title: Hans Rosling

text: Hence, Rosling has been criticized as being Pollyannaist about the global political situation in the face of tragedies such as the long-running conflict in Syria, among others. His work on population growth has also been criticized by Paul R. Ehrlich, the U.S. biologist and Professor of Population Studies at Stanford University, and Anne H. Ehrlich, associate director of the Center for Conservation Biology at Stanford University, in an article, published online by the MAHB, titled "A Confused Statistician." The Ehrlichs also warn that, while some trends that Rosling cites may indeed be positive, there is the possibility of total collapse of those trends if social and political instabilities occur.


title: Rafa Silva

text: Rafael Alexandre Fernandes Ferreira da Silva (; born 17 May 1993) is a Portuguese footballer who plays as an attacking midfielder or winger for Primeira Liga club Benfica.


title: Rafa Silva

text: After starting his professional career at Feirense, he transferred to Braga in 2013 and went on to make 127 appearances for them, scoring 26 goals winning the Taça de Portugal in 2016. He joined Benfica that year for €16.4 million, winning two Primeira Liga championships and another cup during his spell.


title: Rafa Silva

text: A full international from 2014 to 2021, Silva represented Portugal on 25 occasions. He was selected for the 2014 FIFA World Cup, UEFA Euro 2016, the 2019 UEFA Nations League Finals and UEFA Euro 2020, winning the 2016 and 2019 tournaments.


title: Rafa Silva

text: Born in Vila Franca de Xira, Silva grew up in Forte da Casa and started playing football with U.A. Povoense and F.C. Alverca, both in the Lisbon District. In 2011, aged 18, he signed with C.D. Feirense with which he played his last year as a junior.


title: Rafa Silva

text: Silva made his professional debut on 29 July 2012, playing the full 90 minutes in a 2–1 home win against F.C. Penafiel for the season's Taça da Liga. He only missed one game in the league campaign, totalling more than 3,200 minutes of action to help his team finish 13th in the Segunda Liga.


title: Rafa Silva

text: In June 2013, Silva signed a five-year contract with S.C. Braga for an undisclosed fee. On 10 November he scored his first goals, striking twice in a 3–1 win at S.C. Olhanense in the fourth round of the Taça de Portugal; nineteen days later he scored for the first time in the league, in a 4–1 home win over the same opposition. He finished his first year with 23 appearances and three goals to help to a ninth-place finish.


title: Rafa Silva

text: The following season, Silva was ever-present in the league, starting all but one game. He scored three times in the 2014–15 Taça de Portugal, including one in the final which they lost in a penalty shootout to Sporting CP at the Estádio Nacional on 31 May 2015.


title: Rafa Silva

text: Silva scored the only goal on his UEFA Europa League debut on 17 September 2015 to defeat FC Slovan Liberec away, and added two further goals in a run to the quarter-finals. After a 2015–16 season in which Braga won the cup final against FC Porto, he was coveted by S.L. Benfica.


title: Rafa Silva

text: On 1 September 2016, defending champions Benfica announced that Silva had signed a five-year contract with the club, in a deal totalling €16.4 million. His release clause was set at €60 million. He made his debut eight days later, starting in a 2–1 win at F.C. Arouca.


title: Rafa Silva

text: In his 14th game, on 22 January 2017, as a late substitute for Kostas Mitroglou, Silva scored his first goal for the team in a 4–0 win over C.D. Tondela at the Estádio da Luz. In that season, he also played seven minutes in the final of the domestic cup, won after defeating Vitória de Guimarães 2–1.


title: Rafa Silva

text: Silva scored his ninth league goal of 2018–19 on 2 March 2019, in a 2–1 away victory against Porto; Benfica leapfrogged the opposition and took first place with ten rounds remaining. He ended the season with a career-best 17 league goals, bettered only by teammate Haris Seferovic (23 goals) and Bruno Fernandes (20) in the whole division. After the season ended, his contract was renewed until 2024 and its buyout clause increased to €80 million.


title: Rafa Silva

text: On 4 August 2019, Silva scored the opening goal of a 5–0 win for Benfica in the 2019 Supertaça Cândido de Oliveira, against city rivals Sporting at the Estádio Algarve. On 23 October, he scored his first Champions League goal in a 2–1 victory over Lyon. The following 17 January, days after returning from a three-month injury, he scored both goals in a win away to the same team.


title: Rafa Silva

text: On 25 October 2022, he scored a brace in a 4–3 victory over Juventus, which qualified his club to the knockout phase.


title: Rafa Silva

text: Silva made his first appearance for the Portugal under-20 team on 23 April 2013, against Uzbekistan. He was not selected to that year's FIFA U-20 World Cup.


title: Rafa Silva

text: On 28 February 2014, Silva received his first callup for the senior side, for an exhibition game with Cameroon on 5 March. He played the first 45 minutes of the game, in a 5–1 win in Leiria. On 19 May, he was named in the final 23-man squad for the 2014 FIFA World Cup, remaining unused in Brazil as his team were knocked out in the group stage.


title: Rafa Silva

text: After scoring in both games against Azerbaijan in qualification, Silva was part of the under-21 team that finished as runners-up at the 2015 UEFA European Championship in the Czech Republic. He returned to the full side for UEFA Euro 2016, making his only appearance in the competition on 18 June by coming on as an 89th-minute substitute for Nani in a 0–0 draw with Austria at the Parc des Princes.


title: Rafa Silva

text: Silva missed over two years of international football from September 2016 to October 2018, returning against Poland in the UEFA Nations League; he was selected for the final tournament on home soil where he played the final 15 minutes of the 1–0 win over the Netherlands. At UEFA Euro 2020, delayed to 2021, he played two group games off the bench, winning a penalty and assisting Cristiano Ronaldo in a 3–0 win over Hungary.


title: Rafa Silva

text: On 19 September 2022, Silva announced his retirement from international football, citing personal reasons.


title: Elisa Donovan

text: Lisa Adaline Donovan (born February 3, 1971), known professionally as Elisa Donovan, is an American actress. She played the role of Amber in the 1995 teen comedy film "Clueless", and reprised her role in the TV series of the same name (1996-1999). Donovan went on to play the role of Morgan Cavanaugh in the sitcom "Sabrina the Teenage Witch" (2000–2003).


title: Elisa Donovan

text: Elisa was born Lisa Adaline Donovan in Poughkeepsie, New York, the daughter of Charlotte and Jack Donovan, who was a business executive.


title: Elisa Donovan

text: Donovan's breakthrough role was in the 1995 film "Clueless" as Amber, Cher's nemesis. Donovan also reprised her role in the television series of the same name. Other notable roles included the film "A Night at the Roxbury", "Beverly Hills, 90210" as Ginger LaMonica and playing Morgan Cavanaugh on the television sitcom "Sabrina, the Teenage Witch". She also played Sharona on Disney Channel's "Sonny with a Chance", and has appeared in the films "Complacent", "The Dog Who Saved Christmas", and its sequel "The Dog Who Saved the Holidays". Donovan starred in the web series, "The Lake" on TheWB.com. Additionally, Donovan played Gayle, an insurance agent in the small town of Maple Grove, in the web series "In Gayle We Trust" on NBC.com.


title: Elisa Donovan

text: Donovan had a role in the film "A Golden Christmas", where she played Anna, the sister of the main character and she played the lead role in "The 12 Wishes of Christmas."


title: Elisa Donovan

text: She was also featured in 'N Sync's 1999 music video for "Thinking of You (I Drive Myself Crazy)" as the love interest for Joey Fatone.


title: Elisa Donovan

text: She competes on the twenty-fourth season of "Worst Cooks in America", the show's seventh celebrity edition titled "Thats So 90s", airing in April and May 2022.


title: Elisa Donovan

text: Donovan has been married to her husband Charlie Bigelow since October 13, 2012. They have a daughter, Scarlett Avery Bigelow.


title: Lists of Victoria Cross recipients

text: Lists of Victoria Cross recipients are lists of people who have been awarded the Victoria Cross, the highest and most prestigious of the orders, decorations, and medals of the United Kingdom. It is awarded for valour "in the presence of the enemy" to members of the British Armed Forces. It was previously awarded to Commonwealth countries, most of which no longer recommend British honours. The lists are organized alphabetically, by military branch or service, by conflict, by nationality and by other criteria.


title: London Borough of Enfield

text: The London Borough of Enfield () is a London borough in North London. It borders the London boroughs of Barnet to the west, Haringey to the south, and Waltham Forest to the southeast. To the north are the districts of Hertsmere, Welwyn Hatfield and Broxbourne (in Hertfordshire), and to the east is Epping Forest District in Essex. The local authority is Enfield London Borough Council. Enfield's population is estimated to be 333,794; the main towns in the borough are Edmonton, Enfield, Southgate and Palmers Green. Enfield is the northernmost London borough.


title: London Borough of Enfield

text: Enfield was recorded in Domesday Book in 1086 as "Enefelde", and as "Einefeld" in 1214, "Enfeld" in 1293, and "Enfild" in 1564: that is 'open land of a man called Ēana', or 'where lambs are reared', from the Old English "feld" with an Old English personal name or with Old English "ēan" 'lamb'. The "feld" would have been a reference to an area cleared of trees within woodland that would later become known as Enfield Chase.


title: London Borough of Enfield

text: The current borough was created in 1965 from the former areas of the Municipal Borough of Southgate, the Municipal Borough of Enfield and the Municipal Borough of Edmonton. The armorial bearings of these three boroughs were also merged. The heraldic beast on the shield of the Enfield coat of arms is known in heraldry as an "Enfield" (or colloquially as the Enfield beast), and is used extensively as a logo representing Enfield, particularly by the borough council.


title: London Borough of Enfield

text: In Roman times, Enfield was connected to Londinium by Ermine Street, the great Roman road which stretched all the way up to York. Artefacts found in the early 1900s reveal that there were Roman settlements in the areas that are now Edmonton and Bush Hill Park.


title: London Borough of Enfield

text: In 790 King Offa of Mercia was recorded as giving the lands of Edmonton to St Albans Abbey. The area became strategically important as East Anglia was taken over by the Danes. In the 790s strongholds were built by men loyal to King Alfred the Great, in order to keep the Danes to the east of the River Lea.


