In mathematics, when X is a finite set with at least two elements, the permutations of X fall into two classes of equal size: the permutations and the odd permutations. If any total ordering of X is fixed, the parity of a permutation σ of X can be defined as the parity of the number of inversions for σ, i.e. of pairs of elements x, y of X such that x < y and σ > σ. The sign, signature, or signum of a permutation σ is denoted sgn and defined as +1 if σ is and −1 if σ is odd; the signature defines the alternating character of the symmetric group Sn. Another notation for the sign of a permutation is given by the more general Levi-Civita symbol, defined for all maps from X to X, has value zero for non-bijective maps; the sign of a permutation can be explicitly expressed as sgn = Nwhere N is the number of inversions in σ. Alternatively, the sign of a permutation σ can be defined from its decomposition into the product of transpositions as sgn = mwhere m is the number of transpositions in the decomposition.

Although such a decomposition is not unique, the parity of the number of transpositions in all decompositions is the same, implying that the sign of a permutation is well-defined. Consider the permutation σ of the set which turns the initial arrangement 12345 into 34521, it can be obtained by three transpositions: first exchange the numbers 2 and 4 exchange 1 and 3, exchange 1 and 5. This shows. Following the method of the cycle notation article, this could be written as σ = = =. There are many other ways of writing σ as a composition of transpositions, for instance σ =,but it is impossible to write it as a product of an number of transpositions; the identity permutation is an permutation. An permutation can be obtained as the composition of an number and only an number of exchanges of two elements, while an odd permutation can be obtained by an odd number of transpositions; the following rules follow directly from the corresponding rules about addition of integers: the composition of two permutations is the composition of two odd permutations is the composition of an odd and an permutation is oddFrom these it follows that the inverse of every permutation is the inverse of every odd permutation is oddConsidering the symmetric group Sn of all permutations of the set, we can conclude that the map sgn: Sn → that assigns to every permutation its signature is a group homomorphism.

Furthermore, we see that the permutations form a subgroup of Sn. This is the alternating group on n letters, denoted by An, it is the kernel of the homomorphism sgn. The odd permutations cannot form a subgroup, since the composite of two odd permutations is but they form a coset of An. If n > 1 there are just as many permutations in Sn as there are odd ones. A cycle is if and only if its length is odd; this follows from formulas like = or. In practice, in order to determine whether a given permutation is or odd, one writes the permutation as a product of disjoint cycles; the permutation is odd if and only if this factorization contains an odd number of even-length cycles. Another method for determining whether a given permutation is or odd is to construct the corresponding permutation matrix and compute its determinant; the value of the determinant is the same as the parity of the permutation. Every permutation of odd order must be even; the permutation in A4 shows. Every permutation can be produced by a sequence of transpositions: with the first transposition we put the first element of the permutation in its proper place, the second transposition puts the second element right etc.

Since we cannot be left with just a single element in an incorrect position, we must achieve the permutation with our last transposition. Given a permutation σ, we can write it as a product of transpositions in many different ways. We want to show that either all of those decompositions have an number of tran

Sistema Brasileiro de Televisão is a Brazilian free-to-air television network, funded on August 19, 1981 by the businessman and TV host Silvio Santos. The network was established after a public competition of the federal government for the creation of two new television networks, created from revoked concessions of the extinct networks Tupi and Excelsior. SBT was funded in the same day that the concession agreement was signed, that the act was broadcast live by the network, so that this was his first program aired. Is the second more watched television network in Brazil, after Rede Globo. Throughout its existence, the network always occupied this space in the audience ranking, except between 2007 and 2014, when Rede Record took the post. SBT has about 114 owned and operated station throughout the Brazilian territory, is available through pay television operators, by the free-to-air signal available in broadcast and satellite receivers, through streaming media in his mobile application, in apps for smart TVs and in his website.

