We study the 2D Euler equation in a bounded simply-connected domain, and establish local uniqueness of flow whose stream function ψε satisfies{−ε2Δψε=∑i=1k1Bδ(z0,i)(ψε−με,i)+γ,inΩ,ψε=0,on∂Ω, with ε→0+ scale parameter vortices, γ∈(0,∞), Ω⊂R2 simply connected Lipschitz z0,i∈Ω limiting location ith vortex, με,i flux constants unprescribed. Our proof is achieved by detailed description asymptotic b...

First order necessary conditions and duality results for general inexact nonlinear programming problems formulated in nonreflexive spaces are obtained. The Dubovitskii–Milyutin approach is the main tool used. Particular cases of linear and convex programs are also analyzed and some comments about a comparison of the obtained results with those existing in the literature are given. 1 Problem sta...

Let P be a set of n points in Euclidean space and let 0 < ε < 1. A well-known result of Johnson and Lindenstrauss states that there is a projection of P onto a subspace of dimension O(ε−2 logn) such that distances change by a factor of 1 + ε at most. We consider an extension of this result. Our goal is to find an analogous dimension reduction where not only pairs, but all subsets of at most k p...

We consider special numerical approximations to a domain decomposition method for a boundary value problem in the case of singularly perturbed nonlinear convection-diffusion equations, with the perturbation parameter ε. As a rule, a differential problem is approximated by nonlinear grid equations (iteration-free schemes), which are then solved by suitable iterative methods. In the case of ε-uni...

Let C be a class of probability distributions over the discrete domain [n] = {1, . . . , n}. We show that if C satisfies a rather general condition – essentially, that each distribution in C can be well-approximated by a variable-width histogram with few bins – then there is a highly efficient (both in terms of running time and sample complexity) algorithm that can learn any mixture of k unknow...

We present two new results for the problem of approximating a given real m × n matrix A by a rank-k matrix D, where k < min{m, n}, so as to minimize ||A−D||F . It is known that by sampling O(k/ε) rows of the matrix, one can find a low-rank approximation with additive error ε||A||F . Our first result shows that with adaptive sampling in t rounds and O(k/ε) samples in each round, the additive err...

and Applied Analysis 3 Definition 1.3 see 25 . Let X, μ, ν, ∗, be a intuitionistic fuzzy metric space. Let A be any subset of X. Define φ t inf { μ ( x, y, t ) : x, y ∈ A}, ψ t sup{ν(x, y, t) : x, y ∈ A}, 1.2 i A is said to be q-bounded if limt→∞φ t 1 and limt→∞ψ t 0, ii A is said to be semibounded if limt→∞φ t k and limt→∞ψ t 1 − k, 0 < k < 1 iii A is said to be unbounded if limt→∞φ t 0 and li...

This paper revisits the online PCA problem. Given a stream of n vectors xt ∈ R (columns of X) the algorithm must output yt ∈ R (columns of Y ) before receiving xt+1. The goal of online PCA is to simultaneously minimize the target dimension ` and the error ‖X − (XY +)Y ‖. We describe two simple and deterministic algorithms. The first, receives a parameter ∆ and guarantees that ‖X − (XY +)Y ‖ is ...

In this paper, we show the existence of small coresets for the problems of computing k-median and k-means clustering for points in low dimension. In other words, we show that given a point set P in IR, one can compute a weighted set S ⊆ P , of size O(kε−d log n), such that one can compute the k-median/means clustering on S instead of on P , and get an (1 + ε)-approximation. As a result, we impr...

Given a weighted n-vertex graph G with integer edge-weights taken from a range [−M,M ], we show that the minimum-weight simple path visiting k vertices can be found in time Õ(2poly(k)Mn) = O∗(2kM). If the weights are reals in [1,M ], we provide a (1+ε)-approximation which has a running time of Õ(2poly(k)n(log logM + 1/ε)). For the more general problem of k-tree, in which we wish to find a minim...