Theta Functions Research Articles

Cycle integrals of meromorphic modular forms and Siegel theta functions

AbstractWe study meromorphic modular forms associated with positive definite binary quadratic forms and their cycle integrals along closed geodesics in the modular curve. We show that suitable linear combinations of these meromorphic modular forms have rational cycle integrals. Along the way we evaluate the cycle integrals of the Siegel theta function associated with an even lattice of signature (1, 2) in terms of Hecke’s indefinite theta functions.

Some congruences for 12-coloured generalized Frobenius partitions

Abstract In his 1984 AMS Memoir, Andrews introduced the family of functions $c\phi_k(n)$ , the number of k-coloured generalized Frobenius partitions of n. In 2019, Chan, Wang and Yang systematically studied the arithmetic properties of $\textrm{C}\Phi_k(q)$ for $2\leq k\leq17$ by utilizing the theory of modular forms, where $\textrm{C}\Phi_k(q)$ denotes the generating function of $c\phi_k(n)$ . In this paper, we first establish another expression of $\textrm{C}\Phi_{12}(q)$ with integer coefficients, then prove some congruences modulo small powers of 3 for $c\phi_{12}(n)$ by using some parameterized identities of theta functions due to A. Alaca, S. Alaca and Williams. Finally, we conjecture three families of congruences modulo powers of 3 satisfied by $c\phi_{12}(n)$ .

Generalizations of mock theta functions

Generalizations of mock theta functions

Wronskian solutions, bilinear Bäcklund transformation, quasi-periodic waves and asymptotic behaviors for a (3+1)-dimensional generalized Kadomtsev–Petviashvili equation

In this paper, we investigate the integrability of a (3+1)-dimensional generalized Kadomtsev–Petviashvili equation, which is widely used in fluid mechanics and theoretical physics. The N-soliton solution is obtained via the Hirota’s bilinear method. The Wronskian solution is derived by using the Wronskian technique for the bilinear form. Through the exchange formula, we deduce the bilinear Bäcklund transformation consisting of four equations and six parameters. In order to consider the quasi-periodic wave having complex structure, one-, two- and three-periodic waves are investigated systemically by combining the Hirota’s bilinear method with Riemann theta function. Furthermore, the corresponding graphs of periodic wave are presented by considering the geometric properties between the characteristic lines. The propagation characteristics of periodic waves are investigated by virtue of the characteristic lines. Finally, the asymptotic relationships between quasi-periodic wave solutions and soliton solutions are established theoretically under a condition of the small amplitude limit. The analytical method used in this paper can be applied in other integrable systems.

A novel solution to the generalized lattice Liouville equation

One generalization of the lattice Liouville equation (LLE) is investigated. Based on the constructed Lax pair, expressions for potential along discrete flows are calculated by using the theory of Riemann surface. Finally a new solution for the generalized LLE is obtained. It contains both exponential functions and theta functions, different from the usual ones.

Theta Functions and Adiabatic Curvature on an Abelian Variety

Theta Functions and Adiabatic Curvature on an Abelian Variety

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.

Multiple localized waves to the (2+1)-dimensional shallow water waveequation on non-flat constant backgrounds and their applications

In this paper, a new general bilinear Bäcklund transformation and Lax pair for the (2+1)-dimensional shallow water wave equation are given in terms of the binary Bell polynomials. Based on this transformation along with introducing an arbitrary function, the multi-kink soliton, line breather, and multi-line rogue wave solutions on a non-flat constant background plane are derived. Further, we found that the dynamic pattern of line breather on the background of periodic line waves are similar to the two-periodic wave solutions obtained through a multi-dimensional Riemann theta function. Also, the generation mechanism and smooth conditions of the line rogue waves on the periodic line wave background are presented with long-wave limit method. Additionally, a family of new rational solutions, consisting of line rogue waves and line solitons, are derived, which have never been reported before. Furthermore, the present work can be directly applied to other nonlinear equations.

KP solitons and the Riemann theta functions

We show that the τ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au $$\\end{document}-functions of the regular KP solitons from the totally nonnegative Grassmannians can be expressed by the Riemann theta functions on singular curves. We explicitly write the parameters in the Riemann theta function in terms of those of the KP soliton. We give a short remark on the Prym theta function on a double covering of singular curves. We also discuss the KP soliton on quasi-periodic background, which is obtained by applying the vertex operators to the Riemann theta function.

Generating functions of the Hurwitz class numbers associated with certain mock theta functions

Generating functions of the Hurwitz class numbers associated with certain mock theta functions

On MaxCut and the Lovász theta function

In this short note we prove a lower bound for the MaxCut of a graph in terms of the Lovász theta function of its complement. We combine this with known bounds on the Lovász theta function of complements of H H -free graphs to recover many known results on the MaxCut of H H -free graphs. In particular, we give a new, very short proof of a conjecture of Alon, Krivelevich and Sudakov about the MaxCut of graphs with no cycles of length r r .

Near optimal colourability on hereditary graph families

In this paper, we initiate a systematic study on a new notion called near optimal colourability which is closely related to perfect graphs and the Lovász theta function. A graph family G is near optimal colourable if there is a constant number c such that every graph G∈G satisfies χ(G)≤max⁡{c,ω(G)}, where χ(G) and ω(G) are the chromatic number and clique number of G, respectively. The near optimal colourable graph families together with the Lovász theta function are useful for the study of the colouring problems for hereditary graph families. We investigate the near optimal colourability for (H1,H2)-free graphs. Our main result is an almost complete characterisation for the near optimal colourability for (H1,H2)-free graphs with two exceptional cases, one of which is the celebrated Gyárfás conjecture. As an application of our results, we show that the colouring problem for (2K2,P4∨Kn)-free graphs is polynomial time solvable, which solves an open problem in Dabrowski and Paulusma (2018) [6].

