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...

The existence of fixed point in orthogonal metric spaces has been initiated by Eshaghi and et. al [7]. In this paper, we prove existence and uniqueness theorem of fixed point for mappings on $varepsilon$-connected orthogonal metric space. As a consequence of this, we obtain the existence and uniqueness of fixed point for analytic function of one complex variable. The paper concludes with some i...

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.

In this paper we study stationary incompressible Newtonian fluid flow in a thin porous media. The media under consideration is bounded perforated 3D domain confined between two parallel plates. description of the includes small parameters: ε representing distance plates and $a_\varepsilon$ connected to microstructure such that ≪ $a_\varepsilon \ll \varepsilon$ . We consider classical setting me...

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...

Let $$\pi :M\rightarrow B$$ be a Riemannian submersion of two compact smooth manifolds, B is connected. $$M(\varepsilon )$$ denote the manifold M equipped with new metric which obtained from original one by multiplying $$\varepsilon $$ along vertical subspaces (i.e. fibers) and keeping unchanged (orthogonal to them) horizontal subspaces. $$V_i(M(\varepsilon ))$$ ith intrinsic volume. The main r...

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...