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https://en.wikipedia.org/wiki/Largest%20empty%20sphere	In computational geometry, the largest empty sphere problem is the problem of finding a hypersphere of largest radius in d-dimensional space whose interior does not overlap with any given obstacles. Two dimensions The largest empty circle problem is the problem of finding a circle of largest radius in the plane whose interior does not overlap with any given obstacles. A common special case is as follows. Given n points in the plane, find a largest circle centered within their convex hull and enclosing none of them. The problem may be solved using Voronoi diagrams in optimal time . See also Bounding sphere Farthest-first traversal Largest empty rectangle References Geometric algorithms
https://en.wikipedia.org/wiki/Kiwere	Kiwere is an administrative ward in the Iringa Rural district of the Iringa Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 10,229 people in the ward, from 9,776 in 2012. Villages / vitongoji The ward has 5 villages and 21 vitongoji. Kiwere Chapakazi Makondo Mwaya A Mwaya B Mgera Kidete Luganga Mapinduzi Mlangali Itagutwa Itagutwa Kipengele Mapululu Mlenge Kitapilimwa Ikingo Kinyamaduma Lugalo Mfyome Malamba A Malamba B Matembo Mgega Mhefu Msosa References Wards of Iringa Region
https://en.wikipedia.org/wiki/Lumuli	Lumuli is an administrative ward in the Iringa Rural district of the Iringa Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 8,216 people in the ward, from 7,852 in 2012. Villages / vitongoji The ward has 4 villages and 22 vitongoji. Lumuli Kalengachwa Kibalali Kihata Kihesa Kilimahewa Kitemela Lugema Uhopela Vikula Itengulinyi Ipangani Itengulinyi Lukingita Makanyagio A Makanyagio B Isupilo Isupilo Makanyagio Masumbo Usambusi Muwimbi Gezaulole Kibugumo Muwimbi Ulete References Wards of Iringa Region
https://en.wikipedia.org/wiki/Magulilwa	Maguliwa, also known as Magulilwa, is an administrative ward in the Iringa Rural district of the Iringa Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 14,271 people in the ward, from 13,639 in 2012. Villages / vitongoji The ward has 6 villages and 36 vitongoji. Magulilwa Godown A Godown B Godown C Ilala Kihesakilolo Majengo Mbavi Mifugo Sekuse Ng’enza Kinyaminyi Lutengelo Masela Mlevela Muungano Msuluti Igunga Kihesakilolo Milanzi Mlowa Msukwa Mwaya Mlanda Ilembula Mlanda A Mlanda B Msombe Nyalawe Ukang’a Negabihi Igeleke Isoliwaya Kilimahewa Letengano Muungano Ndiwili Kitanzini Madukani Matema Migoli A Migoli B Mivinjeni Msuluti A Msuluti B Mtakuja References Wards of Iringa Region
https://en.wikipedia.org/wiki/Mahuninga	Mahuninga is an administrative ward in the Iringa Rural district of the Iringa Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 4,532 people in the ward, from 4,331 in 2012. Villages / vitongoji The ward has 3 villages and 13 vitongoji. Mahuninga Kitalingolo Majengo A Majengo B Ufyambe Kisilwa Isukutwa Kisilwa Misufi Makifu Isanga Mahove Makambalala-A Makambalala-B Makifu Mkanisoka References Wards of Iringa Region
https://en.wikipedia.org/wiki/Malenga%20Makali	Malengamakali is an administrative ward in the Iringa Rural district of the Iringa Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 8,284 people in the ward, from 7,917 in 2012. Villages / vitongoji The ward has 6 villages and 27 vitongoji. Nyakavangala Ngega Nyakavangala A Nyakavangala B Nyakavangala C Isaka Idari Isaka A Isaka B Makegeke Makadupa Amani Makadupa kati Sokoine Iguluba Bomalang’ombe Iguluba Kati Msumbiji Mkulula Kikuyu Luganga A Luhomelo Lunganga B Mapalagaga Mbuyuni Stendi A Stendi B Usolanga Ihumbiliza A Ihumbiliza B Ihumbiliza C Kawemba Usolanga kati References Wards of Iringa Region
https://en.wikipedia.org/wiki/Mgama	Mgama is an administrative ward in the Iringa Rural district of the Iringa Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 13,143 people in the ward, from 12,561 in 2012. Villages / vitongoji The ward has 5 villages and 44 vitongoji. Ibumila Gezaulole Ibumila A Ibumila B Kilewela Kilimanjaro Kipengele Lugololelo B Lwato Lwato A Lwato B Makete A Makete B Mlenge A Mlenge B Ilandutwa Lugofu Lugololelo A Mayugi Ndolela Nyakatule A nyakatule B Mgama Isombe Katenge ‘A’ Katenge ‘B’ Kihesa Mbalamo Mgama ‘A’ Mgama ‘B’ Mhagati Msichoke Myombwe Wangama Wilolesi Itwaga Ikanavanu Ilyango ‘A’ Ilyango ‘B’ Itwaga Nunumala Ihemi Igunga Ihemi A Kifumbi Kilimahewa Kilimanjaro Mfaranyaki Mjimwema Njiapanda Winome References Wards of Iringa Region
https://en.wikipedia.org/wiki/Mlowa	Mlowa is an administrative ward in the Iringa Rural district of the Iringa Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 9,923 people in the ward, from 9,483 in 2012. Villages / vitongoji The ward has 3 villages and 19 vitongoji. Malizanga Ikonongo Majengo A Majengo B Malizanga Matalawe Mlowa Mtakuja Ndorobo A Ndorobo B Nyamahana Ipwasi Majengo Mbuyuni Mlambalasi Mtakuja Mafuluto Kibuduga Magoya Majengo Mseketule Muungano References Wards of Iringa Region
https://en.wikipedia.org/wiki/Mseke	Mseke is an administrative ward in the Iringa Rural district of the Iringa Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 12,985 people in the ward, from 15,868 in 2012. Villages / vitongoji The ward has 4 villages and 24 vitongoji. Tanangozi Kanisani Kihongolelo Kilindi Kimwanyula Lunguya Stesheni 'A' Stesheni 'B' Mlandege Gezaulole Maumbamatali Mlandege 'A' Mlandege 'B' Ugwachanya Banavanu Igavilo Ismila Lukolela Mseke 'A' Mseke 'B' Njiapanda 'A' Njiapanda 'B' Ulongambi Winome Wenda Wenda 'A; Wenda 'B' Wenda 'C' Wenda 'D' References Wards of Iringa Region
https://en.wikipedia.org/wiki/Nduli	Nduli is an administrative ward in the Iringa Rural district of the Iringa Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 6,933 people in the ward, from 6,626 in 2012. Neighborhoods The ward has 11 neighborhoods. Igungandembwe Kilimahewa Kipululu Kisowele Mapanda Mibata Mji Mwema Msisina Mtalagala Njia Panda Sombeli References Wards of Iringa Region
https://en.wikipedia.org/wiki/Nzihi	Nzihi is an administrative ward in the Iringa Rural district of the Iringa Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 15,562 people in the ward, from 14,872 in 2012. Villages / vitongoji The ward has 5 villages and 45 vitongoji. Nyamihuu Chemchem Igingimali Isala Isupilo Kilimahewa Mabatini Majengo Makanyagio Mbuyuni Mkombe Mlangali Wilolesi Kipera Kipera ofisini Kisombambone Mbulula Mbuyuni Mifugo Mkola Mkwata Mkwawa Mlambalasi Magubike Chelesi Ibogo Igangilonga Kinyang'ama Kinyasaula mtakuja Nzihi Ihanzu Mazombe Mbega Mhanga Mimwema A Mjimwema B Nzihi A Nzihi B Ilalasimba George. Filiakosi Igunga Ndembwe Ilalasimba Ipangani James. Nzani Kalangali Kayugwa Kidamali Songambele Winome References Wards of Iringa Region
https://en.wikipedia.org/wiki/Ulanda	Ulanda is an Ward in the Iringa Rural District of the Iringa Region of Tanzania, East Africa. In 2016 the Tanzania National Bureau of Statistics report there were 9,686 people in the ward, from 9,257 in 2012. The 200 bed Tosamaganga Hospital is located in Ulanda ward. It is operated by a faith based organisation. Villages / vitongoji The ward has 6 villages and 34 vitongoji. Mangalali Itunda A Itunda B Kikongoma Kitoo A Kitoo B Lukwambe A Lukwambe B Mangalali A Mangalali B Kibebe Ikanumgunda Ikanuulime Itamba Kilolo Lugung’unzi Lupange Nyambila Lupalama Lupalama-Kilimani Mjimwema Mlaga Mwangata Ibangamoyo Henge Ibangamoyo Katenge Mlandizi Mwika Mwambao Idete Kilimahewa Mwambao A Mwambao B Weru Imwagamapesa Ipangani Kilangali Magangwe Mseke References Wards of Iringa Region
https://en.wikipedia.org/wiki/Wasa%20%28Tanzanian%20ward%29	Wasa is an administrative ward in the Iringa Rural district of the Iringa Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 11,086 people in the ward, from 10,595 in 2012. Villages / vitongoji The ward has 7 villages and 28 vitongoji. Wasa Itawi Kastamu Nyakigongo Nyamagola Uhepwa Utiga Ikungwe Ikungwe Makanyagio Mkuta Mkuzi Tambalang’ombe Ufyambe Lwamanga Ikonongo Lunguya Usengelindete Igula Itimbo Kigasa Umwaga Ihomasa Ihomasa Lupande Mkondowa Muungano Vikula Ulata Mwefu Ngonamwasi Ulata Mahanzi Kibulilo Kitamba Mahanzi References Wards of Iringa Region
https://en.wikipedia.org/wiki/Wilfred%20Cockcroft	Sir Wilfred Cockcroft (7 June 1923 – 1999) was an eminent mathematics educator from the University of Hull. In 1978 he was commissioned by the then Labour government to chair a comprehensive inquiry into the teaching of mathematics in primary and secondary schools in England and Wales. The committee of inquiry produced its report in 1982, published as Mathematics Counts but widely known as "the Cockcroft report". Cockcroft was knighted in 1983, and in May 1984 was awarded an honorary degree from the Open University as Doctor of the University. External links Tony Crilly (2001), Memories of Sir Wilfred Cockcroft 1923–1999, The Mathematical Gazette 85 (502), 72. Obituary, Bulletin of the London Mathematical Society (2005), 37, 149-155 Short biography, and account of archives at Hull University, Access to Archives 1923 births 1999 deaths 20th-century English mathematicians
https://en.wikipedia.org/wiki/Georg%20Cantor%20Gymnasium	The Georg Cantor Gymnasium is a German gymnasium in Halle (Saale) with a special focus on mathematics and the sciences (specialist school). The all-day school was founded in 1989. It is attended by just over 500 students and has a staff of about 60 teachers. Location Since 2007 the school has been located in the city centre of Halle (Saale). The school building, built in 1905, was used until 2005 by the Torgymnasium. The halls of residence are placed about 30 minutes (by tram) away from the school – next to the former school building, which was in use from 1989 to 2007. As its structural condition worsened, the school moved to its current location. After the renovation between the years 2005–2007 the school got new room setups, technical equipment and a new schoolyard, which is small, but nevertheless offers the students many seats and sports activities like basketball or table tennis. Special Profile The school’s main emphasis is on mathematics, the sciences, and technical education. There are more lessons in these subjects compared with other grammar schools in Germany, and the students also regularly take part in competitions in these fields. Georg-Cantor-Gymnasium has a modified timetable which includes more lessons in the profile subjects (Mathematics, Physics, Chemistry, Biology, Computer Science and Astronomy). That way the students get the opportunity to learn more about additional topics which are not part of the normal curriculum. Students can also join special clubs or work on science projects after school. Furthermore, the students take part in practical science training and have the chance to write a scientific paper on an extra-curricular topic of their choice in year 10. Foreign Languages In year 5 the students continue English as their first foreign language, which they normally start in Primary School. In year 7 they take up a second one. They can choose between Latin and French. Additionally, students can take Spanish as their third foreign language in year 9. Students can choose any language for their A-levels. The school has many language clubs where interested students can take languages like Spanish, Russian, Chinese or Latin as an extra-curricular activity. Admission According to Saxony-Anhalt’s school regulations an entrance exam for schools with special profiles is obligatory. The exam includes a general aptitude test and a mathematical test paper. Places are offered depending on these test results as well as the student’s last school report. Students sit for the exam in the second term of year 4. A typical school day The school day starts at 7.30 am. Typically, years 5 and 6 have 6 lessons a day, with each lesson lasting 45 minutes. Their school day ends at 1.10 pm. At 1.40 pm the afternoon lessons start – mostly the older years take part in these and they finish at about 4 pm at the latest. There is a break after each lesson. In the breaks after the second and fourth lesson students are expected to go to the
https://en.wikipedia.org/wiki/Earned%20run%20average	In baseball statistics, earned run average (ERA) is the average of earned runs allowed by a pitcher per nine innings pitched (i.e. the traditional length of a game). It is determined by dividing the number of earned runs allowed by the number of innings pitched and multiplying by nine. Thus, a lower ERA is better. Runs resulting from passed balls, defensive errors (including pitchers' defensive errors), and runners placed on base at the start of extra innings are recorded as unearned runs and omitted from ERA calculations. Origins Henry Chadwick is credited with devising the statistic, which caught on as a measure of pitching effectiveness after relief pitching came into vogue in the 1900s. Prior to 1900—and, in fact, for many years afterward—pitchers were routinely expected to pitch a complete game, and their win–loss record was considered sufficient in determining their effectiveness. After pitchers like James Otis Crandall and Charley Hall made names for themselves as relief specialists, gauging a pitcher's effectiveness became more difficult using the traditional method of tabulating wins and losses. Some criterion was needed to capture the apportionment of earned-run responsibility for a pitcher in games that saw contributions from other pitchers for the same team. Since pitchers have primary responsibility for striking opposing batters out, they must assume responsibility when a batter they do not retire at the plate moves to base, and eventually reaches home, scoring a run. A pitcher is assessed an earned run for each run scored by a batter (or that batter's pinch-runner) who reaches base while batting against that pitcher. The National League first tabulated official earned run average statistics in 1912 (the outcome was called "Heydler's statistic" for a while, after then-NL secretary John Heydler), and the American League later accepted this standard and began compiling ERA statistics. Baseball encyclopedias will often display ERAs for earlier years, but these were computed retroactively. Negro league pitchers are often rated by RA, or total runs allowed, since the statistics available for Negro league games did not always distinguish between earned and unearned runs. ERA in different decades As with batting average, the definition of a good ERA varies from year to year. During the dead-ball era of the 1900s and 1910s, an ERA below 2.00 (two earned runs allowed per nine innings) was considered good. In the late 1920s and through the 1930s, when conditions of the game changed in a way that strongly favored hitters, a good ERA was below 4.00; only the highest caliber pitchers, for example Dazzy Vance or Lefty Grove, would consistently post an ERA under 3.00 during these years. In the 1960s, sub-2.00 ERAs returned as other influences, such as ballparks with different dimensions, were introduced. Starting with the 2019 season, an ERA under 4.00 is again considered good. The single-season record for the lowest ERA is held by Dutch Leo
https://en.wikipedia.org/wiki/Kihesa	Kihesa is an administrative ward in the Iringa Urban district of the Iringa Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 19,040 people in the ward, from 18,196 in 2012. Neighborhoods The ward has 15 neighborhoods. Dodoma Road A Dodoma Road B Dodoma Road F Ilembula Kilimani Mafifi Mbuma Mfaranyaki Msikitini Mwenge Ngome Ramadhani Waziri Semtema A Semtema B Sokoni References Wards of Iringa Region
https://en.wikipedia.org/wiki/Kitanzini	Kitanzini is an administrative ward in the Iringa Urban district of the Iringa Region of Tanzania. In 2016, the Tanzania National Bureau of Statistics reported that there are 3,785 people in the ward, from 3,617 in 2012. Neighborhoods The ward has 9 neighborhoods. Jamat Kitanzini Legezamwendo Madrasa Maweni Miyomboni Mlimani Polisi Line Stendi Kuu References Wards of Iringa Region Constituencies of Tanzania
