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https://en.wikipedia.org/wiki/History%20of%20Grandi%27s%20series
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Geometry and infinite zeros
Grandi
Guido Grandi (1671–1742) reportedly provided a simplistic account of the series in 1703. He noticed that inserting parentheses into produced varying results: either
or
Grandi's explanation of this phenomenon became well known for its religious overtones:
In fact, the series was not an idle subject for Grandi, and he didn't think it summed to either 0 or 1. Rather, like many mathematicians to follow, he thought the true value of the series was 1⁄2 for a variety of reasons.
Grandi's mathematical treatment of occurs in his 1703 book Quadratura circula et hyperbolae per infinitas hyperbolas geometrice exhibita. Broadly interpreting Grandi's work, he derived through geometric reasoning connected with his investigation of the witch of Agnesi. Eighteenth-century mathematicians immediately translated and summarized his argument in analytical terms: for a generating circle with diameter a, the equation of the witch y = a3/(a2 + x2) has the series expansion
and setting a = x = 1, one has 1 − 1 + 1 − 1 + · · · = 1⁄2.
According to Morris Kline, Grandi started with the binomial expansion
and substituted x = 1 to get . Grandi "also argued that since the sum was both 0 and 1⁄2, he had proved that the world could be created out of nothing."
Grandi offered a new explanation that in 1710, both in the second edition of the Quadratura circula and in a new work, De Infinitis infinitorum, et infinite parvorum ordinibus disquisitio geometrica. Two brothers inherit a priceless gem from their father, whose will forbids them to sell it, so they agree that it will reside in each other's museums on alternating years. If this agreement lasts for all eternity between the brother's descendants, then the two families will each have half possession of the gem, even though it changes hands infinitely often. This argument was later criticized by Leibniz.
The parable of the gem is the first of two additions to the discussion of the corollary that Grandi added to the second edition. The second repeats the link between the series and the creation of the universe by God:
Marchetti
After Grandi published the second edition of the Quadratura, his fellow countryman Alessandro Marchetti became one of his first critics. One historian charges that Marchetti was motivated more by jealousy than any other reason. Marchetti found the claim that an infinite number of zeros could add up to a finite quantity absurd, and he inferred from Grandi's treatment the danger posed by theological reasoning. The two mathematicians began attacking each other in a series of open letters; their debate was ended only by Marchetti's death in 1714.
Leibniz
With the help and encouragement of Antonio Magliabechi, Grandi sent a copy of the 1703 Quadratura to Leibniz, along with a letter expressing compliments and admiration for the master's work. Leibniz received and read this first edition in 1705, and he called it an unoriginal and less-advanced "attempt" at his c
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https://en.wikipedia.org/wiki/International%20Commission%20on%20Mathematical%20Instruction
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The International Commission on Mathematical Instruction (ICMI) is a commission of the International Mathematical Union and is an internationally acting organization focussing on mathematics education. ICMI was founded in 1908 at the International Congress of Mathematicians (ICM) in Rome and aims to improve teaching standards around the world, through programs, workshops and initiatives and publications. It aims to work a great deal with developing countries, to increase teaching standards and education which can improve life quality and aid the country.
History
ICMI was founded at the ICM, and mathematician Felix Klein was elected first president of the organisation. Henri Fehr and Charles Laisant created the international research journal L'Enseignement Mathématique in 1899, and from early on this journal became the official organ of ICMI. A bulletin is published twice a year by ICMI, and from December 1995 this bulletin has been available at the organisation's official website, in their 'digital library'.
In the years between World War I and World War II there was little activity in the organization, but in 1952 ICMI was reconstituted. At this time the organization was reorganized, and it became an official commission of the International Mathematical Union (IMU). As a scientific organization, IMU is a member of the International Council for Science (ICSU). Although ICMI follows the general principles of IMU and ICSU, the organization has a large degree of autonomy.
Structure
All countries that are members of IMU are automatically members of ICMI; membership is also possible for non-IMU members. Currently, there are 90 member states of ICMI. Each member state has the right to appoint a national representative.
As a commission, ICMI has two main bodies:
the Executive Committee (EC), and
the national representatives from the member countries.
Together, these two constitute the General Assembly (GA) of ICMI. The GA is summoned every four years in connection with the International Congress on Mathematical Education, ICME. The executive committee is appointed by the general assembly of IMU for four-year terms.
Affiliate Organisations
These include multi-national organisations, which are independent from ICMI and have interests in the field of mathematics.
There are currently four multinational Mathematical Education Societies:
CIAEM: Inter-American Committee on Mathematics Education (2009)
CIEAEM: International Commission for the Study and Improvement of Mathematics Teaching (2010)
ERME: European Society for Research in Mathematics Education (2010)
MERGA: Mathematics Education Research Group of Australasia (2011)
And six international Study Groups which have obtained affiliation with ICMI:
HPM: The International Study Group on the Relations between the History and Pedagogy of Mathematics (1976)
ICTMA: The International Study Group for Mathematical Modelling and Applications (2003)
IOWME: The International Organization of Women
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https://en.wikipedia.org/wiki/Fedor%20Bogomolov
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Fedor Alekseyevich Bogomolov (born 26 September 1946) (Фёдор Алексеевич Богомолов) is a Russian and American mathematician, known for his research in algebraic geometry and number theory. Bogomolov worked at the Steklov Institute in Moscow before he became a professor at the Courant Institute in New York. He is most famous for his pioneering work on hyperkähler manifolds.
Born in Moscow, Bogomolov graduated from Moscow State University, Faculty of Mechanics and Mathematics, and earned his doctorate ("candidate degree") in 1973, at the Steklov Institute. His doctoral advisor was Sergei Novikov.
Geometry of Kähler manifolds
Bogomolov's Ph.D. thesis was entitled Compact Kähler varieties. In his early papers Bogomolov studied the manifolds which were later called Calabi–Yau and hyperkähler. He proved a decomposition theorem, used for the classification of manifolds with trivial canonical class. It has been re-proven using the Calabi–Yau theorem and Berger's classification of Riemannian holonomies, and is foundational for modern string theory.
In the late 1970s and early 1980s Bogomolov studied the deformation theory for manifolds with trivial canonical class. He discovered what is now known as Bogomolov–Tian–Todorov theorem, proving the smoothness and un-obstructedness of the deformation space for hyperkaehler manifolds (in 1978 paper) and then extended this to all Calabi–Yau manifolds in the 1981 IHES preprint. Some years later, this theorem became the mathematical foundation for Mirror Symmetry.
While studying the deformation theory of hyperkähler manifolds, Bogomolov discovered what is now known as the Bogomolov–Beauville–Fujiki form on . Studying properties of this form, Bogomolov erroneously concluded that compact hyperkaehler manifolds do not exist, with the exception of K3 surfaces, tori, and their products. Almost four years passed since this publication before Akira Fujiki found a counterexample.
Other works in algebraic geometry
Bogomolov's paper on "Holomorphic tensors and vector bundles on projective manifolds" proves what is now known as the Bogomolov–Miyaoka–Yau inequality, and also proves that a stable bundle on a surface, restricted to a curve of sufficiently big degree, remains stable. In "Families of curves on a surface of general type", Bogomolov laid the foundations to the now popular approach to the theory of diophantine equations through geometry of hyperbolic manifolds and dynamical systems. In this paper Bogomolov proved that on any surface of general type with , there is only a finite number of curves of bounded genus. Some 25 years later, Michael McQuillan extended this argument to prove the famous Green–Griffiths conjecture for such surfaces. In "Classification of surfaces of class with ", Bogomolov made the first step in a famously difficult (and still unresolved) problem of classification of surfaces of Kodaira class VII. These are compact complex surfaces with . If they are in addition minimal, they are called c
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https://en.wikipedia.org/wiki/Palmer%20High%20School%20%28Massachusetts%29
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Palmer High School is a public high school located in the city of Palmer, Massachusetts, United States.
Demographics and statistics
For the 20152016 school year, Palmer High School enrolled 489 students in grades 8 through 12. Out of these students, 88.8% were Caucasian, 2.0% were African American, 5.1% were Hispanic, 0.0% were Native American, 0.2% were Native Hawaiian/Pacific Islander and 1.4% were Multi-Ethnic. About 52.1% of the student population is male, while 47.9% is female.
Notable alumni
Todd Smola, member of Massachusetts House of Representatives (class of 1995)
References
High schools in Hampden County, Massachusetts
Public high schools in Massachusetts
Palmer, Massachusetts
1991 establishments in Massachusetts
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https://en.wikipedia.org/wiki/List%20of%20Juventus%20FC%20records%20and%20statistics
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Juventus Football Club is an Italian professional association football club based in Turin, Piedmont that competes in Serie A, the top football league in the country. The club was formed in 1897 as Sport Club Juventus by a group of Massimo d'Azeglio Lyceum young students and played its first competitive match on 11 March 1900, when it entered the Piedmont round of the third Federal Championship.
This list encompasses the major honours won by Juventus and records set by the club, their managers and their players. The individual records section includes details of the club's leading goalscorers and those who have made most appearances in first-team competitions. The club's players have received, among others, a record twelve Serie A Footballer of the Year, the award given by the Italian Footballers' Association (AIC), eight Ballon d'Or awards and four FIFA World Player of the Year awards, more than any other Italian club and third overall in the latter two cases.
Honours
Italy's most successful club of the 20th century with the most title in the history of Italian football, Juventus have won the Italian League Championship, the country's premier football club competition and organised by Lega Nazionale Professionisti Serie A (LNPA), a record 36 times and have the record of consecutive triumphs in that tournament (nine, between 2011–12 and 2019–20). They have also won the Coppa Italia, the country's primary single-elimination competition, a record fourteen times, becoming the first team to retain the trophy successfully with their triumph in the 1959–60 season, and the first to win it in three consecutive seasons from the 2014–15 season to the 2016–17 season, going on to win a fourth consecutive title in 2017–18 (also a record). In addition, the club holds the record for Supercoppa Italiana wins with nine, the most recent coming in 2020.
Overall, Juventus have won 70 official competitions, more than any other club in the country: 59 at national level (which is also a record) and eleven at international stage, making them, in the latter case, the second most successful Italian team. The club is currently sixth in Europe and twelfth in the world with the most international titles won officially recognised by their respective continental football confederation and Fédération Internationale de Football Association (FIFA). In 1977, the Torinese side become the first in Southern Europe to have won the UEFA Cup and the first—and only to date—in Italian football history to achieve an international title with a squad composed by national footballers. In 1993, the club won its third competition's trophy, an unprecedented feat in the continent until then, a confederation record for the next 22 years and the most for an Italian team. Juventus was also the first club in the country to achieve the title in the European Super Cup, having won the competition in 1984, and the first European side to win the Intercontinental Cup in 1985, since it was restructured
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https://en.wikipedia.org/wiki/Subspace%20theorem
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In mathematics, the subspace theorem says that points of small height in projective space lie in a finite number of hyperplanes. It is a result obtained by .
Statement
The subspace theorem states that if L1,...,Ln are linearly independent linear forms in n variables with algebraic coefficients and if ε>0 is any given real number, then
the non-zero integer points x with
lie in a finite number of proper subspaces of Qn.
A quantitative form of the theorem, which determines the number of subspaces containing all solutions, was also obtained by Schmidt, and the theorem was generalised by to allow more general absolute values on number fields.
Applications
The theorem may be used to obtain results on Diophantine equations such as Siegel's theorem on integral points and solution of the S-unit equation.
A corollary on Diophantine approximation
The following corollary to the subspace theorem is often itself referred to as the subspace theorem.
If a1,...,an are algebraic such that 1,a1,...,an are linearly independent over Q and ε>0 is any given real number, then there are only finitely many rational n-tuples (x1/y,...,xn/y) with
The specialization n = 1 gives the Thue–Siegel–Roth theorem. One may also note that the exponent 1+1/n+ε is best possible by Dirichlet's theorem on diophantine approximation.
References
Diophantine approximation
Theorems in number theory
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https://en.wikipedia.org/wiki/Georges%20Ifrah
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Georges Ifrah (1947 – 1 November 2019) was a teacher of mathematics, a French author and a self-taught historian of mathematics, especially numerals.
His work, From One to Zero: A Universal History of Numbers (1985, 1994) was translated into multiple languages, became an international bestseller, and was included in American Scientist'''s list of "100 or so Books that shaped a Century of Science", referring to the 20th century. Despite popular acclaim, it has been broadly criticized by scholars.C. Philipp E. Nothaft: Medieval Europe’s satanic ciphers: on the genesis of a modern myth. British Journal for the History of Mathematics 35, 2020, doi:10.1080/26375451.2020.1726050.
Publications
Several books devoted to numbers and history of numbers and number related topics including:
1981: Histoire Universelle des Chiffres (Paris)
English translation (1985): From one to zero. A universal history of numbers transl. by Lowell Bair. New York: Viking Penguin Inc. XVI, 503 pages. (Zentralblatt review: 0589.01001: "It is the richness in documents from both primitive and advanced cultures, which makes this publication unique.[…]a number of authors mentioned in the text are not cited in this bibliography. And in many cases the sources of illustrations remain anonymous".)
German translation (1986): Universalgeschichte der Zahlen transl. by Alexander von Platen. Frankfurt/New York: Campus Verlag. 580 pages. (Zentralblatt review 0606.01023.)
German translation (1989): Universalgeschichte der Zahlen. 600 pages. (Additional introduction and indices.) (Zentralblatt review: 0686.01001.)
Italian translation (1983): Storia universale dei numeri. Milano: Mondadori. 585 pages.
1985: Les chiffres ou l'histoire d'une grande invention Robert Laffont
"The history of numbers or the history of a great discovery" (abridged version? ≈260 pages)
Polish translation (1990): Dzieje liczby czyli historia wielkiego wynalazku translated by Stanisław Hartman. Wrocław: Zakład Narodowy im. Ossolińskich-Wydawnictwo Polskiej Akademii Nauk (Ossolineum). 260 pages. . (Zentralblatt review: 0758.01017.)
1994: Histoire universelle des chiffres, 2nd edition. (Seghers, puis Bouquins, Robert Laffont, 1994)
Now in two volumes: Vol I 633 pages (Zentralblatt review: 0955.01002), Vol II 412 pages (Zentralblatt review: 0969.68001).
Norwegian translation (1997): All verdens tall. Tallenes kulturhistorie. I, II. Translated by Anne Falken, Guri Haarr, Margrethe Kvarenes and Svanhild Solløs. Oslo: Pax Forlag. 1284 p. (1997). (set); (vol.1); (vol.2). (Zentralblatt review: 0933.01001)
English translation (1998): Universal History of Numbers: From Prehistory to the Invention of the Computer. Translated by David Bellos, E.F. Harding, Sophie Wood and Ian Monk. Harville Press, London, 1998 ().
American edition of English tr., Volume 1 (2000): The Universal History of Numbers: From prehistory to the invention of the computer. Translated by David Bellos, E.F. Harding, Sophie Wood and Ian Monk
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https://en.wikipedia.org/wiki/Heptagonal%20tiling
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In geometry, a heptagonal tiling is a regular tiling of the hyperbolic plane. It is represented by Schläfli symbol of {7,3}, having three regular heptagons around each vertex.
Images
Related polyhedra and tilings
This tiling is topologically related as a part of sequence of regular polyhedra with Schläfli symbol {n,3}.
From a Wythoff construction there are eight hyperbolic uniform tilings that can be based from the regular heptagonal tiling.
Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms.
Hurwitz surfaces
The symmetry group of the tiling is the (2,3,7) triangle group, and a fundamental domain for this action is the (2,3,7) Schwarz triangle. This is the smallest hyperbolic Schwarz triangle, and thus, by the proof of Hurwitz's automorphisms theorem, the tiling is the universal tiling that covers all Hurwitz surfaces (the Riemann surfaces with maximal symmetry group), giving them a tiling by heptagons whose symmetry group equals their automorphism group as Riemann surfaces. The smallest Hurwitz surface is the Klein quartic (genus 3, automorphism group of order 168), and the induced tiling has 24 heptagons, meeting at 56 vertices.
The dual order-7 triangular tiling has the same symmetry group, and thus yields triangulations of Hurwitz surfaces.
See also
Hexagonal tiling
Tilings of regular polygons
List of uniform planar tilings
List of regular polytopes
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
External links
Hyperbolic and Spherical Tiling Gallery
KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
Hyperbolic Planar Tessellations, Don Hatch
Hyperbolic tilings
Isogonal tilings
Isohedral tilings
Regular tilings
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https://en.wikipedia.org/wiki/Order-7%20triangular%20tiling
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In geometry, the order-7 triangular tiling is a regular tiling of the hyperbolic plane with a Schläfli symbol of {3,7}.
Hurwitz surfaces
The symmetry group of the tiling is the (2,3,7) triangle group, and a fundamental domain for this action is the (2,3,7) Schwarz triangle. This is the smallest hyperbolic Schwarz triangle, and thus, by the proof of Hurwitz's automorphisms theorem, the tiling is the universal tiling that covers all Hurwitz surfaces (the Riemann surfaces with maximal symmetry group), giving them a triangulation whose symmetry group equals their automorphism group as Riemann surfaces.
The smallest of these is the Klein quartic, the most symmetric genus 3 surface, together with a tiling by 56 triangles, meeting at 24 vertices, with symmetry group the simple group of order 168, known as PSL(2,7). The resulting surface can in turn be polyhedrally immersed into Euclidean 3-space, yielding the small cubicuboctahedron.
The dual order-3 heptagonal tiling has the same symmetry group, and thus yields heptagonal tilings of Hurwitz surfaces.
Related polyhedra and tiling
It is related to two star-tilings by the same vertex arrangement: the order-7 heptagrammic tiling, {7/2,7}, and heptagrammic-order heptagonal tiling, {7,7/2}.
This tiling is topologically related as a part of sequence of regular polyhedra with Schläfli symbol {3,p}.
This tiling is a part of regular series {n,7}:
From a Wythoff construction there are eight hyperbolic uniform tilings that can be based from the regular heptagonal tiling.
Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms.
See also
Order-7 tetrahedral honeycomb
List of regular polytopes
List of uniform planar tilings
Tilings of regular polygons
Triangular tiling
Uniform tilings in hyperbolic plane
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
External links
Hyperbolic and Spherical Tiling Gallery
KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
Hyperbolic Planar Tessellations, Don Hatch
Hyperbolic tilings
Isogonal tilings
Isohedral tilings
Order-7 tilings
Regular tilings
Triangular tilings
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https://en.wikipedia.org/wiki/Triheptagonal%20tiling
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In geometry, the triheptagonal tiling is a semiregular tiling of the hyperbolic plane, representing a rectified Order-3 heptagonal tiling. There are two triangles and two heptagons alternating on each vertex. It has Schläfli symbol of r{7,3}.
