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https://en.wikipedia.org/wiki/David%20B.%20A.%20Epstein | David Bernard Alper Epstein FRS (born 1937) is a mathematician known for his work in hyperbolic geometry, 3-manifolds, and group theory, amongst other fields. He co-founded the University of Warwick mathematics department with Christopher Zeeman and is founding editor of the journal Experimental Mathematics.
Higher education and early career
In 1954, Epstein came to the UK after completing his bachelor's degree in mathematics in South Africa. Having received the exemption for Mathematical Tripos part I at the University of Cambridge, he completed Mathematical Tripos part II in 1955 and Mathematical Tripos part III in 1957. He completed his Ph.D. on the topic of three-dimensional manifolds under the supervision of Christopher Zeeman in 1960. He then travelled to Princeton University, where he spent one year attending the lectures of Norman Steenrod on cohomology operations, making notes and revisions to them, later published as a book by the Princeton University Press in 1962.
In 1961, Epstein moved to the Institute for Advanced Study (IAS) at Princeton. He returned to the UK in 1962 to become a research fellow of the newly founded Churchill College, Cambridge. In 1964, he moved to the Mathematics Institute of the University of Warwick to take up a Readership position there. He was the first academic at the University of Warwick to move into local accommodation, though many professors were appointed before him.
Awards and honours
Epstein was awarded the Senior Berwick Prize by the London Mathematical Society in 1988.
In 2004 he was elected a Fellow of the Royal Society. In 2012 he became a fellow of the American Mathematical Society.
Personal life
David Epstein was born in 1937 in Pretoria, South Africa to Ben Epstein and Pauline (or Polly) Alper, both Jewish of Lithuanian descent, though Polly was born in South Africa. David finished school at the age of 14, and graduated from the University of the Witwatersrand at the age of 17. He then won a scholarship to the University of Cambridge, where he did Parts II and III of the Mathematical Tripos, graduating in 1957. He married Rona in 1958, after dating her from when he was 16 and she was 14. He did a Ph.D. in Cambridge under Christopher Zeeman, which he completed at the age of 23 in 1960, when he was awarded a Research Fellowship at Trinity College, Cambridge, which he never took up.
After completing his Ph.D., Epstein went to Princeton University for one year, and at the Institute for Advanced Study in Princeton, New Jersey for another year. He returned to Cambridge in 1962, where he was an assistant lecturer at the university and director of studies at the new Churchill College. In 1963 his younger sister Debbie left South Africa when she was considered to be in danger of arrest by the South African apartheid regime. At this stage, his father Ben was also having severe problems with the South African regime as a result of his ethical stand as a doctor. For example, he was instructed by the |
https://en.wikipedia.org/wiki/WaveLab%20%28mathematics%20software%29 | WaveLab is a collection of MATLAB functions for wavelet analysis. Following the success of WaveLab package, there is now the availability of CurveLab and ShearLab.
Wavelets |
https://en.wikipedia.org/wiki/Hitchin%20system | In mathematics, the Hitchin integrable system is an integrable system depending on the choice of a complex reductive group and a compact Riemann surface, introduced by Nigel Hitchin in 1987. It lies on the crossroads of algebraic geometry, the theory of Lie algebras and integrable system theory. It also plays an important role in the geometric Langlands correspondence over the field of complex numbers through conformal field theory.
A genus zero analogue of the Hitchin system, the Garnier system, was discovered by René Garnier somewhat earlier as a certain limit of the Schlesinger equations, and Garnier solved his system by defining spectral curves. (The Garnier system is the classical limit of the Gaudin model. In turn, the Schlesinger equations are the classical limit of the Knizhnik–Zamolodchikov equations).
Almost all integrable systems of classical mechanics can be obtained as particular cases of the Hitchin system or their common generalization defined by Bottacin and Markman in 1994.
Description
Using the language of algebraic geometry, the phase space of the system is a partial compactification of the cotangent bundle to the moduli space of stable G-bundles for some reductive group G, on some compact algebraic curve. This space is endowed with a canonical symplectic form. Suppose for simplicity that , the general linear group; then the Hamiltonians can be described as follows: the tangent space to the moduli space of G-bundles at the bundle F is
which by Serre duality is dual to
where is the canonical bundle, so a pair
called a Hitchin pair or Higgs bundle, defines a point in the cotangent bundle. Taking
one obtains elements in
which is a vector space which does not depend on . So taking any basis in these vector spaces we obtain functions Hi, which are Hitchin's hamiltonians. The construction for general reductive group is similar and uses invariant polynomials on the Lie algebra of G.
For trivial reasons these functions are algebraically independent, and some calculations show that their number is exactly half of the dimension of the phase space. The nontrivial part is a proof of Poisson commutativity of these functions. They therefore define an integrable system in the symplectic or Arnol'd–Liouville sense.
Hitchin fibration
The Hitchin fibration is the map from the moduli space of Hitchin pairs to characteristic polynomials, a higher genus analogue of the map Garnier used to define the spectral curves. used Hitchin fibrations over finite fields in his proof of the fundamental lemma.
See also
Yang–Mills equations
Higgs bundle
Nonabelian Hodge correspondence
Character variety
Hitchin's equations
References
Algebraic geometry
Dynamical systems
Hamiltonian mechanics
Integrable systems
Lie groups
Differential geometry |
https://en.wikipedia.org/wiki/Water%20polo%20at%20the%202009%20World%20Aquatics%20Championships | Water polo at the 2009 World Aquatics Championships were held from July 19 to August 1, 2009, in Rome, Italy.
Medalists
Men
Women
References
External links
FINA Water Polo
Records and statistics (reports by Omega)
2009 in water polo
2009 World Aquatics Championships
2009
2009 |
https://en.wikipedia.org/wiki/Superintegrable%20Hamiltonian%20system | In mathematics, a superintegrable Hamiltonian system is a Hamiltonian system on a -dimensional symplectic manifold for which the following conditions hold:
(i) There exist independent integrals of motion. Their level surfaces (invariant submanifolds) form a fibered manifold over a connected open subset .
(ii) There exist smooth real functions on such that the Poisson bracket of integrals of motion reads
.
(iii) The matrix function is of constant corank on .
If , this is the case of a completely integrable Hamiltonian system. The Mishchenko-Fomenko theorem for superintegrable Hamiltonian systems generalizes the Liouville-Arnold theorem on action-angle coordinates of completely integrable Hamiltonian system as follows.
Let invariant submanifolds of a superintegrable Hamiltonian system be connected compact and mutually diffeomorphic. Then the fibered manifold is a fiber bundle
in tori . There exists an open neighbourhood of which is a trivial fiber bundle provided with the bundle (generalized action-angle) coordinates ,
, such that are coordinates on . These coordinates are the Darboux coordinates on a symplectic manifold . A Hamiltonian of a superintegrable system depends only on the action variables which are the Casimir functions of the coinduced Poisson structure on .
The Liouville-Arnold theorem for completely integrable systems and the Mishchenko-Fomenko theorem for the superintegrable ones are generalized to the case of non-compact invariant submanifolds. They are diffeomorphic to a toroidal cylinder .
See also
Integrable system
Action-angle coordinates
Nambu mechanics
Laplace–Runge–Lenz vector
References
Mishchenko, A., Fomenko, A., Generalized Liouville method of integration of Hamiltonian systems, Funct. Anal. Appl. 12 (1978) 113.
Bolsinov, A., Jovanovic, B., Noncommutative integrability, moment map and geodesic flows, Ann. Global Anal. Geom. 23 (2003) 305; .
Fasso, F., Superintegrable Hamiltonian systems: geometry and perturbations, Acta Appl. Math. 87(2005) 93.
Fiorani, E., Sardanashvily, G., Global action-angle coordinates for completely integrable systems with non-compact invariant manifolds, J. Math. Phys. 48 (2007) 032901; .
Miller, W., Jr, Post, S., Winternitz P., Classical and quantum superintegrability with applications, J. Phys. A 46 (2013), no. 42, 423001,
Giachetta, G., Mangiarotti, L., Sardanashvily, G., Geometric Methods in Classical and Quantum Mechanics (World Scientific, Singapore, 2010) ; .
Hamiltonian mechanics
Dynamical systems
Integrable systems |
https://en.wikipedia.org/wiki/Bojan%20Pavlovi%C4%87%20%28footballer%2C%20born%201986%29 | Bojan Pavlović (; born 8 November 1986) is a Serbian professional footballer who plays as a goalkeeper for Bosnian Premier League club Borac Banja Luka.
Career statistics
Club
Honours
Makedonija Gjorče Petrov
Macedonian First League: 2008–09
Red Star Belgrade
Serbian Cup: 2009–10
Borac Banja Luka
Bosnian Premier League: 2020–21
References
External links
Bojan Pavlović at Sofascore
1986 births
Living people
Sportspeople from Loznica
Serbian men's footballers
Serbian expatriate men's footballers
Serbian SuperLiga players
Azerbaijan Premier League players
Liga Leumit players
Erovnuli Liga players
Premier League of Bosnia and Herzegovina players
FK Bežanija players
Red Star Belgrade footballers
FK Palilulac Beograd players
RFK Grafičar Beograd players
FK Radnički Pirot players
FK Makedonija G.P. players
Qarabağ FK players
Hapoel Ashkelon F.C. players
OFK Beograd players
FC Zestafoni players
FK Sarajevo players
NK Čelik Zenica players
FK Borac Banja Luka players
Expatriate men's footballers in North Macedonia
Expatriate men's footballers in Azerbaijan
Expatriate men's footballers in Bosnia and Herzegovina
Men's association football goalkeepers |
https://en.wikipedia.org/wiki/Computer%20algebra | In mathematics and computer science, computer algebra, also called symbolic computation or algebraic computation, is a scientific area that refers to the study and development of algorithms and software for manipulating mathematical expressions and other mathematical objects. Although computer algebra could be considered a subfield of scientific computing, they are generally considered as distinct fields because scientific computing is usually based on numerical computation with approximate floating point numbers, while symbolic computation emphasizes exact computation with expressions containing variables that have no given value and are manipulated as symbols.
Software applications that perform symbolic calculations are called computer algebra systems, with the term system alluding to the complexity of the main applications that include, at least, a method to represent mathematical data in a computer, a user programming language (usually different from the language used for the implementation), a dedicated memory manager, a user interface for the input/output of mathematical expressions, a large set of routines to perform usual operations, like simplification of expressions, differentiation using chain rule, polynomial factorization, indefinite integration, etc.
Computer algebra is widely used to experiment in mathematics and to design the formulas that are used in numerical programs. It is also used for complete scientific computations, when purely numerical methods fail, as in public key cryptography, or for some non-linear problems.
Terminology
Some authors distinguish computer algebra from symbolic computation using the latter name to refer to kinds of symbolic computation other than the computation with mathematical formulas. Some authors use symbolic computation for the computer science aspect of the subject and "computer algebra" for the mathematical aspect. In some languages the name of the field is not a direct translation of its English name. Typically, it is called calcul formel in French, which means "formal computation". This name reflects the ties this field has with formal methods.
Symbolic computation has also been referred to, in the past, as symbolic manipulation, algebraic manipulation, symbolic processing, symbolic mathematics, or symbolic algebra, but these terms, which also refer to non-computational manipulation, are no longer used in reference to computer algebra.
Scientific community
There is no learned society that is specific to computer algebra, but this function is assumed by the special interest group of the Association for Computing Machinery named SIGSAM (Special Interest Group
on Symbolic and Algebraic Manipulation).
There are several annual conferences on computer algebra, the premier being ISSAC (International Symposium on Symbolic and Algebraic Computation), which is regularly sponsored by SIGSAM.
There are several journals specializing in computer algebra, the top one being Journal of Symbolic Comp |
https://en.wikipedia.org/wiki/Duncan%20McTier | Duncan McTier is an English double bass soloist and professor. He is a member of the Fibonacci Sequence.
Biography
Born in Worcestershire, England, Duncan McTier studied a degree in mathematics at Bristol University before joining the BBC Symphony Orchestra and the Netherlands Chamber Orchestra. McTier won the Isle of Man International Double Bass Competition in 1982 and since then he has performed often with many orchestras, including the Academy of St. Martin in the Fields, Royal Scottish National Orchestra, English Chamber Orchestra, Scottish Chamber Orchestra, BBC Scottish Symphony Orchestra, Concertgebouw Chamber Orchestra, RTVE Symphony Orchestra and the Orchestre de Chambre de Lausanne.
He recorded a series of albums with the pianist Kathron Sturrock.
In November 2014 McTier received a non-custodial sentence after pleading guilty to two indecent assaults and one attempted indecent assault, during the 1980s and 1990s, on former students aged between 17 and 23.
From 1996-2014 McTier was a professor of double bass at the Royal Academy of Music in London and is currently professor at the Queen Sofía College of Music in Madrid, Spain. In 2019, McTier retired from in Zürich.
References
External links
Living people
Academics of the Royal Academy of Music
British double-bassists
Male double-bassists
Academic staff of the Reina Sofía School of Music
Honorary Members of the Royal Academy of Music
Academic staff of the Zurich University of the Arts
English people convicted of indecent assault
21st-century double-bassists
21st-century British male musicians
Year of birth missing (living people)
English expatriates in Switzerland
English expatriates in Spain |
https://en.wikipedia.org/wiki/Vasily%20Vladimirov | Vasily Sergeyevich Vladimirov (; 9 January 1923 – 3 November 2012) was a Soviet and Russian mathematician working in the fields of number theory, mathematical physics, quantum field theory, numerical analysis, generalized functions, several complex variables, p-adic analysis, multidimensional Tauberian theorems.
Life
Vladimirov was born to a peasant family of 5 children, in 1923, Petrograd. Under the impact of food shortage and poverty, he began schooling in 1930. He then went to a 7-year school in 1934, but transferred to the Leningrad Technical School of Hydrology and Meteorology in 1937. In 1939, at the age of sixteen, he enrolled into a night preparatory school for workers, and finally successfully progressed to Leningrad University to study physics.
During the Second World War, Vladimirov took part in defence of Leningrad against German invasion, building defences, working as a tractor driver and as meteorologist in Air Force after training. He served in several different units, mainly as part of air-defense system of Leningrad. He was given the rank of sergeant major in the reserves after the war and permitted to return to his study.
When he returned to university, Vladimirov shifted his focus of interest from physics to number theory. Under the advice of Boris Alekseevich Venkov (1900-1962), an expert on quadratic forms , he started undertaking research in number theory and attained a master's degree in 1948. In the first thesis of his master study in Leningrad, he confirmed the existence of non-extreme perfect quadratic form in six variables in Georgy Fedoseevich Voronoy's conjecture. In his second thesis, he approached packing problems for convex bodies initiated by Hermann Minkowski. Upon graduation, he was appointed as a junior researcher in the Leningrad Branch of the Steklov Mathematical Institute of the USSR Academy of Sciences.
As the Soviet atomic bomb programme ran, Vladimirov was assigned to assist with the development of the bomb, in joint force with many top scientists and industrialists. He worked with Vitalevich Kantorovich calculating critical parameters of certain simple nuclear systems. In 1950, when he was sent to Arzamas-16, he worked under the direction of Nikolai Nikolaevich Bogolyubov, who later became a long-term collaborator with Vladimirov. In Arzamas-16, Vladimirov worked on finding mathematical solutions for problems raised by physicists. He developed new techniques for the numerical solution of boundary value problems, especially for solving the kinetic equation of neutron transfer in nuclear reactors in 1952, which is now known as Vladimirov method.
After the success of the bomb project, Vladimirov was awarded the Stalin Prize in for his contribution 1953. He continued working on mathematics for atomic bomb in the Central Scientific Research Institute for Artillery Armaments, where he served as Senior Researcher in 1955. Vladimirov moved to Steklov Mathematical Institute, Moscow, in 1956, under the supe |
https://en.wikipedia.org/wiki/Gregory%27s%20series | In mathematics, Gregory's series for the inverse tangent function is its infinite Taylor series expansion at the origin:
This series converges in the complex disk except for (where
It was first discovered in the 14th century by Madhava of Sangamagrama (c. 1340 – c. 1425), as credited by Madhava's Kerala school follower Jyeṣṭhadeva's Yuktibhāṣā (c. 1530). In recent literature it is sometimes called the Madhava–Gregory series to recognize Madhava's priority (see also Madhava series). It was independently rediscovered by James Gregory in 1671 and by Gottfried Leibniz in 1673, who obtained the Leibniz formula for as the special case
Proof
If then The derivative is
Taking the reciprocal,
This sometimes is used as a definition of the arctangent:
The Maclaurin series for is a geometric series:
One can find the Maclaurin series for by naïvely integrating term-by-term:
While this turns out correctly, integrals and infinite sums cannot always be exchanged in this manner. To prove that the integral on the left converges to the sum on the right for real can instead be written as the finite sum,
Again integrating both sides,
In the limit as the integral on the right above tends to zero when because
Therefore,
Convergence
The series for and converge within the complex disk , where both functions are holomorphic. They diverge for because when , there is a pole:
When the partial sums alternate between the values and never converging to the value
However, its term-by-term integral, the series for (barely) converges when because disagrees with its series only at the point so the difference in integrals can be made arbitrarily small by taking sufficiently many terms:
Because of its exceedingly slow convergence (it takes five billion terms to obtain 10 correct decimal digits), the Leibniz formula is not a very effective practical method for computing Finding ways to get around this slow convergence has been a subject of great mathematical interest.
History
The earliest person to whom the series can be attributed with confidence is Madhava of Sangamagrama (c. 1340 – c. 1425). The original reference (as with much of Madhava's work) is lost, but he is credited with the discovery by several of his successors in the Kerala school of astronomy and mathematics founded by him. Specific citations to the series for include Nilakantha Somayaji's Tantrasangraha (c. 1500),
Jyeṣṭhadeva's Yuktibhāṣā (c. 1530), and the Yukti-dipika commentary by Sankara Variyar, where it is given in verses 2.206 – 2.209.
See also
List of mathematical series
Madhava series
Notes
References
Mathematical series |
https://en.wikipedia.org/wiki/Maurer%20rose | In geometry, the concept of a Maurer rose was introduced by Peter M. Maurer in his article titled A Rose is a Rose.... A Maurer rose consists of some lines that connect some points on a rose curve.
Definition
Let r = sin(nθ) be a rose in the polar coordinate system, where n is a positive integer. The rose has n petals if n is odd, and 2n petals if n is even.
We then take 361 points on the rose:
(sin(nk), k) (k = 0, d, 2d, 3d, ..., 360d),
where d is a positive integer and the angles are in degrees, not radians.
Explanation
A Maurer rose of the rose r = sin(nθ) consists of the 360 lines successively connecting the above 361 points. Thus a Maurer rose is a polygonal curve with vertices on a rose.
