Algebraic independence of values of exponential and elliptic functions

1976 ◽  
Vol 20 (2) ◽  
pp. 669-673
Author(s):  
A. A. Shmelev
Keyword(s):  
Algebraic Independence ◽  
Elliptic Functions

scholarly journals On the measure of algebraic independence of certain values of elliptic functions

1986 ◽  
Vol 23 (1) ◽  
pp. 60-79 ◽  
Author(s):  
Robert Tubbs
Keyword(s):  
Algebraic Independence ◽  
Elliptic Functions

Algebraic independence properties of values of elliptic functions

1982 ◽  
pp. 360-363 ◽  
Author(s):  
D.W. Masser ◽  
G. Wüstholz
Keyword(s):  
Algebraic Independence ◽  
Elliptic Functions ◽  
Independence Properties

scholarly journals Algebraic independence of values of elliptic functions

1986 ◽  
Vol 276 (1) ◽  
pp. 1-17 ◽  
Author(s):  
D. W. Masser ◽  
G. W�stholz
Keyword(s):  
Algebraic Independence ◽  
Elliptic Functions

scholarly journals On the Algebraic Independence of Certain Numbers Connected with the Exponential and the Elliptic Functions

1986 ◽  
Vol 09 (1) ◽  
pp. 29-40 ◽  
Author(s):  
Masanori TOYODA ◽  
Takeshi YASUDA
Keyword(s):  
Algebraic Independence ◽  
Elliptic Functions

Algebraic independence of certain numbers connected with the exponential and elliptic functions

1981 ◽  
Vol 33 (2) ◽  
pp. 216-220
Author(s):  
A. A. Shmelev
Keyword(s):  
Algebraic Independence ◽  
Elliptic Functions

On a measure of algebraic independence of values of Jacobi elliptic functions

2007 ◽  
Vol 146 (2) ◽  
pp. 5782-5790
Author(s):  
Ya. M. Kholyavka
Keyword(s):  
Algebraic Independence ◽  
Elliptic Functions ◽  
Jacobi Elliptic Functions

Extended F-Expansion Method for Solving the modified Korteweg-de Vries (mKdV) Equation

2020 ◽  
Vol 11 (1) ◽  
pp. 93-100
Author(s):  
Vina Apriliani ◽  
Ikhsan Maulidi ◽  
Budi Azhari
Keyword(s):  
Wave Equation ◽  
Exact Solutions ◽  
Internal Waves ◽  
Water Waves ◽  
Wave Equations ◽  
Elliptic Functions ◽  
Expansion Method ◽  
Mkdv Equation ◽  
Innovative Methods ◽  
De Vries

One of the phenomenon in marine science that is often encountered is the phenomenon of water waves. Waves that occur below the surface of seawater are called internal waves. One of the mathematical models that can represent solitary internal waves is the modified Korteweg-de Vries (mKdV) equation. Many methods can be used to construct the solution of the mKdV wave equation, one of which is the extended F-expansion method. The purpose of this study is to determine the solution of the mKdV wave equation using the extended F-expansion method. The result of solving the mKdV wave equation is the exact solutions. The exact solutions of the mKdV wave equation are expressed in the Jacobi elliptic functions, trigonometric functions, and hyperbolic functions. From this research, it is expected to be able to add insight and knowledge about the implementation of the innovative methods for solving wave equations. 

scholarly journals Robust H∞ Loop Shaping Controller Synthesis for SISO Systems by Complex Modular Functions

2021 ◽  
Vol 26 (1) ◽  
pp. 21
Author(s):  
Ahmad Taher Azar ◽  
Fernando E. Serrano ◽  
Nashwa Ahmad Kamal
Keyword(s):  
Design Methodology ◽  
Closed Loop ◽  
Complex Analysis ◽  
Controller Design ◽  
Filter Design ◽  
Design Procedure ◽  
Elliptic Functions ◽  
Loop System ◽  
Closed Loop System ◽  
Loop Shaping

In this paper, a loop shaping controller design methodology for single input and a single output (SISO) system is proposed. The theoretical background for this approach is based on complex elliptic functions which allow a flexible design of a SISO controller considering that elliptic functions have a double periodicity. The gain and phase margins of the closed-loop system can be selected appropriately with this new loop shaping design procedure. The loop shaping design methodology consists of implementing suitable filters to obtain a desired frequency response of the closed-loop system by selecting appropriate poles and zeros by the Abel theorem that are fundamental in the theory of the elliptic functions. The elliptic function properties are implemented to facilitate the loop shaping controller design along with their fundamental background and contributions from the complex analysis that are very useful in the automatic control field. Finally, apart from the filter design, a PID controller loop shaping synthesis is proposed implementing a similar design procedure as the first part of this study.

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