Related papers: Bounding finite-image sequences of length $\omega^…

This paper quantitatively characterizes the approximation power of deep feed-forward neural networks (FNNs) in terms of the number of neurons. It is shown by construction that ReLU FNNs with width $\mathcal{O}\big(\max\{d\lfloor…

Vizing's theorem states that any $n$-vertex $m$-edge graph of maximum degree $\Delta$ can be edge colored using at most $\Delta + 1$ different colors. Vizing's original proof is easily translated into a deterministic $O(mn)$ time algorithm.…

The hypergraph regularity lemma -- the extension of Szemer\'edi's graph regularity lemma to the setting of $k$-uniform hypergraphs -- is one of the most celebrated combinatorial results obtained in the past decade. By now there are several…

We present an algorithm that, for every fixed genus $g$, will enumerate all hyperelliptic curves of genus $g$ over a finite field $k$ of odd characteristic in quasilinear time; that is, the time required for the algorithm is…

We are interested in characterizing which classes of finite graphs are well-quasi-ordered by the induced subgraph relation. To that end, we devise an algorithm to decide whether a class of finite graphs well-quasi-ordered by the induced…

Given a commutative unital ring $R$, we show that the finiteness length of a group $G$ is bounded above by the finiteness length of the Borel subgroup of rank one $\mathbf{B}_2^\circ(R)=\left( \begin{smallmatrix} * & * \\ 0 & *…

We study the first-order almost-sure theories for classes of finite structures that are specified by homomorphically forbidding a set $\mathcal{F}$ of finite structures. If $\mathcal{F}$ consists of undirected graphs, a full description of…

We investigate the parameterized complexity of Vertex Cover parameterized by the difference between the size of the optimal solution and the value of the linear programming (LP) relaxation of the problem. By carefully analyzing the change…

In this paper we prove a space lower bound of $n^{\Omega(k)}$ for non-deterministic (syntactic) read-once branching programs ({\sc nrobp}s) on functions expressible as {\sc cnf}s with treewidth at most $k$ of their primal graphs. This lower…

For an integer $k$ at least $2$, and a graph $G$, let $f_k(G)$ be the minimum cardinality of a set $X$ of vertices of $G$ such that $G-X$ has either $k$ vertices of maximum degree or order less than $k$. Caro and Yuster (Discrete…

In this paper we shall consider the growth at infinity of a sequence $(P_n)$ of entire functions of bounded orders. Our results extend the results in \cite{trong-tuyen2} for the growth of entire functions of genus zero. Given a sequence of…

Let $\Omega$ be a perfectly normal topological space, let $A$ be a non-empty $G_\delta$-subset of $\Omega$ and let $B_1(A)$ denote the space of all functions $A\to\mathbb{R}$ of Baire-one class on $A$. Let also $\|\cdot\|_\infty$ be the…

In this paper, we strengthen a result by Green about an analogue of Sarkozy's theorem in the setting of polynomial rings $\mathbb{F}_q[x]$. In the integer setting, for a given polynomial $F \in \mathbb{Z}[x]$ with constant term zero, (a…

Extremal graph theory studies the maximum or minimum number of subgraphs isomorphic to a prescribed graph under given constraints. \textit{Localization} has recently emerged as a framework that refines such problems by assigning extremal…

We prove explicit bounds on the radius of a ball centered at the origin which is guaranteed to contain all bounded connected components of a semi-algebraic set $S \subset \mathbbm{R}^k$ defined by a quantifier-free formula involving $s$…

In this paper, we extend the concept of absolutely Ces\`aro boundedness to the fractional case. We construct a weighted shift operator belonging to this class of operators, and we prove that if $T$ is an absolutely Ces\`{a}ro bounded…

This paper establishes an analog of the Erd\H{o}s-Ko-Rado theorem to polynomial rings over finite fields, affirmatively answering a conjecture of C. Tompkins. A $k$-uniform family of subsets of a set of finite size $n$ is $l$-intersecting…

Let $k \geq 3$. We prove the following three bounds for the matching number, $\alpha'(G)$, of a graph, $G$, of order $n$ size $m$ and maximum degree at most $k$. If $k$ is odd, then $\alpha'(G) \ge \left( \frac{k-1}{k(k^2 - 3)} \right) n \,…

If K is an odd-dimensional flag closed manifold, flag generalized homology sphere or a more general flag weak pseudomanifold with sufficiently many vertices, then the maximal number of edges in K is achieved by the balanced join of cycles.…

Let $\textbf{k} := (k_1,\ldots,k_s)$ be a sequence of natural numbers. For a graph $G$, let $F(G;\textbf{k})$ denote the number of colourings of the edges of $G$ with colours $1,\dots,s$ such that, for every $c \in \{1,\dots,s\}$, the edges…