On the solution of heat-conduction problems for thermosensitive bodies

We illustrate methods of constructing analytic-numerical solutions of nonsteady heat-conduction problems for thermosensitive bodies under convective heat transfer, and also two-dimensional steady-state heat-conduction problems for piecewise-homogeneous bodies. Translated fromMatematichni Metodi i Fiziko-Mekhanichni Polya, Vol. 40, No. 1, 1997, pp. 36–44.     

On the solution of heat conduction problems for thermosensitive bodies heated by convective heat exchange

We propose a method of solving heat conduction problems for thermosensitive bodies, in particular for a hollow cylinder from whose surfaces a convective heat exchange with the external environment occurs.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 28, 1988, pp. 83–86.     

Solution of nonstationary heat-conduction problems for thermosensitive bodies under convective heat exchange

We propose a technique for solving nonstationary heat-conduction problems for thermosensitive bodies with simple nonlinearity (the coefficients of thermal conductivity and the heat capacity per unit volume depend on temperature, but the coefficient of thermal diffusivity is constant) heated by convective heat exchange from the surrounding medium. Translated fromMatematichni Metodi ta Fiziko-Mekhanichni Polya, Vol. 40, No. 2, 1997, pp. 148–152.     

Random walks in a convex body and an improved volume algorithm

We give a randomized algorithm using O(n⁷ log² n) separation calls to approximate the volume of a convex body with a fixed relative error. The bound is O(n⁶ log⁴ n) for centrally symmpetric bodies and for polytopes with a polynomial number of facets, and O(n⁵ log⁴ n) for centrally symmetric polytopes with a polynomial number of facets. We also give an O(n⁶ log n) algorithm to sample a point from the uniform distribution over a convex body. Several tools are developed that may be interesting on their own. We extend results of Sinclair–Jerrum [43] and the authors [34] on the mixing rate of Markov chains from finite to arbitrary Markov chains. We also analyze the mixing rate of various random walks on convex bodies, in particular the random walk with steps from the uniform distribution over a unit ball. © 1993 John Wiley & Sons, Inc.     

On the stability of motion of a movable compound structure with a fluid load

We study the stability of uniform rectilinear motion of a system of solid bodies connected by springs and partly filled with an ideal fluid. The frequency equation obtained is studied as a function of the fundamental parameters of the mechanical system. Bibliography: 5 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 26, 1996, pp. 111–116.     

Hyperbolic heat conduction in two semi-infinite bodies in contact

We find, under the viewpoint of the hyperbolic model of heat conduction, the exact analytical solution for the temperature distribution in all points of two semi-infinite homogeneous isotropic bodies that initially are at uniform temperatures T ₀ ¹ and T ₀ ² , respectively, suddenly placed together at time t = 0 and assuming that the contact between the bodies is perfect. We make graphics of the obtained temperature profiles of two bodies at different times and points. And finally, we compare the temperature solution obtained from hyperbolic model to the parabolic or classical solution, for the same problem of heat conduction.This work was partially supported by MEC and FEDER, project MTM-2004-02262 and AVCIT group 03/050.This revised version was published online in April 2005 with a corrected issue number.     

A characterization of parallelepipeds related to weak derivatives

Summary We characterize parallelepipeds in R^m within the family of all convex bodies by a property of special measures on its boundary. We show that these measures are related to weak derivatives (in the sense of [5] and [8]) of convex-valued functions. The results can be applied (see [9]) to derive a generalization of a theorem of Lehmann (see [4]) on the comparison of uniform location experiments.     

Inequalities for mixed p-affine surface area

We prove new Alexandrov-Fenchel type inequalities and new affine isoperimetric inequalities for mixed p-affine surface areas. We introduce a new class of bodies, the illumination surface bodies, and establish some of their properties. We show, for instance, that they are not necessarily convex. We give geometric interpretations of L _p affine surface areas, mixed p-affine surface areas and other functionals via these bodies. The surprising new element is that not necessarily convex bodies provide the tool for these interpretations.     

Classroom note: Using Platonic bodies to generate a random sample from a population having two characteristics linked with a predetermined coefficient of correlation

This note describes a simple method of generating a random sample of N pairs (U _i ,?W _i ) from a population whose elements have two characteristics U  and W associated with a known coefficient of correlation. Although the method described is extremely advantageous when Platonic bodies are used, it can be generalized to include any discrete uniform distribution.     

Coherent random permutations with biased record statistics

We consider random permutations that are defined coherently for all values of n, and for each n have a probability distribution which is conditionally uniform given the set of upper and lower record values. Our central example is a two-parameter family of random permutations that are conditionally uniform given the counts of upper and lower records. This family may be seen as an interpolation between two versions of Ewens’ distribution. We discuss characterisations of the conditionally uniform permutations, their asymptotic properties, constructions and relations to random compositions.     

On Gaussian Marginals of Uniformly Convex Bodies

Recently, Bo’az Klartag showed that arbitrary convex bodies have Gaussian marginals in most directions. We show that Klartag’s quantitative estimates may be improved for many uniformly convex bodies. These include uniformly convex bodies with power type 2, and power type p>2 with some additional type condition. In particular, our results apply to all unit-balls of subspaces of quotients of L _p for 1<p<∞. The same is true when L _p is replaced by S _p ^m , the l _p -Schatten class space. We also extend our results to arbitrary uniformly convex bodies with power type p, for 2≤p<4. These results are obtained by putting the bodies in (surprisingly) non-isotropic positions and by a new concentration of volume observation for uniformly convex bodies. Supported in part by BSF and ISF.     

