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Julie Burchill

Julie Burchill (born 3 July 1959) is an English writer. Beginning as a staff writer at the New Musical Express at the age of 17, she has since contributed to newspapers such as The Daily Telegraph, The Sunday Times and The Guardian. Her writing, which was described by The Observer in 2002 as "outrageously outspoken" and "usually offensive," has been the subject of legal action. Burchill is also a novelist, and her 2004 novel Sugar Rush was adapted for television.

Early life and education
Julie Burchill was born in Bristol and educated at Brislington Comprehensive School. Her father was a Communist union activist who worked in a distillery. Her mother had a job in a cardboard box factory. In 2010, Burchill wrote of her parents: "I don't care much for families. I adored my mum and dad, but to be honest I don't miss them much now they're dead"; three years later she contradicted this when she said she couldn't return to Bristol, as every time she heard someone speaking with her parents' Bristol accent it would remind her how much she missed them. She did not attend university, leaving the A-levels she had started a few weeks earlier to begin writing for the New Musical Express (NME).

At the NME
She began her writing career at the New Musical Express (NME) in 1976, aged 17, after responding (coincidentally with her future husband Tony Parsons) to an advert in that paper seeking "hip young gunslingers" to write about the then emerging punk movement. She gained the job by submitting a "eulogy" of Patti Smith's Horses. She later wrote that at the time she only liked black music, and said: "When I actually heard a punk record, I thought, 'Oh my Lord! This is not music, this is just shouting'." Indeed, she managed to decry the first self-released punk album in the UK, The Outsiders' Calling on Youth featuring Adrian Borland: "Apple-cheeked Ade has a complexion that would turn a Devon milkmaid green with envy." Fortunately for her, as she later said, "Punk was over in two years. That was the only damn good thing about it." She left her position at the NME at the age of 20, and started freelancing to be able to write about other subjects, although she has never completely given up writing about pop music.

1980s
Her main employers after the NME were The Face and The Sunday Times, where she wrote about politics, pop, fashion and society, and was their film critic from 1984 to 1986. She admitted in 2008 to making up film reviews and having "skived" from screenings, and her ex-husband, Cosmo Landesman, has admitted to attending screenings on her behalf.

During the Falklands War in 1982, Burchill argued that the military dictatorship of General Galtieri represented a greater evil. She wrote articles favourable to Margaret Thatcher. Her sympathy for Thatcher helped in gaining a column for The Mail on Sunday, where in 1987 she went against the paper's usual political line by urging its readers to vote Labour. Though she claims to like the MoS, she said of journalists on the Daily Mail in 2008: "Everybody knows that hacks are the biggest bunch of adulterers, the most misbehaving profession in the world – and you have people writing for the Daily Mail writing as though they are vicars ... moralising on single mothers and whatnot."

Into the 1990s
Burchill has spoken repeatedly and frankly of her relationship with drugs, writing that she had "put enough toot up my admittedly sizeable snout to stun the entire Colombian armed forces". She declared that "As one who suffered from chronic shyness and a low boredom threshold ... I simply can't imagine that I could have ever had any kind of social life without [cocaine], let alone have reigned as Queen of the Groucho Club for a good part of the '80s and '90s." While Burchill has frequently drawn on her personal life for her writing, her personal life has been a subject of public comment, especially during this period, when "everything about her – her marriages, her debauchery, her children – seemed to be news."

In 1991, Burchill, Landesman and Toby Young established a short-lived magazine Modern Review through which she met Charlotte Raven, with whom she had a much publicised affair. "[I] was only a lesbian for about six weeks in 1995," she said in an interview with Lynn Barber in 2004, or "my very enjoyable six months of lesbianism" in a 2000 article. Launched under the slogan "Low culture for high brows", the magazine lasted until 1995, when Burchill and her colleagues fell out. It was briefly revived by Burchill, with Raven editing, in 1997. The "Fax war" in 1993 between Burchill and author Camille Paglia, published in the Modern Review, gained much attention.

In 1995, Burchill wrote a column for The Times, titled "I'm a bitch, and I'm proud", in which she argued that women should reclaim the word 'bitch,' used as a slur. She wrote: "it is the nature of these things that, in recent years, the slighted have taken steps to repossess the slight; thus, we have blacks who call each other 'nigger', pansies who call each other 'queer' and upper-class cretins who quite happily call each other 'Henry'."

In 1996, the actor, author, playwright and theatre director Steven Berkoff won a libel action against Burchill in respect of one of her articles, published in the Sunday Times newspaper, which included comments suggesting that he was "hideously ugly". The judge ruled that Burchill's actions "held him to ridicule and contempt." The late 1990s were a turbulent period for Burchill as she has recalled: "I got the heave-ho from my cushy billet at the Sunday Express, where I later learned my nickname had been 'Caligula’s Horse' because my best friend – briefly the editor – had appointed me. For the first time in my brilliant career, no one wanted to hire me. Somehow I limped into a column on the doddering Punch – and then I got the boot from there, too! Surely I had reached the mythical rock bottom at last?"

From 2000 to 2004
A user of cocaine, sharing in the activity in the company of Will Self among others, she was positive about her use in The Guardian in 2000 when defending actress Danniella Westbrook for Westbrook's loss of her nasal septum because of cocaine use. Journalist Deborah Orr, who was then married to Self, was scathing in The Independent of Burchill and her article: "She does not identify herself as a cocaine addict, so she has no pity for Ms Westbrook." In revenge for Deborah Orr's article, Burchill invented a supposedly long-standing crush on Will Self with the intention of upsetting Orr. A letter in The Independent in June 2000 from the head waitress at the Groucho Club at the time, Deborah Bosley, caused a minor stir. Responding to an article by Yvonne Roberts, Bosley, at the time the partner of Richard Ingrams, a long standing critic of Burchill, stated that Burchill was merely "a fat bird in a blue mac sitting in the corner" when ensconced at the Groucho.

The following year's Burchill on Beckham (2001), a short book about Burchill's opinions concerning David Beckham's life, career, and relationship with Victoria Beckham, attracted "some of the worst notices since Jeffrey Archer's heyday. 'Burchill is to football writing what Jimmy Hill is to feminist polemics'," wrote one reviewer. According to Robert Winder in the New Statesman: "The book fits in with Burchill's theme of praising the working class; Burchill presents Beckham as an anti-laddish symbol of old working-class values – he reminds her of those proud men of her childhood, 'paragons of generosity, industry and chastity'."

For five years until 2003, Burchill wrote a weekly column in The Guardian. Appointed in 1998 by Orr, while editor of the Guardian Weekend supplement, Burchill's career was in trouble; she had been sacked by the revived Punch magazine. Burchill frequently thanked Deborah Orr for rescuing her. One of the pieces she wrote for The Guardian was in reaction to the murder of BBC TV presenter Jill Dando in 1999. She compared the shock of Dando's murder to finding a "tarantula in a punnet full of strawberries". In 2002 she narrowly escaped prosecution for incitement to racial hatred, "following a Guardian column where she described Ireland as being synonymous with child molestation, Nazi-sympathising, and the oppression of women". Burchill had expressed anti-Irish sentiment several times throughout her career, announcing in the London journal Time Out that "I hate the Irish, I think they're appalling".

She supported the Iraq War, writing in The Guardian in 2003 that she was "in favour of a smaller war now rather than a far worse war later", and criticised those opposed to the war as "pro-Saddam apologists". She justified her stance by stating that "this war is about freedom, justice – and oil" and that because Britain and the United States sold weapons to Iraq that, "it is our responsibility to redress our greed and ignorance by doing the lion's share in getting rid of him".

Burchill left The Guardian acrimoniously, saying in an interview that they had offered her a sofa in lieu of a pay rise. She stated that she left the newspaper in protest at what she saw as its "vile anti-Semitism".

