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https://en.wikipedia.org/wiki/Bird%E2%80%93Meertens%20formalism
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The Bird–Meertens formalism (BMF) is a calculus for deriving programs from program specifications (in a functional programming setting) by a process of equational reasoning. It was devised by Richard Bird and Lambert Meertens as part of their work within IFIP Working Group 2.1.
It is sometimes referred to in publications as BMF, as a nod to Backus–Naur form. Facetiously it is also referred to as Squiggol, as a nod to ALGOL, which was also in the remit of WG 2.1, and because of the "squiggly" symbols it uses. A less-used variant name, but actually the first one suggested, is SQUIGOL.
Basic examples and notations
Map is a well-known second-order function that applies a given function to every element of a list; in BMF, it is written :
Likewise, reduce is a function that collapses a list into a single value by repeated application of a binary operator. It is written / in BMF.
Taking as a suitable binary operator with neutral element e, we have
Using those two operators and the primitives (as the usual addition), and (for list concatenation), we can easily express the sum of all elements of a list, and the flatten function, as and , in
point-free style. We have:
Similarly, writing for functional composition and for conjunction, it is easy to write a function testing that all elements of a list satisfy a predicate p, simply as :
Bird (1989) transforms inefficient easy-to-understand expressions ("specifications") into efficient involved expressions ("programs") by algebraic manipulation. For example, the specification "" is an almost literal translation of the maximum segment sum problem, but running that functional program on a list of size will take time in general. From this, Bird computes an equivalent functional program that runs in time , and is in fact a functional version of Kadane's algorithm.
The derivation is shown in the picture, with computational complexities given in blue, and law applications indicated in red.
Example instances of the laws can be opened by clicking on [show]; they use lists of integer numbers, addition, minus, and multiplication. The notation in Bird's paper differs from that used above: , , and correspond to , , and a generalized version of above, respectively, while and compute a list of all prefixes and suffixes of its arguments, respectively. As above, function composition is denoted by "", which has lowest binding precedence. In the example instances, lists are colored by nesting depth; in some cases, new operations are defined ad hoc (grey boxes).
The homomorphism lemma and its applications to parallel implementations
A function h on lists is called a list homomorphism if there exists an associative binary operator and neutral element such that the following holds:
The homomorphism lemma states that h is a homomorphism if and only if there exists an operator and a function f such that .
A point of great interest for this lemma is its application to the derivation of highly parallel impl
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https://en.wikipedia.org/wiki/Virtually%20Haken%20conjecture
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In topology, an area of mathematics, the virtually Haken conjecture states that every compact, orientable, irreducible three-dimensional manifold with infinite fundamental group is virtually Haken. That is, it has a finite cover (a covering space with a finite-to-one covering map) that is a Haken manifold.
After the proof of the geometrization conjecture by Perelman, the conjecture was only open for hyperbolic 3-manifolds.
The conjecture is usually attributed to Friedhelm Waldhausen in a paper from 1968, although he did not formally state it. This problem is formally stated as Problem 3.2 in Kirby's problem list.
A proof of the conjecture was announced on March 12, 2012 by Ian Agol in a seminar lecture he gave at the Institut Henri Poincaré. The proof appeared shortly thereafter in a preprint which was eventually published in Documenta Mathematica. The proof was obtained via a strategy by previous work of Daniel Wise and collaborators, relying on actions of the fundamental group on certain auxiliary spaces (CAT(0) cube complexes)
It used as an essential ingredient the freshly-obtained solution to the surface subgroup conjecture by Jeremy Kahn and Vladimir Markovic.
Other results which are directly used in Agol's proof include the Malnormal Special Quotient Theorem of Wise and a criterion of Nicolas Bergeron and Wise for the cubulation of groups.
In 2018 related results were obtained by Piotr Przytycki and Daniel Wise proving that mixed 3-manifolds are also virtually special, that is they can be cubulated into a cube complex with a finite cover where all the hyperplanes are embedded which by the previous mentioned work can be made virtually Haken.
See also
Virtually fibered conjecture
Surface subgroup conjecture
Ehrenpreis conjecture
Notes
References
.
.
External links
3-manifolds
Theorems in topology
Conjectures that have been proved
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https://en.wikipedia.org/wiki/Sequential%20estimation
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In statistics, sequential estimation refers to estimation methods in sequential analysis where the sample size is not fixed in advance. Instead, data is evaluated as it is collected, and further sampling is stopped in accordance with a predefined stopping rule as soon as significant results are observed.
The generic version is called the optimal Bayesian estimator, which is the theoretical underpinning for every sequential estimator (but cannot be instantiated directly). It includes a Markov process for the state propagation and measurement process for each state, which yields some typical statistical independence relations. The Markov process describes the propagation of a probability distribution over discrete time instances and the measurement is the information one has about each time instant, which is usually less informative than the state. Only the observed sequence will, together with the models, accumulate the information of all measurements and the corresponding Markov process to yield better estimates.
From that, the Kalman filter (and its variants), the particle filter, the histogram filter and others can be derived. It depends on the models, which one to use and requires experience to choose the right one. In most cases, the goal is to estimate the state sequence from the measurements. In other cases, one can use the description to estimate the parameters of a noise process for example. One can also accumulate the unmodeled statistical behavior of the states projected in the measurement space (called innovation sequence, which naturally includes the orthogonality principle in its derivations to yield an independence relation and therefore can be also cast into a Hilbert space representation, which makes it very intuitive) over time and compare it with a threshold, which then corresponds to the aforementioned stopping criterion. One difficulty is to setup the initial conditions for the probabilistic models, which is in most cases done by experience, data sheets or precise measurements with a different setup.
The statistical behaviour of the heuristic/sampling methods (e.g. particle filter or histogram filter) depends on many parameters and implementation details and should not be used in safety critical applications (since it is very hard to yield theoretical guarantees or do proper testing), unless one has a very good reason.
If there is a dependence of each state on an overall entity (e.g. a map or simply an overall state variable), one typically uses SLAM (simultaneous localization and mapping) techniques, which include the sequential estimator as a special case (when the overall state variable has just one state). It will estimate the state sequence and the overall entity.
There are also none-causal variants, that have all measurements at the same time, batches of measurements or revert the state evolution to go backwards again. These are then, however, not real time capable (except one uses a really big buffer, that lowers t
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https://en.wikipedia.org/wiki/Bounded%20mean%20oscillation
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In harmonic analysis in mathematics, a function of bounded mean oscillation, also known as a BMO function, is a real-valued function whose mean oscillation is bounded (finite). The space of functions of bounded mean oscillation (BMO), is a function space that, in some precise sense, plays the same role in the theory of Hardy spaces Hp that the space L∞ of essentially bounded functions plays in the theory of Lp-spaces: it is also called John–Nirenberg space, after Fritz John and Louis Nirenberg who introduced and studied it for the first time.
Historical note
According to , the space of functions of bounded mean oscillation was introduced by in connection with his studies of mappings from a bounded set belonging to Rn into Rn and the corresponding problems arising from elasticity theory, precisely from the concept of elastic strain: the basic notation was introduced in a closely following paper by , where several properties of this function spaces were proved. The next important step in the development of the theory was the proof by Charles Fefferman of the duality between BMO and the Hardy space H1, in the noted paper : a constructive proof of this result, introducing new methods and starting a further development of the theory, was given by Akihito Uchiyama.
Definition
The mean oscillation of a locally integrable function u over a hypercube Q in Rn is defined as the value of the following integral:
where
|Q| is the volume of Q, i.e. its Lebesgue measure
uQ is the average value of u on the cube Q, i.e.
A BMO function is a locally integrable function u whose mean oscillation supremum, taken over the set of all cubes Q contained in Rn, is finite.
Note 1. The supremum of the mean oscillation is called the BMO norm of u. and is denoted by ||u||BMO (and in some instances it is also denoted ||u||∗).
Note 2. The use of cubes Q in Rn as the integration domains on which the is calculated, is not mandatory: uses balls instead and, as remarked by , in doing so a perfectly equivalent definition of functions of bounded mean oscillation arises.
Notation
The universally adopted notation used for the set of BMO functions on a given domain is BMO(): when = Rn, BMO(Rn) is simply symbolized as BMO.
The BMO norm of a given BMO function u is denoted by ||u||BMO: in some instances, it is also denoted as ||u||∗.
Basic properties
BMO functions are locally p–integrable
BMO functions are locally Lp if 0 < p < ∞, but need not be locally bounded. In fact, using the John-Nirenberg Inequality, we can prove that
BMO is a Banach space
Constant functions have zero mean oscillation, therefore functions differing for a constant c > 0 can share the same BMO norm value even if their difference is not zero almost everywhere. Therefore, the function ||u||BMO is properly a norm on the quotient space of BMO functions modulo the space of constant functions on the domain considered.
Averages of adjacent cubes are comparable
As the name suggests, the mean or averag
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https://en.wikipedia.org/wiki/Cameron%20Gordon%20%28mathematician%29
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Cameron Gordon (born 1945) is a Professor and Sid W. Richardson Foundation Regents Chair in the Department of Mathematics at the University of Texas at Austin, known for his work in knot theory. Among his notable results is his work with Marc Culler, John Luecke, and Peter Shalen on the cyclic surgery theorem. This was an important ingredient in his work with Luecke showing that knots were determined by their complement. Gordon was also involved in the resolution of the Smith conjecture.
Andrew Casson and Gordon defined and proved basic theorems regarding strongly irreducible Heegaard splittings, an important concept in the modernization of Heegaard splitting theory. They also worked on the slice-ribbon conjecture, inventing the Casson-Gordon invariants in the process.
Gordon was a 1999 Guggenheim Fellow.
In 2005 Gordon was elected a Corresponding Fellow of the Royal Society of Edinburgh.
References
External links
Cameron Gordon's personal webpage, University of Texas at Austin
Cameron McAllan Gordon, Mathematics Genealogy Project
1945 births
20th-century American mathematicians
21st-century American mathematicians
Topologists
Living people
University of Texas at Austin faculty
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https://en.wikipedia.org/wiki/Lebesgue%20integrability
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In mathematics, Lebesgue integrability may refer to:
Whether the Lebesgue integral of a function is defined; this is what is most often meant.
The Lebesgue integrability condition, which determines whether the Riemann integral of a function is defined. Confusingly, this result is due to Lebesgue, but refers to the Riemann integral, not the Lebesgue integral.
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https://en.wikipedia.org/wiki/Victor%20Kac
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Victor Gershevich (Grigorievich) Kac (; born 19 December 1943) is a Soviet and American mathematician at MIT, known for his work in representation theory. He co-discovered Kac–Moody algebras, and used the Weyl–Kac character formula for them to reprove the Macdonald identities. He classified the finite-dimensional simple Lie superalgebras, and found the Kac determinant formula for the Virasoro algebra. He is also known for the Kac–Weisfeiler conjectures with Boris Weisfeiler.
Biography
Kac studied mathematics at Moscow State University, receiving his MS in 1965 and his PhD in 1968. From 1968 to 1976, he held a teaching position at the Moscow Institute of Electronic Machine Building (MIEM). He left the Soviet Union in 1977, becoming an associate professor of mathematics at MIT. In 1981, he was promoted to full professor. Kac received a Sloan Fellowship and the Medal of the Collège de France, both in 1981, and a Guggenheim Fellowship in 1986. He received the Wigner Medal (1996) "in recognition of work on affine Lie algebras that has had wide influence in theoretical physics". In 1978 he was an invited speaker (Highest weight representations of infinite dimensional Lie algebras) at the International Congress of Mathematicians (ICM) in Helsinki. Kac was a plenary speaker at the 1988 American Mathematical Society centennial conference. In 2002 he gave a plenary lecture, Classification of Supersymmetries, at the ICM in Beijing.
Kac is a Fellow of the American Mathematical Society, an honorary member of the Moscow Mathematical Society, Fellow of the American Academy of Arts and Sciences and a Member of the National Academy of Sciences.
The research of Victor Kac primarily concerns representation theory and mathematical physics. His work appears in mathematics and physics and in the development of quantum field theory, string theory and the theory of integrable systems.
Kac has published 13 books and over 200 articles in mathematics and physics journals and is listed as an ISI highly cited researcher. Victor Kac was awarded the 2015 AMS Leroy P. Steele Prize for Lifetime Achievement.
He was married with Michèle Vergne and they have a daughter, Marianne Kac-Vergne, who is a professor of American civilization at the university of Picardie. His brother Boris Katz is a principal research scientist at MIT.
Kac–Moody algebra
"Almost simultaneously in 1967, Victor Kac in the USSR and Robert Moody in Canada developed what was to become Kac–Moody algebra. Kac and Moody noticed that if Wilhelm Killing's conditions were relaxed, it was still possible to associate to the Cartan matrix a Lie algebra which, necessarily, would be infinite dimensional." – A.J. Coleman
Bibliography
References
External links
Victor Kac's home page at MIT
Victor Kac, A Biographical Interview,
1943 births
Living people
People from Buguruslan
20th-century American mathematicians
Russian mathematicians
Soviet mathematicians
Fellows of the American Academy of Arts and Sciences
Amer
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https://en.wikipedia.org/wiki/184%20%28number%29
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184 (one hundred [and] eighty-four) is the natural number following 183 and preceding 185.
In mathematics
There are 184 different Eulerian graphs on eight unlabeled vertices, and 184 paths by which a chess rook can travel from one corner of a 4 × 4 chessboard to the opposite corner without passing through the same square twice. 184 is also a refactorable number.
In other fields
Some physicists have proposed that 184 is a magic number for neutrons in atomic nuclei.
In poker, with one or more jokers as wild cards, there are 184 different straight flushes.
See also
The year AD 184 or 184 BC
List of highways numbered 184
References
Integers
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https://en.wikipedia.org/wiki/National%20Institute%20of%20Statistics%20and%20Census
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National Institute of Statistics and Census (; ) may refer to:
National Institute of Statistics and Census of Argentina
National Institute of Statistics and Census of Costa Rica
National Institute of Statistics and Census of Nicaragua
See also
List of national and international statistical services
National Institute of Statistics (disambiguation)
Instituto Nacional de Estadística (disambiguation)
Instituto Nacional de Estadística e Informática
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https://en.wikipedia.org/wiki/Gesellschaft%20f%C3%BCr%20Angewandte%20Mathematik%20und%20Mechanik
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Gesellschaft für Angewandte Mathematik und Mechanik ("Society of Applied Mathematics and Mechanics"), often referred to by the acronym GAMM, is a German society for the promotion of science, founded in 1922 by the physicist Ludwig Prandtl and the mathematician Richard von Mises. The society awards the Richard von Mises prize annually. The society publishes the journal GAMM-Mitteilungen (Surveys for Applied Mathematics and Mechanics) and Zeitschrift für Angewandte Mathematik und Mechanik (Journal of Applied Mathematics and Mechanics) through Wiley.
According to the statutes, GAMM aims "to maintain and promote scientific work and international cooperation in applied mathematics as well as in all sub-areas of mechanics and physics that are part of the fundamentals the engineerings count." The GAMM pursues this goal primarily by organizing scientific conferences. The GAMM's most important event is the annual conference, which takes place annually in Germany or neighboring countries and is attended by hundreds of scientists, primarily because of its scientific program. The proceedings volume (PAMM) is published every year at the conference. In addition, further conferences on specific areas from the spectrum of disciplines represented in the GAMM take place.
In 1958 the GAMM and the ACM together worked out the "ALGOL 58 Report" at a meeting in Zurich.
Executive board members
Presidents and vice-presidents of the society since inception:
External links
Official site (German)
References
Scientific organisations based in Germany
Mathematical societies
Scientific organizations established in 1922
1922 establishments in Germany
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https://en.wikipedia.org/wiki/United%20States%20immigration%20statistics
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The 1850 United States census was the first federal U.S. census to query respondents about their "nativity"—i.e, where they were born, whether in the United States or outside of it—and is thus the first point at which solid statistics become available. The following chart, based on statistics from the U.S. Census from 1850 on, shows the numbers of non-native residents according to place of birth. Because an immigrant is counted in each census during his or her lifetime, the numbers reflect the cumulative population of living non-native residents.
(NA) Not available.
n.e.c. Not elsewhere classified.
1/ Prior to 1980, Taiwan included with China.
References
History of immigration to the United States
Demographics of the United States
Immigration to the United States
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https://en.wikipedia.org/wiki/Lawrence%20Shepp
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Lawrence Alan Shepp (September 9, 1936 Brooklyn, NY – April 23, 2013, Tucson, AZ) was an American mathematician, specializing in statistics and computational tomography.
Shepp obtained his PhD from Princeton University in 1961 with a dissertation titled Recurrent Sums of Random Variables. His advisor was William Feller. He joined Bell Laboratories in 1962. He joined Rutgers University in 1997. He joined University of Pennsylvania in 2010.
His work in tomography has had biomedical imaging applications, and he has also worked as professor of radiology at Columbia University (1973–1996), as a mathematician in the radiology service of Columbia Presbyterian Hospital.
Awards and honors
2014: IEEE Marie Sklodowska-Curie Award
2012: Became a fellow of the American Mathematical Society.
1992: Elected member of the Institute of Medicine
1989: Elected member of the National Academy of Sciences
1979: IEEE Distinguished Scientist Award in 1979
1979: Lester R. Ford Award (with Joseph Kruskal)
See also
Fishburn–Shepp inequality
Shepp–Logan phantom
Shepp–Olkin conjecture
Coupon collector's problem
Discrete tomography
Dubins path
Gaussian process
Hook length formula
Parallel parking problem
Sieve estimator
Ridge function
References
External links
Obituary at Penn
Princeton University alumni
Rutgers University faculty
University of Pennsylvania faculty
20th-century American mathematicians
21st-century American mathematicians
American statisticians
Probability theorists
Members of the United States National Academy of Sciences
Fellows of the American Mathematical Society
1936 births
2013 deaths
Members of the National Academy of Medicine
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https://en.wikipedia.org/wiki/International%20Society%20for%20Mathematical%20Sciences
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The International Society for Mathematical Sciences is a mathematics society, primarily based in Japan. It was formerly known as the Japanese Association of Mathematical Sciences, and was founded in 1948 by Tatsujiro Shimizu.
The ISMS publishes a bimonthly scientific journal, Scientiae Mathematicae Japonicae (), which was formed in 2001 from the merger of two journals previously published by the same society, Mathematica Japonica, founded in 1948, and Scientiae Mathematicae, which published nine issues over three volumes in 1998, 1999, and 2000. In addition the ISMS holds an annual meeting and publishes a Japanese language mathematics magazine, Kaiho, and a monthly newsletter, Notices from the ISMS.
References
External links
International Society for Mathematical Sciences
Mathematical societies
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https://en.wikipedia.org/wiki/Finite%20map
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A finite map can be one of the following:
In computer science, finite map is a synonym for an associative array.
A finite map in algebraic geometry is a regular map such that the preimage of any point is a finite set, plus a closedness property.
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https://en.wikipedia.org/wiki/Whitehead%27s%20lemma
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Whitehead's lemma is a technical result in abstract algebra used in algebraic K-theory. It states that a matrix of the form
is equivalent to the identity matrix by elementary transformations (that is, transvections):
Here, indicates a matrix whose diagonal block is and entry is .
The name "Whitehead's lemma" also refers to the closely related result that the derived group of the stable general linear group is the group generated by elementary matrices. In symbols,
.
This holds for the stable group (the direct limit of matrices of finite size) over any ring, but not in general for the unstable groups, even over a field. For instance for
one has:
where Alt(3) and Sym(3) denote the alternating resp. symmetric group on 3 letters.
See also
Special linear group#Relations to other subgroups of GL(n,A)
References
Matrix theory
Lemmas in linear algebra
K-theory
Theorems in abstract algebra
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https://en.wikipedia.org/wiki/Integral%20logarithm
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The term integral logarithm may stand for:
Discrete logarithm in algebra,
Logarithmic integral function in calculus.
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https://en.wikipedia.org/wiki/Robert%20Ren%C3%A9%20Kuczynski
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Robert René ('René') Kuczynski (1876–1947) was a left-wing German economist and demographer and is said to be one of the founders of modern vital statistics.
Early life
His father Wilhelm was a successful banker; his mother Lucy (née Brandeis) a progressive thinker who grew up in Paris in exile among French and German intellectuals. Robert married Berta Gradenwitz in 1903. Berta's father was a successful property developer and estate agent in Berlin. Against this wealthy family background, Robert took a decidedly different path as an academic who allied himself with the working class. René studied at the universities of Munich, Freiburg and Strasburg and completed his doctoral dissertation in 1897 under Lujo Brentano.
Career
He moved to the United States in 1899 for an internship at the United States Census Bureau and then worked at the US Bureau of Labor Statistics. During this time he cultivated contacts with people like Eugene V. Debs. He returned to Germany in late 1903, and In 1904 he became director of the Statistical Office in Elberfeld and in 1906 took the same position in Berlin -Schoneberg. He became a strong supporter of the Social Democratic Party of Germany and knew many of its leaders personally.
