title: Survivor: Gabon

text: At the Final Tribal Council, the jury criticized Sugar for her backstabbing, not owning her game, and playing with her emotions. Susie was praised for playing under the radar strategically, while Bob was praised for his physical game, creating two fake immunity idols, and having control of his fate. The jury voted Bob the Sole Survivor in a 4-3-0 vote over Susie and Sugar, respectively.


title: Survivor: Gabon

text: Probst recalled that the live finale of "Gabon" was almost ruined as the production assistant normally in charge of bringing the Final Tribal Council votes from the production offices to the live venue had forgotten to bring them, a fact discovered just as the live show started. Probst explained that another staff member raced back to the offices, breaking into the room where the votes were secured due to having no key, and faxed the votes over to the live venue, where another production staff member retraced the votes by hand with only about 15 minutes left before the live show vote reading. In the end, the original votes arrived at the studio in time and were read.


title: Survivor: Gabon

text: "Survivor: Gabon" was initially met with generally negative reception. Host Jeff Probst originally ranked "Gabon" 14th out of 19 seasons in 2010, thus ranking it as the sixth-worst season of the series. He later went on to say that the show was drifting during "Gabon" and that he nearly quit the show as a consequence. Dalton Ross of "Entertainment Weekly" ranked "Gabon" 28th out of 40, summarizing: "It got better near the end, but it was still a case of too little, too late. The fact that so many unworthy players went so far is simply too damning." "Gabon" is ranked as the second-worst season of the series by Examiner.com (only ahead of ""), and "The Wire" (only ahead of ""). "The Purple Rock Podcast" also ranked "Gabon" 34th out of 40, describing the cast as one of the most "inept Survivor casts ever from a gameplay perspective." In 2012 and 2013, "Survivor" fan site ""Survivor" Oz" consistently ranked "Gabon" as the eighth-worst season of the series in its annual polls ranking all seasons.


title: Survivor: Gabon

text: However, reception for "Gabon" has improved significantly in recent years due to the entertaining personalities of the players, specifically Jessica "Sugar" Kiper, , Randy Bailey, Corinne Kaplan, Matty Whitmore, and . In 2014, it was voted the tenth-best season by ""Survivor" Oz". Its ranking improved substantially again in 2015, when it was ranked the seventh-best season. In 2015, a poll by "Rob Has a Podcast" ranked this season 23rd out of 30, with Rob Cesternino ranking this season 29th. This was updated in 2021 during Cesternino's podcast, "Survivor All-Time Top 40 Rankings", ranking 26th out of 40. In 2020, "Inside Survivor" ranked this season 20th out of 40 saying that "the gameplay is completely baffling" but is "one of the most entertaining seasons of the show from a comedic perspective."


title: Survivor: Gabon

text: "Survivor: Gabon" received some minor criticism from the Parents Television Council for a brief, uncensored glimpse of Marcus's penis when the tip was partially exposed through the fly of his boxer shorts. The exposure occurred on the season opener during the second Immunity Challenge when Marcus raced alongside fellow contestants.


title: Airline hub

text: An airline hub or hub airport is an airport used by one or more airlines to concentrate passenger traffic and flight operations. Hubs serve as transfer (or stop-over) points to help get passengers to their final destination. It is part of the hub-and-spoke system. An airline may operate flights from several non-hub (spoke) cities to the hub airport, and passengers traveling between spoke cities connect through the hub. This paradigm creates economies of scale that allow an airline to serve (via an intermediate connection) city-pairs that could otherwise not be economically served on a non-stop basis. This system contrasts with the point-to-point model, in which there are no hubs and nonstop flights are instead offered between spoke cities. Hub airports also serve origin and destination (O&D) traffic.


title: Airline hub

text: The hub-and-spoke system allows an airline to serve fewer routes, so fewer aircraft are needed. The system also increases passenger loads; a flight from a hub to a spoke carries not just passengers originating at the hub, but also passengers originating at multiple spoke cities. However, the system is costly. Additional employees and facilities are needed to cater to connecting passengers. To serve spoke cities of varying populations and demand, an airline requires several aircraft types, and specific training and equipment are necessary for each type. In addition, airlines may experience capacity constraints as they expand at their hub airports.


title: Airline hub

text: For the passenger, the hub-and-spoke system offers one-stop air service to a wide array of destinations. However, it requires having to regularly make connections en route to their final destination, which increases travel time. Additionally, airlines can come to monopolise their hubs (fortress hubs), allowing them to freely increase fares as passengers have no alternative. High domestic connectivity in the United States is achieved through airport location and hub dominance. The top 10 megahubs in the US are dominated by American Airlines, Delta Air Lines and United Airlines, the three largest United States-based airlines.


title: Airline hub

text: Airlines may operate banks of flights at their hubs, in which several flights arrive and depart within short periods of time. The banks may be known as "peaks" of activity at the hubs and the non-banks as "valleys". Banking allows for short connection times for passengers. However, an airline must assemble many resources to cater to the influx of flights during a bank, and having several aircraft on the ground at the same time can lead to congestion and delays. In addition, banking could result in inefficient aircraft utilisation, with aircraft waiting at spoke cities for the next bank.


title: Airline hub

text: Instead, some airlines have debanked their hubs, introducing a "rolling hub" in which flight arrivals and departures are spread throughout the day. This phenomenon is also known as "depeaking". While costs may decrease, connection times are longer at a rolling hub. American Airlines was the first to depeak its hubs, trying to improve profitability following the September 11 attacks. It rebanked its hubs in 2015, however, feeling the gain in connecting passengers would outweigh the rise in costs.


title: Airline hub

text: For example, the hub of Qatar Airways in Doha Airport has 471 daily movements to 140 destinations by March 2020 with an average of 262 seats per movement; in three main waves: 5h-9h (132 movements), 16h-21h (128) and 23h-3h (132), allowing around 30 million connecting passengers in 2019.


title: Airline hub

text: Before the US airline industry was deregulated in 1978, most airlines operated under the point-to-point system (with a notable exception being Pan Am). The Civil Aeronautics Board dictated which routes an airline could fly. At the same time, however, some airlines began to experiment with the hub-and-spoke system. Delta Air Lines was the first to implement such a system, providing service to remote spoke cities from its Atlanta hub. After deregulation, many airlines quickly established hub-and-spoke route networks of their own.


title: Airline hub

text: In 1974, the governments of Bahrain, Oman, Qatar and the United Arab Emirates took control of Gulf Air from the British Overseas Airways Corporation (BOAC). Gulf Air became the flag carrier of the four Middle Eastern nations. It linked Oman, Qatar and the UAE to its Bahrain hub, from which it offered flights to destinations throughout Europe and Asia. In the UAE, Gulf Air focused on Abu Dhabi rather than Dubai, contrary to the aspirations of UAE Prime Minister Mohammed bin Rashid Al Maktoum to transform the latter into a world-class metropolis. Sheikh Mohammed proceeded to establish a new airline based in Dubai, Emirates, which launched operations in 1985.


title: Airline hub

text: Elsewhere in the Middle East region, Qatar and Oman decided to create their own airlines as well. Qatar Airways and Oman Air were both founded in 1993, with hubs at Doha and Muscat respectively. As the new airlines grew, their home nations relied less on Gulf Air to provide air service. Qatar withdrew its share in Gulf Air in 2002. In 2003, the UAE formed another national airline, Etihad Airways, which is based in Abu Dhabi. The country exited Gulf Air in 2006, and Oman followed in 2007. Gulf Air therefore became fully owned by the government of Bahrain.


title: Airline hub

text: Emirates, Qatar Airways, Saudia and Etihad Airways have since established large hubs at their respective home airports. The hubs, which benefit from their proximity to large population centres, have become popular stopover points on trips between Europe and Asia, for example. Their rapid growth has impacted the development of traditional hubs, such as London, Paris, and New York City.


title: Airline hub

text: A cargo hub is an airport that primarily is operated by a cargo airline that uses the hub-and-spoke system. In the United States, the largest cargo hub airports are traditionally in the midwest for domestic service, like FedEx's Memphis Superhub or UPS Louisville Worldport. FedEx's airline, FedEx Express, established its Memphis hub in 1973, prior to the deregulation of the air cargo industry in the United States. The system has created an efficient delivery system for the airline. UPS Airlines has followed a similar pattern in Louisville. In Europe, TNT Airways, Cargolux and DHL Aviation follow a similar strategy and operate their primary hubs at Liège, Luxembourg and Leipzig respectively.


title: Airline hub

text: Additionally, Ted Stevens International Airport in Anchorage, Alaska is a frequent stop-over hub for many cargo airlines flying between Asia and North America. Most cargo airlines only stop in Anchorage for refueling and customs, but FedEx and UPS frequently use Anchorage to sort trans-pacific packages between regional hubs on each continent in addition to refueling and customs.


title: Airline hub

text: Passenger airlines that operate in a similar manner to the FedEx and UPS hubs are often regarded as scissor hubs, as many flights to one destination all land and deplane passengers simultaneously and, after a passenger transit period, repeat a similar process for departure to the final destination of each plane. Air India operates a scissor hub at London's Heathrow Airport, where passengers from Delhi, Ahmedabad, and Mumbai can continue onto a flight to Newark. Until its grounding, Jet Airways operated a similar scissor hub at Amsterdam Airport Schiphol to transport passengers from Bangalore, Mumbai and Delhi to Toronto-Pearson and vice versa. At the peak of operations at their former scissor hub at Brussels prior to the 2016 shift to Schiphol, flights operated from Mumbai, Delhi, and Chennai and continued onward to Toronto, New York, and Newark after a near-simultaneous stopover in Brussels and vice versa. An international scissor hub could be used for third and fourth freedom flights or it could be used for fifth freedom flights, for which a precursor is a bilateral treaty between two country pairs.


title: Airline hub

text: WestJet uses St. John's as a scissor hub during its summer schedule for flights inbound from Ottawa, Toronto, and Orlando and outbound to Dublin and London–Gatwick. At Los Angeles International Airport, Qantas passengers from Melbourne, Brisbane or Sydney may transfer onto a flight to New York–JFK and vice versa.


title: Airline hub

text: In the airline industry, a focus city is a destination from which an airline operates limited point-to-point routes. Ergo, a focus city primarily caters to the local market rather than to connecting passengers.


title: Airline hub

text: Although the term "focus city" is used to mainly refer to an airport from which an airline operates limited point-to-point routes, its usage has loosely expanded to refer to a small-scale hub as well. For example, JetBlue's New York–JFK focus city runs like a hub, although it is still deemed by JetBlue as a focus city.


title: Airline hub

text: A fortress hub exists when an airline controls a significant majority of the market at one of its hubs. Competition is particularly difficult at fortress hubs. Examples include Delta Air Lines at Atlanta, Detroit, and Minneapolis/St. Paul, American Airlines at Charlotte and Dallas/Fort Worth, and United Airlines at Houston–Intercontinental, Newark and Washington-Dulles.


