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Adaptive Sharpness-Aware Minimization with a Polyak-type Step size: A Theory-Grounded Scheduler
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arxiv.org
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2 days ago
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eng
arXiv:2606.01827v1 Announce Type: new Abstract: Sharpness-Aware Minimization (SAM) has established itself as a powerful and widely adopted optimizer for training machine learning models. By explicitly minimizing the sharpness of the loss landscape, SAM often improves generalization while delivering strong empirical performance. However, SAM and its variants, like most training algorithms, are sensitive to the choice of learning rate, which....
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Infinite Moments of Class Groups for Solvable Fields with a Normal Abelian Subgroup
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arxiv.org
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2 days ago
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eng
arXiv:2606.01831v1 Announce Type: new Abstract: We apply the class field theory and Minkowski bound to obtain an upper bound estimate for the number of solutions to the restricted ramifications when the Galois group is solvable. Together with suitable conditions on the solvable group and the ordering of number fields, we could prove an upper bound on specific field-counting problems, hence the infinite moment of the class groups. In partic..
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MA(1) processes with uniform innovations conditioned to stay positive in the non-expanding regime
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arxiv.org
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2 days ago
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eng
arXiv:2606.01832v1 Announce Type: new Abstract: We study an MA(1)-process with uniform innovations conditioned to stay positive. Representing the model as a Markov chain, we prove the existence of the limiting finite-dimensional distributions under this conditioning and identify the limiting process explicitly as a Doob $h$-transform. In the non-expanding case, i.e. when the coupling parameter $\theta$ satisfies $\theta\in[-1,1)$, we compu..
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arXiv:2606.01866v1 Announce Type: new Abstract: We study a two-dimensional semiflexible membrane model whose formal Hamiltonian is given by $H[\phi]=\sum_x \bigl(\|\nabla \phi_x\|^2 +N^\lambda |\Delta \phi_x|^2\bigr)$, interpolating between the discrete Gaussian free field (DGFF) and the membrane model (MM). We analyze its finite-volume covariance in the bulk as $N\to\infty$, and identify distinct regimes depending on the parameter $\lambd....
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Waterproof Editor: an educational environment for proof assistants and programming languages
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arxiv.org
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2 days ago
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eng
arXiv:2606.01875v1 Announce Type: new Abstract: Waterproof Editor provides an educational environment specifically targeted to teaching with proof assistants or programming languages. It arose from Waterproof, educational software targeted at helping students acquire the skill of giving mathematical proofs. Its original features such as enabling rich formatting and providing clear input areas are now abstracted away in an npm package and c..
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Can the root cluster remain largest forever in random recursive tree percolation?
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arxiv.org
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2 days ago
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eng
arXiv:2606.01881v1 Announce Type: new Abstract: We consider Bernoulli bond percolation with fixed retention parameter $p$ on the random recursive tree, coupled through the natural growth process. We prove that the root cluster has a strictly positive probability of remaining a largest cluster at every time. Equivalently, in the associated Simon-type Chinese restaurant process, the first table has a positive probability of remaining a large..
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Block Jacobi/Gauss-Seidel preconditioning for GLT sequences, and GLH sequences
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arxiv.org
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2 days ago
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eng
arXiv:2606.01888v1 Announce Type: new Abstract: The theory of generalized locally Toeplitz (GLT) sequences is an apparatus for computing the spectral and singular value distribution of sequences of matrices that possess a (possibly hidden) Toeplitz-like structure. These sequences, which are known as GLT sequences, arise in several applications, including the discretization of differential equations. Associated with any GLT sequence is a sp....
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Infinite-dimensionality of the rational homotopy groups of the space of long embeddings of codimension 2
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arxiv.org
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2 days ago
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eng
arXiv:2606.01903v1 Announce Type: new Abstract: In this paper, we study the space of compactly supported embeddings between Euclidean spaces, $\mathrm{Emb}_c(\mathbb{R}^j, \mathbb{R}^n)$. By utilizing hairy graphs, we construct elements in the homotopy groups $\pi_{\bullet}(\overline{\mathrm{Emb}}_c(\mathbb{R}^j, \mathbb{R}^{n})) \otimes \mathbb{Q}$ corresponding to certain uni-trivalent graphs in the model. We then prove that these elemen..
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Three-scale singular limits with applications to rapidly rotating fluids and the hyperbolization of dispersive systems
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arxiv.org
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2 days ago
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eng
arXiv:2606.01913v1 Announce Type: new Abstract: We consider singular problems for a general class of quasilinear hyperbolic systems that involve two a priori independent stiff parameters. We argue that such situations may lead to the rapid development of small-amplitude spatial oscillations of small wavelength starting from arbitrarily smooth initial data. Despite this phenomenon we provide sufficient conditions on initial data that secure..
