LIMITS OF SOLUTIONS TO THE GENERALIZED GINZBURG-LANDAU FUNCTIONAL

ABSTRACT

We discuss the convergence issue for the generalized Ginzburg-Landau functional which penalizes the L^p -energy. We prove the strong convergence for non-integer p and obtain the obstruction for strong convergence for integer p. In the presence of obstruction, we show the limiting couple varifold is a (generalized) stationary (n ? p)-varifold.     

On statistical convergence of difference sequences of fractional order and related Korovkin type approximation theorems

Abstract

Prior to investigating on sequence spaces and their convergence, we study the notion of statistical convergence of difference sequences of fractional order α ∈ ?. As generalizations of previous works, this study includes several special cases under different limiting conditions of α, such as the notion of statistical convergence of difference sequences of zeroth and mth (integer) order. In fact, we study certain new results on statistical convergence via the difference operator Δ^α and interpret them to those of previous works. Also, by using the convergence of Δ^α-summable sequences which is stronger than statistical convergence of difference sequences, we apply classical Bernstein operator and a generalized form of Meyer-Konig and Zeller operator to construct an example in support of our result. Also, we study the rates of Δ^α-statistical convergence of positive linear operators.     

Characterizations of orthonormal scale functions: A probabilistic approach

The construction of a multiresolution analysis starts with the specification of a scale function. The Fourier transform of this function is defined by an infinite product. The convergence of this product is usually discussed in the context of L ²(R).Here, we treat the convergence problem by viewing the partial products as probabilities, converging weakly to a probability defined on an appropriate sequence space. We obtain a sufficient condition for this convergence, which is also necessary in the case where the scale function is continuous. These results extend and clarify those of Cohen [2] and Hernández et al. [4]. The method also applies to more general dilation schemes that commute with translations by Z ^d .     

Mean Convergence of Lagrnage Type Interpolation on an Arbitrary System of Nodes

Abstract   Sufficient conditions of convergence and rate of convergence for Lagrange type interpolation in the weighted L ^p norm on an arbitrary system of nodes are given. Supported by the National Natural Sciences Foundation of China (No.19671082)     

Convergence theorems for semigroup-type families of non-self mappings

Motivated by a paper Chidume and Zegeye [Strong convergence theorems for common fixed points of uniformly L-Lipschitzian pseudocontractive semi-groups, Applicable Analysis, 86 (2007), 353–366], we prove several strong convergence theorems for a family (not necessarily a semigroup) ℱ = {T(t): t ∈ G} of nonexpansive or pseudocontractive non-self mappings in a reflexive strictly convex Banach space with a uniformly Gateaux differentiable norm, where G is an unbounded subset of ℝ⁺. Our results extend and improve the corresponding ones byMatsushita and Takahashi [Strong convergence theorems for nonexpansive nonself-mappings without boundary conditions,Nonlinear Analysis, 68 (2008), 412–419],Morales and Jung [Convergence of paths for pseudo-contractive mappings in Banach spaces, Proceedings of American Mathematical Society, 128 (2000), 3411–3419], Song [Iterative approximation to common fixed points of a countable family of nonexpansive mappings, Applicable Analysis, 86 (2007), 1329–1337], Song and Xu [Strong convergence theorems for nonexpansive semigroup in Banach spaces, Journal of Mathematical Analysis and Applications, 338 (2008), 152–161], Wong, Sahu, and Yao [Solving variational inequalities involving nonexpansive type mappings, Nonlinear Analysis, (2007) doi:10.1016/j.na. 2007.11.025] in the context of a non-semigroup family of non-self mappings.      

On preconditioned MHSS iteration methods for complex symmetric linear systems

We propose a preconditioned variant of the modified HSS (MHSS) iteration method for solving a class of complex symmetric systems of linear equations. Under suitable conditions, we prove the convergence of the preconditioned MHSS (PMHSS) iteration method and discuss the spectral properties of the PMHSS-preconditioned matrix. Numerical implementations show that the resulting PMHSS preconditioner leads to fast convergence when it is used to precondition Krylov subspace iteration methods such as GMRES and its restarted variants. In particular, both the stationary PMHSS iteration and PMHSS-preconditioned GMRES show meshsize-independent and parameter-insensitive convergence behavior for the tested numerical examples.     

