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.

scholarly journals The model theory of separably tame valued fields

2016 ◽  
Vol 447 ◽  
pp. 74-108 ◽  
Author(s):  
Franz-Viktor Kuhlmann ◽  
Koushik Pal
Keyword(s):  
Model Theory ◽  
Valued Fields

Model Theory of Henselian Valued Fields

1989 ◽  
pp. v-vi
Author(s):  
H.-D. Ebbinghaus ◽  
J. Fernandez-Prida ◽  
M. Garrido ◽  
D. Lascar ◽  
M. Rodriquez Artalejo
Keyword(s):  
Model Theory ◽  
Valued Fields ◽  
Henselian Valued Fields

Model theory of valued fields

2018 ◽  
pp. 151-180
Author(s):  
Martin Hils
Keyword(s):  
Model Theory ◽  
Valued Fields

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.

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