title: Charge of the Light Brigade

text: Private William Ellis of the 11th Hussars was erroneously described as the last survivor of the Charge at the time of his funeral at Upper Hale Cemetery in Farnham, Surrey in 1913 after his death aged 82. A number of individuals who died during 1916–17 were thought to be the 'last' survivor of the Charge of the Light Brigade. These include Sergeant James A. Mustard of the 17th Lancers, aged 85, who had his funeral with military honours at Twickenham in early February 1916. In the "Abergavenny Chronicle" news report published on 11 February it was stated:


title: Charge of the Light Brigade

text: He was one of thirty-eight men of the 145 of the 17th Lancers that came out of the charge led by Cardigan, and was always of the opinion that no one sounded the charge at all. He was in the battles of Alma and Mackenzie's Farm, and the storming and taking of Sebastopol, and before leaving for Varna marched with his regiment from Hampton Court to Portsmouth.


title: Charge of the Light Brigade

text: The "The Cambrian News" of 30 June 1916 noted the passing of another 'last', Thomas Warr, who had died the previous day at 85.


title: Charge of the Light Brigade

text: William Henry Pennington of the 11th Hussars, who embarked a relatively successful career as a Shakespearean actor on leaving the Army, died in 1923. The last survivor was Edwin Hughes of the 13th Light Dragoons, who died on 18 May 1927, aged 96.


title: Charge of the Light Brigade

text: Poet Laureate Alfred, Lord Tennyson wrote evocatively about the battle in his poem "The Charge of the Light Brigade". Tennyson's poem, written 2 December and published on 9 December 1854, in "The Examiner", praises the brigade ("When can their glory fade? O the wild charge they made!") while trenchantly mourning the appalling futility of the charge ("Not tho' the soldier knew, someone had blunder'd... Charging an army, while all the world wonder'd"). Tennyson wrote the poem inside only a few minutes after reading an account of the battle in "The Times", according to his grandson Sir Charles Tennyson. It immediately became hugely popular, and even reached the troops in the Crimea, where 1,000 copies were distributed in pamphlet form.


title: Charge of the Light Brigade

text: Nearly 36 years later, Kipling wrote "The Last of the Light Brigade" (1890), commemorating a visit by the last 20 survivors to Tennyson (then aged 80) to reproach him gently for not writing a sequel about the way in which England was treating its old soldiers. Some sources treat the poem as an account of a real event, but other commentators class the destitute old soldiers as allegorical, with the visit invented by Kipling to draw attention to the poverty in which the real survivors were living, in the same way that he evoked Tommy Atkins in "The Absent-Minded Beggar" (1899).


title: Charge of the Light Brigade

text: The 1877 novel "Black Beauty", written in the first person as if by the horse of the title, includes a former cavalry horse named Captain who describes his experience of being in the Charge of the Light Brigade.


title: Scientology and celebrities

text: Recruiting and retaining Scientologist celebrities and getting them to endorse Scientology to the public at large has been important to the Church of Scientology since its early days. The organization has had a written program governing celebrity recruitment since at least 1955, when L. Ron Hubbard created "Project Celebrity", offering rewards to Scientologists who recruited targeted celebrities. Early interested parties included former silent-screen star Gloria Swanson and jazz pianist Dave Brubeck. The Scientology organization has a particular interest in international focus on wealthy businesspeople and influencers to help promote its ideals. A Scientology policy letter of 1976 states that "rehabilitation of celebrities who are just beyond or just approaching their prime" enables the "rapid dissemination" of Scientology.


title: Scientology and celebrities

text: The Church of Scientology operates special Celebrity Centres. Scientology policy governs the Celebrity Centres (the main one in Los Angeles and others in Paris, Nashville, and elsewhere), stating that "one of the major purposes of the Celebrity Centre and its staff is to expand the number of celebrities in Scientology." (Scientology Flag Order 2310) Another order describes Celebrity Centre's Public Clearing Division and its goal, "broad public into Scientology from celebrity dissemination"; this division has departments for planning celebrity events and routing the general public onto Scientology services as a result of celebrity involvement.


title: Scientology and celebrities

text: Celebrities are very Special people and have a very distinct line of dissemination. They have comm[unication] lines that others do not have and many medias [sic] to get their dissemination through (Flag Order 3323, 9 May 1973)


title: Scientology and celebrities

text: Hugh B. Urban, professor of religious studies in the Department of Comparative Studies at Ohio State University said about Scientology's appeal to celebrities in an interview for Beliefnet.com:


title: Scientology and celebrities

text: Journalistic and media sources claim that Scientology is "The Church of the Stars" or a "star-studded sect," although there are likely more Hollywood celebrities in other religious traditions. One reason for this is the church's Celebrity Centres, which is unique to the religion. Most members in these facilities are not celebrities, however, a lot of members are part of the entertainment industry.


title: Scientology and celebrities

text: The Church of Scientology has a long history of seeking out artists, musicians, writers and actors, and states that Scientology can help them in their lives and careers. Celebrity Scientologists include:


title: Hugo Boss

text: Hugo Boss AG, often styled as BOSS, is a luxury fashion house headquartered in Metzingen, Baden-Württemberg, Germany. The company sells clothing, accessories, footwear, and fragrances. Hugo Boss is one of the largest German clothing companies, with global sales of in 2019. Its stock is a component of the MDAX.


title: Hugo Boss

text: The company was founded in 1924 by Hugo Boss and originally produced general-purpose clothing. With the onset of the Great Depression and the rise of Nazism in the early 1930s, Boss began to produce uniforms for the Nazi Party. Boss would eventually supply the wartime German government with military uniforms, resulting in a large boost in sales.


title: Hugo Boss

text: After World War II and the founder's death in 1948, Hugo Boss started to turn its focus from uniforms to men's suits. The company went public in 1988 and introduced a fragrance line that same year, adding men's and women's wear diffusion lines in 1997, a full women's collection in 2000, and children's clothing in 2006–2007. The company has since evolved into a major global fashion house. , it owned more than 1,113 retail stores worldwide.


title: Hugo Boss

text: In 1923, Hugo Boss founded his own clothing company in Metzingen, Germany, where it still operates. In 1924, he started a factory along with two partners. The company produced shirts, jackets, work clothing, sportswear, and raincoats. Due to the economic climate of Germany at the time, Boss was forced into bankruptcy. In 1931, he reached an agreement with his creditors, leaving him with six sewing machines to start again.


title: Hugo Boss

text: That same year, Hugo Boss became a member of the Nazi Party, receiving the membership number 508 889, and a sponsoring member ("Förderndes Mitglied") of the "Schutzstaffel" (SS). He also joined the German Labour Front in 1936, the Reich Air Protection Association in 1939, and the National Socialist People's Welfare in 1941. He was also a member of the Reichskriegerbund and the Reichsbund for physical exercises. After joining these organizations, his sales increased from ($26,993 U.S. dollars in 1932) to over in 1941. Though he claimed in a 1934–35 advertisement that he had been a "supplier for National Socialist uniforms since 1924", it is probable that he did not begin to supply them until 1928 at the earliest. This is the year he became a "Reichszeugmeisterei"-licensed supplier of uniforms to the "Sturmabteilung" (SA), "Schutzstaffel", Hitler Youth, National Socialist Motor Corps, and other party organizations.


title: Hugo Boss

text: By the third quarter of 1932, the all-black SS uniform was designed by SS members Karl Diebitsch (artist) and Walter Heck (graphic designer). The Hugo Boss company was one of the companies that produced these black uniforms for the SS. By 1938, the firm was focused on producing Wehrmacht uniforms and later also uniforms for the Waffen-SS.


title: Hugo Boss

text: During the Second World War, Hugo Boss employed 140 forced laborers, the majority of them women. In addition to these workers, 40 French prisoners of war also worked for the company briefly between October 1940 – April 1941. According to German historian Henning Kober, the company managers were fervent Nazis who were all great admirers of Adolf Hitler. In 1945, Hugo Boss had a photograph in his apartment of him with Hitler, taken at the Berghof, Hitler's Obersalzberg retreat.


title: Hugo Boss

text: Because of his early Nazi Party membership, his financial support of the SS, and the uniforms delivered to the Nazi party, Boss was considered both an "activist" and a "supporter and beneficiary of National Socialism". In a 1946 judgment, he was stripped of his voting rights, his capacity to run a business, and fined "a very heavy penalty" of ($70,553 U.S.) (£54,008 stg). However, Boss appealed, and he was eventually classified as a ‘follower’, a lesser category, which meant that he was not regarded as an active promoter of National Socialism.


title: Hugo Boss

text: He died in 1948, but his business survived. In 2011, the company issued a statement of "profound regret to those who suffered harm or hardship at the factory run by Hugo Boss under National Socialist rule".


title: Hugo Boss

text: As a result of the ban on Boss being in business, his son-in-law Eugen Holy took over ownership and running of the company. In 1950, after a period supplying work uniforms, the company received its first order for men's suits, resulting in an expansion to 150 employees by the end of the year. By 1960, the company was producing ready-made suits. In 1969, Eugen retired, leaving the company to his sons Jochen and Uwe, who began international development. In 1970, the first Boss branded suits were produced, with the brand becoming a registered trademark in 1977. This was followed by the start of the company's long association with motorsport, sponsoring Formula One driver Niki Lauda, and later the McLaren Racing team.


title: Hugo Boss

text: In 1984, the first Boss branded fragrance appeared. This helped the company gain the required growth for listing on the Frankfurt Stock Exchange the following year. The brand began sponsorship of golf with Bernhard Langer in 1986 and tennis with the Davis Cup in 1987. In 1989, Boss launched its first licensed sunglasses. Later that year, the company was bought by a Japanese group.


title: Hugo Boss

text: After the Marzotto textile group acquired a 77.5% stake for $165,000,000 in 1991, the Hugo and Baldessarini brands were introduced in 1993. In 1995, the company launched its footwear range, the first in a now fully developed leather products range across all sub-brands. A partnership with the Solomon R. Guggenheim Foundation was launched in 1995, resulting in the Hugo Boss Prize, an annual $100,000 stipend in modern arts presented since 1996.