He studied rent and income in Berlin before the first World War and found that 600,000 people lived in flats which house five or more people per room.
In 1926, Kuczynski chaired the Kuczynski Committee, working with the German League for Human Rights, which organized the campaign for a referendum on the expropriation of the Prussian landed aristocracy during the Weimar Republic.
In 1928 he led the German delegation to the tenth anniversary of the Russian Revolution.
In 1933, after Hitler had come to power, Kuczynski left Germany and went with around 20,000 books (half of the large family library) to Great Britain. There he lectured at the London School of Economics and became later adviser for the British Colonial Office. His most noted work was in the 1930s when he published figures on the extent of the slave trade between Africa and the Americas over the preceding three centuries. His figure of 15 million slaves became widely used by other researchers, but is no longer thought to be correct.
Family
Kuczynski and wife Berta Gradenwitz had six children, among them the GDR-economist Jürgen Kuczynski, Brigitte Kuczynski, and the Soviet spy and author Ursula Kuczynski. The youngest, Renate, wrote the first history of the PhD degree in England.
Works
Kuczynski R., Munchner Volkswirtschaftliche Studien, Vierundzwanzigstes Stuck 1897
Kuczynski R., Der Zug Nach Der Stadt: Statistische Studien Uber Vorg Nge Der Bevolkerungsbewegung Im Deutschen Reiche, 1897
Lujo Brentano and Robert Kuczynski, Die heutige Grundlage der deutschen Wehrkraft. 1900
Kuczynski R., Das Existenzminimum und verwandte Fragen.1912
Kuczynski R., Arbeitslohn und Arbeitszeit in Europa und Amerika 1870–1909, Springer, 1913
Kuczynski R., Unsere Finanzen nach dem Krie
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https://en.wikipedia.org/wiki/Cyclic%20code
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In coding theory, a cyclic code is a block code, where the circular shifts of each codeword gives another word that belongs to the code. They are error-correcting codes that have algebraic properties that are convenient for efficient error detection and correction.
Definition
Let be a linear code over a finite field (also called Galois field) of block length . is called a cyclic code if, for every codeword from , the word in obtained by a cyclic right shift of components is again a codeword. Because one cyclic right shift is equal to cyclic left shifts, a cyclic code may also be defined via cyclic left shifts. Therefore, the linear code is cyclic precisely when it is invariant under all cyclic shifts.
Cyclic codes have some additional structural constraint on the codes. They are based on Galois fields and because of their structural properties they are very useful for error controls. Their structure is strongly related to Galois fields because of which the encoding and decoding algorithms for cyclic codes are computationally efficient.
Algebraic structure
Cyclic codes can be linked to ideals in certain rings. Let be a polynomial ring over the finite field . Identify the elements of the cyclic code with polynomials in such that
maps to the polynomial
: thus multiplication by corresponds to a cyclic shift. Then is an ideal in , and hence principal, since is a principal ideal ring. The ideal is generated by the unique monic element in of minimum degree, the generator polynomial .
This must be a divisor of . It follows that every cyclic code is a polynomial code.
If the generator polynomial has degree then the rank of the code is .
The idempotent of is a codeword such that (that is, is an idempotent element of ) and is an identity for the code, that is for every codeword . If and are coprime such a word always exists and is unique; it is a generator of the code.
An irreducible code is a cyclic code in which the code, as an ideal is irreducible, i.e. is minimal in , so that its check polynomial is an irreducible polynomial.
Examples
For example, if and , the set of codewords contained in cyclic code generated by is precisely
.
It corresponds to the ideal in generated by .
The polynomial is irreducible in the polynomial ring, and hence the code is an irreducible code.
The idempotent of this code is the polynomial , corresponding to the codeword .
Trivial examples
Trivial examples of cyclic codes are itself and the code containing only the zero codeword. These correspond to generators and respectively: these two polynomials must always be factors of .
Over the parity bit code, consisting of all words of even weight, corresponds to generator . Again over this must always be a factor of .
Quasi-cyclic codes and shortened codes
Before delving into the details of cyclic codes first we will discuss quasi-cyclic and shortened codes which are closely related to the cyclic codes and they all can be con
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https://en.wikipedia.org/wiki/Thomas%20Banchoff
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Thomas Francis Banchoff (born April 7, 1938) is an American mathematician
specializing in geometry. He is a professor at Brown University, where he has taught since 1967. He is known for his research in differential geometry in three and four dimensions, for his efforts to develop methods of computer graphics in the early 1990s, and most recently for his pioneering work in methods of undergraduate education utilizing online resources.
Banchoff graduated from the University of Notre Dame in 1960, receiving his B.A. in Mathematics, and received his Masters and Ph.D. from UC Berkeley in 1962 and 1964, where he was a student of Shiing-Shen Chern. Before going to Brown he taught at Harvard University and the University of Amsterdam. In 2012 he became a fellow of the American Mathematical Society. In addition, he was a president of the Mathematical Association of America.
Selected works
with Stephen Lovett: Differential Geometry of Curves and Surfaces (2nd edition), A. K. Peters 2010
with Terence Gaffney, Clint McCrory: Cusps of Gauss Mappings, Pitman 1982
with John Wermer: Linear Algebra through Geometry, Springer Verlag 1983
Beyond the third dimension: geometry, computer graphics, and higher dimensions, Scientific American Library, Freeman 1990
Triple points and surgery of immersed surfaces. Proc. Amer. Math. Soc. 46 (1974), 407–413. (concerning the number of triple points of immersed surfaces in .)
Critical points and curvature for embedded polyhedra. Journal of Differential Geometry 1 (1967), 245–256. (Theorem of Gauß-Bonnet for Polyhedra)
Teaching Experience
Benjamin Peirce Instructor, Harvard, 1964 - 1966
Research Associate, Universiteit van Amsterdam, 1966 - 1967;
Brown University:
Asst Professor, 1967
Associate Professor 1970
Professor 1973 - 2014
G. Leonard Baker Visiting Professor of Mathematics, Yale, 1998
Visiting Professor, University of Notre Dame, 2001
Visiting Professor, UCLA, 2002
Visiting Professor, University of Georgia, 2006
Visiting Professor, Stanford University, 2010
Visiting Professor, Technical University of Berlin, 2012
Visiting Professor, Sewanee: the University of the South, 2015
Visiting Professor, Carnegie Mellon University, 2015
Visiting Professor, Baylor University, 2016
Paul Halmos Visiting Professor, Santa Clara University, 2018
Further reading
Donald J. Albers & Gerald L. Alexanderson (2011) Fascinating Mathematical People: interviews and memoirs, "Tom Banchoff", pp 57–78, Princeton University Press, .
Illustrating Beyond the Third Dimension by Thomas Banchoff & Davide P. Cervone
References
External links
Personal web page
biography as president of MAA
1938 births
Living people
20th-century American mathematicians
21st-century American mathematicians
Differential geometers
UC Berkeley College of Letters and Science alumni
Harvard University Department of Mathematics faculty
Harvard University faculty
Brown University faculty
Fellows of the American Mathematical Society
Presidents
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https://en.wikipedia.org/wiki/Midsphere
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In geometry, the midsphere or intersphere of a convex polyhedron is a sphere which is tangent to every edge of the polyhedron. Not every polyhedron has a midsphere, but the uniform polyhedra, including the regular, quasiregular and semiregular polyhedra and their duals all have midspheres. The radius of the midsphere is called the midradius. A polyhedron that has a midsphere is said to be midscribed about this sphere.
When a polyhedron has a midsphere, one can form two perpendicular circle packings on the midsphere, one corresponding to the adjacencies between vertices of the polyhedron, and the other corresponding in the same way to its polar polyhedron, which has the same midsphere. The length of each polyhedron edge is the sum of the distances from its two endpoints to their corresponding circles in this circle packing.
For every convex polyhedron there is a combinatorially equivalent polyhedron, the canonical polyhedron, that does have a midsphere, centered at the centroid of the points of tangency of the edges. Numerical approximation algorithms can construct it, but its coordinates cannot be represented exactly as a closed-form expression. Any canonical polyhedron and its polar dual can be used to form two opposite faces of a four-dimensional antiprism.
Definition and examples
A midsphere of a three-dimensional convex polyhedron is defined to be a sphere that is tangent to every edge of the polyhedron. That is to say, each edge must touch it, at an interior point of the edge, without crossing it. Equivalently, it is a sphere that contains the inscribed circle of every face of the polyhedron. When a midsphere exists, it is unique. Not every convex polyhedron has a midsphere; to have a midsphere, every face must have an inscribed circle (that is, it must be a tangential polygon), and all of these inscribed circles must belong to a single sphere. For example, a rectangular cuboid has a midsphere only when it is a cube, because otherwise it has non-square rectangles as faces, and these do not have inscribed circles.
For a unit cube centered at the origin of the Cartesian coordinate system, with vertices at the eight points , the midpoints of the edges are at distance from the origin. Therefore, for this cube, the midsphere is centered at the origin, with radius . This is larger than the radius of the inscribed sphere, , and smaller than the radius of the circumscribed sphere, . More generally, for any Platonic solid of edge length , the midradius is
for a regular tetrahedron,
for a regular octahedron,
for a regular cube,
for a regular icosahedron, where denotes the golden ratio, and
for a regular dodecahedron.
The uniform polyhedra, including the regular, quasiregular and semiregular polyhedra and their duals all have midspheres. In the regular polyhedra, the inscribed sphere, midsphere, and circumscribed sphere all exist and are concentric, and the midsphere touches each edge at its midpoint.
Not every irregular tetrahedron h
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https://en.wikipedia.org/wiki/Quartic%20plane%20curve
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In algebraic geometry, a quartic plane curve is a plane algebraic curve of the fourth degree. It can be defined by a bivariate quartic equation:
with at least one of not equal to zero. This equation has 15 constants. However, it can be multiplied by any non-zero constant without changing the curve; thus by the choice of an appropriate constant of multiplication, any one of the coefficients can be set to 1, leaving only 14 constants. Therefore, the space of quartic curves can be identified with the real projective space It also follows, from Cramer's theorem on algebraic curves, that there is exactly one quartic curve that passes through a set of 14 distinct points in general position, since a quartic has 14 degrees of freedom.
A quartic curve can have a maximum of:
Four connected components
Twenty-eight bi-tangents
Three ordinary double points.
One may also consider quartic curves over other fields (or even rings), for instance the complex numbers. In this way, one gets Riemann surfaces, which are one-dimensional objects over but are two-dimensional over An example is the Klein quartic. Additionally, one can look at curves in the projective plane, given by homogeneous polynomials.
Examples
Various combinations of coefficients in the above equation give rise to various important families of curves as listed below.
Bicorn
Bullet-nose curve
Cartesian oval
Cassini oval
Deltoid curve
Hippopede
Kampyle of Eudoxus
Klein quartic
Lemniscate
Lemniscate of Bernoulli
Lemniscate of Gerono
Limaçon
Lüroth quartic
Spiric section
Squircle
Lamé's special quartic
Toric section
Trott curve
Ampersand curve
The ampersand curve is a quartic plane curve given by the equation:
It has genus zero, with three ordinary double points, all in the real plane.
Bean curve
The bean curve is a quartic plane curve with the equation:
The bean curve has genus zero. It has one singularity at the origin, an ordinary triple point.
Bicuspid curve
The bicuspid is a quartic plane curve with the equation
where a determines the size of the curve.
The bicuspid has only the two cusps as singularities, and hence is a curve of genus one.
Bow curve
The bow curve is a quartic plane curve with the equation:
The bow curve has a single triple point at x=0, y=0, and consequently is a rational curve, with genus zero.
Cruciform curve
The cruciform curve, or cross curve is a quartic plane curve given by the equation
where a and b are two parameters determining the shape of the curve.
The cruciform curve is related by a standard quadratic transformation, x ↦ 1/x, y ↦ 1/y to the ellipse a2x2 + b2y2 = 1, and is therefore a rational plane algebraic curve of genus zero. The cruciform curve has three double points in the real projective plane, at x=0 and y=0, x=0 and z=0, and y=0 and z=0.
Because the curve is rational, it can be parametrized by rational functions. For instance, if a=1 and b=2, then
parametrizes the points on the curve outside of the exceptional cases where a deno
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https://en.wikipedia.org/wiki/Subadditivity%20effect
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The subadditivity effect is the tendency to judge probability of the whole to be less than the probabilities of the parts.
Example
For instance, subjects in one experiment judged the probability of death from cancer in the United States was 18%, the probability from heart attack was 22%, and the probability of death from "other natural causes" was 33%. Other participants judged the probability of death from a natural cause was 58%. Natural causes are made up of precisely cancer, heart attack, and "other natural causes," however, the sum of the latter three probabilities was 73%, and not 58%. According to Tversky and Koehler (1994) this kind of result is observed consistently.
Explanations
In a 2012 article in Psychological Bulletin it is suggested the subadditivity effect can be explained by an information-theoretic generative mechanism that assumes a noisy conversion of objective evidence (observation) into subjective estimates (judgment). This explanation is different than support theory, proposed as an explanation by Tversky and Koehler, which requires additional assumptions. Since mental noise is a sufficient explanation that is much simpler and straightforward than any explanation involving heuristics or behavior, Occam's razor would argue in its favor as the underlying generative mechanism (it is the hypotheses which makes the fewest assumptions).
References
Cognitive biases
Error
Barriers to critical thinking
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https://en.wikipedia.org/wiki/David%20Emmanuel%20%28mathematician%29
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David Emmanuel (31 January 1854 – 4 February 1941) was a Romanian Jewish mathematician and member of the Romanian Academy, considered to be the founder of the modern mathematics school in Romania.
Born in Bucharest, Emmanuel studied at Gheorghe Lazăr and Gheorghe Șincai high schools. In 1873 he went to Paris, where he received his Ph.D. in mathematics from the University of Paris (Sorbonne) in 1879 with a thesis on Study of abelian integrals of the third species, becoming the second Romanian to have a Ph.D. in mathematics from the Sorbonne (the first one was Spiru Haret). The thesis defense committee consisted of Victor Puiseux (advisor), Charles Briot, and Jean-Claude Bouquet.
In 1882, Emmanuel became a professor of superior algebra and function theory at the Faculty of Sciences of the University of Bucharest. Here, in 1888, he held the first courses on group theory and on Galois theory, and introduced set theory in Romanian education. Among his students were Anton Davidoglu, Alexandru Froda, Traian Lalescu, Grigore Moisil, , Miron Nicolescu, Octav Onicescu, Dimitrie Pompeiu, Simion Stoilow, and Gheorghe Țițeica. Emmanuel had an important role in the introduction of modern mathematics and of the rigorous approach to mathematics in Romania.
Emmanuel was the president of the first Congress of Romanian Mathematicians, held in 1929 in Cluj. He died in Bucharest in 1941.
A street in the Dorobanți neighborhood of Bucharest is named after him.
Publications
References
1854 births
1941 deaths
Scientists from Bucharest
Romanian Sephardi Jews
Gheorghe Lazăr National College (Bucharest) alumni
University of Paris alumni
Romanian mathematicians
Academic staff of the University of Bucharest
Honorary members of the Romanian Academy
Mathematical analysts
Romanian expatriates in France
People from the United Principalities of Moldavia and Wallachia
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https://en.wikipedia.org/wiki/Monotone%20class%20theorem
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In measure theory and probability, the monotone class theorem connects monotone classes and -algebras. The theorem says that the smallest monotone class containing an algebra of sets is precisely the smallest -algebra containing It is used as a type of transfinite induction to prove many other theorems, such as Fubini's theorem.
Definition of a monotone class
A is a family (i.e. class) of sets that is closed under countable monotone unions and also under countable monotone intersections. Explicitly, this means has the following properties:
if and then and
if and then
Monotone class theorem for sets
Monotone class theorem for functions
Proof
The following argument originates in Rick Durrett's Probability: Theory and Examples.
Results and applications
As a corollary, if is a ring of sets, then the smallest monotone class containing it coincides with the -ring of
By invoking this theorem, one can use monotone classes to help verify that a certain collection of subsets is a -algebra.
The monotone class theorem for functions can be a powerful tool that allows statements about particularly simple classes of functions to be generalized to arbitrary bounded and measurable functions.
See also
Citations
References
Families of sets
Theorems in measure theory
fr:Lemme de classe monotone
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https://en.wikipedia.org/wiki/Adams%20spectral%20sequence
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In mathematics, the Adams spectral sequence is a spectral sequence introduced by which computes the stable homotopy groups of topological spaces. Like all spectral sequences, it is a computational tool; it relates homology theory to what is now called stable homotopy theory. It is a reformulation using homological algebra, and an extension, of a technique called 'killing homotopy groups' applied by the French school of Henri Cartan and Jean-Pierre Serre.
Motivation
For everything below, once and for all, we fix a prime p. All spaces are assumed to be CW complexes. The ordinary cohomology groups are understood to mean .
The primary goal of algebraic topology is to try to understand the collection of all maps, up to homotopy, between arbitrary spaces X and Y. This is extraordinarily ambitious: in particular, when X is , these maps form the nth homotopy group of Y. A more reasonable (but still very difficult!) goal is to understand the set of maps (up to homotopy) that remain after we apply the suspension functor a large number of times. We call this the collection of stable maps from X to Y. (This is the starting point of stable homotopy theory; more modern treatments of this topic begin with the concept of a spectrum. Adams' original work did not use spectra, and we avoid further mention of them in this section to keep the content here as elementary as possible.)
The set turns out to be an abelian group, and if X and Y are reasonable spaces this group is finitely generated. To figure out what this group is, we first isolate a prime p. In an attempt to compute the p-torsion of , we look at cohomology: send to Hom(H*(Y), H*(X)). This is a good idea because cohomology groups are usually tractable to compute.
The key idea is that is more than just a graded abelian group, and more still than a graded ring (via the cup product). The representability of the cohomology functor makes H*(X) a module over the algebra of its stable cohomology operations, the Steenrod algebra A. Thinking about H*(X) as an A-module forgets some cup product structure, but the gain is enormous: Hom(H*(Y), H*(X)) can now be taken to be A-linear! A priori, the A-module sees no more of [X, Y] than it did when we considered it to be a map of vector spaces over Fp. But we can now consider the derived functors of Hom in the category of A-modules, ExtAr(H*(Y), H*(X)). These acquire a second grading from the grading on H*(Y), and so we obtain a two-dimensional "page" of algebraic data. The Ext groups are designed to measure the failure of Hom's preservation of algebraic structure, so this is a reasonable step.
The point of all this is that A is so large that the above sheet of cohomological data contains all the information we need to recover the p-primary part of [X, Y], which is homotopy data. This is a major accomplishment because cohomology was designed to be computable, while homotopy was designed to be powerful. This is the content of the Adams spectral sequence.
Class
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https://en.wikipedia.org/wiki/Cycle%20index
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In combinatorial mathematics a cycle index is a polynomial in several variables which is structured in such a way that information about how a group of permutations acts on a set can be simply read off from the coefficients and exponents. This compact way of storing information in an algebraic form is frequently used in combinatorial enumeration.
Each permutation π of a finite set of objects partitions that set into cycles; the cycle index monomial of π is a monomial in variables a1, a2, … that describes the cycle type of this partition: the exponent of ai is the number of cycles of π of size i. The cycle index polynomial of a permutation group is the average of the cycle index monomials of its elements. The phrase cycle indicator is also sometimes used in place of cycle index.
Knowing the cycle index polynomial of a permutation group, one can enumerate equivalence classes due to the group's action. This is the main ingredient in the Pólya enumeration theorem. Performing formal algebraic and differential operations on these polynomials and then interpreting the results combinatorially lies at the core of species theory.
Permutation groups and group actions
A bijective map from a set X onto itself is called a permutation of X, and the set of all permutations of X forms a group under the composition of mappings, called the symmetric group of X, and denoted Sym(X). Every subgroup of Sym(X) is called a permutation group of degree |X|. Let G be an abstract group with a group homomorphism φ from G into Sym(X). The image, φ(G), is a permutation group. The group homomorphism can be thought of as a means for permitting the group G to "act" on the set X (using the permutations associated with the elements of G). Such a group homomorphism is formally called a group action and the image of the homomorphism is a permutation representation of G. A given group can have many different permutation representations, corresponding to different actions.
Suppose that group G acts on set X (that is, a group action exists). In combinatorial applications the interest is in the set X; for instance, counting things in X and knowing what structures might be left invariant by G. Little is lost by working with permutation groups in such a setting, so in these applications, when a group is considered, it is a permutation representation of the group which will be worked with, and thus, a group action must be specified. Algebraists, on the other hand, are more interested in the groups themselves and would be more concerned with the kernels of the group actions, which measure how much is lost in passing from the group to its permutation representation.
