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The console was originally planned for a global release through November, but at the start of September the release in Europe and the rest of the world was delayed until March. With it being a somewhat last-minute delay, some companies had taken deposits for pre-orders, at which Sony informed customers that they were e... |
Following speculation that Sony was working on a 'slim' model, Sony officially announced the PS3 CECH-2000 model on August 18, 2009, at the Sony Gamescom press conference. New features included a slimmer form factor, decreased power consumption, and a quieter cooling system. It was released in major territories by Sept... |
In September 2012 at the Tokyo Game Show, Sony announced that a new, slimmer PS3 redesign (CECH-4000) was due for release in late 2012 and that it would be available with either a 250 GB or 500 GB hard drive. Three versions Super Slim model were revealed: one with a 500 GB hard drive, a second with a 250 GB hard drive ... |
PlayStation 3 launched in North America with 14 titles, with another three being released before the end of 2006. After the first week of sales it was confirmed that Resistance: Fall of Man from Insomniac Games was the top-selling launch game in North America. The game was heavily praised by numerous video game website... |
At E3 2007, Sony was able to show a number of their upcoming video games for PlayStation 3, including Heavenly Sword, Lair, Ratchet & Clank Future: Tools of Destruction, Warhawk and Uncharted: Drake's Fortune; all of which were released in the third and fourth quarters of 2007. They also showed off a number of titles t... |
Sony have since launched their budget range of PlayStation 3 titles, known as the Greatest Hits range in North America, the Platinum range in Europe and Australia and The Best range in Japan. Among the titles available in the budget range include Resistance: Fall of Man, MotorStorm, Uncharted: Drakes Fortune, Rainbow S... |
In December 2008, the CTO of Blitz Games announced that it would bring stereoscopic 3D gaming and movie viewing to Xbox 360 and PlayStation 3 with its own technology. This was first demonstrated publicly on PS3 using Sony's own technology in January 2009 at the Consumer Electronics Show. Journalists were shown Wipeout ... |
PS3's hardware has also been used to build supercomputers for high-performance computing. Fixstars Solutions sells a version of Yellow Dog Linux for PlayStation 3 (originally sold by Terra Soft Solutions). RapidMind produced a stream programming package for PS3, but were acquired by Intel in 2009. Also, on January 3, 2... |
PlayStation 3 uses the Cell microprocessor, designed by Sony, Toshiba and IBM, as its CPU, which is made up of one 3.2 GHz PowerPC-based "Power Processing Element" (PPE) and eight Synergistic Processing Elements (SPEs). The eighth SPE is disabled to improve chip yields. Only six of the seven SPEs are accessible to deve... |
At its press conference at the 2007 Tokyo Game Show, Sony announced DualShock 3 (trademarked DUALSHOCK 3), a PlayStation 3 controller with the same function and design as Sixaxis, but with vibration capability included. Hands-on accounts describe the controller as being noticeably heavier than the standard Sixaxis cont... |
The standard PlayStation 3 version of the XrossMediaBar (pronounced Cross Media Bar, or abbreviated XMB) includes nine categories of options. These are: Users, Settings, Photo, Music, Video, TV/Video Services, Game, Network, PlayStation Network and Friends (similar to the PlayStation Portable media bar). TheTV/Video Se... |
PlayStation 3 console protects certain types of data and uses digital rights management to limit the data's use. Purchased games and content from the PlayStation Network store are governed by PlayStation's Network Digital Rights Management (NDRM). The NDRM allows users to access the data from up to 2 different PlayStat... |
Photo Gallery is an optional application to view, create and group photos from PS3, which is installed separately from the system software at 105 MB. It was introduced in system software version 2.60 and provides a range of tools for sorting through and displaying the system's pictures. The key feature of this applicat... |
Since June 2009 VidZone has offered a free music video streaming service in Europe, Australia and New Zealand. In October 2009, Sony Computer Entertainment and Netflix announced that the Netflix streaming service would also be available on PlayStation 3 in the United States. A paid Netflix subscription was required for... |
The 'OtherOS' functionality was not present in the updated PS Slim models, and the feature was subsequently removed from previous versions of the PS3 as part of the machine's firmware update version 3.21 which was released on April 1, 2010; Sony cited security concerns as the rationale. The firmware update 3.21 was man... |
On March 1, 2010 (UTC), many of the original "fat" PlayStation 3 models worldwide were experiencing errors related to their internal system clock. The error had many symptoms. Initially, the main problem seemed to be the inability to connect to the PlayStation Network. However, the root cause of the problem was unrelat... |
