数学

We consider the largest interpoint distance $M_n=\max_{1\le i<j\le n}\|X_i-X_j\|$ among independent random points $X_1,\ldots,X_n$, uniformly distributed on a $d$-dimensional ellipsoid. We assume that the largest semi-axis has length 1 and…

We study compatible Lie algebras from algebraic and representation-theoretic points of view, obtaining counterexamples to some fundamental theorems from classical Lie algebra theory, namely the theorems of Lie, Weyl and Levi. We also…

We propose a semi-Lagrangian adaptive-rank (SLAR) method that combines the large time-step capability of semi-Lagrangian schemes with the efficiency of adaptive-rank tensor representations while simultaneously enforcing local conservation…

The Bethe-Salpeter equation, which has many applications in both theoretical and applied physics, is generally solved via a matrix eigenvalue problem with a rich algebraic structure. The numerical solution of such structured eigenproblem…

We discuss the problem of characterizing equidistant binary codes of a given length $n$ having largest possible distance and the maximum number of codewords. Such characterizations have been studied by several authors over the years and…

We study centralizers of dilations in the quasi-isometry group of the positive real line. We introduce an asymptotic invariant defined via coarsely dense sequences at infinity and establish a rigidity theorem for quasi-isometries that…

We provide an intrinsic characterization of hyperelliptic stable curves of genus $g \geq 2$, independent of admissible covers or auxiliary moduli data. A stable curve is hyperelliptic if it admits an involution yielding a rational tree…

We model Herman Kahn's escalation ladder as an impartial combinatorial game. Reindexing each rung by its distance to the nuclear threshold turns the ladder into a subtraction game, the most tractable class in combinatorial game theory, and…

We analyze via Evolutionary Gamma-convergence a stratified composite structure consisting of a thin adhesive layer with vanishing thickness and undergoing rate-independent damage, as well as two adjacent elastic adherents. As the width of…

Numerical simulation of quantum computing hardware and open quantum systems governed by the Lindblad equation is challenging due to the high dimensionality of the density matrix and the need to preserve fundamental physical properties. In…

Let $A$ be an abelian variety of dimension $g$ over a finite field $\mathbf{F}_q$. We show that if $q$ is sufficiently large relative to $g$, the $g$ point counts $\#A(\mathbf{F}_{q^i})$ for $1 \leq i \leq g$ determine the zeta function of…

Let $k\ge 2$ be fixed. We study the distribution modulo one of the $n^k$ sums \begin{equation*} \sqrt{a_1} + \cdots + \sqrt{a_k}, \qquad 1\le a_1, \dots, a_k \le n, \end{equation*} counted with multiplicity. For \begin{equation*} S(h,n) =…

We propose a model of higher homotopy theory of $L_{\infty}[1]$-morphisms as a natural generalization of the $A_{\infty}$-homotopies defined by Fukaya-Oh-Ohta-Ono \cite{FOOO1}. Within this framework, we show that a filling condition holds…

We give a single compositional setting in which gradient-based learning and Hamiltonian-style mechanics appear as functorial semantics. The syntax is an operad Arr whose objects are input-output interfaces (pairs of manifolds) and whose…

Consider a square matrix $A$ whose all principal minors are equal to $1$. Over a field, this property is inherited by any power of $A$, but this is not the case over an arbitrary commutative ring. We show that it is the case over any…

When datasets contain outliers, robust regression is a well-established alternative to Ordinary Least Squares. A commonly employed robust estimator is Least Trimmed Squares (LTS), which computes the regression coefficients from a subset of…

We prove that the Sinkhorn algorithm converges at the rate of $O(1/k)$ in $\ell_1$-norm marginal error and in joint relative entropy, which is known to be sharp in the asymptotically scalable case. The proof is based on examining the…

An analytic solution has been recently developed for the Maxwell's equation in a medium with zero currents such as vacuum. The solution is attractive in the sense that it is formulated based on the Fourier expansion of the initial value. It…

An explicit cubic Ramanujan--Sato formula for $1/\pi$ on $\Gamma_0(2)^+$ at $D=-163$ is presented. The construction produces a very small cubic CM parameter, giving about $15.01$ decimal digits of geometric contraction per term. This is…

We develop a second-order sensitivity theory for the efficient solution map \(S\) of a parametric vector optimization problem \(\min_C f(p,x)\) subject to \(x\in H(p)\). The main point is the passage from efficient values to efficient…