User:Boris Tsirelson/Sandbox1: Difference between revisions

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The general idea of the Cantor–Bernstein–Schroeder theorem and related results may be formulated as follows. If ''X'' is similar to a part of ''Y'' and at the same time ''Y'' is similar to a part of ''X'' then ''X'' and ''Y'' are similar. In order to be specific one should decide
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* what kind of mathematical objects are ''X'' and ''Y'',
The [[Heisenberg Uncertainty Principle|Heisenberg uncertainty principle]] for a particle does not allow a state in which the particle is simultaneously at a definite location and has also a definite momentum. Instead the particle has a range of momentum and spread in location attributable to quantum fluctuations.
* what is meant by "a part",
* what is meant by "similar".


In the classical Cantor–Bernstein–Schroeder theorem
An uncertainty principle applies to most of quantum mechanical operators that do not commute (specifically, to every pair of operators whose commutator is a non-zero scalar operator).
* ''X'' and ''Y'' are [[Set (mathematics)|sets]] (maybe infinite),
* "a part" is interpreted as a [[subset]],
* "similar" is interpreted as [[Bijective function#Bijections and the concept of cardinality|equinumerous]].
 
Not all statements of this form are true. For example, let
* ''X'' and ''Y'' are [[triangle]]s,
* "a part" means a triangle inside the given triangle,
* "similar" is interpreted as usual in elementary geometry: triangles related by a dilation (in other words, "triangles with the same shape up to a scale factor", or equivalently "triangles with the same angles").
Then the statement fails badly: every triangle ''X'' evidently is similar to some triangle inside ''Y'', and the other way round; however, ''X'' and ''Y'' need no be similar.
 
==Notes==
{{reflist}}
 
==References==
 
{{Citation
| last = Bourbaki
| first = Nicolas
| title = Elements of mathematics: Theory of sets
| year = 1968
| publisher = Hermann (original), Addison-Wesley (translation)
}}.
 
{{Citation
| year = 1992
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| series = Lecture Notes in Mathematics
| volume = 1525
| publisher = Springer-Verlag
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{{Citation
| last = Feynman
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| title = The character of physical law
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}}.
 
{{Citation
| year = 2008
| editor-last = Gowers
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| title = The Princeton companion to mathematics
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}}.
 
{{Citation
| last = Gowers
| first = W.T.
| year = 1996
| title = A solution to the Schroeder-Bernstein problem for Banach spaces
| journal = Bull. London Math. Soc.
| volume = 28
| pages = 297–304
| url = http://blms.oxfordjournals.org/content/28/3/297
}}.
 
{{Citation
| last = Casazza
| first = P.G.
| year = 1989
| title = The Schroeder-Bernstein property for Banach spaces
| journal = Contemp. Math.
| volume = 85
| pages = 61–78
| url = http://www.ams.org/mathscinet/search/publdoc.html?pg1=IID&s1=45945&vfpref=html&r=80&mx-pid=983381
}}.
(Also [http://personnel.univ-reunion.fr/ardm/inefff.pdf here].)

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The Heisenberg uncertainty principle for a particle does not allow a state in which the particle is simultaneously at a definite location and has also a definite momentum. Instead the particle has a range of momentum and spread in location attributable to quantum fluctuations.

An uncertainty principle applies to most of quantum mechanical operators that do not commute (specifically, to every pair of operators whose commutator is a non-zero scalar operator).

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