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arXiv:2606.01917v1 Announce Type: new Abstract: Let $\mathcal P(H)$ be the semigroup obtained by endowing the family of all non-empty subsets of a semigroup $H$ with the setwise operation naturally induced by $H$ on its power set, and denote by $\mathcal P_\text{fin}(H)$ the subsemigroup of $\mathcal P(H)$ consisting of all non-empty finite subsets of $H$. We obtain (as a corollary of a theorem of independent interest) that if $H$ is a gr..
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Exact spectrum of the XX spin chain with constrained non-diagonal boundary fields
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arxiv.org
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2 days ago
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eng
arXiv:2606.01918v1 Announce Type: new Abstract: We study the exact spectrum of the XX spin chain with constrained non-diagonal boundary fields, which can be analyzed by solving the associated Bethe Ansatz equations. In these equations, the number of Bethe roots has a definite parity, and all Bethe roots are located at the zeros of a unary function. We investigate the possible positions of the Bethe roots. Based on numerical observations, w..
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The 1/2-Conjecture for $q$-Binomial Coefficients with Fractional Index
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arxiv.org
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2 days ago
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eng
arXiv:2606.01919v1 Announce Type: new Abstract: For a nonnegative integer $k$ and a rational number $r\in\mathbb{Q}^+$, we define the generalized Gaussian binomial coefficient $\qbinom{r+k}{k} = \frac{(q^{r+1}; q)_k}{(q; q)_k}$. When $r=a/b$ with $a,b$ coprime positive integers and $b\geq 2$, expanding $\qbinom{r+k}{k}$ via the finite $q$-binomial theorem produces fractional powers of $q$, so that $\qbinom{r+k}{k}$ is a \emph{Puiseux serie....
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arXiv:2606.01924v1 Announce Type: new Abstract: In our previous work, we generalized the concept of Galois points for irreducible plane curves to the case of reduced plane curves. We also introduced the concept of simultaneous Galois points, which is an equivalent concept to Galois points, and studied their number when the irreducible components are nonsingular. In this paper, we consider the remaining cases where the irreducible component..
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Stationary fluctuations for an exclusion process with mass and energy conservation
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arxiv.org
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2 days ago
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eng
arXiv:2606.01937v1 Announce Type: new Abstract: We introduce a novel exclusion process with two conservation laws, mass and energy, designed to mimic the essential features of continuous systems like interacting oscillators within the framework of interacting particle systems. This distinguishes our model from conventional multi-species processes where only particle numbers are conserved. As a basis for our fluctuation analysis, we first s....
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Propagation of Chaos for Mean-field Mean Reflected Backward Stochastic Differential Equations
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arxiv.org
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2 days ago
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eng
arXiv:2606.01944v1 Announce Type: new Abstract: In this paper, we establish a propagation of chaos result for mean-field mean reflected backward stochastic differential equations (BSDEs), where both the generator and constraint depend on the distribution of the solution. When the generator does not rely on $z$, under mild Lipschitz and integrability conditions, we prove existence and uniqueness of the solution to the interacting particle s..
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Painlev\'e XXXIV Asymptotics for the Focusing mKdV Equation with Finite-Genus Background and Discrete Spectrum
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arxiv.org
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2 days ago
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eng
arXiv:2606.01977v1 Announce Type: new Abstract: We investigate the Cauchy problem for the focusing modified Korteweg--de Vries (mKdV) equation with finite-genus algebro-geometric quasi-periodic initial data. By applying the nonlinear steepest-descent method of Deift--Zhou to the associated Riemann--Hilbert (RH) problem, we derive the long-time asymptotics of the solution in the critical regime where complex stationary phase points coalesce..
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Generalised eigenvector expansion of infinite Toeplitz matrices with absolutely/completely monotone entries
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arxiv.org
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2 days ago
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eng
arXiv:2606.01986v1 Announce Type: new Abstract: We study the spectral theory of infinite Toeplitz matrices $T = (a_{k - l})$ under the assumption that $(a_k)$ and $(a_{-k})$ are completely monotone sequences. We derive expressions for generalised eigenvectors and prove a generalised eigenvector expansion of $T$. Even if the matrix $T$ is not normal, our expressions involve only eigenvalues and eigenvectors with real entries.