A Maurer rose can be described as a closed route in the polar plane. A walker starts a journey from the origin, (0, 0), and walks along a line to the point (sin(nd), d). Then, in the second leg of the journey, the walker walks along a line to the next point, (sin(n·2d), 2d), and so on. Finally, in the final leg of the journey, the walker walks along a line, from (sin(n·359d), 359d) to the ending point, (sin(n·360d), 360d). The whole route is the Maurer rose of the rose r = sin(nθ). A Maurer rose is a closed curve since the starting point, (0, 0) and the ending point, (sin(n·360d), 360d), coincide.
The following figure shows the evolution of a Maurer rose (n = 2, d = 29°).
Images
The following are some Maurer roses drawn with some values for n and d:
Example implementation
Using Python:
import math, turtle
screen = turtle.Screen()
screen.setup(width=800, height=800, startx=0, starty=0)
screen.bgcolor('black')
pen = turtle.Turtle()
pen.speed(20)
n = 5
d = 97
pen.color('blue')
pen.pensize(0.5)
for theta in range(361):
k = theta * d * math.pi / 180
r = 300 * math.sin(n * k)
x = r * math.cos(k)
y = r * math.sin(k)
pen.goto(x, y)
pen.color('red')
pen.pensize(4)
for theta in range(361):
k = theta * math.pi / 180
r = 300 * math.sin(n * k)
x = r * math.cos(k)
y = r * math.sin(k)
pen.goto(x, y)
References
(Interactive Demonstrations)
Curves
Polygons
Articles with example Python (programming language) code
External links
Interactive Demonstration: https://codepen.io/Igor_Konovalov/full/ZJwPQv/
Explorer: https://filip26.github.io/maurer-rose-explorer/ [source code]
Draw from values and create vector graphics: https://www.sqrt.ch/Buch/Maurer/maurerroses.html |
https://en.wikipedia.org/wiki/K.%20V.%20Sarma | Krishna Venkateswara Sarma (1919–2005) was an Indian historian of science, particularly the astronomy and mathematics of the Kerala school. He was responsible for bringing to light several of the achievements of the Kerala school. He was editor of the Vishveshvaranand Indological Research Series, and published the critical edition of several source works in Sanskrit, including the Aryabhatiya of Aryabhata. He was recognised as "the greatest authority on Kerala's astronomical tradition".
Biography
Sarma's father, Sri S. Krishna Aiyer, was an inspector of schools. Sarma studied chemistry and physics at Maharaja's College of Science in Thiruvananthapuram, receiving his bachelor's degree in 1940. He went on to study Sanskrit at the College of Arts, receiving a master's degree in 1942 from Kerala University.
In 1944 he began his work with palm-leaf manuscripts at Oriental Research Institute & Manuscripts Library where he developed his specialties of manuscriptology and textual criticism.
Sarma joined the Sanskrit department of the University of Madras in 1951 as research assistant in the New Catalogues Project,
In 1962 he became curator of the Vishveshvaranand Research Institute, Hoshiarpur which had a A Vedic Word Concordance. Sarma took an interest in the Kerala school of astronomy and mathematics and assembled a bibliography. In 1965 Panjab University assimilated the institute, and Sarma became a lecturer in Sanskrit with the university. He was named reader in 1972.
Sarma's book A History of the Kerala School of Hindu Astronomy recounted the development of astronomy associated with Kerala. In the preface of A History, Sarma described his research "under the supervision of Prof. Ramaswami Sastri, concurrently with my duties as the Supervising Pundit of the Cataloguing Section of the University Oriental Manuscripts Library, Trivandrum. My intimate association, later, with the compilation of the New Catalogus Cataloguum of Sanskrit Works and Authors at Madras university also proved to be of great help in my work."
Sarma became acting-director of Vishveshvaranand Research Institute in 1975, and director/professor in 1978 when he was awarded Doctor of Letters.
He retired from Panjab University in 1980, but the next year accepted the position of honorary professor of Sanskrit at the Adyar Library Research Center.
He was the author of thirty-five entries in the Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures. For example, in one article he says
Rationale in Hindu mathematics and astronomy is expressed by the terms Yukti and Upapatti, both meaning "the logical principles implied". It is characteristic of Western scientific tradition, from the times of Euclid and Aristotle up to modern times, to enunciate and deduce using step-by-step reasoning. Such a practice is almost absent in the Indian tradition, even though the same background tasks, of collecting and correlating data, identifying and analyzing metho |
https://en.wikipedia.org/wiki/Convex%20graph | In mathematics, a convex graph may be
a convex bipartite graph
a convex plane graph
the graph of a convex function |
https://en.wikipedia.org/wiki/Bureau%20of%20Investigation%20and%20Statistics | The National Bureau of Investigation and Statistics (Military Commission), (NBIS or BIS) (), commonly known as Juntong (), was the military intelligence agency of the Republic of China before 1946. It was devoted to intelligence gathering and covert spying operation for purposes of national security and defense. It was originally headed by Dai Li, and after 1946 he was succeeded by Mao Renfeng. This bureau was largely superseded by the Military Intelligence Bureau under Ministry of National Defense in Taiwan today.
The NBIS had a great influence amongst the Nationalist Government's military, police, administration, and transportation agencies, as well as embassies and consulates abroad during the Political Tutelage period (1928-1946) of Republic of China. It was often criticized by the political dissidents as a "secret police" involved in covert and espionage operation, including surveillance, kidnapping, assassinations, elimination and house-arrest against Chinese communists, Japanese spies as well as political dissidents.
During the Sino-Japanese War, the NBIS was involved in a number of counter-intelligence and covert espionage warfare against the Japanese invaders. There were NBIS agents who defected to the Japanese, and many of the secret police in Wang Jingwei's Japanese-occupied areas were former NBIS agents.
From a historical perspective, NBIS played an important role in Second Sino-Japanese War. Under the leadership of Dai Li, the Nationalist Government had a body of 100,000 active spies involving in espionage warfare against Japanese, as well as against the Wang Jingwei-led puppet Nationalist Government of the Japanese-occupied areas.
History
Early stages
The NBIS was founded in 1932 as the "Military Commission of Clandestine Investigation Section of the National Revolutionary Army" () with the "Special Works Department"() set up in 1932. When the "Investigation and Statistics Bureau" was established under the Military Commission, the "Special Works Department" was incorporated into the Bureau and renamed the "Second Division", and is responsible for intelligence collection and personnel training. All of the bureau's affairs were under the direct command of Chiang Kai-shek. Dai sought to make the Juntong into an extended family with himself as the stern paternal figure, stressing traditional Chinese Confucian values of filial piety, loyalty, benevolence, and righteousness. The Juntong operated as a traditional sworn brotherhood with all of the senior officers taking an oath making themselves into "brothers". The inspiration for the Juntong were the secret sworn brotherhoods portrayed in the classics of Chinese literature like Water Margin and The Romance of the Three Kingdoms. Dai presented himself as a stern Confucian father figure to the men and women of the Juntong and liked to quote from book The Dynastic History of the Han is "Is personal happiness possible before the extermination of the Xiongnu?" Reflecting this mentalit |
https://en.wikipedia.org/wiki/Carol%20Martin%20Gatton%20Academy%20of%20Mathematics%20and%20Science%20in%20Kentucky | The Gatton Academy (Carol Martin Gatton Academy of Mathematics and Science in Kentucky) is a public academy and an early college entrance program funded by the state of Kentucky and located on the campus of Western Kentucky University in Bowling Green, KY, United States.
In 2010 and 2011 the Gatton Academy ranked on [[Newsweek|Newsweek'''s]] Public Elite list, a list of the nation's 20 top public high schools, as graded by scores on standardized tests. The Gatton Academy was recognized by Newsweek magazine as one of the nation's top five high schools. America's Best High Schools 2011 recognized more than 500 schools from across the United States. In June 2012 the Gatton Academy was recognized as Newsweek's Top School in America. For three years in a row, the Gatton Academy was ranked the best in the nation by The Daily Beast''.
Overview
The Gatton Academy began in the 2007–2008 school year.
The Academy admits 95-105 qualifying high school students (aiming for a total of 200 students attending) each year to spend their junior and senior years on the WKU campus taking classes at the university. The students are selected on basis of grades, standardized test scores, extracurricular activities, teacher and community leader recommendations, personal interview, and interest in a science, mathematical, or engineering career, and focus their classes mainly on mathematics and sciences.
Students of the Academy are considered both undergraduates and high school students by Kentucky and federal scholarship programs. As such, they are qualified for undergraduate research programs, scholarships, honors, and even (in exceptional cases) bachelor's degrees, but at the same time must take the classes required by the state of Kentucky for a high school diploma (and, if still enrolled in their home high school, the classes required by their previous school). Students also have the option of being dually-enrolled, or, remaining students at their home high schools while attending the Academy. However, some private schools will not allow students to remain enrolled while attending the Academy. This dual-enrollment option allows students, in some cases, to remain eligible for services offered by their home high school (guidance, textbook funding). However, this option also requires students to meet state graduation requirements, and participate in KPREP testing. The home schools benefit from this arrangement by receiving the test scores from their respective Gatton scholars.
Most of the school's graduates attend four-year colleges (67% of graduates attend either Western Kentucky University, the University of Kentucky or the University of Louisville), while some chose to pursue other opportunities during gap years.
See also
Alabama School of Mathematics and Science
Arkansas School for Mathematics, Sciences, and the Arts
Craft Academy for Excellence in Science and Mathematics
Illinois Mathematics and Science Academy
Indiana Academy for Science, Mathematics, and |
https://en.wikipedia.org/wiki/Aurifeuillean%20factorization | In number theory, an aurifeuillean factorization, named after Léon-François-Antoine Aurifeuille, is factorization of certain integer values of the cyclotomic polynomials. Because cyclotomic polynomials are irreducible polynomials over the integers, such a factorization cannot come from an algebraic factorization of the polynomial. Nevertheless, certain families of integers coming from cyclotomic polynomials have factorizations given by formulas applying to the whole family, as in the examples below.
Examples
Numbers of the form have the following factorization (Sophie Germain's identity): Setting and , one obtains the following aurifeuillean factorization of , where is the fourth cyclotomic polynomial:
Numbers of the form have the following factorization, where the first factor () is the algebraic factorization of sum of two cubes: Setting and , one obtains the following factorization of : Here, the first of the three terms in the factorization is and the remaining two terms provide an aurifeuillean factorization of , where .
Numbers of the form or their factors , where with square-free , have aurifeuillean factorization if and only if one of the following conditions holds:
and
and
Thus, when with square-free , and is congruent to modulo , then if is congruent to 1 mod 4, have aurifeuillean factorization, otherwise, have aurifeuillean factorization.
When the number is of a particular form (the exact expression varies with the base), aurifeuillean factorization may be used, which gives a product of two or three numbers. The following equations give aurifeuillean factors for the Cunningham project bases as a product of F, L and M:
If we let L = C − D, M = C + D, the aurifeuillean factorizations for bn ± 1 of the form F * (C − D) * (C + D) = F * L * M with the bases 2 ≤ b ≤ 24 (perfect powers excluded, since a power of bn is also a power of b) are:
(for the coefficients of the polynomials for all square-free bases up to 199 and up to 998, see )
{|class="wikitable" style="text-align:center"
!b
!Number
!(C − D) * (C + D) = L * M
!F
!C
!D
|-
!2
|24k + 2 + 1
|
|1
|22k + 1 + 1
|2k + 1
|-
!3
|36k + 3 + 1
|
|32k + 1 + 1
|32k + 1 + 1
|3k + 1
|-
!5
|510k + 5 - 1
|
|52k + 1 - 1
|54k + 2 + 3(52k + 1) + 1
|53k + 2 + 5k + 1
|-
!6
|612k + 6 + 1
|
|64k + 2 + 1
|64k + 2 + 3(62k + 1) + 1
|63k + 2 + 6k + 1
|-
!7
|714k + 7 + 1
|
|72k + 1 + 1
|76k + 3 + 3(74k + 2) + 3(72k + 1) + 1
|75k + 3 + 73k + 2 + 7k + 1
|-
!10
|1020k + 10 + 1
|
|104k + 2 + 1
|108k + 4 + 5(106k + 3) + 7(104k + 2) + 5(102k + 1) + 1
|107k + 4 + 2(105k + 3) + 2(103k + 2) + 10k + 1
|-
!11
|1122k + 11 + 1
|
|112k + 1 + 1
|1110k + 5 + 5(118k + 4) - 116k + 3 - 114k + 2 + 5(112k + 1) + 1
|119k + 5 + 117k + 4 - 115k + 3 + 113k + 2 + 11k + 1
|-
!12
|126k + 3 + 1
|
|122k + 1 + 1
|122k + 1 + 1
|6(12k)
|-
!13
|1326k + 13 - 1
|
|132k + 1 - 1
|1312k + 6 + 7(1310k + 5) + 15(138k + 4) + 19(136k + 3) + 15(134k + 2) + 7(132k + 1) + 1
|1311k + 6 + 3(139k + 5) + 5(137k + 4) + 5(1 |
https://en.wikipedia.org/wiki/Hausdorff%20Center%20for%20Mathematics | The Hausdorff Center for Mathematics (HCM) is a research center in Bonn, formed by the four mathematical institutes of the Rheinische Friedrich-Wilhelms-Universität Bonn (Mathematical Institute, Institute for Applied Mathematics, Institute for Numerical Simulation, Research Institute for Discrete Mathematics), the Max Planck Institute for Mathematics (MPIM), and the Institute for Social and Economic Sciences.
History
The Hausdorff Center was established in 2006 as one of the seventeen national Clusters of Excellence that were part of the German government's Excellence Initiative. It was officially opened with a colloquium on 19 and 20 January 2007. In 2012, a second funding period was granted. The Hausdorff Center is the only cluster of excellence in the area of mathematics in Germany. It was ranked 14th in the world for mathematics by the Academic Ranking of World Universities in 2021.
The center is named after the mathematician Felix Hausdorff (born 8 November 1868; died 26 January 1942).
Organization
The coordinator of the HCM is Karl-Theodor Sturm. Altogether, about 70 professors from Bonn are affiliated with the HCM: all professors for Mathematics, of the MPI, and for Theoretical Economy. These include the director of the MPI, Gerd Faltings, who was awarded the Fields Medal in 1986, and Peter Scholze, who was awarded the Fields Medal in 2018.
The Hausdorff Research Institute for Mathematics (HIM), the Bonn International Graduate School in Mathematics (BIGS), and the Hausdorff School for Advanced Studies in Mathematics are part of the Hausdorff Center:
The Hausdorff Research Institute for Mathematics (HIM) organizes international long-term programs and fosters cooperations between German mathematicians and internationally renowned scientists in mathematics and mathematical economics. It also runs specific programs for young scientists. Director of the HIM is Christoph Thiele.
The Bonn International Graduate School in Mathematics (BIGS) supports the scientific development of PhD students. Director of the BIGS is Barbara Niethammer.
The Hausdorff School for Advanced Studies in Mathematics is a training program for postdoctoral researchers based in Germany and worldwide. It fosters the systematic qualification of young scientists. Director is Patrik Ferrari.
References
External links
Hausdorff Center for Mathematics, Official Website
Hausdorff Research Institute for Mathematics
Bonn International Graduate School for Mathematics
Mathematical institutes
Research institutes in Germany
Bonn |
https://en.wikipedia.org/wiki/Calculus%20of%20functors | In algebraic topology, a branch of mathematics, the calculus of functors or Goodwillie calculus is a technique for studying functors by approximating them by a sequence of simpler functors; it generalizes the sheafification of a presheaf. This sequence of approximations is formally similar to the Taylor series of a smooth function, hence the term "calculus of functors".
Many objects of central interest in algebraic topology can be seen as functors, which are difficult to analyze directly, so the idea is to replace them with simpler functors which are sufficiently good approximations for certain purposes.
The calculus of functors was developed by Thomas Goodwillie in a series of three papers in the 1990s and 2000s, and has since been expanded and applied in a number of areas.
Examples
A motivational example, of central interest in geometric topology, is the functor of embeddings of one manifold M into another manifold N, whose first derivative in the sense of calculus of functors is the functor of immersions. As every embedding is an immersion, one obtains an inclusion of functors – in this case the map from a functor to an approximation is an inclusion, but in general it is simply a map.
As this example illustrates, the linear approximation of a functor (on a topological space) is its sheafification, thinking of the functor as a presheaf on the space (formally, as a functor on the category of open subsets of the space), and sheaves are the linear functors.
This example was studied by Goodwillie and Michael Weiss.
Definition
Here is an analogy: with the Taylor series method from calculus, you can approximate the shape of a smooth function f around a point x by using a sequence of increasingly accurate polynomial functions. In a similar way, with the calculus of functors method, you can approximate the behavior of certain kind of functor F at a particular object X by using a sequence of increasingly accurate polynomial functors.
To be specific, let M be a smooth manifold and let O(M) be the category of open subspaces of M, i.e., the category where the objects are the open subspaces of M, and the morphisms are inclusion maps. Let F be a contravariant functor from the category O(M) to the category Top of topological spaces with continuous morphisms. This kind of functor, called a Top-valued presheaf on M, is the kind of functor you can approximate using the calculus of functors method: for a particular open set X∈O(M), you may want to know what sort of a topological space F(X) is, so you can study the topology of the increasingly accurate approximations F0(X), F1(X), F2(X), and so on.
In the calculus of functors method, the sequence of approximations consists of (1) functors , and so on, as well as (2) natural transformations for each integer k. These natural transforms are required to be compatible, meaning that the composition equals the map and thus form a tower
and can be thought of as "successive approximations", just as in a T |
https://en.wikipedia.org/wiki/Paneitz%20operator | In the mathematical field of differential geometry, the Paneitz operator is a fourth-order differential operator defined on a Riemannian manifold of dimension n. It is named after Stephen Paneitz, who discovered it in 1983, and whose preprint was later published posthumously in .
In fact, the same operator was found earlier in the context of conformal supergravity by E. Fradkin and A. Tseytlin in 1982
(Phys Lett B 110 (1982) 117 and Nucl Phys B 1982 (1982) 157 ).
It is given by the formula
where Δ is the Laplace–Beltrami operator, d is the exterior derivative, δ is its formal adjoint, V is the Schouten tensor, J is the trace of the Schouten tensor, and the dot denotes tensor contraction on either index. Here Q is the scalar invariant
where Δ is the positive Laplacian. In four dimensions this yields the Q-curvature.