title: Airline hub

text: Flag carriers have historically enjoyed similar dominance at the main international airport of their countries and some still do. Examples include Aeromexico in Mexico City, Air Canada in Toronto–Pearson, Air France in Paris–Charles de Gaulle, British Airways in London–Heathrow, Cathay Pacific in Hong Kong, Copa Airlines in Panama City, Emirates in Dubai, Ethiopian Airlines in Addis Ababa, Finnair in Helsinki, Iberia in Madrid, Korean Air at Seoul–Incheon, KLM in Amsterdam, Lufthansa in Frankfurt, Qantas in Sydney, Qatar Airways in Doha, Singapore Airlines in Singapore, South African Airways in Johannesburg, Swiss International Air Lines in Zurich, and Turkish Airlines in Istanbul.


title: Airline hub

text: A primary hub is the main hub for an airline. However, as an airline expands operations at its primary hub to the point that it experiences capacity limitations, it may elect to open secondary hubs. Examples of such hubs are British Airways' hub at London–Gatwick, Air India's hub at Mumbai and Lufthansa's hub at Munich. By operating multiple hubs, airlines can expand their geographic reach. They can also better serve spoke–spoke markets, providing more itineraries with connections at different hubs.


title: Airline hub

text: Cargo airlines like FedEx Express and UPS Airlines also operate secondary hubs to an extent, but these are primarily used to serve regional high-demand destinations because shipping packages through its main hub would waste fuel; an example of this would be FedEx transiting a package through Oakland International Airport when shipping packages between destinations near Seattle and Phoenix, Arizona instead of sending deliveries through the Memphis Superhub.


title: Airline hub

text: A given hub's capacity may become exhausted or capacity shortages may occur during peak periods of the day, at which point airlines may be compelled to shift traffic to a reliever hub. A reliever hub has the potential to serve several functions for an airline: it can bypass the congested hub, it can absorb excess demand for flights that could otherwise not be scheduled at the congested hub, and it can schedule new O&D city pairs for connecting traffic.


title: Airline hub

text: One of the most recognized examples of this model is Delta Air Lines' and American Airlines' uses of LaGuardia Airport as a domestic hub in New York City, due to capacity and slot restrictions at their hubs at John F. Kennedy International Airport. Many regional flights operate out of LaGuardia, while most international flights remain at JFK.


title: Airline hub

text: Lufthansa operates a similar model of business with its hubs at Frankfurt Airport and Munich Airport. Generally speaking, a marginal majority of its long-haul flights going to or from Germany have the German terminus of the flight at Frankfurt, while a similarly-sized but smaller minority have their German terminus at Munich.


title: Airline hub

text: In past history, carriers have maintained niche, time-of-day operations at hubs. The most notable was America West's use of McCarran International Airport in Las Vegas as a primary night-flight hub to increase aircraft utilization rates far beyond those of competing carriers.


title: Paul Oakenfold

text: Paul Mark Oakenfold (born 30 August 1963), formerly known mononymously as Oakenfold, is an English record producer, remixer and trance DJ. He has provided over 100 remixes for over 100 artists including U2, Moby, Madonna, Britney Spears, Massive Attack, the Cure, New Order, the Rolling Stones, the Stone Roses and Michael Jackson. Oakenfold was voted the No. 1 DJ in the World twice in 1998 and 1999 by "DJ Magazine".


title: Paul Oakenfold

text: Oakenfold was born on 30 August 1963 at Mile End Hospital. His father delivered the "London Evening News". He lived in Highbury, Greenhithe, then Croydon, attending Archbishop Lanfranc School, then studied to be a chef for four years and worked at the Army and Navy Club.


title: Paul Oakenfold

text: Paul Oakenfold describes his early life as a "bedroom DJ" in a podcasted interview with Vancouver's "24 Hours", stating he grew up listening to the Beatles.


title: Paul Oakenfold

text: Oakenfold's musical career began in the late 1970s, when he met Trevor Fung and began helping him DJ soul music in a Covent Garden wine bar. Here in London, he also met Rumours where he played Earth, Wind and Fire and popular British bands.


title: Paul Oakenfold

text: In 1981, 18-year-old Oakenfold and his friend Ian Paul moved to New York City. Oakenfold worked as a courier in West Harlem. During this time, hip-hop was overtaking dance music as the most popular sound in the area (see 1984 in music). Oakenfold and Fung used fake identification to sneak into various dance clubs, like Studio 54, where they met members of the band Maze, Bobby Womack and Bob Marley, whom they also interviewed, claiming to be "NME" and "Melody Maker" journalists.


title: Paul Oakenfold

text: Returning to London, Oakenfold began breaking into the mainstream, as an A&R man for Champion Records. At that time, he signed DJ Jazzy Jeff and the Fresh Prince, as well as Salt-n-Pepa. Oakenfold appeared on the "Blue Peter" BBC programme for children with a breakdancing crew. He became a promoter and British agent for the Beastie Boys and Run-D.M.C. Since then, he appeared at The Project in Streatham playing soul and jazz music.


title: Paul Oakenfold

text: In 1987, Oakenfold travelled to the island of Ibiza for a week to celebrate his birthday. Trevor Fung, Nicky Holloway, Ian Saint Paul, Danny Rampling and Johnny Walker accompanied him. Oakenfold convinced the owner of a venue in England to host an "Ibiza Reunion" party after-hours. He had previously made an attempt, but it failed as the crowd was not prepared for the acid house style until 1987 when the party was successful. After that, the night became a classic and became one of the UK's major acid house nights, known as "Spectrum" at Heaven in Charing Cross. The party was best known for the "Theatre of Madness", as more than 1,500 people were present on Monday nights, until it went down; with the financial issues it changed its name to the "Land of Oz". Artists like Alex Paterson DJ'd in the VIP chillout area known as the "White Room", which gave Oakenfold more free time, and then he began producing music under the alias "Electra" in 1988. Members included Nick Divaris, John "Johnny" Rocca and Micky. As they continued releasing only four singles as the Balearic beat band Electra, in Full Frequency Range Recordings (FFRR Records) founded and run by Radio 1's Pete Tong, the duo created a new alias under the name Perfecto. Also in 1988 he decided to create a place where new artists could develop their careers. At that moment, Perfecto Records was born.


title: Paul Oakenfold

text: He collaborated with his friend Steve Osborne on various projects. In 1990, he worked with Terry Farley, Andrew Weatherall and Osborne on two remixes for Happy Mondays. The remixes of "Rave On" and "Hallelujah" were released on the "Madchester Rave On" EP, as well as "Step On", a cover version adapted from John Kongos' 1971 hit "He's Gonna Step on You". The song reached the top 5 position in the UK. He was invited as a guest DJ to Spike Island, a gig with the Stone Roses. Pleased with the last single, the Happy Mondays gave Oakenfold and Osborne the opportunity to produce their third studio album, "Pills 'n' Thrills and Bellyaches". The album entered the UK charts at number one with pre-sales of 150,000. The album was named "NME"s 1990 Album of the Year, and both Oakenfold and Osborne won the 1991 Brit Award for Best Producer.


title: Paul Oakenfold

text: In 1992, when U2 released their song "Even Better Than the Real Thing", the Perfecto remix reached a higher charted position than the original song. In 1993 with the success of his last remix as Perfecto, he was hired by U2 to provide the warm-up sonics to their Zoo TV world tour, and replaced BP Fallon in the 1993 legs of Europe and Australia, New Zealand, and Japan, with more than fifty shows in Zooropa '93 and Zoomerang from 7 May to 10 December of the same year.


title: Paul Oakenfold

text: Also in 1993, Oakenfold and Osborne's project Grace was formed as State of Grace and featuring vocals by Patti Low, though by the time Grace had charted a number of singles in the UK charts (around 1996), it had become a 'solo fronted project' with jazz singer Dominique Atkins as the lead vocalist. The project was dissolved in 1997 as Oakenfold was touring as a performance DJ more frequently and could not commit to recording, though a remixed version of "Not Over Yet" was issued under the 'Planet Perfecto featuring Grace' name in 1999.


title: Paul Oakenfold

text: In September 1994 and again in 1998, he teamed up with Steve Osborne and Ben Hillier to remix the Rolling Stones song "You Got Me Rocking" and Duran Duran's "Out of My Mind". He began producing his own tracks as well, continuing to remix songs from popular artists. He began using Goa music, something he discovered on the beaches of Goa in India, fusing it with similar-sounding European tracks to create his own distinct sound. He took this to the mainstream in 1994 and created a pair of two-hour sets for BBC Radio 1's "Essential Mix"; the first of these was broadcast in the early hours of Sunday 20 March 1994, with the second being "The Goa Mix" broadcast on Sunday 18 December 1994. His album "Perfecto Fluoro" became the No. 1 essential dance collection of Boston Beat during 1996 with Jamiroquai's "Travelling Without Moving".


title: Paul Oakenfold

text: On 9 June 1997, Oakenfold created "Global Underground 004: Paul Oakenfold, Live in Oslo" (GU004) which is a double mix CD in the "Global Underground" series. Compiled and mixed by Oakenfold, it is the first work he created for GU. The mix was recorded live at Cosmopolite Club in Oslo, Norway, as part of the official launch of the Quart Festival. It showcased Oakenfold's eclectic taste in music at the time, as the mix combines various forms of dance music. In 1997, Oakenfold mixed one disc of the double album "Fantazia Presents the House Collection 6", a UK house music compilation series. Oakenfold became Cream's resident DJ from 1997–1999. During this time, he began to concentrate on the release of "Tranceport" in 1998.


title: Paul Oakenfold

text: In 1998 and 1999, Oakenfold took the first place in "DJ Mag"s Top 100 DJs. With the two-year contract as a resident in Liverpool's Cream, it was in 1999 that he released "Resident. Two Years of Oakenfold at Cream." on Virgin. Thrive Records, the US distributor for early Global Underground releases had a different numbering scheme for the "Global Underground" albums; due to this "Global Underground 007: Paul Oakenfold, New York" (GU007) was released as (GU002) in the United States only. The compilation was released on 25 May 1998, with the US release on 19 January 1999. This was the second production from Oakenfold with GU and it contained trance, drum and bass, progressive house, progressive trance, breakbeat and downtempo. This became his last work with GU.


title: Paul Oakenfold

text: In 1999, he became the first DJ to play on the main stage at the Glastonbury Festival for 90,000 people, which he considers his favorite gig. Also in that year, he moved to the United States where he went on tour.


title: Paul Oakenfold

text: In 2000, he created fourteen tracks of jazz, soul, house and Goa based styles with Mitchell Oakenfold. Twenty-four FX and scratches loops and sounds were included too, each consisting of six seconds; the album cover says "Only for DJs and Producers" and was released on Music of Life.


title: Paul Oakenfold

text: In March 2000, he teamed up with Steve Osborne, Andy Gray and Bruno Ellingham to remix Moby's song "Natural Blues".


title: Paul Oakenfold

text: Sometime before July 2000, he teamed up with Andy Gray to write and produce the theme for the Channel 4 reality show "Big Brother" under the name Elementfour. The series started airing on 18 July 2000, with the theme later released as a single. The programme and theme moved over to Channel 5 from 18 August 2011 to 5 November 2018.


