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UK Championship

The UK Championship is a professional ranking snooker tournament. It is one of snooker's prestigious Triple Crown events, along with the World Championship and the Masters. It is usually held at the Barbican, York. Ronnie O'Sullivan has won the tournament a record eight times, followed by Steve Davis with six titles and Stephen Hendry with five. O'Sullivan is the reigning champion, winning his eighth title in 2023.

History
The UK Championship was first held in 1977 in Tower Circus, Blackpool as the United Kingdom Professional Snooker Championship, an event open only to British residents and passport holders. Patsy Fagan won the inaugural tournament by defeating Doug Mountjoy by 12 frames to 9 in the final and won the first prize of £2000. The following year the event moved to the Guild Hall, Preston, where it remained until 1997.

The rules were changed in 1984, when the tournament was granted ranking status and all professionals were allowed to enter. Since then, it has carried more ranking points than any tournament other than the World Championship until being overtaken by the International Championship and the China Open, both of which yield a higher number of ranking points based on the amount of money in pounds sterling on offer for winning the event.

The tournament has seen many memorable finals. In 1977 and 1979, it provided Patsy Fagan and John Virgo with their first and only major tournament wins respectively. In 1980, the tournament was won by Steve Davis, as well as in 1981, when the final between Davis and Terry Griffiths set the stage for four more final battles between the two, who were to dominate the rest of the season before their unexpected losses in the first round of the 1982 World Championship. Runner-up Griffiths became champion next year, in 1982, defeating Alex Higgins, who in turn was the winner of the 1983 event, beating Davis 16–15 after having trailed 0–7 at the end of the first session. Between 1984 and 1987, all four tournaments were won by Davis again. In 1985, Willie Thorne led Davis 13–8 at the start of the evening session, only to miss a simple blue off its spot and lose 16–14. The victory regenerated Davis's confidence after his devastating World Championship loss; Thorne, on the other hand, never won another ranking title.

In 1988, Doug Mountjoy became the oldest winner of the tournament at the age of 46, and the second oldest ranking event winner after Ray Reardon (50). Mountjoy produced a stunning display to win 16–12, when he was widely viewed as just making up the numbers against the rising Stephen Hendry. Even more astonishingly, Mountjoy won the Mercantile Credit Classic the following month, which at the time made him only the fourth player to win two ranking tournaments in a row.

Stephen Hendry's 1989 win prefigured his decade of dominance similar to the one prefigured by Davis's win in 1980; its significance was emphasised by the fact that the losing finalist was Davis himself. Hendry's 16–15 win the following year, over Davis again, spoke to his unique qualities of nerve. The final in 1994 against Ken Doherty is considered by many players as one of Hendry's best performance, as he won 10–5 making 7 century breaks along the way, six of which were in the span of eight frames played. Doherty has appeared in two more memorable finals, in 2001, when Ronnie O'Sullivan beat him 10–1, delivering a final's best winning margin since it had become the best of 19 frames in 1993, a feat repeated three years later, in 2004, by Stephen Maguire against David Gray; and in 2002, when Doherty almost won the tournament against Mark Williams, but lost 9–10 in a dramatic deciding frame.

In 1993, the format of the final was reduced from the best of 31 frames to the best of 19 frames. This is the format still used for the final to the current day. Ronnie O'Sullivan became the youngest-ever winner of the tournament (and any ranking tournament) that year, being just 17 years old at the time. Between 1994 and 1996, the championship was won by Stephen Hendry three times in a row, while from 1997 to 2002, the winner was one of the 'Class of '92' each year, with O'Sullivan, John Higgins and Mark Williams all claiming the title twice during this period.

The 2005 tournament saw Steve Davis, aged 48, reach his first ranking tournament final for almost two years and make his highest break in tournament play for 23 years. In a match that featured the widest age gap between finalists in professional tournament history, he lost 6–10 to the 18-year-old Ding Junhui, who became the first non-British or Irish and the second youngest player after O'Sullivan to win the UK Championship. The following year, Peter Ebdon won the title by defeating Stephen Hendry 10-6.

In 2007, the tournament was won by Ronnie O'Sullivan for the fourth time, as he beat Stephen Maguire 10–2 in the final. The tournament was also notable for O'Sullivan's maximum break in the deciding frame of the semi-final, and for the longest televised frame (77 minutes) between Marco Fu and Mark Selby. The 2009 final saw the reigning world champion John Higgins missing an easy brown and the chance to go 8–6 in front, eventually losing to second-time champion Ding Junhui.

The 2010 final turned out to be another dramatic match, instantly described by many commentators as an all-time classic. At one point, John Higgins, playing in his first major tournament after the end of a six-month ban for his involvement in match-fixing discussions, was 5–9 down to Mark Williams. However, he won the next two frames. At 7–9, Williams led by 29 points with only 27 on the table, leaving Higgins requiring a snooker to remain in the tournament. Higgins got the snooker and cleared the colours. Another frame won by Higgins took the match to the decider. Finally, with only brown, blue, pink and black left at the table, Higgins potted the brown into a top pocket by playing cross-double across the long axis of the table and then added a difficult long blue and equally difficult pink, thereby winning the frame and thus the tournament by 10–9. In the emotional post-match interview, he described his win as his finest hour and dedicated it to his terminally ill father.

In 2011 the event returned to the Barbican Centre in York. There was a radical change to the format. All matches up to and including the quarter-finals were changed from the best of 17 frames to the best of 11 frames. In 2013 a 128-player flat draw was used, with all players starting in the first round and all rounds played at the Barbican venue. The tournament was contractually due to stay at the Barbican Centre until 2013, but it also hosted the event in 2014. That year the format was changed once again, with every round up to and including the semi-finals being played over best-of-11 frames. This is the format that is being used to the current day. This tournament saw yet another classic final, as Ronnie O'Sullivan won 10–9 over 2011 winner Judd Trump, who had recovered from 9–4 to take the match into a decider. Besides winning the title for a fifth time, O'Sullivan also produced a maximum in his last 16 match against Matthew Selt, the third consecutive 147 break in the main stages of the tournament since 2012, following the ones made by John Higgins in 2012 and Mark Selby in 2013, and followed by Neil Robertson's maximum in 2015 and Mark Allen's in 2016.

Between 2012 and 2018, the event has been won by either Neil Robertson, Mark Selby or Ronnie O'Sullivan. The final line-ups of 2013 and 2016 featured a combination of these three players, with Robertson defeating defending champion Selby in the former, and Selby beating O'Sullivan in the latter. In 2015, the final featured Australia's Neil Robertson and China's Liang Wenbo, the first time that a UK Championship final had been contested between two overseas players, an occurrence repeated in 2021, when Chinese player Zhao Xintong played against the Belgian Luca Brecel, who at that point was the first player from mainland Europe to compete in a Triple Crown final. This was the first ever UK Championship final between two players ranked outside the top 16. The 2016 final between Selby and O'Sullivan saw five century breaks in the final six frames of the match, as Selby won his second UK title. The next two editions of the tournament in 2017 and 2018 were captured by O'Sullivan, who became the first player since Stephen Hendry in 1996 to successfully defend the title. In addition, O'Sullivan set a new record for most UK titles with seven, that he would extend to eight in 2023 by beating Ding Junhui, who won the tournament for the third time in 2019, and was a runner-up to Mark Allen in 2022. In the 2020 edition of the event, it was Neil Robertson who became a three-time champion of the tournament, defeating Judd Trump 10–9 in a 7-hour long final with a 66-minute long decider frame.

The UK Championship has had many different sponsors over the years, including Super Crystalate, Tennents, StormSeal, Royal Liver Assurance, Liverpool Victoria, PowerHouse, Travis Perkins, Maplin Electronics, Pukka Pies, 12BET.com, williamhill.com, Coral, Betway, Cazoo, and MrQ. It is one of the tournaments televised by the BBC and is held towards the end of each calendar year.

Finalists

 * Active players are shown in bold.

Television coverage
The BBC is the current rights holder and shows the event alongside Eurosport. The BBC covered the inaugural event in 1977 on Grandstand and has broadcast all subsequent tournaments.

title: UK Championship


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Sunset Boulevard (musical)

Sunset Boulevard is a musical with music by Andrew Lloyd Webber, and lyrics and libretto by Don Black and Christopher Hampton. It is based on the 1950 film of the same title.

The plot revolves around Norma Desmond, a faded star of the silent screen era, living in the past in her decaying mansion on the fabled Los Angeles street. When young screenwriter Joe Gillis accidentally crosses her path, she sees in him an opportunity to make her return to the big screen, with romance and tragedy to follow.

Opening first in London in 1993, the musical has had several long runs internationally and enjoyed extensive tours. However, it has been the subject of several legal battles and ultimately lost money due to its extraordinary running costs.

Background
From approximately 1952 to 1956, Gloria Swanson worked with actor Richard Stapley (aka Richard Wyler) and cabaret singer and pianist Dickson Hughes on a musical adaptation originally entitled Starring Norma Desmond, then Boulevard!. It ended on a happier note than the film, with Norma allowing Joe to leave and pursue a happy ending with Betty. Rights holder Paramount Pictures originally had given Swanson verbal permission to proceed with the musical, but there had been no formal legal arrangement. On 20 February 1957, Paramount executive Russell Holman wrote to Swanson asking her to cease work on the project because "it would be damaging for the property to be offered to the entertainment public in another form as a stage musical." In 1994, Hughes incorporated material from the production into Swanson on Sunset, based on his and Stapley's experiences in writing Boulevard!. A recording of the entire score, which had been housed in the Gloria Swanson archives at the University of Texas, was released on CD in 2008.

In the early 1960s, Stephen Sondheim outlined a musical stage adaptation and went so far as to compose the first scene with librettist Burt Shevelove. A chance encounter with Billy Wilder at a cocktail party gave Sondheim the opportunity to introduce himself and ask the original film's co-screenwriter and director his opinion of the project (which was to star Jeanette MacDonald). "You can't write a musical about Sunset Boulevard," Wilder responded, "it has to be an opera. After all, it's about a dethroned queen." Sondheim immediately aborted his plans. A few years later, when he was invited by Hal Prince to write the score for a film remake starring Angela Lansbury as a fading musical comedian rather than a silent film star, Sondheim declined, citing his conversation with Wilder.

