Asymptotic zero distribution of hypergeometric polynomials

We show that the zeros of the hypergeometric polynomials , , cluster on the loop of the lemniscate as . We also state the equations of the curves on which the zeros of , lie asymptotically as . Auxiliary results for the asymptotic zero distribution of other functions related to hypergeometric polynomials are proved, including Jacobi polynomials with varying parameters and associated Legendre functions. Graphical evidence is provided using Mathematica. This revised version was published online in June 2006 with corrections to the Cover Date.     

Some limit relations in the A1 tableau of Dunkl-Cherednik operators

In this paper we study some limit relations involving some q-special functions related with the A₁ (root system) tableau of Dunkl-Cherednik operators. Concretely we consider the limits involving the nonsymmetric q-ultraspherical polynomials (q-Rogers polynomials), ultraspherical polynomials (Gegenbauer polynomials), q-Hermite and Hermite polynomials.     

Asymptotic expansions of some matrix argument hypergeometric functions, with applications to macromolecules

We have studied the asymptotics of two special two-matrix hypergeometric functions. The validity of the asymptotic expressions for these functions is seen in several selected numerical comparisons between the exact and asymptotic results. These hypergeometric functions find applications in configuration statistics of macromolecules as well as multivariate statistics.This work was supported by grant DE-FG06-84ER45123 from the Department of Energy, U.S.A.     

A special identity between three 2F1(a,b;c;4) hypergeometric series

As a by-product of the calculation of eigenvalues of a particular SO(3) invariant operator in the enveloping algebra of SO(5), a three terms relation between three hypergeometric series emerges.     

The young diagrams of a pair of irreducible characters of S n with the same zero set on S n ?

In studying the pairs of irreducible characters of the symmetric group S _n with the same zero set on A _n or S _n ∖ A _n (as well as the pairs of irreducible characters with the same zero set on the alternating group A _n ), the results are important on the connection between the Young diagrams of the characters of these pairs. We prove a theorem that considerably generalizes two previous results of frequent use in this direction. Original Russian Text Copyright ? 2008 Belonogov V. A. The author was supported by the Russian Foundation for Basic Research (Grant 07-01-00148) and the RFBR-NSFC (Grant 05-01-39000). __________ Ekaterinburg. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 49, No. 5, pp. 992–1006, September–October, 2008.     

An everywhere convergent series representation of the distribution of Hotelling's generalized T0

A new series representation of the exact distribution of Hotelling's generalized T₀² statistic is obtained. Unlike earlier work, the series representation given here is everywhere convergent. Explicit formulae are given for both the null and the non-central distributions. Earlier results by [1], 215–225), which are convergent on the interval [0, 1), are also derived quite simply from our formulae. The paper therefore provides a solution to the long standing problem of the exact distribution of the T₀² statistic in the general case.     

Characterization of a class of bivariate distribution functions

Let F_X,Y(x,y) be a bivariate distribution function and P_n(x), Q_m(y), n, m = 0, 1, 2,…, the orthonormal polynomials of the two marginal distributions F_X(x) and F_Y(y), respectively. Some necessary conditions are derived for the co-efficients c_n, n = 0, 1, 2,…, if the conditional expectation E[P_n(X) ∥ Y] = c_nQ_n(Y) holds for n = 0, 1, 2,…. Several examples are given to show the application of these necessary conditions.     

Differential-geometric Newton method for the best rank-(R1, R2, R3) approximation of tensors

An increasing number of applications are based on the manipulation of higher-order tensors. In this paper, we derive a differential-geometric Newton method for computing the best rank-(R ₁, R ₂, R ₃) approximation of a third-order tensor. The generalization to tensors of order higher than three is straightforward. We illustrate the fast quadratic convergence of the algorithm in a neighborhood of the solution and compare it with the known higher-order orthogonal iteration (De Lathauwer et al., SIAM J Matrix Anal Appl 21(4):1324–1342, 2000). This kind of algorithms are useful for many problems. This paper presents research results of the Belgian Network DYSCO (Dynamical Systems, Control, and Optimization), funded by the Interuniversity Attraction Poles Programme, initiated by the Belgian State, Science Policy Office. The scientific responsibility rests with its authors. Research supported by: (1) Research Council K.U.Leuven: GOA-Ambiorics, CoE EF/05/006 Optimization in Engineering (OPTEC), (2) F.W.O.: (a) project G.0321.06, (b) Research Communities ICCoS, ANMMM and MLDM, (3) the Belgian Federal Science Policy Office: IUAP P6/04 (DYSCO, “Dynamical systems, control and optimization”, 2007–2011), (4) EU: ERNSI. M. Ishteva is supported by a K.U.Leuven doctoral scholarship (OE/06/25, OE/07/17, OE/08/007), L. De Lathauwer is supported by “Impulsfinanciering Campus Kortrijk (2007–2012)(CIF1)” and STRT1/08/023.     