Disjoint cycle representation of permutations
Finite permutations are most often represented as group actions on the set X = {1,2, ..., n}. A permutation in this setting can be represented by a two line notation. Thus,
corresponds to a bijection on X = {1, 2, 3, 4, 5} which sends 1 → 2, 2 → 3, 3 → 4, 4 → 5 and 5 → 1. This
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https://en.wikipedia.org/wiki/Solid%20Klein%20bottle
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In mathematics, a solid Klein bottle is a three-dimensional topological space (a 3-manifold) whose boundary is the Klein bottle.
It is homeomorphic to the quotient space obtained by gluing the top disk of a cylinder to the bottom disk by a reflection across a diameter of the disk.
Alternatively, one can visualize the solid Klein bottle as the trivial product , of the möbius strip and an interval . In this model one can see that
the core central curve at 1/2 has a regular neighborhood which is again a trivial cartesian product: and whose boundary is a Klein bottle.
References
3-manifolds
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https://en.wikipedia.org/wiki/First-order%20partial%20differential%20equation
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In mathematics, a first-order partial differential equation is a partial differential equation that involves only first derivatives of the unknown function of n variables. The equation takes the form
Such equations arise in the construction of characteristic surfaces for hyperbolic partial differential equations, in the calculus of variations, in some geometrical problems, and in simple models for gas dynamics whose solution involves the method of characteristics. If a family of solutions
of a single first-order partial differential equation can be found, then additional solutions may be obtained by forming envelopes of solutions in that family. In a related procedure, general solutions may be obtained by integrating families of ordinary differential equations.
General solution and complete integral
The general solution to the first order partial differential equation is a solution which contains an arbitrary function. But, the solution to the first order partial differential equations with as many arbitrary constants as the number of independent variables is called the complete integral. The following n-parameter family of solutions
is a complete integral if . The below discussions on the type of integrals are based on the textbook A Treatise on Differential Equations (Chaper IX, 6th edition, 1928) by Andrew Forsyth.
Complete integral
The solutions are described in relatively simple manner in two or three dimensions with which the key concepts are trivially extended to higher dimensions. A general first-order partial differential equation in three dimensions has the form
where Suppose be the complete integral that contains three arbitrary constants . From this we can obtain three relations by differentiation
Along with the complete integral , the above three relations can be used to eliminate three constants and obtain an equation (original partial differential equation) relating . Note that the elimination of constants leading to the partial differential equation need not be unique, i.e., two different equations can result in the same complete integral, for example, elimination of constants from the relation leads to and .
General integral
Once a complete integral is found, a general solution can be constructed from it. The general integral is obtained by making the constants functions of the coordinates, i.e., . These functions are chosen such that the forms of are unaltered so that the elimination process from complete integral can be utilized. Differentiation of the complete integral now provides
in which we require the right-hand side terms of all the three equations to vanish identically so that elimination of from results in the partial differential equation. This requirement can be written more compactly by writing it as
where
is the Jacobian determinant. The condition leads to the general solution. Whenever , then there exists a functional relation between because whenever a determinant is zero, t
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https://en.wikipedia.org/wiki/Poincar%C3%A9%20inequality
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In mathematics, the Poincaré inequality is a result in the theory of Sobolev spaces, named after the French mathematician Henri Poincaré. The inequality allows one to obtain bounds on a function using bounds on its derivatives and the geometry of its domain of definition. Such bounds are of great importance in the modern, direct methods of the calculus of variations. A very closely related result is Friedrichs' inequality.
Statement of the inequality
The classical Poincaré inequality
Let p, so that 1 ≤ p < ∞ and Ω a subset bounded at least in one direction. Then there exists a constant C, depending only on Ω and p, so that, for every function u of the Sobolev space W01,p(Ω) of zero-trace (a.k.a. zero on the boundary) functions,
Poincaré–Wirtinger inequality
Assume that 1 ≤ p ≤ ∞ and that Ω is a bounded connected open subset of the n-dimensional Euclidean space ℝn with a Lipschitz boundary (i.e., Ω is a Lipschitz domain). Then there exists a constant C, depending only on Ω and p, such that for every function u in the Sobolev space ,
where
is the average value of u over Ω, with |Ω| standing for the Lebesgue measure of the domain Ω. When Ω is a ball, the above inequality is
called a -Poincaré inequality; for more general domains Ω, the above is more familiarly known as a Sobolev inequality.
The necessity to subtract the average value can be seen by considering constant functions for which the derivative is zero while, without subtracting the average, we can have the integral of the function as large as we wish. There are other conditions instead of subtracting the average that we can require in order to deal with this issue with constant functions, for example, requiring trace zero, or subtracting the average over some proper subset of the domain. The constant C in the Poincare inequality may be different from condition to condition. Also note that the issue is not just the constant functions, because it is the same as saying that adding a constant value to a function can increase its integral while the integral of its derivative remains the same. So, simply excluding the constant functions will not solve the issue.
Generalizations
In the context of metric measure spaces, the definition of a Poincaré inequality is slightly different. One definition is: a metric measure space supports a (q,p)-Poincare inequality for some if there are constants C and so that for each ball B in the space,
Here we have an enlarged ball in the right hand side. In the context of metric measure spaces, is the minimal p-weak upper gradient of u in the sense of
Heinonen and Koskela.
Whether a space supports a Poincaré inequality has turned out to have deep connections to the geometry and analysis of the space. For example, Cheeger has shown that a doubling space satisfying a Poincaré inequality admits a notion of differentiation. Such spaces include sub-Riemannian manifolds and Laakso spaces.
There exist other generalizations of the Poincaré inequality to
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https://en.wikipedia.org/wiki/Robin%20Wilson
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Robin Wilson may refer to:
R. N. D. Wilson (1899–1953), Irish poet
Robin Wilson (author) (1928–2013), science fiction author
Robin Wilson (mathematician) (born 1943), head of pure mathematics at the Open University, UK
Robin Wilson (field hockey) (born 1957), New Zealand field hockey player
Robin Wilson (musician) (born 1965), American singer and guitarist, lead vocalist of the Gin Blossoms
Robin Wilson (eco-designer) (born 1969), eco-friendly lifestyle expert
Robin Wilson (curler), Canadian curler
Robin Wilson (psychologist), Canadian-American psychologist
Robin Lee Wilson (1933–2019), British civil engineer
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https://en.wikipedia.org/wiki/Enumerated%20type
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In computer programming, an enumerated type (also called enumeration, enum, or factor in the R programming language, and a categorical variable in statistics) is a data type consisting of a set of named values called elements, members, enumeral, or enumerators of the type. The enumerator names are usually identifiers that behave as constants in the language. An enumerated type can be seen as a degenerate tagged union of unit type. A variable that has been declared as having an enumerated type can be assigned any of the enumerators as a value. In other words, an enumerated type has values that are different from each other, and that can be compared and assigned, but are not specified by the programmer as having any particular concrete representation in the computer's memory; compilers and interpreters can represent them arbitrarily.
For example, the four suits in a deck of playing cards may be four enumerators named Club, Diamond, Heart, and Spade, belonging to an enumerated type named suit. If a variable V is declared having suit as its data type, one can assign any of those four values to it.
Although the enumerators are usually distinct, some languages may allow the same enumerator to be listed twice in the type's declaration. The names of enumerators need not be semantically complete or compatible in any sense. For example, an enumerated type called color may be defined to consist of the enumerators Red, Green, Zebra, Missing, and Bacon. In some languages, the declaration of an enumerated type also intentionally defines an ordering of its members (High, Medium and Low priorities); in others, the enumerators are unordered (English, French, German and Spanish supported languages); in others still, an implicit ordering arises from the compiler concretely representing enumerators as integers.
Some enumerator types may be built into the language. The Boolean type, for example is often a pre-defined enumeration of the values False and True. A unit type consisting of a single value may also be defined to represent null. Many languages allow users to define new enumerated types.
Values and variables of an enumerated type are usually implemented with some integer type as the underlying representation. Some languages, especially system programming languages, allow the user to specify the bit combination to be used for each enumerator, which can be useful to efficiently represent sets of enumerators as fixed-length bit strings. In type theory, enumerated types are often regarded as tagged unions of unit types. Since such types are of the form , they may also be written as natural numbers.
Rationale
Some early programming languages did not originally have enumerated types. If a programmer wanted a variable, for example myColor, to have a value of red, the variable red would be declared and assigned some arbitrary value, usually an integer constant. The variable red would then be assigned to myColor. Other techniques assigned arbitrary values to str
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https://en.wikipedia.org/wiki/Neglect%20of%20probability
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The neglect of probability, a type of cognitive bias, is the tendency to disregard probability when making a decision under uncertainty and is one simple way in which people regularly violate the normative rules for decision making. Small risks are typically either neglected entirely or hugely overrated. The continuum between the extremes is ignored. The term probability neglect was coined by Cass Sunstein.
There are many related ways in which people violate the normative rules of decision making with regard to probability including the hindsight bias, the neglect of prior base rates effect, and the gambler's fallacy. However, this bias is different, in that, rather than incorrectly using probability, the actor disregards it.
"We have no intuitive grasp of risk and thus distinguish poorly among different threats," Dobelli has written. "The more serious the threat and the more emotional the topic (such as radioactivity), the less reassuring a reduction in risk seems to us."
Studies
Adults
In a 1972 experiment, participants were divided into two groups, with the former being told they would receive a mild electric shock and the latter told that there was a 50 percent chance they would receive such a shock. When the subjects' physical anxiety was measured, there was no difference between the two groups. This lack of difference remained even when the second group's chance of being shocked was lowered to 20 percent, then ten, then five. The conclusion: "we respond to the expected magnitude of an event...but not to its likelihood. In other words: We lack an intuitive grasp of probability."
Baron (2000) suggests that the bias manifests itself among adults especially when it comes to difficult choices, such as medical decisions. This bias could make actors drastically violate expected-utility theory in their decision making, especially when a decision must be made in which one possible outcome has a much lower or higher utility but a small probability of occurring (e.g. in medical or gambling situations). In this aspect, the neglect of probability bias is similar to the neglect of prior base rates effect.
Cass Sunstein has cited the history of Love Canal in upstate New York, which became world-famous in the late 1970s owing to widely publicized public concerns about abandoned waste that was supposedly causing medical problems in the area. In response to these concerns, the U.S. federal government set in motion "an aggressive program for cleaning up abandoned hazardous waste sites, without examining the probability that illness would actually occur," and legislation was passed that did not reflect serious study of the actual degree of danger. Furthermore, when controlled studies were publicized showing little evidence that the waste represented a menace to public health, the anxiety of local residents did not diminish.
One University of Chicago study showed that people are as afraid of a 1% chance as of a 99% chance of contamination by poisonous ch
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https://en.wikipedia.org/wiki/Frege%27s%20theorem
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In metalogic and metamathematics, Frege's theorem is a metatheorem that states that the Peano axioms of arithmetic can be derived in second-order logic from Hume's principle. It was first proven, informally, by Gottlob Frege in his 1884 Die Grundlagen der Arithmetik (The Foundations of Arithmetic) and proven more formally in his 1893 Grundgesetze der Arithmetik I (Basic Laws of Arithmetic I). The theorem was re-discovered by Crispin Wright in the early 1980s and has since been the focus of significant work. It is at the core of the philosophy of mathematics known as neo-logicism (at least of the Scottish School variety).
Overview
In The Foundations of Arithmetic (1884), and later, in Basic Laws of Arithmetic (vol. 1, 1893; vol. 2, 1903), Frege attempted to derive all of the laws of arithmetic from axioms he asserted as logical (see logicism). Most of these axioms were carried over from his Begriffsschrift; the one truly new principle was one he called the Basic Law V (now known as the axiom schema of unrestricted comprehension): the "value-range" of the function f(x) is the same as the "value-range" of the function g(x) if and only if ∀x[f(x) = g(x)]. However, not only did Basic Law V fail to be a logical proposition, but the resulting system proved to be inconsistent, because it was subject to Russell's paradox.
The inconsistency in Frege's Grundgesetze overshadowed Frege's achievement: according to Edward Zalta, the Grundgesetze "contains all the essential steps of a valid proof (in second-order logic) of the fundamental propositions of arithmetic from a single consistent principle." This achievement has become known as Frege's theorem.
Frege's theorem in propositional logic
In propositional logic, Frege's theorem refers to this tautology:
(P → (Q→R)) → ((P→Q) → (P→R))
The theorem already holds in one of the weakest logics imaginable, the constructive implicational calculus. The proof under the Brouwer–Heyting–Kolmogorov interpretation reads .
In words:
"Let f denote a reason that P implies that Q implies R. And let g denote a reason that P implies Q. Then given a f, then given a g, then given a reason p for P, we know that both Q holds by g and that Q implies R holds by f. So R holds."
The truth table to the right gives a semantic proof. For all possible assignments of false () or true () to P, Q, and R (columns 1, 3, 5), each subformula is evaluated according to the rules for material conditional, the result being shown below its main operator. Column 6 shows that the whole formula evaluates to true in every case, i.e. that it is a tautology. In fact, its antecedent (column 2) and its consequent (column 10) are even equivalent.
Notes
References
– Edition in modern notation
– Edition in modern notation
Theorems in the foundations of mathematics
Theorems in propositional logic
Metatheorems
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https://en.wikipedia.org/wiki/Australian%20Bureau%20of%20Statistics
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The Australian Bureau of Statistics (ABS) is a government agency that collects and analyzes statistics on economic, population, environmental, and social issues. It provides evidence-based advice to federal, state, and territory governments. The ABS conducts the national Census of Population and Housing every five years and publishes many of its findings on the agency's website.
History
In early 1901, statistics were collected by each state and territory of Australia for their own separate use. Although attempts were made to coordinate collections through an annual Conference of Statisticians, it was decided that a national statistical office would be required to develop nationally comparable statistics.
The Commonwealth Bureau of Census and Statistics (CBCS) was established under the Census and Statistics Act in 1905. Sir George Knibbs was appointed as the first Commonwealth Statistician. Initially, the bureau was located in Melbourne and was attached to the Department of Home Affairs. In 1928, the bureau relocated to Canberra, and in 1932 moved to the Treasury portfolio.
Initially, the states maintained their own statistical offices and worked together with the CBCS to produce national data. However, some states found it difficult to resource a state statistical office to the level required for an adequate statistical service. In 1924, the Tasmanian Statistical Office transferred to the Commonwealth. On 20 August 1957, the New South Wales (NSW) Bureau of Statistics was merged into the Commonwealth Bureau. Unification of the state statistical offices with the CBCS was finally achieved in the late 1950s under the stewardship of Sir Stanley Carver, who was both NSW Statistician and Acting Commonwealth Statistician.
In 1974 the CBCS was abolished and the Australian Bureau of Statistics (ABS) was established in its place. The Australian Bureau of Statistics Act established the ABS as a statutory authority in 1975, headed by the Australian Statistician and responsible to the Treasurer.
Modernisation
In 2015, the Australian government announced a $250 million five-year investment to modernise ABS systems and processes.
Census of Population and Housing
The ABS undertakes the Australian Census of Population and Housing. The census is conducted every five years under federal law as stipulated in the Constitution of Australia.
The most recent Census of Population and Housing was conducted on 10 August 2021. Statistics from the census were published on the ABS website on 28 June 2022.
The census is the largest statistical collection undertaken by the ABS. The census aims to accurately measure the number of people and dwellings in Australia on census night and a range of their key characteristics. This information is used to inform public policy as well as electoral boundaries, infrastructure planning and the provision of community services.
2021 Census
The 2021 Census achieved a response rate above the ABS target, obtaining data from ten million
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https://en.wikipedia.org/wiki/Cwmynyscoy
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Cwmynyscoy is a suburb of Pontypool in Torfaen, South Wales.
Statistics
All figures quoted have been derived from the 2001 Census unless otherwise stated.
Demographic Indicators
Total population of 1283 (Torfaen 90,949)
48.6% Male, 51.4% Female (Torfaen - 48.4% Male, 51.6% Female)
Age Structure; 19.5% aged between 0–15, 36.6% aged between 16 and 44, 25.3% aged 45–59/64 and 18.6% of pensionable age.
Socio-Economic Indicators
Activity Rates (2001)
Male (16-74) economic activity rate 63.4% (Torfaen 67.8%, Wales 67.7%), female (16-74) economic activity rate 48.9% (Torfaen 54.2%, Wales 54.5%), total economic activity rate 56.1% (Torfaen 60.8%, Wales 61.0%)
Unemployment (2004)
Whilst unemployment in the area has declined significantly and only 24 people remain registered unemployed, 16 males and 8 females (June 2004). Of the 24 claimants, 10 are under 24 years of age and 5 are registered as long-term unemployed (unemployed for over 52 weeks).
Home Ownership (2001)
Cwmynyscoy has a lower proportion of owner occupied households at 64.9% than Torfaen 68.3% and Wales as a whole 71.3%. 28.4% of properties are rented from the local authority (Torfaen 22.8%, Wales 13.7%)
Car Ownership (2001)
32.7% of households in Cwmynyscoy do not own a car (Torfaen 27.2%, Wales 26.0%).
Education (2001)
Residents qualified to Level 4/5: 8.9% (Torfaen 13.6%, Wales 17.4%). (Level 4/5: First degree, Higher degree, NVQ levels 4 and 5, HNC, HND, Qualified Teacher Status, Qualified Medical Doctor, Qualified Dentist, Qualified Nurse, Midwife, Health Visitor)
Lone Parent Families (2001)
10.9% of households in Cwmynyscoy are occupied by lone parents (Torfaen 10.8%, Wales 10.6%).
Nature reserve
Cwmynyscoy Quarry is a local nature reserve, within a disused quarry, home to a number of species including noctule bats and barn owls.
References
Suburbs of Pontypool
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https://en.wikipedia.org/wiki/Sasakian%20manifold
|
In differential geometry, a Sasakian manifold (named after Shigeo Sasaki) is a contact manifold equipped with a special kind of Riemannian metric , called a Sasakian metric.
Definition
A Sasakian metric is defined using the construction of the Riemannian cone. Given a Riemannian manifold , its Riemannian cone is the product
of with a half-line ,
equipped with the cone metric
where is the parameter in .
A manifold equipped with a 1-form
is contact if and only if the 2-form
on its cone is symplectic (this is one of the possible
definitions of a contact structure). A contact Riemannian manifold is Sasakian, if its Riemannian cone with the cone metric is a Kähler manifold with
Kähler form
Examples
As an example consider
where the right hand side is a natural Kähler manifold and read as the cone over the sphere (endowed with embedded metric). The contact 1-form on is the form associated to the tangent vector , constructed from the unit-normal vector to the sphere ( being the complex structure on ).
Another non-compact example is with coordinates endowed with contact-form
and the Riemannian metric
As a third example consider:
where the right hand side has a natural Kähler structure, and the group acts by reflection at the origin.
History
Sasakian manifolds were introduced in 1960 by the Japanese geometer Shigeo Sasaki. There was not much activity in this field after the mid-1970s, until the advent of String theory. Since then Sasakian manifolds have gained prominence in physics and algebraic geometry, mostly due to a string of papers by Charles P. Boyer and Krzysztof Galicki and their co-authors.
The Reeb vector field
The homothetic vector field on the cone over a Sasakian manifold is defined to be
As the cone is by definition Kähler, there exists a complex structure J. The Reeb vector field on the Sasaskian manifold is defined to be
It is nowhere vanishing. It commutes with all holomorphic Killing vectors on the cone and in particular with all isometries of the Sasakian manifold. If the orbits of the vector field close then the space of orbits is a Kähler orbifold. The Reeb vector field at the Sasakian manifold at unit radius is a unit vector field and tangential to the embedding.
Sasaki–Einstein manifolds
A Sasakian manifold is a manifold whose Riemannian cone is Kähler. If, in addition, this cone is Ricci-flat, is called Sasaki–Einstein; if it is hyperkähler, is called 3-Sasakian. Any 3-Sasakian manifold is both an Einstein manifold and a spin manifold.
If M is positive-scalar-curvature Kahler–Einstein manifold, then, by an observation of Shoshichi Kobayashi, the circle bundle S in its canonical line bundle admits a Sasaki–Einstein metric, in a manner that makes the projection from S to M into a Riemannian submersion. (For example, it follows that
there exist Sasaki–Einstein metrics on suitable circle bundles over the 3rd through 8th del Pezzo surfaces.) While this Riemannian submersion construction provides a
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https://en.wikipedia.org/wiki/Puppe%20sequence
|
In mathematics, the Puppe sequence is a construction of homotopy theory, so named after Dieter Puppe. It comes in two forms: a long exact sequence, built from the mapping fibre (a fibration), and a long coexact sequence, built from the mapping cone (which is a cofibration). Intuitively, the Puppe sequence allows us to think of homology theory as a functor that takes spaces to long-exact sequences of groups. It is also useful as a tool to build long exact sequences of relative homotopy groups.