The operator is especially important in conformal geometry, because in a suitable sense it depends only on the conformal structure. Another operator of this kind is the conformal Laplacian. But, whereas the conformal Laplacian is second-order, with leading symbol a multiple of the Laplace–Beltrami operator, the Paneitz operator is fourth-order, with leading symbol the square of the Laplace–Beltrami operator. The Paneitz operator is conformally invariant in the sense that it sends conformal densities of weight to conformal densities of weight . Concretely, using the canonical trivialization of the density bundles in the presence of a metric, the Paneitz operator P can be represented in terms of a representative the Riemannian metric g as an ordinary operator on functions that transforms according under a conformal change according to the rule
The operator was originally derived by working out specifically the lower-order correction terms in order to ensure conformal invariance. Subsequent investigations have situated the Paneitz operator into a hierarchy of analogous conformally invariant operators on densities: the GJMS operators.
The Paneitz operator has been most thoroughly studied in dimension four where it appears naturally in connection with extremal problems for the functional determinant of the Laplacian (via the Polyakov formula; see ). In dimension four only, the Paneitz operator is the "critical" GJMS operator, meaning that there is a residual scalar piece (the Q curvature) that can only be recovered by asymptotic analysis. The Paneitz operator appears in extremal problems for the Moser–Trudinger inequality in dimension four as well
CR Paneitz operator
There is a close connection between 4 dimensional Conformal Geometry and 3 dimensional CR geometry associated with the study of CR manifolds. There is a naturally defined fourth order operator on CR manifolds introduced by C. Robin Graham and John Lee that has many properties similar to the classical Paneitz operator defined on 4 dimensional Riemannian manifolds. This operator in CR geometry is called the CR Paneitz operator. The operator defined by Graha |
https://en.wikipedia.org/wiki/SOFA%20Statistics | SOFA Statistics is an open-source statistical package. The name stands for Statistics Open For All. It has a graphical user interface and can connect directly to MySQL, PostgreSQL, SQLite, MS Access (map), and Microsoft SQL Server. Data can also be imported from CSV and Tab-Separated files or spreadsheets (Microsoft Excel, OpenOffice.org Calc, Gumeric, Google Docs). The main statistical tests available are Independent and Paired t-tests, Wilcoxon signed ranks, Mann–Whitney U, Pearson's chi squared, Kruskal Wallis H, one-way ANOVA, Spearman's R, and Pearson's R. Nested tables can be produced with row and column percentages, totals, standard deviation, mean, median, lower and upper quartiles, and sum.
Installation packages are available for several Operating Systems such as Microsoft Windows, Ubuntu, Arch Linux, Linux Mint, and macOS (Leopard upwards).
SOFA Statistics is written in Python, and the widget toolkit used is WxPython. The statistical analyses are based on functions available through the SciPy stats module.
Statistics Features - Workflows
Users are guided through the selection of the appropriate basic statistical methods and assignment of the basic statistical on the table column of the data that should be analyzed.
The features available within SOFA for statistical analysis are limited compared to those found in Open Source R Statistics Software, which contains a large repository of statistics packages.
See also
Comparison of statistical packages
List of statistical packages
List of open-source software for mathematics
References
External links
SOFA Statistics Homepage
SOFA Statistics project page at Source Forge
SOFA Statistics project page at Launchpad
SOFA Statistics page at Show Me Do
Cross-platform free software
Cross-platform software
Free statistical software
Numerical software
Science software for Linux
Science software for macOS
Science software for Windows
Software that uses wxPython
Software using the GNU AGPL license |
https://en.wikipedia.org/wiki/Report%20on%20Probability%20A | Report on Probability A is a science fiction novel by Brian Aldiss. The novel was completed in 1962 and published in 1967 in New Worlds after first being rejected by publishers in the United Kingdom, France and the United States. Report on Probability A has been described as an antinovel. It is a seminal work in the British New Wave of experimental science fiction that began appearing in New Worlds following the appointment of Michael Moorcock as editor in 1964. A revised and extended version was published by Faber and Faber in 1968 and Doubleday in 1969. New Worlds described it as "perhaps [Aldiss'] most brilliant work to date".
According to Aldiss, the idea for the novel came from the Heisenberg uncertainty principle and its corollary that "observation alters what is observed". Aldiss, taking this as his starting point, "sat down to construct a fiction in which everything was observation within observation, and no ultimate reference point existed". The novel incorporates several related concepts in quantum physics, notably the many-worlds interpretation and different frames of reference. and its philosophical theme is indicated in the epigram, which quotes Goethe:
Do not, I beg you, look for anything behind phenomena. They are themselves their own lesson.
The novel has been compared to the work of Samuel Beckett, Jorge Luis Borges, Flann O'Brien and Alain Robbe-Grillet. Its reception has been polarised, with some railing against it and others hailing it as a cult classic. The novel is summarised by Paul Di Filippo who wrote that "an infinite regress of cosmic voyeurs seems to center around an enigmatic painting, as the French nouveau roman movement invades science fiction". Joanna Russ wrote that "Report is a false-narrative, a book full of narrative cues that raise expectations only to thwart them." She concluded that she admired "everything about the novel except its length. Matter organized in the lyrical, not the narrative, mode cannot be sustained for this long. Report would have made a brilliant novelette, but as a novel it is sheer self-indulgence."
Plot summary
The story is divided into three sections:
Part I: G who waits
Part II: S the watchful
Part III: The house and the watchers
In New Worlds these sections are divided into fourteen chapters, with Part II beginning half way through chapter six and ending in chapter eleven. In the Faber edition the three sections are divided into sixteen chapters, with six chapters each in Parts I and II and four chapters in Part III.
The bulk of the novel is the titular report, which describes in objective, repetitive and seemingly trivial detail the bizarre activity, taking place one overcast January day, apparently in England, around a suburban house in which a writer, Mr. Mary, lives with his wife. In the grounds of the house are various outbuildings which are occupied by three of the Marys' ex-employees: the gardener "G" is in a wooden hut, or summerhouse, some ten metres north-west of t |
https://en.wikipedia.org/wiki/Invariant%20set%20postulate | The invariant set postulate concerns the possible relationship between fractal geometry and quantum mechanics and in particular the hypothesis that the former can assist in resolving some of the challenges posed by the latter. It is underpinned by nonlinear dynamical systems theory and black hole thermodynamics.
Author
The proposer of the postulate is climate scientist and physicist Tim Palmer. Palmer completed a PhD at the University of Oxford under Dennis Sciama, the same supervisor that Stephen Hawking had and then worked with Hawking himself at the University of Cambridge on supergravity theory. He later switched to meteorology and has established a reputation pioneering ensemble forecasting. He now works at the European Centre for Medium-Range Weather Forecasts in Reading, England.
Overview
Palmer argues that the postulate may help to resolve some of the paradoxes of quantum mechanics that have been discussed since the Bohr–Einstein debates of the 1920s and 30s and which remain unresolved. The idea backs Einstein's view that quantum theory is incomplete, but also agrees with Bohr's contention that quantum systems are not independent of the observer.
The key idea involved is that there exists a state space for the Universe, and that the state of the entire Universe can be expressed as a point in this state space. This state space can then be divided into "real" and "unreal" sets (parts), where, for example, the states where the Nazis lost WW2 are in the "real" set, and the states where the Nazis won WW2 are in the "unreal" set of points. The partition of state space into these two sets is unchanging, making the sets invariant.
If the Universe is a complex system affected by chaos then its invariant set (a fixed state of rest) is likely to be a fractal. According to Palmer this could resolve problems posed by the Kochen–Specker theorem, which appears to indicate that physics may have to abandon the idea of any kind of objective reality, and the apparent paradox of action at a distance. In a paper submitted to the Proceedings of the Royal Society he indicates how the idea can account for quantum uncertainty and problems of "contextuality". For example, exploring the quantum problem of wave-particle duality, one of the central mysteries of quantum theory, the author claims that "in terms of the Invariant Set Postulate, the paradox is easily resolved, in principle at least". The paper and related talks given at the Perimeter Institute and University of Oxford also explores the role of gravity in quantum physics.
Critical reception
New Scientist quotes Bob Coeke of Oxford University as stating "What makes this really interesting is that it gets away from the usual debates over multiple universes and hidden variables and so on. It suggests there might be an underlying physical geometry that physics has just missed, which is radical and very positive". He added that "Palmer manages to explain some quantum phenomena, but he hasn't yet derived th |
https://en.wikipedia.org/wiki/Moscow%20State%20University%20of%20Economics%2C%20Statistics%2C%20and%20Informatics | Moscow State University of Economics, Statistics, and Informatics (MESI) was a university.
History
Moscow State University of Economics, Statistics and Informatics was founded in 1932 as the Moscow Institute of National Economic Accounting, which in 1948 was reorganized into the Moscow Economic and Statistical Institute (MESI). In 1996 the college received university status and was renamed the Moscow State University of Economics, Statistics and Informatics, while maintaining the same acronym. Plekhanov University acquired the Moscow State University of Economics, Statistics, and Informatics.
Alumni
Boris Nuraliev - one of the founder of 1C Company.
Yuri Ayzenshpis - Russian entertainment promoter.
Puntsagiin Jasrai - former Mongolian prime minister.
Artemy Troitsky - Russian journalist.
Dmitry Kharatyan - famous Russian actor, got his second diploma in finance and investment in 2004.
External links
Dmitrovsky representation MESI
Official website
Moscow State University of Economics, Statistics, and Informatics
Universities and colleges established in 1932
Universities in Moscow
Schools of mathematics
Universities of economics in Europe
Economics schools
1932 establishments in the Soviet Union |
https://en.wikipedia.org/wiki/Finite%20ring | In mathematics, more specifically abstract algebra, a finite ring is a ring that has a finite number of elements.
Every finite field is an example of a finite ring, and the additive part of every finite ring is an example of an abelian finite group, but the concept of finite rings in their own right has a more recent history.
Although rings have more structure than groups do, the theory of finite rings is simpler than that of finite groups. For instance, the classification of finite simple groups was one of the major breakthroughs of 20th century mathematics, its proof spanning thousands of journal pages. On the other hand, it has been known since 1907 that any finite simple ring is isomorphic to the ring – the n-by-n matrices over a finite field of order q (as a consequence of Wedderburn's theorems, described below).
The number of rings with m elements, for m a natural number, is listed under in the On-Line Encyclopedia of Integer Sequences.
Finite field
The theory of finite fields is perhaps the most important aspect of finite ring theory due to its intimate connections with algebraic geometry, Galois theory and number theory. An important, but fairly old aspect of the theory is the classification of finite fields:
The order or number of elements of a finite field equals pn, where p is a prime number called the characteristic of the field, and n is a positive integer.
For every prime number p and positive integer n, there exists a finite field with pn elements.
Any two finite fields with the same order are isomorphic.
Despite the classification, finite fields are still an active area of research, including recent results on the Kakeya conjecture and open problems regarding the size of smallest primitive roots (in number theory).
A finite field F may be used to build a vector space of n-dimensions over F. The matrix ring A of n × n matrices with elements from F is used in Galois geometry, with the projective linear group serving as the multiplicative group of A.
Wedderburn's theorems
Wedderburn's little theorem asserts that any finite division ring is necessarily commutative:
If every nonzero element r of a finite ring R has a multiplicative inverse, then R is commutative (and therefore a finite field).
Nathan Jacobson later discovered yet another condition which guarantees commutativity of a ring: if for every element r of R there exists an integer such that , then R is commutative. More general conditions that imply commutativity of a ring are also known.
Yet another theorem by Wedderburn has, as its consequence, a result demonstrating that the theory of finite simple rings is relatively straightforward in nature. More specifically, any finite simple ring is isomorphic to the ring , the n-by-n matrices over a finite field of order q. This follows from two theorems of Joseph Wedderburn established in 1905 and 1907 (one of which is Wedderburn's little theorem).
Enumeration
(Warning: the enumerations in this section includ |
https://en.wikipedia.org/wiki/Formal%20manifold | In geometry and topology, a formal manifold can mean one of a number of related concepts:
In the sense of Dennis Sullivan, a formal manifold is one whose real homotopy type is a formal consequence of its real cohomology ring; algebro-topologically this means in particular that all Massey products vanish.
A stronger notion is a geometrically formal manifold, a manifold on which all wedge products of harmonic forms are harmonic.
References
Manifolds |
https://en.wikipedia.org/wiki/Special%20functions | Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications.
The term is defined by consensus, and thus lacks a general formal definition, but the list of mathematical functions contains functions that are commonly accepted as special.
Tables of special functions
Many special functions appear as solutions of differential equations or integrals of elementary functions. Therefore, tables of integrals usually include descriptions of special functions, and tables of special functions include most important integrals; at least, the integral representation of special functions. Because symmetries of differential equations are essential to both physics and mathematics, the theory of special functions is closely related to the theory of Lie groups and Lie algebras, as well as certain topics in mathematical physics.
Symbolic computation engines usually recognize the majority of special functions.
Notations used for special functions
Functions with established international notations are the sine (), cosine (), exponential function (), and error function ( or ).
Some special functions have several notations:
The natural logarithm may be denoted , , , or depending on the context.
The tangent function may be denoted , , or ( is used in several European languages).
Arctangent may be denoted , , , or .
The Bessel functions may be denoted
Subscripts are often used to indicate arguments, typically integers. In a few cases, the semicolon (;) or even backslash (\) is used as a separator. In this case, the translation to algorithmic languages admits ambiguity and may lead to confusion.
Superscripts may indicate not only exponentiation, but modification of a function. Examples (particularly with trigonometric and hyperbolic functions) include:
usually means
is typically , but never
usually means , not ; this one typically causes the most confusion, since the meaning of this superscript is inconsistent with the others.
Evaluation of special functions
Most special functions are considered as a function of a complex variable. They are analytic; the singularities and cuts are described; the differential and integral representations are known and the expansion to the Taylor series or asymptotic series are available. In addition, sometimes there exist relations with other special functions; a complicated special function can be expressed in terms of simpler functions. Various representations can be used for the evaluation; the simplest way to evaluate a function is to expand it into a Taylor series. However, such representation may converge slowly or not at all. In algorithmic languages, rational approximations are typically used, although they may behave badly in the case of complex argument(s).
History of special functions
Classical theory
While trigonometry and exponential fu |
https://en.wikipedia.org/wiki/National%20Longitudinal%20Survey%20of%20Children%20and%20Youth | The National Longitudinal Survey of Children and Youth (NLSCY) is a project of Statistics Canada which engages in the long-term study of children.
The NLSCY is implemented by Statistics Canada and Human Resources and Social Development Canada, and charged with identifying and charting longer-term trends in Canadian youth. The survey tracks the progress and development of children from birth through early adulthood, and is meant to identify factors influencing each child's development. Specific areas of study include emotional, social and behavioural development and their implications for the child in adulthood. This is done by observing physical development and overall health, learning ability, behavioural tendencies, family and friend structure, the type of school and community in which the child is raised.
The study is conducted every two years, and was first conducted in 1994. The information gathered is used for policy decisions ranging from university financial aid and enrolment to distribution of educational, medical or family-related funds and education reform. The types of classes offered in schools, for example, may reflect the types of classes shown to best bolster a child's development.
The program targets children raised in each of Canada's 10 provinces, but excludes children living on Indian reserves, Crown lands, residents of institutions, children of full-time members of the Canadian Forces and those in certain remote regions.
The study has been inactive since 2009.
Methodology
The program selects children under age 11 to begin the process, and studying five subject areas: child development and behaviour, childhood, education, training and health. Each biennial report collects data for approximately one year. Evaluations are voluntary, and data is measured by testing. The first three tests are surveys about the child; the "child" component has the person most knowledgeable about the child as a respondent, the "adult" component uses the spouse of the person most knowledgeable about the child as a respondent and the "youth" component is given to the child. The fourth measurement is a cognitive test which incorporates mathematics and other educational subject matter. Finally, a series of tests given to the respondent child several times is used. Which test is given depends on the age of the child: a self-completed questionnaire for ages 12–16, a problem-solving exercise (16–17), a literacy assessment (18–19) and a numeracy assessment (20–21).
Critique
Critics assert that a volunteer-based means of evaluation is not representative; certain respondents will be predisposed to participate or abstain. Moreover, the exclusion of many children ensures that the study will not be wholly representative and may result in neglecting certain groups of children. Military families' children are not included in the survey. Neither are children in remote locations; tracking, administering and accessing these children may be too difficult and c |
https://en.wikipedia.org/wiki/Hudde%27s%20rules | In mathematics, Hudde's rules are two properties of polynomial roots described by Johann Hudde.
1. If r is a double root of the polynomial equation
and if are numbers in arithmetic progression, then r is also a root of
This definition is a form of the modern theorem that if r is a double root of ƒ(x) = 0, then r is a root of ƒ '(x) = 0.
2. If for x = a the polynomial
takes on a relative maximum or minimum value, then a is a root of the equation
This definition is a modification of Fermat's theorem in the form that if ƒ(a) is a relative maximum or minimum value of a polynomial ƒ(x), then ƒ '(a) = 0, where ƒ ' is the derivative of ƒ.
Hudde was working with Frans van Schooten on a Latin edition of La Géométrie of René Descartes. In the 1659 edition of the translation, Hudde contributed two letters: "Epistola prima de Redvctione Ǣqvationvm" (pages 406 to 506), and "Epistola secvnda de Maximus et Minimus" (pages 507 to 16). These letters may be read by the Internet Archive link below.
References
Carl B. Boyer (1991) A History of Mathematics, 2nd edition, page 373, John Wiley & Sons.
Robert Raymond Buss (1979) Newton's use of Hudde's Rule in his Development of the Calculus, Ph.D. Thesis Saint Louis University, ProQuest #302919262
René Descartes (1659) La Géométria, 2nd edition via Internet Archive.
Kirsti Pedersen (1980) §5 "Descartes’s method of determining the normal, and Hudde’s rule", chapter 2: "Techniques of the calculus, 1630-1660", pages 16—19 in From the Calculus to Set Theory edited by Ivor Grattan-Guinness Duckworth Overlook
Rules
Theorems in algebra
Polynomials
Calculus |
https://en.wikipedia.org/wiki/2009%E2%80%9310%20Club%20Nacional%20de%20Football%20season |
Squad
Updated 2 May 2010
Transfers
In
Out
Statistics
Appearances and goals
Last updated on 18 May 2010.
|}
Note
* = Players who left the club mid-season
Top scorers
Includes all competitive matches
Note
* = Players who left the club mid-season
Disciplinary record
Note
* = Players who left the club mid-season
Captains
Penalties Awarded
International players
The following is a list of all squad members who have played for their national sides during the 2009–10 season. Players in bold were in the starting XI for their national side.
