Mathematics
Let \(\vec F(2^{\mathbb Z^2})\) be the directed Schreier graph on the free part of the Bernoulli shift \(\mathbb Z^2\curvearrowright 2^{\mathbb Z^2}\), with arcs in the two coordinate directions. We prove that the continuous oriented…
We investigate the Peres--Schlag nonempty interior problem for orthogonal projections in both the finite-field and Euclidean settings. Over finite fields $\mathbb F_q^n$, we employ the polynomial method to establish sharp projection…
Thermal warpage has become a critical issue in advanced packaging, primarily caused by the mismatch in coefficients of thermal expansion (CTE) among heterogeneously integrated materials. However, only a limited number of studies have…
In this paper, we establish degree obstructions to the equivalence of generalized Airy operators of the same type. As an application, we answer a question posed by Nicholas M. Katz in Inventiones Mathematicae (87, pp. 13-61,1987). The main…
Phase-field models are typically derived from variational principles for a free-energy functional and are widely used to simulate complex multiphase phenomena in science and engineering. A central goal in designing numerical schemes for…
A group is called bidihedral if it can be expressed as a product of two dihedral subgroups. In this paper, a complete classification for all bidihedral groups is given.
We study a min-max bilinear transport problem arising from a two-player zero-sum game with quadratic kinetic and interaction costs. Starting from a dynamic path space formulation, we establish existence of minimax and maximin plans and…
The marked Schottky space records, up to conjugacy, all actions of a free group of fixed rank as a Schottky group on hyperbolic space of fixed dimension. In dimension three it is the classical Schottky space covering the moduli space of…
We provide rigorous error analysis of the mass-preserving time-splitting methods for solving the semiclassical Dirac equation. The scaled Planck constant $\epsilon$ in the equation gives rise to rapid oscillations in both space and time…
We prove a converse theorem for functional equations of Dirichlet $L$-functions. Under mild assumptions, we prove that these functional equations for $L$-series of the form $\sum_{n\ge 1} f(n) n^{-s}$ force the coefficient function $f$ to…
Hybrid switching L\'evy-driven stochastic differential equations with pure-jump noise and state-dependent switching rates are studied under high-frequency observation. A three-stage inference procedure is proposed for the drift, scale, and…
One of the traditional approaches for constructing approximate policies for dynamic assortment optimization problems is to use sampling-based inventory-agnostic policies. Such policies are called sampling-based, as they sample an assortment…
In this paper, we develop the theory of nilpotency and solvability for transposed Novikov-Poisson algebras. We first establish several equivalent conditions for a dialgebra to be nilpotent, and show that the lower central series of a…
We develop a theory of affine algebraic monoids over general base schemes whose unit groups are split reductive groups. Our main result is a classification theorem for such objects, generalizing works of Vinberg and Rittatore over a field.…
A family of permutations is called $t$-intersecting if any two permutations in the family agree on at least $t$ elements. We prove that there exists $n_0 \in \mathbb{N}$ such that for any $n>n_0$ and any $1 \leq t \leq n$, the maximum size…
We study Gibbs measures on high--dimensional Haar--random unimodular lattices, where the energy of a lattice vector is its squared Euclidean norm. The random lattice is viewed as quenched geometric disorder, and $c>0$ denotes the scaled…
We complete the investigation begun in a previous paper to find unitary representations of the non-trivial real $K$-theory elements for the sphere $S^d$ with an involution. Here we consider all involutions except the antipodal involutions.…
Let $\Pi_{0}$ be a cuspidal automorphic representation of $\mathrm{PGL}_{3}(\mathbb{A}_{\mathbb{Q}})$. In this paper, we use Levinson's method to prove that, as $Q\to \infty$, at least $1/9$ of the zeros of the $L$-functions $L(s,…
Simulating incompressible Stokes flow is essential for studies in microfluidics and low-Reynolds number hydrodynamics. However, the computational cost of resolving the associated saddle-point problem grows prohibitively with the…
The Discrete Kirchhoff Triangle (DKT) method for the biharmonic equation is analyzed in the discrete energy norm. The error is bounded by the best approximation of the Hessian by piecewise constants and the oscillation of the right-hand…