title: Martín Cáceres

text: On 25 May 2012, Cáceres joined Juventus on a permanent basis, signing a four-year contract. He missed most of the first half of the 2014–15 campaign, after picking up an injury in a 3–2 home win over Roma in October 2014 which sidelined him for three months. He returned to the starting line-up on 11 January 2015, scoring in a 3–1 defeat of Napoli at the Stadio San Paolo. On 8 March, however, he fractured his ankle during a training session, requiring surgery; as a result, he was ruled out indefinitely.


title: Martín Cáceres

text: Cáceres returned to the starting line-up on 8 August 2015, in Juventus's 2–0 victory over Lazio in the Supercoppa Italiana. In late September, however, he had his driving license revoked for six months and was suspended by the club after crashing his Ferrari into a bus stop in Turin following a wrong turn. He was alleged to have been over the legal alcohol limit for driving.


title: Martín Cáceres

text: Cáceres returned to action on 6 January 2016, starting and playing 84 minutes in a 3–0 home win against Hellas Verona. On 4 February, it was confirmed that he would be sidelined for the rest of the season due to the rupture of his right Achilles tendon during the match the day before against Genoa. In May, Juventus' director of football Giuseppe Marotta announced that the player's contract would not be renewed, and that he would be leaving the club on 30 June.


title: Martín Cáceres

text: On 16 February 2017, Cáceres joined Premier League club Southampton as a free agent, signing an initial contract lasting until the end of the season. Ten days later he was included in a matchday squad for the first time, remaining an unused substitute in the final of the EFL Cup, a 3–2 loss to Manchester United.


title: Martín Cáceres

text: After three months at St Mary's Stadium, Cáceres appeared in his first and only game for Southampton, starting in a 2–1 away win over Middlesbrough on 13 May. He was released on 25 May.


title: Martín Cáceres

text: Cáceres returned to Italy in early August 2017, signing for Hellas Verona on a deal running until 30 June 2018. However, it was reported that Lazio had already expressed an interest in acquiring his services, but were ultimately unable to as their quota of two non-EU signings from abroad had already been filled. He joined Verona instead as a result, although rumours circulated in the media linking him with a possible move to Lazio in January as there were no restrictions impeding the transfer of non-EU players between Italian clubs.


title: Martín Cáceres

text: On 8 January 2018, Cáceres joined Lazio on a one-year contract with the option for a renewal. He scored his only goal of the campaign on 18 April in a 4–3 away defeat of Fiorentina, featuring sparingly for the fifth-placed team.


title: Martín Cáceres

text: On 29 January 2019, Juventus announced the signing of Cáceres on a five-month loan, marking this as his third spell at the club. He made his first appearance since 2016 on 2 February, in a 3–3 home draw against Parma in the domestic league.


title: Martín Cáceres

text: On 30 August 2019, Cáceres joined Fiorentina on a one-year deal with the option for a further season. The following 18 May, he declared that he had tested positive for COVID-19 for 60 days.


title: Martín Cáceres

text: Cáceres moved to Cagliari on a one-year contract on 1 September 2021, where he shared teams with national teammates Diego Godín and Nahitan Nández. He scored his only goal for the Sardinians on 17 October, the second in a 3–1 home victory over Sampdoria.


title: Martín Cáceres

text: Cáceres returned to the Spanish top tier on 30 January 2022, with the free agent agreeing to a one-year deal at Levante with the possibility of a one-year extension. Nine of his appearances were starts, in an eventual relegation; the same happened to his previous side, Cagliari.


title: Martín Cáceres

text: On 24 August 2022, Cáceres joined Major League Soccer club LA Galaxy as a free agent until the end of the season.


title: Martín Cáceres

text: Cáceres played for Uruguay at the 2007 FIFA U-20 World Cup held in Canada, appearing in all the matches in an eventual round-of-16 exit. Previously, he was named the best defender at the 2007 South American U-20 Championship, helping his team to third place.


title: Martín Cáceres

text: Cáceres made his debut for the full side on 12 September 2007, in Johannesburg against South Africa. In 2010, he was selected by manager Óscar Tabárez for his list of 23 players for that year's FIFA World Cup, as back-up to Porto's Jorge Fucile at right-back and taking part in the semi-finals against the Netherlands and the third-place match against Germany (both 3–2 losses).


title: Martín Cáceres

text: Cáceres scored his first international goal on 23 June 2011, in a 3–0 friendly home win over Estonia. With Fucile not being selected to the 2011 Copa América in Argentina, he played five out of six games during the tournament – as right and left back – with the "Charrúas" winning their 15th continental tournament. He notably scored the decisive penalty in a penalty shootout victory over the hosts in the quarter-finals.


title: Martín Cáceres

text: Cáceres made four appearances for Uruguay in the 2013 FIFA Confederations Cup. In the third-place match against Italy, his penalty was saved by Juventus teammate Gianluigi Buffon as the opposition won the shootout by 3–2 after a 2–2 deadlock following extra time. He was also a member of the squad that took part at the 2014 World Cup, featuring four times in the tournament to help the nation reach the last-16 phase, where they suffered a 2–0 defeat to Colombia at the Maracanã Stadium on 28 June.


title: Martín Cáceres

text: Due to an ankle injury suffered in March 2015, Cáceres was excluded from the 2015 Copa América squad. A serious Achilles tendon ailment in February 2016 also prevented his participation in the Copa América Centenario the following summer.


title: Martín Cáceres

text: Cáceres was included in Uruguay's final 23-man squad for the 2018 World Cup in Russia. He played all the matches and minutes in a quarter-final exit.


title: Martín Cáceres

text: In March 2019, Tabárez named Cáceres in the squad for the 2019 Copa América in Brazil. On 16 June, he provided the assist to Luis Suárez's goal in a 4–0 win over Ecuador in the team's opening group match of the tournament. He appeared four times in total throughout the tournament, including the quarter-final tie against Peru in which his team were eliminated from the competition after losing 5–4 on penalties, following a 0–0 draw after regulation time.


title: Martín Cáceres

text: Cáceres earned his 100th cap on 13 October 2020, in a 4–2 loss to Ecuador in the 2022 World Cup qualifiers.


title: Martín Cáceres

text: Cáceres's primary position is that of a central defender, but he can also operate as a left or right-back, being noted for his pace, strength, tackling and aerial ability. He has also been deployed as a wing-back or winger, on either side of the pitch.


title: Martín Cáceres

text: During his time with Juventus, Cáceres' tenacious approach drew comparisons to compatriot Montero, who also played there.


title: Charlie Adam

text: Charles Graham Adam (born 10 December 1985) is a Scottish former professional footballer who played as a midfielder.


title: Charlie Adam

text: Adam started his senior career with Rangers. He spent much of his early senior career on loan to Ross County and St Mirren. While on loan to St Mirren during the 2005–06 season, Adam was part of the team that won both the Scottish Challenge Cup and First Division, playing in over thirty matches for the "Saints". Upon returning to Rangers at the end of the 2006 season, he became a regular under managers Paul Le Guen and Walter Smith. Adam was also part of the Rangers team that reached the 2008 UEFA Cup Final.


title: Charlie Adam

text: After falling out of favour at Rangers during 2008–09, Adam was loaned out to English Championship side Blackpool. The loan was made permanent at the start of the 2009–10 season, and Adam became an integral part of the Blackpool side, captaining the team to a play-off victory over Cardiff City which resulted in their promotion to the Premier League. Adam shone in his first season in the Premier League, his performances being recognised with a nomination for the PFA Players' Player of the Year in April 2011; however, Blackpool were relegated the following month, and he transferred to Liverpool in July.


title: Charlie Adam

text: After one season at Anfield, Adam joined Stoke City in August 2012 for a fee of £4 million. He spent seven seasons with Stoke, making 179 appearances, before joining Reading in July 2019. Adam returned to his boyhood club Dundee in September 2020. He would captain them for two years and led them to promotion to the Scottish Premiership.


title: Charlie Adam

text: Adam was part of the Dundee and Rangers youth set-ups. He won the Scottish Youth Cup with Rangers in 2001–02. He made his senior debut for Rangers on 14 April 2004 against Livingston. He played sporadically for Rangers thereafter, but did not become a first-team regular until the management of Paul Le Guen.


title: Charlie Adam

text: Adam made an appearance at the start of the 2004–05 season, and then joined First Division club Ross County on loan for the rest of that campaign. While in Dingwall, Adam made fifteen appearances, scoring twice, against Raith Rovers and St Mirren. He also played in the 2004 Scottish Challenge Cup Final against Falkirk. County were ahead courtesy of David Winters 56th-minute goal. Adam was substituted in the 60th minute as was teammate Sean Higgins six minutes later. Neil Scally and Darryl Duffy then scored in 70 and 75 minutes respectively for Falkirk to come back to lift the trophy.


title: Charlie Adam

text: Adam then joined First Division side St Mirren on loan for the 2005–06 season. During his spell in Paisley, he again appeared in the 2005 Scottish Challenge Cup Final, however this time he was on the winning side as St Mirren defeated Hamilton Academical 2–1. He was also part of the team that won the First Division title in May 2006. In total he made 37 appearances and scored nine goals for the "Buddies" during a very successful stint in Paisley.


title: Charlie Adam

text: After playing for Rangers during a July 2006 pre-season trip to South Africa, including scoring a hat-trick against a Jomo Select side, Adam began the 2006–07 season with the club, and he started in their first league match against Motherwell. He continued to be a regular for Rangers as the season progressed, and scored his first goal for the club in a 3–2 UEFA Cup win over Livorno on 19 October. This was one of fourteen goals Adam scored that season, including strikes against Israeli sides Hapoel Tel Aviv and Maccabi Haifa, as well as a goal in an end-of-season Old Firm match from a free-kick. On 16 April 2007, at the end of his first season in the first team, Adam was voted by Rangers fans as the club's Young Player of the Year at an award ceremony.


title: Charlie Adam

text: On 28 June 2007, it was announced that Adam had signed a new five-year contract with Rangers. Adam had scored his first UEFA Champions League goal on 19 September against VfB Stuttgart, and his second in the return fixture. Adam played against Panathinaikos and Werder Bremen in Rangers run to the 2008 UEFA Cup Final where he was an unused substitute as Rangers lost 2–0 to Zenit Saint Petersburg. That season Rangers also won the 2007–08 Scottish League Cup and the 2007–08 Scottish Cup, but Adam was suspended for the League Cup final against Dundee United and left out of the squad for the Scottish Cup final against Queen of the South.


