weights admit Gamma-factor cancellations, complete cancellation recovering Piñeiro. For $q\in\{2,3,4\}$, the strict ordering conditions give the only open PBF region, with further lower-dimensional cancellation strata. Rational $(3,2)$ examples provide all factors and the resulting Markov models.
- [424] arXiv:2608.03956 (replaced) [pdf, html, other]
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Title: From Triangular Array Progression to Near-Log-Concave State TransitionsSubjects: Probability (math.PR)
The paper introduces an infinite integer sequence progression that produces a triangular array where the $n$th row sums to $2^n$ for every $n\geq0$. Every row of the triangular array can be realized as the degree sequence of a multigraph without loops. Although the first four rows coincide with those from Pascal's triangle, the proposed array progression becomes asymmetric and diverges from binomial coefficients thereafter. The triangular array hosts infinitely many unimodal near-log-concave sequences with no zeros, and its log-concavity deviation converges to $\log(4/3)$ under logarithmic scaling. The progression introduces near-log-concave random variables, produces state-transitions of an infinite Markov chain, and hosts a totally non-negative Toeplitz matrix of order two. The construct provides a sequence of probability mass functions where the mean and variance decouple, with mean diverging and variance asymptotically approaching a steady-state limit. We study the Shannon entropy dynamics of the construct, and provide a transformation of the triangular array over the Tychonoff cube to produce an infinite family of right-stochastic matrices.
- [425] arXiv:2608.04845 (replaced) [pdf, html, other]
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Title: Entropy Transference for Rainbow-$H$-Free Colourings of Random GraphsSubjects: Combinatorics (math.CO)
Let $H$ be a fixed graph with $e(H)\ge3$ that contains two adjacent edges, and let $\ell\ge e(H)$ be fixed. We establish an entropy-transference principle for rainbow-$H$-free edge-colourings of the binomial random graph at the natural scale $p=n^{-1/m_2(H)}$. Writing $R_{H,\ell}(G)$ for the number of such colourings and $\lambda(H,\ell)$ for the rainbow entropy--Turán density on complete graphs, we show that, with high probability, the per-edge logarithmic counting rate can be made arbitrarily close to $\log\ell$ below a sufficiently small constant multiple of this scale, and arbitrarily close to $\lambda(H,\ell)$ above a sufficiently large constant multiple. Thus the dense-side counting rate on a sparse random host is governed exactly by a deterministic entropy--Turán parameter on complete graphs. We further investigate this parameter, obtaining partial exact evaluations, corresponding counting-stability results, and its first-order asymptotic behaviour as the number of colours tends to infinity. This extends the random Gallai-colouring transition from triangles to every fixed non-matching graph containing at least three edges, and provides a general mechanism for transferring complete-graph template entropy to sparse random hosts.
- [426] arXiv:2608.05621 (replaced) [pdf, html, other]
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Title: An affirmative solution to the generalized Busemann--Petty problem with subspace dimensions $2$ and $3$Comments: 12 pagesSubjects: Functional Analysis (math.FA); Metric Geometry (math.MG)
The generalized Busemann--Petty problem asks whether origin-symmetric convex bodies in $\mathbb{R}^n$ having larger volume of all $m$-dimensional sections necessarily have larger volume. When $m\geq 4$, this is known to be false, but the cases $m=2, 3$ for $n\geq 5$ have remained open since the 1990s. In this paper, we resolve these cases. Together with the known results, the generalized Busemann--Petty problem is completely solved: the answer is affirmative for $m=1, 2, 3$, and negative for $m\geq 4$.
- [427] arXiv:2608.06222 (replaced) [pdf, html, other]
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Title: Nonsofic wreath products of residually finite groupsComments: 16 pages, no figures; v2+v3 updated Theorem ASubjects: Group Theory (math.GR)
This work builds on the breakthrough of OpenAI in finding the first nonsofic group. We analyze the underlying proof mechanism and find further applications. Let $\Gamma<G$ be such that $\{g\in G:g\Gamma g^{-1}\leq\Gamma\}$ generates $G$ as a group, and suppose that both $\Gamma$ and $G$ have property $(T)$. If $\Gamma$ is not normal, then the generalized wreath product $\bigl(\bigoplus_{G/\Gamma}\mathbb Z/2\mathbb Z\bigr)\rtimes G$ and the group double $G \ast_{\Gamma} G$ are nonsofic. These hypotheses hold for explicit pairs of elementary groups over polynomial and Laurent polynomial rings, in which both groups are residually finite and Kazhdan.
- [428] arXiv:2608.06573 (replaced) [pdf, html, other]
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Title: Shannon Capacity and Related Graph Invariants for Lexicographic ProductsComments: Revised version with new technical results (Proposition 3, Remark 3, and Theorem 7)Subjects: Information Theory (cs.IT); Combinatorics (math.CO)
This paper studies the Shannon capacity of lexicographic products of finite confusability graphs, together with the Lovász theta function and the fractional Haemers number. The lexicographic product models zero-error communication in which one graph describes confusability between classes and the other describes confusability within each class. Elementary, self-contained proofs are provided for three known results: multiplicativity of the Lovász theta function and the fractional Haemers number under lexicographic products, and equality of the fractional and ordinary Lovász theta functions. Shannon capacity is proved to be supermultiplicative under lexicographic products in both orders, and these ordered products are compared with the strong product. We explicitly construct a countably infinite family of graphs and demonstrate strict supermultiplicativity of Shannon capacity under the lexicographic product whenever a family member is paired with its complement; the multiplicative gap can be arbitrarily large. These comparisons yield bounds and sufficient conditions for determining Shannon capacities exactly and establish the incomparability of the resulting upper bounds. The Shannon capacities of lexicographic products involving Kneser graphs, their complements, and their $q$-analogues are determined, and a complete outer factor is shown to preserve the inner factor's capacity. Capacities of iterated lexicographic powers are also determined, including powers of self-complementary graphs that are vertex-transitive or strongly regular. Finally, an open problem is posed.
- [429] arXiv:2608.06696 (replaced) [pdf, html, other]
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Title: Relative interval tilting, higher Auslander staircase corners and rational Dyck posetsComments: All comments are welcome!Subjects: Representation Theory (math.RT); Category Theory (math.CT)
We construct an explicit tilting equivalence between the incidence algebra of every rational Dyck staircase and a canonical idempotent corner of a higher Auslander algebra of type~$A$. In the coprime case, this corner identifies with the algebra $B_0$ introduced by Xing. The resulting Dyck-corner equivalence supplies the missing link in the previously known chain of equivalences and thereby proves the Chapoton-Ladkani-Rognerud conjecture for coprime positive integers. The Dyck-corner equivalence itself requires no coprimality hypothesis and is compatible with replicated algebras.
Our main tool is a linear-categorical extension of the interval-tilting mechanism of Chapoton-Ladkani-Rognerud. The relative theorem applies to finite $\kk$-linear categories under finite-global-dimension assumptions on the total category and its fibers. In contrast with the incidence-category setting, it allows arbitrary finite-dimensional $\Hom$ spaces and zero composites of nonzero morphisms, and it does not require the diagonal endomorphism algebras to be semisimple. The tilting object is constructed from exact right Kan extensions of fiberwise representables. We compute its opposite indexed endomorphism category, including all forced-zero compositions, and hence its opposite endomorphism algebra. Iterating this construction one coordinate at a time yields an explicit derived equivalence between the incidence algebra of every finite coordinate staircase and an idempotent corner of a higher Auslander algebra of type~$A$. We further realize the resulting staircase derived categories as triangulated subcategories generated by product Lagrangians in partially wrapped Fukaya categories of stopped-disk symmetric products and, in the coprime Dyck case, as Fukaya-Seidel categories of symmetric Brieskorn--Pham singularities. - [430] arXiv:2608.07388 (replaced) [pdf, html, other]
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Title: The Banach lattice Lean librarySubjects: Functional Analysis (math.FA)
We present a Lean 4 library for the theory of Banach lattices. Its purpose is to support the systematic formalization of contemporary research in Banach lattices and related areas. As evidence of this, we describe three research-level formalizations built using the library. Writing the library at scale was made possible by the use of LLMs with careful human supervision and planning. Unlike autoformalization, this approach allows for an actual understanding of the code. This, in turn, led to new mathematical insights that are also discussed. Judging by the interest expressed by other researchers, we expect the library to become a communal effort in the near future. For this reason, we also describe several parts of the theory that could be added next.
- [431] arXiv:2608.08039 (replaced) [pdf, html, other]
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Title: Shapes and Norms of Random PairsSubjects: Information Theory (cs.IT); Probability (math.PR)
The shape function of a pair of finite-valued random variables was introduced in arXiv:2606.23849, where it was used to derive a spectral bound on the entanglement of the pair, a quantity measuring the extent to which their mutual information can be extracted. In this article, we further develop the theory of shape functions for pairs of random variables. We prove that, when $(X,Y)$ is uniformly supported on the edges of a biregular bipartite graph, the value of the shape function $\mathcal{S}(X,Y)(\alpha,\beta)$ equals the logarithm of the operator norm of the graph's incidence matrix with respect to Lebesgue exponents determined by $(\alpha,\beta)$.This identification, in particular, enables the numerical approximation of the shape function and, by duality, of the extension profile, also known as the tension region, of the pair. We also establish a collection of relations and inequalities satisfied by shape functions, including convexity and monotonicity properties, composition inequalities, and relations describing their behavior under conditioning and the adjoining of variables.
