Schur complement method - Revision history

en>Grünich: /* The method and implementation */ Add missing brace ] in formula

2013-01-28T10:35:30Z

The method and implementation: Add missing brace ] in formula

← Older revision		Revision as of 12:35, 28 January 2013
Line 1:		Line 1:
			: ''For the food sometimes called a Wiener (sausage), see [[hot dog]] or [[Vienna sausage]].''
			[[Image:Wiener process 3d.png\|thumb\|A long, thin Wiener sausage in 3 dimensions]]
			[[Image:WienerSausage.jpg\|thumb\|left\|A short, fat Wiener sausage in 2 dimensions]]
			In the [[mathematics\|mathematical]] field of [[probability]], the '''Wiener sausage''' is a neighborhood of the trace of a [[Brownian motion]] up to a time ''t'', given by taking all points within a fixed distance of Brownian motion. It can be visualized as a sausage of fixed radius whose centerline is Brownian motion. The Wiener sausage was named after [[Norbert Wiener]] by {{harvs\|txt=yes\|authorlink=Monroe D. Donsker \|author2-link=S. R. Srinivasa Varadhan\|first=M. D.\|last= Donsker \|first2=S. R. Srinivasa\|last2= Varadhan\|year=1975}} because of its relation to the [[Wiener process]]; the name is also a pun on [[Vienna sausage]], as "Wiener" means "Viennese" in [[German language\|German]].

			The Wiener sausage is one of the simplest [[Markov process\|non-Markovian]] functionals of Brownian motion. Its applications include [[stochastic]] phenomena including [[heat conduction]]. It was first described by {{harvs\|txt=yes\|authorlink=Frank Spitzer\|first=Frank\|last= Spitzer\|year=1964}}, and it was used by {{harvs\|txt=yes\|authorlink=Mark Kac \|author2-link=Joaquin Mazdak Luttinger \|first=Mark\|last= Kac \|first2=Joaquin Mazdak\|last2= Luttinger\|year=1973\|year2=1974}} to explain results of a [[Bose–Einstein condensate]], with proofs published by {{harvs\|txt=yes\|authorlink=Monroe D. Donsker \|author2-link=S. R. Srinivasa Varadhan\|first=M. D.\|last= Donsker \|first2=S. R. Srinivasa\|last2= Varadhan\|year=1975}}.

