The model of statistical physics on a countable amenable group G is considered. For the natural action of G on the space of configurations S^G, and for any closed invariant set X we prove that there exists pressure which corresponds to a potential with finite norm on X (in the sense of the limit with respect to any Følner sequence of sets in G). The existence of an equilibrium measure is established.
We consider the spaces of function on the circle whose Fourier transform is p-summable. We obtain estimates for the norms of exponential functions deformed by a C1 -smooth phase.
We study the number f of connected components in the complement to a finite set (arrangement) of closed submanifolds of codimension 1 in a closed manifold M. In the case of arrangements of closed geodesics on an isohedral tetrahedron, we find all possible values for the number fof connected components. We prove that the set of numbers that cannot be realized by the number f of an arrangement of n ≥ 71 projective planes in the three-dimensional real projective space is contained in the similar known set of numbers that are not realizable by arrangements of n lines on the projective plane. For Riemannian surfaces M we express the number f via a regular neighbourhood of a union of immersed circles and the multiplicities of their intersection points. For m-dimensional Lobachevskii space we find the set of all possible numbers f for hyperplane arrangements.
Manifolds with locally standard half-dimensional torus actions represent a large and important class of spaces. Cohomology rings of such manifolds are known in particular cases but in general even Betti numbers are difficult to compute. Our approach to this problem is the following: we consider the orbit type filtration on a manifold with locally standard action and study the induced spectral sequence in homology. It collapses at a second page only in the case when the orbit space is homologically trivial. The cohomology ring in this case was already computed. Nevertheless, we can completely describe the spectral sequence under more general assumptions, namely when all proper faces of the orbit space are acyclic. The theory of sheaves and cosheaves on finite partially ordered sets is used in the computation. The second page of the spectral sequence can be described as the cohomology of a certain sheaf on the dual simplicial poset, whose value on a simplex is the homology of the corresponding toric orbit. We study this and related sheaves and establish the generalizations of the Poincare duality and the Zeeman-McCrory spectral sequence for sheaves of ideals of exterior algebras.
We enumerate the local Petrovskii lacunas (that is, the domains of local regularity of the principal fundamental solutions of strictly hyperbolic PDE's with constant coefficients in R^N) at the parabolic singular points of their wavefronts (i.e., at the points of types P_8^1, P_8^2, +X_9, -X_9, X_9^1, X_9^2, J_10^1, J_10^3). These points form the next difficult class of the natural classification of singular points after the so-called simple singularities A_k, D_k, E_6, E_7, E_8, studied previously. Also we promote a computer program counting for topologically different Morsifications of critical points of smooth functions, and hence also for local components of the complement of a generic wavefront at its singular points.
We say that a group G acts infinitely transitively on a set X if for every m ε N the induced diagonal action of G is transitive on the cartesian mth power X m\δ with the diagonals removed. We describe three classes of affine algebraic varieties such that their automorphism groups act infinitely transitively on their smooth loci. The first class consists of normal affine cones over flag varieties, the second of nondegenerate affine toric varieties, and the third of iterated suspensions over affine varieties with infinitely transitive automorphism groups. Bibliography: 42 titles.