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https://en.wikipedia.org/wiki/International%20Society%20for%20Design%20and%20Development%20in%20Education | The International Society for Design and Development in Education (ISDDE) was formed in 2005 with the goal of improving educational design in mathematics and science education around the world. Educational design has been an invisible topic relative to educational research, and there has been very little direct attention focused on design principles and design processes in educational design.
Society goals
This international society, focused on mathematics and science education for strategic reasons, has the following main goals:
broadly improve design and development processes used in educational design
build and support a community among educational designers and create transformational training opportunities for new educational designers
increase the impact of educational designers on educational practice throughout the world
Governance
The society is run by an Executive of approximately 12 members. Three officers have particular duties (such as appointing local chairs of the annual conference, organizing the prize process, recruiting and reviewing new Fellows and members, and directing the journal).
Current executive chair Lynne McClure, University of Cambridge
Secretary Kristen Tripet, Australian Academy of Science
Chairs History
Hugh Burkhardt 2005-2009
Christian Schunn 2010-2014
Susan McKenney 2015-2016
Lynne McClure 2017-2018
Jacquey Barber 2019-2020
Additional details on society governance are described in the society's constitution.
ISDDE Journal
Starting in 2008, the society developed an open access Electronic journal, called the Educational Designer, with roughly annual issues. The editor-in-chief is Kaye Stacey from the University of Melbourne. As an online-only journal, it has the advantage of being able to provide detailed worked examples for other designers.
Annual conference
2005 Oxford, England; Conference chair Hugh Burkhardt
2006 Oxford, England; Conference chair Hugh Burkhardt
2007 Berkeley, California, USA; Conference chair Elizabeth Stage
2008 Egmond aan Zee, the Netherlands; Conference Chair Peter Boon; ISDDE
2009 Cairns, Queensland, Australia; Conference Chair Kaye Stacey Conference – Cairns 2009 – ISDDE
2010 Oxford, England; Conference chair Malcolm Swan Conference – Oxford 2010 – ISDDE
2011 Boston, Massachusetts, USA; Conference Chairs Frank Davis & Christian Schunn;
2012 Utrecht, the Netherlands; Conference Chairs Peter Boon & Frans van Galen; Isdde Utrecht 2012
2013 Berkeley, California, USA; Conference Chairs Rena Dorph & Jacqueline Barber; ISDDE Berkeley 2013: International Society for Design and Development in Education
2014 Cambridge, England; Conference chair Lynne McClure; ISDDE 2014
2015 Boulder, Colorado, USA; Conference chair David Webb; ISDDE2015
2016 Utrecht, the Netherlands; Conference chairs Maarten Pieters, Wout Ottevanger, & Susan McKenney; ISDDE 2016
2017 Berkeley, California, USA; Conference chairs Suzy Loper and Mac Cannady; ISDDE 2017
2018 NUI Galway, Ireland; Confe |
https://en.wikipedia.org/wiki/Dar%20Salah | Dar Salah () is a Palestinian village located east of Bethlehem. The village is in the Bethlehem Governorate Southern West Bank. According to the Palestinian Central Bureau of Statistics, the village had a population of 4,588 in 2017.
As of February 2015, Dar Salah had the only ostrich farm in the Palestinian territories, a unique project by a local farmer.
Footnotes
External links
Welcome To Dar Salah
Survey of Western Palestine, Map 17: IAA, Wikimedia commons
Dar Salah Village (fact sheet), Applied Research Institute–Jerusalem, ARIJ
Dar Salah Village profile, ARIJ
Dar Salah aerial photo, ARIJ
The priorities and needs for development in Dar Salah village based on the community and local authorities’ assessment, ARIJ
Villages in the West Bank
Bethlehem Governorate
Municipalities of the State of Palestine |
https://en.wikipedia.org/wiki/Juhdum | Juhdum is a Palestinian village located eight kilometers east of Bethlehem.The village is in the Bethlehem Governorate Southern West Bank. According to the Palestinian Central Bureau of Statistics, the village had a population of 1,391 in mid-year 2006.
Footnotes
Villages in the West Bank
Bethlehem Governorate
Municipalities of the State of Palestine |
https://en.wikipedia.org/wiki/Rudolf%20Ahlswede | Rudolf F. Ahlswede (15 September 1938 – 18 December 2010) was a German mathematician. Born in Dielmissen, Germany, he studied mathematics, physics, and philosophy. He wrote his Ph.D. thesis in 1966, at the University of Göttingen, with the topic "Contributions to the Shannon information theory in case of non-stationary channels". He dedicated himself in his further career to information theory and became one of the leading representatives of this area worldwide.
Life and work
In 1977, he joined and held a Professorship at the University of Bielefeld, Bielefeld, Germany. In 1988, he received together with Imre Csiszár the Best Paper Award of the IEEE Information Theory Society for work in the area of the hypothesis testing as well as in 1990 together with Gunter Dueck for a new theory of message identification. He has been awarded this prize twice. As an emeritus of Bielefeld University, Ahlswede received the 2006 Claude E. Shannon
Award, one of the first few non-US citizens to receive it. Ahlswede's work began the field of Network coding.
Rudolf Ahlswede died on 18 December 2010, at the age of 72.
Books
R. Ahlswede and I. Wegener, Suchprobleme, Teubner Verlag, Stuttgart, 1979.
R. Ahlswede and I. Wegener, Search Problems, English Edition of "Suchprobleme" with Supplement of recent Literature,
R.L. Graham, J.K. Leenstra, and R.E. Tarjan (Eds.), Wiley-Interscience Series in Discrete Mathematics and Optimization, 1987.
I. Althöfer, N. Cai, G. Dueck, L. Khachatrian, M.S. Pinsker, A. Sárkozy, I. Wegener and Z. Zhang (Eds.),Numbers, Information and Complexity, 50 articles in honour of Rudolf Ahlswede, Kluwer Academic Publishers, Boston, 2000.
http://www.mathematik.uni-bielefeld.de/ahlswede/books/kluwer.html
R. Ahlswede, L. Bäumer, N. Cai, H. Aydinian, V. Blinovsky, C. Deppe, and H. Mashurian (Eds.), General Theory of Information Transfer and Combinatorics, Lecture Notes in Computer Science, Springer-Verlag, Vol. 4123, 2006.
http://www.springer.com/computer/foundations/book/978-3-540-46244-6
R. Ahlswede and V. Blinovsky, Lectures on Advances in Combinatorics, Universitext, Springer-Verlag, 2008.
http://www.springer.com/math/numbers/book/978-3-540-78601-6
See also
Ahlswede–Daykin inequality
Information-theoretic security
Linear network coding
References
Sources
http://www.mathematik.uni-bielefeld.de/ahlswede/
http://www.math.uni-bielefeld.de/ahlswede/homepage/
http://media.itsoc.org/isit2006/ahlswede/
External links
1938 births
2010 deaths
German information theorists
20th-century German mathematicians |
https://en.wikipedia.org/wiki/Brun%E2%80%93Titchmarsh%20theorem | In analytic number theory, the Brun–Titchmarsh theorem, named after Viggo Brun and Edward Charles Titchmarsh, is an upper bound on the distribution of prime numbers in arithmetic progression.
Statement
Let count the number of primes p congruent to a modulo q with p ≤ x. Then
for all q < x.
History
The result was proven by sieve methods by Montgomery and Vaughan; an earlier result of Brun and Titchmarsh obtained a weaker version of this inequality with an additional multiplicative factor of .
Improvements
If q is relatively small, e.g., , then there exists a better bound:
This is due to Y. Motohashi (1973). He used a bilinear structure in the error term in the Selberg sieve, discovered by himself. Later this idea of exploiting structures in sieving errors developed into a major method in Analytic Number Theory, due to H. Iwaniec's extension to combinatorial sieve.
Comparison with Dirichlet's theorem
By contrast, Dirichlet's theorem on arithmetic progressions gives an asymptotic result, which may be expressed in the form
but this can only be proved to hold for the more restricted range q < (log x)c for constant c: this is the Siegel–Walfisz theorem.
References
.
Theorems in analytic number theory
Theorems about prime numbers |
https://en.wikipedia.org/wiki/Thangam%20Thennarasu | Thangam Thenarasu (born 3 June 1966) is an Indian politician and the Minister for Finance, Planning, Human Resources Management, Pensions and Pensionary benefits, Statistics and Archeology Minister of Tamil nadu. He was allocated Electricity and Non-Conventional Energy Development portfolios of minister V. Senthil Balaji who was arrested by the Enforcement Directorate in a job racket case on 16 June 2023 by Tamil Nadu Governor R. N. Ravi on the recommendations of Chief Minister M K Stalin.
He served as Minister for School Education in Tamil Nadu during 2006-2011. He was born in Mallankinaru, Tamil Nadu. He has a bachelor's degree in Engineering. He has been elected to the Tamil Nadu assembly five times. He is the son of the former Member of Legislative Assembly from Aruppukottai, V. Thangapandian
He was elected to the Tamil Nadu legislative assembly as a Dravida Munnetra Kazhagam candidate from Aruppukottai constituency in 1997/98 by-election, and 2006 election. He is the younger brother of recently elected Member of Parliament, Lok Sabha, from the Chennai South constituency, Thamizhachi Thangapandian.
Ministerial roles
Thenarasu, whose father was V. Thangapandian, a Dravida Munnetra Kazhagam (DMK) government minister, himself became Minister for Schools in the DMK government that gained power at the 2006 state assembly elections.
Later career
Thenarasu stood as a candidate in the newly-created constituency of Tiruchuli for the 2011 elections.
He filed a case in the court against the Tamil Nadu Government's decision to form an inquiry into the alleged irregularities in construction of Tamil Nadu legislative assembly-secretariat complex
Elections contested and results
See also
Anna Centenary Library
Iniyavai Naarppathu Parade
Abonded assembly complex
References
External links
First Mega Job Fair
Job Fair in Thiruchirapalli
Job Fair Continued
Dravida Munnetra Kazhagam politicians
1966 births
Living people
Tamil Nadu ministers
Tamil Nadu MLAs 2006–2011
Tamil Nadu MLAs 2011–2016
Tamil Nadu MLAs 2016–2021
Tamil Nadu MLAs 2021–2026 |
https://en.wikipedia.org/wiki/Locally%20normal%20space | In mathematics, particularly topology, a topological space X is locally normal if intuitively it looks locally like a normal space. More precisely, a locally normal space satisfies the property that each point of the space belongs to a neighbourhood of the space that is normal under the subspace topology.
Formal definition
A topological space X is said to be locally normal if and only if each point, x, of X has a neighbourhood that is normal under the subspace topology.
Note that not every neighbourhood of x has to be normal, but at least one neighbourhood of x has to be normal (under the subspace topology).
Note however, that if a space were called locally normal if and only if each point of the space belonged to a subset of the space that was normal under the subspace topology, then every topological space would be locally normal. This is because, the singleton {x} is vacuously normal and contains x. Therefore, the definition is more restrictive.
Examples and properties
Every locally normal T1 space is locally regular and locally Hausdorff.
A locally compact Hausdorff space is always locally normal.
A normal space is always locally normal.
A T1 space need not be locally normal as the set of all real numbers endowed with the cofinite topology shows.
See also
Further reading
References
Topology
Properties of topological spaces |
https://en.wikipedia.org/wiki/Lochs%27s%20theorem | In number theory, Lochs's theorem concerns the rate of convergence of the continued fraction expansion of a typical real number. A proof of the theorem was published in 1964 by Gustav Lochs.
The theorem states that for almost all real numbers in the interval (0,1), the number of terms m of the number's continued fraction expansion that are required to determine the first n places of the number's decimal expansion behaves asymptotically as follows:
.
As this limit is only slightly smaller than 1, this can be interpreted as saying that each additional term in the continued fraction representation of a "typical" real number increases the accuracy of the representation by approximately one decimal place. The decimal system is the last positional system for which each digit carries less information than one continued fraction quotient; going to base-11 (changing to in the equation) makes the above value exceed 1.
The reciprocal of this limit,
,
is twice the base-10 logarithm of Lévy's constant.
A prominent example of a number not exhibiting this behavior is the golden ratio—sometimes known as the "most irrational" number—whose continued fraction terms are all ones, the smallest possible in canonical form. On average it requires approximately 2.39 continued fraction terms per decimal digit.
References
Continued fractions
Theorems in number theory |
https://en.wikipedia.org/wiki/Residuated%20mapping | In mathematics, the concept of a residuated mapping arises in the theory of partially ordered sets. It refines the concept of a monotone function.
If A, B are posets, a function f: A → B is defined to be monotone if it is order-preserving: that is, if x ≤ y implies f(x) ≤ f(y). This is equivalent to the condition that the preimage under f of every down-set of B is a down-set of A. We define a principal down-set to be one of the form ↓{b} = { b' ∈ B : b' ≤ b }. In general the preimage under f of a principal down-set need not be a principal down-set. If all of them are, f is called residuated.
The notion of residuated map can be generalized to a binary operator (or any higher arity) via component-wise residuation. This approach gives rise to notions of left and right division in a partially ordered magma, additionally endowing it with a quasigroup structure. (One speaks only of residuated algebra for higher arities). A binary (or higher arity) residuated map is usually not residuated as a unary map.
Definition
If A, B are posets, a function f: A → B is residuated if and only if the preimage under f of every principal down-set of B is a principal down-set of A.
