Abstract not found

A polynomial P in n complex variables is said to have the “half-plane property” (or Hurwitz property) if it is nonvanishing whenever all the variables lie in the open right half-plane. Such polynomials arise in combinatorics, reliability theory, electrical circuit theory and statistical mechanics. A particularly important case is when the polynomial is homogeneous and multiaffine: then it is the (weighted) generating polynomial of an r-uniform set system. We prove that the support (set of nonzero coefficients) of a homogeneous multiaffine polynomial with the half-plane property is necessarily the set of bases of a matroid. Conversely, we ask: For which matroids M does the basis generating polynomial P B(M) have the half-plane property? Not all matroids have the half-plane property, but we find large classes that do: all sixth-root-of-unity matroids, and a subclass of transversal (or cotransversal) matroids that we call “nice”. Furthermore, the class of matroids with the half-plane property is closed under minors, duality, direct sums, 2-sums, series and parallel connection, full-rank matroid union, and some special cases of principal truncation, principal extension, principal cotruncation and principal coextension. Our positive results depend on two distinct (and apparently unrelated) methods for constructing polynomials with the half-plane property: a determinant construction (exploiting “energy” arguments), and a permanent construction (exploiting the Heilmann–Lieb theorem on matching polynomials). We conclude with a list of open questions. KEY WORDS: Graph, matroid, jump system, abstract simplicial complex, spanning tree, basis, generating polynomial, reliability polynomial, Brown–Colbourn conjecture,

The generalized theta graph \Theta s 1 ;:::;s k consists of a pair of endvertices joined by k internally disjoint paths of lengths s 1 ; : : : ; s k 1. We prove that the roots of the chromatic polynomial (\Theta s 1 ;:::;s k ; z) of a k-ary generalized theta graph all lie in the disc jz \Gamma 1j [1 + o(1)] k= log k, uniformly in the path lengths s i . Moreover, we prove that \Theta 2;:::;2 ' K 2;k indeed has a chromatic root of modulus [1 + o(1)] k= log k. Finally, for k 8 we prove that the generalized theta graph with a chromatic root that maximizes jz \Gamma 1j is the one with all path lengths equal to 2; we conjecture that this holds for all k.

Queue-mergesort is recently introduced by Golin and Sedgewick as an optimal variant of mergesorts in the worst case. In this paper, we present a complete analysis of the cost distribution of queue-mergesort, including the best, average and variance cases. The asymptotic normality of its cost is also established under the uniform permutation model. We address the corresponding optimality problems and show that if we fix the merging scheme then the optimal mergesort as far as the average number of comparisons is concerned is to divide as evenly as possible at each recursive stage (top-down mergesort). On the other hand, the variance of queue-mergesort reaches asymptotically the minimum value. We also characterize a class of mergesorts with the latter property. A comparative discussion is given on the probabilistic behaviors of top-down mergesort, bottom-up mergesort and queue-mergesort. We derive an "invariance principle" for asymptotic linearity of divide-and-conquer recurrences based on general "power-of-two" rules of which the underlying dividing rule of queue-mergesort is a special case. These analyses reveal an interesting algorithmic feature for general "power-of-two" rules.

In this paper, we study the Arneodo-Coullet-Tresser map F(x,y,z) = (ax − b(y − z),bx + a(y − z),cx − dx k + ez) where a,b,c,d,e are real with bd ̸ = 0 and k> 1 is an integer. We obtain stability regions for fixed points of F and symmetric period-2 points while c and e vary as parameters. Varying a and e as parameters, we show that there is a hyperbolic invariant set on which F is conjugate to the full shift on two or three symbols. We also show that chaotic behaviors of F while c and d vary as parameters and F is near an anti-integrable limit. Some numerical results indicates F has Hopf bifurcation, strange attractors, and nested structure of invariant tori. 1