On Heilbronn’s Problem in Higher Dimension

Heilbronn conjectured that given arbitrary n points in the 2-dimensional unit square [0, 1]², there must be three points which form a triangle of area at most O(1/n²). This conjecture was disproved by a nonconstructive argument of Komlós, Pintz and Szemerédi [10] who showed that for every n there is a configuration of n points in the unit square [0, 1]² where all triangles have area at least (log n/n²). Considering a generalization of this problem to dimensions d3, Barequet [3] showed for every n the existence of n points in the d-dimensional unit cube [0, 1]^d such that the minimum volume of every simplex spanned by any (d+1) of these n points is at least (1/n^d). We improve on this lower bound by a logarithmic factor (log n).     

On the geometry of random Cantor sets and fractal percolation

Random Cantor sets are constructions which generalize the classical Cantor set, middle third deletion being replaced by a random substitution in an arbitrary number of dimensions. Two results are presented here. (a) We establish a necessary and sufficient condition for the projection of ad-dimensional random Cantor set in [0,1]^d onto ane-dimensional coordinate subspace to contain ane-dimensional ball with positive probability. The same condition applies to the event that the projection is the entiree-dimensional unit cube [0,1] ^e . This answers a question of Dekking and Meester,⁽⁹⁾ (b) The special case of fractal percolation arises when the substitution is as follows: The cube [0,1] ^d is divided intoM ^d subcubes of side-lengthM ^–, and each such cube is retained with probabilityp independently of all other subcubes. We show that the critical valuep _c(M, d) ofp, marking the existence of crossings of [0,1] ^d contained in the limit set, satisfiesp _c(M, d)p _c(d) asM, wherep _c(d) is the critical probability of site percolation on a latticeL ^d obtained by adding certain edges to the hypercubic lattice ^d . This result generalizes in an unexpected way a finding of Chayes and Chayes,⁽⁴⁾ who studied the special case whend=2.     

An error expansion for cubature with an integrand with homogeneous boundary singularities

Summary A common strategy in the numerical integration over ann-dimensional hypercube or simplex, is to consider a regular subdivision of the integration domain intom ⁿ subdomains and to approximate the integral over each subdomain by means of a cubature formula. An asymptotic error expansion whenm is derived in case of an integrand with homogeneous boundary singularities. The error expansion also copes with the use of different cubature formulas for the boundary subdomains and for the interior subdomains.     

Selecting distances in arrangements of hyperplanes spanned by points

In this paper we consider a problem of distance selection in the arrangement of hyperplanes induced by n given points. Given a set of n points in d-dimensional space and a number k, , determine the hyperplane that is spanned by d points and at distance ranked by k from the origin. For the planar case we present an O(nlog²n) runtime algorithm using parametric search partly different from the usual approach [N. Megiddo, J. ACM 30 (1983) 852]. We establish a connection between this problem in 3-d and the well-known 3SUM problem using an auxiliary problem of counting the number of vertices in the arrangement of n planes that lie between two sheets of a hyperboloid. We show that the 3-d problem is almost 3SUM-hard and solve it by an O(n²log²n) runtime algorithm. We generalize these results to the d-dimensional (d4) space and consider also a problem of enumerating distances.     

On low-dimensional faces that high-dimensional polytopes must have

We prove that every five-dimensional polytope has a two-dimensional face which is a triangle or a quadrilateral. We state and discuss the following conjecture: For every integerk1 there is an integer f(k) such that everyd-polytope,df(k), has ak-dimensional face which is either a simplex or combinatorially isomorphic to thek-dimensional cube.We give some related results concerning facet-forming polytopes and tilings. For example, sharpening a result of Schulte [25] we prove that there is no face to face tiling of ⁵ with crosspolytopes.Supported in part by a BSF Grant and by I.H.E.S, Bures-Sur-Yvette.     

