In the previous lectures, we have defined the Shannon Capacity of a graph, Θ(G). This lecture is concerned with the Shannon Capacity of the disjoint union G +H of two graphs G and H. The union of two graphs is, informally, what you get when you just place one graph next to the other. In 1956, Shannon conjectured [1] that Θ(G + H) = Θ(G) + Θ(H). As we will see, this is a very natural conjecture ...

Say a set H of graphs is heroic if there exists k such that every graph containing no member of H as an induced subgraph has cochromatic number at most k. (The cochromatic number of G is the minimum number of stable sets and cliques with union V (G).) Assuming an old conjecture of Gyárfás and Sumner, we give a complete characterization of the finite heroic sets. This is a consequence of the fol...

Let k an algebraically closed field, P the n-dimensional projective space over k and TPn the tangent vector bundle of P . In this paper I prove the following result : for every integer l, for every non-negative integer s, if Zs is the union of s points in sufficiently general position in P, then the restriction map H(P, TPn(l)) → H 0(Zs, TPn(l)|Zs) has maximal rank. This result implies that the...

In this paper I present a conjecture for a recursive algorithm that finds each permutation of combining two sets of objects (AKA the Shuffle Product). This algorithm provides an efficient way to navigate this problem, as each atomic operation yields a permutation of the union. The permutations of the union of the two sets are represented as binary integers which are then manipulated mathematica...

This paper is concerned with the design of eecient algorithms for generating global name-space communication sets based on execution of array assignment statements on distributed-memory parallel computers. For general cases, although the communication sets can be represented by the union of a functional number of closed forms, these sets cannot be represented by a xed number of closed forms. Cl...

We show that the iterated images of a Jacobian pair f : C 2 → C 2 stabilize; that is, all the sets f k (C 2) are equal for k sufficiently large. More generally, let X be a closed algebraic subset of C N , and f : X → X an open polynomial map with X − f (X) a finite set. We show that the sets f k (X) stabilize, and for any cofinite subset Ω ⊆ X with f (Ω) ⊆ Ω, the sets f k (Ω) stabilize. We appl...

In this paper, we study several propositional team logics that are closed under unions, including inclusion logic. We show all these expressively complete, and introduce sound complete systems of natural deduction for logics. also discuss the locality property its connection with interpolation in