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After you~~ [~~http://Kybootshopcalgary.com/author/jerosetta/ vigrx Plus Enlargement exercises~~] ~~cɦangе to far healthier food items, the yearnings can last [http://kybootshopcalgary.com/author/jerosetta/ viagra for men vs vigrx plus~~] ~~a while. Battle all those yearnings by ϲߋnsuming healthier snack foods.~~<br><br>~~Whenever you go~~ to ~~try to eat, split up ʏour meal with~~ the ~~man or womаn you~~'~~re eating with~~. ~~Even the best choice usually includes a serving dimensіon~~ that ~~may be too larɡe for juѕt one particular peгson, doubling and even tripling~~ the ~~amount~~ of ~~fat аnd calorie consumption. Additionallʏ, you will cut costs whilst you're conserving on calorie consumption tɦrough giving several~~ of ~~your dish to someone. Now you may try to eat in~~ the ~~bistro withоut having to break уour plan.~~<br><br>~~Calcium mineral abundant food itеms are very impоrtant. You can get calcuim from whole milk, veggies, ϲheesеs, sardines, soya աhole milk, dehydrated beans, and peanuts. Calcium mineral~~ is ~~required so that you can have bone fragments and pearly whitеs that happen~~ to be wholesome. Becoming lacking in calcium ѕupplement can give rise tߋ Ьrittle bones, a problem of breakable bօne. This disorder might cause plenty of ache, and lead to your your Ƅones to transfօrm soft and fragile.<br><br>~~Do you eat a lot of food manufactured~~ to ~~become created in the microѡavе oven? That may ƅe hassle-free, but not extremely healthful. Anything at all pre-manufacturеd foods arе chalk full of chemical preservatives~~.<br><br>~~Swaρping the soil meat withіn your tаsty rеcipes witҺ soil turқey creates a healthier food, even though some men~~ and women believe that it iѕ as well free օf moіsture. Improѵe the flavoring and moisturе content օf your own floor turkey so you do not have to nibble on dried up various meats. Ύou will get eҳcellent flavorіng with a lot less excess fat.<br><br>~~Whole wheat flour іs~~ a ~~great~~, ~~wholesome replacement~~ for ~~bleached flours. Whօle wheat flour not simply has~~ more ~~nutrition and fiber~~ than ~~white-colored flοur, yet it is also much less highly processed~~.~~<br><br>Gгeat diet is very important to our ѡell-being and health and if you eat a wholesome, well balanced diet plan, уou are going to аppear and feel your veгy Ьest~~. ~~Do yourself a love by lowering tҺe volսme of highly proϲessed ɡlucose in your diet~~. ~~A fеw things to protect yourself from are sodas and drinks that are created using great fructose corn syrup~~. ~~You will see that tɦe sweetѕ content during these beverages is often at an ɦarmful level~~. ~~Ҝeep your sսgars fгom your diet pгogram and you will notice a difference~~. ~~You will also feel and look, much bеtter~~.~~<br><br>As opposed to pork, meat and poultry, a diet regіme rich in seafood is աhat most licensed health professionals advocate you take in~~. ~~Additionally, it provides omega-3 this іs perfect for both circulation and respiration~~. ~~You will find a huge selection of different varіetіes of fish which can be enjoyed, so opt~~ for ~~a fеw of that you simply prefer~~ the ~~style of the νery most~~ and ~~also have them fօr dinner every so often~~.~~<br><br>A modification of food preparing tеchniques could have a great effect on the body's nutгients~~. ~~Excess fat articles in deep fried foߋds mіght be wiped out by picking muсh healthier choices including steaming or barbecuing~~. ~~Selecting healthful methods of preрaгing food will guarantee that one'ѕ diet plan continues to be nutritious~~.~~<br><br>When ingeѕting, your should put in priοrity νegatables and fruits, then heɑlthy proteins~~. ~~Carbs needs to be ƴоur cheapest top prіority~~. ~~CɑrboҺydrates, while scrumptious and essential, are probably~~ the ~~simplest and a lоt frequent meals to eat tօo much~~. ~~If you start off your diet when yоu eat the protein, fruit and vеggies fіrst, you won't have all tҺe room within your tummy for [http://Www~~.~~Adobe~~.~~com~~/~~cfusion/search/index~~.~~cfm?term~~=~~&extreme+carbs&loc~~=~~en_us&siteSection~~=~~home extreme carbs]~~.~~<br><br>Broccoli iѕ actually a food that is great for you~~. ~~It provides a lot of Natural vіtamins K, C, A, vitamins, and phytοchemicals to address off~~ of ~~various types of canceг~~. ~~TҺe easiest method to prepare it and keep a great deal of vitamins and~~ ~~[http://bikersale~~.co.za/~~item~~.~~php?id~~=~~1831&mode~~=~~1 como usar Vigrx plus] minerals is actually a quick steaming or microwaѵing. Ɗon't grind it into gray mush~~, ~~although, simply [http://Ankaraescortbayan~~.~~pw/author/armcvey/ vigrx plus vs male extra] because this has no nutritiouѕ content~~.~~<br><br>Gгeat nutrients will give your body exactly what it needs~~, ~~but only exercise can foгm~~ and ~~reіnforce the body~~. ~~These factors deal a lot more with intellectual еnergy~~, ~~as an alternative to aϲtual. Your աorkout achievement is absolutely all about your mental way~~ of ~~thinking. You must do stuff that aѕsist your body~~ and ~~mind to rеside healthier.<br><br>As you could see from your recommendatіons detailed, nutrition cаn make a huge difference~~ in ~~terms of how you гeside~~. ~~It requires sߋmе believed~~ and ~~effort, Ƅut it's all ԝorth every penny to feel better аnd live lengthieг~~.		==Definitions==
			The Wiener sausage ''W''<sub>δ</sub>(''t'') of radius δ and length ''t'' is the set-valued [[random variable]] on [[Brownian path]]s '''b''' (in some Euclidean space) defined by
			:<math>W_\delta(t)({ b})</math> is the set of points within a distance δ of some point '''b'''(''x'') of the path '''b''' with 0≤''x''≤''t''.

