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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....
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....
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.
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....
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).
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..
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 ....
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....
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..
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..
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..
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....
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..
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....
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....
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....
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 ..
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 ..
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..
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^..
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....
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....
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.....
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..
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....
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....
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..
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..
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....
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..
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..
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....
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..
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..
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..
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