title: Charlie Adam

text: On 2 February 2009, Adam signed on loan with English Championship club Blackpool until the end of the 2008–09 season, after caretaker manager Tony Parkes initially enquired about having Alan Gow loaned to the club again. Five days later, after an altercation with Richie Wellens, he was sent off on his debut for "the Seasiders" during a 3–2 defeat by Doncaster Rovers at Bloomfield Road.


title: Charlie Adam

text: On 25 February, after serving a three-match suspension, Adam scored twice, including a goal from just inside the halfway line, for the club's reserve team as they beat Accrington Stanley 4–2. Adam scored his first Football League goal for Blackpool in a 2–0 victory over Norwich City at Bloomfield Road on 7 March. Two days later he was named in the Football League's "Championship Team of the Week". His second goal for Blackpool came on 11 April in their 1–0 West Lancashire derby win over Preston North End at Deepdale. Tony Parkes publicly admitted he was keen to sign Adam permanently. He said, "I have spoken to our club secretary and we will be in touch with Rangers to see if we can do a deal for Charlie". At the end of the season, after two goals in 13 appearances, he returned to Rangers.


title: Charlie Adam

text: On 6 July 2009, Rangers confirmed that they had accepted a bid from Blackpool and that he was set to sign for "the Seasiders" the following day, subject to personal terms being agreed. The following day, Blackpool manager Ian Holloway confirmed a deal had been agreed with the Ibrox club, but stressed that talks with Adam would take place on 9 July, once Adam had spoken with Rangers manager Walter Smith. Four weeks later on 2 August, with Adam still a Rangers player and having been on their pre-season trip to Germany, Smith confirmed that the player was in talks with Blackpool, saying, "Adam is talking to Blackpool at the moment. The clubs have agreed a fee and it will be up to Charlie and his representative."


title: Charlie Adam

text: In August 2009 Adam signed a two-year contract with Blackpool with the option of a further year, in a club-record transfer fee of £500,000. His competitive debut as a permanent Blackpool player was in a 1–1 draw with Queens Park Rangers at Loftus Road later that month. His first goal came on 26 August in a 4–1 home win over Wigan Athletic in the League Cup.


title: Charlie Adam

text: In January 2010 he was named in the Press Association's Championship "Team of the Week", along with teammate Neal Eardley, following his performance two days before in the 3–2 home victory over Watford. That same month Adam won the Championship Player of the Month award and was voted the PFA Championship Fans' Player of the Month for January 2010. In March, Adam scored in his 50th league appearance for Blackpool, a 2–0 win at Plymouth Argyle. Just under a month later, Adam was named in the PFA "Championship Team of the Year".


title: Charlie Adam

text: Blackpool qualified for that season's play-offs. Adam scored a match-winning penalty against Nottingham Forest in the semi-final first leg, and Blackpool won the tie on aggregate. In the play-off final Adam scored a free-kick as Blackpool beat Cardiff City 3–2 at Wembley Stadium and gained promotion to the Premier League.


title: Charlie Adam

text: In the fourth week of the 2010–11 Premier League season, Adam was named in the official Team of the Week. He was named again in week seven, alongside teammate Luke Varney. In early December, a tribunal ruled in favour of Adam in a dispute over unpaid bonuses from the previous season. They found that Blackpool were required to pay the player £25,000 for successfully avoiding relegation to League One. The club had argued that the promotion bonus (which had been stipulated in Adam's contract) superseded the 'survival bonus.' The panel upheld Adam's complaint, but did not agree with his second argument that the failure to pay the amount constituted a breach of contract on the club's part. As a result, both parties were responsible for their own legal fees – leaving Adam with a very small net gain from the venture.


title: Charlie Adam

text: In January 2011, Blackpool rejected a £4.5 million bid from Liverpool to buy Adam, an amount described by manager Ian Holloway as "disgraceful". Blackpool then rejected a transfer request that Adam had made to the club. He was one of seven nominees for the 2010–11 PFA Players' Player of the Year. However, Adam and his Blackpool teammates were relegated at the season's end. In May 2011, Blackpool activated an option to increase Adam's contract by a further twelve months.


title: Charlie Adam

text: In July 2011 Liverpool signed Adam from Blackpool for a fee of £6.75 million after having their initial bid rejected. He played the second half of Liverpool's friendly against Guangdong Sunray Cave in a 4–3 win on 13 July. On 13 August 2011, Adam made his full debut in Liverpool's first game of the 2011–12 Premier League, playing the full 90 minutes in a 1–1 draw against Sunderland at Anfield. There, he assisted the opening goal from Luis Suárez from a free kick outside the area.


title: Charlie Adam

text: On 27 August 2011, Adam scored his first Premier League goal for Liverpool in a match against Bolton Wanderers which Liverpool went on to win 3–1. On 18 September 2011, Adam was shown a second yellow card for a challenge on Scott Parker in a 4–0 defeat at Tottenham Hotspur. Adam helped Liverpool beat local rivals Everton 2–0 on 1 October 2011. He then scored a penalty in a 2–0 win away at West Bromwich Albion. Liverpool reached the 2012 Football League Cup Final where they defeated Championship side Cardiff City in a penalty shootout, despite Adam missing his kick. They also made it to the 2012 FA Cup Final, which Adam missed due to injury as they lost 2–1 to Chelsea.


title: Charlie Adam

text: On 31 August 2012 Adam signed a four-year contract at Stoke City for an undisclosed fee believed to be around £4 million. He made his debut for Stoke the following day in a 2–2 draw against Wigan Athletic. After leaving Liverpool, Adam stated he left in order to play more games. He scored his first goal for Stoke in a 1–0 victory against Queens Park Rangers on 10 November 2012. He scored in another 1–0 win for Stoke two weeks later against Fulham. Adam was given compassionate leave after his father died on 17 December 2012. Adam then spent three months in and out of the team before returning to the starting line up in April and scored the winning goal against Norwich City which eased Stoke's relegation worries. He dedicated his goal to his father. Adam ended the 2012–13 season a poor note, being sent off against Tottenham Hotspur for two bookable offences.


title: Charlie Adam

text: Adam scored Stoke's first goal of the 2013–14 season as City beat Crystal Palace 2–1 on 24 August 2013. Adam then scored a penalty in a 3–3 draw with Swansea City and then in the next match against Sunderland in a 2–0 victory. Adam came off the bench to help Stoke to a 2–1 victory over Aston Villa on 21 December 2013 which earned him praise from manager Mark Hughes for his contribution. He scored Stoke's second goal against his former club Liverpool on 12 January 2014 in a 5–3 defeat. On 1 February 2014, Adam scored twice in a 2–1 win over Manchester United to help Stoke record a first league win over the Red Devils since 1984. His first goal however was later ruled by the Premier League as a Michael Carrick own goal. In the next match against Southampton on 8 February, Adam provided assists for both Stoke's goals in a 2–2 draw. He was given a three-match retrospective ban by the FA following an altercation with Olivier Giroud in a 1–0 win over Arsenal on 1 March 2014. On 11 May 2014 Adam scored the winning goal against West Bromwich Albion on the final day of the season which earned Stoke 9th place in the Premier League.


title: Charlie Adam

text: Adam scored his first goals of the 2014–15 against Sunderland and Swansea City. In November 2014 he lost his place in the side due to the form of Bojan Krkić. On 4 April 2015 Adam scored from 65 yards against Chelsea in a 2–1 defeat. Adam described his goal as a "once in a lifetime goal". Adam then went and scored four more goals in April and May including on the final day of the season in a 6–1 victory against his former club Liverpool. Adam ended the season by playing and scoring for Dundee in Julián Speroni's testimonial. Adam signed a new contract with Stoke on 20 June 2015.


title: Charlie Adam

text: Adam made 25 appearances in 2015–16, 11 of which were as a substitute. He scored once, on 30 April 2016 against Crystal Palace. He played 28 times in 2016–17, as Stoke finished in 13th position. Adam again scored one goal which was a penalty against Arsenal in December 2016. Adam struggled for playing time in 2017–18, making 14 appearances, as Stoke suffered relegation to the EFL Championship. Adam made two noticeable errors during Stoke failed survival bid as he saw his last minute penalty saved against Brighton & Hove Albion in February and was needlessly sent-off against Everton in March. Following relegation Adam criticised the behaviour of some of his teammates.


title: Charlie Adam

text: Adam struggled for playing time under Gary Rowett in the first half of the 2018–19 season before he was brought back into the squad by Nathan Jones in January 2019, before a calf injury disrupted his comeback. Adam revealed in April 2019 that he wanted to extend his contract at Stoke. However, after talks with manager Nathan Jones, Adam left Stoke at the end of the season.


title: Charlie Adam

text: Adam met with Blackpool's new owner Simon Sadler about possibly rejoining the club and hopefully finishing his career there. "But, out of the blue, I got an opportunity to go and play for Reading in the Championship."


title: Charlie Adam

text: On 22 July 2019, Adam joined Reading, signing a one-year contract. He scored his first goal for Reading from the penalty spot in a 3–0 victory over Derby County at the Madejski Stadium on 21 December 2019. Adam was released by Reading at the end of the season.


title: Charlie Adam

text: On 15 September 2020, after training on his own and having considered an offer to play in Australia, Adam joined his boyhood club Dundee on a two-year deal, which he described as being a "dream" of his. In an interview with "FourFourTwo", Adam stated he believed Dundee would be his final club, due to his age and happiness at Dens Park.


title: Charlie Adam

text: Adam made his debut for Dundee in a Scottish League Cup tie against Brora Rangers on 10 October 2020. His first goal followed in his first league game in a defeat to Hearts. After a successful start to his time with "the Dee", during which he was made captain, Adam was awarded the SPFL's Championship Player of the Month for December. At the end of the season, Adam was named to PFA Scotland's Championship Team of the Year for 2020–21, as well as being shortlisted as a finalist for their Championship Player of the Year. Adam would captain the club to winning the Premiership play-offs and helping Dundee to a return to the Scottish Premiership. Afterwards, Adam described this achievement as the highlight of his career, stating: "This is the best. I've dreamed of playing for my boyhood club in the top division and we've managed to get there." Adam would also be included in the SPFL's Championship Team of the Season, and would be named the Championship Player of the Year.


title: Charlie Adam

text: Adam sustained a groin injury in a Premiership match against Motherwell in August 2021, and despite avoiding surgery he was ruled out for six weeks. He would make his return in October and be nominated Man of the Match in a home win over Aberdeen. Adam's final few months of the season would feature various highs and lows, with a dive against St Johnstone and an unfortunate "assist" for St Mirren both going viral along with the club being relegated, but also scoring impressive goals against Dundee United and in the club's final home game of the season against Hibernian, with the Dundee derby goal winning Dundee's Goal of the Season. Adam left Dundee following the end of his contract in May 2022.