From 2005 to 2009
Burchill was an early critic of the fashion for denigrating lower social classes as "chavs". In 2005, she presented the Sky One documentary In Defence of Chavs. "Picking on people worse off than you are isn't humour. It's pathetic, it's cowardly and it's bullying," she commented in an interview for The Daily Telegraph at the time. "It's all to do with self-loathing. ... The middle classes can't bear to see people having more fun, so they attack Chavs for things like their cheap jewellery. It's jealousy, because they secretly know Chavs are better than them. They're even better looking."

Following her departure from The Guardian, in early 2005 she moved to The Times, who were more willing to meet her demands, doubling her previous salary. Shortly after starting her weekly column, she referred to George Galloway, but appeared to confuse him with former MP Ron Brown, reporting the misdeeds of Brown as those of Galloway, "he incited Arabs to fight British troops in Iraq." She apologised in her column and The Times paid damages thought to have been £50,000.

In 2006, The Times dropped her Saturday column, and arranged a more flexible arrangement with Burchill writing for the daily paper. Later it emerged, during a Guardian interview published on 4 August 2008, that eventually she "was given the jolly old heave ho" by The Times, and paid off for the last year of her three-year contract, still receiving the £300,000 she would have earned if she had been obliged to provide copy. She later described her columns for her abbreviated Times contract, which ended abruptly in 2007, thus: "I was totally taking the piss. I didn't spend much time on them and they were such arrant crap."

In February 2006, she announced plans for a year's sabbatical from journalism, planning, among other things, to study theology. In June 2007, she announced that she would not be returning to journalism, but instead concentrate on writing books and TV scripts and finally undertake a theology degree, but she returned to writing for The Guardian newspaper.

Burchill's co-written book with Chas Newkey-Burden, Not in My Name: A Compendium of Modern Hypocrisy, appeared in August 2008, and is dedicated "to Arik and Bibi" (Ariel Sharon and Benjamin Netanyahu). According to Gerald Jacobs, writing for The Jewish Chronicle in 2008, "this book does not merely stand up for Israel, it jumps up and down, cheers and waves its arms". The newspaper described her as "Israel's staunchest supporter in the UK media". When asked if Israel has any flaws, she responded: "Yes. They are much too tolerant of their freaking neighbours, much too reasonable".

She declared in 2005, after Ariel Sharon's withdrawal of Israeli settlers from the Gaza Strip, that "Israel is the only country I would fucking die for. He's the enemy of the Jews. Chucking his own people off the Gaza; to me that's disgusting". Besides writing occasional pieces for The Guardian, she wrote four articles for the centre-right politics and culture magazine Standpoint between July and October 2008.

2010s
At the end of June 2010 it was announced Burchill would be writing exclusively for The Independent, contributing a weekly full-page column for the paper. The connection lasted less than 18 months. Burchill wrote her last column for The Independent at the end of October 2011. Admitting he had tried to recruit Burchill for The Sun in the 1980s, Roy Greenslade commented: "my admittedly occasional reading of her columns in recent years has left [me] feeling that she realises her old schtick is no longer working. She has run out of steam – and sympathetic newspaper editors".

Commenting on the Egyptian Revolution of 2011, Burchill wrote in The Independent: "It would be wonderful to think that what replaces Mubarak will be better. But here's the thing about Middle Eastern regimes: they're all vile. The ones that are 'friendly' are vile and the ones that hate us are vile. Revolutions in the region have a habit of going horribly wrong, and this may well have something to do with the fact that Islam and democracy appear to find it difficult to co-exist for long."

On 13 January 2013, Burchill wrote an article for The Observer defending Suzanne Moore after a reference by Moore to transsexuals had been greeted with a great deal of criticism. In Burchill's view, it showed the "chutzpah" of transsexuals to have their "cock cut off and then plead special privileges as women". There were a number of objections to her writing from members of the transgender community and non-transgender community alike. The editor of The Observer, John Mulholland, responded on the comments page to what he described as "many emails protesting about this piece" and stated that he would be looking into the issue. Liberal Democrat MP Lynne Featherstone, formerly a junior Minister for Women and Equalities, called for the dismissal of Burchill and Mulholland in response to the piece. The article was withdrawn from the website the following day and replaced with a message from Mulholland, but reappeared on the Telegraph website. On 18 January, The Observer's Readers Editor Stephen Pritchard defended the decision to remove the article from the newspaper's website, quoting the editor who took that decision as saying "This clearly fell outside what we might consider reasonable. The piece should not have been published in that form. I don't want the Observer to be conducting debates on those terms or with that language. It was offensive, needlessly. We made a misjudgment and we apologise for that".

Religion and philo-semitism
In her 1987 essay collection Damaged Gods: Cults and Heroes Reappraised, Burchill criticised what she called "the anti-Semitism of politicized American blacks" such as Jesse Jackson, who had referred to New York City as "Hymietown." Burchill wrote, "imagine how the blacks would have gnashed their diamond-studded teeth if a Jewish leader had publicly referred to Harlem as 'Nigger-town'!".

In 1999, Burchill said she "found God", and became a Lutheran and later a "self-confessed Christian Zionist". In June 2007, she announced that she would undertake a theology degree, although she subsequently decided to do voluntary work instead as a way to learn more about Christianity.

In June 2009, The Jewish Chronicle reported that Burchill had become a Friend of Brighton and Hove Progressive Synagogue and was again considering a conversion to Judaism. According to TheJC, she had attended Shabbat services for a month, and studying Hebrew, she described herself as an "ex-Christian", pointing out that she had been pondering on her conversion since the age of 25. Burchill said that "At a time of rising and increasingly vicious anti-semitism from both left and right, becoming Jewish especially appeals to me. ... Added to the fact that I admire Israel so much, it does seem to make sense – assuming of course that the Jews will have me". She wrote in November 2012: "The things I love about the Jews are: their religion, their language and their ancient country".

Burchill clashed with Rabbi Elli Tikvah Sarah of the Brighton and Hove Progressive Synagogue, and the Rabbi's lesbian partner, Jess Woods. Among the reasons for their differences was Rabbi Sarah's defence of Muslims and her advocacy of the Palestinian cause. In Burchill's words, the rabbi "respects PIG ISLAM". Rabbi Sarah told The Independent in September 2014: "The problem is [Burchill] doesn’t have any in-depth knowledge. I can imagine her endlessly watching the film Exodus with Paul Newman. She’s got a kind of Hollywood view of Jews. You know, ‘Jews are so clever, we’ve survived ...'."

In 2014, Burchill's crowdfunded book Unchosen: The Memoirs of a Philo-Semite was published. Tel Aviv-based writer Akin Ajayi in Haaretz thought "the reactionary solipsism of Unchosen is far removed from the affectionate warmness that a love of the Jewish people can be". Burchill's ex-husband, Cosmo Landesman, considered it to be an "exhilarating and exasperating mix of the utterly brilliant and the totally bonkers". He observes that "there are plenty of Jews Julie doesn't love" including the "millions of Jews around the world who have ever criticised Israel. Her love is blind, deaf and dumb to such an obvious contradiction". Guardian columnist Hadley Freeman wrote: "Burchill divides up the chosen people into Good Jews (hardliners, Israelites) and Bad Jews (liberal Jews) with the enthusiasm of an antisemite. Hilariously, she sets herself up as the Jewishness Police, railing against Jews who are not Jewish enough". In his review in The Independent, Keith Kahn-Harris described Unchosen as "occasionally touching, sometimes bigoted and sporadically hilarious" but that it "often degenerates into EDL-style abuse that lacks any redeeming wit." In The Guardian Will Self wrote, "I’m afraid I can’t really dignify her latest offering with the ascription 'book', nor the contents therein as 'writing' – rather they are sophomoric, hammy effusions, wrongheaded, rancorous, and pathetically self-aggrandising."