title: Hugo Boss

text: In 2005, Marzotto spun off its fashion brands into the Valentino Fashion Group, which was then sold to Permira private equity group. In March 2015, Permira announced plans to sell the remaining shareholding of 12%. Since the Exit by Permira, 91% of the shares floated on the Börse Frankfurt, and the residual 2% was held by the company. 7% of the shares are owned by the Marzotto family. Hugo Boss has at least 6,102 points of sale in 124 countries. Hugo Boss AG directly owns over 364 shops, 537 mono-brand shops, and over 1,000 franchisee-owned shops.


title: Hugo Boss

text: In 2009, BOSS Hugo Boss was by far the largest segment, consisting of 68% of all sales. The remainder of sales were made up by Boss Orange at 17%, BOSS Selection at 3%, Boss Green at 3% and HUGO at 9%.


title: Hugo Boss

text: In 2010, the company had sales of $2,345,850,000 and a net profit of $262,183,000, with royalties of 42% of total net profit. In June 2013, Jason Wu was named artistic director of Boss Womenswear.


title: Hugo Boss

text: Products are manufactured in a variety of locations, including the company's own production sites in: Metzingen, Germany; Morrovalle, Italy; Radom, Poland; Izmir, Turkey; and Cleveland, United States.


title: Hugo Boss

text: Hugo Boss has invested in technology for its made-to-measure program, using machines for almost all the tailoring traditionally made by hand.


title: Hugo Boss

text: Hugo Boss has licensing agreements with various companies to produce Hugo Boss branded products. These include agreements with Samsung, HTC and Huawei to produce mobile phones; Nike, Inc. to produce sports equipment; C.W.F. Children Worldwide Fashion SAS to produce children's clothing; Coty to produce fragrances and skincare; Movado to produce watches; and Safilo to produce sunglasses and eyewear.


title: Hugo Boss

text: In 2020, Hugo Boss created its first vegan men's suit, using all non-animal materials, dyes, and chemicals.


title: Hugo Boss

text: British comedian and actor Russell Brand was at the 2013 GQ awards, which were sponsored by Hugo Boss. After receiving an award on stage, Brand proceeded to talk about Hugo Boss's Nazi connection and did a goose step. He was later ejected from the ceremony and later apologized.


title: Hugo Boss

text: In March 2010, Hugo Boss was boycotted by actor Danny Glover for the company's plans to close the plant in Brooklyn, Ohio, after 375 employees of the Workers United Union reportedly rejected the Hugo Boss proposal to cut the workers' hourly wage 36% from $13 an hour to $8.30. After an initial statement by CFO Andreas Stockert saying the company had a responsibility to shareholders and would move suit manufacturing from Ohio to other facilities in Turkey, Bulgaria, and Romania, the company capitulated to the boycott and cancelled the project. Renewed plans to close the plant in April 2015 also failed.


title: Hugo Boss

text: In September 2015, Hugo Boss (UK) was fined £1.2m in relation to the death in June 2013 of a child who died four days after suffering fatal head injuries at its store in Bicester, Oxfordshire. The four-year-old boy had been injured when a steel-framed fitting-room mirror weighing fell on him. Oxford Crown Court had earlier been told that it had "negligently been left free-standing without any fixings" and the coroner had said that the death was an "accident waiting to happen". In June 2015, Hugo Boss (UK) had admitted its breach of both the Health and Safety at Work Act 1974 and Management of Health and Safety at Work regulations 1999. The company's legal representative said:


title: Hugo Boss

text: In August 2019, Hugo Boss sent a cease & desist letter, objecting to the trademark application of Boss Brewing, a small brewery based in Swansea, costing the brewery nearly £10,000 in legal fees and compelling them to change the name of several beer brands. In February 2020, professedly as a protest, comedian Joe Lycett changed his legal name to Hugo Boss.


title: Hugo Boss

text: In 2020, Hugo Boss told NBC News it did not use cotton from the Xinjiang area of China to avoid Uyghur forced labor. However, in 2021, the Chinese subsidiary of Hugo Boss stated on its official Sina Weibo account that they had been using cotton from the region and would continue to do so: The statement was later edited to simply saying they have partners "in various regions of China" with a link to an English-language page on their website, which in turn linked to another statement containing the following words: "HUGO BOSS has not procured any goods originating in the Xinjiang region from direct suppliers." Initially attracting thousands of likes, the edited Weibo post received many comments accusing the brand of hypocrisy. A company spokeswoman stated that the original Weibo post was unauthorized and that the company's position has not changed. According to the company's official statement, all materials are only sourced from suppliers that comply with the HUGO BOSS Supplier Code of Conduct.


title: Hugo Boss

text: In September 2021, the European Center for Constitutional and Human Rights filed a complaint with German prosecutors accusing Hugo Boss from abetting and profiting from forced labor in Xinjiang. In 2022, researchers from Nordhausen University of Applied Sciences identified cotton from Xinjiang in Hugo Boss shirts.


title: Chamath Palihapitiya

text: Chamath Palihapitiya (born 3 September 1976) is a Sri Lankan-born Canadian and American venture capitalist, engineer, SPAC sponsor, founder and CEO of Social Capital. Palihapitiya was an early senior executive at Facebook, working at the company from 2007 to 2011. Following his departure from Facebook, Palihapitiya started his fund, The Social+Capital Partnership, through which he invested in several companies, including Yammer and Slack. The Social+Capital Partnership changed its name to Social Capital in 2015. He is a co-host of technology podcast "All In."


title: Chamath Palihapitiya

text: Palihapitiya was born on 3 September 1976 in Sri Lanka. In 1982, Palihapitiya moved with his family to Canada at age five; his father Gamage had been posted to the High Commission of Sri Lanka, Ottawa. Five years later, in 1986, the posting came to an end, and the family applied for refugee status rather than returning home, on the basis that Gamage had been criticized for his views on the violence during the Sri Lankan Civil War.


title: Chamath Palihapitiya

text: Palihapitiya's father was frequently unemployed, and his mother did low-paying housekeeping jobs. At age 14, Palihapitiya worked at a Burger King to support his family. His father died on October 2014.


title: Chamath Palihapitiya

text: He attended Lisgar Collegiate Institute. After graduating from the University of Waterloo in 1999 with a degree in electrical engineering, Palihapitiya worked for a year as a derivatives trader at the investment bank BMO Nesbitt Burns. He then accepted a job offer at Winamp and moved to California.


title: Chamath Palihapitiya

text: Palihapitiya joined Winamp, which was subsequently acquired by AOL, here he became the company's youngest vice president ever, heading its instant messaging division in 2004. In 2005, he left AOL and joined Mayfield Fund; in 2007 he left Mayfield and joined Facebook, which was then a little more than three years old.


title: Chamath Palihapitiya

text: Palihapitiya's work at Facebook in the first year was messy from his account. Palihapitiya led the release of Facebook Beacon, which failed and led to lawsuits. After this failure, Palihapitiya pitched leading Growth, where his team focused entirely on new user growth. After four years, Facebook hit 1B users. Before leaving Facebook, Palihapitiya led the Facebook Phone and Facebook Home projects.


title: Chamath Palihapitiya

text: Steven Levy wrote in "" that Palihapitiya was regarded as a "bully" at Facebook, and that he had made many of his subordinates cry regularly.


title: Chamath Palihapitiya

text: In 2011, he left Facebook and started his own fund, The Social+Capital Partnership, with his then-wife. The firm changed its name to Social Capital in 2015. Through the fund, Palihapitiya invested in a number of companies, including Glooko, Inc, Yammer, SecondMarket, Slack, and Box. As of 2015, the fund had more than $1.1 billion in total assets most of which came from external investors (i.e. LPs).


title: Chamath Palihapitiya

text: In 2018, there was a massive decrease in Social Capital fund's operations and a significant exodus of top management and co-founders. "Axios" reported that Palihapitiya was spending a significant amount of time with his new girlfriend in Italy and rarely showed up to the office or answered emails from employees. The firm returned investor capital and converted into a family office although continued to manage some external capital on a no-fee basis.


title: Chamath Palihapitiya

text: After transitioning to a single GP firm, essentially Palihapitiya's family office, in 2018, Palihapitiya said he wanted to return to first principles and restructure the firm to better align with the long-term interests of entrepreneurs, not just limited partners (LPs).


title: Chamath Palihapitiya

text: Since then, Social Capital has made investments in three areas: climate science, life sciences and biotechnology, and the decentralization of the digital economy through platforms such as blockchain, crypto, and digital assets. Palihapitiya manages from a balance sheet of permanent capital.


title: Chamath Palihapitiya

text: In March 2021 Palihapitiya sold his personal stake Virgin Galactic for around US$213 million. In February 2022, Palihapitiya stepped down as chairman of Virgin Galactic.


title: Chamath Palihapitiya

text: In 2020, Social Capital Hedosophia took Opendoor, an online real estate marketplace, public through a SPAC. OpenDoor raised $1 billion through the merger, $400 million of which came from the SPAC and an additional $600 million through PIPE investors. Palihapitiya accounted for $100 million of the PIPE.


title: Chamath Palihapitiya

text: In 2021, Palihapitiya announced he planned to help take SoFi, a financial services platform, and Clover Health, a Medicare insurance company, public through SPACs. This gained Palihapitiya criticism from the "Financial Times", which said that he is "shilling risky reverse-mergers to retail investors on a almost bimonthly basis".


title: Chamath Palihapitiya

text: Following the Clover Health SPAC, Hindenburg Research, a financial analyst and short-selling specialist firm, issued a report about this transaction accusing Palihapitiya of luring investors into a “broken business”, argumenting that he failed to inform them about an active Department of Justice investigation into Clover's alleged deceptive business practices. Palihapitiya made more than $290 million from the deal based on a $25k investment. In addition the Clover Health co-founder/CEO's previous company, CarePoint Health, a hospital conglomerate in New Jersey, was accused of price gouging customers and according to a NJ state commission siphoning off $150 million to himself and his friends bankrupting the company and causing a hospital crisis in NJ. Regulators in NJ called for an investigation of Clover Health because of the CEO's previous actions. The Securities and Exchange Commission opened an investigation into the allegations set forth in the Hindenburg Research report on February 4.


title: Chamath Palihapitiya

text: During the GameStop short squeeze, Palihapitiya repeatedly attacked Robinhood and its founders for being unethical by selling payment for order flow to HFT firms like Citadel Securities and pushed his fans to switch over to SoFi, which was merging with his SPAC yet failed to mention that SoFi employs the same practice of selling payment for order flow to HFT firms (including to Citadel Securities) and owns a 16% stake in Apex Clearing Corp, a clearing house involved in the controversy.