Sony confirmed that there was an error and stated that they were narrowing down the issue and were continuing to work to restore service. By March 2 (UTC), 2010, owners of original PS3 models could connect to PSN successfully and the clock no longer showed December 31, 1999. Sony stated that the affected models incorre... |
PlayStation Portable can connect with PlayStation 3 in many ways, including in-game connectivity. For example, Formula One Championship Edition, a racing game, was shown at E3 2006 using a PSP as a real-time rear-view mirror. In addition, users are able to download original PlayStation format games from the PlayStation... |
PlayStation Network is the unified online multiplayer gaming and digital media delivery service provided by Sony Computer Entertainment for PlayStation 3 and PlayStation Portable, announced during the 2006 PlayStation Business Briefing meeting in Tokyo. The service is always connected, free, and includes multiplayer su... |
PlayStation Plus (commonly abbreviated PS+ and occasionally referred to as PSN Plus) is a premium PlayStation Network subscription service that was officially unveiled at E3 2010 by Jack Tretton, President and CEO of SCEA. Rumors of such service had been in speculation since Kaz Hirai's announcement at TGS 2009 of a po... |
The PlayStation Store is an online virtual market available to users of Sony's PlayStation 3 (PS3) and PlayStation Portable (PSP) game consoles via the PlayStation Network. The Store offers a range of downloadable content both for purchase and available free of charge. Available content includes full games, add-on cont... |
What's New was announced at Gamescom 2009 and was released on September 1, 2009, with PlayStation 3 system software 3.0. The feature was to replace the existing [Information Board], which displayed news from the PlayStation website associated with the user's region. The concept was developed further into a major PlaySt... |
The system displays the What's New screen by default instead of the [Games] menu (or [Video] menu, if a movie was inserted) when starting up. What's New has four sections: "Our Pick", "Recently Played", latest information and new content available in PlayStation Store. There are four kinds of content the What's New scr... |
PlayStation Home is a virtual 3D social networking service for the PlayStation Network. Home allows users to create a custom avatar, which can be groomed realistically. Users can edit and decorate their personal apartments, avatars or club houses with free, premium or won content. Users can shop for new items or win pr... |
Life with PlayStation, released on September 18, 2008 to succeed Folding@home, was retired November 6, 2012. Life with PlayStation used virtual globe data to display news and information by city. Along with Folding@home functionality, the application provided access to three other information "channels", the first bein... |
On April 20, 2011, Sony shut down the PlayStation Network and Qriocity for a prolonged interval, revealing on April 23 that this was due to "an external intrusion on our system". Sony later revealed that the personal information of 77 million users might have been taken, including: names; addresses; countries; email ad... |
Although its PlayStation predecessors had been very dominant against the competition and were hugely profitable for Sony, PlayStation 3 had an inauspicious start, and Sony chairman and CEO Sir Howard Stringer initially could not convince investors of a turnaround in its fortunes. The PS3 lacked the unique gameplay of t... |
PlayStation 3's initial production cost is estimated by iSuppli to have been US$805.85 for the 20 GB model and US$840.35 for the 60 GB model. However, they were priced at US$499 and US$599 respectively, meaning that units may have been sold at an estimated loss of $306 or $241 depending on model, if the cost estimates ... |
In January 2008, Kaz Hirai, CEO of Sony Computer Entertainment, suggested that the console may start making a profit by early 2009, stating that, "the next fiscal year starts in April and if we can try to achieve that in the next fiscal year that would be a great thing" and that "[profitability] is not a definite commi... |
Since the system's launch, production costs have been reduced significantly as a result of phasing out the Emotion Engine chip and falling hardware costs. The cost of manufacturing Cell microprocessors has fallen dramatically as a result of moving to the 65 nm production process, and Blu-ray Disc diodes have become che... |
Critical and commercial reception to PS3 improved over time, after a series of price revisions, Blu-ray's victory over HD DVD, and the release of several well received titles. Ars Technica's original launch review gave PS3 only a 6/10, but second review of the console in June 2008 rated it a 9/10. In September 2009, IG... |
PS3 was given the number-eight spot on PC World magazine's list of "The Top 21 Tech Screwups of 2006", where it was criticized for being "Late, Expensive and Incompatible". GamesRadar ranked PS3 as the top item in a feature on game-related PR disasters, asking how Sony managed to "take one of the most anticipated game ... |
Developers also found the machine difficult to program for. In 2007, Gabe Newell of Valve said "The PS3 is a total disaster on so many levels, I think it's really clear that Sony lost track of what customers and what developers wanted". He continued "I'd say, even at this late date, they should just cancel it and do a ... |