Other books and television programmes
Burchill has written novels and made television documentaries. Her lesbian-themed novel for teenagers Sugar Rush (2004) was adapted into a television drama series produced by Shine Limited for Channel 4. Lenora Crichlow's portrayal of the central character Maria Sweet inspired the 2007 sequel novel Sweet. Burchill has made television documentaries about the death of her father from asbestosis in 2002 (BBC Four) and Heat magazine broadcast on Sky One in 2006.

Welcome to the Woke Trials
Welcome To The Woke Trials: How #Identity Killed Progressive Politics was planned to be issued by Tabatha Stirling of Stirling Publishing in summer 2021 after being dropped by its original publisher following Burchill's defamatory tweets to Ash Sarkar. On 14 March 2021, when referencing her new publisher Burchill announced that, with Stirling, "I've found someone who's JUST LIKE ME." Stirling is alleged to have written a series of articles for Patriotic Alternative as "Miss Britannia", describing her son's school as "a hellhole for sensible, secure White boys" and claimed "there is one member of staff who is openly gay, and I mean RuPaul extra gay". On 16 March 2021, Burchill announced she would not publish her book with Stirling Publishing, the same day she issued a public apology for libel and harassment of Sarkar. The book was subsequently published by Academica Press.

Responses
Burchill has described her own style as the writing equivalent of screaming and throwing things. For her novel Sugar Rush her publicist described her as "Britain's most famous and controversial journalist". One of her most consistent themes is the championing of the working class against the middle class in most cases, and she has been particularly vocal in defending 'chavs'. According to Will Self, "Burchill's great talent as a journalist is to beautifully articulate the inarticulate sentiments and prejudices of her readers". For Michael Bywater, Burchill's "insights were, and remain, negligible, on the level of a toddler having a tantrum". John Arlidge wrote in The Observer in 2002: "If Burchill is famous for anything it is for being Julie Burchill, the brilliant, unpredictable, outrageously outspoken writer who has an iconoclastic, usually offensive, view on everything.

In November 1980, former Sex Pistols front man John Lydon gave an interview to Ann Louise Bardach in which he referred to Burchill and Tony Parsons as "toss-bag journalists, desperately trying to get in on something" in response to their book, The Boy Looked at Johnny, and described its chapter on amphetamines as "stupidity". Lydon was incensed by Burchill and Parsons attributing his talent to his alleged use of the drug in their book.

In October 1999, in an article for The Guardian, she wrote: "That young men succeed in suicide more often than girls isn't really the point. Indeed, the more callous among us would say that it was quite nice for young men finally to find something that they're better at than girls". After a previous occasion when Burchill wrote "suicides should be left to get on with it", she "received a small number of letters from people whose sons had killed themselves".

In 2002, her life was the subject of a one-woman West End play, Julie Burchill is Away, by Tim Fountain, with Burchill played by her friend Jackie Clune. A sequel by Fountain, Julie Burchill: Absolute Cult, followed in 2014, with Lizzie Roper in the central role.

In 2003, Burchill was ranked number 85 in Channel 4's poll of 100 Worst Britons. The poll was inspired by the BBC series 100 Greatest Britons, though it was less serious in nature. The aim was to discover the "100 worst Britons we love to hate". The poll specified that the nominees had to be British, alive and not currently in prison or pending trial. In 2005, on the 25th anniversary of the Murder of John Lennon, she told The Guardian: "I don't remember where I was but I was really pleased he was dead, as he was a wife-beater, gay-basher, anti-Semite and all-round bully-boy." In the essay "Born Again Cows" published in Damaged Gods (1987), she wrote: "When the sex war is won prostitutes should be shot as collaborators for their terrible betrayal of all women."

On 6 June 2021, and shortly after the announcement of the birth of Lilibet Mountbatten-Windsor, the daughter of the Duke and Duchess of Sussex, Burchill tweeted: "What a missed opportunity. They could have called it Georgina Floydina!”, a reference to George Floyd. Her comments were widely condemned, with racial equality activist Shola Mos-Shogbamimu stating: "She’s (Lilibet) referred to as 'IT'. The utter disrespect & dehumanisation of #HarryandMeghan children because of their proximity to ‘Blackness’ is Racist"; actress Kelechi Okafor wrote: "Likening baby Lilibet to George Floyd is to hone[sic] in on the fact she isn't fully white...She refers to Lilbet as 'it' even though it has been announced that the baby is a girl and she could've addressed her as such...Disgusting scenes." On 8 June, via her Facebook account, Burchill announced that she had been sacked by The Daily Telegraph as a result of her online comments.

Libel
In 2020, Burchill posted a series of defamatory tweets of Ash Sarkar, which included claims that she condones paedophilia and is supportive of Islamist terrorism. Burchill called on her Facebook followers to "wade in on Twitter" against "the Islamists" and the "nonces". As a result of the comments, her publisher, Little, Brown Book Group, cancelled the scheduled publication of Welcome to the Woke Trials, stating that her comments about Islam were “not defensible from a moral or intellectual standpoint”.

In March 2021, after being sued for libel and harassment, Burchill retracted her comments, issued a full apology and paid substantial damages to Sarkar, including her legal costs. On 13 December 2020 I made defamatory statements about @AyoCaesar, which I sincerely regret and retract and have undertaken not to repeat. I have agreed to pay substantial damages to Ash Sarkar and her legal costs. Here is my full and wholehearted apology. 16 March 2021 Burchill stated: "I should not have sent these tweets, some of which included racist and misogynist comments regarding Ms Sarkar's appearance and her sex life." She further apologised for "liking" posts calling on Sarkar to kill herself and promised to refrain from any further harassment of Sarkar.

Personal life
Burchill married Tony Parsons (whom she met at NME) in 1979 at the age of 20. She left Parsons three years later, leaving behind a son, which was followed by years of rancour in the media, described in 2002 as "a steady stream of vitriol in both directions"; she had claimed to have persevered with the "sexual side" of their marriage "by pretending that my husband was my friend Peter York". Her relationships, particularly with Parsons, have featured in her work; Parsons later wrote that "It's like having a stalker. I don't understand her fascination with someone whom she split up with 15 years ago".

Immediately after her relationship with Parsons, Burchill married Cosmo Landesman, the son of Fran and Jay Landesman, with whom she also had a son. The sons from her marriages with Parsons and Landesman lived with their fathers after the separations. After splitting from Landesman in 1992, she married for a third time in 2004, to Daniel Raven, around 13 years her junior, and the brother of her former lover Charlotte Raven. She wrote of the joys of having a "toyboy" in her Times column in 2010. Fellow NME journalist/author Paul Wellings wrote about their friendship in his book I'm A Journalist...Get Me Out of Here. She has written about her lesbian relationships, and declared that "I would never describe myself as 'heterosexual', 'straight' or anything else. Especially not 'bisexual' (it sounds like a sort of communal vehicle missing a mudguard). I like 'spontaneous' as a sexual description". In 2009 she said that she was only attracted to girls in their 20s, and since she was now nearly 50, "I really don't want to be an old perv. So best leave it".

She has lived in Brighton and Hove since 1995 and a book on her adopted home town titled Made in Brighton (Virgin Books) was published in April 2007. Her house in Hove was sold (and demolished for redevelopment as high-density flats) around 2005 for £1.5 million, of which she has given away £300,000, citing Andrew Carnegie: "A man who dies rich, dies shamed."

