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https://en.wikipedia.org/wiki/Kleene%E2%80%93Brouwer%20order
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In descriptive set theory, the Kleene–Brouwer order or Lusin–Sierpiński order is a linear order on finite sequences over some linearly ordered set , that differs from the more commonly used lexicographic order in how it handles the case when one sequence is a prefix of the other. In the Kleene–Brouwer order, the prefix is later than the longer sequence containing it, rather than earlier.
The Kleene–Brouwer order generalizes the notion of a postorder traversal from finite trees to trees that are not necessarily finite. For trees over a well-ordered set, the Kleene–Brouwer order is itself a well-ordering if and only if the tree has no infinite branch. It is named after Stephen Cole Kleene, Luitzen Egbertus Jan Brouwer, Nikolai Luzin, and Wacław Sierpiński.
Definition
If and are finite sequences of elements from , we say that when there is an such that either:
and is defined but is undefined (i.e. properly extends ), or
both and are defined, , and .
Here, the notation refers to the prefix of up to but not including .
In simple terms, whenever is a prefix of (i.e. terminates before , and they are equal up to that point) or is to the "left" of on the first place they differ.
Tree interpretation
A tree, in descriptive set theory, is defined as a set of finite sequences that is closed under prefix operations. The parent in the tree of any sequence is the shorter sequence formed by removing its final element. Thus, any set of finite sequences can be augmented to form a tree, and the Kleene–Brouwer order is a natural ordering that may be given to this tree. It is a generalization to potentially-infinite trees of the postorder traversal of a finite tree: at every node of the tree, the child subtrees are given their left to right ordering, and the node itself comes after all its children. The fact that the Kleene–Brouwer order is a linear ordering (that is, that it is transitive as well as being total) follows immediately from this, as any three sequences on which transitivity is to be tested form (with their prefixes) a finite tree on which the Kleene–Brouwer order coincides with the postorder.
The significance of the Kleene–Brouwer ordering comes from the fact that if is well-ordered, then a tree over is well-founded (having no infinitely long branches) if and only if the Kleene–Brouwer ordering is a well-ordering of the elements of the tree.
Recursion theory
In recursion theory, the Kleene–Brouwer order may be applied to the computation trees of implementations of total recursive functionals. A computation tree is well-founded if and only if the computation performed by it is total recursive. Each state in a computation tree may be assigned an ordinal number , the supremum of the ordinal numbers where ranges over the children of in the tree. In this way, the total recursive functionals themselves can be classified into a hierarchy, according to the minimum value of the ordinal at the root of a computation tree, minimized
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https://en.wikipedia.org/wiki/List%20of%20J%C3%BAbilo%20Iwata%20records%20and%20statistics
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This article contains records and statistics for the Japanese professional football club, Júbilo Iwata.
J. League
Domestic cup competitions
International Competitions
Top scorers by season
Previous record as Yamaha Motor Corporation's team
1Not chosen for new J. League, so a de facto relegation.
References
Júbilo Iwata
Jubilo Iwata
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https://en.wikipedia.org/wiki/Wombling
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In statistics, Wombling is any of a number of techniques used for identifying zones of rapid change, typically in some quantity as it varies across some geographical or Euclidean space. It is named for statistician William H. Womble.
The technique may be applied to gene frequency in a population of organisms, and to evolution of language.
References
William H. Womble 1951. "Differential Systematics". Science vol 114, No. 2961, p315–322.
Fitzpatrick M.C., Preisser E.L., Porter A., Elkinton J., Waller L.A., Carlin B.P. and Ellison A.E. (2010) "Ecological boundary detection using Bayesian areal wombling", Ecology 91:3448–3455
Liang, S., Banerjee, S. and Carlin, B.P. (2009) "Bayesian Wombling for Spatial Point Processes", Biometrics, 65 (11), 1243–1253
Ma, H. and Carlin, B.P. (2007) "Bayesian Multivariate Areal Wombling for Multiple Disease Boundary Analysis", Bayesian Analysis, 2 (2), 281–302
Banerjee, S. and Gelfand, A.E. (2006) "Bayesian Wombling: Curvilinear Gradient Assessment Under Spatial Process Models", Journal of the American Statistical Association, 101(476), 1487–1501.
Quick, H., Banerjee, S. and Carlin, B.P. (2015). "Bayesian Modeling and Analysis for Gradients in Spatiotemporal Processes" Biometrics, 71, 575–584.
Quick, H., Banerjee, S. and Carlin, B.P. (2013). "Modeling temporal gradients in regionally aggregated California asthma hospitalization data" Annals of Applied Statistics, 7(1), 154–176.
Halder, A., Banerjee, S. and Dey, D. K. "Bayesian modeling with spatial curvature processes." Journal of the American Statistical Association (2023): 1-13. Available Software: Git
Change detection
Spatial analysis
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https://en.wikipedia.org/wiki/National%20Centre%20for%20Excellence%20in%20the%20Teaching%20of%20Mathematics
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The National Centre for Excellence in the Teaching of Mathematics (NCETM) is an institution set up in the wake of the Smith Report to improve mathematics teaching in England.
It provides strategic leadership for mathematics-specific CPD and aims to raise the professional status of all those engaged in the teaching of mathematics so that the mathematical potential of learners will be fully realised.
Structure
Its Director until March 2013 was Dame Celia Hoyles, Professor of Mathematics Education at the Institute of Education, University of London and former chief adviser on mathematics education for the government. She was succeeded by the current Director, Charlie Stripp.
An innovative NCETM development is the MatheMaPedia project, masterminded by John Mason, which is a "maths teaching wiki".
Initially headquartered in London, it is headquartered in the south of Sheffield city centre; it is the headquarters of Tribal Education.
It is run by Mathematics in Education and Industry (MEI) and Tribal Education.
Online discussions
Special online events have included the world’s first online discussion of proof.
See also
Centre for Industry Education Collaboration and National Centre for Computing Education, also at York
Count On - maths education initiative
Mathematics education in the United Kingdom
International Congress on Mathematical Education
References
Department for Education
Education in Sheffield
Educational organisations based in the United Kingdom
Mathematics education in the United Kingdom
Mathematics education reform
Mathematics organizations
Mathematics websites
Organisations based in Sheffield
Organizations established in 1996
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https://en.wikipedia.org/wiki/Statistics%20Indonesia
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Statistics Indonesia (), is a non-departmental government institute of Indonesia that is responsible for conducting statistical surveys. Its main customer is the government, but statistical data is also available to the public. Annual surveys cover areas including national and provincial socio-economics, manufacturing establishments, population and the labour force.
Established in 1960 as the Central Bureau of Statistics (), the institute is directly responsible to the president of Indonesia. Its functions include providing data to other governmental institutes as well as to the public and conducting statistical surveys to publish periodic statistics on the economy, social change and development. Statistics Indonesia also assists data processing divisions in other public offices to support and to promote standard statistical methods.
History
In February 1920, the Director of Agriculture and Trade () of the government of the Dutch East Indies, established the Statistical Office based in Bogor. In March 1923, the Commission for Statistics was formed to represent members of each department. It was tasked with planning actions to ensure the achievement of unity in statistical activities in Indonesia. On 24 September 1924, the name of the institution was changed to (CKS) or the Central Statistics Office, and the institution was moved to Jakarta. In June 1942, the Government of Japan reactivated statistical activities focused on meeting the needs of war or military. CKS was renamed . On 26 September 1960 the government of Indonesia enacted Law No. 7 of 1960 on Statistics replacing . Law Number 16 of 1997 concerning Statistics replaced previous laws, and based on it, the Central Bureau on Statistics became the Central Statistics Agency.
Census
Based on Republic of Indonesia Laws No. 6 of 1960 on the Census, Statistics Indonesia organizes a census every 10 years
Demography
A demographic census has been organized every year ending in "0" after 1961 namely in 1970, 1980, 1990, 2000, 2010 and 2020.
Economy
An economic census is held every year ending in "6", namely 1986, 1996, 2006, and 2016.
Agriculture
An agricultural census is held every year ending in "3", namely 1963, 1973, 1983, 1993, 2003, and 2013.
See also
List of national and international statistical services
References
External links
Official site
Indonesia
Government of Indonesia
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https://en.wikipedia.org/wiki/Steiner%27s%20calculus%20problem
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Steiner's problem, asked and answered by , is the problem of finding the maximum of the function
It is named after Jakob Steiner.
The maximum is at , where e denotes the base of the natural logarithm. One can determine that by solving the equivalent problem of maximizing
Applying the first derivative test, the derivative of is
so is positive for and negative for , which implies that – and therefore – is increasing for and decreasing for Thus, is the unique global maximum of
References
Functions and mappings
Mathematical optimization
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https://en.wikipedia.org/wiki/Flexible%20polyhedron
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In geometry, a flexible polyhedron is a polyhedral surface without any boundary edges, whose shape can be continuously changed while keeping the shapes of all of its faces unchanged. The Cauchy rigidity theorem shows that in dimension 3 such a polyhedron cannot be convex (this is also true in higher dimensions).
The first examples of flexible polyhedra, now called Bricard octahedra, were discovered by . They are self-intersecting surfaces isometric to an octahedron. The first example of a flexible non-self-intersecting surface in , the Connelly sphere, was discovered by . Steffen's polyhedron is another non-self-intersecting flexible polyhedron derived from Bricard's octahedra.
Bellows conjecture
In the late 1970s Connelly and D. Sullivan formulated the bellows conjecture stating that the volume of a flexible polyhedron is invariant under flexing. This conjecture was proved for polyhedra homeomorphic to a sphere by
using elimination theory, and then proved for general orientable 2-dimensional polyhedral surfaces by . The proof extends Piero della Francesca's formula for the volume of a tetrahedron to a formula for the volume of any polyhedron. The extended formula shows that the volume must be a root of a polynomial whose coefficients depend only on the lengths of the polyhedron's edges. Since the edge lengths cannot change as the polyhedron flexes, the volume must remain at one of the finitely many roots of the polynomial, rather than changing continuously.
Scissor congruence
Connelly conjectured that the Dehn invariant of a flexible polyhedron is invariant under flexing. This was known as the strong bellows conjecture or (after it was proven in 2018) the strong bellows theorem. Because all configurations of a flexible polyhedron have both the same volume and the same Dehn invariant, they are scissors congruent to each other, meaning that for any two of these configurations it is possible to dissect one of them into polyhedral pieces that can be reassembled to form the other. The total mean curvature of a flexible polyhedron, defined as the sum of the products of edge lengths with exterior dihedral angles, is a function of the Dehn invariant that is also known to stay constant while a polyhedron flexes.
Generalizations
Flexible 4-polytopes in 4-dimensional Euclidean space and 3-dimensional hyperbolic space were studied by . In dimensions , flexible polytopes were constructed by .
See also
Flexagon
Rigid origami
References
Notes
Primary sources
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.
.
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Secondary sources
.
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.
.
External links
Nonconvex polyhedra
Mathematics of rigidity
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https://en.wikipedia.org/wiki/Robert%20Connelly
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Robert Connelly (born July 15, 1942) is a mathematician specializing in discrete geometry and rigidity theory. Connelly received his Ph.D. from University of Michigan in 1969. He is currently a professor at Cornell University.
Connelly is best known for discovering embedded flexible polyhedra. One such polyhedron is in the National Museum of American History. His recent interests include tensegrities and the carpenter's rule problem. In 2012 he became a fellow of the American Mathematical Society.
Asteroid 4816 Connelly, discovered by Edward Bowell at Lowell Observatory 1981, was named after Robert Connelly. The official was published by the Minor Planet Center on 18 February 1992 ().
Author
Connelly has authored or co-authored several articles on mathematics, including Conjectures and open questions in rigidity; A flexible sphere; and A counterexample to the rigidity conjecture for polyhedra.
References
External links
Personal home page
Cornell Mathematics Department web page (with a picture)
Why Things Don't Fall Down – A Lecture About Tensegrity by Robert Connelly
Mathematical Treasures of the Smithsonian Institution – Allyn Jackson, AMS Notices, vol. 46, no. 5 (May 1999), 528–534.
20th-century American mathematicians
21st-century American mathematicians
1942 births
Living people
Geometers
Differential geometers
Researchers in geometric algorithms
University of Michigan alumni
Cornell University faculty
Fellows of the American Mathematical Society
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https://en.wikipedia.org/wiki/List%20of%20Thames%20Ironworks%20F.C.%20records%20and%20statistics
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This is a list of Southern League, FA Cup and Test Match appearances made, and goals scored, by Thames Ironworks F.C. players from 1895 until 1900. This list does not include London League, friendly or reserve statistics.''
Player records
Record victories
Southern League Division One:
Home: 4–0 v Chatham, 18 September 1899
Away: 3–0 v Sheppey United, 20 January 1900
Southern League Division Two:
Home: 10–0 Maidenhead, 15 April 1899
Away: 4–0 Maidenhead F.C., 31 December 1898
London League:
Home: 7–3 v Bromley, 15 January 1898
Away: 5–1 v Bromley F.C., 19 March 1898
FA Cup:
Home: 6–1 v Royal Engineers, 23 September 1899
Away: 7–0 v Dartford, 28 October 1899
Friendly:
Home: 7–1 v Lewisham St. Marys F.C., 28 December 1895
Away: 8–0 v Manor Park F.C., 28 September 1895
Top appearance makers
Most Appearances:
Tommy Moore (61) 1898–1900
Most Appearances in a Season:
Roddy McEachrane (36) 1899–1900
Kenny McKay (36) 1899–1900
Most FA Cup Appearances:
Charlie Dove (10) 1895–1900
Tommy Moore (10) 1898–1900
Top goalscorers
Top Scorer in a Season:
Bill Joyce (18) 1899–1900
Top League Scorer in a Season:
David Lloyd (12) Southern League Division Two 1898–99
Most Goals in One Match:
Henderson (4) v Uxbridge F.C. (h) 18 February 1899
Patrick Leonard (4) v Maidenhead (h) 15 April 1899
References
Thames Ironworks F.C. records and statistics
Thames Ironworks
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https://en.wikipedia.org/wiki/National%20Compensation%20Survey
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The National Compensation Survey (NCS) is produced by the United States Department of Labor's Bureau of Labor Statistics (BLS), measuring occupational earnings, compensation costs, benefit incidence rates, and plan provisions. It is used to adjust the federal wage schedule for all federal employees. Detailed occupational earnings are available for both metropolitan and non-metropolitan areas, broad geographic regions, and on a national basis. The NCS' Employment Cost Index measures changes in labor costs. The average costs of employee compensation per hour worked is presented in the Employer Costs for Employee Compensation (ECEC).
Data
The National Compensation Survey uses data provided by companies, organizations, and government agencies that voluntarily report employee wages, benefit incidence rates, and the costs of employee compensation. All data provided to the BLS is strictly confidential and is used for statistical purposes in accordance with the Confidential Information Protection and Statistical Efficiency Act.
Collection
The National Compensation Survey's data is collected by field economists within the BLS who randomly sample firms and report on the compensation of one to eight occupations within the business over time. Some respondents are also asked to report on the provisions, participation, and costs of benefits offered to employees. Occupations are benchmarked using a four-factor leveling system developed by the BLS to assist in comparing compensation data across industries, professions, and geographic boundaries.
In August 2023, the BLS announced it would stop collecting data on workers' compensation, which provides medical care and wage replacement in exchange for the employee's right to sue their employer for negligence. While this benefit is required by most states, workers' compensation only costs employers an average of $0.46 per hour of an employee's work, representing only 1% of total compensation and 3% of total benefits. The NCS data had been criticized for deviation from reporting by the National Academy of Social Insurance (NASI), potentially due to poor response rates on this section of the NCS. Additionally, the NCS surveyed corporate establishments for their total workers' compensation costs, obscuring differences by occupation.
Publications/Statistics
Employment Cost Index
Employer Costs for Employee Compensation
Employee Benefits in Private Industry
Occupational Pay Relatives
References
External links
National Compensation Survey
Labour economics indices
Reports of the Bureau of Labor Statistics
Workers' compensation
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https://en.wikipedia.org/wiki/Utilization%20distribution
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A utilization distribution is a probability distribution giving the probability density that an animal is found at a given point in space. It is estimated from data sampling the location of an individual or individuals in space over a period of time using, for example, telemetry or GPS based methods.
Estimation of utilization distribution was traditionally based on histograms but newer nonparametric methods based on Fourier transformations, kernel density and local convex hull methods have been developed.
The typical application for this distribution is estimating the home range distribution of animals. According to Lichti & Swihart (2011), kernel density methods provided, in many cases, less biased home-range area estimates compared to convex hull methods.
See also
Home range
Local convex hull
References
Types of probability distributions
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https://en.wikipedia.org/wiki/Bernstein%20inequalities%20%28probability%20theory%29
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In probability theory, Bernstein inequalities give bounds on the probability that the sum of random variables deviates from its mean. In the simplest case, let X1, ..., Xn be independent Bernoulli random variables taking values +1 and −1 with probability 1/2 (this distribution is also known as the Rademacher distribution), then for every positive ,
Bernstein inequalities were proven and published by Sergei Bernstein in the 1920s and 1930s. Later, these inequalities were rediscovered several times in various forms. Thus, special cases of the Bernstein inequalities are also known as the Chernoff bound, Hoeffding's inequality and Azuma's inequality.
The martingale case of the Bernstein inequality
is known as Freedman's inequality and it's refinement
is known as Hoeffding's inequality.
Some of the inequalities
1. Let be independent zero-mean random variables. Suppose that almost surely, for all Then, for all positive ,
2. Let be independent zero-mean random variables. Suppose that for some positive real and every integer ,
Then
3. Let be independent zero-mean random variables. Suppose that
for all integer Denote
Then,
4. Bernstein also proved generalizations of the inequalities above to weakly dependent random variables. For example, inequality (2) can be extended as follows. Let be possibly non-independent random variables. Suppose that for all integers ,
Then
More general results for martingales can be found in Fan et al. (2015).
Proofs
The proofs are based on an application of Markov's inequality to the random variable
for a suitable choice of the parameter .
Generalizations
The Bernstein inequality can be generalized to Gaussian random matrices. Let be a scalar where is a complex Hermitian matrix and is complex vector of size . The vector is a Gaussian vector of size . Then for any , we have
where is the vectorization operation and where is the largest eigenvalue of . The proof is detailed here. Another similar inequality is formulated as
where .
See also
Concentration inequality - a summary of tail-bounds on random variables.
Hoeffding's inequality
References
(according to: S.N.Bernstein, Collected Works, Nauka, 1964)
A modern translation of some of these results can also be found in
Probabilistic inequalities
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https://en.wikipedia.org/wiki/Paul%20St%C3%A4ckel
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Paul Gustav Samuel Stäckel (20 August 1862, Berlin – 12 December 1919, Heidelberg) was a German mathematician, active in the areas of differential geometry, number theory, and non-Euclidean geometry. In the area of prime number theory, he used the term twin prime (in its German form, "Primzahlzwilling") for the first time.
After passing his Abitur in 1880 he studied mathematics and physics at the University of Berlin, but also listened to lectures on philosophy, psychology, education, and history. A year later he qualified for teaching in higher education and then taught at Gymnasien in Berlin. In 1885 he wrote his doctoral dissertation under Leopold Kronecker and Karl Weierstraß. In 1891 he completed his Habilitation at the University of Halle. Later he worked as a professor at the University of Königsberg (außerordentlicher Professor from 1895 to 1897), the University of Kiel (ordentlicher Professor, 1897 to 1905), University of Hannover (1905 to 1908), the Karlsruhe Institute of Technology (1908 to 1913), and the University of Heidelberg (1913 to 1919).
Stäckel worked on both mathematics and the history of mathematics. He edited the letters exchanged between Carl Friedrich Gauss and Wolfgang Bolyai, made contributions to editions of the collected works of Euler and Gauss (for whose works he wrote Gauss als Geometer), and edited the Geometrischen Untersuchungen by Wolfgang and Johann Bolyai (published in 1913). Additionally he translated works of Jacob Bernoulli, Johann Bernoulli, Augustin Louis Cauchy, Leonhard Euler, Joseph-Louis Lagrange, Adrien-Marie Legendre, Carl Gustav Jacobi from French and Latin into German for the series Ostwalds Klassiker der exakten Wissenschaften.
In 1904 he was an invited speaker at the International Congress of Mathematicians in Heidelberg. In 1905 he was the president of the Deutsche Mathematiker-Vereinigung. His doctoral students include Paul Riebesell.