Activision Blizzard CEO Bobby Kotick has criticized PS3's high development costs and inferior attach rate and return to that of Xbox 360 and Wii. He believes these factors are pushing developers away from working on the console. In an interview with The Times Kotick stated "I'm getting concerned about Sony; the PlaySta... |
Despite the initial negative press, several websites have given the system very good reviews mostly regarding its hardware. CNET United Kingdom praised the system saying, "the PS3 is a versatile and impressive piece of home-entertainment equipment that lives up to the hype [...] the PS3 is well worth its hefty price ta... |
The PlayStation 3 Slim received extremely positive reviews as well as a boost in sales; less than 24 hours after its announcement, PS3 Slim took the number-one bestseller spot on Amazon.com in the video games section for fifteen consecutive days. It regained the number-one position again one day later. PS3 Slim also re... |
CNET awarded PS3 Slim four out of five stars praising its Blu-ray capabilities, 120 GB hard drive, free online gaming service and more affordable pricing point, but complained about the lack of backward compatibility for PlayStation 2 games. TechRadar gave PS3 Slim four and a half stars out of five praising its new sma... |
The Super Slim model of PS3 has received positive reviews. Gaming website Spong praised the new Super Slim's quietness, stating "The most noticeable noise comes when the drive seeks a new area of the disc, such as when starting to load a game, and this occurs infrequently." They added that the fans are quieter than tha... |
Tech media website CNET gave new Super Slim 4 out of 5 stars ("Excellent"), saying "The Super Slim PlayStation 3 shrinks a powerful gaming machine into an even tinier package while maintaining the same features as its predecessors: a great gaming library and a strong array of streaming services [...]", whilst also crit... |
Technology magazine T3 gave the Super Slim model a positive review, stating the console is almost "nostalgic" in the design similarities to the original "fat" model, "While we don’t know whether it will play PS3 games or Blu-ray discs any differently yet, the look and feel of the new PS3 Slim is an obvious homage to th... |
Royal assent is sometimes associated with elaborate ceremonies. In the United Kingdom, for instance, the sovereign may appear personally in the House of Lords or may appoint Lords Commissioners, who announce that royal assent has been granted at a ceremony held at the Palace of Westminster. However, royal assent is usu... |
Royal assent is the method by which a country's constitutional monarch (possibly through a delegated official) formally approves an act of that nation's parliament, thus making it a law or letting it be promulgated as law. In the vast majority of contemporary monarchies, this act is considered to be little more than a ... |
Under modern constitutional conventions, the sovereign acts on the advice of his or her ministers. Since these ministers most often maintain the support of parliament and are the ones who obtain the passage of bills, it is highly improbable that they would advise the sovereign to withhold assent. An exception is someti... |
Originally, legislative power was exercised by the sovereign acting on the advice of the Curia Regis, or Royal Council, in which important magnates and clerics participated and which evolved into parliament. The so-called Model Parliament included bishops, abbots, earls, barons, and two knights from each shire and two ... |
The power of parliament to pass bills was often thwarted by monarchs. Charles I dissolved parliament in 1629, after it passed motions critical of and bills seeking to restrict his arbitrary exercise of power. During the eleven years of personal rule that followed, Charles performed legally dubious actions, such as rais... |
During the rule of the succeeding Hanoverian dynasty, power was gradually exercised more by parliament and the government. The first Hanoverian monarch, George I, relied on his ministers to a greater extent than did previous monarchs. Later Hanoverian monarchs attempted to restore royal control over legislation: George... |
Royal assent is the final stage in the legislative process for acts of the Scottish parliament. The process is governed by sections 28, 32, and 33 of the Scotland Act 1998. After a bill has been passed, the Presiding Officer of the Scottish Parliament submits it to the monarch for royal assent after a four-week period,... |
Measures, which were the means by which the National Assembly for Wales passed legislation between 2006 and 2011, were assented to by the Queen by means of an Order in Council. Section 102 of the Government of Wales Act 2006 required the Clerk to the Assembly to present measures passed by the assembly after a four-week... |
Instead, the monarch directly grants royal assent by Order in Council. Assent is granted or refused on the advice of the Lord Chancellor. A recent example when assent was refused (or, more correctly, when the Lord Chancellor declined to present the law for assent) was in 2007, concerning reforms to the constitution of ... |
Royal assent is not sufficient to give legal effect to an Act of Tynwald. By ancient custom, an act did not come into force until it had been promulgated at an open-air sitting of Tynwald, usually held on Tynwald Hill at St John's on St John's Day (24 June), but, since the adoption of the Gregorian calendar in 1753, on... |
