For a function f : X ×Y → Z , the m-fold direct sum is the function f : X × Y → Z, defined by f(〈x1, . . . , xm〉, 〈y1, . . . , ym〉) ∆ = 〈f(x1, y1), . . . , f(xm, ym)〉. We show the following direct sum theorem for classical communication protocols, R(f) = Ω( m k R(f)) where R(f) is the the k-round private coins communication complexity of f and R(f) is the k-round public coin complexity of f . I...

A direct sum form is proposed for constructing a composite game from two 2 x 2 games, prisoner's dilemma and snowdrift game. This kind of direct sum form game is called a multiple roles game. The replicator dynamics of the multiple roles game with will-mixed populations is explored. The dynamical behaviors on square lattice are investigated by numerical simulation. It is found that the dynamica...

In this paper, we present a GPU-accelerated direct-sum boundary integral method to solve the linear Poisson-Boltzmann (PB) equation. In our method, a well-posed boundary integral formulation is used to ensure the fast convergence of Krylov subspace based linear algebraic solver such as the GMRES. The molecular surfaces are discretized with flat triangles and centroid collocation. To speed up ou...

We study a direct-sum question for read-once branching programs. If M(f) denotes the minimum average memory required to compute a function f(x1, x2, . . . , xn) how much memory is required to compute f on k independent inputs that arrive in parallel? We show that when the inputs are sampled independently from some domain X and M(f) = Ω(n), then computing the value of f on k streams requires ave...

1 Alberto Facchini, Dipartimento di Matematica Pura e Applicata, Università di Padova, Via Belzoni 7, I-35131 Padova, Italy, [email protected] 2 Wolfgang Hassler, Institut für Mathematik und Wissenschaftliches Rechnen, Karl-Franzens-Universität Graz, Heinrichstraße 36/IV, A-8010 Graz, Austria, [email protected] 3 Lee Klingler, Department of Mathematical Sciences, Florida Atlanti...

We prove a direct sum theorem for bounded round entanglement-assisted quantum communication complexity. To do so, we use the fully quantum definition for information cost and complexity that we recently introduced, and use both the fact that information is a lower bound on communication, and the fact that a direct sum property holds for quantum information complexity. We then give a protocol fo...

Let R be a ring and C a class of right R-modules closed under finite direct sums. If we suppose that C has a set of representatives, that is, a set V(C) ⊆ C such that every M ∈ C is isomorphic to a unique element [M ] ∈ V(C), then we can view V(C) as a monoid, with the monoid operation [M1] + [M2] = [M1 ⊕M2]. Recent developments in the theory of commutative monoids (e.g., [4], [15]) suggest tha...