Works
Über die Bewegung eines Punktes auf einer Fläche, 1885, Dissertation
Die Integration der Hamilton-Jacobischen Differentialgleichung mittelst Separation der Variablen, 1891, Habilitation
Elementare Dynamik der mathematischen Wissenschaft in: Encyclopädie der mathematischen Wissenschaft IV,1 (1908)
several papers in the Mathematischen Annalen (from 1890 to 1909)
numerous papers in "Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse" from 1896 to 1917.
with Friedrich Engel: Theorie der Parallellinien von Euklid bis auf Gauss, 1895
References
External links
http://geb.uni-giessen.de/geb/volltexte/2005/2127/ Briefnachlass von Paul Stäckel im Nachlass von Friedrich Engel
1862 births
1919 deaths
19th-century German mathematicians
20th-century German mathematicians
German historians of mathematics
Differential geometers
Number theorists
University of Halle alumni
Academic staff of the University of Kiel
Academic staff of the University of Hanover
Academic staff of Heidelberg University
Academic s
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https://en.wikipedia.org/wiki/Dariush%20Yazdani
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Dariush Yazdani (; born June 2, 1977) is an Iranian former footballer who played as a midfielder.
Career statistics
Club
Honors
Bargh Shiraz
Hazfi Cup: 1996–97
Esteghlal
Iranian Football League: 1997–98
Hazfi Cup: 1997–98
Bayer Leverkusen
Bundesliga runner-up: 1999–2000
DFB-Pokal runner-up: 1999–2000
Saipa
Iranian League: 2006–07
Iran
Asian Games: 1998
AFC Asian Cup bronze medal: 1996
References
1977 births
Living people
Iranian men's footballers
Iran men's international footballers
Men's association football midfielders
Bargh Shiraz F.C. players
Esteghlal F.C. players
Bayer 04 Leverkusen players
Bayer 04 Leverkusen II players
Bundesliga players
emirates Club players
Payam Khorasan F.C. players
1996 AFC Asian Cup players
2000 AFC Asian Cup players
Expatriate men's footballers in Germany
Expatriate men's footballers in Belgium
Expatriate men's soccer players in the United States
Iranian expatriate men's footballers
Iranian expatriate football managers
R. Charleroi S.C. players
Pegah F.C. players
Saipa F.C. players
Orange County SC players
Belgian Pro League players
USL Championship players
Paykan F.C. players
Footballers from Shiraz
Asian Games gold medalists for Iran
Asian Games medalists in football
Footballers at the 1998 Asian Games
Orange County SC coaches
Medalists at the 1998 Asian Games
Iranian expatriate sportspeople in Germany
Iranian expatriate sportspeople in Belgium
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https://en.wikipedia.org/wiki/Radovan%20Krivokapi%C4%87
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Radovan Krivokapić (Serbian Cyrillic: Радован Кривокапић; born 14 August 1978) is a Serbian former professional footballer who played as an attacking midfielder.
Managerial statistics
Honours
Red Star Belgrade
First League of Serbia and Montenegro: 2003–04, 2005–06
Serbia and Montenegro Cup: 2003–04, 2005–06
References
External links
Men's association football midfielders
Cypriot First Division players
Enosis Neon Paralimni FC players
Expatriate men's footballers in Cyprus
Expatriate men's footballers in Greece
First League of Serbia and Montenegro players
FK TSC players
OFK Bečej 1918 players
FK Radnički 1923 players
FK Vojvodina players
Football League (Greece) players
Iraklis F.C. (Thessaloniki) players
People from Bačka Topola
Footballers from North Bačka District
Red Star Belgrade footballers
Serbia and Montenegro men's international footballers
Serbia and Montenegro men's under-21 international footballers
Serbian expatriate men's footballers
Serbian expatriate sportspeople in Cyprus
Serbian expatriate sportspeople in Greece
Serbian men's footballers
Serbian people of Montenegrin descent
Serbian SuperLiga players
Super League Greece players
Veria F.C. players
1978 births
Living people
Serbia and Montenegro men's footballers
Serbian football managers
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https://en.wikipedia.org/wiki/Gaussian%20isoperimetric%20inequality
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In mathematics, the Gaussian isoperimetric inequality, proved by Boris Tsirelson and Vladimir Sudakov, and later independently by Christer Borell, states that among all sets of given Gaussian measure in the n-dimensional Euclidean space, half-spaces have the minimal Gaussian boundary measure.
Mathematical formulation
Let be a measurable subset of endowed with the standard Gaussian measure with the density . Denote by
the ε-extension of A. Then the Gaussian isoperimetric inequality states that
where
Proofs and generalizations
The original proofs by Sudakov, Tsirelson and Borell were based on Paul Lévy's spherical isoperimetric inequality.
Sergey Bobkov proved Bobkov's inequality, a functional generalization of the Gaussian isoperimetric inequality, proved from a certain "two point analytic inequality". Bakry and Ledoux gave another proof of Bobkov's functional inequality based on the semigroup techniques which works in a much more abstract setting. Later Barthe and Maurey gave yet another proof using the Brownian motion.
The Gaussian isoperimetric inequality also follows from Ehrhard's inequality.
See also
Concentration of measure
Borell–TIS inequality
References
Probabilistic inequalities
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https://en.wikipedia.org/wiki/Third%20derivative
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In calculus, a branch of mathematics, the third derivative or third-order derivative is the rate at which the second derivative, or the rate of change of the rate of change, is changing. The third derivative of a function can be denoted by
Other notations can be used, but the above are the most common.
Mathematical definitions
Let . Then and . Therefore, the third derivative of f is, in this case,
or, using Leibniz notation,
Now for a more general definition. Let f be any function of x such that f ′′ is differentiable. Then the third derivative of f is given by
The third derivative is the rate at which the second derivative (f′′(x)) is changing.
Applications in geometry
In differential geometry, the torsion of a curve — a fundamental property of curves in three dimensions — is computed using third derivatives of coordinate functions (or the position vector) describing the curve.
Applications in physics
In physics, particularly kinematics, jerk is defined as the third derivative of the position function of an object. It is, essentially, the rate at which acceleration changes. In mathematical terms:
where j(t) is the jerk function with respect to time, and r(t) is the position function of the object with respect to time.
Economic examples
When campaigning for a second term in office, U.S. President Richard Nixon announced that the rate of increase of inflation was decreasing, which has been noted as "the first time a sitting president used the third derivative to advance his case for reelection." Since inflation is itself a derivative—the rate at which the purchasing power of money decreases—then the rate of increase of inflation is the derivative of inflation, opposite in sign to the second time derivative of the purchasing power of money. Stating that a function is decreasing is equivalent to stating that its derivative is negative, so Nixon's statement is that the second derivative of inflation is negative, and so the third derivative of purchasing power is positive.
Nixon's statement allowed for the rate of inflation to increase, however, so his statement was not as indicative of stable prices as it sounds.
See also
Aberrancy (geometry)
Derivative (mathematics)
Second derivative
References
Differential calculus
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https://en.wikipedia.org/wiki/Viktor%20Wagner
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Viktor Vladimirovich Wagner, also Vagner () (4 November 1908 – 15 August 1981) was a Russian mathematician, best known for his work in differential geometry and on semigroups.
Wagner was born in Saratov and studied at Moscow State University, where Veniamin Kagan was his advisor. He became the first geometry chair at Saratov State University. He received the Lobachevsky Medal in 1937.
Wagner was also awarded "the Order of Lenin, the Order of the Red Banner, and the title of Honoured Scientist RSFSR. Moreover, he was also accorded that rarest of privileges in the USSR: permission to travel abroad."
Wagner is credited with noting that the collection of partial transformations on a set X forms a semigroup which is a subsemigroup of the semigroup of binary relations on the same set X, where the semigroup operation is composition of relations. "This simple unifying observation, which is nevertheless an important psychological hurdle, is attributed by Schein (1986) to V.V. Wagner."
See also
Inverse semigroup
Heap
References
External links
Wagner's Biography – in Russian
Soviet mathematicians
Moscow State University alumni
Differential geometers
Algebraists
20th-century Russian mathematicians
1908 births
1981 deaths
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https://en.wikipedia.org/wiki/Moscow%20Mathematical%20Society
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The Moscow Mathematical Society (MMS) is a society of Moscow mathematicians aimed at the development of mathematics in Russia. It was created in 1864, and Victor Vassiliev is the current president.
History
The first meeting of the society was . Nikolai Brashman was the first president of MMO.
The Moscow Mathematical Society was first created in 1810 by extended members of the Muraviev family, but it closed down the year after. In the early 1860s, Nikolai Brashman and August Davidov relaunched the Moscow Mathematical Society at the Moscow University and organized its first meeting on 15 September 1864. The stated goal was «mutual cooperation in the study of the mathematical sciences».
Nikolai Brashman was president, and August Davidov vice-president. 14 members joined the Society, with Pafnuty Chebyshev among them. Each member was in charge of a research project, and the bunch met monthly to share and progress on their projects. The outcome was so valuable that the Society decided in April 1865 to publish its works. The first edition of the journal Matematicheskii Sbornik edited by the Society was released in October 1866.
By 1913, the Moscow Mathematical Society had 112 members. The publication of the Matematicheskii Sbornik ceased from 1919 to 1924.
Former presidents
Nikolai Brashman (1864–1866)
August Davidov (1866–1886)
Vasily Jakovlevich Zinger (1886–1891)
Nikolai Bugaev (1891–1903)
Pavel Alekseevich Nekrasov (1903–1905)
Nikolai Zhukovsky (1905–1921)
Boleslav Mlodzeevskii (1921–1923)
Dmitri Egorov (1923–1930)
Ernst Kolman (1930–1932)
Pavel Alexandrov (1932–1964)
Andrey Kolmogorov (1964–1966, 1973–1985)
Israel Gelfand (1966–1970)
Igor Shafarevich (1970–1973)
Sergei Novikov (1985–1996)
Vladimir Arnold (1996–2010)
Annual Prize
Every year, the Moscow Mathematical Society awards a young mathematician for his/her outstanding work in the field.
1951 : Evgenii Landis
1956 : Friedrich Karpelevich
1967 : Anatoly Stepin
1986 : Victor Vassiliev
2013 : Eugene Gorsky
2015 : Leonid Petrov
See also
List of Mathematical Societies
Matematicheskii Sbornik
References
Further reading
Smilka Zdravkovska, Peter Duren (Herausgeber), The golden ages of Moscow Mathematics, American Mathematical Society, 2007
A.P. Yushkevich, History of mathematics in Russia until 1917, Moscow, 1968
External links
Official website
Mathematical societies
Learned societies of Russia
1864 establishments in the Russian Empire
Organizations established in 1864
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https://en.wikipedia.org/wiki/Homogeneously%20Suslin%20set
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In descriptive set theory, a set is said to be homogeneously Suslin if it is the projection of a homogeneous tree. is said to be -homogeneously Suslin if it is the projection of a -homogeneous tree.
If is a set and is a measurable cardinal, then is -homogeneously Suslin. This result is important in the proof that the existence of a measurable cardinal implies that sets are determined.
See also
Projective determinacy
References
Descriptive set theory
Determinacy
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https://en.wikipedia.org/wiki/Homogeneous%20tree
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In descriptive set theory, a tree over a product set is said to be homogeneous if there is a system of measures such that the following conditions hold:
is a countably-additive measure on .
The measures are in some sense compatible under restriction of sequences: if , then .
If is in the projection of , the ultrapower by is wellfounded.
An equivalent definition is produced when the final condition is replaced with the following:
There are such that if is in the projection of and , then there is such that . This condition can be thought of as a sort of countable completeness condition on the system of measures.
is said to be -homogeneous if each is -complete.
Homogeneous trees are involved in Martin and Steel's proof of projective determinacy.
References
Descriptive set theory
Determinacy
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https://en.wikipedia.org/wiki/Tom%20Snijders
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Tom A. B. Snijders (born 26 September 1949) is professor of Statistics in the Social Sciences at Nuffield College, Oxford, one of the constituent colleges of the University of Oxford (since 1 October 2006). He is also professor of Methodology at the University of Groningen, a position he has held for more than twenty years.
Career
Tom Snijders was born in Tilburg, a son of Jan Snijders, professor of Psychology at Groningen University from 1949 to 1980, and Nan Snijders-Oomen, an internationally well-known child psychologist (author of the SON nonverbal intelligence test, 1943, 1978, 1991).
Snijders grew up in the province of Groningen, in the northern part of the Netherlands. He was awarded a cum laude Ph.D. in Mathematics, specializing in mathematical statistics, for his thesis Asymptotic optimality theory for testing problems with restricted alternatives. Since 1985 he worked as a professor of various forms of mathematics serving social science. He was part-time professor of Mathematical Sociology at Utrecht University from 1989-1992 and is an honorary senior fellow of the University of Melbourne in Australia. Stockholm University made him an honorary doctor in 2005, as did Paris Dauphine University in 2011. In 2007 Snijders became correspondent of the Royal Netherlands Academy of Arts and Sciences. In 2008 Snijders was awarded the Order of Knight of the Netherlands Lion.
Many of his former Ph.D. students and postdocs have found academic positions and are at the forefront of their terrain, including Roel Bosker, Marijtje van Duijn, Roger Leenders, Chris Snijders, Albertine J. (Tineke) Oldehinkel, Rafael Wittek, René Veenstra, Marcel van Assen, Christian Steglich, Miranda Lubbers, Per Block, and Nynke Niezink.
Social networks
Snijders is a prominent researcher in the field of statistic methods in behavioural and social sciences. In 1998, together with others, he created Stocnet, an open software system for the advanced statistical analysis of social networks. Snijders is working especially in social network analysis: statistical methods for analysing social networks and in network evolution, in social science, mathematical sociology and mathematical response theory, and also in multilevel models analysis, about which he wrote a textbook, titled Multilevel Analysis. An Introduction to Basic and Advanced Multilevel Modeling (1999). Apart from this he developed new statistical methods for social science applications, often in combination with the development of computer software to implement these methods. Together with professor Patrick Doreian of the University of Pittsburgh Tom Snijders edited the international scientific journal Social networks. An international journal of structural analysis.
Sources and links
Nuffield College, Sociology Group
University of Groningen, Faculty of Behavioural and Social Sciences
Social Network Analysis page of Tom Snijders
Social networks. An international journal of structural analysis
Website of Tom Snij
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https://en.wikipedia.org/wiki/5-simplex
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In five-dimensional geometry, a 5-simplex is a self-dual regular 5-polytope. It has six vertices, 15 edges, 20 triangle faces, 15 tetrahedral cells, and 6 5-cell facets. It has a dihedral angle of cos−1(), or approximately 78.46°.
The 5-simplex is a solution to the problem: Make 20 equilateral triangles using 15 matchsticks, where each side of every triangle is exactly one matchstick.
Alternate names
It can also be called a hexateron, or hexa-5-tope, as a 6-facetted polytope in 5-dimensions. The name hexateron is derived from hexa- for having six facets and teron (with ter- being a corruption of tetra-) for having four-dimensional facets.
By Jonathan Bowers, a hexateron is given the acronym hix.
As a configuration
This configuration matrix represents the 5-simplex. The rows and columns correspond to vertices, edges, faces, cells and 4-faces. The diagonal numbers say how many of each element occur in the whole 5-simplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element. This self-dual simplex's matrix is identical to its 180 degree rotation.
Regular hexateron cartesian coordinates
The hexateron can be constructed from a 5-cell by adding a 6th vertex such that it is equidistant from all the other vertices of the 5-cell.
The Cartesian coordinates for the vertices of an origin-centered regular hexateron having edge length 2 are:
The vertices of the 5-simplex can be more simply positioned on a hyperplane in 6-space as permutations of (0,0,0,0,0,1) or (0,1,1,1,1,1). These construction can be seen as facets of the 6-orthoplex or rectified 6-cube respectively.
Projected images
Lower symmetry forms
A lower symmetry form is a 5-cell pyramid {3,3,3}∨( ), with [3,3,3] symmetry order 120, constructed as a 5-cell base in a 4-space hyperplane, and an apex point above the hyperplane. The five sides of the pyramid are made of 5-cell cells. These are seen as vertex figures of truncated regular 6-polytopes, like a truncated 6-cube.
Another form is {3,3}∨{ }, with [3,3,2,1] symmetry order 48, the joining of an orthogonal digon and a tetrahedron, orthogonally offset, with all pairs of vertices connected between. Another form is {3}∨{3}, with [3,2,3,1] symmetry order 36, and extended symmetry [[3,2,3],1], order 72. It represents joining of 2 orthogonal triangles, orthogonally offset, with all pairs of vertices connected between.
The form { }∨{ }∨{ } has symmetry [2,2,1,1], order 8, extended by permuting 3 segments as [3[2,2],1] or [4,3,1,1], order 48.
These are seen in the vertex figures of bitruncated and tritruncated regular 6-polytopes, like a bitruncated 6-cube and a tritruncated 6-simplex. The edge labels here represent the types of face along that direction, and thus represent different edge lengths.
The vertex figure of the omnitruncated 5-simplex honeycomb, , is a 5-simplex with a petrie polygon cycle of 5 long edges. It's symmetry is isomophic to dihedral group Dih6 or simple rotation group [6,2
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https://en.wikipedia.org/wiki/Thomas%20A.%20Romberg
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Thomas "Tom" Albert Romberg (born 1932, Burlington, Colorado) was Professor Emeritus of Curriculum and Instruction (mathematics education) at the School of Education, University of Wisconsin–Madison, and former director of the National Center for Improving Student Learning and Achievement in Mathematics and Science, Wisconsin Center for Education Research.
Romberg had a long history of leadership in mathematics curriculum reform. He received his B.S. in mathematics and M.S. in secondary education from the University of Nebraska Omaha. In 1968, he received his Ph.D. in mathematics education from Stanford University.
Romberg joined the faculty at the University of Wisconsin–Madison in 1966. Over the next forty years he became an internationally recognized leader of mathematics education in the U.S. As a researcher, he published 30 books and over 300 research papers. As a professional leader in mathematics education he is best known for two major accomplishments. He was the chair of the "Commission on Standards for School Mathematics" for the National Council of Teachers of Mathematics NCTM that led to the now flourishing standards-based movement in education. Second, from 1987 to 2002, he was Director of the National Center for Research in Mathematical Sciences Education for the U.S. Department of Education. This was the first such national research center devoted to the teaching and learning of mathematics, and became a widely respected research facility. In this role Tom served as an advisor to three U.S. Presidents and their Secretaries of Education and represented the U.S. on several international commissions.
In the 1980s, Romberg was chair of the National Council of Teachers of Mathematics (NCTM) commissions that produced Curriculum and Evaluation Standards NCTM standards for School Mathematics, and the Assessment Standards for School Mathematics. Romberg chaired NCTM's Research Advisory Committee that was responsible for starting the Journal for Research in Mathematics Education and the research pre-sessions at the NCTM's annual meetings.
He was a leader in developing the middle grades mathematics curriculum, Mathematics in Context.
Romberg's research focused on young children's learning of initial mathematical concepts, methods of evaluating students and programs, and integrating research on teaching, curriculum, and student thinking.
For his contributions at the University of Wisconsin–Madison he was awarded an endowed chair, and received a Faculty Distinguished Achievement Award. For his research he received the Review of Research award, the Interpretive Scholarship, and the Professional Service Awards from the American Educational Research Association; was named a Senior Research Fellowship by the Spencer Foundation; and was elected to membership in the National Academy of Education. For his contributions to mathematics education he was awarded the Lifetime Achievement Medal from the National Council of Teachers of Mathematics. He
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https://en.wikipedia.org/wiki/Georgi%20Vladimirov
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Georgi Vladimirov (; born 15 June 1976) is a former Bulgarian footballer who played as a forward and now manager.
Azerbaijan career statistics
References
Bulgarian men's footballers
1976 births
Living people
Men's association football forwards
FC Montana players
PFC Litex Lovech players
Botev Plovdiv players
PFC Slavia Sofia players
PFC Cherno More Varna players
First Professional Football League (Bulgaria) players
Bulgaria men's international footballers
Bulgarian expatriate sportspeople in Azerbaijan
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https://en.wikipedia.org/wiki/Louis%20Denis%20Jules%20Gavarret
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Louis Denis Jules Gavarret, sometimes referred to as Louis Dominique Jules Gavarret (28 January 1809 – 30 August 1890) was a French physician who advocated the use of statistics in medicine.
Life
Gavarret was born in Astaffort, Lot-et-Garonne. He studied at the Ecole Polytechnique in Paris, followed by military service as an artillery officer. In 1833 he resigned his commission and began his studies with Gabriel Andral (1797–1876).
Gavarret is remembered for the systemization and expansion of Pierre Charles Alexandre Louis' (1787-1872) statistical methodology in regards to medicine. Pierre Louis' contention was to make medicine an exact science in diagnosis of a medical condition, and also to refute the "inductive approach" that was prevalent at the time. Gavarret was a major proponent of the statistical method. He emphasized that the process would only work under certain conditions, such as the medical cases must be comparable, and there has to exist enough examples to reach an exact conclusion. Gavarett's precision or "confidence rate" was calculated to be 99.5% or a ratio of 212:1. In essence, the two doctors believed that through knowledge of the aggregate patient data, the disease and treatment would be understood.
