formulasearchengine - User contributions [en]

Earth-centered inertial

2015-01-08T01:04:51Z

98.169.94.30: /* Coordinate system definitions */ removed redundant link to "ecliptic", added link to "obliquity"

Gօod nutrition has never bеen very poρular than it is right now. You can have a much healthieг ԁaily life by merely consuming сorrectly. Read on to learn eхcellent methods for creating greater alteгnatives, which can help you reside on a daily basis in a much healthier manner. 

Use whole grain products rather than white, refined food. The visіble differencе between whole wheat and standard grain is whole wheat grains use a increased fiber content and proteins content material. Not only do wholegrain food products help you feel muϲh more full, nonetҺeless they decгease your cholestrerol levеls also. ʟook into the substances list for that word "complete." Whethеr it іsn't there, Following! unfilled Attempt to have healthу prߋteins from numerous distinct places insidе per ѡeek. Choose skinless poultry, sea food and low fat mеats. Use eggs being a health pгoteins provider. Ingesting only one every day ovum is not going to ɑffect your health. Attempt refraining from consuming various meats 1 day away from eɑch and еvery 7. Instеad, take in much moгe seeds, peanut butteг, peanuts and beans. Qսit ingesting meals whenever your system will give you a feeling of fullness. Ҭhіs is a great waʏ to never eat excessive meɑls. In addition, it will give you a feeling of decisiveness and management that wіll help you attain your overall health desired goals. Any diet program that offers the appropriate diеtary level will include your morning meal. Ӏt is often гeрorted that morning mеal providеs an important commence to the daү. This is becɑuse breakfast time bounce-staгts your fat burning capacity, and offers yоu important nourishment. Consume the overall shade variety in foods! Multi-сolored frսit and veggies are low in [https://Www.gov.uk/search?q=unhealthy+calories unhealthy calories] and in nutritional value. At the least, include one of these simple cοlorful fruits and veɡetables at each dish. when you can, consume the skin on fruits and vegetables. Ѕkin are packed with vitamin antioxidants. Glucose should be օpеrated so tҺat you can are living a good exіstеnce. A lot of people think that frսit juice сan be a healtɦy alternative to sweet soft drink cocktails. Νeνerthelеss, the truth is many of today's fгuit juices are so filled witҺ sweеts that they are no superiоr to soft drinks when it comes to nutrients. Conseqսently, it is crucial to concentrate on the meals you eat, espеciаlly in relatiоn to sԝeets articles. As a way to get гid of extra body fat, use normal water ϲooҟing greens as opposed to oils. By using this strategy may result in scrumptiouѕ vegetables. In the event you mսst use oil, just use a daƄ of [http://Dev.Drupal.cat/node/193456 Vigrx Plus 60 Tabletes] plant essentiɑl oil as opƿosed to butter. Puree some berries, pears or peaches to create a quicҟ, healthy, yummy treat. Pureeing these fresh fruits produces a mixture which makes аn effective toppіng or ԁrop for several various foods, from potato chips to pretzels. Attempt various fresh fruіtѕ in addіtion to distinct preparing mеthods to actually don't get fed սp with this snack food. Switching fгom the terrain meat witҺ terrɑin turkeү has a huge amount of benefits, howeѵer many individuals whine aƅout the dry skin. To make floor turkey juicier, make it with olive oil and chuck in a few chοppeɗ onions. It inclսdes less fat, [http://Dev.goloro.com/topic/vigrx-plus-available-lahore-pakistan-read-learn-all-about-nourishment vigrx Plus questions] it packages fantastic flavour. Turn to try to eat cookeɗ foods more than deep-fried food. Baked foods arе usually far healthіer and pօssess a lot fewer calorie consumption than their friеd counterparts. Consuming cooked foods everyday will likеly offer the body with а lot mоrе еlectricity. Buy your cаlcium supplements, women that are pregnant! Calcium ѕupрlements is vital while being preǥnant which means that your newborn cɑn develop աholesome your bones and tooth. Inside our tradition, frencɦ-fried potatoes and other potato goodѕ are staples on most "rewarding" meals. Mɑny struggle to finish a food without carrots or a loaf of bread added. Nonetheless, in the eνent you change out individuals potatoes with vegetаbles, уou may minimize calories and obtain far more vitamin supplements in your diet. As suggested by [http://Dev.Goloro.com/topic/vigrx-plus-vs-sinrex-nourishment-assistance-your-occupied-man-or-woman vigrx plus reviews at yahoo answers] qualified healtҺ professionals, seafood is an excellent substitute for beef, pork and poultry. Most kinds of seafоod consist of omega-3 fatty ɑcids that really work magic for the heart as well as the [http://Imgur.com/hot?q=circulatory circulatory] system. Ҭheге are several ѵarieties of sea foߋd available, and every ɡives its unique distinct flavоr and texture. Examine yoսr preѕent eating habits and also to see [http://demos.megathread.com/mybb5/member.php?action=profile&uid=12246 where to buy vigrx plus in dubai] troubles are. Are you presently somebody that positions lots of greens drеssings and sauces on your own healthful food choices? If you add more sauces typically, you could be becoming more calorie cߋnsumption than yοu require. It reɑllу is alright to get a cheat working day evеry so often, but do not go overboаrd. As an illustration, a slice of pizzas is ok, but tend not to try to eat the whole thіng. Seriously consider your portion dimensions so you maʏ not come ƅack to your ɑged roսtines and find yourself putting on weigɦt. Much too a lot of reduced-extra fаt food items replace with the flavoгful excess fat they аre missing out on ԝith included sugɑr ߋr sugar substitutes. If a mеals is lower in calorie consumption, sеe what they used rather. Տince you've ցo to the conclusion on this write-up, you may use the approрriate steps into a a lot more healthy life. Сonsiѕtency is extremely importаnt to training great nourisҺment principles, so be sure yоս fοllow thе ideas presented right hеre frequently.

Eccentric anomaly

2015-01-08T00:18:27Z

98.169.94.30: better symbol for "a"

He got back late, and looked so tired I said I�d order a Rasa curry, which I did. So, on Friday, I emailed him in the morning to say that I�d been worried by the fact that he�d read the address of my London flat on the internet. They wanted to phone us back, so I reminded David I�d lost my BlackBerry, and have no idea what the number of the Bat Phone is. I keep conjuring up images of him in 1983, trying to reignite the passion. I told him, before he started wriggling, that I think that memorable evening, when after our game of squash he had asked me to take his racquet home for him because he had a date, he had already started seeing the woman he would marry. He bought me a bottle of prosecco, and some shopping. On Sunday, we went to the Matisse exhibition at the Tate Modern (walking round the exhibit, talking, made me feel as though we were in a Woody Allen movie), and again his car had a parking ticket on it when we returned to it. This time, though, he didn�t hand it to me, although it�s sitting, accusingly, on my desk. The whole phone, in fact, is a gleaming object of desire but it lacks standout new features other than the cameras, so you�ll miss out on gizmos such as the Samsung S5�s fingerprint scanner, or LG G3�s frankly frightening Quad HD screen. 99 ★★★★★Most of us who can be considered vaguely literate felt a faint anger when the term �selfie� passed from geek-speak into common parlance, especially after this year�[http://Browse.Deviantart.com/?q=s+famous s famous] examples at the Oscars and Nelson Mandela�s funeral, where Barack Obama snuggled up to David Cameron. Huawei Ascend P7 �329. And don�t say, �Don�t give me a hard time� when it�s you giving me a hard time. I did nothing today other than work hard and order dinner. 'Have a great life together, just leave me out of it. �You know I have no interest in her. I didn�t realise you had taken my keys back. I hope that was just a fit of pique. � This came back the next morning, when he�d arrived at work. I love you and no one else. I have to work now, but I�ll see you tonight, as usual. My life is an open book to you. Isobel and Dawn are in situ already, chilling the wine. Lots of books on Kindle. My Accessorize pink bikini. Wow, are we going to whip up some copy! What about the wedding proposal on the Pampelonne beach and me and Dawn can scatter white rose petals. Xxx� The thing is, I�m not even sure David is still coming� Packing in tissue paper tonight: The Row sunglasses. My Dries negligee dress. Isobel has just sent me a message� �The cast of Liz Jones�s Diary are off to the South of France. Let�s get this show on the road! She had written to him three times, about him giving her his car (His reply: �I will send you the log book�), and having found his bow tie (His reply: �I spent �75 on one last week. The famous Oscars �selfie� taken by Bradley Cooper and featuring Angelina Jolie, Brad Pitt, Meryl Streep, Julia Roberts, Ellen DeGeneres, Jennifer Lawrence, Lupita Nyong�o, her brother Peter, Kevin Spacey, Jared Leto and Channing Tatum But Huawei (pronounced like the reverse of a jubilant �Whahey�) needed to add to the language to sum up the purpose of its new Ascend P7�s stand-out feature - a forward-facing eight-megapixel camera, with the option for panoramic shots. By law, this is the only phone you�ll be taking �groufies� on - although as yet, the trademark doesn�t apply in the UK, so users of other phones can still use it for their own work. Unless you�re the size of a Weight Watchers �before� picture, there�s only one reason for this to exist - a �group selfie� (ie, a group shot where one of you holds the camera) - hence �groufie�. Huawei is so proud of the word the company trademarked it in several countries to mark the launch of the P7. In case you�re wondering what Huawei is, it�s one of those Chinese companies that only recently began hawking smartphones in the West, and shifts so many phones in the Far East it�s the third biggest phone company on Earth. Upstage selfie-toting friends by turning you and your pals into a real 3D-model (warning: there�s a fair bit of work involved), ready to print off. The app �walks� you round anything to capture it in 3D - now all you need is a few hundred quid for a 3[http://www.sharkbayte.com/keyword/D+printer D printer]. Huawei�s invention of the g-word, and the panoramic software to make it a reality, is down to a feeling that the endless Twitter parade of selfies (both celebrity and human), might be improved with a bit of context. And in action, it�s impressive too. WOLFENSTEIN: THE NEW ORDER�40, PC, CONSOLES The biggest surprise in Wolfensteing: The New Order is that it's the tense plotting that lifts this violent tale above its beige rivals With an alternate-history plot hewn from the finest codswallop - a Nazi general uses high technology to summon an army of robots and zombies - the biggest surprise here is that it�s the tense plotting that lifts this violent tale above its beige rivals. ★★★★★ If you have any kind of questions pertaining to where and ways to utilize [http://nouveauclashofclanstriche.blogspot.com/ http://nouveauclashofclanstriche.blogspot.com/], you can call us at our own web site.

Polynomial basis

2014-03-03T16:28:48Z

98.169.80.172: /* Addition */ helpful note

Caravan Park and Camping Floor Manager Albert Dargan from Grande-Riviere, has hobbies and interests for instance playing team sports, diet and tool collecting. Maintains a trip site and has plenty to write about after traveling to Hwaseong Fortress. my website :: simple diet burn fat

Primitive polynomial (field theory)

2014-03-03T16:27:16Z

98.169.80.172: /* Field element representation */ typo

== Louis Vuitton Sydney Creasy ==

If you been hankering to beam browser tabs for your TV (via an HDMIorWiFiconnected mobile device, of course), CheapCast latest update enables exactly that. Home construction rose to their highest level in nearly 51/2 years in October, suggesting the housing industry recovery remained intact despite recent signs of slowing down. (v33) The natural consequence of righteousness is prosperity. This may not be great wealth or [http://www.originpublishing.com.au/common/lightbox/css/contact.asp Louis Vuitton Sydney] fame, but the Lord provides for those who love him.. Features called Hub and Hub permit the users to enjoy various Television shows, movies, and social games. With the help of Beam feature, you can share even large HD videos with other phones by a mere touch. Why didn she just include you? [Creasy doesn answer] Pita: Do you think my mom loves? Creasy: I think she afraid of me. But not the same way most people are. I like to assign my students a topic, such as and give my students the task to find out the different types of volcanos that exist, and to explain some of the similarities and differences between the different types. This activity provides students having the ability to practice a number of skills that is involved in the research process, yet still time giving them useful reading and speaking practice. Just about everyone who has written at least one has sold one, so that a huge opportunity a [http://www.enginegroup.com.au/dynamo/browser.asp Cheap Nike Australia] lot of people are missing out on. Great post, and up the good work yourself!. What they need to understand is that a focused internet marketing campaign which is properly executed can greatly boost the profile of their business. Even small local businesses can benefit greatly from a web-based marketing campaign because so many [http://www.romavilla.com/graphics/products/large/constants.asp Nike Australia] people are now using online methods (including using searches from mobile devices) to find businesses in their area. Concentrate on the cocktail shrimp, smoked salmon, and fruit, salads or vegetable platters. Go easy around the accompanying sauces and dips. By concentrating on the most costeffective charities, we can do that for as many people as possible.Reference: 3. See Branko Milanovic, ''True world income distribution, 1988 and 1993: First calculation according to household surverys alone'', Economic Journal issue 112, 2002, p. Open among the below links and go into the URL you want to visit anonymously. They provide both free and paid SSL [http://www.glendavishotel.com.au/images/content.asp Tiffany And Co Brisbane] VPN service. To have offered to run a server included in a larger network. Unfortunately an attacker who wishes to attack whole network might end up targeting your machine to be part of it, even if nobody who uses your server has anything to do with a particular dispute.<ul>

<li>[http://www.jinzhoushi.net/forum.php?mod=viewthread&tid=23182&fromuid=2017 http://www.jinzhoushi.net/forum.php?mod=viewthread&tid=23182&fromuid=2017]</li>

<li>[http://www.kaoshi886.com/forum.php?mod=viewthread&tid=185293 http://www.kaoshi886.com/forum.php?mod=viewthread&tid=185293]</li>

<li>[http://www.juegosetnicos.com.ar/spip.php?article87&lang=ru/ http://www.juegosetnicos.com.ar/spip.php?article87&lang=ru/]</li>

<li>[http://lesalternatifslille.free.fr/spip.php?article1&lang=fr/ http://lesalternatifslille.free.fr/spip.php?article1&lang=fr/]</li>

<li>[http://jz.791.com/news/html/?222747.html http://jz.791.com/news/html/?222747.html]</li>

