Determinantal representations of elliptic curves via Weierstrass elliptic functions

2018 ◽  
Vol 34 ◽  
pp. 125-136 ◽  
Author(s):  
Mao-Ting Chien ◽  
Hiroshi Nakazato
Keyword(s):  
Elliptic Curves ◽  
Algebraic Curve ◽  
Theta Functions ◽  
Elliptic Functions ◽  
Determinantal Representation ◽  
Ternary Form ◽  
Riemann Theta Functions

Helton and Vinnikov proved that every hyperbolic ternary form admits a symmetric derminantal representation via Riemann theta functions. In the case the algebraic curve of the hyperbolic ternary form is elliptic, the determinantal representation of the ternary form is formulated by using Weierstrass $\wp$-functions in place of Riemann theta functions. An example of this approach is given.

scholarly journals Inverse Numerical Range and Determinantal Quartic Curves

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2119
Author(s):  
Mao-Ting Chien ◽  
Hiroshi Nakazato
Keyword(s):  
Elliptic Curve ◽  
Vector Function ◽  
Numerical Range ◽  
Symmetric Matrix ◽  
Group Structure ◽  
Theta Functions ◽  
Matrix Pencil ◽  
Determinantal Representation ◽  
Ternary Form ◽  
Quartic Curves

A hyperbolic ternary form, according to the Helton–Vinnikov theorem, admits a determinantal representation of a linear symmetric matrix pencil. A kernel vector function of the linear symmetric matrix pencil is a solution to the inverse numerical range problem of a matrix. We show that the kernel vector function associated to an irreducible hyperbolic elliptic curve is related to the elliptic group structure of the theta functions used in the Helton–Vinnikov theorem.

Theta functions and elliptic functions

2016 ◽  
pp. 384-403
Author(s):  
J. Hietarinta ◽  
N. Joshi ◽  
F. W. Nijhoff
Keyword(s):  
Theta Functions ◽  
Elliptic Functions

scholarly journals Topological control of extreme waves

2019 ◽  
Vol 10 (1) ◽  
Author(s):  
Giulia Marcucci ◽  
Davide Pierangeli ◽  
Aharon J. Agranat ◽  
Ray-Kuang Lee ◽  
Eugenio DelRe ◽  
...  
Keyword(s):  
Nonlinear Waves ◽  
Control Strategies ◽  
Theta Functions ◽  
State Diagram ◽  
Rogue Waves ◽  
Nonlinear Phenomena ◽  
Extreme Waves ◽  
Nonlinear Wave Propagation ◽  
Riemann Theta Functions ◽  
The One

Abstract From optics to hydrodynamics, shock and rogue waves are widespread. Although they appear as distinct phenomena, transitions between extreme waves are allowed. However, these have never been experimentally observed because control strategies are still missing. We introduce the new concept of topological control based on the one-to-one correspondence between the number of wave packet oscillating phases and the genus of toroidal surfaces associated with the nonlinear Schrödinger equation solutions through Riemann theta functions. We demonstrate the concept experimentally by reporting observations of supervised transitions between waves with different genera. Considering the box problem in a focusing photorefractive medium, we tailor the time-dependent nonlinearity and dispersion to explore each region in the state diagram of the nonlinear wave propagation. Our result is the first realization of topological control of nonlinear waves. This new technique casts light on shock and rogue waves generation and can be extended to other nonlinear phenomena.

Elliptic Functions, Theta Functions, and Riemann Surfaces

1974 ◽  
Vol 28 (127) ◽  
pp. 875
Author(s):  
Y. L. L. ◽  
Harry E. Rauch ◽  
Aaron Lebowitz
Keyword(s):  
Riemann Surfaces ◽  
Theta Functions ◽  
Elliptic Functions

scholarly journals Boundary asymptotics of the relative Bergman kernel metric for hyperelliptic curves

2017 ◽  
Vol 4 (1) ◽  
pp. 7-15 ◽  
Author(s):  
Robert Xin Dong
Keyword(s):  
Elliptic Curves ◽  
Bergman Kernel ◽  
Complex Structure ◽  
Elliptic Functions ◽  
Hyperelliptic Curves ◽  
Asymptotic Formulas ◽  
Holomorphic Family ◽  
Structure Information ◽  
Taylor Expansions ◽  
Bergman Kernel Asymptotics

Abstract We survey variations of the Bergman kernel and their asymptotic behaviors at degeneration. For a Legendre family of elliptic curves, the curvature form of the relative Bergman kernel metric is equal to the Poincaré metric on ℂ \ {0,1}. The cases of other elliptic curves are either the same or trivial. Two proofs depending on elliptic functions’ special properties and Abelian differentials’ Taylor expansions are discussed, respectively. For a holomorphic family of hyperelliptic nodal or cuspidal curves and their Jacobians, we announce our results on the Bergman kernel asymptotics near various singularities. For genus-two curves particularly, asymptotic formulas with precise coefficients involving the complex structure information are written down explicitly.

RIEMANN THETA FUNCTIONS SOLUTIONS WITH RATIONAL CHARACTERISTICS FOR (2+1)-DIMENSIONAL SINH-GORDON EQUATION

2010 ◽  
Vol 24 (06) ◽  
pp. 575-584
Author(s):  
YANG FENG ◽  
HONG-QING ZHANG
Keyword(s):  
Gordon Equation ◽  
Theta Functions ◽  
Periodic Wave ◽  
Bilinear Method ◽  
Periodic Wave Solutions ◽  
Wave Solutions ◽  
Wave Numbers ◽  
Riemann Theta Functions ◽  
The Waves ◽  
Hirota Bilinear

In this letter, we use the Riemann theta functions with rational characteristics and the Hirota bilinear method to construct quasi-periodic wave solutions for (2+1)-dimensional sinh-Gordon equation. This method not only conveniently obtains quasi-periodic solutions of nonlinear equations, but also directly gets the explicit expressions of frequencies, wave numbers, phase and amplitudes for the waves.

scholarly journals Computing Riemann theta functions in Sage with applications

2016 ◽  
Vol 127 ◽  
pp. 263-272 ◽  
Author(s):  
Christopher Swierczewski ◽  
Bernard Deconinck
Keyword(s):  
Theta Functions ◽  
Riemann Theta Functions
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