title: Paul Oakenfold

text: In September 2000, he opened the new Digital Radio station Ministry of Sound Radio with a live mix from the famous London club.


title: Paul Oakenfold

text: Oakenfold appeared in the intro scene of EA's "Euro 2000" video game, which featured him using his turntables to activate the video game and control various players. He also composed the game's soundtrack, which featured 7 tracks including a remix of the official anthem of the tournament.


title: Paul Oakenfold

text: After his success in Europe, one of Oakenfold's first major events in America was Fresno, California's Cyberfest on 22 July 2000. The 500 acre indoor and outdoor central California location had the biggest dance capacity ever in America. An estimated 80,000 dancers and music lovers from Seattle, Reno, Phoenix, Los Angeles, and San Diego were welcomed that day. Cyberfest 2000, also known as the "Festival Of The Future" featured other DJs such as Chemical Brothers, BT, and Carl Cox. Cyberfest 2000 paved the way for other major events in the area such as the Electric Daisy Carnival in Los Angeles in 2010.


title: Paul Oakenfold

text: In 2001, Oakenfold took part in the first Area Festival tour. This tour featured Incubus, Carl Cox, the Orb, OutKast, and the Roots. He later released a new compilation album, "Perfecto Presents: Ibiza".


title: Paul Oakenfold

text: 2001 also saw the release of the video game "Frequency", for which Oakenfold produced one track, "See It".


title: Paul Oakenfold

text: Global Underground sold over 150,000 copies of Oakenfold's previous "Global Underground: New York". A spokesperson for the label claimed that in the US, demand for UK dance music had been increasing in the past couple of years, and now made up over two-thirds of the label's sales. The Global Underground New York office opened in on Ninth Avenue in Manhattan. The Mekka Electronic Music Festival, otherwise known as the "electronic Lollapallooza" took place in ten cities in the United States and Canada during August and September, including New York City, Los Angeles, Toronto and San Francisco. The event featured Paul, Armand van Helden, De La Soul, LTJ Bukem, Josh Wink, Derrick Carter, Roni Size, Deep Dish, BT, The Crystal Method, Carl Craig and Überzone. Oakenfold next moved to Los Angeles to work on film soundtracks and to focus his DJing stateside. In 2001 he created the soundtrack for the film "Swordfish", "Swordfish: The Album" contained a transformation of "Planet Rock" into a seven-minute breakbeat trance anthem. Most of the tracks are collaborations with Andy Gray, the remix of N.E.R.D.'s "Lapdance" which gained total notability from other tracks. The soundtrack was produced under Village Roadshow and Warner Bros. and distributed through London-Sire Records.


title: Paul Oakenfold

text: Oakenfold recorded a track with Crazy Town vocalist Shifty Shellshock at the end of the year for his new album. In an interview with "Rolling Stone", Shellshock said that the track known as "Starry Eyed Surprise" was created after the pair met at a Crazy Town show.


title: Paul Oakenfold

text: In 2002, "Bunkka" became his first official studio album when he signed to Maverick. The name "Bunkka" came from Peter Gabriel's studio in the UK, where the album was recorded. An extended play was released featuring live versions of four songs under Peoplesound Records. It is also Oakenfold's best selling album to date, with sales largely exceeding 1,000,000 (1 million) copies worldwide.


title: Paul Oakenfold

text: The album features vocals from Jane's Addiction vocalist Perry Farrell on "Time of Your Life" and Shifty Shellshock of Los Angeles rock and rap band Crazy Town on "Starry Eyed Surprise"; Ice Cube on "Get Em Up"; and Tricky and Nelly Furtado on "The Harder They Come". The album contains appearances by Asher D of So Solid Crew on "Ready Steady Go", and Grant-Lee Philips, founder of the '90s Los Angeles rock band Grant Lee Buffalo is also included with Carmen Rizzo's version of his song "Motion". "Bunkka" also provided the start to three new artists, Carla Werner on the single "Southern Sun", Tiff Lacey on "Hypnotised" and Emiliana Torrini on "Hold Your Hand". Hunter S. Thompson's spoken words are provided on "Nixon's Spirit". Pakistani musician Nusrat Fateh Ali Khan's composition was adapted in an electronic version on the tracks "Zoo York". In 2002, "Q" magazine named Oakenfold in their list of the "50 Bands to See Before You Die". In 2002, Oakenfold revealed he had struggled with dyslexia as a child and announced his intention to help dyslexic children.


title: Paul Oakenfold

text: In 2002, Oakenfold remixed David Arnold's "James Bond Theme"; the song was released under Warner Bros. Records and was followed by the album's next two singles after "Starry Eyed Surprise", "Ready Steady Go" and "Southern Sun". "Southern Sun" with Carla Werner was first issued as a B-side of "Ready Steady Go" until it was included on Tiësto's "" compilation with his own remix of the song. "Southern Sun" became a hit as it was then released as the A-side of "Ready Steady Go" in mid-2002. "Ready Steady Go" was featured in Saab commercials, the 2003 EA Sports game "Tiger Woods PGA Tour", the THQ game "Juiced", the pilot for the television program "Las Vegas", and as the NASCAR theme song for 2006. It appears in the video game series "Dance Dance Revolution" in "DDR Ultramix" for the Xbox, the "Fastlane" episode "Dogtown" and the "Alias" episode "Snowman". "Ready Steady Go" was also listed as the number one song that makes you drive faster by UpVenue. Also in 2003, he remixed another hit soundtrack for 'Elvis Presley' which charted fairly well after his 2003 remix of "Rubberneckin'" made the top three in Australia and top five in the UK. This remix contributed to the Elvis phenomenon of the 21st century. In an earlier episode of "Radio Free Roscoe", "The Imposter", a 2005 series on The N, a character Travis Strong DJed to the song, acting as if it were his own. It has also been used in the of Anthony Horowitz's novel "Stormbreaker", "The Bourne Identity" and it was reproduced with Korean lyrics for the film "Collateral". The song has also been used during the performances of extreme freestyle water ski jumpers, and later became the theme song for the TV show "TRL Italy" from 2003 till 2005. The EP "The Harder They Come" was released on Perfecto and featured other works from Oakenfold and other artists. At the Creamfields event in 2002 at Speke Airfield, DJs such as Oakenfold, Seb Fontaine and Paul van Dyk performed along with Dave Clarke, Jon Carter, Richie Hawtin and Felix Da Housecat. The event also featured live appearances from Basement Jaxx, All Saints, Death in Vegas and Moloko. In 2003, Oakenfold released the fourth single of his album, "Hypnotised" which became successful enough to be included on his next compilation from the "Perfecto Presents..." series, "Perfecto Presents: Great Wall" which also included the Deepsky remix of the song as well as tracks like Motorcycle's "As the Rush Comes", Björk's "Pagan Poetry", UNKLE's remix of Ian Brown's "F.E.A.R." and Oakenfold's remix of Madonna's song "Hollywood". In 2003, Oakenfold produced a remix of "Pourvu qu'elles soient douces", a 1988 hit by French singer Mylène Farmer.


title: Paul Oakenfold

text: With the event of Creamfields that took place in 2004, Oakenfold released a compilation of songs he played during the event as well as tracks influenced by the environment and the vibe of deejays such as Paul van Dyk, Armin van Buuren, Ferry Corsten, Judge Jules, Fergie, Tall Paul, Eddie Halliwell, Chris Lawrence, Adam Sheridan, Shan, and Alex Kidd at the Cream/Goodgreef and "Mixmag" Arena.


title: Paul Oakenfold

text: In 2005, Oakenfold was contacted by the car manufacturer Toyota to create a free promotional CD available from aygo.com to promote a new Toyota car. The CD contained only seven songs which he worked on with Ian Green; the album was entitled "Feed Your Mind".


title: Paul Oakenfold

text: During Oakenfold's career he has remixed a variety of songs from Madonna, like "What It Feels Like for a Girl", "Hollywood", "American Life", "Sorry" and later in 2008 "Give It 2 Me" from her album "Hard Candy", Oakenfold went on tour with Madonna for two months opening her presentation in the Confessions Tour, previously he had supported her in 2004 at Slane Castle in Ireland. His sets lasted for an hour and a half, followed by Madonna's two-hour show.


title: Paul Oakenfold

text: Oakenfold remixed the "Transformers" theme as the theme song for the TV series, "Transformers Cybertron". He also contributed with his single "Beautiful Goal" for the FIFA Football 2005 video game. His single "Ready Steady Go" was composed for the 2005 video game "Juiced". His second studio album, "A Lively Mind" was released on 6 June 2006. Receiving unsuccessful reviews, the first single "Faster Kill Pussycat", a collaboration with the actress Brittany Murphy, was released on 2 May 2006; the second single was "Sex 'n' Money". Both songs stand out from the rest, as most of the album has a more trance-like feel. Gregory Jeffries from AllMusic stated the album might have been in the nominated albums of dance music in 1997 but not in 2006, as the album has guitars with disco sounds that might be only appealing to trance addicts.


title: Paul Oakenfold

text: In 2007, he played live at the Boston Pops which created a piece of orchestral music with electronic music. The event took place in Miami for 10,000 people with a 75-piece orchestra, he wrote a piece of music which he described as "difficult". In 2007 he was nominated to 2 International Dance Music Awards (IDMA) at the Winter Music Conference (WMC), "Best Underground Dance Track" for "Faster Kill Pussycat" and "Best Full Length DJ Mix CD" for "A Lively Mind". 2007 saw the publication of the first official biography of Paul Oakenfold, written by Richard Norris of The Grid and Beyond the Wizard's Sleeve fame. "Paul Oakenfold: The Authorised Biography" was published by Bantam Press on 24 September 2007. Oakenfold is a self-professed supporter of association football clubs Chelsea F.C. and Los Angeles FC. It was thought that he played a zombie in the movie "28 Weeks Later" however this is incorrect, he was offered the opportunity but turned it down. He also scored the soundtrack for the 2007 Japanese CGI anime film "Vexille".


title: Paul Oakenfold

text: In October 2007 he released his "Greatest Hits & Remixes, Vol. 1" which features his best performance tracks. Oakenfold was a judge for the sixth and seventh annual Independent Music Awards. In October 2007 Oakenfold remixed Britney Spears' number one hit "Gimme More", which became the lead remix off the release. Oakenfold has created more than one hundred remixes, and has sold over 5,000,000 (5 million) long plays. He toured in British universities to promote his new album and autobiography. The tribute album was released in November in the United Kingdom with a 2-CD set and a 3 CD version with the same number of songs. It was also released in the United States with only 20 tracks in one CD; it featured some remixes from the original version but it also included two new remixes which are: Justin Timberlake's "My Love" song and his remix of Hans Zimmer's "" which was used for the film "". Exclusive 2008 remixes from Oakenfold were also included from Mark Ronson's "Stop Me" song, Radiohead's "Everything in Its Right Place", Eisbaer from Grovezone, "Missing" by Everything But The Girl and a remix of Paul's own "Southern Sun". Releases with Catalog#: "UL 1602-2" included a DVD of live show and documentary. The compilation consists of tracks which influenced his career and musical persona, with artists such as The Cure, Happy Mondays, The Stone Roses, Massive Attack, U2, Olive, The Smashing Pumpkins, Madonna, and Underworld.


