Convergence Phenomenon with Fourier Series of tg(x2)and Alike

The Fourier series of the 2π-periodic functions tg(x2)and 1sin(x)and some of their relatives (first of their integrals) are investigated and illustrated with respect to their convergence. These functions are Generalized functions and the convergence is weak convergence in the sense of the convergence of continuous linear functionals defining them. The figures show that the approximations of the Fourier series possess oscillations around the function which they represent in a broad band embedding them. This is some analogue to the Gibbs phenomenon. A modification of Fourier series by expansion in powers cosn(x)for the symmetric part of functions and sin(x)cosn−1(x)for the antisymmetric part (analogous to Taylor series) is discussed and illustrated by examples. The Fourier series and their convergence behavior are illustrated also for some 2π-periodic delta-function-like sequences connected with the Poisson theorem showing non-vanishing oscillations around the singularities similar to the Gibbs phenomenon in the neighborhood of discontinuities of functions. .

WEAK CONVERGENCE THEOREMS FOR GENERAL EQUILIBRIUM PROBLEMS AND VARIATIONAL INEQUALITY PROBLEMS AND FIXED POINT PROBLEMS IN BANACH SPACES

In this paper, we introduce two new iterative algorithms for finding a common element of the set of solutions of a general equilibrium problem and the set of solutions of the variational inequality for an inverse-strongly monotone operator and the set of common fixed points of two infinite families of relatively nonexpansive mappings or the set of common fixed points of an infinite family of relatively quasi-nonexpansive mappings in Banach spaces. Then we study the weak convergence of the two iterative sequences. Our results improve and extend the results announced by many others.

UPPER SEMI-CONTINUITY OF RANDOM ATTRACTORS FOR A NON- AUTONOMOUS DYNAMICAL SYSTEM WITH A WEAK CONVERGENCE CONDITION

In this paper,we develop the criterion on the upper semi-continuity of random attractors by a weak-to-weak limit replacing the usual norm-to-norm limit.As an application,we obtain the convergence of random attractors for non-autonomous stochastic reactiondiffusion equations on unbounded domains,when the density of stochastic noises approaches zero.The weak convergence of solutions is proved by means of Alaoglu weak compactness theorem.A differentiability condition on nonlinearity is omitted,which implies that the existence conditions for random attractors are sufficient to ensure their upper semi-continuity.These results greatly strengthen the upper semi-continuity notion that has been developed in the literature.

A Weak Convergence Theorem for A Finite Family of Asymptotically Nonexpansive Mappings

The purpose of this paper is to prove a new weak convergence theorem for a finite family of asymptotically nonexpansive mappings in uniformly convex Banach space.

A Weak Convergence to Rosenblatt Process

A special approximation to Rosenblatt process with the finite-time interval representation was obtained. The construction of approximation family was based on the Poisson process. The proof to the approximation was divided into two aspects. Firstly, the approximation family was tight using the methods given by Billingsley; secondly, the finite-dimension distributions of approximation family converged weakly to the Rosenblatt process by proving the convergence of the corresponding characteristic functions.

SOME LIMIT THEOREMS FOR WEIGHTED SUMS OF ARRAYS OF NOD RANDOM VARIABLES

In this paper the authors study the complete, weak and almost sure convergence for weighted sums of NOD random variables and obtain some new limit theorems for weighted sums of NOD random variables, which extend the corresponding theorems of Stout [1], Thrum [2] and Hu et al. [3].

CONTINUOUS TIME MIXED STATE BRANCHING PROCESSES AND STOCHASTIC EQUATIONS

A continuous time and mixed state branching process is constructed by a scaling limit theorem of two-type Galton-Watson processes.The process can also be obtained by the pathwise unique solution to a stochastic equation system.From the stochastic equation system we derive the distribution of local jumps and give the exponential ergodicity in Wasserstein-type distances of the transition semigroup.Meanwhile,we study immigration structures associated with the process and prove the existence of the stationary distribution of the process with immigration.

GLOBAL SOLUTION TO 1D MODEL OF A COMPRESSIBLE VISCOUS MICROPOLAR HEAT-CONDUCTING FLUID WITH A FREE BOUNDARY

In this paper we consider the nonstationary 1D flow of the compressible viscous and heat-conducting micropolar fluid, assuming that it is in the thermodynamically sense perfect and polytropic. The fluid is between a static solid wall and a free boundary connected to a vacuum state. We take the homogeneous boundary conditions for velocity, microrotation and heat flux on the solid border and that the normal stress, heat flux and microrotation are equal to zero on the free boundary. The proof of the global existence of the solution is based on a limit procedure. We define the finite difference approximate equations system and construct the sequence of approximate solutions that converges to the solution of our problem globally in time.

STABLE SUB-GAUSSIAN MODELS CONSTRUCTED BY POISSON PROCESSES

In this paper, we first prove that one-parameter standard α-stable sub-Gaussian processes can be approximated by processes constructed by integrals based on the Poisson process with random intensity. Then we extend this result to the two-parameter processes. At last, we consider the approximation of the subordinated fractional Brownian motion.

