A Remark on the Uniform Convergence of Some Sequences of Functions

We stress a basic criterion that shows in a simple way how a sequence of real-valued functions can converge uniformly when it is more or less evident that the sequence converges uniformly away from a finite number of points of the closure of its domain. For functions of a real variable, unlike in most classical textbooks our criterion avoids the search of extrema (by differential calculus) of their general term.

On power series statistical convergence and new uniform integrability of double sequences

In the present paper,we mostly focus on P_(p)^(2)-statistical convergence.We will look into the uniform integrability via the power series method and its characterizations for double sequences.Also,the notions of P_(p)^(2)-statistically Cauchy sequence,P_(p)^(2)-statistical boundedness and core for double sequences will be described in addition to these findings.

Lacunary Invariant Statistical Convergence Of Sequences Of Sets With Respect To A Modulus Function

In this paper, we introduce and study the concept of lacunary invariant convergence for sequences of sets with respect to modulus functionfand give some inclusion relations.

About Upper and Lower Strong Fractional Weighted Mean Convergence by Moduli for Triple Sequences

In article, I present a study on upper and lower statistical convergence, and upper and lower strong fractional weighted mean convergence by moduli for triple sequences. One of the generalizations of the discrete operator Cesàro, was weighted mean operators, which are linear operators, too. Given a modulus function f, I established that a triple sequence that is f-upper or lower strong fractional weighted mean convergent, in some supplementary conditions, is also f-lower or upper statistically convergent. The results of this paper adapt the results obtained in [1] and [2] to upper and lower strong fractional weighted mean convergence and to triple sequence concept. Furthermore, new concepts can be applied to the approximation theory, topology, Fourier analysis, analysis interdisciplinary with applications electrical engineering, robotics and other domains.

SOME NEW TYPE OF DIFFERENCE SEQUENCE SPACES DEFINED BY MODULUS FUNCTION AND STATISTICAL CONVERGENCE

In this article we introduce the difference sequence spaces Wo [f, △m], W1 [f, △m],W∞[f ,△m] and S[f, △m], defined by a modulus function f. We obtain a relation between W1 【f, △m] ∩ l∞[f, △m] and S[f, △m]∩ l∞[f, △m] and prove some inclusion results.

COMPLETE CONVERGENCE FOR A CLASS OF MIXING SEQUENCES

Under some conditions on probability, the author obtains some results on the complete convergence for partial sums of not necessary identically distributed p-mixing se- quences, and the complete convergence for partial sums of B-valued martingale differences is also studied. As application the author gives the corresponding results on the complete convergence for randomly indexed partial sums.

Strong Convergence of Partitioning Estimation for Nonparametric Regression Function under Dependence Samples

In this paper, we study the strong consistency for partitioning estimation of regression function under samples that axe φ-mixing sequences with identically distribution.Key words: nonparametric regression function; partitioning estimation; strong convergence;φ-mixing sequences.

Convergence Properties of Generalized Fourier Series on a Parallel Hexagon Domain

A new Rogosinski-type kernel function is constructed using kernel function of partial sums Sn（f; t） of generalized Fourier series on a parallel hexagon domain Ω associating with threedirection partition. We prove that an operator Wn（f; t） with the new kernel function converges uniformly to any continuous function f（t） ∈ Cn（Ω） （the space of all continuous functions with period Ω） on Ω. Moreover, the convergence order of the operator is presented for the smooth approached function.

Convergence Track Based Adaptive Differential Evolution Algorithm(CTbADE)

One of the challenging problems with evolutionary computing algorithms is to maintain the balance between exploration and exploitation capability in order to search global optima.A novel convergence track based adaptive differential evolution(CTbADE)algorithm is presented in this research paper.The crossover rate and mutation probability parameters in a differential evolution algorithm have a significant role in searching global optima.A more diverse population improves the global searching capability and helps to escape from the local optima problem.Tracking the convergence path over time helps enhance the searching speed of a differential evolution algorithm for varying problems.An adaptive powerful parameter-controlled sequences utilized learning period-based memory and following convergence track over time are introduced in this paper.The proposed algorithm will be helpful in maintaining the equilibrium between an algorithm’s exploration and exploitation capability.A comprehensive test suite of standard benchmark problems with different natures,i.e.,unimodal/multimodal and separable/non-separable,was used to test the convergence power of the proposed CTbADE algorithm.Experimental results show the significant performance of the CTbADE algorithm in terms of average fitness,solution quality,and convergence speed when compared with standard differential evolution algorithms and a few other commonly used state-of-the-art algorithms,such as jDE,CoDE,and EPSDE algorithms.This algorithm will prove to be a significant addition to the literature in order to solve real time problems and to optimize computationalmodels with a high number of parameters to adjust during the problem-solving process.

Pringsheim Convergence and the Dirichlet Function

Double sequences have some unexpected properties which derive from the possibility of commuting limit operations. For example, may be defined so that the iterated limits and exist and are equal for all x, and yet the Pringsheim limit does not exist. The sequence is a classic example used to show that the iterated limit of a double sequence of continuous functions may exist, but result in an everywhere discontinuous limit. We explore whether the limit of this sequence in the Pringsheim sense equals the iterated result and derive an interesting property of cosines as a byproduct.

