Brouwer fixed-point theorem: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
← Older editNewer edit →
typo or French-translation error "simplests" changed to "simplest"
en>Dscheink
m Removed the unnecessary and potentially confusing phrase "with certain properties" from the phrase "any continuous function f with certain properties." Continuity is the only property necessary.
Line 1: Line 1:
<!-- The French version of this article is a featured article. Large portions have been translated and inserted here in 2009. -->
== . Mbt Masai Singapore ==
[[Image:poincare.jpg|220px|thumb|right|In 1886, [[Henri Poincaré]] (pictured) proved a result that is equivalent to Brouwer's fixed-point theorem. The three-dimensional case of the exact statement was proved in 1904 by [[Piers Bohl]], and the general case in 1910 by [[Jacques Hadamard]] and [[Luitzen Egbertus Jan Brouwer]].]]
'''Brouwer's fixed-point theorem''' is a [[fixed-point theorem]] in [[topology]], named after [[Luitzen Egbertus Jan Brouwer|Luitzen Brouwer]]. It states that for any continuous function ''f'' with certain properties mapping a compact convex set into itself there is a point ''x''<sub>0</sub> such that ''f''(''x''<sub>0</sub>)&nbsp;=&nbsp;''x''<sub>0</sub>. The simplest forms of Brouwer's theorem are for continuous functions ''f'' from a closed interval ''I'' in the real numbers to itself or from a closed [[Disk (mathematics)|disk]] ''D'' to itself. A more general form than the latter is for continuous functions from a [[Convex set|convex]] [[compactness|compact]] subset ''K'' of [[Euclidean space]] to itself.


Among hundreds of fixed-point theorems,<ref>E.g. F & V Bayart ''[http://www.bibmath.net/dico/index.php3?action=affiche&quoi=./p/pointfixe.html Théorèmes du point fixe]'' on Bibm@th.net</ref> Brouwer's is particularly well known, due in part to its use across numerous fields of mathematics.
Fanhost has been around and [http://www.sandpipermotel.com.au/sandpiper/head.asp?p=79-Mbt-Masai-Singapore Mbt Masai Singapore] we are getting ready to launch a significantly needed redo. This will happen in Feb. Since the current site has a lot of things broken, I would expect to fall below 30k after launch. Type "parkour" into YouTube and thousands of homemade videos of people performing acrobatic feats in cityspaces will likely come up. From the streets of Mumbai to the abandoned government buildings of Cairo, practitioners have [http://www.indopacificmarine.com.au/mail/copyright.asp?page=168-Nike-Air-Australia Nike Air Australia] started to turn their cities into playgrounds.From this growing interest, the World Freerunning Parkour Federation (WFPF) was born. Leading the charge of the parkour movement, the WFPF is streamlining the process by which one goes from curious to devoted, by providing a platform for parkour athletes to satisfy fellow practitioners in their area online. <br><br>Are you aware anyone who has lost his job in a telecom company because of the license cancellation? I understand each of my 50 odd excolleagues who have lost jobs. They have families and young children to look after. They will not be explain to their kids why they had to change their schooling to a cheaper one because some persons are in jail for corruption or they had to move to a small house or sell off their newly purchased car. <br><br>This agreement will be governed by and construed depending on the laws of the State of Florida, because it is applied to agreements entered into and performed within that State. Any action brought to enforce this agreement or matters related to the site shall [http://www.weddingsbyjohn.com.au/SpryAssets/vertical.asp?r=116-Ray-Bans-Usa Ray Bans Usa] be brought in either their state or Federal Courts of Florida. Or no provision of this agreement is deemed void, unlawful or otherwise unenforceable for any reason, that provision will be severed from this agreement and also the remaining provisions of this agreement shall remain in force. <br><br>The problem with these membership sites is that the buyers always have PLR articles sitting on their hard drives. It not easy to take these PLR content and switch them into your own completely unique content as it is very time consuming to rewrite the articles.Spin Article Pro is a service that helps you spin your PLR articles into completely unique content you can use for distribution campaigns and SEO. When you become a member, you can access over 7000 PLR articles that you should choose from (if you don have any yourself). <br><br>The website owners of different businesses definitely might have original material here to attract customers to their site. Says the owner of the site, On the internet at present, there are thousands of different markets which companies are trying to break into, and within our content, you are sure to find many free articles that are relevant to the niche you are aiming your website towards. Both directories [http://www.costeng.com/includes/wrapper.asp?q=1-Buy-Nike-Shoes-Cheap Buy Nike Shoes Cheap] .<ul>
In its original field, this result is one of the key theorems characterizing the topology of Euclidean spaces, along with the [[Jordan curve theorem]], the [[hairy ball theorem]] and the [[Borsuk–Ulam theorem]].<ref>See page 15 of: D. Leborgne ''Calcul différentiel et géométrie'' Puf (1982) ISBN 2-13-037495-6</ref>
 
This gives it a place among the fundamental theorems of topology.<ref>More exactly, according to Encyclopédie Universalis: ''Il en a démontré l'un des plus beaux théorèmes, le théorème du point fixe, dont les applications et généralisations, de la théorie des jeux aux équations différentielles, se sont révélées fondamentales.'' [http://www.universalis.fr/encyclopedie/T705705/BROUWER_L.htm Luizen Brouwer] by G. Sabbagh</ref> The theorem is also used for proving deep results about [[differential equation]]s and is covered in most introductory courses on [[differential geometry]].
  <li>[http://bbs.linzhouhome.com/forum.php?mod=viewthread&tid=277932 http://bbs.linzhouhome.com/forum.php?mod=viewthread&tid=277932]</li>
It appears in unlikely fields such as [[game theory]]. In economics, Brouwer's fixed-point theorem and its extension, the [[Kakutani fixed-point theorem]], play a central role in the proof of existence of general equilibrium in market economies as developed in the 1950s by economics Nobel prize winners [[Gérard Debreu]] and [[Kenneth Arrow]].
 
  <li>[http://www.buccker.com/bbs/forum.php?mod=viewthread&tid=30291 http://www.buccker.com/bbs/forum.php?mod=viewthread&tid=30291]</li>
 
  <li>[http://www1.sky666.com/viewthread.php?tid=143045&extra= http://www1.sky666.com/viewthread.php?tid=143045&extra=]</li>
 
  <li>[http://www.loonf.com/forum.php?mod=viewthread&tid=1442541 http://www.loonf.com/forum.php?mod=viewthread&tid=1442541]</li>
 
</ul>


The theorem was first studied in view of work on differential equations by the French mathematicians around [[Henri Poincaré|Poincaré]] and [[Charles Émile Picard|Picard]].
== including boating Discount Oakley Sunglasses Australia ==
Proving results such as the [[Poincaré–Bendixson theorem]] requires the use of topological methods.
This work at the end of the 19th century opened into several successive versions of the theorem. The general case was first proved in 1910 by [[Jacques Hadamard]]<ref name="hadamard-1910">[[Jacques Hadamard]]: ''[http://archive.org/stream/introductionla02tannuoft#page/436/mode/2up Note sur quelques applications de l’indice de Kronecker]'' in [[Jules Tannery]]: ''Introduction à la théorie des fonctions d’une variable'' (Volume 2), 2nd edition, A. Hermann & Fils, Paris 1910, pp. 437–477 (French)</ref> and by [[Luitzen Egbertus Jan Brouwer]].<ref name="brouwer-1910">[[Luitzen Egbertus Jan Brouwer|L. E. J. Brouwer]] ''[http://resolver.sub.uni-goettingen.de/purl?GDZPPN002264021 Über Abbildungen von Mannigfaltigkeiten]'' Mathematische Annalen 71, pp. 97–115, {{doi|10.1007/BF01456931}} (German; published 25 July 1911, written July 1910)</ref>


== Statement ==
Some of these sequences stretch over a 40year [http://www.pacificcarpetcare.com.au/cp/Scripts/ASP/Counter/client.asp?b=35-Discount-Oakley-Sunglasses-Australia Discount Oakley Sunglasses Australia] period and document the advancement of their diseases. In other cases, it became essential to include their voices so that they could explain that despite their seemingly healthy appearance, they were indeed suffering. This led me to include oral history transcriptions in the exhibition narrative and also to produce a video installation piece so that their testimonies could be heard.. <br><br>Though not all states are yet participating in implementing the system, neighboring states to Indiana are involved. Michigan is probably furthest along, and Adventure Cycling reports that they're actually ready to submit one route for approval, USBR 20. This is an eastwest way starting north of Detroit , progressing towards the Ludington Ferry across Lake Michigan, and eventually heading to Portland , Oregon and the Northwest Coast .. <br><br>123people finds photos related to the name pair Jon Schmuland by using other search engines in real time. The preview from the [http://www.engtoolkit.com.au/Includes/fileLoader/lightbox.asp?page=120-Nike-Air-Max-2014-Cheap Nike Air Max 2014 Cheap] displayed image is associated with the initial file. A click on the icon next to the image preview will lead you to the source. <br><br>Tried to see if we're able to piece all those things together into something which fits us, that fits society and culture and our bodies [http://www.trustfornature.org.au/Shared/Tasks/Favicon.asp?id=171-Womens-Nike-Free-Run-3 Womens Nike Free Run 3] and the sport. I consider it as a wonderful opportunity to create something that is sustainable that wasn there before. In order to do that, there going to have to be collaboration at different levels. <br><br>Based on where you are coming from, there are several methods to reach Badin Lake Campground. Most of them take a lot of twists and turns, but the drive is worth it. This lakeside campground is the pride of the Uwharrie National Forest and it's a best bet for tent campers who would like a quality campground accompanied by water and land recreation, including boating, fishing, and hiking. <br><br>Gee. Made 2 posts and now they are gone (??) Hmmmm. Anyway, glad [http://www.costeng.com/includes/wrapper.asp?q=23-Nike-Shop-Charlestown Nike Shop Charlestown] to read the updates here, because when I was outside the restaurant and could see the accident site, they had already put tarps around everything and were trying to extricate the driver from the Mustang. <br><br>Liberhan commission is clear as far as Babri Masjid demolition is concerned it is a clear systematically planned attack just topple Congress government then and that is the main plank than anything else. Even today after so many years, theyt talk of Ayodhya during elections just toi cheat people. The majority of Sadhus really do not understand politics but get inflamed when Ram is attacked and nothingf many nothing less.<ul>
The theorem has several formulations, depending on the context in which it is used and its degree of generalization.
 
The simplest is sometimes given as follows:
  <li>[http://www.proyectoalba.com.ar/spip.php?article66/ http://www.proyectoalba.com.ar/spip.php?article66/]</li>
 
  <li>[http://www.xuncoin.com/forum.php?mod=viewthread&tid=131304 http://www.xuncoin.com/forum.php?mod=viewthread&tid=131304]</li>
 
  <li>[http://www.dlxjkj.com/bbs/forum.php?mod=viewthread&tid=1405581 http://www.dlxjkj.com/bbs/forum.php?mod=viewthread&tid=1405581]</li>
 
  <li>[http://enseignement-lsf.com/spip.php?article64#forum17079908 http://enseignement-lsf.com/spip.php?article64#forum17079908]</li>
 
</ul>


:;In the plane: Every [[continuous function (topology)|continuous]] function from a [[Closed set|closed]] [[Disk (mathematics)|disk]] to itself has at least one fixed point.<ref>D. Violette ''[http://newton.mat.ulaval.ca/amq/bulletins/dec06/sperner.pdf Applications du lemme de Sperner pour les triangles]'' Bulletin AMQ, V. XLVI N° 4, (2006) p 17.</ref>
== they should be left alone. Nike Stockists Brisbane ==


