In this work we analyze a direct product test in which each of two provers receives a subset of size n of a ground set U , |U | = Θ(n), and the two subsets intersect in about (1 − δ)n elements. We show that if each of the provers provides labels to the n elements it received, and the labels of the two provers agree in the intersection between the subsets with non-negligible probability, then th...

In 1971 Lovász proved the following cancellation law concerning the direct product of digraphs. If A, B and C are digraphs, and C admits no homomorphism into a disjoint union of directed cycles, then A × C ∼= B × C implies A ∼= B. On the other hand, if such a homomorphism exists, then there are pairs A ≁= B for which A×C ∼= B×C . This gives exact conditions on C that governwhether cancellation ...

April 2012 Abstract A manufacturer will often limit competition among downstream partners by authorizing only a select group of retailers to carry its product. However, it is not uncommon for authorized retailers to create an additional competitor by diverting units to an unauthorized seller. Prior research proposes that this diversion will not be optimal unless the diverted units are sold in n...

In this paper we study the direct product decompositions of closure operations and lattices of closed sets. We characterize the direct product decompositions of lattices of closed sets in terms of closure operations, and nd those decompositions of lattices which correspond to the decompositions of closures. If a closure on a nite set is represented by its implication base (i.e. a binary relatio...

Let α be an infinite cardinal. Let Tα be the class of all lattices which are conditionally α-complete and infinitely distributive. We denote by T ′ σ the class of all lattices X such that X is infinitely distributive, σ-complete and has the least element. In this paper we deal with direct factors of lattices belonging to Tα. As an application, we prove a result of Cantor-Bernstein type for latt...

We obtain a strong direct product theorem for two-party bounded round communication complexity. Let sucr(μ, f, C) denote the maximum success probability of an r-round communication protocol that uses at most C bits of communication in computing f(x, y) when (x, y) ∼ μ. Jain et al. [12] have recently showed that if sucr(μ, f, C) ≤ 2 3 and T (C − Ω(r)) · n r , then sucr(μ , f, T ) ≤ exp(−Ω(n/r))....

Various product operations are employed for constructing larger networks from smaller ones, among which direct-product operation is the most frequently employed one. The direct product of two graphs G1 andG2, denoted byG1×G2, is defined on vertex set V G1 ×V G2 , where two vertices x1, x2 and y1, y2 are adjacent to each other in G1 × G2 if and only if x1y1 ∈ E G1 and x2y2 ∈ E G2 . Other names f...

K e y w o r d s D i s t a n c e r e g u l a r graphs, Direct product. 1. I N T R O D U C T I O N E v e r y c o n n e c t e d g r a p h is r e p r e s e n t a b l e by m e a n s o f a level diagram (cf. [1]) as follows. C h o o s e a v e r t e x u, a n d le t i t be t h e sole r e s iden t of level zero. T h e ve r t i c e s on level i a re p rec i se ly t h o s e w h o s e d i s t a n c e f r o...

The direct product of two words is a naturally defined word on the alphabet of pairs of symbols. An infinite word is uniformly recurrent if each its subword occurs in it with bounded gaps. An infinite word is strongly recurrent if the direct product of it with each uniformly recurrent word is also uniformly recurrent. We prove that fixed points of the expanding binary symmetric morphisms are st...

Direct product theorems are formal statements of the intuition, “It is harder to solve multiple instances of a problem than to solve a single instance.” For this talk, the amplification of hardness will be in reducing the probability of success for a feasible algorithm. So a direct product result for circuits might say, “If no small circuit can compute Boolean function f(x) with probability at ...