Characterizations of generalized Hermite and sieved ultraspherical polynomials

A new characterization of the generalized Hermite polyno-
mials and of the orthogonal polynomials with respect to the measure
is derived which is based on a ``reversing property" of the coefficients in the corresponding recurrence formulas and does not use the representation in terms of Laguerre and Jacobi polynomials. A similar characterization can be obtained for a generalization of the sieved ultraspherical polynomials of the first and second kind. These results are applied in order to determine the asymptotic limit distribution for the zeros when the degree and the parameters tend to infinity with the same order.

     

Separation of r-hirudin from similar substances by capillary electrophoresis

The thrombin inhibitor r-hirudin is a peptide of 65 amino acids and with a pI of 4.4. Capillary electrophoresis (CE) was used to separate r-hirudin from seven closely related substances which may be found as by-products or from degradation. Possibly two of these substances differ only in an isoaspartyl instead of an aspartyl binding. A baseline separation was possible with an acetate buffer (pH 4.4, 60 mM) containing 0.3% (m/m) PEG 20 000 and 0.1 mM Zn²⁺. The possibilities to prevent wall adsorption are discussed.     

Testing strict monotonicity in nonparametric regression

A new test for strict monotonicity of the regression function is proposed which is based on a composition of an estimate of the inverse of the regression function with a common regression estimate. This composition is equal to the identity if and only if the “true” regression function is strictly monotone, and a test based on an L ²-distance is investigated. The asymptotic normality of the corresponding test statistic is established under the null hypothesis of strict monotonicity.      

Identifying shifts between two regression curves

Annals of the Institute of Statistical Mathematics - This article studies the problem whether two convex (concave) regression functions modelling the relation between a response and covariate in...     

Precise quantitative results by capillary electrophoresis (CE): instrumental aspects

Summary The precision of different CE-instruments was tested by a standard procedure. These results and other interesting features are commented and presented in a table. The high technical standard of these instruments is demonstrated by obtainable day-to-day precisions of below 1%. Thermostating is very important to provide a reproducible dosage.     

Uniform approximation of eigenvalues in Laguerre and Hermite -ensembles by roots of orthogonal polynomials

We derive strong uniform approximations for the eigenvalues in general Laguerre and Hermite -ensembles by showing that the maximal discrepancy between the suitably scaled eigenvalues and roots of orthogonal polynomials converges almost surely to zero when the dimension converges to infinity. We also provide estimates of the rate of convergence. In the special case of a normalized real Wishart matrix , where denotes the dimension and the degrees of freedom, the rate is , if with , and the rate is , if with . In the latter case we also show the a.s. convergence of the largest eigenvalue of to the corresponding quantile of the Marcenko-Pastur law.

     

First return probabilities of birth and death chains and associated orthogonal polynomials

For a birth and death chain on the nonnegative integers, integral representations for first return probabilities are derived. While the integral representations for ordinary transition probabilities given by Karlin and McGregor (1959) involve a system of random walk polynomials and the corresponding measure of orthogonality, the formulas for the first return probabilities are based on the corresponding systems of associated orthogonal polynomials. Moreover, while the moments of the measure corresponding to the random walk polynomials give the ordinary return probabilities to the origin, the moments of the measure corresponding to the associated polynomials give the first return probabilities to the origin.

As a by-product we obtain a new characterization in terms of canonical moments for the measure of orthogonality corresponding to the first associated orthogonal polynomials. The results are illustrated by several examples.

     

Rapid crystal growth without inherent supersaturation induced by nanoscale fluid flows?

Crystal growth is a process that only takes place under non‐equilibrium conditions and a necessary prerequisite is that the crystal is exposed to a phase that is supersaturated in the material the crystal is composed of, be it a solution, a vapour or a supercooled melt. In industrial mass crystallization the growth rate for a population of crystals (in suspension growth processes [1]) rarely exceeds mean linear velocities of 10^‐7 ms^‐1. Here we present a mass crystallization process which is accompanied by rapid crystal growth several orders of magnitude faster and into a region of solution that is without inherent supersaturation. The material investigated is a solid hydrate that exhibits a solution mediated phase transition to its anhydrous form in the presence of methanol [2]. The phase transition is initiated simply by placing an amount of hydrate crystals into the solvent and is characterized by the rapid emergence of needle‐shaped crystals. The needles emanate from the crystal faces of the hydrate crystals and grow into the solution, which is nominally free of the substance to be crystallized. The high growth rate of the crystals, which of the order of up to 10^‐4 ms^‐1 is surprising. Although rapid needle growth has been observed before [3‐9], to date a satisfactory explanation for needles growing under the abovementioned conditions is still outstanding. Based upon the topology of the crystals we propose a tentative mechanism for this phenomenon capable of explaining the unusually rapid growth and highlight those questions that need addressing in order to verify this mechanism. X‐ray powder diffraction is used to characterize the crystal phase of the needles; confocal fluorescence microscopy reveals that the needles are hollow. The width of these needles is between 0.5 and 5 μm, their length appears to be limited only by the amount of hydrate available for their formation. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Kolmogorov–Smirnov‐type testing for the partial homogeneity of Markov processes—with application to credit risk

In banking, the default behaviour of the counterpart is not only of interest for the pricing of transactions under credit risk but also for the assessment of a portfolio credit risk. We develop a test against the hypothesis that default intensities are chronologically constant within a group of similar counterparts, e.g. a rating class. The Kolmogorov–Smirnov‐type test builds up on the asymptotic normality of counting processes in event history analysis. The right censoring accommodates for Markov processes with more than one no‐absorbing state. A simulation study and two examples of rating systems demonstrate that partial homogeneity can be assumed, however occasionally, certain migrations must be modelled and estimated inhomogeneously. Copyright © 2007 John Wiley & Sons, Ltd.     

Some new asymptotic properties for the zeros of Jacobi,Laguerre, and Hermite polynomials

For the generalized Jacobi, Laguerre, and Hermite polynomials $P_n^{\left( {\alpha _n ,\beta _n } \right)} \left( x \right),L_n^{\left( {\alpha _n } \right)} \left( x \right),H_n^{\left( {\gamma _n } \right)} \left( x \right)$ , the limit distributions of the zeros are found, when the sequences α _n or β _n tend to infinity with a larger order thann. The derivation uses special properties of the sequences in the corresponding recurrence formulas. The results are used to give second-order approximations for the largest and smallest zero which improve (and generalize) the limit statements in a paper by Moak, Saff, and Varga [11].