title: Charlie Adam

text: On 21 September 2022, Adam officially announced his retirement from professional football at the age of 36.


title: Charlie Adam

text: Adam was called up by national manager Alex McLeish to the Scotland squad for the first time on 11 May 2007 for a friendly against Austria and a Euro 2008 qualifying Group B match against the Faroe Islands. He made his debut at the Gerhard Hanappi Stadium in Vienna on 30 May as a 67th-minute substitute, in a 1–0 victory. He then made his competitive debut as a 77th-minute substitute in a 2–0 win over the Faroe Islands at the Svangaskarð stadium in Toftir on 6 June.


title: Charlie Adam

text: Adam was recalled to the Scotland squad by manager George Burley for a friendly against Japan on 10 October 2009 at the Nissan Stadium, Yokohama, Japan. On 10 November 2009, Adam was drafted into the Scotland squad for the friendly against Wales four days later.


title: Charlie Adam

text: He replaced his former Rangers teammate Kevin Thomson, who pulled out due to illness. George Burley said of Adam: "He scored a cracking goal for Blackpool last weekend. He is a very good replacement. He has been playing consistently, is a quality player, and is getting a real benefit from playing regularly."


title: Charlie Adam

text: Adam is the son of former professional player Charlie Adam, a midfielder who played for various Scottish clubs in the 1980s and 1990s. His father died on 17 December 2012, aged 50; the cause of death was publicly revealed as suicide in 2015. Adam's mother, Eleanor, fell ill with cancer, and was a big factor in his move back to Scotland. She died in December 2020. Adam's brother, Grant, is a goalkeeper who is currently playing for fellow Dundee-based side Lochee United. He has four other siblings: brothers Gary, Connor and Grant, and sister Nicola.


title: Charlie Adam

text: Adam grew up supporting Dundee. He attended Dundee's Braeview Academy, alongside fellow future footballers Garry Kenneth and Scott Robertson (the latter of which he would be reunited with at Dundee).


title: Charlie Adam

text: Adam is married to Sophie Anderson, whom he met in September 2009 in her hometown of Poulton-le-Fylde during a night out with some of his Blackpool teammates. They have two sons, Jack and Louis, and a daughter, Anabella. He moved with his family to Glasgow after signing for Dundee, having lived in Poulton-le-Fylde since his days at Blackpool.


title: Empty set

text: In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other theories, its existence can be deduced. Many possible properties of sets are vacuously true for the empty set.


title: Empty set

text: In some textbooks and popularizations, the empty set is referred to as the "null set". However, null set is a distinct notion within the context of measure theory, in which it describes a set of measure zero (which is not necessarily empty). The empty set may also be called the void set.


title: Empty set

text: Common notations for the empty set include "{}", "formula_1", and "∅". The latter two symbols were introduced by the Bourbaki group (specifically André Weil) in 1939, inspired by the letter Ø in the Danish and Norwegian alphabets. In the past, "0" was occasionally used as a symbol for the empty set, but this is now considered to be an improper use of notation.


title: Empty set

text: The symbol ∅ is available at Unicode point U+2205. It can be coded in HTML as and as . It can be coded in LaTeX as . The symbol formula_1 is coded in LaTeX as .


title: Empty set

text: When writing in languages such as Danish and Norwegian, where the empty set character may be confused with the alphabetic letter Ø (as when using the symbol in linguistics), the Unicode character U+29B0 REVERSED EMPTY SET ⦰ may be used instead.


title: Empty set

text: In standard axiomatic set theory, by the principle of extensionality, two sets are equal if they have the same elements. As a result, there can be only one set with no elements, hence the usage of "the empty set" rather than "an empty set".


title: Empty set

text: By the definition of subset, the empty set is a subset of any set "A". That is, element "x" of formula_10 belongs to "A". Indeed, if it were not true that every element of formula_10 is in "A", then there would be at least one element of formula_10 that is not present in "A". Since there are elements of formula_10 at all, there is no element of formula_10 that is not in "A". Any statement that begins "for every element of formula_10" is not making any substantive claim; it is a vacuous truth. This is often paraphrased as "everything is true of the elements of the empty set."


title: Empty set

text: When speaking of the sum of the elements of a finite set, one is inevitably led to the convention that the sum of the elements of the empty set is zero. The reason for this is that zero is the identity element for addition. Similarly, the product of the elements of the empty set should be considered to be one (see empty product), since one is the identity element for multiplication.


title: Empty set

text: A derangement is a permutation of a set without fixed points. The empty set can be considered a derangement of itself, because it has only one permutation (formula_19), and it is vacuously true that no element (of the empty set) can be found that retains its original position.


title: Empty set

text: Since the empty set has no member when it is considered as a subset of any ordered set, every member of that set will be an upper bound and lower bound for the empty set. For example, when considered as a subset of the real numbers, with its usual ordering, represented by the real number line, every real number is both an upper and lower bound for the empty set. When considered as a subset of the extended reals formed by adding two "numbers" or "points" to the real numbers (namely negative infinity, denoted formula_20 which is defined to be less than every other extended real number, and positive infinity, denoted formula_21 which is defined to be greater than every other extended real number), we have that:


title: Empty set

text: That is, the least upper bound (sup or supremum) of the empty set is negative infinity, while the greatest lower bound (inf or infimum) is positive infinity. By analogy with the above, in the domain of the extended reals, negative infinity is the identity element for the maximum and supremum operators, while positive infinity is the identity element for the minimum and infimum operators.


title: Empty set

text: In any topological space "X", the empty set is open by definition, as is "X". Since the complement of an open set is closed and the empty set and "X" are complements of each other, the empty set is also closed, making it a clopen set. Moreover, the empty set is compact by the fact that every finite set is compact.


title: Empty set

text: If formula_24 is a set, then there exists precisely one function formula_25 from formula_10 to formula_27 the empty function. As a result, the empty set is the unique initial object of the category of sets and functions.


title: Empty set

text: The empty set can be turned into a topological space, called the empty space, in just one way: by defining the empty set to be open. This empty topological space is the unique initial object in the category of topological spaces with continuous maps. In fact, it is a strict initial object: only the empty set has a function to the empty set.


title: Empty set

text: In the von Neumann construction of the ordinals, 0 is defined as the empty set, and the successor of an ordinal is defined as formula_28. Thus, we have formula_29, formula_30, formula_31, and so on. The von Neumann construction, along with the axiom of infinity, which guarantees the existence of at least one infinite set, can be used to construct the set of natural numbers, formula_32, such that the Peano axioms of arithmetic are satisfied.


title: Empty set

text: In Zermelo set theory, the existence of the empty set is assured by the axiom of empty set, and its uniqueness follows from the axiom of extensionality. However, the axiom of empty set can be shown redundant in at least two ways:


title: Empty set

text: While the empty set is a standard and widely accepted mathematical concept, it remains an ontological curiosity, whose meaning and usefulness are debated by philosophers and logicians.


title: Empty set

text: The empty set is not the same thing as ; rather, it is a set with nothing it and a set is always . This issue can be overcome by viewing a set as a bag—an empty bag undoubtedly still exists. Darling (2004) explains that the empty set is not nothing, but rather "the set of all triangles with four sides, the set of all numbers that are bigger than nine but smaller than eight, and the set of all opening moves in chess that involve a king."


title: Empty set

text: is often used to demonstrate the philosophical relation between the concept of nothing and the empty set. Darling writes that the contrast can be seen by rewriting the statements "Nothing is better than eternal happiness" and "[A] ham sandwich is better than nothing" in a mathematical tone. According to Darling, the former is equivalent to "The set of all things that are better than eternal happiness is formula_10" and the latter to "The set {ham sandwich} is better than the set formula_10". The first compares elements of sets, while the second compares the sets themselves.


title: Empty set

text: George Boolos argued that much of what has been heretofore obtained by set theory can just as easily be obtained by plural quantification over individuals, without reifying sets as singular entities having other entities as members.


title: Deep sea

text: The deep sea is broadly defined as the ocean depth where light begins to fade, at an approximate depth of 200 metres (656 feet) or the point of transition from continental shelves to continental slopes. Conditions within the deep sea are a combination of low temperatures, darkness and high pressure The deep sea is considered the least explored Earth biome, with the extreme conditions making the environment difficult to access and explore.


title: Deep sea

text: Organisms living within the deep sea have a variety of adaptations to survive in these conditions. Organisms can survive in the deep sea through a number of feeding methods including scavenging, predation and filtration, with a number of organisms surviving by feeding on marine snow. Marine snow is organic material that has fallen from upper waters into the deep sea.


title: Deep sea

text: In 1960, the bathyscaphe "Trieste" descended to the bottom of the Mariana Trench near Guam, at , the deepest known spot in any ocean. If Mount Everest () were submerged there, its peak would be more than beneath the surface. After the "Trieste" was retired, the Japanese remote-operated vehicle (ROV) "Kaikō" was the only vessel capable of reaching this depth until it was lost at sea in 2003. In May and June 2009, the hybrid-ROV "Nereus" returned to the Challenger Deep for a series of three dives to depths exceeding .


title: Deep sea

text: Natural light does not penetrate the deep ocean, with the exception of the upper parts of the mesopelagic. Since photosynthesis is not possible, plants and phytoplankton cannot live in this zone, and as these are the primary producers of almost all of earth's ecosystems, life in this area of the ocean must depend on energy sources from elsewhere. Except for the areas close to the hydrothermal vents, this energy comes from organic material drifting down from the photic zone. The sinking organic material is composed of algal particulates, detritus, and other forms of biological waste, which is collectively referred to as marine snow.


title: Deep sea

text: Because pressure in the ocean increases by about 1 atmosphere for every 10 meters of depth, the amount of pressure experienced by many marine organisms is extreme. Until recent years, the scientific community lacked detailed information about the effects of pressure on most deep sea organisms because the specimens encountered arrived at the surface dead or dying and weren't observable at the pressures at which they lived. With the advent of traps that incorporate a special pressure-maintaining chamber, undamaged larger metazoan animals have been retrieved from the deep sea in good condition.


title: Deep sea

text: Salinity is remarkably constant throughout the deep sea, at about 35 parts per thousand. There are some minor differences in salinity, but none that are ecologically significant, except in the Mediterranean and Red Seas.


title: Deep sea

text: The two areas of greatest temperature gradient in the oceans are the transition zone between the surface waters and the deep waters, the thermocline, and the transition between the deep-sea floor and the hot water flows at the hydrothermal vents. Thermoclines vary in thickness from a few hundred meters to nearly a thousand meters. Below the thermocline, the water mass of the deep ocean is cold and far more homogeneous. Thermoclines are strongest in the tropics, where the temperature of the epipelagic zone is usually above 20 °C. From the base of the epipelagic, the temperature drops over several hundred meters to 5 or 6 °C at 1,000 meters. It continues to decrease to the bottom, but the rate is much slower. The cold water stems from sinking heavy surface water in the polar regions.