On your website, its programming is available in video on demand for free available from the video-sharing site YouTube since 2010. SBT broadcast in your programming a wide variety of television genres, whereas its own material stand adjacent to the entertainment. Foreign programming the telenovelas produced by the networks owned by the Mexican Televisa, make up much of the schedule, it is the only commercial television broadcaster in Brazil which airs children's programming arranging a partnership with the U. S; the Walt Disney Company, in which the company provides two hours of daily programming for the network. SBT possess times for the television news, producing in all three daily newscasts, a weekly news program and a weekly newscast. Bake Off Brasil: Mão na Massa BBQ Brasil: Churrasco na Brasa Corre e Costura com Alexandre Herchcovitch Esquadrão da Moda Hell's Kitchen: Cozinha sob Pressão Máquina da Fama Pra Ganhar É Só Rodar Roda a Roda Conexão Repórter Jornal da Semana SBT Primeiro Impacto SBT Brasil SBT Praça SBT Notícias Acelerados Brasil Caminhoneiro Casos de Família Domingo Legal Eliana Fofocalizando A Praça É Nossa Programa do Ratinho Programa Raul Gil Programa Silvio Santos Sabadão com Celso Portiolli Turismo e Aventura The Noite com Danilo Gentili Bom Dia & Companhia Carrossel Animado Mundo Disney Sábado Animado Feriadão SBT Retrospectiva SBT Folia Teleton Troféu Imprensa Big Bang: A Teoria Chaves Crimes Graves Diários de um Vampiro Dois Homens e Meio Uma Família Perdida no Meio do Nada A Garota da Moto How to Rock Kenan & Kel Mike & Molly O Mentalista Revolução Sullivan & Filho The Following Cine Belas Artes Cine Espetacular Tela de Sucessos Caldeirão da Sorte Tele Sena Abismo de Paixão Carrossel Cúmplices de um Resgate A Gata Mar de Amor Acontece lá em Casa As Aventuras de Pollyanna e João Feijão Carinha de Anjo Chiquititas Jornal do SBT Mundo Disney Bake Off Brasil Júnior Bake Off Brasil Bake Off Brasil Celebrity BBQ Brasil Hell's Kitchen: Cozinha sob Pressão Roda a Roda Jequiti Programa Silvio Santos Passa ou Repassa - Pra Ganhar É Só Rodar Fábrica de Casamentos Jogo das Fichas Nada além de Um Minuto Rola ou Enrola?

- Fenômenos - Esquadrão da Moda Bomba! - Cabelo Pantene - O Reality Qual é o Seu Talento? Caldeirão da Sorte Duelo de Mães Dance se Puder - Máquina da Fama Esse Artista Sou Eu Festival Sertanejo Menino de Ouro Famoso Quem? Vamos Brincar de Forca Cante se Puder Se Ela Dança, Eu Danço Esquadrão do Amor Um Milhão na Mesa Cantando no SBT SOS Casamento Romance no Escuro - Solitários Meu Pai é Melhor que Seu Pai Topa ou Não Topa Um Contra Cem Você Se Lembra? 10 Anos Mais Jovem Identidade Secreta Só Falta Esposa Astros Supernanny Nada Além da Verdade Quem Manda É o Chefe High School Musical: A Seleção Tentação Você É mais Esperto que um Aluno da Quinta Série? Quem Perde, Ganh

Mitcham is an inner-southern suburb of Adelaide in the City of Mitcham. Created as a village separate from Adelaide, it was ancillary to a sheep station at Brown Hill Creek belonging to the South Australia Company. Prior to British colonisation, the area was inhabited by an Aboriginal people. A group of about 150 Kaurna camped at "Wirraparinga", now Mitcham reserve. Mitcham is located in the federal electorate of Boothby and the state electorate of Waite, which both tend to be safe Liberal seats, it is the seat of the Mitcham Council. The area is affluent. Theodore Ambrose medical practitionerMajor Rupert Downes surgeon and soldierHedley Herbert Finlayson conservationist and mammalogistJohn Harvey Finlayson newspaper editorLaura Margaret Hope medical practitionerDoris Egerton Jones writerEllen Thornber schoolmistressJoseph Garnett Wood botanist

