Related papers: The frequency and the structure of large character…

We establish new bounds on character values and character ratios for finite groups $G$ of Lie type, which are considerably stronger than previously known bounds, and which are best possible in many cases. These bounds have the form…

Consider an irreducible, admissible representation $\pi$ of GL(2,$F$) whose restriction to GL(2,$F)^+$ breaks up as a sum of two irreducible representations $\pi_+ + \pi_-$. If $\pi=r_{\theta}$, the Weil representation of GL(2,$F$) attached…

Let $\xi_1, \xi_2,\ldots$ be a sequence of independent and identically distributed random variables with zero mean, finite second moment and regularly varying right distribution tail. Motivated by a stop-loss insurance model, we consider a…

In 1887, Minkowski determined the least common multiple of the orders of all finite subgroups of $GL_n(\mathbb{Q})$; we refer to this number as $M(n)$. In (Katznelson, 1994), Katznelson provides the asymptotic behaviour of $M(n)$, with a…

We consider a well known model of random directed acyclic graphs of order $n$, obtained by recursively adding vertices, where each new vertex has a fixed outdegree $d\ge2$ and the endpoints of the $d$ edges from it are chosen uniformly at…

For $f$ a Steinhaus random multiplicative function, we prove convergence in distribution of the appropriately normalised partial sums \[ \frac{{(\log \log x)}^{1/4}}{\sqrt{x}} \sum_{\substack{n \leq x \\ P(n) > \sqrt{x}}} f(n), \] where…

We use finite-size scaling to investigate the critical behavior of the quasiespecies model of molecular evolution in the single-sharp-peak replication landscape. This model exhibits a sharp threshold phenomenon at Q=Q_c=1/a, where Q is the…

Scattered subspaces and $h$-scattered subspaces have been extensively studied in recent decades for both theoretical purposes and their connections to various applications. While numerous constructions of scattered subspaces exist,…

It is well-known that each statistic in the family of power divergence statistics, across $n$ trials and $r$ classifications with index parameter $\lambda\in\mathbb{R}$ (the Pearson, likelihood ratio and Freeman-Tukey statistics correspond…

We establish a new bound for short character sums in finite fields, particularly over two-dimensional grids in $\mathbb{F}_{p^3}$ and higher-dimensional lattices in $\mathbb{F}_{p^d}$, extending an earlier work of Mei-Chu Chang on Burgess…

A subspace partition $\Pi$ of $V=V(n,q)$ is a collection of subspaces of $V$ such that each 1-dimensional subspace of $V$ is in exactly one subspace of $\Pi$. The size of $\Pi$ is the number of its subspaces. Let $\sigma_q(n,t)$ denote the…

We use a chiral random matrix model with 2N_f flavors to mock up the QCD Dirac spectrum at finite chemical potential. We show that the 1/N approximation breaks down in the quenched state with spontaneously broken chiral symmetry. The…

Let $M_n$ denote a random symmetric $n \times n$ matrix whose upper diagonal entries are independent and identically distributed Bernoulli random variables (which take values $1$ and $-1$ with probability $1/2$ each). It is widely…

Let $\xi_1, \xi_2, \dots$ be i.i.d. non-negative random variables whose tail varies regularly with index $-1$, let $S_n$ be the sum and $M_n$ the largest of the first $n$ values. We clarify for which sequences $x_n\to\infty$ we have…

Let $\chi$ be a non-principal Dirichlet character modulo a prime $p$. Let $q_1<q_2$ denote the two smallest prime non-residues of $\chi$. We give explicit upper bounds on $q_2$ that improve upon all known results. We also provide a good…

From known effective bounds on the prime counting function of the form \[ |\pi(x)-\mathrm{Li}(x)| < a \;x \;(\ln x)^{b} \; \exp\left(-{c}\; \sqrt{\ln x}\right); \qquad (x \geq x_0); \] it is possible to establish exponentially tight…

Let $n$ and $t$ be positive integers with $t<n$, and let $q$ be a prime power. A partial $(t-1)$-spread of ${\rm PG}(n-1,q)$ is a set of $(t-1)$-dimensional subspaces of ${\rm PG}(n-1,q)$ that are pairwise disjoint. Let $r\equiv n\pmod{t}$…

Let $g$ be a random matrix distributed according to uniform probability measure on the finite general linear group $\mathrm{GL}_n(\mathbb{F}_q)$. We show that $\mathrm{Tr}(g^k)$ equidistributes on $\mathbb{F}_q$ as $n \to \infty$ as long as…

Exact expressions are given for the distribution function of the ratio of a weighted sum of independent chi-squared variables to a single chi-square variable, scaled appropriately. This distribution is the generalization of the classical F…

This paper discusses the approximate distributions of eigenvalues of a singular Wishart matrix. We give the approximate joint density of eigenvalues by Laplace approximation for the hyper-geometric functions of matrix arguments.…