THE SPHERICAL FUNCTIONS AND THE CENTRAL LIMIT THEOREM ON SU（2,2）/S（U（2） × U（2））

Let G = SU（2, 2）, K = S（U（2） × U（2））, and for l ∈ Z, let {Tl}l∈z be a one-dimensional K-type and let El be the line bundle over G/K associated to Tl. It is shown that the Tl-spherical function on G is given by the hypergeometric functions of several variables. By applying this result, a central limit theorem for the space G/K is obtained.     

The SU(v) Calogero spin system

A one-dimensional quantum particle system in which particles with su(v) spins interact through inverse square interactions is introduced. We refer to it as the SU(v) Calogero spin system. Using the quantum inverse scattering method, we reveal algebraic structures of the system: hidden symmetry is the U(v) − SU(v) U(1) current algebra. This is consistent with the fact that the ground-state wave function is a solution of the Knizhnik-Zamolodchikov equation. Furthermore we show that the system has a higher symmetry, known as the w_{1 + ∞}-algebra. With this W-algebra we have a unified viewpoint on the integrable quantum particle systems with long-range interactions such as the Calogero type (1/x²-interactions) and Sutherland type (1/sin²x-interactions). The Yangian symmetry is briefly discussed.     

The Dichotomy Problem for Orbital Measures of SU(n)

For many orbital measures μ, on SU(n), we show that either μ^k ∈ L² or μ^k is singular to L¹. The least k for which μ^k ∈ L² is determined and is shown to be the minimum k for which the k-fold product of the conjugacy class supporting the measure has positive measure. It would be interesting to know if all orbital measures satisfy this dichotomy.     

The Local Index Formula for SU q (2)

We discuss the local index formula of Connes–Moscovici for the isospectral noncommutative geometry that we have recently constructed on quantum SU(2). We work out the cosphere bundle and the dimension spectrum as well as the local cyclic cocycles yielding the index formula. (Received: January 2005)     

Whittaker functions for generalized principal series representations of GSp(2,R)

We study Whittaker functions for generalized principal series representations of $\mathrm{GSp}(2,\mathbf{R})$ induced from Siegel parabolic subgroup. We give Mellin–Barnes type integral representations of Whittaker functions belonging to certain K-types. As an application of our explicit formulas, we compute the archimedean parts of Novodvorsky's zeta integrals. As a consequence we can show the entireness of the spinor L-functions for generic cusp forms on $\mathrm{GSp}(2)$.     

Fourier-Jacobi type spherical functions for principal series representations of Sp(2, R)

In this paper, we study the Fourier-Jacobi type spherical functions on Sp(2, R) for irreducible principal series representations. We give the multiplicity theorem and an explicit formula for this function.     

Smoothness of convolution powers of orbital measures on the symmetric space SU(n)/SO(n)

We prove that if m_a = m_K*d_a*m_K{\mu _{a}\,{=}\,m_{K}*\delta _{a}*m_{K}} is the K-bi-invariant measure supported on the double coset KaK í SU(n){KaK\subseteq SU(n)} , for K = SO(n), then m_a^k{\mu _{a}^{k}} is absolutely continuous with respect to the Haar measure on SU(n) for all a not in the normalizer of K if and only if k ≥ n. The measure, μ _a , supported on the minimal dimension double coset has the property that m_a^n-1{\mu _{a}^{n-1}} is singular to the Haar measure.     

Differential equations for septic theta functions

We demonstrate that quotients of septic theta functions appearing in Ramanujan’s Notebooks and in Klein’s work satisfy a new coupled system of nonlinear differential equations with symmetric form. This differential system bears a close resemblance to an analogous system for quintic theta functions. The proof extends an elementary technique used by Ramanujan to prove the classical differential system for normalized Eisenstein series on the full modular group. In the course of our work, we show that Klein’s quartic relation induces symmetric representations for low-weight Eisenstein series in terms of weight one modular forms of level seven.     

-symmetry for (r,s)-hypergeometric series

Contiguous operators for a two-parameter analogue of hypergeometric series are constructed. These represent a two-parameter quantum enveloping algebra introduced by Takeuchi.     