title: Paul Oakenfold

text: In 2008, he released the last single from his last studio album, "Not Over". This was a new version "Not Over Yet" which he produced while working with Osborne as Grace and in collaboration with Ryan Tedder from OneRepublic. The song was covered by Klaxons as "It's Not Over Yet". In 2008, Oakenfold worked on film scores for various films and television programs such as "Californication", "The Heavy", "Fever", "The Bourne Identity", and "Speed Racer". In mid-July Oakenfold ended his world tour promoting his greatest hits album. He also joined Madonna again in her Sticky & Sweet Tour in London, Santiago, Buenos Aires, Rio de Janeiro and São Paulo. After touring, he began his residency in Las Vegas on 30 August. In the fall of 2008, he started his first Resident DJ position in the United States. "Paul Oakenfold Presents: Perfecto Las Vegas" was conceptualized specifically for Rain Nightclub, the legendary nightclub and concert venue at the Palms Casino Resort known for its special effects and international headliner acts. From July to August 2009, he again opened Madonna's concerts for the Sticky & Sweet Tour in Europe. In 2009 Paul's third studio album was announced to be actually named "Pop Killer", and not "Decade of Dance" as the article from "The Sun" stated.


title: Paul Oakenfold

text: He also worked with Madonna on her third greatest hits collection, "Celebration", and her final release with Warner Bros. Records. The first single was released on 3 August 2009, it is also titled "Celebration" and was produced by Oakenfold. He also remixed a past demo "It's So Cool" from "American Life", and it is included as a bonus track on iTunes.


title: Paul Oakenfold

text: Oakenfold remixed a song called "Firefly" from the Matt Goss album "Gossy". In 2010, Oakenfold returned to the US as a DJ at Rain Nightclub.


title: Paul Oakenfold

text: In February 2012 Oakenfold headlined a huge one-off event for club brand Goodgreef at the famous Tall Trees venue in the UK. Later in the year, he performed alongside Californian band Train at the Google I/O June 2012 After Party at Moscone Center in San Francisco.


title: Paul Oakenfold

text: Oakenfold co-wrote a song for Cher's twenty-fifth studio album "Closer to the Truth": "Woman's World". It was released on 18 June 2013 as the album's lead-single.


title: Paul Oakenfold

text: Oakenfold remixed a song by British pop singer Sophie Ellis-Bextor and up-and-coming new electronic project DedRekoning on their single "Only Child", released on 8 September 2014 through Oakenfold's label Perfecto Records.


title: Paul Oakenfold

text: Since May 2015, Paul is heading the department of DJs and EDM vocalists at Isina, a worldwide talent search and development mentorship program. On September 25, 2020, Oakenfold launched the English-language electronic single, "The Perfect Song" featuring Mexican pop icon Fey, to celebrate the 25th anniversary of her career. His album "Shine On" is set to be released in 2022.


title: Paul Oakenfold

text: Oakenfold is the opening act for the Pet Shop Boys and New Order Unity Tour in September and October 2022.


title: Paul Oakenfold

text: The BMI Awards are annual award ceremonies for songwriters in various genres organized by Broadcast Music, Inc.


title: Paul Oakenfold

text: The DJ Awards organizes the annual electronic music DJ awards event it is the only international ceremony for DJs and also the oldest, the awards are held once a year at Pacha club in Ibiza Spain it is one of the most important accolades an artist can win or be honoured by.


title: Paul Oakenfold

text: Oakenfold has won the "Best Trance DJ Award" two times, and received eleven nominations. He won the "Best Techno DJ Award" one time, and he honored with an "Outstanding Contribution Award" in 2004.


title: Paul Oakenfold

text: Artists are nominated to the DJ Magazine, The Top 100 list each year the public votes to decide who they rank as the World's No 1 DJ at the end of the poll. Oakenfold has been named Worlds Best DJ two times.


title: Paul Oakenfold

text: The Grammy Award (originally called Gramophone Award), or Grammy, is an honor awarded by The Recording Academy to recognize outstanding achievement in the mainly English-language music industry. The annual presentation ceremony features performances by prominent artists, and the presentation of those awards that have a more popular interest. It shares recognition of the music industry as that of the other as performance awards such as the Emmy Awards (television), the Tony Awards (stage performance), and the Academy Awards (motion pictures)


title: Paul Oakenfold

text: The IDMA is a major event which is part of the Winter Music Conference (WMC) a weeklong electronic music conference, held every March since 1985 in Miami Beach, Florida, United States.


title: Paul Oakenfold

text: The World Music Awards is an international awards show founded in 1989 under the high patronage of Albert II, Prince of Monaco and is based in Monte-Carlo. Awards are presented to the world's best-selling artists in the various categories and to the best-selling artists from each major territory.


title: Bob Stoops

text: Robert Anthony Stoops (born September 9, 1960) is an American football coach. He was the head football coach at the University of Oklahoma from 1999 through the 2016 season, and on an interim basis during the 2021 Alamo Bowl. He led the Oklahoma Sooners to a record of 191–48 over his career. His 2000 Oklahoma Sooners football team won the 2001 Orange Bowl, which served as the BCS National Championship Game, and earned a consensus national championship. Since 2020, Stoops has been a head coach with the XFL, coaching the Arlington Renegades in 2020 and has been re-signed for 2023.


title: Bob Stoops

text: Stoops played college football at the University of Iowa as a defensive back from 1979 to 1982. Prior to his tenure at Oklahoma, he held various assistant coaching positions at the University of Iowa, Kent State University, Kansas State University, and the University of Florida. Stoops was awarded the Paul "Bear" Bryant Award in 2000 and the Walter Camp Coach of the Year Award in both 2000 and 2003. Stoops has been nicknamed "Big Game Bob" by both supporters and detractors.


title: Bob Stoops

text: Stoops is one of six children born to Ron Sr. and Evelyn "Dee Dee" Stoops in Youngstown, Ohio. He is a 1978 graduate of Cardinal Mooney High School, where his father was the long-time defensive coordinator of the football team. Bob and his three brothers (Ron Jr., Mike, and Mark) were all coached by Ron Sr. at Mooney. During a game in 1988 against the team coached by Ron Jr., Ron Sr. began experiencing chest pains. He was placed in an ambulance following the game and died en route to the hospital.


title: Bob Stoops

text: While at Iowa, Stoops was a four-year starter, and one-time All-Big Ten selection at defensive back at the University of Iowa. He was named Team MVP in 1982.


title: Bob Stoops

text: After graduating with a marketing degree in 1983, Stoops began his coaching career as a volunteer coach and graduate assistant in the Iowa Hawkeyes program under Hayden Fry. He next was an assistant at Kent State University under Dick Crum in 1988, and then joined the coaching staff at Kansas State University the following year. Stoops was named co-defensive coordinator at Kansas State under Bill Snyder in 1991 and assistant head coach and co-defensive coordinator in 1995. During his tenure on the Wildcats staff, Stoops played a key role in their impressive turnaround, helping take what many considered to be the worst program in Division I-A to national contention. During his final four seasons there, Kansas State was 35–12 with three bowl appearances.


title: Bob Stoops

text: Stoops then left for the University of Florida, serving three years as Steve Spurrier's defensive coordinator. Hired after Florida allowed 62 points to Nebraska in the 1996 Fiesta Bowl, he was granted authority over the Gators defense; the team responded by winning the national championship over Florida State in the 1997 Sugar Bowl.


title: Bob Stoops

text: Due to his success leading defenses at Kansas State and Florida, Stoops was mentioned as a future head coach.


title: Bob Stoops

text: The University of Oklahoma named Stoops its head coach in 1999. OU won seven games in Stoops' first year, taking the Sooners to their first bowl game since the 1994 season.


title: Bob Stoops

text: In his 18 years as head coach of the Sooners, Stoops had a combined record of 190–48 (). On November 16, 2013, Stoops notched his 157th win as Oklahoma's head coach with a victory over Iowa State, tying him with Barry Switzer for the most wins in Sooners history. A week later, on November 23, 2013, he surpassed Switzer's record with a 41–31 victory over Kansas State. Stoops accumulated a home winning streak of 39 consecutive games from 2005 to 2011. The streak was ended on October 22, 2011 when Texas Tech defeated Oklahoma 41–38. He also had the most wins of the decade of any BCS school with 110 (2000–2009). Along with Switzer, Bud Wilkinson and Bennie Owen, he is one of four coaches to win over 100 games at the University of Oklahoma; no other college football program has had more than three coaches accomplish such a feat. Overall, Oklahoma was 4–6 in BCS games and 9–9 in bowl games under Stoops.


title: Bob Stoops

text: Stoops led the Sooners to the 2000 BCS National Championship and finished the season undefeated, outscoring thirteen opponents by a combined 481–194. His Oklahoma teams again earned the opportunity to play in the BCS National Championship Game in 2004, 2005 and 2009, losing to LSU, 21–14, in the 2004 Sugar Bowl, and to USC, 55–19 in the 2005 Orange Bowl, and Florida, 24–14, in the 2009 BCS National Championship Game. Under Stoops, Oklahoma had four BCS National Championship Game appearances, a record shared with Florida State.


title: Bob Stoops

text: Stoops' teams finished the season ranked in the Top 10 of the polls for 11 of his 18 seasons, seven times finishing in the top five.


title: Bob Stoops

text: Stoops led his team to bowl games in each of his 17 years at Oklahoma, ten of which were Bowl Championship Series (BCS) bowls, including the Big 12 Conference's first Rose Bowl victory as the Sooners upended Washington State, 34–14, in the 2003 Rose Bowl. With Oklahoma's victory over Alabama at the 2014 Sugar Bowl, Stoops became the first and only coach to win all four BCS bowl games and a BCS National Championship.


title: Bob Stoops

text: Stoops' penchant for winning big games early in his career earned him the nickname "Big Game Bob", From 1999 to 2003, Oklahoma under Stoops was 18–2 (0.900) vs. ranked opponents and 3–1 (0.750) in bowl games, with one national title and three Big 12 titles.


title: Bob Stoops

text: Late in the 2003 season, however, Bob's brother Mike Stoops left his position of defensive coordinator and associate head coach at Oklahoma to accept the head coaching job at Arizona. The Sooners promptly lost two games in a row against ranked teams after Mike's departure that season (a 35–7 loss against #13 Kansas State in the Big 12 Title Game, and a 21–14 loss to #3 LSU in the BCS National Title Game). Since then (2004–2016), Stoops' teams went 17–13 vs. ranked opponents, and 3–4 in Bowl Games with no National Titles (although they played for 3 more), and five Big 12 Titles. Stoops' teams did finish with two Heisman Trophy winners during this time, however, and two runners-up.


