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https://en.wikipedia.org/wiki/Cofinal%20%28mathematics%29
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In mathematics, a subset of a preordered set is said to be cofinal or frequent in if for every it is possible to find an element in that is "larger than " (explicitly, "larger than " means ).
Cofinal subsets are very important in the theory of directed sets and nets, where “cofinal subnet” is the appropriate generalization of "subsequence". They are also important in order theory, including the theory of cardinal numbers, where the minimum possible cardinality of a cofinal subset of is referred to as the cofinality of
Definitions
Let be a homogeneous binary relation on a set
A subset is said to be or with respect to if it satisfies the following condition:
For every there exists some that
A subset that is not frequent is called .
This definition is most commonly applied when is a directed set, which is a preordered set with additional properties.
Final functions
A map between two directed sets is said to be if the image of is a cofinal subset of
Coinitial subsets
A subset is said to be (or in the sense of forcing) if it satisfies the following condition:
For every there exists some such that
This is the order-theoretic dual to the notion of cofinal subset.
Cofinal (respectively coinitial) subsets are precisely the dense sets with respect to the right (respectively left) order topology.
Properties
The cofinal relation over partially ordered sets ("posets") is reflexive: every poset is cofinal in itself. It is also transitive: if is a cofinal subset of a poset and is a cofinal subset of (with the partial ordering of applied to ), then is also a cofinal subset of
For a partially ordered set with maximal elements, every cofinal subset must contain all maximal elements, otherwise a maximal element that is not in the subset would fail to be any element of the subset, violating the definition of cofinal. For a partially ordered set with a greatest element, a subset is cofinal if and only if it contains that greatest element (this follows, since a greatest element is necessarily a maximal element). Partially ordered sets without greatest element or maximal elements admit disjoint cofinal subsets. For example, the even and odd natural numbers form disjoint cofinal subsets of the set of all natural numbers.
If a partially ordered set admits a totally ordered cofinal subset, then we can find a subset that is well-ordered and cofinal in
If is a directed set and if is a cofinal subset of then is also a directed set.
Examples and sufficient conditions
Any superset of a cofinal subset is itself cofinal.
If is a directed set and if some union of (one or more) finitely many subsets is cofinal then at least one of the set is cofinal. This property is not true in general without the hypothesis that is directed.
Subset relations and neighborhood bases
Let be a topological space and let denote the neighborhood filter at a point
The superset relation is a partial order on : explic
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https://en.wikipedia.org/wiki/Hensel%27s%20lemma
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In mathematics, Hensel's lemma, also known as Hensel's lifting lemma, named after Kurt Hensel, is a result in modular arithmetic, stating that if a univariate polynomial has a simple root modulo a prime number , then this root can be lifted to a unique root modulo any higher power of . More generally, if a polynomial factors modulo into two coprime polynomials, this factorization can be lifted to a factorization modulo any higher power of (the case of roots corresponds to the case of degree for one of the factors).
By passing to the "limit" (in fact this is an inverse limit) when the power of tends to infinity, it follows that a root or a factorization modulo can be lifted to a root or a factorization over the -adic integers.
These results have been widely generalized, under the same name, to the case of polynomials over an arbitrary commutative ring, where is replaced by an ideal, and "coprime polynomials" means "polynomials that generate an ideal containing ".
Hensel's lemma is fundamental in -adic analysis, a branch of analytic number theory.
The proof of Hensel's lemma is constructive, and leads to an efficient algorithm for Hensel lifting, which is fundamental for factoring polynomials, and gives the most efficient known algorithm for exact linear algebra over the rational numbers.
Modular reduction and lifting
Hensel's original lemma concerns the relation between polynomial factorization over the integers and over the integers modulo a prime number and its powers. It can be straightforwardly extended to the case where the integers are replaced by any commutative ring, and is replaced by any maximal ideal (indeed, the maximal ideals of have the form where is a prime number).
Making this precise requires a generalization of the usual modular arithmetic, and so it is useful to define accurately the terminology that is commonly used in this context.
Let be a commutative ring, and an ideal of . Reduction modulo refers to the replacement of every element of by its image under the canonical map For example, if is a polynomial with coefficients in , its reduction modulo , denoted is the polynomial in obtained by replacing the coefficients of by their image in Two polynomials and in are congruent modulo , denoted if they have the same coefficients modulo , that is if If a factorization of modulo consists in two (or more) polynomials in such that
The lifting process is the inverse of reduction. That is, given objects depending on elements of the lifting process replaces these elements by elements of (or of for some ) that maps to them in a way that keeps the properties of the objects.
For example, given a polynomial and a factorization modulo expressed as lifting this factorization modulo consists of finding polynomials such that and Hensel's lemma asserts that such a lifting is always possible under mild conditions; see next section.
Statement
Originally, Hensel's lemma was stated (and proved)
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https://en.wikipedia.org/wiki/Richard%20P.%20Stanley
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Richard Peter Stanley (born June 23, 1944) is an Emeritus Professor of Mathematics at the Massachusetts Institute of Technology, in Cambridge, Massachusetts. From 2000 to 2010, he was the Norman Levinson Professor of Applied Mathematics. He received his Ph.D. at Harvard University in 1971 under the supervision of Gian-Carlo Rota. He is an expert in the field of combinatorics and its applications to other mathematical disciplines.
Contributions
Stanley is known for his two-volume book Enumerative Combinatorics (1986–1999). He is also the author of Combinatorics and Commutative Algebra (1983) and well over 200 research articles in mathematics. He has served as thesis advisor to 60 doctoral students, many of whom have had distinguished careers in combinatorial research. Donald Knuth named Stanley as one of his combinatorial heroes in a 2023 interview.
Awards and honors
Stanley's distinctions include membership in the National Academy of Sciences (elected in 1995), the 2001 Leroy P. Steele Prize for Mathematical Exposition, the 2003 Schock Prize, a plenary lecture at the International Congress of Mathematicians (in Madrid, Spain), and election in 2012 as a fellow of the American Mathematical Society. In 2022 he was awarded the Leroy P. Steele Prize for Lifetime Achievement.
Selected publications
Stanley, Richard P. (1996). Combinatorics and Commutative Algebra, 2nd ed. .
Stanley, Richard P. (1997, 1999). Enumerative Combinatorics, Volumes 1 and 2. Cambridge University Press. , 0-521-56069-1.
See also
Exponential formula
Order polynomial
Stanley decomposition
Stanley's reciprocity theorem
References
External links
Richard Stanley's Homepage
1944 births
Living people
Members of the United States National Academy of Sciences
Fellows of the American Mathematical Society
20th-century American mathematicians
21st-century American mathematicians
Combinatorialists
Harvard University alumni
Massachusetts Institute of Technology School of Science faculty
Rolf Schock Prize laureates
Educators from New York City
Mathematicians from New York (state)
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https://en.wikipedia.org/wiki/Grothendieck%20group
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In mathematics, the Grothendieck group, or group of differences, of a commutative monoid is a certain abelian group. This abelian group is constructed from in the most universal way, in the sense that any abelian group containing a homomorphic image of will also contain a homomorphic image of the Grothendieck group of . The Grothendieck group construction takes its name from a specific case in category theory, introduced by Alexander Grothendieck in his proof of the Grothendieck–Riemann–Roch theorem, which resulted in the development of K-theory. This specific case is the monoid of isomorphism classes of objects of an abelian category, with the direct sum as its operation.
Grothendieck group of a commutative monoid
Motivation
Given a commutative monoid , "the most general" abelian group that arises from is to be constructed by introducing inverse elements to all elements of . Such an abelian group always exists; it is called the Grothendieck group of . It is characterized by a certain universal property and can also be concretely constructed from .
If does not have the cancellation property (that is, there exists and in such that and ), then the Grothendieck group cannot contain . In particular, in the case of a monoid operation denoted multiplicatively that has a zero element satisfying for every the Grothendieck group must be the trivial group (group with only one element), since one must have
for every .
Universal property
Let M be a commutative monoid. Its Grothendieck group is an abelian group K with a monoid homomorphism satisfying the following universal property: for any monoid homomorphism from M to an abelian group A, there is a unique group homomorphism such that
This expresses the fact that any abelian group A that contains a homomorphic image of M will also contain a homomorphic image of K, K being the "most general" abelian group containing a homomorphic image of M.
Explicit constructions
To construct the Grothendieck group K of a commutative monoid M, one forms the Cartesian product . The two coordinates are meant to represent a positive part and a negative part, so corresponds to in K.
Addition on is defined coordinate-wise:
.
Next one defines an equivalence relation on , such that is equivalent to if, for some element k of M, m1 + n2 + k = m2 + n1 + k (the element k is necessary because the cancellation law does not hold in all monoids). The equivalence class of the element (m1, m2) is denoted by [(m1, m2)]. One defines K to be the set of equivalence classes. Since the addition operation on M × M is compatible with our equivalence relation, one obtains an addition on K, and K becomes an abelian group. The identity element of K is [(0, 0)], and the inverse of [(m1, m2)] is [(m2, m1)]. The homomorphism sends the element m to [(m, 0)].
Alternatively, the Grothendieck group K of M can also be constructed using generators and relations: denoting by the free abelian group generated by the set M, th
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https://en.wikipedia.org/wiki/Semiset
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In set theory, a semiset is a proper class that is a subclass of a set. In the typical foundations of Zermelo–Fraenkel set theory, semisets are impossible due to the axiom schema of specification.
The theory of semisets was proposed and developed by Czech mathematicians Petr Vopěnka and Petr Hájek (1972). It is based on a modification of the von Neumann–Bernays–Gödel set theory; in standard NBG, the existence of semisets is precluded by the axiom of separation.
The concept of semisets opens the way for a formulation of an alternative set theory.
In particular, Vopěnka's Alternative Set Theory (1979) axiomatizes the concept of semiset, supplemented with several additional principles.
Semisets can be used to represent sets with imprecise boundaries. Novák (1984) studied approximation of semisets by fuzzy sets, which are often more suitable for practical applications of the modeling of imprecision.
Vopěnka's alternative set theory
Vopěnka's "Alternative Set Theory" builds on some ideas of the theory of semisets, but also introduces more radical changes: for example, all sets are "formally" finite, which means that sets in AST satisfy the law of mathematical induction for set-formulas (more precisely: the part of AST that consists of axioms related to sets only is equivalent to the Zermelo–Fraenkel (or ZF) set theory, in which the axiom of infinity is replaced by its negation). However, some of these sets contain subclasses that are not sets, which makes them different from Cantor (ZF) finite sets and they are called infinite in AST.
References
Vopěnka, P., and Hájek, P. The Theory of Semisets. Amsterdam: North-Holland, 1972.
Vopěnka, P. Mathematics in the Alternative Set Theory. Teubner, Leipzig, 1979.
Holmes, M.R. Alternative Axiomatic Set Theories, §9.2, Vopenka's alternative set theory. In E. N. Zalta (ed.): The Stanford Encyclopedia of Philosophy (Fall 2014 Edition).
Novák, V. "Fuzzy sets—the approximation of semisets." Fuzzy Sets and Systems 14 (1984): 259–272.