Exact Puppe sequence
Let be a continuous map between pointed spaces and let denote the mapping fibre (the fibration dual to the mapping cone). One then obtains an exact sequence:
where the mapping fibre is defined as:
Observe that the loop space injects into the mapping fibre: , as it consists of those maps that both start and end at the basepoint . One may then show that the above sequence extends to the longer sequence
The construction can then be iterated to obtain the exact Puppe sequence
The exact sequence is often more convenient than the coexact sequence in practical applications, as Joseph J. Rotman explains:
(the) various constructions (of the coexact sequence) involve quotient spaces instead of subspaces, and so all maps and homotopies require more scrutiny to ensure that they are well-defined and continuous.
Examples
Example: Relative homotopy
As a special case, one may take X to be a subspace A of Y that contains the basepoint y0, and f to be the inclusion of A into Y. One then obtains an exact sequence in the category of pointed spaces:
where the are the homotopy groups, is the zero-sphere (i.e. two points) and denotes the homotopy equivalence of maps from U to W. Note that . One may then show that
is in bijection to the relative homotopy group , thus giving rise to the relative homotopy sequence of pairs
The object is a group for and is abelian for .
Example: Fibration
As a special case, one may take f to be a fibration . Then the mapping fiber Mp has the homotopy lifting property and it follows that Mp and the fiber have the same homotopy type. It follows trivially that maps of the sphere into Mp are homotopic to maps of the sphere to F, that is,
From this, the Puppe sequence gives the homotopy sequence of a fibration:
Example: Weak fibration
Weak fibrations are strictly weaker than fibrations, however, the main result above still holds, although the proof must be altered. The key observation, due to Jean-Pierre Serre, is that, given a weak fibration , and the fiber at the basepoint given by , that there is a bijection
.
This bijection can be used in the relative homotopy sequence above, to obtain the homotopy sequence of a weak fibration, having the same form as the fibration sequence, although with a different connecting map.
Coexact Puppe sequence
Let be a continuous map between CW complexes and let denote a mapping cone of f, (i.e., the cofiber of the map f), so that we have a (cofiber) sequence:
.
Now
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https://en.wikipedia.org/wiki/Symplectic%20cut
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In mathematics, specifically in symplectic geometry, the symplectic cut is a geometric modification on symplectic manifolds. Its effect is to decompose a given manifold into two pieces. There is an inverse operation, the symplectic sum, that glues two manifolds together into one. The symplectic cut can also be viewed as a generalization of symplectic blow up. The cut was introduced in 1995 by Eugene Lerman, who used it to study the symplectic quotient and other operations on manifolds.
Topological description
Let be any symplectic manifold and
a Hamiltonian on . Let be any regular value of , so that the level set is a smooth manifold. Assume furthermore that is fibered in circles, each of which is an integral curve of the induced Hamiltonian vector field.
Under these assumptions, is a manifold with boundary , and one can form a manifold
by collapsing each circle fiber to a point. In other words, is with the subset removed and the boundary collapsed along each circle fiber. The quotient of the boundary is a submanifold of of codimension two, denoted .
Similarly, one may form from a manifold , which also contains a copy of . The symplectic cut is the pair of manifolds and .
Sometimes it is useful to view the two halves of the symplectic cut as being joined along their shared submanifold to produce a singular space
For example, this singular space is the central fiber in the symplectic sum regarded as a deformation.
Symplectic description
The preceding description is rather crude; more care is required to keep track of the symplectic structure on the symplectic cut. For this, let be any symplectic manifold. Assume that the circle group acts on in a Hamiltonian way with moment map
This moment map can be viewed as a Hamiltonian function that generates the circle action. The product space , with coordinate on , comes with an induced symplectic form
The group acts on the product in a Hamiltonian way by
with moment map
Let be any real number such that the circle action is free on . Then is a regular value of , and is a manifold.
This manifold contains as a submanifold the set of points with and ; this submanifold is naturally identified with . The complement of the submanifold, which consists of points with , is naturally identified with the product of
and the circle.
The manifold inherits the Hamiltonian circle action, as do its two submanifolds just described. So one may form the symplectic quotient
By construction, it contains as a dense open submanifold; essentially, it compactifies this open manifold with the symplectic quotient
which is a symplectic submanifold of of codimension two.
If is Kähler, then so is the cut space ; however, the embedding of is not an isometry.
One constructs , the other half of the symplectic cut, in a symmetric manner. The normal bundles of in the two halves of the cut are opposite each other (meaning symplectically anti-isomorphic). The symplectic sum of and along
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https://en.wikipedia.org/wiki/NAR%201
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NAR 1 or just NAR (Serbian Nastavni Računar, en. Educational Computer) was a theoretical model of a computer created by Faculty of Mathematics of University of Belgrade professor Nedeljko Parezanović (In Serbian:Недељко Парезановић). It was used for Assembly language and Computer architecture courses.
Specifications
NAR 1 processor has a 5-bit address bus (32 bytes of addressable memory) and 8-bit data bus. Machine instructions were single-byte with three most significant bits specifying the opcode and 5 least significant bits the parameter - memory address. A single 8-bit accumulator register was available and there were no flags or flag registers. Only absolute addressing mode was available and all others were achieved by self-modifying code.
Even though this is only a theoretical computer the following physical characteristics were given:
Memory cycle: 1μs
Arithmetic operation (SABF) cycle: 0.9μs (900ns)
Control panel facilitates power on and off, memory data entry and readout, instruction counter entry and selection of either program execution mode or control panel mode.
Instruction coding and set
SABF (001aaaaa, sr. , en. Add Fixed point) loads the content of memory location specified by the address parameter, adds it to the current value of the accumulator and stores the result into the accumulator
PZAF (010xxxxx, sr. , en. Change the sign of the accumulator in fixed point) Negates the fixed point (such as integer) value in the accumulator
AUM (011aaaaa, sr. , en. Accumulator Into Memory) stores the content of the accumulator into memory location specified by the address parameter
MUA (100aaaaa, sr. , en. Memory Into Accumulator) loads the content of memory location specified by the address parameter into the accumulator
NES (101aaaaa, sr. , en. Negative Jump) performs a conditional jump to the address specified by the parameter if the current value of the accumulator is negative
ZAR (110xxxxx, sr. , en. Stop the Computer) stops any further processing.
Two more instructions were not specified but were commonly present in simulators and took instruction codes 000aaaaa and 111aaaaa:
BES (sr. , en. Unconditional Jump) performs an unconditional jump to the address specified by the parameter
NUS (sr. , en. Zero Jump) performs a conditional jump to the address specified by the parameter if the current value of the accumulator is zero
Example programs
A sample program that sums up an array of 8-bit integers:
00: 0 ; input: 0 or value 22, output: result
01..21: 0,0,0... ; input: values 1..21
22: MUA 0 ; Start of program; Load accumulator from address 0
23: SABF 1 ; Add value from address 1 to accumulator
24: AUM 0 ; Store accumulator to address 0
25: MUA 23 ; Load instruction at address 23 (SABF)
26: SABF 31 ; Add value from address 31 (+1) to accumulator
27: AUM 23 ; Store accumulator to address 23 (modifies SABF instruction)
28: SABF 30 ; Add value from address 30 t
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https://en.wikipedia.org/wiki/NAR%202
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NAR 2 (Serbian Nastavni Računar 2, en. Educational Computer 2) is a theoretical model of a 32-bit word computer created by Faculty of Mathematics of University of Belgrade professor Nedeljko Parezanović as an enhancement to its predecessor, NAR 1. It was used for Assembly language and Computer architecture courses. The word "nar" means Pomegranate in Serbian. Many NAR 2 simulators have been created — for instance, one was named "Šljiva" (en. plum) as that fruit grows in Serbia, while "nar" does not.
Instruction structure
The NAR 2 processor uses 32-bit machine words. Each Machine instruction contains:
opcode in 8 most significant bits (bits 24 to 31)
4 bits (20 to 23) specifying the Index register to use with indexed addressing modes
4 bits (16 to 19) containing address mode flags:
bit 19: P (sr. Posredno, en. mediated) - indexed
bit 18: R (sr. Relativno) - relative to program counter
bit 17: I (sr. Indirektno) - multi-level memory indirect (note: the address is loaded from specified location and, should it also specify "I" flag the indirect address calculation continues)
bit 16: N (sr. Neposredno) - immediate
16 bit signed parameter value
Registers
NAR 2 has four registers:
a Program counter called BN (sr. Brojač Naredbi, en. Counter of Instructions)
Single 32-bit accumulator that can be treated either as integer (fixed point) or real (floating point) number
Up to 16 Index registers are specifiable, X0 to X15. However, X0 was never used, possibly because it was reserved as program counter (BN)
There were no flags or flag registers
Mnemonics
Following opcodes were available (actual codes were not specified, only mnemonics):
Memory/register access
MUA (sr. , en. Memory Into Accumulator) loads the value into accumulator
AUM (sr. , en. Accumulator Into Memory) stores the content of the accumulator
PIR (sr. , en. Load Index Register) Loads the value into the index register
Integer arithmetic
Note: all mnemonics in this group end with letter "F" indicating "Fiksni zarez" (en. Fixed point) arithmetic. However, this is only true for addition, subtraction and negation (sign change). Multiplication and division assume that the "point" is fixed to the right of least significant bit - that is that the numbers are integer.
SABF (sr. , en. Add, Fixed Point) - adds parameter to the accumulator
ODUF (sr. , en. Subtract, Fixed Point) - subtracts the parameter from the accumulator
MNOF (sr. , en. Multiply, Fixed Point) - Multiples the accumulator with the parameter
DELF (sr. , en. Divide, Fixed Point) - Divides the accumulator by the parameter
PZAF (sr. , en. Change the Sign of Accumuator, Fixed Point) - Changes (flips) the sign of the accumulator
Floating point arithmetic
SAB (sr. , en. Add) - adds parameter to the accumulator
ODU (sr. , en. Subtract) - subtracts the parameter from the accumulator
MNO (sr. , en. Multiply) - Multiples the accumulator with the parameter
DEL (sr. , en. Divide) - Divides the accumulator by the param
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https://en.wikipedia.org/wiki/189%20%28number%29
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189 (one hundred [and] eighty-nine) is the natural number following 188 and preceding 190.
In mathematics
189 is a centered cube number and a heptagonal number.
The centered cube numbers are the sums of two consecutive cubes, and 189 can be written as sum of two cubes in two ways: and The smallest number that can be written as the sum of two positive cubes in two ways is 1729.
There are 189 zeros among the decimal digits of the positive integers with at most three digits.
The largest prime number that can be represented in 256-bit arithmetic is the "ultra-useful prime" used in quasi-Monte Carlo methods and in some cryptographic systems.
See also
The year AD 189 or 189 BC
List of highways numbered 189
References
Integers
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https://en.wikipedia.org/wiki/Hyperplane%20separation%20theorem
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In geometry, the hyperplane separation theorem is a theorem about disjoint convex sets in n-dimensional Euclidean space. There are several rather similar versions. In one version of the theorem, if both these sets are closed and at least one of them is compact, then there is a hyperplane in between them and even two parallel hyperplanes in between them separated by a gap. In another version, if both disjoint convex sets are open, then there is a hyperplane in between them, but not necessarily any gap. An axis which is orthogonal to a separating hyperplane is a separating axis, because the orthogonal projections of the convex bodies onto the axis are disjoint.
The hyperplane separation theorem is due to Hermann Minkowski. The Hahn–Banach separation theorem generalizes the result to topological vector spaces.
A related result is the supporting hyperplane theorem.
In the context of support-vector machines, the optimally separating hyperplane or maximum-margin hyperplane is a hyperplane which separates two convex hulls of points and is equidistant from the two.
Statements and proof
In all cases, assume to be disjoint, nonempty, and convex subsets of . The summary of the results are as follows:
The number of dimensions must be finite. In infinite-dimensional spaces there are examples of two closed, convex, disjoint sets which cannot be separated by a closed hyperplane (a hyperplane where a continuous linear functional equals some constant) even in the weak sense where the inequalities are not strict.
Here, the compactness in the hypothesis cannot be relaxed; see an example in the section Counterexamples and uniqueness. This version of the separation theorem does generalize to infinite-dimension; the generalization is more commonly known as the Hahn–Banach separation theorem.
The proof is based on the following lemma:
Since a separating hyperplane cannot intersect the interiors of open convex sets, we have a corollary:
Case with possible intersections
If the sets have possible intersections, but their relative interiors are disjoint, then the proof of the first case still applies with no change, thus yielding:
in particular, we have the supporting hyperplane theorem.
Converse of theorem
Note that the existence of a hyperplane that only "separates" two convex sets in the weak sense of both inequalities being non-strict obviously does not imply that the two sets are disjoint. Both sets could have points located on the hyperplane.
Counterexamples and uniqueness
If one of A or B is not convex, then there are many possible counterexamples. For example, A and B could be concentric circles. A more subtle counterexample is one in which A and B are both closed but neither one is compact. For example, if A is a closed half plane and B is bounded by one arm of a hyperbola, then there is no strictly separating hyperplane:
(Although, by an instance of the second theorem, there is a hyperplane that separates their interiors.) Another type
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https://en.wikipedia.org/wiki/Thomas%20M.%20Cover
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Thomas M. Cover [ˈkoʊvər] (August 7, 1938 – March 26, 2012) was an American information theorist and professor jointly in the Departments of Electrical Engineering and Statistics at Stanford University. He devoted almost his entire career to developing the relationship between information theory and statistics.
Early life and education
He received his B.S. in Physics from MIT in 1960 and Ph.D. in electrical engineering from Stanford University in 1964.
Career
Cover was President of the IEEE Information Theory Society and was a Fellow of the Institute of Mathematical Statistics and of the Institute of Electrical and Electronics Engineers. He received the Outstanding Paper Award in Information Theory for his 1972 paper "Broadcast Channels"; he was selected in 1990 as the Shannon Lecturer, regarded as the highest honor in information theory; in 1997 he received the IEEE Richard W. Hamming Medal; and in 2003 he was elected to the American Academy of Arts and Sciences.
During his 48-year career as a professor of Electrical Engineering and Statistics at Stanford University, he graduated 64 PhD students, authored over 120 journal papers in learning, information theory, statistical complexity, pattern recognition, and portfolio theory; and he partnered with Joy A. Thomas to coauthor the book Elements of Information Theory, which has become the most widely used textbook as an introduction to the topic since the publication of its first edition in 1991. He was also coeditor of the book Open Problems in Communication and Computation.
Selected works
Van Campenhout, Jan. and Cover, T. (1981). Maximum entropy and conditional probability. Information Theory, IEEE Transactions on
Cover, T. (1974). The Best Two Independent Measurements Are Not the Two Best. Systems, Man and Cybernetics, IEEE Transactions on
Cover, T. and Hart, P. (1967). Nearest neighbor pattern classification.] Information Theory, IEEE Transactions on.
Cover, T. (1965). [http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=4038449 Geometrical and Statistical Properties of Systems of Linear Inequalities with Applications in Pattern Recognition. Electronic Computers, IEEE Transactions on
See also
k-nearest neighbors algorithm
Cover's theorem
References
External links
Thomas M. Cover Stanford University homepage
1938 births
2012 deaths
American statisticians
Fellow Members of the IEEE
Fellows of the Institute of Mathematical Statistics
Members of the United States National Academy of Engineering
People from San Bernardino, California
Probability theorists
Stanford University School of Engineering alumni
Stanford University School of Engineering faculty
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https://en.wikipedia.org/wiki/Ambiguity%20effect
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The ambiguity effect is a cognitive bias where decision making is affected by a lack of information, or "ambiguity". The effect implies that people tend to select options for which the probability of a favorable outcome is known, over an option for which the probability of a favorable outcome is unknown. The effect was first described by Daniel Ellsberg in 1961.
Example
As an example, consider a bucket containing 30 balls. The balls are either red, black or white. Ten of the balls are red, and the remaining 20 are either black or white, with all combinations of black and white being equally likely. In option X, drawing a red ball wins a person $100, and in option Y, drawing a black ball wins them $100. The probability of picking a winning ball is the same for both options X and Y. In option X, the probability of selecting a winning ball is 1 in 3 (10 red balls out of 30 total balls). In option Y, despite the fact that the number of black balls is uncertain, the probability of selecting a winning ball is also 1 in 3. This is because the number of black balls is equally distributed among all possibilities between 0 and 20. The difference between the two options is that in option X, the probability of a favorable outcome is known, but in option Y, the probability of a favorable outcome is unknown ("ambiguous").
In spite of the equal probability of a favorable outcome, people have a greater tendency to select a ball under option X, where the probability of selecting a winning ball is perceived to be more certain. The uncertainty as to the number of black balls means that option Y tends to be viewed less favorably. Despite the fact that there could possibly be twice as many black balls as red balls, people tend not to want to take the opposing risk that there may be fewer than 10 black balls. The "ambiguity" behind option Y means that people tend to favor option X, even when the probability is the same.
Explanation
One possible explanation of the effect is that people have a rule of thumb (heuristic) to avoid options where information is missing. This will often lead them to seek out the missing information. In many cases, though, the information cannot be obtained. The effect is often the result of calling some particular missing piece of information to the person's attention.
See also
Ambiguity aversion
Black swan theory
Choice under uncertainty
Ellsberg paradox
Prospect theory
Risk aversion
References
Cognitive biases
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https://en.wikipedia.org/wiki/E0
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E0 or E00 can refer to :
ε0, in mathematics, the smallest member of the epsilon numbers, a type of ordinal number
ε0, in physics, vacuum permittivity, the absolute dielectric permittivity of classical vacuum
E0 (cipher), a cipher used in the Bluetooth protocol
E0 (robot), a 1986 humanoid robot by Honda
Eo, in electrochemistry, the standard electrode potential, measuring individual potential of a reversible electrode at standard state
E0, the digital carrier for audio, specified in G.703
E0, Eos Airlines IATA code
E0, ethanol-free gasoline, see REC-90
e0, in demographics, the life expectancy of an individual at birth (age zero)
E00, Cretinism ICD-10 code
E00, ECO code for certain variations of the Queen's Pawn Game chess opening
Enemy Zero, a 1996 Japanese horror video game for the Sega Saturn
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https://en.wikipedia.org/wiki/Bol%20loop
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In mathematics and abstract algebra, a Bol loop is an algebraic structure generalizing the notion of group. Bol loops are named for the Dutch mathematician Gerrit Bol who introduced them in .
A loop, L, is said to be a left Bol loop if it satisfies the identity
, for every a,b,c in L,
while L is said to be a right Bol loop if it satisfies
, for every a,b,c in L.
These identities can be seen as weakened forms of associativity, or a strengthened form of (left or right) alternativity.
A loop is both left Bol and right Bol if and only if it is a Moufang loop. Alternatively, a right or left Bol loop is Moufang if and only if it satisfies the flexible identity a(ba) = (ab)a . Different authors use the term "Bol loop" to refer to either a left Bol or a right Bol loop.
Properties
The left (right) Bol identity directly implies the left (right) alternative property, as can be shown by setting b to the identity.
It also implies the left (right) inverse property, as can be seen by setting b to the left (right) inverse of a, and using loop division to cancel the superfluous factor of a. As a result, Bol loops have two-sided inverses.
Bol loops are also power-associative.
Bruck loops
A Bol loop where the aforementioned two-sided inverse satisfies the automorphic inverse property, (ab)−1 = a−1 b−1 for all a,b in L, is known as a (left or right) Bruck loop or K-loop (named for the American mathematician Richard Bruck). The example in the following section is a Bruck loop.
Bruck loops have applications in special relativity; see Ungar (2002). Left Bruck loops are equivalent to Ungar's (2002) gyrocommutative gyrogroups, even though the two structures are defined differently.
Example
Let L denote the set of n x n positive definite, Hermitian matrices over the complex numbers. It is generally not true that the matrix product AB of matrices A, B in L is Hermitian, let alone positive definite. However, there exists a unique P in L and a unique unitary matrix U such that AB = PU; this is the polar decomposition of AB. Define a binary operation * on L by A * B = P. Then (L, *) is a left Bruck loop. An explicit formula for * is given by A * B = (A B2 A)1/2, where the superscript 1/2 indicates the unique positive definite Hermitian square root.
Bol algebra
A (left) Bol algebra is a vector space equipped with a binary operation and a ternary operation that satisfies the following identities:
and
and
and
.
Note that {.,.,.} acts as a Lie triple system.
If is a left or right alternative algebra then it has an associated Bol algebra , where is the commutator and is the Jordan associator.
References
Chapter VI is about Bol loops.
Non-associative algebra
Group theory
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https://en.wikipedia.org/wiki/Institute%20of%20Statistical%20Mathematics
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The Institute of Statistical Mathematics is Japan's national research institute for statistical science. In October 2009, it relocated from the Azabu district of Tokyo to Tachikawa. Founded in 1944, since 2004 it has been part of the Research Organization of Information and Systems ().
