Let P be a set of n points in Euclidean space and let 0< ε< 1. A wellknown 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 at most a factor of 1+ ε. 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 points...

An (α, β)-spanner of a graph G is a subgraph H that approximates distances in G within a multiplicative factor α and an additive error β, ensuring that for any two nodes u, v, dH(u, v) ≤ α ·dG(u, v)+β. This paper concerns algorithms for the distributed deterministic construction of a sparse (α, β)-spanner H for a given graph G and distortion parameters α and β. It first presents a generic distr...

Let X be a smooth projective variety defined over field k of characteristic 0 and let L nef line bundle k. We prove that if x∈X is k-rational point then the Seshadri constant ε(X,L,x) k‾ same as show, by constructing families examples, there are varieties whose global ε(X) zero. also result on existence curve with natural (and necessary) hypothesis.

Nonlinear approximation has usually been studied under deterministic assumptions and complete information about the underlying functions. In the present paper we assume only partial information, e.g., function values at some points, and we are interested in the average case error and complexity of approximation. We show that the problem can be essentially split into two independent problems rel...

This project outlines a collection of problems which combine techniques of model reduction and filtering. The basis of this work is a collection of limit theories for stochastic processes which model dynamical systems with multiple time scales. Multiple time scales occur in many real systems, and reflect different orders of magnitudes of rates of change of different variables. These different t...

The class FORMULA[s]∘G consists of Boolean functions computable by size- s De Morgan formulas whose leaves are any from a G. We give lower bounds and (SAT, Learning, pseudorandom generators ( PRG )) algorithms for FORMULA[n 1.99 ]∘G, classes G with low communication complexity . Let R (k) be the maximum k -party number-on-forehead randomized function in Among other results, we show following: •...

We prove that for every k and every ε > 0, there exists g such that every graph with tree-width at most k and odd-girth at least g has circular chromatic number at most 2 + ε.

ε-support vector regression (ε-SVR) can be converted into an unconstrained convex and non-smooth quadratic programming problem. It is not solved by the traditional algorithm. In order to solve this non-smooth problem, a class of piecewise smooth functions is introduced to approximate the ε-insensitive loss function of ε-SVR, which generates a ε-piecewise smooth support vector regression (ε-dPWS...