Using Forward Accumulation for Automatic Differentiation of Implicitly-Defined Functions

This paper deals with the calculation of partial derivatives (w.r.t. the independent variables, x) of a vec of dependent variables y which satisfy a system of nonlinear equations g(u(x), y) = 0 . A number of authors have suggested that the forward accumulation method of automatic differentiation can be applied to a suitable iterative scheme for solving the nonlinear system with a view to giving simultaneous convergence both to the correct value y and also to its Jacobian matrix y _x . It is known, however, that convergence of the derivatives may not occur at the same rate as the convergence of the y values. In this paper we avoid both the difficulty and the potential cost of iterating the gradient part of the calculation to sufficient accuracy. We do this by observing that forward accumulation need only be applied to the functions g after the dependent variables, y, have been computed in standard real arithmetic usin g any appropriate method. This so-called Post-Differentiation (PD) technique is shown, on a number of examples, to have an advantage in terms of both accuracy and speed over approaches where forward accumulation is applied over the entire iterative process. Moreover, the PD technique can be implemented in such a way as to provide a friendly interface for non-specialist users.     

Some difference sequence spaces defined by a sequence of <Emphasis Type="Italic">ϕ</Emphasis> -functions

Abstract  In this paper, we introduce new difference sequence spaces combining with de la Vallee-Poussin mean using by a sequence of modulus functions and ϕ -functions. We also studied connections between statistically convergence related with this space. Keywords Difference sequence, Modulus function, ϕ -function, De la Vallee-Poussin means, Statistical convergence Mathematics Subject Classification (2000) 46A45, 40F05, 46A80     

Weak and Strong Convergence Theorems for Maximal Monotone Operators in a Banach Space

In this paper, we introduce an iterative sequence for finding a solution of a maximal monotone operator in a uniformly convex Banach space. Then we first prove a strong convergence theorem, using the notion of generalized projection. Assuming that the duality mapping is weakly sequentially continuous, we next prove a weak convergence theorem, which extends the previous results of Rockafellar [SIAM J. Control Optim. 14 (1976), 877–898] and Kamimura and Takahashi [J. Approx. Theory 106 (2000), 226–240]. Finally, we apply our convergence theorem to the convex minimization problem and the variational inequality problem.     

Numerical Solutions of Stochastic Differential Delay Equations with Jumps

Abstract

In this article, we investigate the strong convergence of the Euler–Maruyama method and stochastic theta method for stochastic differential delay equations with jumps. Under a global Lipschitz condition, we not only prove the strong convergence, but also obtain the rate of convergence. We show strong convergence under a local Lipschitz condition and a linear growth condition. Moreover, it is the first time that we obtain the rate of the strong convergence under a local Lipschitz condition and a linear growth condition, i.e., if the local Lipschitz constants for balls of radius R are supposed to grow not faster than log R.     

Thin plate spline interpolation on the unit interval

It is known that the unique thin plate spline interpolant to a function f ∈ C ³ IR sampled at the scaled integers h Z converges at an optimal rate of h ³. In this paper we present results from a recent numerical investigation of the case where the function is sampled at equally spaced points on the unit interval. In this setting the known theoretical error bounds predict a drop in the convergence rate from h ³ to h. However, numerical experiments show that the usual rate of convergence is h ^3/2 and that the deterioration occurs near the end points of the interval. We will examine the effect of the boundary on the accuracy of the interpolant and also the effect of the smoothness of the target function. We will show that there exists functions which enjoy an even faster order of convergence of h ^5/2.     

Exact Rate of Convergence of Some Approximation Schemes Associated to SDEs Driven by a Fractional Brownian Motion

In this article, we derive the exact rate of convergence of some approximation schemes associated to scalar stochastic differential equations driven by a fractional Brownian motion with Hurst index H. We consider two cases. If H>1/2, the exact rate of convergence of the Euler scheme is determined. We show that the error of the Euler scheme converges almost surely to a random variable, which in particular depends on the Malliavin derivative of the solution. This result extends those contained in J. Complex. 22(4), 459–474, 2006 and C.R. Acad. Sci. Paris, Ser. I 340(8), 611–614, 2005. When 1/6<H<1/2, the exact rate of convergence of the Crank-Nicholson scheme is determined for a particular equation. Here we show convergence in law of the error to a random variable, which depends on the solution of the equation and an independent Gaussian random variable.     