A method for the resolution of the Jacobi equation Y ″ + RY = 0 on the manifold Sp (2)/ SU (2)

In this paper a method for the resolution of the differential equation of the Jacobi vector fields in the manifold V ₁ = Sp(2)/SU(2) is exposed. These results are applied to determine areas and volumes of geodesic spheres and balls.     

Affine functions on CAT (κ)-spaces

We describe affine functions on spaces with an upper curvature bound.     

Quasi-permutation Representations of the Groups SU (3, q2) and PSU (3, q2)

A square matrix over the complex field with non-negative integral trace is called a quasi-permutation matrix. For a finite group G the minimal degree of a faithful permutation representation of G is denoted by p(G). The minimal degree of a faithful representation of G by quasi-permutation matrices over the rational and the complex numbers are denoted by q(G) and c(G) respectively. Finally r(G) denotes the minimal degree of a faithful rational valued complex character of G. In this paper p(G), q(G), c(G) and r(G) are calculated for the groups PSU (3, q²) and SU (3, q²).AMS Subject Classification (2000): 20C15     

Some remarks on holomorphic functions and taylor series in C

Some previous results on convergence of Taylor series in C^n [3] are improved by indicating outside the domain of convergence the points where the series diverges and simplifying some proofs. These results contain the Cauchy-Hadamard theorem in C. Some Cauchy integral formulas of a holomorphic function on a closed ball in C^n are constructed and the Taylor series expansion is deduced.     

Differential equations and recursion relations for Laguerre functions on symmetric cones

We obtain the differential equation and recurrence relations satisfied by the Laguerre functions on an arbitrary symmetric cone .

     

Harnack Differential Inequalities for the Parabolic Equation ut=LF(u) on Riemannian Manifolds and Applications

In this paper, let(M~n, g) be an n-dimensional complete Riemannian manifold with the mdimensional Bakry–mery Ricci curvature bounded below. By using the maximum principle, we first prove a Li–Yau type Harnack differential inequality for positive solutions to the parabolic equation u_t= LF(u)=ΔF(u)-f·F(u),on compact Riemannian manifolds Mn, where F∈C~2(0, ∞), F0 and f is a C~2-smooth function defined on M~n. As application, the Harnack differential inequalities for fast diffusion type equation and porous media type equation are derived. On the other hand, we derive a local Hamilton type gradient estimate for positive solutions of the degenerate parabolic equation on complete Riemannian manifolds. As application, related local Hamilton type gradient estimate and Harnack inequality for fast dfiffusion type equation are established. Our results generalize some known results.     

Quasiconvex functions, SO(n) and two elastic wells

We use W^1,∞ approximations of minimizing sequences to study the growth of some quasiconvex functions near their zero sets. We show that for SO(n), the quasiconvexification of the distance function dist²(·, SO(n)) can be bounded below by the distance function itself. In certain cases of the incompatible two elastic well structure, we establish a similar result. We also prove that for small Lipschitz perturbations of SO(n) and of the two well structure, the Young measure limits of gradients supported on these perturbed sets are Dirac masses.     

A study of some discrete Dirac equations with principal functions

We prove that, under CH, any space with a regular \({G_{\delta}}\)-diagonal and caliber \({\omega_1}\) is separable; a corollary of this result answers, under CH, a question of Buzyakova. For any Urysohn space X, we establish the inequality \({|X|\leqq wL{(X)}^{s\Delta_2(X)\cdot{\dot(X)}}}\) which represents a generalization of a theorem of Basile, Bella, and Ridderbos. We also show that if X is a Hausdorff space, then \({|X| \leqq {(\pi\chi(X)\cdot d(X))}^{{\rm ot}(X)\cdot\psi_c(X)}}\); this result implies ?apirovski?’s inequality \({|X|\leqq \pi\chi{(X)}^{c(X)\cdot\psi(X)}}\) which only holds for regular spaces. It is also proved that \({|X|\leqq \pi\chi{(X)}^{{\rm ot}(X)\cdot\psi_c(X)\cdot aL_c(X)}}\) for any Hausdorff space X; this gives one more generalization of the famous Arhangel’skii’s inequality \({|X|\le 2^{\chi(X)\cdot L(X)}}\).