(a) X is said to be a Loeb space i® the family of all non-empty, closed subsets of X has a choice function. (b) X is said to be selective i® the family of all non-empty, open subsets of X has a choice function. (c) X is said to be locally compact i® each point in X has a neighborhood base consisting of compact sets. (d) X is called second countable if it has a countable base for T . (e) X is ca...

In the topology of coordinatewise convergence ∇∞ is a compact, metrizable and separable space. Denote by C(∇∞) the algebra of real continuous functions on ∇∞ with pointwise operations and the supremum norm. In C(∇∞) there is a distinguished dense subspace F := R [q1, q2, . . . ] generated (as a commutative unital algebra) by algebraically independent continuous functions qk(x) := ∑∞ i=1 x k+1 i...

We study a general Kishimoto’s problem for automorphisms on simple C∗-algebras with tracial rank zero. Let A be a unital separable simple C∗-algebra with tracial rank zero and let α be an automorphism. Under the assumption that α has certain Rokhlin property, we present a proof that A ⋊α Z has tracial rank zero. We also show that if the induced map α∗0 on K0(A) fixes a “dense” subgroup of K0(A)...

This paper deals with a certain class of optimization methods, based on conservative convex separable approximations (CCSA), for solving inequality-constrained nonlinear programming problems. Each generated iteration point is a feasible solution with lower objective value than the previous one, and it is proved that the sequence of iteration points converges toward the set of Karush–Kuhn–Tucker...

The main results of the paper: (1) The dual Banach space X∗ contains a linear subspace A ⊂ X∗ such that the set A of all limits of weak∗ convergent bounded nets in A is a proper norm-dense subset of X∗ if and only if X is a non-quasi-reflexive Banach space containing an infinite-dimensional subspace with separable dual. (2) Let X be a non-reflexive Banach space. Then there exists a convex subse...

In this article, we propose to investigate the extension of the SEDT (Squared Euclidean Distance Transformation) on irregular isothetic grids. We give two algorithms to handle different structurations of grids. We first describe a simple approach based on the complete Voronoi diagram of the background irregular pixels, very fast on sparse grids. Then, we extend the separable algorithm defined o...

In this paper we study Banakh’s quarter-stratifiability among generalized ordered (GO)-spaces. All quarterstratifiable GO-spaces have a σ-closed-discrete dense set and therefore are perfect, and have a Gδ-diagonal. We characterize quarter-stratifiability among GO-spaces and show that, unlike the situation in general topological spaces, quarter-stratifiability is a hereditary property in GO-spac...

We prove there is a countable dense homogeneous subspace of R of size א1. The proof involves an absoluteness argument using an extension of the Lω1ω(Q) logic obtained by adding predicates for Borel sets. A separable topological space X is countable dense homogeneous (CDH) if, given any two countable dense subsets D and D′ of X, there is a homeomorphism h of X such that h[D] = D′. The main purpo...

Block encoding lies at the core of many existing quantum algorithms. Meanwhile, efficient and explicit block encodings dense operators are commonly acknowledged as a challenging problem. This paper presents comprehensive study rich family operators: pseudo-differential (PDOs). First, scheme for generic PDOs is developed. Then we propose more with separable structure. Finally, demonstrate an alg...