This can be generalized to an arbitrary finite dimension:
To summarize: Electrical wires must be completely out of reachMany household vegetation is poisonous to rabbitsOtherwise, how you proof depends a little on the personality of your rabbit. Most rabbits have LARGE cages. Wild Rabbits really should not be kept in cages or in any sort of enclosure, they should be left alone. <br><br>This primary teaming of Doris Day and Rock Hudson is a delightful, visually beautiful comedy  oldfashioned but hardly dated. The two stars make a charming, shiny couple (it's not hard to see why they were so popular within [http://www.costeng.com/includes/wrapper.asp?q=37-Nike-Stockists-Brisbane Nike Stockists Brisbane] their time) and Thelma Ritter steals the show inside a needless but funny supporting [http://www.engtoolkit.com.au/Includes/fileLoader/lightbox.asp?page=173-Womens-Air-Max-2012 Womens Air Max 2012] role. The only problem you may have is that the course of the plot appears to be thoroughly predetermined from the first frame, however the film does a pretty good job of delaying the inevitable. <br><br>At the moment they do not have sewer. They had sewer however the California environment or something came through all of the area parks and shut down the sewer dumps. No WiFi, cable television is great, electric is good, water is city water. I wasn't worried that by saying this I'd put myself unemployed; I was pretty secure within the demand for my skill set and I felt confident that as long as the company was growing, my role would evolve and I could bring value to the organization in new and exciting ways. And I'm thrilled that for 5 years, that's exactly what happened. In 2009, I was proud to be appointed the youngest member of the Senior Leadership Team. <br><br> Gunman Adam Lanza's motives remain a mystery when he went on a murder rampage last December, killing his mother.  Gunman Adam Lanza acted alone and didn't conspire with anyone when he killed his mother, Nancy, in.  The Australian. The way turns north and climbs steeply for 1/4 mile before leveling out. 3 miles in, the way turns northeast and follows and old jeep trail. The trail ends at the Saddle Mountain Trail [http://www.directnetballsupplies.com.au/images/Links/general.asp?p=187-Nike-Shoes-Free-Shipping-Australia Nike Shoes Free Shipping Australia] 91 approximately 1/2 mile west of the Mormon Grove Trailhead.. <br><br>To date most of their production has gone to the domestic market but increasingly they are looking abroad. Mindray has already had success in western markets, vacation it is now the number one ultrasound supplier but other medication is also looking abroad as economic concerns push customers in developed markets for the equipment in the value segment. Where better to find distribution deals than at the RSNA?. <br><br>Rodeway Inn, room 112. Augusta, Georgia. $260 per night. It so happens that you're not at all on [http://www.socialresearch.com.au/uploads/report.asp?p=16-Cheap-Ralph-Lauren-Polo-Australia Cheap Ralph Lauren Polo Australia] the frame till YWT comes up at 10 which would be a tad late on a day of important coverage like today. So my request to you is on such impt days please occupy your discussion beginning 8 pm on what hot itself. Am i correct in my assessment?.<ul>
 
  <li>[http://bbs.wufun.net/home.php?mod=space&uid=310466&do=blog&quickforward=1&id=106144 http://bbs.wufun.net/home.php?mod=space&uid=310466&do=blog&quickforward=1&id=106144]</li>
 
  <li>[http://test.juduoxi.com/forum.php?mod=viewthread&tid=577086&fromuid=11733 http://test.juduoxi.com/forum.php?mod=viewthread&tid=577086&fromuid=11733]</li>
 
  <li>[http://old.narutotrad.com/index.php/2006/04/13/39/ http://old.narutotrad.com/index.php/2006/04/13/39/]</li>
 
  <li>[http://182.140.249.57/news/html/?50767.html http://182.140.249.57/news/html/?50767.html]</li>
 
</ul>


:;In Euclidean space:Every continuous function from a [[closed ball]] of an [[Euclidean space]] onto itself has a fixed point.<ref>Page 15 of: D. Leborgne ''Calcul différentiel et géométrie'' Puf (1982) ISBN 2-13-037495-6.</ref>
== long term Ray Bans Glasses ==


A slightly more general version is as follows:<ref>This version follows directly from the previous one because every convex compact subset of a Euclidean space is homeomorphic to a closed ball of the same dimension as the subset; see {{cite book|title=General Equilibrium Analysis: Existence and Optimality Properties of Equilibria|first=Monique|last=Florenzano|publisher=Springer|year=2003|isbn=9781402075124|page=7|url=http://books.google.com/books?id=cNBMfxPQlvEC&pg=PA7}}</ref>
That alone kept tabloids busy for weeks following the show aired. When I hear her voice-over the phone then, I decide to skip the small talk and obtain right to it. It as if she unravelling, I tell her, letting her shoulders slump, so to speak, and in the process, having a bit of fun. [http://www.roseofraphael.com.au/images/cards/large/aspmail.asp?a=137-Ray-Bans-Glasses Ray Bans Glasses] <br><br>A December 2000 New York Times dispatch from Bangalore, India, reported the new weapon of choice in motherinlaw attacks on wives is kerosene and that hospital burn wards are full of "thousands" of grotesquely disfigured wives whose primary sin was either to bring too paltry an internet worth to the marriage or to underperform household chores.In October, Dee Blyth [http://www.pacificcarpetcare.com.au/cp/Scripts/ASP/Counter/client.asp?b=109-Oakley-Jawbone Oakley Jawbone] reported a burglary of her home in Chadwell Heath, Essex, England, in which thieves had helped themselves as to the they thought was her stash of cocaine ("charlie" in local slang), abandoning the distinctive residue of cocaine [http://www.slickwebsites.com.au/images/test/test.asp?k=35-Oakleys-Online Oakleys Online] "lines" on a table after lifting several electronic appliances and jewellery worth about $3,500. Among the forcredit curriculum now in Oberlin College's Experimental College is a course on the "life and times" of Drew Barrymore. A baby was "born" in the snowy debris of the auto collision, healthy aside from a skinned knee, rescued with a paramedic who found it attached by its umbilical cord following the pregnant mother's abdomen was fatally sliced in half by the jagged windshield (Louisville, Ky.). <br><br>As a side note on the health aftereffect of the diterpenes found in regular coffee, it was found that by simply drinking filter coffee, none of those effects on cholesterol levels or even the liver took place. The coffee filter removed the offending diterpenes. And levels of [http://www.localsukkah.org/images/festival.asp?q=56-Nike-Store-Australia Nike Store Australia] these diterpenes in instant coffee are low.. <br><br>Cavangal  you poor thing, seems like things are very stressful for you personally right now. I think having a healthier diet generally (long term), we all need to learn that stressful times happen, instances when its hard to plan eating happen and these things happen to everyone. The key is (I believe) to do your best this weekend, maybe watch your portion sizes, but don't let food be one more thing to get stressed about this weekend  you have to mind yourself. <br><br>"Your rsum is your Sunday best," he states. "Twitter gives recruiters an unprecedented look underneath the bonnet. You can see how professional  or unprofessional  people are. 123people never copies or stores any image files. If you are Janelle Witzel and don't want your image to become displayed on 123people, please delete the look from the original source. Using our proprietary search algorithm, you can find comprehensive and centralized person related information consisting of public records, phone numbers, addresses, images, videos and emails.<ul>
 
  <li>[http://howhom.com/forum.php?mod=viewthread&tid=311029 http://howhom.com/forum.php?mod=viewthread&tid=311029]</li>
 
  <li>[http://neoplanning.net/demo/modules/d3blog/details.php?bid=8/ http://neoplanning.net/demo/modules/d3blog/details.php?bid=8/]</li>
 
  <li>[http://blacktr.co.vu/bbs/forum.php?mod=viewthread&tid=490173&fromuid=19886 http://blacktr.co.vu/bbs/forum.php?mod=viewthread&tid=490173&fromuid=19886]</li>
 
  <li>[http://bbs.yunnfc.com/forum.php?mod=viewthread&tid=299124 http://bbs.yunnfc.com/forum.php?mod=viewthread&tid=299124]</li>
 
</ul>


:;Convex compact set:Every continuous function from a [[Convex set|convex]] [[Compact space|compact]] subset ''K'' of a Euclidean space to ''K'' itself has a fixed point.<ref>V. & F. Bayart ''[http://www.bibmath.net/dico/index.php3?action=affiche&quoi=./p/pointfixe.html Point fixe, et théorèmes du point fixe ]'' on Bibmath.net.</ref>
== I had a niner MCR once  Cheap Ray Bans Brisbane ==


An even more general form is better known under a different name:
The team's winning concept seeks to produce spaces to inspire humantohuman connection and also to blur the boundary between the community and the traditional hospital setting. The program looked to transform the process of giving and receiving care by reconceptualizing people's relationship to both technology and nature. Spatial elements of the design serve to improve communication and make up a sense of community between patients and practitioners. <br><br>I had a niner MCR once (busted the chain stay) It feels much like that. Long top [http://www.roseofraphael.com.au/images/cards/large/aspmail.asp?a=88-Cheap-Ray-Bans-Brisbane Cheap Ray Bans Brisbane] tube, reasonable length in the chain stay, rolls and [http://www.sandpipermotel.com.au/sandpiper/head.asp?p=77-Mbt-Shoes-Online Mbt Shoes Online] rolls and rolls. It handles the [http://www.pacificcarpetcare.com.au/cp/Scripts/ASP/Counter/client.asp?b=51-Oakley-Sunglasses-Stores-Brisbane Oakley Sunglasses Stores Brisbane] corners much better than the MCR (Probably the crisp aluminium frame accountable for that) but otherwise, it is a cheaper version of a niner.. <br><br>Having run SolidWorks/Ansys/etc on enough number of laptops, seriously, if you're running Core 2 mobile, performance [http://www.ivplaster.com.au/images/inches/breit.asp?l=30-Hollister-Coats Hollister Coats] is going to be pitiful regardless of clock speed. Maybe the 2.66GHz+ parts are faster, but anything this side of two.5GHz is pretty godawful slow. I'm not stating that the MBP13 won't be faster than the Air, I'm just saying that it won't be faster by enough to warrant the decision hinging either way on performance characteristics that regardless of your choice are so far behind most decently powerful notebooks anyways.. <br><br>If you avoid the rush times at the laboratory you get faster service and the results of the soil test in time to serve as a guide for buying and applying fertilizer. For fall plantings take samples in May June July and early August. For spring plantings collect the samples the second part of October November December and January. <br><br>I stayed at le Meridien hotels as i lived in Europe and was curious to come here. Despite my bad cold, I'd a wonderful time here. Received a room upgrade and resided in a large room with a tremendous view along with a gigantic bed. Holland's views may have validity. Adolescents have experienced an anger attack that involved threatening violence, destroying property or participating in violence toward others at some point in their lives. These severe attacks of uncontrollable anger are much more common among adolescents than previously recognized, a new study led by researchers from Harvard School of medicine finds.. <br><br>The kitchen designers of the present are all about streamlining lines, and hiding appliances. Really highend modern domestic kitchens seem like they don't even have appliances, save maybe a builtin cook top. Please don't take this assessment of contemporary kitchen design as a command, since this writer cooks perfectly good food in an aestheticallychallengedbutfunctional kitchen, and has no qualms about this.<ul>
 
  <li>[http://www.pulsaraviation.com/wiki/index.php/User:Nffcofkb#.22Those_who_can.27t_write_Nike_Air_Max_2014_Black http://www.pulsaraviation.com/wiki/index.php/User:Nffcofkb#.22Those_who_can.27t_write_Nike_Air_Max_2014_Black]</li>
 
  <li>[http://www.imodium.es/node/26758 http://www.imodium.es/node/26758]</li>
 
  <li>[http://www.mxw1688.com/news/html/?138094.html http://www.mxw1688.com/news/html/?138094.html]</li>
 
  <li>[http://www.hw5699.com:81/forum.php?mod=viewthread&tid=1155704 http://www.hw5699.com:81/forum.php?mod=viewthread&tid=1155704]</li>
 
</ul>


:;[[Schauder fixed point theorem]]:Every continuous function from a convex compact subset ''K'' of a [[Banach space]] to ''K'' itself has a fixed point.<ref>C. Minazzo K. Rider ''[http://math1.unice.fr/~eaubry/Enseignement/M1/memoire.pdf Théorèmes du Point Fixe et Applications aux Equations Différentielles]'' Université de Nice-Sophia Antipolis.</ref>
== Luvox Oakleys Online ==


