On the convergence of some Procrustean averaging algorithms

Euclidean “(size-and-)shape spaces” are spaces of configurations of points in R ^N modulo certain equivalences. In many applications one seeks to average a sample of shapes, or sizes-and-shapes, thought of as points in one of these spaces. This averaging is often done using algorithms based on generalized Procrustes analysis (GPA). These algorithms have been observed in practice to converge rapidly to the Procrustean mean (size-and-)shape, but proofs of convergence have been lacking. We use a general Riemannian averaging (RA) algorithm developed in [Groisser, D. (2004) “Newton's method, zeroes of vector fields, and the Riemannian center of mass”, Adv. Appl. Math. 33, pp. 95–135] to prove convergence of the GPA algorithms for a fairly large open set of initial conditions, and estimate the convergence rate. On size-and-shape spaces the Procrustean mean coincides with the Riemannian average, but not on shape spaces; in the latter context we compare the GPA and RA algorithms and bound the distance between the averages to which they converge.     

Approximation by means of nonlinear Kantorovich sampling type operators in Orlicz spaces

In this paper we introduce a nonlinear version of the Kantorovich sampling type series in a nonuniform setting. By means of the above series we are able to reconstruct signals (functions) which are continuous or uniformly continuous. Moreover, we study the problem of the convergence in the setting of Orlicz spaces: this allows us to treat signals which are not necessarily continuous. Our theory applies to L^p-spaces, interpolation spaces, exponential spaces and many others. Several graphical examples are provided.     

On convergence theory in fuzzy topological spaces and its applications

In this paper we introduce and study new concepts of convergence and adherent points for fuzzy filters and fuzzy nets in the light of the Q-relation and the Q-neighborhood of fuzzy points due to Pu and Liu [28]. As applications of these concepts we give several new characterizations of the closure of fuzzy sets, fuzzy Hausdorff spaces, fuzzy continuous mappings and strong Q-compactness. We show that there is a relation between the convergence of fuzzy filters and the convergence of fuzzy nets similar to the one which exists between the convergence of filters and the convergence of nets in topological spaces.     

A New Uniform Convergence for Partial Functions

Suppose X, Y are topological spaces. In this paper maps are not necessarily continuous. A map f from a non-empty subset of X to Y is called a partial map. Partial maps occur as inverse functions in elementary analysis, as solution of ordinary differential equations, as utility functions in mathematical economics, etc. In many applications, X and Y are metric spaces and there is a need to have a uniform convergence on a family of partial functions. Since partial maps do not have a common domain, the usual uniform convergence (u.c.) is not available. Noting that in many situations, all maps of a family under consideration, have a common range, we define a new uniform convergence (n.u.c.) that is complementary to the usual one. This n.u.c. does not preserve continuity but preserves (uniform) openness. Its usefulness stems from the fact that it can be used when u.c. cannot be defined. Moreover, in some situations where both u.c. and n.u.c. are available, the latter satisfies our intuition but not the former. We give applications to ODE's and throw some light on earlier literature.     

PreT 2 Objects in Topological Categories

In previous papers, two notions of pre-Hausdorff (PreT ₂) objects in a topological category were introduced and compared. The main objective of this paper is to show that the full subcategory of PreT ₂ objects is a topological category and all of T ₀, T ₁, and T ₂ objects in this topological category are equivalent. Furthermore, the characterizations of pre-Hausdorff objects in the categories of filter convergence spaces, (constant) local filter convergence spaces, and (constant) stack convergence spaces are given and as a consequence, it is shown that these categories are homotopically trivial.     

Convergence of a Hybrid Projection Algorithm in Banach Spaces

In this paper, we consider a hybrid projection algorithm for two families of quasi-φ-nonexpansive mappings. We establish strong convergence theorems of common fixed points in the framework of Banach spaces. Our results improve and extend the corresponding results announced by many others.     

Subcategories of Filter Tower Spaces

The concept of a convergence tower space, or equivalently, a convergence approach space is formulated here in the context of a Cauchy setting in order to include a completion theory. Subcategories of filter tower spaces are defined in terms of axioms involving a general t-norm, T, in order to include a broad range of spaces. A T-regular sequence for a filter tower space is defined and, moreover, it is shown that the category of T-regular objects is a bireflective subcategory of all filter tower spaces. A completion theory for subcategories of filter tower spaces is given.     

