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| [[Probability theory]] routinely uses results from other fields of mathematics (mostly, analysis). The opposite cases, collected below, are relatively rare; however, probability theory is used systematically in combinatorics via the [[probabilistic method]]. They are particularly used for [[non-constructive]] proofs.
| | Electrical Design Draftsperson Whitney from Baie-Saint-Paul, usually spends time with hobbies and interests such as wargames, [http://www.militiajointops.com/Qustodian_C%C3%B3mo_conseguir_dinero_Con_Tu_M%C3%B3vil_Android las mejores aplicaciones android, juegos android, liberar movil] and walking. Has finished a great around the world tour that included going to the Mt Siguniang and Jiajin Mountains. |
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| ==Analysis==
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| * [[Normal number]]s exist. Moreover, [[Computable number|computable]] normal numbers [[Normal number#Properties and examples|exist]]. These non-probabilistic existence theorems follow from probabilistic results: (a) a number chosen at random (uniformly on (0,1)) is normal almost surely (which follows easily from the [[strong law of large numbers]]); (b) some probabilistic inequalities behind the strong law. The existence of a normal number follows from (a) immediately. The proof of the existence of computable normal numbers, based on (b), involves additional arguments. All known proofs use probabilistic arguments.
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| * [[Dvoretzky's theorem]] which states that high-dimensional convex bodies have ball-like slices is proved probabilistically. No deterministic construction is known, even for many specific bodies.
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| * The diameter of the [[Banach–Mazur compactum]] was calculated using a probabilistic construction. No deterministic construction is known.
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| * The original proof that the [[Hausdorff–Young inequality]] cannot be extended to <math>p > 2</math> is probabilistic. The proof of the de Leeuw–Kahane–Katznelson theorem (which is a stronger claim) is partially probabilistic.<ref>Karel de Leeuw, Yitzhak Katznelson and Jean-Pierre Kahane, Sur les coefficients de Fourier des fonctions continues. (French) C. R. Acad. Sci. Paris Sér. A–B 285:16 (1977), A1001–A1003.</ref>
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| * The first construction of a Salem set was probabilistic.<ref>Raphaël Salem, On singular monotonic functions whose spectrum has a given Hausdorff dimension. Ark. Mat. 1, (1951). 353–365.</ref> Only in 1981 did Kaufman give a deterministic construction.<ref>Robert Kaufman, On the theorem of Jarník and Besicovitch. Acta Arith. 39:3 (1981), 265–267</ref>
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| * Every continuous function on an interval can be uniformly approximated by polynomials, which is the [[Weierstrass approximation theorem]]. A [[Bernstein polynomial#Proof|probabilistic proof]] uses the [[weak law of large numbers]]. Non-probabilistic proofs were available earlier.
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| * Existence of a nowhere differentiable continuous function follows easily from properties of [[Wiener process]]. A [[Weierstrass function|non-probabilistic proof]] was available earlier.
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| * [[Stirling's formula]] was first discovered by [[Abraham de Moivre]] in his `[[The Doctrine of Chances]]' (with a constant identified later by Stirling) in order to be used in probability theory. Several probabilistic proofs of Stirling's formula (and related results) were found in the 20th century.<ref>{{citation|last1=Blyth|first1=Colin R.|last2=Pathak|first2=Pramod K.
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| |year=1986|title=A note on easy proofs of Stirling's theorem|journal=American Mathematical Monthly|volume=93|issue=5|pages=376–379|doi=10.2307/2323600|jstor=2323600}}.</ref><ref>{{citation|last=Gordon|first=Louis|year=1994
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| |title=A stochastic approach to the gamma function
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| |journal=American Mathematical Monthly
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| |volume=101|issue=9|pages=858–865|doi=10.2307/2975134|jstor=2975134}}.</ref>
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| * The only bounded harmonic functions defined on the whole plane are constant functions by [[Potential_theory#Inequalities|Liouville's theorem]]. A probabilistic proof via two-dimensional Brownian motion is well-known.<ref name="RY">{{citation|last1=Revuz|first1=Daniel|last2=Yor|first2=Marc|year=1994|title=Continuous martingales and Brownian motion|edition=2nd|publisher=Springer}} (see Exercise (2.17) in Section V.2, page 187).</ref> Non-probabilistic proofs were available earlier.
