A short proof of the Approximation Conjecture for amenable groups

We give a short proof of the Approximation Conjecture with complex coefficients for amenable groups [G. Elek, The Strong Approximation Conjecture holds for amenable groups, J. Funct. Anal. 239 (1) (2006) 345–355].     

Two new modified Gauss–Seidel methods for linear system with -matrices

In 2002, H. Kotakemori et al. proposed the modified Gauss–Seidel (MGS) method for solving the linear system with the preconditioner [H. Kotakemori, K. Harada, M. Morimoto, H. Niki, A comparison theorem for the iterative method with the preconditioner () J. Comput. Appl. Math. 145 (2002) 373–378]. Since this preconditioner is constructed by only the largest element on each row of the upper triangular part of the coefficient matrix, the preconditioning effect is not observed on the nth row. In the present paper, to deal with this drawback, we propose two new preconditioners. The convergence and comparison theorems of the modified Gauss–Seidel methods with these two preconditioners for solving the linear system are established. The convergence rates of the new proposed preconditioned methods are compared. In addition, numerical experiments are used to show the effectiveness of the new MGS methods.     

On Bernstein–Markov-type inequalities for multivariate polynomials in -norm

Bernstein–Markov-type inequalities provide estimates for the norms of derivatives of algebraic and trigonometric polynomials. They play an important role in Approximation Theory since they are widely used for verifying inverse theorems of approximation. In the past decades these inequalities were extended to the multivariate setting, but the main emphasis so far was on the uniform norm. It is considerably harder to derive Bernstein–Markov-type inequalities in the L_q-norm, and it requires introduction of new methods. In this paper we verify certain Bernstein–Markov-type inequalities in L_q-norm on convex and star-like domains. Special attention is given to the question of how the geometry of the domain affects the corresponding estimates.     

A regularization semismooth Newton method based on the generalized Fischer–Burmeister function for -NCPs

We consider a regularization method for nonlinear complementarity problems with F being a P₀-function which replaces the original problem with a sequence of the regularized complementarity problems. In this paper, this sequence of regularized complementarity problems are solved approximately by applying the generalized Newton method for an equivalent augmented system of equations, constructed by the generalized Fischer–Burmeister (FB) NCP-functions φ_p with p>1. We test the performance of the regularization semismooth Newton method based on the family of NCP-functions through solving all test problems from MCPLIB. Numerical experiments indicate that the method associated with a smaller p, for example p[1.1,2], usually has better numerical performance, and the generalized FB functions φ_p with p[1.1,2) can be used as the substitutions for the FB function φ₂.     

A new exclusion test for finding the global minimum

Exclusion algorithms have been used recently to find all solutions of a system of nonlinear equations or to find the global minimum of a function over a compact domain. These algorithms are based on a minimization condition that can be applied to each cell in the domain. In this paper, we consider Lipschitz functions of order α and give a new minimization condition for the exclusion algorithm. Furthermore, convergence and complexity results are presented for such algorithm.     

The well-posedness and stability of a repairable standby human–machine system

In this paper, the solution of a standby human–machine system is investigated. By using the method of functional analysis, especially, the linear operator theory and the C₀ semigroup theory on Banach space, we prove the well-posedness and the existence of a positive solution of the system. And under some appropriate hypotheses, we study the asymptotic stability of solution of the system.     

The behavior of best -approximations as . A counterexample of convergence

In the space of summable sequences we give an example of a one-dimensional affine subspace C such that the best L_p-approximations of 0 from C fail to converge as p↓1. We thus give an answer to this problem of convergence in infinite measure spaces.     

A completeness result for the simply typed -calculus

In this paper, we define a realizability semantics for the simply typed λμ-calculus. We show that, if a term is typable, then it inhabits the interpretation of its type. This result serves to give characterizations of the computational behavior of some closed typed terms. We also prove a completeness result of our realizability semantics using a particular term model.     

A new proof of convergence of MCMC via the ergodic theorem

A key result underlying the theory of MCMC is that any η-irreducible Markov chain having a transition density with respect to η and possessing a stationary distribution π is automatically positive Harris recurrent. This paper provides a short self-contained proof of this fact using the ergodic theorem in its standard form as the most advanced tool.     

