相关论文: The distribution of solutions to xy = N mod a with…

The norm of an integer partition is defined as the product of its parts. This statistic was recently introduced by Schneider in connection to partition zeta functions. In this note, we use the method of moments to study the distribution of…

We obtain uniform in time bounds for the solutions to a class of thermo-diffusive systems with classical and fractional diffusions. In the classical diffusion case, the nonlinearities are assumed to be at most exponentially growing, while…

In this article we study a broad class of integer programming problems in variable dimension. We show that these so-termed {\em n-fold integer programming problems} are polynomial time solvable. Our proof involves two heavy ingredients…

We study the problem of estimating the covariance matrix of a high-dimensional distribution when a small constant fraction of the samples can be arbitrarily corrupted. Recent work gave the first polynomial time algorithms for this problem…

An \emph{indexing} of a finite set $S$ is a bijection $D : \{1,...,|S|\} \rightarrow S$. We present an indexing for the set of quadratic residues modulo $N$ that is decodable in polynomial time on the size of $N$, given the factorization of…

In this work the asymptotic properties of $Q_t(N)$ ,the probability of the number of renewals ($N$), that occur during time $t$ are explored. While the forms of the distribution at very long times, i.e. $t\to\infty$, are very well known and…

The distribution of the sum of independent identically distributed uniform random variables is well-known. However, it is sometimes necessary to analyze data which have been drawn from different uniform distributions. By inverting the…

It has been conjectured that the sequence $(3/2)^n$ modulo $1$ is uniformly distributed. The distribution of this sequence is signifcant in relation to unsolved problems in number theory including the Collatz conjecture. In this paper, we…

We study the distribution of the time to explosion for one-dimensional diffusions. We relate this question to computing the expectations of suitable nonnegative local martingales, and to the distributions of related diffusions with unit…

In this paper, an alternating direction implicit (ADI) difference scheme for two-dimensional time-fractional wave equation of distributed-order with a nonlinear source term is presented. The unique solvability of the difference solution is…

We consider a two-color P\'{o}lya urn in the case when a fixed number $S$ of balls is added at each step. Assume it is a large urn that is, the second eigenvalue $m$ of the replacement matrix satisfies $1/2<m/S\leq1$. After $n$ drawings,…

We examine the short and long-time behaviors of time-fractional diffusion equations with variable space-dependent order. More precisely, we describe the time-evolution of the solution to these equations as the time parameter goes either to…

We overview our recently introduced theory of n-fold integer programming which enables the polynomial time solution of fundamental linear and nonlinear integer programming problems in variable dimension. We demonstrate its power by…

We investigate the approximation for computing the sum $a_1+...+a_n$ with an input of a list of nonnegative elements $a_1,..., a_n$. If all elements are in the range $[0,1]$, there is a randomized algorithm that can compute an…

In this note, we represent integers in a type of factoradic notation. Rather than use the corresponding Lehmer code, we will view integers as permutations. Given a pair of integers n and k, we give a formula for n mod k in terms of the…

In this paper we characterize all distributional limits of the random quadratic form $T_n =\sum_{1\le u< v\le n} a_{u, v} X_u X_v$, where $((a_{u, v}))_{1\le u,v\le n}$ is a $\{0, 1\}$-valued symmetric matrix with zeros on the diagonal and…

In this note we introduce the notion of factorial moment distance for non-negative integer-valued random variables and we compare it with the total variation distance. Furthermore, we study the rate of convergence in the classical matching…

The scaling invariance for chaotic orbits near a transition from unlimited to limited diffusion in a dissipative standard mapping is explained via the analytical solution of the diffusion equation. It gives the probability of observing a…

A set ${\cal A} \subseteq \Set{1,...,N}$ is of type $B_2$ if all sums $a+b$, with $a\ge b$, $a,b\in {\cal A}$, are distinct. It is well known that the largest such set is of size asymptotic to $N^{1/2}$. For a $B_2$ set ${\cal A}$ of this…

The partial differential equation of Gaussian diffusion is generalized by using the time-fractional derivative of distributed order between 0 and 1, in both the Riemann-Liouville (R-L) and the Caputo (C) sense. For a general distribution of…