Proceedings of the 1st Symposium Mathematics in the Alternative Set Theory. JSMF, Bratislava, 1989.
Systems of set theory
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https://en.wikipedia.org/wiki/Radical%20of%20an%20integer
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In number theory, the radical of a positive integer n is defined as the product of the distinct prime numbers dividing n. Each prime factor of n occurs exactly once as a factor of this product:
The radical plays a central role in the statement of the abc conjecture.
Examples
Radical numbers for the first few positive integers are
1, 2, 3, 2, 5, 6, 7, 2, 3, 10, 11, 6, 13, 14, 15, 2, 17, 6, 19, 10, 21, 22, 23, 6, 5, 26, 3, 14, 29, 30, 31, 2, 33, 34, 35, 6, 37, 38, 39, 10, 41, 42, 43, 22, 15, 46, 47, 6, 7, 10, ... .
For example,
and therefore
Properties
The function is multiplicative (but not completely multiplicative).
The radical of any integer is the largest square-free divisor of and so also described as the square-free kernel of . There is no known polynomial-time algorithm for computing the square-free part of an integer.
The definition is generalized to the largest -free divisor of , , which are multiplicative functions which act on prime powers as
The cases and are tabulated in and .
The notion of the radical occurs in the abc conjecture, which states that, for any , there exists a finite such that, for all triples of coprime positive integers , , and satisfying ,
For any integer , the nilpotent elements of the finite ring are all of the multiples of .
References
Multiplicative functions
de:Zahlentheoretische Funktion#Multiplikative Funktionen
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https://en.wikipedia.org/wiki/List%20of%20alternative%20set%20theories
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In mathematical logic, an alternative set theory is any of the alternative mathematical approaches to the concept of set and any alternative to the de facto standard set theory described in axiomatic set theory by the axioms of Zermelo–Fraenkel set theory.
Alternative set theories
Alternative set theories include:
Vopěnka's alternative set theory
Von Neumann–Bernays–Gödel set theory
Morse–Kelley set theory
Tarski–Grothendieck set theory
Ackermann set theory
Type theory
New Foundations
Positive set theory
Internal set theory
Naive set theory
S (set theory)
Kripke–Platek set theory
Scott–Potter set theory
Constructive set theory
Zermelo set theory
General set theory
See also
Non-well-founded set theory
Notes
Systems of set theory
Mathematics-related lists
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https://en.wikipedia.org/wiki/Quadratic%20assignment%20problem
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The quadratic assignment problem (QAP) is one of the fundamental combinatorial optimization problems in the branch of optimization or operations research in mathematics, from the category of the facilities location problems first introduced by Koopmans and Beckmann.
The problem models the following real-life problem:
There are a set of n facilities and a set of n locations. For each pair of locations, a distance is specified and for each pair of facilities a weight or flow is specified (e.g., the amount of supplies transported between the two facilities). The problem is to assign all facilities to different locations with the goal of minimizing the sum of the distances multiplied by the corresponding flows.
Intuitively, the cost function encourages facilities with high flows between each other to be placed close together.
The problem statement resembles that of the assignment problem, except that the cost function is expressed in terms of quadratic inequalities, hence the name.
Formal mathematical definition
The formal definition of the quadratic assignment problem is as follows:
Given two sets, P ("facilities") and L ("locations"), of equal size, together with a weight function w : P × P → R and a distance function d : L × L → R. Find the bijection f : P → L ("assignment") such that the cost function:
is minimized.
Usually weight and distance functions are viewed as square real-valued matrices, so that the cost function is written down as:
In matrix notation:
where is the set of permutation matrices, is the weight matrix and is the distance matrix.
Computational complexity
The problem is NP-hard, so there is no known algorithm for solving this problem in polynomial time, and even small instances may require long computation time. It was also proven that the problem does not have an approximation algorithm running in polynomial time for any (constant) factor, unless P = NP. The travelling salesman problem (TSP) may be seen as a special case of QAP if one assumes that the flows connect all facilities only along a single ring, all flows have the same non-zero (constant) value and all distances are equal to the respective distances of the TSP instance. Many other problems of standard combinatorial optimization problems may be written in this form.
Applications
In addition to the original plant location formulation, QAP is a mathematical model for the problem of placement of interconnected electronic components onto a printed circuit board or on a microchip, which is part of the place and route stage of computer aided design in the electronics industry.
See also
Quadratic bottleneck assignment problem
References
Notes
Sources
A2.5: ND43, pg.218.
External links
https://www.opt.math.tugraz.at/qaplib/ QAPLIB - A Quadratic Assignment Problem Library
http://www.wiomax.com/team/xie/maos-qap-quadratic-assignment-problem-project-portal/ MAOS-QAP - Java-based Quadratic Assignment Problem Solver
https://CRAN.R-project.org/pack
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https://en.wikipedia.org/wiki/Bernstein%E2%80%93Sato%20polynomial
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In mathematics, the Bernstein–Sato polynomial is a polynomial related to differential operators, introduced independently by and , . It is also known as the b-function, the b-polynomial, and the Bernstein polynomial, though it is not related to the Bernstein polynomials used in approximation theory. It has applications to singularity theory, monodromy theory, and quantum field theory.
gives an elementary introduction, while and give more advanced accounts.
Definition and properties
If is a polynomial in several variables, then there is a non-zero polynomial and a differential operator with polynomial coefficients such that
The Bernstein–Sato polynomial is the monic polynomial of smallest degree amongst such polynomials . Its existence can be shown using the notion of holonomic D-modules.
proved that all roots of the Bernstein–Sato polynomial are negative rational numbers.
The Bernstein–Sato polynomial can also be defined for products of powers of several polynomials . In this case it is a product of linear factors with rational coefficients.
generalized the Bernstein–Sato polynomial to arbitrary varieties.
Note, that the Bernstein–Sato polynomial can be computed algorithmically. However, such computations are hard in general. There are implementations of related algorithms in computer algebra systems RISA/Asir, Macaulay2, and SINGULAR.
presented algorithms to compute the Bernstein–Sato polynomial of an affine variety together with an implementation in the computer algebra system SINGULAR.
described some of the algorithms for computing Bernstein–Sato polynomials by computer.
Examples
If then
so the Bernstein–Sato polynomial is
If then
so
The Bernstein–Sato polynomial of x2 + y3 is
If tij are n2 variables, then the Bernstein–Sato polynomial of det(tij) is given by
which follows from
where Ω is Cayley's omega process, which in turn follows from the Capelli identity.
Applications
If is a non-negative polynomial then , initially defined for s with non-negative real part, can be analytically continued to a meromorphic distribution-valued function of s by repeatedly using the functional equation
It may have poles whenever b(s + n) is zero for a non-negative integer n.
If f(x) is a polynomial, not identically zero, then it has an inverse g that is a distribution; in other words, f g = 1 as distributions. If f(x) is non-negative the inverse can be constructed using the Bernstein–Sato polynomial by taking the constant term of the Laurent expansion of f(x)s at s = −1. For arbitrary f(x) just take times the inverse of
The Malgrange–Ehrenpreis theorem states that every differential operator with constant coefficients has a Green's function. By taking Fourier transforms this follows from the fact that every polynomial has a distributional inverse, which is proved in the paragraph above.
showed how to use the Bernstein polynomial to define dimensional regularization rigorously, in the massive Euclidean case.
T
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https://en.wikipedia.org/wiki/Karl%20Mahlburg
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Karl Mahlburg is an American mathematician whose research interests lie in the areas of modular forms, partitions, combinatorics and number theory. He is the author of over 40 peer-reviewed journal articles. Mahlburg received his PhD in 2006 from the University of Wisconsin–Madison. Between 2011 and 2021 he was an assistant professor and an associate professor of mathematics at Louisiana State University.
Work
Mahlburg's paper Partition Congruences and the Andrews-Garvan-Dyson Crank, published in 2005 in the Proceedings of the National Academy of Sciences (PNAS), was deemed an important advance in the theory of integer partitions. In the paper, Mahlburg proved a conjecture made earlier by Ken Ono regarding a family of congruences satisfied by a mathematical function known as the crank, extending a line of results dating back to the work of Srinivasa Ramanujan on partition congruences. A commentary on Mahlburg's paper, written by George Andrews and Ken Ono and published concurrently with the paper, states: "The achievement of Karl Mahlburg in this issue of PNAS adds a lustrous chapter to a unique mathematical object: the crank. [...] The story of the crank is a long romantic tale, one that [...] has now reached a satisfying and unexpected conclusion with the work of Mahlburg." Mahlburg's paper received the PNAS Paper of the Year prize, being selected from among 3,000 papers published in the journal in 2005.
References
External links
Mahlburg makes math history
Article from PhysOrg.com
Mathematician untangles legendary problem
Classic maths puzzle cracked at last. New Scientist, March 21, 2005
Year of birth missing (living people)
Living people
21st-century American mathematicians
Combinatorialists
Number theorists
Harvey Mudd College alumni
University of Wisconsin–Madison College of Letters and Science alumni
Louisiana State University faculty
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https://en.wikipedia.org/wiki/Flanders%20Mathematics%20Olympiad
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The Flanders Mathematics Olympiad (; VWO) is a Flemish mathematics competition for students in grades 9 through 12. Two tiers of this competition exist: one for 9th- and 10th-graders (; JWO), and one for 11th- and 12th-graders. It is a feeder competition for the International Mathematical Olympiad.
History
The Olympiad was founded in 1985, replacing a system previously used since 1969 in which Flemish students were nominated to the IMO by their teachers. , 20,000 students participate annually.
In 2015, the founders of the Olympiad, Paul Igodt of the Katholieke Universiteit Leuven and Frank De Clerck of Ghent University, were given the career award for science communication of the Royal Flemish Academy of Belgium for Science and the Arts for their work.
Procedure
The competition lasts three rounds. During the first and second rounds, students must answer 30 multiple-choice mathematics problems. The first round occurs in schools, and the second round is organized by province, and is administered at various universities. The first round has a three-hour time limit for completion, the second round has a two-hour time limit.
The final round consists of four problems which require a detailed and coherent essay-type response. After the final round, three contestants are selected to compete in the International Mathematical Olympiad, making up half of the team from Belgium; the other half of the team comes from Wallonia.
References
External links
Official site (in Dutch)
International Mathematical Olympiad
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https://en.wikipedia.org/wiki/VWO
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VWO may refer to:
Vlaamse Wiskunde Olympiade, a Flemish mathematics competition
Voorbereidend wetenschappelijk onderwijs, a Dutch school system
Voluntary welfare organisation, charitable organisation
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https://en.wikipedia.org/wiki/Geometric%20invariant%20theory
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In mathematics, geometric invariant theory (or GIT) is a method for constructing quotients by group actions in algebraic geometry, used to construct moduli spaces. It was developed by David Mumford in 1965, using ideas from the paper in classical invariant theory.
