Izumi Kubo

Impact of local topological information on random walks on finite graphs

It is just amazing that both of the mean hitting time and the cover time of a random walk on a finite graph, in which the vertex visited next is selected from the adjacent vertices at random with the same probability, are bounded by O(n3) for any undirected graph with order n, despite of the lack of global topological information. Thus a natural guess is that a better transition matrix is designable if more topological information is available. For any undirected connected graph G = (V,E), let P(β) = (puvβ)u,v∈V be a transition matrix defined by puvβ = exp [-βU(u, v)]/Σw∈N(u) exp [-βU(u, w)] for u∈V, v∈N(u), where β is a real number, N(u) is the set of vertices adjacent to a vertex u, deg(u) = |N(u)|, and U(., .) is a potential function defined as U(u, v) = log (max {deg(u), deg(v)}) for u∈V, v∈N(u). In this paper, we show that for any undirected graph with order n, the cover time and the mean hitting time with respect to P(1) are bounded by O(n2 log n) and O(n2), respectively. We further show that P(1) is best possible with respect to the mean hitting time, in the sense that the mean hitting time of a path graph of order n, with respect to any transition matrix, is Ω(n2).

INTERPOLATION OF CHEBYSHEV POLYNOMIALS AND INTERACTING FOCK SPACES

We discover a family of probability measures μa, 0 < a ≤ 1, which contains the arcsine distribution (a = 1) and semi-circle distribution (a = 1/2). We show that the multiplicative renormalization method can be used to produce orthogonal polynomials, called Chebyshev polynomials with one parameter a, which reduce to Chebyshev polynomials of the first and second kinds when a = 1 and 1/2 respectively. Moreover, we derive the associated Jacobi–Szego parameters. This one-parameter family of probability measures coincides with the vacuum distribution of the field operator of the interacting Fock spaces related to the Anderson model.

GENERATING FUNCTION METHOD FOR ORTHOGONAL POLYNOMIALS AND JACOBI-SZEG ˜ O PARAMETERS

Let μ be a probability measure on R with finite moments of all orders. Suppose μ is not supported by a finite set of points. Then there exists a unique sequence {Pn(x)}n=0 of orthogonal polynomials such that Pn(x) is a polynomial of degree n with leading coefficient 1 and the equality (x−αn)Pn(x) = Pn+1(x)+ωnPn−1(x) holds. The numbers {αn, ωn}n=0 are called the Jacobi-Szegö parameters of μ. The family {Pn(x), αn, ωn}n=0 determines the interacting Fock space of μ. In this paper we use the concept of generating function to give several methods for computing the orthogonal polynomials Pn(x) and the Jacobi-Szegö parameters αn and ωn. We also describe how to identify the orthogonal polynomials in terms of differential or difference operators.

MRM-FACTORS FOR THE PROBABILITY MEASURES IN THE MEIXNER CLASS

It is known that the gamma distribution γκ is MRM-applicable for h(x) = ex and for some hypergeometric functions also. We are interested in the problem to determine all possible MRM-factors of probability measures which are MRM-applicable for ex. We may say that the measures are in Meixner class. Such typical measures are Gaussian, Poisson, gamma, negative binomial and Meixner distributions and others are obtained from their modifications by affine transforms. We will give the complete list of MRM-factors different from ex up to trivial deformation: (1) for gamma distribution γκ. (2) for gamma distribution γκ. (3) for standard Gaussian distribution. (4) for shifted negative binomial distribution. σβNegBin(κ,p) with κ = 2, β = 1, for Meixner distribution Mκ,η with κ = 2 and for gamma distribution γκ with κ = 2, which is a special case of (2) with c = 1.

RENORMALIZATION, ORTHOGONALIZATION, AND GENERATING FUNCTIONS

Let μ be a probability measure on the real line with finite moments of all orders. Apply Gram-Schmidt orthogonalization process to the system {1, x, · · · , xn, . . . } to get a sequence {Pn}∞ n=0 of orthogonal polynomials with respect to μ. In this paper we explain a method of deriving a generating function ψ(t, x) for μ. The power series expansion of ψ(t, x) in t produces the explicit form of polynomials Pn, n ≥ 0. 1. Gaussian measure and Hermite polynomials Let μ be the Gaussian measure on the real line with mean 0 and variance σ, dμ(x) = 1 √ 2π σ e /2σ dx. (1.1) It is well-known that the Hermite polynomials

THE BRENKE TYPE GENERATING FUNCTIONS AND EXPLICIT FORMS OF MRM-TRIPLES BY MEANS OF Q-HYPERGEOMETRIC SERIES

An MRM-triple (h(x), ρ(t), B(t)) gives a generating function B(t)h(ρ(t)x) of some orthogonal polynomials on ℝ. In particular, B(t)h(tx) is called the Brenke type.19 In this paper, we shall determine all MRM-triples and associated Jacobi-Szego parameters of this type with showing very careful computations in detail. (h(x), t, B(t)) is classified into four categories. In any case, h(x) and B(t) can be expressed in terms of two kinds of q-hypergeometric series, old basic and basic hypergeometric series, rΦs and rϕs, respectively. As examples, our results contain generating functions of the Al-Salam-Carlitz (I and II), little q-Laguerre, q-Laguerre, and discrete q-Hermite (I and II) polynomials. Our results are more complete and general than those of Refs. 20 and 21 by Chihara. The following are special cases of our results in each class. Here {αn, ωn} are the Jacobi-Szego parameters.

A New Nonrecursive Pseudorandom Number Generator Based on Chaos Mappings

Abstract We introduce a new pseudorandom number generator SSR (the Simplified Shift-Real random number generator) which generates the k-th random number nonrecursively (directly) based on chaos mappings on the interval [1, 2). We investigate properties of SSR random numbers and give the theoretical background of generation of random numbers. A practical integral (all-integer) version SSI of SSR, which is suitable for parallel computation, is also provided.