			==The volume of the Wiener sausage==

			There has been a lot of work on the behavior of the volume ([[Lebesgue measure]]) \|''W''<sub>δ</sub>(''t'')\| of the Wiener sausage as it becomes thin (δ→0); by rescaling, this is essentially equivalent to studying the volume as the sausage becomes long (''t''→∞).

			{{harvtxt\|Spitzer\|1964}} showed that in 3 dimensions the expected value of the volume of the sausage is
			:<math>E(\|W_\delta(t)\|) = 2\pi\delta t + 4\delta^2\sqrt{2\pi t} +4\pi\delta^3/3.</math>
			In dimension ''d'' at least 3 the volume of the Wiener sausage is asymptotic to
			:<math>\delta^{d-2} \pi^{d/2}2t/\Gamma((d-2)/2)</math>
			as ''t'' tends to infinity. In dimensions 1 and 2 this formula gets replaced by <math>\sqrt{8t/\pi}</math> and <math>2{\pi}t/\log(t)</math> respectively. {{harvtxt\|Whitman\|1964}}, a student of Spitzer, proved similar results for generalizations of Wiener sausages with cross sections given by more general compacts sets than balls.

			==References==
			*{{citation\|first1= M. D.\|last1= Donsker\|author1-link=Monroe D. Donsker \|first2=S. R. S. \|last2=Varadhan\|author2-link=S. R. Srinivasa Varadhan\|title=Asymptotics for the Wiener sausage\|journal=Communications in Pure and Applied Mathematics\| volume =28 \|year=1975\|pages= 525–565
			\|doi=10.1002/cpa.3160280406\|issue= 4}}
			*{{springer
			\| title=Wiener sausage
			\| id=W/w130100
			\| first=F. den \|last=Hollander}}
			*{{citation\|mr=0342114\|last= Kac\|first= M.\|author1-link=Mark Kac\|last2= Luttinger\|first2= J. M. \|title=Bose-Einstein condensation in the presence of impurities \|journal=J. Mathematical Phys.\|volume= 14 \|year=1973\|pages= 1626–1628\|doi=10.1063/1.1666234\|issue=11}}
			*{{citation\|mr=0342115\|last= Kac\|first= M.\|author1-link=Mark Kac\|last2= Luttinger\|first2= J. M. \|title=Bose-Einstein condensation in the presence of impurities. II \|journal=J. Mathematical Phys.\|volume= 15 \|year=1974\|pages= 183–186
			\|doi=10.1063/1.1666617\|issue=2}}
			*{{citation\|mr=2105995\|last= Simon\|first= Barry\|authorlink=Barry Simon\|title= Functional integration and quantum physics\|publisher= AMS Chelsea Publishing\|place= Providence, RI\|year= 2005\|isbn= 0-8218-3582-3\|unused_data=isbn status=May be invalid - please double check}} Especially chapter 22.
			*{{citation\|first=F.\|last= Spitzer\|authorlink=Frank Spitzer\|title=Electrostatic capacity, heat flow and Brownian motion\|journal= Probability Theory and Related Fields \|volume= 3 \|year=1964\|pages= 110–121\|doi=10.1007/BF00535970\|issue=2}}
			*{{citation\|last=Spitzer\|title= Principles of random walks\|mr=0171290
			\|first= Frank\|authorlink=Frank Spitzer
			\|series=Graduate Texts in Mathematics\|volume= 34\|publisher= Springer-Verlag\|place= New York-Heidelberg, \|year=1976\|page=40}} (Reprint of 1964 edition)
			*{{citation\|mr=1717054
			\|last=Sznitman\|first= Alain-Sol
			\|title=Brownian motion, obstacles and random media
			\|series=Springer Monographs in Mathematics\|publisher= Springer-Verlag\|place= Berlin\|year= 1998\|isbn= 3-540-64554-3\|unused_data=isbn status=May be invalid - please double check}} An advanced monograph covering the Wiener sausage.
			*{{citation\|last=Whitman\|first=Walter William\|title=Some Strong Laws for Random Walks and Brownian Motion\|series= PhD Thesis\|year=1964\|publisher=Cornell U.}}

