Mathematics

We prove a stronger version of the log-convexity inequality for the Bell numbers $B_n$. In particular, for $n\ge 5$, we have \[ B_{n+1}B_{n-1} - (B_n)^2 \ge \sum_{i=1}^{n} F_i (B_{n-i})^2, \] where $F_i$ is the $i$-th Fibonacci number with…

We prove the surgery exact triangle for monopole (Seiberg--Witten) Floer homology over integer coefficients, extending the work of Kronheimer--Mrowka--Ozsv\'{a}th--Szab\'{o} over $\mathbb{Z}/2$, Lin--Ruberman--Saveliev over $\mathbb{Q}$,…

Data-driven material modeling techniques have gained significant attention due to their ability to capture complex constitutive behaviors beyond the limitations of classical material models. Physics-augmented neural networks (PANNs), which…

The \textit{girth} of a graph $G$, denoted $\mathrm{g}(G)$, is the length of a shortest cycle in $G$. If $G$ contains no cycle, we define $\mathrm{g}(G)=\infty$. Sivaraman (2020) asked for the optimal $\chi$-bounding function for the class…

We prove that an arbitrary finite group $G$ having the same order and same set of conjugacy class sizes as an alternating or symmetric group $S$ must be isomorphic to $S$. From this and previously known results it follows that the same…

In this article, we introduce two new algebraic structures associated with the triplet group on $n$ strands, $L_n$: the singular triplet monoid $SLM_n$ and its virtual extension $VSLM_n$, defined in analogy with the singular braid monoid…

Let $G$ be a split reductive group over a $p$-adic field. We give germ expansions of Kloosterman integrals for $G$. As an application, we prove that Bessel distributions are regular for all generic representations on $G$ provided that…

In this paper, we study two problems concerning holomorphic flows on $\mathbb C^n$. First, we prove Runge-type results for positive-time flow invariant domains. For a linear flow $e^{tA}$, where $A\in GL(n,\mathbb C)$, let $E^s$, $E^u$, and…

I construct a compact subset of the plane whose visible parts are $\tfrac{3}{2}$-dimensional in all directions. This disproves the visibility conjecture. The value $\tfrac{3}{2}$ cannot be increased, as shown in recent collaboration with A.…

We study the noise sensitivity of Boolean functions on the symmetric group, where noise is induced by running a Markov chain on the symmetric group $S_n$, focusing in particular on the case where the underlying chain is an interchange…

This article proves non-trivial estimates for a bilinear sum involving the Fourier coefficients of a Hecke-holomorphic or Hecke-Maass cusp form for $\mathrm{SL}(2,\mathbb{Z})$. As corollaries, we draw interesting results related to…

In this paper, we study symbolic defect functions of edge ideals through finite antichains of exponent vectors. Let $G$ be a finite simple graph and let $I(G)$ be its edge ideal. For each symbolic degree $s$, we define the symbolic exponent…

The classical Bohr inequality states that if $f(z)=\sum_{n=0}^{\infty} a_n z^n$ is analytic and $|f(z)|<1$ in the unit disk $\mathbb{D}$, then $\sum_{n=0}^{\infty} |a_n| r^n \le 1$ for $|z|=r \le 1/3$, where $1/3$ is sharp. Extending this…

We study the positivity properties of finite flat quotients of a normal projective variety. The numerical groups and the positive cones of these quotient varieties are related to those of the original variety.

In this note, we give some generalizations of G\"{o}del's second incompleteness theorem and study their surroundings. We revisit it from two perspectives. One perspective is the relationship between the definable complexity of a theory and…

A classical theorem of Frucht states that every finite group occurs as the automorphism group of a finite graph. We prove an embedded analogue for regular graphs of arbitrary degree. In particular, we show that for every $d\geq 3$ and every…

A simple undirected graph $M$ is called a discrete $d$-pseudomanifold if, for every vertex $v$, the induced subgraph $N_M(v)$ on the neighbors of $v$ is a discrete $(d-1)$-pseudomanifold, where a discrete $1$-pseudomanifold is defined to be…

We investigate a variant of Nim called Halve Nim, which in addition to the standard moves of Nim, we allow replacing each pile of coins with half its amount. We determine the P-positions of all two-pile games of Halve Nim. Also, we…

Let $M$ be a closed, connected, smooth $n$-dimensional manifold. We prove that $M$ is dominated by the underlying space of the $n$-skeleton of a finite simplicial complex. Furthermore, the total number of simplices in the $n$-skeleton is…

Motivated by the correspondence between ideal cotorsion pairs in Frobenius exact categories and those in their stable categories, we introduce the notion of an ideal $n$-cotorsion pair in an extriangulated category. We study the…