Burchill's second son, Jack Landesman, died by suicide in late June 2015, aged 29. In an article for The Sunday Times Magazine, she wrote of his inability over many years to experience pleasure and the serious mental health issues from which he suffered.

title: Julie Burchill


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Talk:Recursive

Alright, whose sick joke is it to redirect Recursive to Recursive? 128.175.112.225
 * Fixed... but it's a bit of an in-joke hehe Dysprosia 03:54, 9 Aug 2003 (UTC)

Well, check out the latest change to Recursion... I transfered the joke there in a nonannoying way. 128.175.112.225

What the...? It's a dabpage that doesn't actually disambiguate. If there's no article that actually bears the title "recursive" (which is proper since it's an adjective), there's no need for a dabpage.

Peter Isotalo 18:46, 5 January 2006 (UTC)

I'm a big fan of the "for other uses, see recursive (disambiguation)" joke. It should stay. -kslays (talk • contribs) 05:06, 17 May 2011 (UTC)

good one. — Preceding unsigned comment added by 75.111.79.48 (talk) 04:47, 25 January 2012 (UTC)

Redirecting to Recursion
Peter Isotalo is right: this page does not disambiguate anything, and should not merely list titles that contain "recursive". If a valid article titled Recursive is created in future (for example, a book or album of that name), this page can be turned back into a disambiguation page again, but until then, it should redirect to Recursion.

Relevant policy:

 * WP:PTM
 * A disambiguation page is not a search index. Do not add a link that merely contains part of the page title, or a link that includes the page title in a longer proper name, where there is no significant risk of confusion.

 * WP:NOUN
 * Adjective and verb forms should redirect to articles titled with the corresponding noun.

--Piet Delport (talk) 03:53, 2 April 2012 (UTC)

 * The main problem is that "recursive" also means "computable", but if someone searches for "recursive" in an effort to find that out, the redirect to recursion is not going to help them. Note the "risk of confusion" line in the quote from PTM, and the rest of that section in the linked guideline. — Carl (CBM · talk) 10:56, 2 April 2012 (UTC)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Talk:Recursive


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Skinheads Against Racial Prejudice

Skinheads Against Racial Prejudice (SHARP) are anti-racist skinheads who oppose white power skinheads, neo-fascists and other political racists, particularly if they identify themselves as skinheads. SHARPs claim to reclaim the original multicultural identity of the original skinheads, hijacked by white power skinheads, who they sometimes deride as "boneheads".

SHARP professes no political ideology or affiliation beyond the common opposition to racism. The group stresses the importance of the black Jamaican influence in the original late-1960s skinhead movement, much akin to Trojan skinheads.

Background
The original skinhead subculture started in the United Kingdom in the late 1960s, and had heavy British mod and Jamaican rude boy influences, including a love for ska and soul music. Although some skinheads (including black skinheads) had engaged in "Paki bashing" (random violence against Pakistanis and other South Asian immigrants), skinheads were not associated with an organized racist political movement in the 1960s. However, in the late 1970s, a skinhead revival in the UK included a sizable white nationalist faction, involving organizations such as the National Front, British Movement, Rock Against Communism and in the late eighties Blood and Honour. Because of this, the mainstream media began to label the whole skinhead identity as neo-fascist. This new white power skinhead movement then spread to other countries, including the United States.

Emergence
Skinheads Against Racial Prejudice was founded in 1987 by Marcus, a skinhead from New York City. It emerged as a response by suburban adolescents to the bigotry of the growing White Power Movement in 1982. Traditional skinheads (Trads) formed as a way to show that the skinhead subculture was not based on racism and political extremism. NYC Oi! band The Press and Jason O'Toole (vocalist of the hardcore punk group Life's Blood) were among SHARP's early supporters. In 1989, Roddy Moreno of the Welsh Oi! band The Oppressed visited New York City and met a few SHARP members. On his return to the United Kingdom, he designed a new SHARP logo based on the Trojan Reggae labels design and started promoting SHARP ideals to British skinheads.

SHARP then spread throughout Europe and in other continents. In the UK and other European countries, the SHARP attitude was more based on the individual than on organized groups. In the 2000s, SHARP is thought to have become more of an individual designation than an official organization.

Skinheads, especially in the United States and ASEAN countries like Malaysia, Singapore & Indonesia align themselves with groups and organizations to this day. Most of these would designate themselves as crews. Many strive for an individualist presentation with collectivist goals. As well they are generally imposed into community service, protesting, activism both violent and peaceful. SHARPs often take part of local mutual aid or activist groups such as Black Lives Matter or Anti-Racist Action, in which the latter was even in part founded by skinheads, the most well known of which being Mic Crenshaw.

The United States SHARPs scene has been entirely agitated by the racist overture and have resorted to all forms of anti-racism and anti-fascism to redeem their style and culture.

Violence has been rampant within either of the Skinhead factions for decades now. Between fighting in Clubs and Venues as well as the streets, from mosh pit shuffles to murder. The American scene has been alive and vibrant since the 80's. Some of the most well known Anti-Racist Skinhead crews include The Baldies Syndicate and American S.H.A.R.P (Colloquially known as A.M.S). Additional groups maintain active membership globally.

Many people may confuse SHARP members with racists, since their appearance is superficially similar: shaved heads, denim, lace up boots, button-down shirts and suspenders (called braces). One glib differentiation that might be imagined to separate the two would be music interests. SHARPs may listen to culturally influenced music such as: soul, reggae, and ska; but also punk, hardcore and Oi!. Racist skinheads would disagree with some or all of these musical choices; but may listen to punk, hardcore, Oi!, as well as Nazi punk and National Socialist black metal.

In a deliberate attempt to reject the growing racist subculture, since the early eighties SHARPs promulgated an anti-racist identity through small amateur fanzine publications like Hard As Nails. During the pre-Internet era, these publication established a network of likeminded individuals with similar musical and stylistic attitudes, who considered anti-racism an indispensable part of a living skinhead scene. Another strand of the same Trad revival sought to affirm explicit links with the foundation of Mod subculture and its apolitical, black-positive standards of fashion. The scooter scene, with its runs and Northern Soul dances, had never gone entirely away; and in the post-punk rediscovery of the past, under the influence of The Jam and Quadrophenia, it seemed a fresh and self-renewing direction for skinhead itself to go in.

By 1989, this Trad scene was ripe for the injection of a cultural influence like SHARP, much as its own appearance had been symptomatic of an American internal revolution in US skinheads' attitudes to race and their own subculture.

An outgrowth of SHARP, Red and Anarchist Skinheads (RASH), formed in the United States in 1993 against anti-gay sentiment in the nonracist skinhead community.

Image
The original logo was an American flag surrounded by the SHARP lettering. The second SHARP logo is based on the logo of Trojan Records, which originally mainly released black Jamaican ska, rocksteady, and reggae artists. Some variants of this design also incorporate the checkerboard motif of 2 Tone Records, known for its multiracial roster of ska- and reggae-influenced bands.

The way in which SHARPs dress is to project an image that looks hard and smart, in an evolving continuity with style ideals established in the middle-to-late 1960s. This style and demeanour originated from the UK, growing out of the pre-existing mod movement, taking cues and influences from Jamaican ska and rude boy culture. They remain true to the style's original purpose of enjoying life, clothes, attitude and music.

title: Skinheads Against Racial Prejudice


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Combinatorial game theory

Combinatorial game theory is a branch of mathematics and theoretical computer science that typically studies sequential games with perfect information. Study has been largely confined to two-player games that have a position that the players take turns changing in defined ways or moves to achieve a defined winning condition. Combinatorial game theory has not traditionally studied games of chance or those that use imperfect or incomplete information, favoring games that offer perfect information in which the state of the game and the set of available moves is always known by both players. However, as mathematical techniques advance, the types of game that can be mathematically analyzed expands, thus the boundaries of the field are ever changing. Scholars will generally define what they mean by a "game" at the beginning of a paper, and these definitions often vary as they are specific to the game being analyzed and are not meant to represent the entire scope of the field.