title: Chamath Palihapitiya

text: In April 2021, John Coates, acting director of the SEC's corporate-finance division, criticized Palihapitiya's views on the benefits of SPACs over traditional IPOs:


title: Chamath Palihapitiya

text: Coates clarified that a judge could rule a SPAC is similar enough to an IPO that the lesser securities law liability would be void.


title: Chamath Palihapitiya

text: In June 2021, Social Capital Survetta filed for four new SPACs which would focus on biotechnology companies, under the stock tickers DNAA through DNAD.


title: Chamath Palihapitiya

text: Palihapitiya has donated to the Democratic Party. As of February 2021, he has reportedly donated $1.3 million to the party over the past decade. He donated $7,500 to Republican Ted Cruz in 2011. In March 2020, Palihapitiya told "The New York Times" that he would like to see Michael Bloomberg at the top of the Democratic ticket in the 2020 Democratic Party presidential primaries, paired with Amy Klobuchar or Elizabeth Warren.


title: Chamath Palihapitiya

text: Palihapitiya was listed as one of the "Founders" of the lobbying group FWD.us. The group launched on 11 April 2013, and its goals include immigration reform, improving education, and enabling technological innovation, all in a United States context. An article in "The New Republic" stated that Palihapitiya received a weekly report about FWD.us and also quoted him as saying, in response to controversy around the FWD.us political lobbying strategy: "The folks that are actually people that run that day to day are sophisticated and understand the nuances of how to affect it... It's a really gnarly, gnarly thing having to deal with Washington. And to be honest with you, my perspective was, it's a really good investment because it's a good way to pay it forward, and I'm really glad there are other people other than me who are dealing with it who have the patience and resolve to figure it out."


title: Chamath Palihapitiya

text: At Bloomberg's "Next Big Thing" conference in Sausalito, California, Palihapitiya made remarks critical of San Francisco's then mayor, Ed Lee, and proposed that the city provide subsidized housing to low-income residents funded by an equity tax on startups, with the tax-and-subsidy schemes potentially restricted to particular zones of the city. This led to a heated debate between Palihapitiya and super angel Ron Conway. Conway, a supporter of Lee, defended the city's policies, argued that things would get better for all residents, and noted that Palihapitiya lives in Palo Alto rather than in the city. In a later clarification to "TechCrunch", Palihapitiya outlined his vision in more detail and described how his views on inequality and social mobility were shaped by his experience growing up with relatively poor immigrant parents in Canada.


title: Chamath Palihapitiya

text: In November 2017, Palihapitiya said that, for ethical reasons, he regretted helping Facebook to become the largest social media platform. He said,


title: Chamath Palihapitiya

text: The short-term, dopamine-driven feedback loops that we have created are destroying how society works: no civil discourse, no collaboration, misinformation, mistruth and it's not an American problem. This is not about Russian ads. This is a global problem. It is eroding the core foundations of how people behave by and between each other.


title: Chamath Palihapitiya

text: I genuinely believe that Facebook is a force for good in the world, so I'd like to expand on my comments... My comments were meant to start an important conversation, not to criticize one company—particularly one I love. In 2017, many of us have grappled with the unintended consequences of the products we've built. Social media platforms in particular have been used and abused in ways that we, their architects, never imagined. Much blame has been thrown and guilt felt, but the important thing is what we as an industry do now to ensure that our impact on society continues to be a positive one.


title: Chamath Palihapitiya

text: On 25 January 2021, Palihapitiya announced he would challenge incumbent California Governor Gavin Newsom in the event Newsom were recalled. If elected, Palihapitiya said he would cut the state income tax rate from 16 to 0 percent, provide free education vouchers, allow for a two thousand dollar credit for every child born in California, and work to make California the center of climate and technology jobs. He shared a campaign website that was created by a supporter.


title: Chamath Palihapitiya

text: In January 2022, Palihapitiya said on the podcast "All-In with Chamath, Jason, Sacks & Friedberg" to co-host Jason Calacanis that the Uyghur genocide does not concern him:


title: Chamath Palihapitiya

text: Nobody cares about what's happening to the Uyghurs, okay. You bring it up because you care and I think it's nice that you care. The rest of us don't care. I'm just telling you a very hard, ugly truth. Of all the things that I care about, yes, it is below my line.


title: Chamath Palihapitiya

text: Palihapitiya said he and most Americans care more about domestic economic issues than the human rights abuses of China's Uyghur minority. In response, the Golden State Warriors issued a statement saying that Palihapitiya "does not speak on behalf of our franchise, and his views certainly don't reflect those of our organization." Palihapitiya made further remarks in a tweet, saying that he recognized he came across as "lacking empathy" and that "To be clear, my belief is that human rights matter, whether in China, the United States, or elsewhere. Full stop."


title: Chamath Palihapitiya

text: During his time at Facebook, Palihapitiya invested in several startups through Embarcadero Ventures, a venture capital fund.


title: Chamath Palihapitiya

text: In 2010, Palihapitiya helped to buy the Golden State Warriors for $450 million; he remains a minority stakeholder and board member of the team. According to the team as of 2022 he was a "limited investor with no day-to-day operating functions". In 2022, Palihapitiya’s 10% stake in the team was worth $520 million - a 2000% return on his $25 million initial investment.


title: Chamath Palihapitiya

text: Palihapitiya has donated consistently to his alma mater, the University of Waterloo, including a $25 million donation towards the engineering department in 2018.


title: Chamath Palihapitiya

text: In 2021, he donated $7 million to provide clean drinking water to 1,000 families in California's central valley through a partnership with one2one, an American foundation and Source Global, the maker of solar-powered hydropanels that extract potable water from the air.


title: Chamath Palihapitiya

text: After graduating from college, Palihapitiya followed his future wife, Brigette Lau, to California. They had three children together, and divorced in 2018.


title: Chamath Palihapitiya

text: Palihapitiya now lives in California with Nathalie Dompe, an Italian pharmaceutical heiress, model, and CEO of Dompe Holdings, whom he started dating in 2018, and with whom he has one child.


title: Chamath Palihapitiya

text: In a December 2017 interview, Palihapitiya said that he keeps his children as far away from technology as possible, except for the occasional movie, explaining "I don't like this co-dependency of 'they need to rely on me, and when they can't, I feed them a device because that becomes a babysitter'."


title: Chamath Palihapitiya

text: Palihapitiya has three World Series of Poker (WSOP) and two World Poker Tour (WPT) cashes for a total of $175,801. In 2011, he finished 101st out of 6,865 entries in the World Series of Poker's Main Event.


title: Baldr

text: Baldr (also Balder, Baldur) is a god in Germanic mythology. In Norse mythology, Baldr (Old Norse: ) is a son of the god Odin and the goddess Frigg, and has numerous brothers, such as Thor and Váli. In wider Germanic mythology, the god was known in Old English as ', and in Old High German as ', all ultimately stemming from the Proto-Germanic theonym ('hero' or 'prince').


title: Baldr

text: During the 12th century, Danish accounts by Saxo Grammaticus and other Danish Latin chroniclers recorded a euhemerized account of his story. Compiled in Iceland during the 13th century, but based on older Old Norse poetry, the "Poetic Edda" and the "Prose Edda" contain numerous references to the death of Baldr as both a great tragedy to the Æsir and a harbinger of Ragnarök.


title: Baldr

text: According to "Gylfaginning", a book of Snorri Sturluson's Prose Edda, Baldr's wife is Nanna and their son is Forseti. Baldr had the greatest ship ever built, Hringhorni, and there is no place more beautiful than his hall, Breidablik.


title: Baldr

text: The Old Norse theonym "Baldr" ('brave, defiant'; 'lord, prince') and its various Germanic cognates – including Old English "Bældæg" and Old High German "Balder" (or "Palter") – probably stems from Proto-Germanic "*Balðraz" ('Hero, Prince'; cf. Old Norse "mann-baldr" 'great man', Old English "bealdor" 'prince, hero'), itself a derivative of "*balþaz", meaning 'brave' (cf. Old Norse "ballr" 'hard, stubborn', Gothic "balþa*" 'bold, frank', Old English "beald" 'bold, brave, confident', Old Saxon "bald" 'valiant, bold', Old High German "bald" 'brave, courageous').


title: Baldr

text: This etymology was originally proposed by Jacob Grimm (1835), who also speculated on a comparison with the Lithuanian "báltas" ('white', also the name of a light-god) based on the semantic development from 'white' to 'shining' then 'strong'. According to linguist Vladimir Orel, this could be linguistically tenable. Philologist Rudolf Simek also argues that the Old English "Bældæg" should be interpreted as meaning 'shining day', from a Proto-Germanic root *"bēl"- (cf. Old English "bæl", Old Norse "bál" 'fire') attached to "dæg" ('day').


title: Baldr

text: Old Norse also shows the usage of the word as an honorific in a few cases, as in "baldur î brynju" (Sæm. 272b) and "herbaldr" (Sæm. 218b), in general epithets of heroes. In continental Saxon and Anglo-Saxon tradition, the son of Woden is called not "Bealdor" but "Baldag" (Saxon) and "Bældæg, Beldeg" (Anglo-Saxon), which shows association with "day", possibly with Day personified as a deity. This, as Grimm points out, would agree with the meaning "shining one, white one, a god" derived from the meaning of Baltic "baltas", further adducing Slavic "Belobog" and German "Berhta".


title: Baldr

text: One of the two Merseburg Incantations names "Baldere", but also mentions a figure named "Phol", considered to be a byname for Baldr (as in Scandinavian "Falr", "Fjalarr"; (in Saxo) "Balderus" : "Fjallerus"). The incantation relates of "Phol ende Odin" riding to the woods, where the foot of Baldr's foal is sprained. Sinthgunt (the sister of the sun), Freyja and Odin sing to the foot in order for it to heal.


title: Baldr

text: Unlike the Prose Edda, in the Poetic Edda the tale of Baldr's death is referred to rather than recounted at length. Baldr is mentioned in "Völuspá", in "Lokasenna", and is the subject of the Eddic poem "Baldr's Dreams".


