R-matrices and Tensor Product Graph Method

A systematic method for constructing trigonometric R-matrices corresponding to the (multiplicity-free) tensor product of any two affinizable representations of a quantum algebra or superalgebra has been developed by the Brisbane group and its collaborators. This method has been referred to as the Tensor Product Graph Method. Here we describe applications of this method to untwisted and twisted quantum affine superalgebras.Comment: LaTex 7 pages. Contribution to the APCTP-Nankai Joint Symposium on "Lattice Statistics and Mathematical Physics", 8-10 October 2001, Tianjin, Chin

WFIRST Ultra-Precise Astrometry II: Asteroseismology

WFIRST microlensing observations will return high-precision parallaxes, sigma(pi) < 0.3 microarcsec, for the roughly 1 million stars with H<14 in its 2.8 deg^2 field toward the Galactic bulge. Combined with its 40,000 epochs of high precision photometry (~0.7 mmag at H_vega=14 and ~0.1 mmag at H=8), this will yield a wealth of asteroseismic data of giant stars, primarily in the Galactic bulge but including a substantial fraction of disk stars at all Galactocentric radii interior to the Sun. For brighter stars, the astrometric data will yield an external check on the radii derived from the two asteroseismic parameters, and nu_max, while for the fainter ones, it will enable a mass measurement from the single measurable asteroseismic parameter nu_max. Simulations based on Kepler data indicate that WFIRST will be capable of detecting oscillations in stars from slightly less luminous than the red clump to the tip of the red giant branch, yielding roughly 1 million detections.Comment: 13 pages, 6 figures, submitted to JKA

Lax Operator for the Quantised Orthosymplectic Superalgebra U_q[osp(2|n)]

Each quantum superalgebra is a quasi-triangular Hopf superalgebra, so contains a \textit{universal $R$-matrix} in the tensor product algebra which satisfies the Yang-Baxter equation. Applying the vector representation $\pi$, which acts on the vector module $V$, to one side of a universal $R$-matrix gives a Lax operator. In this paper a Lax operator is constructed for the $C$-type quantum superalgebras $U_q[osp(2|n)]$. This can in turn be used to find a solution to the Yang-Baxter equation acting on $V \otimes V \otimes W$ where $W$ is an arbitrary $U_q[osp(2|n)]$ module. The case $W=V$ is included here as an example.Comment: 15 page

Unitarity and Complete Reducibility of Certain Modules over Quantized Affine Lie Algebras

Let $U_q(\hat{\cal G})$ denote the quantized affine Lie algebra and $U_q({\cal G}^{(1)})$ the quantized {\em nontwisted} affine Lie algebra. Let ${\cal O}_{\rm fin}$ be the category defined in section 3. We show that when the deformation parameter $q$ is not a root of unit all integrable representations of $U_q(\hat{\cal G})$ in the category ${\cal O}_{\rm fin}$ are completely reducible and that every integrable irreducible highest weight module over $U_q({\cal G}^{(1)})$ corresponding to $q>0$ is equivalent to a unitary module.Comment: 17 pages (minor errors corrected

Punctuated Equilibrium in Software Evolution

The approach based on paradigm of self-organized criticality proposed for experimental investigation and theoretical modelling of software evolution. The dynamics of modifications studied for three free, open source programs Mozilla, Free-BSD and Emacs using the data from version control systems. Scaling laws typical for the self-organization criticality found. The model of software evolution presenting the natural selection principle is proposed. The results of numerical and analytical investigation of the model are presented. They are in a good agreement with the data collected for the real-world software.Comment: 4 pages, LaTeX, 2 Postscript figure

Generalised Perk--Schultz models: solutions of the Yang-Baxter equation associated with quantised orthosymplectic superalgebras

The Perk--Schultz model may be expressed in terms of the solution of the Yang--Baxter equation associated with the fundamental representation of the untwisted affine extension of the general linear quantum superalgebra $U_q[sl(m|n)]$, with a multiparametric co-product action as given by Reshetikhin. Here we present analogous explicit expressions for solutions of the Yang-Baxter equation associated with the fundamental representations of the twisted and untwisted affine extensions of the orthosymplectic quantum superalgebras $U_q[osp(m|n)]$. In this manner we obtain generalisations of the Perk--Schultz model.Comment: 10 pages, 2 figure