26 September 2009
29 January 2009
2 October 2009
7 October 2009
15 November 2009
18 November 2009
Starting 11
Overall
{|class="wikitable"
|-
|Games played || 41 (33 Primera División Uruguaya, 8 Copa Libertadores)
|-
|Games won || 25 (22 Primera División Uruguaya, 3 Copa Libertadores)
|-
|Games drawn || 7 (4 Primera División Uruguaya, 3 Copa Libertadores)
|-
|Games lost || 9 (7 Primera División Uruguaya, 2 Copa Libertadores)
|-
|Goals scored || 76
|-
|Goals conceded || 38
|-
|Goal difference || +38
|-
|Yellow cards || 130
|-
|Red cards || 13
|-
|Worst discipline || Mario Regueiro (12 , 2 )
|-
|Best result || 6-0 (A) v Atenas – Primera División Uruguaya 2010.03.13
|-
|Worst result || 0-4 (A) v Montevideo Wanderers – Primera División Uruguaya 2009.10.17
|-
|Most appearances || Rodrigo Muñoz (38 appearances)
|-
|Top scorer || Sergio Blanco (13 goals)
|-
Club
Coaching staff
Friendlies
Copa Amistad
Copa Bimbo
Semi-finals
Final
Primera División Uruguaya
Apertura's table
The apertura's winner qualifies for the semifinal of the Primera División.
Matches
Clausura's table
The Clausura winner qualifies for the semifinal of the Primera División.
Matches
Aggregate table
The Aggregate winner qualify for the final of Primera División and 2011 Copa Libertadores group stage.
The Aggregate runners-up (if not qualified for the final) qualify for the 2011 Copa Libertadores first stage.
The Aggregate third place (if not qualified for the final) qualify for the 2010 Copa Sudamericana first stage.
The Aggregate fourth place (if not qualified for the final) qualify for the 2010 Copa Sudamericana first stage.
Results by round
Relegation
The three clubs with lowest points are relegated.
Semi-final
Final
First leg
Second leg
Peñarol won 2–1 on aggregate
Copa Libertadores
Group stage
Round of 16
First leg
Second leg
Cruzeiro won 6–1 on aggregate.
Records
Doubles achieved
Comeback
Nacional have conceded the first goal in a match 10 times this season in the Primera División and the Copa Libertadores, recorded 5 wins, 1 draw and 5 loss.
{| class="wikitable"
|-
!Opponent
!H/A
!Result
!Scoreline
|-
|Liverpool
|align=center|H
|align=center|2–1
|Alfaro 66, Matías Rodríguez 73', Lembo 86'
|-
|Racing
|align=center|H
|align=center|3–2
|Cauteruccio 8', Quiñones 14''', Balsas 45+3' (pen.), 49, Ferro 79'
|-
|Central Español
|align=center|H
|align=center|5–1
|Espiga 49, Aranda 56', Gar |
https://en.wikipedia.org/wiki/Lasse%20Linjala | Lasse Linjala (born 15 August 1987) is a Finnish former footballer who played as a striker.
External links
#17 Lasse Linjala
Veikkausliiga Player statistics
1987 births
Living people
Finnish men's footballers
Kokkolan Palloveikot players
JJK Jyväskylä players
FC KooTeePee players
FC Haka players
Mikkelin Palloilijat players
Vaasan Palloseura players
Vasa IFK players
Veikkausliiga players
Kakkonen players
Ykkönen players
Men's association football forwards
People from Pieksämäki
Sportspeople from South Savo |
https://en.wikipedia.org/wiki/German%20casualties%20in%20World%20War%20II | Statistics for German World War II military casualties are divergent. The wartime military casualty figures compiled by German High Command, up until January 31, 1945, are often cited by military historians when covering individual campaigns in the war. A study by German historian Rüdiger Overmans found that the German military casualties were 5.3 million, including 900,000 men conscripted from outside of Germany's 1937 borders, in Austria and in east-central Europe, higher than those originally reported by the German high command. The German government reported that its records list 4.3 million dead and missing military personnel.
Civilian deaths during the war include air raid deaths, estimates of German civilians killed only by Allied strategic bombing have ranged from around 350,000 to 500,000.
Civilian deaths, due to the flight and expulsion of Germans, Soviet war crimes and the forced labor of Germans in the Soviet Union are disputed and range from 500,000 to over 2.0 million. According to the German government Suchdienste (Search Service) there were 300,000 German victims (including Jews) of Nazi racial, political and religious persecution. This statistic does not include 200,000 German people with disabilities who were murdered in the Action T4 and Action 14f13 euthanasia programs.
German sources for military casualties
Records of German military search service
In the post-war era the military search service Deutsche Dienststelle (WASt) has been responsible for providing information for the families of those military personnel who were killed or went missing in the war. They maintain the files of over 18 million men who served in the war. By the end of 1954, they had identified approximately 4 million military dead and missing (2,730,000 dead and 1,240,629 missing). (Since German reunification, the records in the former GDR (East Germany) have become available to the WASt). The German Red Cross reported in 2005 that the records of the military search service WAS list total Wehrmacht losses at 4.3 million men (3.1 million dead and 1.2 million missing) in World War II. Their figures include Austria and conscripted ethnic Germans from Eastern Europe.
The German historian Rüdiger Overmans used the files of WASt) to conduct his research project on German military casualties.
Wartime statistics compiled by German High Command (OKW)
The German military system for reporting casualties was based on a numerical reporting of casualties by individual units and a separate listing of the names of individual casualties. The system was not uniform because various military branches such as the Army, Air Force, Navy, Waffen SS and the military hospitals each had different systems of reporting. In early 1945 the German High Command (OKW) prepared a summary of total losses up to January 31, 1945. The German historian Rüdiger Overmans believes, based on his research, that these figures are incomplete and unreliable. According to Overmans, the casualty r |
https://en.wikipedia.org/wiki/Guido%20Zappa | Guido Zappa (7 December 1915 – 17 March 2015) was an Italian mathematician and a noted group theorist: his other main research interests were geometry and also the history of mathematics. Zappa was particularly known for some examples of algebraic curves that strongly influenced the ideas of Francesco Severi.
Life and work
Honors
He was elected ordinary non-resident member of the Accademia Pontaniana on June 16, 1949.
On June 3, 1951, he was elected the corresponding member to the class of mathematical sciences of the Società Nazionale di Scienze Lettere e Arti in Napoli: subsequently, he became an ordinary member (2 June 1951) and ordinary non-resident member (15 December 1953).
On 14 October 1960 he was elected corresponding member of the Accademia Nazionale dei Lincei: he became national member of the same academy on March 21, 1977.
Selected publications
. "Fundamentals of group theory. First volume" (English translation of the title) is the first part of monograph in group theory dealing extensively with many of its aspects.
. "Fundamentals of group theory. Second volume" (English translation of the title) is the second part of monograph in group theory dealing extensively with many of its aspects.
. This work describes the research activity at the Sapienza University of Rome and at the (at that time newly created) "Istituto Nazionale di Alta Matematica Francesco Severi" from the end of the 1930s to the early 1940s.
See also
Algebraic geometry
Group theory
Italian school of algebraic geometry
Francesco Severi
Zappa-Szép product
Notes
References
Biographical and general references
. The "Yearbook" of the renowned Italian scientific institution, including an historical sketch of its history, the list of all past and present members as well as a wealth of informations about its academic and scientific activities.
. The "Yearbook 2015" of the Accademia Pontaniana, published by the academy itself and describing its past and present hierarchies and its activities. It also gives some notes on its history, the full list of its members and other useful information.
.
. The biographical and bibliographical entry (updated up to 1976) on Guido Zappa, published under the auspices of the Accademia dei Lincei in a book collecting many profiles of its members living members up to 1976.
. The "Yearbook 2014" of the Società Nazionale di Scienze Lettere e Arti in Napoli, published by the society itself and describing its past and present hierarchies, and its activities. It also reports some notes on its history, the full list of its members and other useful information.
Scientific references
. Guido Zappa and combinatorial geometry (English translation of the title), a paper from the Atti del Convegno Internazionale di Teoria dei Gruppi e Geometria Combinatoria - Firenze, Ottobre 23–26 1986, in onore di Guido Zappa (Proceedings of the international conference on group theory and combinatorial geometry held in Florence on October 23–26, 19 |
https://en.wikipedia.org/wiki/A.%20M.%20Mathai | Arakaparampil Mathai "Arak" Mathai (born 28 April 1935) is an Indian mathematician who has worked in Statistics, Applied Analysis, Applications of special functions and Astrophysics. Mathai established the Centre for Mathematical Sciences, Palai, Kerala, India.
He has published more than 25 books and more than 300 research publications. In 1998 he received the Founder Recognition Award from the Statistical Society of Canada. He is a Fellow of the National Academy of Sciences, India and a Fellow of the Institute of Mathematical Statistics.
Early years
Mathai was born in Arakulam near Palai in the Idukki district of Kerala, India as the eldest son of Aley and Arakaparampil Mathai. After completing his high school education in 1953 from St. Thomas High School, Palai, with record marks he joined St. Thomas College, Palai, and obtained his B Sc. degree in mathematics in 1957. In 1959 he completed his master's degree in statistics from University of Kerala, Thiruvananthapuram, Kerala, India, with first class, first rank and gold medal. Then he joined St. Thomas College, Palai, University of Kerala, as a lecturer in Statistics and served there until 1961. He obtained Canadian Commonwealth scholarship in 1961 and went to University of Toronto, Canada, for completing his MA degree in mathematics in 1962. He was awarded his PhD from the University of Toronto in 1964, then joined McGill University as an assistant professor until 1968. From 1968 to 1978 he was an associate professor there. He became a full professor at McGill in 1979 and served the department of mathematics and statistics until 2000. From 2000 onwards he was an emeritus professor of McGill University.
Selected publications
Random p-content of a p-parallelotope in Euclidean n-space, Advances in Applied Probability, 31(2), 343–354 (1999).
An Introduction to Geometrical Probability: Distributional Aspects with Applications, Gordon and Breach, Newark, (1999).
Jacobians of Matrix Transformations and Functions of Matrix Argument, World Scientific Publishing, New York, (1997).
Appell's and Humbert's functions of matrix arguments, Linear Algebra and Its Applications, 183, 202–221, (1993).
On non-central generalized Laplacianness of quadratic forms in normal variables, Journal of Multivariate Analysis, 45, 239–246, (1993).
(With S.B. Provost), Quadratic Forms in random Variables: Theory and Applications, Marcel Dekker, New York, (1992).
On a system of differential equations connected with the gravitational instability in a multi-component medium in Newtonian cosmology, Studies in Applied Mathematics, 80, 75–93, (1989).
(With H.J. Haubold), Modern Problems in Nuclear and Neutrino Astrophysics, Akademie-Verlag, Berlin.
On a conjecture in geometric probability regarding asymptotic normality of a random simplex, Annals of Probability, 10, 247–251, (1982).
(With R.S. Katiyar), Exact percentage points for testing independence, Biometrika, 66, 353–356, (1979).
(With P.N. Rathie), Recent contributions t |
https://en.wikipedia.org/wiki/Lucio%20Lombardo-Radice | Lucio Lombardo-Radice (Catania, 10 July 1916; Brussels, 21 November 1982) was an Italian mathematician. A student of Gaetano Scorza, Lombardo-Radice contributed to finite geometry and geometric combinatorics together with Guido Zappa and Beniamino Segre, and wrote important works concerning the Non-Desarguesian plane. He was also a leading member of the Italian Communist Party and a member of its central committee.
Lombardo-Radice's parents were Giuseppe Lombardo Radice and Gemma Harasim. His children included the writer and actor Giovanni Lombardo Radice.
The Istituto Tecnico Statale Commerciale "Lucio Lombardo Radice" per Programmatori, a school in Rome, Italy, founded in 1982 as the XXV Istituto Tecnico Commerciale per Programmatori, was in 1992 renamed after Lombardo-Radice. It is now named Istituto di Istruzione Superiore Lombardo Radice
References
External links
1916 births
1982 deaths
20th-century Italian mathematicians
Italian communists
Combinatorialists
People from Catania
Mathematicians from Sicily |
https://en.wikipedia.org/wiki/Right%20conoid | In geometry, a right conoid is a ruled surface generated by a family of straight lines that all intersect perpendicularly to a fixed straight line, called the axis of the right conoid.
Using a Cartesian coordinate system in three-dimensional space, if we take the to be the axis of a right conoid, then the right conoid can be represented by the parametric equations:
where is some function for representing the height of the moving line.
Examples
A typical example of right conoids is given by the parametric equations
The image on the right shows how the coplanar lines generate the right conoid.
Other right conoids include:
Helicoid:
Whitney umbrella:
Wallis's conical edge:
Plücker's conoid:
hyperbolic paraboloid: (with x-axis and y-axis as its axes).
See also
Conoid
Helicoid
Whitney umbrella
Ruled surface
External links
Right Conoid from MathWorld.
Plücker's conoid from MathWorld
Surfaces
Geometric shapes |
https://en.wikipedia.org/wiki/Wallis%27s%20conical%20edge | In geometry, Wallis's conical edge is a ruled surface given by the parametric equations
where , and are constants.
Wallis's conical edge is also a kind of right conoid. It is named after the English mathematician John Wallis, who was one of the first to use Cartesian methods to study conic sections.
See also
Ruled surface
Right conoid
References
A. Gray, E. Abbena, S. Salamon, Modern differential geometry of curves and surfaces with Mathematica, 3rd ed. Boca Raton, Florida:CRC Press, 2006. ()
External links
Wallis's Conical Edge from MathWorld.
Surfaces
Geometric shapes |
https://en.wikipedia.org/wiki/Complex%20lamellar%20vector%20field | In vector calculus, a complex lamellar vector field is a vector field which is orthogonal to a family of surfaces. In the broader context of differential geometry, complex lamellar vector fields are more often called hypersurface-orthogonal vector fields. They can be characterized in a number of different ways, many of which involve the curl. A lamellar vector field is a special case given by vector fields with zero curl.
The adjective "lamellar" derives from the noun "lamella", which means a thin layer. The lamellae to which "lamellar vector field" refers are the surfaces of constant potential, or in the complex case, the surfaces orthogonal to the vector field. This language is particularly popular with authors in rational mechanics.
Complex lamellar vector fields
In vector calculus, a complex lamellar vector field is a vector field in three dimensions which is orthogonal to its own curl. That is,
The term lamellar vector field is sometimes used as a synonym for the special case of an irrotational vector field, meaning that
Complex lamellar vector fields are precisely those that are normal to a family of surfaces. An irrotational vector field is locally the gradient of a function, and is therefore orthogonal to the family of level surfaces (the equipotential surfaces). Any vector field can be decomposed as the sum of an irrotational vector field and a complex lamellar field.
Hypersurface-orthogonal vector fields
In greater generality, a vector field on a pseudo-Riemannian manifold is said to be hypersurface-orthogonal if through an arbitrary point there is a smoothly embedded hypersurface which, at all of its points, is orthogonal to the vector field. By the Frobenius theorem this is equivalent to requiring that the Lie bracket of any smooth vector fields orthogonal to is still orthogonal to .
The condition of hypersurface-orthogonality can be rephrased in terms of the differential 1-form which is dual to . The previously given Lie bracket condition can be reworked to require that the exterior derivative , when evaluated on any two tangent vectors which are orthogonal to , is zero. This may also be phrased as the requirement that there is a smooth 1-form whose wedge product with equals .
Alternatively, this may be written as the condition that the differential 3-form is zero. This can also be phrased, in terms of the Levi-Civita connection defined by the metric, as requiring that the totally anti-symmetric part of the 3-tensor field is zero. Using a different formulation of the Frobenius theorem, it is also equivalent to require that is locally expressible as for some functions and .
In the special case of vector fields on three-dimensional Euclidean space, the hypersurface-orthogonal condition is equivalent to the complex lamellar condition, as seen by rewriting in terms of the Hodge star operator as , with being the 1-form dual to the curl vector field.
Hypersurface-orthogonal vector fields are particularly important in ge |
https://en.wikipedia.org/wiki/Beltrami%20vector%20field | In vector calculus, a Beltrami vector field, named after Eugenio Beltrami, is a vector field in three dimensions that is parallel to its own curl. That is, F is a Beltrami vector field provided that
Thus and are parallel vectors in other words, .
If is solenoidal - that is, if such as for an incompressible fluid or a magnetic field, the identity becomes and this leads to
and if we further assume that is a constant, we arrive at the simple form
Beltrami vector fields with nonzero curl correspond to Euclidean contact forms in three dimensions.
The vector field
is a multiple of the standard contact structure −z i + j, and furnishes an example of a Beltrami vector field.
Beltrami fields and fluid mechanics
Beltrami fields with a constant proportionality factor are a distinct category of vector fields that act as eigenfunctions of the curl operator. In essence, they are functions that map points in a three-dimensional space, either in (Euclidean space) or on a flat torus , to other points in the same space. Mathematically, this can be represented as:
(for Euclidean space) or (for the flat torus).
These vector fields are unique due to the special relationship between the curl of the vector field and the field itself. This relationship can be expressed using the following equation:
In this equation, is a non-zero constant, which indicates that the curl of the vector field is proportional to the field itself.
Beltrami fields are relevant in fluid dynamics, as they offer a classical family of stationary solutions to the Euler equation in three dimensions. The Euler equations describe the motion of an ideal, incompressible fluid and can be written as a system of two equations:
For stationary flows, where the velocity field does not change with time, i.e. , we can introduce the Bernoulli function, , and the vorticity, . These new variables simplify the Euler equations into the following system:
The simplification is possible due to a vector identity, which relates the convective term to the gradient of the kinetic energy and the cross product of the velocity field and its curl:
When the Bernoulli function is constant, Beltrami fields become valid solutions to the simplified Euler equations. Note that we do not need the proportionality factor to be constant for the proof to work.
Beltrami fields and complexity in fluid mechanics
Beltrami fields have a close connection to Lagrangian turbulence, as shown by V.I. Arnold's work on stationary Euler flows.
Arnold's "conjecture"
Arnold's quote from his aforementioned work highlights the probable complicated topology of the streamlines in Beltrami fields, drawing parallels with celestial mechanics:
Il est probable que les écoulements tels que rot , , ont des lignes de courant à la topologie compliquée. De telles complications interviennent en mécanique céleste. La topologie des lignes de courant des écoulements stationnaires des fluides visqueux peut être semblable à celle de mé |
https://en.wikipedia.org/wiki/Bryan%20Barley | Bryan Barley was a former England international rugby union centre.
He was educated at Normanton Grammar School and Leeds University where he studied economics and mathematics.
He joined Wakefield RFCin 1978 converting thirteen of Wakefield's seventeen tries (a club record) on his debut. He continued to play for the club until 1993, playing in over 300 games and he was the first Wakefield player to be selected for England direct from the club since Jack Ellis in 1939. On leaving Wakefield he joined Sandal RFC.