https://en.wikipedia.org/wiki/Kitwiru	Kitwiru is an administrative ward in the Iringa Urban district of the Iringa Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 11,992 people in the ward, from 11,461 in 2012. Neighborhoods The ward has 14 neighborhoods. Cagliero Kibwabwa 'A' Kibwabwa 'B' Kisiwani Kitwiru 'A' Kitwiru 'B' Mosi Nyamhanga 'A' Nyamhanga 'B' Nyamhanga 'C' Nyamhanga 'D' Nyamhanga 'E' Uyole 'A' Uyole 'B' References Wards of Iringa Region Constituencies of Tanzania
https://en.wikipedia.org/wiki/Kwakilosa	Kwakilosa is an administrative ward in the Iringa Urban district of the Iringa Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 8,317 people in the ward, from 7,948 in 2012. Neighborhoods The ward has 10 neighborhoods. Beira Frelimo 'C' Jangwani Kidunda Kijiweni Kisiwani Muungano 'A' Muungano 'B' Samora Shule References Wards of Iringa Region
https://en.wikipedia.org/wiki/Makorongoni	Makorongoni is an administrative ward in the Iringa Urban district of the Iringa Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 8,151 people in the ward, from 7,790 in 2012. Neighborhoods The ward has 13 neighborhoods. Baniani Kaguo Kibwana Mahagi Mahiwa Mkwawa Road Msichoke Muhimba "A" Muhimba "B" Pangani Pawaga Road Sekondari Tandamti References Wards of Iringa Region
https://en.wikipedia.org/wiki/Mivinjeni	Mivinjeni is an administrative ward in the Iringa Urban district of the Iringa Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 5,002 people in the ward, from 4,780 in 2012. Neighborhoods The ward has 7 neighborhoods. Darajani Frelimo 'A' Idunda Kanisani Kondoa Migombani Mjimwema References Wards of Iringa Region
https://en.wikipedia.org/wiki/Mlandege	Mlandege is an administrative ward in the Iringa Urban district of the Iringa Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 4,855 people in the ward, from 4,640 in 2012. Neighborhoods The ward has 12 neighborhoods. Kalenga Road Kota Lubida Mafuruto Makondeko Mapinduzi Mlambalazi Msanya Ngeng'ena Ngulika Nguzo Sokoni References Wards of Iringa Region
https://en.wikipedia.org/wiki/Mshindo	Mshindo is an administrative ward in the Iringa Urban district of the Iringa Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 1,980 people in the ward, from 1,892 in 2012. Neighborhoods The ward has 7 neighborhoods. Benki Mshindo 'A' Mshindo 'B' Msikiti Mtwa 'A' Mtwa 'B' Ruaha References Wards of Iringa Region
https://en.wikipedia.org/wiki/Mtwivila	Mtwivila is an administrative ward in the Iringa Urban district of the Iringa Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 11,174 people in the ward, from 21,017 in 2012. Neighborhoods The ward has 10 neighborhoods. Dodoma Road "C" Dodoma Road "D" Dodoma Road "E" Idunda Mtwivila "C" Mtwivila 'A' Mtwivila 'B' Mwautwa Semkini Viziwi References Wards of Iringa Region
https://en.wikipedia.org/wiki/Mwangata	Mwangata is an administrative ward in the Iringa Urban district of the Iringa Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 14,111 people in the ward, from 13,486 in 2012. Neighborhoods The ward has 11 neighborhoods. Isoka 'A' Isoka 'B' Kigamboni Kisiwani Mawelewele Muungano Mwangata 'A' Mwangata 'B' Mwangata 'C' Mwangata 'D' Ngelewala References Wards of Iringa Region
https://en.wikipedia.org/wiki/Ruaha%20%28Iringa%20Urban%20ward%29	Ruaha is an administrative ward in the Iringa Urban district of the Iringa Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 13,401 people in the ward, from 16,984 in 2012. Neighborhoods The ward has 13 neighborhoods. Buguruni Chuo Ipogolo 'A' Ipogolo 'B' Ipogolo 'C' Ipogolo 'D' Ipogolo 'E' Kinegamgosi 'A' Kinegamgosi 'B' Kinegamgosi 'C' Mwagongo Ngeleli Tagamenda References Wards of Iringa Region
https://en.wikipedia.org/wiki/Ilala%20%28Iringa%20Urban%20ward%29	Ilala is an administrative ward in the Iringa Urban district of the Iringa Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 4,654 people in the ward, from 4,448 in 2012. Neighborhoods The ward has 7 neighborhoods. Dabobado Embakasi Kajificheni Lami A Lami B Mlamke Nyumba Tatu References Wards of Iringa Region
https://en.wikipedia.org/wiki/Mkwawa%20%28Tanzanian%20ward%29	Mkwawa is an administrative ward in the Iringa Urban district of the Iringa Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 10,122 people in the ward, from 9,673 in 2012. Neighborhoods The ward has 15 neighborhoods. Bwawani "A" Bwawani 'B' Don Bosco 'A' Don Bosco 'B' Hoho Ikonongo 'A' Ikonongo 'B' Imalanongwa A Imalanongwa B Itamba Lukosi Mgera Mkwawa Chuo Wazo 'A' Wazo 'B' References Wards of Iringa Region
https://en.wikipedia.org/wiki/List%20of%20metropolitan%20areas%20in%20Switzerland	List of metropolitan areas in Switzerland. Switzerland has five metropolitan areas as defined by Swiss Federal Statistics Office: Metropolitan areas Basel metropolitan area Bern metropolitan area (Espace Mittelland) Geneva metropolitan area (≈ Grand Genève) Lausanne metropolitan area Zürich metropolitan area See also List of cities in Switzerland References External links Polycentricity and metropolitan governance in Switzerland, Vienna University of Economics and Business Metropolitan governance and the "democratic deficit", Swiss Federal Institute of Technology in Lausanne Switzerland Metropolitan areas
https://en.wikipedia.org/wiki/MMPC	MMPC may refer to: Michigan Mathematics Prize Competition, a math competition held in Michigan, U.S. Milwaukee Motion Picture Commission, the former film censor board of the city of Milwaukee, Wisconsin, U.S. Mitsubishi Motors Philippines, the Philippine operation of Mitsubishi Motors Corporation Monday Morning Podcast, a podcast by American comedian Bill Burr Multimedia PC, a recommended configuration for a personal computer with a CD-ROM drive The gene (mmpC) and protein (MmpC) involved in biosynthesis of Mupirocin
https://en.wikipedia.org/wiki/Lamination%20%28topology%29	In topology, a branch of mathematics, a lamination is a : "topological space partitioned into subsets" decoration (a structure or property at a point) of a manifold in which some subset of the manifold is partitioned into sheets of some lower dimension, and the sheets are locally parallel. A lamination of a surface is a partition of a closed subset of the surface into smooth curves. It may or may not be possible to fill the gaps in a lamination to make a foliation. Examples A geodesic lamination of a 2-dimensional hyperbolic manifold is a closed subset together with a foliation of this closed subset by geodesics. These are used in Thurston's classification of elements of the mapping class group and in his theory of earthquake maps. Quadratic laminations, which remain invariant under the angle doubling map. These laminations are associated with quadratic maps. It is a closed collection of chords in the unit disc. It is also topological model of Mandelbrot or Julia set. See also train track (mathematics) Orbit portrait Notes References Conformal Laminations Thesis by Vineet Gupta, California Institute of Technology Pasadena, California 2004 Topology Manifolds
https://en.wikipedia.org/wiki/Ladder%20operator	In linear algebra (and its application to quantum mechanics), a raising or lowering operator (collectively known as ladder operators) is an operator that increases or decreases the eigenvalue of another operator. In quantum mechanics, the raising operator is sometimes called the creation operator, and the lowering operator the annihilation operator. Well-known applications of ladder operators in quantum mechanics are in the formalisms of the quantum harmonic oscillator and angular momentum. Terminology There is some confusion regarding the relationship between the raising and lowering ladder operators and the creation and annihilation operators commonly used in quantum field theory. The creation operator ai† increments the number of particles in state i, while the corresponding annihilation operator ai decrements the number of particles in state i. This clearly satisfies the requirements of the above definition of a ladder operator: the incrementing or decrementing of the eigenvalue of another operator (in this case the particle number operator). Confusion arises because the term ladder operator is typically used to describe an operator that acts to increment or decrement a quantum number describing the state of a system. To change the state of a particle with the creation/annihilation operators of QFT requires the use of both annihilation and creation operators. An annihilation operator is used to remove a particle from the initial state and a creation operator is used to add a particle to the final state. The term "ladder operator" is also sometimes used in mathematics, in the context of the theory of Lie algebras and in particular the affine Lie algebras, to describe the su(2) subalgebras, from which the root system and the highest weight modules can be constructed by means of the ladder operators. In particular, the highest weight is annihilated by the raising operators; the rest of the positive root space is obtained by repeatedly applying the lowering operators (one set of ladder operators per subalgebra). General formulation Suppose that two operators X and N have the commutation relation, for some scalar c. If is an eigenstate of N with eigenvalue equation, then the operator X acts on in such a way as to shift the eigenvalue by c: In other words, if is an eigenstate of N with eigenvalue n then is an eigenstate of N with eigenvalue n + c or it is zero. The operator X is a raising operator for N if c is real and positive, and a lowering operator for N if c is real and negative. If N is a Hermitian operator then c must be real and the Hermitian adjoint of X obeys the commutation relation: In particular, if X is a lowering operator for N then X† is a raising operator for N and vice versa. Angular momentum A particular application of the ladder operator concept is found in the quantum mechanical treatment of angular momentum. For a general angular momentum vector, J, with components, Jx, Jy and Jz one defines the two
https://en.wikipedia.org/wiki/Metropolitan%20areas%20in%20Belgium	National statistics differ between five Metropolitan areas in Belgium. These five metropolitan areas (Dutch: Agglomeratie, French: Agglomération) are also covered by Eurostat statistics as separate Larger Urban Zones (LUZ). Metropolitan areas See also List of cities and towns in Belgium List of metropolitan areas in European Union References External links Directorate-general Statistics Belgium, FPS Economy Belgium Belgium Belgium geography-related lists
https://en.wikipedia.org/wiki/Genus%E2%80%93degree%20formula	In classical algebraic geometry, the genus–degree formula relates the degree d of an irreducible plane curve with its arithmetic genus g via the formula: Here "plane curve" means that is a closed curve in the projective plane . If the curve is non-singular the geometric genus and the arithmetic genus are equal, but if the curve is singular, with only ordinary singularities, the geometric genus is smaller. More precisely, an ordinary singularity of multiplicity r decreases the genus by . Proof The proof follows immediately from the adjunction formula. For a classical proof see the book of Arbarello, Cornalba, Griffiths and Harris. Generalization For a non-singular hypersurface of degree d in the projective space of arithmetic genus g the formula becomes: where is the binomial coefficient. Notes See also Thom conjecture References Enrico Arbarello, Maurizio Cornalba, Phillip Griffiths, Joe Harris. Geometry of algebraic curves. vol 1 Springer, , appendix A. Phillip Griffiths and Joe Harris, Principles of algebraic geometry, Wiley, , chapter 2, section 1. Robin Hartshorne (1977): Algebraic geometry, Springer, . Algebraic curves
https://en.wikipedia.org/wiki/Novak%20Djokovic%20career%20statistics	This is a list of the main career statistics of the Serbian professional tennis player Novak Djokovic. All statistics are based on data from the Association of Tennis Professionals (ATP). Performance timelines {{Performance key\|short=yes}} Singles Current through the 2023 Davis Cup Group stage Doubles * not held due to COVID-19 pandemic. Grand Slam tournaments Djokovic has won an all-time record 24 Grand Slam titles and he is the only man to achieve a triple Career Grand Slam by winning each of the four majors at least three time. Djokovic is also the only man in tennis history to be the reigning champion of all four majors at once across three different surfaces. He is the record holder for the most Grand Slam final played (36) and the only player to reach at least 7 finals at each of the four majors. Grand Slam tournament finals: 36 (24 titles, 12 runner-ups) Year–end championships Djokovic has won a record six year-end championships (tied with Federer). He also holds a record streak of four titles from 2012 to 2015. Year–end championship finals: 8 (6 titles, 2 runner-ups) ATP Masters Djokovic has won a record 39 Masters titles and he is the only player to complete the Career Golden Masters by winning all Masters tournaments of the tennis calendar, a feat he achieved twice. He also holds the record for most Masters won in a season with six titles in 2015. Finals: 57 (39 titles, 18 runner-ups) ATP career finals Singles: 136 (96 titles, 40 runner-ups) Doubles: 3 (1 title, 2 runner-ups) Summer Olympics Singles: 3 (1 bronze medal, 2 fourth places) Mixed doubles: 1 (1 fourth place) ATP Challengers & ITF Futures Singles: 6 (6 titles) Doubles: 1 (1 title) ATP ranking Djokovic has spent the most weeks as ATP world No. 1, a record total of 397. He had been ranked No. 1 in a record 12 different years and finished as year-end No. 1 a record seven times. Djokovic also holds the record for most points accumulated at the top of the rankings (16,950). Timeline Weeks statistics *. ATP world No. 1 Weeks at No. 1 by span World No. 1 ranking records Span holding the No. 1 ranking Age at first and last dates at No. 1 Head-to-head records Record against top-10 players Djokovic's match record against those who have been ranked in the top 10. Active players are in boldface. Record against players ranked No. 11–20 Active players are in boldface. Viktor Troicki 13–1 Andreas Seppi 12–0 Philipp Kohlschreiber 12–2 Feliciano López 9–1 Sam Querrey 9–2 Alexandr Dolgopolov 6–0 Albert Ramos Viñolas 6–0 Bernard Tomic 6–0 Kyle Edmund 6–1 Paul-Henri Mathieu 6–1 Jarkko Nieminen 6–1 Robby Ginepri 5–0 Florian Mayer 5–0 Igor Andreev 4–0 Borna Ćorić 4–0 Marcel Granollers 4–0 Cristian Garín 3–0 Lorenzo Musetti 3–1 Nikoloz Basilashvili 2–0 Juan Ignacio Chela 2–0 Aslan Karatsev 2–1 Benoît Paire 2–1 Pablo Cuevas 1–0 Alex de Minaur 1–0 Dominik Hrbatý 1–0 Jerzy Janowicz 1–0 Tommy Paul 1–0 Andrei Pavel 1–0 Guido Pella 1–0 Ben S