Compare to trihexagonal tiling with vertex configuration 3.6.3.6.
Images
7-3 Rhombille
In geometry, the 7-3 rhombille tiling is a tessellation of identical rhombi on the hyperbolic plane. Sets of three and seven rhombi meet two classes of vertices.
7-3 rhombile tiling in band model
Related polyhedra and tilings
The triheptagonal tiling can be seen in a sequence of quasiregular polyhedrons and tilings:
From a Wythoff construction there are eight hyperbolic uniform tilings that can be based from the regular heptagonal tiling.
Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms.
See also
Trihexagonal tiling - 3.6.3.6 tiling
Rhombille tiling - dual V3.6.3.6 tiling
Tilings of regular polygons
List of uniform tilings
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
External links
Hyperbolic and Spherical Tiling Gallery
KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
Hyperbolic Planar Tessellations, Don Hatch
Hyperbolic tilings
Isogonal tilings
Isotoxal tilings
Quasiregular polyhedra
Semiregular tilings
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https://en.wikipedia.org/wiki/Truncated%20triheptagonal%20tiling
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In geometry, the truncated triheptagonal tiling is a semiregular tiling of the hyperbolic plane. There is one square, one hexagon, and one tetradecagon (14-sides) on each vertex. It has Schläfli symbol of
Uniform colorings
There is only one uniform coloring of a truncated triheptagonal tiling. (Naming the colors by indices around a vertex: 123.)
Symmetry
Each triangle in this dual tiling, order 3-7 kisrhombille, represent a fundamental domain of the Wythoff construction for the symmetry group [7,3].
Related polyhedra and tilings
This tiling can be considered a member of a sequence of uniform patterns with vertex figure (4.6.2p) and Coxeter-Dynkin diagram . For p < 6, the members of the sequence are omnitruncated polyhedra (zonohedrons), shown below as spherical tilings. For p > 6, they are tilings of the hyperbolic plane, starting with the truncated triheptagonal tiling.
From a Wythoff construction there are eight hyperbolic uniform tilings that can be based from the regular heptagonal tiling.
Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms.
See also
Tilings of regular polygons
List of uniform planar tilings
References
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 19, The Hyperbolic Archimedean Tessellations)
External links
Hyperbolic and Spherical Tiling Gallery
KaleidoTile 3: Educational software to create spherical, planar and hyperbolic tilings
Hyperbolic Planar Tessellations, Don Hatch
Hyperbolic tilings
Isogonal tilings
Semiregular tilings
Truncated tilings
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https://en.wikipedia.org/wiki/Neil%20Shephard
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Neil Shephard (born 8 October 1964), FBA, is an econometrician, currently Frank B. Baird Jr., Professor of Science in the Department of Economics and the Department of Statistics at Harvard University.
His most well known contributions are: (i) the formalisation of the econometrics of realised volatility, which nonparametrically estimates the volatility of asset prices, (ii) the introduction of the auxiliary particle filter (signal extraction), (iii) the nonparametric identification of jumps in financial economics, through multipower variation, (iv) stochastic volatility models based on non-Gaussian Ornstein-Uhlenbeck processes, known as 'Barndorff-Nielsen-Shephard' models.
Early life and education
Neil Shephard was born in Plymouth, England, but moved to Norfolk, England, aged one. His mother was Tydfil Shephard (1930-1972), who was a high school teacher.
His father was Tom Shephard (1930-2023), who was a Norfolk high school head. Since 1975 Gillian Shephard has been his step-mother. He attended the Marshland High School West Walton, King Edward VII Grammar School in King's Lynn and 1981-1983 City of Norwich School (he studied pure mathematics & statistics, economics and politics at A-level). He studied economics and statistics as an undergraduate at the University of York in the UK 1983-1986, awarded a first class degree with distinction. He did his M.Sc. (awarded in 1987, with distinction) and Ph.D. (examined in 1989 and graduated in 1990) at the LSE.
Academic career
He was a lecturer in statistics at the LSE from 1988 to 1993. He moved to Nuffield College, Oxford in 1991 to join the economics group as the Gatsby Prize Research Fellow in Econometrics (funded by the Gatsby Foundation). In 1993 he became an Official Fellow in Economics at Nuffield College, Oxford.
He has been Professor of Economics and of Statistics at Harvard University since 2013.
He was elected a Fellow of the British Academy in 2006, a Fellow of the Econometric Society in 2004. He was awarded an honorary doctorate in economics by Aarhus University in 2009, the 2012 Richard Stone Prize in Applied Econometrics and the 2017 Guy Medal in Silver of the Royal Statistical Society.
With David F. Hendry he founded the Econometrics Journal in 1998. With Colin Mayer he founded Oxford University's Masters in Financial Economics. In 2007 he co-founded the Oxford-Man Institute, which he directed from 2007 to 2011. He chaired the Statistics Department at Harvard from 2015 to 2022. With Computer Science and Statistical colleagues, he founded Harvard University's Masters in Data Science in 2018 and the Harvard Data Science Initiative.
Publications
Representative articles
Iavor Bojinov and Neil Shephard (2019) "Time Series Experiments and Causal Estimands: Exact Randomization Tests and Trading", Journal of the American Statistical Association, 114, 1665-1682.
Luke Bornn, Neil Shephard and Reza Solgi (2019) "Moment conditions and Bayesian nonparametrics", Journal of Royal
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https://en.wikipedia.org/wiki/Test%20Valley%20School
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Test Valley School is a comprehensive secondary school with Specialist Status in Mathematics and Computing located in Stockbridge, Hampshire, England. Due to its rural location, it has a wide catchment area, with significant numbers of students travelling from Andover, Salisbury, Romsey, The Wallops and other small villages near to Stockbridge.
Exam Results
5 GCSEs at A*–C: 64% (2012)
5 GCSEs at A*–C: 69% (2014)
5 GCSEs at A*–C: 67% (2015)
In 2018, 57% of pupils achieved 5 or more 9–4 grades, compared to a 60% national average. 38% of pupils achieved grade 5 (a strong pass) in both maths and English, compared to a 40% national average.
References
Report, OFSTED
External links
Secondary schools in Hampshire
Community schools in Hampshire
Stockbridge, Hampshire
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https://en.wikipedia.org/wiki/Rebound%20rate
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In basketball statistics, rebound rate or rebound percentage is a statistic to gauge how effective a player is at gaining possession of the basketball after a missed field goal or free throw. Rebound rate is an estimate of the percentage of missed shots a player rebounded while he was on the floor. Using raw rebound totals to evaluate rebounding fails to take into account external factors unrelated to a player's ability, such as the number of shots taken in games and the percentage of those shots that are made. Both factors affect the number of missed shots that are available to be rebounded. Rebound rate takes these factors into account.
The formula are:
In the National Basketball Association (NBA), the statistic is available for seasons since the 1970–71 season. The highest career rebound rate by a player is 23.4, by Dennis Rodman. The highest rebound rate for one season is 29.7, also by Dennis Rodman, which he achieved during the season. He also owned seven of the top ten rebound percentage seasons (four of the top five) in NBA history, all time.
References
Basketball statistics
Rates
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https://en.wikipedia.org/wiki/Golod%E2%80%93Shafarevich%20theorem
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In mathematics, the Golod–Shafarevich theorem was proved in 1964 by Evgeny Golod and Igor Shafarevich. It is a result in non-commutative homological algebra which solves the class field tower problem, by showing that class field towers can be infinite.
The inequality
Let A = K⟨x1, ..., xn⟩ be the free algebra over a field K in n = d + 1 non-commuting variables xi.
Let J be the 2-sided ideal of A generated by homogeneous elements fj of A of degree dj with
2 ≤ d1 ≤ d2 ≤ ...
where dj tends to infinity. Let ri be the number of dj equal to i.
Let B=A/J, a graded algebra. Let bj = dim Bj.
The fundamental inequality of Golod and Shafarevich states that
As a consequence:
B is infinite-dimensional if ri ≤ d2/4 for all i
Applications
This result has important applications in combinatorial group theory:
If G is a nontrivial finite p-group, then r > d2/4 where d = dim H1(G,Z/pZ) and r = dim H2(G,Z/pZ) (the mod p cohomology groups of G). In particular if G is a finite p-group with minimal number of generators d and has r relators in a given presentation, then r > d2/4.
For each prime p, there is an infinite group G generated by three elements in which each element has order a power of p. The group G provides a counterexample to the generalised Burnside conjecture: it is a finitely generated infinite torsion group, although there is no uniform bound on the order of its elements.
In class field theory, the class field tower of a number field K is created by iterating the Hilbert class field construction. The class field tower problem asks whether this tower is always finite; attributed this question to Furtwangler, though Furtwangler said he had heard it from Schreier. Another consequence of the Golod–Shafarevich theorem is that such towers may be infinite (in other words, do not always terminate in a field equal to its Hilbert class field). Specifically,
Let K be an imaginary quadratic field whose discriminant has at least 6 prime factors. Then the maximal unramified 2-extension of K has infinite degree.
More generally, a number field with sufficiently many prime factors in the discriminant has an infinite class field tower.
References
(in Russian)
(in Russian)
See Chapter 8.
Johnson, D.L. (1980). "Topics in the Theory of Group Presentations" (1st ed.). Cambridge University Press. . See chapter VI.
Serre, J.-P. (2002), "Galois Cohomology," Springer-Verlag. . See Appendix 2. (Translation of Cohomologie Galoisienne, Lecture Notes in Mathematics 5, 1973.)
Class field theory
Theorems in group theory
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https://en.wikipedia.org/wiki/Bertrand%27s%20box%20paradox
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Bertrand's box paradox is a veridical paradox in elementary probability theory. It was first posed by Joseph Bertrand in his 1889 work Calcul des Probabilités.
There are three boxes:
a box containing two gold coins,
a box containing two silver coins,
a box containing one gold coin and one silver coin.
The question is to calculate the probability, after choosing a box at random and withdrawing one coin at random, if that happens to be a gold coin, of the next coin drawn from the same box also being a gold coin.
A veridical paradox is when the correct solution to a puzzle appears to be counterintuitive. It may seem intuitive that the probability that the remaining coin is gold should be , but the probability is actually . However, this is not the paradox Bertrand referred to. He showed that if were correct, it would lead to a contradiction, so cannot be correct.
This simple but counterintuitive puzzle is used as a standard example in teaching probability theory. The solution illustrates some basic principles, including the Kolmogorov axioms.
Solution
The problem can be reframed by describing the boxes as each having one drawer on each of two sides. Each drawer contains a coin. One box has a gold coin on each side (GG), one a silver coin on each side (SS), and the other a gold coin on one side and a silver coin on the other (GS). A box is chosen at random, a random drawer is opened, and a gold coin is found inside it. What is the chance of the coin on the other side being gold?
The following faulty reasoning appears to give a probability of :
Originally, all three boxes were equally likely to be chosen.
The chosen box cannot be box SS.
So it must be box GG or GS.
The two remaining possibilities are equally likely. So the probability that the box is GG, and the other coin is also gold, is .
The flaw is in the last step. While those two cases were originally equally likely, the fact that you are certain to find a gold coin if you had chosen the GG box, but are only 50% sure of finding a gold coin if you had chosen the GS box, means they are no longer equally likely given that you have found a gold coin. Specifically:
The probability that GG would produce a gold coin is 1.
The probability that SS would produce a gold coin is 0.
The probability that GS would produce a gold coin is .
Initially GG, SS and GS are equally likely . Therefore, by Bayes' rule the conditional probability that the chosen box is GG, given we have observed a gold coin, is:
The correct answer of can also be obtained as follows:
Originally, all six coins were equally likely to be chosen.
The chosen coin cannot be from drawer S of box GS, or from either drawer of box SS.
So it must come from the G drawer of box GS, or either drawer of box GG.
The three remaining possibilities are equally likely, so the probability that the drawer is from box GG is .
Alternatively, one can simply note that the chosen box has two coins of the same type of the time. So, regardless of
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https://en.wikipedia.org/wiki/List%20of%20Urawa%20Red%20Diamonds%20records%20and%20statistics
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This article contains records and statistics for the Japanese professional football club, Urawa Red Diamonds.
J.League
Domestic cup competitions
Major international competitions
Top scorers by season
International Games
Key:
aet - after extra time
PS - after penalty shootout
References
Urawa Red Diamonds
Urawa Red Diamonds
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https://en.wikipedia.org/wiki/Simple%20polytope
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In geometry, a -dimensional simple polytope is a -dimensional polytope each of whose vertices are adjacent to exactly edges (also facets). The vertex figure of a simple -polytope is a -simplex.
Simple polytopes are topologically dual to simplicial polytopes. The family of polytopes which are both simple and simplicial are simplices or two-dimensional polygons. A simple polyhedron is a three-dimensional polyhedron whose vertices are adjacent to three edges and three faces. The dual to a simple polyhedron is a simplicial polyhedron, in which all faces are triangles.
Examples
Three-dimensional simple polyhedra include the prisms (including the cube), the regular tetrahedron and dodecahedron, and, among the Archimedean solids, the truncated tetrahedron, truncated cube, truncated octahedron, truncated cuboctahedron, truncated dodecahedron, truncated icosahedron, and truncated icosidodecahedron. They also include the Goldberg polyhedra and fullerenes, including the chamfered tetrahedron, chamfered cube, and chamfered dodecahedron. In general, any polyhedron can be made into a simple one by truncating its vertices of valence four or higher. For instance, truncated trapezohedrons are formed by truncating only the high-degree vertices of a trapezohedron; they are also simple.
Four-dimensional simple polytopes include the regular 120-cell and tesseract. Simple uniform 4-polytope include the truncated 5-cell, truncated tesseract, truncated 24-cell, truncated 120-cell, and duoprisms. All bitruncated, cantitruncated, or omnitruncated four-polytopes are simple.
Simple polytopes in higher dimensions include the d-simplex, hypercube, associahedron, permutohedron, and all omnitruncated polytopes.
Unique reconstruction
Micha Perles conjectured that a simple polytope is completely determined by its 1-skeleton; his conjecture was proven in 1987 by Roswitha Blind and Peter Mani-Levitska. Gil Kalai shortly after provided a simpler proof of this result based on the theory of unique sink orientations.
References
Euclidean geometry
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https://en.wikipedia.org/wiki/Simplicial%20polytope
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In geometry, a simplicial polytope is a polytope whose facets are all simplices. For example, a simplicial polyhedron in three dimensions contains only triangular faces and corresponds via Steinitz's theorem to a maximal planar graph.
They are topologically dual to simple polytopes. Polytopes which are both
simple and simplicial are either simplices or two-dimensional polygons.
Examples
Simplicial polyhedra include:
Bipyramids
Gyroelongated dipyramids
Deltahedra (equilateral triangles)
Platonic
tetrahedron, octahedron, icosahedron
Johnson solids:
triangular bipyramid, pentagonal bipyramid, snub disphenoid, triaugmented triangular prism, gyroelongated square dipyramid
Catalan solids:
triakis tetrahedron, triakis octahedron, tetrakis hexahedron, disdyakis dodecahedron, triakis icosahedron, pentakis dodecahedron, disdyakis triacontahedron
Simplicial tilings:
Regular:
triangular tiling
Laves tilings:
tetrakis square tiling, triakis triangular tiling, kisrhombille tiling
Simplicial 4-polytopes include:
convex regular 4-polytope
4-simplex, 16-cell, 600-cell
Dual convex uniform honeycombs:
Disphenoid tetrahedral honeycomb
Dual of cantitruncated cubic honeycomb
Dual of omnitruncated cubic honeycomb
Dual of cantitruncated alternated cubic honeycomb
Simplicial higher polytope families:
simplex
cross-polytope (Orthoplex)
See also
Simplicial complex
Delaunay triangulation
Notes
References
Euclidean geometry
Polytopes
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https://en.wikipedia.org/wiki/List%20of%20Kashima%20Antlers%20records%20and%20statistics
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This article contains records and statistics for the Japanese professional football club, Kashima Antlers.
J.League
Domestic cup competitions
Major international competitions
Top scorers by season
References
Kashima Antlers
Kashima Antlers
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https://en.wikipedia.org/wiki/Andr%C3%A9-Michel%20Guerry
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André-Michel Guerry (; December 24, 1802 – April 9, 1866) was a French lawyer and amateur statistician. Together with Adolphe Quetelet he may be regarded as the founder of moral statistics which led to the development of criminology, sociology and ultimately, modern social science.
Early life and education
Guerry was born in Tours, Indre-et-Loire, the only child of Michel Guerry, a building contractor, whose family had a long history as innkeepers, merchants, farmers and trades-people. About 1817-1820 he studied at the Imperial College of Tours (now the Lyceum Descartes, founded in 1807) and subsequently studied law at the University of Poitiers.
About 18241825 he moved to Paris and was admitted to the bar as a royal advocate. Shortly after, he was employed by the Ministry of Justice. Guerry worked with the data on crime statistics in France collected as part of the General office for administration of criminal justice in France, the first centralized national system of crime reporting. Guerry was so fascinated with these data, and the possibility to discover empirical regularities and laws that might govern them, that he gave up the active practice of law to devote the rest of his life to study crime and its relation to other moral variables.
Moral statistics and criminology
Guerry's first work on what would come to be called moral statistics was a large, one page sheet containing three shaded maps of France, prepared together with the Venetian geographer, Adriano Balbi in 1829. These showed the departments of France, shaded according to crimes against persons, crimes against
property, and school instruction. Such statistical maps, now called choropleth maps had just been invented in 1826 by Baron Charles Dupin.
Guerry is best known for his Essay on moral statistics of France, presented to the French Academy of Sciences on July 2, 1832, and published in 1833 after it was awarded the Prix Montyon in statistics. His presentation, in tables and thematic maps, showed that rates of crime and suicide remained remarkably stable over time, when broken down by age, sex, region of France and even season of the year. Yet, these numbers also varied systematically across departments of France. This regularity of social numbers created the possibility to conceive that human actions could be described by social laws, just as inanimate actions were governed by physical laws.
Throughout his career, Guerry was particularly interested in uncovering the relation between social and moral variables. How are personal crime and property related to each other, and to suicide, donations to the poor, illegitimate births, wealth, and so forth? How do different types of crimes vary with age of the accused? Statistical methods (correlation and regression) were still in their infancy, so Guerry relied on graphic comparisons of maps and semi-graphic tables. Shown below are three of the six thematic maps that Guerry included in his Essay.