Geometric invariant theory studies an action of a group on an algebraic variety (or scheme) and provides techniques for forming the 'quotient' of by as a scheme with reasonable properties. One motivation was to construct moduli spaces in algebraic geometry as quotients of schemes parametrizing marked objects. In the 1970s and 1980s the theory developed interactions with symplectic geometry and equivariant topology, and was used to construct moduli spaces of objects in differential geometry, such as instantons and monopoles.
Background
Invariant theory is concerned with a group action of a group on an algebraic variety (or a scheme) . Classical invariant theory addresses the situation when is a vector space and is either a finite group, or one of the classical Lie groups that acts linearly on . This action induces a linear action of on the space of polynomial functions on by the formula
The polynomial invariants of the -action on are those polynomial functions on which are fixed under the 'change of variables' due to the action of the group, so that for all in . They form a commutative algebra , and this algebra is interpreted as the algebra of functions on the 'invariant theory quotient' because any one of these functions gives the same value for all points that are equivalent (that is, for all ). In the language of modern algebraic geometry,
Several difficulties emerge from this description. The first one, successfully tackled by Hilbert in the case of a general linear group, is to prove that the algebra is finitely generated. This is necessary if one wanted the quotient to be an affine algebraic variety. Whether a similar fact holds for arbitrary groups was the subject of Hilbert's fourteenth problem, and Nagata demonstrated that the answer was negative in general. On the other hand, in the course of development of representation theory in the first half of the twentieth century, a large class of groups for which the answer is positive was identified; these are called reductive groups and include all finite groups and all classical groups.
The finite generation of the algebra is but the first step towards the complete description of , and progress in resolving this more delicate question was rather modest. The invariants had classically been described only in a restricted range of situations, and the complexity of this description beyond the first few cases held out little hope for full understanding of the algebras of invariants in general. Furthermore, it may happen that any polynomial invariant takes the same value on a given pair of points and in , yet these points are in different orbits of the -action. A simple example is provided by the m
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https://en.wikipedia.org/wiki/Subquotient
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In the mathematical fields of category theory and abstract algebra, a subquotient is a quotient object of a subobject. Subquotients are particularly important in abelian categories, and in group theory, where they are also known as sections, though this conflicts with a different meaning in category theory.
In the literature about sporadic groups wordings like " is involved in " can be found with the apparent meaning of " is a subquotient of ."
A quotient of a subrepresentation of a representation (of, say, a group) might be called a subquotient representation; e.g., Harish-Chandra's subquotient theorem.
Examples
Of the 26 sporadic groups, the 20 subquotients of the monster group are referred to as the "Happy Family", whereas the remaining 6 are called "pariah groups."
Order relation
The relation subquotient of is an order relation.
Proof of transitivity for groups
Notation
For group , subgroup of and normal subgroup of the quotient group is a subquotient of
Let be subquotient of , furthermore be subquotient of and be the canonical homomorphism. Then all vertical () maps
with suitable are surjective for the respective pairs
The preimages and are both subgroups of containing and it is and , because every has a preimage with Moreover, the subgroup is normal in
As a consequence, the subquotient of is a subquotient of in the form
Relation to cardinal order
In constructive set theory, where the law of excluded middle does not necessarily hold, one can consider the relation subquotient of as replacing the usual order relation(s) on cardinals. When one has the law of the excluded middle, then a subquotient of is either the empty set or there is an onto function . This order relation is traditionally denoted
If additionally the axiom of choice holds, then has a one-to-one function to and this order relation is the usual on corresponding cardinals.
See also
Homological algebra
Subcountable
References
Category theory
Abstract algebra
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https://en.wikipedia.org/wiki/Mathematical%20Kangaroo
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Mathematical Kangaroo (also known as Kangaroo challenge, or jeu-concours Kangourou in French) is an international mathematics competition in over 77 countries. There are six levels of participation, ranging from grade 1 to grade 12. The competition is held annually on the third Thursday of March. The challenge consists of problems in multiple-choice form that are not standard notebook problems and come from a variety of topics. Besides basic computational skills, they require inspiring ideas, perseverance, creativity and imagination, logical thinking, and other problem-solving strategies. Often there are small stories, intriguing problems, and surprising results, which encourage discussions with friends and family.
It had over 6 million participants from 57 countries in 2014. In 2022, it has 84 participants countries and claims to be the largest competition for school students in the world.
History
Mathematicians in Australia came up with the idea to organize a competition that underlines the joy of mathematics and encourages mathematical problem-solving. A multiple-choice competition was created, which has been taking place in Australia since 1978. At the same time, both in France and all over the world, a widely supported movement emerged towards the popularization of mathematics. The idea of a multiple-choice competition then sprouted from two French teachers, André Deledicq and Jean Pierre Boudine, who visited their Australian colleagues Peter O’Holloran and Peter Taylor and witnessed their competition. In 1990, they decided to start a challenge in France under the name Kangourou des Mathématiques in order to pay tribute to their Australian colleagues. The particularity of this challenge was the desire for massive distribution of documentation, offering a gift to each participant (books, small games, fun objects, scientific and cultural trips). The first Kangaroo challenge took place on May 15, 1991. Since it was immediately very successful, shortly afterward they spread the idea in Europe.
In May 1993, three teams of teachers from Romania, Poland and Bulgaria participated in Kangaroo together with France. After that, Kangourou des Mathématiques invited mathematicians and organizers of mathematical competitions from several European countries. All of them were impressed by the increasing number of participants in the Kangaroo challenge in France: 120 000 in 1991, 300 000 in 1992, half a million in 1993. In seven countries – Belarus, Hungary, The Netherlands, Poland, Romania, Russia, and Spain – teams of teachers decided to also organize the contest in 1994. It was a great success in all of these countries. An international competition promoting the dissemination of basic mathematical culture was born.
Since then, the competition has spread around the world. Pupils from Sweden first took part in 1999. By 2011, 860,000 pupils from 9,000 schools took part in Germany, having grown rapidly from 549,000 in 2007. In 2014, the competition was
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https://en.wikipedia.org/wiki/Row
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Row or ROW may refer to:
Exercise
Rowing, or a form of aquatic movement using oars
Row (weight-lifting), a form of weight-lifting exercise
Mathematics and informatics
Row vector, a 1 × n matrix in linear algebra
Row(s) in a table (information), a data arrangement with rows and columns
Row (database), a single, implicitly structured data item in a database table
Tone row, an arrangement of the twelve notes of the chromatic scale
Places
Rów, Pomeranian Voivodeship, north Poland
Rów, Warmian-Masurian Voivodeship, north Poland
Rów, West Pomeranian Voivodeship, northwest Poland
Roswell International Air Center's IATA code
Row, a former spelling of Rhu, Dunbartonshire, Scotland
The Row (Lyme, New York), a set of historic homes
The Row, Virginia, an unincorporated community
Rest of the world (RoW)
The Row or The Row Fulton Market, 900 West Randolph, a Chicago Skyscraper on Chicago's Restaurant Row
Other
Reality of Wrestling, an American professional wrestling promotion founded in 2005
Row (album), an album by Gerard
Right-of-way (transportation), ROW, also often R/O/W.
The Row (fashion label)
The Row (film), a 2018 Canadian-American film
See also
Skid row (disambiguation)
Rowing (disambiguation)
Rowe (disambiguation)
Roe (disambiguation)
Rho (disambiguation)
Line (disambiguation)
Column (disambiguation)
Controversy
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https://en.wikipedia.org/wiki/Quasisimple%20group
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In mathematics, a quasisimple group (also known as a covering group) is a group that is a perfect central extension E of a simple group S. In other words, there is a short exact sequence
such that , where denotes the center of E and [ , ] denotes the commutator.
Equivalently, a group is quasisimple if it is equal to its commutator subgroup and its inner automorphism group Inn(G) (its quotient by its center) is simple (and it follows Inn(G) must be non-abelian simple, as inner automorphism groups are never non-trivial cyclic). All non-abelian simple groups are quasisimple.
The subnormal quasisimple subgroups of a group control the structure of a finite insoluble group in much the same way as the minimal normal subgroups of a finite soluble group do, and so are given a name, component.
The subgroup generated by the subnormal quasisimple subgroups is called the layer, and along with the minimal normal soluble subgroups generates a subgroup called the generalized Fitting subgroup.
The quasisimple groups are often studied alongside the simple groups and groups related to their automorphism groups, the almost simple groups. The representation theory of the quasisimple groups is nearly identical to the projective representation theory of the simple groups.
Examples
The covering groups of the alternating groups are quasisimple but not simple, for
See also
Almost simple group
Schur multiplier
Semisimple group
References
External links
http://mathworld.wolfram.com/QuasisimpleGroup.html
Notes
Properties of groups
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https://en.wikipedia.org/wiki/Symmetric%20space
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In mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of symmetries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geometry, leading to consequences in the theory of holonomy; or algebraically through Lie theory, which allowed Cartan to give a complete classification. Symmetric spaces commonly occur in differential geometry, representation theory and harmonic analysis.
In geometric terms, a complete, simply connected Riemannian manifold is a symmetric space if and only if its curvature tensor is invariant under parallel transport. More generally, a Riemannian manifold (M, g) is said to be symmetric if and only if, for each point p of M, there exists an isometry of M fixing p and acting on the tangent space as minus the identity (every symmetric space is complete, since any geodesic can be extended indefinitely via symmetries about the endpoints). Both descriptions can also naturally be extended to the setting of pseudo-Riemannian manifolds.
From the point of view of Lie theory, a symmetric space is the quotient G/H of a connected Lie group G by a Lie subgroup H which is (a connected component of) the invariant group of an involution of G. This definition includes more than the Riemannian definition, and reduces to it when H is compact.
Riemannian symmetric spaces arise in a wide variety of situations in both mathematics and physics. Their central role in the theory of holonomy was discovered by Marcel Berger. They are important objects of study in representation theory and harmonic analysis as well as in differential geometry.
Geometric definition
Let M be a connected Riemannian manifold and p a point of M. A diffeomorphism f of a neighborhood of p is said to be a geodesic symmetry if it fixes the point p and reverses geodesics through that point, i.e. if γ is a geodesic with then It follows that the derivative of the map f at p is minus the identity map on the tangent space of p. On a general Riemannian manifold, f need not be isometric, nor can it be extended, in general, from a neighbourhood of p to all of M.
M is said to be locally Riemannian symmetric if its geodesic symmetries are in fact isometric. This is equivalent to the vanishing of the covariant derivative of the curvature tensor.
A locally symmetric space is said to be a (globally) symmetric space if in addition its geodesic symmetries can be extended to isometries on all of M.
Basic properties
The Cartan–Ambrose–Hicks theorem implies that M is locally Riemannian symmetric if and only if its curvature tensor is covariantly constant, and furthermore that every simply connected, complete locally Riemannian symmetric space is actually Riemannian symmetric.
Every Riemannian symmetric space M is complete and Riemannian homogeneous (meaning that the isometry group of M acts transitively on M). In fact, already the identity component of the isometry group acts
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https://en.wikipedia.org/wiki/K%C5%8Dsaku%20Yosida
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was a Japanese mathematician who worked in the field of functional analysis. He is known for the Hille-Yosida theorem concerning C0-semigroups. Yosida studied mathematics at the University of Tokyo, and held posts at Osaka and Nagoya Universities. In 1955, Yosida returned to the University of Tokyo.
