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- [1] arXiv:2512.22132 [pdf, html, other]
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Title: $\mathcal{X}$-Gorenstein projective and $\mathcal{Y}$-Gorenstein injective modules over tensor ringsComments: 16 pages. arXiv admin note: substantial text overlap with arXiv:2504.21349 by other authorsSubjects: Rings and Algebras (math.RA)
Let $T_R(M)$ be a tensor ring and $\mathcal{X}$, $\mathcal{Y}$ be two classes of $R$-modules. Under certain conditions, we prove that a $T_R(M)$-module $(A, u)$ is $Ind(\mathcal{X})$-Gorenstein projective if and only if $u$ is monomorphic and $coker(u)$ is an $\mathcal{X}$-Gorenstein projective $R$-module. $\mathcal{Y}$-Gorenstein injective $T_R(M)$-modules are also explicitly described. As a consequence, the characterizations of Ding projective and Ding injective modules over $T_R(M)$ are obtained. Some applications to trivial ring extensions and Morita context rings are given.
- [2] arXiv:2512.22133 [pdf, other]
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Title: A Course in Ring TheorySubjects: Rings and Algebras (math.RA)
An introductory textbook on ring theory, including ideals and homomorphisms, Euclidean domains, PIDs, and UFDs, with examples and exercises.
- [3] arXiv:2512.22134 [pdf, html, other]
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Title: A Mnemonic Matrix Rule for (Split) Octonionic Multiplication and its Extension to the Cayley--Dickson TowerComments: 3 pagesSubjects: Rings and Algebras (math.RA)
We present a compact mnemonic device for computing the product of two (split) octonions represented in quaternionic form q+ l p. The rule is expressed as a simple (R+L) pattern of ``right-ordered'' and ``left-ordered'' quaternionic products, visible within a 2 X 2 quaternionic matrix representation of the Cayley--Dickson construction. This formulation extends verbatim to all algebras in the Cayley--Dickson tower, and provides an efficient computational tool in non-associative settings. To our knowledge, this explicit mnemonic pattern does not appear in the classical literature on octonions or composition algebras.
- [4] arXiv:2512.22138 [pdf, html, other]
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Title: Liouvillian integrability of rational vector fields: The case of algebraic extensionsComments: 26 pagesSubjects: Rings and Algebras (math.RA)
As shown in a previous paper, whenever a rational vector field on $\mathbb C^n$, $n>2$, is Liouvillian integrable, then it admits a first integral obtained by two successive integrations from a one-form with coefficients in a finite algebraic extension $L$ of the rational function field $K$. In the present work we discuss and characterize exceptional vector fields in this class, for which -- by definition -- the choice $L=K$ is not possible. In particular we show that exceptional vector field exist, giving explicit constructions in dimension three.
- [5] arXiv:2512.22162 [pdf, html, other]
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Title: Exchangeability and randomness for infinite and finite sequencesComments: 15 pages, 1 figureSubjects: Statistics Theory (math.ST); Methodology (stat.ME)
Randomness (in the sense of being generated in an IID fashion) and exchangeability are standard assumptions in nonparametric statistics and machine learning, and relations between them have been a popular topic of research. This note draws the reader's attention to the fact that, while for infinite sequences of observations the two assumptions are almost indistinguishable, the difference between them becomes very significant for finite sequences of a given length.
- [6] arXiv:2512.22191 [pdf, html, other]
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Title: Research projects and Moscow Mathematical Conference for high school studentsComments: 13 pages, in Russian language, no figuresSubjects: History and Overview (math.HO)
This paper shares some experience in advanced mathematical education. We show how a high school student can be naturally and gradually introduced to basic steps of scientific research: developing intuition by finding and correcting mistakes through discussions and writing a paper, (transparent) anonymous peer review, recognition and award. We show that most of this can be done in research projects not aiming at scientific novelty. We share the experience (both principles and examples) of the Moscow Mathematical Conference of High School Students.
- [7] arXiv:2512.22204 [pdf, html, other]
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Title: Smarandache curves and their properties on null curves in lightlike cone space $\mathbb{Q}_{2}^{3}$Comments: 22 page, no figuresSubjects: General Mathematics (math.GM)
This study investigates the differential geometric properties of Smarandache curves derived from null curves defined in the ligtlike cone space $% Q_{3}^{2}\subset E_{2}^{4}$. The indefinite metric structure causes the null vectors, and hence the null curves, to have a richer geometry in this space than in Euclidean or Minkowski spaces. In this study, we analyse the kinematic properties of the null curve using the null natural Frenet frame $% \{x,\xi ,N,W\}$. We then investigate the bending, torsion, and other geometric invariants of Smarandache curves constructed as linear combinations of these frame vectors (i.e., combinations of tangent, normal, or binormal vectors). The findings reveal how the original properties of the null curve are transferred to the Smarandache curves and how the metric of this particular space affects the characteristics of the Smarandache curves. This analysis provides a new perspective on the relationships between constrained and degenerate structures in differential geometry and light-like particle dynamics in theoretical physics.
- [8] arXiv:2512.22224 [pdf, html, other]
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Title: A New Special Function and Its Application in ProbabilityComments: 17 pages,12 figuresJournal-ref: International Journal of Mathematics and Mathematical Sciences Volume 2018Issue 1Subjects: General Mathematics (math.GM)
In this note we present a new special function that behaves like the error function and we provide an approximated accurate closed form for its CDF in terms of both Chebyshev polynomials of the first kind and the error function. Also, we provide its series representation using Padé approximant. We show convincing numerical evidence of an accuracy of $10^{-6}$ for the approximants in the sense of the quadratic mean norm. A similar approach may be applied to other probability distributions, for example, the Maxwell--Boltzmann distribution and the normal distribution, and we show its application using both of those distributions.
- [9] arXiv:2512.22253 [pdf, html, other]
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Title: A New Class of Fuzzy Inner Products and Fuzzy Norms over Ordered IntervalsSubjects: General Mathematics (math.GM)
In this article, we first define the concept of ordered intervals, then introduce ordered fuzzy inner product and describe some of its properties.
- [10] arXiv:2512.22273 [pdf, html, other]
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Title: Orbifold Chern classes and Bogomolov-Gieseker inequalitiesComments: This note is splitted from arXiv:2401.07273Subjects: Algebraic Geometry (math.AG)
Assume that $X$ is a compact complex analytic variety which has quotient singularities in codimension 2, and that $\mathcal{F}$ is a reflexive sheaf on $X$. Using orbifold modifications, we can define first and second homological Chern classes for $\mathcal{F}$. If in addition $X$ has a Kähler form $\omega$ and $\mathcal{F}$ is $\omega$-stable, then we deduce Bogomolov-Gieseker inequality on the orbifold Chern classes of $\mathcal{F}$.
- [11] arXiv:2512.22312 [pdf, html, other]
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Title: Necessary and sufficient conditions for high dimensional Central Limit Theorem under moment conditionsSubjects: Probability (math.PR)
High dimensional central limit theorems (the CLTs) have been extensively studied in recent years under a variety of sufficient moment conditions connecting the dimension growth rate with the tail decay rate. In this article, we investigate whether the existing moment conditions are also necessary under the independence of the components. We consider four exhaustive classes, viz. when underlying random variables (I) have all polynomial moments, (II) have some polynomial moment of order higher than two, (III) have only second moment but no polynomial moment higher than two exists, and (IV) have infinite second moment, but belong to the domain of attraction of normal distribution. We find the optimal growth rate of the dimension with respect to sample size in the high dimensional CLTs over hyper-rectangles. More precisely, we derive necessary and sufficient moment conditions for the validity of the the CLT over hyper-rectangles in each of the four regimes listed above, showing that the CLT may hold under much weaker conditions compared to those considered in the existing literature.
- [12] arXiv:2512.22314 [pdf, other]
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Title: Skands and coskands (The non-founded set theory with individuals and its model in the Field of all Conway numbers)Subjects: Logic (math.LO)
The basic one in this work is the axiomatic set theory $NBG$ (von Neumann-Bernays-G{ö}del), which is a first-order theory with its own axioms, including in particular the axiom of choice ${\bf AC}$ and the axiom of regularity ${\bf RA}$. The universal class ${\bf V}$ of all sets in this theory exactly coincides with the class of all founded sets, i.e., such $X\in{\bf V}$ that {\it does not exist} an infinitely descending $\in$-sequence $X\ni X_1\ni X_2\ni...\ni X_n\ni...$ of sets $X_n$, $n=1,2,3,...\,\,$. In the first part of the paper, a new concept of {\it skand} is introduced -- a random aggregate, or \grqq decreasing\grqq\, tuple composed of founded sets, e.g., $X=\{1,\{2,\{3,\{...\,\,\,...\}\}\}\}$, and the theory of $NBG^-=NBG-{\bf RA}$, i.e., the theory of $NBG$ without the axiom of regularity ${\bf RA}$, to which is added the new axiom ${\bf SEA}$ of the existence of infinite-length skands and the pseudo-founding axiom ${\bf PFA}$. These new axioms are a negation of the axiom of regularity and are thus less restrictive than the axiom of regularity ${\bf RA}$ in the sense that they admit the existence of non-founded sets, and the axiom of regularity excludes the existence of such sets. At the same time of course the axiom of extensionality ${\bf EA}$ is replaced by a more accurate axiom of extensionality ${\bf EEA}$, since it takes into account the equality of new objects. In the second part of the paper, a new concept of {\it coskand} is introduced, which is dual to a notion of skand and is a random aggregate, or \grqq increasing\grqq\, tuple composed of founded sets and the theory of $NBG$ and actually is a theory $NBG[\cal U]$ with individuals as limiting coskands, e.g., $X=...\{3,\{2,\{1,\{0\}\}\}\}...\,\,$.
- [13] arXiv:2512.22319 [pdf, other]
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Title: Amicable numbers and their connection to the Euler totient functionComments: 13 pagesSubjects: History and Overview (math.HO)
A pair of numbers is amicable if each number equals the sum of the proper divisors of the other. This paper after exploring the history and evolution of amicable numbers, introduces a novel characterization of amicable pairs whose greatest common divisor is a power of two, using their distinct prime factorizations. Specifically, we examine pairs of the forms $A=2^n ab, B=2^n cd$, $A=2^n abc, B=2^n de$, and $A=2^n abc, B=2^n def$. From these configurations, we establish explicit symmetric identities that relate the sum $\varphi(A)+\varphi(B)$ of Euler's totient functions directly to the odd prime factors of $A$ and $B$.
- [14] arXiv:2512.22325 [pdf, other]
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Title: Quadratic-Phase Dunkl Transform: Fundamental properties, translation operators, convolution product and HUPComments: 24 pagesSubjects: General Mathematics (math.GM)
In this paper, we introduce and study the quadratic-phase Dunkl transform, a novel integral transform on the real line parameterized by five real numbers $(a, b, c, d, e)$ and a multiplicity parameter $\mu\geq -1/2$. We define the transform and establish its fundamental properties, including continuity, a Riemann--Lebesgue lemma, linearity, scaling, and most importantly, a reversibility theorem and an associated Parseval formula. We show that this novel quadratic-phase integral type transform generalizes a wide class of known transforms, such as the quadratic-phase Fourier-Bessel transform, the quadratic-phase Fourier transform, the linear canonical Dunkl transform, the fractional Dunkl transform, and the classical Dunkl transform, by choosing the appropriate specialization of its parameters. Furthermore, we introduce and investigate a corresponding quadratic-phase Dunkl translation operator and a convolution structure, proving their basic properties and a Young's inequality. Finally, we establish a new Heisenberg-type uncertainty principle for the quadratic-phase Dunkl transform, which extends the classical uncertainty principle for a large class of integral type transforms.
- [15] arXiv:2512.22327 [pdf, html, other]
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Title: Hawksmoor's Ceiling, Mercator's Projection and the Roman PantheonComments: 30 pages, 16 figuresSubjects: History and Overview (math.HO); Differential Geometry (math.DG); Popular Physics (physics.pop-ph)
The ceiling of the Buttery in All Souls College, Oxford, designed by the English Baroque architect Nicholas Hawksmoor, has a vaulted form on an oval base. It is coffered with an array of approximately square sunken lacunaria, whose sizes and positions vary so as to accommodate the constraints of the curved surface and its boundaries. A similar design appears in the dome of the Roman Pantheon. Using methods of differential geometry, we hypothesise that these cofferings should be the images under conformal mappings of regular square tilings of a rectangle or finite cylinder. This guarantees that the coffer ribs meet exactly at right angles and the coffers are close to being square. These mappings are simply the inverse of Mercator's projection of the curved surface onto a plane. For a ceiling which is a general surface of revolution, we derive formulae for the dimensions and location of each coffer. Our results, taking into account camera distortion, are in excellent agreement with photographs of the Hawksmoor ceiling and the Pantheon dome, as well as with recent direct measurements of the latter. We also describe a protocol by which Hawksmoor's ceiling might have been constructed without advanced mathematics.
- [16] arXiv:2512.22329 [pdf, html, other]
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Title: A Continuous-Order Integral Operator for Maclaurin-type ReconstructionComments: 15 pages, 6 figures, Continuous-order Maclaurin-type reconstruction operatorSubjects: General Mathematics (math.GM)
I introduce a continuous-order analog of the Maclaurin expansion that reconstructs analytic functions. This continuous-order Maclaurin-type operator replaces the discrete sum of integer-order derivatives in the classical Maclaurin expansion with an integral over fractional derivative orders, weighted at the evaluation point by x^r/Gamma(r+1). Numerical experiments on a representative set of analytic functions f show that the uncorrected operator reliably tracks the global shape of f, with a systematic, largely constant offset and an additional deviation localized at the origin. Low-order correction terms, motivated by the classical Euler-Maclaurin summation formula, reduce this discrepancy. With these corrections, the operator reconstructs f accurately in the tested domains. The operator reproduces monomials exactly, reflecting the collapse of derivative information to a single order, as in the classical Maclaurin expansion of monomials. This singular collapse motivates a Taylor-centered extension away from the origin, where the order dependence is predicted to be smooth. Taken together, these results suggest that the continuous-order integration operator, with low-order corrections, provides a coherent framework for generalizing the classical Taylor-Maclaurin expansion.
- [17] arXiv:2512.22330 [pdf, html, other]
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Title: Revisiting De Moivre-LaplaceSubjects: Probability (math.PR)
We revisit the proof of the de Moivre--Laplace theorem, which is the ancestor of the central limit theorem for the binomial distribution. Our goal is to provide a proof that can be reasonably presented to undergraduate students within a basic course of probability theory. We follow the strategies presented in two classical references, the books of Breiman and Feller, but we replace the arguments involving series expansions of the logarithm or the exponential by the basic inequality $\exp(t)\geq 1+t$. This way we avoid completely the use of uniform convergence and power series. We also avoid using Stirling's formula, instead we use the exact formula for the Wallis integral. As a by product of the proof, we also obtain a non-asymptotic inequality linking the binomial and the Gaussian distributions.
- [18] arXiv:2512.22344 [pdf, html, other]
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Title: Integration of an arbitrary linear ODEComments: 21 pagesSubjects: Classical Analysis and ODEs (math.CA)
The standard text book theory of ODEs lacks a general method to solve linear equations having variable coefficients, providing instead a collection of special techniques for particular classes of equations. The present article addresses this shortcoming in the basic theory. We introduce the multex integral operator, generalizing to several input functions the standard exponential primitive operator that is inverse to the logarithmic derivative. The multex operator serves to integrate in explicit form an arbitrary linear ordinary differential equation.
- [19] arXiv:2512.22346 [pdf, html, other]
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Title: Higher Order Dualities between Prime IdealsComments: Will be submitted for publication soonSubjects: Number Theory (math.NT)
Extending the works of Alladi and Sweeting and Woo, we state and prove the general higher order duality between prime ideals in number rings. We then use the second order duality to obtain the a new formula for the Chebotarev Density involving sums of the generalized Möbius function and the prime ideal counting function. We also provide two estimates of such sums as an application of the duality identity. A discussion of the duality in a slightly more general setting is done at the end.
- [20] arXiv:2512.22347 [pdf, html, other]
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Title: Reinforcement Learning for Optimal Stopping in POMDPs with Application to Quickest Change DetectionComments: 24 pages, 9 figures. To appear, IEEE Trans. Auto. ControlSubjects: Optimization and Control (math.OC)
The field of quickest change detection (QCD) focuses on the design and analysis of online algorithms that estimate the time at which a significant event occurs. In this paper, design and analysis are cast in a Bayesian framework, where QCD is formulated as an optimal stopping problem with partial observations. An approximately optimal detection algorithm is sought using techniques from reinforcement learning. The contributions of the paper are summarized as follows: (i) A Q-learning algorithm is proposed for the general partially observed optimal stopping problem. It is shown to converge under linear function approximation, given suitable assumptions on the basis functions. An example is provided to demonstrate that these assumptions are necessary to ensure algorithmic stability. (ii) Prior theory motivates a particular choice of features in applying Q-learning to QCD. It is shown that, in several scenarios and under ideal conditions, the resulting class of policies contains one that is approximately optimal. (iii) Numerical experiments show that Q-learning consistently produces policies that perform close to the best achievable within the chosen function class.
- [21] arXiv:2512.22353 [pdf, html, other]
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Title: Tableaux and orbit harmonics quotients for finite transformation monoidsComments: Comments welcomeSubjects: Representation Theory (math.RT); Combinatorics (math.CO)
We extend Grood's tableau construction of irreducible representations of the rook monoid and Steinberg's analogous result for the full transformation monoid. Our approach is characteristic-free and applies to any submonoid $\mathcal{M}(n)$ of the partial transformation monoid on an $n$-element set that contains the symmetric group. To achieve this, we introduce and study a functor from the category of rational representations of the monoid of $n \times n$ matrices to the category of finite dimensional representations of $\mathcal{M}(n)$. We establish two branching rules. Our main results describe graded module structures of orbit harmonics quotients for the rook, partial transformation, and full transformation monoids. This yields analogs of the Cauchy decomposition for polynomial rings in $n\times n$ variables.
- [22] arXiv:2512.22360 [pdf, html, other]
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Title: Generalized K-theoretic invariants and wall-crossing via non-abelian localizationComments: 76 pagesSubjects: Algebraic Geometry (math.AG); K-Theory and Homology (math.KT)
Given an abelian category and a stability condition satisfying appropriate conditions, we define generalized $K$-theoretic invariants and prove that they satisfy wall-crossing formulas. For this, we introduce a new associative algebra structure on the $K$-homology of the stack of objects of an abelian category, which we call the $K$-Hall algebra. We first define $\delta$-invariants directly coming from the stack of semistable objects and use the $K$-Hall algebra to take a formal logarithm and construct $\varepsilon$-invariants. We prove that these satisfy appropriate wall-crossing formulas using the non-abelian localization theorem. Based on work of Joyce in the cohomological setting, Liu had previously defined similar invariants assuming the existence of a framing functor; we show that when their definition of invariants makes sense it agrees with ours. Our results extend Joyce--Liu wall-crossing to non-standard hearts of $D^b(X)$, for which framing functors are not known to exist.
- [23] arXiv:2512.22362 [pdf, html, other]
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Title: On the number of words of $N=3 \,M$ letters with a three-letter alphabetComments: 7 pagesSubjects: Combinatorics (math.CO)
In this paper we address the well-known problem of counting the number of $3M$-letter words that can be formed from a three-letter alphabet by decomposing it into four possible cases based on its remainder when divided by three. The solution to the problem also gives us some sums of trinomial coefficients.
- [24] arXiv:2512.22366 [pdf, html, other]
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Title: Time Reparametrization, Not Fractional Calculus: A Reassessment of the Conformable DerivativeComments: 20 pages, 6 figures, 1 table. Critical reassessment of conformable derivative as time reparametrization; includes theoretical equivalence proofs, ODE/PDE reformulations, Lorenz system analysis, and numerical comparisons with Caputo derivativeSubjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph); Dynamical Systems (math.DS); Numerical Analysis (math.NA)
The conformable derivative has been promoted in numerous publications as a new fractional derivative operator. This article provides a critical reassessment of this claim. We demonstrate that the conformable derivative is not a fractional operator but a useful computational tool for systems with power-law time scaling, equivalent to classical differentiation under a nonlinear time reparametrization. Several results presented in the literature as novel fractional contributions can be reinterpreted within a classical framework. We show that problems formulated using the conformable derivative can be transformed into classical formulations via a change of variable. The solution is derived classically and then transformed back, this reformulation highlights the absence of genuinely nonlocal fractional effects. We provide a theoretical analysis, numerical simulations comparing conformable, classical, and truly fractional (Caputo) models, and discuss the reasons why this misconception persists. Our results suggest that classical derivatives, as well as established fractional derivatives, offer a more faithful framework for modeling memory-dependent phenomena.
- [25] arXiv:2512.22391 [pdf, html, other]
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Title: Derived Gamma Geometry II: Stable $\infty$-Categories of Gamma-Modules, Derived Monoidal Structures, and Obstructions to Binary ShadowsChandrasekhar Gokavarapu (Government College (Autonomous), Rajahmundry, Andhra Pradesh, India)Subjects: Rings and Algebras (math.RA)
Let \(\T\) be a commutative ternary \(\Gm\)-semiring in the sense of the triadic, \(\Gm\)-parametrized multiplication \(\{a,b,c\}_{\gamma}\). Building on the affine \(\Gm\)-spectrum \(\SpecG(\T)\), the structure sheaf, and the equivalence between \(\Gm\)-modules and quasi-coherent \(\Gm\)-sheaves on affine \(\Gm\)-schemes, we construct and organize the derived formalism at the level of stable \(\infty\)-categories.
Our first contribution is a technically explicit construction of a stable \(\infty\)-category \(\Dinfty(\T,\Gm)\) enhancing the unbounded derived category of \(\Gm\)-modules, obtained by dg-nerve and \(\infty\)-localization of chain complexes. We further explain the derived monoidal structure induced by the ternary \(\Gm\)-tensor product and the corresponding internal \(\RHom\), under standard exactness/projectivity hypotheses.
Our second contribution is an obstruction theory to \emph{binary reduction}: we formalize the nonexistence of any conservative ``binary module shadow'' compatible with the cubic localization calculus intrinsic to ternary \(\Gm\)-semirings. In particular, any attempt to represent the triadic \(\Gm\)-action by binary scalars forces \(\Gm\)-mode data to be absorbed into the scalars, hence ceases to be a genuine reduction.
Finally, we give a detailed affine derived equivalence between derived quasi-coherent \(\Gm\)-sheaves on \(X=\SpecG(\T)\) and \(\Dinfty(\T,\Gm)\), and we include worked examples illustrating the cubic localization relation and its derived consequences. - [26] arXiv:2512.22394 [pdf, html, other]
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Title: Preliminaries on Pre-Hilbert Structures on Polynomial Spaces and Associated LaplaciansSubjects: General Mathematics (math.GM)
We study orthogonal polynomial systems arising from general pre-Hilbert inner products on polynomial spaces, beyond the classical framework of measures. To each such inner product we associate a canonical Laplacian defined from an abstract derivation, and we investigate the operator-theoretic structures induced by this construction.
Our main contribution is the introduction of a resolvent-based distance between polynomial Hilbert geometries, and the proof of quantitative stability results for finite-degree orthogonalization procedures. In particular, we show that norm-resolvent closeness of the associated Laplacians implies stability of Gram--Schmidt orthogonal bases, orthogonal projectors and reproducing kernels on all finite-dimensional polynomial subspaces.
The general theory is illustrated by several explicit examples. We analyze in detail the case of orthogonal polynomials on the unit circle, comparing classical $L^2$ geometries associated with finite Radon measures and Sobolev-type regularizations via Fourier methods. We also revisit the thin annulus problem, showing that its asymptotic regime admits a natural interpretation as a resolvent limit of polynomial geometries.
These results provide a unified operator-theoretic framework for the study of stability, degenerations and geometric limits of orthogonal polynomial systems. - [27] arXiv:2512.22401 [pdf, other]
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Title: $U_q(\mathfrak{gl}(m|n))$ bounds on the minimal genus of virtual linksComments: 49 pages, 39 figures, comments welcome. Supplemental Mathematica notebook available at: this https URLSubjects: Geometric Topology (math.GT)
For links $L \subset \Sigma \times [0,1]$, where $\Sigma$ is a closed orientable surface, we define a $U_q(\mathfrak{gl}(1|1))$ Reshetikhin-Turaev invariant with coefficients in $\mathbb{Z}[H_1(\Sigma)]$. This invariant turns out to be equivalent to an infinite cyclic version of the Carter-Silver-Williams (CSW) polynomial. The importance of the CSW polynomial is that half its symplectic rank gives strong lower bounds on the virtual genus. Recall that the virtual genus of a virtual link $J$ is the smallest genus of all closed orientable surfaces $\Sigma$ on which $J$ can be represented by a link diagram on $\Sigma$. Here we generalize the CSW lower bound to all quantum supergroups $U_q(\mathfrak{gl}(m|n))$ with $m,n>0$. For $(m,n)=(1,1)$, the $U_q(\mathfrak{gl}(m|n))$ bound is the same as the CSW bound. However, changing the value of the pair $(m,n)$ can give lower bounds better than those available from other known methods. We compare the $U_q(\mathfrak{gl}(m|n))$ lower bounds to those coming from the CSW polynomial, the surface bracket, the arrow polynomial, hyperbolicity, and the Gordon-Litherland determinant test. As a first application, we show that the Seifert genus of homologically trivial knots in thickened surfaces is not additive under the connected sum operation of virtual knots. As a second application, we prove that the Jaeger-Kauffman-Saleur invariant of a virtual knot is always realizable as the Alexander polynomial of an infinite cyclic cover of a knot complement in some $\Sigma \times [0,1]$, but is not always so on a surface of minimal genus. This is accomplished with a generalization of the $Zh$-construction, called the homotopy $Zh$-construction.
- [28] arXiv:2512.22403 [pdf, html, other]
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Title: Active Nonparametric Two-Sample Testing by Betting on Heterogeneous Data SourcesSubjects: Statistics Theory (math.ST); Information Theory (cs.IT)
We study the problem of active nonparametric sequential two-sample testing over multiple heterogeneous data sources. In each time slot, a decision-maker adaptively selects one of $K$ data sources and receives a paired sample generated from that source for testing. The goal is to decide as quickly as possible whether the pairs are generated from the same distribution or not. The gain achieved by such adaptive sampling (in terms of smaller expected stopping time or larger error exponents) has been well-characterized for parametric models via Chernoff's adaptive MLE selection rule [1]. However, analogous results are not known for the case of nonparametric problems, such as two-sample testing, where we place no restrictions on the distributions.
Our main contribution is a general active nonparametric testing procedure that combines an adaptive source-selecting strategy within the testing-by-betting framework of [2] that works under minimal distributional assumptions. In each time slot, our scheme proceeds by selecting a source according to a probability that mixes exploitation, favoring sources with the largest empirical distinguishability, and exploration via a vanishing greedy strategy. The (paired) observations so collected are then used to update the "betting-wealth process", which is a stochastic process guaranteed to be a nonnegative martingale under the null. The procedure stops and rejects the null when the wealth process exceeds an appropriate threshold; an event that is unlikely under the null. We show that our test controls the type-I error at a prespecified level-$\alpha$ under the null, and establish its power-one property and a bound on its expected sample size under the alternative. Our results provide a precise characterization of the improvements achievable by a principled adaptive sampling strategy over its passive analog. - [29] arXiv:2512.22410 [pdf, html, other]
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Title: Connectivity of $p$-subgroup posets with irreducible charactersComments: 16 pagesSubjects: Group Theory (math.GR); Combinatorics (math.CO)
Let $G$ be a finite group. For a prime $p$ and an integer $e \geq 0$, we denote by $\Gamma_{p,e}(G)$ the set of all pairs $(H, \varphi)$, where $H$ is a $p$-subgroup of $G$ of order greater than $p^e$ and $\varphi$ is a complex irreducible character of $H$. In this paper, we investigate the connected components of the poset $\Gamma_{p,e}(G)$. For the case $e = 0$, we prove that $\Gamma_{p,0}(G)$ is disconnected if and only if either $G$ has a strongly $p$-embedded subgroup, or every Sylow $p$-subgroup of $G$ contains a unique subgroup of order $p$. Furthermore, for $e = 1$ and $G$ a $p$-group, we show that the number of connected components of $\Gamma_{p,1}(G)$ equals the order of the intersection of all subgroups of $G$ of order $p^2$.
- [30] arXiv:2512.22412 [pdf, html, other]
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Title: Sequential change-point detection for generalized Ornstein-Uhlenbeck processesSubjects: Statistics Theory (math.ST)
In this article, we study sequential change-point methods for discretely observed generalized Ornstein-Uhlenbeck processes with periodic drift. Two detection methods are proposed, and their respective performance is studied through numerical experiments for several choices of parameters.
- [31] arXiv:2512.22413 [pdf, html, other]
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Title: Sierpinski's Hypothesis H1Matt Visser (Victoria University of Wellington)Comments: 15 pagesSubjects: Number Theory (math.NT)
Sierpinski's Hypothesis H1, formulated in 1958, is the conjecture that (provided $n\geq 2$), when the first $n^2$ counting numbers, $1, 2,3,\dots n^2$, are arranged in a square, then each row contains at least one prime. This conjecture is particularly interesting in that it subsumes and is stronger than both the Oppermann and Legrendre conjectures. Herein I shall verify Sierpinski's Hypothesis H1 for (at least) the first $n \leq \hbox{4 553 432 387} \approx 4.5 \hbox{ billion}$ of these Sierpinski matrices. I shall also demonstrate some partial but more general results. For example: Even for arbitrary $n\geq \hbox{4 553 432 388}$ at least one quarter of the rows of the $n$th Sierpinski matrix contain at least one prime. Furthermore, even for arbitrary $n\geq \hbox{4 553 432 388}$ at least the first $\hbox{131 294}$ rows of the $n$th Sierpinski matrix always contain at least one prime. These and related results are obtained largely by using the locations and values of the known maximal prime gaps, the pigeonhole principle, and some recent bounds on the first Chebyshev function.
- [32] arXiv:2512.22419 [pdf, html, other]
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Title: A Decomposition Method for Solving Sensitivity-Based Distributed Optimal Power FlowSubjects: Optimization and Control (math.OC); Systems and Control (eess.SY)
Efficiently solving large-scale optimal power flow (OPF) problems is challenging due to the high dimensionality and interconnectivity of modern power systems. Decomposition methods offer a promising solution via partitioning large problems into smaller subproblems that can be solved in parallel, often with local information. These approaches reduce computational burden and improve flexibility by allowing agents to manage their local models. This article introduces a decomposition method that enables a distributed solution to OPF problems. The proposed method solves OPF problems with a sensitivity-based formulation using the alternating direction method of multipliers (ADMM) algorithm. We also propose a distributed method to compute system-wide sensitivities without sharing local parameters. This approach facilitates scalable optimization while satisfying global constraints and limiting data sharing. We demonstrate the effectiveness of the proposed approach using a large set of test systems and compare its performance against existing decomposition methods. The results show that the proposed method significantly outperforms the typical phase-angle formulation with a 14-times faster computation speed on average.
- [33] arXiv:2512.22421 [pdf, html, other]
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Title: Differentiable Inverse Modeling with Physics-Constrained Latent Diffusion for Heterogeneous Subsurface Parameter FieldsComments: 33 pages, 16 figuresSubjects: Numerical Analysis (math.NA); Machine Learning (cs.LG); Geophysics (physics.geo-ph)
We present a latent diffusion-based differentiable inversion method (LD-DIM) for PDE-constrained inverse problems involving high-dimensional spatially distributed coefficients. LD-DIM couples a pretrained latent diffusion prior with an end-to-end differentiable numerical solver to reconstruct unknown heterogeneous parameter fields in a low-dimensional nonlinear manifold, improving numerical conditioning and enabling stable gradient-based optimization under sparse observations. The proposed framework integrates a latent diffusion model (LDM), trained in a compact latent space, with a differentiable finite-volume discretization of the forward PDE. Sensitivities are propagated through the discretization using adjoint-based gradients combined with reverse-mode automatic differentiation. Inversion is performed directly in latent space, which implicitly suppresses ill-conditioned degrees of freedom while preserving dominant structural modes, including sharp material interfaces. The effectiveness of LD-DIM is demonstrated using a representative inverse problem for flow in porous media, where heterogeneous conductivity fields are reconstructed from spatially sparse hydraulic head measurements. Numerical experiments assess convergence behavior and reconstruction quality for both Gaussian random fields and bimaterial coefficient distributions. The results show that LD-DIM achieves consistently improved numerical stability and reconstruction accuracy of both parameter fields and corresponding PDE solutions compared with physics-informed neural networks (PINNs) and physics-embedded variational autoencoder (VAE) baselines, while maintaining sharp discontinuities and reducing sensitivity to initialization.
- [34] arXiv:2512.22432 [pdf, html, other]
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Title: Divisorial fans and algebraic torus actions over arbitrary fieldsComments: 54 pages. Comments are welcome:)Subjects: Algebraic Geometry (math.AG)
We provide a algebro-geometric combinatorial description of geometrically integral geometrically normal varieties endowed with an effective action of an algebraic torus over arbitrary fields. This description is achieved in terms of divisorial fans endowed with a Galois semilinear action. This work concludes the description of normal $T$-varieties over fields.
- [35] arXiv:2512.22436 [pdf, html, other]
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Title: Regularity of solutions of the Navier-Stokes-αβ equations with wall-eddy boundary conditionsComments: 24 pagesSubjects: Analysis of PDEs (math.AP)
We establish global well-posedness and regularity for the Navier-Stokes-{\alpha}{\beta} system endowed with the wall-eddy boundary conditions proposed by Fried and Gurtin (2008). These conditions introduce a tangential vorticity traction proportional to wall vorticity and provide a continuum-mechanical model for near-wall turbulence. Our analysis begins with a variational formulation of the stationary fourth-order system, where we prove symmetry and a Gårding inequality for the associated bilinear form. We then verify Douglis-Nirenberg ellipticity and the Lopatinskii-Shapiro covering condition, establishing full Agmon-Douglis-Nirenberg regularity for the coupled system. Building on this framework, we derive a hierarchy of energy estimates for the nonlinear evolution equation, which yields global regularity, uniqueness, and stability. To our knowledge, this provides the first complete analytical treatment of the wall-eddy boundary model of Fried and Gurtin.
- [36] arXiv:2512.22444 [pdf, html, other]
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Title: Three-Dimensional Almost Contact Metric Manifolds Revisited via the Newman-Penrose FormalismComments: 19pages, 1 figure, comment welcomeSubjects: Differential Geometry (math.DG)
This paper applies the Newman-Penrose formalism-a technique primarily used in General Relativity-to the analysis of three-dimensional almost contact metric (ACM) manifolds. We reformulate and discuss several known notions and properties within the Newman-Penrose framework, demonstrating the applicability of the method in this geometric context. Furthermore, as an application showcasing the utility of the formalism, we address the classification of three-dimensional compact normal ACM manifolds, or equivalently trans-Sasakian manifolds, that admit an $\eta$-Einstein metric.
- [37] arXiv:2512.22451 [pdf, html, other]
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Title: Zeros of Polynomials in Derivatives of Automorphic $L$-functionsComments: 29 pages, 1 figureSubjects: Number Theory (math.NT)
Let $\mathfrak{F}_m$ be the set of all cuspidal automorphic representations of $\mathrm{GL}_m(\mathbb{A}_{\mathbb{Q}})$, and let $F(s,\boldsymbol{\pi})$ be a polynomial in the derivatives of $L$-functions associated with representations $\pi_u \in \cup_{m=1}^{\infty} \mathfrak{F}_m$. We establish an asymptotic formula for the number of nontrivial zeros of $F(s,\boldsymbol{\pi})$ with $0 < \operatorname{Im}(s) < T$. We explicitly determine the main term of this formula in terms of the degrees, the ranks, the arithmetic conductors, and the orders of differentiation of the component $L$-functions. Furthermore, we show that, under certain conditions, almost all nontrivial zeros of $F(s,\boldsymbol{\pi})$ lie near the critical line $\operatorname{Re}(s)=1/2$.
- [38] arXiv:2512.22456 [pdf, html, other]
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Title: The Burness-Giudici Conjecture on Primitive Groups of Lie-type with Rank One: Part (II)Subjects: Group Theory (math.GR)
It was conjectured by Burness and Giudici that every primitive permutation group $G$ containing some regular suborbits has the property that $\Gamma \cap \Gamma^g\neq \emptyset$ for any $g\in G$, where $\Gamma$ is the union of all regular suborbits of $G$ relative to $\alpha$. We focused on proving the conjecture for all primitive groups $G$ whose socle is a simple group of Lie-type of rank $1$, that is, those with $soc(G)\in \{PSL(2,q), PSU(3,q), Ree(q),Sz(q)\}$. The case of $soc(G)=PSL(2,q)$ has been published in two papers. The case of $soc(G)=PSU(3,q)$ is divided into two parts:Part (I) addressed all primitive groups $G$ with socle $PSU(3,q)$ whose point stabilizers contain $PSO(3,q)$ and based on it, this Part (II) will finish all cases of $soc(G)=PSU(3,q)$. The cases for $soc(G)\in\{Ree(q),Sz(q)\}$ will be treated in Part (III).
- [39] arXiv:2512.22458 [pdf, html, other]
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Title: CR Yamabe Equation on the Heisenberg Group via the method of moving spheresComments: 21 pagesSubjects: Analysis of PDEs (math.AP)
In this paper, we classify positive solutions to the CR Yamabe equation on the Heisenberg group $\mathbb{H}^n$. We show that all such solutions are Jerison-Lee bubbles, without imposing any finite-energy or a priori symmetry assumptions. This result can be regarded as an analogue for $\mathbb{H}^n$ of the celebrated Caffarelli-Gidas-Spruck classification theorem in $\mathbb{R}^n$. To establish this, we develop a systematic approach to implement the method of moving spheres in the setting of the Heisenberg group.
- [40] arXiv:2512.22459 [pdf, html, other]
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Title: The Burness-Giudici Conjecture on Primitive Groups of Lie-type with Rank One: Part (I)Subjects: Group Theory (math.GR)
It was conjectured by Burness and Giudici that every primitive permutation group $G$ containing some regular suborbits has the property that $\Gamma \cap \Gamma^g\neq \emptyset$ for any $g\in G$, where $\Gamma$ is the union of all regular suborbits of $G$ relative to $\alpha$. We focused on proving the conjecture for all primitive groups $G$ whose socle is a simple group of Lie-type of rank $1$, that is, those with $soc(G)\in \{PSL(2,q), PSU(3,q), Ree(q),Sz(q)\}$. The case of $soc(G)=PSL(2,q)$ has been published in two papers. The case of $soc(G)=PSU(3,q)$ is divided into two parts: this paper constitutes Part (I), addressing all primitive groups $G$ with socle $PSU(3,q)$ whose point stabilizers contain $PSO(3,q)$; and the remaining primitive actions will be covered in Part (II). The cases for $soc(G)\in\{Ree(q),Sz(q)\}$ are treated in Part (III).
To finsh this work, we draw on methods from abstract- and permutation- group theory, finite unitary geometry, probabilistic approach, number theory (employing Weil's bound), and, most importantly, algebraic combinatorics, which provides us some key ideas. - [41] arXiv:2512.22461 [pdf, html, other]
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Title: The Burness-Giudici Conjecture on Primitive Groups of Lie-type with Rank One: Part (III)Subjects: Group Theory (math.GR)
It was conjectured by Burness and Giudici that every primitive permutation group $G$ containing some regular suborbits has the property that $\Gamma \cap \Gamma^g\neq \emptyset$ for any $g\in G$, where $\Gamma$ is the union of all regular suborbits of $G$ relative to $\alpha$. We focused on proving the conjecture for all primitive groups $G$ whose socle is a simple group of Lie-type of rank $1$, that is, those with $soc(G)\in \{PSL(2,q), PSU(3,q), Ree(q), Sz(q)\}$. The case of $soc(G)=PSL(2,q)$ has been published in two papers. The other three cases are divided into three parts, where the case of $soc(G)=PSU(3,q)$ has been submitted in Part (I) and (II); and the case of $soc(G)\in\{Ree(q),Sz(q)\}$ will be finished in this paper (Part III).
- [42] arXiv:2512.22480 [pdf, html, other]
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Title: Inverse scattering for waveguides in topological insulatorsSubjects: Mathematical Physics (math-ph); Analysis of PDEs (math.AP); Numerical Analysis (math.NA)
This paper concerns the inverse scattering problem of a topologically non-trivial waveguide separating two-dimensional topological insulators. We consider the specific model of a Dirac system. We show that a short-range perturbation can be fully reconstructed from scattering data in a linearized setting and in a finite-dimensional setting under a smallness constraint. We also provide a stability result in appropriate topologies. We then solve the problem numerically by means of a standard adjoint method and illustrate our theoretical findings with several numerical simulations.
- [43] arXiv:2512.22482 [pdf, html, other]
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Title: Spectral supersaturation for color-critical graphsComments: 31 pages. Any suggestions are welcomeSubjects: Combinatorics (math.CO)
A graph is color-critical if it contains an edge whose deletion reduces its chromatic number. This class of graphs, including cliques and odd cycles, plays a central role in extremal graph theory. In this paper, following an influential line of research initiated by Bollobás-Nikiforov, we study the spectral supersaturation problem for color-critical graphs. Let $T_{n,r}$ be the $r$-partite Turán graph, let $\mathcal{T}_{n,r,q}$ denote the family of graphs obtained from $T_{n,r}$ by adding $q$ edges, and let $\lambda(G)$ be the spectral radius of a graph $G$. We first prove that for any color-critical graph $F$ with chromatic number $r+1$, there exists $\delta_F > 0$ such that for sufficiently large $n$ and all $1 \leq q \leq \delta_F \sqrt{n}$, any $n$-vertex graph $G$ with $\lambda(G) \ge \min_{T \in \mathcal{T}_{n,r,q}} \lambda(T)$ contains at least $q \cdot c(n,F)$ copies of $F$, where $c(n,F)$ denotes the minimum number of copies of $F$ created by adding a single edge to $T_{n,r}$; moreover, any extremal graph $G$ must belong to $ \mathcal{T}_{n,r,q}$.Next, we prove a spectral supersaturation result for the analogous condition $\lambda(G) \ge \max_{T \in \mathcal{T}_{n,r,q}} \lambda(T)$, valid for all $1 \leq q \leq \delta_F n$. Together, these results provide a complete resolution to a problem proposed by Ning-Zhai, and establish a spectral counterpart to the well-known results of Mubayi and Pikhurko-Yilma in the extremal supersaturation setting. A notable feature of our first result is that the restriction $q = O(\sqrt{n})$ is tight up to a constant factor, in contrast to the linear bounds provided by other settings discussed above. As applications, we extend a result of Liu-Mubayi, and solve a related conjecture by Li-Lu-Peng.
- [44] arXiv:2512.22493 [pdf, html, other]
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Title: Finite propagation and saturation in reaction-diffusion-advection equations governed by p-Laplacian operatorSubjects: Analysis of PDEs (math.AP)
The paper concerns front propagation for the following mono-stable reaction-diffusion-advection equation \[f(u)u_x + g(u)u_\tau = [d(u)|u_x|^{p-2} u_x]_x+ \rho(u), \quad (x,\tau)\in \R\times [0,+\infty).\] Besides existence and non-existence results for traveling wave solutions, the main focus is their classification: we provide criteria to establish if they attain one or both the equilibria at a finite time and in this case, if they are continuable as $C^1$-solutions or if they are sharp solutions.
- [45] arXiv:2512.22494 [pdf, html, other]
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Title: On the Limiting Density of a gcd MapSubjects: Number Theory (math.NT); Rings and Algebras (math.RA)
The function \[f(a,b)=\frac{\gcd(a+b,ab)}{\gcd(a,b)}\] is of interest in this paper. We then ask a natural question regarding how often $f(a,b)=1$ is. We yield the limiting density $\rho=\prod_{p}\left(1-\frac{1}{p^2(p+1)}\right)\approx 0.88151$ which is an Euler product that unexpectedly matches the quadratic class number constant from the theory of real quadratic fields. We also consider its higher-order analogue $f_r$, where the problem collapses to coprimality and the density becomes $1/\zeta(2)=6/\pi^2$.
- [46] arXiv:2512.22497 [pdf, html, other]
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Title: Projections in the Algebra generated by an n-Potent OperatorSubjects: Functional Analysis (math.FA)
This paper investigates the projection operators that lie in the algebra generated by powers of an $n$-potent operator $T$ on a complex Banach space, where $T^n = T$. We give a complete description of all projections in the algebra $\operatorname{comb}(T) = \text{span}\{T, T^2, \dots, T^{n-1}\}$, and prove that each such projection is uniquely determined by, and in bijection with, a subset of the nonzero spectrum of $T$. As a consequence, the family of projections in $\operatorname{comb}(T)$ forms a Boolean algebra isomorphic to the power set of $\sigma(T)\setminus\{0\}$. We also establish a spectral decomposition for $n$-potent operators in terms of their Riesz projections and derive explicit formulas for the associated Riesz projections using resolvent expansions. We give an illustration of the theory for $5$-potent operators, which highlights the algebraic and spectral structure of finite-order operators on Banach spaces.
- [47] arXiv:2512.22498 [pdf, html, other]
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Title: Solvability of Dirichlet boundary value problems governed by non-monotone differential operatorsSubjects: Classical Analysis and ODEs (math.CA)
We prove existence results for Dirichlet boundary value problems for equations of the type
\begin{align*}
\left( \Phi(k(t) x'(t) ) \right)' = f(t, x(t) , x'(t) ) \qquad \text{for a.e. } t \in I:=[0,T] ,
\end{align*}
where $\Phi : J \to \mathbb{R} $ is a generic possibly non-monotone differential operator defined in a open interval $J\subseteq \mathbb{R}$,
$k:I \to \mathbb{R}$, $k$ is measurable with $k(t) >0$ for a.e. $t \in I$ and $f: \mathbb{R}^3 \to \mathbb{R}$ is a Carathéodory function.
Under very mild assumptions, we prove the existence of solutions for suitably prescribed boundary conditions, and we also address the study of the existence of heteroclinic solutions on the half-line $[0,+\infty)$. - [48] arXiv:2512.22509 [pdf, html, other]
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Title: First Moment of Quadratic Hecke $L$-Functions with Lower Order TermComments: 15 pagesSubjects: Number Theory (math.NT)
We evaluate the first moment of the family of primitive quadratic Hecke $L$-functions in the Gaussian field using the method of double Dirichlet series under the Riemann hypothesis and the Lindelöf hypothesis. We obtain asymptotic formulas with secondary main terms and error terms of size that is one quarter of that of the main term.
- [49] arXiv:2512.22512 [pdf, html, other]
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Title: Small-time approximate controllability for the nonlinear complex Ginzburg-Landau equation with bilinear controlComments: 26 pagesSubjects: Optimization and Control (math.OC); Analysis of PDEs (math.AP)
In this paper, we consider the bilinear approximate controllability for the complex Ginzburg-Landau (CGL) equation with a power-type nonlinearity of any integer degree on a torus of arbitrary space dimension. Under a saturation hypothesis on the control operator, we show the small-time global controllability of the CGL equation. The proof is obtained by developing a multiplicative version of a geometric control approach, introduced by Agrachev and Sarychev in \cite{AS05,AS06}.
- [50] arXiv:2512.22516 [pdf, html, other]
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Title: On Oscillatory Integral Operators Satisfying the cinematic curvature conditionSubjects: Classical Analysis and ODEs (math.CA)
We study the oscillatory integral operators satisfying the cinematic curvature condition. First, we formulate a conjecture for this class of operators, motivated by certain necessary conditions arising from counterexamples. We then establish an estimate for these operators by combining Wolff's two-ends reduction with refined decoupling inequalities.
- [51] arXiv:2512.22517 [pdf, html, other]
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Title: The $\mathrm{L}^p$-index of the Hodge-Dirac operator on compact Riemannian manifoldsComments: 40 pagesSubjects: Functional Analysis (math.FA); Differential Geometry (math.DG); K-Theory and Homology (math.KT)
We investigate the spectral and index-theoretic properties of the Hodge-Dirac operator $D = \mathrm{d} + \mathrm{d}^*$ acting on the Banach space $\mathrm{L}^p(\Omega^\bullet(M))$ of differential forms over an oriented compact Riemannian manifold $M$. Relying on the compactness of $M$, we establish the existence of a bounded bisectorial $\mathrm{H}^\infty$ functional calculus for $D$, without curvature assumptions. This result enables us to prove that the triple $(\mathrm{C}(M), \mathrm{L}^p(\Omega^\bullet(M)), D)$ constitutes a compact Banach spectral triple. We then define a consistent pairing between the Banach K-homology and the K-theory of the manifold, proving the invariance of the Fredholm index with respect to $p$. We recover the classical Euler characteristic and the Hirzebruch signature as $\mathrm{L}^p$-indices, demonstrating the effectiveness of Banach noncommutative geometry for geometric analysis, beyond the Hilbertian setting.
- [52] arXiv:2512.22518 [pdf, html, other]
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Title: Notes on model structures on preordersComments: 42 pages, 6 figures. Relevant codes are linked within the paperSubjects: Category Theory (math.CT); Algebraic Topology (math.AT)
Given subsets $\mathcal{C},\mathcal{F}$ of a preorder $\mathcal{A}$, we give necessary and sufficient conditions for $\mathcal{A}$ to admit the structure of a model category whose cofibrant objects are $\mathcal{C}$ and whose fibrant objects are $\mathcal{F}$. We give various classification results for model structures on preorders by describing model structures in terms of their fibrant and cofibrant objects, or in terms of their (co)fibrant replacment (co)monads. This leads to a construction which takes topologies and matroids as input, and produces model structures on Boolean algebras. We carry out some detailed case studies, calculating all model structures on small Boolean algebras, and all the Bousfield localization and colocalization relations between them.
- [53] arXiv:2512.22520 [pdf, html, other]
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Title: The $L$-function of the surface parametrizing cuboidsComments: 9 pagesSubjects: Number Theory (math.NT); Algebraic Geometry (math.AG)
In this note, we compute the $L$-function of the projective smooth surface $S$ over $\mathbb{Q}$ that parametrizes cuboids whose geometric properties are studied in detailby Stoll and Testa. As a byproduct, we completely determine the structure of ${\rm Pic}(S_{\overline{\mathbb{Q}}})$ as a ${\rm Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$-module.
- [54] arXiv:2512.22533 [pdf, html, other]
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Title: RIS, Active RIS or RDARS: A Comparative Insight Through the Lens of Energy EfficiencySubjects: Information Theory (cs.IT)
Multiplicative fading is a major limitation of reconfigurable intelligent surfaces (RIS), restricting their effective coverage in both existing sub-6GHz systems and future mmWave networks. Although active RIS architectures mitigate this issue, they require high power consumption and introduce practical challenges due to the need for integrated amplifiers. Recently, reconfigurable distributed antenna and reflecting surfaces (RDARS) have been proposed to alleviate multiplicative fading through connected modes. In this work, we compare RIS, active RIS, and RDARS in terms of coverage and energy efficiency (EE) in both sub-6GHz and mmWave bands, and we investigate the impact of placement and the number of elements of reconfigurable surface (RS) on EE and coverage. The simulation results show that RDARS offers a highly energy-efficient alternative of enhancing coverage in sub-6GHz systems, while active RIS is significantly more energy-efficient in mmWave systems. Additionally, for a lower number of RS elements and for near UEs, RIS remains considerably more energy-efficient than both active RIS and RDARS.
- [55] arXiv:2512.22543 [pdf, html, other]
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Title: Wave dynamics governing vortex breakdown in smooth Euler flowsSubjects: Analysis of PDEs (math.AP)
We consider the three-dimensional incompressible Euler equations in a setting where a vortex is transported by a prescribed Lagrangian flow. We show that vortex breakdown is governed by wave dynamics generated by the underlying transport flow. The key idea is to avoid any singular integral representation of the pressure term and instead construct an effective Lagrangian coordinate system in which the associated Lie bracket vanishes identically.
- [56] arXiv:2512.22544 [pdf, other]
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Title: The Stochastic Six Vertex model and discrete Orthogonal Polynomial ensemblesComments: 159 pages, 11 figuresSubjects: Mathematical Physics (math-ph); Probability (math.PR)
Stochastic growth models in the Kardar-Parisi-Zhang (KPZ) universality class exhibit remarkable fluctuation phenomena. While a variety of powerful methods have led to a detailed understanding of their typical fluctuations or large deviations, much less is known about behavior on intermediate, or moderate deviation, scales. Addressing this problem requires refined asymptotic control of the integrable structures underlying KPZ models.
Motivated by this perspective, we study multiplicative statistics of discrete orthogonal polynomial ensembles (dOPEs) in different scaling regimes, with a particular focus on applications to tail probabilities of the height function in the stochastic six-vertex model. For a large class of dOPEs, we obtain robust singular asymptotic estimates for multiplicative statistics critically scaled near a saturated-to-band transition. These asymptotics exhibit universal crossover behavior, interpolating between Airy, Painlevé XXXIV, and Bessel-type regimes. Our proofs employ the Riemann-Hilbert Problem (RHP) approach to obtain asymptotics for the correlation kernel of a deformed version of the dOPE across the critical scaling windows. These asymptotics are then used on a double integral formula relating this kernel to partition function ratios, which may be of independent interest. At the technical level, the RHP analysis requires a novel parameter-dependent local parametrix, which needs a separate asymptotic analysis of its own.
Using these results, together with a known identity relating a Laplace-type transform of the stochastic six-vertex model height function to a multiplicative statistic of the Meixner point process, we derive moderate deviation estimates for the height function in both the upper and lower tail regimes, with sharp exponents and constants. - [57] arXiv:2512.22546 [pdf, other]
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Title: Projection onto the parabolaSubjects: General Mathematics (math.GM)
In this note, we provide explicit expressions for the projections onto the graph of a quadratic polynomial. The projections are obtained by examining the critical points of the associated quartic polynomial, that is, the roots of the cubic polynomial defining its derivative. We also focus on the case where the point we project lies on the vertical line defined by the parabola. Lastly, an explicit formula for the projection onto a higher dimensional parabola is derived.
- [58] arXiv:2512.22547 [pdf, other]
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Title: The fibre operators in the Bloch-Floquet decomposition of periodic magnetic pseudo-differential operatorsSubjects: Analysis of PDEs (math.AP)
We study the structure of the fibre operators corresponding to periodic magnetic pseudo-differential operators having periodic magnetic potentials. We obtain explicit formulas for their distribution kernel, both when these fibres are seen as operators on the $d$-dimensional torus, and also when they are seen as infinite matrices acting on a discrete $\ell^2$ space via a discrete Fourier transform. Moreover, using these distribution kernels we prove that the fibre operators are toroidal pseudo-differential operators.
- [59] arXiv:2512.22551 [pdf, html, other]
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Title: Asymptotics of local height pairingComments: 46 pagesSubjects: Algebraic Geometry (math.AG); Number Theory (math.NT)
We discuss the asymptotics of the Archimedean part of the Arakelov intersection number. The theorem is motivated by recent conjectures and their proof strategy by Gao and Zhang on the Northcott property of the Beilinson--Bloch height pairing. Our method involves a homological algebra interpretation of the Archimedean height by Hain. This interpretation allows us to introduce motivic viewpoints using Deligne cohomology, cycle class maps and higher Chow groups.
Especially, we compare the biextension by Hain and Brosnan--Pearlstein over $\mathbb{C}$ based on Poincaré line bundle and Hodge theory with the $\mathbb{G}_{\mathrm{m}}$-biextension of Bloch and Seibold defined by two families of homologically trivial cycles on a generically smooth family of projective varieties over a smooth curve. Our comparison, a relative version of the work of Gorchinskiy, enhances his derived viewpoint on these biextensions. Especially when the family of varieties are smooth, the two constructions are related via derived regulator maps to Deligne cohomology, reinterpreted similarly to Beilinson's absolute Hodge cohomology, as well as the derived description of Hardouin's biextension that generalizes Poincaré line bundle by Hain. The comparison when the family defined over a smooth curve has a strongly semistable reduction further involves a simple monodromy computation using mixed Hodge modules.
Along our discussion, we simplify the discussion of Bloch and Seibold, partly in the style of Gorchinskiy. For example, the symmetry of their biextension is proved more easily than their work. - [60] arXiv:2512.22554 [pdf, html, other]
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Title: On the duality between consensus problems and Markov processes, with application to delay systemsJournal-ref: Markov Processes and Related Fields, 22(3):537--553, 2016Subjects: Probability (math.PR); Mathematical Physics (math-ph)
We consider consensus of multi-agent systems as a dual problem to Markov processes. Based on an exchange of relevant notions and results between the two fields, we present a uniform framework which admits the introduction and treatment of time delays in a common setting. We study both information propagation and information processing delays, and for each case derive conditions for reaching consensus and calculate the consensus value.
- [61] arXiv:2512.22556 [pdf, html, other]
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Title: Distinctive power and comparability of Harary polynomialComments: 23 pagesSubjects: Combinatorics (math.CO)
Let $\mathcal{P}$ be a graph property. A $\mathcal{P}$-coloring with at most $k$ colors is a coloring of the vertices of a simple graph $G$ such that each color class induces a graph in $\mathcal{P}$. Harary polynomials are generalizations of the chromatic polynomial for simple graphs based on conditional colorings. We denote by $\chi_{\mathcal{P}}(G; k)$ the number of $\mathcal{P}$-colorings of $G$ with at most $k$ colors. $\chi_{\mathcal{P}}(G; k)$ is a polynomial in $\Z[k]$. A first paper studying Harary polynomials systematically was published in 2021 by this http URL, J.A. Makowsky and V. Rakita. It studies under which conditions on $\mathcal{P}$ is $\chi_{\mathcal{P}}(G; k)$ definable in Monadic Second Order Logic and under which conditions is $\chi_{\mathcal{P}}(G; k)$ a chromatic invariant. Let $\mathcal{P}, \mathcal{Q}$ be two graph properties. Two graphs $G, H$ are $\mathcal{P}$-mates if $\chi_{\mathcal{P}}(G; k) = \chi_{\mathcal{P}}(H; k)$. $\chi_{\mathcal{Q}}$ is at least as distinctive as $\chi_{\mathcal{P}}$, $\chi_{\mathcal{P}} \leq \chi_{\mathcal{Q}}$, if for all graphs $G, H$ we have that $\chi_{\mathcal{Q}}(G; k) = \chi_{\mathcal{Q}}(H; k)$ implies $\chi_{\mathcal{P}}(G; k) = \chi_{\mathcal{P}}(H; k)$. In this paper we study under which conditions on $\mathcal{P}$ are there any (many) $\mathcal{P}$-mates and under which conditions on $\mathcal{P}, \mathcal{Q}$ is $\chi_{\mathcal{Q}}$ is at least as distinctive as $\chi_{\mathcal{P}}$.
- [62] arXiv:2512.22557 [pdf, html, other]
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Title: Sharp Non-Asymptotic Bounds for the Star Discrepancy of Double-Infinite Random Matrices via Optimal Covering NumbersSubjects: Statistics Theory (math.ST)
We establish sharp non-asymptotic probabilistic bounds for the star discrepancy of double-infinite random matrices -- a canonical model for sequences of random point sets in high dimensions. By integrating the recently proved \textbf{optimal covering numbers for axis-parallel boxes} (Gnewuch, 2024) into the dyadic chaining framework, we achieve \textbf{explicitly computable constants} that improve upon all previously known bounds.
For dimension $d \ge 3$, we prove that with high probability, \[ D_N^d \le \sqrt{\alpha A_d + \beta B \frac{\ln \log_2 N}{d}} \sqrt{\frac{d}{N}}, \] where $A_d$ is given by an explicit series and satisfies $A_3 \le 745$, a \textbf{14\% improvement} over the previous best constant of 868 (Fiedler et al., 2023). For $d=2$, we obtain the currently smallest known constant $A_2 \le 915$.
Our analysis reveals a \textbf{precise trade-off} between the dimensional dependence and the logarithmic factor in $N$, highlighting how optimal covering estimates directly translate to tighter discrepancy bounds. These results immediately yield improved error guarantees for \textbf{quasi-Monte Carlo integration, uncertainty quantification, and high-dimensional sampling}, and provide a new benchmark for the probabilistic analysis of geometric discrepancy.
\textbf{Keywords:} Star discrepancy, double-infinite random matrices, covering numbers, dyadic chaining, high-dimensional integration, quasi-Monte Carlo, probabilistic bounds. - [63] arXiv:2512.22561 [pdf, html, other]
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Title: Robust generalized S-ProcedureComments: 11 pagesSubjects: Optimization and Control (math.OC)
We introduce in this paper the so-called robust generalized S-procedure associated with a given robust optimization problem. We provide a primal characterization for the validity of this procedure as well as a dual characterization under the assumption that the decision space is locally convex. We also analyze an extension of the mentioned robust S-procedure that incorporates a right-hand side function.
- [64] arXiv:2512.22567 [pdf, html, other]
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Title: ROM for Viscous, Incompressible Flow in Polygons -- exponential $n$-width bounds and convergence rateSubjects: Numerical Analysis (math.NA)
We demonstrate exponential convergence of Reduced Order Model (ROM) approximations for mixed boundary value problems of the stationary, incompressible Navier-Stokes equations in plane, polygonal domains $\Omega$. Admissible boundary conditions comprise mixed BCs, no-slip, slip and open boundary conditions, subject to corner-weighted analytic boundary data and volume forcing. The small data hypothesis is assumed to ensure existence of a unique weak solution in the sense of Leray-Hopf. Recent results on corner-weighted, analytic regularity of velocity and pressure fields in $\Omega$, imply exponential convergence rates of so-called mixed $hp$-Finite Element Methods in $H^1(\Omega)^2\times L^2(\Omega)$ on sequences of geometric partitions of $\Omega$, with corner-refinement. Based on these exponential convergence rate bounds, we infer exponential bounds for the Kolmogorov $n$-widths of solution sets for analytic forcing and boundary data. This implies corresponding exponential convergence rates of POD Galerkin methods that are based on truth solutions which are obtained offline from low-order, divergence stable mixed Finite Element discretizations. Numerical experiments confirm the exponential rates and the theoretical results.
- [65] arXiv:2512.22580 [pdf, html, other]
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Title: On the growth rate of the Stanley-Wilf limit of blockable permutationsSubjects: Combinatorics (math.CO)
Given a permutation $\pi$, let $\text{Av}_n(\pi)$ be the number of permutations of length $n$ that avoid $\pi$ as a subpermutation. The celebrated resolution of the Stanley-Wilf conjecture by Marcus and Tardos confirmed that the limit $L(\pi) = \lim_{n \to \infty} |\text{Av}_n(\pi)|^{1/n}$ exists. A central and challenging question concerns the behavior of $L(\pi)$ as a function of the pattern length $|\pi|$. While Fox proved that $L(\pi)$ is exponential in $|\pi|$ for almost all permutations, it is known that $L(\pi)$ grows polynomially for specific structural classes. For instance, $L(\pi)$ is known to be quadratic in $|\pi|$ when $\pi$ is a monotone or a layered permutation. In this paper, we address this question for {\it blockable} permutations $\pi$.
- [66] arXiv:2512.22584 [pdf, html, other]
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Title: Characterization of Matrix $K$-Positivity Preserver for $K=\mathbb{R}^n$ and for Compact Sets $K\subseteq\mathbb{R}^n$Subjects: Functional Analysis (math.FA); Algebraic Geometry (math.AG)
For any closed $K\subseteq\mathbb{R}^n$, in [P.\ J.\ di\,Dio, K.\ Schmüdgen: $K$-Positivity Preserver and their Generators, SIAM J.\ Appl.\ Algebra Geom.\ 9 (2025), 794--824] all $K$-positivity preserver have been characterized, i.e., all linear maps $T:\mathbb{R}[x_1,\dots,x_n]\to\mathbb{R}[x_1,\dots,x_n]$ such that $Tp\geq 0$ on $K$ for all $p\geq 0$ on $K$. An important extension of polynomials $\mathbb{R}[x_1,\dots,x_n]$ with real coefficients are polynomials $\mathbb{R}^{m\times m}[x_1,\dots,x_n]$ with matrix coefficients. Non-negativity on $K$ for matrix polynomials with Hermitian coefficients $\mathrm{Herm}_m$ is then $p(x)\succeq 0$ for all $x\in K$. In the current work, we investigate linear maps $T:\mathrm{Herm}_m[x_1,\dots,x_n]\to\mathrm{Herm}_m[x_1,\dots,x_n]$. We focus on matrix $K$-positivity preserver, i.e., $Tp\succeq 0$ on $K$ for all $p\succeq 0$ on $K$. For $K=\mathbb{R}^n$ and compact sets $K\subseteq\mathrm{R}^n$, we give characterizations of matrix $K$-positivity preservers. We discuss the difference between the real and the matrix coefficient case and where our proof fails for general sets $K\subseteq\mathbb{R}^n$ with $K\neq \mathbb{R}^n$ and $K$ non-compact.
- [67] arXiv:2512.22585 [pdf, html, other]
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Title: On a Thermodynamically Consistent Diffuse-Interface Model for Incompressible Two-Phase Flows with Chemotaxis and Mass TransportSubjects: Analysis of PDEs (math.AP)
We investigate a hydrodynamic system of Navier--Stokes/Cahn--Hilliard type, which describes the motion of a two-phase flow of two incompressible fluids with unmatched densities coupled with a soluble chemical species. Derived from Onsager's variational principle, this thermodynamically consistent diffuse-interface model incorporates both the chemotaxis effects induced by the chemical species and the mass transport processes within the mixture. For the two-dimensional initial-boundary value problem, we establish the existence of global finite energy solutions and global weak solutions, using a suitable approximation scheme combined with compactness methods. Next, by carefully analyzing three decoupled subsystems and employing a bootstrap argument, we prove the existence and uniqueness of a global strong solution for sufficiently regular initial data, as well as the propagation of regularity for global weak solutions. In particular, we show that the density of the chemical substance stays bounded for all time if its initial datum is bounded. This implies a significant distinction from the classical Keller--Segel system: diffusion driven by the chemical potential gradient can prevent the formation of concentration singularities.
- [68] arXiv:2512.22592 [pdf, html, other]
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Title: Limit theorems for critical branching processes in an extremely unfavorable random environmentComments: 26 pagesSubjects: Probability (math.PR)
Let $\{Z_{m},m\geq 0\}$ be a critical branching process in random environment and $\{S_{m},m\geq 0\}$ be its associated random walk. Assuming that the increments distribution of the associated random walk belongs without centering to the domain of attraction of an $\alpha $-stable law we prove conditional limit theorems describing, as $n\rightarrow \infty $, the distribution the number of particles in the process $\{Z_{m},0\leq m\leq n\}$ given $Z_{n}>0$ and $S_{n}\leq const$.
- [69] arXiv:2512.22595 [pdf, html, other]
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Title: Modules whose minimal free resolutions are self-dual or eventually periodicComments: 7 pagesSubjects: Commutative Algebra (math.AC)
Let $R$ be a commutative noetherian local ring. In this paper, we study the self-duality and eventual periodicity of minimal free resolutions of finitely generated $R$-modules in terms of their syzygy modules and Ext modules. As an application, we recover theorems of Dey.
- [70] arXiv:2512.22598 [pdf, html, other]
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Title: Complete hypersurfaces in $R^{n+1}$ with constant mean and scalar curvatureSubjects: Differential Geometry (math.DG)
In this paper, we investigate the rigidity problems of complete hypersurfaces with constant mean curvature and constant scalar curvature in Euclidean spaces. Firstly, under some conditions of Gaussian-Kronecker curvature, we provide characterizations for the unsolved cases of Núñez's theorems in dimensions 4 and 5, as well as several rigidity results under some conditions of $r$-th mean curvatures. Moreover, for the case of dimension 6, we also present analogous rigidity results. Finally, for general dimensions, we offer a rigidity theorem under similar pinching conditions.
- [71] arXiv:2512.22610 [pdf, html, other]
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Title: Statistical order convergence of operators on Riesz SpacesComments: 13Subjects: Functional Analysis (math.FA)
This paper introduces statistical order convergence and its pointwise variant for sequences of order bounded operators between Riesz spaces. We establish fundamental properties: uniqueness of the limit, stability under lattice operations, and a characterization via natural density linking it to classical order convergence. Explicit examples show that statistical order convergence is strictly weaker than order convergence, confirming that this concept provides a proper extension of operator-theoretic convergence notions. The results preserve essential lattice structures and open avenues for further research in unbounded convergence and Banach lattice theory.
- [72] arXiv:2512.22613 [pdf, html, other]
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Title: Dispersive estimates for discrete Klein-Gordon equations on one-dimensional lattice with quasi-periodic potentialsComments: 29 pagesSubjects: Analysis of PDEs (math.AP)
We prove $\ell^{1}\!\to\!\ell^{\infty}$ dispersive estimates for the discrete Klein--Gordon equation on $\mathbb Z$ with small real-analytic quasi-periodic potentials, showing that the time-decay rate persists as $(\tfrac13)^{-}$. As applications, we derive the corresponding Strichartz estimates and establish small-data global well-posedness for the associated nonlinear discrete Klein--Gordon equation.
- [73] arXiv:2512.22614 [pdf, other]
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Title: van Hamel-Lichtenbaum duality for singular varieties over $p$-adic fieldsComments: 38 pagesSubjects: Number Theory (math.NT); Algebraic Geometry (math.AG)
In this article, we extend the van Hamel-Lichtenbaum duality theorem to (not necessarily smooth) proper and geometrically integral varieties defined over a $p$-adic field $k$. More precisely, we prove that for such variety $X$ there exists a natural continuous perfect pairing \[ \mathrm{Br}_1(X)\times H_0(X,\mathbb{Z})_\tau^{\wedge} \to \mathbb{Q}/\mathbb{Z}, \] where $\mathrm{Br}_1(X):=\ker(\mathrm{Br}(X)\to\mathrm{Br}(\overline{X}))$ is the algebraic Brauer group of $X$, $H_0(X,\mathbb{Z})_\tau$ is the zeroth group of truncated homology $\mathrm{Hom}_{D(k_{\mathrm{sm}})}(\tau_{\leq 1}R\phi_*\mathbb{G}_{m,X},\mathbb{G}_{m,k})$, $\phi$ is the structure morphism of $X$, and $(-)^{\wedge}$ is the profinite completion functor.
