Jacobi Theta Functions Research Articles

Comparaison des résultats cliniques et radiologiques d’une réparation arthroscopique de la coiffe des rotateurs chez des patients de moins de 50 ans et de plus de 70 ans sur une série prospective continue multicentrique de 312 cas

The Euler theorem in partition theory and its generalization are derived from a non-interacting quantum field theory in which each bosonic mode with a given frequency is equivalent to a sum of bosonic mode whose frequency is twice (s-times) as much, and a fermionic (parafermionic) mode with the same frequency. Explicit formulas for the graded parafermionic partition functions are obtained, and the inverse of the graded partition function (IGPPF), turns out to be bosonic (fermionic) partition function depending on the parity of the order s of the parafermions. It is also shown that these partition functions are generating functions of partitions of integers with restrictions, the Euler generating function is identified with the inverse of the graded parafermionic partition function of order 2. As a result we obtain new sequences of partitions of integers with given restrictions. If the parity of the order s is even, then mixing a system of parafermions with a system whose partition function is (IGPPF), results in a system of fermions and bosons. On the other hand, if the parity of s is odd, then, the system we obtain is still a mixture of fermions and bosons but the corresponding Fock space of states is truncated. It turns out that these partition functions are given in terms of the Jacobi theta function θ4, and generate sequences in partition theory. Our partition functions coincide with the overpartitions of Corteel and Lovejoy, and jagged partitions in conformal field theory. Also, the partition functions obtained are related to the Ramond characters of the superconformal minimal models, and in the counting of the Moore–Read edge spectra that appear in the fractional quantum Hall effect. The different partition functions for the Riemann gas that are the counter parts of the Euler gas are obtained by a simple change of variables. In particular the counter part of the Jacobi theta function is ζ(2t)ζ(t)2. Finally, we propose two formulas which brings the additive number theory and the multiplicative number theory closer.

The t-coefficient method II: A new series expansion formula of theta function products and its implications

By means of Jacobiʼs triple product identity and the t-coefficient method, we establish a general series expansion formula with five free parameters for the product of arbitrary two Jacobi theta functions. It embodies the triple, quintuple, sextuple and septuple theta function product identities and the generalized Schröter formula. As further applications, we also set up a series expansion formula for the product of three theta functions. It not only generalizes Ewellʼs and Chen–Chen–Huangʼs octuple product identities, but also contains three cubic theta function identities due to Farkas–Kra and Ramanujan respectively and the Macdonald identity for the root system A2 as special cases. In the meantime, many other new identities including a new short expression of the triple theta series of Andrews are also presented.

Connection coefficients for basic Harish-Chandra series

Basic Harish-Chandra series are asymptotically free meromorphic solutions of the system of basic hypergeometric difference equations associated to root systems. The associated connection coefficients are explicitly computed in terms of Jacobi theta functions. We interpret the connection coefficients as the transition functions for asymptotically free meromorphic solutions of Cherednikʼs root system analogs of the quantum Knizhnik–Zamolodchikov equations. They thus give rise to explicit elliptic solutions of root system analogs of dynamical Yang–Baxter and reflection equations. Applications to quantum c-functions, basic hypergeometric functions, reflectionless difference operators and multivariable Baker–Akhiezer functions are discussed.

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].

Cubic theta functions

The theory of the cubic theta function ∑qm2+mn+n2z1mz2n is developed analogously to the theory of the classical Jacobian theta functions. Many inversion formulas, addition formulas, and modular equations are derived, and these results extend previous work on cubic theta functions.

Analytic continuation of the doubly-periodic Barnes zeta function

The aim of this work is to study the analytic continuation of the doubly-periodic Barnes zeta function. By using a suitable complex integral representation as a starting point we find the meromorphic extension of the doubly periodic Barnes zeta function to the entire complex plane in terms of a real integral containing the Hurwitz zeta function and the first Jacobi theta function. These allow us to explicitly give expressions for the derivative at all non-positive integer points.