title: Deep sea

text: At any given depth, the temperature is practically unvarying over long periods of time, without seasonal changes and with very little interannual variability. No other habitat on earth has such a constant temperature.


title: Deep sea

text: In hydrothermal vents the temperature of the water as it emerges from the "black smoker" chimneys may be as high as 400 °C (it is kept from boiling by the high hydrostatic pressure) while within a few meters it may be back down to 2 to 4 °C.


title: Deep sea

text: Regions below the epipelagic are divided into further zones, beginning with the "bathyal zone" (also considered the "continental slope") which spans from 200 to 3000 meters below sea level and is essentially transitional, containing elements from both the shelf above and the abyss below. Below this zone, the deep sea consists of the "abyssal zone" which occurs between the ocean depths of 3000 and 6000 metres and the "hadal zone" (6000 - 11,000 meters). Food consists of falling organic matter known as 'marine snow' and carcasses derived from the productive zone above, and is scarce both in terms of spatial and temporal distribution.


title: Deep sea

text: Instead of relying on gas for their buoyancy, many deep-sea species have jelly-like flesh consisting mostly of glycosaminoglycans, which provides them with very low density. It is also common among deep water squid to combine the gelatinous tissue with a flotation chamber filled with a coelomic fluid made up of the metabolic waste product ammonium chloride, which is lighter than the surrounding water.


title: Deep sea

text: The midwater fish have special adaptations to cope with these conditions—they are small, usually being under ; they have slow metabolisms and unspecialized diets, preferring to sit and wait for food rather than waste energy searching for it. They have elongated bodies with weak, watery muscles and skeletal structures. They often have extendable, hinged jaws with recurved teeth. Because of the sparse distribution and lack of light, finding a partner with which to breed is difficult, and many organisms are hermaphroditic.


title: Deep sea

text: Because light is so scarce, fish often have larger than normal, tubular eyes with only rod cells. Their upward field of vision allows them to seek out the silhouette of possible prey. Prey fish however also have adaptations to cope with predation. These adaptations are mainly concerned with reduction of silhouettes, a form of camouflage. The two main methods by which this is achieved are reduction in the area of their shadow by lateral compression of the body, and counter illumination via bioluminescence. This is achieved by production of light from ventral photophores, which tend to produce such light intensity to render the underside of the fish of similar appearance to the background light. For more sensitive vision in low light, some fish have a retroreflector behind the retina. Flashlight fish have this plus photophores, which combination they use to detect eyeshine in other fish (see "tapetum lucidum").


title: Deep sea

text: Organisms in the deep sea are almost entirely reliant upon sinking living and dead organic matter which falls at approximately 100 meters per day. In addition, only about 1 to 3% of the production from the surface reaches the sea bed mostly in the form of marine snow. Larger food falls, such as whale carcasses, also occur and studies have shown that these may happen more often than currently believed. There are many scavengers that feed primarily or entirely upon large food falls and the distance between whale carcasses is estimated to only be 8 kilometers. In addition, there are a number of filter feeders that feed upon organic particles using tentacles, such as "Freyella elegans".


title: Deep sea

text: Marine bacteriophages play an important role in cycling nutrients in deep sea sediments. They are extremely abundant (between 5×10 and 1×10 phages per square meter) in sediments around the world.


title: Deep sea

text: Despite being so isolated deep sea organisms have still been harmed by human interaction with the oceans. The London Convention aims to protect the marine environment from dumping of wastes such as sewage sludge and radioactive waste. A study found that at one region there had been a decrease in deep sea coral from 2007 to 2011, with the decrease being attributed to global warming and ocean acidification, and biodiversity estimated as being at the lowest levels in 58 years. Ocean acidification is particularly harmful to deep sea corals because they are made of aragonite, an easily soluble carbonate, and because they are particularly slow growing and will take years to recover. Deep sea trawling is also harming the biodiversity by destroying deep sea habitats which can take years to form. Another human activity that has altered deep sea biology is mining. One study found that at one mining site fish populations had decreased at six months and at three years, and that after twenty six years populations had returned to the same levels as prior to the disturbance.


title: Deep sea

text: There are a number of species that do not primarily rely upon dissolved organic matter for their food. These species and communities are found at hydrothermal vents at sea-floor spreading zones. One example is the symbiotic relationship between the tube worm "Riftia" and chemosynthetic bacteria. It is this chemosynthesis that supports the complex communities that can be found around hydrothermal vents. These complex communities are one of the few ecosystems on the planet that do not rely upon sunlight for their supply of energy.


title: Deep sea

text: Deep sea fish have different adaptations in their proteins, anatomical structures, and metabolic systems to survive in the Deep sea, where the inhabitants have to withstand great amount of hydrostatic pressure. While other factors like food availability and predator avoidance are important, the deep-sea organisms must have the ability to maintain well-regulated metabolic system in the face of high pressures. In order to adjust for the extreme environment, these organisms have developed unique characteristics.


title: Deep sea

text: Proteins are affected greatly by the elevated hydrostatic pressure, as they undergo changes in water organization during hydration and dehydration reactions of the binding events. This is due to the fact that most enzyme-ligand interactions form through charged or polar non-charge interactions. Because hydrostatic pressure affects both protein folding and assembly and enzymatic activity, the deep sea species must undergo physiological and structural adaptations to preserve protein functionality against pressure.


title: Deep sea

text: Actin is a protein that is essential for different cellular functions. The α-actin serves as a main component for muscle fiber, and it is highly conserved across numerous different species. Some Deep-sea fish developed pressure tolerance through the change in mechanism of their α-actin. In some species that live in depths greater than 5000m, "C.armatus" and "C.yaquinae" have specific substitutions on the active sites of α-Actin, which serves as the main component of muscle fiber. These specific substitutions, Q137K and V54A from "C.armatus" or I67P from "C.yaquinae" are predicted to have importance in pressure tolerance. Substitution in the active sites of actin result in significant changes in the salt bridge patterns of the protein, which allows for better stabilization in ATP binding and sub unit arrangement, confirmed by the free energy analysis and molecular dynamics simulation. It was found that deep sea fish have more salt bridges in their actins compared to fish inhabiting the upper zones of the sea.


title: Deep sea

text: In relations to protein substitution, specific osmolytes were found to be abundant in deep sea fish under high hydrostatic pressure. For certain chondrichtyans, it was found that Trimethylamine N-oxide (TMAO) increased with depth, replacing other osmolytes and urea. Due to the ability of TMAO being able to protect proteins from high hydrostatic pressure destabilizing proteins, the osmolyte adjustment serves are an important adaptation for deep sea fish to withstand high hydrostatic pressure.


title: Deep sea

text: Deep-sea organisms possess molecular adaptations to survive and thrive in the deep oceans. Mariana hadal snailfish developed modification in the Osteocalcin("burlap") gene, where premature termination of the gene was found. Osteocalcin gene regulates bone development and tissue mineralization, and the frameshift mutation seems to have resulted in the open skull and cartilage-based bone formation. Due to high hydrostatic pressure in the deep sea, closed skulls that organisms living on the surface develop cannot withstand the enforcing stress. Similarly, common bone developments seen in surface vertebrates cannot maintain their structural integrity under constant high pressure.


title: Deep sea

text: It has been suggested that more is known about the Moon than the deepest parts of the ocean. This is a common misconception based on a 1953 statement by George E.R. Deacon published in the "Journal of Navigation", and largely refers to the scarce amount of seafloor bathymetry available at the time. The similar idea that more people have stood on the moon than have been to the deepest part of the ocean is likewise problematic.


title: Deep sea

text: Still the deep-sea remains one of the least explored regions on planet Earth. Pressures even in the mesopelagic become too great for traditional exploration methods, demanding alternative approaches for deep-sea research. Baited camera stations, small manned submersibles, and ROVs (remotely operated vehicles) are three methods utilized to explore the ocean's depths. Because of the difficulty and cost of exploring this zone, current knowledge is limited. Pressure increases at approximately one atmosphere for every 10 meters meaning that some areas of the deep sea can reach pressures of above 1,000 atmospheres. This not only makes great depths very difficult to reach without mechanical aids, but also provides a significant difficulty when attempting to study any organisms that may live in these areas as their cell chemistry will be adapted to such vast pressures.


title: Kamui Kobayashi

text: Kobayashi was born in Amagasaki, near Kobe. His father owns a sushi restaurant. He began his career in motorsport in 1996 when he was nine years old, finishing third in his first season of karting in the "SL Takarazuka Tournament Cadet Class". During the following seven years, Kobayashi took four karting titles, winning the "Toyota SL All Japan Tournament Cadet Class" series twice.


title: Kamui Kobayashi

text: In 2004, he signed for Toyota's Driver Academy and soon began his career in open wheel racing. His next step was Formula Renault, entering the Asian, German, Italian and Dutch championships and taking two race victories in the Italian championship. Kobayashi continued in the Formula Renault class, entering the Italian and European championships and with six wins in both championships, he won both titles.


title: Kamui Kobayashi

text: In 2006, Kobayashi entered the Formula 3 Euro Series with ASM Formule 3 alongside Paul di Resta, Giedo van der Garde and Sebastian Vettel. He took three podium positions in his debut season, coming eighth in the Drivers' Championship and first in the Rookie's Championship. Kobayashi also entered the Macau Grand Prix and the Masters of Formula 3, which are annual Formula Three events. Kobayashi started in 10th place and finished the race a place lower in 11th, while at the Macau Grand Prix, he started the race in pole position but finished in 19th place.


title: Kamui Kobayashi

text: At the beginning of 2007, Kobayashi, along with Kohei Hirate, was named as one of the Toyota Formula One team's test drivers. He stayed in the Euro Series for the upcoming season and had an impressive start, taking two podiums in the first four rounds. He achieved his first race victory in Formula 3 at Magny-Cours, in the tenth round, a support race for the Formula One French Grand Prix. Kobayashi finished fourth in the Drivers' Championship.


title: Kamui Kobayashi

text: Following a successful GP2 Asia Series campaign in early 2008, Kobayashi won his first GP2 Series race in only the second race of the season. After a strong start from pole in the sprint race at the Circuit de Catalunya, Kobayashi took the chequered flag in first place. This was after a controversial piece of defensive driving from his former Euro Series teammate Romain Grosjean after a safety car period. At the end of the pit straight, Kobayashi attempted a pass on Grosjean. However, the Frenchman moved more than once to defend his position, forcing Kobayashi into evasive action. This resulted in a drive through penalty for the Frenchman, handing the win to Kobayashi. However, he only finished in the points on one further occasion, restricting him to sixteenth in the championship.