title: Bob Stoops

text: Under Stoops, the Sooners won ten Big 12 Conference championships, the most of any Big 12 team. Oklahoma is also the only Big 12 team to win back-to-back-to-back Big 12 championships. Stoops posted a 121-29 (.807) conference record during his career, and was named the Big 12 Coach of the Year six times.


title: Bob Stoops

text: In his 18 seasons as Sooners' head coach, Stoops was 11–7 against the Texas Longhorns in the Red River Rivalry. During a five-game winning streak in that rivalry from 2000 to 2004, his Sooners handed the Longhorns two of their worst defeats in school history, 63–14 in 2000 and 65–13 in 2003. Since then, he led the team to additional large-margin wins of 55–17 in 2011 and 63–21 in 2012.


title: Bob Stoops

text: On July 11, 2007, Oklahoma was placed on probation for two years by the NCAA for a rules violation involving quarterback Rhett Bomar and offensive lineman JD Quinn, whom the university had previously suspended from the team due to the players' efforts to obtain payment for hours not worked. While the students who violated the rules were reinstated by the NCAA and allowed to play for other schools, the university, which had self-reported the violations, was initially directed to vacate all wins during the 2005–06 season, which included a 17–14 win over the University of Oregon in the Holiday Bowl. Oklahoma appealed the NCAA's ruling of a "failure to monitor" the employment of players in the program, as well as a reduction in scholarships and probation lasting until May, 2010. On February 22, 2008, the NCAA reversed part of the decision and reinstated the vacated wins.


title: Bob Stoops

text: Stoops' 2008 team went down in the history books as the highest scoring team in college football history, scoring a total of 716 points, averaging 51 points per game. Over five consecutive games, the Sooners scored 60 points or more, another record (in the game prior to the streak, the Sooners scored 58 points). After a four-week layoff, the offense was held in check against the nation's best defense of the Florida Gators in the National Championship game, scoring only 14 points and committing two turnovers near the Florida goal line. The Sooners were without one of their star offensive playmakers in DeMarco Murray, who sat out with an injury. However, the Sooners' 2008 defense, which was much maligned during the season for allowing a Stoops'-worst 25 points per game average, held the Florida Gators' high-powered Tim Tebow-led offense to only 24 points, 21 points below their season average.


title: Bob Stoops

text: Stoops' success at Oklahoma made him the frequent subject of head coach searches by NFL teams and college programs alike, which he repeatedly turned down. He was reportedly the top-paid coach in Division I-A football with annual compensation in excess of $3 million until Nick Saban was signed by the University of Alabama for $4 million per year in 2007. However, Stoops did receive a "longevity bonus" of $3 million at the end of the 2008 season (his 10th), making his annual salary in 2008 approximately $6.1 million (equivalent to $ million in ) .


title: Bob Stoops

text: In the 2012 season, he led the Sooners back to the top 25 and went to the Cotton Bowl, losing to Heisman Trophy-winning Johnny Manziel and the Texas A&M Aggies, 41–13. In his career at Oklahoma, several of Stoops' assistants became head coaches at other Division I-A programs, including his brother Mike Stoops (Arizona), Mark Mangino (Kansas), Mike Leach (Texas Tech, Washington State and Mississippi State), Chuck Long (San Diego State), Bo Pelini (Nebraska and Youngstown State), Kevin Sumlin (Texas A&M), Kevin Wilson (Indiana), and his eventual successor, Lincoln Riley (USC, Oklahoma).


title: Bob Stoops

text: Stoops is the only head coach in the BCS era to have won the Rose Bowl, Orange Bowl, Fiesta Bowl, and Sugar Bowl.


title: Bob Stoops

text: On June 7, 2017, Stoops announced his retirement from college coaching. He was succeeded by Lincoln Riley as Oklahoma's head football coach.


title: Bob Stoops

text: On February 7, 2019, Stoops announced his plans to come out of retirement, as he was named head coach/general manager of the upcoming Dallas Renegades in the XFL. He served in this role until the league ceased operations on April 14, 2020 due to bankruptcy stemming from the COVID-19 outbreak. On April 13, 2022, Stoops re-signed with the XFL.


title: Bob Stoops

text: On November 28, 2021, Stoops was named interim head coach for Oklahoma's bowl game after Lincoln Riley took the USC Trojans head coaching job. The head coaching position was filled by former Stoops assistant Brent Venables. Stoops coached the Sooners in the 2021 Alamo Bowl; the Sooners won 47–32.


title: Bob Stoops

text: Stoops is married to Carol Stoops, a Mary Kay National Sales Director. They have three children: a daughter, Mackenzie, who now attends the University of Oklahoma, and twin sons, Isaac and Drake. Drake Stoops plays wide receiver for the Oklahoma Sooners. Isaac Stoops has coached at the high school level as a wide receivers coach for Moore (OK) High School and took a position as a volunteer analyst with the University of Oklahoma Sooners for the 2021-22 football season.


title: Bob Stoops

text: Stoops' younger brother, Mike, is the former defensive coordinator for the Sooners and was previously head football coach at the University of Arizona. He currently serves as inside linebackers coach for the football program of University of Kentucky. Another brother (the youngest), Mark, became the head coach at the University of Kentucky in November 2012. Stoops' older brother, Ron Jr., is an assistant football coach at Youngstown State University in Youngstown, Ohio.


title: Breastaurant

text: A breastaurant is a restaurant that requires female waiting staff to be skimpily-dressed. The term dates from the early 1990s, after restaurant chain Hooters opened in the United States. The format has since been adopted by other restaurants, including Redneck Heaven, Tilted Kilt Pub & Eatery, Twin Peaks, Bombshells, Bone Daddy's, Ojos Locos, Chula's, Bikinis Sports Bar & Grill, Racks, Show-Me's, Mugs & Jugs, and The WingHouse Bar & Grill.


title: Breastaurant

text: These restaurants often use a sexual double-entendre brand name and may also have appropriate themes, both in decoration and menu. The restaurants may offer perks for customers, such as alcoholic drinks and flirty servers.


title: Breastaurant

text: Hooters is credited as the first breastaurant, having operated since 1983. Other companies soon adopted the format. According to food industry research firm Technomic, the top three breastaurant chains in the United States after Hooters each had sales growth of 30% or more in 2011.


title: Breastaurant

text: In October 2012, Bikinis Sports Bar & Grill successfully registered the term "breastaurant" as a trademark with the United States Patent and Trademark Office; but as of May 24, 2019, the trademark lapsed under section 8, "Continued use not filed within Grace Period". Bikinis Sports Bar & Grill had closed its last restaurant on December 23, 2018.


title: Breastaurant

text: Restaurants staffed by males, with a similar focus on server appearance, include Tallywackers, featuring scantily clad men, which opened in Dallas, Texas, in May 2015 and closed in August 2016. In Japan, there are establishments such as Macho Cafe and Macho Meat Shop, where brawny men serve food and drinks.


title: Nelson's Column

text: Nelson's Column is a monument in Trafalgar Square in the City of Westminster, Central London, built to commemorate Vice-Admiral Horatio Nelson's decisive victory at the Battle of Trafalgar over the combined French and Spanish navies, during which he lost his life. The monument was constructed between 1840 and 1843 to a design by William Railton at a cost of . It is a column of the Corinthian order built from Dartmoor granite. The statue of Nelson was carved from Craigleith sandstone by sculptor Edward Hodges Baily. The four bronze lions around its base, designed by Sir Edwin Landseer, were added in 1867.


title: Nelson's Column

text: The pedestal is decorated with four bronze relief panels, each square, cast from captured French guns. They depict the Battle of Cape St. Vincent, the Battle of the Nile, the Battle of Copenhagen and the death of Nelson at Trafalgar. The sculptors were Musgrave Watson, William F. Woodington, John Ternouth and John Edward Carew, respectively. The ornate capital upon which Nelson stands is by Charles Harriott Smith.


title: Nelson's Column

text: It was refurbished in 2006 at a cost of , at which time it was surveyed and found to be shorter than previously supposed. The whole monument is tall from the bottom of the pedestal to the top of Nelson's hat.


title: Nelson's Column

text: In February 1838, a group of 121 peers, Members of Parliament (MPs) and other gentry formed a committee to raise a monument to Lord Nelson, funded by public subscription, and the Government agreed to provide a site in Trafalgar Square, in front of the newly completed National Gallery. A competition was held for designs with an estimated budget of between £20,000 and £30,000. The deadline for submissions was 31 January 1839.


title: Nelson's Column

text: The winning entry, chosen by the sub-committee headed by the Duke of Wellington was a design by William Railton for a Corinthian column, surmounted by a statue of Nelson, and flanked by four sculpted lions. Flights of steps would lead up between the lions


title: Nelson's Column

text: to the pedestal of the column. Several other entrants also submitted schemes for columns. The second prize was won by Edward Hodges Baily who suggested an obelisk surrounded by sculptures.


title: Nelson's Column

text: Criticism of the organisation of the competition caused it to be re-run. Railton submitted a slightly revised design, and was once again declared the winner, with the stipulation that the statue of Nelson should be made by EH Baily. The original plan was for a column high, including the base and statue, but this was reduced to with a shaft of due to concerns over stability. The base was to have been of granite and the shaft of Craigleith sandstone, but before construction began it was decided that the shaft should also be of granite.


title: Nelson's Column

text: Excavations for the brick foundations had begun by July 1840. On 30 September 1840, the first stone of the column was laid by Charles Davison Scott, honorary secretary of the committee (and son of Nelson's secretary, John Scott), at a ceremony conducted, according to the "Nautical Magazine", "in a private manner, owing to the noblemen and gentlemen comprising the committee being absent from town". Construction of the monument, by the contractors Grissell and Peto, progressed slowly, and the stonework, ready for the installation of the statue, was not completed until November 1843.


title: Nelson's Column

text: In 1844, the Nelson Memorial Committee ran out of money, having only raised £20,485 in public subscriptions, and the Government, in the form of the Office of Woods and Forests took over the project.


title: Nelson's Column

text: Installation of the bronze reliefs on the pedestal did not begin until late 1849, when John Edward Carew's depiction of the death of Nelson was put in place on the side facing Whitehall. This was followed early the next year by William F. Woodington's relief of the Battle of the Nile on the opposite side. Carew's relief was cast by Adams, Christie and Co. of Rotherhithe. The other three were cast by Moore, Fressange and Moore. The last to be made, "The Battle of Cape St. Vincent" became the subject of legal action, when it was discovered that the bronze had been adulterated with iron. The partners in the company were jailed for fraud and the relief was completed by Robinson and Cottam. It was finally put in place in May 1854.


title: Nelson's Column

text: The statue at the top was sculpted by Edward Hodges Baily R.A. from three pieces of Craigleith sandstone donated by the Duke of Buccleuch, former chairman of the Nelson Memorial Committee, from his own quarries.


title: Nelson's Column

text: The statue stands on a fluted column built from solid blocks of granite from the Foggintor quarries on Dartmoor. The Corinthian capital is made of bronze elements, cast from cannon salvaged from the wreck of HMS "Royal George" at the Woolwich Arsenal foundry. It is based on the Temple of Mars Ultor in Rome, and was modelled by C.H. Smith.. The bronze pieces, some weighing as much as are fixed to the column by the means of three large belts of metal lying in grooves in the stone.