David Ingall in the club history book describes how "he brought both power and subtlety to the outside centre position, with an eye for the smallest gap in the defence or the pace and strength to carry him round his opponent. His tackling was devastating, and he kicked accurately and purposefully from hand"
He played for Yorkshire and England at both 16 and 19 groups and toured Australia in his second year with the England Under 19 group in 1979.
He first played for Yorkshire in 1979, touring with them to France in 1980. He played for England under 23's in 1980 and 1982 and England Students in 1983.
he won the first of his seven England caps against Ireland in the 1984 Five Nations championships. He toured South Africa (1984), New Zealand (1985) and Australia and Fiji (1988) with England.
He also played twice for the Barbarians and represented the North of England against Australia (1983), USSR (1989) and South Africa (1992).
References
Wakefield Rugby Football Club—1901-2001 A Centenary History. Written and compiled by David Ingall in 2001.
Wakefield RFC programmes - various dates.
Wakefield Express newspaper - various dates.
England v Wales official match programme 17 March 1985.
Yorkshire Post 12 June 2004
Sporting Heroes.net
1960 births
Living people
Barbarian F.C. players
England international rugby union players
English rugby union players
Rugby union players from Wakefield
Wakefield RFC players
Yorkshire County RFU players
Rugby union centres |
https://en.wikipedia.org/wiki/Total%20bases | In baseball statistics, total bases is the number of bases a player gains with hits. It is a weighted sum with values of 1 for a single, 2 for a double, 3 for a triple and 4 for a home run. For example, three singles is three total bases, while a double and a home run is six total bases.
Only bases attained from hits count toward this total. Reaching base by other means (such as a base on balls) or advancing further after the hit (such as when a subsequent batter gets a hit) does not increase the player's total bases. In box scores and other statistical summaries, total bases is often denoted by the abbreviation TB.
The total bases divided by the number of at bats is the player's slugging percentage.
Records
Hank Aaron's 6,856 career total bases make him the all-time MLB record holder. Having spent the majority of his career playing in the National League, he also holds that league's record with 6,591 total bases. Aaron hit for 300 or more total bases in a record 15 different seasons. Aaron regarded this record as his proudest accomplishment, over his career home run record, because he felt it better reflected his performance as a team player. Ty Cobb's 5,854 total bases constitute the American League record. Albert Pujols is the active leader and 2nd all-time with 6,144 total bases, as of August 22 of the 2022 MLB season.
The single season MLB and American League records are held by Babe Ruth, who hit for 457 TB in the 1921 season. The following season saw Rogers Hornsby set the National League record when he hit for 450 total bases.
Shawn Green holds the single game total bases record of 19 TB. Green hit four home runs, a single and a double for the Los Angeles Dodgers against the Milwaukee Brewers on May 23, 2002. The equivalent American League record is held by Josh Hamilton, who hit four home runs and a double (18 TB) for the Texas Rangers in a May 8, 2012, game versus the Baltimore Orioles.
Dustin Pedroia collected the most total bases in a single interleague game during the regular season, with 15. Pedroia hit three home runs, a single and a double for the Boston Red Sox on June 24, 2010, in a game against the Colorado Rockies at Coors Field.
The 2003 Boston Red Sox and 2019 Minnesota Twins jointly hold the American League single season team record with 2,832 total bases; the National League record is held by the 2001 Colorado Rockies (2,748 TB). The Red Sox also have the record for most total bases by a team in one game: they hit for 60 TB in a 29–4 victory over the St. Louis Browns on June 8, 1950.
Among major league pitchers, Phil Niekro gave up the most total bases in a career (7,473), while Robin Roberts (555 TB allowed in 1956) holds the single season record. The record number of total bases allowed in a single game by one pitcher is 42, by Allan Travers of the Detroit Tigers.
Postseason
Two players have hit for 14 total bases in a postseason game. Albert Pujols is the only player to accomplish this in the World Series, doi |
https://en.wikipedia.org/wiki/J.%20Halcombe%20Laning | J. Halcombe "Hal" Laning Jr. (February 14, 1920, in Kansas City, Missouri – May 29, 2012) was a Massachusetts Institute of Technology computer pioneer who in 1952 invented an algebraic compiler called George (also known as the Laning and Zierler system after the authors of the published paper) that ran on the MIT Whirlwind, the first real-time computer. Laning designed George to be an easier-to-use alternative to assembly language for entering mathematical equations into a computer. He later became a key contributor to the 1960s race to the Moon, with pioneering work on space-based guidance systems for the Apollo Moon missions. From 1955 to 1980, he was deputy associate director of the MIT Instrumentation Laboratory.
In 1956 he published the book Random Processes in Automatic Control (McGraw-Hill Series on Control System Engineering), with Richard Battin as a coauthor.
In collaboration with Phil Hankins and Charlie Werner of MIT, he initiated work on MAC (MIT
Algebraic Compiler), an algebraic programming language for the IBM 650, which was completed by early spring of 1958.
Career
Laning received his PhD from MIT in 1947 with a dissertation titled "Mathematical Theory of Lubrication-Type Flow". His undergraduate degree in Chemical Engineering (1940) was also from MIT.
He was elected to the National Academy of Engineering in 1983 for his work in aerospace engineering, particularly his "unique pioneering achievements in missile guidance and computer science—the Q-guidance system for Thor and Polaris [missiles] and George". He was also an honorary member of the American Mathematical Society.
Laning features prominently in the third episode of the Science Channel's documentary miniseries titled Moon Machines which aired in June 2008.
Apollo Program
He later worked in the MIT Draper Lab, with Richard H. Battin, to develop a scheme for doing onboard navigation on the Apollo program's command/service module guidance system. He designed the Executive and Waitlist operating system for the LGC (Lunar Module Guidance Computer) in the mid 1960s; he built it up from scratch with no examples to guide him, and the design is still valid. The allocation of functions among a sensible number of asynchronous processes, under control of a rate and priority-driven preemptive executive, still represents the state of the art in real-time GN&C computers for spacecraft. His design saved the Apollo 11 landing mission when the rendezvous radar interface program began using more register core sets and "Vector Accumulator" areas than were physically available in memory, causing the infamous 1201 and 1202 errors. Had it not been for Laning's design the landing would have been aborted for lack of a stable guidance computer.
References
External links
A portrait of J. Halcombe Laning, taken in 1997
NASA people
MIT School of Engineering alumni
1920 births
Members of the United States National Academy of Engineering
2012 deaths |
https://en.wikipedia.org/wiki/Annals%20of%20Mathematical%20Statistics | The Annals of Mathematical Statistics was a peer-reviewed statistics journal published by the Institute of Mathematical Statistics from 1930 to 1972. It was superseded by the Annals of Statistics and the Annals of Probability. In 1938, Samuel Wilks became editor-in-chief of the Annals and recruited a remarkable editorial staff: Fisher, Neyman, Cramér, Hotelling, Egon Pearson, Georges Darmois, Allen T. Craig, Deming, von Mises, H. L. Rietz, and Shewhart.
References
External links
Annals of Mathematical Statistics at Project Euclid
Statistics journals
Probability journals |
https://en.wikipedia.org/wiki/Statistical%20dispersion | In statistics, dispersion (also called variability, scatter, or spread) is the extent to which a distribution is stretched or squeezed. Common examples of measures of statistical dispersion are the variance, standard deviation, and interquartile range. For instance, when the variance of data in a set is large, the data is widely scattered. On the other hand, when the variance is small, the data in the set is clustered.
Dispersion is contrasted with location or central tendency, and together they are the most used properties of distributions.
Measures of statistical dispersion
A measure of statistical dispersion is a nonnegative real number that is zero if all the data are the same and increases as the data become more diverse.
Most measures of dispersion have the same units as the quantity being measured. In other words, if the measurements are in metres or seconds, so is the measure of dispersion. Examples of dispersion measures include:
Standard deviation
Interquartile range (IQR)
Range
Mean absolute difference (also known as Gini mean absolute difference)
Median absolute deviation (MAD)
Average absolute deviation (or simply called average deviation)
Distance standard deviation
These are frequently used (together with scale factors) as estimators of scale parameters, in which capacity they are called estimates of scale. Robust measures of scale are those unaffected by a small number of outliers, and include the IQR and MAD.
All the above measures of statistical dispersion have the useful property that they are location-invariant and linear in scale. This means that if a random variable has a dispersion of then a linear transformation for real and should have dispersion , where is the absolute value of , that is, ignores a preceding negative sign .
Other measures of dispersion are dimensionless. In other words, they have no units even if the variable itself has units. These include:
Coefficient of variation
Quartile coefficient of dispersion
Relative mean difference, equal to twice the Gini coefficient
Entropy: While the entropy of a discrete variable is location-invariant and scale-independent, and therefore not a measure of dispersion in the above sense, the entropy of a continuous variable is location invariant and additive in scale: If is the entropy of a continuous variable and , then .
There are other measures of dispersion:
Variance (the square of the standard deviation) – location-invariant but not linear in scale.
Variance-to-mean ratio – mostly used for count data when the term coefficient of dispersion is used and when this ratio is dimensionless, as count data are themselves dimensionless, not otherwise.
Some measures of dispersion have specialized purposes. The Allan variance can be used for applications where the noise disrupts convergence. The Hadamard variance can be used to counteract linear frequency drift sensitivity.
For categorical variables, it is less common to measure dispersion by a single |
https://en.wikipedia.org/wiki/Basic%20Element | Basic element is a term in algebra.
Basic Element may refer to:
Basic Element (company)
Basic Element (music group)
See also
Element (disambiguation) |
https://en.wikipedia.org/wiki/Postselection | In probability theory, to postselect is to condition a probability space upon the occurrence of a given event. In symbols, once we postselect for an event , the probability of some other event changes from to the conditional probability .
For a discrete probability space, , and thus we require that be strictly positive in order for the postselection to be well-defined.
See also PostBQP, a complexity class defined with postselection. Using postselection it seems quantum Turing machines are much more powerful: Scott Aaronson proved PostBQP is equal to PP.
Some quantum experiments use post-selection after the experiment as a replacement for communication during the experiment, by post-selecting the communicated value into a constant.
References
Conditional probability
Theoretical computer science
Quantum complexity theory |
https://en.wikipedia.org/wiki/Vanuatu%20National%20Statistics%20Office | The Vanuatu National Statistics Office (VNSO) is Vanuatu's official statistical agency. The agency compiles and publishes statistics about the Pacific island nation.
See also
2009 Vanuatu Census
2016 Vanuatu Mini-Census (in response to Cyclone Pam)
References
External links
VNSO home page
Government of Vanuatu |
https://en.wikipedia.org/wiki/North%20American%20Transportation%20Statistics%20Interchange | The North American Transportation Statistics Interchange, established in 1991, is a trilateral forum of government officials from transportation and statistical agencies of the United States, Canada, and Mexico. The purpose of the NATS Interchange is to share information on how to collect, analyze, and publish transportation data; to improve comparability among data programs of the three nations; and to provide North American statistics on transportation. To accomplish this, officials meet on an annual basis in the context of Interchange meetings being held alternately in Canada, Mexico, and the United States. Participating agencies include the U.S. Census Bureau, the U.S. Department of Transportation, the U.S. Army Corps of Engineers, Statistics Canada, Transport Canada, the Mexican Ministry of Communications and Transport (SCT), the Mexican Institute of Transportation (IMT) (http://www.imt.mx), and the National Institute of Statistics and Geography (INEGI) of Mexico.
The North American Transportation Statistics (NATS) On-line Database
The NATS On-line Database is a product of the NATS Interchange. Publications of the Interchange are available at: http://nats.sct.gob.mx/en/
See also
Bureau of Transportation Statistics
North American Industry Classification System
North American Product Classification System
References
National statistical services
Transport in North America
Trilateral relations of Canada, Mexico, and the United States |
https://en.wikipedia.org/wiki/Quaternionic%20analysis | In mathematics, quaternionic analysis is the study of functions with quaternions as the domain and/or range. Such functions can be called functions of a quaternion variable just as functions of a real variable or a complex variable are called.
As with complex and real analysis, it is possible to study the concepts of analyticity, holomorphy, harmonicity and conformality in the context of quaternions. Unlike the complex numbers and like the reals, the four notions do not coincide.
Properties
The projections of a quaternion onto its scalar part or onto its vector part, as well as the modulus and versor functions, are examples that are basic to understanding quaternion structure.
An important example of a function of a quaternion variable is
which rotates the vector part of q by twice the angle represented by u.
The quaternion multiplicative inverse is another fundamental function, but as with other number systems, and related problems are generally excluded due to the nature of dividing by zero.
Affine transformations of quaternions have the form
Linear fractional transformations of quaternions can be represented by elements of the matrix ring operating on the projective line over . For instance, the mappings where and are fixed versors serve to produce the motions of elliptic space.
Quaternion variable theory differs in some respects from complex variable theory. For example: The complex conjugate mapping of the complex plane is a central tool but requires the introduction of a non-arithmetic, non-analytic operation. Indeed, conjugation changes the orientation of plane figures, something that arithmetic functions do not change.
In contrast to the complex conjugate, the quaternion conjugation can be expressed arithmetically, as
This equation can be proven, starting with the basis {1, i, j, k}:
.
Consequently, since is linear,
The success of complex analysis in providing a rich family of holomorphic functions for scientific work has engaged some workers in efforts to extend the planar theory, based on complex numbers, to a 4-space study with functions of a quaternion variable. These efforts were summarized in .
Though appears as a union of complex planes, the following proposition shows that extending complex functions requires special care:
Let be a function of a complex variable, . Suppose also that is an even function of and that is an odd function of . Then is an extension of to a quaternion variable where and .
Then, let represent the conjugate of , so that . The extension to will be complete when it is shown that . Indeed, by hypothesis
one obtains
Homographies
In the following, colons and square brackets are used to denote homogeneous vectors.
The rotation about axis r is a classical application of quaternions to space mapping.
In terms of a homography, the rotation is expressed
where is a versor. If p * = −p, then the translation is expressed by
Rotation and translation xr along the axis of rotation is |
https://en.wikipedia.org/wiki/Wada%20%28house%29 | Wada is a type of dwelling found in Maharashtra, western India. Wada is a Marathi word for denoting a large mansion. The term, in all probability, is derived from the Sanskrit word Vata, meaning a plot or a piece of land meant for a house. Over time it came to denote the house built on that plot. Wadi, an extended meaning of wada, denotes a cluster of huts. Typically, wada refers to a house with courtyards found in Maharashtra and surrounding regions in India.
Origin
The courtyard houses developed in medieval India and were prevalent all over the sub-continent, varying regionally, under different names. They were called Wada in Maharastra (western India), Haveli in Rajasthan (North India), Deori in Hyderabad (southern Indian plateau), Nalukettu in Kerala (southernmost coastal India), and Rajbadi in Bengal (east India).
In Maharashtra, the wada house form received patronage from the Maratha rulers in the 17th century and later from the Peshwas and their successors. The latter were responsible for the sitewide spread and its expansion to the adjoining regions of Malwa, parts of Gujarat and Karnataka.
Types of Wada
Size based
Garhis were fortified wadas with bastions and ramparts in the village's focus. The village or town grew around the Garhi with peripheral clusters of wada houses all around. They are introverted structures built in brick and stone with a series of courtyards inside. Examples are Shaniwar Wada in Pune, Hilkar Wada in Chandwad, and Vafgaon.
Rajwada or Palace wada houses were also the central focus of the town but with the absence of bastions and ramparts. Examples are the Bhor Rajwada, with a single courtyard, and the Satara Rajwada, with two courtyards.
Smaller wada houses formed clusters around the nucleus of the town, which is either a Garhi or a Rajwada. While Garhi and Rajwada were isolated buildings built on larger areas, these houses were narrow buildings built along the streets with their narrow side facing the street and sharing walls with adjacent buildings.
There was another class of buildings, usually, poor rural dwellings called the wadi, with an extroverted structure with temporary boundaries merely existing to mark the boundary. The central space is not defined as in the other houses.
Courtyard based
Integrated open courtyard wada are the most popular ones with a courtyard in the center, semi-open spaces around it and then enclosed spaces. These are mostly Garhi or Rajwada types.
Front and rear yard type are the one with open courts in the front and back. These are the comman man's houses around the periphery of the Garhi or Rajwada.
Wada without an open inegrated courtyard are a special type built in places with heavy rainfall like in the coastal Konkan region.
Location based spread
Most of the Garhi and Rajwada are found in the Desh and Vidharba regions, the center of power of the Peshwas. They were built with elaborate wooden carvings, three or four courtyards, four to five floors and a more dive |
https://en.wikipedia.org/wiki/C.D.%20%C3%81guila%20records%20and%20statistics | C.D. Aguila is a Salvadorian professional association football club based in San Miguel. The club was formed in and played its first competitive match on July 27, 1958 when it played its first season in the Primera División. Aguila currently plays in the Primera División, the top tier of El Salvador football, and is one of three clubs, including Alianza F.C. and C.D. FAS, never to have been relegated from the league.
Honours
As of 11 June 2014 Aguila have won 15 Primera División, one Copa Presidente and one CONCACAF Champions League trophies.
Domestic competitions
League
Primera División
Winners (15): 1959, 1960-61, 1963-64, 1964, 1967-68, 1972, 1975-76, 1976-77, 1983, 1987-88, 1999 Apertura, 2000 Apertura, Clausura 2001, Clausura 2006, Clausura 2012
Cup
Copa Presidente
Winners (1): 2000
CONCACAF competitions
Official titles
CONCACAF Champions League
Winners (1): 1976
Players
Appearances
Competitive, professional matches only including substitution, number of appearances as a substitute appears in brackets.
Last updated - 18 July 2023
Others
Youngest first-team player: – Robin Borjas v Isidro Metapan, Primera Division, 1 December 2019
Oldest first team player: – Luis Ramírez Zapata v TBD, Primera Division, 1992
Most appearances in Primera Division: TBD – TBD
Most appearances in Copa Presidente: TBD – TBD
Most appearances in International competitions: TBD – TBD
Most appearances in CONCACAF competitions: TBD – TBD
Most appearances in UNCAF competitions: TBD – TBD
Most appearances in CONCACAF Champions League: TBD – TBD
Most appearances in UNCAF Copa: TBD TBD
Most appearances in FIFA Club World Cup: 2
Zózimo
Most appearances as a foreign player in all competitions: TBD – TBD
Most appearances as a foreign player in Primera Division: TBD – TBD
Most consecutive League appearances: TBD – TBD – from Month Day, Year at Month Day, Year
Shortest appearance: –
Goalscorers
Competitive, professional matches only. Appearances, including substitutes, appear in brackets.
