A counting polynomial P(x) is a description of a graph property P(G) in terms of a sequence of numbers so that the exponents express the extent of its partitions while the coefficients are related to the number of partitions of a given extent. Basic definitions and some properties are given for two classes of polynomials, called here polynomials of vertex proximity and edge proximity, respectiv...

An introduction is given to the use orthogonal polynomials in distance regular graphs and enumeration. Some examples in each area are given, along with open problems.

Let K be a local non-Archimedean field of positive characteristic and let L be the degree-n unramified extension of K. Let θ be a smooth character of L× such that for each nontrivial γ ∈ Gal(L/K), θ and θ/θ have the same level. Via the local Langlands and Jacquet-Langlands correspondences, θ corresponds to an irreducible representation ρθ of D×, where D is the central division algebra over K wi...

Let G be a reductive Lie group and O the coadjoint orbit of a hyperbolic element of g. By π is denoted the unitary irreducible representation of G associated with O by the orbit method. We give geometric interpretations in terms of concepts related to O of the constant π(g), for g ∈ Z(G). In the spirit of the orbit method we interpret geometrically the infinitesimal character of the differentia...

We continue the study of the distribution of closed geodesics on nilmanifolds Γ\N , constructed from a simply connected 2-step nilpotent Lie group N with a left invariant metric and a lattice Γ in N . We consider a Lie group N with associated 2-step nilpotent Lie algebra N constructed from an irreducible representation of a compact semisimple Lie algebra g0 on a real finite dimensional vector s...

Problem For a given class of Boolean functions, such as polynomial-size Boolean circuits or Disjunctive Normal Form ( ? @FA ) formulas, the goal is to design polynomial-time learning algorithms for any unknown I*J and ask a polynomial number of queries. The output of the learning algorithm should be a function of polynomial size satisfying KL M . The polynomial functions bounding the running ti...

We prove that the genus polynomials of the graphs called iterated claws are real-rooted. This continues our work directed toward the 25-year-old conjecture that the genus distribution of every graph is log-concave. We have previously established log-concavity for sequences of graphs constructed by iterative vertexamalgamation or iterative edge-amalgamation of graphs that satisfy a commonly obse...

The goal of this paper is to present a data model to manage highly variable data of the real world. The management of such data raises new database problems. These problems are the data volume, the great number of states that database operators have to manipulate, as well as the deficiency of versions that leads to non-determinist temporal queries. To this end, a new approach to represent and t...

In the context of second-order polynomial-time computability, we prove that there is no general function space construction. We proceed to identify restrictions on the domain or the codomain that do provide a function space with polynomial-time function evaluation containing all polynomial-time computable functions of that type. As side results we show that a polynomial-time counterpart to admi...

We consider representation classes of polynomial query complexity in Angluin's exact learning model with all three possible combinations of membership and (proper) equivalence queries allowed. We show that in all three cases, polynomial query complexity implies already polynomial-time learnability where the learner additionally has access to an oracle in p 2. As an application, it follows that ...