Combinatorial games include well-known games such as chess, checkers, and Go, which are regarded as non-trivial, and tic-tac-toe, which is considered trivial, in the sense of being "easy to solve". Some combinatorial games may also have an unbounded playing area, such as infinite chess. In combinatorial game theory, the moves in these and other games are represented as a game tree.

Combinatorial games also include one-player combinatorial puzzles such as Sudoku, and no-player automata, such as Conway's Game of Life, (although in the strictest definition, "games" can be said to require more than one participant, thus the designations of "puzzle" and "automata". )

Game theory in general includes games of chance, games of imperfect knowledge, and games in which players can move simultaneously, and they tend to represent real-life decision making situations.

Combinatorial game theory has a different emphasis than "traditional" or "economic" game theory, which was initially developed to study games with simple combinatorial structure, but with elements of chance (although it also considers sequential moves, see extensive-form game). Essentially, combinatorial game theory has contributed new methods for analyzing game trees, for example using surreal numbers, which are a subclass of all two-player perfect-information games. The type of games studied by combinatorial game theory is also of interest in artificial intelligence, particularly for automated planning and scheduling. In combinatorial game theory there has been less emphasis on refining practical search algorithms (such as the alpha–beta pruning heuristic included in most artificial intelligence textbooks), but more emphasis on descriptive theoretical results (such as measures of game complexity or proofs of optimal solution existence without necessarily specifying an algorithm, such as the strategy-stealing argument).

An important notion in combinatorial game theory is that of the solved game. For example, tic-tac-toe is considered a solved game, as it can be proven that any game will result in a draw if both players play optimally. Deriving similar results for games with rich combinatorial structures is difficult. For instance, in 2007 it was announced that checkers has been weakly solved—optimal play by both sides also leads to a draw—but this result was a computer-assisted proof. Other real world games are mostly too complicated to allow complete analysis today, although the theory has had some recent successes in analyzing Go endgames. Applying combinatorial game theory to a position attempts to determine the optimum sequence of moves for both players until the game ends, and by doing so discover the optimum move in any position. In practice, this process is torturously difficult unless the game is very simple.

It can be helpful to distinguish between combinatorial "mathgames" of interest primarily to mathematicians and scientists to ponder and solve, and combinatorial "playgames" of interest to the general population as a form of entertainment and competition. However, a number of games fall into both categories. Nim, for instance, is a playgame instrumental in the foundation of combinatorial game theory, and one of the first computerized games. Tic-tac-toe is still used to teach basic principles of game AI design to computer science students.

History
Combinatorial game theory arose in relation to the theory of impartial games, in which any play available to one player must be available to the other as well. One such game is Nim, which can be solved completely. Nim is an impartial game for two players, and subject to the normal play condition, which means that a player who cannot move loses. In the 1930s, the Sprague–Grundy theorem showed that all impartial games are equivalent to heaps in Nim, thus showing that major unifications are possible in games considered at a combinatorial level, in which detailed strategies matter, not just pay-offs.

In the 1960s, Elwyn R. Berlekamp, John H. Conway and Richard K. Guy jointly introduced the theory of a partisan game, in which the requirement that a play available to one player be available to both is relaxed. Their results were published in their book Winning Ways for your Mathematical Plays in 1982. However, the first work published on the subject was Conway's 1976 book On Numbers and Games, also known as ONAG, which introduced the concept of surreal numbers and the generalization to games. On Numbers and Games was also a fruit of the collaboration between Berlekamp, Conway, and Guy.

Combinatorial games are generally, by convention, put into a form where one player wins when the other has no moves remaining. It is easy to convert any finite game with only two possible results into an equivalent one where this convention applies. One of the most important concepts in the theory of combinatorial games is that of the sum of two games, which is a game where each player may choose to move either in one game or the other at any point in the game, and a player wins when his opponent has no move in either game. This way of combining games leads to a rich and powerful mathematical structure.

Conway stated in On Numbers and Games that the inspiration for the theory of partisan games was based on his observation of the play in Go endgames, which can often be decomposed into sums of simpler endgames isolated from each other in different parts of the board.

Examples
The introductory text Winning Ways introduced a large number of games, but the following were used as motivating examples for the introductory theory:
 * Blue–Red Hackenbush - At the finite level, this partisan combinatorial game allows constructions of games whose values are dyadic rational numbers. At the infinite level, it allows one to construct all real values, as well as many infinite ones that fall within the class of surreal numbers.
 * Blue–Red–Green Hackenbush - Allows for additional game values that are not numbers in the traditional sense, for example, star.
 * Toads and Frogs - Allows various game values. Unlike most other games, a position is easily represented by a short string of characters.
 * Domineering - Various interesting games, such as hot games, appear as positions in Domineering, because there is sometimes an incentive to move, and sometimes not. This allows discussion of a game's temperature.
 * Nim - An impartial game. This allows for the construction of the nimbers. (It can also be seen as a green-only special case of Blue-Red-Green Hackenbush.)

The classic game Go was influential on the early combinatorial game theory, and Berlekamp and Wolfe subsequently developed an endgame and temperature theory for it (see references). Armed with this they were able to construct plausible Go endgame positions from which they could give expert Go players a choice of sides and then defeat them either way.

Another game studied in the context of combinatorial game theory is chess. In 1953 Alan Turing wrote of the game, "If one can explain quite unambiguously in English, with the aid of mathematical symbols if required, how a calculation is to be done, then it is always possible to programme any digital computer to do that calculation, provided the storage capacity is adequate." In a 1950 paper, Claude Shannon estimated the lower bound of the game-tree complexity of chess to be 10120, and today this is referred to as the Shannon number. Chess remains unsolved, although extensive study, including work involving the use of supercomputers has created chess endgame tablebases, which shows the result of perfect play for all end-games with seven pieces or less. Infinite chess has an even greater combinatorial complexity than chess (unless only limited end-games, or composed positions with a small number of pieces are being studied).

Overview
A game, in its simplest terms, is a list of possible "moves" that two players, called left and right, can make. The game position resulting from any move can be considered to be another game. This idea of viewing games in terms of their possible moves to other games leads to a recursive mathematical definition of games that is standard in combinatorial game theory. In this definition, each game has the notation {L|R}. L is the set of game positions that the left player can move to, and R is the set of game positions that the right player can move to; each position in L and R is defined as a game using the same notation.

Using Domineering as an example, label each of the sixteen boxes of the four-by-four board by A1 for the upper leftmost square, C2 for the third box from the left on the second row from the top, and so on. We use e.g. (D3, D4) to stand for the game position in which a vertical domino has been placed in the bottom right corner. Then, the initial position can be described in combinatorial game theory notation as

 * $$\{(\mathrm{A}1,\mathrm{A}2),(\mathrm{B}1,\mathrm{B}2),\dots|(\mathrm{A}1,\mathrm{B}1), (\mathrm{A}2,\mathrm{B}2),\dots\}.$$

In standard Cross-Cram play, the players alternate turns, but this alternation is handled implicitly by the definitions of combinatorial game theory rather than being encoded within the game states.

 * [[Image:20x20square.png]][[Image:20x20square.png]]
 * [[Image:20x20square.png]]

 * $$\{(\mathrm{A}1,\mathrm{A}2) | (\mathrm{A}1,\mathrm{B}1)\} = \{ \{|\} | \{|\} \}.$$

The above game describes a scenario in which there is only one move left for either player, and if either player makes that move, that player wins. (An irrelevant open square at C3 has been omitted from the diagram.) The {|} in each player's move list (corresponding to the single leftover square after the move) is called the zero game, and can actually be abbreviated 0. In the zero game, neither player has any valid moves; thus, the player whose turn it is when the zero game comes up automatically loses.

