We describe the homotopy types of complexes of partite graphs and hypergraphs with a fixed set of vertices covered by their edges
We study the cohomology of complexes of ordinary (non-decorated) graphs, introduced by M. Kontsevich. We construct spectral sequences converging to zero whose first page contains the graph cohomology. In particular, these spectral sequences may be used to show the existence of an infinite series of previously unknown and provably non-trivial cohomology classes, and put constraints on the structure of the graph cohomology as a whole.
I describe a combinatorial construction of the cohomology classes in compactified moduli spaces of curves ZˆI∈H∗(barM_g,n) starting from the following data: an odd derivation I, whose square is non-zero in general, I2≠0, acting on a ℤ/2ℤ-graded associative algebra with odd scalar product. The constructed cocycles were first described in the theorem 2 in the author's paper "Noncommmutative Batalin-Vilkovisky geometry and Matrix integrals". Comptes Rendus Mathematique, 348, pp. 359-362, arXiv:0912.5484 , preprint HAL-00102085 (09/2006). By the theorem 3 from loc.cit. the family of the cohomology classes obtained in the case of the algebra Q(N) and the derivation I=[Λ,⋅] coincided with the generating function of products of ψ−classes. This was the first nontrivial computation of categorical Gromov-Witten invariants of higher genus. The result matched with the mirror symmetry prediction, i.e. with the classical (non-categorical) Gromov-Witten descendent invariants of a point for all genus. As a byproduct of that computation a new combinatorial formula for products of ψ-classes ψi=c1(T∗pi) in the cohomology H∗(barM_g,n) is written out.
A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.