- [432] arXiv:2608.08234 (replaced) [pdf, html, other]
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Title: Generalized Quantum Minors Generate Quantized Coordinate RingsComments: Some small changes were made, and some parts were rewrittenSubjects: Quantum Algebra (math.QA); Representation Theory (math.RT)
Let $G$ be a simply connected simple complex algebraic group. It is proved by Oya, Qin, and Yakimov that the quantized coordinate ring $\mathcal{O}_q(G)$ is generated by generalized quantum minors, and therefore carries a quantized cluster algebra structure, for all $G$ but type $F_4$. In this article, we settle the $F_4$ case by an argument uniform across $G_2$, $F_4$, and $E_8$. The main idea is to bootstrap the existing proof in type $E_8$, which relies on Lusztig's canonical basis of the quantum adjoint representation, and replace it with the crystal combinatorics of the quasi-minuscule representation. As a consequence, we prove that $\mathcal{O}_q(F_4)$ also has a quantized cluster algebra structure.
- [433] arXiv:2608.08461 (replaced) [pdf, html, other]
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Title: Isometries in the symmetrized bidisc, IISubjects: Complex Variables (math.CV)
We prove that every map $f:U\to\GG$ preserving the Poincaré distance and the Kobayashi distance is holomorphic or anti-holomorphic, where $U$ is a connected open subset of the unit disc. We also study the nonroyal automorphism orbits of $\GG$. A $C^1$ map on a connected relatively open part of such an orbit which preserves the restriction of the ambient Kobayashi--Royden metric is the restriction of a global automorphism or anti-automorphism of $\GG$. The same conclusion holds for maps preserving the ambient Kobayashi distance.
- [434] arXiv:2608.10742 (replaced) [pdf, html, other]
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Title: Finite monodromy groups in degenerating families and the stability degree of curvesComments: Revised version. Theorem 3.4 and its proof have been clarified. Several typos and notation issues have been corrected. The main results are unchangedSubjects: Algebraic Geometry (math.AG); Number Theory (math.NT)
We compute the stability degree of curves of genus g and show that it is equal to the classical Minkowsky bound. We also compute the stability degree of semi-abelian varieties with prescribed toric and abelian ranks. The proof uses a specialization result for finite monodromy groups in degenerating families, obtained from a variant of Krasner's lemma, together with an explicit construction at the prime 2.
- [435] arXiv:2608.11487 (replaced) [pdf, html, other]
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Title: Deletion-contraction properties of graphically stable spacesComments: Fixed font size and a few typos. Comments welcome!Subjects: Algebraic Geometry (math.AG); Combinatorics (math.CO)
Graphically stable spaces $\overline{\mathcal{M}}_{g,G}$ parametrize marked nodal curves whose permitted collisions of markings are determined by a graph. We study intersection numbers of $\psi$-classes on $\overline{\mathcal{M}}_{g,G}$, as well as the classes $[M_{g,G}]$ in the Grothendieck ring of varieties. In both settings, we show that the geometry is governed by an underlying graphical structure, expressed through deletion-contraction relations. As consequences, we derive string and dilaton equations and express several families of $\psi$-class integrals in terms of the chromatic polynomial. We also express the Grothendieck class of $M_{0,G}$ over an arbitrary field in terms of the chromatic polynomial and identify various Euler characteristics with combinatorial quantities. Along the way, we obtain a new formula for Crapo's $\beta$-invariant of graphs. Finally, we extend these relations to genus one and, under a chromatic condition, to higher genus.
- [436] arXiv:2608.12089 (replaced) [pdf, html, other]
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Title: Complexity rank one implies real rank zeroSubjects: Operator Algebras (math.OA)
We show that $C^*$-algebras with complexity rank one have real rank zero, as an application, the uniform Roe algebra $C_u^*|\mathbb{Z}|$ has real rank zero.
- [437] arXiv:2608.13302 (replaced) [pdf, html, other]
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Title: Input-to-state stability of chemical reaction networks with application to molecular computationComments: 28 pages, 3 figures; Revised version; an illustrative example has been added to the Appendix. The main theoretical results remain unchangedSubjects: Dynamical Systems (math.DS)
In biological reaction systems, reaction rates may vary over time due to environmental fluctuations, regulation, or coupling with other reaction modules. Input-to-state stability (ISS) provides a useful tool for analyzing the robustness of time-varying chemical reaction networks (CRNs). Existing ISS results for CRNs typically rely on restrictive structural assumptions, such as zero deficiency, a single linkage class, or weak reversibility. This paper makes two main contributions. First, we establish ISS for a broader class of weakly reversible CRNs, allowing nonzero deficiency and multiple linkage classes. Second, we extend the analysis to certain CRNs that are not weakly reversible by using network transformation techniques (linear conjugacy and reconstruction). Together, these results enlarge the class of CRNs for which robustness under time-varying reaction-rate inputs can be certified. Since CRNs are a standard framework for biomolecular computation, our results further enable the stability analysis of parallel molecular computing systems, an important problem in biomolecular computation where multiple CRN-based computing modules operate simultaneously and perturb one another through time-varying effective reaction rates.
- [438] arXiv:2608.13397 (replaced) [pdf, html, other]
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Title: Polynomial gaps below linear growth for Kreiss bounded semigroups and operatorsComments: Substantially expanded version: we add the positive L^p and discrete cases and give a unified proof of the four results. Title and exposition revised accordingly. Added a new section on individually eventually positive Kreiss bounded C0-semigroups on Lp-spaces, establishing a non-quantitative polynomial gap below linear growth. 17 pagesSubjects: Functional Analysis (math.FA)
We prove that every Kreiss bounded $C_0$-semigroup $(T_t)_{t\geq0}$ on a Hilbert space satisfies \[ \|T_t\|\leq C(1+t)^{1-\varepsilon_K}, \qquad t\geq0, \] where $\varepsilon_K>0$ depends explicitly only on the Kreiss constant. The same conclusion is obtained for positive Kreiss bounded $C_0$-semigroups on $L^p$-spaces, $1<p<\infty$, and in discrete time for Kreiss bounded operators on Hilbert spaces and positive Kreiss bounded operators on $L^p$-spaces. Finally, we obtain a non-quantitative polynomial gap for individually eventually positive Kreiss bounded $C_0$-semigroups on $L^p$-spaces.
- [439] arXiv:2608.13637 (replaced) [pdf, html, other]
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Title: More than two thirds of the zeta zeros are simple and on the critical lineComments: 21 pages. Proof discovered autonomously by Claude (Anthropic); verified and communicated by the listed authors. See §1 for provenance. Lean formalization availableSubjects: Number Theory (math.NT)
We prove unconditionally that at least two thirds of the nontrivial zeros of the Riemann zeta function, counted with multiplicity, are simple and lie on the critical line, and that at least five sixths are distinct; the previous unconditional records are $\frac{5}{12}$ and $0.6603$. With the Montgomery--Taylor window the constants become $0.6725$ and $0.8362$. The argument makes Montgomery's 1973 deduction unconditional: the Riemann hypothesis, classically needed to read the zero side as a positive sum over real ordinates, is replaced by a rank-trace inequality applied to a finite compression of Weil's Hermitian form, with Sylvester's law of inertia handling off-line pairs. The analytic inputs are those of Aryan and of Baluyot, Goldston, Suriajaya and Turnage-Butterbaugh. The results extend to primitive Dirichlet $L$-functions and are formally verified in Lean 4.
