A Bound for the Modulus of Continuity for Metric Projections in a Uniformly Convex and Uniformly Smooth Banach Space

In 1979, Bjornestal obtained a local estimate for a modulus of uniform continuity of the metric projection operator on a closed subspace in a uniformly convex and uniformly smooth Banach spaceB. In the present paper we give the global version of this result for the projection operator on an arbitrary closed convex set inB.     

A hybrid of the extragradient method and proximal point algorithm for inverse strongly monotone operators and maximal monotone operators in Banach spaces

In the paper, we introduce two iterative sequences for finding a point in the intersection of the zero set of a inverse strongly monotone or inverse-monotone operator and the zero set of a maximal monotone operator in a uniformly smooth and uniformly convex Banach space. We prove weak convergence theorems under appropriate conditions, respectively.     

Conditional computability of real functions with respect to a class of operators

For any class of operators which transform unary total functions in the set of natural numbers into functions of the same kind, we define what it means for a real function to be uniformly computable or conditionally computable with respect to this class. These two computability notions are natural generalizations of certain notions introduced in a previous paper co-authored by Andreas Weiermann and in another previous paper by the same authors, respectively. Under certain weak assumptions about the class in question, we show that conditional computability is preserved by substitution, that all conditionally computable real functions are locally uniformly computable, and that the ones with compact domains are uniformly computable. The introduced notions have some similarity with the uniform computability and its non-uniform extension considered by Katrin Tent and Martin Ziegler, however, there are also essential differences between the conditional computability and the non-uniform computability in question.     

Halpern-Type Iterative Algorithm for an Infinite Family of Relatively Quasi-Nonexpansive Multivalued Mappings and Equilibrium Problem in Banach Spaces

In this paper, we introduce Halpern-type iterative algorithm for finding a common element of the set of common fixed points of an infinite family of relatively quasi-nonexpansive multivalued mappings and the set of solutions of an equilibrium problem in uniformly smooth and uniformly convex Banach spaces. Strong convergence to common elements of two set is established. Our results improved and extend the corresponding results announced by many others.     

Regularization methods for a class of variational inequalities in banach spaces

In this paper, we introduce two regularization methods, based on the Browder-Tikhonov and iterative regularizations, for finding a solution of variational inequalities over the set of common fixed points of an infinite family of nonexpansive mappings on real reflexive and strictly convex Banach spaces with a uniformly Gateaux differentiate norm.     

Generating a Random Collection of Discrete Joint Probability Distributions Subject to Partial Information

In this paper, we develop a practical and flexible methodology for generating a random collection of discrete joint probability distributions, subject to a specified information set, which can be expressed as a set of linear constraints (e.g., marginal assessments, moments, or pairwise correlations). Our approach begins with the construction of a polytope using this set of linear constraints. This polytope defines the set of all joint distributions that match the given information; we refer to this set as the “truth set.” We then implement a Monte Carlo procedure, the Hit-and-Run algorithm, to sample points uniformly from the truth set. Each sampled point is a joint distribution that matches the specified information. We provide guidelines to determine the quality of this sampled collection. The sampled points can be used to solve optimization models and to simulate systems under different uncertainty scenarios.     

Strong convergence theorems for infinite family of relatively quasi nonexpansive mappings and systems of equilibrium problems

In this paper, we construct a new iterative scheme by hybrid method for approximation of common element of set of common fixed points of countably infinite family of relatively quasi-nonexpansive mappings and set of common solutions to a system of equilibrium problems in a uniformly convex and uniformly smooth real Banach space using the properties of generalized f-projection operator. Then, we prove strong convergence of the scheme to a common element of the two sets. Furthermore, we apply our results to solve convex minimization problem. Our results extend important recent results.     

Sensitivity analysis of a railway station track layout with respect to a given timetable

An important step in the process of designing a railway station track layout is the verification of the robustness of the layout with respect to the timetables it is based on. For this purpose we develop in this paper an algorithm to randomly perturb a given timetable such that the perturbation is feasible and has the same structure as the given timetable. Mathematically, in this paper we study the problem of, given a set of integer variables and a set of binary relations stating minimal and maximal differences between the variables, to generate solutions uniformly at random. The algorithm involves the simulation of a Markov chain whose state space is a particular subset of the set of feasible timetables and whose limiting and equilibrium distribution is the uniform distribution. Whereas this idea seems simple, some technical pitfalls need to be overcome to make it sound.     

Strong convergence theorems for variational inequality problems and fixed point problems in uniformly smooth and uniformly convex Banach spaces

In this paper, we introduce a new iterative algorithm for finding a common element of the set of solutions of a general variational inequality problem for finite inverse-strongly accretive mappings and the set of common fixed points for a nonexpansive mapping in a uniformly smooth and uniformly convex Banach space. We obtain a strong convergence theorem under some suitable conditions. Our results improve and extend the recent ones announced by many others in the literature.     

A new explicit iteration method for a class of variational inequalities

In this paper, we propose a new simple explicit iterative algorithm to find a solution for variational inequalities over the set of common fixed points of an infinite family of nonexpansive mappings on real reflexive and strictly convex Banach spaces with a uniformly Gâteaux differentiable norm. Two numerical examples also are given for illustration.     

一致连续映射的两个扩张定理

把文[1],[2]中的两个定理进行了拓广和改进,给出两个一致连续映射的扩张定理:(i)任一集合A到一个完备度量空间的一致连续映射可一致连续扩张到A珡上.(ii)任一闭集A到Rn的一致连续映射可一致连续扩张到A与任一紧集的并.     

