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Involution (mathematics)

In mathematics, an involution, or an involutory function, is a function f, its own inverse, f = xfor all x in the domain of f. Equivalently, applying f twice produces the original value; the term anti-involution refers to involutions based on antihomomorphisms f = f fsuch that xy = f = f = f f = xy. Any involution is a bijection; the identity map is a trivial example of an involution. Common examples in mathematics of nontrivial involutions include multiplication by −1 in arithmetic, the taking of reciprocals, complementation in set theory and complex conjugation. Other examples include circle inversion, rotation by a half-turn, reciprocal ciphers such as the ROT13 transformation and the Beaufort polyalphabetic cipher; the number of involutions, including the identity involution, on a set with n = 0, 1, 2... elements is given by a recurrence relation found by Heinrich August Rothe in 1800: a0 = a1 = 1. The first few terms of this sequence are 1, 1, 2, 4, 10, 26, 76, 232; the composition g ∘ f of two involutions f and g is an involution if and only if they commute: g ∘ f = f ∘ g.

Every involution on an odd number of elements has at least one fixed point. More for an involution on a finite set of elements, the number of elements and the number of fixed points have the same parity. Basic examples of involutions are the functions: f 1 = − x, or f 2 = 1 x, as well as their composition = = f 3 = − 1 x; these are not the only pre-calculus involutions. Another one within the positive reals is: f = ln ⁡; the graph of an involution is line-symmetric over the line y = x. This is due to the fact that the inverse of any general function will be its reflection over the 45° line y = x; this can be seen by "swapping" x with y. If, in particular, the function is an involution it will serve as its own reflection. Other elementary involutions are useful in solving functional equations. A simple example of an involution of the three-dimensional Euclidean space is reflection through a plane. Performing a reflection twice brings a point back to its original coordinates. Another involution is reflection through the origin.

These transformations are examples of affine involutions. An involution is a projectivity of period 2, that is, a projectivity that interchanges pairs of points. Any projectivity that interchanges two points is an involution; the three pairs of opposite sides of a complete quadrangle meet any line in three pairs of an involution. This theorem has been called Desargues's Involution Theorem, its origins can be seen in Lemma IV of the lemmas to the Porisms of Euclid in Volume VII of the Collection of Pappus of Alexandria. If an involution has one fixed point, it has another, consists of the correspondence between harmonic conjugates with respect to these two points. In this instance the involution is termed "hyperbolic", while if there are no fixed points it is "elliptic". In the context of projectivities, fixed points are called double points. Another type of involution occurring in projective geometry is a polarity, a correlation of period 2. In linear algebra, an involution is a linear operator T on a vector space, such that T 2 = I.

Except for in characteristic 2, such operators are diagonalizable for a given basis with just 1s and −1s on the diagonal of the corresponding matrix. If the operator is orthogonal, it is orthonormally diagonalizable. For example, suppose that a basis for a vector space V is chosen, that e1 and e2 are basis elements. There exists a linear transformation f which sends e1 to e2, sends e2 to e1, and, the identity on all other basis vectors, it can be checked that f = x for all x in V. That is, f is an involution of V. For a specific basis, any linear operator can be represented by a matrix T; every matrix has a transpose, obtained by swapping rows for columns. This transposition is an involution on the set of matrices; the definition of involution extends to modules. Given a module M over a ring R, an R endomorphism f of M is called an involution if f 2 is the identity homomorphism on M. Involutions are related to idempotents. In a quaternion algebra, an involution is defined b

Castrol Honda Superbike 2000

Castrol HONDA -World Superbike Team- Superbike 2000 is a licensed motorcycle racing game, developed by Interactive Entertainment, Ltd. and published by Midas Interactive. The game features the Honda RVF750 RC45 racing motorcycle; the game only licenses the Castrol Honda team and the names of both riders, Aaron Slight and Colin Edwards from the 2000 Superbike World Championship season. However, it is possible to edit the names of both riders of 12 teams; the player controls a motorcyclist in races on various international race tracks. Game types are "Trainer", "Practice", "Weekend" and "Championship"; each race has three parts: "Practice Session", "Qualifying" and "Race". The degree of "Realism" can be modified. In the bike setup, gearbox transmission, final drive speed for the 6 gears, the gear sprockets can be modified; the game features randomly changing weather conditions. The game received a score of 72.29% on GameRankings, based on seven individual reviews. The game was reviewed by the German magazine "PC Games" and received a rating of 66%.