Since 1993, the Sodor and Man Diocesan Synod has had power to enact measures making provision "with respect to any matter concerning the Church of England in the Island". If approved by Tynwald, a measure "shall have the force and effect of an Act of Tynwald upon the Royal Assent thereto being announced to Tynwald". Be... |
If the Governor General of Canada is unable to give assent, it can be done by either the Deputy of the Governor General of Canada—the Chief Justice of Canada—or another justice of the Supreme Court of Canada. It is not actually necessary for the governor general to sign a bill passed by a legislature, the signature bei... |
While royal assent has not been withheld in the United Kingdom since 1708, it has often been withheld in British colonies and former colonies by governors acting on royal instructions. In the United States Declaration of Independence, colonists complained that George III "has refused his Assent to Laws, the most wholes... |
In Australia, a technical issue arose with the royal assent in both 1976 and 2001. In 1976, a bill originating in the House of Representatives was mistakenly submitted to the Governor-General and assented to. However, it was later discovered that it had not been passed by each house. The error arose because two bills o... |
Special procedures apply to legislation passed by Tynwald, the legislature of the Isle of Man. Before the lordship of the Island was purchased by the British Crown in 1765 (the Revestment), the assent of the Lord of Mann to a bill was signified by letter to the governor. After 1765, royal assent was at first signified ... |
In Commonwealth realms other than the UK, royal assent is granted or withheld either by the realm's sovereign or, more frequently, by the representative of the sovereign, the governor-general. In federated realms, assent in each state, province, or territory is granted or withheld by the representatives of the sovereig... |
Since the Balfour Declaration of 1926 and the Statute of Westminster 1931, the all Commonwealth realms have been sovereign kingdoms, the monarch and governors-general acting solely on the advice of the local ministers who generally maintain the support of the legislature and are the ones who secure the passage of bills... |
In the United Kingdom, a bill is presented for royal assent after it has passed all the required stages in both the House of Commons and the House of Lords. Under the Parliament Acts 1911 and 1949, the House of Commons may, under certain circumstances, direct that a bill be presented for assent despite lack of passage ... |
The Clerk of the Parliaments, an official of the House of Lords, traditionally states a formula in Anglo-Norman Law French, indicating the sovereign's decision. The granting of royal assent to a supply bill is indicated with the words "La Reyne remercie ses bons sujets, accepte leur benevolence, et ainsi le veult", tra... |
Before the reign of Henry VIII, the sovereign always granted his or her assent in person. The sovereign, wearing the Imperial State Crown, would be seated on the throne in the Lords chamber, surrounded by heralds and members of the royal court—a scene that nowadays is repeated only at the annual State Opening of Parlia... |
A new device for granting assent was created during the reign of King Henry VIII. In 1542, Henry sought to execute his fifth wife, Catherine Howard, whom he accused of committing adultery; the execution was to be authorised not after a trial but by a bill of attainder, to which he would have to personally assent after ... |
When granting assent by commission, the sovereign authorises three or more (normally five) lords who are Privy Counsellors to grant assent in his or her name. The Lords Commissioners, as the monarch's representatives are known, wear scarlet parliamentary robes and sit on a bench between the throne and the Woolsack. The... |
During the 1960s, the ceremony of assenting by commission was discontinued and is now only employed once a year, at the end of the annual parliamentary session. In 1960, the Gentleman Usher of the Black Rod arrived to summon the House of Commons during a heated debate and several members protested against the disruptio... |
Under the Royal Assent Act 1967, royal assent can be granted by the sovereign in writing, by means of letters patent, that are presented to the presiding officer of each house of parliament. Then, the presiding officer makes a formal, but simple statement to the house, acquainting each house that royal assent has been ... |
When the act is assented to by the sovereign in person, or by empowered Royal Commissioners, royal assent is considered given at the moment when the assent is declared in the presence of both houses jointly assembled. When the procedure created by the Royal Assent Act 1967 is followed, assent is considered granted when... |
Independently of the method used to signify royal assent, it is the responsibility of the Clerk of the Parliaments, once the assent has been duly notified to both houses, not only to endorse the act in the name of the monarch with the formal Norman French formula, but to certify that assent has been granted. The clerk ... |