title: London Borough of Enfield

text: After the Norman Conquest, both Enfield and Edmonton were mentioned in Domesday Book. Both had churches, and Enfield had 400 inhabitants, Edmonton 300. Enfield is also described as having a "parc". This parc—a heavily forested area for hunting—was key to Enfield's existence in the Middle Ages (see Enfield Old Park). Wealthy Londoners came to Enfield first to hunt, and then to build houses in the green, wooded surroundings. In 1303, Edward I of England granted Enfield a charter to hold a weekly market, which has continued up to this day. The old market cross was removed in the early 20th century to make way for a monument to the coronation of King Edward VII, but was preserved by the horticulturalist E. A. Bowles for his garden at nearby Myddelton House, where it remains today.


title: London Borough of Enfield

text: Enfield Grammar School with its Tudor Old Hall stands next to the Enfield Town Market Place and St. Andrew's Church, the school having been extended several times since 1586. A new hall and further additions were completed shortly before World War II.


title: London Borough of Enfield

text: Nearby historically was the palace of Edward VI, where Elizabeth I lived while a princess, including during the final illness of Henry VIII. Edward was taken there to join her, so that in the company of his sister, Edward Seymour, 1st Earl of Hertford could break the news to Edward, formally announcing the death of their royal father in the presence chamber at Enfield, on his knees to make formal obeisance to the boy as King. Later Elizabeth held court there when she was queen (this was remembered in the name Palace Gardens that was a street running behind Pearsons department store and is still recalled in the name of Enfield's shopping centre).


title: London Borough of Enfield

text: Enfield has a history of armaments manufacture—see Royal Small Arms Factory. The Lee–Enfield .303 rifle was standard issue for the British Army until 1957, although its usage carried on afterwards for some time. Other firearms manufactured there include the Bren and Sten machine guns—the "en" in both cases denoting the place of manufacture.


title: London Borough of Enfield

text: The world's first solid state circuitry colour televisions were manufactured by Ferguson at their now closed plant in Enfield.


title: London Borough of Enfield

text: The Barclays Bank branch in Enfield was the first place in the world to have an ATM or cash machine; it was officially opened in June 1967 by Reg Varney, a television actor and personality most famous for his lead role in the comedy series "On the Buses". This historical event was marked by a silver plaque on the wall of the bank, and later by an English Heritage Blue plaque.


title: London Borough of Enfield

text: A fine example of a grade II listed art deco factory building can be found along Southbury Road, with the former Ripaults Factory, now an office building for Travis Perkins.


title: London Borough of Enfield

text: The borough's Civic Centre is in Silver Street, Enfield Town, and is home to the council. Enfield Town is also home to the local credit union, North London Credit Union.


title: London Borough of Enfield

text: In 2007, Enfield Town centre completed a major redevelopment project under the name PalaceXchange while retaining the Palace Gardens Shopping Centre. An extension was added to the existing retail area with many new shops, and a second multi-storey car park was built along with a new road layout.


title: London Borough of Enfield

text: A major redevelopment of Edmonton Green including the shopping centre, and adjacent municipal housing over a wide area, started in 1999. This is still on-going, and provides new housing, health facilities, a new leisure centre, a supermarket, and many other civic features.


title: London Borough of Enfield

text: Many local activities are located around the A10 road, on the sites of former industrial enterprises, which has a number of large retail outlets and a large multiplex Cineworld cinema. The cinema also plays host to Jubilee Church on a Sunday morning; whilst cinemagoers continue to watch films in the other screens, a charismatic church which draws its worshippers from a wide area, hiring several auditoria for worship.


title: London Borough of Enfield

text: The western part of Enfield is largely residential, with shopping centres in Southgate, Palmers Green and Cockfosters.


title: London Borough of Enfield

text: Parts of Enfield experienced rioting in August 2011, in which a private car and a van were set alight and completely destroyed, a police car vandalised (smashed windows) and a number of shops in Enfield Town Centre (HMV, G. Mantella, Pearsons, Argos etc.) as well as others in the Enfield Retail Park being broken into and looted. The Sony Distribution Centre in the Innova Business Park, near Waltham Cross, was burnt to the ground. Although early reports questioned whether this was an arson or a coincidental incident arrests were later made in connection. In September 2012, a year after the attack, a rebuilt Sony Distribution Centre was opened by the prime minister, David Cameron.


title: London Borough of Enfield

text: Elections to Enfield Council take place every four years, there are 21 wards in the borough and three councillors are returned to each ward. Of the current 63 councillors there are 46 Labour members, 17 Conservatives meaning the Labour Group currently controls the council. The next Local Government Elections affecting Enfield will take place in 2022.


title: London Borough of Enfield

text: Two major NHS hospitals, Chase Farm Hospital operated by the Royal Free London NHS Foundation Trust and North Middlesex Hospital operated by the North Middlesex University Hospital NHS Trust, are located in the borough. NHS Enfield Clinical Commissioning Group is responsible for local primary health care, taking over this role from Enfield Primary Care NHS Trust in 2013. Another hospital in the borough, Highlands, was closed in 1993.


title: London Borough of Enfield

text: The London Borough of Enfield is the education authority for the district. Education is provided in a mix of community, voluntary aided and state schools. A number of private schools are also located in the borough.


title: London Borough of Enfield

text: The borough is home to the well-renowned Enfield Grammar School, founded in 1558, which still uses its Tudor building which is now often referred to as the Old Hall. Enfield Grammar School is based in the center of Enfield Town. St Ignatius' College is the borough's largest Catholic school and one of the best-performing non-selective schools in the borough, at both GCSE and A-Level. Bishop Stopford's School is Enfield's largest school, and The College of Haringey, Enfield and North East London also has a campus in the borough.


title: London Borough of Enfield

text: In 2007 a new school named Oasis Academy Enfield was opened, following the takeover of Albany by the Oasis Company as Oasis Academy Hadley.


title: London Borough of Enfield

text: Enfield County School is an all girls comprehensive school which sits opposite Enfield Grammar Boys school in Enfield town. It is a split site school with the lower years situated towards the area of Forty Hall.


title: London Borough of Enfield

text: Middlesex University, the former polytechnic, used to have two campuses in Enfield: Cat Hill and Trent Park. These sites both closed in 2011 and courses were relocated to the Hendon campus.


title: London Borough of Enfield

text: Oak Hill Theological College, an associate college of Middlesex University, is located in the borough.


title: London Borough of Enfield

text: The London Borough of Enfield has three fire stations; Edmonton, Enfield and Southgate. However, if an incident requires a higher attendance, appliances from throughout London can be mobilised to meet the needs of the area if required.


title: London Borough of Enfield

text: The three fire stations operate a total of five pumping appliances, one rescue tender, a command unit and a USAR unit.


title: London Borough of Enfield

text: Of the three fire stations within the borough, Enfield is mobilised to the most incidents and has the largest station ground (the area of which Enfield is the first appliance to attend).


title: London Borough of Enfield

text: The borough's museums include Forty Hall and the garden exhibitions at Capel Manor, Myddleton House (part of the Lee Valley Park) and the Whitewebbs Museum of Transport.


title: London Borough of Enfield

text: Scouting and Guiding have many groups within Enfield. The annual St George's Day Parade has around a thousand young people marching through Enfield Town, led by the Enfield District Scout Band .


title: London Borough of Enfield

text: The Boys' and Girls' Brigade are also quite active within the borough, participating in various ceremonies and events throughout the year, including the Enfield Town Show, St Georges Day Parade and on Remembrance Day Parade.


title: London Borough of Enfield

text: The Air Training Corps has three units in the area, No.85 (Enfield & Southgate) Squadron based in Winchmore Hill & Enfield Town, No.1159 (Edmonton) Squadron based in Edmonton, No.1571 (Ayleward) Squadron based in Aylward School and No.2473 (Highlands & Southgate) Squadron in Highlands School, Southgate.


title: London Borough of Enfield

text: The Red Room music and film project ran from 2008 - 2014 based in Ponders End, aimed at 13 - 19 year olds supported and funded by Enfield Council.


title: London Borough of Enfield

text: Enfield's King George's Field, named in memory of King George V, includes the Queen Elizabeth II athletics stadium, the Enfield Ignatians Rugby Club and numerous football, rugby and baseball diamonds. The playing fields were used as a POW camp for Italians during World War II.


title: London Borough of Enfield

text: The second largest playing fields are at Firs Farm on Firs Lane. There are a handful of rugby pitches along with more than a dozen football pitches. These are used by local amateur football clubs including Winchmore Hill Football Club, Mayfield Athletic FC and Southgate County FC. The pitches drain reasonably well but are generally in poor condition with old, rusted goalposts. The changing facilities are also very dilapidated, with some changing rooms being no more than concrete sheds with corrugated roofs.


title: London Borough of Enfield

text: Many sports teams and clubs are located in Enfield; Winchmore Hill Sports Club, Winchmore Hill Football Club, Enfield Town F.C., Enfield 1893 F.C., Cockfosters F.C., Enfield Borough F.C. and Futsal Club Enfield are the most prominent. The once successful football club, Enfield FC, were forced into a nomadic existence after the sale of their Southbury Road ground and became defunct at the end of the 2006/2007 season after years of ground sharing at many Hertfordshire based clubs. Enfield 1893 were formed after Enfield FC's demise and have spent time groundsharing at Ware FC and Broxbourne Borough FC. They finally found a ground in the borough when they merged with Brimsdown Rovers for the start of the 2011/2012 season. Saracens Rugby Football Club used to play in Enfield and still train at Bramley Road sports ground near Cockfosters. Enfield Ignatians R.F.C are currently the area's most successful Rugby club.


title: London Borough of Enfield

text: The principal cricket clubs in the area are Enfield, Winchmore Hill Cricket Club , Edmonton and Southgate, with many others playing cricket in and around the borough such as, Enfield Invicta, Mayfield CC, Myddleton House, North Enfield CC, Botany Bay and Holtwhites Trinibis CC. Many clubs play in either the Middlesex Cricket League or the Hertfordshire League. Enfield, Southgate, and Winchmore Hill have long been the area's most successful cricket clubs, regularly competing in the Middlesex Premier League. Enfield (in 1988) and Southgate (in 1977) are both past winners of the ECB National Club Cricket Championship


title: London Borough of Enfield

text: The main swimming club in the borough is Enfield Swim Squad, which was created from an amalgamation of Edmonton Phoenix S.C. and Griffins S.C. of Enfield.


