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For $m, d in {mathbb N}$, a jittered sampling point set $P$ having $N = m^d$ points in $[0,1)^d$ is constructed by partitioning the unit cube $[0,1)^d$ into $m^d$ axis-aligned cubes of equal size and then placing one point independently and uniformly at random in each cube. We show that there are constants $c ge 0$ and $C$ such that for all $d$ and all $m ge d$ the expected non-normalized star discrepancy of a jittered sampling point set satisfies [c ,dm^{frac{d-1}{2}} sqrt{1 + log(tfrac md)} le {mathbb E} D^*(P) le C, dm^{frac{d-1}{2}} sqrt{1 + log(tfrac md)}.] This discrepancy is thus smaller by a factor of $Thetabig(sqrt{frac{1+log(m/d)}{m/d}},big)$ than the one of a uniformly distributed random point set of $m^d$ points. This result improves both the upper and the lower bound for the discrepancy of jittered sampling given by Pausinger and Steinerberger (Journal of Complexity (2016)). It also removes the asymptotic requirement that $m$ is sufficiently large compared to $d$.
In the stochastic online vector balancing problem, vectors $v_1,v_2,ldots,v_T$ chosen independently from an arbitrary distribution in $mathbb{R}^n$ arrive one-by-one and must be immediately given a $pm$ sign. The goal is to keep the norm of the discr
Consider a unit interval $[0,1]$ in which $n$ points arrive one-by-one independently and uniformly at random. On arrival of a point, the problem is to immediately and irrevocably color it in ${+1,-1}$ while ensuring that every interval $[a,b] subsete
For the $h$-finite-element method ($h$-FEM) applied to the Helmholtz equation, the question of how quickly the meshwidth $h$ must decrease with the frequency $k$ to maintain accuracy as $k$ increases has been studied since the mid 80s. Nevertheless,
Let $mathcal{H}$ be a $t$-regular hypergraph on $n$ vertices and $m$ edges. Let $M$ be the $m times n$ incidence matrix of $mathcal{H}$ and let us denote $lambda =max_{v perp overline{1},|v| = 1}|Mv|$. We show that the discrepancy of $mathcal{H}$ is
Let Pd denote the space of all real polynomials of degree at most d. It is an old result of Stein and Wainger that for every polynomial P in Pd: |p.v.int_R {e^{iP(t)} dt/t} | < C(d) for some constant C(d) depending only on d. On the other hand, Car