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| * Non-tangential boundary values<ref>See [[Fatou's theorem]].</ref> of an [[Holomorphic function|analytic]] or [[harmonic function|harmonic]] function exist at almost all boundary points of non-tangential boundedness. This result ([[Ivan Privalov|Privalov]]'s theorem), and several results of this kind, are deduced from [[Doob's martingale convergence theorems|martingale convergence]].<ref>{{citation|last=Durrett|first=Richard|author-link=Rick Durrett|year=1984|title=Brownian motion and martingales in analysis|place=California|publisher=Wadsworth|isbn=0-534-03065-3}}.</ref> Non-probabilistic proofs were available earlier.
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| * The [[boundary Harnack principle]] is proved using Brownian motion<ref>{{citation|last1=Bass|first1=R.F.|author1-link=Richard F. Bass|last2=Burdzy|first2=K.|contribution=A probabilistic proof of the boundary Harnack principle|year=1989|publication-date=1990|title=Seminar on Stochastic Processes|place=Boston|publisher=Birkhäuser|pages=1–16|id={{hdl|1773/2249}}}}.</ref> (see also<ref>{{citation|last=Bass|first=Richard F.|authorlink=Richard F. Bass|year=1995|title=Probabilistic techniques in analysis|publisher=Springer|page=228}}.</ref>). Non-probabilistic proofs were available earlier.
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| * [[Basel problem|Euler's Basel sum]], <math>
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| \qquad \sum_{n=1}^\infin \frac{1}{n^2} = \frac{\pi^2}{6},
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| </math> can be demonstrated by considering the expected exit time of planar Brownian motion from an infinite strip. A number of other less well-known identities can be deduced in a similar manner.<ref>{{citation|last=Markowsky|first=Greg T.|year=2011|title=On the expected exit time of planar Brownian motion from simply connected domains|journal=Electronic communication in probability|volume=16|pages=652–663|url=http://ecp.ejpecp.org/article/view/1653}}.</ref>
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| ==Combinatorics==
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| * A number of theorems stating existence of graphs (and other discrete structures) with desired properties are proved by the [[probabilistic method]]. Non-probabilistic proofs are available for some of them.
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| * The [[maximum-minimums identity]] admits a probabilistic proof.
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| ==Algebra==
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| * The [[fundamental theorem of algebra]] can be proved using two-dimensional Brownian motion.<ref name="RY" /> Non-probabilistic proofs were available earlier.
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| * The [[Atiyah–Singer index theorem#Generalizations|index theorem for elliptic complexes]] is proved using probabilistic methods<ref>{{citation|authorlink=Jean-Michel Bismut|first=Jean-Michel|last=Bismut|title=The Atiyah–Singer Theorems: A Probabilistic Approach. I. The index theorem|url=http://www.sciencedirect.com/science/article/pii/0022123684901010|journal=J. Funct. Analysis|year=1984|volume=57|pages=56–99|doi=10.1016/0022-1236(84)90101-0}}.</ref> (rather than heat equation methods). A non-probabilistic proof was available earlier.