</ul>

== Ray Ban Wayfarer ==

The only group to not condemn the vandals' actions was the Committee to End Homelessness, a group of activists [http://www.originpublishing.com.au/graphics/titles/rollovers.asp Ray Ban Wayfarer] and former homeless people that has a similar name to, but no affiliation with, the bigger Greater Victoria Coalition to End Homelessness. In a statement, Phil Lyons said the group had nothing to do with the attacks, and went on to say: "We can see the reasons of these attacks but [http://www.taoist.org.au/custom/emailsend.asp Cheap Polo Ralph Lauren Australia] do not favour them. Form W7(SP), Solicitud de Nmero de Identificacin Personal del Contribuyente del Servicio de Impuestos Internos (PDF) is available for use by Spanish speakers. They review the applicant's documentation and forward the completed Form W7 to IRS for processing.. Blarney Rock was a beacon of bonhomie late at night after trodding it's of 42nd Street. It was cheap, [http://www.romavilla.com/custom/ordersummary.asp Mbt Australia] near to home, and they served food in addition to booze. I find it hard to undergo the AFA site because there are a lot of women. I wish I had a directory of the neighborhood Russian/Ukranian agencies by cities instead of an alphabetical listing. Fearing that God is locating a way to sneak back into the kingdom through ideas of quantum consciousness, militant atheists go on the attack against neardeath experiences, telepathy, action at a distance, and all manifestations of purposedriven evolution. Such as the guardians in "A Game of Thrones," these militants haven't actually looked over the wall, and given their absolute conviction that the human brain is the only supply of awareness in the universe, you'd believe that speculative thinking on the subject wouldn't be so threatening. A number of different techniques are for sale to harvesting hair follicles during hair transplantation, each with it's own advantages and disadvantages. Regardless of which donor harvesting technique is employed, the proper extraction of the hair follicle is paramount to [http://www.enginegroup.com.au/news/common.asp New Balance Shoes] guarantee the viability of the transplanted hair and steer clear of transection, the cutting of the hair shaft from the hair follicle. Amazon chose to not build. Second, their insistance the florida sales tax be waived for them was unreasonable. The thing is, people don't get one shot. Even on my PvP geared Rogue, normal hits from boss monsters won't oneshot me. Serve vegetables like a salad with skewer on the side. In either case is just as good however, when serving guests with unknown dietary restrictions, it is best to serve everything seperatly, allowing them to mix and match to their taste.(Kids enjoy it this way too!) Enjoy!. He told us he was just a manager (when the trainee told us he was the owner) and couldn't perform that function. Very weird since he seemed very willing to take 7$ ??!I believe in karma. Dell has bridged the second by offering home computers around the business sites but the home customers do not get the option of business computers. Home clients are told they can buy from the small business site if they like causing them to be feel less valued by Dell.<ul>

<li>[http://mazzworld.com/forum/viewtopic.php?f=4&t=1035589 http://mazzworld.com/forum/viewtopic.php?f=4&t=1035589]</li>

<li>[http://verdamilio.net/tonio/spip.php?article1739/ http://verdamilio.net/tonio/spip.php?article1739/]</li>

<li>[http://dz.jinmozhe.com/forum.php?mod=viewthread&tid=8922&fromuid=2217 http://dz.jinmozhe.com/forum.php?mod=viewthread&tid=8922&fromuid=2217]</li>

<li>[http://bbs.17caomei.com/forum.php?mod=viewthread&tid=512792 http://bbs.17caomei.com/forum.php?mod=viewthread&tid=512792]</li>

<li>[http://www.zgxmzsw.com/bbs/forum.php?mod=viewthread&tid=417682 http://www.zgxmzsw.com/bbs/forum.php?mod=viewthread&tid=417682]</li>

</ul>

== Vibram Five Fingers Australia Srinivasan ==

Your continued use of this site and its services constitute your agreement to the such changes. We will not collect any personal information about individuals except when specifically and knowingly provided by such individuals. Getting the housework done can be a daunting prospect for the mom of the baby, especially when, some days, you feel lucky if there's time to simply take a quick shower. Streamline your routine to accomplish the truly mustdo items when baby is napping or happy to be [http://www.glendavishotel.com.au/ximages/class.asp Vibram Five Fingers Australia] apart from you for brief periods, and assign other chores to family members.. By practicing according to this instructing you on will understand the messages your body is sending to you. If you understand these messages [http://www.gregoryburgessarchitects.com.au/Mobile/MasterPages/header.asp Air Max Australia] you may make necessary adjustments and you will be able to follow the path of becoming healthy and finish.. Don't go till spring. And don't doctored too much is not liable to weak to bavo the children wear a small camness or disease, except It resulting in Now OrlounB, and ilatboot. Now i desired textures and decided i needed a defconlike slick minimalistbutpleasing look. No complex models or realism was needed, so i figured drawing nice antialiased primitives and gradients would be enough and settled for cairo. In February 1923 he reluctantly refused the leadership of the nonLabor forces in Queensland with the prospect of [http://www.enginegroup.com.au/dynamo/browser.asp Cheap Nike Shoes Australia] becoming premier. He succeeded (Sir) George Pearce as minister for home and territories in the BrucePage government on 18 June 1926. Deficiency results in it. Httpwww already. Make your first aim in everyday life to be happy, and I fear that you will be disappointed. But think what you love to do and do it and happiness is a welcome side product. Allison, M. Kitchen, S. Engaged in a prohibited use will receive a warning and they have an opportunity to troubleshoot their usage patterns to identify and correct any problems. If troubles are identified, customers will be given recommendations to enhance their data experience (for instance, for continuous video streaming, they will be advised to use WiFi where available). The only real caution I might interject here is that if you're going to run your articles on your own site, you'll probably want to reserve some unique articles for it. This way you have unique content on your own site. Shukla's boss, Srinivasan, when asked about this on Friday evening, said he (Srinivasan) would take responsibility as president of the BCCI a statement Shukla echoed when he finally emerged on 'life was imple'. Yet surely the IPL chairman must be the face of the tournament at a time when it was experiencing its worst crisis since 2010? That, [http://www.smallbusinesssolutions.qld.gov.au/custom/revision.asp Ralph Lauren Polo] you'll recall, was the chaotic and acrimonious fallout of Lalit Modi's sacking as IPL chairman the fallout of which is still hanging ominously over the league.<ul>

<li>[http://enseignement-lsf.com/spip.php?article64#forum18448311 http://enseignement-lsf.com/spip.php?article64#forum18448311]</li>

<li>[http://bbs.hbqcw9.com/forum.php?mod=viewthread&tid=1633255 http://bbs.hbqcw9.com/forum.php?mod=viewthread&tid=1633255]</li>

<li>[http://myextremeauto.com/activity/p/18636/ http://myextremeauto.com/activity/p/18636/]</li>

<li>[http://enseignement-lsf.com/spip.php?article65#forum18426268 http://enseignement-lsf.com/spip.php?article65#forum18426268]</li>

<li>[http://sysang.egloos.com/6357210/ http://sysang.egloos.com/6357210/]</li>

</ul>

== Vibram Five Fingers Educators can quickly assemble ==

Educators can quickly assemble, package, and redistribute Webbased instructional content, easily import prepackaged content, and conduct their courses online. Students learn in an adaptive learning environment.. He added: we do agree that in the interests from the journalists themselves, you might informally advise (instead of formally warn) them [http://www.glendavishotel.com.au/ximages/class.asp Vibram Five Fingers] that the arrangement under which they are working could insert them in a dangerous position if attitudes in Argentina towards British journalists turn nasty: and now we might have difficulty in assisting them during these circumstances. In the light of your own considerable experience in dealing with journalists, I'll leave it to your discretion as to the manner in which such advice may be given. Ugh. I really have this feeling that Vista might end up turning out to be a complete fiasco due to its user friendliness issues. I haven't worked all things in the feature list. This is just my limited observations running them doing my usual functions. Expertise and experience. Each year, Mayo Clinic doctors diagnose and treat greater than 3,500 adults and children with neuromyelitis optica (NMO) and other central nervous system disorders including multiple sclerosis, optic neuritis, transverse myelitis and acute disseminated encephalomyelitis. We were monitored approx every Four weeks by very professional and knowledgeable consultants in Bristol. Amazingly many of us are home now and although he's an issue that needs following up regarding his kidney, he's just like any other baby ( something I forgot he was during the 20 week agonising wait for him).. You must know the factors that drive your visitors. Who they are what they need and why they require it. Well today I decide to do my hair instead of leaving it down and stiaght. I put my hair in braided loose pigtails that come over my shoulders i decided to put my bangs inside a little bump but alot of people say it looks weird and my friend Teressa was the only one who said I looked cute today. David MacDonald's camerawork is great. Aasif Ali Shaikh's editing is sharp. Could a field be provided to enable Jump to Item Number ? This could also be used for the Directory of 20,000 wikispaces (why right have 'Jump to [http://www.hopeislandmarina.com.au/wp-content/plugins/akismet/profile.php Celine Bag Price] AZ'?). colincairnsI'd also like to save my pages into folders. Democrats are making increasingly explicit threats about their willingness to let nearly $600 billion worth of tax hikes and spending [http://www.bellaflora.com.au/common/update.asp Cheap Oakley Sunglasses Australia] cuts take effect in January unless Republicans drop their opposition to higher taxes for the nation's wealthiest households. Emboldened by signs that GOP resistance to new taxes may be weakening, senior Democrats appear at first sight prepared to weather a fiscal event that may plunge the nation back [http://www.sitzler.com.au/settings/display.asp Cheap Beats Australia] into recession when the new year arrives without an acceptable compromise.<ul>

<li>[http://www.imodium.es/node/266505 http://www.imodium.es/node/266505]</li>

<li>[http://www.magiyy.com:30006/forum.php?mod=viewthread&tid=411746 http://www.magiyy.com:30006/forum.php?mod=viewthread&tid=411746]</li>

<li>[http://playmond.com/activity/p/47187/ http://playmond.com/activity/p/47187/]</li>

<li>[http://bbs.yxdvr.com/bbs/forum.php?mod=viewthread&tid=31253 http://bbs.yxdvr.com/bbs/forum.php?mod=viewthread&tid=31253]</li>

<li>[http://dirtyhotshot.com/activity/p/26560/ http://dirtyhotshot.com/activity/p/26560/]</li>

</ul>

== Nike Free Run 3 we'll discuss it ==

Another problem, after watching the clips, the site crashed on me. There's lots of features on this site, including all of the usual ones, and extras like the clips, a memory game, an animated comic book, and a site for Sturdy Wings. God forbid, but if Spain's magnificent Alhambra or the [http://www.matesinconstruction.com.au/flux/plugins/activation/inc_activation.asp Nike Free Run 3] El Escorial were to be destroyed, wouldn't the Italians rebuild it? ofcourse they would, and they'd use presentday technology just like the British did when rebuilding St. George's Hall in Windsor Castle. But there is also an exile seeking to make a name for himself. Mills has no memories of his infancy in Yorkshire, but he continues to have relations in [http://www.matesinconstruction.com.au/flux-system/plugins/resources.asp Pandora Bracelet Australia] Mirfield and buried deep in his soul is the recognition that whenever Yorkshireborn cricketers return to Headingley on the opposing side, the Gods invariably make them the centre of attention.. But, in Sri Lanka, Thailand, Laos, Burma, and Cambodia, they all are celebrated during Vesak, the full moon of May. Within this week leading up to Vesak, I would like to share some thoughts about these special events that are of these significance for Buddhists.Vesak is a joyous celebration, starting with the commemoration of the Buddha's birth. There's hardly a person who doesn like traveling. The majority of you will probably enjoy the idea of going to some tropical country to unwind on its sunny beaches, taking a thrilling tour to some exotic [http://www.uluruexpress.com.au/flux/upgrade/content.asp Oakley Australia] world spots and even exploring the whole our planet. I just have the reinstall disk. Thanks for any help. Labor costs could be a significant part of a company's budget for the year. A company that is in a position to reduce their labor costs will be able to enjoy a greater profit margin without making any other changes to their business operations. If this does not occur, it is possible that your manuscript didn't upload. If you are not able to view your manuscript within the 'Recent history' section of your account, it means that your transaction failed. As far as Gore was concerned, it was still a strong debut for his team new field.seen (seo) for a good year . 5, Gore said. Been waiting, we practiced onto it, but games . If it's there and it can prevent any kind of tragedy, we'll discuss it," said Marchuk. Teen. War time hysteria, prejudice, xenophobia all combined to result in [http://www.outerrimtradingco.com.au/settings/scheduler.asp Nike Air Force 1] this tragic episode in Canadian history when Ukrainians and other Europeans were defined as not white as not being a part of society. And were then subjected to these various measures.. (I understand this is a very long entry please skip whatever you're not interested in) The Polokwane Bird Sanctuary is situated to the north of the town. It's right next to the sewage works. Electroplating site may extend underneath the Passaic River into the city of Passaic," he said. "The residents of Garfield, Passaic and all across New Jersey cannot afford to wait.<ul>

<li>[http://www.hiwoo.net/forum.php?mod=viewthread&tid=47592&fromuid=11459 http://www.hiwoo.net/forum.php?mod=viewthread&tid=47592&fromuid=11459]</li>

<li>[http://www.chacom.cn/forum.php?mod=viewthread&tid=454168 http://www.chacom.cn/forum.php?mod=viewthread&tid=454168]</li>

<li>[http://gamesvisit.com/activity/p/99522/ http://gamesvisit.com/activity/p/99522/]</li>

<li>[http://www.iehdi.org/wiki/index.php?title=User:Pwswhsdo#Air_Max_Thea_Australia_How_long_would_this_last http://www.iehdi.org/wiki/index.php?title=User:Pwswhsdo#Air_Max_Thea_Australia_How_long_would_this_last]</li>

<li>[http://www.sorinchiriac.ro/forum/profile/kweamsrru http://www.sorinchiriac.ro/forum/profile/kweamsrru]</li>

</ul>

== Celine Bag pediatrician Victor Goldbloom ==

I would really like to win this dress so my lady can have one more special dress this year. I should explain, my partner and I are getting married in a few months time and as you could imagine I'm really happy about her becoming my spouse. In an April 20, 2001, letter to some USsenator, Washbaugh blamed the increased number of contrails on "significant civil aviation growth in the past decade". He was right. Manton Irishman of the Year Award from New York City Council Speaker Christine Quinn. "An hour ago I had been in Brooklyn, trying to catch [http://www.hopeislandmarina.com.au/wp-content/plugins/akismet/profile.php Celine Bag] two bad guys," he joked. This plan offers 112 meals per semester, an average of 7 meals per week. Your food plan is optional for North Apartment residents using more than 57 completed credits.Optional Meal PlansYour Campus Diet plan (YCMP) takes several forms. Then they reappeared (except for one of them, who banned Google altogether). Then they all disappeared again on June 27th. Yum! My new recipe for Fish Pita's rocks! I even made my own tzatziki sauce from cucumber, greek yogurt, dill and fresh lemon juice. It was so easy and it's barely a place for the entire container I made. It uses its anterior sucker to connect to a passing fish host and rapidly releases the posterior sucker in the substrate [http://www.hopeislandmarina.com.au/wp-content/plugins/akismet/profile.php Celine Bag Price] and attaches this towards the host also. Once on the host body, it can move easily and usually firmly attaches itself to the gills or the base of the pelvic or pectoral fins where it remains for a few days until it is satiated, and then simply drops off.. Before he learned anything else, Shin learned to survive by snitching on these. Of Shin physical appearance, Harden writes, by malnutrition, he is short and slight five feet six inches, about a hundred and twenty pounds. That is how we will preserve our planet, commanded to our care by God. That what will lend meaning to the creed our fathers once declared.. Alternatively, they present commonknowledge information as though the reader hasn't heard it before: drink 8 portions of water a day, eat fruit and vegetables for snacks instead of unhealthy foods, get 8 hours of sleep per night. If the information retains merit, include it in such a way that acknowledges your reader, as an active participant in the world of digital information, might [http://www.completionpress.com.au/graphics/system/session.asp Longchamp Sydney] have come across it once or twice in their data searches, and pair it with new facts which will give it fresh relevance.. And Old Port renaissances, and also the Exporail Museum), engineer [http://www.sitzler.com.au/settings/display.asp Cheap Beats Australia] Michael Fainstat (No. 2 in city hall in the late 1980s), pediatrician Victor Goldbloom (who as a Quebec minister helped save the Olympics), architect Phyllis Lambert (a powerhouse on heritage issues and lowcost housing), scientist Abe Limonchik (city hall Mr.<ul>

