Elliptic Theta Functions Research Articles

Impulsive impact of a twin hull

An impulsively starting motion of two cylindrical bodies floating on a free liquid surface is considered. The shape of the cross-section of each body and the distance between them are arbitrary. The integral hodograph method is advanced to derive the complex velocity potential defined in a rectangular parameter region in terms of the elliptic quasi-doubly periodic Jacobi theta functions. A system of singular integral equations in the velocity magnitude on the free surface and in the slope of the wetted part of each body is derived using the kinematic boundary condition, which is then solved numerically. The velocity field, the pressure impulse on the bodies and the added mass coefficients of each body immediately after the impact are determined in a wide range of distances between the bodies and for cross-sectional shapes such as the flat plate and half-circle.

On the Applications of the Generalized Littlewood Theorem Concerning Integrals of the Logarithm of Analytical Functions to Elliptic Functions

Recently, we have established and used the generalized Littlewood theorem concerning contour integrals of the logarithm of analytical function to obtain new criteria equivalent to the Riemann hypothesis. Later, the same theorem was applied to calculate certain infinite sums and study the properties of zeroes of a few analytical functions. In this study, we apply this approach to elliptic functions of Jacobi and Weierstrass. Numerous sums over inverse powers of zeroes and poles are calculated, including some results for the Jacobi elliptic functions sn(z, k) and others understood as functions of the index k. The consideration of the case of the derivative of the Weierstrass rho-function, ℘z(z,τ), leads to quite easy and transparent proof of numerous equalities between the sums over inverse powers of the lattice points m+nτ and “demi-lattice” points m+1/2+nτ, m+(n+1/2)τ, m+1/2+(n+1/2)τ. We also prove theorems showing that, in most cases, fundamental parallelograms contain exactly one simple zero for the first derivative θ1′(z|τ) of the elliptic theta-function and the Weierstrass ζ-function, and that far from the origin of coordinates such zeroes of the ζ-function tend to the positions of the simple poles of this function.

Several identities, estimates and special values for the elliptic theta functions

In this short note, we first give several new infinite product representations for the elliptic theta functions. These will then be used to deduce some special values and estimations for these functions on the real line.

Exploration of non-trivial relations for the non-steady state nucleation rate: usefulness of the elliptic theta functions for its experimental estimations

It is known that the non-steady state nucleation rate can be expressed as the product of the steady state nucleation rate and an elliptic theta function. However, there has been no research where rich mathematical properties of the elliptic theta functions were utilized to analyze the non-steady state nucleation processes. In this paper, we achieved two objectives by using the properties. The first one is to derive non-trivial relations for the non-steady state nucleation rate by using the rich mathematical properties of the elliptic theta functions, which could not be discovered in the conventional classical nucleation theory because of its complexities. The second one is to find solutions to two problems by solving a difference equation for the non-steady state nucleation rate: (i) it requires a large amount of effort and cost to estimate the time evolution of the non-steady state nucleation rate under some special conditions, and (ii) it is impossible to measure the non-steady state nucleation rate under some special cases. It is shown that our result help reduce the estimation cost of the non-steady state nucleation rate and makes mechanical estimation of the non-steady state nucleation rate possible from the past to the future.Article highlightsNon-trivial relations for the non-steady state nucleation rate are found from the mathematical properties of the elliptic theta functions.It is shown that our result help reduce the estimation cost of the non-steady state nucleation rate.It is shown that our result makes mechanical estimation of the non-steady state nucleation rate possible from the past to the future.

The pentagram map, Poncelet polygons, and commuting difference operators

The pentagram map takes a planar polygon $P$ to a polygon $P'$ whose vertices are the intersection points of consecutive shortest diagonals of $P$. This map is known to interact nicely with Poncelet polygons, that is, polygons which are simultaneously inscribed in a conic and circumscribed about a conic. A theorem of Schwartz states that if $P$ is a Poncelet polygon, then the image of $P$ under the pentagram map is projectively equivalent to $P$. In the present paper, we show that in the convex case this property characterizes Poncelet polygons: if a convex polygon is projectively equivalent to its pentagram image, then it is Poncelet. The proof is based on the theory of commuting difference operators, as well as on properties of real elliptic curves and theta functions.

A counting problem on lattice points in two-, three- and four-dimensional spaces

A counting problem on lattice points in n-dimensional space with rectangular coordinate system was considered. Lattice points are all the points having integer coordinates. The distances of these lattice points to the origin O are denoted from small to large as rn,0<rn,1<rn,2<…, where rn,0=0 denotes the origin itself. All the lattice points scatter on a series of concentric spherical surfaces with the center O and the radii rn,k,k=0,1,2,…. The number of lattice points on the spherical surface with the radius rn,k is denoted as Nn,k. Several properties about the sequences rn,k and Nn,k,k=0,1,2,…, were investigated. The relevant generating function was derived in terms of the elliptic theta functions for convenient calculation of rn,k and Nn,k. We proved that for the 2-dimensional case, the number of lattice points is 4 on each circles with the radii satisfying r2,s2=2h,h=0,1,2,…; for the 3-dimensional case, the number of lattice points is 6 on each spherical surfaces with the radii satisfying r3,s2=4h,h=0,1,2,…; for the 4-dimensional case, the number of lattice points is 24 on each hyper-spherical surfaces with the radii satisfying r4,s2=2h,h=1,2,….