https://en.wikipedia.org/wiki/Recursive%20language	In mathematics, logic and computer science, a formal language (a set of finite sequences of symbols taken from a fixed alphabet) is called recursive if it is a recursive subset of the set of all possible finite sequences over the alphabet of the language. Equivalently, a formal language is recursive if there exists a Turing machine that, when given a finite sequence of symbols as input, always halts and accepts it if it belongs to the language and halts and rejects it otherwise. In Theoretical computer science, such always-halting Turing machines are called total Turing machines or algorithms (Sipser 1997). Recursive languages are also called decidable. The concept of decidability may be extended to other models of computation. For example, one may speak of languages decidable on a non-deterministic Turing machine. Therefore, whenever an ambiguity is possible, the synonym used for "recursive language" is Turing-decidable language, rather than simply decidable. The class of all recursive languages is often called R, although this name is also used for the class RP. This type of language was not defined in the Chomsky hierarchy of . All recursive languages are also recursively enumerable. All regular, context-free and context-sensitive languages are recursive. Definitions There are two equivalent major definitions for the concept of a recursive language: A recursive formal language is a recursive subset in the set of all possible words over the alphabet of the language. A recursive language is a formal language for which there exists a Turing machine that, when presented with any finite input string, halts and accepts if the string is in the language, and halts and rejects otherwise. The Turing machine always halts: it is known as a decider and is said to decide the recursive language. By the second definition, any decision problem can be shown to be decidable by exhibiting an algorithm for it that terminates on all inputs. An undecidable problem is a problem that is not decidable. Examples As noted above, every context-sensitive language is recursive. Thus, a simple example of a recursive language is the set L={abc, , , ...}; more formally, the set is context-sensitive and therefore recursive. Examples of decidable languages that are not context-sensitive are more difficult to describe. For one such example, some familiarity with mathematical logic is required: Presburger arithmetic is the first-order theory of the natural numbers with addition (but without multiplication). While the set of well-formed formulas in Presburger arithmetic is context-free, every deterministic Turing machine accepting the set of true statements in Presburger arithmetic has a worst-case runtime of at least , for some constant c>0 . Here, n denotes the length of the given formula. Since every context-sensitive language can be accepted by a linear bounded automaton, and such an automaton can be simulated by a deterministic Turing machine with worst-case
https://en.wikipedia.org/wiki/Bruno%20Pinheiro%20%28footballer%29	Bruno Filipe Tavares Pinheiro (born 21 August 1987 in Paranhos (Porto)) is a Portuguese professional footballer who plays as a centre-back or a defensive midfielder for Maia Lidador. Club statistics References External links National team data 1987 births Living people Portuguese men's footballers Footballers from Porto Men's association football defenders Men's association football midfielders Primeira Liga players Liga Portugal 2 players Segunda Divisão players Boavista F.C. players G.D. Ribeirão players Gil Vicente F.C. players A.R. São Martinho players Cypriot First Division players Aris Limassol FC players Ekstraklasa players Widzew Łódź players Israeli Premier League players Maccabi Netanya F.C. players Hapoel Haifa F.C. players Football League (Greece) players Niki Volos F.C. players Apollon Smyrnis F.C. players Indian Super League players FC Goa players Bruno Pinheiro Bruno Pinheiro Hong Kong Premier League players Lee Man FC players Portugal men's youth international footballers Portugal men's under-21 international footballers Portuguese expatriate men's footballers Expatriate men's footballers in Cyprus Expatriate men's footballers in Poland Expatriate men's footballers in Israel Expatriate men's footballers in Greece Expatriate men's footballers in India Expatriate men's footballers in Thailand Expatriate men's footballers in Hong Kong Portuguese expatriate sportspeople in Cyprus Portuguese expatriate sportspeople in Poland Portuguese expatriate sportspeople in Israel Portuguese expatriate sportspeople in Greece Portuguese expatriate sportspeople in India Portuguese expatriate sportspeople in Thailand Portuguese expatriate sportspeople in Hong Kong
https://en.wikipedia.org/wiki/Cyrus%20Colton%20MacDuffee	Cyrus Colton MacDuffee (June 29, 1895 – August 21, 1961) from Oneida, New York was a professor of mathematics at University of Wisconsin. He wrote a number of influential research papers in abstract algebra. MacDuffee served on the Council of the American Mathematical Society (A.M.S.), was editor of the Transactions of the A.M.S., and served as president of the Mathematical Association of America (M.A.A). MacDuffee obtained his B.S. degree in 1917 from Colgate University and a Ph.D. in 1922 from the University of Chicago ; his thesis was on Nonassociative algebras under the direction of Leonard E. Dickson. In 1935, MacDuffee joined the University of Wisconsin, where he remained until his death in 1961. He served as chair of the department (1951–56). Later, Wisconsin endowed a university chair under his name. Prior to joining the University of Wisconsin, he served at Princeton and Ohio State. He guided 30 Ph.D. students, among them D. R. Fulkerson, H. J. Ryser, and Bonnie Stewart. MacDuffee's daughter Helen became a statistician at Oregon State University and the mayor of Corvallis, Oregon. Bibliography (2nd ed., 1946). See also Latimer–MacDuffee theorem References 1895 births 1961 deaths 20th-century American mathematicians Algebraists University of Chicago alumni Colgate University alumni University of Wisconsin–Madison faculty Ohio State University faculty Presidents of the Mathematical Association of America People from Oneida, New York Mathematicians from New York (state)
https://en.wikipedia.org/wiki/Eli%20Turkel	Eli L. Turkel (hebrew אלי טורקל) (born January 22, 1944) is an Israeli applied mathematician and currently an emeritus professor of applied mathematics at the School of Mathematical Sciences, Tel Aviv University. He is known for his contributions to numerical analysis of Partial Differential equations particularly in the fields of computational fluid dynamics, computational electromagnetics, acoustics, elasticity and image processing with applications to first Temple ostraca and recently deep earning for forward and inverse problems in PDEs, Research His research interests include algorithms for scattering and inverse scattering, image processing, and crack propagation. His most quoted paper is with Jameson and Schmidt (JST) on a Runge-Kutta scheme to solve the Euler equations. His other main contributions include fast algorithms for the Navier-Stokes equations based on preconditioning techniques, radiation boundary conditions and high order accuracy for wave propagation in general shaped domains. His recent work is on reading ostraca from the first Temple period. Algorithmic handwriting analysis of Judah’s military correspondence sheds light on the composition of biblical texts, which appeared in PNAS was quoted by numerous sources including the front page of the NY Times. Later articles deal with ostraca at both Samaria and Arad. Other research includes high order numerical methods for hyperbolic equations, including the Helmholtz equation, acoustics and Maxwell's equations, using Cartesian grids but general shaped boundaries and interfaces. Other research uses deep learning to detect sources and obstacles underwater using the acoustic wave equation and data at a few sensors. Other applications of deep learning include using large time steps and improving the accuracy of finite differences for high frequencies on coarse grids. Deep learning algorithms include, HINTS, VITO and DITTO. He has also authored articles in Tradition and the Journal of Contemporary Halacha. Turkel was listed as an ISI highly cited researcher in mathematics. Google Scholar lists over 20,000 citations. Education Turkel was born in New York City, United States. He received his B.A. degree from the Yeshiva University in 1965, M.S. degree from the New York University in 1967, and Ph.D. degree from the Courant Institute at New York University in 1970; all in mathematics. His Ph.D. thesis advisors were J. J. Stoker and Eugene Isaacson. He received rabbinical ordination from Rabbi J.B. Soloveitchik. References External links Home page: http://www.math.tau.ac.il/~turkel/ Index book for the Rav Soloveitchik https://www.otzar.org/wotzar/book.aspx?191322&&lang=eng Yeshiva University alumni New York University alumni Academic staff of Tel Aviv University Israeli mathematicians Living people Computational fluid dynamicists 1944 births
https://en.wikipedia.org/wiki/Venezuela%20national%20football%20team%20records%20and%20statistics	The following is a list of the Venezuela national football team's competitive records and statistics. Individual records Player records Players in bold are still active with Venezuela. Most capped players Top goalscorers Manager records Team records Competition records FIFA World Cup Copa América Champions Runners-up Third place Fourth place Pan American Games Head-to-head record The list shows the Venezuela national football team all-time international record against opposing nations. As of 17 October 2023. References Venezuela national football team records and statistics National association football team records and statistics
https://en.wikipedia.org/wiki/Ponte%20Tron%2C%20Venice	{ "type": "FeatureCollection", "features": [ { "type": "Feature", "properties": {}, "geometry": { "type": "Point", "coordinates": [ 12.336361, 45.434603 ] } } ] }The Ponte Tron or , near Piazza San Marco in Venice spans the Rio Orseolo near the Bacino Orseolo. It is one of more than 400 bridges that connect the archipelago of 118 islands divided by about 150 canals and narrow rivers in the shallow Venetian lagoon. The lion's head of Saint Mark appears on the cartouche at the center of the arch. In spite of its traditional construction in the salt-white Istrian stone used elsewhere in Venice and the classic vase profiles of the balustrade, this bridge, called , the "doll's bridge" in Venetian, for its diminutive size, was built at the late date of 1840. Notes Bridges in Venice
https://en.wikipedia.org/wiki/David%20Origanus	David Origanus or David Tost (9 July 1558 – 11 July 1628/29) was a German astronomer and professor for Greek language and Mathematics at the Viadrina University in Frankfurt (Oder), where he had also studied. Tost was born in Glatz (Kladsko), Bohemia (now Kłodzko in southern Poland). During his scientific career he observed numerous comets and published about Ephemeris in 1599 and 1609. In contrast to Tycho Brahe, he was convinced that the Earth rotates. He died in Frankfurt (Oder), aged 71. Works Novae motuum coelestium ephemerides Brandenburgicae, Frankfurt aO: Eichornius, 1609, „Praefatio" Ephemerides Novae Annorum XXXVI, Incipientes Ab Anno ... 1595, Quo Joannis Stadii maxime aberrare incipiunt, & desinentes in annum 1630: Quibus praemissa est Introductio Seu Compendiaria Ephemeridum Enarratio ... Eichornius, 1599 Notes External links 16th-century German astronomers 16th-century German mathematicians People from Kłodzko People from Austrian Silesia European University Viadrina alumni Academic staff of European University Viadrina 1558 births 1628 deaths 17th-century German astronomers
https://en.wikipedia.org/wiki/Bordiga%20surface	In algebraic geometry, a Bordiga surface is a certain sort of rational surface of degree 6 in P4, introduced by Giovanni Bordiga. A Bordiga surface is isomorphic to the projective plane blown up in 10 points, the embedding into P4 is given by the 5-dimensional space of quartics passing through the 10 points. White surfaces are the generalizations using more points. References Complex surfaces Algebraic surfaces
https://en.wikipedia.org/wiki/White%20surface	In algebraic geometry, a White surface is one of the rational surfaces in Pn studied by , generalizing cubic surfaces and Bordiga surfaces, which are the cases n = 3 or 4. A White surface in Pn is given by the embedding of P2 blown up in n(n + 1)/2 points by the linear system of degree n curves through these points. References Complex surfaces Algebraic surfaces
https://en.wikipedia.org/wiki/Bicentric%20quadrilateral	In Euclidean geometry, a bicentric quadrilateral is a convex quadrilateral that has both an incircle and a circumcircle. The radii and centers of these circles are called inradius and circumradius, and incenter and circumcenter respectively. From the definition it follows that bicentric quadrilaterals have all the properties of both tangential quadrilaterals and cyclic quadrilaterals. Other names for these quadrilaterals are chord-tangent quadrilateral and inscribed and circumscribed quadrilateral. It has also rarely been called a double circle quadrilateral and double scribed quadrilateral. If two circles, one within the other, are the incircle and the circumcircle of a bicentric quadrilateral, then every point on the circumcircle is the vertex of a bicentric quadrilateral having the same incircle and circumcircle. This is a special case of Poncelet's porism, which was proved by the French mathematician Jean-Victor Poncelet (1788–1867). Special cases Examples of bicentric quadrilaterals are squares, right kites, and isosceles tangential trapezoids. Characterizations A convex quadrilateral with sides is bicentric if and only if opposite sides satisfy Pitot's theorem for tangential quadrilaterals and the cyclic quadrilateral property that opposite angles are supplementary; that is, Three other characterizations concern the points where the incircle in a tangential quadrilateral is tangent to the sides. If the incircle is tangent to the sides at respectively, then a tangential quadrilateral is also cyclic if and only if any one of the following three conditions holds: is perpendicular to The first of these three means that the contact quadrilateral is an orthodiagonal quadrilateral. If are the midpoints of respectively, then the tangential quadrilateral is also cyclic if and only if the quadrilateral is a rectangle. According to another characterization, if is the incenter in a tangential quadrilateral where the extensions of opposite sides intersect at and , then the quadrilateral is also cyclic if and only if is a right angle. Yet another necessary and sufficient condition is that a tangential quadrilateral is cyclic if and only if its Newton line is perpendicular to the Newton line of its contact quadrilateral . (The Newton line of a quadrilateral is the line defined by the midpoints of its diagonals.) Construction There is a simple method for constructing a bicentric quadrilateral: It starts with the incircle around the centre with the radius and then draw two to each other perpendicular chords and in the incircle . At the endpoints of the chords draw the tangents to the incircle. These intersect at four points , which are the vertices of a bicentric quadrilateral. To draw the circumcircle, draw two perpendicular bisectors on the sides of the bicentric quadrilateral respectively . The perpendicular bisectors intersect in the centre of the circumcircle with the distance to the centre of the incircle
https://en.wikipedia.org/wiki/Canonical%20map	In mathematics, a canonical map, also called a natural map, is a map or morphism between objects that arises naturally from the definition or the construction of the objects. Often, it is a map which preserves the widest amount of structure. A choice of a canonical map sometimes depends on a convention (e.g., a sign convention). A closely related notion is a structure map or structure morphism; the map or morphism that comes with the given structure on the object. These are also sometimes called canonical maps. A canonical isomorphism is a canonical map that is also an isomorphism (i.e., invertible). In some contexts, it might be necessary to address an issue of choices of canonical maps or canonical isomorphisms; for a typical example, see prestack. For a discussion of the problem of defining a canonical map see Kevin Buzzard's talk at the 2022 Grothendieck conference. Examples If N is a normal subgroup of a group G, then there is a canonical surjective group homomorphism from G to the quotient group G/N, that sends an element g to the coset determined by g. If I is an ideal of a ring R, then there is a canonical surjective ring homomorphism from R onto the quotient ring R/I, that sends an element r to its coset I+r. If V is a vector space, then there is a canonical map from V to the second dual space of V, that sends a vector v to the linear functional fv defined by fv(λ) = λ(v). If is a homomorphism between commutative rings, then S can be viewed as an algebra over R. The ring homomorphism f is then called the structure map (for the algebra structure). The corresponding map on the prime spectra is also called the structure map. If E is a vector bundle over a topological space X, then the projection map from E to X is the structure map. In topology, a canonical map is a function f mapping a set X → X/R (X modulo R), where R is an equivalence relation on X, that takes each x in X to the equivalence class [x] modulo R. References Mathematical terminology