The Institute is represented on the national Coordinating Committee for Earthquake Prediction.
Notable faculty
Hirotugu Akaike
Motosaburo Masuyama
Joichi Suetsuna
Genichi Taguchi
Publications
The Institute publishes the scientific journal Annals of the Institute of Statistical Mathematics and the time series analysis software package TIMSAC.
References
External links
Official website
Mathematical institutes
Research institutes in Japan
National statistical services
1944 establishments in Japan
Organizations established in 1944
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https://en.wikipedia.org/wiki/Occupational%20Outlook%20Handbook
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The Occupational Outlook Handbook (OOH) is a publication of the United States Department of Labor's Bureau of Labor Statistics that includes information about the nature of work, working conditions, training and education, earnings and job outlook for hundreds of different occupations in the United States. It is released biennially with a companion publication, the Career Guide to Industries and is available free of charge from the Bureau of Labor Statistics' website. The 2012–13 edition was released in November 2012 and the 2014–15 edition in March 2014.
Because it is a work by the United States federal government, the Handbook is not under copyright and is reproduced in various forms by other publishers, often with additional information or features.
The first edition was published in 1948.
See also
Career development
Global Career Development Facilitator (GCDF)
Holland Codes
Lists of occupations
Myers–Briggs Type Indicator
Standard Occupational Classification System
References
External links
BLS News release
JIST Publishing - America's Career Publisher
ocouha: Occupational Outlook Handbook plus
Occupational Outlook Handbook, digitized and available on FRASER
Occupations
Economic data
United States Department of Labor publications
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https://en.wikipedia.org/wiki/Yuri%20Linnik
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Yuri Vladimirovich Linnik (; January 8, 1915 – June 30, 1972) was a Soviet mathematician active in number theory, probability theory and mathematical statistics.
Biography
Linnik was born in Bila Tserkva, in present-day Ukraine. He went to Saint Petersburg University where his supervisor was Vladimir Tartakovsky, and later worked at that university and the Steklov Institute. He was a member of the Academy of Sciences of the Soviet Union, as was his father, Vladimir Pavlovich Linnik. He was awarded both Stalin and Lenin Prizes. He died in Leningrad.
Work in number theory
Linnik's theorem in analytic number theory
The dispersion method (which allowed him to solve the Titchmarsh problem).
The large sieve (which turned out to be extremely influential).
An elementary proof of the Hilbert-Waring theorem; see also Schnirelmann density.
The Linnik ergodic method, see , which allowed him to study the distribution properties of the representations of integers by integral ternary quadratic forms.
Work in probability theory and statistics
Infinitely divisible distributions
Linnik obtained numerous results concerning infinitely divisible distributions. In particular, he proved the following generalisation of Cramér's theorem: any divisor of a convolution of Gaussian and Poisson random variables is also a convolution of Gaussian and Poisson.
He has also coauthored the book on the arithmetics of infinitely divisible distributions.
Central limit theorem
Linnik zones (zones of asymptotic normality)
Information-theoretic proof of the central limit theorem
Statistics
Behrens–Fisher problem
Selected publications
Notes
External links
Acta Arithmetica: Linnik memorial issue (1975)
List of books by Linnik provided by National Library of Australia
1915 births
1972 deaths
20th-century Russian mathematicians
People from Bila Tserkva
Full Members of the USSR Academy of Sciences
Members of the Royal Swedish Academy of Sciences
Saint Petersburg State University alumni
Heroes of Socialist Labour
Recipients of the Stalin Prize
Recipients of the Lenin Prize
Recipients of the Order of the Badge of Honour
Recipients of the Order of Lenin
Recipients of the Order of the Red Banner of Labour
Mathematical statisticians
Number theorists
Russian statisticians
Soviet mathematicians
Russian scientists
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https://en.wikipedia.org/wiki/%C3%85ke%20Pleijel
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Åke Vilhelm Carl Pleijel (10 August 1913 – 24 September 1989) was a Swedish mathematician.
He completed his Ph.D. in mathematics at Stockholm University in 1940 (with Torsten Carleman as supervisor), and later became Professor of Mathematics at Uppsala University.
Åke Vilhelm Carl Pleijel published the paper in which the Minakshisundaram–Pleijel zeta function was introduced.
References
External links
1913 births
1989 deaths
Stockholm University alumni
20th-century Swedish mathematicians
Members of the Royal Swedish Academy of Sciences
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https://en.wikipedia.org/wiki/Topological%20half-exact%20functor
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In mathematics, a topological half-exact functor F is a functor from a fixed topological category (for example CW complexes or pointed spaces) to an abelian category (most frequently in applications, category of abelian groups or category of modules over a fixed ring) that has a following property: for each sequence of spaces, of the form:
X → Y → C(f)
where C(f) denotes a mapping cone, the sequence:
F(X) → F(Y) → F(C(f))
is exact. If F is a contravariant functor, it is half-exact if for each sequence of spaces as above,
the sequence F(C(f)) → F(Y) → F(X) is exact.
Homology is an example of a half-exact functor, and
cohomology (and generalized cohomology theories) are examples of contravariant half-exact functors.
If B is any fibrant topological space, the (representable) functor F(X)=[X,B] is half-exact.
Homotopy theory
Homological algebra
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https://en.wikipedia.org/wiki/Lies%2C%20Damn%20Lies%20and%20Statistics
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Lies, Damn Lies and Statistics may refer to:
"Lies, Damn Lies and Statistics" (The West Wing), a first-season episode of the TV series The West Wing
Lies, damned lies, and statistics, a phrase describing the persuasive power of numbers
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https://en.wikipedia.org/wiki/Elation
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Elation is an emotion of happiness.
Elation may also refer to:
Carnival Elation, cruise ship
A type of collineation in perspective geometry where the center lies on the axis
Elation (album), a 2012 studio album by the band Great White
Groove Elation, a 1995 album by jazz musician John Scofield
See also
Euphoria (disambiguation)
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https://en.wikipedia.org/wiki/Michel%20Raynaud
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Michel Raynaud (; 16 June 1938 – 10 March 2018) was a French mathematician working in algebraic geometry and a professor at Paris-Sud 11 University.
Early life and education
He was born in Riom, France as a single son to a modest household. His father was a carpenter and his mother cleaned houses. He attended the local primary school at Châtel Guyon and Riom, and attended high school at the boarding school in Clermont-Ferrand.
Raynaud entered the École normale supérieur where he studied from 1958 to 1962, while being first of the class in the "agrégation" exam where the new high school teachers were selected in 1961. In 1962, he entered the French National Centre for Scientific Research where he studied together with his future wife Michèle Chaumartin. Both had the same doctoral advisor in Alexander Grothendieck. Raynaud received his doctoral degree in 1967.
Career
Raynaud was hired as professor at the Orsay Faculty of Sciences in Paris where he was a employed until 2001, when he retired.
Raynaud died on 10 March 2018 in Rueil-Malmaison, France.
Research
In 1983, Raynaud published a proof of the Manin–Mumford conjecture. In 1985, he proved Raynaud's isogeny theorem on Faltings heights of isogenous elliptic curves. With David Harbater and following the work of Jean-Pierre Serre, Raynaud proved Abhyankar's conjecture in 1994.
The Raynaud surface was named after him by William E. Lang in 1979.
Honors and awards
In 1970 Raynaud was an invited speaker at the International Congress of Mathematicians in Nice. In 1987 he received the Prize Ampère from the French Academy of Sciences. In 1995 he received the Cole Prize, together with David Harbater, for his solution of the Abhyankar conjecture.
Personal life
He practiced skiing (especially in Val-d'Isère), tennis, and rock climbing (in Fontainebleau). He was married to the mathematician Michèle Raynaud () who also worked with Alexander Grothendieck.
References
External links
Cole Prize citation for Michel Raynaud
1938 births
2018 deaths
People from Riom
20th-century French mathematicians
21st-century French mathematicians
École Normale Supérieure alumni
Academic staff of Paris-Sud University
Institute for Advanced Study visiting scholars
Members of the French Academy of Sciences
Algebraic geometers
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https://en.wikipedia.org/wiki/Weight%20%28disambiguation%29
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Weight is a measurement of the gravitational force acting on an object.
Weight or The Weight may also refer to:
Mathematics
Weight (graph theory) a number associated to an edge or to a vertex of a graph
Weight (representation theory), a type of function
Weight (strings), the number of times a letter occurs in a string
Weight, an integer associated to each variable of a quasi-homogeneous polynomial
Weight of a topological space; see base
Weighting, making some data contribute to a result more than others
Weight function
Weighted mean and weighted average, the importance can vary on each piece of data
Weighting filter
Science and technology
Weight (unit), a former English unit
Weight, a connection strength, or coefficient in a linear combination, as in an artificial neural network
Weight, a measure of paper density
Body weight, a commonly used term for the mass of an organism's body
Font weight
Line weight in contour line construction in cartography
Specific weight, the weight per unit volume of a material
In underwater diving, a dense object used for ballast in a diving weighting system
Balance weights, part of a weighing scale
Weight (object), object whose chief task is to exert weight
Film and television
The Weight (film), a 2012 South Korean film
"Weight" (Justified), a 2014 television episode
"The Weight" (The Sopranos), a 2002 television episode
Music
Weight (album), by Rollins Band, 1994
Weight (EP), by the Kindred, 2017
"Weight" (song), by Latrice Royale, 2014
"The Weight", a song by The Band, 1968
"Weight", a song by Alexz Johnson, 2020
"Weight", a song by Brockhampton from Iridescence, 2018
"Weight", a song by Isis from Oceanic, 2002
Sports
Weight, an object of known mass used in weight training
Weight, the object thrown in a weight throw
Draw weight of a bow
Weight class, a competition division used to match competitors against others of their own size
Other uses
Weight (surname)
Weight (wine) or "body", a quality of wine
vi:Tương tác hấp dẫn#Trọng lực
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https://en.wikipedia.org/wiki/Pakistan%20Bureau%20of%20Statistics
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The Pakistan Bureau of Statistics (, abbreviated as PBS) is a federal agency under the Government of Pakistan. It is an attached department of the Ministry of Planning, Development & Special Initiatives. It works for collecting statistics in the country.
History
In 1947, the Central Statistical Office (CSO) was set up by the government of Prime Minister Liaquat Ali Khan. In 1950, CSO became an attached department of the Economic Affairs Division. In 1972, on the recommendation of IBRD Mission, Prime Minister Zulfikar Ali Bhutto upgraded the Central Statistical Office to a full-fledged government division. In 1981, the bureau was reorganized and its technical wing (CSO) was converted into the then Federal Bureau of Statistics. Former Finance Minister Dr. Mahbub ul Haq further reorganized the bureau.
See also
Government of Pakistan
Politics of Pakistan
Statistics
References
External links
Federal Bureau of Statistics
Pakistan federal departments and agencies
Pakistan
1950 establishments in Pakistan
Government agencies established in 1950
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https://en.wikipedia.org/wiki/Response%20amplitude%20operator
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In the field of ship design and design of other floating structures, a response amplitude operator (RAO) is an engineering statistic, or set of such statistics, that are used to determine the likely behavior of a ship when operating at sea. Known by the acronym of RAO, response amplitude operators are usually obtained from models of proposed ship designs tested in a model basin, or from running specialized CFD computer programs, often both. RAOs are usually calculated for all ship motions and for all wave headings.
Usage
RAOs are effectively transfer functions used to determine the effect that a sea state will have upon the motion of a ship through the water, and therefore, for example, whether or not (in the case of cargo vessels) the addition of cargo to the vessel will require measures to be taken to improve stability and prevent the cargo from shifting within the vessel. Generation of extensive RAOs at the design phase allows shipbuilders to determine the modifications to a design that may be required for safety reasons (i.e., to make the design robust and resistant to capsizing or sinking in highly adverse sea conditions) or to improve performance (e.g., improve top speed, fuel consumption, stability in rough seas). RAOs are computed in tandem with the generation of a hydrodynamic database, which is a model of the effects of water pressure upon the ship's hull under a wide variety of flow conditions. Together, the RAOs and hydrodynamic database provide (inasmuch as this is possible within modelling and engineering constraints) certain assurances about the behavior of a proposed ship design. They also allow the designer to dimension the ship or structure so it will hold up to the most extreme sea states it will likely be subjected to (based on sea state statistics).
Additionally, RAOs are amplitude operators which enables to determine amplitude of motion based on a unitary wave. They are superposable, thus effective on sea state based probability solutions.
RAOs in ship design
Different modelling and design criteria will affect the nature of the 'ideal' RAO curves (as plotted graphically) being sought for a particular ship: for example, an ocean cruise liner will have a considerable emphasis placed upon minimizing accelerations to ensure the comfort of the passengers, while the stability concerns for a naval warship will be concentrated upon making the ship an effective weapons platform.
Finding the forces on the ship when it is restrained from motion and subjected to regular waves. The forces acting on the body are:
The Froude–Krylov force, which is the pressure in the undisturbed waves integrated over the wetted surface of the floating vessel.
The Diffraction forces, which are pressures that occur due to the disturbances in the water because of a body being present.
Finding the forces on the ship when it is forced to oscillate in still water conditions. The forces are divided into:
Added mass forces due to having to accelerate th
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https://en.wikipedia.org/wiki/Purely%20inseparable%20extension
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In algebra, a purely inseparable extension of fields is an extension k ⊆ K of fields of characteristic p > 0 such that every element of K is a root of an equation of the form xq = a, with q a power of p and a in k. Purely inseparable extensions are sometimes called radicial extensions, which should not be confused with the similar-sounding but more general notion of radical extensions.
Purely inseparable extensions
An algebraic extension is a purely inseparable extension if and only if for every , the minimal polynomial of over F is not a separable polynomial. If F is any field, the trivial extension is purely inseparable; for the field F to possess a non-trivial purely inseparable extension, it must be imperfect as outlined in the above section.
Several equivalent and more concrete definitions for the notion of a purely inseparable extension are known. If is an algebraic extension with (non-zero) prime characteristic p, then the following are equivalent:
E is purely inseparable over F.
For each element , there exists such that .
Each element of E has minimal polynomial over F of the form for some integer and some element .
It follows from the above equivalent characterizations that if (for F a field of prime characteristic) such that for some integer , then E is purely inseparable over F. (To see this, note that the set of all x such that for some forms a field; since this field contains both and F, it must be E, and by condition 2 above, must be purely inseparable.)
If F is an imperfect field of prime characteristic p, choose such that a is not a pth power in F, and let f(X) = Xp − a. Then f has no root in F, and so if E is a splitting field for f over F, it is possible to choose with . In particular, and by the property stated in the paragraph directly above, it follows that is a non-trivial purely inseparable extension (in fact, , and so is automatically a purely inseparable extension).
Purely inseparable extensions do occur naturally; for example, they occur in algebraic geometry over fields of prime characteristic. If K is a field of characteristic p, and if V is an algebraic variety over K of dimension greater than zero, the function field K(V) is a purely inseparable extension over the subfield K(V)p of pth powers (this follows from condition 2 above). Such extensions occur in the context of multiplication by p on an elliptic curve over a finite field of characteristic p.
Properties
If the characteristic of a field F is a (non-zero) prime number p, and if is a purely inseparable extension, then if , K is purely inseparable over F and E is purely inseparable over K. Furthermore, if [E : F] is finite, then it is a power of p, the characteristic of F.
Conversely, if is such that and are purely inseparable extensions, then E is purely inseparable over F.
An algebraic extension is an inseparable extension if and only if there is some such that the minimal polynomial of over F is not a separable polynomial (i
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https://en.wikipedia.org/wiki/Hyperboloid%20model
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In geometry, the hyperboloid model, also known as the Minkowski model after Hermann Minkowski, is a model of n-dimensional hyperbolic geometry in which points are represented by points on the forward sheet S+ of a two-sheeted hyperboloid in (n+1)-dimensional Minkowski space or by the displacement vectors from the origin to those points, and m-planes are represented by the intersections of (m+1)-planes passing through the origin in Minkowski space with S+ or by wedge products of m vectors. Hyperbolic space is embedded isometrically in Minkowski space; that is, the hyperbolic distance function is inherited from Minkowski space, analogous to the way spherical distance is inherited from Euclidean distance when the n-sphere is embedded in (n+1)-dimensional Euclidean space.
Other models of hyperbolic space can be thought of as map projections of S+: the Beltrami–Klein model is the projection of S+ through the origin onto a plane perpendicular to a vector from the origin to specific point in S+ analogous to the gnomonic projection of the sphere; the Poincaré disk model is a projection of S+ through a point on the other sheet S− onto perpendicular plane, analogous to the stereographic projection of the sphere; the Gans model is the orthogonal projection of S+ onto a plane perpendicular to a specific point in S+, analogous to the orthographic projection; the band model of the hyperbolic plane is a conformal “cylindrical” projection analogous to the Mercator projection of the sphere; Lobachevsky coordinates are a cylindrical projection analogous to the equirectangular projection (longitude, latitude) of the sphere.
Minkowski quadratic form
If (x0, x1, ..., xn) is a vector in the -dimensional coordinate space Rn+1, the Minkowski quadratic form is defined to be
The vectors such that form an n-dimensional hyperboloid S consisting of two connected components, or sheets: the forward, or future, sheet S+, where x0>0 and the backward, or past, sheet S−, where x0<0. The points of the n-dimensional hyperboloid model are the points on the forward sheet S+.
The Minkowski bilinear form B is the polarization of the Minkowski quadratic form Q,
(This is sometimes also written using scalar product notation )
Explicitly,
The hyperbolic distance between two points u and v of S+ is given by the formula
where is the inverse function of hyperbolic cosine.
Choice of metric signature
The bilinear form also functions as the metric tensor over the space. In n+1 dimensional Minkowski space, there are two choices for the metric with opposite signature, in the 3-dimensional case either (+, −, −) or (−, +, +).
If the signature (−, +, +) is chosen, then the scalar square of chords between distinct points on the same sheet of the hyperboloid will be positive, which more closely aligns with conventional definitions and expectations in mathematics. Then n-dimensional hyperbolic space is a Riemannian space and distance or length can be defined as the square root of the sca
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https://en.wikipedia.org/wiki/Mathematical%20sociology
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Mathematical sociology or the sociology of mathematics is an interdisciplinary field of research concerned both with the use of mathematics within sociological research as well as research into the relationships that exist between maths and society.
Because of this, mathematical sociology can have a diverse meaning depending on the authors in question and the kind of research being carried out. This creates contestation over whether mathematical sociology is a derivative of sociology, an intersection of the two disciplines, or a discipline in its own right. This is a dynamic, ongoing academic development that leaves mathematical sociology sometimes blurred and lacking in uniformity, presenting grey areas and need for further research into developing its academic merit.
History
Starting in the early 1940s, Nicolas Rashevsky, and subsequently in the late 1940s, Anatol Rapoport and others, developed a relational and probabilistic approach to the characterization of large social networks in which the nodes are persons and the links are acquaintanceship. During the late 1940s, formulas were derived that connected local parameters such as closure of contacts – if A is linked to both B and C, then there is a greater than chance probability that B and C are linked to each other – to the global network property of connectivity.
Moreover, acquaintanceship is a positive tie, but what about negative ties such as animosity among persons? To tackle this problem, graph theory, which is the mathematical study of abstract representations of networks of points and lines, can be extended to include these two types of links and thereby to create models that represent both positive and negative sentiment relations, which are represented as signed graphs. A signed graph is called balanced if the product of the signs of all relations in every cycle (links in every graph cycle) is positive. Through formalization by mathematician Frank Harary, this work produced the fundamental theorem of this theory. It says that if a network of interrelated positive and negative ties is balanced, e.g. as illustrated by the psychological principle that "my friend's enemy is my enemy", then it consists of two sub-networks such that each has positive ties among its nodes and there are only negative ties between nodes in distinct sub-networks. The imagery here is of a social system that splits into two cliques. There is, however, a special case where one of the two sub-networks is empty, which might occur in very small networks.
In another model, ties have relative strengths. 'Acquaintanceship' can be viewed as a 'weak' tie and 'friendship' is represented as a strong tie. Like its uniform cousin discussed above, there is a concept of closure, called strong triadic closure. A graph satisfies strong triadic closure If A is strongly connected to B, and B is strongly connected to C, then A and C must have a tie (either weak or strong).
In these two developments we have mathematical
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https://en.wikipedia.org/wiki/Thiophosgene
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Thiophosgene is a red liquid with the formula . It is a molecule with trigonal planar geometry. There are two reactive C–Cl bonds that allow it to be used in diverse organic syntheses.
Preparation
is prepared in a two-step process from carbon disulfide. In the first step, carbon disulfide is chlorinated to give trichloromethanesulfenyl chloride (perchloromethyl mercaptan), :
The chlorination must be controlled as excess chlorine converts trichloromethanesulfenyl chloride into carbon tetrachloride. Steam distillation separates the trichloromethanesulfenyl chloride, a rare sulfenyl chloride, and hydrolyzes the disulfur dichloride. Reduction of trichloromethanesulfenyl chloride produces thiophosgene:
Tin and dihydroanthracene have been used for the reducing agents.