Suicide
In addition to a m
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https://en.wikipedia.org/wiki/PRISM%20%28website%29
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Portal Resources for Indiana Science and Mathematics (PRISM) is a free website originally designed for Indiana middle school math, science, and technology teachers. It links Indiana Academic Standards for middle school science, technology, pre-engineering, and math (STEM fields) to appropriate, teacher-reviewed online learning activities. Users may either browse materials by academic standard or use the keyword search engine to find appropriate sites.
With the integration of the Moodle open source Learning Management System in 2006, PRISM now serves a much larger audience. Teachers from all grades may use Moodle to establish online classroom courses.
Typical PRISM reviewed resources include web-delivered simulations, visualizations, modeling packages, and resource sites providing access to live data or collaborative experiments. PRISM endeavors to encourage interactive learning, foster new liaisons among students, parents, and teachers, and foster alternative pedagogical approaches.
Membership in PRISM is free and is open to parents, teachers, and pre-service personnel. Student names and/or usernames are not displayed publicly on the site.
The PRISM Project is funded by a grant from the Lilly Endowment and hosted at Rose–Hulman Institute of Technology (RHIT). PRISM is the West Central Regional Coordinator for the I-STEM Network. Dr. Patricia A. Carlson (RHIT) is the Program Director of the project.
References
External links
PRISM Website
Lilly Endowment Grant Supporting PRISM in Providing Digital Tools and Resources to Indiana K-12 Teachers
PRISM Becomes Go-To Resource for Indiana School Teachers
American educational websites
Mathematics websites
Rose–Hulman Institute of Technology
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https://en.wikipedia.org/wiki/Antun%20Karlo%20Bakoti%C4%87
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Antun Karlo Bakotić (4 November 1831 in Kaštel Gomilica – 13 January 1887 in Zadar) was a Croatian writer and physicist.
He studied mathematics and physics at Vienna and Venice. He worked as a professor in Rijeka, as well as managing the Velika Gimnazija in Split and acting as school superintendent. He was one of the first in Croatia to publish an academic book about nature. In 1862 he arranged for printing the book Pojavi iz prirode za pouku prostoga naroda from the works of an Italian author.
He wrote popular articles in his field in the magazine Književnik. He collaborated with Bogoslav Šulek on Riječnik znanstvenog nazivlja (Dictionary of Scientific Terms). Bakotić also published the book Vinarstvo (1867). He was one of the chief members of the Croatian National Revival in southern Croatia. He campaigned for teaching Croatian in Dalmatian schools and was also a member of the Narodni list newspaper.
His novel about Bosnian life with national liberation themes, Raja (Dhimmis, 1890), was published in installments in Iskra and Hrvatska and in entirety in Dom i svijet.
Sources
Info - vremeplov
Bakotić, Antun Karlo at enciklopedija.hr
1831 births
1887 deaths
People from Kaštela
Croatian writers
Croatian physicists
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https://en.wikipedia.org/wiki/Iteration%20%28disambiguation%29
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Iteration means the act of repeating in the contexts of mathematics, computing and project management, particularly software development.
It can also refer to:
Iterated function, in mathematics
"Iteration", a song from Potemkin City Limits, the fourth full-length album by the punk rock band Propagandhi
Iteration, a 2017 album by American electronic music producer Seth Haley, released under his alias Com Truise
Iterations, a 2002 collection of short stories by Canadian science fiction author Robert J. Sawyer as well as the title of a story in that collection
Iterative and incremental development, a term in new product development that came from software development
Iterative aspect in linguistics
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https://en.wikipedia.org/wiki/Josep%20Guia
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Josep Guia i Marín (; born 1947, in Valencia) is a Spanish writer, mathematics professor of University of Valencia and political activist within PSAN party. In 1986, he was awarded by Fundació Jaume I.
Some of his most relevant essays about Catalan nationalism are: Països Catalans i Llibertat ("Catalan Countries and Freedom) (1983), És molt senzill, digueu-li Catalunya ("It's very easy, call it Catalonia") (1985), Des de la Catalunya del Sud ("From Southern Catalonia") (1987), València, 750 anys de nació catalana (Valencia, 750 years of Catalan Nation) (1988)) and Catalunya descoberta ("Catalonia, discovered") (1990). He is an editor of Lluita magazine.
External links
Biographical information in Catalan language writers association webpage
Catalan-language writers
Politics of Catalonia
People from Valencia
Politicians from the Valencian Community
1947 births
Living people
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https://en.wikipedia.org/wiki/Amoeba%20%28mathematics%29
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In complex analysis, a branch of mathematics, an amoeba is a set associated with a polynomial in one or more complex variables. Amoebas have applications in algebraic geometry, especially tropical geometry.
Definition
Consider the function
defined on the set of all n-tuples of non-zero complex numbers with values in the Euclidean space given by the formula
Here, log denotes the natural logarithm. If p(z) is a polynomial in complex variables, its amoeba is defined as the image of the set of zeros of p under Log, so
Amoebas were introduced in 1994 in a book by Gelfand, Kapranov, and Zelevinsky.
Properties
Any amoeba is a closed set.
Any connected component of the complement is convex.
The area of an amoeba of a not identically zero polynomial in two complex variables is finite.
A two-dimensional amoeba has a number of "tentacles", which are infinitely long and exponentially narrow towards infinity.
Ronkin function
A useful tool in studying amoebas is the Ronkin function. For p(z), a polynomial in n complex variables, one defines the Ronkin function
by the formula
where denotes Equivalently, is given by the integral
where
The Ronkin function is convex and affine on each connected component of the complement of the amoeba of .
As an example, the Ronkin function of a monomial
with is
References
.
Further reading
External links
Amoebas of algebraic varieties
Algebraic geometry
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https://en.wikipedia.org/wiki/Ryan%20Daut
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Ryan Daut (born April 18, 1984) is an American professional poker player from New Jersey. After earning a degree in mathematics and computer science from the University of Richmond in 2006, Daut began work on a doctorate in mathematics at Penn State University. He dropped out of the program after one semester to play poker professionally.
Ryan Daut started out competitive gaming in StarCraft and StarCraft: Brood War. He was an avid poster on teamliquid.net posting strategies and guides. Around 2003 when the poker boom hit he shifted into poker along with many other notable starcraft players, Bertrand "Elky" Grosspellier, Guillaume "Grrrr" Patry, Dan "Rekrul" Schreiber, Hevad "Rain" Khan, etc.
On January 10, 2007 Daut won a World Poker Tour event at the PokerStars Caribbean Poker Adventure. He defeated Isaac Haxton heads up and won $1,535,255 for first place.
As of 2010, his total live tournament winnings exceed $1,800,000.
References
External links
Hendon Mob tournament results
Interview with Ryan Daut
American poker players
World Poker Tour winners
1984 births
Living people
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https://en.wikipedia.org/wiki/Wigan%20urban%20area
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The Wigan Urban Area is an area of land defined by the Office for National Statistics consisting of the built-up, or 'urbanised' area containing Wigan in Greater Manchester and Skelmersdale in West Lancashire.
The Urban Area includes the integrated conurbation around Wigan, (containing the contiguous areas of Ince-in-Makerfield and Wigan itself), along with the outlying areas of Standish, Abram and the West Lancashire town of Skelmersdale.
The Wigan Urban Area has a total population of 175,405. This is an increase of 5% on the 2001 figure of 166,840.
Constituent parts
The historic town of Wigan forms an integrated conurbation along with the Metropolitan Borough of Wigan district of Ince-in-Makerfield, this is connected by ribbon development to Standish, Platt Bridge and Abram. These areas, together with Skelmersdale in West Lancashire, are defined by the Office for National Statistics as the Wigan Urban Area.
Breakdown
The ONS figures were broken down into constituent parts, and an individual population figure given for each. These were:
Notes
Orrell is included under the Wigan subdivision in the 2011 census data.
The Abram subdivision was renamed Platt Bridge in the 2011 census.
The Worthington subdivision was part of the Standish subdivision in the 2001 census.
References
Office for National Statistics: Census 2001, Key Statistics for urban areas
Errata
Urban areas of England
Geography of the Metropolitan Borough of Wigan
Geography of Lancashire
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https://en.wikipedia.org/wiki/Multivariate%20probit%20model
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In statistics and econometrics, the multivariate probit model is a generalization of the probit model used to estimate several correlated binary outcomes jointly. For example, if it is believed that the decisions of sending at least one child to public school and that of voting in favor of a school budget are correlated (both decisions are binary), then the multivariate probit model would be appropriate for jointly predicting these two choices on an individual-specific basis. J.R. Ashford and R.R. Sowden initially proposed an approach for multivariate probit analysis. Siddhartha Chib and Edward Greenberg extended this idea and also proposed simulation-based inference methods for the multivariate probit model which simplified and generalized parameter estimation.
Example: bivariate probit
In the ordinary probit model, there is only one binary dependent variable and so only one latent variable is used. In contrast, in the bivariate probit model there are two binary dependent variables and , so there are two latent variables: and .
It is assumed that each observed variable takes on the value 1 if and only if its underlying continuous latent variable takes on a positive value:
with
and
Fitting the bivariate probit model involves estimating the values of and . To do so, the likelihood of the model has to be maximized. This likelihood is
Substituting the latent variables and in the probability functions and taking logs gives
After some rewriting, the log-likelihood function becomes:
Note that is the cumulative distribution function of the bivariate normal distribution. and in the log-likelihood function are observed variables being equal to one or zero.
Multivariate Probit
For the general case, where we can take as choices and as individuals or observations, the probability of observing choice is
Where and,
The log-likelihood function in this case would be
Except for typically there is no closed form solution to the integrals in the log-likelihood equation. Instead simulation methods can be used to simulated the choice probabilities. Methods using importance sampling include the GHK algorithm (Geweke, Hajivassilou, McFadden and Keane), AR (accept-reject), Stern's method. There are also MCMC approaches to this problem including CRB (Chib's method with Rao-Blackwellization), CRT (Chib, Ritter, Tanner), ARK (accept-reject kernel), and ASK (adaptive sampling kernel). A variational approach scaling to large datasets is proposed in Probit-LMM (Mandt, Wenzel, Nakajima et al.).
References
Further reading
Greene, William H., Econometric Analysis, seventh edition, Prentice-Hall, 2012.
Regression models
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https://en.wikipedia.org/wiki/Undergraduate%20Texts%20in%20Mathematics
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Undergraduate Texts in Mathematics (UTM) () is a series of undergraduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are small yellow books of a standard size.
The books in this series tend to be written at a more elementary level than the similar Graduate Texts in Mathematics series, although there is a fair amount of overlap between the two series in terms of material covered and difficulty level.
There is no Springer-Verlag numbering of the books like in the Graduate Texts in Mathematics series.
The books are numbered here by year of publication.
List of books
External links
Springer-Verlag's Summary of Undergraduate Texts in Mathematics
Series of mathematics books
Mathematics
Mathematics textbooks
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https://en.wikipedia.org/wiki/Graduate%20Studies%20in%20Mathematics
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Graduate Studies in Mathematics (GSM) is a series of graduate-level textbooks in mathematics published by the American Mathematical Society (AMS). The books in this series are published in hardcover and e-book formats.
List of books
1 The General Topology of Dynamical Systems, Ethan Akin (1993, )
2 Combinatorial Rigidity, Jack Graver, Brigitte Servatius, Herman Servatius (1993, )
3 An Introduction to Gröbner Bases, William W. Adams, Philippe Loustaunau (1994, )
4 The Integrals of Lebesgue, Denjoy, Perron, and Henstock, Russell A. Gordon (1994, )
5 Algebraic Curves and Riemann Surfaces, Rick Miranda (1995, )
6 Lectures on Quantum Groups, Jens Carsten Jantzen (1996, )
7 Algebraic Number Fields, Gerald J. Janusz (1996, 2nd ed., )
8 Discovering Modern Set Theory. I: The Basics, Winfried Just, Martin Weese (1996, )
9 An Invitation to Arithmetic Geometry, Dino Lorenzini (1996, )
10 Representations of Finite and Compact Groups, Barry Simon (1996, )
11 Enveloping Algebras, Jacques Dixmier (1996, )
12 Lectures on Elliptic and Parabolic Equations in Hölder Spaces, N. V. Krylov (1996, )
13 The Ergodic Theory of Discrete Sample Paths, Paul C. Shields (1996, )
14 Analysis, Elliott H. Lieb, Michael Loss (2001, 2nd ed., )
15 Fundamentals of the Theory of Operator Algebras. Volume I: Elementary Theory, Richard V. Kadison, John R. Ringrose (1997, )
16 Fundamentals of the Theory of Operator Algebras. Volume II: Advanced Theory, Richard V. Kadison, John R. Ringrose (1997, )
17 Topics in Classical Automorphic Forms, Henryk Iwaniec (1997, )
18 Discovering Modern Set Theory. II: Set-Theoretic Tools for Every Mathematician, Winfried Just, Martin Weese (1997, )
19 Partial Differential Equations, Lawrence C. Evans (2010, 2nd ed., )
20 4-Manifolds and Kirby Calculus, Robert E. Gompf, András I. Stipsicz (1999, )
21 A Course in Operator Theory, John B. Conway (2000, )
22 Growth of Algebras and Gelfand-Kirillov Dimension, Günter R. Krause, Thomas H. Lenagan (2000, Revised ed., )
23 Foliations I, Alberto Candel, Lawrence Conlon (2000, )
24 Number Theory: Algebraic Numbers and Functions, Helmut Koch (2000, )
25 Dirac Operators in Riemannian Geometry, Thomas Friedrich (2000, )
26 An Introduction to Symplectic Geometry, Rolf Berndt (2001, )
27 A Course in Differential Geometry, Thierry Aubin (2001, )
28 Notes on Seiberg-Witten Theory, Liviu I. Nicolaescu (2000, )
29 Fourier Analysis, Javier Duoandikoetxea (2001, )
30 Noncommutative Noetherian Rings, J. C. McConnell, J. C. Robson (1987, ); 2001 pbk reprint with corrections
31 Option Pricing and Portfolio Optimization: Modern Methods of Financial Mathematics, Ralf Korn, Elke Korn (2001, )
32 A Modern Theory of Integration, Robert G. Bartle (2001, )
33 A Course in Metric Geometry, Dmitri Burago, Yuri Burago, Sergei Ivanov (2001, )
34 Differential Geometry, Lie Groups, and Symmetric Spaces, Sigurdur Helgason (2001, )
35 Lecture Notes in Algebraic Topology, James F. Davis, Paul Kirk (2001, )
36 Principles of Functional Analysis, M
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https://en.wikipedia.org/wiki/Lucky%20number%20%28disambiguation%29
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A lucky number, in number theory, is a natural number generated by a particular sieve algorithm.
Lucky number may also refer to:
Film and television
The Lucky Number, a 1933 British comedy film
Lucky Number (film), a 1951 Donald Duck cartoon
Lucky Numbers, a 2000 American comedy film
#Lucky Number, a 2015 film starring Tom Pelphrey
Lucky Numbers (TV series), a 1995–1997 British game show
Music
Lucky Number (album) or the title song, by Jolin Tsai, 2001
Lucky Numbers (album), by Frank Sinatra, 1998
Lucky Number: The Best of Lene Lovich, an album by Lene Lovich, 2004
"Lucky Number" (song), by Lene Lovich, 1979
"Lucky Number", a song by Saves the Day from Saves the Day, 2013
Other uses
Lucky numbers of Euler, producing prime-generating polynomials
A number believed to affect one's luck
Lucky number combinations, an element of Chinese numerology
Lucky Numbers, a discontinued Cadbury product
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https://en.wikipedia.org/wiki/Brian%20Pink
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Brian Pink was the Australian Statistician, the head of the Australian Bureau of Statistics (ABS), between 5 March 2007 and 12 January 2014.
Prior to September 1999, Brian Pink was ABS's Statistical Support Group Manager, when he was appointed as the Government Statistician for New Zealand and Chief Executive of Statistics New Zealand.
Biography
Pink's career in official statistics began in Australia with the then Commonwealth Bureau of Census and Statistics in Sydney in 1966, followed by postings to various state offices of its successor, the Australian Bureau of Statistics. He was Government Statistician and Chief Executive of Statistics New Zealand from late October 2000 to March 2007.
As well as his duties as ex officio member of the Australian Statistics Advisory Council, Pink is Vice Chairman of the OECD Committee on Statistics, and Australia's Head of Delegation to the United Nations Statistical Commission. He was President of the International Association for Official Statistics from 2005 to 2007.
Brian Pink professes strong views about the importance of the role of official statistics in society, beyond necessary government use.
Controversy
Brian Pink has been involved in a number of controversies relating to employees of the ABS.
Bulletin
In April 2008, the distribution of a staff bulletin by union employees led Brian Pink to issue a letter which advised employees that continued distribution of bulletins would be in breach of the Australian Public Service Code of Conduct. This led to Federal Court action which was settled in mediation.
Sackings
In April 2009, the ABS was taken to the Australian Industrial Relations Commission by the Community and Public Sector Union for the manner in which staff were terminated, which resulted in a decision against the ABS.
References
Australian public servants
Government Statisticians of New Zealand
Living people
Australian statisticians
Year of birth missing (living people)
21st-century Australian public servants
21st-century New Zealand public servants
ja:デニス・トレウィン
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https://en.wikipedia.org/wiki/Turmus%20Ayya
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Turmus Ayya () is a Palestinian town located in the Ramallah and al-Bireh Governorate in the West Bank, in Palestine. According to the Palestinian Central Bureau of Statistics (PCBS), it had a population of 2,464 in 2017. An estimated 80% of the residents are Palestinian binationals with US citizenship.
Turmus Ayya is the frequent target of Israeli settler violence.
Geography
Turmus Ayya is located northeast of the city of Ramallah. Its surrounding villages are Sinjil and Khirbet Abu Falah as well as the Israeli settlement of Shilo. Its jurisdiction is about . Turmus Ayya is 720 m above sea level. It is also the northernmost town in the Ramallah District. Turmus Ayya's climate is similar to that of the central West Bank, which is rainy in the winter, and hot and humid in the summer.
History
Potsherds from the late Iron Age (8 -7th century B.C.E.) period and later have been found, and it is estimated that the village has existed continuously since then.
