We describe deterministic algorithms which for a given depth-2 circuit $F$ approximate the probability that on a random input $F$ outputs a specific value $\alpha$. Our approach gives an algorithm which for a given $GF[2]$ multivariate polynomial $p$ and given $\varepsilon >0$ approximates the number of zeros of $p$ within a multiplicative factor $1+ \varepsilon$. The algorithm runs in time $ex...

We study transport properties in a simple model of two-dimensional roll convection under a slow periodic (period of order 1/ varepsilon >>1) perturbation. The problem is considered in terms of conservation of the adiabatic invariant. It is shown that the adiabatic invariant is well conserved in the system. It results in almost regular dynamics on large time scales (of order approximately vareps...

In convex stochastic optimization, convergence rates in terms of minimizing the objective have been well-established. However, in terms of making the gradients small, the best known convergence rate was $O(\varepsilon^{-8/3})$ and it was left open how to improve it. In this paper, we improve this rate to $\tilde{O}(\varepsilon^{-2})$, which is optimal up to log factors.

This paper is an attempt to prove the following result:Let $n>1$ be an integer and let $mathcal{R}$ be a $n!$-torsion-free ring with the identity element. Suppose that $d, delta, varepsilon$ are additive mappings satisfyingbegin{equation}d(x^n) = sum^{n}_{j=1}x^{n-j}d(x)x^{j-1}+sum^{n-1}_{j=1}sum^{j}_{i=1}x^{n-1-j}Big(delta(x)x^{j-i}varepsilon(x)+varepsilon(x)x^{j-i}delta(x)Big)x^{i-1}quadend{e...

The problem of finding \emph{distance} between \emph{pattern} of length $m$ and \emph{text} of length $n$ is a typical way of generalizing pattern matching to incorporate dissimilarity score. For both Hamming and $L_1$ distances only a super linear upper bound $\widetilde{O}(n\sqrt{m})$ are known, which prompts the question of relaxing the problem: either by asking for $1 \pm \varepsilon$ appro...

We study the problem of computing the diameter of a  set of $n$ points in $d$-dimensional Euclidean space for a fixed dimension $d$, and propose a new $(1+varepsilon)$-approximation algorithm with $O(n+ 1/varepsilon^{d-1})$ time and $O(n)$ space, where $0 < varepsilonleqslant 1$. We also show that the proposed algorithm can be modified to a $(1+O(varepsilon))$-approximation algorithm with $O(n+...

The ratio $\epsilon'/\epsilon$ measures the size of direct CP violation in $K_L\to\pi\pi$ decays $(\epsilon^\prime)$ relative to indirect one described by $\epsilon$ and is very sensitive new sources violation. As such it played a prominent role particle physics already for 45 years. Due smallness its measurement required heroic efforts 1980s 1990s on both sides Atlantic with final results pres...

We present a systemic theoretical study of the electronic properties of the quantum dots inserted in quantum dot infrared photodetectors (QDIPs). The strain distribution of three different shaped quantum dots (QDs) with a same ratio of the base to the vertical aspect is calculated by using the short-range valence-force-field (VFF) approach. The calculated results show that the hydrostatic strai...

The dynamics of a globally coupled, logistic map lattice is explored over a parameter plane consisting of the coupling strength, varepsilon, and the map parameter, a. By considering simple periodic orbits of relatively small lattices, and then an extensive set of initial-value calculations, the phenomenology of solutions over the parameter plane is broadly classified. The lattice possesses many...