The type of game in the diagram above also has a simple name; it is called the star game, which can also be abbreviated ∗. In the star game, the only valid move leads to the zero game, which means that whoever's turn comes up during the star game automatically wins.

An additional type of game, not found in Domineering, is a loopy game, in which a valid move of either left or right is a game that can then lead back to the first game. Checkers, for example, becomes loopy when one of the pieces promotes, as then it can cycle endlessly between two or more squares. A game that does not possess such moves is called loopfree.

Numbers
Numbers represent the number of free moves, or the move advantage of a particular player. By convention positive numbers represent an advantage for Left, while negative numbers represent an advantage for Right. They are defined recursively with 0 being the base case.
 * 0 = {|}
 * 1 = {0|}, 2 = {1|}, 3 = {2|}
 * −1 = {|0}, −2 = {|−1}, −3 = {|−2}

The zero game is a loss for the first player.

The sum of number games behaves like the integers, for example 3 + −2 = 1.

Star
Star, written as ∗ or {0|0}, is a first-player win since either player must (if first to move in the game) move to a zero game, and therefore win.

 * ∗ + ∗ = 0, because the first player must turn one copy of ∗ to a 0, and then the other player will have to turn the other copy of ∗ to a 0 as well; at this point, the first player would lose, since 0 + 0 admits no moves.

The game ∗ is neither positive nor negative; it and all other games in which the first player wins (regardless of which side the player is on) are said to be fuzzy with or confused with 0; symbolically, we write ∗ || 0.

Up
Up, written as ↑, is a position in combinatorial game theory. In standard notation, ↑ = {0|∗}.

 * −↑ = ↓ (down)

Up is strictly positive (↑ > 0), but is infinitesimal. Up is defined in Winning Ways for your Mathematical Plays.

Down
Down, written as ↓, is a position in combinatorial game theory. In standard notation, ↓ = {∗|0}.

 * −↓ = ↑ (up)

Down is strictly negative (↓ < 0), but is infinitesimal. Down is defined in Winning Ways for your Mathematical Plays.

"Hot" games
Consider the game {1|−1}. Both moves in this game are an advantage for the player who makes them; so the game is said to be "hot;" it is greater than any number less than −1, less than any number greater than 1, and fuzzy with any number in between. It is written as ±1. It can be added to numbers, or multiplied by positive ones, in the expected fashion; for example, 4 ± 1 = {5|3}.

Nimbers
An impartial game is one where, at every position of the game, the same moves are available to both players. For instance, Nim is impartial, as any set of objects that can be removed by one player can be removed by the other. However, domineering is not impartial, because one player places horizontal dominoes and the other places vertical ones. Likewise Checkers is not impartial, since the players own different colored pieces. For any ordinal number, one can define an impartial game generalizing Nim in which, on each move, either player may replace the number with any smaller ordinal number; the games defined in this way are known as nimbers. The Sprague–Grundy theorem states that every impartial game is equivalent to a nimber.

The "smallest" nimbers – the simplest and least under the usual ordering of the ordinals – are 0 and ∗.

title: Combinatorial game theory


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Talk:Topologist's sine curve

This page is useless without the figure. Could I scan it from a book? I guess a mathematical figure cannot be copyrighted. wshun 03:50, 9 Aug 2003 (UTC)

Image crashes browser when viewed full-size
Seems like this space is too weird for Iceweasel 2.0.0.3 (Debian version of Firefox) ;) Works fine in Konqueror. Functor salad 19:39, 19 July 2007 (UTC)

-- Image looks inconsistant with definition as the plot goes to negative x and in definition we take x from (0;1]. I may be just wrong though. 83.21.141.16 (talk) 10:36, 26 August 2016 (UTC)

Something's very very wrong here
In the article, it says: You take the closure of the graph of sin(1/x) with x\in ]0,1]. The function is bounded, the domain is bounded, hence the graph is bounded. The closure of a bounded set w.r.t. the topology of a finite dimensional euclidean space is always compact. => Therefore, the topologist's sine curve is always compact, hence locally compact. But in the article, it says: "T is the continuous image of a locally compact space (namely, let V be the space {−1} ∪ (0, 1], and use the map f from V to T defined by f(−1) = (0,0) and f(x) = (x, sin(1/x)) for x > 0), but is not locally compact itself."

Later, it says in the article, that you may a variation, named "closed topologist's sine curve", which is now exactly the closure of the graph and therefore - by defintion - equal to the topologist's sine curve. So, the original topologist's sine curve is already the closed one...

I guess that some of the statements in this article refer to another sort of sine curve, where you just add (0,0) to the graph of sin(1/x). Then it would make sense to take the closure of it and then it would not be locally compact, but the image of a locally compact set (even a compact set)

The question is now: When topologists talk about "topologist's sine curve" do they mean the one with the interval or the one with just a point? --131.234.106.197 (talk) 16:42, 26 November 2008 (UTC)

 * Fixed. I don't know the answer to your last question. In Munkres, the closure is used. –Pomte 16:31, 11 December 2008 (UTC)

 * thanx--131.234.106.197 (talk) 12:26, 15 December 2008 (UTC)
 * The current terminology of the article (with 'topologist's sine curve' denoting a non-closed set) is that of Counterexamples in Topology. Algebraist 22:55, 15 January 2009 (UTC)

Hausdorff dimension of this space
Is the Hausdorff dimension of this space known exactly or just approximately? Does it exceed 1? 92.105.139.80 (talk) 22:08, 20 January 2010 (UTC)

Graph improvement?
I'm not sure how to do this, but it would be neat to come up with a graph whose color becomes darker and/or more saturated to suggest increasing density approaching 0. As it is, at a certain point it just turns into a solid block. --Dfeuer (talk) 17:05, 12 January 2013 (UTC)

Definition: including origin vs interval
An anonymous contributor in the last month changed the formula of the definition of the set T to include not just the origin but the whole interval $$\{0\} \times [-1,1]$$. This contradicts with the text, and it is a bit confusing, since later in the article the closed version of this set is referenced.

I have thus reverted the change: I think both definitions are used in the literature, so it does not really matter which one we give first, but we should be consistent in the article.

make page: Fourier analysis on topological space
My title is very generic. If we make the page we then have to update the title. But the page should be created! — Preceding unsigned comment added by 85.75.194.216 (talk) 22:10, 22 July 2019 (UTC)

title: Talk:Topologist's sine curve


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Transitive closure

In mathematics, the transitive closure $R+$ of a homogeneous binary relation $R$ on a set $X$ is the smallest relation on $X$ that contains $R$ and is transitive. For finite sets, "smallest" can be taken in its usual sense, of having the fewest related pairs; for infinite sets $R+$ is the unique minimal transitive superset of $R$.

For example, if $X$ is a set of airports and $x R y$ means "there is a direct flight from airport $x$ to airport $y$" (for $x$ and $y$ in $X$), then the transitive closure of $R$ on $X$ is the relation $R+$ such that $x R+ y$ means "it is possible to fly from $x$ to $y$ in one or more flights".

More formally, the transitive closure of a binary relation $R$ on a set $X$ is the smallest (w.r.t. ⊆) transitive relation $R+$ on $X$ such that $R$ ⊆ $R+$; see. We have $R+$ = $R$ if, and only if, $R$ itself is transitive.

Conversely, transitive reduction adduces a minimal relation $S$ from a given relation $R$ such that they have the same closure, that is, $S+ = R+$; however, many different $S$ with this property may exist.

Both transitive closure and transitive reduction are also used in the closely related area of graph theory.