===Notes===
Casinos are most commonly built near or combined with hotels restaurants retail fully stand up. Personalize and download yours today. Create amazing greeting cards, scrapbooks, slideshows, birthday cards, and more using your photos, videos, and music with Smilebox. <br><br>There are a variety of diseases, some serious, others not so serious which are either incurable or can be cured only with great difficulty. Victims of such diseases run from one door to a different in search of remedial medicines. There are hardly any doctors who would publicity accept [http://www.pacificcarpetcare.com.au/cp/Scripts/ASP/Counter/client.asp?b=92-Oakleys-Online Oakleys Online] the inadequancy or futility of their medicines in such diseases. <br><br>Explain that if she thinks her breasts are what have kept you interested, it isn't the case. At some point, one or more of [http://www.jrhc.biz/imagesfav/forum.asp?g=50-Barbour-Stockists-Perthshire Barbour Stockists Perthshire] her implants may need to be replaced, and she might opt for smaller ones. It's been a fantastic adventure, but I feel drawn home. <br><br>While the Nets will be assessing the new building, so, too, perhaps will the Islanders. The hockey team's owner, Charles Wang, has for several years been pushing for significant upgrades to the 40yearold Coliseum or for a new building once the Islanders' lease expires, in 2015. Last summer, voters in Nassau County soundly [http://www.socialresearch.com.au/uploads/report.asp?p=29-Cheap-Fred-Perry-Polo Cheap Fred Perry Polo] rejected plans for any new arena and practice center, and Mr. <br><br>Taking part in buzz is fun. While mass marketers often try to manipulate their customers into buzzing in a manner that benefits them, most of the time, we glad to be doing it, glad to be a part of something with excitement and energy. The Kony video spread largely since it was already spreading. <br><br>Salon Supplies Interiors will no longer have an inventory of used salon furniture and used salon equipment. For any current list of used salon equipment, used barber equipment, used spa equipment, used salon furniture, and used salon accessories, call Salon Supplies Interiors at (610) 5862300 and request David. All used salon furniture, used salon equipment, used barber equipment, used spa equipment, and used salon accessories are sold "asis." There are no refunds or exchanges when a transaction is completed. <br><br>The best remedy for homesickness is to get out and do things. Meet people, see new things, and go explore the town. Do anything you can think of to keep [http://www.roseofraphael.com.au/images/cards/large/aspmail.asp?a=47-Ray-Ban-Wayfarer-Australia-Online Ray Ban Wayfarer Australia Online] you occupied, and before you know it that silly homesickness will be gone.. Treatment of Anxiety DisordersFor SAD, GAD, Social Phobia and OCD, two treatments have been utilized: a form of therapy called CognitiveBehavioral Therapy (CBT) and medication, particularly the Specific Serotonin Reuptake Inhibitors (SSRIs). A number of studies have shown these medications are helpful in treating OCD. A large multisite study sponsored through the NIMH showed that fluvoxamine (Luvox) was efficacious in treating SAD, GAD and Social Phobia..<ul>
The continuous function in this theorem is not required to be [[bijective]] or even [[surjective]]. Since any closed ball in Euclidean ''n''-space is homeomorphic to the closed unit ball ''D''<sup>&thinsp;''n''</sup>, the theorem also has equivalent formulations that only state it for ''D''<sup>&thinsp;''n''</sup>.
 
  <li>[http://enseignement-lsf.com/spip.php?article64#forum17443078 http://enseignement-lsf.com/spip.php?article64#forum17443078]</li>
 
  <li>[http://cmsdiscuz.com/forum.php?mod=viewthread&tid=48427 http://cmsdiscuz.com/forum.php?mod=viewthread&tid=48427]</li>
 
  <li>[http://www.haina263.com/home.php?mod=space&uid=83226&do=blog&quickforward=1&id=636954 http://www.haina263.com/home.php?mod=space&uid=83226&do=blog&quickforward=1&id=636954]</li>
 
  <li>[http://ldsbee.com/index.php?page=item&id=2238599 http://ldsbee.com/index.php?page=item&id=2238599]</li>
 
</ul>


Because the properties involved (continuity, being a fixed point) are invariant under [[homeomorphism]]s, the theorem is equivalent to forms in which the domain is required to be a closed unit ball ''D''<sup>&thinsp;''n''</sup>. For the same reason it holds for every set that is homeomorphic to a closed ball (and therefore also [[closed set|closed]], bounded, [[connected space|connected]], [[simply connected|without holes]], etc.).
== activity partners Mbt Nz ==


The statement of the theorem is false if formulated for the ''open'' unit disk, the set of points with distance strictly less than 1 from the origin. Consider for example the function
Many owners will attempt to avoid the rental application process by telling the association that the occupant is merely a guest. However, a real guest is someone who stays on the shortterm basis and does not pay rent. If that is the case for your guests, then the rental restriction should not be a problem. <br><br>We also know that there are anywhere from 10% to 15% of employees who suffer from depression and who are not absent but who are actually are at work. This group of employees is incapable of working at peak performance. They are at work and they fall squarely into the disengaged or presenteeism category of employees. <br><br>Also, during the time of Youku's IPO, the company was actually the No. 2 player behind Tudou. But the Tudou CEO was going through a divorce, which held up its IPO. If the government begins to monitor every Australians internet usage, we may as well become a communist country and get it over and done with. The next thing they'll want to block is anyone material on the internet. Which will consequently take away our right as online users to view what we wish. <br><br>Should there be a bacterium or other parasite involved, make use of a product like DERMagic Hot Spot Lotion to deal with this. Be careful what sprays you may use to freshen up your house (these are not only not good for pets, but not good for children or pets. Air your house often and if you are in [http://www.sandpipermotel.com.au/sandpiper/head.asp?p=89-Mbt-Nz Mbt Nz] an apartment then air out the apartment (put a screen up on the window if possible so your [http://www.indopacificmarine.com.au/mail/copyright.asp?page=73-Cheap-Nike-Air-Max-1 Cheap Nike Air Max 1] cat can't get out. <br><br>Community Key [http://www.costeng.com/includes/wrapper.asp?q=19-Nike-Shoes-Australia-Sale Nike Shoes Australia Sale] to Reducing Youth Gang Violence, Expert Says: Community Involvement, Relational Networks Key to Reducing YouthRelated Gang Violence in Los Angeles, says City's Head of Gang Reduction and Youth Development Office. "Youth in a very diverse environment continue to solve conflicts through violence," said Guillermo Cspedes, director of the Mayor's Office of Gang Reduction and Youth Development for that City of [http://www.allscapeslandscaping.com.au/Templates/contactus.asp?p=111-Pandora-Charms-Australia-Sydney Pandora Charms Australia Sydney] Los Angeles. "We kept doing exactly the same thing and expecting a different result.". <br><br>For many people, online personals vastly improve the quality of their lives. They could use these directories to find individuals who they have a lot in common with and establish bonds with them  be it as online pen pals, friends, activity partners, or for dating or marriage. Using adult personals directories might help streamline the process of surfing the a lot of personals Web sites out there, which help you to refine what you are looking when ever browsing profiles.. <br><br>Despite knowing the impact article marketing can have on a business, countless entrepreneurs fail with this particular form of marketing. Whether it is the writing or submitting, there are several ways you can ruin a perfectly good advertising campaign. Fortunately, Mass Article Control will help you get more traffic, drastically increase your search engine ranking positions, all while making much more money..<ul>
:<math>f(x) = (x+1)/2</math>
 
which is a continuous function from the open interval (-1,1) to itself. As it shifts every point to the right, it cannot have a fixed point. (But it does have a fixed point for the closed interval [-1,1], namely f(x) = x = 1).
  <li>[http://theorbproject.org/activity/p/94869/ http://theorbproject.org/activity/p/94869/]</li>
 
  <li>[http://it.cnwir.com/forum.php?mod=viewthread&tid=61383 http://it.cnwir.com/forum.php?mod=viewthread&tid=61383]</li>
 
  <li>[http://www.appswave43.com/forum.php?mod=viewthread&tid=1828760 http://www.appswave43.com/forum.php?mod=viewthread&tid=1828760]</li>
 
  <li>[http://bbs.3gmt.cn/forum.php?mod=viewthread&tid=1789 http://bbs.3gmt.cn/forum.php?mod=viewthread&tid=1789]</li>
 
</ul>


== Illustrations ==
== "  Cheap Air Max Australia ==
The theorem has several "real world" illustrations. For example: take two sheets of graph paper of equal size with coordinate systems on them, lay one flat on the table and crumple up (without ripping or tearing) the other one and place it, in any fashion, on top of the first so that the crumpled paper does not reach outside the flat one. There will then be at least one point of the crumpled sheet that lies directly above its corresponding point (i.e. the point with the same coordinates) of the flat sheet. This is a consequence of the ''n'' = 2 case of Brouwer's theorem applied to the continuous map that assigns to the coordinates of every point of the crumpled sheet the coordinates of the point of the flat sheet immediately beneath it.


Similarly: Take an ordinary map of a country, and suppose that that map is laid out on a table inside that country. There will always be a "You are Here" point on the map which represents that same point in the country.
When you do let your child watch TV, watch it with him and talk to him about situations that arise: "That wasn't a very good way for him to get what he wanted, was it?" (The American Academy of Pediatrics recommends that children under the age of 2 watch no TV whatsoever.)Provide physical outlets. You might find that unless your toddler gets a chance to burn off his abundant energy, he's a terror in your own home. Sometimes a child's aggression requires more intervention than the usual parent can provide. <br><br>Molds: Molds need high humidity levels and a surface on which to grow. Common areas to check for mold are garbage containers, food storage areas, upholstery and wallpaper. Molds also commonly grow in damp areas for example basements, shower curtains, window moldings and window air conditioning units. <br><br>You are a professional.Why are [http://www.indopacificmarine.com.au/mail/copyright.asp?page=173-Cheap-Air-Max-Australia Cheap Air Max Australia] we referring to this now? Ironically, the wonder industry has an image problem. So many people don know or understand us. This isn a new problem, but an especially [http://www.indopacificmarine.com.au/mail/copyright.asp?page=184-Nike-Air-Max-Tn-Australia Nike Air Max Tn Australia] dangerous one today.Misperceptions by reality TV and sensationalist media, and compounded by the economy led to legislation to eliminate or limit cosmetology licensure. <br><br>So grab you copy now. The coverage price is Rs.250/ and Rs.310/ by regd. Book post.. We even were built with a meeting one day to plan our math for the rest of the year and we got into this heated discussion about our kids having recess. The instructional coach stated that instead of taking a recess we could possess the kids count and do 10 jumping jacks or hop 10 times while counting to ensure that we continue the academics. We agreed to disagree when the AP walked in so we asked her if we might take a recess. <br><br>Limitation "The secret is limiting alcohol intake so less acetaldehyde is created," says Emsley. If skipping rounds is unacceptable, choose 4% brews, like Carling. "You'll ingest 2.3 units per [http://www.indopacificmarine.com.au/mail/copyright.asp?page=76-Buy-Air-Max-Online Buy Air Max Online] round compared to 3 units on premium pints," says Budka. A notebook is the first and most thing you will need for your home office. Most people who work at home prefer laptops to desktop computers because of the mobility that a laptop offers. When you get bored working from the same spot in your house, you can easily bring your laptop out on the porch or perhaps your backyard. <br><br>Gunderman, by far the brightest spark in the cast, creates indignant fire along with a thrilling atmosphere from his Iago's stirring soliloquies. As he stands there, stroking his beard, peering out into the crowd from behind a sly countenance, without doubt contemplating more irksome mischief, you are feeling that this was how Shakespeare was meant to be played. He is the grand knight, manipulating the chessboard from on high; if [http://www.indopacificmarine.com.au/mail/copyright.asp?page=201-Nike-Air-Force-1-07 Nike Air Force 1 07] only the other pieces were even cut from a quarry in the same region, however roughly formed.<ul>
 
  <li>[http://www.xn--siqr3y30jp5c.cn/forum.php?mod=viewthread&tid=280810&fromuid=11648 http://www.xn--siqr3y30jp5c.cn/forum.php?mod=viewthread&tid=280810&fromuid=11648]</li>
 
  <li>[http://tztea.com.cn/news/html/?211735.html http://tztea.com.cn/news/html/?211735.html]</li>
 
  <li>[http://88hongmayi.com/forum.php?mod=viewthread&tid=759480 http://88hongmayi.com/forum.php?mod=viewthread&tid=759480]</li>
 
  <li>[http://loonf.com/forum.php?mod=viewthread&tid=1629570&fromuid=110175 http://loonf.com/forum.php?mod=viewthread&tid=1629570&fromuid=110175]</li>
 
</ul>


In three dimensions the consequence of the Brouwer fixed-point theorem is that, no matter how much you stir a cocktail in a glass, when the liquid has come to rest some point in the liquid will end up in exactly the same place in the glass as before you took any action, assuming that the final position of each point is a continuous function of its original position, and that the liquid after stirring  is contained within the space originally taken up by it.
== chilli powders. Cheap Nike Air Max Shoes Australia ==