title: London Borough of Enfield

text: Futsal is also played in the area with Futsal Club Enfield who were founded in 2007 playing in the National Futsal League Premiership South Division for 2019/20.


title: London Borough of Enfield

text: Gaelic Football is also played in the area. The club North London Shamrocks play at Enfield Playing Fields.


title: London Borough of Enfield

text: Enfield Phoenix is the local basketball club with teams playing in the National League and the Central London League. The team play at the Edmonton Leisure Centre, Edmonton Green with seating for 400 spectators.


title: London Borough of Enfield

text: Theatre goers are catered for by the Intimate Theatre, the Millfield Theatre, the Dugdale Centre and the Chickenshed Theatre Company, with amateur and professional shows, dance pieces, musicals and live music on show. Amateur dramatic companies include Saint Monica's Players, The Capel Players and The London Pantomimers. Talkies Community Cinema screens British Independent and specialist films in a range of venues across the borough.


title: London Borough of Enfield

text: It is also home to Cheerleading squads- Sapphire Elite Cheerleaders, North London Wildcats and Enfield Blaze All-star Cheerleaders.


title: London Borough of Enfield

text: The first international branch of the WWE Performance Center was opened in Enfield's Great Cambridge Industrial Estate on 11 January 2019. The center is used by the WWE as a professional wrestling school to train potential future wrestlers from across the globe, as well as a training centre for current WWE wrestlers.


title: London Borough of Enfield

text: In 1840 the first section of the Northern and Eastern Railway was opened from Stratford to Broxbourne, with stations at Water Lane (Angel Road) and Ponders End. Further stations were added in 1855 at Enfield Lock (Royal Small Arms Factory) and 1884 at Brimsdown. A branch line from Water Lane to Enfield Town was opened in 1849 serving Edmonton Low Level and Enfield Town. A further station was added in 1880 at Bush Hill Park. The direct line from London to Enfield Town was opened in 1872 with stations at Silver Street and Lower Edmonton (now called Edmonton Green). In 1891, a loop from Edmonton serving Southbury (Churchbury) and Turkey Street (Forty Hill) to Cheshunt on the main line was added. The Cheshunt and Enfield Town services are currently served by London Overground.


title: London Borough of Enfield

text: In 1871 the Great Northern Railway opened its station on Windmill Hill. This was later replaced in 1910 when the line was extended to Cuffley. This section of railway is now part of the line commonly known as the Hertford Loop.


title: London Borough of Enfield

text: Enfield is served by the Piccadilly line of the London Underground with stations at Arnos Grove, Southgate, Oakwood (named as Enfield West when it first opened) and Cockfosters (where the line terminates). Suburban London Overground and National Rail Great Northern services also run within the borough.


title: London Borough of Enfield

text: The main roads running through the borough include the North Circular Road, the M25, Green Lanes, the A10 and the old Hertford Road (A1010).


title: London Borough of Enfield

text: In March 2011, the main forms of transport that residents used to travel to work were: driving a car or van, 26.1% of all residents aged 16–74; underground, metro, light rail, tram, 8.2%; bus, minibus or coach, 8.2%; train, 7.3%; on foot, 4.0%; work mainly at or from home, 2.7%; passenger in a car or van, 1.6%.


title: Tweenies

text: Tweenies is a British live action children's television series created by Will Brenton and Iain Lauchlan. The programme is focused on four pre-school aged characters, known as the "Tweenies", playing, singing, dancing, and learning in a fictional playgroup in England. They are cared for by two adult Tweenies and two dogs.


title: Tweenies

text: 390 episodes were broadcast between 6 September 1999 and 25 July 2002. In 2000, the show won a BAFTA award for Best Pre-school Live Action series, and singles featuring exclusive songs spent some weeks in the charts during the early 2000s. The series premiered on the BBC's children's block, and aired reruns on CBeebies from the channel's launch in 2002 until 2016. In the United States, it was shown on Noggin, a sister channel to Nickelodeon. It also aired on Nickelodeon's Nick Jr. block from 18 July to 25 September 2003.


title: Tweenies

text: The idea for the programme came from Will Brenton and Iain Lauchlan, a pair with a track record of being involved in BBC children's programming. Together they set up Tell-Tale Productions at Elstree Film Studios in Hertfordshire. Iain Lauchlan was a presenter on "Play School", "Fingermouse" and "Playdays", meeting Brenton, a director, writer and also a presenter, during the latter. Together they started producing two of the "Playdays" strands before forming their own production companies Tell-Tale Productions and, more recently, Wish Films. They also created "The Fun Song Factory", "Boo!", "BB3B",


title: Tweenies

text: "Tweenies" is a production of Tell-Tale Productions in association with the BBC. Animation for the show was produced by A Productions, an animation studio based in Bristol, England, with Ealing Animation providing some animation in a few early episodes. Computer animation for the series was provided by "Clockwork Digital", with Ben Mars animating Mungo, the computer creature who appears in some later episodes.


title: Tweenies

text: Episodes consist of a mixture of story, song, and creative activity. Music plays an important part in the programme and children are encouraged to join in with songs and actions.


title: Tweenies

text: The programme is set in a nursery in England attended by the four Tweenies themselves: Jake, Fizz, Milo and Bella. They are in the care of two adults, Max and Judy, and two dogs, Doodles and Izzles.


title: Tweenies

text: The show includes a "Tweenie Clock", with five circular lights arranged in a pentagonal shape with the lights denoting "News Time" (Circle Time in the US) – Orange, "Messy Time" – Blue, "Song Time" – Yellow, "Telly Time" (Video Time in the US) – Green and "Story Time" – Red. "Surprise Time" is a special time determined when all five lights glow. A button at the centre of the clock (Purple) is pressed to select the activity that will be undertaken next.


title: Tweenies

text: It was formerly shown on CBeebies, from 11 February 2002 until 1 April 2016. Like a number of other CBeebies programmes, a live stage version of the show has toured in the UK. The most recent tour, "Top of the Tots", toured the British Isles during 2009, the series' 10th anniversary. In addition, the tour also played several shows in Hong Kong in late September and early October 2009. The Tweenies were also regulars on the annual CBeebies Live tours around the British Isles. Between 2003 and 2005, there was also a "Tweenies" theme park, at Alton Towers in the Cred Street section aimed at younger children, which replaced the "Barney" section and then was removed in favour of "Bob the Builder". In 2000, "Tweenies" won two awards: Best Pre-School Educational Program and Best Live Action Pre-School Programme.


title: Tweenies

text: The original prototypes for the Tweenies characters were designed by Sally Preisig of Mimics Productions; they were later re-scaled into two sizes for the characters shown on the series. In addition, Preisig also designed the character costumes and was co-constructor/developer for the Tweenies' full-size puppets. The costume fabric comes from the UK and was imported and dyed into their present colours; it is about the same thickness as fleece jumpers.


title: Tweenies

text: The "Tweenies" aired its first episode "Tweenie Band" on 6 September 1999 on its original programme block CBBC and last aired on 25 July 2002 with "What Makes Summer?" on its former channel CBeebies. In total, there are 390 episodes. There have been special episodes, such as a series of "Be Safe with the Tweenies" and expanded forty minute episodes.


title: Tweenies

text: Episode 252, "Favourite Songs" (first aired in March 2001) was removed from further airing in January 2013, after the BBC received 216 complaints about a scene within the episode, themed as a parody of an episode of "Top of the Pops" where Max appears dressed as Jimmy Savile. The episode aired three months after claims of sexual abuse committed by Savile came to light. The BBC removed "Tweenies" from the CBeebies schedule altogether for four months. The episodes then came back to the schedule from June 2013 until April 2016.


title: Tweenies

text: In the USA, it was shown on Noggin, a sister channel to Nickelodeon. It also aired on Nickelodeon's Nick Jr. block from 18 July to 25 September 2003.


title: Tweenies

text: In November 2000, the Tweenies appeared on a few episodes of the show, performing their single "No. 1".


title: Tweenies

text: In December 2000, the Tweenies starred in their own dedicated special of the series which featured more of their singles like "Do The Lollipop" and "Have Fun Go Mad!". This episode was later released on VHS and DVD as "Music is Pop-a-Rooney!" in July 2002, with new wrap-around segments recorded in the Playgroup.


title: Tweenies

text: Beginning in 2001, the first "Tweenies Live!" show went on tour around arenas around the United Kingdom, produced by Tell-Tale Productions and BBC Worldwide, the show featured the Tweenies singing their favourite songs. A live-recording of this tour from the London Arena was released on VHS in July 2001. A follow-up show - "Tweenies Live! 2: The Fab-a-Rooney Tour", toured from February–July 2002, and featured the live debut of Izzles.


title: Tweenies

text: In December 2001, a follow-up show titled "Tweenies Live! The Christmas Present" was presented at the Royal Albert Hall for the Christmas period, and was later given a wide arena tour in December 2003-January 2004, where Izzles joined the cast. A recording of the 2001 version was released on VHS in November 2002 and was later released on DVD in November 2006 as part of the "Tweenies: The Ultimate Christmas Collection" DVD compilation.


title: Tweenies

text: The fourth tour - "No Sleep 'til Bedtime", toured the UK from March–May 2005. This tour was notable for featuring covers of licensed tracks sung by the Tweenies.


title: Tweenies

text: The fifth tour - "The Enchanted Toyshop" toured the UK in the Spring and Summer of 2007. It was the first Tweenies live tour to be solely produced by BBC Worldwide and the first to be shown in theatres instead of arenas.


title: Tweenies

text: The most recent tour - "Top of the Tots", toured the UK in Early 2009 to celebrate the programme's 10th anniversary.


title: Tweenies

text: Many of the above videos have also been dubbed into other languages. In addition, several of the "Tweenies" videos were produced in special versions for retailers with additional footage (for example, the "Party Games, Laughs & Giggles" video distributed at Marks & Spencer stores carried two additional segments, adding ten minutes to the video's running time). Many of the above videos were also released as DVDs under different names. iTunes has released a "Best of Tweenies volume 1" which consists of the first six episodes of the series.