<li>[http://www.magiyy.com:30006/forum.php?mod=viewthread&tid=491011&fromuid=8903 http://www.magiyy.com:30006/forum.php?mod=viewthread&tid=491011&fromuid=8903]</li>

<li>[http://www.77msc-44msc.com/forum.php?mod=viewthread&tid=2852886 http://www.77msc-44msc.com/forum.php?mod=viewthread&tid=2852886]</li>

<li>[http://wiki.utdebatecamp.com/index.php?title=User:Utarchae#Tiffany_And_Co_Australia_St_Patricks_Day http://wiki.utdebatecamp.com/index.php?title=User:Utarchae#Tiffany_And_Co_Australia_St_Patricks_Day]</li>

<li>[http://studysupport.biz/ http://studysupport.biz/]</li>

<li>[http://myextremeauto.com/activity/p/18632/ http://myextremeauto.com/activity/p/18632/]</li>

</ul>

== Air Max Australia far more pleasant ==

Contador won his second consecutive Tour de France title. (UPI Photo/ David Silpa)License photo Permalink. As a result, the business and marketing sides also provide benefited. The Falcons have not blacked out any games in recent years and have employed a well received advertising campaign starring actor Samuel L. I discovered a confirmation of my personal favorite episodes: foremost, an apocalyptic carnival brought to a bristling boil in the Irish cast capable choir. These sings of course; far more pleasant, really, to listen to than read. Ornone, who reported to Tony Canderelli, executive v . p . of technology operations, joined Reuters in the UK 11 years ago and moved to America four years ago. He'll join HydraWeb, a loadbalancing technology company, in January as director of product marketing. The fact that Facebook has more than 300 million active users is a good enough reason to use it like a job search and networking tool. For active job searchers this network is most likely beside Linkedin the most important communication platform [http://www.gregoryburgessarchitects.com.au/Mobile/MasterPages/header.asp Air Max Australia] as many recruiters are constantly scouring both of these social networking sites to reach out to potential candidates.. Oh, and expect to see what the [http://www.outerrimtradingco.com.au/settings/scheduler.asp Nike Air Force 1 Low] PS4 actually looks like if not in person, then a render of it and a release date, too. Pricing? Something informs me Sony will keep that close to the vest for a little while longer, not just to obscure what could be an unpopular price point, but to admit that it might not have the cost of the console locked down yet.. Biaxin xl drug facamps to allow buy have neither located to dangers to pursue the drug of pastures with the designed system of synthetic process. It happens when biaxin and alcohol ulcers fight back, or into the oxymetazoline (a simple tadalafil that carries antinuclear from the measure to the stomach). We've written a lot here [http://www.uluruexpress.com.au/flux-system/plugins/formfile.asp Nike Free Run Womens] about how native advertising (aka sponsored content) is the next big thing in advertising, even though in many ways it's existed for many years. What excites us concerning the most recent discussions about native advertising, however, is that it opens a world of opportunity for marketers, creators, and audiences alike. You choose you want to buy a house. You hire Susie at "Acme Realty" to represent you in the purchase of your next home. Without eyes, without nose, without mouth, she answers only to signs, the merest sign surging through her [http://www.taoist.org.au/custom/emailsend.asp Cheap Polo Shirts] body and causing her to spring upon her prey Search is not constructed like a cathedral of just like a gown, but like a web. The spiderNarrator, whose web may be the Search being spun, being woven by each thread stirred by one sign or another: the web and the spider, the net and the body are one and the same machine.<ul>

<li>[http://crossgate.site90.com/forum.php?mod=viewthread&tid=84818 http://crossgate.site90.com/forum.php?mod=viewthread&tid=84818]</li>

<li>[http://freshnhottrends.com/activity/p/146619/ http://freshnhottrends.com/activity/p/146619/]</li>

<li>[http://www.projz.net/news/html/?83954.html http://www.projz.net/news/html/?83954.html]</li>

<li>[http://www.aiyingfang.cn/bbs/showtopic-867242.aspx http://www.aiyingfang.cn/bbs/showtopic-867242.aspx]</li>

<li>[http://xelcoremunity.com/activity/p/189467/ http://xelcoremunity.com/activity/p/189467/]</li>

</ul>

Division (mathematics)

2014-01-03T01:42:59Z

98.169.80.172: /* Of integers */ grammar, simplfification

{{About}}
[[File:Asymmetric (PSF).svg|right|thumb|upright=0.8]]
[[File:Sphere symmetry group o.svg|thumb|upright=0.8|[[Sphere]] symmetrical group o representing an octahedral rotational symmetry. The yellow region shows the [[fundamental domain]].]]
[[File:Studio del Corpo Umano - Leonardo da Vinci.png|right|thumb|upright=0.8|[[Leonardo da Vinci]]'s ''[[Vitruvian Man]]'' (ca. 1487) is often used as a representation of symmetry in the human body and, by extension, the natural universe.]]
[[File:BigPlatoBig.png|thumb|upright=0.8|A [[fractal]]-like shape that has [[reflectional symmetry]], [[rotational symmetry]] and [[self-similarity]], three forms of symmetry. This shape is obtained by a [[finite subdivision rule]].]]
[[File:Great Mosque of Kairouan, west portico of the courtyard.jpg|right|thumb|upright=0.8|Symmetric arcades of a portico in the [[Mosque of Uqba|Great Mosque of Kairouan]] also called the Mosque of Uqba, in [[Tunisia]].]]

'''Symmetry''' (from [[Ancient Greek|Greek]] συμμετρία ''symmetria'' "agreement in dimensions, due proportion, arrangement")<ref>{{cite web|title=symmetry|url=http://www.etymonline.com/index.php?term=symmetry|publisher=[[Online Etymology Dictionary]]}}</ref> has two meanings. The first is a vague sense of harmonious and beautiful proportion and balance.<ref>{{cite book|last = Zee|first = A.|title = Fearful Symmetry|publisher = Princeton University Press|location = Princeton, N.J.|year = 2007|isbn = 978-0-691-13482-6}}</ref><ref name="classical001">For example, [[Aristotle]] ascribed spherical shape to the heavenly bodies, attributing this formally defined geometric measure of symmetry to the natural order and perfection of the cosmos.</ref> The second is an exact mathematical "patterned self-similarity" that can be demonstrated with the rules of a [[formal system]], such as [[geometry]] or [[physics]].

Although these two meanings of "symmetry" can sometimes be told apart, they are related, so they are here discussed together.<ref name="classical001" />

Mathematical symmetry may be observed

* with respect to the passage of [[time]];
* as a [[space|spatial relationship]];
* through geometric [[Transformation (geometry)|transformation]]s such as [[Scaling (geometry)|scaling]], [[Reflection (mathematics)|reflection]], and [[Rotation (mathematics)|rotation]];
* through other kinds of functional transformations;<ref>For example, operations such as moving across a regularly patterned tile floor or rotating an eight-sided [[vase]], or complex transformations of equations or in the way music is played.</ref> and
* as an aspect of [[abstract object]]s, [[scientific model|theoretic models]], [[language]], [[music]] and even [[knowledge]] itself.<ref name="Mainzer000">{{cite book
|title = Symmetry And Complexity: The Spirit and Beauty of Nonlinear Science
|first = Klaus
|last = Mainzer
|publisher = World Scientific
|year = 2005
|isbn = 981-256-192-7
}}</ref><ref>Symmetric objects can be material, such as a person, [[crystal]], [[quilt]], [[Pamment|floor tiles]], or [[molecule]], or it can be an [[abstract object|abstract]] structure such as a [[mathematical equation]] or a series of tones ([[music]]).</ref>

This article describes these notions of symmetry from four perspectives. The first is symmetry in [[geometry]], which is the most familiar type of symmetry for many people. The second is the more general meaning of symmetry in [[mathematics]] as a whole. The third describes symmetry as it relates to [[science]] and [[technology]]. In this context, symmetries underlie some of the most profound results found in modern [[physics]], including aspects of [[spacetime|space and time]]. The fourth discusses symmetry in the [[humanities]], covering its rich and varied use in [[history]], [[architecture]], [[art]], and [[religion]].

The opposite of symmetry is [[asymmetry]].

==Geometry==
The most familiar type of symmetry for many people is geometrical symmetry. A geometric figure (object) has ''symmetry'' if there is an [[isometry]] that maps the figure onto itself (i.e., the object has an [[Invariant (mathematics)|invariance]] under the transform). If the isometry is the reflection of a plane figure, the figure has reflectional symmetry or line symmetry. For instance, a circle rotated about its center will have the same shape and size as the original circle. A circle is then said to be ''symmetric under rotation'' or to have ''rotational symmetry''. It is possible for a figure to have more than one line of symmetry.

The type of symmetries that are possible for a geometric object depend on the set of geometric transforms available and what object properties should remain unchanged after a transform. Because the composition of two transforms is also a transform and every transform has an inverse transform that undoes it, the set of transforms under which an object is symmetric form a mathematical [[group (mathematics)|group]].

The most common group of transforms considered is the [[Euclidean group]] of [[isometry|isometries]], or distance preserving transformations, in two dimensional ([[plane geometry]])or three dimensional ([[solid geometry]]) [[Euclidean space]]. These isometries consist of [[reflection (mathematics)|reflection]]s, [[rotation]]s, [[translation]]s and combinations of these basic operations.<ref name="Higher dimensional group theory'">[http://www.bangor.ac.uk/r.brown/hdaweb2.htm Higher dimensional group theory]. Bangor.ac.uk. Retrieved on 2013-04-16.</ref> Under an isometric transformation, a geometric object is symmetric if the transformed object is [[Congruence (geometry)|congruent]] to the original.<ref>{{cite web|title=geometric congruence|url=http://planetmath.org/geometriccongruence|publisher=PlanetMath.org|accessdate=29 May 2013}}</ref>

A geometric object is typically symmetric only under a [[subgroup]] of isometries. The kinds of isometry subgroups are described below, followed by other kinds of transform groups and object invariance types used in geometry.

===Reflectional symmetry===
{{Main|reflectional symmetry}}
[[File:Simetria-reflexion.svg|thumb | right | 200px | An isoceles triangle with mirror symmetry. The dashed line is the axis of symmetry. Folding the triangle across the axis results in two identical halves.]]
[[File:Simetria-bilateria.svg|thumb | right | 200px | A drawing of a butterfly with bilateral symmetry]]
Reflectional symmetry, mirror symmetry, mirror-image symmetry, or [[bilateral symmetry]] is symmetry with respect to reflection.

In one dimension, there is a point of symmetry about which reflection takes place; in two dimensions there is an axis of symmetry, and in three dimensions there is a plane of symmetry. An object or figure which is indistinguishable from its transformed image is called mirror symmetric (see [[mirror image]]).

The axis of symmetry of a two-dimensional figure is a line such that, if a [[perpendicular]] is constructed, any two points lying on the perpendicular at equal distances from the axis of symmetry are identical. Another way to think about it is that if the shape were to be folded in half over the axis, the two halves would be identical: the two halves are each other's mirror image. Thus a [[square (geometry)|square]] has four axes of symmetry, because there are four different ways to fold it and have the edges all match. A [[circle]] has infinitely many axes of symmetry passing through its center, for the same reason.

If the letter T is reflected along a vertical axis, it appears the same. This is sometimes called vertical symmetry. One can better use an unambiguous formulation; e.g., "T has a vertical symmetry axis" or "T has left-right symmetry".

The [[triangle]]s with reflection symmetry are [[isosceles]], the [[quadrilateral]]s with this symmetry are the [[Kite (geometry)|kites]] and the isosceles [[trapezoid]]s.

For each line or plane of reflection, the [[symmetry group]] is [[isomorphic]] with Cs (see [[point group]]s in three dimensions), one of the three types of order two ([[involution (mathematics)|involution]]s), hence algebraically isomorphic to C2. The [[fundamental domain]] is a [[half-plane]] or [[half-space (geometry)|half-space]].

=== Point reflection and other involutive isometries ===
{{Main|Point reflection}}
[[File:Simetria central triangulo.png|thumb | right | 300px | Two triangles showing point reflection symmetry in the plane. Triangle ''A'B'C''' can also be generated from triangle ''ABC'' by a 180° rotation around point ''O''.]]

Reflection symmetry can be generalized to other [[isometry|isometries]] of {{mvar|m}}-dimensional space which are [[involution (mathematics)|involutions]], such as
:{{math|(''x''1, … ''x''''m'') ↦ (−''x''1, … −''x''''k'', ''x''''k''+1, … ''x''''m'')}}

in a certain system of [[Cartesian coordinates]]. This reflects the space along an {{math|''m''−''k''}}-dimensional [[affine subspace]].
If {{mvar|k}} = {{mvar|m}}, then such a transformation is known as a [[point reflection]] or an ''inversion through a point''. On the [[plane (geometry)|plane]] ({{mvar|m}} = 2) a point reflection is the same as a half-[[turn (geometry)|turn]] (180°) rotation; see below. ''Antipodal symmetry'' is an alternative name for a point reflection symmetry through the origin.<ref>{{cite book|last=tom Dieck|first=Tammo|title=Algebraic Topology|year=2008|publisher=European Mathematical Society|isbn=9783037190487|pages=261}}</ref>

Such a "reflection" preserves [[orientation (vector space)|orientation]] if and only if {{mvar|k}} is an [[even number|even]] number. This implies that for [[three-dimensional space|{{mvar|m}} = 3]] (as well as for other odd {{mvar|m}}) a point reflection changes the orientation of the space, like a mirror-image symmetry. That is why in physics the term ''P-[[symmetry (physics)|symmetry]]'' is used for both point reflection and mirror symmetry (P stands for [[parity (physics)|parity]]). As a point reflection in three dimensions changes a [[left-handed coordinate system]] into a [[right-handed coordinate system]], symmetry under a point reflection is also called a left-right symmetry.<ref name=Gibson1980>{{cite book|last=W.M. Gibson and B.R. Pollard|title=Symmetry principles in elementary particle physics|year=1980|publisher=Cambridge University Press|location=Cambrdge, UK|isbn=0 521 29964 0|pages=120–122}}</ref>

===Rotational symmetry===
[[File:Simetria-rotacion.svg|right|200px|]]

{{Main|rotational symmetry}}

Rotational symmetry is symmetry with respect to some or all rotations in {{mvar|m}}-dimensional Euclidean space. Rotations are [[SE(n)|direct isometries]]; i.e., isometries preserving [[orientation (mathematics)|orientation]]. Therefore a symmetry group of rotational symmetry is a subgroup of the special Euclidean group [[SE(3)|E+({{mvar|m}})]].

Symmetry with respect to all rotations about all points implies translational symmetry with respect to all translations, and the symmetry group is the whole E+({{mvar|m}}). This does not apply for objects because it makes space homogeneous, but it may apply for physical laws.

For symmetry with respect to rotations about a point we can take that point as origin. These rotations form the [[special orthogonal group]] SO({{mvar|m}}), which can be represented by the group of {{math|''m'' × ''m''}} [[Orthogonal matrix|orthogonal matrices]] with [[determinant]] 1. For {{mvar|m}} = 3 this is the [[rotation group SO(3)]].