- [74] arXiv:2512.22619 [pdf, html, other]
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Title: Ground states of the Schrödinger equation coupled with fourth-order gravitation -- Part 1: the case $K_{a, b} \leq 0$Subjects: Analysis of PDEs (math.AP)
We are interested in the existence and asymptotic behavior of ground states of the following normalized nonlocal semilinear problem: \[ \begin{cases} - \Delta u + (V - \omega) u + (K_{a, b} \ast u^2) u = 0 &\text{in} ~ \mathbb{R}^3; \\ \|u\|_{\mathscr{L}^2}^2 = \mu, \end{cases} \] where \[ K_{a, b} (x) := \frac{1}{|x|} \left( \frac{4}{3} e^{- b |x|} - \frac{1}{3} e^{- a |x|} - 1 \right); \] $0 \leq a, b \leq \infty$; $V$ denotes a singular potential that vanishes at infinity and the unknowns are $\omega \in \mathbb{R}$, $u \colon \mathbb{R}^3 \to \mathbb{R}$. This problem is obtained by looking for standing waves of the Schrödinger equation coupled with the nonrelativistic gravitational potential prescribed by a family of fourth-order gravity theories. In this paper, (i) we obtain a complete picture of the existence/nonexistence of ground states of the associated autonomous problem for every possible geometry of $K_{a, b}$, (ii) we obtain conditions that ensure the existence of ground states of the nonautonomous problem when $K_{a, b} \leq 0$ and (iii) we prove that as \[ (a, b) \to (A, B) \in \left\{(0, 0), (\infty, \infty), (0, \infty)\right\}, \] ground states of this problem respectively converge to a ground state of (1) the Schrödinger equation, (2) the Choquard equation and (3) a rescaling of the Choquard equation.
- [75] arXiv:2512.22620 [pdf, html, other]
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Title: Optimal Beamforming Design for Multi-user MIMO Near-Field ISAC Systems with Movable AntennasComments: 13 pages, 6 figuresSubjects: Information Theory (cs.IT)
Integrated sensing and communication (ISAC) has been recognized as one of the key technologies capable of simultaneously improving communication and sensing services in future wireless networks. Moreover, the introduction of recently developed movable antennas (MAs) has the potential to further increase the performance gains of ISAC systems. Although the gains of MA-enabled ISAC systems are relatively well studied in the far field, they remain almost unexplored in near-field scenarios. Motivated by this, in this paper we maximize the weighted sum rate (WSR) for communication users while maintaining a minimum sensing requirement in an MA-enabled near-field ISAC system. To achieve this goal, we propose algorithms that optimize the communication precoding matrices, the sensing transmit beamformer, the sensing receive combiner, the positions of the users' MAs and the positions of the base station (BS) transmit MAs in an alternating manner for the considered ISAC system, for the cases where linear procoding and zero-forcing (ZF) precoding are employed at the BS. Simulation results show that using MAs in near-field ISAC systems provides a substantial performance advantage compared to near-field ISAC systems equipped with fixed antennas only. We show that the scheme with linear precoding achieves larger WSR for unequal users' weight rates, while the scheme with ZF precoding maintains an approximately constant WSR for all users' weight rates. Additionally, we demonstrate that the WSRs of the proposed schemes are highly dependent on the inter-antenna interference between different user's MAs, and that the sensing performance is significantly more affected by the minimum sensing signal-to-interference-plus-noise ratio (SINR) threshold compared to the communication performance.
- [76] arXiv:2512.22640 [pdf, html, other]
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Title: Truncation StructuresSubjects: Logic (math.LO)
We characterize intrinsically the truncation structures on valued fields arising from embeddings into Hahn fields with truncation closed image.
- [77] arXiv:2512.22642 [pdf, html, other]
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Title: A Survey of Machine-Learning-Based Scheduling: From Solver-Centric to Data-Centric ParadigmsSubjects: Optimization and Control (math.OC)
Scheduling problems are a fundamental class of combinatorial optimization problems that underpin operational efficiency in manufacturing, logistics, and service systems. While operations research has traditionally developed solver-centric methods emphasizing model structure and optimality, recent advances in machine learning are reshaping scheduling into a data-centric discipline that learns from experience and adapts to dynamic environments. This paper provides a comprehensive and comparative review of this methodological transition. We first revisit classical optimization-based approaches and summarize how ML has been integrated within them to improve computational efficiency. We then review end-to-end learning approaches that generate scheduling solutions directly from data, highlighting how they shift decision-making from explicit optimization to learned inference. Adopting a systematic, method-oriented perspective, we compare these paradigms and their underlying learning algorithms in terms of principles, scalability, interpretability, and generalization. Finally, we discuss key research challenges and outline future directions along three interdependent dimensions, scalability, reliability, and universality, that together define a pathway toward adaptive, intelligent, and trustworthy scheduling systems for data-driven operations management.
- [78] arXiv:2512.22645 [pdf, html, other]
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Title: Divisibility of generalized Mersenne numbersComments: 6 pagesSubjects: General Mathematics (math.GM)
In this article, I present a theorem determining a criterion for divisibility of two generalized Mersenne numbers, which are repunits of the same length in base-$a^m$ and base-$a^k$. In addition to the general proof, I present an alternative proof for a special case of the theorem.
- [79] arXiv:2512.22662 [pdf, html, other]
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Title: Extending Fubini MeasuresSubjects: Logic (math.LO)
Let $C\subseteq M$ be stably embedded in a structure $\cM=(M;\dots)$. We consider {\em Fubini measures} on the subcategory $\Def(C)$ of the category $\Def(\cM)$ of definable sets in $\cM$, with ``Fubini" signaling good behaviour in definable families. We show that such a Fubini measure extends uniquely to the larger subcategory of $\Def(\cM)$ whose objects are the sets that are ``fiberable over $C$". In cases of interest ``fiberable over $C$" coincides with ``co-analyzable relative to $C$." This applies in particular to the differential field $\T$ of transseries with $C=\R$, and to differentially closed fields with constant field $C$.
- [80] arXiv:2512.22663 [pdf, html, other]
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Title: Auslander-Yorke Dichotomy and Its Generalizations for Non-Autonomous Dynamical SystemsComments: 24 pagesSubjects: Dynamical Systems (math.DS)
We investigate the dynamics of periodic non-autonomous discrete dynamical systems on uniform spaces and topological spaces, focusing on the extension of the classical Auslander-Yorke dichotomy to these settings. We prove various dichotomy theorems in the uniform space framework, showing that a minimal periodic non-autonomous system is either sensitive or equicontinuous, and prove some more refined versions involving syndetic equicontinuity and thick sensitivity and eventual sensitivity versus equicontinuity on compact uniform spaces. We further introduce topological analogues like topological equicontinuity, Hausdorff sensitivity, and their syndetic and multi-sensitive variants and prove corresponding Auslander-Yorke-type dichotomies on T3 spaces.
- [81] arXiv:2512.22677 [pdf, html, other]
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Title: Asymptotic behavior of a nonlinear shallow shell model when the shell becomes a plateSubjects: Analysis of PDEs (math.AP)
This paper studies a nonlinear shallow shell model proposed by Donnell, Vlasov, Mushtari, Galimov, and Koiter. More specifically, we address the question concerning the asymptotic behavior of minimizing solutions. Our result can be applied to general applied forces. Thus, it substantially extends the one given in \cite{oana2} whereby the tangential components of the applied forces are assumed to vanish.
- [82] arXiv:2512.22691 [pdf, html, other]
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Title: An Improved Lower Bound on Cardinality of Support of the Amplitude-Constrained AWGN ChannelSubjects: Information Theory (cs.IT); Statistics Theory (math.ST)
We study the amplitude-constrained additive white Gaussian noise channel. It is well known that the capacity-achieving input distribution for this channel is discrete and supported on finitely many points. The best known bounds show that the support size of the capacity-achieving distribution is lower-bounded by a term of order $A$ and upper-bounded by a term of order $A^2$, where $A$ denotes the amplitude constraint. It was conjectured in [1] that the linear scaling is optimal. In this work, we establish a new lower bound of order $A\sqrt{\log A}$, improving the known bound and ruling out the conjectured linear scaling.
To obtain this result, we quantify the fact that the capacity-achieving output distribution is close to the uniform distribution in the interior of the amplitude constraint. Next, we introduce a wrapping operation that maps the problem to a compact domain and develop a theory of best approximation of the uniform distribution by finite Gaussian mixtures. These approximation bounds are then combined with stability properties of capacity-achieving distributions to yield the final support-size lower bound. - [83] arXiv:2512.22696 [pdf, other]
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Title: Tiling Triangles with $2π/3$ AnglesComments: 16 pages, 12 figuresSubjects: Combinatorics (math.CO)
Motivated by a question of Erdös and inquiries by Beeson and Laczkovich, we explore the possible $N$ for which a triangle $T$ can tile into $N$ congruent copies of a triangle $R$. The \emph{reptile} cases (where $T$ is similar to $R$) and the \emph{commensurable-angles} cases (where all angles of $R$ are rational multiples of $\pi$) are well-understood. We tackle the most interesting remaining case, which is when $R$ contains an angle of $2\pi/3$ and when $T$ is one of $6$ ``sporadic'' specific triangles, of which only $2$ were known to have constructions. For each of these, we create a family of constructions and conjecture that they are the only possible $N$ that occur for these triangles.
- [84] arXiv:2512.22700 [pdf, html, other]
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Title: Infinitesimal moments in free and c-free probability and Motzkin pathsComments: 36 pages, 4 figuresSubjects: Operator Algebras (math.OA); Combinatorics (math.CO); Probability (math.PR)
Infinitesimal moments associated with infinitesimal freeness and infinitesimal conditional freeness are studied. For free random variables, we consider continuous deformations of moment functionals associated with Motzkin paths $w$, which provide a decomposition of their moments, and we compute their derivatives at zero. We show that the first-order derivative of each functional vanishes unless the path has exactly one local maximum. Geometrically, this means that $w$ is a pyramid path, which is consistent with the characteristic formula for alternating moments of infinitesimally free centered random variables. In this framework, infinitesimal Boolean independence is also obtained and it corresponds to flat paths. A similar approach is developed for infinitesimal conditional freeness, for which we show that the only moment functionals that have a non-zero first-order derivative are associated with concatenations of a pyramid path and a flat path. This charaterization leads to a Leibniz-type definition of infinitesimal conditional freeness at the level of moments.
- [85] arXiv:2512.22708 [pdf, html, other]
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Title: A high-order method for the numerical approximation of fractional nonlinear Schrödinger equationsSubjects: Numerical Analysis (math.NA)
In this paper, the periodic initial-value problem for the fractional nonlinear Schrödinger (fNLS) equation is discretized in space by a Fourier spectral Galerkin method and in time by diagonally implicit, high-order Runge-Kutta schemes, based on the composition with the implicit midpoint rule (IMR). Some properties and error estimates for the semidiscretization in space and for the full discretization are proved. The convergence results and the general performance of the scheme are illustrated with several numerical experiments.
- [86] arXiv:2512.22713 [pdf, html, other]
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Title: Computing quaternionic representations via twisted forms of Bruhat-Tits treesComments: Comments are welcomeSubjects: Representation Theory (math.RT); Number Theory (math.NT); Rings and Algebras (math.RA)
This work is devoted to the study of representations of finite subgroups of the group of units of quaternion division algebras over a global or local field arising from the inclusion via extension of scalars splitting the algebra. Following a question by Serre, we study the set $\mathrm{IF}$ of conjugacy classes of integral representations that are conjugates of the given representation over the field. The set $\mathrm{IF}$ is often called the set of integral forms in the literature. In previous works we have seen that, for a given representation, the set $\mathrm{IF}$ can be indexed by the vertex set of a suitable subgraph of the Bruhat-Tits tree for the special linear group. In this work, we describe a construction that allows the simultaneous study of the set $\mathrm{IF}$ over different splitting fields. For this, we devise and use a theory of twisted Galois form of Bruhat-Tits trees. With this tool, we explicitly compute, in most cases, the cardinality of $\mathrm{IF}$ for the representation of the classical quaternion group of order $8$ studied by Serre, Feit and others, as much as for other similar groups.
- [87] arXiv:2512.22714 [pdf, html, other]
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Title: Polynomial-Time Near-Optimal Estimation over Certain Type-2 Convex BodiesSubjects: Statistics Theory (math.ST); Methodology (stat.ME)
We develop polynomial-time algorithms for near-optimal minimax mean estimation under $\ell_2$-squared loss in a Gaussian sequence model under convex constraints. The parameter space is an origin-symmetric, type-2 convex body $K \subset \mathbb{R}^n$, and we assume additional regularity conditions: specifically, we assume $K$ is well-balanced, i.e., there exist known radii $r, R > 0$ such that $r B_2 \subseteq K \subseteq R B_2$, as well as oracle access to the Minkowski gauge of $K$. Under these and some further assumptions on $K$, our procedures achieve the minimax rate up to small factors, depending poly-logarithmically on the dimension, while remaining computationally efficient.
We further extend our methodology to the linear regression and robust heavy-tailed settings, establishing polynomial-time near-optimal estimators when the constraint set satisfies the regularity conditions above. To the best of our knowledge, these results provide the first general framework for attaining statistically near-optimal performance under such broad geometric constraints while preserving computational tractability. - [88] arXiv:2512.22718 [pdf, html, other]
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Title: Resurgence and perverse sheavesSubjects: Algebraic Geometry (math.AG); High Energy Physics - Theory (hep-th); Complex Variables (math.CV); Quantum Algebra (math.QA); Symplectic Geometry (math.SG)
We propose a point of view on resurgence theory based on the study of perverse sheaves on the complex line carrying an algebraic structure with respect to additive convolution. In particular, we lift the concept of alien derivatives introduced originally by J. Écalle, to the framework of perverse sheaves and study its behavior under sheaf-theoretic convolution. The full fledged resurgence theory needs a (yet undeveloped) generalization of the concept of perverse sheaves allowing infinite, possibly dense, sets of singularities. We discuss possible approaches to defining such objects and some potential examples of them coming from Cohomological Hall Algebras, wall-crossing structures and Chern-Simons theory.
- [89] arXiv:2512.22719 [pdf, html, other]
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Title: Global Martingale Entropy Solutions to the Stochastic Isentropic Euler EquationsComments: 88 pagesSubjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph); Functional Analysis (math.FA); Probability (math.PR)
We establish the existence and compactness of global martingale entropy solutions with finite relative-energy for the stochastically forced system of isentropic Euler equations governed by a general pressure law. To achieve these, a stochastic compensated compactness framework in $L^p$ is developed to overcome the difficulty that the uniform $L^{\infty}$ bound for the stochastic approximate solutions is unavailable, owing to the stochastic forcing term. The convergence of the vanishing viscosity method is established by employing the stochastic compactness framework, along with careful uniform estimates of the stochastic approximate solutions, to obtain the existence of global martingale entropy solutions with finite relative-energy. In particular, in the polytropic pressure case for all adiabatic exponents, we prove that the global solutions satisfy the local mechanical energy inequality when the initial data are only required to have finite relative-energy (while the higher moment estimates for entropy are not required here, as needed in the earlier work). Higher-order relative energy estimates for approximate solutions are also derived to establish the entropy inequality for more convex entropy pairs and to then prove the compactness of solutions to the stochastic isentropic Euler system. The stochastic compensated compactness framework and the uniform estimate techniques for approximate solutions developed in this paper should be useful in the study of other similar problems.
- [90] arXiv:2512.22724 [pdf, html, other]
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Title: Involutions on S^4Subjects: Geometric Topology (math.GT)
This paper studies locally linear involutions on S^4. Our main theorem shows that any such involution with a 1-dimensional fixed-point set is necessarily linear, provided the fixed-point set admits an equivariant tubular neighborhood. The proof combines modified surgery theory with an equivariant version of the Schoenflies theorem, which we establish here. We also show that equivariant tubular neighborhoods of 1-dimensional fixed-point sets, when they exist, are not unique, in contrast to the nonequivariant case. Our results combine with earlier work to provide a classification of all locally linear involutions on S^4. As a further application, we obtain that strongly negative amphichiral knots with trivial Alexander polynomial are equivariantly topologically slice with respect to the linear action, strengthening a previous result of the first two authors. Finally, we also prove that when the fixed-point set is 2-dimensional, the involution is linear if and only if the fixed-point set is an unknotted 2-knot.
- [91] arXiv:2512.22731 [pdf, html, other]
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Title: Iterative Channel Estimation, Detection and Decoding for Multi-Antenna Systems with RISComments: 15 pages, 12 figuresSubjects: Information Theory (cs.IT)
This work proposes an iterative channel estimation, detection and decoding (ICEDD) scheme for the uplink of multi-user multi-antenna systems assisted by multiple reconfigurable intelligent surfaces (RIS)}. A novel iterative code-aided channel estimation (ICCE) technique is developed that uses low-density parity-check (LDPC) codes and iterative processing to enhance estimation accuracy while reducing pilot overhead. The core idea is to exploit encoded pilots (EP), enabling the use of both pilot and parity bits to iteratively refine channel estimates. To further improve performance, an iterative channel tracking (ICT) method is proposed that takes advantage of the temporal correlation of the channel. An analytical evaluation of the proposed estimator is provided in terms of normalized mean-squared error (NMSE), along with a study of its computational complexity and the impact of the code rate. Numerical results validate the performance of the proposed scheme in a sub-6 GHz multi-RIS scenario with non-sparse propagation, under both LOS and NLOS conditions, and different RIS architectures.
- [92] arXiv:2512.22750 [pdf, html, other]
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Title: A Single-loop Stochastic Riemannian ADMM for Nonsmooth OptimizationSubjects: Optimization and Control (math.OC)
We study a class of nonsmooth stochastic optimization problems on Riemannian manifolds. In this work, we propose MARS-ADMM, the first stochastic Riemannian alternating direction method of multipliers with provable near-optimal complexity guarantees. Our algorithm incorporates a momentum-based variance-reduced gradient estimator applied exclusively to the smooth component of the objective, together with carefully designed penalty parameter and dual stepsize updates. Unlike existing approaches that rely on computationally expensive double-loop frameworks, MARS-ADMM operates in a single-loop fashion and requires only a constant number of stochastic gradient evaluations per iteration. Under mild assumptions, we establish that MARS-ADMM achieves an iteration complexity of \(\tilde{\mathcal{O}}(\varepsilon^{-3})\), which improves upon the previously best-known bound of \(\mathcal{O}(\varepsilon^{-3.5})\) for stochastic Riemannian operator-splitting methods. As a result, our analysis closes the theoretical complexity gap between stochastic Riemannian operator-splitting algorithms and stochastic methods for nonsmooth optimization with nonlinear constraints. Notably, the obtained complexity also matches the best-known bounds in deterministic nonsmooth Riemannian optimization, demonstrating that deterministic-level accuracy can be achieved using only constant-size stochastic samples.
- [93] arXiv:2512.22754 [pdf, html, other]
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Title: Tilings of $\mathcal{H}_{q}(n,w)$ with optimal $(n,d,w)_{q}$-codesComments: 21 pagesSubjects: Combinatorics (math.CO)
The metric space $\mathcal{H}_{q}(n,w)$ is the set of all words of length $n$ with weight $w$ over the alphabet $\mathbb{Z}_{q}$, under the Hamming distance metric. A $q$-ary constant-weight code, as a nonempty subset of $\mathcal{H}_{q}(n,w)$, has always been a fundamental topic in coding theory. This paper investigates the tiling problem of $\mathcal{H}_{q}(n,w)$ with optimal $(n,d,w)_{q}$-codes, simply denoted by $\mathrm{TOC}_{q}(n,d,w)$, meaning a partition of $\mathcal{H}_{q}(n,w)$ into mutually disjoint optimal $q$-ary constant-weight codes with distance $d$. When the distance $d$ is odd, we investigate large sets of generalized Steiner systems. When $d$ is even, we define large sets of generalized maximum H-packings. We present several general construction approaches for generating $\mathrm{TOC}_{q}(n,d,w)$s via $t$-resolvable Steiner systems and almost-regular edge-colorings of complete hypergraphs. For the cases $d=2$ and $d=2w$, we completely resolve the existence problem of $\mathrm{TOC}_{q}(n,d,w)$s for all parameters $q,n$ and $w$. Particularly, we pay attention to tilings for weight three. For binary case and weight three, the existence problem of $\mathrm{TOC}_{2}(n,d,3)$s is totally resolved. For specific alphabet size $q\ge 3$, we obtain many infinite families of $\mathrm{TOC}_{q}(n,d,3)$s for distances $d=3,4,5$.
- [94] arXiv:2512.22755 [pdf, html, other]
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Title: On the abstract wrapped Floer setupsComments: 22 pages, to appear in HHASubjects: Symplectic Geometry (math.SG)
The wrapped Fukaya category of a Liouville sector is defined via an axiomatic construction from the associated abstract wrapped Floer setup. In this paper, we propose a modified axiomatic construction, removing the irrelevant choices and the factorization axiom from the abstract wrapped Floer setup. Based on our modification, we reformulate and then prove a conjecture which essentially claims that the wrapped Fukaya category is obtained as the infinity categorical localization along continuation maps. This amounts to compare the two axiomatic constructions.
- [95] arXiv:2512.22761 [pdf, html, other]
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Title: Graphs with large maximum forcing numberSubjects: Combinatorics (math.CO)
For a graph $G$ with order $2n$ and a perfect matching, let $f(G)$ and $F(G)$ denote the minimum and maximum forcing number of $G$ respectively. Then $0\leq f(G)\leq F(G)\leq n-1$. Liu and Zhang [10] ever proposed a conjecture: $e(G)\geq \frac{n^2}{n-F(G)}$, where $e(G)$ denotes the number of edges of $G$. In this paper we confirm this conjecture and obtain $F(G)\leq n-\frac{n^2}{e(G)}$. If $F(G)=n-1$, Liu and Zhang [9] proved that any two perfect matchings of $G$ can be obtained from each other by a series of matching switches along 4-cycles. If $G$ is bipartite and $F(G)\geq n-k$, $1\leq k\leq n-1$, we show that any two perfect matchings of $G$ can be obtained from each other by a series of matching switches along even cycles of length at most $2(k+1)$. Finally, we ask whether $f(G)\geq \lceil\frac{n}{k}\rceil-1$ holds for such bipartite graphs $G$, and give positive answers for the cases $k=1,2$. Further we show all minimum forcing numbers of the bipartite graphs $G$ of order $2n$ and with $F(G)=n-2$ form an integer interval $[\lfloor\frac{n}{2}\rfloor, n-2]$.
- [96] arXiv:2512.22769 [pdf, other]
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Title: The Grothendieck Group of the Variety of Spanning Line ConfigurationsComments: 35 pagesSubjects: Combinatorics (math.CO); Algebraic Geometry (math.AG)
We study the Grothendieck group of the variety $X_{n,k}$ of spanning line configurations introduced by Pawlowski--Rhoades [arXiv:1711.08301] as a geometric model for the generalized coinvariant algebra $R_{n,k}$. Our first result is a localization statement in $K$-theory for the complements of cell closures in smooth cellular varieties. Combining with the Fulton--Lascoux degeneracy loci formula, we prove that $K_0(X_{n,k})$ is canonically isomorphic to $R_{n,k}$, extending classical isomorphisms for the flag variety. We next identify the classes of the Pawlowski--Rhoades varieties with Grothendieck polynomials associated to words $w \in [k]^n$. Motivated by this identification, we develop models of classical and bumpless pipe dreams for words. We show that Schubert and Grothendieck polynomials of words are monomial-weight generating functions for these pipe dreams, extending the classical story from permutations to words and ordered set partitions.
- [97] arXiv:2512.22773 [pdf, html, other]
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Title: Exact Recovery in the Geometric SBMComments: 38 pagesSubjects: Probability (math.PR); Statistics Theory (math.ST)
Community detection is the problem of identifying dense communities in networks. Motivated by transitive behavior in social networks ("thy friend is my friend"), an emerging line of work considers spatially-embedded networks, which inherently produce graphs containing many triangles. In this paper, we consider the problem of exact label recovery in the Geometric Stochastic Block Model (GSBM), a model proposed by Baccelli and Sankararaman as the spatially-embedded analogue of the well-studied Stochastic Block Model. Under mild technical assumptions, we completely characterize the information-theoretic threshold for exact recovery, generalizing the earlier work of Gaudio, Niu, and Wei.
- [98] arXiv:2512.22776 [pdf, html, other]
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Title: Surjective Mappings in the Hyers--Ulam Theorem and the Gromov--Hausdorff DistanceComments: 12 pagesSubjects: Metric Geometry (math.MG)
A topological space is said to be cardinality homogeneous if every nonempty open subset has the same cardinality as the space itself. Let $X$ and $Y$ be cardinality homogeneous metric spaces of the same cardinality. If there exists a $\delta$-surjective $d$-isometry between such equicardinal cardinality homogeneous metric spaces $X$ and $Y$, then there exists a bijective $(d+2\delta)$-isometry between $X$ and $Y$. This result allows us to reduce the Dilworth--Tabor theorem to the Gevirtz--Omladič--Šemrl theorem on approximation by isometries and, in particular, to questions concerning the isometry of Banach spaces.
- [99] arXiv:2512.22788 [pdf, html, other]
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Title: Degeneration of the archimedean height pairing of algebraically trivial cyclesComments: 40 pages; comments are welcome!Subjects: Algebraic Geometry (math.AG); Number Theory (math.NT)
We consider the limiting behaviour of the archimedean height pairing for homologically trivial algebraic cycles in a degenerating one-parameter family of smooth projective complex varieties. We conjecture that the limit is controlled by the non-archimedean geometric height pairing of the cycles on the generic fiber and verify this for algebraically trivial cycles, assuming a conjecture of Griffiths on incidence equivalence. Our work offers a more geometric understanding of a related asymptotic result of Brosnan--Pearlstein and suggests a new perspective on the positivity of the Beilinson--Bloch height pairing over a one-variable complex function field.
- [100] arXiv:2512.22794 [pdf, other]
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Title: Pita factorisation in operadic categoriesComments: 39 pages. Not a final versionSubjects: Category Theory (math.CT)
In strictly factorisable operadic categories, every morphism $f$
factors uniquely as $f=\eta_f \circ \pi_f$ where $\eta_f$ is
order-preserving and $\pi_f$ is a quasi\-bijection that is
order-preserving on the fibres of $\eta_f$. We call it the pita
factorisation. In this paper we develop some general theory to
compensate for the fact that generally pita factorisations do not form
an orthogonal factorisation system. The main technical result states
that a certain simplicial object in Cat, called the pita nerve, is
oplax (rather than strict as it would be for an orthogonal
factorisation system). The main application is the result that the
so-called operadic nerve of any operadic category is coherent. This
result is a key ingredient in the simplicial approach to operadic
categories developed in the `main paper'
\cite{Batanin-Kock-Weber:mainpaper}, which motivated the present paper.
We also show that in the important case where quasibijections are
invertible, the pita nerve is a decomposition space. - [101] arXiv:2512.22797 [pdf, other]
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Title: 3-Crossed modules, Quasi-categories, and the Moore complexSubjects: Category Theory (math.CT); Algebraic Topology (math.AT)
The established equivalence between 2-crossed modules and Gray 3-groups [M. Sarikaya and E. Ulualan, 2024] serves as a benchmark for higher-dimensional algebraic models. However, to the best of our knowledge, the established definitions of 3-crossed modules [Z. Arvasi, T. S. Kuzpinari, and E. Ö. Uslu, 2009] are not clearly suited for extending this equivalence. In this paper, we propose an alternative formulation of a 3-crossed module, equipped with a new type of lifting, which is specifically designed to serve as a foundation for this higher-order categorical correspondence. As the primary results of this paper, we validate this new structure. We prove that the simplicial set induced by our 3-crossed module forms a quasi-category. Furthermore, we show that the Moore complex of length 3 associated with a simplicial group naturally admits the structure of our 3-crossed module. This work establishes our definition as a robust candidate for modeling the next level in this algebraic-categorical program.
- [102] arXiv:2512.22798 [pdf, html, other]
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Title: Primes in simultaneous arithmetic progressionsComments: 49 pages. All comments are welcome!Subjects: Number Theory (math.NT)
We prove a new mean value theorem on the distribution of primes in two simultaneous arithmetic progressions. Our approach builds on previous arguments of Bombieri, Fouvry, Friedlander, and Iwaniec appealing to spectral theory of Kloosterman sums, as well as the $q$-analogue of van der Corput method. In particular, we need estimates for exponential sums coming from the spectral theory of automorphic forms (sums of Kloosterman sums) and from algebraic geometry (Weil--Deligne bound for algebraic exponential sums). As an application, we show that the greatest prime factor of $p + 6$ for Chen prime $p$ is infinitely often greater than $p^{0.217}$.
- [103] arXiv:2512.22803 [pdf, html, other]
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Title: Fast mixing in Ising models with a negative spectral outlier via Gaussian approximationComments: 42 pagesSubjects: Probability (math.PR); Data Structures and Algorithms (cs.DS); Mathematical Physics (math-ph)
We study the mixing time of Glauber dynamics for Ising models in which the interaction matrix contains a single negative spectral outlier. This class includes the anti-ferromagnetic Curie-Weiss model, the anti-ferromagnetic Ising model on expander graphs, and the Sherrington-Kirkpatrick model with disorder of negative mean. Existing approaches to rapid mixing rely crucially on log-concavity or spectral width bounds and therefore can break down in the presence of a negative outlier.
To address this difficulty, we develop a new covariance approximation method based on Gaussian approximation. This method is implemented via an iterative application of Stein's method to quadratic tilts of sums of bounded random variables, which may be of independent interest. The resulting analysis provides an operator-norm control of the full correlation structure under arbitrary external fields. Combined with the localization schemes of Eldan and Chen, these estimates lead to a modified logarithmic Sobolev inequality and near-optimal mixing time bounds in regimes where spectral width bounds fail. As a complementary result, we prove exponential lower bounds on the mixing time for low temperature anti-ferromagnetic Ising models on sparse Erdös-Rényi graphs, based on the existence of gapped states as in the recent work of Sellke. - [104] arXiv:2512.22805 [pdf, html, other]
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Title: Proper conflict-free choosability of planar graphsSubjects: Combinatorics (math.CO); Discrete Mathematics (cs.DM)
A proper conflict-free coloring of a graph is a proper vertex coloring wherein each non-isolated vertex's open neighborhood contains at least one color appearing exactly once. For a non-negative integer $k$, a graph $G$ is said to be proper conflict-free (degree+$k$)-choosable if given any list assignment $L$ for $G$ where $|L(v)| = d(v) + k$ holds for every vertex $v \in V(G)$, there exists a proper conflict-free coloring $\phi$ of $G$ such that $\phi(v) \in L(v)$ for all $v \in V(G)$. Recently, Kashima, Škrekovski, and Xu proposed two related conjectures on proper conflict-free choosability: the first asserts the existence of an absolute constant $k$ such that every graph is proper conflict-free (degree+$k$)-choosable, while the second strengthens this claim by restricting to connected graphs other than the cycle of length 5 and reducing the constant to $k=2$. In this paper, we confirm the second conjecture for three graph classes: $K_4$-minor-free graphs with maximum degree at most 4, outer-1-planar graphs with maximum degree at most 4, and planar graphs with girth at least 12; we also confirm the first conjecture for these same graph classes, in addition to all outer-1-planar graphs (without degree constraints). Moreover, we prove that planar graphs with girth at least 12 and outer-1-planar graphs are proper conflict-free $6$-choosable.
- [105] arXiv:2512.22806 [pdf, html, other]
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Title: On Composite Foster Functions for a Class of Singularly Perturbed Stochastic Hybrid InclusionsSubjects: Optimization and Control (math.OC)
We study sufficient conditions for stability and recurrence in a class of singularly perturbed stochastic hybrid dynamical systems. The systems considered combine multi-time-scale deterministic continuous-time dynamics, modeled by constrained differential inclusions, with discrete-time dynamics described by constrained difference inclusions subject to random disturbances. Under suitable regularity assumptions on the dynamics and causality of the associated solutions, we develop a family of composite nonsmooth Lagrange-Foster and Lyapunov-Foster functions that certify stability and recurrence properties by leveraging simpler functions related to the slow and fast subsystems. Stability is characterized with respect to compact sets, while recurrence is established for bounded open sets. The proposed framework is illustrated through several examples and applications, including the stability analysis of singularly perturbed switching systems with stochastic spontaneous mode transitions, feedback optimization problems with stochastically switching plants, and momentum-based feedback optimization algorithms with stochastic restarting.
- [106] arXiv:2512.22807 [pdf, html, other]
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Title: On the Ando-Hiai property for spectral geometric meansComments: Submitted to Linear Algebra and its ApplicationsSubjects: Functional Analysis (math.FA)
In this paper, we consider a two-variable operator function that includes two weighted spectral geometric means, and show fundamental properties of the operator function. Moreover, it satisfies the Ando-Hiai type inequality under some restricted conditions. As an application, we show the log-majorization relations and norm inequalities for the spectral geometric means of positive definite matrices.
- [107] arXiv:2512.22809 [pdf, html, other]
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Title: Fast algorithm for $S$-packing coloring of Halin graphsSubjects: Combinatorics (math.CO); Discrete Mathematics (cs.DM)
Motivated by frequency assignment problems in wireless broadcast networks, Goddard, Hedetniemi, Hedetniemi, Harris, and Rall introduced the notion of $S$-packing coloring in 2008. Given a non-decreasing sequence $S = (s_1, s_2, \ldots, s_k)$ of positive integers, an $S$-packing coloring of a graph $G$ is a partition of its vertex set into $k$ subsets $\{V_1, V_2, \ldots, V_k\}$ such that for each $1 \leq i \leq k$, the distance between any two distinct vertices $u, v \in V_i$ is at least $s_i + 1$. In this paper, we study the $S$-packing coloring problem for Halin graphs with maximum degree $\Delta \leq 5$. Specifically, we present a linear-time algorithm that constructs a $(1,1,2,2,2)$-packing coloring for any Halin graph satisfying $\Delta \leq 5$. It is worth noting that there are Halin graphs that are not $(1,2,2,2)$-packing colorable.
- [108] arXiv:2512.22813 [pdf, html, other]
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Title: Exact rainbow numbers of cycle-related graphs in multi-hubbed wheelsSubjects: Combinatorics (math.CO); Discrete Mathematics (cs.DM)
The rainbow number ${\rm rb}(G, H)$ is the minimum number of colors $k$ for which any edge-coloring of $G$ with at least $k$ colors guarantees a rainbow subgraph isomorphic to $H$. The rainbow number has many applications in diverse fields such as wireless communication networks, cryptography, bioinformatics, and social network analysis. In this paper, we determine the exact rainbow number $\mathrm{rb}(G, H)$ where $G$ is a multi-hubbed wheel graph $W_d(s)$, defined as the join of $s$ isolated vertices and a cycle $C_d$ of length $d$ (i.e., $W_d(s) = \overline{K_s} + C_d$), and $H = \theta_{t,\ell}$ represents a cycle $C_t$ of length $t$ with $0 \leq \ell \leq t-3$ chords emanating from a common vertex, by establishing \[ {\rm rb}(W_{d}(s), \theta_{t,\ell}) = \begin{cases} \left\lfloor \dfrac{2t - 5}{t - 2}d \right\rfloor + 1, & \text{if } \ell=t-3,~s = 1 \text{ and } t\ge 4, \\[10pt] \left\lfloor \dfrac{3t-10}{t - 3}d \right\rfloor + 1, & \text{if } \ell=t-3,~s = 2\text{ and } t\ge 6,\\[10pt] \left\lfloor \dfrac{(s + 1)t - (3s + 4)}{t - 3}d \right\rfloor + 1, & \text{if } \ell=t-3,~s \geq 3\text{ and } t\ge 7,\\[10pt] \left\lfloor \dfrac{2t - 7}{t - 3}d \right\rfloor + 1, & \text{if } s = 1 \text{ and } t\ge \max\{5,\ell+4\}, \end{cases} \] when $d\geq 3t-5$, with all bounds for the parameter $t$ presented here being tight. This addresses the problems proposed by Jakhar, Budden, and Moun (2025), which involve investigating the rainbow numbers of large cycles and large chorded cycles in wheel graphs (specifically corresponding to the cases in our framework where $s=1$ and $\ell\in \{0,1\}$). Furthermore, it completely determines the rainbow numbers of cycles of arbitrary length in large wheel graphs, thereby generalizing a result of Lan, Shi, and Song (2019).
- [109] arXiv:2512.22816 [pdf, other]
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Title: Field Theory via Higher Geometry II: Thickened Smooth Sets as Synthetic FoundationsComments: 73 pagesSubjects: Mathematical Physics (math-ph); High Energy Physics - Theory (hep-th); Differential Geometry (math.DG)
This is the second in a series of papers that aim to develop rigorous and most encompassing foundations for field theory, where in the first installment, we laid out the natural formulation of bosonic variational field theory via the functorial geometry of smooth sets. Here, we extend this to the category ThickenedSmoothSets of infinitesimally thickened smooth sets. We first describe the Cahiers topos in a simplified, but fully rigorous, $\mathbb{R}$-algebraic setting -- which should serve as a more accessible introduction to the theory of Synthetic Differential Geometry to both physicists and mathematicians. Then, we formulate local Lagrangian field theory in this rigorous setting in which infinitesimal spaces exist and interact correctly with the field-theoretic spaces of infinite jet bundles, off-shell and on-shell spaces of fields etc.
This setting subsumes all previous constructions and further recovers all the relevant tangent bundles of traditional (off-shell and on-shell) field theory considerations via the synthetic tangent bundle construction, i.e., as ``infinitesimal curves'' in those spaces, which were previously defined only in an ad-hoc manner. Beyond finally establishing a firm foundation for such aspects of the theory, this approach recognizes the variational principle of local Lagrangian field theory, equivalently, as the intersection of thickened smooth sets. It also suggests the rigorous formalization of perturbative field theory as the restriction to a (synthetic) infinitesimal neighborhood around a field configuration. Furthermore, our context naturally accommodates more general, rigorous considerations, in which the manifolds may have boundaries and corners, a situation that has recently been attracting greater attention in the field-theoretical literature. - [110] arXiv:2512.22817 [pdf, html, other]
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Title: Baillon-Bruck-Reich revisited: divergent-series parameters and strong convergence in the linear caseComments: 5 pages, 0 figuresSubjects: Optimization and Control (math.OC)
The Krasnoselskii-Mann iteration is an important algorithm in optimization and variational analysis for finding fixed points of nonexpansive mappings. In the general case, it produces a sequence converging \emph{weakly} to a fixed point provided the parameter sequence satisfies a divergent-series condition.
In this paper, we show that \emph{strong} convergence holds provided the underlying nonexpansive mapping is \emph{linear}. This improves on a celebrated result by Baillon, Bruck, and Reich from 1978, where the parameter sequence was assumed to be constant as well as on recent work where the parameters were bounded away from $0$ and $1$. - [111] arXiv:2512.22821 [pdf, html, other]
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Title: Blowup rate for rotational NLS with a repulsive potentialSubjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph)
In this paper we give an analytical proof of the ``$\log$-$\log$'' blowup rate for mass-critical nonlinear Schrödinger equation (NLS) with a rotation ($\Omega \neq 0$) and a repulsive harmonic potential $V_{\gamma}(x) = \textrm{sgn}(\gamma) \gamma^2 |x|^2$, $\gamma < 0$ when the initial data has a mass slightly above that of $Q$, the ground state solution to the free NLS. The proof is based on a virial identity and an $\mathcal{R}_{\gamma}$-transform, a pseudo-conformal transform in this setting. Further, we obtain a limiting behavior description concerning the mass concentration near blowup time. A remarkable finding is that increasing the value $|\gamma|$ for the repulsive potential $V_{\gamma}$ can give rise to global in time solution for the focusing RNLS, which is in contrast to the case where $\gamma$ is positive. This kind of phenomenon was earlier observed in the non-rotational case $\Omega = 0$ in Carles' work. In addition, we provide numerical simulations to partially illustrate the blowup profile along with the blowup rate using dynamic rescaling and adaptive mesh refinement method.
- [112] arXiv:2512.22828 [pdf, html, other]
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Title: Beyond Beam Sweeping: One-Shot Satellite Acquisition with Doppler-Aware Rainbow BeamformingComments: 16 pages, 5 figuresSubjects: Information Theory (cs.IT)
High-gain beamforming (BF) is essential for low Earth orbit (LEO) satellite communications to overcome severe path loss, but this requires acquiring precise satellite positions. Conventional satellite acquisition typically relies on time-domain beam sweeping, which incurs substantial overhead and latency. In this correspondence, we propose an efficient one-shot satellite acquisition framework that capitalizes on two phenomena traditionally regarded as impairments: i) Doppler effects and ii) beam-squint effects. Specifically, we derive a closed-form \emph{rainbow beamformer} that leverages beam-squint effects to align frequency-dependent beam directions with satellite positions inferred from their Doppler shifts. This approach enables reception from multiple satellites at once without requiring beam sweeping. To extract satellite position information, we develop three Doppler-aware angle estimation algorithms based on received signals. Simulation results demonstrate that the proposed method significantly outperforms conventional beam sweeping approaches in both acquisition accuracy and required time slots. These gains stem from the ability of the proposed rainbow BF to exploit the \emph{angle-dependent nature of Doppler shifts}, enabling full angular-domain coverage with a single pilot transmission and reception.
- [113] arXiv:2512.22831 [pdf, other]
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Title: Convergent numerical schemes for the viscoelastic Giesekus model in two dimensionsSubjects: Numerical Analysis (math.NA); Analysis of PDEs (math.AP)
In this work, we develop a class of stable and convergent numerical methods for the approximate solution of the viscoelastic Giesekus model in two space dimensions. The model couples the incompressible Navier--Stokes equations with an evolution equation for an additional stress tensor accounting for elastic effects. This coupled evolution equation is stated here in terms of the elastic deformation gradient and models transport and nonlinear relaxation effects. In the existing literature, numerical schemes for such models often suffer from accuracy limitations and convergence problems, usually due to the lack of rigorous existence results or inherent limitations of the discretization. Therefore, our main goal is to prove the (subsequence) convergence of the proposed numerical method to a large-data global weak solution in two dimensions, without relying on cut-offs or additional regularization. This also provides an alternative proof of the recent existence result by Bul\'ıček et al.~(Nonlinearity, 2022). Finally, we verify the practicality of the proposed method through numerical experiments, including convergence studies and typical benchmark problems.
- [114] arXiv:2512.22835 [pdf, html, other]
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Title: The Hilton-Milner type results of $(k, \ell)$-sum-free sets in $\mathbb F_p^n$Comments: 40 pages, 0 figureSubjects: Combinatorics (math.CO)
For a prime $p \equiv 2 \pmod 3$, it is well known that the largest sum-free subsets of $\mathbb{F}_p^n$ have size $\frac{p+1}{3} p^{n-1}$, and the extremal sets must be a cuboid of the form $\{\frac{p+1}{3}, \frac{p+1}{3}+1, \ldots, \frac{2p-1}{3}\} \times \mathbb{F}_p^{n-1}$ up to isomorphism. Recently, Reiner and Zotova proved a Hilton--Milner type stability result showing that for large $p$, any sum-free set not contained in the extremal cuboid has size at most $\frac{p-2}{3} p^{n-1}$, and all possible structures attaining this bound were classified.
In this paper, we develop a general Hilton--Milner theory for $(k,\ell)$-sum-free sets in $\mathbb{F}_p^n$ for $k > \ell \ge 1$. We determine the maximum size of such sets for all $p \equiv \mu \pmod{k+\ell}$ with $2 \le \mu \le k+\ell-1$, and show that the extremal configurations are precisely $\lceil (\mu-1)/2 \rceil$ non-isomorphic cuboids. Beyond the extremal regime, we prove sharp Hilton--Milner type stability results showing that, for all sufficiently large $p$, a $(k,\ell)$-sum-free set not contained in any of these extremal cuboids is uniformly bounded away from the maximum by a gap of order $p^{n-1}$, and we determine the full structure of all sets achieving this second-best bound in several broad parameter ranges. In particular, when $2 \le \mu \le k+\ell-3$ (which is tight), only two structural types occur for all $k+\ell \ge 5$; and when $\mu = 2$ or $3$, we obtain a complete classification for all $k > \ell \ge 1$.
Our arguments combine additive combinatorics and Fourier-analytic methods, and make use of recent progress toward the long-standing $3k-4$ conjecture, highlighting new connections between inverse additive number theory and extremal problems over finite vector spaces. - [115] arXiv:2512.22836 [pdf, html, other]
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Title: Submartingale Condition for Weak Convergence for Semi-Markov ProcessesSubjects: Probability (math.PR)
In this paper, we consider a modified version of a well-known submartingale condition fortheweak convergence of probabilitymeasures, adapted to the semi-Markov case. In this setting, it is convenient to work with an embedded Markov chain and the filtration generated by jump times. We demonstrate that a straightforward restatement of the classical result is not valid, and that an additional condition is required.
- [116] arXiv:2512.22841 [pdf, html, other]
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Title: Undecidability of epimorphisms onto products of hyperbolic groupsComments: 11 pages, Comments welcomeSubjects: Group Theory (math.GR)
We exhibit examples of finitely presented subgroups $P$ of direct products of hyperbolic groups for which there is no algorithm that detects whether a finitely presented group has a quotient isomorphic to $P$. For any torsion-free, linear, hyperbolic group $Q$ that maps onto the free group of rank $2$ and $m\geq 2$, we construct a recursive sequence $(\Gamma_n)_{n\in \mathbb{N}}$ of torsion-free, hyperbolic $C'(\frac{1}{6})$ small cancellation groups, with the property that there is no algorithm determining the values $n\in \mathbb{N}$ such that $\Gamma_n$ has a quotient isomorphic to the direct product $Q^{m}$ of $m$-copies of $Q$.
- [117] arXiv:2512.22844 [pdf, html, other]
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Title: Discrete Feynman-Kac approximation for parabolic Anderson model using random walksComments: 30 pages, 1 figureSubjects: Probability (math.PR)
In this paper, we introduce a natively positive approximation method based on the Feynman-Kac representation using random walks, to approximate the solution to the one-dimensional parabolic Anderson model of Skorokhod type, with either a flat or a Dirac delta initial condition. Assuming the driving noise is a fractional Brownian sheet with Hurst parameters $H \geq \frac{1}{2}$ and $H_* \geq \frac{1}{2}$ in time and space, respectively, we also provide an error analysis of the proposed method. The error in $L^p (\Omega)$ norm is of order \[ O \big(h^{\frac{1}{2}[(2H + H_* - 1) \wedge 1] - \epsilon}\big), \] where $h > 0$ is the step size in time (resp. $\sqrt{h}$ in space), and $\epsilon > 0$ can be chosen arbitrarily small. This error order matches the Hölder continuity of the solution in time with a correction order $\epsilon$, making it `almost' optimal. Furthermore, these results provide a quantitative framework for convergence of the partition function of directed polymers in Gaussian environments to the parabolic Anderson model.
- [118] arXiv:2512.22847 [pdf, html, other]
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Title: The Moduli Stack of Compact Metric SpacesComments: 28 pagesSubjects: Metric Geometry (math.MG)
In this paper, we introduce a Grothendieck topology on the category of totally bounded metric spaces and develop a theory of stacks with respect to this topology. We further define the fine moduli stack of compact metric spaces and prove that its coarse moduli space is isometric to the Gromov--Hausdorff space.
- [119] arXiv:2512.22849 [pdf, html, other]
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Title: Statistics of bad parts of class groupsSubjects: Number Theory (math.NT)
Let $p$ be an odd prime. We give a formula for the bad part of $p$-class groups that is valid for $100\%$ of the abelian $p$-extensions when ordered by product of ramified primes.
- [120] arXiv:2512.22852 [pdf, html, other]
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Title: More on Goldstern's principleSubjects: Logic (math.LO)
This paper is a continuation of the paper [Got25]. The main result is that the Hausdorff measure version of Goldstern's principle for $\boldsymbol{\Pi}^1_1$ sets fails in $L$, despite the fact that the Lebesgue measure version is true. Other various results regarding Goldstern's principle are established.
- [121] arXiv:2512.22861 [pdf, html, other]
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Title: On the Hausdorff Dimension of Measures for a Non-Uniquely Ergodic Family of Interval Exchange TransformationsSubjects: Dynamical Systems (math.DS)
In this paper, based on a construction by J. Fickenscher, we construct a family of non-uniquely ergodic interval exchange transformations on $n$ intervals with the maximal possible number of measures, $\left\lfloor \frac{n}{2} \right\rfloor$. Subsequently, we generalize J. Chaika's result on estimating the Hausdorff dimension of the two measures from M. Keane's example to the family of interval exchange transformations with the maximal possible number of measures.
- [122] arXiv:2512.22865 [pdf, html, other]
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Title: Improved Erdős-Pósa inequalities for odd cycles in planar graphsSubjects: Combinatorics (math.CO)
In an undirected graph, the odd cycle packing number is the maximum number of pairwise vertex-disjoint odd cycles. The odd cycle transversal number is the minimum number of vertices that hit every odd cycle. The maximum ratio between transversal and packing number is called Erdős-Pósa ratio. We show that in planar graphs, this ratio does not exceed 4. This improves on the previously best known bound of 6 by Král', Sereni and Stacho.
- [123] arXiv:2512.22866 [pdf, other]
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Title: A Recursive Exponential-Gamma Mixture: a New Generalized of the Lindley DistributionSubjects: Statistics Theory (math.ST)
The Lindley distribution was first introduced by Lindley in 1958 for Bayesian computations. Over the past years, various generalizations of this distribution have been proposed by different authors. The generalized Lindley distributions sometimes have many parameters, and although they show good flexibility, their statistical form becomes complicated. In this article, we propose a new and simple distribution determined by the recursive relation of the Lindley distribution and the Gamma distribution with specific weights. Subsequently, some statistical properties of this distribution are examined, and with real numerical examples, its superiority over the Lindley generalizations is demonstrated.
- [124] arXiv:2512.22870 [pdf, other]
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Title: Crystalline Motion of discrete interfaces in the Blume-Emery-Griffiths Model: partial wettingComments: 38 pages, 20 figuresSubjects: Analysis of PDEs (math.AP); Soft Condensed Matter (cond-mat.soft)
We continue the variational study of the discrete-to-continuum evolution of lattice systems of Blume-Emery-Griffith type which model two immiscible phases in the presence of a surfactant. In our previous work \cite{CFS}, we analyzed the case of a completely wetted crystal and described how the interplay between surfactant evaporation and mass conservation leads to a transition between crystalline mean curvature flow and pinned evolutions. In the present paper, we extend the analysis to the regime of partial wetting, where the surfactant occupies only a portion of the interface. Within the minimizing-movements scheme, we rigorously derive the continuum evolution and show how partial wetting introduces a complex coupling between interfacial motion and redistribution of surfactant. The resulting evolution exhibits new features absent in the fully wetted case, including the coexistence of moving and pinned facets or the emergence and long-lived metastable states. This provides, to our knowledge, the first discrete-to-continuum variational description of partially wetted crystalline interfaces, bridging the gap between microscopic lattice models and experimentally observed surfactant-induced pinning phenomena in immiscible systems.
- [125] arXiv:2512.22885 [pdf, html, other]
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Title: Scaling inequalities for Steklov eigenvalues in space forms and sharp eigenvalue estimates on warped product manifoldsComments: 44 pages, 2 figures. All comments are welcome!Subjects: Differential Geometry (math.DG); Analysis of PDEs (math.AP); Spectral Theory (math.SP)
In the first part, we derive monotonicity of the normalized spectra for the second-order Steklov problem and two fourth-order Steklov problems on the $2$-dimensional geodesic disks with respect to the geodesic radius in the sphere and the hyperbolic space. The normalizations are made using four natural geometric factors. As corollaries, we get Escobar-type bounds for Steklov eigenvalues on $2$-dimensional geodesic disks with varying curvature in space forms. We also get two monotonicity results for higher-dimensional cases. In the second part, we obtain some sharp bounds concerning the spectra of the two fourth-order Steklov problems on warped product manifolds with non-negative Ricci curvature and a strictly convex boundary. In particular, we confirm Qiaoling Wang and Changyu Xia's conjecture (2018) on the sharp lower bound of the first non-zero eigenvalue of a fourth-order Steklov problem in the case of $3$-dimensional warped product manifolds.
- [126] arXiv:2512.22887 [pdf, html, other]
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Title: Non-SUSY physics and the Atiyah-Singer index theoremSubjects: Mathematical Physics (math-ph)
The Atiyah-Singer index theorem, a cornerstone of modern mathematics, has traditionally been derived from supersymmetric (SUSY) physics. This paper demonstrates a direct derivation from non-supersymmetric quantum statistics by establishing a fundamental correspondence: the grand partition functions of non-interacting bosonic and fermionic systems are precisely the Chern characters of certain vector bundles. Furthermore, we generalize this correspondence to infinite dimensions, where we construct a novel mathematical framework of spectral-sheaf pairs. Within this framework, we formulate a generalized index theorem, identifying the topological index with a regularized spectral product. This work not only circumvents the need for supersymmetry but also provides a deeper unifying perspective, revealing quantum statistics as a sufficient foundation for topological invariants.
- [127] arXiv:2512.22900 [pdf, html, other]
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Title: A Note on Lagrange Subsets of Finite GroupsComments: 8 pagesSubjects: Group Theory (math.GR)
In a finite group, a subset is called a Lagrange subset if its size divides the group order, and a factor if it admits a complementary subset. We provide a new and comparatively direct proof of the classification of groups in which every Lagrange subset is a factor. We show that any nontrivial such group must be a cyclic group of prime order, the cyclic group of order 4, or an elementary abelian group of order 4, 8, or 9.
- [128] arXiv:2512.22902 [pdf, html, other]
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Title: Equidistribution of Diophantine pairs among the equivalence classes of quadratic formsComments: 16 pagesSubjects: Number Theory (math.NT); Dynamical Systems (math.DS)
For a fixed integer n, a pair of nonzero integers {a, c} is called a D(n)-pair if the product ac plus n is a perfect square.
In this short note we prove that D(n)-pairs are asymptotically equidistributed (via their associated quadratic forms) among proper SL_2(Z)-equivalence classes of binary quadratic forms of discriminant 4n with fixed content. As a consequence, we obtain a more streamlined and simpler proof of Badesa's asymptotic formula for the number of D(n)-pairs. - [129] arXiv:2512.22906 [pdf, html, other]
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Title: Further q-Supercongruences from Singh's Quadratic TransformationSubjects: Combinatorics (math.CO)
In this paper, we investigate some q-congruences for truncated ${}_{4}\phi_3$ series by using Singh's quadratic transformation and the creative microscoping method (introduced by Victor J. W. Guo and Zudilin in 2019).
- [130] arXiv:2512.22907 [pdf, html, other]
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Title: A point in the interior of the convex hullsSubjects: Combinatorics (math.CO)
Steinitz's theorem states that if a point $a \in \mathrm{int\,conv\,} X$ for a set $X \subset \mathbb{R}^d$, then $X$ contains a subset $Y$ of size at most $2d$ such that $a \in \mathrm{int\,conv\,}Y$. The bound $2d$ is best possible here. We prove the colourful version of this theorem and characterize the cases when exactly $2d$ sets are needed.
- [131] arXiv:2512.22909 [pdf, html, other]
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Title: A first-order method for nonconvex-strongly-concave constrained minimax optimizationComments: Accepted by Optimization Methods and SoftwareSubjects: Optimization and Control (math.OC); Machine Learning (cs.LG); Numerical Analysis (math.NA); Machine Learning (stat.ML)
In this paper we study a nonconvex-strongly-concave constrained minimax problem. Specifically, we propose a first-order augmented Lagrangian method for solving it, whose subproblems are nonconvex-strongly-concave unconstrained minimax problems and suitably solved by a first-order method developed in this paper that leverages the strong concavity structure. Under suitable assumptions, the proposed method achieves an \emph{operation complexity} of $O(\varepsilon^{-3.5}\log\varepsilon^{-1})$, measured in terms of its fundamental operations, for finding an $\varepsilon$-KKT solution of the constrained minimax problem, which improves the previous best-known operation complexity by a factor of $\varepsilon^{-0.5}$.
- [132] arXiv:2512.22911 [pdf, html, other]
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Title: Covering in Hamming and Grassmann Spaces: New Bounds and Reed--Solomon-Based ConstructionsComments: 14 pages, 6 figuresSubjects: Information Theory (cs.IT); Signal Processing (eess.SP)
We study covering problems in Hamming and Grassmann spaces through a unified coding-theoretic and information-theoretic framework. Viewing covering as a form of quantization in general metric spaces, we introduce the notion of the average covering radius as a natural measure of average distortion, complementing the classical worst-case covering radius. By leveraging tools from one-shot rate-distortion theory, we derive explicit non-asymptotic random-coding bounds on the average covering radius in both spaces, which serve as fundamental performance benchmarks.
On the construction side, we develop efficient puncturing-based covering algorithms for generalized Reed--Solomon (GRS) codes in the Hamming space and extend them to a new family of subspace codes, termed character-Reed--Solomon (CRS) codes, for Grassmannian quantization under the chordal distance. Our results reveal that, despite poor worst-case covering guarantees, these structured codes exhibit strong average covering performance. In particular, numerical results in the Hamming space demonstrate that RS-based constructions often outperform random codebooks in terms of average covering radius. In the one-dimensional Grassmann space, we numerically show that CRS codes over prime fields asymptotically achieve average covering radii within a constant factor of the random-coding bound in the high-rate regime. Together, these results provide new insights into the role of algebraic structure in covering problems and high-dimensional quantization. - [133] arXiv:2512.22918 [pdf, html, other]
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Title: Refined Limiting Profiles of the Principal Eigenvalue Problems with Large AdvectionSubjects: Analysis of PDEs (math.AP)
In this paper, we are concerned with the following eigenvalue problem with an advection term: \begin{equation}\label{0.1} \left\{ \begin{split} -\epsilon\Delta \phi-2\alpha\nabla m(x)\cdot\nabla \phi+V(x)\phi&=\lambda \phi\ \ \text{in}\ \ \Omega,\\ \phi&=0\ \ \hbox{on}\ \ \partial\Omega, ~~~\text{(0.1)} \end{split} \right. \end{equation} where $\Omega\subset\mathbb{R}^N~(N\geq1)$ satisfying $\partial\Omega\in C^{2}$ is a bounded domain and contains the origin as an interior point, the constants $\epsilon>0$ and $\alpha>0$ are the diffusive and advection coefficients, respectively, and $m(x)\in C^{2}(\bar{\Omega})$, $V (x)\in C^{\gamma}(\bar{\Omega})~(0<\gamma<1)$ are given functions. We analyze the refined limiting profiles of the principal eigenpair $(\lambda, \phi)$ for (0.1) as $\alpha\rightarrow\infty$, which display the visible effect of the large advection on $(\lambda, \phi)$. It expects that our argument is applicable to investigating the refined expansions of the general principal eigenvalue problems.
- [134] arXiv:2512.22921 [pdf, html, other]
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Title: Diffusion wave phenomena and optimal time decay for incompressible viscoelastic flowsComments: 32 pagesSubjects: Analysis of PDEs (math.AP)
Motivated by the work of D. Hoff and K. Zumbrun (Indiana Univ. Math. J. 44: 603-676, 1995), we investigate the diffusion wave phenomena in three-dimensional incompressible viscoelastic flows. By employing the representation formula of the wave equation and the stationary phase methods on the sphere $\mathbb{S}^{d-1}$, we establish $L^p$ decay estimates for the solution over the whole range $1\leq p \leq \infty$, which reveals the hyperbolic nature of the incompressible viscoelastic flows.
- [135] arXiv:2512.22935 [pdf, html, other]
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Title: Convergence rates for the $p$-Wasserstein distance of the empirical measures of an ergodic Markov processComments: 25 pagesSubjects: Probability (math.PR)
Let $X:=(X_t)_{t\geq 0}$ be an ergodic Markov process on $\real^d$, and $p>0$. We derive upper bounds of the $p$-Wasserstein distance between the invariant measure and the empirical measures of the Markov process $X$. For this we assume, e.g.\ that the transition semigroup of $X$ is exponentially contractive in terms of the $1$-Wasserstein distance, or that the iterated Poincaré inequality holds together with certain moment conditions on the invariant measure. Typical examples include diffusions and underdamped Langevin dynamics.
- [136] arXiv:2512.22940 [pdf, html, other]
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Title: Waldschmidt constant of monomial ideals and Simis idealsSubjects: Commutative Algebra (math.AC); Combinatorics (math.CO)
In 2017, Cooper et al. proposed a conjecture providing a lower bound for the Waldschmidt constant of monomial ideals. We confirm this conjecture for some classes of monomial ideals. Recently, Méndez, Pinto, and Villarreal formulated a conjecture stating that if $I$ is a monomial ideal without embedded associated primes, whose irreducible decomposition is minimal and which is a Simis ideal, then there exist a Simis squarefree monomial ideal $J$ and a standard linear weighting $w$ such that $I = J_{w}.$ In this work, we verify this conjecture for some classes of monomial ideals.
- [137] arXiv:2512.22943 [pdf, html, other]
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Title: A Note on the Legendre TransformationSubjects: History and Overview (math.HO); Symplectic Geometry (math.SG)
We present the Legendre transformation in a geometric way based on the procedure of the Legendrian lift. This approach allows us to understand some interesting properties of it, in particular, the reason for the appearance of singularities of dual curves. Also we consider application of the Legendre transformation to the Clairaut differential equation. Finally, we say a few words class of contact transformations and present an infinite group of contact transformations different from the Legendre transformation.
Keywords: Legendre transformation, duality, contact transformation, contact structure, pedal curve, singular point, Clairaut equation - [138] arXiv:2512.22946 [pdf, html, other]
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Title: Determining habitat anomalies in cross-diffusion predator-prey chemotaxis modelsComments: 27 pages Key words:Inverse boundary value problems, multi-species cross-diffusion predator-prey chemotaxis models, habitat degradation, anomaly detection, unique identifiabilitySubjects: Analysis of PDEs (math.AP); Cell Behavior (q-bio.CB)
This paper addresses an open inverse problem at the interface of mathematical analysis and spatial ecology: the unique identification of unknown spatial anomalies -- interpreted as zones of habitat degradation -- and their associated ecological parameters in multi-species predator-prey systems with multiple chemical signals, using only boundary measurements. We formulate the problem as the simultaneous recovery of an unknown interior subdomain and discontinuous ecological interaction rules across its boundary. A unified theooretical framework is developed that unique determines both the anomaly's geometry and discontinuous coefficients characterizing the altered interactions within the degraded region. Our results cover smooth anomalies in time-dependent systems and are extended to non-smooth polyhedral inclusions in stationary regimes. This work bridges a gap between ecological sensing and the quantitative inference of internal habitat heterogeneity, offering a mathamtical basis for detecting and characterizing habitat degradation from limited external data.
- [139] arXiv:2512.22948 [pdf, html, other]
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Title: Generalized Hyperderivative Reed-Solomon CodesSubjects: Information Theory (cs.IT); Rings and Algebras (math.RA)
This article introduces Generalized Hyperderivative Reed-Solomon codes (GHRS codes), which generalize NRT Reed-Solomon codes. Its main results are as follows: 1) every GHRS code is MDS, 2) the dual of a GHRS code is also an GHRS code, 3) determine subfamilies of GHRS codes whose members are low-density parity-check codes (LDPCs), and 4) determine a family of GHRS codes whose members are quasi-cyclic. We point out that there are GHRS codes having all of these properties.
- [140] arXiv:2512.22954 [pdf, html, other]
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Title: Lovász--Saks--Schrijver Ideals and the Irreducible Components of the Variety of Orthogonal Representations of a GraphSubjects: Combinatorics (math.CO); Commutative Algebra (math.AC); Algebraic Geometry (math.AG)
Given a finite simple graph $G$ and a positive integer $d$, one can associate to $G$ the Lovász--Saks--Schrijver ideal $L_{G}(d)$, an ideal generated by quadratic polynomials coming from orthogonality conditions. The corresponding variety $\mathbb{V}(L_{G}(d))$, denoted $\mathrm{OR}_{d}(\overline{G})$, is the variety of orthogonal representations of the complement graph $\overline{G}$: its points are maps from the vertex set of $G$ to $\mathbb{K}^{d}$ that send adjacent vertices of $G$ to orthogonal vectors. In this paper we study the irreducible decomposition of $\mathrm{OR}_{d}(\overline{G})$ and the primary decomposition of $L_{G}(d)$. Our main focus is the case in which $G$ is a forest. Under this assumption, we determine the irreducible components of $\mathrm{OR}_{d}(\overline{G})$, compute their dimensions, and describe their defining equations, thereby obtaining the primary decomposition of $L_{G}(d)$. The key ingredient is a matroid-theoretic framework in which we associate to every forest $G$ a paving matroid $\mathcal{M}(G)$.
- [141] arXiv:2512.22960 [pdf, html, other]
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Title: Standing waves of the Anderson-Gross-Pitaevskii equationSubjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph); Probability (math.PR)
In this paper, we study standing waves for the Anderson-Gross-Pitaevskii equation in dimension 1 and 2. The Anderson-Gross-Pitaevskii equation is a nonlinear Schrödinger equation with a confining potential and a multiplicative spatial white noise. Standing waves are characterized by a profile which is invariant by the dynamic and solves a nonlinear elliptic equation with spatial white noise potential. We construct such solutions via variational methods and obtain some results on their regularity, localization and stability.
- [142] arXiv:2512.22961 [pdf, html, other]
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Title: Deep Learning for the Multiple Optimal Stopping ProblemSubjects: Optimization and Control (math.OC); Machine Learning (cs.LG)
This paper presents a novel deep learning framework for solving multiple optimal stopping problems in high dimensions. While deep learning has recently shown promise for single stopping problems, the multiple exercise case involves complex recursive dependencies that remain challenging. We address this by combining the Dynamic Programming Principle with neural network approximation of the value function. Unlike policy-search methods, our algorithm explicitly learns the value surface. We first consider the discrete-time problem and analyze neural network training error. We then turn to continuous problems and analyze the additional error due to the discretization of the underlying stochastic processes. Numerical experiments on high-dimensional American basket options and nonlinear utility maximization demonstrate that our method provides an efficient and scalable method for the multiple optimal stopping problem.
- [143] arXiv:2512.22967 [pdf, other]
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Title: Panhandle polynomials of torus links and geometric applicationsComments: 40 pages, 12 figures, 5 tablesSubjects: Geometric Topology (math.GT); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Quantum Algebra (math.QA)
We use a decomposition of the tensor of the fundamental representation of the quantum group $U_q(\mathfrak{sl}_N)$ and the Rosso-Jones formula to establish a peculiar ``panhandle'' shape of the HOMFLY-PT polynomial of the reverse parallel of torus knots and links. Due to their panhandle-like intrinsic properties, the HOMFLY-PT polynomial is referred to as a ``panhandle polynomial''. With the help of the $\ell$-invariant, this extends to links the Etnyre-Honda result about the arc index and maximal Thurston-Bennequin invariant of torus knots. It has further geometric consequences, related to the braid index, the existence of minimal string Bennequin surfaces for banded and Whitehead doubled links, the Bennequin sharpness problem, and the equivalence of their quasipositivity and strong quasipositivity. We extend these properties to torus links, which relate to the classification of their component-wise Thurston-Bennequin invariants. Finally, we discuss the definition of the $\ell$-invariant for general links.
- [144] arXiv:2512.22970 [pdf, html, other]
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Title: Delta-Unknotting Number for Two-Bridge KnotsComments: 15 pages, 7 figures, 2 tablesSubjects: Geometric Topology (math.GT)
The $\Delta$-unknotting number for a knot is defined as the minimum number of $\Delta$-moves needed to deform the knot into the trivial knot. We determine the $\Delta$-unknotting numbers for two-bridge knots of type $C(2\beta_1, 2\beta_2, ... , 2\beta_n)$ and type $C(2\beta_1, 2\beta_2, ... , 2\beta_{n-1}, 2\beta_n-1)$, where $\beta_i$ is a positive integer for $1 \leq i \leq n$. We also discuss two-bridge knots whose $\Delta$-unknotting number is equal to one.
- [145] arXiv:2512.22977 [pdf, other]
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Title: A solution to Godsil's conjecture on the edge-connectivity of graphs in association schemesComments: 27 pgaes, 8 figuresSubjects: Combinatorics (math.CO)
A graph $G$ is called equiarboreal if the number of spanning trees containing a given edge in $G$ is independent of the choice of edge. In [Combinatorica 1(2) (1981) 163--167], Godsil proved that any graph which is a colour class in an association scheme is equiarboreal, and further conjectured that the edge-connectivity of a connected graph which is a colour class in an association scheme equals its vertex degree. In this paper, we confirm this long-standing conjecture. More generally, we prove an even stronger result that the edge-connectivity of a connected regular equiarboreal graph equals its degree by combinatorial and electrical network approaches. As a consequence, we show that every connected regular equiarboreal graph on an even number of vertices has a perfect matching.
- [146] arXiv:2512.22985 [pdf, html, other]
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Title: Tensor Power Asymptotics for Linearly Reductive GroupsComments: 8 pagesSubjects: Representation Theory (math.RT)
Given a finite-dimensional faithful representation $V$ of a linearly reductive group $G$ over a field $K=\bar K$, we consider the growth of the number of irreducible factors of $V^{\otimes n}$ when $n$ is large. We prove that there exist upper and lower bounds which are constant multiples of $n^{-u/2} (\dim V)^n$, where $u$ is the dimension of any maximal unipotent subgroup of $G$.
- [147] arXiv:2512.22986 [pdf, html, other]
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Title: Risk-Averse Learning with Varying Risk LevelsSubjects: Optimization and Control (math.OC); Machine Learning (cs.LG)
In safety-critical decision-making, the environment may evolve over time, and the learner adjusts its risk level accordingly. This work investigates risk-averse online optimization in dynamic environments with varying risk levels, employing Conditional Value-at-Risk (CVaR) as the risk measure. To capture the dynamics of the environment and risk levels, we employ the function variation metric and introduce a novel risk-level variation metric. Two information settings are considered: a first-order scenario, where the learner observes both function values and their gradients; and a zeroth-order scenario, where only function evaluations are available. For both cases, we develop risk-averse learning algorithms with a limited sampling budget and analyze their dynamic regret bounds in terms of function variation, risk-level variation, and the total number of samples. The regret analysis demonstrates the adaptability of the algorithms in non-stationary and risk-sensitive settings. Finally, numerical experiments are presented to demonstrate the efficacy of the methods.
- [148] arXiv:2512.22988 [pdf, html, other]
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Title: Positive commutators of positive square-zero operatorsComments: 14 pagesSubjects: Functional Analysis (math.FA)
In this paper we first consider the question which nonnegative matrices are commutators of nonnegative square-zero matrices. Then, we treat infinite-dimensional analogues of these results for operators on the Banach lattices $L^p[0,1]$ and $\ell^p$ ($ 1 \leq p < \infty$). In the last setting we need to extend the notion of the nonnegative rank of a nonnegative matrix.
- [149] arXiv:2512.23001 [pdf, html, other]
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Title: Around the Fejér-Jackson inequality: Tight bounds for certain oscillatory functions via Laplace transform representationsComments: 26 pp, 2 figuresSubjects: Classical Analysis and ODEs (math.CA)
The error of approximation of the $2\pi$-periodic sawtooth function $(\pi-x)/2$, $0\leq x<2\pi$, by its $n$-th Fourier polynomial is shown to be bounded by arccot$((2n+1)\sin(x/2))$. Related asymptotically tight inequalities with explicit constants are given for the integral of the Dirichlet kernel interpolated to non-integer values of frequency parameter and for the Taylor series remainder of the logarithmic function $\log(1-z)$ in the unit circle. The proofs are based on the Laplace transform representation of the Lerch Zeta function with $s=1$.