The p-step iterative algorithm for a system of generalized mixed quasi-variational inclusions with -accretive operators in q-uniformly smooth banach spaces

In this paper, we introduce and study a new system of generalized mixed quasi-variational inclusions with (A,η)-accretive operators in q-uniformly smooth Banach spaces. By using the resolvent operator technique associated with (A,η)-accretive operators, we construct a new p-step iterative algorithm for solving this system of generalized mixed quasi-variational inclusions in real q-uniformly smooth Banach spaces. We also prove the existence of solutions for the generalized mixed quasi-variational inclusions and the convergence of iterative sequences generated by algorithm. Our results improve and generalize many known corresponding results.     

Hybrid Projection Algorithm for Two Finite Families of Asymptotically Quasi ϕ-Nonexpansive Mappings in Reflexive Banach Spaces

In this paper, we introduce a hybrid projection algorithm for two finite families of asymptotically quasi ?-nonexpansive mappings to establish a convergence theorem in smooth, strictly convex, and reflexive Banach spaces. As applications, we state the convergence of hybrid projection algorithm for two finite families of asymptotically quasi-nonexpansive mappings in Hilbert spaces and the convergence for a system of equilibrium problems.     

Noncommutative integration for traces with values in Kantorovich-Pinsker spaces

In this paper we consider traces on a von Neumann algebra M with values in complex Kantorovich-Pinsker spaces. We establish the connection between the convergence with respect to the trace and the convergence locally in measure in the algebra S(M) of measurable operators affiliated with M. We define the (bo)-complete lattice-normed spaces of integrable operators in S(M) and prove that they are decomposable if the trace possesses the Maharam property.     

On the order conditions of fuzzy convergence classes

By means of the order structure of the related lattice, the LIMINF condition of fuzzy convergence classes is proposed in this paper, which reflects the essential difference between fuzzy convergence classes and ordinary convergence classes. The relationship between the LIMINF condition and two related conditions proposed by Liu and Wang respectively are discussed. The theory of fuzzy convergence classes based on LIMINF condition is established for topological molecular lattices, L-topological spaces (in the sense of Chang or Lowen), weakly induced spaces, and induced spaces.     

Boundedly Compatible Webs and Strict Mackey Convergence

We look for characterizations of those locally convex spaces that satisfy the strict Mackey convergence condition within the context of spaces with webs. We will say that a locally convex space has a boundedly compatible web if it has a web of absolutely convex sets whose members behave like zero neighborhoods in a metrizable locally convex space. It will be shown that these locally convex spaces satisfy the strict Mackey convergence condition. One consequence of this result will be a characterization of boundedly retractive inductive limits. We will also prove that if E is locally complete and webbed, then the strict Mackey convergence condition is equivalent to E having a boundedly compatible web.     

Convergence of Some Greedy Algorithms in Banach Spaces

We consider some theoretical greedy algorithms for approximation in Banach spaces with respect to a general dictionary. We prove convergence of the algorithms for Banach spaces which satisfy certain smoothness assumptions. We compare the algorithms and their rates of convergence when the Banach space is L_p(\mathbbT^d)L_p(\mathbb{T}^d) ($1相似文献   

Convergence for a family of neural network operators in Orlicz spaces

In this paper, we develop the theory for a family of neural network (NN) operators of the Kantorovich type, in the general setting of Orlicz spaces. In particular, a modular convergence theorem is established. In this way, we study the above family of operators in many instances of useful spaces by a unique general approach. The above NN operators provide a constructive approximation process, in which the coefficients, the weights, and the thresholds of the networks needed in order to approximate a given function f , are known. At the end of the paper, several examples of Orlicz spaces, and of sigmoidal activation functions for which the present theory can be applied, are studied in details.     

