In this paper we give new extremal bounds on polynomial threshold function (PTF) representations of Boolean functions. Our results include the following: • Almost every Boolean function has PTF degree at most n2 + O( √ n log n). Together with results of Anthony and Alon, this establishes a conjecture of Wang and Williams [32] and Aspnes, Beigel, Furst, and Rudich [4] up to lower order terms. • ...

Indeed, G must have an independent set of size at least n/x, and hence it has 2”‘X subsets of it whose average size is n/2x. On the other hand, standard estimates imply that the total number of subsets of cardinality not exceeding n/10 x log x of an n element set does not exceed 2”lK. Therefore, even if all these small subsets of vertices are independent, still the average size of an independen...

The full-information model was introduced by Ben-Or and Linial in 1985 to study collective coin-flipping: the problem of generating a common bounded-bias bit in a network of n players with t = t(n) faults. They showed that the majority protocol, in which each player sends a random bit and the output is the majority of the players’ bits, can tolerate t(n) = O( √ n) even in the presence of adapti...

This paper shows necessary and sufficient conditions on subsets of variable exponent spaces Lp(?)(?) in order to be weakly compact. Useful criteria are given extending Andô results for Orlicz spaces. As application, we prove that all separable Banach-Saks. Also, L-weakly compact inclusions between studied.

In a finite partially ordered set, Prob(x > y) denotes the proportion of linear extensions in which element x appears above element y. In 1969, S. S. Kislitsyn conjectured that in every finite poset which is not a chain, there exists a pair (x, y) for which 1/3 ≤ Prob(x > y) ≤ 2/3. In 1984, J. Kahn and M. Saks showed that there exists a pair (x, y) with 3/11 < Prob(x > y) < 8/11, but the full 1...

A discrete Morse function for a simplicial complex describes how to construct a homotopy equivalent CW-complex with hopefully fewer cells. We associate a boolean function with the simplicial complex and construct a discrete Morse function using its Fourier transform. Methods from theoretical computer science by O’Donnell, Saks, Schramm, and Servedio, together with experimental data on complexes...

OF THE DISSERTATION Information theory methods in communication complexity by Nikolaos Leonardos Dissertation Director: Michael Saks This dissertation is concerned with the application of notions and methods from the field of information theory to the field of communication complexity. It consists of two main parts. In the first part of the dissertation, we prove lower bounds on the randomized ...