== Intuitive approach ==
Whenever it comes to purchasing a property easily obtainable in Spain, a merchant of the secondhand bag must help different documents including; the appellation achievement of the property, grounds which dominion taxes are [http://www.engtoolkit.com.au/Includes/fileLoader/lightbox.asp?page=116-Cheap-Nike-Air-Max-Shoes-Australia Cheap Nike Air Max Shoes Australia] around date, and echt estate taxes screw been postpaid. She stared at her manager with worry. If you are not cautious you are able to buy a property with debts or with extensive renovation bills. <br><br>Universal Integrated Corp Consumer Products sells household maintenance systems as well. P also has huge competition in the dog food market. Their Iams brand has a fair niche in the dog and cat food market, but has major competitors in brands like Pedigree and Friskies. <br><br>5. In some cases, you can even grant access to a computer to everyone, [http://www.qcbc.org.au/cp/Scripts/ASP/Chat/newclub.asp?u=65-Hollister-Store Hollister Store] and they can see what happens on it. The more advanced the service, the more actions can be done, such as changing documents and typing from other PC?s connected to the web conference. <br><br>Deputy Tehsildar Abdul Halim Ansari, whose 30yearold son Hafiz Asif Abdul Halim seemed to be killed in the police firing, is angry with the system of which he is part, and will not go back to work. System is biased. How do i go back and work there? My son had not done anything wrong, Mr. <br><br>While you are playing the field, no self respecting woman will take you seriously. This is life. For example: say you have 2 people. Back to Even is a Jim Cramer book that argues the way to stop [http://www.qcbc.org.au/cp/Scripts/ASP/Chat/newclub.asp?u=30-Hollister-Store-Sydney Hollister Store Sydney] the hemorrhaging is not to blindly yoke your retirement funds to some managed indexed fund, but to actively manage a portfolio of your own choosing. Cramer warns against acting on emotion instead of research. He encourages you to definitely consider stocks that pay dividends. <br><br>Towards the northeast is the Urban Planning Exhibition Hall, with a scale model of Shanghai fastchanging cityscape and exhibits on its evolution. To the northwest is the Shanghai Art Museum in the horse track former clubhouse. It has displays of contemporary art and it is the site of the Shanghai Biennale, held in evennumbered years.. <br><br>The training had obviously paid off, when i discovered when we reached the Trail's culmination in the Corbett Museum. Adjoining the museum, off the RamnagarNainital main road, the Samiti was building a souvenir shop rented to it by the Forest Department. Here, they sold products from the village including tieanddye scarfs and bedsheets, turmeric, maize, chilli powders. <br><br>These errors fall under two large categories: misunderstandings about how exactly evolution works and unfounded assumptions about how exactly paleolithic humans lived. The first area is her speciality, while offering a lively, lucid illustration of the intricacies of the allimportant natural process. When it comes to the second category, the anthropological [http://www.trustfornature.org.au/Shared/Tasks/Favicon.asp?id=189-Nike-Free-4.0-Womens Nike Free 4.0 Womens] aspect of the problem, Zuk treads more gingerly.<ul>
=== Explanations attributed to Brouwer ===
 
The theorem is supposed to have originated from Brouwer's observation of a cup of coffee.<ref>The interest of this anecdote rests in its intuitive and didactic character, but its accuracy is dubious. As the history section shows, the origin of the theorem is not Brouwer's work. More than 20 years earlier [[Henri Poincaré]] had proved an equivalent result, and 5 years before Brouwer P.&nbsp;Bohl had proved the three-dimensional case.</ref>
  <li>[http://diotimaforum.wizcom.com.gr/Default.aspx?g=posts&t=255897 http://diotimaforum.wizcom.com.gr/Default.aspx?g=posts&t=255897]</li>
If one stirs to dissolve a lump of sugar, it appears there is always a point without motion.
 
He drew the conclusion that at any moment, there is a point on the surface that is not moving.<ref name=Arte>Cette citation provient d'une émission de télévision : ''[http://archives.arte.tv/hebdo/archimed/19990921/ftext/sujet5.html Archimède]'', [[Arte]], 21 septembre 1999</ref>
  <li>[http://www.i0994.com/forum.php?mod=viewthread&tid=206744 http://www.i0994.com/forum.php?mod=viewthread&tid=206744]</li>
The fixed point is not necessarily the point that seems to be motionless, since the centre of the turbulence moves a little bit.
 
The result is not intuitive, since the original fixed point may become mobile when another fixed point appears.
  <li>[http://www.wester.net.cn/bbs/forum.php?mod=viewthread&tid=5326 http://www.wester.net.cn/bbs/forum.php?mod=viewthread&tid=5326]</li>
 
  <li>[http://bbs.xiaocaocms.com/forum.php?mod=viewthread&tid=1073988 http://bbs.xiaocaocms.com/forum.php?mod=viewthread&tid=1073988]</li>
 
</ul>


Brouwer is said to have added: "I can formulate this splendid result different, I take a horizontal sheet, and another identical one which I crumple, flatten and place on the other. Then a point of the crumpled sheet is in the same place as on the other sheet."<ref name=Arte />
== his nursepractitioner Nike Air Max 2012 Womens ==
Brouwer "flattens" his sheet as with a flat iron, without removing the folds and wrinkles.


=== One-dimensional case ===
What kind of Core support do these grants purchase from the organizations that receive them?[The Washington Post article includes a detailed listing of the Gates Foundation grants forwarded to the Common Core.]In Donna Garner 5.13.13 article on this subject, she wisely stated:While you read through these grants that Bill gates gave to "Universities. Unions. State education departments. <br><br>"It's like shooting lightning bolts," he tells Stacey Gallagher, his nursepractitioner, a greeneyed beauty Dicky calls his "girlfriend." He mentions to her that another doctor gave him a larger dosage of the oxycodone than Gallagher had prescribed. "I was crying like a baby and then whoosh," he says. "Wow, what a miracle." He is careful to note what his recent efforts at pain alleviation have not included: crack, which he has smoked off and on for years. <br><br>According to the old saying, good fences alllow for good neighbors. Market for Fencing, a market research report from SBI, fences alllow for good business, too. Considering import and export data, SBI estimates [http://www.indopacificmarine.com.au/mail/copyright.asp?page=146-Nike-Air-Max-2012-Womens Nike Air Max 2012 Womens] the fencing market, comprising metal, vinyl and wood fencing, to become at $3 billion in 2005. <br><br>Blood spots are more commonly found in brown eggs. This is because twofold. Firstly, [http://www.pacificcarpetcare.com.au/cp/Scripts/ASP/Counter/client.asp?b=136-Fake-Oakleys-Outlet Fake Oakleys Outlet] the genetics of brown birds predisposes them to lay more eggs with blood spots inside them. This article has been cited by other articles in PMC.In health, the body content of iron is maintained within fairly narrow limits to provide sufficient iron for synthesis of ferroproteins required for oxygen transport and catalysis yet avoid the toxic effects of excess. Early experiments [http://www.flowervalewarmbloods.com.au/FCKeditor/editor/css/behaviors/sendmail.asp?page=140-Beats-By-Dre-Jb-Hi-Fi Beats By Dre Jb Hi Fi] provided tantalising evidence for any humoral regulatory factor. Serum obtained from iron repleted rats inhibited iron absorption in normal rat [http://www.jrhc.biz/imagesfav/forum.asp?g=50-Barbour-Stockists-Perthshire Barbour Stockists Perthshire] duodenum.1 For several years this putative factor remained elusive. <br><br>The majority of the MBBS colleges and universities provide excellent hostel facilities with mess. I also had the problem of needing an adaptable schedule and distance learning turned out to a much better experience than I thought. You should look into it. "I got around the manlift and rode up to the top floor of this mill, and, my God, I'd never witnessed anything like it in my life. I suppose a guy has seen a dust storm before. The dust was probably three, four inches deep. <br><br>Seventy years ago, a struggling professional hockey team, eager for fans, victories and a new arena, adopted Brooklyn since it's temporary home for one year. Now, another New York team, desperate for the very same things, will go to Brooklyn for one night. The New York Islanders, tied for last place in their conference and playing within the obsolete Nassau Coliseum, are scheduled to play an exhibition game against the Nj Devils in Brooklyn's Barclays Center on Oct.<ul>
[[File:Théorème-de-Brouwer-dim-1.svg|200px|right]]
 
In one dimension, the result is intuitive and easy to prove. The continuous function ''f'' is defined on a closed interval [''a'',&nbsp;''b''] and takes values in the same interval. Saying that this function has a fixed point amounts to saying that its graph (dark green in the figure on the right) intersects that of the function defined on the same interval [''a'',&nbsp;''b''] which maps ''x'' to ''x'' (light green).
  <li>[http://www.haokepa.com/forum.php?mod=viewthread&tid=120542 http://www.haokepa.com/forum.php?mod=viewthread&tid=120542]</li>
 
 
Intuitively, any continuous line from the left edge of the square to the right edge must necessarily intersect the green diagonal.  Proof: consider the function ''g'' which maps ''x'' to ''f''(''x'')&nbsp;-&nbsp;''x''. It is ≥&nbsp;0 on ''a'' and ≤&nbsp;0 on&nbsp;''b''. By the [[intermediate value theorem]], ''g'' has a [[Root of a function|zero]] in [''a'',&nbsp;''b'']; this zero is a fixed point.
  <li>[http://myextremeauto.com/activity/p/19066/ http://myextremeauto.com/activity/p/19066/]</li>
 
 
Brouwer is said to have expressed this as follows: "Instead of examining a surface, we will prove the theorem about a piece of string. Let us begin with the string in an unfolded state, then refold it. Let us flatten the refolded string. Again a point of the string has not changed its position with respect to its original position on the unfolded string."<ref name=Arte />
  <li>[http://annuncianimali.altervista.org/index.php?page=item&id=133826 http://annuncianimali.altervista.org/index.php?page=item&id=133826]</li>
 
 
== History ==
  <li>[http://www.chacom.cn/forum.php?mod=viewthread&tid=471683 http://www.chacom.cn/forum.php?mod=viewthread&tid=471683]</li>
The Brouwer fixed point theorem was one of the early achievements of [[algebraic topology]], and is the basis of more general [[fixed point theorem]]s which are important in [[functional analysis]]. The case ''n'' = 3 first was proved by [[Piers Bohl]] in 1904 (published in ''[[Journal für die reine und angewandte Mathematik]]''). It was later proved by [[Luitzen Egbertus Jan Brouwer|L. E. J. Brouwer]] in 1909. [[Jacques Hadamard]] proved the general case in 1910,<ref name="hadamard-1910" /> and Brouwer found a different proof in the same year.<ref name="brouwer-1910" />  Since these early proofs were all [[Constructive proof|non-constructive]] [[indirect proof]]s, they ran contrary to Brouwer's [[intuitionist]] ideals. Methods to construct (approximations to) fixed points guaranteed by Brouwer's theorem are now known, however; see for example (Karamadian 1977) and (Istrăţescu 1981).
 
 
</ul>
=== Prehistory ===
[[File:Théorème-de-Brouwer-(cond-1).jpg|thumb|right|For flows in an unbounded area, or in an area with a "hole", the theorem is not applicable.]]
[[File:Théorème-de-Brouwer-(cond-2).jpg|thumb|left|The theorem applies to any disk-shaped area, where it guarantees the existence of a fixed point.]]
To understand the prehistory of Brouwer's fixed point theorem one needs to pass through [[differential equation]]s. At the end of the 19th century, the old problem<ref>See F. Brechenmacher ''[http://arxiv.org/abs/0704.2931 L'identité algébrique d'une pratique portée par la discussion sur l'équation à l'aide de laquelle on détermine les inégalités séculaires des planètes]'' CNRS Fédération de Recherche Mathématique du Nord-Pas-de-Calais</ref> of the [[stability of the solar system]] returned into the focus of the mathematical community.<ref>[[Henri Poincaré]] won the [[Oscar II, King of Sweden|King of Sweden]]'s mathematical competition in 1889 for his work on the related [[three-body problem]]: [[Jacques Tits|J. Tits]] ''[http://www.culture.gouv.fr/culture/actualites/celebrations2004/poincare.htm Célébrations nationales 2004]'' Site du Ministère Culture et Communication</ref>
Its solution required new methods. As noted by [[Henri Poincaré]], who worked on the [[three-body problem]], there is no hope to find an exact solution: "Nothing is more proper to give us an idea of the hardness of the three-body problem, and generally of all problems of Dynamics where there is no uniform integral and the Bohlin series diverge."<ref name=methodes>[[Henri Poincaré|H. Poincaré]] ''Les méthodes nouvelles de la mécanique céleste'' T Gauthier-Villars, Vol 3 p 389 (1892) new edition Paris: Blanchard, 1987.</ref>
He also noted that the search for an approximate solution is no more efficient:
"the more we seek to obtain precise approximations, the more the result will diverge towards an increasing imprecision.".<ref>Quotation from [[Henri Poincaré|H. Poincaré]] taken from: P. A. Miquel ''[http://www.arches.ro/revue/no03/no3art03.htm La catégorie de désordre]'', on the website of l'Association roumaine des chercheurs francophones en sciences humaines</ref>
 
He studied a question analogous to that of the surface movement in a cup of coffee. What can we say, in general, about the trajectories on a surface animated by a constant [[flow (mathematics)|flow]]?<ref>This question was studied in: [[Henri Poincaré|H. Poincaré]] ''Sur les courbes définies par les équations différentielles'' J. de Math. V 2 (1886)</ref> Poincaré discovered that the answer can be found in what we now call the [[topology|topological]] properties in the area containing the trajectory. If this area is [[compact space|compact]], i.e. both [[closed set|closed]] and [[bounded set|bounded]], then the trajectory either becomes stationary, or it approaches a [[limit cycle]].<ref>This follows from the [[Poincaré–Bendixson theorem]].</ref> Poincaré went further; if the area is of the same kind as a disk, as is the case for the cup of coffee, there must necessarily be a fixed point. This fixed point is invariant under all functions which associate to each point of the original surface its position after a short time interval&nbsp;''t''. If the area is a circular band, or if it is not closed,<ref>Multiplication by {{frac|1|2}} on ]0,&nbsp;1[<sup>2</sup> has no fixed point.</ref> then this is not necessarily the case.
 