Turmus Ayya is generally accepted as being the Turbasaim in Crusader sources. A little northeast of Turmus Ayya is Khirbet Ras ad-Deir/Deir el-Fikia, believed to be the Crusader village of Dere. In 1145, half of the income from both villages were given to the Abbey of Mount Tabor, so that they could maintain the church at Sinjil. In 1175, all three villages; Turmus Ayya, Dere and Sinjil, were transferred to the Church of the Holy Sepulchre.
Ottoman era
In 1517, Turmus Ayya was incorporated into the Ottoman Empire with the rest of Palestine, and in 1596 it appeared in the tax registers as being in the Nahiya of Quds of the Liwa of Quds. It had a population of 43 households, all Muslim, and paid taxes on wheat, barley, olive trees, vineyards, fruit trees, goats and/or beehives; a total of 7,200 akçe. 11/24 of the revenue went to a Waqf.
In 1838, Edward Robinson noted that Turmus Aya was within the province of Jerusalem, but the province of Nablus was just north of it. It was further noted that it was situated "on a low rocky mound in the level valley."
In Turmus Ayya's cemetery, several graves have headstones that date back to the Ottoman Era.
French explorer Victor Guérin visited the village in 1870 and found ancient cisterns, cut stones built up in the houses, a broken lintel with a garland carved upon, and the fragments of a column. He further noted that the village had about seven hundred inhabitants, and was administered by two sheikhs and divided into two different areas. Some ancient cisterns were almost completely dry, and women were forced to fetch water either from Ain Siloun, or Ain Sindjel. An official Ottoman village list from about 1870 showed that "Turmus Aya" had a total of 88 houses and a population of 301, though the population count included men only.
In 1882, the PEF's Survey of Western Palestine Turmus Aya was described as "a village on a low knoll, in a fertile plain, with a spring to the south. The village is of moderate size, and surrounded by fruit trees. On the
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https://en.wikipedia.org/wiki/Cutler%27s%20bar%20notation
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In mathematics, Cutler's bar notation is a notation system for large numbers, introduced by Mark Cutler in 2004. The idea is based on iterated exponentiation in much the same way that exponentiation is iterated multiplication.
Introduction
A regular exponential can be expressed as such:
However, these expressions become arbitrarily large when dealing with systems such as Knuth's up-arrow notation. Take the following:
Cutler's bar notation shifts these exponentials counterclockwise, forming . A bar is placed above the variable to denote this change. As such:
This system becomes effective with multiple exponents, when regular denotation becomes too cumbersome.
At any time, this can be further shortened by rotating the exponential counterclockwise once more.
The same pattern could be iterated a fourth time, becoming . For this reason, it is sometimes referred to as Cutler's circular notation.
Advantages and drawbacks
The Cutler bar notation can be used to easily express other notation systems in exponent form. It also allows for a flexible summarization of multiple copies of the same exponents, where any number of stacked exponents can be shifted counterclockwise and shortened to a single variable. The bar notation also allows for fairly rapid composure of very large numbers. For instance, the number would contain more than a googolplex digits, while remaining fairly simple to write with and remember.
However, the system reaches a problem when dealing with different exponents in a single expression. For instance, the expression could not be summarized in bar notation. Additionally, the exponent can only be shifted thrice before it returns to its original position, making a five degree shift indistinguishable from a one degree shift. Some have suggested using a double and triple bar in subsequent rotations, though this presents problems when dealing with ten- and twenty-degree shifts.
Other equivalent notations for the same operations already exist without being limited to a fixed number of recursions, notably Knuth's up-arrow notation and hyperoperation notation.
See also
Mathematical notation
References
Mark Cutler, Physical Infinity, 2004
Daniel Geisler, tetration.org
R. Knobel. "Exponentials Reiterated." American Mathematical Monthly 88, (1981)
Mathematical notation
Large numbers
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https://en.wikipedia.org/wiki/Camille-Christophe%20Gerono
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Camille-Christophe Gerono (1799 in Paris, France – 1891 in Paris) was a French mathematician. He concerned himself above all with geometry. The Lemniscate of Gerono or figure-eight curve was named after him. With Olry Terquem, he was founding co-editor in 1842 of the scientific journal Nouvelles Annales de Mathématiques.
References
1799 births
1891 deaths
19th-century French mathematicians
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https://en.wikipedia.org/wiki/Function%20series
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In calculus, a function series is a series, where the summands are not just real or complex numbers but functions.
Examples
Examples of function series include power series, Laurent series, Fourier series, etc.
Convergence
There exist many types of convergence for a function series, such as uniform convergence, pointwise convergence, almost everywhere convergence, etc.
The Weierstrass M-test is a useful result in studying convergence of function series.
See also
Function space
References
Chun Wa Wong (2013) Introduction to Mathematical Physics: Methods & Concepts Oxford University Press p. 655
Mathematical analysis
Mathematical series
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https://en.wikipedia.org/wiki/Aspherical
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Aspherical may refer to:
Aspherical space, a concept in topology
Aspherical lens, a type of lens assembly used in photography which contains an aspheric lens, sometimes referred to as ASPH
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https://en.wikipedia.org/wiki/Pillai%20prime
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In number theory, a Pillai prime is a prime number p for which there is an integer n > 0 such that the factorial of n is one less than a multiple of the prime, but the prime is not one more than a multiple of n. To put it algebraically, but . The first few Pillai primes are
23, 29, 59, 61, 67, 71, 79, 83, 109, 137, 139, 149, 193, ...
Pillai primes are named after the mathematician Subbayya Sivasankaranarayana Pillai, who studied these numbers. Their infinitude has been proved several times, by Subbarao, Erdős, and Hardy & Subbarao.
References
.
.
https://planetmath.org/pillaiprime, PlanetMath
Classes of prime numbers
Eponymous numbers in mathematics
Factorial and binomial topics
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https://en.wikipedia.org/wiki/Perfect%20totient%20number
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In number theory, a perfect totient number is an integer that is equal to the sum of its iterated totients. That is, one applies the totient function to a number n, apply it again to the resulting totient, and so on, until the number 1 is reached, and adds together the resulting sequence of numbers; if the sum equals n, then n is a perfect totient number.
Examples
For example, there are six positive integers less than 9 and relatively prime to it, so the totient of 9 is 6; there are two numbers less than 6 and relatively prime to it, so the totient of 6 is 2; and there is one number less than 2 and relatively prime to it, so the totient of 2 is 1; and , so 9 is a perfect totient number.
The first few perfect totient numbers are
3, 9, 15, 27, 39, 81, 111, 183, 243, 255, 327, 363, 471, 729, 2187, 2199, 3063, 4359, 4375, ... .
Notation
In symbols, one writes
for the iterated totient function. Then if c is the integer such that
one has that n is a perfect totient number if
Multiples and powers of three
It can be observed that many perfect totient are multiples of 3; in fact, 4375 is the smallest perfect totient number that is not divisible by 3. All powers of 3 are perfect totient numbers, as may be seen by induction using the fact that
Venkataraman (1975) found another family of perfect totient numbers: if is prime, then 3p is a perfect totient number. The values of k leading to perfect totient numbers in this way are
0, 1, 2, 3, 6, 14, 15, 39, 201, 249, 1005, 1254, 1635, ... .
More generally if p is a prime number greater than 3, and 3p is a perfect totient number, then p ≡ 1 (mod 4) (Mohan and Suryanarayana 1982). Not all p of this form lead to perfect totient numbers; for instance, 51 is not a perfect totient number. Iannucci et al. (2003) showed that if 9p is a perfect totient number then p is a prime of one of three specific forms listed in their paper. It is not known whether there are any perfect totient numbers of the form 3kp where p is prime and k > 3.
References
Integer sequences
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https://en.wikipedia.org/wiki/Negative%20pedal%20curve
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In geometry, a negative pedal curve is a plane curve that can be constructed from another plane curve C and a fixed point P on that curve. For each point X ≠ P on the curve C, the negative pedal curve has a tangent that passes through X and is perpendicular to line XP. Constructing the negative pedal curve is the inverse operation to constructing a pedal curve.
Definition
In the plane, for every point X other than P there is a unique line through X perpendicular to XP. For a given curve in the plane and a given fixed point P, called the pedal point, the negative pedal curve is the envelope of the lines XP for which X lies on the given curve.
Parameterization
For a parametrically defined curve, its negative pedal curve with pedal point (0; 0) is defined as
Properties
The negative pedal curve of a pedal curve with the same pedal point is the original curve.
See also
Fish curve, the negative pedal curve of an ellipse with squared eccentricity 1/2
External links
Negative pedal curve on Mathworld
Curves
Differential geometry
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https://en.wikipedia.org/wiki/List%20of%20Omiya%20Ardija%20records%20and%20statistics
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This article contains records and statistics for the Japanese professional football club, Omiya Ardija.
J.League
Domestic cup competitions
Top scorers by season
References
Omiya Ardija
Omiya Ardija
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https://en.wikipedia.org/wiki/List%20of%20Kashiwa%20Reysol%20records%20and%20statistics
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This article contains records and statistics for the Japanese professional football club, Kashiwa Reysol.
J.League
Domestic cup competitions
Top scorers by season
References
Kashiwa Reysol
Kashiwa Reysol
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https://en.wikipedia.org/wiki/National%20Statistics%20Socio-economic%20Classification
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The National Statistics Socio-economic Classification (often abbreviated to NS-SEC) is the official socio-economic classification in the United Kingdom. It is an adaptation of the Goldthorpe schema which was first known as the Nuffield Class Schema developed in the 1970s. It was developed using the Standard Occupational Classification 1990 (SOC90) and rebased on the Standard Occupational Classification 2000 (SOC2000) before its first major use on the 2001 UK census. The NS-SEC replaced two previous social classifications: Socio-economic Groups (SEG) and Social Class based on Occupation (SC, formerly known as Registrar General's Social Class, RGSC). The NS-SEC was rebased on the Standard Occupational Classification 2010 prior to the 2011 UK census and it will be further rebased on the new Standard Occupational Classification 2020 for use on the 2021 UK census.
The NS-SEC is a nested classification. It has 14 operational categories, with some sub-categories, and is commonly used in eight-class, five-class, and three-class versions. Only the three-category version is intended to represent any form of hierarchy. The version intended for most users (the analytic version) has eight classes:
Higher managerial and professional occupations
Lower managerial and professional occupations
Intermediate occupations (clerical, sales, service)
Small employers and own account workers
Lower supervisory and technical occupations
Semi-routine occupations
Routine occupations
Never worked or long-term unemployed
The three-class version is reduced to following:
Higher occupations
Intermediate occupations
Lower occupations
See also
ACORN (demographics)
NRS social grade
References
Demographics of the United Kingdom
Office for National Statistics
Socio-economic mobility
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https://en.wikipedia.org/wiki/Caccioppoli%20set
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In mathematics, a Caccioppoli set is a set whose boundary is measurable and has (at least locally) a finite measure. A synonym is set of (locally) finite perimeter. Basically, a set is a Caccioppoli set if its characteristic function is a function of bounded variation.
History
The basic concept of a Caccioppoli set was first introduced by the Italian mathematician Renato Caccioppoli in the paper : considering a plane set or a surface defined on an open set in the plane, he defined their measure or area as the total variation in the sense of Tonelli of their defining functions, i.e. of their parametric equations, provided this quantity was bounded. The measure of the boundary of a set was defined as a functional, precisely a set function, for the first time: also, being defined on open sets, it can be defined on all Borel sets and its value can be approximated by the values it takes on an increasing net of subsets. Another clearly stated (and demonstrated) property of this functional was its lower semi-continuity.
In the paper , he precised by using a triangular mesh as an increasing net approximating the open domain, defining positive and negative variations whose sum is the total variation, i.e. the area functional. His inspiring point of view, as he explicitly admitted, was those of Giuseppe Peano, as expressed by the Peano-Jordan Measure: to associate to every portion of a surface an oriented plane area in a similar way as an approximating chord is associated to a curve. Also, another theme found in this theory was the extension of a functional from a subspace to the whole ambient space: the use of theorems generalizing the Hahn–Banach theorem is frequently encountered in Caccioppoli research. However, the restricted meaning of total variation in the sense of Tonelli added much complication to the formal development of the theory, and the use of a parametric description of the sets restricted its scope.
Lamberto Cesari introduced the "right" generalization of functions of bounded variation to the case of several variables only in 1936: perhaps, this was one of the reasons that induced Caccioppoli to present an improved version of his theory only nearly 24 years later, in the talk at the IV UMI Congress in October 1951, followed by five notes published in the Rendiconti of the Accademia Nazionale dei Lincei. These notes were sharply criticized by Laurence Chisholm Young in the Mathematical Reviews.
In 1952 Ennio de Giorgi presented his first results, developing the ideas of Caccioppoli, on the definition of the measure of boundaries of sets at the Salzburg Congress of the Austrian Mathematical Society: he obtained this results by using a smoothing operator, analogous to a mollifier, constructed from the Gaussian function, independently proving some results of Caccioppoli. Probably he was led to study this theory by his teacher and friend Mauro Picone, who had also been the teacher of Caccioppoli and was likewise his friend. De Giorgi me
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https://en.wikipedia.org/wiki/Richard%20Earl%20Block
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Richard Earl Block (born 1931) is a mathematician at the University of California, Riverside who works on Lie algebras over fields of prime characteristic.
Block earned his Ph.D. from the University of Chicago in 1956 under the supervision of Abraham Adrian Albert. He was the first to discover the central extension of the Witt algebra that gives the Virasoro algebra, though his discovery went unnoticed for many years. With Robert Lee Wilson he classified the simple Lie algebras over "well behaved" fields of finite characteristic.
In 2012 he became a fellow of the American Mathematical Society.
Selected publications
Richard E. Block, On the Mills–Seligman axioms for Lie algebras of classical type, Transactions of the American Mathematical Society, 121 (1966), pp. 378–392. (The paper that gives the central extension defining the Virasoro algebra.)
Richard E. Block and Robert Lee Wilson, The Restricted Simple Lie Algebras are of Classical or Cartan Type, Proceedings of the National Academy of Sciences of the United States of America August 15, 1984 vol. 81, no. 16, 5271–5274.
References
Living people
1931 births
20th-century American mathematicians
21st-century American mathematicians
University of Chicago alumni
University of California, Riverside faculty
Fellows of the American Mathematical Society
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https://en.wikipedia.org/wiki/Zonal%20polynomial
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In mathematics, a zonal polynomial is a multivariate symmetric homogeneous polynomial. The zonal polynomials form a basis of the space of symmetric polynomials.
They appear as zonal spherical functions of the Gelfand pairs
(here, is the hyperoctahedral group) and , which means that they describe canonical basis of the double class
algebras and .
They are applied in multivariate statistics.
The zonal polynomials are the case of the C normalization of the Jack function.
References
Robb Muirhead, Aspects of Multivariate Statistical Theory, John Wiley & Sons, Inc., New York, 1984.
Homogeneous polynomials
Symmetric functions
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https://en.wikipedia.org/wiki/Discrete%20measure
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In mathematics, more precisely in measure theory, a measure on the real line is called a discrete measure (in respect to the Lebesgue measure) if it is concentrated on an at most countable set. The support need not be a discrete set. Geometrically, a discrete measure (on the real line, with respect to Lebesgue measure) is a collection of point masses.
Definition and properties
A measure defined on the Lebesgue measurable sets of the real line with values in is said to be discrete if there exists a (possibly finite) sequence of numbers
such that
The simplest example of a discrete measure on the real line is the Dirac delta function One has and
More generally, if is a (possibly finite) sequence of real numbers, is a sequence of numbers in of the same length, one can consider the Dirac measures defined by
for any Lebesgue measurable set Then, the measure
is a discrete measure. In fact, one may prove that any discrete measure on the real line has this form for appropriately chosen sequences and
Extensions
One may extend the notion of discrete measures to more general measure spaces. Given a measurable space and two measures and on it, is said to be discrete in respect to if there exists an at most countable subset of such that
All singletons with are measurable (which implies that any subset of is measurable)
Notice that the first two requirements are always satisfied for an at most countable subset of the real line if is the Lebesgue measure, so they were not necessary in the first definition above.
As in the case of measures on the real line, a measure on is discrete in respect to another measure on the same space if and only if has the form
where the singletons are in and their measure is 0.
One can also define the concept of discreteness for signed measures. Then, instead of conditions 2 and 3 above one should ask that be zero on all measurable subsets of and be zero on measurable subsets of
References
External links
Measures (measure theory)
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https://en.wikipedia.org/wiki/Conservative%20system
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In mathematics, a conservative system is a dynamical system which stands in contrast to a dissipative system. Roughly speaking, such systems have no friction or other mechanism to dissipate the dynamics, and thus, their phase space does not shrink over time. Precisely speaking, they are those dynamical systems that have a null wandering set: under time evolution, no portion of the phase space ever "wanders away", never to be returned to or revisited. Alternately, conservative systems are those to which the Poincaré recurrence theorem applies. An important special case of conservative systems are the measure-preserving dynamical systems.
Informal introduction
Informally, dynamical systems describe the time evolution of the phase space of some mechanical system. Commonly, such evolution is given by some differential equations, or quite often in terms of discrete time steps. However, in the present case, instead of focusing on the time evolution of discrete points, one shifts attention to the time evolution of collections of points. One such example would be Saturn's rings: rather than tracking the time evolution of individual grains of sand in the rings, one is instead interested in the time evolution of the density of the rings: how the density thins out, spreads, or becomes concentrated. Over short time-scales (hundreds of thousands of years), Saturn's rings are stable, and are thus a reasonable example of a conservative system and more precisely, a measure-preserving dynamical system. It is measure-preserving, as the number of particles in the rings does not change, and, per Newtonian orbital mechanics, the phase space is incompressible: it can be stretched or squeezed, but not shrunk (this is the content of Liouville's theorem).
Formal definition
Formally, a measurable dynamical system is conservative if and only if it is non-singular, and has no wandering sets.
A measurable dynamical system (X, Σ, μ, τ) is a Borel space (X, Σ) equipped with a sigma-finite measure μ and a transformation τ. Here, X is a set, and Σ is a sigma-algebra on X, so that the pair (X, Σ) is a measurable space. μ is a sigma-finite measure on the sigma-algebra. The space X is the phase space of the dynamical system.
A transformation (a map) is said to be Σ-measurable if and only if, for every σ ∈ Σ, one has . The transformation is a single "time-step" in the evolution of the dynamical system. One is interested in invertible transformations, so that the current state of the dynamical system came from a well-defined past state.