Consequences
With A, B posets, the set of functions A → B can be ordered by the pointwise order f ≤ g ↔ (∀x ∈ A) f(x) ≤ g(x).
It can be shown that f is residuated if and only if there exists a (necessarily unique) monotone function f +: B → A such that f o f + ≤ idB and f + o f ≥ idA, where id is the identity function. The function f + is the residual of f. A residuated function and its residual form a Galois connection under the (more recent) monotone definition of that concept, and for every (monotone) Galois connection the lower adjoint is residuated with the residual being the upper adjoint. Therefore, the notions of monotone Galois connection and residuated mapping essentially coincide.
Additionally, we have f -1(↓{b}) = ↓{f +(b)}.
If B° denotes the dual order (opposite poset) to B then f : A → B is a residuated mapping if and only if there exists an f * such that f : A → B° and f *: B° → A form a Galois connection under the original antitone definition of this notion.
If f : A → B and g : B → C are residuated mappings, then so is the function composition fg : A → C, with residual (fg) + = g +f +. The antitone Galois connections do not share this property.
The set of monotone transformations (functions) over a poset is an ordered monoid with the pointwise order, and so is the set of residuated transformations.
Examples
The ceiling function from R to Z (with the usual order in each case) is residuated, with residual mapping the natural embedding of Z into R.
The embedding of Z into R is also residuated. Its residual is the floor function .
Residuated binary operators
If • : P × Q → R is a binary map and P, Q, and R are posets, then one may define residuation component-wise for the left and right translations, i.e. multiplication by a fixed element. For an element x in |
https://en.wikipedia.org/wiki/Maximising%20measure | In mathematics — specifically, in ergodic theory — a maximising measure is a particular kind of probability measure. Informally, a probability measure μ is a maximising measure for some function f if the integral of f with respect to μ is "as big as it can be". The theory of maximising measures is relatively young and quite little is known about their general structure and properties.
Definition
Let X be a topological space and let T : X → X be a continuous function. Let Inv(T) denote the set of all Borel probability measures on X that are invariant under T, i.e., for every Borel-measurable subset A of X, μ(T−1(A)) = μ(A). (Note that, by the Krylov-Bogolyubov theorem, if X is compact and metrizable, Inv(T) is non-empty.) Define, for continuous functions f : X → R, the maximum integral function β by
A probability measure μ in Inv(T) is said to be a maximising measure for f if
Properties
It can be shown that if X is a compact space, then Inv(T) is also compact with respect to the topology of weak convergence of measures. Hence, in this case, each continuous function f : X → R has at least one maximising measure.
If T is a continuous map of a compact metric space X into itself and E is a topological vector space that is densely and continuously embedded in C(X; R), then the set of all f in E that have a unique maximising measure is equal to a countable intersection of open dense subsets of E.
References
Ergodic theory
Measures (measure theory) |
https://en.wikipedia.org/wiki/Plancherel%20theorem%20for%20spherical%20functions | In mathematics, the Plancherel theorem for spherical functions is an important result in the representation theory of semisimple Lie groups, due in its final form to Harish-Chandra. It is a natural generalisation in non-commutative harmonic analysis of the Plancherel formula and Fourier inversion formula in the representation theory of the group of real numbers in classical harmonic analysis and has a similarly close interconnection with the theory of differential equations.
It is the special case for zonal spherical functions of the general Plancherel theorem for semisimple Lie groups, also proved by Harish-Chandra. The Plancherel theorem gives the eigenfunction expansion of radial functions for the Laplacian operator on the associated symmetric space X; it also gives the direct integral decomposition into irreducible representations of the regular representation on . In the case of hyperbolic space, these expansions were known from prior results of Mehler, Weyl and Fock.
The main reference for almost all this material is the encyclopedic text of .
History
The first versions of an abstract Plancherel formula for the Fourier transform on a unimodular locally compact group G were due to Segal and Mautner. At around the same time, Harish-Chandra and Gelfand & Naimark derived an explicit formula for SL(2,R) and complex semisimple Lie groups, so in particular the Lorentz groups. A simpler abstract formula was derived by Mautner for a "topological" symmetric space G/K corresponding to a maximal compact subgroup K. Godement gave a more concrete and satisfactory form for positive definite spherical functions, a class of special functions on G/K. Since when G is a semisimple Lie group these spherical functions φλ were naturally labelled by a parameter λ in the quotient of a Euclidean space by the action of a finite reflection group, it became a central problem to determine explicitly the Plancherel measure in terms of this parametrization. Generalizing the ideas of Hermann Weyl from the spectral theory of ordinary differential equations, Harish-Chandra introduced his celebrated c-function c(λ) to describe the asymptotic behaviour of the spherical functions φλ and proposed c(λ)−2 dλ as the Plancherel measure. He verified this formula for the special cases when G is complex or real rank one, thus in particular covering the case when G/K is a hyperbolic space. The general case was reduced to two conjectures about the properties of the c-function and the so-called spherical Fourier transform. Explicit formulas for the c-function were later obtained for a large class of classical semisimple Lie groups by Bhanu-Murthy. In turn these formulas prompted Gindikin and Karpelevich to derive a product formula for the c-function, reducing the computation to Harish-Chandra's formula for the rank 1 case. Their work finally enabled Harish-Chandra to complete his proof of the Plancherel theorem for spherical functions in 1966.
In many special cases, for example for com |
https://en.wikipedia.org/wiki/Hypograph | Hypograph may refer to:
Hypograph (mathematics), the set of points lying below the graph of a function
Hypograph, or hypogram, something written at the end of a document (for example, a postscript)
See also
Hypergraph, in mathematics
Hypographa, a genus of moths
Hypographia, another, obsolete, genus of moths
Hippogriff, a legendary creature |
https://en.wikipedia.org/wiki/Constant%20problem | In mathematics, the constant problem is the problem of deciding whether a given expression is equal to zero.
The problem
This problem is also referred to as the identity problem or the method of zero estimates. It has no formal statement as such but refers to a general problem prevalent in transcendental number theory. Often proofs in transcendence theory are proofs by contradiction. Specifically, they use some auxiliary function to create an integer n ≥ 0, which is shown to satisfy n < 1. Clearly, this means that n must have the value zero, and so a contradiction arises if one can show that in fact n is not zero.
In many transcendence proofs, proving that n ≠ 0 is very difficult, and hence a lot of work has been done to develop methods that can be used to prove the non-vanishing of certain expressions. The sheer generality of the problem is what makes it difficult to prove general results or come up with general methods for attacking it. The number n that arises may involve integrals, limits, polynomials, other functions, and determinants of matrices.
Results
In certain cases, algorithms or other methods exist for proving that a given expression is non-zero, or of showing that the problem is undecidable. For example, if x1, ..., xn are real numbers, then there is an algorithm for deciding whether there are integers a1, ..., an such that
If the expression we are interested in contains an oscillating function, such as the sine or cosine function, then it has been shown that the problem is undecidable, a result known as Richardson's theorem. In general, methods specific to the expression being studied are required to prove that it cannot be zero.
See also
Integer relation algorithm
References
Analytic number theory
Undecidable problems |
https://en.wikipedia.org/wiki/Sohrab%20Entezari | Sohrab Entezari (; born April 21, 1977) is a retired Iranian footballer.
Club career statistics
Last update: May 21, 2011
External links
Persian League Profile
1977 births
Living people
Sportspeople from Babol
Iranian men's footballers
Persian Gulf Pro League players
Persepolis F.C. players
Shahrdari Tabriz F.C. players
Rah Ahan Tehran F.C. players
Shamoushak Noshahr F.C. players
Men's association football forwards
Footballers from Mazandaran province |
https://en.wikipedia.org/wiki/Waterford%20county%20hurling%20team%20records%20and%20statistics | The following is a list of All-Ireland Senior Hurling Championship matches in which Waterford has competed from the 1989 Championship to present.
County hurling team records and statistics
Records and statistics |
https://en.wikipedia.org/wiki/Quadrant%20%28plane%20geometry%29 | The axes of a two-dimensional Cartesian system divide the plane into four infinite regions, called quadrants, each bounded by two half-axes.
These are often numbered from 1st to 4th and denoted by Roman numerals: I (where the signs of the (x; y) coordinates are I (+; +), II (−; +), III (−; −), and IV (+; −). When the axes are drawn according to the mathematical custom, the numbering goes counter-clockwise starting from the upper right ("northeast") quadrant.
Mnemonic
In the above graphic, the words in quotation marks are a mnemonic for remembering which three trigonometric functions (sine, cosine and tangent) are positive in each quadrant. The expression reads "All Science Teachers Crazy" and proceeding counterclockwise from the upper right quadrant, we see that "All" functions are positive in quadrant I, "Science" (for sine) is positive in quadrant II, "Teachers" (for tangent) is positive in quadrant III, and "Crazy" (for cosine) is positive in quadrant IV. There are several variants of this mnemonic.
See also
Orthant
Octant (solid geometry)
External links
Euclidean plane geometry |
https://en.wikipedia.org/wiki/Ludwikowo%2C%20Gmina%20Dobre | Ludwikowo is a village in the administrative district of Gmina Dobre, within Radziejów County, Kuyavian-Pomeranian Voivodeship, in north-central Poland.
Statistics
Population-26
Children-5
Houses-6 (7)
References
Ludwikowo |
https://en.wikipedia.org/wiki/Paul%20Cohn | Paul Moritz Cohn FRS (8 January 1924 – 20 April 2006) was Astor Professor of Mathematics at University College London, 1986–1989, and author of many textbooks on algebra. His work was mostly in the area of algebra, especially non-commutative rings.
Ancestry and early life
He was the only child of Jewish parents, James (or Jakob) Cohn, owner of an import business, and Julia (née Cohen), a schoolteacher.
Both of his parents were born in Hamburg, as were three of his grandparents. His ancestors came from various parts of Germany. His father fought in the German army in World War I; he was wounded several times and awarded the Iron Cross. A street in Hamburg is named in memory of his mother.
When he was born, his parents were living with his mother's mother in Isestraße. After her death in October 1925, the family moved to a rented flat in a new building in Lattenkamp, in the Winterhude quarter. He attended a kindergarten then, in April 1930, moved to Alsterdorfer Straße School. After a while, he had a new teacher, a National Socialist, who picked on him and punished him without cause. Thus in 1931, he moved to the Meerweinstraße School where his mother taught.
Following the rise of the Nazis in 1933, his father's business was confiscated and his mother dismissed. He moved to the Talmud-Tora-Schule, a Jewish school. In mid-1937, the family moved to Klosterallee. This was nearer the school, the synagogue and other pupils, being in the Jewish area. His German teacher was Dr. Ernst Loewenberg, the son of the poet Jakob Loewenberg.
On the night of 9/10 November 1938 (Kristallnacht), his father was arrested and sent to Sachsenhausen concentration camp. He was released after four months but told to emigrate. Cohn went to Britain in May 1939 on the Kindertransport to work on a chicken farm, and never saw his parents again. He corresponded regularly with them until late 1941. At the end of the War, he learned that they were deported to Riga on 6 December 1941 and never returned. At the end of 1941, the farm closed. He trained as a precision engineer, acquired a work permit and worked in a factory for 4½ years. He passed the Cambridge Scholarship Examination, and won an exhibition to Trinity College, Cambridge.
Career
He received a B.A in Mathematics from Cambridge University in 1948 and a Ph.D. (supervised by Philip Hall) in 1951. He then spent a year as a Chargé de Recherches at the University of Nancy. On his return, he became a lecturer in mathematics at Manchester University. He was a visiting professor at Yale University in 1961–1962, and for part of 1962 was at the University of California at Berkeley. On his return, he became Reader at Queen Mary College. He was a visiting professor at the University of Chicago in 1964 and at the State University of New York at Stony Brook in 1967. By then, he was regarded as one of the world's leading algebraists.
Also in 1967, he became head of the Department of Mathematics at Bedford College. He held sever |
https://en.wikipedia.org/wiki/Rashidjon%20Gafurov | Rashidjon Aleksandrovich Gafurov (born 26 September 1977) is a former Uzbekistani International footballer who played as a midfielder.
Career statistics
Club
International
Statistics accurate as of 17 March 2016
International goals
Honours
Navbahor Namangan
Uzbek Cup (1): 1998
Bunyodkor
Uzbek League (1): 2008
Uzbek Cup (1): 2008
References
External links
1977 births
Living people
Uzbekistani men's footballers
Uzbekistan men's international footballers
Navbahor Namangan players
Footballers from Tashkent
Men's association football midfielders
Footballers at the 1998 Asian Games
Asian Games competitors for Uzbekistan |
https://en.wikipedia.org/wiki/Copa%20Libertadores%20records%20and%20statistics | This page details the records and statistics of the Copa Libertadores. The Copa Libertadores is an international premier club tournament played annually by the top football clubs of South America. It includes 3–5 teams from all ten CONMEBOL members plus Mexico, whose clubs are sometimes invited as guests to the tournament. It is now held from January to November and it consists of eight stages.
The data below does not include the 1948 South American Championship of Champions, as it is not listed by CONMEBOL either as a Copa Libertadores edition or as an official competition. It must be pointed out, however, that at least in the years 1996 and 1997, CONMEBOL entitled equal status to both the Copa Libertadores and the 1948 tournament, in that the 1948 champions (Vasco da Gama) were allowed to participate in the Supercopa Libertadores, a CONMEBOL official competition that allowed participation for former Libertadores champions only (for example, not admitting participation for champions of other CONMEBOL official competitions, such as the Copa CONMEBOL).