See also
Einar Carl Hille
Functional analysis
References
Kôsaku Yosida: Functional analysis. Grundlehren der mathematischen Wissenschaften 123, Springer-Verlag, 1971 (3rd ed.), 1974 (4th ed.), 1978 (5th ed.), 1980 (6th ed.)
External links
Photo
Kosaku Yosida / School of Mathematics and Statistics University of St Andrews, Scotland
94. Normed Rings and Spectral Theorems, II. By Kôsaku YOSIDA. Mathematical Inlstitute, Nagoya Imperial University. (Comm. by T.TAKAGMI, M.I.A. Oct.12,1943.)
Kosaku Yosida (1909 - 1990) - Biography - MacTutor
1909 births
1990 deaths
20th-century Japanese mathematicians
Mathematical analysts
Functional analysts
Operator theorists
Approximation theorists
University of Tokyo alumni
Academic staff of the University of Tokyo
Academic staff of Osaka University
Academic staff of Nagoya University
Laureates of the Imperial Prize
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https://en.wikipedia.org/wiki/%C3%98ystein%20Ore
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Øystein Ore (7 October 1899 – 13 August 1968) was a Norwegian mathematician known for his work in ring theory, Galois connections, graph theory, and the history of mathematics.
Life
Ore graduated from the University of Oslo in 1922, with a Cand.Real.degree in mathematics. In 1924, the University of Oslo awarded him the Ph.D. for a thesis titled Zur Theorie der algebraischen Körper, supervised by Thoralf Skolem. Ore also studied at Göttingen University, where he learned Emmy Noether's new approach to abstract algebra. He was also a fellow at the Mittag-Leffler Institute in Sweden, and spent some time at the University of Paris. In 1925, he was appointed research assistant at the University of Oslo.
Yale University’s James Pierpont went to Europe in 1926 to recruit research mathematicians. In 1927, Yale hired Ore as an assistant professor of mathematics, promoted him to associate professor in 1928, then to full professor in 1929. In 1931, he became a Sterling Professor (Yale's highest academic rank), a position he held until he retired in 1968.
Ore gave an American Mathematical Society Colloquium lecture in 1941 and was a plenary speaker at the International Congress of Mathematicians in 1936 in Oslo. He was also elected to the American Academy of Arts and Sciences and the Oslo Academy of Science. He was a founder of the Econometric Society.
Ore visited Norway nearly every summer. During World War II, he was active in the "American Relief for Norway" and "Free Norway" movements. In gratitude for the services rendered to his native country during the war, he was decorated in 1947 with the Order of St. Olav.
In 1930, Ore married Gudrun Lundevall. They had two children. Ore had a passion for painting and sculpture, collected ancient maps, and spoke several languages.
Work
Ore is known for his work in ring theory, Galois connections, and most of all, graph theory.
His early work was on algebraic number fields, how to decompose the ideal generated by a prime number into prime ideals. He then worked on noncommutative rings, proving his celebrated theorem on embedding a domain into a division ring. He then examined polynomial rings over skew fields, and attempted to extend his work on factorisation to non-commutative rings. The Ore condition, which (if true) allows a ring of fractions to be defined, and the Ore extension, a non-commutative analogue of rings of polynomials, are part of this work. In more elementary number theory, Ore's harmonic numbers are the numbers whose divisors have an integer harmonic mean.
As a teacher, Ore is notable for supervising two doctoral students who would make contributions to science and mathematics: Grace Hopper, who eventually became a United States rear admiral and computer scientist and who was a pioneer in developing the first computers, and Marshall Hall, Jr., an American mathematician who did important research in group theory and combinatorics.
In 1930, the Collected Works of Richard Dedekind were publish
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https://en.wikipedia.org/wiki/Gabriel%20Andrew%20Dirac
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Gabriel Andrew Dirac (13 March 1925 – 20 July 1984) was a Hungarian-British mathematician who mainly worked in graph theory. He served as Erasmus Smith's Professor of Mathematics at Trinity College Dublin from 1964 to 1966. In 1952, he gave a sufficient condition for a graph to contain a Hamiltonian circuit. The previous year, he conjectured that n points in the plane, not all collinear, must span at least two-point lines, where is the largest integer not exceeding . This conjecture was proven true when n is sufficiently large by Green and Tao in 2012.
Education
Dirac started his studies at St John's College, Cambridge in 1942, but in that same year the war saw him serving in the aircraft industry. He received his MA in 1949, and moved to the University of London, getting his Ph.D. "On the Colouring of Graphs: Combinatorial topology of Linear Complexes" there under Richard Rado.
Career
Dirac's main academic positions were at the King's College London (1948-1954), University of Toronto (1952-1953), University of Vienna (1954-1958), University of Hamburg (1958-1963), Trinity College Dublin (Erasmus Smith's Professor of Mathematics, 1964-1966), University of Wales at Swansea (1967-1970), and Aarhus University (1970-1984).
Family
He was born Balázs Gábor in Budapest, to Richárd Balázs, a military officer and businessman, and Margit "Manci" Wigner (sister of Eugene Wigner). When his mother married Paul Dirac in 1937, he and his sister resettled in England and were formally adopted, changing their family name to Dirac. He married Rosemari Dirac and they had four children together: Meike, Barbara, Holger and Annette.
See also
Dirac's theorem on Hamiltonian cycles
Dirac's theorem on chordal graphs
Dirac's theorem on cycles in -connected graphs
Notes
References
L. Døvling Andersen, I. Tafteberg Jakobsen, C. Thomassen, B. Toft, and P. Vestergaard (eds.), Graph Theory in Memory of G.A. Dirac, Annals of Discrete Mathematics, volume 41, North-Holland, 1989. .
20th-century Hungarian mathematicians
Graph theorists
Alumni of St John's College, Cambridge
Alumni of the University of London
1925 births
1984 deaths
Hungarian Jews
Paul Dirac
Academic staff of Aarhus University
Academics of Trinity College Dublin
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https://en.wikipedia.org/wiki/Einar%20Hille
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Carl Einar Hille (28 June 1894 – 12 February 1980) was an American mathematics professor and scholar. Hille authored or coauthored twelve mathematical books and a number of mathematical papers.
Early life and education
Hille was born in New York City. His parents were both immigrants from Sweden who separated before his birth. His father, Carl August Heuman, was a civil engineer. He was brought up by his mother, Edla Eckman, who took the surname Hille. When Einar was two years old, he and his mother returned to Stockholm. Hille spent the next 24 years of his life in Sweden, returning to the United States when he was 26 years old. Hille entered the University of Stockholm in 1911. Hille was awarded his first degree in mathematics in 1913 and the equivalent of a master's degree in the following year. He received a Ph.D. from Stockholm in 1918 for a doctoral dissertation entitled Some Problems Concerning Spherical Harmonics.
Career
In 1919 Hille was awarded the Mittag-Leffler Prize and was given the right to teach at the University of Stockholm. He subsequently taught at Harvard University, Princeton University, Stanford University and the University of Chicago. In 1933, he became an endowed professor on mathematics in the Graduate School of Yale University, retiring in 1962.
Hille's main work was on integral equations, differential equations, special functions, Dirichlet series and Fourier series. Later in his career his interests turned more towards functional analysis. His name persists among others in the Hille–Yosida theorem. Hille was a member of the London Mathematical Society and the Circolo Matematico di Palermo. Hille served as president of the American Mathematical Society (1947–48) and was the Society's Colloquium lecturer in 1944. He received many honours including election to the United States National Academy of Sciences (1953) and the Swedish Royal Academy of Sciences. He was awarded by Sweden with the Order of the Polar Star.
Personal life
Hille was married to Kirsti Ore Hille (1906–2001) in 1937, sister of Norwegian mathematician Øystein Ore. They had two sons, Harald and Bertil Hille.
Works
with Ralph Phillips: Functional Analysis and Semi-Groups. 1948, 1957.
Analytic Function Theory. 2 vols., 1959, 1964.
Analysis. 2 vols., 1964, 1966.
Lectures on Ordinary Differential Equations. 1969.
Methods in Classical and Functional Analysis. 1972.
Ordinary Differential Equations in the Complex Domain. 1976.
In Retrospect. Mathematical Intelligencer, Vol.3, 1980/81, No.1, pp. 3–13.
References
Other sources
External links
1894 births
1980 deaths
20th-century American mathematicians
Mathematical analysts
American people of Swedish descent
Members of the United States National Academy of Sciences
Order of the Polar Star
Stockholm University alumni
Presidents of the American Mathematical Society
Members of the Royal Swedish Academy of Sciences
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https://en.wikipedia.org/wiki/C0-semigroup
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{{DISPLAYTITLE:C0-semigroup }}
In mathematics, a C0-semigroup, also known as a strongly continuous one-parameter semigroup, is a generalization of the exponential function. Just as exponential functions provide solutions of scalar linear constant coefficient ordinary differential equations, strongly continuous semigroups provide solutions of linear constant coefficient ordinary differential equations in Banach spaces. Such differential equations in Banach spaces arise from e.g. delay differential equations and partial differential equations.
Formally, a strongly continuous semigroup is a representation of the semigroup (R+, +) on some Banach space X that is continuous in the strong operator topology. Thus, strictly speaking, a strongly continuous semigroup is not a semigroup, but rather a continuous representation of a very particular semigroup.
Formal definition
A strongly continuous semigroup on a Banach space is a map
such that
, (the identity operator on )
, as .
The first two axioms are algebraic, and state that is a representation of the semigroup ; the last is topological, and states that the map is continuous in the strong operator topology.
Infinitesimal generator
The infinitesimal generator A of a strongly continuous semigroup T is defined by
whenever the limit exists. The domain of A, D(A), is the set of x∈X for which this limit does exist; D(A) is a linear subspace and A is linear on this domain. The operator A is closed, although not necessarily bounded, and the domain is dense in X.
The strongly continuous semigroup T with generator A is often denoted by the symbol (or, equivalently, ). This notation is compatible with the notation for matrix exponentials, and for functions of an operator defined via functional calculus (for example, via the spectral theorem).
Uniformly continuous semigroup
A uniformly continuous semigroup is a strongly continuous semigroup T such that
holds. In this case, the infinitesimal generator A of T is bounded and we have
and
Conversely, any bounded operator
is the infinitesimal generator of a uniformly continuous semigroup given by
.
Thus, a linear operator A is the infinitesimal generator of a uniformly continuous semigroup if and only if A is a bounded linear operator. If X is a finite-dimensional Banach space, then any strongly continuous semigroup is a uniformly continuous semigroup. For a strongly continuous semigroup which is not a uniformly continuous semigroup the infinitesimal generator A is not bounded. In this case, does not need to converge.