A measurable transformation is called non-singular when if and only if . In this case, the system (X, Σ, μ, τ) is called a non-singular dynamical system. The condition of being non-singular is necessary for a dynamical system to be suitable for modeling (non-equilibrium) systems. That is, if a certain configuration of the system is "impossible" (i.e. ) then it must stay "impossible" (was always impossible: ), but otherwise, the system can e
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https://en.wikipedia.org/wiki/Alperin%E2%80%93Brauer%E2%80%93Gorenstein%20theorem
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In mathematics, the Alperin–Brauer–Gorenstein theorem characterizes the finite simple groups with quasidihedral or wreathed Sylow 2-subgroups. These are isomorphic either to three-dimensional projective special linear groups or projective special unitary groups over a finite field of odd order, depending on a certain congruence, or to the Mathieu group . proved this in the course of 261 pages. The subdivision by 2-fusion is sketched there, given as an exercise in , and presented in some detail in .
Notes
References
Theorems about finite groups
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https://en.wikipedia.org/wiki/Edward%20Copson
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Edward Thomas Copson FRSE (21 August 1901 – 16 February 1980) was a British mathematician who contributed widely to the development of mathematics at the University of St Andrews, serving as Regius Professor of Mathematics amongst other positions.
Life
He was born in Coventry, and was a pupil at King Henry VIII School, Coventry. He studied at St John's College, Oxford. He was appointed by E. T. Whittaker as a lecturer at the University of Edinburgh, where he was later awarded a DSc.
He married Beatrice, the elder daughter of E. T. Whittaker, and moved to the University of St Andrews where he was Regius Professor of Mathematics, and later Dean of Science, then Master of the United College. He was instrumental in the construction of the new Mathematics Institute building at the University.
He was elected a Fellow of the Royal Society of Edinburgh in 1924, his proposers being Sir Edmund Taylor Whittaker, Herbert Stanley Allen, Bevan Braithwaite Baker and A. Crichton Mitchell. He was awarded the Keith Medal by the Royal Society of Edinburgh in 1942 for his research in mathematics. He served as the Society's Vice President from 1950-53.
Work
Copson's primary focus was in classical analysis, asymptotic expansions, differential and integral equations, and applications to problems in theoretical physics. His first book "The theory of functions of a complex variable" was published in 1935.
Publications
Copson, E. T., An Introduction to the Theory of Functions of A Complex Variable (1935)
Baker, Bevan Braithwaite; Copson, E. T., "The Mathematical Theory of Huygens' Principle" (1939); 2nd edition 1950; 3rd edition 1987 with several reprints
Copson, E. T., Asymptotic Expansions (1965); reprint 1976; 2nd edition 2004
Copson, E. T., Metric Spaces (1968); reprint with corrections 1972; reprint 1979; pbk. reprint 1988
Copson, E. T., Partial Differential Equations (1975)
References
Alumni of St John's College, Oxford
Alumni of the University of Edinburgh
20th-century British mathematicians
People educated at King Henry VIII School, Coventry
1901 births
1980 deaths
Academics of the University of St Andrews
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https://en.wikipedia.org/wiki/Environmental%20statistics
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Environment statistics is the application of statistical methods to environmental science. It covers procedures for dealing with questions concerning the natural environment in its undisturbed state, the interaction of humanity with the environment, and urban environments. The field of environmental statistics has seen rapid growth in the past few decades as a response to increasing concern over the environment in the public, organizational, and governmental sectors.
The United Nations' Framework for the Development of Environment Statistics (FDES) defines the scope of environment statistics as follows:
The scope of environment statistics covers biophysical aspects of the environment and those aspects of the socio-economic system that directly influence and interact with the environment.
The scope of environment, social and economic statistics overlap. It is not easy – or necessary – to draw a clear line dividing these areas. Social and economic statistics that describe processes or activities with a direct impact on, or direct interaction with, the environment are used widely in environment statistics. They are within the scope of the FDES.
Uses
Statistical analysis is essential to the field of environmental sciences, allowing researchers to gain an understanding of environmental issues through researching and developing potential solutions to the issues they study. The applications of statistical methods to environmental sciences are numerous and varied. Environmental statistics are used in many fields including; health and safety organizations, standard bodies, research institutes, water and river authorities, meteorological organizations, fisheries, protection agencies, and in risk, pollution, regulation and control concerns.
Environmental statistics is especially pertinent and widely used in the academic, governmental, regulatory, technological, and consulting industries.
Specific applications of statistical analysis within the field of environmental science include earthquake risk analysis, environmental policymaking, ecological sampling planning, environmental forensics.
Within the scope of environmental statistics, there are two main categories of their uses.
Descriptive statistics is not used to make inferences about data, but simply to describe its characteristics.
Inferential statistics is used to make inferences about data, test hypotheses or make predictions.
Types of studies covered in environmental statistics include:
Baseline studies to document the present state of an environment to provide background in case of unknown changes in the future;
Targeted studies to describe the likely impact of changes being planned or of accidental occurrences;
Regular monitoring to attempt to detect changes in the environment.
Sources
Sources of data for environmental statistics are varied and include surveys related to human populations and the environment, records from agencies managing environmental resources, maps and images, equ
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https://en.wikipedia.org/wiki/Null%20distribution
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In statistical hypothesis testing, the null distribution is the probability distribution of the test statistic when the null hypothesis is true.
For example, in an F-test, the null distribution is an F-distribution.
Null distribution is a tool scientists often use when conducting experiments. The null distribution is the distribution of two sets of data under a null hypothesis. If the results of the two sets of data are not outside the parameters of the expected results, then the null hypothesis is said to be true.
Examples of application
The null hypothesis is often a part of an experiment. The null hypothesis tries to show that among two sets of data, there is no statistical difference between the results of doing one thing as opposed to doing a different thing. For an example of this, a scientist might be trying to prove that people who walk two miles a day have healthier hearts than people who walk less than two miles a day. The scientist would use the null hypothesis to test the health of the hearts of people who walked two miles a day against the health of the hearts of the people who walked less than two miles a day. If there was no difference between their heart rate, then the scientist would be able to say that the test statistics would follow the null distribution. Then the scientists could determine that if there was significant difference that means the test follows the alternative distribution.
Obtaining the null distribution
In the procedure of hypothesis testing, one needs to form the joint distribution of test statistics to conduct the test and control type I errors. However, the true distribution is often unknown and a proper null distribution ought to be used to represent the data. For example, one sample and two samples tests of means can use t statistics which have Gaussian null distribution, while F statistics, testing k groups of population means, which have Gaussian quadratic form the null distribution. The null distribution is defined as the asymptotic distributions of null quantile-transformed test statistics, based on marginal null distribution. During practice, the test statistics of the null distribution is often unknown, since it relies on the unknown data generating distribution. Resampling procedures, such as non-parametric or model-based bootstrap, can provide consistent estimators for the null distributions. Improper choice of the null distribution poses significant influence on type I error and power properties in the testing process. Another approach to obtain the test statistics null distribution is to use the data of generating null distribution estimation.
Null distribution with large sample size
The null distribution plays a crucial role in large scale testing. Large sample size allows us to implement a more realistic empirical null distribution. One can generate the empirical null using an MLE fitting algorithm. Under a Bayesian framework, the large-scale studies allow the null distribution to be put int
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https://en.wikipedia.org/wiki/Thomas%20William%20K%C3%B6rner
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Thomas William Körner (born 17 February 1946) is a British pure mathematician and the author of three books on popular mathematics. He is titular Professor of Fourier Analysis in the University of Cambridge and a Fellow of Trinity Hall. He is the son of the philosopher Stephan Körner and of Edith Körner.
He studied at Trinity Hall, Cambridge, and wrote his PhD thesis Some Results on Kronecker, Dirichlet and Helson Sets there in 1971, studying under Nicholas Varopoulos. In 1972 he won the Salem Prize.
He has written academic mathematics books aimed at undergraduates:
Fourier Analysis
Exercises for Fourier Analysis
A Companion to Analysis
Vectors, Pure and Applied
Calculus for the Ambitious
He has also written three books aimed at secondary school students, the popular 1996 title The Pleasures of Counting, Naive Decision Making (published 2008) on probability, statistics and game theory, and Where Do Numbers Come From? (published October 2019).
External links
Professor Körner's website
References
1946 births
Living people
Alumni of Trinity Hall, Cambridge
Fellows of Trinity Hall, Cambridge
20th-century British mathematicians
21st-century British mathematicians
Mathematical analysts
Cambridge mathematicians
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https://en.wikipedia.org/wiki/David%20Bergamini
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David Howland Bergamini (11 October 1928 – 3 September 1983, in Tokyo) was an American author who wrote books on 20th-century history and popular science, notably mathematics.
Bergamini was interned as an Allied civilian in a Japanese concentration camp in the Philippines with his mother and father, John Van Wie Bergamini, an architect who worked for the American Episcopal Mission in China, Japan, the Philippines and Africa, and younger sister for the duration of World War II.
From 1949 to 1951 Bergamini studied at Merton College, Oxford on a Rhodes Scholarship. In 1951 he joined Time as a reporter; in 1961 he was appointed Assistant Editor of Life magazine.
According to Professor Charles Sheldon of the University of Cambridge, his 1971 book on Japan's Imperial Conspiracy "is a polemic which, to our knowledge, contradicts all previous scholarly work.... Specialists on Japan have unanimously demolished Bergamini's thesis and his pretensions to careful scholarship.
Partial bibliography
The Fleet in the Window (a novel published in 1961)
The Universe (Life Nature Library) (1962; revised 1966, 1967)
Mathematics (Life Science Library) (1963)
Australia, Its Land and Wildlife (1964)
The Scientist (Life Science Library) (1965)
Japan's Imperial Conspiracy (1971),
Venus Development (a novel published in 1976)
References
1928 births
1983 deaths
20th-century American novelists
20th-century American male writers
American science writers
American historical novelists
American war novelists
Dartmouth College alumni
20th-century American historians
American male novelists
American male non-fiction writers
Alumni of Merton College, Oxford
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https://en.wikipedia.org/wiki/Goodman%20and%20Kruskal%27s%20gamma
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In statistics, Goodman and Kruskal's gamma is a measure of rank correlation, i.e., the similarity of the orderings of the data when ranked by each of the quantities. It measures the strength of association of the cross tabulated data when both variables are measured at the ordinal level. It makes no adjustment for either table size or ties. Values range from −1 (100% negative association, or perfect inversion) to +1 (100% positive association, or perfect agreement). A value of zero indicates the absence of association.
This statistic (which is distinct from Goodman and Kruskal's lambda) is named after Leo Goodman and William Kruskal, who proposed it in a series of papers from 1954 to 1972.
Definition
The estimate of gamma, G, depends on two quantities:
Ns, the number of pairs of cases ranked in the same order on both variables (number of concordant pairs),
Nd, the number of pairs of cases ranked in reversed order on both variables (number of reversed pairs),
where "ties" (cases where either of the two variables in the pair are equal) are dropped.
Then
This statistic can be regarded as the maximum likelihood estimator for the theoretical quantity , where
and where Ps and Pd are the probabilities that a randomly selected pair of observations will place in the same or opposite order respectively, when ranked by both variables.
Critical values for the gamma statistic are sometimes found by using an approximation, whereby a transformed value, t of the statistic is referred to Student t distribution, where
and where n is the number of observations (not the number of pairs):
Yule's Q
A special case of Goodman and Kruskal's gamma is Yule's Q, also known as the Yule coefficient of association, which is specific to 2×2 matrices. Consider the following contingency table of events, where each value is a count of an event's frequency:
Yule's Q is given by:
Although computed in the same fashion as Goodman and Kruskal's gamma, it has a slightly broader interpretation because the distinction between nominal and ordinal scales becomes a matter of arbitrary labeling for dichotomous distinctions. Thus, whether Q is positive or negative depends merely on which pairings the analyst considers to be concordant, but is otherwise symmetric.
Q varies from −1 to +1. −1 reflects total negative association, +1 reflects perfect positive association and 0 reflects no association at all. The sign depends on which pairings the analyst initially considered to be concordant, but this choice does not affect the magnitude.
In term of the odds ratio OR, Yule's Q is given by
and so Yule's Q and Yule's Y are related by
See also
Kendall tau rank correlation coefficient
Goodman and Kruskal's lambda
Yule's Y, also known as the coefficient of colligation
References
Further reading
Sheskin, D.J. (2007) The Handbook of Parametric and Nonparametric Statistical Procedures. Chapman & Hall/CRC,
Rankings
Statistical tests
Summary statistics for contingency tables
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https://en.wikipedia.org/wiki/Los%20Lagos%2C%20Chile
|
Los Lagos () is a Chilean city and commune in Valdivia Province, Los Ríos Region. San Pedro River passes by the city.
Demographics
According to the 2002 census of the National Statistics Institute, Los Lagos spans an area of and has 20,168 inhabitants (10,370 men and 9,798 women). Of these, 9,479 (47%) lived in urban areas and 10,689 (53%) in rural areas. The population grew by 8.6% (1,604 persons) between the 1992 and 2002 censuses.
It has a large German presence, like the entire Los Rios and Los Lagos Regions.
Administration
As a commune, Los Lagos is a third-level administrative division of Chile administered by a municipal council, headed by an alcalde who is directly elected every four years. The 2008-2012 alcalde is Samuel Torres Sepúlveda (Ind.).
Within the electoral divisions of Chile, Los Lagos is represented in the Chamber of Deputies by Enrique Jaramillo (PDC) and Gastón Von Mühlenbrock (UDI) as part of the 54th electoral district, together with Panguipulli, Futrono, Lago Ranco, Río Bueno, La Unión and Paillaco. The commune is represented in the as part of the 16th senatorial constituency (Los Ríos Region).
References
External links
Municipality of Los Lagos
Populated places in Valdivia Province
Communes of Chile
Populated places established in 1891
1891 establishments in Chile
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https://en.wikipedia.org/wiki/List%20of%20Kawasaki%20Frontale%20records%20and%20statistics
|
This article contains records and statistics for the Japanese professional football club, Kawasaki Frontale.
J.League
Key
Domestic cup competitions
Japan Football League (JFL)
Key
Points: 90min win – 3 points, Extra Time win (sudden death) – 2 points, PK win after Extra Time – 1 point, Any lost – 0 point
In 1998, Kawasaki attended ""J1 Participation Tournament"", and lost at 1st round.
Major international competitions
Continental record
Top scorers by season
References
Kawasaki Frontale
Kawasaki Frontale
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https://en.wikipedia.org/wiki/Aczel%27s%20anti-foundation%20axiom
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In the foundations of mathematics, Aczel's anti-foundation axiom is an axiom set forth by , as an alternative to the axiom of foundation in Zermelo–Fraenkel set theory. It states that every accessible pointed directed graph corresponds to exactly one set. In particular, according to this axiom, the graph consisting of a single vertex with a loop corresponds to a set that contains only itself as element, i.e. a Quine atom. A set theory obeying this axiom is necessarily a non-well-founded set theory.
Accessible pointed graphs
An accessible pointed graph is a directed graph with a distinguished vertex (the "root") such that for any node in the graph there is at least one path in the directed graph from the root to that node.
The anti-foundation axiom postulates that each such directed graph corresponds to the membership structure of exactly one set. For example, the directed graph with only one node and an edge from that node to itself corresponds to a set of the form x = {x}.
See also
von Neumann universe
References
Axioms of set theory
Directed graphs
de:Fundierungsaxiom#Mengenlehren ohne Fundierungsaxiom
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https://en.wikipedia.org/wiki/Finite%20von%20Neumann%20algebra
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In mathematics, a finite von Neumann algebra is a von Neumann algebra in which every isometry is a unitary. In other words, for an operator V in a finite von Neumann algebra if , then . In terms of the comparison theory of projections, the identity operator is not (Murray-von Neumann) equivalent to any proper subprojection in the von Neumann algebra.
Properties
Let denote a finite von Neumann algebra with center . One of the fundamental characterizing properties of finite von Neumann algebras is the existence of a center-valued trace. A von Neumann algebra is finite if and only if there exists a normal positive bounded map with the properties:
,
if and then ,
for ,
for and .
Examples
Finite-dimensional von Neumann algebras
The finite-dimensional von Neumann algebras can be characterized using Wedderburn's theory of semisimple algebras.
Let Cn × n be the n × n matrices with complex entries. A von Neumann algebra M is a self adjoint subalgebra in Cn × n such that M contains the identity operator I in Cn × n.
Every such M as defined above is a semisimple algebra, i.e. it contains no nilpotent ideals. Suppose M ≠ 0 lies in a nilpotent ideal of M. Since M* ∈ M by assumption, we have M*M, a positive semidefinite matrix, lies in that nilpotent ideal. This implies (M*M)k = 0 for some k. So M*M = 0, i.e. M = 0.
The center of a von Neumann algebra M will be denoted by Z(M). Since M is self-adjoint, Z(M) is itself a (commutative) von Neumann algebra. A von Neumann algebra N is called a factor if Z(N) is one-dimensional, that is, Z(N) consists of multiples of the identity I.
Theorem Every finite-dimensional von Neumann algebra M is a direct sum of m factors, where m is the dimension of Z(M).
Proof: By Wedderburn's theory of semisimple algebras, Z(M) contains a finite orthogonal set of idempotents (projections) {Pi} such that PiPj = 0 for i ≠ j, Σ Pi = I, and
where each Z(M)Pi is a commutative simple algebra. Every complex simple algebras is isomorphic to
the full matrix algebra Ck × k for some k. But Z(M)Pi is commutative, therefore one-dimensional.
The projections Pi "diagonalizes" M in a natural way. For M ∈ M, M can be uniquely decomposed into M = Σ MPi. Therefore,
One can see that Z(MPi) = Z(M)Pi. So Z(MPi) is one-dimensional and each MPi is a factor. This proves the claim.
For general von Neumann algebras, the direct sum is replaced by the direct integral. The above is a special case of the central decomposition of von Neumann algebras.
Abelian von Neumann algebras
Type factors
References
Linear algebra
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https://en.wikipedia.org/wiki/Of%20the%20form
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In mathematics, the phrase "of the form" indicates that a mathematical object, or (more frequently) a collection of objects, follows a certain pattern of expression. It is frequently used to reduce the formality of mathematical proofs.
Example of use
Here is a proof which should be appreciable with limited mathematical background:
Statement:
The product of any two even natural numbers is also even.
Proof:
Any even natural number is of the form 2n, where n is a natural number. Therefore, let us assume that we have two even numbers which we will denote by 2k and 2l. Their product is (2k)(2l) = 4(kl) = 2(2kl). Since 2kl is also a natural number, the product is even.
