Related papers: Asymptotic repetitive threshold of balanced sequen…

Richomme, Saari and Zamboni (J. Lond. Math. Soc. 83: 79-95, 2011) proved that at every position of a Sturmian word starts an abelian power of exponent $k$ for every $k > 0$. We improve on this result by studying the maximum exponents of…

In a recent work on the bipartite Erd\H{o}s-R\'{e}nyi graph, Do et al. (2023) established upper bounds on the number of connected labeled bipartite graphs with a fixed surplus. We use some recent encodings of bipartite random graphs in…

In this paper we are mainly concerned with nontrivial positive solutions to the Dirichlet problem for the degenerate elliptic equation \begin{gather} -\frac{\partial^2 u}{\partial x^2} -\left|x\right|^{2k}\frac{\partial^2 u}{\partial…

Replacing the spectral measure by a random vector $\bfZ$ allows the representation of a max-stable distribution on $\R^d$ with standard negative margins via a norm, called \emph{$D$-norm}, whose generator is $\bfZ$. The set of $D$-norms can…

In this paper, we study the large-time behavior of solutions to a class of partially dissipative linear hyperbolic systems with applications in velocity-jump processes in several dimensions. Given integers $n,d\ge 1$, let $\mathbf…

The prefix palindromic length $PPL_u(n)$ of an infinite word $u$ is the minimal number of concatenated palindromes needed to express the prefix of length $n$ of $u$. In a 2013 paper with Puzynina and Zamboni we stated the conjecture that…

Let $G$ be a finite group. By a sequence over $G$, we mean a finite unordered sequence of terms from $G$, where repetition is allowed, and we say that it is a product-one sequence if its terms can be ordered such that their product equals…

A word $u$ defined over an alphabet $\mathcal{A}$ is $c$-balanced ($c\in\mathbb{N}$) if for all pairs of factors $v$, $w$ of $u$ of the same length and for all letters $a\in\mathcal{A}$, the difference between the number of letters $a$ in…

Let $D$ be a strong balanced digraph on $2a$ vertices. Adamus et al. have proved that $D$ is hamiltonian if $d(u)+d(v)\ge 3a$ whenever $uv\notin A(D)$ and $vu\notin A(D)$. The lower bound $3a$ is tight. In this paper, we shall show that the…

We present two short proofs giving the best known asymptotic lower bound for the maximum element in a set of $n$ positive integers with distinct subset sums.

Fix an elliptic curve $E/\Q$, and assume the generalized Riemann hypothesis for the $L$-function $ L(E_D, s) $ for every quadratic twist $E_D$ of $E$ by $D\in\Z$. We combine Weil's explicit formula with techniques of Heath-Brown to derive…

In this work, we study the problem of finding the asymptotic growth rate of the number of of $d$-dimensional arrays with side length $n$ over a given alphabet which avoid a list of one-dimensional "forbidden" words along all cardinal…

We study the quantity $$ \tau_{n,k}:=\frac{|T_n^{(k)}(\omega_{n,k})|}{T_n^{(k)}(1)}\,, $$ where $T_n$ is the Chebyshev polynomial of degree $n$, and $\omega_{n,k}$ is the rightmost zero of $T_n^{(k+1)}$. Since the absolute values of the…

We define a new class of ternary sequences that are 2-balanced. These sequences are obtained by colouring of Sturmian sequences. We show that the class contains sequences of any given letter frequencies. We provide an upper bound on factor…

Using recent developments on the theory of locally decodable codes, we prove that the critical size for Szemer\'edi's theorem with random differences is bounded from above by $N^{1-\frac{2}{k} + o(1)}$ for length-$k$ progressions. This…

We study $c$-crossing-critical graphs, which are the minimal graphs that require at least $c$ edge-crossings when drawn in the plane. For every fixed pair of integers with $c\ge 13$ and $d\ge 1$, we give first explicit constructions of…

We consider a variant of the classical Erd\H{o}s-R\'enyi random graph, where components with surplus are slowed down to prevent the apparition of complex components. The sizes of the components of this process undergo a similar phase…

In many iterative optimization methods, fixed-point theory enables the analysis of the convergence rate via the contraction factor associated with the linear approximation of the fixed-point operator. While this factor characterizes the…

An asymptotic expansion with respect to a small parameter of the solution of the Cauchy problem is constructed for a system of three transfer equations, two of which are singularly perturbed by the degeneracy of the entire senior part of…

Given a sequence of integers $a_j, j\ge 1,$ a multiset is a combinatorial object composed of unordered components, such that there are exactly $a_j$ one-component multisets of size $j.$ When $a_j\asymp j^{r-1} y^j$ for some $r>0$, $y\geq…