Related papers: Sampling Goldbach Numbers at Random

For every even integer N, denote by D_{1,2}(N) the number of representations of N as a sum of a prime and an integer having at most two prime factors. In this paper, we give a new lower bound for D_{1,2}(N).

We obtain the minimax rate for a mean location model with a bounded star-shaped set $K \subseteq \mathbb{R}^n$ constraint on the mean, in an adversarially corrupted data setting with Gaussian noise. We assume an unknown fraction $\epsilon…

For non-negative integers $(d_n(k))_{k \ge 1}$ such that $\sum_{k \ge 1} d_n(k) = n$, we sample a bipartite planar map with $n$ faces uniformly at random amongst those which have $d_n(k)$ faces of degree $2k$ for every $k \ge 1$ and we…

We show that for any set $D$ of at least two digits in a given base $b$, there exists a $\delta(D,b)>0$ such that within the set $\mathcal{A}$ of numbers whose digits base $b$ are exclusively from $D$, the number of even integers in…

We consider Gaussian distributions on certain Riemannian symmetric spaces. In contrast to the Euclidean case, it is challenging to compute the normalization factors of such distributions, which we refer to as partition functions. In some…

Let $(X_i)_{1 \le i \le n}$ be independent and identically distributed (i.i.d.) standard Gaussian random variables, and denote by $X_{(n)} = \max_{1 \le i \le n} X_i$ the maximum order statistic. It is well-known in extreme value theory…

We establish two new Waring--Goldbach type representations: every sufficiently large odd integer $n$ can be expressed as \[ n = p_1^2 + p_2^2 + p_3^3 + p_4^3 + p_5^5 + p_6^6 + p_7^c, \] where each $p_i$ is prime and $c \in \{6,7\}$.

Let $\{X_i,i=1,2,...\}$ be i.i.d. standard gaussian variables. Let $S_n=X_1+...+X_n$ be the sequence of partial sums and $$ L_n=\max_{0\leq i<j\leq n}\frac{S_j-S_i}{\sqrt{j-i}}. $$ We show that the distribution of $L_n$, appropriately…

This work is based on the proposal of a deterministic randomness extractor of a random Diffie-Hellman element defined over two prime order multiplicative subgroups of a finite fields $\mathbb{F}_{p^n}$, $G_1$ and $G_2$. We show that the…

A central problem in computational statistics is to convert a procedure for sampling combinatorial from an objects into a procedure for counting those objects, and vice versa. Weconsider sampling problems coming from *Gibbs distributions*,…

We give a deterministic, nearly logarithmic-space algorithm for mild spectral sparsification of undirected graphs. Given a weighted, undirected graph $G$ on $n$ vertices described by a binary string of length $N$, an integer $k\leq \log n$,…

For the Erd\H{o}s-R\'enyi random graph G(n,p), we give a precise asymptotic formula for the size of a largest vertex subset in G(n,p) that induces a subgraph with average degree at most t, provided that p = p(n) is not too small and t =…

We study an asymptotic formula for average orders of Goldbach representations of an integer as the sum of k primes. We extend the existing result for k=2 to a general k, for which we obtain a better error term. Moreover, we prove an…

Several proofs of the monotonicity of the non-Gaussianness (divergence with respect to a Gaussian random variable with identical second order statistics) of the sum of n independent and identically distributed (i.i.d.) random variables were…

We investigate the problem of characterizing the optimal variance proxy for sub-Gaussian random variables,whose moment-generating function exhibits bounded growth at infinity. We apply a general characterization method to discrete random…

Let $a_n$ be the random increasing sequence of natural numbers which takes each value independently with decreasing probability of order $n^{-\alpha}$, $0 < \alpha < 1/2$. We prove that, almost surely, for every measure-preserving system…

Consider the uniform random graph $G(n,M)$ with $n$ vertices and $M$ edges. Erd\H{o}s and R\'enyi (1960) conjectured that the limit $$ \lim_{n \to \infty} \Pr\{G(n,\textstyle{n\over 2}) is planar}} $$ exists and is a constant strictly…

We provide a numerical scheme to approximate as closely as desired the Gaussian or exponential measure $\mu(\om)$ of (not necessarily compact) basic semi-algebraic sets$\om\subset\R^n$. We obtain two monotone (non increasing and non…

We prove that if $A$ is a subset of those primes which are congruent to $1 \pmod{3}$ such that the relative density of $A$ in this residue class is larger than $\frac{1}{2},$ then every sufficiently large odd integer $n$ which satisfies $n…

For a set $S$ of vertices of a graph $G$, we define its density $0 \leq \sigma(S) \leq 1$ as the ratio of the number of edges of $G$ spanned by the vertices of $S$ to ${|S| \choose 2}$. We show that, given a graph $G$ with $n$ vertices and…