title: Tweenies

text: From 1999 until 2006, various books were released from "Tweenies", based on the episodes that were mostly aired. There were also annuals that were released from 2001 to 2006 and other books, based on each character from "Tweenies".


title: Tweenies

text: From the late 1990s into the early 2000s, the Tweenies were one of the most popular set of children's TV characters at the time, which led to a wide range merchandise being sold. Many toys were sold in various retailers, such as Toys "R" Us, Argos, & Woolworths. Merchandise included soft toys, playsets, collectible figures, board games, jigsaws and puzzles, talking toys, the video games which were "Tweenies: Doodles' Bones" for Game Boy Color and "Tweenies: Game Time" for PlayStation, and many other products.


title: Tweenies

text: Tweenies: Game Time is an educational kids video game released in Europe on 30 March 2001 for PlayStation, published by BBC Multimedia. It takes the form of a minigame collection. The player controls Milo, Jake, Fizz and Bella through a series of 4 scenarios inspired by the show, one for each character.


title: Tweenies

text: The game was developed by Intelligent Games. The direction of the game was partially based on research by BBC Worldwide of children who watched the Television show, which found that "singing and dancing are major factors in the success of the show". according to a BBC article from before the game launched.


title: Sheep dog

text: A sheep dog or sheepdog is generally a dog or breed of dogs historically used in connection with the raising of sheep. These include livestock guardian dogs used to guard sheep and other livestock and herding dogs used to move, manage and control sheep and other livestock.


title: Isabella Boylston

text: Hildur Isabella Boylston (born October 13, 1986) is an American ballet dancer who is currently a principal dancer with the American Ballet Theatre (ABT).


title: Isabella Boylston

text: Boylston was born Hildur Isabella Boylston, named for an Icelandic great-grandmother, in Sun Valley, Idaho. Her father Mike was an American country-blues drummer and "ski bum" while her mother Cornelia was a Swedish engineer.


title: Isabella Boylston

text: When she was seven, her family moved to Boulder, Colorado, where she started training at the Boulder Ballet. By the age of 12, she had started studies at the Colorado Ballet Academy. During that time, she won a gold medal at the 2001 Youth America Grand Prix Finals in New York City.


title: Isabella Boylston

text: In 2002, she received a full scholarship to train at the HARID Conservatory in Boca Ratón, Florida. During her time there she worked with choreographer Mark Godden and danced leading roles such as Medora in "Le Corsaire", the Paquita pas de trois, Lise in "La fille mal gardée" and the Sugarplum Fairy in "The Nutcracker". In 2004, she received the Reuger Scholarship for excellence in dance. She participated in summer programs at the School of American Ballet, the Boston Ballet and the American Ballet Theatre.


title: Isabella Boylston

text: In 2005, Boylston joined the ABT Studio Company and became an apprentice with the main company in May 2006. She joined the corps de ballet in March 2007, was promoted to soloist in June 2011 and principal in August 2014. Lead roles she danced include Nikiya in "La Bayadère", Kitri in "Don Quixote" and Columbine in "Harlequinade". Boylston won the Princess Grace Award in 2009 and was nominated for the 2010 Prix Benois de la Danse.


title: Isabella Boylston

text: Outside of ABT, she also designed costumes for the Pacific Northwest Ballet's 2010 production of Benjamin Millepied's "3 Movements", a ballet set to Steve Reich's "Three Movements for Orchestra". She has appeared as a guest artist with the Mariinsky Ballet in St. Petersburg and the Royal Danish Ballet. Boylston also served as Jennifer Lawrence's dance double in 2018 film "Red Sparrow", choreographed by Justin Peck. In 2019, she helped breaking the Guinness World Record for the most dancers ever to go on pointe at the same time, alongside James Whiteside on "Live with Kelly and Ryan".


title: Isabella Boylston

text: Alastair Macaulay of the "New York Times" noted Boylston portrayal of Lise in "La Fille mal gardée" is "intimate, heartfelt, rapturous." On "Swan Lake", Macaulay praised Boylston's musicality and phrasing.


title: Isabella Boylston

text: In 2020, Boylston participated in Misty Copeland's fundraiser, Swans for Relief, by dancing "The Swan", in light of the impacts of the 2019-20 coronavirus pandemic on the dance community. The fund will go to participating dancers' companies and other related relief funds.


title: Altair 8800

text: The Altair 8800 is a microcomputer designed in 1974 by MITS and based on the Intel 8080 CPU. Interest grew quickly after it was featured on the cover of the January 1975 issue of Popular Electronics and was sold by mail order through advertisements there, in "Radio-Electronics", and in other hobbyist magazines. The Altair is widely recognized as the spark that ignited the microcomputer revolution as the first commercially successful personal computer. The computer bus designed for the Altair was to become a "de facto" standard in the form of the S-100 bus, and the first programming language for the machine was Microsoft's founding product, Altair BASIC.


title: Altair 8800

text: While serving at the Air Force Weapons Laboratory at Kirtland Air Force Base, Ed Roberts and Forrest M. Mims III decided to use their electronics background to produce small kits for model rocket hobbyists. In 1969, Roberts and Mims, along with Stan Cagle and Robert Zaller, founded Micro Instrumentation and Telemetry Systems (MITS) in Roberts' garage in Albuquerque, New Mexico, and started selling radio transmitters and instruments for model rockets.


title: Altair 8800

text: The model rocket kits were a modest success and MITS wanted to try a kit that would appeal to more hobbyists. The November 1970 issue of "Popular Electronics" featured the Opticom, a kit from MITS that would send voice over an LED light beam. As Mims and Cagle were losing interest in the kit business, Roberts bought his partners out, then began developing a calculator kit. Electronic Arrays had just announced the EAS100, a set of six large scale integrated (LSI) circuit chips that would make a four-function calculator. The MITS 816 calculator kit used the chipset and was featured on the November 1971 cover of "Popular Electronics". This calculator kit sold for , or $275 assembled. Forrest Mims wrote the assembly manual for this kit and many others over the next several years. As payment for each manual he often accepted a copy of the kit.


title: Altair 8800

text: The calculator was successful and was followed by several improved models. The MITS 1440 calculator was featured in the July 1973 issues of "Radio-Electronics". It had a 14-digit display, memory, and square root function. The kit sold for and the assembled version was . MITS later developed a programmer unit that would connect to the 816 or 1440 calculator and allow programs of up to 256 steps.


title: Altair 8800

text: In addition to calculators, MITS made a line of test equipment kits. These included an IC tester, a waveform generator, a digital voltmeter, and several other instruments. To keep up with the demand, MITS moved into a larger building at 6328 Linn NE in Albuquerque in 1973. They installed a wave soldering machine and an assembly line at the new location. In 1972, Texas Instruments developed its own calculator chip and started selling complete calculators at less than half the price of other commercial models. MITS and many other companies were devastated by this, and Roberts struggled to reduce his quarter-million-dollar debt.


title: Altair 8800

text: In January 1972, "Popular Electronics" merged with another Ziff-Davis magazine, "Electronics World". The change in editorial staff upset many of their authors, and they started writing for a competing magazine, "Radio-Electronics". In 1972 and 1973, some of the best construction projects appeared in "Radio-Electronics".


title: Altair 8800

text: In 1974, Art Salsberg became editor of "Popular Electronics". It was Salsberg's goal to reclaim the lead in electronics projects. He was impressed with Don Lancaster's TV Typewriter ("Radio Electronics", September 1973) article and wanted computer projects for "Popular Electronics". Don Lancaster did an ASCII keyboard for "Popular Electronics" in April 1974. They were evaluating a computer trainer project by Jerry Ogden when the Mark-8 8008-based computer by Jonathan Titus appeared on the July 1974 cover of "Radio-Electronics". The computer trainer was put on hold and the editors looked for a real computer system. ("Popular Electronics" gave Jerry Ogden a column, "Computer Bits", starting in June 1975.)


title: Altair 8800

text: One of the editors, Les Solomon, knew MITS was working on an Intel 8080 based computer project and thought Roberts could provide the project for the always popular January issue. The TV Typewriter and the Mark-8 computer projects were just a detailed set of plans and a set of bare printed circuit boards. The hobbyist faced the daunting task of acquiring all of the integrated circuits and other components. The editors of "Popular Electronics" wanted a complete kit in a professional-looking enclosure.


title: Altair 8800

text: Ed Roberts and his head engineer, Bill Yates, finished the first prototype in October 1974 and shipped it to "Popular Electronics" in New York via the Railway Express Agency. However, it never arrived due to a strike by the shipping company. Solomon already had a number of pictures of the machine and the article was based on them. Roberts got to work on building a replacement. The computer on the magazine cover is an empty box with just switches and LEDs on the front panel. The finished Altair computer had a completely different circuit board layout than the prototype shown in the magazine. The January 1975 issues appeared on newsstands a week before Christmas of 1974 and the kit was officially (if not yet practically) available for sale.


title: Altair 8800

text: The typical MITS product had a generic name like the "" or the "". Ed Roberts was busy finishing the design and left the naming of the computer to the editors of "Popular Electronics".


title: Altair 8800

text: One explanation of the Altair name, which editor Les Solomon later told the audience at the first Altair Computer Convention (March 1976), is that the name was inspired by Les's 12-year-old daughter, Lauren. "She said why don't you call it Altair – that's where the Enterprise is going tonight." The "" episode is probably "Amok Time", as this is the only one from "The Original Series" which takes the Enterprise crew to Altair (Six).


title: Altair 8800

text: Another explanation is that the Altair was originally going to be named the PE-8 (Popular Electronics 8-bit), but Les Solomon thought this name to be rather dull, so Les, Alexander Burawa (associate editor), and John McVeigh (technical editor) decided that: "It's a stellar event, so let's name it after a star." McVeigh suggested "Altair", the twelfth brightest star in the sky.