The reviewer concluded. There are four games in the Castrol HONDA Superbike series, each one was produced by Interactive Entertainment Ltd: Castrol HONDA SuperBike World Champions Castrol HONDA -World Superbike Team- Superbike Racing Castrol HONDA -World Superbike Team- Superbike 2000 Castrol HONDA -World Superbike Team- VTR Note: Some sites erroneously state that "Castrol HONDA Superbike Racing" and Castrol HONDA Superbike 2000 are versions of the same game. However, game titles and content are quite different, making up for two different games

British National Party

The British National Party is a far-right, fascist political party in the United Kingdom. It is headquartered in Wigton and its current leader is Adam Walker. A minor party, it has no elected representatives at any level of UK government. Founded in 1982, the party reached its greatest level of success in the 2000s, when it had over fifty seats in local government, one seat on the London Assembly, two Members of the European Parliament. Taking its name from that of a defunct 1960s far-right party, the BNP was created by John Tyndall and other former members of the fascist National Front. During the 1980s and 1990s, the BNP placed little emphasis on contesting elections, in which it did poorly. Instead, it focused on street marches and rallies, creating the Combat 18 paramilitary—its name a coded reference to Nazi German leader Adolf Hitler—to protect its events from anti-fascist protesters. A growing'moderniser' faction was frustrated by Tyndall's leadership, ousted him in 1999; the new leader Nick Griffin sought to broaden the BNP's electoral base by presenting a more moderate image, targeting concerns about rising immigration rates, emphasising localised community campaigns.

This resulted in increased electoral growth throughout the 2000s, to the extent that it became the most electorally successful far-right party in British history. Concerns regarding financial mismanagement resulted in Griffin being removed from office in 2014. By this point the BNP's membership and vote share had declined groups like Britain First and National Action had splintered off, the English Defence League had supplanted it as the UK's foremost far-right group. Ideologically positioned on the extreme-right or far-right of British politics, the BNP has been characterised as fascist or neo-fascist by political scientists. Under Tyndall's leadership, it was more regarded as neo-Nazi; the party is ethnic nationalist, it espouses the view that only white people should be citizens of the United Kingdom. It calls for an end to non-white migration into the UK and for non-white Britons to be stripped of citizenship and removed from the country, it called for the compulsory expulsion of non-whites, although since 1999 has advocated voluntary removals with financial incentives.

It promotes biological racism and the white genocide conspiracy theory, calling for global racial separatism and condemning interracial relationships. Under Tyndall, the BNP emphasised anti-semitism and Holocaust denial, promoting the conspiracy theory that Jews seek to dominate the world through both communism and international capitalism. Under Griffin, the party's focus switched from anti-semitism towards Islamophobia, it promotes economic protectionism, a transformation away from liberal democracy, while its social policies oppose feminism, LGBT rights, societal permissiveness. Operating around a centralised structure that gave its chair near total control, the BNP built links with far-right parties across Europe and created various sub-groups, including a record label and trade union; the BNP attracted most support from within White British working-class communities in northern and eastern England among middle-aged and elderly men. Polls suggested, it faced much opposition from anti-fascists, religious organisations, the mainstream media, most politicians, BNP members were banned from various professions.

