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Gradient estimates and Liouville theorems for the \(\Phi\)-Laplacian equations on Riemannian manifolds
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arxiv.org
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2 days ago
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eng
arXiv:2606.01579v1 Announce Type: new Abstract: This paper establishes gradient estimates and Liouville-type theorems for the \(\Phi\)-Laplacian equation \(\Delta_{\Phi}(u) = G(|\nabla u|^2)\) on complete Riemannian manifolds and its parabolic counterpart \(\partial_t u = \Lambda_{\Phi}(u)\) on compact Riemannian manifolds. Using a nonlinear \(\Phi\)-Bochner formula and the Nash-Moser iteration technique, we prove local gradient bounds un..
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Connectivities for k-knitted graphs and for minimal counterexamples to Hadwiger's Conjecture
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arxiv.org
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2 days ago
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eng
arXiv:2606.01586v1 Announce Type: new Abstract: For a given subset $S\subseteq V(G)$ of a graph $G$, the pair $(G,S)$ is \emph{knitted} if for every partition of $S$ into non-empty subsets $S_1, S_2, \ldots, S_t$, there exist pairwise disjoint connected subgraphs $C_1, C_2, \ldots, C_t$ in $G$ such that $S_i\subseteq V(C_i)$ for all $1 \le i \le t$. A graph $G$ is \emph{$\ell$-knitted} if $(G,S)$ is knitted for every subset $S\subseteq V(G....
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arXiv:2606.01587v1 Announce Type: new Abstract: Strata of holomorphic $1$-forms have an absolute period foliation given by varying $1$-forms while keeping their integrals along closed loops fixed. In this paper, we classify the leaf closures of this foliation when the stratum is connected, outside of a subvariety of high codimension.
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arXiv:2606.01596v1 Announce Type: new Abstract: Learning chaotic dynamical systems from data requires more than short-term predictive accuracy: the learned model must preserve the attractor geometry and its invariant statistics. Trajectory (zero-order) and Jacobian (first-order) matching supervise the values and tangent structure of the vector field, but neither constrains how the field bends away from its tangent plane. A model can thus m....
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arXiv:2606.01609v1 Announce Type: new Abstract: Let $G$ be a connected reductive algebraic group defined over $\mathbb{R}$. In the 1980s, Arthur conjectured the existence of certain packets of irreducible admissible representations of $G(\mathbb{R})$ satisfying various remarkable properties. These packets were given a precise definition in the book of Adams, Barbasch, and Vogan in terms of microlocal geometry on a space of Langlands parame....
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Stabilization-free virtual element methods based on finite element interpolation
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arxiv.org
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2 days ago
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eng
arXiv:2606.01614v1 Announce Type: new Abstract: In this paper, we introduce a new framework for designing stabilization-free virtual element methods (VEMs) based on an finite element interpolation-based strategy, where we can simultaneously eliminate the stabilization terms in the discretizations of diffusion and reaction terms. The core idea is to construct a computable, polynomial-preserving, and norm-equivalent interpolation operator fr....
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Stability of $N$-soliton solutions for the modified Camassa--Holm equation
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arxiv.org
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2 days ago
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eng
arXiv:2606.01618v1 Announce Type: new Abstract: In this work, we address the stability of $N$-soliton solutions to the completely integrable modified Camassa--Holm (mCH) equation. Recently, Li, Liu, and Zhu (Math. Ann. 392 (2025), 899--932) established the orbital stability of 2-soliton solutions in $H^4(\mathbb{R})$ with respect to the solution $u$ and highlighted the stability of mCH $N$-soliton solutions remains an urgent challenge. Mot....
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An alternating learning-based collocation method for solving inverse elliptic problems
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arxiv.org
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2 days ago
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eng
arXiv:2606.01622v1 Announce Type: new Abstract: We propose the Alternating Learning-Based Collocation (ALBC) method for solving inverse elliptic problems. Our approach employs sinusoidal shallow networks as adaptive basis generators. By alternately updating the state variable and the unknown parameter, we decompose the original nonconvex joint optimization problem into a sequence of tractable linear subproblems. This strategy effectively o....
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arXiv:2606.01627v1 Announce Type: new Abstract: Consider a random geometric graph $G_M(n;r)$ whose vertex set consists of $n$ points chosen independently and uniformly from a Riemannian manifold $M$, with edges joining pairs of vertices whose distance in the metric $d_M$ is at most $r$. Let $\Delta$ denote the expected average degree of the graph. As is the case for Erd\H{o}s-R\'enyi graphs, there is a critical value $\Delta_c$, depending ....
