Filtrations for which All ℋ<Superscript>2</Superscript> Martingales Are of Integrable Variation; Distances between <Emphasis Type="Italic">σ</Emphasis>-Algebras

We consider filtrations for which all ℋ² martingales are of integrable variation. We find a sufficient condition and a necessary condition for this property. Both these conditions are stated in terms the volume of a filtration. The volume of a filtration is defined using a metric on a space of σ-algebras. To obtain the aforementioned conditions we use two equivalent metrics introduced by Boylan and Rogge. We also prove that the original definitions of these metrics can be simplified.      

Paracompactness of Metric Spaces and the Axiom of Multiple Choice

The axiom of multiple choice implies that metric spaces are paracompact but the reverse implication cannot be proved in set theory without the axiom of choice.     

Linear Restriction Estimates for Schrödinger Equation on Metric Cones

In this paper, we study some modified linear restriction estimates of the dynamics generated by Schrödinger operator on metric cone M, where the metric cone M is of the form M = (0, ∞) _r  × Σ, with the cross section Σ being a compact (n ? 1)-dimensional Riemannian manifold (Σ, h) and the equipped metric being g = dr ² + r ² h. Assuming the initial data possesses additional regularity in angular variable θ ∈ Σ, we show some linear restriction estimates for the solutions. In terms of their applications, we obtain global-in-time Strichartz estimates for radial initial data and show small initial data scattering theory for the mass-critical nonlinear Schrödinger equation on two-dimensional metric cones.     

Probability measures and Milyutin maps between metric spaces

We prove that the functor of Radon probability measures transforms any open map between completely metrizable spaces into a soft map. This result is applied to establish some properties of Milyutin maps between completely metrizable spaces.     

On Regularity for Constrained Extremum Problems. Part 1: Sufficient Optimality Conditions

The main aspect of the paper consists in the application of a particular theorem of separation between two sets to the image associated with a constrained extremum problem. In the image space, the two sets are a convex cone, which depends on the constraints (equalities or inequalities) of the given problem, and its image. In this way, a condition for the existence of a regular saddle point (i.e., a sufficient optimality condition) is obtained. This regularity condition is compared with those existing in the literature.     

关于随机真度的若干注记

以随机真度为基础,在三值R_0命题逻辑系统中给出了三种不同的近似推理模式并讨论了它们之间的关系,其次利用根的性质得出误差定义的若干推理结果.     

Randomized algorithm for the k-server problem on decomposable spaces

We study the randomized k-server problem on metric spaces consisting of widely separated subspaces. We give a method which extends existing algorithms to larger spaces with the growth rate of the competitive quotients being at most O(logk). This method yields o(k)-competitive algorithms solving the randomized k-server problem for some special underlying metric spaces, e.g. HSTs of “small” height (but unbounded degree). HSTs are important tools for probabilistic approximation of metric spaces.     

Improving bounds on the minimum Euclidean distance for block codes by inner distance measure optimization

The minimum Euclidean distance is a fundamental quantity for block coded phase shift keying (PSK). In this paper we improve the bounds for this quantity that are explicit functions of the alphabet size q, block length n and code size |C|. For q=8, we improve previous results by introducing a general inner distance measure allowing different shapes of a neighborhood for a codeword. By optimizing the parameters of this inner distance measure, we find sharper bounds for the outer distance measure, which is Euclidean.The proof is built upon the Elias critical sphere argument, which localizes the optimization problem to one neighborhood. We remark that any code with q=8 that fulfills the bound with equality is best possible in terms of the minimum Euclidean distance, for given parameters n and |C|. This is true for many multilevel codes.     

Maps of several variables of finite total variation. II. E. Helly-type pointwise selection principles

Given a=(a₁,…,an), b=(b₁,…,bn)∈Rn with a<b componentwise and a map f from the rectangle into a metric semigroup M=(M,d,+), denote by the Hildebrandt-Leonov total variation of f on , which has been recently studied in [V.V. Chistyakov, Yu.V. Tretyachenko, Maps of several variables of finite total variation. I, J. Math. Anal. Appl. (2010), submitted for publication]. The following Helly-type pointwise selection principle is proved: If a sequence{fj}_j∈Nof maps frominto M is such that the closure in M of the set{fj(x)}_j∈Nis compact for eachandis finite, then there exists a subsequence of{fj}_j∈N, which converges pointwise onto a map f such that. A variant of this result is established concerning the weak pointwise convergence when values of maps lie in a reflexive Banach space (M,‖⋅‖) with separable dual M^∗.     

A selection theorem in metric trees

In this paper, we show that nonempty closed convex subsets of a metric tree enjoy many properties shared by convex subsets of Hilbert spaces and admissible subsets of hyperconvex spaces. Furthermore, we prove that a set-valued mapping of a metric tree with convex values has a selection for which for each . Here by we mean the Hausdroff distance. Many applications of this result are given.