https://en.wikipedia.org/wiki/Clasper%20%28mathematics%29	In the mathematical field of low-dimensional topology, a clasper is a surface (with extra structure) in a 3-manifold on which surgery can be performed. Motivation Beginning with the Jones polynomial, infinitely many new invariants of knots, links, and 3-manifolds were found during the 1980s. The study of these new `quantum' invariants expanded rapidly into a sub-discipline of low-dimensional topology called quantum topology. A quantum invariant is typically constructed from two ingredients: a formal sum of Jacobi diagrams (which carry a Lie algebra structure), and a representation of a ribbon Hopf algebra such as a quantum group. It is not clear a-priori why either of these ingredients should have anything to do with low-dimensional topology. Thus one of the main problems in quantum topology has been to interpret quantum invariants topologically. The theory of claspers comes to provide such an interpretation. A clasper, like a framed link, is an embedded topological object in a 3-manifold on which one can perform surgery. In fact, clasper calculus can be thought of as a variant of Kirby calculus on which only certain specific types of framed links are allowed. Claspers may also be interpreted algebraically, as a diagram calculus for the braided strict monoidal category Cob of oriented connected surfaces with connected boundary. Additionally, most crucially, claspers may be roughly viewed as a topological realization of Jacobi diagrams, which are purely combinatorial objects. This explains the Lie algebra structure of the graded vector space of Jacobi diagrams in terms of the Hopf algebra structure of Cob. Definition A clasper is a compact surface embedded in the interior of a 3-manifold equipped with a decomposition into two subsurfaces and , whose connected components are called the constituents and the edges of correspondingly. Each edge of is a band joining two constituents to one another, or joining one constituent to itself. There are four types of constituents: leaves, disk-leaves, nodes, and boxes. Clasper surgery is most easily defined (after elimination of nodes, boxes, and disk-leaves as described below) as surgery along a link associated to the clasper by replacing each leaf with its core, and replacing each edge by a right Hopf link. Clasper calculus The following are the graphical conventions used when drawing claspers (and may be viewed as a definition for boxes, nodes, and disk-leaves): Habiro found 12 moves which relate claspers along which surgery gives the same result. These moves form the core of clasper calculus, and give considerable power to the theory as a theorem-proving tool. Cn-equivalence Two knots, links, or 3-manifolds are said to be -equivalent if they are related by -moves, which are the local moves induced by surgeries on a simple tree claspers without boxes or disk-leaves and with leaves. For a link , a -move is a crossing change. A -move is a Delta move. Most applications of claspers use
https://en.wikipedia.org/wiki/Butcher%20group	In mathematics, the Butcher group, named after the New Zealand mathematician John C. Butcher by , is an infinite-dimensional Lie group first introduced in numerical analysis to study solutions of non-linear ordinary differential equations by the Runge–Kutta method. It arose from an algebraic formalism involving rooted trees that provides formal power series solutions of the differential equation modeling the flow of a vector field. It was , prompted by the work of Sylvester on change of variables in differential calculus, who first noted that the derivatives of a composition of functions can be conveniently expressed in terms of rooted trees and their combinatorics. pointed out that the Butcher group is the group of characters of the Hopf algebra of rooted trees that had arisen independently in their own work on renormalization in quantum field theory and Connes' work with Moscovici on local index theorems. This Hopf algebra, often called the Connes–Kreimer algebra, is essentially equivalent to the Butcher group, since its dual can be identified with the universal enveloping algebra of the Lie algebra of the Butcher group. As they commented: Differentials and rooted trees A rooted tree is a graph with a distinguished node, called the root, in which every other node is connected to the root by a unique path. If the root of a tree t is removed and the nodes connected to the original node by a single bond are taken as new roots, the tree t breaks up into rooted trees t1, t2, ... Reversing this process a new tree t = [t1, t2, ...] can be constructed by joining the roots of the trees to a new common root. The number of nodes in a tree is denoted by \|t\|. A heap-ordering of a rooted tree t is an allocation of the numbers 1 through \|t\| to the nodes so that the numbers increase on any path going away from the root. Two heap orderings are equivalent, if there is an automorphism of rooted trees mapping one of them on the other. The number of equivalence classes of heap-orderings on a particular tree is denoted by α(t) and can be computed using the Butcher's formula: where St denotes the symmetry group of t and the tree factorial is defined recursively by with the tree factorial of an isolated root defined to be 1 The ordinary differential equation for the flow of a vector field on an open subset U of RN can be written where x(s) takes values in U, f is a smooth function from U to RN and x0 is the starting point of the flow at time s = 0. gave a method to compute the higher order derivatives x(m)(s) in terms of rooted trees. His formula can be conveniently expressed using the elementary differentials introduced by Butcher. These are defined inductively by With this notation giving the power series expansion As an example when N = 1, so that x and f are real-valued functions of a single real variable, the formula yields where the four terms correspond to the four rooted trees from left to right in Figure 3 above. In a single variable this
https://en.wikipedia.org/wiki/2009%E2%80%9310%20Galatasaray%20S.K.%20season	The 2009–10 season was Galatasarays 106th in existence and the 52nd consecutive season in the Süper Lig. This article shows statistics of the club's players in the season, and also lists all matches that the club have played in the season. Club The Board of Directors Technical Staff Medical Staff Squad Transfers In Out Loan out Squad statistics Statistics accurate as of match played May 16, 2010 Pre-season and friendlies All times at CET Competitions Süper Lig League table Results summary Results by round Matches All times in EEST Turkish Cup Kick-off listed in local time (EEST) Play-off round Group stage Quarter-finals UEFA Europa League All times at CET Second qualifying round Third qualifying round Play-off round Group stage Knockout phase Round of 32 Attendance References External links Galatasaray Sports Club Official Website Turkish Football Federation - Galatasaray A.Ş. uefa.com - Galatasaray AŞ 2009-10 Turkish football clubs 2009–10 season 2000s in Istanbul 2010 in Istanbul Galatasaray Sports Club 2009–10 season
https://en.wikipedia.org/wiki/N-ary%20group	In mathematics, and in particular universal algebra, the concept of an n-ary group (also called n-group or multiary group) is a generalization of the concept of a group to a set G with an n-ary operation instead of a binary operation. By an operation is meant any map f: Gn → G from the n-th Cartesian power of G to G. The axioms for an group are defined in such a way that they reduce to those of a group in the case . The earliest work on these structures was done in 1904 by Kasner and in 1928 by Dörnte; the first systematic account of (what were then called) polyadic groups was given in 1940 by Emil Leon Post in a famous 143-page paper in the Transactions of the American Mathematical Society. Axioms Associativity The easiest axiom to generalize is the associative law. Ternary associativity is the polynomial identity , i.e. the equality of the three possible bracketings of the string abcde in which any three consecutive symbols are bracketed. (Here it is understood that the equations hold for all choices of elements a, b, c, d, e in G.) In general, associativity is the equality of the n possible bracketings of a string consisting of distinct symbols with any n consecutive symbols bracketed. A set G which is closed under an associative operation is called an n-ary semigroup. A set G which is closed under any (not necessarily associative) operation is called an n-ary groupoid. Inverses / unique solutions The inverse axiom is generalized as follows: in the case of binary operations the existence of an inverse means has a unique solution for x, and likewise has a unique solution. In the ternary case we generalize this to , and each having unique solutions, and the case follows a similar pattern of existence of unique solutions and we get an n-ary quasigroup. Definition of n-ary group An n-ary group is an semigroup which is also an quasigroup. Identity / neutral elements In the case, there can be zero or one identity elements: the empty set is a 2-ary group, since the empty set is both a semigroup and a quasigroup, and every inhabited 2-ary group is a group. In groups for n ≥ 3 there can be zero, one, or many identity elements. An groupoid (G, f) with , where (G, ◦) is a group is called reducible or derived from the group (G, ◦). In 1928 Dörnte published the first main results: An groupoid which is reducible is an group, however for all n > 2 there exist inhabited groups which are not reducible. In some n-ary groups there exists an element e (called an identity or neutral element) such that any string of n-elements consisting of all e's, apart from one place, is mapped to the element at that place. E.g., in a quaternary group with identity e, eeae = a for every a. An group containing a neutral element is reducible. Thus, an group that is not reducible does not contain such elements. There exist groups with more than one neutral element. If the set of all neutral elements of an group is non-empty it forms an subgr
https://en.wikipedia.org/wiki/Segre%20surface	In algebraic geometry, a Segre surface, studied by and , is an intersection of two quadrics in 4-dimensional projective space. They are rational surfaces isomorphic to a projective plane blown up in 5 points with no 3 on a line, and are del Pezzo surfaces of degree 4, and have 16 rational lines. The term "Segre surface" is also occasionally used for various other surfaces, such as a quadric in 3-dimensional projective space, or the hypersurface References Algebraic surfaces Complex surfaces
https://en.wikipedia.org/wiki/Sarti%20surface	In algebraic geometry, a Sarti surface is a degree-12 nodal surface with 600 nodes, found by . The maximal possible number of nodes of a degree-12 surface is not known (as of 2015), though Yoichi Miyaoka showed that it is at most 645. Sarti has also found sextic, octic and dodectic nodal surfaces with high numbers of nodes and high degrees of symmetry. See also Nodal surface References External links Algebraic surfaces Complex surfaces
https://en.wikipedia.org/wiki/Ch%C3%A2telet%20surface	In algebraic geometry, a Châtelet surface is a rational surface studied by given by an equation where P has degree 3 or 4. They are conic bundles. References Algebraic surfaces Complex surfaces
https://en.wikipedia.org/wiki/Zeynal%20Zeynalov%20%28footballer%29	Zeynal Zeynalov (born 6 December 1979) is an Azerbaijani professional footballer & futsal player. As of 2009, he plays for Standard Sumgayit. Career statistics National team statistics International goals References External links 1979 births Living people Footballers from Baku Azerbaijani men's footballers Azerbaijan men's international footballers FK Standard Sumgayit players Mil-Muğan FK players Men's association football midfielders Neftçi PFK players European Games competitors for Azerbaijan Beach soccer players at the 2015 European Games
https://en.wikipedia.org/wiki/Doob%20decomposition%20theorem	In the theory of stochastic processes in discrete time, a part of the mathematical theory of probability, the Doob decomposition theorem gives a unique decomposition of every adapted and integrable stochastic process as the sum of a martingale and a predictable process (or "drift") starting at zero. The theorem was proved by and is named for Joseph L. Doob. The analogous theorem in the continuous-time case is the Doob–Meyer decomposition theorem. Statement Let be a probability space, with or a finite or an infinite index set, a filtration of , and an adapted stochastic process with for all . Then there exist a martingale and an integrable predictable process starting with such that for every . Here predictable means that is -measurable for every . This decomposition is almost surely unique. Remark The theorem is valid word by word also for stochastic processes taking values in the -dimensional Euclidean space or the complex vector space . This follows from the one-dimensional version by considering the components individually. Proof Existence Using conditional expectations, define the processes and , for every , explicitly by and where the sums for are empty and defined as zero. Here adds up the expected increments of , and adds up the surprises, i.e., the part of every that is not known one time step before. Due to these definitions, (if ) and are -measurable because the process is adapted, and because the process is integrable, and the decomposition is valid for every . The martingale property a.s. also follows from the above definition (), for every }. Uniqueness To prove uniqueness, let be an additional decomposition. Then the process is a martingale, implying that a.s., and also predictable, implying that a.s. for any }. Since by the convention about the starting point of the predictable processes, this implies iteratively that almost surely for all , hence the decomposition is almost surely unique. Corollary A real-valued stochastic process is a submartingale if and only if it has a Doob decomposition into a martingale and an integrable predictable process that is almost surely increasing. It is a supermartingale, if and only if is almost surely decreasing. Proof If is a submartingale, then a.s. for all }, which is equivalent to saying that every term in definition () of is almost surely positive, hence is almost surely increasing. The equivalence for supermartingales is proved similarly. Example Let be a sequence in independent, integrable, real-valued random variables. They are adapted to the filtration generated by the sequence, i.e. for all . By () and (), the Doob decomposition is given by and If the random variables of the original sequence have mean zero, this simplifies to and hence both processes are (possibly time-inhomogeneous) random walks. If the sequence consists of symmetric random variables taking the values and , then is bounded