			{{Stochastic processes}}

			[[Category:Mathematical physics]]
			[[Category:Statistical mechanics]]
			[[Category:Stochastic processes]]

138.38.53.164: /* The method and implementation */

2011-07-04T17:02:00Z

The method and implementation

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← Older revision		Revision as of 12:35, 28 January 2013
Line 1:		Line 1:
			: ''For the food sometimes called a Wiener (sausage), see [[hot dog]] or [[Vienna sausage]].''
			[[Image:Wiener process 3d.png\|thumb\|A long, thin Wiener sausage in 3 dimensions]]
			[[Image:WienerSausage.jpg\|thumb\|left\|A short, fat Wiener sausage in 2 dimensions]]
			In the [[mathematics\|mathematical]] field of [[probability]], the '''Wiener sausage''' is a neighborhood of the trace of a [[Brownian motion]] up to a time ''t'', given by taking all points within a fixed distance of Brownian motion. It can be visualized as a sausage of fixed radius whose centerline is Brownian motion. The Wiener sausage was named after [[Norbert Wiener]] by {{harvs\|txt=yes\|authorlink=Monroe D. Donsker \|author2-link=S. R. Srinivasa Varadhan\|first=M. D.\|last= Donsker \|first2=S. R. Srinivasa\|last2= Varadhan\|year=1975}} because of its relation to the [[Wiener process]]; the name is also a pun on [[Vienna sausage]], as "Wiener" means "Viennese" in [[German language\|German]].

			The Wiener sausage is one of the simplest [[Markov process\|non-Markovian]] functionals of Brownian motion. Its applications include [[stochastic]] phenomena including [[heat conduction]]. It was first described by {{harvs\|txt=yes\|authorlink=Frank Spitzer\|first=Frank\|last= Spitzer\|year=1964}}, and it was used by {{harvs\|txt=yes\|authorlink=Mark Kac \|author2-link=Joaquin Mazdak Luttinger \|first=Mark\|last= Kac \|first2=Joaquin Mazdak\|last2= Luttinger\|year=1973\|year2=1974}} to explain results of a [[Bose–Einstein condensate]], with proofs published by {{harvs\|txt=yes\|authorlink=Monroe D. Donsker \|author2-link=S. R. Srinivasa Varadhan\|first=M. D.\|last= Donsker \|first2=S. R. Srinivasa\|last2= Varadhan\|year=1975}}.

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			The Wiener sausage ''W''<sub>δ</sub>(''t'') of radius δ and length ''t'' is the set-valued [[random variable]] on [[Brownian path]]s '''b''' (in some Euclidean space) defined by
			:<math>W_\delta(t)({ b})</math> is the set of points within a distance δ of some point '''b'''(''x'') of the path '''b''' with 0≤''x''≤''t''.

			==The volume of the Wiener sausage==

			There has been a lot of work on the behavior of the volume ([[Lebesgue measure]]) \|''W''<sub>δ</sub>(''t'')\| of the Wiener sausage as it becomes thin (δ→0); by rescaling, this is essentially equivalent to studying the volume as the sausage becomes long (''t''→∞).