title: Baldr

text: Among the visions which the Völva sees and describes in Völuspá is Baldr's death. In stanza 32, the Völva says she saw the fate of Baldr "the bleeding god":


title: Baldr

text: In the next two stanzas, the Völva refers to Baldr's killing, describes the birth of Váli for the slaying of Höðr and the weeping of Frigg:


title: Baldr

text: In stanza 62 of Völuspá, looking far into the future, the Völva says that Höðr and Baldr will come back, with the union, according to Bellows, being a symbol of the new age of peace:


title: Baldr

text: Baldr is mentioned in two stanzas of Lokasenna, a poem which describes a flyting between the gods and the god Loki. In the first of the two stanzas, Frigg, Baldr's mother, tells Loki that if she had a son like Baldr, Loki would be killed:


title: Baldr

text: In the next stanza, Loki responds to Frigg, and says that he is the reason Baldr "will never ride home again":


title: Baldr

text: The Eddic poem "Baldr's Dreams" opens with the gods holding a council discussing why Baldr had had bad dreams:


title: Baldr

text: Odin then rides to Hel to a Völva's grave and awakens her using magic. The Völva asks Odin, who she does not recognize, who he is, and Odin answers that he is Vegtam ("Wanderer"). Odin asks the Völva for whom are the benches covered in rings and the floor covered in gold. The Völva tells him that in their location mead is brewed for Baldr, and that she spoke unwillingly, so she will speak no more:


title: Baldr

text: Odin asks the Völva to not be silent and asks her who will kill Baldr. The Völva replies and says that Höðr will kill Baldr, and again says that she spoke unwillingly, and that she will speak no more:


title: Baldr

text: Odin again asks the Völva to not be silent and asks her who will avenge Baldr's death. The Völva replies that Váli will, when he will be one night old. Once again, she says that she will speak no more:


title: Baldr

text: Odin again asks the Völva to not be silent and says that he seeks to know who the women that will then weep be. The Völva realizes that Vegtam is Odin in disguise. Odin says that the Völva is not a Völva, and that she is the mother of three giants. The Völva tells Odin to ride back home proud, because she will speak to no more men until Loki escapes his bounds.


title: Baldr

text: Apart from this description, Baldr is known primarily for the story of his death, which is seen as the first in a chain of events that will ultimately lead to the destruction of the gods at Ragnarök. According to "Völuspá", Baldr will be reborn in the new world.


title: Baldr

text: He had a dream of his own death and his mother had the same dream. Since dreams were usually prophetic, this depressed him, so his mother Frigg made every object on earth vow never to hurt Baldr. All objects made this vow except mistletoe—a detail which has traditionally been explained with the idea that it was too unimportant and nonthreatening to bother asking it to make the vow, but which Merrill Kaplan has instead argued echoes the fact that young people were not eligible to swear legal oaths, which could make them a threat later in life.


title: Baldr

text: When Loki, the mischief-maker, heard of this, he made a magical spear from this plant (in some later versions, an arrow). He hurried to the place where the gods were indulging in their new pastime of hurling objects at Baldr, which would bounce off without harming him. Loki gave the spear to Baldr's brother, the blind god Höðr, who then inadvertently killed his brother with it (other versions suggest that Loki guided the arrow himself). For this act, Odin and the asynja Rindr gave birth to Váli, who grew to adulthood within a day and slew Höðr.


title: Baldr

text: Baldr was ceremonially burnt upon his ship, Hringhorni, the largest of all ships. As he was carried to the ship, Odin whispered in his ear. This was to be a key riddle asked by Odin (in disguise) of the giant Vafthrudnir (and which was unanswerable) in the poem "Vafthrudnismal". The riddle also appears in the riddles of Gestumblindi in "Hervarar saga".


title: Baldr

text: The dwarf Litr was kicked by Thor into the funeral fire and burnt alive. Nanna, Baldr's wife, also threw herself on the funeral fire to await Ragnarök when she would be reunited with her husband (alternatively, she died of grief). Baldr's horse with all its trappings was also burned on the pyre. The ship was set to sea by Hyrrokin, a giantess, who came riding on a wolf and gave the ship such a push that fire flashed from the rollers and all the earth shook.


title: Baldr

text: Upon Frigg's entreaties, delivered through the messenger Hermod, Hel promised to release Baldr from the underworld if all objects alive and dead would weep for him. All did, except a giantess, Þökk (often presumed to be the god Loki in disguise), who refused to mourn the slain god. Thus Baldr had to remain in the underworld, not to emerge until after Ragnarök, when he and his brother Höðr would be reconciled and rule the new earth together with Thor's sons.


title: Baldr

text: Besides descriptions of Baldr, the Prose Edda also explicitly links "Baldr" to the Anglo-Saxon "Beldeg" in its prologue.


title: Baldr

text: Writing during the end of the 12th century, the Danish historian Saxo Grammaticus tells the story of Baldr (recorded as "Balderus") in a form that professes to be historical. According to him, Balderus and Høtherus were rival suitors for the hand of Nanna, daughter of Gewar, King of Norway. Balderus was a demigod and common steel could not wound his sacred body. The two rivals encountered each other in a terrific battle. Though Odin and Thor and the other gods fought for Balderus, he was defeated and fled away, and Høtherus married the princess.


title: Baldr

text: Nevertheless, Balderus took heart of grace and again met Høtherus in a stricken field. But he fared even worse than before. Høtherus dealt him a deadly wound with a magic sword, named Mistletoe, which he had received from Mimir, the satyr of the woods; after lingering three days in pain Balderus died of his injury and was buried with royal honours in a barrow.


title: Baldr

text: There are also two lesser known Danish Latin chronicles, the "Chronicon Lethrense" and the "Annales Lundenses" of which the latter is included in the former. These two sources provide a second euhemerized account of Höðr's slaying of Baldr.


title: Baldr

text: It relates that Hother was the king of the Saxons and son of Hothbrodd and Hadding. Hother first slew Othen's (i.e. Odin) son Balder in battle and then chased Othen and Thor. Finally, Othen's son Both killed Hother. Hother, Balder, Othen and Thor were incorrectly considered to be gods.


title: Baldr

text: A Latin votive inscription from Utrecht, from the 3rd or 4th century C.E., has been theorized as containing the dative form "Baldruo", pointing to a Latin nominative singular *"Baldruus", which some have identified with the Norse/Germanic god, although both the reading and this interpretation have been questioned.


title: Baldr

text: In the Anglo-Saxon Chronicle Baldr is named as the ancestor of the monarchy of Kent, Bernicia, Deira, and Wessex through his supposed son Brond.


title: Baldr

text: As referenced in "Gylfaginning", in Sweden and Norway, the scentless mayweed ("Matricaria perforata") and the similar sea mayweed ("Matricaria maritima") are both called "baldursbrá" "Balder's brow" and regionally in northern England ("baldeyebrow"). In Iceland only the former is found. In Germany valerian is known as "Baldrian"; variations using or influenced by reflexes of "Phol" include "Faltrian" (upper Austria), Villum"fallum" (Salzburg), and "Fildron" or "Faldron" (Tyrol).


title: Baldr

text: There are a few old place names in Scandinavia that contain the name "Baldr". The most certain and notable one is the (former) parish name Balleshol in Hedmark county, Norway: "a Balldrshole" 1356 (where the last element is "hóll" m "mound; small hill"). Others may be (in Norse forms) "Baldrsberg" in Vestfold county, "Baldrsheimr" in Hordaland county "Baldrsnes" in Sør-Trøndelag county—and (very uncertain) the Balsfjorden fjord and Balsfjord municipality in Troms county.


title: Baldr

text: In Copenhagen, there is also a Baldersgade, or "Balder's Street". A street in downtown Reykjavík is called Baldursgata (Baldur's Street).


title: Baldr

text: In Sweden there is a Baldersgatan (Balder's Street) in Stockholm. There is also Baldersnäs (Balder's isthmus), Baldersvik (Balder's bay), Balders udde (Balder's headland) and Baldersberg (Balder's mountain) at various places.


title: Baldr

text: Balder the Brave is a fictional character based on Baldr. He has appeared in comic books published by Marvel Comics as the half-brother of Thor, and son of Odin, ruler of the gods.


title: Baldr

text: Baldr is featured in a number of video games. In Ensemble Studios' 2002 video game "Age of Mythology" Baldr is one of nine minor gods Norse players can worship. Baldr (spelled Baldur in-game) is also the main antagonist in Santa Monica Studio's 2018 video game "God of War". However, he differs greatly in the game from the Baldr depicted in Norse writings and traditional artistic depictions as he is much more aggressive, crude, and rugged in appearance.


title: Baldr

text: In BioWare's 1998 role-playing video game series the Bhaalspawn Saga and the Dark Alliance, Balduran was a legendary seafaring explorer who founded the city of Baldur's Gate, after which many of the games are named.


title: Flamingo

text: Flamingos or flamingoes are a type of wading bird in the family Phoenicopteridae, which is the only extant family in the order Phoenicopteriformes. There are four flamingo species distributed throughout the Americas (including the Caribbean), and two species native to Afro-Eurasia.


title: Flamingo

text: The name "flamingo" comes from Portuguese or Spanish ("flame-colored"), which in turn comes from Provençal – a combination of ("flame") and a Germanic-like suffix "-ing". The word may also have been influenced by the Spanish ethnonym ("Fleming" or "Flemish").


title: Flamingo

text: The name of the genus, "Phoenicopterus", is from the Greek , ); other genera names include "Phoeniconaias," which means "crimson/red water nymph (or naiad)", and "Phoenicoparrus," which means "crimson/red bird (though, an unknown bird of omen)".


title: Flamingo

text: The family Phoenicopteridae was introduced by the French zoologist Charles Lucien Bonaparte in 1831, with "Phoenicopterus" as the type genus.


title: Flamingo

text: Traditionally, the long-legged Ciconiiformes, probably a paraphyletic assemblage, have been considered the flamingos' closest relatives and the family was included in the order. Usually, the ibises and spoonbills of the Threskiornithidae were considered their closest relatives within this order. Earlier genetic studies, such as those of Charles Sibley and colleagues, also supported this relationship. Relationships to the waterfowl were considered as well, especially as flamingos are parasitized by feather lice of the genus "Anaticola", which are otherwise exclusively found on ducks and geese. The peculiar presbyornithids were used to argue for a close relationship between flamingos, waterfowl, and waders. A 2002 paper concluded they are waterfowl, but a 2014 comprehensive study of bird orders found that flamingos and grebes are not waterfowl, but rather are part of Columbea, along with doves, sandgrouse, and mesites.


