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Let $P$ and $T$ be disjoint sets of prime numbers with $T$ finite. A simple formula is given for the natural density of the set of square-free numbers which are divisible by all of the primes in $T$ and by none of the primes in $P$. If $P$…

We consider ensembles of $N \times N$ Hermitian Wigner matrices, whose entries are (up to the symmetry constraints) independent and identically distributed random variables. Assuming sufficient regularity for the probability density…

We consider Gaussian ensembles of m N x N complex matrices. We identify an enhanced symmetry in the system and the resultant closed subsector, which is naturally associated with the radial sector of the theory. The density of radial…

This article examines the value distribution of $S_{N}(f, \alpha) := \sum_{n=1}^N f(n\alpha)$ for almost every $\alpha$ where $N \in \mathbb{N}$ is ranging over a long interval and $f$ is a $1$-periodic function with discontinuities or…

We consider an ensemble of $2\times 2$ normal matrices with complex entries representing operators in the quantum mechanics of 2 - level parity-time reversal (PT) symmetric systems. The randomness of the ensemble is endowed by obtaining…

We introduce a new canonical form of lattices called the systematic normal form (SNF). We show that for every lattice there is an efficiently computable "nearby" SNF lattice, such that for any lattice one can solve lattice problems on its…

We address the construction of stable random matrix ensembles as the generalization of the stable random variables (Levy distributions). With a simple method we derive the Cauchy case, which is known to have remarkable properties. These…

For a fixed rational number g different from -1,0,1 and integers a and d the set N_g(a,d) of primes p for which the order of g(mod p) is congruent to a(mod d) is considered. It is shown, assuming the Generalized Riemann Hypothesis (GRH),…

We study the distribution of entries of a random permutation matrix under a "randomized basis," i.e., we conjugate the random permutation matrix by an independent random orthogonal matrix drawn from Haar measure. It is shown that under…

We define a general method for finding a quasi-best approximant in sup-norm to a target density belonging to a given model, based on independent samples drawn from distributions which average to the target (which does not necessarily belong…

By a mixture density is meant a density of the form $\pi_{\mu}(\cdot)=\int\pi_{\theta}(\cdot)\times\mu(d\theta)$, where $(\pi_{\theta})_{\theta\in\Theta}$ is a family of probability densities and $\mu$ is a probability measure on $\Theta$.…

Let $r > 0$ be an integer, let $\mathbb{F}_q$ be a finite field of $q$ elements, and let $\mathcal{A}$ be a nonempty proper subset of $\mathbb{F}_q$. Moreover, let $\mathbf{M}$ be a random $m \times n$ rank-$r$ matrix over $\mathbb{F}_q$…

Tractable generalizations of the Gaussian distribution play an important role for the analysis of high-dimensional data. One very general super-class of Normal distributions is the class of $\nu$-spherical distributions whose random…

We study the uniform distribution of the polynomial sequence $\lambda(P)=(\lfloor P(k) \rfloor )_{k\geq 1}$ modulo integers, where $P(x)$ is a polynomial with real coefficients. In the nonlinear case, we show that $\lambda(P)$ is uniformly…

A probabilistic characterization of the dominance partial order on the set of partitions is presented. This extends work in "Symmetric polynomials and symmetric mean inequalities". Electron. J. Combin., 20(3): Paper 34, 2013. Let $n$ be a…

Let $\Sigma \subset \mathbb{C}$ be a compact subset of the complex plane, and $\mu$ be a probability distribution on $\Sigma$. We give necessary and sufficient conditions for $\mu$ to be the weak* limit of a sequence of uniform probability…

This work proposes a very simple random matrix model, the Flip Matrix Model, liable to approximate the behavior of a two dimensional electron in a weak random potential. Its construction is based on a phase space analysis, a suitable…

Given any fixed $N \times N$ positive semi-definite diagonal matrix $G\ge 0$ we derive the explicit formula for the density of complex eigenvalues for random matrices $A$ of the form $A=U\sqrt{G}$} where the random unitary matrices $U$ are…

While score-based generative models (SGMs) have achieved remarkable success in enormous image generation tasks, their mathematical foundations are still limited. In this paper, we analyze the approximation and generalization of SGMs in…

We propose a generalization of the random matrix theory following the basic prescription of the recently suggested concept of superstatistics. Spectral characteristics of systems with mixed regular-chaotic dynamics are expressed as weighted…