In this paper we describe an operation on directed graphs which produces a graph with fewer vertices, such that the C-algebra of the new graph is Morita equivalent to that of the original graph. We unify and generalize several related constructions, notably delays and desingularizations of directed graphs.

Knowledge representation (KR) is used to store and retrieve meaningful data. This data is saved using dynamic data structures that are suitable for the style of KR being implemented. The KR allows the system to manipulate the knowledge in the data by using reasoning operations. The data structure, together with the contents of the transformed knowledge, is known as the knowledge base (KB). An a...

Abstract We consider the interlacement Poisson point process on the space of doubly-infinite Zd -valued trajectories modulo time-shift, tending to infinity at positive and negative infinite times. The set of vertices and edges visited by at least one of these trajectories is the graph induced by the random interlacements at level u of Sznitman [9]. We prove that for any u> 0, almost surely, the...

Some improved bounds on the number of directions not determined by a point set in the affine space AG(k, q) are presented. More precisely, if there are more than p(q−1) directions not determined by a set of qk−1 points S then every hyperplane meets S in 0 modulo p points. This bound is shown to be tight in the case p = q and when q = p sets of qk−1 points that do not meet every hyperplane in 0 ...

In [Ha] Haviv gave a way of assigning Lie tensors to directed trivalent graphs. Weight systems on oriented chord idagrams modulo 6T can then be constructed from such tensors. In this paper we give explicit combinatorial formulas of weight systems using Manin triples constrcted from classical Lie algebras. We then compose these oriented weight systems with the averaging map to get weight systems...

We prove a higher dimensional generalization of Gross and Zagier’s theorem on the factorization of differences of singular moduli. Their result is proved by giving a counting formula for the number of isomorphisms between elliptic curves with complex multiplication by two different imaginary quadratic fields K and K′, when the curves are reduced modulo a supersingular prime and its powers. Equi...

Tits [ 191 has initiated the study of geometries having properties strongly resembling those of buildings (compare [6]). His main result is that, in general, such a geometry is the image of a building under a suitable type of morphism. These geometries that are almost buildings were called GABS in [lo], and finite examples were described in [2, 8, 10, 15, 161 that are not buildings but have hig...

Solving Diophantine equation is one of the main problem in number theory for a long time. It is very difficult but wonderful. For example, it took over 300 years to see that Xn + Y n = Zn has no nontrivial integers solution when n ≥ 3. We would like to consider an easier problem: solving the Diophantine equation modulo p, where p is a prime number. We expect that this problem is easy enough to ...

A set of nite structures is in Binary NP if it can be characterized by existential second order formulas in which second order quan-tiication is over relations of arity 2. In DLS95] subclasses of Binary NP were considered, in which the second order quantiiers range only over certain classes of relations. It was shown that many of these subclasses coincide and that all of them can be ordered in ...

Let α : Zn → Zn be a permutation and Ψ(α) be the number of collinear triples modulo n in the graph of α. Cooper and Solymosi had given by induction the bound minα Ψ(α) ≥ ⌈(n − 1)/4⌉ when n is a prime number. The main purpose of this paper is to give a direct proof of that bound. Besides, the expected number of collinear triples a permutation can have is also been determined.