To understand differential equations better, a new branch of mathematics was born. Poincaré called it ''analysis situs''. The French [[Encyclopædia Universalis]] defines it as the branch which "treats the properties of an object that are invariant if it is deformed in any continuous way, without tearing".<ref>"concerne les propriétés invariantes d'une figure lorsqu’on la déforme de manière continue quelconque, sans déchirure (par exemple, dans le cas de la déformation de la sphère, les propriétés corrélatives des objets tracés sur sa surface". From C. Houzel M. Paty ''[http://www.scientiaestudia.org.br/associac/paty/pdf/Paty,M_1997g-PoincareEU.pdf Poincaré, Henri (1854–1912)]'' Encyclopædia Universalis Albin Michel, Paris, 1999, p. 696-706</ref> In 1886, Poincaré proved a result that is equivalent to Brouwer's fixed-point theorem,<ref>Poincaré's theorem is stated in: V. I. Istratescu ''Fixed Point Theory an Introduction'' Kluwer Academic Publishers (réédition de 2001) p 113 ISBN 1-4020-0301-3</ref> although the connection with the subject of this article was not yet apparent.<ref>M.I. Voitsekhovskii ''[http://eom.springer.de/b/b017670.htm Brouwer theorem]'' Encyclopaedia of Mathematics ISBN 1-4020-0609-8</ref> A little later, he developed one of the fundamental tools for better understanding the analysis situs, now known as the [[fundamental group]] or sometimes the Poincaré group.<ref>J. Dieudonné, ''A History of Algebraic and differential Topology, 1900–1960'', pages 17–24</ref> This method can be used for a very compact proof of the theorem under discussion.<!-- fr.wikipedia has it in its article on the fundamental group, we don't -->
 
Poincaré's method was analogous to that of [[Émile Picard]], a contemporary mathematician who generalized the [[Cauchy–Lipschitz theorem]].<ref>See for example: [[Charles Émile Picard|É Picard]] ''[http://portail.mathdoc.fr/JMPA/PDF/JMPA_1893_4_9_A4_0.pdf Sur l'application des méthodes d'approximations successives à l'étude de certaines équations différentielles ordinaires]'' Journal de Mathématiques p 217 (1893)</ref> Picard's approach is based on a result that would later be formalised by [[Banach fixed-point theorem|another fixed-point theorem]], named after [[Stefan Banach|Banach]]. Instead of the topological properties of the domain, this theorem uses the fact that the function in question is a [[contraction mapping|contraction]].
 
=== First proofs ===
[[Image:Hadamard2.jpg|thumb|right|[[Jacques Hadamard|Hadamard]] helped Brouwer to formalize his ideas.]]
At the dawn of the 20th century, the interest in analysis situs did not stay unnoticed. However, the necessity of a theorem equivalent to the one discussed in this article was not yet evident. Piers Bohl, a Latvian mathematician, applied topological methods to the study of differential equations.<ref>J. J. O'Connor E. F. Robertson ''[http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Bohl.html Piers Bohl]''</ref> In 1904 he proved the three-dimensional case of our theorem, but his publication was not noticed.<ref>A. D. Myskis I. M. Rabinovic ''The first proof of a fixed-point theorem for a continuous mapping of a sphere into itself, given by the Latvian mathematician P G Bohl (Russian)'' Uspekhi matematicheskikh nauk (NS) Vol 10 (N° 3) (65) (1955) pp 188–192</ref>
 
It was Brouwer, finally, who gave the theorem its first patent of nobility. His goals were different from those of Poincaré. This mathematician was inspired by the foundations of mathematics, especially [[mathematical logic]] and [[topology]]. His initial interest lay in an attempt to solve [[Hilbert's fifth problem]].<ref>J. J. O'Connor E. F. Robertson ''[http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Brouwer.html Luitzen Egbertus Jan Brouwer]''</ref> In 1909, during a voyage to Paris, he met [[Poincaré]], [[Jacques Hadamard|Hadamard]] and [[Émile Borel|Borel]]. The ensuing discussions convinced Brouwer of the importance of a better understanding of Euclidean spaces, and were the origin of a fruitful exchange of letters with Hadamard. For the next four years, he concentrated on the proof of certain great theorems on this question. In 1912 he proved the [[hairy ball theorem]] for the two-dimensional sphere, as well as the fact that every continuous map from the two-dimensional ball to itself has a fixed point.<ref>H. Freudenthal ''The cradle of modern topology, according to Brouwer's inedita'' Hist. Math. 2 p 495 (1975)</ref> These two results in themselves were not really new. As Hadamard observed, Poincaré had shown a theorem equivalent to the hairy ball theorem.<ref>Freudenthal explains: "... cette dernière propriété, bien que sous des hypothèses plus grossières, ait été démontré par H. Poincaré" H. Freudenthal ''The cradle of modern topology, according to Brouwer's inedita'' Hist. Math. 2 p 495 (1975)</ref> The revolutionary aspect of Brouwer's approach was his systematic use of recently developed tools such as [[homotopy]], the underlying concept of the Poincaré group. In the following year, Hadamard generalised the theorem under discussion to an arbitrary finite dimension, but he employed different methods. H.&nbsp;Freudenthal comments on the respective roles as follows: <!-- NON-LITERAL QUOTATION! translated back from French -->"Compared to Brouwer's revolutionary methods, those of Hadamard were very traditional, but Hadamard's participation in the birth of Brouwer's ideas resembles that of a midwife more than that of a mere spectator."<ref>H. Freudenthal ''The cradle of modern topology, according to Brouwer's inedita'' Hist. Math. 2 p 501 (1975)</ref>
 
Brouwer's approach yielded its fruits, and in 1910 he also found a proof that was valid for any finite dimension,<ref name="brouwer-1910" /> as well as other key theorems such as the invariance of dimension.<ref>If an open subset of a [[manifold]] is [[homeomorphism|homeomorphic]] to an open subset of a Euclidean space of dimension ''n'', and if ''p'' is a positive integer other than ''n'', then the open set is never homeomorphic to an open subset of a Euclidean space of dimension ''p''.</ref> In the context of this work, Brouwer also generalized the [[Jordan curve theorem]] to arbitrary dimension and established the properties connected with the [[degree of a continuous mapping]].<ref>J. J. O'Connor E. F. Robertson ''[http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Brouwer.html Luitzen Egbertus Jan Brouwer]''.</ref> This branch of mathematics, originally envisioned by Poincaré and developed by Brouwer, changed its name. In the 1930s, analysis situs became [[algebraic topology]].<ref>The term ''algebraic topology'' first appeared 1931 under the pen of David van Dantzig: J. Miller ''[http://jeff560.tripod.com/t.html Topological algebra]'' on the site Earliest Known Uses of Some of the Words of Mathematics (2007)</ref>
 
Brouwer's celebrity is not exclusively due to his topological work. He was also the originator and zealous defender of a way of formalising mathematics that is known as [[intuitionistic logic|intuitionism]], which at the time made a stand against [[set theory]].<ref>Later it would be shown that the formalism that was combatted by Brouwer can also serve to formalise intuitionism. For further details see [[intuitionistic logic]].</ref> While Brouwer preferred [[constructive proof]]s, ironically, the original proofs of his great topological theorems were not constructive,<ref>For a long explanation, see: J.P. Dubucs''[http://www.persee.fr/web/revues/home/prescript/article/rhs_0151-4105_1988_num_41_2_4094 L.J.E. Brouwer : Topologie et constructivisme ]'' Revue d'histoire des sciences V. 41 N°41-2 pp 133–155 (1988)</ref> and it took until 1967 for constructive proofs to be found.<ref>H. Scarf found the first algorithmic proof: M.I. Voitsekhovskii ''[http://eom.springer.de/b/b017670.htm Brouwer theorem]'' Encyclopaedia of Mathematics ISBN 1-4020-0609-8.</ref>
 
===Reception===
[[Image:John f nash 20061102 2.jpg|thumb|220px|left|[[John Forbes Nash|John Nash]] used the theorem in [[game theory]] to prove the existence of an equilibrium strategy profile.]]
The theorem proved its worth in more than one way. During the 20th century numerous fixed-point theorems were developed, and even a branch of mathematics called fixed-point theory.<ref>V. I. Istratescu ''Fixed Point Theory. An Introduction'' Kluwer Academic Publishers (new edition 2001) ISBN 1-4020-0301-3.</ref>
Brouwer's theorem is probably the most important.<ref>"... Brouwer's fixed point theorem, perhaps the most important fixed point theorem." p xiii V. I. Istratescu ''Fixed Point Theory an Introduction'' Kluwer Academic Publishers (new edition 2001) ISBN 1-4020-0301-3.</ref> It is also among the foundational theorems on the topology of [[topological manifold]]s and is often used to prove other important results such as the [[Jordan curve theorem]].<ref>E.g.: S. Greenwood J. Cao'' [http://www.math.auckland.ac.nz/class750/section5.pdf Brouwer’s Fixed Point Theorem and the Jordan Curve Theorem]'' University of Auckland, New Zealand.</ref>
 
Besides the fixed-point theorems for more or less [[contraction mapping|contracting]] functions, there are many that have emerged directly or indirectly from the result under discussion. A continuous map from a closed ball of Euclidean space to its boundary cannot be the identity on the boundary. Similarly, the [[Borsuk–Ulam theorem]] says that a continuous map from the ''n''-dimensional sphere to '''R'''<sup>n</sup> has a pair of antipodal points that are mapped to the same point. In the finite-dimensional case, the [[Lefschetz fixed-point theorem]] provided from 1926 a method for counting fixed points. In 1930, Brouwer's fixed-point theorem was generalized to [[Banach space]]s.<ref>{{cite journal |first=J. |last=Schauder |title=Der Fixpunktsatz in Funktionsräumen |journal=[[Studia Mathematica|Studia. Math.]] |volume=2 |year=1930 |issue= |pages=171–180 |doi= }}</ref> This generalization is known as [[Fixed-point theorems in infinite-dimensional spaces|Schauder's fixed-point theorem]], a result generalized further by S. Kakutani to [[multivalued function]]s.<ref>{{cite journal |first=S. |last=Kakutani |title=A generalization of Brouwer's Fixed Point Theorem |journal=Duke Math. Journal |volume=8 |year=1941 |issue=3 |pages=457–459 |doi=10.1215/S0012-7094-41-00838-4 }}</ref> One also meets the theorem and its variants outside topology. It can be used to prove the [[Hartman-Grobman theorem]], which describes the qualitative behaviour of certain differential equations near certain equilibria. Similarly, Brouwer's theorem is used for the proof of the [[Central Limit Theorem]]. The theorem can also be found in existence proofs for the solutions of certain [[partial differential equation]]s.<ref>These examples are taken from: F. Boyer ''[http://www.cmi.univ-mrs.fr/~fboyer/ter_fboyer2.pdf Théorèmes de point fixe et applications]'' CMI Université Paul Cézanne (2008–2009)</ref>
 
Other areas are also touched. In [[game theory]], [[John Forbes Nash|John Nash]] used the theorem to prove that in the game of [[Hex (board game)|Hex]] there is a winning strategy for white.<ref>For context and references see the article [[Hex (board game)]].</ref> In economy, P. Bich explains that certain generalizations of the theorem show that its use is helpful for certain classical problems in game theory and generally for equilibria ([[Hotelling's law]]), financial equilibria and incomplete markets.<ref>P. Bich ''[http://www.ann.jussieu.fr/~plc/code2007/bich.pdf Une extension discontinue du théorème du point fixe de Schauder, et quelques applications en économie]'' Institut Henri Poincaré, Paris (2007)</ref>
 
== Proof outlines ==
===A  proof using homology===
The proof uses the observation that the [[boundary (topology)|boundary]] of ''D''<sup>&thinsp;''n''</sup> is ''S''<sup>&thinsp;''n''&thinsp;−&thinsp;1</sup>, the (''n'' − 1)-[[sphere]].
 