In 1840, Gavarret and Gabriel Andral were the first to show that blood composition varied depending on the pathological condition of the subject. Their research demonstrated the value of blood chemistry as a means of confirming diagnoses.
His later work largely dealt with topics in the fields of biophysics and physiology, that included research of acoustic and phonation phenomena.
Writings
Sur les modifications de properties de quelques principes du sang (fibrine, globules, materiaux solides du sérum, et eau) dans les maladies. Ann Chim 1840 (with G. Andral).
Recherches sur la quantité d’acide carbonique exhalé par le poumon dans l’espèce humaine. written with G. Andral 30 pages, 1 pl. Paris, Masson & cie., 1843. Ext. Annales de chimie et de physique.
Principes généraux de statistique médicale, Béchet Jne et Labé, L. Gavarret, Paris, 1840.
Lois générales de l'électricité dynamique (1843)
Physique médicale. De la chaleur produite par les êtres vivants, (1855)
Les phénomènes physiques de la vie, (1869).
References
History of Statistical Thinking in Medicine
19th-century French physicians
1809 births
1890 deaths
People from Lot-et-Garonne
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https://en.wikipedia.org/wiki/Dvoretzky%27s%20theorem
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In mathematics, Dvoretzky's theorem is an important structural theorem about normed vector spaces proved by Aryeh Dvoretzky in the early 1960s, answering a question of Alexander Grothendieck. In essence, it says that every sufficiently high-dimensional normed vector space will have low-dimensional subspaces that are approximately Euclidean. Equivalently, every high-dimensional bounded symmetric convex set has low-dimensional sections that are approximately ellipsoids.
A new proof found by Vitali Milman in the 1970s was one of the starting points for the development of asymptotic geometric analysis (also called asymptotic functional analysis or the local theory of Banach spaces).
Original formulations
For every natural number k ∈ N and every ε > 0 there exists a natural number N(k, ε) ∈ N such that if (X, ‖·‖) is any normed space of dimension N(k, ε), there exists a subspace E ⊂ X of dimension k and a positive definite quadratic form Q on E such that the corresponding Euclidean norm
on E satisfies:
In terms of the multiplicative Banach-Mazur distance d the theorem's conclusion can be formulated as:
where denotes the standard k-dimensional Euclidean space.
Since the unit ball of every normed vector space is a bounded, symmetric, convex set and the unit ball of every Euclidean space is an ellipsoid, the theorem may also be formulated as a statement about ellipsoid sections of convex sets.
Further developments
In 1971, Vitali Milman gave a new proof of Dvoretzky's theorem, making use of the concentration of measure on the sphere to show that a random k-dimensional subspace satisfies the above inequality with probability very close to 1. The proof gives the sharp dependence on k:
where the constant C(ε) only depends on ε.
We can thus state: for every ε > 0 there exists a constant C(ε) > 0 such that for every normed space (X, ‖·‖) of dimension N, there exists a subspace E ⊂ X of dimension
k ≥ C(ε) log N and a Euclidean norm |·| on E such that
More precisely, let SN − 1 denote the unit sphere with respect to some Euclidean structure Q on X, and let σ be the invariant probability measure on SN − 1. Then:
there exists such a subspace E with
For any X one may choose Q so that the term in the brackets will be at most
Here c1 is a universal constant. For given X and ε, the largest possible k is denoted k*(X) and called the Dvoretzky dimension of X.
The dependence on ε was studied by Yehoram Gordon, who showed that k*(X) ≥ c2 ε2 log N. Another proof of this result was given by Gideon Schechtman.
Noga Alon and Vitali Milman showed that the logarithmic bound on the dimension of the subspace in Dvoretzky's theorem can be significantly improved, if one is willing to accept a subspace that is close either to a Euclidean space or to a Chebyshev space. Specifically, for some constant c, every n-dimensional space has a subspace of dimension k ≥ exp(c) that is close either to ℓ or to ℓ.
Important related results were proved by Tadeusz Figiel,
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https://en.wikipedia.org/wiki/Young%E2%80%93Laplace%20equation
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In physics, the Young–Laplace equation () is an algebraic equation that describes the capillary pressure difference sustained across the interface between two static fluids, such as water and air, due to the phenomenon of surface tension or wall tension, although use of the latter is only applicable if assuming that the wall is very thin. The Young–Laplace equation relates the pressure difference to the shape of the surface or wall and it is fundamentally important in the study of static capillary surfaces. It's a statement of normal stress balance for static fluids meeting at an interface, where the interface is treated as a surface (zero thickness):
where is the Laplace pressure, the pressure difference across the fluid interface (the exterior pressure minus the interior pressure), is the surface tension (or wall tension), is the unit normal pointing out of the surface, is the mean curvature, and and are the principal radii of curvature. Note that only normal stress is considered, this is because it has been shown that a static interface is possible only in the absence of tangential stress.
The equation is named after Thomas Young, who developed the qualitative theory of surface tension in 1805, and Pierre-Simon Laplace who completed the mathematical description in the following year. It is sometimes also called the Young–Laplace–Gauss equation, as Carl Friedrich Gauss unified the work of Young and Laplace in 1830, deriving both the differential equation and boundary conditions using Johann Bernoulli's virtual work principles.
Soap films
If the pressure difference is zero, as in a soap film without gravity, the interface will assume the shape of a minimal surface.
Emulsions
The equation also explains the energy required to create an emulsion. To form the small, highly curved droplets of an emulsion, extra energy is required to overcome the large pressure that results from their small radius.
The Laplace pressure, which is greater for smaller droplets, causes the diffusion of molecules out of the smallest droplets in an emulsion and drives emulsion coarsening via Ostwald ripening.
Capillary pressure in a tube
In a sufficiently narrow (i.e., low Bond number) tube of circular cross-section (radius a), the interface between two fluids forms a meniscus that is a portion of the surface of a sphere with radius R. The pressure jump across this surface is related to the radius and the surface tension γ by
This may be shown by writing the Young–Laplace equation in spherical form with a contact angle boundary condition and also a prescribed height boundary condition at, say, the bottom of the meniscus. The solution is a portion of a sphere, and the solution will exist only for the pressure difference shown above. This is significant because there isn't another equation or law to specify the pressure difference; existence of solution for one specific value of the pressure difference prescribes it.
The radius of the sphere will be a function
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https://en.wikipedia.org/wiki/List%20of%20Nagoya%20Grampus%20records%20and%20statistics
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This article contains records and statistics for the Japanese professional football club, Nagoya Grampus.
Key
P = Played
W = Games won
D = Games drawn
L = Games lost
F = Goals for
A = Goals against
Pts = Points
Pos = Final position
J1 = J1 League
J2 = J2 League
F = Final
Group = Group stage
QF = Quarter-finals
QR1 = First Qualifying Round
QR2 = Second Qualifying Round
QR3 = Third Qualifying Round
QR4 = Fourth Qualifying Round
RInt = Intermediate Round
R1 = Round 1
R2 = Round 2
R3 = Round 3
R4 = Round 4
R5 = Round 5
R6 = Round 6
SF = Semi-finals
Seasons
AFC history
References
Nagoya Grampus
Nagoya Grampus
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https://en.wikipedia.org/wiki/ESAIM%3A%20Control%2C%20Optimisation%20and%20Calculus%20of%20Variations
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ESAIM: Control, Optimisation and Calculus of Variations is a scientific journal in the field of applied mathematics.
External links
Mathematics journals
EDP Sciences academic journals
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https://en.wikipedia.org/wiki/Generalized%20assignment%20problem
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In applied mathematics, the maximum generalized assignment problem is a problem in combinatorial optimization. This problem is a generalization of the assignment problem in which both tasks and agents have a size. Moreover, the size of each task might vary from one agent to the other.
This problem in its most general form is as follows: There are a number of agents and a number of tasks. Any agent can be assigned to perform any task, incurring some cost and profit that may vary depending on the agent-task assignment. Moreover, each agent has a budget and the sum of the costs of tasks assigned to it cannot exceed this budget. It is required to find an assignment in which all agents do not exceed their budget and total profit of the assignment is maximized.
In special cases
In the special case in which all the agents' budgets and all tasks' costs are equal to 1, this problem reduces to the assignment problem. When the costs and profits of all tasks do not vary between different agents, this problem reduces to the multiple knapsack problem. If there is a single agent, then, this problem reduces to the knapsack problem.
Explanation of definition
In the following, we have n kinds of items, through and m kinds of bins through . Each bin is associated with a budget . For a bin , each item has a profit and a weight . A solution is an assignment from items to bins. A feasible solution is a solution in which for each bin the total weight of assigned items is at most . The solution's profit is the sum of profits for each item-bin assignment. The goal is to find a maximum profit feasible solution.
Mathematically the generalized assignment problem can be formulated as an integer program:
Complexity
The generalized assignment problem is NP-hard, However, there are linear-programming relaxations which give a -approximation.
Greedy approximation algorithm
For the problem variant in which not every item must be assigned to a bin, there is a family of algorithms for solving the GAP by using a combinatorial translation of any algorithm for the knapsack problem into an approximation algorithm for the GAP.
Using any -approximation algorithm ALG for the knapsack problem, it is possible to construct a ()-approximation for the generalized assignment problem in a greedy manner using a residual profit concept.
The algorithm constructs a schedule in iterations, where during iteration a tentative selection of items to bin is selected.
The selection for bin might change as items might be reselected in a later iteration for other bins.
The residual profit of an item for bin is if is not selected for any other bin or – if is selected for bin .
Formally: We use a vector to indicate the tentative schedule during the algorithm. Specifically, means the item is scheduled on bin and means that item is not scheduled. The residual profit in iteration is denoted by , where if item is not scheduled (i.e. ) and if item is scheduled on bin (i.e. ).
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https://en.wikipedia.org/wiki/Abstract%20model%20checking
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In computer science and in mathematics, abstraction model checking is for systems where an actual representation is too complex in developing the model alone. So, the design undergoes a kind of translation to scaled down "abstract" version.
The set of variables are partitioned into visible and invisible depending on their change of values. The real state space is summarized into a smaller set of the visible ones.
Galois connected
The real and the abstract state spaces are Galois connected. This means that if we take an element from the abstract space, concretize it and abstract the concretized version, the result will be equal to the original. On the other hand, if you pick an element from the real space, abstract it and concretize the abstract version, the final result will be a super set of the original.
That is,
((abstract)) = abstract
((real)) real
See also
References
Model checking
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https://en.wikipedia.org/wiki/Paulo%20Ribenboim
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Paulo Ribenboim (born March 13, 1928) is a Brazilian-Canadian mathematician who specializes in number theory.
Biography
Ribenboim was born into a Jewish family in Recife, Brazil. He received his BSc in mathematics from the University of São Paulo in 1948, and won a fellowship to study with Jean Dieudonné in France at the University of Nancy in the early 1950s, where he became a close friend of Alexander Grothendieck.
He has contributed to the theory of ideals and of valuations.
Ribenboim has authored 246 publications including 13 books. He has been at Queen's University in Kingston, Ontario, since the 1960s, where he remains a professor emeritus.
Jean Dieudonné was one of his doctoral advisors. Andrew Granville, Jan Minac, Karl Dilcher and Aron Simis have been a doctoral students of Ribenboim.
The Ribenboim Prize of the Canadian Number Theory Association is named in his honor.
Personal life
In 1951, Ribenboim married Huguette Demangelle, a French Catholic woman whom he met in France. The couple have two children and five grandchildren, and have lived in Canada since 1962.
Bibliography
Paulo Ribenboim (1964) Functions, Limits, and Continuity , John Wiley & Sons, Inc.
References
External links
The Canadian Number Theory Association Ribenboim Prize
1928 births
Living people
People from Recife
Brazilian Jews
Number theorists
Brazilian emigrants to Canada
Brazilian expatriates in France
Academic staff of Queen's University at Kingston
20th-century Brazilian mathematicians
21st-century Canadian mathematicians
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https://en.wikipedia.org/wiki/Gelfand%E2%80%93Mazur%20theorem
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In operator theory, the Gelfand–Mazur theorem is a theorem named after Israel Gelfand and Stanisław Mazur which states that a Banach algebra with unit over the complex numbers in which every nonzero element is invertible is isometrically isomorphic to the complex numbers, i. e., the only complex Banach algebra that is a division algebra is the complex numbers C.
The theorem follows from the fact that the spectrum of any element of a complex Banach algebra is nonempty: for every element a of a complex Banach algebra A there is some complex number λ such that λ1 − a is not invertible. This is a consequence of the complex-analyticity of the resolvent function. By assumption, λ1 − a = 0. So a = λ · 1. This gives an isomorphism from A to C.
The theorem can be strengthened to the claim that there are (up to isomorphism) exactly three real Banach division algebras: the field of reals R, the field of complex numbers C, and the division algebra of quaternions H. This result was proved first by Stanisław Mazur alone, but it was published in France without a proof, when the author refused the editor's request to shorten his proof. Gelfand (independently) published a proof of the complex case a few years later.
References
Banach algebras
Theorems in functional analysis
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https://en.wikipedia.org/wiki/List%20of%20United%20States%20commuter%20rail%20systems
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The following is a list of commuter rail systems in the United States, ranked by ridership. All figures come from the American Public Transportation Association's (APTA) Ridership Reports Statistics for the fourth quarter of 2022, unless otherwise indicated.
List
Systems excluded from ridership table
See also
Commuter rail in North America
List of rail transit systems in the United States
List of United States light rail systems by ridership
List of United States local bus agencies by ridership
List of United States rapid transit systems by ridership
Notes
References
Commuter rail systems
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https://en.wikipedia.org/wiki/Advances%20in%20Applied%20Clifford%20Algebras
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Advances in Applied Clifford Algebras is a peer-reviewed scientific journal that publishes original research papers and also notes, expository and survey articles, book reviews, reproduces abstracts and also reports on conferences and workshops in the area of Clifford algebras and their applications to other branches of mathematics and physics, and in certain cognate areas. There is a vibrant and interdisciplinary community around Clifford and Geometric Algebras with a wide range of applications. The main conferences in this subject include the The International Conference on Clifford Algebras and Their Applications in Mathematical Physics (ICCA) and Applications of Geometric Algebra in Computer Science and Engineering (AGACSE) series.
The journal was established in 1991 by Jaime Keller who was its editor-in-chief until his death in 2011. The second editor-in-chief of the journal was Waldyr Alves Rodrigues Jr. (Universidade Estadual de Campinas), and the current editor-in-chief is Uwe Kähler from University of Aveiro. The journal is published by Springer Science+Business Media under its Birkhäuser Verlag imprint.
References
External links
Mathematics journals
Springer Science+Business Media academic journals
Quarterly journals
Academic journals established in 1991
English-language journals
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https://en.wikipedia.org/wiki/Uniformly%20convex%20space
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In mathematics, uniformly convex spaces (or uniformly rotund spaces) are common examples of reflexive Banach spaces. The concept of uniform convexity was first introduced by James A. Clarkson in 1936.
Definition
A uniformly convex space is a normed vector space such that, for every there is some such that for any two vectors with and the condition
implies that:
Intuitively, the center of a line segment inside the unit ball must lie deep inside the unit ball unless the segment is short.
Properties
The unit sphere can be replaced with the closed unit ball in the definition. Namely, a normed vector space is uniformly convex if and only if for every there is some so that, for any two vectors and in the closed unit ball (i.e. and ) with , one has (note that, given , the corresponding value of could be smaller than the one provided by the original weaker definition).
The "if" part is trivial. Conversely, assume now that is uniformly convex and that are as in the statement, for some fixed . Let be the value of corresponding to in the definition of uniform convexity. We will show that , with .
If then and the claim is proved. A similar argument applies for the case , so we can assume that . In this case, since , both vectors are nonzero, so we can let and . We have and similarly , so and belong to the unit sphere and have distance . Hence, by our choice of , we have . It follows that and the claim is proved.
The Milman–Pettis theorem states that every uniformly convex Banach space is reflexive, while the converse is not true.
Every uniformly convex Banach space is a Radon–Riesz space, that is, if is a sequence in a uniformly convex Banach space that converges weakly to and satisfies then converges strongly to , that is, .
A Banach space is uniformly convex if and only if its dual is uniformly smooth.
Every uniformly convex space is strictly convex. Intuitively, the strict convexity means a stronger triangle inequality whenever are linearly independent, while the uniform convexity requires this inequality to be true uniformly.
Examples
Every inner-product space is uniformly convex.
Every closed subspace of a uniformly convex Banach space is uniformly convex.
Hanner's inequalities imply that Lp spaces are uniformly convex.
Conversely, is not uniformly convex.
See also
Modulus and characteristic of convexity
Uniformly convex function
Uniformly smooth space
References
Citations
General references
.
.
Lindenstrauss, Joram and Benyamini, Yoav. Geometric nonlinear functional analysis. Colloquium publications, 48. American Mathematical Society.
Convex analysis
Banach spaces
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https://en.wikipedia.org/wiki/Geodynamics
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Geodynamics is a subfield of geophysics dealing with dynamics of the Earth. It applies physics, chemistry and mathematics to the understanding of how mantle convection leads to plate tectonics and geologic phenomena such as seafloor spreading, mountain building, volcanoes, earthquakes, faulting. It also attempts to probe the internal activity by measuring magnetic fields, gravity, and seismic waves, as well as the mineralogy of rocks and their isotopic composition. Methods of geodynamics are also applied to exploration of other planets.
Overview
Geodynamics is generally concerned with processes that move materials throughout the Earth. In the Earth's interior, movement happens when rocks melt or deform and flow in response to a stress field. This deformation may be brittle, elastic, or plastic, depending on the magnitude of the stress and the material's physical properties, especially the stress relaxation time scale. Rocks are structurally and compositionally heterogeneous and are subjected to variable stresses, so it is common to see different types of deformation in close spatial and temporal proximity. When working with geological timescales and lengths, it is convenient to use the continuous medium approximation and equilibrium stress fields to consider the average response to average stress.
Experts in geodynamics commonly use data from geodetic GPS, InSAR, and seismology, along with numerical models, to study the evolution of the Earth's lithosphere, mantle and core.
Work performed by geodynamicists may include:
Modeling brittle and ductile deformation of geologic materials
Predicting patterns of continental accretion and breakup of continents and supercontinents
Observing surface deformation and relaxation due to ice sheets and post-glacial rebound, and making related conjectures about the viscosity of the mantle
Finding and understanding the driving mechanisms behind plate tectonics.
Deformation of rocks
Rocks and other geological materials experience strain according to three distinct modes, elastic, plastic, and brittle depending on the properties of the material and the magnitude of the stress field. Stress is defined as the average force per unit area exerted on each part of the rock. Pressure is the part of stress that changes the volume of a solid; shear stress changes the shape. If there is no shear, the fluid is in hydrostatic equilibrium. Since, over long periods, rocks readily deform under pressure, the Earth is in hydrostatic equilibrium to a good approximation. The pressure on rock depends only on the weight of the rock above, and this depends on gravity and the density of the rock. In a body like the Moon, the density is almost constant, so a pressure profile is readily calculated. In the Earth, the compression of rocks with depth is significant, and an equation of state is needed to calculate changes in density of rock even when it is of uniform composition.
Elastic
Elastic deformation is always reversible, whi
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https://en.wikipedia.org/wiki/Angelo%20Genocchi
|
Angelo Genocchi (5 March 1817 – 7 March 1889) was an Italian mathematician who specialized in number theory. He worked with Giuseppe Peano. The Genocchi numbers are named after him.
Genocchi was President of the Academy of Sciences of Turin.
Notes
References
Obituary in:
19th-century Italian mathematicians
1817 births
1889 deaths
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https://en.wikipedia.org/wiki/Axis%E2%80%93angle%20representation
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In mathematics, the axis–angle representation parameterizes a rotation in a three-dimensional Euclidean space by two quantities: a unit vector indicating the direction (geometry) of an axis of rotation, and an angle of rotation describing the magnitude and sense (e.g., clockwise) of the rotation about the axis. Only two numbers, not three, are needed to define the direction of a unit vector rooted at the origin because the magnitude of is constrained. For example, the elevation and azimuth angles of suffice to locate it in any particular Cartesian coordinate frame.
By Rodrigues' rotation formula, the angle and axis determine a transformation that rotates three-dimensional vectors. The rotation occurs in the sense prescribed by the right-hand rule.
The rotation axis is sometimes called the Euler axis. The axis–angle representation is predicated on Euler's rotation theorem, which dictates that any rotation or sequence of rotations of a rigid body in a three-dimensional space is equivalent to a pure rotation about a single fixed axis.
It is one of many rotation formalisms in three dimensions.