title: Kamui Kobayashi

text: Another successful GP2 Asia campaign followed over the winter months of 2008 and 2009, with Kobayashi winning two races en route to the championship, with a round to spare. Kobayashi could not repeat his form in the main series, finishing sixteenth again.


title: Kamui Kobayashi

text: On November 16, 2007, it was confirmed that Kobayashi would replace the departing Franck Montagny as the Toyota F1 team's third driver. He was the team's test and reserve driver during the and seasons.


title: Kamui Kobayashi

text: At the 2009 Japanese Grand Prix, Kobayashi competed in the first two free practice sessions in place of Timo Glock, who was ill. Glock recovered in time to take part in the third free practice session and qualifying, but was injured after crashing in the final session and had to miss the race. Toyota asked the Fédération Internationale de l'Automobile (FIA) for permission to run Kobayashi in the race, but this was refused as the regulations state a driver must run in at least one session on Saturday to be eligible to start the race.


title: Kamui Kobayashi

text: Kobayashi made his Formula One debut at the 2009 Brazilian Grand Prix, following a complication to Glock's injury that was initially not detected. He qualified 11th in a chaotic session that lasted for over two and half hours and was red-flagged twice due to accidents caused by torrential rain. Early in the race, while running in sixth place, he held off for several laps a challenge by Jenson Button, who needed to finish well to clinch the world championship. He finished the race in tenth place, and was later promoted to ninth when Heikki Kovalainen was penalised. Button jokingly described Kobayashi as "absolutely crazy, very aggressive". He also competed in the 2009 Abu Dhabi Grand Prix, as Glock's injury had not healed sufficiently. Kobayashi qualified 12th and finished sixth, scoring his first World Championship points, in the inaugural day-night race in Abu Dhabi.


title: Kamui Kobayashi

text: Before Toyota decided to withdraw from Formula One, Kobayashi was expected to be given a full-time seat at Toyota for the 2010 Formula One Season.


title: Kamui Kobayashi

text: Following Toyota's withdrawal, Kobayashi faced an uncertain future, but he was mentioned in lists of probable drivers for the series' new teams for 2010. After weeks of speculation, it was confirmed on December 17, 2009, he would drive for Sauber for the 2010 season. Despite the team having been sold by BMW back to founder Peter Sauber after the 2009 season, and no longer using BMW components, the team was still named BMW Sauber as it had been known for the past four seasons. His teammate was former McLaren tester Pedro de la Rosa. Kobayashi completed his first laps in the new Sauber C29 chassis during F1 winter testing on 2 February.


title: Kamui Kobayashi

text: During the , a front wing failure on his Sauber caused him to hit the barrier, rebounding off it to cause a three-car crash taking out Nico Hülkenberg and Sébastien Buemi. In the next race in Malaysia he qualified in ninth place, his best grid position up to that point, however he suffered an engine failure early in the race. In China, Kobayashi was involved in a three-way collision with Buemi and Vitantonio Liuzzi on the first lap, making him the only driver to retire from the first four races. In Turkey, he won his first points of the season, coming home tenth after being promoted a place due to Vettel's retirement after a collision with teammate Webber and Petrov's puncture. In Valencia, he finished seventh by passing both Fernando Alonso and Sébastien Buemi in the final laps on fresh tyres, after driving the majority of the race in third position on his first set of tyres. He followed that with sixth place in Silverstone, eleventh in Germany, ninth in Hungary and eighth in Belgium.


title: Kamui Kobayashi

text: At the , Kobayashi suffered a gearbox failure and retired from the after hitting a track-side barrier. Kobayashi's teammate changed in Singapore as Pedro de la Rosa was removed in favour of Nick Heidfeld. In Japan, Kobayashi qualified 14th and finished seventh, passing several drivers along the way including his teammate, in a very impressive fashion. He finished eighth in Korea and tenth in Brazil, eventually finishing the season with 32 points. In his review of the season, former TV commentator Murray Walker stated that Kobayashi is "without a doubt Japan's best [F1 driver] yet". Kobayashi gained a reputation during the season as a highly skilled overtaker, being able to outbrake drivers several car lengths in front of him. His aggressive overtaking style was described by Martin Brundle as, "He gets to the normal braking point and then goes, 'Now, which one is the brake again? That's right, it's on the left,' and he just sails past people!" He qualified well against his more experienced teammates, outqualifying de la Rosa and Heidfeld 11 times to 8 over the season.


title: Kamui Kobayashi

text: Kobayashi remained with Sauber (renamed from BMW Sauber to Sauber F1 Team) in , where he was partnered by GP2 graduate Sergio Pérez. Kobayashi finished eighth in the season opening , but he and Pérez – who had finished seventh – were disqualified after the race due to an irregularity with the car's rear-wing. The next race of the season in Malaysia was another strong showing for Kobayashi, finishing eighth in the race, eventually classified seventh after Lewis Hamilton received a penalty. He finished tenth in his next three races, before a career-high fifth place in an incident-packed Monaco Grand Prix. In an extremely wet , Kobayashi worked his way up from 13th place to 2nd having not decided to change to extreme wet tyres before the race was red-flagged, as many other drivers had. This essentially gave him a free pit stop while the race was suspended. After the restart, the track began to dry out, and after changing to intermediate tyres and finally slicks, Kobayashi dropped several places, including having a spin whilst lapping a backmarker and being rear-ended by Nick Heidfeld. He eventually finished seventh, 0.045 seconds behind Felipe Massa, who passed him on the final straight.


title: Kamui Kobayashi

text: On 28 July 2011, it was announced that Kobayashi would remain with Sauber into the season, alongside teammate Pérez.


title: Kamui Kobayashi

text: Kobayashi started the season with sixth place at the , and a retirement at the , due to a problem with his car's brakes.


title: Kamui Kobayashi

text: He then started third at the behind the Mercedes of Nico Rosberg and Michael Schumacher. He dropped to tenth but managed to set the fastest lap. At the he finished fifth after overtaking Jenson Button and Nico Rosberg. Kobayashi finished in the points once in the next four races, finishing ninth in Canada. Kobayashi finished in a career-best fourth place at the  – having finished fifth on-the-road – as he was helped by a post-race penalty for second-placed Sebastian Vettel. After retiring late in the race at the , Kobayashi qualified a career-best second for the but was caught in a first-corner accident along with four other drivers. Kobayashi was the only one of the five to continue in the race, and finished thirteenth.


title: Kamui Kobayashi

text: Kobayashi took his maiden podium in Formula One with third place at the , after lasting through race-long pressure from Jenson Button. Kobayashi became the first Japanese driver to finish on a Formula One podium in Japan in 22 years, after Aguri Suzuki in the 1990 Japanese Grand Prix, and was the third Japanese driver to finish on a Formula One podium after Suzuki and Takuma Sato in the 2004 United States Grand Prix.


title: Kamui Kobayashi

text: On 23 November 2012, Sauber announced that Kobayashi would not be a part of the team's line-up for the season, as Nico Hülkenberg and Esteban Gutiérrez would form the race team and Robin Frijns as reserve driver. Kobayashi ultimately finished the season in twelfth place in the Drivers' Championship, with 60 points. Despite raising around €8 million in sponsorship, Kobayashi elected to focus on gaining a competitive drive for the season rather than a drive.


title: Kamui Kobayashi

text: On 11 March 2013 it was confirmed that Kobayashi would drive for AF Corse in the 2013 FIA World Endurance Championship season. He competed in the LMGTE-Pro class in the Ferrari 458 GT for what is expected to be all of the season's 8 rounds, including the 2013 24 Hours of Le Mans.


title: Kamui Kobayashi

text: Kobayashi also tested a 2010 Formula One Ferrari in preparation for a promotional event in Moscow, where he crashed in the wet.


title: Kamui Kobayashi

text: In the 81st edition of the 24 Hours of Le Mans 2013, Kobayashi and the AF Corse team scored fifth place in the GTE-Pro class along with his co-drivers Olivier Beretta and Toni Vilander, their Ferrari 458 GT covered a total of 312 laps in the Circuit de la Sarthe. The race was run in very difficult weather conditions and several serious accidents bringing out a record of twelve safety car caution periods.


title: Kamui Kobayashi

text: On 21 January 2014, it was confirmed that Kobayashi would return to Formula One with the Caterham F1 team for the 2014 season partnering Swedish driver Marcus Ericsson after considerable speculation about the team's all new line-up for the new season.


title: Kamui Kobayashi

text: At the first race of the season, the , Kobayashi crashed into Massa at the start due to a brake failure. However, in the following , he was running as high as eighth, ahead of his teammate, the Marussias, and several other cars. He however finished the race in 13th, promoting Caterham to 10th in the Constructors' standings. However, later in the season Caterham were demoted to 11th due to Jules Bianchi achieving Marussia's first points finish in the .


title: Kamui Kobayashi

text: On 20 August 2014, it was announced that German driver André Lotterer would replace Kobayashi for the race weekend. He returned to racing action at the after Lotterer declined a further offer due to the seat being taken in practice by Roberto Merhi, who was attempting to qualify for an FIA Super Licence. Kobayashi declared his unhappiness at the situation, with the team's driver plans changing at short notice and his own future uncertain.


title: Kamui Kobayashi

text: On 30 January 2015, it was confirmed that Kobayashi would drive for Team LeMans in the 2015 Super Formula season. He scored three podiums on his way to a fifth-place finish in the drivers' championship during his first year in the series.


title: Kamui Kobayashi

text: His second year with Team LeMans was less successful, only managing to score one point and finishing 17th in the championship.


title: Kamui Kobayashi

text: On 4 February 2016, Kobayashi was confirmed as a LMP1 driver for Toyota Gazoo Racing in the FIA World Endurance Championship.


title: Kamui Kobayashi

text: Kobayashi scored his first WEC race victory at the 2016 6 Hours of Fuji, finishing ahead of the No. 8 Audi and the No. 1 Porsche.


title: Kamui Kobayashi

text: Kobayashi won the 2021 24 Hours of Le Mans from pole after numerous attempts, alongside Mike Conway and Jose Maria Lopez. Kobayashi is the fourth Japanese driver to win the 24 Hours of Le Mans, the first being Masanori Sekiya, and is only the second to do so for a Japanese manufacturer.


title: Kamui Kobayashi

text: In December 2021, Toyota announced that Kobayashi would succeed Hisatake Murata as team principal of the manufacturer's WEC programme, combining the management position with his role as a driver for the team.


title: Kamui Kobayashi

text: He was named after Kamuy, a divine being in Ainu mythology, and the letter of the name imitated the sound citing three Kanji from the sentence "Enabling great dream(s)".


