Mathematics

Let $(X,d,\mu)$ be a metric measure space whose measure $\mu$ is uniformly locally doubling and which supports a local weak $(1,p)$-Poincar\'e inequality for some $p\in[1,\infty)$. Given $\theta\in(0,p)$ and an Ahlfors--David…

Let $N_{\alpha}(d)$ denote the maximum number of equiangular lines in $\mathbb{R}^d$ with common angle $\arccos(\alpha)$. Balla conjectured that, if the spectral radius order $\kappa_{\frac{1-\alpha}{2\alpha}}$ of $\frac{1-\alpha}{2\alpha}$…

This dissertation develops a framework for embedding arbitrary connected graphs isometrically into Cayley graphs of abelian groups, with applications to harmonic analysis on networks. It addresses representing irregular graph-structured…

For integers $1\le r\le n+1$, let $N(n,r)$ denote the least number of chains in the Boolean lattice $B_n=2^{[n]}$ that cover every strict $r$-term chain. The case $r=1$ is the classical chain-decomposition problem and is generalizing…

We consider self-maps of a sphere in the critical Sobolev space with a given Brouwer degree. Our main result is that the (directed) distance between maps of different degrees is equal to an explicit constant times the difference in degrees.…

We formulate a notion of oriented polytope, including Street's oriented simplices and Gray's oriented cubes, and use this to prove an oriented version of the Street--Roberts conjecture, presenting $(\infty,\infty)$-categories as sheaves on…

Let $V \subset \mathbb{A}^2(\mathbb{C})$ be an algebraic curve such that $\mathrm{deg} X \neq \mathrm{deg} Y$, where $X, Y$ denote the coordinate functions on $\mathbb{A}^2(\mathbb{C})$ restricted to $V$. We prove there exists an…

In this work, we propose a multi-level machine learning framework for solving inverse scattering problems with multi-frequency data. The multi-level neural network is built along the frequency axis of the scattering problem, wherein at each…

Balance-oriented multi-warehouse inventory allocation is a recurring decision problem in large-scale e-commerce supply chains, in which a fixed replenishment quantity is distributed across warehouses to balance post-allocation inventory…

We construct a Hermitian covariance form on hyperbolic components in parameter spaces of complex H\'enon maps, associated to the full complex unstable derivative cocycle. The form measures infinitesimal variations in the marked complex…

Let $f_3(N)$ be the least integer such that every set $A\subseteq\{1,\ldots,N\}$ of size at least $f_3(N)$ contains distinct elements $a,b,c\in A$ such that $a+b\in A$, $a+c\in A$, and $b+c\in A$. We prove that $f_3(N)\le 5N/8+O(1)$.…

Let $X$ be a $PD_4$-complex and let $\pi=\pi_1(X)$. If $\pi$ is torsion-free and $\pi_2(X)$ is a finitely generated projective $\mathbb{Z}[\pi]$-module then either $\pi$ is free or $\pi$ is $FP$ and $c.d.\pi=4$. If, moreover,…

In this paper, we study the initial-boundary value problem of the 1D compressible Navier--Stokes/Cahn--Hilliard system with vacuum. We establish the global existence and uniqueness of strong solutions to this initial-boundary value problem.…

In this paper, we study the long-time behaviour of the two-dimensional stochastic damped Euler equation on the torus driven by bounded random forcing. Unlike stochastic Navier-Stokes or fractionally dissipative Euler equations, the model…

This paper addresses the existence of nontrivial solutions to a class of mixed local-nonlocal problems involving a mixed interpolated Hardy potential. We first establish a concentration-compactness principle for mixed local and nonlocal…

We study a class of mixed-integer bilevel stochastic programs where the leader commits to a first-stage decision before uncertainty is realized, and the follower solves a subsequent mixed-integer optimization problem for each revealed…

We study long-time optimal control of control-affine semiautonomous neural ordinary differential equations (SA-NODEs) with $\ell^1$-regularized controls. Three results are established. First, optimal state-control pairs satisfy an…

In this paper, we introduce controlled semiautonomous neural ordinary differential equations (controlled SA-NODEs) for the approximation and learning of nonlinear controlled dynamical systems. The proposed framework extends semiautonomous…

We study the double affine Hecke algebra (DAHA) of type $(C_n^\vee,C_n)$ from the perspective of deformation theory. First, we provide a zeros-and-residues realization of this algebra, extending the construction of Ginzburg, Kapranov, and…

Let $K=\mathbb{Q}(\sqrt{D})$ with $D>1$ squarefree, and let $\varepsilon_+$ be the totally positive fundamental unit of $\mathcal{O}_K$. B. M. Kim proved in 2000 that the octonary diagonal form \[…