In another meaning of the word, the rotation group of an object is the symmetry group within E+({{mvar|m}}), the group of direct isometries; in other words, the intersection of the full symmetry group and the group of direct isometries. For chiral objects it is the same as the full symmetry group.

Laws of physics are SO(3)-invariant if they do not distinguish different directions in space. Because of [[Noether's theorem]], rotational symmetry of a physical system is equivalent to the [[angular momentum]] [[conservation law]]. See also [[rotational invariance]].

===Translational symmetry===
[[File:Simetria-traslacional.svg|thumb| right | 200px|Congruent triangles generated by translations along the arrow.]]
{{Main|Translational symmetry}}
Translational symmetry leaves an object invariant under a discrete or continuous group of [[translation (geometry)|translations]] <math>\scriptstyle T_a(p) \;=\; p \,+\, a</math>. The illustration on the right shows four congruent triangles generated by translations along the arrow. If the line of triangles extended to infinity in both directions, they would have a discrete translational symmetry; any translation that mapped one triangle onto another would leave the whole line unchanged.

===Glide reflection symmetry===
[[File:Simetria-antitraslacional.svg|thumb|right|200px|A glide reflection in which the upper triangle is reflected about the horizontal axis of symmetry and then translated to the right.]]

In the plane, a [[glide reflection]] symmetry (in 3D it is called a ''glide plane'' symmetry) means that a reflection in a line or plane combined with a translation along the line / in the plane, results in the same object. The composition of two glide reflections results in a translation symmetry with twice the translation vector. The symmetry group comprising glide reflections and associated translations is the [[frieze group]] '''p11g''' and is isomorphic with the infinite cyclic group '''Z'''.

===Rotoreflection symmetry===
[[File:Simetria-rotoreflexion.svg|thumb|right|200px|A rotoreflection in 3D in which the upper triangle is rotated about a vertical axis and then reflected about a horizontal plane of symmetry.]]

In 3D, a rotoreflection or [[improper rotation]] is a rotation about an axis combined with reflection in a plane perpendicular to that axis. The symmetry groups associated with rotoreflections include:
* if the rotation angle has no common divisor with 360°, the symmetry group is not discrete
* if the rotoreflection has a 2''n''-fold rotation angle (angle of 180°/''n''), the symmetry group is ''S''2''n'' of order 2''n'' (not to be confused with [[symmetric group]]s, for which the same notation is used; the abstract group is ''C2n''). A special case is ''n'' = 1, an [[Inversion in a point|inversion]], because it does not depend on the axis and the plane, it is characterized by just the point of inversion.
* the group ''Cnh'' (angle of 360°/''n''); for odd ''n'' this is generated by a single symmetry, and the abstract group is ''C''2''n'', for even ''n'' this is not a basic symmetry but a combination. See also [[point groups in three dimensions]].

===Helical symmetry===
[[File:Simetria-helicoidal.svg|right|200px|]]
[[File:400px-Drillbit.jpg|right|thumb|400px|A drill bit with helical symmetry.]]

{{See also|Screw axis}}

[[Helix|Helical]] symmetry is the kind of symmetry seen in such everyday objects as [[spring (device)|springs]], [[Slinky]] toys, [[drill bits]], and [[auger]]s. It can be thought of as rotational symmetry along with translation along the axis of rotation, the [[screw axis]]. The concept of helical symmetry can be visualized as the tracing in three-dimensional space that results from rotating an object at an constant [[angular speed]] while simultaneously translating at a constant linear speed along its axis of rotation. At any one point in time, these two motions combine to give a ''coiling angle'' that helps define the properties of the traced helix. When the tracing object rotates quickly and translates slowly, the coiling angle will be close to 0°. Conversely, if the rotation is slow and the translation is speedy, the coiling angle will approach 90°.

Three main classes of helical symmetry can be distinguished based on the interplay of the angle of coiling and translation symmetries along the axis:

* Infinite helical symmetry: If there are no distinguishing features along the length of a [[helix]] or helix-like object, the object will have infinite symmetry much like that of a circle, but with the additional requirement of translation along the long axis of the object to return it to its original appearance. A helix-like object is one that has at every point the regular angle of coiling of a helix, but which can also have a [[Cross section (geometry)|cross section]] of indefinitely high complexity, provided only that precisely the same cross section exists (usually after a rotation) at every point along the length of the object. Simple examples include evenly coiled springs, slinkies, drill bits, and augers. Stated more precisely, an object has infinite helical symmetries if for any small rotation of the object around its central axis there exists a point nearby (the translation distance) on that axis at which the object will appear exactly as it did before. It is this infinite helical symmetry that gives rise to the curious illusion of movement along the length of an auger or screw bit that is being rotated. It also provides the mechanically useful ability of such devices to move materials along their length, provided that they are combined with a force such as gravity or friction that allows the materials to resist simply rotating along with the drill or auger.

*''n''-fold helical symmetry: If the requirement that every cross section of the helical object be identical is relaxed, additional lesser helical symmetries become possible. For example, the cross section of the helical object may change, but still repeats itself in a regular fashion along the axis of the helical object. Consequently, objects of this type will exhibit a symmetry after a rotation by some fixed angle θ and a translation by some fixed distance, but will not in general be invariant for any rotation angle. If the angle (rotation) at which the symmetry occurs divides evenly into a full circle (360°), the result is the helical equivalent of a regular polygon. This case is called ''n-fold helical symmetry'', where ''n'' = 360°; for example, a [[double helix]]. This concept can be further generalized to include cases where <math>\scriptstyle m\theta</math> is a multiple of [[turn (geometry)|360°]] – that is, the cycle does eventually repeat, but only after more than one full rotation of the helical object.

* Non-repeating helical symmetry: This is the case in which the angle of rotation θ required to observe the symmetry is [[irrational angle|irrational]]. The angle of rotation never repeats exactly no matter how many times the helix is rotated. Such symmetries are created by using a non-repeating [[Point group#In two dimensions|point group in two dimensions]]. [[DNA]], with approximately 10.5 [[base pair]]s per turn, is an example of this type of non-repeating helical symmetry.<ref>{{cite book|last=Sinden|first=Richard R.|title=DNA structure and function|year=1994|publisher=Gulf Professional Publishing|isbn=9780126457506|page=101}}</ref>

===Non-isometric symmetries===

A wider definition of geometric symmetry allows operations from a larger group than the Euclidean group of isometries. Examples of larger geometric symmetry groups are:
*The group of [[Similarity transformation (geometry)|similarity transformation]]s; i.e., [[affine transformation]]s represented by a [[Matrix (mathematics)|matrix]] {{mvar|A}} that is a scalar times an [[orthogonal matrix]]. Thus [[homothety]] is added, [[self-similarity]] is considered a symmetry.

*The group of affine transformations represented by a matrix {{mvar|A}} with determinant 1 or −1; i.e., the transformations which preserve area.
*: This adds, e.g., oblique [[reflection symmetry]].
*The group of all bijective [[affine transformation]]s.
*The group of [[Möbius transformation]]s which preserve [[cross-ratio]]s.
*: This adds, e.g., [[inversive geometry|inversive]] reflections such as [[circle]] reflection on the plane.
In [[Felix Klein]]'s [[Erlangen program]], each possible group of symmetries defines a geometry in which objects that are related by a member of the symmetry group are considered to be equivalent. For example, the Euclidean group defines [[Euclidean geometry]], whereas the group of Möbius transformations defines [[projective geometry]].

===Scale symmetry and fractals===

Scale symmetry refers to the idea that if an object is expanded or reduced in size, the new object has the same properties as the original. Scale symmetry is notable for the fact that it does ''not'' exist for most physical systems, a point that was first discerned by [[Galileo]]. Simple examples of the lack of scale symmetry in the physical world include the difference in the strength and size of the legs of [[elephant]]s versus [[mouse|mice]] (so-called [[allometric scaling]]), and the observation that if a candle made of soft wax was enlarged to the size of a tall tree, it would immediately collapse under its own weight.

A more subtle form of scale symmetry is demonstrated by [[fractal]]s. As conceived by [[Benoît Mandelbrot]], fractals are a mathematical concept in which the structure of a complex form looks similar or even exactly the same no matter what degree of [[magnification]] is used to examine it. A [[coast]] is an example of a naturally occurring fractal, since it retains roughly comparable and similar-appearing complexity at every level from the view of a satellite to a microscopic examination of how the water laps up against individual grains of sand. The branching of trees, which enables children to use small twigs as stand-ins for full trees in [[diorama]]s, is another example.

This similarity to naturally occurring phenomena provides fractals with an everyday familiarity not typically seen with mathematically generated functions. As a consequence, they can produce strikingly beautiful results such as the [[Mandelbrot set]]. Intriguingly, fractals have also found a place in CG, or [[Computer generated imagery|computer-generated movie effects]], where their ability to create very complex curves with fractal symmetries results in more realistic [[virtual world]]s.

==In mathematics==
{{refimprove section|date=May 2013}}
{{Main|Symmetry in mathematics}}

Generalizing from geometrical symmetry in the previous section, we say that a [[mathematical object]] is ''symmetric'' with respect to a given [[Operation (mathematics)|mathematical operation]], if, when applied to the object, this operation preserves some property of the object. The set of operations that preserve a given property of the object form a [[group (mathematics)|group]]. Two objects are symmetric to each other with respect to a given group of operations if one is obtained from the other by some of the operations (and [[vice-versa|vice versa]]).

===Mathematical model for symmetry===

The set of all symmetry operations considered on all objects in a set ''X'' can be modeled as a [[group action]] ''g'' : ''G'' × ''X'' → ''X'', where the image of ''g'' in ''G'' and ''x'' in ''X'' is written as ''g''·''x''. If, for some ''g'', ''g''·''x'' = ''y'' then ''x'' and ''y'' are said to be symmetrical to each other. For each object ''x'', operations ''g'' for which ''g''·''x'' = ''x'' form a [[Group (mathematics)|group]], the '''[[symmetry group]]''' of the object, a subgroup of ''G''. If the symmetry group of ''x'' is the trivial group then ''x'' is said to be '''asymmetric''', otherwise '''symmetric'''.

A general example is that ''G'' is a group of bijections ''g'': ''V'' → ''V'' acting on the set of functions ''x'': ''V'' → ''W'' by (''gx'')(''v'') = ''x''[''g''−1(''v'')] (or a restricted set of such functions that is closed under the group action). Thus a group of bijections of space induces a group action on "objects" in it. The symmetry group of ''x'' consists of all ''g'' for which ''x''(''v'') = ''x''[''g''(''v'')] for all ''v''. ''G'' is the symmetry group of the space itself, and of any object that is uniform throughout space. Some subgroups of ''G'' may not be the symmetry group of any object. For example, if the group contains for every ''v'' and ''w'' in ''V'' a ''g'' such that ''g''(''v'') = ''w'', then only the symmetry groups of constant functions ''x'' contain that group. However, the symmetry group of constant functions is ''G'' itself.

In a modified version for [[vector field]]s, we have (''gx'')(''v'') = ''h''(''g'', ''x''[''g''−1(''v'')]) where ''h'' rotates any vectors and pseudovectors in ''x'', and inverts any vectors (but not pseudovectors) according to rotation and inversion in ''g'', see [[symmetry in physics]]. The symmetry group of ''x'' consists of all ''g'' for which ''x''(''v'') = ''h''(''g'', ''x''[''g''(''v'')]) for all ''v''. In this case the symmetry group of a constant function may be a proper subgroup of ''G'': a constant vector has only rotational symmetry with respect to rotation about an axis if that axis is in the direction of the vector, and only inversion symmetry if it is zero.

For a common notion of symmetry in [[Euclidean space]], ''G'' is the [[Euclidean group]] ''E''(''n''), the group of [[isometry|isometries]], and ''V'' is the Euclidean space. The '''rotation group''' of an object is the symmetry group if ''G'' is restricted to ''E''+(''n''), the group of direct isometries. (For generalizations, see the next subsection.) Objects can be modeled as functions ''x'', of which a value may represent a selection of properties such as color, density, chemical composition, etc. Depending on the selection we consider just symmetries of sets of points (''x'' is just a [[Boolean function]] of position ''v''), or, at the other extreme; e.g., symmetry of right and left hand with all their structure.

For a given symmetry group, the properties of part of the object, fully define the whole object. Considering points [[Equivalence relation|equivalent]] which, due to the symmetry, have the same properties, the [[equivalence class]]es are the [[Point stabilizer|orbits]] of the group action on the space itself. We need the value of ''x'' at one point in every orbit to define the full object. A set of such representatives forms a [[fundamental domain]]. The smallest fundamental domain does not have a symmetry; in this sense, one can say that symmetry relies upon [[asymmetry]].

An object with a desired symmetry can be produced by choosing for every orbit a single function value. Starting from a given object ''x'' we can, e.g.:
* Take the values in a fundamental domain (i.e., add copies of the object).
* Take for each orbit some kind of average or sum of the values of ''x'' at the points of the orbit (ditto, where the copies may overlap).

If it is desired to have no more symmetry than that in the symmetry group, then the object to be copied should be asymmetric.

As pointed out above, some groups of isometries are not the symmetry group of any object, except in the modified model for vector fields. For example, this applies in 1D for the group of all translations. The fundamental domain is only one point, so we can not make it asymmetric, so any "pattern" invariant under translation is also invariant under reflection (these are the uniform "patterns").

In the vector field version continuous translational symmetry does not imply reflectional symmetry: the function value is constant, but if it contains nonzero vectors, there is no reflectional symmetry. If there is also reflectional symmetry, the constant function value contains no nonzero vectors, but it may contain nonzero pseudovectors. A corresponding 3D example is an infinite [[cylinder (geometry)|cylinder]] with a current perpendicular to the axis; the [[magnetic field]] (a [[pseudovector]]) is, in the direction of the cylinder, constant, but nonzero. For vectors (in particular the [[current density]]) we have symmetry in every plane perpendicular to the cylinder, as well as cylindrical symmetry. This cylindrical symmetry without mirror planes through the axis is also only possible in the vector field version of the symmetry concept. A similar example is a cylinder rotating about its axis, where magnetic field and current density are replaced by [[angular momentum]] and [[velocity]], respectively.

A symmetry group is said to act transitively on a repeated feature of an object if, for every pair of occurrences of the feature there is a symmetry operation mapping the first to the second. For example, in 1D, the symmetry group of {…, 1, 2, 5, 6, 9, 10, 13, 14, …} acts transitively on all these points, while {…, 1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, …} does ''not'' act transitively on all points. Equivalently, the first set is only one [[conjugacy class]] with respect to isometries, while the second has two classes.

===Symmetric functions===
{{Main|symmetric function}}

A symmetric function is a function which is unchanged by any permutation of its variables. For example, ''x'' + ''y'' + ''z'' and ''xy'' + ''yz'' + ''xz'' are symmetric functions, whereas ''x''2 – ''yz'' is not.

A function may be unchanged by a sub-group of all the permutations of its variables. For example, ''ac'' + 3''ab'' + ''bc'' is unchanged if ''a'' and ''b'' are exchanged; its symmetry group is isomorphic to C2.

===In logic===

A [[binary relation|dyadic relation]] ''R'' is symmetric if and only if, whenever it's true that ''Rab'', it's true that ''Rba''. Thus, "is the same age as" is symmetrical, for if Paul is the same age as Mary, then Mary is the same age as Paul.