title: Altair 8800

text: Ed Roberts had designed and manufactured programmable calculators and was familiar with the microprocessors available in 1974. He thought the Intel 4004 and Intel 8008 were not powerful enough (in fact several microcomputers based on Intel chips were already on the market: the Canadian company Microsystems International's CPS-1 built-in 1972 used a MIL MF7114 chip modeled on the 4004, the Micral marketed in January 1973 by the French company R2E and the MCM/70 marketed in 1974 by the Canadian company Micro Computer Machines); the National Semiconductor IMP-8 and IMP-16 required external hardware; the Motorola 6800 was still in development. So he chose the 8-bit Intel 8080. At that time, Intel's main business was selling memory chips by the thousands to computer companies. They had no experience in selling small quantities of microprocessors. When the 8080 was introduced in April 1974, Intel set the single unit price at (About in 2014 dollars). "That figure had a nice ring to it," recalled Intel's Dave House in 1984. "Besides, it was a computer, and they usually cost thousands of dollars, so we felt it was a reasonable price." Ed Roberts had experience in buying OEM quantities of calculator chips and he was able to negotiate a price (about in 2014 dollars) for the 8080 microprocessor chips.


title: Altair 8800

text: Intel made the Intellec-8 Microprocessor Development System that typically sold for a very profitable . It was functionally similar to the Altair 8800 but it was a commercial grade system with a wide selection of peripherals and development software. Customers would ask Intel why their Intellec-8 was so expensive when that Altair was only . Some salesmen said that MITS was getting cosmetic rejects or otherwise inferior chips. In July 1975, Intel sent a letter to its sales force stating that the MITS Altair 8800 computer used standard Intel 8080 parts. The sales force should sell the Intellec system based on its merits and that no one should make derogatory comments about valued customers like MITS. The letter was reprinted in the August 1975 issue of MITS Computer Notes. The "cosmetic defect" rumor has appeared in many accounts over the years although both MITS and Intel issued written denials in 1975.


title: Altair 8800

text: For a decade, colleges had required science and engineering majors to take a course in computer programming, typically using the FORTRAN or BASIC languages. This meant there was a sizable customer base who knew about computers. In 1970, electronic calculators were not seen outside of a laboratory, but by 1974 they were a common household item. Calculators and video games like "Pong" introduced computer power to the general public. Electronics hobbyists were moving on to digital projects such as digital voltmeters and frequency counters. The Altair had enough power to be actually useful, and was designed as an expandable system that opened it up to all sorts of applications.


title: Altair 8800

text: Ed Roberts optimistically told his banker that he could sell 800 computers, while in reality they needed to sell 200 over the next year just to break even. When readers got the January issue of "Popular Electronics", MITS was flooded with inquiries and orders. They had to hire extra people just to answer the phones. In February MITS received 1,000 orders for the Altair 8800. The quoted delivery time was 60 days but it was months before they could meet that. Roberts focused on delivering the computer; all of the options would wait until they could keep pace with the orders. MITS claimed to have delivered 2,500 Altair 8800s by the end of May. The number was over 5,000 by August 1975. MITS had under 20 employees in January but had grown to 90 by October 1975.


title: Altair 8800

text: The Altair 8800 computer was a break-even sale for MITS. They needed to sell additional memory boards, I/O boards and other options to make a profit. The system came with a "1024 word" (1024 byte) memory board populated with 256 bytes. The BASIC language was announced in July 1975 and it required one or two 4096 word memory boards and an interface board.


title: Altair 8800

text: MITS had no competition in the US for the first half of 1975. Their 4K memory board used dynamic RAM and it had several design problems. The delay in shipping optional boards and the problems with the 4K memory board created an opportunity for outside suppliers.


title: Altair 8800

text: An enterprising Altair owner, Robert Marsh, designed a 4K static memory that was plug-in compatible with the Altair 8800 and sold for $255. His company was Processor Technology, one of the most successful Altair compatible board suppliers. Their advertisement in the July 1975 issue of "Popular Electronics" promised interface and PROM boards in addition to the 4K memory board. They would later develop a popular video display board that would plug directly into the Altair.


title: Altair 8800

text: A consulting company in San Leandro, California, IMS Associates, Inc., wanted to purchase several Altair computers, but the long delivery time convinced them that they should build their own computers. In the October 1975 issue of "Popular Electronics", a small advertisement announced the IMSAI 8080 computer. The ad noted that all boards were "plug compatible" with the Altair 8800. The computer cost $439 as a kit. The first 50 IMSAI computers shipped in December 1975. The IMSAI 8080 computer improved on the original Altair design in several areas. It was easier to assemble: The Altair required 60 wire connections between the front panel and the motherboard (backplane.) The IMSAI motherboard had 18 slots. The MITS motherboard consisted of 4 slots segments that had to be connected together with 100 wires. The IMSAI also had a larger power supply to handle the increasing number of expansion boards used in typical systems. The IMSAI advantage was short lived because MITS had recognized these shortcomings and developed the Altair 8800B, which was introduced in June 1976.


title: Altair 8800

text: In 1977, Pertec Computer Corporation purchased MITS and began to market the computer, without changes (except for branding), as the PCC 8800 in 1978.


title: Altair 8800

text: In the first design of the Altair, the parts needed to make a complete machine would not fit on a single motherboard, and the machine consisted of four boards stacked on top of each other with stand-offs. Another problem facing Roberts was that the parts needed to make a truly useful computer were not available, or would not be designed in time for the January launch date. So during the construction of the second model, he decided to build most of the machine on removable cards, reducing the motherboard to nothing more than an interconnect between the cards, a backplane. The basic machine consisted of five cards, including the CPU on one and memory on another. He then looked for a cheap source of connectors and came across a supply of 100-pin edge connectors. The S-100 bus was eventually acknowledged by the professional computer community and adopted as the IEEE-696 computer bus standard.


title: Altair 8800

text: The Altair bus consists of the pins of the Intel 8080 run out onto the backplane. No particular level of thought went into the design, which led to disasters such as shorting from various power lines of differing voltages being located next to each other. Another oddity was that the system included two unidirectional 8-bit data buses, when the normal practice was for a single bidirectional bus (this oddity did, however, allow a later expansion of the S-100 standard to 16 bits bidirectional by using both 8-bit buses in parallel). A deal on power supplies led to the use of +8 V and +18 V, which had to be locally regulated on the cards to TTL (+5 V) or RS-232 (+12 V) standard voltage levels.


title: Altair 8800

text: The Altair shipped in a two-piece case. The backplane and power supply were mounted on a base plate, along with the front and rear of the box. The "lid" was shaped like a C, forming the top, left, and right sides of the box. The front panel, which was inspired by the Data General Nova minicomputer, included a large number of toggle switches to feed binary data directly into the memory of the machine, and a number of red LEDs to read those values back out.


title: Altair 8800

text: Programming the Altair via the front panel could be a tedious and time-consuming process. Programming required the toggling of the switches to positions corresponding to the desired 8080 microprocessor instruction or opcode in binary, then used the "DEPOSIT NEXT" switch to load that instruction into the next address of the machine's memory. This step was repeated until all the opcodes of a presumably complete and correct program were in place. The only output from the programs was the patterns of lights on the panel. Nevertheless, many were sold in this form. Development was already underway on additional cards, including a paper tape reader for storage, additional RAM cards, and an RS-232 interface to connect to a proper teletype terminal.


title: Altair 8800

text: Ed Roberts received a letter from Traf-O-Data asking whether he would be interested in buying what would eventually be the BASIC programming language for the machine. He called the company and reached a private home, where no one had heard of anything like BASIC. In fact the letter had been sent by Bill Gates and Paul Allen from the Boston area, and they had no BASIC yet to offer. When they called Roberts to follow up on the letter he expressed his interest, and the two started work on their BASIC interpreter using a self-made simulator for the 8080 on a PDP-10 mainframe computer. They figured that they had 30 days before someone else beat them to the punch, and once they had a version working on the simulator, Allen flew to Albuquerque to deliver the program, Altair BASIC (aka MITS 4K BASIC), on a paper tape. The first time it was run, it displayed "READY", then Allen typed "PRINT 2+2", and it immediately printed the correct answer: "4". The game "Lunar Lander" was entered in, and this worked as well. Gates soon joined Allen and formed Microsoft, then spelled "Micro-Soft".


title: Stephen Perkins

text: Stephen Andrew Perkins (September 13, 1967) is an American musician and songwriter. A drummer and percussionist, he currently plays with Jane's Addiction and Hellride.


title: Stephen Perkins

text: Following the dissolution of Jane's Addiction, Perkins continued to play with frontman Perry Farrell in the rock band Porno for Pyros. He also has a solo project called Banyan, and in the 1990s was involved in Lil' Pit with Mike Watt. In 1992, he appeared as percussionist on Rage Against the Machine's "Know Your Enemy" and in 1995, contributed percussion to the Red Hot Chili Peppers songs "One Big Mob" and "Warped" on their album "One Hot Minute" (fellow Jane's Addiction member Dave Navarro was the guitarist for the Chili Peppers during the making of that album). Perkins also played drums on the Nine Inch Nails album "The Downward Spiral" on the track "I Do Not Want This". In addition to Banyan, Perkins worked with Jane's Addiction guitarist Dave Navarro, Jane's Addiction bass player Chris Chaney and singer/guitarist Steve Isaacs on a project entitled The Panic Channel from 2004 until 2007.


title: Stephen Perkins

text: Perkins was also involved with members of Suicidal Tendencies in their Infectious Grooves side project, as well as Hellride, a Los Angeles-based Stooges cover band. Jane's Addiction reunited and played at the "NME Awards" in April 2008, and recorded the album "The Great Escape Artist" in 2011, and produced a live album "Live in NYC", from material recorded during the follow-up tour. The band has played sporadically since 2013, as members have worked on various side projects. Perkins most recently played with the band in February 2020 during a show with the Red Hot Chili Peppers; Perkins also filled in on drums for the Red Hot Chili Peppers at the same show.


title: 1996 UEFA Champions League Final

text: The 1996 UEFA Champions League Final was a football match played on 22 May 1996 between Ajax of the Netherlands and Juventus of Italy. The match ended in a 1–1 draw after extra time, forcing a penalty shoot-out, which Juventus won 4–2. It was the club's second triumph in the competition.


title: 1996 UEFA Champions League Final

text: Although Juventus won the 1996 Champions League Final, the victory remains controversial because of accusations of doping. The Juventus team has been accused of using erythropoietin (EPO) and the matter went to trial in 2004. In November 2004, club doctor Riccardo Agricola was given a 22-month prison sentence and fined €2,000 for sporting fraud by providing performance-enhancing drugs, specifically EPO, to players between 1994 and 1998, Leading hematologist Giuseppe d'Onofrio said that it was "practically certain" that midfielders Antonio Conte and Alessio Tacchinardi had taken EPO to overcome brief bouts of anemia, and that it was "very probable" that seven other players – Alessandro Birindelli, Alessandro Del Piero, Didier Deschamps, Dimas, Paolo Montero, Gianluca Pessotto and Moreno Torricelli – had taken EPO in small doses.


title: 1996 UEFA Champions League Final

text: In April 2005, the Court of Arbitration for Sport gave the following advisory opinion, in part: "The use of pharmaceutical substances which are not expressly prohibited by sports law, and which cannot be considered as substances similar or related to those expressly prohibited, is not to be sanctioned by disciplinary measures. However, regardless of the existence or not of any judgement rendered by a State court, sports authorities are under the obligation to prosecute the use of pharmaceutical substances which are prohibited by sports law or any other anti-doping rule violation in order to adopt disciplinary measures." In December 2005, Agricola was acquitted of the charges by Turin's court of appeal. In March 2007, in the final verdict by the Supreme Court of Cassation, stated: "that in the years of 1994 to 1998 there was no ascertained positive case of doping substances by Juventus players, that the purchase of erythropoietin or its administration to the athletes of the club does not emerge from any act of the trial, and that the same expert had identified the possibility of an administration of erythropoietin in distant terms from the sure evidence ("very probable" and in two cases "practically certain"): it is that there fore, the judgement of probability and not of certainty, did not allow for a statement of responsibility." The verdict also went on to say: "In response to the conclusion taken, the territorial court notes that there were no deferred values higher than the limits set in the various antidoping protocols and that the situation of the Juventus players, both with reference to the average hematological values, and in relation to that of material balance, did not differ from the national average population.


title: Distribution (mathematics)

text: Distributions, also known as Schwartz distributions or generalized functions, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative.


title: Distribution (mathematics)

text: Distributions are widely used in the theory of partial differential equations, where it may be easier to establish the existence of distributional solutions than classical solutions, or where appropriate classical solutions may not exist. Distributions are also important in physics and engineering where many problems naturally lead to differential equations whose solutions or initial conditions are singular, such as the Dirac delta function.


title: Distribution (mathematics)

text: A function formula_1 is normally thought of as on the in the function domain by "sending" a point formula_2 in its domain to the point formula_3 Instead of acting on points, distribution theory reinterprets functions such as formula_1 as acting on in a certain way.


title: Distribution (mathematics)

text: In applications to physics and engineering, are usually infinitely differentiable complex-valued (or real-valued) functions with compact support that are defined on some given non-empty open subset formula_5. (Bump functions are examples of test functions.) The set of all such test functions forms a vector space that is denoted by formula_6 or formula_7


title: Distribution (mathematics)

text: Most commonly encountered functions, including all continuous maps formula_8 if using formula_9 can be canonically reinterpreted as acting via "integration against a test function." Explicitly, this means that formula_1 "acts on" a test function formula_11 by "sending" it to the number formula_12 which is often denoted by formula_13 This new action formula_14 of formula_1 is a scalar-valued map, denoted by formula_16 whose domain is the space of test functions formula_17 This functional formula_18 turns out to have the two defining properties of what is known as a : it is linear and also continuous when formula_19 is given a certain topology called . The action of this distribution on a test function can be interpreted as a weighted average of the distribution on the support of the test function, even if the values of the distribution at a single point are not well-defined. Distributions like formula_18 that arise from functions in this way are prototypical examples of distributions, but many cannot be defined by integration against any function. Examples of the latter include the Dirac delta function and distributions defined to act by integration of test functions against certain measures. It is nonetheless still possible to reduce any arbitrary distribution down to a simpler of related distributions that do arise via such actions of integration.


title: Distribution (mathematics)

text: More generally, a is by definition a linear functional on formula_6 that is continuous when formula_6 is given a topology called the . This leads to space of (all) distributions on formula_23, usually denoted by formula_24 (note the prime), which by definition is the space of all distributions on formula_23 (that is, it is the continuous dual space of formula_6); it is these distributions that are the main focus of this article.


title: Distribution (mathematics)

text: Definitions of the appropriate topologies on spaces of test functions and distributions are given in the article on spaces of test functions and distributions. This article is primarily concerned with the definition of distributions, together with their properties and some important examples.


title: Distribution (mathematics)

text: The practical use of distributions can be traced back to the use of Green functions in the 1830s to solve ordinary differential equations, but was not formalized until much later. According to , generalized functions originated in the work of on second-order hyperbolic partial differential equations, and the ideas were developed in somewhat extended form by Laurent Schwartz in the late 1940s. According to his autobiography, Schwartz introduced the term "distribution" by analogy with a distribution of electrical charge, possibly including not only point charges but also dipoles and so on. comments that although the ideas in the transformative book by were not entirely new, it was Schwartz's broad attack and conviction that distributions would be useful almost everywhere in analysis that made the difference.


title: Distribution (mathematics)

text: The Lemma combined with the fact that the formal transpose of the formal transpose is the original differential operator, that is, formula_50 enables us to arrive at the correct definition: the formal transpose induces the (continuous) canonical linear operator formula_51 defined by formula_52 We claim that the transpose of this map, formula_53 can be taken as formula_54 To see this, for every formula_55 compute its action on a distribution of the form formula_18 with formula_57:


title: Distribution (mathematics)

text: We call the continuous linear operator formula_59 the . Its action on an arbitrary distribution formula_60 is defined via:


title: Distribution (mathematics)

text: A differential operator of order 0 is just multiplication by a smooth function. And conversely, if formula_1 is a smooth function then formula_67 is a differential operator of order 0, whose formal transpose is itself (that is, formula_68). The induced differential operator formula_69 maps a distribution formula_70 to a distribution denoted by formula_71 We have thus defined the multiplication of a distribution by a smooth function.


title: Distribution (mathematics)

text: We now give an alternative presentation of the multiplication of a distribution formula_70 on formula_23 by a smooth function formula_74 The product formula_75 is defined by


title: Distribution (mathematics)

text: This definition coincides with the transpose definition since if formula_77 is the operator of multiplication by the function formula_78 (that is, formula_79), then


title: Distribution (mathematics)

text: Under multiplication by smooth functions, formula_24 is a module over the ring formula_83 With this definition of multiplication by a smooth function, the ordinary product rule of calculus remains valid. However, some unusual identities also arise. For example, if formula_84 is the Dirac delta distribution on formula_85 then formula_86 and if formula_87 is the derivative of the delta distribution, then


title: Distribution (mathematics)

text: The bilinear multiplication map formula_89 given by formula_90 is continuous; it is however, hypocontinuous.


title: Distribution (mathematics)

text: Example. The product of any distribution formula_70 with the function that is identically on formula_23 is equal to formula_93


title: Distribution (mathematics)

text: Example. Suppose formula_94 is a sequence of test functions on formula_23 that converges to the constant function formula_96 For any distribution formula_70 on formula_98 the sequence formula_99 converges to formula_100


title: Distribution (mathematics)

text: If formula_62 converges to formula_63 and formula_94 converges to formula_57 then formula_105 converges to formula_106


title: Distribution (mathematics)

text: It is easy to define the product of a distribution with a smooth function, or more generally the product of two distributions whose singular supports are disjoint. With more effort, it is possible to define a well-behaved product of several distributions provided their wave front sets at each point are compatible. A limitation of the theory of distributions (and hyperfunctions) is that there is no associative product of two distributions extending the product of a distribution by a smooth function, as has been proved by Laurent Schwartz in the 1950s. For example, if formula_107 is the distribution obtained by the Cauchy principal value


title: Distribution (mathematics)

text: so the product of a distribution by a smooth function (which is always well-defined) cannot be extended to an associative product on the space of distributions.


title: Distribution (mathematics)

text: Thus, nonlinear problems cannot be posed in general and thus are not solved within distribution theory alone. In the context of quantum field theory, however, solutions can be found. In more than two spacetime dimensions the problem is related to the regularization of divergences. Here Henri Epstein and Vladimir Glaser developed the mathematically rigorous (but extremely technical) . This does not solve the problem in other situations. Many other interesting theories are non-linear, like for example the Navier–Stokes equations of fluid dynamics.


title: Distribution (mathematics)

text: Several not entirely satisfactory theories of algebras of generalized functions have been developed, among which Colombeau's (simplified) algebra is maybe the most popular in use today.


title: Distribution (mathematics)

text: Inspired by Lyons' rough path theory, Martin Hairer proposed a consistent way of multiplying distributions with certain structures (regularity structures), available in many examples from stochastic analysis, notably stochastic partial differential equations. See also Gubinelli–Imkeller–Perkowski (2015) for a related development based on Bony's paraproduct from Fourier analysis.


title: Distribution (mathematics)

text: Let formula_70 be a distribution on formula_113 Let formula_114 be an open set in formula_115 and formula_116 If formula_117 is a submersion then it is possible to define


title: Distribution (mathematics)

text: The pullback is often denoted formula_120 although this notation should not be confused with the use of '*' to denote the adjoint of a linear mapping.


title: Distribution (mathematics)

text: The condition that formula_117 be a submersion is equivalent to the requirement that the Jacobian derivative formula_122 of formula_117 is a surjective linear map for every formula_124 A necessary (but not sufficient) condition for extending formula_125 to distributions is that formula_117 be an open mapping. The Inverse function theorem ensures that a submersion satisfies this condition.


title: Distribution (mathematics)

text: If formula_117 is a submersion, then formula_125 is defined on distributions by finding the transpose map. The uniqueness of this extension is guaranteed since formula_125 is a continuous linear operator on formula_7 Existence, however, requires using the change of variables formula, the inverse function theorem (locally), and a partition of unity argument.


title: Distribution (mathematics)

text: In the special case when formula_117 is a diffeomorphism from an open subset formula_114 of formula_115 onto an open subset formula_23 of formula_115 change of variables under the integral gives:


title: Distribution (mathematics)

text: Under some circumstances, it is possible to define the convolution of a function with a distribution, or even the convolution of two distributions.