- [150] arXiv:2512.23003 [pdf, html, other]
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Title: MSO logic of the real order with the set quantifier ranging over Borel setsSubjects: Logic (math.LO)
A celebrated 1969 theorem of Michael Rabin is that the MSO theory of the real order where the monadic quantifier is allowed only to range over the sets of rational numbers, is decidable. In 1975 Saharon Shelah proved that if the monadic quantifier is allowed to range over all subsets of the reals, the resulting MSO theory is undecidable. He asked what happens when we allow the monadic quantifier to range over the Borel subsets of the reals. We answer that question in the affirmative. Namely, the MSO theory of the real order where the set quantifier is allowed to range over the Borel sets, is decidable.
- [151] arXiv:2512.23006 [pdf, html, other]
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Title: On subdivisions of the permutahedron and flags of lattice path matroidsComments: 22 pages, 9 figuresSubjects: Combinatorics (math.CO)
In this manuscript we study the subdivisions of the permutahedron $\Pi_n$ into two subpolytopes corresponding to flags of positroids, which are in particular flags of lattice path matroids (LPFMs). A subpolytope $P_{[u,v]}$ of $\Pi_n$ is a Bruhat Interval Polytope (BIP) if $P_{[u,v]}$ is the convex hull of all the permutations (viewed as points in $\RR^n$) in the interval $[u,v]$ in the Bruhat order of $§_n$. We show that the coarsest subdivisions we obtain into LPFMs are the only subdivisions of $\Pi_n$ via hyperplane splits, into subpolytopes corresponding to BIPs. More specifically, we describe the hyperplanes whose intersection with $\Pi_n$ give rise to BIPs. Hence, these subdivisions are polytopes coming from points in the complete nonnegative flag variety.
- [152] arXiv:2512.23007 [pdf, html, other]
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Title: Derivation of nonlinear time-dependent macroscopic conductivity for an electropermeabilization model via homogenizationComments: 40 pages, 9 figuresSubjects: Analysis of PDEs (math.AP)
We study a phenomenological electropermeabilization model in a periodic medium representing biological tissue. Starting from a cell-level model describing the electric potential and the degree of porosity, we perform dimension analysis to identify a relevant scaling in terms of a small parameter $\ve$ - the ratio between the cell and the tissue size. The electric potential satisfies electrostatic equations in the extra- and intracellular domains, while its jump across the cell membrane evolves according to a nonlinear law coupled with an ordinary differential equation for the porosity degree. We prove the well-posedness of the microscopic problem, derive a priori estimates, obtain formal asymptotics, and rigorously justify the expansion combining two-scale convergence with monotonicity arguments. The resulting macroscopic model exhibits memory effects and a nonlinear, time-dependent effective current. It captures the nontrivial evolution of effective conductivity, including a characteristic drop reflecting the capacitive behavior of the lipid bilayer, in agreement with experimental data. Numerical computations of the effective conductivity confirm that, although microscopic conductivity is constant, tissue conductivity varies nonlinearly with electric field strength, showing a sigmoid trend. This suggests a rigorous mathematical explanation for experimentally observed conductivity dynamics.
- [153] arXiv:2512.23011 [pdf, html, other]
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Title: The codegree Turán density of tight cyclesComments: 32 pages, 3 figuresSubjects: Combinatorics (math.CO)
The codegree Turán density $\gamma(F)$ of a $k$-uniform hypergraph $F$ is the minimum real number $\gamma \ge 0$ such that every $k$-uniform hypergraph on sufficiently many $n$ vertices, in which every set of $k-1$ vertices is contained in at least $(\gamma+o(1))n$ edges, contains a copy of $F$. A recent result of Piga, Sanhueza-Matamala, and Schacht determines that $\gamma(C_{\ell}^3)=\frac13$ for every $3$-uniform tight cycle $C_\ell^3$ of length $\ell$, where $\ell \ge \ell_0$ and $\ell$ is not divisible by $3$. In this paper, we investigate the codegree Turán density of $k$-uniform tight cycles $C_\ell^k$. We establish improved upper and lower bounds on $\gamma(C_{\ell}^k)$ for general $\ell$ not divisible by $k$. These results yield the following consequences:
1). For any prime $k \ge 3$, we show that $\gamma(C_{\ell}^k)=\frac13$ for all sufficiently large $\ell$ not divisible by $k$, generalizing the above theorem of Piga et al.
2). For all $k \ge 3$, we determine the exact value of $\gamma(C_{\ell}^k)$ for integers $\ell$ not divisible by $k$ in a set of (natural) density at least $\frac{\varphi(k)}{k}$, where $\varphi(\cdot)$ denotes Euler's totient function.
3). We give a complete answer to a question of Han, Lo, and Sanhueza-Matamala concerning the tightness of their construction for $\gamma(C_{\ell}^k)$.
Moreover, our results also determine the codegree Turán density of $C_\ell^{k-}$, that is, the $k$-uniform tight cycle of length $\ell$ with one edge removed, for a new set of integers $\ell$ of positive density for every $k \ge 3$. Our upper bound result is based on a structural characterization of $C_{\ell}^k$-free $k$-uniform hypergraphs with high minimum codegree, while the lower bounds are derived from a novel construction model, coupled with the arithmetic properties of the integers $k$ and $\ell$. - [154] arXiv:2512.23012 [pdf, other]
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Title: Wall-crossing for invariants of equivariant 3CY categoriesComments: 175 pages, 3 figures. Comments welcomeSubjects: Algebraic Geometry (math.AG); High Energy Physics - Theory (hep-th)
We provide a wall-crossing framework for operational enumerative invariants of equivariant 3-Calabi--Yau categories arising from virtual cycles. The strategy follows ideas of Joyce's ``universal'' wall-crossing framework arXiv:2111.04694, using the authors' symmetrized pullback technique to preserve the symmetry of the (almost-perfect) obstruction theories throughout. As an application, we define and study wall-crossings of simple type between operational equivariant Donaldson--Thomas (DT), Pandharipande--Thomas (PT), and Bryan--Steinberg (BS) vertices. In particular, we give an explicit DT/PT descendent vertex correspondence in the Calabi--Yau limit. As another application, we construct and prove wall-crossing formulas for operational refined semistable Vafa--Witten invariants.
- [155] arXiv:2512.23018 [pdf, html, other]
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Title: Many critical points for discrete Riesz energy on $\mathbb{T}^2$Subjects: Classical Analysis and ODEs (math.CA)
It is widely believed that the energy functional $E_p:(\mathbb{S}^2)^n \rightarrow \mathbb{R}$ $$ E_p = \sum_{i,j=1 \atop i \neq j}^{n} \frac{1}{\|x_i-x_j\|^p}$$ has a number of critical points, $\nabla E(x) = 0$, that grows exponentially in $n$. Despite having been extensively tested and being physically well motivated, no rigorous result in this direction exists. We prove a version of this result on the two-dimensional flat torus $\mathbb{T}^2$ and show that there are infinitely many $n \in \mathbb{N}$ such that the number of critical points of $E_p: (\mathbb{T}^2)^n \rightarrow \mathbb{R}$ is at least $\exp(c \sqrt{n})$ provided $p \geq 5 \log{n}$. We also investigate the special cases $n=3,4,5$ which turn out to be surprisingly interesting.
- [156] arXiv:2512.23022 [pdf, other]
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Title: A Generalization of the "Brouwer-Schauder-Tychonoff" Fixed-Point TheoremComments: 5 pagesSubjects: Functional Analysis (math.FA)
We prove a new fixed - point result for the image Im(j) of any continuous function j from K to (K x K), where K is a compact convex subset of a Hausdorff locally convex space, provided that the projection of Im(j) to the first factor is onto, and a condition on the convex hull of Im(j) holds. A special case of our result is the Brouwer - Schauder - Tychonoff fixed point theorem for continuous functions from K to K.
- [157] arXiv:2512.23046 [pdf, html, other]
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Title: User-Centric Cell-Free Massive MIMO Enhanced by Fluid-Antenna Access Points: Uplink AnalysisComments: Submitted to an IEEE JournalSubjects: Information Theory (cs.IT); Signal Processing (eess.SP)
In this paper, we investigate cell-free massive MIMO (CF-mMIMO) systems in which access points (APs) are equipped with fluid antennas (FAs) and develop a comprehensive framework for channel estimation, antenna port selection, and uplink spectral efficiency (SE) optimization. We propose a generalized LMMSE-based uplink channel estimation scheme that dynamically activates FA ports during pilot transmission, efficiently exploiting antenna reconfigurability under practical training constraints. Building on this, we design a distributed port selection strategy that minimizes per-AP channel estimation error by exploiting spatial correlation among FA ports. We systematically analyze the impact of antenna geometry and spatial correlation using the Jakes' channel model for different AP array configurations, including uniform linear and planar arrays. We then derive SINR expressions for centralized and distributed uplink processing and obtain a closed-form uplink SE expression for centralized maximum-ratio combining using the use-and-then-forget bound. Finally, we propose an alternating-optimization framework to select FA port configurations that maximize the uplink sum SE. Numerical results show that the proposed FA-aware channel estimation and port optimization strategies greatly reduce channel estimation error and significantly improve sum-SE over fixed-antenna and non-optimized FA baselines, confirming FAs as a key enabler for scalable, adaptive CF-mMIMO networks.
- [158] arXiv:2512.23047 [pdf, html, other]
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Title: Bayesian Effective Dimension: A Mutual Information PerspectiveSubjects: Statistics Theory (math.ST)
High-dimensional Bayesian procedures often exhibit behavior that is effectively low dimensional, even when the ambient parameter space is large or infinite-dimensional. This phenomenon underlies the success of shrinkage priors, regularization, and approximate Bayesian methods, yet it is typically described only informally through notions such as sparsity, intrinsic dimension, or degrees of freedom. In this paper we introduce the \emph{Bayesian effective dimension}, a model- and prior-dependent quantity defined through the mutual information between parameters and data. This notion quantifies the expected information gain from prior to posterior and provides a coordinate-free measure of how many directions in parameter space are statistically learnable at a given sample size. In regular parametric models the effective dimension coincides with the usual parameter dimension, while in high-dimensional, ill-posed, or strongly regularized settings it can be substantially smaller. We develop basic properties of the effective dimension and present explicit calculations for Gaussian location models and linear models with general design, revealing close connections with spectral complexity and effective rank. These examples illustrate how shrinkage and regularization mechanisms directly control the growth of effective dimension. The framework offers a unifying perspective on dimension reduction in Bayesian inference and provides insight into uncertainty quantification and the behavior of approximate posterior distributions.
- [159] arXiv:2512.23052 [pdf, html, other]
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Title: A regularized theta lift on the symmetric space of $SL_N$Subjects: Number Theory (math.NT)
We define a regularized lift from harmonic weak Maass forms of weight $2-N$ to differential forms of degree $N-1$ on the symmetric space $\SL_N(\R)/\SO(N)$, that are smooth outside of certain modular symbols. We show that this lift is adjoint to the derivative of a theta lift. We compute periods of the regularized lift over tori and relate them to Fourier coefficients of Hilbert-Eisenstein series.
- [160] arXiv:2512.23058 [pdf, html, other]
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Title: Lê modules and hypersurfaces with one-dimensional singular setsComments: 8 pagesSubjects: Algebraic Geometry (math.AG)
By using our previous results on Lê modules and an upper-bound on the betti numbers which we proved with Lê, we investigate the cohomology of Milnor fibers and the internal local systems given by the vanishing cycles of hypersurfaces with one-dimensional singular sets.
- [161] arXiv:2512.23064 [pdf, other]
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Title: Bounding the integral of the argument of the Riemann Zeta functionComments: 17 pages, 8 tablesSubjects: Number Theory (math.NT)
This article improves the estimate of $|S_1(t_2)-S_1(t_1)|$, which is the definite integral of the argument of the Riemann zeta-function between $t_1$ and $t_2$. Estimates of this quantity are needed to apply Turing's method to compute the exact number of zeta zeros up to a given height.
- [162] arXiv:2512.23069 [pdf, html, other]
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Title: Robustness of OLS to sample removals: Theoretical analysis and implicationsSubjects: Statistics Theory (math.ST)
For learned models to be trustworthy, it is essential to verify their robustness to perturbations in the training data. Classical approaches involve uncertainty quantification via confidence intervals and bootstrap methods. In contrast, recent work proposes a more stringent form of robustness: stability to the removal of any subset of $k$ samples from the training set. In this paper, we present a theoretical study of this criterion for ordinary least squares (OLS). Our contributions are as follows: (1) Given $n$ i.i.d. training samples from a general misspecified model, we prove that with high probability, OLS is robust to the removal of any $k \ll n $ samples. (2) For data of dimension $p$, OLS can withstand up to ${k\ll \sqrt{np}/\log n}$ sample removals while remaining robust and achieving the same error rate as OLS applied to the full dataset. Conversely, if $k$ is proportional to $n$, OLS is provably non-robust. (3) We revisit prior analyses that found several econometric datasets to be highly non-robust to sample removals. While this appears to contradict our results in (1), we demonstrate that the sensitivity is due to either heavy-tailed responses or correlated samples. Empirically, this sensitivity is considerably attenuated by classical robust methods, such as linear regression with a Huber loss.
- [163] arXiv:2512.23074 [pdf, html, other]
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Title: Cohomology and deformation theory of Reynolds--Nijenhuis associative algebrasSubjects: Rings and Algebras (math.RA)
In this paper, we introduce and study Reynolds--Nijenhuis operators on associative algebras a novel hybrid structure that simultaneously satisfies the defining identities of both Reynolds and Nijenhuis operators. We investigate their connections with Rota-Baxter and modified Rota-Baxter operators. We develop a representation theory for Reynolds--Nijenhuis associative algebras and introduce a corresponding cohomology theory. Furthermore, we establish a one-parameter formal deformation theory for these algebras, examining the role of infinitesimals, rigidity, and equivalence in the context of deformations.
- [164] arXiv:2512.23079 [pdf, html, other]
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Title: An exceptional set of uniformly spread Kakutani tilings of the lineComments: 14 pages, 6 figuresSubjects: Dynamical Systems (math.DS); Combinatorics (math.CO); Metric Geometry (math.MG); Number Theory (math.NT)
The {\alpha}-Kakutani substitution rule splits the unit interval into two subintervals of lengths alpha and 1 - {\alpha}, for a fixed {\alpha} in (0,1). A simple inflation-substitution procedure produces tilings of the real line and their associated Delone sets. We show that there are precisely five distinct values of min({\alpha}, 1 - {\alpha}) for which these sets are uniformly spread, meaning that they are a bounded displacement of a lattice. The proof of this surprising fact combines the construction and analysis of a related family of primitive substitution tilings, Solomon's criterion for uniform spreadness, and a classification of Pisot-Vijayaraghavan polynomials.
- [165] arXiv:2512.23083 [pdf, html, other]
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Title: Growth of (α,\b{eta},γ)-order solutions of linear differential equations with analytic coefficients in the unit discComments: 18 pagesSubjects: Complex Variables (math.CV)
In this paper, we study the growth of solutions to higher-order complex linear differential equations in the unit disc, where the analytic coefficients are of finite ({\alpha},\b{eta},{\gamma})-order. By employing the concepts of ({\alpha},\b{eta},{\gamma})-order and ({\alpha},\b{eta},{\gamma})-type, we establish new results concerning the growth of such solutions. These results extend and generalize previous work by the second author and by Biswas.
- [166] arXiv:2512.23091 [pdf, html, other]
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Title: Singular MeandersComments: 12 pages, 7 figuresSubjects: Combinatorics (math.CO); Geometric Topology (math.GT)
The problem of enumerating meanders -- pairs of simple plane curves with transverse intersections -- was formulated about forty years ago and is still far from solved. Recently, it was discovered that meanders admit a factorization into prime components. This factorization naturally leads to a broader class of objects, which we call singular meanders, in which tangential intersections between the curves are also allowed. In the present paper we initiate a systematic study of singular meanders: we develop a basic combinatorial framework, point out connections with other combinatorial objects and known integer sequences, and completely enumerate several natural families of singular meanders.
- [167] arXiv:2512.23092 [pdf, html, other]
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Title: On $T$-avoiding spherical codes and designs in $\mathbb{R}^{32}$Comments: 10 pagesSubjects: Combinatorics (math.CO)
In this article, we show that the minimal vectors of the extremal even unimodular lattices in $\mathbb{R}^{32}$ are $T$-avoiding universally optimal for suitable sets $T$. Moreover, they are minimal $T$-avoiding spherical designs and maximal $T$-avoiding codes for appropriate choices of $T$.
- [168] arXiv:2512.23099 [pdf, html, other]
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Title: Lectures on Gauge theories and Many-Body systemsComments: 52 pages, 9 problemsSubjects: Mathematical Physics (math-ph); High Energy Physics - Theory (hep-th)
These lectures discuss two correspondences between gauge theories and integrable many-body systems. The first correspondence goes back to the work of many mathematicians and physicists in the 1980-1990's. It is realized by an infinite dimensional Hamiltonian reduction and its quantum counterpart. In this approach the quantization parameters of the gauge theory coincide with the quantization parameters of the many-body system. The second correspondence emerged in the mid-1990's, it involves non-trivial dualities, relating classical problems on one side to quantum on another and vice versa. This duality has various reincarnations: Fourier and Legendre transforms, Langlands duality, etc. The quantization parameters are mapped to geometric parameters. Simple questions on one side solve complicated ones on the other and vice versa.
- [169] arXiv:2512.23101 [pdf, html, other]
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Title: On the Rational Hyperbolicity problemComments: 25 pages, 1 figureSubjects: Differential Geometry (math.DG)
We prove that a compact simply connected manifold $M$ with a variationally complete $G$-action satisfying certain mild conditions (e.g. trivial principal isotropy, or simply connected principal orbits) is rationally elliptic if and only if $M/G$ is flat. This answers several conjectures and problems regarding the rational homotopy of manifolds with symmetries. On the other hand, without the extra conditions we find examples of rationally elliptic $G$-manifolds $M$ where $M/G$ admits a hyperbolic metric.
- [170] arXiv:2512.23106 [pdf, html, other]
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Title: Guillarmou's Normal Operator for Magnetic and Thermostat FlowsComments: 29 pagesSubjects: Analysis of PDEs (math.AP); Differential Geometry (math.DG); Dynamical Systems (math.DS)
Guillarmou's normal operator over a closed Anosov manifold is analogous to the classical normal operator of the geodesic X-ray transform over manifolds with boundary. In this paper, we generalize this normal operator, under some dynamical assumptions, to thermostat flows as well as to the case of the magnetic flows. In particular, we show that these generalized normal operators are elliptic pseudodifferential operators of order -1 in each case. As an application, we prove a stability estimate for the magnetic X-ray transform.
- [171] arXiv:2512.23108 [pdf, html, other]
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Title: Vietoris thickenings and complexes of manifolds are homotopy equivalentComments: 14 pagesSubjects: Geometric Topology (math.GT); General Topology (math.GN)
We show that if $X$ is a finite-dimensional Polish metric space, then the natural bijection $\mathrm{VR}(X;r)\to \mathrm{VR^m}(X;r)$ from the (open) Vietoris-Rips complex to the Vietoris-Rips metric thickening is a homotopy equivalence. This occurs, for example, if $X$ is a Riemannian manifold. The same is true for the map $\mathrm{\check{C}}(X;r)$ to $\mathrm{\check{C}}^\mathrm{m}(X;r)$ from the Čech complex to the Čech metric thickening, and more generally, for the natural bijection $\mathrm{V}(\mathcal W)\to \mathrm{V^m}(\mathcal W)$ from the Vietoris complex to the Vietoris metric thickening of any uniformly bounded cover $\mathcal W$ of a finite dimensional Polish metric space. We also show that if $X$ is a compact metrizable space, then $\mathrm{V^m}(\mathcal W)$ is strongly locally contractible.
- [172] arXiv:2512.23115 [pdf, html, other]
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Title: The Design of Optimal Dependency and RewardsSubjects: Optimization and Control (math.OC)
We analyze a two-period principal-agent model in which the principal faces a budget constraint, and the agent's private costs of performing tasks across the two periods may be correlated. We examine the optimal design of the reward scheme and the cost correlation structure. Our findings reveal that when the budget is low, the optimal reward scheme employs \textit{sufficient performance targeting}, rewarding the agent's first performance. Conversely, when the principal's budget is high, the focus shifts to \textit{sustained performance targeting}, compensating the agent's second performance. Introducing a negative cost correlation proves particularly beneficial in both scenarios: it increases the likelihood of the agent performing at least once under low budgets and balances the agent's total costs to facilitate consistent performance under high budgets. However, the optimal cost correlation structure can be more elaborate, especially for intermediate budget levels. Our results offer valuable insights for real-world applications, such as research funding allocation.
- [173] arXiv:2512.23134 [pdf, html, other]
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Title: Difference-of-Convex Elastic Net for Compressed SensingSubjects: Optimization and Control (math.OC)
This work proposes a novel and unified sparse recovery framework, termed the difference of convex Elastic Net (DCEN). This framework effectively balances strong sparsity promotion with solution stability, and is particularly suitable for high-dimensional variable selection involving highly correlated features. Built upon a difference-of-convex (DC) structure, DCEN employs two continuously tunable parameters to unify classical and state-of-the-art models--including Lasso, Elastic Net, Ridge, and $\ell_1-\alpha\ell_2$--as special cases. Theoretically, sufficient conditions for exact and stable recovery are established under the restricted isometry property (RIP), and a closed-form expression of the DCEN regularization proximal operator is derived. Moreover, two efficient optimization algorithms are developed based on the DC algorithm (DCA) and the alternating direction method of multipliers (ADMM). Within the Kurdyka-Lojasiewicz (KL) framework, the global convergence of DCA and its linear convergence rate are rigorously established. Furthermore, DCEN is extended to image reconstruction by incorporating total variation (TV) regularization, yielding the DCEN-TV model, which is efficiently solved via the Split Bregman method. Numerical experiments demonstrate that DCEN consistently outperforms state-of-the-art methods in sparse signal recovery, high-dimensional variable selection under strong collinearity, and Magnetic Resonance Imaging (MRI) image reconstruction, achieving superior recovery accuracy and robustness.
- [174] arXiv:2512.23143 [pdf, html, other]
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Title: $x$ Plays Pokemon, for Almost-Every $x$Subjects: History and Overview (math.HO); Formal Languages and Automata Theory (cs.FL); Combinatorics (math.CO)
This paper provides a brief write-up showing that for any finite state game, a disjunctive number $x$ will eventually win that game. The proof techniques here are well known and this result follows immediately from folklore results in graph theory and cellular automata. This short paper primarily serves as an expositional piece to collect this proof with the fun context of $\pi$ Plays Pokémon serving as motivation.
- [175] arXiv:2512.23148 [pdf, other]
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Title: On construction of differential $\mathbb Z$-graded varietiesComments: 45 pagesSubjects: Mathematical Physics (math-ph); Commutative Algebra (math.AC); Differential Geometry (math.DG)
Given a commutative unital algebra $\mathcal O$, a proper ideal $\mathcal I$ in $\mathcal O$, and a positively graded differential variety over $\mathcal O/\mathcal I$, we provide a $\mathbb Z$-graded extension, whose negative part is an arborescent Koszul-Tate resolution of $\mathcal O/ \mathcal I$. This extension is obtained through an algorithm exploiting the explicit homotopy retract data of the arborescent Koszul-Tate resolution, so that the number of homological computations in the construction is significantly reduced. For a positively graded differential variety over $\mathcal O$ that preserves the ideal $\mathcal I$, the extension admits a manifest description in terms of decorated trees and computed data.
As a by-product, to every Lie--Rinehart algebra over the coordinate ring of an affine variety $ W \subseteq M = \mathbb{C}^d $, one associates an explicit differential $\mathbb{Z}$-graded variety over $M$ whose negative component is the arborescent Koszul--Tate resolution of the coordinate ring $\mathbb C[x_1, \ldots, x_d]/\mathcal I_W$ of $W$, and whose positive component is the universal dg-variety of the given Lie--Rinehart algebra. Concrete examples are given. - [176] arXiv:2512.23149 [pdf, html, other]
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Title: Graph Limits via QuotientsSubjects: Combinatorics (math.CO); Probability (math.PR)
We present a new notion of limits of weighted directed graphs of growing size based on convergence of their random quotients. These limits are specified in terms of random exchangeable measures on the unit square. We call our limits grapheurs and show that these are dual to graphons in a precise sense. Grapheurs are well-suited to modeling hubs and connections between them in large graphs; previous notions of graph limits based on subgraph densities fail to adequately model such global structures as subgraphs are inherently local. Using our framework, we present an edge-based sampling approach for testing properties pertaining to hubs in large graphs. This method relies on an edge-based analog of the Szemerédi regularity lemma, whereby we show that sampling a small number of edges from a large graph approximately preserves its quotients. Finally, we observe that the random quotients of a graph are related to each other by equipartitions, and we conclude with a characterization of such random graph models.
- [177] arXiv:2512.23150 [pdf, html, other]
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Title: Constraint programming model and biased random-key genetic algorithm for the single-machine coupled task scheduling problem with exact delays to minimize the makespanSubjects: Optimization and Control (math.OC); Artificial Intelligence (cs.AI)
We consider the strongly NP-hard single-machine coupled task scheduling problem with exact delays to minimize the makespan. In this problem, a set of jobs has to be scheduled, each composed of two tasks interspersed by an exact delay. Given that no preemption is allowed, the goal consists of minimizing the completion time of the last scheduled task. We model the problem using constraint programming (CP) and propose a biased random-key genetic algorithm (BRKGA). Our CP model applies well-established global constraints. Our BRKGA combines some successful components in the literature: an initial solution generator, periodical restarts and shakes, and a local search algorithm. Furthermore, the BRKGA's decoder is focused on efficiency rather than optimality, which accelerates the solution space exploration. Computational experiments on a benchmark set containing instances with up to 100 jobs (200 tasks) indicate that the proposed BRKGA can efficiently explore the problem solution space, providing high-quality approximate solutions within low computational times. It can also provide better solutions than the CP model under the same computational settings, i.e., three minutes of time limit and a single thread. The CP model, when offered a longer running time of 3600 seconds and multiple threads, significantly improved the results, reaching the current best-known solution for 90.56% of these instances. Finally, our experiments highlight the importance of the shake and local search components in the BRKGA, whose combination significantly improves the results of a standard BRKGA.
- [178] arXiv:2512.23164 [pdf, html, other]
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Title: Infinite divisibility of $α$-Cauchy and related variablesComments: 14 pagesSubjects: Probability (math.PR)
We study the infinite divisibility of the $\alpha$-Cauchy variable $\mathcal{C}_\alpha$, $\alpha > 1$. The distribution of $\mathcal{C}_2$ is the well-known Cauchy distribution, which is infinitely divisible and even stable. But when $\alpha \neq 2$, there is no known result on the infinite divisibility of $\mathcal{C}_\alpha$. In this paper, we prove that $\mathcal{C}_\alpha$ is infinitely divisible if $1 < \alpha \leq 6/5$, and we give some sufficient conditions for $|\mathcal{C}_\alpha|^p, \, p\in \mathbb{R},$ to be infinitely divisible, which partially answers the open questions raised by Yano, Yano and Yor in 2009. In the proofs, a class of positive random variables having moments of Gamma type plays an important role, and we investigate the conditions for their existence.
- [179] arXiv:2512.23166 [pdf, html, other]
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Title: A Proximal-Gradient Method for Solving Regularized Optimization Problems with General ConstraintsComments: 1 figureSubjects: Optimization and Control (math.OC)
We propose, analyze, and test a proximal-gradient method for solving regularized optimization problems with general constraints. The method employs a decomposition strategy to compute trial steps and uses a merit function to determine step acceptance or rejection. Under various assumptions, we establish a worst-case iteration complexity result, prove that limit points are first-order KKT points, and show that manifold identification and active-set identification properties hold. Preliminary numerical experiments on a subset of the CUTEst test problems and sparse canonical correlation analysis problems demonstrate the promising performance of our approach.
- [180] arXiv:2512.23172 [pdf, other]
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Title: Qualitative analysis on the critical points of the Kirchhoff-Routh functionComments: 77 pagesSubjects: Analysis of PDEs (math.AP)
In this paper, we study the number of critical points of the Kirchhoff-Routh function \begin{equation*} \mathcal{KR}_D(x,y)=\Lambda_1^2\mathcal{R}_D(x)+\Lambda_2^2\mathcal{R}_D(y)-2\Lambda_1\Lambda_2G_D(x,y), \end{equation*} where $D$ is a bounded domain in $\mathbb{R}^2$, $x,y\in D$, $\Lambda_1,\Lambda_2>0$, $\mathcal{R}_D$ is the Robin function, and $G_D$ is the Green function of the operator $-\Delta$ with $0$ Dirichlet boundary condition on $D$. This function arises from concentration phenomena in nonlinear elliptic problems and from the de-singularization problem for the steady Euler equation. For domains with a small hole, we establish not only the exact number and the location of the critical points of $\mathcal{KR}_D$, but also their nondegeneracy. We show that the location of the hole plays a crucial role. Finally in the context of elliptic problems, we establish the existence of multiple two-peak solutions.
- [181] arXiv:2512.23174 [pdf, html, other]
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Title: q-Opers and Bethe Ansatz for Open Spin Chains IComments: 26 pages, 1 figureSubjects: Algebraic Geometry (math.AG); High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Representation Theory (math.RT)
In in a nutshell, the classical geometric $q$-Langlands duality can be viewed as a correspondence between the space of $(G,q)$-opers and the space of solutions of $^L\mathfrak{g}$ XXZ Bethe Ansatz equations. The latter describe spectra of closed spin chains with twisted periodic boundary conditions and, upon the duality, the twist elements are identified with the $q$-oper connections on a projective line in a certain gauge. In this work, we initiate the geometric study of Bethe Ansatz equations for spin chains with open boundary conditions. We introduce the space of $q$-opers whose defining sections are invariant under reflection through the unit circle in a selected gauge. The space of such reflection-invariant $q$-opers in the presence of certain nondegeneracy conditions is thereby described by the corresponding Bethe Ansatz problem. We compare our findings with the existing results in integrable systems and representation theory. This paper discusses the type-A construction leaving the general case for the upcoming work.
- [182] arXiv:2512.23178 [pdf, html, other]
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Title: Clipped Gradient Methods for Nonsmooth Convex Optimization under Heavy-Tailed Noise: A Refined AnalysisComments: Part of this work is in submissionSubjects: Optimization and Control (math.OC); Machine Learning (cs.LG); Machine Learning (stat.ML)
Optimization under heavy-tailed noise has become popular recently, since it better fits many modern machine learning tasks, as captured by empirical observations. Concretely, instead of a finite second moment on gradient noise, a bounded ${\frak p}$-th moment where ${\frak p}\in(1,2]$ has been recognized to be more realistic (say being upper bounded by $\sigma_{\frak l}^{\frak p}$ for some $\sigma_{\frak l}\ge0$). A simple yet effective operation, gradient clipping, is known to handle this new challenge successfully. Specifically, Clipped Stochastic Gradient Descent (Clipped SGD) guarantees a high-probability rate ${\cal O}(\sigma_{\frak l}\ln(1/\delta)T^{1/{\frak p}-1})$ (resp. ${\cal O}(\sigma_{\frak l}^2\ln^2(1/\delta)T^{2/{\frak p}-2})$) for nonsmooth convex (resp. strongly convex) problems, where $\delta\in(0,1]$ is the failure probability and $T\in\mathbb{N}$ is the time horizon. In this work, we provide a refined analysis for Clipped SGD and offer two faster rates, ${\cal O}(\sigma_{\frak l}d_{\rm eff}^{-1/2{\frak p}}\ln^{1-1/{\frak p}}(1/\delta)T^{1/{\frak p}-1})$ and ${\cal O}(\sigma_{\frak l}^2d_{\rm eff}^{-1/{\frak p}}\ln^{2-2/{\frak p}}(1/\delta)T^{2/{\frak p}-2})$, than the aforementioned best results, where $d_{\rm eff}\ge1$ is a quantity we call the $\textit{generalized effective dimension}$. Our analysis improves upon the existing approach on two sides: better utilization of Freedman's inequality and finer bounds for clipping error under heavy-tailed noise. In addition, we extend the refined analysis to convergence in expectation and obtain new rates that break the known lower bounds. Lastly, to complement the study, we establish new lower bounds for both high-probability and in-expectation convergence. Notably, the in-expectation lower bounds match our new upper bounds, indicating the optimality of our refined analysis for convergence in expectation.
- [183] arXiv:2512.23179 [pdf, html, other]
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Title: An example of a non-log-concave distribution where the difference has a log-concave densityComments: 2 pagesJournal-ref: Theory of Probability and Its Applications; Vol. 69, No. 3 (2024), pp.503-504Subjects: Probability (math.PR)
By the Prékopa-Leindler inequality, the difference $X-X'$ has a log-concave density provided that $X$ has a log-concave density and $X, X'$ are independent and identically distributed. We prove that the opposite direction does not always hold true by giving an explicit example.
- [184] arXiv:2512.23181 [pdf, html, other]
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Title: An index formula for hemispheres of a $C^2$-regular convex closed surface in Euclidean $3$-spaceSubjects: Differential Geometry (math.DG)
Carathéodory's conjecture has long been regarded as one of the central problems in the classical theory of convex surfaces. In this paper, we establish an index formula for hemispheres of convex closed surfaces under $C^2$-regularity. The proof is based on studying a vertical section of the null hypersurfaces in Lorentz--Minkowski $4$-space associated with the originally given convex surface. As a consequence, the conjecture is affirmatively solved in the $C^2$-case.
- [185] arXiv:2512.23182 [pdf, html, other]
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Title: A Two-Stage Finite Element Approach for High-precision Guaranteed Lower Eigenvalue BoundsComments: 45 pages, 9 figuresSubjects: Numerical Analysis (math.NA); Spectral Theory (math.SP)
Obtaining high-precision guaranteed lower eigenvalue bounds remains difficult, even though the standard high-order conforming finite element (FEM) easily yields extremely sharp upper bounds. Recently developed rigorous approaches using such as Crouzeix--Raviart or linear conforming elements do not extend well to high-order FEM. Some non-standard FEM approaches can provide sharp eigenvalue bounds but are technically involved. This persistent gap between accurate upper bounds and equally sharp rigorous lower bounds via standard high-order conforming FEMs makes the problem technically demanding and highly competitive. In this paper, we propose a new two-stage rigorous algorithm that closes this gap by employing high-order FEM on graded meshes and producing rigorous lower eigenvalue bounds as sharp as the corresponding high-order upper bounds, as demonstrated in our numerical examples. Numerical experiments for the Laplacian and Steklov eigenvalue problems on square and dumbbell domains show the accuracy and efficiency of the method, particularly on graded or highly nonuniform meshes. These results confirm that the proposed approach provides a practical and competitive solution to the long-standing difficulty of obtaining sharp, reliable lower eigenvalue bounds.
- [186] arXiv:2512.23188 [pdf, other]
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Title: Incorporating Authority Perception, Economic Status, and Behavioral Response in Infectious Disease ControlSubjects: Optimization and Control (math.OC)
We introduce a multi-population mean field game framework to examine how economic status and authority perception shape vaccination and social distancing decisions under different epidemic control policies. We carried out a survey to inform our model and stratify the population into six groups based on income and perception of authority, capturing behavioral heterogeneity. Individuals adjust their socialization and vaccination levels to optimize objectives such as minimizing treatment costs, complying with social-distancing guidelines if they are authority-followers, or reducing losses from decreased social interactions if they are authority-indifferents, alongside economic costs. Public health authorities influence behavior through social-distancing guidelines and vaccination costs. We characterize the Nash equilibrium via a forward-backward differential equation system, provide its mathematical analysis, and develop a numerical algorithm to solve it. Our findings reveal a trade-off between social-distancing and vaccination decisions. Under stricter guidelines that target both susceptible and infected individuals, followers reduce both socialization and vaccination levels, while indifferents increase socialization due to followers' preventative measures. Adaptive guidelines targeting infected individuals effectively reduce infections and narrow the gap between low- and high-income groups, even when susceptible individuals socialize more and vaccinate less. Lower vaccination costs incentivize vaccination among low-income groups, but their impact on disease spread is smaller than when they are coupled with social-distancing guidelines. Trust-building emerges as a critical factor in epidemic mitigation, underscoring the importance of data-informed, game-theoretical models that aim to understand complex human responses to mitigation policies.
- [187] arXiv:2512.23194 [pdf, html, other]
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Title: A New Family of Binary Sequences via Elliptic Function Fields over Finite Fields of Odd CharacteristicsComments: arXiv admin note: substantial text overlap with arXiv:2407.18570 by other authorsSubjects: Information Theory (cs.IT)
Motivated by the constructions of binary sequences by utilizing the cyclic elliptic function fields over the finite field $\mathbb{F}_{2^{n}}$ by Jin \textit{et al.} in [IEEE Trans. Inf. Theory 71(8), 2025], we extend the construction to the cyclic elliptic function fields with odd characteristic by using the quadratic residue map $\eta$ instead of the trace map used therein. For any cyclic elliptic function field with $q+1+t$ rational points and any positive integer $d$ with $\gcd(d, q+1+t)=1$, we construct a new family of binary sequences of length $q+1+t$, size $q^{d-1}-1$, balance upper bounded by $(d+1)\cdot\lfloor2\sqrt{q}\rfloor+|t|+d,$ the correlation upper bounded by $(2d+1)\cdot\lfloor2\sqrt{q}\rfloor+|t|+2d$ and the linear complexity lower bounded by $\frac{q+1+2t-d-(d+1)\cdot\lfloor2\sqrt{q}\rfloor}{d+d\cdot\lfloor2\sqrt{q}\rfloor}$ where $\lfloor x\rfloor$ stands for the integer part of $x\in\mathbb{R}$.
- [188] arXiv:2512.23195 [pdf, html, other]
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Title: Uniqueness of Replica-symmetric Saddle Point for Ising PerceptronComments: 21 pages. Even though the proof is entirely analytic, we independently perform numerical checks (see this https URL)Subjects: Probability (math.PR); Statistical Mechanics (cond-mat.stat-mech)
We study the replica-symmetric saddle point equations for the Ising perceptron with Gaussian disorder and margin $\kappa\ge 0$. We prove that for each $\kappa\ge 0$ there is a critical capacity $\alpha_c(\kappa)=\frac{2}{\pi\,\mathbb E[(\kappa-Z)_+^2]}$, where $Z$ is a standard normal and $(x)_+=\max\{x,0\}$, such that the saddle point equation has a unique solution for $\alpha\in(0,\alpha_c(\kappa))$ and has no solution when $\alpha\ge \alpha_c(\kappa)$. When $\alpha\uparrow \alpha_c(\kappa)$ and $\kappa>0$, the replica-symmetric free energy at this solution diverges to $-\infty$. In the zero-margin case $\kappa=0$, Ding and Sun obtained a conditional uniqueness result, with one step verified numerically. Our argument gives a fully analytic proof without computer assistance. We used GPT-5 to help develop intermediate proof steps and to perform sanity-check computations.
- [189] arXiv:2512.23197 [pdf, html, other]
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Title: Global strong solutions for non-isothermal compressible nematic liquid crystal flows under a scaling-invariant smallness conditionComments: 20 pagesSubjects: Analysis of PDEs (math.AP)
We study the three-dimensional Cauchy problem for a non-isothermal compressible nematic liquid crystal system with far-field vacuum. By deriving refined energy estimates and exploiting the coupled structure of the equations, we establish the global existence and uniqueness of strong solutions, provided that the following scaling-invariant quantity is sufficiently small:
$$
\big(1+\bar{\rho}+\tfrac{1}{\bar{\rho}}\big)
\big[\|\rho_{0}\|_{L^{3}}+(\bar{\rho}^{2}+\bar{\rho})\big(\|\sqrt{\rho_{0}}u_{0}\|_{L^{2}}^{2}+\|\nabla d_{0}\|_{L^{2}}^{2}\big)\big]
\big[\|\nabla u_{0}\|_{L^{2}}^{2}+(\bar{\rho}+1)\|\sqrt{\rho_{0}}\theta_{0}\|_{L^{2}}^{2}
+\|\nabla^{2} d_{0}\|_{L^{2}}^{2}+\|\nabla d_{0}\|_{L^{4}}^{4}\big].
$$
In particular, our result identifies a new scaling-invariant quantity and does not impose additional restrictions on the viscosity coefficients, which improves previous work (Commun. Math. Sci. 21 (2023), 1455--1486). - [190] arXiv:2512.23198 [pdf, other]
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Title: Asymptotics aspects of Teichmüller TQFT for generalized FAMED semi-geometric triangulationsComments: 47 pages, 4 figuresSubjects: Geometric Topology (math.GT); Mathematical Physics (math-ph)
We introduce a generalized FAMED property for ideal triangulations of hyperbolic knot complements in $\mathbb{S}^3$. Given a hyperbolic knot $K$ in $\mathbb{S}^3$ and a semi-geometric triangulation $X$ of $\mathbb{S}^3 \setminus K$ that is generalized FAMED with respect to the longitude. We prove that in the semi-classical limit $\hbar \to 0^+$, for any angle structure $\alpha$, the partition function $\mathscr{Z}_\hbar(X,\alpha)$ in Teichmüller TQFT decays exponentially with decrease rate the volume of $\mathbb{S}^3 \setminus K$ equipped with a hyperbolic cone structure determined by $\alpha$, and that the 1-loop invariant of Dimofte-Garoufalidis emerges as the 1-loop term. With additional combinatorial conditions on the triangulations, we prove the existence of the Jones function and show that its decay rate is governed by the Neumann-Zagier potential function. In particular, the Andersen-Kashaev volume conjecture holds for every hyperbolic knot whose complement admits such kinds of triangulations.
- [191] arXiv:2512.23199 [pdf, html, other]
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Title: On extremal graphs with respect to the ABS indexSubjects: Combinatorics (math.CO)
Recently, Ali et al. posed several open problems concerning extremal graphs with respect to the ABS index. These problems involve characterizing graphs that attain the maximum ABS index within specific graph classes, including: connected graphs with n vertices and p cut-vertices; (ii) connected graphs of order n with vertex k-partiteness; and (iii) connected bipartite graphs of order n with a fixed vertex connectivity \kappa. In this paper, we provide complete solutions to all of these problems.
- [192] arXiv:2512.23201 [pdf, html, other]
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Title: Local well-posedness of the Schrödinger flow into $\mathbb{S}^2$ with natural boundary conditionsComments: 52 pages, comments welcome!Subjects: Analysis of PDEs (math.AP); Differential Geometry (math.DG)
In this paper, we develop a new approximation scheme to resolve the local well-posedness problem for the Landau-Lifshitz equation (i.e., the Schrödinger flow into the standard unit 2-sphere $\mathbb{S}^2\subset \mathbb{R}^3$) with natural boundary conditions.
- [193] arXiv:2512.23203 [pdf, html, other]
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Title: Output feedback stabilization of linear port-Hamiltonian descriptor systemsComments: 20 pagesSubjects: Optimization and Control (math.OC)
This paper presents a structure-preserving method for the stabilization of linear port-Hamiltonian (pH) descriptor systems via output feedback. The stabilization problem is NP-hard for general descriptor systems. Existing approaches often rely on explicit knowledge of the structure-defining matrix $Q$, which is difficult to determine in practice. When $Q$ is unknown, we derive necessary and sufficient conditions under which proportional output feedback ensures that the closed-loop system is regular, impulse-free, asymptotically stable, and retains the port-Hamiltonian structure. These conditions also allow any positive definite matrix to serve as the feedback matrix. The framework is further extended to proportional and derivative output feedback, enabling the assignment of a desired dynamical order. The proposed method thus generalizes existing stabilization results from the special case $Q = I$ to systems with an unknown $Q$, offering a systematic method to structure-preserving stabilization of pH descriptor systems.
- [194] arXiv:2512.23204 [pdf, html, other]
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Title: Counting rational points near manifolds: a refined estimate, a conjecture and a variantComments: 32 pages. Comments are welcomeSubjects: Number Theory (math.NT)
Refining an argument of the second author, we improve the known bounds for the number of rational points near a submanifold of $\mathbb{R}^d$ of intermediate dimension under a natural curvature condition. Furthermore, in the codimension $2$ case we formulate a conjecture concerning this count. The conjecture is motivated in part by interpreting certain codimension $2$ submanifolds of $\mathbb{R}^{2m+2}$ as complex hypersurfaces in $\mathbb{C}^{m+1}$ and using the complex structure to provide a natural reformulation of the curvature condition. Finally, we provide further evidence for the conjecture by proving a natural variant for $n \geq 2$ in which rationals are replaced with Gaussian rationals.
- [195] arXiv:2512.23207 [pdf, html, other]
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Title: Infinitely many positive solutions to nonlinear scalar field equation with nonsmooth nonlinearitySubjects: Analysis of PDEs (math.AP)
This paper investigates the existence of infinitely many positive solutions for the logarithmic scalar field equation
\begin{equation}
\tag{$P$} \label{equ1}
-\Delta u+ V(x) u= u\log u^2, \quad u\in H^1(\mathbb{R}^N),
\end{equation}
and its counterpart with prescribed $L^2$-norms
\begin{align}\label{equ2} \tag{$P_N$}
& -\Delta u+ V(x) u +\lambda u= u\log u^2, \quad u\in H^1(\mathbb{R}^N),
&\int_{\mathbb{R}^N} u^2 ~\mathrm{d}x=a^2>0,
\end{align}
which come from physically relevant situations. Here, $N\geq 2$, $V:\mathbb{R}^N\to \mathbb{R}$ is a non-symmetric and non-periodic potential satisfying certain decay conditions, $ a $ is prescribed constant, and $\lambda$ arises as an unknown Lagrange multipliers. For problem \eqref{equ1}, using purely variational methods, we establish the existence of multi-bump positive solutions with either finitely or infinitely many bumps. For normalized problem \eqref{equ2}, we prove the existence of normalized multi-bump positive solutions with a finite number of bumps.
The main difficulty comes from the nonsmooth nature of logarithmic nonlinearity, which introduces some challenges to the variational framework.
In particular, the corresponding energy functional is not of class $C^1$ on $H^1(\mathbb{R}^N)$, which prevents the direct application of standard critical point theory for $C^1$ functional or any reduction methods for $C^{1+\sigma}$ nonlinearity. The main ingredients in this paper are nonsmooth critical point theory, localized variational methods and a max-min argument.
To the best of our knowledge, this paper appears to be the first successful application of the localized variational method to nonsmooth functionals. - [196] arXiv:2512.23209 [pdf, html, other]
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Title: Extremal $ABS$ Spectral Radius in Bicyclic and Bipartite Unicyclic GraphsSubjects: Combinatorics (math.CO)
The ABS spectral radius of a graph G is defined as the largest eigenvalue of its $ABS$ matrix. Motivated by recent studies on this parameter, in this paper, we determine the bipartite unicyclic graphs that attain the largest $ABS$ spectral radius. Furthermore, we characterize the bicyclic graphs that attain the largest and the second largest $ABS$ spectral radii.
- [197] arXiv:2512.23218 [pdf, html, other]
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Title: Aubert duals of strongly positive representations for metaplectic groupsSubjects: Number Theory (math.NT); Representation Theory (math.RT)
We determine the Aubert duals of strongly positive representations of the metaplectic group \(\widetilde{Sp}(n)\) over a non-Archimedean local field $F$ of characteristic different from two. Using the classification of Matić and an explicit analysis of Jacquet modules, we describe these duals in terms of precise inducing data. Our results extend known descriptions for classical groups to the metaplectic groups case and clarify the role of Aubert duality for non-linear covering groups, providing a foundation for future applications to the study of unitary representations for those cases. Furthermore, We are able to show that the same method applies to odd general spin groups $GSpin(2n+1)$, yielding an explicit description of Aubert duals in that setting as well.
- [198] arXiv:2512.23223 [pdf, html, other]
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Title: The five-vertex model as a discrete log-gasComments: 30 pages, 3 figuresSubjects: Mathematical Physics (math-ph); Statistical Mechanics (cond-mat.stat-mech); Probability (math.PR)
We consider the five-vertex model on a rectangular domain of the square lattice, with the so-called `scalar-product' boundary conditions. We address the evaluation of the free-energy density of the model in the scaling limit, that is when the number of sites is sent to infinity and the mesh of the lattice to zero, while keeping the size of the domain constant. To this aim, we reformulate the partition function of the model in terms of a discrete log-gas, and study its behaviour in the thermodynamic limit. We reproduce previous results, obtained by using a differential equation approach. Moreover, we provide the explicit form of the resolvent in all possible regimes. This work is preliminary to further studies of limit shape phenomena in the model.
- [199] arXiv:2512.23224 [pdf, html, other]
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Title: Quantum $K$-theoretic Whitney relations for type $C$ flag manifoldsComments: 27 pagesSubjects: Quantum Algebra (math.QA); Algebraic Geometry (math.AG); Combinatorics (math.CO); K-Theory and Homology (math.KT); Representation Theory (math.RT)
We study relations of $\lambda_{y}$-classes associated to tautological bundles over the flag manifold of type $C$ in the quantum $K$-ring. These relations are called the quantum $K$-theoretic Whitney relations. The strategy of the proof of the quantum $K$-theoretic Whitney relations is based on the method of semi-infinite flag manifolds and the Borel-type presentation. In addition, we observe that the quantum $K$-theoretic Whitney relations give a complete set of the defining relations of the quantum $K$-ring. This gives a presentation of the quantum $K$-ring of the flag manifold of type $C$, called the Whitney-type presentation, as a quotient of a polynomial ring, different from the Borel-type presentation.
- [200] arXiv:2512.23225 [pdf, html, other]
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Title: On topology estimation of submanifolds in Riemannian manifolds by random pointsSubjects: Differential Geometry (math.DG)
We show that, by sampling a sufficiently large number of random points in a neighborhood of a compact submanifold M of a Riemannian manifold N, one can recover the topology of M with high confidence. This holds under the assumptions on the curvatures of M and N.
- [201] arXiv:2512.23229 [pdf, html, other]
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Title: A note on the possibility of a motion without crossing a given subsetSubjects: Differential Geometry (math.DG)
We study the fractal dimension of a given subset X of R^{n} such that a motion is possible without crossing X.
- [202] arXiv:2512.23233 [pdf, html, other]
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Title: Kurosh Subgroup Theorem in Free Pro-p ProductsSubjects: Group Theory (math.GR)
We provide a sufficient condition under which a closed subgroup of a restricted free pro-p product is itself a free pro-p product.
- [203] arXiv:2512.23238 [pdf, html, other]
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Title: Frenet Immersed Finite Element Spaces on Triangular MeshesSubjects: Numerical Analysis (math.NA)
In this paper, we develop geometry-conforming immersed finite element (GC-IFE) spaces on triangular meshes for elliptic interface problems. These IFE spaces are constructed via a Frenet-Serret mapping that transforms the interface curve into a straight line, allowing the interface jump conditions to be imposed exactly. Extending the framework of [7,8] from rectangular to triangular meshes, we introduce three procedures for constructing high-degree Frenet-IFE spaces: an initial construction based on monomial bases, a generalized construction using orthogonal polynomials, and reconstruction methods aimed at improving the conditioning of the associated mass matrix. The optimal approximation capability of the proposed IFE spaces is demonstrated through numerical examples. We further incorporate these spaces into interior penalty discontinuous Galerkin methods for elliptic interface problems and observe optimal convergence rates in the $H^1$ and $L^2$ norms.
- [204] arXiv:2512.23242 [pdf, html, other]
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Title: Sum Rate optimization for RIS-Aided RSMA system with Movable AntennaSubjects: Information Theory (cs.IT)
Rate-Splitting Multiple Access (RSMA) is regarded as a key enabling technique for sixth-generation (6G) wireless systems for its powerful interference management substantially enhancing link throughput. Reconfigurable Intelligent Surface (RIS) can effectively shape the wireless propagation to match the environment and improve communication performance. However, in conventional RSMA-RIS architectures, the antenna elements are fixed, which underutilizes spatial degrees of freedom and hence constrains system performance. To address this limitation, we propose a movable-antenna (MA) assisted RSMA-RIS framework and formulate a sum-rate maximization problem that jointly optimizes the transmit beamforming matrix, the RIS reflection matrix, the common-rate partition, and the MA positions. The original problem is equivalently transformed by employing the fractional programming (FP) method, and a closed-form solution for the common rate splitting is derived. Leveraging the Karush-Kuhn-Tucker (KKT) conditions, we obtain iterative updates for the Lagrange multipliers together with a closed-form expression for the beamforming matrix. We then develop an update rule for the RIS reflection matrix via the dual problem, and finally determine the optimal antenna locations using a gradient-ascent procedure. Numerical results indicate that, even in the presence of RIS assistance, incorporating MAs yields additional performance improvements for RSMA. Moreover, relative to space-division multiple access (SDMA), the assistance of MA yields a greater performance gain for RSMA.
- [205] arXiv:2512.23249 [pdf, html, other]
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Title: A horofunction counterpart to Teichmüller distanceComments: 24 pagesSubjects: Geometric Topology (math.GT)
We generalize the horofunction compactification to maps that are not distance functions. As an application we define a horofunction counterpart to the Teichmüller distance, and discuss its properties.
- [206] arXiv:2512.23259 [pdf, other]
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Title: Galilei and Huygens: Music and scienceAthanase Papadopoulos (IRMA)Journal-ref: In: Vincenzo Galilei, the Renaissance between music and science, ed. Natacha Fabbri and Ferdinando Abbri, Leo S. Olschki editore, Coll. Biblioteca di Galil{\ae}ana, 12, Firenze, 2025, 9788822269904Subjects: History and Overview (math.HO)
Vincenzo Galilei and Constantijn Huygens were both humanists and eminent musicians, the former from the late Renaissance and the latter from the early Modern era. Their respective sons, Galileo and Christiaan, were scientists whose importance cannot be overestimated. My aim in this chapter is to set the scene for a parallel presentation of the legacy of the Galilei on the one hand, and the Huygens on the other. This will give us an opportunity to talk about mathematics, music and acoustics, but also about science in general, at this time of birth of the Modern era.
- [207] arXiv:2512.23263 [pdf, html, other]
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Title: Global stability and asymptotic behavior for incompressible ideal MHD equations with velocity damping termSubjects: Analysis of PDEs (math.AP)
In this article, we study the stability and large time behavior for an multi-dimensional incompressible magnetohydrodynamical system with a velocity damping term, for small perturbations near a steady-state of magnetic field fulfilling the Diophantine condition. Our results mathematically characterize the background magnetic field exerts the stabilizing effect, and bridge the gap left by previous work with respect to the asymptotic behavior in time. Our proof approach mainly relies on the Fourier analysis and energy estimates. In addition, we provide a versatile analytical framework applicable to many other partially dissipative fluid models.
- [208] arXiv:2512.23266 [pdf, other]
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Title: Average-weight percolation on the complete graphElie Aïdékon, Yueyun Hu (LAGA)Subjects: Probability (math.PR)
Attach to each edge of the complete graph on $n$ vertices, i.i.d. exponential random variables with mean $n$. Aldous [1] proved that the longest path with average weight below $p$ undergoes a phase transition at $p=\frac{1}{e}$: it is $o(n)$ when $p<\frac{1}{e}$ and of order $n$ if $p>\frac1e$. Later, Ding [4] revealed a finer phase transition around $\frac{1}{e}$: there exist $c'>c>0$ such that the length of the longest path is of order $\ln^3 n$ if $ p \le \frac{1}{e}+\frac{c}{\ln^2 n}$ and is polynomial if $p\ge \frac{1}{e}+\frac{c'}{\ln^2 n}$. We identify the location of this phase transition and obtain sharp asymptotics of the length near criticality. The proof uses an exploration mechanism mimicking a branching random walk with selection introduced by Brunet and Derrida [3].
- [209] arXiv:2512.23268 [pdf, html, other]
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Title: Dynamics of the Morse vector fieldSubjects: Differential Geometry (math.DG)
The (negative) gradient vector fields of Morse functions on a compact manifold provide an important example in dynamical system. In this note we prove two important properties of this kind of vector field: Connectedness of critical points through orbits and exponential shrinkage of the flow on stable submanifolds. We also find applications in showing some vanishing results of maps or curvature operators.
- [210] arXiv:2512.23271 [pdf, html, other]
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Title: On blow-up rate for the Hénon parabolic equation with Sobolev supercritical nonlinearityComments: 32 pagesSubjects: Analysis of PDEs (math.AP)
We discuss the Hénon parabolic equation $\partial_t u = \Delta u + |x|^\sigma u^p$ in a finite ball in $\mathbb{R}^N$ under the Dirichlet boundary condition, where $N\ge1$, $p>1$, and $\sigma>0$. We assume that the exponent $p$ is supercritical in the Sobolev sense. Since the spatial potential term $|x|^\sigma$ vanishes at the origin, solutions seem less likely to blow up at the origin. We construct a solution that blows up at the origin and also carry out an analysis of blow-up rate of solutions. In particular, if $p$ is less than the Joseph--Lundgren exponent, all blow-ups are shown to be of Type I. The lower bound corresponding to Type I rate is also shown for some particular blow-up solutions. As by products, we present a basic result on classification to threshold solutions for every $p>1+\sigma/N$.
- [211] arXiv:2512.23276 [pdf, html, other]
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Title: Chamber zeta function and closed galleries in the standard non-uniform complex from $\operatorname{PGL}_3$Comments: 22 pages, 6 figuresSubjects: Number Theory (math.NT); Combinatorics (math.CO); Dynamical Systems (math.DS)
We introduce the \emph{chamber zeta function} for a complex of groups, defined via an Euler product over primitive tailless chamber galleries, extending the Ihara--Bass framework from weighted graphs to higher-rank settings. Let $\mathcal{B}$ be the Bruhat--Tits building of $\mathrm{PGL}_{3}(F)$ for a non-archimedean local field $F$ with residue field $\mathbb{F}_{q}$. For the standard arithmetic quotient $\Gamma\backslash\mathcal{B}$ with $\Gamma=\mathrm{PGL}_{3}(\mathbb{F}_{q}[t])$, we prove an Ihara--Bass type \emph{determinant formula} expressing the chamber zeta function as the reciprocal of a characteristic polynomial of a naturally defined chamber transfer operator. In particular, the chamber zeta function is \emph{rational} in its complex parameter. As an application of the determinant formula, we obtain explicit counting results for closed gallery classes arising from tailless galleries in $\mathcal{B}$, including exact identities and spectral asymptotics governed by the chamber operator.
- [212] arXiv:2512.23281 [pdf, other]
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Title: Spectral properties of magnetic fields on sub-Riemannian contact manifoldsRiccardo Bonalli (L2S), Dario Prandi (L2S, CNRS)Subjects: Differential Geometry (math.DG)
Motivated by some recent studies of the magnetic Laplacian on Riemannian manifolds, we focus on the first eigenvalue of the magnetic horizontal Laplacian on contact manifolds. We characterize conditions for positive spectral shift, and provide some sharp upper bounds. In the Riemannian setting, a genus 1 assumption is known to force the underlying metric to be flat when equality holds in the sharp upper bounds. Interestingly, we show that the equivalent topological condition in the three--dimensional contact setting consists of having first Betti number equal to 2. In this case, equality in our upper bounds implies that the structure is that of a Heisenberg left--invariant nilmanifold. We conclude by showing that, in some specific three--dimensional contact settings, the knowledge of the first eigenvalue of the magnetic Laplacian uniquely determines the manifold Chern class, fully determining the topology of the underlying manifold.
- [213] arXiv:2512.23285 [pdf, html, other]
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Title: Schur--Weyl duality for diagonalizing a Markov chain on the hypercubeComments: Please feel free to make comments! This is a companion paper split off from arXiv:2511.01245, which originally contained the content of this workSubjects: Representation Theory (math.RT); Combinatorics (math.CO); Probability (math.PR)
We show how the tools of modern algebraic combinatorics -- representation theory, Murphy elements, and particularly Schur--Weyl duality -- can be used to give an explicit orthonormal basis of eigenfunctions for a "curiously slowly mixing Markov chain" on the space of binary $n$-tuples. The basis is used to give sharp rates of convergence to stationarity.
- [214] arXiv:2512.23286 [pdf, other]
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Title: Ground States for the Nonlinear Schr{ö}dinger Equation on Open Books and Dimensional Reduction to Metric GraphsStefan Le Coz (IMT, CIMI), Boris Shakarov (IMT)Subjects: Analysis of PDEs (math.AP)
In this work, we study the dimensional reduction of stationary states in the shrinking limit for a broad class of two-dimensional domains, called open books, to their counterparts on metric graphs. An open book is a two-dimensional structure formed by rectangular domains sharing common boundaries. We first develop a functional-analytic framework suited to variational problems on open books and establish the existence of solutions as constrained action minimizers. For graph-based open books (i.e., those isomorphic to the product of a graph with an interval) we prove the existence of a sharp transition in the dimensionality of ground states. Specifically, there exists a critical transverse width: below this threshold, all ground states coincide with the ground states on the underlying graph trivially extended in the transverse direction; above it, ground states become genuinely two-dimensional.
- [215] arXiv:2512.23287 [pdf, html, other]
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Title: Interpolation of classical Lorentz spaces measuring oscillationComments: 27 pages, 0 figuresSubjects: Functional Analysis (math.FA)
We obtain an explicit characterization of the $K$-functional of a pair of weighted classical Lorentz spaces of type $S$. We develop a method for obtaining such characterization based on a relation between the desired quantity and the $K$-functional of a specific couple of spaces of type $\Lambda$, which are substantially more manageable than their companions of type $S$. The core of our techniques is a subtle manipulation with respective fundamental functions. We present several applications, in particular we nail down a formula for the $K$-functional of a Lebesgue space and a classical Lorentz space of type $S$ with a power weight, and using this formula we establish an inequality of a reverse Marchaud type.
- [216] arXiv:2512.23288 [pdf, other]
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Title: Bismut-Elworthy-Li Formulae for Forward-Backward SDEs with Jumps and ApplicationsComments: 49 pagesSubjects: Probability (math.PR)
Under nondegeneracy assumptions on the diffusion coefficients, we establish the derivative formulae of Bismut-Elworthy-Li's type for forward-backward stochastic differential equations with respect to Poisson random measure using the lent particle method created by Bouleau and Denis, which is not given before. Applying this formula, the existence and uniqueness of a solution of nonlocal quasi-linear integral partial differential equations, which are differentiable with respect to the space variable, are obtained, even if the initial datum and coefficients of this equation are not.
- [217] arXiv:2512.23293 [pdf, html, other]
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Title: Curvature criteria of A-simple singularities R,0-> R^2,0 and their parallel curvesSubjects: Differential Geometry (math.DG)
We introduce the notion of curvature parameters for singular plane curves with finite multiplicities and define the notion of curvatures for them. We then provide criteria to determine their singularity types for A-simple singularities. As an application, we investigate singularity types of their parallel curves.
- [218] arXiv:2512.23296 [pdf, other]
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Title: Evaluation of Volume Variation Partition: the Breathing Coefficient and its Application on Uniaxial Monosized Disc Packing SwellingSubjects: Metric Geometry (math.MG)
An analysis of the general concept of volume variation partition of a porous body is presented, introducing the breathing coefficient, defined as the ratio of two volume variations. Considering a total volume of a porous body, composed of solid volume and ``void'' volume, this ratio can be used to evaluate the distribution of a volume variation into both others. A full description of its physical interpretation is detailed, together with an uncertainty analysis that specifies precautions about its use. As an example of application, a case study of 2D monosized disc packing swelling is developed. The analytical model reveals the presence of minimisation points of the breathing coefficient dependent on the initial granular organisation, showing possible ways to minimise the breathing of a granular material.
- [219] arXiv:2512.23302 [pdf, other]
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Title: Chebyshev's bias without linear independenceMounir Hayani (IMB)Subjects: Number Theory (math.NT)
We confirm Chebyshev's observation that primes are strikingly more abundant in non-square residue classes modulo a fixed integer under the Generalized Riemann Hypothesis (GRH) by proving a (natural) density 1 statement for prime counting functions in residue classes where each prime is weighted by its inverse square root. In contrast to the majority of the existing literature on the subject, we do not need to restrict to logarithmic densities to measure Chebyshev's bias and we do not rely on any hypothesis related to L-function zeros that is stronger than GRH.
- [220] arXiv:2512.23303 [pdf, html, other]
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Title: Two-colorings of finite grids: variations on a theorem of Tibor GallaiSubjects: Combinatorics (math.CO); Discrete Mathematics (cs.DM)
A celebrated but non-effective theorem of Tibor Gallai states that for any finite set $A$ of $\Z^n$ and for any finite number of colors $c$ there is a minimal $m$ such that no coloring of the finite $m^n$-grid can avoid that a homothetic image of $A$ is monochromatic. We find (or confirm) $m$ for equilateral triangles, squares, and various types of rectangles. Also, we extend the problem from homothety to general similarity, or to similarity generated using some special rotations. In particular, we compute Gallai similarity numbers for lattice rectangles similar to $1\times k$ (in all orientations) for $k=2,3,4$. The solutions have been found in the framework of the Satisfiability Problem in Propositional Logic (SAT). While some questions were solved using managed brute force, for the more computationally intensive questions we used modern SAT solvers together with symmetry breaking techniques. Some other minor questions are solved for triangles and squares, and new lower bounds are found for regular hexagons on the triangular lattice and for three-dimensional cubes in $\Z^3$.
- [221] arXiv:2512.23308 [pdf, html, other]
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Title: Conformal Prediction = Bayes?Subjects: Statistics Theory (math.ST); Methodology (stat.ME)
Conformal prediction (CP) is widely presented as distribution-free predictive inference with finite-sample marginal coverage under exchangeability. We argue that CP is best understood as a rank-calibrated descendant of the Fisher-Dempster-Hill fiducial/direct-probability tradition rather than as Bayesian conditioning in disguise.
We establish four separations from coherent countably additive predictive semantics. First, canonical conformal constructions violate conditional extensionality: prediction sets can depend on the marginal design P(X) even when P(Y|X) is fixed. Second, any finitely additive sequential extension preserving rank calibration is nonconglomerable, implying countable Dutch-book vulnerabilities. Third, rank-calibrated updates cannot be realized as regular conditionals of any countably additive exchangeable law on Y^infty. Fourth, formalizing both paradigms as families of one-step predictive kernels, conformal and Bayesian kernels coincide only on a Baire-meagre subset of the space of predictive laws.
We further show that rank- and proxy-based reductions are generically Blackwell-deficient relative to full-data experiments, yielding positive Le Cam deficiency for suitable losses. Extending the analysis to prediction-powered inference (PPI) yields an analogous message: bias-corrected, proxy-rectified estimators can be valid as confidence devices while failing to define transportable belief states across stages, shifts, or adaptive selection. Together, the results sharpen a general limitation of wrappers: finite-sample calibration guarantees do not by themselves supply composable semantics for sequential updating or downstream decision-making. - [222] arXiv:2512.23309 [pdf, html, other]
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Title: Structure preservation and emergent dissipation in stochastic wave equations with transport noiseComments: 27 pagesSubjects: Probability (math.PR)
We study nonlinear wave equations perturbed by transport noise acting either on the displacement or on the velocity. Such noise models random advection and, under suitable scaling of space covariance, may generate an effective dissipative term. We establish well-posedness in both cases and analyse the associated scaling limits. When the noise acts on the displacement, the system preserves its original structure and converges to the deterministic nonlinear wave equation, whereas if it acts on the velocity, the rescaled dynamics produce an additional Laplacian damping term, leading to a stochastic derivation of a Westervelt-type acoustic model.
- [223] arXiv:2512.23316 [pdf, html, other]
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Title: Information Inequalities for Five Random VariablesSubjects: Information Theory (cs.IT)
The entropic region is formed by the collection of the Shannon entropies of all subvectors of finitely many jointly distributed discrete random variables. For four or more variables the structure of the entropic region is mostly unknown. We utilize a variant of the Maximum Entropy Method to delimit the five-variable entropy region. This method adds copies of some of the random variables in generations. A significant reduction in computational complexity, achieved through theoretical considerations and by harnessing the inherent symmetries, allowed us to calculate all five-variable non-Shannon inequalities provided by the first nine generations. Based on the results, we define two infinite collections of such inequalities, and prove them to be entropy inequalities. We investigate downward closed subsets of non-negative lattice points that parameterize these collections, based on which we develop an algorithm to enumerate all extremal inequalities. The discovered set of entropy inequalities is conjectured to characterize the applied method completely.
- [224] arXiv:2512.23317 [pdf, html, other]
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Title: Essential Convergence Rates of Continuous-Time Models for Optimization MethodsSubjects: Optimization and Control (math.OC)
Designing and analyzing optimization methods via continuous-time models expressed as ordinary differential equations (ODEs) is a promising approach for its intuitiveness and simplicity. A key concern, however, is that the convergence rates of such models can be arbitrarily modified by time rescaling, rendering the task of seeking ODEs with ``fast'' convergence meaningless. To eliminate this ambiguity of the rates, we introduce the notion of the essential convergence rate. We justify this notion by proving that, under appropriate assumptions on discretization, no method obtained by discretizing an ODE can achieve a faster rate than its essential convergence rate.
- [225] arXiv:2512.23321 [pdf, html, other]
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Title: Normalized solutions of nonlinear magnetic Schrödinger equations on metric graphsComments: 36 pagesSubjects: Analysis of PDEs (math.AP)
In this paper we first establish the theory of a magnetic Sobolev space $H^1_A(\mathcal{G},\mathbb{C})$ on metric graphs $\mathcal{G}$ and we prove the self-adjointness of its corresponding magnetic Schrödinger operator. Then, in this setting, we investigate the existence and multiplicity of normalized solutions to nonlinear magnetic Schrödinger equations on compact metric graphs and on noncompact metric graphs with localized nonlinearities or nonlinearities acting on whole metric graphs, covering the mass-subcritical, mass-critical, and mass-supercritical cases.
- [226] arXiv:2512.23331 [pdf, html, other]
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Title: Solutions of the singular Yamabe problem near singular boundariesComments: 25 pagesSubjects: Analysis of PDEs (math.AP)
In this paper, we investigate the asymptotic behaviors of solutions to the singular Yamabe problem with negative constant scalar curvature near singular boundaries and derive optimal estimates, where the background metrics are not assumed to be conformally flat. Specifically, we demonstrate that for a wide class of Lipschitz domains with asymptotic conical structure, the local positive solutions are well approximated by the positive solutions in the tangent cones at singular boundary points. This extends the results of [10, 12, 26].
- [227] arXiv:2512.23334 [pdf, html, other]
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Title: On a Class of Partitions with Lower Expected Star Discrepancy and Its Upper Bound than Jittered SamplingSubjects: Probability (math.PR)
We investigate the expected star discrepancy under a newly designed class of convex equivolume partition models. The main contributions are two-fold. First, we establish a strong partition principle for the star discrepancy, showing that our newly designed partitions yield stratified sampling point sets with lower expected star discrepancy than both classical jittered sampling and simple random sampling. Specifically, we prove that $\mathbb{E}(D^{*}_{N}(Z))\leq\mathbb{E}(D^{*}_{N}(Y))<\mathbb{E}(D^{*}_{N}(X))$, where $X$, $Y$, and $Z$ represent simple random sampling, jittered sampling, and our new partition sampling, respectively. Second, we derive explicit upper bounds for the expected star discrepancy under our partition models, which improve upon existing bounds for jittered sampling. Our results resolve Open Question 2 posed in Kiderlen and Pausinger (2021) regarding the strong partition principle for star discrepancy.