title: Flamingo

text: Recent molecular studies have suggested a relation with grebes, while morphological evidence also strongly supports a relationship between flamingos and grebes. They hold at least 11 morphological traits in common, which are not found in other birds. Many of these characteristics have been previously identified on flamingos, but not on grebes. The fossil palaelodids can be considered evolutionarily, and ecologically, intermediate between flamingos and grebes.


title: Flamingo

text: For the grebe-flamingo clade, the taxon Mirandornithes ("miraculous birds" due to their extreme divergence and apomorphies) has been proposed. Alternatively, they could be placed in one order, with Phoenocopteriformes taking priority.


title: Flamingo

text: Six extant flamingo species are recognized by most sources, and were formerly placed in one genus (have common characteristics) – "Phoenicopterus". As a result of a 2014 publication, the family was reclassified into two genera. Currently, the family has three recognized genera, according to HBW.


title: Flamingo

text: Flamingos usually stand on one leg with the other tucked beneath the body. The reason for this behaviour is not fully understood. One theory is that standing on one leg allows the birds to conserve more body heat, given that they spend a significant amount of time wading in cold water. However, the behaviour also takes place in warm water and is also observed in birds that do not typically stand in water. An alternative theory is that standing on one leg reduces the energy expenditure for producing muscular effort to stand and balance on one leg. A study on cadavers showed that the one-legged pose could be held without any muscle activity, while living flamingos demonstrate substantially less body sway in a one-legged posture. As well as standing in the water, flamingos may stamp their webbed feet in the mud to stir up food from the bottom.


title: Flamingo

text: Flamingos are capable flyers, and flamingos in captivity often require wing clipping to prevent escape. A pair of African flamingos which had not yet had their wings clipped escaped from the Wichita, Kansas, zoo in 2005. One was spotted in Texas 14 years later. It had been seen previously by birders in Texas, Wisconsin and Louisiana.


title: Flamingo

text: Young flamingos hatch with grayish-red plumage, but adults range from light pink to bright red due to aqueous bacteria and beta-carotene obtained from their food supply. A well-fed, healthy flamingo is more vibrantly colored, thus a more desirable mate; a white or pale flamingo, however, is usually unhealthy or malnourished. Captive flamingos are a notable exception; they may turn a pale pink if they are not fed carotene at levels comparable to the wild.


title: Flamingo

text: The greater flamingo is the tallest of the six different species of flamingos, standing at with a weight up to , and the shortest flamingo species (the lesser) has a height of and weighs . Flamingos can have a wingspan as small as to as big as .


title: Flamingo

text: Flamingos filter-feed on brine shrimp and blue-green algae as well as insect larvae, small insects, mollusks and crustaceans, making them omnivores. Their bills are specially adapted to separate mud and silt from the food they eat, and are uniquely used upside-down. The filtering of food items is assisted by hairy structures called lamellae, which line the mandibles, and the large, rough-surfaced tongue. The pink or reddish color of flamingos comes from carotenoids in their diet of animal and plant plankton. American flamingos are a brighter red color because of the beta carotene availability in their food while the lesser flamingos are a paler pink due to ingesting a smaller amount of this pigment. These carotenoids are broken down into pigments by liver enzymes. The source of this varies by species, and affects the color saturation. Flamingos whose sole diet is blue-green algae are darker than those that get it second-hand by eating animals that have digested blue-green algae.


title: Flamingo

text: Flamingos are considered very noisy birds with their noises and vocalizations ranging from grunting or growling to nasal honking. Vocalizations play an important role in parent-chick recognition, ritualized displays, and keeping large flocks together. Variations in vocalizations exist in the voices of different species of flamingos.


title: Flamingo

text: Flamingos are very social birds; they live in colonies whose population can number in the thousands. These large colonies are believed to serve three purposes for the flamingos: avoiding predators, maximizing food intake, and using scarce suitable nesting sites more efficiently. Before breeding, flamingo colonies split into breeding groups of about 15 to 50 birds. Both males and females in these groups perform synchronized ritual displays. The members of a group stand together and display to each other by stretching their necks upwards, then uttering calls while head-flagging, and then flapping their wings. The displays do not seem directed towards an individual, but occur randomly. These displays stimulate "synchronous nesting" (see below) and help pair up those birds that do not already have mates.


title: Flamingo

text: Flamingos form strong pair bonds, although in larger colonies, flamingos sometimes change mates, presumably because more mates are available to choose. Flamingo pairs establish and defend nesting territories. They locate a suitable spot on the mudflat to build a nest (the female usually selects the place). Copulation usually occurs during nest building, which is sometimes interrupted by another flamingo pair trying to commandeer the nesting site for their use. Flamingos aggressively defend their nesting sites. Both the male and the female contribute to building the nest, and to protecting the nest and egg. Same-sex pairs have been reported.


title: Flamingo

text: After the chicks hatch, the only parental expense is feeding. Both the male and the female feed their chicks with a kind of crop milk, produced in glands lining the whole of the upper digestive tract (not just the crop). The hormone prolactin stimulates production. The milk contains fat, protein, and red and white blood cells. (Pigeons and doves—Columbidae—also produce crop milk (just in the glands lining the crop), which contains less fat and more protein than flamingo crop milk.)


title: Flamingo

text: For the first six days after the chicks hatch, the adults and chicks stay in the nesting sites. At around 7–12 days old, the chicks begin to move out of their nests and explore their surroundings. When they are two weeks old, the chicks congregate in groups, called "microcrèches", and their parents leave them alone. After a while, the microcrèches merge into "crèches" containing thousands of chicks. Chicks that do not stay in their crèches are vulnerable to predators.


title: Flamingo

text: The first flamingo hatched in a European zoo was a Chilean flamingo at Zoo Basel in Switzerland in 1958. Since then, over 389 flamingos have grown up in Basel and been distributed to other zoos around the globe.


title: Flamingo

text: Greater, an at least 83-year-old greater flamingo, believed to be the oldest in the world, died at the Adelaide Zoo in Australia in January 2014.


title: Flamingo

text: Zoos have used mirrors to improve flamingo breeding behaviour. The mirrors are thought to give the flamingos the impression that they are in a larger flock than they actually are.


title: Flamingo

text: While many different kinds of birds were valued items in Roman food, flamingos were among the most prized in Ancient Roman cuisine. An early reference to their consumption, and especially of their tongues, is found in Pliny the Elder, who states in the "Natural History":


title: Flamingo

text: Although a few recipes for flamingos are found in Apicius' extant works, none refer specifically to flamingo tongues.


title: Flamingo

text: There is also a mention of flamingo brains in a later, highly contentious source, detailing, in the life of Elagabalus, a food item not apparently to his liking as much as camels' heels and parrot tongues, in the belief that the latter was a prophylactic:


title: En passant

text: En passant (, "in passing") is a special method of capturing in chess that occurs when a pawn captures a horizontally adjacent enemy pawn that has just made an initial two-square advance. The capturing pawn moves to the square that the enemy pawn passed over, as if the enemy pawn had advanced only one square. The rule ensures that a pawn cannot use its two-square move to safely skip past an enemy pawn.


title: En passant

text: Capturing "en passant" is permitted only on the turn immediately after the two-square advance; it cannot be done on a later turn. The capturing move is sometimes notated by appending the abbreviation e.p.


title: En passant

text: If these conditions are met, the capturing pawn can move diagonally forward to the square that the enemy pawn passed, capturing the enemy pawn as if it had moved only one square. If the right to capture "en passant" is not exercised immediately, it is subsequently lost. Making the capture is optional, unless there is no other legal move.


title: En passant

text: Only pawns may capture or be captured "en passant"; other pieces with the ability to capture diagonally—the king, queen, and bishop—cannot perform the capture. The "en passant" capture is the only capturing move in chess where the capturing piece moves to a square not occupied by the captured piece.


title: En passant

text: In algebraic notation, an "en passant" capture is notated using the capturing pawn's destination (not the captured pawn's location). In both algebraic and descriptive notation, the move may optionally be denoted by appending "e.p." or similar. For example, in algebraic notation, bxa3 or bxa3 e.p. may be used to represent a black pawn on b4 capturing a white pawn on a4 "en passant".


title: En passant

text: Some chess openings feature the "en passant" capture. In the following line from Petrov's Defence, White captures the pawn on d5 "en passant" on move 6:


title: En passant

text: An "en passant" capture can occur as early as move 3. For example, in the French Defence after 1. e4 e6 2. e5, if Black responds with 2... d5, White can play 3. exd6 e.p. (diagram). This occurred in the game Steinitz–Fleissig, Vienna 1882.


title: En passant

text: In the diagram, the move 1... g5+ may seem to checkmate White, but it is in fact a blunder: White can counter this check with the "en passant" capture 2. fxg6 e.p., which cross-checks and checkmates Black. (Black can draw in the diagrammed position by playing 1...Qxf2+.)


title: En passant

text: In a game between Gunnar Gundersen and Albert H. Faul, Black played 12...f7-f5. White could have captured the black f-pawn "en passant" with his e-pawn, but he instead played:


title: En passant

text: The "en passant" capture places Black in double check and checkmate (in fact, White's bishop is not necessary for the mate). An "en passant" capture is the only way a double check can be delivered without one of the checking pieces moving, as in this case.


title: En passant

text: The largest known number of "en passant" captures in one game is three. This record is shared by three games; in none of them were all three captures by the same player. The earliest known example is a 1980 game between Alexandru Segal and Karl-Heinz Podzielny.


title: En passant

text: In old versions of chess, the pawn could not advance two squares on its first move. The two-square advance was introduced later, between the 13th and 16th centuries, to speed up games. The "en passant" capture may have been introduced at that time, or may have come later; the earliest references to this rule date to the 16th century.


title: En passant

text: The "en passant" capture was one of the last major additions to European chess. In some parts of Europe, particularly in Italy, the rule was excluded; this exclusion was known as "passar battaglia". In 1880, Italy adopted the rules used by the rest of the world, including the "en passant" capture, in preparation for the 1881 Milan tournament.


title: En passant

text: In the context of threefold and fivefold repetition, two positions are considered different if the opportunity to perform a given "en passant" capture exists in one position but not the other.


