Projecting <Emphasis Type="Italic">l</Emphasis><Subscript>∞</Subscript> onto Classical Spaces

We describe an explicit construction of a linear projection of a symmetric conical section of the n-dimensional cube onto a (1+ε)-isomorphic version of the Euclidean ball of proportional dimension, or more generally onto a (1+ε)-isomorphic image of an l _p ^m -ball. Allowing non-linear projections (of logarithmic polynomial nonlinearity) we may even project the full n-dimensional cube onto the same images. This is done by gluing together explicit projections onto two-dimensional spaces, interpreting and modifying a paper of Ben-Tal and Nemirowski on polynomial reductions of conic quadratic programming problems to linear programming problems in terms of Banach spaces.      

On the Asymmetry Constant of a Body with Few Vertices

In this note, we show that a nondegenerated polytope in IR ⁿ with n + k, 1 k < n, vertices is far from any symmetric body. We provide the asymptotically sharp estimates for the asymmetry constant of such polytopes.     

Inequalities for the Gamma function and estimates for the volume of sections of

Let and let be a -dimensional subspace of . We prove that , for and whenever . We also consider and other related cases. We obtain sharp inequalities involving Gamma function in order to get these results.

     

Universal Polynomial Majorants on Convex Bodies

Let K be a convex body in ^d (d2), and denote by B_n(K) the set of all polynomials p_n in ^d of total degree n such that |p_n|1 on K. In this paper we consider the following question: does there exist a p*_nB_n(K) which majorates every element of B_n(K) outside of K? In other words can we find a minimal γ1 and p*_nB_n(K) so that |p_n(x)|γ |p*_n(x)| for every p_nB_n(K) and x ^d\K? We discuss the magnitude of γ and construct the universal majorants p*_n for evenn. It is shown that γ can be 1 only on ellipsoids. Moreover, γ=O(1) on polytopes and has at most polynomial growth with respect to n, in general, for every convex body K.     

A scanning and transmission electron microscopic study of the development of the surface structure of neuroepithelial bodies in the mouse lung

Neuroepithelial bodies (NEBs) are groups of neuroepithelial (NE) cells that are localized on mounds on the bronchiolar epithelium of the lung. The present study examined NEBs in mice ranging in age from 2 days before birth to 80 days after birth. The position and surface architecture of NEBs was examined using scanning electron microscopy (SEM) and transmission electron microscopy (TEM). In foetal mice, 2 days before birth, NEBs were distinguished from the rest of the bronchiolar epithelium by a slight elevation of non-ciliated Clara-like cells arranged in a cobbestone-like pattern. The exposed surface of the NEB was identified by small protrusions with regular microvilli intermittenly located at the base of deep clefts between the Clara-like cells. The surface of the Clara-like cells had fewer and smaller microvilli and could be easily distinguished from the apical surface of the NEB. Before birth, the surface of all of the apical cells was covered by regularly placed microville, however after birth some of the more prominently positional apical cells revealed a bare patch at the centre of the portion of apical cell exposed to the lumen of the lung. As the mice aged there was an increase in the number of apical cell protusions observed with centrally positioned bare patches. These two morphologically distinct surfaces of apical cells may have separate specialized functions. The exposed surfaces of apical cells were often observed in pairs and this feature has been observed in various sensory organs providing support for chemoreceptive function. However small bright spheres resembling vesicles were frequently observed on the lumenal surface of apical cells of the centrally placed bare patch. Transmission electron microscopy confirmed the presence of vesicles on the surface of apical cells and due to their location these vesicles were thought to contain a substance secreted into the lumen of the lung by apical cells. The significance of the bare region on the apical cells is not clear in terms of the proposed chemoreceptive function usually attributed to NEBs. It may be possible that the morphological changes observed in apical cells after birth are more appropriate for secretion of than for chemoreception. This is supported by the observation in the present study of vesicles lying on the lumenal surface of the bare region of the apical cell, however the mechanism for secretion of whoel vesicles is not clear and requires further investigation.     

Numerical solution of the dynamic problem of elasticity for bodies with concentrated masses

On the basis of the virtual-displacement and the DAlembert principles, a new initial-boundary-value problem is formulated for the system of dynamic equilibrium equations of two bodies contacting with friction over a constant area. Their masses are assumed to be distributed with certain densities over the contact surfaces. An equivalent generalized problem is formulated. Classes of discontinuous functions of the finite-element method are used to construct computational schemes of high order of accuracy.__________Translated from Prikladnaya Mekhanika, Vol. 40, No. 12, pp. 65–75, December 2004.     

The topological structure of maximal lattice free convex bodies: The general case

Given a genericm × n matrixA, the simplicial complexK(A) is defined to be the collection of simplices representing maximal lattice point free convex bodies of the form {x : Ax b}. The main result of this paper is that the topological space associated withK(A) is homeomorphic withR ^m–1 . © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.Supported by NSF grant SES-9121936 and the program in Discrete Mathematics at Yale University.Partially supported by the Hungarian NSF grant 1909 and the program in Discrete Mathematics at Yale University.     

The complex of maximal lattice free simplices

The simplicial complexK(A) is defined to be the collection of simplices, and their proper subsimplices, representing maximal lattice free bodies of the form (x: Axb), withA a fixed generic (n + 1) ×n matrix. The topological space associated withK(A) is shown to be homeomorphic to ⁿ , and the space obtained by identifying lattice translates of these simplices is homeorphic to then-torus.Corresponding author.The first author was partially supported by Hungarian NSF grants 1907 and 1909, and also by U.S. NSF grant CCR-9111491. The research of the second author was supported by DMS9103608 and the third author by NSF grant SES9121936.     

RHEOLOGICAL FRACTURE ANALYSIS OF CABLE BY CONSIDERING THE SECOND ORDER EFFECTS OF RANDOM SLACK

In this paper,we consider the cable as a bundle consisting of n sub-bundles,with mparallel tension members per sub-bundle,and the tension members themselves are polymeric yarns im-pregnated with a resin matrix.The nonfailed members at any instant must share an applied system loadaccording to some rule,since there is a clearly expressed dependence of the fracture on the durationand character of the loading.So then,the fracture of cable is a process of nonlinear dynamic evolu-tion,which accommodates to the non-equilibrium thermodynamics of irreversible processes by itself.Let us assume that the polymeric yarns are as viscoelastic solid,under certain probabilistic assump-tions,according to the principles of theology of bodies with defects,the relationship between the singlemember loading and failure and the bundle loading are investigated.It can be shown that the bundlefailure time is asymptotically normally distributed as the number of members grows large.After astudy of the second order effects of random slack,it is known that the asymptotic mean and varianceare functions of the parameters of loading and single member rhcological behavior.Hence the loss inthe asymptotic bundle strength mean brought about by random member and sub-bundle slack,L.,andthe loss in the asymptotic bundle strength variance caused by random member slack,Δ_(?),are deter-mined.And finally,it is known that the asymptotic time of failure can make up a considerable part ofthe fracture of cable,and the fracture of cable is a time-dependent process of rheological fracture.