Constructive equivalence relations on computable probability measures

A central object of study in the field of algorithmic randomness are notions of randomness for sequences, i.e., infinite sequences of zeros and ones. These notions are usually defined with respect to the uniform measure on the set of all sequences, but extend canonically to other computable probability measures. This way each notion of randomness induces an equivalence relation on the computable probability measures where two measures are equivalent if they have the same set of random sequences.In what follows, we study the equivalence relations induced by Martin-Löf randomness, computable randomness, Schnorr randomness and Kurtz randomness, together with the relations of equivalence and consistency from probability theory. We show that all these relations coincide when restricted to the class of computable strongly positive generalized Bernoulli measures. For the case of arbitrary computable measures, we obtain a complete and somewhat surprising picture of the implications between these relations that hold in general.     

Jensen's inequality for multivariate medians

Given a probability measure μ on Borel sigma-field of Rd, and a function f:Rd?R, the main issue of this work is to establish inequalities of the type f(m)?M, where m is a median (or a deepest point in the sense explained in the paper) of μ and M is a median (or an appropriate quantile) of the measure μf=μ○f−1. For the most popular choice of halfspace depth, we prove that the Jensen's inequality holds for the class of quasi-convex and lower semi-continuous functions f. To accomplish the task, we give a sequence of results regarding the “type D depth functions” according to classification in [Y. Zuo, R. Serfling, General notions of statistical depth function, Ann. Statist. 28 (2000) 461-482], and prove several structural properties of medians, deepest points and depth functions. We introduce a notion of a median with respect to a partial order in Rd and we present a version of Jensen's inequality for such medians. Replacing means in classical Jensen's inequality with medians gives rise to applications in the framework of Pitman's estimation.     

On bounds for the mode and median of the generalized hyperbolic and related distributions

Except for certain parameter values, a closed form formula for the mode of the generalized hyperbolic (GH) distribution is not available. In this paper, we exploit results from the literature on modified Bessel functions and their ratios to obtain simple but tight two-sided inequalities for the mode of the GH distribution for general parameter values. As a special case, we deduce tight two-sided inequalities for the mode of the variance-gamma (VG) distribution, and through a similar approach we also obtain tight two-sided inequalities for the mode of the McKay Type I distribution. The analogous problem for the median is more challenging, but we conjecture some monotonicity results for the median of the VG and McKay Type I distributions, from we which we conjecture some tight two-sided inequalities for their medians. Numerical experiments support these conjectures and also lead us to a conjectured tight lower bound for the median of the GH distribution.     

Ultrahigh-gain bulk solid-state stimulated Brillouin scattering phase-conjugation material

We report what is believed to be the first observation of phase conjugation by stimulated Brillouin scattering (SBS) in TeO2 single crystal. The observed very low threshold for phase-conjugate mirror (PCM) formation, high PCM reflectivities in this initial experiment, and commercial availability of material hold promise for a host of practical applications in the near future. The resultant steady-state gain parameter, approximately 100 cm/GW, is to our knowledge the largest ever reported for any SBS material.     

Structure of peptide solutions: A light scattering and numerical study

We investigated the interactions between protein molecules in solution, in particular for low salt concentrations and thus strong electrostatic interactions where a treatment based on the second virial coefficient is not sufficient. Static and dynamic light scattering experiments on solutions containing the peptide human calcitonin (hCT) were combined with calculations based on the Ornstein-Zernike equation with the hypernetted chain (HNC) closure and computer simulations within the primitive electrolyte model. The simulation illustrates the distribution of proteins in solution and the formation of (transient) protein aggregates. It furthermore allows us to predict the physical stability of hCT solutions in dependence of ionic strength, pH and hCT concentration.Received: 26 December 2003, Published online: 25 March 2004PACS: 87.15.Nn Properties of solutions; aggregation and crystallization of macromolecules - 61.20.Qg Structure of associated liquids: electrolytes, molten salts, etc. - 87.14.Ee ProteinsV. Lobaskin: Present address: Max-Planck-Institut für Polymerforschung, 55128 Mainz, Germany     

Comparison of an instrumental and modified Kjeldahl technique for determination of nitrogen in niobium and tantalum alloys

Results obtained for the determination of nitrogen in two tantalum alloys and six niobium alloys by modified Kjeldahl and Leco TC-30 nitrogen—oxygen determinator are compared. In the 5–25 ppm range, for tantalum alloys, the relative standard deviation was 3–9% by the Kjeldahl procedure and 9–11% by the instrumental technique. In the range 30–80 ppm, for niobium alloys, the relative standard deviation was 2–8% by the Kjeldahl procedure and 5–7% by the instrumental technique.