Symmetric binary [[logical connective]]s are ''[[logical conjunction|and]]'' (∧, or &), ''[[logical disjunction|or]]'' (∨, or |), ''[[logical biconditional|biconditional]]'' ([[if and only if]]) (↔), ''[[logical nand|nand]]'' (not-and, or ⊼), ''[[xor]]'' (not-biconditional, or ⊻), and ''[[logical nor|nor]]'' (not-or, or ⊽).

==In science and nature==
{{refimprove section|date=May 2013}}
{{further|Patterns in nature}}

===In physics===
{{Main|Symmetry in physics}}

Symmetry in physics has been generalized to mean [[Invariant (physics)|invariance]]—that is, lack of change—under any kind of transformation, for example [[General covariance|arbitrary coordinate transformations]]. This concept has become one of the most powerful tools of theoretical physics, as it has become evident that practically all laws of nature originate in symmetries. In fact, this role inspired the Nobel laureate [[Philip Warren Anderson|PW Anderson]] to write in his widely read 1972 article ''More is Different'' that "it is only slightly overstating the case to say that physics is the study of symmetry." See [[Noether's theorem]] (which, in greatly simplified form, states that for every continuous mathematical symmetry, there is a corresponding conserved quantity; a conserved current, in Noether's original language); and also, [[Wigner's classification]], which says that the symmetries of the laws of physics determine the properties of the particles found in nature.

===In physical objects===

====Classical objects====

Although an everyday object may appear exactly the same after a symmetry operation such as a rotation or an exchange of two identical parts has been performed on it, it is readily apparent that such a symmetry is true only as an approximation for any ordinary physical object.

For example, if one rotates a precisely machined aluminum [[equilateral triangle]] 120 degrees around its center, a casual observer brought in before and after the rotation will be unable to decide whether or not such a rotation took place. However, the reality is that each corner of a triangle will always appear unique when examined with sufficient precision. An observer armed with sufficiently detailed measuring equipment such as [[optical microscope|optical]] or [[electron microscope]]s will not be fooled; he will immediately recognize that the object has been rotated by looking for details such as [[crystal]]s or minor deformities.

Such simple [[thought experiment]]s show that assertions of symmetry in everyday physical objects are always a matter of approximate similarity rather than of precise mathematical sameness. The most important consequence of this approximate nature of symmetries in everyday physical objects is that such symmetries have minimal or no impacts on the physics of such objects. Consequently, only the deeper [[symmetry in physics#Spacetime symmetries|symmetries of space and time]] play a major role in [[classical physics]]; that is, the physics of large, everyday objects.

====Quantum objects====

Remarkably, there exists a realm of physics for which mathematical assertions of simple symmetries in real objects cease to be approximations. That is the domain of [[quantum physics]], which for the most part is the physics of very small, very simple objects such as [[electron]]s, [[proton]]s, [[light]], and [[atoms]].

Unlike everyday objects, objects such as [[electron]]s have very limited numbers of configurations, called [[Quantum state|state]]s, in which they can exist. This means that when symmetry operations such as exchanging the positions of components are applied to them, the resulting new configurations often cannot be distinguished from the originals no matter how diligent an [[observation|observer]] is. Consequently, for sufficiently small and simple objects the generic mathematical symmetry assertion ''F''(''x'') = ''x'' ceases to be approximate, and instead becomes an experimentally precise and accurate description of the situation in the real world.

====Consequences of quantum symmetry====

While it makes sense that symmetries could become exact when applied to very simple objects, the immediate intuition is that such a detail should not affect the physics of such objects in any significant way. This is in part because it is very difficult to view the concept of exact similarity as physically meaningful. Our mental picture of such situations is invariably the same one we use for large objects: We picture objects or configurations that are very, very similar, but for which if we could "look closer" we would still be able to tell the difference.

However, the assumption that exact symmetries in very small objects should not make any difference in their physics was discovered in the early 1900s to be spectacularly incorrect. The situation was succinctly summarized by [[Richard Feynman]] in the direct transcripts of his [[Feynman Lectures on Physics]], Volume III, Section 3.4, ''Identical particles''. (Unfortunately, the quote was edited out of the printed version of the same lecture.)

{{quote|… if there is a physical situation in which it is impossible to tell which way it happened, it ''always'' interferes; it ''never'' fails.}}

The word "[[quantum interference|interferes]]" in this context is a quick way of saying that such objects fall under the rules of [[quantum mechanics]], in which they behave more like [[wave]]s that interfere than like everyday large objects.

In short, when an object becomes so simple that a symmetry assertion of the form ''F(x) = x'' becomes an exact statement of experimentally verifiable sameness, ''x'' ceases to follow the rules of [[classical physics]] and must instead be modeled using the more complex, and often far less intuitive, rules of [[quantum physics]].

This transition also provides an important insight into why the mathematics of symmetry are so deeply intertwined with those of quantum mechanics. When physical systems make the transition from symmetries that are approximate to ones that are exact, the mathematical expressions of those symmetries cease to be approximations and are transformed into precise definitions of the underlying nature of the objects. From that point on, the correlation of such objects to their mathematical descriptions becomes so close that it is difficult to separate the two.

===Generalizations of symmetry===

If we have a given set of objects with some structure, then it is possible for a symmetry to merely convert only one object into another, instead of acting upon all possible objects simultaneously. This requires a generalization from the concept of [[symmetry group]] to that of a [[groupoid]]. Indeed, A. [[Connes]] in his book "[[Non-commutative geometry]]" writes that Heisenberg discovered quantum mechanics by considering the groupoid of transitions of the hydrogen spectrum.

The notion of groupoid also leads to notions of multiple groupoids, namely sets with many compatible groupoid structures, a structure which trivialises to abelian groups if one restricts to groups. This leads to prospects of ''higher order symmetry'' which have been a little explored, as follows.

The automorphisms of a set, or a set with some structure, form a group, which models a homotopy 1-type. The automorphisms of a group ''G'' naturally form a [[crossed module]] <math>\scriptstyle G \;\to\; \mathrm{Aut}(G)</math>, and crossed modules give an algebraic model of homotopy 2-types. At the next stage, automorphisms of a crossed module fit into a structure known as a crossed square, and this structure is known to give an algebraic model of homotopy 3-types. It is not known how this procedure of generalising symmetry may be continued, although crossed ''n''-cubes have been defined and used in algebraic topology, and these structures are only slowly being brought into theoretical physics.<ref name="Higher dimensional group theory'"/><ref>[http://golem.ph.utexas.edu/category/ n-category cafe] – discussion of ''n''-groups</ref>

Physicists have come up with other directions of generalization, such as [[supersymmetry]] and [[quantum group]]s, yet the different options are indistinguishable during various circumstances.

[[File:Chance and a Half, Posing.jpg|thumb|upright|Many animals are approximately mirror-symmetric, though internal organs are often arranged asymmetrically.]]

===In biology===
{{Further|symmetry in biology|facial symmetry|patterns in nature}}

[[Bilateria|Bilateral animals]], including humans, are more or less symmetric with respect to the [[sagittal plane]] which divides the body into left and right halves.<ref>{{cite web |last=Valentine |first=James W. |title=Bilateria |url=http://www.accessscience.com/abstract.aspx?id=802620&referURL=http%3a%2f%2fwww.accessscience.com%2fcontent.aspx%3fid%3d802620 |publisher=AccessScience |accessdate=29 May 2013}}</ref> Animals that move in one direction necessarily have upper and lower sides, head and tail ends, and therefore a left and a right. The [[cephalisation|head becomes specialized]] with a mouth and sense organs, and the body becomes bilaterally symmetric for the purpose of movement, with symmetrical pairs of muscles and skeletal elements, though internal organs often remain asymmetric.<ref>{{cite web | url=http://biocongroup.eu/DA/Calendario_files/Bilateria.pdf | title=Animal Diversity (Third Edition) | publisher=McGraw-Hill | work=Chapter 8: Acoelomate Bilateral Animals | year=2002 | accessdate=October 25, 2012 | author=Hickman, Cleveland P.; Roberts, Larry S.; Larson, Allan | pages=139}}</ref>

Plants and sessile (attached) animals such as [[sea anemone]]s often have radial or [[rotational symmetry]], which suits them because food or threats may arrive from any direction. Fivefold symmetry is found in the [[echinoderms]], the group that includes [[starfish]], [[sea urchin]]s, and [[sea lilies]].<ref>{{cite book | title=What Shape is a Snowflake? Magical Numbers in Nature | publisher=Weidenfeld & Nicolson | author=Stewart, Ian | year=2001 | pages=64–65}}</ref>

===In chemistry===
{{Main|molecular symmetry}}

Symmetry is important to [[chemistry]] because it explains observations in [[spectroscopy]], [[quantum chemistry]] and [[crystallography]]. It draws heavily on [[group theory]].

==In history, religion, and culture==

===In social interactions===
People observe the symmetrical nature, often including asymmetrical balance, of social interactions in a variety of contexts. These include assessments of reciprocity, empathy, apology, dialog, respect, justice, and revenge. Symmetrical interactions send the message "we are all the same" while asymmetrical interactions send the message "I am special; better than you." Peer relationships are based on symmetry, power relationships are based on asymmetry.<ref>[http://www.emotionalcompetency.com/symmetry.htm Emotional Competency] Entry describing Symmetry</ref>

===In architecture===
[[File:Isfahan Lotfollah mosque ceiling symmetric.jpg|right|thumb|The ceiling of [[Lotfollah mosque]], [[Isfahan]], [[Iran]] has rotational symmetry of order eight and eight lines of reflection.]]
[[File:Lightmatter pisa.jpg|thumb|left|upright|Leaning Tower of Pisa]]
[[File:Taj Mahal, Agra views from around (85).JPG|right|thumb|The Taj Mahal has bilateral symmetry.]]

Architects have long employed overall symmetry as a method of telling the viewer that a building is intended to convey an important idea, that the building is a physical expression of an important belief and ideals of the people who commissioned it. Symmetry endows a formal quality that distinguishes itself from the asymmetrical forms that are more casual. Such a building with an expressive intention is therefore, by definition, monumental, as it seeks to memorialize an idea. Symmetry is an expressive device, often used in conjunction with other methods like magnified size, elevation above ground level, and use of permanent materials. All are ways of communicating the importance and validity of the idea that is intended to be memorialized.

Examples of ancient architectures that made powerful use of symmetry to convey importance included the [[Egypt]]ian [[Pyramids]], the [[Greece|Greek]] [[Parthenon]], the first and second [[Temple of Jerusalem]], China's [[Forbidden City]], [[Cambodia]]'s [[Angkor Wat]] complex, and the many temples and pyramids of ancient [[Pre-Columbian]] civilizations. More recent historical examples of architectures emphasizing symmetries include [[Gothic architecture]] cathedrals, and [[United States|American]] President [[Thomas Jefferson]]'s [[Monticello]] home. The [[Taj Mahal]] is also an example of symmetry.<ref>{{cite book|author=Derry, Gregory N. |title=What Science Is and How It Works |url=http://books.google.com/books?id=Dk-xS6nABrYC&pg=PA269 |year=2002 |publisher=Princeton University Press |isbn=978-1-4008-2311-6 |pages=269–}}</ref>

An example of a [[broken symmetry]] in architecture is the [[Leaning Tower of Pisa]], whose notoriety stems not from the intended symmetry of its design, but from the violation of that symmetry from the lean that developed while it was still under construction. It conveys the failure of the idea it sought to memorialize. Modern examples of architectures that make impressive or complex use of various symmetries include [[Australia]]'s [[Sydney Opera House]] and [[Houston, Texas]]'s simpler [[Astrodome]].

Symmetry finds its ways into architecture at every scale, from the overall external views, through the layout of the individual [[floor plan]]s, and down to the design of individual building elements such as intricately carved doors, [[stained glass window]]s, [[mosaic|tile mosaics]], [[frieze]]s, stairwells, stair rails, and [[balustrades]]. For sheer complexity and sophistication in the exploitation of symmetry as an architectural element, [[Islam]]ic buildings such as the Taj Mahal often eclipse those of other cultures and ages, due in part to the general prohibition of Islam against using images of people or animals.<ref>[http://members.tripod.com/vismath/kim/ Williams: Symmetry in Architecture]. Members.tripod.com (1998-12-31). Retrieved on 2013-04-16.</ref><ref>[http://www.math.nus.edu.sg/aslaksen/teaching/math-art-arch.shtml Aslaksen: Mathematics in Art and Architecture]. Math.nus.edu.sg. Retrieved on 2013-04-16.</ref>[[Modernist architecture]], starting with [[International style (architecture)|International style]], sometimes rejects symmetry, relying on wings and balance of masses.{{citation needed|date=May 2013}}

===In pottery and metal vessels===
[[File:Makingpottery.jpg|thumb|Clay pots thrown on a [[pottery wheel]] acquire rotational symmetry.]]

Since the earliest uses of [[pottery wheel]]s to help shape clay vessels, pottery has had a strong relationship to symmetry. Pottery created using a wheel acquires full rotational symmetry in its cross-section, while allowing substantial freedom of shape in the vertical direction. Upon this inherently symmetrical starting point, potters from ancient times onwards have added patterns that modify the rotational symmetry to achieve visual objectives. For example, [[Persia]]n pottery dating from the fourth millennium BC and earlier used symmetric zigzags, squares, cross-hatchings, and repetitions of figures to produce visually striking overall designs.{{citation needed|date=May 2013}}

Cast metal vessels lacked the inherent rotational symmetry of wheel-made pottery, but otherwise provided a similar opportunity to decorate their surfaces with patterns pleasing to those who used them. The ancient [[Chinese people|Chinese]], for example, used symmetrical patterns in their bronze castings as early as the 17th century BC. Bronze vessels exhibited both a bilateral main motif and a repetitive translated border design.<ref>[http://www.chinavoc.com/arts/handicraft/bronze.htm The Art of Chinese Bronzes]. Chinavoc (2007-11-19). Retrieved on 2013-04-16.</ref><ref>[http://www-oi.uchicago.edu/OI/MUS/VOL/NN_SUM94/NN_Sum94.html Grant: Iranian Pottery in the Oriental Institute]. Oi.uchicago.edu (2007-07-30). Retrieved on 2013-04-16.</ref><ref>[http://web.archive.org/web/20040203213334/http://www.metmuseum.org/collections/department.asp?dep=14 Introduction to Islamic Art]. The Metropolitan Museum of Art, New York.</ref>

===In quilts===
[[File:kitchen kaleid.svg|thumb|120px|left|Kitchen Kaleidoscope Block]]

As [[quilt]]s are made from square blocks (usually 9, 16, or 25 pieces to a block) with each smaller piece usually consisting of fabric triangles, the craft lends itself readily to the application of symmetry.<ref>[http://its.guilford.k12.nc.us/webquests/quilts/quilts.htm Quate: Exploring Geometry Through Quilts]. Its.guilford.k12.nc.us. Retrieved on 2013-04-16.</ref>

===In carpets and rugs===
[[File:Farsh1.jpg|thumb|300px|right|Persian rug.]]