AN ANALYSIS ABOUT BEHAVIOR OF EVOLUTIONARY ALGORITHMS:A KIND OF THEORETICAL DESCRIPTION BASED ON GLOBAL RANDOM SEARCH METHODS

Evolutionary computation is a kind of adaptive non--numerical computation method which is designed tosimulate evolution of nature. In this paper, evolutionary algorithm behavior is described in terms of theconstruction and evolution of the sampling distributions over the space of candidate solutions. Iterativeconstruction of the sampling distributions is based on the idea of the global random search of generationalmethods. Under this frame, propontional selection is characterized as a gobal search operator, and recombination is characerized as the search process that exploits similarities. It is shown-that by properly constraining the search breadth of recombination operators, weak convergence of evolutionary algorithms to aglobal optimum can be ensured.

Freidlin-Wentzell’s Large Deviations for Stochastic Evolution Equations with Poisson Jumps

We establish a Freidlin-Wentzell’s large deviation principle for general stochastic evolution equations with Poisson jumps and small multiplicative noises by using weak convergence method.

Discretization error of irregular sampling approximations of stochastic integrals

This paper studies the limit distributions for discretization error of irregular sam- pling approximations of stochastic integral. The irregular sampling approximation was first presented in Hayashi et al.[3], which was more general than the sampling approximation in Lindberg and Rootzen [10]. As applications, we derive the asymptotic distribution of hedging error and the Euler scheme of stochastic differential equation respectively.

APPROXIMATING SOLUTION OF 0 ∈T(x) FOR AN H-MONOTONE OPERATOR IN HILBERT SPACES

The purpose of this paper is to study the solution of 0∈ T（x） for an H- monotone operator introduced in [Fang and Huang, Appl. Math. Comput. 145（2003）795- 803] in Hilbert spaces, which is the first proposal of it＇s kind. Some strong and weak convergence results are presented and the relations between maximal monotone operators and H-monotone operators are analyzed. Simultaneously, we apply these results to the minimization problem for T = δf and provide some numerical examples to support the theoretical findings.

SOME LARGE SAMPLE PROPERTIES OF AN ESTIMATOR OF THE HAZARD FUNCTION FROM RANDOMLY CENSORED DATA

In this paper, A nonparametric hazard estimator is introduced. Weak convergence and strong uniformly consistency of the proposed estimator lambda(n)(t) are investigated on a bounded interval, respectively. An asymptotic representation of lambda(n)(t) is also given, and the asymptotic representation is used to prove asymptotic normality of the hazard estimator.

DOUBLE INERTIAL PROXIMAL GRADIENT ALGORITHMS FOR CONVEX OPTIMIZATION PROBLEMS AND APPLICATIONS

In this paper, we propose double inertial forward-backward algorithms for solving unconstrained minimization problems and projected double inertial forward-backward algorithms for solving constrained minimization problems. We then prove convergence theorems under mild conditions. Finally, we provide numerical experiments on image restoration problem and image inpainting problem. The numerical results show that the proposed algorithms have more efficient than known algorithms introduced in the literature.

NOTES ON ERDOS' CONJECTURE

Let Xn,n ≥ 1, be a sequence of independent random variables satisfying P(Xn = 0) = 1 - P(Xn = an) = 1 - 1/Pn, where an,n ≥ 1, is a sequence of real numbers, and Pn is the nth prime,set FN(x) = P (N Xn ≤ x). The authors investigate a conjecture of Erdos in probabilistic number theory and show that in order for the sequence FN to be weakly convergent, it is both sufficient and necessary that there exist three numbers X0 and X1 < X2 such that limsup(FN(X2) - FN(X1)) > 0 holds, and Lo = N→ ∞ lim FN(X0) exists. Moreover, the authors point out that they can also obtain the same result in the weakened case of lim inf P(Xn = 0) > 0.

ON A CLASS OF KTHE SEQUENCE SPACES WITH NORMAL STRUCTURE

In this note, we investigate the generalized modulus of convexity δ（λ） and the generalized modulus smoothness p（λ）. We obtain some estimates of these moduli for X=lp. We obtain inequalities between WCS coefficient of a KSthe sequence space X and δX（λ）. We show that, for a wide class of class the sequence spaces X, if for some ε∈（0, 9/10] holds δx（e） 〉 1/3（1- √3/2）ε, then X has normal structure.

A CLASS OF REACTION-DIFFUSION EQUATIONS WITH HYSTERESIS DIFFERENTIAL OPERATOR

In this paper, the classical and weak derivatives with respect to spatial variable of a class of hysteresis functional are discussed. Some conclusions about solutions of a class of reaction-diffusion equations with hysteresis differential operator are given.

A LARGE DEVIATION PRINCIPLE FOR THE STOCHASTIC GENERALIZED GINZBURG-LANDAU EQUATION DRIVEN BY JUMP NOISE

In this paper,we establish a large deviation principle for the stochastic generalized Ginzburg-Landau equation driven by jump noise.The main difficulties come from the highly non-linear coefficient and the jump noise.Here,we adopt a new sufficient condition for the weak convergence criterion of the large deviation principle,which was initially proposed by Matoussi,Sabbagh and Zhang(2021).

On discrete time hedging errors in a fractional Black-Scholes model

In this paper we investigate asymptotic behavior of error of a discrete time hedging strategy in a fractional Black-Scholes model in the sense of Wick-ItS-Skorohod integration. The rate of convergence of the hedging error due to discrete-time trading when the true strategy is known for the trader, is investigated. The result provides new statistical tools to study and detect the effect of the long-memory and the Hurst parameter for the error of discrete time hedging.