Asymptotics of generalized Galois numbers via affine Kac-Moody algebras

Generalized Galois numbers count the number of flags in vector spaces over finite fields. Asymptotically, as the dimension of the vector space becomes large, we give their exponential growth and determine their initial values. The initial values are expressed analytically in terms of theta functions and Euler’s generating function for the partition numbers. Our asymptotic enumeration method is based on a Demazure module limit construction for integrable highest weight representations of affine Kac-Moody algebras. For the classical Galois numbers that count the number of subspaces in vector spaces over finite fields, the theta functions are Jacobi theta functions. We apply our findings to the asymptotic number of linearqq-ary codes and conclude with some final remarks about possible future research concerning asymptotic enumerations via limit constructions for affine Kac-Moody algebras and modularity of characters of integrable highest weight representations.

D-type anticipatory iterative learning control for a class of inhomogeneous heat equations

In this paper, a D-type anticipatory iterative learning control (ILC) scheme is applied to the boundary control of a class of inhomogeneous heat equations, where the heat flux at one side is the control input while the temperature measurement at the other side is the control output. By transforming the inhomogeneous heat equation into its integral form and exploiting the properties of the embedded Jacobi Theta functions, the learning convergence of ILC is guaranteed through rigorous analysis, without any simplification or discretization of the 3D dynamics in the time, space as well as iteration domains. The adopted ILC scheme makes full use of the process repetition and deals with state-independent or state-dependent uncertainties. Meanwhile, due to the feedforward characteristic of ILC, the proposed scheme not only makes anticipatory compensation possible to overcome the heat conduction delay in boundary output tracking, but also eliminates the gain margin limitation encountered in feedback control. In the end, an illustrative example is presented to demonstrate the performance of the proposed ILC scheme.

Duplication formulae involving Jacobi theta functions and Gosper’s 𝑞-trigonometric functions

Using the q q -trigonometric definitions of Gosper, we devise a new q q -exponential function. Based on this concept, we derive a number of identities involving the Jacobi theta functions. These considerations lead to the answers to Gosper’s “mysteries”.

The Euler–Riemann gases, and partition identities

The Euler theorem in partition theory and its generalization are derived from a non-interacting quantum field theory in which each bosonic mode with a given frequency is equivalent to a sum of bosonic mode whose frequency is twice (s-times) as much, and a fermionic (parafermionic) mode with the same frequency. Explicit formulas for the graded parafermionic partition functions are obtained, and the inverse of the graded partition function (IGPPF), turns out to be bosonic (fermionic) partition function depending on the parity of the order s of the parafermions. It is also shown that these partition functions are generating functions of partitions of integers with restrictions, the Euler generating function is identified with the inverse of the graded parafermionic partition function of order 2. As a result we obtain new sequences of partitions of integers with given restrictions. If the parity of the order s is even, then mixing a system of parafermions with a system whose partition function is (IGPPF), results in a system of fermions and bosons. On the other hand, if the parity of s is odd, then, the system we obtain is still a mixture of fermions and bosons but the corresponding Fock space of states is truncated. It turns out that these partition functions are given in terms of the Jacobi theta function θ4, and generate sequences in partition theory. Our partition functions coincide with the overpartitions of Corteel and Lovejoy, and jagged partitions in conformal field theory. Also, the partition functions obtained are related to the Ramond characters of the superconformal minimal models, and in the counting of the Moore–Read edge spectra that appear in the fractional quantum Hall effect. The different partition functions for the Riemann gas that are the counter parts of the Euler gas are obtained by a simple change of variables. In particular the counter part of the Jacobi theta function is ζ(2t)ζ(t)2. Finally, we propose two formulas which brings the additive number theory and the multiplicative number theory closer.

On the log-concavity of a Jacobi theta function

A new proof of the log-concavity of the Jacobi theta function, appearing in the Fourier representation of the Riemann $\Xi$ function, is presented. An open problem, involving the normalized moments of log-concave kernels, is investigated. In particular, several Turán-type inequalities are established.

Cubic analogues of the Jacobian theta functions and the Jacobian elliptic functions

In this paper, the Jacobian theta functions, the Jacobian elliptic functions, and their cubic analogues are introduced. Then, the properties satisfied the cubic analogues are discussed.

Iterative Learning Control for a Class of Inhomogeneous Heat Equations

Abstract This paper addresses the boundary control problem of heat equations under the framework of iterative learning control (ILC). Without any simplification or discretization of the 3D dynamics in the time, space as well as iteration domains, the learning convergence of ILC is guaranteed through rigorous analysis by transforming the inhomogeneous heat equation into its integral form and exploiting the properties of the embedded Jacobi Theta functions. The proposed scheme not only makes anticipatory compensation possible to overcome the heat conduction delay in boundary output tracking, but also eliminates the gain margin limitation encountered in feedback control. In the end, an illustrative example is presented to demonstrate the performance of the proposed ILC scheme.