Rotation vector
The axis–angle representation is equivalent to the more concise rotation vector, also called the Euler vector. In this case, both the rotation axis and the angle are represented by a vector codirectional with the rotation axis whose length is the rotation angle ,
It is used for the exponential and logarithm maps involving this representation.
Many rotation vectors correspond to the same rotation. In particular, a rotation vector of length , for any integer , encodes exactly the same rotation as a rotation vector of length . Thus, there are at least a countable infinity of rotation vectors corresponding to any rotation. Furthermore, all rotations by are the same as no rotation at all, so, for a given integer , all rotation vectors of length , in all directions, constitute a two-parameter uncountable infinity of rotation vectors encoding the same rotation as the zero vector. These facts must be taken into account when inverting the exponential map, that is, when finding a rotation vector that corresponds to a given rotation matrix. The exponential map is onto but not one-to-one.
Example
Say you are standing on the ground and you pick the direction of gravity to be the negative direction. Then if you turn to your left, you will rotate radians (or -90°) about the axis. Viewing the axis-angle representation as an ordered pair, this would be
The above example can be represented as a rotation vector with a magnitude of pointing in the direction,
Uses
The axis–angle representation is convenient when dealing with rigid body dynamics. It is useful to both characterize rotations, and also for converting between different representations of rigid body motion, such as homogeneous transformations and twists.
When a rigid body rotates around a fixed axis, its axis–angle data are a constant rotation axis and the rotation angle continuou
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https://en.wikipedia.org/wiki/Two-dimensional%20singular-value%20decomposition
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In linear algebra, two-dimensional singular-value decomposition (2DSVD) computes the low-rank approximation of a set of matrices such as 2D images or weather maps in a manner almost identical to SVD (singular-value decomposition) which computes the low-rank approximation of a single matrix (or a set of 1D vectors).
SVD
Let matrix contains the set of 1D vectors which have been centered. In PCA/SVD, we construct covariance matrix and Gram matrix
,
and compute their eigenvectors and . Since and we have
If we retain only principal eigenvectors in , this gives low-rank approximation of .
2DSVD
Here we deal with a set of 2D matrices . Suppose they are centered . We construct row–row and column–column covariance matrices
and
in exactly the same manner as in SVD, and compute their eigenvectors and . We approximate as
in identical fashion as in SVD. This gives a near optimal low-rank approximation of with the objective function
Error bounds similar to Eckard–Young theorem also exist.
2DSVD is mostly used in image compression and representation.
References
Chris Ding and Jieping Ye. "Two-dimensional Singular Value Decomposition (2DSVD) for 2D Maps and Images". Proc. SIAM Int'l Conf. Data Mining (SDM'05), pp. 32–43, April 2005. http://ranger.uta.edu/~chqding/papers/2dsvdSDM05.pdf
Jieping Ye. "Generalized Low Rank Approximations of Matrices". Machine Learning Journal. Vol. 61, pp. 167–191, 2005.
Singular value decomposition
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https://en.wikipedia.org/wiki/Milman%E2%80%93Pettis%20theorem
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In mathematics, the Milman–Pettis theorem states that every uniformly convex Banach space is reflexive.
The theorem was proved independently by D. Milman (1938) and B. J. Pettis (1939). S. Kakutani gave a different proof in 1939, and John R. Ringrose published a shorter proof in 1959.
Mahlon M. Day (1941) gave examples of reflexive Banach spaces which are not isomorphic to any uniformly convex space.
References
S. Kakutani, Weak topologies and regularity of Banach spaces, Proc. Imp. Acad. Tokyo 15 (1939), 169–173.
D. Milman, On some criteria for the regularity of spaces of type (B), C. R. (Doklady) Acad. Sci. U.R.S.S, 20 (1938), 243–246.
B. J. Pettis, A proof that every uniformly convex space is reflexive, Duke Math. J. 5 (1939), 249–253.
J. R. Ringrose, A note on uniformly convex spaces, J. London Math. Soc. 34 (1959), 92.
Banach spaces
Theorems in functional analysis
fr:Théorème de Milman-Pettis
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https://en.wikipedia.org/wiki/Albert%20algebra
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In mathematics, an Albert algebra is a 27-dimensional exceptional Jordan algebra. They are named after Abraham Adrian Albert, who pioneered the study of non-associative algebras, usually working over the real numbers. Over the real numbers, there are three such Jordan algebras up to isomorphism. One of them, which was first mentioned by and studied by , is the set of 3×3 self-adjoint matrices over the octonions, equipped with the binary operation
where denotes matrix multiplication. Another is defined the same way, but using split octonions instead of octonions. The final is constructed from the non-split octonions using a different standard involution.
Over any algebraically closed field, there is just one Albert algebra, and its automorphism group G is the simple split group of type F4. (For example, the complexifications of the three Albert algebras over the real numbers are isomorphic Albert algebras over the complex numbers.) Because of this, for a general field F, the Albert algebras are classified by the Galois cohomology group H1(F,G).
The Kantor–Koecher–Tits construction applied to an Albert algebra gives a form of the E7 Lie algebra. The split Albert algebra is used in a construction of a 56-dimensional structurable algebra whose automorphism group has identity component the simply-connected algebraic group of type E6.
The space of cohomological invariants of Albert algebras a field F (of characteristic not 2) with coefficients in Z/2Z is a free module over the cohomology ring of F with a basis 1, f3, f5, of degrees 0, 3, 5. The cohomological invariants with 3-torsion coefficients have a basis 1, g3 of degrees 0, 3. The invariants f3 and g3 are the primary components of the Rost invariant.
See also
Euclidean Jordan algebra for the Jordan algebras considered by Jordan, von Neumann and Wigner
Euclidean Hurwitz algebra for details of the construction of the Albert algebra for the octonions
Notes
References
Further reading
Albert algebra at Encyclopedia of Mathematics.
Non-associative algebras
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https://en.wikipedia.org/wiki/Hyperbolization%20theorem
|
In geometry, Thurston's geometrization theorem or hyperbolization theorem implies that closed atoroidal Haken manifolds are hyperbolic, and in particular satisfy the Thurston conjecture.
Statement
One form of Thurston's geometrization theorem states:
If M is a compact irreducible atoroidal Haken manifold whose boundary has zero Euler characteristic, then the interior of M has a complete hyperbolic structure of finite volume.
The Mostow rigidity theorem implies that if a manifold of dimension at least 3 has a hyperbolic structure of finite volume, then it is essentially unique.
The conditions that the manifold M should be irreducible and atoroidal are necessary, as hyperbolic manifolds have these properties. However the condition that the manifold be Haken is unnecessarily strong. Thurston's hyperbolization conjecture states that a closed irreducible atoroidal 3-manifold with infinite fundamental group is hyperbolic, and this follows from Perelman's proof of the Thurston geometrization conjecture.
Manifolds with boundary
showed that if a compact 3 manifold is prime, homotopically atoroidal, and has non-empty boundary, then it has a complete hyperbolic structure unless it is homeomorphic to a certain manifold (T2×[0,1])/Z/2Z with boundary T2.
A hyperbolic structure on the interior of a compact orientable 3-manifold has finite volume if and only if all boundary components are tori, except for the manifold T2×[0,1] which has a hyperbolic structure but none of finite volume .
Proofs
Thurston never published a complete proof of his theorem for reasons that he explained in , though parts of his argument are contained in . and gave summaries of Thurston's proof. gave a proof in the case of manifolds that fiber over the circle, and and gave proofs for the generic case of manifolds that do not fiber over the circle. Thurston's geometrization theorem also follows from Perelman's proof using Ricci flow of the more general Thurston geometrization conjecture.
Manifolds that fiber over the circle
Thurston's original argument for this case was summarized by .
gave a proof in the case of manifolds that fiber over the circle.
Thurston's geometrization theorem in this special case states that if M is a 3-manifold that fibers over the circle and whose monodromy is a pseudo-Anosov diffeomorphism, then the interior of M has a complete hyperbolic metric of finite volume.
Manifolds that do not fiber over the circle
and gave proofs of Thurston's theorem for the generic case of manifolds that do not fiber over the circle.
The idea of the proof is to cut a Haken manifold M along an incompressible surface, to obtain a new manifold N. By induction one assumes that the interior of N has a hyperbolic structure, and the problem is to modify it so that it can be extended to the boundary of N and glued together. Thurston showed that this follows from the existence of a fixed point for a map of Teichmuller space called the skinning map. The core of t
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https://en.wikipedia.org/wiki/Odd%20number%20theorem
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The odd number theorem is a theorem in strong gravitational lensing which comes directly from differential topology.
The theorem states that the number of multiple images produced by a bounded transparent lens must be odd.
Formulation
The gravitational lensing is a thought to mapped from what's known as image plane to source plane following the formula :
.
Argument
If we use direction cosines describing the bent light rays, we can write a vector field on plane .
However, only in some specific directions , will the bent light rays reach the observer, i.e., the images only form where . Then we can directly apply the Poincaré–Hopf theorem .
The index of sources and sinks is +1, and that of saddle points is −1. So the Euler characteristic equals the difference between the number of positive indices and the number of negative indices . For the far field case, there is only one image, i.e., . So the total number of images is , i.e., odd. The strict proof needs Uhlenbeck's Morse theory of null geodesics.
References
Perlick V., Gravitational lensing from a geometric viewpoint, in B. Schmidt (ed.) "Einstein's field equations and their physical interpretations" Selected Essays in Honour of Jürgen Ehlers, Springer, Heidelberg (2000) pp. 373–425
Gravitational lensing
Physics theorems
Equations of astronomy
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https://en.wikipedia.org/wiki/Andr%C3%A9%20Joyal
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André Joyal (; born 1943) is a professor of mathematics at the Université du Québec à Montréal who works on category theory. He was a member of the School of Mathematics at the Institute for Advanced Study in 2013, where he was invited to join the Special Year on Univalent Foundations of Mathematics.
Research
He discovered Kripke–Joyal semantics, the theory of combinatorial species and with Myles Tierney a generalization of the Galois theory of Alexander Grothendieck in the setup of locales. Most of his research is in some way related to category theory, higher category theory and their applications. He did some work on quasi-categories, after their invention by Michael Boardman and Rainer Vogt, in particular conjecturing and proving the existence of a Quillen model structure on sSet whose weak equivalences generalize both equivalence of categories and Kan equivalence of spaces. He co-authored the book "Algebraic Set Theory" with Ieke Moerdijk and recently started a web-based expositional project Joyal's CatLab on categorical mathematics.
Personal life
Joyal was born in Drummondville (formerly Saint-Majorique). He has three children and lives in Montreal.
Bibliography
; ;
André Joyal, Ieke Moerdijk, Algebraic set theory. London Mathematical Society Lecture Note Series 220. Cambridge Univ. Press 1995. viii+123 pp.
André Joyal, Myles Tierney, Notes on simplicial homotopy theory, CRM Barcelona, Jan 2008 pdf
André Joyal, Disks, duality and theta-categories, preprint (1997) (contains an original definition of a weak n-category: for a short account see Leinster's , 10.2).
References
External links
Interview with André Joyal (in French)
Official Web page at UQAM
Living people
1943 births
Category theorists
20th-century Canadian mathematicians
21st-century Canadian mathematicians
Academic staff of the Université du Québec à Montréal
People from Drummondville
French Quebecers
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https://en.wikipedia.org/wiki/Jackknife%20resampling
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In statistics, the jackknife (jackknife cross-validation) is a cross-validation technique and, therefore, a form of resampling.
It is especially useful for bias and variance estimation. The jackknife pre-dates other common resampling methods such as the bootstrap. Given a sample of size , a jackknife estimator can be built by aggregating the parameter estimates from each subsample of size obtained by omitting one observation.
The jackknife technique was developed by Maurice Quenouille (1924–1973) from 1949 and refined in 1956. John Tukey expanded on the technique in 1958 and proposed the name "jackknife" because, like a physical jack-knife (a compact folding knife), it is a rough-and-ready tool that can improvise a solution for a variety of problems even though specific problems may be more efficiently solved with a purpose-designed tool.
The jackknife is a linear approximation of the bootstrap.
A simple example: mean estimation
The jackknife estimator of a parameter is found by systematically leaving out each observation from a dataset and calculating the parameter estimate over the remaining observations and then aggregating these calculations.
For example, if the parameter to be estimated is the population mean of random variable , then for a given set of i.i.d. observations the natural estimator is the sample mean:
where the last sum used another way to indicate that the index runs over the set .
Then we proceed as follows: For each we compute the mean of the jackknife subsample consisting of all but the -th data point, and this is called the -th jackknife replicate:
It could help to think that these jackknife replicates give us an approximation of the distribution of the sample mean and the larger the the better this approximation will be. Then finally to get the jackknife estimator we take the average of these jackknife replicates:
One may ask about the bias and the variance of . From the definition of as the average of the jackknife replicates one could try to calculate explicitly, and the bias is a trivial calculation but the variance of is more involved since the jackknife replicates are not independent.
For the special case of the mean, one can show explicitly that the jackknife estimate equals the usual estimate:
This establishes the identity . Then taking expectations we get , so is unbiased, while taking variance we get . However, these properties do not hold in general for other parameters than the mean.
This simple example for the case of mean estimation is just to illustrate the construction of a jackknife estimator, while the real subtleties (and the usefulness) emerge for the case of estimating other parameters, such as higher moments than the mean or other functionals of the distribution.
could be used to construct an empirical estimate of the bias of , namely with some suitable factor , although in this case we know that so this construction does not add any meaningful knowledge, but it gives the
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https://en.wikipedia.org/wiki/CO1
|
CO1 may refer to:
CO postcode area
Conway group Co1 in mathematics
Carbon monoxide in chemistry
Cytochrome Oxidase Subunit 1
Min'an Electric CO1, a vehicle made by Min'an Electric
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https://en.wikipedia.org/wiki/Dirichlet-multinomial%20distribution
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In probability theory and statistics, the Dirichlet-multinomial distribution is a family of discrete multivariate probability distributions on a finite support of non-negative integers. It is also called the Dirichlet compound multinomial distribution (DCM) or multivariate Pólya distribution (after George Pólya). It is a compound probability distribution, where a probability vector p is drawn from a Dirichlet distribution with parameter vector , and an observation drawn from a multinomial distribution with probability vector p and number of trials n. The Dirichlet parameter vector captures the prior belief about the situation and can be seen as a pseudocount: observations of each outcome that occur before the actual data is collected. The compounding corresponds to a Pólya urn scheme. It is frequently encountered in Bayesian statistics, machine learning, empirical Bayes methods and classical statistics as an overdispersed multinomial distribution.
It reduces to the categorical distribution as a special case when n = 1. It also approximates the multinomial distribution arbitrarily well for large α. The Dirichlet-multinomial is a multivariate extension of the beta-binomial distribution, as the multinomial and Dirichlet distributions are multivariate versions of the binomial distribution and beta distributions, respectively.
Specification
Dirichlet-multinomial as a compound distribution
The Dirichlet distribution is a conjugate distribution to the multinomial distribution. This fact leads to an analytically tractable compound distribution.
For a random vector of category counts , distributed according to a multinomial distribution, the marginal distribution is obtained by integrating on the distribution for p which can be thought of as a random vector following a Dirichlet distribution:
which results in the following explicit formula:
where is defined as the sum . Another form for this same compound distribution, written more compactly in terms of the beta function, B, is as follows:
The latter form emphasizes the fact that zero count categories can be ignored in the calculation - a useful fact when the number of categories is very large and sparse (e.g. word counts in documents).
Observe that the pdf is the Beta-binomial distribution when . It can also be shown that it approaches the multinomial distribution as approaches infinity. The parameter governs the degree of overdispersion or burstiness relative to the multinomial. Alternative choices to denote found in the literature are S and A.
Dirichlet-multinomial as an urn model
The Dirichlet-multinomial distribution can also be motivated via an urn model for positive integer values of the vector α, known as the Polya urn model. Specifically, imagine an urn containing balls of K colors numbering for the ith color, where random draws are made. When a ball is randomly drawn and observed, then two balls of the same color are returned to the urn. If this is performed n times, then the p
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https://en.wikipedia.org/wiki/Jacques%20Dixmier
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Jacques Dixmier (born 24 May 1924) is a French mathematician. He worked on operator algebras, especially C*-algebras, and wrote several of the standard reference books on them, and introduced the Dixmier trace and the Dixmier mapping.
Biography
Dixmier received his Ph.D. in 1949 from the University of Paris, and his students include Alain Connes.
In 1949 upon the initiative of Jean-Pierre Serre and Pierre Samuel, Dixmier became a member of Bourbaki, in which he made essential contributions to the Bourbaki volume on Lie algebras. After retiring as professor emeritus from the University of Paris VI, he spent five years at the Institut des Hautes Études Scientifiques.
Often, there is made the erroneous claim that Dixmier originated the name von Neumann algebra for the operator algebras introduced by John von Neumann, but Dixmier said in an interview that the name originated from a proposal by Jean Dieudonné.
Dixmier was an invited speaker at the International Congress of Mathematicians in 1966 in Moscow with the talk Espace dual d'une algèbre, ou d'un groupe localement compact and again in 1978 in Helsinki with the talk Algèbres enveloppantes.
Publications
J. Dixmier, C*-algebras. Translated from the French by Francis Jellett. North-Holland Mathematical Library, Vol. 15. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. xiii+492 pp.
A translation of Les C*-algèbres et leurs représentations, Gauthier-Villars, 1969.
A translation of Algèbres enveloppantes, Cahiers Scientifiques, Fasc. XXXVII. Gauthier-Villars Éditeur, Paris-Brussels-Montreal, Que., 1974. ii+349 pp.
J. Dixmier, von Neumann algebras, Translated from the second French edition by F. Jellett. North-Holland Mathematical Library, 27. North-Holland Publishing Co., Amsterdam-New York, 1981. xxxviii+437 pp.
A translation of Les algèbres d'opérateurs dans l'espace hilbertien: algèbres de von Neumann, Gauthier-Villars (1957), the first book about von Neumann algebras.
Books
Notes
1924 births
Living people
École Normale Supérieure alumni
20th-century French mathematicians
21st-century French mathematicians
Mathematical analysts
University of Paris alumni
Nicolas Bourbaki
Scientists from Saint-Étienne
Academic staff of the University of Paris
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https://en.wikipedia.org/wiki/Fran%C3%A7ois%20Bruhat
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François Georges René Bruhat (; 8 April 1929 – 17 July 2007) was a French mathematician who worked on algebraic groups. The Bruhat order of a Weyl group, the Bruhat decomposition, and the Schwartz–Bruhat functions are named after him.
He was the son of physicist (and associate director of the École Normale Supérieure during the occupation) Georges Bruhat, and brother of physicist Yvonne Choquet-Bruhat.
See also
Hadamard space
References
1929 births
2007 deaths
École Normale Supérieure alumni
Members of the French Academy of Sciences
Nicolas Bourbaki
20th-century French mathematicians
21st-century French mathematicians
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https://en.wikipedia.org/wiki/Squaring
|
Squaring may refer to:
Square (algebra), the result of multiplying something by itself
Quadrature (mathematics), the process of determining the area of a plane figure
|
https://en.wikipedia.org/wiki/Second%20degree
|
Second degree may refer to:
A postgraduate degree or a professional degree in postgraduate education
Second-degree burn
Second-degree polynomial, in mathematics
Second-degree murder, actual definition varies from country to country
The second degree in Freemasonry
See also
First degree (disambiguation)
Third degree (disambiguation)
Minute and second of arc, a second of arc being of a degree
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https://en.wikipedia.org/wiki/Braid%20algebra
|
A braid algebra can be:
A Gerstenhaber algebra (also called an antibracket algebra).
An algebra related to the braid group.
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https://en.wikipedia.org/wiki/Schatten%20norm
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In mathematics, specifically functional analysis, the Schatten norm (or Schatten–von-Neumann norm)
arises as a generalization of p-integrability similar to the trace class norm and the Hilbert–Schmidt norm.
Definition
Let , be Hilbert spaces, and a (linear) bounded operator from
to . For , define the Schatten p-norm of as
where
,
using the
operator square root.
If is compact and are separable, then
for
the singular values of , i.e. the eigenvalues of the Hermitian operator .
Properties
In the following we formally extend the range of to with the convention that is the operator norm. The dual index to is then .
The Schatten norms are unitarily invariant: for unitary operators and and ,
They satisfy Hölder's inequality: for all and such that , and operators defined between Hilbert spaces and respectively,
If satisfy , then we have
.
The latter version of Hölder's inequality is proven in higher generality (for noncommutative spaces instead of Schatten-p classes) in.
(For matrices the latter result is found in .)
Sub-multiplicativity: For all and operators defined between Hilbert spaces and respectively,
Monotonicity: For ,
Duality: Let be finite-dimensional Hilbert spaces, and such that , then
where denotes the Hilbert–Schmidt inner product.
