Zorich's theorem: Difference between revisions

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In mathematical analysis, Zorich's theorem was proved by Vladimir A. Zorich in 1967. The result was conjectured by M. A. Lavrentev in 1938.^{[citation needed]}

Theorem

Every locally homeomorphic quasiregular mapping $f:R^{n}\rightarrow R^{n}$ for $n\geq 3$ , is a homeomorphism of $R^{n}$ .^[1]

The fact that there is no such result for $n=2$ is easily shown using the exponential function.^{[citation needed]}

References

^ Zorich, Vladimir A. (1992). "The global homeomorphism theorem for space quasiconformal mappings, its development and related open problems". In Vuorinen, Matti (ed.). Quasiconformal Space Mappings: A collection of surveys 1960-1990. Germany: Springer-Verlag. pp. 132–148. doi:10.1007/BFB0094243. ISBN 3-540-55418-1. LCCN 92012192. OCLC 25675026. S2CID 116148715. Retrieved February 10, 2024.

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[1] Zorich, Vladimir A. (1992). "The global homeomorphism theorem for space quasiconformal mappings, its development and related open problems". In Vuorinen, Matti (ed.). Quasiconformal Space Mappings: A collection of surveys 1960-1990. Germany: Springer-Verlag. pp. 132–148. doi:10.1007/BFB0094243. ISBN 3-540-55418-1. LCCN 92012192. OCLC 25675026. S2CID 116148715. Retrieved February 10, 2024.

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-In [[mathematical analysis]],  '''Zorich's theorem''' was proved by [[Vladimir A. Zorich]] in 1967. The result was conjectured by [[Mikhail Alekseevich Lavrentev|M. A. Lavrentev]] in 1938.
+In [[mathematical analysis]],  '''Zorich's theorem''' was proved by [[Vladimir A. Zorich]] in 1967. The result was conjectured by [[Mikhail Alekseevich Lavrentev|M. A. Lavrentev]] in 1938.{{citation needed|date=February 2024}}
 == Theorem ==
+Every locally [[homeomorphism|homeomorphic]] [[quasiregular map]]ping <math>f : R^{n} \rightarrow R^{n}</math> for <math>n \geq 3</math>, is a homeomorphism of <math>R^{n}</math>.<ref>{{cite book |last=Zorich |first=Vladimir A. |author-link=Vladimir A. Zorich |editor-last=Vuorinen |editor-first=Matti |editor-link=Matti Vuorinen |year=1992 |chapter=The global homeomorphism theorem for space quasiconformal mappings, its development and related open problems |pages=132–148 |title=Quasiconformal Space Mappings: A collection of surveys 1960-1990 |publisher=[[Springer-Verlag]] |publication-place=Germany |doi=10.1007/BFB0094243 |isbn=3-540-55418-1 |lccn=92012192 |oclc=25675026 |s2cid=116148715 |chapter-url={{GBurl|fo58CwAAQBAJ|p=132}} |access-date=February 10, 2024}}</ref>
-Every locally [[homeomorphism|homeomorphic]] [[quasiregular map]]ping ''ƒ''&nbsp;:&nbsp;'''R'''<sup>''n''</sup>&nbsp;→&nbsp;'''R'''<sup>''n''</sup> for ''n''&nbsp;≥&nbsp;3, is a homeomorphism of&nbsp;'''R'''<sup>''n''</sup>.
-The fact that there is no such result for ''n''&nbsp;=&nbsp;2 is easily shown using the [[exponential function]].
+The fact that there is no such result for <math>n = 2</math> is easily shown using the [[exponential function]].{{citation needed|date=February 2024}}
 ==References==
+{{Reflist}}
-* V.A. Zorich, "The global homeomorphism theorem for space quasiconformal mappings, its development and related open problems", M. Vuorinen (ed.), ''Quasiconformal Space Mappings'', ''Lecture Notes in Mathematics'', 1508 (1992) pp.&nbsp;132–148
 [[Category:General topology]]
+{{mathanalysis-stub}}