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arXiv:2606.01633v1 Announce Type: new Abstract: B\"uy\"uka\c{s}ik et al. [Publ. Math. Debrecen, 2024] introduced a family of generalizations of Euler's totient function $\varphi(n)$, by setting $\varphi_k(n) = \sum_{a} a^k$ for $a \in [1, n]$ such that $(a, n) = 1$, with $\varphi_0(n) = \varphi(n)$. Letting $\mathcal{D}_{s} = \{ k \geq s : \forall n \geq 1 \ \varphi_s(n) \mid \varphi_k(n) \}$, B\"uy\"uka\c{s}ik et al. proved that $\mathcal..
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arXiv:2606.01646v1 Announce Type: new Abstract: Let $\K$ denote a field. Extending the structural frameworks established in \cite{JZ2025-2}, this paper introduces novel techniques utilizing non-commutative reduction orders, factorization theory, and the generalized lifting methodology. We establish a definitive necessary and sufficient criterion for the affineness of Noetherian Hopf algebras, thereby providing a significant advancement tow....
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arXiv:2606.01655v1 Announce Type: new Abstract: The Bayesian paradigm offers principled tools for sequential decision-making under uncertainty, but its reliance on a probabilistic model for all parameters can hinder the incorporation of complex structural constraints. We introduce a minimalist Bayesian framework that places a prior only on the location of the optimum, while eliminating nuisance parameters through profile likelihood. This y..
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Brown measure convergence for the spectrum of polynomials in Ginibre matrices
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arxiv.org
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2 days ago
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eng
arXiv:2606.01664v1 Announce Type: new Abstract: Fix a multivariate polynomial $\mathfrak{p}$ in $n$ non-commuting variables of arbitrary degree, and consider $n$ independent $N\times N$ complex Ginibre matrices $X_1^N,\cdots,X_n^N$. We prove that the empirical spectral distribution of $P^N=\mathfrak{p}(X_1^N,\cdots,X_n^N)$ converges as $N$ tends to infinity to the so-called Brown measure of $\mathfrak{p}$ evaluated at free circular variabl....
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$\varrho$: The Self-Referential Fixed Point of the Complex Exponential
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arxiv.org
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2 days ago
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eng
arXiv:2606.01668v1 Announce Type: new Abstract: I was taught that $e^x = x$ has no solution, and taught to leave it at that. But in mathematics "no solution" has usually meant "not on this line yet": $x^2 = -1$ waited for the complex plane, and $e^x = x$ turns out to be waiting there too. Over $\mathbb{C}$ the exponential has a fixed point $\varrho = 0.318\ldots + 1.337\ldots\,i$, the unique solution of $\exp(z) = z$ in the strip $0 < \ope....
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arXiv:2606.01675v1 Announce Type: new Abstract: The computation of the numerical index of classical Banach spaces is one of the original problems in the theory. In this paper, we compute the numerical index of two-dimensional real \(\ell_p\)-spaces for all \(p\ge1\). More precisely, we prove that \[ n(\ell_p^2)=v(J), \qquad J= \begin{pmatrix} 0&1 -1&0 \end{pmatrix}, \] confirming the conjectured formula in the two-dimensional real case.
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arXiv:2606.01683v1 Announce Type: new Abstract: We consider a generalization of the spectral gap of reversible Markov generators to nonreversible processes, following the recent work arXiv:2310.10876 on nonreversible finite-state Markov chains. Extending Chatterjee's observations, we find that this spectral quantity that we call the \textit{singular-value gap} characterizes the convergence of empirical averages, providing upper and lower b....
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arXiv:2606.01684v1 Announce Type: new Abstract: The theory of submodular flows, introduced by Edmonds and Giles, is a cornerstone of combinatorial optimization, unifying network flows, matroid intersections and directed cut coverings. In this paper, we establish a measurable-space version of this framework, addressing the structural existence and duality questions raised as part of Problem~10.6 by Lov\'asz in \textit{Submodular setfunction....