Let be two orthonormal basis of the Hilbert spaces , then for
.
Remarks
Notice that is the Hilbert–Schmidt norm (see Hilbert–Schmidt operator), is the trace class norm (see trace class), and is the operator norm (see operator norm).
For the function is an example of a quasinorm.
An operator which has a finite Schatten norm is called a Schatten class operator and the space of such operators is denoted by . With this norm, is a Banach space, and a Hilbert space for p = 2.
Observe that , the algebra of compact operators. This follows from the fact that if the sum is finite the spectrum will be finite or countable with the origin as limit point, and hence a compact operator (see compact operator on Hilbert space).
The case p = 1 is often referred to as the nuclear norm (also known as the trace norm, or the Ky Fan 'n'-norm)
See also
Matrix Norms
References
Rajendra Bhatia, Matrix analysis, Vol. 169. Springer Science & Business Media, 1997.
John Watrous, Theory of Quantum Information, 2.3 Norms of operators, lecture notes, University of Waterloo, 2011.
Joachim Weidmann, Linear operators in Hilbert spaces, Vol. 20. Springer, New York, 1980.
Operator theory
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https://en.wikipedia.org/wiki/Schatten%20class%20operator
|
In mathematics, specifically functional analysis, a pth Schatten-class operator is a bounded linear operator on a Hilbert space with finite pth Schatten norm. The space of pth Schatten-class operators is a Banach space with respect to the Schatten norm.
Via polar decomposition, one can prove that the space of pth Schatten class operators is an ideal in B(H). Furthermore, the Schatten norm satisfies a type of Hölder inequality:
If we denote by the Banach space of compact operators on H with respect to the operator norm, the above Hölder-type inequality even holds for . From this it follows that , is a well-defined contraction. (Here the prime denotes (topological) dual.)
Observe that the 2nd Schatten class is in fact the Hilbert space of Hilbert–Schmidt operators. Moreover, the 1st Schatten class is the space of trace class operators.
Operator theory
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https://en.wikipedia.org/wiki/Operator%20system
|
Given a unital C*-algebra , a *-closed subspace S containing 1 is called an operator system. One can associate to each subspace of a unital C*-algebra an operator system via .
The appropriate morphisms between operator systems are completely positive maps.
By a theorem of Choi and Effros, operator systems can be characterized as *-vector spaces equipped with an Archimedean matrix order.
See also
Operator space
References
Operator theory
Operator algebras
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https://en.wikipedia.org/wiki/Harmonic%20wavelet%20transform
|
In the mathematics of signal processing, the harmonic wavelet transform, introduced by David Edward Newland in 1993, is a wavelet-based linear transformation of a given function into a time-frequency representation. It combines advantages of the short-time Fourier transform and the continuous wavelet transform. It can be expressed in terms of repeated Fourier transforms, and its discrete analogue can be computed efficiently using a fast Fourier transform algorithm.
Harmonic wavelets
The transform uses a family of "harmonic" wavelets indexed by two integers j (the "level" or "order") and k (the "translation"), given by , where
These functions are orthogonal, and their Fourier transforms are a square window function (constant in a certain octave band and zero elsewhere). In particular, they satisfy:
where "*" denotes complex conjugation and is Kronecker's delta.
As the order j increases, these wavelets become more localized in Fourier space (frequency) and in higher frequency bands, and conversely become less localized in time (t). Hence, when they are used as a basis for expanding an arbitrary function, they represent behaviors of the function on different timescales (and at different time offsets for different k).
However, it is possible to combine all of the negative orders (j < 0) together into a single family of "scaling" functions where
The function φ is orthogonal to itself for different k and is also orthogonal to the wavelet functions for non-negative j:
In the harmonic wavelet transform, therefore, an arbitrary real- or complex-valued function (in L2) is expanded in the basis of the harmonic wavelets (for all integers j) and their complex conjugates:
or alternatively in the basis of the wavelets for non-negative j supplemented by the scaling functions φ:
The expansion coefficients can then, in principle, be computed using the orthogonality relationships:
For a real-valued function f(t), and so one can cut the number of independent expansion coefficients in half.
This expansion has the property, analogous to Parseval's theorem, that:
Rather than computing the expansion coefficients directly from the orthogonality relationships, however, it is possible to do so using a sequence of Fourier transforms. This is much more efficient in the discrete analogue of this transform (discrete t), where it can exploit fast Fourier transform algorithms.
References
Time–frequency analysis
Transforms
Wavelets
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https://en.wikipedia.org/wiki/Bundle%20metric
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In differential geometry, the notion of a metric tensor can be extended to an arbitrary vector bundle, and to some principal fiber bundles. This metric is often called a bundle metric, or fibre metric.
Definition
If M is a topological manifold and : E → M a vector bundle on M, then a metric on E is a bundle map k : E ×M E → M × R from the fiber product of E with itself to the trivial bundle with fiber R such that the restriction of k to each fibre over M is a nondegenerate bilinear map of vector spaces. Roughly speaking, k gives a kind of dot product (not necessarily symmetric or positive definite) on the vector space above each point of M, and these products vary smoothly over M.
Properties
Every vector bundle with paracompact base space can be equipped with a bundle metric. For a vector bundle of rank n, this follows from the bundle charts : the bundle metric can be taken as the pullback of the inner product of a metric on ; for example, the orthonormal charts of Euclidean space. The structure group of such a metric is the orthogonal group O(n).
Example: Riemann metric
If M is a Riemannian manifold, and E is its tangent bundle TM, then the Riemannian metric gives a bundle metric, and vice versa.
Example: on vertical bundles
If the bundle :P → M is a principal fiber bundle with group G, and G is a compact Lie group, then there exists an Ad(G)-invariant inner product k on the fibers, taken from the inner product on the corresponding compact Lie algebra. More precisely, there is a metric tensor k defined on the vertical bundle E = VP such that k is invariant under left-multiplication:
for vertical vectors X, Y and Lg is left-multiplication by g along the fiber, and Lg* is the pushforward. That is, E is the vector bundle that consists of the vertical subspace of the tangent of the principal bundle.
More generally, whenever one has a compact group with Haar measure μ, and an arbitrary inner product h(X,Y) defined at the tangent space of some point in G, one can define an invariant metric simply by averaging over the entire group, i.e. by defining
as the average.
The above notion can be extended to the associated bundle where V is a vector space transforming covariantly under some representation of G.
In relation to Kaluza–Klein theory
If the base space M is also a metric space, with metric g, and the principal bundle is endowed with a connection form ω, then *g+kω is a metric defined on the entire tangent bundle E = TP.
More precisely, one writes *g(X,Y) = g(*X, *Y) where * is the pushforward of the projection , and g is the metric tensor on the base space M. The expression kω should be understood as (kω)(X,Y) = k(ω(X),ω(Y)), with k the metric tensor on each fiber. Here, X and Y are elements of the tangent space TP.
Observe that the lift *g vanishes on the vertical subspace TV (since * vanishes on vertical vectors), while kω vanishes on the horizontal subspace TH (since the horizontal subspace is defined as that part of the tangent sp
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https://en.wikipedia.org/wiki/Scott%20W.%20Williams
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Scott Williams (born April 22, 1943, in Staten Island, New York) is a professor of mathematics at the University at Buffalo, SUNY. He was recognized by Mathematically Gifted & Black as a Black History Month 2017 Honoree.
Education
Raised in Baltimore, Maryland, Williams attended Morgan State University and earned his bachelor degree of Science in mathematics.
Before earning his bachelor's degree he was already able to solve four advanced problems in The Mathematical Monthly and co-authored two papers on Non-Associative Algebra with his undergraduate advisor Dr. Volodymir Bohun-Chudyniv. Scott Williams earned his Master's and Ph.D. in mathematics from Lehigh University in 1967 and 1969, respectively.
Career
Williams served as a Research Associate in the Department of Mathematics at Pennsylvania State University - University Park, from 1969 to 1971. In 1971, he was chosen to be assistant professor of mathematics at the University at Buffalo and in 1985 was promoted to Full Professor at the university. In 1982, he won the New York Chancellor Award for Excellence in Teaching. In 2004, he was named one of the 50 Most Important Blacks in Research Science by Science Spectrum Magazine and Career Communications Group.
Williams primarily focused on topology and the field of mathematics. In 1975, he was the first topologist to apply the concept of scales (now known as b=d) to give a partial solution of the famous Box Product problem, which is still unsettled today. Dr. Williams is one of two founders of Black and Third World Mathematicians, which in 1971 became the National Association of Mathematicians. Together with Willam Massey of Lucent Technologies, Dr. Williams founded the Committee for African American Researchers in the Mathematical Sciences in 1997.
In 1997 Williams created the website Mathematicians of the African Diaspora (MAD) dedicated to promoting and highlighting the contributions of members of the African diaspora to mathematics, especially contributions to current mathematical research.
Publications
Non-research publications
Williams, Scott W. "Million-buck problems". Math. Intelligencer 24 (2002), no. 3, 17–20.
Williams, Scott W. "Compact! A tutorial", Contemp. Math., 275, Amer. Math. Soc., 161–171, Providence, RI, 2001.
Williams, Scott W. "Black research mathematicians in the United States". African Americans in mathematics, II (Houston, TX, 1998), Contemp. Math., 252, 165–168 Amer. Math. Soc., Providence, RI, 1999.
Williams, Scott W. "Some dynamics on the irrationals. African Americans in mathematics" (Piscataway, NJ, 1996), 83–103, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 34, Amer. Math. Soc., Providence, RI, 1997.
Williams, Scott W. "Box products". Handbook of set-theoretic topology, 169–200, North-Holland, Amsterdam-New York, 1984.
Research publications
Williams, Scott W.; Zhou, Haoxuan Order-like structure of monotonically normal spaces. Comment. Math. Univ. Carolin. 39 (1998), no. 1, 207–217.
Pelant, Ja
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https://en.wikipedia.org/wiki/Armero
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Armero is a municipality in the Tolima Department, Colombia. According to the National Department of Statistics of Colombia, 12,852 lived in the town in 2005. Its median temperature is 27 °C. It was founded in 1895, but was not officially recognized as the seat of the region until 29 September 1908, by President Rafael Reyes. The town was originally named San Lorenzo. In 1930, the name was changed to Armero in memory of José León Armero, a national martyr.
Because the region became the main cotton producer in the country, the city was called Colombia's White City. It was a prosperous agricultural area until 1985.
The original seat of the region was destroyed on 13 November 1985, after an eruption of the Nevado del Ruiz Volcano produced lahars that buried the town and killed about 23,000 people. Approximately 31,000 people lived in the area at the time. The incident became known as the Armero tragedy. While the destruction of the town made world news in its own right, the best known victim was Omayra Sánchez, a young girl who died after being trapped by water and concrete up to her neck for three days. After this event, the town of Guayabal was assigned as the seat of the municipality of Armero, rendering Armero a ghost town.
The survivors were relocated to the towns of Guayabal and Lérida where they received housing and money, although little was done in aiding the survivors in reconstructing their lives.
In the area where the city was located, survivors created an extensive cemetery. Where each one had a house, they constructed a tomb with an epitaph. In this way, they constructed a new symbolic city called Camposanto.
Armando Armero is a foundation set up to bring social and economic development to a zone that has been devastated in the aftermath of the last eruption of Ruiz. It has created the Centro de Interpretación de la Memoria y la Tragedia de Armero, the first Memory Interpretation Center of a Natural Catastrophe in the world located exactly where the events occurred. There are memorial sites at each of the important places of the city (such as hospitals, parks, and theaters) near the ruins. In those, visitors can learn about the city as they existed before the tragedy.
References
External links
MSN Encarta: Armero Guayabal (Archived 31 October 2009)
Armando Armero organization
Bogotá, Ediciones Bartleby,
Municipalities of Tolima Department
Populated places established in 1895
Natural disaster ghost towns
1895 establishments in Colombia
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https://en.wikipedia.org/wiki/Thorold%20Gosset
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John Herbert de Paz Thorold Gosset (16 October 1869 – December 1962) was an English lawyer and an amateur mathematician. In mathematics, he is noted for discovering and classifying the semiregular polytopes in dimensions four and higher, and for his generalization of Descartes' theorem on tangent circles to four and higher dimensions.
Biography
Thorold Gosset was born in Thames Ditton, the son of John Jackson Gosset, a civil servant and statistical officer for HM Customs, and his wife Eleanor Gosset (formerly Thorold). He was admitted to Pembroke College, Cambridge as a pensioner on 1 October 1888, graduated BA in 1891, was called to the bar of the Inner Temple in June 1895, and graduated LLM in 1896. In 1900 he married Emily Florence Wood, and they subsequently had two children, named Kathleen and John.
Mathematics
According to H. S. M. Coxeter, after obtaining his law degree in 1896 and having no clients, Gosset amused himself by attempting to classify the regular polytopes in higher-dimensional (greater than three) Euclidean space. After rediscovering all of them, he attempted to classify the "semi-regular polytopes", which he defined as polytopes having regular facets and which are vertex-uniform, as well as the analogous honeycombs, which he regarded as degenerate polytopes. In 1897 he submitted his results to James W. Glaisher, then editor of the journal Messenger of Mathematics. Glaisher was favourably impressed and passed the results on to William Burnside and Alfred Whitehead. Burnside, however, stated in a letter to Glaisher in 1899 that "the author's method, a sort of geometric intuition" did not appeal to him. He admitted that he never found the time to read more than the first half of Gosset's paper. In the end Glaisher published only a brief abstract of Gosset's results.
Gosset's results went largely unnoticed for many years. His semiregular polytopes were rediscovered by Elte in 1912 and later by H.S.M. Coxeter who gave both Gosset and Elte due credit.
Coxeter introduced the term Gosset polytopes for three semiregular polytopes in 6, 7, and 8 dimensions discovered by Gosset: the 221, 321, and 421 polytopes. The vertices of these polytopes were later seen to arise as the roots of the exceptional Lie algebras E6, E7 and E8.
A new and more precise definition of the Gosset Series of polytopes has been given by Conway in 2008.
See also
Gosset graph
Scott Vorthmann with David Richter in this article are displaying and presenting computerized vZome images of Gosset's Polytopes built with vZome program and which are including the 3_21 polytope of Coxeter of 27 nodes which interested Pierre Etevenon in France.
References
Amateur mathematicians
19th-century English mathematicians
20th-century English mathematicians
1869 births
1962 deaths
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https://en.wikipedia.org/wiki/Waterman%20polyhedron
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In geometry, the Waterman polyhedra are a family of polyhedra discovered around 1990 by the mathematician Steve Waterman. A Waterman polyhedron is created by packing spheres according to the cubic close(st) packing (CCP), also known as the face-centered cubic (fcc) packing, then sweeping away the spheres that are farther from the center than a defined radius, then creating the convex hull of the sphere centers.
Waterman polyhedra form a vast family of polyhedra. Some of them have a number of nice properties such as multiple symmetries, or interesting and regular shapes. Others are just a collection of faces formed from irregular convex polygons.
The most popular Waterman polyhedra are those with centers at the point (0,0,0) and built out of hundreds of polygons. Such polyhedra resemble spheres. In fact, the more faces a Waterman polyhedron has, the more it resembles its circumscribed sphere in volume and total area.
With each point of 3D space we can associate a family of Waterman polyhedra with different values of radii of the circumscribed spheres. Therefore, from a mathematical point of view we can consider Waterman polyhedra as 4D spaces W(x, y, z, r), where x, y, z are coordinates of a point in 3D, and r is a positive number greater than 1.
Seven origins of cubic close(st) packing (CCP)
There can be seven origins defined in CCP, where n = {1, 2, 3, …}:
Origin 1: offset 0,0,0, radius
Origin 2: offset ,,0, radius
Origin 3: offset ,,, radius
Origin 3*: offset ,,, radius
Origin 4: offset ,,, radius
Origin 5: offset 0,0,, radius
Origin 6: offset 1,0,0, radius
Depending on the origin of the sweeping, a different shape and resulting polyhedron are obtained.
Relation to Platonic and Archimedean solids
Some Waterman polyhedra create Platonic solids and Archimedean solids. For this comparison of Waterman polyhedra they are normalized, e.g. has a different size or volume than but has the same form as an octahedron.
Platonic solids
Tetrahedron: W1 O3*, W2 O3*, W1 O3, W1 O4
Octahedron: W2 O1, W1 O6
Cube: W2 O6
Icosahedron and dodecahedron have no representation as Waterman polyhedra.
Archimedean solids
Cuboctahedron: W1 O1, W4 O1
Truncated octahedron: W10 O1
Truncated tetrahedron: W4 O3, W2 O4
The other Archimedean solids have no representation as Waterman polyhedra.
The W7 O1 might be mistaken for a truncated cuboctahedron, as well W3 O1 = W12 O1 mistaken for a rhombicuboctahedron, but those Waterman polyhedra have two edge lengths and therefore do not qualify as Archimedean solids.
Generalized Waterman polyhedra
Generalized Waterman polyhedra are defined as the convex hull derived from the point set of any spherical extraction from a regular lattice.
Included is a detailed analysis of the following 10 lattices – bcc, cuboctahedron, diamond, fcc, hcp, truncated octahedron, rhombic dodecahedron, simple cubic, truncated tet tet, truncated tet truncated octahedron cuboctahedron.
Each of the 10 lattices were examined to isolat
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https://en.wikipedia.org/wiki/List%20of%20Gillingham%20F.C.%20records%20and%20statistics
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Gillingham Football Club is an English professional association football club based in Gillingham, Kent, playing in League One, the third level of the English football league system, as of the 2019–20 season. The club was formed in 1893 as New Brompton F.C., a name which was retained until 1913, and has played home matches at Priestfield Stadium throughout its history. The club joined the Football League in 1920, was voted out of the league in favour of Ipswich Town at the end of the 1937–38 season, but returned to the league 12 years later after it was expanded from 88 to 92 clubs. Between 2000 and 2005, Gillingham played in the second tier of the English league for the only time in the club's history, achieving a highest league finish of eleventh place in 2002–03.
The record for most games played for the club is held by Ron Hillyard, who made 655 appearances between 1974 and 1991. Brian Yeo is the club's record goalscorer, scoring 149 goals during his Gillingham career. Andrew Crofts holds the record for the most international caps gained as a Gillingham player, having made 12 appearances for Wales. The highest transfer fee ever paid by the club is the £600,000 paid to Reading for Carl Asaba in 1998, and the highest fee received is the £1,500,000 paid by Manchester City for Robert Taylor in 1999. The highest attendance recorded at Priestfield was 23,002 for the visit of Queens Park Rangers in 1948. The club holds one Football League record, having conceded the fewest goals in a 46-match season, when the team conceded only 20 goals during the 1995–96 season.
All figures are correct as of 2022.
Honours and achievements
Gillingham have won two major honours in English football; first the Football League Fourth Division title in the 1963–64 season and then the Football League Two title in the 2012–13 season. The club has also achieved promotion on four other occasions, most recently in the 2008–09 season, when a 1–0 victory over Shrewsbury Town in the 2009 Football League Two play-off final secured a return to League One following relegation the previous season.
Gillingham's only previous victory at Wembley Stadium came in the 1999–2000 season, when a 3–2 victory over Wigan Athletic in the Second Division play-off final clinched promotion to the second tier of English football for the first time in Gillingham's history. Between 1938 and 1950, when the club played outside the Football League, Gillingham won the Southern Football League championship on two occasions and the Kent League once.
The Football League
Second Division (level 3):
Promotion (1): 1999–2000
Third Division / Fourth Division / Football League Two (level 4):
Winners (2): 1963–64, 2012–13
Promotion (3): 1973–74, 1995–96, 2008–09
Other honours
Southern League:
Division One champions (2): 1946–47, 1948–49
Division Two champions (1): 1894–95
Southern League Cup winners (1): 1946–47
Kent League:
Champions (1): 1945–46
Kent League Cup winners (1): 1945–46
National cup compe
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https://en.wikipedia.org/wiki/Dinei%20%28footballer%2C%20born%201983%29
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Telmário de Araújo Sacramento, known by his nickname Dinei (born November 11, 1983 in São Domingos, Bahia), is a Brazilian former football striker who played for Vitória.
Club statistics
Updated to 31 August 2018.
References
External links
Profile at Shonan Bellmare
Dinei at Furacao
Loan to Palmeiras at GloboEsporte
1983 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
Campeonato Brasileiro Série A players
Campeonato Brasileiro Série B players
La Liga players
Segunda División players
J1 League players
J2 League players
Club Athletico Paranaense players
Clube Atlético Bragantino players
Esporte Clube Vitória players
Associação Ferroviária de Esportes players
Esporte Clube Noroeste players
Guaratinguetá Futebol players
Sociedade Esportiva Palmeiras players
RC Celta de Vigo players
CD Tenerife players
Kashima Antlers players
Shonan Bellmare players
Ventforet Kofu players
Matsumoto Yamaga FC players
Esporte Clube Água Santa players
Esporte Clube Jacuipense players
Expatriate men's footballers in Spain
Expatriate men's footballers in Japan
Brazilian expatriate sportspeople in Spain
Brazilian expatriate sportspeople in Japan
Men's association football forwards
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https://en.wikipedia.org/wiki/Football%20records%20and%20statistics%20in%20Italy
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This page details football records and statistics in Italy.
