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Combinatorial Algebraic Topology is concerned with computing algebraic invari-ants of combinatorially given cell complexes by combinatorial means. It arises from the quest of explicit descriptions of invariants in Algebraic Topology, and has applications in Discrete Mathematics. In this talk we shall outline the general philosophy of Combinatorial Algebraic… (More)

To every directed graph G one can associate a complex (G) consisting of directed subforests. This construction, suggested to us by R. Stanley, is especially important in the case of a complete double directed graph Gn, where it leads to studying some interesting representations of the symmetric group and corresponds (via Stanley-Reisner correspondence) to… (More)

We introduce the notion of nonevasive reduction, and show that for any monotone poset map ϕ : P → P , the simplicial complex ∆(P) NE-reduces to ∆(Q), for any Q ⊇ Fix ϕ. As a corollary, we prove that for any order-preserving map ϕ : P → P satisfying ϕ(x) ≥ x, for any x ∈ P , the simplicial complex ∆(P) collapses to ∆(ϕ(P)). We also obtain a generalization of… (More)

We extend the combinatorial Morse complex construction to the arbitrary free chain complexes, and give a short, self-contained, and elementary proof of the quasi-isomorphism between the original chain complex and its Morse complex. Even stronger, the main result states that, if C * is a free chain complex, and M an acyclic matching, then C * = C M * ⊕ T * ,… (More)

This paper concerns itself with a family of simplicial complexes, which we call the view complexes. Our choice of objects of study is motivated by theoretical distributed computing, since the view complex is a key simpli-cial construction used for protocol complexes in the snapshot computational model. We show that the view complex View n can be collapsed… (More)

Given a set of m coins out of a collection of coins of k unknown distinct weights, we wish to decide if all the m given coins have the same weight or not using the minimum possible number of weighings in a regular balance beam. Let m(n; k) denote the maximum possible number of coins for which the above problem can be solved in n weighings. It is known that… (More)

We discuss the problem of maximizing the number of coins, for which, using just n weighings, one can tell whether all of them are of the same weight or not, under condition that the weights of the coins are generic. The first purpose of the paper is to show the connection between this problem and a problem in lattice geometry. Using this approach, we are… (More)