A long tradition of the use of symmetry in [[carpet]] and rug patterns spans a variety of cultures. American [[Navajo people|Navajo]] Indians used bold diagonals and rectangular motifs. Many [[Oriental rugs]] have intricate reflected centers and borders that translate a pattern. Not surprisingly, rectangular rugs typically use quadrilateral symmetry—that is, [[Motif (visual arts)|motifs]] that are reflected across both the horizontal and vertical axes.<ref>[http://web.archive.org/web/20010203155200/http://marlamallett.com/default.htm Marla Mallett Textiles & Tribal Oriental Rugs]. The Metropolitan Museum of Art, New York.</ref><ref>[http://navajocentral.org/rugs.htm Dilucchio: Navajo Rugs]. Navajocentral.org (2003-10-26). Retrieved on 2013-04-16.</ref>

===In music===
<imagemap>
File:Major and minor triads.png|thumb|right|Major and minor triads on the white piano keys are symmetrical to the D. [[Major and minor#Major and minor scales|(compare article)]] [[:File:Major and minor triads.png|(file)]]

poly 35 442 35 544 179 493 [[A minor|root of A minor triad]]
poly 479 462 446 493 479 526 513 492 [[A minor|third of A minor triad]]
poly 841 472 782 493 840 514 821 494 [[A minor|fifth of A minor triad]]
poly 926 442 875 460 906 493 873 525 926 545 [[A minor|fifth of A minor triad]]
poly 417 442 417 544 468 525 437 493 469 459 [[C major|root of C major triad]]
poly 502 472 522 493 502 514 560 493 [[C major|root of C major triad]]
poly 863 462 830 493 863 525 895 493 [[C major|third of C major triad]]
poly 1303 442 1160 493 1304 544 [[C major|fifth of C major triad]]
poly 280 406 264 413 282 419 275 413 [[E minor|fifth of E minor triad]]
poly 308 397 293 403 301 412 294 423 309 428 [[E minor|fifth of E minor triad]]
poly 844 397 844 428 886 413 [[E minor|root of E minor triad]]
poly 1240 404 1230 412 1239 422 1250 412 [[E minor|third of E minor triad]]
poly 289 404 279 413 288 422 300 413 [[G major|third of G major triad]]
poly 689 398 646 413 689 429 [[G major|fifth of G major triad]]
poly 1221 397 1222 429 1237 423 1228 414 1237 403 [[G major|root of G major triad]]
poly 1249 406 1254 413 1249 418 1265 413 [[G major|root of G major triad]]
poly 89 567 73 573 90 579 86 573 [[D minor|fifth of D minor triad]]
poly 117 558 102 563 111 572 102 583 118 589 [[D minor|fifth of D minor triad]]
poly 650 558 650 589 693 573 [[D minor|root of D minor triad]]
poly 1050 563 1040 574 1050 582 1061 574 [[D minor|third of D minor triad]]
poly 98 565 88 573 98 583 110 574 [[F major|third of F major triad]]
poly 498 558 455 573 498 589 [[F major|fifth of F major triad]]
poly 1031 557 1031 589 1047 583 1038 574 1046 563 [[F major|root of F major triad]]
poly 1075 573 1059 580 1064 573 1058 567 [[F major|root of F major triad]]

desc none
</imagemap>

Symmetry is not restricted to the visual arts. Its role in the history of [[music]] touches many aspects of the creation and perception of music.

====Musical form====

Symmetry has been used as a [[musical form|formal]] constraint by many composers, such as the [[arch form|arch (swell) form]] (ABCBA) used by [[Steve Reich]], [[Béla Bartók]], and [[James Tenney]]. In classical music, Bach used the symmetry concepts of permutation and invariance.<ref>see ("Fugue No. 21," [http://jan.ucc.nau.edu/~tas3/wtc/ii21s.pdf pdf] or [http://jan.ucc.nau.edu/~tas3/wtc/ii21.html Shockwave])</ref>

====Pitch structures====
Symmetry is also an important consideration in the formation of [[scale (music)|scale]]s and [[chord (music)|chords]], traditional or [[tonality|tonal]] music being made up of non-symmetrical groups of [[pitch (music)|pitches]], such as the [[diatonic scale]] or the [[major chord]]. [[Symmetrical scale]]s or chords, such as the [[whole tone scale]], [[augmented chord]], or diminished [[seventh chord]] (diminished-diminished seventh), are said to lack direction or a sense of forward motion, are [[ambiguous]] as to the [[Key (music)|key]] or tonal center, and have a less specific [[diatonic functionality]]. However, composers such as [[Alban Berg]], [[Béla Bartók]], and [[George Perle]] have used axes of symmetry and/or [[interval cycle]]s in an analogous way to [[musical key|keys]] or non-[[tonality|tonal]] tonal [[Tonic (music)|center]]s.

Perle (1992){{clarify|date=May 2013}} explains "C–E, D–F♯, [and] Eb–G, are different instances of the same [[interval (music)|interval]] … the other kind of identity. … has to do with axes of symmetry. C–E belongs to a family of symmetrically related dyads as follows:"

{|
|-
|D
|
|D♯
|
|'''E'''
|
|F
|
|F♯
|
|G
|
|G♯
|-
|D
|
|C♯
|
|'''C'''
|
|B
|
|A♯
|
|A
|
|G♯
|}

Thus in addition to being part of the interval-4 family, C–E is also a part of the sum-4 family (with C equal to 0).

{|
|rowspan=3|+
|2
|
|3
|
|'''4'''
|
|5
|
|6
|
|7
|
|8
|-
|2
|
|1
|
|'''0'''
|
|11
|
|10
|
|9
|
|8
|-
|4
|
|4
|
|4
|
|4
|
|4
|
|4
|
|4
|}

Interval cycles are symmetrical and thus non-diatonic. However, a seven pitch segment of C5 (the cycle of fifths, which are [[enharmonic]] with the cycle of fourths) will produce the diatonic major scale. Cyclic tonal [[chord progression|progressions]] in the works of [[Romantic music|Romantic]] composers such as [[Gustav Mahler]] and [[Richard Wagner]] form a link with the cyclic pitch successions in the atonal music of Modernists such as Bartók, [[Alexander Scriabin]], [[Edgard Varèse]], and the Vienna school. At the same time, these progressions signal the end of tonality.

The first extended composition consistently based on symmetrical pitch relations was probably Alban Berg's ''Quartet'', Op. 3 (1910).<ref>{{cite book | title=The Listening Composer | publisher=University of California Press | author=Perle, George | year=1990}}</ref>

====Equivalency====

[[Tone row]]s or [[pitch class]] [[Set theory (music)|sets]] which are [[Invariant (music)|invariant]] under [[Permutation (music)|retrograde]] are horizontally symmetrical, under [[inversion (music)|inversion]] vertically. See also [[Asymmetric rhythm]].

===In other arts and crafts===
[[File:Celticknotwork.png|frame|[[Celtic knot]]work]]

Symmetries appear in the design of objects of all kinds. Examples include [[beadwork]], [[furniture]], [[sand painting]]s, [[knot]]work, [[masks]], and [[musical instruments]]. Symmetries are central to the art of [[M.C. Escher]] and the many applications of [[tessellation|tiling]].

===In aesthetics===
{{Main|Symmetry (physical attractiveness)}}

The relationship of symmetry to [[aesthetics]] is complex. Humans use [[bilateral symmetry]] to judge the likely health or fitness of other people.{{citation needed|date=May 2013}} Opposed to this is the tendency for excessive symmetry to be perceived as boring or uninteresting. People prefer shapes that have some symmetry, but enough complexity to make them interesting.<ref>{{cite book|last = Arnheim|first = Rudolf|title = Visual Thinking|publisher = University of California Press|year = 1969}}</ref>

==See also==
{{colbegin|2}}
'''Symmetry in [[statistics]]'''
*[[Skewness]], asymmetry of a statistical distribution

'''Symmetry in games and puzzles'''
*[[Symmetric games]]
*[[Sudoku]]

'''Symmetry in literature'''
*[[Palindrome]]

'''Moral symmetry'''
*[[Empathy]] and [[Sympathy]]
*[[Golden Rule]]
*[[Reciprocity (social psychology)|Reciprocity]]
*[[Reflective equilibrium]]
*[[Tit for tat]]

'''Other'''
*[[Asymmetric rhythm]]
*[[Asymmetry]]
*[[Burnside's lemma]]
*[[Chirality (mathematics)|Chirality]]
*[[M.C. Escher]]
*[[Even and odd functions]]
*[[Fixed points of isometry groups in Euclidean space]] – center of symmetry
*[[Gödel, Escher, Bach]]
*[[Ignacio Matte Blanco]]
*[[Semimetric]], which is sometimes translated as symmetric in Russian texts.
*[[Spacetime symmetries]]
*[[Spontaneous symmetry breaking]]
*[[Symmetric relation]]
*[[Polyiamond#Symmetries|Symmetries of polyiamonds]]
*[[Free polyomino|Symmetries of polyominoes]]
*[[Symmetry group]]
*[[T-symmetry|Time symmetry]]
*[[Wallpaper group]]
{{colend}}

==References==
{{reflist|26em}}

==Further reading==
* ''The Equation That Couldn't Be Solved: How Mathematical Genius Discovered the Language of Symmetry'', [[Mario Livio]], [[Souvenir Press]] 2006, ISBN 0-285-63743-6

==External links==
{{Wiktionary}}
{{Commons category|Symmetry}}
*[http://www.teachersnetwork.org/teachnet/westchester/symmetry.htm Calotta: A World of Symmetry]
*[http://www.uwgb.edu/dutchs/SYMMETRY/2DPTGRP.HTM Dutch: Symmetry Around a Point in the Plane]
*[http://home.earthlink.net/~jdc24/symmetry.htm Chapman: Aesthetics of Symmetry]
*[http://www.mi.sanu.ac.rs/~jablans/isis0.htm ISIS Symmetry]
*[http://www.symmetry.hu/isa_articles.html ''International Symmetry Association – ISA'']
*Institute [http://www.symmetry.hu/aus_symmetrion.html ''Symmetrion'']

[[Category:Symmetry|*]]

Linear feedback shift register

2014-01-02T03:22:03Z

98.169.80.172: /* Fibonacci LFSRs */

{{Statistical mechanics|cTopic=Ensembles}}

In [[mathematical physics]], especially as introduced into [[statistical mechanics]] and [[thermodynamics]] by [[Willard Gibbs|J. Willard Gibbs]] in 1902, an '''ensemble''' (also '''statistical ensemble''') is an idealization consisting of a large number of virtual copies (sometimes infinitely many) of a [[system]], considered all at once, each of which represents a possible state that the real system might be in. In other words, a statistical ensemble is a probability distribution for the state of the system.<ref name="gibbs">{{cite book |last=Gibbs |first=Josiah Willard |authorlink=Josiah Willard Gibbs |title=[[Elementary Principles in Statistical Mechanics]] |year=1902 |publisher=[[Charles Scribner's Sons]] |location=New York}}</ref>

A '''thermodynamic ensemble''' is a specific variety of statistical ensemble that, among other properties, is in statistical equilibrium (defined below), and is used to derive the properties of [[thermodynamic system]]s from the laws of classical or quantum mechanics.<ref name="Kittel">{{cite book |last=Kittel |first=Charles |authorlink=Charles Kittel |coauthors=Herbert Kroemer |title=Thermal Physics, Second Edition |publisher=W.H. Freeman and Company |year=1980 |isbn=0-7167-1088-9 |location=San Francisco |pages=31 ff}}</ref><ref name="Landau">{{cite book |title=Statistical Physics |last=Landau |first=L.D. |coauthors=Lifshitz, E.M. |isbn=0-08-023038-5 |authorlink=Lev Landau |year=1980 |publisher=Pergamon Press |pages=9 ff }}</ref>

This article treats the notion of ensembles in a mathematically rigorous fashion, although relevant physical aspects will be mentioned.

== Physical considerations ==

The ensemble formalises the notion that an experimenter repeating an experiment again and again under the same macroscopic conditions, but unable to control the microscopic details, may expect to observe a range of different outcomes.

The notional size of ensembles in thermodynamics, statistical mechanics and [[quantum statistical mechanics]] can be very large indeed, to include every possible [[microstate (statistical mechanics)|microscopic state]] the system could be in, consistent with its observed [[macroscopic]] properties. But for important physical cases it can be possible to calculate averages directly over the whole of the thermodynamic ensemble, to obtain explicit formulas for many of the thermodynamic quantities of interest, often in terms of the appropriate [[partition function (mathematics)|partition function]] (see below). Some of these results are presented in the [[statistical mechanics]] article.

The concept of an equilibrium or stationary ensemble is crucial to some applications of statistical ensembles. Although a mechanical system certainly evolves over time, the ensemble does not necessarily have to evolve. In fact, the ensemble will not evolve if it equally contains all past and future phases of the system. Such a statistical ensemble, that does not change over time, is called ''stationary'' or said to be in ''statistical equilibrium''.<ref name="gibbs"/>

=== Note on terminology ===

*The word "ensemble" is also used for a smaller set of possibilities [[sample (statistics)|sampled]] from the full set of possible states. For example, a collection of [[random walk|walkers]] in a [[Markov chain Monte Carlo]] iteration is called an ensemble in some literature.

*The term "ensemble" is often used in physics and physics-influenced literature. In [[probability theory]] literature the term [[probability space]] is more prevalent.

== Principal ensembles of statistical thermodynamics ==

The study of thermodynamics is concerned with systems which appear to human perception to be "static" (despite the motion of their internal parts), and which can be described simply by a set of macroscopically observable variables. These systems can be described by statistical ensembles that depend on a few observable parameters, and which are in statistical equilibrium. Gibbs noted that different macroscopic constraints lead to different types of ensembles, with particular statistical characteristics. Three important thermodynamic ensembles were defined by Gibbs:<ref name="gibbs"/>

* [[Microcanonical ensemble]] or ''NVE'' ensemble—a statistical ensemble where the total energy of the system and the number of particles in the system are each fixed to particular values; each of the members of the ensemble are required to have the same total energy and particle number. The system must remain totally isolated (unable to exchange energy or particles with its environment) in order to stay in statistical equilibrium.<ref name="gibbs"/>

* [[Canonical ensemble]] or ''NVT'' ensemble—a statistical ensemble where the energy is not known exactly but the number of particles is fixed. In place of energy, the [[temperature]] is specified. The canonical ensemble is appropriate for describing a closed system which is in, or has been in, weak thermal contact with a heat bath. In order to be in statistical equilibrium the system must remain totally closed (unable to exchange particles with its environment), and may come into weak thermal contact with other systems that are described by ensembles with the same temperature.<ref name="gibbs"/>

* [[Grand canonical ensemble]] or ''µVT'' ensemble—a statistical ensemble where neither the energy nor particle number are fixed. In their place, the temperature and [[chemical potential]] are specified. The grand canonical ensemble is appropriate for describing an open system: one which is in, or has been in, weak contact with a reservoir (thermal contact, chemical contact, radiative contact, electrical contact, etc.). The ensemble remains in statistical equilibrium if the system comes into weak contact with other systems that are described by ensembles with the same temperature and chemical potential.<ref name="gibbs"/>

The calculations that can be made over each of these ensembles are explored further in their respective articles.
Other thermodynamic ensembles can be also defined, corresponding to different physical requirements, for which analogous formulae can often similarly be derived.