Team records
Most championships won
Overall
36, Juventus
Consecutive titles
9, Juventus (2011–12 season to 2019–20 season)
5, Juventus (1930–31 season to 1934–35 season)
5, Torino (1942–43 season and the 1945–46 season to 1948–49 season)
5, Internazionale (2005–06 season to 2009–10 season)
Most seasons in Serie A
91, Internazionale
Most seasons in Serie B
63, Brescia
Most points in a season
2 Teams in Final Round (2 points per win) 1928–29
4, Bologna
6 Teams in Final Round (2 points per win) 1926–27
14, Torino
8 Teams in Final Round (2 points per win) 1927–28 - 1945–46
22, Torino
16 Teams (2 points per win) 1934–35 to 1942–43 - 1967–68 to 1987–88
51, Juventus 1976–77
18 Teams (2 points per win) 1929–30 to 1933–34 - 1952–53 to 1966–67 - 1988–89 to 1993–94
58, Internazionale 1988–89
18 Teams (3 points per win) 1994–95 to 2003–04
82, Milan 2003–04
20 Teams (2 points per win) 1946–47 - 1948–49 to 1951–52
63, Torino 1946–47
20 Teams (3 points per win) 2004–05 to present
102, Juventus 2013–14
21 Teams (2 points per win) 1947–48
65, Torino
Most consecutive wins
17, Internazionale, 2006–07
15, Juventus, 2015–16
13, Napoli, 2016–17 to 2017–18
13, Juventus, 2013–14 to 2014–15
12, Juventus, 2013–14 and 2017–18
11, Roma, 2005–06 and 2012–13 to 2013–14
11, Lazio, 2019–20
11, Internazionale, 2020–21
11, Napoli, 2022–23
10, Juventus, 1931–32 and 2015–16
10, Milan, 1950–51
10, Bologna, 1963–64
10, Napoli, 2017–18
Most consecutive home wins
33, Juventus, 2015–16 to 2016–17
Most consecutive away wins
12, Roma, 2016–17 to 2017–18
Longest win streak from the start of a Serie A season
10, Roma, 2013–14
Longest win streak without conceding from the start of a Serie A season
5, Juventus, 2014–15
Longest win streak from the start of the second half of a Serie A season
11, Internazionale, 2020–21
Most wins in a single season
33, Juventus, 2013–14 (38 matches)
30, Internazionale, 2006–07 (38 matches)
30, Juventus, 2017–18 (38 matches)
29, Juventus, 2015–16 and 2016–17 (38 matches)
29, Torino, 1947–48 (40 matches)
28, Torino, 1946–47 (38 matches)
28, Juventus, 1949–50 and 2018–19 (38 matches)
28, Milan, 2005–06 (38 matches)
28, Roma, 2016–17 (38 matches)
28, Napoli, 2017–18 and 2022–23 (38 matches)
28, Internazionale, 2020–21 (38 matches)
Most home wins in a season
19, Juventus, 2013–14 (19 matches)
Most away wins in a season
16, Milan, 2020–21 (19 matches)
Most matches won
1,696, Juventus
1,568, Internazionale
1,485, Milan
1,298, Roma
1,143, Fiorentina
Most goals scored
5,387, Juventus
5,294, Internazionale
5,010, Milan
4,510, Roma
4,026, Fiorentina
Most goals in a season
21 Teams
125, Torino, 1947–48
20 Teams
118, Milan, 1949–50
18 Teams
95, Fiorentina, 1958–59
16 Teams
75, Juventus, 1942–43
Longest unbeaten streak
58, Milan, 1990–91 to 1992–93 (26 May 1991, 0–0 v Parma; 21 March 1993, 0–1 v Parma)
Longest unbeaten strea
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https://en.wikipedia.org/wiki/Time-varying%20covariate
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A time-varying covariate (also called time-dependent covariate) is a term used in statistics, particularly in survival analysis. It reflects the phenomenon that a covariate is not necessarily constant through the whole study Time-varying covariates are included to represent time-dependent within-individual variation to predict individual responses. For instance, if one wishes to examine the link between area of residence and cancer, this would be complicated by the fact that study subjects move from one area to another. The area of residency could then be introduced in the statistical model as a time-varying covariate. In survival analysis, this would be done by splitting each study subject into several observations, one for each area of residence. For example, if a person is born at time 0 in area A, moves to area B at time 5, and is diagnosed with cancer at time 8, two observations would be made. One with a length of 5 (5 − 0) in area A, and one with a length of 3 (8 − 5) in area B.
References
Survival analysis
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https://en.wikipedia.org/wiki/Holm%E2%80%93Bonferroni%20method
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In statistics, the Holm–Bonferroni method, also called the Holm method or Bonferroni–Holm method, is used to counteract the problem of multiple comparisons. It is intended to control the family-wise error rate (FWER) and offers a simple test uniformly more powerful than the Bonferroni correction. It is named after Sture Holm, who codified the method, and Carlo Emilio Bonferroni.
Motivation
When considering several hypotheses, the problem of multiplicity arises: the more hypotheses are checked, the higher the probability of obtaining Type I errors (false positives). The Holm–Bonferroni method is one of many approaches for controlling the FWER, i.e., the probability that one or more Type I errors will occur, by adjusting the rejection criteria for each of the individual hypotheses.
Formulation
The method is as follows:
Suppose you have p-values, sorted into order lowest-to-highest , and their corresponding hypotheses （null hypotheses). You want the FWER to be no higher than a certain pre-specified significance level .
Is ? If so, reject and continue to the next step, otherwise EXIT.
Is ? If so, reject also, and continue to the next step, otherwise EXIT.
And so on: for each P value, test whether . If so, reject and continue to examine the larger P values, otherwise EXIT.
This method ensures that the FWER is at most , in the strong sense.
Rationale
The simple Bonferroni correction rejects only null hypotheses with p-value less than , in order to ensure that the FWER, i.e., the risk of rejecting one or more true null hypotheses (i.e., of committing one or more type I errors) is at most . The cost of this protection against type I errors is an increased risk of failing to reject one or more false null hypotheses (i.e., of committing one or more type II errors).
The Holm–Bonferroni method also controls the FWER at , but with a lower increase of type II error risk than the classical Bonferroni method. The Holm–Bonferroni method sorts the p-values from lowest to highest and compares them to nominal alpha levels of to (respectively), namely the values .
The index identifies the first p-value that is not low enough to validate rejection. Therefore, the null hypotheses are rejected, while the null hypotheses are not rejected.
If then no p-values were low enough for rejection, therefore no null hypotheses are rejected.
If no such index could be found then all p-values were low enough for rejection, therefore all null hypotheses are rejected (none are accepted).
Proof
Let be the family of hypotheses sorted by their p-values . Let be the set of indices corresponding to the (unknown) true null hypotheses, having members.
Claim: If we wrongly reject some true hypothesis, there is a true hypothesis for which at most .
First note that, in this case, there is at least one true hypothesis, so . Let be such that is the first rejected true hypothesis. Then are all rejected false hypotheses. It follows that and, hence, (1). Since is
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https://en.wikipedia.org/wiki/Convolution%20random%20number%20generator
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In statistics and computer software, a convolution random number generator is a pseudo-random number sampling method that can be used to generate random variates from certain classes of probability distribution. The particular advantage of this type of approach is that it allows advantage to be taken of existing software for generating random variates from other, usually non-uniform, distributions. However, faster algorithms may be obtainable for the same distributions by other more complicated approaches.
A number of distributions can be expressed in terms of the (possibly weighted) sum of two or more random variables from other distributions. (The distribution of the sum is the convolution of the distributions of the individual random variables).
Example
Consider the problem of generating a random variable with an Erlang distribution, . Such a random variable can be defined as the sum of k random variables each with an exponential distribution . This problem is equivalent to generating a random number for a special case of the Gamma distribution, in which the shape parameter takes an integer value.
Notice that:
One can now generate samples using a random number generator for the exponential distribution:
if then
Non-uniform random numbers
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https://en.wikipedia.org/wiki/Honeycomb%20structure
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Honeycomb structures are natural or man-made structures that have the geometry of a honeycomb to allow the minimization of the amount of used material to reach minimal weight and minimal material cost. The geometry of honeycomb structures can vary widely but the common feature of all such structures is an array of hollow cells formed between thin vertical walls. The cells are often columnar and hexagonal in shape. A honeycomb shaped structure provides a material with minimal density and relative high out-of-plane compression properties and out-of-plane shear properties.
Man-made honeycomb structural materials are commonly made by layering a honeycomb material between two thin layers that provide strength in tension. This forms a plate-like assembly. Honeycomb materials are widely used where flat or slightly curved surfaces are needed and their high specific strength is valuable. They are widely used in the aerospace industry for this reason, and honeycomb materials in aluminum, fibreglass and advanced composite materials have been featured in aircraft and rockets since the 1950s. They can also be found in many other fields, from packaging materials in the form of paper-based honeycomb cardboard, to sporting goods like skis and snowboards.
Introduction
Natural honeycomb structures include beehives, honeycomb weathering in rocks, tripe, and bone.
Man-made honeycomb structures include sandwich-structured composites with honeycomb cores. Man-made honeycomb structures are manufactured by using a variety of different materials, depending on the intended application and required characteristics, from paper or thermoplastics, used for low strength and stiffness for low load applications, to high strength and stiffness for high performance applications, from aluminum or fiber reinforced plastics. The strength of laminated or sandwich panels depends on the size of the panel, facing material used and the number or density of the honeycomb cells within it. Honeycomb composites are used widely in many industries, from aerospace industries, automotive and furniture to packaging and logistics.
The material takes its name from its visual resemblance to a bee's honeycomb – a hexagonal sheet structure.
History
The hexagonal comb of the honey bee has been admired and wondered about from ancient times. The first man-made honeycomb, according to Greek mythology, is said to have been manufactured by Daedalus from gold by lost wax casting more than 3000 years ago. Marcus Varro reports that the Greek geometers Euclid and Zenodorus found that the hexagon shape makes most efficient use of space and building materials. The interior ribbing and hidden chambers in the dome of the Pantheon in Rome is an early example of a honeycomb structure.
Galileo Galilei discusses in 1638 the resistance of hollow solids: "Art, and nature even more, makes use of these in thousands of operations in which robustness is increased without adding weight, as is seen in the bones of bir
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https://en.wikipedia.org/wiki/List%20of%20Delta%20Sigma%20Phi%20chapters
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Undergraduate Chapters & New Chapters
This is a list of chapters, active, new and inactive, of Delta Sigma Phi fraternity.
Undergraduate chapter statistics
Since 1899, Delta Sigma Phi has issued 238 charters in 41 states (United States of America), Washington, D.C., and 3 provinces in Canada. Currently, the Fraternity has active chapters and new chapters in 32 states and Washington, D.C. All three former chapters in Canada are dormant. States that have never had the presence of the Fraternity are Alaska, Delaware, Hawaii, Maine, Mississippi, New Hampshire, North Dakota, Vermont, West Virginia.
Alumni associations
There are 20 alumni associations throughout the United States. These are located in:
Atlanta, GA
Boston, MA
Charlotte, NC
Chicago, IL
Denver, CO
Detroit, MI
Houston, TX
Inland Empire, CA (Riverside and San Bernardino counties)
La Verne, CA
New York City, NY
Orange County, CA
Phoenix, AZ
Portland, OR
Salt Lake City, UT
San Diego, CA
San Jose, CA (Silicon Valley)
San Francisco, CA (Bay Area)
St. Louis, MO
Raleigh / Durham, NC (Triangle Area)
Washington, DC
Notes
chapters
Lists of chapters of North American Interfraternity Conference members by society
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https://en.wikipedia.org/wiki/Fatou%27s%20theorem
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In mathematics, specifically in complex analysis, Fatou's theorem, named after Pierre Fatou, is a statement concerning holomorphic functions on the unit disk and their pointwise extension to the boundary of the disk.
Motivation and statement of theorem
If we have a holomorphic function defined on the open unit disk , it is reasonable to ask under what conditions we can extend this function to the boundary of the unit disk. To do this, we can look at what the function looks like on each circle inside the disk centered at 0, each with some radius . This defines a new function:
where
is the unit circle. Then it would be expected that the values of the extension of onto the circle should be the limit of these functions, and so the question reduces to determining when converges, and in what sense, as , and how well defined is this limit. In particular, if the norms of these are well behaved, we have an answer:
Theorem. Let be a holomorphic function such that
where are defined as above. Then converges to some function pointwise almost everywhere and in norm. That is,
Now, notice that this pointwise limit is a radial limit. That is, the limit being taken is along a straight line from the center of the disk to the boundary of the circle, and the statement above hence says that
The natural question is, with this boundary function defined, will we converge pointwise to this function by taking a limit in any other way? That is, suppose instead of following a straight line to the boundary, we follow an arbitrary curve converging to some point on the boundary. Will converge to ? (Note that the above theorem is just the special case of ). It turns out that the curve needs to be non-tangential, meaning that the curve does not approach its target on the boundary in a way that makes it tangent to the boundary of the circle. In other words, the range of must be contained in a wedge emanating from the limit point. We summarize as follows:
Definition. Let be a continuous path such that . Define
That is, is the wedge inside the disk with angle whose axis passes between and zero. We say that converges non-tangentially to , or that it is a non-tangential limit, if there exists such that is contained in and .
Fatou's Theorem. Let Then for almost all
for every non-tangential limit converging to where is defined as above.
Discussion
The proof utilizes the symmetry of the Poisson kernel using the Hardy–Littlewood maximal function for the circle.
The analogous theorem is frequently defined for the Hardy space over the upper-half plane and is proved in much the same way.
See also
Hardy space
References
John B. Garnett, Bounded Analytic Functions, (2006) Springer-Verlag, New York
Walter Rudin. Real and Complex Analysis (1987), 3rd Ed., McGraw Hill, New York.
Elias Stein, Singular integrals and differentiability properties of functions (1970), Princeton University Press, Princeton.
Theorems in complex analysis
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https://en.wikipedia.org/wiki/Leung%20Tsz%20Chun
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Leung Tsz Chun (; born 19 May 1985 in Hong Kong) is a former Hong Kong professional footballer. He played as a striker and as a right-winger.
Career statistics
International
Hong Kong
As of 18 August 2012
Hong Kong U-23
''As of 18 April 2007
Honours
Eastern
Hong Kong Senior Shield: 2007–08
Hong Kong FA Cup: 2013–14
Pegasus
Hong Kong FA Cup: 2009–10
Sun Hei
Hong Kong Senior Shield: 2011–12
External links
1985 births
Living people
Hong Kong men's footballers
Men's association football midfielders
South China AA players
Hong Kong Rangers FC players
Southern District FC players
Hong Kong Sapling players
Hong Kong First Division League players
Hong Kong Premier League players
Eastern Sports Club footballers
Hong Kong Pegasus FC players
Tai Po FC players
Hong Kong men's international footballers
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https://en.wikipedia.org/wiki/Li%20Hang%20Wui
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Li Hang Wui (; born 15 February 1985) is a Hong Kong football coach and a former professional footballer.
He was the captain of Hong Kong Olympic football team in 2007.
Career statistics
International
Hong Kong U-23
As of 21 November 2009
Hong Kong
As of 4 October 2011
External links
Profile at HKFA.com
Profile at doha-2006.com
1985 births
Living people
Hong Kong men's footballers
Hong Kong First Division League players
Hong Kong Premier League players
Hong Kong men's international footballers
Men's association football defenders
Citizen AA players
Kitchee SC players
Resources Capital FC players
Hong Kong FC (football) players
Sun Hei SC players
Metro Gallery FC players
Hong Kong football managers
Footballers at the 2006 Asian Games
Asian Games competitors for Hong Kong
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https://en.wikipedia.org/wiki/Ershad%20Yousefi
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Ershad Yousefi (, born September 19, 1981, in Mashhad, Iran) is an Iranian football goalkeeper who most recently plays for Foolad in Iran Pro League.
Club career
Club Career Statistics
Last Update 25 May 2015
International career
He was the goalkeeper of Iran national under-20 football team at the 2001 FIFA World Youth Championship held in Argentina. In 2002, he was the reserve goalkeeper for Iran national football team at the West Asian Football Federation Championship, but did not make an appearance. In 2005, he was goalkeeper for Iran Under-23 team that participated in the 2005 Islamic Solidarity Games in Saudi Arabia.
Honours
Foolad
Iran Pro League (1): 2013–14
Sepahan
Hazfi Cup (1}: 2003–04
References
Iran Pro League Stats
Iranian men's footballers
Persian Gulf Pro League players
F.C. Aboomoslem players
Sepahan S.C. footballers
Saipa F.C. players
Rahian Kermanshah F.C. players
Foolad F.C. players
Saba Qom F.C. players
Footballers from Mashhad
1981 births
Living people
Asian Games gold medalists for Iran
Asian Games medalists in football
Footballers at the 2002 Asian Games
Medalists at the 2002 Asian Games
Men's association football goalkeepers
21st-century Iranian people
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https://en.wikipedia.org/wiki/Abel%E2%80%93Jacobi%20map
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In mathematics, the Abel–Jacobi map is a construction of algebraic geometry which relates an algebraic curve to its Jacobian variety. In Riemannian geometry, it is a more general construction mapping a manifold to its Jacobi torus.
The name derives from the theorem of Abel and Jacobi that two effective divisors are linearly equivalent if and only if they are indistinguishable under the Abel–Jacobi map.
Construction of the map
In complex algebraic geometry, the Jacobian of a curve C is constructed using path integration. Namely, suppose C has genus g, which means topologically that
Geometrically, this homology group consists of (homology classes of) cycles in C, or in other words, closed loops. Therefore, we can choose 2g loops generating it. On the other hand, another more algebro-geometric way of saying that the genus of C is g is that
where K is the canonical bundle on C.
By definition, this is the space of globally defined holomorphic differential forms on C, so we can choose g linearly independent forms . Given forms and closed loops we can integrate, and we define 2g vectors
It follows from the Riemann bilinear relations that the generate a nondegenerate lattice (that is, they are a real basis for ), and the Jacobian is defined by
The Abel–Jacobi map is then defined as follows. We pick some base point and, nearly mimicking the definition of define the map
Although this is seemingly dependent on a path from to any two such paths define a closed loop in and, therefore, an element of so integration over it gives an element of Thus the difference is erased in the passage to the quotient by . Changing base-point does change the map, but only by a translation of the torus.
The Abel–Jacobi map of a Riemannian manifold
Let be a smooth compact manifold. Let be its fundamental group. Let be its abelianisation map. Let be the torsion subgroup of . Let be the quotient by torsion. If is a surface, is non-canonically isomorphic to , where is the genus; more generally, is non-canonically isomorphic to , where is the first Betti number. Let be the composite homomorphism.
Definition. The cover of the manifold corresponding to the subgroup is called the universal (or maximal) free abelian cover.
Now assume M has a Riemannian metric. Let be the space of harmonic 1-forms on , with dual canonically identified with . By integrating an integral harmonic 1-form along paths from a basepoint , we obtain a map to the circle .
Similarly, in order to define a map without choosing a basis for cohomology, we argue as follows. Let be a point in the universal cover of . Thus is represented by a point of together with a path from to it. By integrating along the path , we obtain a linear form on :
This gives rise a map
which, furthermore, descends to a map
where is the universal free abelian cover.
Definition. The Jacobi variety (Jacobi torus) of is the torus
Definition. The Abel–Jacobi map
is obtained from
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https://en.wikipedia.org/wiki/Nicholas%20J.%20Higham
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Nicholas J. Higham may refer to:
Nicholas Higham (Nicholas John Higham), professor of mathematics at the University of Manchester (UK)
N. J. Higham (Nicholas John 'Nick' Higham), professor emeritus of history at the University of Manchester (UK)
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https://en.wikipedia.org/wiki/Reeb%20vector%20field
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In mathematics, the Reeb vector field, named after the French mathematician Georges Reeb, is a notion that appears in various domains of contact geometry including:
in a contact manifold, given a contact 1-form , the Reeb vector field satisfies ,
in particular, in the context of Sasakian manifold#The Reeb vector field.
References
Contact geometry
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https://en.wikipedia.org/wiki/Generalised%20circle
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In geometry, a generalized circle, sometimes called a cline or circline, is a straight line or a circle.