Note:
In this case, both exhaustivity and exclusivity were needed. That is, it was not only necessary that every even number is of the form 2n (exhaustivity), but also that every expression of the form 2n is an even number (exclusivity). This will not be the case in every proof, but normally, at least exhaustivity is implied by the phrase of the form.
References
External links
Mathematical proofs
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https://en.wikipedia.org/wiki/Rahul%20Pandharipande
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Rahul Pandharipande (born 1969) is a mathematician who is currently a professor of mathematics at the Swiss Federal Institute of Technology Zürich (ETH) working in algebraic geometry. His particular interests
concern moduli spaces, enumerative invariants associated to moduli spaces, such as Gromov–Witten invariants and Donaldson–Thomas invariants, and the cohomology of the moduli space of curves. His father Vijay Raghunath Pandharipande was a renowned theoretical physicist who worked in the area of nuclear physics.
Educational and professional history
He received his A.B. from Princeton University in 1990 and his PhD from Harvard University in 1994 with a thesis entitled `A Compactification over the Moduli Space of Stable Curves of the Universal Moduli Space of Slope-Semistable Vector Bundles'. His thesis advisor at Harvard was Joe Harris. After teaching at the University of Chicago and the California Institute of Technology, he joined the faculty as Professor of Mathematics at Princeton University in 2002. In 2011, he accepted a Professorship at ETH Zürich. In 2022, he was awarded an honorary degree - Doctor of Science from the University of Illinois Urbana-Champaign.
References
External links
1969 births
20th-century Indian mathematicians
Academic staff of ETH Zurich
Indian emigrants to the United States
Algebraic geometers
21st-century Indian mathematicians
Harvard University faculty
Living people
Princeton University alumni
Princeton University faculty
People from Amravati
American people of Marathi descent
Harvard University alumni
American people of Indian descent
University Laboratory High School (Urbana, Illinois) alumni
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https://en.wikipedia.org/wiki/Operational%20calculus
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Operational calculus, also known as operational analysis, is a technique by which problems in analysis, in particular differential equations, are transformed into algebraic problems, usually the problem of solving a polynomial equation.
History
The idea of representing the processes of calculus, differentiation and integration, as operators
has a long history that goes back to Gottfried Wilhelm Leibniz. The mathematician Louis François Antoine Arbogast was one of the first to manipulate these symbols independently of the function to which they were applied.
This approach was further developed by Francois-Joseph Servois who developed convenient notations. Servois was followed by a school of British and Irish mathematicians including Charles James Hargreave, George Boole, Bownin, Carmichael, Doukin, Graves, Murphy, William Spottiswoode and Sylvester.
Treatises describing the application of operator methods to ordinary and partial differential equations were written by Robert Bell Carmichael in 1855 and by Boole in 1859.
This technique was fully developed by the physicist Oliver Heaviside in 1893, in connection with his work in telegraphy.
Guided greatly by intuition and his wealth of knowledge on the physics behind his circuit studies, [Heaviside] developed the operational calculus now ascribed to his name.
At the time, Heaviside's methods were not rigorous, and his work was not further developed by mathematicians.
Operational calculus first found applications in electrical engineering problems, for
the calculation of transients in linear circuits after 1910, under the impulse of Ernst Julius Berg, John Renshaw Carson and Vannevar Bush.
A rigorous mathematical justification of Heaviside's operational methods came only
after the work of Bromwich that related operational calculus with
Laplace transformation methods (see the books by Jeffreys, by Carslaw or by MacLachlan for a detailed exposition).
Other ways of justifying the operational methods of Heaviside were introduced in the mid-1920s using
integral equation techniques (as done by Carson) or Fourier transformation (as done by Norbert Wiener).
A different approach to operational calculus was developed in the 1930s by Polish mathematician
Jan Mikusiński, using algebraic reasoning.
Norbert Wiener laid the foundations for operator theory in his review of the existential status of the operational calculus in 1926:
The brilliant work of Heaviside is purely heuristic, devoid of even the pretense to mathematical rigor. Its operators apply to electric voltages and currents, which may be discontinuous and certainly need not be analytic. For example, the favorite corpus vile on which he tries out his operators is a function which vanishes to the left of the origin and is 1 to the right. This excludes any direct application of the methods of Pincherle…
Although Heaviside’s developments have not been justified by the present state of the purely mathematical theory of operators, there i
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https://en.wikipedia.org/wiki/CricketGraph
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CricketGraph was a graphic software program for the Apple Macintosh by Cricket Software sold until 1996. It could take tabulated data and create common business and statistics graphs such as bar chart, pie chart, scatter plots and radial plots. These graphs could be saved in common image formats such as PICT and EPS and added to other documents. It did not have the same capabilities as a spreadsheet.
Competition
The main competitor was Visual Business, as well as the built-in graphing packages in Microsoft Excel, Informix Wingz and specialty statistics software such as Systat.
Although this package was written cleanly enough to run on much later versions of Classic Mac OS, the feature set was eventually superseded by packages such as DataGraph, DeltaGraph and KaleidaGraph.
See also
List of information graphics software
References
Plotting software
Classic Mac OS software
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https://en.wikipedia.org/wiki/Equiareal%20map
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In differential geometry, an equiareal map, sometimes called an authalic map, is a smooth map from one surface to another that preserves the areas of figures.
Properties
If M and N are two Riemannian (or pseudo-Riemannian) surfaces, then an equiareal map f from M to N can be characterized by any of the following equivalent conditions:
The surface area of f(U) is equal to the area of U for every open set U on M.
The pullback of the area element μN on N is equal to μM, the area element on M.
At each point p of M, and tangent vectors v and w to M at p,</p><p>where denotes the Euclidean wedge product of vectors and df denotes the pushforward along f.
Example
An example of an equiareal map, due to Archimedes of Syracuse, is the projection from the unit sphere to the unit cylinder outward from their common axis. An explicit formula is
for (x, y, z) a point on the unit sphere.
Linear transformations
Every Euclidean isometry of the Euclidean plane is equiareal, but the converse is not true. In fact, shear mapping and squeeze mapping are counterexamples to the converse.
Shear mapping takes a rectangle to a parallelogram of the same area. Written in matrix form, a shear mapping along the -axis is
Squeeze mapping lengthens and contracts the sides of a rectangle in a reciprocal manner so that the area is preserved. Written in matrix form, with λ > 1 the squeeze reads
A linear transformation multiplies areas by the absolute value of its determinant .
Gaussian elimination shows that every equiareal linear transformation (rotations included) can be obtained by composing at most two shears along the axes, a squeeze and (if the determinant is negative), a reflection.
In map projections
In the context of geographic maps, a map projection is called equal-area, equivalent, authalic, equiareal, or area-preserving, if areas are preserved up to a constant factor; embedding the target map, usually considered a subset of R2, in the obvious way in R3, the requirement above then is weakened to:
for some not depending on and .
For examples of such projections, see equal-area map projection.
See also
Jacobian matrix and determinant
References
Differential geometry
Functions and mappings
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https://en.wikipedia.org/wiki/List%20of%20Yokohama%20F.%20Marinos%20records%20and%20statistics
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This article contains records and statistics for the Japanese professional football club, Yokohama F. Marinos.
J.League
Domestic cup competitions
Major International Competitions
Top scorers by season
References
Yokohama F. Marinos
Yokohama F. Marinos
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https://en.wikipedia.org/wiki/GRJ
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GRJ may refer to:
George Airport, an airport in George, South Africa
Gradshteyn and Ryzhik (and Jeffrey) aka Table of Integrals, Series, and Products, a classical book in mathematics
Jabo language, by ISO-639 code
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https://en.wikipedia.org/wiki/English%20cricket%20team%20in%20Sri%20Lanka%20in%201992%E2%80%9393
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Squads
Test series
Only Test
ODI series
1st ODI
2nd ODI
Records and statistics
Batting
Bowling
External links
Series home
References
1993 in English cricket
1993 in Sri Lankan cricket
1992-93
International cricket competitions from 1991–92 to 1994
Sri Lankan cricket seasons from 1972–73 to 1999–2000
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https://en.wikipedia.org/wiki/List%20of%20Plymouth%20Argyle%20F.C.%20records%20and%20statistics
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Plymouth Argyle Football Club are an English professional association football club based in Plymouth, Devon. They compete in the EFL Championship, the second tier of English football, following promotion from the 2022–23 EFL League One. The club was formed in 1886 as Argyle Football Club, a name which was retained until 1903 when the club became professional and were elected to the Southern Football League. The club also entered English football's premier knockout competition, the Football Association Challenge Cup, for the first time that same year. The club joined the Football League in 1920, and have competed there since then, achieving multiple league titles, promotions, and relegations.
This list encompasses the major honours won by Plymouth Argyle, records set by the club, their managers and their players. The player records section includes details of the club's leading goalscorers and those who have made most appearances in first-team competitions. It also records notable achievements by Plymouth Argyle players on the international stage, and the highest transfer fees paid and received by the club. Attendance records at Home Park, the club's ground since 1901, are also included in the list.
Honours
Plymouth Argyle have won multiple titles in domestic competition. They have won the third tier of English football four times, which is a record. In their second season as a professional club they won their first title, the Western League, in 1905. The club's most recent title came in 2004 when they won the Football League Second Division.
Domestic
League
Football League Second Division
Winners (1): 2003–04
Football League Third Division
Winners (2): 1958–59, 2001–02
Runners-up (2): 1974–75, 1985–86
Football League Third Division South
Winners (2): 1929–30, 1951–52
Runners-up (6): 1921–22, 1922–23, 1923–24, 1924–25, 1925–26, 1926–27
Southern Football League
Winners (1): 1912–13
Runners-up (2): 1907–08, 1911–12
Western Football League
Winners (1): 1904–05
Runners-up (1): 1906–07
Cups
Football League Third Division Play-offs
Winners (1): 1996
Player records
Appearances
Plymouth Argyle's appearance record is held by Kevin Hodges, who played 620 times for the club over the course of 15 seasons from 1978 to 1992. Hodges also holds the records for appearances made in league, FA Cup, and League Cup competition. The player to have made the most appearances in the current squad is Luke McCormick, with 350 (including 4 as a substitute) as at 19 February 2022.
Most appearances in all competitions: Kevin Hodges, 620.
Most League appearances: Kevin Hodges, 530.
Most FA Cup appearances: Kevin Hodges, 39.
Most League Cup appearances: Kevin Hodges, 35.
Youngest first-team player: Freddie Issaka, 15 years and 34 days (against Newport County, 31 August 2021).
Oldest first-team player: Peter Shilton, 44 years and 21 days (against Burnley, 9 October 1993).
Oldest debutant: Peter Shilton, 42 years and 199 days (against Charlton A
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https://en.wikipedia.org/wiki/Category%20utility
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Category utility is a measure of "category goodness" defined in and . It attempts to maximize both the probability that two objects in the same category have attribute values in common, and the probability that objects from different categories have different attribute values. It was intended to supersede more limited measures of category goodness such as "cue validity" (; ) and "collocation index" . It provides a normative information-theoretic measure of the predictive advantage gained by the observer who possesses knowledge of the given category structure (i.e., the class labels of instances) over the observer who does not possess knowledge of the category structure. In this sense the motivation for the category utility measure is similar to the information gain metric used in decision tree learning. In certain presentations, it is also formally equivalent to the mutual information, as discussed below. A review of category utility in its probabilistic incarnation, with applications to machine learning, is provided in .
Probability-theoretic definition of category utility
The probability-theoretic definition of category utility given in and is as follows:
where is a size- set of -ary features, and is a set of categories. The term designates the marginal probability that feature takes on value , and the term designates the category-conditional probability that feature takes on value given that the object in question belongs to category .
The motivation and development of this expression for category utility, and the role of the multiplicand as a crude overfitting control, is given in the above sources. Loosely , the term is the expected number of attribute values that can be correctly guessed by an observer using a probability-matching strategy together with knowledge of the category labels, while is the expected number of attribute values that can be correctly guessed by an observer the same strategy but without any knowledge of the category labels. Their difference therefore reflects the relative advantage accruing to the observer by having knowledge of the category structure.
Information-theoretic definition of category utility
The information-theoretic definition of category utility for a set of entities with size- binary feature set , and a binary category is given in as follows:
where is the prior probability of an entity belonging to the positive category (in the absence of any feature information), is the conditional probability of an entity having feature given that the entity belongs to category , is likewise the conditional probability of an entity having feature given that the entity belongs to category , and is the prior probability of an entity possessing feature (in the absence of any category information).
The intuition behind the above expression is as follows: The term represents the cost (in bits) of optimally encoding (or transmitting) feature information when it is known that the objects to be
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https://en.wikipedia.org/wiki/List%20of%20Shimizu%20S-Pulse%20records%20and%20statistics
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This article contains records and statistics for the Japanese professional football club, Shimizu S-Pulse.
J.League
Domestic cup competitions
International Competitions
Top scorers by season
References
Shimizu S-Pulse
Shimizu S-Pulse
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https://en.wikipedia.org/wiki/Mathematical%20formalism
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Mathematical formalism can mean:
Formalism (philosophy of mathematics), a general philosophical approach to mathematics
Formal logical systems, in mathematical logic, a particular system of formal logical reasoning
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https://en.wikipedia.org/wiki/John%20Fitch%20%28computer%20scientist%29
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John Peter Fitch (also known as John ffitch) is a computer scientist, mathematician and composer, who has worked on relativity, planetary astronomy, computer algebra and Lisp. Alongside Victor Lazzarini and Steven Yi, he is the project leader for audio programming language Csound, having a leading role in its development since the early 1990s; and he was a director of Codemist Ltd, which developed the Norcroft C compiler.
Education and early life
Born in Barnsley, Yorkshire, England in December 1945, Fitch was educated at St John's College, Cambridge where he gained a PhD from the University of Cambridge in 1971 supervised by David Barton.
Career and research
Fitch spent six years at Cambridge as a postdoctoral researcher - winning the Adams Prize for Mathematics in 1975 for a joint essay with David Barton on Applications of algebraic manipulative systems to physics.
Fitch was a visiting professor the University of Utah for a year, then lectured at the University of Leeds for 18 months, before becoming professor and then chair of software engineering at the University of Bath, which his biography claims is "a subject about which he knows little"; his 31-year career there lasted April 1980 – September 2011, after which he was named an adjunct professor of music at Maynooth University.
Fitch lectured for the module CM20029: The Essence of Compilers, as well as optional modules involving computer music and digital signal processing. According to his biography, "despite his long hair and beard, and the uncertain spelling of his name, [he] was never a hippie".
His former doctoral students include James Davenport and Tom Crick.
Personal life
Fitch is married to historian Audrey Fitch.
References
1945 births
English computer scientists
English mathematicians
Academics of the University of Bath
Living people
People from Barnsley
Alumni of St John's College, Cambridge
University of Utah faculty
Academics of the University of Leeds
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https://en.wikipedia.org/wiki/Coercive%20function
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In mathematics, a coercive function is a function that "grows rapidly" at the extremes of the space on which it is defined. Depending on the context different exact definitions of this idea are in use.
Coercive vector fields
A vector field is called coercive if
where "" denotes the usual dot product and denotes the usual Euclidean norm of the vector x.
A coercive vector field is in particular norm-coercive since for , by Cauchy–Schwarz inequality.
However a norm-coercive mapping is not necessarily a coercive vector field. For instance the rotation by 90° is a norm-coercive mapping which fails to be a coercive vector field since for every .
Coercive operators and forms
A self-adjoint operator where is a real Hilbert space, is called coercive if there exists a constant such that
for all in
A bilinear form is called coercive if there exists a constant such that
for all in
It follows from the Riesz representation theorem that any symmetric (defined as for all in ), continuous ( for all in and some constant ) and coercive bilinear form has the representation
for some self-adjoint operator which then turns out to be a coercive operator. Also, given a coercive self-adjoint operator the bilinear form defined as above is coercive.
If is a coercive operator then it is a coercive mapping (in the sense of coercivity of a vector field, where one has to replace the dot product with the more general inner product). Indeed, for big (if is bounded, then it readily follows); then replacing by we get that is a coercive operator.
One can also show that the converse holds true if is self-adjoint. The definitions of coercivity for vector fields, operators, and bilinear forms are closely related and compatible.
Norm-coercive mappings
A mapping between two normed vector spaces and is called norm-coercive if and only if
More generally, a function between two topological spaces and is called coercive if for every compact subset of there exists a compact subset of such that
The composition of a bijective proper map followed by a coercive map is coercive.
(Extended valued) coercive functions
An (extended valued) function is called coercive if
A real valued coercive function is, in particular, norm-coercive. However, a norm-coercive function is not necessarily coercive.
For instance, the identity function on is norm-coercive but not coercive.
See also
Radially unbounded functions
References
Functional analysis
General topology
Types of functions
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https://en.wikipedia.org/wiki/Dynamic%20mutation
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In genetics, a dynamic mutation is an unstable heritable element where the probability of expression of a mutant phenotype is a function of the number of copies of the mutation. That is, the replication product (progeny) of a dynamic mutation has a different likelihood of mutation than its predecessor. These mutations, typically short sequences repeated many times, give rise to numerous known diseases, including the trinucleotide repeat disorders.
Robert I. Richards and Grant R. Sutherland called these phenomena, in the framework of dynamical genetics, dynamic mutations. Triplet expansion is caused by slippage during DNA replication. Due to the repetitive nature of the DNA sequence in these regions , 'loop out' structures may form during DNA replication while maintaining complementary base pairing between the parent strand and daughter strand being synthesized. If the loop out structure is formed from sequence on the daughter strand this will result in an increase in the number of repeats. However, if the loop out structure is formed on the parent strand a decrease in the number of repeats occurs. It appears that expansion of these repeats is more common than reduction. Generally the larger the expansion the more likely they are to cause disease or increase the severity of disease. This property results in the characteristic of anticipation seen in trinucleotide repeat disorders. Anticipation describes the tendency of age of onset to decrease and severity of symptoms to increase through successive generations of an affected family due to the expansion of these repeats.
Common features
Most of these diseases have neurological symptoms.
Anticipation/The Sherman paradox refers to progressively earlier or more severe expression of the disease in more recent generations.
Repeats are usually polymorphic in copy number, with mitotic and meiotic instability.
Copy number related to the severity and/or age of onset
Imprinting effects
Reverse mutation - The mutation can revert to normal or to a premutation carrier state.