General performances
By club
By country
By department, province or state
By city
All-time top ten table
The list is current as of the end of 2021 edition. Last updated 6 December 2021.
CONMEBOL club ranking
This ranking is used for seeding in the qualifying and group stage draws of the Copa Libertadores, and is based on a club's performance in the last 10 years of the Copa Libertadores, its historic performance in the competition, and its performance in local championship tournaments. Starting from 2021, the CONMEBOL ranking of the Copa Libertadores was updated to also include Copa Sudamericana performances, and thus was rebranded as the CONMEBOL Clubs Ranking.
This list is current as of 9 December 2022.
Number of participating clubs by country
The following is a list of the 217 clubs that have played at least one match in the Copa Libertadores, updated to the 2023 edition.
Teams in bold: winner of the edition.
Teams in italics: runner-up of the edition.
Clubs
By semi-final appearances
Clubs were finalists in years that are in bold.
By country
By quarter-final appearances
Note: 1) In 1960 and 1961, the tournament started in this round, so teams are not marked as quarter-finalists in the table. 2) From 1962 to 1965, no quarter-finals were played as the tournament had a first stage which consisted of three groups where the winners of each group advanced to semi-finals with the winners of the previous edition. 3) In 1966 and 1967, no quarter-finals were played as the tournament had a first stage with several groups of four, five, six or even seven teams, where the two best teams of each group advanced to semi-finals with the winner of the previous edition. 4) From 1968 to 1970, no quarter-finals were played as the tournament had a first stage with several groups of four or six teams, where the two best teams of each group advanced to the second stage with several groups of two, three or |
https://en.wikipedia.org/wiki/Rectified%205-simplexes | In five-dimensional geometry, a rectified 5-simplex is a convex uniform 5-polytope, being a rectification of the regular 5-simplex.
There are three unique degrees of rectifications, including the zeroth, the 5-simplex itself. Vertices of the rectified 5-simplex are located at the edge-centers of the 5-simplex. Vertices of the birectified 5-simplex are located in the triangular face centers of the 5-simplex.
Rectified 5-simplex
In five-dimensional geometry, a rectified 5-simplex is a uniform 5-polytope with 15 vertices, 60 edges, 80 triangular faces, 45 cells (30 tetrahedral, and 15 octahedral), and 12 4-faces (6 5-cell and 6 rectified 5-cells). It is also called 03,1 for its branching Coxeter-Dynkin diagram, shown as .
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S.
Alternate names
Rectified hexateron (Acronym: rix) (Jonathan Bowers)
Coordinates
The vertices of the rectified 5-simplex can be more simply positioned on a hyperplane in 6-space as permutations of (0,0,0,0,1,1) or (0,0,1,1,1,1). These construction can be seen as facets of the rectified 6-orthoplex or birectified 6-cube respectively.
As a configuration
This configuration matrix represents the rectified 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 rectified 5-simplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element.
The diagonal f-vector numbers are derived through the Wythoff construction, dividing the full group order of a subgroup order by removing one mirror at a time.
Images
Related polytopes
The rectified 5-simplex, 031, is second in a dimensional series of uniform polytopes, expressed by Coxeter as 13k series. The fifth figure is a Euclidean honeycomb, 331, and the final is a noncompact hyperbolic honeycomb, 431. Each progressive uniform polytope is constructed from the previous as its vertex figure.
Birectified 5-simplex
The birectified 5-simplex is isotopic, with all 12 of its facets as rectified 5-cells. It has 20 vertices, 90 edges, 120 triangular faces, 60 cells (30 tetrahedral, and 30 octahedral).
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as S.
It is also called 02,2 for its branching Coxeter-Dynkin diagram, shown as . It is seen in the vertex figure of the 6-dimensional 122, .
Alternate names
Birectified hexateron
dodecateron (Acronym: dot) (For 12-facetted polyteron) (Jonathan Bowers)
Construction
The elements of the regular polytopes can be expressed in a configuration matrix. Rows and columns reference vertices, edges, faces, and cells, with diagonal element their counts (f-vectors). The nondiagonal elements represent the number of row elements are incident to the column element.
The diagonal f-vector numbers are derived through the Wythoff construction, dividing the full group order of a subgroup order by removing one mirror |
https://en.wikipedia.org/wiki/10-demicube | In geometry, a 10-demicube or demidekeract is a uniform 10-polytope, constructed from the 10-cube with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as HM10 for a ten-dimensional half measure polytope.
Coxeter named this polytope as 171 from its Coxeter diagram, with a ring on one of the 1-length branches, and Schläfli symbol or {3,37,1}.
Cartesian coordinates
Cartesian coordinates for the vertices of a demidekeract centered at the origin are alternate halves of the dekeract:
(±1,±1,±1,±1,±1,±1,±1,±1,±1,±1)
with an odd number of plus signs.
Images
References
H.S.M. Coxeter:
Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, , p.296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, p.296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
(Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, (Chapter 26. pp. 409: Hemicubes: 1n1)
Norman Johnson Uniform Polytopes, Manuscript (1991)
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
External links
Multi-dimensional Glossary
10-polytopes |
https://en.wikipedia.org/wiki/Alternated%20hypercubic%20honeycomb | In geometry, the alternated hypercube honeycomb (or demicubic honeycomb) is a dimensional infinite series of honeycombs, based on the hypercube honeycomb with an alternation operation. It is given a Schläfli symbol h{4,3...3,4} representing the regular form with half the vertices removed and containing the symmetry of Coxeter group for n ≥ 4. A lower symmetry form can be created by removing another mirror on an order-4 peak.
The alternated hypercube facets become demihypercubes, and the deleted vertices create new orthoplex facets. The vertex figure for honeycombs of this family are rectified orthoplexes.
These are also named as hδn for an (n-1)-dimensional honeycomb.
References
Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition,
pp. 122–123, 1973. (The lattice of hypercubes γn form the cubic honeycombs, δn+1)
pp. 154–156: Partial truncation or alternation, represented by h prefix: h{4,4}={4,4}; h{4,3,4}={31,1,4}, h{4,3,3,4}={3,3,4,3}
p. 296, Table II: Regular honeycombs, δn+1
Kaleidoscopes: Selected Writings of H. S. M. Coxeter, editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
Honeycombs (geometry)
Polytopes |
https://en.wikipedia.org/wiki/Censo%20General%20de%20Poblaci%C3%B3n%20y%20Vivienda | The Censo de Población y Vivienda (Population and Housing Census) is the main national population census for Mexico. It is compiled by the National Institute of Statistics and Geography (INEGI), a decentralized agency of the Mexican Federal government, with the purpose of collating and reporting detailed demographic, socioeconomic and geographical data from across the nation, and is conducted every ten years.
As of 2021, there have been a total of 14 national population censuses, the most recent completed in 2020.
History
Pre-Columbian era
The practice of census-taking in Mexico may have precedents dating back to the late pre-Columbian period. According to traditions recorded in several of the post-conquest historical sources, Xólotl, a 12th-century ruler of a Chichimec polity in the Valley of Mexico, ordered a review to be undertaken to enumerate the populace under his control. This survey was carried out close to its capital, Tenayuca, at a locality subsequently named Nepōhualco in Classical Nahuatl, meaning "place of enumeration".
The count was conducted by adding stones to a pile representing each person counted, giving a total of 3,200,000 residents. The retelling of this tradition was documented in the late 18th century by Francesco Clavigero, based on Fray Juan de Torquemada's Monarchia Indiana, first published in 1615. Clavigero himself goes on to doubt some of what Torquemada wrote on the tale, citing aspects of it as "incredible". Nepohualco and the survey is also referenced in the codex Historia Tolteca-Chichimeca, folio 33R.
During the later Aztec Empire, it is known that written census-like records were used to keep track of land ownership and the tribute obligations of individual city-states (altepetl) across central Mexico.
Spanish rule
In the decades after the conquest and Spanish colonial expansion, the administrators and missionaries for the Real Audiencia of Mexico began the systematic collection of population data for the new territories. One such was the document known as the Suma de visitas de pueblos por orden alfabético from 1548, which contained a survey and description of 907 villages and settlements in central Mexico.
A census taken twenty years later in 1568, covering about 90% of the towns and villages of Central Mexico, is probably the most comprehensive of the 16th-century recorded enumerations. During the later Colonial period in the 17th century a number of other demographic counts and compilations were made. In general the data from these, likely incomplete and rudimentary, are no longer preserved.
It was not until the late 18th century that an accounting of the population was conducted, known as the Revillagigedo census, the first poll to resemble a national census. Conducted under viceroy Juan de Güemes Padilla, Count of Revillagigedo between 1790 and 1791, some forty volumes of data from this census are conserved in the Mexican national archives.
Independence and modern era
After the Mexican indepe |
https://en.wikipedia.org/wiki/QE3 | QE3 may refer to:
MS Queen Elizabeth, third Cunard passenger ship of the name
Round 3 of quantitative easing
(5346) 1981 QE3, an asteroid
Qe3, the algebraic chess notation for a move of the queen to square e3
QE3, the boat Don Allum used for his atlantic crossing
See also
QE (disambiguation)
QE1 (disambiguation)
QE2 (disambiguation) |
https://en.wikipedia.org/wiki/Aggregated%20indices%20randomization%20method | In applied mathematics and decision making, the aggregated indices randomization method (AIRM) is a modification of a well-known aggregated indices method, targeting complex objects subjected to multi-criteria estimation under uncertainty. AIRM was first developed by the Russian naval applied mathematician Aleksey Krylov around 1908.
The main advantage of AIRM over other variants of aggregated indices methods is its ability to cope with poor-quality input information. It can use non-numeric (ordinal), non-exact (interval) and non-complete expert information to solve multi-criteria decision analysis (MCDM) problems. An exact and transparent mathematical foundation can assure the precision and fidelity of AIRM results.
Background
Ordinary aggregated indices method allows comprehensive estimation of complex (multi-attribute) objects’ quality. Examples of such complex objects (decision alternatives, variants of a choice, etc.) may be found in diverse areas of business, industry, science, etc. (e.g., large-scale technical systems, long-time projects, alternatives of a crucial financial/managerial decision, consumer goods/services, and so on). There is a wide diversity of qualities under evaluation too: efficiency, performance, productivity, safety, reliability, utility, etc.
The essence of the aggregated indices method consists in an aggregation (convolution, synthesizing, etc.) of some single indices (criteria) q(1),...,q(m), each single index being an estimation of a fixed quality of multiattribute objects under investigation, into one aggregated index (criterion) Q=Q(q(1),...,q(m)).
In other words, in the aggregated indices method single estimations of an object, each of them being made from a single (specific) “point of view” (single criterion), is synthesized by aggregative function Q=Q(q(1),...,q(m)) in one aggregated (general) object's estimation Q, which is made from the general “point of view” (general criterion).
Aggregated index Q value is determined not only by single indices’ values but varies depending on non-negative weight-coefficients w(1),...,w(m). Weight-coefficient (“weight”) w(i) is treated as a measure of relative significance of the corresponding single index q(i) for general estimation Q of the quality level.
Summary
It is well known that the most subtle and delicate stage in a variant of the aggregated indices method is the stage of weights estimation because of usual shortage of information about exact values of weight-coefficients. As a rule, we have only non-numerical (ordinal) information, which can be represented by a system of equalities and inequalities for weights, and/or non-exact (interval) information, which can be represented by a system of inequalities, which determine only intervals for the weight-coefficients possible values. Usually ordinal and/or interval information is incomplete (i.e., this information is not enough for one-valued estimation of all weight-coefficients). So, one can say that there |
https://en.wikipedia.org/wiki/St%20Catherine%27s%20College%2C%20Eastbourne | St Catherine's College (previously The Bishop Bell Church of England Mathematics & Computing Specialist School) is a coeducational Church of England secondary school situated on the south coast of England in Eastbourne. The school is part of the Diocese of Chichester Academy Trust.
History
Formerly Bedewell School on Whitley Road, Eastbourne, (the town's Fire Station now stands on the old site) it was reopened in its current location in Priory Road on 25 May 1959 by Princess Margaret. It was named after Bishop George Bell, who ordered its construction and of whom there is a painting in the school. The old site was commemorated with the addition of a Science and Technology building across the road from the main site, which is named Bedewell. The two sites are joined by a skywalk which cost £800,000 in 2004, replacing an outdoor metal bridge which had been deemed impractical.
There have been several ecclesiastical visits from Bishops and members of the Christian faith. The school has had visits from the Quicken Trust, a Christian organisation which works with people in Kabubu, Africa. Bishop Bell has links with Schlenker Secondary school from Freetown, Sierra Leone, within which it helped to implement an IT centre in 2008.
In January 2016 the school announced that it would shortly be renamed. This was after the Diocese of Chichester paid compensation and apologised after sex abuse allegations were made against Bishop George Bell in a civil claim. The school was renamed St Catherine's College.
Performance
Following a period of poor performance, the school's educational achievement improved when Terry Boatwright became head teacher in 1995. There was an increase in the number of pupils achieving 5 A*-C grade GCSEs for eleven consecutive years. By 1999 it was one of the top improving schools in the country; this was attributed to Boatwright by the local MP. By 2006 the school was oversubscribed. In 2014, Boatwright retired as headteacher, with Mark Talbot replacing him.
Since then, school performance has dropped severely, and now it is the second-worst school in Eastbourne, in front of only The Causeway School.