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| ==Topology and geometry==
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| * A smooth [[Boundary (topology)|boundary]] is evidently two-sided, but a non-smooth (especially, fractal) boundary can be quite complicated. It was conjectured to be two-sided in the sense that the natural projection of the [[Martin boundary]]<ref>As long as we have no article on [[Martin boundary]], see [[Compactification (mathematics)#Other compactification theories]].</ref> to the topological boundary is at most 2 to 1 almost everywhere.<ref name="Bishop">{{citation|last=Bishop|first=C.|year=1991|title=A characterization of Poissonian domains|journal=Arkiv för Matematik|volume=29|issue=1|pages=1–24|doi=10.1007/BF02384328}} (see Section 6).</ref> This conjecture is proved using [[Wiener process|Brownian motion]], [[Local time (mathematics)|local time]], [[Stochastic calculus|stochastic integration]], [[Coupling (probability)|coupling]], hypercontractivity etc.<ref>{{citation|last=Tsirelson|first=Boris|author-link=Boris Tsirelson|year=1997|title=Triple points: from non-Brownian filtrations to harmonic measures|journal=GAFA, Geometric and functional analysis|publisher=Birkhauser|volume=7|issue=6|pages=1096–1142|doi=10.1007/s000390050038|url=http://www.springerlink.com/content/56rd92tcftkf1c75/?p=43028bb7776b469abef8e7439f8ca086&pi=2}}. [http://www.tau.ac.il/~tsirel/Research/Recent/triple.html author's site]</ref> (see also<ref>{{citation|last=Tsirelson|first=Boris|author-link=Boris Tsirelson|year=1998|contribution=Within and beyond the reach of Brownian innovation|title=Proceedings of the international congress of mathematicians|series=Documenta mathematica|volume=Extra Volume ICM 1998, III|place=Berlin|publisher=der Deutschen Mathematiker-Vereinigung|pages=311–320|issn=1431-0635|url=http://www.mathematik.uni-bielefeld.de/documenta/xvol-icm/12/Tsirelson.MAN.html}}.</ref>). Known non-probabilistic approaches give weaker results:<ref name="Bishop">{{citation|last=Bishop|first=C.|year=1991|title=A characterization of Poissonian domains|journal=Arkiv för Matematik|volume=29|issue=1|pages=1–24|doi=10.1007/BF02384328}} (see Section 6).</ref> at most 10 sides in four (and more) dimensions; at most 4 sides in three dimensions; and 2 sides on the plane.
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| * The [[Introduction to systolic geometry#Loewner's torus inequality|Loewner's torus inequality]] relates the area of a [[compact surface]] (topologically, a torus) to its [[Introduction to systolic geometry#Notion of systole|systole]]. It can be [[Loewner's torus inequality#Proof of Loewner's torus inequality|proved most easily]] by using the probabilistic notion of [[variance]].<ref>Charles Horowitz, Karin Usadi Katz and [[Mikhail Katz|Mikhail G. Katz]] (2008), Loewner's torus inequality with isosystolic defect, Journal of Geometric Analysis 19 (2009), no. 4, 796-808. See [http://arxiv.org/abs/0803.0690 arXiv:0803.0690].</ref> A non-probabilistic proof was available earlier.
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| * The weak halfspace theorem for [[minimal surface]]s states that any complete minimal surface of bounded curvature which is not a plane is not contained in any halfspace. This theorem is proved using a [[coupling]] between Brownian motions on minimal surfaces.<ref>{{citation | last=Neel | first=Robert W. |year=2008 | title=A martingale approach to minimal surfaces | journal=Journal of Functional Analysis | publisher=Elsevier | volume=256 | issue=8 | pages=2440–2472 | doi=10.1016/j.jfa.2008.06.033}}. Also [http://arxiv.org/abs/0805.0556 arXiv:0805.0556].</ref> A non-probabilistic proof was available earlier.
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| ==Number theory==
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| * The [[normal number]] theorem (1909), due to [[Émile Borel]], could be one of the first examples of the [[probabilistic method]], providing the first proof of existence of normal numbers, with the help of the first version of the strong [[Law of large numbers#Strong law|law of large numbers]] (see also the first item of the section [[#Analysis|Analysis]]).
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| * The [[Rogers–Ramanujan identities]] are proved using [[Markov chain]]s.<ref>{{citation|last=Fulman|first=Jason|year=2001|title=A probabilistic proof of the Rogers–Ramanujan identities|journal=Bulletin of the London Mathematical Society|volume=33|issue=4|pages=397–407|url=http://blms.oxfordjournals.org/cgi/content/abstract/33/4/397|doi=10.1017/S0024609301008207}}. Also [http://arxiv.org/abs/math.CO/0001078/ arXiv:math.CO/0001078].</ref> A non-probabilistic proof was available earlier.
