An Institute affiliate and former director, Robert H. Haveman is John Bascom Professor of Economics and Director of the La Follette Institute of Public Affairs at the University of Wisconsin-Madison. This essay is based on the Daniel Saks Memorial Lecture, delivered at Vanderbilt University on March 28,1988, and is drawn from the author's forthcoming book, Starting Even: An Equal Opportunity Pr...

We give a simple proof of the OSSS inequality (O’Donnell, Saks, Schramm, Servedio, FOCS 2005). The inequality states that for any decision tree T calculating a Boolean function f : {0,1} → {−1,1}, we have Var[ f ]≤ ∑i δi(T ) Infi( f ), where δi(T ) is the probability that the input variable xi is read by T and Infi( f ) is the influence of the ith variable on f . ACM Classification: G.2, G.3, F...

This paper concerns the open problem of Lovasz and Saks regarding the relationship between the communication complexity of a boolean function and the rank of the associated matrix. We rst give an example exhibiting the largest gap known. We then prove two related theorems. y Extended Abstract appeared in FOCS 1994. subject "MAIL ME CLEAR", body "pub/eccc/ftpmail.txt" followed by an empty line, ...

almost everywhere on A. At the conclusion of this paper we state a number of interesting problems which are closely connected with the theorems which we prove. We wish to thank Dr. S. Saks for many helpful suggestions. 2. Measurability of second partial derivatives. We require the following lemma. Lemma 1. Let u(x, y) be a function of. Baire, and let Anbe the point set on which the following in...

The celebrated Dilworth theorem (Ann. of Math. 5 1 (1950), 161-166) on the decomposition of finite posets was extended by Greene and Kleitman (J. Combin. Theory Ser. A 20 (1976), 41-68). Using the Gallai-Milgram theorem (Acta Sci. Math. 2 1 (1960), 18 l-186) we prove a theorem on acyclic digraphs which contains the Greene-Kleitman theorem. The method of proof is derived from M. Saks’ elegant pr...

We show lower bounds of Ω( √ n) and Ω(n) on the randomized and quantum communication complexity, respectively, of all nvariable read-once Boolean formulas. Our results complement the recent lower bound ofΩ(n/8) by Leonardos and Saks [LS09] andΩ(n/2 log ) by Jayram, Kopparty and Raghavendra [JKR09] for randomized communication complexity of read-once Boolean formulas with depth d. We obtain our ...

The present paper deals with the study of the notion of an atom of a function m defined on an effect algebra L with values in [0,∞]; a few examples of atoms for null-additive as well as for non-null-additive functions are also given. We have proved a Saks type decomposition theorem for an element a with m(a) > 0 (for a suitable m), which does not contain any atom of m, in a σ-complete effect al...

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