The British National Party was founded by the extreme-right political activist John Tyndall. Tyndall had been involved in neo-Nazi groups since the late 1950s before leading the far-right National Front throughout most of the 1970s. Following an argument with senior party member Martin Webster, he resigned from the NF in 1980. In June 1980 Tyndall established the New National Front. At the recommendation of Ray Hill—who was secretly an anti-fascist spy seeking to sow disharmony among Britain's far-right—Tyndall decided to unite an array of extreme-right groups as a single party. To this end, Tyndall established a Committee for Nationalist Unity in January 1982. In March 1982, the CNU held a conference at the Charing Cross Hotel in London, at which 50 far-right activists agreed to the formation of the BNP; the BNP was formally launched on 7 April 1982 at a press conference in Victoria. Led by Tyndall, most of its early members came from the NNF, although others were defectors from the NF, British Movement, British Democratic Party, Nationalist Party.

Tyndall remarked that there was "scarcely any difference in ideology or policy save in the minutest detail", most of the BNP's leading activists had been senior NF figures. Under Tyndall's leadership the party was neo-Nazi in orientation and engaged in nostalgia for Nazi Germany, it adopted the NF's tactic of holding street marches and rallies, believing that these boosted morale and attracted new recruits. Their first march took place in London on St. George's Day 1982; these marches involved clashes with anti-fascist protesters and resulted in multiple arrests, helping to cement the BNP's association with political violence and older fascist groups in the public eye. As a result, BNP organisers began to favour indoor rallies, although street marches continued to be held throughout the mid-to-late 1980s. In its early years, the BNP's involvement in elections was "irregular and intermittent", for its first two decades it faced consistent electoral failure, it suffered from low finances and few personnel, its leadership was aware that its electoral viability was weakened by the anti-immigration rhetoric of Conservative Party Prime Minister Margaret Thatcher.

In the 1983 general election the BNP stood 54 candidates, although it only ca

Alan Morton (footballer, born 1942)

Alan Morton is an English former professional footballer who scored 28 goals from 94 appearances in the Football League playing as an inside forward for Peterborough United, Lincoln City and Chesterfield. Morton was born in Peterborough, he began his football career with Arsenal, but never played for the first team, made his debut in the 1961–62 Football League season with Peterborough United. In 1963 he moved on to Lincoln City, where he was the club's top scorer in the 1963–64 season with 21 goals in all competitions, he played only infrequently the following season, moved on to Chesterfield where he finished his professional career playing non-League football for Wisbech Town

The Sound of Music (soundtrack)

The soundtrack of the film The Sound of Music was released in 1965 by RCA Victor and is one of the most successful soundtrack albums in history, having sold more than 20 million copies worldwide. The label has issued the soundtrack in German, Italian and French editions; the soundtrack reached the number one position on the Billboard 200 that year in the United States, remained in the top ten for a record 109 weeks, from May 1, 1965 to July 16, 1967, remained on the Billboard 200 chart for 238 weeks. In 2015, Billboard named the original soundtrack album the second-best charting album of all time, it was the best-selling album in the United Kingdom in 1965, 1966 and 1968 and the second best-selling of the decade, spending a total of 70 weeks at number one on the UK Album Charts. The album stayed for 73 weeks on the Norwegian charts, as of December 2017 it is the tenth best-charting album of all time in that country; the album has been reissued several times, including anniversary editions in 1995, 2000, 2005, 2010, 2015.

These CD editions incorporate musical material from the film that would not fit on the original LP. Three songs from the original Broadway production, "An Ordinary Couple", "How Can Love Survive?", "No Way to Stop It" were replaced, in the film, with two new songs, "I Have Confidence" and "Something Good". For the original Broadway show, the music was written by Richard Rodgers with lyrics by Oscar Hammerstein II. All songs were conducted for the soundtrack by Irwin Kostal. In 2018, it was selected for preservation in the National Recording Registry by the Library of Congress as being "culturally or artistically significant." Hischak, Thomas. The Rodgers and Hammerstein Encyclopedia. Greenwood Publishing Group. ISBN 0-313-34140-0

