105 research outputs found
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
and the Higgs mass in the Standard Model. The one loop
equation for is non linear and it is of the Riccati type which we
numerically and analytically solve in the energy range where
is the mass of the top quark and GeV. We find that
depending on the value of the solution for
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 occurs. We find
that for the Standard Model is valid in
the whole range . 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
correspond to the following Higgs mass relation 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 , and
indicates the existence of a phase transition in the standard model. For the
effective potential this phase transition occurs at the mass range
GeV and for the tree level mass relations at 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 -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
Mixed Hodge polynomials of character varieties
We calculate the E-polynomials of certain twisted GL(n,C)-character varieties
M_n of Riemann surfaces by counting points over finite fields using the
character table of the finite group of Lie-type GL(n,F_q) and a theorem proved
in the appendix by N. Katz. We deduce from this calculation several geometric
results, for example, the value of the topological Euler characteristic of the
associated PGL(n,C)-character variety. The calculation also leads to several
conjectures about the cohomology of M_n: an explicit conjecture for its mixed
Hodge polynomial; a conjectured curious Hard Lefschetz theorem and a conjecture
relating the pure part to absolutely indecomposable representations of a
certain quiver. We prove these conjectures for n = 2.Comment: with an appendix by Nicholas M. Katz; 57 pages. revised version: New
definition for homogeneous weight in Definition 4.1.6, subsequent arguments
modified. Some other minor changes. To appear in Invent. Mat
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
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