title: Kamui Kobayashi

text: In April 2013, he was awarded the Outstanding Achievement in Sport Award at The Asian Awards in London.


title: Stanislaw Ulam

text: Stanisław Marcin Ulam (; 13 April 1909 – 13 May 1984) was a Polish-American scientist in the fields of mathematics and nuclear physics. He participated in the Manhattan Project, originated the Teller–Ulam design of thermonuclear weapons, discovered the concept of the cellular automaton, invented the Monte Carlo method of computation, and suggested nuclear pulse propulsion. In pure and applied mathematics, he proved some theorems and proposed several conjectures.


title: Stanislaw Ulam

text: Born into a wealthy Polish Jewish family, Ulam studied mathematics at the Lwów Polytechnic Institute, where he earned his PhD in 1933 under the supervision of Kazimierz Kuratowski and Włodzimierz Stożek. In 1935, John von Neumann, whom Ulam had met in Warsaw, invited him to come to the Institute for Advanced Study in Princeton, New Jersey, for a few months. From 1936 to 1939, he spent summers in Poland and academic years at Harvard University in Cambridge, Massachusetts, where he worked to establish important results regarding ergodic theory. On 20 August 1939, he sailed for the United States for the last time with his 17-year-old brother Adam Ulam. He became an assistant professor at the University of Wisconsin–Madison in 1940, and a United States citizen in 1941.


title: Stanislaw Ulam

text: In October 1943, he received an invitation from Hans Bethe to join the Manhattan Project at the secret Los Alamos Laboratory in New Mexico. There, he worked on the hydrodynamic calculations to predict the behavior of the explosive lenses that were needed by an implosion-type weapon. He was assigned to Edward Teller's group, where he worked on Teller's "Super" bomb for Teller and Enrico Fermi. After the war he left to become an associate professor at the University of Southern California, but returned to Los Alamos in 1946 to work on thermonuclear weapons. With the aid of a cadre of female "computers", including his wife Françoise Aron Ulam, he found that Teller's "Super" design was unworkable. In January 1951, Ulam and Teller came up with the Teller–Ulam design, which became the basis for all thermonuclear weapons.


title: Stanislaw Ulam

text: Ulam considered the problem of nuclear propulsion of rockets, which was pursued by Project Rover, and proposed, as an alternative to Rover's nuclear thermal rocket, to harness small nuclear explosions for propulsion, which became Project Orion. With Fermi, John Pasta, and Mary Tsingou, Ulam studied the Fermi–Pasta–Ulam–Tsingou problem, which became the inspiration for the field of non-linear science. He is probably best known for realising that electronic computers made it practical to apply statistical methods to functions without known solutions, and as computers have developed, the Monte Carlo method has become a common and standard approach to many problems.


title: Stanislaw Ulam

text: Ulam was born in Lemberg, Galicia, on 13 April 1909. At this time, Galicia was in the Kingdom of Galicia and Lodomeria of the Austro-Hungarian Empire, which was known to Poles as the Austrian partition. In 1918, it became part of the newly restored Poland, the Second Polish Republic, and the city took its Polish name again, Lwów.


title: Stanislaw Ulam

text: The Ulams were a wealthy Polish Jewish family of bankers, industrialists, and other professionals. Ulam's immediate family was "well-to-do but hardly rich". His father, Józef Ulam, was born in Lwów and was a lawyer, and his mother, Anna (née Auerbach), was born in Stryj. His uncle, Michał Ulam, was an architect, building contractor, and lumber industrialist. From 1916 until 1918, Józef's family lived temporarily in Vienna. After they returned, Lwów became the epicenter of the Polish–Ukrainian War, during which the city experienced a Ukrainian siege.


title: Stanislaw Ulam

text: In 1919, Ulam entered Lwów Gymnasium Nr. VII, from which he graduated in 1927. He then studied mathematics at the Lwów Polytechnic Institute. Under the supervision of Kazimierz Kuratowski, he received his Master of Arts degree in 1932, and became a Doctor of Science in 1933. At the age of 20, in 1929, he published his first paper "Concerning Function of Sets" in the journal "Fundamenta Mathematicae". From 1931 until 1935, he traveled to and studied in Wilno (Vilnius), Vienna, Zurich, Paris, and Cambridge, England, where he met G. H. Hardy and Subrahmanyan Chandrasekhar.


title: Stanislaw Ulam

text: Along with Stanisław Mazur, Mark Kac, Włodzimierz Stożek, Kuratowski, and others, Ulam was a member of the Lwów School of Mathematics. Its founders were Hugo Steinhaus and Stefan Banach, who were professors at the Jan Kazimierz University. Mathematicians of this "school" met for long hours at the Scottish Café, where the problems they discussed were collected in the Scottish Book, a thick notebook provided by Banach's wife. Ulam was a major contributor to the book. Of the 193 problems recorded between 1935 and 1941, he contributed 40 problems as a single author, another 11 with Banach and Mazur, and an additional 15 with others. In 1957, he received from Steinhaus a copy of the book, which had survived the war, and translated it into English. In 1981, Ulam's friend R. Daniel Mauldin published an expanded and annotated version.


title: Stanislaw Ulam

text: In 1935, John von Neumann, whom Ulam had met in Warsaw, invited him to come to the Institute for Advanced Study in Princeton, New Jersey, for a few months. In December of that year, Ulam sailed to the US. At Princeton, he went to lectures and seminars, where he heard Oswald Veblen, James Alexander, and Albert Einstein. During a tea party at von Neumann's house, he encountered G. D. Birkhoff, who suggested that he apply for a position with the Harvard Society of Fellows. Following up on Birkhoff's suggestion, Ulam spent summers in Poland and academic years at Harvard University in Cambridge, Massachusetts from 1936 to 1939, where he worked with John C. Oxtoby to establish results regarding ergodic theory. These appeared in Annals of Mathematics in 1941. In 1938, Stanislaw's mother Anna hanna Ulam (maiden name Auerbach) died of cancer.


title: Stanislaw Ulam

text: On 20 August 1939, in Gdynia, Józef Ulam, along with his brother Szymon, put his two sons, Stanislaw and 17 year old Adam, on a ship headed for the US. Eleven days later, the . Within two months, the Germans completed their occupation of western Poland, and the Soviets and occupied eastern Poland. Within two years, Józef Ulam and the rest of his family, including Stanislaw's sister Stefania Ulam, were victims of the Holocaust, Hugo Steinhaus was in hiding, Kazimierz Kuratowski was lecturing at the underground university in Warsaw, Włodzimierz Stożek and his two sons had been killed in the massacre of Lwów professors, and the last problem had been recorded in the Scottish Book. Stefan Banach survived the Nazi occupation by feeding lice at Rudolf Weigl's typhus research institute. In 1963, Adam Ulam, who had become an eminent kremlinologist at Harvard, received a letter from George Volsky, who hid in Józef Ulam's house after deserting from the Polish army. This reminiscence gave a chilling account of Lwów's chaotic scenes in late 1939. In later life Ulam described himself as "an agnostic. Sometimes I muse deeply on the forces that are for me invisible. When I am almost close to the idea of God, I feel immediately estranged by the horrors of this world, which he seems to tolerate".


title: Stanislaw Ulam

text: In 1940, after being recommended by Birkhoff, Ulam became an assistant professor at the University of Wisconsin–Madison. Here, he became a United States citizen in 1941. That year, he married Françoise Aron. She had been a French exchange student at Mount Holyoke College, whom he met in Cambridge. They had one daughter, Claire. In Madison, Ulam met his friend and colleague C. J. Everett, with whom he collaborated on a number of papers.


title: Stanislaw Ulam

text: In early 1943, Ulam asked von Neumann to find him a war job. In October, he received an invitation to join an unidentified project near Santa Fe, New Mexico. The letter was signed by Hans Bethe, who had been appointed as leader of the theoretical division of Los Alamos National Laboratory by Robert Oppenheimer, its scientific director. Knowing nothing of the area, he borrowed a New Mexico guide book. On the checkout card, he found the names of his Wisconsin colleagues, Joan Hinton, David Frisch, and Joseph McKibben, all of whom had mysteriously disappeared. This was Ulam's introduction to the Manhattan Project, which was the US's wartime effort to create the atomic bomb.


title: Stanislaw Ulam

text: A few weeks after Ulam reached Los Alamos in February 1944, the project experienced a crisis. In April, Emilio Segrè discovered that plutonium made in reactors would not work in a gun-type plutonium weapon like the "Thin Man", which was being developed in parallel with a uranium weapon, the "Little Boy" that was dropped on Hiroshima. This problem threatened to waste an enormous investment in new reactors at the Hanford site and to make slow uranium isotope separation the only way to prepare fissile material suitable for use in bombs. To respond, Oppenheimer implemented, in August, a sweeping reorganization of the laboratory to focus on development of an implosion-type weapon and appointed George Kistiakowsky head of the implosion department. He was a professor at Harvard and an expert on precise use of explosives.


title: Stanislaw Ulam

text: The basic concept of implosion is to use chemical explosives to crush a chunk of fissile material into a critical mass, where neutron multiplication leads to a nuclear chain reaction, releasing a large amount of energy. Cylindrical implosive configurations had been studied by Seth Neddermeyer, but von Neumann, who had experience with shaped charges used in armor-piercing ammunition, was a vocal advocate of spherical implosion driven by explosive lenses. He realized that the symmetry and speed with which implosion compressed the plutonium were critical issues, and enlisted Ulam to help design lens configurations that would provide nearly spherical implosion. Within an implosion, because of enormous pressures and high temperatures, solid materials behave much like fluids. This meant that hydrodynamical calculations were needed to predict and minimize asymmetries that would spoil a nuclear detonation. Of these calculations, Ulam said:


title: Stanislaw Ulam

text: Nevertheless, with the primitive facilities available at the time, Ulam and von Neumann did carry out numerical computations that led to a satisfactory design. This motivated their advocacy of a powerful computational capability at Los Alamos, which began during the war years, continued through the cold war, and still exists. Otto Frisch remembered Ulam as "a brilliant Polish topologist with a charming French wife. At once he told me that he was a pure mathematician who had sunk so low that his latest paper actually contained numbers with decimal points!"


title: Stanislaw Ulam

text: Even the inherent statistical fluctuations of neutron multiplication within a chain reaction have implications with regard to implosion speed and symmetry. In November 1944, David Hawkins and Ulam addressed this problem in a report entitled "Theory of Multiplicative Processes". This report, which invokes probability-generating functions, is also an early entry in the extensive literature on statistics of branching and multiplicative processes. In 1948, its scope was extended by Ulam and Everett.


