Kind Of Convergence Research Articles

Greek and Lydian Evidence of Diversity, Erasure, and Convergence in Western Asia Minor

After the Persian, Greek, and Roman conquests of Lydia, a separate Lydian culture and language gradually disappeared under the influence of the kind of socioeconomic and sociolinguistic forces that have more recently reduced the diversity of the world’s cultures and languages. There is evidence, however, that a convergence between Greek and Anatolian (particularly Lydian) cultures stretched back into the Bronze Age. This kind of convergence would explain Herodotus’ remarkable statement that Greeks and Lydians follow much the same customs.

Projection estimates of point processes boundaries

We present a method for estimating the edge of a two-dimensional bounded set, given a finite random set of points drawn from the interior. The estimator is based both on projections on C 1 bases and on extreme points of the point process. We give conditions on the Dirichlet's kernel associated to the C 1 bases for various kinds of convergence and asymptotic normality. We propose a method for reducing the negative bias and illustrate it by a simulation.

Theory of hypernumbers and extrafunctions: Functional spaces and differentiation

The theory of hypernumbers and extrafunctions is a novel approach in functional analysis aimed at problems of mathematical and computational physics. The new technique allows operations with divergent integrals and series and makes it possible to distinct different kinds of convergence and divergence. Although, it resembles nonstandard analysis, there are several distinctions between these theories. For example, while nonstandard analysis changes spaces of real and complex numbers by injecting into them infinitely small numbers and other nonstandard entities, the theory of extrafunctions does not change the inner structure of spaces of real and complex numbers, but adds to them infinitely big and oscillating numbers as external objects. In this paper, we consider a simplified version of hypernumbers, but a more general version of extrafunctions and their extraderivatives in comparison with previous works.

A New Approach to Quantitative Domain Theory

This paper introduces a new approach to the theory of Ω-categories enriched by a frame. The approach combines ideas from various areas such as generalized ultrametric domains, Ω-categories, constructive analysis, and fuzzy mathematics. As the basic framework, we use the Wagner's Ω-category [18,19] with a frame instead of a quantale with unit. The objects and morphisms in the category will be called L-Fuzzy posets and L-Fuzzy monotone mappings, respectively. Moreover, we introduce concepts of adjoints and a kind of convergence in an L-Fuzzy poset that makes the theory “constructive” or “computable”.

Computational uncertainty principle in nonlinear ordinary differential equations

The error propagation for general numerical method in ordinary differential equations ODEs is studied. Three kinds of convergence, theoretical, numerical and actual convergences, are presented. The various components of round-off error occurring in floating-point computation are fully detailed. By introducing a new kind of recurrent inequality, the classical error bounds for linear multistep methods are essentially improved, and joining probabilistic theory the “normal” growth of accumulated round-off error is derived. Moreover, a unified estimate for the total error of general method is given. On the basis of these results, we rationally interpret the various phenomena found in the numerical experiments in part I of this paper and derive two universal relations which are independent of types of ODEs, initial values and numerical schemes and are consistent with the numerical results. Furthermore, we give the explicitly mathematical expression of the computational uncertainty principle and expound the intrinsic relation between two uncertainties which result from the inaccuracies of numerical method and calculating machine.

On the incompressible limit of inviscid compressible fluids

We consider the Euler equations of barotropic inviscid compressible fluids in a bounded domain. It is well known that, as the Mach number goes to zero, the compressible flows approximate the solution of the equations of motion of inviscid, incompressible fluids. Here we discuss, for the boundary case, the different kinds of convergence under various assumptions on the data, in particular the weak convergence in the case of uniformly bounded initial data and the strong convergence in the norm of the data space.

On the Singular Incompressible Limit of Inviscid Compressible Fluids

We consider the Euler equations of barotropic inviscid compressible fluids in a bounded domain. It is well known that, as the Mach number goes to zero, the compressible flows approximate the solution of the equations of motion of inviscid, incompressible fluids. In this paper we discuss, for the boundary case, the different kinds of convergence under various assumptions on the data, in particular the weak convergence in the case of uniformly bounded initial data and the strong convergence in the norm of the data space.

Notes on the Unconditional Convergence of Multidimensional Functional Series

Abstract The convergence of a multidimensional functional series essentially depends on the way its partial sums are formed. Different ways of definition of partial sums lead to different kinds of convergence. In the paper relations between various kinds of unconditional almost everywhere convergence of multidimensional functional series are studied.

On convergences of probability measures in different Prohorov-type metrics

In the article “Convergence of conditional expectations given the random variables of a Skorohod representation” we proved a theorem that gives sufficient conditions for weakly converging distributions to have a certain convergence of conditional expectations. The aim is to find additional conditions to give implications between the cited assumptions and a kind of convergence in Prohorov metric.

On the convergence and representation of random fuzzy number integrals

In this paper, a new kind of convergence, w-s convergence is introduced for certain type of fuzzy numbers taking values in separable reflexive Banach spaces. The convergence theorem of random fuzzy number integrals in the w-s sense is given and a condition, under which a fuzzy number function can be represented by random fuzzy number integrals, is obtained.