Volumetric bounds for intersections of congruent balls in Euclidean spaces

We investigate the intersections of balls of radius r, called r-ball bodies, in Euclidean d-space. An r-lense (resp., r-spindle) is the intersection of two balls of radius r (resp., balls of radius r containing a given pair of points). We prove that among r-ball bodies of a given volume, the r-lense (resp., r-spindle) has the smallest inradius (resp., largest circumradius). In general, we upper (resp., lower) bound the intrinsic volumes of r-ball bodies of a given inradius (resp., circumradius). This complements and extends some earlier results on volumetric estimates for r-ball bodies.

     

Local properties of intertwining operators on the sphere

Blaschke?s original question regarding the local determination of zonoids (or projection bodies) has been the subject of much research over the years. In recent times this research has been extended to include intersection bodies and it has been shown that neither zonoids nor intersection bodies have local characterizations. However, it has also been proved that both these classes of bodies admit equatorial characterizations in odd dimensions, but not in even dimensions. The proofs of these results were mostly analytic using properties of associated spherical integral transforms, the Cosine transform and the Radon transform.Here we elaborate a general principle, showing that such local or equatorial characterization problems are equivalent to corresponding support properties of the spherical operators. We discuss this within a general framework, for intertwining operators on C^∞-functions, and apply the results to further geometric constructions, namely to certain mean section bodies, to Lq-centroid bodies and to k-intersection bodies.     

EMBEDDINGS INTO THE CATEGORY OF SUPER UNIFORM SPACES

Abstract

The category US of uniform spaces has been generalised in various ways. The category FUS, of fuzzy uniform spaces and the category GUS, of generalised uniform spaces have both been shown to be good extensions in the sense that US can be embedded into them. We show here that the category SUS, of super uniform spaces also enjoys this property and furthermore, the categories FUS and GUS can be embedded into SUS.     

Order Types of Convex Bodies

We prove a Hadwiger transversal-type result, characterizing convex position on a family of non-crossing convex bodies in the plane. This theorem suggests a definition for the order type of a family of convex bodies, generalizing the usual definition of order type for point sets. This order type turns out to be an oriented matroid. We also give new upper bounds on the Erdős–Szekeres theorem in the context of convex bodies.     

On neighborly families of convex bodies

A family of convex bodies in E^d is called neighborly if the intersection of every two of them is (d-1)-dimensional. In the present paper we prove that there is an infinite neighborly family of centrally symmetric convex bodies in E^d, d 3, such that every two of them are affinely equivalent (i.e., there is an affine transformation mapping one of them onto another), the bodies have large groups of affine automorphisms, and the volumes of the bodies are prescribed. We also prove that there is an infinite neighborly family of centrally symmetric convex bodies in E^d such that the bodies have large groups of symmetries. These two results are answers to a problem of B. Grünbaum (1963). We prove also that there exist arbitrarily large neighborly families of similar convex d-polytopes in E^d with prescribed diameters and with arbitrarily large groups of symmetries of the polytopes.     

Lipschitz Estimates in Almost‐Periodic Homogenization

We establish uniform Lipschitz estimates for second‐order elliptic systems in divergence form with rapidly oscillating, almost‐periodic coefficients. We give interior estimates as well as estimates up to the boundary in bounded C^1,α domains with either Dirichlet or Neumann data. The main results extend those in the periodic setting due to Avellaneda and Lin for interior and Dirichlet boundary estimates and later Kenig, Lin, and Shen for the Neumann boundary conditions. In contrast to these papers, our arguments are constructive (and thus the constants are in principle computable) and the results for the Neumann conditions are new even in the periodic setting, since we can treat nonsymmetric coefficients. We also obtain uniform W^1,p estimates.© 2016 Wiley Periodicals, Inc.     

James' Theorem Fails for Starlike Bodies

Starlike bodies are interesting in nonlinear functional analysis because they are strongly related to bump functions and to n-homogeneous polynomials on Banach spaces, and their geometrical properties are thus worth studying. In this paper we deal with the question whether James' theorem on the characterization of reflexivity holds for (smooth) starlike bodies, and we establish that a feeble form of this result is trivially true for starlike bodies in nonreflexive Banach spaces, but a reasonable strong version of James' theorem for starlike bodies is never true, even in the smooth case. We also study the related question as to how large the set of gradients of a bump function can be, and among other results we obtain the following new characterization of smoothness in Banach spaces: a Banach space X has a C¹ Lipschitz bump function if and only if there exists another C¹ smooth Lipschitz bump function whose set of gradients contains the unit ball of the dual space X*. This result might also be relevant to the problem of finding an Asplund space with no smooth bump functions.     

Strong consistency of MLE for finite uniform mixtures when the scale parameters are exponentially small

We consider maximum likelihood estimation of finite mixture of uniform distributions. We prove that maximum likelihood estimator is strongly consistent, if the scale parameters of the component uniform distributions are restricted from below by exp(−n ^d ), 0<d<1, wheren is the sample size.