- [440] arXiv:2608.13869 (replaced) [pdf, html, other]
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Title: Exactness and Drinfeld-Sokolov Realization of the McRae-Yang Tensor FunctorsSubjects: Representation Theory (math.RT)
Let $p,q\geq2$ be coprime, set $k=-2+p/q$, and let $c_{p,q}=1-6(p-q)^2/(pq)$. McRae and Yang constructed right exact braided tensor functors from the non-semisimple Kazhdan-Lusztig categories of $V^{-2+p/q}(\mathfrak{sl}_2)$ and $V^{-2+q/p}(\mathfrak{sl}_2)$ to the Virasoro category $\mathcal{O}_{c_{p,q}}$, and conjectured that these functors are exact and agree with the corresponding quantum Drinfeld-Sokolov reductions. We prove this conjecture. For the $(p,q)$ branch, we determine the images of all indecomposable projective objects without assuming exactness, using projective tensor-product recursions and exact generalized conformal-residue projections. Projective faithfulness is then combined with a one-row Virasoro extension analysis to identify the non-wall images and to prove full faithfulness on projectives. The Virasoro input is matched objectwise with Nakano's logarithmic extension theorem away from the vacuum edge; the exceptional vacuum edge is handled directly by the staggered-module theory of Kytölä-Ridout, including the higher prime singular-vector branch. Independently, principal Drinfeld-Sokolov reduction is shown to be exact and faithful on the whole finite-length affine category and to have the same Weyl, simple, and projective images. Compatibility with the non-standard affine twist identifies the canonical nilpotent endomorphisms on projectives and removes the remaining scalar ambiguity in the comparison of adjacent projective morphisms. Projective density for right exact functors then yields a natural isomorphism \[
F_{p,q}\cong
H^0_{DS,+}\big|_{{KL}^k(\mathfrak{sl}_2)}. \] The same argument after interchanging $p$ and $q$ identifies the second McRae-Yang functor with the transposed Drinfeld-Sokolov reduction. - [441] arXiv:2608.14453 (replaced) [pdf, html, other]
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Title: New Congruences Involving $p$-adic dual sequencesSubjects: Number Theory (math.NT); Combinatorics (math.CO)
Let $(a_n)_{n\geqslant 0}$ be a sequence of integers. Its dual sequence $(a_n^*)_{n\geqslant 0}$ is defined by \begin{equation*}
a_n^* := \sum_{k=0}^{n} \binom{n}{k}(-1)^k a_k. \end{equation*}
Let $p>3$ be a prime. In this paper we mainly investigate congruences modulo $p^2$ involving central binomial coefficients and $p$-adic dual sequences. For example, we prove that for any sequence $(a_k)_{k\ge0}$ of $p$-adic integers, \begin{align*} \sum^{(p-1)/2}_{k=0}\binom{2k}{k}^2\frac{a_{2k}}{16^k}\equiv\left( \frac{-1}{p}\right) \sum_{k=0}^{p-1}\frac{\mathcal{P}_{k}}{16 ^{k}}a_{k}^*\pmod{p^2}, \end{align*} where $(\mathcal{P}_n)_{n\ge0}$ are the Catalan--Larcombe--French numbers given by \begin{equation*}
\mathcal{P}_0=1,\quad \mathcal{P}_1=8, \quad n^2 \mathcal{P}_n = 8(3n^2-3n+1)\mathcal{P}_{n-1}-128(n-1)^2\mathcal{P}_{n-2}
\quad (n\ge2). \end{equation*}
We also establish a new formula for $\sum_{k=0}^{(p-1)/2}\binom{2k}{k}a_{2k}^*/4^k \pmod{p^2}$ and as a consequence we confirm some conjectures of Z.-W. Sun \cite{Sun2014CANT} on the generalized central trinomial coefficients $T_{2k}(b,c)$, i.e., the coefficient of $x^{2k}$ in $(x^2+bx+c)^{2k}$, where $b,c$ are integers. - [442] arXiv:2608.15020 (replaced) [pdf, html, other]
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Title: Monoidal seeds of the categories $\mathcal{C}_{wv}$ over quiver Hecke algebrasComments: 63 pagesSubjects: Representation Theory (math.RT); Quantum Algebra (math.QA)
In this paper, when the quiver Hecke algebra R is symmetric, we present a new construction of quantum monoidal seeds $ \mathscr{S}_{w,v}$ for $\mathcal{C}_{wv}$ using the reflection functors $\mathcal{F}_i$ and the newly introduced operators $\mathcal{K}_i$. The monoidal seed $ \mathscr{S}_{w,v}$ is obtained as a subseed of the monoidal seed of $\mathcal{C}_w$ constructed by applying $\mathcal{K}_i$ and $\mathcal{F}_i$ along the special KF sequence determined by a reduced expression of $w$ and $v$. We further prove that the monoidal seed $\mathscr{S}_{w,v}$ coincides with the set of all prime factors of the determinantial modules $M(w_{\le k } \Lambda_{i_k}, v_{\le k} \Lambda_{i_k} )$. We prove that the Grothendieck ring $K(\mathcal{C}_{wv}) $ lies between the cluster algebra and the upper cluster algebra.
- [443] arXiv:2608.15682 (replaced) [pdf, html, other]
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Title: Variations of colored multiset Eulerian polynomials and applicationsComments: 17 pagesSubjects: Combinatorics (math.CO)
Deligeorgaki, Han and Solus introduced colored multiset Eulerian polynomials, derived a generating function identity which generalizes MacMahon's identity, proved their self-interlacing under suitable parameter conditions, and identified these polynomials as the h^*-polynomials of direct products of dilated lattice simplices. In this paper, we introduce an ascent analogue of the colored multiset Eulerian polynomial, derive an explicit generating function identity for this polynomial, show that it is equal to the h^*-polynomial of a family of half-open lattice polytopes, and verify that this ascent polynomial also satisfies self-interlacing under the same parameter conditions. By establishing recurrence relations, we prove that both polynomials are real-rooted for all positive integer parameters. The obtained identities are further applied to interpret combinatorially the h^*-polynomials of Pitman--Stanley polytopes, composition polytopes and a family of reflexive lattice polytopes defined from preorders.
- [444] arXiv:2608.15724 (replaced) [pdf, html, other]
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Title: A method to identify the ordinary edges for symmetric traveling salesman problem based on frequency $K_i$sComments: 29 pages, 4 figuresSubjects: Combinatorics (math.CO); Discrete Mathematics (cs.DM)
The frequency $K_i$s ($i\in[4,n]$) are studied for symmetric traveling salesman problem ($TSP$) to characterize the structure properties of the edges inside and outside the optimal Hamiltonian cycle ($OHC$). Given a $K_i$ in $K_n$ where $i\in [4,n]$, the frequency $K_i$ is computed with the set of ${{i}\choose{2}}$ optimal $i$-vertex paths with fixed endpoints (optimal $i$-vertex paths) in the $K_i$. Given an $OHC$ edge in a $K_i$, it has a frequency bigger than $\frac{1}{2}{{i}\choose{2}}$ in the frequency $K_i$, and that of an ordinary edge outside the $OHC$ is smaller than $\frac{1}{2}{{i}\choose{2}}$. As the frequency of an edge is computed with the frequency $K_i$s, an $OHC$ edge of $K_n$ has an average frequency bigger than $\frac{1}{2}{{i}\choose{2}}$. It indicates an $OHC$ edge of $K_n$ is also one $OHC$ edge of a $K_i$ containing it. It also found that the probability that an $OHC$ edge has the frequency bigger than $\frac{1}{2}{{i}\choose{2}}$ increases according to $i\in [4, n]$ based on the frequency $K_i$s. For an ordinary edge outside the $OHC$, the probability that it has a frequency smaller than $\frac{1}{2}{{i}\choose{2}}$ increases according to $i$. Based on the findings, a method is given to identify the ordinary edges for $TSP$.
- [445] arXiv:2608.15808 (replaced) [pdf, html, other]
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Title: Uniform sine-kernel determinant asymptotics, tail-side quantiles, and prolate eigenvalue boundsComments: 69 pages, no figures. Revised and substantially expanded version. Adds tail-side quantile results and their lower-half, full-plunge, and tensor-product consequences; expands the signed sine-kernel determinant analysis; and clarifies the uniformity ranges and references. The main conclusions of the previous version are retainedSubjects: Functional Analysis (math.FA)
Let $S_c=P_{(0,c)}QP_{(0,c)}$ be the one-dimensional sinc-kernel concentration operator, let $N_a(c)=\#{n:\lambda_n(c)>a}$, and set $\bar L=\log((1-\delta)/\delta)$. We prove, uniformly for each fixed $A>0$, the tail-side quantile formula $N_\delta(c)=c+\pi^{-2}\bar L\log(4\pi^2c/\bar L)+O_A(\log c+\bar L)$ for $6\le\bar L\le A\log c$. It yields corresponding additive formulas for the lower half and full plunge, with main terms respectively $\pi^{-2}\bar L\log(4\pi^2c/\bar L)$ and twice this quantity. An exact one-tail-coordinate selection gives, for fixed $A>0$, $d\ge1$, and $q\in(1/2,1)$, the one-sided tensor-product bound $\Lambda_\delta(c;d)\ge\pi^{-2}d c^{d-1}\bar L\log(4\pi^2c/\bar L)-O_{A,d,q}(c^{d-1}(\log c+\bar L))$ for $L_{d,q}\le\bar L\le A\log c$, where $L_{d,q}=\log(q^{-(d-1)}(e^6+1)-1)$; the tensor content is nontrivial for $d\ge2$. The analytic input is a signed growing-parameter sine-kernel determinant asymptotic: uniformly for $0\le\omega\le A\log s$, $\log\det(I+(e^{2\omega}-1)K_s)=4\omega s/\pi+2\pi^{-2}\omega^2\log(4s)+2\log|G(1+i\omega/\pi)|^2+O_A((1+\omega)^4\log^2s/s)$, where $G$ is the Barnes $G$-function. We prove this negative-coupling counterpart of the Bothner--Deift--Its--Krasovsky theorem by direct IIKS steepest descent. We also retain the uniform head-side results and use a two-way determinant reduction to obtain the moving-depth lower-half bridge bound with constant $1/(32\pi^2)$; extending it to the deeper range uses Kulikov--Dam Larsen and may require a smaller constant. These counting formulas are additive. Their errors become uniformly relative when $\bar L$ tends uniformly to infinity; fixed thresholds are covered separately by Landau--Widom. A Lambert-$W_{-1}$ formula is recorded only for the continuous main term, not for individual eigenvalues.
- [446] arXiv:2608.15845 (replaced) [pdf, html, other]
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Title: Weighted singular vectors in common-base self-similar setsComments: 15 pages, 2 figures, 1 table. Corrected cross-reference labels for definitions, lemmas, and corollaries; no changes to mathematical resultsSubjects: Number Theory (math.NT)
We prove a lower bound for the Hausdorff dimension of weighted totally irrational singular vectors in affine-spanning common-base integral self-similar sets satisfying the open set condition. For the middle-third Cantor square, the bound improves the previously known explicit lower bound.
