Elliptic Theta Functions Research Articles

On the theory of reflectivity and transmissivity of a lossless nonlinear dielectric slab

For the transverse electric polarization case (TE) we present a treatment of the optical reflectivity and transmissivity of a slab whose dielectric coefficient is a real valued function of the light intensity. If this function is numerically integrable with respect to the light intensity, our treatment can serve as an algorithm for a numerical solution of the nonlinear wave equation. If the dielectric function is proportional to the intensity, an analytical solution of the cubic wave equation is given for the electric field strength and for the phase of the field in terms of Weierstrass' elliptic functions and first elliptic theta functions, respectively. Evaluating this solution by means of a computer algebra system yields the reflectivity, transmissivity and phase dependency on the incident field intensity and on parameters characteristic for the problem. Certain combinations of the parameters lead to bistable and multivalued behavior. The solution found is used to determine the relative extrema of the reflectivity and the critical values of the thickness and of the incident intensity. The results are a generalization of linear optics results. Application of the analysis to the cubic-quintic wave equation yields the general analytic solution which is used to detemine the reflectivity of a semi-infinite nonlinear medium.

Quantum affine algebras and holonomic difference equations

We derive new holonomicq-difference equations for the matrix coefficients of the products of intertwining operators for quantum affine algebra representations of levelk. We study the connection opertors between the solutions with different asymptotics and show that they are given by products of elliptic theta functions. We prove that the connection operators automatically provide elliptic solutions of Yang-Baxter equations in the “face” formulation for any type of Lie algebra $$\mathfrak{g}$$ and arbitrary finite-dimensional representations of . We conjecture that these solutions of the Yang-Baxter equations cover all elliptic solutions known in the contexts of IRF models of statistical mechanics. We also conjecture that in a special limit whenq→1 these solutions degenerate again into solutions with $$q' = \exp \left( {\frac{{2\pi i}}{{k + g}}} \right)$$ . We also study the simples examples of solutions of our holonomic difference equations associated to $$U_q (\widehat{\mathfrak{s}\mathfrak{l}(2)})$$ and find their expressions in terms of basic (orq−)-hypergeometric series. In the special case of spin −1/2 representations, we demonstrate that the connection matrix yields a famous Baxter solution of the Yang-Baxter equation corresponding to the solid-on-solid model of statistical mechanics.

Exact solution of solid-on-solid models for twisted affine Lie algebras A 2 n(2) and A 2 n−1 (2)

We report new exact solutions of restricted and unrestricted solid-on-solid (SOS) models associated with the twisted affine Lie algebras X n (2) = A 2 n (2) and A 2 n−1 (2). The Boltzmann weights are parametrized by elliptic theta functions and fulfill the Yang-Baxter equation. The trigonometric case of the solutions corresponds to the SOS counterpart of the solvable vertex models arising from the X n (2) quantum R-matrices. The restricted models (RSOS) are labeled by the level l as well as the rank n. We show that the level- l A 2 n−1 (2) RSOS model is equivalent to the level- n C l (1) RSOS model, which is a manifestation of the level-rank duality. One-point functions of the A 2 n−1 (2) RSOS model are obtained by using the coset pair C n (1) ⊕ A 2 n−1 (1) ⊃ C n (1). We also determine the one-point functions of the X n (2) vertex models in terms of X n (2) string functions.

Solvable face models related to the lie superalgebra sl(m|n)

A new series of solvable face models is presented. The Boltzmann weights are parametrized in terms of elliptic theta functions. This series generalizes the face models of type A n (1) introduced in [I].

Solvable lattice models related to the vector representation of classical simple Lie algebras

A series of solvable lattice models with face interaction are introduced on the basis of the affine Lie algebraX (1) =A (1) ,B (1) ,C (1) ,D (1) . The local states taken on by the fluctuation variables are the dominant integral weights ofX (1) of a fixed level. Adjacent local states are subject to a condition related to the vector representation ofX n . The Boltzmann weights are parametrized by elliptic theta functions and solve the star-triangle relation.

Lattice gas generalization of the hard hexagon model. III.q-Trinomial coefficients

In the first two papers in this series we considered an extension of the hard hexagon model to a solvable two-dimensional lattice gas with at most two particles per pair of adjacent sites, and we described the local densities in terms of elliptic theta functions. Here we present the mathematical theory behind our derivation of the local densities. Our work centers onq-analogs of trinomial coefficients.

First passage time distributions for finite one-dimensional random walks

We present closed expressions for the characteristic function of the first passage time distribution for biased and unbiased random walks on finite chains and continuous segments with reflecting boundary conditions. Earlier results on mean first passage times for one-dimensional random walks emerge as special cases. The divergences that result as the boundary is moved out to infinity are exhibited explicitly. For a symmetric random walk on a line, the distribution is an elliptic theta function that goes over into the known Levy distribution with exponent 1/2 as the boundary tends to ∞.

Voltage response of distributed RC lines with a discontinuous initial potential distribution

Prior work by this author has shown how the voltage step response of an initially quiescent RC line can be concisely expressed in terms of a set of higher transcendental functions. For convenience, the latter were defined in termsof integrals of elliptic theta functions and are termed “diffusion functions”. These same diffusion functions are also found to be useful in expressing closed-form solutions for the voltage response of distributed RC lines with a non-quiescent and discontinuous initial potential distribution. Results herein provide functional expressions for this voltage response with either open-circuited or short-circuited terminations. A variety of solution plots show how the responses change for different positions of the discontinuity. Also, plots of the diffusion functions are supplied showing their behavior over one complete cycle and more information on their symmetry properties is provided than was included in the initial paper.

Elliptic Functions, Theta Functions, and Riemann Surfaces

Elliptic Functions, Theta Functions, and Riemann Surfaces

Statistical Mechanics of Dimers on a Quadratic Lattice

The partition function for the square lattice completely filled with dimers is analyzed for a finite n × m rectangular lattice with edges and for the corresponding lattice with periodic boundary conditions. The total free energy is calculated asymptotically for fixed ξ = n/m up to terms o(1/n2−δ) for any δ > 0. The bulk terms proportional to nm, the surface terms proportional to (n + m) which vanish with periodic boundary conditions, and the constant terms which reveal a parity and shape dependence are expressed explicitly using dilogarithms and elliptic theta functions.

On Diophantine approximation and the generalized elliptic theta function

On Diophantine approximation and the generalized elliptic theta function

The Behaviour of Certain Series Associated with Limiting Cases of Elliptic Theta-Functions: (III)

The Behaviour of Certain Series Associated with Limiting Cases of Elliptic Theta-Functions: (III)

The Behaviour of Certain Series Associated with Limiting Cases of Elliptic Theta-Functions: II

The Behaviour of Certain Series Associated with Limiting Cases of Elliptic Theta-Functions: II

The Behaviour of Certain Series Associated with Limiting Cases of Elliptic Theta-Functions

The Behaviour of Certain Series Associated with Limiting Cases of Elliptic Theta-Functions

Some problems of diophantine approximation: An additional note on the trigonometrical series associated with the elliptic theta-functions

Some problems of diophantine approximation: An additional note on the trigonometrical series associated with the elliptic theta-functions

An image in four-dimensional lattice space of the theory of the elliptic theta functions

An image in four-dimensional lattice space of the theory of the elliptic theta functions