Transitive relations and examples
A relation R on a set X is transitive if, for all x, y, z in X, whenever x R y and y R z then x R z. Examples of transitive relations include the equality relation on any set, the "less than or equal" relation on any linearly ordered set, and the relation "x was born before y" on the set of all people. Symbolically, this can be denoted as: if x < y and y < z then x < z.

One example of a non-transitive relation is "city x can be reached via a direct flight from city y" on the set of all cities. Simply because there is a direct flight from one city to a second city, and a direct flight from the second city to the third, does not imply there is a direct flight from the first city to the third. The transitive closure of this relation is a different relation, namely "there is a sequence of direct flights that begins at city x and ends at city y". Every relation can be extended in a similar way to a transitive relation.

An example of a non-transitive relation with a less meaningful transitive closure is "x is the day of the week after y". The transitive closure of this relation is "some day x comes after a day y on the calendar", which is trivially true for all days of the week x and y (and thus equivalent to the Cartesian square, which is "x and y are both days of the week").

Existence and description
For any relation R, the transitive closure of R always exists. To see this, note that the intersection of any family of transitive relations is again transitive. Furthermore, there exists at least one transitive relation containing R, namely the trivial one: X × X. The transitive closure of R is then given by the intersection of all transitive relations containing R.

For finite sets, we can construct the transitive closure step by step, starting from R and adding transitive edges. This gives the intuition for a general construction. For any set X, we can prove that transitive closure is given by the following expression
 * $$R^{+}=\bigcup_{i = 1}^{\infty} R^i.$$

where $$R^i$$ is the i-th power of R, defined inductively by
 * $$R^1 = R$$

and, for $$i>0$$,
 * $$R^{i+1} = R \circ R^i$$

where $$\circ$$ denotes composition of relations.

To show that the above definition of R+ is the least transitive relation containing R, we show that it contains R, that it is transitive, and that it is the smallest set with both of those characteristics.

 * $$R \subseteq R^{+}$$: $$ R^+$$ contains all of the $$ R^i$$, so in particular $$ R^+$$ contains $$ R$$.
 * $$ R^{+}$$ is transitive: If $$(s_1, s_2), (s_2, s_3)\in R^+$$, then $$(s_1, s_2)\in R^j$$ and $$(s_2, s_3)\in R^k$$ for some $$j,k$$ by definition of $$R^+$$. Since composition is associative, $$R^{j+k} = R^j \circ R^k$$; hence $$(s_1, s_3)\in R^{j+k} \subseteq R^+$$ by definition of $$\circ$$ and $$R^+$$.
 * $$R^{+}$$ is minimal, that is, if $$T$$ is any transitive relation containing $$R$$, then $$R^{+} \subseteq T$$: Given any such $$T$$, induction on $$i$$ can be used to show $$R^i\subseteq T$$ for all $$i$$ as follows: Base: $$R^1 = R \subseteq T$$ by assumption. Step: If $$R^i\subseteq T$$ holds, and $$(s_1, s_3)\in R^{i+1} = R \circ R^i$$, then $$(s_1, s_2) \in R$$ and $$(s_2, s_3)\in R^i$$ for some $$s_2$$, by definition of $$\circ$$. Hence, $$(s_1, s_2), (s_2, s_3)\in T$$ by assumption and by induction hypothesis. Hence $$(s_1, s_3)\in T$$ by transitivity of $$T$$; this completes the induction. Finally, $$R^i\subseteq T$$ for all $$i$$ implies $$R^{+} \subseteq T$$ by definition of $$R^{+}$$.

Properties
The intersection of two transitive relations is transitive.

The union of two transitive relations need not be transitive. To preserve transitivity, one must take the transitive closure. This occurs, for example, when taking the union of two equivalence relations or two preorders. To obtain a new equivalence relation or preorder one must take the transitive closure (reflexivity and symmetry—in the case of equivalence relations—are automatic).

In graph theory
In computer science, the concept of transitive closure can be thought of as constructing a data structure that makes it possible to answer reachability questions. That is, can one get from node a to node d in one or more hops? A binary relation tells you only that node a is connected to node b, and that node b is connected to node c, etc. After the transitive closure is constructed, as depicted in the following figure, in an O(1) operation one may determine that node d is reachable from node a. The data structure is typically stored as a Boolean matrix, so if matrix[1][4] = true, then it is the case that node 1 can reach node 4 through one or more hops.

The transitive closure of the adjacency relation of a directed acyclic graph (DAG) is the reachability relation of the DAG and a strict partial order.

The transitive closure of an undirected graph produces a cluster graph, a disjoint union of cliques. Constructing the transitive closure is an equivalent formulation of the problem of finding the components of the graph.

In logic and computational complexity
The transitive closure of a binary relation cannot, in general, be expressed in first-order logic (FO). This means that one cannot write a formula using predicate symbols R and T that will be satisfied in any model if and only if T is the transitive closure of R. In finite model theory, first-order logic (FO) extended with a transitive closure operator is usually called transitive closure logic, and abbreviated FO(TC) or just TC. TC is a sub-type of fixpoint logics. The fact that FO(TC) is strictly more expressive than FO was discovered by Ronald Fagin in 1974; the result was then rediscovered by Alfred Aho and Jeffrey Ullman in 1979, who proposed to use fixpoint logic as a database query language. With more recent concepts of finite model theory, proof that FO(TC) is strictly more expressive than FO follows immediately from the fact that FO(TC) is not Gaifman-local.

In computational complexity theory, the complexity class NL corresponds precisely to the set of logical sentences expressible in TC. This is because the transitive closure property has a close relationship with the NL-complete problem STCON for finding directed paths in a graph. Similarly, the class L is first-order logic with the commutative, transitive closure. When transitive closure is added to second-order logic instead, we obtain PSPACE.

In database query languages
Since the 1980s Oracle Database has implemented a proprietary SQL extension  that allows the computation of a transitive closure as part of a declarative query. The SQL 3 (1999) standard added a more general  construct also allowing transitive closures to be computed inside the query processor; as of 2011 the latter is implemented in IBM Db2, Microsoft SQL Server, Oracle, PostgreSQL, and MySQL (v8.0+). SQLite released support for this in 2014.

Datalog also implements transitive closure computations.

MariaDB implements Recursive Common Table Expressions, which can be used to compute transitive closures. This feature was introduced in release 10.2.2 of April 2016.

Algorithms
Efficient algorithms for computing the transitive closure of the adjacency relation of a graph can be found in. Reducing the problem to multiplications of adjacency matrices achieves the time complexity of matrix multiplication, $$O(n^{2.3728596})$$. However, this approach is not practical since both the constant factors and the memory consumption for sparse graphs are high. The problem can also be solved by the Floyd–Warshall algorithm in $$O(n^3)$$, or by repeated breadth-first search or depth-first search starting from each node of the graph.

For directed graphs, Purdom's algorithm solves the problem by first computing its condensation DAG and its transitive closure, then lifting it to the original graph. Its runtime is $$O(m+\mu n)$$, where $$\mu$$ is the number of edges between its strongly connected components.

More recent research has explored efficient ways of computing transitive closure on distributed systems based on the MapReduce paradigm.

title: Transitive closure


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Canadian Radio Broadcasting Commission

The Canadian Radio Broadcasting Commission (CRBC/CCR; Commission canadienne de radiodiffusion), also referred to as the Canadian Radio Commission (CRC), was Canada's first public broadcaster and the immediate precursor to the Canadian Broadcasting Corporation.

Origins
The CRBC was established in 1932 by the government of R.B. Bennett based on the recommendations of the 1929 Royal Commission on Radio Broadcasting and as a result of the lobbying efforts of the Canadian Radio League. The network was created on May 26, 1932 and existed until November 2, 1936 when it was reorganized as a Crown corporation becoming the Canadian Broadcasting Corporation. Daily national broadcasting began in May 1933 with an hour's programming a day which was increased over time.