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On the threshold Ramsey multiplicity conjectures for paths and even cycles
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arxiv.org
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2 days ago
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eng
arXiv:2606.01996v1 Announce Type: new Abstract: The Ramsey number $r(H)$ of a graph $H$ is the minimum positive integer $n$ such that every red/blue edge-coloring of the complete graph $K_n$ on $n$ vertices contains a monochromatic copy of $H$. The threshold Ramsey multiplicity $m(H)$ of $H$ is the minimum number of monochromatic copies of $H$ over all red/blue edge-colorings of $K_{r(H)}$. Let $P_t$ and $C_t$ be a path and a cycle on $t$ ..
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Second-Order Optimality Conditions for Sparse Differentiable Optimization Problems via Limiting Second-Order Subdifferentials
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arxiv.org
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2 days ago
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eng
arXiv:2606.01998v1 Announce Type: new Abstract: In this paper, we investigate a class of sparse optimization problems in which both the objective and constraint functions are Fr\'echet differentiable and possess locally Lipschitz continuous gradient mappings. More precisely, by utilizing the limiting (Mordukhovich) second-order subdifferential of the associated Lagrangian function, we establish new second-order necessary and sufficient opt..
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Higher-Order Multifractional Stable Motion: Definition and Fundamental Properties
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arxiv.org
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2 days ago
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eng
arXiv:2606.02003v1 Announce Type: new Abstract: This paper introduces the $n$-th order multifractional stable motion ($n$-MFSM), a novel stochastic process that simultaneously unifies three key modelling features: heavy-tailed distributions ($\alpha$-stable with $\alpha\in(1,2]$), time-varying local regularity via a functional Hurst parameter $H(t)\in(n-1,n)$, and extended scaling behaviour of order $n\geq1$. No existing framework combines..
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arXiv:2606.02023v1 Announce Type: new Abstract: We introduce two families of quandles arising from Coxeter quandles. One, which we call a double covering, is the set of roots with binary operation defined by using the negatives of reflections. The double covering is realized as a conjugation quandle in a Pin group. The other, which we call a rotational $D_n$ quandle, is the set of some right angle rotations in the Coxeter group of type $D_..
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arXiv:2606.02025v1 Announce Type: new Abstract: For an internal category $\mathbb{C}$ in a cartesian category $\mathcal{C}$ we define, naturally in objects $X$ of $\mathcal{C}$, $Prin_{\mathbb{C}}(X)$. This is a category whose objects are principal $c \mathbb{C}$-bundles over $X$ and whose morphisms are principal $c(\mathbb{C}^{\uparrow})$-bundles. Here $c(\_)$ denotes taking the core groupoid of a category (same objects but only isomorphi....
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arXiv:2606.02026v1 Announce Type: new Abstract: The aim of the paper is to survey results on Jordan types for pairs of nilpotent commuting matrices. We also review recent proof of the Box Conjecture on Jordan types that have equal dense orbit in the nilpotent commutator.
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A Domatic Analogue of $\chi$-Bounded Graph Classes and the Gy\'{a}rf\'{a}s-Sumner Conjecture
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arxiv.org
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2 days ago
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eng
arXiv:2606.02030v1 Announce Type: new Abstract: Given a graph $G$, a dominating set is a subset $X\subseteq V(G)$ such that $N[X]=V(G)$. The \emph{domatic number} of $G$, denoted ${\rm dom}(G)$, is the maximum size of a partition of $V(G)$ into dominating sets. In analogy with the lower bound of the chromatic number by the clique number, the domatic number satisfies the upper bound ${\rm dom}(G)\le \delta(G)+1$ where $\delta(G)$ is the min....
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Self-intersection local times for Volterra Gaussian processes in stochastic flows with interaction
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arxiv.org
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2 days ago
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eng
arXiv:2606.02036v1 Announce Type: new Abstract: In this paper, we study self-intersection local times for a stochastic process $x(u(\cdot),t)$, where $u$ is a Gaussian process of the form $u(t)=\int^t_0k(t,s)\mathrm{d}{w(s)}$, $k$ is a deterministic kernel of the Volterra type, $w$ is a Wiener process, and $x$ is a solution to the \emph{equation with interaction}. Equations with interaction are a class of interacting particle system descri....
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arXiv:2606.02040v1 Announce Type: new Abstract: A fundamental open problem in the homological theory of Banach spaces is the calculation of the injective dimension of the Banach space $c_0$. We make a contribution to the study of this problem by proving that, if the Continuum Hypothesis ($\mathsf{CH}$) holds, then the injective dimension of $c_0$ is at least 3. In the course of proving this result, we introduce the notion of an \emph{almos..