When Lloyd Webber saw the film in the early 1970s, he was inspired to write what he pictured as the title song for a theatrical adaptation, fragments of which he instead incorporated into Gumshoe. In 1976, after a conversation with Hal Prince, who had the theatrical rights to Sunset, Lloyd Webber wrote "an idea for the moment when Norma Desmond returns to Paramount Studios"; Lloyd Webber did no further work on the play until after 1989's Aspects of Love. At that point, Lloyd Webber "felt it was the subject [he] had to compose next", though by February 1990 he had announced plans to turn Really Useful Group private so he could "make movies rather than musicals."

In 1991, Lloyd Webber asked Amy Powers, a lawyer from New York with hardly any professional lyric-writing experience, to write the lyrics for Sunset Boulevard. Don Black was later brought in to work with Powers; the two wrote the version that was performed in 1991 at Lloyd Webber's Sydmonton Festival. This original version starred Ria Jones as Norma and Michael Ball as Joe Gillis, but it was still in the experimental stage and not ready for potential producers. A revised version, written by Black and Christopher Hampton had a complete performance at the 1992 Sydmonton Festival, now with Patti LuPone playing Norma and Kevin Anderson as Joe Gillis. This "met with great success". Lloyd Webber borrowed several of the tunes from his 1986 mini-musical Cricket, written with Tim Rice, which had been performed at Windsor Castle and later at the 1986 Sydmonton Festival.

Act I
The place: A mansion on Sunset Boulevard, Los Angeles, 5 a.m. A homicide has been reported. Joe Gillis sets the scene ("Prologue"), noting that "an old-time movie star is involved / Maybe the biggest star of all", and that, if you want to know the "real facts", "you've come to the right party."

Flashback to... Hollywood, 1949 - where a down-on-his-luck screenwriter, Joe Gillis, is trying to hustle up some work at Paramount Studios ("Let's Have Lunch"). His appointment with a producer goes poorly when the executive rejects both Joe's proposed script and a loan to bring his car payments up to date. Joe does, however, meet Betty Schaefer, a pretty, young script editor who suggests they collaborate to rework one of his earlier screenplays ("Every Movie's a Circus"). As they chat, car repossession agents spot Joe, who quickly escapes.

During the ensuing chase, Joe evades his pursuers by pulling in to the garage of a palatial but dilapidated mansion on Sunset Boulevard. Beckoned inside the house, Joe encounters Norma Desmond ("Surrender"), a star actress of the silent-film era. Taken aback, Joe comments, "You used to be in pictures; you used to be big," to which Norma retorts, "I am big – it's the pictures that got small!" ("With One Look")

The gloomy estate is inhabited only by Norma and Max von Mayerling, her loyal butler and chauffeur. Although decades past her prime and mostly forgotten by the public, Norma is convinced she is as beautiful and popular as ever. Max perpetuates this illusion by shielding her from the realities of life out of the limelight and by writing her letters purportedly from still-devoted fans. Norma informs Joe that she plans to make her comeback with Salome, a script she has written for Cecil B. DeMille to direct with her in the starring role as the teenage biblical temptress ("Salome"). Dubious but sensing opportunity, Joe accepts her offer to work on editing the script. Norma insists that Joe stay in her home while they collaborate on Salome ("The Greatest Star of All").

Joe immediately realizes the script is incoherent, but because Norma won't allow a major rewrite, the revision drags on for months. During this time Joe is virtually imprisoned within the house, but he does break away to fulfill his commitment to Betty. Their working relationship blossoms into a romance that has her reconsidering her engagement to Joe's best friend, Artie Green ("Girl Meets Boy").

Blind to Joe's opportunism, Norma lavishes him with gifts that include a wardrobe makeover and he becomes her kept man ("The Lady's Paying"). She declares her love for him and turns quite possessive ("The Perfect Year"); when he leaves her to attend Artie's New Year's Eve party ("This Time Next Year"), she is distraught and attempts suicide. As a conciliatory gesture, Joe reluctantly returns to work on Salome.

Act II
Joe is now living in luxury at Norma Desmond's mansion, for reasons he bluntly states are mercenary ("Sunset Boulevard"). A cryptic message from Paramount has Norma certain that DeMille is eager to discuss her script ("There's Been A Call"). She drops in on the set of his current film and is greeted warmly by former colleagues and the famed director himself, but he is non-committal about Salome ("As If We Never Said Goodbye"). Meanwhile, Max discovers the studio had called to ask about Norma's Isotta Fraschini, not her screenplay. However, a delusional Norma leaves the lot convinced she'll soon be back in front of the cameras and begins to prepare for the role ("Eternal Youth Is Worth a Little Suffering").

Increasingly paranoid, Norma deduces that Joe and Betty are more than just friends ("Too Much in Love to Care"). She calls Betty to reveal Joe's secret life at the mansion, but he overhears and grabs the phone to tell Betty to come see for herself. Realizing their affair is doomed, Joe brusquely tells her he enjoys being Norma's pet and that she should go back to Artie. Betty departs, confused and brokenhearted, and Joe tells Norma he is leaving her and returning to his hometown of Dayton, Ohio. He also bluntly informs her that Salome is an unfilmable script and her fans have long abandoned her. Furious and grief-stricken, Norma fatally shoots Joe three times as he storms out of the house.

Now completely insane, Norma mistakes the swarms of police and reporters who arrive for studio personnel. Imagining herself on the set of Salome, she slowly descends her grand staircase and utters "And now, Mr. DeMille, I am ready for my close-up."

Major characters

 * Norma Desmond — Mezzo-soprano; a faded, eccentric, former silent screen star
 * Joe Gillis — Baritenor; a struggling young screenwriter
 * Max von Mayerling – Bass-baritone; Norma's first husband and butler
 * Betty Schaefer – Soprano; A budding writer and Joe's love interest
 * Artie Green – Baritenor; Betty's fiancé
 * Cecil B. DeMille – Bass; the famous director

Notable replacements

 * West End (1993-97)
 * Norma Desmond: Betty Buckley, Elaine Paige, Petula Clark, Rita Moreno
 * Joe Gillis: John Barrowman, Alexander Hanson, Graham Bickley, Glyn Kerslake

 * Broadway (1994-97)
 * Norma Desmond: Betty Buckley, Elaine Paige, Karen Mason (s/b), Maureen Moore (s/b)
 * Joe Gillis: John Barrowman, Bryan Batt (u/s)
 * Cecil B. DeMille: Rod Loomis

 * UK Tour (2001-02)
 * Joe Gillis: Ramin Karimloo (u/s)
 * Artie Green: Ramin Karimloo

Musical numbers

 * Act I
 * "Overture" / "Prologue" – Joe
 * "Let's Have Lunch" – Joe, Artie, Sheldrake, Betty, Actors, Actress & Scriptwriters
 * "Every Movie's A Circus" † – Betty, Joe, Finance Man 1 & Finance Man 2
 * "Surrender" – Norma
 * "With One Look" – Norma
 * "Salome" – Norma & Joe
 * "The Greatest Star of All" – Max
 * "Every Movie's a Circus (Reprise)" ∞ † – Artie, Joe, Betty, Barman, Actors, Actress & Waiters
 * "Girl Meets Boy" – Joe & Betty
 * "New Ways to Dream" – Norma & Joe
 * "The Lady's Paying" ^ – Norma, Joe, Manfred & Men's Shop Salesmen
 * "The Perfect Year" – Norma & Joe
 * "This Time Next Year" – Joe, Betty, Artie & Ensemble

 * Act II
 * "Entr'acte" – Orchestra
 * "Sunset Boulevard" – Joe
 * "There's Been a Call" – Norma & Joe
 * "As If We Never Said Goodbye" – Norma
 * "Surrender (Reprise)" – Cecil B. DeMile
 * "Girl Meets Boy (Reprise)" – Joe & Betty
 * "Eternal Youth Is Worth a Little Suffering" ^ – Norma, Astrologer & Beauticians
 * "Too Much in Love to Care" – Betty & Joe
 * "New Ways to Dream (Reprise)" – Max
 * "The Final Scene" – Joe, Betty, Norma & Max

† Song added for Los Angeles production

^ - cut from the 2023 West End and 2024 Broadway productions

∞ Originally a reprise of "Let's Have Lunch"

Original West End production
The original West End production, directed by Trevor Nunn and choreographed by Bob Avian, with costumes from Anthony Powell, opened on 12 July 1993 at the Adelphi Theatre. The cast featured Patti LuPone as Norma Desmond, Kevin Anderson as Joe Gillis, Meredith Braun as Betty Schaefer, and Daniel Benzali as Norma's ex-husband, Max.

Billy Wilder and his wife Audrey were joined by Nancy Olson, who had played Betty Schaefer in the original film, at the opening night performance. Of it, Wilder observed, "The best thing they did was leave the script alone," and of Patti LuPone he exclaimed, "She's a star from the moment she walks on stage".

Reviews were mixed, according to the Associated Press (AP) review summary. That summary quoted, for example, the review by Michael Kuchwara for the AP: "Some reviewers felt Lloyd Webber took the sting out of a cynical tale. 'Wilder's bitter brew has been diluted,' wrote AP Drama Critic Michael Kuchwara. He added: 'When LuPone is off stage, the show sags.'" Frank Rich wrote "Much of the film's plot, dialogue and horror-movie mood are preserved, not to mention clips used to illustrate those sequences in which the faded silent-film star, Norma Desmond ...and her kept, young screenwriter, Joe Gillis, ... travel by car. The lyricist, Don Black ..., and the playwright, Christopher Hampton ..., smartly tailor their jokes to the original screenplay's style. At times, even Lloyd Webber gets into the Wilder swing. Both acts open with joltingly angry diatribes about Hollywood, part exposition-packed recitative and part song, in which the surprisingly dark, jazz-accented music, the most interesting I've yet encountered from this composer, meshes perfectly with the cynical lyrics. Anderson makes the sardonic Wilder voice an almost physical presence in Sunset Boulevard, but he is too often drowned out by both LuPone's Broadway belt and mechanical efforts of Lloyd Webber and his director, Trevor Nunn, to stamp the proven formulas of Phantom and Les Miz on even an intimate tale. At odd moments, the mammoth set advances like a glacier toward the audience or retreats, or, most dramatically, rises partly up into the flies, actors in tow."

