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## Voronoi's Conjecture for Extensions of Voronoi Parallelohedra

In 1908 Voronoi conjectured that every parallelohedron is a Voronoi parallelohedron for some Euclidean metric in E^d . Although the conjecture is still neither proved, nor disproved, there are several positive results for some special classes of parallelohedra. In this paper we extend the list of such classes by one new case. Let I be a segment in the d-dimensional Euclidean space E^d . Let P and P + I be parallelohedra in E^d , where the plus sign denotes the Minkowski sum. We prove that, if Voronoi’s Conjecture holds for P , then Voronoi’s Conjecture holds for P + I as well.

We show that the Voronoi conjecture is true for parallelohedra with simply connected δ-surfaces. That is, we show that if the boundary of parallelohedron P remains simply connected after removing closed nonprimitive faces of codimension 2, then P is affinely equivalent to a Dirichlet–Voronoi domain of some lattice. Also, we construct the π-surface associated with a parallelohedron and give another condition in terms of a homology group of the constructed surface. Every parallelohedron with a simply connected δ-surface also satisfies the condition on the homology group of the π-surface.

We show that the Voronoi conjecture is true for parallelohedra with simply connected δδ-surfaces. That is, we show that if the boundary of parallelohedron PP remains simply connected after removing closed nonprimitive faces of codimension 2, then PP is affinely equivalent to a Dirichlet–Voronoi domain of some lattice. Also, we construct the ππ-surface associated with a parallelohedron and give another condition in terms of a homology group of the constructed surface. Every parallelohedron with a simply connected δδ-surface also satisfies the condition on the homology group of the ππ-surface.

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.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.