title: En passant

text: When a player is not in check, and capturing "en passant" is their only legal move, they are forbidden to "claim" a draw by stalemate; they must either perform the move or end the game on their turn via normal means. In his book on chess organization and rules, International Arbiter Kenneth Harkness wrote that people frequently asked if this is the case. Chess players debated this point in the 19th century, with some arguing that the right to capture "en passant" is a "privilege" that one cannot be compelled to exercise. In his 1860 book "Chess Praxis", Howard Staunton wrote that the "en passant" capture is mandatory in such a position; the rules of chess were amended to make this clear.


title: En passant

text: The "en passant" capture is often used as a theme in chess problems. According to Kenneth S. Howard, "En passant pawn captures frequently produce striking effects in the opening and closing of lines, both for white and black." By retrograde analysis convention, a pawn may be captured "en passant" only if it can be proven to have advanced two squares on the previous move.


title: En passant

text: introduces the threat of 2.d5+ cxd5 3.Bxd5#. Black can capture the d4-pawn "en passant" in either of two ways:


title: En passant

text: shifting the e4-pawn from the e- to the d-file, preventing an "en passant" capture after White plays 2.f4. To stop the threat of 3.f5#, Black can advance 2...f5, but this allows White to play 3.exf6 e.p.# due to the decisive opening of the .


title: En passant

text: threatens 2.Qf2#. The moves of the black e-pawn are restricted in an unusual manner. The "en passant" capture 1...exd3 e.p.+ is illegal (it exposes Black’s king to check), but


title: En passant

text: In most chess variants, pawns move as in standard chess, so the "en passant" capture is the same. Some larger variants allow pawns to make an initial move of more than two squares. For example, a pawn can make an initial move of up to four squares in double chess, and up to six squares in chess on a really big board. Such games usually allow an "en passant" capture on any square the pawn passes.


title: En passant

text: In some three-dimensional variants, such as millennium 3D chess or Alice chess, capturing "en passant" is allowed, though in the former case, the captured pawn's two-square move cannot have been purely vertical. In "5D Chess with Multiverse Time Travel", capturing "en passant" is allowed within the spatial dimensions but not across time or between timelines.


title: En passant

text: The "en passant" capture is not present in some chess variants, such as Dragonchess and Raumschach. Traditional Asian games of the chess family such as shogi, xiangqi, and janggi have no two-square pawn advance and therefore no "en passant" capture.


title: Determinant

text: In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if and only if the matrix is invertible and the linear map represented by the matrix is an isomorphism. The determinant of a product of matrices is the product of their determinants (the preceding property is a corollary of this one).


title: Determinant

text: The determinant of a matrix can be defined in several equivalent ways. Leibniz formula expresses the determinant as a sum of signed products of matrix entries such that each summand is the product of different entries, and the number of these summands is formula_3 the factorial of (the product of the first positive integers). The Laplace expansion expresses the determinant of a matrix as a linear combination of determinants of formula_4 submatrices. Gaussian elimination express the determinant as the product of the diagonal entries of a diagonal matrix that is obtained by a succession of elementary row operations.


title: Determinant

text: Determinants can also be defined by some of their properties: the determinant is the unique function defined on the matrices that has the four following properties. The determinant of the identity matrix is ; the exchange of two rows (or of two columns) multiplies the determinant by ; multiplying a row (or a column) by a number multiplies the determinant by this number; and adding to a row (or a column) a multiple of another row (or column) does not change the determinant.


title: Determinant

text: Determinants occur throughout mathematics. For example, a matrix is often used to represent the coefficients in a system of linear equations, and determinants can be used to solve these equations (Cramer's rule), although other methods of solution are computationally much more efficient. Determinants are used for defining the characteristic polynomial of a matrix, whose roots are the eigenvalues. In geometry, the signed -dimensional volume of a -dimensional parallelepiped is expressed by a determinant. This is used in calculus with exterior differential forms and the Jacobian determinant, in particular for changes of variables in multiple integrals.


title: Determinant

text: The determinant of a matrix formula_5 is denoted either by "" or by vertical bars around the matrix, and is defined as


title: Determinant

text: The determinant has several key properties that can be proved by direct evaluation of the definition for formula_8-matrices, and that continue to hold for determinants of larger matrices. They are as follows: first, the determinant of the identity matrix formula_9 is 1.


title: Determinant

text: Finally, if any column is multiplied by some number formula_12 (i.e., all entries in that column are multiplied by that number), the determinant is also multiplied by that number:


title: Determinant

text: If the matrix entries are real numbers, the matrix can be used to represent two linear maps: one that maps the standard basis vectors to the rows of , and one that maps them to the columns of . In either case, the images of the basis vectors form a parallelogram that represents the image of the unit square under the mapping. The parallelogram defined by the rows of the above matrix is the one with vertices at , , , and , as shown in the accompanying diagram.


title: Determinant

text: The absolute value of is the area of the parallelogram, and thus represents the scale factor by which areas are transformed by . (The parallelogram formed by the columns of is in general a different parallelogram, but since the determinant is symmetric with respect to rows and columns, the area will be the same.)


title: Determinant

text: The absolute value of the determinant together with the sign becomes the "oriented area" of the parallelogram. The oriented area is the same as the usual area, except that it is negative when the angle from the first to the second vector defining the parallelogram turns in a clockwise direction (which is opposite to the direction one would get for the identity matrix).


title: Determinant

text: To show that is the signed area, one may consider a matrix containing two vectors and representing the parallelogram's sides. The signed area can be expressed as for the angle "θ" between the vectors, which is simply base times height, the length of one vector times the perpendicular component of the other. Due to the sine this already is the signed area, yet it may be expressed more conveniently using the cosine of the complementary angle to a perpendicular vector, e.g. , so that , which can be determined by the pattern of the scalar product to be equal to :


title: Determinant

text: Thus the determinant gives the scaling factor and the orientation induced by the mapping represented by "A". When the determinant is equal to one, the linear mapping defined by the matrix is equi-areal and orientation-preserving.


title: Determinant

text: The object known as the "bivector" is related to these ideas. In 2D, it can be interpreted as an "oriented plane segment" formed by imagining two vectors each with origin , and coordinates and . The bivector magnitude (denoted by ) is the "signed area", which is also the determinant .


title: Determinant

text: This means that formula_17 maps the unit "n"-cube to the "n"-dimensional parallelotope defined by the vectors formula_18 the region formula_19


title: Determinant

text: The determinant gives the signed "n"-dimensional volume of this parallelotope, formula_20 and hence describes more generally the "n"-dimensional volume scaling factor of the linear transformation produced by "A". (The sign shows whether the transformation preserves or reverses orientation.) In particular, if the determinant is zero, then this parallelotope has volume zero and is not fully "n"-dimensional, which indicates that the dimension of the image of "A" is less than "n". This means that "A" produces a linear transformation which is neither onto nor one-to-one, and so is not invertible.


title: Determinant

text: The entries formula_22 etc. are, for many purposes, real or complex numbers. As discussed below, the determinant is also defined for matrices whose entries are in a commutative ring.


title: Determinant

text: The determinant of "A" is denoted by det("A"), or it can be denoted directly in terms of the matrix entries by writing enclosing bars instead of brackets:


title: Determinant

text: There are various equivalent ways to define the determinant of a square matrix "A", i.e. one with the same number of rows and columns: the determinant can be defined via the Leibniz formula, an explicit formula involving sums of products of certain entries of the matrix. The determinant can also be characterized as the unique function depending on the entries of the matrix satisfying certain properties. This approach can also be used to compute determinants by simplifying the matrices in question.


title: Determinant

text: The rule of Sarrus is a mnemonic for the expanded form of this determinant: the sum of the products of three diagonal north-west to south-east lines of matrix elements, minus the sum of the products of three diagonal south-west to north-east lines of elements, when the copies of the first two columns of the matrix are written beside it as in the illustration. This scheme for calculating the determinant of a matrix does not carry over into higher dimensions.


title: Determinant

text: In higher dimension, the Leibniz formula expresses the determinant of an formula_25-matrix as an expression involving permutations and their "signatures". A permutation of the set formula_26 is a function formula_27 that reorders this set of integers. The value in the formula_28-th position after the reordering formula_27 is denoted below by formula_30. The set of all such permutations, called the symmetric group, is commonly denoted formula_31. The signature formula_32 of a permutation formula_27 is formula_34 if the permutation can be obtained with an even number of exchanges of two entries; otherwise, it is formula_35


title: Determinant

text: The Levi-Civita symbol formula_39 is defined on the -tuples of integers in formula_40 as if two of the integers are equal, and, otherwise, as the signature of the permutation defined by the tuple of integers. With the Levi-Civita symbol, Leibniz formula may be written as


title: Determinant

text: The determinant can be characterized by the following three key properties. To state these, it is convenient to regard an formula_25-matrix "A" as being composed of its formula_44 columns, so denoted as


title: Determinant

text: where the column vector formula_46 (for each "i") is composed of the entries of the matrix in the "i"-th column.


title: Determinant

text: If the determinant is defined using the Leibniz formula as above, these three properties can be proved by direct inspection of that formula. Some authors also approach the determinant directly using these three properties: it can be shown that there is exactly one function that assigns to any formula_25-matrix "A" a number that satisfies these three properties. This also shows that this more abstract approach to the determinant yields the same definition as the one using the Leibniz formula.


title: Determinant

text: To see this it suffices to expand the determinant by multi-linearity in the columns into a (huge) linear combination of determinants of matrices in which each column is a standard basis vector. These determinants are either 0 (by property 9) or else ±1 (by properties 1 and 12 below), so the linear combination gives the expression above in terms of the Levi-Civita symbol. While less technical in appearance, this characterization cannot entirely replace the Leibniz formula in defining the determinant, since without it the existence of an appropriate function is not clear.


title: Determinant

text: These characterizing properties and their consequences listed above are both theoretically significant, but can also be used to compute determinants for concrete matrices. In fact, Gaussian elimination can be applied to bring any matrix into upper triangular form, and the steps in this algorithm affect the determinant in a controlled way. The following concrete example illustrates the computation of the determinant of the matrix formula_17 using that method:


title: Determinant

text: This can be proven by inspecting the Leibniz formula. This implies that in all the properties mentioned above, the word "column" can be replaced by "row" throughout. For example, viewing an matrix as being composed of "n" rows, the determinant is an "n"-linear function.