Reactions
is mainly used to prepare compounds with the connectivity where X = OR, NHR. Such reactions proceed via intermediate such as CSClX. Under certain conditions, one can convert primary amines into isothiocyanates.
also serves as a dienophile to give, after reduction 5-thiacyclohexene derivatives. Thiophosgene is also known as the appropriate reagent in Corey-Winter synthesis for stereospecific conversion of 1,2-diols into alkenes.
It forms a head-to-tail dimer upon irradiation with UV light:
Unlike thiophosgene monomer, a red liquid, the photodimer, an example of a 1,3-dithietane, is a colourless solid.
Safety considerations
is considered highly toxic.
References
Further reading
Inorganic carbon compounds
Inorganic sulfur compounds
Thiochlorides
Thiocarbonyl compounds
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https://en.wikipedia.org/wiki/Clifford%20theory
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In mathematics, Clifford theory, introduced by , describes the relation between representations of a group and those of a normal subgroup.
Alfred H. Clifford
Alfred H. Clifford proved the following result on the restriction of finite-dimensional irreducible representations from a group G to a normal subgroup N of finite index:
Clifford's theorem
Theorem. Let π: G → GL(n,K) be an irreducible representation with K a field. Then the restriction of π to N breaks up into a direct sum of irreducible representations of N of equal dimensions. These irreducible representations of N lie in one orbit for the action of G by conjugation on the equivalence classes of irreducible representations of N. In particular the number of pairwise nonisomorphic summands is no greater than the index of N in G.
Clifford's theorem yields information about the restriction of a complex irreducible character of a finite group G to a normal subgroup N. If μ is a complex character of N, then for a fixed element g of G, another character, μ(g), of N may be constructed by setting
for all n in N. The character μ(g) is irreducible if and only if μ is. Clifford's theorem states that if χ is a complex irreducible character of G, and μ is an irreducible character of N with
then
where e and t are positive integers, and each gi is an element of G. The integers e and t both divide the index [G:N]. The integer t is the index of a subgroup of G, containing N, known as the inertial subgroup of μ. This is
and is often denoted by
The elements gi may be taken to be representatives of all the right cosets of the subgroup IG(μ) in G.
In fact, the integer e divides the index
though the proof of this fact requires some use of Schur's theory of projective representations.
Proof of Clifford's theorem
The proof of Clifford's theorem is best explained in terms of modules (and the module-theoretic version works for irreducible modular representations). Let K be a field, V be an irreducible K[G]-module, VN be its restriction to N and U be an irreducible K[N]-submodule of VN. For each g in G, U.g is an irreducible K[N]-submodule of VN, and is an K[G]-submodule of V, so must be all of V by irreducibility. Now VN is expressed as a sum of irreducible submodules, and this expression may be refined to a direct sum. The proof of the character-theoretic statement of the theorem may now be completed in the case K = C. Let χ be the character of G afforded by V and μ be the character of N afforded by U. For each g in G, the C[N]-submodule U.g affords the character μ(g) and . The respective equalities follow because χ is a class-function of G and N is a normal subgroup. The integer e appearing in the statement of the theorem is this common multiplicity.
Corollary of Clifford's theorem
A corollary of Clifford's theorem, which is often exploited, is that the irreducible character χ appearing in the theorem is induced from an irreducible character of the inertial subgroup IG(μ). If, for example, th
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https://en.wikipedia.org/wiki/Elementary%20number
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An elementary number is one formalization of the concept of a closed-form number. The elementary numbers form an algebraically closed field containing the roots of arbitrary expressions using field operations, exponentiation, and logarithms. The set of the elementary numbers is subdivided into the explicit elementary numbers and the implicit elementary numbers.
References
Algebraic number theory
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https://en.wikipedia.org/wiki/Hyperbolic%20trigonometry
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In mathematics, hyperbolic trigonometry can mean:
The study of hyperbolic triangles in hyperbolic geometry (traditional trigonometry is the study of triangles in plane geometry)
The use of the hyperbolic functions
The use of gyrotrigonometry in hyperbolic geometry
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https://en.wikipedia.org/wiki/Generalized%20Wiener%20process
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In statistics, a generalized Wiener process (named after Norbert Wiener) is a continuous time random walk with drift and random jumps at every point in time. Formally:
where a and b are deterministic functions, t is a continuous index for time, x is a set of exogenous variables that may change with time, dt is a differential in time, and η is a random draw from a standard normal distribution at each instant.
See also
Wiener process
Wiener process
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https://en.wikipedia.org/wiki/White%20test
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In statistics, the White test is a statistical test that establishes whether the variance of the errors in a regression model is constant: that is for homoskedasticity.
This test, and an estimator for heteroscedasticity-consistent standard errors, were proposed by Halbert White in 1980. These methods have become widely used, making this paper one of the most cited articles in economics.
In cases where the White test statistic is statistically significant, heteroskedasticity may not necessarily be the cause; instead the problem could be a specification error. In other words, the White test can be a test of heteroskedasticity or specification error or both. If no cross product terms are introduced in the White test procedure, then this is a test of pure heteroskedasticity.
If cross products are introduced in the model, then it is a test of both heteroskedasticity and specification bias.
Testing constant variance
To test for constant variance one undertakes an auxiliary regression analysis: this regresses the squared residuals from the original regression model onto a set of regressors that contain the original regressors along with their squares and cross-products. One then inspects the R2. The Lagrange multiplier (LM) test statistic is the product of the R2 value and sample size:
This follows a chi-squared distribution, with degrees of freedom equal to P − 1, where P is the number of estimated parameters (in the auxiliary regression).
The logic of the test is as follows. First, the squared residuals from the original model serve as a proxy for the variance of the error term at each observation. (The error term is assumed to have a mean of zero, and the variance of a zero-mean random variable is just the expectation of its square.) The independent variables in the auxiliary regression account for the possibility that the error variance depends on the values of the original regressors in some way (linear or quadratic). If the error term in the original model is in fact homoskedastic (has a constant variance) then the coefficients in the auxiliary regression (besides the constant) should be statistically indistinguishable from zero and the R2 should be “small". Conversely, a “large" R2 (scaled by the sample size so that it follows the chi-squared distribution) counts against the hypothesis of homoskedasticity.
An alternative to the White test is the Breusch–Pagan test, where the Breusch-Pagan test is designed to detect only linear forms of heteroskedasticity. Under certain conditions and a modification of one of the tests, they can be found to be algebraically equivalent.
If homoskedasticity is rejected one can use heteroskedasticity-consistent standard errors.
Software implementations
In R, White's Test can be implemented using the white function of the skedastic package.
In Python, White's Test can be implemented using the het_white function of the statsmodels.stats.diagnostic.het_white
In Stata, the test can be implemented using th
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https://en.wikipedia.org/wiki/Alfred%20Burne
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Alfred Higgins Burne DSO (1886–1959) was a soldier and military historian. He invented the concept of Inherent Military Probability; in battles and campaigns where there is some doubt over what action was taken, Burne believed that the action taken would be one which a trained staff officer of the twentieth century would take.
Career
Alfred Burne was educated at Winchester School and RMA Woolwich, before being commissioned into the Royal Artillery in 1906. He was awarded the DSO during the First World War and, during World War II, was Commandant of the 121st Officer Cadet Training Unit. He retired as a Lieutenant-Colonel.
He was Military Editor of Chambers Encyclopedia from 1938 to 1957 and became an authority on the history of land warfare. He was a contributor to the Dictionary of National Biography.
Burne lived in Kensington and his funeral was held at St Mary Abbots there.
Inherent Military Probability
Burne introduced the concept of Inherent Military Probability (IMP) to the study of military history. He himself defined it thus :
My method here is to start with what appear to be undisputed facts, then to place myself in the shoes of each commander in turn, and to ask myself in each case what I would have done. This I call working on Inherent Military Probability. I then compare the resulting action with the existing record in order to see whether it discloses any incompatibility with the existing facts. If not, I then go on to the next debatable or obscure point in the battle and repeat the operation
More succinctly, John Keegan defined IMP as
The solution of an obscurity by an estimate of what a trained soldier would have done in the circumstances
Bibliography
Mesopotamia, The Last Phase (1936)
Lee, Grant and Sherman (1938)
The Art of War on Land (1944)
Strategy as Exemplified in the Second World War (1946)
The Noble Duke of York (1949)
The Battlefields of England (1950)
More Battlefields of England (1953)
The Woolwich Mess (1954)
The Crecy War (1954)
The Agincourt War (1956)
A Military History of the First Civil War (1642-1646) (with Peter Young, 1959)
References
1886 births
1959 deaths
People educated at Winchester College
Graduates of the Royal Military Academy, Woolwich
Royal Artillery officers
Companions of the Distinguished Service Order
British military historians
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https://en.wikipedia.org/wiki/Steven%20Strogatz
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Steven Henry Strogatz (), born August 13, 1959, is an American mathematician and the Susan and Barton Winokur Distinguished Professor for the Public Understanding of Science and Mathematics at Cornell University.
He is known for his work on nonlinear systems, including contributions to the study of synchronization in dynamical systems, and for his research in a variety of areas of applied mathematics, including mathematical biology and complex network theory.
Strogatz is the host of Quanta Magazine'''s The Joy of Why podcast. He previously hosted The Joy of x podcast, named after his book of the same name.
Education
Strogatz attended high school at Loomis Chaffee from 1972 to 1976. He then attended Princeton University, graduating summa cum laude with a B.A. in mathematics. Strogatz completed his senior thesis, titled "The mathematics of supercoiled DNA: an essay in geometric biology", under the supervision of Frederick J. Almgren, Jr. Strogatz then studied as a Marshall Scholar at Trinity College, Cambridge, from 1980 to 1982, and then received a Ph.D. in applied mathematics from Harvard University in 1986 for his research on the dynamics of the human sleep-wake cycle. He completed his postdoc under Nancy Kopell at Boston University.
Career
After spending three years as a National Science Foundation Postdoctoral Fellow at Harvard and Boston University, Strogatz joined the faculty of the department of mathematics at MIT in 1989. His research on dynamical systems was recognized with a Presidential Young Investigator Award from the National Science Foundation in 1990. In 1994 he moved to Cornell where he is a professor of mathematics. From 2007 to 2023 he was the Jacob Gould Schurman Professor of Applied Mathematics, and in 2023 he was named the inaugural holder of the Susan and Barton Winokur Distinguished Professorship for the Public Understanding of Science and Mathematics. From 2004 to 2010, he was also on the external faculty of the Santa Fe Institute.
Research
Early in his career, Strogatz worked on a variety of problems in mathematical biology, including the geometry of supercoiled DNA, the topology of three-dimensional chemical waves, and the collective behavior of biological oscillators, such as swarms of synchronously flashing fireflies. In the 1990s, his work focused on nonlinear dynamics and chaos applied to physics, engineering, and biology. Several of these projects dealt with coupled oscillators, such as lasers, superconducting Josephson junctions, and crickets that chirp in unison. His more recent work examines complex systems and their consequences in everyday life, such as the role of crowd synchronization in the wobbling of London's Millennium Bridge on its opening day, and the dynamics of structural balance in social systems.
Perhaps his best-known research contribution is his 1998 Nature paper with Duncan Watts, entitled "Collective dynamics of small-world networks". This paper is widely regarded as a foundational contri
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https://en.wikipedia.org/wiki/Representation%20theorem
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In mathematics, a representation theorem is a theorem that states that every abstract structure with certain properties is isomorphic to another (abstract or concrete) structure.
Examples
Algebra
Cayley's theorem states that every group is isomorphic to a permutation group.
Representation theory studies properties of abstract groups via their representations as linear transformations of vector spaces.
Stone's representation theorem for Boolean algebras states that every Boolean algebra is isomorphic to a field of sets.
A variant, Stone's representation theorem for distributive lattices, states that every distributive lattice is isomorphic to a sublattice of the power set lattice of some set.
Another variant, Stone's duality, states that there exists a duality (in the sense of an arrow-reversing equivalence) between the categories of Boolean algebras and that of Stone spaces.
The Poincaré–Birkhoff–Witt theorem states that every Lie algebra embeds into the commutator Lie algebra of its universal enveloping algebra.
Ado's theorem states that every finite-dimensional Lie algebra over a field of characteristic zero embeds into the Lie algebra of endomorphisms of some finite-dimensional vector space.
Birkhoff's HSP theorem states that every model of an algebra A is the homomorphic image of a subalgebra of a direct product of copies of A.
In the study of semigroups, the Wagner–Preston theorem provides a representation of an inverse semigroup S, as a homomorphic image of the set of partial bijections on S, and the semigroup operation given by composition.
Category theory
The Yoneda lemma provides a full and faithful limit-preserving embedding of any category into a category of presheaves.
Mitchell's embedding theorem for abelian categories realises every small abelian category as a full (and exactly embedded) subcategory of a category of modules over some ring.
Mostowski's collapsing theorem states that every well-founded extensional structure is isomorphic to a transitive set with the ∈-relation.
One of the fundamental theorems in sheaf theory states that every sheaf over a topological space can be thought of as a sheaf of sections of some (étalé) bundle over that space: the categories of sheaves on a topological space and that of étalé spaces over it are equivalent, where the equivalence is given by the functor that sends a bundle to its sheaf of (local) sections.
Functional analysis
The Gelfand–Naimark–Segal construction embeds any C*-algebra in an algebra of bounded operators on some Hilbert space.
The Gelfand representation (also known as the commutative Gelfand–Naimark theorem) states that any commutative C*-algebra is isomorphic to an algebra of continuous functions on its Gelfand spectrum. It can also be seen as the construction as a duality between the category of commutative C*-algebras and that of compact Hausdorff spaces.
The Riesz–Markov–Kakutani representation theorem is actually a list of several theorems; one of them
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https://en.wikipedia.org/wiki/Werner%20Fenchel
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Moritz Werner Fenchel (; 3 May 1905 – 24 January 1988) was a mathematician known for his contributions to geometry and to optimization theory. Fenchel established the basic results of convex analysis and nonlinear optimization theory which would, in time, serve as the foundation for nonlinear programming. A German-born Jew and early refugee from Nazi suppression of intellectuals, Fenchel lived most of his life in Denmark. Fenchel's monographs and lecture notes are considered influential.
Biography
Early life and education
Fenchel was born on 3 May 1905 in Berlin, Germany, his younger brother was the Israeli film director and architect Heinz Fenchel.
Fenchel studied mathematics and physics at the University of Berlin between 1923 and 1928. He wrote his doctorate thesis in geometry (Über Krümmung und Windung geschlossener Raumkurven) under Ludwig Bieberbach.
Professorship in Germany
From 1928 to 1933, Fenchel was Professor E. Landau's Assistant at the University of Göttingen. During a one-year leave (on Rockefeller Fellowship) between 1930 and 1931, Fenchel spent time in Rome with Levi-Civita, as well as in Copenhagen with Harald Bohr and Tommy Bonnesen.
He visited Denmark again in 1932.
Professorship in exile
Fenchel taught at Göttingen until 1933, when the Nazi discrimination laws led to mass-firings of Jews.
Fenchel emigrated to Denmark somewhere between April and September 1933, ultimately obtaining a position at the University of Copenhagen. In December 1933, Fenchel married fellow German refugee mathematician Käte Sperling.
When Germany occupied Denmark, Fenchel and roughly eight-thousand other Danish Jews received refuge in Sweden, where he taught (between 1943 and 1945) at the Danish School in Lund. After the Allied powers' liberation of Denmark, Fenchel returned to Copenhagen.
Professorship postwar
In 1946, Fenchel was elected a member of the Royal Danish Academy of Sciences and Letters.
On leave between 1949 and 1951, Fenchel taught in the U.S. at the University of Southern California, Stanford University, and Princeton University.
From 1952 to 1956 Fenchel was the professor in mechanics at the Polytechnic in Copenhagen.
From 1956 to 1974 he was the professor in mathematics at the University of Copenhagen.
Last years, death, legacy
Professor Fenchel died on 24 January 1988.
Geometric contributions
Convex geometry
Optimization theory
Fenchel lectured on "Convex Sets, Cones, and Functions" at Princeton University in the early 1950s. His lecture notes shaped the field of convex analysis, according to the monograph Convex Analysis of R. T. Rockafellar.
Hyperbolic geometry
Books
See also
Convex analysis
Convex cone
Convex function
Convex set
Legendre–Fenchel transformation
Convex minimization
Fenchel's duality theorem
Geometry
Convex geometry
Brunn–Minkowski theorem
Differential geometry
Fenchel's theorem
Hyperbolic geometry
Jakob Nielsen
Fenchel–Nielsen coordinates
Nonlinear programming
References
External links
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https://en.wikipedia.org/wiki/Vladimir%20Smirnov%20%28mathematician%29
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Vladimir Ivanovich Smirnov () (10 June 1887 – 11 February 1974) was a mathematician who made significant contributions in both pure and applied mathematics, and also in the history of mathematics.
Smirnov worked on diverse areas of mathematics, such as complex functions and conjugate functions in Euclidean spaces. In the applied field his work includes the propagation of waves in elastic media with plane boundaries (with Sergei Sobolev) and the oscillations of elastic spheres.
His pioneering approach to solving the initial-boundary value problem to the wave equation formed the basis of the spacetime triangle diagram (STTD) technique for wave motion developed by his follower Victor Borisov (also known as the Smirnov method of incomplete separation of variables).
Smirnov was a Ph.D. student of Vladimir Steklov. Among his notable students were Sergei Sobolev, Solomon Mikhlin and Nobel prize winner Leonid Kantorovich.
Smirnov is also widely known among students for his five volume series (in seven books) A Course in Higher Mathematics (Курс высшей математики) (the first volume was written jointly with Jacob Tamarkin).
References
External links
1887 births
1974 deaths
Soviet mathematicians
Mathematicians from Saint Petersburg
Saint Petersburg State University alumni
Academic staff of Perm State University
Full Members of the USSR Academy of Sciences
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https://en.wikipedia.org/wiki/2-sided
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In mathematics, specifically in topology of manifolds, a compact codimension-one submanifold of a manifold is said to be 2-sided in when there is an embedding
with for each and
.
In other words, if its normal bundle is trivial.
This means, for example that a curve in a surface is 2-sided if it has a tubular neighborhood which is a cartesian product of the curve times an interval.
A submanifold which is not 2-sided is called 1-sided.
Examples
Surfaces
For curves on surfaces, a curve is 2-sided if and only if it preserves orientation, and 1-sided if and only if it reverses orientation: a tubular neighborhood is then a Möbius strip. This can be determined from the class of the curve in the fundamental group of the surface and the orientation character on the fundamental group, which identifies which curves reverse orientation.
An embedded circle in the plane is 2-sided.
An embedded circle generating the fundamental group of the real projective plane (such as an "equator" of the projective plane – the image of an equator for the sphere) is 1-sided, as it is orientation-reversing.
Properties
Cutting along a 2-sided manifold can separate a manifold into two pieces – such as cutting along the equator of a sphere or around the sphere on which a connected sum has been done – but need not, such as cutting along a curve on the torus.
Cutting along a (connected) 1-sided manifold does not separate a manifold, as a point that is locally on one side of the manifold can be connected to a point that is locally on the other side (i.e., just across the submanifold) by passing along an orientation-reversing path.
Cutting along a 1-sided manifold may make a non-orientable manifold orientable – such as cutting along an equator of the real projective plane – but may not, such as cutting along a 1-sided curve in a higher genus non-orientable surface,
maybe the simplest example of this is seen when one cut a mobius band along its core curve.
References
Geometric topology
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https://en.wikipedia.org/wiki/Lattice%20reduction
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In mathematics, the goal of lattice basis reduction is to find a basis with short, nearly orthogonal vectors when given an integer lattice basis as input. This is realized using different algorithms, whose running time is usually at least exponential in the dimension of the lattice.
Nearly orthogonal
One measure of nearly orthogonal is the orthogonality defect. This compares the product of the lengths of the basis vectors with the volume of the parallelepiped they define. For perfectly orthogonal basis vectors, these quantities would be the same.
Any particular basis of vectors may be represented by a matrix , whose columns are the basis vectors . In the fully dimensional case where the number of basis vectors is equal to the dimension of the space they occupy, this matrix is square, and the volume of the fundamental parallelepiped is simply the absolute value of the determinant of this matrix . If the number of vectors is less than the dimension of the underlying space, then volume is . For a given lattice , this volume is the same (up to sign) for any basis, and hence is referred to as the determinant of the lattice or lattice constant .
The orthogonality defect is the product of the basis vector lengths divided by the parallelepiped volume;
From the geometric definition it may be appreciated that with equality if and only if the basis is orthogonal.