Asymptotic convergence of genetic algorithms

We study a markovian evolutionary process which encompasses the classical simple genetic algorithm. This process is obtained by randomly perturbing a very simple selection scheme. Using the Freidlin-Wentzell theory, we carry out a precise study of the asymptotic dynamics of the process as the perturbations disappear. We show how a delicate interaction between the perturbations and the selection pressure may force the convergence toward the global maxima of the fitness function. We put forward the existence of a critical population size, above which this kind of convergence can be achieved. We compute upper bounds of this critical population size for several examples. We derive several conditions to ensure convergence in the homogeneous case and these provide the first mathematically well-founded convergence results for genetic algorithms.

Asymptotic convergence of genetic algorithms

We study a markovian evolutionary process which encompasses the classical simple genetic algorithm. This process is obtained by randomly perturbing a very simple selection scheme. Using the Freidlin-Wentzell theory, we carry out a precise study of the asymptotic dynamics of the process as the perturbations disappear. We show how a delicate interaction between the perturbations and the selection pressure may force the convergence toward the global maxima of the fitness function. We put forward the existence of a critical population size, above which this kind of convergence can be achieved. We compute upper bounds of this critical population size for several examples. We derive several conditions to ensure convergence in the homogeneous case and these provide the first mathematically well-founded convergence results for genetic algorithms.

ON SEQUENCES OF MONOTONE FUNCTIONS

Several kinds of convergence (including pointwise, monotone, a.c., uniform, $\ldots$) in the family of monotone functions are investigated

Ergodic problem for the Hamilton-Jacobi-Bellman equation. I. Existence of the ergodic attractor

The problem of the convergence of the terms λuλ(x), ≠ u(x,T) in the Hamilton-Jacobi-Bellman equations (HJBs) as λ tends to +0, T goes to +∞, to the unique number is called the ergodic problem of the HJBs. We show in this paper what kind of qualitative properties exist behind this kind of convergence. The existence of the ergodic attractor is shown in Theorems 1 and 2. Our solutions of HJBs satisfy the equations in the viscosity solutions sense.

Convergence on closed convex sets in a locally convex space

The purpose of this paper is to compare several kinds of convergences on the space C(X) of nonempty closed convex subsets of a locally convex space X. First we verify that the AW-convergence on C(X) is weaker than the metric Attouch-Wets convergence on C(X) of a metrizable locally convex space X. Moreover, we show that X is normable if and only if the two convergences on C(X × R) are equivalent. Secondly we define two convergences on C(X) analogous to the corresponding ones in a normed linear space, and investigate some basic properties of these convergences and compare them.

Convergence of retention for small solutes in reversed-phase liquid chromatography

It is shown theoretically that when the concentration of organic solvent in the mobile phase increases, or solute size decreases, log k′ values of small solutes in reversed-phase liquid chromatography (RPLC) will tend to have a minimum value called the convergence point. A theoretical model for evaluating the convergent coordinates of small solutes is presented by using a stoichiometric displacement model for retention (SMDR). The physical meaning of the coordinates of each kind of convergence are also elucidated. The convergence points have either two-dimensional coordinates with a common ordinate (the logarithm of the phase ratio of the column, log φ) or threedimensional corrdinates with two common axes: — log φ and the logarithm of the molar concentration of the pure displacing agent in mobile phase, log aD. The other axis relates to the nature of the solutes, such as carbon number of a homolog, van der Waal's surface area, hydrophobic fragment constant etc. for the latter and those and/or concentration axis for the former. The model was tested with published data and found to give a good fit.

Convergent evolution: the need to be explicit

Convergence as a phenomenon in molecular evolution is an issue that confuses many discussions. Often the problem is that not enough care is taken to state exactly what kind of convergence one has in mind. Functional and mechanistic convergence are both common, and some structural convergence has probably occurred, but a convincing case for genuine sequence convergence has yet to be made.

Optimization algorithm with probabilistic estimation

The authors present a stochastic optimization algorithm based on the idea of the gradient method which incorporates a novel adaptive-precision technique. Unlike recent methods, the proposed algorithm adaptively selects the precision without any need for prior knowledge on the speed of convergence of the generated sequence. The algorithm can avoid increasing the estimation precision unnecessarily, yet it retains its favorable convergence properties. In fact, it tries to maintain a nice balance between the requirements for computational accuracy and those for computational expediency. The authors present two types of convergence results delineating under what assumptions various kinds of convergence can be obtained for the proposed algorithm. They also present a parallel version of the proposed algorithm. >

Convergence properties of descent methods for unconstrained minization

This paper studies the convergence properties of a class of descent methods, proposes the necessary and sufficient conditions under which the general descent methods can have a linearly convergent rate, proves some kind of convergence of the general descent method without assuming that the point sequence {x k ,} is bounded.

Fuzzy equality and convergences for $F$-observables in $F$-quantum spaces

We introduce a fuzzy equality for $F$-observables on an $F$-quantum space which enables us to characterize different kinds of convergences, and to represent them by pointwise functions on an appropriate measurable space.