The Kronecker theta function and a decomposition theorem for theta functions I

The Kronecker theta function is a quotient of the Jacobi theta functions, which is also a special case of Ramanujan’s \({}_{1}\psi _{1}\) summation. Using the Kronecker theta function as building blocks, we prove a decomposition theorem for theta functions. This decomposition theorem is the common source of a large number of theta function identities. Many striking theta function identities, both classical and new, are derived from this decomposition theorem with ease. A new addition formula for theta functions is established. Several known results in the theory of elliptic theta functions due to Ramanujan, Weierstrass, Kiepert, Winquist and Shen among others are revisited. A curious trigonometric identity is proved.

An application of hypergeometric functions to heat kernels on rectangular and hexagonal tori and a \u201cWeltkonstante\u201d-or-how Ramanujan split temperatures

In this work we investigate the heat kernel of the Laplace–Beltrami operator on a rectangular torus and the according temperature distribution. We compute the minimum and the maximum of the temperature on rectangular tori of fixed area by means of Gauss’ hypergeometric function _2F_1 and the elliptic modulus. In order to be able to do this, we employ a beautiful result of Ramanujan, connecting hypergeometric functions, the elliptic modulus and theta functions. Also, we investigate the temperature distribution of the heat kernel on hexagonal tori and use Ramanujan’s corresponding theory of signature 3 to derive analogous results to the rectangular case. Lastly, we show connections to the problem of finding the exact value of Landau’s “Weltkonstante”, a universal constant arising in the theory of extremal holomorphic mappings; and for a related, restricted extremal problem we show that the conjectured solution is the second lemniscate constant.

Theta Functions and Brownian Motion

A theta function for an arbitrary connected and simply connected compact simple Lie group is defined as an infinite determinant that is naturally related to the transformation of a family of independent Gaussian random variables associated with a pinned Brownian motion in the Lie group. From this definition of a theta function, the equality of the product and the sum expressions for a theta function is obtained. This equality for an arbitrary connected and simply connected compact simple Lie group is known as a Macdonald identity which generalizes the Jacobi triple product for the elliptic theta function associated with su(2).

Calculation of radii and atom numbers of different coordination shells in cubic crystals

Computational problems of multilayered coordination radii and coordination numbers in cubic crystals were considered. The Diophantine equation models and the generating functions for computing the coordination radii and coordination numbers were derived for the simple cubic (SC), body-centered cubic (BCC) and face-centered cubic (FCC) crystals, respectively. Properties of the sequences of coordination radii and coordination numbers were discussed, and the results for the three types of crystals were compared. We find that generating functions can be expressed in terms of the elliptic theta functions. These results provide efficient algorithms for the multilayered coordination radii and coordination numbers by solving the Diophantine equations, or expanding the powers of series in the generating functions, or directly expanding the elliptic theta functions into power series using a computing software such as Mathematica.

Winding of a Brownian particle around a point vortex

We derive the asymptotic winding law for a Brownian particle in the plane subjected to a tangential drift due to a point vortex. For winding around a point, the normalized winding angle converges to an inverse Gamma distribution. For winding around a disc, the angle converges to a distribution given by an elliptic theta function. For winding in an annulus, the winding angle is asymptotically Gaussian with a linear drift term. We validate our results with numerical simulations.This article is part of the theme issue ‘Topological and geometrical aspects of mass and vortex dynamics’.

Exact finite-size corrections in the dimer model on a planar square lattice

We consider the dimer model on the rectangular lattice with free boundary conditions. We derive exact expressions for the coefficients in the asymptotic expansion of the free energy in terms of the elliptic theta functions () and the elliptic integral of second kind (E), up to 22nd order. Surprisingly we find that ratio of the coefficients f p in the free energy expansion for strip () and square () geometries in the limit of large p tends to 1/2. Furthermore, we predict that the ratio of the coefficients f p in the free energy expansion for rectangular () for aspect ratio to the coefficients of the free energy for square geometries, multiplied by , that is , is also equal to 1/2 in the limit of . With these results, one can find the asymptotic behavior of the from that of the , whose asymptotic behavior is derived explicitly here. We find that the corner contribution to the free energy for the dimer model on rectangular lattices with free boundary conditions is equal to zero and explain that result in the framework of conformal field theory, with two consistent values of the central charge, namely, c = −2 for the construction of a conformal field theory using a mapping of spanning trees and c = 1 for the height function description. We also derive a simple exact expression for the free energy of open strips of arbitrary width.