As of January 2022
By competition
Most goals scored in all competitions: TBD – TBD, Year–Yesr
Most goals scored in Primera Division: TBD – TBD, Year–Yesr
Most goals scored in Copa Presidente: TBD – TBD, Year–Yesr
Most goals scored in International competitions: TBD'' – TBD, Year–Yesr
Most goals scored in CONCACAF competitions: TBD – TBD, Year–Yesr
Most goals scored in UNCAF competitions: TBD – TBD, Year–Yesr
Most goals scored in CONCACAF Champions League: TBD – TBD, Year–Yesr
Most goals scored in UNCAF Cup: TBD – TBD, Year–Yesr
Most goals scored in FIFA World Cup: 1 – TBD, 1982
In a single season
Most goals scored in a season in all competitions: TBD – TBD, Year–Year
Most goals scored in a single Primera Division season: TBD – TBD, Year–Year
Most goals scored in a single Apertura/Clausura season: TBD – TBD, Year–Year
Most goals scored in a single Copa Presidente season: TBD – TBD, Year–Year
Most goals scored in a single CONCACAF Champio |
https://en.wikipedia.org/wiki/Pl%C3%BCcker%27s%20conoid | In geometry, Plücker's conoid is a ruled surface named after the German mathematician Julius Plücker. It is also called a conical wedge or cylindroid; however, the latter name is ambiguous, as "cylindroid" may also refer to an elliptic cylinder.
Plücker's conoid is the surface defined by the function of two variables:
This function has an essential singularity at the origin.
By using cylindrical coordinates in space, we can write the above function into parametric equations
Thus Plücker's conoid is a right conoid, which can be obtained by rotating a horizontal line about the with the oscillatory motion (with period 2π) along the segment of the axis (Figure 4).
A generalization of Plücker's conoid is given by the parametric equations
where denotes the number of folds in the surface. The difference is that the period of the oscillatory motion along the is . (Figure 5 for )
See also
Ruled surface
Right conoid
References
A. Gray, E. Abbena, S. Salamon, Modern differential geometry of curves and surfaces with Mathematica, 3rd ed. Boca Raton, Florida:CRC Press, 2006. ()
Vladimir Y. Rovenskii, Geometry of curves and surfaces with MAPLE ()
External links
Surfaces
Geometric shapes |
https://en.wikipedia.org/wiki/List%20of%20Mexican%20league%20top%20scorers | The following article contains a year-by-year list and statistics of football topscorers in Liga MX (Mexican First Division).
Amateur Era Topscorers (1902-1943)
Professional Era (1943-Currently)
References
See also
Football in Mexico
Footballers in Mexico
Mexico
Association football player non-biographical articles |
https://en.wikipedia.org/wiki/Rejecta%20Mathematica | Rejecta Mathematica was an online journal for publishing papers that had been rejected by other mathematics journals. Each paper was accompanied by an open letter describing why the paper was rejected, how the topic has been developed since and why it is worthy of publication. The first issue was published in July 2009 containing topics such as image enhancement and condition numbers. The quality of the contributions in the first issue was seen as mixed. The editors were Michael Wakin, Christopher Rozell, Mark Davenport and Jason Laska.
After almost two years since the inaugural issue, the second issue was published in June 2011 and contains topics such as subspace classification and distributions of pseudoprimes.
, the original website is no longer online, but an archival copy is hosted on GitHub.
A similar operating model is implemented by unconventional journals like Annals of Improbable Research, the Null Hypothesis: The Journal of Unlikely Science, the Journal of Irreproducible Results or, in different contexts, by Health Promotion International.
See also
Deletionpedia
References
External links
Rejecta Mathematica ()
Mathematics journals
Academic journals established in 2009
Publications disestablished in 2013 |
https://en.wikipedia.org/wiki/Ak%20singularity | In mathematics, and in particular singularity theory, an singularity, where is an integer, describes a level of degeneracy of a function. The notation was introduced by V. I. Arnold.
Let be a smooth function. We denote by the infinite-dimensional space of all such functions. Let denote the infinite-dimensional Lie group of diffeomorphisms and the infinite-dimensional Lie group of diffeomorphisms The product group acts on in the following way: let and be diffeomorphisms and any smooth function. We define the group action as follows:
The orbit of , denoted , of this group action is given by
The members of a given orbit of this action have the following fact in common: we can find a diffeomorphic change of coordinate in and a diffeomorphic change of coordinate in such that one member of the orbit is carried to any other. A function is said to have a type -singularity if it lies in the orbit of
where and is an integer.
By a normal form we mean a particularly simple representative of any given orbit. The above expressions for give normal forms for the type -singularities. The type -singularities are special because they are amongst the simple singularities, this means that there are only a finite number of other orbits in a sufficiently small neighbourhood of the orbit of .
This idea extends over the complex numbers where the normal forms are much simpler; for example: there is no need to distinguish from .
References
Singularity theory |
https://en.wikipedia.org/wiki/Catalan%20surface | In geometry, a Catalan surface, named after the Belgian mathematician Eugène Charles Catalan, is a ruled surface all of whose rulings are parallel to a fixed plane.
Equations
The vector equation of a Catalan surface is given by
r = s(u) + v L(u),
where r = s(u) is the space curve and L(u) is the unit vector of the ruling at u = u. All the vectors L(u) are parallel to the same plane, called the directrix plane of the surface. This can be characterized by the condition: the mixed product [L(u), L' (u), L" (u)] = 0.
The parametric equations of the Catalan surface are
Special cases
If all the rulings of a Catalan surface intersect a fixed line, then the surface is called a conoid.
Catalan proved that the helicoid and the plane were the only ruled minimal surfaces.
See also
Generalized helicoid
References
A. Gray, E. Abbena, S. Salamon, Modern differential geometry of curves and surfaces with Mathematica, 3rd ed. Boca Raton, Florida:CRC Press, 2006. ()
V. Y. Rovenskii, Geometry of curves and surfaces with MAPLE ()
Surfaces
Geometric shapes |
https://en.wikipedia.org/wiki/Lorraine%20Foster | Lorraine Lois Foster (December 25, 1938, Culver City, California) is an American mathematician. In 1964 she became the first woman to receive a Ph.D. in mathematics from California Institute of Technology. Her thesis advisor at Caltech was Olga Taussky-Todd. Foster's Erdos number is 2.
Born Lorraine Lois Turnbull, she attended Occidental College where she majored in physics. She was admitted to Caltech after receiving a Woodrow Wilson Foundation fellowship. In 1964 she joined the faculty of California State University, Northridge. She works in number theory and the theory of mathematical symmetry.
Selected bibliography
Foster, L. (1966). On the characteristic roots of the product of certain rational integral matrices of order two. Pacific Journal of Mathematics, 18(1), 97–110. http://doi.org/10.2140/pjm.1966.18.97
Brenner, J. L., & Foster, L. L. (1982). Exponential diophantine equations. Pacific Journal of Mathematics, 101(2), 263–301.
Alex, L. J., & Foster, L. L. (1983). On diophantine equations of the form . Rocky Mountain Journal of Mathematics, 13(2), 321–332. http://doi.org/10.1216/RMJ-1983-13-2-321
Alex, L. J., & Foster, L. L. (1985). On the Diophantine equation . Rocky Mountain Journal of Mathematics, 15(3), 739–762. http://doi.org/10.1216/RMJ-1985-15-3-739
L. Foster (1989). Finite Symmetry Groups in Three Dimensions, CSUN Instructional Media Center, Jan. 1989 (video, 27 minutes).
L. Foster (1990). Archimedean and Archimedean Dual Polyhedra, CSUN Instructional Media Center, Feb. 1990 (video, 47 minutes). https://www.worldcat.org/title/archimedean-and-archimedean-dual-polyhedra/oclc/63936926&referer=brief_results
Foster, L. L. (1990). On the symmetry group of the dodecahedron. Mathematics Magazine, 63, 106–107.
Foster, L. L. (1991). Convex Polyhedral Models for the Finite Three-Dimensional Isometry Groups. The Mathematical Heritage of CF Gauss, pp 267–281.
L. Foster (1991). The Alhambra Past and Present—a Geometer’s Odyssey Part 1, CSUN Instructional Media Center, December 1991 (video, 40 minutes).
L. Foster (1991). The Alhambra Past and Present—a Geometer’s Odyssey Part 2, CSUN Instructional Media Center, December 1991 (video, 40 minutes). https://www.worldcat.org/title/alhambra-past-and-present-a-geometers-odyssey-parts-1-and-2/oclc/28680624?loc=94043&tab=holdings&start_holding=7
Foster, L. L. (1991). Convex polyhedral models for the finite three-dimensional isometry groups. In G. M. Rassias (Ed.), The Mathematical Heritage of C F Gauss (pp. 267–281). Singapore: World Scientific.
L. Foster (1992). Regular-Faced Polyhedra—an Introduction, CSUN Instructional Media Center, Dec. 1992 (video, 47 minutes)
Alex, L. J., & Foster, L. L. (1992). On the Diophantine equation . Rocky Mountain Journal of Mathematics, 22(1), 11–62. http://doi.org/10.1216/rmjm/1181072793
Alex, L. J., & Foster, L. L. (1995). On the Diophantine equation , with . Rev. Mat. Univ. Complut. Madrid, 8(1), 13–48.
References
20th-century American mathematicians
21st-c |
https://en.wikipedia.org/wiki/Matthias%20Flach%20%28mathematician%29 | Matthias Flach is a German mathematician, professor and former executive officer for mathematics (department chair) at California Institute of Technology.
Professional overview
Research interests includes:
Arithmetic algebraic geometry (see Glossary of arithmetic and Diophantine geometry).
Special values of L-functions.
Conjectures of:
Bloch
Beilinson
Deligne
Bloch–Kato conjecture (see also List of conjectures).
Galois module theory.
Motivic cohomology.
Education overview
Ph.D. University of Cambridge UK 1991 Dissertation: Selmer groups for the Symmetric Square of an Elliptic Curve – Algebraic geometry
Diplom, Goethe University Frankfurt, Germany, 1986
Publications
Iwasawa Theory and Motivic L-functions (2009) – Flach, Matthias
On Galois structure invariants associated to Tate motives – Matthias Flach and D. Burns, King's College London
On the Equivariant Tamagawa Number Conjecture for Tate Motives, Part II. (2006) – Burns, David; Flach, Matthias.
Euler characteristics in relative K-groups – Matthias Flach
The equivariant Tamagawa number conjecture: A survey (with an appendix by C. Greither) – Matthias Flach
A geometric example of non-abelian Iwasawa theory, June 2004, Canadian Number Theory Association VIII Meeting – Flach, Matthias.
The Tamagawa number conjecture of adjoint motives of modular forms (2004) – Diamond, Fred; Flach, Matthias; Guo, Li.
Adjoint motives of modular forms and the Tamagawa number conjecture (2001) – Fred Diamond; Matthias Flach; Li Guo.
Notes
References
ScientificCommons Publication List
Seminar on Fermat's last theorem By Vijaya Kumar Murty, Fields Institute for Research in Mathematical Sciences
The Fermat diary By Charles J. Mozzochi
External links
Flach's Homepage at Caltech
Living people
1963 births |
https://en.wikipedia.org/wiki/Simple%20random%20sample | In statistics, a simple random sample (or SRS) is a subset of individuals (a sample) chosen from a larger set (a population) in which a subset of individuals are chosen randomly, all with the same probability. It is a process of selecting a sample in a random way. In SRS, each subset of k individuals has the same probability of being chosen for the sample as any other subset of k individuals. A simple random sample is an unbiased sampling technique. Simple random sampling is a basic type of sampling and can be a component of other more complex sampling methods.
Introduction
The principle of simple random sampling is that every set of items has the same probability of being chosen. For example, suppose N college students want to get a ticket for a basketball game, but there are only X < N tickets for them, so they decide to have a fair way to see who gets to go. Then, everybody is given a number in the range from 0 to N-1, and random numbers are generated, either electronically or from a table of random numbers. Numbers outside the range from 0 to N-1 are ignored, as are any numbers previously selected. The first X numbers would identify the lucky ticket winners.
In small populations and often in large ones, such sampling is typically done "without replacement", i.e., one deliberately avoids choosing any member of the population more than once. Although simple random sampling can be conducted with replacement instead, this is less common and would normally be described more fully as simple random sampling with replacement.
Sampling done without replacement is no longer independent, but still satisfies exchangeability, hence many results still hold. Further, for a small sample from a large population, sampling without replacement is approximately the same as sampling with replacement, since the probability of choosing the same individual twice is low.
An unbiased random selection of individuals is important so that if many samples were drawn, the average sample would accurately represent the population. However, this does not guarantee that a particular sample is a perfect representation of the population. Simple random sampling merely allows one to draw externally valid conclusions about the entire population based on the sample.
Conceptually, simple random sampling is the simplest of the probability sampling techniques. It requires a complete sampling frame, which may not be available or feasible to construct for large populations. Even if a complete frame is available, more efficient approaches may be possible if other useful information is available about the units in the population.
Advantages are that it is free of classification error, and it requires minimum advance knowledge of the population other than the frame. Its simplicity also makes it relatively easy to interpret data collected in this manner. For these reasons, simple random sampling best suits situations where not much information is available about the population and data |
https://en.wikipedia.org/wiki/Timba%C3%BAba | Timbaúba is a city in Pernambuco, Brazil. According to the Brazilian Institute of Geography and Statistics, it has an estimated population of 52,802 inhabitants as of 2020.
Geography
State - Pernambuco
Region - Zona da mata Pernambucana
Boundaries - Paraiba state (N); Vicência (S); Macaparana (W); Aliança, Ferreiros and Camutanga (E)
Area - 289.51 km2
Elevation - 102 m
Hydrography - Goiana River
Vegetation - Subcaducifólia forest
Climate - Hot tropical and humid
Annual average temperature - 24.6 c
Distance to Recife - 96 km
Economy
The main economic activities in Timbaúba are based in commerce and agribusiness, especially growing sugarcane and bananas, and raising livestock such as cattle, sheep and goats.
Economic indicators
Economy by sector
2006
Health indicators
Sports
The main sport in Timbaúba is football, which is represented by Timbaúba Futebol Clube, currently playing the Campeonato Pernambucano's Série A2.
References
Municipalities in Pernambuco |
https://en.wikipedia.org/wiki/Dry%20gas%20seal | Dry gas seals are non-contacting, dry-running mechanical face seals that consist of a mating (rotating) ring and a primary (stationary) ring. When operating, lifting geometry in the rotating ring generates a fluid-dynamic lifting force causing the stationary ring to separate and create a gap between the two rings. Dry gas seals are mechanical seals but use other chemicals and functions so that they do not contaminate a process. These seals are typically used in a harsh working environment such as oil exploration, extraction and refining, petrochemical industries, gas transmission and chemical processing.
Machined-in lift profiles on one side of the seal face direct gas inward toward an extremely flat portion of the face. The gas that is flowing across the face generates a pressure that maintains a minute gap between the faces, optimizing fluid film stiffness and providing the highest possible degree of protection against face contact. The seal's film stiffness compensates for varying operations by adjusting gap and pressure to maintain stability.
Design and use
Grooves or machined ramps on the seal direct gas inward toward the non-grooved portion. The action of the gas flowing across the seal generates pressure that keeps a minute gap, therefore optimizing fluid film stiffness and providing protection against face contact.
The use of these seals in centrifugal compressors has increased significantly in the last two decades because they eliminate contamination and do not use lubricating oil. Non-contacting dry gas seals are often used on compressors for pipelines, off-shore applications, oil refineries, petrochemical and gas processing plants.
Types
There are many dry gas seal configurations based on their application:
Single seal
Tandem seal - Broadly used in the petroleum industry
Tandem seal with intermediate labyrinth
Double opposed seal - Used when the processed gas is abrasive (like hydrogen) and lower pressure designs.
All designs use buffering with "dry" gas, supplied through control and purification systems.
All Dry Gas Seals need additional protection from the process and the bearing lubrication sides of the seal
History
The first dry gas seal for a compressor was patented by Kaydon Ring & Seal in 1951 when it was known as Koppers Corporation. Field applications of dry gas seal designs were completed in 1952. The original patent was for Kaydon's "Tapered Ramp" lift geometry, a constant diameter / variable depth dynamic lift design. From that first dry gas seal ever manufactured for a centrifugal compressor in 1951, Kaydon Ring & Seal has been instrumental in developing the dry gas seal into one of the most reliable and maintenance free sealing solution available today.
John Crane Inc. issued a patent for dry gas seals in 1968 with field applications beginning in 1975, though the technology is now widely available among seal manufacturers. When the technology is aimed at correcting the problems with dry gas film environments |
https://en.wikipedia.org/wiki/Arnold%20diffusion | In applied mathematics, Arnold diffusion is the phenomenon of instability of integrable Hamiltonian systems. The phenomenon is named after Vladimir Arnold who was the first to publish a result in the field in 1964. More precisely, Arnold diffusion refers to results asserting the existence of solutions to nearly integrable Hamiltonian systems that exhibit a significant change in the action variables.
Arnold diffusion describes the diffusion of trajectories due to the ergodic theorem in a portion of phase space unbound by any constraints (i.e. unbounded by Lagrangian tori arising from constants of motion) in Hamiltonian systems. It occurs in systems with more than N=2 degrees of freedom, since the N-dimensional invariant tori do not separate the 2N-1 dimensional phase space any more. Thus, an arbitrarily small perturbation may cause a number of trajectories to wander pseudo-randomly through the whole portion of phase space left by the destroyed tori.
Background and statement
For integrable systems, one has the conservation of the action variables. According to the KAM theorem if we perturb an integrable system slightly, then many, though certainly not all, of the solutions of the perturbed system stay close, for all time, to the unperturbed system. In particular, since the action variables were originally conserved, the theorem tells us that there is only a small change in action for many solutions of the perturbed system.
However, as first noted in Arnold's paper, there are nearly integrable systems for which there exist solutions that exhibit arbitrarily large growth in the action variables. More precisely, Arnold considered the example of nearly integrable Hamiltonian system with Hamiltonian
The first three terms of this Hamiltonian describe a rotator-pendulum system.
Arnold showed that for this system, for any choice of , and for , there is a solution to the system for which
for some time
His proof relies on the existence of `transition chains' of `whiskered' tori, that is, sequences of
tori with transitive dynamics such that the unstable manifold
(whisker) of one of these tori intersects transversally the stable manifold (whisker) of the next
one. Arnold conjectured that "the mechanism of 'transition chains' which guarantees that nonstability in our example is also applicable to the general case (for example, to the problem of three bodies)."
A background on the KAM theorem can be found in
and a compendium of rigorous mathematical results, with insight from physics, can be found in.
General Case
In Arnold's model the perturbation term is of a special type. The general case of Arnold's diffusion problem concerns Hamiltonian systems of one of the forms
where , , and describes a rotator-pendulum system, or
where ,
For systems as in , the unperturbed Hamiltonian possesses smooth families of invariant tori that have hyperbolic stable and unstable manifolds; such systems are referred to as a priori unstable. |
https://en.wikipedia.org/wiki/2009%20Club%20Atlas%20season | F.C. Atlas debuted the Apertura 2009 on July 25, 2009, with a 1–0 win over Pumas.