- [447] arXiv:2608.16378 (replaced) [pdf, html, other]
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Title: Mathematics in the age of the reproduction of intelligenceComments: 11 pages. In this revision, the essay has been trimmed and polishedSubjects: History and Overview (math.HO)
Drawing on Walter Benjamin's account of technological reproducibility and aura in relation to art, together with insights from Yehuda Rav and Ludwig Wittgenstein, this speculative essay reflects on the impact of AI systems on the practice of pure mathematics.
- [448] arXiv:2608.16383 (replaced) [pdf, html, other]
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Title: A Concavity Inequality for Hessian Quotient EquationsComments: v2. Added a remark on related work by Li and WuSubjects: Analysis of PDEs (math.AP)
We prove a concavity inequality for the Hessian quotient operator $\sigma_k/\sigma_{k-1}$ using a change of basis for symmetric polynomials. This removes an additional structural concavity assumption previously imposed in the interior $C^2$ estimate for convex solutions of the $\sigma_k/\sigma_{k-1}$ equation. The method also yields several useful inequalities for elementary symmetric polynomials.
- [449] arXiv:2608.17199 (replaced) [pdf, html, other]
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Title: Shifted second moment of Gaussian Hecke $L$-functions $L(s,λ^k)$Subjects: Number Theory (math.NT)
We establish a uniform asymptotic formula for the smoothly weighted shifted second moment of the Gaussian angular Hecke $L$-functions $L(s,\lambda^k)$. The completed moment is expressed as the sum of four explicit main terms, corresponding to the four functional-equation swaps, with an error of size $O_{\Phi,\epsilon}(K^{1/2+\epsilon})$.After the Archimedean factors are removed, the resulting formula agrees with the numerator-only specialization of the four-swap prediction of the $L$-functions Ratios Conjecture. The proof transforms the off-diagonal contribution into Weyl sums over the roots of $r^2\equiv-1\pmod C$, realizes these sums spectrally through incomplete Poincaré series evaluated at $i$, and separates the Eisenstein and Maaß spectra. The two $v$-type main terms arise respectively from the zero frequency and from the combined residues of two moving Eisenstein poles, while the cuspidal spectrum is absorbed into the square-root error term.
- [450] arXiv:2608.17924 (replaced) [pdf, html, other]
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Title: From complex-step differentiation to a general reconstruction frameworkSubjects: Numerical Analysis (math.NA); Geophysics (physics.geo-ph)
The complex-step method is traditionally derived from the Taylor expansion of an analytic function and is widely used as a numerical technique for derivative approximation. We present an alternative formulation based on the Cauchy--Riemann equations and show that the classical complex-step relation arises naturally from the harmonic structure of holomorphic functions. In particular, the complex-step method admits two complementary harmonic interpretations: as a Cauchy problem, in which the derivative is identified with the normal datum of the imaginary component on the real axis, and as a reconstruction problem in a strip, in which the finite imaginary perturbation provides the upper-boundary data. The latter formulation leads explicitly to the strip Poisson and conjugate Poisson kernels and their derivatives. A related harmonic reconstruction framework in the upper half-plane leads to the Poisson, conjugate Poisson, and Cauchy kernels as elementary reconstruction operators for harmonic and holomorphic functions. Extending this reconstruction from ordinary boundary functions to finite measures yields the classical Stieltjes transform and its inversion formula. The same measure-theoretic structure appears in spectral theory, where scalar matrix elements of the resolvent are Stieltjes transforms of the associated spectral measures. These results establish a common complex-analytic structure connecting complex-step differentiation, harmonic reconstruction, Stieltjes inversion, and spectral reconstruction, while distinguishing the boundary-value problems through which the corresponding information is recovered.
- [451] arXiv:2608.18024 (replaced) [pdf, html, other]
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Title: An equidistribution conjecture for quotient-closed and submodule-closed subcategoriesComments: 61 pages. v2: A link to the author's essay added only. This work is a collaboration between the author, Fable 5, and GPT-5.6 Sol. Comments welcome! Full Lean 4 formalization of this paper is available at this https URLSubjects: Representation Theory (math.RT); Combinatorics (math.CO); Rings and Algebras (math.RA)
We study subcategories of the module category of a finite-dimensional algebra that are closed under quotients or submodules. We propose the quotient--submodule equidistribution conjecture: over a representation-finite algebra, the number of quotient-closed subcategories of size $i$ is equal to that of submodule-closed subcategories of size $i$ for every $i$, where size is the number of indecomposable modules in the subcategory. We prove the following cases of the conjecture: (1) the five smallest and five largest values of $i$, hence all algebras with at most nine indecomposable modules; (2) Nakayama algebras; (3) algebras with radical square zero; and (4) representation-directed algebras. In the last case, we classify quotient-closed subcategories by a lower Bruhat interval in a Coxeter group constructed from the Auslander--Reiten quiver, extending the classification of Oppermann--Reiten--Thomas for Dynkin quivers. We also prove that quotient closure defines a finitary convex geometry and classify the functorially finite quotient-closed subcategories by Gen-minimal modules.
- [452] arXiv:2608.18236 (replaced) [pdf, html, other]
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Title: Exceptional eigenvalue density for thin groupsComments: 7 pages; comments welcomeSubjects: Spectral Theory (math.SP); Geometric Topology (math.GT)
We prove a limit multiplicity conjecture of Hee Oh for principal congruence covers of geometrically finite hyperbolic manifolds, with an explicit power-saving rate. The rate is governed by the return of Patterson--Sullivan shadows through a fixed compact core, giving a geometric interpretation to the exceptional eigenvalue density. For convex-cocompact groups, a packing argument for enlarged shadows gives a stronger rate.
- [453] arXiv:2608.18722 (replaced) [pdf, html, other]
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Title: Two-sided coherent algebras over any field with $\mathcal{PGF}(R)\subsetneq\mathcal{GP}(R)$Comments: The construction is extended from fields of characteristic (2) to arbitrary fields. A new corollary section collects consequences of the main theorem obtained from existing results in Gorenstein homological algebraSubjects: Rings and Algebras (math.RA)
For a ring $R$, let $\mathcal{GP}(R)$, $\mathcal{GF}(R)$, and $\mathcal{PGF}(R)$ denote the classes of Gorenstein projective, Gorenstein flat, and projectively coresolved Gorenstein flat left $R$-modules, respectively. We answer negatively the question whether $\mathcal{GP}(R)=\mathcal{PGF}(R)$ for every ring. More precisely, over every field $k$ we construct a left and right coherent central $k$-algebra $T$ and a strongly Gorenstein projective left $T$-module which is not Gorenstein flat; hence $\mathcal{PGF}(T)\subsetneq\mathcal{GP}(T)$.
- [454] arXiv:2608.18766 (replaced) [pdf, html, other]
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Title: A shifted energy barrier approach for phase-field modeling of tensile-dominated brittle fractureSubjects: Numerical Analysis (math.NA); Materials Science (cond-mat.mtrl-sci)
The classical AT1 phase-field model contains an intrinsic energy barrier for crack nucle ation, which makes the predicted strength depend on the fracture toughness and the regularization length. For tensile-dominated brittle fracture, this barrier is shifted by mapping the Rankine criterion, evaluated on the effective stress, onto a state-dependent active-energy threshold. The prescribed tensile strength then controls crack nucleation, while the AT1 crack-density functional, stiffness degradation, and degraded stress response remain unchanged. Since the threshold depends on the current stress state, the field equations are derived from a restricted variational principle. A microforce formulation identifies the barrier shift as a dissipative resistance and provides the corresponding lower bound on the regularization length. In one-dimensional tension, closed-form solutions recover the prescribed peak strength and give a cosine-type localization profile that ap proaches the classical AT1 profile as the shift vanishes. Numerical examples show that, within the admissible range, the nucleation load is nearly insensitive to the regularization length and the predicted multiaxial nucleation states follow the Rankine envelope. Under overall compression, crack nucleation remains associated with local tensile stress concen trations. The formulation also captures the transition from strength-controlled failure for small flaws to the LEFM limit for large cracks.
- [455] arXiv:2608.19189 (replaced) [pdf, html, other]
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Title: Entry growth in Gaussian eliminationSubjects: Numerical Analysis (math.NA)
Gaussian elimination is one of the oldest algorithms in mathematics, and the most popular method for solving an unstructured linear system. Its stability in finite precision is controlled by its growth factor, which measures how large the entries produced during elimination can become. Understanding the worst-case behavior of this quantity has been a central problem in numerical analysis since the 1940s.
Here we make a significant leap in that understanding, settling several open problems. In particular, we determine the asymptotic behavior of the maximum growth factor under complete and rook pivoting, proving that both are quasi-polynomial in dimension. We also show that the exponential growth under partial pivoting persists for sparse matrices and that randomized partial pivoting suffers the same instability. By contrast, we show that every matrix has a row permutation with polynomial growth, though finding the optimal row permutation is NP-hard. - [456] arXiv:2103.04021 (replaced) [pdf, html, other]
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Title: Asymptotic Theory for IV-Based Reinforcement Learning with Potential EndogeneitySubjects: Machine Learning (stat.ML); Machine Learning (cs.LG); Econometrics (econ.EM); Optimization and Control (math.OC)
In the standard data analysis framework, data is collected (once and for all), and then data analysis is carried out. However, with the advancement of digital technology, decision-makers constantly analyze past data and generate new data through their decisions. We model this as a Markov decision process and show that the dynamic interaction between data generation and data analysis leads to a new type of bias -- reinforcement bias -- that exacerbates the endogeneity problem in standard data analysis. We propose a class of instrument variable (IV)-based reinforcement learning (RL) algorithms to correct for the bias and establish their theoretical properties by incorporating them into a stochastic approximation (SA) framework. Our analysis accommodates iterate-dependent Markovian structures and, therefore, can be used to study RL algorithms with policy improvement. We also provide formulas for inference on optimal policies of the IV-RL algorithms. These formulas highlight how intertemporal dependency of the Markovian environment affects the inference.