[[Image:Brouwer fixed point theorem retraction.svg|thumb|right|Illustration of the retraction ''F'']]
The argument proceeds by contradiction, supposing that a continuous function ''f''&nbsp;:&nbsp;''D''<sup>&thinsp;''n''</sup>&nbsp;→&nbsp;''D''<sup>&thinsp;''n''</sup> has ''no'' fixed point, and then attempting to derive an inconsistency, which proves that the function must in fact have a fixed point. For each ''x'' in ''D''<sup>&thinsp;''n''</sup>, there is only one straight line that passes through ''f''(''x'') and ''x'', because it must be the case that ''f''(''x'') and ''x'' are distinct by hypothesis (recall that ''f'' having no fixed points means that ''f''(''x'') ≠ ''x''). Following this line from ''f''(''x'') through ''x'' leads to a point on ''S''<sup>&thinsp;''n''&thinsp;−&thinsp;1</sup>, denoted by ''F''(''x''). This defines a continuous function ''F''&nbsp;:&nbsp;''D''<sup>&thinsp;''n''</sup>&nbsp;→&nbsp;''S''<sup>&thinsp;''n''&thinsp;−&thinsp;1</sup>, which is a special type of continuous function known as a [[retract]]ion: every point of the [[codomain]] (in this case ''S''<sup>&thinsp;''n''&thinsp;−&thinsp;1</sup>) is a fixed point of the function.
 
Intuitively it seems unlikely that there could be a retraction of ''D''<sup>&thinsp;''n''</sup> onto ''S''<sup>&thinsp;''n''&thinsp;−&thinsp;1</sup>, and in the case ''n'' = 1 it is obviously impossible because ''S''<sup>&thinsp;0</sup> (i.e., the endpoints of the closed interval ''D''<sup>&thinsp;1</sup>) is not even connected. The case ''n'' = 2 is less obvious, but can be proven by using basic arguments involving the [[fundamental group]]s of the respective spaces: the retraction would induce an injective [[group homomorphism]] from the fundamental group of ''S''<sup>&thinsp;1</sup> to that of ''D''<sup>&thinsp;2</sup>, but the first group is isomorphic to '''Z''' while the latter group is trivial, so this is impossible. The case ''n'' = 2 can also be proven by contradiction based on a theorem about non-vanishing [[vector field]]s.
 
For ''n'' > 2, however, proving the impossibility of the retraction is more difficult. One way is to make use of [[Homology (mathematics)|homology groups]]:  the homology ''H''<sub>''n''&thinsp;−&thinsp;1</sub>(''D''<sup>&thinsp;''n''</sup>) is trivial, while ''H''<sub>''n''&thinsp;−&thinsp;1</sub>(''S''<sup>&nbsp;''n''&thinsp;−&thinsp;1</sup>) is infinite [[cyclic group|cyclic]]. This shows that the retraction is impossible, because again the retraction would induce an injective group homomorphism from the latter to the former group.
 
===A proof using Stokes's theorem===
To prove that a map has fixed points, one can assume that it is smooth, because if a map has no fixed points then convolving it with a smooth function of sufficiently small support produces a smooth function with no fixed points. As in the proof using homology, one is reduced to proving that there is no smooth retraction ''f'' from the ball ''B'' onto its boundary ''&part;B''. If ω is a volume form on the boundary then by [[Stokes Theorem]],
:<math>0<\int_{\partial B}\omega = \int_{\partial B}f^*(\omega) = \int_Bdf^*(\omega)= \int_Bf^*(d\omega)=\int_Bf^*(0) = 0</math>
giving a contradiction.
 
More generally, this shows that there is no smooth retraction from any non-empty smooth orientable compact manifold onto its boundary. The proof using Stokes's theorem is closely related to the proof using homology (or rather cohomology), because the form ω generates the de Rham cohomology group ''H''<sup>''n''−1</sup>(''&part;B'') used in the cohomology proof.
 
===A combinatorial proof===
There is also a more elementary [[combinatorial proof]], whose main step consists in establishing [[Sperner's lemma]] in ''n'' dimensions.
 
===A proof by Hirsch===
There is also a quick proof, by [[Morris Hirsch]] (indeed by [[Elon Lages Lima]]), based on the impossibility of a differentiable retraction.  The [[indirect proof]] starts by noting that the map ''f'' can be approximated by a smooth map retaining the property of not fixing a point; this can be done by using the [[Weierstrass approximation theorem]], for example.  One then defines a retraction as above which must now be differentiable.  Such a retraction must have a non-singular value, by [[Sard's theorem]], which is also non-singular for the restriction to the boundary (which is just the identity).  Thus the inverse image would be a 1-manifold with boundary. The boundary would have to contain at least two end points, both of which would have to lie on the boundary of the original ball—which is impossible in a retraction.
 
Kellogg, Li, and Yorke turned Hirsch's proof into a constructive proof by observing that the retract is in fact defined everywhere except at the fixed points. For almost any point, q, on the boundary, (assuming it is not a fixed point) the one manifold with boundary mentioned above does exist and the only possibility is that it leads from q to a fixed point. It is an easy numerical task to follow such a path from q to the fixed point so the method is essentially constructive. Chow, Mallet-Paret, and Yorke gave a conceptually similar path-following version of the homotopy proof which extends to a wide variety of related problems.
 
===A proof using the ''oriented area''===
A variation of the preceding proof  does not employ the Sard's theorem, and  goes as follows. If ''r'' : ''B''→∂''B'' &nbsp; is a smooth retraction, one considers the smooth deformation ''g<sup>t</sup>(x)&nbsp;:=&nbsp;t&nbsp;r(x)&nbsp;+&nbsp;(1-t)x,'' and the smooth function
:<math>\varphi(t):=\int_B \operatorname{det} D g^t(x) dx</math>
Differentiating under the sign of integral it is not difficult to check that ''&phi;&prime;(t)=0'' for all ''t'', so ''&phi;'' is a constant function, which is a contradiction because ''&phi;(0)'' is the ''n''-dimensional volume of the ball, while ''&phi;(1)'' is zero.  The geometric idea is that ''&phi;(t)'' is the oriented area of ''g<sup>t</sup>(B)'' (that is, the Lebesgue measure of the image of the ball via ''g<sup>t</sup>'', taking into account multiplicity and orientation), and should remain constant (as it is very clear in the one-dimensional case). On the other hand, as the parameter ''t'' passes form ''0'' to ''1'' the map ''g<sup>t</sup>'' transforms continuously from the identity map of the ball, to the retraction ''r'', which is a contradiction since the oriented area of the identity coincides with the volume of the ball, while the oriented area of ''r'' is necessarily ''0'', as its image is the boundary of the ball, a set of null measure.
 
===A proof using the game hex===
A quite different proof given by [[David Gale]] is based on the game of [[Hex (board game)|Hex]].  The basic theorem about Hex is that no game can end in a draw.  This is equivalent to the Brouwer fixed-point theorem for dimension 2.  By considering ''n''-dimensional versions of Hex, one can prove in general that Brouwer's theorem is equivalent to the [[determinacy]] theorem for Hex.<ref>{{cite journal|author=David Gale |year=1979|title=The Game of Hex and Brouwer Fixed-Point Theorem | journal=The American Mathematical Monthly | volume=86 | pages=818–827|doi=10.2307/2320146|jstor=2320146|issue=10}}</ref>
 
===A proof using the Lefschetz fixed-point theorem===
The Lefschetz fixed-point theorem says that if a continuous map ''f'' from a finite simplicial complex ''B'' to itself has only isolated fixed points, then the number of fixed points counted with multiplicities (which may be negative) is equal to the Lefschetz number
:<math>\displaystyle \sum_n(-1)^nTr(f|H_n(B))</math>
and in particular if the Lefschetz number is nonzero then ''f'' must have a fixed point. If ''B'' is a ball (or more generally is contractible) then the Lefschetz number is one because the only non-zero homology group is ''H''<sup>0</sup>(''B''), so ''f'' has a fixed point.
 
===A proof in a weak logical system===
In [[reverse mathematics]], Brouwer's theorem can be proved in the system [[Weak König's lemma|WKL<sub>0</sub>]], and conversely over the base system [[reverse mathematics|RCA<sub>0</sub>]] Brouwer's theorem for a square implies the [[weak König's lemma]], so this gives a precise description of the strength of Brouwer's theorem.
 
== Generalizations ==
The Brouwer fixed-point theorem forms the starting point of a number of more general [[fixed-point theorem]]s.
 
The straightforward generalization to infinite dimensions, i.e. using the unit ball of an arbitrary [[Hilbert space]] instead of Euclidean space, is not true. The main problem here is that the unit balls of infinite-dimensional Hilbert spaces are not [[compact space|compact]]. For example, in the Hilbert space [[Lp space|&#8467;<sup>2</sup>]] of square-summable real (or complex) sequences, consider the map ''f'' : &#8467;<sup>2</sup> → &#8467;<sup>2</sup> which sends a sequence (''x''<sub>''n''</sub>) from the closed unit ball of &#8467;<sup>2</sup> to the sequence (''y''<sub>''n''</sub>) defined by
:<math>y_0=\sqrt{1-\|x\|_2^2}\qquad\mbox{ and }\qquad y_n=x_{n-1}\quad\mbox{ for }\quad n\geq 1.</math>
It is not difficult to check that this map is continuous, has its image in the unit sphere of &#8467;<sup>&thinsp;2</sup>, but does not have a fixed point.
 
The generalizations of the Brouwer fixed-point theorem to infinite dimensional spaces therefore all include a compactness assumption of some sort, and in addition also often an assumption of [[Convex set|convexity]]. See [[fixed-point theorems in infinite-dimensional spaces]] for a discussion of these theorems.
 
There is also finite-dimensional generalization to a larger class of spaces: If <math>X</math> is a product of finitely many chainable continua, then every continuous function <math>f:X\rightarrow X</math> has a fixed point,<ref>{{cite journal|author=Eldon Dyer |year=1956|title=A fixed point theorem | journal=Proceedings of the American Mathematical Society| volume=7 | pages=662–672|doi=10.1090/S0002-9939-1956-0078693-4|url=http://www.ams.org/journals/proc/1956-007-04/S0002-9939-1956-0078693-4/home.html|issue=4}}</ref> where a chainable continuum is a (usually but in this case not necessarily [[Metric space|metric]]) [[Compact space|compact]] [[Hausdorff space]] of which every [[open cover]] has a finite open refinement <math>\{U_1,\ldots,U_m\}</math>, such that <math>U_i \cap U_j \neq \emptyset</math> if and only if <math>|i-j| \leq 1</math>. Examples of chainable continua include compact connected linearly ordered spaces and in particular closed intervals of real numbers.
 
The [[Kakutani fixed point theorem]] generalizes the Brouwer fixed-point theorem in a different direction: it stays in '''R'''<sup>''n''</sup>, but considers upper [[hemi-continuous]] [[correspondence (mathematics)|correspondences]] (functions that assign to each point of the set a subset of the set). It also requires compactness and convexity of the set.
 
The [[Lefschetz fixed-point theorem]] applies to (almost) arbitrary compact topological spaces, and gives a condition in terms of [[singular homology]] that guarantees the existence of fixed points; this condition is trivially satisfied for any map in the case of ''D''<sup>&thinsp;''n''</sup>.
 