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arXiv:2606.01685v1 Announce Type: new Abstract: We prove that the optimal pebbling number of the $n$-dimensional hypercube is \[ o(Q_n)\,=\,\Theta\!\left(\left(\frac43\right)^n\right). \]
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Sharp median testing and sparse criteria for generalized \(BMO\) spaces
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arxiv.org
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2 days ago
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eng
arXiv:2606.01688v1 Announce Type: new Abstract: We study generalized \(BMO\)-type spaces associated with a normalized family of local quasi-Banach function spaces \(\mathbb X=\{X_Q\}_{Q\subset\mathbb R^n}\). For such a family we consider two oscillation seminorms: the mean-based seminorm \(BMO_{\mathbb X}\) and the best-constant seminorm \(BMO_{\mathbb X}^{*}\). The main purpose of the paper is to separate the two mechanisms that govern th....
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arXiv:2606.01690v1 Announce Type: new Abstract: Condorcet domains are subsets of permutations that ensure pairwise majority voting yields acyclic outcomes, and they form an active area of research at the intersection of social choice theory and combinatorics. In this paper, we extend the theory of Condorcet domains to the broader setting of arbitrary finite Coxeter groups. The core contribution of our approach is the introduction of Condor....
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On Dancer-type solutions for the Lane--Emden equation via semivirial-vanishing geometry
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arxiv.org
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2 days ago
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eng
arXiv:2606.01692v1 Announce Type: new Abstract: Aubin--Talenti bubbles describe the decaying positive solutions of the zero-frequency critical Lane--Emden equation in Euclidean space. By appealing to bifurcation methods, Dancer constructed in his seminar paper \cite{DancerSolution} positive-frequency solutions to the Lane--Emden equation which decay in the noncompact directions and are periodic in one direction. Alternatively, we give in t....
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arXiv:2606.01715v1 Announce Type: new Abstract: We show that any entire, capillary minimal graph in a half-space must be linear in low-dimensions or, more generally, when some tangent cone at infinity does not split off a vertical line. We also show that the regular set of any entire, capillary-minimizing hypersurface must be connected, and we discuss connections with the one-phase Bernoulli problem.
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Asymptotic Recovery in Fourier Spectral Methods for the Schr\"odinger Equation with Point Singularities
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arxiv.org
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2 days ago
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eng
arXiv:2606.01718v1 Announce Type: new Abstract: This paper studies the Fourier spectral method (FSM) for the Schr\"odinger equation with singular potentials $V \in H^{s}$, where $s > \max\{d/2-2,-1\}$ and $d$ denotes the spatial dimension. This setting includes a broad class of singular potentials, such as the 3D Coulomb potential and the 1D Dirac-delta potential. First, we combine the Feshbach-Schur map with a refined perturbation argumen....
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An Upper Bound For Hausdorff Distance Between Finite-Dimensional Hyperbolic Space and Its Medianization
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arxiv.org
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2 days ago
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eng
arXiv:2606.01721v1 Announce Type: new Abstract: We use de Sitter space to construct a concrete model for the measured wall structure of finite-dimensional hyperbolic space in hyperbolic model I^n, which will induce a medianization M(I^n). We get an upper bound for the Hausdorff distance between I^n and M(I^n).
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arXiv:2606.01726v1 Announce Type: new Abstract: We investigate spaces of prime congruences on tropical algebras and study their geometry, inspired by classical scheme theory. Our main strategy is to use tropical algebras associated to ordered monoids, which play the role of the monomial structure of these algebras. Using the space of prime congruences as local models, we introduce a tropical toric scheme which contains the usual tropical t..
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A Unified Variational Design of Predictive Mirror Descent in Convex Games under Stochastic Feedback
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arxiv.org
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2 days ago
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eng
arXiv:2606.01729v1 Announce Type: new Abstract: Mirror descent provides a geometric framework for learning in games, but its last-iterate behavior can fail in weakly stable regimes, where the dynamics may exhibit rotational or recurrent transients. Predictive mirror methods mitigate this issue by modifying the feedback entering the mirror update, yet standard predictive variants are typically introduced algorithmically and analyzed one at ....
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arXiv:2606.01739v1 Announce Type: new Abstract: We document the evolution of Karl Marx's take on infinitesimals. We contrast his initial favorable stance with later criticisms, and examine the differing perspectives of Marx and Engels on the subject. Marx's favorable assessment was based on his study of Sauri's textbook. Later, influenced by Boucharlat's textbook, Marx reversed his position to an unfavorable stance, describing belief in in....
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arXiv:2606.01740v1 Announce Type: new Abstract: Massera and Schaffer [\textit{Ann. Math. (2), 1958}] derived a breakthrough upper bound for the Clarkson angle between two nonzero vectors in a normed linear space, which was later improved by Maligranda [\textit{Am. Math. Mon., 2006}]. Pecaric and Rajic [\textit{Math. Inequal. Appl., 2007}] extended Maligranda's inequality to finitely many nonzero vectors. We derive a non-Archimedean version..