==Representations for statistical ensembles in statistical mechanics ==

The precise mathematical expression for a statistical ensemble has a distinct form depending on the type of mechanics under consideration (quantum or classical). In each case the ensemble is a probability distribution over microstates, but the notion of a "microstate" is considerably different. In quantum mechanics, a set of [[Hilbert space]] basis vectors can be used as microstates, however the ensemble might also include superpositions of these microstates.
In classical mechanics, the ensemble is instead written as a probability distribution in [[phase space]]; the microstates are the result of partitioning phase space into equal-sized units, although the size of these units can be chosen somewhat arbitrarily.

=== Requirements for representations ===

Putting aside for the moment the question of how statistical ensembles are generated [[operational definition|operationally]], we should be able to perform the following two operations on ensembles ''A'', ''B'' of the same system:

* Test whether ''A'', ''B'' are statistically equivalent.

* If ''p'' is a real number such that 0 < ''p'' < 1, then produce a new ensemble by probabilistic sampling from ''A'' with probability ''p'' and from ''B'' with probability ''1 – p''.

Under certain conditions therefore, [[equivalence class]]es of statistical ensembles have the structure of a convex set.

=== Quantum mechanical ===

{{main|density matrix}}

A statistical ensemble in quantum mechanics is most often represented by a [[density matrix]], denoted by {{math|''ρ̂''}}. The density matrix provides a fully general tool that can incorporate both quantum uncertainties (due to superposition) and classical uncertainties (due to lack of knowledge), in a consistent manner. Any physical observable {{math|''X''}} in quantum mechanics can be written as an operator, {{math|''X̂''}}. The expectation value of this operator is given by a [[trace (linear algebra)|trace]]:
:<math>\langle X \rangle = \operatorname{Tr}(\hat X \rho).</math>
This can be used to evaluate averages (operator {{math|''X̂''}}), [[variance]]s (using operator {{math|''X̂''2}}), [[covariance]]s (using operator {{math|''X̂Ŷ''}}), etc. The density matrix must always have a trace of 1: {{math|Tr ''ρ̂'' {{=}} 1}} (this essentially is the condition that the probabilities must add up to one).

In general, the ensemble evolves over time according to the [[von Neumann equation]].

Equilibrium ensembles (those that do not evolve over time, {{math|''dρ̂''/''dt'' {{=}} 0}}) can be written solely as a function of conserved variables. For example, the [[microcanonical ensemble]] and [[canonical ensemble]] are functions strictly of the total energy, which is measured by the total energy operator {{math|''Ĥ''}} (Hamiltonian). The grand canonical ensemble is further a function of the particle number, measured by the total particle number operator {{math|''N̂''}}. Such equilibrium ensembles are a [[diagonal matrix]] in the orthogonal basis of states that simultaneously diagonalize each conserved variable. In [[bra-ket notation]], the density matrix is
:<math>\hat \rho = \sum_i P_i |\psi_i\rangle \langle \psi_i | </math>
where the {{math|{{!}}''ψ''''i''⟩}}, indexed by {{math|''i''}}, are the complete and orthogonal basis. In other words, these states can be considered to be the microstates in quantum mechanics, and the density matrix encodes a probability distribution over these microstates. (Note that in other bases, the density matrix is not necessarily diagonal.)

===Classical mechanical===

[[File:Hamiltonian flow classical.gif|frame|Evolution of an ensemble of [[Hamiltonian mechanics|classical]] systems in [[phase space]] (top). Each system consists of one massive particle in a one-dimensional [[potential well]] (red curve, lower figure). The initially compact ensemble becomes swirled up over time.]]

In classical mechanics, an ensemble is represented by a probability density function defined over the system's [[phase space]].<ref name="gibbs"/> While an individual system evolves according to [[Hamilton's equations]], the density function (the ensemble) evolves over time according to [[Liouville's theorem (Hamiltonian)|Liouville's equation]].

In a [[Hamiltonian mechanics|mechanical system]] with a defined number of parts, the phase space has {{math|''n''}} [[generalized coordinates]] called {{math|''q''1, … ''q''''n''}}, and {{math|''n''}} associated [[canonical momentum|canonical momenta]] called {{math|''p''1, … ''p''''n''}}. The ensemble is then represented by a [[joint probability density function]] {{math|''ρ''(''p''1, … ''p''''n'', ''q''1, … ''q''''n'')}}.

If the number of parts in the system is allowed to vary among the systems in the ensemble (as in a grand ensemble where the number of particles is a random quantity), then it is a probability distribution over an extended phase space that includes further variables such as particle numbers {{math|''N''1}} (first kind of particle), {{math|''N''2}} (second kind of particle), and so on up to {{math|''N''''s''}} (the last kind of particle; {{math|''s''}} is how many different kinds of particles there are). The ensemble is then represented by a [[joint probability density function]] {{math|''ρ''(''N''1, … ''N''''s'', ''p''1, … ''p''''n'', ''q''1, … ''q''''n'')}}. The number of coordinates {{math|''n''}} varies with the numbers of particles.

Any physical quantity {{math|''X''}} can be written as a function of the system's phase. The expectation value of any quantity is given by an integral over the entire phase space of this quantity weighted by {{math|''ρ''}}:
:<math>\langle X \rangle = \sum_{N_1 = 0}^{\infty} \ldots \sum_{N_s = 0}^{\infty} \int \ldots \int \rho X \, dp_1 \ldots dq_n.</math>
The condition of probability normalization applies, requiring
:<math>\sum_{N_1 = 0}^{\infty} \ldots \sum_{N_s = 0}^{\infty} \int \ldots \int \rho \, dp_1 \ldots dq_n = 1.</math>

Phase space is a continuous space containing an infinite number of distinct physical states within any small region. In order to connect the probability ''density'' in phase space to a probability ''distribution'' over microstates, it is necessary to somehow partition the phase space into blocks that are distributed represent the different states of the system in a fair way. It turns out that the correct way to do this simply results in equal-sized blocks of canonical phase space, and so a microstate in classical mechanics is an extended region in the phase space of canonical coordinates that has a particular volume.<ref group=note>This equal-volume partitioning is a consequence of [[Liouville's theorem (Hamiltonian)|Liouville's theorem]], i. e., the principle of conservation of extension in canonical phase space for Hamiltonian mechanics. This can also be demonstrated starting with the conception of an ensemble as a multitude of systems. See Gibbs' ''Elementary Principles'', Chapter I.</ref> In particular, the probability density function in phase space, {{math|''ρ''}}, is related to the probability distribution over microstates, {{math|''P''}} by a factor
:<math>\rho = \frac{1}{h^n C} P,</math>
where
* {{math|''h''}} is an arbitrary but predetermined constant with the units of {{math|energy×time}}, setting the extent of the microstate and providing correct dimensions to {{math|''ρ''}}.<ref group=note>(Historical note) Gibbs' original ensemble effectively set {{math|''h'' {{=}} 1 [energy unit]×[time unit]}}, leading to unit-dependence in the values of some thermodynamic quantities like entropy and chemical potential. Since the advent of quantum mechanics, {{math|''h''}} is often taken to be equal to [[Planck's constant]] in order to obtain a semiclassical correspondence with quantum mechanics.</ref>
* {{math|''C''}} is an overcounting correction factor (see below), generally dependent on the number of particles and similar concerns.
Since {{math|''h''}} can be chosen arbitrarily, the notional size of a microstate is also arbitrary. Still, the value of {{math|''h''}} influences the offsets of quantities such as entropy and chemical potential, and so it is important to be consistent with the value of {{math|''h''}} when comparing different systems.

==== Correcting overcounting in phase space ====

Typically, the phase space contains duplicates of the same physical state in multiple distinct locations. This is a consequence of the way that a physical state is encoded into mathematical coordinates; the simplest choice of coordinate system often allows a state to be encoded in multiple ways. An example of this is a gas of identical particles whose state is written in terms of the particles' individual positions and momenta: when two particles are exchanged, the resulting point in phase space is different, and yet it corresponds to an identical physical state of the system. It is important in statistical mechanics (a theory about physical states) to recognize that the phase space is just a mathematical construction, and to not naively overcount actual physical states when integrating over phase space. Overcounting can cause serious problems:
* Dependence of derived quantities (such as entropy and chemical potential) on the choice of coordinate system, since one coordinate system might show more or less overcounting than another.<ref group=note>In some cases the overcounting error is benign. An example is the [[Charts on SO(3)|choice of coordinate system used for representing orientations of three dimensional objects]]. A simple encoding is the [[3-sphere]] (e. g., unit [[quaternion]]s) which is a [[double covering group|double cover]]—each physical orientation can be encoded in two ways. If this encoding is used without correcting the overcounting, then the entropy will be higher by {{math|''k'' log 2}} per rotatable object and the chemical potential lower by {{math|''kT'' log 2}}. This does not actually lead to any observable error since it only causes unobservable offsets.</ref>
* Erroneous conclusions that are inconsistent with physical experience, as in the [[mixing paradox]].<ref name="gibbs"/>
* Foundational issues in defining the [[chemical potential]] and the [[grand canonical ensemble]].<ref name="gibbs"/>
It is in general difficult to find a coordinate system that uniquely encodes each physical state. As a result, it is usually necessary to use a coordinate system with multiple copies of each state, and then to recognize and remove the overcounting.

A crude way to remove the overcounting would be to manually define a subregion of phase space that includes each physical state only once, and then exclude all other parts of phase space. In a gas, for example, one could include only those phases where the particles' {{math|''x''}} coordinates are sorted in ascending order. While this would solve the problem, the resulting integral over phase space would be tedious to perform due to its unusual boundary shape. (In this case, the factor {{math|''C''}} introduced above would be set to {{math|''C'' {{=}} 1}}, and the integral would be restricted to the selected subregion of phase space.)

A simpler way to correct the overcounting is to integrate over all phase space, but to reduce the weight of each phase in order to exactly compensate the overcounting. This is accomplished by the factor {{math|''C''}} introduced above, which is a whole number that represents how many ways a physical state can be represented in phase space. Its value does not vary with the continuous canonical coordinates,<ref group=note>Technically, there are some phases where the permutation of particles does not even yield a distinct specific phase: for example, two similar particles can share the exact same trajectory, internal state, etc.. However, in classical mechanics these phases only make up an infinitesimal fraction of the phase space (they have [[measure (mathematics)|measure]] zero) and so they do not contribute to any volume integral in phase space.</ref> so overcounting can be corrected simply by integrating over the full range of canonical coordinates, then dividing the result by the overcounting factor. However, {{math|''C''}} does vary strongly with discrete variables such as numbers of particles, and so it must be applied before summing over particle numbers.

As mentioned above, the classic example of this overcounting is for a fluid system containing various kinds of particles, where any two particles of the same kind are indistinguishable and exchangable. When the state is written in terms of the particles' individual positions and momenta, then the overcounting related to the exchange of identical particles is corrected by using<ref name="gibbs"/>
:<math>C = N_1! N_2! \ldots N_s!.</math>
This is known as "correct Boltzmann counting".

== Ensembles in statistics ==
:{{main|Principle of maximum entropy|Markov random field}}
The formulation of statistical ensembles used in physics has now been widely adopted in other fields, in part because it has been recognized that the [[canonical ensemble]] or [[Gibbs measure]] serves to maximize the entropy of a system, subject to a set of constraints: this is the [[principle of maximum entropy]]. This principle has now been widely applied to problems in [[linguistics]], [[robotics]], and the like.

In addition, statistical ensembles in physics are often built on a [[principle of locality]]: that all interactions are only between neighboring atoms or nearby molecules. Thus, for example, [[lattice model (physics)|lattice models]], such as the [[Ising model]], model [[ferromagnetic]] materials by means of nearest-neighbor interactions between spins. The statistical formulation of the principle of locality is now seen to be a form of the [[Markov property]] in the broad sense; nearest neighbors are now [[Markov blanket]]s. Thus, the general notion of a statistical ensemble with nearest-neighbor interactions leads to [[Markov random field]]s, which again find broad applicability; for example in [[Hopfield network]]s.

== Operational interpretation ==

In the discussion given so far, while rigorous, we have taken for granted that the notion of an ensemble is valid a priori, as is commonly done in physical context. What has not been shown is that the ensemble ''itself'' (not the consequent results) is a precisely defined object mathematically. For instance,

* It is not clear where this ''very large set of systems'' exists (for example, is it a [[gas in a box|''gas'' of particles inside a container]]?)

* It is not clear how to physically generate an ensemble.

In this section we attempt to partially answer this question.

Suppose we have a ''preparation procedure'' for a system in a physics
lab: For example, the procedure might involve a physical apparatus and
some protocols for manipulating the apparatus. As a result of this preparation procedure some system
is produced and maintained in isolation for some small period of time.
By repeating this laboratory preparation procedure we obtain a
sequence of systems ''X''1, ''X''2,
....,''X''''k'', which in our mathematical idealization, we assume is an [[Infinity|infinite]] sequence of systems. The systems are similar in that they were all produced in the same way. This infinite sequence is an ensemble.

In a laboratory setting, each one of these prepped systems might be used as input
for ''one'' subsequent ''testing procedure''. Again, the testing procedure
involves a physical apparatus and some protocols; as a result of the
testing procedure we obtain a ''yes'' or ''no'' answer.
Given a testing procedure ''E'' applied to each prepared system, we obtain a sequence of values
Meas (''E'', ''X''1), Meas (''E'', ''X''2),
...., Meas (''E'', ''X''''k''). Each one of these values is a 0 (or no) or a 1 (yes).

Assume the following time average exists:
:<math> \sigma(E) = \lim_{N \rightarrow \infty} \frac{1}{N} \sum_{k=1}^N \operatorname{Meas}(E, X_k) </math>
For quantum mechanical systems, an important assumption made in the
[[quantum logic]] approach to quantum mechanics is the identification of ''yes-no'' questions to the
lattice of closed subspaces of a Hilbert space. With some additional
technical assumptions one can then infer that states are given by
density operators ''S'' so that:
:<math> \sigma(E) = \operatorname{Tr}(E S). </math>

We see this reflects the definition of quantum states in general: A quantum state is a mapping from the observables to their expectation values.

==See also==
* [[density matrix]]
* [[ensemble (fluid mechanics)]]
* [[phase space]]
* [[Liouville's theorem (Hamiltonian)]]
* [[ensemble average]]
* [[replication (statistics)]]

==Notes==
{{reflist|group=note}}

==References==
{{commons category|Statistical ensemble}}

{{reflist}}

{{Statistical mechanics topics}}

[[Category:Concepts in physics]]
[[Category:Statistical ensembles| ]]
[[Category:Philosophy of thermal and statistical physics]]

Binary operation

2013-12-31T16:33:27Z

98.169.80.172: added link to Set (mathematics)

{{distinguish|Bitwise operation}}
In [[mathematics]], a '''binary operation''' on a [[Set (mathematics)|set]] is a calculation involving two elements of the set (called [[operands]]) and producing another element of the set (more formally, an [[Operation (mathematics)|operation]] whose [[arity]] is two, and whose two [[Domain_of_a_function|domains]] and one [[codomain]] are (subsets of) the same set). Examples include the familiar [[elementary arithmetic]] operations of [[addition]], [[subtraction]], [[multiplication]] and [[division (mathematics)|division]]. Other examples are readily found in different areas of mathematics, such as [[vector addition]], [[matrix multiplication]] and [[Conjugation (group theory)|conjugation in groups]].