A Learning Algorithm for the Control of Continuous Action Set-Point Regulator Systems

The convergence properties for reinforcement learning approaches, such as temporal differences and Q-learning, have been established under moderate assumptions for discrete state and action spaces. In practice, however, many systems have either continuous action spaces or a large number of discrete elements. This paper presents an approximate dynamic programming approach to reinforcement learning for continuous action set-point regulator problems, which learns near-optimal control policies based on scalar performance measures. The continuous-action space (CAS) algorithm uses derivative-free line search methods to obtain the optimal action in the continuous space. The theoretical convergence properties of the algorithm are presented. Several heuristic stopping criteria are investigated and practical application is illustrated by two example problems—the inverted pendulum balancing problem and the power system stabilization problem.     

Convergence to equilibrium for linear spatially homogeneous Boltzmann equation with hard and soft potentials: A semigroup approach in L1‐spaces

We investigate the large time behavior of solutions to the spatially homogeneous linear Boltzmann equation from a semigroup viewpoint. Our analysis is performed in some (weighted) L¹‐spaces. We deal with both the cases of hard and soft potentials (with angular cut‐off). For hard potentials, we provide a new proof of the fact that, in weighted L¹‐spaces with exponential or algebraic weights, the solutions converge exponentially fast towards equilibrium. Our approach uses weak‐compactness arguments combined with recent results of the second author on positive semigroups in L¹‐spaces. For soft potentials, in L¹‐spaces, we exploit the convergence to ergodic projection for perturbed substochastic semigroup to show that, for very general initial datum, solutions to the linear Boltzmann equation converges to equilibrium in large time. Moreover, for a large class of initial data, we also prove that the convergence rate is at least algebraic. Notice that, for soft potentials, no exponential rate of convergence is expected because of the absence of spectral gap.     

Completion of Semiuniform Convergence Spaces

Semiuniform convergence spaces form a common generalization of filter spaces (including symmetric convergence spaces [and thus symmetric topological spaces] as well as Cauchy spaces) and uniform limit spaces (including uniform spaces) with many convenient properties such as cartesian closedness, hereditariness and the fact that products of quotients are quotients. Here, for each semiuniform convergence space a completion is constructed, called the simple completion. This one generalizes Császár's -completion of filter spaces. Thus, filter spaces are characterized as subspaces of convergence spaces. Furthermore, Wyler's completion of separated uniform limit spaces can be easily derived from the simple completion.     

Connectedness for lattice-valued subsets in lattice-valued convergence spaces

In this paper, a connectedness in stratified L-generalized convergence spaces, is defined and discussed. This connectedness is di?erent from Jäger’s connectedness, in particular, it is defined for L-subsets however Jäger’s connectedness is defined for crisp subsets. Thus this give a partly answer to Jäger’s question in 2016: define a connectedness for L-subsets in lattice-valued convergence spaces. Then the basic properties of this connectedness are discussed.     

A cyclic iterative method for solving Multiple Sets Split Feasibility Problems in Banach Spaces

In this paper, we construct an iterative scheme and prove strong convergence theorem of the sequence generated to an approximate solution to a multiple sets split feasibility problem in a p-uniformly convex and uniformly smooth real Banach space. Some numerical experiments are given to study the efficiency and implementation of our iteration method. Our result complements the results of F. Wang (A new algorithm for solving the multiple-sets split feasibility problem in Banach spaces, Numerical Functional Anal. Optim. 35 (2014), 99–110), F. Scho¨pfer et al. (An iterative regularization method for the solution of the split feasibility problem in Banach spaces, Inverse Problems 24 (2008), 055008) and many important recent results in this direction.     

Weighted Statistical Relative Invariant Mean in Modular Function Spaces with Related approximation Results

Abstract

The idea of statistical relative convergence on modular spaces has been introduced by Orhan and Demirci. The notion of σ-statistical convergence was introduced by Mursaleen and Edely and further extended based on a fractional order difference operator by Kadak. The concern of this paper is to define two new summability methods for double sequences by combining the concepts of statistical relative convergence and σ-statistical convergence in modular spaces. Furthermore, we give some inclusion relations involving the newly proposed methods and present an illustrative example to show that our methods are nontrivial generalizations of the existing results in the literature. We also prove a Korovkin-type approximation theorem and estimate the rate of convergence by means of the modulus of continuity. Finally, using the bivariate type of Stancu-Schurer-Kantorovich operators, we display an example such that our approximation results are more powerful than the classical, statistical, and relative modular cases of Korovkin-type approximation theorems.