- [457] arXiv:2202.08870 (replaced) [pdf, html, other]
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Title: A Linear-Time Algorithm for Product Structure in Planar GraphsComments: 21 pages, 8 figuresSubjects: Data Structures and Algorithms (cs.DS); Combinatorics (math.CO)
The \emph{Product Structure Theorem} for planar graphs (Dujmović et al.\ \emph{JACM}, \textbf{67}(4):22) states that any planar graph is contained in the strong product of a planar $3$-tree, a path, and a $3$-cycle. We give a simple linear-time algorithm for finding this partition as well as several related partitions. This improves on the previous $O(n\log n)$ time algorithm for finding this partition (Morin.\ \emph{Algorithmica}, \textbf{85}(5):1544--1558). The algorithm is practical; we provide open-source C and C++ implementations that, even on modest hardware, can process graphs with millions of vertices in a matter of seconds.
- [458] arXiv:2310.17641 (replaced) [pdf, html, other]
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Title: Criteria for Davies Irreducibility of Markovian Quantum DynamicsComments: 16 pages; minor improvements; published version; fixed a typo in abstract and introduction: "K=iH+\sum_a L^+_aL_a" -> "K=iH+\sum_a L^+_aL_a/2"Journal-ref: J. Phys. A: Math. Theor. 57, 115301 (2024)Subjects: Quantum Physics (quant-ph); Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph)
The dynamics of Markovian open quantum systems are described by Lindblad master equations, generating a quantum dynamical semigroup. An important concept for such systems is (Davies) irreducibility, i.e., the question whether there exist non-trivial invariant subspaces. Steady states of irreducible systems are unique and faithful, i.e., they have full rank. In the 1970s, Frigerio showed that a system is irreducible if the Lindblad operators span a self-adjoint set with trivial commutant. We discuss a more general and powerful algebraic criterion, showing that a system is irreducible if and only if the multiplicative algebra generated by the Lindblad operators $L_a$ and the operator $K=iH+\sum_a L^\dagger_aL_a/2$, involving the Hamiltonian $H$, is the entire operator space. Examples for two-level systems, show that a change of Hamiltonian terms as well as the addition or removal of dissipators can render a reducible system irreducible and vice versa. Examples for many-body systems show that a large class of spin chains can be rendered irreducible by dissipators on just one or two sites. Additionally, we discuss the decisive differences between (Davies) reducibility and Evans reducibility for quantum channels and dynamical semigroups which has lead to some confusion in the recent physics literature, especially, in the context of boundary-driven systems. We give a criterion for quantum reducibility in terms of associated classical Markov processes and, lastly, discuss the relation of the main result to the stabilization of pure states and argue that systems with local Lindblad operators cannot stabilize pure Fermi-sea states.
- [459] arXiv:2504.02256 (replaced) [pdf, html, other]
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Title: A direct algebraic proof for the non-positivity of Liouvillian spectral values and controlling the dissipative gap of systems with normal Lindblad operatorsComments: 19 pages, 2 figures; added argument for infinite-dimensional Hilbert spaces; added an application of the algebraic approach, showing how to impose a strict minimum for the dissipative gap of any Markovian spin-s or fermionic system with normal Lindblad operators by adding certain single-site dissipators; further minor improvementsSubjects: Quantum Physics (quant-ph); Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph)
Markovian open quantum systems are described by the Lindblad master equation $\partial_t\rho =\mathcal{L}(\rho)$, where $\rho$ denotes the system's density operator and $\mathcal{L}$ the Liouville super-operator, which is also known as the Liouvillian. For systems with a finite-dimensional Hilbert space, it is a fundamental property of the Liouvillian that the real parts of all its eigenvalues are non-positive. Analogously, for infinite-dimensional Hilbert spaces, the Liouvillian as a map on trace-class operators only has spectral values with non-positive real parts. The usual arguments for these properties are indirect, using that $\mathcal{L}$ generates a quantum channel and that quantum channels are contractive. We provide a direct algebraic proof based on the Lindblad form of Liouvillians. Subtleties for infinite-dimensional systems are highlighted. For example, unbounded operators can then be eigenvectors of the adjoint Liouvillian with positive eigenvalues, corresponding to a physical instability of the system. As an illustrative application of the algebraic approach, we prove that a strict minimum can be imposed on the dissipative gap of any Markovian spin-$s$ or fermionic many-particle system with normal Lindblad operators by adding simple local dissipators. The added dissipators comprise Pauli or generalized Pauli Lindblad operators for spin systems and Majorana Lindblad operators for fermionic systems.
- [460] arXiv:2506.09726 (replaced) [pdf, html, other]
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Title: Don't be Afraid of Cell Complexes! An Introduction from an Applied PerspectiveComments: 51 pages, 16 figures (plus appendix)Subjects: Signal Processing (eess.SP); Computational Geometry (cs.CG); Social and Information Networks (cs.SI); Algebraic Topology (math.AT)
Cell complexes (CCs) are a higher-order network model deeply rooted in algebraic topology that has gained interest in signal processing and network science recently. The processing of signals supported on CCs can be described in terms of easily-accessible algebraic or combinatorial notions. However, the commonly presented definition of CCs is grounded in abstract concepts from topology and remains disconnected from the signal processing methods developed for CCs. In this paper, we aim to bridge this gap by providing a simplified definition of CCs that is accessible to a wider audience and can be used in practical applications. Specifically, we first introduce a simplified notion of abstract regular cell complexes (ARCCs). These ARCCs only rely on notions from algebra and are equivalent to regular cell complexes for most practical applications. Second, using this new definition we provide an accessible introduction to (abstract) cell complexes from a perspective of network science and signal processing, including an introduction to central methods and recent developments. Furthermore, as many practical applications work with CCs of dimension 2 and below, we provide an even simpler definition for this case that significantly simplifies understanding and working with CCs in practice.
- [461] arXiv:2507.02082 (replaced) [pdf, html, other]
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Title: Adjusting Higher Chern-Simons TheoryComments: 48 pages, presentation improved, typos fixed, published versionJournal-ref: J. Math. Phys. 67, 082302 (2026)Subjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
A fundamental problem in formulating higher Chern-Simons theories is the construction of a consistent higher gauge theory that circumvents the fake-flatness constraint. Here, we propose a solution to this problem using adjusted higher connections. Concretely, we shall demonstrate that there is an obstruction to the direct construction of such action functionals since, generically, adjusted higher gauge algebras do not admit an inner product. To overcome this obstruction, we introduce half-adjusted higher Chern-Simons theories. These theories have both well-defined underlying kinematic data as well as the expected properties of a higher generalisation of Chern-Simons theory. We develop the general construction of these theories in arbitrary dimensions and provide explicit details for the four-dimensional case. We also present the complete differential cohomological framework for principal 2-bundles with half-adjusted connections. Finally, we discuss an alternative approach introducing additional trivial symmetries.
- [462] arXiv:2509.11408 (replaced) [pdf, html, other]
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Title: The $r$-matrix structure on the moduli space of framed Higgs pairsComments: 24 pages. Ver 2.: 25 pages. Added relevant bibliography, added examples, typos fixed. Ver 3: there were bad signs. Ver 3: minor corrections and improved bibliography. Ver 4: 26 pages, improved grammar and bibliographySubjects: Exactly Solvable and Integrable Systems (nlin.SI); Symplectic Geometry (math.SG)
On the space of matrices with rational (trigonometric/elliptic) entries there is a well-known Lie-Poisson $r$-matrix structure. The known $r$-matrices are defined on the Riemann sphere (rational), the cylinder (trigonometric), or the torus (elliptic). We extend the formalism to the case of a Riemann surface $\mathcal C$ of higher genus $g$: we consider the moduli space of framed vector bundles of rank $n$ and degree $ng$, where the framing consists of a choice of basis of $n$ independent holomorphic sections that trivialize the fiber at a given point $\infty\in \mathcal C$.
The cotangent space is known to be identified with the set of Higgs fields, i.e., one-forms on $\mathcal C$ with values in the endomorphisms of the vector bundle, with an additional simple pole at $\infty$. The natural symplectic structure on the cotangent bundle of the moduli space induces a Poisson structure on the Higgs fields. Building on Dolgushev's higher genus $r$-matrix construction we identify the kernel with an explicitly computable non-abelian Cauchy kernel and derive the complete dynamical Poisson algebra, including the mixed bracket between the Higgs field and the kernel.
A detailed discussion of the elliptic case including a comparison with the literature, is also provided. - [463] arXiv:2509.13570 (replaced) [pdf, html, other]
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Title: Gen AI in Proof-based Math Courses: A Pilot StudyComments: 20 pages, 5 figures, Revised Version. Comments welcome!Subjects: Artificial Intelligence (cs.AI); History and Overview (math.HO)
With the rapid rise of generative AI in higher education, understanding how students use AI is increasingly important. This exploratory study examines student use and perceptions of generative AI across three proof-based undergraduate mathematics courses: a first-semester abstract algebra course, a topology course, and a second-semester abstract algebra course. In each case, course policy permitted some use of generative AI. Drawing on survey responses and student interviews, we analyze how students engaged with AI tools as well as their perceptions of generative AI's usefulness, accuracy and limitations.