Convergence theorems for mixed equilibrium problems, variational inequality problem and uniformly quasi-?-asymptotically nonexpansive mappings

We introduce a modified block hybrid projection algorithm for finding a common element of the set of common fixed points of an infinite family of closed and uniformly quasi-?-asymptotically nonexpansive mappings, the set of the variational inequality for an α-inverse-strongly monotone operator, the set of solutions of the mixed equilibrium problems. Then, we obtain strong convergence theorems for the sequences generated by this process in a 2-uniformly convex and uniformly smooth Banach space. Our results extend and improve ones from several earlier works.     

Convergence of a non-failable mean-field particle system

The existing literature contains many examples of mean-field particle systems converging to the distribution of a Markov process conditioned to not hit a given set. In many situations, these mean-field particle systems are failable, meaning that they are not well defined after a given random time. Our first aim is to introduce an original mean-field particle system, which is always well defined and whose large number particle limit is, in all generality, the distribution of a process conditioned to not hit a given set. Under natural conditions on the underlying process, we also prove that the convergence holds uniformly in time as the number of particles goes to infinity. As an illustration, we show that our assumptions are satisfied in the case of a piece-wise deterministic Markov process.     

Eigenfunction Statistics for a Point Scatterer on a Three-Dimensional Torus

In this paper, we study eigenfunction statistics for a point scatterer (the Laplacian perturbed by a delta-potential) on a three-dimensional flat torus. The eigenfunctions of this operator are the eigenfunctions of the Laplacian which vanish at the scatterer, together with a set of new eigenfunctions (perturbed eigenfunctions). We first show that for a point scatterer on the standard torus all of the perturbed eigenfunctions are uniformly distributed in configuration space. Then we investigate the same problem for a point scatterer on a flat torus with some irrationality conditions, and show uniform distribution in configuration space for almost all of the perturbed eigenfunctions.     

Asymptotics of the moments of the number of cycles of a random A-permutation

We consider random permutations uniformly distributed on the set of all permutations of degree n whose cycle lengths belong to a fixed set A (the so-called A-permutations). In the present paper, we establish an asymptotics of the moments of the total number of cycles and of the number of cycles of given length of this random permutation as n → ∞.     

Exponential stability for set dynamic equations on time scales

A new theory known as set dynamic equations on time scales has been built. The criteria for the equistability, equiasymptotic stability, uniform and uniformly asymptotic stability were developed in Hong (2010) [1]. In this paper, we consider the exponential stability, exponentially asymptotic stability, uniform and uniformly exponentially asymptotic stability for the trivial solution of set dynamic equations on time scales by using Lyapunov-like functions.     

Attractivity properties of α-contractions

It is known that if T:X → X is completely continuous where X is a Banach space, then point dissipative and compact dissipative are equivalent, and imply the existence of a maximal compact invariant set which is uniformly asymptotically stable and attracts bounded sets uniformly. If T is an α-contraction, it is not known whether point dissipative and compact dissipative are equivalent. However, it is known that if T is an α-contraction and compact dissipative, then there exists a maximal compact invariant set which is uniformly asymptotically stable and attracts a neighborhood of any compact set uniformly. In this paper we show that for most practical examples which give rise to α-contraction, point dissipative and compact dissipative are equivalent. For example, we show this is true for stable neutral functional differential equations, retarded functional differential equations of infinite delay, and strongly damped nonlinear wave equations. We conjecture that this should be true for almost any practical application which gives rise to an α-contraction.     

UNIFORM APPROXIMATION OF A MULTIOBJECTIVE OPTIMIZATION SEQUENCE

In this paper the Pareto efficiency of a uniformly convergent multiobjective optimization sequence is studied. We obtain some relation between the Pareto efficient solutions of a given multiobjective optimization problem and those of its uniformly convergent optimization sequence and also some relation between the weak Pareto efficient solutions of the same optimization problem and those of its uniformly convergent optimization sequence. Besides, under a compact convex assumption for constraints set and a certain convex assumption for both objective and constraint functions, we also get some sufficient and necessary conditions that the limit of solutions of a uniformly convergent multiobjective optimization sequence is the solution of a given multiobjective optimization problem.     

Randomized methods based on new Monte Carlo schemes for control and optimization

We address randomized methods for control and optimization based on generating points uniformly distributed in a set. For control systems this sets are either stability domain in the space of feedback controllers, or quadratic stability domain, or robust stability domain, or level set for a performance specification. By generating random points in the prescribed set one can optimize some additional performance index. To implement such approach we exploit two modern Monte Carlo schemes for generating points which are approximately uniformly distributed in a given convex set. Both methods use boundary oracle to find an intersection of a ray and the set. The first method is Hit-and-Run, the second is sometimes called Shake-and-Bake. We estimate the rate of convergence for such methods and demonstrate the link with the center of gravity method. Numerical simulation results look very promising.     

A criterion for the equivalence of the Euclidean and the geodesic metric on compact subsets of {\mathbb{R}^N}

In (Studia Math 170:89–111, 2005), Bland and Feinstein show in two theorems that a compact plane set with pointwise (uniformly) regular boundary is itself pointwise (uniformly) regular. However, these statements do not cover simple sets as annular-like compact plane sets. In this note we prove that under suitable circumstances the statements keep true if one assumes the boundary to consist of finitely many pointwise (uniformly) regular sets.