Apollonian packings and Kac-Moody root systems

We study Apollonian circle packings using the properties of a certain rank 4 indefinite Kac-Moody root system Φ \Phi . We introduce the generating function Z ( s ) Z(\mathbf {s}) of a packing, an exponential series in four variables with an Apollonian symmetry group, which is a symmetric function for Φ \Phi . By exploiting the presence of affine and Lorentzian hyperbolic root subsystems of Φ \Phi , with automorphic Weyl denominators, we express Z ( s ) Z(\mathbf {s}) in terms of Jacobi theta functions and the Siegel modular form Δ 5 \Delta _5 . We also show that the domain of convergence of Z ( s ) Z(\mathbf {s}) is the Tits cone of Φ \Phi , and discover that this domain inherits the intricate geometric structure of Apollonian packings.

Infinite families of congruences modulo $2$ for $(\ell, k)$-regular partitions

Let $b_{\ell, k}(n)$ denote the number of $(\ell, k)$-regular partition of $n$. Recently, some congruences modulo $2$ for $ (3, 8), (4, 7)$-regular partition and modulo $8$, modulo $9$ and modulo $12$ for $(4, 9)$-regular partition has been studied. In this paper, we use theta function identities and Newman results to prove some infinite families of congruences modulo $2$ for $(2, 7)$, $(5, 8)$, $(4, 11)$-regular partition and modulo $4$ for $(4, 5)$-regular partition.

High moments of theta functions and character sums

AbstractAssuming the Generalised Riemann Hypothesis, we prove a sharp upper bound on moments of shifted Dirichlet L‐functions. We use this to obtain conditional upper bounds on high moments of theta functions. Both of these results strengthen theorems of Munsch, who proved almost sharp upper bounds for these quantities. The main new ingredient of our proof comes from a paper of Harper, who showed the related result for all under the Riemann Hypothesis. Finally, we obtain a sharp conditional upper bound on high moments of character sums of arbitrary length.

Gluon scattering in AdS at finite string coupling from localization

We consider gluons scattering in Type IIB string theory on AdS5 × S5/ℤ2 in the presence of D7 branes, which is dual to the flavor multiplet correlator in a certain 4d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 USp(2N) gauge theory with SO(8) flavor symmetry and complexified coupling τ. We compute this holographic correlator in the large N and finite τ expansion using constraints from derivatives of the mass deformed sphere free energy, which we compute to all orders in 1/N and finite τ using supersymmetric localization. In particular, we fix the F4 higher derivative correction to gluon scattering on AdS at finite string coupling τs = τ in terms of Jacobi theta functions, which feature the expected relations between the SL(2, ℤ) duality and the SO(8) triality of the CFT, and match it to the known flat space term. We also use the flat space limit to compute D2F4 corrections of the correlator at finite τ in terms of a non-holomorphic Eisenstein series. At weak string coupling, we find that the AdS correlator takes a form which is remarkably similar to that of the flat space Veneziano amplitude.

Quasi-periodic waves to the defocusing nonlinear Schrödinger equation

A direct approach for the quasi-periodic wave solutions to the defocusing nonlinear Schrödinger equation is proposed based on the theta functions and Hirota’s bilinear method. We transform the problem into a system of algebraic equations, which can be formulated into a least squares problem and then solved by using numerical iterative methods. A rigorous asymptotic analysis demonstrates that these solutions can be classified into two categories: quasi-periodic oscillatory waves and quasi-periodic dark solitons. Singular behaviors may arise in the former case. The numerical results obtained for both the (1+1) -dimensional and (2+1) -dimensional equations are consistent with the theoretical results. Additionaly, the system of algebraic equations can be further extended to address the Riemann–Schottky problem for hyperelliptic curves with 2 infinite points.

On mock theta functions of Gordon and McIntosh

On mock theta functions of Gordon and McIntosh

Differential Equations for Approximate Solutions of Painlevé Equations: Application to the Algebraic Solutions of the Painlevé-III $({\rm D}_7)$ Equation

It is well known that the Painlevé equations can formally degenerate to autonomous differential equations with elliptic function solutions in suitable scaling limits. A way to make this degeneration rigorous is to apply Deift-Zhou steepest-descent techniques to a Riemann-Hilbert representation of a family of solutions. This method leads to an explicit approximation formula in terms of theta functions and related algebro-geometric ingredients that is difficult to directly link to the expected limiting differential equation. However, the approximation arises from an outer parametrix that satisfies relatively simple conditions. By applying a method that we learned from Alexander Its, it is possible to use these simple conditions to directly obtain the limiting differential equation, bypassing the details of the algebro-geometric solution of the outer parametrix problem. In this paper, we illustrate the use of this method to relate an approximation of the algebraic solutions of the Painlevé-III (D$_7$) equation valid in the part of the complex plane where the poles and zeros of the solutions asymptotically reside to a form of the Weierstraß equation.

Large N and large representations of Schur line defect correlators

We study the large N and large representation limits of the Schur line defect correlators of the Wilson line operators transforming in the (anti)symmetric, hook and rectangular representations for \U0001d4a9 = 4 U(N) super Yang-Mills theory. By means of the factorization property, the large N correlators of the Wilson line operators in arbitrary representations can be exactly calculated in principle. In the large representation limit they turn out to be expressible in terms of certain infinite series such as Ramanujan’s general theta functions and the q-analogues of multiple zeta values (q-MZVs). Several generating functions for combinatorial objects, including partitions with non-negative cranks and conjugacy classes of general linear groups over finite fields, emerge from the large N correlators. Also we find conjectured properties of the automorphy and the hook-length expansion satisfied by the large N correlators.