In this paper we prove that a polynomial system can be always mapped by some rational mapping into a general quadratic system. This result is useful for studies of systems with a complex dynamics. One generalization is discussed. Our approach is based on theorems on algebraic dependent polynomials. Key-Words:Polynomial, polynomial system, quadratic system, rational mapping, algebraic dependence...

An independent set Ic is a critical independent set if |Ic|−|N(Ic)| ≥ |J | − |N(J)|, for any independent set J . The critical independence number of a graph is the cardinality of a maximum critical independent set. This number is a lower bound for the independence number and can be computed in polynomial-time. Any graph can be decomposed into two subgraphs where the independence number of one s...

In this paper we study unimodality problems for the independence polynomial of a graph, including unimodality, log-concavity and reality of zeros. We establish recurrence relations and give factorizations of independence polynomials for certain classes of graphs. As applications we settle some unimodality conjectures and problems. © 2010 Elsevier Ltd. All rights reserved.

A scaling vector = (1 ; ; r) T is a compactly supported and vector-valued distribution vector that satisses a matrix reenement equation (x) = P P k (2x ? k); where (P k) is a nite matrix sequence. We call P (z) = 1 2 P P k z k the symbol of. A symbol P (z) is said two-scale similar to a polynomial matrix Q(z) if there is an invertible polynomial matrix T (z) that satisses P (z) = T (z 2)Q(z)T ?...

We show that there are polynomial-time algorithms to compute maximum independent sets in the categorical products of two cographs and two splitgraphs. The ultimate categorical independence ratio of a graph G is defined as limk→∞ α(G )/n. The ultimate categorical independence ratio is polynomial for cographs, permutation graphs, interval graphs, graphs of bounded treewidth and splitgraphs. When ...

Suppose that an independence system (E, ) is characterized by a subroutine which indicates in unit time whether or not a given subset of E is independent. It is shown that there is no algorithm for generating all the K maximal independent sets of such an independence system in time polynomial in IEI and K, unless V. However, it is possible to apply ideas of Paull and Unger and of Tsukiyama et a...