title: Stanislaw Ulam

text: Early in the Manhattan project, Enrico Fermi's attention was focused on the use of reactors to produce plutonium. In September 1944, he arrived at Los Alamos, shortly after breathing life into the first Hanford reactor, which had been poisoned by a xenon isotope. Soon after Fermi's arrival, Teller's "Super" bomb group, of which Ulam was a part, was transferred to a new division headed by Fermi. Fermi and Ulam formed a relationship that became very fruitful after the war.


title: Stanislaw Ulam

text: In September 1945, Ulam left Los Alamos to become an associate professor at the University of Southern California in Los Angeles. In January 1946, he suffered an acute attack of encephalitis, which put his life in danger, but which was alleviated by emergency brain surgery. During his recuperation, many friends visited, including Nicholas Metropolis from Los Alamos and the famous mathematician Paul Erdős, who remarked: "Stan, you are just like before." This was encouraging, because Ulam was concerned about the state of his mental faculties, for he had lost the ability to speak during the crisis. Another friend, Gian-Carlo Rota, asserted in a 1987 article that the attack changed Ulam's personality: afterwards, he turned from rigorous pure mathematics to more speculative conjectures concerning the application of mathematics to physics and biology; Rota also cites Ulam's former collaborator Paul Stein as noting that Ulam was sloppier in his clothing afterwards, and John Oxtoby as noting that Ulam before the encephalitis could work for hours on end doing calculations, while when Rota worked with him, was reluctant to solve even a quadratic equation. This assertion was not accepted by Françoise Aron Ulam.


title: Stanislaw Ulam

text: By late April 1946, Ulam had recovered enough to attend a secret conference at Los Alamos to discuss thermonuclear weapons. Those in attendance included Ulam, von Neumann, Metropolis, Teller, Stan Frankel, and others. Throughout his participation in the Manhattan Project, Teller's efforts had been directed toward the development of a "super" weapon based on nuclear fusion, rather than toward development of a practical fission bomb. After extensive discussion, the participants reached a consensus that his ideas were worthy of further exploration. A few weeks later, Ulam received an offer of a position at Los Alamos from Metropolis and Robert D. Richtmyer, the new head of its theoretical division, at a higher salary, and the Ulams returned to Los Alamos.


title: Stanislaw Ulam

text: Late in the war, under the sponsorship of von Neumann, Frankel and Metropolis began to carry out calculations on the first general-purpose electronic computer, the ENIAC at the Aberdeen Proving Ground in Maryland. Shortly after returning to Los Alamos, Ulam participated in a review of results from these calculations. Earlier, while playing solitaire during his recovery from surgery, Ulam had thought about playing hundreds of games to estimate statistically the probability of a successful outcome. With ENIAC in mind, he realized that the availability of computers made such statistical methods very practical. John von Neumann immediately saw the significance of this insight. In March 1947 he proposed a statistical approach to the problem of neutron diffusion in fissionable material. Because Ulam had often mentioned his uncle, Michał Ulam, "who just had to go to Monte Carlo" to gamble, Metropolis dubbed the statistical approach "The Monte Carlo method". Metropolis and Ulam published the first unclassified paper on the Monte Carlo method in 1949.


title: Stanislaw Ulam

text: Fermi, learning of Ulam's breakthrough, devised an analog computer known as the Monte Carlo trolley, later dubbed the FERMIAC. The device performed a mechanical simulation of random diffusion of neutrons. As computers improved in speed and programmability, these methods became more useful. In particular, many Monte Carlo calculations carried out on modern massively parallel supercomputers are embarrassingly parallel applications, whose results can be very accurate.


title: Stanislaw Ulam

text: On 29 August 1949, the Soviet Union tested its first fission bomb, the RDS-1. Created under the supervision of Lavrentiy Beria, who sought to duplicate the US effort, this weapon was nearly identical to Fat Man, for its design was based on information provided by spies Klaus Fuchs, Theodore Hall, and David Greenglass. In response, on 31 January 1950, President Harry S. Truman announced a crash program to develop a fusion bomb.


title: Stanislaw Ulam

text: To advocate an aggressive development program, Ernest Lawrence and Luis Alvarez came to Los Alamos, where they conferred with Norris Bradbury, the laboratory director, and with George Gamow, Edward Teller, and Ulam. Soon, these three became members of a short-lived committee appointed by Bradbury to study the problem, with Teller as chairman. At this time, research on the use of a fission weapon to create a fusion reaction had been ongoing since 1942, but the design was still essentially the one originally proposed by Teller. His concept was to put tritium and/or deuterium in close proximity to a fission bomb, with the hope that the heat and intense flux of neutrons released when the bomb exploded, would ignite a self-sustaining fusion reaction. Reactions of these isotopes of hydrogen are of interest because the energy per unit mass of fuel released by their fusion is much larger than that from fission of heavy nuclei.


title: Stanislaw Ulam

text: Because the results of calculations based on Teller's concept were discouraging, many scientists believed it could not lead to a successful weapon, while others had moral and economic grounds for not proceeding. Consequently, several senior people of the Manhattan Project opposed development, including Bethe and Oppenheimer. To clarify the situation, Ulam and von Neumann resolved to do new calculations to determine whether Teller's approach was feasible. To carry out these studies, von Neumann decided to use electronic computers: ENIAC at Aberdeen, a new computer, MANIAC, at Princeton, and its twin, which was under construction at Los Alamos. Ulam enlisted Everett to follow a completely different approach, one guided by physical intuition. Françoise Ulam was one of a cadre of women "computers" who carried out laborious and extensive computations of thermonuclear scenarios on mechanical calculators, supplemented and confirmed by Everett's slide rule. Ulam and Fermi collaborated on further analysis of these scenarios. The results showed that, in workable configurations, a thermonuclear reaction would not ignite, and if ignited, it would not be self-sustaining. Ulam had used his expertise in combinatorics to analyze the chain reaction in deuterium, which was much more complicated than the ones in uranium and plutonium, and he concluded that no self-sustaining chain reaction would take place at the (low) densities that Teller was considering. In late 1950, these conclusions were confirmed by von Neumann's results.


title: Stanislaw Ulam

text: In January 1951, Ulam had another idea: to channel the mechanical shock of a nuclear explosion so as to compress the fusion fuel. On the recommendation of his wife, Ulam discussed this idea with Bradbury and Mark before he told Teller about it. Almost immediately, Teller saw its merit, but noted that soft X-rays from the fission bomb would compress the thermonuclear fuel more strongly than mechanical shock and suggested ways to enhance this effect. On 9 March 1951, Teller and Ulam submitted a joint report describing these innovations. A few weeks later, Teller suggested placing a fissile rod or cylinder at the center of the fusion fuel. The detonation of this "spark plug" would help to initiate and enhance the fusion reaction. The design based on these ideas, called staged radiation implosion, has become the standard way to build thermonuclear weapons. It is often described as the "Teller–Ulam design".


title: Stanislaw Ulam

text: In September 1951, after a series of differences with Bradbury and other scientists, Teller resigned from Los Alamos, and returned to the University of Chicago. At about the same time, Ulam went on leave as a visiting professor at Harvard for a semester. Although Teller and Ulam submitted a joint report on their design and jointly applied for a patent on it, they soon became involved in a dispute over who deserved credit. After the war, Bethe returned to Cornell University, but he was deeply involved in the development of thermonuclear weapons as a consultant. In 1954, he wrote an article on the history of the H-bomb, which presents his opinion that both men contributed very significantly to the breakthrough. This balanced view is shared by others who were involved, including Mark and Fermi, but Teller persistently attempted to downplay Ulam's role. "After the H-bomb was made," Bethe recalled, "reporters started to call Teller the father of the H-bomb. For the sake of history, I think it is more precise to say that Ulam is the father, because he provided the seed, and Teller is the mother, because he remained with the child. As for me, I guess I am the midwife."


title: Stanislaw Ulam

text: With the basic fusion reactions confirmed, and with a feasible design in hand, there was nothing to prevent Los Alamos from testing a thermonuclear device. On 1 November 1952, the first thermonuclear explosion occurred when Ivy Mike was detonated on Enewetak Atoll, within the US Pacific Proving Grounds. This device, which used liquid deuterium as its fusion fuel, was immense and utterly unusable as a weapon. Nevertheless, its success validated the Teller–Ulam design, and stimulated intensive development of practical weapons.


title: Stanislaw Ulam

text: When Ulam returned to Los Alamos, his attention turned away from weapon design and toward the use of computers to investigate problems in physics and mathematics. With John Pasta, who helped Metropolis to bring MANIAC on line in March 1952, he explored these ideas in a report "Heuristic Studies in Problems of Mathematical Physics on High Speed Computing Machines", which was submitted on 9 June 1953. It treated several problems that cannot be addressed within the framework of traditional analytic methods: billowing of fluids, rotational motion in gravitating systems, magnetic lines of force, and hydrodynamic instabilities.


title: Stanislaw Ulam

text: Soon, Pasta and Ulam became experienced with electronic computation on MANIAC, and by this time, Enrico Fermi had settled into a routine of spending academic years at the University of Chicago and summers at Los Alamos. During these summer visits, Pasta, Ulam, and Mary Tsingou, a programmer in the MANIAC group, joined him to study a variation of the classic problem of a string of masses held together by springs that exert forces linearly proportional to their displacement from equilibrium. Fermi proposed to add to this force a nonlinear component, which could be chosen to be proportional to either the square or cube of the displacement, or to a more complicated "broken linear" function. This addition is the key element of the Fermi–Pasta–Ulam–Tsingou problem, which is often designated by the abbreviation FPUT.


title: Stanislaw Ulam

text: A classical spring system can be described in terms of vibrational modes, which are analogous to the harmonics that occur on a stretched violin string. If the system starts in a particular mode, vibrations in other modes do not develop. With the nonlinear component, Fermi expected energy in one mode to transfer gradually to other modes, and eventually, to be distributed equally among all modes. This is roughly what began to happen shortly after the system was initialized with all its energy in the lowest mode, but much later, essentially all the energy periodically reappeared in the lowest mode. This behavior is very different from the expected equipartition of energy. It remained mysterious until 1965, when Kruskal and Zabusky showed that, after appropriate mathematical transformations, the system can be described by the Korteweg–de Vries equation, which is the prototype of nonlinear partial differential equations that have soliton solutions. This means that FPUT behavior can be understood in terms of solitons.


title: Stanislaw Ulam

text: Starting in 1955, Ulam and Frederick Reines considered nuclear propulsion of aircraft and rockets. This is an attractive possibility, because the nuclear energy per unit mass of fuel is a million times greater than that available from chemicals. From 1955 to 1972, their ideas were pursued during Project Rover, which explored the use of nuclear reactors to power rockets. In response to a question by Senator John O. Pastore at a congressional committee hearing on "Outer Space Propulsion by Nuclear Energy", on January 22, 1958, Ulam replied that "the future as a whole of mankind is to some extent involved inexorably now with going outside the globe."