The show closed for three weeks, re-opening on 19 April 1994, revamped to follow the Los Angeles production, with a second official "opening". The revamped musical had a new song, "Every Movie's a Circus", a new set, and new stars, Betty Buckley and John Barrowman. Anita Louise Combe took over the role of Betty Schaeffer and, as a result of her performance, was invited to play the role in the original Canadian Premiere production in Toronto the following year. Michael Bauer, who had originally played DeMille, replaced Benzali as Max, a role he played until the end of the London run (and subsequently on the UK tour and in the BBC concert). Buckley and the production garnered rave reviews. David Lister of The Independent, for example wrote: "The show looked an improvement on the one that got decidedly mixed reviews last summer."

Betty Buckley went on to replace Glenn Close as Norma in the second year of the Broadway production. Elaine Paige, who had filled in when Buckley was ill in 1994, took over the part in the West End in May 1995 before joining the Broadway production for the end of its run between 1996 and 1997. Petula Clark filled in for Paige during her holiday in September/October 1995, before taking over the role in January 1996 when Paige departed for the United States. The last actress to play Norma in London was Rita Moreno, who filled in for a vacationing Clark in September and October 1996. John Barrowman played Joe until 1995, when he was replaced by Alexander Hanson. Graham Bickley played the role for the final year of the London run.

The show closed on 5 April 1997 after 1,530 performances.

Los Angeles production
The American premiere was at the Shubert Theatre in Century City, Los Angeles, California, on 9 December 1993, with Close as Norma and Alan Campbell as Joe. Featured were George Hearn as Max and Judy Kuhn as Betty. Lloyd Webber had reworked both the book and score, tightening the production, better organising the orchestrations, and adding the song "Every Movie's a Circus". This new production was better received by the critics and was an instant success, running for 369 performances. The Los Angeles production also recorded a new cast album that is well regarded. It is also the only unabridged cast recording of the show, since the original London recording was trimmed by over thirty minutes.

A controversy arose with this production after Faye Dunaway was hired to replace Glenn Close. Dunaway went into rehearsals with Rex Smith as Joe and Jon Cypher as Max. Tickets went on sale for Dunaway's engagement but shortly after rehearsals started the producers announced that Dunaway was unable to sing the role to their standards and the production would shut down when Close left. Dunaway "claims that when advance sales slumped, Sir Andrew decided to save money by sending the LA cast of Sunset Boulevard - based on the story of a clapped-out Hollywood actress - to Broadway, where it is due to open in November. " However, the Los Angeles Times reported that "The cancellation came despite advance ticket sales for the Los Angeles production 'way in excess of $4 million,' said Peter Brown, a spokesman for Lloyd Webber."

Dunaway filed a lawsuit claiming her reputation had been damaged by the producer's claims. Dunaway's lawsuit was settled and the producers paid her a settlement but no other terms of the agreement have ever been disclosed.

Original Broadway production
The musical opened on Broadway at the Minskoff Theatre on 17 November 1994 with Close, Campbell, and Hearn recreating their roles from the Los Angeles production and Alice Ripley joining the cast as Betty. Also in the cast were Alan Oppenheimer as Cecil B. DeMille and Vincent Tumeo making his Broadway debut as Artie Green. The production opened with the highest advance in the history of Broadway ticket sales at that time. It closed on March 22, 1997, after playing 977 performances. Billy Wilder was in attendance on opening night and was coaxed onstage by Close for the curtain call. In a season with only one other musical nominated for Best Musical, the production won several Tony Awards; Glenn Close, with only one other nominee as Best Actress in a musical, won the Tony for Best Performance by an Actress in a Leading Role.

The New York Times theatre critic Vincent Canby commented about the Tony Awards for this year: "Awards don't really tell you much when the competition is feeble or simply non-existent, as was the case the year that Sunset Boulevard won its Tony. Such prizes are for use in advertising and promotion and to impress the folks back home."

Patti LuPone, who initially had been promised the Broadway run, sued Lloyd Webber and received a settlement reported to be $1 million. Frank Rich, in his book The Hot Seat, noted that these lawsuits contributed to Sunset Boulevard setting the record for the most money lost by a theatrical endeavour in the history of the United States. According to The New York Times, operating costs soared far beyond the budget, and the "Broadway production has earned back, at best, 80% of the initial $13 million". For example, during the week of 2 July 1995, "it cost $731,304 to run Sunset Boulevard, including... advertising fees of $138,352 (which had been budgeted at $40,000 a week)." The road companies also generated large financial losses. Rich puts the final figure near or above US$20 million lost, making the show what he termed a "flop-hit", as it ran more than two years. The musical sold over a million tickets on Broadway.

Touring productions
The first national US tour in 1996 starring Linda Balgord ended in early 1997 after only a handful of venues due to exorbitant costs involved in transporting the set. Lloyd Webber called in director Susan H. Schulman to design a scaled-down production, with Petula Clark once again in the lead opposite Lewis Cleale as Joe. This production featured Anthony Powell's Tony Award nominated costumes, a slightly modified libretto by Schulman and Don Black and a new, more tour-friendly set by Derek McLane. The revised production, opening in Pittsburgh about a year after the closing of the original tour in Chicago, went on the road for almost two years, though it avoided the cities covered by the previous tour.

In August 2001, a UK tour commenced in Plymouth starring Faith Brown as Norma, opposite Earl Carpenter as Joe, and Michael Bauer reprising his West End performance as Max. The production had a completely new set, much simpler than the original London set, although the overall production remained closer to the original staging than the revamped US national tour. Carpenter left midway through the tour and was replaced by Jeremy Finch, who had previously understudied the role. The tour finished in late 2002 in Manchester and met with both excellent reviews and respectable ticket sales.

Ria Jones, who originated the role of Norma Desmond in the 1991 Sydmonton Workshop and understudied Glenn Close in the 2016 West End revival, led a new production opening at Leicester's Curve Theatre on 16 September 2017 for a two-week run before embarking on a national tour around the United Kingdom. Starring opposite Jones as Norma was Danny Mac as Joe Gillis. The tour, directed by Curve artistic by Nikolai Foster, transferred the musical into a Hollywood sound stage setting, designed by Colin Richmond, with acclaimed use of vintage archive film and projections designed by Douglas O'Connell.

West End & London revivals
An eight-week engagement of a minimalist production, in which the actors used musical instruments, enjoyed a good run at the Watermill Theatre in Newbury over the summer of 2008. Directed and choreographed by Craig Revel Horwood, the cast featured Kathryn Evans as Norma Desmond and Ben Goddard as Joe Gillis. A West End transfer of the Watermill production began on 4 December 2008 prior to an official opening 15 December at the Comedy Theatre, with Evans and Goddard reprising their roles, and Dave Willetts joining the cast as Max. The production received rave reviews and extended its run to September 2009. However, the production closed just after initially planned on 30 May 2009. It had originally been booking until 19 September 2009. There were plans for a UK Tour and also talks of bringing the show to Broadway For their performances, Evans and Goddard were nominated for the 2009 Laurence Olivier Award for Best Actress in a Musical and Best Actor in a Supporting Role in a Musical, respectively.

Opening on 4 April 2016, English National Opera (ENO) presented a five-week 'semi-staged' run at the London Coliseum. Glenn Close reprised her role as Norma, making her London theatre debut, along with Michael Xavier as Joe, Siobhan Dillon as Betty and Fred Johanson as Max. Michael Linnit and Michael Grade for Gate Ventures PLC worked with ENO to present the show; Johnny Hon was the executive producer. For her performance, Close won the 2016 Evening Standard Theatre Award for Best Musical Performance and was nominated for the 2017 Laurence Olivier Award for Best Actress in a Musical. It was announced on 25 October 2016 that the production would transfer to Broadway at the Palace Theatre in a limited engagement.

The third revival of the show opened at the Savoy Theatre from September 2023 for a 16-week limited run. Starring Nicole Scherzinger as Norma, the piece was reimagined in a modern and minimalist staging directed by Jamie Lloyd and produced by Lloyd Webber Harrison Musicals. On 14 August 2023, complete casting was announced with Tom Francis, David Thaxton, and Grace Hodgett Young (making her professional debut) starring as Joe, Max, and Betty, respectively. Rachel Tucker guest starred as Norma on October 12, 2023, and played Monday evenings beginning 16 October. Lloyd's production featured many changes to the score, including two songs being cut and substantial lyrical rewrites. "Let's Have Lunch" became "Let's Do Lunch" and the "Exit Music" was more substantively incorporated into the show. The revival won both of its two 2023 Evening Standard Theatre Awards nominations, including Best Musical Performance for Scherzinger and Best Director for Lloyd. It also received eleven nominations at the 2024 Laurence Olivier Awards and won seven, the most for any production of the season, including Best Musical Revival, Best Actress in a Musical for Scherzinger, Best Actor in a Musical for Francis, and Best Director for Lloyd. Thaxton and Hodgett Young were nominated for Best Actor in a Supporting Role in a Musical and Best Actress in a Supporting Role in a Musical awards. Recorded live at the Savoy Theatre and featuring Nicole Scherzinger and the rest of the West End cast, SUNSET BLVD: THE ALBUM is slated for release in October 2024. On 4 January 2024, it was announced that the West End production will transfer to Broadway later that year.

Broadway revivals
Glenn Close reprised her performance as Norma Desmond in the first revival on Broadway. Featuring a 40-piece onstage orchestra and a relatively minimalist set, the production began performances at the Palace Theatre on 2 February 2017 before opening officially on 9 February for a limited run, with tickets on sale through 25 June 2017. The cast featured Michael Xavier as Joe Gillis, Siobhan Dillon as Betty Schaefer, and Fred Johanson as Max von Mayerling, all reprising their roles from the 2016 London ENO production. The 2017 Broadway revival was directed by Lonny Price.

The 2023 West End revival is set to transfer to Broadway at the St. James Theatre, with previews beginning on 28 September 2024 and opening night planned for 20 October. Nicole Scherzinger will reprise her role as Norma and will be joined by her West End co-stars Tom Francis as Joe, David Thaxton as Max, and Grace Hodgett Young as Betty. Mandy Gonzalez will play Norma Desmond for select performances.