In March 2018, Mark Talbot announced that he was resigning as Principal of the school, after being appointed CEO of the Diocese of Chichester Academy Trust. In May 2018, the school announced that Solomon Berhane would be replacing Mr Talbot starting September that year.
Curriculum
The school educates students from Year 7 through to Year 11. The school offers pupils a range of voluntary and compulsory GCSE subjects which are taken from Year 9 to Year 11. There was a change after the academic year (2007–2008) in which the school started GCSEs at Year 10, with pupils deciding their subject choices the year before.
The school educates all of its pupils in Citizenship and Personal, Social and Health Education. English, Mathematics, Science, Religious Education and Physical Education are core subjects taught to all students throughout the en |
https://en.wikipedia.org/wiki/Truncated%20tetrahedral%20prism | In geometry, a truncated tetrahedral prism is a convex uniform polychoron (four-dimensional polytope). This polychoron has 10 polyhedral cells: 2 truncated tetrahedra connected by 4 triangular prisms and 4 hexagonal prisms. It has 24 faces: 8 triangular, 18 square, and 8 hexagons. It has 48 edges and 24 vertices.
It is one of 18 uniform polyhedral prisms created by using uniform prisms to connect pairs of parallel Platonic solids and Archimedean solids.
Net
Alternative names
Truncated-tetrahedral dyadic prism (Norman W. Johnson)
Tuttip (Jonathan Bowers: for truncated-tetrahedral prism)
Truncated tetrahedral hyperprism
External links
4-polytopes |
https://en.wikipedia.org/wiki/Joseph%20Gallian | Joseph A. Gallian (born January 5, 1942) is an American mathematician, the Morse Alumni Distinguished University Professor of Teaching in the Department of Mathematics and Statistics at the University of Minnesota Duluth.
Professional career
Gallian completed his Ph.D. thesis, entitled Two-Step Centralizers in Finite p-Groups, at the University of Notre Dame in 1971 under the supervision of Karl Kronstein. He has been a professor at the University of Minnesota Duluth since 1972. In addition to teaching math classes, he taught a humanities course called "The Lives and Music of the Beatles" for 33 years and a liberal arts course on math and sports.
Gallian has authored or edited six books (Contemporary Abstract Algebra, Taylor & Francis 10th edition; For All Practical Purposes, W.H. Freeman (coauthor); Principles and Practices of Mathematics, Springer-Verlag; Proceedings of the Conference on Summer Undergraduate Mathematics Research Programs, Editor, American Mathematical Society; Proceedings of the Conference on Promoting Undergraduate Research in Mathematics (editor), American Mathematical Society; Mathematics and Sports, Mathematical Association of America) and over 100 articles. He earned media attention in 1991 when he determined the methods used by Minnesota and many other states for assigning drivers' license numbers.
Between 1977 and 2022, Gallian ran forty-three Research Experience for Undergraduates (REU) programs at the University of Minnesota Duluth. The program has been funded by the University of Minnesota Duluth grants from the National Science Foundation (40+ years) and the National Security Agency (30+ years). It is one of the oldest and longest-running REUs in the country. As of the end of 2019, the program has had 254 undergraduate participants and has produced more than 240 publications in mainstream professional journals. More than 150 Duluth REU students have received a PhD degree.
Gallian served a 2-year term as the President of the Mathematical Association of America starting in January 2007. In addition, he was co-director of Project NExT from 1998 to 2012, Associate Editor of MAA OnLine since 1997, a member of the advisory board of Math Horizons from 1993 to 2013, a member of the editorial board of the Mathematics Magazine for five years and the American Mathematical Monthly for 15 years.
In 2021 he became a co-editor of the MAA Mugs to Donuts e-newsletter for Putnam competition students.
Awards and honors
Gallian has won both the Allendoerfer and Evans awards for exposition from the Mathematical Association of America (MAA) and was the Pólya lecturer for the MAA from 1999 to 2001.
His excellence in teaching earned him the Haimo Award for distinguished teaching from the MAA in 1993 and he was the Carnegie Foundation for the Advancement of Teaching Minnesota Professor of the Year in 2003.
In 2019 he received the MAA's Mary P. Dolciani Award for making a distinguished contribution to the mathematical education o |
https://en.wikipedia.org/wiki/Identity%20problem | Identity problem may refer to:
An additional psychiatric condition from DSM-IV, code 313.82. See also: Identity disorder;
A constant problem in mathematics, an indistinctness over whether an expression equals zero. |
https://en.wikipedia.org/wiki/Ornstein%E2%80%93Uhlenbeck%20operator | In mathematics, the Ornstein–Uhlenbeck operator is a generalization of the Laplace operator to an infinite-dimensional setting. The Ornstein–Uhlenbeck operator plays a significant role in the Malliavin calculus.
Introduction: the finite-dimensional picture
The Laplacian
Consider the gradient operator ∇ acting on scalar functions f : Rn → R; the gradient of a scalar function is a vector field v = ∇f : Rn → Rn. The divergence operator div, acting on vector fields to produce scalar fields, is the adjoint operator to ∇. The Laplace operator Δ is then the composition of the divergence and gradient operators:
,
acting on scalar functions to produce scalar functions. Note that A = −Δ is a positive operator, whereas Δ is a dissipative operator.
Using spectral theory, one can define a square root (1 − Δ)1/2 for the operator (1 − Δ). This square root satisfies the following relation involving the Sobolev H1-norm and L2-norm for suitable scalar functions f:
The Ornstein–Uhlenbeck operator
Often, when working on Rn, one works with respect to Lebesgue measure, which has many nice properties. However, remember that the aim is to work in infinite-dimensional spaces, and it is a fact that there is no infinite-dimensional Lebesgue measure. Instead, if one is studying some separable Banach space E, what does make sense is a notion of Gaussian measure; in particular, the abstract Wiener space construction makes sense.
To get some intuition about what can be expected in the infinite-dimensional setting, consider standard Gaussian measure γn on Rn: for Borel subsets A of Rn,
This makes (Rn, B(Rn), γn) into a probability space; E will denote expectation with respect to γn.
The gradient operator ∇ acts on a (differentiable) function φ : Rn → R to give a vector field ∇φ : Rn → Rn.
The divergence operator δ (to be more precise, δn, since it depends on the dimension) is now defined to be the adjoint of ∇ in the Hilbert space sense, in the Hilbert space L2(Rn, B(Rn), γn; R). In other words, δ acts on a vector field v : Rn → Rn to give a scalar function δv : Rn → R, and satisfies the formula
On the left, the product is the pointwise Euclidean dot product of two vector fields; on the right, it is just the pointwise multiplication of two functions. Using integration by parts, one can check that δ acts on a vector field v with components vi, i = 1, ..., n, as follows:
The change of notation from "div" to "δ" is for two reasons: first, δ is the notation used in infinite dimensions (the Malliavin calculus); secondly, δ is really the negative of the usual divergence.
The (finite-dimensional) Ornstein–Uhlenbeck operator L (or, to be more precise, Lm) is defined by
with the useful formula that for any f and g smooth enough for all the terms to make sense,
The Ornstein–Uhlenbeck operator L is related to the usual Laplacian Δ by
The Ornstein–Uhlenbeck operator for a separable Banach space
Consider now an abstract Wiener space E with Cameron-Martin Hilbert |
https://en.wikipedia.org/wiki/Icosahedral%20prism | In geometry, an icosahedral prism is a convex uniform 4-polytope (four-dimensional polytope). This 4-polytope has 22 polyhedral cells: 2 icosahedra connected by 20 triangular prisms. It has 70 faces: 30 squares and 40 triangles. It has 72 edges and 24 vertices.
It can be constructed by creating two coinciding icosahedra in 3-space, and translating each copy in opposite perpendicular directions in 4-space until their separation equals their edge length.
It is one of 18 convex uniform polyhedral prisms created by using uniform prisms to connect pairs of parallel Platonic solids or Archimedean solids.
Alternate names
Icosahedral dyadic prism Norman W. Johnson
Ipe for icosahedral prism/hyperprism (Jonathan Bowers)
Snub tetrahedral prism/hyperprism
Related polytopes
Snub tetrahedral antiprism - = ht0,1,2,3{3,3,2} or , a related nonuniform 4-polytope
External links
4-polytopes |
https://en.wikipedia.org/wiki/Davenport%E2%80%93Schmidt%20theorem | In mathematics, specifically the area of Diophantine approximation, the Davenport–Schmidt theorem tells us how well a certain kind of real number can be approximated by another kind. Specifically it tells us that we can get a good approximation to irrational numbers that are not quadratic by using either quadratic irrationals or simply rational numbers. It is named after Harold Davenport and Wolfgang M. Schmidt.
Statement
Given a number α which is either rational or a quadratic irrational, we can find unique integers x, y, and z such that x, y, and z are not all zero, the first non-zero one among them is positive, they are relatively prime, and we have
If α is a quadratic irrational we can take x, y, and z to be the coefficients of its minimal polynomial. If α is rational we will have x = 0. With these integers uniquely determined for each such α we can define the height of α to be
The theorem then says that for any real number ξ which is neither rational nor a quadratic irrational, we can find infinitely many real numbers α which are rational or quadratic irrationals and which satisfy
where C is any real number satisfying C > 160/9.
While the theorem is related to Roth's theorem, its real use lies in the fact that it is effective, in the sense that the constant C can be worked out for any given ξ.
Notes
References
Wolfgang M. Schmidt. Diophantine approximation. Lecture Notes in Mathematics 785. Springer. (1980 [1996 with minor corrections])
Wolfgang M. Schmidt.Diophantine approximations and Diophantine equations, Lecture Notes in Mathematics, Springer Verlag 2000
External links
Diophantine approximation
Theorems in number theory |
https://en.wikipedia.org/wiki/Mexico%20national%20football%20team%20records%20and%20statistics | This is a list of Mexico national football team's all kinds of competitive records.
Individual records
Player records
Players in bold are still active with Mexico.
Most capped players
Top goalscorers
Competition records
For the all-time record of the national team against opposing nations, see the team's all-time record page.
FIFA World Cup
CONCACAF Gold Cup
CONCACAF Nations League
Copa América
FIFA Confederations Cup
Olympic Games
Head-to-head record
The list shown below shows the Mexico national football team's all-time international record against opposing nations. The statistics are composed of FIFA World Cup, FIFA World Cup Qualifying, FIFA Confederations Cup, CONCACAF Gold Cup (including CONCACAF Championship), CONCACAF Cup, Summer Olympics, Copa America, and CONCACAF Nations League matches, as well as international friendly matches.
After the match against on 17 October 2023.
Record against former nations
Penalty shootouts
FIFA World Ranking
Last update was on 25 August 2022.
Source:
Best Ranking Worst Ranking Best Mover Worst Mover
Notes
References
Mexico national football team
National association football team records and statistics |
https://en.wikipedia.org/wiki/C63 | C63 or C-63 may refer to:
Caldwell 63, a planetary nebula
Convention concerning Statistics of Wages and Hours of Work, 1938 of the International Labour Organization
JNR Class C63, a proposed Japanese steam locomotive
Lockheed C-63 Hudson, an American military transport aircraft
Mercedes-AMG C 63, a German automobile
Ruy Lopez, a chess opening |
https://en.wikipedia.org/wiki/K-topology | In mathematics, particularly topology, the K-topology is a topology that one can impose on the set of all real numbers which has some interesting properties. Relative to the set of all real numbers carrying the standard topology, the set K = {1/n | n is a positive integer} is not closed since it doesn't contain its (only) limit point 0. Relative to the K-topology however, the set K is automatically decreed to be closed by adding ‘more’ basis elements to the standard topology on R. Basically, the K-topology on R is strictly finer than the standard topology on R. It is mostly useful for counterexamples in basic topology.
Formal definition
Let R be the set of all real numbers and let K = {1/n | n is a positive integer}. Generate a topology on R by taking basis as all open intervals (a, b) and all sets of the form (a, b) – K (the set of all elements in (a, b) that are not in K). The topology generated is known as the K-topology on R.
The sets described in the definition form a basis (they satisfy the conditions to be a basis).
Properties and examples
Throughout this section, T will denote the K-topology and (R, T) will denote the set of all real numbers with the K-topology as a topological space.
1. The topology T on R is strictly finer than the standard topology on R but not comparable with the lower limit topology on R
2. From the previous example, it follows that (R, T) is not compact
3. (R, T) is Hausdorff but not regular. The fact that it is Hausdorff follows from the first property. It is not regular since the closed set K and the point {0} have no disjoint neighbourhoods about them
4. Surprisingly enough, (R, T) is a connected topological space. However, (R, T) is not path connected; it has precisely two path components: (−∞, 0] and (0, +∞)
5. (R, T) is not locally path connected (since its path components are not equal to its components). It is also not locally connected at {0} but it is locally connected everywhere else
6. The closed interval [0,1] is not compact as a subspace of (R, T) since it is not even limit point compact (K is an infinite subspace of [0,1] that has no limit point in [0,1])
7. In fact, no subspace of (R, T) containing K can be compact. If A were a subspace of (R, T) containing K, K would have no limit point in A so that A can not be limit point compact. Therefore, A cannot be compact
8. The quotient space of (R, T) obtained by collapsing K to a point is not Hausdorff. K is distinct from 0, but can't be separated from 0 by disjoint open sets.
