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.

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 Theta Surfaces

2020 ◽  
Author(s):  
Daniele Agostini ◽  
Türkü Özlüm Çelik ◽  
Julia Struwe ◽  
Bernd Sturmfels
Keyword(s):  
Algebraic Geometry ◽  
Theta Functions ◽  
Minkowski Sum ◽  
Zero Set ◽  
Analytic Surface ◽  
Numerical Algebraic Geometry ◽  
Abelian Integrals ◽  
Space Curves ◽  
Quartic Curves ◽  
Special Plane

Abstract A theta surface in affine 3-space is the zero set of a Riemann theta function in genus 3. This includes surfaces arising from special plane quartics that are singular or reducible. Lie and Poincaré showed that any analytic surface that is the Minkowski sum of two space curves in two different ways is a theta surface. The four space curves that generate such a double translation structure are parametrized by abelian integrals, so they are usually not algebraic. This paper offers a new view on this classical topic through the lens of computation. We present practical tools for passing between quartic curves and their theta surfaces, and we develop the numerical algebraic geometry of degenerations of theta functions.

Fast Elliptic Curve Point Multiplication Algorithm Optimization

2013 ◽  
Vol 441 ◽  
pp. 1044-1048
Author(s):  
Hai Bin Zhang ◽  
Xiao Ping Ji ◽  
Bo Ying Wu ◽  
Guang Yu Li
Keyword(s):  
Positive Integer ◽  
Elliptic Curve ◽  
Hot Spot ◽  
Algorithm Optimization ◽  
Point Multiplication ◽  
Multiplication Algorithm ◽  
Multiplication Operation ◽  
Ternary Form ◽  
Main Ideas ◽  
Operation Cost

Scalar point multiplication operation on elliptic curve is the most expensive part of the elliptic curve cryptosystem, also has always been the hot spot of the research. Recoding the positive integer and reducing the amount of inversion in the operation are the two main ideas. In this article, we use the balanced ternary form to recode the positive integer, at the same time, improve the part of calculation way of, reducing the amount of inversion, decreasing operation cost, and improving operation efficiency

Inverse numerical range and determinantal representation

2018 ◽  
Vol 558 ◽  
pp. 79-100 ◽  
Author(s):  
Mao-Ting Chien ◽  
Hiroshi Nakazato ◽  
Lina Yeh
Keyword(s):  
Numerical Range ◽  
Determinantal Representation

On a strong limit-point condition and an integral inequality associated with a symmetric matrix differential expression

1977 ◽  
Vol 76 (2) ◽  
pp. 155-159 ◽  
Author(s):  
D. A. R. Rigler
Keyword(s):  
Differential Expression ◽  
Vector Function ◽  
Real Line ◽  
Integral Inequality ◽  
Limit Point ◽  
Symmetric Matrix ◽  
Strong Limit ◽  
The Matrix ◽  
Point Condition ◽  
Matrix Differential

SynopsisThis paper is concerned with some properties of an ordinary symmetric matrix differential expression M, denned on a certain class of vector-functions, each of which is defined on the real line. For such a vector-function F we have M[F] = −F“ + QF on R, where Q is an n × n matrix whose elements are reasonably behaved on R. M is classified in an equivalent of the limit-point condition at the singular points ± ∞, and conditions on the matrix coefficient Q are given which place M, when n> 1, in the equivalent of the strong limit-point for the case n = 1. It is also shown that the same condition on Q establishes the integral inequality for a certain class of vector-functions F.

Theta Functions and Adiabatic Curvature on an Elliptic Curve

2022 ◽  
Vol 32 (3) ◽  
Author(s):  
Ching-Hao Chang ◽  
Jih-Hsin Cheng ◽  
I-Hsun Tsai
Keyword(s):  
Elliptic Curve ◽  
Theta Functions

The Generalized Rayleigh Quotient

1974 ◽  
Vol 17 (2) ◽  
pp. 251-256 ◽  
Author(s):  
M. V. Pattabhiraman
Keyword(s):  
Banach Space ◽  
Symmetric Matrix ◽  
Matrix Pencil ◽  
Complex Banach Space ◽  
Rayleigh Quotient ◽  
Positive Definite ◽  
Characteristic Vector ◽  
Characteristic Value ◽  
Image Position ◽  
Generalized Rayleigh Quotient

In this paper we generalize the concept of the Rayleigh quotient to a complex Banach space. Lord Rayleigh investigated the quotient(1)considered as a function of the components of q, in the case of a symmetric matrix pencil Aλ+C with A positive definite. It is known that R(q) has a stationary value when q is a characteristic vector of Aλ+C and that(2)where qi is a characteristic vector corresponding to the characteristic value λi

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