In this paper, we discuss the relationships between spaces with locally countable weak-bases and spaces with various locally countable networks, establish the relationships between spaces with locally countable weak-bases and locally separable metric spaces, and show that spaces have locally countable weak-bases if and only if they are locally Lindelöf, g-metrizable spaces. These are improvemen...

In this article I will analyze the impact of forcing with a measure algebra on various topological statements. In particular our interest will focus on the study of hereditary separability and the hereditary Lindelöf property in the classes of compact, extremally disconnected, and cometrizable spaces.

We use the theory of quantization to introduce non-commutative versions of metric on state space and Lipschitz seminorm. We show that a lower semicontinuous matrix Lipschitz seminorm is determined by their matrix metrics on the matrix state spaces. A matrix metric comes from a lower semicontinuous matrix Lip-norm if and only if it is convex, midpoint balanced, and midpoint concave. The operator...

Given two Banach spaces X and Y, we analyze when the projective tensor product X?ˆ?Y has Corson's property (C) or is weakly Lindelöf determined (WLD), subspace of a compactly generated (WCG) space Hilbert space. For instance, show that: (i) WLD if only both Y are all operators from to Y? X? have separable range; (ii) WCG same holds for under assumption that every operator compact; (iii) ?p(?)?ˆ...

The objective of this paper is to introduce a general scheme for deriving a posteriori error estimates by using duality theory of the calculus of variations. We consider variational problems of the form inf v∈V {F (v) +G(Λv)}, where F : V → R is a convex lower semicontinuous functional, G : Y → R is a uniformly convex functional, V and Y are reflexive Banach spaces, and Λ : V → Y is a bounded l...

*Correspondence: [email protected] School of Science, Sichuan University of Science and Engineering, Zigong, Sichuan 643000, China Abstract This paper introduces the notion of exceptional family of elements for generalized variational inequalities in Hilbert spaces. The set-valued mapping is assumed to be upper semi-continuous compact with nonempty closed convex values. Based on topological deg...

It is known that the subdifferential of a lower semicontinuous convex function f over a Banach space X determines this function up to an additive constant in the sense that another function of the same type g whose subdifferential coincides with that of f at every point is equal to f plus a constant, i.e., g = f + c for some real constant c. Recently, Thibault and Zagrodny introduced a large cl...

We study the distribution of zeros of holomorphic Hecke cusp forms in several “thin” sets as the weight, k, tends to infinity. We obtain unconditional results for slowly shrinking (with k) hyperbolic balls. This relies on a new, effective, proof of Rudnick’s theorem and on an effective version of Quantum Unique Ergodicity for holomorphic forms, which we obtain in this paper. In addition, assumi...

Let V be an algebraic variety in Cn . We say that V satisfies the strong Phragmén-Lindelöf property (SPL) or that the classical Phragmén-Lindelöf Theorem holds on V if the following is true: There exists a positive constant A such that each plurisubharmonic function u on V which is bounded above by |z| + o(|z|) on V and by 0 on the real points in V already is bounded by A| Im z|. For algebraic ...