On the Helmholtz decomposition in general unbounded domains

It is well known that the Helmholtz decomposition of L^q-spaces fails to exist for certain unbounded smooth planar domains unless q = 2, see [2], [9]. As recently shown [6], the Helmholtz projection does exist for general unbounded domains of uniform C²-type in if we replace the space L^q, 1 < q < ∞, by L² ∩ L^q for q > 2 and by L^q + L² for 1 < q < 2. In this paper, we generalize this new approach from the three-dimensional case to the n-dimensional case, n ≥ 2. By these means it is possible to define the Stokes operator in arbitrary unbounded domains of uniform C²-type. Received: 15 February 2006     

Punktprozesse mit Wechselwirkung

We consider classical, continuous systems of particles in r dimensions described by infinite system equilibrium states which have been defined by Dobrushin [5] and Lanford/Ruelle [24]. For a large class of potentials we prove the theorem of Lee/Yang [43] together with a variational characterizafor these equilibrium states. The main idea stems from Föllmer [9] who showed that in the case of lattice systems, the theorem of Lee/Yang is intimately related to Birkhoff's ergodic theorem and McMillan's theorem (ergodic theorem of information theory). Following this idea we obtain as main results an r-dimensional ergodic theorem for random measures in ^r , limit theorems concerning energy and entropy and an r-dimensional version of Breiman's theorem showing that there is almost sure convergence behind McMillan's theorem.

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Danken möchten wir Klaus Krickeberg, der diese Arbeit durch eine Fülle wertvoller Hinweise und Anregungen gefördert hat.     

New families of ovoids in O + 8

We construct four new infinite families of ovoids in the 8-dimensional orthogonal geometry O _inf8 ^sup+ . We determine the automorphism groups of these ovoids and we show that the two sporadic ovoids recently found by Cooperstein [2] and Shult [11] are members of our families.     

Three-dimensional hyperelliptic manifolds

Let X³ = H³, E³, S³, H² × E¹, S² × E¹, T₁(H²), Nil of Solv be one of the eight 3-dimensional geometrics of Thurston [10] and G be a discrete group of isometrics of X³ acting without fixed points. A manifold M³ = X³/G is said to be hyperelliptic if there is an isometric involution on it such that the factor space M³/<> is diffeomorphic to the 3-sphere S³. In analogy with the theory of Riemann surfaces we call involution.In the present paper the existence of hyperelliptic manifolds in each light of the eight 3-dimensional geometrics will be obtained. All the proofs given there will be written in the language of orbifolds whose basic facts can be found in [9].     

Some constructions of systematic authentication codes using galois rings

For q = p ^m and m ≥ 1, we construct systematic authentication codes over finite field using Galois rings. We give corrections of the construction of [2]. We generalize corresponding systematic authentication codes of [6] in various ways.     

Algorithms for approximate calculation of the minimum of a convex function from its values

The paper deals with a numerical minimization problem for a convex function defined on a convexn-dimensional domain and continuous (but not necessarily smooth). The values of the function can be calculated at any given point. It is required to find the minimum with desired accuracy. A new algorithm for solving this problem is presented, whose computational complexity asn is considerably less than that of similar algorithms known to the author. In fact, the complexity is improved fromCn ⁷ ln²(n+1) [4] toCn ² ln(n+1).Translated fromMatematicheskie Zametki, Vol. 59, No. 1, pp. 95–102, January, 1996.     

Estimates on Eigenvalues of Laplacian

In this paper, we study eigenvalues of Laplacian on either a bounded connected domain in an n-dimensional unit sphere Sⁿ(1), or a compact homogeneous Riemannian manifold, or an n-dimensional compact minimal submanifold in an N-dimensional unit sphere S^N(1). We estimate the k+1-th eigenvalue by the first k eigenvalues. As a corollary, we obtain an estimate of difference between consecutive eigenvlaues. Our results are sharper than ones of P. C. Yang and Yau [25], Leung [19], Li [20] and Harrel II and Stubbe [12], respectively. From Weyls asymptotical formula, we know that our estimates are optimal in the sense of the order of k for eigenvalues of Laplacian on a bounded connected domain in an n-dimensional unit sphere Sⁿ(1).Mathematics Subject Classification (2000): 35P15, 58G25, 53C42Research was partially supported by a Grant-in-Aid for Scientific Research from the Japan Society for the Promotion of Science.Research was partially Supported by SF of CAS, Chinese NSF and NSF of USA.     

