scholarly journals A characterization of postcritically minimal Newton maps of complex exponential functions

2018 ◽  
Vol 39 (10) ◽  
pp. 2855-2880
Author(s):  
KHUDOYOR MAMAYUSUPOV
Keyword(s):  
Exponential Function ◽  
Julia Sets ◽  
Exponential Functions ◽  
Postcritically Finite ◽  
One To One ◽  
Complex Exponential

We obtain a unique, canonical one-to-one correspondence between the space of marked postcritically finite Newton maps of polynomials and the space of postcritically minimal Newton maps of entire maps that take the form $p(z)\exp (q(z))$ for $p(z)$, $q(z)$ polynomials and $\exp (z)$, the complex exponential function. This bijection preserves the dynamics and embedding of Julia sets and is induced by a surgery tool developed by Haïssinsky.

JULIA SET OF THE NEWTON METHOD FOR SOLVING SOME COMPLEX EXPONENTIAL EQUATION

2009 ◽  
Vol 09 (02) ◽  
pp. 153-169 ◽  
Author(s):  
XINGYUAN WANG ◽  
WENJING SONG ◽  
LIXIAN ZOU
Keyword(s):  
Newton Method ◽  
Exponential Function ◽  
Julia Set ◽  
Julia Sets ◽  
Exponential Equation ◽  
Distribution Structure ◽  
Complex Exponential

We extend Kim's complex exponential function and come up with a theory about Julia sets of Newton method for general exponential equation. We analyze the behavior of the roots of some complex exponential equation, and prove the Julia Set's symmetry, boundedness and embedding topology distribution structure of attraction regions in theory.

THE ACCURACY OF COMPUTER ALGORITHMS IN DYNAMICAL SYSTEMS

1991 ◽  
Vol 01 (03) ◽  
pp. 625-639 ◽  
Author(s):  
MARILYN B. DURKIN
Keyword(s):  
Dynamical Systems ◽  
Exponential Function ◽  
Julia Set ◽  
Julia Sets ◽  
Simple Algorithm ◽  
Computer Algorithms ◽  
Simple Expansion ◽  
Mathematical Accuracy ◽  
Complex Exponential

We study the mathematical accuracy of computer algorithms used to produce pictures of Julia sets by analyzing two representatives cases of the complex exponential function. We first define the Julia set and give the simple algorithm used for the exponential function. We then define what it means for a picture to be "right" and consider the two totally different Julia sets of E0.3(z) = 0.3ez and E(z) = ez. We use a simple expansion argument together with the properties of the exponential function to show that each of these pictures is correct.

Julia sets of Newton’s method for a class of complex-exponential function F(z)=P(z)e Q(z)

2010 ◽  
Vol 62 (4) ◽  
pp. 955-966 ◽  
Author(s):  
Xing-Yuan Wang ◽  
Yi-Ke Li ◽  
Yuan-Yuan Sun ◽  
Jun-Mei Song ◽  
Feng-Dan Ge
Keyword(s):  
Exponential Function ◽  
Newton’S Method ◽  
Newton's Method ◽  
Julia Sets ◽  
Complex Exponential

scholarly journals Exponential Functions in Cartesian Differential Categories

2020 ◽  
Author(s):  
Jean-Simon Pacaud Lemay
Keyword(s):  
Differential Equations ◽  
Exponential Function ◽  
Differential Calculus ◽  
Exponential Map ◽  
Dual Numbers ◽  
Exponential Functions ◽  
Smooth Functions ◽  
Exponential Maps ◽  
Differential Categories ◽  
Complex Exponential

Abstract In this paper, we introduce differential exponential maps in Cartesian differential categories, which generalizes the exponential function $$e^x$$ e x from classical differential calculus. A differential exponential map is an endomorphism which is compatible with the differential combinator in such a way that generalizations of $$e^0 = 1$$ e 0 = 1 , $$e^{x+y} = e^x e^y$$ e x + y = e x e y , and $$\frac{\partial e^x}{\partial x} = e^x$$ ∂ e x ∂ x = e x all hold. Every differential exponential map induces a commutative rig, which we call a differential exponential rig, and conversely, every differential exponential rig induces a differential exponential map. In particular, differential exponential maps can be defined without the need of limits, converging power series, or unique solutions of certain differential equations—which most Cartesian differential categories do not necessarily have. That said, we do explain how every differential exponential map does provide solutions to certain differential equations, and conversely how in the presence of unique solutions, one can derivative a differential exponential map. Examples of differential exponential maps in the Cartesian differential category of real smooth functions include the exponential function, the complex exponential function, the split complex exponential function, and the dual numbers exponential function. As another source of interesting examples, we also study differential exponential maps in the coKleisli category of a differential category.

Approximation of Lightning Current Waveforms Using Complex Exponential Functions

2016 ◽  
Vol 58 (5) ◽  
pp. 1686-1689 ◽  
Author(s):  
Mirko Yanque Tomasevich ◽  
Antonio C. S. Lima ◽  
Robson F. S. Dias
Keyword(s):  
Exponential Functions ◽  
Lightning Current ◽  
Complex Exponential

scholarly journals Structure of perfect rings

1970 ◽  
Vol 2 (1) ◽  
pp. 117-124 ◽  
Author(s):  
Vlastimil Dlab
Keyword(s):  
Present Note ◽  
Matrix Representation ◽  
Simple Characterization ◽  
One To One ◽  
The Way

In the present note, we offer a simple characterization of perfect rings in terms of their components and socle sequences, which is subsequently used to establish a one-to-one correspondence between perfect rings and certain finite additive categories. This correspondence is effected by means of a matrix representation, which describes the way in which perfect rings are built from local perfect rings.

Sign in / Sign up

Export Citation Format

Share Document