Some addition formulae for Abelian functions for elliptic and hyperelliptic curves of cyclotomic type

We discuss a family of multi-term addition formulae for Weierstrass functions on specialized curves of genus one and two with many automorphisms. In the genus one case we find new addition formulae for the equianharmonic and lemniscate cases, and in genus two we find some new addition formulae for a number of curves, including the Burnside curve.Comment: 19 pages. We have extended the Introduction, corrected some typos and tidied up some proofs, and inserted extra material on genus 3 curve

Precise bounds on the Higgs boson mass

We study the renormalization group evolution of the Higgs quartic coupling $\lambda_{H}$ and the Higgs mass $m_{H}$ in the Standard Model. The one loop equation for $\lambda_{H}$ is non linear and it is of the Riccati type which we numerically and analytically solve in the energy range $[m_{t},E_{GU}]$ where $m_{t}$ is the mass of the top quark and $E_{GU}=10^{14}$ GeV. We find that depending on the value of $\lambda_{H}(m_{t})$ the solution for $\lambda_{H}(E)$ may have singularities or zeros and become negative in the former energy range so the ultra violet cut off of the standard model should be below the energy where the zero or singularity of $\lambda_{H}$ occurs. We find that for $0.369\leq\lambda_{H}(m_{t})\leq0.613$ the Standard Model is valid in the whole range $[m_{t},E_{GU}]$. We consider two cases of the Higgs mass relation to the parameters of the standard model: (a) the effective potential method and (b) the tree level mass relations. The limits for $\lambda_{H}(m_{t})$ correspond to the following Higgs mass relation $150\leq m_{H}\lessapprox 193$ GeV. We also plot the dependence of the ultra violet cut off on the value of the Higgs mass. We analyze the evolution of the vacuum expectation value of the Higgs field and show that it depends on the value of the Higgs mass. The pattern of the energy behavior of the VEV is different for the cases (a) and (b). The behavior of $\lambda_{H}(E)$, $m_{H}(E)$ and $v(E)$ indicates the existence of a phase transition in the standard model. For the effective potential this phase transition occurs at the mass range $m_{H}\approx 180$ GeV and for the tree level mass relations at $m_{H}\approx 168$ GeV.Comment: 14 pages, 7 figures. Expanded the discussion of the Higgs mass relation between the parameters of the Standard Model. Included the method of the Higgs effective potentia

Elliptic (N,N^\prime)-Soliton Solutions of the lattice KP Equation

Elliptic soliton solutions, i.e., a hierarchy of functions based on an elliptic seed solution, are constructed using an elliptic Cauchy kernel, for integrable lattice equations of Kadomtsev-Petviashvili (KP) type. This comprises the lattice KP, modified KP (mKP) and Schwarzian KP (SKP) equations as well as Hirota's bilinear KP equation, and their successive continuum limits. The reduction to the elliptic soliton solutions of KdV type lattice equations is also discussed.Comment: 18 page

Two-Center Integrals for r_{ij}^{n} Polynomial Correlated Wave Functions

All integrals needed to evaluate the correlated wave functions with polynomial terms of inter-electronic distance are included. For this form of the wave function, the integrals needed can be expressed as a product of integrals involving at most four electrons

On conformal measures and harmonic functions for group extensions

We prove a Perron-Frobenius-Ruelle theorem for group extensions of topological Markov chains based on a construction of $\sigma$-finite conformal measures and give applications to the construction of harmonic functions.Comment: To appear in Proceedings of "New Trends in Onedimensional Dynamics, celebrating the 70th birthday of Welington de Melo

Sign Rules for Anisotropic Quantum Spin Systems

We present new and exact ``sign rules'' for various spin-s anisotropic spin-lattice models. It is shown that, after a simple transformation which utilizes these sign rules, the ground-state wave function of the transformed Hamiltonian is positive-definite. Using these results exact statements for various expectation values of off-diagonal operators are presented, and transitions in the behavior of these expectation values are observed at particular values of the anisotropy. Furthermore, the effects of sign rules in variational calculations and quantum Monte Carlo calculations are considered. They are illustrated by a simple variational treatment of a one-dimensional anisotropic spin model.Comment: 4 pages, 1 ps-figur

Quantum chaos, random matrix theory, and statistical mechanics in two dimensions - a unified approach

We present a theory where the statistical mechanics for dilute ideal gases can be derived from random matrix approach. We show the connection of this approach with Srednicki approach which connects Berry conjecture with statistical mechanics. We further establish a link between Berry conjecture and random matrix theory, thus providing a unified edifice for quantum chaos, random matrix theory, and statistical mechanics. In the course of arguing for these connections, we observe sum rules associated with the outstanding counting problem in the theory of braid groups. We are able to show that the presented approach leads to the second law of thermodynamics.Comment: 23 pages, TeX typ

Integrable Time-Discretisation of the Ruijsenaars-Schneider Model

An exactly integrable symplectic correspondence is derived which in a continuum limit leads to the equations of motion of the relativistic generalization of the Calogero-Moser system, that was introduced for the first time by Ruijsenaars and Schneider. For the discrete-time model the equations of motion take the form of Bethe Ansatz equations for the inhomogeneous spin-1/2 Heisenberg magnet. We present a Lax pair, the symplectic structure and prove the involutivity of the invariants. Exact solutions are investigated in the rational and hyperbolic (trigonometric) limits of the system that is given in terms of elliptic functions. These solutions are connected with discrete soliton equations. The results obtained allow us to consider the Bethe Ansatz equations as ones giving an integrable symplectic correspondence mixing the parameters of the quantum integrable system and the parameters of the corresponding Bethe wavefunction.Comment: 27 pages, latex, equations.st

Controllability on infinite-dimensional manifolds

Following the unified approach of A. Kriegl and P.W. Michor (1997) for a treatment of global analysis on a class of locally convex spaces known as convenient, we give a generalization of Rashevsky-Chow's theorem for control systems in regular connected manifolds modelled on convenient (infinite-dimensional) locally convex spaces which are not necessarily normable.Comment: 19 pages, 1 figur