==See also==
* [[Fixed-point theorem]]s
* [[Banach fixed-point theorem]]
* [[Schauder fixed-point theorem]]
* [[Lefschetz fixed-point theorem]]
* [[Tucker's lemma]]
* [[Kakutani fixed-point theorem]]
* [[Topological combinatorics]]
* [[Nash_equilibrium#Alternate_proof_using_the_Brouwer_fixed-point_theorem|Nash equilibrium]]
 
==Notes==
{{reflist|35em}}
 
==References==
*{{cite journal |first=S. N. |last=Chow |first2=J. |last2=Mallet-Paret |first3=J. A. |last3=Yorke |title=Finding zeroes of maps: Homotopy methods that are constructive with probability one |journal=Math. of Comp. |volume=32 |year=1978 |issue= |pages=887–899 |doi=10.1090/S0025-5718-1978-0492046-9 }}
*{{cite journal|author=Gale, D. |year=1979|title=The Game of Hex and Brouwer Fixed-Point Theorem | journal=The American Mathematical Monthly | volume=86 | pages=818–827|doi=10.2307/2320146|jstor=2320146|issue=10}}
*{{cite book |first=Morris W. |last=Hirsch |title=Differential Topology |location=New York |publisher=Springer |year=1988 |isbn=0-387-90148-5 }} (see p.&nbsp;72–73 for Hirsch's proof utilizing non-existence of a differentiable retraction)
*{{cite book |first=V. I. |last=Istrăţescu |title=Fixed Point Theory |location= |publisher=Reidel |year=1981 |isbn=90-277-1224-7 }}
*{{cite book |editor-first=S. |editor-last=Karamadian |title=Fixed Points: Algorithms and Applications |location= |publisher=Academic Press |year=1977 |isbn=0-12-398050-X }}
*{{cite journal |first=R. B. |last=Kellogg |first2=T. Y. |last2=Li |first3=J. A. |last3=Yorke |title=A constructive proof of the Brouwer fixed point theorem and computational results |journal=SIAM J. Numer. Anal. |volume=13 |year=1976 |issue=4 |pages=473–483 |doi=10.1137/0713041 }}
*{{springer | title=Brouwer theorem | id=B/b017670 | last=Sobolev | first=V.&nbsp;I. | author-link=<!--Vladimir Ivanovich Sobolev-->}}
 
==External links==
* [http://www.cut-the-knot.org/do_you_know/poincare.shtml#brouwertheorem Brouwer's Fixed Point Theorem for Triangles] at [[cut-the-knot]]
* [http://planetmath.org/encyclopedia/BrouwerFixedPointTheorem.html Brouwer theorem], from [[PlanetMath]] with attached proof.
* [http://www.mathpages.com/home/kmath262/kmath262.htm Reconstructing Brouwer] at MathPages
 
{{DEFAULTSORT:Brouwer Fixed Point Theorem}}
[[Category:Fixed-point theorems]]
[[Category:Continuous mappings]]
[[Category:Mathematical and quantitative methods (economics)]]
[[Category:Theorems in topology]]
 
{{Link GA|fr}}

Revision as of 17:16, 23 February 2014

. Mbt Masai Singapore

Fanhost has been around and Mbt Masai Singapore we are getting ready to launch a significantly needed redo. This will happen in Feb. Since the current site has a lot of things broken, I would expect to fall below 30k after launch. Type "parkour" into YouTube and thousands of homemade videos of people performing acrobatic feats in cityspaces will likely come up. From the streets of Mumbai to the abandoned government buildings of Cairo, practitioners have Nike Air Australia started to turn their cities into playgrounds.From this growing interest, the World Freerunning Parkour Federation (WFPF) was born. Leading the charge of the parkour movement, the WFPF is streamlining the process by which one goes from curious to devoted, by providing a platform for parkour athletes to satisfy fellow practitioners in their area online.

Are you aware anyone who has lost his job in a telecom company because of the license cancellation? I understand each of my 50 odd excolleagues who have lost jobs. They have families and young children to look after. They will not be explain to their kids why they had to change their schooling to a cheaper one because some persons are in jail for corruption or they had to move to a small house or sell off their newly purchased car.

This agreement will be governed by and construed depending on the laws of the State of Florida, because it is applied to agreements entered into and performed within that State. Any action brought to enforce this agreement or matters related to the site shall Ray Bans Usa be brought in either their state or Federal Courts of Florida. Or no provision of this agreement is deemed void, unlawful or otherwise unenforceable for any reason, that provision will be severed from this agreement and also the remaining provisions of this agreement shall remain in force.

The problem with these membership sites is that the buyers always have PLR articles sitting on their hard drives. It not easy to take these PLR content and switch them into your own completely unique content as it is very time consuming to rewrite the articles.Spin Article Pro is a service that helps you spin your PLR articles into completely unique content you can use for distribution campaigns and SEO. When you become a member, you can access over 7000 PLR articles that you should choose from (if you don have any yourself).

The website owners of different businesses definitely might have original material here to attract customers to their site. Says the owner of the site, On the internet at present, there are thousands of different markets which companies are trying to break into, and within our content, you are sure to find many free articles that are relevant to the niche you are aiming your website towards. Both directories Buy Nike Shoes Cheap .

including boating Discount Oakley Sunglasses Australia

Some of these sequences stretch over a 40year Discount Oakley Sunglasses Australia period and document the advancement of their diseases. In other cases, it became essential to include their voices so that they could explain that despite their seemingly healthy appearance, they were indeed suffering. This led me to include oral history transcriptions in the exhibition narrative and also to produce a video installation piece so that their testimonies could be heard..

Though not all states are yet participating in implementing the system, neighboring states to Indiana are involved. Michigan is probably furthest along, and Adventure Cycling reports that they're actually ready to submit one route for approval, USBR 20. This is an eastwest way starting north of Detroit , progressing towards the Ludington Ferry across Lake Michigan, and eventually heading to Portland , Oregon and the Northwest Coast ..

123people finds photos related to the name pair Jon Schmuland by using other search engines in real time. The preview from the Nike Air Max 2014 Cheap displayed image is associated with the initial file. A click on the icon next to the image preview will lead you to the source.

Tried to see if we're able to piece all those things together into something which fits us, that fits society and culture and our bodies Womens Nike Free Run 3 and the sport. I consider it as a wonderful opportunity to create something that is sustainable that wasn there before. In order to do that, there going to have to be collaboration at different levels.

Based on where you are coming from, there are several methods to reach Badin Lake Campground. Most of them take a lot of twists and turns, but the drive is worth it. This lakeside campground is the pride of the Uwharrie National Forest and it's a best bet for tent campers who would like a quality campground accompanied by water and land recreation, including boating, fishing, and hiking.

Gee. Made 2 posts and now they are gone (??) Hmmmm. Anyway, glad Nike Shop Charlestown to read the updates here, because when I was outside the restaurant and could see the accident site, they had already put tarps around everything and were trying to extricate the driver from the Mustang.

Liberhan commission is clear as far as Babri Masjid demolition is concerned it is a clear systematically planned attack just topple Congress government then and that is the main plank than anything else. Even today after so many years, theyt talk of Ayodhya during elections just toi cheat people. The majority of Sadhus really do not understand politics but get inflamed when Ram is attacked and nothingf many nothing less.

they should be left alone. Nike Stockists Brisbane

To summarize: Electrical wires must be completely out of reachMany household vegetation is poisonous to rabbitsOtherwise, how you proof depends a little on the personality of your rabbit. Most rabbits have LARGE cages. Wild Rabbits really should not be kept in cages or in any sort of enclosure, they should be left alone.

This primary teaming of Doris Day and Rock Hudson is a delightful, visually beautiful comedy oldfashioned but hardly dated. The two stars make a charming, shiny couple (it's not hard to see why they were so popular within Nike Stockists Brisbane their time) and Thelma Ritter steals the show inside a needless but funny supporting Womens Air Max 2012 role. The only problem you may have is that the course of the plot appears to be thoroughly predetermined from the first frame, however the film does a pretty good job of delaying the inevitable.

At the moment they do not have sewer. They had sewer however the California environment or something came through all of the area parks and shut down the sewer dumps. No WiFi, cable television is great, electric is good, water is city water. I wasn't worried that by saying this I'd put myself unemployed; I was pretty secure within the demand for my skill set and I felt confident that as long as the company was growing, my role would evolve and I could bring value to the organization in new and exciting ways. And I'm thrilled that for 5 years, that's exactly what happened. In 2009, I was proud to be appointed the youngest member of the Senior Leadership Team.

Gunman Adam Lanza's motives remain a mystery when he went on a murder rampage last December, killing his mother. Gunman Adam Lanza acted alone and didn't conspire with anyone when he killed his mother, Nancy, in. The Australian. The way turns north and climbs steeply for 1/4 mile before leveling out. 3 miles in, the way turns northeast and follows and old jeep trail. The trail ends at the Saddle Mountain Trail Nike Shoes Free Shipping Australia 91 approximately 1/2 mile west of the Mormon Grove Trailhead..

To date most of their production has gone to the domestic market but increasingly they are looking abroad. Mindray has already had success in western markets, vacation it is now the number one ultrasound supplier but other medication is also looking abroad as economic concerns push customers in developed markets for the equipment in the value segment. Where better to find distribution deals than at the RSNA?.

Rodeway Inn, room 112. Augusta, Georgia. $260 per night. It so happens that you're not at all on Cheap Ralph Lauren Polo Australia the frame till YWT comes up at 10 which would be a tad late on a day of important coverage like today. So my request to you is on such impt days please occupy your discussion beginning 8 pm on what hot itself. Am i correct in my assessment?.

long term Ray Bans Glasses

That alone kept tabloids busy for weeks following the show aired. When I hear her voice-over the phone then, I decide to skip the small talk and obtain right to it. It as if she unravelling, I tell her, letting her shoulders slump, so to speak, and in the process, having a bit of fun. Ray Bans Glasses

A December 2000 New York Times dispatch from Bangalore, India, reported the new weapon of choice in motherinlaw attacks on wives is kerosene and that hospital burn wards are full of "thousands" of grotesquely disfigured wives whose primary sin was either to bring too paltry an internet worth to the marriage or to underperform household chores.In October, Dee Blyth Oakley Jawbone reported a burglary of her home in Chadwell Heath, Essex, England, in which thieves had helped themselves as to the they thought was her stash of cocaine ("charlie" in local slang), abandoning the distinctive residue of cocaine Oakleys Online "lines" on a table after lifting several electronic appliances and jewellery worth about $3,500. Among the forcredit curriculum now in Oberlin College's Experimental College is a course on the "life and times" of Drew Barrymore. A baby was "born" in the snowy debris of the auto collision, healthy aside from a skinned knee, rescued with a paramedic who found it attached by its umbilical cord following the pregnant mother's abdomen was fatally sliced in half by the jagged windshield (Louisville, Ky.).

As a side note on the health aftereffect of the diterpenes found in regular coffee, it was found that by simply drinking filter coffee, none of those effects on cholesterol levels or even the liver took place. The coffee filter removed the offending diterpenes. And levels of Nike Store Australia these diterpenes in instant coffee are low..

Cavangal you poor thing, seems like things are very stressful for you personally right now. I think having a healthier diet generally (long term), we all need to learn that stressful times happen, instances when its hard to plan eating happen and these things happen to everyone. The key is (I believe) to do your best this weekend, maybe watch your portion sizes, but don't let food be one more thing to get stressed about this weekend you have to mind yourself.

"Your rsum is your Sunday best," he states. "Twitter gives recruiters an unprecedented look underneath the bonnet. You can see how professional or unprofessional people are. 123people never copies or stores any image files. If you are Janelle Witzel and don't want your image to become displayed on 123people, please delete the look from the original source. Using our proprietary search algorithm, you can find comprehensive and centralized person related information consisting of public records, phone numbers, addresses, images, videos and emails.

I had a niner MCR once Cheap Ray Bans Brisbane

The team's winning concept seeks to produce spaces to inspire humantohuman connection and also to blur the boundary between the community and the traditional hospital setting. The program looked to transform the process of giving and receiving care by reconceptualizing people's relationship to both technology and nature. Spatial elements of the design serve to improve communication and make up a sense of community between patients and practitioners.

I had a niner MCR once (busted the chain stay) It feels much like that. Long top Cheap Ray Bans Brisbane tube, reasonable length in the chain stay, rolls and Mbt Shoes Online rolls and rolls. It handles the Oakley Sunglasses Stores Brisbane corners much better than the MCR (Probably the crisp aluminium frame accountable for that) but otherwise, it is a cheaper version of a niner..

Having run SolidWorks/Ansys/etc on enough number of laptops, seriously, if you're running Core 2 mobile, performance Hollister Coats is going to be pitiful regardless of clock speed. Maybe the 2.66GHz+ parts are faster, but anything this side of two.5GHz is pretty godawful slow. I'm not stating that the MBP13 won't be faster than the Air, I'm just saying that it won't be faster by enough to warrant the decision hinging either way on performance characteristics that regardless of your choice are so far behind most decently powerful notebooks anyways..

If you avoid the rush times at the laboratory you get faster service and the results of the soil test in time to serve as a guide for buying and applying fertilizer. For fall plantings take samples in May June July and early August. For spring plantings collect the samples the second part of October November December and January.