The following is a list of mayors of the city of Porto Velho, in Rondônia state, Brazil. Fernando Guapindaia de Souza Brejense, 1915-1917 Joaquim Augusto Tanajura, 1917-1920 Raimundo Oliveira, 1920-1922 Álvaro Maia, 1922-1923 Joaquim Augusto Tanajura, 1923-1925 Fernando Corrêa, 1925-1929 Salustiano Liberato, 1929 Tófilo Marinho, 1929-1930 Raimundo Gonzaga Pinheiro, 1930-1931 Arthur Napoleão Lebre, 1931-1932 Ariosto Lopes Braga, 1932 Francisco Plínio Coelho, 1932-1933 Bohemundo Álvares Afonso, 1933 José Ferreira Sobrinho, 1933-1938 Francisco Guedes L. Fonseca, 1938 Bohemundo Álvares Afonso, 1938-1943 José Marques Galvão, 1943 Mário Monteiro, 1943-1946 Carlos Augusto de Mendonça, 1946-1947 José Otino de Freitas, 1947-1948 Celso Pinheiro, 1948 Flamínio de Júlio de Albuquerque, 1948 Rui Brasil Cantanhede, 1948-1951 Rafael Jaime Castiel, 1951 Balduíno Guedes de Lira, 1951-1954 José Saleh Moreb, 1954-1955 Renato Climaco Borralho de Medeiros, 1955-1956 Walter Montezuma de Oliveira, 1956-1958 Thomas Miguel Chaquian, 1958 Rubens Cantanhede, 1958-1961 Floriano Rodrigues Riva, 1961 Hamilton Raulino Gondim, 1961-1962 Homero Martins, 1962-1963 Odacir Soares, 1963-1965, 1969-1972 Paulo Trajano de Medeiros, 1965-1967 Irineu Martins de Farias, 1967 Hebert Alencar de Souza, 1967 Hércules Lima de Carvalho, 1967 Walter Paula de Sales, 1967-1969 Jacob Freitas Atallah, 1972-1974 Emanuel Pontes Pinto, 1974-1975 Antônio Carlos Carpinteiro, 1975-1976 Luis Gonzaga Farias Ferreira, 1976-1979 Reditario Cassol, 1979-1985 José Guedes, 1985-1986, 1993-1996 Jerônimo Santana, 1986 Tomaz Corrêa, 1986-1988 Chiquilito Erse, 1989-1992, 1997-1998 Carlos Camurça, 1998-2004 Roberto Eduardo Sobrinho, 2005-2012 Mauro Nazif, 2013-2016 Hildon Chaves, 2017- Porto Velho municipal election, 2012 List of mayors of largest cities in Brazil List of mayors of capitals of Brazil This article incorporates information from the Portuguese Wikipedia

A Sense of Purpose Tour was a concert tour by Swedish melodic death metal band In Flames in support of the act's ninth studio album, A Sense of Purpose, released in April 2008. The tour began in May 2008 with a slot on Gigantour in North American markets, headlined by Megadeth. Following runs through Europe and a second North American leg, the tour continued through 2009 with the band's first visit to South America, as well as dates in Australia and Japan. In February 2009, it was confirmed that lead guitarist Jesper Strömblad would be taking leave from future touring, due to his focus on alcohol rehabilitation, he was replaced with former member Niklas Engelin. In the month, an upcoming spring tour of the U. K. and Ireland was cancelled after it was announced that a second member, bassist Peter Iwers, would be taking leave due to the anticipated birth of his child. The band, minus Strömblad, fulfilled all touring commitments up to early April and resumed performing in the summer. In September 2009, the band commenced a Canadian co-headlining trek with Killswitch Engage.

Following the dates in Canada, the group toured the United States through early October. In November 2009, the group teamed up once again with Killswitch Engage for the European leg of the "Taste of Chaos" tour. In January 2010, the band played their final leg of the tour. 1^ Headline show. 2^ Date supporting Lamb of God. 3^ Date featuring co-headliners Lamb of God. 4^ Date featuring co-headliners Killswitch Engage. 5^ Date part of the "Taste of Chaos" tour featuring co-headliners Killswitch Engage. Cancelled dates Official website

The Paonia First Christian Church, at 235 Box Elder Ave. in Paonia, was built in. It was listed on the National Register of Historic Places in 2011, it has been known as Paonia Christian Fellowship. A local history website describes it:The Paonia First Christian Church is an excellent example of Romanesque style church as interpreted by local craftsmen; the building, along with several other churches, is in the heart of the residential area of Paonia, which has a long-standing religious identity marked by the high density of churches in the small town. The building is constructed of coursed, rusticated sandstone with distinctive architectural features such as the round, three-story, crenellated tower with a graduated buttress. Media related to Paonia First Christian Church at Wikimedia Commons