Quantum diffusion during inflation and primordial black holes

We calculate the full probability density function (PDF) of inflationary curvature perturbations, even in the presence of large quantum backreaction. Making use of the stochastic-δ N formalism, two complementary methods are developed, one based on solving an ordinary differential equation for the characteristic function of the PDF, and the other based on solving a heat equation for the PDF directly. In the classical limit where quantum diffusion is small, we develop an expansion scheme that not only recovers the standard Gaussian PDF at leading order, but also allows us to calculate the first non-Gaussian corrections to the usual result. In the opposite limit where quantum diffusion is large, we find that the PDF is given by an elliptic theta function, which is fully characterised by the ratio between the squared width and height (in Planck mass units) of the region where stochastic effects dominate. We then apply these results to the calculation of the mass fraction of primordial black holes from inflation, and show that no more than ∼ 1 e-fold can be spent in regions of the potential dominated by quantum diffusion. We explain how this requirement constrains inflationary potentials with two examples.

Vortex motion around a circular cylinder above a plane

The study of vortex flows around solid obstacles is of considerable interest from both a theoretical and practical perspective. One geometry that has attracted renewed attention recently is that of vortex flows past a circular cylinder placed above a plane wall, where a stationary recirculating eddy can form in front of the cylinder, in contradistinction to the usual case (without the plane boundary) for which a vortex pair appears behind the cylinder. Here we analyze the problem of vortex flows past a cylinder near a wall through the lenses of the point-vortex model. By conformally mapping the fluid domain onto an annular region in an auxiliary complex plane, we compute the vortex Hamiltonian analytically in terms of certain special functions related to elliptic theta functions. A detailed analysis of the equilibria of the model is then presented. The location of the equilibrium in front of the cylinder is shown to be in qualitative agreement with recent experimental findings. We also show that a topological transition occurs in phase space as the parameters of the systems are varied.

Analysis of multimode interferometers.

A general analytical formula describing the transfer function between the input and output of an arbitrary rectangular multimode interference (MMI) coupler has been derived using the elliptic theta function ϑ(x', z'). This formula provides the positions, amplitudes and relative phases of all the self-images of a given source. It is shown how the transfer function can be used as a propagation matrix for any rectangular NxM MMI. Specific simplified solutions for NxN, symmetric and paired MMIs are also derived from the general formula.

The Ihara Zeta Function of the Infinite Grid

The infinite grid is the Cayley graph of $\mathbb{Z} \times \mathbb{Z}$ with the usual generators. In this paper, the Ihara zeta function for the infinite grid is computed using elliptic integrals and theta functions. The zeta function of the grid extends to an analytic, multivalued function which satisfies a functional equation. The set of singularities in its domain is finite.The grid zeta function is the first computed example which is non-elementary, and which takes infinitely many values at each point of its domain. It is also the limiting value of the normalized sequence of Ihara zeta functions for square grid graphs and torus graphs.

Correlation functions of the half-infinite XXZ spin chain with a triangular boundary

The half-infinite XXZ spin chain with a triangular boundary is considered in the massive regime. Two integral representations of correlation functions are proposed using bosonization. Sufficient conditions such that the expressions for triangular boundary conditions coincide with those for diagonal boundary conditions are identified. As an application, summation formulae of the boundary expectation values 〈σ1a〉 with a=z,± are obtained. Exploiting the spin-reversal property, relations between n-fold integrals of elliptic theta functions are extracted.

Wilker- and Huygens-type inequalities for Jacobian elliptic and theta functions

Wilker- and Huygens-type inequalities for Jacobian elliptic functions and classical theta functions are established. For the limiting values of the modulus parameter of the elliptic functions obtained results simplify to known ones which have been established earlier for circular and hyperbolic functions. The main results in this paper are derived with the aid of two inequalities proven in Neuman [Inequalities for weighted sums of powers and their applications. Math Inequal Appl. 2012;15(4):995–1005].

Construction of cyclic calorons

Analytic Nahm data of calorons, i.e., Yang-Mills instantons on ℝ3 × S1, with spatial CN-symmetries are considered. The Nahm equations for the bulk data are reduced to the periodic Toda equation by applying the CN-symmetric ansatz for monopoles by Sutcliffe. It is found that the bulk data are solved by elliptic theta functions which enjoy the hermiticity and the reality conditions. The defining relation to the boundary data are given and the C3-symmetric Nahm data are found as an example.

Cyclic calorons

The Nahm data of periodic instantons, often called calorons, with spatial CN-symmetries are considered, by applying Sutcliffeʼs ansatz for the monopoles with CN-symmetries. The bulk data of calorons are shown to enjoy the periodic Toda lattice, and the solutions are given in terms of elliptic theta functions. The case of N=3 calorons are investigated in detail. It is found that the “scale parameters” of these calorons have upper bounds in their values, so that they do not have the large scale, or monopole, limits. The instanton limit of the C3-symmetric caloron is obtained.