Many-particle problem with logatithmic potentials and its application to quark bound states

Theoretical and Mathematical Physics -     

Estimating densities,quantiles, quantile densities and density quantiles

To estimate the quantile density function (the derivative of the quantile function) by kernel means, there are two alternative approaches. One is the derivative of the kernel quantile estimator, the other is essentially the reciprocal of the kernel density estimator. We give ways in which the former method has certain advantages over the latter. Various closely related smoothing issues are also discussed.     

On weighted densities

The continuity of densities given by the weight functions n ^α , α ∈ [−1, ∞[, with respect to the parameter α is investigated. This work is supported by MIUR Italy, Program Barrande n. 2003-009-2, MSM6198898701 and GA ČR no. 201/04/0381.     

Deconvolving compactly supported densities

This paper addresses the statistical problem of density deconvolution under the condition that the density to be estimated has compact support. We introduce a new estimation procedure, which establishes faster rates of convergence for smooth densities as compared to the optimal rates for smooth densities with unbounded support. This framework also allows us to relax the usual condition of known error density with non-vanishing Fourier transform, so that a nonparametric class of densities is valid; therefore, even the shape of the noise density need not be assumed. These results can also be generalized for fast decaying densities with unbounded support. We prove optimality of the rates in the underlying experiment and study the practical performance of our estimator by numerical simulations.      

Edge-force densities and second-order powers

By means of balanced virtual powers, an axiomatic approach is developed, in the spirit of Noll, to second-gradient continua. The measure-theoretical formulation allows a considerable simplification since the existence of an edge stress density is regarded as a special case of a surface stress which is a singular measure with respect to the area. To prove our results, we introduce a particular class of subbodies, namely the sets with curvature measure.     

Estimating copula densities through wavelets

Wavelet analysis is used to construct a rank-based estimator of a copula density. The procedure, which can be easily implemented with ready-to-use wavelet packages, is based on an algorithm that handles boundary effects automatically. The resulting estimator provides a non-parametric benchmark for the selection of a parametric copula family. From a theoretical point of view, the estimation procedure is shown to be optimal in the minimax sense on a large functional class of regular copula densities. The approach is illustrated with actuarial and financial data.     

Spectral densities describing off-white noises

For the white noise, the spectral density is constant, and the past (restriction to (−∞,0)) is independent from the future (restriction to (0,+∞)). If the spectral density is not too far from being constant, then dependence between the past and the future can be eliminated by an equivalent measure change. A necessary and sufficient condition for a spectral density to have such a property (in other words, to describe an off-white noise) is derived here from well-known results.     

Asymptotic densities of Maass newforms

We define the counting function for Maass newforms of Hecke congruence groups and calculate the three main terms of this counting function. We then give necessary and sufficient conditions for this expansion to have the same shape as if it were counting eigenvalues related to cocompact surfaces. We relate the result to classical instances of the Jacquet-Langlands correspondence.     

Geometry of manifolds with densities

We study the geometry of complete Riemannian manifolds endowed with a weighted measure, where the weight function is of quadratic growth. Assuming the associated Bakry–Émery curvature is bounded from below, we derive a new Laplacian comparison theorem and establish various sharp volume upper and lower bounds. We also obtain some splitting type results by analyzing the Busemann functions. In particular, we show that a complete manifold with nonnegative Bakry–Émery curvature must split off a line if it is not connected at infinity and its weighted volume entropy is of maximal value among linear growth weight functions.     

The K-moment problem with densities

Given a compact basic semi-algebraic set ${\mathbf{K}} \subset {\mathbb{R}}^n$ , a rational fraction $f:{\mathbb{R}}^n\to{\mathbb{R}}$ , and a sequence of scalars y = (y _α), we investigate when $y_\alpha =\int_{\mathbf{K}} x^\alpha f\,d\mu$ holds for all $\alpha\in{\mathbb{N}}^n$ , i.e., when y is the moment sequence of some measure fdμ, supported on K. This yields a set of (convex) linear matrix inequalities (LMI). We also use semidefinite programming to develop a converging approximation scheme to evaluate the integral ∫ fdμ when the moments of μ are known and f is a given rational fraction. Numerical expreriments are also provided. We finally provide (again LMI) conditions on the moments of two measures $\nu,\mu$ with support contained in K, to have $d\nu=f d\mu$ for some rational fraction f.     

Category product densities and liftings

In this paper we investigate two main problems. One of them is the question on the existence of category liftings in the product of two topological spaces. We prove, that if X×Y is a Baire space, then, given (strong) category liftings ρ and σ on X and Y, respectively, there exists a (strong) category lifting π on the product space such that π is a product of ρ and σ and satisfies the following section property: