Liédson is the number of measurements and formula_64 consists of intervals of formula_36 in the variables, subject to the constraint that formula_66 and formula_67 are constant, where formula_66 is the mean resultant: and formula_70 is the mean angle: Note that product term in parentheses is just the distribution of the mean for a circular uniform distribution. This means that the distribution of the mean direction formula_1 of a von Mises distribution formula_73 is a von Mises distribution formula_74, or, equivalently, formula_75. Entropy. By definition, the information entropy of the von Mises distribution is where formula_24 is any interval of length formula_36	Von Mises distribution
14633724	. The logarithm of the density of the Von Mises distribution is straightforward: The characteristic function representation for the Von Mises distribution is: where formula_81. Substituting these expressions into the entropy integral, exchanging the order of integration and summation, and using the orthogonality of the cosines, the entropy may be written: For formula_83, the von Mises distribution becomes the circular uniform distribution and the entropy attains its maximum value of formula_84. Notice that the Von Mises distribution maximizes the entropy when the real and imaginary parts of the first circular moment are specified or, equivalently, the circular mean and circular variance 	Von Mises distribution
14633725	are specified. Von Mises distribution In probability theory and directional statistics, the von Mises distribution (also known as the circular normal distribution or Tikhonov distribution) is a continuous probability distribution on the circle. It is a close approximation to the wrapped normal distribution, which is the circular analogue of the normal distribution. A freely diffusing angle formula_7 on a circle is a wrapped normally distributed random variable with an unwrapped variance that grows linearly in time. On the other hand, the von Mises distribution is the stationary distribution of a drift and diffusion process on the circle in a harmonic 	Von Mises distribution
14633726	Muhammad ibn Yusuf al-Harawi Muhammad ibn Yusuf al-Harawi Muhammad ibn Yusuf al-Harawi (, fl. 1492-1518 and died 1542) was a Persian late 15th century physician from Herat, Safavid Iran, now part of Afghanistan. In 1518 he composed, in Arabic, an alphabetical medical dictionary and encyclopedia. It covers anatomical and pathological terms and concepts, medicinal substances, and prominent physicians, with all the entries arranged alphabetically. This medical lexicon, titled 'Jawahir al-lughah wa-Bahr al-jawahir', has three chapters: the first explaining terminology for parts of the body (in alphabetical order), the second on the names of simple and compound drugs (also in 	Muhammad ibn Yusuf al-Harawi
14633727	alphabetical order), and the third on names of diseases, presented in order from head to toe according to their locations. An autograph copy of Jawahir al-lughah exists in which the author states that he completed the correction of the treatise in 1492 (London, Wellcome Library for the History and Understanding of Medicine, MS Arab. 143). His other work, “Ainul Hayat” is based on Ageing. The original manuscript of Ainul Hayat was written in 1532 by the author in Herat. There are four copies of this old manuscript reported in various libraries of the world. It is stated that this 	Muhammad ibn Yusuf al-Harawi
14633728	is the first text on ageing in the world. After collation of these four copies of the manuscript, Hakim Syed Zillur Rahman edited and translated the manuscript in 2007. In the edited book, one can find how the author 500 years back discussed all types of behavioral and lifestyle factors including diet, environment and housing conditions related to ageing. He also discussed what drugs could increase and decrease ageing. Muhammad ibn Yusuf al-Harawi Muhammad ibn Yusuf al-Harawi (, fl. 1492-1518 and died 1542) was a Persian late 15th century physician from Herat, Safavid Iran, now part of Afghanistan	Muhammad ibn Yusuf al-Harawi
14633729	Pontop Pike transmitting station Pontop Pike transmitting station The Pontop Pike transmitting station is a facility for telecommunications and broadcasting situated on a 312-metre (1,024-ft) high hill of the same name between Stanley and Consett, County Durham, near the village of Dipton, England. The mast is high, giving an average antenna height of above sea level. It is owned and operated by Arqiva. History. The mast was built in 1953, by BICC with Rowridge (also ) and North Hessary Tor in Devon (). Its construction was brought forward by the BBC so that people in North East England could watch the Coronation of Queen 	Pontop Pike transmitting station
14633730	Elizabeth II live on the 405-line television system VHF then in use in the UK. Test transmissions from a low-power temporary aerial began on Monday, 20 April 1953, and the first programmes were transmitted on Friday, 1 May 1953, in plenty of time for the Coronation on 2 June. UHF transmissions began in 1966 with the first colour transmissions in 1970, and the VHF television signal was switched off in 1985. Coverage. The Pontop Pike transmitter provides digital television transmissions to Tyne and Wear, County Durham, Tees Valley, most of Northumberland and parts of North Yorkshire. It also 	Pontop Pike transmitting station
14633731	carries the national BBC Radio FM signals, covering the whole North East, as well as 95.4 MHz FM BBC Newcastle. It was one of the first national FM transmitters in December 1955. All of its television output is within the C/D aerial group. Digital TV switchover. Analogue TV transmissions from this mast began to close from 12 September 2012 and completely ceased on 26 September that year, making Pontop Pike, alongside Bilsdale and Chatton, the last-but-one transmitter group in the United Kingdom to complete digital switchover (DSO) with Northern Ireland being the last area to switch. In 	Pontop Pike transmitting station
14633732	July 2007 it was confirmed by Ofcom that Pontop Pike would remain a C/D group after DSO. Pontop Pike underwent its 700 MHz clearance between September and November 2019 when its output moved down the band from a C/D group to a K group. Thus people with original C/D group aerials, who live in poor reception areas, may struggle to pick up some channels (see graph). Services listed by frequency. Analogue television. Analogue television from Pontop Pike has now ceased permanently. BBC Two analogue was switched off on 12 September 2012 and the remaining three on 26 	Pontop Pike transmitting station
14633733	September 2012. Pontop Pike did not transmit analogue Channel 5. Instead it was transmitted from Burnhope on UHF 68. Relay services. Being the main broadcasting transmitter, there are also a number of relays (or repeaters) to cover patches where this transmitter can't properly serve. Digital television. The following is a list of the television relays served by Pontop Pike: Pontop Pike transmitting station The Pontop Pike transmitting station is a facility for telecommunications and broadcasting situated on a 312-metre (1,024-ft) high hill of the same name between Stanley and Consett, County Durham, near the village of Dipton	Pontop Pike transmitting station
14633734	Liédson Liédson Liédson da Silva Muniz (born 17 December 1977), known simply as Liédson, is a former footballer who played as a striker. Nicknamed Levezinho" ("Slender one") due to his light frame (63 kg), he spent the bulk of his professional career in Portugal representing Sporting CP, scoring nearly 200 goals in more than 300 competitive matches, winning four major titles with the club and helping it to four Champions League participations. Born in Brazil, Liédson became a Portuguese citizen in 2009 and appeared with the national team at the 2010 World Cup, already in his 30s. Club career. Early years . Born in Cairu, Bahia, Liédson started his professional career at Coritiba Foot Ball Club, aged 22. Before that he played amateur football on weekends, working at a supermarket during the week. Liédson continued his career at Clube de Regatas do Flamengo and Sport Club Corinthians Paulista, traditional big clubs at Rio de Janeiro and São Paulo respectively, with relative individual success. Sporting. Liédson was signed by Sporting CP for €2 million on 31 August 2003, giving manager Fernando Santos a new option following injuries and sales. On his debut, the "S" in his name was printed in reverse, which  he took as a good omen and maintained it like that for the rest of his career. Liédson was crowned top scorer in the Primeira Liga in his second season, scoring 25 goals while also netting nine in 14 games as his team reached the final of the UEFA Cup – this included a hat-trick in a 4–0 win at FC Dinamo Tbilisi in the group stage. The following year he finished second in the scoring race, scoring 15 times from 31 appearances (28 of which he played the full 90 minutes) with an impressive mark of 2,899 minutes played ; he was top scorer again in 2006–07, but the "Lions" only conquered one domestic cup during that timeframe. Considered by many as one of the greatest strikers in Sporting history since the early performances of Fernando Peyroteo, Héctor Yazalde, Manuel Fernandes, Rui Jordão, Fernando Gomes, Alberto Acosta and Mário Jardel, Liédson was also regarded as the club's top trademark player since the departures of Luís Figo, Cristiano Ronaldo, Nani and Ricardo Quaresma. He also ranked team top scorer in the UEFA Champions League, after scoring the lonely goal in the 2008–09 group stage contest against FC Shakhtar Donetsk on  22 October 2008. In the last group stage match, he also found the net – in a 2–5 home loss to FC Barcelona – for the 19th time in European competition, another club best; on 1 November he appeared in his 150th league match with the green-and-white striped jersey, 213 overall. On 17 January 2010, after having missed one month due to injury, Liédson came from the bench to score twice, helping Sporting overcome C.D. Nacional 3–2 at home. On 7 March, all of the game's goals were his as his team beat last-placed C.F. Os Belenenses 4–0  away. Liédson entered Sporting's history books on 21 October 2010 as he scored the club's 200th goal in European competitions, netting twice in a 5–1 home win against K.A.A. Gent for the Europa League. Return to Brazil. On 31 January 2011, after having appeared in 310 official matches with Sporting, scoring 174 goals, 33-year-old Liédson returned to his homeland, signing with former club Corinthians. He was still allowed to play with the "Lions" the following Friday, starting and scoring twice against Associação Naval 1º de Maio, in a 3–3 home draw. Liédson contributed with ten games  in the 2012 Copa Libertadores campaign, scoring once in the 6–0 group stage home win against Deportivo Táchira of Venezuela (immediately after missing a penalty kick) as the "Timão" won the continental competition for the first time in its history. On 31 July, he left Corinthians after both club and player decided not to renew his contract. On 2 August 2012, Liédson agreed on a return to Flamengo, signing until December of the following year. In late January of the following year he came back to Portugal, joining FC Porto on a loan until the end of the season. On  11 May, he was involved in the goal that clinched the title for the "Dragons" against rivals S.L. Benfica in "O Clássico"; put on as a substitute by manager Vítor Pereira alongside Kelvin, he assisted the youngster's added-time winner. International career. Liédson expressed interest in playing for the Portuguese national team despite being Brazilian-born. He stated: "If one day I was called up to play, I wouldn’t hesitate for a second, as I feel right at home in Portugal." After the retirement of striker Pauleta, Liédson was one of the main candidates in line for succession , especially after countrymen Deco and Pepe's previous switches. Liédson's naturalization process created some controversy, as the Portuguese Football Players' Union in August 2009 publicly objected Portuguese Football Federation's (FPF) plans of expediting his citizenship. After six years of residence in the country, Liédson received Portuguese nationality, being eligible to play for the country. On 26 August 2009, he was called up by national coach Carlos Queiroz for the 2010 FIFA World Cup qualifiers against Denmark and Hungary, and made his debut against the former on 5 September, coming on as a substitute for Tiago at half-time : after having been booked, he scored the final 1–1 in the 85th minute with a header from a corner. Liédson started the first game in the final stages in South Africa, a 0–0 against Ivory Coast. He lost his place to Hugo Almeida for the following match, but both found the net in the 7–0 routing of North Korea on 21 June 2010, in Cape Town. Career statistics. Club. Includes the Taça da Liga and Supertaça Cândido de Oliveira. Honours. Club. Corinthians Sporting Porto Liédson Liédson da Silva Muniz (born 17 December 1977), known simply as Liédson, is a former