https://en.wikipedia.org/wiki/Alyaksandr%20Pawlaw	Alyaksandr Valeryevich Pawlaw (; ; born 18 August 1984) is a Belarusian professional football coach and former player. Career statistics Club International Honours BATE Borisov Belarusian Premier League (6): 2009, 2010, 2011, 2012, 2013, 2014 Belarusian Cup (1): 2009–10 Belarusian Super Cup (4): 2010, 2011, 2013, 2014 References External links 1984 births Living people Belarusian men's footballers Men's association football midfielders Belarus men's international footballers Belarusian expatriate men's footballers Expatriate men's footballers in Kazakhstan FC Dnepr Mogilev players FC BATE Borisov players FC Okzhetpes players FC Shakhtyor Soligorsk players Belarusian football managers FC Vitebsk managers
https://en.wikipedia.org/wiki/Appell%E2%80%93Humbert%20theorem	In mathematics, the Appell–Humbert theorem describes the line bundles on a complex torus or complex abelian variety. It was proved for 2-dimensional tori by and , and in general by Statement Suppose that is a complex torus given by where is a lattice in a complex vector space . If is a Hermitian form on whose imaginary part is integral on , and is a map from to the unit circle , called a semi-character, such that then is a 1-cocycle of defining a line bundle on . For the trivial Hermitian form, this just reduces to a character. Note that the space of character morphisms is isomorphic with a real torusif since any such character factors through composed with the exponential map. That is, a character is a map of the formfor some covector . The periodicity of for a linear gives the isomorphism of the character group with the real torus given above. In fact, this torus can be equipped with a complex structure, giving the dual complex torus. Explicitly, a line bundle on may be constructed by descent from a line bundle on (which is necessarily trivial) and a descent data, namely a compatible collection of isomorphisms , one for each . Such isomorphisms may be presented as nonvanishing holomorphic functions on , and for each the expression above is a corresponding holomorphic function. The Appell–Humbert theorem says that every line bundle on can be constructed like this for a unique choice of and satisfying the conditions above. Ample line bundles Lefschetz proved that the line bundle , associated to the Hermitian form is ample if and only if is positive definite, and in this case is very ample. A consequence is that the complex torus is algebraic if and only if there is a positive definite Hermitian form whose imaginary part is integral on See also Complex torus for a treatment of the theorem with examples References Abelian varieties Theorems in algebraic geometry Theorems in complex geometry
https://en.wikipedia.org/wiki/Cayley%27s%20ruled%20cubic%20surface	In differential geometry, Cayley's ruled cubic surface is the ruled cubic surface It contains a nodal line of self-intersection and two cuspital points at infinity. In projective coordinates it is . References External links Cubical ruled surface Algebraic surfaces Differential geometry
https://en.wikipedia.org/wiki/Completely-S%20matrix	In linear algebra, a completely-S matrix is a square matrix such that for every principal submatrix R there exists a positive vector u such that Ru > 0. Notes Matrices
https://en.wikipedia.org/wiki/Matching%20%28statistics%29	Matching is a statistical technique which is used to evaluate the effect of a treatment by comparing the treated and the non-treated units in an observational study or quasi-experiment (i.e. when the treatment is not randomly assigned). The goal of matching is to reduce bias for the estimated treatment effect in an observational-data study, by finding, for every treated unit, one (or more) non-treated unit(s) with similar observable characteristics against which the covariates are balanced out. By matching treated units to similar non-treated units, matching enables a comparison of outcomes among treated and non-treated units to estimate the effect of the treatment reducing bias due to confounding. Propensity score matching, an early matching technique, was developed as part of the Rubin causal model, but has been shown to increase model dependence, bias, inefficiency, and power and is no longer recommended compared to other matching methods. A simple, easy-to-understand, and statistically powerful method of matching known as Coarsened Exact Matching or CEM. Matching has been promoted by Donald Rubin. It was prominently criticized in economics by LaLonde (1986), who compared estimates of treatment effects from an experiment to comparable estimates produced with matching methods and showed that matching methods are biased. Dehejia and Wahba (1999) reevaluated LaLonde's critique and showed that matching is a good solution. Similar critiques have been raised in political science and sociology journals. Analysis When the outcome of interest is binary, the most general tool for the analysis of matched data is conditional logistic regression as it handles strata of arbitrary size and continuous or binary treatments (predictors) and can control for covariates. In particular cases, simpler tests like paired difference test, McNemar test and Cochran-Mantel-Haenszel test are available. When the outcome of interest is continuous, estimation of the average treatment effect is performed. Matching can also be used to "pre-process" a sample before analysis via another technique, such as regression analysis. Overmatching Overmatching, or post-treatment bias, is matching for an apparent mediator that actually is a result of the exposure. If the mediator itself is stratified, an obscured relation of the exposure to the disease would highly be likely to be induced. Overmatching thus causes statistical bias. For example, matching the control group by gestation length and/or the number of multiple births when estimating perinatal mortality and birthweight after in vitro fertilization (IVF) is overmatching, since IVF itself increases the risk of premature birth and multiple birth. It may be regarded as a sampling bias in decreasing the external validity of a study, because the controls become more similar to the cases in regard to exposure than the general population. See also Propensity score matching References Further reading Bias Design of exper
https://en.wikipedia.org/wiki/Regressive%20discrete%20Fourier%20series	In applied mathematics, the regressive discrete Fourier series (RDFS) is a generalization of the discrete Fourier transform where the Fourier series coefficients are computed in a least squares sense and the period is arbitrary, i.e., not necessarily equal to the length of the data. It was first proposed by Arruda (1992a, 1992b). It can be used to smooth data in one or more dimensions and to compute derivatives from the smoothed curve, surface, or hypersurface. Technique One-dimensional regressive discrete Fourier series The one-dimensional RDFS proposed by Arruda (1992a) can be formulated in a very straightforward way. Given a sampled data vector (signal) , one can write the algebraic expression: Typically , but this is not necessary. The above equation can be written in matrix form as The least squares solution of the above linear system of equations can be written as: where is the conjugate transpose of , and the smoothed signal is obtained from: The first derivative of the smoothed signal can be obtained from: Two-dimensional regressive discrete Fourier series (RDFS) The two-dimensional, or bidimensional RDFS proposed by Arruda (1992b) can also be formulated in a straightforward way. Here the equally spaced data case will be treated for the sake of simplicity. The general non-equally-spaced and arbitrary grid cases are given in the reference (Arruda, 1992b). Given a sampled data matrix (bi dimensional signal) one can write the algebraic expression: The above equation can be written in matrix form for a rectangular grid. For the equally spaced sampling case : we have: The least squares solution may be shown to be: and the smoothed bidimensional surface is given by: where is the conjugate, and is the transpose of . Differentiation with respect to can be easily implemented analogously to the one-dimensional case (Arruda, 1992b). Current applications Spatially dense data condensation applications: Arruda, J.R.F. [1993] applied the RDFS to condense spatially dense spatial measurements made with a laser Doppler vibrometer prior to applying modal analysis parameter estimation methods. More recently, Vanherzeele et al. (2006,2008a) proposed a generalized and an optimized RDFS for the same kind of application. A review of optical measurement processing using the RDFS was published by Vanherzeele et al. (2009). Spatial derivative applications: Batista et al. [2009] applied RDFS to obtain spatial derivatives of bi dimensional measured vibration data to identify material properties from transverse modes of rectangular plates. SHM applications: Vanherzeele et al. [2009] applied a generalized version of the RDFS to tomography reconstruction. Software Recently, a package that includes one and two-dimensional RDFS was developed in order to make easier its use in the free and open source software R: A R package for RDFS at Github See also Discrete Fourier transform Fourier series References Arruda, J.R.F., 1992a: Analysis of n
https://en.wikipedia.org/wiki/Clebsch%20surface	In mathematics, the Clebsch diagonal cubic surface, or Klein's icosahedral cubic surface, is a non-singular cubic surface, studied by and , all of whose 27 exceptional lines can be defined over the real numbers. The term Klein's icosahedral surface can refer to either this surface or its blowup at the 10 Eckardt points. Definition The Clebsch surface is the set of points (x0:x1:x2:x3:x4) of P4 satisfying the equations Eliminating x0 shows that it is also isomorphic to the surface in P3. Properties The symmetry group of the Clebsch surface is the symmetric group S5 of order 120, acting by permutations of the coordinates (in P4). Up to isomorphism, the Clebsch surface is the only cubic surface with this automorphism group. The 27 exceptional lines are: The 15 images (under S5) of the line of points of the form (a : −a : b : −b : 0). The 12 images of the line though the point (1:ζ: ζ2: ζ3: ζ4) and its complex conjugate, where ζ is a primitive 5th root of 1. The surface has 10 Eckardt points where 3 lines meet, given by the point (1 : −1 : 0 : 0 : 0) and its conjugates under permutations. showed that the surface obtained by blowing up the Clebsch surface in its 10 Eckardt points is the Hilbert modular surface of the level 2 principal congruence subgroup of the Hilbert modular group of the field Q(). The quotient of the Hilbert modular group by its level 2 congruence subgroup is isomorphic to the alternating group of order 60 on 5 points. Like all nonsingular cubic surfaces, the Clebsch cubic can be obtained by blowing up the projective plane in 6 points. described these points as follows. If the projective plane is identified with the set of lines through the origin in a 3-dimensional vector space containing an icosahedron centered at the origin, then the 6 points correspond to the 6 lines through the icosahedron's 12 vertices. The Eckardt points correspond to the 10 lines through the centers of the 20 faces. References External links Clebsch Surface, John Baez, 1 March, 2016, AMS Visual Insight Blog Algebraic surfaces
https://en.wikipedia.org/wiki/Walter%20Mebane	Walter Richard Mebane, Jr. (born November 30, 1958) is a University of Michigan professor of political science and statistics and an expert on detecting electoral fraud. He has authored numerous articles on potentially fraudulent election results, including a series of notes on the results of the Iranian presidential election, 2009. He authored a paper disputing the Organization of American States's claim of fraud in the 2019 Bolivian general election as well. References External links Walter Mebane's page at the University of Michigan CV at University of Michigan 1958 births Living people People from Long Branch, New Jersey American political scientists Harvard College alumni Yale University alumni University of Michigan faculty
https://en.wikipedia.org/wiki/Georgi%20Takhokhov	Georgi Borisovich Takhokhov (; born 26 September 1970) is a Tajikistani former professional footballer. Club career In 1992, he moved from Pamir Dushanbe to FC Spartak Vladikavkaz. Career statistics International Statistics accurate as of 1 March 2016 References External links 1970 births Living people Soviet men's footballers Tajikistani men's footballers Men's association football forwards Tajikistani expatriate men's footballers Tajikistan men's international footballers Russian Premier League players FC Spartak Vladikavkaz players CSKA Pamir Dushanbe players Expatriate men's footballers in Russia Tajikistani football managers FC Slavyansk Slavyansk-na-Kubani players FC Avtodor Vladikavkaz players
https://en.wikipedia.org/wiki/Weighted%20Voronoi%20diagram	In mathematics, a weighted Voronoi diagram in n dimensions is a generalization of a Voronoi diagram. The Voronoi cells in a weighted Voronoi diagram are defined in terms of a distance function. The distance function may specify the usual Euclidean distance, or may be some other, special distance function. In weighted Voronoi diagrams, each site has a weight that influences the distance computation. The idea is that larger weights indicate more important sites, and such sites will get bigger Voronoi cells. In a multiplicatively weighted Voronoi diagram, the distance between a point and a site is divided by the (positive) weight of the site. In the plane under the ordinary Euclidean distance, the multiplicatively weighted Voronoi diagram is also called circular Dirichlet tessellation and its edges are circular arcs and straight line segments. A Voronoi cell may be non-convex, disconnected and may have holes. This diagram arises, e.g., as a model of crystal growth, where crystals from different points may grow with different speed. Since crystals may grow in empty space only and are continuous objects, a natural variation is the crystal Voronoi diagram, in which the cells are defined somewhat differently. In an additively weighted Voronoi diagram, weights are subtracted from the distances. In the plane under the ordinary Euclidean distance this diagram is also known as the hyperbolic Dirichlet tessellation and its edges are arcs of hyperbolas and straight line segments. The power diagram is defined when weights are subtracted from the squared Euclidean distance. It can also be defined using the power distance defined from a set of circles. References External links Adam Dobrin: A review of properties and variations of Voronoi diagrams Discrete geometry Geometric algorithms Diagrams
