Research
Research
Research
Research
1,679+ open-access research outputs.
Let $k\ge 2$ be fixed integer, $0<c<1$ a constant. Consider a graph $G$ with $n$ vertices and average degree $cn$. We answer a question of Simon Griffiths by showing that $G$ has $k$ vertices such tha…
We investigate several aspects of the Bialynicki-Birula decomposition of a smooth complete $\mathbb{G}_m$-variety with finite fixed locus. Our results include novel characterizations of when the Bialy…
We prove the $\frac{2d}{d+1}$-Sidon inequality for a system of functions representing the most general extension of the Rademacher $d$-chaos to the $p$-ary case.…
A finite set $S \subset \mathbb{Z}$ is a \emph{Sidon set} if its pairwise differences are distinct. A \emph{perfect difference set} (PDS) of order $n$ is a set $B \subset \mathbb{Z}_v$ ($v = n^2 - n +…
The Lie group SE(3) of isometric orientation preserving transformation is used for modeling multibody systems, robots, and Cosserat continua. The use of these models in numerical simulation and optimi…
We study the existence of negative eigenvalues for two-dimensional Schr\"odinger operators with real-valued potentials in the weak coupling regime. In his pioneering paper [Simon 1976] from half a cen…
A conjecture of Simon Donaldson is that on a compact $4$-manifold $X^4$ one can flow from a hypersymplectic structure to a hyperk\"ahler structure while remaining in the same cohomology class. To this…
For positive integers $n$, $d$, $k$ and $h$, let $[n]^d$ be the $d$-dimensional grid of order $n$, and we refer to the equation $\sum_{i=1}^{h}x_{1,i}=\cdots =\sum_{i=1}^{h}x_{k,i}$ as the {\it $B_{k,…
We consider a nonlinear constrained heat flow evolving on the manifold $\mathcal{M}=\{v\in L^{2}:\|v\|_{L^{2}}=1\}$ over bounded smooth domains. It is known that the solution corresponding to any nonn…
Oblivious subspace injection (OSI) was introduced by Cama\~no, Epperly, Meyer, and Tropp in 2025 as a much weaker sketching property than oblivious subspace embedding (OSE) that still yields constant-…
We prove that for a residual (and hence dense) subset $\mathcal{G}$ of Riemannian metrics on $S^{n+1}$ in the $C^{3}$ topology, no area-minimizing integral $n$-current that is a boundary admits a sing…
We study the hydrodynamic limit of the Chern--Simons--Higgs system, a relativistic gauge field model involving the Chern--Simons interaction. We introduce a single scaling parameter capturing both the…
We classify extended Abelian Chern-Simons theories with gauge group $U(1)^n$ as extended $(2+1)$-dimensional topological quantum field theories. For an even integral nondegenerate lattice $(\Lambda,K)…
We prove a natural isomorphism between toral Chern-Simons theory with gauge group $\mathbb T=\mathfrak t/\Lambda\cong U(1)^n$ and the Reshetikhin-Turaev theory associated with the finite quadratic mod…
A common paradigm in phase-field models with singular potentials is that global-in-time weak solutions converge to a single equilibrium only after undergoing asymptotic regularization. However, in arX…
This constant is the maximum of the sum $|c_0|+|c_1|+|c_2|+|c_3|$ of the moduli of the coefficients of a trigonometric polynomial $c_0+c_1e^{it}+c_2e^{2it}+c_3e^{3it}$ bounded by 1. Its value is still…
We establish the equivalence between $U(1)$ Chern-Simons and Reshetikhin-Turaev TQFTs associated with finite quadratic modules. For gauge group $U(1)$ and even level $k$, we prove that the correspondi…
We provide a positive answer to an old problem of Jonathan K. Simon: if $K$ and $K'$ are two knots such that there is an epimorphism from the knot group of $K$ to the knot group of $K'$, then the genu…
We consider the initial-boundary value problem of a thermodynamically consistent diffuse interface model for incompressible two-phase flows with unmatched densities in a bounded domain $\Omega\subset\…
Let $f$ be a newform of even weight at least $4$, level $N$ and trivial character. Let $p\nmid N$ be an odd prime number that is ordinary for $f$ and let $K$ be an imaginary quadratic field satisfying…
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