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arXiv:2606.02046v1 Announce Type: new Abstract: We explicitly construct dual bimodules in a semistrict monoidal 2-category, using Frobenius algebra structure. The main result shows that a coherent dual of the underlying object can be promoted to a coherent dual of the bimodule, with zigzag 2-isomorphisms additionally require special Frobenius structures. We also prove that every special Frobenius algebra in $\mathbf{2Vect}$ is rigid, via a..
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arXiv:2606.02055v1 Announce Type: new Abstract: We study exact community recovery in the two-community stochastic block model on $n$ vertices under limited and noisy access to network data. The learner may query a noisy neighborhood oracle that reveals each true neighbor of a queried vertex independently with fixed probability and never returns non-neighbors, subject to a finite query budget. We consider both oracle-only access and a combi....
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arXiv:2606.02056v1 Announce Type: new Abstract: We introduce the class of \emph{Special--Dirichlet processes}, consisting of c\`adl\`ag adapted processes admitting a decomposition \[ X=M+\Gamma, \] where \(M\) is a local martingale and \(\Gamma\) is an adapted c\`adl\`ag process with vanishing continuous quadratic variation whose jumps are predictable and \(\mathcal F_{s-}\)-measurable. This class arises naturally from transformations of ....
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A first-optimize-then-reduce framework for optimal control with the shifted proper orthogonal decomposition
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arxiv.org
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2 days ago
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eng
arXiv:2606.02064v1 Announce Type: new Abstract: Overcoming the Kolmogorov barrier for constructing efficient reduced-order models (ROMs) for transport-dominated problems remains a challenge. This has impeded their use as computationally cheap surrogates for partial differential equations (PDEs) associated with optimal control problems. Since such problems require multiple computations of the full PDE, employing their reduced-order surrogat....
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Critical mass threshold for the 2D Patlak-Keller-Segel-Navier-Stokes system
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arxiv.org
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2 days ago
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eng
arXiv:2606.02070v1 Announce Type: new Abstract: In this paper, we investigate critical mass threshold for the Patlak-Keller-Segel-Navier-Stokes system on the two-dimensional whole space and obtain global existence of strong solutions if the initial mass is less than or equal to $8\pi$, regardless of the initial norm of the velocity. One new observation is that the local mass of the density function rearrangement satisfies a good inequality..
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arXiv:2606.02071v1 Announce Type: new Abstract: In this article, we consider the words with cyclic indices. For given $s$, we consider the pair $(\iota,\kappa)$ of indices such that the word of length $s$ from $\iota$ is equal to the word of length $s$ from $\kappa$. We give a characterization of such pairs for a cyclic Fibonacci word, and give the number of them.
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Decision-calibrated prediction sets for robust power system operations
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arxiv.org
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2 days ago
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eng
arXiv:2606.02081v1 Announce Type: new Abstract: Robust optimization offers a tractable approach to balance operating costs and reliability in power systems dominated by weather-dependent renewable uncertainty, but its performance depends critically on the uncertainty set. Standard data-driven approaches often calibrate uncertainty sets to attain predictive coverage, which can produce unnecessarily large sets and costly operating decisions.....
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$p$-Robust Trace Liftings for Discrete Harmonic Extensions and Boundary-Preserving $hp$ Interpolation on Tetrahedral Meshes
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arxiv.org
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2 days ago
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eng
arXiv:2606.02086v1 Announce Type: new Abstract: We construct p-robust polynomial trace liftings on three-dimensional tetrahedral meshes. The prescribed trace is a continuous piecewise polynomial function on a boundary face patch; the tetrahedra touching this patch have one common degree, while the interior degrees may be arbitrary. The lifting is degree-preserving, supported in the corresponding boundary layer, and satisfies both an H^1 es..
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Generalization of Marion's Theorem: Volumes of Central Polytopes Obtained by Trisecting the Edges of Simplices
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arxiv.org
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2 days ago
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eng
arXiv:2606.02087v1 Announce Type: new Abstract: The classical Marion's theorem states that the area of the central hexagon obtained by dividing each side of a triangle into three equal parts and connecting the interior division points is exactly $1/10$ of the area of the original triangle. In this paper, the construction is extended to an arbitrary n-dimensional simplex: each edge is divided into three equal parts, and through each n-1 ver..
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Modified scattering type asymptotic behavior for a quadratic nonlinear Schr\"odinger system under the mass-resonance condition in two dimensions
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arxiv.org
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2 days ago
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eng
arXiv:2606.02088v1 Announce Type: new Abstract: We study a system of nonlinear Schrodinger equations under the mass-resonance condition and provide a complete description of the asymptotic behavior of small solutions in two space dimensions. Our analysis is based on a detailed study of an associated system of ordinary differential equations governing the asymptotic profile. We establish a phase-amplitude representation for this ODE with ar....
