In this paper we prove two results. The first theorem uses a paper of Kim [8] to show that for fixed primes p1, . . . , pk, and for fixed integers m1, . . . ,mk, with pi 6 |mi, the numbers (ep1(n), . . . , epk(n)) are uniformly distributed modulo (m1, . . . ,mk), where ep(n) is the order of the prime p in the factorization of n!. That implies one of Sander’s conjecture from [9], for any set of ...

We present an elementary proof that the qualitative picture of a linear hyperbolic flow is insensitive to slight measurements errors in its associated vector field. 1 Robustness of the linear hyperbolic flows We start considering linear flows e , where the matrix H is hyperbolic, i.e., all its eigenvalues have nonzero real part. The set H of all hyperbolic matrices is generic, in the sense that...

This paper is concerned with the study of the wave equation on compact surfaces and locally distributed damping, described by utt −∆Mu+ a(x) g(ut) = 0 on M× ]0,∞[ , where M ⊂ R is a smooth (of class C) oriented embedded compact surface without boundary, such that M = M0 ∪M1, where M1 := {x ∈ M;m(x) · ν(x) > 0} and M0 = M\M1. Here, m(x) := x − x, (x ∈ R fixed) and ν is the exterior unit normal v...

Variational inference using the reparameterization trick has enabled large-scale approximate Bayesian inference in complex probabilistic models, leveraging stochastic optimization to sidestep intractable expectations. The reparameterization trick is applicable when we can simulate a random variable by applying a (differentiable) deterministic function on an auxiliary random variable whose distr...

Building an atlas from a set of anatomical data relies on (1) the construction of a mean anatomy (called template or prototype) and (2) the estimation of the variations of this template within the population. To avoid biases introduced by separate processing, we jointly estimate the template and its deformation, based on a consistent statistical model. We use here a forward model that considers...

Single crystals of nonlinear optical material bis (guanidinium) hydrogen phosphate monohydrate (G2HP) belonging to non centrosymmetric space group P 21c were successfully grown by the slow evaporation method. Optical transmittance and second harmonic generation of the grown crystals have been studied by UV-vis-NIR spectrum and Kurtz powder technique respectively. The transmittance of G2HP cryst...

Motivated by a question of Krzysztof Oleszkiewicz we study a notion of weak tail domination of random vectors. We show that if the dominating random variable is sufficiently regular weak tail domination implies strong tail domination. In particular positive answer to Oleszkiewicz question would follow from the so-called Bernoulli conjecture. Introduction. This note is motivated by the following...

Consider an autoregressive model with measurement error: we observe Zi = Xi + εi, where Xi is a stationary solution of the autoregressive equation Xi = fθ0(Xi−1) + ξi. The regression function fθ0 is known up to a finite dimensional parameter θ. The distributions of X0 and ξ1 are unknown whereas the distribution of ε0 is completely known. We want to estimate the parameter θ by using the observat...

The celebrated Littlewood-Offord problem asks for an upper bound on the probability that random variable ε1v1+⋯+εnvn lies in Euclidean unit ball, where ε1,…,εn∈{−1,1} are independent Rademacher variables and v1,…,vn∈Rd fixed vectors of at least length. We extend some known results to case εi obtained from a Markov chain, including general bounds first shown by Erdős scalar Kleitman vector case,...