title: Distribution (mathematics)

text: Recall that if formula_1 and formula_140 are functions on formula_115 then we denote by formula_142 defined at formula_143 to be the integral


title: Distribution (mathematics)

text: provided that the integral exists. If formula_145 are such that formula_146 then for any functions formula_147 and formula_148 we have formula_149 and formula_150 If formula_1 and formula_140 are continuous functions on formula_153 at least one of which has compact support, then formula_154 and if formula_155 then the value of formula_142 on formula_157 do depend on the values of formula_1 outside of the Minkowski sum formula_159


title: Distribution (mathematics)

text: Importantly, if formula_160 has compact support then for any formula_161 the convolution map formula_162 is continuous when considered as the map formula_163 or as the map formula_164


title: Distribution (mathematics)

text: Given formula_165 the translation operator formula_166 sends formula_167 to formula_168 defined by formula_169 This can be extended by the transpose to distributions in the following way: given a distribution formula_170 is the distribution formula_171 defined by formula_172


title: Distribution (mathematics)

text: Given formula_173 define the function formula_174 by formula_175 Given a distribution formula_170 let formula_177 be the distribution defined by formula_178 The operator formula_179 is called .


title: Distribution (mathematics)

text: Convolution of formula_1 with a distribution formula_184 can be defined by taking the transpose of formula_185 relative to the duality pairing of formula_186 with the space formula_187 of distributions. If formula_188 then by Fubini's theorem


title: Distribution (mathematics)

text: An alternative way to define the convolution of a test function formula_1 and a distribution formula_70 is to use the translation operator formula_195 The convolution of the compactly supported function formula_1 and the distribution formula_70 is then the function defined for each formula_143 by


title: Distribution (mathematics)

text: It can be shown that the convolution of a smooth, compactly supported function and a distribution is a smooth function. If the distribution formula_70 has compact support, and if formula_1 is a polynomial (resp. an exponential function, an analytic function, the restriction of an entire analytic function on formula_202 to formula_153 the restriction of an entire function of exponential type in formula_202 to formula_115), then the same is true of formula_206 If the distribution formula_70 has compact support as well, then formula_208 is a compactly supported function, and the Titchmarsh convolution theorem implies that:


title: Distribution (mathematics)

text: Let formula_212 and formula_184 and assume that at least one of formula_1 and formula_70 has compact support. The of formula_1 and formula_170 denoted by formula_218 or by formula_219 is the smooth function:


title: Distribution (mathematics)

text: Let formula_223 be the map formula_224. If formula_70 is a distribution, then formula_223 is continuous as a map formula_227. If formula_70 also has compact support, then formula_223 is also continuous as the map formula_230 and continuous as the map formula_231


title: Distribution (mathematics)

text: If formula_232 is a continuous linear map such that formula_233 for all formula_234 and all formula_235 then there exists a distribution formula_184 such that formula_237 for all formula_238


title: Distribution (mathematics)

text: Let formula_84 be the Dirac measure at 0 and let formula_244 be its derivative as a distribution. Then formula_245 and formula_246 Importantly, the associative law fails to hold:


title: Distribution (mathematics)

text: It is also possible to define the convolution of two distributions formula_60 and formula_70 on formula_153 provided one of them has compact support. Informally, to define formula_251 where formula_70 has compact support, the idea is to extend the definition of the convolution formula_253 to a linear operation on distributions so that the associativity formula


title: Distribution (mathematics)

text: It is also possible to provide a more explicit characterization of the convolution of distributions. Suppose that formula_60 and formula_70 are distributions and that formula_60 has compact support. Then the linear maps


title: Distribution (mathematics)

text: This common value is called and it is a distribution that is denoted by formula_251 or formula_263 It satisfies formula_264 If formula_60 and formula_70 are two distributions, at least one of which has compact support, then for any formula_165 formula_268 If formula_70 is a distribution in formula_115 and if formula_84 is a Dirac measure then formula_272; thus formula_84 is the identity element of the convolution operation. Moreover, if formula_1 is a function then formula_275 where now the associativity of convolution implies that formula_276 for all functions formula_1 and formula_278


title: Distribution (mathematics)

text: It can be readily shown that this defines a smooth function of formula_282 which moreover has compact support. The convolution of formula_60 and formula_70 is defined by


title: Distribution (mathematics)

text: This generalizes the classical notion of convolution of functions and is compatible with differentiation in the following sense: for every multi-index formula_286


title: Distribution (mathematics)

text: The convolution of a finite number of distributions, all of which (except possibly one) have compact support, is associative.


title: Distribution (mathematics)

text: This definition of convolution remains valid under less restrictive assumptions about formula_60 and formula_93


title: Distribution (mathematics)

text: The convolution of distributions with compact support induces a continuous bilinear map formula_290 defined by formula_291 where formula_292 denotes the space of distributions with compact support. However, the convolution map as a function formula_293 is continuous although it is separately continuous. The convolution maps formula_294 and formula_295 given by formula_296 both to be continuous. Each of these non-continuous maps is, however, separately continuous and hypocontinuous.


title: Distribution (mathematics)

text: In general, regularity is required for multiplication products, and locality is required for convolution products. It is expressed in the following extension of the Convolution Theorem which guarantees the existence of both convolution and multiplication products. Let formula_297 be a rapidly decreasing tempered distribution or, equivalently, formula_298 be an ordinary (slowly growing, smooth) function within the space of tempered distributions and let formula_117 be the normalized (unitary, ordinary frequency) Fourier transform. Then, according to ,


title: Distribution (mathematics)

text: hold within the space of tempered distributions. In particular, these equations become the Poisson Summation Formula if formula_301 is the Dirac Comb. The space of all rapidly decreasing tempered distributions is also called the space of formula_302 and the space of all ordinary functions within the space of tempered distributions is also called the space of formula_303 More generally, formula_304 and formula_305 A particular case is the Paley-Wiener-Schwartz Theorem which states that formula_306 and formula_307 This is because formula_308 and formula_309 In other words, compactly supported tempered distributions formula_292 belong to the space of formula_302 and


title: Distribution (mathematics)

text: Paley-Wiener functions formula_312 better known as bandlimited functions, belong to the space of formula_303


title: Distribution (mathematics)

text: For example, let formula_314 be the Dirac comb and formula_315 be the Dirac delta;then formula_316 is the function that is constantly one and both equations yield the Dirac-comb identity. Another example is to let formula_140 be the Dirac comb and formula_318 be the rectangular function; then formula_319 is the sinc function and both equations yield the Classical Sampling Theorem for suitable formula_320 functions. More generally, if formula_140 is the Dirac comb and formula_322 is a smooth window function (Schwartz function), for example, the Gaussian, then formula_323 is another smooth window function (Schwartz function). They are known as mollifiers, especially in partial differential equations theory, or as regularizers in physics because they allow turning generalized functions into regular functions.


title: Distribution (mathematics)

text: Let formula_324 and formula_325 be open sets. Assume all vector spaces to be over the field formula_326 where formula_327 or formula_328 For formula_329 define for every formula_330 and every formula_331 the following functions:


title: Distribution (mathematics)

text: These definitions associate every formula_338 and formula_339 with the (respective) continuous linear map:


title: Distribution (mathematics)

text: Moreover, if either formula_60 (resp. formula_70) has compact support then it also induces a continuous linear map of formula_343 (resp.


title: Distribution (mathematics)

text: For all formula_348 and all formula_349 every one of the following canonical injections is continuous and has an image (also called the range) that is a dense subset of its codomain:


title: Distribution (mathematics)

text: where the topologies on formula_351 (formula_352) are defined as direct limits of the spaces formula_353 in a manner analogous to how the topologies on formula_354 were defined (so in particular, they are not the usual norm topologies). The range of each of the maps above (and of any composition of the maps above) is dense in its codomain.


title: Distribution (mathematics)

text: Suppose that formula_355 is one of the spaces formula_354 (for formula_357) or formula_358 (for formula_359) or formula_360 (for formula_361). Because the canonical injection formula_362 is a continuous injection whose image is dense in the codomain, this map's transpose formula_363 is a continuous injection. This injective transpose map thus allows the continuous dual space formula_364 of formula_355 to be identified with a certain vector subspace of the space formula_24 of all distributions (specifically, it is identified with the image of this transpose map). This transpose map is continuous but it is necessarily a topological embedding.


title: Distribution (mathematics)

text: A linear subspace of formula_24 carrying a locally convex topology that is finer than the subspace topology induced on it by formula_368 is called .


title: Distribution (mathematics)

text: Almost all of the spaces of distributions mentioned in this article arise in this way (for example, tempered distribution, restrictions, distributions of order formula_369 some integer, distributions induced by a positive Radon measure, distributions induced by an formula_370-function, etc.) and any representation theorem about the continuous dual space of formula_355 may, through the transpose formula_372 be transferred directly to elements of the space formula_373


title: Distribution (mathematics)

text: The inclusion map formula_374 is a continuous injection whose image is dense in its codomain, so the transpose formula_375 is also a continuous injection.


title: Distribution (mathematics)

text: Note that the continuous dual space formula_376 can be identified as the space of Radon measures, where there is a one-to-one correspondence between the continuous linear functionals formula_377 and integral with respect to a Radon measure; that is,


title: Distribution (mathematics)

text: Through the injection formula_385 every Radon measure becomes a distribution on . If formula_1 is a locally integrable function on then the distribution formula_387 is a Radon measure; so Radon measures form a large and important space of distributions.


title: Distribution (mathematics)

text: The following is the theorem of the structure of distributions of Radon measures, which shows that every Radon measure can be written as a sum of derivatives of locally formula_388 functions on :


title: Distribution (mathematics)

text: A linear function formula_70 on a space of functions is called if whenever a function formula_1 that belongs to the domain of formula_70 is non-negative (that is, formula_1 is real-valued and formula_393) then formula_394 One may show that every positive linear functional on formula_382 is necessarily continuous (that is, necessarily a Radon measure).