title: Determinant

text: The determinant is a "multiplicative map", i.e., for square matrices formula_17 and formula_71 of equal size, the determinant of a matrix product equals the product of their determinants:


title: Determinant

text: This key fact can be proven by observing that, for a fixed matrix formula_71, both sides of the equation are alternating and multilinear as a function depending on the columns of formula_17. Moreover, they both take the value formula_75 when formula_17 is the identity matrix. The above-mentioned unique characterization of alternating multilinear maps therefore shows this claim.


title: Determinant

text: A matrix formula_17 is invertible precisely if its determinant is nonzero. This follows from the multiplicativity of formula_78 and the formula for the inverse involving the adjugate matrix mentioned below. In this event, the determinant of the inverse matrix is given by


title: Determinant

text: In particular, products and inverses of matrices with non-zero determinant (respectively, determinant one) still have this property. Thus, the set of such matrices (of fixed size formula_44) forms a group known as the general linear group formula_81 (respectively, a subgroup called the special linear group formula_82. More generally, the word "special" indicates the subgroup of another matrix group of matrices of determinant one. Examples include the special orthogonal group (which if "n" is 2 or 3 consists of all rotation matrices), and the special unitary group.


title: Determinant

text: The Cauchy–Binet formula is a generalization of that product formula for "rectangular" matrices. This formula can also be recast as a multiplicative formula for compound matrices whose entries are the determinants of all quadratic submatrices of a given matrix.


title: Determinant

text: Laplace expansion expresses the determinant of a matrix formula_17 in terms of determinants of smaller matrices, known as its minors. The minor formula_84 is defined to be the determinant of the formula_85-matrix that results from formula_17 by removing the formula_28-th row and the formula_88-th column. The expression formula_89 is known as a cofactor. For every formula_28, one has the equality


title: Determinant

text: which is called the "Laplace expansion along the th row". For example, the Laplace expansion along the first row (formula_92) gives the following formula:


title: Determinant

text: Unwinding the determinants of these formula_8-matrices gives back the Leibniz formula mentioned above. Similarly, the "Laplace expansion along the formula_88-th column" is the equality


title: Determinant

text: Laplace expansion can be used iteratively for computing determinants, but this approach is inefficient for large matrices. However, it is useful for computing the determinants of highly symmetric matrix such as the Vandermonde matrix


title: Determinant

text: This determinant has been applied, for example, in the proof of Baker's theorem in the theory of transcendental numbers.


title: Determinant

text: The formula for the determinant of a formula_8-matrix above continues to hold, under appropriate further assumptions, for a block matrix, i.e., a matrix composed of four submatrices formula_103 of dimension formula_25, formula_105, formula_106 and formula_107, respectively. The easiest such formula, which can be proven using either the Leibniz formula or a factorization involving the Schur complement, is


title: Determinant

text: If the blocks are square matrices of the "same" size further formulas hold. For example, if formula_115 and formula_110 commute (i.e., formula_117), then there holds


title: Determinant

text: This formula has been generalized to matrices composed of more than formula_8 blocks, again under appropriate commutativity conditions among the individual blocks.


title: Determinant

text: For formula_120 and formula_121, the following formula holds (even if formula_17 and formula_71 do not commute)


title: Determinant

text: Sylvester's determinant theorem states that for "A", an matrix, and "B", an matrix (so that "A" and "B" have dimensions allowing them to be multiplied in either order forming a square matrix):


title: Determinant

text: The determinant of the sum formula_126 of two square matrices of the same size is not in general expressible in terms of the determinants of "A" and of "B". However, for positive semidefinite matrices formula_17, formula_71 and formula_115 of equal size, formula_130 with the corollary formula_131 Conversely, if formula_17 and formula_71 are Hermitian, positive-definite, and size formula_134, then the determinant has concave formula_44 root; this implies formula_136 by homogeneity.


title: Determinant

text: For the special case of formula_137 matrices with complex entries, the determinant of the sum can be written in terms of determinants and traces in the following identity:


title: Determinant

text: This has an application to formula_137 matrix algebras. For example, consider the complex numbers as a matrix algebra. The complex numbers have a representation as matrices of the form


title: Determinant

text: with formula_141 and formula_142 real. Since formula_143, taking formula_144 and formula_145 in the above identity gives


title: Determinant

text: The determinant is closely related to two other central concepts in linear algebra, the eigenvalues and the characteristic polynomial of a matrix. Let formula_17 be an formula_25-matrix with complex entries with eigenvalues formula_151. (Here it is understood that an eigenvalue with algebraic multiplicity occurs times in this list.) Then the determinant of is the product of all eigenvalues,


title: Determinant

text: Here, formula_154 is the indeterminate of the polynomial and formula_48 is the identity matrix of the same size as formula_17. By means of this polynomial, determinants can be used to find the eigenvalues of the matrix formula_17: they are precisely the roots of this polynomial, i.e., those complex numbers formula_158 such that


title: Determinant

text: A Hermitian matrix is positive definite if all its eigenvalues are positive. Sylvester's criterion asserts that this is equivalent to the determinants of the submatrices


title: Determinant

text: The trace tr("A") is by definition the sum of the diagonal entries of and also equals the sum of the eigenvalues. Thus, for complex matrices ,


title: Determinant

text: Here exp() denotes the matrix exponential of , because every eigenvalue of corresponds to the eigenvalue exp() of exp(). In particular, given any logarithm of , that is, any matrix satisfying


title: Determinant

text: cf. Cayley-Hamilton theorem. Such expressions are deducible from combinatorial arguments, Newton's identities, or the Faddeev–LeVerrier algorithm. That is, for generic , the signed constant term of the characteristic polynomial, determined recursively from


title: Determinant

text: The formula can be expressed in terms of the complete exponential Bell polynomial of "n" arguments "s" = −("l" – 1)! tr("A") as


title: Determinant

text: This formula can also be used to find the determinant of a matrix with multidimensional indices and . The product and trace of such matrices are defined in a natural way as


title: Determinant

text: An important arbitrary dimension identity can be obtained from the Mercator series expansion of the logarithm when the expansion converges. If every eigenvalue of "A" is less than 1 in absolute value,


title: Determinant

text: is expanded as a formal power series in then all coefficients of for are zero and the remaining polynomial is .


title: Determinant

text: For a positive definite matrix , the trace operator gives the following tight lower and upper bounds on the log determinant


title: Determinant

text: with equality if and only if . This relationship can be derived via the formula for the KL-divergence between two multivariate normal distributions.


title: Determinant

text: These inequalities can be proved by expressing the traces and the determinant in terms of the eigenvalues. As such, they represent the well-known fact that the harmonic mean is less than the geometric mean, which is less than the arithmetic mean, which is, in turn, less than the root mean square.


title: Determinant

text: The Leibniz formula shows that the determinant of real (or analogously for complex) square matrices is a polynomial function from formula_178 to formula_179. In particular, it is everywhere differentiable. Its derivative can be expressed using Jacobi's formula:


title: Determinant

text: where formula_98 denotes the adjugate of formula_17. In particular, if formula_17 is invertible, we have


title: Determinant

text: This identity is used in describing Lie algebras associated to certain matrix Lie groups. For example, the special linear group formula_190 is defined by the equation formula_191. The above formula shows that its Lie algebra is the special linear Lie algebra formula_192 consisting of those matrices having trace zero.


title: Determinant

text: Writing a formula_193-matrix as formula_194 where formula_195 are column vectors of length 3, then the gradient over one of the three vectors may be written as the cross product of the other two:


title: Determinant

text: Historically, determinants were used long before matrices: A determinant was originally defined as a property of a system of linear equations.


title: Determinant

text: The determinant "determines" whether the system has a unique solution (which occurs precisely if the determinant is non-zero).


title: Determinant

text: In this sense, determinants were first used in the Chinese mathematics textbook "The Nine Chapters on the Mathematical Art" (九章算術, Chinese scholars, around the 3rd century BCE). In Europe, solutions of linear systems of two equations were expressed by Cardano in 1545 by a determinant-like entity.


title: Determinant

text: Determinants proper originated from the work of Seki Takakazu in 1683 in Japan and parallelly of Leibniz in 1693. stated, without proof, Cramer's rule. Both Cramer and also were led to determinants by the question of plane curves passing through a given set of points.


title: Determinant

text: Vandermonde (1771) first recognized determinants as independent functions. gave the general method of expanding a determinant in terms of its complementary minors: Vandermonde had already given a special case. Immediately following, Lagrange (1773) treated determinants of the second and third order and applied it to questions of elimination theory; he proved many special cases of general identities.


title: Determinant

text: Gauss (1801) made the next advance. Like Lagrange, he made much use of determinants in the theory of numbers. He introduced the word "determinant" (Laplace had used "resultant"), though not in the present signification, but rather as applied to the discriminant of a quantic. Gauss also arrived at the notion of reciprocal (inverse) determinants, and came very near the multiplication theorem.


title: Determinant

text: The next contributor of importance is Binet (1811, 1812), who formally stated the theorem relating to the product of two matrices of "m" columns and "n" rows, which for the special case of reduces to the multiplication theorem. On the same day (November 30, 1812) that Binet presented his paper to the Academy, Cauchy also presented one on the subject. (See Cauchy–Binet formula.) In this he used the word "determinant" in its present sense, summarized and simplified what was then known on the subject, improved the notation, and gave the multiplication theorem with a proof more satisfactory than Binet's. With him begins the theory in its generality.


title: Determinant

text: The study of special forms of determinants has been the natural result of the completion of the general theory. Axisymmetric determinants have been studied by Lebesgue, Hesse, and Sylvester; persymmetric determinants by Sylvester and Hankel; circulants by Catalan, Spottiswoode, Glaisher, and Scott; skew determinants and Pfaffians, in connection with the theory of orthogonal transformation, by Cayley; continuants by Sylvester; Wronskians (so called by Muir) by Christoffel and Frobenius; compound determinants by Sylvester, Reiss, and Picquet; Jacobians and Hessians by Sylvester; and symmetric gauche determinants by Trudi. Of the textbooks on the subject Spottiswoode's was the first. In America, Hanus (1886), Weld (1893), and Muir/Metzler (1933) published treatises.


title: Determinant

text: Determinants can be used to describe the solutions of a linear system of equations, written in matrix form as formula_197. This equation has a unique solution formula_198 if and only if formula_199 is nonzero. In this case, the solution is given by Cramer's rule:


title: Determinant

text: where formula_201 is the matrix formed by replacing the formula_28-th column of formula_17 by the column vector formula_142. This follows immediately by column expansion of the determinant, i.e.