The natural setting for generalized circles is the extended plane, a plane along with one point at infinity through which every straight line is considered to pass. Given any three distinct points in the extended plane, there exists precisely one generalized circle passing through all three.
Generalized circles sometimes appear in Euclidean geometry, which has a well-defined notion of distance between points, and where every circle has a center and radius: the point at infinity can be considered infinitely distant from any other point, and a line can be considered as a degenerate circle without a well-defined center and with infinite radius (zero curvature). A reflection across a line is a Euclidean isometry (distance-preserving transformation) which maps lines to lines and circles to circles; but an inversion in a circle is not, distorting distances and mapping any line to a circle passing through the reference circles's center, and vice-versa.
However, generalized circles are fundamental to inversive geometry, in which circles and lines are considered indistinguishable, the point at infinity is not distinguished from any other point, and the notions of curvature and distance between points are ignored. In inversive geometry, reflections, inversions, and more generally their compositions, called Möbius transformations, map generalized circles to generalized circles, and preserve the inversive relationships between objects.
The extended plane can be identified with the sphere using a stereographic projection. The point at infinity then becomes an ordinary point on the sphere, and all generalized circles become circles on the sphere.
Extended complex plane
The extended Euclidean plane can be identified with the extended complex plane, so that equations of complex numbers can be used to describe lines, circles and inversions.
Bivariate linear equation
A circle is the set of points in a plane that lie at radius from a center point
In the complex plane, is a complex number and is a set of complex numbers. Using the property that a complex number multiplied by its conjugate is the square of its modulus (its Euclidean distance from the origin), an implicit equation for is:
This is a homogeneous bivariate linear polynomial equation in terms of the complex variable and its conjugate of the form
where coefficients and are real, and and are complex conjugates.
By dividing by and then reversing the steps above, the radius and center can be recovered from any equation of this form. The equation represents a generalized circle in the plane when is real, which occurs when so that the squared radius is positive. When is zero, the equation defines a straight line.
Complex reciprocal
That the reciprocal transformation maps generalized circles to generalized circles is straight-forward to verify:
Lines through the origin
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https://en.wikipedia.org/wiki/Semiautomaton
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In mathematics and theoretical computer science, a semiautomaton is a deterministic finite automaton having inputs but no output. It consists of a set Q of states, a set Σ called the input alphabet, and a function T: Q × Σ → Q called the transition function.
Associated with any semiautomaton is a monoid called the characteristic monoid, input monoid, transition monoid or transition system of the semiautomaton, which acts on the set of states Q. This may be viewed either as an action of the free monoid of strings in the input alphabet Σ, or as the induced transformation semigroup of Q.
In older books like Clifford and Preston (1967) semigroup actions are called "operands".
In category theory, semiautomata essentially are functors.
Transformation semigroups and monoid acts
A transformation semigroup or transformation monoid is a pair consisting of a set Q (often called the "set of states") and a semigroup or monoid M of functions, or "transformations", mapping Q to itself. They are functions in the sense that every element m of M is a map . If s and t are two functions of the transformation semigroup, their semigroup product is defined as their function composition .
Some authors regard "semigroup" and "monoid" as synonyms. Here a semigroup need not have an identity element; a monoid is a semigroup with an identity element (also called "unit"). Since the notion of functions acting on a set always includes the notion of an identity function, which when applied to the set does nothing, a transformation semigroup can be made into a monoid by adding the identity function.
M-acts
Let M be a monoid and Q be a non-empty set. If there exists a multiplicative operation
which satisfies the properties
for 1 the unit of the monoid, and
for all and , then the triple is called a right M-act or simply a right act. In long-hand, is the right multiplication of elements of Q by elements of M. The right act is often written as .
A left act is defined similarly, with
and is often denoted as .
An M-act is closely related to a transformation monoid. However the elements of M need not be functions per se, they are just elements of some monoid. Therefore, one must demand that the action of be consistent with multiplication in the monoid (i.e. ), as, in general, this might not hold for some arbitrary , in the way that it does for function composition.
Once one makes this demand, it is completely safe to drop all parenthesis, as the monoid product and the action of the monoid on the set are completely associative. In particular, this allows elements of the monoid to be represented as strings of letters, in the computer-science sense of the word "string". This abstraction then allows one to talk about string operations in general, and eventually leads to the concept of formal languages as being composed of strings of letters.
Another difference between an M-act and a transformation monoid is that for an M-act Q, two distinct elements of the monoid m
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https://en.wikipedia.org/wiki/St%20Marylebone%20School
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Saint Marylebone School is a secondary school for girls in Marylebone, London. It specialises in Performing Arts, General Arts, Maths & Computing. In the sixth form, boys can attend as well. The school then became a converter academy, having previously been judged as "outstanding in every respect" by Ofsted.
Founded in 1791, Saint Marylebone Church of England School is now a multi-faith comprehensive school for girls aged from eleven to eighteen. The main site is located just behind St Marylebone Parish Church, with the Sixth Form Centre based in another building nearby at Blandford Street.
History
The St Marylebone School began as the Marylebone "Day School of Industry," founded in 1791 in what was then Paradise Street, now Moxon Street, to educate the children of the poor in the parish. Boys and girls were taught skills such as needlework and straw plaiting. The school was funded by donations, charity sermons and income from the children's handiwork. In 1808, with the support of local philanthropist and social reformer Sir Thomas Bernard, the school moved to 82 Marylebone High Street, which is now a boutique store. Subsequently, to make room for growing numbers, it moved to a site on Paddington Street, which is identifiable today as a Mission Church. Then in 1858 the 5th Duke of Portland bought a plot of ground near the top of Marylebone High Street and covenanted the site to be used for a girls' school in perpetuity. The main site of the school has been there ever since.
The Day British School of Industry had been incorporated with Sir Thomas Bernard's school under the direction of the Governor of the Church of England's United National Schools. In 1858, it became known as Central National School, to distinguish it from the Eastern (now All Souls CE Primary) and Western National Schools (now St Mary's Bryanston Square CE Primary) founded in 1824 at nearby parishes.
The boys' section was eventually closed and it became a girls' school, adopting its current name. In the 1960s-70s the school used a building in Penfold Street, about 15 minutes from the main site, for domestic science lessons; this building is now used by the Westminster Youth Service. In 2005, the sixth form moved to part of a building that had housed a convent; in 2008-9 this was demolished and rebuilt as a five-story, university-style Sixth Form Centre.
During the school's grant-maintained period, it was highly selective and the school used to interview parents and prospective pupils.
Between 2005 and 2010, the main site saw extensive building and refurbishment work. Major new facilities were opened in 2007, including a below-ground gymnasium and dance studios as well as a music recording studio space and a three-story visual and performing arts space. Since 2013, the school's studio has been the main filming spot for Spirit Young Performers Company. Popular videos shot at this location include "Little Miss High and Mighty" and "Hard Knock Life".
Houses and local connect
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https://en.wikipedia.org/wiki/Goro%20Azumaya
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was a Japanese mathematician who introduced the notion of Azumaya algebra in 1951. His advisor was Shokichi Iyanaga. At the time of his death he was an emeritus professor at Indiana University.
References
External links
Biography of Azumaya by BiRep, Bielefeld University
1920 births
20th-century Japanese mathematicians
21st-century Japanese mathematicians
Algebraists
Indiana University faculty
2010 deaths
Nagoya University alumni
Japanese expatriates in the United States
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https://en.wikipedia.org/wiki/Real%20structure
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In mathematics, a real structure on a complex vector space is a way to decompose the complex vector space in the direct sum of two real vector spaces. The prototype of such a structure is the field of complex numbers itself, considered as a complex vector space over itself and with the conjugation map , with , giving the "canonical" real structure on , that is .
The conjugation map is antilinear: and .
Vector space
A real structure on a complex vector space V is an antilinear involution . A real structure defines a real subspace , its fixed locus, and the natural map
is an isomorphism. Conversely any vector space that is the complexification
of a real vector space has a natural real structure.
One first notes that every complex space V has a realification obtained by taking the same vectors as in the original set and restricting the scalars to be real. If and then the vectors and are linearly independent in the realification of V. Hence:
Naturally, one would wish to represent V as the direct sum of two real vector spaces, the "real and imaginary parts of V". There is no canonical way of doing this: such a splitting is an additional real structure in V. It may be introduced as follows. Let be an antilinear map such that , that is an antilinear involution of the complex space V.
Any vector can be written ,
where and .
Therefore, one gets a direct sum of vector spaces where:
and .
Both sets and are real vector spaces. The linear map , where , is an isomorphism of real vector spaces, whence:
.
The first factor is also denoted by and is left invariant by , that is . The second factor is
usually denoted by . The direct sum reads now as:
,
i.e. as the direct sum of the "real" and "imaginary" parts of V. This construction strongly depends on the choice of an antilinear involution of the complex vector space V. The complexification of the real vector space , i.e.,
admits
a natural real structure and hence is canonically isomorphic to the direct sum of two copies of :
.
It follows a natural linear isomorphism between complex vector spaces with a given real structure.
A real structure on a complex vector space V, that is an antilinear involution , may be equivalently described in terms of the linear map from the vector space to the complex conjugate vector space defined by
.
Algebraic variety
For an algebraic variety defined over a subfield of the real numbers,
the real structure is the complex conjugation acting on the points of the variety in complex projective or affine space.
Its fixed locus is the space of real points of the variety (which may be empty).
Scheme
For a scheme defined over a subfield of the real numbers, complex conjugation
is in a natural way a member of the Galois group of the algebraic closure of the base field.
The real structure is the Galois action of this conjugation on the extension of the
scheme over the algebraic closure of the base field.
The real points are the points whose resi
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https://en.wikipedia.org/wiki/Weitzenb%C3%B6ck%27s%20inequality
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In mathematics, Weitzenböck's inequality, named after Roland Weitzenböck, states that for a triangle of side lengths , , , and area , the following inequality holds:
Equality occurs if and only if the triangle is equilateral. Pedoe's inequality is a generalization of Weitzenböck's inequality. The Hadwiger–Finsler inequality is a strengthened version of Weitzenböck's inequality.
Geometric interpretation and proof
Rewriting the inequality above allows for a more concrete geometric interpretation, which in turn provides an immediate proof.
Now the summands on the left side are the areas of equilateral triangles erected over the sides of the original triangle and hence the inequation states that the sum of areas of the equilateral triangles is always greater than or equal to threefold the area of the original triangle.
This can now can be shown by replicating area of the triangle three times within the equilateral triangles. To achieve that the Fermat point is used to partition the triangle into three obtuse subtriangles with a angle and each of those subtriangles is replicated three times within the equilateral triangle next to it. This only works if every angle of the triangle is smaller than , since otherwise the Fermat point is not located in the interior of the triangle and becomes a vertex instead. However if one angle is greater or equal to it is possible to replicate the whole triangle three times within the largest equilateral triangle, so the sum of areas of all equilateral triangles stays greater than the threefold area of the triangle anyhow.
Further proofs
The proof of this inequality was set as a question in the International Mathematical Olympiad of 1961. Even so, the result is not too difficult to derive using Heron's formula for the area of a triangle:
First method
It can be shown that the area of the inner Napoleon's triangle, which must be nonnegative, is
so the expression in parentheses must be greater than or equal to 0.
Second method
This method assumes no knowledge of inequalities except that all squares are nonnegative.
and the result follows immediately by taking the positive square root of both sides. From the first inequality we can also see that equality occurs only when and the triangle is equilateral.
Third method
This proof assumes knowledge of the AM–GM inequality.
As we have used the arithmetic-geometric mean inequality, equality only occurs when and the triangle is equilateral.
Fourth method
Write so the sum and i.e. . But , so .
See also
List of triangle inequalities
Isoperimetric inequality
Hadwiger–Finsler inequality
Notes
References & further reading
Claudi Alsina, Roger B. Nelsen: When Less is More: Visualizing Basic Inequalities. MAA, 2009, , pp. 84-86
Claudi Alsina, Roger B. Nelsen: Geometric Proofs of the Weitzenböck and Hadwiger–Finsler Inequalities. Mathematics Magazine, Vol. 81, No. 3 (Jun., 2008), pp. 216–219 (JSTOR)
D. M. Batinetu-Giurgiu, Nicusor Minculete, N
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https://en.wikipedia.org/wiki/Mathematical%20maturity
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In mathematics, mathematical maturity is an informal term often used to refer to the quality of having a general understanding and mastery of the way mathematicians operate and communicate. It pertains to a mixture of mathematical experience and insight that cannot be directly taught. Instead, it comes from repeated exposure to mathematical concepts. It is a gauge of mathematics students' erudition in mathematical structures and methods, and can overlap with other related concepts such as mathematical intuition and mathematical competence. The topic is occasionally also addressed in literature in its own right.
Definitions
Mathematical maturity has been defined in several different ways by various authors, and is often tied to other related concepts such as comfort and competence with mathematics, mathematical intuition and mathematical beliefs.
One definition has been given as follows:
A broader list of characteristics of mathematical maturity has been given as follows:
Finally, mathematical maturity has also been defined as an ability to do the following:
It is sometimes said that the development of mathematical maturity requires a deep reflection on the subject matter for a prolonged period of time, along with a guiding spirit which encourages exploration.
Progression
Mathematician Terence Tao has proposed a three-stage model of mathematics education that can be interpreted as a general framework of mathematical maturity progression. The stages are summarized in the following table:
See also
Logical intuition
Four stages of competence
References
Mathematics and culture
Skills
Literacy
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https://en.wikipedia.org/wiki/Christianity%20in%20Colombia
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The National Administrative Department of Statistics (DANE) does not collect religious statistics, and accurate reports are difficult to obtain. However, based on various studies and a survey, about 90% of the population adheres to Christianity, the majority of which (70.9%) are Roman Catholic, while a significant minority (16.7%) adhere to Protestantism (primarily Evangelicalism) and other Christian groups.
Roman Catholic archdioceses and other dioceses (in brackets)
Barranquilla: (El Banco, Riohacha, Santa Marta, Valledupar)
Bogotá: (Engativá, Facatativá, Fontibón, Girardot, Soacha, Zipaquirá)
Bucaramanga: (Barrancabermeja, Málaga-Soatá, Socorro y San Gil, Vélez)
Cali: (Buenaventura, Buga, Cartago, Palmira)
Cartagena: (Magangué, Montelibano, Montería, Sincelejo)
Ibagué: (Espinal, Florencia, Garzón, Líbano-Honda, Neiva)
Manizales: (Armenia, La Dorada-Guaduas, Pereira)
Medellín: (Caldas, Girardota, Jericó, Sonsón-Rionegro)
Nueva Pamplona: (Arauca, Cúcuta, Ocaña, Tibú)
Popayán: (Ipiales, Mocoa-Sibundoy, Pasto, Tumaco)
Santa Fe de Antioquia: (Apartadó, Istmina-Tadó, Quibdó, Santa Rosa de Osos)
Tunja: (Chiquinquirá, Duitama-Sogamoso, Garagoa, Yopal)
Villavicencio: (Granada en Colombia, San José del Guaviare)
Other Churches
Protestantism, primarily Evangelicalism, represents 14% of the population in 2022; international NGOs have stated that indigenous Protestants face threats, harassment and arbitrary detention in their communities due to their religious beliefs.
The Episcopal Diocese of Colombia is a part of Province 9 of the Episcopal Church in the United States of America.
The Church of Jesus Christ of Latter-day Saints in Colombia claims 209,985 members in Colombia.
There is a small Greek Orthodox community in the country.
Freedom of religion
The constitution provides for freedom of religion. However, international NGOs have noted difficulties for indigenous Christians; in particular, indigenous authorities in the Pizarro and Litoral de San Juan municipalities in the Chocó Department have banned the practice of Christianity, and Protestants in particular face threats, harassment and arbitrary detention in their communities due to their religious beliefs.
In 2023, the country was scored 4 out of 4 for religious freedom.
In the same year, the country was rank as the 22nd most difficult place in the world to be a Christian.
See also
Religion in Colombia
References
Sources
Status of religious freedom in Colombia article
Colombia article
US State Dept 2004 report
Catholic Hierarchy website
ar:المسيحية في بنما
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https://en.wikipedia.org/wiki/Fagnano%27s%20problem
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In geometry, Fagnano's problem is an optimization problem that was first stated by Giovanni Fagnano in 1775:
The solution is the orthic triangle, with vertices at the base points of the altitudes of the given triangle.
Solution
The orthic triangle, with vertices at the base points of the altitudes of the given triangle, has the smallest perimeter of all triangles inscribed into an acute triangle, hence it is the solution of Fagnano's problem. Fagnano's original proof used calculus methods and an intermediate result given by his father Giulio Carlo de' Toschi di Fagnano. Later however several geometric proofs were discovered as well, amongst others by Hermann Schwarz and Lipót Fejér. These proofs use the geometrical properties of reflections to determine some minimal path representing the perimeter.
Physical principles
A solution from physics is found by imagining putting a rubber band that follows Hooke's Law around the three sides of a triangular frame , such that it could slide around smoothly. Then the rubber band would end up in a position that minimizes its elastic energy, and therefore minimize its total length. This position gives the minimal perimeter triangle.
The tension inside the rubber band is the same everywhere in the rubber band, so in its resting position, we have, by Lami's theorem,
Therefore, this minimal triangle is the orthic triangle.
See also
Set TSP problem, a more general task of visiting each of a family of sets by the shortest tour
References
Heinrich Dörrie: 100 Great Problems of Elementary Mathematics: Their History and Solution. Dover Publications 1965, p. 359-360. , problem 90 (restricted online version (Google Books))
Paul J. Nahin: When Least is Best: How Mathematicians Discovered Many Clever Ways to Make Things as Small (or as Large) as Possible. Princeton University Press 2004, , p. 67
Coxeter, H. S. M.; Greitzer, S. L.:Geometry Revisited. Washington, DC: Math. Assoc. Amer. 1967, pp. 88–89.
H.A. Schwarz: Gesammelte Mathematische Abhandlungen, vol. 2. Berlin 1890, pp. 344-345. (online at the Internet Archive, German)
External links
Fagnano's problem at cut-the-knot
Fagnano's problem in the Encyclopaedia of Mathematics
Fagnano's problem at a website for triangle geometry
Triangle problems
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https://en.wikipedia.org/wiki/Convex%20metric%20space
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In mathematics, convex metric spaces are, intuitively, metric spaces with the property any "segment" joining two points in that space has other points in it besides the endpoints.
Formally, consider a metric space (X, d) and let x and y be two points in X. A point z in X is said to be between x and y if all three points are distinct, and
that is, the triangle inequality becomes an equality. A convex metric space is a metric space (X, d) such that, for any two distinct points x and y in X, there exists a third point z in X lying between x and y.
Metric convexity:
does not imply convexity in the usual sense for subsets of Euclidean space (see the example of the rational numbers)
nor does it imply path-connectedness (see the example of the rational numbers)
nor does it imply geodesic convexity for Riemannian manifolds (consider, for example, the Euclidean plane with a closed disc removed).
Examples
Euclidean spaces, that is, the usual three-dimensional space and its analogues for other dimensions, are convex metric spaces. Given any two distinct points and in such a space, the set of all points satisfying the above "triangle equality" forms the line segment between and which always has other points except and in fact, it has a continuum of points.
Any convex set in a Euclidean space is a convex metric space with the induced Euclidean norm. For closed sets the converse is also true: if a closed subset of a Euclidean space together with the induced distance is a convex metric space, then it is a convex set (this is a particular case of a more general statement to be discussed below).
A circle is a convex metric space, if the distance between two points is defined as the length of the shortest arc on the circle connecting them.
Metric segments
Let be a metric space (which is not necessarily convex). A subset of is called a metric segment between two distinct points and in if there exists a closed interval on the real line and an isometry
such that and
It is clear that any point in such a metric segment except for the "endpoints" and is between and As such, if a metric space admits metric segments between any two distinct points in the space, then it is a convex metric space.
The converse is not true, in general. The rational numbers form a convex metric space with the usual distance, yet there exists no segment connecting two rational numbers which is made up of rational numbers only. If however, is a convex metric space, and, in addition, it is complete, one can prove that for any two points in there exists a metric segment connecting them (which is not necessarily unique).
Convex metric spaces and convex sets
As mentioned in the examples section, closed subsets of Euclidean spaces are convex metric spaces if and only if they are convex sets. It is then natural to think of convex metric spaces as generalizing the notion of convexity beyond Euclidean spaces, with usual linear segments replaced by
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https://en.wikipedia.org/wiki/Pac-12%20Conference%20football%20statistics
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Historical statistics for football in the Pacific Coast Conference (PCC, 1915-1959), Athletic Association of Western Universities (AAWU, 1959–68), Pacific-8 (1968–78), Pacific-10 (1978-2011), and Pac-12 Conference (2011–present).
