Necessary and sufficient conditions for mean convergence of orthogonal expansions for Freud weights

A classic 1970 paper of B. Muckenhoupt established necessary and sufficient conditions for weightedL _p convergence of Hermite series, that is, orthogonal expansions corresponding to the Hermite weight. We generalize these to orthogonal expansions for a class of Freud weightsW ²:=e ^–2Q , by first proving a bound for the difference of the orthonormal polynomials of degreen+1 andn–1 of the weightW ². Our identical necessary and sufficient conditions close a slight gap in Muckenhoupt's conditions for the Hermite weight at least forp>1. Moreover, our necessary conditions apply whenQ(x)=|x|, >1 while our sufficient conditions apply at least for =2,4,6,....Communicated by Vilmos Totik.     

A counterexample to subexponential growth of orthogonal polynomials

We construct an example of polynomialsp _n orthonormal with respect to a measure such that the sequencep _n (x) has an exponential lower bound for a pointx in the support of .Communicated by Paul Nevai.     

Covering with blocks in the non-symmetric case

Considering an infinite string of i.i.d. random letters drawn from a finite alphabet we define the cover timeW _n as the number of random letters needed until each pattern of lenghtn appears at least once as a substring. Sharp weak and a.s. limit results onW _n are known in the symmetric case, i.e., when the random letters are uniformly distributed over the alphabet. In this paper we determine the limit distribution ofW _n in the nonsymmetric case asn. Generalizations in terms of point processes are also proved.Dedicated to Endre Csáki on his 60th birthday.     

On Lipschitz property of solutions of functional equations

In this work it is proved that under certain conditions the continuous vector-valued solutionsf of the functional equation

Lehmer pairs of zeros,the de Bruijn-Newman constant Λ, and the Riemann Hypothesis

We give here a rigorous formulation for a pair of consecutive simple positive zeros of the functionH ₀ (which is closely related to the Riemann -function) to be a Lehmer pair of zeros ofH ₀. With this formulation, we establish that each such pair of zeros gives a lower bound for the de Bruijn-Newman constant (where the Riemann Hypothesis is equivalent to the assertion that 0). We also numerically obtain the following new lower bound for :

Convex polynomial and spline approximation in C[−1, 1]

We prove that a convex functionf C[–1, 1] can be approximated by convex polynomialsp _n of degreen at the rate of ₃(f, 1/n). We show this by proving that the error in approximatingf by C² convex cubic splines withn knots is bounded by ₃(f, 1/n) and that such a spline approximant has anL third derivative which is bounded by n³₃(f, 1/n). Also we prove that iff C²[–1, 1], then it is approximable at the rate ofn ^–2 (f, 1/n) and the two estimates yield the desired result.Communicated by Ronald A. DeVore.     

Orientation-reversing involutions on handlebodies

The observation that the quotient orbifold of an orientation- reversing involution on a 3-dimensional handlebody has the structure of a compression body leads to a strong classification theorem, and general structure theorems. The structure theorems decompose the action along invariant discs into actions on handlebodies which preserve the -fibers of some -bundle structure. As applications, various results of R. Nelson are proved without restrictive hypotheses.

     

HRGC of wine free amino acids as a tool for elementary wine characterization

In this paper we show that wine free amino acids derivatized as isopropyl N-heptafluorobutyryl esters can be used as a tool for wine characterization. Elementary wines of eight Vitis vinifera varieties were studied during a seven year period. The characteristic wine amino acids for each variety were assessed by means of pattern recognition techniques. A “star symbol plot” was used for graphic representation.     

EHMO和模式识别法研究芬太尼衍生物的构效关系

自Hansch等创立QSAR法以来,药物构效关系的研究已获得较大的进展。本文将模式识别法用于芬太尼衍生物构效关系的研究,将该类药物的分子结构看作与生物活性具有对应关系的表现形式——模式,以药物分子结构中有关取代基的多个量子化学参数为数量化的模式向量成分。将已知生物活性的药物作为模式识别训练点,则所得模式识别分类图反映了该类药物的生物活性与其量子化学结构特征参数间的统计学意义的关系。它既可用来探寻高效药物的结构参数,又能预测新设计药物生物活性的等级或类别。     

Electrosorption of sodium cellulose sulfates with different substitution patterns

Various types of sodium cellulose sulfate (SCS), dissolved in a 1M LiCl solution, were investigated by alternating current (AC)polarography. The SCS samples differed in the degree of substitution (DS), thedistribution of substituents within the anhydroglucose unit (AGU), and alongthechain, due to the method of synthesis. The goal was to study theelectrosorptionbehavior, characterized by the shape of the desorption wave in the polarogram,as a function of the chemical structure of the SCS samples. The shape of thedesorption wave reflects domains of different substitution. A superimpositionofparameters like DS and pattern of substitution on the electrosorption behaviorwas observed. The AC polarography method described can be used as a tool todistinguish between an even or uneven distribution of substituents along thecellulose chain.