On the notion of stability of order convergence in vector lattices

In the theory of Banach lattices the following criterion for a norm to be order continuous is established: a norm is order continuous if and only if every order bounded sequence of positive pairwise disjoint elements in a lattice converges to zero in norm. In this paper we give a criterion for order convergence to be stable in a rather wide class of vector lattices which includes all Köthe spaces. The formulation of the criterion is analogous to that of the above-mentioned criterion for a norm to be order continuous. Namely, under certain conditions imposed on a vector lattice, stability of order convergence is equivalent to the condition that every order bounded sequence of positive pairwise disjoint elements converges relatively uniformly to zero. Furthermore, we study some types of order ideals in vector lattices. In terms of these ideals we give clarified versions of the above-stated criterions. As for notation and terminology, see for example [1,2].Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 35, No. 5, pp. 1026–1031, September–October, 1994.     

A note on the upper bound in SA AMG convergence analysis

In [M. Brezina, P. Vaněk and P. S. Vassilevski, An improved convergence of smoothed aggregation algebraic multigrid, Numer. Linear Algebra Appl., 19 (2012), pp. 441–469], a uniform convergence bound for smoothed aggregation algebraic multigrid with aggressive coarsening and massive polynomial prolongator and multigrid smoothers is established provided that the number of smoothing steps is equal to the coarsening ratio parameter ν. The final convergence estimate needs the uniform bound for the constant C_ν ∕ (2ν + 1). In this note, we give an improved upper bound. Copyright © 2013 John Wiley & Sons, Ltd.     

Almost uniform convergence

The almost uniform convergence is between uniform and quasi-uniform one. We give some necessary and sufficient conditions under which the almost uniform convergence coincides on compact sets with uniform, quasi-uniform or uniform convergence, respectively. In the second section continuity of almost uniform limits is considered. Finally we characterize the set of all points at which a net of functions is almost uniformly convergent to a given function.     

On the exact rate of convergence of frequencies of digits in self-similar sets

We study the exact rate of convergence of frequencies of digits of “normal” points of a self-similar set. Our results have applications to metric number theory. One particular application gives the following surprising result: there is an uncountable family of pairwise disjoint and exceptionally big subsets of ?^d that do not obey the law of the iterated logarithm. More precisely, we prove that there is an uncountable family of pairwise disjoint and exceptionally big sets of points x in ?^d—namely, sets with full Hausdorff dimension—for which the rate of convergence of frequencies of digits in the N-adic expansion of x is either significantly faster or significantly slower than the typical rate of convergence predicted by the law of the iterated logarithm.     

Statistical convergence of a non-positive approximation process

Starting from a general sequence of linear and positive operators of discrete type, we associate its r-th order generalization. This construction involves high order derivatives of a signal and it looses the positivity property. Considering that the initial approximation process is A-statistically uniform convergent, we prove that the property is inherited by the new sequence. Also, our result includes information about the uniform convergence. Two applications in q-Calculus are presented. We study q-analogues both of Meyer-König and Zeller operators and Stancu operators.     

The strong Schur property in Banach lattices

We prove that in the class of discrete Banach lattices the strong Schur property is equivalent to the disjoint strong Schur property (Theorem 3.1). Roughly speaking the strong Schur property holds iff an appropriate condition concerning sequences with positive pairwise disjoint terms is satisfied.      

Multiplier convergent series and uniform convergence of mapping series

In this paper, we introduce the frame property of complex sequence sets and study the uniform convergence of nonlinear mapping series in β-dual of spaces consisting of multiplier convergent series.     

First order convergence of matroids

The model theory based notion of the first order convergence unifies the notions of the left-convergence for dense structures and the Benjamini–Schramm convergence for sparse structures. It is known that every first order convergent sequence of graphs with bounded tree-depth can be represented by an analytic limit object called a limit modeling. We establish the matroid counterpart of this result: every first order convergent sequence of matroids with bounded branch-depth representable over a fixed finite field has a limit modeling, i.e., there exists an infinite matroid with the elements forming a probability space that has asymptotically the same first order properties. We show that neither of the bounded branch-depth assumption nor the representability assumption can be removed.     

The Menger property for infinite ordered sets

It is shown that, if an ordered set P contains at most k pairwise disjoint maximal chains, where k is finite, then every finite family of maximal chains in P has a cutset of size at most k. As a corollary of this, we obtain the following Menger-type result that, if in addition, P contains k pairwise disjoint complete maximal chains, then the whole family, M (P), of maximal chains in P has a cutset of size k. We also give a direct proof of this result. We give an example of an ordered set P in which every maximal chain is complete, P does not contain infinitely many pairwise disjoint maximal chains (but arbitrarily large finite families of pairwise disjoint maximal chains), and yet M (P) does not have a cutset of size <x, where x is any given (infinite) cardinal. This shows that the finiteness of k in the above corollary is essential and disproves a conjecture of Zaguia.     

Statistical convergence and ideal convergence for sequences of functions

Let I⊂P(N) stand for an ideal containing finite sets. We discuss various kinds of statistical convergence and I-convergence for sequences of functions with values in R or in a metric space. For real valued measurable functions defined on a measure space (X,M,μ), we obtain a statistical version of the Egorov theorem (when μ(X)<∞). We show that, in its assertion, equi-statistical convergence on a big set cannot be replaced by uniform statistical convergence. Also, we consider statistical convergence in measure and I-convergence in measure, with some consequences of the Riesz theorem. We prove that outer and inner statistical convergences in measure (for sequences of measurable functions) are equivalent if the measure is finite.     