title: Distribution (mathematics)

text: One particularly important class of Radon measures are those that are induced locally integrable functions. The function formula_396 is called if it is Lebesgue integrable over every compact subset of . This is a large class of functions that includes all continuous functions and all Lp space formula_370 functions. The topology on formula_398 is defined in such a fashion that any locally integrable function formula_1 yields a continuous linear functional on formula_398 – that is, an element of formula_24 – denoted here by formula_402 whose value on the test function formula_403 is given by the Lebesgue integral:


title: Distribution (mathematics)

text: Conventionally, one abuses notation by identifying formula_405 with formula_35 provided no confusion can arise, and thus the pairing between formula_405 and formula_403 is often written


title: Distribution (mathematics)

text: If formula_1 and formula_140 are two locally integrable functions, then the associated distributions formula_405 and formula_413 are equal to the same element of formula_24 if and only if formula_1 and formula_140 are equal almost everywhere (see, for instance, ). Similarly, every Radon measure formula_379 on formula_23 defines an element of formula_24 whose value on the test function formula_403 is formula_421 As above, it is conventional to abuse notation and write the pairing between a Radon measure formula_379 and a test function formula_403 as formula_424 Conversely, as shown in a theorem by Schwartz (similar to the Riesz representation theorem), every distribution which is non-negative on non-negative functions is of this form for some (positive) Radon measure.


title: Distribution (mathematics)

text: The test functions are themselves locally integrable, and so define distributions. The space of test functions formula_6 is sequentially dense in formula_24 with respect to the strong topology on formula_427 This means that for any formula_428 there is a sequence of test functions, formula_429 that converges to formula_63 (in its strong dual topology) when considered as a sequence of distributions. Or equivalently,


title: Distribution (mathematics)

text: The inclusion map formula_432 is a continuous injection whose image is dense in its codomain, so the transpose map formula_433 is also a continuous injection. Thus the image of the transpose, denoted by formula_434 forms a space of distributions.


title: Distribution (mathematics)

text: The elements of formula_435 can be identified as the space of distributions with compact support. Explicitly, if formula_70 is a distribution on then the following are equivalent,


title: Distribution (mathematics)

text: Compactly supported distributions define continuous linear functionals on the space formula_441; recall that the topology on formula_441 is defined such that a sequence of test functions formula_446 converges to 0 if and only if all derivatives of formula_446 converge uniformly to 0 on every compact subset of . Conversely, it can be shown that every continuous linear functional on this space defines a distribution of compact support. Thus compactly supported distributions can be identified with those distributions that can be extended from formula_6 to formula_83


title: Distribution (mathematics)

text: Let formula_450 The inclusion map formula_451 is a continuous injection whose image is dense in its codomain, so the transpose formula_452 is also a continuous injection. Consequently, the image of formula_453 denoted by formula_454 forms a space of distributions. The elements of formula_455 are ' The distributions of order formula_456 which are also called ' are exactly the distributions that are Radon measures (described above).


title: Distribution (mathematics)

text: For formula_457 a is a distribution of order formula_458 that is not a distribution of order formula_459.


title: Distribution (mathematics)

text: A distribution is said to be of if there is some integer formula_31 such that it is a distribution of order formula_461 and the set of distributions of finite order is denoted by formula_462 Note that if formula_463 then formula_464 so that formula_465 is a vector subspace of formula_24, and furthermore, if and only if formula_467


title: Distribution (mathematics)

text: Every distribution with compact support in is a distribution of finite order. Indeed, every distribution in is a distribution of finite order, in the following sense: If is an open and relatively compact subset of and if formula_468 is the restriction mapping from to , then the image of formula_24 under formula_468 is contained in formula_471


title: Distribution (mathematics)

text: The following is the theorem of the structure of distributions of finite order, which shows that every distribution of finite order can be written as a sum of derivatives of Radon measures:


title: Distribution (mathematics)

text: Example. (Distributions of infinite order) Let formula_472 and for every test function formula_35 let formula_474


title: Distribution (mathematics)

text: Then formula_60 is a distribution of infinite order on . Moreover, formula_60 can not be extended to a distribution on formula_477; that is, there exists no distribution formula_70 on formula_477 such that the restriction of formula_70 to is equal to formula_481


title: Distribution (mathematics)

text: Defined below are the , which form a subspace of formula_482 the space of distributions on formula_29 This is a proper subspace: while every tempered distribution is a distribution and an element of formula_482 the converse is not true. Tempered distributions are useful if one studies the Fourier transform since all tempered distributions have a Fourier transform, which is not true for an arbitrary distribution in formula_485


title: Distribution (mathematics)

text: The Schwartz space formula_486 is the space of all smooth functions that are rapidly decreasing at infinity along with all partial derivatives. Thus formula_487 is in the Schwartz space provided that any derivative of formula_488 multiplied with any power of formula_489 converges to 0 as formula_490 These functions form a complete TVS with a suitably defined family of seminorms. More precisely, for any multi-indices formula_234 and formula_492 define:


title: Distribution (mathematics)

text: The family of seminorms formula_496 defines a locally convex topology on the Schwartz space. For formula_497 the seminorms are, in fact, norms on the Schwartz space. One can also use the following family of seminorms to define the topology:


title: Distribution (mathematics)

text: The Schwartz space is a Fréchet space (that is, a complete metrizable locally convex space). Because the Fourier transform changes formula_501 into multiplication by formula_502 and vice versa, this symmetry implies that the Fourier transform of a Schwartz function is also a Schwartz function.


title: Distribution (mathematics)

text: A sequence formula_503 in formula_486 converges to 0 in formula_486 if and only if the functions formula_506 converge to 0 uniformly in the whole of formula_153 which implies that such a sequence must converge to zero in formula_508


title: Distribution (mathematics)

text: formula_186 is dense in formula_510 The subset of all analytic Schwartz functions is dense in formula_486 as well.


title: Distribution (mathematics)

text: The Schwartz space is nuclear and the tensor product of two maps induces a canonical surjective TVS-isomorphisms


title: Distribution (mathematics)

text: where formula_513 represents the completion of the injective tensor product (which in this case is identical to the completion of the projective tensor product).


title: Distribution (mathematics)

text: The inclusion map formula_514 is a continuous injection whose image is dense in its codomain, so the transpose formula_515 is also a continuous injection. Thus, the image of the transpose map, denoted by formula_516 forms a space of distributions.


title: Distribution (mathematics)

text: The space formula_517 is called the space of . It is the continuous dual space of the Schwartz space. Equivalently, a distribution formula_70 is a tempered distribution if and only if


title: Distribution (mathematics)

text: The derivative of a tempered distribution is again a tempered distribution. Tempered distributions generalize the bounded (or slow-growing) locally integrable functions; all distributions with compact support and all square-integrable functions are tempered distributions. More generally, all functions that are products of polynomials with elements of Lp space formula_520 for formula_521 are tempered distributions.


title: Distribution (mathematics)

text: The can also be characterized as , meaning that each derivative of formula_70 grows at most as fast as some polynomial. This characterization is dual to the behaviour of the derivatives of a function in the Schwartz space, where each derivative of formula_403 decays faster than every inverse power of formula_524 An example of a rapidly falling function is formula_525 for any positive formula_526


title: Distribution (mathematics)

text: To study the Fourier transform, it is best to consider complex-valued test functions and complex-linear distributions. The ordinary continuous Fourier transform formula_527 is a TVS-automorphism of the Schwartz space, and the is defined to be its transpose formula_528 which (abusing notation) will again be denoted by formula_529 So the Fourier transform of the tempered distribution formula_70 is defined by formula_531 for every Schwartz function formula_532 formula_533 is thus again a tempered distribution. The Fourier transform is a TVS isomorphism from the space of tempered distributions onto itself. This operation is compatible with differentiation in the sense that


title: Distribution (mathematics)

text: and also with convolution: if formula_70 is a tempered distribution and formula_536 is a smooth function on formula_153 formula_538 is again a tempered distribution and


title: Distribution (mathematics)

text: is the convolution of formula_533 and formula_541 In particular, the Fourier transform of the constant function equal to 1 is the formula_84 distribution.


title: Distribution (mathematics)

text: If formula_543 is a tempered distribution, then there exists a constant formula_544 and positive integers formula_223 and formula_546 such that for all Schwartz functions formula_547


title: Distribution (mathematics)

text: This estimate, along with some techniques from functional analysis, can be used to show that there is a continuous slowly increasing function formula_117 and a multi-index formula_234 such that


title: Distribution (mathematics)

text: If formula_552 then for any compact set formula_553 there exists a continuous function formula_117compactly supported in formula_115 (possibly on a larger set than itself) and a multi-index formula_234 such that formula_557 on formula_558


title: Distribution (mathematics)

text: The success of the theory led to an investigation of the idea of hyperfunction, in which spaces of holomorphic functions are used as test functions. A refined theory has been developed, in particular Mikio Sato's algebraic analysis, using sheaf theory and several complex variables. This extends the range of symbolic methods that can be made into rigorous mathematics, for example, Feynman integrals.


title: Press Play (film)

text: Press Play is a 2022 science fiction romantic drama film written and directed by Greg Björkman in his directorial debut, from a story by Josh Boone. It stars Clara Rugaard, Lewis Pullman, Lyrica Okano, Christina Chang, Matt Walsh, and Danny Glover. It was shot on Oahu. The film was released in the United States on June 24, 2022, by The Avenue. It received mixed reviews from critics.


title: Press Play (film)

text: Laura's friend Chloe introduces her to her stepbrother, Harrison, who works at the record store Lost & Found. They immediately connect and attend a Japanese Breakfast show together. She meets the record store's owner, Cooper. At the beach, Harrison gifts Laura a cassette so they can create a mixtape. They celebrate after learning an art-related mentorship has accepted Laura. He mentions his intentions of moving to attend medical school but is interrupted by an earthquake. At her house, she shows him her art. Harrison's parents tell Laura that Harrison is going to a medical school on the other side of the country. Harrison privately tells Laura he does not want to go so he can stay with her. She tells him she does not want to be the reason he gives up on his dreams; he reassures her of his decision to stay. They attend her art exhibition. On his birthday, they finish painting a mural. The next day, Harrison is hit and killed by a car. Laura destroys the mural and leaves behind her mixtape at Lost & Found.


title: Press Play (film)

text: Four years later, Laura attends Chloe's wedding. Cooper gives Laura her mixtape back. She goes home to listen to it. After pressing play, Laura is transported to her first date with Harrison at the Japanese Breakfast concert. After a brief moment, she returns to the present. Every time she presses play, the mixtape sends her back in time. She does it again and appears on their date at the beach. She warns him about his future death and convinces him by predicting the earthquake seconds before it happens. In the present, she learns she is affecting the future after learning Chloe has married a different, obnoxious man instead of her soulmate. She learns Harrison had died the same day from falling off a cliff.