On surface integrals related to distributions of random matrices

We construct the first quadratic form and the volume element of the surface consisting of all positive semidefinite m × m matrices of rank r with r distinct positive eigenvalues. We give the density function of the singular gamma distribution.     

On surface area measures of convex bodies

The set L _j of jth-order surface area measures of convex bodies in d-space is well known for j=d–1. A characterization of L _j was obtained by Firey and Berg. The determination of L _j, for j{2, ..., d–2}, is an open problem. Here we show some properties of L _j concerning convexity, closeness, and size. Especially we prove that the difference set L _j–L _j is dense (in the weak topology) in the set of signed Borel measures on the unit sphere which have barycentre 0.     

On surface area integration and the related mapping

This paper concerns the computation of a surface area given by a parametric representation on an arbitrary domain D. The basic approach is to construct an analytic mapping with nonzero Jacobian from a simply connected domain; say a unit square, to D. Then the integration used to determine the area can be easily performed on the unit square. For computational purposes the mapping is estimated by a standard relaxation method and an algorithm is developed to estimate the desired area by a finite sum. With sufficiently large sums the error of this estimation can be made arbitrarily small.     

On the volume and surface area of hyperbolic polyhedra

A truncated tetrahedron is a building block of hyperbolic 3-manifolds with totally geodesic boundary. We study the relation between the volume of a truncated tetrahedron and the area of its faces which form the boundary of manifolds.     

On the expected surface area of the Wiener sausage

For parallel neighborhoods of the paths of the d ‐dimensional Brownian motion, so‐called Wiener sausages, formulae for the expected surface area are given for any dimension d ≥ 2. It is shown by means of geometric arguments that the expected surface area is equal to the first derivative of the mean volume of the Wiener sausage with respect to its radius (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Geominimal surface area

Geometriae Dedicata -     

On methods for generating uniform random points on the surface of a sphere

Some methods for generating random points uniformly distributed on the surface of ann-sphere have been proposed to simulate spherical processes on computer. A standard method is to normalize random points inside of the sphere, see M. Muller [5]. Improved methods were given by J. M. Cook [1] and G. Marsaglia [4] in three and four dimensions, and computational methods in higher dimensions by J. S. Hicks and R. F. Wheeling [3] and M. Sibuya [6]. In this paper we shall offer direct methods for generating uniform random points on the surface of a unitn-sphere, which can be easily combined with Marsaglia's idea for getting more improved methods. Our method in even dimensions was obtained by M. Sibuya [6], but a differential-geometric view-point will make analyses simpler, even in odd dimensions.     

On random partitions

Let S be a finite set with n labeled elements. One of the partitions of S is selected at random each of them has the same probability. Harper determined the expected number of subsets in the random partition. Haigh studied the probability that the random partition has (at least one) subset of a given size. The present paper considers the expected number of subsets in a given size. Average and maximum are also determined. Results are generalized for the case if the number of subsets in the partition is also fixed.     

On the area and perimeter of a random convex hull in a bounded convex set

Suppose K is a compact convex set in ℝ² and X _i , 1≤i≤n, is a random sample of points in the interior of K. Under general assumptions on K and the distribution of the X _i we study the asymptotic properties of certain statistics of the convex hull of the sample. Received: 24 July 1996/Revised version: 24 February 1998     

On the random max-closure for heavy-tailed random variables

In this paper, we study the random max-closure property for not necessarily identically distributed real-valued random variables X ₁ ,X ₂ , . . . , which states that, given distributions \( {F}_{X_1} \) , \( {F}_{X_2} \) , . . . from some class of heavy-tailed distributions, the distribution of the random maximum X( _η) := max{0,X ₁ , . . . , X _η } or random maximum S _(η) := max{0, S ₁ , . . . , S _η } belongs to the same class of heavy-tailed distributions. Here, S _n = X ₁ + · · · + X _n , n ≥ 1, and η is a counting random variable, independent of {X ₁ ,X ₂ , . . . }. We provide the conditions for the random max-closure property in the case of classes Open image in new window and Open image in new window .     

On distributions of random variables chosen at random

The existence of typical distributions for random variables chosen at random from a finite-dimensional random variable vector space of high dimension is established. Possible typical distributions are described, and conditions for the typical distribution to be standard Gaussian are given. Bibliography: 2 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 216, 1994, pp. 153–160. Translated by the author.     

On random cartesian trees

Cartesian trees are binary search trees in which the nodes exhibit the heap property according to a second (priority) key. If the search key and the priority key are independent, and the trees is built based on n independent copies, Cartesian trees basically behave like ordinary random binary search trees. In this article, we analyze the expected behavior when the keys are dependent: in most cases, the expected search, insertion, and deletion times are Φ(√n). We indicate how these results can be used in the analysis of divide-and-conguer algorithms for maximal vectors and convex hulls. Finally, we look at distributions for which the expected time per operation grows like n^a for a ?[1/2, 1]. © 1994 John Wiley & Sons, Inc.     

On random least-square analysis

Dispersion in the output data of a system can be analyzed as either noise fluctuations about a deterministic model or as the noise with added fluctuations due to randomness in the model itself. This latter interpretation finds applications in the identification of inherently random systems which provide rational models for systems such as biological and economic systems. It is shown that the computational procedure is closely related to traditional least-square analysis. Both linear and nonlinear models are considered. Results of computer simulations are presented for some simple cases.     

Dynamical localization for continuum random surface models

We prove Anderson localization and strong dynamical localization for random surface models in \mathbbR^d \mathbb{R}^d .     

On invariant random approximations

A result regarding invariant random approximation is proved. Supported by Kuwait University Research Grant No. SM-119.     

On random frequent universality

We give a probabilistic construction of frequently hypercyclic or universal entire functions using Birkhoff?s ergodic Theorem. We apply the same ideas to construct random frequently universal vectors for the polynomials of the weighted backward shift on the classical real or complex ?^p${?}^p$ space.