Examples
Multiplication semigroup
Consider the Banach space endowed with the sup norm . Let be a continuous function with . The operator with domain is a closed densely defined operator and generates the multiplication semigroup where Multiplication operators can be viewed as the infinite dimensional generalisation of diagonal matrices and a lot of the properties of can be derived by properties of . For example is
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https://en.wikipedia.org/wiki/Statistical%20Methods%20for%20Research%20Workers
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Statistical Methods for Research Workers is a classic book on statistics, written by the statistician R. A. Fisher. It is considered by some to be one of the 20th century's most influential books on statistical methods, together with his The Design of Experiments (1935). It was originally published in 1925, by Oliver & Boyd (Edinburgh); the final and posthumous 14th edition was published in 1970.
Reviews
According to Denis Conniffe:
Ronald A. Fisher was "interested in application and in the popularization
of statistical methods and his early book Statistical Methods for Research Workers, published in 1925, went through many editions and
motivated and influenced the practical use of statistics in many fields of
study. His Design of Experiments (1935) [promoted] statistical technique and application. In that book he
emphasized examples and how to design experiments systematically from
a statistical point of view. The mathematical justification of the methods
described was not stressed and, indeed, proofs were often barely sketched
or omitted altogether ..., a fact which led H. B. Mann to fill the gaps with a rigorous mathematical treatment in his well-known treatise, ."
Chapters
Prefaces
Introduction
Diagrams
Distributions
Tests of Goodness of Fit, Independence and Homogeneity; with table of χ2
Tests of Significance of Means, Difference of Means, and Regression Coefficients
The Correlation Coefficient
Intraclass Correlations and the Analysis of Variance
Further Applications of the Analysis of Variance
SOURCES USED FOR DATA AND METHODS INDEX
In the second edition of 1928 a chapter 9 was added: The Principles of Statistical Estimation.
See also
The Design of Experiments
Notes
Further reading
The March 1951 issue of the Journal of the American Statistical Association contains articles celebrating the 25th anniversary of the publication of the first edition.
A.W.F. Edwards (2005) "R. A. Fisher, Statistical Methods for Research Workers, 1925," in I. Grattan-Guinness (ed) Landmark Writings in Western Mathematics: Case Studies, 1640-1940, Amsterdam: Elsevier.
Reviews
Nature anonymous review of Fisher’s Statistical Methods
BMJ anonymous review of Fisher’s Statistical Methods
Student’s review of Fisher’s Statistical Methods
Egon Pearson’s reviews of Fisher’s Statistical Methods
Harold Hotelling’s review of Fishers’ Statistical Methods
Leon Isserlis’s review of Fishers’ Statistical Methods
W. P. Elderton’sreview of Fisher’s Statistical Methods
External links
Text of first edition
The 14th edition (prepared from notes left by Fisher when he died in 1962) is reprinted as the first part of Statistical Methods, Experimental Design and Scientific Inference
History of probability and statistics
Statistics books
1925 non-fiction books
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https://en.wikipedia.org/wiki/Trigonal%20pyramidal%20molecular%20geometry
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In chemistry, a trigonal pyramid is a molecular geometry with one atom at the apex and three atoms at the corners of a trigonal base, resembling a tetrahedron (not to be confused with the tetrahedral geometry). When all three atoms at the corners are identical, the molecule belongs to point group C3v. Some molecules and ions with trigonal pyramidal geometry are the pnictogen hydrides (XH3), xenon trioxide (XeO3), the chlorate ion, , and the sulfite ion, . In organic chemistry, molecules which have a trigonal pyramidal geometry are sometimes described as sp3 hybridized. The AXE method for VSEPR theory states that the classification is AX3E1.
Trigonal pyramidal geometry in ammonia
The nitrogen in ammonia has 5 valence electrons and bonds with three hydrogen atoms to complete the octet. This would result in the geometry of a regular tetrahedron with each bond angle equal to cos−1(−) ≈ 109.5°. However, the three hydrogen atoms are repelled by the electron lone pair in a way that the geometry is distorted to a trigonal pyramid (regular 3-sided pyramid) with bond angles of 107°. In contrast, boron trifluoride is flat, adopting a trigonal planar geometry because the boron does not have a lone pair of electrons. In ammonia the trigonal pyramid undergoes rapid nitrogen inversion.
See also
VSEPR theory#AXE method
Molecular geometry
References
External links
Chem| Chemistry, Structures, and 3D Molecules
Indiana University Molecular Structure Center
Interactive molecular examples for point groups
Molecular Modeling
Animated Trigonal Planar Visual
Stereochemistry
Molecular geometry
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https://en.wikipedia.org/wiki/Uniform%204-polytope
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In geometry, a uniform 4-polytope (or uniform polychoron) is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons.
There are 47 non-prismatic convex uniform 4-polytopes. There are two infinite sets of convex prismatic forms, along with 17 cases arising as prisms of the convex uniform polyhedra. There are also an unknown number of non-convex star forms.
History of discovery
Convex Regular polytopes:
1852: Ludwig Schläfli proved in his manuscript Theorie der vielfachen Kontinuität that there are exactly 6 regular polytopes in 4 dimensions and only 3 in 5 or more dimensions.
Regular star 4-polytopes (star polyhedron cells and/or vertex figures)
1852: Ludwig Schläfli also found 4 of the 10 regular star 4-polytopes, discounting 6 with cells or vertex figures {5/2,5} and {5,5/2}.
1883: Edmund Hess completed the list of 10 of the nonconvex regular 4-polytopes, in his book (in German) Einleitung in die Lehre von der Kugelteilung mit besonderer Berücksichtigung ihrer Anwendung auf die Theorie der Gleichflächigen und der gleicheckigen Polyeder .
Convex semiregular polytopes: (Various definitions before Coxeter's uniform category)
1900: Thorold Gosset enumerated the list of nonprismatic semiregular convex polytopes with regular cells (Platonic solids) in his publication On the Regular and Semi-Regular Figures in Space of n Dimensions. In four dimensions, this gives the rectified 5-cell, the rectified 600-cell, and the snub 24-cell.
1910: Alicia Boole Stott, in her publication Geometrical deduction of semiregular from regular polytopes and space fillings, expanded the definition by also allowing Archimedean solid and prism cells. This construction enumerated 45 semiregular 4-polytopes, corresponding to the nonprismatic forms listed below. The snub 24-cell and grand antiprism were missing from her list.
1911: Pieter Hendrik Schoute published Analytic treatment of the polytopes regularly derived from the regular polytopes, followed Boole-Stott's notations, enumerating the convex uniform polytopes by symmetry based on 5-cell, 8-cell/16-cell, and 24-cell.
1912: E. L. Elte independently expanded on Gosset's list with the publication The Semiregular Polytopes of the Hyperspaces, polytopes with one or two types of semiregular facets.
Convex uniform polytopes:
1940: The search was expanded systematically by H.S.M. Coxeter in his publication Regular and Semi-Regular Polytopes.
Convex uniform 4-polytopes:
1965: The complete list of convex forms was finally enumerated by John Horton Conway and Michael Guy, in their publication Four-Dimensional Archimedean Polytopes, established by computer analysis, adding only one non-Wythoffian convex 4-polytope, the grand antiprism.
1966 Norman Johnson completes his Ph.D. dissertation The Theory of Uniform Polytopes and Honeycombs under advisor Coxeter, completes the basic theory of uniform polytopes for dimensions 4 and higher.
1986 Coxeter publis
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https://en.wikipedia.org/wiki/Generalized%20Kac%E2%80%93Moody%20algebra
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In mathematics, a generalized Kac–Moody algebra is a Lie algebra that is similar to a Kac–Moody algebra, except that it is allowed to have imaginary simple roots.
Generalized Kac–Moody algebras are also sometimes called GKM algebras, Borcherds–Kac–Moody algebras, BKM algebras, or Borcherds algebras. The best known example is the monster Lie algebra.
Motivation
Finite-dimensional semisimple Lie algebras have the following properties:
They have a nondegenerate symmetric invariant bilinear form (,).
They have a grading such that the degree zero piece (the Cartan subalgebra) is abelian.
They have a (Cartan) involution w.
(a, w(a)) is positive if a is nonzero.
For example, for the algebras of n by n matrices of trace zero, the bilinear form is (a, b) = Trace(ab), the Cartan involution is given by minus the transpose, and the grading can be given by "distance from the diagonal" so that the Cartan subalgebra is the diagonal elements.
Conversely one can try to find all Lie algebras with these properties (and satisfying a few other technical conditions). The answer is that one gets sums of finite-dimensional and affine Lie algebras.
The monster Lie algebra satisfies a slightly weaker version of the conditions above:
(a, w(a)) is positive if a is nonzero and has nonzero degree, but may be negative when a has degree zero. The Lie algebras satisfying these weaker conditions are more or less generalized Kac–Moody algebras.
They are essentially the same as algebras given by certain generators and relations (described below).
Informally, generalized Kac–Moody algebras are the Lie algebras that behave like finite-dimensional semisimple Lie algebras. In particular they have a Weyl group, Weyl character formula, Cartan subalgebra, roots, weights, and so on.
Definition
A symmetrized Cartan matrix is a (possibly infinite) square matrix with entries such that
if
is an integer if
The universal generalized Kac–Moody algebra with given symmetrized Cartan matrix is defined by generators and and and relations
if , 0 otherwise
,
for applications of or if
if
These differ from the relations of a (symmetrizable) Kac–Moody algebra mainly by allowing the diagonal entries of the Cartan matrix to be non-positive.
In other words, we allow simple roots to be imaginary, whereas in a Kac–Moody algebra simple roots are always real.
A generalized Kac–Moody algebra is obtained from a universal one by changing the Cartan matrix, by the operations of killing something in the center, or taking a central extension, or adding outer derivations.
Some authors give a more general definition by removing the condition that the Cartan matrix should be symmetric. Not much is known about these non-symmetrizable generalized Kac–Moody algebras, and there seem to be no interesting examples.
It is also possible to extend the definition to superalgebras.
Structure
A generalized Kac–Moody algebra can be graded by giving ei degree 1, fi degree −1, and hi deg
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https://en.wikipedia.org/wiki/Opposite
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Opposite or Opposites may refer to:
Opposite (semantics), a word that means the reverse of a word
Opposite (leaf), an arrangement of leaves on a stem
Opposite (mathematics), the negative of a number; numbers that, when added, yield zero
"The Opposite", a 1994 episode of Seinfeld
Music
The Opposites, Dutch rap group
Opposites (album), 2013 album by Scottish alternative rock band Biffy Clyro
"Opposite" (song), 2013 song by Biffy Clyro
Opposites (EP), 2010 album by Tracey Thorn
"The Opposite", 1964 song by Johnny Burnette
See also
Opposite hitter, a position in volleyball
Antinomy, opposites in a certain form from Kant
Anti (disambiguation)
Contrary (disambiguation)
Flipside (disambiguation)
Inverse (disambiguation)
Opposite sex (disambiguation)
Opposition (disambiguation)
Polar opposite (disambiguation)
The House Opposite (disambiguation)
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https://en.wikipedia.org/wiki/Trigonal%20planar%20molecular%20geometry
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In chemistry, trigonal planar is a molecular geometry model with one atom at the center and three atoms at the corners of an equilateral triangle, called peripheral atoms, all in one plane. In an ideal trigonal planar species, all three ligands are identical and all bond angles are 120°. Such species belong to the point group D3h. Molecules where the three ligands are not identical, such as H2CO, deviate from this idealized geometry. Examples of molecules with trigonal planar geometry include boron trifluoride (BF3), formaldehyde (H2CO), phosgene (COCl2), and sulfur trioxide (SO3). Some ions with trigonal planar geometry include nitrate (), carbonate (), and guanidinium (). In organic chemistry, planar, three-connected carbon centers that are trigonal planar are often described as having sp2 hybridization.