title: Nelson's Column

text: The four identical bronze Barbary lions at the column's base were not added until 1867. At one stage, they were intended to be of granite, and the sculptor John Graham Lough was chosen to carve them. However, in 1846, after consultations with Railton, he turned down the commission, unwilling to work under the restrictions imposed by the architect. The sculptures eventually installed, commissioned in 1858, were designed by Sir Edwin Landseer in collaboration with Baron Marochetti. Their design may have been influenced by Marschalko János's lions at each abutment to the Széchenyi Lánchíd (Chain Bridge) in Budapest, installed 6 years before the Trafalgar Square lions were commissioned. Landseer was paid £6,000 for his services, and Marochetti £11,000. In 2011, consultants for the Greater London Authority reported that tourists climbing onto the backs of the lions have caused considerable damage and recommended banning tourists from climbing them.


title: Nelson's Column

text: The column also had a symbolic importance to Adolf Hitler. If Hitler's plan to invade Britain, Operation Sea Lion, had been successful, he planned to move it to Berlin.


title: Nelson's Column

text: The column was refurbished in 2006, during which time it was scaffolded from top to bottom for access. Steam cleaning was used together with gentle abrasives to minimise any harmful impact on the bronze and stonework. The £420,000 cost was covered by Zurich Financial Services, which advertised on the scaffolding for the duration of the work. Before restoration began, laser surveys were taken during which it was found that the column was significantly shorter than the usually quoted . In fact, it measures from the bottom of the first step to the tip of the admiral's hat.


title: Nelson's Column

text: John Noakes of the BBC TV children's programme "Blue Peter" climbed the column in 1977. Television presenter and entertainer Gary Wilmot climbed the column in 1989 for LWT's "Six O' Clock Show" to recreate the 'topping out' ceremony of 1843. Dressed in Victorian attire and sporting a boater hat, Wilmott enjoyed tea and sandwiches at the top of the column before climbing down.


title: Nelson's Column

text: The column has also been climbed on several occasions as a publicity stunt to draw attention to social or political causes. Ed Drummond made the first such climb in 1978 for the Anti-Apartheid Movement, making use of the lightning conductor "en route". On 30 March 1988, Joe Simpson and John Stevenson climbed the column as part of a Greenpeace Campaign against Acid Rain. On 14 June 1992, it was climbed by Martin Cotterrel, Joe Simpson and John Stevenson on behalf of Greenpeace to protest against the first Earth Summit meeting in Brazil. On 13 April 1995, Simon Nadin free-climbed Nelson's Column with Noel Craine, Jerry Moffatt and Johnny Dawes following on top rope, and graded the climb as "E6 6b/5a". This protest time was on behalf of Survival International to publicize the plight of Canadian Inuit. On 13 May 1998, the Column was climbed by Al Baker, Peter Morris and John Cunningham on behalf of Greenpeace to protest against Old growth logging activity in British Columbia. In May 2003, BASE jumper and stuntman Gary Connery parachuted from the top of the column, in a stunt designed to draw attention to the Chinese policies in Tibet.


title: Nelson's Column

text: In December 2015, Disney paid £24,000 to cover it in lights to make it resemble a giant lightsaber, to promote "".


title: Nelson's Column

text: On 18 April 2016, in the early hours of the morning, Greenpeace activists climbed up the column and placed a breathing mask on Admiral Lord Nelson in protest of dangerous air pollution levels.


title: Nelson's Column

text: The first civic monument to be erected in Nelson's honour was the Nelson Monument, a 44-metre high obelisk on Glasgow Green in Glasgow, Scotland, in 1806. Also in Scotland, the foundation stone for at Forres in Moray was laid in 1806 and it was completed in 1812; while the Nelson Monument stands on top of Calton Hill, Edinburgh. In Dublin, Ireland, Nelson's Pillar was erected in 1808 but was destroyed by republicans in 1966, and in the Bull Ring, Birmingham, England, there is a Grade II* listed bronze statue of Nelson by Richard Westmacott, dating from 1809. Sir Richard Westmacott also designed the elaborate monument to Nelson in Liverpool. In Portsmouth, Nelson's Needle, on top of Portsdown Hill, was paid for by the company of after arriving back in Portsmouth. There is a column topped with a decorative urn in the Castle Green, Hereford – a statue was planned in place of the urn, but insufficient money was raised. The Britannia Monument, Great Yarmouth, England (1819), is a 144-foot-high doric column design.


title: Nelson's Column

text: Elsewhere in the world, Nelson's Column in Montreal was erected by both Britons and Canadians in 1809, and there is also a Mount Nelson, near Invermere, British Columbia. As at London, the column in Montreal has the admiral standing with his back to the waves. A much shorter statue of Lord Nelson in Trafalgar Square, Bridgetown, Barbados, is older than its counterpart in London.


title: Mulholland Drive

text: Mulholland Drive is a street and road in the eastern Santa Monica Mountains of Southern California. It is named after pioneering Los Angeles civil engineer William Mulholland. The western rural portion in Los Angeles and Ventura Counties is named Mulholland Highway. The road is featured in a significant number of films, songs, and novels. David Lynch, who wrote and directed a film named after Mulholland Drive, has said that one can feel "the history of Hollywood" on it. Academy Award–winning actor Jack Nicholson has resided at Mulholland Drive for many years and still lives there today.


title: Mulholland Drive

text: The main portion of the road, from Cahuenga Pass in Hollywood westward past Sepulveda Pass, was originally called Mulholland Highway and was opened in 1924. It was built by a consortium of developers investing in the Hollywood Hills. DeWitt Reaburn, the construction engineer responsible for the project, said while it was being built, "The Mulholland Highway is destined to be one of the heaviest traveled and one of the best known scenic roads in the United States."


title: Mulholland Drive

text: The long mostly two-lane, minor arterial road loosely follows the ridgeline of the eastern Santa Monica Mountains and the Hollywood Hills, connecting two sections of U.S. Route 101, and crossing Sepulveda Boulevard, Beverly Glen Boulevard, Coldwater Canyon Avenue, Laurel Canyon Boulevard, Nichols Canyon Road, and Outpost Drive.


title: Mulholland Drive

text: The road offers outstanding views of the Los Angeles Basin, the San Fernando Valley, Downtown Los Angeles and the Hollywood Sign.


title: Mulholland Drive

text: Mulholland Drive has some of the most exclusive and expensive homes in the world, housing mainly Hollywood celebrities.


title: Mulholland Drive

text: The eastern terminus of Mulholland Drive is at its intersection with Cahuenga Boulevard at the Cahuenga Pass over the Santa Monica Mountains (at this point Cahuenga Boulevard runs parallel to Highway 101/The Hollywood Freeway). The road continues to the west, offering vistas of the Hollywood Sign, Downtown Los Angeles, and then Burbank, Universal City, and the rest of the San Fernando Valley with the San Gabriel, Verdugo, and Santa Susana Mountains.


title: Mulholland Drive

text: The road winds along the top of the mountains until a few miles west of the San Diego/Interstate 405 Freeway. Just west of the intersection with Encino Hills Drive, it becomes an unpaved road not open to motor vehicles. This part is known by many as "Dirt Mulholland". This portion connects with other unpaved roads and bike trails and allows access to a decommissioned Project Nike command post that is now a Cold War memorial park.


title: Mulholland Drive

text: The road opens again east of Topanga Canyon Boulevard (State Route 27) at Santa Maria Road but remains dirt until it reaches Saltillo Street. Shortly thereafter, the thoroughfare splits into Mulholland Drive and Mulholland Highway. Mulholland Drive terminates at U.S. Highway 101 (the Ventura Freeway), where it becomes Valley Circle Boulevard. Mulholland Highway continues to the southwest until it terminates at State Route 1 (PCH) in Leo Carrillo State Park at the Pacific Ocean coast and the border of Los Angeles and Ventura counties.


title: Yurizan Beltran

text: Yurizan Beltrán (November 2, 1986 – December 13, 2017) was an American pornographic actress, model, and mainstream actress. In addition to her successful pornographic and modeling career, Beltran also starred in the 2006 horror film "Werewolf in a Women's Prison" and later in 2017 the romantic comedy "White on Rice."


title: Yurizan Beltran

text: During her fairly successful pornographic career, Beltran received positive attention from her critics and peers. Over the course of her career, Beltran received multiple high industry award nominations from Adult Video News, NightMoves, XBIZ, and the X-Rated Critics Organization.


title: Yurizan Beltran

text: Beginning in 2003, Beltran began doing feature modeling and appearances in Southern California as a Hooters girl. In 2005, she began adult modeling when she decided to launch her first website, "SweetYurizan.com". In 2006, she appeared in her first mainsteam film a low budget horror film "Werewolf in a Women's Prison."


title: Yurizan Beltran

text: In 2005, Beltran was featured in her first adult film, a non-sex solo performance for JB Video titled "Yurizan." In 2006 and 2007, Beltran only modeled, she was featured by Hustler, Playboy, and Penthouse. She was also a covergirl for Lowrider, posing in a gold bikini with a Chevrolet convertible in the magazine's December issue. In 2009, she returned to porn only shooting lesbian, solo or non-sex films. She shot with New Sensations, Reality Kings, Digital Playground, and Twistys.com.


title: Yurizan Beltran

text: In the 2010 film "Big Tits Round Asses 20" by Bang Bros, Beltran made her boy/girl scene debut. In 2011, she was nominated for Best Web Star 28th AVN Awards. Also in 2011, Beltran began working with Brazzers she shot six films with them that year. In 2012, she continued her success and was nominated for Unsung Starlet of the Year at the 29th AVN Awards as well as Best Latina Performer from NightMoves Awards.


title: Yurizan Beltran

text: At the 30th AVN Awards, Beltran was nominated for Best All-Girl Group Sex Scene for her work in "Training Day: a XXX Parody" with her co-stars Chanel Preston and Nyomi Banxxx.


title: Yurizan Beltran

text: Beltran died of a drug overdose at her apartment in Bellflower, California on December 13, 2017. She was found by her landlord, lying in her bed, beside a spilled pill bottle. After about five months, the LA medical examiner revealed her official cause of death was bronchopneumoniainflammation of the lungscaused by an overdose of the prescription opioid hydrocodone.


title: Yurizan Beltran

text: The adult studio PornStar Platinum, which Beltran was a part of, set up a GoFundMe page for Beltran's family. The page closed after raising over $9,000 for funeral expenses.


title: Yurizan Beltran

text: There were a number of other high-profile deaths in the porn industry around the same time: Shyla Stylez died in November 2017; August Ames committed suicide eight days prior to Beltran's death; and in January 2018, Olivia Lua and Olivia Nova died. Their deaths generated discussion regarding whether the adult industry is doing enough for performers’ mental health and focused on the common knowledge that most adult stars are ostracized from much of polite society; while critics of the industry conversely argued how the tragically premature death comported with longstanding trends perhaps evincing, rather, that such negative outcomes could be interpreted as implicit to participation in the industry itself, and thus are not possible to remediate by actionable measures provided by said industry.