If the lattice reduction problem is defined as finding the basis with the smallest possible defect, then the problem is NP-hard . However, there exist polynomial time algorithms to find a basis with defect
where c is some constant depending only on the number of basis vectors and the dimension of the underlying space (if different). This is a good enough solution in many practical applications.
In two dimensions
For a basis consisting of just two vectors, there is a simple and efficient method of reduction closely analogous to the Euclidean algorithm for the greatest common divisor of two integers. As with the Euclidean algorithm, the method is iterative; at each step the larger of the two vectors is reduced by adding or subtracting an integer multiple of the smaller vector.
The pseudocode of the algorithm, often known as Lagrange's algorithm or the Lagrange-Gauss algorithm, is as follows:
Input: a basis for the lattice . Assume that , otherwise swap them.
Output: A basis with .
While :
# Round to nearest integer
See the section on Lagrange's algorithm in for further details.
Applications
Lattice reduction algorithms are used in a number of modern number theoretical applications, including in the discovery of a spigot algorithm for . Although determining the shortest basis is possibly an NP-complete problem, algorithms such as the LLL algorithm can find a short (not necessarily shortest) basis in polynomial time with guaranteed worst-case performance. LLL is widely used in the cryptanalysis of public key cryptosystems.
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https://en.wikipedia.org/wiki/Pointwise
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In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value of some function An important class of pointwise concepts are the pointwise operations, that is, operations defined on functions by applying the operations to function values separately for each point in the domain of definition. Important relations can also be defined pointwise.
Pointwise operations
Formal definition
A binary operation on a set can be lifted pointwise to an operation on the set of all functions from to as follows: Given two functions and , define the function by
Commonly, o and O are denoted by the same symbol. A similar definition is used for unary operations o, and for operations of other arity.
Examples
where .
See also pointwise product, and scalar.
An example of an operation on functions which is not pointwise is convolution.
Properties
Pointwise operations inherit such properties as associativity, commutativity and distributivity from corresponding operations on the codomain.
If is some algebraic structure, the set of all functions to the carrier set of can be turned into an algebraic structure of the same type in an analogous way.
Componentwise operations
Componentwise operations are usually defined on vectors, where vectors are elements of the set for some natural number and some field . If we denote the -th component of any vector as , then componentwise addition is .
Componentwise operations can be defined on matrices. Matrix addition, where is a componentwise operation while matrix multiplication is not.
A tuple can be regarded as a function, and a vector is a tuple. Therefore, any vector corresponds to the function such that , and any componentwise operation on vectors is the pointwise operation on functions corresponding to those vectors.
Pointwise relations
In order theory it is common to define a pointwise partial order on functions. With A, B posets, the set of functions A → B can be ordered by f ≤ g if and only if (∀x ∈ A) f(x) ≤ g(x). Pointwise orders also inherit some properties of the underlying posets. For instance if A and B are continuous lattices, then so is the set of functions A → B with pointwise order. Using the pointwise order on functions one can concisely define other important notions, for instance:
A closure operator c on a poset P is a monotone and idempotent self-map on P (i.e. a projection operator) with the additional property that idA ≤ c, where id is the identity function.
Similarly, a projection operator k is called a kernel operator if and only if k ≤ idA.
An example of an infinitary pointwise relation is pointwise convergence of functions—a sequence of functions
with
converges pointwise to a function if for each in
Notes
References
For order theory examples:
T. S. Blyth, Lattices and Ordered Algebraic Structures, Springer, 2005, .
G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. Mislove, D. S. Scott: Continuous Lattic
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https://en.wikipedia.org/wiki/Numena%20%2B%20Geometry
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Numena + Geometry (1997) is an album by the American ambient musician Robert Rich. It is a two-disc set containing Rich’s albums Numena (1987) and Geometry (1991).
Track listing
Disc one: Numena
”The Other Side of Twilight” – 25:04
”Moss Dance” – 5:45
”Numen” – 11:51
”The Walled Garden” – 10:32
Disc two: Geometry
”Primes, Part 1” – 5:20
”Primes, Part 2” – 6:34
”Interlocking Circles” – 8:35
”Geometry of the Skies” – 13:48
”Nesting Ground” – 6:13
”Geomancy” – 10:35
”Amrita (Water of Life)” – 6:39
”Logos” – 9:57
External links
Hearts of Space Records Album Page
Robert Rich (musician) albums
1997 compilation albums
Hearts of Space Records albums
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https://en.wikipedia.org/wiki/Legendre%27s%20equation
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In mathematics, Legendre's equation is the Diophantine equation
The equation is named for Adrien-Marie Legendre who proved in 1785 that it is solvable in integers x, y, z, not all zero, if and only if
−bc, −ca and −ab are quadratic residues modulo a, b and c, respectively, where a, b, c are nonzero, square-free, pairwise relatively prime integers, not all positive or all negative .
References
L. E. Dickson, History of the Theory of Numbers. Vol.II: Diophantine Analysis, Chelsea Publishing, 1971, . Chap.XIII, p. 422.
J.E. Cremona and D. Rusin, "Efficient solution of rational conics", Math. Comp., 72 (2003) pp. 1417-1441.
Diophantine equations
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https://en.wikipedia.org/wiki/Meager
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Meager or Meagre may refer to:
Meagre set (also meager set) in mathematics
Mount Meager (British Columbia) in British Columbia, Canada
Mount Meager massif in British Columbia, Canada
Meager Creek, a creek in British Columbia, Canada
Meagre, Argyrosomus regius, a fish
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https://en.wikipedia.org/wiki/Non-abelian%20group
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In mathematics, and specifically in group theory, a non-abelian group, sometimes called a non-commutative group, is a group (G, ∗) in which there exists at least one pair of elements a and b of G, such that a ∗ b ≠ b ∗ a. This class of groups contrasts with the abelian groups, where all pairs of group elements commute.
Non-abelian groups are pervasive in mathematics and physics. One of the simplest examples of a non-abelian group is the dihedral group of order 6. It is the smallest finite non-abelian group. A common example from physics is the rotation group SO(3) in three dimensions (for example, rotating something 90 degrees along one axis and then 90 degrees along a different axis is not the same as doing them in reverse order).
Both discrete groups and continuous groups may be non-abelian. Most of the interesting Lie groups are non-abelian, and these play an important role in gauge theory.
See also
Associative algebra
Noncommutative geometry
Niels Henrik Abel
References
Properties of groups
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https://en.wikipedia.org/wiki/Positive%20map
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The term positive map may refer to:
Positive-definite functions in classical analysis
Choi's theorem on completely positive maps between C*-algebras (pronounced "C-star algebra")
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https://en.wikipedia.org/wiki/W-algebra
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In conformal field theory and representation theory, a W-algebra is an associative algebra that generalizes the Virasoro algebra. W-algebras were introduced by Alexander Zamolodchikov, and the name "W-algebra" comes from the fact that Zamolodchikov used the letter W for one of the elements of one of his examples.
Definition
A W-algebra is an associative algebra that is generated by the modes of a finite number of meromorphic fields , including the energy-momentum tensor . For , is a primary field of conformal dimension . The generators of the algebra are related to the meromorphic fields by the mode expansions
The commutation relations of are given by the Virasoro algebra, which is parameterized by a central charge . This number is also called the central charge of the W-algebra. The commutation relations
are equivalent to the assumption that is a primary field of dimension .
The rest of the commutation relations can in principle be determined by solving the Jacobi identities.
Given a finite set of conformal dimensions (not necessarily all distinct), the number of W-algebras generated by may be zero, one or more. The resulting W-algebras may exist for all , or only for some specific values of the central charge.
A W-algebra is called freely generated if its generators obey no other relations than the commutation relations. Most commonly studied W-algebras are freely generated, including the W(N) algebras. In this article, the sections on representation theory and correlation functions apply to freely generated W-algebras.
Constructions
While it is possible to construct W-algebras by assuming the existence of a number of meromorphic fields and solving the Jacobi identities, there also exist systematic constructions of families of W-algebras.
Drinfeld-Sokolov reduction
From a finite-dimensional Lie algebra , together with an embedding , a W-algebra may be constructed from the universal enveloping algebra of the affine Lie algebra by a kind of BRST construction.
Then the central charge of the W-algebra is a function of the level of the affine Lie algebra.
Coset construction
Given a finite-dimensional Lie algebra , together with a subalgebra , a W-algebra may be constructed from the corresponding affine Lie algebras . The fields that generate are the polynomials in the currents of and their derivatives that commute with the currents of . The central charge of is the difference of the central charges of and , which are themselves given in terms of their level by the Sugawara construction.
Commutator of a set of screenings
Given a holomorphic field with values in , and a set of vectors , a W-algebra may be defined as the set of polynomials of and its derivatives that commute with the screening charges . If the vectors are the simple roots of a Lie algebra , the resulting W-algebra coincides with an algebra that is obtained from by Drinfeld-Sokolov reduction.
The W(N) algebras
For any integer , the W(N) algebra is a W-a
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https://en.wikipedia.org/wiki/Supermathematics
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Supermathematics is the branch of mathematical physics which applies the mathematics of Lie superalgebras to the behaviour of bosons and fermions. The driving force in its formation in the 1960s and 1970s was Felix Berezin.
Objects of study include superalgebras (such as super Minkowski space and super-Poincaré algebra), superschemes, supermetrics/supersymmetry, supermanifolds, supergeometry, and supergravity, namely in the context of superstring theory.
References
"The importance of Lie algebras"; Professor Isaiah Kantor, Lund University
External links
Felix Berezin, The Life and Death of the Mastermind of Supermathematics, edited by Mikhail Shifman, World Scientific, Singapore, 2007,
Mathematical physics
Supersymmetry
Lie algebras
String theory
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https://en.wikipedia.org/wiki/Tau%20%28disambiguation%29
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Tau (Τ or τ) is the 19th letter of the Greek alphabet.
Tau may also refer to:
Mathematics
Tau (mathematical constant), a circle constant equal to (6.28318...)
Tau test in statistics (tau-a, tau-b and tau-c tests or Kendall tau rank correlation coefficient)
Tau function (disambiguation), several
Geography
Tau, Norway, a small town in Strand municipality, Rogaland county, Norway
Tău (disambiguation), two villages in Romania
Ta‘ū, an island in the Manua Island Group of American Samoa
Ta'u County, a county in American Samoa
Tau (Tongatapu), an island of the Tongatapu group in Tonga
Tau (Haapai), an island of the Haapai group in Tonga
Tau (Botswana), a village at the base of the Tswapong Hills in Botswana
Science and technology
TAU (spacecraft), a proposal to send a space probe to a thousand astronomical units from the Earth
Tau (particle), also called Tau lepton, an elementary particle in particle physics
Tau emerald, a species of dragonfly
Tau neutrino a subatomic elementary particle
Tau protein, a biochemical protein associated with microtubules
Tau, the standard astronomical abbreviation for Taurus (constellation)
Tau, a mutation in the Casein kinase 1 epsilon protein, in circadian biology
Rational Tau, a UML and SysML modeling tool
Opsanus tau, the scientific name for the oyster toadfish
Arts and media
Tau (film), a 2018 thriller film starring Maika Monroe
Tău (Negură Bunget album), a 2015 album by Romanian black metal band Negură Bunget
Tau, an alien race from Warhammer 40,000
Tau, a character in the Battle Arena Toshinden fighting game series
Pan Tau, hero of Czech films and television series
Tau Zero, a novel by Poul Anderson
Tau Volantis, a planet shown in Dead Space 3
Tau Cannon, an energy weapon from the video game Half-Life
Tau Films, an American visual effects and animation company
Colony Tau, a location from Xenoblade Chronicles 3
Education
Tama Art University
Tel Aviv University
Texila American University
Anna University of Technology, Tiruchirappalli
People
Abel Tau, South African politician
Jimmy Tau, South African football (soccer) player
Max Tau, German-Norwegian writer, editor, and publisher
Nicolae Țâu, Moldovan politician who was Foreign Minister of Moldova between 1990 and 1993
Parks Tau, South African politician who was mayor of Johannesburg from 2011-2016
Tau (rapper), Polish rapper
Devi Lal (politician), Indian politician who was Deputy Prime Minister from 1989 to 1991
Other
Tau (mythology), an evil spirit in Guaraní mythology
Tau effect, a sensory illusion relating to the perception of space
Cross of Tau, a Christian symbol so called due to its resemblance to the letter Tau
Tourist Association of Ukraine
The Greek letter representing Expiration (options)
See also
Tau function (disambiguation)
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https://en.wikipedia.org/wiki/Beltrami%E2%80%93Klein%20model
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In geometry, the Beltrami–Klein model, also called the projective model, Klein disk model, and the Cayley–Klein model, is a model of hyperbolic geometry in which points are represented by the points in the interior of the unit disk (or n-dimensional unit ball) and lines are represented by the chords, straight line segments with ideal endpoints on the boundary sphere.
The Beltrami–Klein model is named after the Italian geometer Eugenio Beltrami and the German Felix Klein while "Cayley" in Cayley–Klein model refers to the English geometer Arthur Cayley.
The Beltrami–Klein model is analogous to the gnomonic projection of spherical geometry, in that geodesics (great circles in spherical geometry) are mapped to straight lines.
This model is not conformal, meaning that angles and circles are distorted, whereas the Poincaré disk model preserves these.
In this model, lines and segments are straight Euclidean segments, whereas in the Poincaré disk model, lines are arcs that meet the boundary orthogonally.
History
This model made its first appearance for hyperbolic geometry in two memoirs of Eugenio Beltrami published in 1868, first for dimension and then for general n, these essays proved the equiconsistency of hyperbolic geometry with ordinary Euclidean geometry.
The papers of Beltrami remained little noticed until recently and the model was named after Klein ("The Klein disk model"). This happened as follows. In 1859 Arthur Cayley used the cross-ratio definition of angle due to Laguerre to show how Euclidean geometry could be defined using projective geometry. His definition of distance later became known as the Cayley metric.
In 1869, the young (twenty-year-old) Felix Klein became acquainted with Cayley's work. He recalled that in 1870 he gave a talk on the work of Cayley at the seminar of Weierstrass and he wrote:
"I finished with a question whether there might exist a connection between the ideas of Cayley and Lobachevsky. I was given the answer that these two systems were conceptually widely separated."
Later, Felix Klein realized that Cayley's ideas give rise to a projective model of the non-Euclidean plane.
As Klein puts it, "I allowed myself to be convinced by these objections and put aside this already mature idea." However, in 1871, he returned to this idea, formulated it mathematically, and published it.
Distance formula
The distance function for the Beltrami–Klein model is a Cayley–Klein metric. Given two distinct points p and q in the open unit ball, the unique straight line connecting them intersects the boundary at two ideal points, a and b, label them so that the points are, in order, a, p, q, b , so that and .
The hyperbolic distance between p and q is then:
The vertical bars indicate Euclidean distances between the points in the model, ln is the natural logarithm and the factor of one half is needed to give the model the standard curvature of −1.
When one of the points is the origin and Euclidean distance between
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https://en.wikipedia.org/wiki/Metric%20connection
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In mathematics, a metric connection is a connection in a vector bundle E equipped with a bundle metric; that is, a metric for which the inner product of any two vectors will remain the same when those vectors are parallel transported along any curve. This is equivalent to:
A connection for which the covariant derivatives of the metric on E vanish.
A principal connection on the bundle of orthonormal frames of E.
A special case of a metric connection is a Riemannian connection; there exists a unique such connection which is torsion free, the Levi-Civita connection. In this case, the bundle E is the tangent bundle TM of a manifold, and the metric on E is induced by a Riemannian metric on M.
Another special case of a metric connection is a Yang–Mills connection, which satisfies the Yang–Mills equations of motion. Most of the machinery of defining a connection and its curvature can be worked through without requiring any compatibility with the bundle metric. However, once one does require compatibility, this metric connection defines an inner product, Hodge star (which additionally needs a choice of orientation), and Laplacian, which are required to formulate the Yang–Mills equations.
Definition
Let be any local sections of the vector bundle E, and let X be a vector field on the base space M of the bundle. Let define a bundle metric, that is, a metric on the vector fibers of E. Then, a connection D on E is a metric connection if:
Here d is the ordinary differential of a scalar function. The covariant derivative can be extended so that it acts as a map on E-valued differential forms on the base space:
One defines for a function , and
where is a local smooth section for the vector bundle and is a (scalar-valued) p-form. The above definitions also apply to local smooth frames as well as local sections.
Metric versus dual pairing
The bundle metric imposed on E should not be confused with the natural pairing of a vector space and its dual, which is intrinsic to any vector bundle. The latter is a function on the bundle of endomorphisms so that
pairs vectors with dual vectors (functionals) above each point of M. That is, if is any local coordinate frame on E, then one naturally obtains a dual coordinate frame on E* satisfying .
By contrast, the bundle metric is a function on
giving an inner product on each vector space fiber of E. The bundle metric allows one to define an orthonormal coordinate frame by the equation
Given a vector bundle, it is always possible to define a bundle metric on it.
Following standard practice, one can define a connection form, the Christoffel symbols and the Riemann curvature without reference to the bundle metric, using only the pairing They will obey the usual symmetry properties; for example, the curvature tensor will be anti-symmetric in the last two indices and will satisfy the second Bianchi identity. However, to define the Hodge star, the Laplacian, the first Bianchi identity, and the Yang–
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https://en.wikipedia.org/wiki/Holomorphic%20vector%20bundle
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In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold such that the total space is a complex manifold and the projection map is holomorphic. Fundamental examples are the holomorphic tangent bundle of a complex manifold, and its dual, the holomorphic cotangent bundle. A holomorphic line bundle is a rank one holomorphic vector bundle.
By Serre's GAGA, the category of holomorphic vector bundles on a smooth complex projective variety X (viewed as a complex manifold) is equivalent to the category of algebraic vector bundles (i.e., locally free sheaves of finite rank) on X.
Definition through trivialization
Specifically, one requires that the trivialization maps
are biholomorphic maps. This is equivalent to requiring that the transition functions
are holomorphic maps. The holomorphic structure on the tangent bundle of a complex manifold is guaranteed by the remark that the derivative (in the appropriate sense) of a vector-valued holomorphic function is itself holomorphic.
The sheaf of holomorphic sections
Let be a holomorphic vector bundle. A local section is said to be holomorphic if, in a neighborhood of each point of , it is holomorphic in some (equivalently any) trivialization.
This condition is local, meaning that holomorphic sections form a sheaf on . This sheaf is sometimes denoted , or abusively by . Such a sheaf is always locally free of the same rank as the rank of the vector bundle. If is the trivial line bundle then this sheaf coincides with the structure sheaf of the complex manifold .
Basic examples
There are line bundles over whose global sections correspond to homogeneous polynomials of degree (for a positive integer). In particular, corresponds to the trivial line bundle. If we take the covering then we can find charts defined byWe can construct transition functions defined byNow, if we consider the trivial bundle we can form induced transition functions . If we use the coordinate on the fiber, then we can form transition functionsfor any integer . Each of these are associated with a line bundle . Since vector bundles necessarily pull back, any holomorphic submanifold has an associated line bundle , sometimes denoted .
Dolbeault operators
Suppose is a holomorphic vector bundle. Then there is a distinguished operator defined as follows. In a local trivialisation of , with local frame , any section may be written for some smooth functions .
Define an operator locally by
where is the regular Cauchy–Riemann operator of the base manifold. This operator is well-defined on all of because on an overlap of two trivialisations with holomorphic transition function , if where is a local frame for on , then , and so
because the transition functions are holomorphic. This leads to the following definition: A Dolbeault operator on a smooth complex vector bundle is an -linear operator
such that
(Cauchy–Riemann condition) ,
(Leibniz rule) For any section and functio
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https://en.wikipedia.org/wiki/Canonical%20connection
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In geometry (more specifically differential geometry), a canonical connection can mean either
A canonical connection on a symmetric space that is canonically defined (as described in Ch XI of Kobayashi and Nomizu, Foundations of Differential Geometry Vol II.).
A Chern connection, a connection of a holomorphic vector bundle with a Hermitian structure, is the unique metric connection D, such that the part which increases the anti-holomorphic type D`` annihilates holomorphic sections.
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https://en.wikipedia.org/wiki/Balian%E2%80%93Low%20theorem
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In mathematics, the Balian–Low theorem in Fourier analysis is named for Roger Balian and Francis E. Low.
The theorem states that there is no well-localized window function (or Gabor atom) g either in time or frequency for an exact Gabor frame (Riesz Basis).
Statement
Suppose g is a square-integrable function on the real line, and consider the so-called Gabor system
for integers m and n, and a,b>0 satisfying ab=1. The Balian–Low theorem states that if
is an orthonormal basis for the Hilbert space
then either
Generalizations
The Balian–Low theorem has been extended to exact Gabor frames.