Nitrogen inversion is the distortion of pyramidal amines through a transition state that is trigonal planar.
Pyramidalization is a distortion of this molecular shape towards a tetrahedral molecular geometry. One way to observe this distortion is in pyramidal alkenes.
See also
AXE method
Molecular geometry
VSEPR theory
References
External links
3D Chem Chemistry, Structures, and 3D Molecules
Indiana University Molecular Structure Center
Interactive molecular examples for point groups
Molecular Modeling
Animated Trigonal Planar Visual
Stereochemistry
Molecular geometry
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https://en.wikipedia.org/wiki/Mathematics%2C%20Engineering%2C%20Science%20Achievement
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Mathematics, Engineering, Science Achievement (MESA) is an academic preparation program for pre-college, community college and university-level students. Established in 1970 in California, the program provides academic support to students from educationally disadvantaged backgrounds throughout the education pathway so they will excel in math and science and ultimately attain four-year degrees in science, technology, engineering or math (STEM) fields. The program has successfully been replicated in over a dozen other states.
Locations and partners
MESA, while administered by the University of California, is an intersegmental program, with centers located at all major statewide education institutions (California Department of Education, University of California, California State University, California Community Colleges, the Association of Independent California Colleges and Universities). MESA has established an active partnership with industry and STEM leaders such as AT&T, Chevron, Google, HP, Sempra Energy, and PG&E. These partners supply expertise, volunteers, internship and opportunities for students to visit companies and learn about career options in STEM fields. The strong relationship with industry has resulted in MESA incorporating many elements of industry culture into its approaches and activities.
A partnership of MESA programs in eleven states (Arizona, California, Colorado, Maryland, New Mexico, Nevada, Oregon, Utah and Washington) has established a network called MESA USA.
Program components
The MESA School Program (MSP) supports pre-college students (mostly in middle and senior high schools) to excel in math and science and go on to college as majors in STEM fields. Advising and academic assistance is provided through a MESA advisor, usually a math or science teacher. Components of the MSP include development of individual academic plans to ensure timely completion of college preparatory classes, study skills training, local and regional competitions in hands-on activities, career and college exploration, and parent leadership development. MSP also offers professional development opportunities for math and science teachers to learn innovative approaches and hands-on activities that can be replicated in schools with limited resources.
The MESA Community College Program (MCCP) provides academic preparation for community college students who are interested in transferring to four-year institutions to attain baccalaureate degrees in STEM fields. MESA establishes an on-campus peer community to reinforce and support academic achievement while providing academic assistance and transfer guidance. Components of the MCCP include Academic Excellence Workshops that teach collaborative learning techniques that help students to master complex concepts; a special orientation course for STEM students; a dedicated study center; career advising and exploration of STEM options; transfer assistance; scholarships, and links with student and profe
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https://en.wikipedia.org/wiki/No-arbitrage%20bounds
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In financial mathematics, no-arbitrage bounds are mathematical relationships specifying limits on financial portfolio prices. These price bounds are a specific example of good–deal bounds, and are in fact the greatest extremes for good–deal bounds.
The most frequent nontrivial example of no-arbitrage bounds is put–call parity for option prices. In incomplete markets, the bounds are given by the subhedging and superhedging prices.
The essence of no-arbitrage in mathematical finance is excluding the possibility of "making money out of nothing" in the financial market. This is necessary because the existence of arbitrage is not only unrealistic, but also contradicts the possibility of an economic equilibrium. All mathematical models of financial markets have to satisfy a no-arbitrage condition to be realistic models.
See also
Box spread
Indifference price
References
Mathematical finance
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https://en.wikipedia.org/wiki/Unimodular%20lattice
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In geometry and mathematical group theory, a unimodular lattice is an integral lattice of determinant 1 or −1. For a lattice in n-dimensional Euclidean space, this is equivalent to requiring that the volume of any fundamental domain for the lattice be 1.
The E8 lattice and the Leech lattice are two famous examples.
Definitions
A lattice is a free abelian group of finite rank with a symmetric bilinear form (·, ·).
The lattice is integral if (·,·) takes integer values.
The dimension of a lattice is the same as its rank (as a Z-module).
The norm of a lattice element a is (a, a).
A lattice is positive definite if the norm of all nonzero elements is positive.
The determinant of a lattice is the determinant of the Gram matrix, a matrix with entries (ai, aj), where the elements ai form a basis for the lattice.
An integral lattice is unimodular if its determinant is 1 or −1.
A unimodular lattice is even or type II if all norms are even, otherwise odd or type I.
The minimum of a positive definite lattice is the lowest nonzero norm.
Lattices are often embedded in a real vector space with a symmetric bilinear form. The lattice is positive definite, Lorentzian, and so on if its vector space is.
The signature of a lattice is the signature of the form on the vector space.
Examples
The three most important examples of unimodular lattices are:
The lattice Z, in one dimension.
The E8 lattice, an even 8-dimensional lattice,
The Leech lattice, the 24-dimensional even unimodular lattice with no roots.
Properties
An integral lattice is unimodular if and only if its dual lattice is integral. Unimodular lattices are equal to their dual lattices, and for this reason, unimodular lattices are also known as self-dual.
Given a pair (m,n) of nonnegative integers, an even unimodular lattice of signature (m,n) exists if and only if m−n is divisible by 8, but an odd unimodular lattice of signature (m,n) always exists. In particular, even unimodular definite lattices only exist in dimension divisible by 8. Examples in all admissible signatures are given by the IIm,n and Im,n constructions, respectively.
The theta function of a unimodular positive definite lattice is a modular form whose weight is one half the rank. If the lattice is even, the form has level 1, and if the lattice is odd the form has Γ0(4) structure (i.e., it is a modular form of level 4). Due to the dimension bound on spaces of modular forms, the minimum norm of a nonzero vector of an even unimodular lattice is no greater than ⎣n/24⎦ + 1. An even unimodular lattice that achieves this bound is called extremal. Extremal even unimodular lattices are known in relevant dimensions up to 80, and their non-existence has been proven for dimensions above 163,264.
Classification
For indefinite lattices, the classification is easy to describe.
Write Rm,n for the m + n dimensional vector space Rm+n with the inner product of
(a1, ..., am+n) and (b1, ..., bm+n) given by
In Rm,n there is
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https://en.wikipedia.org/wiki/Random%20matrix
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In probability theory and mathematical physics, a random matrix is a matrix-valued random variable—that is, a matrix in which some or all elements are random variables. Many important properties of physical systems can be represented mathematically as matrix problems. For example, the thermal conductivity of a lattice can be computed from the dynamical matrix of the particle-particle interactions within the lattice.
Applications
Physics
In nuclear physics, random matrices were introduced by Eugene Wigner to model the nuclei of heavy atoms. Wigner postulated that the spacings between the lines in the spectrum of a heavy atom nucleus should resemble the spacings between the eigenvalues of a random matrix, and should depend only on the symmetry class of the underlying evolution. In solid-state physics, random matrices model the behaviour of large disordered Hamiltonians in the mean-field approximation.
In quantum chaos, the Bohigas–Giannoni–Schmit (BGS) conjecture asserts that the spectral statistics of quantum systems whose classical counterparts exhibit chaotic behaviour are described by random matrix theory.
In quantum optics, transformations described by random unitary matrices are crucial for demonstrating the advantage of quantum over classical computation (see, e.g., the boson sampling model). Moreover, such random unitary transformations can be directly implemented in an optical circuit, by mapping their parameters to optical circuit components (that is beam splitters and phase shifters).
Random matrix theory has also found applications to the chiral Dirac operator in quantum chromodynamics, quantum gravity in two dimensions, mesoscopic physics, spin-transfer torque, the fractional quantum Hall effect, Anderson localization, quantum dots, and superconductors
Mathematical statistics and numerical analysis
In multivariate statistics, random matrices were introduced by John Wishart, who sought to estimate covariance matrices of large samples. Chernoff-, Bernstein-, and Hoeffding-type inequalities can typically be strengthened when applied to the maximal eigenvalue (i.e. the eigenvalue of largest magnitude) of a finite sum of random Hermitian matrices. Random matrix theory is used to study the spectral properties of random matrices—such as sample covariance matrices—which is of particular interest in high-dimensional statistics. Random matrix theory also saw applications in neuronal networks and deep learning, with recent work utilizing random matrices to show that hyper-parameter tunings can be cheaply transferred between large neural networks without the need for re-training.
In numerical analysis, random matrices have been used since the work of John von Neumann and Herman Goldstine to describe computation errors in operations such as matrix multiplication. Although random entries are traditional "generic" inputs to an algorithm, the concentration of measure associated with random matrix distributions implies that random matrices wi
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https://en.wikipedia.org/wiki/List%20of%20Guggenheim%20Fellowships%20awarded%20in%201970
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List of Guggenheim Fellowship winners for 1970.
United States and Canadian fellows
Patrick Ahern, professor of mathematics, University of Wisconsin–Madison.
Michael M. Ames, former director and professor emeritus, Museum of Anthropology, University of British Columbia.
Albert K. Ando, professor of economics and finance, University of Pennsylvania.
Jon Howard Appleton, composer; Arthur R. Virgin Professor of Music, Dartmouth College.
Giuseppe Attardi, professor of biology, California Institute of Technology: 1970, 1986.
James M. Banner, Jr., independent historian, Washington, D.C..
Thomas G. Barnes, professor of history, University of California, Berkeley.
Samuel Haskell Baron, alumni distinguished professor emeritus of history, University of North Carolina at Chapel Hill: 1970.
Romare Bearden, deceased. Fine arts.
Max Beberman, deceased. Education.
Jonathan Beckwith, American Cancer Society Research Professor of Microbiology and Molecular Genetics, Harvard Medical School.
Charles Franklin Bennett, professor of biogeography, University of California, Los Angeles.
Malcolm Bersohn, associate professor of chemistry, University of Toronto.
Alexander M. Bickel, deceased. Law.
Peter J. Bickel, chair, professor of statistics, University of California, Berkeley.
James Bishop, artist, New York City.
Ronald Bladen, deceased. Fine Arts.
John McDonald Blakely, professor of materials science and engineering, Cornell University.
Henry David Block, deceased. Computer science.
Derk Bodde, professor emeritus of Chinese studies, University of Pennsylvania.
Peter Boerner, emeritus professor of Germanic languages, of comparative literature, and of West European studies, Indiana University.
Robert Earl Boles, novelist.
Stephen Booth, professor of English, University of California, Berkeley.
Daniel Branton, Higgins Research Professor of Biology, Harvard University.