In Australia, the formal ceremony of granting assent in parliament has not been regularly used since the early 20th century. Now, the bill is sent to the governor-general's residence by the house in which it originated. The governor-general then signs the bill, sending messages to the President of the Senate and the Sp... |
In Canada, the traditional ceremony for granting assent in parliament was regularly used until the 21st century, long after it had been discontinued in the United Kingdom and other Commonwealth realms. One result, conceived as part of a string of royal duties intended to demonstrate Canada's status as an independent ki... |
The Royal Assent ceremony takes place in the Senate, as the sovereign is traditionally barred from the House of Commons. On the day of the event, the Speaker of the Senate will read to the chamber a notice from the secretary to the governor general indicating when the viceroy or a deputy thereof will arrive. The Senate... |
In Belgium, the sanction royale has the same legal effect as royal assent; the Belgian constitution requires a theoretically possible refusal of royal sanction to be countersigned—as any other act of the monarch—by a minister responsible before the House of Representatives. The monarch promulgates the law, meaning that... |
The constitution of Jordan grants its monarch the right to withhold assent to laws passed by its parliament. Article 93 of that document gives the Jordanian sovereign six months to sign or veto any legislation sent to him from the National Assembly; if he vetoes it within that timeframe, the assembly may override his v... |
After the House of Representatives has debated the law, it either approves it and sends it to the Senate with the text "The Second Chamber of the States General sends the following approved proposal of law to the First Chamber", or it rejects it and returns it to the government with the text "The Second Chamber of the ... |
The government, consisting of the monarch and the ministers, will then usually approve the proposal and the sovereign and one of the ministers signs the proposal with the addition of an enacting clause, thereafter notifying the States General that "The King assents to the proposal." It has happened in exceptional circu... |
Articles 77–79 of the Norwegian Constitution specifically grant the monarch of Norway the right to withhold royal assent from any bill passed by the Storting. Should the sovereign ever choose to exercise this privilege, Article 79 provides a means by which his veto may be over-ridden: "If a Bill has been passed unalter... |
Title IV of the 1978 Spanish constitution invests the Consentimiento Real (Royal Assent) and promulgation (publication) of laws with the monarch of Spain, while Title III, The Cortes Generales, Chapter 2, Drafting of Bills, outlines the method by which bills are passed. According to Article 91, within fifteen days of p... |
No provision within the constitution grants the monarch an ability to veto legislation directly; however, no provision prohibits the sovereign from withholding royal assent, which effectively constitutes a veto. When the Spanish media asked King Juan Carlos if he would endorse the bill legalising same-sex marriages, he... |
If the Spanish monarch ever refused in conscience to grant royal assent, a procedure similar to the Belgian handling of King Baudouin's objection would not be possible under the current constitution. If the sovereign were ever declared incapable of discharging royal authority, his or her powers would not be transferred... |
Articles 41 and 68 of the constitution empower the sovereign to withhold royal assent from bills adopted by the Legislative Assembly. In 2010, the kingdom moved towards greater democracy, with King George Tupou V saying that he would be guided by his prime minister in the exercising of his powers. Nonetheless, this doe... |
In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element. The operation satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility. One of the most familiar exampl... |
Groups share a fundamental kinship with the notion of symmetry. For example, a symmetry group encodes symmetry features of a geometrical object: the group consists of the set of transformations that leave the object unchanged and the operation of combining two such transformations by performing one after the other. Lie... |
The concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s. After contributions from other fields such as number theory and geometry, the group notion was generalized and firmly established around 1870. Modern group theory—an active mathematical discipline—studies gr... |
The set G is called the underlying set of the group (G, •). Often the group's underlying set G is used as a short name for the group (G, •). Along the same lines, shorthand expressions such as "a subset of the group G" or "an element of group G" are used when what is actually meant is "a subset of the underlying set G ... |
These symmetries are represented by functions. Each of these functions sends a point in the square to the corresponding point under the symmetry. For example, r1 sends a point to its rotation 90° clockwise around the square's center, and fh sends a point to its reflection across the square's vertical middle line. Compo... |
The modern concept of an abstract group developed out of several fields of mathematics. The original motivation for group theory was the quest for solutions of polynomial equations of degree higher than 4. The 19th-century French mathematician Évariste Galois, extending prior work of Paolo Ruffini and Joseph-Louis Lagr... |
