Böttcher's equation: Difference between revisions
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'''Bloch's Principle''' is a [[philosophy|philosophical]] principle in [[mathematics]] | |||
stated by [[André Bloch (mathematician)|André Bloch]].<ref>{{cite article|first=A.|last=Bloch| | |||
title=La conception actuelle de la theorie de fonctions entieres et meromorphes|journal=Enseignement math.|year=1926| volume=25|pages=83–103}}</ref> | |||
Bloch states the principle in Latin as: ''Nihil est in infinito quod non prius fueris in finito,'' and explains this as follows: Every proposition in whose statement the [[actual infinity]] occurs can be always considered a consequence, almost immediate, of a proposition where it does not occur, a proposition in ''finite terms''. | |||
Bloch mainly applied this principle to the theory of [[function (mathematics)|functions]] of a [[complex variable]]. Thus, for example, according to this principle, [[Picard's theorem]] corresponds to [[Schottky's theorem]], and [[Bloch's theorem (complex variables)|Valiron's theorem]] corresponds to [[Bloch's theorem (complex variables)|Bloch's theorem]]. | |||
Based on his Principle, Bloch was able to predict or conjecture several | |||
important results such as the [[Ahlfors theory|Ahlfors's Five Islands theorem]], | |||
[[Henri cartan|Cartan]]'s theorem on holomorphic curves omitting hyperplanes,<ref>{{cite book|ref name="la"|first=S.|last=Lang| | |||
title=Introduction to complex hyperbolic spaces|publisher=[[Springer Verlag]]|year=1987}}</ref> [[Walter Hayman|Hayman]]'s result that an exceptional set of radii is unavoidable in [[Nevanlinna theory]]. | |||
In the more recent times several general theorems were proved which can be | |||
regarded as rigorous statements in the spirit of the Bloch Principle. | |||
==Zalcman's lemma== | |||
Let <math>\{f_n\}</math> be a sequence of meromorphic functions in a region ''D'', which is not a [[normal family]]. | |||
Then there exist a sequence of points <math>z_n </math> in ''D'' and positive numbers <math>\rho_n </math> with <math>\lim_{n\rightarrow\infty}\rho_{n}=0</math> such that | |||
: <math>f_n(z_n+\rho_nz)\to f, \, </math> | |||
where ''f'' is a non-constant meromorphic function in the complex plane.<ref>{{cite paper|first=L.|last=Zalcman| | |||
title=Heuristic principle in complex function theory|journal=Amer. Math. Monthly| | |||
volume=82|year=1975|pages=813–817}}</ref> | |||
==Brody's lemma== | |||
Let ''X'' be a [[compact space|compact]] [[complex analytic manifold]], such that every [[holomorphic map]] from the [[complex plane]] | |||
to ''X'' is constant. Then there exists a [[metric (mathematics)|metric]] on ''X'' such that every holomorphic map from the unit disc with the [[Poincaré metric]] to ''X'' does not increase distances.<ref>Lang (1987).</ref> | |||
==References== | |||
<references /> | |||
[[Category:Mathematical principles]] | |||
[[Category:Philosophy of mathematics]] | |||
Revision as of 15:34, 27 December 2013
Bloch's Principle is a philosophical principle in mathematics stated by André Bloch.[1]
Bloch states the principle in Latin as: Nihil est in infinito quod non prius fueris in finito, and explains this as follows: Every proposition in whose statement the actual infinity occurs can be always considered a consequence, almost immediate, of a proposition where it does not occur, a proposition in finite terms.
Bloch mainly applied this principle to the theory of functions of a complex variable. Thus, for example, according to this principle, Picard's theorem corresponds to Schottky's theorem, and Valiron's theorem corresponds to Bloch's theorem.
Based on his Principle, Bloch was able to predict or conjecture several important results such as the Ahlfors's Five Islands theorem, Cartan's theorem on holomorphic curves omitting hyperplanes,[2] Hayman's result that an exceptional set of radii is unavoidable in Nevanlinna theory.
In the more recent times several general theorems were proved which can be regarded as rigorous statements in the spirit of the Bloch Principle.
Zalcman's lemma
Let be a sequence of meromorphic functions in a region D, which is not a normal family. Then there exist a sequence of points in D and positive numbers with such that
where f is a non-constant meromorphic function in the complex plane.[3]
Brody's lemma
Let X be a compact complex analytic manifold, such that every holomorphic map from the complex plane to X is constant. Then there exists a metric on X such that every holomorphic map from the unit disc with the Poincaré metric to X does not increase distances.[4]
References
- ↑ Template:Cite article
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ Template:Cite paper
- ↑ Lang (1987).