- [464] arXiv:2510.02545 (replaced) [pdf, html, other]
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Title: Mean-field analysis of a neural network with stochastic STDPJournal-ref: Phys. Rev. E, 114, 2, 2026Subjects: Biological Physics (physics.bio-ph); Disordered Systems and Neural Networks (cond-mat.dis-nn); Probability (math.PR); Neurons and Cognition (q-bio.NC)
Analysing biological spiking neural network models with synaptic plasticity has proven to be challenging both theoretically and numerically. In a network with N all-to-all connected neurons, the number of synaptic connections is on the order of $N^2$, making these models computationally demanding. Furthermore, the intricate coupling between neuron and synapse dynamics, along with the heterogeneity generated by plasticity, hinder the use of classic theoretical tools such as mean-field or slow-fast analyses. To address these challenges, we introduce a new variable which we term a typical neuron X. Viewed as a post-synaptic neuron, X is composed of the activity state V , the time since its last spike S, and the empirical distribution $\xi$ of the triplet V , S and W (incoming weight) associated to the pre-synaptic neurons. In particular, we study a stochastic spike-timing-dependent plasticity (STDP) model of connection in a probabilistic Wilson-Cowan spiking neural network model, which features binary neural activity. Taking the large N limit, we obtain from the empirical distribution of the typical neuron a simplified yet accurate representation of the original spiking network. This mean-field limit is a piecewise deterministic Markov process (PDMP) of McKean-Vlasov type, where the typical neuron dynamics depends on its own distribution. We term this analysis McKean-Vlasov mean-field (MKV-MF). Our approach not only reduces computational complexity but also provides insights into the dynamics of this spiking neural network with plasticity. The model obtained is mathematically exact and capable of tracking transient changes. This analysis marks the first exploration of MKV-MF dynamics in a network of spiking neurons interacting with STDP.
- [465] arXiv:2510.07263 (replaced) [pdf, html, other]
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Title: Painlevé-Gullstrand coordinates for Kiselev black holesComments: 17 pages. 20 figures. Updated referencesJournal-ref: J. Theor. Exp. Appl. Phys. 2(3), 01-14 (2026)Subjects: General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
We investigate the implications provided by the modified Painlevé-Gullstrand coordinates in the context of quintessence for the Kiselev black hole. In this regard, we set up a fully static line element in terms of lapse and shift functions, apart from including the deformation parameter signaling deviation from the standard Painlevé-Gullstrand metric. We address two specific issues pertaining to the problems of radiation and dust furnished by the corresponding barotropic index parameter and study the related consequences by performing a range of analyses to explore the influence imposed by quintessence. We also discuss the thermodynamical consequences by evaluating the expressions of the Hawking temperature and the entropy function in closed forms.
- [466] arXiv:2601.06369 (replaced) [pdf, html, other]
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Title: Exact Solutions for Compactly Supported Parabolic and Landau Quantum BarriersComments: 27 pages, 9 figures: For published full text go to this https URLJournal-ref: International Journal of Theoretical Physics, (2026) 65:242Subjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph)
We derive exact solutions to the one-dimensional Schrödinger equation for compact support parabolic and hyperbolic secant potential barriers, along with combinations of these types of potential barriers. We give the expressions for transmission and reflection coefficients and calculate some dwell times of interest.
- [467] arXiv:2602.00511 (replaced) [pdf, html, other]
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Title: Partition of Unity Neural Networks for Interpretable Classification with Explicit Class RegionsComments: Revised version with expanded theoretical discussion, geometry-informed gate parameterizations, and updated numerical experimentsSubjects: Machine Learning (cs.LG); Optimization and Control (math.OC)
We introduce \emph{Partition of Unity Neural Networks} (PUNNs), a neural-network architecture for multiclass classification based on the classical mathematical notion of a partition of unity. The starting point is the observation that the characteristic functions of ideal class regions form a partition of unity. PUNNs replace these discontinuous indicators by learned continuous functions \[ h_1,\ldots,h_C:\mathcal X\to[0,1] \] whose sum is identically one and whose values are interpreted directly as class probabilities.
The partition functions are generated through a recursive family of input-dependent gates. This construction guarantees nonnegative class probabilities summing to one without a separate normalization layer such as softmax, while providing an explicit ordered factorization of each probability in terms of the gate values. The resulting gate trace gives an interpretable representation of how individual class probabilities are formed. The framework also allows multiple partition functions to represent a single class and admits both neural-network and geometry-informed realizations of the gates.
We prove that PUNNs are dense in the space of continuous probability maps from compact subsets of $\mathbb R^d$ into the probability simplex. Thus, the recursive partition-of-unity structure retains universal approximation of continuous probabilistic classifiers, including maps whose components may vanish.
Numerical experiments on synthetic datasets, MNIST, and CIFAR-100 illustrate the learned partitions, the effect of class ordering, and the use of multiple partition components per class. We also develop shape-informed gates that incorporate geometric information directly; when the chosen geometry is well matched to the class regions, these models achieve comparable accuracy with substantially fewer trainable parameters. - [468] arXiv:2602.07867 (replaced) [pdf, html, other]
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Title: Minimal nonintegrable models with three-site interactionsComments: 18 pages, 3 figures, 6 tablesSubjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph); Exactly Solvable and Integrable Systems (nlin.SI)
We study integrability breaking in translationally invariant spin-$1/2$ chains with genuine three-site interactions. Using a two-qubit composite representation, we prove that the deformed Fredkin spin chain is nonintegrable for any nonzero deformation parameter, although its Hamiltonian decomposes into coupled integrable building blocks. We then extract the minimal injective models, defined as the simplest models needed for a rigorous nonintegrability test, from the Hamiltonian density of the deformed Fredkin spin chain. Among the sixteen such models, six satisfy the Reshetikhin condition and are integrable, while two satisfy Hokkyo's criterion and are nonintegrable away from special coefficient values. Returning to one-site translation-invariant spin-$1/2$ chains leaves four genuine three-site models: two integrable models and two generically nonintegrable models. These results give a compact classification of minimal three-site models obtained from the deformed Fredkin spin chain and identify a local mechanism by which coupling individually integrable structures can destroy integrability beyond nearest-neighbor interactions.
- [469] arXiv:2602.17430 (replaced) [pdf, html, other]
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Title: Tight any-shot quantum decouplingComments: v2: Theorem 3 slightly modifiedSubjects: Quantum Physics (quant-ph); Information Theory (cs.IT); Mathematical Physics (math-ph)
Quantum information decoupling is a fundamental primitive in quantum information theory, underlying various applications in quantum physics. We prove a novel one-shot decoupling theorem formulated in terms of quantum relative entropy distance, with the decoupling error bounded by two sandwiched Rényi conditional entropies. In the asymptotic i.i.d. setting of standard information decoupling via partial trace, we show that this bound is ensemble-tight in quantum relative entropy distance and thereby yields a characterization of the associated decoupling error exponent in the low-cost-rate regime.
Leveraging this framework, we derive several operational applications formulated in terms of purified distance: (i) single-letter expressions for the exact error exponents of quantum state merging in the low-entanglement-cost and high-entanglement-distillation-rate regimes, in terms of Petz-Rényi conditional entropies, and (ii) regularized expressions for achievable error exponents of entanglement distillation and quantum channel coding in terms of Petz-Rényi coherent informations. We further prove that these achievable bounds are tight for maximally correlated states and generalized dephasing channels, respectively, for the high distillation-rate/coding-rate regimes. - [470] arXiv:2603.22026 (replaced) [pdf, html, other]
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Title: A robust method to identify chimera statesSubjects: Pattern Formation and Solitons (nlin.PS); Mathematical Physics (math-ph); Adaptation and Self-Organizing Systems (nlin.AO); Chaotic Dynamics (nlin.CD); Computational Physics (physics.comp-ph)
Chimera states are one of the most intriguing phenomena in nonlinear dynamics, characterized by the coexistence of coherent and incoherent behavior in systems of coupled identical oscillators. Despite extensive studies and numerous observations in different settings, the development of reliable and systematic methods to classify chimera states and distinguish them from other dynamical patterns remains a challenging task. Existing approaches are often limited in scope and lack robustness. In this work, we propose a method based on Fourier analysis combined with statistical classification to identify chimera behavior. The method is applied to a system of topological signals coupled via the Dirac operator, where it successfully captures the rich dynamical regimes exhibited by the model. We demonstrate that the proposed approach is robust with respect to variations in network topology and system parameters. Beyond the specific model considered, the framework provides a general and automated tool for distinguishing different dynamical regimes in complex systems.