- [228] arXiv:2512.23338 [pdf, html, other]
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Title: Quantum Dilogarithms and New Integrable Lattice Models in Three DimensionsComments: 18 pagesSubjects: Mathematical Physics (math-ph); Statistical Mechanics (cond-mat.stat-mech); High Energy Physics - Theory (hep-th)
In this paper we introduce a new class of integrable 3D lattice models, possessing continuous families of commuting layer-to-layer transfer matrices. Algebraically, this commutativity is based on a very special construction of local Boltzmann weights in terms of quantum dilogarithms satisfying the inversion and pentagon identities. We give three examples of such quantum dilogarithms, leading to integrable 3D lattice models. The partition function per site in these models can be exactly calculated in the limit of an infinite lattice by using the functional relations, symmetry and factorization properties of the transfer matrix. The results of such calculations for 3D models associated with the Faddeev modular quantum dilogarithm are briefly presented.
- [229] arXiv:2512.23339 [pdf, html, other]
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Title: Small-time global controllability of a class of bilinear fourth-order parabolic equationsComments: 29 pages, Comments are welcome!Subjects: Optimization and Control (math.OC); Analysis of PDEs (math.AP)
In this work, we investigate the small-time global controllability properties of a class of fourth-order nonlinear parabolic equations driven by a bilinear control posed on the one-dimensional torus. The controls depend only on time and act through a prescribed family of spatial profiles. Our first result establishes the small-time global approximate controllability of the system using three scalar controls, between states that share the same sign. This property is obtained by adapting the geometric control approach to the fourth-order setting, using a finite family of frequency-localized controls. We then study the small-time global exact controllability to non-zero constant states for the concerned system. This second result is achieved by analyzing the null controllability of an appropriate linearized fourth-order system and by deducing the controllability of the nonlinear bilinear model through a fixed-point argument together with the small-time global approximate control property.
- [230] arXiv:2512.23342 [pdf, html, other]
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Title: Multi-resolution deconvolutionSubjects: Spectral Theory (math.SP)
We extend the classical deconvolution framework in Rn to the case with a pseudodifferential-like solution operator with a symbol depending on both the base and cotangent variable. Our framework enables deconvolu- tion with spatially varying resolution while maintaining a set global stability, and it additionally allows rather general distributional convolution kernels. We provide consistency, convergence and stability results, as well as conver- gence rates. Finally, we include numerical examples supporting our results and demonstrating advantages of the generalized framework.
- [231] arXiv:2512.23346 [pdf, html, other]
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Title: Backward Stochastic Volterra integral equations driven by G-Brownian motionSubjects: Probability (math.PR)
In this paper, we study the Backward stochastic Volterra integral equation driven by G-Brownian motion (G-BSVIE). By adopting a different backward iteration method, we construct the approximating sequences on each local interval. With the help of G-stochastic analysis techniques and the monotone convergence theorem, the existence, uniqueness, and continuity of the solution over the entire interval are established. Moreover, we derive the comparison theorem.
- [232] arXiv:2512.23348 [pdf, html, other]
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Title: Persistent Homology via Finite Topological SpacesSubjects: Algebraic Topology (math.AT); Computational Geometry (cs.CG); Machine Learning (cs.LG)
We propose a functorial framework for persistent homology based on finite topological spaces and their associated posets. Starting from a finite metric space, we associate a filtration of finite topologies whose structure maps are continuous identity maps. By passing functorially to posets and to simplicial complexes via crosscut constructions, we obtain persistence modules without requiring inclusion relations between the resulting complexes. We show that standard poset-level simplifications preserve persistent invariants and prove stability of the resulting persistence diagrams under perturbations of the input metric in a density-based instantiation.
- [233] arXiv:2512.23357 [pdf, html, other]
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Title: $L^2$ and $L^\infty$ rational approximationSubjects: Numerical Analysis (math.NA); Complex Variables (math.CV)
Using recently developed algorithms, we compute and compare best $L^2$ and $L^\infty$ rational approximations of analytic functions on the unit disk. Although there is some theory for these problems going back decades, this may be the first computational study. To compute the $L^2$ best approximations, we employ a new formulation of TF-IRKA in barycentric form.
- [234] arXiv:2512.23359 [pdf, other]
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Title: Regularity for mixed-order nonlinear fractional equations with degenerate coefficientsSubjects: Analysis of PDEs (math.AP)
We consider a class of nonlinear integro-differential equations whose leading operator is obtained as a superposition of $(-\Delta_{p})^{s}$ and $(-\Delta_{p})^{t}$, where $0<s<t<1<p<\infty$, weighted via two possibly degenerate coefficients $a(\cdot,\cdot),b(\cdot,\cdot) \ge 0$. We prove local boundedness and Hölder regularity of its weak solutions under natural assumptions on the coefficients $a(\cdot,\cdot)$, $b(\cdot,\cdot)$ and the powers $s,t$, and $p$. Moreover, when $a(\cdot,\cdot) \equiv 1$, we also prove a Harnack inequality for weak solutions.
- [235] arXiv:2512.23362 [pdf, html, other]
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Title: A Data-Driven Approach to Solving First-Kind Fredholm Integral Equations and Their Convergence AnalysisSubjects: Numerical Analysis (math.NA); Mathematical Physics (math-ph)
We investigate the statistical recovery of solutions to first-kind Fredholm integral equations with discrete, scattered, and noisy pointwise measurements. Assuming the forward operator's range belongs to the Sobolev space of order $m$, which implies algebraic singular-value decay $s_j\le Cj^{-m}$, we derive optimal upper bounds for the reconstruction error in the weak topology under an a priori choice of the regularization parameter. For bounded-variance noise, we establish mean-square error rates that explicitly quantify the dependence on sample size $n$, noise level $\sigma$, and smoothness index $m$; under sub-Gaussian noise, we strengthen these to exponential concentration bounds. The analysis yields an explicit a priori and a posteriori rule for the regularization parameter. Numerical experiments validate the theoretical results and demonstrate the efficiency of our practical parameter choice.
- [236] arXiv:2512.23363 [pdf, html, other]
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Title: High-order implicit Runge-Kutta time integrators for component-based model reduction of FSI problemsSubjects: Numerical Analysis (math.NA)
We propose a model order reduction framework for incompressible fluid-structure interaction (FSI) problems based on high-order implicit Runge-Kutta (IRK) methods. We consider separate reduced spaces for fluid velocity, fluid pressure and solid displacement; we enrich the velocity space with supremizer modes to ensure the inf-sup stability of the fluid subproblem; we consider bubble-port decomposition of fluid velocity and solid displacement to satisfy the kinematic conditions at the fluid structure interface. We resort to Galerkin projection to define the semi-discrete reduced-order model and we consider a Radau-IIA IRK method for time integration: the resulting algebraic system is solved using static condensation of the interface degrees of freedom. The reduced-order model preserves a semi-discrete energy balance inherited from the full-order model, and avoids the need for additional interface enrichment. Numerical experiments demonstrate that the proposed combination of high-order IRK schemes with bubble-port decoupling of velocity and displacement degrees of freedom yields stable and accurate reduced-order model for long-time integration of strongly-coupled parametric FSI problems.
- [237] arXiv:2512.23377 [pdf, html, other]
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Title: Faster-than-Nyquist Signaling for Next-Generation Wireless: Principles, Applications, and ChallengesSubjects: Information Theory (cs.IT)
Future wireless networks are expected to deliver ultra-high throughput for supporting emerging applications. In such scenarios, conventional Nyquist signaling may falter. As a remedy, faster-than-Nyquist (FTN) signaling facilitates the transmission of more symbols than Nyquist signaling without expanding the time-frequency resources. We provide an accessible and structured introduction to FTN signaling, covering its core principles, theoretical foundations, unique advantages, open facets, and its road map. Specifically, we present promising coded FTN results and highlight its compelling advantages in integrated sensing and communications (ISAC), an increasingly critical function in future networks. We conclude with a discussion of open research challenges and promising directions.
- [238] arXiv:2512.23382 [pdf, html, other]
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Title: Turán number of disjoint Berge pathsSubjects: Combinatorics (math.CO)
For a graph $F$, an $r$-uniform hypergraph $\mathcal{H}$ is a $\text{Berge-} F$ if there is a bijection $\phi: E(F)\to E(\mathcal{H})$ such that $e\subseteq \phi(e)$ for each $e\in E(F)$. When $F$ is a path, we call $\text{Berge-} F$ as Berge path. Let $\mathcal{F}$ be a family of $r$-graphs. An $r$-graph $\mathcal{H}$ is called $\mathcal{F}$-free if $\mathcal{H}$ does not contain any member in $\mathcal{F}$ as a subhypergraph. The Turán number $\mathrm{ex}_{r}(n,\mathcal{F})$ is the maximum number of hyperedges in an $\mathcal{F}$-free $r$-graph on $n$ vertices. The Turán number of Berge paths has received widespread attention. In this paper, we determine the exact value of $\mathrm{ex}_r(n,\text{Berge-} kP_{\ell})$ when $n$ is large enough for $k\geq 2$, $r\ge 3$, $\ell'\geq r$ and $2\ell'\geq r+7$, where $\ell'=\left\lfloor\frac{\ell+1}{2}\right\rfloor$.
- [239] arXiv:2512.23390 [pdf, html, other]
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Title: On the density of Sylow numbersComments: 12 pagesSubjects: Group Theory (math.GR)
Let $p$ be a prime number. We say that a positive integer $n$ is a Sylow $p$-number if there exists a finite group having exactly $n$ Sylow $p$-subgroups. When $p=2$, every odd integer is a Sylow $2$-number. In contrast, when $p$ is odd, there exist two positive constants $c_p$ and $c_p^\prime$ such that, denoting by $\beta(p,x)$ the number of Sylow $p$-numbers less than or equal to $x$,
\[c_p\,x(\log x)^{\frac{1}{p-1}-1}
\leq \beta(p,x)\leq
c_p^\prime\,x(\log x)^{\frac{1}{p-1}-1}.
\]
Moreover if $\beta_s(p,x)$ is the number of positive integers $n\le x$ such that $n$ is the Sylow $p$-number of some finite solvable group then $$\beta_s(p,x)\sim c_p\,x(\log x)^{\,\frac{1}{p-1}-1}
\qquad\text{as } x\to\infty.$$
In particular, when $p$ is odd, the natural density of Sylow $p$-numbers is $0$. - [240] arXiv:2512.23391 [pdf, html, other]
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Title: Two-color partitions with evens in one colorComments: Submitted for publications. 10 pagesSubjects: Number Theory (math.NT)
We consider sequences counting integer partitions in two colors (red and blue) in which the even parts occur only in blue color. We focus on subsequences defined by constraints on the parity and color of the summands. We establish formulas for our sequences and deduce identities of integer partitions.
- [241] arXiv:2512.23402 [pdf, html, other]
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Title: Hausdorff dimension of intersections between the Jarník sets and Diophantine fractalsComments: 16 pages, no figureSubjects: Number Theory (math.NT); Dynamical Systems (math.DS)
The irrationality exponent of a real number measures how well that number can be approximated by rationals. Real numbers with irrationality exponent strictly greater than $2$ are transcendental numbers, and form a set with rich fractal structure. We show that this set intersects the limit set of any parabolic iterated function system arising from the backward continued fraction in a set of full Hausdorff dimension. As a corollary, we show that the set of irrationals whose irrationality exponents are strictly bigger than $2$ and whose backward continued fraction expansions have bounded partial quotients is of Hausdorff dimension $1$. This is a sharp contrast to the fact that there exists no irrational whose irrationality exponent is strictly greater than $2$ and whose regular continued fraction expansion has bounded partial quotients.
- [242] arXiv:2512.23414 [pdf, html, other]
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Title: On the existence of the KMS spectral gap in Gaussian quantum Markov semigroupsComments: 32 pagesSubjects: Functional Analysis (math.FA); Quantum Physics (quant-ph)
In arXiv:2405.04947, it was shown that the GNS spectral gap of a Gaussian quantum Markovian generator is strictly positive if and only if there exists a maximal number of linearly independent noise operators, under the assumption that the generated semigroup admits a unique faithful normal invariant state. In this paper, we provide a necessary and sufficient condition for the existence of the KMS spectral gap, which also depends only on the noise operators of the generator. We further show that the existence of the GNS spectral gap implies the existence of the KMS spectral gap.
- [243] arXiv:2512.23425 [pdf, html, other]
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Title: A general framework for deep learningSubjects: Statistics Theory (math.ST); Machine Learning (cs.LG); Machine Learning (stat.ML)
This paper develops a general approach for deep learning for a setting that includes nonparametric regression and classification. We perform a framework from data that fulfills a generalized Bernstein-type inequality, including independent, $\phi$-mixing, strongly mixing and $\mathcal{C}$-mixing observations. Two estimators are proposed: a non-penalized deep neural network estimator (NPDNN) and a sparse-penalized deep neural network estimator (SPDNN). For each of these estimators, bounds of the expected excess risk on the class of Hölder smooth functions and composition Hölder functions are established. Applications to independent data, as well as to $\phi$-mixing, strongly mixing, $\mathcal{C}$-mixing processes are considered. For each of these examples, the upper bounds of the expected excess risk of the proposed NPDNN and SPDNN predictors are derived. It is shown that both the NPDNN and SPDNN estimators are minimax optimal (up to a logarithmic factor) in many classical settings.
- [244] arXiv:2512.23446 [pdf, html, other]
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Title: On the Non-Semipositivity of a Nef and Big Line Bundle on Grauert's ExampleComments: 9 pages, 1 figureSubjects: Algebraic Geometry (math.AG); Complex Variables (math.CV)
We study the relation between semipositivity, nefness, and bigness of line bundles on compact Kähler manifolds. Every nef and big line bundle on a compact Kähler manifold $X$ is positive when ${\rm dim}\,X = 1$. Kim constructed an explicit example of a nef and big line bundle that is not semipositive in the case ${\rm dim}\,X \ge 3$. Motivated by a conjecture of Filip and Tosatti, we then focus on the case of dimension two. In this talk, we show that the line bundle on Grauert's example is nef and big but not semipositive, by explicitly computing its first obstruction class, which was originally introduced by Koike as a generalization of the Ueda class.
- [245] arXiv:2512.23459 [pdf, html, other]
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Title: Extremal orthogonal arraysComments: 24 pagesSubjects: Combinatorics (math.CO)
It is known that a Delsarte $t$-design in a $Q$-polynomial association scheme has degree at least $\left \lceil{\frac{t}{2}}\right \rceil $. Following Ionin and Shrikhande who studied combinatorial $(2s-1)$-designs (i.e., Delsarte designs in Johnson association schemes) having exactly $s$ block intersection numbers, we call a Delsarte $(2s-1)$-design with degree $s$ extremal and study extremal orthogonal arrays, which are Delsarte designs in Hamming association schemes.
It was shown by Delsarte that a $t$-design with degree $s$ and $t\geq 2s-2$ in a Hamming association scheme induces an $s$-class association scheme. We prove that an extremal orthogonal array gives rise to a fission scheme of the latter one, which has $2s-1$ or $2s$ classes. As a corollary, a new necessary condition for the existence of tight orthogonal arrays of strength $3$ is obtained.
Furthermore, as a counterpart to a result of Ionin and Shrikhande, we prove an inequality for Hamming distances in extremal orthogonal arrays. The inequality is tight as shown by examples related to the Golay codes. - [246] arXiv:2512.23470 [pdf, html, other]
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Title: Dynamic Channel Knowledge Map Construction in MIMO-OFDM SystemsSubjects: Information Theory (cs.IT)
Channel knowledge map (CKM) is a promising paradigm for environment-aware communications by establishing a deterministic mapping between physical locations and channel parameters. Existing CKM construction methods focus on quasi-static propagation environment. This paper develops a dynamic CKM construction method for multiple-input multiple-output orthogonal frequency division multiplexing (MIMO-OFDM) systems. We establish a dynamic channel model that captures the coexistence of quasi-static and dynamic scatterers, as well as the impacts of antenna rotation and synchronization errors. Based on this model, we formulate the problem of dynamic CKM construction within a Bayesian inference framework and design a two-stage approximate Bayesian inference algorithm. In stage I, a high-performance algorithm is developed to jointly infer quasi-static channel parameters and calibrate synchronization errors from historical measurements. In stage II, by leveraging the quasi-static parameters as informative priors, a low-complexity algorithm is designed to estimate dynamic parameters from limited real-time measurements. Simulation results validate the superiority of the proposed method and demonstrate its effectiveness in enabling low-overhead, high-performance channel estimation in dynamic environments.
- [247] arXiv:2512.23476 [pdf, html, other]
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Title: Sensitivity Analysis on the Sphere and a Spherical ANOVA DecompositionSubjects: Numerical Analysis (math.NA)
We establish sensitivity analysis on the sphere. We present formulas that allow us to decompose a function $f\colon \mathbb S^d\rightarrow \mathbb R$ into a sum of terms $f_{\boldsymbol u,\boldsymbol \xi}$. The index $\boldsymbol u$ is a subset of $\{1,2,\ldots,d+1\}$, where each term $f_{\boldsymbol u,\boldsymbol \xi}$ depends only on the variables with indices in $\boldsymbol u$. In contrast to the classical analysis of variance (ANOVA) decomposition, we additionally use the decomposition of a function into functions with different parity, which adds the additional parameter $\boldsymbol \xi$. The natural geometry on the sphere naturally leads to the dependencies between the input variables. Using certain orthogonal basis functions for the function approximation, we are able to model high-dimensional functions with low-dimensional variable interactions.
- [248] arXiv:2512.23478 [pdf, html, other]
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Title: Bethe subspaces and wonderful models for toric arrangementsComments: 23 pagesSubjects: Algebraic Geometry (math.AG); Mathematical Physics (math-ph)
We study the family of commutative subspaces in the trigonometric holonomy Lie algebra $t^{\mathrm{trig}}_{\Phi}$, introduced by Toledano Laredo, for an arbitrary root system $\Phi$. We call these subspaces \emph{Bethe subspaces} because they can be regarded as quadratic components of \emph{Bethe subalgebras} in the Yangian corresponding to the root system $\Phi$, that are responsible for integrals of the generalized XXX Heisenberg spin chain. Bethe subspaces are naturally parametrized by the complement of the corresponding toric arrangement . We prove that this family extends regularly to the minimal wonderful model $X_{\Phi}$ of the toric arrangement described by De Concini and Gaiffi, thus giving a compactification of the parameter space for Bethe subspaces. For classical types $A_n, B_n, C_n, D_n$, we show that this extension is faithful. As a special case, when $\Phi$ is of type $A_n$, our construction agrees with the main result of Aguirre--Felder--Veselov on the closure of the set of quadratic Gaudin subalgebras. Our work is also closely related to, and refines in this root system setting, a parallel compactification result of J. Peters obtained for more general toric arrangements arising from quantum multiplication on hypertoric varieties. Next, we show that the Bethe subspaces assemble into a vector bundle over $X_{\Phi}$, which we identify with the logarithmic tangent bundle of $X_{\Phi}$. Finally, we formulate conjectures extending these results to the setting of Bethe subalgebras in Yangians and to the quantum cohomology rings of Springer resolutions. We plan to address this in our next papers.
- [249] arXiv:2512.23479 [pdf, html, other]
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Title: Families of cyclic curve coverings with maximal monodromySubjects: Algebraic Geometry (math.AG)
We study the algebraic monodromy of families of cyclic Galois coverings of curves. Under a condition on the $G$-decomposition of the associated variation of Hodge structures, we prove a criterion for the maximality of the monodromy. The proof combines the genus-zero case with a degeneration argument involving Prym varieties of certain admissible coverings. As a consequence of our criterion, we show that for $g\geq 8$ there exists no special family of Galois covers of the type we consider, providing new evidence towards the Coleman-Oort conjecture. Finally, we determine when the loci of double and triple Galois covers are totally geodesic.
- [250] arXiv:2512.23503 [pdf, other]
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Title: On Hopf Ideals, Integrality, and Automorphisms of Quantum Groups at Roots of 1Comments: 158 pagesSubjects: Quantum Algebra (math.QA)
We consider skew-commutative subalgebras in Drinfeld-Jimbo quantum groups at a root of unity $\zeta$ generated by primitive power elements. We classify the centrality and commutativity of these skew-polynomial algebras depending on the Lie type and the order of $\zeta$ modulo 8. We describe Hopf ideals in the quantum group induced from ideals in these subalgebras, including the non-commutative cases.
Among these, we construct and analyze a family of Hopf ideals that depend on the choice of an element in the Weyl group. We show that they arise naturally both in the construction of (partial) $R$-matrices and as vanishing ideals of Bruhat subgroups. Specialization to the maximal element yields a rigorous construction of restricted quantum groups as pre-triangular Hopf algebras, independent of any choices.
Our treatment also includes even orders of $\zeta$, non-simply laced Lie types, and minimal ground rings. Consequently, we extend some results of De Concini-Kac-Procesi, whose work focuses on odd orders of $\zeta$, which forces the subalgebra to be strictly central, and complex ground fields. This includes the identification of the subalgebras for Lie types $\mathsf{A}_n$ and $\mathsf{B}_2$ with the coordinate rings of associated algebraic groups in the commutative cases, even if $\zeta$ has even order. Our descriptions are computationally explicit and do not utilize Poisson structures.
As technical preparations, we discuss PBW bases over minimal rings, dependencies on choices of convex orderings, as well as various new constructions of, and relations among, automorphisms on quantum groups. The latter include formulae for the Garside element in the Lustzig-Artin group action and the family of Che-transformations. - [251] arXiv:2512.23506 [pdf, html, other]
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Title: Affine-Projection Recovery of Continuous Angular Power Spectrum: Geometry and ResolutionComments: 6 pages, 1 figureSubjects: Information Theory (cs.IT); Signal Processing (eess.SP)
This paper considers recovering a continuous angular power spectrum (APS) from the channel covariance. Building on the projection-onto-linear-variety (PLV) algorithm, an affine-projection approach introduced by Miretti \emph{et. al.}, we analyze PLV in a well-defined \emph{weighted} Fourier-domain to emphasize its geometric interpretability. This yields an explicit fixed-dimensional trigonometric-polynomial representation and a closed-form solution via a positive-definite matrix, which directly implies uniqueness. We further establish an exact energy identity that yields the APS reconstruction error and leads to a sharp identifiability/resolution characterization: PLV achieves perfect recovery if and only if the ground-truth APS lies in the identified trigonometric-polynomial subspace; otherwise it returns the minimum-energy APS among all covariance-consistent spectra.
- [252] arXiv:2512.23507 [pdf, html, other]
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Title: Encoding higher-order argumentation frameworks with supports to propositional logic systemsComments: 39 pagesSubjects: Logic (math.LO)
Argumentation frameworks ($AF$s) have been extensively developed, but existing higher-order bipolar $AF$s suffer from critical limitations: attackers and supporters are restricted to arguments, multi-valued and fuzzy semantics lack unified generalization, and encodings often rely on complex logics with poor interoperability. To address these gaps, this paper proposes a higher-order argumentation framework with supports ($HAFS$), which explicitly allows attacks and supports to act as both targets and sources of interactions. We define a suite of semantics for $HAFS$s, including extension-based semantics, adjacent complete labelling semantics (a 3-valued semantics), and numerical equational semantics ([0,1]-valued semantics). Furthermore, we develop a normal encoding methodology to translate $HAFS$s into propositional logic systems ($\mathcal{PLS}$s): $HAFS$s under complete labelling semantics are encoded into Łukasiewicz's three-valued propositional logic ($\mathcal{PL}_3^L$), and those under equational semantics are encoded into fuzzy $\mathcal{PLS}$s ($\mathcal{PL}_{[0,1]}$) such as Gödel and Product fuzzy logics. We prove model equivalence between $HAFS$s and their encoded logical formulas, establishing the logical foundation of $HAFS$ semantics. Additionally, we investigate the relationships between 3-valued complete semantics and fuzzy equational semantics, showing that models of fuzzy encoded semantics can be transformed into complete semantics models via ternarization, and vice versa for specific t-norms. This work advances the formalization and logical encoding of higher-order bipolar argumentation, enabling seamless integration with lightweight computational solvers and uniform handling of uncertainty.
- [253] arXiv:2512.23521 [pdf, html, other]
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Title: On the continuity of the product of distributions in local Sobolev spacesSubjects: Functional Analysis (math.FA)
We consider the space $\mathscr{H}_L ^{s,r} (O)$ consisting of all local Sobolev distributions of order $s$ on an open set $O$ whose Sobolev wave front set of order $r$ is contained in the closed conic set $L\subseteq O\times(\mathbb{R}^m\backslash\{0\})$. We introduce a locally convex topology on $\mathscr{H}_L ^{s,r} (O)$ and show that the ordinary product of smooth functions uniquely extends to a continuous bilinear mapping $\mathscr{H}_{L_1} ^{r_1,r'} (O) \times \mathscr{H}_{L_2} ^{r_2,r''} (O) \to \mathscr{H}_{L} ^{s,r} (O)$, for appropriate $s$ and $r$ when $L_1$ and $L_2$ are in a favorable position. The key ingredient in our proof is to employ Hörmander's idea of considering the pullback by the diagonal map $x\mapsto (x,x)$ of the tensor product of two distributions.
- [254] arXiv:2512.23522 [pdf, html, other]
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Title: Defect of projective hypersurfaces with isolated singularitiesSubjects: Algebraic Geometry (math.AG)
Let $X$ be a hypersurface with isolated singularities defined by $f$ in ${\bf P^{n+1}}$ with $n>1$. The difference ${\rm def}(X):=h^{n+1}(X)-h^{n-1}(X)$ is called the defect of $X$ (for self-duality of the cohomology of $X$). It is known that its vanishing is closely related to ${\bf Q}$-factoriality of $X$ in the rational singularity case with $n=3$. This number coincides with the dimension of the cokernel of the inclusion $H^{n-1}(X)\to{\rm IH}^{n-1}(X)$, the rank of the morphism from the vanishing cohomologies of $X$ to $H^{n+1}(X)$ for a one-parameter smoothing of $X$ with total space smooth, and also with the dimension of the unipotent monodromy part of the Milnor fiber cohomology of $f$ with degree $n$. In the case $X$ has only weighted homogeneous isolated singularities, the defect ${\rm def}(X)$ is then given by the $E_2$-term of the spectral sequence of the double complex with differentials ${\rm d}f\wedge$ and $\rm d$ by the $E_2$-degeneration of the pole order spectral sequence. It can be calculated explicitly using a computer even for analogues of the Hirzebruch quintic threefold with more than one hundred ordinary double points found by B.\ van Geemen and J.\ Werner in a compatible way with their computation. We give also an example with ${\rm def}(X)>0$ and $|{\rm Sing}\,X|=1$ where $n=3$.
- [255] arXiv:2512.23527 [pdf, html, other]
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Title: Identifying faulty edges in resistive electrical networksSubjects: Optimization and Control (math.OC); Discrete Mathematics (cs.DM); Information Theory (cs.IT); Combinatorics (math.CO)
Given a resistive electrical network, we would like to determine whether all the resistances (edges) in the network are working, and if not, identify which edge (or edges) are faulty. To make this determination, we are allowed to measure the effective resistance between certain pairs of nodes (which can be done by measuring the amount of current when one unit of voltage difference is applied at the chosen pair of nodes). The goal is to determine which edge, if any, is not working in the network using the smallest number of measurements. We prove rigorous upper and lower bounds on this optimal number of measurements for different classes of graphs. These bounds are tight for several of these classes showing that our measurement strategies are optimal.
- [256] arXiv:2512.23528 [pdf, html, other]
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Title: On the Brown measure of $x + i y$, with $x,y$ selfadjoint and $y$ free PoissonComments: Preliminary version, comments welcomeSubjects: Operator Algebras (math.OA); Mathematical Physics (math-ph); Probability (math.PR)
Let $x,y$ be freely independent selfadjoint elements in a $W^{*}$-probability space, where $y$ has free Poisson distribution of parameter $p$. We pursue a methodology for computing the absolutely continuous part of the Brown measure of $x + i y$, which relies on the matrix-valued subordination function $\Omega$ of the Hermitization of $x + i y$, and on the fact that $\Omega$ has an explicitly described left inverse $H$. Our main point is that the Brown measure of $x + i y$ becomes more approachable when it is reparametrized via a certain change of variable $h : \mathcal{D} \to \mathcal{M}$, with $\mathcal{D}, \mathcal{M}$ open subsets of $\mathbb{C}$, where $\mathcal{D}$ and $h$ are defined in terms of the aforementioned left inverse $H$, and $\mathrm{cl} \,(\mathcal{M})$ contains the support of the Brown measure. More precisely, we find (with some conditions on the distribution of $x$, which have to be imposed for certain values of the parameter $p$) the following formula: \[ f(s + i \, t) =\frac{1}{4\pi}\left[\frac{2}{t}\left(\frac{\partial \alpha}{\partial s} +\frac{\partial \beta}{\partial t}\right)-\frac{2}{t}-\frac{2\beta}{t^2}\right], \ \ s + i \, t \in \mathcal{M}, \] where $f$ is the density of the absolutely continuous part of the Brown measure and the functions $\alpha, \beta : \mathcal{M} \to \mathbb{R}$ are the real and respectively the imaginary part of $h^{-1}$.
- [257] arXiv:2512.23529 [pdf, html, other]
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Title: An overview on curve semistable and numerically flat Higgs bundlesComments: 11 pagesSubjects: Algebraic Geometry (math.AG)
After recalling the basic notions concerning Higgs-Grassmannian schemes, I review how these latter can be used to define generalisations of the notion of positivity conditions, such as numerically flatness, which "feel" the Higgs field. Then I prove several properties of Higgs bundles, over smooth projective varieties defined over an algebraically closed field of characteristic $0$, satisfying these conditions. Finally, I discuss how one can relate them to semistability of the so-called "curve semistable" Higgs bundles.
- [258] arXiv:2512.23534 [pdf, html, other]
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Title: On Goldbach numbers in short intervalsSubjects: Number Theory (math.NT)
Assuming the Riemann Hypothesis, we prove that for all $x\geq 2$, there exists at least one even integer within the interval $(x, x+123\log^2x]$, that can be expressed as the sum of two primes. This result is an improvement over the recent work of Cully-Hugill and Dudek, who obtained the constant $9696$ instead of $123$.
- [259] arXiv:2512.23540 [pdf, html, other]
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Title: Error Estimates for Gauss--Christoffel Quadrature under Reduced Regularity ConditionsComments: 14 pages, 3 tablesSubjects: Numerical Analysis (math.NA)
Gauss--Christoffel quadrature is a fundamental method for numerical integration, and its convergence analysis is closely related to the decay of Chebyshev expansion coefficients. Classical estimates, including those due to Trefethen, are based on weighted bounded variation assumptions involving the singular weight $(1-x^{2})^{-1/2}$, which may be too restrictive for functions with limited regularity at the endpoints.
In this paper, we establish a new error bound for Gauss--Christoffel quadrature under weakened regularity assumptions. The analysis relies on a new identity for higher-order derivatives of Chebyshev polynomials. As a consequence, we obtain an improved decay estimate for Chebyshev coefficients, where the classical weighted condition \[ V_{r}=\int_{-1}^{1}\frac{|f^{(r+1)}(x)|}{\sqrt{1-x^{2}}}\,dx \] is replaced by the weaker condition \[ U_{r}=\int_{-1}^{1}|f^{(r+1)}(x)|\,dx. \]
This result leads to a corresponding error estimate for the Gauss--Christoffel quadrature rule, which is less restrictive than previous bounds. The approach is also extended to the Gauss--Gegenbauer case. Numerical experiments are provided to illustrate the theoretical results. - [260] arXiv:2512.23543 [pdf, html, other]
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Title: Complex structures on 2-step nilpotent Lie algebras arising from graphsComments: Comments are welcome!Subjects: Differential Geometry (math.DG); Rings and Algebras (math.RA)
This work investigates the existence of complex structures on 2-step nilpotent Lie algebras arising from finite graphs. We introduce the notion of adapted complex structure, namely a complex structure that maps vertices and edges of the graph to vertices and edges, and we analyze in depth the restrictions imposed by the integrability condition. We completely characterize the graphs that admit abelian adapted complex structures, showing that they belong to a small family of graphs that we call basic. We prove that any graph endowed with an adapted complex structure $J$ contains a unique $J$-invariant basic spanning subgraph, and conversely, that every such graph can be constructed through a systematic expansion procedure starting from a basic graph. We also explore geometric and combinatorial consequences, including the existence of special Hermitian metrics as well as other graph-theoretic properties.
- [261] arXiv:2512.23544 [pdf, html, other]
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Title: All spaces of countable spread can be smallComments: 13 pagesSubjects: General Topology (math.GN); Logic (math.LO)
The main result of this paper is the proof of the simultaneous consistency, modulo a weakly compact cardinal, of the equality $2^{< \mathfrak{c}} = \mathfrak{c}$ with the following property (*) of partitions of pairs of $\mathfrak{c}$:
\smallskip
(*) For any coloring (or partition) $k : [\mathfrak{c}]^2 \rightarrow 2$ either there is a homogeneous set of size $\mathfrak{c}$ in color $0$ or there is a set $S \in [\mathfrak{c}]^\mathfrak{c}$ such that for every countable $A \subset S$ there is $\beta \in \mathfrak{c}$ for which $A \subset \beta$ and $k(\{\alpha, \beta\}) = 1$ for all $\alpha \in A$.
\smallskip
(*) plus $2^{< \mathfrak{c}} = \mathfrak{c}$ together then imply that for every topological space $X$ of countable
spread, i.e. not containing any uncountable discrete subset,
$|X| \le \mathfrak{c}$ if it is
Hausdorff and
$o(X) = \mathfrak c$ if it is also infinite and regular. Here $o(X)$ denotes the number of all open subsets of $X$. - [262] arXiv:2512.23549 [pdf, html, other]
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Title: A note on the hypergeometric datum $\big((\frac{1}{2},\frac{1}{6},\frac{5}{6}),(1,1)\big)$ and symmetric squares of elliptic curvesComments: 11 pagesSubjects: Number Theory (math.NT)
This is an expository note on a mod $p$ congruence relating the truncated hypergeometric sums associated to $\big((\frac{1}{2},\frac{1}{6},\frac{5}{6}),(1,1)\big)$ to symmetric squares of elliptic curves.
- [263] arXiv:2512.23566 [pdf, html, other]
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Title: From geometry to dynamics: Learning overdamped Langevin dynamics from sparse observations with geometric constraintsComments: 12+50 pages, 6 figures; An earlier account of this work has previously appeared in arXiv:2301.08102 and arXiv:2304.00423 ; main methodology remains the same, this version includes additional numerical experiments and theorySubjects: Dynamical Systems (math.DS); Statistical Mechanics (cond-mat.stat-mech); Machine Learning (cs.LG); Optimization and Control (math.OC); Machine Learning (stat.ML)
How can we learn the laws underlying the dynamics of stochastic systems when their trajectories are sampled sparsely in time? Existing methods either require temporally resolved high-frequency observations, or rely on geometric arguments that apply only to conservative systems, limiting the range of dynamics they can recover. Here, we present a new framework that reconciles these two perspectives by reformulating inference as a stochastic control problem. Our method uses geometry-driven path augmentation, guided by the geometry in the system's invariant density to reconstruct likely trajectories and infer the underlying dynamics without assuming specific parametric models. Applied to overdamped Langevin systems, our approach accurately recovers stochastic dynamics even from extremely undersampled data, outperforming existing methods in synthetic benchmarks. This work demonstrates the effectiveness of incorporating geometric inductive biases into stochastic system identification methods.
- [264] arXiv:2512.23574 [pdf, html, other]
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Title: Intersections of sumsets in additive number theoryComments: 6 pagesSubjects: Number Theory (math.NT)
Let $A$ be a subset of an additive abelian semigroup and let $hA$ be the $h$-fold sumset of $A$. The following question is considered: Let $(A_q)_{q=1}^{\infty}$ be a strictly decreasing sequence of sets in the semigroup and let $A = \bigcap_{q=1}^{\infty} A_q$. When does one have \[ hA = \bigcap_{q=1}^{\infty} hA_q \] for some or all $h \geq 2$?
- [265] arXiv:2512.23579 [pdf, html, other]
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Title: Torsion-Free Bimodule Connections and the Maximal Prolongation of a First-Order Differential CalculusComments: Preliminary version. Extension from the quantum Grassmannians to all irreducible quantum flag manifolds, and extension from the anti-holomorphic calculus to the entire Dolbeault double complex will followSubjects: Quantum Algebra (math.QA)
We give an unexpectedly simple presentation of the maximal prolongation of a first-order differential calculus in terms of the bimodule map of a torsion-free bimodule connection. We then show that in the quantum homogeneous space case this simplifies even further. More explicitly, we show that the bimodule map associated to a bimodule connection, for any relative left Hopf module endowed with its canonical right module structure, admits a concise formula, given in terms of the adjont action of a Hopf algebra on a bimodule. %{\color{red} We also have the dual tangent space formula.} This is then used to derive sufficient conditions, in terms of the first-order differential forms, for the extendability of a first-order almost-complex structure. These results are applied to the quantum Grassmannian Heckenberger--Kolb calculi, yielding a simple uniform presentation of their degree two anti-holomorphic relations.
- [266] arXiv:2512.23580 [pdf, html, other]
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Title: Analysis of kinetic-diffusion Monte Carlo simulation and source term estimation scheme in nuclear fusion applicationsComments: Section 5.6 of this article builds upon the numerical results presented in arXiv:2509.11883 and these articles contain some textual overlapSubjects: Numerical Analysis (math.NA)
In plasma edge simulations, the behavior of neutral particles is often described by a Boltzmann--BGK equation. Solving this kinetic equation and estimating the moments of its solution are essential tasks, typically carried out using Monte Carlo (MC) methods. However, for large-sized reactors, like ITER and DEMO, high collision rates lead to a substantial computational cost. To accelerate the calculation, an asymptotic-preserving kinetic-diffusion Monte Carlo (KDMC) simulation method (Mortier et al., SIAM J. Sci. Comput., 2022) and a corresponding fluid estimation technique (Mortier et al., Contrib. Plasma Phys., 2022) have recently been proposed. In this work, we present a comprehensive analysis of the convergence of KDMC combined with the associated fluid estimation. The analysis consists of proving theoretical upper bounds for both KDMC and the fluid estimation, and numerical verifications of these bounds. In addition, we compare the analyzed algorithm with a purely fluid-based method using the fully kinetic MC method as a reference. The algorithm consistently achieves lower error than the fluid-based method, and even one order of magnitude lower in a fusion-relevant test case. Moreover, the algorithm exhibits a significant speedup compared to the reference kinetic MC method. Overall, our analysis confirms the effectiveness of KDMC with the associated fluid estimation in nuclear fusion applications.
- [267] arXiv:2512.23582 [pdf, html, other]
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Title: The Time-Periodic Cahn-Hilliard-Gurtin System on the Half Space as a Mixed-Order System with General Boundary ConditionsSubjects: Analysis of PDEs (math.AP)
A well-posedness and maximal regularity result for the time-periodic Cahn-Hilliard-Gurtin system in the half space is proved. For this purpose, we introduce a novel class of complementing boundary conditions, extending the classical Lopatinski\uı-Shapiro conditions from elliptic and parabolic theory to time-periodic mixed-order systems with general boundary conditions. Moreover, we show that the classical Lopatinski\uı-Shapiro conditions are in general insufficient for well-posedness of mixed-order systems.
- [268] arXiv:2512.23584 [pdf, html, other]
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Title: Set Valued Riemann-Liouville integral and some Regular SelectionsComments: 16 Pages. Suggestions and comments are appreciatedSubjects: Dynamical Systems (math.DS)
In this article, we introduce the notion of the Riemann-Liouville fractional integral of set-valued mappings via integrable selections. We establish fundamental properties of this fractional integral, including convexity, boundedness, and continuity with respect to the Hausdorff metric. The investigation of preservation of regularity under fractional integration with respect to the Hausdorff metric is given. We show that bounded variation and Lipschitz continuity of a set-valued mapping are inherited by its Riemann-Liouville fractional integral. We discuss the existence of regular selections for the fractional integral under the corresponding regularity assumptions on the original mapping. In the scalar case, we further identify extremal selections given by the pointwise minimum and maximum of the fractional integral and show that they possess the same regularity properties. Finally, we discuss possible applications in differential inclusion and directions for future research.
- [269] arXiv:2512.23598 [pdf, html, other]
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Title: On regions of mixed unitarity for semigroups of unital quantum channelsComments: 28 pages. All comments are welcomeSubjects: Operator Algebras (math.OA); Mathematical Physics (math-ph); Functional Analysis (math.FA)
It is established that both discrete and continuous semigroups of unital quantum channels are eventually mixed unitary. However, after introducing the mixed unitary index of a unital quantum channel as the least time beyond which all subsequent powers of the channel are mixed unitary, we demonstrate that for any fixed finite dimension $d\geq 3$, there exists no universal upper bound for this index. Furthermore, for a continuous semigroup that is not mixed unitary for some $t>0$, we prove it remains non-mixed unitary for all times $t>0$, sufficiently close to the origin. Finally, a necessary and sufficient condition is derived for a quantum dynamical semigroup to be a convex combination of maps implemented solely by Weyl unitaries.
- [270] arXiv:2512.23614 [pdf, html, other]
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Title: On the origin of the Jacobian conjectureSubjects: Algebraic Geometry (math.AG); Complex Variables (math.CV)
The Jacobian conjecture is thought to have been proposed by O. H. Keller in 1939. However, we have found that the statement of the conjecture is precisely the main result of a paper published by L. Kraus in 1884. Although the final step of Kraus's proof is flawed, the ideas he introduced anticipated approaches to the problem that would only emerge more than a century later. Interestingly, the root of Kraus's error remains the principal obstacle to algebro-geometric approaches: controlling the ramification at infinity.
- [271] arXiv:2512.23615 [pdf, other]
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Title: Galois Realisations of $\operatorname{PSL}_2(\mathbb{F}_{p^2})$ via non-unirational Hilbert IrreducibilityComments: 31 pagesSubjects: Number Theory (math.NT); Algebraic Geometry (math.AG)
We establish non-unirational versions of Hilbert Irreducibility for all Hilbert modular surfaces which are of K3 type. As an application we prove new instances of the regular Inverse Galois Problem for the simple groups $\operatorname{PSL}_2(\mathbb{F}_{p^2})$ subject to congruence conditions on $p$.
- [272] arXiv:2512.23621 [pdf, html, other]
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Title: Learning Lévy density via adaptive RKHS regression with bi-level optimizationSubjects: Numerical Analysis (math.NA)
We propose a nonparametric method to learn the Lévy density from probability density data governed by a nonlocal Fokker-Planck equation. We recast the problem as identifying the kernel in a nonlocal integral operator from discrete data, which leads to an ill-posed inverse problem. To regularize it, we construct an adaptive reproducing kernel Hilbert space (RKHS) whose kernel is built directly from the data. Under standard source and spectral decay conditions, we show that the reconstruction error decays in the mesh size at a near optimal rate. Importantly, we develop a generalized singular value decomposition (GSVD)-based bilevel optimization algorithm to choose the regularization parameter, leading to efficient and robust computation of the regularized estimator. Numerical experiments for several Lévy densities, drift fields and data types (PDE-based densities and sample ensemble-based KDE reconstructions) demonstrate that our bilevel RKHS method outperforms classical L-curve and generalized cross-validation strategies and that the adaptive RKHS norm is more accurate and robust than $L^2_\rho$- and $\ell^2$-based regularization.
- [273] arXiv:2512.23623 [pdf, html, other]
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Title: Rotationally symmetric translating solitons of fully nonlinear extrinsic geometric flows: Classification and ApplicationsComments: Comment are welcome!Subjects: Differential Geometry (math.DG); Analysis of PDEs (math.AP)
We study rotationally symmetric translators for fully nonlinear extrinsic geometric flows driven by a curvature function, and we establish the fine asymptotics of bowl-type evolutions and, when admissible, the construction and classification of catenoidal-type solutions, together with their asymptotic behavior. Under natural structural and convexity assumptions, we also prove rigidity and uniqueness results within appropriate classes of graphical translators of such curvature flows.
- [274] arXiv:2512.23643 [pdf, html, other]
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Title: Simultaneous Approximation of the Score Function and Its Derivatives by Deep Neural NetworksComments: 38 pagesSubjects: Numerical Analysis (math.NA); Machine Learning (cs.LG); Statistics Theory (math.ST); Machine Learning (stat.ML)
We present a theory for simultaneous approximation of the score function and its derivatives, enabling the handling of data distributions with low-dimensional structure and unbounded support. Our approximation error bounds match those in the literature while relying on assumptions that relax the usual bounded support requirement. Crucially, our bounds are free from the curse of dimensionality. Moreover, we establish approximation guarantees for derivatives of any prescribed order, extending beyond the commonly considered first-order setting.
- [275] arXiv:2512.23648 [pdf, html, other]
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Title: A High-Order Spectral Element Solver for Steady-State Free Surface FlowsComments: 23 pages, 17 figuresSubjects: Numerical Analysis (math.NA)
We present a spectral element solver for the steady incompressible Navier-Stokes equations subject to a free surface. Utilizing the kinematic behaviour of the free surface boundary, an iterative pseudo-time procedure is proposed to determine the a priori unknown free surface profile. The numerical model is implemented in the open-source finite element framework Firedrake, which enables the use of a high-order polynomial basis on unstructured meshes through weak formulations. Additionally, the curvature of the free surface and submerged bodies is incorporated through curvilinear elements obtained via transfinite linear blending, which conserves the high-order convergent properties of the overall scheme. The model is applied to several benchmark cases in two spatial dimensions. Initially, it addresses fixed-domain problems, including the lid-driven cavity flow and flows around bodies such as a cylinder and a NACA airfoil. Subsequently, with the presence of a free surface, it is extended to determine the flow around a bathymetry bump and a submerged NACA airfoil. The results confirm the high-order accuracy of the model through convergence studies and demonstrate a substantial speed-up over low-order numerical schemes.
- [276] arXiv:2512.23651 [pdf, html, other]
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Title: Notes on non-separable arrangements of convex bodiesComments: 14 pages, 3 figuresSubjects: Metric Geometry (math.MG)
A problem posed by Erdős in 1945 initiated the study of non-separable arrangements of convex bodies. A finite collection of convex bodies in Euclidean $d$-space is called a non-separable family (or NS-family) if every hyperplane intersecting their convex hull also intersects at least one member of the family. Recent work has focused on minimal coverings of NS-families consisting of positive homothetic convex bodies. In this paper, we strengthen these results by establishing their analogues for weakly non-separable families of convex polytopes. We further obtain stability results and analyze maximal weakly non-separable families of cubes. As an additional extension, we also examine weakly $k$-impassable families of convex $d$-polytopes for $0<k<d-1$.
- [277] arXiv:2512.23668 [pdf, html, other]
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Title: Diamond lift of Hirose--Sato's formula involving the Hoffman basisComments: 9 pagesSubjects: Number Theory (math.NT)
In this paper, we give a new proof of Hirose--Sato's formula for the expansion of $\zeta(\{2\}^{a_1-1},3,\dots,\{2\}^{a_r-1},3,\{2\}^{c-1},1,\{2\}^{b_1},\dots, 1,\{2\}^{b_s})$ in the Hoffman basis, using the drop 1 relation.
- [278] arXiv:2512.23673 [pdf, html, other]
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Title: Spectral norm of matrices with independent entries up to polyloglogSubjects: Probability (math.PR); Functional Analysis (math.FA)
In this paper, we study the expectation of the operator norm of the random matrix (a_{ij} X_{ij}) for i,j <= n, under the assumption that the random variables (X_{ij}) are independent, symmetric and satisfy the moment growth condition ||X_{ij}||{2p} <= C ||X_{ij}||{p} for every p >= 1. We derive an upper bound expressed in terms of quantities that can be explicitly computed in many cases. This bound implies a two-sided estimate, up to a factor given by a power of an iterated logarithm. This factor is considerably smaller than the natural scale of the problem. Our result thus provides positive evidence supporting a conjecture formulated by Rafal Latala and Jan Swiatkowski.
- [279] arXiv:2512.23674 [pdf, html, other]
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Title: Groups with fast-growing conjugator length functionsComments: 23 pages, no figuresSubjects: Group Theory (math.GR)
We construct the first examples of finitely presented groups where the conjugator length function is exponential; these are central extensions of groups of the form $F_m \rtimes F_2$. Further, we use a fibre product construction to exhibit a family of finitely presented groups $\Gamma_k$ where, for each $k$, the conjugator length function of $\Gamma_k$ grows like functions in the $k$-th level of the Grzegorczyk hierarchy of primitive recursive functions.
- [280] arXiv:2512.23677 [pdf, html, other]
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Title: Prime Splitting and Common $N$-Index Divisors in Radical Extensions: Part $p=2$Comments: 26 pages including many examples and constructions, comments welcome!Subjects: Number Theory (math.NT)
Following work of Vélez, we explicitly describe the splitting of the integral prime 2 in the radical extension $\mathbb{Q}(\sqrt[n]{a})$, where $x^n-a$ is an irreducible polynomial in $\mathbb{Z}[x]$. With previous work of the second author, this fully describes the splitting of any prime in $\mathbb{Q}(\sqrt[n]{a})$. Using this description, we classify common index divisors (the primes whose splitting prevents the existence of a power integral basis for the ring of integers). Using work of Pleasants, we extend this to describe common $N$-index divisors (primes that divide the index of any order generated over $\mathbb{Z}$ by $N$ elements). We also present two novel constructions of non-monogenic fields with no common index divisors as well as constructions of number rings requiring $N$ ring generators for any $N>1$. Examples are provided throughout.
- [281] arXiv:2512.23679 [pdf, html, other]
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Title: Asymptotics of the shifted finite differences of the overpatition function and a problem of Wang--Xie--ZhangSubjects: Number Theory (math.NT)
Let $\overline{p}(n)$ denote the overpartition function, and for $j\in \mathbb{N}$, $\Delta^r_j$ denote the $r$-fold applications of the shifted difference operator $\Delta_j$ defined by $\Delta_j(a)(n):=a(n)-a(n-j)$. The main goal of this paper is to derive an asymptotic expansion of $\Delta^r_j(\overline{p})(n)$ with an effective error bound which subsequently gives an answer to a problem of Wang, Xie, and Zhang. In order to get the asymptotics of $\Delta^r_j(\overline{p})(n)$, we derive an asymptotic expansion of the shifted overpartition function $\overline{p}(n+k)$ for any integer $k\neq 0$.
- [282] arXiv:2512.23681 [pdf, html, other]
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Title: Better than squareroot cancellation in number theoryComments: 27 pages. A version of this paper will appear in the Proceedings of the ICM 2026Subjects: Number Theory (math.NT)
We give a short survey of the phenomenon of better than squareroot cancellation, specifically as it applies to averages of multiplicative character sums (such as $\frac{1}{r-1} \sum_{\chi \; \text{mod} \; r} |\sum_{n \leq x} \chi(n)|^{2q}$) thanks to their connection with so-called multiplicative chaos.
We focus on the number theoretic aspects of the arguments, and also touch on some possible applications. - [283] arXiv:2512.23695 [pdf, html, other]
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Title: A dimension reduction procedure for the selection of Two-spring lattice-spring topologies with minimal fabrication cost and required weighted force-resistance performanceComments: 8 pages, 5 figuresSubjects: Optimization and Control (math.OC)
Starting from a problem in elastoplasticity, we consider an optimization problem $C(c_1,c_2)=c_1+c_2\to \min$ under constraints $F_R^k(c_1,c_2)=a\cdot F^k(c_1,c_2)+b\cdot R^k(c_1,c_2)\ge 1$ and $F^k(c_1,c_2)\ge 1$, where both $F^k$ and $R^k$ non-linear, $a,b$ are constants, and $i\in\{1,2\}$ is an index. For each $(a,b)$ we determine which of the two values of $i\in\{1,2\}$ leads to the smaller minimum of the optimization problem. This way we obtain an interesting curve bounding the region where $k=1$ outperforms $k=2$.
- [284] arXiv:2512.23698 [pdf, other]
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Title: Non-compact 3D TQFT and non-semisimplicityComments: PhD thesis, 75 pages, comments welcome!Subjects: Quantum Algebra (math.QA); Category Theory (math.CT); Geometric Topology (math.GT)
We define a once extended non-compact 3-dimensional TQFT $\mathcal{Z}$ from the data of a (potentially) non-semisimple modular tensor category. This is in the framework of generators and relations of [Bartlett et al., arXiv:1509.06811 (2015)], having disallowed generating 2-morphisms whose source is the empty. Moreover, we show that the projective mapping class group representations this TQFT gives rise to, are dual to those of [Lyubashenko, arXiv:hep-th/9405167 (1994)] and [De Renzi et al., arXiv:2010.14852 (2020)]. We develop a method to decompose a closed 3-manifold in terms of 2-morphism generators. We use this to compute the value of $\mathcal{Z}$ on 3-manifolds, explaining why it should recover Lyubashenko's 3-manifold invariants [Lyubashenko, arXiv:hep-th/9405167 (1994)]. Finally, we explain that the value of the non-compact TQFT on the solid torus recovers the data of a modified trace [Geer et al., arXiv:0711.4229 (2007)].
- [285] arXiv:2512.23700 [pdf, other]
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Title: Quantum Invariants and FiberednessComments: 39 pages, 9 figures, 1 .tsv fileSubjects: Geometric Topology (math.GT); Quantum Algebra (math.QA)
We explore the topological significance of the knot two-variable series $F_K$, proposed by Gukov--Manolescu and defined by Park for a class of `nice' knots. We show that the leading coefficient of $F_K$ is a monomial and express its exponent in terms of the Hopf invariant for all homogeneous braid knots and fibered knots up to 12 crossings. As an application, we deduce an explicit formula for the Hopf invariant in terms of colored Jones polynomials.
For non-fibered strongly quasipositive knots, we study a relation between $F_K$ and the stability series of the colored Jones function, and explore similarities between $F_K$ and knot Floer homology. Finally, we propose a slope conjecture for $F_K$, relating it to the boundary slopes of the knot.
New submissions (showing 285 of 285 entries)
- [286] arXiv:2512.22124 (cross-list from physics.comp-ph) [pdf, html, other]
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Title: The Solution of Potential-Driven, Steady-State Nonlinear Network Flow Equations via Graph PartitioningComments: 6 pages, 2 figuresSubjects: Computational Physics (physics.comp-ph); Optimization and Control (math.OC)
The solution of potential-driven steady-state flow in large networks is required in various engineering applications, such as transport of natural gas or water through pipeline networks. The resultant system of nonlinear equations depends on the network topology, and its solution grows more challenging as the network size increases. We present an algorithm that utilizes a given partition of a network into tractable sizes to compute a global solution for the full nonlinear system through local solution of smaller subsystems induced by the partitions. When the partitions are induced by interconnects or transfer points corresponding to networks owned by different operators, the method ensures data is shared solely at the interconnects, leaving network operators free to solve the network flow system corresponding to their own domain in any manner of their choosing. The proposed method is shown to be connected to the Schur complement and the method's viability demonstrated on some challenging test cases.
- [287] arXiv:2512.22126 (cross-list from eess.SY) [pdf, other]
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Title: Validation methodology on real data of reversible Kalman Filter for state estimation with ManifoldSubjects: Systems and Control (eess.SY); Optimization and Control (math.OC)
This work extends a previous study that introduced an algorithm for state estimation on manifolds within the framework of the Kalman filter. Its objective is to address the limitations of the earlier approach. The reversible Kalman filter was designed to provide a methodology for evaluating the accuracy of existing Kalman filter variants with arbitrary precision on synthetic data. It has favorable numerical properties on synthetic data, achieving arbitrary precision without relying on the small-velocity assumption and depending only on sensor noise. However, its application to real data encountered difficulties related to measurement noise, which was mitigated using a heuristic. In particular, the heuristic involved an event detection step switching between reversible Kalman filter and classical Kalman variant at chosen moments. In the present work, we propose a study of this detection step and propose a methodology to prove at which moment the reversible Kalman approach improves on classical multiplicative variant. In particular, we propose a metric allowing one to discriminate situations in real-world scenarios where it behaves better than classical approach.
- [288] arXiv:2512.22246 (cross-list from q-bio.PE) [pdf, html, other]
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Title: A nonconservative kinetic framework with logistic growth for modeling the coexistence in a multi-species ecological systemSubjects: Populations and Evolution (q-bio.PE); Mathematical Physics (math-ph)
Kinetic theory frameworks are widely used for modeling stochastic interacting systems, where the evolution primarily depends on binary interactions. Recently, in this framework the action of the external force field has been introduction in order to gain a more realistic picture of some phenomena. In this paper, we introduce nonconservative kinetic equations where a particular shape external force field acts on the overall system. Then, this framework is used in an ecological context for modeling the evolution of a system composed of two species interacting with a prey-predator mechanism. The linear stability analysis concerned with the coexistence equilibrium point is provided, and a case where a Hopf bifurcations occurs is discussed. Finally, some relevant scenarios are numerically simulated.
- [289] arXiv:2512.22265 (cross-list from cond-mat.mtrl-sci) [pdf, html, other]
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Title: Representations of the symmetry groups of infinite crystalsComments: 27 pagesSubjects: Materials Science (cond-mat.mtrl-sci); Mathematical Physics (math-ph); Group Theory (math.GR)
We investigate the representations of the symmetry groups of infinite crystals. Crystal symmetries are usually described as the finite symmetry group of a finite crystal with periodic boundary conditions, for which the Brillouin zone is a finite set of points. However, to deal with the continuous crystal momentum $\mathbf{k}$ required to discuss the continuity, singularity or analyticity of band energies $\epsilon_n(\mathbf{k})$ and Bloch states $\psi_{\mathbf{k}}$, we need to consider infinite crystals. The symmetry groups of infinite crystals belong to the category of infinite non-compact groups, for which many standard tools of group theory break down. For example, character theory is no longer available for these groups and we use harmonic analysis to build the group algebra, the regular representation, the induction of irreducible representations of the crystallographic group from projective representations of the point groups and the decomposition of a representation into its irreducible parts. We deal with magnetic and non-magnetic groups in arbitrary dimensions. In the last part of the paper, we discuss Mackey's restriction of an induced representation to a subgroup, the tensor product of induced representations and the symmetric and antisymmetric squares of induced representations.
- [290] arXiv:2512.22276 (cross-list from nlin.PS) [pdf, html, other]
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Title: Discrete equations and auto-traveling kinks of the $ϕ^6$ modelComments: 7 pages, 2 figures, 38 referencesSubjects: Pattern Formation and Solitons (nlin.PS); Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph)
We study the $\phi^{6}$ model and derive two broad classes of lattice discretizations that admit static, translationally invariant kinks; that is, stationary kink profiles that can be centered at an arbitrary position relative to the lattice. These discretizations are constructed using a one-dimensional map, $\phi_{n+1}=F(\phi_{n})$, which provides a direct and systematic algorithm for generating such models. Numerical computations for two representative cases show that the discrete kinks do not possess internal modes, consistent with the continuum theory, although an additional high-frequency mode may appear above the phonon band. We also show that generic discretizations of the $\phi^{6}$ model do not support static kink solutions. Instead, the resulting dynamics produce auto-traveling and self-accelerating kinks that propagate at the maximal group velocity while continuously emitting radiation.
- [291] arXiv:2512.22282 (cross-list from stat.ML) [pdf, html, other]
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Title: A review of NMF, PLSA, LBA, EMA, and LCA with a focus on the identifiability issueSubjects: Machine Learning (stat.ML); Machine Learning (cs.LG); Optimization and Control (math.OC); Statistics Theory (math.ST)
Across fields such as machine learning, social science, geography, considerable attention has been given to models that factorize a nonnegative matrix into the product of two or three matrices, subject to nonnegative or row-sum-to-1 constraints. Although these models are to a large extend similar or even equivalent, they are presented under different names, and their similarity is not well known. This paper highlights similarities among five popular models, latent budget analysis (LBA), latent class analysis (LCA), end-member analysis (EMA), probabilistic latent semantic analysis (PLSA), and nonnegative matrix factorization (NMF). We focus on an essential issue-identifiability-of these models and prove that the solution of LBA, EMA, LCA, PLSA is unique if and only if the solution of NMF is unique. We also provide a brief review for algorithms of these models. We illustrate the models with a time budget dataset from social science, and end the paper with a discussion of closely related models such as archetypal analysis.
- [292] arXiv:2512.22357 (cross-list from nlin.SI) [pdf, html, other]
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Title: Dispersionless version of multi-component Pfaff-Toda hierarchyComments: 28 pages, no figuresSubjects: Exactly Solvable and Integrable Systems (nlin.SI); Mathematical Physics (math-ph)
We consider the dispersionless limit of the recently introduced multi-component Pfaff-Toda hierarchy. Its dispersionless version is a set of nonlinear differential equations for the dispersionless limit of logarithm of the tau-function (the F-function). They are obtained as limiting cases of bilinear equations of the Hirota-Miwa type. The analysis of the Pfaff-Toda hierarchy is substantially simplified by using the observation that the full (not only dispersionless) N-component Pfaff-Toda hierarchy is actually equivalent to the 2N-component DKP hierarchy. In the dispersionless limit, there is an elliptic curve built in the structure of the hierarchy, with the elliptic modular parameter being a dynamical variable. This curve can be uniformized by elliptic functions, and in the elliptic parametrization the hierarchy acquires a compact and especially nice form.
- [293] arXiv:2512.22388 (cross-list from cs.LG) [pdf, html, other]
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Title: BLISS: Bandit Layer Importance Sampling Strategy for Efficient Training of Graph Neural NetworksComments: Accepted for 5th Muslims in ML Workshop co-located with NeurIPS 2025. OpenReview: this https URL Code: this https URLSubjects: Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Social and Information Networks (cs.SI); Optimization and Control (math.OC); Machine Learning (stat.ML)
Graph Neural Networks (GNNs) are powerful tools for learning from graph-structured data, but their application to large graphs is hindered by computational costs. The need to process every neighbor for each node creates memory and computational bottlenecks. To address this, we introduce BLISS, a Bandit Layer Importance Sampling Strategy. It uses multi-armed bandits to dynamically select the most informative nodes at each layer, balancing exploration and exploitation to ensure comprehensive graph coverage. Unlike existing static sampling methods, BLISS adapts to evolving node importance, leading to more informed node selection and improved performance. It demonstrates versatility by integrating with both Graph Convolutional Networks (GCNs) and Graph Attention Networks (GATs), adapting its selection policy to their specific aggregation mechanisms. Experiments show that BLISS maintains or exceeds the accuracy of full-batch training.
- [294] arXiv:2512.22484 (cross-list from cs.RO) [pdf, other]
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Title: Asymmetric Friction in Geometric LocomotionComments: 23 pages, 15 figuresSubjects: Robotics (cs.RO); Differential Geometry (math.DG)
Geometric mechanics models of locomotion have provided insight into how robots and animals use environmental interactions to convert internal shape changes into displacement through the world, encoding this relationship in a ``motility map''. A key class of such motility maps arises from (possibly anisotropic) linear drag acting on the system's individual body parts, formally described via Riemannian metrics on the motions of the system's individual body parts. The motility map can then be generated by invoking a sub-Riemannian constraint on the aggregate system motion under which the position velocity induced by a given shape velocity is that which minimizes the power dissipated via friction. The locomotion of such systems is ``geometric'' in the sense that the final position reached by the system depends only on the sequence of shapes that the system passes through, but not on the rate with which the shape changes are made.
In this paper, we consider a far more general class of systems in which the drag may be not only anisotropic (with different coefficients for forward/backward and left/right motions), but also asymmetric (with different coefficients for forward and backward motions). Formally, including asymmetry in the friction replaces the Riemannian metrics on the body parts with Finsler metrics. We demonstrate that the sub-Riemannian approach to constructing the system motility map extends naturally to a sub-Finslerian approach and identify system properties analogous to the constraint curvature of sub-Riemannian systems that allow for the characterization of the system motion capabilities. - [295] arXiv:2512.22510 (cross-list from quant-ph) [pdf, html, other]
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Title: Quasi-harmonic spectra from branched HamiltoniansSubjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph); Exactly Solvable and Integrable Systems (nlin.SI)
We revisit the canonical quantization to assess the spectrum of the modified Emden equation $\ddot{x} + kx\dot{x} + \omega^2 x + \frac{k^2}{9}x^3 = 0$, which is an isochronous case of the Liénard-Kukles equation. While its classical isochronicity and canonical quantization, leading to polynomial solutions with an exactly-equispaced spectrum have been discussed earlier, including in the recent paper [Int. J. Theor. Phys. 64, 212 (2025)], the present study focuses on the quantization of its branched Hamiltonians. For small $k$, we show numerically that the resulting energy spectrum is no longer perfectly harmonic but only approximately equispaced, exhibiting quasi-harmonic behavior characterized by deviations from uniform spacing. Our numerical results are precisely validated by analytical calculations based on perturbation theory.
- [296] arXiv:2512.22542 (cross-list from quant-ph) [pdf, html, other]
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Title: Quantum preferential attachmentTingyu Zhao, Balázs Maga, Pierfrancesco Dionigi, Gergely Ódor, Kyle Soni, Anastasiya Salova, Bingjie Hao, Miklós Abért, István A. KovácsSubjects: Quantum Physics (quant-ph); Probability (math.PR); Adaptation and Self-Organizing Systems (nlin.AO)
The quantum internet is a rapidly developing technological reality, yet, it remains unclear what kind of quantum network structures might emerge. Since indirect quantum communication is already feasible and preserves absolute security of the communication channel, a new node joining the quantum network does not need to connect directly to its desired target. Instead, in our proposed quantum preferential attachment model, it uniformly randomly connects to any node within the proximity of the target, including, but not restricted to, the target itself. This local flexibility is found to qualitatively change the global network behavior, leading to two distinct classes of complex network architectures, both of which are small-world, but neither of which is scale-free. Our numerical findings are supported by rigorous analytic results, in a framework that incorporates quantum and classical variants of preferential attachment in a unified phase diagram. Besides quantum networks, we expect that our results will have broad implications for classical scenarios where there is flexibility in establishing new connections.
- [297] arXiv:2512.22622 (cross-list from cs.DM) [pdf, html, other]
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Title: Roman domination in weighted graphsSubjects: Discrete Mathematics (cs.DM); Combinatorics (math.CO)
A Roman dominating function for a (non-weighted) graph $G=(V,E)$, is a function $f:V\rightarrow \{0,1,2\}$ such that every vertex $u\in V$ with $f(u)=0$ has at least {one} neighbor $v\in V$ such that $f(v)=2$. The minimum weight $\sum_{v\in V}f(v)$ of a Roman {dominating function} $f$ on $G$ is called the Roman domination number of $G$ and is denoted by $\gamma_{R}(G)$. A graph {$G= (V,E)$} together with a positive real-valued weight-function $w:V\rightarrow \mathbf{R}^{>0}$ is called a {\it weighted graph} and is denoted by $(G;w)$. The minimum weight $\sum_{v\in V}f(v)w(v)$ of a Roman {dominating function} $f$ on $G$ is called the weighted Roman domination number of $G$ and is denoted by $\gamma_{wR}(G)$. The domination and Roman domination numbers of unweighted graphs have been extensively studied, particularly for their applications in bioinformatics and computational biology. However, graphs used to model biomolecular structures often require weights to be biologically meaningful. In this paper, we initiate the study of the weighted Roman domination number in weighted graphs. We first establish several bounds for this parameter and present various realizability results. Furthermore, we determine the exact values for several well-known graph families and demonstrate an equivalence between the weighted Roman domination number and the differential of a weighted graph.
- [298] arXiv:2512.22623 (cross-list from cs.LG) [pdf, html, other]
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Title: Communication Compression for Distributed Learning with Aggregate and Server-Guided FeedbackSubjects: Machine Learning (cs.LG); Signal Processing (eess.SP); Optimization and Control (math.OC)
Distributed learning, particularly Federated Learning (FL), faces a significant bottleneck in the communication cost, particularly the uplink transmission of client-to-server updates, which is often constrained by asymmetric bandwidth limits at the edge. Biased compression techniques are effective in practice, but require error feedback mechanisms to provide theoretical guarantees and to ensure convergence when compression is aggressive. Standard error feedback, however, relies on client-specific control variates, which violates user privacy and is incompatible with stateless clients common in large-scale FL. This paper proposes two novel frameworks that enable biased compression without client-side state or control variates. The first, Compressed Aggregate Feedback (CAFe), uses the globally aggregated update from the previous round as a shared control variate for all clients. The second, Server-Guided Compressed Aggregate Feedback (CAFe-S), extends this idea to scenarios where the server possesses a small private dataset; it generates a server-guided candidate update to be used as a more accurate predictor. We consider Distributed Gradient Descent (DGD) as a representative algorithm and analytically prove CAFe's superiority to Distributed Compressed Gradient Descent (DCGD) with biased compression in the non-convex regime with bounded gradient dissimilarity. We further prove that CAFe-S converges to a stationary point, with a rate that improves as the server's data become more representative. Experimental results in FL scenarios validate the superiority of our approaches over existing compression schemes.
- [299] arXiv:2512.22638 (cross-list from stat.ML) [pdf, html, other]
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Title: Likelihood-Preserving Embeddings for Statistical InferenceSubjects: Machine Learning (stat.ML); Machine Learning (cs.LG); Statistics Theory (math.ST); Methodology (stat.ME)
Modern machine learning embeddings provide powerful compression of high-dimensional data, yet they typically destroy the geometric structure required for classical likelihood-based statistical inference. This paper develops a rigorous theory of likelihood-preserving embeddings: learned representations that can replace raw data in likelihood-based workflows -- hypothesis testing, confidence interval construction, model selection -- without altering inferential conclusions. We introduce the Likelihood-Ratio Distortion metric $\Delta_n$, which measures the maximum error in log-likelihood ratios induced by an embedding. Our main theoretical contribution is the Hinge Theorem, which establishes that controlling $\Delta_n$ is necessary and sufficient for preserving inference. Specifically, if the distortion satisfies $\Delta_n = o_p(1)$, then (i) all likelihood-ratio based tests and Bayes factors are asymptotically preserved, and (ii) surrogate maximum likelihood estimators are asymptotically equivalent to full-data MLEs. We prove an impossibility result showing that universal likelihood preservation requires essentially invertible embeddings, motivating the need for model-class-specific guarantees. We then provide a constructive framework using neural networks as approximate sufficient statistics, deriving explicit bounds connecting training loss to inferential guarantees. Experiments on Gaussian and Cauchy distributions validate the sharp phase transition predicted by exponential family theory, and applications to distributed clinical inference demonstrate practical utility.