International productions
The original Canadian production opened in Toronto on October 15, 1995, with Diahann Carroll in the lead role. Her performance was praised by critics. It also starred Rex Smith as Joe, Walter Charles as Max and Anita Louise Combe, who had played Betty in the London production, repeating in this role. Toronto performances ended in August 1996, with the production later moving to Vancouver where it ran from November 1996 to March 1997. A highlights recording of this production was released on CD.

A German production of the musical opened 7 December 1995 at the newly built Rhein-Main Theater in Niedernhausen near Wiesbaden, starring Helen Schneider and Sue Mathys (matinees) as Norma and Uwe Kröger as Joe. A cast recording (with Schneider and Kröger) was released in 1996. The role of Norma was later played by Daniela Ziegler and Christina Grimandi, with Schneider and, for the last few months, Sue Mathys both returning to play the lead. The production closed in May 1998.

In October 1996, the original Australian production of the musical opened at Melbourne's newly restored Regent Theatre. The cast included Debra Byrne as Norma, Hugh Jackman as Joe, and Catherine Porter as Betty. Maria Mercedes starred as the alternate Norma, performing two of the eight shows each week. Amanda Harrison took over the role of Betty for the final months of the show's run. The production ran until 14 June 1997.

A year-long Dutch tour commenced in the Netherlands on 10 October 2008, with Simone Kleinsma and Pia Douwes alternating as Norma and Antonie Kamerling as Joe, using the same modified libretto that was first used in the 2001 UK tour. Kleinsma went on to win the Best Actress Award for the role in the 2009 Dutch Musical Awards and also Best Actress for the Flemish Musical Prizes. An official cast album was released, with Kleinsma appearing on the main album and with a four track bonus CD of Pia Douwes singing Norma's main arias.

The Swedish premiere took place at the Värmlandsoperan in September 2009, to mostly positive reviews. The role of Norma was played by Maria Lundqvist. A second much more elaborate production opened in October 2010, at the Gothenburg Opera House, with Gunilla Backman (who previously understudied the role of Betty in the original German production) starring as Norma.

The first Danish production of Sunset Boulevard opened 2012 at Nørregadeteatret, Maribo, starring Karen-Lise Mynster as Norma.

A South African production starring Angela Kilian as Norma and Jonathan Roxmouth as Joe, was shown at the Pieter Toerien Theatre at Montecasino in Johannesburg from late August 2013 to mid October 2013 and at Theatre on the Bay in Cape Town from late October 2013 until early January 2014.

The Czech production with Hana Fialová and Katarína Hasprová in the role of Norma had its premiere in National Moravian-Silesian Theatre in Ostrava. The premiere took place on 19 February 2015. The show closed in March 2017.

Another German production, starring Katharina Scherer as Norma, Philippe J. Kayser as Joe, Antonia Crames as Betty, and Stephan Vanecek as Max, opened to sold-out seats on 15 September 2017 and had its fourteenth and final performance on 8 October 2017 at the Tuchfabrik in Trier.

The Spanish premiere production opened 27 December 2017 at the Auditorio de Tenerife, directed by Jaime Azpilicueta and starring Paloma San Basilio as Norma Desmond, Gerónimo Rauch as Joe Gillis, Inma Mira as Betty Schaefer, and Gonzalo Montes as Max von Mayerling.

Another Danish production by Den Jyske Opera opened 12 August 2022 at Musikhuset Aarhus, directed by Philipp Kochheim and starring Tammi Øst as Norma, Randy Diamond as Joe, Katharina Maria Abt as Betty, and Martin Loft as Max. The production received mixed reviews from critics who praised the performances of Øst and the Aarhus Symphony Orchestra while criticizing the direction and the casting of Diamond as Joe due to his advanced age.

Sarah Brightman stars as Norma Desmond in a new production of Sunset Boulevard that premiered in Melbourne on May 21, 2024, at the Princess Theatre with an official opening night on May 29. This will be followed by an engagement at the Sydney Opera House beginning August 28. This is Brightman's first theatrical role in more than three decades.

Other productions
In 2004, the first regional production of Sunset Boulevard was staged in the round at the Marriott Theatre in Chicago for a limited period and starred Paula Scrofano as Norma. It was the first and only regional production to be licensed by the Really Useful Group (RUG) for the next six years. However, in the spring of 2010, the leasing rights were finally released to regional companies and numerous productions have been staged around the United States.

The Ogunquit Playhouse production ran from 28 July through 14 August 2010 and starred Stefanie Powers as Norma and Todd Gearhart as Joe. This was the first fully staged production in the U.S. in nearly a decade. The Ogunquit production was directed by Shaun Kerrison with choreography by Tom Kosis, featured costumes by Anthony Powell and an all new set designed exclusively for Ogunquit by Todd Ivins.

The Arvada Center production ran from 14 September to 10 October 2010, in Denver, Colorado, the same city that launched the ill-fated first US tour in 1996. The show starred Ann Crumb as Norma and Kevin Earley as Joe. The production was directed by Rod A. Landsberry.

The Kennedy Center production ran from February 1-8, 2023 in Washington, D.C., with Stephanie J. Block starring as Norma, Derek Klena as Joe, Nathan Gunn as Max and Auli'i Cravalho as Betty. The show ran as a limited engagement as part of the Kennedy Center's Broadway Center Stage series.

Concert productions
In April 2004, Petula Clark reprised her role as Norma opposite Michael Ball as Joe in a concert production of the show that ran for two nights at the Cork Opera House in Ireland, which was later broadcast on BBC Radio 2. The cast also included Michael Bauer (Max), Emma Williams (Betty), Michael Xavier (Artie) and the BBC Concert Orchestra was conducted by Martin Yates.

Another two-day concert engagement took place in 2004 in Sydney by the Production Company; Judi Connelli starred as Norma, Michael Cormick played Joe and Anthony Warlow was Max. The Production Company staged a slightly more elaborate version of the concert for a week in Melbourne during 2005. Connelli again starred as Norma, and David Campbell took the role of Joe. The State Theatre was sold out for every performance.

A concert production of Sunset Boulevard was performed Off West End at the Alexandra Palace in June 2021. Ramin Karimloo starred as Joe Gillis, with Mazz Murray as Norma Desmond, Zizi Strallen as Betty Schaeffer, and Jeremy Secomb as Max. The production transferred to the Royal Albert Hall in London, England on December 3, 2021. Produced by Alpha, the concert was a fully staged production with direction by Jordan Murphy, choreography by Joanna Goodwin, and conducted by Alex Parker. Karimloo, Murray, Strallen, and Secomb reprised their performances.

Film adaptation
Paramount Pictures and the Relevant Picture Company announced in 2005 that they were developing a film adaptation of the musical. In 2007, The Daily Telegraph reported that actresses considered for the role of Norma Desmond included Close, Paige, Meryl Streep, Liza Minnelli, and Barbra Streisand. In a 2008 interview, Andrew Lloyd Webber said that there were no plans for a film adaptation to be made in the near future, but remained hopeful that one would be made at some point. In 2011, Lloyd Webber indicated that he would have wanted Madonna to star in the film.

In a 2013 interview with Elaine Paige during her BBC Radio 2 show Elaine Paige on Sunday, Lloyd Webber was asked about the progress of the film, to which he replied:

"I would love a film to be made of Sunset, of course, but unfortunately the rights are owned by Paramount who own the original movie and, so far, talks with them have never led to anything. And it's sad for me because I think in many ways Sunset is, I think, the most complete musical I have written, I mean in the sense that the book and the music and everything come together I think in a way that perhaps even some of the others, they don't. Anyway that's my sadness at the moment and maybe, one day, the whole thing will get brokered. I'm doing something else with Paramount because after I've done this I'm producing School of Rock on stage, and that's a Paramount picture, so maybe if they like what I do with that they'll let me do Sunset."

In February 2019, it was announced that Rob Ashford had signed on to direct the film, with Close reprising her role as Norma Desmond and Tom MacRae penning the script, with production slated to begin on October of that year, but was then delayed to the fall of 2020. In an October 2020 interview, Close, along with her confirmation of co-producing the film alongside Lloyd Webber, said that she hoped filming would commence in early 2021, citing the COVID-19 pandemic as the reason for the delay. In May 2021, Close stated in an interview that production was "very, very close" to beginning and that they were "ready to go", while also adding that Lloyd Webber was still working on new music for the film. However, in October 2021, Lloyd Webber announced that production had been stalled, stating, "I wish I could say it's going into production tomorrow morning, but it's not. Paramount has not wanted to go ahead with it. It's not for want of trying. Glenn Close has been absolutely doggedly trying to get it made." On May 2, 2022, at the 2022 Met Gala, Close revealed that "We're getting closer, believe it or not. We're getting closer," without disclosing any additional information.

In May 2024, during Variety's 2024 Power of Women luncheon, Close revealed that the film is still moving forward, but with Ashford no longer attached as director. She went on to remark on how the character of Norma Desmond still resonates in modern times, saying "That character of Norma is so heartbreakingly beautiful and will never not have relevance, it's been a great blessing to play her and I'm not willing to let her go yet without one more tribute."

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

title: Sunset Boulevard (musical)


text:
Two-phase flow

In fluid mechanics, two-phase flow is a flow of gas and liquid — a particular example of multiphase flow. Two-phase flow can occur in various forms, such as flows transitioning from pure liquid to vapor as a result of external heating, separated flows, and dispersed two-phase flows where one phase is present in the form of particles, droplets, or bubbles in a continuous carrier phase (i.e. gas or liquid).

Categorization
The widely accepted method to categorize two-phase flows is to consider the velocity of each phase as if there is not other phases available. The parameter is a hypothetical concept called Superficial velocity.

Examples and applications
Historically, probably the most commonly studied cases of two-phase flow are in large-scale power systems. Coal and gas-fired power stations used very large boilers to produce steam for use in turbines. In such cases, pressurised water is passed through heated pipes and it changes to steam as it moves through the pipe. The design of boilers requires a detailed understanding of two-phase flow heat-transfer and pressure drop behaviour, which is significantly different from the single-phase case. Even more critically, nuclear reactors use water to remove heat from the reactor core using two-phase flow. A great deal of study has been performed on the nature of two-phase flow in such cases, so that engineers can design against possible failures in pipework, loss of pressure, and so on (a loss-of-coolant accident (LOCA)).