https://en.wikipedia.org/wiki/Energy%20in%20the%20Democratic%20Republic%20of%20the%20Congo	The Democratic Republic of the Congo was a net energy exporter in 2008. Most energy was consumed domestically in 2008. According to the IEA statistics the energy export was in 2008 small and less than from the Republic of Congo. 2010 population figures were 3.8 million for the RC compared to CDR 67.8 Million. Electricity The Democratic Republic of the Congo has reserves of petroleum, natural gas, coal, and a potential hydroelectric power generating capacity of around 100,000 MW. The Inga Dam on the Congo River has the potential capacity to generate 40,000 to 45,000 MW of electric power, sufficient to supply the electricity needs of the whole Southern Africa region. Ongoing uncertainties in the political arena, and a resulting lack of interest from investors has meant that the Inga Dam's potential has been limited. In 2001, the dam was estimated to have an installed generating capacity of 2,473 MW. It is estimated that the dam is capable of producing no more than 650–750 MW, because two-thirds of the facility's turbines do not work. The African Development bank agreed to supply $8 million towards dam expansion. The government has also agreed to strengthen the Inga-kolwezi and Inga-South Africa interconnections and to construct a 2nd power line to supply power to Kinshasa. In 2007, the DR Congo had a gross production of public and self-produced electricity of 8,302 million kWh. The DR Congo imported 78 million kWh of electricity in 2007. The DR Congo is also an exporter of electric power. In 2003, electric power exports came to 1.3 TWh, with power transmitted to the Republic of Congo and its capital, Brazzaville, as well as to Zambia and South Africa. There were plans to build the Western Power Corridor (Westcor) to supply electricity from Inga III hydroelectric power plant to the Democratic Republic of the Congo, Angola, Namibia, Botswana and South Africa. The national power company is Société nationale d'électricité (SNEL). Only 13% of the country has access to electricity. As of 2003, 98.2% of electricity was produced by hydroelectric power. The DRC a member of three electrical power pools: SAPP (Southern African Power Pool), EAPP (East African Power Pool), and CAPP (Central African Power Pool). Hydropower The country has vast potential in hydroelectricity. The second stage of the hydroelectric dam was completed in 1982 on the lower Congo River at Inga Falls, with a large portion of its power production supplying hydroelectricity to the mining industry and Kinshasa. Further plans are to build the proposed 11,050 MW Inga III hydropower project with the construction of two dams. There will be approximately 2,000 km and 3,000 km of transmissions lines within the DRC and across its borders respectively. The Inga III hydropower project is expected to electrify Kinshasa, lead to the development of the DRC’s mining sector, and exported hydroelectricity. Petroleum The DROC has crude oil reserves that are second only to Angola's in southern Afr
https://en.wikipedia.org/wiki/Studies%20in%20Applied%20Mathematics	The journal Studies in Applied Mathematics is published by Wiley–Blackwell on behalf of the Massachusetts Institute of Technology. It features scholarly articles on mathematical applications in allied fields, notably computer science, mechanics, astrophysics, geophysics, biophysics and high-energy physics. Its pedigree came from the Journal of Mathematics and Physics which was founded by the MIT Mathematics Department in 1920. The Journal changed to its present name in 1969. The journal was edited from 1969 by David Benney of the Department of Mathematics, Massachusetts Institute of Technology. According to ISI Journal Citation Reports, in 2020 it ranked 26th among the 265 journals in the Applied Mathematics category. Notes External links Journal Home Page MIT Faculty Page of Dr. David Benney Mathematics journals Wiley-Blackwell academic journals English-language journals 8 times per year journals
https://en.wikipedia.org/wiki/B.%20V.%20Shah	Babubhai V. Shah (born 6 February 1935), professor of statistics, was a chief scientist at Research Triangle Institute (RTI) from 1966 till he retired in 2003. He held several positions with RTI and Research Triangle Park for over four decades. B. V. Shah was responsible for the development of the SUDAAN software; to recognize his contributions, RTI has dedicated the current release of SUDAAN (9.0) to B. V. Shah. During that time, he was also part of the academia of biostatistics department, University of North Carolina at Chapel Hill, N.C. Shah received his Ph.D. in statistics from University of Mumbai in 1960 under the guidance of professor M. C. Chakrabarti. Honors American Statistical Association, Fellow International Statistical Institute, Fellow Royal Statistical Society, Fellow Selected peer-reviewed publications 1958, On balancing in factorial experiments. Annals of Mathematical Statistics 29: 766–779. 1969. A note on predicting failures in a future time period. The Institute of Electrical and Electronics Engineers Transactions on Reliability, p. 203-204. 1979. Language design for survey data analysis. Bulletin of the International Statistical Institute, p. 273-292. (with R.W. Haley), 1981. Nosocomial infections in U.S. Hospitals, 1975–1976: estimated frequency by selected characteristics of patients. The American Journal of Medicine 70: 947–959 Books and book chapters Myers, L.E., N. Adams, L. Kier, T.K. Rao, B. Shah, and L. Williams (1991). Microcomputer Software for Data Management and Statistical Analysis of the Ames/Salmonella Test. In Krewski, D. and C. Franklin (Eds.), Statistics in Toxicology. New York: Gordon and Breach Science Publishers, pp. 265–279. LaVange, L.M., B.V. Shah, B.G. Barnwell, and J.F. Killinger (1990). SUDAAN: A comprehensive package for survey data analysis. In G.E. Liepins and V.R.R. Uppuluri (Eds.), Data Quality Control. New York: Marcell Dekker, Inc., pp. 209–227. Shah, B.V. and R.P. Moore (1977). National Longitudinal Study of the High School Class of 1972: Sample Design Efficiency Study—Effects of Stratification, Clustering, and Unequal Weighting on the Variances of NLS Statistics. National Center for Education Statistics. Sponsored Reports Series, Stock No. 017-080-01692-3, GPO. Shah, B.V. (1974). On Mathematics of Population Simulation Models. In B. Dyke and J.W. MacCluer (Eds.), Computer Simulation in Human Population Studies. New York: Academic Press, Inc., pp. 421–434. Shah, B.V., A.V. Rao, Q.W. Lindsey, R.C. Bhavsar, D.G. Horvitz, and J.R. Batts (1974). The Evaluation of Four Alternative Family Planning Programs for Poland, a Less Developed Country. In B. Dyke and J.W. MacCluer (Eds.), Computer Simulation in Human Population Studies. New York: Academic Press, Inc., pp. 261–304. Shah, B.V., D.G. Horvitz, P.A. Lachenbruch, and F.G. Giesbrecht (1971). POPSIM, A Demographic Microsimulation Model. Carolina Population Center, University of North Carolina at Chapel
https://en.wikipedia.org/wiki/Prime%20end	In mathematics, the prime end compactification is a method to compactify a topological disc (i.e. a simply connected open set in the plane) by adding the boundary circle in an appropriate way. Historical notes The concept of prime ends was introduced by Constantin Carathéodory to describe the boundary behavior of conformal maps in the complex plane in geometric terms. The theory has been generalized to more general open sets. The expository paper of provides a good account of this theory with complete proofs: it also introduces a definition which make sense in any open set and dimension. gives an accessible introduction to prime ends in the context of complex dynamical systems. Formal definition The set of prime ends of the domain is the set of equivalence classes of chains of arcs converging to a point on the boundary of . In this way, a point in the boundary may correspond to many points in the prime ends of , and conversely, many points in the boundary may correspond to a point in the prime ends of . Applications Carathéodory's principal theorem on the correspondence between boundaries under conformal mappings can be expressed as follows: If maps the unit disk conformally and one-to-one onto the domain , it induces a one-to-one mapping between the points on the unit circle and the prime ends of . Notes References . , , Compactification (mathematics)
https://en.wikipedia.org/wiki/List%20of%20St%20Kilda%20Football%20Club%20records%20and%20statistics	St Kilda Football Club is a club from Melbourne, Australia. AFL home and away seasons The St Kilda Football Club has fared well since the league became known as the AFL prior to the start of the 1990 premiership season (formerly the VFL – first season 1897). The team have made the finals series eleven times. St Kilda finished second in both 1997 and 2009 after qualifying for the finals series in first position and winning the minor premierships. They were eliminated in preliminary finals in 2004, 2005 and 2008. The club has also played in 5 Pre-Season Cup finals, winning 3 (1996, 2004, 2008) and losing 2 (1998, 2010). Bold text indicates that the club qualified for the finals series in that year. AFL finals series St Kilda Football Club's finals series records since the league changed its name to the AFL: AFL pre-season cups St Kilda Football Club's pre-season cup records: Statistics do not include 1988 to 1992 as data is not available. ^ One win in the pre-season cup was a forfeit by Essendon after they failed to arrive for the game within the designated allowable time. Most goals Most goals kicked by a player while playing with St Kilda Football Club. Record home game attendances In home and away season games: Highest Home Game Attendance during a Home and Away Season at the MCG in St Kilda FC History. Record away game attendances Record crowds when St Kilda were the away team in the club's home city of Melbourne for matches in home and away season games: Record crowds when St Kilda were the away team against teams based outside the state of Victoria in home and away season games: Record scores Highest scores Record highest scores against each opponent: Lowest scores Record lowest scores against each opponent: Lowest scores since 1919 Lowest post-1919 scores against all pre-1919 opponents except University (who disbanded in 1915): Record winning margins Record winning margins in the Australian Football League (formerly the Victorian Football League established 1897). Records set in 2010 are in bold. Most club best and fairest awards: 5 Nick Riewoldt (2002, 2004, 2006–07, 2009) Most consecutive games: 123 Ian Synman (1961–1968) Most seasons as leading goalkicker: 12 Bill Mohr (1929–1940) Most goals: 898 Tony Lockett (1983–1994) Most goals in a season: 132 Tony Lockett (1992) Most goals kicked in a game: 15 Tony Lockett (1992, v Sydney Swans) Most games: 383 Robert Harvey (1988–2008) Most matches as coach: 332 Allan Jeans (1961–1976) Notes References External links St Kilda Football Club official website Records and statistics Australian rules football-related lists ca:Saint Kilda Football Club it:St.Kilda Football Club pl:St Kilda Football Club simple:St Kilda Football Club
https://en.wikipedia.org/wiki/Q0	Q0 may refer to: a graphics file format with extension .q0 Q0, a formulation of higher-order typed logic in mathematics a variable used in a digital counter
https://en.wikipedia.org/wiki/Semi-major%20and%20semi-minor%20axes	In geometry, the major axis of an ellipse is its longest diameter: a line segment that runs through the center and both foci, with ends at the two most widely separated points of the perimeter. The semi-major axis (major semiaxis) is the longest semidiameter or one half of the major axis, and thus runs from the centre, through a focus, and to the perimeter. The semi-minor axis (minor semiaxis) of an ellipse or hyperbola is a line segment that is at right angles with the semi-major axis and has one end at the center of the conic section. For the special case of a circle, the lengths of the semi-axes are both equal to the radius of the circle. The length of the semi-major axis of an ellipse is related to the semi-minor axis's length through the eccentricity and the semi-latus rectum , as follows: The semi-major axis of a hyperbola is, depending on the convention, plus or minus one half of the distance between the two branches. Thus it is the distance from the center to either vertex of the hyperbola. A parabola can be obtained as the limit of a sequence of ellipses where one focus is kept fixed as the other is allowed to move arbitrarily far away in one direction, keeping fixed. Thus and tend to infinity, faster than . The major and minor axes are the axes of symmetry for the curve: in an ellipse, the minor axis is the shorter one; in a hyperbola, it is the one that does not intersect the hyperbola. Ellipse The equation of an ellipse is where (h, k) is the center of the ellipse in Cartesian coordinates, in which an arbitrary point is given by (x, y). The semi-major axis is the mean value of the maximum and minimum distances and of the ellipse from a focus — that is, of the distances from a focus to the endpoints of the major axis In astronomy these extreme points are called apsides. The semi-minor axis of an ellipse is the geometric mean of these distances: The eccentricity of an ellipse is defined as so Now consider the equation in polar coordinates, with one focus at the origin and the other on the direction: The mean value of and , for and is In an ellipse, the semi-major axis is the geometric mean of the distance from the center to either focus and the distance from the center to either directrix. The semi-minor axis of an ellipse runs from the center of the ellipse (a point halfway between and on the line running between the foci) to the edge of the ellipse. The semi-minor axis is half of the minor axis. The minor axis is the longest line segment perpendicular to the major axis that connects two points on the ellipse's edge. The semi-minor axis is related to the semi-major axis through the eccentricity and the semi-latus rectum , as follows: A parabola can be obtained as the limit of a sequence of ellipses where one focus is kept fixed as the other is allowed to move arbitrarily far away in one direction, keeping fixed. Thus and tend to infinity, faster than . The length of the semi-minor axis could also b
https://en.wikipedia.org/wiki/Al-Dimas	Al-Dimas (), also known as Ad-Dimas, is a town in Syria, located west of the capital city of Damascus. According to the Syria Central Bureau of Statistics, the town had a population of 14,574 in the 2004 census. Al-Dimas is the location where a Canadian peacekeeping aircraft crashed after being shot down by three Syrian surface-to-air missiles on August 9, 1974. Climate In Ad Dimas, there is a Mediterranean climate. Rainfall is higher in winter than in summer. The Köppen-Geiger climate classification is Csa. The average annual temperature in Ad Dimas is . About of precipitation falls annually. References Populated places in Qudsaya District Towns in Syria
https://en.wikipedia.org/wiki/Saccheri%E2%80%93Legendre%20theorem	In absolute geometry, the Saccheri–Legendre theorem states that the sum of the angles in a triangle is at most 180°. Absolute geometry is the geometry obtained from assuming all the axioms that lead to Euclidean geometry with the exception of the axiom that is equivalent to the parallel postulate of Euclid. The theorem is named after Giovanni Girolamo Saccheri and Adrien-Marie Legendre. The existence of at least one triangle with angle sum of 180 degrees in absolute geometry implies Euclid's parallel postulate. Similarly, the existence of at least one triangle with angle sum of less than 180 degrees implies the characteristic postulate of hyperbolic geometry. Max Dehn gave an example of a non-Legendrian geometry where the angle sum of a triangle is greater than 180 degrees, and a semi-Euclidean geometry where there is a triangle with an angle sum of 180 degrees but Euclid's parallel postulate fails. In Dehn's geometries the Archimedean axiom does not hold. Notes Euclidean geometry Theorems about triangles Non-Euclidean geometry