- [300] arXiv:2512.22644 (cross-list from physics.flu-dyn) [pdf, html, other]
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Title: On the Reynolds-number scaling of Poisson solver complexitySubjects: Fluid Dynamics (physics.flu-dyn); Mathematical Physics (math-ph); Computational Physics (physics.comp-ph)
We aim to answer the following question: is the complexity of numerically solving the Poisson equation increasing or decreasing for very large simulations of incompressible flows? Physical and numerical arguments are combined to derive power-law scalings at very high Reynolds numbers. A theoretical convergence analysis for both Jacobi and multigrid solvers defines a two-dimensional phase space divided into two regions depending on whether the number of solver iterations tends to decrease or increase with the Reynolds number. Numerical results indicate that, for Navier-Stokes turbulence, the complexity decreases with increasing Reynolds number, whereas for the one-dimensional Burgers equation it follows the opposite trend. The proposed theoretical framework thus provides a unified perspective on how solver convergence scales with the Reynolds number and offers valuable guidance for the development of next-generation preconditioning and multigrid strategies for extreme-scale simulations.
- [301] arXiv:2512.22676 (cross-list from eess.SP) [pdf, html, other]
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Title: Synthesis of signal processing algorithms with constraints on minimal parallelism and memory spaceComments: English translation of PhD thesis (Candidate of Physical and Mathematical Sciences), defended at Saint Petersburg State University (2017). 191 pagesSubjects: Signal Processing (eess.SP); Hardware Architecture (cs.AR); Distributed, Parallel, and Cluster Computing (cs.DC); Numerical Analysis (math.NA)
This thesis develops signal-processing algorithms and implementation schemes under constraints of minimal parallelism and memory space, with the goal of improving energy efficiency of low-power computing hardware. We propose (i) a power/energy consumption model for clocked CMOS logic that supports selecting optimal parallelism, (ii) integer-friendly approximation methods for elementary functions that reduce lookup-table size via constrained piecewise-polynomial (quasi-spline) constructions with accuracy guarantees, (iii) provably conflict-free data placement and execution order for mixed-radix streaming FFT on multi-bank and single-port memories, including a self-sorting FFT variant, and (iv) a parallelism/memory analysis of the fast Schur algorithm for superfast Toeplitz system solving, motivated by echo-cancellation workloads. The results provide constructive theorems, schedules, and design trade-offs enabling efficient specialized accelerators.
- [302] arXiv:2512.22697 (cross-list from econ.EM) [pdf, html, other]
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Title: Canonical correlation regression with noisy dataComments: 45 pages, 5 figuresSubjects: Econometrics (econ.EM); Statistics Theory (math.ST); Machine Learning (stat.ML)
We study instrumental variable regression in data rich environments. The goal is to estimate a linear model from many noisy covariates and many noisy instruments. Our key assumption is that true covariates and true instruments are repetitive, though possibly different in nature; they each reflect a few underlying factors, however those underlying factors may be misaligned. We analyze a family of estimators based on two stage least squares with spectral regularization: canonical correlations between covariates and instruments are learned in the first stage, which are used as regressors in the second stage. As a theoretical contribution, we derive upper and lower bounds on estimation error, proving optimality of the method with noisy data. As a practical contribution, we provide guidance on which types of spectral regularization to use in different regimes.
- [303] arXiv:2512.22704 (cross-list from gr-qc) [pdf, other]
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Title: Solving the constraint equation for general free dataSubjects: General Relativity and Quantum Cosmology (gr-qc); Analysis of PDEs (math.AP); Differential Geometry (math.DG)
We revisit the problem of solving the Einstein constraint equations in vacuum by a new method, which allows us to prescribe four scalar quantities, representing the full dynamical degrees of freedom of the constraint system. We show that once appropriate gauge conditions have been chosen and four scalars freely specified (modulo $\ell\leq 1$ modes), we can rewrite the constraint equations as a well-posed system of coupled transport and elliptic equations on $2$-spheres, which we solve by an iteration procedure. Our method provides a large class of exterior solutions of the constraint equations that can be matched to given interior solutions, according to the existing gluing techniques. As such, it can be applied to provide a large class of initial Cauchy data sets evolving to black holes, generalizing the well-known result of the formation of trapped surfaces due to Li and Yu. Though in our main theorem, we only specify conditions consistent with $g-g_{Schw}=O(r^{-1-\delta})$, $k=O(r^{-2-\delta})$, the method is flexible enough to be applied in many other situations. It can, in particular, be easily adapted to construct arbitrarily fast decaying data. We expect, moreover, that our method can also be applied to construct data with slower decay, such as that used by Shen. In fact, an important motivation for developing our method is to show that the result of Shen is sharp, i.e., construct small, smooth initial data sets which violate Shen's decay conditions, and for which the stability of the Minkowski space result is wrong.
- [304] arXiv:2512.22757 (cross-list from cs.RO) [pdf, html, other]
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Title: Active Constraint Learning in High Dimensions from DemonstrationsComments: Under review, 25 pages, 11 figuresSubjects: Robotics (cs.RO); Artificial Intelligence (cs.AI); Machine Learning (cs.LG); Systems and Control (eess.SY); Optimization and Control (math.OC)
We present an iterative active constraint learning (ACL) algorithm, within the learning from demonstrations (LfD) paradigm, which intelligently solicits informative demonstration trajectories for inferring an unknown constraint in the demonstrator's environment. Our approach iteratively trains a Gaussian process (GP) on the available demonstration dataset to represent the unknown constraints, uses the resulting GP posterior to query start/goal states, and generates informative demonstrations which are added to the dataset. Across simulation and hardware experiments using high-dimensional nonlinear dynamics and unknown nonlinear constraints, our method outperforms a baseline, random-sampling based method at accurately performing constraint inference from an iteratively generated set of sparse but informative demonstrations.
- [305] arXiv:2512.22846 (cross-list from econ.EM) [pdf, html, other]
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Title: Causal-Policy Forest for End-to-End Policy LearningSubjects: Econometrics (econ.EM); Machine Learning (cs.LG); Statistics Theory (math.ST); Methodology (stat.ME); Machine Learning (stat.ML)
This study proposes an end-to-end algorithm for policy learning in causal inference. We observe data consisting of covariates, treatment assignments, and outcomes, where only the outcome corresponding to the assigned treatment is observed. The goal of policy learning is to train a policy from the observed data, where a policy is a function that recommends an optimal treatment for each individual, to maximize the policy value. In this study, we first show that maximizing the policy value is equivalent to minimizing the mean squared error for the conditional average treatment effect (CATE) under $\{-1, 1\}$ restricted regression models. Based on this finding, we modify the causal forest, an end-to-end CATE estimation algorithm, for policy learning. We refer to our algorithm as the causal-policy forest. Our algorithm has three advantages. First, it is a simple modification of an existing, widely used CATE estimation method, therefore, it helps bridge the gap between policy learning and CATE estimation in practice. Second, while existing studies typically estimate nuisance parameters for policy learning as a separate task, our algorithm trains the policy in a more end-to-end manner. Third, as in standard decision trees and random forests, we train the models efficiently, avoiding computational intractability.
- [306] arXiv:2512.22851 (cross-list from cs.LO) [pdf, other]
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Title: Many-valued coalgebraic dynamic logics: Safety and strong completeness via reducibilitySubjects: Logic in Computer Science (cs.LO); Logic (math.LO)
We present a coalgebraic framework for studying generalisations of dynamic modal logics such as PDL and game logic in which both the propositions and the semantic structures can take values in an algebra $\mathbf{A}$ of truth-degrees. More precisely, we work with coalgebraic modal logic via $\mathbf{A}$-valued predicate liftings where $\mathbf{A}$ is a $\mathsf{FL}_{\mathrm{ew}}$-algebra, and interpret actions (abstracting programs and games) as $\mathsf{F}$-coalgebras where the functor $\mathsf{F}$ represents some type of $\mathbf{A}$-weighted system. We also allow combinations of crisp propositions with $\mathbf{A}$-weighted systems and vice versa. We introduce coalgebra operations and tests, with a focus on operations that are reducible in the sense that modalities for composed actions can be reduced to compositions of modalities for the constituent actions. We prove that reducible operations are safe for bisimulation and behavioural equivalence, and prove a general strong completeness result, from which we obtain new strong completeness results for $2$-valued iteration-free PDL with $\mathbf{A}$-valued accessibility relations when $\mathbf{A}$ is a finite chain, and for many-valued iteration-free game logic with many-valued strategies based on finite Lukasiewicz logic.
- [307] arXiv:2512.22863 (cross-list from quant-ph) [pdf, html, other]
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Title: A Counterexample to the Optimality Conjecture in Convex Quantum Channel OptimizationSubjects: Quantum Physics (quant-ph); Optimization and Control (math.OC)
This paper presents a counterexample to the optimality conjecture in convex quantum channel optimization proposed by Coutts et al. The conjecture posits that for nuclear norm minimization problems in quantum channel optimization, the dual certificate of an optimal solution can be uniquely determined via the spectral calculus of the Choi matrix. By constructing a counterexample in 2-dimensional Hilbert spaces, we disprove this conjecture.
- [308] arXiv:2512.22942 (cross-list from quant-ph) [pdf, html, other]
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Title: Random matrix prediction of average entanglement entropy in non-Abelian symmetry sectorsComments: 22 pages, 4 figures, 2 tablesSubjects: Quantum Physics (quant-ph); Statistical Mechanics (cond-mat.stat-mech); Mathematical Physics (math-ph)
We study the average bipartite entanglement entropy of Haar-random pure states in quantum many-body systems with global $\mathrm{SU}(2)$ symmetry, constrained to fixed total spin $J$ and magnetization $J_z = 0$. Focusing on spin-$\tfrac12$ lattices and subsystem fractions $f < \frac{1}{2}$, we derive a asymptotic expression for the average entanglement entropy up to constant order in the system volume $V$. In addition to the expected leading volume law term, we prove the existence of a $\frac{1}{2}\log V$ finite-size correction resulting from the scaling of the Clebsch-Gordon coefficients and compute explicitly the $O(1)$ contribution reflecting angular-momentum coupling within magnetization blocks. Our analysis uses features of random matrix ensembles and provides a fully analytical treatment for arbitrary spin densities, thereby extending Page type results to non-Abelian sectors and clarifying how $\mathrm{SU}(2)$ symmetry shapes average entanglement.
- [309] arXiv:2512.22975 (cross-list from cs.DS) [pdf, html, other]
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Title: Computing parameters that generalize interval graphs using restricted modular partitionsComments: 28 pages, 7 figures, 3 appendicesSubjects: Data Structures and Algorithms (cs.DS); Discrete Mathematics (cs.DM); Combinatorics (math.CO)
Recently, Lafond and Luo [MFCS 2023] defined the $\mathcal{G}$-modular cardinality of a graph $G$ as the minimum size of a partition of $V(G)$ into modules that belong to a graph class $\mathcal{G}$. We analyze the complexity of calculating parameters that generalize interval graphs when parameterized by the $\mathcal{G}$-modular cardinality, where $\mathcal{G}$ corresponds either to the class of interval graphs or to the union of complete graphs. Namely, we analyze the complexity of computing the thinness and the simultaneous interval number of a graph.
We present a linear kernel for the Thinness problem parameterized by the interval-modular cardinality and an FPT algorithm for Simultaneous Interval Number when parameterized by the cluster-modular cardinality plus the solution size. The interval-modular cardinality of a graph is not greater than the cluster-modular cardinality, which in turn generalizes the neighborhood diversity and the twin-cover number. Thus, our results imply a linear kernel for Thinness when parameterized by the neighborhood diversity of the input graph, FPT algorithms for Thinness when parameterized by the twin-cover number and vertex cover number, and FPT algorithms for Simultaneous Interval Number when parameterized by the neighborhood diversity plus the solution size, twin-cover number, and vertex cover number. To the best of our knowledge, prior to our work no parameterized algorithms (FPT or XP) for computing the thinness or the simultaneous interval number were known.
On the negative side, we observe that Thinness and Simultaneous Interval Number parameterized by treewidth, pathwidth, bandwidth, (linear) mim-width, clique-width, modular-width, or even the thinness or simultaneous interval number themselves, admit no polynomial kernels assuming NP $\not\subseteq$ coNP/poly. - [310] arXiv:2512.22997 (cross-list from hep-th) [pdf, html, other]
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Title: Generalised Entanglement Entropies from Unit-Invariant Singular Value DecompositionComments: 53 pages, 7 figuresSubjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Probability (math.PR); Quantum Physics (quant-ph)
We introduce generalisations of von Neumann entanglement entropy that are invariant with respect to certain scale transformations. These constructions are based on Unit-Invariant Singular Value Decomposition (UISVD) with its right-, left-, and bi-invariant incarnations, which itself are variations of the standard Singular Value Decomposition (SVD) that remain invariant under (appropriate set of) diagonal transformations. These measures are naturally defined for non-Hermitian or rectangular operators and remain useful when the input and output spaces possess different dimensions or metric weights. We apply the UISVD entropy and discuss its advantages in the physically interesting framework of Biorthogonal Quantum Mechanics, whose important aspect is indeed the behavior under scale transformations. Further, we illustrate features of UISVD-based entropies in other representative settings, from simple quantum mechanical bipartite states to random matrices relevant to quantum chaos and holography, and in the context of Chern-Simons theory. In all cases, the UISVD yields stable, physically meaningful entropic spectra that are invariant under rescalings and normalisations.
- [311] arXiv:2512.22998 (cross-list from quant-ph) [pdf, html, other]
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Title: Fast chiral resolution with optimal controlJournal-ref: Physical Review A 112, 023116 (2025)Subjects: Quantum Physics (quant-ph); Optimization and Control (math.OC)
In this work, we formulate the problem of achieving in minimum-time perfect chiral resolution with bounded control fields, as an optimal control problem on two non-interacting spins-$1/2$. We assume the same control bound for the two Raman fields (pump and Stokes) and a different bound for the field connecting directly the two lower-energy states. Using control theory, we show that the optimal fields can only take the boundary values or be zero, the latter corresponding to singular control. Subsequently, using numerical optimal control and intuitive arguments, we identify some three-stage symmetric optimal pulse-sequences, for relatively larger values of the ratio between the two control bounds, and analytically calculate the corresponding pulse timings as functions of this ratio. For smaller values of the bounds ratio, numerical optimal control indicates that the optimal pulse-sequence loses its symmetry and the number of stages increases in general. In all cases, the analytical or numerical optimal protocol achieves a faster perfect chiral resolution than other pulsed protocols, mainly because of the simultaneous action of the control fields. The present work is expected to be useful in the wide spectrum of applications across the natural sciences where enantiomer separation is a crucial task.
- [312] arXiv:2512.23002 (cross-list from nlin.CD) [pdf, html, other]
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Title: On the efficient numerical computation of covariant Lyapunov vectorsComments: 10 pages, 12 figuresSubjects: Chaotic Dynamics (nlin.CD); Mathematical Physics (math-ph); Dynamical Systems (math.DS)
Covariant Lyapunov vectors (CLVs) are useful in multiple applications, but the optimal time windows needed to accurately compute these vectors are yet unclear. To remedy this, we investigate two methods for determining when to safely terminate the forward and backward transient phases of the CLV computation algorithm by Ginelli et al.~\cite{GinelliEtAl2007} when applied to chaotic orbits of conservative Hamiltonian systems. We perform this investigation for two prototypical Hamiltonian systems, namely the well-known Hénon-Heiles system of two degrees of freedom and a system of three nonlinearly coupled harmonic oscillators having three degrees of freedom, finding very similar results for the two methods and thus recommending the more efficient one. We find that the accuracy of two-dimesnional center subspace computations is significantly reduced when the backward evolution stages of the algorithm are performed over long time intervals. We explain this observation by examining the tangent dynamics of the center subspace wherein CLVs tend to align/anti-align, and we propose an adaptation of the algorithm that improves the accuracy of such computations over long times by preventing this alignment/anti-alignment of CLVs in the center subspace.
- [313] arXiv:2512.23027 (cross-list from cs.CE) [pdf, html, other]
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Title: A Domain Decomposition-based Solver for Acoustic Wave propagation in Two-Dimensional Random MediaComments: 29 pages, 55 figures. Based on the author's thesis submitted to Carleton University (2023). This research was performed while the author was at the Department of Civil and Environmental Engineering, Carleton UniversitySubjects: Computational Engineering, Finance, and Science (cs.CE); Distributed, Parallel, and Cluster Computing (cs.DC); Numerical Analysis (math.NA)
An acoustic wave propagation problem with a log normal random field approximation for wave speed is solved using a sampling-free intrusive stochastic Galerkin approach. The stochastic partial differential equation with the inputs and outputs expanded using polynomial chaos expansion (PCE) is transformed into a set of deterministic PDEs and further to a system of linear equations. Domain decomposition (DD)-based solvers are utilized to handle the overwhelming computational cost for the resulting system with increasing mesh size, time step and number of random parameters. A conjugate gradient iterative solver with a two-level Neumann-Neumann preconditioner is applied here showing their efficient scalabilities.
- [314] arXiv:2512.23041 (cross-list from hep-th) [pdf, html, other]
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Title: The topological life of Dynkin indices: universal scaling and matter selectionComments: 32 pages + appendices + references, 3 tables, and a figureSubjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Representation Theory (math.RT)
For simple, simply-connected compact Lie groups, Dynkin embedding indices obey a universal scaling law with a direct topological meaning. Given an inclusion $f:G\hookrightarrow H$, the Dynkin embedding index $j_f$ is characterized equivalently by the induced maps on $\pi_3$ and on the canonical generators of $H^3$, $H^4(B{-})$, and $H^4(\Sigma{-})$. Consequently, $j_f$ controls instanton-number scaling, the quantization levels of Chern--Simons and Wess--Zumino--Witten terms, and the matching of gauge couplings and one-loop RG scales. We connect this picture to representation theory via the $\beta$-construction in topological $K$-theory, relating Dynkin indices to Chern characters through Harris' degree--$3$ formula and Naylor's suspended degree--$4$ refinement. Finally, we apply these results to F-theory to explain the prevalence of index-one matter: we propose a ``genericity heuristic'' where geometry favors regular embeddings (typically $j_f=1$) associated with minimal singularity enhancements, while higher-index embeddings require non-generic tuning.
- [315] arXiv:2512.23045 (cross-list from eess.SP) [pdf, html, other]
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Title: Flexible Intelligent Metasurface for Downlink Communications under Statistical CSIVaibhav Kumar, Anastasios Papazafeiropoulos, Pandelis Kourtessis, John Senior, Marwa Chafii, Dimitra I. Kaklamani, Iakovos S. VenierisComments: 5 pages, 4 figures, accepted in IEEE WCLSubjects: Signal Processing (eess.SP); Information Theory (cs.IT)
Flexible intelligent metasurface (FIM) is a recently developed, groundbreaking hardware technology with promising potential for 6G wireless systems. Unlike conventional rigid antenna array (RAA)-based transmitters, FIM-assisted transmitters can dynamically alter their physical surface through morphing, offering new degrees of freedom to enhance system performance. In this letter, we depart from prior works that rely on instantaneous channel state information (CSI) and instead address the problem of average sum spectral efficiency maximization under statistical CSI in a FIM-assisted downlink multiuser multiple-input single-output setting. To this end, we first derive the spatial correlation matrix for the FIM-aided transmitter and then propose an iterative FIM optimization algorithm based on the gradient projection method. Simulation results show that with statistical CSI, the FIM-aided system provides a significant performance gain over its RAA-based counterpart in scenarios with strong spatial channel correlation, whereas the gain diminishes when the channels are weakly correlated.
- [316] arXiv:2512.23075 (cross-list from cs.LG) [pdf, html, other]
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Title: Trust Region Masking for Long-Horizon LLM Reinforcement LearningSubjects: Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Information Theory (cs.IT); Machine Learning (stat.ML)
Policy gradient methods for large language models optimize a surrogate objective computed from samples of a rollout policy $\pi_{\text{roll}}$. When $\pi_{\text{roll}} \ne \pi_{\theta}$, there is approximation error between the surrogate and the true objective. Prior work has shown that this off-policy mismatch is unavoidable in modern LLM-RL due to implementation divergence, mixture-of-experts routing discontinuities, and distributed training staleness. Classical trust region bounds on the resulting error scale as $O(T^2)$ with sequence length $T$, rendering them vacuous for long-horizon tasks. We derive two tighter bounds: a Pinsker-Marginal bound scaling as $O(T^{3/2})$ and a Mixed bound scaling as $O(T)$. Crucially, both bounds depend on $D_{kl}^{tok,max}$ -- the maximum token-level KL divergence across all positions in a sequence. This is inherently a sequence-level quantity: it requires examining the entire trajectory to compute, and therefore cannot be controlled by token-independent methods like PPO clipping. We propose Trust Region Masking (TRM), which excludes entire sequences from gradient computation if any token violates the trust region, providing the first non-vacuous monotonic improvement guarantees for long-horizon LLM-RL.
- [317] arXiv:2512.23095 (cross-list from hep-th) [pdf, html, other]
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Title: Torus Knots in Adjoint RepresentationComments: 7 pagesSubjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph); Geometric Topology (math.GT)
We derive a closed-form expression for the adjoint polynomials of torus knots and investigate their special properties. The results are presented in the very explicit double sum form and provide a deeper insight into the structure of adjoint invariants essential for the Vogel's universality of Chern-Simons theory.
- [318] arXiv:2512.23100 (cross-list from hep-th) [pdf, html, other]
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Title: Phase Space Formulation of S-matrixComments: 62 pages, 5 figuresSubjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
We establish an exact relation between the S-symplectomorphism and the S-matrix by means of the phase space formulation of quantum mechanics. The adjoint action of the S-matrix defines a fuzzy diffeomorphism on phase space whose classical limit is the S-symplectomorphism. The relation between classical and quantum eikonals is immediate via $\hbar$-deformation of each Poisson bracket in the Magnus formula. Diagrammatic computation of quantum eikonal is illustrated for quantizations in both symmetric and normal orderings.
- [319] arXiv:2512.23104 (cross-list from gr-qc) [pdf, html, other]
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Title: Scalar-Field Wave Dynamics and Quasinormal Modes of the Teo Rotating WormholeSubjects: General Relativity and Quantum Cosmology (gr-qc); Mathematical Physics (math-ph)
We analyze scalar field perturbations of the rotating Teo wormhole and compute its quasinormal mode (QNM) spectrum using WKB methods in a fully horizonless geometry. The Klein Gordon equation separates and yields a Schrödinger type radial equation with a single, smooth potential barrier shaped by the localized frame dragging profile of the wormhole throat. This barrier supports damped oscillatory modes across the full spin range examined. The resulting QNM spectrum exhibits a coherent and monotonic dependence on rotation. As the spin increases, both the oscillation frequency and the damping rate decrease, indicating progressively longer-lived modes in the absence of horizon induced absorption. In the eikonal limit, we extract the photon-ring radius, orbital frequency, and Lyapunov exponent, and verify the standard QNM-Eikonal correspondence. Comparison with Kerr black holes reveals qualitative differences. Whereas Kerr QNMs are governed by horizon absorption and exhibit symmetric prograde/retrograde mode splitting, the Teo wormhole displays a stronger but spatially confined spin response, partial reflection at the throat, and a distinctive one-sided splitting that saturates rapidly with increasing spin. Although the rotating Teo wormhole admits an ergoregion and superradiant compatible frequency kinematics, the absence of an event horizon or dissipative boundary prevents classical superradiant amplification. These results demonstrate how rotation and boundary conditions jointly shape wave propagation in horizonless compact objects and provide characteristic spectral signatures distinguishing rotating wormholes from Kerr black holes.
- [320] arXiv:2512.23122 (cross-list from hep-th) [pdf, other]
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Title: Torus Knots and Minimal Models Revisited : Rational VOA characters from non-hyperbolic knotsComments: 51 pages (plus appendices)Subjects: High Energy Physics - Theory (hep-th); Geometric Topology (math.GT); Number Theory (math.NT)
In 2003, Hikami and Kirillov uncovered an intriguing connection between torus knots $\mathcal{K}_{(P,Q)}$ and Virasoro minimal models $\mathcal{M}(P,Q)$ by relating the Kashaev invariants of the knots to the characters of the corresponding minimal models. In this work, we recover and extend this connection by combining the 3D--3D correspondence with a bulk--boundary correspondence. More concretely, we study the 3D $\mathcal{N}=2$ gauge theories associated with torus-knot complements via the Dimofte--Gaiotto--Gukov construction and show that, in the infrared, these theories either flow to a unitary TQFT (when $|P-Q| = 1$), whose boundary chiral algebra reproduces that of the associated unitary minimal model, or to a 3D $\mathcal{N}=4$ rank-0 SCFT (when $|P-Q| > 1$), which realizes the corresponding non-unitary chiral minimal model at the boundary after an appropriate topological twist. This framework yields new Nahm-sum-like expressions for the characters of Virasoro minimal models and other related rational conformal field theories, providing a systematic algorithm for constructing characters of rational VOAs directly from the combinatorial data of an ideal triangulation of a non-hyperbolic knot complement.
- [321] arXiv:2512.23139 (cross-list from q-fin.MF) [pdf, html, other]
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Title: Lambda Expected ShortfallSubjects: Mathematical Finance (q-fin.MF); Probability (math.PR); Risk Management (q-fin.RM)
The Lambda Value-at-Risk (Lambda$-VaR) is a generalization of the Value-at-Risk (VaR), which has been actively studied in quantitative finance. Over the past two decades, the Expected Shortfall (ES) has become one of the most important risk measures alongside VaR because of its various desirable properties in the practice of optimization, risk management, and financial regulation. Analogously to the intimate relation between ES and VaR, we introduce the Lambda Expected Shortfall (Lambda-ES), as a generalization of ES and a counterpart to Lambda-VaR. Our definition of Lambda-ES has an explicit formula and many convenient properties, and we show that it is the smallest quasi-convex and law-invariant risk measure dominating Lambda-VaR under mild assumptions. We examine further properties of Lambda-ES, its dual representation, and related optimization problems.
- [322] arXiv:2512.23152 (cross-list from eess.SP) [pdf, html, other]
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Title: Unscented and Higher-Order Linear Covariance Fidelity Checks and Measures of Non-GaussianitySubjects: Signal Processing (eess.SP); Probability (math.PR)
Linear covariance (LinCov) techniques have gained widespread traction in the modeling of uncertainty, including in the preliminary study of spacecraft navigation performance. While LinCov methods offer improved computational efficiency compared to Monte Carlo based uncertainty analysis, they inherently rely on linearization approximations. Understanding the fidelity of these approximations and identifying when they are deficient is critically important for spacecraft navigation and mission planning, especially when dealing with highly nonlinear systems and large state uncertainties. This work presents a number of computational techniques for assessing linear covariance performance. These new LinCov fidelity measures are formulated using higher-order statistics, constrained optimization, and the unscented transform.
- [323] arXiv:2512.23190 (cross-list from cs.LG) [pdf, html, other]
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Title: A Simple, Optimal and Efficient Algorithm for Online Exp-Concave OptimizationSubjects: Machine Learning (cs.LG); Optimization and Control (math.OC); Machine Learning (stat.ML)
Online eXp-concave Optimization (OXO) is a fundamental problem in online learning. The standard algorithm, Online Newton Step (ONS), balances statistical optimality and computational practicality, guaranteeing an optimal regret of $O(d \log T)$, where $d$ is the dimension and $T$ is the time horizon. ONS faces a computational bottleneck due to the Mahalanobis projections at each round. This step costs $\Omega(d^\omega)$ arithmetic operations for bounded domains, even for the unit ball, where $\omega \in (2,3]$ is the matrix-multiplication exponent. As a result, the total runtime can reach $\tilde{O}(d^\omega T)$, particularly when iterates frequently oscillate near the domain boundary. For Stochastic eXp-concave Optimization (SXO), computational cost is also a challenge. Deploying ONS with online-to-batch conversion for SXO requires $T = \tilde{O}(d/\epsilon)$ rounds to achieve an excess risk of $\epsilon$, and thereby necessitates an $\tilde{O}(d^{\omega+1}/\epsilon)$ runtime. A COLT'13 open problem posed by Koren [2013] asks for an SXO algorithm with runtime less than $\tilde{O}(d^{\omega+1}/\epsilon)$.
This paper proposes a simple variant of ONS, LightONS, which reduces the total runtime to $O(d^2 T + d^\omega \sqrt{T \log T})$ while preserving the optimal $O(d \log T)$ regret. LightONS implies an SXO method with runtime $\tilde{O}(d^3/\epsilon)$, thereby answering the open problem. Importantly, LightONS preserves the elegant structure of ONS by leveraging domain-conversion techniques from parameter-free online learning to introduce a hysteresis mechanism that delays expensive Mahalanobis projections until necessary. This design enables LightONS to serve as an efficient plug-in replacement of ONS in broader scenarios, even beyond regret minimization, including gradient-norm adaptive regret, parametric stochastic bandits, and memory-efficient online learning. - [324] arXiv:2512.23211 (cross-list from econ.EM) [pdf, html, other]
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Title: Nonparametric Identification of Demand without Exogenous Product CharacteristicsSubjects: Econometrics (econ.EM); Statistics Theory (math.ST); Methodology (stat.ME)
We study the identification of differentiated product demand with exogenous supply-side instruments, allowing product characteristics to be endogenous. Past analyses have argued that exogenous characteristic-based instruments are essentially necessary given a sufficiently flexible demand model with a suitable index restriction. We show, however, that price counterfactuals are nonparametrically identified by recentered instruments -- which combine exogenous shocks to prices with endogenous product characteristics -- under a weaker index restriction and a new condition we term faithfulness. We argue that faithfulness, like the usual completeness condition for nonparametric identification with instruments, can be viewed as a technical requirement on the richness of identifying variation rather than a substantive economic restriction, and we show that it holds under a variety of non-nested conditions on either price-setting or the index.
- [325] arXiv:2512.23283 (cross-list from gr-qc) [pdf, html, other]
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Title: Motion of extended fluid bodies in the Newtonian limit of $f(R)$ gravityComments: 41 pages, 0 figuresSubjects: General Relativity and Quantum Cosmology (gr-qc); Mathematical Physics (math-ph)
In the Newtonian limit of $f(R)$ gravity, for an isolated self-gravitating system consisting of $N$ extended fluid bodies, the inter-body dynamics are studied by applying the symmetric and trace-free formalism in terms of irreducible Cartesian tensors. The multipole expansion of each body's center-of-mass acceleration is derived, and the expansion comprises the Coulomb-type part and the Yukawa-type part, where the former, identical to that in General Relativity, is encoded by the products of the mass multipole moments of the body with those of other bodies, and the latter, as the modification introduced by $f(R)$ gravity, is encoded by the products of the scalar multipole moments of the body with those of other bodies. As an essential component of the system's orbital dynamics, the multipole expansion for the total gravitational potential energy is provided, and the expression for the total conserved energy in terms of the mass and scalar multipole moments of the bodies is offered. To investigate the system's spin dynamics, the equation of motion for each body's spin angular momentum is further deduced and presented in the form of multipole expansion. These findings constitute the main content of the coarse-grained description of inter-body dynamics for the system within the framework of the Newtonian limit of $f(R)$ gravity. As a by-product, for a two-body system, the effective one-body equation governing the relative motion between the two bodies and the total energy of this system are achieved.
- [326] arXiv:2512.23284 (cross-list from eess.SY) [pdf, html, other]
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Title: Revealing design archetypes and flexibility in e-molecule import pathways using Modeling to Generate Alternatives and interpretable machine learningSubjects: Systems and Control (eess.SY); Machine Learning (cs.LG); Optimization and Control (math.OC)
Given the central role of green e-molecule imports in the European energy transition, many studies optimize import pathways and identify a single cost-optimal solution. However, cost optimality is fragile, as real-world implementation depends on regulatory, spatial, and stakeholder constraints that are difficult to represent in optimization models and can render cost-optimal designs infeasible. To address this limitation, we generate a diverse set of near-cost-optimal alternatives within an acceptable cost margin using Modeling to Generate Alternatives, accounting for unmodeled uncertainties. Interpretable machine learning is then applied to extract insights from the resulting solution space. The approach is applied to hydrogen import pathways considering hydrogen, ammonia, methane, and methanol as carriers. Results reveal a broad near-optimal space with great flexibility: solar, wind, and storage are not strictly required to remain within 10% of the cost optimum. Wind constraints favor solar-storage methanol pathways, while limited storage favors wind-based ammonia or methane pathways.
- [327] arXiv:2512.23355 (cross-list from cs.SI) [pdf, html, other]
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Title: A new adaptive two-layer model for opinion spread in hypergraphs: parameter sensitivity and estimationComments: 21 pages, 12 figuresSubjects: Social and Information Networks (cs.SI); Probability (math.PR)
When opinion spread is studied, peer pressure is often modeled by interactions of more than two individuals (higher-order interactions). In our work, we introduce a two-layer random hypergraph model, in which hyperedges represent households and workplaces. Within this overlapping, adaptive structure, individuals react if their opinion is in majority in their groups. The process evolves through random steps: individuals can either change their opinion, or quit their workplace and join another one in which their opinion belongs to the majority. Based on computer simulations, our first goal is to describe the effect of the parameters responsible for the probability of changing opinion and quitting workplace on the homophily and speed of polarization. We also analyze the model as a Markov chain, and study the frequency of the absorbing states. Then, we quantitatively compare how different statistical and machine learning methods, in particular, linear regression, xgboost and a convolutional neural network perform for estimating these probabilities, based on partial information from the process, for example, the distribution of opinion configurations within households and workplaces. Among other observations, we conclude that all methods can achieve the best results under appropriate circumstances, and that the amount of information that is necessary to provide good results depends on the strength of the peer pressure effect.
- [328] arXiv:2512.23358 (cross-list from physics.comp-ph) [pdf, html, other]
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Title: A space-time extension of a conservative two-fluid cut-cell diffusion method for moving geometriesComments: 25 pages, 11 figuresSubjects: Computational Physics (physics.comp-ph); Numerical Analysis (math.NA)
We present a space-time extension of a conservative Cartesian cut-cell finite-volume method for two-phase diffusion in prescribed-motion geometries. The formulation follows a two-fluid approach: one scalar field is solved in each phase with discontinuous material properties, coupled by sharp interface conditions enforcing flux continuity and jump laws. To handle moving boundaries on a fixed Cartesian grid, the discrete balance is written over phase-restricted space-time control volumes, whose geometric moments (swept volumes and apertures) are used as weights in the finite-volume operators. This construction naturally accounts for the creation and destruction of cut cells (fresh/dead-cell events) and yields strict discrete conservation. The resulting scheme retains the algebraic structure of the static cut-cell formulation while incorporating motion through local geometric weights and interface coupling operators. A series of verification and validation tests in two and three dimensions demonstrate super-linear accuracy in space, robust behavior under repeated topology changes and conservation across strong coefficient jumps and moving interfaces. The proposed space-time cut-cell framework provides a conservative building block for multiphase transport in evolving geometries and a foundation for future free-boundary extensions such as Stefan-type phase change.
- [329] arXiv:2512.23395 (cross-list from stat.ME) [pdf, html, other]
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Title: Intrinsic Whittle--Matérn fields and sparse spatial extremesSubjects: Methodology (stat.ME); Probability (math.PR); Statistics Theory (math.ST); Computation (stat.CO)
Intrinsic Gaussian fields are used in many areas of statistics as models for spatial or spatio-temporal dependence, or as priors for latent variables. However, there are two major gaps in the literature: first, the number and flexibility of existing intrinsic models are very limited; second, theory, fast inference, and software are currently underdeveloped for intrinsic fields. We tackle these challenges by introducing the new flexible class of intrinsic Whittle--Matérn Gaussian random fields obtained as the solution to a stochastic partial differential equation (SPDE). Exploiting sparsity resulting from finite-element approximations, we develop fast estimation and simulation methods for these models. We demonstrate the benefits of this intrinsic SPDE approach for the important task of kriging under extrapolation settings. Leveraging the connection of intrinsic fields to spatial extreme value processes, we translate our theory to an SPDE approach for Brown--Resnick processes for sparse modeling of spatial extreme events. This new paradigm paves the way for efficient inference in unprecedented dimensions. To demonstrate the wide applicability of our new methodology, we apply it in two very different areas: a longitudinal study of renal function data, and the modeling of marine heat waves using high-resolution sea surface temperature data.
- [330] arXiv:2512.23409 (cross-list from econ.TH) [pdf, html, other]
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Title: Axiomatic Foundations of Bayesian PersuasionSubjects: Theoretical Economics (econ.TH); Computer Science and Game Theory (cs.GT); Information Theory (cs.IT)
In this paper, we study axiomatic foundations of Bayesian persuasion, where a principal (i.e., sender) delegates the task of choice making after informing a biased agent (i.e., receiver) about the payoff relevant uncertain state (see, e.g., Kamenica and Gentzkow (2011)). Our characterizations involve novel models of Bayesian persuasion, where the principal can steer the agent's bias after acquiring costly information. Importantly, we provide an elicitation method using only observable menu-choice data of the principal, which shows how to construct the principal's subjective costs of acquiring information even when he anticipates managing the agent's bias.
- [331] arXiv:2512.23509 (cross-list from cs.LO) [pdf, other]
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Title: Modelling of logical systems by means of their fragmentsComments: Doctor of Sciences dissertation. In RussianSubjects: Logic in Computer Science (cs.LO); Logic (math.LO)
This work investigates the algorithmic complexity of non-classical logics, focusing on superintuitionistic and modal systems. It is shown that propositional logics are usually polynomial-time reducible to their fragments with at most two variables (often to the one-variable or even variable-free fragments). Also, it is proved that predicate logics are usually reducible to their fragments with one or two unary predicate letters and two or three individual variables. The work describes conditions sufficient for such reductions and provides examples where they fail, establishing non-reducibility in those cases. Furthermore, the work provides new complexity bounds for several logics, results on Kripke-incompleteness of predicate calculi, and analogues of the classical theorems of Church and Trakhtenbrot for the logic of quasiary predicates.
- [332] arXiv:2512.23589 (cross-list from cond-mat.mtrl-sci) [pdf, other]
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Title: The Fundamental Lemma of Altermagnetism: Emergence of AlterferrimagnetismComments: Chanchal K. Barman and Bishal Das contributed equally to this work. 38 pages (27 pages main, 11 pages supplement), 17 figures (11 figures main, 6 figures supplement), 2 tables (all in main)Subjects: Materials Science (cond-mat.mtrl-sci); Strongly Correlated Electrons (cond-mat.str-el); Mathematical Physics (math-ph); Computational Physics (physics.comp-ph); Quantum Physics (quant-ph)
Recent years have seen a proliferation in investigations on Altermagnetism due to its exciting prospects both from an applications perspective and theoretical standpoint. Traditionally, altermagnets are distinguished from collinear antiferromagnets using the central concept of halving subgroups within the spin space group formalism. In this work, we propose the Fundamental Lemma of Altermagnetism (FLAM) deriving the exact conditions required for the existence of altermagnetic phase in a magnetic material on the basis of site-symmetry groups and halving subgroups for a given crystallographic space group. The spin group formalism further clubs ferrimagnetism with ferromagnetism since the same-spin and opposite-spin sublattices lose their meaning in the presence of multiple magnetic species. As a consequence of FLAM, we further propose a class of fully compensated ferrimagnets, termed as Alterferrimagnets (AFiMs), which can show alternating momentum-dependent spin-polarized non-relativistic electronic bands within the first Brillouin zone. We show that alterferrimagnetism is a generalization of traditional collinear altermagnetism where multiple magnetic species are allowed to coexist forming fully compensated magnetic-sublattices, each with individual up-spin and down-spin sublattices.
- [333] arXiv:2512.23617 (cross-list from cs.LG) [pdf, html, other]
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Title: Le Cam Distortion: A Decision-Theoretic Framework for Robust Transfer LearningSubjects: Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Statistics Theory (math.ST); Methodology (stat.ME); Machine Learning (stat.ML)
Distribution shift is the defining challenge of real-world machine learning. The dominant paradigm--Unsupervised Domain Adaptation (UDA)--enforces feature invariance, aligning source and target representations via symmetric divergence minimization [Ganin et al., 2016]. We demonstrate that this approach is fundamentally flawed: when domains are unequally informative (e.g., high-quality vs degraded sensors), strict invariance necessitates information destruction, causing "negative transfer" that can be catastrophic in safety-critical applications [Wang et al., 2019].
We propose a decision-theoretic framework grounded in Le Cam's theory of statistical experiments [Le Cam, 1986], using constructive approximations to replace symmetric invariance with directional simulability. We introduce Le Cam Distortion, quantified by the Deficiency Distance $\delta(E_1, E_2)$, as a rigorous upper bound for transfer risk conditional on simulability. Our framework enables transfer without source degradation by learning a kernel that simulates the target from the source. Across five experiments (genomics, vision, reinforcement learning), Le Cam Distortion achieves: (1) near-perfect frequency estimation in HLA genomics (correlation $r=0.999$, matching classical methods), (2) zero source utility loss in CIFAR-10 image classification (81.2% accuracy preserved vs 34.7% drop for CycleGAN), and (3) safe policy transfer in RL control where invariance-based methods suffer catastrophic collapse. Le Cam Distortion provides the first principled framework for risk-controlled transfer learning in domains where negative transfer is unacceptable: medical imaging, autonomous systems, and precision medicine. - [334] arXiv:2512.23619 (cross-list from cs.RO) [pdf, html, other]
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Title: The N-5 Scaling Law: Topological Dimensionality Reduction in the Optimal Design of Fully-actuated MultirotorsSubjects: Robotics (cs.RO); Geometric Topology (math.GT); Optimization and Control (math.OC)
The geometric design of fully-actuated and omnidirectional N-rotor aerial vehicles is conventionally formulated as a parametric optimization problem, seeking a single optimal set of N orientations within a fixed architectural family. This work departs from that paradigm to investigate the intrinsic topological structure of the optimization landscape itself. We formulate the design problem on the product manifold of Projective Lines \RP^2^N, fixing the rotor positions to the vertices of polyhedral chassis while varying their lines of action. By minimizing a coordinate-invariant Log-Volume isotropy metric, we reveal that the topology of the global optima is governed strictly by the symmetry of the chassis. For generic (irregular) vertex arrangements, the solutions appear as a discrete set of isolated points. However, as the chassis geometry approaches regularity, the solution space undergoes a critical phase transition, collapsing onto an N-dimensional Torus of the lines tangent at the vertexes to the circumscribing sphere of the chassis, and subsequently reducing to continuous 1-dimensional curves driven by Affine Phase Locking. We synthesize these observations into the N-5 Scaling Law: an empirical relationship holding for all examined regular planar polygons and Platonic solids (N <= 10), where the space of optimal configurations consists of K=N-5 disconnected 1D topological branches. We demonstrate that these locking patterns correspond to a sequence of admissible Star Polygons {N/q}, allowing for the exact prediction of optimal phases for arbitrary N. Crucially, this topology reveals a design redundancy that enables optimality-preserving morphing: the vehicle can continuously reconfigure along these branches while preserving optimal isotropic control authority.
- [335] arXiv:2512.23671 (cross-list from stat.ML) [pdf, html, other]
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Title: Calibrated Multi-Level Quantile ForecastingSubjects: Machine Learning (stat.ML); Machine Learning (cs.LG); Optimization and Control (math.OC); Methodology (stat.ME)
We present an online method for guaranteeing calibration of quantile forecasts at multiple quantile levels simultaneously. A sequence of $\alpha$-level quantile forecasts is calibrated if the forecasts are larger than the target value at an $\alpha$-fraction of time steps. We introduce a lightweight method called Multi-Level Quantile Tracker (MultiQT) that wraps around any existing point or quantile forecaster to produce corrected forecasts guaranteed to achieve calibration, even against adversarial distribution shifts, while ensuring that the forecasts are ordered -- e.g., the 0.5-level quantile forecast is never larger than the 0.6-level forecast. Furthermore, the method comes with a no-regret guarantee that implies it will not worsen the performance of an existing forecaster, asymptotically, with respect to the quantile loss. In experiments, we find that MultiQT significantly improves the calibration of real forecasters in epidemic and energy forecasting problems.
- [336] arXiv:2512.23680 (cross-list from cs.CC) [pdf, html, other]
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Title: Coloring Hardness on Low Twin-Width GraphsComments: 12 pages, 4 figuresSubjects: Computational Complexity (cs.CC); Discrete Mathematics (cs.DM); Data Structures and Algorithms (cs.DS); Combinatorics (math.CO)
As the class $\mathcal T_4$ of graphs of twin-width at most 4 contains every finite subgraph of the infinite grid and every graph obtained by subdividing each edge of an $n$-vertex graph at least $2 \log n$ times, most NP-hard graph problems, like Max Independent Set, Dominating Set, Hamiltonian Cycle, remain so on $\mathcal T_4$. However, Min Coloring and k-Coloring are easy on both families because they are 2-colorable and 3-colorable, respectively.
We show that Min Coloring is NP-hard on the class $\mathcal T_3$ of graphs of twin-width at most 3. This is the first hardness result on $\mathcal T_3$ for a problem that is easy on cographs (twin-width 0), on trees (whose twin-width is at most 2), and on unit circular-arc graphs (whose twin-width is at most 3). We also show that for every $k \geqslant 3$, k-Coloring is NP-hard on $\mathcal T_4$. We finally make two observations: (1) there are currently very few problems known to be in P on $\mathcal T_d$ (graphs of twin-width at most $d$) and NP-hard on $\mathcal T_{d+1}$ for some nonnegative integer $d$, and (2) unlike $\mathcal T_4$, which contains every graph as an induced minor, the class $\mathcal T_3$ excludes a fixed planar graph as an induced minor; thus it may be viewed as a special case (or potential counterexample) for conjectures about classes excluding a (planar) induced minor. These observations are accompanied by several open questions. - [337] arXiv:2512.23687 (cross-list from cs.DS) [pdf, html, other]
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Title: The Minimum Subgraph Complementation ProblemSubjects: Data Structures and Algorithms (cs.DS); Combinatorics (math.CO)
Subgraph complementation is an operation that toggles all adjacencies inside a selected vertex set. Given a graph \(G\) and a target class \(\mathcal{C}\), the Minimum Subgraph Complementation problem asks for a minimum-size vertex set \(S\) such that complementing the subgraph induced by \(S\) transforms \(G\) into a graph belonging to \(\mathcal{C}\). While the decision version of Subgraph Complementation has been extensively studied and is NP-complete for many graph classes, the algorithmic complexity of its optimization variant has remained largely unexplored.
In this paper, we study MSC from an algorithmic perspective. We present polynomial-time algorithms for MSC in several nontrivial settings. Our results include polynomial-time solvability for transforming graphs between bipartite, co-bipartite, and split graphs, as well as for complementing bipartite regular graphs into chordal graphs. We also show that MSC to the class of graphs of fixed degeneracy can be solved in polynomial time when the input graph is a forest. Moreover, we investigate MSC with respect to connectivity and prove that MSC to the class of disconnected graphs and to the class of 2-connected graphs can be solved in polynomial time for arbitrary inputs.
Cross submissions (showing 52 of 52 entries)
- [338] arXiv:0908.3671 (replaced) [pdf, other]
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Title: A Short Note on Disjointness Conditions for Triples of Group Subsets Satisfying the Triple Product PropertyComments: Not significant enoughSubjects: Group Theory (math.GR); Combinatorics (math.CO)
We deduce some elementary pairwise disjointness and semi-disjointness conditions on triples of subsets in arbitrary groups satisfying the so-called triple product property (TPP) as originally defined by H. Cohn and C. Umans in 2003. This property TPP for a triple of group subsets, called a TPP triple, allows the group to "realize" matrix multiplication of dimensions the sizes of the subsets, with the subsets acting as indexing sets for input matrices which are embedded into the regular algebra of the group. We derive nine different disjointness casetypes for an arbitrary TPP triple, and classify these into four different disjointness classes based on an integer measure of the degree of pairwise disjointness among the subsets. Finally, we derive lower and upper bounds for the sum of sizes of the subsets forming a TPP triple, which is the additive equivalent of the multiplicative bounds originally derived by Cohn and Umans for the product of sizes of subsets forming a TPP triple.
- [339] arXiv:1005.0270 (replaced) [pdf, html, other]
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Title: Ground state representations of loop algebrasComments: v3: minor correction. 22 pages, no figureJournal-ref: Ann. Henri Poincare Vol. 12, No. 4 (2011), 805-827Subjects: Representation Theory (math.RT); Mathematical Physics (math-ph)
Let g be a simple Lie algebra, Lg be the loop algebra of g. Fixing a point in S^1 and identifying the real line with the punctured circle, we consider the subalgebra Sg of Lg of rapidly decreasing elements on R. We classify the translation-invariant 2-cocycles on Sg. We show that the ground state representation of Sg is unique for each cocycle. These ground states correspond precisely to the vacuum representations of Lg.
- [340] arXiv:2009.14814 (replaced) [pdf, html, other]
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Title: Dependence Balance and Capacity Bounds for Multiterminal Communication and Wiretap ChannelsSubjects: Information Theory (cs.IT)
An information measure based on fractional partitions of a set is used to develop a general dependence balance inequality for communication. This inequality is used to obtain new upper bounds on reliable and secret rates for multiterminal channels. For example, we obtain a new upper bound on the rate of shared randomness generated among terminals, a counterpart of the cut-set bound for reliable communication. The bounds for reliable communication utilize the concept of auxiliary receivers, and we show the bounds are optimized by Gaussian distributions for Gaussian channels. The bounds are applied to multiaccess channels with generalized feedback and relay channels, and improve the cut-set bound for scalar Gaussian channels. The improvement for Gaussian relay channels complements results obtained with other methods.
- [341] arXiv:2111.01060 (replaced) [pdf, other]
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Title: Exponential Lower Bounds for Locally Decodable and Correctable Codes for Insertions and DeletionsComments: Accepted to the 62nd Annual Symposium on Foundations of Computer Science (FOCS)Subjects: Information Theory (cs.IT); Computational Complexity (cs.CC)
Locally Decodable Codes (LDCs) are error-correcting codes for which individual message symbols can be quickly recovered despite errors in the codeword. LDCs for Hamming errors have been studied extensively in the past few decades, where a major goal is to understand the amount of redundancy that is necessary and sufficient to decode from large amounts of error, with small query complexity.
In this work, we study LDCs for insertion and deletion errors, called Insdel LDCs. Their study was initiated by Ostrovsky and Paskin-Cherniavsky (Information Theoretic Security, 2015), who gave a reduction from Hamming LDCs to Insdel LDCs with a small blowup in the code parameters. On the other hand, the only known lower bounds for Insdel LDCs come from those for Hamming LDCs, thus there is no separation between them. Here we prove new, strong lower bounds for the existence of Insdel LDCs. In particular, we show that $2$-query linear Insdel LDCs do not exist, and give an exponential lower bound for the length of all $q$-query Insdel LDCs with constant $q$. For $q \ge 3$ our bounds are exponential in the existing lower bounds for Hamming LDCs. Furthermore, our exponential lower bounds continue to hold for adaptive decoders, and even in private-key settings where the encoder and decoder share secret randomness. This exhibits a strict separation between Hamming LDCs and Insdel LDCs.
Our strong lower bounds also hold for the related notion of Insdel LCCs (except in the private-key setting), due to an analogue to the Insdel notions of a reduction from Hamming LCCs to LDCs.
Our techniques are based on a delicate design and analysis of hard distributions of insertion and deletion errors, which depart significantly from typical techniques used in analyzing Hamming LDCs. - [342] arXiv:2111.14835 (replaced) [pdf, html, other]
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Title: Smooth solutions to the Schrödinger flow for maps from smooth bounded domains in Euclidean spaces into $\mathbb{S}^2$Comments: The final version, to appear in Comm. Anal. GeomSubjects: Analysis of PDEs (math.AP)
The results of this paper are twofold. One is that we show the local existence and uniqueness of very regular or smooth solution to the initial-Neumann boundary value problem of the Schrödinger flow for maps from a smooth bounded domain $\Omega\subset \mathbb{R}^m$ with $m=1,2,3$ into $\mathbb{S}^2$ in the scale of Sobolev spaces. In this part, we provide a precise description of the compatibility conditions at the boundary for the initial data. The other is that we further prove that the locally smooth solution to the initial-Neumann boundary value problem of the 1-dimensional Schrödinger flow can be extended to a global smooth one.
- [343] arXiv:2112.10289 (replaced) [pdf, html, other]
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Title: Prime Factorization of MeandersComments: 26 pages, 15 figures; v5: minor revision. To appear in The Electronic Journal of CombinatoricsSubjects: Combinatorics (math.CO)
In this paper, we introduce a prime factorization of open meanders, articulated through the framework of 2-colored operads. We demonstrate that each open meander can be canonically constructed from building blocks of two types: iterated snakes and irreducible meanders. We find out that iterated snakes allow efficient enumeration, and thus the problem of enumerating meanders reduces to the problem of enumerating irreducible meanders. Additionally, we present some results concerning the asymptotic of meanders of both classes.
- [344] arXiv:2201.05235 (replaced) [pdf, other]
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Title: Well-posedness of a fully nonlinear evolution inclusion of second orderSubjects: Analysis of PDEs (math.AP); Functional Analysis (math.FA)
The well-posedness of the abstract \textsc{Cauchy} problem for the doubly nonlinear evolution inclusion equation of second order \begin{align*} \begin{cases} u''(t)+\partial \Psi(u'(t))+B(t,u(t))\ni f(t), &\quad t\in (0,T),\, T>0,\\ u(0)=u_0, \quad u'(0)=v_0 \end{cases} \end{align*} in a real separable \textsc{Hilbert} space $\mathscr{H}$, where $u_0\in \mathscr{H}, v_0\in \overline{D(\partial \Psi)}\cap D(\Psi), f\in L^2(0,T;\mathscr{H})$. The functional $\Psi: \mathscr{H} \rightarrow (-\infty,+\infty]$ is supposed to be proper, lower semicontinuous, and convex and the nonlinear operator $B:[0,T]\times \mathscr{H}\rightarrow \mathscr{H}$ is supposed to satisfy a (local) \textsc{Lipschitz} condition. Existence and uniqueness of strong solutions $u\in H^2(0,T^*;\mathscr{H})$ as well as the continuous dependence of solutions from the data re shown by employing the theory of nonlinear semigroups and the Banach fixed-point theorem. If $B$ satisfies a local Lipschitz condition, then the existence of strong local solutions are obtained.
- [345] arXiv:2205.03368 (replaced) [pdf, html, other]
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Title: An Efficient Minimax Optimal Estimator For Multivariate Convex RegressionComments: Minor corrections and improved presentation (appeared at COLT 2022)Subjects: Statistics Theory (math.ST); Machine Learning (cs.LG); Metric Geometry (math.MG); Computation (stat.CO)
This work studies the computational aspects of multivariate convex regression in dimensions $d \ge 5$. Our results include the \emph{first} estimators that are minimax optimal (up to logarithmic factors) with polynomial runtime in the sample size for both $L$-Lipschitz convex regression, and $\Gamma$-bounded convex regression under polytopal support. Our analysis combines techniques from empirical process theory, stochastic geometry, and potential theory, and leverages recent algorithmic advances in mean estimation for random vectors and in distribution-free linear regression. These results provide the first efficient, minimax-optimal procedures for non-Donsker classes for which their corresponding least-squares estimator is provably minimax-suboptimal.
- [346] arXiv:2209.13010 (replaced) [pdf, html, other]
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Title: Iterating sum of power divisor function and New equivalence to the Riemann hypothesisComments: We thank Pedro Caceres for his contribution and the arXiv administrators; the English, overlap, and bounds with a new chaotic spectral approach are improvedSubjects: General Mathematics (math.GM)
This paper investigates the dynamics of the iterated sum-of-divisors function $\sigma_k(m)$ and its behaviour modulo $m$, motivated by classical questions on perfect and multiperfect numbers and by the congruences $\sigma_k(m) \equiv 0 \pmod m$. Perfect and multiperfect numbers remain extremely rare; odd perfect numbers are still unknown and must be astronomically large. Here, the emphasis is on the dynamical and statistical structure of the iterates rather than on isolated examples.
Three main results are obtained. First, it is proved that no integer $m>1$ can satisfy $\sigma_k(m) \equiv 0 \pmod m$ for all $k \ge 0$, thereby ruling out the existence of "metaperfect" numbers and showing that the iteration of $\sigma$ cannot remain permanently trapped in the residue class $0$ modulo $m$. Second, for certain explicit integers such as $m=6,12,24$, the sequence $\sigma_k(m) \bmod m$ is strictly periodic with small period dividing $L=\mathrm{lcm}(e_i+1)$, where the $e_i$ are the prime exponents of $m$. Bifurcation plots and distributional analysis reveal a transition from rigid two-cycle structure to more complex residue dynamics as $m$ increases. Third, a new equivalence with the Riemann Hypothesis is established: RH holds if and only if, for every even non-squarefree $m \ge 5041$ containing a prime fifth power, \[ \frac{\sigma_k(m)}{\sigma_{k-1}(m)\log\log\sigma_{k-1}(m)} \le e^\gamma, \] and the sequence $\sigma_k(m) \bmod m$ is eventually periodic, uniformly in $k \ge 0$. Extensive computations support these periodicity phenomena, yield non-normal discrete distribution models for the residues, and suggest a connection with a newly proposed Schrodinger-type "Caceres" operator whose spectrum numerically reproduces key statistical features of the nontrivial zeros of the Riemann zeta function. - [347] arXiv:2210.06062 (replaced) [pdf, html, other]
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Title: The Specular DerivativeComments: 34 pages, 11 figuresSubjects: Classical Analysis and ODEs (math.CA)
In this paper, we introduce a new generalized derivative, which we term the specular derivative. We establish the Quasi-Rolles' Theorem, the Quasi-Mean Value Theorem, and the Fundamental Theorem of Calculus in light of the specular derivative. We also investigate various analytic and geometric properties of specular derivatives and apply these properties to several differential equations.
- [348] arXiv:2210.11626 (replaced) [pdf, html, other]
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Title: Optimal plug-in Gaussian processes for modeling derivativesSubjects: Statistics Theory (math.ST)
Derivatives are a key nonparametric functional in wide-ranging applications where the rate of change of an unknown function is of interest. In the Bayesian paradigm, Gaussian processes (GPs) are routinely used as a flexible prior for unknown functions, and are arguably one of the most popular tools in many areas. However, little is known about the optimal modeling strategy and theoretical properties when using GPs for derivatives. In this article, we study a plug-in strategy by differentiating the posterior distribution with GP priors for derivatives of any order. This practically appealing plug-in GP method has been previously perceived as suboptimal and degraded, but this is not necessarily the case. We provide posterior contraction rates for plug-in GPs and establish that they achieve optimal rates simultaneously for all derivative orders. We show that the posterior measure of the regression function and its derivatives, with the same choice of hyperparameter that does not depend on the order of derivatives, converges at the minimax optimal rate up to a logarithmic factor for functions in certain classes. We analyze a data-driven hyperparameter tuning method based on empirical Bayes, and show that it satisfies the optimal rate condition while maintaining computational efficiency. This article to the best of our knowledge provides the first positive result for plug-in GPs in the context of inferring derivative functionals, and leads to a practically simple nonparametric Bayesian method with optimal and adaptive hyperparameter tuning for simultaneously estimating the regression function and its derivatives. Simulations show competitive finite sample performance of the plug-in GP method. A climate change application for analyzing the global sea-level rise is discussed.
- [349] arXiv:2211.07908 (replaced) [pdf, other]
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Title: Nonamenable subforests of multi-ended quasi-pmp graphsComments: 36 pages, 3 figures. Improved and revised results, changed the exposition of the cluster graphing construction, polished presentationSubjects: Dynamical Systems (math.DS); Logic (math.LO); Probability (math.PR)
We prove the a.e. nonamenability of locally finite quasi-pmp Borel graphs whose every component admits at least three nonvanishing ends with respect to the underlying Radon--Nikodym cocycle. We witness their nonamenability by constructing Borel subforests with at least three nonvanishing ends per component, and then applying Tserunyan and Tucker-Drob's recent characterization of amenability for acyclic quasi-pmp Borel graphs. Our main technique is a weighted cycle-cutting algorithm, which yields a weight-maximal spanning forest. We also introduce a random version of this forest, which generalizes the Free Minimal Spanning Forest, to capture nonunimodularity in the context of percolation theory.
- [350] arXiv:2301.12610 (replaced) [pdf, html, other]
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Title: To Define the Core Entropy for All Polynomials Having a Connected Julia SetComments: 40 pages, 1 figureSubjects: Dynamical Systems (math.DS)
For all polynomials $f$ with ${\rm deg}(f)\ge2$ that have a connected filled Julia set $K$, we introduce a new quantity $h_{\rm GCE}(f)$, such that $h_{\rm GCE}\left(f^n\right)=n\cdot h_{\rm GCE}(f)$ for all $n\ge1$ and $h_{\rm GCE}(f)=h_{\rm GCE}(g)$ for $J$-equivalent $f$ and $g$. When the coefficients and the critical points of $f$ are real, $h_{\rm GCE}(f)=h(K\cap\mathbb{R},f)$. When $f$ is post-critically finite, $h_{\rm GCE}(f)$ equals the core entropy $h(\mathcal{H}(f),f)$, where $\mathcal{H}(f)$ is the Hubbard tree. For $f_c(z)=z^2+c$ with $c$ varying in the Mandelbrot set $\mathcal{M}$, the entropy map $c\mapsto h_{\rm GCE}(f_c)$ is not continuous. However, its lower envelope $h_{\rm core}:\mathcal{M}\rightarrow\mathbb{R}$ given by $h_{\rm core}(c)=\inf\left\{t:\ \exists\ c_n\ne c\ \text{with}\ c_n\rightarrow c\ \text{and}\ t=\lim\limits_{n\rightarrow\infty}h_{\rm GCE}\left(f_{c_n}\right)\right\}$ is continuous over $\mathcal{M}$ and has three properties. First, every $h_{\rm core}^{-1}([0,s])$ with $s\ge0$ is connected. In particular, $h_{\rm core}^{-1}(0)$ coincides with the central molecule. Second, $h_{\rm core}(c)=h(\mathbb{R},f_c)$ for $c\in[-2,\frac14]$. Third, $h_{\rm core}(c)=h(\mathcal{H}(f_c),f_c)$ for post-critically finite $f_c$.
- [351] arXiv:2303.07328 (replaced) [pdf, html, other]
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Title: Perturbations of Fefferman spaces over CR three-manifoldsComments: 50 pages; v2: 42 pages, some results removed and transferred to a separate preprint, namely "Perturbations of Fefferman spaces over almost CR manifolds"; v3: 47 pages, revised version to be published in Transactions of the American Mathematical Society (Presentation improved, clarifications and references added)Subjects: Differential Geometry (math.DG); General Relativity and Quantum Cosmology (gr-qc); Mathematical Physics (math-ph); Complex Variables (math.CV)
We introduce a generalisation of Fefferman's conformal circle bundle over a contact Cauchy-Riemann three-manifold. These can be viewed as exact `perturbations' of Fefferman's structure by a semi-basic one-form, which encodes additional data on the CR three-manifold.
We find conditions on the Weyl tensor and the Bach tensor for a Lorentzian conformal four-manifold equipped with a twisting non-shearing congruence of null geodesics to be locally conformally isometric to such a perturbed Fefferman space.
We investigate the existence of metrics in the perturbed Fefferman conformal class satisfying appropriate sub-conditions of the Einstein equations, such as the so-called pure radiation equations. These metrics are defined only off cross-sections of Fefferman's circle bundle, and are conveniently expressed in terms of densities that generalise Gover's notion of almost Einstein scales. Our setup allows us to reduce the prescriptions on the Ricci tensor to the underlying CR three-manifold in terms of differential constraints on a complex density and the CR data of the perturbation one-form. One such constraint turns out to arise from a non-linear, or gauged, analogue of a second-order differential operator on densities. A solution thereof provides a criterion for the existence of a CR function and, under certain conditions, for the realisability of the CR three-manifold. These findings are consistent with previous works by Lewandowski, Nurowski, Tafel, Hill, and independently, by Mason.
We also provide an analysis of the Weyl curvature of such conformal structures in terms of the underlying CR data. In particular, we arrive at a CR formulation of the asymptotic Einstein condition by viewing conformal infinity as a cross-section of Fefferman's circle bundle. - [352] arXiv:2304.05317 (replaced) [pdf, html, other]
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Title: Moduli stacks of generalized phi-modulesComments: 28 pagesSubjects: Number Theory (math.NT)
Let $F$ be an arbitrary $p$-adic field and let $G$ be an arbitrary reductive group over $F$ with Langlands dual group $^LG$. We show that the change-of-group morphism of Emerton-Gee stacks $\mathcal{X}_{^LG}\to\mathcal{X}_{GL_d}$ is relatively representable by algebraic stacks of finite presentation over $\operatorname{Spf}\mathbf{Z}_p$ for any embedding $^LG\to GL_d$, which improves the result of \cite{Min25} which says the morphism is representable by locally Noetherian formal algebraic stacks.
- [353] arXiv:2306.06628 (replaced) [pdf, html, other]
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Title: Contraction 2.0 Natural Metrics in Contraction AnalysisSubjects: Dynamical Systems (math.DS); Quantum Physics (quant-ph)
Contraction analysis establishes exponential incremental convergence of a nonlinear system by solving a linear matrix inequality for a contraction metric, and has become a standard resource for solving problems in nonlinear control and estimation. This paper shows that, for a general nonlinear time-varying system, a contraction metric can be systematically derived by rewriting the system dynamics as a complex natural gradient dynamics. In this form, the variational dynamics can be modally decomposed with geodesic coordinates, and exact exponential convergence rates can be computed. The results are extended to systems with nonlinear inequality constraints. All derivations are tensor-based, and the computed eigenvalues themselves are coordinate-invariant, i.e., the contraction rates are independent of the chosen coordinate system.
Simple examples including a gravity pendulum, gradient descent with non-convex cost, Schuler dynamics, and a two-link manipulator, illustrate that the computation of the decomposed convergence rates is straightforward. The role of inequality constraints is illustrated for a controller confined to an operational envelope. - [354] arXiv:2307.04629 (replaced) [pdf, html, other]
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Title: Slope filtrations of log $p$-divisible groupsComments: 24 pagesSubjects: Algebraic Geometry (math.AG)
Oort-Zink proved that a $p$-divisible group over a normal base in characteristic $p$ with constant Newton polygon is isogenous to a $p$-divisible group admitting a slope filtration. In this paper, we generalize this result to log $p$-divisible groups.
- [355] arXiv:2307.08725 (replaced) [pdf, html, other]
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Title: Real exponential sums over primes and prime gapsComments: 27 pages, submitted to Annals of MathematicsSubjects: Number Theory (math.NT)
We prove that given $\lambda \in \mathbb{R}$ such that $0 < \lambda < 1$, then $\pi(x + x^\lambda) - \pi(x) \sim \displaystyle \frac{x^\lambda}{\log(x)}$. This solves a long-standing problem concerning the existence of primes in short intervals. In particular, we give a positive answer (for all sufficiently large number) to some old conjectures about prime numbers, such as Legendre's conjecture about the existence of at least two primes between two consecutive squares.
- [356] arXiv:2307.10081 (replaced) [pdf, other]
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Title: The Containment Game in the plane: between the Firefighter Problem and Conway's Angel ProblemComments: 34 pages, 6 figures, 1 tableSubjects: Combinatorics (math.CO)
The containment game is a full information game for two players, initialised with a set of occupied vertices in an infinite connected graph $G$. On the $t$-th turn, the first player, called Spreader, extends the occupied set to $g(t)$ adjacent vertices, and then the second player, called Container, removes $q$ unoccupied vertices from the graph. If the spreading process continues perpetually -- Spreader wins, and otherwise -- Container wins. For $g=\infty$ this game reduces to a solitaire game for Container, known as the Firefighter Problem. On $\mathbb{Z}^2$, for $q=1/k$ and $g\equiv 1$ it is equivalent to Conway's Angel Problem.
We introduce the game, and writing $q(G,g)$ for the set of $q$ values for which Container wins against a given $g(t)$, we study the minimal asymptotics of $g(t)$ such that $q(G,g)=q(G,\infty)$, i.e. for which defeating Spreader is as hard as winning the Firefighter Problem solitaire. We show, by providing explicit winning strategies, a sub-linear upper bound $g(t)=O(t^{6/7})$ and a lower bound of $g(t)=\Omega(t^{1/2})$. - [357] arXiv:2307.15135 (replaced) [pdf, html, other]
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Title: $THH$ of the Morava $E$-theory Spectrum $E_{2}$Comments: 30 pages, minor edits, typos corrected. Accepted in JHRSSubjects: Algebraic Topology (math.AT)
The Morava $E$-theories, $E_{n}$, are complex-oriented $2$-periodic ring spectra, with homotopy groups $\mathbb{W}_{\mathbb{F}_{p^{n}}}[[u_{1}, u_{2}, ... , u_{n-1}]][u,u^{-1}]$. Here $\mathbb{W}$ denotes the Witt vector ring. $E_{n}$ is a Landweber exact spectrum and hence uniquely determined by this ring as $BP_{\ast}$-algebra. Algebraic $K$-theory of $E_{n}$ is a key ingredient towards analyzing the layers in the $p$-complete Waldhausen $K$-theory chromatic tower. One hopes to use the machinery of trace methods to get results towards $K$-theory once the computation for $THH(E_{n})$ is known.
In this paper we describe $THH(E_{2})$ as part of consecutive chain of cofiber sequences where each cofiber sits in the next cofiber sequence and the first term of each cofiber sequence is describable completely in terms of suspensions and localizations of $E_{2}$. For these results, we first calculate $K(i)$-homology of $THH(E_{2})$ using a Bökstedt spectral sequence and then lift the generating classes of $K(1)$-homology to fundamental classes in homotopy group of $THH(E_{2})$. These lifts allow us to construct terms of the cofiber sequence and explicitly understand how they map to $THH(E_{2})$. - [358] arXiv:2309.03673 (replaced) [pdf, html, other]
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Title: Semistable refined Vafa-Witten invariantsComments: 33 pages. v3: the nonzero H1 case of the main theorem has been weakened due to a mistake in previous versions, plus minor revisions based on referee comments. Journal versionSubjects: Algebraic Geometry (math.AG)
For any smooth complex projective surface $S$, we construct semistable refined Vafa-Witten invariants of $S$ which prove the main conjecture of arXiv:1810.00078. This is done by extending part of Joyce's universal wall-crossing formalism to equivariant K-theory, and to moduli stacks with symmetric obstruction theories, particularly moduli stacks of sheaves on Calabi-Yau threefolds. An important technical tool which we introduce is the symmetrized pullback, along smooth morphisms, of symmetric obstruction theories.
- [359] arXiv:2309.11023 (replaced) [pdf, html, other]
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Title: Preconditioning for time-harmonic Maxwell's equations using the Laguerre transformComments: 12 pages, 4 figuresSubjects: Numerical Analysis (math.NA)
A method of numerically solving the Maxwell equations is considered for modeling harmonic electromagnetic fields. The vector finite element method makes it possible to obtain a physically consistent discretization of the differential equations. However, solving large systems of linear algebraic equations with indefinite ill-conditioned matrices is a challenge. The high order of the matrices limits the capabilities of the Gaussian method to solve such systems, since this requires large RAM and much calculation. To reduce these requirements, an iterative preconditioned algorithm based on integral Laguerre transform in time is used. This approach allows using multigrid algorithms and, as a result, needs less RAM compared to the direct methods of solving systems of linear algebraic equations.
