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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..
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..
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....
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..
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....
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..
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..
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.
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 $t$ ..
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..
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..
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_..
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....
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.
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....
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....
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..
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..
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....
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 ....
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....
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..
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.
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.....
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..
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..
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....
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..
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....
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..
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..
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 ....
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.
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..
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....
arXiv:2606.02144v1 Announce Type: new Abstract: We investigate support thresholds for fully smeary and directionally smeary absolutely continuous probability measures on the sphere \(\mathbb{S}^m\). The motivation is inferential: smeariness is caused by degeneracy of the Hessian of the Fr\'echet function, and such degeneracy can invalidate the classical central limit theorem (CLT) for Fr\'echet means and the corresponding Wald-type \(\chi^....
arXiv:2606.02150v1 Announce Type: new Abstract: We consider the reciprocal cost function $ J(x)=\frac12(x+x^{-1})-1 $ and its $n$-dimensional extension $$J(x_1,\ldots,x_n) = \frac12(R+R^{-1})-1, \qquad R=\prod_{i=1}^n x_i^{\alpha_i}, \qquad \alpha=(\alpha_1,\ldots,\alpha_n)\in\mathbb{R}^n\setminus\{0\}.$$ In logarithmic coordinates $t_i=\log x_i$, the Hessian of $J$ has rank one at every point. The associated Hessian geometry is degenerate....
arXiv:2606.02159v1 Announce Type: new Abstract: Lambert-Cole and Meier showed that the elliptic surface $E(n)$ admits a $(12n-2,0)$-trisection, considering the property that $E(n)$ is a certain double branched cover of $S^2 \times S^2$, which is a minimal genus trisection. In this paper, we clarify a way to construct an explicit $(12n-2,0)$-trisection diagram of $E(n)$ from its handle diagram arising from its Lefschetz fibration.
arXiv:2606.02160v1 Announce Type: new Abstract: Resolving a conjecture of Espuny D\'iaz and Gir\~ao [Random Structures Algorithms, 2023], we determine the sharp minimum-degree threshold for Hamiltonicity in graphs perturbed by a uniformly random $K_r$-factor. In fact, we prove the stronger pancyclic statement. More generally, for each fixed connected graph $F$, we study the union of an arbitrary deterministic graph of linear minimum degree....
arXiv:2606.02164v1 Announce Type: new Abstract: We consider the skein algebra of an unpunctured marked surface. Thurston previously constructed its band basis topologically. We show that this band basis coincides with the common triangular basis, which is a Kazhdan-Lusztig type basis for quantum cluster algebras analogous to the dual canonical basis of quantum groups. Our result confirms a conjecture of Thurston. It also provides new cases..
arXiv:2606.02165v1 Announce Type: new Abstract: In wholesale electricity markets, electricity producers and the independent system operator (ISO) play a central role. The ISO is responsible for minimizing production costs while satisfying supply-demand balance and capacity constraints. In this paper, we study a continuous-time problem in which the ISO seeks to minimize the joint cost of operation and investment in an electricity network. T....
arXiv:2606.02176v1 Announce Type: new Abstract: This work introduces L0PathBrex, a novel method for estimating the solution path of L0-regularized problems through the use of L0 Bregman relaxations (B-rex). Recently introduced and analyzed in the literature, these relaxations provide continuous reformulations of the original objective, are applicable to possibly non-quadratic data fidelity terms, and depend on a family of functions designe....
arXiv:2606.02180v1 Announce Type: new Abstract: In this paper, we prove universality of the distribution of the cokernels of a random $p$-adic matrix with inhomogeneously balanced columns. More precisely, let $u \ge 0$ be an integer and $A(n)$ be a random $n \times (n+u)$ matrix over $\mathbb{Z}_p$ whose $i$-th column is $\alpha_n(i)$-balanced. We prove that if $\sum_{i=1}^{n+u} \exp(-\epsilon \alpha_n(i)n) \to 0$ as $n \to \infty$ for eve..
arXiv:2606.02186v1 Announce Type: new Abstract: We study $O(2)$-equivariant spectral flow for Dirac operators on a finite warped cylinder equipped with fixed admissible regularized APS boundary conditions. The twisting bundle is a real higher-rank orthogonal bundle, and reflection symmetry is implemented by a fiber involution. After complexifying the twisting bundle and diagonalizing the orthogonal twist, the Dirac equation decomposes into....
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