Transcendence of Binomial and Lucas' Formal Power Series

The formal power series[formula]is transcendental over (X) whentis an integer ≥ 2. This is due to Stanley forteven, and independently to Flajolet and to Woodcock and Sharif for the general case. While Stanley and Flajolet used analytic methods and studied the asymptotics of the coefficients of this series, Woodcock and Sharif gave a purely algebraic proof. Their basic idea is to reduce this series modulo prime numbersp, and to use thep-Lucas property: ifn = ∑n_ipⁱis the basepexpansion of the integern, then[equation]The series reduced modulopis then proved algebraic over _p(X), the field of rational functions over the Galois field _p, but its degree is not a bounded function ofp. We generalize this method to characterize all formal power series that have thep-Lucas property for “many” prime numbersp, and that are furthermore algebraic over (X).     

Vector addition theorems and Baker-Akhiezer functions

Functional equations that arise naturally in various problems of modern mathematical physics are discussed. We introduce the concepts of anN-dimensional addition theorem for functions of a scalar argument and Cauchy equations of rankN for a function of ag-dimensional argument that generalize the classical functional Cauchy equation. It is shown that forN=2 the general analytic solution of these equations is determined by the Baker—Akhiezer function of an algebraic curve of genus 2. It is also shown that functions give solutions of a Cauchy equation of rankN for functions of ag-dimensional argument withN2 ^g in the case of a generalg-dimensional Abelian variety andNg in the case of a Jacobian variety of an algebra curve of genusg. It is conjectured that a functional Cauchy equation of rankg for a function of ag-dimensional argument is characteristic for functions of a Jacobian variety of an algebraic curve of genusg, i.e., solves the Riemann—Schottky problem.In memory of M. K. PolivanovResearch Institute of Physicotechnical and Electronic Measurements; L. D. Landau Institute of Theoretical Physics. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 94, No. 2, pp. 200–212, February, 1993.     

Some characterizing properties of the simplex

We shall prove that a convex body in ^d (d2) is a simplex if, and only if, each of its Steiner symmetrals is a convex double cone over the symmetrization space or, equivalently, has exactly two extreme points outside of this hyperplane. In [3] it is shown that every Steiner symmetral of an arbitrary d-simplex is such a double cone, more precisely a bipyramid. Therefore our main aim is to prove that a convex body which is not a simplex has Steiner symmetrals with more than two extreme points not in the symmetrization space. Some equivalent properties of simplices will also be given.     

Solution of matrix Riemann-Hilbert problems with quasi-permutation monodromy matrices

In this paper we solve an arbitrary matrix Riemann-Hilbert (inverse monodromy) problem with irreducible quasi-permutation monodromy representation outside of a divisor in the space of monodromy data. This divisor is characterized in terms of the theta-divisor on the Jacobi manifold of an auxiliary compact Riemann surface realized as an appropriate branched covering of P1 . The solution is given in terms of a generalization of Szegö kernel on the Riemann surface. In particular, our construction provides a new class of solutions of the Schlesinger system. The isomonodromy tau-function of these solutions is computed up to a nowhere vanishing factor independent of the elements of monodromy matrices. Results of this work generalize the results of papers [13] and [14] where the 2× 2 case was solved.Mathematics Subject Classification (1991): 35Q15, 30F60, 32G81     