- [471] arXiv:2604.17810 (replaced) [pdf, html, other]
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Title: Memory Centric Power Allocation for Multi-Agent Embodied Question AnsweringChengyang Li, Shuai Wang, Kejiang Ye, Weijie Yuan, Boyu Zhou, Yik-Chung Wu, Chengzhong Xu, Huseyin ArslanComments: 6 pages, accepted by IEEE GLOBECOM 2026Subjects: Robotics (cs.RO); Information Theory (cs.IT)
This paper considers multi-agent embodied question answering (MA-EQA), which enables robot teams to answer queries based on their long-horizon observations. In contrast to existing edge resource management methods that optimize sensing, communication, or computation performance metrics, MA-EQA focuses on the quality of aggregated memory. To address this paradigm shift, we propose a quality of memory (QoM) model based on generative adversarial exam (GAE), which leverages forward simulation to evaluate memory retrieval and utilizes the resulting exam scores to quantify QoM. Based on the QoM model, we develop a memory-centric power allocation (MCPA) scheme that maximizes memory quality under communication resource constraints. Through analytical characterization in the noise-limited regime, we reveal a GAE-augmented capped water-filling structure for MCPA. Extensive experiments demonstrate that MCPA achieves significant improvements over existing benchmarks across diverse metrics and scenarios.
- [472] arXiv:2604.26367 (replaced) [pdf, html, other]
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Title: A Category-Theoretic Framework from Biological Mechanics to Engineered Stimulus-Response SystemsSubjects: Soft Condensed Matter (cond-mat.soft); Materials Science (cond-mat.mtrl-sci); Computational Engineering, Finance, and Science (cs.CE); Category Theory (math.CT)
Natural materials achieve adaptive behavior through hierarchical organization and coupled mechanisms across scales. Their translation into engineering, however, remains largely heuristic. What is missing is a formal translation framework that carries biological design logic into engineered realization while preserving physical consistency across levels of abstraction. Here we present a category theoretic compositional framework for verified nature-derived design. The framework defines a category of stimulus response dynamical systems with natural and artificial subcategories. It introduces a structure preserving implementation functor from biological mechanics to engineered systems. It also formalizes a machine agnostic specification layer that links behavioral intent to executable fabrication programs. We instantiate the framework on the hygromorphic pinecone hierarchy as a representative biological case. We implement the full pipeline in Grasshopper, where formal specifications are translated into modular parametric scripts that preserve the compositional structure of the model. The resulting designs are fabricated by fused filament fabrication, evaluated experimentally, and tested against model predictions derived from the pipeline. The current implementation generates four actuator classes spanning two stimulus types and two kinematic responses. One actuator arises purely through composition from previously validated components, without additional manual derivation. The results show that compositionality can function not just as a descriptive language, but as a generative and system level verifiable method for mechanical material design. More broadly, the work provides a concrete route for embedding formal multiscale reasoning within increasingly computational, generative, and physics-driven design workflows.
- [473] arXiv:2605.00075 (replaced) [pdf, html, other]
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Title: Essential Duality and Quantitative Channel Composability in Algebraic Quantum Field TheoryComments: 28 pages, no figures. Corrected a typo in the titleSubjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph)
Let \(O\mapsto A(O)\) be a local Haag--Kastler net and let \(A^d(O)=A(O')'\) be its dual net. Normal unital completely positive maps on \(B(H)\) that fix \(A(O')\) pointwise are exactly the channels with Kraus operators in \(A^d(O)\). Thus Haag duality at \(O\) is equivalent to equality with the \(A(O)\)-inner channels, while essential duality is equivalent to order-independence of the corresponding causal-complement channel classes at spacelike separation. For self-adjoint generators \(a\in M\) and \(b\in N\), the small-parameter order defect has coefficient \(\operatorname{diam}\sigma(-i[a,b])\). Uniform optimization gives \(\Gamma(M,N)=2\Delta_{\mathrm{sa}}(M,N)\), and reversible inner-channel cb balls recover the same invariant through \(\lim_{\varepsilon\downarrow0}\Omega^{\mathrm{rev}}_\varepsilon(M,N)/\varepsilon^2=\Gamma(M,N)/4\). The full inner-channel class admits a normalized coefficient with the same commutativity zero set. With a positive reference observable \(G\), the mean-input-energy-constrained coefficient \(\Gamma_{G,E}\) has the same zero criterion and increases to \(\Gamma\) as \(E\to\infty\), while no model-independent recovery rate follows from the general von Neumann-algebraic hypotheses. For bosonic second-quantization nets the spacelike defect is either \(0\) or \(4\); generalized free-field examples realize both branches. In the translation-covariant case with positive one-particle time generator, failure of essential duality admits extremal bounded witness flows that are energy-limited relative to the second-quantized time-translation energy.
- [474] arXiv:2605.22950 (replaced) [pdf, other]
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Title: Diffusion-based Denoising Beats Vanilla Score Matching in Parameter Estimation: A Theoretical ExplanationSubjects: Machine Learning (stat.ML); Machine Learning (cs.LG); Statistics Theory (math.ST); Methodology (stat.ME)
Score matching is an alternative to maximum likelihood estimation when the normalizing constant is unknown or too costly to evaluate. However, vanilla score matching has shown to be inefficient relative to maximum likelihood estimation for multimodal distributions with well-separated modes, which are commonly encountered in practical applications. We compare a novel diffusion-based denoising score matching estimator (DDSME) to the vanilla score matching estimator (SME) in this scenario. In particular, we prove statistical guarantees for both estimators, showing that the error bound for the vanilla SME worsens when the separation between the modes increases, which can be avoided in case of the DDSME with suitable hyperparameter tuning. This provides a novel theoretical explanation for the superior behavior of diffusion-based score matching over the vanilla version. We support our theoretical findings by numerical experiments.
- [475] arXiv:2606.20247 (replaced) [pdf, html, other]
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Title: The auxiliary-metric formulation of Born-Infeld New Massive GravityComments: 14 pages, v2 matches the published versionJournal-ref: Phys.Rev.D 114 (2026) 4, 044025Subjects: General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
Born-Infeld New Massive Gravity (BINMG) completes New Massive Gravity to all orders in curvature through the determinant of the metric shifted by the Einstein tensor. We recast it with an independent auxiliary metric $q_{\mu\nu}$, whose algebraic equation of motion $q_{\mu\nu}=g_{\mu\nu}+\frac{\sigma}{m^2}G_{\mu\nu}(g)$ recovers the determinant action exactly on the regular branch and resums the infinite curvature series into a single relation. In the densitized variable $P^{\mu\nu}=\sqrt{-q}\,q^{\mu\nu}$ the three-dimensional action is polynomial, with all derivative dependence carried by the coupling $P^{\mu\nu}G_{\mu\nu}(g)$. The formulation makes known properties follow with substantially less algebra: the unique vacuum follows in one line, and the quadratic action yields a single Pauli-Fierz massive spin-2 field with the Fierz-Pauli tuning generated rather than imposed. On locally AdS backgrounds the conserved charges, BTZ mass and angular momentum, central charge, and entropy reduce to the Einstein results times a common factor. The formulation also isolates the nonlinear degree-of-freedom problem in the right variables, leaving the full Dirac count to separate work.
- [476] arXiv:2606.24038 (replaced) [pdf, other]
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Title: Sim-to-Real Betting on the E-Process: Bringing "simulators" to anytime-valid confidence sequencesComments: A more complete version has been developed and will replace this work under a new title with extended scopeSubjects: Robotics (cs.RO); Probability (math.PR)
This note describes an integration of the sim-to-real performance estimate with betting (from Chen et al.) and the safe anytime-valid inference (from Ramdas et al.). Using the scaled simulators. The method produces efficient, reliable certificates for the mean estimate, an approach that is especially valuable in robot performance testing. This note gives a primary, self-contained account of the construction; preliminaries of the respective methods are kept at a minimum, and one shall refer to the original works for full detail. Some synthetic examples demonstrating the proposed algorithm can be found at this https URL.
- [477] arXiv:2607.18652 (replaced) [pdf, html, other]
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Title: The Price of Hidden Curvature: Improved Lower Bounds for Bandit Convex OptimizationComments: 44 pages, 2 figuresSubjects: Machine Learning (stat.ML); Information Theory (cs.IT); Machine Learning (cs.LG)
We establish improved lower bounds on the minimax expected regret of stochastic bandit convex optimization for $1$-Lipschitz functions on the $d$-dimensional Euclidean ball. For time horizons $n\ge d^{10/3}$, we prove a lower bound of $\Omega(d^{4/3}\sqrt{n})$, the first nontrivial bound that exceeds the $d\sqrt{n}$ dependence of linear bandits, showing that stochastic bandit convex optimization is fundamentally harder than linear bandits. For $d^2\le n\le d^{10/3}$, we obtain a lower bound of $\Omega(\sqrt{d}n^{3/4})$, matching the regret of the algorithm of Flaxman et al. (2005), establishing its optimality in this regime.
The hard class of convex functions we construct takes the following form in dimension $2d$: for an action $a=(a^1,a^2)\in \mathbb{B}^{2d}$, each function is the scaled soft maximum of a "tube", $r^{-1}\|W^\star a^1-\frac{r}{8\varepsilon}a^2 \|$ (hyperparameterized by $\varepsilon,r$), and a squared distance function, $\frac12\|a^1-u^\star\|^2-\frac12\|u^\star\|^2$. Here $u^\star\in\mathbb{R}^d$ is the unknown target determining the minimizer, while $W^\star\in\mathbb{R}^{d\times d}$ hides the region in which the quadratic curvature is observable. Indeed, observations reveal substantial information about $u^\star$ only when the learner acts near the hidden tube $a^2\approx \frac{8\varepsilon}{r}W^\star a^1$; away from it, the tube branch masks the quadratic branch. Thus the learner must pay to uncover the geometry encoded by $W^\star$ before it can effectively exploit the curvature that identifies $u^\star$. Formalizing this tradeoff yields a sample complexity lower bound of $\Omega(\frac{d^{5/2}}{\varepsilon^2}\wedge\frac{d^2}{\varepsilon^4})$ for finding an $\varepsilon$-optimal action, and ultimately the $\Omega(d^{4/3}\sqrt{n}\wedge\sqrt{d}n^{3/4})$ regret lower bound.
