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In a multi-base representation of an integer (in contrast to, for example, the binary or decimal representation) the base (or radix) is replaced by products of powers of single bases. The resulting numeral system has desirable properties…

We propose a novel way of generalizing the class of interval graphs, via a graph width parameter called the simultaneous interval number. This parameter is related to the simultaneous representation problem for interval graphs and defined…

We develop numerical tools for Diagrammatic Monte-Carlo simulations of non-Abelian lattice field theories in the t'Hooft large-N limit based on the weak-coupling expansion. First we note that the path integral measure of such theories…

In this paper, we derive closed-form exact expressions for the main statistics of the ratio of squared alpha-mu random variables, which are of interest in many scenarios for future wireless networks where generalized distributions are more…

Let $f(t_1,\ldots,t_n)$ be a nondegenerate integral quadratic form. We analyze the asymptotic behavior of the function $D_f(X)$, the number of integers of absolute value up to $X$ represented by $f$. When $f$ is isotropic or $n$ is at least…

Frameproof codes are used to fingerprint digital data. It can prevent copyrighted materials from unauthorized use. In this paper, we study upper and lower bounds for $w$-frameproof codes of length $N$ over an alphabet of size $q$. The upper…

A nonparametric variant of the Kiefer--Weiss problem is proposed and investigated. In analogy to the classical Kiefer--Weiss problem, the objective is to minimize the maximum expected sample size of a sequential test. However, instead of…

We show that problems of existence and characterization of wavelets for non-expanding dilations are intimately connected with the geometry of numbers; more specifically, with a bound on the number of lattice points in balls dilated by the…

We use rigorous numerical techniques to compute a lower bound for the exponent of expansivity outside a neighborhood of the critical point for thousands of intervals of parameter values in the quadratic family. We compute a possibly small…

Recently, designs of pseudorandom number generators (PRNGs) using integer-valued variants of logistic maps and their applications to some cryptographic schemes have been studied, due mostly to their ease of implementation and performance.…

Flat optics have been proposed as an attractive approach for the implementation of new imaging and sensing modalities to replace and augment refractive optics. However, chromatic aberrations impose fundamental limitations on diffractive…

Zaremba's conjecture concerns a formation of continued fraction expansions for rational numbers with partial quotient bounded by an absolute constant. We present asymptotic estimates for the size of $\epsilon$-thickening of certain fractal…

We give an asymptotic for the number of prime solutions to $Q(x_1,\dots, x_8) = N$, subject to a mild non-degeneracy condition on the homogeneous quadratic form $Q$. The argument initially proceeds via the circle method, but this does not…

We establish arithmetical properties and provide essential bounds for bi-sequences of approximation coefficients associated with the natural extension of maps, leading to continued fraction-like expansions. These maps are realized as the…

Let $x \in [0,1)$ be an irrational number with continued fraction expansion $[a_1(x),a_2(x), \cdots,a_n(x),\cdots]$ and $q_n(x)$ be the denominator of its $n$-th convergent. We establish, for any $\alpha,\beta$ in $[0,+\infty]$, the…

We study "dead ends" in square-free digit walks: square-free integers $N$ such that, in base $b$, every one-digit extension $bN+d$ is non-square-free. In base $10$, the stochastic independence model of Miller et al. suggests that infinite…

Distributed Arithmetic Coding (DAC) proves to be an effective implementation of Slepian-Wolf Coding (SWC), especially for short data blocks. To study the property of DAC codewords, the author has proposed the concept of DAC codeword…

Let $G$ be a finite abelian group with $\exp(G)$ the exponent of $G$. Then $\mathsf W(G)$ denotes the set of cross numbers of minimal zero-sum sequences over $G$ and $\mathsf w(G)$ denotes the set of all cross numbers of non-trivial…

Let $\boldsymbol W=\{\boldsymbol W_n:n\in\mathbb N\}$ be a sequence of random vectors in $\mathbb R^d$, $d\ge 1$. This paper considers the logarithmic asymptotics of the extremes of $\boldsymbol W$, that is, for any vector $\boldsymbol…

In this paper we derive results concerning the connected components and the diameter of random graphs with an arbitrary i.i.d. degree sequence. We study these properties primarily, but not exclusively, when the tail of the degree…