We prove that for every convex body K with the center of mass at the origin and every ε ∈ ( 0, 12 ) , there exists a convex polytope P with at most eO(d)ε− d−1 2 vertices such that (1− ε)K ⊂ P ⊂ K.

The process Kδ is a blurred version of K , polluted by a Gaussian operator white noise Ḃ with a noise level δ > 0. The operator K acting on f is unknown and treated as a nuisance parameter. However, preliminary statistical inference about K is possible, with an accuracy governed by δ. Another equivalent approach is to consider that for experimental reasons we never have access to K in practice,...

Mohammad Zaki, MN State Univ, Mankato We define an algebraic number α as a root of an algebraic equation, a0∗x+a1∗x+· · ·+an−1∗x+an = 0 where ao, a1, · · · , an are rational integers, not all zero. We say α is an algebraic integer if a0 = 0. If an algebraic number α satisfies an algebraic equation of degree n with rational coefficients, and none of lower degree, then we say α is of degree n. If...

In this paper we extend the study of bipartite graphs with the induced ε-density property introduced by Frankl, Rödl, and the author. For a given kpartite k-uniform hypergraph G we say that a k-partite k-uniform hypergraph R = (W1, . . . ,Wk,F) has the induced ε-density property if every subhypergraph ofR with at least ε|F| edges contains a copy of G which is an induced subhypergraph of R. We s...

We explore a number of problems related to learning and covering structured distributions. Hypothesis Selection: We provide an improved and generalized algorithm for selecting a good candidate distribution from among competing hypotheses. Namely, given a collection of N hypotheses containing at least one candidate that is ε-close to an unknown distribution, our algorithm outputs a candidate whi...

We prove that for any positive integer k and for ε = 1/((k + 2)(3k2 + 1)), the edges of any graph whose fractional arboricity is at most k + ε can be decomposed into k forests and a matching.

We give a pedagogical review of the theory of CP violation with emphasis on the implications of recent experimental results. The review includes: (i) A detailed description of how CP violation arises in the Standard Model and in its extension that allows for neutrino masses; (ii) The formalism of CP violation in meson decays and its application to various K decays (ε K , ε ′ /ε and K → πνν), D ...

We consider the boundary value problem ∆u+ε k(x) e = 0 in a bounded, smooth domain Ω in R with homogeneous Dirichlet boundary conditions. Here ε > 0, k(x) is a non-negative, not identically zero function. We find conditions under which there exists a solution uε which blows up at exactly m points as ε→ 0 and satisfies ε ∫ Ω keε → 8mπ. In particular, we find that if k ∈ C(Ω̄), infΩ k > 0 and Ω is...