Another case where two-phase flow can occur is in pump cavitation. Here a pump is operating close to the vapor pressure of the fluid being pumped. If pressure drops further, which can happen locally near the vanes for the pump, for example, then a phase change can occur and gas will be present in the pump. Similar effects can also occur on marine propellers; wherever it occurs, it is a serious problem for designers. When the vapor bubble collapses, it can produce very large pressure spikes, which over time will cause damage on the propeller or turbine.

The above two-phase flow cases are for a single fluid occurring by itself as two different phases, such as steam and water. The term 'two-phase flow' is also applied to mixtures of different fluids having different phases, such as air and water, or oil and natural gas. Sometimes even three-phase flow is considered, such as in oil and gas pipelines where there might be a significant fraction of solids. Although oil and water are not strictly distinct phases (since they are both liquids) they are sometimes considered as a two-phase flow; and the combination of oil, gas and water (e.g. the flow from an offshore oil well) may also be considered a three-phase flow.

Other interesting areas where two-phase flow is studied includes water electrolysis, climate systems such as clouds, and in groundwater flow, in which the movement of water and air through the soil is studied.

Other examples of two-phase flow include bubbles, rain, waves on the sea, foam, fountains, mousse, cryogenics, and oil slicks. One final example is in the electrical explosion of metal.

Characteristics of two-phase flow
Several features make two-phase flow an interesting and challenging branch of fluid mechanics:

 * Surface tension makes all dynamical problems nonlinear (see Weber number)
 * In the case of air and water at standard temperature and pressure, the density of the two phases differs by a factor of about 1000. Similar differences are typical of water liquid/water vapor densities
 * The sound speed changes dramatically for materials undergoing phase change, and can be orders of magnitude different. This introduces compressible effects into the problem
 * The phase changes are not instantaneous, and the liquid vapor system will not necessarily be in phase equilibrium
 * The change of phase means flow-induced pressure drops can cause further phase-change (e.g. water can evaporate through a valve) increasing the relative volume of the gaseous, compressible medium and increasing exit velocities, unlike single-phase incompressible flow where closing a valve would decrease exit velocities
 * Can give rise to other counter-intuitive, negative resistance-type instabilities, like Ledinegg instability, geysering, chugging, relaxation instability, and flow maldistribution instabilities as examples of static instabilities, and other dynamic instabilities

Acoustics
Gurgling is a characteristic sound made by unstable two-phase fluid flow, for example, as liquid is poured from a bottle, or during gargling.

Modelling
Modelling of two phase flow is still under development. Known methods are
 * Volume of fluid method
 * Level-set method
 * Front tracking by Gretar Tryggvason
 * Lattice Boltzmann methods
 * Smoothed-particle hydrodynamics (SPH)

title: Two-phase flow


text:
Greatest element and least element

In mathematics, especially in order theory, the greatest element of a subset $$S$$ of a partially ordered set (poset) is an element of $$S$$ that is greater than every other element of $$S$$. The term least element is defined dually, that is, it is an element of $$S$$ that is smaller than every other element of $$S.$$

Definitions
Let $$(P, \leq)$$ be a preordered set and let $$S \subseteq P.$$ An element $$g \in P$$ is said to be if $$g \in S$$ and if it also satisfies:
 * $$s \leq g$$ for all $$s \in S.$$

By switching the side of the relation that $$s$$ is on in the above definition, the definition of a least element of $$S$$ is obtained. Explicitly, an element $$l \in P$$ is said to be if $$l \in S$$ and if it also satisfies:
 * $$l \leq s$$ for all $$s \in S.$$

If $$(P, \leq)$$ is also a partially ordered set then $$S$$ can have at most one greatest element and it can have at most one least element. Whenever a greatest element of $$S$$ exists and is unique then this element is called  greatest element of $$S$$. The terminology  least element of $$S$$ is defined similarly.

If $$(P, \leq)$$ has a greatest element (resp. a least element) then this element is also called (resp. ) of $$(P, \leq).$$

Relationship to upper/lower bounds
Greatest elements are closely related to upper bounds.

Let $$(P, \leq)$$ be a preordered set and let $$S \subseteq P.$$ An  is an element $$u$$ such that $$u \in P$$ and $$s \leq u$$ for all $$s \in S.$$ Importantly, an upper bound of $$S$$ in $$P$$ is  required to be an element of $$S.$$

If $$g \in P$$ then $$g$$ is a greatest element of $$S$$ if and only if $$g$$ is an upper bound of $$S$$ in $$(P, \leq)$$ $$g \in S.$$ In particular, any greatest element of $$S$$ is also an upper bound of $$S$$ (in $$P$$) but an upper bound of $$S$$ in $$P$$ is a greatest element of $$S$$ if and only if it  to $$S.$$ In the particular case where $$P = S,$$ the definition of "$$u$$ is an upper bound of $$S$$ " becomes: $$u$$ is an element such that $$u \in S$$ and $$s \leq u$$ for all $$s \in S,$$ which is to the definition of a greatest element given before. Thus $$g$$ is a greatest element of $$S$$ if and only if $$g$$ is an upper bound of $$S$$.

If $$u$$ is an upper bound of $$S$$ that is not an upper bound of $$S$$  (which can happen if and only if $$u \not\in S$$) then $$u$$ can  be a greatest element of $$S$$ (however, it may be possible that some other element  a greatest element of $$S$$). In particular, it is possible for $$S$$ to simultaneously have a greatest element  for there to exist some upper bound of $$S$$.

Even if a set has some upper bounds, it need not have a greatest element, as shown by the example of the negative real numbers. This example also demonstrates that the existence of a least upper bound (the number 0 in this case) does not imply the existence of a greatest element either.

Contrast to maximal elements and local/absolute maximums

A greatest element of a subset of a preordered set should not be confused with a maximal element of the set, which are elements that are not strictly smaller than any other element in the set.

Let $$(P, \leq)$$ be a preordered set and let $$S \subseteq P.$$ An element $$m \in S$$ is said to be a  if the following condition is satisfied:

 * whenever $$s \in S$$ satisfies $$m \leq s,$$ then necessarily $$s \leq m.$$

If $$(P, \leq)$$ is a partially ordered set then $$m \in S$$ is a maximal element of $$S$$ if and only if there does exist any $$s \in S$$ such that $$m \leq s$$ and $$s \neq m.$$ A is defined to mean a maximal element of the subset $$S := P.$$

A set can have several maximal elements without having a greatest element. Like upper bounds and maximal elements, greatest elements may fail to exist.

In a totally ordered set the maximal element and the greatest element coincide; and it is also called maximum; in the case of function values it is also called the absolute maximum, to avoid confusion with a local maximum. The dual terms are minimum and absolute minimum. Together they are called the absolute extrema. Similar conclusions hold for least elements.

 * Role of (in)comparability in distinguishing greatest vs. maximal elements

One of the most important differences between a greatest element $$g$$ and a maximal element $$m$$ of a preordered set $$(P, \leq)$$ has to do with what elements they are comparable to. Two elements $$x, y \in P$$ are said to be if $$x \leq y$$ or $$y \leq x$$; they are called  if they are not comparable. Because preorders are reflexive (which means that $$x \leq x$$ is true for all elements $$x$$), every element $$x$$ is always comparable to itself. Consequently, the only pairs of elements that could possibly be incomparable are pairs. In general, however, preordered sets (and even directed partially ordered sets) may have elements that are incomparable.

By definition, an element $$g \in P$$ is a greatest element of $$(P, \leq)$$ if $$s \leq g,$$ for every $$s \in P$$; so by its very definition, a greatest element of $$(P, \leq)$$ must, in particular, be comparable to element in $$P.$$ This is not required of maximal elements. Maximal elements of $$(P, \leq)$$ are required to be comparable to every element in $$P.$$ This is because unlike the definition of "greatest element", the definition of "maximal element" includes an important statement. The defining condition for $$m \in P$$ to be a maximal element of $$(P, \leq)$$ can be reworded as:

 * For all $$s \in P,$$  $$m \leq s$$ (so elements that are incomparable to $$m$$ are ignored) then $$s \leq m.$$

 * Example where all elements are maximal but none are greatest

Suppose that $$S$$ is a set containing (distinct) elements and define a partial order $$\,\leq\,$$ on $$S$$ by declaring that $$i \leq j$$ if and only if $$i = j.$$ If $$i \neq j$$ belong to $$S$$ then neither $$i \leq j$$ nor $$j \leq i$$ holds, which shows that all pairs of distinct (i.e. non-equal) elements in $$S$$ are comparable. Consequently, $$(S, \leq)$$ can not possibly have a greatest element (because a greatest element of $$S$$ would, in particular, have to be comparable to element of $$S$$ but $$S$$ has no such element). However, element $$m \in S$$ is a maximal element of $$(S, \leq)$$ because there is exactly one element in $$S$$ that is both comparable to $$m$$ and $$\geq m,$$ that element being $$m$$ itself (which of course, is $$\leq m$$).

In contrast, if a preordered set $$(P, \leq)$$ does happen to have a greatest element $$g$$ then $$g$$ will necessarily be a maximal element of $$(P, \leq)$$ and moreover, as a consequence of the greatest element $$g$$ being comparable to element of $$P,$$ if $$(P, \leq)$$ is also partially ordered then it is possible to conclude that $$g$$ is the  maximal element of $$(P, \leq).$$ However, the uniqueness conclusion is no longer guaranteed if the preordered set $$(P, \leq)$$ is also partially ordered. For example, suppose that $$R$$ is a non-empty set and define a preorder $$\,\leq\,$$ on $$R$$ by declaring that $$i \leq j$$ holds for all $$i, j \in R.$$ The directed preordered set $$(R, \leq)$$ is partially ordered if and only if $$R$$ has exactly one element. All pairs of elements from $$R$$ are comparable and element of $$R$$ is a greatest element (and thus also a maximal element) of $$(R, \leq).$$ So in particular, if $$R$$ has at least two elements then $$(R, \leq)$$ has multiple  greatest elements.