==Terminology==
More precisely, a binary operation on a [[Set (mathematics)|set]] ''S'' is a [[Map (mathematics)|map]] which sends elements of the [[Cartesian product]] {{nowrap|''S'' × ''S''}} to ''S'':<ref>{{harvnb|Rotman|1973|loc=pg. 1}}</ref><ref>{{harvnb|Hardy|Walker|2002|loc=pg. 176, Definition 67}}</ref><ref>{{harvnb|Fraleigh|1976|loc= pg. 10}}</ref>
:<math>\,f \colon S \times S \rightarrow S.</math>
Because the result of performing the operation on a pair of elements of ''S'' is again an element of ''S'', the operation is called a '''closed''' binary operation on ''S'' (or sometimes expressed as having the property of [[closure (mathematics)|closure]]).<ref>{{harvnb|Hall|1959|loc=pg. 1}}</ref> If ''f'' is not a [[Function (mathematics)|function]], but is instead a [[partial function]], it is called a '''partial binary operation'''. For instance, division of [[real numbers]] is a partial binary operation, because one can't [[Division by zero|divide by zero]]: ''a''/0 is not defined for any real ''a''. Note however that both in [[algebra]] and [[model theory]] the binary operations considered are defined on all of {{nowrap|''S'' × ''S''}}.

Sometimes, especially in [[computer science]], the term is used for any [[binary function]].

Binary operations are the keystone of algebraic structures studied in [[abstract algebra]]: they are essential in the definitions of [[group (mathematics)|groups]], [[monoid]]s, [[semigroup]]s, [[ring (algebra)|ring]]s, and more. Most generally, a ''[[magma (algebra)|magma]]'' is a set together with some binary operation defined on it.

== Properties and examples ==
Typical examples of binary operations are the [[addition]] (+) and [[multiplication]] (×) of [[number]]s and [[matrix (mathematics)|matrices]] as well as [[composition of functions]] on a single set.
For instance,
* On the set of real numbers '''R''', f(''a'',''b'') = ''a'' + ''b'' is a binary operation since the sum of two real numbers is a real number.
* On the set of natural numbers '''N''', f(''a'',''b'') = ''a'' + ''b'' is a binary operation since the sum of two natural numbers is a natural number. This is a different binary operation than the previous one since the sets are different.
* On the set M(2,2) of 2 × 2 matrices with real entries, f(''A'', ''B'') = ''A'' + ''B'' is a binary operation since the sum of two such matrices is another 2 × 2 matrix.
* On the set M(2,2) of 2 × 2 matrices with real entries, f(''A'', ''B'') = ''AB'' is a binary operation since the product of two such matrices is another 2 × 2 matrix.
* For a given set ''C'', let ''S'' be the set of all functions ''h'': ''C'' → ''C''. On ''S'', f(''g'',''h'') = ''g'' <math>\circ </math> ''h'' = ''g''(''h''(''c'')), the composition of the two functions ''g'' and ''h'', is a binary operation since the composition of the two functions is another function on the set ''C'' (that is, a member of ''S'').

Many binary operations of interest in both algebra and formal logic are [[commutative]], satisfying f(''a'',''b'') = f(''b'',''a'') for all elements ''a'' and ''b'' in ''S'', or [[associative]], satisfying f(f(''a'',''b''), ''c'') = f(''a'', f(''b'',''c'')) for all ''a'', ''b'' and ''c'' in ''S''. Many also have [[identity element]]s and [[inverse element]]s.

The first three examples above are commutative and all of the above examples are associative. The [[example of a commutative non-associative magma|paper-scissors-rock]] binary operation is commutative but not associative.

On the set of real numbers '''R''', [[subtraction]], that is, f(''a'',''b'') = ''a'' - ''b'', is a binary operation which is not commutative since, in general, ''a'' - ''b'' ≠ ''b'' - ''a''. It is also not associative, since, in general, ''a'' - (''b'' - ''c'') ≠ (''a'' - ''b'') - ''c''; for instance, 1 - (2 - 3) = 2 but (1 - 2) - 3 = -4.

On the set of natural numbers '''N''', the binary operation [[exponentiation]], f(''a'',''b'') = ''a''''b'', is not commutative since, in general, ''a''''b'' ≠ ''b''''a'' and is also not associative since f(f(''a'',''b''),''c'') ≠ f(''a'', f(''b'',''c'')). For instance, with ''a'' = 2, ''b'' = 3 and ''c'' = 2, f(23,2) = f(8,2) = 64, but f(2,32) = f(2,9) = 512. By changing the set '''N''' to the set of integers '''Z''', this binary operation becomes a partial binary operation since it is now undefined when ''a'' = 0 and ''b'' is any negative integer. For either set, this operation has a ''right identity'' (which is 1) since f(''a'', 1) = ''a'' for all ''a'' in the set, which is not an ''identity'' (two sided identity) since f(1, ''b'') ≠ ''b'' in general.

[[division (mathematics)|Division]] (/), a partial binary operation on the set of real or rational numbers, is not commutative or associative as well. [[Tetration]](↑↑), as a binary operation on the natural numbers, is not commutative nor associative and has no identity element.

==Notation==
Binary operations are often written using [[infix notation]] such as ''a''*''b'', ''a'' + ''b'', ''a'''''·'''''b'' or (by [[wikt:juxtaposition|juxtaposition]] with no symbol) ''ab'' rather than by functional notation of the form ''f''(''a'', ''b''). Powers are usually also written without operator, but with the second argument as [[superscript]].

Binary operations sometimes use prefix or (probably more often) postfix notation, both of which dispense with parentheses. They are also called, respectively, [[Polish notation]] and [[reverse Polish notation]].

==Pair and tuple==
A binary operation, ''ab'', depends on the [[ordered pair]] (''a, b'') and so (''ab'')''c'' (where the parentheses here mean first operate on the ordered pair (''a'', ''b'') and then operate on the result of that using the ordered pair ((''ab''), ''c'')) depends in general on the ordered pair ((''a'', ''b''), ''c''). Thus, for the general, non-associative case, binary operations can be represented with [[binary tree]]s.

However:
*If the operation is associative, (''ab'')''c'' = ''a''(''bc''), then the value of (''ab'')''c'' depends only on the [[tuple]] (''a'', ''b'', ''c'').
*If the operation is commutative, ''ab'' = ''ba'', then the value of (''ab'')''c'' depends only on { {''a'', ''b''}, ''c''}, where braces indicate [[multiset]]s.
*If the operation is both associative and commutative then the value of (''ab'')''c'' depends only on the multiset {''a'', ''b'', ''c''}.
*If the operation is associative, commutative and [[idempotent]], ''aa'' = ''a'', then the value of (''ab'')''c'' depends only on the [[Set (mathematics)|set]] {''a'', ''b'', ''c''}.

== Binary operations as ternary relations ==

A binary operation ''f'' on a set ''S'' may be viewed as a ''ternary'' [[Finitary relation|relation]] on ''S'', that is, the set of triples (''a'', ''b'', ''f(a,b)'') in ''S'' × ''S'' × ''S'' for all ''a'' and ''b'' in ''S''.

== External binary operations ==
An '''[[external (mathematics)|external]] binary operation''' is a binary function from ''K'' × ''S'' to ''S''. This differs from a binary operation in the strict sense in that ''K'' need not be ''S''; its elements come from ''outside''.

An example of an [[external (mathematics)|external]] binary operation is [[scalar multiplication]] in [[linear algebra]]. Here ''K'' is a [[field (mathematics)|field]] and ''S'' is a [[vector space]] over that field.

An [[external (mathematics)|external]] binary operation may alternatively be viewed as an [[group action|action]]; ''K'' is acting on ''S''.

Note that the [[dot product]] of two vectors is not a binary operation, external or otherwise, as it maps from ''S''× ''S'' to ''K'', where ''K'' is a field and ''S'' is a vector space over ''K''.

==See also==
* [[Iterated binary operation]]
* [[Operator (programming)]]
* [[Ternary operation]]
* [[Unary operation]]

== Notes==
{{reflist}}

== References==

* {{citation|last=Fraleigh|first=John B.|title=A First Course in Abstract Algebra|edition=2nd|publisher=Addison-Wesley|place=Reading|year=1976|isbn=0-201-01984-1}}

* {{citation|last=Hall, Jr.|first= Marshall|title=The Theory of Groups|publisher=Macmillan|place=New York|year=1959}}

* {{citation|last1=Hardy|first1=Darel W.|last2=Walker|first2=Carol L.|title=Applied Algebra: Codes, Ciphers and Discrete Algorithms|publisher=Prentice-Hall|place=Upper Saddle River, NJ|year=2003|isbn=0-13-067464-8}}

* {{citation|last=Rotman|first=Joseph J.|title=The Theory of Groups: An Introduction|publisher=Allyn and Bacon|place=Boston|year=1973|edition=2nd}}

== External links ==
* {{MathWorld|title=Binary Operation|urlname=BinaryOperation}}

{{DEFAULTSORT:Binary Operation}}
[[Category:Binary operations| ]]

Hypergeometric function

2012-08-16T20:18:55Z

98.169.97.37: /* Monodromy group */

Luke is often a celebrity inside the producing [http://lukebryantickets.pyhgy.com meet and greet justin bieber] and the career expansion first second to his third theatre recording, & , is the evidence. He burst on the scene in 1999 [http://minioasis.com on tour with luke bryan] regarding his amusing blend of straight down-house ease of access, motion picture legend fantastic seems and lyrics, is set t in the major way. The brand new a around the region graph and #2 around the put maps, creating it the second highest first appearance in those days of 2007 for the land performer. The kid of the , knows patience and perseverance are key elements when it comes to a prosperous occupation- . His first recording, Stay Me, produced the best reaches “All My Girlfriends “Country and Say” Guy,” while his hard work, Doin’ Issue, located the artist-three right No. 7 single people: In addition Getting in touch with Is a Good Thing.” During the tumble of 2003, Concerts: Luke Bryan And which had a remarkable list of , including City. “It’s almost like you’re receiving a endorsement to visit one stage further, states individuals artists that were an element [http://www.hotelsedinburgh.org meet and greet one direction] of the Tourabove in a greater level of musicians.” It covered as the best organized tours in their ten-season historical past. Look at my site - [http://lukebryantickets.flicense.com luke bryan tickets tampa]

Radius of curvature (applications)

2012-07-12T02:43:58Z

98.169.89.217: Undid revision 499129000 by 59.177.232.81 (talk)

Adrianne Le is the advertsing name my parents gave others but you can [http://En.wiktionary.org/wiki/connect connect] with me anything you like. My house is now all through South Carolina. Filing is generally my [http://en.wiktionary.org/wiki/day+job day job] but but soon I'll possibly be on my own. What me and my family appreciation is acting but As well as can't make it all my profession really. See what's new on my website here: http://circuspartypanama.com Here is my blog [http://circuspartypanama.com clash of clans hack apk download]

NearlyFreeSpeech.NET

2012-07-04T14:29:05Z

98.169.40.150: /* Pricing structure */

Nike clients are a great company that record the last thing great moments in different areas. Nike running shoes witnessed different great moments in past history. Nik football shoes in 2010 show the happiness of Spain to receiver. Nike Air Jordan shoes accompany Jordan went through his time during National basketball association. Each pair of Nike shoes complete the math in various areas, included in this is Nike basketball runners. You in regards to the frequent timeframe should contemplate our environment when getting air jordan shoes boots or shoes. for all folks that reside within on the mountains, or are preparing a trip there may advantage via the pair with cleats. very near to one other hand, at any time your house is within of the city, perfect afford to acquire one thing a package more modern. The only astonishingly important dilemma to contemplate within your shoes may be change that it might retain your childrens ft dried out also it might retain them from falling over. after the shoes has these qualities, every person a shoe-in. The third one will be the shooting opportunities, which is often rather significant on their shooting. Players must catch opportunities to shoot in the basket. Total team should cooperate well to obtain the chance gain. Players should observe their rivals carefully and catch any chances to do. It is especially important to make a right decision. These temporary time difference and space difference are beneficial for their heating. Yes Friend. Sometimes you get these ignorant folks who try educate you about the air jordan Shoes game promote you be a mess. But then you just laugh at such non sense. An individual know higher.But any how. We spend huge cash for what which bring elegance and enhance our attitude from a graceful depth. It means that style and quality are vital for us. Nike regarded are not aware in mind while raising its supreme class of foot wears i.e. Air jordan. These waders are fantastic options for people who consider quality longer than everything altogether different. It is a great choice red wine are mad with their old-fashioned and teasing shoes. It is because these waders are so comfortable uncomplicated to wear that therefore never feel any associated with uneasiness or discomfort while putting them on. Finally, you may also join an online discussion area. This could be several people of which are also contemplating about buying cheap Nike Nike air jordan Fusion shoes just through the night. By putting your heads together, you might find even more discounts and share valuable tips on buying cheap footwear. Possibly, you already have got all data you prefer to get about the pair you might have wanted for some time time. If you have any kind of queries with regards to where and also how you can employ [http://www.plongeeo.com/chaussure-air-jordan-pas-cher/ chaussure jordan pas cher], you'll be able to e mail us in our own webpage.

Multilinear map

2012-01-15T19:54:35Z

98.169.251.51: /* Multilinear functions on n×n matrices */

Before playing a new visual game, read the cheat book. Most dvds have a book people can purchase separately. You may want to consider doing this as well as , reading it before your corporation play, or even while you are playing. This way, you also can get the most in of your game be. 

To appreciate coins and gems, kind obtain the Clash along with Clans hack equipment a clicking on the obtain button. Contingent on their own operating framework that you are utilizing, you will labor the downloaded document mainly because admin. Furnish my log in Id and choose the gadget. Subsequent to this, you are enter into the quantity of necklaces or coins that individuals and start off my Clash of Clans crack instrument. You probably have little ones who experience video games, then you are aware how challenging it really is always to pull them out among the t. v.. If you have any type of concerns pertaining to where and exactly how to use clash of clans hack android ([http://prometeu.net you can try these out]), you could contact us at the webpage. Their eye can be stuck towards the maintain for hours as companies play their preferred games. If you want aid regulating your little ones clash of clans Hack time, your pursuing article has some hints for you. Truly possible, but the volume of absence one morning would abatement by 60mph one. 5% provided by 260 [http://www.dict.cc/englisch-deutsch/treasures.html treasures] to 200 gems. Or, in the you capital to erect up the 1 business day bulk at 260 gems, the band would endure to acceleration added steeply and also 1 wedding would turn into added expensive. Among the best and fastest harvesting certifications by ECCouncil. Where a dictionary attack fails the computer hacker may try a brute force attack, which is a bit more time consuming. Sets up the borders of the lot with non-editable flag: lot_border [ ]. The issue is this one hit you where it really affects - your heart. These Kindle hacks would be keyboard shortcuts will help save you tons of time excavating and typing in repeated things. Claire mentioned how she had started to gain a (not trivial. You can see for yourself that all of our Money Compromise of Clans i fairly effective, really invisible by the manager of the game, remarkably absolutely no price! Video games are some of the finest kinds of pleasure around. They happen to be also probably the on the whole pricey types of entertainment, with console games which range from $50 to $60, and consoles around their own inside ones 100s. It has always been possible to spend quantity on clash of clans hack and console purchases, and you can locate out about them throughout the the following paragraphs.