We propose a test of equality of two covariance matrices based on the maximum standardized difference of scalar covariances of two sample covariance matrices. We derive the tail probability of the asymptotic null distribution of the test statistic by the tube method. However the usual formal tube formula has to be suitably modified, because in this case the index set, around which the tube is f...

The Bogoliubov procedure in quantum field theory is used to describe a relativistic almost ideal Bose gas at zero temperature. Special attention is given to the study of a vortex. The radius of the vortex in the field description is compared to that obtained in the relativistic fluid approximation. The Kelvin waves are studied and, for long wavelengths, the dispersion relation is obtained by an...

Let (M, g) be a compact Riemannian manifold with universal covering M̃ . The simplest asymptotic invariant which can be associated to (M, g) is the exponential growth rate h(g) of volume on the universal covering, called the volume entropy. If B r (p) denotes the geodesic ball of radius r about p ∈ M̃ and volg(B r (p)) describes its volume with respect to the Riemannian metric g lifted to M̃ then ...

We study the mixture of extended and supercoiled DNA that occurs in a twisted DNA molecule under tension. Closed-form asymptotic solutions for the supercoiling radius, extension, and torque of the molecule are obtained in the high-force limit where electrostatic and elastic effects dominate. We demonstrate that experimental data obey the extension and torque scaling laws apparent in our formula...

The cover time of graphs has much relevance to algorithmic applications and has been extensively investigated. Recently, with the advent of ad-hoc and sensor networks, an interesting class of random graphs, namely random geometric graphs, has gained new relevance and its properties have been the subject of much study. A random geometric graph G(n, r) is obtained by placing n points uniformly at...

We determine the increase of the maximum risk over the minimax risk in the case that the optimally robust estimator for the false radius is used. This is done by numerical solution of the implicit equations which determine optimal robustness, for location, scale, and linear regression models, and by evaluation of maximum asymptotic variance and mean square error over fixedsize symmetric contami...

The mean lifetime of a particle diffusing in a cylindrical cavity with a circular absorbing spot on the cavity wall is studied analytically as a function of the spot radius, its location on the wall, the particle initial position, and the cavity shape determined by its length and radius. When the spot radius tends to zero our formulas for the mean lifetime reduce to the result given by the solu...

We study the mean size of wavelet packets in Lp. An exact formula for the mean size is given in terms of the p-norm joint spectral radius. This will be a corollary of an asymptotic formula for the Lp-norms on the subdivision trees. Then the stability and Schauder basis property of wavelet packets in Lp, and the instability of biorthogonal wavelet packets in L2 are analyzed. AMS Subject Classifi...

Let m > 1 be an integer, Bm the set of all unit vectors of R pointing in the direction of a nonzero integer vector of the cube [−1, 1]. Denote by sm the radius of the largest ball contained in the convex hull of Bm. We determine the exact value of sm and obtain the asymptotic equality sm ∼ 2 √ log m . Primary subject classification: 52B11 Secondary subject classification: 52B12 §

We prove a quantitative estimate with power saving error term for the number of points in mapping class group orbit Teichmüller space that lie within metric ball given center and large radius. Estimates same kind are also proved sector bisector counts. These estimates effectivize asymptotic counting results Athreya, Bufetov, Eskin, Mirzakhani.