Rotation, scale, and translation resilient digital watermarking based on complex exponential function

2005 ◽  
Author(s):  
S. Pholsomboon ◽  
S. Vongpradhip
Keyword(s):  
Digital Watermarking ◽  
Exponential Function ◽  
Complex Exponential

Periodic Point at Real Axis for a Generalized 3x+1 Function

2011 ◽  
Vol 55-57 ◽  
pp. 1670-1674 ◽  
Author(s):  
Shuai Liu ◽  
Zheng Xuan Wang
Keyword(s):  
Periodic Point ◽  
Real Axis ◽  
Exponential Function ◽  
Periodic Points ◽  
Divergence Points ◽  
Fractal Character ◽  
Convergence And Divergence ◽  
Complex Exponential

In order to study the fractal character of representative complex exponential function just as generalized 3x+1 function T(x). In this essay, we proved that T(x) has periodic points of every period in bound (n, n+1) when n>1 in real axis. Then, we found the distribution of 2-periods points of T(x) in real axis. We put forward the bottom bound of 2-periodic point’s number and proved it. Moreover, we found the number of T(x)’s 2-periodic points in different bounds to validate our conclusion. Then, we extended the conclusion to i-periods points and find similar conclusion. Finally, we proved there exist endless convergence and divergence points of T(x) in real axis.

scholarly journals An explosion point for the set of endpoints of the Julia set of λ exp (z)

1990 ◽  
Vol 10 (1) ◽  
pp. 177-183 ◽  
Author(s):  
John C. Mayer
Keyword(s):  
Exponential Function ◽  
Complex Plane ◽  
Riemann Sphere ◽  
Julia Set ◽  
Periodic Points ◽  
Real Parameter ◽  
Topological Dimension ◽  
Complex Exponential ◽  
Totally Disconnected ◽  
Topological Sense

AbstractThe Julia set Jλ of the complex exponential function Eλ: z → λez for a real parameter λ(0 < λ < 1/e) is known to be a Cantor bouquet of rays extending from the set Aλ of endpoints of Jλ to ∞. Since Aλ contains all the repelling periodic points of Eλ, it follows that Jλ = Cl (Aλ). We show that Aλ is a totally disconnected subspace of the complex plane ℂ, but if the point at ∞ is added, then is a connected subspace of the Riemann sphere . As a corollary, Aλ has topological dimension 1. Thus, ∞ is an explosion point in the topological sense for Âλ. It is remarkable that a space with an explosion point occurs ‘naturally’ in this way.

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.

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.

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.

scholarly journals Mappings by the complex exponential function

2020 ◽  
Vol 1674 (1) ◽  
pp. 012006
Author(s):  
J G Triana ◽  
J A Fuentes ◽  
P Ramirez
Keyword(s):  
Exponential Function ◽  
Complex Plane ◽  
Conformal Mappings ◽  
Complex Functions ◽  
Complex Exponential ◽  
Analytic Domain

Abstract In mathematics, engineering, and physics, some problems can be solved through complex functions; in many cases, with geometric inconveniences or complicated domains. Conformal mappings are essential to transform a complicated analytic domain onto a simple domain. Physical approaches to visualization of complex functions can be used to represent conformal mappings, here we use the transformation of regions of the complex plane. This paper provides a graphical overview of the transformation of a set of regions by the complex exponential function.

Model Theory of Analytic Functions: Some Historical Comments

2012 ◽  
Vol 18 (3) ◽  
pp. 368-381
Author(s):  
Deirdre Haskell
Keyword(s):  
Exponential Function ◽  
Model Theory ◽  
Analytic Functions ◽  
Open Problems ◽  
Valued Fields ◽  
Seminal Work ◽  
Complex Exponential

AbstractModel theorists have been studying analytic functions since the late 1970s. Highlights include the seminal work of Denef and van den Dries on the theory of the p-adics with restricted analytic functions, Wilkie's proof of o-minimality of the theory of the reals with the exponential function, and the formulation of Zilber's conjecture for the complex exponential. My goal in this talk is to survey these main developments and to reflect on today's open problems, in particular for theories of valued fields.

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