See also
Connected space
List of topologies
Locally connected space
Lower limit topology
Natural topology
Sequence
References
Topological spaces |
https://en.wikipedia.org/wiki/McLaughlin%20sporadic%20group | In the area of modern algebra known as group theory, the McLaughlin group McL is a sporadic simple group of order
27 ⋅ 36 ⋅ 53 ⋅ 7 ⋅ 11 = 898,128,000
≈ 9.
History and properties
McL is one of the 26 sporadic groups and was discovered by as an index 2 subgroup of a rank 3 permutation group acting on the McLaughlin graph with vertices. It fixes a 2-2-3 triangle in the Leech lattice and thus is a subgroup of the Conway groups , , and . Its Schur multiplier has order 3, and its outer automorphism group has order 2. The group 3.McL:2 is a maximal subgroup of the Lyons group.
McL has one conjugacy class of involution (element of order 2), whose centralizer is a maximal subgroup of type 2.A8. This has a center of order 2; the quotient modulo the center is isomorphic to the alternating group A8.
Representations
In the Conway group Co3, McL has the normalizer McL:2, which is maximal in Co3.
McL has 2 classes of maximal subgroups isomorphic to the Mathieu group M22. An outer automorphism interchanges the two classes of M22 groups. This outer automorphism is realized on McL embedded as a subgroup of Co3.
A convenient representation of M22 is in permutation matrices on the last 22 coordinates; it fixes a 2-2-3 triangle with vertices the origin and the type 2 points and '. The triangle's edge is type 3; it is fixed by a Co3. This M22 is the monomial, and a maximal, subgroup of a representation of McL.
(p. 207) shows that the subgroup McL is well-defined. In the Leech lattice, suppose a type 3 point v is fixed by an instance of . Count the type 2 points w such that the inner product v·w = 3 (and thus v-w is type 2). He shows their number is and that this Co3 is transitive on these w.
|McL| = |Co3|/552 = 898,128,000.
McL is the only sporadic group to admit irreducible representations of quaternionic type. It has 2 such representations, one of dimension 3520 and one of dimension 4752.
Maximal subgroups
found the 12 conjugacy classes of maximal subgroups of McL as follows:
U4(3) order 3,265,920 index 275 – point stabilizer of its action on the McLaughlin graph
M22 order 443,520 index 2,025 (two classes, fused under an outer automorphism)
U3(5) order 126,000 index 7,128
31+4:2.S5 order 58,320 index 15,400
34:M10 order 58,320 index 15,400
L3(4):22 order 40,320 index 22,275
2.A8 order 40,320 index 22,275 – centralizer of involution
24:A7 order 40,320 index 22,275 (two classes, fused under an outer automorphism)
M11 order 7,920 index 113,400
5+1+2:3:8 order 3,000 index 299,376
Conjugacy classes
Traces of matrices in a standard 24-dimensional representation of McL are shown. The names of conjugacy classes are taken from the Atlas of Finite Group Representations.
Cycle structures in the rank 3 permutation representation, degree 275, of McL are shown.
Generalized Monstrous Moonshine
Conway and Norton suggested in their 1979 paper that monstrous moonshine is not limited to the monster. Larissa Queen and others subsequently |
https://en.wikipedia.org/wiki/Truncation%20%28statistics%29 | In statistics, truncation results in values that are limited above or below, resulting in a truncated sample. A random variable is said to be truncated from below if, for some threshold value , the exact value of is known for all cases , but unknown for all cases . Similarly, truncation from above means the exact value of is known in cases where , but unknown when .
Truncation is similar to but distinct from the concept of statistical censoring. A truncated sample can be thought of as being equivalent to an underlying sample with all values outside the bounds entirely omitted, with not even a count of those omitted being kept. With statistical censoring, a note would be recorded documenting which bound (upper or lower) had been exceeded and the value of that bound. With truncated sampling, no note is recorded.
Applications
Usually the values that insurance adjusters receive are either left-truncated, right-censored, or both. For example, if policyholders are subject to a policy limit u, then any loss amounts that are actually above u are reported to the insurance company as being exactly u because u is the amount the insurance company pays. The insurer knows that the actual loss is greater than u but they don't know what it is. On the other hand, left truncation occurs when policyholders are subject to a deductible. If policyholders are subject to a deductible d, any loss amount that is less than d will not even be reported to the insurance company. If there is a claim on a policy limit of u and a deductible of d, any loss amount that is greater than u will be reported to the insurance company as a loss of because that is the amount the insurance company has to pay. Therefore, insurance loss data is left-truncated because the insurance company doesn't know if there are values below the deductible d because policyholders won't make a claim. The insurance loss is also right-censored if the loss is greater than u because u is the most the insurance company will pay. Thus, it only knows that your claim is greater than u, not the exact claim amount.
Probability distributions
Truncation can be applied to any probability distribution. This will usually lead to a new distribution, not one within the same family. Thus, if a random variable X has F(x) as its distribution function, the new random variable Y defined as having the distribution of X truncated to the semi-open interval (a, b] has the distribution function
for y in the interval (a, b], and 0 or 1 otherwise. If truncation were to the closed interval [a, b], the distribution function would be
for y in the interval [a, b], and 0 or 1 otherwise.
Data analysis
The analysis of data where observations are treated as being from truncated versions of standard distributions can be undertaken using maximum likelihood, where the likelihood would be derived from the distribution or density of the truncated distribution. This involves taking account of the factor in the modified density fu |
https://en.wikipedia.org/wiki/Minimal%20model%20%28set%20theory%29 | In set theory, a branch of mathematics, the minimal model is the minimal standard model of ZFC.
The minimal model was introduced by and rediscovered by .
The existence of a minimal model cannot be proved in ZFC, even assuming that ZFC is consistent, but follows from the existence of a standard model as follows. If there is a set W in the von Neumann universe V that is a standard model of ZF, and the ordinal κ is the set of ordinals that occur in W, then Lκ is the class of constructible sets of W. If there is a set that is a standard model of ZF, then the smallest such set is such a Lκ. This set is called the minimal model of ZFC, and also satisfies the axiom of constructibility V=L. The downward Löwenheim–Skolem theorem implies that the minimal model (if it exists as a set) is a countable set. More precisely, every element s of the minimal model can be named; in other words there is a first-order sentence φ(x) such that s is the unique element of the minimal model for which φ(s) is true.
gave another construction of the minimal model as the strongly constructible sets, using a modified form of Gödel's constructible universe.
Of course, any consistent theory must have a model, so even within the minimal model of set theory there are sets that are models of ZFC (assuming ZFC is consistent). However, these set models are non-standard. In particular, they do not use the normal membership relation and they are not well-founded.
If there is no standard model then the minimal model cannot exist as a set. However in this case the class of all constructible sets plays the same role as the minimal model and has similar properties (though it is now a proper class rather than a countable set).
The minimal model of set theory has no inner models other than itself. In particular it is not possible to use the method of inner models to prove that any given statement true in the minimal model (such as the continuum hypothesis) is not provable in ZFC.
References
Constructible universe |
https://en.wikipedia.org/wiki/List%20of%20designated%20places%20in%20British%20Columbia | A designated place is a type of geographic unit used by Statistics Canada to disseminate census data. It is usually "a small community that does not meet the criteria used to define incorporated municipalities or Statistics Canada population centres (areas with a population of at least 1,000 and no fewer than 400 persons per square kilometre)." Provincial and territorial authorities collaborate with Statistics Canada in the creation of designated places so that data can be published for sub-areas within municipalities. Starting in 2016, Statistics Canada allowed the overlapping of designated places with population centres.
In the 2021 Census of Population, British Columbia had 332 designated places, an increase from 326 in 2016. Designated place types in British Columbia include 55 Indian reserves, 13 island trusts, 5 Nisga'a villages, 5 retired population centres, and 254 unincorporated places. In 2021, the 332 designated places had a cumulative population of 258,060 and an average population of . British Columbia's largest designated place is Walnut Grove with a population of 28,420.
List
See also
List of census agglomerations in British Columbia
List of population centres in British Columbia
Notes
References
Designated places |
https://en.wikipedia.org/wiki/Mathematical%20Society%20of%20Japan | The Mathematical Society of Japan (MSJ, ) is a learned society for mathematics in Japan.
In 1877, the organization was established as the Tokyo Sugaku Kaisha and was the first academic society in Japan. It was re-organized and re-established in its present form in 1946.
The MSJ has more than 5,000 members. They have the opportunity to participate in programs at MSJ meetings which take place in spring and autumn each year. They also have the opportunity to announce their own research at these meetings.
Prizes
Iyanaga Prize
The Iyanaga Prize was a mathematics award granted by the Mathematical Society of Japan. The prize was funded through an endowment given by Shokichi Iyanaga. Since 1988, it has been replaced by the Spring Prize.
1973 - Yasutaka Ihara
1974 - Reiko Sakamoto
1975 - Motoo Takahashi
1976 -
1977 - Takahiro Kawai
1978 - Takuro Shintani
1979 - Goro Nishida
1980 - Katsuhiro Shiohama
1981 - Masaki Kashiwara
1982 - Shigeru Iitaka
1983 - Shigefumi Mori
1984 - Yukio Matsumoto
1985 - Toshio Oshima
1986 - Shinichi Kotani
1987 - Toshikazu Sunada
Geometry Prize
The Geometry Prize is a mathematics award granted by the Mathematical Society of Japan to recognise significant or long-time research work in the field of geometry, including differential geometry, topology, and algebraic geometry. It was established in 1987.
Takebe Prize
In the context of its 50th anniversary celebrations, the Mathematical Society of Japan established the Takebe Prize for the encouragement of those who show promise as mathematicians. The award is named after Edo period mathematician (also known as Takebe Kenkō).
Spring Prize
Autumn Prize
English Publications from MSJ
MSJ publishes the following journals in English.
Journal of the Mathematical Society of Japan (JMSJ)
Japanese Journal of Mathematics (JJM)
Publications of the Mathematical Society of Japan
Advanced Studies in Pure Mathematics
MSJ Memoirs
See also
Japan Society for Industrial and Applied Mathematics
List of mathematical societies
Notes
References
Mathematical Society of Japan
External links
Official website
Geometry Prize homepage
1877 establishments in Japan
Mathematical societies
Learned societies of Japan |
https://en.wikipedia.org/wiki/Essential%20range | In mathematics, particularly measure theory, the essential range, or the set of essential values, of a function is intuitively the 'non-negligible' range of the function: It does not change between two functions that are equal almost everywhere. One way of thinking of the essential range of a function is the set on which the range of the function is 'concentrated'.
Formal definition
Let be a measure space, and let be a topological space. For any -measurable , we say the essential range of to mean the set
Equivalently, , where is the pushforward measure onto of under and denotes the support of
Essential values
We sometimes use the phrase "essential value of " to mean an element of the essential range of
Special cases of common interest
Y = C
Say is equipped with its usual topology. Then the essential range of f is given by
In other words: The essential range of a complex-valued function is the set of all complex numbers z such that the inverse image of each ε-neighbourhood of z under f has positive measure.
(Y,T) is discrete
Say is discrete, i.e., is the power set of i.e., the discrete topology on Then the essential range of f is the set of values y in Y with strictly positive -measure:
Properties
The essential range of a measurable function, being the support of a measure, is always closed.
The essential range ess.im(f) of a measurable function is always a subset of .
The essential image cannot be used to distinguish functions that are almost everywhere equal: If holds -almost everywhere, then .
These two facts characterise the essential image: It is the biggest set contained in the closures of for all g that are a.e. equal to f:
.
The essential range satisfies .
This fact characterises the essential image: It is the smallest closed subset of with this property.
The essential supremum of a real valued function equals the supremum of its essential image and the essential infimum equals the infimum of its essential range. Consequently, a function is essentially bounded if and only if its essential range is bounded.
The essential range of an essentially bounded function f is equal to the spectrum where f is considered as an element of the C*-algebra .
Examples
If is the zero measure, then the essential image of all measurable functions is empty.
This also illustrates that even though the essential range of a function is a subset of the closure of the range of that function, equality of the two sets need not hold.
If is open, continuous and the Lebesgue measure, then holds. This holds more generally for all Borel measures that assign non-zero measure to every non-empty open set.
Extension
The notion of essential range can be extended to the case of , where is a separable metric space.
If and are differentiable manifolds of the same dimension, if VMO and if , then .
See also
Essential supremum and essential infimum
measure
Lp space
References
Real analysis
Measure theory |
https://en.wikipedia.org/wiki/Richard%20Swan | Richard Gordon Swan (; born 1933) is an American mathematician who is known for the Serre–Swan theorem relating the geometric notion of vector bundles to the algebraic concept of projective modules, and for the Swan representation, an l-adic projective representation of a Galois group. His work has mainly been in the area of algebraic K-theory.
Education and career
As an undergraduate at Princeton University, Swan was one of five winners in the William Lowell Putnam Mathematical Competition in 1952. He earned his Ph.D. in 1957 from Princeton University under the supervision of John Coleman Moore.
In 1969 he proved in full generality what is now known as the Stallings-Swan theorem. He is the Louis Block Professor Emeritus of Mathematics at the University of Chicago.
His doctoral students at Chicago include Charles Weibel, also known for his work in K-theory.
Awards and honors
In 1970 Swan was awarded the American Mathematical Society's Cole Prize in Algebra.
Books
References
External links
Swan's homepage at Chicago.
