Ubiquitous Angles in Equiangular Sets of Lines

A set of lines through the origin in R ^d is called equiangular if all pairs intersect in the same angle. For a given dimension d , there are finitely many nonisometric configurations of (d+1) equiangular lines in R ^d . In particular, there are finitely many possible angles. Such angles can occur in only few dimensions or in infinitely many dimensions. We show that there are infinitely many ubiquitous angles θ occurring among sets of equiangular lines, i.e., for all d large enough, there are d+1 θ -equiangular lines in R ^d . We also show that there are infinitely many frequent angles, those which occur for infinitely many dimensions d , but are not ubiquitous. Received March 31, 1999, and in revised form August 12, 1999. Online publication May 15, 2000.     

ILLUMINATION PROBLEMS AND CODES

It is shown that if there exists a binary code C of length d and covering radius k then a zonotope in the d-dimensional Euclidean space can be illuminated by |C| affine subspaces of dimension k. Applying results from coding theory, the exact value of the illumination numbers of d-dimensional parallelotopes is determined in some special cases. This revised version was published online in August 2006 with corrections to the Cover Date.     

Orthogonal Polynomials on the Ball and the Simplex for Weight Functions with Reflection Symmetries

Generalized classical orthogonal polynomials on the unit ball B ^d and the standard simplex T ^d are orthogonal with respect to weight functions that are reflection-invariant on B ^d and, after a composition, on T ^d , respectively. They are also eigenfunctions of a second-order differential—difference operator that is closely related to Dunkl's h -Laplacian for the reflection groups. Under a proper limit, the generalized classical orthogonal polynomials on B ^d converge to the generalized Hermite polynomials on R ^d , and those on T ^d converge to the generalized Laguerre polynomials on R ^d ₊ . The latter two are related to the Calogero—Sutherland models associated to the Weyl groups of type A and type B . February 14, 2000. Date revised: July 26, 2000. Date accepted: August 4, 2000.     

Infinite-dimensional Langevin equations: uniqueness and rate of convergence for finite-dimensional approximations

The paper deals with the infinite-dimensional stochastic equation dX= B(t, X) dt + dW driven by a Wiener process which may also cover stochastic partial differential equations. We study a certain finite dimensional approximation of B(t, X) and give a qualitative bound for its rate of convergence to be high enough to ensure the weak uniqueness for solutions of our equation. Examples are given demonstrating the force of the new condition. Received: 6 November 1999 / Revised version: 21 August 2000 / Published online: 6 April 2001     

Edgewise Subdivision of a Simplex

In this paper we introduce the abacus model of a simplex and use it to subdivide a d -simplex into k ^d d -simplices all of the same volume and shape characteristics. The construction is an extension of the subdivision method of Freudenthal [3] and has been used by Goodman and Peters [4] to design smooth manifolds. Received June 24, 1999, and in revised form January 13, 2000. Online publication August\/ 11, 2000.     

Discrepancy Estimates on the Sphere

 For measures on the unit sphere in ℝ ^d , d≥3, we derive discrepancy estimates in terms of the quality of corresponding quadrature formulas and in terms of bounds for potential differences. (Received 1 August 1998; in revised form 30 December 1998)     

Chaotic time dependence in a disordered spin system

Stochastic Ising and voter models on ℤ ^d are natural examples of Markov processes with compact state spaces. When the initial state is chosen uniformly at random, can it happen that the distribution at time t has multiple (subsequence) limits as t→∞? Yes for the d = 1 Voter Model with Random Rates (VMRR) – which is the same as a d = 1 rate-disordered stochastic Ising model at zero temperature – if the disorder distribution is heavy-tailed. No (at least in a weak sense) for the VMRR when the tail is light or d≥ 2. These results are based on an analysis of the “localization” properties of Random Walks with Random Rates. Received: 10 August 1998     

Inradii of Simplices

We study the following generalization of the inradius: For a convex body K in the d-dimensional Euclidean space and a linear k-plane L we define the inradius of K with respect to L by , where r(K;x+L) denotes the ordinary inradius of with respect to the affine plane x+L. We show how to determine for polytopes and use the result to estimate for the regular d-simplex T_r ^d . These estimates are optimal for all k in infinitely many dimensions and for certain k in the remaining dimensions. Received July 5, 1996, and in revised form August 8, 1996.     

The size of the shadow boundary projection

Among all embedded closed manifolds with positive exterior curvature ≤k the ratio between the (d-1)-Hausdorff measure of the shadow boundary projection and the volume of M ^d is maximized by the sphere of radius 1/k. Received: 22 August 1997 / Revised version: 2 December 1997     

ALMOST EQUIDISTANT POINTS ON S D-1

Let S ^d-1 denote the (d − 1)-dimensional unit sphere centered at the origin of the d-dimensional Euclidean space. Let 0 < α < π. A set P of points in S ^d-1 is called almost α-equidistant if among any three points of P there is at least one pair lying at spherical distance α. In this note we prove upper bounds on the cardinality of P depending only on d. This revised version was published online in August 2006 with corrections to the Cover Date.     