Examples
Fragile X syndromes
Huntington's disease
Myotonic dystrophy
Spinal and bulbar muscular atrophy
Spinocerebellar ataxia type 3
Friedreich ataxia
Ocularpharyngeal muscular dystrophy
Progressive myoclonus epilepsy
References
Mutation
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https://en.wikipedia.org/wiki/SU2
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SU2 may refer to:
SU-2, a scout version of the Vought O2U Corsair biplane
SU(2), a special unitary group in mathematics
SU2 code, a suite of open-source software tools written in C++ for the numerical solution of partial differential equations
Sukhoi Su-2, a Soviet reconnaissance and light bomber aircraft used in the early stages of World War II
Asteroids
3158 Anga (provisionally 1976 SU), a minor planet discovered on September 24, 1976
5499 (provisionally 1981 SU), a minor planet discovered on September 29, 1981
7887 Bratfest (provisionally 1993 SU), a minor planet discovered on September 18, 1993
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https://en.wikipedia.org/wiki/Quantum%20nonlocality
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In theoretical physics, quantum nonlocality refers to the phenomenon by which the measurement statistics of a multipartite quantum system do not admit an interpretation in terms of a local realistic theory. Quantum nonlocality has been experimentally verified under different physical assumptions. Any physical theory that aims at superseding or replacing quantum theory should account for such experiments and therefore cannot fulfill local realism; quantum nonlocality is a property of the universe that is independent of our description of nature.
Quantum nonlocality does not allow for faster-than-light communication, and hence is compatible with special relativity and its universal speed limit of objects. Thus, quantum theory is local in the strict sense defined by special relativity and, as such, the term "quantum nonlocality" is sometimes considered a misnomer. Still, it prompts many of the foundational discussions concerning quantum theory.
History
Einstein, Podolsky and Rosen
In the 1935 EPR paper, Albert Einstein, Boris Podolsky and Nathan Rosen described "two spatially separated particles which have both perfectly correlated positions and momenta" as a direct consequence of quantum theory. They intended to use the classical principle of locality to challenge the idea that the quantum wavefunction was a complete description of reality, but instead they sparked a debate on the nature of reality.
Afterwards, Einstein presented a variant of these ideas in a letter to Erwin Schrödinger, which is the version that is presented here. The state and notation used here are more modern, and akin to David Bohm's take on EPR. The quantum state of the two particles prior to measurement can be written as
where .
Here, subscripts “A” and “B” distinguish the two particles, though it is more convenient and usual to refer to these particles as being in the possession of two experimentalists called Alice and Bob. The rules of quantum theory give predictions for the outcomes of measurements performed by the experimentalists. Alice, for example, will measure her particle to be spin-up in an average of fifty percent of measurements. However, according to the Copenhagen interpretation, Alice's measurement causes the state of the two particles to collapse, so that if Alice performs a measurement of spin in the z-direction, that is with respect to the basis , then Bob's system will be left in one of the states . Likewise, if Alice performs a measurement of spin in the x-direction, that is, with respect to the basis , then Bob's system will be left in one of the states . Schrödinger referred to this phenomenon as "steering". This steering occurs in such a way that no signal can be sent by performing such a state update; quantum nonlocality cannot be used to send messages instantaneously and is therefore not in direct conflict with causality concerns in special relativity.
In the Copenhagen view of this experiment, Alice's measurement—and particularly her measurem
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https://en.wikipedia.org/wiki/Boom%20Boom%20Rocket
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Boom Boom Rocket (BBR) is a downloadable video game for Xbox 360's Xbox Live Arcade service. Boom Boom Rocket is the first rhythm game for Xbox Live Arcade and was developed by Geometry Wars creators Bizarre Creations and published by the Pogo division of Electronic Arts. The game was made backwards compatible on Xbox One on July 26, 2016.
Gameplay
The objective of Boom Boom Rocket is to trigger fireworks explosions in time with music, in a gameplay style very similar to that of Dance Dance Revolution, Guitar Hero and Fantavision. Each rocket is color-mapped to one of the colored buttons on the Xbox 360 controller. A life gauge, which also serves as a score multiplier meter, fills with each successful shot and drains with each missed shot, and players are graded on overall hit accuracy. If the life meter drains completely, the player fails the song and the game is over. Each song has three unlockable firework types, one for each difficulty level. If the player successfully triggers a prescribed number of fireworks a rocket with a wavy tail appears. If this special rocket is triggered, the firework is unlocked and it will randomly replace other firework types on subsequent songs. If the rocket with the wavy tail is missed, or the song is not completed, the firework type remains locked.
An update that was released in November 2007 allows the game to recognize other controllers like guitars and dance pads. When using a guitar the rockets need to be "strummed" just as in Guitar Hero to be exploded in time.
Boom Boom Rocket includes several single-player modes and a local two-player mode. Single-player modes include the basic game, Endurance Mode (in which the song loops continuously and gradually speeds up, with the player attempting to complete as many "laps" as possible), and Practice Mode. Additionally, the game provides a Visualizer mode, which creates a fireworks display timed to the rhythm of audio files stored on the player's console.
The game provides twelve achievements (worth 200 Gamerscore points), which focus mainly on unlocking fireworks and attaining high grade levels and hit ratios. It also supports two-player mode on the same system, but does not support online multiplayer. As with most Xbox Live Arcade games, the title includes online leaderboards.
Boom Boom Rocket features ten music tracks (fifteen with the update), with three difficulty levels per track. Each track is a classical song that has been remixed into a modern style, such as ska, funk or techno. The game's music was composed by Ian Livingstone (Batman Returns and Project Gotham Racing 2 game soundtracks). While users cannot create their own custom soundtracks or utilize music from other sources (apart from the music visualizer mode), the game does support downloadable content including new tracks composed by Chris Chudley from Audioantics. (Geometry Wars, Project Gotham Racing 3) which should have been released on November 29, 2007; but were a day late.
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https://en.wikipedia.org/wiki/Dirichlet%20algebra
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In mathematics, a Dirichlet algebra is a particular type of algebra associated to a compact Hausdorff space X. It is a closed subalgebra of C(X), the uniform algebra of bounded continuous functions on X, whose real parts are dense in the algebra of bounded continuous real functions on X. The concept was introduced by .
Example
Let be the set of all rational functions that are continuous on ; in other words functions that have no poles in . Then
is a *-subalgebra of , and of . If is dense in , we say is a Dirichlet algebra.
It can be shown that if an operator has as a spectral set, and is a Dirichlet algebra, then has a normal boundary dilation. This generalises Sz.-Nagy's dilation theorem, which can be seen as a consequence of this by letting
References
Completely Bounded Maps and Operator Algebras Vern Paulsen, 2002
.
Functional analysis
C*-algebras
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https://en.wikipedia.org/wiki/2007%20in%20Japanese%20football
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Japanese football in 2007
J.League Division 1
J.League Division 2
Japan Football League
Japanese Regional Leagues
Emperor's Cup
J.League Cup
National team (Men)
Results
Players statistics
National team (Women)
Results
Players statistics
External links
Seasons in Japanese football
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https://en.wikipedia.org/wiki/2006%20in%20Japanese%20football
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Japanese football in 2006
J.League Division 1
J.League Division 2
Japan Football League
Japanese Regional Leagues
Emperor's Cup
J.League Cup
National team (Men)
Results
Players statistics
National team (Women)
Results
Players statistics
External links
Seasons in Japanese football
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https://en.wikipedia.org/wiki/2005%20in%20Japanese%20football
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Japanese football in 2005
J.League Division 1
J.League Division 2
Japan Football League
Japanese Regional Leagues
Emperor's Cup
J.League Cup
National team (Men)
Results
Players statistics
National team (Women)
Results
Players statistics
External links
Seasons in Japanese football
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https://en.wikipedia.org/wiki/2004%20in%20Japanese%20football
|
Japanese football in 2004
J.League Division 1
J.League Division 2
Japan Football League
Japanese Regional Leagues
Emperor's Cup
J.League Cup
National team (Men)
Results
Players statistics
National team (Women)
Results
Players statistics
External links
Seasons in Japanese football
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https://en.wikipedia.org/wiki/Kan%20fibration
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In mathematics, Kan complexes and Kan fibrations are part of the theory of simplicial sets. Kan fibrations are the fibrations of the standard model category structure on simplicial sets and are therefore of fundamental importance. Kan complexes are the fibrant objects in this model category. The name is in honor of Daniel Kan.
Definitions
Definition of the standard n-simplex
For each n ≥ 0, recall that the standard -simplex, , is the representable simplicial set
Applying the geometric realization functor to this simplicial set gives a space homeomorphic to the topological standard -simplex: the convex subspace of ℝn+1 consisting of all points such that the coordinates are non-negative and sum to 1.
Definition of a horn
For each k ≤ n, this has a subcomplex , the k-th horn inside , corresponding to the boundary of the n-simplex, with the k-th face removed. This may be formally defined in various ways, as for instance the union of the images of the n maps corresponding to all the other faces of . Horns of the form sitting inside look like the black V at the top of the adjacent image. If is a simplicial set, then maps
correspond to collections of -simplices satisfying a compatibility condition, one for each . Explicitly, this condition can be written as follows. Write the -simplices as a list and require that
for all with .
These conditions are satisfied for the -simplices of sitting inside .
Definition of a Kan fibration
A map of simplicial sets is a Kan fibration if, for any and , and for any maps and such that (where is the inclusion of in ), there exists a map such that and
. Stated this way, the definition is very similar to that of fibrations in topology (see also homotopy lifting property), whence the name "fibration".
Technical remarks
Using the correspondence between -simplices of a simplicial set and morphisms (a consequence of the Yoneda lemma), this definition can be written in terms of simplices. The image of the map can be thought of as a horn as described above. Asking that factors through corresponds to requiring that there is an -simplex in whose faces make up the horn from (together with one other face). Then the required map corresponds to a simplex in whose faces include the horn from . The diagram to the right is an example in two dimensions. Since the black V in the lower diagram is filled in by the blue -simplex, if the black V above maps down to it then the striped blue -simplex has to exist, along with the dotted blue -simplex, mapping down in the obvious way.
Kan complexes defined from Kan fibrations
A simplicial set is called a Kan complex if the map from , the one-point simplicial set, is a Kan fibration. In the model category for simplicial sets, is the terminal object and so a Kan complex is exactly the same as a fibrant object. Equivalently, this could be stated as: if every map from a horn has an extension to , meaning there is a lift such thatfor the inclusion map , th
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https://en.wikipedia.org/wiki/Augustin%20Banyaga
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Augustin Banyaga (born March 31, 1947) is a Rwandan-born American mathematician whose research fields include symplectic topology and contact geometry. He is currently a Professor of Mathematics at Pennsylvania State University.
Biography
He earned his Ph.D. degree in 1976 at the University of Geneva under the supervision of André Haefliger. (Banyaga was the first person from Rwanda to obtain a Ph.D. in mathematics.) He was a member of the Institute for Advanced Study in Princeton, New Jersey (1977–1978), Benjamin Peirce Assistant Professor at Harvard University (1978–1982), and assistant professor at Boston University (1982–1984), before joining the faculty at Pennsylvania State University in 1984 as associate professor. He was promoted to full professor in 1992.
In 2009 Banyaga was elected a Fellow of the African Academy of Sciences, and in 2015 he was named a Distinguished Senior Scholar by Pennsylvania State University.
He has made significant contributions in symplectic topology, especially on the structure of groups of diffeomorphisms preserving a symplectic form (symplectomorphisms). One of his best-known results states that the group of Hamiltonian diffeomorphisms of a compact, connected, symplectic manifold is a simple group; in particular, it does not admit any non-trivial homomorphism to the real line.
Banyaga is an editor of Afrika Matematica, the journal of the African Mathematical Union, and an editor of the African Journal of Mathematics. He has supervised the theses of 9 Ph.D. students.
Bibliography
Articles
Augustin Banyaga, Sur la structure du groupe des difféomorphismes qui préservent une forme symplectique, Commentarii Mathematici Helvetici 53 (1978), no. 2, 174–227.
Augustin Banyaga, On fixed points of symplectic maps, Inventiones Mathematicae 56 (1980), no. 3, 215–229.
Augustin Banyaga, On Isomorphic Classical Diffeomorphism Groups. I., Proceedings of the American Mathematical Society 98 (1986), no. 1, 113–118.
Augustin Banyaga, On Isomorphic Classical Diffeomorphism Groups. II., Journal of Differential Geometry 28 (1988), no. 1, 23–35.
Augustin Banyaga, A note on Weinstein's conjecture, Proceedings of the American Mathematical Society 123 (1990), no. 12, 3901–3906.
Books
References
External links
People from Kigali
20th-century American mathematicians
21st-century American mathematicians
Topologists
Pennsylvania State University faculty
Harvard University Department of Mathematics faculty
Harvard University faculty
Boston University faculty
Institute for Advanced Study visiting scholars
Living people
1947 births
Rwandan emigrants to the United States
University of Geneva alumni
Fellows of the African Academy of Sciences
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https://en.wikipedia.org/wiki/Mathematics%20%28Mos%20Def%20song%29
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"Mathematics" is a b-side single from Mos Def's solo debut album, Black on Both Sides. It contains lyrics about various social issues and asks the listener to add them up and come to conclusions about them. Many references to numbers are found in this song and at times, Mos Def rhymes statistics in numerical order.
Background
The song highlights the differences between the White and African-American citizens of the US and uses the lyrics "Do your math..." - telling young African-Americans to 'do their maths' so they can avoid being part of the numerous degrading statistics he raps about in the opening and third verses of the song. The song is produced by DJ Premier whose famous scratch samples make up the song's bridge. Premier has called it one of his favorite beats.
Premier also revealed that Scarface originally wanted the beat. He was recording his album The Last of a Dying Breed and wanted Premier to produce a song on it. However, Mos Def took the track and recorded something to it. Scarface later met up with Mos Def to tell him that he really wanted the track.
Samples
The bridge of "Mathematics" contains DJ Premier's signature scratched vocals from various hip hop songs. The lyrics of those samples as well as information about their origin can be found below:
"The Mighty Mos Def..." (from Mos Def's "Body Rock"),
"It's simple mathematics" (from Fat Joe's "John Blaze"),
"Check it out" (The Lady of Rage's vocals from Snoop Dogg's "For All My Niggaz & Bitches"),
"I revolve around science..." (Ghostface Killah's vocals from Raekwon's "Criminology"),
"What are we talking about here?" (Art Seigner of Flying Dutchman Records interviewing Angela Davis),
"Do your math.." (from Erykah Badu's "On & On"), and
"One, two, three, four" (from James Brown's "Funky Drummer")
The instrumental from "Baby I'm-a Want You" by The Fatback Band is also sampled.
In popular culture
"Mathematics" can be found on the soundtrack of Madden NFL 2002. The song is also played briefly in the CSI: Crime Scene Investigation episode "Crate 'n' Burial".
Single track list
Vinyl 12"
A-Side
Ms. Fat Booty (Clean)
Ms. Fat Booty (Dirty)
Ms. Fat Booty (Instrumental)
Ms. Fat Booty (A Capella)
B-Side
Mathematics (Clean)
Mathematics (Dirty)
Mathematics (Instrumental)
Mathematics (A Capella)
CD/Maxi single
Ms. Fat Booty (4:08)
Ms. Fat Booty (Instrumental) (4:06)
Mathematics (4:08)
References
1999 songs
Mos Def songs
Song recordings produced by DJ Premier
Songs written by Mos Def
Songs written by DJ Premier
1999 singles
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https://en.wikipedia.org/wiki/What%20Is%20Mathematics%3F
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What Is Mathematics? is a mathematics book written by Richard Courant and Herbert Robbins, published in England by Oxford University Press. It is an introduction to mathematics, intended both for the mathematics student and for the general public.
First published in 1941, it discusses number theory, geometry, topology and calculus. A second edition was published in 1996 with an additional chapter on recent progress in mathematics, written by Ian Stewart.
Authorship
The book was based on Courant's course material. Although Robbins assisted in writing a large part of the book, he had to fight for authorship. Nevertheless, Courant alone held the copyright for the book. This resulted in Robbins receiving a smaller share of the royalties.
Title
Michael Katehakis remembers Robbins' interest in the literature and Tolstoy in particular and he is convinced that the title of the book is most likely due to Robbins, who was inspired by the title of the essay What Is Art? by Leo Tolstoy. Robbins did the same in the book Great Expectations: The Theory of Optimal Stopping he co-authored with Yuan-Shih Chow and David Siegmund, where one can not miss the connection with the title of the novel Great Expectations by Charles Dickens.
According to Constance Reid, Courant finalized the title after a conversation with Thomas Mann.
Translations
The first Russian translation Что такое математика? was published in 1947; there were 5 translations since then, the last one in 2010.
The first Italian translation, Che cos'è la matematica?, was published in 1950. А translation of the second edition was issued in 2000.
The first German translation Was ist Mathematik? by Iris Runge was published in 1962.
A Spanish translation of the second edition, ¿Qué Son Las Matemáticas?, was published in 2002.
The first Bulgarian translation, Що е математика?, was published in 1967. А second translation appeared in 1985.
The first Romanian translation, Ce este matematica?, was published in 1969.
The first Polish translation, Co to jest matematyka, was published in 1959. А second translation appeared in 1967. А translation of the second edition was published in 1998.
The first Hungarian translation, Mi a matematika?, was published in 1966.
The first Serbian translation, Šta je matematika?, was published in 1973.
The first Japanese translation, , was published in 1966. А translation of the second edition was published in 2001.
A Korean translation of the second edition, , was published in 2000.
A Portuguese translation of the second edition, O que é matemática?, was published in 2000.
Reviews
What is Mathematics? An Elementary Approach to Ideas and Methods, book review by Brian E. Blank, Notices of the American Mathematical Society 48, #11 (December 2001), pp. 1325–1330
What is Mathematics?, book review by Leonard Gillman, The American Mathematical Monthly 105, #5 (May 1998), pp. 485–488.
Editions
Reprinted several times with a few corrections of minor errors and misprints as a "S
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https://en.wikipedia.org/wiki/Blind%20equalization
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Blind equalization is a digital signal processing technique in which the transmitted signal is inferred (equalized) from the received signal, while making use only of the transmitted signal statistics. Hence, the use of the word blind in the name.
Blind equalization is essentially blind deconvolution applied to digital communications. Nonetheless, the emphasis in blind equalization is on online estimation of the equalization filter, which is the inverse of the channel impulse response, rather than the estimation of the channel impulse response itself. This is due to blind deconvolution common mode of usage in digital communications systems, as a means to extract the continuously transmitted signal from the received signal, with the channel impulse response being of secondary intrinsic importance.
The estimated equalizer is then convolved with the received signal to yield an estimation of the transmitted signal.
