Volume 6 (2006), Number 4. Abstracts
F. Aicardi. On Trigroups and Semigroups of Binary Quadratic Forms Values and of their Associated Linear Operators [PDF]
Some algebraic properties of binary quadratic forms are presented, related to the fact that the product of any three values attained by a binary quadratic form with integer coefficients on the integer points of the plane is always a value attained by the same form. A sufficient condition on the form coefficients for the set of values to be a semigroup is also given.
Keywords. Integer binary quadratic forms, multiplicative semigroups
2000 Mathematics Subject Classification. 11E12, 15A63
A. Borodin and G. Olshanski. Meixner Polynomials and Random Partitions [PDF]
The paper deals with a 3-parameter family of probability measures on the set of partitions, called the z-measures. The z-measures first emerged in connection with the problem of harmonic analysis on the infinite symmetric group. They are a special and distinguished case of Okounkov's Schur measures. It is known that any Schur measure determines a determinantal point process on the 1-dimensional lattice. In the particular case of z-measures, the correlation kernel of this process, called the discrete hypergeometric kernel, has especially nice properties. The aim of the paper is to derive the discrete hypergeometric kernel by a new method, based on a relationship between the z-measures and the Meixner orthogonal polynomial ensemble. In another paper (Prob. Theory Rel. Fields 135 (2006), 84–152) we apply the same approach to a dynamical model related to the z-measures.
Keywords. Random partitions, random Young diagrams, determinantal point processes, correlation functions, correlation kernels, orthogonal polynomial ensembles, Meixner polynomials, Krawtchouk polynomials
2000 Mathematics Subject Classification. 60C05, 33C45
B. Fayad and M. Lemańczyk. On the Ergodicity of Cylindrical Transformations given by the Logarithm [PDF]
Given α ∈ [0,1] and φ: T → R measurable, the cylindrical cascade Sα,φ is the map from T×R to itself given by Sα,φ(x,y) = (x+α,y+φ(x)), which naturally appears in the study of some ordinary differential equations on R3. In this paper, we prove that for a set of full Lebesgue measure of α ∈ [0,1] the cylindrical cascades Sα,φ are ergodic for every smooth function φ with a logarithmic singularity, provided that the average of φ vanishes.
Closely related to Sα,φ are the special flows constructed above Rα and under φ+c, where c ∈ R is such that φ+c > 0. In the case of a function φ with an asymmetric logarithmic singularity, our result gives the first examples of ergodic cascades Sα,φ with the corresponding special flows being mixing. Indeed, if the latter flows are mixing, then the usual techniques used to prove the essential value criterion for Sα,φ, which is equivalent to ergodicity, fail, and we devise a new method to prove this criterion, which we hope could be useful in tackling other problems of ergodicity for cocycles preserving an infinite measure.
Keywords. Cylindrical cascade, essential value, logarithmic singularity
2000 Mathematics Subject Classification. 37C40, 37A20, 37C10
E. Gutkin and S. Tabachnikov. Complexity of Piecewise Convex Transformations in Two Dimensions, with Applications to Polygonal Billiards on Surfaces of Constant Curvature [PDF]
We introduce piecewise convex transformations, and develop geometric tools to study their complexity. We apply the results to the complexity of polygonal inner and outer billiards on surfaces of constant curvature.
Keywords. Geodesic polygon, constant curvature, complexity, inner billiard, outer billiard
2000 Mathematics Subject Classification. 53D25, 37E99, 37B10
Yu. Neretin. Central Extensions of Groups of Symplectomorphisms [PDF]
We construct canonically defined central extensions of groups of symplectomorphisms. We show that these central extensions are nontrivial for tori of dimension ≥6 and for two-dimensional surfaces of genus ≥3.
Keywords. 57R57, 58D05, 19C09, 22E40
2000 Mathematics Subject Classification. Central extensions, symplectomorphisms, mapping class groups, Teichmüller group, Dehn twist, Calabi invariant, flux homomorphism
R. Uribe-Vargas. A Projective Invariant for Swallowtails and Godrons, and Global Theorems on the Flecnodal Curve [PDF]
We show some generic (robust) properties of smooth surfaces immersed in the real 3-space (Euclidean, affine or projective), in the neighbourhood of a godron (called also cusp of Gauss): an isolated parabolic point at which the (unique) asymptotic direction is tangent to the parabolic curve. With the help of these properties and a projective invariant that we associate to each godron we present all possible local configurations of the tangent plane, the self-intersection line, the cuspidal edge and the flecnodal curve at a generic swallowtail in R3. We present some global results, for instance: In a hyperbolic disc of a generic smooth surface, the flecnodal curve has an odd number of transverse self-intersections (hence at least one self-intersection).
Keywords. Geometry of surfaces, tangential singularities, swallowtail, parabolic curve, flecnodal curve, cusp of Gauss, godron, wave front, Legendrian singularities
2000 Mathematics Subject Classification. 14B05, 32S25, 58K35, 58K60, 53A20, 53A15, 53A05, 53D99, 70G45
Moscow Mathematical Journal