https://en.wikipedia.org/wiki/Iterative%20proportional%20fitting	The iterative proportional fitting procedure (IPF or IPFP, also known as biproportional fitting or biproportion in statistics or economics (input-output analysis, etc.), RAS algorithm in economics, raking in survey statistics, and matrix scaling in computer science) is the operation of finding the fitted matrix which is the closest to an initial matrix but with the row and column totals of a target matrix (which provides the constraints of the problem; the interior of is unknown). The fitted matrix being of the form , where and are diagonal matrices such that has the margins (row and column sums) of . Some algorithms can be chosen to perform biproportion. We have also the entropy maximization, information loss minimization (or cross-entropy) or RAS which consists of factoring the matrix rows to match the specified row totals, then factoring its columns to match the specified column totals; each step usually disturbs the previous step’s match, so these steps are repeated in cycles, re-adjusting the rows and columns in turn, until all specified marginal totals are satisfactorily approximated. However, all algorithms give the same solution. In three- or more-dimensional cases, adjustment steps are applied for the marginals of each dimension in turn, the steps likewise repeated in cycles. History IPF has been "re-invented" many times, the earliest by Kruithof in 1937 in relation to telephone traffic ("Kruithof’s double factor method"), Deming and Stephan in 1940 for adjusting census crosstabulations, and G.V. Sheleikhovskii for traffic as reported by Bregman. (Deming and Stephan proposed IPFP as an algorithm leading to a minimizer of the Pearson X-squared statistic, which Stephan later reported it does not). Early proofs of uniqueness and convergence came from Sinkhorn (1964), Bacharach (1965), Bishop (1967), and Fienberg (1970). Bishop's proof that IPFP finds the maximum likelihood estimator for any number of dimensions extended a 1959 proof by Brown for 2x2x2... cases. Fienberg's proof by differential geometry exploits the method's constant crossproduct ratios, for strictly positive tables. Csiszár (1975). found necessary and sufficient conditions for general tables having zero entries. Pukelsheim and Simeone (2009) give further results on convergence and error behavior. An exhaustive treatment of the algorithm and its mathematical foundations can be found in the book of Bishop et al. (1975). Idel (2016) gives a more recent survey. Other general algorithms can be modified to yield the same limit as the IPFP, for instance the Newton–Raphson method and the EM algorithm. In most cases, IPFP is preferred due to its computational speed, low storage requirements, numerical stability and algebraic simplicity. Applications of IPFP have grown to include trip distribution models, Fratar or Furness and other applications in transportation planning (Lamond and Stewart), survey weighting, synthesis of cross-classified demographic data,
https://en.wikipedia.org/wiki/Burkhardt%20quartic	In mathematics, the Burkhardt quartic is a quartic threefold in 4-dimensional projective space studied by , with the maximum possible number of 45 nodes. Definition The equations defining the Burkhardt quartic become simpler if it is embedded in P5 rather than P4. In this case it can be defined by the equations σ1 = σ4 = 0, where σi is the ith elementary symmetric function of the coordinates (x0 : x1 : x2 : x3 : x4 : x5) of P5. Properties The automorphism group of the Burkhardt quartic is the Burkhardt group U4(2) = PSp4(3), a simple group of order 25920, which is isomorphic to a subgroup of index 2 in the Weyl group of E6. The Burkhardt quartic is rational and furthermore birationally equivalent to a compactification of the Siegel modular variety A2(3). References External links 3-folds
https://en.wikipedia.org/wiki/Igusa%20quartic	In algebraic geometry, the Igusa quartic (also called the Castelnuovo–Richmond quartic CR4 or the Castelnuovo–Richmond–Igusa quartic) is a quartic hypersurface in 4-dimensional projective space, studied by . It is closely related to the moduli space of genus 2 curves with level 2 structure. It is the dual of the Segre cubic. It can be given as a codimension 2 variety in P5 by the equations References 3-folds
https://en.wikipedia.org/wiki/Segre%20cubic	In algebraic geometry, the Segre cubic is a cubic threefold embedded in 4 (or sometimes 5) dimensional projective space, studied by . Definition The Segre cubic is the set of points (x0:x1:x2:x3:x4:x5) of P5 satisfying the equations Properties The intersection of the Segre cubic with any hyperplane xi = 0 is the Clebsch cubic surface. Its intersection with any hyperplane xi = xj is Cayley's nodal cubic surface. Its dual is the Igusa quartic 3-fold in P4. Its Hessian is the Barth–Nieto quintic. A cubic hypersurface in P4 has at most 10 nodes, and up to isomorphism the Segre cubic is the unique one with 10 nodes. Its nodes are the points conjugate to (1:1:1:−1:−1:−1) under permutations of coordinates. The Segre cubic is rational and furthermore birationally equivalent to a compactification of the Siegel modular variety A2(2). References 3-folds
https://en.wikipedia.org/wiki/Barth%E2%80%93Nieto%20quintic	In algebraic geometry, the Barth–Nieto quintic is a quintic 3-fold in 4 (or sometimes 5) dimensional projective space studied by that is the Hessian of the Segre cubic. Definition The Barth–Nieto quintic is the closure of the set of points (x0:x1:x2:x3:x4:x5) of P5 satisfying the equations Properties The Barth–Nieto quintic is not rational, but has a smooth model that is a modular Calabi–Yau manifold with Kodaira dimension zero. Furthermore, it is birationally equivalent to a compactification of the Siegel modular variety A1,3(2). References 3-folds
https://en.wikipedia.org/wiki/Mehdi%20Zarghamee	Dr. Mehdi Shaghaghi Zarghamee () is a former Chancellor of Arya Mehr University of Technology (currently Sharif University of Technology) in Iran, former professor at the Department of Mathematics and Computer Sciences, and founder of the Isfahan University of Technology. Dr. Zarghamee currently works as Senior Principal in the Division of Engineering Mechanics and Infrastructure of Simpson Gumpertz & Heger Inc. Most notably, was the principal investigator for the structural modeling of the 9/11 collapse of the World Trade Center Towers for the NIST. Dr. Zarghamee earned his Ph.D. in Structural Engineering from the University of Illinois and his S.M. in Mathematics from the Massachusetts Institute of Technology. References Massachusetts Institute of Technology School of Science alumni University of Illinois alumni Academic staff of Sharif University of Technology Living people Chancellors of the Sharif University of Technology Year of birth missing (living people)
https://en.wikipedia.org/wiki/Coble%20surface	In algebraic geometry, a Coble surface was defined by to be a smooth rational projective surface with empty anti-canonical linear system \|−K\| and non-empty anti-bicanonical linear system \|−2K\|. An example of a Coble surface is the blowing up of the projective plane at the 10 nodes of a Coble curve. References Algebraic surfaces Complex surfaces
https://en.wikipedia.org/wiki/Coble%20curve	In algebraic geometry, a Coble curve is an irreducible degree-6 planar curve with 10 double points (some of them may be infinitely near points). They were studied by . See also Coble surface References Sextic curves
https://en.wikipedia.org/wiki/Coble%20hypersurface	In algebraic geometry, a Coble hypersurface is one of the hypersurfaces associated to the Jacobian variety of a curve of genus 2 or 3 by Arthur Coble. There are two similar but different types of Coble hypersurfaces. The Kummer variety of the Jacobian of a genus 3 curve can be embedded in 7-dimensional projective space under the 2-theta map, and is then the singular locus of a 6-dimensional quartic hypersurface , called a Coble hypersurface. Similarly the Jacobian of a genus 2 curve can be embedded in 8-dimensional projective space under the 3-theta map, and is then the singular locus of a 7-dimensional cubic hypersurface , also called a Coble hypersurface. See also Coble curve (dimension 1) Coble surface (dimension 2) Coble variety (dimension 4) References Abelian varieties
https://en.wikipedia.org/wiki/Coble%20variety	In mathematics, the Coble variety is the moduli space of ordered sets of 6 points in the projective plane, and can be represented as a double cover of the projective 4-space branched over the Igusa quartic. It is a 4-dimensional variety that was first studied by Arthur Coble. See also Coble curve Coble surface Coble hypersurface References Algebraic varieties
https://en.wikipedia.org/wiki/Corner-point%20grid	In geometry, a corner-point grid is a tessellation of a Euclidean 3D volume, where the base cell has 6 faces (hexahedron). A set of straight lines defined by their end points define the pillars of the corner-point grid. The pillars have a lexicographical ordering that determines neighbouring pillars. On each pillar, a constant number of nodes (corner-points) is defined. A corner-point cell is now the volume between 4 neighbouring pillars and two neighbouring points on each pillar. Each cell can be identified by integer coordinates , where the coordinate runs along the pillars, and and span each layer. The cells are ordered naturally, where the index runs the fastest and the slowest. Data within the interior of such cells can be computed by trilinear interpolation from the boundary values at the 8 corners, 12 edges, and 6 faces. In the special case of all pillars being vertical, the top and bottom face of each corner-point cell are described by bilinear surfaces and the side faces are planes. Corner-point grids are supported by most reservoir simulation software, and has become an industry standard. Degeneracy A main feature of the format is the ability to define erosion surfaces in geological modelling, effectively done by collapsing nodes along each pillar. This means that the corner-point cells degenerate and may have less than 6 faces. For the corner-point grids, non-neighboring connections are supported, meaning that grid cells that are not neighboring in ijk-space can be defined as neighboring. This feature allows for representation of faults with significant throw/displacement. Moreover, the neighboring grid cells do not need to have matching cell faces (just overlap). References Corner Point Grid. Open Porous Media Initiative Aarnes J, Krogstad S and Lie KA (2006). Multiscale Mixed/Mimetic Methods on Corner Point Grids. SINTEF ICT, Dept. Applied Mathematics Tessellation Geometry
https://en.wikipedia.org/wiki/Dana%20Tomlin	Charles Dana Tomlin is an author, professor, and originator of Map Algebra, a vocabulary and conceptual framework for classifying ways to combine map data to produce new maps. Tomlin's teaching and research focus on the development and application of geographic information systems (GIS). He is currently a professor at the University of Pennsylvania School of Design and an adjunct professor at the Yale School of Forestry and Environmental Studies, having also taught at the Harvard Graduate School of Design and the Ohio State University School of Natural Resources. His coursework in Landscape Architecture has extensively included GIS and cartographic modeling applications. Contributions to GIS Tomlin's contributions to GIS extend across a number of years and a wide variety of applications. As a student at Harvard University in the mid-1970s, he developed the Tomlin Subsystem of IMGRID as a master's thesis. Many analytical functions in IMGRID were later integrated into Imagine, a satellite image processing application developed by ERDAS. As a doctoral student at Yale University in the late 1970s, and as a junior faculty member at Harvard in the early 1980s, Tomlin developed MAP (the Map Analysis Package), one of the most widely used programs of its kind. The open source GRASS application derives many of its raster analytical capabilities directly from MAP and was extensively used by the U.S. Army Corps of Engineers and other federal agencies throughout the late 1980s. Tomlin's work on MAP has also been directly inherited by a long list of other software packages, including, OSUMAP, MAP II, MapFactory, MFWorks, MacGIS, IDRISI, MapBox, pMap, and MGE. In 1990, Tomlin led an informal group of City and Regional Planning doctoral students at the University of Pennsylvania in founding the Cartographic Modeling Laboratory. The Cartographic Modeling Lab conducts academic research and urban and social policy analysis using GIS and spatial research applications. Tomlin has been co-director of the lab since 1995. Map algebra Tomlin's landmark book, Geographic Information Systems and Cartographic Modeling, was published in 1990 to expand on his earlier dissertation work on Map Algebra. A significantly revised version was released as GIS and Cartographic Modeling in 2012. Map Algebra is used for a broad array of GIS modeling applications, including suitability modeling, surface analysis, density analysis, statistics, hydrology, landscape ecology, real estate and geographic prioritization. Early in its development, Tomlin made the decision to openly share all of the source code, documentation and algorithms associated with Map Algebra. Consequently, the overall concepts of Map Algebra are still used today in every GIS application that supports raster calculations. Esri’s Spatial Analyst solution, as well as its predecessor, the GRID module of ArcInfo, has incorporated most of the Map Algebra concepts. While primarily applied to raster data, Map Alg
https://en.wikipedia.org/wiki/Kummer%20variety	In mathematics, the Kummer variety of an abelian variety is its quotient by the map taking any element to its inverse. The Kummer variety of a 2-dimensional abelian variety is called a Kummer surface. References Abelian varieties
https://en.wikipedia.org/wiki/Isserlis%27%20theorem	In probability theory, Isserlis' theorem or Wick's probability theorem is a formula that allows one to compute higher-order moments of the multivariate normal distribution in terms of its covariance matrix. It is named after Leon Isserlis. This theorem is also particularly important in particle physics, where it is known as Wick's theorem after the work of . Other applications include the analysis of portfolio returns, quantum field theory and generation of colored noise. Statement If is a zero-mean multivariate normal random vector, thenwhere the sum is over all the pairings of , i.e. all distinct ways of partitioning into pairs , and the product is over the pairs contained in . More generally, if is a zero-mean complex-valued multivariate normal random vector, then the formula still holds. The expression on the right-hand side is also known as the hafnian of the covariance matrix of . Odd case If is odd, there does not exist any pairing of . Under this hypothesis, Isserlis' theorem implies that This also follows from the fact that has the same distribution as , which implies that . Even case In his original paper, Leon Isserlis proves this theorem by mathematical induction, generalizing the formula for the order moments, which takes the appearance If is even, there exist (see double factorial) pair partitions of : this yields terms in the sum. For example, for order moments (i.e. random variables) there are three terms. For -order moments there are terms, and for -order moments there are terms. Proof Since the formula is linear on both sides, if we can prove the real case, we get the complex case for free. Let be the covariance matrix, so that we have the zero-mean multivariate normal random vector . Since both sides of the formula are continuous with respect to , it suffices to prove the case when is invertible. Using quadratic factorization , we get Differentiate under the integral sign with to obtain . That is, we need only find the coefficient of term in the Taylor expansion of . If is odd, this is zero. So let , then we need only find the coefficient of term in the polynomial . Expand the polynomial and count, we obtain the formula. Generalizations Gaussian integration by parts An equivalent formulation of the Wick's probability formula is the Gaussian integration by parts. If is a zero-mean multivariate normal random vector, then This is a generalization of Stein's lemma. The Wick's probability formula can be recovered by induction, considering the function defined by . Among other things, this formulation is important in Liouville conformal field theory to obtain conformal Ward identities, BPZ equations and to prove the Fyodorov-Bouchaud formula. Non-Gaussian random variables For non-Gaussian random variables, the moment-cumulants formula replaces the Wick's probability formula. If is a vector of random variables, then where the sum is over all the partitions of , the product is over the