The proof was developed by GPT-5.5 Pro and GPT-5.6 Sol Pro under the authors' guidance. - [478] arXiv:2607.23287 (replaced) [pdf, html, other]
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Title: Learning Asymptotics with Convergence-Rate Guarantees using Linear Least SquaresComments: 62 pages, 7 tables, 5 figures; additional referencesSubjects: Machine Learning (stat.ML); Machine Learning (cs.LG); Combinatorics (math.CO); Numerical Analysis (math.NA)
We introduce a new research area that is called Asymptotics Learning Theory (ALT) and combines optimization with asymptotic analysis. In particular, ALT provides a unified approach for computing unknown constants/parameters in proven asymptotic expansions using optimization theory. In this paper, we focus on a general asymptotic form which includes a broad class of asymptotics. Furthermore, we study two powerful numerical methods, namely, sliding Linear Least Squares (sLLSQ) and sliding Tikhonov Linear Least Squares (sT-LLSQ). For these techniques we rigorously prove asymptotic estimates that lead to sufficient conditions for convergence (to the correct values of unknown parameters) and convergence-rate guarantees. Despite their strengths, both methods have also limitations, e.g., slow convergence---or even, counterintuitively, divergence---in some cases. Moreover, we present fundamental applications in analytic combinatorics, a beautiful field of mathematics that deals with asymptotic enumeration of discrete structures using complex analysis. The proposed techniques complement existing approaches, such as the ratio method and its variants. Numerical examples also verify the theoretical results. Finally, we discuss interesting research directions in ALT.
- [479] arXiv:2607.25646 (replaced) [pdf, html, other]
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Title: Conserved Gravitational Charges as Curvature FluxesComments: 37 pages, 1 figure, references addedSubjects: General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
Conserved gravitational charges are commonly expressed as surface integrals of the metric perturbation and its first derivatives. We show that, in the usual asymptotically AdS and asymptotically flat settings, the standard charges admit equivalent representatives as fluxes of linearized curvature. The construction uses a divergence-free rank-four tensor whose trace is proportional to the cosmological Einstein tensor. On a maximally symmetric AdS background, it reproduces the known curvature representation of the Abbott-Deser charges. The asymptotically flat construction is not obtained by taking a naive $\Lambda \rightarrow 0$ limit, since the Killing two-form vanishes for translations. We instead introduce an antisymmetric Poincaré Killing potential. A representative Killing potential adapted to the algebraic Bianchi identity converts the linearized Einstein current into a total divergence and yields a single curvature-flux formula. Its translation sector reproduces the ADM energy--momentum, while in four dimensions its Lorentz sector reproduces the angular momentum and boost/center-of-mass charges under the standard Regge-Teitelboim falloff and parity conditions. The normalization is checked explicitly for Schwarzschild, boosted Schwarzschild, and Kerr data. On non-maximally symmetric Einstein backgrounds, background Weyl curvature generates additional terms, so the maximally symmetric construction does not directly extend to a pure codimension-two curvature-flux formula.
- [480] arXiv:2608.03459 (replaced) [pdf, html, other]
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Title: Genus-One Rigidity of Bounded Virasoro Pairings for 3D GravityComments: 48 pages, 3 figures; ancillary code included. v2: revised title and physical interpretation to correct an overclaim in v1; added referencesSubjects: High Energy Physics - Theory (hep-th); General Relativity and Quantum Cosmology (gr-qc); Mathematical Physics (math-ph)
Motivated by the proposed relation between 3D gravity and the doubled Virasoro TQFT, and by the associated program of ensemble holography, we ask whether ensemble holography in AdS$_3$/CFT$_2$ can be understood as an average over absolute 2D CFTs obtained by varying the topological boundary condition of the doubled Virasoro TQFT. We show that, at genus one and in the ordinary nondegenerate sector, every vacuum-normalized, modular-invariant genus-one pairing that acts boundedly on the auxiliary $L^2$ label space is diagonal. Under the ordinary block-diagonal vacuum--continuum ansatz, the same conclusion holds for entrywise-positive, modular-invariant Borel-measure pairings satisfying the two vacuum marginals, without assuming absolute continuity or boundedness. The bounded class and this positive-measure class therefore contain no distinct nontrivial torus pairings to average. Any nontrivial Virasoro boundary ensemble must use genus-one pairings outside these classes, involve additional sectors or vacuum--continuum couplings, or distinguish its members through information not captured by the genus-one pairing. Technically, the proof identifies the Virasoro $S$ and $T$ transformations with the even Weil representation of the metaplectic group and applies the Cowling--Steger lattice-restriction theorem. We also give elementary proofs for several explicit classes of candidate boundary conditions and perform numerical checks of the general result.
- [481] arXiv:2608.06796 (replaced) [pdf, html, other]
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Title: Online Multi-Level Aggregation with Per-Batch Maximum DelaySubjects: Data Structures and Algorithms (cs.DS); Discrete Mathematics (cs.DM); Optimization and Control (math.OC)
We study online multi-level aggregation on finite rooted trees with a per-batch maximum-delay objective. A service pays for a rooted subtree and for the maximum waiting time among the requests cleared by that service. We show that the offline optimum admits a consecutive-arrival-block normal form and can be computed by a polynomial-time dynamic program. The same dynamic program defines the deadlines of a family of online algorithms, which we call DP-Envelope. Its deterministic endpoint is $2$-competitive. Sampling one global parameter with density $e^\theta/(e-1)$ leads to an $e/(e-1)$-competitive randomized algorithm against an oblivious adversary. The deterministic guarantee matches the known fixed-node lower bound, and we prove a matching randomized lower bound. Thus, both guarantees are optimal on every nondegenerate rooted tree. We first develop the line metric as a warm-up, where the algorithm and its nested block partitions have a direct geometric interpretation. Finally, we show that the upper bounds extend to every realizable static service system with a normalized, nondecreasing, submodular joint service cost.
- [482] arXiv:2608.15052 (replaced) [pdf, html, other]
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Title: Andy: A Mathematical Agent for Rigorous Proof and Autonomous ResearchComments: 20 pages, 4 figures, 3 tables, and 3 algorithms. Research logs, reports, and simulation code are available at this https URLSubjects: Artificial Intelligence (cs.AI); Optimization and Control (math.OC)
Andy is an autonomous mathematical research agent that turns a mathematical problem into a traceable proof. It solves or verifies a submitted problem, formulates a literature-grounded new problem through a research-value gate, and carries it through proof construction and final verification. It organizes proof steps in an executable DAG, verifies each step independently and binds the result to a certificate, retains verified work whose interfaces remain unchanged during local repair, and records the full path from problem formulation to final proof. The system separates proof generation from correctness evaluation and can acquire, retain, retrieve, and reuse knowledge from existing results. Starting from a self-triggered impulsive consensus result, Andy formulates a global exponential leader-follower synchronization problem for delayed heterogeneous networks with switching communication topologies. The proposed hybrid control combines self-triggered impulses with execution delay and continuous feedback over a recovery window. After each delayed impulse, the feedback cancels the delayed error channel until the pre-impulse history leaves the active delay interval. Sufficient conditions for global exponential synchronization are established, Zeno behavior is excluded for both timing sequences, and a numerical example illustrates the result.
- [483] arXiv:2608.19094 (replaced) [pdf, html, other]
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Title: Realizing Logical Diagonal Gates via Transversal Physical $Z$-Rotations in CSS CodesComments: 53 pagesSubjects: Quantum Physics (quant-ph); Information Theory (cs.IT)
Calderbank-Shor-Steane (CSS) codes, constructed from nested classical codes $C_2 \subseteq C_1$, are typically optimized for good code parameters. However, practical quantum computing equally demands fault-tolerant logical gates. In this work, we characterize nested pairs $(C_1, C_2)$ whose resulting CSS codes realize a target logical diagonal gate via transversal physical $Z$-rotations. In doing so, we recover a result of Camps-Moreno et al. that CSS codes can realize only logical single-qubit $Z$-rotations and multi-qubit controlled-$Z$ rotations via transversal physical $Z$-rotations. Building on our characterization, we develop the ''appending construction'', that takes as input an $[[n',k']]$ CSS code $Q'$ and a target logical $Z$-rotation (single-qubit or multi-controlled) $U_L$, and extends $Q'$ by systematically appending $n''$ physical qubits to obtain an $[[n,k]]$ CSS code $Q$ with $n = n'+n''$ and $k=k'$. The target logical gate $U_L$ is realized in $Q$ by applying a well-chosen physical transversal $Z$-rotation to the $n''$ appended physical qubits. Moreover, any logical gate realized via transversal physical $Z$-rotations in the input code $Q'$ remains transversally realizable in the extended code $Q$. The CSS code $Q$ may incur a loss in minimum distance, but the loss can be controlled through the parameter choices made in the construction. By repeatedly applying the appending construction, we can extend any CSS code $Q'$ to obtain a CSS code $Q$ that supports fault-tolerant implementations of multiple desired logical $Z$-rotations. The cost to be paid for this is the increased physical qubit overhead as the number of target logical gates grows.