Jacobi Spectral Methods for Volterra-Urysohn Integral Equations of Second Kind with Weakly Singular Kernels

Abstract

In this article, we discuss Jacobi spectral Galerkin and iterated Jacobi spectral Galerkin methods for Volterra-Urysohn integral equations with weakly singular kernels and obtain the convergence results in both the infinity and weighted L²-norm. We show that the order of convergence in iterated Jacobi spectral Galerkin method improves over Jacobi spectral Galerkin method. We obtain the convergence results in two cases when the exact solution is sufficiently smooth and non-smooth. For finding the improved convergence results, we also discuss Jacobi spectral multi-Galerkin and iterated Jacobi spectral multi-Galerkin method and obtain the convergence results in weighted L²-norm. In fact, we prove that the iterated Jacobi spectral multi-Galerkin method improves over iterated Jacobi spectral Galerkin method. We provide numerical results to verify the theoretical results.     

On a theorem of Stein-Rosenberg type in interval analysis

Summary In classical numerical analysis the asymptotic convergence factor (R ₁-factor) of an iterative processx ^m+1=Ax^m+b coincides with the spectral radius of then×n iteration matrixA. Thus the famous Theorem of Stein and Rosenberg can at least be partly reformulated in terms of asymptotic convergence factor. Forn×n interval matricesA with irreducible upper bound and nonnegative lower bound we compare the asymptotic convergence factor ( _T ) of the total step method in interval analysis with the factor _S of the corresponding single step method. We derive a result similar to that of the Theorem of Stein and Rosenberg. Furthermore we show that _S can be less than the spectral radius of the real single step matrix corresponding to the total step matrix |A| where |A| is the absolute value ofA. This answers an old question in interval analysis.     

Several New Third-Order Iterative Methods for Solving Nonlinear Equations

In this paper, we present some new third-order iterative methods for finding a simple root α of nonlinear scalar equation f(x)=0 in R. A geometric approach based on the circle of curvature is used to construct the new methods. Analysis of convergence shows that the new methods have third-order convergence, that is, the sequence {x _n }₀^∞ generated by each of the presented methods converges to α with the order of convergence three. The efficiency of the methods are tested on several numerical examples. It is observed that our methods can compete with Newton’s method and the classical third-order methods.     

Limiting properties of some measures of information

In this paper we investigate the limiting behaviour of the measures of information due to Csiszár, Rényi and Fisher. Conditions for convergence of measures of information and for convergence of Radon-Nikodym derivatives are obtained. Our results extend the results of Kullback (1959,Information Theory and Statistics, Wiley, New York) and Kirmani (1971,Ann. Inst. Statist. Math.,23, 157–162).     

A New Characterization of the Continuous Functions on a Locale

Within the category W of archimedean lattice-ordered groups with weak order unit, we show that the objects of the form C(L), the set of continuous real-valued functions on a locale L, are precisely those which are divisible and complete with respect to a variant of uniform convergence, here termed indicated uniform convergence. We construct the corresponding completion of a W-object A purely algebraically in terms of Cauchy sequences. This completion can be variously described as c³A, the ``closed under countable composition hull of A,' as C(Y_lA), where Y_lA is the Yosida locale of A, and as the largest essential reflection of A.     

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.     

The theoretical and empirical rate of convergence for geometric branch-and-bound methods

Geometric branch-and-bound solution methods, in particular the big square small square technique and its many generalizations, are popular solution approaches for non-convex global optimization problems. Most of these approaches differ in the lower bounds they use which have been compared empirically in a few studies. The aim of this paper is to introduce a general convergence theory which allows theoretical results about the different bounds used. To this end we introduce the concept of a bounding operation and propose a new definition of the rate of convergence for geometric branch-and-bound methods. We discuss the rate of convergence for some well-known bounding operations as well as for a new general bounding operation with an arbitrary rate of convergence. This comparison is done from a theoretical point of view. The results we present are justified by some numerical experiments using the Weber problem on the plane with some negative weights.     

On Adaptive Combination of Regression Estimators

Consider the problem of choosing between two estimators of the regression function, where one estimator is based on stronger assumptions than the other and thus the rates of convergence are different. We propose a linear combination of the estimators where the weights are estimated by Mallows' C _L . The adaptive estimator retains the optimal rates of convergence and is an extension of Stein-type estimators considered by Li and Hwang (1984, Ann. Statist., 12, 887-897) and related to an estimator in Burman and Chaudhuri (1999, Ann. Inst. Statist. Math. (to appear)).