Orthogonal hyperspheres

Umbilical projection ([12], [14]) is a process suggested to derive results rather quickly in regard to four intersecting spheres [17] andn+1 intersecting hyperspheres in ann-space [18]. The same has been used with an advantage to deduce a porism on 2n+5 hyperspheres in ann-space [23]. The purpose of this paper is to concentrate on mutually orthogonal hyperspheres only and to illustrate simultaneously once again the utility and facility of this tool to arrive at a number of new and interesting results as follows:The 2(n+1) intersections ofn+1 mutually orthogonal hyperspheres in ann-space, takenn at a time, give rise to 2 ⁿ pairs ofsemi-inverse [22] simplexes, perspective from their radical centreH, such that the 2 ⁿ primes of perspectivity coincide with their 2 ⁿ hyperplanes of similitude and form anS-configuration (S-C) [15] with theircentral simplex S(A) as itsdiagonal simplex. Everysimplex of intersection introduced here isisodynamic [25] such that itstangential simplex, circumscribed to it along circumhypersphere, is perspective to it from itsLemoine point L. ItsLemoine hyperplane l, as the polar prime ofL w. r. t. it, is the same as that of itscomplementary simplex of intersection and coincides whith their prime of perspectivity such that their 2(n+1) altitudes are met by their commonBrocard diameter through their Lemoine points. The 2 ⁿ Brocard diameters of the 2 ⁿ pairs of complementary simplexes of intersection concur atH. The hyperspheres of antisimilitude of the given hyperspheres, having centres in a prime of similitude, form the commonNeuberg hyperspheres of the pair of semi-inverse simplexes, having this prime as their common Lemoine hyperplane, are consequently orthogonal to their cirumhyperspheres whose radical hyperplane, too, coincides whith this prime, and therefore belong to acoaxal net [15] passing through the pair of their commonNeuberg points on their common Brocard diameter. The second centres of similitude of the 2 ⁿ pairs ofcomplementary hyperspheres of intersection form the 2 ⁿ vertices of the dual [15] of the (S-C), whithS(A) as common diagonal simplex, as its polar reciprocal w. r. t. the common orthogonal hypersphere of then+1 hyperspheres, the first centres of similitude coinciding atH.Due inspiration is derived from the works ofCourt ([2]–[9]) on mutually orthogonal circles and spheres. Presented by G. Hajós     

A variational problem for submanifolds in a sphere

Let be an n-dimensional submanifold in an (n + p)-dimensional unit sphere S ^{n + p} , M is called a Willmore submanifold (see [11], [16]) if it is a critical submanifold to the Willmore functional , where is the square of the length of the second fundamental form, H is the mean curvature of M. In [11], the second author proved an integral inequality of Simons’ type for n-dimensional compact Willmore submanifolds in S ^{n + p} . In this paper, we discover that a similar integral inequality of Simons’ type still holds for the critical submanifolds of the functional . Moreover, it has the advantage that the corresponding Euler-Lagrange equation is simpler than the Willmore equation.     

On three-dimensional conformally flat quasi-Sasakian manifolds

LetM be a 3-dimensional quasi-Sasakian manifold. On such a manifold, the so-called structure function is defined. With the help of this function, we find necessary and sufficient conditions forM to be conformally flat. Next it is proved that ifM is additionally conformally flat with = const., then (a)M is locally a product ofR and a 2-dimensional Kählerian space of constant Gauss curvature (the cosymplectic case), or (b)M is of constant positive curvature (the non cosymplectic case; here the quasi-Sasakian structure is homothetic to a Sasakian structure). An example of a 3-dimensional quasi-Sasakian structure being conformally flat with nonconstant structure function is also described. For conformally flat quasi-Sasakian manifolds of higher dimensions see [O₁]     

On the Central Limit Theorem for Nonuniform φ-Mixing Random Fields

The partial-sum processes, indexed by sets, of a stationary nonuniform -mixing random field on the d-dimensional integer lattice are considered. A moment inequality is given from which the convergence of the finite-dimensional distributions to a Brownian motion on the Borel subsets of [0, 1] ^d is obtained. A Uniform CLT is proved for classes of sets with a metric entropy restriction and applied to certain Gibbs fields. This extends some results of Chen⁽⁵⁾ for rectangles. In this case and when the variables are bounded a simpler proof of the uniform CLT is given.