The convergence of these various sources into a uniform theory of groups started with Camille Jordan's Traité des substitutions et des équations algébriques (1870). Walther von Dyck (1882) introduced the idea of specifying a group by means of generators and relations, and was also the first to give an axiomatic definit... |
The University of Chicago's 1960–61 Group Theory Year brought together group theorists such as Daniel Gorenstein, John G. Thompson and Walter Feit, laying the foundation of a collaboration that, with input from numerous other mathematicians, classified all finite simple groups in 1982. This project exceeded previous ma... |
To understand groups beyond the level of mere symbolic manipulations as above, more structural concepts have to be employed.c[›] There is a conceptual principle underlying all of the following notions: to take advantage of the structure offered by groups (which sets, being "structureless", do not have), constructions r... |
Two groups G and H are called isomorphic if there exist group homomorphisms a: G → H and b: H → G, such that applying the two functions one after another in each of the two possible orders gives the identity functions of G and H. That is, a(b(h)) = h and b(a(g)) = g for any g in G and h in H. From an abstract point of ... |
In the example above, the identity and the rotations constitute a subgroup R = {id, r1, r2, r3}, highlighted in red in the group table above: any two rotations composed are still a rotation, and a rotation can be undone by (i.e. is inverse to) the complementary rotations 270° for 90°, 180° for 180°, and 90° for 270° (n... |
In many situations it is desirable to consider two group elements the same if they differ by an element of a given subgroup. For example, in D4 above, once a reflection is performed, the square never gets back to the r2 configuration by just applying the rotation operations (and no further reflections), i.e. the rotati... |
This set inherits a group operation (sometimes called coset multiplication, or coset addition) from the original group G: (gN) • (hN) = (gh)N for all g and h in G. This definition is motivated by the idea (itself an instance of general structural considerations outlined above) that the map G → G / N that associates to ... |
Quotient groups and subgroups together form a way of describing every group by its presentation: any group is the quotient of the free group over the generators of the group, quotiented by the subgroup of relations. The dihedral group D4, for example, can be generated by two elements r and f (for example, r = r1, the r... |
Sub- and quotient groups are related in the following way: a subset H of G can be seen as an injective map H → G, i.e. any element of the target has at most one element that maps to it. The counterpart to injective maps are surjective maps (every element of the target is mapped onto), such as the canonical map G → G / ... |
Groups are also applied in many other mathematical areas. Mathematical objects are often examined by associating groups to them and studying the properties of the corresponding groups. For example, Henri Poincaré founded what is now called algebraic topology by introducing the fundamental group. By means of this connec... |
In modular arithmetic, two integers are added and then the sum is divided by a positive integer called the modulus. The result of modular addition is the remainder of that division. For any modulus, n, the set of integers from 0 to n − 1 forms a group under modular addition: the inverse of any element a is n − a, and 0... |
For any prime number p, there is also the multiplicative group of integers modulo p. Its elements are the integers 1 to p − 1. The group operation is multiplication modulo p. That is, the usual product is divided by p and the remainder of this division is the result of modular multiplication. For example, if p = 5, the... |
In the groups Z/nZ introduced above, the element 1 is primitive, so these groups are cyclic. Indeed, each element is expressible as a sum all of whose terms are 1. Any cyclic group with n elements is isomorphic to this group. A second example for cyclic groups is the group of n-th complex roots of unity, given by compl... |
Symmetry groups are groups consisting of symmetries of given mathematical objects—be they of geometric nature, such as the introductory symmetry group of the square, or of algebraic nature, such as polynomial equations and their solutions. Conceptually, group theory can be thought of as the study of symmetry.t[›] Symme... |
Likewise, group theory helps predict the changes in physical properties that occur when a material undergoes a phase transition, for example, from a cubic to a tetrahedral crystalline form. An example is ferroelectric materials, where the change from a paraelectric to a ferroelectric state occurs at the Curie temperatu... |
Finite symmetry groups such as the Mathieu groups are used in coding theory, which is in turn applied in error correction of transmitted data, and in CD players. Another application is differential Galois theory, which characterizes functions having antiderivatives of a prescribed form, giving group-theoretic criteria ... |
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