A q -Raabe formula and an integral of the fourth Jacobi theta function

We generalize the Raabe-formula to the q -loggamma function by giving an integral formula for log Γ q when q > 1 . As a consequence, we get that the integral of the logarithm of the fourth Jacobi theta function between its least imaginary zeros is connected to the partition function and the Riemann zeta function.

Properties of finite Gaussians and the discrete-continuous transition

Weyl's formulation of quantum mechanics opened the possibility of studying the dynamics of quantum systems both in infinite-dimensional and finite-dimensional systems. Based on Weyl's approach, generalized by Schwinger, a self-consistent theoretical framework describing physical systems characterized by a finite-dimensional space of states has been created. The used mathematical formalism is further developed by adding finite-dimensional versions of some notions and results from the continuous case. Discrete versions of the continuous Gaussian functions have been defined by using the Jacobi theta functions. We continue the investigation of the properties of these finite Gaussians by following the analogy with the continuous case. We study the uncertainty relation of finite Gaussian states, the form of the associated Wigner quasi-distribution and the evolution under free-particle and quantum harmonic oscillator Hamiltonians. In all cases, a particular emphasis is put on the recovery of the known continuous-limit results when the dimension d of the system increases.

Reduction Formulae for Products of Theta Functions.

In four cases it is already known that the product of two distinct Jacobian theta functions having the same variable z and the same nome q is a multiple of a single Jacobian theta function, with the multiple independent of z. The main purpose of the present note is to show that this property also applies in the remaining two cases.

Critical phenomena and phase sequence in a classical bilayer Wigner crystal at zero temperature

We study the ground-state properties of a system of identical classical Coulombic point particles, evenly distributed between two equivalently charged parallel plates at distance $d$; the system as a whole is electroneutral. It was previously shown that upon increasing d from 0 to infinity, five different structures of the bilayer Wigner crystal become energetically favored, starting from a hexagonal lattice (phase I, d=0) and ending at a staggered hexagonal lattice (phase V, d -> infinity). In this paper, we derive new series representations of the ground-state energy for all five bilayer structures. The derivation is based on a sequence of transformations for lattice sums of Coulomb two-particle potentials plus the neutralizing background, having their origin in the general theory of Jacobi theta functions. The new series provide convenient starting points for both analytical and numerical progress. Its convergence properties are indeed excellent: Truncation at the fourth term determines in general the energy correctly up to 17 decimal digits. The accurate series representations are used to improve the specification of transition points between the phases and to solve a controversy in previous studies. In particular, it is shown both analytically and numerically that the hexagonal phase I is stable only at d=0, and not in a finite interval of small distances between the plates as was anticipated before. The expansions of the structure energies around second-order transition points can be done analytically, which enables us to show that the critical behavior is of the Ginzburg-Landau type, with a mean-field critical index beta=1/2 for the growth of the order parameters.

Descent for differential Galois theory of difference equations: confluence andq-dependence

The present paper contains two results that generalize and improve constructions of Hardouin and Singer. In the case of a derivation, we prove that the parametrized Galois theory for difference equations constructed by Hardouin and Singer can be descended from a differentially closed to an algebraically closed field. In the second part of the paper, we show that the theory can be applied to deformations of q-series to study the differential dependence with respect to x d d x and q d dq . We show that the parametrized difference Galois group (with respect to a convenient derivation defined in the text) of the Jacobi Theta function can be considered as the Galoisian counterpart of the heat equation.

On Andrica’s Conjecture, Cramér’s Conjecture, Gaps between Primes and Jacobi Theta Functions I

The main objective of this paper is to develop upper and lower bound for the Andrica conjecture, gaps between primes, using Jacobi elliptic functions.

On Andrica’s Conjecture, Cramér’s Conjecture, Gaps between Primes and Jacobi Theta Functions II: A Simple Proof of Asymptotic for Andrica’s Conjecture

Bulletin of Mathematical Sciences and Applications (Volume 4): On Andrica’s Conjecture, Cramér’s Conjecture, Gaps between Primes and Jacobi Theta Functions II: A Simple Proof of Asymptotic for Andrica’s Conjecture