Year of birth missing (living people)
Living people
20th-century American mathematicians
21st-century American mathematicians
Princeton University alumni
University of Chicago faculty
Members of the United States National Academy of Sciences
Putnam Fellows |
https://en.wikipedia.org/wiki/Semantic%20mapping%20%28statistics%29 | Semantic mapping (SM) in statistics is a method for dimensionality reduction (the transformation of data from a high-dimensional space into a low-dimensional space). SM can be used in a set of multidimensional vectors of features to extract a few new features that preserves the main data characteristics.
SM performs dimensionality reduction by clustering the original features in semantic clusters and combining features mapped in the same cluster to generate an extracted feature. Given a data set, this method constructs a projection matrix that can be used to map a data element from a high-dimensional space into a reduced dimensional space.
SM can be applied in construction of text mining and information retrieval systems, as well as systems managing vectors of high dimensionality. SM is an alternative to random mapping, principal components analysis and latent semantic indexing methods.
See also
Dimensionality reduction
Principal components analysis
Latent semantic indexing
Unification (logic reduction)
References
CORRÊA, R. F.; LUDERMIR, T. B. Improving Self Organization of Document Collections by Semantic Mapping. Neurocomputing(Amsterdam), v. 70, p. 62-69, 2006. doi:10.1016/j.neucom.2006.07.007
CORRÊA, R. F. and LUDERMIR, T. B. (2007) "Dimensionality Reduction of very large document collections by Semantic Mapping". Proceedings of 6th Int. Workshop on Self-Organizing Maps (WSOM). .
External links
Full list of publications about Semantic Mapping method
Dimension reduction |
https://en.wikipedia.org/wiki/Nine-point%20hyperbola | In Euclidean geometry with triangle , the nine-point hyperbola is an instance of the nine-point conic described by American mathematician Maxime Bôcher in 1892. The celebrated nine-point circle is a separate instance of Bôcher's conic:
Given a triangle and a point in its plane, a conic can be drawn through the following nine points:
the midpoints of the sides of ,
the midpoints of the lines joining to the vertices, and
the points where these last named lines cut the sides of the triangle.
The conic is an ellipse if lies in the interior of or in one of the regions of the plane separated from the interior by two sides of the triangle; otherwise, the conic is a hyperbola. Bôcher notes that when is the orthocenter, one obtains the nine-point circle, and when is on the circumcircle of , then the conic is an equilateral hyperbola.
Allen
An approach to the nine-point hyperbola using the analytic geometry of split-complex numbers was devised by E. F. Allen in 1941. Writing , , he uses split-complex arithmetic to express a hyperbola as
It is used as the circumconic of triangle Let Then the nine-point conic is
Allen's description of the nine-point hyperbola followed a development of the nine-point circle that Frank Morley and his son published in 1933. They requisitioned the unit circle in the complex plane as the circumcircle of the given triangle.
In 1953 Allen extended his study to a nine-point conic of a triangle inscribed in any central conic.
Yaglom
For Yaglom, a hyperbola is a Minkowskian circle as in the Minkowski plane. Yaglom's description of this geometry is found in the "Conclusion" chapter of a book that initially addresses Galilean geometry. He considers a triangle inscribed in a "circumcircle" which is in fact a hyperbola. In the Minkowski plane the nine-point hyperbola is also described as a circle:
… the midpoints of the sides of a triangle and the feet of its altitudes (as well as the midpoints of the segments joining the orthocenter of to its vertices) lie on a [Minkowskian] circle whose radius is half the radius of the circumcircle of the triangle. It is natural to refer to S as the six- (nine-) point circle of the (Minkowskian) triangle ; if has an incircle , then the six- (nine-) point circle of touches its incircle (Fig.173).
Others
In 2005 J. A. Scott used the unit hyperbola as the circumconic of triangle ABC and found conditions for it to include six triangle centers: the centroid X(2), the orthocenter X(4), the Fermat points X(13) and X(14), and the Napoleon points X(17) and X(18) as listed in the Encyclopedia of Triangle Centers. Scott’s hyperbola is a Kiepert hyperbola of the triangle.
Christopher Bath describes a nine-point rectangular hyperbola passing through these centers: incenter X(1), the three excenters, the centroid X(2), the de Longchamps point X(20), and the three points obtained by extending the triangle medians to twice their cevian length.
References
Maxime Bôcher (1892) Nine-point Con |
https://en.wikipedia.org/wiki/Standard%20model%20%28set%20theory%29 | In set theory, a standard model for a theory is a model for where the membership relation is the same as the membership relation of the set theoretical universe (restricted to the domain of ). In other words, is a substructure of . A standard model that satisfies the additional transitivity condition that implies is a standard transitive model (or simply a transitive model).
Usually, when one talks about a model of set theory, it is assumed that is a set model, i.e. the domain of is a set in . If the domain of is a proper class, then is a class model. An inner model is necessarily a class model.
References
Set theory
Model theory |
https://en.wikipedia.org/wiki/Moji%20Station | is a railway station on the Kagoshima Main Line and the Sanyō Main Line, operated by Kyushu Railway Company in Moji-ku, Kitakyushu, Japan.
Passenger statistics
In fiscal 2016, the station was used by an average of 6,392 passengers daily (boarding passengers only), and it ranked 29th among the busiest stations of JR Kyushu.
Station number code
for the Kagoshima Main Line:
for the Sanyō Main Line:
References
Railway stations in Fukuoka Prefecture
Railway stations in Japan opened in 1891 |
https://en.wikipedia.org/wiki/Nielsen%20transformation | In mathematics, especially in the area of abstract algebra known as combinatorial group theory, Nielsen transformations, named after Jakob Nielsen, are certain automorphisms of a free group which are a non-commutative analogue of row reduction and one of the main tools used in studying free groups, . They were introduced in to prove that every subgroup of a free group is free (the Nielsen–Schreier theorem), but are now used in a variety of mathematics, including computational group theory, k-theory, and knot theory. The textbook devotes all of chapter 3 to Nielsen transformations.
Definitions
One of the simplest definitions of a Nielsen transformation is an automorphism of a free group, but this was not their original definition. The following gives a more constructive definition.
A Nielsen transformation on a finitely generated free group with ordered basis [ x1, ..., xn ] can be factored into elementary Nielsen transformations of the following sorts:
Switch x1 and x2
Cyclically permute x1, x2, ..., xn, to x2, ..., xn, x1.
Replace x1 with x1−1
Replace x1 with x1·x2
These transformations are the analogues of the elementary row operations. Transformations of the first two kinds are analogous to row swaps, and cyclic row permutations. Transformations of the third kind correspond to scaling a row by an invertible scalar. Transformations of the fourth kind correspond to row additions.
Transformations of the first two types suffice to permute the generators in any order, so the third type may be applied to any of the generators, and the fourth type to any pair of generators.
When dealing with groups that are not free, one instead applies these transformations to finite ordered subsets of a group. In this situation,
compositions of the elementary transformations are called regular. If one allows removing elements of the subset that are the identity element, then the transformation is called singular.
The image under a Nielsen transformation (elementary or not, regular or not) of a generating set of a group G is also a generating set of G. Two generating sets are called Nielsen equivalent if there is a Nielsen transformation taking one to the other. If the generating sets have the same size, then it suffices to consider compositions of regular Nielsen transformations.
Examples
The dihedral group of order 10 has two Nielsen equivalence classes of generating sets of size 2. Letting x be an element of order 2, and y being an element of order 5, the two classes of generating sets are represented by [ x, y ] and [ x, yy ], and each class has 15 distinct elements. A very important generating set of a dihedral group is the generating set from its presentation as a Coxeter group. Such a generating set for a dihedral group of order 10 consists of any pair of elements of order 2, such as [ x, xy ]. This generating set is equivalent to [ x, y ] via:
[ x−1, y ], type 3
[ y, x−1 ], type 1
[ y−1, x−1 ], type 3
[ y−1x−1, x−1 ], type 4
[ xy, x−1 |
https://en.wikipedia.org/wiki/Completeness | Complete may refer to:
Logic
Completeness (logic)
Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable
Mathematics
The completeness of the real numbers, which implies that there are no "holes" in the real numbers
Complete metric space, a metric space in which every Cauchy sequence converges
Complete uniform space, a uniform space where every Cauchy net in converges (or equivalently every Cauchy filter converges)
Complete measure, a measure space where every subset of every null set is measurable
Completion (algebra), at an ideal
Completeness (cryptography)
Completeness (statistics), a statistic that does not allow an unbiased estimator of zero
Complete graph, an undirected graph in which every pair of vertices has exactly one edge connecting them
Complete category, a category C where every diagram from a small category to C has a limit; it is cocomplete if every such functor has a colimit
Completeness (order theory), a notion that generally refers to the existence of certain suprema or infima of some partially ordered set
Complete variety, an algebraic variety that satisfies an analog of compactness
Complete orthonormal basis—see Orthonormal basis#Incomplete orthogonal sets
Complete sequence, a type of integer sequence
Computing
Complete (complexity), a notion referring to a problem in computational complexity theory that all other problems in a class reduce to
Turing complete set, a related notion from recursion theory
Completeness (knowledge bases), found in knowledge base theory
Complete search algorithm, a search algorithm that is guaranteed to find a solution if there is one
Music
Completeness, a 1998 collection of Miki Nakatani music videos
"Complete" (Jaimeson song), a 2003 song by the British electronic music artist Jaimeson
Complete (Lila McCann album), the third album by country music artist Lila McCann
Complete (News from Babel album), a three-CD box set by the English avant-rock band News from Babel
Complete (The Smiths album), a box set released by British band The Smiths on 26 September 2011
Complete (The Veronicas album), 2009
"Complete", a song by Kutless from To Know That You're Alive
"Complete", a 2007 song by Girls' Generation from the album Girls' Generation
Complete (BtoB album), 2015
Other uses
Complete set of commuting operators (or CSCO), a set of commuting operators in quantum mechanics whose eigenvalues are sufficient to specify the physical state of a system
Complete flower, a flower with both male and female reproductive structures as well as petals and sepals. See Sexual reproduction in plants
Complete market, a market with negligible transaction costs and a price for every asset
Completion (oil and gas wells), the process of making a well ready for production
See also
Completion (disambiguation)
Completely (disambiguation)
Compleat (disambiguation)
Wholeness (disambiguation) |
https://en.wikipedia.org/wiki/Nielsen%E2%80%93Schreier%20theorem | In group theory, a branch of mathematics, the Nielsen–Schreier theorem states that every subgroup of a free group is itself free. It is named after Jakob Nielsen and Otto Schreier.
Statement of the theorem
A free group may be defined from a group presentation consisting of a set of generators with no relations. That is, every element is a product of some sequence of generators and their inverses, but these elements do not obey any equations except those trivially following from = 1. The elements of a free group may be described as all possible reduced words, those strings of generators and their inverses in which no generator is adjacent to its own inverse. Two reduced words may be multiplied by concatenating them and then removing any generator-inverse pairs that result from the concatenation.
The Nielsen–Schreier theorem states that if H is a subgroup of a free group G, then H is itself isomorphic to a free group. That is, there exists a set S of elements which generate H, with no nontrivial relations among the elements of S.
The Nielsen–Schreier formula, or Schreier index formula, quantifies the result in the case where the subgroup has finite index: if G is a free group of rank n (free on n generators), and H is a subgroup of finite index [G : H] = e, then H is free of rank .
Example
Let G be the free group with two generators , and let H be the subgroup consisting of all reduced words of even length (products of an even number of letters ). Then H is generated by its six elements A factorization of any reduced word in H into these generators and their inverses may be constructed simply by taking consecutive pairs of letters in the reduced word. However, this is not a free presentation of H because the last three generators can be written in terms of the first three as . Rather, H is generated as a free group by the three elements which have no relations among them; or instead by several other triples of the six generators. Further, G is free on n = 2 generators, H has index e = [G : H] = 2 in G, and H is free on 1 + e(n–1) = 3 generators. The Nielsen–Schreier theorem states that like H, every subgroup of a free group can be generated as a free group, and if the index of H is finite, its rank is given by the index formula.
Proof
[[File:Covering-Graph.png|thumb|300x300px|The free group G = π1(X) has n = 2 generators corresponding to loops a,b from the base point P in X. The subgroup H of even-length words, with index e = [G : H] = 2, corresponds to the covering graph Y with two vertices corresponding to the cosets H and H''' = aH = bH = a−1H = b−1H, and two lifted edges for each of the original loop-edges a,b. Contracting one of the edges of Y gives a homotopy equivalence to a bouquet of three circles, so that H = π1(Y) is a free group on three generators, for example aa, ab, ba.|alt=]]
A short proof of the Nielsen–Schreier theorem uses the algebraic topology of fundamental groups and covering spaces. A free group G on a set of gene |
https://en.wikipedia.org/wiki/Eugen%20P%C3%B3lya | Jenő Sándor Pólya, , (April 30, 1876 – 1944) was a Hungarian surgeon who was a native of Budapest. He was the brother of George Pólya (1887–1985), who was a professor of mathematics at Stanford University.
He studied in Budapest, and in 1898 earned his medical doctorate. In 1909 he was habilitated for surgical anatomy at Budapest, attaining the title of professor of 1914. Reportedly, he was murdered by the Nazis during the Siege of Budapest, although his body was never recovered.
Jenö Pólya is remembered for a surgical procedure known as the "Reichel-Pólya operation", a type of posterior gastroenterostomy that is a modification of the Billroth II operation. The operation is named in conjunction with German surgeon Friedrich Paul Reichel (1858–1934).