À Nous la Liberté

À nous la liberté is a 1931 French film directed by René Clair. With a score by Georges Auric, the film has more music than any of Clair's early works. Praised for its scenic design and use of sound, À nous la liberté has been called Clair's "crowning achievement"; the film opens with an image of a wooden toy horse. We observe that this is an assembly line in a prison, staffed by prisoners, they sing. Close-ups of two prisoners indicate; the prisoner next to Louis looks on, looking somewhat bored. After dinner everyone goes back to their cell. After feigning sleep during a guard's nightly rounds, Louis and Émile sing the title song as they resume a project of sawing off the prison window. Émile cuts himself, Louis kindly mends the wound with a handkerchief. The window breaks free and they attempt to escape. Louis is able to get over the retaining wall. Louis escapes, accidentally knocks someone off a bicycle, rides off on the bicycle. Meanwhile, we hear a chorus suggesting. Louis heads into a village emblazoned with the words "Finishing Line" - the cyclist he knocked over was in a bicycle race, Louis has won first prize.

Louis enters a store to purchase some handkerchiefs. While the proprietor is looking in a backroom, he hears Louis's muffled cries for help. After being unbound, Louis explains that someone made off with the money, he points the direction and a group of people run after the thief, leaving Louis alone, revealing that it was he who stole the money, feigning the story. A montage sequence follows in which we see Louis transform himself from a poor record merchant, to the well-attired and well-mannered head of an industrial factory that produces record players. Interior shots of the assembly line bear a strong resemblance to the assembly line seen at the beginning of the film. Meanwhile, behind the factory we see an open field. Émile has been sleeping, wakes up to a beautiful day. A flower sings "Ami, l'ombre de la prison a cédé la place au soleil" = "Friend, the shadow of prison has given way to the sun." A policeman tells Émile he must get to work. But he is put in a cell for resisting arrest. Through the prison window, he sees more flowers surrounding an apartment window and thinks he hears them singing.

A lovely woman, appears at the window and appears to be the source of the singing. Sad at his predicament, he tries to hang himself from the prison window, but the gate comes loose and falls on Émile's head, enabling him to escape. He stands by the apartment entrance, looking up at the flower-covered window, but is disappointed to realize that the flowers are not singing, that it's only a recording, he realizes that Jeanne has emerged from the apartment with her Uncle, who appears to be overly-protective and pulls her away from Émile, kicks him. A commotion and chase ensue in which Émile runs as we realize that Jeanne has a boyfriend, Paul. Émile makes his way into the lands in the recruitment department. A recorded song instructs him on how to be measured and fingerprinted. In the next scene we see Émile back on an assembly line, this time assembling phonographs. Upon seeing Jeanne working in the factory, Émile becomes absent minded, causing great consternation and humor on the assembly line A guard tries to stop Émile from talking to Jeanne and they chase after him.

He is stopped by guards near the foot of a grand staircase at the doorway of an office. Louis emerges with aides, Émile gets his attention. At first Louis does not appear to remember his prison friend, but takes him into his office, thinking this is an extortion plot. After a scuffle, Émile cuts himself; as Louis tends to the wound with his handkerchief, he recalls that he did when the two tried to break out of prison at the film's outset. His attitude changes to one of friendship, as he embraces Émile and sings a brief reprise of the title song; the scene is a dinner party at Louis's house. Neither his guests nor his wife Maud. Meanwhile, Louis and Émile don't care, enjoy sending up the haughtiness of his life style, climaxing in another reprise of the title song, as the two friends dance in front of a painting of Louis which he has damaged with a bottle of wine. With her suitcases packed, Maud leaves the house. Émile leaves, encounters an ex-prisoner on the street. Louis is thrilled to look out the door at the departing Maud, but does not see the ex-prisoner, who makes a nod of understanding to himself.

Back at the factory, Émile tries to make overtures to Jeanne, but is thwarted by guards finding himself in Louis's office. Louis has been explaining that his new factory will increasing productivity. Upon seeing his friend interrupt, Louis is annoyed until Émile explains that he wants to court Jeanne, another worker. From his office card files, Louis is able to produce a picture profile of Jeanne, but this is automatically followed by a profile of her Uncle. Louis invites both Jeanne and her Uncle into his office to explain Émile's interest, offers some mo