title: Blood Money (Breaking Bad)

text: "Blood Money" is the ninth episode of the fifth season of the American television drama series "Breaking Bad", and the 55th overall episode of the series. Written by Peter Gould and directed by Bryan Cranston, it aired on AMC in the United States and Canada on August 11, 2013, as the midseason premiere.


title: Blood Money (Breaking Bad)

text: The episode received high critical acclaim, with critics praising the flashforward scene in the beginning, the ending scene and the performances from Bryan Cranston and Dean Norris. For his performance in the episode, Norris was called "Performer of the Week" by TVLine.


title: Blood Money (Breaking Bad)

text: In a flashforward, a 52-year-old Walter White arrives at his abandoned, dilapidated, and fenced-off house. He enters and sees "HEISENBERG" spray-painted on the living room wall. Observing a group of teenagers using his empty pool to skateboard, he retrieves the hidden vial of ricin from his bedroom. As he leaves, he greets his former neighbor, Carol, who is terrified by his presence.


title: Blood Money (Breaking Bad)

text: In the present, Hank Schrader reels from finding Gale Boetticher's handwritten dedication in Walt's copy of "Leaves of Grass". Realizing that his brother-in-law was Heisenberg all along, Hank states that he is feeling unwell to excuse himself and his wife Marie from the party at Walt's house. While driving home, Hank suffers a panic attack and swerves off the road into a yard. Feigning illness to work from home, Hank reviews the DEA's case files on Heisenberg and Gus Fring, linking people, events, and circumstances, as well as matching the handwriting in the "Leaves of Grass" dedication with that in Gale's lab notebook, to confirm that Walt is Heisenberg.


title: Blood Money (Breaking Bad)

text: Walt, who has left the meth business, discusses with Skyler White ways to expand their car wash business and launder his drug money faster. Lydia shows up at the car wash looking distressed and pleads for Walt's help, saying the quality of the meth since his retirement has fallen below acceptable standards, jeopardizing their deal. Walt dismisses her, and Skyler firmly warns her never to come back. It is later revealed that Walt's cancer has returned, but he keeps this from his family and undergoes chemotherapy again.


title: Blood Money (Breaking Bad)

text: Meanwhile, Jesse Pinkman feels guilty over his role in Walt's meth business, and is particularly distraught over the deaths of Drew Sharp and (he assumes) Mike Ehrmantraut. He gives all the money he received from Walt to Saul Goodman and asks him to deliver half to Mike's granddaughter and the other half to Drew's parents. Saul refuses, advising it would raise suspicions, and reports this to Walt, who visits Jesse to return his money. Walt lies to Jesse, telling him that Mike is probably still alive and does not need help taking care of his granddaughter. Jesse is still distressed and later gives a $10,000 bundle to a homeless man. He then drives through a neighborhood, throwing bundles of cash onto front lawns.


title: Blood Money (Breaking Bad)

text: In his bathroom, Walt finds his copy of "Leaves of Grass" missing. Alarmed by the coincidental timing of Hank's apparent illness, his suspicions are deepened when he discovers a GPS tracker on his car that is identical to the one Hank used while attempting to track Gus Fring. He shows up at Hank's garage and eventually asks about the tracker; an enraged Hank punches Walt and accuses him of being Heisenberg, which Walt neither confirms nor denies. Walt tells Hank that he would have difficulty proving his allegations; in any case, Walt says his cancer has returned and would probably kill him before he could be jailed. Hank demands Walt leave his children in Hank and Marie's care before he will consider Walt's argument, but Walt refuses. Hank states that he does not know who Walt is anymore, to which Walt replies, "If that's true, if you don't know who I am, then maybe your best course would be to tread lightly."


title: Blood Money (Breaking Bad)

text: The episode is dedicated to Kevin Cordasco, a 16-year-old fan of "Breaking Bad" who had met several members of the series' cast and crew; Cordasco died earlier in 2013 from neuroblastoma. This was the fourth dedication over the course of the series.


title: Blood Money (Breaking Bad)

text: The term "blood money" has two meanings; it means money obtained at the cost of another's life, as well as money paid to the family of a person who has been killed, usually by the killer or the killer's clan.


title: Blood Money (Breaking Bad)

text: In the opening flashforward scene, Walt is seen by his former neighbor Carol, who drops a bag of oranges (which spill and roll across the ground) in fear. Some critics interpret this as a reference to "The Godfather", in which oranges represent death. For example, in one scene of "The Godfather", after Vito Corleone (Marlon Brando) is shot, he drops a bag of oranges, which subsequently roll across the ground.


title: Blood Money (Breaking Bad)

text: "Blood Money" contains several other continuity references to past episodes. Hank's montage, in which he investigates the revelation that Walt is Heisenberg, sees him looking through old case files and pictures. These reference Gus Fring, Gale Boetticher and his notebook ("Bullet Points"), Dennis Markowski, Jesse's car in Seasons 1 and 2, the bullet-ridden car from his shoot-out with Tuco Salamanca ("Grilled"), Combo's corpse ("Mandala"), Hector Salamanca, Mike Ehrmantraut, Duane Chow ("Full Measure"/"Madrigal"), Ron Forenall, Tyrus Kitt, Gonzo and No-Doze's corpses ("Grilled"), the gas mask Walt took from the school he worked at, Walt and Jesse stealing a barrel of methylamine ("A No-Rough-Stuff-Type Deal"), the burned lab ("Live Free or Die"), and finally, the Heisenberg drawing ("No Más"). When Hank confronts Walt about his identity, he mentions Walt's purposefully crashing the car on their way to Fring's laundromat ("Crawl Space"), killing the imprisoned members of Fring's drug empire ("Gliding Over All"), calling him about Marie to distract him from Jesse and the RV ("Sunset"), and bombing a nursing home ("Face Off").


title: Blood Money (Breaking Bad)

text: Walt also acts like Gus in some scenes. His conversation with Lydia at the car wash references Walt and Gus' earlier conversations at Los Pollos Hermanos, with Walt's role being reversed. In another scene, Walt goes to the bathroom to vomit, turning the sink on to mask the noise and laying a towel down on the floor to kneel on, just as Gus had done in "Salud". Hank's neighbor's son's remote-controlled car ("Seven Thirty-Seven") also makes an appearance. The GPS tracker Hank places under Walt's car is the same one he planted on Gus' car in the fourth season. When Jesse opens a pack of cigarettes in front of Huell, Huell seems to remember the ricin-filled cigarette he stole from Jesse in "End Times".


title: Blood Money (Breaking Bad)

text: The episode originally aired on August 11, 2013, in the United States and Canada on AMC. It aired the next day in the United Kingdom on the web streaming service Netflix. According to the Nielsen ratings system, "Blood Money" was watched in the United States by an estimated 5.92 million viewers, the most in series history at the time.


title: Blood Money (Breaking Bad)

text: The episode received widespread critical acclaim, with extensive praise for the performances of Dean Norris and Bryan Cranston. For his performance, Norris was named "Performer of the Week" by TVLine.


title: Blood Money (Breaking Bad)

text: Seth Amitin of "IGN" called the episode a satisfying preparation and set-up for the endgame of the series. Amitin also praised the confrontation at the end between Walt and Hank and the tense dialogue that fueled it. ""Blood Money" was an amalgamation of a bunch of little things to love."


title: Blood Money (Breaking Bad)

text: Mark Berman of "The Washington Post" said the episode paid off plot points set up in previous episodes while simultaneously laying the groundwork for future episodes. Berman was also surprised at how much ground was covered in Hank's plotline in just one episode.


title: Blood Money (Breaking Bad)

text: David Berry of "National Post" called Walt and Hank's showdown abrupt, menacing and cathartic without relieving any of the tension of the storyline. He also praised the acting.


title: Blood Money (Breaking Bad)

text: After reading other critics' reviews, Alex Fletcher of "Digital Spy" wrote that "['Blood Money'] exceeded the hype and reached new heights."


title: Blood Money (Breaking Bad)

text: In 2019, "The Ringer" ranked "Blood Money" as the 21st best out of the 62 total "Breaking Bad" episodes.


title: Cayley–Hamilton theorem

text: In linear algebra, the Cayley–Hamilton theorem (named after the mathematicians Arthur Cayley and William Rowan Hamilton) states that every square matrix over a commutative ring (such as the real or complex numbers or the integers) satisfies its own characteristic equation.


title: Cayley–Hamilton theorem

text: If is a given matrix and is the identity matrix, then the characteristic polynomial of is defined as formula_1, where is the determinant operation and is a variable for a scalar element of the base ring. Since the entries of the matrix formula_2 are (linear or constant) polynomials in , the determinant is also a degree- monic polynomial in , formula_3 One can create an analogous polynomial formula_4 in the matrix instead of the scalar variable , defined as formula_5 The Cayley–Hamilton theorem states that this polynomial expression is equal to the zero matrix, which is to say that formula_6. The theorem allows to be expressed as a linear combination of the lower matrix powers of . When the ring is a field, the Cayley–Hamilton theorem is equivalent to the statement that the minimal polynomial of a square matrix divides its characteristic polynomial.


title: Cayley–Hamilton theorem

text: The theorem was first proven in 1853 in terms of inverses of linear functions of quaternions, a "non-commutative" ring, by Hamilton. This corresponds to the special case of certain real or complex matrices. The theorem holds for general quaternionic matrices. Cayley in 1858 stated it for and smaller matrices, but only published a proof for the case. The general case was first proved by Ferdinand Frobenius in 1878.


title: Cayley–Hamilton theorem

text: For a general invertible matrix , i.e., one with nonzero determinant, can thus be written as an order polynomial expression in : As indicated, the Cayley–Hamilton theorem amounts to the identity


title: Cayley–Hamilton theorem

text: The coefficients are given by the elementary symmetric polynomials of the eigenvalues of . Using Newton identities, the elementary symmetric polynomials can in turn be expressed in terms of power sum symmetric polynomials of the eigenvalues:


title: Cayley–Hamilton theorem

text: In general, the formula for the coefficients is given in terms of complete exponential Bell polynomials as


title: Cayley–Hamilton theorem

text: In particular, the determinant of equals . Thus, the determinant can be written as the trace identity:


title: Cayley–Hamilton theorem

text: and, by multiplying both sides by (note ), one is led to an expression for the inverse of as a trace identity,


title: Cayley–Hamilton theorem

text: Another method for obtaining these coefficients for a general matrix, provided no root be zero, relies on the following alternative expression for the determinant,


title: Cayley–Hamilton theorem

text: where the exponential "only" needs be expanded to order , since is of order , the net negative powers of automatically vanishing by the C–H theorem. (Again, this requires a ring containing the rational numbers.) Differentiation of this expression with respect to allows one to express the coefficients of the characteristic polynomial for general as determinants of matrices,


title: Cayley–Hamilton theorem

text: It is apparent from the general formula for "c", expressed in terms of Bell polynomials, that the expressions


title: Cayley–Hamilton theorem

text: always give the coefficients of and of in the characteristic polynomial of any matrix, respectively. So, for a matrix , the statement of the Cayley–Hamilton theorem can also be written as


title: Cayley–Hamilton theorem

text: where the right-hand side designates a matrix with all entries reduced to zero. Likewise, this determinant in the case, is now


title: Cayley–Hamilton theorem

text: and so on for larger matrices. The increasingly complex expressions for the coefficients is deducible from Newton's identities or the Faddeev–LeVerrier algorithm.


