Related papers: Random sparse unary predicates

In this paper, we consider a well-known sparse optimization problem that aims to find a sparse solution of a possibly noisy underdetermined system of linear equations. Mathematically, it can be modeled in a unified manner by minimizing…

Under sufficiently strong assumptions about the first term in an arithmetic progression, we prove that for any integer $a$, there are infinitely many $n\in \mathbb N$ such that for each prime factor $p|n$, we have $p-a|n-a$. This can be…

It is shown that the behavior of the solutions of the nonlinear recursion $y(\ell + 1) = [1-y(\ell)]^p$ -- where the dependent variable $y(l)$ is a real number, $\ell= 0; 1; 2...$ is the independent variable, and $p$ is an arbitrary…

We prove an L^1 subsequence ergodic theorem for sequences chosen by independent random selector variables, thereby showing the existence of universally L^1-good sequences nearly as sparse as the set of squares. In the process, we prove that…

We answer several questions of P. Erdos and R. Graham by showing that for any rational number r > 0, there exist integers n1, n2, ..., nk, k variable, where N < n1 < n2 < ... < nk < (e^r + o_r(1) ) N, such that r = 1/n1 + 1/n2 + ... + 1/nk.…

In a 2011 paper published in the journal "Asian Journal of Algebra"(see reference[1]), the authors consider, among other equations,the diophantine equations 2xy=n(x+y) and 3xy=n(x+y). For the first equation, with n being an odd positive…

We give simple proofs, under minimal hypotheses, of the Weak Law of Large Numbers and the Central Limit Theorem for independent identically distributed random variables. These proofs use only the elementary calculus, together with the most…

Let $(F_n)$ be the sequence of Fibonacci numbers and, for each positive integer $k$, let $\mathcal{P}_k$ be the set of primes $p$ such that $\gcd(p - 1, F_{p - 1}) = k$. We prove that the relative density $\text{r}(\mathcal{P}_k)$ of…

We say that a convergence law holds for a sequence of random combinatorial objects if, for any first-order sentence $\varphi$, the proportion of objects satisfying $\varphi$ converges to a limiting value as the size of the objects tends to…

Let p_n denote the persistence probability that the first n iterated partial sums of integrable, zero-mean, i.i.d. random variables X_k, are negative. We show that p_n is bounded above up to universal constant by the square root of the…

Given a finite set of roots of unity, we show that all power sums are non-negative integers iff the set forms a group under multiplication. The main argument is purely combinatorial and states that for an arbitrary finite set system the…

Holroyd and Propp used Hall's marriage theorem to show that, given a probability distribution pi on a finite set S, there exists an infinite sequence s_1,s_2,... in S such that for all integers k >= 1 and all s in S, the number of i in…

Motivated by classical nontransitivity paradoxes, we call an $n$-tuple $(x_1,\dots,x_n) \in[0,1]^n$ \textit{cyclic} if there exist independent random variables $U_1,\dots, U_n$ with $P(U_i=U_j)=0$ for $i\not=j$ such that…

It is known that limit theorems for triangular arrays with identically distributed rows yields convergence of densities rather than just convergence in distribution. We show that this superconvergence result holds -- at least at points at…

In this paper we prove the Random Van der Waerden Theorem: For $q_1 \geq q_2 \geq \dotsb \geq q_r \geq 3 \in \mathbb{N}$ there exist $c,C >0$ such that \[ \lim_{n \to \infty} \mathbb{P}([n]_p \rightarrow (q_1,\dotsc, q_r)) = \begin{cases} 1…

Consider the binomial model $G^{d+1}(n,p)$ of the random $(d+1)$-uniform hypergraph on $n$ vertices, where each edge is present, independently of one another, with probability $p:\mathbb{N}\to[0,1]$. We prove that, for all…

Given k>1, let a_n be the sequence defined by the recurrence a_n=c_1a_{n-1}+c_2a_{n-2}+...+c_ka_{n-k} for n>=k, with initial values a_0=a_1=...=a_{k-2}=0 and a_{k-1}= 1. We show under a couple of assumptions concerning the constants c_i…

Let $w$ be a finite word over the alphabet $\{0,1\}$. For any natural number $n$, let $s_w(n)$ denote the number of occurrence of $w$ in the binary expansion of $n$ as a scattered subsequence. We study the behavior of the partial sum…

In the local, characteristic 0, non archimedean case, we consider distributions on GL(n+1) which are invariant under the adjoint action of GL(n). We prove that such distributions are invariant by transposition. This implies that an…

Let $\{X, X_n, n\geq 1\}$ be a sequence of independent identically distributed non-degenerate random variables. Put $S_0=0, S_n = \sum^n_{i=1} X_i$ and $V_n^2=\sum^n_{i=1} X_i^2, n\ge 1.$ A weak convergence theorem is established for the…