Complete convergence of ρ-mixing sequence

Let {Xn, n≥1} be a sequence of random variables and let Sn=∑1≤i≤nXi,n-=σ（Xi1≤i≤n）,n+=σ（Xi,i≥n）,n≥1.

Weak and Strong Convergence for Fixed Points of Asymptotically Non-expansive Mappings

A few weak and strong convergence theorems of the modified three-step iterative sequence with errors and the modified Ishikawa iterative sequence with errors for asymptotically non-expansive mappings in any non-empty closed convex subsets of uniformly convex Banach spaces are established. The results presented in this paper substantially extend the results due to Chang (2001), Osilike and Aniagbosor (2000), Rhoades (1994) and Schu (1991).

Exponential Convergence in Probability for Empirical Means of Lévy Processes

Let （Xt）t≥0 be a Lévy process taking values in R^d with absolutely continuous marginal distributions. Given a real measurable function f on R^d in Kato＇s class, we show that the empirical mean 1/t ∫ f（Xs）ds converges to a constant z in probability with an exponential rate if and only if f has a uniform mean z. This result improves a classical result of Kahane et al. and generalizes a similar result of L. Wu from the Brownian Motion to general Lévy processes.

On some difference sequence spaces defined by a sequence of Orlicz functions

The idea of difference sequence spaces was introduced in （Klzmaz, 1981） and this concept was generalized in （Et and Colak, 1995）. In this paper we define some difference sequence spaces by a sequence of Orlicz functions and establish some inclusion relations.

The k Nearest Neighbors Estimator of the M-Regression in Functional Statistics

It is well known that the nonparametric estimation of the regression function is highly sensitive to the presence of even a small proportion of outliers in the data.To solve the problem of typical observations when the covariates of the nonparametric component are functional,the robust estimates for the regression parameter and regression operator are introduced.The main propose of the paper is to consider data-driven methods of selecting the number of neighbors in order to make the proposed processes fully automatic.We use thek Nearest Neighbors procedure(kNN)to construct the kernel estimator of the proposed robust model.Under some regularity conditions,we state consistency results for kNN functional estimators,which are uniform in the number of neighbors(UINN).Furthermore,a simulation study and an empirical application to a real data analysis of octane gasoline predictions are carried out to illustrate the higher predictive performances and the usefulness of the kNN approach.

迭代函数方程的同胚解

已有的迭代函数方程解结论主要是关于它的连续性和光滑性.然而在函数方程与动力系统的共轭和线性化理论研究中,解的同胚性质起到重要作用.对这一问题虽然已取得部分结论,但还未有一个完整的描述.该文首先利用构造和逼近的方法考虑一般类型的迭代函数方程的同胚解,然后将得到的结论应用到研究多项式型迭代方程.

NUMERICAL METHOD FOR THE SYSTEM OF REACTION-DIFFUSION EQUATIONS WITH A SMALL PARAMETER

This paper deals with the numerical method for the system of reaction-diffusion equations with a small parameter. It is difficult to solve the problems of this kind numerically because of the boundary layer efect Besed on singular perturbed theory and Greens function, we have established the difference scheme that is suited for the solution to the problems. We introduce an idea of feasitbe equidistant degree a here. And this proves that if a>2 the scheme converges in norm with speed uniformly.

Statistically Convergent Double Sequence Spaces in 2-Normed Spaces Defined by Orlicz Function

The concept of statistical convergence was introduced by Stinhauss [1] in 1951. In this paper, we study con- vergence of double sequence spaces in 2-normed spaces and obtained a criteria for double sequences in 2-normed spaces to be statistically Cauchy sequence in 2-normed spaces.

THE UNIQUENESS AND EXISTENCE OF SOLUTION OF THE CHABACTERISTIC PROBLEM ON THE GENERALIZED KdV EQUATION

The generalized KdV equation is a typical integr-able equation. It is derived studying the dissemination of magnet sound wave in coldplasma￣[2], Ihe isolated wave in transmission line￣[3], and the isolated wave in the bound-ary surface of the divided layer fluid￣[4]. For the characteristic problem of the gene-ralized KdV equation, this paper, based on the Riemann function, designs a suitablestructure, then changes the characteristic problem to an equivalent integral and dif-ferential equation whose corresponding fixed point, the above integral differential equ-ation has a unique regular solution, so the characteristic problem of the generalizedKdV equation has a. unique solution. The iteration solution derived from the integraldifferential equation sequence is uniformly convegent in.

(<i>f</i>, <i>p</i>)-Asymptotically Lacunary Equivalent Sequences with Respect to the Ideal <i>I</i>

In this study, we define (f, p)-Asymptotically Lacunary Equivalent Sequences with respect to the ideal I using a non-trivial ideal , a lacunary sequence , a strictly positive sequence , and a modulus function f, and obtain some revelent connections between these notions.