Problem statement
Noiseless model
Assuming a linear time invariant channel with impulse response , the noiseless model relates the received signal to the transmitted signal via
The blind equalization problem can now be formulated as follows; Given the received signal , find a filter , called an equalization filter, such that
where is an estimation of .
The solution to the blind equalization problem is not unique. In fact, it may be determined only up to a signed scale factor and an arbitrary time delay. That is, if are estimates of the transmitted signal and channel impulse response, respectively, then give rise to the same received signal for any real scale factor and integral time delay . In fact, by symmetry, the roles of and are Interchangeable.
Noisy model
In the noisy model, an additional term, , representing additive noise, is included. The model is therefore
Algorithms
Many algorithms for the solution of the blind equalization problem have been suggested over the years.
However, as one usually has access to only a finite number of samples from the received signal , further restrictions must be imposed over the above models to render the blind equalization problem tractable.
One such assumption, common to all algorithms described below is to assume that the channel has finite impulse response, , where is an arbitrary natural number.
This assumption may be justified on physical grounds, since the energy of any real signal must be finite, and therefore its impulse response must tend to zero. Thus it may be assumed that all coefficients beyond a certain point are negligibly small.
Minimum phase
If the channel impulse response is assumed to be minimum phase, the problem becomes trivial.
Bussgang methods
Bussgang methods make use of the Least mean squares filter algorithm
with
where is an appropriate positive adaptation step and is a suitable nonlinear function.
Polyspectra techniques
Polyspectra techniques utilize higher order statistics in order to compute the equalizer.
See also
Independent component analy
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https://en.wikipedia.org/wiki/Robbins%20pentagon
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In geometry, a Robbins pentagon is a cyclic pentagon whose side lengths and area are all rational numbers.
History
Robbins pentagons were named by after David P. Robbins, who had previously given a formula for the area of a cyclic pentagon as a function of its edge lengths. Buchholz and MacDougall chose this name by analogy with the naming of Heron triangles after Hero of Alexandria, the discoverer of Heron's formula for the area of a triangle as a function of its edge lengths.
Area and perimeter
Every Robbins pentagon may be scaled so that its sides and area are integers. More strongly, Buchholz and MacDougall showed that if the side lengths are all integers and the area is rational, then the area is necessarily also an integer, and the perimeter is necessarily an even number.
Diagonals
Buchholz and MacDougall also showed that, in every Robbins pentagon, either all five of the internal diagonals are rational numbers or none of them are. If the five diagonals are rational (the case called a Brahmagupta pentagon by ), then the radius of its circumscribed circle must also be rational, and the pentagon may be partitioned into three Heron triangles by cutting it along any two non-crossing diagonals, or into five Heron triangles by cutting it along the five radii from the circle center to its vertices.
Buchholz and MacDougall performed computational searches for Robbins pentagons with irrational diagonals but were unable to find any. On the basis of this negative result they suggested that Robbins pentagons with irrational diagonals may not exist.
References
.
.
.
Arithmetic problems of plane geometry
Circles
Types of polygons
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https://en.wikipedia.org/wiki/Robert%20Tijdeman
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Robert Tijdeman (born 30 July 1943 in Oostzaan, North Holland) is a Dutch mathematician. Specializing in number theory, he is best known for his Tijdeman's theorem. He is a professor of mathematics at the Leiden University since 1975, and was chairman of the department of mathematics and computer science at Leiden from 1991 to 1993. He was also president of the Dutch Mathematical Society from 1984 to 1986.
Tijdeman received his PhD in 1969 from the University of Amsterdam, and received an honorary doctorate from Kossuth Lajos University in 1999. In 1987 he was elected to the Royal Netherlands Academy of Arts and Sciences.
References
External links
Tijdeman's web site at Leiden.
1943 births
Living people
People from Oostzaan
20th-century Dutch mathematicians
21st-century Dutch mathematicians
Number theorists
Members of the Royal Netherlands Academy of Arts and Sciences
Academic staff of Leiden University
University of Amsterdam alumni
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https://en.wikipedia.org/wiki/Frances%20Kirwan
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Dame Frances Clare Kirwan, (born 21 August 1959) is a British mathematician, currently Savilian Professor of Geometry at the University of Oxford. Her fields of specialisation are algebraic and symplectic geometry.
Education
Kirwan was educated at Oxford High School, and studied maths as an undergraduate at Clare College in the University of Cambridge. She took a D.Phil at Oxford in 1984, with the dissertation title The Cohomology of Quotients in Symplectic and Algebraic Geometry, which was supervised by Michael Atiyah.
Research
Kirwan's research interests include moduli spaces in algebraic geometry, geometric invariant theory (GIT), and in the link between GIT and moment maps in symplectic geometry. Her work endeavours to understand the structure of geometric objects by investigation of their algebraic and topological properties. She introduced the Kirwan map.
From 1983 to 1985 she held a junior fellowship at Harvard. From 1983 to 1986 she held a Fellowship at Magdalen College, Oxford, before becoming a Fellow of Balliol College, Oxford. She is an honorary fellow of Clare College, Cambridge and also at Magdalen College.
In 1996, she was awarded the Title of Distinction of Professor of Mathematics. From 2004 to 2006 she was president of the London Mathematical Society, the second-youngest president in the society's history and only the second woman to be president. In 2005, she received a five-year EPSRC Senior Research Fellowship, to support her research on the moduli spaces of complex algebraic curves.
In 2017, she was elected Savilian Professor of Geometry, becoming the first woman to hold the post. While this entailed a move to New College, Oxford she was elected an emeritus fellow at Balliol. She was the convenor of the 2008–9 meeting of European Women in Mathematics and deputy convenor of the following meeting in 2010–11.
Prizes, awards and scholarships
London Mathematical Society Whitehead Prize, 1989
Fellow of the Royal Society, 2001
President, London Mathematical Society, 2003–2005
EPSRC Senior Research Fellowship, 2005–2010, for her work in algebraic geometry
Fellow of the American Mathematical Society, 2012
London Mathematical Society Senior Whitehead Prize, 2013
DBE for services to mathematics, 2014
Maths and Computing Suffrage Science award, 2016
Member of Academia Europaea
Chairman of the United Kingdom Mathematics Trust
Sylvester Medal of The Royal Society, 2021
Honorary degree, University of York, 2020
Honorary degree, University of St Andrews, 2022
L'Oréal-UNESCO For Women in Science Awards (Laureate for Europe – Mathematics), 2023
Kirwan served on the medal-selection committee that awarded the Fields medal to Maryam Mirzakhani.
Publications
with Jonathan Woolf:
References
External links
Profile at the University of Oxford Mathematical Institute
1959 births
Living people
Place of birth missing (living people)
20th-century British mathematicians
21st-century British mathematicians
Algebraic geometers
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https://en.wikipedia.org/wiki/Polar%20rose
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Polar rose may refer to:
Rose (mathematics), a mathematical curve
Polar Rose (facial recognition), a company out of Malmö, Sweden which makes facial recognition software.
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https://en.wikipedia.org/wiki/Queue%20jump
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"Queue jump" may also refer to cutting in line.
A queue jump is a type of roadway geometry used to provide preference to buses at intersections, often found in bus rapid transit systems. It consists of an additional travel lane on the approach to a signalised intersection. This lane is often restricted to transit vehicles only. A queue jump lane is usually accompanied by a signal which provides a phase specifically for vehicles within the queue jump. Vehicles in the queue jump lane get a "head-start" over other queued vehicles and can therefore merge into the regular travel lanes immediately beyond the signal. The intent of the lane is to allow the higher-capacity vehicles to cut to the front of the queue, reducing the delay caused by the signal and improving the operational efficiency of the transit system.
Design considerations
Queue jumps are only effective in certain situations. First, there has to be an existing source of delay or roadway congestion; if there is no congestion and the normal traffic signal is usually green, then the bus driver has no reason to move into the queue jump. The length of the queue jump lane needs to be long enough to provide a meaningful time savings. Queue jumps can also be used in situations such as bus stop pullouts or at the end of a bus-only lane, in order to help expedite the bus merge into traffic. Queue jumps may not work well where there are high volumes of right turning vehicles (left turning in UK) that might get in the way of the bus through movement, although in some cases these turning vehicles can be provided a separate lane and/or a protected signal phase at the same time as the advanced transit phase. Available right-of-way is needed to provide the bypass lane.
Bus stop location is another important consideration. Where there are far-side bus stops, an advance signal does not provide any benefit to the bus because it will not be able to merge into traffic during the advance signal phase. Where near-side stops are present, an advance signal can be highly effective in giving the bus a head start, however the bus stop location needs to consider the detection strategy used for the advance signal phase so that the bus is detected only after it is done serving the bus stop.
It is preferable to provide a receiving lane on the far side of the intersection to provide an acceleration/merging area, however this is not always a requirement when an advance signal is used.
Detection strategies and timing
When an advance signal is used, it should be actuated by an approaching bus to avoid needlessly delaying other traffic when there is no bus present. If the queue jump lane is designated as bus-only, then standard traffic signal detection such as loop detectors or video detection can be used. If there is a limited amount of other traffic in the lane, then two or more loop detectors can be used and configured with AND logic such that only a long vehicle will actuate the advance phase. If the queue jump
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https://en.wikipedia.org/wiki/Preorder%20%28disambiguation%29
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The term preorder may refer to:
In mathematics:
Preorder, a reflexive, transitive relation
Preorder field, a field of sets structure on a set with preorder
Preordered field, a field with a preorder
Preordering, a vertex ordering from a tree or other graph traversal; see Depth-first search#Vertex orderings
In marketing:
Pre-order – an order placed for an item which has not yet been released.
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https://en.wikipedia.org/wiki/Exponential%20polynomial
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In mathematics, exponential polynomials are functions on fields, rings, or abelian groups that take the form of polynomials in a variable and an exponential function.
Definition
In fields
An exponential polynomial generally has both a variable x and some kind of exponential function E(x). In the complex numbers there is already a canonical exponential function, the function that maps x to ex. In this setting the term exponential polynomial is often used to mean polynomials of the form P(x, ex) where P ∈ C[x, y] is a polynomial in two variables.
There is nothing particularly special about C here; exponential polynomials may also refer to such a polynomial on any exponential field or exponential ring with its exponential function taking the place of ex above. Similarly, there is no reason to have one variable, and an exponential polynomial in n variables would be of the form P(x1, ..., xn, ex1, ..., exn), where P is a polynomial in 2n variables.
For formal exponential polynomials over a field K we proceed as follows. Let W be a finitely generated Z-submodule of K and consider finite sums of the form
where the fi are polynomials in K[X] and the exp(wi X) are formal symbols indexed by wi in W subject to exp(u + v) = exp(u) exp(v).
In abelian groups
A more general framework where the term 'exponential polynomial' may be found is that of exponential functions on abelian groups. Similarly to how exponential functions on exponential fields are defined, given a topological abelian group G a homomorphism from G to the additive group of the complex numbers is called an additive function, and a homomorphism to the multiplicative group of nonzero complex numbers is called an exponential function, or simply an exponential. A product of additive functions and exponentials is called an exponential monomial, and a linear combination of these is then an exponential polynomial on G.
Properties
Ritt's theorem states that the analogues of unique factorization and the factor theorem hold for the ring of exponential polynomials.
Applications
Exponential polynomials on R and C often appear in transcendental number theory, where they appear as auxiliary functions in proofs involving the exponential function. They also act as a link between model theory and analytic geometry. If one defines an exponential variety to be the set of points in Rn where some finite collection of exponential polynomials vanish, then results like Khovanskiǐ's theorem in differential geometry and Wilkie's theorem in model theory show that these varieties are well-behaved in the sense that the collection of such varieties is stable under the various set-theoretic operations as long as one allows the inclusion of the image under projections of higher-dimensional exponential varieties. Indeed, the two aforementioned theorems imply that the set of all exponential varieties forms an o-minimal structure over R.
Exponential polynomials appear in the characteristic equation associat
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https://en.wikipedia.org/wiki/Roman%20Catholic%20Diocese%20of%20Tarn%C3%B3w
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The Diocese of Tarnów () is a Latin diocese of the Catholic Church in Poland. According to Church statistics, it is the most religious diocese in Poland, with 72.5% weekly Mass attendance.
References
Roman Catholic dioceses in Poland
Tarnów
Roman Catholic dioceses and prelatures established in the 18th century
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https://en.wikipedia.org/wiki/Hermite%20normal%20form
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In linear algebra, the Hermite normal form is an analogue of reduced echelon form for matrices over the integers Z. Just as reduced echelon form can be used to solve problems about the solution to the linear system Ax=b where x is in Rn, the Hermite normal form can solve problems about the solution to the linear system Ax=b where this time x is restricted to have integer coordinates only. Other applications of the Hermite normal form include integer programming, cryptography, and abstract algebra.
Definition
Various authors may prefer to talk about Hermite normal form in either row-style or column-style. They are essentially the same up to transposition.
Row-style Hermite normal form
An m by n matrix A with integer entries has a (row) Hermite normal form H if there is a square unimodular matrix U where H=UA and H has the following restrictions:
H is upper triangular (that is, hij = 0 for i > j), and any rows of zeros are located below any other row.
The leading coefficient (the first nonzero entry from the left, also called the pivot) of a nonzero row is always strictly to the right of the leading coefficient of the row above it; moreover, it is positive.
The elements below pivots are zero and elements above pivots are nonnegative and strictly smaller than the pivot.
The third condition is not standard among authors, for example some sources force non-pivots to be nonpositive or place no sign restriction on them. However, these definitions are equivalent by using a different unimodular matrix U. A unimodular matrix is a square invertible integer matrix whose determinant is 1 or −1.
Column-style Hermite normal form
A m-by-n matrix A with integer entries has a (column) Hermite normal form H if there is a square unimodular matrix U where H=AU and H has the following restrictions:
H is lower triangular, hij = 0 for i < j, and any columns of zeros are located on the right.
The leading coefficient (the first nonzero entry from the top, also called the pivot) of a nonzero column is always strictly below of the leading coefficient of the column before it; moreover, it is positive.
The elements to the right of pivots are zero and elements to the left of pivots are nonnegative and strictly smaller than the pivot.
Note that the row-style definition has a unimodular matrix U multiplying A on the left (meaning U is acting on the rows of A), while the column-style definition has the unimodular matrix action on the columns of A. The two definitions of Hermite normal forms are simply transposes of each other.
Existence and uniqueness of the Hermite normal form
Every m-by-n matrix A with integer entries has a unique m-by-n matrix H, such that H=UA for some square unimodular matrix U.
Examples
In the examples below, is the Hermite normal form of the matrix , and is a unimodular matrix such that .
If has only one row then either or , depending on whether the single row of has a positive or negative leading coefficient.
Algorithms
There are many al
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https://en.wikipedia.org/wiki/Spanish%20Sign%20Language
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Spanish Sign Language () is a sign language used mainly by deaf people in Spain and the people who live with them. Although there are not many reliable statistics, it is estimated that there are over 100,000 speakers, 20-30% of whom use it as a second language.
From a strictly linguistic point of view, Spanish Sign Language refers to a sign language variety employed in an extensive central-interior area of the Iberian Peninsula, having Madrid as a cultural and linguistic epicenter, with other varieties used in regions such as Asturias, Aragon, Murcia, parts of western Andalusia and near the Province of Burgos.
Mutual intelligibility with the rest of the sign languages used in Spain is generally high due to a highly shared lexicon. However, Catalan Sign Language, Valencian Sign Language as well as the Spanish Sign Language dialects used in eastern Andalusia, Canary Islands, Galicia and Basque Country are the most distinctive lexically (between 10 and 30% difference in the use of nouns, depending on the case). Only the Catalan and Valencian Sign Languages share less than 75% of their vocabulary with the rest of the Spanish dialects, which makes them particularly marked, distinct dialects or even languages separate from Spanish Sign Language, depending on the methods used to determine language versus dialect. Some linguists consider both these and the Spanish Sign language three variants of a polymorphic sign language.
See also
Signed Spanish
References
External links
Alphabet
Sign language isolates
Languages of Spain
Sign languages of Spain
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https://en.wikipedia.org/wiki/Hafner%E2%80%93Sarnak%E2%80%93McCurley%20constant
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The Hafner–Sarnak–McCurley constant is a mathematical constant representing the probability that the determinants of two randomly chosen square integer matrices will be relatively prime. The probability depends on the matrix size, n, in accordance with the formula
where pk is the kth prime number. The constant is the limit of this expression as n approaches infinity. Its value is roughly 0.3532363719... .
References
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External links
Mathematical constants
Infinite products
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https://en.wikipedia.org/wiki/F%C3%A9lix%20Pollaczek
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Félix Pollaczek (1 December 1892 in Vienna – 29 April 1981 at Boulogne-Billancourt) was an Austrian-French engineer and mathematician, known for numerous contributions to number theory, mathematical analysis, mathematical physics and probability theory. He is best known for the Pollaczek–Khinchine formula in queueing theory (1930), and the Pollaczek polynomials.
Education and career
Pollaczek studied at the Technical University of Vienna, got a M.Sc. in electrical engineering from Technical University of Brno (1920), and his Ph.D. in mathematics from University of Berlin (1922) with a dissertation titled Über die Kreiskörper der l-ten und l2-ten Einheitswurzeln, advised by Issai Schur and based on results published first in 1917.
Pollaczek was employed by AEG in Berlin (1921–23), worked for Reichspost (1923–33). In 1933, he was fired because he was Jewish. He moved to Paris, where
he was consulting teletraffic engineer to various institutions from 1933 onwards, including the Société d’Études pour Liaisons Téléphoniques et Télégraphiques (SELT) and the French National Centre for Scientific Research (CNRS).
In 1977, Pollaczek was awarded the John von Neumann Theory Prize, although his age prevented him from receiving the prize in person. He was posthumously elected to the 2002 class of Fellows of the Institute for Operations Research and the Management Sciences.
Personal life
He married mathematician Hilda Geiringer in 1921, and they had a child, Magda, in 1922. However, their marriage did not last, and Magda was brought up by Hilda. Pollaczek became physicist László Tisza's father-in-law due to Magda's marriage.
References
Austrian mathematicians
20th-century French mathematicians
Queueing theorists
Scientists from Vienna
Scientists from Paris
TU Wien alumni
Humboldt University of Berlin alumni
1892 births
1981 deaths
Mathematicians from Austria-Hungary
Austro-Hungarian Jews
Austrian Jews
John von Neumann Theory Prize winners
Fellows of the Institute for Operations Research and the Management Sciences
Austrian emigrants to France
French National Centre for Scientific Research scientists
AEG people
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