Season finishes
References
statistics
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https://en.wikipedia.org/wiki/Residue%20field
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In mathematics, the residue field is a basic construction in commutative algebra. If R is a commutative ring and m is a maximal ideal, then the residue field is the quotient ring k = R/m, which is a field. Frequently, R is a local ring and m is then its unique maximal ideal.
This construction is applied in algebraic geometry, where to every point x of a scheme X one associates its residue field k(x). One can say a little loosely that the residue field of a point of an abstract algebraic variety is the 'natural domain' for the coordinates of the point.
Definition
Suppose that R is a commutative local ring, with maximal ideal m. Then the residue field is the quotient ring R/m.
Now suppose that X is a scheme and x is a point of X. By the definition of scheme, we may find an affine neighbourhood U = Spec(A), with A some commutative ring. Considered in the neighbourhood U, the point x corresponds to a prime ideal p ⊆ A (see Zariski topology). The local ring of X in x is by definition the localization R = Ap, with the maximal ideal m = p·Ap. Applying the construction above, we obtain the residue field of the point x :
k(x) := Ap / p·Ap.
One can prove that this definition does not depend on the choice of the affine neighbourhood U.
A point is called K-rational for a certain field K, if k(x) = K.
Example
Consider the affine line A1(k) = Spec(k[t]) over a field k. If k is algebraically closed, there are exactly two types of prime ideals, namely
(t − a), a ∈ k
(0), the zero-ideal.
The residue fields are
, the function field over k in one variable.
If k is not algebraically closed, then more types arise, for example if k = R, then the prime ideal (x2 + 1) has residue field isomorphic to C.
Properties
For a scheme locally of finite type over a field k, a point x is closed if and only if k(x) is a finite extension of the base field k. This is a geometric formulation of Hilbert's Nullstellensatz. In the above example, the points of the first kind are closed, having residue field k, whereas the second point is the generic point, having transcendence degree 1 over k.
A morphism Spec(K) → X, K some field, is equivalent to giving a point x ∈ X and an extension K/k(x).
The dimension of a scheme of finite type over a field is equal to the transcendence degree of the residue field of the generic point.
References
Further reading
, section II.2
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https://en.wikipedia.org/wiki/Janko%20group%20J1
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{{DISPLAYTITLE:Janko group J1}}
In the area of modern algebra known as group theory, the Janko group J1 is a sporadic simple group of order
233571119 = 175560
≈ 2.
History
J1 is one of the 26 sporadic groups and was originally described by Zvonimir Janko in 1965. It is the only Janko group whose existence was proved by Janko himself and was the first sporadic group to be found since the discovery of the Mathieu groups in the 19th century. Its discovery launched the modern theory of sporadic groups.
In 1986 Robert A. Wilson showed that J1 cannot be a subgroup of the monster group. Thus it is one of the 6 sporadic groups called the pariahs.
Properties
The smallest faithful complex representation of J1 has dimension 56. J1 can be characterized abstractly as the unique simple group with abelian 2-Sylow subgroups and with an involution whose centralizer is isomorphic to the direct product of the group of order two and the alternating group A5 of order 60, which is to say, the rotational icosahedral group. That was Janko's original conception of the group.
In fact Janko and Thompson were investigating groups similar to the Ree groups 2G2(32n+1), and showed that if a simple group G has abelian Sylow 2-subgroups and a centralizer of an involution of the form Z/2Z×PSL2(q) for q a prime power at least 3, then either q is a power of 3 and G has the same order as a Ree group (it was later shown that G must be a Ree group in this case) or q is 4 or 5. Note that PSL2(4)=PSL2(5)=A5. This last exceptional case led to the Janko group J1.
J1 is the automorphism group of the Livingstone graph, a distance-transitive graph with 266 vertices and 1463 edges.
J1 has no outer automorphisms and its Schur multiplier is trivial.
J1 is contained in the O'Nan group as the subgroup of elements fixed by an outer automorphism of order 2.
Construction
Janko found a modular representation in terms of 7 × 7 orthogonal matrices in the field of eleven elements, with generators given by
and
Y has order 7 and Z has order 5. Janko (1966) credited W. A. Coppel for recognizing this representation as an embedding into Dickson's simple group G2(11) (which has a 7-dimensional representation over the field with 11 elements).
There is also a pair of generators a, b such that
a2=b3=(ab)7=(abab−1)10=1
J1 is thus a Hurwitz group, a finite homomorphic image of the (2,3,7) triangle group.
Maximal subgroups
Janko (1966) found the 7 conjugacy classes of maximal subgroups of J1 shown in the table. Maximal simple subgroups of order 660 afford J1 a permutation representation of degree 266. He found that there are 2 conjugacy classes of subgroups isomorphic to the alternating group A5, both found in the simple subgroups of order 660. J1 has non-abelian simple proper subgroups of only 2 isomorphism types.
The notation A.B means a group with a normal subgroup A with quotient B, and
D2n is the dihedral group of order 2n.
Number of elements of each order
The greatest order of any ele
|
https://en.wikipedia.org/wiki/Janko%20group%20J3
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{{DISPLAYTITLE:Janko group J3}}
In the area of modern algebra known as group theory, the Janko group J3 or the Higman-Janko-McKay group HJM is a sporadic simple group of order
273551719 = 50232960.
History and properties
J3 is one of the 26 Sporadic groups and was predicted by Zvonimir Janko in 1969 as one of two new simple groups having 21+4:A5 as a centralizer of an involution (the other is the Janko group J2).
J3 was shown to exist by .
In 1982 R. L. Griess showed that J3 cannot be a subquotient of the monster group. Thus it is one of the 6 sporadic groups called the pariahs.
J3 has an outer automorphism group of order 2 and a Schur multiplier of order 3, and its triple cover has a unitary 9-dimensional representation over the finite field with 4 elements. constructed it via an underlying geometry. It has a modular representation of dimension eighteen over the finite field with 9 elements.
It has a complex projective representation of dimension eighteen.
Presentations
In terms of generators a, b, c, and d its automorphism group J3:2 can be presented as
A presentation for J3 in terms of (different) generators a, b, c, d is
Constructions
J3 can be constructed by many different generators. Two from the ATLAS list are 18x18 matrices over the finite field of order 9, with matrix multiplication carried out with finite field arithmetic:
and
Maximal subgroups
found the 9 conjugacy classes of maximal subgroups of J3 as follows:
PSL(2,16):2, order 8160
PSL(2,19), order 3420
PSL(2,19), conjugate to preceding class in J3:2
24: (3 × A5), order 2880
PSL(2,17), order 2448
(3 × A6):22, order 2160 - normalizer of subgroup of order 3
32+1+2:8, order 1944 - normalizer of Sylow 3-subgroup
21+4:A5, order 1920 - centralizer of involution
22+4: (3 × S3), order 1152
References
R. L. Griess, Jr., The Friendly Giant, Inventiones Mathematicae 69 (1982), 1-102. p. 93: proof that J3 is a pariah.
Z. Janko, Some new finite simple groups of finite order, 1969 Symposia Mathematica (INDAM, Rome, 1967/68), Vol. 1 pp. 25–64 Academic Press, London, and in The theory of finite groups (Edited by Brauer and Sah) p. 63-64, Benjamin, 1969.
External links
MathWorld: Janko Groups
Atlas of Finite Group Representations: J3 version 2
Atlas of Finite Group Representations: J3 version 3
Sporadic groups
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https://en.wikipedia.org/wiki/Janko%20group%20J4
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{{DISPLAYTITLE:Janko group J4}}
In the area of modern algebra known as group theory, the Janko group J4 is a sporadic simple group of order
22133571132329313743
= 86775571046077562880
≈ 9.
History
J4 is one of the 26 Sporadic groups. Zvonimir Janko found J4 in 1975 by studying groups with an involution centralizer of the form 21 + 12.3.(M22:2). Its existence and uniqueness was shown using computer calculations by Simon P. Norton and others in 1980. It has a modular representation of dimension 112 over the finite field with 2 elements and is the stabilizer of a certain 4995 dimensional subspace of the exterior square, a fact which Norton used to construct it, and which is the easiest way to deal with it computationally. and gave computer-free proofs of uniqueness. and gave a computer-free proof of existence by constructing it as an amalgams of groups 210:SL5(2) and (210:24:A8):2 over a group 210:24:A8.
The Schur multiplier and the outer automorphism group are both trivial.
Since 37 and 43 are not supersingular primes, J4 cannot be a subquotient of the monster group. Thus it is one of the 6 sporadic groups called the pariahs.
Representations
The smallest faithful complex representation has dimension 1333; there are two complex conjugate representations of this dimension. The smallest faithful representation over any field is a 112 dimensional representation over the field of 2 elements.
The smallest permutation representation is on 173067389 points, with point stabilizer of the form 211M24. These points can be identified with certain "special vectors" in the 112 dimensional representation.
Presentation
It has a presentation in terms of three generators a, b, and c as
Maximal subgroups
found the 13 conjugacy classes of maximal subgroups of J4 as follows:
211:M24 - containing Sylow 2-subgroups and Sylow 3-subgroups; also containing 211:(M22:2), centralizer of involution of class 2B
21+12.3.(M22:2) - centralizer of involution of class 2A - containing Sylow 2-subgroups and Sylow 3-subgroups
210:PSL(5,2)
23+12.(S5 × PSL(3,2)) - containing Sylow 2-subgroups
U3(11):2
M22:2
111+2:(5 × GL(2,3)) - normalizer of Sylow 11-subgroup
PSL(2,32):5
PGL(2,23)
U3(3) - containing Sylow 3-subgroups
29:28 Frobenius group
43:14 Frobenius group
37:12 Frobenius group
A Sylow 3-subgroup is a Heisenberg group: order 27, non-abelian, all non-trivial elements of order 3.
References
D.J. Benson The simple group J4, PhD Thesis, Cambridge 1981, https://web.archive.org/web/20110610013308/http://www.maths.abdn.ac.uk/~bensondj/papers/b/benson/the-simple-group-J4.pdf
Z. Janko, A new finite simple group of order 86,775,570,046,077,562,880 which possesses M24 and the full covering group of M22 as subgroups, J. Algebra 42 (1976) 564-596. (The title of this paper is incorrect, as the full covering group of M22 was later discovered to be larger: center of order 12, not 6.)
S. P. Norton The construction of J4 in The Santa Cruz conference on
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https://en.wikipedia.org/wiki/Janko%20group%20J2
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{{DISPLAYTITLE:Janko group J2}}
In the area of modern algebra known as group theory, the Janko group J2 or the Hall-Janko group HJ is a sporadic simple group of order
2733527 = 604800
≈ 6.
History and properties
J2 is one of the 26 Sporadic groups and is also called Hall–Janko–Wales group. In 1969 Zvonimir Janko predicted J2 as one of two new simple groups having 21+4:A5 as a centralizer of an involution (the other is the Janko group J3). It was constructed by as a rank 3 permutation group on 100 points.
Both the Schur multiplier and the outer automorphism group have order 2. As a permutation group on 100 points J2 has involutions moving all 100 points and involutions moving just 80 points. The former involutions are
products of 25 double transportions, an odd number, and hence lift to 4-elements in the double cover 2.A100. The double cover 2.J2 occurs as a subgroup of the Conway group Co0.
J2 is the only one of the 4 Janko groups that is a subquotient of the monster group; it is thus part of what Robert Griess calls the Happy Family. Since it is also found in the Conway group Co1, it is therefore part of the second generation of the Happy Family.
Representations
It is a subgroup of index two of the group of automorphisms of the Hall–Janko graph, leading to a permutation representation of degree 100. It is also a subgroup of index two of the group of automorphisms of the Hall–Janko Near Octagon, leading to a permutation representation of degree 315.
It has a modular representation of dimension six over the field of four elements; if in characteristic two we have , then J2 is generated by the two matrices
and
These matrices satisfy the equations
(Note that matrix multiplication on a finite field of order 4 is defined slightly differently from ordinary matrix multiplication. See for the specific addition and multiplication tables, with w the same as a and w the same as 1 + a.)
J2 is thus a Hurwitz group, a finite homomorphic image of the (2,3,7) triangle group.
The matrix representation given above constitutes an embedding into Dickson's group G2(4). There is only one conjugacy class of J2 in G2(4). Every subgroup J2 contained in G2(4) extends to a subgroup J2:2= Aut(J2) in G2(4):2= Aut(G2(4)) (G2(4) extended by the field automorphisms of F4). G2(4) is in turn isomorphic to a subgroup of the Conway group Co1.
Maximal subgroups
There are 9 conjugacy classes of maximal subgroups of J2. Some are here described in terms of action on the Hall–Janko graph.
U3(3) order 6048 – one-point stabilizer, with orbits of 36 and 63
Simple, containing 36 simple subgroups of order 168 and 63 involutions, all conjugate, each moving 80 points. A given involution is found in 12 168-subgroups, thus fixes them under conjugacy. Its centralizer has structure 4.S4, which contains 6 additional involutions.
3.PGL(2,9) order 2160 – has a subquotient A6
21+4:A5 order 1920 – centralizer of involution moving 80 points
22+4:(3 × S3) order 1152
A4 × A5 or
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https://en.wikipedia.org/wiki/Karamata%27s%20inequality
|
In mathematics, Karamata's inequality, named after Jovan Karamata, also known as the majorization inequality, is a theorem in elementary algebra for convex and concave real-valued functions, defined on an interval of the real line. It generalizes the discrete form of Jensen's inequality, and generalizes in turn to the concept of Schur-convex functions.
Statement of the inequality
Let be an interval of the real line and let denote a real-valued, convex function defined on . If and are numbers in such that majorizes , then
Here majorization means that and satisfies
and we have the inequalities
and the equality
If is a strictly convex function, then the inequality () holds with equality if and only if we have for all .
Remarks
If the convex function is non-decreasing, then the proof of () below and the discussion of equality in case of strict convexity shows that the equality () can be relaxed to
The inequality () is reversed if is concave, since in this case the function is convex.
Example
The finite form of Jensen's inequality is a special case of this result. Consider the real numbers and let
denote their arithmetic mean. Then majorizes the -tuple , since the arithmetic mean of the largest numbers of is at least as large as the arithmetic mean of all the numbers, for every . By Karamata's inequality () for the convex function ,
Dividing by gives Jensen's inequality. The sign is reversed if is concave.
Proof of the inequality
We may assume that the numbers are in decreasing order as specified in ().
If for all , then the inequality () holds with equality, hence we may assume in the following that for at least one .
If for an , then the inequality () and the majorization properties () and () are not affected if we remove and . Hence we may assume that for all .
It is a property of convex functions that for two numbers in the interval the slope
of the secant line through the points and of the graph of is a monotonically non-decreasing function in for fixed (and vice versa). This implies that
for all . Define and
for all . By the majorization property (), for all and by (), . Hence,
which proves Karamata's inequality ().
To discuss the case of equality in (), note that by () and our assumption for all . Let be the smallest index such that , which exists due to (). Then . If is strictly convex, then there is strict inequality in (), meaning that . Hence there is a strictly positive term in the sum on the right hand side of () and equality in () cannot hold.
If the convex function is non-decreasing, then . The relaxed condition () means that , which is enough to conclude that in the last step of ().
If the function is strictly convex and non-decreasing, then . It only remains to discuss the case . However, then there is a strictly positive term on the right hand side of () and equality in () cannot hold.
References
External links
An explanation of Karamata's inequality and
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https://en.wikipedia.org/wiki/Inferential
|
Inferential may refer to:
Inferential statistics; see statistical inference
Inference (logic)
Inferential mood (grammar)
Inferential programming
Inferential role semantics
Inferential theory of learning
Informal inferential reasoning
Simple non-inferential passage
|
https://en.wikipedia.org/wiki/Zuckerman%20functor
|
In mathematics, a Zuckerman functor is used to construct representations of real reductive Lie groups from representations of Levi subgroups. They were introduced by Gregg Zuckerman (1978). The Bernstein functor is closely related.
Notation and terminology
G is a connected reductive real affine algebraic group (for simplicity; the theory works for more general groups), and g is the Lie algebra of G. K is a maximal compact subgroup of G.
L is a Levi subgroup of G, the centralizer of a compact connected abelian subgroup, and *l is the Lie algebra of L.
A representation of K is called K-finite if every vector is contained in a finite-dimensional representation of K. Denote by WK the subspace of K-finite vectors of a representation W of K.
A (g,K)-module is a vector space with compatible actions of g and K, on which the action of K is K-finite.
R(g,K) is the Hecke algebra of G of all distributions on G with support in K that are left and right K finite. This is a ring which does not have an identity but has an approximate identity, and the approximately unital R(g,K)- modules are the same as (g,K) modules.
Definition
The Zuckerman functor Γ is defined by
and the Bernstein functor Π is defined by
References
David A. Vogan, Representations of real reductive Lie groups,
Anthony W. Knapp, David A. Vogan, Cohomological induction and unitary representations, prefacereview by Dan Barbasch
David A. Vogan, Unitary Representations of Reductive Lie Groups. (AM-118) (Annals of Mathematics Studies)
Gregg J. Zuckerman, Construction of representations via derived functors, unpublished lecture series at the Institute for Advanced Study, 1978.
Representation theory
Functors
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https://en.wikipedia.org/wiki/Rathowen
|
Rathowen () is a small village in County Westmeath, Ireland, on the N4 national primary route. Rathowen was designated as a census town by the Central Statistics Office for the first time in the 2016 census, at which time it had a population of 150 people.
The village is around 20 km northwest of Mullingar, 20 km southeast of Longford Town, and 100 km northwest of Dublin city centre.
Transport
Street and Rathowen railway station was opened on 1 August 1877 and finally closed on 17 June 1963.
See also
List of towns and villages in Ireland
References
Towns and villages in County Westmeath
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https://en.wikipedia.org/wiki/Abbas%20Edalat
|
Abbas Edalat () is a British-Iranian academic who is a professor of computer science and mathematics at the Department of Computing, Imperial College London and a political activist. In a 2018 letter to The Guardian, 129 experts in computer science, mathematics and machine learning described him as "a prominent academic, making fundamental contributions to mathematical logic and theoretical computer science" Edalat also founded SAF and CASMII, a campaign against sanctions and military intervention in Iran.
Edalat has appeared on BBC News on numerous occasions.
Academic career
Edalat is Professor of Computer Science and Mathematics at Imperial College, London, since 1997. Before this he was a lecturer in the Department of Mathematical Sciences at Sharif University of Technology, Tehran (1987–88). He completed his PhD in Mathematics at Warwick University (UK) in 1985 advised by Christopher Zeeman. His research interests include Exact Computation in Differential and Integral Calculus, Computational Geometry, Computation in Logical Form, Optimisation Theory, Game Theory and Computational Psychiatry.
At Imperial College, Professor Edalat serves as the head of both the Algorithmic Human Development and Continuous Data-Types and Exact Computing research groups. His 1997 paper on "Bisimulation for Labelled Markov Processes" received the IEEE LICS Test of Time Award in 2017.
Science and Arts Foundation
In 1999, Edalat founded the Science and Arts Foundation (SAF), a UK registered charity with the mission "to provide the youth of the developing world with educational opportunities particularly in information technology and internet enjoyed in the industrial world." The foundation's president was Dr. Mohammad Reza Haeri-Yazdi, faculty member of the University of Tehran. The foundation raised over US$1 million toward technology projects in Iranian middle and high schools, in partnership with institutions of higher learning, such as Sharif University of Technology, University of Guilan, Shahid Chamran University and University of Kashan. According to Fars News, SAF "established the first modern computer sites with internet access for some 250 schools in various provinces in Iran."
Campaign Against Sanctions and Military Intervention in Iran (CASMII)
Edalat founded the educational peace organization CASMII, on 1 December 2005 in London, UK. It expanded to the US the following year. The organization's membership is described as a group of academics, students and professionals of "diverse range of political and ideological viewpoints", formed to oppose sanctions or military action against Iran. Edalat and CASMII have been involved in numerous anti-war events, news programs and speaking engagements.
Personal life
Edalat was arrested in Tehran on 15 April 2018 by officers of the intelligence department of IRGC for unknown reasons. He was transferred to Evin Prison. Edalat had come to Iran to attend educational workshops. He returned to the UK in December 201
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https://en.wikipedia.org/wiki/Milan%20Kolibiar
|
Milan Kolibiar (born 14 February 1922 in Detvianska Huta, died 9 July 1994 in Bratislava) was a Slovak mathematician.
He worked mostly in lattice theory and universal algebra.
External links
Milan Kolibiar's entry at biographies of Slovak mathematicians on the website of Mathematical Institute of Slovak academy of science
References
T. Katriňák: Professor Milan Kolibiar šesťdesiatročný, Math. Slovaca 32 (2), 1982, 189–194
1922 births
1994 deaths
Slovak mathematicians
Comenius University alumni
Czechoslovak mathematicians
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