Properties
Throughout, let $$(P, \leq)$$ be a partially ordered set and let $$S \subseteq P.$$
 * A set $$S$$ can have at most greatest element. Thus if a set has a greatest element then it is necessarily unique.
 * If it exists, then the greatest element of $$S$$ is an upper bound of $$S$$ that is also contained in $$S.$$
 * If $$g$$ is the greatest element of $$S$$ then $$g$$ is also a maximal element of $$S$$ and moreover, any other maximal element of $$S$$ will necessarily be equal to $$g.$$
 * Thus if a set $$S$$ has several maximal elements then it cannot have a greatest element.
 * If $$P$$ satisfies the ascending chain condition, a subset $$S$$ of $$P$$ has a greatest element if, and only if, it has one maximal element.
 * When the restriction of $$\,\leq\,$$ to $$S$$ is a total order ($$S = \{ 1, 2, 4 \}$$ in the topmost picture is an example), then the notions of maximal element and greatest element coincide.
 * However, this is not a necessary condition for whenever $$S$$ has a greatest element, the notions coincide, too, as stated above.
 * If the notions of maximal element and greatest element coincide on every two-element subset $$S$$ of $$P,$$ then $$\,\leq\,$$ is a total order on $$P.$$

Sufficient conditions

 * A finite chain always has a greatest and a least element.

Top and bottom
The least and greatest element of the whole partially ordered set play a special role and are also called bottom (⊥) and top (⊤), or zero (0) and unit (1), respectively. If both exist, the poset is called a bounded poset. The notation of 0 and 1 is used preferably when the poset is a complemented lattice, and when no confusion is likely, i.e. when one is not talking about partial orders of numbers that already contain elements 0 and 1 different from bottom and top. The existence of least and greatest elements is a special completeness property of a partial order.

Further introductory information is found in the article on order theory.

Examples

 * The subset of integers has no upper bound in the set $$\mathbb{R}$$ of real numbers.
 * Let the relation $$\,\leq\,$$ on $$\{ a, b, c, d \}$$ be given by $$a \leq c,$$ $$a \leq d,$$ $$b \leq c,$$ $$b \leq d.$$ The set $$\{ a, b \}$$ has upper bounds $$c$$ and $$d,$$ but no least upper bound, and no greatest element (cf. picture).
 * In the rational numbers, the set of numbers with their square less than 2 has upper bounds but no greatest element and no least upper bound.
 * In $$\mathbb{R},$$ the set of numbers less than 1 has a least upper bound, viz. 1, but no greatest element.
 * In $$\mathbb{R},$$ the set of numbers less than or equal to 1 has a greatest element, viz. 1, which is also its least upper bound.
 * In $$\mathbb{R}^2$$ with the product order, the set of pairs $$(x, y)$$ with $$0 < x < 1$$ has no upper bound.
 * In $$\mathbb{R}^2$$ with the lexicographical order, this set has upper bounds, e.g. $$(1, 0).$$ It has no least upper bound.

title: Greatest element and least element


text:
User:Chuq/Maps/Tasmania

Here are some locator maps I have done for Local Government Areas of Tasmania. Original images are 370-400px wide. Best displayed as a 250-300wide thumbnail I have found. The images on this page are resized to 180px. The only ones that need work are Kingborough (missing half of Bruny Island) and Devonport (maybe a little too small). All maps have been drawn by me (User:Chuq) using public domain outline maps from the internet.

Some maps of Australian states at various points in history. I'm not sure where is the best place to put them yet:

I used a page I found (I forget where) on the net for ideas (particularly colours) however these maps were created from this existing Wikipedia image:

title: User:Chuq/Maps/Tasmania


text:
Frank Quitely

Vincent Patrick Deighan (born 1968), better known by the pen name Frank Quitely, is a Scottish comic book artist. He is best known for his frequent collaborations with Grant Morrison on titles such as New X-Men, We3, All-Star Superman, and Batman and Robin, as well as his work with Mark Millar on The Authority and Jupiter's Legacy.

Early life
Deighan was raised in Rutherglen, although attended St. Bride's High School in East Kilbride (as his father worked there as a PE teacher). He studied drawing at the Glasgow School of Art.

Career
Deighan worked up the Scottish underground comics title Electric Soup in 1990, writing and drawing The Greens, a parody of The Broons strip published by D. C. Thomson. It was in working on this book that he adopted the pseudonym of Frank Quitely (a spoonerism of "quite frankly"), as he did not want his family to know it was his work, worried that they may have found it upsetting.

Initially Electric Soup was only distributed locally in Glasgow, then it was picked up by John Brown Publishing for widespread national UK distribution. This brought Quitely's work to the attention of Judge Dredd Megazine editor David Bishop. He was given work on Shimura, written by Robbie Morrison, and Missionary Man, by Gordon Rennie, quickly rising to prominence. He drew various stories in Paradox Press' series of The Big Book Of graphic novels, as well as work in Dark Horse Presents for Dark Horse Comics.

His first major work in American comics was Flex Mentallo in 1996, a Doom Patrol spin-off written by fellow Glaswegian Grant Morrison for DC Comics' Vertigo imprint. Initially he worked on strips for anthology titles such as Weird War Tales, and drew three issues of Jamie Delano's 2020 Visions, as well as various covers for DC. He later drew his first full length graphic novel, Batman: The Scottish Connection, with writer Alan Grant in which The Greens make a cameo appearance. Quitely and Grant worked on a one-shot titled Lobo: The Hand-to-Hand Job later retitled as It's a Man's World. Although Quitely did all the pencils, the story was not released.

2000 saw Quitely and Morrison collaborate again, on JLA: Earth 2. The graphic novel was met with positive critical response, and later that year Quitely took over from Bryan Hitch as artist on The Authority, with Mark Millar as writer.

Quitely left The Authority to draw New X-Men. Quitely illustrated a Destiny story for Neil Gaiman's The Sandman: Endless Nights hardcover graphic novel in 2003. After leaving New X-Men, Quitely drew the mini-series We3 in 2004, again in collaboration with Morrison. He shared the 2005 Best Penciller/Inker or Penciller/Inker Team Eisner Award in a tie with artist John Cassaday for his work on the book. That same year, he and Morrison were nominated for Best Limited Series for that book, and Quitely additionally was nominated for the Best Cover Artist Eisner for both We3 and Bite Club. He wrote and drew new instalments of The Greens for the Scottish underground comic Northern Lightz, and in 2005 Morrison and Quitely designed a series of tarot cards for Intensive Care, an album by popstar Robbie Williams.

In December 2004, Quitely signed to a two-year exclusive contract with DC Comics, where he illustrated All-Star Superman. The twelve issue series, yet another collaboration with Morrison, began publication in November 2005. Quitely and Morrison's work on the series won them the Eisner Award for Best New Series in 2006, with Quitely collecting another nomination for Best Penciller/Inker. The series won Best Continuing Series in 2007 and 2009. In 2008 Quitely illustrated the cover for the debut issue of Scottish underground comic Wasted.

In early 2009, Frank Quitely collaborated with the Scottish rock band The Phantom Band in designing artwork for a limited edition 7" single for their song "The Howling", which plays on Quitely's interest in occultism and esoterica. This was released as a limited run collector's pressing by Glasgow's Chemikal Underground Records. That same year, Quitely again teamed with Morrison, illustrated the first three issues of Batman and Robin title, which debuted in June 2009 after the "Battle for the Cowl" storyline. He provided covers through issue No. 16. Quitely was one of the artists of Batman No. 700 (Aug. 2010).

On 9 April 2011, Quitely was one of 62 comics creators who appeared at the IGN stage at the Kapow! convention in London to set two Guinness World Records, the "Fastest Production of a Comic Book" and "Most Contributors to a Comic Book". With Guinness officials on hand to monitor their progress, writer Mark Millar began work at 9 AM scripting a 20-page black and white Superior comic book, with Quitely and the other artists appearing on stage throughout the day to work on the pencils, inks, and lettering, including Dave Gibbons, John Romita Jr., Jock, Adi Granov, Doug Braithwaite, Ian Churchill, Olivier Coipel, Duncan Fegredo, Simon Furman, David Lafuente, John McCrea, Sean Phillips and Liam Sharp, who all drew a panel each, with regular Superior artist Leinil Yu creating the book's front cover. The book was completed in 11 hours, 19 minutes, and 38 seconds, and was published through Icon on 23 November 2011, with all royalties being donated to Yorkhill Children's Foundation.

That same month, he mentioned during a panel at WonderCon that although he had split his time equally between illustrating covers and interiors, he had recently been devoting more of his time to covers, due to back problems, and the difficulty that presented in meeting his deadlines.

In 2012, Quitely was one of several artists to illustrate a variant cover for Robert Kirkman's The Walking Dead No. 100, which was released on 11 July at the San Diego Comic-Con.

Quitely was the artist on Jupiter's Legacy, a ten-issue, creator-owned miniseries published by Image Comics that premiered in September 2012. It is written by Mark Millar, who described the project as "his Star Wars", and a cross between The Lord of the Rings and a large-scale superhero crossover, albeit one that did not require the in-depth knowledge normally required of such stories, as it features entirely new characters. On July 17, 2018, it was announced that Netflix had given a series order for a television adaptation of Jupiter's Legacy.

Quitely drew the fourth issue of Grant Morrison's The Multiversity limited series which was published in November 2014.

In March 2017, an exhibition of his work was displayed at Kelvingrove Art Gallery and Museum in Glasgow. A few months later, Quitely received an honorary degree as a Doctor of Letters from the University of Glasgow in recognition of his achievements.

Personal life
Quitely married his high school sweetheart, Ann Jane Docherty, in 1994. They live in Rutherglen with their sons Vincent and Joseph, and their daughter, Orla. Quitely's son Vincent is also an artist who created art for local St Columbkille's RC Church in 2011 while a pupil at Trinity High School.

Quitely used to design his own hats and clothing. For a long time, he eschewed social media, noting that the Twitter account @Frank_Quitely is unconnected with him. In 2021, Quitely established official accounts on both Twitter and Instagram.

Technique and materials
In a 2014 BBC News interview, Deighan, having developed a reputation for difficulty in meeting deadlines, stated while he did his drawing digitally, it did not reduce the time it took to complete his art, saying, "I have never been on a project where I've thought 'ach, this is rubbish', and not tried hard," he says, "It's not in my nature. The downside of that is that things take a while."