- [360] arXiv:2310.12926 (replaced) [pdf, html, other]
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Title: Locally Integral Involutive PO-SemigroupsSubjects: Logic (math.LO)
We show that every locally integral involutive partially ordered semigroup (ipo-semigroup) $\mathbf A = (A,\le, \cdot, \sim,-)$, and in particular every locally integral involutive semiring, decomposes in a unique way into a family $\{\mathbf A_p : p\in A^+\}$ of integral ipo-monoids, which we call its integral components. In the semiring case, the integral components are unital semirings. Moreover, we show that there is a family of monoid homomorphisms $\Phi = \{\varphi_{pq}: \mathbf A_p\to \mathbf A_q : p\le q\}$, indexed over the positive cone $(A^+,\le)$, so that the structure of $\mathbf A$ can be recovered as a glueing $\int_\Phi \mathbf A_p$ of its integral components along $\Phi$. Reciprocally, we give necessary and sufficient conditions so that the Płonka sum of any family of integral ipo-monoids $\{\mathbf A_p : p\in D\}$, indexed over a join-semilattice $(D,\lor)$ along a family of monoid homomorphisms $\Phi$ is an ipo-semigroup.
- [361] arXiv:2310.14115 (replaced) [pdf, other]
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Title: Moduli Spaces of Positive Curvature Metrics in Dimension Four and BeyondComments: 24 pages, v2: improved results and more detailed presentation in Section 2. To appear in Mathematische ZeitschriftSubjects: Differential Geometry (math.DG); Algebraic Geometry (math.AG); Geometric Topology (math.GT)
We construct non-trivial elements in the homotopy groups of the observer moduli space of positive sectional curvature metrics on $\mathbb{C}P^n$ and non-trivial elements in the homotopy groups of the observer moduli space of positive scalar curvature metrics on $\mathbb{C}P^2 \sharp M^4$.
- [362] arXiv:2311.12402 (replaced) [pdf, html, other]
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Title: Examples of cubulable groups with fixed-point propertiesComments: 26 pages. Version 2: Theorem 1.1 generalised from virtually abelian groups to groups with no non-abelian free subgroupsSubjects: Group Theory (math.GR); Metric Geometry (math.MG)
For every $n \geq 1$, let $(\mathrm{FW}_n)$ denote the fixed-point property for median graphs of cubical dimension $n$ (or equivalently, for CAT(0) cube complexes of dimension $n$). In this article, we construct explicit examples of groups satisfying $(\mathrm{FW}_n)$ but with good cubical properties in higher dimensions. First, we prove that, for a finitely generated group $G$ with no non-abelian free subgroup, $G$ satisfies $(\mathrm{FW}_n)$ if and only if no subgroup $H \leq G$ of index $\leq n$ can be mapped to $\mathbb{D}_\infty$ with an infinite image. For instance, the affine Coxeter group $\tilde{A}_n$ satisfies $(\mathrm{FW}_n)$ but not $(\mathrm{FW}_{n+1})$. In another direction, we investigate virtually graph products of finite groups. As an application of our constructions, we find explicit examples, for every $n \geq 1$, of acylindrically hyperbolic groups that are cocompactly cubulable but satisfy $(\mathrm{FW}_n)$. Several conjectures and open questions are included.
- [363] arXiv:2312.05522 (replaced) [pdf, html, other]
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Title: The cyclic flats of $\mathcal{L}$-polymatroidsComments: 36 pagesSubjects: Combinatorics (math.CO)
We consider structural properties of $\mathcal{L}$-polymatroids, especially those defined on a finite complemented modular lattice $\mathcal{L}$. We introduce a set of cover-weight axioms and establish a cryptomorphism between these axioms and the rank axioms of an $\mathcal{L}$-polymatroid. We introduce the notion of a cyclic flat of an $\mathcal{L}$-polymatroid and study properties of its lattice of cyclic flats. We show that the weighted lattice of cyclic flats of an $\mathcal{L}$-polymatroid $\mathcal{P}$, along with the atomic weights of $\mathcal{P}$, is sufficient to define its rank function on $\mathcal{L}$. In our main result, we characterize those weighted lattices $(\mathcal{Z},\lambda)$ such that $\mathcal{Z}\subseteq\mathcal{L}$ is the collection of cyclic flats of an $\mathcal{L}$-polymatroid.
- [364] arXiv:2312.09834 (replaced) [pdf, other]
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Title: Anisotropic Proximal Point AlgorithmSubjects: Optimization and Control (math.OC)
In this paper we study a nonlinear dual space preconditioning approach for the relaxed Proximal Point Algorithm (PPA) with application to monotone and relatively cohypomonotone inclusions, called anisotropic PPA. The algorithm is an instance of Luque's nonlinear PPA wherein the nonlinear preconditioner is chosen as the gradient of a Legendre convex function. Since the preconditioned operator is nonmonotone in general, convergence cannot be shown using standard arguments, unless the preconditioner exhibits isotropy (preserves directions) as in existing literature. To address the broader applicability we show convergence along subsequences invoking a Bregman version of Fejér-monotonicity in the dual space. Via a nonlinear generalization of Moreau's decomposition for operators, we provide a dual interpretation of the algorithm in terms of a forward iteration applied to a $D$-firmly nonexpansive mapping which involves the Bregman resolvent. For a suitable preconditioner, convergence rates of arbitrary order are derived under a mild Hölder growth condition. Finally, we discuss an anisotropic generalization of the proximal augmented Lagrangian method obtained via the proposed scheme. This aligns with Rockafellar's generalized and sharp Lagrangian functions.
- [365] arXiv:2401.00527 (replaced) [pdf, html, other]
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Title: Sub-Poissonian estimates for exponential moments of additive functionals over pairs of particles with respect to determinantal and symplectic Pfaffian point processes governed by entire functionsComments: 18 pages; references have been updatedJournal-ref: Moscow Mathematical Journal, 23:4(2023), 463-478Subjects: Probability (math.PR); Mathematical Physics (math-ph); Classical Analysis and ODEs (math.CA); Dynamical Systems (math.DS)
The aim of this note is to estimate the tail of the distribution of the number of particles in an interval under determinantal and Pfaffian point processes. The main result of the note is that the square of the number of particles under the determinantal point process whose correlation kernel is an entire function of finite order has sub-Poissonian tails. The same result also holds in the symplectic Pfaffian case. As a corollary, sub-Poissonian estimates are also obtained for exponential moments of additive functionals over pairs of particles.
- [366] arXiv:2403.07679 (replaced) [pdf, html, other]
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Title: Directional testing for one-way MANOVA in divergent dimensionsComments: 55 pages, 15 figuresSubjects: Statistics Theory (math.ST)
Testing the equality of mean vectors across $g$ different groups plays an important role in many scientific fields. In regular frameworks, likelihood-based statistics under the normality assumption offer a general solution to this task. However, the accuracy of standard asymptotic results is not reliable when the dimension $p$ of the data is large relative to the sample size $n_i$ of each group. We propose here an exact directional test for the equality of $g$ normal mean vectors with identical unknown covariance matrix in a high dimensional setting, provided that $\sum_{i=1}^g n_i \ge p+g+1$. In the case of two groups ($g=2$), the directional test coincides with the Hotelling's $T^2$ test. In the more general situation where the $g$ independent groups may have different unknown covariance matrices, although exactness does not hold, simulation studies show that the directional test is more accurate than most commonly used likelihood{-}based solutions, at least in a moderate dimensional setting in which $p=O(n_i^\tau)$, $\tau \in (0,1)$. Robustness of the directional approach and its competitors under deviation from the assumption of multivariate normality is also numerically investigated. Our proposal is here applied to data on blood characteristics of male athletes and to microarray data storing gene expressions in patients with breast tumors.
- [367] arXiv:2404.01003 (replaced) [pdf, html, other]
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Title: On the Brun--Titchmarsh theorem. IComments: 51 pages. Minor revisions following the referee's reportSubjects: Number Theory (math.NT)
The classical Brun--Titchmarsh theorem gives an upper bound, which is of correct order of magnitude in the full range, for the number of primes $p\leqslant x$ satisfying $p\equiv a\bmod q$. We strengthen this inequality for different ranges of $\log q/\log x$, improving upon previous works by Motohashi, Goldfeld, Iwaniec, Friedlander and Iwaniec, and Maynard for general or special moduli. In particular, we are able to beat Iwaniec's barrier $q<x^{9/20-}$, and improve all existing inequalities in the range $x^{9/20}\ll q<x^{1/2-}$ by utilizing bilinear or trilinear structures in the remainder terms of linear sieve. The proof is based on various estimates for character and exponential sums, which we derive by appealing to arithmetic exponent pairs and bilinear forms with algebraic trace functions from $\ell$-adic cohomology, trilinear forms with Kloosterman fractions, and sums of Kloosterman sums from spectral theory of automorphic forms, as well as large value theorem for Dirichlet polynomials.
- [368] arXiv:2404.07469 (replaced) [pdf, html, other]
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Title: On the stability of the spherically symmetric solution to an inflow problem for an isentropic model of compressible viscous fluidComments: 31 pagesSubjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph)
We investigate an inflow problem for the multi-dimensional isentropic compressible Navier-Stokes equations. The fluid under consideration occupies the exterior domain of unit ball, $\Omega=\{x\in\mathbb{R}^n\,\vert\, |x|\ge 1\}$, and a constant stream of mass is flowing into the domain from the boundary $\partial\Omega=\{|x|=1\}$. It is shown in Hashimoto-Matsumura(2021) that if the fluid velocity at the far-field is assumed to be zero, then there exists a unique spherically symmetric stationary solution, denoted as $(\tilde{\rho},\tilde{u})(r)$ with $r\equiv |x|$. In this paper, we show that either $\tilde{\rho}$ is monotone increasing or $\tilde{\rho}$ attains a unique global minimum, and this is classified by the boundary condition of density. In addition, we also derive a set of spatial decay rates for $(\tilde{\rho},\tilde{u})$ which allows us to prove the time-asymptotic stability of $(\tilde{\rho},\tilde{u})$ using the energy method. More specifically, we prove this under small initial perturbation on $(\tilde{\rho},\tilde{u})$, provided that the density at the far-field is supposed to be strictly positive but suitably small, in other words, the far-field state of the fluid is not vacuum but suitably rarefied. The main difficulty for the proof is the boundary terms that appears in the a-priori estimates. We resolve this issue by reformulating the problem in Lagrangian coordinate system.
- [369] arXiv:2405.02900 (replaced) [pdf, html, other]
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Title: Weighted Ehrhart theory via equivariant toric geometryComments: final version, to appear in Advances in MathematicsSubjects: Algebraic Geometry (math.AG)
We give a $K$-theoretic and geometric interpretation for a generalized weighted Ehrhart theory of a full-dimensional lattice polytope $P$, depending on a given homogeneous polynomial function $\varphi$ on $P$, and with Laurent polynomial weights $f_Q(y)\in \mathbb{Z}[y^{\pm 1}]$ associated to the faces $Q \preceq P$ of the polytope. For this purpose, we calculate equivariant $K$-theoretic Hodge-Chern classes of a torus-equivariant mixed Hodge module $\mathcal{M}$ on the toric variety $X_P$ associated to $P$. For any integer $\ell$, we introduce an equivariant Hodge $\chi_y$-polynomial $\chi_y(X_P, \ell D_P; [\mathcal{M}])$, with $D_P$ the corresponding ample Cartier divisor on $X_P$ (defined by the facet presentation of $P$). Motivic properties of the Hodge-Chern classes are used to express this equivariant Hodge polynomial in terms of weighted character sums fitting with a generalized weighted Ehrhart theory. The equivariant Hodge polynomials are shown to satisfy a reciprocity and purity formula fitting with the duality for equivariant mixed Hodge modules, and implying similar properties for the generalized weighted Ehrhart polynomials. In the special case of the equivariant intersection cohomology mixed Hodge module, with the weight function given by Stanley's $g$-function of the polar polytope of $P$, we recover in geometric terms a recent combinatorial formula of Beck-Gunnells-Materov. More generally, motivated by the analogy to the Kazhdan-Lusztig theory, we introduce a duality involution on the free $ \mathbb{Z}[y^{\pm 1}]$-module of weight functions corresponding to the duality of equivariant mixed Hodge modules, and prove a new reciprocity formula in terms of this duality. This unifies and generalizes the classical reciprocity formula of Brion-Vergne in Ehrhart theory as well as the above-mentioned more recent combinatorial formula of Beck-Gunnells-Materov.
- [370] arXiv:2405.04676 (replaced) [pdf, html, other]
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Title: Measures of maximal entropy for non-uniformly hyperbolic mapsComments: To appear in Journal of the European Mathematical SocietySubjects: Dynamical Systems (math.DS)
For $C^{1+}$ maps, possibly non-invertible and with singularities, we prove that each homoclinic class of an ergodic adapted hyperbolic measure carries at most one adapted hyperbolic measure of maximal entropy. We then apply this to study the finiteness/uniqueness of such measures in several different settings: finite horizon dispersing billiards, codimension one partially hyperbolic endomorphisms with ``large'' entropy, robustly non-uniformly hyperbolic volume-preserving endomorphisms as in Andersson-Carrasco-Saghin (2025), and Viana maps (1997).
- [371] arXiv:2405.07375 (replaced) [pdf, other]
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Title: Lie superalgebra invariants and almost classical knotsComments: 42 pages, 29 figures, comments are welcome; v2-title changed to reflect focus of the paper on almost classical tangles. Abstract and introduction also rewritten and references added. Minor typos corrected. Pictures of virtual slice knots in Section 7 removed to decrease page length. No change to results of the paperSubjects: Geometric Topology (math.GT)
A virtual link is said to be almost classical (AC) if it has a homologically trivial representative in some thickened surface $\Sigma \times [0,1]$, where $\Sigma$ is a closed orientable surface. AC links provide a useful window for observing the geometric topology of virtual knots. Here we take a different approach and look at AC links through the lens of quantum topology. Two adjustments are needed to the existing theory. First, it is necessary to generalize the definition of AC to include virtual tangles and, in particular, virtual braids. Secondly, to distinguish AC and non-AC tangles, the additional structure of quantum supergroups is required. For each Lie superalgebra $\mathfrak{gl}(m|n)$, we define a pair of $U_q(\mathfrak{gl}(m|n))$ Reshetikhin-Turaev functors $Q^{m|n}$, $\widetilde{Q}^{m|n} \circ Zh$ on framed virtual tangles. Here $Zh$ denotes the Bar-Natan $Zh$ construction. These functors unify the Alexander polynomial (AP) of AC links and the generalized Alexander polynomial (GAP) of all virtual links into a single quantum model: $Q^{1|1}$ recovers the AP of an AC link and for any virtual link $K$, $\widetilde{Q}^{1|1}\circ Zh(K)$ is the 2-variable GAP. However, when $(m,n) \ne (1,1)$, these invariants are generally distinct from the AP and GAP. Furthermore, in contrast to the classical case, they are not determined by $m-n$. For example, there are virtual knots with trivial GAP but nontrivial $U_q(\mathfrak{gl}(2|2))$ and $U_q(\mathfrak{gl}(3|3))$ invariants.
Silver and Williams proved that the GAP vanishes on all AC links. Our main result is a generalization of this theorem to almost classical tangles and the $U_q(\mathfrak{gl}(m|n))$ Reshetikhin-Turaev functors. We prove that if $T$ is an almost classical tangle, then $\widetilde{Q}^{m|n}\circ Zh(T)$ is conjugate to $Q^{m|n}(T)$, with conjugation determined by an Alexander numbering of $T$. - [372] arXiv:2405.10854 (replaced) [pdf, html, other]
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Title: Strong log-convexity of genus sequencesComments: Accepted for publication in JCTB (2025)Subjects: Combinatorics (math.CO)
For a graph $G$, and a nonnegative integer $g$, let $a_g(G)$ be the number of $2$-cell embeddings of $G$ in an orientable surface of genus $g$ (counted up to the combinatorial homeomorphism equivalence). In 1989, Gross, Robbins, and Tucker [Genus distributions for bouquets of circles, J. Combin. Theory Ser. B 47 (1989), 292-306] proposed a conjecture that the sequence $a_0(G),a_1(G),a_2(G),\dots$ is log-concave for every graph $G$. This conjecture is reminiscent to the Heron-Rota-Welsh Log Concavity Conjecture that was recently resolved in the affirmative by June Huh et al., except that it is closer to the notion of $\Delta$-matroids than to the usual matroids. In this short paper, we disprove the Log Concavity Conjecture of Gross, Robbins, and Tucker by providing examples that show strong deviation from log-concavity at multiple terms of their genus sequences.
- [373] arXiv:2405.11435 (replaced) [pdf, html, other]
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Title: Time-inhomogeneous random walks on finite groups and cokernels of random integer block matricesComments: Weaker condition on block size in Theorem 1.2Subjects: Probability (math.PR); Group Theory (math.GR); Number Theory (math.NT)
We study time-inhomogeneous random walks on finite groups in the case where each random walk step need not be supported on a generating set of the group. When the supports of the random walk steps satisfy a natural condition involving normal subgroups of quotients of the group, we show that the random walk converges to the uniform distribution on the group and give bounds for the convergence rate using spectral properties of the random walk steps. As an application, we use the moment method of Wood to prove a universality theorem for cokernels of random integer matrices allowing some dependence between entries.
- [374] arXiv:2405.12782 (replaced) [pdf, other]
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Title: Erdős' problem and $(n, \frac{1}{3})$-separated setsComments: We find a very simple proof of the main problem in the manuscript. So, we feel the idea in the manuscript is of no help to combinatoricsSubjects: Dynamical Systems (math.DS); Combinatorics (math.CO)
Inspired by the Erdős' problem in Ramsey theory, we propose a dynamical version of the problem and answer it positively for circle maps.
- [375] arXiv:2405.13741 (replaced) [pdf, html, other]
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Title: Existence in NSOP$_1$ theoriesSubjects: Logic (math.LO)
We show that Kim-forking satisfies existence in all NSOP$_1$ theories.
- [376] arXiv:2407.04838 (replaced) [pdf, html, other]
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Title: The semi-simple theory of higher rank acylindricityComments: An unproven claim and changes to the existing literature affected our work. All of the results herein are sill true, but addressing the changes required many additional pages of writing. So we have split this work into two parts for readability. Part I is available as arXiv:2512.21936 and Part II will be released shortly. For details of the changes, we refer the reader to Section 2 of Part 1Subjects: Group Theory (math.GR); General Topology (math.GN)
We present a new notion of non-positively curved groups: the collection of discrete countable groups acting (AU-)acylindrically on finite products of $\delta$-hyperbolic spaces with general type factors. Inspired by the classical theory of ($S$-arithmetic) lattices and the flourishing theory of acylindrically hyperbolic groups, we show that, up to virtual isomorphism, finitely generated groups in this class enjoy a strongly canonical product decomposition. This semi-simple decomposition also descends to the outer-automorphism group, allowing us to give a partial resolution to a recent conjecture of Sela. We also develop various structure results including a free vs abelian Tits' Alternative, and connections to lattice envelopes. Along the way we give representation-theoretic proofs of various results about acylindricity -- some methods are new even in the rank-1 setting.
The vastness of this class of groups is exhibited by recognizing that it contains, for example, $S$-arithmetic lattices with rank-1 factors, acylindrically hyperbolic groups, HHGs, groups with property (QT), and is closed under direct products, passing to (totally general type) subgroups, and finite index over-groups. - [377] arXiv:2407.06521 (replaced) [pdf, other]
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Title: Two Birds With One Stone: Beamforming Design for Target Sensing and Proactive EavesdroppingComments: 16 pages, 8 figures, submitted to IEEE Journal for reviewSubjects: Information Theory (cs.IT); Signal Processing (eess.SP)
This work studies the beamforming design in the joint proactive eavesdropping (PE) and target sensing (TS) systems. The base station (BS) wiretaps the information transmitted by the illegal transmitter and sends the waveform for TS. The shared waveform also serves as artificial noise to interfere with the illegal receiver, thereby achieving successful this http URL firstly optimize the transmitting beampattern of the BS only to maximize the eavesdropping rate or only to minimize the Cram{é}r-Rao bound, respectively. Then, the joint design of PE and TS is investigated by formulating the PE-centric, the TS-centric, and the normalized weighted optimization problems. The formulated problems are solved by the semi-definite relaxation technique and the sequential rank-one constraint relaxation method to address the complexity of the original problem. Furthermore, the scenario in which the quality of the eavesdropping channel is stronger than that of the illegal channel is considered. Numerical results demonstrate that the proposed algorithm can effectively realize PE and TS simultaneously.
- [378] arXiv:2407.07242 (replaced) [pdf, other]
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Title: Tensor network approximation of Koopman operatorsDimitrios Giannakis, Mohammad Javad Latifi Jebelli, Michael Montgomery, Philipp Pfeffer, Jörg Schumacher, Joanna SlawinskaComments: 53 pages, 11 figuresSubjects: Dynamical Systems (math.DS); Quantum Physics (quant-ph)
We propose a tensor network framework for approximating the evolution of observables of measure-preserving ergodic systems. Our approach is based on a spectrally-convergent approximation of the skew-adjoint Koopman generator by a diagonalizable, skew-adjoint operator $W_\tau$ that acts on a reproducing kernel Hilbert space $\mathcal H_\tau$ with coalgebra structure and Banach algebra structure under the pointwise product of functions. Leveraging this structure, we lift the unitary evolution operators $e^{t W_\tau}$ (which can be thought of as regularized Koopman operators) to a unitary evolution group on the Fock space $F(\mathcal H_\tau)$ generated by $\mathcal H_\tau$ that acts multiplicatively with respect to the tensor product. Our scheme also employs a representation of classical observables ($L^\infty$ functions of the state) by quantum observables (self-adjoint operators) acting on the Fock space, and a representation of probability densities in $L^1$ by quantum states. Combining these constructions leads to an approximation of the Koopman evolution of observables that is representable as evaluation of a tree tensor network built on a tensor product subspace $\mathcal H_\tau^{\otimes n} \subset F(\mathcal H_\tau)$ of arbitrarily high grading $n \in \mathbb N$. A key feature of this quantum-inspired approximation is that it captures information from a tensor product space of dimension $(2d+1)^n$, generated from a collection of $2d + 1$ eigenfunctions of $W_\tau$. Furthermore, the approximation is positivity preserving. The paper contains a theoretical convergence analysis of the method and numerical applications to two dynamical systems on the 2-torus: an ergodic torus rotation as an example with pure point Koopman spectrum and a Stepanoff flow as an example with topological weak mixing.
- [379] arXiv:2407.16359 (replaced) [pdf, html, other]
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Title: EM++: A parameter learning framework for stochastic switching systemsSubjects: Optimization and Control (math.OC); Systems and Control (eess.SY)
This paper proposes a general switching dynamical system model, and a custom majorization-minimization-based algorithm EM++ for identifying its parameters. For certain families of distributions, such as Gaussian distributions, this algorithm reduces to the well-known expectation-maximization method. We prove global convergence of the algorithm under suitable assumptions, thus addressing an important open issue in the switching system identification literature. The effectiveness of both the proposed model and algorithm is validated through extensive numerical experiments.
- [380] arXiv:2409.08624 (replaced) [pdf, html, other]
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Title: Borel graphable equivalence relationsComments: 62 pages, updated to fix typos and add some remarksSubjects: Logic (math.LO); Dynamical Systems (math.DS)
This paper is devoted to the study of analytic equivalence relations which are Borel graphable, i.e. which can be realized as the connectedness relation of a Borel graph. Our main focus is the question of which analytic equivalence relations are Borel graphable. First, we study an equivalence relation arising from the theory of countable admissible ordinals and show that it is Borel graphable if and only if there is a non-constructible real. As a corollary of the proof, we construct an analytic equivalence relation which is (provably in ZFC) not Borel graphable and an effectively analytic equivalence relation which is Borel graphable but not effectively Borel graphable. Next, we study analytic equivalence relations given by the isomorphism relation for some class of countable structures. We show that all such equivalence relations are Borel graphable, which implies that for every Borel action of $S_\infty$, the associated orbit equivalence relation is Borel graphable. This leads us to study the class of Polish groups whose Borel actions always give rise to Borel graphable orbit equivalence relations; we refer to such groups as graphic groups. We show that besides $S_\infty$, the class of graphic groups includes all connected Polish groups and is closed under countable products. We finish by studying structural properties of the class of Borel graphable analytic equivalence relations and by considering two variations on Borel graphability: a generalization with hypegraphs instead of graphs and an analogue of Borel graphability in the setting of computably enumerable equivalence relations.
- [381] arXiv:2409.16741 (replaced) [pdf, html, other]
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Title: On generic $3$-rigidity of graphsSubjects: Combinatorics (math.CO)
We aim to give an exact condition of generic $3$ -rigidity of graphs relying on partitioning the edges into $3$ subsets; such that each subset-pair gives a generically $2$-rigid graph, either by themselves or after an appropriate edge-deletion.
- [382] arXiv:2410.01090 (replaced) [pdf, html, other]
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Title: Parametrized Families of Resolvent CompositionsSubjects: Optimization and Control (math.OC); Functional Analysis (math.FA)
This paper presents an in-depth analysis of a parametrized version of the resolvent composition, an operation that combines a set-valued operator and a linear operator. We provide new properties and examples, and show that resolvent compositions can be interpreted as parallel compositions of perturbed operators. Additionally, we establish new monotonicity results, even in cases when the initial operator is not monotone. Finally, we derive asymptotic results regarding operator convergence, specifically focusing on graph-convergence and the $\rho$-Hausdorff distance.
- [383] arXiv:2410.21887 (replaced) [pdf, html, other]
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Title: Graphs with positive Lin-Lu-Yau curvature without quadrilateralsSubjects: Combinatorics (math.CO); Differential Geometry (math.DG)
The definition of Ricci curvature on graphs was given in Lin-Lu-Yau, Tohoku Math., 2011, which is a variation of Ollivier, J. Funct. Math., 2009. Recently, a powerful limit-free formulation of Lin-Lu-Yau curvature using the graph Laplacian has been given in Münch-Wojciechowski, Adv. Math., 2019. Let $F_k$ be the friendship graph obtained from $k$ triangles by sharing a common vertex and $T$ be the graph obtained from a triangle and $K_{1,3}$ by adding a matching between every leaf of $K_{1,3}$ and a vertex of the triangle. In this paper, we classify all the simple connected $C_4$-free graphs with positive Lin-Lu-Yau curvature for minimum degree at least 2: the cycles $C_3,C_5$, the friendship graphs $F_2,F_3$, the line graph of Peterson graph, and $T$.
- [384] arXiv:2410.23406 (replaced) [pdf, other]
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Title: On a Rigidity Result in Positive Scalar Curvature GeometryComments: The result is far from optimal; the radius of the geodesic ball that can be rigid under large deformation is not clear at all, making the result almost useless despite some new techniques such as using the biorthogonal spin system. Therefore, it is best to retract thisSubjects: Differential Geometry (math.DG); General Relativity and Quantum Cosmology (gr-qc); Mathematical Physics (math-ph)
I prove a scalar curvature rigidity theorem for spheres. In particular, I prove that geodesic balls of radii strictly less than $\frac{\pi}{2}$ in $n+1~(n\geq 2)$ dimensional unit sphere can be rigid under smooth deformations that increase scalar curvature preserving the intrinsic geometry and the mean curvature of the boundary, and such rigidity result fails for the hemisphere. The proof of this assertion requires the notion of a real Killing connection and solution of the boundary value problem associated with its Dirac operator. The result serves as the sharpest refinement of the now-disproven Min-Oo conjecture.
- [385] arXiv:2411.01392 (replaced) [pdf, html, other]
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Title: Well-posedeness for the non-isotropic Schrödinger equations on cylinders and periodic domainsComments: 24 pagesSubjects: Analysis of PDEs (math.AP)
The initial value problem (IVP) for the non-isotropic Schrödinger equation posed on the two-dimensional cylinders and $\mathbb{T}^2$ is considered. The IVP is shown to be locally well-posed for small initial data in $H^s(\mathbb{T}\times\mathbb{R})$ if $s\geq0$. For the IVP posed on $\mathbb{R}\times\mathbb{T}$, given data are considered in the anisotropic Sobolev spaces thereby obtaining the local well-posedness result in $H^{s_1, s_2}(\mathbb{R}\times\mathbb{T})$, if $s_1\geq0$ and $s_2>\frac12$. In the purely periodic case, a particular case of the IVP is shown to be locally well-posed for any given initial data in $H^s(\mathbb{T}^2)$ if $s>\frac14$. In some cases, ill-posedness issues are also considered showing that the IVP posed on $\mathbb{T}\times \mathbb{R}$, in the focusing case, is ill-posed in the sense that the application data-solution fails to be uniformly continuous for data in $H^s(\mathbb{T}\times\mathbb{R})$ if $-\frac12\leq s<0$.
- [386] arXiv:2411.02961 (replaced) [pdf, html, other]
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Title: The Cone Restriction: An Old Approach RevisitedComments: Optimize the layoutSubjects: Classical Analysis and ODEs (math.CA)
We consider the Ou-Wang's approach to cone restriction via polynomial partitioning. By restructuring their induction arguments into a recursive algorithm and applying the nested polynomial Wolff axioms, we refine the bounds on cone restriction estimate in higher dimensions.
- [387] arXiv:2411.07103 (replaced) [pdf, html, other]
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Title: On a connection between total positivity and Bernoulli stopping problemsComments: 15 pages, 1 figure. Fixing some errors and adding new resultsSubjects: Probability (math.PR)
Consider a discrete-time optimal selection problem where one observes a sequence of independent Bernoulli trials and receives a nonnegative reward upon stopping on a success. The aim is to find a single-choice strategy that maximises the expected payoff. These Bernoulli stopping problems are characterised by two key properties: (i) a recurrence relation connecting the reward sequence to the continuation payoff sequence, and (ii) the total positivity of the Markov chain embedded in success epochs of the trials. The recurrence is fundamental in proving the optimality of the myopic strategy under unimodal continuation payoff sequence, while the total positivity ensures that the expectation of a quasi-unimodal function of the chain remains quasi-unimodal with respect to the initial state. In particular, if the number of successes is finite almost surely, the quasi-unimodality of the reward sequence is sufficient for the myopic rule to be optimal. Illustrative examples are given in various last-success settings.
- [388] arXiv:2411.08372 (replaced) [pdf, html, other]
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Title: Equitable list coloring of sparse graphsSubjects: Combinatorics (math.CO)
A proper vertex coloring of a graph is equitable if the sizes of all color classes differ by at most $1$. For a list assignment $L$ of $k$ colors to each vertex of an $n$-vertex graph $G$, an equitable $L$-coloring of $G$ is a proper coloring of vertices of $G$ from their lists such that no color is used more than $\lceil n/k\rceil$ times. Call a graph equitably $k$-choosable if it has an equitable $L$-coloring for every $k$-list assignment $L$. A graph $G$ is $(a,b)$-sparse if for every $A\subseteq V(G)$, the number of edges in the subgraph $G[A]$ of $G$ induced by $A$ is at most $a|A|+b$.
Our first main result is that every $(\frac{7}{6},\frac{1}{3})$-sparse graph with minimum degree at least $2$ is equitably $3$-colorable and equitably $3$-choosable. This is sharp. Our second main result is that every $(\frac{5}{4},\frac{1}{2})$-sparse graph with minimum degree at least $2$ is equitably $4$-colorable and equitably $4$-choosable. This is also sharp.
One of the tools in the proof is the new notion of strongly equitable (SE) list coloring. This notion is both stronger and more natural than equitable list coloring; and our upper bounds are for SE list coloring. - [389] arXiv:2411.11084 (replaced) [pdf, html, other]
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Title: Integral filtered Sen theory and applicationsComments: v3: substantial revision; title changed. Contents in sections 6 and 10-12 are completely new. Comments are welcome!Subjects: Number Theory (math.NT)
We study Nygaard-, conjugate-, and Hodge filtrations on the many variants of Breuil--Kisin modules associated to integral semi-stable Galois representations. This leads to an integral Sen operator satisfying certain ``$1$-degree shrinking" on the increasing conjugate filtration, and (in special cases) a mod $p$ Sen operator satisfying certain ``$p$-degree shrinking". These constructions are related with prismatic $F$-crystals, Hodge--Tate crystals and $F$-gauges, and have explicit relations with classical (non-prismatic) operators. As applications, we obtain vanishing and torsion bound results on graded of the integral Hodge filtration; our explicit methods also recover results of Gee--Kisin and Bhatt--Gee--Kisin concerning the mod $p$ Hodge filtrations and Frobenius structures.
- [390] arXiv:2411.12316 (replaced) [pdf, html, other]
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Title: Behaviors of the Tate--Shafarevich group of elliptic curves under quadratic field extensionsComments: 19 pages. Improved expositionSubjects: Number Theory (math.NT)
Let $E/\mathbb{Q}$ be an elliptic curve. We study the behavior of the Tate--Shafarevich group of $E$ under quadratic extensions $\mathbb{Q}(\sqrt{D})/\mathbb{Q}$. By analyzing the cokernel of the restriction map, without assuming the finiteness of the Tate--Shafarevich group, we prove that the ratio $\frac{\#\Sha(E/\mathbb{Q}(\sqrt{D}))[4]}{\#\Sha(E_D/\mathbb{Q})[2]}$ and $\#\Sha(E_D/\mathbb{Q})[2]$ can, under some conditions on $E/\mathbb{Q}$, grow arbitrarily large simultaneously, where $E_D$ denotes the quadratic twist of $E$ by $D$. For elliptic curves of the form $E : y^2 = x^3 + px$ with $p\equiv 1 \bmod 4$ being an odd prime, assuming the finiteness of the relevant Tate--Shafarevich groups, we prove that $\#\Sha(E/\mathbb{Q}(\sqrt{D}))[2] \leq 4$ and $\Sha(E_D/\mathbb{Q})[2] = 0$ for infinitely many square-free integers $D$ with $-D$ being a prime number. Additionally, $\Sha(E/\mathbb{Q}(\sqrt{-D}))[2]\neq 0$ for all $D$ when $p=257$.
- [391] arXiv:2411.15803 (replaced) [pdf, html, other]
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Title: A Proof of Ramanujan's Classic $π$ FormulaSubjects: Number Theory (math.NT)
In 1914, Ramanujan presented a collection of 17 elegant and rapidly converging formulae for $\pi$. Among these, one of the most celebrated is the following series: \[\frac{1}{\pi}=\frac{2\sqrt{2}}{9801}\sum_{n=0}^{\infty}\frac{26390n+1103}{\left(n!\right)^4}\cdot \frac{\left(4n\right)!}{396^{4n}}\] In this paper, we give a full proof of this classic formula using hypergeometric series and a special type of lattice sums due to Zucker and Robertson. We will also use some results by Dirichlet and Edwards in algebraic number theory.
- [392] arXiv:2411.18427 (replaced) [pdf, other]
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Title: Brick chain filtrationsComments: The new version restricts the attention to the brick chain filtrations mentioned in the title. The further considerations have been deleted. There are no longer references to $τ$-tilting theorySubjects: Representation Theory (math.RT)
We deal with the category of finitely generated modules over an artin algebra $A$. Recall that an object in an abelian category is said to be a brick provided its endomorphism ring is a division ring. Simple modules are, of course, bricks, but in case $A$ is connected and not local, there do exist bricks which are not simple. The aim of this survey is to focus the attention to filtrations of modules where all factors are bricks, with bricks being ordered in some definite way.
In general, a module category will have many oriented cycles. Recently, Demonet has proposed to look at so-called brick chains in order to deal with a very interesting directedness feature of a module category. These are the orderings of bricks which we will use.
This is a survey which relies on recent investigations by a quite large group of mathematicians. We have singled out some important observations and have reordered them in order to obtain a completely self-contained (and elementary) treatment of the relevance of bricks in a module category. (Most of the papers we rely on are devoted to what is called $\tau$-tilting theory, but for the results we are interested in, there is no need to deal with $\tau$-tilting, or even with the Auslander-Reiten translation $\tau$). - [393] arXiv:2412.02483 (replaced) [pdf, html, other]
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Title: Actions of diagonalizable $p$-groups and Chern numbers modulo $p$Comments: 20 pagesSubjects: Algebraic Geometry (math.AG); K-Theory and Homology (math.KT)
We obtain lower bounds for the dimension of fixed loci of diagonalizable $p$-groups acting on smooth projective varieties. Those bounds depend on the modulo $p$ Chern numbers of the ambient variety, and are expressed in a natural way by introducing an appropriate filtration on the "modulo $p$ cobordism ring" (for $p=2$ this is Thom's unoriented cobordism ring $MO^*$). They are obtained using equivariant localization methods, via the concentration theorem for the Chow ring, and by a technique of "partition dividing". As applications we derive statements in the spirit of Boardman's Five-Halves Theorem for involutions on manifolds.
- [394] arXiv:2412.07968 (replaced) [pdf, html, other]
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Title: Weighted cscK metrics on Kähler varietiesComments: 35 pages; v3: corrected inaccuracies in the earlier version, accepted by J. Lond. Math. SocSubjects: Differential Geometry (math.DG); Analysis of PDEs (math.AP); Complex Variables (math.CV)
We study the weighted constant scalar curvature Kähler equations on mildly singular Kähler varieties. Assuming the existence of a suitable resolution of singularities, we establish the existence of singular weighted cscK metrics when the weighted Mabuchi functional is coercive for an extremal weight. This extends the works of Chen-Cheng and He to the singular weighted setting. Moreover, we provide a method for constructing examples of singular cscK metrics inspired by the work of Arezzo-Pacard. In contrast to the usual gluing techniques, our approach does not require a precise understanding about of the metric behavior near the singular locus.
- [395] arXiv:2412.08379 (replaced) [pdf, other]
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Title: Determining superconvergence points for $L2-1_σ$ scheme of variable-exponent subdiffusion and error estimateComments: 22 pages, 2 figuresSubjects: Numerical Analysis (math.NA)
We develop a numerical scheme for subdiffusion of variable exponent by combining the $L2-1_\sigma$ temporal discretization with finite element spatial approximation. In existing works, determining the superconvergence points requires solving a nonlinear equation related to the variable exponent at each time step. This work relaxes the selection criterion of superconvergence points without affecting the numerical accuracy, which may reduce the cost of determining superconvergence points. To handle the initial singularity of the solution, we employ a graded temporal mesh. Then we prove the stability and error estimates with a convergence rate $O\left(N^{-\min\{r\delta,2\}}+h^{\mu}\right)$ for the $L2-1_\sigma$ scheme of variable-exponent subdiffusion. Numerical results are performed to substantiate the theoretical findings.
- [396] arXiv:2501.03906 (replaced) [pdf, html, other]
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Title: A regularized transportation cost stemming from entropic approximationSubjects: Optimization and Control (math.OC); Analysis of PDEs (math.AP); Probability (math.PR)
We study the entropic regularizations of optimal transport problems under suitable summability assumptions on the point-wise transport cost. These summability assumptions already appear in the literature. However, we show that the weakest compactness conditions that can be derived are already enough to obtain the convergence of the regularized functionals. This approach allows us to characterize the variational limit of the regularization even when it does not converge to the original problem. The results apply also to problems with more than two marginals.
- [397] arXiv:2501.10218 (replaced) [pdf, html, other]
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Title: The density of maximal IC-plane graphs and maximal NIC-plane graphsSubjects: Combinatorics (math.CO)
In this paper, we show that any maximal IC-plane graph of order $n$ has at least $\left\lceil\frac{7}{3}n-\frac{14}{3}\right\rceil$ edges, and any maximal NIC-plane graph of order $n$ has at least $\left\lceil\frac{11}{5}n-\frac{18}{5}\right\rceil$ edges. Moreover, we show that both results are tight for infinitely many integers $n$.
- [398] arXiv:2502.10370 (replaced) [pdf, html, other]
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Title: Explicit Formulas for the Alexander Polynomial of Pretzel KnotsComments: 9 pages, 3 figures; v3: minor correctionsSubjects: Geometric Topology (math.GT)
We provide explicit formulas for the Alexander polynomial of pretzel knots and establish several immediate corollaries, including the characterization of pretzel knots with a trivial Alexander polynomial.
- [399] arXiv:2502.12037 (replaced) [pdf, html, other]
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Title: Information geometry of tempered stable processesComments: 19 pagesSubjects: Differential Geometry (math.DG); Information Theory (cs.IT); Probability (math.PR)
We find the information geometry of tempered stable processes. Beginning with the derivation of $\alpha$-divergence between two tempered stable processes, we obtain the corresponding Fisher information matrices and the $\alpha$-connections on their statistical manifolds. Furthermore, we explore statistical applications of this geometric framework. Various tempered stable processes such as generalized tempered stable processes, classical tempered stable processes, and rapidly-decreasing tempered stable processes are presented as illustrative examples.
- [400] arXiv:2502.12269 (replaced) [pdf, other]
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Title: Joint typical periodic optimizationComments: 71 pages. In this version we introduce the Joint TPO problem, significantly extending the first version, with a different focus. Some of the material on beta-transformations is based on, and supersedes, the results presented in the first versionSubjects: Dynamical Systems (math.DS)
We prove a generalised Yuan--Hunt--Mañé Conjecture: if $\mathcal{F}$ is the Banach space of $\alpha$-Hölder functions, and $\mathcal{T}$ is either a space of Lipschitz expanding maps, or of Anosov diffeomorphisms, or the family of beta-transformations on the interval, there is an open dense subset of $\mathcal{T}\times\mathcal{F}$ consisting of map-function pairs whose maximizing invariant measure is unique and supported on a periodic orbit.
- [401] arXiv:2502.16692 (replaced) [pdf, other]
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Title: Effective stability of negatively curved Einstein metrics in dimensions at least $4$Comments: 19 pages. Version 2 removes the restriction on the dimensionSubjects: Differential Geometry (math.DG)
We show that if a closed manifold of dimension at least four admits a negatively curved metric that is almost Einstein in a suitable sense, then it admits a genuine Einstein metric of negative sectional curvature. Importantly, the pinching constant measuring the almost-Einstein condition neither depends on an upper bound for the diameter or volume, nor on a lower bound for the injectivity radius.
- [402] arXiv:2502.20559 (replaced) [pdf, html, other]
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Title: Topologies on abelian groups and a topological five-lemmaComments: 19 pages. Hypotheses on Lemma 2.13 and, consequently, Proposition 3.22 are fixed (now they are more restrictive)Subjects: General Topology (math.GN); Number Theory (math.NT)
In this article we establish some results that allow to deduce the continuity of homomorphisms of (topological) abelian groups from commutative diagrams. In particular, we present a new topological version of the classical Five-Lemma. These results aim to be applied in duality results between cohomology groups in arithmetical contexts. In such a topological-arithmetical context, Pontryagin duality plays a central role and it becomes necessary to know whether certain homomorphisms are continuous.
- [403] arXiv:2502.21155 (replaced) [pdf, html, other]
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Title: The generalised Mukai conjecture for spherical varietiesComments: A question of Gongyo on nef partitions is answered. There are new examples, and the previous version has been split into two separate papersSubjects: Algebraic Geometry (math.AG)
We prove the generalised Mukai conjecture for $\mathbb{Q}$-factorial spherical Fano varieties. In this case, a stronger inequality holds featuring an extra term - the minimum absolute complexity of a log Calabi-Yau pair - which measures how close the Fano variety is to being toric.
- [404] arXiv:2503.01719 (replaced) [pdf, html, other]
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Title: On the Hauptvermutung of Causal Set TheoryComments: 7 pagesSubjects: Differential Geometry (math.DG)
We formulate the Hauptvermutung of Causal Set Theory in two mathematically well-defined but different ways one of which turns out to be wrong and the other one turns out to be true. A further result is that the Hauptvermutung is true if we replace finite by countable sets.
- [405] arXiv:2503.01775 (replaced) [pdf, html, other]
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Title: Structure-Preserving Neural Ordinary Differential Equations for Stiff SystemsComments: 19 pages, 8 figuresSubjects: Numerical Analysis (math.NA)
Neural ordinary differential equations (NODEs) are an effective approach for data-driven modeling of dynamical systems arising from simulations and experiments. One of the major shortcomings of NODEs, especially when coupled with explicit integrators, is its long-term stability, which impedes their efficiency and robustness when encountering stiff problems. In this work we present a structure-preserving NODE approach that learns a transformation into a system with a linear and nonlinear split. It is then integrated using an exponential integrator, which is an explicit integrator with stability properties comparable to implicit methods. We demonstrate that our model has advantages in both learning and deployment over standard explicit or even implicit NODE methods. The long-time stability is further enhanced by the Hurwitz matrix decomposition that constrains the spectrum of the linear operator, therefore stabilizing the linearized dynamics. When combined with a Lipschitz-controlled neural network treatment for the nonlinear operator, we show the nonlinear dynamics of the NODE are provably stable near a fixed point in the sense of Lyapunov. For high-dimensional data, we further rely on an autoencoder performing dimensionality reduction and Higham's algorithm for the matrix-free application of the matrix exponential on a vector. We demonstrate the effectiveness of the proposed NODE approach in various examples, including the Robertson chemical reaction problem and the Kuramoto-Sivashinky equation.
- [406] arXiv:2503.03318 (replaced) [pdf, html, other]
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Title: Linear-quadratic optimal control for non-exchangeable mean-field SDEs and applications to systemic riskSubjects: Optimization and Control (math.OC); Probability (math.PR)
We study the linear-quadratic control problem for a class of non-exchangeable mean-field systems, which model large populations of heterogeneous interacting agents. We explicitly characterize the optimal control in terms of a new infinite-dimensional system of Riccati equations, for which we establish existence and uniqueness. To illustrate our results, we apply this framework to a systemic risk model involving heterogeneous banks, demonstrating the impact of agent heterogeneity on optimal risk mitigation strategies.
- [407] arXiv:2503.04294 (replaced) [pdf, html, other]
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Title: On some triangulated categories over group algebrasComments: 19 this http URL in Applied Categorical Structures. Comments are welcome!Journal-ref: Appl Categor Struct 34, 6 (2026)Subjects: Category Theory (math.CT)
In this paper, we introduce the cofibrant derived category of a group algebra $kG$ and study its relation to the derived category of $kG$. We also define the cofibrant singularity category of $kG$, whose triviality characterizes the regularity of $kG$ with respect to the cofibrant dimension, and examine its significance as a measure of the obstruction to the equality between the classes of Gorenstein projective and cofibrant modules. We show that the same obstruction can be measured by certain localization sequences between stable categories.
- [408] arXiv:2503.06262 (replaced) [pdf, html, other]
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Title: Representations of shifted affine quantum groups and Coulomb branchesComments: 58 pagesSubjects: Representation Theory (math.RT)
We compare the integral category O of shifted affine quantum groups of symmetric and non symmetric types. To do so we compute the K-theoretic analog of the Coulomb branches with symmetrizers introduced by Nakajima and Weekes. This yields an equivalence of the category O with a module category over a new type of quiver Hecke algebras. At the decategorified level, this establishes a connection between the Grothendieck group of O and a finite-dimensional module over a simple Lie algebra of unfolded symmetric type. We compute this module in certain cases and give a combinatorial rule for its crystal.
- [409] arXiv:2503.07571 (replaced) [pdf, other]
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Title: Concentration via metastable mixing, with applications to the supercritical exponential random graph modelComments: 43 pages, 8 figures. The main result has been strengthened, more simulations have been added, and the exposition has been streamlinedSubjects: Probability (math.PR); Statistical Mechanics (cond-mat.stat-mech); Discrete Mathematics (cs.DM); Mathematical Physics (math-ph); Statistics Theory (math.ST)
Folklore belief holds that metastable wells in low-temperature statistical mechanics models exhibit high-temperature behavior. We make this rigorous in the exponential random graph model (ERGM) through the lens of concentration of measure. We make use of the supercritical (low-temperature) metastable mixing which was recently proven by Bresler, Nagaraj, and Nichani, and obtain a novel concentration inequality for Lipschitz observables of the ERGM in a large metastable well, answering a question posed by those authors. To achieve this, we prove a new connectivity property for metastable mixing in the ERGM and introduce a new general result yielding concentration inequalities, which extends a result of Chatterjee. We also use a result of Barbour, Brightwell, and Luczak to cover all cases of interest. Our work extends a result of Ganguly and Nam from the subcritical (high-temperature) regime to metastable wells, and we also extend applications of this concentration, namely a central limit theorem for small subcollections of edges and a bound on the Wasserstein distance between the ERGM and the Erdős-Rényi random graph. Finally, to supplement the mathematical content of the article, we present a simulation study of metastable wells in the supercritical ERGM.
- [410] arXiv:2503.12618 (replaced) [pdf, other]
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Title: Stable homotopy theory of invertible gapped quantum spin systems I: Kitaev's $Ω$-spectrumComments: 107 pages. Minor revisions in v2. Comments are welcome!Subjects: Mathematical Physics (math-ph); Strongly Correlated Electrons (cond-mat.str-el); Algebraic Topology (math.AT); Operator Algebras (math.OA); Quantum Physics (quant-ph)
We provide a mathematical realization of a conjecture by Kitaev, on the basis of the operator-algebraic formulation of infinite quantum spin systems. Our main results are threefold. First, we construct an $\Omega$-spectrum $\mathit{IP}_*$ whose homotopy groups are isomorphic to the smooth homotopy group of invertible gapped quantum systems on Euclidean spaces. Second, we develop a model for the homology theory associated with the $\Omega$-spectrum $\mathit{IP}_*$, describing it in terms of the space of quantum systems placed on an arbitrary subspace of a Euclidean space. This involves introducing the concept of localization flow, a semi-infinite path of quantum systems with decaying interaction range, inspired by Yu's localization C*-algebra in coarse index theory. Third, we incorporate spatial symmetries given by a crystallographic group $\Gamma $ and define the $\Omega$-spectrum $\mathit{IP}_*^\Gamma$ of $\Gamma$-invariant invertible phases. We propose a strategy for computing the homotopy group $\pi_n(\mathit{IP}_d^\Gamma )$ that uses the Davis--Lück assembly map and its description by invertible gapped localization flow. In particular, we show that the assembly map is split injective, and hence $\pi_n(\mathit{IP}_d^\Gamma)$ contains a computable direct summand.
- [411] arXiv:2503.12629 (replaced) [pdf, html, other]
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Title: Quasilinearization with regularizing tensor paraproductsComments: 21 pages, 1 figureSubjects: Analysis of PDEs (math.AP); Discrete Mathematics (cs.DM)
We extend Bony's celebrated work on paraproducts to continous and multiscale \emph{tensor} paraproducts. For $A \in \mathcal{C}^2(\mathbb{R})$ and $f \in \Lambda_{\alpha}([0,1]^2, d_d(x,y)^{\alpha} \times d'_d(x',y')^{\alpha})$, we construct an approximation, $\tilde{A}_{(N,N')}(f)$ to $A(f)$, replacing the operator $T: f \to A(f)$ with the continous tensor paraproduct, $\Pi^{(t,t')}_{(A',A'')}$, and the multiscale tensor paraproduct $\Pi^{(N,N')}_{(A',A'')}:f \to \tilde{A}_{(N,N')}(f) + \Delta_{ (N,N')}(A,f)$. In the multiscale case, we provide estimates on the residual, $\Delta_{(N,N')}(A,f)$, and show it has twice the regularity of $f$ such that $\Delta_{(N,N')}(A,f) \in \Lambda_{2 \alpha}([0,1]^2)$ and $\lVert \Delta_{(N,N')}(A,f) \rVert_{\Lambda_{2\alpha}([0,1]^2)} \leq C_A \lVert f \rVert_{\Lambda_{\alpha}([0,1]^2)} $. Our theoretical findings are supplemented with a computational example.
- [412] arXiv:2503.12980 (replaced) [pdf, other]
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Title: Numerical modeling of flocking dynamics with topological interactionsJournal-ref: Communications in Computational Physics, 2025, 39 (1), pp.240-260Subjects: Analysis of PDEs (math.AP)
In this paper, we propose a numerical investigation of topological interactions in flocking dynamics. Starting from a microscopic description of the phenomena, mesoscopic and macroscopic models have been previously derived under specific assumptions. We explore the role of topological interactions by describing the convergence speed to consensus in both microscopic and macroscopic dynamics, considering different forms of topological interactions. Additionally, we compare mesoscopic and macroscopic dynamics for monokinetic and non-monokinetic initial data. Finally, we illustrate with some simulations in one- and two-dimensional domains the sensitive dependence of solutions on initial conditions, including the case where the system exhibits two solutions starting with the same initial data.
- [413] arXiv:2503.16360 (replaced) [pdf, other]
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Title: A "cubist" decomposition of the Handel-Mosher axis bundleComments: v3 - 30 pages, 25 figures. Added Conjecture 1.2 and Question 1.4Subjects: Group Theory (math.GR); Geometric Topology (math.GT)
We show that the axis bundle of a nongeometric fully irreducible outer automorphism admits a canonical "cubist" decomposition into branched cubes that fit together with special combinatorics. From this structure, we locate a canonical finite collection of periodic fold lines in each axis bundle. This can be considered as an analogue of results of Hamenstädt and Agol from the surface setting, which state that the set of trivalent train tracks carrying the unstable lamination of a pseudo-Anosov map can be given the structure of a CAT(0) cube complex, and that there is a canonical periodic fold line in this cube complex. This work also gives an answer to questions of Handel-Mosher and Bridson-Vogtmann regarding the geometry of the axis bundle and a solution of a new flavor to the fully irreducible conjugacy problem in $\mathrm{Out}(F_r)$.
- [414] arXiv:2503.16763 (replaced) [pdf, html, other]
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Title: On uniqueness of free boundary minimal annuli in geodesic balls of $\mathbb{S}^3_+$ and $\mathbb{H}^3$Subjects: Differential Geometry (math.DG)
We consider $\Sigma$ an embedded free boundary minimal annulus in a geodesic ball in the round hemisphere $\mathbb{S}^3_+$ or in the hyperbolic space $\mathbb{H}^3$. Under the hypothesis of invariance due to an antipodal map on the geodesic ball and using the fact that this surface satisfies the Steklov problem with frequency, we prove that $\Sigma$ is congruent to a critical rotational annulus.
- [415] arXiv:2503.18183 (replaced) [pdf, html, other]
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Title: On the relative Nullstellensatz in nonarchimedean geometryComments: 9 pages, accepted version, to appear in Research in Number TheorySubjects: Algebraic Geometry (math.AG); Number Theory (math.NT)
We establish a relative version of the Nullstellensatz for algebras topologically of finite type over a given Banach Tate ring $A$, under the assumption that the corresponding statement holds for rational localizations of $A$. This applies in particular to pseudoaffinoid algebras and to the coordinate rings of affinoid subspaces of a Fargues--Fontaine curve.
- [416] arXiv:2504.00209 (replaced) [pdf, html, other]
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Title: A new iterated Tikhonov regularization method for Fredholm integral equation of first kindSubjects: Numerical Analysis (math.NA)
We consider Fredholm integral equation of the first kind, present an efficient new iterated Tikhonov method to solve it. The new Tikhonov iteration method has been proved which can achieve the optimal order under a-priori assumption. In numerical experiments, the new iterated Tikhonov regularization method is compared with the classical iterated Tikhonov method, Landweber iteration method to solve the corresponding discrete problem, which indicates the validity and efficiency of the proposed method.
- [417] arXiv:2504.01927 (replaced) [pdf, html, other]
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Title: Characterisation of distributions via record-like observationsComments: 31 pages, 2 figuresSubjects: Probability (math.PR)
We characterise probability distributions via a martingale property associated with a natural generalisation of record values, known as $\delta$-records. For an independent and identically distributed sequence $(X_n)$ with running maximum $M_n$, let $N_n$ be the number of $\delta$-records (those $X_k$ with $X_k>M_{k-1}+\delta$). We determine distributions for which $N_n-cM_n$ is a martingale, and show that this property uniquely determines the underlying distribution within broad classes.
We show that the problem can be reformulated in terms of a delay-integrated Cauchy functional equation. A distinctive feature of this equation is that it is required to hold on a set that depends on the unknown distribution itself, which both complicates the analysis and allows for a rich variety of solutions.
A complete characterisation is obtained when $\delta<0$. For $\delta>0$, all solutions with bounded support are identified. In the case of $\delta>0$ and unbounded support, we consider both continuous and lattice distributions. In the continuous case, the characterisation reduces to a delay differential equation, which admits classical exponential-type solutions as well as broader families, including mixtures of exponential and gamma distributions. An analogous discrete analysis leads to difference equations whose solutions include mixtures of geometric and negative binomial distributions. In particular, this yields a new characterisation of the geometric distribution based on weak records. - [418] arXiv:2504.03066 (replaced) [pdf, html, other]
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Title: A Lanczos-Based Algorithmic Approach for Spike Detection in Large Sample Covariance MatricesSubjects: Statistics Theory (math.ST); Probability (math.PR); Computation (stat.CO)
We introduce a new approach for estimating the number of spikes in a general class of spiked covariance models without directly computing the eigenvalues of the sample covariance matrix. This approach is based on the Lanczos algorithm and the asymptotic properties of the associated Jacobi matrix and its Cholesky factorization. A key aspect of the analysis is interpreting the eigenvector spectral distribution as a perturbation of its asymptotic counterpart. The specific exponential-type asymptotics of the Jacobi matrix enables an efficient approximation of the Stieltjes transform of the asymptotic spectral distribution via a finite continued fraction. As a consequence, we also obtain estimates for the density of the asymptotic distribution and the location of outliers. We provide consistency guarantees for our proposed estimators, proving their convergence in the high-dimensional regime. We demonstrate that, when applied to standard spiked covariance models, our approach outperforms existing methods in computational efficiency and runtime, while still maintaining robustness to exotic population covariances.
- [419] arXiv:2504.07616 (replaced) [pdf, html, other]
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Title: Isometric Splitting of Metrics Without Conjugate Points on $Σ\times S^1$Subjects: Differential Geometry (math.DG)
We establish a global rigidity theorem for Riemannian metrics without conjugate points on three-manifolds of the form $M = \Sigma \times S^1$, where $\Sigma$ is a compact orientable surface of genus at least 2. The main result states that any such metric must be a Riemannian product, with universal cover isometric to $(\mathbb{H}^2, g_0) \times (\mathbb{R}, dt^2)$. This extends the classical Hopf conjecture from tori to this natural class of manifolds with non-abelian fundamental group containing a central $\mathbb{Z}$ factor. We provide two independent proofs: one utilizing the regularity of Busemann functions and stability theory of the Riccati equation along Killing flows, and another based on a detailed analysis of the curvature operator acting on Jacobi fields. We derive sharp geometric inequalities, analyze the deformation space of such metrics, and discuss several geometric and dynamical consequences, including marked length spectrum rigidity, constraints on topological entropy, and incompatibility with non-trivial Sasakian structures.
- [420] arXiv:2504.14199 (replaced) [pdf, html, other]
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Title: Canonical bases of tensor products of integrable highest weight modules arising from framed constructionsComments: Final version. Published in International Mathematics Research Notices (2025)Subjects: Quantum Algebra (math.QA); Representation Theory (math.RT)
Given a quantum group, we prove that the canonical bases of the tensor products of its integrable highest weight modules can be obtained from the canonical bases of the integrable highest weight modules of a bigger quantum group. As a result, based on the positivity of the canonical bases of the integrable highest weight modules due to Lusztig, we prove that the canonical bases of the tensor products have the positivity.
- [421] arXiv:2504.14447 (replaced) [pdf, html, other]
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Title: Scaling Limit of Dependent Random WalkSubjects: Probability (math.PR)
Recently, a generalized Bernoulli process (GBP) was developed as a stationary binary sequence that can have long-range dependence. In this paper, we find the scaling limit of a random walk that follows GBP. The result is a new class of non-Markovian diffusion processes. The limiting processes include continuous-time stochastic processes with stationary increments whose correlation decays with an exponential rate, a power law, or an exponentially tempered power law. The limit densities solve a time tempered fractional diffusion equation or time fractional diffusion equation. The second-family of Mittag-Leffler distribution and exponential distribution arise as special cases of the limiting distributions. Subordinated processes are considered as time-changed Lévy processes, and the governing equations and dependence structure of the subordinated processes are discussed.
- [422] arXiv:2504.20517 (replaced) [pdf, html, other]
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Title: Boundary Control and Calderón type Inverse Problems in Non-local heat equationSaumyajit Das (Harish-Chandra Research Institute, Prayagraj, India)Comments: 31 pagesSubjects: Analysis of PDEs (math.AP)
We examine various density results related to the solutions of the non-local heat equation at a specific time slice, focusing on two distinct models: one with homogeneous Dirichlet boundary condition and the other with singular boundary data. In both the cases, we assume the non-local exponent $a\in(\frac{1}{2},1)$. We explore both the qualitative and quantitative aspects of the approximations. Additionally, we address Calderón-type inverse problems for these parabolic models, where we recover the potentials by analyzing the solutions either on the boundary or at a particular time slice. In both the density results and the Calderón type inverse problems, the Pohozaev identity plays a crucial role. Finally, in the last section, we apply the Pohozaev identity to a specific elliptic eigenvalue problem and demonstrate that the eigenfunctions, when divided by an appropriate power of the distance function, can not vanish on any non-empty open subset of the boundary. This particular eigenvalue problem does not need any restriction on the non-local exponent.
- [423] arXiv:2504.21835 (replaced) [pdf, html, other]
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Title: Patch bubbles for advection-dominated steady and unsteady problemsSubjects: Numerical Analysis (math.NA)
A novel variant of the \emph{residual-free bubble} method (RFB) for advection dominated problems is presented. Since the usual RFB still suffers from oscillations and strong under/overshoots, the bubble space is enriched by \emph{patch bubbles}, giving more freedom to the bubble space.
We use a recursive and efficient approach to accurately compute the bubbles. Numerical experiments clearly demonstrate the superiority of our method compared to the standard RFB.
The method is similar to the \emph{enhanced residual-free bubble} method (eRFB) proposed by Cangiani and Süli in 2005, but differs in the definition of the additional bubbles. - [424] arXiv:2505.05135 (replaced) [pdf, html, other]
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Title: Generalized modular equations and the CM values of HauptmodulnSubjects: Number Theory (math.NT)
Monstrous moonshine relates the representation of the Monster finite sporadic simple group to the distinguished modular functions, called Hauptmoduln. Chen-Yui~\cite{Chen-Yui} showed that the CM values of Hauptmoduln which appeare in monstrous moonshine (but not all) are algebraic integers, which is similar to the singular moduli of the $j$-function. In this paper, we generalize this result to Hauptmoduln whose $q$-coefficients are cyclotomic integers. A main idea for our proof is the use of generalized modular equations for Hauptmoduln, which was introduced by Cummins-Gannon~\cite{Cummins-Gannon} in the study of monstrous moonshine. As an application, we show that if a formal $q$-series satisfies the special combinatoric property called complete replicability, its CM values are algebraic integers, without assuming the modular invariance.
- [425] arXiv:2505.08461 (replaced) [pdf, html, other]
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Title: An Optimal and Robust Nonconforming Finite Element Method for the Strain Gradient ElasticityComments: 25 pagesSubjects: Numerical Analysis (math.NA)
An optimal and robust low-order nonconforming finite element method is developed for the strain gradient elasticity (SGE) model in arbitrary dimension. An $H^2$-nonconforming quadratic vector-valued finite element in arbitrary dimension is constructed, which together with the Nitsche's technique, is applied for solving the SGE model. The resulting nonconforming finite element method is optimal and robust with respect to the Lamé coefficient $\lambda$ and the size parameter $\iota$, as confirmed by numerical results. Additionally, nonconforming finite element discretization of the smooth Stokes complex in two and three dimensions is devised.
- [426] arXiv:2505.09629 (replaced) [pdf, html, other]
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Title: Primes in arithmetic progressions to smooth moduli: A minorant versionComments: 6 pages. Corrected an error in the proofSubjects: Number Theory (math.NT)
The author prove that there exists a function $\rho(n)$ which is a minorant for the prime indicator function $\mathbb{1}_{p}(n)$ and has distribution level $\frac{10}{19}$ in arithmetic progressions to smooth moduli. This refines the previous results of Baker--Irving and Stadlmann.
- [427] arXiv:2505.13106 (replaced) [pdf, other]
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Title: How to optimise tournament draws: The case of the FIFA World CupComments: 32 pages, 8 figures, 6 tablesSubjects: Optimization and Control (math.OC); Physics and Society (physics.soc-ph); Applications (stat.AP)
The organisers of major sports competitions use different policies with respect to constraints in the group draw. Our paper aims to rationalise these choices by analysing the trade-off between attractiveness (the number of games played by teams from the same geographic zone) and fairness (the departure of the draw mechanism from a uniform distribution). A parametric optimisation model is formulated and applied to the 2018 and 2022 FIFA World Cup draws. A flaw of the draw procedure is identified: the pre-assignment of the host to a group unnecessarily increases the distortions. All Pareto efficient sets of draw constraints are determined via simulations. The proposed framework can be used to find the optimal draw rules and justify the non-uniformity of the draw procedure for the stakeholders.
- [428] arXiv:2505.15871 (replaced) [pdf, html, other]
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Title: The strong hull property for affine irreducible Coxeter groups of rank 3Comments: This paper is based on the author's bachelor's thesis at Shandong University defended May 20, 2025Subjects: Combinatorics (math.CO); Group Theory (math.GR)
A conjecture proposed by Gaetz and Gao asserts that the Cayley graph of any Coxeter group possesses the strong hull property. In this paper, we prove this conjecture for all affine irreducible Coxeter groups of rank 3. Our approach exploits the geometry of affine buildings to reduce the analysis of convex hulls to finitely many manageable configurations. These geometric reduction techniques offer a novel framework that may be applicable to higher-rank cases.
- [429] arXiv:2506.01012 (replaced) [pdf, html, other]
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Title: Stability and rigidity results of space-like hypersurface in the Minkowski spaceSubjects: Differential Geometry (math.DG)
In this paper, we establish some rigidity theorems for space-like hypersurfaces in Minkowski space by using a Weinberger-type approach with P-functions and integral identities. Firstly, for space-like hypersurfaces $M$ represented as graphs $x_{n+1}=u(x)$ over domain $\Omega\subset\mathbb R^n$, if higher-order mean curvature ratio $\frac{H_{k}}{H_l}(l<k)$ is constant and the boundary $\partial M$ lies on a hyperplane intersecting with constant angles, then the hypersurface must be a part of hyperboloid. Secondly, for convex space-like hypersurfaces with boundaries on a hyperboloid or light cone, if higher-order mean curvature ratio $\frac{H_{k}}{H_l}(l<k)$ is constant and the angle function between the normal vectors of the hypersurface and the hyperboloid (or the lightcone) on the boundary is constant, then such hypersurfaces must be a part of hyperboloid. These results significantly extend Gao's previous work presented in \cite{Gao1,Gao2}.
Furthermore, we derive two fundamental integral identities for constant mean curvature (CMC) graphical hypersurfaces $x_{n+1}=u(x)$, $x\in\Omega\subset\mathbb R^n$, and the boundary lies on a hyperplane. As some applications: we obtain complete equivalence conditions for hyperboloid identification through curvature properties. We also
establish a geometric stability estimate demonstrating that the square norm of the trace-free second fundamental form $\bar h$ of $M$ is quantitatively controlled by geometric quantities of $\partial\Omega$, as expressed by the inequality: $$ ||\bar h||_{L^2(\Omega)}\leq C(n,K)||H_{\partial\Omega}-H_0||_{L^1(\partial\Omega)}^{1/2}. $$
Here, $H_{\partial\Omega}$ is the mean curvature of $\partial\Omega$, $H_0$ is some reference constant and $C$ is a constant.
Finally, analogous estimates are established. - [430] arXiv:2506.12559 (replaced) [pdf, html, other]
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Title: On the cross-correlation properties of large-size families of Costas arraysSubjects: Information Theory (cs.IT); Combinatorics (math.CO)
Costas arrays have been an interesting combinatorial object for decades because of their optimal aperiodic auto-correlation properties. Meanwhile, it is interesting to find families of Costas arrays or extended arrays with small maximal cross-correlation values, since for applications in multi-user systems, the cross-interferences between different signals should also be small. The objective of this paper is to study several large-size families of Costas arrays or extended arrays, and their values of maximal crosscorrelation are partially bounded for some cases of horizontal shifts $u$ and vertical shifts $v$. Given a prime $p \geq 5$, a large-size family of Costas arrays over $\{1, \ldots, p-1\}$ is investigated, including both the exponential and logarithmic Welch Costas arrays. An upper bound on the maximal cross-correlation of this family for arbitrary $u$ and $v$ is given. We also show that the maximal cross-correlation of the family of power permutations over $\{1, \ldots, p-1\}$ for $u=0$ and $v \neq 0$ is bounded by $\frac{1}{2}+\sqrt{p-1}$. Furthermore, we give the first nontrivial upper bound on the maximal cross-correlation of the larger family including both exponential Welch Costas arrays and power permutations over $\{1, \ldots, p-1\}$ for arbitrary $u$ and $v=0$ that it equals $(p-1) / t$ where $t$ is the smallest prime divisor of $(p-1) / 2$ if p is not a safe prime and is at most $(p-1)^{\frac{1}{2}}+(p-1)^{\frac{1}{4}}+\frac{1}{2}$ otherwise.
- [431] arXiv:2506.13587 (replaced) [pdf, html, other]
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Title: Non-exchangeable mean-field theory for adaptive weights: propagation of dissociatedness and graphon sampling lemmaComments: arXiv v3: Major revision. Title changed from "propagation of chaos..." to "propagation of dissociatedness..." to better reflect the probabilistic structure. Revised the setup using the Aldous-Hoover representation theorem. Notation and layout updated for improved readability, while the core proof arguments remain consistent with the previous version. Proof of the Sampling Lemma moved to AppendixSubjects: Probability (math.PR); Analysis of PDEs (math.AP); Combinatorics (math.CO)
We develop a mean-field theory for large, non-exchangeable particle (agent) systems where the states and interaction weights co-evolve in a coupled system of SDEs. A first main result is the establishment of the propagation of dissociatedness, a conceptual generalization of the classical propagation of chaos that accommodates the intrinsic local correlations between particles and their weights. The limiting McKean-Vlasov process is characterized by an Aldous-Hoover representation on a filtered probability space, beyond the standard one-particle law (or a family thereof). Paralleling the classical equivalence between propagation of chaos and the convergence of empirical measures to the one-particle law, we show that the propagation of dissociatedness corresponds to the convergence of the empirical structure under a distance unifying the Wasserstein distance for particles and the cut distance for weights. This quantitative stability is grounded in an adaptation of the sampling lemma from dense graph theory, analogous to the classical concentration results for empirical measures in the Wasserstein distance.