I stayed at le Meridien hotels as i lived in Europe and was curious to come here. Despite my bad cold, I'd a wonderful time here. Received a room upgrade and resided in a large room with a tremendous view along with a gigantic bed. Holland's views may have validity. Adolescents have experienced an anger attack that involved threatening violence, destroying property or participating in violence toward others at some point in their lives. These severe attacks of uncontrollable anger are much more common among adolescents than previously recognized, a new study led by researchers from Harvard School of medicine finds..

The kitchen designers of the present are all about streamlining lines, and hiding appliances. Really highend modern domestic kitchens seem like they don't even have appliances, save maybe a builtin cook top. Please don't take this assessment of contemporary kitchen design as a command, since this writer cooks perfectly good food in an aestheticallychallengedbutfunctional kitchen, and has no qualms about this.

Luvox Oakleys Online

Casinos are most commonly built near or combined with hotels restaurants retail fully stand up. Personalize and download yours today. Create amazing greeting cards, scrapbooks, slideshows, birthday cards, and more using your photos, videos, and music with Smilebox.

There are a variety of diseases, some serious, others not so serious which are either incurable or can be cured only with great difficulty. Victims of such diseases run from one door to a different in search of remedial medicines. There are hardly any doctors who would publicity accept Oakleys Online the inadequancy or futility of their medicines in such diseases.

Explain that if she thinks her breasts are what have kept you interested, it isn't the case. At some point, one or more of Barbour Stockists Perthshire her implants may need to be replaced, and she might opt for smaller ones. It's been a fantastic adventure, but I feel drawn home.

While the Nets will be assessing the new building, so, too, perhaps will the Islanders. The hockey team's owner, Charles Wang, has for several years been pushing for significant upgrades to the 40yearold Coliseum or for a new building once the Islanders' lease expires, in 2015. Last summer, voters in Nassau County soundly Cheap Fred Perry Polo rejected plans for any new arena and practice center, and Mr.

Taking part in buzz is fun. While mass marketers often try to manipulate their customers into buzzing in a manner that benefits them, most of the time, we glad to be doing it, glad to be a part of something with excitement and energy. The Kony video spread largely since it was already spreading.

Salon Supplies Interiors will no longer have an inventory of used salon furniture and used salon equipment. For any current list of used salon equipment, used barber equipment, used spa equipment, used salon furniture, and used salon accessories, call Salon Supplies Interiors at (610) 5862300 and request David. All used salon furniture, used salon equipment, used barber equipment, used spa equipment, and used salon accessories are sold "asis." There are no refunds or exchanges when a transaction is completed.

The best remedy for homesickness is to get out and do things. Meet people, see new things, and go explore the town. Do anything you can think of to keep Ray Ban Wayfarer Australia Online you occupied, and before you know it that silly homesickness will be gone.. Treatment of Anxiety DisordersFor SAD, GAD, Social Phobia and OCD, two treatments have been utilized: a form of therapy called CognitiveBehavioral Therapy (CBT) and medication, particularly the Specific Serotonin Reuptake Inhibitors (SSRIs). A number of studies have shown these medications are helpful in treating OCD. A large multisite study sponsored through the NIMH showed that fluvoxamine (Luvox) was efficacious in treating SAD, GAD and Social Phobia..

activity partners Mbt Nz

Many owners will attempt to avoid the rental application process by telling the association that the occupant is merely a guest. However, a real guest is someone who stays on the shortterm basis and does not pay rent. If that is the case for your guests, then the rental restriction should not be a problem.

We also know that there are anywhere from 10% to 15% of employees who suffer from depression and who are not absent but who are actually are at work. This group of employees is incapable of working at peak performance. They are at work and they fall squarely into the disengaged or presenteeism category of employees.

Also, during the time of Youku's IPO, the company was actually the No. 2 player behind Tudou. But the Tudou CEO was going through a divorce, which held up its IPO. If the government begins to monitor every Australians internet usage, we may as well become a communist country and get it over and done with. The next thing they'll want to block is anyone material on the internet. Which will consequently take away our right as online users to view what we wish.

Should there be a bacterium or other parasite involved, make use of a product like DERMagic Hot Spot Lotion to deal with this. Be careful what sprays you may use to freshen up your house (these are not only not good for pets, but not good for children or pets. Air your house often and if you are in Mbt Nz an apartment then air out the apartment (put a screen up on the window if possible so your Cheap Nike Air Max 1 cat can't get out.

Community Key Nike Shoes Australia Sale to Reducing Youth Gang Violence, Expert Says: Community Involvement, Relational Networks Key to Reducing YouthRelated Gang Violence in Los Angeles, says City's Head of Gang Reduction and Youth Development Office. "Youth in a very diverse environment continue to solve conflicts through violence," said Guillermo Cspedes, director of the Mayor's Office of Gang Reduction and Youth Development for that City of Pandora Charms Australia Sydney Los Angeles. "We kept doing exactly the same thing and expecting a different result.".

For many people, online personals vastly improve the quality of their lives. They could use these directories to find individuals who they have a lot in common with and establish bonds with them be it as online pen pals, friends, activity partners, or for dating or marriage. Using adult personals directories might help streamline the process of surfing the a lot of personals Web sites out there, which help you to refine what you are looking when ever browsing profiles..

Despite knowing the impact article marketing can have on a business, countless entrepreneurs fail with this particular form of marketing. Whether it is the writing or submitting, there are several ways you can ruin a perfectly good advertising campaign. Fortunately, Mass Article Control will help you get more traffic, drastically increase your search engine ranking positions, all while making much more money..

" Cheap Air Max Australia

When you do let your child watch TV, watch it with him and talk to him about situations that arise: "That wasn't a very good way for him to get what he wanted, was it?" (The American Academy of Pediatrics recommends that children under the age of 2 watch no TV whatsoever.)Provide physical outlets. You might find that unless your toddler gets a chance to burn off his abundant energy, he's a terror in your own home. Sometimes a child's aggression requires more intervention than the usual parent can provide.

Molds: Molds need high humidity levels and a surface on which to grow. Common areas to check for mold are garbage containers, food storage areas, upholstery and wallpaper. Molds also commonly grow in damp areas for example basements, shower curtains, window moldings and window air conditioning units.

You are a professional.Why are Cheap Air Max Australia we referring to this now? Ironically, the wonder industry has an image problem. So many people don know or understand us. This isn a new problem, but an especially Nike Air Max Tn Australia dangerous one today.Misperceptions by reality TV and sensationalist media, and compounded by the economy led to legislation to eliminate or limit cosmetology licensure.

So grab you copy now. The coverage price is Rs.250/ and Rs.310/ by regd. Book post.. We even were built with a meeting one day to plan our math for the rest of the year and we got into this heated discussion about our kids having recess. The instructional coach stated that instead of taking a recess we could possess the kids count and do 10 jumping jacks or hop 10 times while counting to ensure that we continue the academics. We agreed to disagree when the AP walked in so we asked her if we might take a recess.

Limitation "The secret is limiting alcohol intake so less acetaldehyde is created," says Emsley. If skipping rounds is unacceptable, choose 4% brews, like Carling. "You'll ingest 2.3 units per Buy Air Max Online round compared to 3 units on premium pints," says Budka. A notebook is the first and most thing you will need for your home office. Most people who work at home prefer laptops to desktop computers because of the mobility that a laptop offers. When you get bored working from the same spot in your house, you can easily bring your laptop out on the porch or perhaps your backyard.

Gunderman, by far the brightest spark in the cast, creates indignant fire along with a thrilling atmosphere from his Iago's stirring soliloquies. As he stands there, stroking his beard, peering out into the crowd from behind a sly countenance, without doubt contemplating more irksome mischief, you are feeling that this was how Shakespeare was meant to be played. He is the grand knight, manipulating the chessboard from on high; if Nike Air Force 1 07 only the other pieces were even cut from a quarry in the same region, however roughly formed.

chilli powders. Cheap Nike Air Max Shoes Australia

Whenever it comes to purchasing a property easily obtainable in Spain, a merchant of the secondhand bag must help different documents including; the appellation achievement of the property, grounds which dominion taxes are Cheap Nike Air Max Shoes Australia around date, and echt estate taxes screw been postpaid. She stared at her manager with worry. If you are not cautious you are able to buy a property with debts or with extensive renovation bills.

Universal Integrated Corp Consumer Products sells household maintenance systems as well. P also has huge competition in the dog food market. Their Iams brand has a fair niche in the dog and cat food market, but has major competitors in brands like Pedigree and Friskies.

5. In some cases, you can even grant access to a computer to everyone, Hollister Store and they can see what happens on it. The more advanced the service, the more actions can be done, such as changing documents and typing from other PC?s connected to the web conference.

Deputy Tehsildar Abdul Halim Ansari, whose 30yearold son Hafiz Asif Abdul Halim seemed to be killed in the police firing, is angry with the system of which he is part, and will not go back to work. System is biased. How do i go back and work there? My son had not done anything wrong, Mr.

While you are playing the field, no self respecting woman will take you seriously. This is life. For example: say you have 2 people. Back to Even is a Jim Cramer book that argues the way to stop Hollister Store Sydney the hemorrhaging is not to blindly yoke your retirement funds to some managed indexed fund, but to actively manage a portfolio of your own choosing. Cramer warns against acting on emotion instead of research. He encourages you to definitely consider stocks that pay dividends.

Towards the northeast is the Urban Planning Exhibition Hall, with a scale model of Shanghai fastchanging cityscape and exhibits on its evolution. To the northwest is the Shanghai Art Museum in the horse track former clubhouse. It has displays of contemporary art and it is the site of the Shanghai Biennale, held in evennumbered years..

The training had obviously paid off, when i discovered when we reached the Trail's culmination in the Corbett Museum. Adjoining the museum, off the RamnagarNainital main road, the Samiti was building a souvenir shop rented to it by the Forest Department. Here, they sold products from the village including tieanddye scarfs and bedsheets, turmeric, maize, chilli powders.

These errors fall under two large categories: misunderstandings about how exactly evolution works and unfounded assumptions about how exactly paleolithic humans lived. The first area is her speciality, while offering a lively, lucid illustration of the intricacies of the allimportant natural process. When it comes to the second category, the anthropological Nike Free 4.0 Womens aspect of the problem, Zuk treads more gingerly.

his nursepractitioner Nike Air Max 2012 Womens

What kind of Core support do these grants purchase from the organizations that receive them?[The Washington Post article includes a detailed listing of the Gates Foundation grants forwarded to the Common Core.]In Donna Garner 5.13.13 article on this subject, she wisely stated:While you read through these grants that Bill gates gave to "Universities. Unions. State education departments.

"It's like shooting lightning bolts," he tells Stacey Gallagher, his nursepractitioner, a greeneyed beauty Dicky calls his "girlfriend." He mentions to her that another doctor gave him a larger dosage of the oxycodone than Gallagher had prescribed. "I was crying like a baby and then whoosh," he says. "Wow, what a miracle." He is careful to note what his recent efforts at pain alleviation have not included: crack, which he has smoked off and on for years.

According to the old saying, good fences alllow for good neighbors. Market for Fencing, a market research report from SBI, fences alllow for good business, too. Considering import and export data, SBI estimates Nike Air Max 2012 Womens the fencing market, comprising metal, vinyl and wood fencing, to become at $3 billion in 2005.

Blood spots are more commonly found in brown eggs. This is because twofold. Firstly, Fake Oakleys Outlet the genetics of brown birds predisposes them to lay more eggs with blood spots inside them. This article has been cited by other articles in PMC.In health, the body content of iron is maintained within fairly narrow limits to provide sufficient iron for synthesis of ferroproteins required for oxygen transport and catalysis yet avoid the toxic effects of excess. Early experiments Beats By Dre Jb Hi Fi provided tantalising evidence for any humoral regulatory factor. Serum obtained from iron repleted rats inhibited iron absorption in normal rat Barbour Stockists Perthshire duodenum.1 For several years this putative factor remained elusive.

The majority of the MBBS colleges and universities provide excellent hostel facilities with mess. I also had the problem of needing an adaptable schedule and distance learning turned out to a much better experience than I thought. You should look into it. "I got around the manlift and rode up to the top floor of this mill, and, my God, I'd never witnessed anything like it in my life. I suppose a guy has seen a dust storm before. The dust was probably three, four inches deep.

Seventy years ago, a struggling professional hockey team, eager for fans, victories and a new arena, adopted Brooklyn since it's temporary home for one year. Now, another New York team, desperate for the very same things, will go to Brooklyn for one night. The New York Islanders, tied for last place in their conference and playing within the obsolete Nassau Coliseum, are scheduled to play an exhibition game against the Nj Devils in Brooklyn's Barclays Center on Oct.

Retrieved from "https://en.formulasearchengine.com/index.php?title=Brouwer_fixed-point_theorem&oldid=218774"