Surfaces with pg = 2, K2 = 3 and a pencil of curves of genus 2

We classify minimal smooth surfaces of general type with K ² = 3, p _g = 2 which admit a fibration of curves of genus 2.We prove that they form an irreducible set of dimension 22 in their moduli space.      

K2 of a Ring via the G-Construction

We interpret the Steinberg symbols x_i,j(a) as homotopies contracting the elementary matrices e_i,j(a), the latters being represented by certain arcs in a simplicial model of the K-theory. We further prove the Steinberg relations for these homotopies. This provides an explicit map from K₂ of a ring, defined classically as ker(St(R) → GL(R)), to π₂ of the G-construction assigned to R. Though the two groups are known to be isomorphic, a certain work is to be done to prove that this explicit map is an isomorphism. Mathematics Subject Classification 1991: Primary 19B99, 19D99; secondary 18E10, 18F25.     

A probabilistic algorithm for determining the fundamental matrix of a block M/G/1 markov chain

A simple probabilistic algorithm is given for the determination of the fundamental matrix of a block M/G/1 Markov chain. The algorithm does not require the chain to be positive recurrent. A prima facie case for its value is made with a standard testbench example.     

Classification of holomorphic spheres of constant curvature in complex Grassmann manifold G2,5

In this paper, we classify all nonsingular holomorphic spheres in complex Grassmann manifolds G_2,5 with the induced constant curvatures K=4, 2, 4/3,1 and 4/5 into some classes, up to unitary equivalence, in which none of the spheres are congruent; At the same time we also prove that there does not exist the nonsingular holomorphic sphere in G_2,5 with constant curvature 2/3.     

Generalizations of Whipple's theorem on the sum of a 3F2

Thirty-eight summation closely related to Whipple's theorem, in the theory of the generalized hypergeometric series, are obtained. Some limiting cases are also considered.     

K2 of a Fermat quotient and the value of its L-function

Two linearly independent elements of k ₂ of a certain quotient of a Fermat curve is exhibited in an explicit manner. The covolume of the regulator and the value of the L-function of the curve is numerically computed, and their ratio is nearly equal to a simple rational number.     

A Van Kampen theorem for π2

If the path connected topological space X has a countable open cover $\text{U}$ with path connected elements, then π₂(X,1) is computed as a colimit determined by the second homotopy groups of the intersection of elements of $\text{U}$ and the indices of the fundamental group injections of these intersections into the fundamental group of X. Aside from assuming that the inclusions induce such monomorphisms, certain other inclusions are also required to induce monomorphisms of fundamental groups and restrictions are placed on the arrangement of the elements of $\text{U}$.     

Quasirecognizability by the Set of Element Orders for Groups 3D4(q) and F4(q), for q Odd

It is proved that if L is one of the simple groups ³D₄(q) or F₄(q), where q is odd, and G is a finite group with the set of element orders as in L, then the derived subgroup of G/F(G) is isomorphic to L and the factor group G/G′ is a cyclic {2, 3}-group. __________ Translated from Algebra i Logika, Vol. 44, No. 5, pp. 517–539, September–October, 2005. Supported by RFBR grant No. 04-01-00463.     

A Generalized φ-Divergence for Asymptotically Multivariate Normal Models

I. Csiszár's (Magyar. Tud. Akad. Mat. Kutató Int. Közl8 (1963), 85–108) -divergence, which was considered independently by M. S. Ali and S. D. Silvey (J. R. Statist. Soc. Ser. B28 (1966), 131–142) gives a goodness-of-fit statistic for multinomial distributed data. We define a generalized φ-divergence that unifies the -divergence approach with that of C. R. Rao and S. K. Mitra (“Generalized Inverse of Matrices and Its Applications,” Wiley, New York, 1971) and derive weak convergence to a χ² distribution under the assumption of asymptotically multivariate normal distributed data vectors. As an example we discuss the application to the frequency count in Markov chains and thereby give a goodness-of-fit test for observations from dependent processes with finite memory.     

A conjecture of Abe-Iiyori and the Ree groups 2F4(q), II

In this paper the author has solved a problem of Abe and liyori for the finite simple groups ² F ₄(q) and ² F ₄(2)′.