- [432] arXiv:2506.17527 (replaced) [pdf, html, other]
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Title: Detection and Reconstruction of a Random Hypergraph from Noisy Graph ProjectionComments: 19 pages, 1 figure; minor updatesSubjects: Statistics Theory (math.ST); Combinatorics (math.CO); Probability (math.PR)
For a $d$-uniform random hypergraph on $n$ vertices in which hyperedges are included i.i.d.\ so that the average degree in the hypergraph is $n^{\delta+o(1)}$, the projection of such a hypergraph is a graph on the same $n$ vertices where an edge connects two vertices if and only if they belong to a same hyperedge. In this work, we study the inference problem where the observation is a \emph{noisy} version of the graph projection where each edge in the projection is kept with probability $p=n^{-1+\alpha+o(1)}$ and each edge not in the projection is added with probability $q=n^{-1+\beta+o(1)}$. For all constant $d$, we establish sharp thresholds for both detection (distinguishing the noisy projection from an Erdős-Rényi random graph with edge density $q$) and reconstruction (estimating the original hypergraph). Notably, our results reveal a \emph{detection-reconstruction gap} phenomenon in this problem. Our work also answers a problem raised in \cite{BGPY25+}.
- [433] arXiv:2506.19790 (replaced) [pdf, html, other]
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Title: On Bott's residue formula for toric complete intersectionsComments: Accepted in Forum MathematicumSubjects: Algebraic Geometry (math.AG); Complex Variables (math.CV)
We determine the number of singularities - counted whit multiplicities - of generic distributions of dimension and codimension one on smooth complete intersections in compact toric orbifolds with isolated singularities. We also present some applications of this results. First, we analyze the case of regular distributions. As a second application, we establish a Poincaré-type inequality that relates the multidegree of a foliation to the multidegrees of an invariant smooth complete intersection curve.
- [434] arXiv:2506.23961 (replaced) [pdf, html, other]
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Title: Boundary Value Problems in graph Lipschitz domains in the plane with $A_{\infty}$-measures on the boundaryComments: 31 pages, 2 figures. V2: updated version addressing referee comments. References updated. Accepted for publication in Adv. MathJournal-ref: Adv. Math. Volume 486, February 2026, 110734Subjects: Analysis of PDEs (math.AP); Classical Analysis and ODEs (math.CA)
We prove several results for the Dirichlet, Neumann and Regularity problems for the Laplace equation in graph Lipschitz domains in the plane, considering $A_{\infty}$-measures on the boundary. More specifically, we study the $L^{p,1}$-solvability for the Dirichlet problem, complementing results of Kenig (1980) and Carro and Ortiz-Caraballo (2018). Then, we study $L^p$-solvability of the Neumann problem, obtaining a range of solvability which is empty in some cases, a clear difference with the arc-length case. When it is not empty, it is an interval, and we consider solvability at its endpoints, establishing conditions for Lorentz space solvability when $p>1$ and atomic Hardy space solvability when $p=1$. Solving the Lorentz endpoint leads us to a two-weight Sawyer-type inequality, for which we give a sufficient condition. Finally, we show how to adapt to the Regularity problem the results for the Neumann problem.
- [435] arXiv:2507.03754 (replaced) [pdf, html, other]
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Title: On a magneto-spectral invariant on finite graphsSubjects: Spectral Theory (math.SP); Combinatorics (math.CO)
In this paper, we introduce a magneto-spectral invariant for finite graphs. This invariant vanishes on trees and is maximized by complete graphs. We compute this invariant for cycles, complete graphs, wheel graphs, hypercubes, complete bipartite graphs and suspensions of trees and derive various lower and upper bounds. In particular, we provide a sharp upper bound for regular bipartite graphs and derive a direct relation between the class of graphs assuming this upper bound and the class of unit weighing matrices, which are generalizations of complex Hadamard matrices. Moreover, this class of bipartite graphs has non-negative magnetic Bakry-Émery curvature and is preserved under both the Cartesian product and a partial tensor product for bipartite graphs. The study of our invariant for certain pairs of cospectral graphs indicates also that this invariant allows us to distinguish between them. Finally, we discuss the behaviour of this invariant under various graph operations and investigate relations to the spectral gap.
- [436] arXiv:2507.04526 (replaced) [pdf, html, other]
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Title: On the theories classified by an étendueComments: Substantial changes, fixing an error in the previous version. 19 pagesSubjects: Logic (math.LO); Category Theory (math.CT)
We give a model-theoretic characterisation of the geometric theories classified by étendues -- the `locally localic' topoi. They are the theories where each model is determined, syntactically and semantically, by any witness of a fixed collection of formulae.
- [437] arXiv:2507.05225 (replaced) [pdf, html, other]
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Title: Extremal Behavior of ideals of minorsComments: are welcome! 16 pagesSubjects: Commutative Algebra (math.AC)
Let $(R,\mathfrak m,\mathsf k)$ be either a fiber product or an artinian stretched Gorenstein ring, with $\operatorname{ch}(\mathsf k)\neq 2$ in the latter case. We prove that the ideals of minors of the minimal free resolution of any finitely generated $R$-module are eventually 2-periodic. Moreover, if the embedding dimension of $R$ is at least 3, eventually the ideals of minors become the powers of the maximal ideal, yielding the 1-periodicity. These are analogs of results obtained over complete intersections and Golod rings by Brown, Dao, and Sridhar. We also study the transfer of periodicity between rings. Specifically, we prove that for any local ring $(R,\mathfrak m)$, if $x\in \mathfrak m$ is a super-regular element and $M$ is an $R/(x)$ module whose ideals of minors are asymptotically the powers of the maximal ideal over $R/(x)$, then the same holds for the ideals of minors of $M$ over $R$.
- [438] arXiv:2507.05430 (replaced) [pdf, html, other]
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Title: Toric reduction of singularities for Newton nondegenerate $p$-formsComments: 14 pages, 2 figures. Addition of two new results in v2: the inverse implication for the main theorem in Section 3 and an application. Fixed typos and some minor changes in the presentationSubjects: Algebraic Geometry (math.AG)
We study a class of holomorphic $p$-forms satisfying nondegeneracy conditions expressed through their Newton polyhedron and called Newton nondegenerate (NND). We give a characterization of NND $p$-forms by their toric reduction of singularities defined through a regular refinement of their dual fan. We then present an application of this result to the study of singularities of $(n-1)$-forms on $\mathbb{C}^n$.
- [439] arXiv:2507.06524 (replaced) [pdf, html, other]
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Title: Unique and Stable Recovery of Space-Variable Order in Multidimensional SubdiffusionComments: 23 pages, 1 figureSubjects: Analysis of PDEs (math.AP)
In this work we investigate the unique identifiability and stable recovery of a spatially dependent variable-order in the subdiffusion model from the boundary flux measurement. We establish several new unique identifiability results from the observation at one point on the boundary without / with the knowledge of medium properties, and a conditional Lipschitz stability estimate when the observation is available on the whole boundary. The analysis crucially employs resolvent estimates in the $L^r(\Omega)$ ($r>d$) spaces, solution representation in the Laplace domain and novel asymptotic expansions of the Laplace transform of the boundary flux at $p= 0$ and $p=1$.
- [440] arXiv:2507.09304 (replaced) [pdf, html, other]
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Title: Counting fixed-point-free Cayley permutationsComments: 25 pages, 6 figures, 4 tablesSubjects: Combinatorics (math.CO)
Two-sort species yield differential equations for functional digraphs of Cayley permutations. From these we obtain an explicit formula for fixed-point-free Cayley permutations and conjecture that their proportion tends to $1/e$, as for permutations and endofunctions. Our approach also yields counting formulas when the functional digraph is a tree, forest, or connected.
- [441] arXiv:2507.10531 (replaced) [pdf, other]
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Title: Quantitative central limit theorems for exponential random graphsComments: 56 pages, 3 figures. Abstract shortened to meet arXiv requirements. The statement of Theorem 2.3 has been updated to reflect a change in the literatureSubjects: Probability (math.PR); Statistical Mechanics (cond-mat.stat-mech); Discrete Mathematics (cs.DM); Mathematical Physics (math-ph); Statistics Theory (math.ST)
Ferromagnetic exponential random graph models (ERGMs) are nonlinear exponential tilts of Erdős-Rényi models, under which the presence of certain subgraphs such as triangles may be emphasized. These models are mixtures of metastable wells which each behave macroscopically like new Erdős-Rényi models themselves, exhibiting the same laws of large numbers for the overall edge count as well as all subgraph counts. However, the microscopic fluctuations of these quantities remained elusive for some time. Building on a recent breakthrough by Fang, Liu, Shao and Zhao [FLSZ24] driven by Stein's method, we prove quantitative central limit theorems (CLTs) for these quantities and more in metastable wells under ferromagnetic ERGMs. One main novelty of our results is that they apply also in the supercritical (low temperature) regime of parameters, which has previously been relatively unexplored. To accomplish this, we develop a novel probabilistic technique based on the careful analysis of the evolution of relevant quantities under the ERGM Glauber dynamics. Our technique allows us to deliver the main input to the method developed by [FLSZ24], which is the fact that the fluctuations of subgraph counts are driven by those of the overall edge count. This was first shown for the triangle count by Sambale and Sinulis [SS20] in the Dobrushin (very high temperature) regime via functional-analytic methods. We feel our technique clarifies the underlying mechanisms at play, and it also supplies improved bounds on the Wasserstein and Kolmogorov distances between the observables at hand and the limiting Gaussians, as compared to the results of [FLSZ24] in the subcritical (high temperature) regime beyond the Dobrushin regime. Moreover, our technique is flexible enough to also yield quantitative CLTs for vertex degrees and local subgraph counts, which have not appeared before in any parameter regime.
- [442] arXiv:2507.10545 (replaced) [pdf, other]
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Title: KPZ equation from a class of nonlinear SPDEs in infinite volumeComments: minor revisions, comments welcome!Subjects: Probability (math.PR); Mathematical Physics (math-ph); Analysis of PDEs (math.AP)
We study a general class of nonlinear Ginzburg-Landau SPDEs in infinite volume under weak nonlinearity scaling and with non-equilibrium initial data. We derive the KPZ equation as a continuum limit of these equations. This makes rigorous the original derivation of the KPZ equation from physics in the full-space setting, which was a problem posed by Hairer-Quastel '18. Our analysis is based on a stochastic heat kernel for a linearization of said SPDEs.
- [443] arXiv:2507.11496 (replaced) [pdf, html, other]
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Title: Variants of a theorem of Macbeath in finite dimensional normed spacesComments: 17 pages, 2 figuresSubjects: Metric Geometry (math.MG)
A classical theorem of Macbeath states that for any integers $d \geq 2$, $n \geq d+1$, $d$-dimensional Euclidean balls are hardest to approximate, in terms of volume difference, by inscribed convex polytopes with $n$ vertices. In this paper we investigate normed variants of this problem: we intend to find the extremal values of the Busemann volume, Holmes-Thompson volume, Gromov's mass and Gromov's mass$^*$ of a largest volume convex polytope with $n$ vertices, inscribed in the unit ball of a $d$-dimensional normed space.
- [444] arXiv:2507.11835 (replaced) [pdf, html, other]
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Title: Asymptotically optimal Ramsey goodness of sparse graphs versus odd cycles and pathsComments: 22 pagesSubjects: Combinatorics (math.CO)
A fundamental problem in graph Ramsey theory is to determine, for sparse graphs $G$ on $n$ vertices, the minimal $n$ such that $G$ is Ramsey-good for odd cycles $C_k$ and paths $P_k$. Burr, Erdős, Faudree, Rousseau, and Schelp (Trans. AMS 1982) addressed this problem, establishing bounds requiring $n = \Omega(k^{10})$ for odd cycles and $n = \Omega(k^{12})$ for paths. We settle the asymptotic version of this problem, proving that these bounds are essentially tight: $n = \Omega(k)$ suffices for odd cycles and $n = \Omega(k^2)$ (or $n = \Omega(k)$ under additional conditions) for paths. Specifically, we prove:
(1) For odd cycles $C_k$ ($k\ge3$), we prove $r(G, C_k) = 2n-1$ for any connected $n$-vertex graph $G$ satisfying the relaxed conditions $n = \Omega(k)$ and $e(G) \le (1 + O(1/k^2)) n$.
(2) For paths $P_k$ ($k\ge2$), we prove $r(G, P_k) = \max\{ n + \lfloor k/2\rfloor - 1, n + k - 2 - \alpha' - \gamma \}$ for any connected $n$-vertex graph $G$ satisfying one of the following:
(i) $n = \Omega(k^2)$ and $e(G) \le (1 + O(1/k^2)) n$;
(ii) $n = \Omega(k)$, $\delta(G)\ge2$, $\alpha'\geq k/2$, and $e(G) \le (1 + O(1/k)) n$.
In the above, $\alpha'$ is the independence number of an appropriate subgraph of $G$ and $\gamma=0$ if $k-1$ divides $n+k-3-\alpha'$, and $\gamma=1$ otherwise.
Consequently, our results unify and generalize classical theorems on odd cycles due to Bondy and Erdős (1973), Faudree and Schelp (1974), and Rosta (1973), and on paths due to Gerencsér and Gyárfás (1967), Faudree, Lawrence, Parsons and Schelp (1974), and Parsons (1974). The proofs feature two key innovations: a novel reconstruction of the end-edge matching and an enhancement of Burr et al.'s dichotomy lemma. - [445] arXiv:2507.16313 (replaced) [pdf, html, other]
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Title: Hyperbolicity and Schwarz Lemmas in Calibrated GeometryComments: 38 pages. An error in the previous version has been correctedSubjects: Differential Geometry (math.DG)
This paper has two main objectives. First, for an arbitrary calibrated manifold $(X,\phi)$, we define notions of $R_\phi$-hyperbolicity and $\phi$-hyperbolicity, which respectively generalize the notions of Kobayashi and Brody hyperbolicity from complex geometry. To make sense of the former, we introduce the "KR $\phi$-metric," a decreasing Finsler pseudo-metric that specializes to the Kobayashi-Royden pseudo-metric in the Kahler case. We prove that $R_\phi$-hyperbolicity implies $\phi$-hyperbolicity, and give examples showing that the converse fails in general. Moreover, for constant-coefficient, inner Mobius rigid calibrations $\phi$ in $\mathbb{R}^n$, we completely characterize those domains that are $\phi$-hyperbolic.
Second, we derive a Schwarz lemma for Smith immersions (a.k.a. conformal $\phi$-curves) into an arbitrary calibrated manifold $(X, \phi)$, thereby extending the Schwarz lemma for holomorphic curves into Kahler manifolds. The relevant Bochner formula features the "$\phi$-sectional curvature," a new notion that includes both the scalar and holomorphic sectional curvatures as special cases. As an application, we prove that calibrated geometries with $\phi$-sectional curvature bounded above by a negative constant are $R_\phi$-hyperbolic, generalizing the corresponding result from complex geometry. As another application, we calculate the KR $\phi$-metric of real, complex, and quaternionic hyperbolic spaces equipped with their natural calibrations. - [446] arXiv:2507.16646 (replaced) [pdf, html, other]
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Title: On the interplay between inverse scattering for asymptotically hyperbolic manifolds and the Calderón problem for the Conformal LaplacianComments: Corrected an hypothesis on the main result. It is valid on constant sectional curvatureSubjects: Analysis of PDEs (math.AP); Differential Geometry (math.DG); Spectral Theory (math.SP)
In this short note, we use the relation obtained by Guillarmou--Guillopé and Chang--González between the generalized eigenvalue problem for asymptotically hyperbolic (AH) manifolds and the Conformal Laplacian, to obtain a new inverse scattering result: on an AH manifold of dimension $n+1$ with constant scalar curvature $-n(n+1)$, we show that the scattering matrix at energy $\frac{n+1}{2}$ determines the jet of the metric on the boundary, up to a diffeomorphism and conformal factor.
- [447] arXiv:2507.17097 (replaced) [pdf, html, other]
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Title: On local rings of finite syzygy representation typeComments: 23 pages, comments are welcome! Make some changes to the introductionSubjects: Commutative Algebra (math.AC); Representation Theory (math.RT)
Let $R$ be a commutative Noetherian local ring. We characterize when its completion has an isolated singularity, thereby strengthening the Dao-Takahashi refinement of the Auslander-Huneke-Leuschke-Wiegand theorem. We investigate the ascent and descent of finite and countable syzygy representation type along the canonical map from $R$ to its completion. One consequence is a complete affirmative answer to Schreyer's conjecture. We explore analogues of Chen's questions in the context of finite Cohen-Macaulay representation type over Cohen-Macaulay rings. The main result in this direction shows that if $R$ is Cohen-Macaulay and there are only finitely many non-isomorphic indecomposable maximal Cohen-Macaulay modules that are locally free on the punctured spectrum, then either $R$ is a hypersurface or every Gorenstein projective module is projective; moreover, every Gorenstein projective module over the completion of $R$ is a direct sum of finite generated ones. Finally, we study dominant local rings, introduced by Takahashi, under certain finite representation type conditions, and identify a new class of virtually Gorenstein rings.
- [448] arXiv:2508.01722 (replaced) [pdf, html, other]
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Title: Ladder Operators for Laguerre-type and Jacobi-type Orthogonal PolynomialsSubjects: Classical Analysis and ODEs (math.CA); Mathematical Physics (math-ph)
In the literature concerning the Laguerre-type weight function $x^\lambda w_0(x), x\in[0,+\infty)$, the Jacobi-type weight function $(1-x)^{\alpha}(1+x)^{\beta}w_0(x),x\in[-1,1]$, and the shifted Jacobi-type weight function $x^{\alpha}(1-x)^{\beta}w_0(x), x\in[0,1]$, with $w_0(x)$ continuously differentiable, the parameters $\lambda,\alpha,\beta$ are usually constrained to be strictly positive to ensure the validity of the results. Recently, in [C. Min and P. Fang, Physica D 473 (2025), 134560 (9pp)], the ladder operators for the monic Laguerre-type orthogonal polynomials with $\lambda>-1$ were derived by exploiting the orthogonality properties. The quantities $A_n$ and $B_n$, which appear as coefficients in the ladder operators, exhibit different expressions compared with the previous ones for $\lambda>0$. In this paper, we construct an alternative deduction by making use of the Riemann-Hilbert problem satisfied by the orthogonal polynomials. Moreover, we employ both derivation strategies mentioned above to produce the ladder operators for the monic standard and shifted Jacobi-type orthogonal polynomials with $\alpha,\beta>-1$. When $\lambda,\alpha,\beta$ are restricted to positive values, our expressions of $A_n$ and $B_n$ are consistent with those in prior work. We present examples to validate our findings and generalize the existing conclusions, established by using the three compatibility conditions of the ladder operators and differentiating the orthogonality relations for the monic orthogonal polynomials, from $\lambda,\alpha,\beta>0$ to $\lambda,\alpha,\beta>-1$.
- [449] arXiv:2508.03390 (replaced) [pdf, html, other]
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Title: Two operator splitting methods for three-dimensional stochastic Maxwell equations with multiplicative noiseSubjects: Numerical Analysis (math.NA)
In this paper, we develop two energy-preserving splitting methods for solving three-dimensional stochastic Maxwell equations driven by multiplicative noise. We use operator splitting methods to decouple stochastic Maxwell equations into simple one-dimensional subsystems and construct two stochastic splitting methods, Splitting Method I and Splitting Method II, through a combination of spatial compact difference methods and the midpoint rule in time discretization for the deterministic parts, and exact unitary analytical solutions for the stochastic parts. Theoretical proofs show that both methods strictly preserve the discrete energy conservation law. Finally, numerical experiments fully verify the energy conservation of the methods and demonstrate that the temporal convergence order of the two splitting methods is first-order.
- [450] arXiv:2508.06198 (replaced) [pdf, html, other]
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Title: Distribution Dependent Birth-Death Processes: $\mathbb{W}_p$-Estimate, Ergodicity and Propagation of ChaosSubjects: Probability (math.PR)
For a class of time inhomogenous distribution dependent birth-death processes, we derive the well-posedness, $\mathbb{W}_p$-estimate, exponential ergodicity, and uniform in time propagation of chaos. These extend the corresponding results derived for distribution dependent SDEs and mean field particle systems. As preparation, a criterion on the well-posedness of inhomogenous jump process is presented in the end of the paper, which should be interesting by itself.
- [451] arXiv:2508.06933 (replaced) [pdf, other]
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Title: Global well-posedness and asymptotic behavior of large strong solutions to the 3D full compressible Navier-Stokes equations with temperature-dependent coefficientsComments: 72 pages, all comments are welcomeSubjects: Analysis of PDEs (math.AP)
It is well known that the global well-posedness of the Navier-Stokes equations with temperature-dependent coefficients is a challenging problem, especially in multi-dimensional space. In this paper, we study the 3D Navier-Stokes equations with temperature-dependent coefficients in the whole space. When the initial density and the initial temperature are linearly equivalent to some large constant states, we establish the first result on the global existence of large strong solution. Moreover, the optimal decay rates of the solution to its associated equilibrium are established when the initial data belong to $L^{p_0}(\mathbb{R}^3)$ for some $p_0\in[1,2]$.
- [452] arXiv:2508.07734 (replaced) [pdf, html, other]
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Title: Selberg orthogonality for half-integral weight modular formsComments: 11 pagesSubjects: Number Theory (math.NT)
The Keating--Snaith conjecture for orthogonal families may be viewed as analogous to a Gaussian distribution with a negative mean, and the possibility that mixed moments resemble a composition of independent moments, these two insights were combined and applied in Lester and Radziwi{łł}'s proof of quantum unique ergodicity for half-integral weight automorphic forms, via Soundararajan's method under the Generalized Riemann Hypothesis (GRH). This observation also yields a crucial and nontrivial saving in the resolution of certain arithmetic problems.
Inspired by this, we select a series of typical mixed orthogonal families of $L$-functions: $\mathrm{GL}_2$ quadratic twisted families, Gao and Zhao established a sharp upper bound by building upon Harper's method, and one can replace square-free numbers with primes in this argument. Under the assumptions of the GRH and the Generalized Ramanujan Conjecture, we present the following three arithmetic applications:
i) The decorrelation of Fourier coefficients of half-integral weight modular forms, specifically, a variant of Selberg orthogonality for distinct half-integral weight modular forms.
ii) The decorrelation of automorphic periods averaged over prime imaginary quadratic fields.
iii) The decorrelation of the analytic orders of isotropy subgroups of Tate--Shafarevich groups of elliptic curves under prime quadratic twists. - [453] arXiv:2508.12369 (replaced) [pdf, html, other]
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Title: Prüfer Transformation and Spectral Analysis for a Sturm--Liouville-Type EquationSubjects: Classical Analysis and ODEs (math.CA); Spectral Theory (math.SP)
We study a second-order differential equation involving a quasi-derivative, leading to a non-self-adjoint Sturm--Liouville-type problem with four coefficient functions. To analyze this equation, we develop a generalized Prüfer transformation that expresses solutions in terms of amplitude and phase variables. We further prove the monotonicity of eigenfunction zeros with respect to the spectral parameter and derive upper and lower bounds for the eigenvalues.
- [454] arXiv:2508.17910 (replaced) [pdf, html, other]
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Title: Quasi-likelihood inference for SDE with mixed-effects observed at high frequencySubjects: Statistics Theory (math.ST)
We consider statistical inference for a class of dynamic mixed-effect models described by stochastic differential equations whose drift and diffusion coefficients simultaneously depend on fixed- and random-effect parameters. Assuming that each process is observed at high frequency and the number of individuals goes to infinity, we propose a stepwise inference procedure and prove its theoretical properties. The methodology is based on suitable quasi-likelihood functions by profiling the random effect in the diffusion coefficient at the first stage, and then taking the marginal distribution in the drift coefficient in the second stage, resulting in a fully explicit and computationally convenient method.
- [455] arXiv:2508.19741 (replaced) [pdf, html, other]
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Title: Two-color partitions and overpartitions: a combinatorial proofComments: updated acknowledgmentsSubjects: Number Theory (math.NT); Combinatorics (math.CO)
George Andrews and Mohamed El Bachraoui recently explored identities for two-color partitions. In particular, they studied the connection between two-colored partitions and overpartitions. Their proofs were analytical, but they conjectured combinatorial proofs of their results. In this paper we use two-modular diagrams to give a combinatorial proof of their main result.
- [456] arXiv:2509.01000 (replaced) [pdf, html, other]
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Title: Colourful Carathéodory's Theorem plus a constraintComments: Dedicated to the memory of Martin Aigner, a great man and an excellent mathematicianSubjects: Algebraic Topology (math.AT); Metric Geometry (math.MG)
We develop a topological framework in an attempt to generalize the classical colourful Caratheodory theorem by imposing an additional constraint. For that we introduce the notion of zero-avoding complexes and covering criteria for the existence of colourful transversals. Using the developed method in combination with the homological Nerve theorem of Meshulam we recover all known versions of the colourful Caratheodory's theorem and prove a constraint extension which, in particular, implies an extension of the original (affine) Tverberg result.
- [457] arXiv:2509.03777 (replaced) [pdf, html, other]
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Title: Quadrature Domains and the Faber TransformComments: 60 Pages, 21 FiguresSubjects: Complex Variables (math.CV); Mathematical Physics (math-ph); Analysis of PDEs (math.AP)
We present a framework for reconstructing any simply connected, bounded or unbounded, quadrature domain $\Omega$ from its quadrature function $h$. Using the Faber transform, we derive formulae directly relating $h$ to the Riemann map for $\Omega$. Through this approach, we obtain a complete classification of one point quadrature domains with complex charge. We proceed to develop a theory of weighted quadrature domains with respect to weights of the form $\rho_a(w)=|w|^{2(a-1)}$ when $a > 0$ ("power-weighted" quadrature domains) and the limiting case of when $a=0$ ("log-weighted" quadrature domains). Furthermore, we obtain Faber transform formulae for reconstructing weighted quadrature domains from their respective quadrature functions. Several examples are presented throughout to illustrate this approach both in the simply connected setting and in the presence of rotational symmetry.
- [458] arXiv:2509.03992 (replaced) [pdf, html, other]
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Title: Divergence-kernel method for linear responses of densities and generative modelsComments: Revised. Expanded discussion of generative models, renamed the method DK-SDE, added 20D experiments and hyperparameter sweeps, and updated implementation/runtimes (JAX)Subjects: Dynamical Systems (math.DS); Machine Learning (cs.LG); Probability (math.PR)
We derive the divergence-kernel formula for the linear response of random dynamical systems. Specifically, the pathwise expression is for the parameter-derivative of the marginal or stationary density, not an averaged observable. Our formula works for multiplicative and parameterized noise over any period of time; it does not require hyperbolicity. Then we derive a Monte-Carlo algorithm for linear responses.
We develop a new framework of generative models, DK-SDE, where the model is a parameterized SDE, that (1) directly uses the KL divergence between the empirical data distribution and the marginal density of the SDE as the training objective, and (2) accommodates parametrizations in both drift and diffusion over a long time span, allowing prior structural knowledge to be incorporated explicitly. The optimization is done by gradient-descent enabled by the divergence-kernel method, which involves only forward processes and therefore substantially reduces memory cost. We demonstrate the new model on a 20-dimensional Lorenz system. - [459] arXiv:2509.07524 (replaced) [pdf, html, other]
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Title: Nagell-Lutz Theorem for Imaginary Quadratic Fields and class groups of quadratic fieldsSubjects: Number Theory (math.NT)
We prove the Nagell-Lutz theorem for the imaginary quadratic fields of class number one.
- [460] arXiv:2509.07537 (replaced) [pdf, html, other]
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Title: Two-dimensional fractional Brownian motion: Analysis in time and frequency domainsComments: 30 pages, 7 figuresSubjects: Probability (math.PR); Statistical Mechanics (cond-mat.stat-mech)
This article introduces a novel construction of the two-dimensional fractional Brownian motion (2D fBm) with dependent components. Unlike similar models discussed in the literature, our approach uniquely accommodates the full range of model parameters and explicitly incorporates cross-dependencies and anisotropic scaling through a matrix-valued Hurst operator. We thoroughly analyze the theoretical properties of the proposed causal and well-balanced 2D fBm versions, deriving their auto- and cross-covariance structures in both time and frequency domains. In particular, we present the power spectral density of these processes and their increments. Our analytical findings are validated with numerical simulations. This work provides a comprehensive framework for modeling anomalous diffusion phenomena in multidimensional systems where component interdependencies are crucial.
- [461] arXiv:2509.11887 (replaced) [pdf, html, other]
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Title: Frame redundancy and Beurling densityComments: The main results have been improved by showing that the upper Beurling density is close to critical density. In addition, new statements about frame bounds have been addedSubjects: Functional Analysis (math.FA); Classical Analysis and ODEs (math.CA)
We show that the frame measure function of a frame in certain reproducing kernel Hilbert spaces on metric measure spaces is given by the reciprocal of the Beurling density of its index set. In addition, we show that each such frame with Beurling density greater than one contains a subframe with Beurling density arbitrary close to one. This confirms that the concept of frame measure function as introduced by Balan and Landau is a meaningful quantitative definition for the redundancy of a large class of infinite frames. In addition, it shows that the necessary density conditions for sampling in reproducing kernel Hilbert spaces obtained by Führ, Gröchenig, Haimi, Klotz and Romero are optimal. As an application, we also settle the open questions of the existence of frames near the critical density for exponential frames on unbounded sets and for nonlocalized Gabor frames. The techniques used in this paper combine a selector form of Weaver's conjecture and various methods for quantifying the overcompleteness of frames.
- [462] arXiv:2509.12354 (replaced) [pdf, html, other]
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Title: Toroidal Cartesian Products Where One Factor is 3-ConnectedComments: 13 pages, 2 figuresSubjects: Combinatorics (math.CO)
In this paper, we show that if $G$ is $3$-connected, then the Cartesian product of graphs $G \square H$ embeds on the torus if and only if $G$ is outer-cylindrical and $H$ is a path on two vertices, $P_2$. As a by-product of our work, we also show that $K_{4} \square P_{3}$ has genus two.
- [463] arXiv:2509.16148 (replaced) [pdf, html, other]
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Title: On Tent Spaces for the Gaussian MeasureSubjects: Analysis of PDEs (math.AP); Classical Analysis and ODEs (math.CA)
Following the scheme of tent spaces in classical harmonic analysis developed by R. Coifman, Y. Meyer, and E. Stein in \cite{cms}, we succeed in doing so for the Gaussian setting. In \cite{MNP}, part of this theory (an atomic decomposition) is developed for a specific tent space where functions are defined just in a proper subset of $\mathbb{R}^{n+1}_+,$ and without the use of an area function. In the present paper, using a variation of the area function considered in \cite{FSU}, we define the Gaussian area function and Gaussian tent spaces and prove both their atomic decompositions and the characterization of their dual spaces. Some applications are also considered.
- [464] arXiv:2509.18628 (replaced) [pdf, html, other]
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Title: A New Approach to Defining Cochain Complexes for Dendriform and Pre-Lie AlgebrasSubjects: Rings and Algebras (math.RA)
Our constructions provide a systematic way to study cohomology pre-algebraic structures via classical cohomology, simplifying computations and enabling the use of established techniques.
- [465] arXiv:2509.19950 (replaced) [pdf, html, other]
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Title: Stäckel and Eisenhart lifts, Haantjes geometry and GravitationComments: 26 pages, no figuresSubjects: Mathematical Physics (math-ph); Differential Geometry (math.DG)
We study lifts of integrable systems by means of generalized Stäckel geometry. To this aim, we present the notion of Stäckel lift as a unified setting for the construction of new classes of integrable Hamiltonian systems of physical interest. The Stäckel lift extends the geometric framework underlying both the Riemannian and the Lorentzian-type classical Eisenhart lifts. Moreover, we prove that Hamiltonian systems constructed through momentum-dependent Stäckel matrices are naturally endowed with a non-trivial symplectic-Haantjes structure.
We further illustrate applications to magnetic systems separable in cylindrical coordinates; we describe them within the Stäckel framework by means of modified Stäckel basis.
We also show that explicitly momentum-dependent lifting matrices produce systems interpretable as gravitational waves, or momentum-dependent metrics of Hamilton and Finsler geometries, with potential applications in modified gravity theories. - [466] arXiv:2509.20213 (replaced) [pdf, html, other]
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Title: SU(N) integrals and tau functionsComments: 12 pagesSubjects: Mathematical Physics (math-ph)
We present a family of solvable multi-matrix models associated with an arbitrary embedded graph $\Gamma$ with a single vertex. The graph with $n$ edges is equipped with $2n$ corner matrices. The partition function of each member of the family depends on the set of eigenvalues of monodromies of corner matrices around the vertices of the dual graph $\Gamma^*$ and sets of parameters attached to each vertex of $\Gamma$. We select the cases where the partition function of a model is a tau function of KP, 2KP and BKP hiearachies. We compare integrals over ${U}(N)$ and over ${SU}(N)$ groups. In $U(N)$ case there is no restriction on the number of vertices of $\Gamma$. We also consider mixed ensembles of matrices from $GL(N),U(N)$ and $SU(N)$.
- [467] arXiv:2509.21744 (replaced) [pdf, html, other]
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Title: Various Diamond Properties in Combinatorial Game TheorySubjects: Combinatorics (math.CO)
We investigate conditions under which positions in combinatorial games admit simple values. We introduce a unified diamond framework, the $\Diamond_A$-property ($A\in\{\mathbb{Z},\mathbb{D}$), for sets of positions closed under options. Under certain conditions, this framework guarantees that all values are integers, dyadic rationals, or pairs $\{m|n\}$ (on $\mathbb{Z}$ or $\mathbb{D}$). As an application, we establish that every position in \textsc{Yashima} game on bipartite graphs has an integer pair value.
- [468] arXiv:2509.22470 (replaced) [pdf, html, other]
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Title: Geometric inequalities for convex spacelike hypersurface in de Sitter spaceComments: 15 pagesSubjects: Differential Geometry (math.DG)
In this paper, the long-time existence and convergence results are derived for locally constrained flows with initial value some compact spacelike hypersurface that is suitably pinched in the de Sitter space. As applications, geometric inequalities related to the quermassintegrals as well as the weighted curvature integrals are established.
- [469] arXiv:2509.24645 (replaced) [pdf, html, other]
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Title: Strong BSD for abelian surfaces and the Bloch-Beilinson conjectureComments: 12 pages, certain incomplete arguments have been addressedSubjects: Algebraic Geometry (math.AG); Number Theory (math.NT)
In this paper, we prove the Bloch-Beilinson conjecture for certain abelian surfaces over $\mathbb{Q}$, provided that the BSD is known for these abelian surfaces.
- [470] arXiv:2510.01015 (replaced) [pdf, html, other]
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Title: Quantifying the noise sensitivity of the Wasserstein metric for imagesSubjects: Statistics Theory (math.ST)
Wasserstein metrics are increasingly being used as similarity scores for images treated as discrete measures on a grid, yet their behavior under noise remains poorly understood. In this work, we consider the sensitivity of the signed Wasserstein distance with respect to pixel-wise additive noise and derive non-asymptotic upper bounds. Among other results, we prove that the error in the signed 2-Wasserstein distance scales with the square root of the noise standard deviation, whereas the Euclidean norm scales linearly. We present experiments that support our theoretical findings and point to a peculiar phenomenon where increasing the level of noise can decrease the Wasserstein distance. A case study on cryo-electron microscopy images demonstrates that the Wasserstein metric can preserve the geometric structure even when the Euclidean metric fails to do so.
- [471] arXiv:2510.03009 (replaced) [pdf, html, other]
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Title: Homogeneous steady states for the generalized surface quasi-geostrophic equationsComments: 51 pages, 10 figuresSubjects: Analysis of PDEs (math.AP)
We consider homogeneous (stationary self-similar) solutions to the generalized surface quasi-geostrophic (gSQG) equations parametrized by the constant $0<s<1$, representing the 2D Euler equations ($s=1$), the SQG equations $(s=1/2)$, and stationary equations ($s=0$); namely, solutions whose stream function $\psi$ and advected scalar $\omega$ are of the form \begin{align*} \psi=\frac{w(\theta)}{r^{\beta}},\quad \omega=\frac{g(\theta)}{r^{\beta+2s}}, \end{align*} in polar coordinates $(r,\theta)$ with parameter $\beta\in \mathbb{R}$. We classify homogeneous steady states across the full parameter space, and we identify the limiting singular regimes assuming an odd symmetric profile $(w,g)$ with Fourier modes larger than $m_0\geq 1$. Specifically, we show existence of such solutions for $-m_0-2s<\beta<-2s$ and $0<\beta<m_0+2$ ($1/2-s<\beta< m_0+2$ for $0<s<1/2$) and non-existence of such solutions for $-2s\leq \beta\leq 0$. The main result provides examples of self-similar solutions which belong to critical and supercritical regimes for the local well-posedness of the gSQG equations for $0<s<1$ and the first examples of self-similar solutions for the SQG equations and the more singular equations $0<s\leq 1/2$ in the stationary setting. We also complement our findings with a numerical illustration of the solutions.
- [472] arXiv:2510.04244 (replaced) [pdf, html, other]
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Title: Spectral gap for the signed interchange process with arbitrary setsSubjects: Probability (math.PR); Group Theory (math.GR); Representation Theory (math.RT)
In 2020, F. Cesi introduced a random walk on the hyperoctahedral group $B_n$ and analysed its spectral gap when the allowed generators are transpositions and diagonal elements corresponding to singletons. In this paper we extend the allowed generators to transpositions and any diagonal elements, and characterise completely the set of representations from which the spectral gap arises. This settles a conjecture posed in Cesi's paper.
- [473] arXiv:2510.16283 (replaced) [pdf, html, other]
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Title: Weakly localized states of one dimensional Schrodinger equations have localized energyComments: 31 pages, comments welcome!Subjects: Analysis of PDEs (math.AP)
We study the asymptotics of the Schrödinger equation with time-dependent potential in dimension one. Assuming that the potential decays sufficiently rapidly as $|x| \to \infty$, we prove that the solution can be written as the sum of a free wave $e^{-it\Delta} u_+$ and a weakly bound component $u_{\text{wb}}(t)$. Moreover, we show that the weakly bound part decomposes as $u_{\text{wb}}(t) = u_{\text{loc}}(t) + o_{\dot{H}^1}(1)$, where $\partial_x u_\text{loc}(t)$ is localized near the origin uniformly in time. Since decay conditions on the potential do not preclude resonances unless $d \geq 5$, our results can be seen as a natural extension of [Terence Tao. Dynamics of Partial Differential Equations, 5(2), 2008] and [Avy Soffer, Xiaoxu Wu. arXiv:2304.04245] to the lower-dimensional case.
- [474] arXiv:2510.18758 (replaced) [pdf, html, other]
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Title: Quasilinear Elliptic Cooperative and Competitive SystemsComments: Keywords: Subcritical nonlinearities, gradient elliptic systems, least energy solutions, mixed cooperation and competition, Dirichlet boundary conditions, quasilinear elliptic equations, nonsmooth critical point theorySubjects: Analysis of PDEs (math.AP)
We study the existence and multiplicity of weak solutions for the following quasilinear elliptic system: \[ \begin{cases} -\mathrm{div}(A_1(x,u_1)\nabla u_1) + \displaystyle\frac{1}{2} D_{u_1}A_1(x,u_1)\nabla u_1 \cdot \nabla u_1 = \lambda_1 u_1 + g_{\beta,1}(u) & \text{in } \Omega, \\[3mm] -\mathrm{div}(A_2(x,u_2)\nabla u_2) + \displaystyle\frac{1}{2} D_{u_2}A_2(x,u_2)\nabla u_2 \cdot \nabla u_2 = \lambda_2 u_2 + g_{\beta,2}(u) & \text{in } \Omega, \\[2mm] u_1 = u_2 = 0 & \text{on } \partial\Omega, \end{cases} \] where $\lambda_1, \lambda_2 < \mu_1$, the first Dirichlet eigenvalue of the Laplacian, and $\Omega$ is a bounded domain. The nonlinearity derives from a potential $G_\beta$ with subcritical growth.
Due to the lack of differentiability of the associated energy functional, we employ nonsmooth critical point theory and variational methods based on the concept of weak slope. We prove the existence of least energy solutions in both the cooperative ($\beta > 0$) and competitive ($\beta < 0$) regimes. - [475] arXiv:2510.20282 (replaced) [pdf, html, other]
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Title: On 4-dimensional 3-handle attachmentsSubjects: Geometric Topology (math.GT)
Kirby diagrams for smooth four-dimensional manifolds typically depict only the 1- and 2-handles, omitting the 3-handles. In this work, we undertake a study of 3-handle attachments and provide tools to explicitly include them in handle diagrams. We show a set of moves involving 3-handles to extend the classical Kirby calculus. Under the condition that the number of 3-handles equals the rank of the spherical part of the specific boundary's second homology group, we establish a homological criterion that identifies a geometric basis of disjoint embedded spheres in the boundary corresponding to 3-handle attachments, yielding a uniqueness theorem for 3-handle attachments.
- [476] arXiv:2510.23952 (replaced) [pdf, html, other]
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Title: A Fixed Point Theorem for Generalized Strictly Nonexpansive Mappings on Bounded Sets in Complete Metric SpacesComments: 15pagesSubjects: Functional Analysis (math.FA)
This paper offers substantial advances in the theory of fixed points for generalized strictly nonexpansive mappings. We develop a novel proof technique based on nonstandard analysis to establish a new fixed-point theorem. The core result demonstrates that, in a complete metric space, every continuous generalized strictly nonexpansive mapping with a bounded orbit possesses a unique fixed point to which all iterative sequences converge. The significance of this theorem lies in its substantial relaxation of the classical framework: it entirely dispenses with compactness and convexity requirements, which are typically indispensable in the study of nonexpansive mappings (such as in the Browder--Göhde theorem), replacing them solely with a boundedness condition.
- [477] arXiv:2510.24170 (replaced) [pdf, html, other]
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Title: SymMaP: Improving Computational Efficiency in Linear Solvers through Symbolic PreconditioningSubjects: Numerical Analysis (math.NA); Artificial Intelligence (cs.AI)
Matrix preconditioning is a critical technique to accelerate the solution of linear systems, where performance heavily depends on the selection of preconditioning parameters. Traditional parameter selection approaches often define fixed constants for specific scenarios. However, they rely on domain expertise and fail to consider the instance-wise features for individual problems, limiting their performance. In contrast, machine learning (ML) approaches, though promising, are hindered by high inference costs and limited interpretability. To combine the strengths of both approaches, we propose a symbolic discovery framework-namely, Symbolic Matrix Preconditioning (SymMaP)-to learn efficient symbolic expressions for preconditioning parameters. Specifically, we employ a neural network to search the high-dimensional discrete space for expressions that can accurately predict the optimal parameters. The learned expression allows for high inference efficiency and excellent interpretability (expressed in concise symbolic formulas), making it simple and reliable for deployment. Experimental results show that SymMaP consistently outperforms traditional strategies across various benchmarks.
- [478] arXiv:2511.00752 (replaced) [pdf, html, other]
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Title: Model-free source seeking of exponentially convergent unicycle: theoretical and robotic experimental resultsSubjects: Optimization and Control (math.OC); Robotics (cs.RO)
This paper introduces a novel model-free, real-time unicycle-based source seeking design. This design autonomously steers the unicycle dynamic system towards the extremum point of an objective function or physical/scalar signal that is unknown expression-wise, but accessible via measurements. A key contribution of this paper is that the introduced design converges exponentially to the extremum point of objective functions (or scalar signals) that behave locally like a higher-degree power function (e.g., fourth-degree polynomial function) as opposed to locally quadratic objective functions, the usual case in literature. We provide theoretical results and design characterization, supported by a variety of simulation results that demonstrate the robustness of the proposed design, including cases with different initial conditions and measurement delays/noise. Also, for the first time in the literature, we provide experimental robotic results that demonstrate the effectiveness of the proposed design and its exponential convergence ability.
- [479] arXiv:2511.00938 (replaced) [pdf, html, other]
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Title: Persistence-Based Statistics for Detecting Structural Changes in High-Dimensional Point CloudsComments: 45 pages, 3 figures, under reviewSubjects: Statistics Theory (math.ST); Algebraic Topology (math.AT); Probability (math.PR)
We study the probabilistic behavior of persistence-based statistics and propose a novel nonparametric framework for detecting structural changes in high-dimensional random point clouds. We establish moment bounds and tightness results for classical persistence statistics-total and maximum persistence-under general distributions, with explicit variance-scaling behavior derived for Gaussian mixture models. Building on these results, we introduce a bounded and normalized statistic based on persistence landscapes combined with the Jensen-Shannon divergence, and we prove its Holder continuity with respect to perturbations of the input point clouds. The resulting measure is stable, scale- and shift-invariant, and well suited for finite-sample nonparametric inference via permutation testing. An illustrative numerical study using dynamic attribute vectors from decentralized governance data demonstrates the practical applicability of the proposed method. Overall, this work provides a statistically rigorous and computationally stable approach to change-point detection in complex, high-dimensional data.
- [480] arXiv:2511.01245 (replaced) [pdf, html, other]
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Title: A curiously slowly mixing Markov chainComments: Please feel free to make comments! (The connection to Schur--Weyl duality has been moved to a separate paper, and some results have been slightly improved.)Subjects: Probability (math.PR); Combinatorics (math.CO); Representation Theory (math.RT)
We study a Markov chain with very different mixing rates depending on how mixing is measured. The chain is the "Burnside process on the hypercube $C_2^n$." Started at the all-zeros state, it mixes in a bounded number of steps, no matter how large $n$ is, in $\ell^1$ and in $\ell^2$. And started at general $x$, it mixes in at most $\log n$ steps in $\ell^1$. But, in $\ell^2$, it takes $\frac{n}{\log n}$ steps for most starting $x$. The $\ell^2$ mixing results follow from an explicit diagonalization of the Markov chain into binomial-coefficient-valued eigenvectors.
- [481] arXiv:2511.02813 (replaced) [pdf, html, other]
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Title: A Construction of Infinite Families of Self-Orthogonal Quasi-Cyclic Codes Using Constituent CodesComments: 20 pagesSubjects: Information Theory (cs.IT)
Quasi-cyclic codes have been recently employed in the constructions of quantum error-correcting codes. In this paper, we propose a construction of infinite families of quasi-cyclic codes which are self-orthogonal with respect to the Euclidean and Hermitian inner products. In particular, their dimension and a lower bound for their minimum distance are computed using their constituent codes defined over field extensions of $\mathbb{F}_q$. We also show that the lower bound for the minimum distance satisfies the square-root-like lower bound and also show how self-dual quasi-cyclic codes can arise from our construction. Using the CSS construction, we show the existence of quantum error-correcting codes with good parameters.
- [482] arXiv:2511.08870 (replaced) [pdf, other]
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Title: Gaussian Approximation for High-Dimensional Second-Order $U$- and $V$-statistics with Size-Dependent Kernels under i.n.i.d. SamplingSubjects: Statistics Theory (math.ST)
We develop Gaussian approximations for high-dimensional vectors formed by second-order $U$- and $V$-statistics whose kernels depend on sample size under independent but not identically distributed (i.n.i.d.) sampling. Our results hold irrespective of which component of the Hoeffding decomposition is dominant, thereby covering both non-degenerate and degenerate regimes as special cases. By allowing i.n.i.d.~sampling, the class of statistics we analyze includes weighted $U$- and $V$-statistics and two-sample $U$- and $V$-statistics as special cases, which cover estimators of parameters in regression models with many covariates, many-weak instruments as well as a broad class of smoothed two-sample tests and the separately exchangeable arrays, among others. In addition, we extend sharp maximal inequalities for high-dimensional $U$-statistics with size-dependent kernels from the i.i.d.~to the i.n.i.d.~setting, which may be of independent interest.
- [483] arXiv:2511.09156 (replaced) [pdf, html, other]
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Title: Zero-Order Sharpness-Aware MinimizationSubjects: Statistics Theory (math.ST)
Prompt learning has become a key method for adapting large language models to specific tasks with limited data. However, traditional gradient-based optimization methods for tuning prompts are computationally intensive, posing challenges for efficiency. We introduce ZOSA (Zero-Order Sharpness-Aware Minimization), a novel optimization framework that integrates zero-order optimization with sharpness-aware minimization to enhance prompt tuning. ZOSA employs Rademacher perturbation vectors to estimate gradients without requiring backpropagation. By incorporating sharpness-aware principles, it targets flat minima in the loss landscape, improving generalization. An adaptive learning rate, guided by loss variability, further ensures stable convergence. Experiments on few-shot learning tasks, such as text classification and natural language inference, show that ZOSA significantly outperforms existing methods. With its theoretical foundation and computational efficiency, ZOSA offers a practical solution for prompt-based learning in resource-limited settings.
- [484] arXiv:2511.12862 (replaced) [pdf, html, other]
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Title: Word Length Formulae and Normal Forms of Conjugacy Classes in Surface GroupsComments: 61 pages, 6 figures; Section 9 added; all other content unchangedSubjects: Geometric Topology (math.GT); Group Theory (math.GR)
In this paper, we primarily investigate the following symmetric presentation of the surface group $\pi_1(\Sigma_g)=\left\langle c_1,\dots, c_{2g}\mid c_1\cdots c_{2g}c_1^{-1}\cdots c_{2g}^{-1}\right\rangle$. For every nontrivial element $x\in \pi_1(\Sigma_g)$, we obtain a uniform representation of the normal forms of $x^k$ under the length-lexicographical order. Based on this, we find a new relation among these normal forms, and then derive the following three formulae related to the word length: $|x^2|>|x|$; $|x^k|=(k-1)(|x^2|-|x|)+|x|$; $\lim_{k\to\infty}\frac{|x^k|}{k}=|x^2|-|x|$. Moreover, we extend these results to obtain analogous but less precise formulae for every minimal geometric presentation. Then, we define the normal forms of conjugacy classes in $\pi_1(\Sigma_g)$ and give a criterion for determining the conjugacy of elements. As a consequence, we give efficient algorithms for solving the root-finding and conjugacy problems. Finally, we present applications concerning the computation of some growth rates.
- [485] arXiv:2511.14985 (replaced) [pdf, html, other]
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Title: On the existence of universal links in three-manifoldsFrancisco González-Acuña, Araceli Guzmán-Tristán, Jesús Rodríguez-Viorato, José Andrés Rodríguez MiguelesComments: 8 pagesSubjects: Geometric Topology (math.GT)
We study the existence of branched coverings between closed $3$-manifolds, with emphasis on universal knots and links. We prove that the only closed $3$-manifolds that admit a universal link are spherical. Furthermore, we distinguish between universal links and complement universal links and show that these notions do not coincide in general, by exhibiting infinitely many examples of complement universal links that are not universal. Also, we prove that there is no closed aspherical $3$-manifold, such that every closed, aspherical $3$-manifold is a branched covering over it. Finally, we characterize the closed $3$-manifolds admitting branching coverings from $P^3 \# P^3$, and deduce that there is no closed reducible $3$-manifold, such that every closed reducible $3$-manifold is a branched covering over it.
- [486] arXiv:2511.16994 (replaced) [pdf, html, other]
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Title: Kovalevskaya exponents of the Riccati hierarchySubjects: Classical Analysis and ODEs (math.CA); Dynamical Systems (math.DS)
We study the Kovalevskaya exponents of the Riccati hierarchy, deriving the general indicial loci and exponents. A unique commuting vector field is identified, and the general solution is obtained via symmetry reduction. Finally, Laurent series expansions in annular regions are analyzed to reveal the singular structure of the solutions.
- [487] arXiv:2511.20827 (replaced) [pdf, html, other]
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Title: Hyperbolicity of adjoint linear series on varieties with positive tangent bundleComments: 29 pages. v2: added more results about abelian varieties, removed results about Fano varietiesSubjects: Algebraic Geometry (math.AG)
Let $X$ be a smooth projective variety of dimension $n\geq 3$, and let $L$ be an ample line bundle on $X$. In this article, we study the algebraic hyperbolicity of a very general section of the adjoint linear series $|K_X+mL|$ when the tangent bundle $T_X$ of $X$ has suitable positivity properties. As a consequence, we show that the linear system $|K_X+mL|$ is hyperbolic (or pseudo-hyperbolic) for $m\geq 3n+1$, for various classes of polarized pairs $(X,L)$, thus providing new evidence of a conjecture that was proposed by the second and fourth authors. Moreover, when $X$ is abelian, we show that the linear system $|mL|$ is hyperbolic for $m\geq n$, and the same holds when $m\geq n-1$, if $|L|$ has no base divisors. It turns out that these bounds for abelian varieties are sharp. We also prove analogous statements for Kummer varieties and certain classes of hyperelliptic varieties.
- [488] arXiv:2511.21632 (replaced) [pdf, other]
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Title: Dynamics of generalized abcd Boussinesq solitary waves under a slowly variable bottomComments: v2: 82 pp., corrected typos, simplified some computations, expanded references, submitted versionSubjects: Analysis of PDEs (math.AP); Mathematical Physics (math-ph)
The Boussinesq $abcd$ system is a 4-parameter set of equations posed in $\mathbb R_t\times\mathbb R_x$, originally derived by Bona, Chen and Saut as first-order 2-wave approximations of the incompressible and irrotational, two-dimensional water wave equations in the shallow water wave regime, in the spirit of the original Boussinesq derivation. Among the various particular regimes, each determined by the values of the parameters $(a, b, c, d)$ appearing in the equations, the \emph{generic} regime is characterized by the conditions $b, d > 0$ and $a, c < 0$. If additionally $b=d$, the $abcd$ system is Hamiltonian.
In this paper, we investigate the existence of generalized solitary waves and the corresponding collision problem in the physically relevant \emph{variable bottom regime}, introduced by M.\ Chen. More precisely, the bottom is represented by a smooth space-time dependent function $h=\varepsilon h_0(\varepsilon t,\varepsilon x)$, where $\varepsilon$ is a small parameter and $h_0$ is a fixed smooth profile. This formulation allows for a detailed description of weak long-range interactions and the evolution of the solitary wave without its destruction. We establish this result by constructing a new approximate solution that captures the interaction between the solitary wave and the slowly varying bottom. - [489] arXiv:2511.22614 (replaced) [pdf, html, other]
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Title: Differential graded algebras with divided powers and homotopy Lie algebrasComments: 46 pages. Introduction and abstract updated, references added, redaction improved, section 5 updatedSubjects: Representation Theory (math.RT); Commutative Algebra (math.AC)
Given a commutative algebra $A$ and a quotient $A$-algebra $A/I$, we construct a resolution of $A/I$ as an $A$-module such that it is also a differential graded (dg) algebra with divided powers (PD). This construction makes use of symmetric tensors in the symmetric tensor category of dg $A$-modules and does not require a Noetherian assumption on $A$. Moreover, the resolution has many lifting properties which we leverage to study the homotopy Lie algebra associated to the pair $(A,A/I)$, which is defined as the cohomology of the PD derivations of this PD dg algebra. Finally we investigate the complete intersection case in more details as well as connect it to the finite generation of the Yoneda algebra.
- [490] arXiv:2512.00460 (replaced) [pdf, html, other]
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Title: (Quasi-)admissible modules over symmetrizable Kac-Moody superalgebrasSubjects: Representation Theory (math.RT); Mathematical Physics (math-ph)
The theory of admissible modules over symmetrizable anisotropic Kac-Moody superalgebras, introduced by Kac and Wakimoto in late 80's, is a well-developed subject with many applications, including representation theory of vertex algebras. Recently this theory was developed in a more general setup by Gorelik and Serganova. In the present paper we develop in this more general setup the theory of admissible modules over arbitrary symmetrizable Kac-Moody superalgebras.
- [491] arXiv:2512.00758 (replaced) [pdf, html, other]
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Title: Movable Antenna Empowered Near-Field Sensing via Antenna Position OptimizationSubjects: Information Theory (cs.IT); Signal Processing (eess.SP)
Movable antenna (MA) technology exhibits great promise for enhancing the sensing capabilities of future sixth-generation (6G) networks due to its capability to alter antenna array geometry. With the growing prevalence of near-field propagation at ultra-high frequencies, this paper focuses on the application of one-dimensional (1D) and two-dimensional (2D) MA arrays for near-field sensing to jointly estimate the angle and distance information about a target. First, for the 1D MA array scenario, to gain insights into MA-enhanced near-field sensing, we investigate two simplified cases with only angle-of-arrival (AoA) or distance estimation, respectively, assuming that the other information is already known. The worst-case Cramer-Rao bounds (CRBs) on the mean square errors (MSEs) of the AoA estimation and the distance estimation are derived in these two cases. Then, we jointly optimize the positions of the MAs within the 1D array to minimize these CRBs and derive their closed-form solutions, which yield an identical array geometry to MA-enhanced far-field sensing. For the more challenging joint AoA and distance estimation, since the associated worst-case CRB is a highly complex and non-convex function with respect to the MA positions, a discrete sampling-based approach is proposed to sequentially update the MA positions and obtain an efficient suboptimal solution. Furthermore, we investigate the worst-case CRB minimization problems for a 2D MA array under various conditions and extend our proposed algorithms to solve them efficiently. Numerical results demonstrate that the proposed MA-enhanced near-field sensing scheme dramatically outperforms conventional fixed-position antennas (FPAs). Moreover, the joint angle and distance estimation results in a different array geometry from that in the individual estimation of angle/distance or far-field sensing.
- [492] arXiv:2512.01002 (replaced) [pdf, html, other]
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Title: Coarse spaces using extended generalized eigenproblems for heterogeneous Helmholtz problemsSubjects: Numerical Analysis (math.NA)
An abstract construction of coarse spaces for non-Hermitian problems and non-Hermitian domain decomposition preconditioners based on extended generalized eigenproblems was proposed in [Nataf and Parolin, arXiv:2404.02758] and analyzed on the matrix formulation. Building upon this work, we consider instead here the specific case of heterogeneous Helmholtz problems, and the derivation and analysis is performed at the continuous level. Albeit different from its derivation, its use of oversampling and the underlying eigenproblems, our approach shares similarities with the methods of Hu and Li [SIAM J. Numer. Anal, 63(2), 716-743, 2025] and Ma, Alber, Scheichl and Zhang [J. Sci. Comput., 105(3), No. 99, 2025].
- [493] arXiv:2512.01409 (replaced) [pdf, html, other]
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Title: Localization of spectral Turán-type theoremsComments: Updated references and corrected some errorsSubjects: Combinatorics (math.CO)
Let $G$ be a graph, and let $v$ and $e$ be a vertex and an edge of $G$, respectively. Define $c(v)$ (resp. $c(e)$) to be the order of the largest clique in $G$ containing $v$ (resp. $e$). Denote the adjacency eigenvalues of $G$ by $\lambda_1 \ge \cdots \ge \lambda_n$. We study localized refinements of spectral Turán-type theorems by replacing global parameters such as the clique number $\omega(G)$, size $m$ and order $n$ of $G$ with local quantities $c(v)$ and $c(e)$.
Motivated by a conjecture of Elphick, Linz and Wocjan (2024), we first propose a vertex-localized strengthening of Wilf's inequality: \[ \sqrt{s^{+}(G)} \le \sum_{v\in V(G)}\left(1-\frac{1}{c(v)}\right), \] where $s^+(G) = \sum_{\lambda_i > 0}\lambda_i^2$. Inspired by the Bollobás-Nikiforov conjecture (2007) on the first two eigenvalues, we then introduce an edge-localized analogue: \[\lambda_1^2(G) + \lambda_2^2(G) \le \sum_{e\in E(G)} 2\left(1-\frac{1}{c(e)}\right).\] As evidence of their validity, we verify the above conjectures for diamond-free graphs and random graphs. We also propose strengthening of the spectral versions of the Erdős, Stone and Simonovits Theorem by replacing the spectral radius with $\sqrt{s^{+}(G)}$ and establish it for all $F$-free graphs with $\chi(F)=3$. A key ingredient in our proofs is a general upper bound relating $\sqrt{s^{+}(G)}$ to the triangle count $t(G)$. Finally, we prove a localized version of Nikiforov's walk inequality and conjecture a stronger localized version. These results contribute to the broader program of localizing spectral extremal inequalities. - [494] arXiv:2512.03572 (replaced) [pdf, html, other]
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Title: Global embeddings of weakly pseudoconvex complex spaces and refined approximation theoremsComments: v2: 19 pages; added Examples 2.11, 2.12 and 2.13, Theorem 5.4, and Corollary 5.5. v1: 19 pagesSubjects: Complex Variables (math.CV)
In this paper, by refining approximation theorems for holomorphic sections of adjoint line bundles, it is proved that the regular locus of a weakly pseudoconvex complex space admitting a positive line bundle can be holomorphically embedded into a complex projective space. As an application of approximation theorems, it is shown that the Union problem can be solved for weakly pseudoconvex manifolds.
- [495] arXiv:2512.07870 (replaced) [pdf, html, other]
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Title: Mixed exponential statistical structures and their approximation operatorsComments: 12 pagesSubjects: Statistics Theory (math.ST)
The paper examines the construction and analysis of a new class of mixed exponential statistical structures that combine the properties of stochastic models and linear positive operators. The relevance of the topic is driven by the growing need to develop a unified theoretical framework capable of describing both continuous and discrete random structures that possess approximation properties. The aim of the study is to introduce and analyze a generalized family of mixed exponential statistical structures and their corresponding linear positive operators, which include known operators as particular cases. We define auxiliary statistical structures B and H through differential relations between their elements, and construct the main Phillips-type structure. Recurrent relations for the central moments are obtained, their properties are established, and the convergence and approximation accuracy of the constructed operators are investigated. The proposed approach allows mixed exponential structures to be viewed as a generalization of known statistical systems, providing a unified analytical and stochastic description. The results demonstrate that mixed exponential statistical structures can be used to develop new classes of positive operators with controllable preservation and approximation properties. The proposed methodology forms a basis for further research in constructing multidimensional statistical structures, analyzing operators in weighted spaces, and studying their asymptotic characteristics.
- [496] arXiv:2512.09170 (replaced) [pdf, html, other]
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Title: Magic Gems: A Polyhedral Framework for Magic SquaresComments: Connecting Combinatorics, Geometry, and Linear Algebra. 8 figures, ancillary code included. Interactive visualization: this https URLSubjects: Combinatorics (math.CO); Computational Geometry (cs.CG); Discrete Mathematics (cs.DM); Metric Geometry (math.MG)
We introduce Magic Gems, a geometric representation of magic squares as three-dimensional polyhedra. By mapping an n times n magic square onto a centered coordinate grid with cell values as vertical displacements, we construct a point cloud whose convex hull defines the Magic Gem. Building on prior work connecting magic squares to physical properties such as moment of inertia, this construction reveals an explicit statistical structure: we show that magic squares have vanishing covariances between position and value. We develop a covariance energy functional (the sum of squared covariances with individual row, column, and diagonal indicator variables) and prove that for all orders of n greater than or equal to three, an arrangement is a magic square if and only if this complete energy vanishes. This characterization transforms the classical line-sum definition into a statistical orthogonality condition. We also study a simpler low-mode relaxation using only four aggregate position indicators; this coincides with the complete characterization for n equals three (verified exhaustively) but defines a strictly larger class for n greater than or equal to four (explicit counterexamples computed). Perturbation analysis demonstrates that magic squares are isolated local minima in the energy landscape. The representation is invariant under dihedral symmetry D4, yielding canonical geometric objects for equivalence classes.
- [497] arXiv:2512.09330 (replaced) [pdf, html, other]
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Title: Complex exponential integral means spectrums of univalent functions and the Brennan conjectureComments: 40 pagesSubjects: Complex Variables (math.CV)
In this paper we investigate the complex exponential integral means spectrums of univalent functions in the unit disk. We show that all integral means spectrum (IMS) functionals for complex exponents on the universal Teichmüller space, the closure of the universal Teichmüller curve, and the universal asymptotic Teichmüller space are continuous. We also show that the complex exponential integral means spectrum of any univalent function admitting a quasiconformal extension to the extended complex plane is strictly less than the universal integral means spectrum. These extend some related results in our recent work \cite{Jin}. Here we employ a different and more direct approach to prove the continuity of IMS functional on the universal asymptotic Teichmüller space. Additionally, we completely determine the integral means spectrums of all univalent rational functions in the unit disk. As a consequence, we show that the Brennan conjecture is true for this class of univalent functions. Finally, we present some remarks and raise some problems and conjectures regarding IMS functionals on Teichmüller spaces, univalent rational functions, and a multiplier operator whose norm is closely related to the Brennan conjecture.
- [498] arXiv:2512.10368 (replaced) [pdf, html, other]
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Title: Löwner equations and de Branges--Rovnyak spacesComments: Revised 29 December 2025Subjects: Functional Analysis (math.FA); Complex Variables (math.CV)
We study de Branges--Rovnyak spaces parametrized by Löwner equations. A new approach based on the Löwner theory to problem ``Find concrete elements in de Branges--Rovnyak spaces'' is given.
- [499] arXiv:2512.12335 (replaced) [pdf, html, other]
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Title: Hulls of Free Linear Codes over a Non-Unital RingComments: 27Subjects: Information Theory (cs.IT)
This paper investigates the hull codes of free linear codes over a non-unital ring $ E= \langle \kappa,\tau \mid 2 \kappa =2 \tau=0,~ \kappa^2=\kappa,~ \tau^2=\tau,~ \kappa \tau=\kappa,~ \tau \kappa=\tau \rangle$. Initially, we examine the residue and torsion codes of various hulls of $E$-linear codes and obtain an explicit form of the generator matrix of the hull of a free $E$-linear code. Then, we propose four build-up construction methods to construct codes with a larger length and hull-rank from codes with a smaller length and hull-rank. Some illustrative examples are also given to support our build-up construction methods. Subsequently, we study the permutation equivalence of two free $E$-linear codes and discuss the hull-variation problem. As an application, we classify optimal free $E$-linear codes for lengths up to $8$.
- [500] arXiv:2512.12862 (replaced) [pdf, html, other]
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Title: Reversibility in finite-dimensional collapse dynamicsSubjects: Mathematical Physics (math-ph); Dynamical Systems (math.DS)
We study finite dimensional quantum systems with arbitrary collapse events, establishing, under no-information-erasure conditions, a structural no-go for trajectory-level operational irreversibility. More precisely, we fix a realization map (a physically admissible selector of the collapse dynamics) and do not rely on any regularity of the induced dynamics. We prove that, for every realization of the collapse dynamics, there exists a topologically closed, forward-invariant subset of the projective state space on which any two states can be connected with arbitrarily fine Fubini-Study precision and arbitrarily small integrated energetic cost. This shows that the preservation of information along a realized branch guarantees islands of quasi-reversibility, while genuine irreversibility requires additional ingredients such as non-compactness, explicit erasure, or coupling to reservoirs.
KEYWORDS: Quantum collapse dynamics; Quasi-reversibility; Chain-recurrence; Information non-erasure. - [501] arXiv:2512.14607 (replaced) [pdf, html, other]
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Title: Minimal multiplicity of fiber components in abelian fibrationsComments: 9 pages, v. 2.1, added more reference, corrected an errorSubjects: Algebraic Geometry (math.AG)
An abelian fibration is a proper projective surjective map of complex varieties with general fiber an abelian variety. Consider a multiple fiber of an abelian fibration, and let $m_1, ..., m_k$ be the multiplicities of its irreducible components. We prove that the minimum of $m_i$ is equal to their greatest common divisor $gcd(m_1, ..., m_k)$
- [502] arXiv:2512.15356 (replaced) [pdf, html, other]
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Title: A complete dichotomy theorem on the sparse $t$-Uniform Hypergraphicality ProblemComments: 18 pagesSubjects: Combinatorics (math.CO)
We prove a complete dichotomy theorem for the parameterized sparse $t$-uniform hypergraphic degree sequence problem, $\mathrm{sparse}\text{-}t\text{-}\mathrm{uni}\text{-}\mathrm{HDS}_{\alpha',\alpha}$. For any fixed $t \ge 3$, given parameters $0 \le \alpha' \le \alpha < t-1$, the input consists of degree sequences $D$ of length $n$ with degrees between $n^{\alpha'}$ and $6n^{\alpha}$. We show that the problem is NP-complete whenever $\alpha' \le \frac{t(\alpha - 1) + 1}{t - 1}$, and solvable in linear time when $\alpha' > \frac{t(\alpha - 1) + 1}{t - 1}$. This establishes a sharp boundary between polynomial-time solvable and NP-complete instances, thereby characterizing the computational complexity across all degree exponent regimes. The result extends the earlier NP-completeness of dense hypergraphicality to a unified framework covering both sparse and dense regimes, revealing that even extremely sparse instances (with maximum degree $o(n)$ but $\Omega(n^{\frac{t-1}{t}})$) remain NP-complete. On the other hand, the $t$-uniform hypergraphicality solvable in linear time when the maximum degree is $o(n^{\frac{t-1}{t}})$. This dichotomy provides a comprehensive classification of the complexity landscape for hypergraphic degree sequences.
- [503] arXiv:2512.15407 (replaced) [pdf, html, other]
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Title: No exact on average additive complements of squaresComments: The main result is improvedSubjects: Number Theory (math.NT)
Let $\mathbb{N}$ be the set of natural numbers and $\mathcal{S}=\big\{1^2, 2^2, 3^2,\cdots\big\}$ the set of squares. Let $\mathcal{W}$ be an additive complement of $\mathcal{S}$ and $$ f(n)=\#\big\{(w,m^2)\in \mathcal{W}\times \mathcal{S}: n=w+m^2\big\}. $$ It is proved that there is an absolute constant $c_0>0$ such that for all large $N$ we have $$ \sum_{n\le N}f(n)-N\ge c_{0}N, $$ which makes some further progress on a 1993 conjecture of Cilleruelo. As consequences of this result, we answer negatively a 2001 problem of Ruzsa as well as a 2017 problem of Ben Green.
- [504] arXiv:2512.16412 (replaced) [pdf, html, other]
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Title: Higher-order Volterra-type integral operator on Hardy and Bergman spacesComments: 19 pages, 1 figureSubjects: Complex Variables (math.CV)
We investigate the higher-order Volterra-type integral operator $T_{g,n}$ on the unit disk, defined for $n\in\mathbb N$ by \[ T_{g,n}[f](z) := \underbrace{\int_{0}^{z}\int_{0}^{t_1}\cdots\int_{0}^{t_{n-1}}}_{n\ \text{times}} f(t_n)g'(t_n)\,dt_n\cdots dt_1,\quad z\in\mathbb D, \] where $f$ and $g$ are analytic in the unit disk $\mathbb D$. We establish sharp norm and essential norm estimates, and give complete characterizations of boundedness and compactness of $T_{g,n}$ on Hardy spaces $H^p$ and weighted Bergman spaces $A_\alpha^p$, in terms of (vanishing) Carleson measure conditions determined by $|g'|$.