The convergence of three spline methods for scattered data interpolation and fitting

The convergences of three L₁ spline methods for scattered data interpolation and fitting using bivariate spline spaces are studied in this paper. That is, L₁ interpolatory splines, splines of least absolute deviation, and L₁ smoothing splines are shown to converge to the given data function under some conditions and hence, the surfaces from these three methods will resemble the given data values.     

-adic refinable functions and MRA-based wavelets

The main goal of this paper is the development of the MRA theory in . We described a wide class of p-adic refinement equations generating p-adic multiresolution analyses. A method for the construction of p-adic orthogonal wavelet bases within the framework of the MRA theory is suggested. A realization of this method is illustrated by an example which gives a new 3-adic wavelet basis. Another realization leads to the p-adic Haar bases which were known before.     

The Leray–Schauder approach to the degree theory for -perturbations of maximal monotone operators in separable reflexive Banach spaces

The purpose of this paper is to demonstrate the fact that the topological degree theory of Leray and Schauder may be used for the development of the topological degree theory for bounded demicontinuous (S₊)-perturbations f of strongly quasibounded maximal monotone operators T in separable reflexive Banach spaces. Certain basic homotopy properties and the extension of this degree theory to (possibly unbounded) strongly quasibounded perturbations f are shown to hold. This work uses the well known embedding of Browder and Ton, and extends the work of Berkovits who developed this theory for the case T=0. Besides being an interesting mathematical problem, the existence of such a degree theory may, conceivably, become useful in situations where the use of the Leray–Schauder degree (via infinite dimensional compactness) is necessary.     

A quantitative version of Trotter's approximation theorem

A quantitative version, based on modified K-functionals, of the classical Trotter's theorem concerning the approximation of C₀-semigroups is presented. The result is applied to the study of the degree of convergence of the iterated Bernstein operators on the N-dimensional simplex to their limiting semigroup.     

Approximation theorems for localized szsz–Mirakjan operators

In the present paper, we investigate the convergence and the approximation order of the localized Szsz–Mirakjan operators, and obtain some new results to improve the results due to Omey [Note on operators of Szsz–Mirakjan type, J. Approx. Theory 47 (1986) 246–254].     

Characterizations and construction of Poisson/symplectic and symmetric multi-revolution implicit Runge–Kutta methods of high order

This paper deals with the characterizations and construction of Poisson/symplectic and (φ⁻¹)-symmetric implicit high-order multi-revolution Runge–Kutta methods (MRRKMs). The basic tool is a modified W-transformation based on quadrature formulas and orthogonal polynomials. Two sufficient conditions can be obtained under which MRRKMs are Poisson/symplectic or (φ⁻¹)-symmetric. We construct two classes of high order implicit MRRKMs by using these sufficient conditions. Our results can be considered as an extension of related results of the standard Runge–Kutta methods in some references.     

Complexity of the min–max (regret) versions of min cut problems

This paper investigates the complexity of the min–max and min–max regret versions of the min s–t cut and min cut problems. Even if the underlying problems are closely related and both polynomial, the complexities of their min–max and min–max regret versions, for a constant number of scenarios, are quite contrasted since they are respectively strongly NP-hard and polynomial. However, for a non-constant number of scenarios, these versions become strongly NP-hard for both problems. In the interval scenario case, min–max versions are trivially polynomial. Moreover, for min–max regret versions, we obtain the same contrasted results as for a constant number of scenarios: min–max regret min s–t cut is strongly NP-hard whereas min–max regret min cut is polynomial.     

Linear information versus function evaluations for -approximation

We study algorithms for the approximation of functions, the error is measured in an L₂ norm. We consider the worst case setting for a general reproducing kernel Hilbert space of functions. We analyze algorithms that use standard information consisting in n function values and we are interested in the optimal order of convergence. This is the maximal exponent b for which the worst case error of such an algorithm is of order n^-b.Let p be the optimal order of convergence of all algorithms that may use arbitrary linear functionals, in contrast to function values only. So far it was not known whether p>b is possible, i.e., whether the approximation numbers or linear widths can be essentially smaller than the sampling numbers. This is (implicitly) posed as an open problem in the recent paper [F.Y. Kuo, G.W. Wasilowski, H. Woźniakowski, On the power of standard information for multivariate approximation in the worst case setting, J. Approx. Theory, to appear] where the authors prove that implies . Here we prove that the case and b=0 is possible, hence general linear information can be exponentially better than function evaluation. Since the case is quite different, it is still open whether b=p always holds in that case.