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| ==Quantum theory==
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| * Non-commutative dynamics (called also quantum dynamics) is formulated in terms of [[Von Neumann algebra]]s and continuous [[Tensor product of Hilbert spaces|tensor products of Hilbert spaces]].<ref>{{citation|last=Arveson|first=William|author-link=William Arveson|year=2003|title=Noncommutative dynamics and E-semigroups|place=New York|publisher=Springer|isbn=0-387-00151-4}}.</ref> Several results (for example, a continuum of mutually non-isomorphic models) are obtained by probabilistic means ([[random compact set]]s and [[Wiener process|Brownian motion]]).<ref>{{citation|last=Tsirelson|first=Boris|author-link=Boris Tsirelson|year=2003|contribution=Non-isomorphic product systems|editor-last=Price|editor-first=Geoffrey|title=Advances in quantum dynamics|series=Contemporary mathematics|volume=335|publisher=American mathematical society|pages=273–328|isbn=0-8218-3215-8}}. Also [http://arxiv.org/abs/math.FA/0210457/ arXiv:math.FA/0210457].</ref><ref>{{citation|last=Tsirelson|first=Boris|author-link=Boris Tsirelson|year=2008|title=On automorphisms of type II Arveson systems (probabilistic approach)|journal=New York Journal of Mathematics|volume=14|pages=539–576|url=http://nyjm.albany.edu/j/2008/14-25.html}}.</ref> One part of this theory (so-called type III systems) is translated into the analytic language<ref>{{citation|last1=Bhat|first1=B.V.Rajarama|last2=Srinivasan|first2=Raman|year=2005|title=On product systems arising from sum systems|journal=Infinite Dimensional Analysis, Quantum Probability and Related Topics (IDAQP)|volume=8|issue=1|pages=1–31|doi=10.1142/S0219025705001834|url=http://www.worldscinet.com/cgi-bin/details.cgi?id=pii:S0219025705001834&type=html}}. Also [http://arxiv.org/abs/math.OA/0405276/ arXiv:math.OA/0405276].</ref> and is developing analytically;<ref>{{citation|last=Izumi|first=Masaki|last2=Srinivasan|first2=Raman|year=2008|title=Generalized CCR flows|journal=Communications in Mathematical Physics|volume=281|issue=2|pages=529–571|url=http://www.springerlink.com/content/8642264k2064213v/|doi=10.1007/s00220-008-0447-z}}. Also [http://arxiv.org/abs/0705.3280 arXiv:0705.3280].</ref> the other part (so-called type II systems) exists still in the probabilistic language only.
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| * Tripartite quantum states can lead to arbitrary large violations of [[Bell inequalities]]<ref>{{citation|last1=Perez-Garcia|first1=D.|last2=Wolf|first2=M.M.|first3=Palazuelos|last3=C.|last4=Villanueva|first4=I.|last5=Junge|first5=M.|year=2008|title=Unbounded violation of tripartite Bell inequalities|journal=Communications in mathematical physics|publisher=Springer|volume=279|issue=2|pages=455–486|doi=10.1007/s00220-008-0418-4|url=http://www.springerlink.com/content/728263187124v762/}}</ref> (in sharp contrast to the bipartite case). The proof uses random unitary matrices. No other proof is available.
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| == Information theory ==
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| * The proof of [[Claude Shannon|Shannon]]'s [[Noisy-channel coding theorem|channel coding theorem]] uses random coding to show the existence of a code that achieves [[channel capacity]].
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| == See also ==
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| * [[Probabilistic method]]
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| ==Notes==
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| <references />
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| ==External links==
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| * [http://mathoverflow.net/questions/9218/probabilistic-proofs-of-analytic-facts Probabilistic Proofs of Analytic Facts] at [[MathOverflow]]
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| [[Category:Mathematical proofs]]
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| [[Category:Probabilistic arguments]]
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Electrical Design Draftsperson Whitney from Baie-Saint-Paul, usually spends time with hobbies and interests such as wargames, las mejores aplicaciones android, juegos android, liberar movil and walking. Has finished a great around the world tour that included going to the Mt Siguniang and Jiajin Mountains.