Uniform solution for uniform polyhedra

An arbitrary precision solution of uniform polyhedra and their duals is presented. The solution is uniform for all polyhedra given by their kaleidoscopic construction, with no need to examine each polyhedron separately.In memoriam of my father, Gershon Har'El, who introduced me to spatial structures.     

A variational problem determined by probability measures

ABSTRACT

An optimization problem of maximizing an integral of a function over a family of probability measures is considered. The problem is a generalization of a well-studied variational problem in mathematical economics, concerning optimal allocations. The specific generalization that we examine arises also in the limit of singularly perturbed optimal control problems. We examine the mathematical problem and allude to the singular perturbation motivation.     

Transport Maps for $${\beta}$$-Matrix Models and Universality

We construct approximate transport maps for non-critical \({\beta}\)-matrix models, that is, maps so that the push forward of a non-critical \({\beta}\)-matrix model with a given potential is a non-critical \({\beta}\)-matrix model with another potential, up to a small error in the total variation distance. One of the main features of our construction is that these maps enjoy regularity estimates that are uniform in the dimension. In addition, we find a very useful asymptotic expansion for such maps which allows us to deduce that local statistics have the same asymptotic behavior for both models.     

Optimal location with equitable loads

The problem considered in this paper is to find p locations for p facilities such that the weights attracted to each facility will be as close as possible to one another. We model this problem as minimizing the maximum among all the total weights attracted to the various facilities. We propose solution procedures for the problem on a network, and for the special cases of the problem on a tree or on a path. The complexity of the problem is analyzed, O(n) algorithms and an O(pn ³) dynamic programming algorithm are proposed for the problem on a path respectively for p=2 and p>2 facilities. Heuristic algorithms (two types of a steepest descent approach and tabu search) are proposed for its solution. Extensive computational results are presented.     

Efficient algorithms for finding minimum spanning trees in undirected and directed graphs

Recently, Fredman and Tarjan invented a new, especially efficient form of heap (priority queue). Their data structure, theFibonacci heap (or F-heap) supports arbitrary deletion inO(logn) amortized time and other heap operations inO(1) amortized time. In this paper we use F-heaps to obtain fast algorithms for finding minimum spanning trees in undirected and directed graphs. For an undirected graph containingn vertices andm edges, our minimum spanning tree algorithm runs inO(m logβ (m, n)) time, improved fromO(mβ(m, n)) time, whereβ(m, n)=min {i|log⁽ⁱ⁾ n ≦m/n}. Our minimum spanning tree algorithm for directed graphs runs inO(n logn + m) time, improved fromO(n log n +m log log log_(m/n+2) n). Both algorithms can be extended to allow a degree constraint at one vertex. Research supported in part by National Science Foundation Grant MCS-8302648. Research supported in part by National Science Foundation Grant MCS-8303139. Research supported in part by National Science Foundation Grant MCS-8300984 and a United States Army Research Office Program Fellowship, DAAG29-83-GO020.     

On the complexity of the exchange algorithm for minimax optimization problems

We present an exchange algorithm for the solution of minimax optimization problems involving convex functions. For a certain class of functions, the complexity of this algorithm is shown to be either linear in the number of functions, or at least squared in that number.     

On groups with conjugacy classes of distinct sizes

A finite group G is called an ah-group if any two distinct conjugacy classes of G have distinct cardinality. We show that if G is an ah-group, then the non-abelian socle of G is isomorphic to one of the following:

1. , for 1a5, a≠2.

2. A₈.

3. PSL(3,4)^e, for 1e10.

4. A₅×PSL(3,4)^e, for 1e10.

Based on this result, we virtually show that if G is an ah-group with π(G) 2,3,5,7 , then F(G)≠1, or equivalently, that G has an abelian normal subgroup.In addition, we show that if G is an ah-group of minimal size which is not isomorphic to S₃, then the non-abelian socle of G is either trivial or isomorphic to one of the following:

1. , for 3a5.

2. PSL(3,4)^e, for 1e10.

Our research lead us to interesting results related to transitivity and homogeneousity in permutation groups, and to subgroups of wreath products of form Z₂S_n. These results are of independent interest and are located in appendices for greater autonomy.     

Twist impeded self-diffusion in a cholesteric

Molecular self-diffusion along the pitch axis of a twisted nematic is measured by its motional averaging effect on the deuterium quadrupole interaction. The diffusion constant is found to decrease as the pitch length is decreased. This result is explained by simple phenomenological considerations and is consistent with published viscosity measurements.     

Projections on Convex Sets in the Relaxed Limit

The paper establishes the continuity of the best approximation, or the projection, of a function in L _p for p[1,), on a closed convex set in the space, when the set varies and converges to a limit set in the Young-measure relaxation of the space. To this end a strong-type convergence and a convexity structure are identified on the space of Young measures. The appropriate convergence of sets with respect to which the continuity holds is the Mosco convergence of sets associated with the strong-type convergence of functions.     

Continuum regularization of gauge theory with fermions

The covariant-derivative regularization program is discussed ford-dimensional gauge theory coupled to fermions in an arbitrary representation.