Between World War I and World War II, he was visited in Budapest by several American surgeons who came to observe his surgical technique. Consequently, in 1939, Pólya was elected an honorary member of the American College of Surgeons.
References
Jenö (Eugen) Alexander Pólya @ Who Named It
1876 births
1944 deaths
20th-century Hungarian physicians
Hungarian surgeons
Hungarian people of Jewish descent
Physicians from Budapest
20th-century surgeons |
https://en.wikipedia.org/wiki/Commutation%20theorem%20for%20traces | In mathematics, a commutation theorem for traces explicitly identifies the commutant of a specific von Neumann algebra acting on a Hilbert space in the presence of a trace.
The first such result was proved by Francis Joseph Murray and John von Neumann in the 1930s and applies to the von Neumann algebra generated by a discrete group or by the dynamical system associated with a measurable transformation preserving a probability measure.
Another important application is in the theory of unitary representations of unimodular locally compact groups, where the theory has been applied to the regular representation and other closely related representations. In particular this framework led to an abstract version of the Plancherel theorem for unimodular locally compact groups due to Irving Segal and Forrest Stinespring and an abstract Plancherel theorem for spherical functions associated with a Gelfand pair due to Roger Godement. Their work was put in final form in the 1950s by Jacques Dixmier as part of the theory of Hilbert algebras.
It was not until the late 1960s, prompted partly by results in algebraic quantum field theory and quantum statistical mechanics due to the school of Rudolf Haag, that the more general non-tracial Tomita–Takesaki theory was developed, heralding a new era in the theory of von Neumann algebras.
Commutation theorem for finite traces
Let H be a Hilbert space and M a von Neumann algebra on H with a unit vector Ω such that
M Ω is dense in H
M ' Ω is dense in H, where M ' denotes the commutant of M
(abΩ, Ω) = (baΩ, Ω) for all a, b in M.
The vector Ω is called a cyclic-separating trace vector. It is called a trace vector because the last condition means that the matrix coefficient corresponding to Ω defines a tracial state on M. It is called cyclic since Ω generates H as a topological M-module. It is called separating
because if aΩ = 0 for a in M, then aMΩ= (0), and hence a = 0.
It follows that the map
for a in M defines a conjugate-linear isometry of H with square the identity, J2 = I. The operator J is usually called the modular conjugation operator.
It is immediately verified that JMJ and M commute on the subspace M Ω, so that
The commutation theorem of Murray and von Neumann states that
{| border="1" cellspacing="0" cellpadding="5"
|
|}
One of the easiest ways to see this is to introduce K, the closure of the real
subspace Msa Ω, where Msa denotes the self-adjoint elements in M. It follows that
an orthogonal direct sum for the real part of the inner product. This is just the real orthogonal decomposition for the ±1 eigenspaces of J.
On the other hand for a in Msa and b in Msa, the inner product (abΩ, Ω) is real, because ab is self-adjoint. Hence K is unaltered if M is replaced by M '.
In particular Ω is a trace vector for M and J is unaltered if M is replaced by M '. So the opposite inclusion
follows by reversing the roles of M and M.
Examples
One of the simplest cases of the commutation theorem, wh |
https://en.wikipedia.org/wiki/Lo%20Chun%20Kit | Toby Lo Chun Kit (, born 13 November 1985) is a former Hong Kong professional footballer who played as a right back.
Career statistics
Club
As at 20 September 2008
References
External links
SouthChinaFC.com, 20. 盧俊傑
Accolades
1985 births
Living people
Hong Kong men's footballers
Men's association football defenders
Hong Kong First Division League players
Hong Kong Premier League players
Citizen AA players
Eastern Sports Club footballers
South China AA players
Resources Capital FC players
Hong Kong Pegasus FC players |
https://en.wikipedia.org/wiki/Boole%20%28disambiguation%29 | George Boole (1815–1864) was a British mathematician and philosopher, and originator of Boolean algebra.
Boole may also refer to:
Boole (band), an electronic music group from the United States
Boole (crater), a lunar crater
Boole (tree), a giant sequoia tree in Sequoia National Forest
People with the name
Alicia Boole or Alicia Boole Stott (1860–1940), mathematician and daughter of Mary Everest and George Boole
Mary Everest Boole (1832–1916), mathematician and wife of George Boole
Sasha Boole (born 1988), Ukrainian singer and songwriter
William H. Boole (1827–1896), pastor and prohibitionist in New York
See also
Bool (disambiguation)
Boolean (disambiguation)
Boule (disambiguation) |
https://en.wikipedia.org/wiki/Legendre%E2%80%93Clebsch%20condition |
In the calculus of variations the Legendre–Clebsch condition is a second-order condition which a solution of the Euler–Lagrange equation must satisfy in order to be a minimum.
For the problem of minimizing
the condition is
Generalized Legendre–Clebsch
In optimal control, the situation is more complicated because of the possibility of a singular solution. The generalized Legendre–Clebsch condition, also known as convexity, is a sufficient condition for local optimality such that when the linear sensitivity of the Hamiltonian to changes in u is zero, i.e.,
The Hessian of the Hamiltonian is positive definite along the trajectory of the solution:
In words, the generalized LC condition guarantees that over a singular arc, the Hamiltonian is minimized.
See also
Bang–bang control
References
Further reading
Optimal control
Calculus of variations |
https://en.wikipedia.org/wiki/Pingo%20%28footballer%2C%20born%201980%29 | Erison Carlos dos Santos Silva (born May 22, 1980) is a former Brazilian football player.
Club statistics
References
https://web.archive.org/web/20110813091225/http://cerezo.co.jp/news_detail_backnum.asp?c_idx=2655&contents_code=100&date_s=2006%2F01&iPage=
External links
1980 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
Sport Club Corinthians Paulista players
Associação Desportiva São Caetano players
Cerezo Osaka players
Associação Atlética Ponte Preta players
Busan IPark players
Avaí FC players
Campeonato Brasileiro Série A players
Campeonato Brasileiro Série B players
J1 League players
K League 1 players
Brazilian expatriate sportspeople in Japan
Brazilian expatriate sportspeople in South Korea
Expatriate men's footballers in Japan
Expatriate men's footballers in South Korea
Men's association football midfielders
People from Barra do Piraí
Footballers from Rio de Janeiro (state) |
https://en.wikipedia.org/wiki/Shayle%20R.%20Searle | Shayle Robert Searle PhD (26 April 1928 – 18 February 2013) was a New Zealand mathematician who was professor emeritus of biological statistics at Cornell University. He was a leader in the field of linear and mixed models in statistics, and published widely on the topics of linear models, mixed models, and variance component estimation.
Searle was one of the first statisticians to use matrix algebra in statistical methodology, and was an early proponent of the use of applied statistical techniques in animal breeding.
He died at his home in Ithaca, New York.
Education
BA – Victoria University of Wellington – 1949
MSc – Victoria University of Wellington – 1950
PhD – Cornell University – 1958
DSc (h.c.) – Victoria University of Wellington – 2005
Employment
Research statistician – New Zealand Dairy Board – 1953 to 1955, 1959 to 1962
Statistician – University Computing Center, Cornell University – 1962 to 1965
Professor of biological statistics – Cornell University – 1965 to 1996
Honours
Winner, Humboldt Research Award of the Alexander von Humboldt Foundation
Fellow, American Statistical Association
Fellow, Royal Statistical Society
Honorary Fellow, Royal Society of New Zealand
Bibliography
Books
Selected journal articles
2000s
1990s
1980s
1970s
1960s
1950s
References
Further reading
External links
Cornell University alumni
Cornell University faculty
Victoria University of Wellington alumni
New Zealand mathematicians
New Zealand statisticians
Biostatisticians
1928 births
2013 deaths
Fellows of the American Statistical Association
Writers from Ithaca, New York |
https://en.wikipedia.org/wiki/Wright%20omega%20function | In mathematics, the Wright omega function or Wright function, denoted ω, is defined in terms of the Lambert W function as:
Uses
One of the main applications of this function is in the resolution of the equation z = ln(z), as the only solution is given by z = e−ω(π i).
y = ω(z) is the unique solution, when for x ≤ −1, of the equation y + ln(y) = z. Except on those two rays, the Wright omega function is continuous, even analytic.
Properties
The Wright omega function satisfies the relation .
It also satisfies the differential equation
wherever ω is analytic (as can be seen by performing separation of variables and recovering the equation ), and as a consequence its integral can be expressed as:
Its Taylor series around the point takes the form :
where
in which
is a second-order Eulerian number.
Values
Plots
Notes
References
"On the Wright ω function", Robert Corless and David Jeffrey
Special functions |
https://en.wikipedia.org/wiki/Mykhaylo%20Kopolovets | Mykhaylo Kopolovets (born 29 January 1984) is a Ukrainian former professional football midfielder who is most known for playing for Zakarpattia Uzhhorod, Karpaty Lviv and Mynai.
Career statistics
External links
Profile on Official Karpaty Lviv Website
1984 births
Living people
Ukrainian men's footballers
Ukrainian Premier League players
Ukrainian First League players
Ukrainian Second League players
Ukrainian Amateur Football Championship players
Belarusian Premier League players
NOFV-Oberliga players
FC Karpaty Lviv players
FC Hoverla Uzhhorod players
FC Zakarpattia-2 Uzhhorod players
FC Belshina Bobruisk players
FC Einheit Rudolstadt players
FC Mynai players
Expatriate men's footballers in Belarus
Ukrainian expatriate men's footballers
Expatriate men's footballers in Germany
Ukrainian expatriate sportspeople in Belarus
Ukrainian expatriate sportspeople in Germany
Men's association football midfielders
Footballers from Zakarpattia Oblast |
https://en.wikipedia.org/wiki/1929%20Ekstraklasa | Statistics of Ekstraklasa for the 1929 season.
Overview
The championship was contested by 13 teams and Warta Poznań won the title.
League table
Results
References
Poland - List of final tables (RSSSF)
Ekstraklasa seasons
1
Pol
Pol |
https://en.wikipedia.org/wiki/1930%20Ekstraklasa | Statistics of Ekstraklasa for the 1930 season.
Overview
It contested by 12 teams, and KS Cracovia won the championship.
League table
Results
References
Poland - List of final tables (RSSSF)
Ekstraklasa seasons
1
Pol
Pol |
https://en.wikipedia.org/wiki/1931%20Ekstraklasa | Statistics of Ekstraklasa for the 1931 season.
Overview
It was contested by 12 teams, and Garbarnia Kraków won the championship.
League table
Results
References
Poland - List of final tables (RSSSF)
Ekstraklasa seasons
1
Pol
Pol |
https://en.wikipedia.org/wiki/1932%20Ekstraklasa | Statistics of Ekstraklasa for the 1932 season.
Overview
It was contested by 12 teams, and Cracovia won its third title.
League table
Results
References
Poland - List of final tables (RSSSF)
Ekstraklasa seasons
1
Pol
Pol |
https://en.wikipedia.org/wiki/1933%20Ekstraklasa | Statistics of Ekstraklasa for the 1933 season.
Overview
It was contested by 12 teams, and Ruch Chorzów won the championship.
First phase
Eastern Group
Results
Western Group
Results
Final phase
Championship group
Results
Relegation group
Results
Promotion/relegation playoffs
References
Poland - List of final tables (RSSSF)
Ekstraklasa seasons
1
Pol
Pol |
https://en.wikipedia.org/wiki/1934%20Ekstraklasa | Statistics of Ekstraklasa for the 1934 season.
Overview
It was contested by 12 teams, and Ruch Chorzów won the championship.
League table
Results
References
Poland - List of final tables (RSSSF)
Ekstraklasa seasons
1
Pol
Pol |
https://en.wikipedia.org/wiki/1935%20Ekstraklasa | Statistics of Ekstraklasa for the 1935 season.
Overview
It was contested by 11 teams, and Ruch Chorzów won the championship.
League table
Results
References
Poland - List of final tables (RSSSF)
Ekstraklasa seasons
1
Pol
Pol |
https://en.wikipedia.org/wiki/1936%20Ekstraklasa | Statistics of Ekstraklasa for the 1936 season.
Overview
The championship was contested by 10 teams, and Ruch Hajduki Wielkie (currently Ruch Chorzów) won the title.
Season's Legia Warsaw relegation from Ekstraklasa, was its only decline from the first tier of the Polish football league system in its more than 100-year history.
League table
Results
References
Poland - List of final tables (RSSSF)
Ekstraklasa seasons
1
Pol
Pol |
https://en.wikipedia.org/wiki/1937%20Ekstraklasa | Statistics of Ekstraklasa for the 1937 season.
Overview
The championship was contested by 10 teams, and the title went to Cracovia.
League table
Results
References
Poland - List of final tables (RSSSF)
Ekstraklasa seasons
1
Pol
Pol |
https://en.wikipedia.org/wiki/1938%20Ekstraklasa | Statistics of Ekstraklasa for the 1938 season.
Overview
The championship was contested by 10 teams, and Ruch Chorzów won the title.
League table
Results
References
Poland - List of final tables (RSSSF)
Ekstraklasa seasons
1
Pol
Pol |
https://en.wikipedia.org/wiki/1939%20Ekstraklasa | Statistics of the Ekstraklasa for the 1939 season. The championship was unfinished because of the Nazi German attack on Poland which triggered the Second World War.
League table
Results
References
Poland - List of final tables (RSSSF)
Ekstraklasa seasons
1
Pol
Pol |
https://en.wikipedia.org/wiki/1948%20Ekstraklasa | Statistics of Ekstraklasa for the 1948 season.
