Jacobi Theta Research Articles

Relations and positivity results for the derivatives of the Riemann ξ function

We present and evaluate the integer-order derivatives of the Riemann xi function. These derivatives contain logarithmic integrals of powers multiplying a specific Jacobi theta function and as such can be alternatively viewed as certain Mellin transforms at integer argument. We describe how the derivatives at s=0, s= 1 2 , and s=1 can be evaluated exactly. We further show, based upon a novel representation, that the even order derivatives at s= 1 2 are all positive, as are all derivatives at s=1. An expression is presented for the derivatives on the critical line, which may be useful in studying the zeros of the function Ξ(t)=ξ( 1 2 + it) .

Jacobi Theta Functions over Number Fields

We use Jacobi theta functions to construct examples of Jacobi forms over number fields. We determine the behavior under modular transformations by regarding certain coefficients of the Jacobi theta functions as specializations of symplectic theta functions. In addition, we show how sums of those Jacobi theta functions appear as a single coefficient of a symplectic theta function.

Yukawa couplings in intersecting D-brane models

We compute the Yukawa couplings among chiral fields in toroidal Type II compactifications with wrapping D6-branes intersecting at angles. Those models can yield realistic standard model spectrum living at the intersections. The Yukawa couplings depend both on the Kahler and open string moduli but not on the complex structure. They arise from worldsheet instanton corrections and are found to be given by products of complex Jacobi theta functions with characteristics. The Yukawa couplings for a particular intersecting brane configuration yielding the chiral spectrum of the MSSM are computed as an example. We also show how our methods can be extended to compute Yukawa couplings on certain classes of elliptically fibered CY manifolds which are mirror to complex cones over del Pezzo surfaces. We find that the Yukawa couplings in intersecting D6-brane models have a mathematical interpretation in the context of homological mirror symmetry. In particular, the computation of such Yukawa couplings is related to the construction of Fukaya's category in a generic symplectic manifold.

Cyclic identities for Jacobi elliptic and related functions

Identities involving cyclic sums of terms composed from Jacobi elliptic functions evaluated at p equally shifted points on the real axis were recently found. These identities played a crucial role in discovering linear superposition solutions of a large number of important nonlinear equations. We derive four master identities, from which the identities discussed earlier are derivable as special cases. Master identities are also obtained which lead to cyclic identities with alternating signs. We discuss an extension of our results to pure imaginary and complex shifts as well as to the ratio of Jacobi theta functions.

Splitting the shadow

We derive formulae for the theta series of the two translates of the even sublattice L 0 of an odd unimodular lattice L that constitute the shadow of L. The proof rests on special evaluations of the Jacobi theta series attached to L and to a certain vector. We produce an analogous theorem for codes. Additionally, we construct non-linear formally self-dual codes and relate them to lattices.

First passage time distribution in random walks with absorbing boundaries

We calculate the first passage time distribution in simple, unbiased random walks in presence of absorbing boundaries of various shapes. We obtain explicit solutions for the following geometries of the boundaries—a box in one dimension, circular, square and triangular boundaries in two dimensions and cubical box and spherical shell in three dimensions. The distribution in all cases shows scaling and the scaling function can be expressed in terms of the Jacobi Theta functions.

Développement asymptotique [formula omitted]-Gevrey et fonction thêta de Jacobi

Some notions of q-Gevrey asymptotic expansion have been studied in [6,7]. Recently we became interested in a new notion of asymptotic expansion [8]: it is related to a Jacobi theta function and allows one to establish the natural link between the asymptotics of q-difference equations and the theory of elliptic functions. The purpose of this Note is to give some new results related to this notion of asymptotic expansion. To cite this article: J.-P. Ramis, C. Zhang, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 899–902.

Elliptic Ruijsenaars operators and functional equations

We study mutually commutative difference operators introduced by Ruijsenaars, which are regarded as elliptic analogs of Macdonald operators and consist of the Jacobi theta functions and shift operators. These operators are constructed in terms of R-operators due to Shibukawa and Ueno or root algebras introduced by Cherednik, which give rise to a set of functional equations. We investigate solutions of these functional equations for generalizations of elliptic Ruijsenaars operators.

The equipotential surfaces of cubic lattices

The areas of zero electrostatic potential within an array of electric charges in an ionic crystal can be represented as a zero equipotential surface (ZEPS), which separates space into domains of positive and negative potential. ZEPS are conventionally expressed in terms of nodal surfaces. We express the coordinates and charge distribution of several ZEPS in terms of the Jacobi theta functions, and provide their graphical visualisation.

Nodal Surface Approximations to the Zero Equipotential Surfaces for Cubic Lattices

Models of electrostatic surfaces in atomic crystals rely on equations involving the Jacobi theta functions. Numerical integration of these is prohibitively time consuming, making it difficult to examine the properties of the fields which give rise to the surfaces. We give simple expressions for the key electrostatic surfaces using Fourier expansions in basis sets of nodal surfaces. Any surface may be computed in seconds in a form ammenable to further analysis. The distribution of the mean and Gaussian curvatures over each surface has been visualised by assigning colours so that the range from minimum to maximum value spans blue to red. We similarly explore the mean and Gaussian scalar fields over a range of triply periodic surfaces of the same morphology.