Oscar G. Brockett, Z. T. Scott Family Chair and Professor of Drama, University of Texas at Austin.
Bertram Neville Brockhouse, professor of physics, McMaster University.
Charles Jacob Brokaw, emeritus professor of biology, California Institute of Technology.
James Broughton, deceased. Filmmaker; retired lecturer in film, College of the San Francisco Art Institute and California State University, San Francisco: 1970, 1973.
Frederick Brown, professor of French & Italian, State University of New York at Stony Brook: 1970, 1984.
Jan Harold Brunvand, emeritus professor of English, University of Utah.
Alma Lyman Burlingame, professor of chemistry and pharmaceutical chemistry, University of California, San Francisco.
Ben Caldwell, playwright, New York City.
David Cass, professor of economics, Carnegie Mellon University
Richard W. Castenholz, professor of biology, University of Oregon.
Joseph Chaikin, Theatre Arts. Founder of The Open Theater.
Hung Cheng, professor of applied mathematics, Massachusetts Institute of Technology.
Dorrit Cohn, Ernest Bernbaum Professor Emeritus
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https://en.wikipedia.org/wiki/Appell%20sequence
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In mathematics, an Appell sequence, named after Paul Émile Appell, is any polynomial sequence satisfying the identity
and in which is a non-zero constant.
Among the most notable Appell sequences besides the trivial example are the Hermite polynomials, the Bernoulli polynomials, and the Euler polynomials. Every Appell sequence is a Sheffer sequence, but most Sheffer sequences are not Appell sequences. Appell sequences have a probabilistic interpretation as systems of moments.
Equivalent characterizations of Appell sequences
The following conditions on polynomial sequences can easily be seen to be equivalent:
For ,
and is a non-zero constant;
For some sequence of scalars with ,
For the same sequence of scalars,
where
For ,
Recursion formula
Suppose
where the last equality is taken to define the linear operator on the space of polynomials in . Let
be the inverse operator, the coefficients being those of the usual reciprocal of a formal power series, so that
In the conventions of the umbral calculus, one often treats this formal power series as representing the Appell sequence . One can define
by using the usual power series expansion of the and the usual definition of composition of formal power series. Then we have
(This formal differentiation of a power series in the differential operator is an instance of Pincherle differentiation.)
In the case of Hermite polynomials, this reduces to the conventional recursion formula for that sequence.
Subgroup of the Sheffer polynomials
The set of all Appell sequences is closed under the operation of umbral composition of polynomial sequences, defined as follows. Suppose and are polynomial sequences, given by
Then the umbral composition is the polynomial sequence whose th term is
(the subscript appears in , since this is the th term of that sequence, but not in , since this refers to the sequence as a whole rather than one of its terms).
Under this operation, the set of all Sheffer sequences is a non-abelian group, but the set of all Appell sequences is an abelian subgroup. That it is abelian can be seen by considering the fact that every Appell sequence is of the form
and that umbral composition of Appell sequences corresponds to multiplication of these formal power series in the operator .
Different convention
Another convention followed by some authors (see Chihara) defines this concept in a different way, conflicting with Appell's original definition, by using the identity
instead.
Hypergeometric Appell polynomials
The enormous class of Appell polynomials can be obtained in terms of the generalized hypergeometric function.
Let denote the array of ratios
Consider the polynomial
where is the generalized hypergeometric function.
Theorem.
The polynomial family is the Appell sequence for any natural parameters .
For example, if then the polynomials become the Gould-Hopper polynomials and if they become the Hermite polynomials .
See also
She
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https://en.wikipedia.org/wiki/Pr%C3%BCfer%20sequence
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In combinatorial mathematics, the Prüfer sequence (also Prüfer code or Prüfer numbers) of a labeled tree is a unique sequence associated with the tree. The sequence for a tree on n vertices has length n − 2, and can be generated by a simple iterative algorithm. Prüfer sequences were first used by Heinz Prüfer to prove Cayley's formula in 1918.
Algorithm to convert a tree into a Prüfer sequence
One can generate a labeled tree's Prüfer sequence by iteratively removing vertices from the tree until only two vertices remain. Specifically, consider a labeled tree T with vertices {1, 2, ..., n}. At step i, remove the leaf with the smallest label and set the ith element of the Prüfer sequence to be the label of this leaf's neighbour.
The Prüfer sequence of a labeled tree is unique and has length n − 2.
Both coding and decoding can be reduced to integer radix sorting and parallelized.
Example
Consider the above algorithm run on the tree shown to the right. Initially, vertex 1 is the leaf with the smallest label, so it is removed first and 4 is put in the Prüfer sequence. Vertices 2 and 3 are removed next, so 4 is added twice more. Vertex 4 is now a leaf and has the smallest label, so it is removed and we append 5 to the sequence. We are left with only two vertices, so we stop. The tree's sequence is {4,4,4,5}.
Algorithm to convert a Prüfer sequence into a tree
Let {a[1], a[2], ..., a[n]} be a Prüfer sequence:
The tree will have n+2 nodes, numbered from 1 to n+2.
For each node set its degree to the number of times it appears in the sequence plus 1.
For instance, in pseudo-code:
Convert-Prüfer-to-Tree(a)
1 n ← length[a]
2 T ← a graph with n + 2 isolated nodes, numbered 1 to n + 2
3 degree ← an array of integers
4 for each node i in T do
5 degree[i] ← 1
6 for each value i in a do
7 degree[i] ← degree[i] + 1
Next, for each number in the sequence a[i], find the first (lowest-numbered) node, j, with degree equal to 1, add the edge (j, a[i]) to the tree, and decrement the degrees of j and a[i]. In pseudo-code:
8 for each value i in a do
9 for each node j in T do
10 if degree[j] = 1 then
11 Insert edge[i, j] into T
12 degree[i] ← degree[i] - 1
13 degree[j] ← degree[j] - 1
14 break
At the end of this loop two nodes with degree 1 will remain (call them u, v). Lastly, add the edge (u,v) to the tree.
15 u ← v ← 0
16 for each node i in T
17 if degree[i] = 1 then
18 if u = 0 then
19 u ← i
20 else
21 v ← i
22 break
23 Insert edge[u, v] into T
24 degree[u] ← degree[u] - 1
25 degree[v] ← degree[v] - 1
26 return T
Cayley's formula
The Prüfer sequence of a labeled tree on n vertices is a unique sequence of length n − 2 on the labels 1 to n. For a given sequence S of length n − 2 on the labels 1 to n, there is a unique labeled tree whose Prüfer sequence is S.
The immediate consequence is tha
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https://en.wikipedia.org/wiki/Locally%20compact%20group
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In mathematics, a locally compact group is a topological group G for which the underlying topology is locally compact and Hausdorff. Locally compact groups are important because many examples of groups that arise throughout mathematics are locally compact and such groups have a natural measure called the Haar measure. This allows one to define integrals of Borel measurable functions on G so that standard analysis notions such as the Fourier transform and spaces can be generalized.
Many of the results of finite group representation theory are proved by averaging over the group. For compact groups, modifications of these proofs yields similar results by averaging with respect to the normalized Haar integral. In the general locally compact setting, such techniques need not hold. The resulting theory is a central part of harmonic analysis. The representation theory for locally compact abelian groups is described by Pontryagin duality.
Examples and counterexamples
Any compact group is locally compact.
In particular the circle group T of complex numbers of unit modulus under multiplication is compact, and therefore locally compact. The circle group historically served as the first topologically nontrivial group to also have the property of local compactness, and as such motivated the search for the more general theory, presented here.
Any discrete group is locally compact. The theory of locally compact groups therefore encompasses the theory of ordinary groups since any group can be given the discrete topology.
Lie groups, which are locally Euclidean, are all locally compact groups.
A Hausdorff topological vector space is locally compact if and only if it is finite-dimensional.
The additive group of rational numbers Q is not locally compact if given the relative topology as a subset of the real numbers. It is locally compact if given the discrete topology.
The additive group of p-adic numbers Qp is locally compact for any prime number p.
Properties
By homogeneity, local compactness of the underlying space for a topological group need only be checked at the identity. That is, a group G is a locally compact space if and only if the identity element has a compact neighborhood. It follows that there is a local base of compact neighborhoods at every point.
A topological group is Hausdorff if and only if the trivial one-element subgroup is closed.
Every closed subgroup of a locally compact group is locally compact. (The closure condition is necessary as the group of rationals demonstrates.) Conversely, every locally compact subgroup of a Hausdorff group is closed. Every quotient of a locally compact group is locally compact. The product of a family of locally compact groups is locally compact if and only if all but a finite number of factors are actually compact.
Topological groups are always completely regular as topological spaces. Locally compact groups have the stronger property of being normal.
Every locally compact group which is first-count
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https://en.wikipedia.org/wiki/Henk%20Barendregt
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Hendrik Pieter (Henk) Barendregt (born 18 December 1947, Amsterdam) is a Dutch logician, known for his work in lambda calculus and type theory.
Life and work
Barendregt studied mathematical logic at Utrecht University, obtaining his master's degree in 1968 and his PhD in 1971, both cum laude, under Dirk van Dalen and Georg Kreisel. After a postdoctoral position at Stanford University, he taught at Utrecht University.
Since 1986, Barendregt has taught at Radboud University Nijmegen, where he now holds the Chair of Foundations of Mathematics and Computer Science. His research group works on Constructive Interactive Mathematics. He is also Adjunct Professor at Carnegie Mellon University, Pittsburgh, USA. He has been a visiting scholar at Darmstadt, ETH Zürich, Siena, and Kyoto.
Barendregt was elected a member of Academia Europaea in 1992. In 1997 Barendregt was elected member of the Royal Netherlands Academy of Arts and Sciences. On 6 February 2003 Barendregt was awarded the Spinozapremie for 2002, the highest scientific award in the Netherlands. In 2002 he was knighted in the Orde van de Nederlandse Leeuw.
Barendregt received an honorary doctorate from Heriot-Watt University in 2015.
Selected publications
— See Errata
References
External links
Barendregt's homepage
Author profile in the database zbMATH
1947 births
Living people
Dutch computer scientists
Mathematical logicians
Members of Academia Europaea
Members of the Royal Netherlands Academy of Arts and Sciences
Academic staff of Radboud University Nijmegen
Spinoza Prize winners
Utrecht University alumni
Scientists from Amsterdam
Academic staff of Technische Universität Darmstadt
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https://en.wikipedia.org/wiki/Ordinary%20least%20squares
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In statistics, ordinary least squares (OLS) is a type of linear least squares method for choosing the unknown parameters in a linear regression model (with fixed level-one effects of a linear function of a set of explanatory variables) by the principle of least squares: minimizing the sum of the squares of the differences between the observed dependent variable (values of the variable being observed) in the input dataset and the output of the (linear) function of the independent variable.
Geometrically, this is seen as the sum of the squared distances, parallel to the axis of the dependent variable, between each data point in the set and the corresponding point on the regression surface—the smaller the differences, the better the model fits the data. The resulting estimator can be expressed by a simple formula, especially in the case of a simple linear regression, in which there is a single regressor on the right side of the regression equation.