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Large deviation principles for SPDEs with locally Lipschitz coefficients
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arxiv.org
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2 days ago
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eng
arXiv:2606.01742v1 Announce Type: new Abstract: Consider the stochastic partial differential equation, \begin{align*} \partial_t u^{\varepsilon}(t\,,x) = \frac{1}{2} \partial^2_x u^{\varepsilon}(t\,,x) + b(t\,,u^{\varepsilon}(t\,,x)) + \sqrt{\varepsilon}\sigma(t\,,u^{\varepsilon}(t\,,x)) \dot{W}(t\,,x), \end{align*} where $(t\,,x)\in(0\,,\infty)\times\mathbb{R}$, and $\dot{W}$ denotes space-time white noise. Foondun, Khoshnevisan, and..
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arXiv:2606.01749v1 Announce Type: new Abstract: In this article we study periodic orbits of an electron attracted by a proton subject to Lorentz, electric, and Euler forces where each of them is allowed to depend periodically on time. This setup is motivated by the elliptic restricted three-body-problem where the Lorentz force corresponds to Coriolis force, the Coulomb force is replaced by the gravitational force, and the electric force of..
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An Improved Lower Bound for the Three-Dimensional Blaschke--Lebesgue Problem from Spectral and Dual Perspectives
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arxiv.org
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2 days ago
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eng
arXiv:2606.01754v1 Announce Type: new Abstract: The Blaschke--Lebesgue problem asks for convex bodies of minimum volume among all convex bodies of prescribed constant width. In the plane, the minimizer is the Reuleaux triangle, whereas the corresponding three-dimensional problem remains open and is also known as Meissner's conjecture. In this paper, we establish the lower bound $(4\pi/33)d^3 \simeq 0.380799\,d^3$ for the volume of any thre....
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On cyclotomic matrices related to Kloosterman sums over finite fields
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arxiv.org
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2 days ago
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eng
arXiv:2606.01759v1 Announce Type: new Abstract: Let $p$ be an odd prime and $\mathbb{F}_p$ be the finite field with $p$ elements. For any $a,b\in\mathbb{F}_p$, it is known that the Kloosterman sum $$K_p(a,b)=\sum_{x\in\mathbb{F}_p\setminus\{0\}}e^{\frac{2\pi i}{p}(ax+\frac{b}{x})}$$ can be viewed as a finite field analogue of certain Bessel function. In this paper, using the arithmetic properties of character sums over $\mathbb{F}_p$, we..
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A note on the Ratio and Inertia Bounds for the $k$-Independence Number
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arxiv.org
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2 days ago
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eng
arXiv:2606.01761v1 Announce Type: new Abstract: The $k$-th power $G^k$ of a graph $G$ is the graph on the same vertex set where the edge set consists of those pairs of distinct vertices of $G$ that are at distance at most $k$ from each other. A. Abiad, G. Coutinho, and M. A. Fiol [On the $k$-independence number of graphs, Discrete Mathematics 342 (2019), 2875--2885] proposed extensions of the classical ratio (for regular graphs) and inerti....
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High-Velocity Inverse Scattering for Nonlinear Schr\"odinger Equations with Spatially Dependent Nonlinearities
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arxiv.org
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2 days ago
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eng
arXiv:2606.01762v1 Announce Type: new Abstract: We study a high-velocity inverse scattering problem for nonlinear Schr\"odinger equations with spatially dependent nonlinearities in dimensions $d\ge3$. We consider the whole mass-supercritical and energy-subcritical range, including the endpoint cases. By introducing a moving frame adapted to highly boosted initial data, we construct the scattering operator for a class of large incoming stat....
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arXiv:2606.01764v1 Announce Type: new Abstract: We revisit the convergence guarantees of the Extragradient (EG) method for unconstrained biaffine min-max optimization. It is known that EG with a fixed stepsize achieves a $\Theta(T^{-1/2})$ last-iterate convergence rate, which is slower than the optimal $\mathcal{O}(T^{-1})$ rate attainable by incorporating additional mechanisms such as anchoring. Motivated by recent advances showing that d....