Awards

 * 2002: Won "Best Artist in Comics Today" National Comics Award
 * 2005: Won "Best Penciller/Inker" Eisner Award for We3 (tied with John Cassaday)
 * 2006: Won "Best New Series" Eisner Award for All-Star Superman with Grant Morrison
 * 2007:
 * Won "Best Continuing Series" Eisner Award for All-Star Superman with Grant Morrison
 * Won "Best Artist" Harvey Award for All-Star Superman
 * 2008: Won "Best Artist" Harvey Award for All-Star Superman
 * 2009:
 * Won "Best Continuing Series" Eisner Award for All-Star Superman with Grant Morrison
 * Won "Best Artist" Harvey Award for All-Star Superman

Nominations

 * 2001:
 * "Best Penciller/Inker Team" Eisner Award for The Authority with Trevor Scott
 * "Best Serialized Story" Eisner Award for The Authority: The Nativity with Mark Millar
 * 2002: "Best Serialized Story" Eisner Award for New X-Men: E is for Extinction with Grant Morrison
 * 2003: "Best Artist in Comics Today" National Comics Award
 * 2004: "Best Painter/Multimedia Artist" for The Sandman: Endless Nights: "Destiny"
 * 2005:
 * "Best Cover Artist" Eisner Award for We3 and Bite Club (comics)
 * "Best Limited Series" Eisner Award for We3
 * 2006: "Best Penciller/Inker" Eisner Award for All-Star Superman
 * 2010: "Best Artist" Harvey Award for Batman and Robin

Covers only

 * Judge Dredd Megazine vol. 3 #19, 32 (Fleetway, 1996–1997)
 * Negative Burn #38 (Caliber, 1996)
 * DNA Swamp #1 (DNA Design, 1997)
 * JLA Gallery #1 (DC Comics, 1997)
 * Transmetropolitan #4–6 (Vertigo, 1997–1998)
 * Vamps: Pumpkin Time #1–3 (Vertigo, 1998–1999)
 * James Murphy's August #1 (Bigger Betty, 1998)
 * The Foot Soldiers vol. 2 #3 (Image, 1998)
 * Avengers Forever #4 (Marvel, 1999)
 * The Authority #17–18, 21 (Wildstorm, 2000–2001)
 * Gen-Active #5 (Wildstorm, 2001)
 * Wizard #117, 124, 140 (Wizard, 2001–2003)
 * New X-Men #117–120, 127, 132–133 (Marvel, 2001–2003)
 * Wizard Ace Edition: Uncanny X-Men #94 (Marvel, 2002)
 * Wizard Special Edition: X-Men (Marvel, 2002)
 * X-Statix #4 (Marvel, 2003)
 * Puffed #1 (Image, 2003)
 * Youngblood: Bloodsport #1 (Arcade, 2003)
 * Anarchy for the Masses sc (Mad Yak Press, 2003)
 * Masters of the Universe vol. 3 #1 (MVCreations, 2004)
 * Bite Club #1–6 + Vampire Crime Unit #1–2 (Vertigo, 2004; 2006)
 * Books of Magick: Life During Wartime #1–5, 7, 9, 11 (Vertigo, 2004–2005)
 * Transformers: Last Stand hc (Titan Books, 2005)
 * Wanted #6 (Top Cow, 2005)
 * Jonah Hex vol. 2 #1 (DC Comics, 2006)
 * American Virgin #1–3 (Vertigo, 2006)
 * All-Star Batman and Robin, the Boy Wonder #10 (DC Comics, 2008)
 * Wasted #1 (Bad Press, 2008)
 * Olympus #2 (Image, 2009)
 * Birds of Prey #125 (DC Comics, 2009)
 * Madame Xanadu #6 (Vertigo, 2009)
 * Final Crisis Secret Files (DC Comics, 2009)
 * Wonder Woman vol. 3 #27 (DC Comics, 2009)
 * Batman and Robin #4–16 (DC Comics, 2009–2010)
 * DCU Holiday Special 2009 (DC Comics, 2009)
 * Judge Dredd Megazine #301 (Rebellion, 2010)
 * Superman/Batman #75 (DC Comics, 2010)
 * Green Lantern vol. 4 #60 (DC Comics, 2011)
 * Teen Titans vol. 3 #89 (DC Comics, 2011)
 * Liberty Annual '11 (Image, 2011)
 * T.H.U.N.D.E.R. Agents vol. 3 #1 (DC Comics, 2011)
 * Batman Incorporated vol. 2 #1 (DC Comics, 2012)
 * The Walking Dead #100 (Image, 2012)
 * Channel Evil #3 (Renegade, 2012)
 * Happy! #4 (Image, 2013)
 * Django Unchained #5 (Vertigo, 2013)
 * Elephantmen #50 (Image, 2013)
 * House of Gold and Bones #4 (Dark Horse, 2013)
 * Jupiter's Circle vol. 1 #1–6 + vol. 2 #1–6 (Image, 2015–2016)
 * Super Sons #1 (DC Comics, 2017)
 * Kid Lobotomy #1 (IDW Publishing, 2017)
 * Kingsman: The Red Diamond #1 (Image, 2017)
 * The Dark Knight III: The Master Race #9 (DC Comics, 2017)
 * Klaus and the Crisis in Xmasville #1 (Boom! Studios, 2017)
 * The Edge Off #1 (Cabal, 2018)
 * Kick-Ass vol. 4 #1 (Image, 2018)
 * JLA/Doom Patrol Special #1 (DC's Young Animal, 2018)
 * Mother Panic/Batman Special #1 (DC's Young Animal, 2018)
 * Shade/Wonder Woman Special #1 (DC's Young Animal, 2018)
 * Prodigy #1 (Image, 2018)
 * The Green Lantern #1 (DC Comics, 2019)
 * Sharkey the Bounty Hunter #1 (Image, 2019)
 * Action Comics #1013 (DC Comics, 2019)
 * Killtopia Volume 2 (BHP Comics, 2019)
 * American Jesus: The New Messiah #1 (Image, 2019)
 * The Batman's Grave #7 (DC Comics, 2020)
 * Detective Comics #1027 (DC Comics, 2020)
 * The Joker vol. 2 #1 (DC Comics, 2021)
 * Fire Power #12 (Image, 2021)
 * Jupiter's Legacy: Requien #1 (Image, 2021)
 * Gotham City Villains Anniversary Giant #1 (DC Comics, 2022)
 * Shaolin Cowboy: Cruel to be Kin #2 (Dark Horse, 2022)
 * Sgt. Rock Vs. The Army of the Dead #1 (DC Comics, 2022)
 * Nemesis: Reloaded #1 (Image, 2023)
 * The Ambassadors #2 (Image, 2023)
 * Batman: Dark Age #1 (DC Comics, 2024)
 * Batman: Gargoyle of Gotham #3 (DC Comics, 2024)

Interviews

 * An Audio Interview with Frank Quitely by SiDEBAR (Part I) and (Part II)

title: Frank Quitely


text:
Cook–Levin theorem

In computational complexity theory, the Cook–Levin theorem, also known as Cook's theorem, states that the Boolean satisfiability problem is NP-complete. That is, it is in NP, and any problem in NP can be reduced in polynomial time by a deterministic Turing machine to the Boolean satisfiability problem.

The theorem is named after Stephen Cook and Leonid Levin. The proof is due to Richard Karp, based on an earlier proof (using a different notion of reducibility) by Cook.

An important consequence of this theorem is that if there exists a deterministic polynomial-time algorithm for solving Boolean satisfiability, then every NP problem can be solved by a deterministic polynomial-time algorithm. The question of whether such an algorithm for Boolean satisfiability exists is thus equivalent to the P versus NP problem, which is still widely considered the most important unsolved problem in theoretical computer science.

Contributions
The concept of NP-completeness was developed in the late 1960s and early 1970s in parallel by researchers in North America and the USSR. In 1971, Stephen Cook published his paper "The complexity of theorem proving procedures" in conference proceedings of the newly founded ACM Symposium on Theory of Computing. Richard Karp's subsequent paper, "Reducibility among combinatorial problems", generated renewed interest in Cook's paper by providing a list of 21 NP-complete problems. Karp also introduced the notion of completeness used in the current definition of NP-completeness (i.e., by polynomial-time many-one reduction). Cook and Karp each received a Turing Award for this work.

The theoretical interest in NP-completeness was also enhanced by the work of Theodore P. Baker, John Gill, and Robert Solovay who showed, in 1975, that solving NP-problems in certain oracle machine models requires exponential time. That is, there exists an oracle A such that, for all subexponential deterministic-time complexity classes T, the relativized complexity class NPA is not a subset of TA. In particular, for this oracle, PA ≠ NPA.

In the USSR, a result equivalent to Baker, Gill, and Solovay's was published in 1969 by M. Dekhtiar. Later Leonid Levin's paper, "Universal search problems", was published in 1973, although it was mentioned in talks and submitted for publication a few years earlier.

Levin's approach was slightly different from Cook's and Karp's in that he considered search problems, which require finding solutions rather than simply determining existence. He provided six such NP-complete search problems, or universal problems. Additionally he found for each of these problems an algorithm that solves it in optimal time (in particular, these algorithms run in polynomial time if and only if P = NP).

Definitions
A decision problem is in NP if it can be decided by a non-deterministic Turing machine in polynomial time.

An instance of the Boolean satisfiability problem is a Boolean expression that combines Boolean variables using Boolean operators. Such an expression is satisfiable if there is some assignment of truth values to the variables that makes the entire expression true.

Idea
Given any decision problem in NP, construct a non-deterministic machine that solves it in polynomial time. Then for each input to that machine, build a Boolean expression that computes whether when that specific input is passed to the machine, the machine runs correctly, and the machine halts and answers "yes". Then the expression can be satisfied if and only if there is a way for the machine to run correctly and answer "yes", so the satisfiability of the constructed expression is equivalent to asking whether or not the machine will answer "yes".

Proof
This proof is based on the one given by .

There are two parts to proving that the Boolean satisfiability problem (SAT) is NP-complete. One is to show that SAT is an NP problem. The other is to show that every NP problem can be reduced to an instance of a SAT problem by a polynomial-time many-one reduction.

