Polytopes passing through circles

Suppose a convex body wants to pass through a circular hole in a wall. Does its ability to do so depend on the thickness of the wall? In fact in most cases it does, and in this paper we present a sufficient criterion for a polytope to allow an affirmative answer to the question.     

On edge-antipodal <Emphasis Type="Italic">d</Emphasis>-polytopes

A convex d-polytope in ℝ ^d is called edge-antipodal if any two vertices that determine an edge of the polytope lie on distinct parallel supporting hyperplanes of the polytope. We introduce a program for investigating such polytopes, and examine those that are simple.      

Asymptotic behavior of geophysical fluids in highly rotating balls

In this note we study the convergence in the limit of small Ekman and Rossby numbers of the magnetohydrodynamics equations relevant to describe the flow in the Earth core. In particular, we prove the nonlinear stability of Ekman-Hartmann type boundary layers in a spherical geometry for some class of well-prepared initial data.     

Scaling Limits of Bipartite Planar Maps are Homeomorphic to the 2-Sphere

We prove that scaling limits of random planar maps which are uniformly distributed over the set of all rooted 2k-angulations are a.s. homeomorphic to the two-dimensional sphere. Our methods rely on the study of certain random geodesic laminations of the disk. Received: December 2006, Revision: August 2007, Accepted: October 2007     

Homological Invariance For Asymptotic Invariants and Systolic Inequalities

We show that the systolic constant, the minimal volume entropy, and the spherical volume of a manifold depend only on the image of the fundamental class under the classifying map of the universal covering. Moreover, we compute the systolic constant of manifolds with fundamental group of order two (modulo the value for the real projective space) and derive an inequality between the minimal volume entropy and the systolic constant. Received: February 2007, Revised: November 2007, Accepted: December 2007     

Asymptotic behavior of the irrational factor

We study the irrational factor function I(n) introduced by Atanassov and defined by , where is the prime factorization of n. We show that the sequence {G(n)/n} _n≧1, where G(n) = Π _ν=1 ⁿ I(ν)^1/n , is convergent; this answers a question of Panaitopol. We also establish asymptotic formulas for averages of the function I(n). Research of the third author is supported in part by NSF grant number DMS-0456615.     

Expectation of intrinsic volumes of random polytopes

Let K be a convex body in ℝ ^d , let j ∈ {1, …, d−1}, and let K(n) be the convex hull of n points chosen randomly, independently and uniformly from K. If ∂K is C ₊², then an asymptotic formula is known due to M. Reitzner (and due to I. Bárány if ∂K is C ₊³) for the difference of the jth intrinsic volume of K and the expectation of the jth intrinsic volume of K(n). We extend this formula to the case when the only condition on K is that a ball rolls freely inside K. Funded by the Marie-Curie Research Training Network “Phenomena in High-Dimensions” (MRTN-CT-2004-511953).     

Minimax Problems Related to Cup Powers and Steenrod Squares

If F is a family of mod 2 k-cycles in the unit n-ball, we lower bound the maximal volume of any cycle in F in terms of the homology class of F in the space of all cycles. We give examples to show that these lower bounds are fairly sharp. Received: February 2007, Revision: January 2008, Accepted: January 2008     

Number Variance of Random Zeros on Complex Manifolds

We show that the variance of the number of simultaneous zeros of m i.i.d. Gaussian random polynomials of degree N in an open set with smooth boundary is asymptotic to , where is a universal constant depending only on the dimension m. We also give formulas for the variance of the volume of the set of simultaneous zeros in U of k < m random degree-N polynomials on . Our results hold more generally for the simultaneous zeros of random holomorphic sections of the N-th power of any positive line bundle over any m-dimensional compact K?hler manifold. Received: August 2006 Revision: March 2007 Accepted: April 2007     

The averaging method and the asymptotic behavior of solutions to differential inclusions

We prove one version of the first Bogolyubov theorem for differential inclusions with multivalued mappings that satisfy certain one-sided constraints. We study the dependence of solutions to differential inclusions on the parameters.     

A lower bound for the number of function evaluations in an error estimate for numerical integration

A popular practical way to estimate the error in numerical integration is to use two cubature formulae. In this paper we give a lower bound for the number of function evaluations necessary to approximate the integral and the error.Communicated by Ronald A. DeVore.AMS classification: 65D30, 65G99.     

On the universal A.S. central limit theorem

Let (X _k ) be a sequence of independent r.v.’s such that for some measurable functions gk : R ^k → R a weak limit theorem of the form

The operad of finite labeled tournaments

We study the operad of finite labeled tournaments. We describe the structure of suboperads of this operad generated by simple tournaments. We prove that a suboperad generated by a tournament with two vertices (i.e., the operad of finite linearly ordered sets) is isomorphic to the operad of symmetric groups, and a suboperad generated by a simple tournament with more that two vertices is isomorphic to the quotient operad of the free operad with respect to a certain congruence. We obtain this congruence explicitly.     

Boundedness of characters in locally a-convex algebras

We characterize advertibly complete topological algebras among those whose completions are algebras. We show that the set of non-zero characters in a commutative, advertibly complete,M-complete locallyA-convex algebra is equibounded whenever every element of the algebra is bounded.     

Collapsing monoids consisting of permutations and constants

In this paper we determine all collapsing transformation monoids that contain at least one unary constant operation and whose nonconstant operations are permutations. Furthermore, we find an infinite family of transformation monoids that consist of at least three unary constant operations and some permutations for which the corresponding monoidal intervals are 2-element chains. This research is supported by Hungarian National Foundation for Scientific Research grant nos. T 37877 and K 60148.     

A characterization of affine length and asymptotic approximation of convex discs

It is shown that every equi-affine invariant and upper semicontinuous valuation on the space of convex discs is a linear combination of the Euler characteristic, area, and affine length. Asymptotic formulae for approximation of convex discs by polygons are derived, extending results of L. Fejes Tóth from smooth convex discs to general convex discs.     

Complete unital-derived arcs in the hall planes

We prove the existence of complete (q ² -q + 1)-arcs in each Hall plane of orderq ²,q ² > 9. Work performed under the auspicies of the G.N.S.A.G.A. of the C.N.R. (National Research Council) of Italy and supported by M.U.R.S.T. progetto “Strutture Geometriche Combinatoria e Loro Applicazioni”.     

The effect of curvature on the best constant in the Hardy–Sobolev inequalities

We address the question of attainability of the best constant in the following Hardy–Sobolev inequality on a smooth domain Ω of :

An application of index forms in cryptography

We investigate the possibility of using index forms as basic ingredients of cryptographically important functions. We suggest the use of a hash function based on index forms and we prove some important properties of the suggested function. The research was supported in part by the Hungarian Academy of Sciences, by grants T048791 and K67580 of the Hungarian NFSR, by the National Office for Research and Technology and by the grant JP-26/2006.     

Random walks on diestel-leader graphs

We investigate various features of a quite new family of graphs, introduced as a possible example of vertex-transitive graph not roughly isometric with a Cayley graph of some finitely generated group. We exhibit a natural compactification and study a large class of random walks, proving theorems concerning almost sure convergence to the boundary, a strong law of large numbers and a central limit theorem. The asymptotic type of then-step transition probabilities of the simple random walk is determined.