https://en.wikipedia.org/wiki/Francesco%20Stelluti	Francesco Stelluti (12 January 1577, in Fabriano – November 1652, in Rome) was an Italian polymath who worked in the fields of mathematics, microscopy, literature, and astronomy. Along with Federico Cesi, Anastasio de Filiis and Johannes van Heeck, he founded the Accademia dei Lincei in August 1603. Early life Francesco was the son of Bernardino Stelluti and his wife Lucrezia Corradini. He went to Rome at a young age to study law. Once he had completed his studies he began to work in the law, which he practised all his life, while also dedicating himself to literary and scientific studies. In the Accademia, he was first appointed Grand Counsellor, with the task of teaching mathematics, geometry and astronomy to the other members. Later he was appointed proponitor (lecturer) in machines and mathematical instruments, and provisor (supervisor) and calculator of the motions of the stars. His pseudonym in the Accademia was Tardigrado (slow stepper), reflecting his character as a quiet, studious man, careful and versatile. His protecting star was Saturn, from which he was said to draw his capacity for reflection and speculation, as well as his motto, Quo serius eo citius (“the slower, the swifter”). In 1604 he authored the Logicae Physicae et Metaphysicae Brevissimum Compendium. Like his companions he faced hostility from the family of Prince Cesi because of the creation of the Accademia dei Lincei and was compelled to leave Rome for several years, going first to Fabriano and then to the Farnese court in Parma. He returned to Rome in 1609, and took an active role in the development of the Accademia. In 1610 he went to Naples with Cesi to establish a branch of the Accademia there, to be run by Giambattista della Porta and he also began his life’s work of editing the “Tesoro Messicano”, which contained the records gathered in Mexico by naturalist Francisco Hernández de Toledo in the 1570s. In 1612 he was elected procuratore generale of the Accademia. Later life In 1625 he and Federico Cesi printed in broadsheet (or broadside) form the work Apiarium in 1625, marking the first published microscopic revelations of biological structures. The broadsheet contained an illustration of three bees - a design of three bees was the family crest of the new pope, Urban VIII. Stelluti’s Persio tradotto in verso sciolto e dichiarato ("[Works of Aulus] Persius [Flaccus] translated into light verse and annotated [lit. 'declared' in the sense of 'remarked/commented upon']"), published in Rome in 1630, is the first book published in codex form to contain images of organisms viewed through the microscope. Subsequently, in 1637 he published a work on fossilised wood, apparently also with the aid of magnifying instruments and he finally published the “Tesoro Messicano” in 1651. The Tesoro Messicano (“Mexican Treasure”) or more precisely the Rerum Medicarum Novae Hispaniae Thesaurus, was the final work of the Accademia. It represented nearly half a century of collaborati
https://en.wikipedia.org/wiki/Mixed%20motive	Mixed motive may refer to: Mixed motives (math), Voevodsky's construction of the derived category of algebraic geometric mixed motives "Mixed motive" discrimination, in constitutional law
https://en.wikipedia.org/wiki/Weddle%20surface	In algebraic geometry, a Weddle surface, introduced by , is a quartic surface in 3-dimensional projective space, given by the locus of vertices of the family of cones passing through 6 points in general position. Weddle surfaces have 6 nodes and are birational to Kummer surfaces. References Algebraic surfaces Complex surfaces
https://en.wikipedia.org/wiki/Pl%C3%BCcker%20surface	In algebraic geometry, a Plücker surface, studied by , is a quartic surface in 3-dimensional projective space with a double line and 8 nodes. Construction For any quadric line complex, the lines of the complex in a plane envelop a quadric in the plane. A Plücker surface depends on the choice of a quadric line complex and a line, and consists of points of the quadrics associated to the planes through the chosen line. References Algebraic surfaces
https://en.wikipedia.org/wiki/3D	3-D, 3D, or 3d may refer to: Science, technology, and mathematics Relating to three-dimensionality Three-dimensional space 3D computer graphics, computer graphics that use a three-dimensional representation of geometric data 3D film, a motion picture that gives the illusion of three-dimensional perception 3D modeling, developing a representation of any three-dimensional surface or object 3D printing, making a three-dimensional solid object of a shape from a digital model 3D display, a type of information display that conveys depth to the viewer 3D television, television that conveys depth perception to the viewer Stereoscopy, any technique capable of recording three-dimensional visual information or creating the illusion of depth in an image Other uses in science and technology or commercial products 3D projection 3D rendering 3D scanning, making a digital representation of three-dimensional objects 3D video game (disambiguation) 3-D Secure, a secure protocol for online credit and debit card transactions Biela's Comet, a lost periodic comet discovered in 1826 British Rail Class 207, sometimes known as 3Ds Music Artists Robert Del Naja (born 1965), also known as 3D, English artist and musician in the band Massive Attack The 3Ds, a rock band Albums 3D (Go West album), 2010 "Weird Al" Yankovic in 3-D, 1984, sometimes simply referred to as 3-D 3D (The Three Degrees album), 1979 3-D (I See Stars album), the debut album from the band I See Stars 3-D (SPC ECO album) 3-D (TLC album), 2002 3-D (Wrathchild America album) Songs "3D" (song), a 2023 song by Jungkook featuring Jack Harlow "3-D", a song by Cheap Trick from their 1983 album Next Position Please Other uses 3D Aerobatics, a form of flying using flying aircraft to perform specific aerial maneuvers 3D (Long Island bus), bus service in New York State 3D Test of Antisemitism, put forth by Israeli politician and human rights activist Natan Sharansky Middle finger, the third digit (abbreviated 3D) of the hand Three-dimensional chess Threepence (disambiguation), a coin used in several countries, abbreviated as '3d' The 3D ("Dudley Death Drop"), a professional wrestling double-team maneuver See also 3ds (disambiguation) D3 (disambiguation) DDD (disambiguation) 3rd (disambiguation)
https://en.wikipedia.org/wiki/Specht%27s%20theorem	In mathematics, Specht's theorem gives a necessary and sufficient condition for two complex matrices to be unitarily equivalent. It is named after Wilhelm Specht, who proved the theorem in 1940. Two matrices A and B with complex number entries are said to be unitarily equivalent if there exists a unitary matrix U such that B = U AU. Two matrices which are unitarily equivalent are also similar. Two similar matrices represent the same linear map, but with respect to a different basis; unitary equivalence corresponds to a change from an orthonormal basis to another orthonormal basis. If A and B are unitarily equivalent, then tr AA = tr BB, where tr denotes the trace (in other words, the Frobenius norm is a unitary invariant). This follows from the cyclic invariance of the trace: if B = U AU, then tr BB* = tr U AUU AU = tr AUU AUU = tr AA, where the second equality is cyclic invariance. Thus, tr AA = tr BB* is a necessary condition for unitary equivalence, but it is not sufficient. Specht's theorem gives infinitely many necessary conditions which together are also sufficient. The formulation of the theorem uses the following definition. A word in two variables, say x and y, is an expression of the form where m1, n1, m2, n2, …, mp are non-negative integers. The degree of this word is Specht's theorem: Two matrices A and B are unitarily equivalent if and only if tr W(A, A) = tr W(B, B) for all words W. The theorem gives an infinite number of trace identities, but it can be reduced to a finite subset. Let n denote the size of the matrices A and B. For the case n = 2, the following three conditions are sufficient: For n = 3, the following seven conditions are sufficient: For general n, it suffices to show that tr W(A, A) = tr W(B, B) for all words of degree at most It has been conjectured that this can be reduced to an expression linear in n. Notes References . . . . . . Matrix theory Combinatorics on words Theorems in linear algebra
https://en.wikipedia.org/wiki/Line%20complex	In algebraic geometry, a line complex is a 3-fold given by the intersection of the Grassmannian G(2, 4) (embedded in projective space P5 by Plücker coordinates) with a hypersurface. It is called a line complex because points of G(2, 4) correspond to lines in P3, so a line complex can be thought of as a 3-dimensional family of lines in P3. The linear line complex and quadric line complex are the cases when the hypersurface has degree 1 or 2; they are both rational varieties. References Algebraic varieties 3-folds
https://en.wikipedia.org/wiki/Spherical%20image	In differential geometry, the spherical image of a unit-speed curve is given by taking the curve's tangent vectors as points, all of which must lie on the unit sphere. The movement of the spherical image describes the changes in the original curve's direction If is a unit-speed curve, that is , and is the unit tangent vector field along , then the curve is the spherical image of . All points of must lie on the unit sphere because . References Differential geometry
https://en.wikipedia.org/wiki/Restricted%20isometry%20property	In linear algebra, the restricted isometry property (RIP) characterizes matrices which are nearly orthonormal, at least when operating on sparse vectors. The concept was introduced by Emmanuel Candès and Terence Tao and is used to prove many theorems in the field of compressed sensing. There are no known large matrices with bounded restricted isometry constants (computing these constants is strongly NP-hard, and is hard to approximate as well), but many random matrices have been shown to remain bounded. In particular, it has been shown that with exponentially high probability, random Gaussian, Bernoulli, and partial Fourier matrices satisfy the RIP with number of measurements nearly linear in the sparsity level. The current smallest upper bounds for any large rectangular matrices are for those of Gaussian matrices. Web forms to evaluate bounds for the Gaussian ensemble are available at the Edinburgh Compressed Sensing RIC page. Definition Let A be an m × p matrix and let 1 ≤ s ≤ p be an integer. Suppose that there exists a constant such that, for every m × s submatrix As of A and for every s-dimensional vector y, Then, the matrix A is said to satisfy the s-restricted isometry property with restricted isometry constant . This condition is equivalent to the statement that for every m × s submatrix As of A we have where is the identity matrix and is the operator norm. See for example for a proof. Finally this is equivalent to stating that all eigenvalues of are in the interval . Restricted Isometric Constant (RIC) The RIC Constant is defined as the infimum of all possible for a given . It is denoted as . Eigenvalues For any matrix that satisfies the RIP property with a RIC of , the following condition holds: . The tightest upper bound on the RIC can be computed for Gaussian matrices. This can be achieved by computing the exact probability that all the eigenvalues of Wishart matrices lie within an interval. See also Compressed sensing Mutual coherence (linear algebra) Terence Tao's website on compressed sensing lists several related conditions, such as the 'Exact reconstruction principle' (ERP) and 'Uniform uncertainty principle' (UUP) Nullspace property, another sufficient condition for sparse recovery Generalized restricted isometry property, a generalized sufficient condition for sparse recovery, where mutual coherence and restricted isometry property are both its special forms. Johnson-Lindenstrauss lemma References Signal processing Linear algebra
https://en.wikipedia.org/wiki/Siberian%20Mathematical%20Journal	The Siberian Mathematical Journal (abbreviated as Sib. Math. J.) is a cover-to-cover English translation of the Russian peer-reviewed mathematics journal Sibirskii Matematicheskii Zhurnal, a publication of the Sobolev Institute of Mathematics of the Siberian Division of the Russian Academy of Sciences (Novosibirsk). Sibirskii Matematicheskii Zhurnal was established in 1960 and the Siberian Mathematical Journal was launched in 1966. It is published by Springer Science+Business Media. The journal publishes research papers in all branches of mathematics, including functional analysis, differential equations, algebra and logic, geometry and topology, probability theory and mathematical statistics, ill-posed problems of mathematical physics, computational methods of linear algebra, etc. External links Print: Online: Mathematics journals Academic journals established in 1960 Magazines published in Novosibirsk
https://en.wikipedia.org/wiki/Ralph%20Button	Ralph Button (died 1680) was an English academic and clergyman, Gresham Professor of Geometry, canon of Christ Church, Oxford under the Commonwealth, and later a nonconformist schoolmaster. Life He was the son of Robert Button of Bishopstone, Wiltshire, and was educated at Exeter College, Oxford. He proceeded B.A. in 1630; in 1633 the Rector of Exeter, John Prideaux, recommended him to Sir Nathaniel Brent, the Warden of Merton College, for a fellowship in his college. The fellowship was conferred on him, and he became known in the university as a successful tutor. Among his pupils were Zachary Bogan, Anthony à Wood, and John Murcot. On the outbreak of the First English Civil War in 1642, Button, who sympathised with the parliamentarians, moved to London, and on 15 November 1643 was elected Professor of Geometry at Gresham College, in the place of John Greaves. In 1647 he was nominated a delegate to aid the parliamentary visitors at Oxford in their work of reform, and apparently resumed his tutorship at Merton. On 18 February. 1648 Button was appointed by the visitors junior proctor; on 11 April he pronounced a Latin oration before Philip Herbert, 4th Earl of Pembroke, the new chancellor of the university, and on 13 June he resigned his Gresham professorship. On 4 August he was made canon of Christ Church and public orator of the university, in the place of Henry Hammond, who had been removed by the parliamentary commission. At the same time Button declined to supplicate for the degree of D.D. on the ground of the expense; Wood says that he had then lately married. Button showed independence in successfully resisting the endeavour of the visitors to expel Edward Pocock from the Hebrew and Arabic lectureship on the ground of political disaffection. At the Restoration Button was ejected from all his offices and his place at Christ Church was taken by John Fell. Leaving Oxford, he went to Brentford, where he kept a school; he taught alongside Thomas Pakeman who was his neighbour. Richard Baxter says that he was soon afterwards imprisoned for six months for teaching and not having taken the Oxford oath. At the date of the Declaration of Indulgence (1672) Button moved to Islington, and Joseph Jekyll lived with him as his pupil. He died at Islington in October 1680, and was buried in the parish church. A son died and was buried at the same time. Baxter in Reliquiae Baxterianae speaks highly of him. He left a daughter, who married Dr. Boteler of London. Notes References Year of birth missing 1680 deaths Ejected English ministers of 1662 Dissenting academy tutors Alumni of Exeter College, Oxford Fellows of Merton College, Oxford

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