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arXiv:2606.01771v1 Announce Type: new Abstract: This paper investigates an isoperimetric-type problem posed by L. A. Santal\'{o}, concerning convex surfaces in hyperbolic 3-space that minimize total mean curvature among all convex surfaces with fixed surface area. The problem aims to characterize the geometry of such minimizers and to determine the optimal form of a Minkowski type inequality in hyperbolic 3-space. In this work, we propose ..
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Strong Localization of the Kobayashi-Eisenman Volume Element and Its Boundary Asymptotics
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arxiv.org
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2 days ago
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eng
arXiv:2606.01773v1 Announce Type: new Abstract: We establish a quantitative version of strong localization of the Kobayashi-Eisenman volume element and the quotient invariant near plurisubharmonic peak points of domains in $\mathbb{C}^n$. As an application of this strong localization result, we derive the non-tangential asymptotic limit of the Kobayashi-Eisenman volume element at exponentially flat infinite type boundary points of domains ..
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Long Time Well-posedness for the Benjamin-Ono Equation with Quasi-Periodic Initial Data
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arxiv.org
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2 days ago
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eng
arXiv:2606.01778v1 Announce Type: new Abstract: This paper investigates the long time well-posedness of the Benjamin-Ono (BO) equation with quasi-periodic initial data in analytic space. We demonstrate that the Lax operator of the BO equation exhibits a trivial spectrum, and elucidate the fundamental mathematical obstacles to establishing global well-posedness. Furthermore, by employing Tao's gauge transformation in conjunction with the B..
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arXiv:2606.01780v1 Announce Type: new Abstract: For positive integers $N$ and $k$, let $f(N,k)$ be the minimum size of a set $A\subseteq\{0,1,\ldots,N-1\}$ which intersects every $k$-term arithmetic progression contained in $\{0,1,\ldots,N-1\}$. Brown and Freedman introduced this hitting problem for arithmetic progressions and studied it for growing $k$. The square-root scale $k=\sqrt N$ is a natural transition point. Truss proved \[ f(n^..
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On the product of cross-intersecting families with small covering number
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arxiv.org
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2 days ago
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eng
arXiv:2606.01798v1 Announce Type: new Abstract: A central problem in extremal set theory is to determine or estimate $m(n,k,t),n>2k\geq 2t$, the maximum size of an intersecting $k$-graph and covering number at least $t$(see the paper for the definitions). For $t=1$ and $2$ the classical Erd\H{o}s-Ko-Rado Theorem and the Hilton-Milner Theorem provide the answer.The complete solution for $t=3$ was only achieved recently . There are some part....
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Uniqueness of an Inverse Coefficient Problem for a Time-Fractional Damped Wave Equation from Boundary Measurements
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arxiv.org
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2 days ago
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eng
arXiv:2606.01812v1 Announce Type: new Abstract: This paper studies an inverse coefficient problem for a time-fractional damped wave equation on a finite time interval. The aim is to determine two spatially varying coefficients, namely the fractional damping coefficient and the zeroth-order potential, from the associated Dirichlet-to-Neumann (DtN) map. We first prove the well-posedness of the forward problem for boundary data with sufficien....
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The Immersed Discontinuous Galerkin Method for Elliptic Interface Problems
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arxiv.org
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2 days ago
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eng
arXiv:2606.01814v1 Announce Type: new Abstract: This paper is devoted to construction and convergence analysis of the linear explicit immersed finite element (IFE) function. For the interface elements, the proposed IFE functions precisely satisfy the interface conditions on the actual interface. The IFE functions are constructed in an explicit form and can be obtained directly without solving any auxiliary problems or local linear systems.....
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On the product of cross-intersecting families with maximal covering number
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arxiv.org
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2 days ago
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eng
arXiv:2606.01817v1 Announce Type: new Abstract: For integers $k ,\ell \geq 2$ let $m(k,\ell)$ denote the maximum of $|\mathcal{F}| |\mathcal{G}|$ where the maximum is taken over all pairs of cross-intersecting families, $\mathcal{F}$ being a $k$-graph with covering number $\ell$ and $\mathcal{G}$ a $\ell$-graph with covering number $k$ (see the paper for the definitions). Erdos and Lovasz initiated the study of the one family version. That..
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arXiv:2606.01821v1 Announce Type: new Abstract: Pivoted Cholesky factorizations construct low-rank approximations of symmetric positive definite matrices by sequentially selecting pivots from the residual diagonal. Classical greedy and randomized rules, such as randomly pivoted Cholesky, target the algebraic trace-norm error of the residual. In many applications, however, the matrix enters a nonlinear matrix functional whose value, not the....
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