- [505] arXiv:2512.17399 (replaced) [pdf, html, other]
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Title: Minimizing numerical radius of weighted cyclic matrices under permutation of the weightsComments: 20 pagesSubjects: Functional Analysis (math.FA)
In this article we answer a question asked by Chien et al. in arXiv:2304.06050 in which they study the numerical range of weighted cyclic matrices under permutation of their entries. Namely, we are interested in how $w(A_\sigma)$ fluctuates for various permutations $\sigma\in S_n$ and fixed $0\leq a_1<\cdots<a_n$ with $A_\sigma=\begin{pmatrix} 0&a_{\sigma(1)}&{}&{}&{}\cr {}&0&a_{\sigma(2)}&{}&{}\cr {}&{}&\ddots&\ddots&{}\cr {}&{}&{}&\ddots&a_{\sigma(n-1)}\cr a_{\sigma(n)}&{}&{}&{}&0 \end{pmatrix}$. Previous results of Gau \cite{gau2024proof} and Chang and Wang \cite{chang2012maximizing} made clear the case when $w(A_\sigma)$ is maximal among all the $w(A_\mu)$ with $\mu\in S_n$. Chien et al. in arXiv:2304.06050 ask what the permutation which makes $w(A_\sigma)$ minimal for $n\geq 6$ could be. Answering this question is the aim of this note.
- [506] arXiv:2512.18654 (replaced) [pdf, html, other]
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Title: Hierarchical filtrations of vector bundles and birational geometryComments: The author would appreciate any comments or suggestions that may help improve this workSubjects: Algebraic Geometry (math.AG); Commutative Algebra (math.AC)
We introduce and systematically study \emph{hierarchical filtrations} of vector bundles on smooth projective varieties. These are filtrations by saturated subsheaves of equal rank whose successive quotients are torsion sheaves supported in codimension one. The associated numerical invariant, called \emph{hierarchical depth}, measures the maximal length of such filtrations.
We establish general bounds for hierarchical depth in terms of the determinant class and provide exact formulas for smooth curves and varieties of Picard rank one. A key technical result concerns the commutativity of elementary transforms along disjoint divisors and their role in constructing filtrations.
For surfaces, we analyze the behavior of hierarchical depth under birational morphisms and prove that it transforms additively along the steps of the minimal model program. In particular, we obtain an explicit formula relating the depth on a surface to that on its minimal model via exceptional divisor contributions.
As an application, we connect hierarchical depth to degeneracies in algebraic--geometric codes and show that birational simplification via the MMP leads to effective improvements of code parameters. This establishes hierarchical depth as a new bridge between birational geometry, vector bundle theory, and coding theory. - [507] arXiv:2512.19156 (replaced) [pdf, html, other]
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Title: Classical billiards can computeComments: 16 pages, 8 figures. Appendix addedSubjects: Dynamical Systems (math.DS); Computational Complexity (cs.CC); Mathematical Physics (math-ph)
We show that two-dimensional billiard systems are Turing complete by encoding their dynamics within the framework of Topological Kleene Field Theory. Billiards serve as idealized models of particle motion with elastic reflections and arise naturally as limits of smooth Hamiltonian systems under steep confining potentials. Our results establish the existence of undecidable trajectories in physically natural billiard-type models, including billiard-type models arising in hard-sphere gases and in collision-chain limits of celestial mechanics.
- [508] arXiv:2512.19282 (replaced) [pdf, html, other]
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Title: Local Topological Constraints on Berry Curvature in Spin--Orbit Coupled BECsComments: v2: 23 pages, established the structural stability of the topological bound under metric deformations; expanded Section 4 with new lemmas on Kaluza-Klein geometry; added a concrete SOC BEC example demonstrating curvature persistence even for total Chern number c1=0Subjects: Differential Geometry (math.DG); Algebraic Topology (math.AT)
We establish a local topological obstruction to flattening Berry curvature in spin-orbit-coupled Bose-Einstein condensates (SOC BECs), valid even when the global Chern number vanishes. For a generic two-component SOC BEC, the extended parameter space $M=T^{2}_{\mathrm{BZ}}\times S^{1}_{\phi_{+}}\times S^{1}_{\phi_{-}}$ carries a natural metric connection $\nabla^{C}$ whose torsion 3-form encodes the synthetic gauge fields. Its harmonic part defines a mixed cohomology class $ [\omega]\in\bigl(H^{2}(T^{2}_{\mathrm{BZ}})\otimes H^{1}(S^{1}_{\phi_{+}})\bigr)\oplus\bigl(H^{2}(T^{2}_{\mathrm{BZ}})\otimes H^{1}(S^{1}_{\phi_{-}})\bigr), $ whose mixed tensor rank equals one. Using a general geometric bound for metric connections with totally skew torsion on product manifolds, we show that the obstruction kernel $\mathcal{K}$ vanishes, yielding the sharp inequality $\dim\mathfrak{hol}^{\mathrm{off}}(\nabla^{C})\geq 1$. This forces at least one off-diagonal curvature operator, preventing complete gauging-away of Berry phases even when the total Chern number is zero. This provides the first cohomological lower bound certifying locally irremovable curvature in SOC BECs beyond the Chern-number paradigm.
- [509] arXiv:2512.19308 (replaced) [pdf, other]
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Title: A Spinorial Heat Flow Framework for Geometric Degeneration on $3$-ManifoldsComments: 12 pages, 1 figuresSubjects: Differential Geometry (math.DG); Geometric Topology (math.GT)
We study a spinor-driven formulation of geometric evolution on closed $3$-manifolds, in which the spinor field is treated as the primary dynamical variable and the Riemannian metric is induced conformally by the spinor amplitude. We introduce a spinorial heat flow governed by the squared Dirac operator, \[ \partial_t \psi = - D_{g(\psi)}^{\,2} \psi , \] where the metric $g(\psi)$ depends nonlinearly on the evolving spinor field. As a consequence, the resulting system is quasi-linear and parabolic away from the nodal set $\{\psi=0\}$, while exhibiting degenerate behavior at vanishing spinor amplitude.
We show that degeneration of the induced metric corresponds analytically to nodal behavior of the spinor field, rather than to curvature blow-up of the spinor evolution itself. This observation motivates an interpretation of geometric singularities as spinorial nodal transitions, across which the spinor field remains locally bounded in a weak or weighted sense. The induced metric evolution is derived explicitly and shown to be purely conformal, capturing only the trace component of curvature evolution and containing additional gradient terms that are not controlled \emph{a priori}. Accordingly, the proposed flow should not be identified with the Ricci flow, and any analogy with curvature smoothing is understood at a heuristic level.
The present work establishes a coherent analytical framework for studying geometric degeneration via spinor dynamics and highlights several open problems in degenerate parabolic theory, including rigorous existence results and the precise role of nodal structures in geometric and topological evolution. - [510] arXiv:2512.19446 (replaced) [pdf, html, other]
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Title: An alternative approach to well-posedness of McKean-Vlasov equations arising in Consensus-Based OptimizationSubjects: Optimization and Control (math.OC); Analysis of PDEs (math.AP); Probability (math.PR)
In this work we study the mean-field description of Consensus-Based Optimization (CBO), a derivative-free particle optimization method. Such a description is provided by a non-local SDE of McKean-Vlasov type, whose fields lack of global Lipschitz continuity. We propose a novel approach to prove the well-posedness of the mean-field CBO equation based on a truncation argument. The latter is performed through the introduction of a cut-off function, defined on the space of probability measures, acting on the fields. This procedure allows us to study the well-posedness problem in the classical framework of Sznitman. Through this argument, we recover the established result on the existence of strong solutions, and we extend the class of solutions for which pathwise uniqueness holds.
- [511] arXiv:2512.19831 (replaced) [pdf, html, other]
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Title: Epsilon dichotomy via root numbers of intertwining periodsComments: We strengthened the globalization lemma 7.3, the previous version not being precise enough to support our conclusionSubjects: Number Theory (math.NT); Representation Theory (math.RT)
We give a new proof of the epsilon dichotomy conjecture, stated by Prasad and Takloo-Bighash, for non Archimedean local fields of characteristic zero, when the twisting character is trivial. Our method relies on the functional equation and the analytic properties of intertwining periods, instead of trace formula and type theory. It removes the odd residual characteristic restriction in the previous proof, coming from type theory.
- [512] arXiv:2512.20391 (replaced) [pdf, other]
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Title: Contingency Model-based Control (CMC) for Communicationless Cooperative Collision Avoidance in Robot SwarmsSubjects: Optimization and Control (math.OC); Robotics (cs.RO); Systems and Control (eess.SY)
Cooperative collision avoidance between robots in swarm operations remains an open challenge. Assuming a decentralized architecture, each robot is responsible for making its own control decisions, including motion planning. To this end, most existing approaches mostly rely some form of (wireless) communication between the agents of the swarm. In reality, however, communication is brittle. It may be affected by latency, further delays and packet losses, transmission faults, and is subject to adversarial attacks, such as jamming or spoofing. This paper proposes Contingency Model-based Control (CMC) as a communicationless alternative. It follows the implicit cooperation paradigm, under which the design of the robots is based on consensual (offline) rules, similar to traffic rules. They include the definition of a contingency trajectory for each robot, and a method for construction of mutual collision avoidance constraints. The setup is shown to guarantee the recursive feasibility and collision avoidance between all swarm members in closed-loop operation. Moreover, CMC naturally satisfies the Plug \& Play paradigm, i.e., for new robots entering the swarm. Two numerical examples demonstrate that the collision avoidance guarantee is intact and that the robot swarm operates smoothly under the CMC regime.
- [513] arXiv:2512.20484 (replaced) [pdf, html, other]
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Title: Supersonic sonic patch solution for the two-dimensional Euler equations with a van der Waals equation of stateComments: 31 pages, 2 figuresSubjects: Analysis of PDEs (math.AP)
We investigate supersonic transonic phenomena in the two-dimensional compressible Euler equations governed by a polytropic van der Waals equation of state. In contrast to the ideal gas setting, the non-ideal pressure law introduces stronger nonlinear effects and modifies the degeneracy structure near sonic states, which significantly complicates the analytical treatment of transonic flows. Within the self-similar framework associated with the four-state Riemann problem, we construct a supersonic sonic patch solution that connects a strictly supersonic region to a sonic boundary along a pseudo streamline. The analysis is based on a characteristic decomposition combined with a partial hodograph transformation, through which the problem is reformulated as a degenerate hyperbolic system. We establish the existence of a globally defined supersonic solution and prove its uniform regularity up to the sonic curve. In addition, we investigate the regularity properties of the resulting sonic boundary. Our results extend the theory of supersonic sonic patches from polytropic gases to a realistic non-ideal gas model.
- [514] arXiv:2512.21016 (replaced) [pdf, html, other]
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Title: On the Euclidean Distance Degree of Quadratic Two-Neuron Neural NetworksComments: Comments are welcome!Subjects: Algebraic Geometry (math.AG)
We study the Euclidean Distance degree of algebraic neural network models from the perspective of algebraic geometry. Focusing on shallow networks with two neurons, quadratic activation, and scalar output, we identify the associated neurovariety with the second secant variety of a quadratic Veronese embedding. We introduce and analyze the virtual Euclidean Distance degree, a projective invariant defined as the sum of the polar degrees of the variety, which coincides with the usual Euclidean Distance degree for a generic choice of scalar product. Using intersection theory, Chern-Mather classes, and the Nash blow-up provided by Kempf's resolution, we reduce the computation of the virtual Euclidean Distance degree to explicit intersection numbers on a Grassmannian. Applying equivariant localization, we prove that this invariant depends stably polynomially on the input dimension. Numerical experiments based on homotopy continuation illustrate the dependence of the Euclidean Distance degree on the chosen metric and highlight the distinction between the generic and nongeneric cases, such as the Bombieri-Weyl metric.
- [515] arXiv:2512.21134 (replaced) [pdf, html, other]
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Title: The monoid of monotone and decreasing partial transformations on a finite chainComments: The results in this paper were obtained in October 2024 during a postdoctoral visit to Sultan Qaboos University, by the first author. The paper was first submitted to the Semigroup Forum, Prof. via Victoria Gould in December 2024, and subsequently other Journals. It is imperative that we have to arxiv these results pending the outcome of review after submission to other JournalsSubjects: Group Theory (math.GR)
In this article, we consider the monoid of all monotone and order-decreasing partial transformations denoted as $\mathcal{DORP}_{n}$ on an $n$ ordered chain $[n]=\{1, \ldots,n\}$, its two-sided ideal $I(n,p)= \{\rho \in \mathcal{DORP}_{n} : \, |Im \, \rho| \leq p\}$ and the Rees quotient ${RQ}_{p}(n)$ of the ideal $I(n,p)$. We compute the order of the monoid $\mathcal{DORP}_{n}$ and show that for any semigroup $S$ in $\{\mathcal{DORP}_{n}, \, I(n,p), \, {RQ}_{p}(n)\}$, $S$ is abundant for all values of $n$. In particular, we show that the Rees quotient ${RQ}_{p}(n)$, is a non-regular $0-*$bisimple abundant semigroup. In addition, we compute the ranks of the Rees quotient ${RQ}_{p}(n)$ and the two-sided ideal $I(n,p)$. Finally, the rank of $\mathcal{DORP}_{n}$ is determined to be $3n-2$.
- [516] arXiv:2512.21164 (replaced) [pdf, html, other]
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Title: Mixed Precision General Alternating-Direction Implicit Method for Solving Large Sparse Linear SystemsSubjects: Numerical Analysis (math.NA)
In this article, we introduce a three-precision formulation of the General Alternating-Direction Implicit method (GADI) designed to accelerate the solution of large-scale sparse linear systems $Ax=b$. GADI is a framework that can represent many existing Alternating-Direction Implicit (ADI) methods. These methods are a class of linear solvers based on a splitting of $A$ such that the solution of the original linear system can be decomposed into the successive computation of easy-to-solve structured subsystems. Our proposed mixed precision scheme for GADI solves these subsystems in low precision to reduce the overall execution time while computing the residual and solution update in high precision to enable the solution to converge to high accuracy. We develop a rounding error analysis of mixed precision GADI that establishes the rates of convergence of the forward and backward errors to certain limiting accuracies. Our analysis also highlights the conditions on the splitting matrices under which mixed precision GADI is guaranteed to converge for a given set of precisions. We then discuss a systematic and robust strategy for selecting the GADI regularization parameter $\alpha$, whose adjustment is critical for performance. Specifically, our proposed strategy makes use of a Gaussian Process Regression (GPR) model trained on a dataset of low-dimensional problems to initialize $\alpha$. Finally, we proceed to a performance analysis of mixed precision GADI on an NVIDIA A100 GPU to validate our approach. Using low precision (Bfloat16 or FP32) to solve the subsystems, we obtain speedups of $2.6\times$, $1.7\times$, and $3.1\times$ over a full double precision GADI implementation on large-scale 2D, 3D convection-diffusion and complex reaction-diffusion problems (up to $1.3\times 10^{8}$ unknowns), respectively.
- [517] arXiv:2512.21269 (replaced) [pdf, html, other]
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Title: The Dynamical Anatomy of Anderson Acceleration:From Adaptive Momentum to Variable-Mass ODEsSubjects: Optimization and Control (math.OC); Numerical Analysis (math.NA)
This paper provides a rigorous derivation and analysis of accelerated optimization algorithms through the lens of High-Resolution Ordinary Differential Equations (ODEs). While classical Nesterov acceleration is well-understood via asymptotic vanishing damping, the dynamics of Anderson Acceleration (AA) remain less transparent. This work makes significant theoretical contributions to AA by bridging discrete acceleration algorithms with continuous dynamical systems, while also providing practical algorithmic innovations. Our work addresses fundamental questions about the physical nature of Anderson Acceleration that have remained unanswered since its introduction in 1965. Firstly, we prove that AA can be exactly rewritten as an adaptive momentum method and, in the high-resolution limit, converges to a second-order ODE with Variable Effective Mass. Through a Lyapunov energy analysis, we reveal the specific instability mechanism of standard AA: unchecked growth in effective mass acts as negative damping, physically injecting energy into the system and violating dissipation constraints. Conversely, high-resolution analysis identifies an implicit Hessian-driven damping term that provides stabilization in stiff regimes. Leveraging these dynamical insights, we then propose Energy-Guarded Anderson Acceleration (EG-AA), an algorithm that acts as an inertial governor to enforce thermodynamic consistency. Morevoer, our convergence analysis, formulated via the Acceleration Gain Factor, proves that EG-AA improves upon gradient descent by maximizing the geometric contraction of the linear subspace projection while actively suppressing nonlinear approximation errors. Theoretical bounds confirm that EG-AA is no worse than standard AA, and numerical experiments demonstrate strictly improved convergence stability and rates in ill-conditioned convex composite problems compared to standard Anderson mixing.
- [518] arXiv:2306.03869 (replaced) [pdf, html, other]
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Title: Connecting classical finite exchangeability to quantum theorySubjects: Quantum Physics (quant-ph); Probability (math.PR)
Exchangeability is a fundamental concept in probability theory and statistics. It allows to model situations where the order of observations does not matter. The classical de Finetti's theorem provides a representation of infinitely exchangeable sequences of random variables as mixtures of independent and identically distributed variables. The quantum de Finetti theorem extends this result to symmetric quantum states on tensor product Hilbert spaces. It is well known that both theorems do not hold for finitely exchangeable sequences. The aim of this work is to investigate two lesser-known representation theorems, which were developed in classical probability theory to extend de Finetti's theorem to finitely exchangeable sequences by using quasi-probabilities and quasi-expectations. With the aid of these theorems, we illustrate how a de Finetti-like representation theorem for finitely exchangeable sequences can be formulated through a mathematical representation which is formally equivalent to quantum theory (with boson-symmetric density matrices). We then show a promising application of this connection to the challenge of defining entanglement for indistinguishable bosons.
- [519] arXiv:2310.16284 (replaced) [pdf, html, other]
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Title: Bayesian Image Mediation AnalysisSubjects: Methodology (stat.ME); Statistics Theory (math.ST); Computation (stat.CO)
Mediation analysis aims to separate the indirect effect through mediators from the direct effect of the exposure on the outcome. It is challenging to perform mediation analysis with neuroimaging data which involves high dimensionality, complex spatial correlations, sparse activation patterns and relatively low signal-to-noise ratio. To address these issues, we develop a new spatially varying coefficient structural equation model for Bayesian Image Mediation Analysis (BIMA). We define spatially varying mediation effects within the potential outcomes framework, employing a soft-thresholded Gaussian process prior for functional parameters. We establish posterior consistency for spatially varying mediation effects along with selection consistency on important regions that contribute to the mediation effects. We develop an efficient posterior computation algorithm scalable to analysis of large-scale imaging data. Through extensive simulations, we show that BIMA can improve the estimation accuracy and computational efficiency for high-dimensional mediation analysis over existing methods. We apply BIMA to analyze behavioral and fMRI data in the Adolescent Brain Cognitive Development (ABCD) study with a focus on inferring the mediation effects of the parental education level on the children's general cognitive ability that are mediated through the working memory brain activity.
- [520] arXiv:2312.07520 (replaced) [pdf, other]
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Title: Estimating Counterfactual Matrix Means with Short Panel DataComments: 100 pages, 7 figures, 3 tablesSubjects: Econometrics (econ.EM); Statistics Theory (math.ST); Methodology (stat.ME)
We develop a spectral approach for identifying and estimating average counterfactual outcomes under a low-rank factor model with short panel data and general outcome missingness patterns. Applications include event studies and studies of outcomes of "matches" between agents of two types, e.g. people and places, typically conducted using less-flexible Two-Way Fixed Effects (TWFE) models of outcomes. Given finite observed outcomes per unit, we show our approach identifies all counterfactual outcome means, including those not identified by existing methods, if a particular graph algorithm determines that units' sets of observed outcomes have sufficient overlap. Our analogous, computationally efficient estimation procedure yields consistent, asymptotically normal estimates of counterfactual outcome means under fixed-$T$ (number of outcomes), large-$N$ (sample size) asymptotics. When estimating province-level averages of held-out wages from an Italian matched employer-employee dataset, our estimator outperforms a TWFE-model-based estimator.
- [521] arXiv:2312.08531 (replaced) [pdf, html, other]
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Title: Revisiting the Last-Iterate Convergence of Stochastic Gradient MethodsComments: The preliminary version has been accepted at ICLR 2024. For the update history, please refer to the PDFSubjects: Machine Learning (cs.LG); Optimization and Control (math.OC); Machine Learning (stat.ML)
In the past several years, the last-iterate convergence of the Stochastic Gradient Descent (SGD) algorithm has triggered people's interest due to its good performance in practice but lack of theoretical understanding. For Lipschitz convex functions, different works have established the optimal $O(\log(1/\delta)\log T/\sqrt{T})$ or $O(\sqrt{\log(1/\delta)/T})$ high-probability convergence rates for the final iterate, where T is the time horizon and \delta is the failure probability. However, to prove these bounds, all the existing works are either limited to compact domains or require almost surely bounded noise. It is natural to ask whether the last iterate of SGD can still guarantee the optimal convergence rate but without these two restrictive assumptions. Besides this important question, there are still lots of theoretical problems lacking an answer. For example, compared with the last-iterate convergence of SGD for non-smooth problems, only few results for smooth optimization have yet been developed. Additionally, the existing results are all limited to a non-composite objective and the standard Euclidean norm. It still remains unclear whether the last-iterate convergence can be provably extended to wider composite optimization and non-Euclidean norms. In this work, to address the issues mentioned above, we revisit the last-iterate convergence of stochastic gradient methods and provide the first unified way to prove the convergence rates both in expectation and in high probability to accommodate general domains, composite objectives, non-Euclidean norms, Lipschitz conditions, smoothness, and (strong) convexity simultaneously. Additionally, we extend our analysis to obtain the last-iterate convergence under heavy-tailed noise.
- [522] arXiv:2408.07936 (replaced) [pdf, html, other]
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Title: Quantum-Classical Hybrid Algorithm for Solving the Learning-With-Errors Problem on NISQ DevicesMuxi Zheng, Jinfeng Zeng, Wentao Yang, Pei-Jie Chang, Quanfeng Lu, Bao Yan, Haoran Zhang, Min Wang, Shijie Wei, Gui-Lu LongJournal-ref: Commun Phys 8, 208 (2025)Subjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph)
The Learning-With-Errors (LWE) problem is a fundamental computational challenge with implications for post-quantum cryptography and computational learning theory. Here we propose a quantum-classical hybrid algorithm with Ising model to address LWE, transforming it into the Shortest Vector Problem and using variable qubits to encode lattice vectors into an Ising Hamiltonian. By identifying low-energy Hamiltonian levels, the solution is extracted, making the method suitable for noisy intermediate-scale quantum devices. The required number of qubits is less than $m(m+1)$, where $m$ is the number of samples. Our heuristic algorithm's time complexity depends on the specific quantum eigensolver used to find low-energy levels, and the performance when using the Quantum Approximate Optimization Algorithm is investigated. We validate the algorithm by solving a $2$-dimensional LWE problem on a $5$-qubit quantum device, demonstrating its potential for solving meaningful LWE instances on near-term quantum devices.
- [523] arXiv:2410.00232 (replaced) [pdf, html, other]
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Title: Preconditioning for Accelerated Gradient Descent Optimization and RegularizationComments: 21 pagesSubjects: Machine Learning (cs.LG); Numerical Analysis (math.NA); Machine Learning (stat.ML)
Accelerated training algorithms, such as adaptive learning rates (or preconditioning) and various normalization methods, are widely used but not fully understood. When regularization is introduced, standard optimizers like adaptive learning rates may not perform effectively. This raises the need for alternative regularization approaches such as AdamW and the question of how to properly combine regularization with preconditioning. In this paper, we address these challenges using the theory of preconditioning as follows: (1) We explain how AdaGrad, RMSProp, and Adam accelerates training through improving Hessian conditioning; (2) We explore the interaction between $L_2$-regularization and preconditioning, demonstrating that AdamW amounts to selecting the underlying intrinsic parameters for regularization, and we derive a generalization for the $L_1$-regularization; and (3) We demonstrate how various normalization methods such as input data normalization, batch normalization, and layer normalization accelerate training by improving Hessian conditioning. Our analysis offers a unified mathematical framework for understanding various acceleration techniques or deriving appropriate regularization schemes.
- [524] arXiv:2410.15155 (replaced) [pdf, html, other]
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Title: On the Convergence Theory of Pipeline Gradient-based Analog In-memory TrainingSubjects: Machine Learning (cs.LG); Hardware Architecture (cs.AR); Optimization and Control (math.OC)
Aiming to accelerate the training of large deep neural networks (DNN) in an energy-efficient way, analog in-memory computing (AIMC) emerges as a solution with immense potential. AIMC accelerator keeps model weights in memory without moving them from memory to processors during training, reducing overhead dramatically. Despite its efficiency, scaling up AIMC systems presents significant challenges. Since weight copying is expensive and inaccurate, data parallelism is less efficient on AIMC accelerators. It necessitates the exploration of pipeline parallelism, particularly asynchronous pipeline parallelism, which utilizes all available accelerators during the training process. This paper examines the convergence theory of stochastic gradient descent on AIMC hardware with an asynchronous pipeline (Analog-SGD-AP). Although there is empirical exploration of AIMC accelerators, the theoretical understanding of how analog hardware imperfections in weight updates affect the training of multi-layer DNN models remains underexplored. Furthermore, the asynchronous pipeline parallelism results in stale weights issues, which render the update signals no longer valid gradients. To close the gap, this paper investigates the convergence properties of Analog-SGD-AP on multi-layer DNN training. We show that the Analog-SGD-AP converges with iteration complexity $O(\varepsilon^{-2}+\varepsilon^{-1})$ despite the aforementioned issues, which matches the complexities of digital SGD and Analog SGD with synchronous pipeline, except the non-dominant term $O(\varepsilon^{-1})$. It implies that AIMC training benefits from asynchronous pipelining almost for free compared with the synchronous pipeline by overlapping computation.
- [525] arXiv:2410.16250 (replaced) [pdf, html, other]
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Title: Cups and Gates I: Cohomology invariants and logical quantum operationsComments: v2:typos and minor errors fixed; 40 pagesSubjects: Quantum Physics (quant-ph); Strongly Correlated Electrons (cond-mat.str-el); Mathematical Physics (math-ph)
We take initial steps towards a general framework for constructing logical gates in general quantum CSS codes. Viewing CSS codes as cochain complexes, we observe that cohomology invariants naturally give rise to diagonal logical gates. We show that such invariants exist if the quantum code has a structure that relaxes certain properties of a differential graded algebra. We show how to equip quantum codes with such a structure by defining cup products on CSS codes. The logical gates obtained from this approach can be implemented by a constant-depth unitary circuit. In particular, we construct a $\Lambda$-fold cup product that can produce a logical operator in the $\Lambda$-th level of the Clifford hierarchy on $\Lambda$ copies of the same quantum code, which we call the copy-cup gate. For any desired $\Lambda$, we can construct several families of quantum codes that support gates in the $\Lambda$-th level with various asymptotic code parameters.
- [526] arXiv:2412.03110 (replaced) [pdf, html, other]
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Title: On palindromic numerators of bigraded symmetric orbifold Hilbert series and Kostka-Foulkes polynomialsComments: 14 pages, minor changes to match published versionSubjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
From our work on partition functions in log gravity, we show that the palindromic numerators in two variables of bigraded symmetric orbifold Hilbert series take the form of sums of products of Kostka-Foulkes polynomials associated with a pair of partition $\lambda$ and $\mu=(1^n)$. The log partition function also being a KP $\tau$-function, our work gives a new description of Hall-Littlewood and Kostka-Foulkes polynomials as palindromic numerators of quotient expansions in the moduli space of formal power series solutions of the KP hierarchy. Using the structure and properties of the log partition function, we also show that the palindromic polynomials are eigenvalues of a differential operator arising from a recurrence relation and acting on the Hilbert series.
- [527] arXiv:2412.10721 (replaced) [pdf, html, other]
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Title: A Two-Step Projection-Based Goodness-of-Fit Test for Ultra-High Dimensional Sparse RegressionsSubjects: Methodology (stat.ME); Statistics Theory (math.ST)
This paper proposes a novel two-step strategy for testing the goodness-of-fit of parametric regression models in ultra-high dimensional sparse settings, where the predictor dimension far exceeds the sample size. This regime usually renders existing goodness-of-fit tests for regressions infeasible, primarily due to the curse of dimensionality or their reliance on the asymptotic linearity and normality of parameter estimators -- properties that may no longer hold under ultra-high dimensional settings. To address these limitations, our strategy first constructs multiple test statistics based on projected predictors from distinct projections and establishes their asymptotic properties under both the null and alternative hypotheses. This projection-based approach significantly mitigates the dimensionality problem, enabling our tests to detect local alternatives converging to the null at the rate as if the predictor were univariate. An important finding is that the resulting test statistics based on linearly independent projections are asymptotically independent under the null hypothesis. Based on this, our second step employs powerful $p$-value combination procedures, such as the minimum $p$-value and the Fisher combination of $p$-value, to form our final tests and enhance power. Theoretically, our tests only require the standard convergence rate of parameter estimators to derive their limiting distributions, thereby circumventing the need for asymptotic linearity or normality of parameter estimators. Simulations and real-data applications confirm that our approach provides robust and powerful goodness-of-fit testing in ultra-high dimensional settings.
- [528] arXiv:2412.18432 (replaced) [pdf, html, other]
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Title: Gaussian entropic optimal transport: Schrödinger bridges and the Sinkhorn algorithmComments: Accepted to Foundations of Data Science (FoDS)Subjects: Machine Learning (stat.ML); Machine Learning (cs.LG); Probability (math.PR); Computation (stat.CO)
Entropic optimal transport problems are regularized versions of optimal transport problems. These models play an increasingly important role in machine learning and generative modelling. For finite spaces, these problems are commonly solved using Sinkhorn algorithm (a.k.a. iterative proportional fitting procedure). However, in more general settings the Sinkhorn iterations are based on nonlinear conditional/conjugate transformations and exact finite-dimensional solutions cannot be computed.
This article presents a finite-dimensional recursive formulation of the iterative proportional fitting procedure for general Gaussian multivariate models. As expected, this recursive formulation is closely related to the celebrated Kalman filter and related Riccati matrix difference equations, and it yields algorithms that can be implemented in practical settings without further approximations. We extend this filtering methodology to develop a refined and self-contained convergence analysis of Gaussian Sinkhorn algorithms, including closed form expressions of entropic transport maps and Schrödinger bridges. - [529] arXiv:2412.19555 (replaced) [pdf, html, other]
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Title: Asymptotic Properties of the Maximum Likelihood Estimator for Markov-switching Observation-driven ModelsSubjects: Econometrics (econ.EM); Statistics Theory (math.ST)
A Markov-switching observation-driven model is a stochastic process $((S_t,Y_t))_{t \in \mathbb{Z}}$ where $(S_t)_{t \in \mathbb{Z}}$ is an unobserved Markov chain on a finite set and $(Y_t)_{t \in \mathbb{Z}}$ is an observed stochastic process such that the conditional distribution of $Y_t$ given $(Y_\tau)_{\tau \leq t-1}$ and $(S_\tau)_{\tau \leq t}$ depends on $(Y_\tau)_{\tau \leq t-1}$ and $S_t$. In this paper, we prove consistency and asymptotic normality of the maximum likelihood estimator for such model. As a special case, we also give conditions under which the maximum likelihood estimator for the widely applied Markov-switching generalised autoregressive conditional heteroscedasticity model introduced by Haas, Mittnik, and Paolella (2004b) is consistent and asymptotically normal.
- [530] arXiv:2412.20553 (replaced) [pdf, html, other]
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Title: Edge of Stochastic Stability: Revisiting the Edge of Stability for SGDComments: 83 pages, 36 figuresSubjects: Machine Learning (cs.LG); Optimization and Control (math.OC); Machine Learning (stat.ML)
Recent findings by Cohen et al., 2021, demonstrate that when training neural networks using full-batch gradient descent with a step size of $\eta$, the largest eigenvalue $\lambda_{\max}$ of the full-batch Hessian consistently stabilizes around $2/\eta$. These results have significant implications for convergence and generalization. This, however, is not the case for mini-batch optimization algorithms, limiting the broader applicabilityof the consequences of these findings. We show mini-batch Stochastic Gradient Descent (SGD) trains in a different regime we term Edge of Stochastic Stability (EoSS). In this regime, what stabilizes at $2/\eta$ is Batch Sharpness: the expected directional curvature of mini-batch Hessians along their corresponding stochastic gradients. As a consequence $\lambda_{\max}$ -- which is generally smaller than Batch Sharpness -- is suppressed, aligning with the long-standing empirical observation that smaller batches and larger step sizes favor flatter minima. We further discuss implications for mathematical modeling of SGD trajectories.
- [531] arXiv:2502.05684 (replaced) [pdf, html, other]
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Title: Machine Unlearning via Information Theoretic RegularizationComments: 59 pages, 4 figuresSubjects: Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Information Theory (cs.IT); Machine Learning (stat.ML)
How can we effectively remove or ''unlearn'' undesirable information, such as specific features or the influence of individual data points, from a learning outcome while minimizing utility loss and ensuring rigorous guarantees? We introduce a unified mathematical framework based on information-theoretic regularization to address both data point unlearning and feature unlearning. For data point unlearning, we introduce the $\textit{Marginal Unlearning Principle}$, an auditable and provable framework inspired by memory suppression studies in neuroscience. Moreover, we provide formal information-theoretic unlearning definition based on the proposed principle, named marginal unlearning, and provable guarantees on sufficiency and necessity of marginal unlearning to the existing approximate unlearning definitions. We then show the proposed framework provide natural solution to the marginal unlearning problems. For feature unlearning, the framework applies to deep learning with arbitrary training objectives. By combining flexibility in learning objectives with simplicity in regularization design, our approach is highly adaptable and practical for a wide range of machine learning and AI applications. From a mathematical perspective, we provide an unified analytic solution to the optimal feature unlearning problem with a variety of information-theoretic training objectives. Our theoretical analysis reveals intriguing connections between machine unlearning, information theory, optimal transport, and extremal sigma algebras. Numerical simulations support our theoretical finding.
- [532] arXiv:2502.09832 (replaced) [pdf, html, other]
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Title: Computational Lower Bounds for Correlated Random Graphs via Algorithmic ContiguityComments: This substantially improves the results and simplifies the proofs in an earlier versionSubjects: Machine Learning (stat.ML); Data Structures and Algorithms (cs.DS); Machine Learning (cs.LG); Probability (math.PR); Statistics Theory (math.ST)
In this paper, assuming the low-degree conjecture, we provide evidence of computational hardness for two problems: (1) the (partial) matching recovery problem in the sparse correlated Erdős-Rényi graphs $\mathcal G(n,q;\rho)$ when the edge-density $q=n^{-1+o(1)}$ and the correlation $\rho<\sqrt{\alpha}$ lies below the Otter's threshold, this resolves a remaining problem in \cite{DDL23+}; (2) the detection problem between a pair of correlated sparse stochastic block models $\mathcal S(n,\tfrac{\lambda}{n};k,\epsilon;s)$ and a pair of independent stochastic block models $\mathcal S(n,\tfrac{\lambda s}{n};k,\epsilon)$ when $\epsilon^2 \lambda s<1$ lies below the Kesten-Stigum (KS) threshold and $s<\sqrt{\alpha}$ lies below the Otter's threshold, this resolves a remaining problem in \cite{CDGL24+}.
One of the main ingredient in our proof is to derive certain forms of \emph{algorithmic contiguity} between two probability measures based on bounds on their low-degree advantage. To be more precise, consider the high-dimensional hypothesis testing problem between two probability measures $\mathbb{P}$ and $\mathbb{Q}$ based on the sample $\mathsf Y$. We show that if the low-degree advantage $\mathsf{Adv}_{\leq D} \big( \frac{\mathrm{d}\mathbb{P}}{\mathrm{d}\mathbb{Q}} \big)=O(1)$, then (assuming the low-degree conjecture) there is no efficient algorithm $\mathcal A$ such that $\mathbb{Q}(\mathcal A(\mathsf Y)=0)=1-o(1)$ and $\mathbb{P}(\mathcal A(\mathsf Y)=1)=\Omega(1)$. This framework provides a useful tool for performing reductions between different inference tasks, without requiring a strengthened version of the low-degree conjecture as in \cite{MW23+, DHSS25+}. - [533] arXiv:2502.11289 (replaced) [pdf, html, other]
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Title: A large data semi-global existence and convergence theorem for vacuum Einstein's equationsComments: Comments welcome, 46 pages, 1 figureSubjects: General Relativity and Quantum Cosmology (gr-qc); Mathematical Physics (math-ph); Analysis of PDEs (math.AP); Differential Geometry (math.DG)
We prove a semi-global existence and convergence theorem for the $(3+1)$-dimensional vacuum Einstein equations with positive cosmological constant on spacetimes $\widetilde{M} \sim M \times \mathbb{R}$, where $M$ is a closed, connected, oriented three-manifold of negative Yamabe type. In constant mean curvature transported spatial coordinates, we show that solutions arising from a class of arbitrarily large initial data converge to a Riemannian metric of constant negative scalar curvature in infinite Newtonian-like `time'. A main novelty is to uncover a new weak null-type structure (different from the well known null structure in the literature) in the field equations induced by the positive cosmological constant in constant mean curvature gauge that is absent in pure vacuum. As a consequence, the Einstein-$\Lambda$ flow generically fails to produce geometrization in the sense of Thurston. Our results affirm a conjecture of Ringström concerning the asymptotic in-distinguishability of spatial topology in the large data regime of Einstein-$\Lambda$ dynamics. A related result is established for positive Yamabe type under a technical condition.
- [534] arXiv:2503.17809 (replaced) [pdf, html, other]
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Title: Poisson-Process Topic Model for Integrating Knowledge from Pre-trained Language ModelsComments: 96 pagesSubjects: Machine Learning (stat.ML); Machine Learning (cs.LG); Statistics Theory (math.ST)
Topic modeling is traditionally applied to word counts without accounting for the context in which words appear. Recent advancements in large language models (LLMs) offer contextualized word embeddings, which capture deeper meaning and relationships between words. We aim to leverage such embeddings to improve topic modeling.
We use a pre-trained LLM to convert each document into a sequence of word embeddings. This sequence is then modeled as a Poisson point process, with its intensity measure expressed as a convex combination of $K$ base measures, each corresponding to a topic. To estimate these topics, we propose a flexible algorithm that integrates traditional topic modeling methods, enhanced by net-rounding applied before and kernel smoothing applied after. One advantage of this framework is that it treats the LLM as a black box, requiring no fine-tuning of its parameters. Another advantage is its ability to seamlessly integrate any traditional topic modeling approach as a plug-in module, without the need for modifications
Assuming each topic is a $\beta$-Hölder smooth intensity measure on the embedded space, we establish the rate of convergence of our method. We also provide a minimax lower bound and show that the rate of our method matches with the lower bound when $\beta\leq 1$. Additionally, we apply our method to several datasets, providing evidence that it offers an advantage over traditional topic modeling approaches. - [535] arXiv:2504.07341 (replaced) [pdf, html, other]
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Title: Learning to erase quantum states: thermodynamic implications of quantum learning theoryComments: 7 pages + 1 figure + 12 pages of appendices. We have added a detailed discussion of related works in Discussion and Appendix A, a pedagogical introduction to information thermodynamics in formal quantum information language in Appendix B, and detailed proofs of our main results in Appendices C and D. Numerous remarks and clarifications have been made throughout the manuscriptSubjects: Quantum Physics (quant-ph); Statistical Mechanics (cond-mat.stat-mech); Computational Complexity (cs.CC); Information Theory (cs.IT); Machine Learning (cs.LG)
The energy cost of erasing quantum states depends on our knowledge of the states. We show that learning algorithms can acquire such knowledge to erase many copies of an unknown state at the optimal energy cost. This is proved by showing that learning can be made fully reversible and has no fundamental energy cost itself. With simple counting arguments, we relate the energy cost of erasing quantum states to their complexity, entanglement, and magic. We further show that the constructed erasure protocol is computationally efficient when learning is efficient. Conversely, under standard cryptographic assumptions, we prove that the optimal energy cost cannot be achieved efficiently in general. These results also enable efficient work extraction based on learning. Together, our results establish a concrete connection between quantum learning theory and thermodynamics, highlighting the physical significance of learning processes and enabling provably-efficient learning-based protocols for thermodynamic tasks.
- [536] arXiv:2505.08128 (replaced) [pdf, html, other]
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Title: Beyond Basic A/B testing: Improving Statistical Efficiency for Business GrowthSubjects: Methodology (stat.ME); Machine Learning (cs.LG); Statistics Theory (math.ST); Computation (stat.CO)
The standard A/B testing approaches are mostly based on t-test in large scale industry applications. These standard approaches however suffers from low statistical power in business settings, due to nature of small sample-size or non-Gaussian distribution or return-on-investment (ROI) consideration. In this paper, we (i) show the statistical efficiency of using estimating equation and U statistics, which can address these issues separately; and (ii) propose a novel doubly robust generalized U that allows flexible definition of treatment effect, and can handles small samples, distribution robustness, ROI and confounding consideration in one framework. We provide theoretical results on asymptotics and efficiency bounds, together with insights on the efficiency gain from theoretical analysis. We further conduct comprehensive simulation studies, apply the methods to multiple real A/B tests at LinkedIn, and share results and learnings that are broadly useful.
- [537] arXiv:2505.20784 (replaced) [pdf, html, other]
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Title: Colouring Probe $H$-Free GraphsSubjects: Data Structures and Algorithms (cs.DS); Discrete Mathematics (cs.DM); Combinatorics (math.CO)
The NP-complete problems Colouring and k-Colouring $(k\geq 3$) are well studied on $H$-free graphs, i.e., graphs that do not contain some fixed graph $H$ as an induced subgraph. We research to what extent the known polynomial-time algorithms for $H$-free graphs can be generalized if we only know some of the edges of the input graph. We do this by considering the classical probe graph model introduced in the early nineties. For a graph $H$, a partitioned probe $H$-free graph $(G,P,N)$ consists of a graph $G=(V,E)$, together with a set $P\subseteq V$ of probes and an independent set $N=V\setminus P$ of non-probes, such that $G+F$ is $H$-free for some edge set $F\subseteq \binom{N}{2}$. We first fully classify the complexity of Colouring on partitioned probe $H$-free graphs and show that this dichotomy is different from the known dichotomy of Colouring for $H$-free graphs. Our main result is a dichotomy of $3$-Colouring for partitioned probe $P_t$-free graphs: we prove that the problem is polynomial-time solvable if $t\leq 5$ but NP-complete if $t\geq 6$. In contrast, $3$-Colouring on $P_t$-free graphs is known to be polynomial-time solvable if $t\leq 7$ and quasi polynomial-time solvable for $t\geq 8$.
- [538] arXiv:2507.00340 (replaced) [pdf, html, other]
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Title: Chiral higher-spin symmetry of the celestial twistor sphereComments: v2. Revised and improved presentation, with correctionsSubjects: High Energy Physics - Theory (hep-th); Mathematical Physics (math-ph)
We study the chiral higher-spin symmetry algebras $\mathfrak{ca}$ of various twistorial higher-spin theories. These symmetries play the roles of asymptotic symmetries on the celestial twistor sphere, which constrain the observables of twistorial theories. To first order in quantum correction, we show that the chiral algebras associated with anomaly-free holomorphic twistorial higher-spin theories are associative themselves. On the other hand, the chiral algebras associated with anomalous holomorphic twistorial higher-spin theories only become associative upon including suitable axionic currents. When computing $4d$ form factors in terms of correlation functions between higher-spin currents on the celestial twistor sphere, we observe that there are some non-vanishing higher-spin form factors. This observation, however, is only well justified for the case of theories with Yang-Mills-like interactions. We also give some brief comments on the case of higher-derivative interactions.
- [539] arXiv:2507.01687 (replaced) [pdf, html, other]
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Title: Neural Measures for learning distributions of Random PDEsSubjects: Machine Learning (stat.ML); Machine Learning (cs.LG); Numerical Analysis (math.NA)
The integration of Scientific Machine Learning (SciML) techniques with uncertainty quantification (UQ) represents a rapidly evolving frontier in computational science. This work advances Physics-Informed Neural Networks (PINNs) by incorporating probabilistic frameworks to effectively model uncertainty in complex systems. Our approach enhances the representation of uncertainty in forward problems by combining generative modeling techniques with PINNs. This integration enables in a systematic fashion uncertainty control while maintaining the predictive accuracy of the model. We demonstrate the utility of this method through applications to random differential equations and random partial differential equations (PDEs).
- [540] arXiv:2507.01771 (replaced) [pdf, html, other]
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Title: Higher-Order Tensor-Based Deferral of Gaussian Splitting for Orbit Uncertainty PropagationSubjects: Signal Processing (eess.SP); Probability (math.PR)
Accurate propagation of orbital uncertainty is essential for a range of applications within space domain awareness. Adaptive Gaussian mixture-based approaches offer tractable nonlinear uncertainty propagation through splitting mixands to increase resolution in areas of stronger nonlinearities, as well as by reducing mixands to prevent unnecessary computational effort. Recent work introduced principled heuristics that incorporate information from the system dynamics and initial uncertainty to determine optimal directions for splitting. This paper develops adaptive uncertainty propagation methods based on these robust splitting techniques. A deferred splitting algorithm tightly integrated with higher-order splitting techniques is proposed and shown to offer substantial gains in computational efficiency without sacrificing accuracy. Second-order propagation of mixand moments is also seen to improve accuracy while retaining significant computational savings from deferred splitting. Different immediate and deferred splitting methods are compared in four representative test cases, including a low Earth orbit, a geostationary orbit, a Molniya orbit, and a multi-body cislunar orbit.
- [541] arXiv:2507.15567 (replaced) [pdf, html, other]
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Title: Well-posed geometric boundary data in General Relativity, III: conformal-volume boundary dataComments: 20 pages. Minor changes to introduction to improve exposition. References addedSubjects: General Relativity and Quantum Cosmology (gr-qc); Analysis of PDEs (math.AP); Differential Geometry (math.DG)
In this third work in a series, we prove the local-in-time well-posedness of the IBVP for the vacuum Einstein equations in general relativity with twisted DIrichlet boundary conditions on a finite timelike boundary. The boundary conditions consist of specification of the pointwise conformal class of the boundary metric, together with a scalar density involving a combination of the volume form of the bulk metric restricted to the boundary together with the volume form of the boundary metric itself.
- [542] arXiv:2507.20621 (replaced) [pdf, html, other]
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Title: Sequential Operation of Residential Energy Hubs using Physics-Based Economic Nonlinear MPCDarío Slaifstein (1), Gautham Ram Chandra Mouli (1), Laura Ramirez-Elizondo (1), Pavol Bauer (1) ((1) Delft University of Technology)Subjects: Systems and Control (eess.SY); Optimization and Control (math.OC)
The operation of residential energy hubs with multiple energy carriers (electricity, heat, mobility) poses a significant challenge due to different carrier dynamics, hybrid storage coordination and high-dimensional action-spaces. Energy management systems oversee their operation, deciding the set points of the primary control layer. This paper presents a novel 2-stage economic model predictive controller for electrified buildings including physics-based models of the battery degradation and thermal systems. The hierarchical control operates in the Dutch sequential energy markets. In particular common assumptions regarding intra-day markets (auction and continuous-time) are discussed as well as the coupling of the different storage systems. The best control policy it is best to follow continuous time intra-day in the summer and the intra-day auction in the winter. This sequential operation comes at the expense of increased battery degradation. Lastly, under our controller, the realized short-term flexibility of the thermal energy storage is marginal compared to the flexibility delivered by stationary battery pack and electric vehicles with bidirectional charging.
- [543] arXiv:2508.16110 (replaced) [pdf, html, other]
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Title: Estimating the growth rate of a birth and death process using data from a small sampleSubjects: Methodology (stat.ME); Probability (math.PR)
The problem of estimating the growth rate of a birth and death processes based on the coalescence times of a sample of $n$ individuals has been considered by several authors (\cite{stadler2009incomplete, williams2022life, mitchell2022clonal, Johnson2023}). This problem has applications, for example, to cancer research, when one is interested in determining the growth rate of a clone.
Recently, \cite{Johnson2023} proposed an analytical method for estimating the growth rate using the theory of coalescent point processes, which has comparable accuracy to more computationally intensive methods when the sample size $n$ is large. We use a similar approach to obtain an estimate of the growth rate that is not based on the assumption that $n$ is large.
We demonstrate, through simulations using the R package \texttt{cloneRate}, that our estimator of the growth rate performs well in comparison with previous approaches when $n$ is small. - [544] arXiv:2508.19582 (replaced) [pdf, html, other]
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Title: Approximating mixed volumes to arbitrary accuracySubjects: Computational Geometry (cs.CG); Combinatorics (math.CO)
We study the problem of approximating the mixed volume $V(P_1^{(\alpha_1)}, \dots, P_k^{(\alpha_k)})$ of an $k$-tuple of convex polytopes $(P_1, \dots, P_k)$, each of which is defined as the convex hull of at most $m_0$ points in $\mathbb{Z}^n$. We design an algorithm that produces an estimate that is within a multiplicative $1 \pm \epsilon$ factor of the true mixed volume with a probability greater than $1 - \delta.$ Let the constant $ \prod_{i=2}^{k} \frac{(\alpha_{i}+1)^{\alpha_{i}+1}}{\alpha_{i}^{\,\alpha_{i}}}$ be denoted by $\tilde{A}$. When each $P_i \subseteq B_\infty(2^L)$, we show in this paper that the time complexity of the algorithm is bounded above by a polynomial in $n, m_0, L, \tilde{A}, \epsilon^{-1}$ and $\log \delta^{-1}$. In fact, a stronger result is proved in this paper, with slightly more involved terminology.
In particular, we provide the first randomized polynomial time algorithm for computing mixed volumes of such polytopes when $k$ is an absolute constant, but $\alpha_1, \dots, \alpha_k$ are arbitrary. Our approach synthesizes tools from convex optimization, the theory of Lorentzian polynomials, and polytope subdivision. - [545] arXiv:2509.18205 (replaced) [pdf, html, other]
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Title: Structure-Fair Quantum Circuit Complexity: An Auditable Information-Theoretic Lower BoundComments: 36 pages , Comments welcomeSubjects: Quantum Physics (quant-ph); Information Theory (cs.IT)
Quantifying the complexity of quantum states that possess intrinsic structure, such as symmetry or encoding, in a fair manner constitutes a core challenge in the benchmarking of quantum technologies. This paper introduces the Reference-Contingent Complexity (RCC), an information-theoretic measure calibrated by the available quantum operations. The core idea is to leverage the quantum relative entropy to quantify the deviation of a quantum state from its "structured vacuum"-namely, the maximum entropy state within its constrained subspace-thereby only pricing the process of creating non-trivial information. Our central result is a key theorem that rigorously proves the RCC serves as a lower bound for the complexity of any universal quantum circuit. This lower bound is comprised of a linear dominant term, a universal logarithmic correction, and a precise physical correction term that accounts for non-uniformity in the spectral distribution. Crucially, we establish a set of operational protocols, grounded in tasks like quantum hypothesis testing, which make this theoretical lower bound experimentally "auditable." This work provides a "ruler" for quantum technology that is structure-fair and enables cross-platform comparison, thereby establishing a strictly verifiable constraint between the computational cost of the process and the structured information of the final state.
- [546] arXiv:2510.17031 (replaced) [pdf, html, other]
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Title: Modified open Toda chain and quasi-integrabilityComments: 22 pages, Latex, 4 figures. Appendix on numerical simulation addedSubjects: Exactly Solvable and Integrable Systems (nlin.SI); Mathematical Physics (math-ph)
We present a study of a quasi-integrable deformation of the three-particle open Toda chain, constructed by introducing a translation-invariant three-body interaction terms. Although this modification explicitly breaks the exact integrability of the standard Toda model, it retains fundamental structural properties, including energy and momentum conservation. Furthermore, we show that under a specific time-reflection and discrete symmetry among the chain coordinates, the system admits a quasi-conserved higher-order integral. Through analytic and numerical analysis of the deformed dynamics, we demonstrate the emergence and long-time persistence of quasi-conserved quantities, thereby establishing a controlled realization of quasi-integrability in a minimal nonlinear chain. Given the central role of integrable systems in elucidating the dynamics of classical and quantum models, this framework provides a concrete setting to investigate the mechanisms underlying the gradual breakdown of integrability and the onset of quasi-integrability in few-body systems.
- [547] arXiv:2510.22464 (replaced) [pdf, html, other]
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Title: Robust Spatial Confounding Adjustment via Basis VotingComments: 61 pages, 9 figures, supplementary appendix includedSubjects: Methodology (stat.ME); Statistics Theory (math.ST)
Estimating effects of spatially structured exposures is complicated by unmeasured spatial confounders, which undermine identifiability in spatial linear regression models unless structural assumptions are imposed. We develop a general framework for effect estimation in spatial regression models that relaxes the commonly assumed requirement that exposures contain higher-frequency variation than confounders. We propose basis voting, a plurality-rule estimator - novel in the spatial literature - that consistently identifies causal effects only under the assumption that, in a spatial basis expansion of the exposure and confounder, there exist several basis functions in the support of the exposure but not the confounder. This assumption generalizes existing assumptions of differential basis support used for identification of the causal effect under spatial confounding, and does not require prior knowledge of which basis functions satisfy this support condition. We design this estimator as the mode of several candidate estimators each computed based on a single working basis function. We also show that the standard projection-based candidate estimator typically used in other plurality-rule based methods is inefficient, and provide a more efficient novel candidate. Extensive simulations and a real-world application demonstrate that our approach reliably recovers unbiased causal estimates whenever exposure and confounder signals are separable on a plurality of basis functions. By not relying on higher-frequency variation, our method remains applicable to settings where exposures are smooth spatial functions, such as distance to pollution sources or major roadways, common in environmental studies.
- [548] arXiv:2510.23434 (replaced) [pdf, html, other]
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Title: Learning What to Learn: Experimental Design when Combining Experimental with Observational EvidenceSubjects: Econometrics (econ.EM); Statistics Theory (math.ST); Methodology (stat.ME)
Experiments deliver credible treatment-effect estimates but, because they are costly, are often restricted to specific sites, small populations, or particular mechanisms. A common practice across several fields is therefore to combine experimental estimates with reduced-form or structural external (observational) evidence to answer broader policy questions such as those involving general equilibrium effects or external validity. We develop a unified framework for the design of experiments when combined with external evidence, i.e., choosing which experiment(s) to run and how to allocate sample size under arbitrary budget constraints. Because observational evidence may suffer bias unknown ex-ante, we evaluate designs using a minimax proportional-regret criterion that compares any candidate design to an oracle that knows the observational study bias and jointly chooses the design and estimator. This yields a transparent bias-variance trade-off that does not require the researcher to specify a bias bound and relies only on information already needed for conventional power calculations. We illustrate the framework by (i) designing cash-transfer experiments aimed at estimating general equilibrium effects and (ii) optimizing site selection for microfinance interventions.
- [549] arXiv:2511.03142 (replaced) [pdf, html, other]
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Title: A Theory of Saving under Risk Preference DynamicsComments: 48 pages, 3 tables, 3 figuresSubjects: Theoretical Economics (econ.TH); Optimization and Control (math.OC)
Empirical evidence shows that wealthy households have substantially higher saving rates and markedly lower marginal propensity to consume (MPC) than other groups. Existing theory cannot account for this pattern unless under restrictive assumptions on returns, discounting, and preferences. This paper develops a general theory of optimal savings with preference shocks, allowing risk aversion to vary across states and over time. We show that incorporating such heterogeneity in risk attitudes fundamentally alters the asymptotic dynamics of consumption and saving. In particular, we provide an analytical characterization of the asymptotic MPCs and show that zero asymptotic MPCs, corresponding to a 100% asymptotic saving rate, arise under markedly weaker conditions than in existing theory. Strikingly, such outcomes occur whenever there is a positive probability that agents become less risk averse in the future. As a result, the vanishing MPC emerges as a generic feature rather than a knife-edge result of the optimal savings model, offering a more theoretically robust and empirically consistent account of the saving behavior of wealthy households.
- [550] arXiv:2511.04645 (replaced) [pdf, html, other]
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Title: On the foundations and applications of Lorentz-Finsler GeometryComments: Improved and expanded version with new contents related to Mechanics, Contact Geometry (Sect. 3.2 and new Section 2.5) and anisotropic connections (Sect. 2.4). Other minor modifications and 16 new references. 62 pages, 18 figuresSubjects: General Relativity and Quantum Cosmology (gr-qc); Differential Geometry (math.DG)
Finslerian extensions of Special and General Relativity -- commonly referred to as Very Special and Very General Relativity -- necessitate the development of a unified Lorentz-Finsler geometry. However, the scope of this geometric framework extends well beyond relativistic physics. Indeed, it offers powerful tools for modeling wave propagation in classical mechanics, discretizing spacetimes in classical and relativistic settings, and supporting effective theories in fundamental physics. Moreover, Lorentz-Finsler geometry provides a versatile setting that facilitates the resolution of problems within Riemannian, Lorentzian, and Finslerian geometries individually. This work presents a plain introduction to the subject, reviewing foundational concepts, key applications, and future prospects. The reviewed topics include (i) basics on the setting of cones, Finsler and Lorentz-Finsler metrics and their (nonlinear, anisotropic and linear) connections, (ii) the global structure of Lorentz-Finsler manifolds and its space of null geodesics, (iii) links among Riemannian, Finsler and Lorentz geometries, (iv) applications in classical settings as wildfires and seisms propagation, and discretization in classical and relativistic settings with quantum prospects, and (v) Finslerian variational approach to Einstein equations. The new results include the splitting of globally hyperbolic Finsler spacetimes, in addition to the analysis of several extensions of the Lorentz setting, as the case of timelike boundaries.
- [551] arXiv:2511.06153 (replaced) [pdf, html, other]
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Title: Topologically Invariant Permutation TestComments: 24 pages, 8 figures, 3 tablesSubjects: Neurons and Cognition (q-bio.NC); Algebraic Topology (math.AT); Methodology (stat.ME)
Functional brain networks exhibit topological structures that reflect neural organization; however, statistical comparison of these networks is challenging for several reasons. This paper introduces a topologically invariant permutation test for detecting topological inequivalence. Under topological equivalence, topological features can be permuted separately between groups without distorting individual network structures. The test statistic uses $2$-Wasserstein distances on persistent diagrams, computed in closed form. To reduce variability in brain connectivities while preserving topology, heat kernel expansion on the Hodge Laplacian is applied with bandwidth $t$ controlling diffusion intensity. Theoretical results guarantee variance reduction through optimal Hilbert space projection. Simulations across diverse network topologies show superior performance compared to conventional two-sample tests and alternative metrics. Applied to resting-state fMRI data from the Multimodal Treatment of ADHD study, the method detects significant topological differences between cannabis users and non-users.
- [552] arXiv:2511.13688 (replaced) [pdf, html, other]
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Title: Geometric Theory of Quark ConfinementComments: 16 pages, one figure. Significant new result. Exact solution for the loop equation leading to linear Regge trajectories in $\bar q q$ sectorSubjects: High Energy Physics - Theory (hep-th); Complex Variables (math.CV); Differential Geometry (math.DG)
We present the nonperturbative solution of the loop equation in quenched QCD (one quark loop in full gluon vacuum, including nonplanar graphs). This solution relies on a specific local minimum of the Plateau problem -- one that is additive over the closed parts of the bounding loop formed at self-intersections. This surface applies to large loops, leading to quark confinement via a factor $\exp(-\kappa S[C])$ multiplying the perturbative Wilson loop $W_{pert}[C]$. Crucially, the confinement mechanism relies on the self-duality of the area derivative -- a property that exists exclusively in four dimensions. This geometric constraint ensures stability only in four dimensions, distinguishing the resulting spectrum from standard string models, which are stable only in higher embedding dimensions. We compute the high-energy meson spectrum resulting from this novel confinement mechanism. The result is a usual linear Regge trajectory with the slope $\alpha' = \frac{1}{2 \pi \sigma}$ in our normalization of string tension $\sigma = 2 \sqrt{2} \kappa$. However, there are no string modes to be added to the spectrum in our solution, which amounts to the static linear potential for the quark pair. As we argue, the fluctuations of ``flux tube'' between quarks are accounted for in the gluon diagrams in $W_{pert}[C]$, and do not contribute to the confining force. This eliminates the problems of quantization of string in four dimensions.
- [553] arXiv:2512.01328 (replaced) [pdf, html, other]
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Title: Beyond Single-Device Constraints: A System-Level Theoretical Framework for High-Performance Single-Photon Detection at Room TemperatureComments: 19 pagesSubjects: Quantum Physics (quant-ph); Mathematical Physics (math-ph); Optimization and Control (math.OC)
Photon detection, a fundamental quantum technology, is traditionally treated as a fixed device-level operation governed by intrinsic properties of single-photon detectors (SPDs). High-performance detection has therefore largely relied on superconducting technologies whose requirement for cryogenic operation imposes substantial infrastructure constraints, limiting scalable deployments. Here, the enhanced single-photon detection (ESPD) framework is presented as a system-level theoretical paradigm that shifts photon detection from device-centric optimization to an integrated quantum-information-processing task, by reformulating it as an iteratively enhanced process integrating state-preparation, controlled operations, projective measurements, and multi-copy analysis. ESPD enables systematic performance enhancement through architectural design rather than material modification, thereby circumventing superconducting components, allowing high-performance detection using exclusively room-temperature hardware. Numerical simulations grounded in physically motivated parameters indicate that the ESPD framework can upgrade a conventional room-temperature SPD to effective DE exceeding 93\% and DCR below $10^{-9}$, which are comparable to state-of-the-art superconducting SPDs and can significantly relax the minimal tolerable channel transmission rate in quantum communications. While physical realization would require further component integration, this work establishes a rigorous theoretical framework for enhancing detection performance through architectural quantum-information principles, providing a general blueprint for transcending device-level constraints and guiding the development of next-generation room-temperature quantum technologies.
- [554] arXiv:2512.09057 (replaced) [pdf, other]
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Title: The ${\cal N}=1$ supersymmetric Pati-Salam models with extra $SU(2)_{L_2/R_2}$ gauge symmetry from intersecting D6-branesComments: This paper has been withdrawn by the authors as it was uploaded by mistake due to a miscommunication with a collaborator, and a revised manuscript will be resubmittedSubjects: High Energy Physics - Theory (hep-th); High Energy Physics - Phenomenology (hep-ph); Mathematical Physics (math-ph)
By introducing an extra stack of D6-branes to standard ${\cal N}=1$ supersymmetric Pati-Salam models, we extend the landscape of its complete search. In this construction, the $d$-stack of D6-branes is introduced besides the standard $a,~b,~c$-stacks. More intersections from the extra stacks of D6-branes appear, and thus Higgs/Higgs-like particles arise from more origins. Among these models, we find eight new classes of ${\cal N}=1$ supersymmetric Pati-Salam models with gauge symmetries $SU(4)_C\times SU(2)_L\times SU(2)_{R_1}\times SU(2)_{R_2}$ and $SU(4)_C\times SU(2)_{L_1}\times SU(2)_{R}\times SU(2)_{L_2}$, where $d$-stack of D6-branes carries the gauge symmetries $SU(2)_{R_2}$ and $SU(2)_{L_2}$, respectively. The $SU(2)_{L_1/R_1} \times SU(2)_{L_2/R_2}$ can be broken down to the diagonal $SU(2)_{L/R}$ gauge symmetry via bifundamental Higgs fields. In such a way, we for the first time successfully constructed three-family supersymmetric Pati-Salam models from non-rigid D6-branes with extra $d$-stacks of D6-branes as visible sectors. Interestingly, by introducing extra stack of D6-branes to the standard supersymmetric Pati-Salam models, the number of filler brane reduces in general, and eventually the models without any $USp(N)$ gauge symmetry present. This reduces the exotic particles from filler brane intersection yet provides more vector-like particles from ${\cal N}=2$ subsector that are useful in renormalization group equation evolution as an advantage. Moreover, interesting degeneracy behavior with the same gauge coupling ratio exists in certain class of models.
- [555] arXiv:2512.11459 (replaced) [pdf, html, other]
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Title: A mini-review on combinatorial solutions to the Marcus-Lushnikov irreversible aggregationComments: 42 pages, 15 figures, a mini-reviewSubjects: Statistical Mechanics (cond-mat.stat-mech); Earth and Planetary Astrophysics (astro-ph.EP); Soft Condensed Matter (cond-mat.soft); Mathematical Physics (math-ph); Chemical Physics (physics.chem-ph)
Over the past decade, a combinatorial framework for discrete, finite, and irreversibly aggregating systems has emerged. This work reviews its progress, practical applications, and limitations. We outline the approach's assumptions and foundations, based on direct enumeration of system states, contrasting with classical Smoluchowski and Marcus-Lushnikov methods. Using the constant kernel as an example, we derive combinatorial expressions for the average number of clusters of a given size and their standard deviation, and present the complete probability distribution for cluster counts. The method is then extended to several kernels (additive, product, linear-chain, condensation) by explicitly enumerating ways to form clusters of a given size. For general kernels, approximate solutions are obtained via recursive expressions, enabling predictions without explicit solutions. Applications to aerosol growth and planetesimal formation are demonstrated, with comparisons to numerical results. We summarize issues of validity and precision and propose open problems. The appendix includes partial Bell polynomials, generating functions, Lagrange inversion, potential applications, and links between combinatorial and scaling solutions of the Smoluchowski equation.
- [556] arXiv:2512.15771 (replaced) [pdf, html, other]
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Title: TENG++: Time-Evolving Natural Gradient for Solving PDEs With Deep Neural Nets under General Boundary ConditionsComments: 7 pages, 2 figuresSubjects: Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Numerical Analysis (math.NA); Machine Learning (stat.ML)
Partial Differential Equations (PDEs) are central to modeling complex systems across physical, biological, and engineering domains, yet traditional numerical methods often struggle with high-dimensional or complex problems. Physics-Informed Neural Networks (PINNs) have emerged as an efficient alternative by embedding physics-based constraints into deep learning frameworks, but they face challenges in achieving high accuracy and handling complex boundary conditions. In this work, we extend the Time-Evolving Natural Gradient (TENG) framework to address Dirichlet boundary conditions, integrating natural gradient optimization with numerical time-stepping schemes, including Euler and Heun methods, to ensure both stability and accuracy. By incorporating boundary condition penalty terms into the loss function, the proposed approach enables precise enforcement of Dirichlet constraints. Experiments on the heat equation demonstrate the superior accuracy of the Heun method due to its second-order corrections and the computational efficiency of the Euler method for simpler scenarios. This work establishes a foundation for extending the framework to Neumann and mixed boundary conditions, as well as broader classes of PDEs, advancing the applicability of neural network-based solvers for real-world problems.
- [557] arXiv:2512.17203 (replaced) [pdf, html, other]
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Title: Learning solution operator of dynamical systems with diffusion maps kernel ridge regressionSubjects: Machine Learning (cs.LG); Numerical Analysis (math.NA)
In this work, we propose a simple kernel ridge regression (KRR) framework with a dynamic-aware validation strategy for long-term prediction of complex dynamical systems. By employing a data-driven kernel derived from diffusion maps, the proposed Diffusion Maps Kernel Ridge Regression (DM-KRR) method implicitly adapts to the intrinsic geometry of the system's invariant set, without requiring explicit manifold reconstruction or attractor modeling, procedures that often limit predictive performance. Across a broad range of systems, including smooth manifolds, chaotic attractors, and high-dimensional spatiotemporal flows, DM-KRR consistently outperforms state-of-the-art random feature, neural-network and operator-learning methods in both accuracy and data efficiency. These findings underscore that long-term predictive skill depends not only on model expressiveness, but critically on respecting the geometric constraints encoded in the data through dynamically consistent model selection. Together, simplicity, geometry awareness, and strong empirical performance point to a promising path for reliable and efficient learning of complex dynamical systems.
- [558] arXiv:2512.19196 (replaced) [pdf, html, other]
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Title: Adaptive Probability Flow Residual Minimization for High-Dimensional Fokker-Planck EquationsSubjects: Computational Physics (physics.comp-ph); Machine Learning (cs.LG); Numerical Analysis (math.NA)
Solving high-dimensional Fokker-Planck (FP) equations is a challenge in computational physics and stochastic dynamics, due to the curse of dimensionality (CoD) and the bottleneck of evaluating second-order diffusion terms. Existing deep learning approaches, such as Physics-Informed Neural Networks, face computational challenges as dimensionality increases, driven by the $O(d^2)$ complexity of automatic differentiation for second-order derivatives. While recent probability flow approaches bypass this by learning score functions or matching velocity fields, they often involve serial operations or depend on sampling efficiency in complex distributions. To address these issues, we propose the Adaptive Probability Flow Residual Minimization (A-PFRM) method. We reformulate the second-order FP equation into an equivalent first-order deterministic Probability Flow ODE (PF-ODE) constraint, which avoids explicit Hessian computation. Unlike score matching or velocity matching, A-PFRM solves this problem by minimizing the residual of the continuity equation induced by the PF-ODE. We leverage Continuous Normalizing Flows combined with the Hutchinson Trace Estimator to reduce the training complexity to linear scale $O(d)$, achieving an effective $O(1)$ wall-clock time on GPUs. To address data sparsity in high dimensions, we apply a generative adaptive sampling strategy and theoretically prove that dynamically aligning collocation points with the evolving probability mass is a necessary condition to bound the approximation error. Experiments on diverse benchmarks -- ranging from anisotropic Ornstein-Uhlenbeck (OU) processes and high-dimensional Brownian motions with time-varying diffusion terms, to Geometric OU processes featuring non-Gaussian solutions -- demonstrate that A-PFRM effectively mitigates the CoD, maintaining high accuracy and constant temporal cost for problems up to 100 dimensions.
- [559] arXiv:2512.19493 (replaced) [pdf, html, other]
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Title: Fare Zone Assignment on TreesSubjects: Data Structures and Algorithms (cs.DS); Computer Science and Game Theory (cs.GT); Optimization and Control (math.OC)
Tariff setting in public transportation networks is an important challenge. A popular approach is to partition the network into fare zones ("zoning") and fix journey prices depending on the number of traversed zones ("pricing"). In this paper, we focus on finding revenue-optimal solutions to the zoning problem for a given concave pricing function. We consider tree networks with $n$ vertices, since trees already pose non-trivial algorithmic challenges. Our main results are efficient algorithms that yield a simple $\mathcal{O}(\log n)$-approximation as well as a more involved $\mathcal{O}(\log n/\log \log n)$-approximation. We show how to solve the problem exactly on rooted instances, in which all demand arises at the same source. For paths, we prove strong NP-hardness and outline a PTAS. Moreover, we show that computing an optimal solution is in FPT or XP for several natural problem parameters.
- [560] arXiv:2512.20068 (replaced) [pdf, html, other]
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Title: Change Point Detection and Mean-Field Dynamics of Variable Productivity Hawkes ProcessesSubjects: Other Statistics (stat.OT); Statistics Theory (math.ST)
Many self-exciting systems change because endogenous amplification, as opposed to exogenous forcing, varies. We study a Hawkes process with fixed background rate and kernel, but piecewise time-varying productivity. For exponential kernels we derive closed-form mean-field relaxation after a change and a deterministic surrogate for post-change Fisher information, revealing a boundary layer in which change time information localises and saturates, while post-change level information grows linearly beyond a short transient. These results motivate a Bayesian change point procedure that stabilizes inference on finite windows. We illustrate the method on invasive pneumococcal disease incidence in The Gambia, identifying a decline in productivity aligned with pneumococcal conjugate vaccine rollout.