Overview
It was contested by 14 teams, and Cracovia won the championship.
League table
Results
Final
Cracovia 3-1 Wisła Kraków
Top goalscorers
References
External links
Poland – List of final tables at RSSSF
List of Polish football championships
List of Polish football championships
Ekstraklasa seasons
1
Pol
Pol |
https://en.wikipedia.org/wiki/1949%20Ekstraklasa | Statistics of Ekstraklasa for the 1949 season.
Overview
It was contested by 12 teams, and Wisła Kraków won the championship.
League table
Results
Top goalscorers
References
Poland – List of final tables at RSSSF
Ekstraklasa seasons
1
Pol
Pol |
https://en.wikipedia.org/wiki/1950%20Ekstraklasa | Statistics of Ekstraklasa for the 1950 season.
Overview
It was contested by 12 teams, and Wisła Kraków won the championship.
League table
Results
Top goalscorers
References
Poland – List of final tables at RSSSF
Ekstraklasa seasons
1
Pol
Pol |
https://en.wikipedia.org/wiki/1951%20Ekstraklasa | Statistics of Ekstraklasa for the 1951 season.
Overview
12 teams played in the league and Wisła Kraków finished in the first position and became the league champion. In the 1951 season, the Ekstraklasa was not a competition for the title of the Polish Champion. Before the season Polish Football Association decided that Champion of Poland title will be awarded to the winner of the Polish Cup, which was later Ruch Chorzów.
League table
Results
Top goalscorers
References
External links
Poland – List of final tables at RSSSF
Ekstraklasa seasons
1
Pol
Pol |
https://en.wikipedia.org/wiki/1952%20Ekstraklasa | Statistics of Ekstraklasa for the 1952 season.
Overview
It was contested by 12 teams, and Ruch Chorzów won the championship.
Regular season
Group 1
Results
Group 2
Results
Final Games
Ruch Chorzów 7-0 Polonia Bytom
Polonia Bytom 0-0 Ruch Chorzów
Top goalscorers
References
Poland – List of final tables at RSSSF
Ekstraklasa seasons
1
Pol
Pol |
https://en.wikipedia.org/wiki/1953%20Ekstraklasa | Statistics of Ekstraklasa for the 1953 season.
Overview
12 teams competed in the 1953 season. Ruch Chorzów won the championship.
League table
Results
Top goalscorers
References
Poland – List of final tables at RSSSF
Ekstraklasa seasons
1
Pol
Pol |
https://en.wikipedia.org/wiki/1954%20Ekstraklasa | Statistics of Ekstraklasa for the 1954 season.
Overview
It was contested by 12 teams, and Polonia Bytom won the championship.
League table
Results
Top goalscorers
References
Poland – List of final tables at RSSSF
Ekstraklasa seasons
1
Pol
Pol |
https://en.wikipedia.org/wiki/1955%20Ekstraklasa | Statistics of Ekstraklasa for the 1955 season.
Overview
It was contested by 12 teams, and Legia Warsaw won the championship.
League table
Results
Top goalscorers
References
Poland – List of final tables at RSSSF
Ekstraklasa seasons
1
Pol
Pol |
https://en.wikipedia.org/wiki/1956%20Ekstraklasa | Statistics of Ekstraklasa for the 1956 season.
Overview
12 teams competed in the league and the championship was won by Legia Warsaw.
League table
Results
Top goalscorers
References
Poland – List of final tables at RSSSF
Ekstraklasa seasons
1
Pol
Pol |
https://en.wikipedia.org/wiki/1957%20Ekstraklasa | Statistics of Ekstraklasa for the 1957 season.
Overview
12 teams competed in the 1957 Ekstraklasa. Górnik Zabrze won the championship.
League table
Results
Top goalscorers
References
Poland – List of final tables at RSSSF
Ekstraklasa seasons
1
Pol
Pol |
https://en.wikipedia.org/wiki/1958%20Ekstraklasa | Statistics of Ekstraklasa for the 1958 season.
Overview
12 teams competed in the 1958 season. ŁKS Łódź won the championship.
League table
Results
Top goalscorers
References
Poland – List of final tables at RSSSF
Ekstraklasa seasons
1
Pol
Pol |
https://en.wikipedia.org/wiki/1959%20Ekstraklasa | Statistics of Ekstraklasa for the 1959 season.
Overview
It was contested by 12 teams, and Górnik Zabrze won the championship.
League table
Results
Top goalscorers
References
Poland – List of final tables at RSSSF
Ekstraklasa seasons
1
Pol
Pol |
https://en.wikipedia.org/wiki/1960%20Ekstraklasa | Statistics of Ekstraklasa for the 1960 season.
Overview
It was contested by 12 teams, and Ruch Chorzów won the championship.
League table
Results
Top goalscorers
References
Poland – List of final tables at RSSSF
Ekstraklasa seasons
1
Pol
Pol |
https://en.wikipedia.org/wiki/1961%20Ekstraklasa | Statistics of Ekstraklasa for the 1961 season.
Overview
It was contested by 14 teams, and Górnik Zabrze won the championship.
League table
Results
Top goalscorers
References
Poland – List of final tables at RSSSF
Ekstraklasa seasons
1
Pol
Pol |
https://en.wikipedia.org/wiki/1962%20Ekstraklasa | Statistics of Ekstraklasa for the 1962 season.
Overview
It was contested by 14 teams, and Polonia Bytom won the championship.
Regular season
Group A
Results
Group B
Results
Playoff stage
1st place playoff
Górnik Zabrze 1-4 ; 2-2 Polonia Bytom
3rd place playoff
Odra Opole 1-0 ; 0-1 Zagłębie Sosnowiec
5th place playoff
Wisła Kraków 1-1 ; 1-4 Legia Warsaw
7th place playoff
Arkonia Szczecin 2-0 ; 1-2 Ruch Chorzów
9th place playoff
Lechia Gdańsk 2-0 ; 1-3 ŁKS Łódź
11th place playoff
Gwardia Warszawa 2-6 ; 1-2 Lech Poznań
13th place playoff
Stal Mielec 3-0 ; 1-0 KS Cracovia
Top goalscorers
References
Poland – List of final tables at RSSSF
Ekstraklasa seasons
1
Pol
Pol |
https://en.wikipedia.org/wiki/Median%20algebra | In mathematics, a median algebra is a set with a ternary operation satisfying a set of axioms which generalise the notions of medians of triples of real numbers and of the Boolean majority function.
The axioms are
The second and third axioms imply commutativity: it is possible (but not easy) to show that in the presence of the other three, axiom (3) is redundant. The fourth axiom implies associativity.
There are other possible axiom systems: for example the two
also suffice.
In a Boolean algebra, or more generally a distributive lattice, the median function satisfies these axioms, so that every Boolean algebra and every distributive lattice forms a median algebra.
Birkhoff and Kiss showed that a median algebra with elements 0 and 1 satisfying is a distributive lattice.
Relation to median graphs
A median graph is an undirected graph in which for every three vertices , , and there is a unique vertex that belongs to shortest paths between any two of , , and . If this is the case, then the operation defines a median algebra having the vertices of the graph as its elements.
Conversely, in any median algebra, one may define an interval to be the set of elements such that . One may define a graph from a median algebra by creating a vertex for each algebra element and an edge for each pair such that the interval contains no other elements. If the algebra has the property that every interval is finite, then this graph is a median graph, and it accurately represents the algebra in that the median operation defined by shortest paths on the graph coincides with the algebra's original median operation.
References
External links
Median Algebra Proof
Algebra
Ternary operations |
https://en.wikipedia.org/wiki/1962%E2%80%9363%20Ekstraklasa | Statistics of Ekstraklasa for the 1962–63 season.
Overview
It was contested by 14 teams, and Górnik Zabrze won the championship.
League table
Results
Top goalscorers
References
Poland – List of final tables at RSSSF
Ekstraklasa seasons
1962–63 in Polish football
Pol |
https://en.wikipedia.org/wiki/1963%E2%80%9364%20Ekstraklasa | Statistics of Ekstraklasa for the 1963–64 season.
Overview
It was contested by 14 teams, and Górnik Zabrze won the championship.
League table
Results
Top goalscorers
References
- List of final tables (mogiel.net)
Ekstraklasa seasons
1963–64 in Polish football
Pol |
https://en.wikipedia.org/wiki/1964%E2%80%9365%20Ekstraklasa | Statistics of Ekstraklasa for the 1964–65 season.
Overview
14 teams played in the league and the championship went to Górnik Zabrze.
League table
Results
Top goalscorers
References
Poland – List of final tables at RSSSF
Ekstraklasa seasons
1964–65 in Polish football
Pol |
https://en.wikipedia.org/wiki/1965%E2%80%9366%20Ekstraklasa | Statistics of Ekstraklasa for the 1965–66 season.
Overview
14 teams competed in the 1965-66 season with Górnik Zabrze winning the championship.
League table
Results
Top goalscorers
References
Poland – List of final tables at RSSSF
Ekstraklasa seasons
1965–66 in Polish football
Pol |
https://en.wikipedia.org/wiki/1966%E2%80%9367%20Ekstraklasa | Statistics of Ekstraklasa for the 1966–67 season.
Overview
It was contested by 14 teams, and Górnik Zabrze won the championship.
League table
Results
Top goalscorers
References
Poland – List of final tables at RSSSF
Ekstraklasa seasons
1966–67 in Polish football
Pol |
https://en.wikipedia.org/wiki/1967%E2%80%9368%20Ekstraklasa | Statistics of Ekstraklasa for the 1967–68 season.
Overview
It was contested by 14 teams, and Ruch Chorzów won the championship.
League table
Results
Top goalscorers
References
Poland – List of final tables at RSSSF
Ekstraklasa seasons
1967–68 in Polish football
Pol |
https://en.wikipedia.org/wiki/1968%E2%80%9369%20Ekstraklasa | Statistics of Ekstraklasa for the 1968–69 season.
Overview
It was contested by 14 teams, and Legia Warsaw won the championship.
League table
Results
Top goalscorers
References
Poland – List of final tables at RSSSF
Ekstraklasa seasons
1968–69 in Polish football
Pol |
https://en.wikipedia.org/wiki/1982%E2%80%9383%20Ekstraklasa | Statistics of Ekstraklasa for the 1982–83 season.
Overview
It was contested by 16 teams, and Lech Poznań won the championship.
League table
Results
Top goalscorers
References
External links
Poland – List of final tables at RSSSF
Ekstraklasa seasons
1982–83 in Polish football
Pol |
https://en.wikipedia.org/wiki/1983%E2%80%9384%20Ekstraklasa | Statistics of Ekstraklasa for the 1983–84 season.
Overview
It was contested by 16 teams, and Lech Poznań won the championship.
League table
Results
Top goalscorers
References
External links
Poland – List of final tables at RSSSF
Ekstraklasa seasons
1983–84 in Polish football
Pol |
https://en.wikipedia.org/wiki/1984%E2%80%9385%20Ekstraklasa | Statistics of Ekstraklasa for the 1984–85 season.
Overview
The league was contested by 16 teams, and Górnik Zabrze won the championship.
League table
Results
Top goalscorers
References
External links
Poland – List of final tables at RSSSF
Ekstraklasa seasons
1984–85 in Polish football
Pol |
https://en.wikipedia.org/wiki/1985%E2%80%9386%20Ekstraklasa | Statistics of Ekstraklasa for the 1985–86 season.
Overview
It was contested by 16 teams, and Górnik Zabrze won the championship.
League table
Results
Top goalscorers
References
External links
Poland – List of final tables at RSSSF
Ekstraklasa seasons
1985–86 in Polish football
Pol |
https://en.wikipedia.org/wiki/1986%E2%80%9387%20Ekstraklasa | Statistics of Ekstraklasa for the 1986–87 season.
Overview
It was contested by 16 teams, and Górnik Zabrze won the championship.
League table
Results
Relegation playoffs
The matches were played on 28 June and 1 July 1987.
Top goalscorers
References
External links
Poland – List of final tables at RSSSF
Ekstraklasa seasons
1986–87 in Polish football
Pol |
https://en.wikipedia.org/wiki/1987%E2%80%9388%20Ekstraklasa | Statistics of the Ekstraklasa for the 1987–88 season.
Overview
It was contested by 16 teams, and Górnik Zabrze won the championship.
League table
Results
Relegation playoffs
The matches were played on 25 and 28 June 1988.
Top goalscorers
References
External links
Poland – List of final tables at RSSSF
Ekstraklasa seasons
1987–88 in Polish football
Pol |
https://en.wikipedia.org/wiki/1988%E2%80%9389%20Ekstraklasa | Statistics of Ekstraklasa for the 1988–89 season.
Overview
It was contested by 16 teams, and Ruch Chorzów won the championship.
League table
Results
Relegation playoffs
After the end of the season, play-offs played for two places in the first league in the 1989–90 season between teams from places 13-14 in the first league and runners-up in the 2nd league groups:
13th team of the I league and 2nd team of the II league of group II - Pogoń Szczecin and Motor Lublin,
14th team of the I league and 2nd team of the II league of group I - GKS Jastrzębie and Zawisza Bydgoszcz.
Pogoń Szczecin and GKS Jastrzębie did not hold places at the highest league level.
Top goalscorers
References
External links
Poland – List of final tables at RSSSF
Ekstraklasa seasons
1988–89 in Polish football
Pol |
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