Ruijsenaars’ commuting difference operators and invariant subspace spanned by theta functions

We study a family of mutually commutative difference operators introduced by Ruijsenaars. The conjugations of these operators with an appropriate function give the Hamiltonians of some relativistic quantum systems. These operators can be regarded as elliptic analogs of the Macdonald operators and their coefficients consist of the Jacobi theta functions. We show that these operators act on the space of meromorphic functions on the Cartan subalgebra of affine Lie algebras and that the space spanned by characters of a fixed positive level is invariant under the action of these operators.

Probability laws related to the Jacobi theta and Riemann zeta functions, and Brownian excursions

This paper reviews known results which connect Riemann’s integral representations of his zeta function, involving Jacobi’s theta function and its derivatives, to some particular probability laws governing sums of independent exponential variables. These laws are related to one-dimensional Brownian motion and to higher dimensional Bessel processes. We present some characterizations of these probability laws, and some approximations of Riemann’s zeta function which are related to these laws.

Classical Yang-Mills vacua onT3:Explicit constructions

Flat connections for unitary gauge groups on a 3-torus with twisted boundary conditions as well as recently discovered periodic nontrivial flat connections with ''nondiagonalizable'' triples of holonomies for higher orthogonal and exceptional groups are constructed explicitly in terms of Jacobi theta functions with rational characteristics. The (fractional) Chern-Simons numbers of these vacuum gauge field configurations are verified by direct computation.

Discrete Normal distribution and its relationship with Jacobi Theta functions

We introduce new, natural parameters in a formula defining a family of discrete Normal distributions. One of the parameters is closely related to the expectation and the other to the variance of that family. We show that under such a parametrization, uniformly for all sufficiently large variances and all expectations, discrete Normal distributions and their first two moments are given by very simple formulae. We indicate the relation between our results and Jacobi Theta functions and Jacobi summation formulae.

Vector potential due to a circular loop in a toroidal cavity in a high-permeability core

We present a Green's function for a circular loop in a toroidal cavity in a high-permeability material. We show that under certain conditions, Green's function may be expressed in terms of Jacobi's theta functions. We demonstrate an application of such a Green's function in analyzing the performance of pot core transformers. Results compare well with the values obtained by other analytical methods and by the measurement.

On nonlinear equations integrable in theta functions of nonprincipally polarized Abelian varieties

The formula of expanding the Abel variety theta function restricted to Abel subvariety into theta functions of this subvariety is obtained. With the help of this formula the solution of differential equations with Jacobi theta functions, restricted on a nonprincipally polarized Abel subvariety and their translations are rewritten in terms of the theta functions of these subvarieties. This is exemplified by the CKP equations, the Bogoyavlenskij system, and the Toda $g_2^{(1)}$-chain.

Quotients of theta series as rational functions of j1,4

Let Q( n, 1) be the set of even unimodular positive definite integral quadratic forms in n-variables. Then n is divisible by 8. For A[ x] in Q( n, 1), the theta series 0 A(z): = ∑ XϵZ ne πizA¦X¦ (z ϵ h , the complex upper half plane) is a modular form of weight n 2 for the congruence group Γ 1(4) = {γ ϵ SL 2(Z)¦γ  1 ∗ 0 1 mod 4} . If n ≥ 24 and A[ X], B[ X] are two quadratic forms in Q( n, 1), then the quotient θ A(z) θ B(z) is a modular function for Γ 1(4). Since we can identify the field of modular functions for Γ 1(4) with the function field K( X 1(4)) over the modular curve x 1(4) = Γ 1(4) h∗ (the extended plane of h ) with genus 0, in this paper, we express it as a rational function j 1,4 which is a field generator over C of K( X 1(4)) and defined by j 1.4(z): = θ 2(2z) 4 (θ 3(2z) 4 . Here, θ 2 and θ 3 denote the classical Jacobi theta functions.

Projections in Rotation Algebras and Theta Functions

For each $\alpha \in (0,1)$, $A_\alpha$ denotes the universal $C^*$-algebra generated by two unitaries $u$ and $v$, which satisfy the commutation relation $uv=\exp (2\pi i\alpha)vu$. We consider the order four automorphism $\sigma$ of $A_\alpha$ defined by $\sigma (u)=v$, $\sigma (v)=u^{-1}$ and describe a method for constructing projections in the fixed point algebra $A_\alpha^\sigma$, using Rieffel's imprimitivity bimodules and Jacobi's theta functions. In the case $\alpha =q^{-1}$, $q\in {\mathbf Z}$, $q\geq 2$, we give explicit formulae for such projections and find a lower bound for the norm of the Harper operator $u+u^* +v+v^*$.

Elliptic functions, theta function and hypersurfaces satisfying a basic equality

In previous papers [4, 6], B.-Y. Chen introduced a Riemannian invariant δM for a Riemannian n-manifold Mn, namely take the scalar curvature and subtract at each point the smallest sectional curvature. He proved that every submanifold Mn in a Riemannian space form Rm(ε) satisfies: δM[les ][n2(n−2)]/ 2(n−1)H2+[half](n+1)(n−2)ε. In this paper, first we classify constant mean curvature hypersurfaces in a Riemannian space form which satisfy the equality case of the inequality. Next, by utilizing Jacobi's elliptic functions and theta function we obtain the complete classification of conformally flat hypersurfaces in Riemannian space forms which satisfy the equality.

A tight upper bound on discrete entropy

The standard upper bound on discrete entropy was derived based on the differential entropy bound for continuous random variables. A tighter discrete entropy bound is derived using the transformation formula of Jacobi theta function. The new bound is applicable only when the probability mass function of the discrete random variable satisfies certain conditions. Its application to the class of binomial random variables is presented as an example.