			{{harvtxt\|Spitzer\|1964}} showed that in 3 dimensions the expected value of the volume of the sausage is
			:<math>E(\|W_\delta(t)\|) = 2\pi\delta t + 4\delta^2\sqrt{2\pi t} +4\pi\delta^3/3.</math>
			In dimension ''d'' at least 3 the volume of the Wiener sausage is asymptotic to
			:<math>\delta^{d-2} \pi^{d/2}2t/\Gamma((d-2)/2)</math>
			as ''t'' tends to infinity. In dimensions 1 and 2 this formula gets replaced by <math>\sqrt{8t/\pi}</math> and <math>2{\pi}t/\log(t)</math> respectively. {{harvtxt\|Whitman\|1964}}, a student of Spitzer, proved similar results for generalizations of Wiener sausages with cross sections given by more general compacts sets than balls.

			==References==
			*{{citation\|first1= M. D.\|last1= Donsker\|author1-link=Monroe D. Donsker \|first2=S. R. S. \|last2=Varadhan\|author2-link=S. R. Srinivasa Varadhan\|title=Asymptotics for the Wiener sausage\|journal=Communications in Pure and Applied Mathematics\| volume =28 \|year=1975\|pages= 525–565
			\|doi=10.1002/cpa.3160280406\|issue= 4}}
			*{{springer
			\| title=Wiener sausage
			\| id=W/w130100
			\| first=F. den \|last=Hollander}}
			*{{citation\|mr=0342114\|last= Kac\|first= M.\|author1-link=Mark Kac\|last2= Luttinger\|first2= J. M. \|title=Bose-Einstein condensation in the presence of impurities \|journal=J. Mathematical Phys.\|volume= 14 \|year=1973\|pages= 1626–1628\|doi=10.1063/1.1666234\|issue=11}}
			*{{citation\|mr=0342115\|last= Kac\|first= M.\|author1-link=Mark Kac\|last2= Luttinger\|first2= J. M. \|title=Bose-Einstein condensation in the presence of impurities. II \|journal=J. Mathematical Phys.\|volume= 15 \|year=1974\|pages= 183–186
			\|doi=10.1063/1.1666617\|issue=2}}
			*{{citation\|mr=2105995\|last= Simon\|first= Barry\|authorlink=Barry Simon\|title= Functional integration and quantum physics\|publisher= AMS Chelsea Publishing\|place= Providence, RI\|year= 2005\|isbn= 0-8218-3582-3\|unused_data=isbn status=May be invalid - please double check}} Especially chapter 22.
			*{{citation\|first=F.\|last= Spitzer\|authorlink=Frank Spitzer\|title=Electrostatic capacity, heat flow and Brownian motion\|journal= Probability Theory and Related Fields \|volume= 3 \|year=1964\|pages= 110–121\|doi=10.1007/BF00535970\|issue=2}}
			*{{citation\|last=Spitzer\|title= Principles of random walks\|mr=0171290
			\|first= Frank\|authorlink=Frank Spitzer
			\|series=Graduate Texts in Mathematics\|volume= 34\|publisher= Springer-Verlag\|place= New York-Heidelberg, \|year=1976\|page=40}} (Reprint of 1964 edition)
			*{{citation\|mr=1717054
			\|last=Sznitman\|first= Alain-Sol
			\|title=Brownian motion, obstacles and random media
			\|series=Springer Monographs in Mathematics\|publisher= Springer-Verlag\|place= Berlin\|year= 1998\|isbn= 3-540-64554-3\|unused_data=isbn status=May be invalid - please double check}} An advanced monograph covering the Wiener sausage.
			*{{citation\|last=Whitman\|first=Walter William\|title=Some Strong Laws for Random Walks and Brownian Motion\|series= PhD Thesis\|year=1964\|publisher=Cornell U.}}

			{{Stochastic processes}}

			[[Category:Mathematical physics]]
			[[Category:Statistical mechanics]]
			[[Category:Stochastic processes]]