Summer transfers
In:
Out:
Current roster
Statistics
Goalkeepers
2009 Mexico Apertura
Atlas debuted the Apertura 2009 with a 1–0 victory over Pumas. Atlas showed a great performance in the first half. At the 45' minute marked the first goal of the game and season for Atlas, scored by Edgar Ivan Pacheco with a header, assisted by a long pass from Daniel Osorno. Apertura 2009 will be coached by Ricardo La Volpe.
Jornada 2, Atlas visited Monterrey. A poor performance by Atlas led them to lose to Monterrey 3–0. Goals scored by Monterrey were an own goal by Edgar Pacheco, Sergio Santana, and Humberto Suazo. Atlas' defense left many open spots in the defense which led Monterrey to shoot many shots. Canales was a huge factor in Monterrey's third, in which he came out and ended up leaving the goal, open.
Jornada 3, Atlas at home takes a 2–1 victory over Santos. Atlas with a much better improvements from last week. On Atlas' side, Ismael Fuentes scored the first on a corner, passed by Dario Botinelli. Gerardo Espinoza scored a very well goal. Dario Botinelli with a long pass for Daniel Osorno who makes a fast run for the ball and then re-passes the ball into the area where Espinoza gets the ball and shoots into the far angle of the goal. For Santos, Juan Carlos Mosqueda scored, but it wasn't enough for Atlas won.
Jornada 4, Atlas loses against the home team America. Atlas showed some poor performances on the defense. Now starting to show in need of a forward. Cabanas scored first for Club América with a very good goal, with errors by Hugo Ayala and Luis Robles. Atlas tied the game with a corner by Edgar Pacheco and Fuentes scores with a hard header and injures himself by bumping his head into Mosquera's head. Second Half, America again taking the lead. After a shot bounced off the crossbar, Beausejour picks up the ball and scores on Barbosa with a weak shot. Atlas had chances to score like a shot by Botinelli in which goalkeeper Guillermo Ochoa blocks and hits the crossbar. Another good opportunity by Pacheco which hits the crossbar as well.
Jornada 5, Atlas takes a loss at home against Morelia, 2–0. Sabah scored 2 goals for Morelia. Atlas, who showed great work on the field, but made few mistakes in which Morelia capitalized in.
Jornada 6, Atlas takes an away tie, 1–1 with Indios. It was a well-deserved result. Both sides gave their all and Pacheco scored a header, assisted by Mario Pérez. Indios scored a penalty, which was kicked by Edwin Santibanez.
Group standings
Top 3 goalscorers
References
2009
Atlas |
https://en.wikipedia.org/wiki/2003%E2%80%9304%20Real%20Madrid%20CF%20season | The 2003–04 season was Real Madrid CF's 73rd season in La Liga. This article lists all matches that the club played in the 2003–04 season, and also shows statistics of the club's players. The club played the season wearing their classic white home and teal blue away kits.
Season summary
In spite of the arduous pre-season, the team got off to a good start. They won the Supercopa de España against Mallorca with a 3–0 victory on 27 August in the second leg, avenging their loss to the same side in the 2002–03 Copa del Rey. By the time half of the season had passed, Madrid topped the league table and was still in contention for the Copa del Rey and Champions League trophies. However, the team was eliminated in the quarter-finals of the Champions League on away goals by Monaco and finished as runners-up in the domestic cup, losing to Zaragoza after extra time. They also lost their final five La Liga matches and finished in fourth place, which gave Valencia the title. It was the last time that Real Madrid finished below second place until the 2013–14 season, which brought a long-awaited La Décima.
Transfers
In
Total spending: €35,000,000
Out
Total income: €16,000,000
Squad
Left club during season
(transferred to Chelsea in August 2003)
Pre-season
The team embarked on a summer tour in Asia, for 18 days, to cash in on the worldwide appeal of their new signing, David Beckham. It included exhibition matches in Beijing, Tokyo, Hong Kong and Bangkok, which alone earned the club €10 million. This was compared by popular contrary with the tour with the first visit of The Beatles to the United States in 1964. Although lucrative and generating wide publicity, the preparation value of the Asia was questionable, considering that the long 2003–04 season which lay ahead. It was exhausting for the players, due to endless rounds of publicity engagements and restrictions on the players' freedom of movement (due to the team hotel being besieged by fans). Most players admitted that they would have preferred a low-profile training camp and/or to have been home in Spain for the pre-season, instead of playing meaningless show matches against low quality opponents. The Asia tour has been said to have catered more to the needs of the club's marketing than to its players' preparations.
Pre-season
Results
2003 Supercopa de España
La Liga
Classification
Results by round
Matches
Copa del Rey
Round of 64
Round of 32
Round of 16
Quarter-finals
Semi-finals
Final
UEFA Champions League
Group stage
Knockout phase
Round of 16
Quarter-finals
Statistics
Players statistics
References
Spanish football clubs 2003–04 season
Real Madrid CF seasons |
https://en.wikipedia.org/wiki/Non-squeezing%20theorem | The non-squeezing theorem, also called Gromov's non-squeezing theorem, is one of the most important theorems in symplectic geometry. It was first proven in 1985 by Mikhail Gromov.
The theorem states that one cannot embed a ball into a cylinder via a symplectic map unless the radius of the ball is less than or equal to the radius of the cylinder. The theorem is important because formerly very little was known about the geometry behind symplectic maps.
One easy consequence of a transformation being symplectic is that it preserves volume. One can easily embed a ball of any radius into a cylinder of any other radius by a volume-preserving transformation: just picture squeezing the ball into the cylinder (hence, the name non-squeezing theorem). Thus, the non-squeezing theorem tells us that, although symplectic transformations are volume-preserving, it is much more restrictive for a transformation to be symplectic than it is to be volume-preserving.
Background and statement
We start by considering the symplectic spaces
the ball of radius R:
and the cylinder of radius r:
each endowed with the symplectic form
Note: The choice of axes for the cylinder are not arbitrary given the fixed symplectic form above; namely the circles of the cylinder each lie in a symplectic subspace of .
The non-squeezing theorem tells us that if we can find a symplectic embedding φ : B(R) → Z(r) then R ≤ r.
The “symplectic camel”
Gromov's non-squeezing theorem has also become known as the principle of the symplectic camel since Ian Stewart referred to it by alluding to the parable of the camel and the eye of a needle. As Maurice A. de Gosson states:
Similarly:
De Gosson has shown that the non-squeezing theorem is closely linked to the Robertson–Schrödinger–Heisenberg inequality, a generalization of the Heisenberg uncertainty relation. The Robertson–Schrödinger–Heisenberg inequality states that:
with Q and P the canonical coordinates and var and cov the variance and covariance functions.
References
Further reading
Maurice A. de Gosson: The symplectic egg, arXiv:1208.5969v1, submitted on 29 August 2012 – includes a proof of a variant of the theorem for case of linear canonical transformations
Dusa McDuff: What is symplectic geometry?, 2009
Symplectic geometry
Theorems in geometry |
https://en.wikipedia.org/wiki/British%20Journal%20of%20Mathematical%20and%20Statistical%20Psychology | The British Journal of Mathematical and Statistical Psychology is a British scientific journal founded in 1947. It covers the fields of psychology, statistics, and mathematical psychology. It was established as the British Journal of Psychology (Statistical Section), was renamed the British Journal of Statistical Psychology in 1953, and was renamed again to its current title in 1965.
Abstracting and indexing
The journal is indexed in ''Current Index to Statistics, PsycINFO, Social Sciences Citation Index, Current Contents / Social & Behavioral Sciences, Science Citation Index Expanded, and Scopus.
Academic journals established in 1947
Statistics journals
Mathematical and statistical psychology journals
Wiley-Blackwell academic journals
British Psychological Society academic journals
Triannual journals
Past Editors
Thom Baguley
Matthias von Davier |
https://en.wikipedia.org/wiki/Erd%C5%91s%20Prize | The Anna and Lajos Erdős Prize in Mathematics is a prize given by the Israel Mathematical Union to an Israeli mathematician (in any field of mathematics and computer science), "with preference to candidates up to the age of 40." The prize was established by Paul Erdős in 1977 in honor of his parents, and is awarded annually or biannually. The name was changed from "Erdős Prize" in 1996, after Erdős's death, to reflect his original wishes.
Erdős Prize recipients
See also
List of things named after Paul Erdős
List of mathematics awards
References
Mathematics awards
Awards established in 1977
Israeli awards
Lists of Israeli award winners
Israeli science and technology awards |
https://en.wikipedia.org/wiki/Homogeneous%20coordinate%20ring | In algebraic geometry, the homogeneous coordinate ring R of an algebraic variety V given as a subvariety of projective space of a given dimension N is by definition the quotient ring
R = K[X0, X1, X2, ..., XN] /I
where I is the homogeneous ideal defining V, K is the algebraically closed field over which V is defined, and
K[X0, X1, X2, ..., XN]
is the polynomial ring in N + 1 variables Xi. The polynomial ring is therefore the homogeneous coordinate ring of the projective space itself, and the variables are the homogeneous coordinates, for a given choice of basis (in the vector space underlying the projective space). The choice of basis means this definition is not intrinsic, but it can be made so by using the symmetric algebra.
Formulation
Since V is assumed to be a variety, and so an irreducible algebraic set, the ideal I can be chosen to be a prime ideal, and so R is an integral domain. The same definition can be used for general homogeneous ideals, but the resulting coordinate rings may then contain non-zero nilpotent elements and other divisors of zero. From the point of view of scheme theory these cases may be dealt with on the same footing by means of the Proj construction.
The irrelevant ideal J generated by all the Xi corresponds to the empty set, since not all homogeneous coordinates can vanish at a point of projective space.
The projective Nullstellensatz gives a bijective correspondence between projective varieties and homogeneous ideals I not containing J.
Resolutions and syzygies
In application of homological algebra techniques to algebraic geometry, it has been traditional since David Hilbert (though modern terminology is different) to apply free resolutions of R, considered as a graded module over the polynomial ring. This yields information about syzygies, namely relations between generators of the ideal I. In a classical perspective, such generators are simply the equations one writes down to define V. If V is a hypersurface there need only be one equation, and for complete intersections the number of equations can be taken as the codimension; but the general projective variety has no defining set of equations that is so transparent. Detailed studies, for example of canonical curves and the equations defining abelian varieties, show the geometric interest of systematic techniques to handle these cases. The subject also grew out of elimination theory in its classical form, in which reduction modulo I is supposed to become an algorithmic process (now handled by Gröbner bases in practice).
There are for general reasons free resolutions of R as graded module over K[X0, X1, X2, ..., XN]. A resolution is defined as minimal if the image in each module morphism of free modules
φ:Fi → Fi − 1
in the resolution lies in JFi − 1, where J is the irrelevant ideal. As a consequence of Nakayama's lemma, φ then takes a given basis in Fi to a minimal set of generators in Fi − 1. The concept of minimal free resolution is well-defined in |
https://en.wikipedia.org/wiki/Arithmetic%20topology | Arithmetic topology is an area of mathematics that is a combination of algebraic number theory and topology. It establishes an analogy between number fields and closed, orientable 3-manifolds.
Analogies
The following are some of the analogies used by mathematicians between number fields and 3-manifolds:
A number field corresponds to a closed, orientable 3-manifold
Ideals in the ring of integers correspond to links, and prime ideals correspond to knots.
The field Q of rational numbers corresponds to the 3-sphere.
Expanding on the last two examples, there is an analogy between knots and prime numbers in which one considers "links" between primes. The triple of primes are "linked" modulo 2 (the Rédei symbol is −1) but are "pairwise unlinked" modulo 2 (the Legendre symbols are all 1). Therefore these primes have been called a "proper Borromean triple modulo 2" or "mod 2 Borromean primes".
History
In the 1960s topological interpretations of class field theory were given by John Tate based on Galois cohomology, and also by Michael Artin and Jean-Louis Verdier based on Étale cohomology. Then David Mumford (and independently Yuri Manin) came up with an analogy between prime ideals and knots which was further explored by Barry Mazur. In the 1990s Reznikov and Kapranov began studying these analogies, coining the term arithmetic topology for this area of study.
See also
Arithmetic geometry
Arithmetic dynamics
Topological quantum field theory
Langlands program
Notes
Further reading
Masanori Morishita (2011), Knots and Primes, Springer,
Masanori Morishita (2009), Analogies Between Knots And Primes, 3-Manifolds And Number Rings
Christopher Deninger (2002), A note on arithmetic topology and dynamical systems
Adam S. Sikora (2001), Analogies between group actions on 3-manifolds and number fields
Curtis T. McMullen (2003), From dynamics on surfaces to rational points on curves
Chao Li and Charmaine Sia (2012), Knots and Primes
External links
Mazur’s knotty dictionary
Number theory
3-manifolds
Knot theory |
https://en.wikipedia.org/wiki/Gordan%27s%20lemma | Gordan's lemma is a lemma in convex geometry and algebraic geometry. It can be stated in several ways.
Let be a matrix of integers. Let be the set of non-negative integer solutions of . Then there exists a finite subset of vectors in , such that every element of is a linear combination of these vectors with non-negative integer coefficients.
The semigroup of integral points in a rational convex polyhedral cone is finitely generated.
An affine toric variety is an algebraic variety (this follows from the fact that the prime spectrum of the semigroup algebra of such a semigroup is, by definition, an affine toric variety).
The lemma is named after the mathematician Paul Gordan (1837–1912). Some authors have misspelled it as "Gordon's lemma".
Proofs
There are topological and algebraic proofs.
Topological proof
Let be the dual cone of the given rational polyhedral cone. Let be integral vectors so that Then the 's generate the dual cone ; indeed, writing C for the cone generated by 's, we have: , which must be the equality. Now, if x is in the semigroup
then it can be written as
where are nonnegative integers and . But since x and the first sum on the right-hand side are integral, the second sum is a lattice point in a bounded region, and so there are only finitely many possibilities for the second sum (the topological reason). Hence, is finitely generated.
Algebraic proof
The proof is based on a fact that a semigroup S is finitely generated if and only if its semigroup algebra is a finitely generated algebra over . To prove Gordan's lemma, by induction (cf. the proof above), it is enough to prove the following statement: for any unital subsemigroup S of ,
If S is finitely generated, then , v an integral vector, is finitely generated.
Put , which has a basis . It has -grading given by
.
By assumption, A is finitely generated and thus is Noetherian. It follows from the algebraic lemma below that is a finitely generated algebra over . Now, the semigroup is the image of S under a linear projection, thus finitely generated and so is finitely generated. Hence, is finitely generated then.
Lemma: Let A be a -graded ring. If A is a Noetherian ring, then is a finitely generated -algebra.
Proof: Let I be the ideal of A generated by all homogeneous elements of A of positive degree. Since A is Noetherian, I is actually generated by finitely many , homogeneous of positive degree. If f is homogeneous of positive degree, then we can write with homogeneous. If f has sufficiently large degree, then each has degree positive and strictly less than that of f. Also, each degree piece is a finitely generated -module. (Proof: Let be an increasing chain of finitely generated submodules of with union . Then the chain of the ideals stabilizes in finite steps; so does the chain ) Thus, by induction on degree, we see is a finitely generated -algebra.
Applications
A multi-hypergraph over a certain set is a multiset of subsets of (it is ca |
https://en.wikipedia.org/wiki/Ranklet | In statistics, a ranklet is an orientation-selective non-parametric feature which is based on the computation of Mann–Whitney–Wilcoxon (MWW) rank-sum test statistics. Ranklets achieve similar response to Haar wavelets as they share the same pattern of orientation-selectivity, multi-scale nature and a suitable notion of completeness. There were invented by Fabrizio Smeralhi in 2002.
Rank-based (non-parametric) features have become popular in the field of image processing for their robustness in detecting outliers and invariance to monotonic transformations such as brightness, contrast changes and gamma correction.
The MWW is a combination of Wilcoxon rank-sum test and Mann–Whitney U-test. It is a non-parametric alternative to the t-test used to test the hypothesis for the comparison of two independent distributions. It assesses whether two samples of observations, usually referred as Treatment T and Control C, come from the same distribution but do not have to be normally distributed.
The Wilcoxon rank-sum statistics Ws is determined as:
Subsequently, let MW be the Mann–Whitney statistics defined by:
where m is the number of Treatment values.
A ranklet R is defined as the normalization of MW in the range [−1, +1]:
where a positive value means that the Treatment region is brighter than the Control region, and a negative value otherwise.
Example
Suppose and then
Hence, in the above example the Control region was a little bit brighter than the Treatment region.
Method
Since Ranklets are non-linear filters, they can only be applied in the spatial domain. Filtering with Ranklets involves dividing an image window W into Treatment and Control regions as shown in the image below:
Subsequently, Wilcoxon rank-sum test statistics are computed in order to determine the intensity variations among conveniently
chosen regions (according to the required orientation) of the samples in W. The intensity values of both regions are then
replaced by the respective ranking scores. These ranking scores determine a pairwise comparison between the T and C regions. This means that a ranklet essentially counts the number of TxC pairs which are brighter in the T set. Hence a positive value means that the Treatment values are brighter than the Control values, and vice versa.
References
External links
Matlab RankletFilter.m -> source file to decompose an image into Intensity Ranklets
Nonlinear filters
Nonparametric statistics
Spatial analysis |
https://en.wikipedia.org/wiki/Asymptotic%20homogenization | In mathematics and physics, homogenization is a method of studying partial differential equations with rapidly oscillating coefficients, such as
where is a very small parameter and
is a 1-periodic coefficient:
,
.
It turns out that the study of these equations is also of great importance in physics and engineering, since equations of this type govern the physics of inhomogeneous or heterogeneous materials. Of course, all matter is inhomogeneous at some scale, but frequently it is convenient to treat it as homogeneous. A good example is the continuum concept which is used in continuum mechanics. Under this assumption, materials such as fluids, solids, etc. can be treated as homogeneous materials and associated with these materials are material properties such as shear modulus, elastic moduli, etc.
Frequently, inhomogeneous materials (such as composite materials) possess microstructure and therefore they are subjected to loads or forcings which vary on a length scale which is far bigger than the characteristic length scale of the microstructure. In this situation, one can often replace the equation above with an equation of the form
where is a constant tensor coefficient and is known as the effective property associated with the material in question. It can be explicitly computed as
from 1-periodic functions satisfying:
This process of replacing an equation with a highly oscillatory coefficient with one with a homogeneous (uniform) coefficient is known as homogenization. This subject is inextricably linked with the subject of micromechanics for this very reason.
