Rademacher chaos: tail estimates versus limit theorems

We study Rademacher chaos indexed by a sparse set which has a fractional combinatorial dimension. We obtain tail estimates for finite sums and a normal limit theorem as the size tends to infinity. The tails for finite sums may be much larger than the tails of the limit.     

A polynomial algorithm for an integer quadratic non-separable transportation problem

We study the problem of minimizing the total weighted tardiness when scheduling unti-length jobs on a single machine, in the presence of large sets of identical jobs. Previously known algorithms, which do not exploit the set structure, are at best pseudo-polynomial, and may be prohibitively inefficient when the set sizes are large. We give a polynomial algorithm for the problem, whose number of operations is independent of the set sizes. The problem is reformulated as an integer program with a quadratic, non-separable objective and transportation constraints. Employing methods of real analysis, we prove a tight proximity result between the integer solution to that problem and a fractional solution of a related problem. The related problem is shown to be polynomially solvable, and a rounding algorithm applied to its solution gives the optimal integer solution to the original problem.Supported in part by the National Science Foundation under grant ECS-85-01988, and by the Office of Naval Research under grant N00014-88-K-0377.Supported in part by Allon Fellowship, by Air Force grants 89-0512 and 90-0008 and by DIMACS (Center for Discrete Mathematics and Theoretical Computer Science), a National Science Foundation Science and Technology Center—NSF-STC88-09648. Part of the research of this author was performed in DIMACS Center, Rutgers University.Supported in part by Air Force grant 84-0205.     

Infinite,highly connected digraphs with no two arc-disjoint spanning trees

We present a construction of countably infinite, highly connected graphs and digraphs, which shows that several basic connectivity results on finite graphs, including Edmondsapos;s branching theorem, cannot be extended to the infinite case.     

Recurrence relations for Tchebycheffian B-splines

Four-term recurrence relations with constant coefficients are derived for a wide class of T chebycheffian B-splines, LB-splines and complex B-splines. Such a relation exists whenever the differential operator defining the underlying “polynomial” space can be factored in two essentially different ways. The four lower order B-splines in the recurrence relation appear in two pairs, each pair corresponding to one of these factorization. It is shown that the two-term recurrence relations for polynomial, trigonometric and hyperbolic B-splines as well as other known two-term recurrence relations are obtained directly from the four-term recurrence relations in a unified and systematic way. The above derivation also yields two different two-term recurrence relations for Green’s functions of these “polynomial” spaces In this context the special examples of exponential functions and rational functions are analyzed in detail.     

Bounds for arrays of dots with distinct slopes or lengths

Ann×m sonar sequence is a subset of then×m grid with exactly one point in each column, such that the vectors determined by them are all distinct. We show that for fixedn the maximalm for which a sonar sequence exists satisfiesn–Cn ^11/20<m<n+4n ^2/3 for alln andm>n+c logn log logn for infinitely manyn.Another problem concerns the maximal numberD of points that can be selected from then×m grid so that all the vectors have slopes. We proven ^1/2Dn ^4/5 Supported by Hungarian National Foundation for Scientific Research, Grant No. 1901Research conducted by Herbert Taylor was sponsored in part by the Office of Naval Research under ONR Contract No. N00014-90-J-1341.     

Counting colorful multi-dimensional trees

LetV be a disjoint union ofr finite setsV ₁,...,V _r (colors). A collectionT of subsets ofV iscolorful if each member ifT contains at most one point of each color. Ak-dimensional colorful tree is a colorful collectionT of subsets ofV, each of sizek+1, such that if we add toT all the colorful subsets ofV of sizek or less, we get aQ-acyclic simplicial complex _T We count (using the Binet-Cauchy theorem) thek-dimensional colorful trees onV (for allk), where each treeT is counted with weight . The result confirms, in a way, a formula suggested by Bolker. (fork-r–1). It extends, on one hand, a result of Kalai on weighted counting ofk-dimensional trees and, on the other hand, enumeration formulas for multi-partite (1-dimensional) trees. All these results are extensions of Cayley's celebrated treecounting formula, now 100 years old.     

On the product of sign vectors and unit vectors

For a fixed unit vectora=(a ₁,...,a _n )S ^n-1, consider the 2 ⁿ sign vectors=(₁,..., _n ){±1{ ⁿ and the corresponding scalar products^·a = _n ⁱ⁼¹ = _i a _i . The question that we address is: for how many of the sign vectors must^.a lie between–1 and 1. Besides the straightforward interpretation in terms of the sums ±a ₂ , this question has appealing reformulations using the language of probability theory or of geometry.The natural conjectures are that at least 1/2 the sign vectors yield |.a|1 and at least 3/8 of the sign vectors yield |.a|<1 (the latter excluding the case when |a _i |=1 for somei). These conjectured lower bounds are easily seen to be the best possible. Here we prove a lower bound of 3/8 for both versions of the problem, thus completely solving the version with strict inequality. The main part of the proof is cast in a more general probabilistic framework: it establishes a sharp lower bound of 3/8 for the probability that |X+Y|<1, whereX andY are independent random variables, each having a symmetric distribution with variance 1/2.We also consider an asymptotic version of the question, wheren along a sequence of instances of the problem satisfying ||a||0. Our result, best expressed in probabilistic terms, is that the distribution of .a converges to the standard normal distribution, and in particular the fraction of sign vectors yielding .a between –1 and 1 tends to 68%.This research was supported in part by the Institute for Mathematics and its Applications with funds provided by the National Science Foundation.     

Greene-Kleitman's theorem for infinite posets

It is proved that if (P) is a poset with no infinite chain and k is a positive integer, then there exist a partition of P into disjoint chains C _i and disjoint antichains A ₁, A ₂, ..., A _k, such that each chain C _i meets min (k, |C _i|) antichains A _j. We make a dual conjecture, for which the case k=1 is: if (P) is a poset with no infinite antichain, then there exist a partition of P into antichains A _i and a chain C meeting all A _i. This conjecture is proved when the maximal size of an antichain in P is 2.