An expansion of the Jones representation of genus 2 and the Torelli group

We study the algebraic property of the representation of the mapping class group of a closed oriented surface of genus 2 constructed by VFR Jones [Annals of Math. 126 (1987) 335-388]. It arises from the Iwahori-Hecke algebra representations of Artin's braid group of 6 strings, and is defined over integral Laurent polynomials Z[t, t^{-1}]. We substitute the parameter t with -e^{h}, and then expand the powers e^h in their Taylor series. This expansion naturally induces a filtration on the Torelli group which is coarser than its lower central series. We present some results on the structure of the associated graded quotients, which include that the second Johnson homomorphism factors through the representation. As an application, we also discuss the relation with the Casson invariant of homology 3-spheres.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol1/agt-1-3.abs.htm

Rapidity Dependence of HBT radii based on a hydrodynamical model

We calculate two-pion correlation functions at finite rapidities based on a hydrodynamical model which does not assume explicit boost invariance along the collision axis. Extracting the HBT radii through $\chi^2$ fits in both Cartesian and Yano-Koonin-Podgoretski\u{\i} parametrizations, we compare them with the experimental results obtained by the PHOBOS. Based on the results, we discuss longitudinal expansion dynamics.Comment: 8 pages, 10 figures, talk given at II Workshop on Particle Correlation and Femtoscopy (WPCF 2006), Sep 9-11, Sao Paulo, Brazil. Revised version to be published in Braz.J.Phy

A Connection Formula of the Hahn-Exton qq-Bessel Function

We show a connection formula of the Hahn-Exton $q$-Bessel function around the origin and the infinity. We introduce the $q$-Borel transformation and the $q$-Laplace transformation following C. Zhang to obtain the connection formula. We consider the limit $p\to 1^-$ of the connection formula