In homogenization one equation is replaced by another if
for small enough , provided
in some appropriate norm as .
As a result of the above, homogenization can therefore be viewed as an extension of the continuum concept to materials which possess microstructure. The analogue of the differential element in the continuum concept (which contains enough atom, or molecular structure to be representative of that material), is known as the "Representative Volume Element" in homogenization and micromechanics. This element contains enough statistical information about the inhomogeneous medium in order to be representative of the material. Therefore averaging over this element gives an effective property such as above.
Classical results of homogenization theory were obtained for media with periodic microstructure modeled by partial differential equations with periodic coefficients. These results were later generalized to spatially homogeneous random media modeled by differential equations with random coefficients which statistical properties are the same at every point in space. In practice, many applications require a more general way of modeling that is neither periodic nor statistically homogeneous. For this end the methods of the homogenization theory have been extended to partial differential equations, which coefficients are neither periodic nor statistically homogeneous ( |
https://en.wikipedia.org/wiki/Mirko%20Radovanovi%C4%87 | Mirko Radovanović (; born 5 April 1986 in Čačak) is a Serbian footballer.
Career statistics
External links
Profile and stats at Srbijafudbal.
1986 births
Living people
Footballers from Čačak
Serbian men's footballers
Men's association football defenders
FK Borac Čačak players
FK Remont Čačak players
FK Mladi Radnik players
FK Mladost Lučani players
FK Radnički 1923 players
FK Smederevo 1924 players
Serbian SuperLiga players
Serbian expatriate men's footballers
FK Željezničar Sarajevo players
Expatriate men's footballers in Slovakia
Slovak First Football League players
AS Trenčín players
Serbian expatriate sportspeople in Slovakia
Expatriate men's footballers in Finland
Serbian expatriate sportspeople in Finland
Ykkönen players
Oulun Palloseura players
FK Sloga Petrovac na Mlavi players |
https://en.wikipedia.org/wiki/Spectral%20triple | In noncommutative geometry and related branches of mathematics and mathematical physics, a spectral triple is a set of data which encodes a geometric phenomenon in an analytic way. The definition typically involves a Hilbert space, an algebra of operators on it and an unbounded self-adjoint operator, endowed with supplemental structures. It was conceived by Alain Connes who was motivated by the Atiyah-Singer index theorem and sought its extension to 'noncommutative' spaces. Some authors refer to this notion as unbounded K-cycles or as unbounded Fredholm modules.
Motivation
A motivating example of spectral triple is given by the algebra of smooth functions on a compact spin manifold, acting on the Hilbert space of L2-spinors, accompanied by the Dirac operator associated to the spin structure. From the knowledge of these objects one is able to recover the original manifold as a metric space: the manifold as a topological space is recovered as the spectrum of the algebra, while the (absolute value of) Dirac operator retains the metric. On the other hand, the phase part of the Dirac operator, in conjunction with the algebra of functions, gives a K-cycle which encodes index-theoretic information. The local index formula expresses the pairing of the K-group of the manifold with this K-cycle in two ways: the 'analytic/global' side involves the usual trace on the Hilbert space and commutators of functions with the phase operator (which corresponds to the 'index' part of the index theorem), while the 'geometric/local' side involves the Dixmier trace and commutators with the Dirac operator (which corresponds to the 'characteristic class integration' part of the index theorem).
Extensions of the index theorem can be considered in cases, typically when one has an action of a group on the manifold, or when the manifold is endowed with a foliation structure, among others. In those cases the algebraic system of the 'functions' which expresses the underlying geometric object is no longer commutative, but one may able to find the space of square integrable spinors (or, sections of a Clifford module) on which the algebra acts, and the corresponding 'Dirac' operator on it satisfying certain boundedness of commutators implied by the pseudo-differential calculus.
Definition
An odd spectral triple is a triple (A, H, D) consisting of a Hilbert space H, an algebra A of operators on H (usually closed under taking adjoints) and a densely defined self adjoint operator D satisfying ‖[a, D]‖ < ∞ for any a ∈ A. An even spectral triple is an odd spectral triple with a Z/2Z-grading on H, such that the elements in A are even while D is odd with respect to this grading. One could also say that an even spectral triple is given by a quartet (A, H, D, γ) such that γ is a self adjoint unitary on H satisfying a γ = γ a for any a in A and D γ = - γ D.
A finitely summable spectral triple is a spectral triple (A, H, D) such that a.D for any a in A has a compact resolvent |
https://en.wikipedia.org/wiki/Triunfo%2C%20Pernambuco | Triunfo is a municipality in the Northeastern Brazilian state of Pernambuco. The estimated population in 2020, according to the Brazilian Institute of Geography and Statistics (IBGE) was 15,243. The area of the municipality is 191.52 km2, and in 2010 the population density was 78 inhabitants/km2.
Triunfo sits at an elevation of in a forested part of the Sertão, and is the highest municipality in Pernambuco.
Far inland and with a milder climate than the surrounding semi-arid plateau, the city has the nickname "Oasis of the Sertão", and tourism is a significant part of the economy. Its current mayor is Luciano Fernando de Sousa (better known as Luciano Bonfim) of the Avante party, elected in 2020.
Geography
Region – Sertão of Pernambuco
Boundaries – state of Paraíba (N); Calumbi (S); Flores (E); Santa Cruz da Baixa Verde (W)
Area – 191.52 km2
Elevation – 1004 m
Drainage basin – Pajeú River
Vegetation – Semi-deciduous forest
Climate – Tropical wet and dry (Köppen Aw)
Annual average temperature – 20.4 °C
Distance to Recife – 403 km
Climate
Economy
The main economic activities in Triunfo are tourism, commerce and agribusiness, especially farming of goats, cattle, sheep, ; and plantations of guavas and sugarcane.
Economic indicators
Economy by sector
2006
Health indicators
References
External links
Official website of the Prefeitura (mayor and city hall) of Triunfo (in Portuguese)
Official website of the Câmara Municipal (city council) of Triunfo (in Portuguese)
Municipalities in Pernambuco |
https://en.wikipedia.org/wiki/Stefan%20Lehner | Stefan Lehner (born 1957 in St. Gallen, Switzerland) is a Swiss designer who lives and works in Utrecht, Netherlands. He studied Philosophy, Mathematics, Language and Communication Coach in Enterprises.
From 1978–2004 he lived and worked in Fribourg, Switzerland, and also studied philosophy, mathematics, language and Communication Coach in Enterprises, and was member of the Committee of the Art Laboratory Belluard Bollwerk International.
Design career
His career as a designer started in 1986, with the first construction of a metal furniture's based on recycled materials from the industry. In 1988 he founded the Atelier En-Fer and work with Cristina Lanzos until 1999. In 2004 he transferred the Atelier En-Fer to the old town of Utrecht, The Netherlands.
The peculiar characteristic of his work is that the furniture, objects and interiors are based on used materials and reuse former or hidden functions of the recycled materials. The transformations are made with complex materials (industry), but also with cheap trash (packaging). The metamorphose of the materials should be useful and comfortable, surprising and make smile. His projects bring to the houses the sober beauty of industrial objects but simultaneously take profits from their former function: a spring damps weights (banc, office chair), a car seat has a good ergonomics (arm chair, sofa, reception room), a chain tracks and stays flexible (arm chair, bed, couch) and a supermarket trolley rolls and can be pushed together for storage (seat, child car, coat rack). The main topics related to his work are: research recycling and function reuse (studies and constructions re-using functions in new objects), furniture prototypes (authentic materials for personalised use), interior design, international projects (collaboration with designers and artists in Brazil).
Fascination for inventions and industrial materials
As a child he was always drawing and inventing machines and collected a lot of thrown objects. Since this time he has visited scrap yards and factories and collected interesting used materials. Searching for a new life for those objects, he started in 1985 with the construction of furniture prototypes. These principles are also applied for bigger installations: kitchen (Spain, Switzerland), bathroom (Spain), bar (Fribourg), reception (Bern), office (Fribourg), jewellery shop (Zurich), flower shop (Fribourg). Serial production and new row materials - For Chesterfield he developed a recycling ashtray and produced 1001 pieces for sponsored trendy restaurants, bars and concert rooms. Later he started to use also other recycled row materials in combination with metals: rubber, wood, glass and animal bones.
Exhibitions
Personal Expositions
2006 - Personal Exposition in the Sociale Verzakeringsbank Amstelveen
1999 - Designshop „Einzigart“ in Zürich
1992 - Forum d’Art Contemporain in Siders
1990 - Galerie Delikt in Freiburg
Group expositions
2006 - 100%design Rotterdam
2006 - Woonbe |
https://en.wikipedia.org/wiki/Complex%20squaring%20map | In mathematics, the complex squaring map, a polynomial mapping of degree two, is a simple and accessible demonstration of chaos in dynamical systems. It can be constructed by performing the following steps:
Choose any complex number on the unit circle whose argument (angle) is not a rational multiple of π,
Repeatedly square that number.
This repetition (iteration) produces a sequence of complex numbers that can be described alone by their arguments. Any choice of starting angle that satisfies (1) above will produce an extremely complicated sequence of angles, that belies the simplicity of the steps. It can be shown that the sequence will be chaotic, i.e. it is sensitive to the detailed choice of starting angle.
Chaos and the complex squaring map
The informal reason why the iteration is chaotic is that the angle doubles on every iteration and doubling grows very quickly as the angle becomes ever larger, but angles which differ by multiples of 2π (radians) are identical. Thus, when the angle exceeds 2π, it must wrap to the remainder on division by 2π. Therefore, the angle is transformed according to the dyadic transformation (also known as the 2x mod 1 map). As the initial value z0 has been chosen so that its argument is not a rational multiple of π, the forward orbit of zn cannot repeat itself and become periodic.
More formally, the iteration can be written as
where is the resulting sequence of complex numbers obtained by iterating the steps above, and represents the initial starting number. We can solve this iteration exactly:
Starting with angle θ, we can write the initial term as so that . This makes the successive doubling of the angle clear. (This is equivalent to the relation by Euler's formula.)
Generalisations
This map is a special case of the complex quadratic map, which has exact solutions for many special cases. The complex map obtained by raising the previous number to any natural number power is also exactly solvable as . In the case p = 2, the dynamics can be mapped to the dyadic transformation, as described above, but for p > 2, we obtain a shift map in the number base p. For example, p = 10 is a decimal shift map.
See also
Logistic map
Dyadic transformation
References
Chaotic maps |
https://en.wikipedia.org/wiki/Bodoc%C3%B3 | Bodocó is a municipality in the state of Pernambuco, Brazil. Its population in 2020, according to the Brazilian Institute of Geography and Statistics (IBGE), was an estimated 38,378 and its area is 1621.79 km². Bodocó was established in 1909 from territory of the municipality of Granito.
Its current mayor () is Otávio Augusto Tavares Pedrosa Cavalcante of the Brazilian Socialist Party, elected in 2020.
Geography
Region – Sertão of Pernambuco
Boundaries – state of Ceará (N); Parnamirim (S); Exu and Granito (E); Ouricuri and Ipubi (W)
Area – 1553.85 km²
Elevation – 443 m
Drainage basin – Brigida River
Vegetation – Caatinga (shrubland)
Climate – semi-arid, hot and dry, Köppen: BSh
Annual average temperature – 25.6°C
Distance to Recife – 642.6 km
Economy
The main economic activities in Bodocó are based in commerce and agribusiness, especially the farming of goats, cattle, sheep, horses, donkeys, pigs, and honey, and cultivation of corn and manioc.
Economic indicators
Economy by sector (as of 2013)
Health indicators
References
External links
Official site of the Prefeitura (mayor and city hall) (in Portuguese)
Official site of the Câmara Municipal (city council) (in Portuguese)
Municipalities in Pernambuco |
https://en.wikipedia.org/wiki/Subnormal | Subnormal may refer to:
Subnormal body temperature, a common term for hypothermia
Subnormal operator, a type of operator in operator theory in mathematics
Subnormal number, another name for a denormal number in floating point arithmetic
Subnormal profit, which is negative profit (economics)
Subnormal series, a type of subgroup series in group theory in mathematics
Subnormal subgroup, a type of subgroup in group theory in mathematics
The projection of a normal of a curve onto the x-axis; see subtangent |
https://en.wikipedia.org/wiki/Ferenc%20R%C3%A1cz | Ferenc Rácz (born 28 March 1991) is a Hungarian football player who plays for Dorog.
Club statistics
Updated to games played as of 31 March 2018.
References
External links
1991 births
Living people
Footballers from Szombathely
Hungarian men's footballers
Hungary men's youth international footballers
Men's association football forwards
Szombathelyi Haladás footballers
FC Ajka players
Kozármisleny SE footballers
MTK Budapest FC players
Pécsi MFC players
Mezőkövesdi SE footballers
Győri ETO FC players
Balmazújvárosi FC players
Kisvárda FC players
Dorogi FC footballers
Nemzeti Bajnokság I players
Nemzeti Bajnokság II players |
https://en.wikipedia.org/wiki/M%C3%A1t%C3%A9%20Skriba | Máté Skriba (born 13 March 1992 in Celldömölk) is a Hungarian football player who currently plays for FC Ajka on loan from Haladás.
Career statistics
References
Haladas FC
Illes Academia
HLSZ
UEFA Official Website
1992 births
Living people
People from Celldömölk
Hungarian men's footballers
Men's association football forwards
Szombathelyi Haladás footballers
MTK Budapest FC players
Tatabányai SC players
FC Ajka players
FC Veszprém footballers
Budafoki MTE footballers
Nemzeti Bajnokság I players
Sportspeople from Vas County |
https://en.wikipedia.org/wiki/1963%20Liberian%20Premier%20League | Statistics of Liberian Premier League in season 1963.
Overview
Invincible Eleven won the championship.
References
Liberia - List of final tables (RSSSF)
Football competitions in Liberia |
https://en.wikipedia.org/wiki/1964%20Liberian%20Premier%20League | Statistics of Liberian Premier League in season 1964.
Overview
Invincible Eleven won the championship.
References
Liberia - List of final tables (RSSSF)
Football competitions in Liberia |
https://en.wikipedia.org/wiki/1965%20Liberian%20Premier%20League | Statistics of Liberian Premier League in season 1965.
Overview
Invincible Eleven won the championship.
References
Liberia - List of final tables (RSSSF)
Football competitions in Liberia |
https://en.wikipedia.org/wiki/1966%20Liberian%20Premier%20League | Statistics of Liberian Premier League in season 1966.
Overview
Invincible Eleven won the championship.
References
Liberia - List of final tables (RSSSF)
Football competitions in Liberia |
https://en.wikipedia.org/wiki/1967%20Liberian%20Premier%20League | Statistics of Liberian Premier League in season 1967.
Overview
Mighty Barrolle won the championship.
References
Liberia - List of final tables (RSSSF)
Football competitions in Liberia |
https://en.wikipedia.org/wiki/1972%20Liberian%20Premier%20League | Statistics of Liberian Premier League in season 1972.
Overview
Mighty Barrolle won the championship.
References
Liberia - List of final tables (RSSSF)
Football competitions in Liberia |
https://en.wikipedia.org/wiki/1973%20Liberian%20Premier%20League | Statistics of Liberian Premier League in season 1973.
Overview
Mighty Barrolle won the championship.
References
Liberia - List of final tables (RSSSF)
Football competitions in Liberia |
https://en.wikipedia.org/wiki/1974%20Liberian%20Premier%20League | Statistics of Liberian Premier League in season 1974.
Overview
Mighty Barrolle won the championship.
References
Liberia - List of final tables (RSSSF)
Football competitions in Liberia |
https://en.wikipedia.org/wiki/1976%20Liberian%20Premier%20League | Statistics of Liberian Premier League in season 1976.
Overview
Saint Joseph Warriors won the championship.
References
Liberia - List of final tables (RSSSF)
Football competitions in Liberia |
https://en.wikipedia.org/wiki/1978%20Liberian%20Premier%20League | Statistics of Liberian Premier League in season 1978.
Overview
Saint Joseph Warriors won the championship.
References
Liberia - List of final tables (RSSSF)
Football competitions in Liberia |
https://en.wikipedia.org/wiki/1979%20Liberian%20Premier%20League | Statistics of Liberian Premier League in season 1979.
Overview
Saint Joseph Warriors won the championship.
References
Liberia - List of final tables (RSSSF)
Football competitions in Liberia |
https://en.wikipedia.org/wiki/1980%20Liberian%20Premier%20League | Statistics of Liberian Premier League in season 1980.
Overview
Invincible Eleven won the championship.
References
Liberia - List of final tables (RSSSF)
Football competitions in Liberia |
https://en.wikipedia.org/wiki/1981%20Liberian%20Premier%20League | Statistics of Liberian Premier League in season 1981.
Overview
Invincible Eleven won the championship.
References
Liberia - List of final tables (RSSSF)
Football competitions in Liberia |
https://en.wikipedia.org/wiki/1983%20Liberian%20Premier%20League | Statistics of Liberian Premier League in season 1983.
Overview
Invincible Eleven won the championship.
References
Liberia - List of final tables (RSSSF)
Football competitions in Liberia |
https://en.wikipedia.org/wiki/1984%20Liberian%20Premier%20League | Statistics of Liberian Premier League in season 1984.
Overview
It was contested by 8 teams, and Invincible Eleven won the championship.
References
Liberia - List of final tables (RSSSF)
Football competitions in Liberia |
https://en.wikipedia.org/wiki/1985%20Liberian%20Premier%20League | Statistics of Liberian Premier League in season 1985.
Overview
Invincible Eleven won the championship.
References
Liberia - List of final tables (RSSSF)
Football competitions in Liberia |
https://en.wikipedia.org/wiki/1986%20Liberian%20Premier%20League | Statistics of Liberian Premier League in season 1986.
Overview
It was contested by 12 teams, and Mighty Barrolle won the championship.
References
Liberia - List of final tables (RSSSF)
Football competitions in Liberia |
https://en.wikipedia.org/wiki/1987%20Liberian%20Premier%20League | Statistics of Liberian Premier League in season 1987.
Overview
Invincible Eleven won the championship.
References
Liberia - List of final tables (RSSSF)
Football competitions in Liberia |
https://en.wikipedia.org/wiki/1988%20Liberian%20Premier%20League | Statistics of Liberian Premier League in season 1988.
Overview
Mighty Barrolle won the championship.
References
Liberia - List of final tables (RSSSF)
Football competitions in Liberia |
https://en.wikipedia.org/wiki/1989%20Liberian%20Premier%20League | Statistics of Liberian Premier League in season 1989.
Overview
The 1989 Liberian Premier League comprised 16 teams, and Mighty Barrolle won the championship.
League standings
References
Liberia - List of final tables (RSSSF)
Football competitions in Liberia |
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