See also
Gabor filter (in image processing)
References
Theorems in Fourier analysis
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https://en.wikipedia.org/wiki/Riesz%20sequence
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In mathematics, a sequence of vectors (xn) in a Hilbert space is called a Riesz sequence if there exist constants such that
for all sequences of scalars (an) in the ℓp space ℓ2. A Riesz sequence is called a Riesz basis if
.
Alternatively, one can define the Riesz basis as a family of the form , where is an orthonormal basis for and is a bounded bijective operator.
Paley-Wiener criterion
Let be an orthonormal basis for a Hilbert space and let be "close" to in the sense that
for some constant , , and arbitrary scalars . Then is a Riesz basis for . Hence, Riesz bases need not be orthonormal.
Theorems
If H is a finite-dimensional space, then every basis of H is a Riesz basis.
Let be in the Lp space L2(R), let
and let denote the Fourier transform of . Define constants c and C with . Then the following are equivalent:
The first of the above conditions is the definition for () to form a Riesz basis for the space it spans.
See also
Orthonormal basis
Hilbert space
Frame of a vector space
References
Functional analysis
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https://en.wikipedia.org/wiki/Megarock%20Records
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Megarock Records was a Swedish record label which focused on heavy metal music. It was located in Stockholm.
Bands
Abstrakt Algebra
Ace's High
Alien
Backyard Babies
Bad Habit
Candlemass
Criss
Crossroad Jam
It's Alive
Jester
Landberk
Machine Gun Kelly
Megaton
Misha Calvin
Nocturnal Rites
Passion Street
Pole Position
The Quill
Renegade
Schizophrenic Circus
Shadowseeds
Slam St Joan
Sphinx
Ten Foot Pole (Sweden)
Therion
Tungsten
Walk the Wire
See also
List of record labels
External links
Megarock Records
MEGAROCK @ anormus - list of all releases
Swedish record labels
Heavy metal record labels
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https://en.wikipedia.org/wiki/Bicomplex%20number
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In abstract algebra, a bicomplex number is a pair of complex numbers constructed by the Cayley–Dickson process that defines the bicomplex conjugate , and the product of two bicomplex numbers as
Then the bicomplex norm is given by
a quadratic form in the first component.
The bicomplex numbers form a commutative algebra over C of dimension two, which is isomorphic to the direct sum of algebras .
The product of two bicomplex numbers yields a quadratic form value that is the product of the individual quadratic forms of the numbers:
a verification of this property of the quadratic form of a product refers to the Brahmagupta–Fibonacci identity. This property of the quadratic form of a bicomplex number indicates that these numbers form a composition algebra. In fact, bicomplex numbers arise at the binarion level of the Cayley–Dickson construction based on with norm z2.
The general bicomplex number can be represented by the matrix , which has determinant . Thus, the composing property of the quadratic form concurs with the composing property of the determinant.
Bicomplex numbers feature two distinct imaginary units. Multiplication being associative and commutative, the product of these imaginary units must have positive one for its square. Such an element as this product has been called a hyperbolic unit.
As a real algebra
Bicomplex numbers form an algebra over C of dimension two, and since C is of dimension two over R, the bicomplex numbers are an algebra over R of dimension four. In fact the real algebra is older than the complex one; it was labelled tessarines in 1848 while the complex algebra was not introduced until 1892.
A basis for the tessarine 4-algebra over R specifies z = 1 and z = −i, giving the matrices
, which multiply according to the table given. When the identity matrix is identified with 1, then a tessarine t = w + z j .
History
The subject of multiple imaginary units was examined in the 1840s. In a long series "On quaternions, or on a new system of imaginaries in algebra" beginning in 1844 in Philosophical Magazine, William Rowan Hamilton communicated a system multiplying according to the quaternion group. In 1848 Thomas Kirkman reported on his correspondence with Arthur Cayley regarding equations on the units determining a system of hypercomplex numbers.
Tessarines
In 1848 James Cockle introduced the tessarines in a series of articles in Philosophical Magazine.
A tessarine is a hypercomplex number of the form
where
Cockle used tessarines to isolate the hyperbolic cosine series and the hyperbolic sine series in the exponential series. He also showed how zero divisors arise in tessarines, inspiring him to use the term "impossibles". The tessarines are now best known for their subalgebra of real tessarines ,
also called split-complex numbers, which express the parametrization of the unit hyperbola.
Bicomplex numbers
In a 1892 Mathematische Annalen paper, Corrado Segre introduced bicomplex numbers, which form an al
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https://en.wikipedia.org/wiki/Arithmetica%20Universalis
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Arithmetica Universalis ("Universal Arithmetic") is a mathematics text by Isaac Newton. Written in Latin, it was edited and published by William Whiston, Newton's successor as Lucasian Professor of Mathematics at the University of Cambridge. The Arithmetica was based on Newton's lecture notes.
Whiston's original edition was published in 1707. It was translated into English by Joseph Raphson, who published it in 1720 as the Universal Arithmetick. John Machin published a second Latin edition in 1722.
None of these editions credit Newton as author; Newton was unhappy with the publication of the Arithmetica, and so refused to have his name appear. In fact, when Whiston's edition was published, Newton was so upset he considered purchasing all of the copies so he could destroy them.
The Arithmetica touches on algebraic notation, arithmetic, the relationship between geometry and algebra, and the solution of equations. Newton also applied Descartes' rule of signs to imaginary roots. He also offered, without proof, a rule to determine the number of imaginary roots of polynomial equations. A rigorous proof of Newton's counting formula for equations up to and including the fifth degree was published by James Joseph Sylvester in 1864.
References
The Arithmetica Universalis from the Grace K. Babson Collection, including links to PDFs of English and Latin versions of the Arithmetica
Centre College Library information on Newton's works
External links
Arithmetica Universalis (1707), first edition
Universal Arithmetick (1720), English translation by Joseph Raphson
Arithmetica Universalis (1722), second edition
1707 books
1720 books
Mathematics books
Books by Isaac Newton
18th-century Latin books
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https://en.wikipedia.org/wiki/Simcoe%20Composite%20School
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Simcoe Composite School is a high school in Simcoe, Ontario, Canada.
More than 800 students attend this rural secondary school and courses range from English, French, Art, Music, and Mathematics to Computer Sciences, Business, Athletics, Cosmetology, Tech, World History, Civics, and Drama class. Megan Timpf, a representative for the 2008 Canadian softball team at the Olympic Games in Beijing attended this high school.
Other notable alumni include the late saxophonist Margo Davidson, one of the founding members of The Parachute Club, which achieved international success in the 1980s, and Rick Danko, the bassist of The Band. Rob Blake, the former captain of the Los Angeles Kings of the NHL and Olympic Gold Medal winner for men's ice hockey, also attended Simcoe Composite School starting in 1983 and ending around 1987. Dr. Robert Gardner emigrated to this school from Glasgow, Scotland and graduated as a member of the Class of 1956.
See also
List of high schools in Ontario
References
External links
Grand Erie District School Board
Weather Network
Educational institutions established in 1893
High schools in Norfolk County, Ontario
School buildings in Canada destroyed by arson
1893 establishments in Ontario
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https://en.wikipedia.org/wiki/Rand%20index
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The Rand index or Rand measure (named after William M. Rand) in statistics, and in particular in data clustering, is a measure of the similarity between two data clusterings. A form of the Rand index may be defined that is adjusted for the chance grouping of elements, this is the adjusted Rand index. The Rand index is the accuracy of determining if a link belongs within a cluster or not.
Rand index
Definition
Given a set of elements and two partitions of to compare, , a partition of S into r subsets, and , a partition of S into s subsets, define the following:
, the number of pairs of elements in that are in the same subset in and in the same subset in
, the number of pairs of elements in that are in different subsets in and in different subsets in
, the number of pairs of elements in that are in the same subset in and in different subsets in
, the number of pairs of elements in that are in different subsets in and in the same subset in
The Rand index, , is:
Intuitively, can be considered as the number of agreements between and and as the number of disagreements between and .
Since the denominator is the total number of pairs, the Rand index represents the frequency of occurrence
of agreements over the total pairs, or the probability that and
will agree on a randomly chosen pair.
is calculated as .
Similarly, one can also view the Rand index as a measure of the percentage of correct decisions made by the algorithm. It can be computed using the following formula:
where is the number of true positives, is the number of true negatives, is the number of false positives, and is the number of false negatives.
Properties
The Rand index has a value between 0 and 1, with 0 indicating that the two data clusterings do not agree on any pair of points and 1 indicating that the data clusterings are exactly the same.
In mathematical terms, a, b, c, d are defined as follows:
, where
, where
, where
, where
for some
Relationship with classification accuracy
The Rand index can also be viewed through the prism of binary classification accuracy over the pairs of elements in . The two class labels are " and are in the same subset in and " and " and are in different subsets in and ".
In that setting, is the number of pairs correctly labeled as belonging to the same subset (true positives), and is the number of pairs correctly labeled as belonging to different subsets (true negatives).
Adjusted Rand index
The adjusted Rand index is the corrected-for-chance version of the Rand index. Such a correction for chance establishes a baseline by using the expected similarity of all pair-wise comparisons between clusterings specified by a random model. Traditionally, the Rand Index was corrected using the Permutation Model for clusterings (the number and size of clusters within a clustering are fixed, and all random clusterings are generated by shuffling the elements between the fixed clusters). However, the premises
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https://en.wikipedia.org/wiki/Alternating%20permutation
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In combinatorial mathematics, an alternating permutation (or zigzag permutation) of the set {1, 2, 3, ..., n} is a permutation (arrangement) of those numbers so that each entry is alternately greater or less than the preceding entry. For example, the five alternating permutations of {1, 2, 3, 4} are:
1, 3, 2, 4 because 1 < 3 > 2 < 4,
1, 4, 2, 3 because 1 < 4 > 2 < 3,
2, 3, 1, 4 because 2 < 3 > 1 < 4,
2, 4, 1, 3 because 2 < 4 > 1 < 3, and
3, 4, 1, 2 because 3 < 4 > 1 < 2.
This type of permutation was first studied by Désiré André in the 19th century.
Different authors use the term alternating permutation slightly differently: some require that the second entry in an alternating permutation should be larger than the first (as in the examples above), others require that the alternation should be reversed (so that the second entry is smaller than the first, then the third larger than the second, and so on), while others call both types by the name alternating permutation.
The determination of the number An of alternating permutations of the set {1, ..., n} is called André's problem. The numbers An are known as Euler numbers, zigzag numbers, or up/down numbers. When n is even the number An is known as a secant number, while if n is odd it is known as a tangent number. These latter names come from the study of the generating function for the sequence.
Definitions
A permutation is said to be alternating if its entries alternately rise and descend. Thus, each entry other than the first and the last should be either larger or smaller than both of its neighbors. Some authors use the term alternating to refer only to the "up-down" permutations for which , calling the "down-up" permutations that satisfy by the name reverse alternating. Other authors reverse this convention, or use the word "alternating" to refer to both up-down and down-up permutations.
There is a simple one-to-one correspondence between the down-up and up-down permutations: replacing each entry with reverses the relative order of the entries.
By convention, in any naming scheme the unique permutations of length 0 (the permutation of the empty set) and 1 (the permutation consisting of a single entry 1) are taken to be alternating.
André's theorem
The determination of the number An of alternating permutations of the set {1, ..., n} is called André's problem. The numbers An are variously known as Euler numbers, zigzag numbers, up/down numbers, or by some combinations of these names. The name Euler numbers in particular is sometimes used for a closely related sequence. The first few values of An are 1, 1, 1, 2, 5, 16, 61, 272, 1385, 7936, 50521, ... .
These numbers satisfy a simple recurrence, similar to that of the Catalan numbers: by splitting the set of alternating permutations (both down-up and up-down) of the set { 1, 2, 3, ..., n, n + 1 } according to the position k of the largest entry , one can s
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https://en.wikipedia.org/wiki/Eduard%20Heis
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Eduard Heis (18 February 1806, Cologne – 30 June 1877 in Münster) was a German mathematician and astronomer.
He completed his education at the University of Bonn in 1827, then taught mathematics at a school in Cologne. In 1832 he taught at Aachen, and remained there until 1852. He was then appointed by King Frederick William IV to a chair position at the Academy of Münster in 1852. In 1869 he became rector of the Academy.
While at the academy he made a series of observations of the night sky, including the Milky Way, zodiacal light, stars, and shooting stars. These were published in the following works, among others:
Atlas Coelestis Novus, Cologne, 1872.
Zodiakal-Beobachtungen.
Sternschnuppen-Beobachtungen.
De Magnitudine, 1852.
His star atlas, which was based on Argelander's Uranometria Nova (1843), helped define the selection of constellations in the northern sky that was officially adopted by the International Astronomical Union in 1922. His other publications included a treatise on the eclipses during the Peloponnesian war, Halley's comet, and some mathematical text books.
He was also the first person to record a count of the Perseid meteor shower in 1839, giving an hourly rate of 160. Observers have recorded the hourly count every year since that time.
Awards and honors
Order of the Red Eagle, 1870.
Awarded doctor honoris causa by Bonn University, 1852.
Foreign associate, Royal Astronomical Society of London, 1874.
Honorary member, Leopoldine Academy, 1877.
Honorary member, Scientific Society of Brussels, 1877.
The crater Heis on the Moon is named after him.
References
External links
Atlas Coelestis Eclipticus, Coloniae ad Rhenum, 1878 da www.atlascoelestis.com
The Atlas Coelestis Novus of Eduard Heis
1806 births
1877 deaths
19th-century German astronomers
German Roman Catholics
19th-century German mathematicians
Scientists from Cologne
Scientists from the Rhine Province
University of Bonn alumni
Academic staff of the University of Münster
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https://en.wikipedia.org/wiki/Primary%20cyclic%20group
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In mathematics, a primary cyclic group is a group that is both a cyclic group and a p-primary group for some prime number p.
That is, it is a cyclic group of order p, C, for some prime number p, and natural number m.
Every finite abelian group G may be written as a finite direct sum of primary cyclic groups, as stated in the fundamental theorem of finite abelian groups:
This expression is essentially unique: there is a bijection between the sets of groups in two such expressions, which maps each group to one that is isomorphic.
Primary cyclic groups are characterised among finitely generated abelian groups as the torsion groups that cannot be expressed as a direct sum of two non-trivial groups. As such they, along with the group of integers, form the building blocks of finitely generated abelian groups.
The subgroups of a primary cyclic group are linearly ordered by inclusion. The only other groups that have this property are the quasicyclic groups.
Finite groups
Abelian group theory
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https://en.wikipedia.org/wiki/Records%20and%20statistics%20of%20the%20Rugby%20World%20Cup
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Rugby World Cup records have been accumulating since the first Rugby World Cup tournament was held in 1987.
Team records
Titles
Title win rate
Most finals
Most semi-finals
Most quarter-finals
Most appearances
10 teams appeared in every World Cup:
/Western Samoa also qualified for every World Cup but was not invited to the 1987 Rugby World Cup.
was banned from competing in 1987 and 1991 due to the sporting boycott of South Africa but appeared in every World Cup since the ban was lifted. South Africa is the only nation to win all World Cup finals they played.
Points
<small>Last updated: 28 October 2023</small>
Margins
Tries
Player records
Points
Key: App = Appearances. Con = conversions. Pen = penalties. Drop = drop goals.
Tries Youngest try scorer in a World Cup gameGeorge North (), aged (2 tries v , 26 September 2011)Oldest try scorer in a World Cup gameDiego Ormaechea (), aged (v , 2 October 1999)
Conversions
Penalty goals
Drop goals
Appearance statistics Oldest player to appear in a World Cup matchDiego Ormaechea, , aged (v , 15 October 1999)For the specific match where Ormaeches established the current record, see the list of his Test matches at ESPN Scrum.Oldest player to appear in a World Cup finalDuane Vermeulen, , aged (v , 28 October 2023)Oldest player to win a World Cup finalSchalk Brits, , aged (v , 2 November 2019)Youngest player to appear in a World Cup matchVasil Lobzhanidze, , aged (v , 19 September 2015)Youngest player to appear in a World Cup finalJonah Lomu, , aged (v , 24 June 1995)Youngest player to win a World Cup finalFrançois Steyn, , aged (v , 20 October 2007)
By tournament
Note: *''' denotes an all-time record
Miscellaneous
Winning coaches and captains
A foreign coach has never managed a World Cup-winning team.
Draws
Nil points
Highest attendance
89,267 – v , 27 September 2015 at Wembley Stadium, London, England, 2015.
Lowest attendance
2,000 – v , 28 May 1987 at Lancaster Park, Christchurch, New Zealand, 1987.
Hosting
Eden Park in Auckland Park was the first stadium to host the Rugby World Cup Final twice, with the 1987 and 2011 finals having been held there. Twickenham Stadium also hosted the final twice in 1991 and 2015.
The record for the city that has been a part of most Rugby World Cups is currently four and is held by Cardiff that hosted matches in 1991, 1999, 2007 and 2015. If the definition of "city" includes its metropolitan area, Paris has also hosted matches in four tournaments. The city of Paris hosted matches in 1991, its adjacent suburb of Saint-Denis hosted matches in 1999 and 2023, and both cities hosted matches in 2007. Edinburgh and Toulouse hosted matches in three tournaments.
Head-to-Head
The highest number of Head-to-Head matches between two nations currently stands at eight meetings, encompassing four teams (Australia, France, New Zealand, and Wales) in two Rugby World Cup rivalries. On the other end of the table, there are currently 7
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https://en.wikipedia.org/wiki/Grassmann%E2%80%93Cayley%20algebra
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In mathematics, a Grassmann–Cayley algebra is the exterior algebra with an additional product, which may be called the shuffle product or the regressive product.
It is the most general structure in which projective properties are expressed in a coordinate-free way.
The technique is based on work by German mathematician Hermann Grassmann on exterior algebra, and subsequently by British mathematician Arthur Cayley's work on matrices and linear algebra.
It is a form of modeling algebra for use in projective geometry.
The technique uses subspaces as basic elements of computation, a formalism which allows the translation of synthetic geometric statements into invariant algebraic statements. This can create a useful framework for the modeling of conics and quadrics among other forms, and in tensor mathematics. It also has a number of applications in robotics, particularly for the kinematical analysis of manipulators.
References
External links
Geometric Algebra FAQ
Multilinear algebra
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https://en.wikipedia.org/wiki/Metatheory
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A metatheory or meta-theory is a theory the subject matter of which is theory itself, for example as an analysis or description of existing theory. For mathematics and mathematical logic, a metatheory is a mathematical theory about another mathematical theory. Meta-theoretical investigations are part of the philosophy of science. The topic of metascience is an attempt to use scientific knowledge to improve the practice of science itself.
The study of metatheory became widespread during the 20th century after its application to various topics, including scientific linguistics and its concept of metalanguage.
Examples of metatheories
Metascience
Metascience is the use of scientific method to study science itself. Metascience is an attempt to increase the quality of scientific research while reducing wasted activity; it uses research methods to study how research is done or can be improved. It has been described as "research on research", "the science of science", and "a bird's eye view of science". In the words of John Ioannidis, "Science is the best thing that has happened to human beings ... but we can do it better."
In 1966, an early meta-research paper examined the statistical methods of 295 papers published in ten well-known medical journals. It found that, "in almost 73% of the reports read ... conclusions were drawn when the justification for these conclusions was invalid". Meta-research during the ensuing decades found many methodological flaws, inefficiencies, and bad practices in the research of numerous scientific topics. Many scientific studies could not be reproduced, particularly those involving medicine and the so-called soft sciences. The term "replication crisis" was invented during the early 2010s as part of an increasing awareness of the problem.
Measures have been implemented to address the issues revealed by metascience. These measures include the pre-registration of scientific studies and clinical trials as well as the founding of organizations such as CONSORT and the EQUATOR Network that issue guidelines for methods and reporting. There are continuing efforts to reduce the misuse of statistics, to eliminate perverse incentives from academia, to improve the peer review process, to reduce bias in scientific literature, and to increase the overall quality and efficiency of the scientific process.
A major criticism of metatheory is that it is theory based on other theory.
Metamathematics
Introduced in 20th-century philosophy as a result of the work of the German mathematician David Hilbert, who in 1905 published a proposal for proof of the consistency and completeness of mathematics, creating the topic of metamathematics. His hopes for the success of this proof were disappointed by the work of Kurt Gödel, who in 1931, used his incompleteness theorems to prove the goal of consistency and completeness to be unattainable. Nevertheless, his program of unsolved mathematical problems influenced mathematics for the rest of the
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https://en.wikipedia.org/wiki/Lacunary%20value
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In complex analysis, a subfield of mathematics, a lacunary value or gap of a complex-valued function defined on a subset of the complex plane is a complex number which is not in the image of the function.
More specifically, given a subset X of the complex plane C and a function f : X → C, a complex number z is called a lacunary value of f if z ∉ image(f).
Note, for example, that 0 is the only lacunary value of the complex exponential function. The two Picard theorems limit the number of possible lacunary values of certain types of holomorphic functions.
References
Complex analysis
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