title: Cayley–Hamilton theorem

text: The Cayley–Hamilton theorem always provides a relationship between the powers of (though not always the simplest one), which allows one to simplify expressions involving such powers, and evaluate them without having to compute the power or any higher powers of .


title: Cayley–Hamilton theorem

text: Notice that we have been able to write the matrix power as the sum of two terms. In fact, matrix power of any order can be written as a matrix polynomial of degree at most , where is the size of a square matrix. This is an instance where Cayley–Hamilton theorem can be used to express a matrix function, which we will discuss below systematically.


title: Cayley–Hamilton theorem

text: and the characteristic polynomial of degree of an matrix , the function can be expressed using long division as


title: Cayley–Hamilton theorem

text: Thus, the analytic function of the matrix can be expressed as a matrix polynomial of degree less than .


title: Cayley–Hamilton theorem

text: This amounts to a system of linear equations, which can be solved to determine the coefficients . Thus, one has


title: Cayley–Hamilton theorem

text: When the eigenvalues are repeated, that is for some , two or more equations are identical; and hence the linear equations cannot be solved uniquely. For such cases, for an eigenvalue with multiplicity , the first derivatives of vanish at the eigenvalue. This


title: Cayley–Hamilton theorem

text: Finding a polynomial that passes through the points is essentially an interpolation problem, and can be solved using Lagrange or Newton interpolation techniques, leading to Sylvester's formula.


title: Cayley–Hamilton theorem

text: The characteristic polynomial is , and the eigenvalues are . Let . Evaluating at the eigenvalues, one obtains two linear equations, and .


title: Cayley–Hamilton theorem

text: As before, evaluating the function at the eigenvalues gives us the linear equations and ; the solution of which gives, and . Thus, for this case,


title: Cayley–Hamilton theorem

text: Standard examples of such usage is the exponential map from the Lie algebra of a matrix Lie group into the group. It is given by a matrix exponential,


title: Cayley–Hamilton theorem

text: which is Rodrigues' rotation formula. For the notation, see 3D rotation group#A note on Lie algebras.


title: Cayley–Hamilton theorem

text: More recently, expressions have appeared for other groups, like the Lorentz group , and , as well as . The group is the conformal group of spacetime, its simply connected cover (to be precise, the simply connected cover of the connected component of ). The expressions obtained apply to the standard representation of these groups. They require knowledge of (some of) the eigenvalues of the matrix to exponentiate. For (and hence for ), closed expressions have been obtained for "all" irreducible representations, i.e. of any spin.


title: Cayley–Hamilton theorem

text: The Cayley–Hamilton theorem is an effective tool for computing the minimal polynomial of algebraic integers. For example, given a finite extension formula_54 of formula_55 and an algebraic integer formula_56 which is a non-zero linear combination of the formula_57 we can compute the minimal polynomial of formula_58 by finding a matrix representing the formula_55-linear transformation


title: Cayley–Hamilton theorem

text: If we call this transformation matrix formula_61, then we can find the minimal polynomial by applying the Cayley–Hamilton theorem to formula_61.


title: Cayley–Hamilton theorem

text: The Cayley–Hamilton theorem is an immediate consequence of the existence of the Jordan normal form for matrices over algebraically closed fields, see . In this section, direct proofs are presented.


title: Cayley–Hamilton theorem

text: requires two steps: first the coefficients of the characteristic polynomial are determined by development as a polynomial in of the determinant


title: Cayley–Hamilton theorem

text: and then these coefficients are used in a linear combination of powers of that is equated to the zero matrix:


title: Cayley–Hamilton theorem

text: The left-hand side can be worked out to an matrix whose entries are (enormous) polynomial expressions in the set of entries of , so the Cayley–Hamilton theorem states that each of these expressions equals . For any fixed value of , these identities can be obtained by tedious but straightforward algebraic manipulations. None of these computations, however, can show why the Cayley–Hamilton theorem should be valid for matrices of all possible sizes , so a uniform proof for all is needed.


title: Cayley–Hamilton theorem

text: which is the zero vector since (the eigenvalues of are precisely the roots of ). This holds for all possible eigenvalues , so the two matrices equated by the theorem certainly give the same (null) result when applied to any eigenvector. Now if admits a basis of eigenvectors, in other words if is diagonalizable, then the Cayley–Hamilton theorem must hold for , since two matrices that give the same values when applied to each element of a basis must be equal.


title: Cayley–Hamilton theorem

text: Consider now the function formula_72 which maps matrices to matrices given by the formula formula_73, i.e. which takes a matrix formula_61 and plugs it into its own characteristic polynomial. Not all matrices are diagonalizable, but for matrices with complex coefficients many of them are: the set formula_75 of diagonalizable complex square matrices of a given size is dense in the set of all such square matrices (for a matrix to be diagonalizable it suffices for instance that its characteristic polynomial not have any multiple roots). Now viewed as a function formula_76(since matrices have formula_77entries) we see that this function is continuous. This is true because the entries of the image of a matrix are given by polynomials in the entries of the matrix. Since


title: Cayley–Hamilton theorem

text: and since the set formula_75 is dense, by continuity this function must map the entire set of matrices to the zero matrix. Therefore, the Cayley–Hamilton theorem is true for complex numbers, and must therefore also hold for formula_80- or formula_81-valued matrices.


title: Cayley–Hamilton theorem

text: While this provides a valid proof, the argument is not very satisfactory, since the identities represented by the theorem do not in any way depend on the nature of the matrix (diagonalizable or not), nor on the kind of entries allowed (for matrices with real entries the diagonalizable ones do not form a dense set, and it seems strange one would have to consider complex matrices to see that the Cayley–Hamilton theorem holds for them). We shall therefore now consider only arguments that prove the theorem directly for any matrix using algebraic manipulations only; these also have the benefit of working for matrices with entries in any commutative ring.


title: Cayley–Hamilton theorem

text: There is a great variety of such proofs of the Cayley–Hamilton theorem, of which several will be given here. They vary in the amount of abstract algebraic notions required to understand the proof. The simplest proofs use just those notions needed to formulate the theorem (matrices, polynomials with numeric entries, determinants), but involve technical computations that render somewhat mysterious the fact that they lead precisely to the correct conclusion. It is possible to avoid such details, but at the price of involving more subtle algebraic notions: polynomials with coefficients in a non-commutative ring, or matrices with unusual kinds of entries.


title: Cayley–Hamilton theorem

text: All proofs below use the notion of the adjugate matrix of an matrix , the transpose of its cofactor matrix. This is a matrix whose coefficients are given by polynomial expressions in the coefficients of (in fact, by certain determinants), in such a way that the following fundamental relations hold,


title: Cayley–Hamilton theorem

text: These relations are a direct consequence of the basic properties of determinants: evaluation of the entry of the matrix product on the left gives the expansion by column of the determinant of the matrix obtained from by replacing column by a copy of column , which is if and zero otherwise; the matrix product on the right is similar, but for expansions by rows.


title: Cayley–Hamilton theorem

text: Being a consequence of just algebraic expression manipulation, these relations are valid for matrices with entries in any commutative ring (commutativity must be assumed for determinants to be defined in the first place). This is important to note here, because these relations will be applied below for matrices with non-numeric entries such as polynomials.


title: Cayley–Hamilton theorem

text: This proof uses just the kind of objects needed to formulate the Cayley–Hamilton theorem: matrices with polynomials as entries. The matrix whose determinant is the characteristic polynomial of is such a matrix, and since polynomials form a commutative ring, it has an adjugate


title: Cayley–Hamilton theorem

text: Since is also a matrix with polynomials in as entries, one can, for each , collect the coefficients of in each entry to form a matrix of numbers, such that one has


title: Cayley–Hamilton theorem

text: (The way the entries of are defined makes clear that no powers higher than occur). While this "looks" like a polynomial with matrices as coefficients, we shall not consider such a notion; it is just a way to write a matrix with polynomial entries as a linear combination of constant matrices, and the coefficient has been written to the left of the matrix to stress this point of view.


title: Cayley–Hamilton theorem

text: one obtains an equality of two matrices with polynomial entries, written as linear combinations of constant matrices with powers of as coefficients.


title: Cayley–Hamilton theorem

text: Such an equality can hold only if in any matrix position the entry that is multiplied by a given power is the same on both sides; it follows that the constant matrices with coefficient in both expressions must be equal. Writing these equations then for from down to 0, one finds


title: Cayley–Hamilton theorem

text: The left-hand sides form a telescoping sum and cancel completely; the right-hand sides add up to formula_90:


title: Cayley–Hamilton theorem

text: This proof is similar to the first one, but tries to give meaning to the notion of polynomial with matrix coefficients that was suggested by the expressions occurring in that proof. This requires considerable care, since it is somewhat unusual to consider polynomials with coefficients in a non-commutative ring, and not all reasoning that is valid for commutative polynomials can be applied in this setting.


title: Cayley–Hamilton theorem

text: Notably, while arithmetic of polynomials over a commutative ring models the arithmetic of polynomial functions, this is not the case over a non-commutative ring (in fact there is no obvious notion of polynomial function in this case that is closed under multiplication). So when considering polynomials in with matrix coefficients, the variable must not be thought of as an "unknown", but as a formal symbol that is to be manipulated according to given rules; in particular one cannot just set to a specific value.


title: Cayley–Hamilton theorem

text: Let formula_93 be the ring of matrices with entries in some ring "R" (such as the real or complex numbers) that has as an element. Matrices with as coefficients polynomials in , such as formula_94 or its adjugate "B" in the first proof, are elements of formula_95.


title: Cayley–Hamilton theorem

text: By collecting like powers of , such matrices can be written as "polynomials" in with constant matrices as coefficients; write formula_96 for the set of such polynomials. Since this set is in bijection with formula_95, one defines arithmetic operations on it correspondingly, in particular multiplication is given by


title: Cayley–Hamilton theorem

text: respecting the order of the coefficient matrices from the two operands; obviously this gives a non-commutative multiplication.


title: Cayley–Hamilton theorem

text: At this point, it is tempting to simply set equal to the matrix , which makes the first factor on the left equal to the zero matrix, and the right hand side equal to ; however, this is not an allowed operation when coefficients do not commute. It is possible to define a "right-evaluation map" ev : M["t" ] → M, which replaces each by the matrix power of , where one stipulates that the power is always to be multiplied on the right to the corresponding coefficient. But this map is not a ring homomorphism: the right-evaluation of a product differs in general from the product of the right-evaluations. This is so because multiplication of polynomials with matrix coefficients does not model multiplication of expressions containing unknowns: a product formula_101 is defined assuming that commutes with , but this may fail if is replaced by the matrix .