title: Stanislaw Ulam

text: Ulam and C. J. Everett also proposed, in contrast to Rover's continuous heating of rocket exhaust, to harness small nuclear explosions for propulsion. Project Orion was a study of this idea. It began in 1958 and ended in 1965, after the Partial Nuclear Test Ban Treaty of 1963 banned nuclear weapons tests in the atmosphere and in space. Work on this project was spearheaded by physicist Freeman Dyson, who commented on the decision to end Orion in his article, "Death of a Project".


title: Stanislaw Ulam

text: Bradbury appointed Ulam and John H. Manley as research advisors to the laboratory director in 1957. These newly created positions were on the same administrative level as division leaders, and Ulam held his until he retired from Los Alamos. In this capacity, he was able to influence and guide programs in many divisions: theoretical, physics, chemistry, metallurgy, weapons, health, Rover, and others.


title: Stanislaw Ulam

text: In addition to these activities, Ulam continued to publish technical reports and research papers. One of these introduced the Fermi–Ulam model, an extension of Fermi's theory of the acceleration of cosmic rays. Another, with Paul Stein and Mary Tsingou, titled "Quadratic Transformations", was an early investigation of chaos theory and is considered the first published use of the phrase "chaotic behavior".


title: Stanislaw Ulam

text: During his years at Los Alamos, Ulam was a visiting professor at Harvard from 1951 to 1952, MIT from 1956 to 1957, the University of California, San Diego, in 1963, and the University of Colorado at Boulder from 1961 to 1962 and 1965 to 1967. In 1967, the last of these positions became permanent, when Ulam was appointed Professor and Chairman of the Department of Mathematics at the University of Colorado. He kept a residence in Santa Fe, which made it convenient to spend summers at Los Alamos as a consultant. He was an elected member of the American Academy of Arts and Sciences, the United States National Academy of Sciences, and the American Philosophical Society.


title: Stanislaw Ulam

text: In Colorado, where he rejoined his friends Gamow, Richtmyer, and Hawkins, Ulam's research interests turned toward biology. In 1968, recognizing this emphasis, the University of Colorado School of Medicine appointed Ulam as Professor of Biomathematics, and he held this position until his death. With his Los Alamos colleague Robert Schrandt he published a report, "Some Elementary Attempts at Numerical Modeling of Problems Concerning Rates of Evolutionary Processes", which applied his earlier ideas on branching processes to evolution. Another, report, with William Beyer, Temple F. Smith, and M. L. Stein, titled "Metrics in Biology", introduced new ideas about numerical taxonomy and evolutionary distances.


title: Stanislaw Ulam

text: When he retired from Colorado in 1975, Ulam began to spend winter semesters at the University of Florida, where he was a graduate research professor. Except for sabbaticals at the University of California, Davis from 1982 to 1983, and at Rockefeller University from 1980 to 1984, this pattern of spending summers in Colorado and Los Alamos and winters in Florida continued until Ulam died of an apparent heart attack in Santa Fe on 13 May 1984.


title: Stanislaw Ulam

text: Paul Erdős noted that "he died suddenly of heart failure, without fear or pain, while he could still prove and conjecture." In 1987, Françoise Ulam deposited his papers with the American Philosophical Society Library in Philadelphia. She continued to live in Santa Fe until she died in 2011, at the age of 93. Both Françoise and her husband were buried with her family in Montparnasse Cemetery in Paris.


title: Stanislaw Ulam

text: Alfred Marshall and his disciples dominated economic theory until the end of WWII. With the Cold War, the theory changed, emphasizing that a market economy was superior and the only sensible way. In Paul Samuelson's "Economics: An Introductory Analysis", 1948, Adam Smith's "invisible hand" was only a footnote. In later editions, it became the central theme. As Samuelson remembers, all this was challenged by Stanislaw Ulam: "[Y]ears ago... I was in the Society of Fellows at Harvard along with the mathematician Stanislaw Ulam. Ulam, who was to become an originator of the Monte Carlo method and co-discoverer of the hydrogen-bomb... used to tease me by saying, 'Name me one proposition in all of the social sciences which is both true and non-trivial.' This was the test that I always failed. But now, some thirty years later ... an appropriate answer occurs to me: The Ricardian theory of comparative advantage ... That it is logically true need not be argued before a mathematician; that it is not trivial is attested by the thousands of important and intelligent men who have never been able to grasp the doctrine for themselves or to believe it after it was explained to them."


title: Stanislaw Ulam

text: Participated in the creation of a hydrogen bomb as part of the Los Alamos Laboratory nuclear project. From the publication of his first paper as a student in 1929 until his death, Ulam was constantly writing on mathematics. The list of Ulam's publications includes more than 150 papers. Topics represented by a significant number of papers are: set theory (including measurable cardinals and abstract measures), topology, transformation theory, ergodic theory, group theory, projective algebra, number theory, combinatorics, and graph theory. In March 2009, the Mathematical Reviews database contained 697 papers with the name "Ulam".


title: Stanislaw Ulam

text: With his pivotal role in the development of thermonuclear weapons, Stanislaw Ulam changed the world. According to Françoise Ulam: "Stan would reassure me that, barring accidents, the H-bomb rendered nuclear war impossible." In 1980, Ulam and his wife appeared in the television documentary "The Day After Trinity".


title: Stanislaw Ulam

text: The Monte Carlo method has become a ubiquitous and standard approach to computation, and the method has been applied to a vast number of scientific problems. In addition to problems in physics and mathematics, the method has been applied to finance, social science, environmental risk assessment, linguistics, radiation therapy, and sports.


title: Stanislaw Ulam

text: The Fermi–Pasta–Ulam–Tsingou problem is credited not only as "the birth of experimental mathematics", but also as inspiration for the vast field of Nonlinear Science. In his Lilienfeld Prize lecture, David K. Campbell noted this relationship and described how FPUT gave rise to ideas in chaos, solitons, and dynamical systems. In 1980, Donald Kerr, laboratory director at Los Alamos, with the strong support of Ulam and Mark Kac, founded the Center for Nonlinear Studies (CNLS). In 1985, CNLS initiated the "Stanislaw M. Ulam Distinguished Scholar" program, which provides an annual award that enables a noted scientist to spend a year carrying out research at Los Alamos.


title: Stanislaw Ulam

text: The fiftieth anniversary of the original FPUT paper was the subject of the March 2005 issue of the journal Chaos, and the topic of the 25th Annual International Conference of CNLS. The University of Southern Mississippi and the University of Florida supported the "Ulam Quarterly", which was active from 1992 to 1996, and which was one of the first online mathematical journals. Florida's Department of Mathematics has sponsored, since 1998, the annual "Ulam Colloquium Lecture", and in March 2009, the "Ulam Centennial Conference".


title: Stanislaw Ulam

text: Ulam's work on non-Euclidean distance metrics in the context of molecular biology made a significant contribution to sequence analysis and his contributions in theoretical biology are considered watersheds in the development of cellular automata theory, population biology, pattern recognition, and biometrics generally (David Sankoff, however, challenged conclusions of Walter by writing that Ulam had only modest influence on early development of sequence alignment methods.). Colleagues noted that some of his greatest contributions were in clearly identifying problems to be solved and general techniques for solving them.


title: Stanislaw Ulam

text: In 1987, Los Alamos issued a special issue of its "Science" publication, which summarized his accomplishments, and which appeared, in 1989, as the book "From Cardinals to Chaos". Similarly, in 1990, the University of California Press issued a compilation of mathematical reports by Ulam and his Los Alamos collaborators: "Analogies Between Analogies". During his career, Ulam was awarded honorary degrees by the Universities of New Mexico, Wisconsin, and Pittsburgh.


title: Stanislaw Ulam

text: In 2021, German film director Thorsten Klein made the of the book "Adventures of a Mathematician" about Ulam's life


title: Carlos Corberán

text: Carlos Corberán Vallet (born 7 April 1983) is a Spanish football coach who is currently head coach of EFL Championship club West Bromwich Albion.


title: Carlos Corberán

text: Previously the head coach of Doxa and Ermis in the Cypriot First Division, Corberán was first team assistant coach and manager of the under-23s side at Leeds United. He then managed in his own right at Huddersfield Town and Olympiacos.


title: Carlos Corberán

text: Born in Cheste, Valencian Community, Corberán represented Valencia CF as a youth. At the age of 23, however, after only representing the team's reserves and playing no higher than Tercera División, he decided to retire to pursue his passion for coaching.


title: Carlos Corberán

text: After retiring, Corberán subsequently started working at Villarreal CF's C and B-teams, as a fitness coach. In 2011, after the appointment of Juan Carlos Garrido in the first team, he was named fitness coach of the main squad.


title: Carlos Corberán

text: In February 2012, Corberán signed for Saudi Arabian team Al-Ittihad Club alongside head coach Raúl Caneda, having been recommended by Pep Guardiola. Whilst at the club, he was a fitness coach and helped get the club to the semi-finals of the AFC Champions League.


title: Carlos Corberán

text: In July 2013, Corberán was appointed manager of AD Alcorcón's "Juvenil A" squad, but was relieved from his duties and subsequently replaced by José María Rico the following January. In 2014, he returned to Saudi Arabia and joined Al-Nassr FC and was named Caneda's assistant. They were runners-up in the Super Cup, and reached the group phase in the AFC Champions League and were also finalists in the King's Cup.


title: Carlos Corberán

text: On 29 November 2016, Corberán had his first senior managerial experience after being appointed in charge of Doxa Katokopias FC in the Cypriot First Division. The following 24 January, however, he was sacked.


title: Carlos Corberán

text: Corberán was appointed the new head coach of Ermis Aradippou FC also in the Cypriot top tier on 30 January 2017, where he helped guide them to a seventh place finish before being replaced by Nicos Panayiotou.


title: Carlos Corberán

text: On 21 June 2017, Corberán was announced as the new Leeds United under-23 manager, replacing the departed Jason Blunt.


title: Carlos Corberán

text: After the appointment of new head coach Marcelo Bielsa in June 2018, Corberán was promoted to first team coach as well as continuing his role as the head coach of the Under 23s. In October 2018, Corberán was described as "very talented" by Bielsa, with Bielsa stating he values Corberán's opinion "more than his own".


title: Carlos Corberán

text: Corberán's side won the PDL Northern League 2018–19 season by winning the league, they then became the national Professional Development League Champions by beating Birmingham City in the final.