Today Nand Flash memory is not only used in hand hold electronic devices but also as secondary storage medium. It serves as an alternative to Hard Disk Drives (HDDs) in the form of Solid State Dives (SDDs). However, unlike HDD flash memory does not support in place update, i. e. for updating data, old data can not be replaced by new data. Data can be written only at clean i. e. already erased p...

In Section 5, we saw that for a Brownian motion process in n _ 3 dimensions, P (limtxIX, = o0) = 1 for all x. In sharp contrast to this situation, a planar Brownian motion is certain to hit any nonpolar set. THEOREM 8.1. Let B be a Borel set. Then PX(VB < cX) is either identically 1 or identically 0. PROOF. A simple computation shows that for any x e R2, 1' p(s, x) ds T co as t T oc. Thus, for ...

Let M be a reductive monoid with unit group G. Let denote the idempotent cross-section of the G × G-orbits on M . If W is the Weyl group of G and e, f ∈ with e ≤ f , we introduce a projection map from WeW to WfW. We use these projection maps to obtain a new description of the Bruhat-Chevalley order on the Renner monoid of M . For the canonical compactification X of a semisimple group G0 with Bo...

A Borel equivalence relation on a Polish space is countable if all of its equivalence classes are countable. Standard examples of countable Borel equivalence relations (on the space of subsets of the integers) that occur in recursion theory are: recursive isomorphism, Turing equivalence, arithmetic equivalence, etc. There is a canonical hierarchy of complexity of countable Borel equivalence rel...

We prove that any countable index, universally measurable subgroup of a Polish group is open. By consequence, any universally measurable homomorphism from a Polish group into the infinite symmetric group S∞ is continuous. We also show that a universally measure homomorphism from a Polish group into a second countable, locally compact group is necessarily continuous. The present work is motivate...

for suitable constants 0 < c 1 < c 2 < +o% 1 < p < n. For every ge Lq(f2), 1/p + 1/q = 1, we denote by mh(g) and Mh(g) respectively the minimum value and the set of all minimum points of problem (0.1). We shall prove the following compactness theorem (Section 6): for every sequence (Eh) of closed subsets of f2 there exist a subsequence (Ec(h)) and a non-negative Borel measure g, vanishing on ev...

For a class K of countable relational structures, a countable Borel equivalence relation E is said to be K-structurable if there is a Borel way to put a structure in K on each E-equivalence class. We study in this paper the global structure of the classes of K-structurable equivalence relations for various K. We show that K-structurability interacts well with several kinds of Borel homomorphism...

We characterize the structural impediments to the existence of Borel perfect matchings for acyclic locally countable Borel graphs admitting a Borel selection of finitely many ends from their connected components. In particular, this yields the existence of Borel matchings for such graphs of degree at least three. As a corollary, it follows that acyclic locally countable Borel graphs of degree a...

Since the prox-mapping of stochastic mirror descent is intractable when directly being applied to the optimization problem (1), we propose the -inexact prox-mapping within the stochastic mirror descent framework in Section 3. Instead of solving the prox-mapping exactly, we approximate the solution with error. In this section, we will show as long as the approximation error is tolerate, the stoc...

For any countable Borel equivalence relation E on a standard Borel space X, there is a Borel function θ from X to the 2-generated groups such that xEy ⇔ θ(x) ∼= θ(y) .