title: Determinant

text: Cramer's rule can be implemented in formula_208 time, which is comparable to more common methods of solving systems of linear equations, such as LU, QR, or singular value decomposition.


title: Determinant

text: Determinants can be used to characterize linearly dependent vectors: formula_209 is zero if and only if the column vectors (or, equivalently, the row vectors) of the matrix formula_17 are linearly dependent. For example, given two linearly independent vectors formula_211, a third vector formula_212 lies in the plane spanned by the former two vectors exactly if the determinant of the formula_193-matrix consisting of the three vectors is zero. The same idea is also used in the theory of differential equations: given functions formula_214 (supposed to be formula_215 times differentiable), the Wronskian is defined to be


title: Determinant

text: It is non-zero (for some formula_198) in a specified interval if and only if the given functions and all their derivatives up to order formula_215 are linearly independent. If it can be shown that the Wronskian is zero everywhere on an interval then, in the case of analytic functions, this implies the given functions are linearly dependent. See the Wronskian and linear independence. Another such use of the determinant is the resultant, which gives a criterion when two polynomials have a common root.


title: Determinant

text: The determinant can be thought of as assigning a number to every sequence of "n" vectors in R, by using the square matrix whose columns are the given vectors. For instance, an orthogonal matrix with entries in R represents an orthonormal basis in Euclidean space. The determinant of such a matrix determines whether the orientation of the basis is consistent with or opposite to the orientation of the standard basis. If the determinant is +1, the basis has the same orientation. If it is −1, the basis has the opposite orientation.


title: Determinant

text: More generally, if the determinant of "A" is positive, "A" represents an orientation-preserving linear transformation (if "A" is an orthogonal or matrix, this is a rotation), while if it is negative, "A" switches the orientation of the basis.


title: Determinant

text: As pointed out above, the absolute value of the determinant of real vectors is equal to the volume of the parallelepiped spanned by those vectors. As a consequence, if formula_219 is the linear map given by multiplication with a matrix formula_17, and formula_221 is any measurable subset, then the volume of formula_222 is given by formula_223 times the volume of formula_224. More generally, if the linear map formula_225 is represented by the formula_106-matrix formula_17, then the formula_44-dimensional volume of formula_222 is given by:


title: Determinant

text: By calculating the volume of the tetrahedron bounded by four points, they can be used to identify skew lines. The volume of any tetrahedron, given its vertices formula_231, formula_232, or any other combination of pairs of vertices that form a spanning tree over the vertices.


title: Determinant

text: For a general differentiable function, much of the above carries over by considering the Jacobian matrix of "f". For


title: Determinant

text: Its determinant, the Jacobian determinant, appears in the higher-dimensional version of integration by substitution: for suitable functions "f" and an open subset "U" of R (the domain of "f"), the integral over "f"("U") of some other function is given by


title: Determinant

text: When applied to the field of Cartography, the determinant can be used to measure the rate of expansion of a map near the poles.


title: Determinant

text: The above identities concerning the determinant of products and inverses of matrices imply that similar matrices have the same determinant: two matrices "A" and "B" are similar, if there exists an invertible matrix "X" such that . Indeed, repeatedly applying the above identities yields


title: Determinant

text: The determinant is therefore also called a similarity invariant. The determinant of a linear transformation


title: Determinant

text: for some finite-dimensional vector space "V" is defined to be the determinant of the matrix describing it, with respect to an arbitrary choice of basis in "V". By the similarity invariance, this determinant is independent of the choice of the basis for "V" and therefore only depends on the endomorphism "T".


title: Determinant

text: The above definition of the determinant using the Leibniz rule holds works more generally when the entries of the matrix are elements of a commutative ring formula_238, such as the integers formula_239, as opposed to the field of real or complex numbers. Moreover, the characterization of the determinant as the unique alternating multilinear map that satisfies formula_240 still holds, as do all the properties that result from that characterization.


title: Determinant

text: A matrix formula_241 is invertible (in the sense that there is an inverse matrix whose entries are in formula_238) if and only if its determinant is an invertible element in formula_238. For formula_244, this means that the determinant is +1 or −1. Such a matrix is called unimodular.


title: Determinant

text: between the general linear group (the group of invertible formula_25-matrices with entries in formula_238) and the multiplicative group of units in formula_238. Since it respects the multiplication in both groups, this map is a group homomorphism.


title: Determinant

text: Given a ring homomorphism formula_249, there is a map formula_250 given by replacing all entries in formula_238 by their images under formula_252. The determinant respects these maps, i.e., the identity


title: Determinant

text: For example, the determinant of the complex conjugate of a complex matrix (which is also the determinant of its conjugate transpose) is the complex conjugate of its determinant, and for integer matrices: the reduction modulo formula_254 of the determinant of such a matrix is equal to the determinant of the matrix reduced modulo formula_254 (the latter determinant being computed using modular arithmetic). In the language of category theory, the determinant is a natural transformation between the two functors formula_81 and formula_257. Adding yet another layer of abstraction, this is captured by saying that the determinant is a morphism of algebraic groups, from the general linear group to the multiplicative group,


title: Determinant

text: The determinant of a linear transformation formula_237 of an formula_44-dimensional vector space formula_261 or, more generally a free module of (finite) rank formula_44 over a commutative ring formula_238 can be formulated in a coordinate-free manner by considering the formula_44-th exterior power formula_265 of formula_261. The map formula_267 induces a linear map


title: Determinant

text: As formula_265 is one-dimensional, the map formula_270 is given by multiplying with some scalar, i.e., an element in formula_238. Some authors such as use this fact to "define" the determinant to be the element in formula_238 satisfying the following identity (for all formula_273):


title: Determinant

text: This definition agrees with the more concrete coordinate-dependent definition. This can be shown using the uniqueness of a multilinear alternating form on formula_44-tuples of vectors in formula_276.


title: Determinant

text: For this reason, the highest non-zero exterior power formula_265 (as opposed to the determinant associated to an endomorphism) is sometimes also called the determinant of formula_261 and similarly for more involved objects such as vector bundles or chain complexes of vector spaces. Minors of a matrix can also be cast in this setting, by considering lower alternating forms formula_279 with formula_280.


title: Determinant

text: Determinants as treated above admit several variants: the permanent of a matrix is defined as the determinant, except that the factors formula_32 occurring in Leibniz's rule are omitted. The immanant generalizes both by introducing a character of the symmetric group formula_31 in Leibniz's rule.


title: Determinant

text: For any associative algebra formula_17 that is finite-dimensional as a vector space over a field formula_284, there is a determinant map


title: Determinant

text: This definition proceeds by establishing the characteristic polynomial independently of the determinant, and defining the determinant as the lowest order term of this polynomial. This general definition recovers the determinant for the matrix algebra formula_286, but also includes several further cases including the determinant of a quaternion,


title: Determinant

text: the norm formula_288 of a field extension, as well as the Pfaffian of a skew-symmetric matrix and the reduced norm of a central simple algebra, also arise as special cases of this construction.


title: Determinant

text: For matrices with an infinite number of rows and columns, the above definitions of the determinant do not carry over directly. For example, in the Leibniz formula, an infinite sum (all of whose terms are infinite products) would have to be calculated. Functional analysis provides different extensions of the determinant for such infinite-dimensional situations, which however only work for particular kinds of operators.


title: Determinant

text: The Fredholm determinant defines the determinant for operators known as trace class operators by an appropriate generalization of the formula


title: Determinant

text: For operators in a finite factor, one may define a positive real-valued determinant called the Fuglede−Kadison determinant using the canonical trace. In fact, corresponding to every tracial state on a von Neumann algebra there is a notion of Fuglede−Kadison determinant.


title: Determinant

text: For matrices over non-commutative rings, multilinearity and alternating properties are incompatible for , so there is no good definition of the determinant in this setting.


title: Determinant

text: For square matrices with entries in a non-commutative ring, there are various difficulties in defining determinants analogously to that for commutative rings. A meaning can be given to the Leibniz formula provided that the order for the product is specified, and similarly for other definitions of the determinant, but non-commutativity then leads to the loss of many fundamental properties of the determinant, such as the multiplicative property or that the determinant is unchanged under transposition of the matrix. Over non-commutative rings, there is no reasonable notion of a multilinear form (existence of a nonzero with a regular element of "R" as value on some pair of arguments implies that "R" is commutative). Nevertheless, various notions of non-commutative determinant have been formulated that preserve some of the properties of determinants, notably quasideterminants and the Dieudonné determinant. For some classes of matrices with non-commutative elements, one can define the determinant and prove linear algebra theorems that are very similar to their commutative analogs. Examples include the "q"-determinant on quantum groups, the Capelli determinant on Capelli matrices, and the Berezinian on supermatrices (i.e., matrices whose entries are elements of formula_290-graded rings). Manin matrices form the class closest to matrices with commutative elements.


title: Determinant

text: Determinants are mainly used as a theoretical tool. They are rarely calculated explicitly in numerical linear algebra, where for applications like checking invertibility and finding eigenvalues the determinant has largely been supplanted by other techniques. Computational geometry, however, does frequently use calculations related to determinants.


title: Determinant

text: While the determinant can be computed directly using the Leibniz rule this approach is extremely inefficient for large matrices, since that formula requires calculating formula_291 (formula_44 factorial) products for an formula_25-matrix. Thus, the number of required operations grows very quickly: it is of order formula_291. The Laplace expansion is similarly inefficient. Therefore, more involved techniques have been developed for calculating determinants.


title: Determinant

text: Some methods compute formula_295 by writing the matrix as a product of matrices whose determinants can be more easily computed. Such techniques are referred to as decomposition methods. Examples include the LU decomposition, the QR decomposition or the Cholesky decomposition (for positive definite matrices). These methods are of order formula_208, which is a significant improvement over formula_297.


title: Determinant

text: of a permutation matrix formula_300 (which has exactly a single formula_162 in each column, and otherwise zeros), a lower triangular matrix formula_302 and an upper triangular matrix formula_303.


title: Determinant

text: The determinants of the two triangular matrices formula_302 and formula_303 can be quickly calculated, since they are the products of the respective diagonal entries. The determinant of formula_300 is just the sign formula_307 of the corresponding permutation (which is formula_308 for an even number of permutations and is formula_309 for an odd number of permutations). Once such a LU decomposition is known for formula_17, its determinant is readily computed as