Comparison principles applied to obstacle problems with penalties

Penalty methods form a well known technique to embed elliptic variational inequality problems into a family of variational equations (cf. [6], [13], [17]). Using the specific inverse monotonicity properties of these problems L _∞-bounds for the convergence can be derived by means of comparison solutions. Lagrange duality is applied to estimate parameters involved.

For piecewise linear finite elements applied on weakly acute triangulations in combination with mass lumping the inverse monotonicity of the obstacle problems can be transferred to its discretization. This forms the base of similar error estimations in the maximum norm for the penalty method applied to the discrete problem.

The technique of comparison solutions combined with the uniform boundedness of the Lagrange multipliers leads to decoupled convergence estimations with respect to the discretization and penalization parameters.     

Weak set convergence and its application to projection methods

There are many examples in Numerical Analysis where convergence of approximate solutions to a solution of the original problem can not be shown in the sense of a norm topology but in the sense of weak convergence ([6], [9], [10]).

Moreover, (global) solutions are often not unique such that a concept of set convergence instead of convergence in the usual sense is more convenient and reasonable ([1], [2]). This particularly holds if weakly formulated problems are under consideration.

When dealing with problems where both situations coincide, a concept of weak set convergence seems to be adequate. Such a concept is developed and will be applied to certain projections methods.     

Strong convergence of pairwise NQD random sequences

Strong limit theory is one of the most important problems in probability theory. Some results on the convergence of pairwise NQD random sequences have been presented. This paper further analyzes the strong convergence of pairwise NQD sequences and generalizes partial results of Wu [Q.Y. Wu, Convergence properties of pairwise NQD random sequences, Acta Math. Sinica 45 (3) (2002) 617-624 (in Chinese)]. Since no general moment inequalities are given as so far, we avoid this problem and obtain a class of strong limit theorem for NQD sequences and some corresponding conclusions by use of truncation methods and generalized three series theorem, which are the supplements to the previous fruits.     

Markov Chains with finite convergence time

We study the properties of finite ergodic Markov Chains whose transition probability matrix P is singular. The results establish bounds on the convergence time of P^m to a matrix where all the rows are equal to the stationary distribution of P. The results suggest a simple rule for identifying the singular matrices which do not have a finite convergence time. We next study finite convergence to the stationary distribution independent of the initial distribution. The results establish the connection between the convergence time and the order of the minimal polynomial of the transition probability matrix. A queuing problem and a maintenance Markovian decision problem which possess the property of rapid convergence are presented.     

On conditional weak convergence

In this paper we discuss a number of technical issues associated with conditional weak convergence. The main modes of convergence of conditional probability distributions areuniform, probability, andalmost sure convergence in the conditioning variable. General results regarding conditional convergence are obtained, including details of sufficient conditions for each mode of convergence, and characterization theorems for uniform conditional convergence.     

Quadratic convergence for cell-centered grids

This work discusses some of the convergence properties of approximations defined on standard cell-centered finite difference grids. It is shown that the order of convergence is quadratic in the grid spacing for both uniform and nonuniform grids. This order of convergence cannot be improved upon, even if uniform point-distributed grids are used. It is concluded that order of convergence arguments do not favor point-distributed grid construction over the more physically reasonable cell-centered construction. The techniques used are elementary and rely entirely on Taylor series expansions. Other applications of these techniques, such as to local grid refinement, are indicated.     

Zur Gleichwertigkeit zweier Arten der Randomisierung

For the choice of a mathematical model describing randomized decisions it is important to make sure that given a ‘behavioral decision rule’ there exists an equivalent ‘randomized decision function’ (cf. [2], p. 24–26). This fact seems to be known only in some special cases (cf. [1], Th. 8.3.1 and [3]). Here we give a simple proof for the general case.     

Bitopologies on Totally Ordered Sets

We study the category of ray bispaces, that is, the category whose objects are totally ordered sets with two topologies, each having a subbase of rays and so that the resulting bitopological space is pairwise weakly symmetric, and whose morphisms are the pairwise continuous functions. In contrast with the purely topological results of [5], we show that, (1) such spaces are utterly normal and hence monotonically normal (in the sense of [6]), and (2) (Intermediate Value Theorem) the pairwise continuous image of a pairwise connected bitopological space in a selective ray bispace is an interval. We also obtain conditions for the equality of the de Groot dual (see [4]) and the ray dual (see [5]) of a ray topology and show that a selective ray topology is compact if and only if it is skew compact.     

The weak convergence of uniform measures

This paper presents a necessary and sufficient condition for the weak convergence of uniform measures on an arbitrary Hausdorff uniform space in terms of their projections in metric spaces. This result was inspired by and extends a result of Bartoszynski which characterizes the weak convergence of countably additive measures on C[0,1] in terms of their projections in finite dimensional spaces.     

Normalized convergence in stochastic optimization

A new concept of (normalized) convergence of random variables is introduced. This convergence is preserved under Lipschitz transformations, follows from convergence in mean and itself implies convergence in probability. If a sequence of random variables satisfies a limit theorem then it is a normalized convergent sequence. The introduced concept is applied to the convergence rate study of a statistical approach in stochastic optimization.