Lower Bounds on the Transversal Numbers of d -Intervals

Let l ₁ ,l ₂ ,\ldots,l _d be disjoint parallel lines in the plane. A d-interval is a subset of their union that intersects each l _i in a closed interval. Kaiser [4] showed that any system of d -intervals containing no subsystem of k+1 pairwise disjoint d -intervals can be pierced by at most (d ² -d)k points. We show that this bound is close to being optimal, by proving a lower bound of const(d ² /log ² d)k . The construction involves an extension of a construction due to Sgall [8] of certain systems of set pairs. Received April 4, 2000, and in revised form January 4, 2001. Online publication August 28, 2001.     

Smoothing properties of transition semigroups relative to SDEs with values in Banach spaces

In the present paper we consider the transition semigroup P _t related to some stochastic reaction-diffusion equations with the nonlinear term f having polynomial growth and satisfying some dissipativity conditions. We are proving that it has a regularizing effect in the Banach space of continuous functions , where ⊂ℝ ^d is a bounded open set. In L ²() the only result proved is the strong Feller property, for d=1. Here we are able to prove that if f∈C ^∞(ℝ) and d≤3, then for any and t>0. An important application is to the study of the ergodic properties of the system. These results are also of interest for some problem in stochastic control. Received: 20 August 1997 / Revised version: 27 May 1998     

VORONOI POLYHEDRA OF UNIT BALL PACKINGS WITH SMALL SURFACE AREA

In this note, first, we give a very short new proof of the theorem which yields a lower bound for the surface area of Voronoi cells of unit ball packings in E ^d and implies Rogers' upper bound for the density of unit ball packings in E ^d for all d ≥ 2. Second we sharpen locally a classical result of Gauss by finding the locally smallest surface area Voronoi cells of lattice unit ball packings in E ³. This revised version was published online in August 2006 with corrections to the Cover Date.     

A Helly-Type Theorem for Unions of Convex Sets

We prove that for any d, k ≥ 1 there are numbers q = q(d,k) and h = h(d,k) such that the following holds: Let be a family of subsets of the d-dimensional Euclidean space, such that the intersection of any subfamily of consisting of at most q sets can be expressed as a union of at most k convex sets. Then the Helly number of is at most h. We also obtain topological generalizations of some cases of this result. The main result was independently obtained by Alon and Kalai, by a different method. Received April 14, 1995, and in revised form August 1, 1995.     

Isomorphism rigidity of irreducible algebraic ℤd-actions

An irreducible algebraic ℤ ^d -actionα on a compact abelian group X is a ℤ ^d -action by automorphisms of X such that every closed, α-invariant subgroup Y⊊X is finite. We prove the following result: if d≥2, then every measurable conjugacy between irreducible and mixing algebraic ℤ ^d -actions on compact zero-dimensional abelian groups is affine. For irreducible, expansive and mixing algebraic ℤ ^d -actions on compact connected abelian groups the analogous statement follows essentially from a result by Katok and Spatzier on invariant measures of such actions (cf. [4] and [3]). By combining these two theorems one obtains isomorphism rigidity of all irreducible, expansive and mixing algebraic ℤ ^d -actions with d≥2. Oblatum 30-IX-1999 & 4-V-2000?Published online: 16 August 2000     

Additive patterns in a multiplicative group in a finite field

 Let F _q be a field with q elements, let d>1 be a divisor of q−1 and U _d be the subgroup of F _q ^× of index d. Under some growth conditions, we show that the distribution of s-tuples of elements of U _d which follow a given additive pattern approaches a Poissonian distribution. Received: 28 August 2002 Published online: 20 March 2003 Mathematics Subject Classification (2000): 11T99     

The complexification and degree of a semi-algebraic set

The complexification of a semi-algebraic set is the smallest complex algebraic set containing S. Let S be defined by s polynomials of degrees less than d. We prove that the geometric degree of the complexification is less than . Received: 9 January 1997; in final form: 11 August 2000 / Published online: 17 May 2001     

Properties of the Flows Generated by Stochastic Equations with Reflection

We consider the properties of a random set ϕ _t (ℝ ₊ ^d ), where ϕ _t (x) is a solution of a stochastic differential equation in ℝ ₊ ^d with normal reflection from the boundary that starts from a point x. We characterize inner and boundary points of the set ϕ _t (ℝ ₊ ^d ) and prove that the Hausdorff dimension of the boundary ∂ϕ _t (ℝ ₊ ^d ) does not exceed d − 1. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 8, pp. 1069 – 1078, August, 2005.     

Homomorphisms of Products of Graphs into Graphs Without Four Cycles

Given two graphs A and G, we write if there is a homomorphism of A to G and if there is no such homomorphism. The graph G is -free if, whenever both a and c are adjacent to b and d, then a = c or b = d. We will prove that if A and B are connected graphs, each containing a triangle and if G is a -free graph with and , then (here " denotes the categorical product). Received August 31, 1998/Revised April 19, 2000 RID="†" ID="†" Supported by NSERC of Canada Grant #691325.     

On the mixing time of a simple random walk on the super critical percolation cluster

 We study the robustness under perturbations of mixing times, by studying mixing times of random walks in percolation clusters inside boxes in Z ^d . We show that for d≥2 and p>p _c (Z ^d ), the mixing time of simple random walk on the largest cluster inside is Θ(n ² ) – thus the mixing time is robust up to a constant factor. The mixing time bound utilizes the Lovàsz-Kannan average conductance method. This is the first non-trivial application of this method which yields a tight result. Received: 16 December 2001 / Revised version: 13 August 2002 / Published online: 19 December 2002