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arXiv:2606.02110v1 Announce Type: new Abstract: Let $\mathcal{H}$ be an $r$-uniform hypergraph. For $S\in \binom{V(\mathcal{H})}{j}$, let $\mathrm{deg}(S)$ be the number of edges of $\mathcal{H}$ containing $S$, and define the $(j,p)$-norm of $\mathcal{H}$ by $\|\mathcal{H}\|_{j,p}=\left(\sum_{S\in \binom{V(\mathcal{H})}{j}}\mathrm{deg}(S)^p\right)^{1/p}$. Motivated by a problem of Chao, Dong, Shen and Yang, we determine the maximum number..
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Switched Event-Triggered Adaptive Control of Reaction-Diffusion PDE-ODE with Neural Operator Implementation
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arxiv.org
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2 days ago
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eng
arXiv:2606.02114v1 Announce Type: new Abstract: This paper develops a switched event-triggered adaptive boundary control for a class of reaction-diffusion PDE-ODE cascade systems, where the system and input matrices in the ODE as well as the spatially-varying reaction coefficient in the PDE are uncertain. A two-step backstepping transformation is constructed to derive the continuous-time control law. Then a novel dynamic event-triggered co....
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arXiv:2606.02116v1 Announce Type: new Abstract: Geodesic regression generalizes classical regression models to manifold-valued data by replacing affine models in Euclidean spaces with geodesic models on Riemannian manifolds. In this paper, we set up a framework for regression based on retractions instead of the Riemannian exponential map and its corresponding retraction-based distance. The associated optimization problem is posed on a subs..
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Applications of a formula of Maesaka-Seki-Watanabe type for multiple harmonic $q$-sums
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arxiv.org
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2 days ago
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eng
arXiv:2606.02123v1 Announce Type: new Abstract: Maesaka, Seki and Watanabe proved a formula for multiple harmonic sums. Yamamoto generalized it to Schur-type multiple harmonic sums, and the second author proved a $q$-analogue of this generalization. In this paper, we give two applications of the $q$-analogue formula. The first is an alternative proof of the duality of a $q$-analogue of multiple zeta values. The second is a proof of an iden..
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Trace-Preserving hp Interpolation and Polynomial Liftings on Conforming Hexahedral Meshes
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arxiv.org
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2 days ago
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eng
arXiv:2606.02125v1 Announce Type: new Abstract: Trace-compatible polynomial extensions are a recurring local ingredient in high-order finite element analysis on conforming hexahedral meshes. They are needed whenever prescribed edge and face traces must be preserved while a polynomial is extended into a neighboring cell or boundary patch. The main contribution of this paper is the construction of p-robust polynomial liftings on nonsingular ....
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arXiv:2606.02135v1 Announce Type: new Abstract: We exhibit, in a systematic way, connections between hypergraph Tur\'an problems and extremal set theory. More specifically, we construct natural families of uniform hypergraphs for which the upper bounds on their Tur\'an densities reduce to classical problems in extremal set theory, including the Erd\H{o}s--Ko--Rado theorem, $L$-intersecting families, and the Erd\H{o}s matching problem.
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QEC and EAQEC Codes from Hermitian Sums and Hulls of Cyclic Codes over $\mathbb{F}_2 \times (\mathbb{F}_2+v\mathbb{F}_2)$
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arxiv.org
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2 days ago
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eng
arXiv:2606.02137v1 Announce Type: new Abstract: In this work, we determine the generator polynomials for the Hermitian hulls and Hermitian sums of cyclic codes defined over the composite ring $\mathbb{F}_2 \times (\mathbb{F}_2 + v\mathbb{F}_2)$, where $v^2 = v$. Based on these structures, we develop quantum error-correcting (QEC) codes by applying the Hermitian dual version of Quantum Construction~X to the obtained Hermitian hulls and sums..
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Normalized solutions to an exponential growth Choquard equation driven by mixed local-nonlocal operator in $\mathbb{R}^2$
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arxiv.org
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2 days ago
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eng
arXiv:2606.02141v1 Announce Type: new Abstract: In this article, we study the existence of normalized solutions to the following mixed nonlinear Choquard equation with exponential growth \begin{align*} \left\{ \begin{aligned} \mathcal{L}u+\lambda u \; &=\; \Lambda(I_{\alpha}\ast F(u))F'(u), \quad \text{in }\mathbb{R}^{2}, \int_{\mathbb{R}^{2}}|u|^{2}\,dx \; &=\; a^{2}, \end{aligned} \right. \end{align*} where $\mathcal{L}= -\Delta....
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