SAT is in NP because any assignment of Boolean values to Boolean variables that is claimed to satisfy the given expression can be verified in polynomial time by a deterministic Turing machine. (The statements verifiable in polynomial time by a deterministic Turing machine and solvable in polynomial time by a non-deterministic Turing machine are equivalent, and the proof can be found in many textbooks, for example Sipser's Introduction to the Theory of Computation, section 7.3., as well as in the Wikipedia article on NP).

Now suppose that a given problem in NP can be solved by the nondeterministic Turing machine $$M = (Q, \Sigma, s, F, \delta)$$, where $$Q$$ is the set of states, $$\Sigma$$ is the alphabet of tape symbols, $$s \in Q$$ is the initial state, $$F \subseteq Q$$ is the set of accepting states, and $$\delta \subseteq ((Q \setminus F) \times \Sigma) \times (Q \times \Sigma \times \{-1, +1\})$$ is the transition relation. Suppose further that $$M$$ accepts or rejects an instance of the problem after at most $$p(n)$$ computation steps, where $$n$$ is the size of the instance and $$p$$ is a polynomial function.

For each input, $$I$$, specify a Boolean expression $$B$$ that is satisfiable if and only if the machine $$M$$ accepts $$I$$.

The Boolean expression uses the variables set out in the following table. Here, $$q \in Q$$ is a machine state, $$-p(n) \leq i \leq p(n)$$ is a tape position, $$j \in \Sigma$$ is a tape symbol, and $$0 \leq k \leq p(n)$$ is the number of a computation step.

Define the Boolean expression $$B$$ to be the conjunction of the sub-expressions in the following table, for all $$-p(n) \leq i \leq p(n)$$ and $$0 \leq k \leq p(n)$$:

If there is an accepting computation for $$M$$ on input $$I$$, then $$B$$ is satisfiable by assigning $$T_{i,j,k}$$, $$H_{i,k}$$ and $$Q_{i,k}$$ their intended interpretations. On the other hand, if $$B$$ is satisfiable, then there is an accepting computation for $$M$$ on input $$I$$ that follows the steps indicated by the assignments to the variables.

There are $$O(p(n)^2)$$ Boolean variables, each encodable in space $$O(\log p(n))$$. The number of clauses is $$O(p(n)^3)$$ so the size of $$B$$ is $$O(\log(p(n)) p(n)^3)$$. Thus the transformation is certainly a polynomial-time many-one reduction, as required.

Only the first table row ($$T_{i,j,0}$$) actually depends on the input string $$I$$. The remaining lines depend only on the input length $$n$$ and on the machine $$M$$; they formalize a generic computation of $$M$$ for up to $$p(n)$$ steps.

The transformation makes extensive use of the polynomial $$p(n)$$. As a consequence, the above proof is not constructive: even if $$M$$ is known, witnessing the membership of the given problem in NP, the transformation cannot be effectively computed, unless an upper bound $$p(n)$$ of $$M$$'s time complexity is also known.

Complexity
While the above method encodes a non-deterministic Turing machine in complexity $$O(\log(p(n))p(n)^3)$$, the literature describes more sophisticated approaches in complexity $$O(p(n)\log(p(n)))$$. The quasilinear result first appeared seven years after Cook's original publication.

The use of SAT to prove the existence of an NP-complete problem can be extended to other computational problems in logic, and to completeness for other complexity classes. The quantified Boolean formula problem (QBF) involves Boolean formulas extended to include nested universal quantifiers and existential quantifiers for its variables. The QBF problem can be used to encode computation with a Turing machine limited to polynomial space complexity, proving that there exists a problem (the recognition of true quantified Boolean formulas) that is PSPACE-complete. Analogously, dependency quantified boolean formulas encode computation with a Turing machine limited to logarithmic space complexity, proving that there exists a problem that is NL-complete.

Consequences
The proof shows that every problem in NP can be reduced in polynomial time (in fact, logarithmic space suffices) to an instance of the Boolean satisfiability problem. This means that if the Boolean satisfiability problem could be solved in polynomial time by a deterministic Turing machine, then all problems in NP could be solved in polynomial time, and so the complexity class NP would be equal to the complexity class P.

The significance of NP-completeness was made clear by the publication in 1972 of Richard Karp's landmark paper, "Reducibility among combinatorial problems", in which he showed that 21 diverse combinatorial and graph theoretical problems, each infamous for its intractability, are NP-complete.

Karp showed each of his problems to be NP-complete by reducing another problem (already shown to be NP-complete) to that problem. For example, he showed the problem 3SAT (the Boolean satisfiability problem for expressions in conjunctive normal form (CNF) with exactly three variables or negations of variables per clause) to be NP-complete by showing how to reduce (in polynomial time) any instance of SAT to an equivalent instance of 3SAT.

Garey and Johnson presented more than 300 NP-complete problems in their book Computers and Intractability: A Guide to the Theory of NP-Completeness, and new problems are still being discovered to be within that complexity class.

Although many practical instances of SAT can be solved by heuristic methods, the question of whether there is a deterministic polynomial-time algorithm for SAT (and consequently all other NP-complete problems) is still a famous unsolved problem, despite decades of intense effort by complexity theorists, mathematical logicians, and others. For more details, see the article P versus NP problem.

title: Cook–Levin theorem


text:
Earthlight

Earthlight is a science fiction novel by British writer Arthur C. Clarke, published in 1955. It is an expansion to novel length of a novella of the same name that he had published four years earlier.

Overview
Earthlight is a science fiction adventure story set on the Moon, where a government agent is looking for a suspected spy at a major observatory on the Moon. The context is strong tension between Earth (which controls the Moon) and independent settlers elsewhere in the Solar System. The year is not given, but it is some time in the 22nd century. There have been no wars for the last 200 years.

Events are low-key: the government agent is a mild-mannered accountant who does not like the task. He notices the beauty of the Moon under 'earthlight'; the Earth in the sky far bigger than the Moon in the skies of Earth.

The story proceeds with very few violent incidents, though it does climax in a space battle. There is also an enigma - the apparent sighting of a 'beam of light', that should not be possible on the airless world. This is explained later in the story as a weapons beam that included metal particulates moving at high velocity.

Even though many of Clarke's science fiction novels take place in rather similar futures—Earthlight, A Fall of Moondust, The Sands of Mars, Rendezvous with Rama—the human background is never quite the same and they do not form a series.

Plot summary
The plot describes how political tension between the government of a politically united Earth (which maintains sovereignty over the Moon) and independent settlers and traders elsewhere in the Solar System who have formed a federation, erupts into warfare over the terms for the availability to the Federation of scarce heavy metals.

The trigger for hostilities is the publication of a research paper suggesting that the Moon may have previously unsuspected heavy metal resources which Earth proposes to monopolise. The Earth government's intelligence agency suspects that confidential information concerning the exploitation of these mineral riches may be being leaked to the Federation and presses an accountant, Bertram Sadler, into service. Sadler is sent to the Moon's main astronomical observatory located near the crater of Plato as a tip off has suggested that information is being routed through that location. Sadler's cover story is that he is carrying out an investigation of waste in government spending.

The rising political tension is accompanied by the observatory staff enjoying the good fortune of observing a nearby supernova explosion in the constellation of Draco.

Despite a relatively long preceding era of peace, Earth and the Federation each prepare technologically for war. The Federation develops a new method of spacedrive propulsion while Earth develops new shielding technology and a weapon which uses an electromagnet-propelled bayonet of liquid metal. (The weapon mistaken for a beam of light).

A climactic battle between three Federation cruisers and the fortified mining installation ("Project Thor") is played out near Mount Pico close to the lunar observatory. Two astronomers who have delivered a top Earth scientist to Pico with only a couple of hours to spare, witness the battle. Sadler, whose investigations have had no pay off except for the unmasking of an embezzling store manager, relinquishes his cover by going to debrief the two astronomers.

Of the three Federal cruisers, two are destroyed along with the mine in the battle. The third cruiser, named The Acheron, is terminally damaged and retreats towards Mars, but has little chance of reaching it before her nuclear reactor explodes. However, her new drive gives her the capability of a rendezvous with a passenger liner, The Pegasus, which is able to rescue all but one of the crew who have to make the 40 second crossing without space suits.

This inconclusive duel between mother planet and formerly dependent colonists, with each side suffering stiffer resistance than anticipated, discredits the governments on both sides. Sadler is able to return to civilian life but suffers nagging frustration that he never found out whether the spy that he was searching for existed or not. Many years later the commander of the Acheron writes his memoirs and reveals that information had reached the Federation from "One of Earth's most distinguished astronomers, now living in honoured retirement on the Moon". With this hint, Sadler is able to confirm the spy's identity as Robert Molton, the first one of the observatory staff to greet him on his way to the observatory. The novel concludes with Molton enlightening Sadler and the reader as to the brilliant technical subterfuge with which he transmitted information, namely that he used the observatory's main telescope as a transmitter by placing a modulated ultra-violet source at its prime focus. The signal was received by a Federation spaceship a few million kilometers away.

Reception
Groff Conklin characterized Earthlight as "a fairly standard type of melodrama [but] developed with all of the author's abundant ability to make even melodrama plausible." Floyd C. Gale stated that the novel had "some of the most inspired descriptive writing in or out of science fiction ... a thoroughgoing delight ... Worth reading and rereading". Anthony Boucher praised the novel as a convincingly real, scientifically detailed story of the near future, yet infused with that sense of wonder and excitement that we sometimes think vanished from literature about the time our voices changed."

At the time of the film 2001: A Space Odyssey, Lester del Rey expressed regret in his review of the film that Earthlight had not been filmed instead.

The weapon developed in the story by Earth, which uses an electromagnet-propelled bayonet of liquid metal, is said to have inspired DARPA to develop a weapon along the same lines.

title: Earthlight


text:
The Call of Duty (comics)

The Call of Duty was a series of short-lived Marvel Comics series featuring firefighters and emergency service workers dealing with paranormal phenomena in the Marvel Universe.

It was conceived in the wake of the September 11, 2001 attack and consists of three limited series (The Call of Duty: The Brotherhood #1–6, by Chuck Austen and David Finch; The Call of Duty: The Precinct #1–5, by Bruce Jones and Tom Mandrake; The Call of Duty: The Wagon #1–4 by Chuck Austen and Danijel Zezelj ) and a short unsuccessful run as an ongoing series in 2002–2003, that lasted four issues (by Chuck Austen and Pat Olliffe).

title: The Call of Duty (comics)