BERNOULLI CONVOLUTIONS ASSOCIATED WITH CERTAIN NON-PISOT NUMBERS

1　IntroductionForany 0 <λ <1 ,letνλ　denotethedistributionof ∞n=0 εnλn wherethecoefficientsεnareeither0or1 ,chosenindependentlywithprobability12 foreach .Itistheinfiniteconvo lutionproductofthedistributions 12 (δ0 +δλn) ,givingrisetotheterm“infiniteBernoulliconvolution”orsimply“Bernoulliconvolution” .TheBernoulliconvolutioncanbeexpressedasaself similarmeasureνλsatisfyingtheequationνλ =12 νλ φ- 10 + 12 νλ φ- 11,( 1 .1 )whereφ0 (x) =λxand φ1(x) =λx + 1 .Thisme…     

An approximation problem of the finite rank operator in Banach spaces

By the method of geometry of Banach spaces, we have proven that a bounded linear operator in Banach space is a compact linear one iff it can be uniformly approximated by a sequence of the finite rank bounded homogeneous operators, which reveals the essence of the counter example given by Enflo.     

Weighted exponential polynomial approximation

A necessary and sufficient condition for completeness of systems of exponentials with a weight in Lp is established and a quantitative relation between the weight and the system of exponential in Lp is obtained by using a generalization of Malliavin's uniqueness theorem about Watson's problem.     

THE INVERSE PROBLEM FOR PART SYMMETRIC MATRICES ON A SUBSPACE

In this paper, the following two problems are considered:Problem Ⅰ. Given S∈E Rn×p,X,B 6 Rn×m, find A ∈ SRs,n such that AX = B, where SR8,n = {A∈ Rn×n|xT(A - AT) = 0, for all x ∈ R(S)}.Problem Ⅱ. Given A* ∈ Rn×n, find A ∈ SE such that ||A-A*|| = minA∈sE||A-A*||, where SE is the solution set of Problem Ⅰ.The necessary and sufficient conditions for the solvability of and the general form of the solutions of problem Ⅰ are given. For problem Ⅱ, the expression for the solution, a numerical algorithm and a numerical example are provided.     

Theoretical and numerical analysis for the quasi-continuum approximation of a material particle model

In many applications materials are modeled by a large number of particles (or atoms) where any one of particles interacts with all others. Near or nearest neighbor interaction is expected to be a good simplification of the full interaction in the engineering community. In this paper we shall analyze the approximate error between the solution of the simplified problem and that of the full-interaction problem so as to answer the question mathematically for a one-dimensional model. A few numerical methods have been designed in the engineering literature for the simplified model. Recently much attention has been paid to a finite-element-like quasicontinuum (QC) method which utilizes a mixed atomistic/continuum approximation model. No numerical analysis has been done yet. In the paper we shall estimate the error of the QC method for this one-dimensional model. Possible ill-posedness of the method and its modification are discussed as well.

     

Finite element approximation of spectral problems with Neumann boundary conditions on curved domains

This paper deals with the finite element approximation of the spectral problem for the Laplace equation with Neumann boundary conditions on a curved nonconvex domain . Convergence and optimal order error estimates are proved for standard piecewise linear continuous elements on a discrete polygonal domain in the framework of the abstract spectral approximation theory.

     

Small Solutions of Quadratic Diophantine Equations

We consider quadratic diophantine equations of the shape for a polynomial Q(X₁, ..., X_s) Z[X₁, ..., X_s] of degree 2.Let H be an upper bound for the absolute values of the coefficientsof Q, and assume that the homogeneous quadratic part of Q isnon-singular. We prove, for all s 3, the existence of a polynomialbound _s(H) with the following property: if equation (1) hasa solution x Z^s at all, then it has one satisfying For s = 3 and s = 4 no polynomial bounds _s(H) were previouslyknown, and for s 5 we have been able to improve existing boundsquite significantly. 2000 Mathematics Subject Classification11D09, 11E20, 11H06, 11P55.     

On certain positive linear operators in polynomial weight spaces

We consider certain modified Szász-Mirakyan operators A _n (f;r) in polynomial weight spaces of functions of one variable and we study approximation properties of these operators. This revised version was published online in June 2006 with corrections to the Cover Date.     

Blind Channel Identification Based on Noisy Observation by Stochastic Approximation Method

A stochastic approximation algorithm for estimating multichannel coefficients is proposed, and the estimate is proved to converge to the true parameters a.s. up-to a constant scaling factor. The estimate is updated after receiving each new observation, so the output data need not be collected in advance. The input signal is allowed to be dependent and the observation is allowed to be corrupted by noise, but no noise statistics are used in the estimation algorithm.     

Duality Theory in Fuzzy Linear Programming Problems with Fuzzy Coefficients

The concept of fuzzy scalar (inner) product that will be used in the fuzzy objective and inequality constraints of the fuzzy primal and dual linear programming problems with fuzzy coefficients is proposed in this paper. We also introduce a solution concept that is essentially similar to the notion of Pareto optimal solution in the multiobjective programming problems by imposing a partial ordering on the set of all fuzzy numbers. We then prove the weak and strong duality theorems for fuzzy linear programming problems with fuzzy coefficients.