<Emphasis Type="Italic">p</Emphasis>-Nuclearity in a New Perspective

In this Letter we try to settle some confused points concerning the use of the notion of p-nuclearity in the mathematical and physical literature, pointing out that the nuclearity index in the physicists’ sense vanishes for any p> 1. Our discussion of these issues suggests a new perspective, in terms of ε-entropy and operator spaces, which might permit connections to be drawn between phase space criteria and quantum energy inequalities.Mathematics Subject Classification (2000): 81T05, 47B10, 47L25.     

Examples of non-conjugated holomorphic vector fields and foliations

We consider germs of holomorphic vector fields near the origin of with a saddle-node singularity, and the induced singular foliations. In a previous article we described the invariants addressing the analytical classification of these vector fields. They split into three parts: a formal, an orbital and a tangential component. For a fixed formal class, the orbital invariant (associated to the foliation) was obtained by Martinet and Ramis; we give it an integral representation. We then derive examples of non-orbitally conjugated foliations by the use of a “first-step” normal form, whose first-significative jet is an invariant. The tangential invariant also admits an integral representation, hence we derive explicit examples of vector fields, inducing the same foliation, that are not mutually conjugated. In addition, we provide a family of normal forms for vector fields orbitally equivalent to the model of Poincaré-Dulac.     

Classifying G-protein coupled receptors with bagging classification tree

G-protein coupled receptors (GPCRs) play a key role in different biological processes, such as regulation of growth, death and metabolism of cells. They are major therapeutic targets of numerous prescribed drugs. However, the ligand specificity of many receptors is unknown and there is little structural information available. Bioinformatics may offer one approach to bridge the gap between sequence data and functional knowledge of a receptor. In this paper, we use a bagging classification tree algorithm to predict the type of the receptor based on its amino acid composition. The prediction is performed for GPCR at the sub-family and sub-sub-family level. In a cross-validation test, we achieved an overall predictive accuracy of 91.1% for GPCR sub-family classification, and 82.4% for sub-sub-family classification. These results demonstrate the applicability of this relative simple method and its potential for improving prediction accuracy.     

Absence of Positive Eigenvalues for the Linearized Elasticity System

In this paper, we prove that the linearized elasticity system has no eigenvalues in two geometric situations: the whole space and a local perturbation of the half-space. We consider the Lamé coefficients and the density varying in an unbounded part of the domain. For the whole space, we use the operations curl and div to reduce our system to a scalar problem and use a limiting absorption principle for the reduced scalar equation given by the partial Fourier transform. For the perturbed half-space, this decompositions being no longer valid, we give an other method based on a pseudo-decomposition using the operations div and curl in the horizontal direction. In contrast to the whole space case, the reduced problems depend strongly on the dual Fourier variable which do not enable us to use same techniques. To study these reduced problems, we use the analytic theory of linear operators.     

On C classifications of hyperbolic vector fields

In this paper we study smooth classification of hyperbolic vector fields based on their linear approximations only and obtain the following. On Rⁿ, n?5, with only two kinds of exceptions, any two hyperbolic vector fields with generic nonlinear parts and where A_i are n×n matrices, are C¹ conjugate to each other if and only if A₁ and A₂ are strictly similar, and they are C¹ orbitally equivalent if and only if A₁ and A₂ are similar.     

On zeros of polynomials orthogonal over a convex domain

We establish a discrepancy theorem for signed measures, with a given positive part, which are supported on an arbitrary convex curve. As a main application, we obtain a result concerning the distribution of zeros of polynomials orthogonal on a convex domain.     

Polynomial inequalities on measurable sets and their applications

We study Pólya-and Remez-type inequalities for univariate and multivariate polynomials and discuss their applications to Nikolskii-type inequalities and upper estimates of trigonometric integrals.     

Unitary representations of superconformal algebra

superconformal algebra is algebra with two Virasoro operators. The Kac determinant is calculated and the complete list of unitary representations is determined. Two types of extensions of algebra are discussed. A new approach to construction of algebras from rational conformal field theories is proposed.     

Level and Center Frequency of the Singer''s Formant

The "singer's formant" is a prominent spectrum envelope peak near 3 kHz, typically found in voiced sounds produced by classical operatic singers. According to previous research, it is mainly a resonatory phenomenon produced by a clustering of formants 3, 4, and 5. Its level relative to the first formant peak varies depending on vowel, vocal loudness, and other factors. Its dependence on vowel formant frequencies is examined. Applying the acoustic theory of voice production, the level difference between the first and third formant is calulated for some standard vowels. The difference between observed and calculated levels is determined for various voices. It is found to vary considerably more between vowels sung by professional singers than by untrained voices. The center frequency of the singer's formant as determined from long-term spectrum analysis of commercial recordings is found to increase slightly with the pitch range of the voice classification.