At its creation, Prime Minister Bennett spoke of the need for public control of radio saying:
 * "This country must be assured of complete Canadian control of broadcasting from Canadian sources. Without such control, broadcasting can never be the agency by which national consciousness may be fostered and sustained and national unity still further strengthened."

The three-man commission had a mandate to regulate, control all broadcasting in Canada as well as to create and transmit its own programming and to lease, purchase or construct stations for its dissemination. Its ability to carry out its mandate was severely restricted by the financial constraints of the Great Depression and it was forced to rely on private stations to carry its programs to much of the country.

Bennett named Hector Charlesworth, editor of Saturday Night magazine, as chairman of the CRBC. The other members of the Commission were Thomas Maher, an unsuccessful Quebec Conservative candidate in 1930 federal election and director of a private radio station in Quebec City and Lieutenant Colonel William Arthur Steel (1890–1968) former Chief Radio Officer with the Canadian Corps during World War I.

Regulator
In addition to being a creator of programming and a broadcaster, the CRBC was responsible under the Canadian Radio Broadcasting Act for regulating and controlling all broadcasting in Canada, including privately owned radio stations that did not carry CRBC programming. The CRBC's regulatory responsibilities included determining the number, location and transmitting power of radio stations as well as the amount of time spent carrying local and national programming.

Operations
The CRBC acquired radio stations in Ottawa, Vancouver and Moncton from the CNR Radio network for $50,000 as well as CNR Radio studios in various other cities and leased or established additional stations in Montreal, Chicoutimi, Quebec City, Toronto and Windsor. The stations had been created in the 1920s by Canadian National Railways to provide broadcasting for railway passengers but were also heard by the general public and functioned, along with up two dozen stations across the country on which CNR Radio leased time, as an early national radio network. The CRBC also hired private stations across the country to carry at least three hours a day of CRBC programming. These affiliates were known as "basic stations". CRBC-owned or leased stations had the prefix "CRC" in their call-letters.

E. Austin Weir, formerly of the CNR radio network, became the CRBC's program director. He was terminated, however, for not providing enough programming. Weir was replaced by Ernie Bushnell who became director of CRBC programming in Ontario and Western Canada and Arthur Dupont who was responsible for Quebec and the Maritimes.

Programming
Network programming included orchestral music, live Metropolitan Opera radio broadcasts from NBC and New York Philharmonic Orchestra broadcasts from CBS, dramatized informational programs such as The Youngbloods of Beaver Band which serialized life on a western Canadian farm. One of the best known national drama series was Radio Theatre Guild produced in Montreal by Rupert Caplan. Under program director Ernie Bushnell, CRBC increased the number of weekly English language series being broadcast over the network to 17 by the time of the network's demise.

Another CRBC program, one which originated on the CNR's network in 1931, was Hockey Night in Canada under the names Saturday Night Hockey, General Motors Hockey Broadcast and then, starting in 1934, The Imperial Oil Hockey Broadcast. Other programming including fare such as Bible Dramas from CRCM Montreal, Canadian Press News - a 15-minute nightly newscast from CRCT Toronto presented by Charles Jennings (the father of Peter Jennings), Western Radio Players a weekly dramatic half-hour from CKY. Winnipeg

The Northern Messenger, a weekly program originating from Toronto late Saturday nights but aimed at Canadians living in the far north including personal messages to RCMP officers, missionaries, trappers and others from family and friends, began in December 1933. The show was made up of listener letters, important messages from family and friends in other parts of the country, news and recorded music, and would run from November to May on CRBC's stations as well as several Canadian shortwave stations. During its first season, the program relayed 1,745 messages; a figure that increased sixfold within four years. The program would be continued by the CBC into the 1970s.

CRBC's national radio coverage of the 1935 federal election was the first time Canadian election results were broadcast nationwide.

All programs were live as there were no recording facilities. The network had six stations of its own and relied largely on private affiliates to provide studios, equipment and staff.

In April 1936, CRBC provided round the clock coverage of the Moose River Mine Disaster in Nova Scotia with announcer J. Frank Willis broadcasting live reports from the mine head every half hour for five days as rescue crews attempted to recover the lost miners. The reports were broadcast throughout Canada as well as to 650 stations in the United States and the BBC.

In July 1936, the network broadcast live coverage from France of the unveiling by King Edward VIII of the Canadian National Vimy Memorial, which was relayed across the ocean by shortwave radio.

The CRBC turned to radio advertising in order to make up a shortfall between government grants and the amount of money needed to run the network.

Demise
The network was criticized by the Canadian Radio League for having poor programming while the Liberal Party opposition accused the network of being biased towards the governing Conservatives.

During the election campaign, the CRBC broadcast a series of 15 minutes soap operas called Mr. Sage which were critical of Opposition leader William Lyon Mackenzie King and the Liberal Party. Decried as political propaganda, the incident was one factor in King's decision to replace the CRBC with a new entity when the Liberals took office following the election.

New legislation was introduced creating the Canadian Broadcasting Corporation as a crown corporation with an arm's length relationship with the government. The management of the CRBC, including Charlesworth, were dismissed and the new CBC was launched on November 2, 1936 with a new management team.

At its demise, the CRBC was made up of eight network owned-and-operated stations and 14 privately owned network affiliates.

The CRBC's regulatory powers were largely transferred to the CBC while the role of licensing stations and allocating wavelengths was assumed by the newly created Department of Transport. In 1958, the Board of Broadcast Governors assumed responsibilities for regulating public and private broadcasters from the CBC and the Department of Transport and in 1968, with the adoption of the Broadcasting Act, regulation became the responsibility of the Canadian Radio-television Commission (CRTC).

CRBC owned or leased basic stations

 * CRCA Moncton (closed October 31, 1933 due to plans to build more powerful transmitters at Sackville, New Brunswick. The new station was eventually opened by the CBC as CBA.)
 * CRCK Quebec City AM 1050 - now CBV-FM
 * CRCS Chicoutimi AM 950 - now CBJ-FM
 * CRCM Montreal AM 910 - now CBME-FM
 * CRCO Ottawa AM 880 - now CBO-FM
 * CRCT Toronto AM 840 - (leased) - now CBLA-FM
 * CRCY Toronto AM 1030 and AM 1420 (leased) - now CJBC AM 860
 * CRCW Windsor AM 600 - now CBEW-FM
 * CRCV Vancouver AM 1100 - now CBU AM 690

All but the Moncton station later became CBC Radio or Radio Canada stations.

The CRBC also leased and operated shortwave radio station CRCX (formerly VE9GW) in Bowmanville, Ontario, broadcasting on 6095 kHz. The station was leased from Gooderham & Worts with Toronto station CRCT (formerly CKGW), which transmitted from the same Bowmanville site. CBC bought CRCX and CRCT from Gooderham & Worts in 1937, after leasing them for a year, but would close the shortwave station in 1938.

CRBC affiliates

 * CHNS Halifax
 * CHSJ Saint John
 * CHNC New Carlisle, Quebec
 * CKCH Hull (now defunct)
 * CFRC Kingston
 * CFPL London
 * CKY Winnipeg
 * CKCK Regina
 * CFQC Saskatoon
 * CKBI Prince Albert
 * CHAB Moose Jaw
 * CJCA Edmonton
 * CFAC Calgary
 * CHWK Chilliwack
 * CFCJ Kamloops
 * CKOV Kelowna
 * CJAT Trail

With the exception of CKY (now CBW, unrelated to today's CKY-FM), the remaining affiliates are no longer connected the CBC.

title: Canadian Radio Broadcasting Commission