In a seminal paper from 1983, Burr and Erd1⁄2os started the systematic study of Ramsey numbers of cliques vs. large sparse graphs, raising a number of problems. In this paper we develop a new approach to such Ramsey problems using a mix of the Szemerédi regularity lemma, embedding of sparse graphs, Turán type stability, and other structural results. We give exact Ramsey numbers for various clas...

A hybrid control system is a control theoretic model for a computer controlled engineering system. A de nition of a hybrid control system is formulated that consists of a product of a nite state automaton and of a family of continuous control systems. An example of a transportation system consisting of a line of conveyor belts is used as a running example. The realization problem for this class...

Chen, Lih, and Wu conjectured that for r ≥ 3, the only connected graphs with maximum degree at most r that are not equitably r-colorable are Kr,r (for odd r) and Kr+1. If true, this would be a joint strengthening of the Hajnal-Szemerédi Theorem and Brooks' Theorem. Chen, Lih, and Wu proved that their conjecture holds for r = 3. In this paper we study properties of the hypothetical minimum count...

Let A = {a1, a2, . . . } (a1 < a2 < . . . ) be a xed in nite sequence of positive integers, and let Rk(n) denote the number of solutions of ai1 + ai2 + · · · + aik = n, ai1 ∈ A, . . . , aik ∈ A. For k = 2, P. Erd®s and A. Sárközy proved if F (n) is a nice arithmetic function then there exists a sequence A such that |R2(n)−F (n)| (F (n) log n)1/2. The aim of this paper is to extend their result ...

The Shapley-Shubik power index in a voting situation depends on the number of orderings in which each player is pivotal. The Banzhaf power index depends on the number of ways in which each voter can e¤ect a swing. We introduce a combinatorial method based in generating functions for computing these power indices e¢ciently and we study the time complexity of the algorithms. We also analyze the m...

Assume that a cellular automaton (CA) rule enlarges subsets of Z 2 , and does so in such a way that a larger set results in a larger outcome. Such models are called monotone solidi cation CA. In the critical case, these dynamics cannot cover the lattice starting from any nite set, but are able to do so from any set with nite complement. We assume that the initial set is a product measure with s...

We show that a fast algorithm for the QR factorization of a Toeplitz or Hankel matrix A is weakly stable in the sense that RR is close to AA. Thus, when the algorithm is used to solve the semi-normal equations RRx = A b, we obtain a weakly stable method for the solution of a nonsingular Toeplitz or Hankel linear systemAx = b. The algorithm also applies to the solution of the full-rank Toeplitz ...

A hybrid control system is a control theoretic model for a computer controlled engineering system. A de nition of a hybrid control system is formulated that consists of a product of a nite state automaton and of a family of continuous control systems. An example of a transportation system consisting of a line of conveyor belts is used as a running example. The realization problem for this class...

In his paper Motivic equivalence of quadratic forms , Izhboldin modi es a conjecture of Lam and asks whether two quadratic forms, each of which isomorphic to the product of an Albert form and a k-fold P ster form, are similar provided they are equivalent modulo I k+3 . We relate this conjecture to another conjecture on the dimensions of anisotropic forms in I k+3 . As a consequence, we obtain t...

There are many di erent ways of proving formulas in proposition logic. Many of these can easily be characterized as forms of resolution (e.g. [12] and [9]). Others use so-called binary decision diagrams (BDDs) [2, 10]. Experimental evidence suggests that BDDs and resolution based techniques are fundamentally di erent, in the sense that their performance can di er very much on benchmarks [14]. I...