The OLS estimator is consistent for the level-one fixed effects when the regressors are exogenous and forms perfect colinearity (rank condition), consistent for the variance estimate of the residuals when regressors have finite fourth moments and—by the Gauss–Markov theorem—optimal in the class of linear unbiased estimators when the errors are homoscedastic and serially uncorrelated. Under these conditions, the method of OLS provides minimum-variance mean-unbiased estimation when the errors have finite variances. Under the additional assumption that the errors are normally distributed with zero mean, OLS is the maximum likelihood estimator that outperforms any non-linear unbiased estimator.
Linear model
Suppose the data consists of observations . Each observation includes a scalar response and a column vector of parameters (regressors), i.e., . In a linear regression model, the response variable, , is a linear function of the regressors:
or in vector form,
where , as introduced previously, is a column vector of the -th observation of all the explanatory variables; is a vector of unknown parameters; and the scalar represents unobserved random variables (errors) of the -th observation. accounts for the influences upon the responses from sources other than the explanatory variables . This model can also be written in matrix notation as
where and are vectors of the response variables and the errors of the observations, and is an matrix of regressors, also sometimes called the design matrix, whose row is and contains the -th observations on all the explanatory variables.
Typically, a constant term is included in the set of regressors , say, by taking for all . The coefficient corresponding to this regressor is called the intercept. Without the intercept, the fitted line is forced to cross the origin when .
Regressors do not have to be independent: there can be any desired relationship between the regressors (so long as it is not a linear relationship). For instance, we might s
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https://en.wikipedia.org/wiki/Niemeier%20lattice
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In mathematics, a Niemeier lattice is one of the 24
positive definite even unimodular lattices of rank 24,
which were classified by . gave a simplified proof of the classification. In the 1970s, has a sentence mentioning that he found more than 10 such lattices in the 1940s, but gives no further details. One example of a Niemeier lattice is the Leech lattice found in 1967.
Classification
Niemeier lattices are usually labelled by the Dynkin diagram of their
root systems. These Dynkin diagrams have rank either 0 or 24, and all of their components have the same Coxeter number. (The Coxeter number, at least in these cases, is
the number of roots divided by the dimension.) There are exactly 24 Dynkin diagrams with these properties, and there turns out to be a unique Niemeier
lattice for each of these Dynkin diagrams.
The complete list of Niemeier lattices is given in the following table.
In the table,
G0 is the order of the group generated by reflections
G1 is the order of the group of automorphisms fixing all components of the Dynkin diagram
G2 is the order of the group of automorphisms of permutations of components of the Dynkin diagram
G∞ is the index of the root lattice in the Niemeier lattice, in other words, the order of the "glue code". It is the square root of the discriminant of the root lattice.
G0×G1×G2 is the order of the automorphism group of the lattice
G∞×G1×G2 is the order of the automorphism group of the corresponding deep hole.
The neighborhood graph of the Niemeier lattices
If L is an odd unimodular lattice of dimension 8n and M its sublattice of even vectors, then M is contained in exactly 3 unimodular lattices, one of which is L and the other two of which are even. (If L has a norm 1 vector then the two even lattices are isomorphic.) The Kneser neighborhood graph in 8n dimensions has a point for each even lattice, and a line joining two points for each odd 8n dimensional lattice with no norm 1 vectors, where the vertices of each line are the two even lattices associated to the odd lattice. There may be several lines between the same pair of vertices, and there may be lines from a vertex to itself. Kneser proved that this graph is always connected. In 8 dimensions it has one point and no lines, in 16 dimensions it has two points joined by one line, and in 24 dimensions it is the following graph:
Each point represents one of the 24 Niemeier lattices, and the lines joining them represent the 24 dimensional odd unimodular lattices with no norm 1 vectors. (Thick lines represent multiple lines.) The number on the right is the Coxeter number of the Niemeier lattice.
In 32 dimensions the neighborhood graph has more than a billion vertices.
Properties
Some of the Niemeier lattices are related to sporadic simple groups.
The Leech lattice is acted on by a double cover of the Conway group,
and the lattices A124 and A212
are acted on by the Mathieu groups M24 and M12.
The Niemeier lattices, other than the Leech lattice, cor
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https://en.wikipedia.org/wiki/Toru%20Kumon
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was a Japanese mathematics educator, born in Kōchi Prefecture, Japan. He graduated from the College of Science at Osaka University with a degree in mathematics and taught high school mathematics in his home town of Osaka. In 1954, his son, Takeshi, performed poorly in a Year 2 mathematics test. Prompted by his wife, Teiko, Toru closely examined Takeshi's textbooks and believed they lacked the proper opportunity for a child to practice and master a topic. As a result, he began to handwrite worksheets each day for his son. By the time Takeshi was in Year 6, he was able to solve differential and integral calculus usually seen in the final years of high school. This was the beginning of the Kumon Method of Learning.
As a result of Takeshi's progress, other parents became interested in Kumon's ideas, and in 1955, the first Kumon Center was opened in Osaka, Japan. In 1958, Toru Kumon founded the Kumon Institute of Education, which set the standards for the Kumon Centers that began to open around the world.
The Kumon Programs are designed to strengthen a student's fundamental maths and language skills by studying worksheets tailored to a student's ability. The method also aims for students to learn independently and to study advanced material beyond their school grade level.
Students progress once they demonstrate mastery of a topic. Kumon defined mastery as being able to achieve an excellent score on the material in a given time. Kumon strongly emphasised the concepts of time and accuracy.
Even in his later years, Toru Kumon gave lectures on his method of learning including the importance of having students learn material that is suited to their ability and not their age and the benefits of allowing students to learn material well ahead of their grade level.
Toru Kumon died in Osaka on July 25, 1995, at the age of 81 from pneumonia. There is a Toru Kumon museum in Osaka, Japan, and a Kumon Foundation Day celebrated on October 20 each year. Asteroid 3569 Kumon is named after him.
References
1914 births
1995 deaths
Japanese educational theorists
20th-century Japanese mathematicians
People from Kōchi Prefecture
Osaka University alumni
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https://en.wikipedia.org/wiki/Symbolic%20integration
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In calculus, symbolic integration is the problem of finding a formula for the antiderivative, or indefinite integral, of a given function f(x), i.e. to find a differentiable function F(x) such that
This is also denoted
Discussion
The term symbolic is used to distinguish this problem from that of numerical integration, where the value of F is sought at a particular input or set of inputs, rather than a general formula for F.
Both problems were held to be of practical and theoretical importance long before the time of digital computers, but they are now generally considered the domain of computer science, as computers are most often used currently to tackle individual instances.
Finding the derivative of an expression is a straightforward process for which it is easy to construct an algorithm. The reverse question of finding the integral is much more difficult. Many expressions which are relatively simple do not have integrals that can be expressed in closed form. See antiderivative and nonelementary integral for more details.
A procedure called the Risch algorithm exists which is capable of determining whether the integral of an elementary function (function built from a finite number of exponentials, logarithms, constants, and nth roots through composition and combinations using the four elementary operations) is elementary and returning it if it is. In its original form, Risch algorithm was not suitable for a direct implementation, and its complete implementation took a long time. It was first implemented in Reduce in the case of purely transcendental functions; the case of purely algebraic functions was solved and implemented in Reduce by James H. Davenport; the general case was solved by Manuel Bronstein, who implemented almost all of it in Axiom, though to date there is no implementation of the Risch algorithm which can deal with all of the special cases and branches in it.
However, the Risch algorithm applies only to indefinite integrals, while most of the integrals of interest to physicists, theoretical chemists, and engineers are definite integrals often related to Laplace transforms, Fourier transforms, and Mellin transforms. Lacking a general algorithm, the developers of computer algebra systems have implemented heuristics based on pattern-matching and the exploitation of special functions, in particular the incomplete gamma function. Although this approach is heuristic rather than algorithmic, it is nonetheless an effective method for solving many definite integrals encountered by practical engineering applications. Earlier systems such as Macsyma had a few definite integrals related to special functions within a look-up table. However this particular method, involving differentiation of special functions with respect to its parameters, variable transformation, pattern matching and other manipulations, was pioneered by developers of the Maple system and then later emulated by Mathematica, Axiom, MuPAD and other systems.
Recent
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https://en.wikipedia.org/wiki/Inverse-gamma%20distribution
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In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to the gamma distribution.
Perhaps the chief use of the inverse gamma distribution is in Bayesian statistics, where the distribution arises as the marginal posterior distribution for the unknown variance of a normal distribution, if an uninformative prior is used, and as an analytically tractable conjugate prior, if an informative prior is required. It is common among some Bayesians to consider an alternative parametrization of the normal distribution in terms of the precision, defined as the reciprocal of the variance, which allows the gamma distribution to be used directly as a conjugate prior. Other Bayesians prefer to parametrize the inverse gamma distribution differently, as a scaled inverse chi-squared distribution.
Characterization
Probability density function
The inverse gamma distribution's probability density function is defined over the support
with shape parameter and scale parameter . Here denotes the gamma function.
Unlike the Gamma distribution, which contains a somewhat similar exponential term, is a scale parameter as the distribution function satisfies:
Cumulative distribution function
The cumulative distribution function is the regularized gamma function
where the numerator is the upper incomplete gamma function and the denominator is the gamma function. Many math packages allow direct computation of , the regularized gamma function.
Moments
Provided that , the -th moment of the inverse gamma distribution is given by
Characteristic function
in the expression of the characteristic function is the modified Bessel function of the 2nd kind.
Properties
For and ,
and
The information entropy is
where is the digamma function.
The Kullback-Leibler divergence of Inverse-Gamma(αp, βp) from Inverse-Gamma(αq, βq) is the same as the KL-divergence of Gamma(αp, βp) from Gamma(αq, βq):
where are the pdfs of the Inverse-Gamma distributions and are the pdfs of the Gamma distributions, is Gamma(αp, βp) distributed.
Related distributions
If then , for
If then (inverse-chi-squared distribution)
If then (scaled-inverse-chi-squared distribution)
If then (Lévy distribution)
If then (Exponential distribution)
If (Gamma distribution with rate parameter ) then (see derivation in the next paragraph for details)
Note that If (Gamma distribution with scale parameter ) then
Inverse gamma distribution is a special case of type 5 Pearson distribution
A multivariate generalization of the inverse-gamma distribution is the inverse-Wishart distribution.
For the distribution of a sum of independent inverted Gamma variables see Witkovsky (2001)
Derivation from Gamma distribution
Let , and recall that the pdf of the gamma distribution is
, .
Note that is the ra
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https://en.wikipedia.org/wiki/List%20of%20lakes%20of%20Ontario
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This is an incomplete list of lakes in Ontario, a province of Canada. There are over 250,000 lakes in Ontario, constituting around 20% of the world's fresh water supply.
Larger lake statistics
This is a list of lakes of Ontario with an area larger than .
#
24 Mile Lake
A
B
C
D
E
F
G
Gananoque Lake
Garson Lake
Gathering Lake
Gibson Lake (disambiguation), multiple lakes
Gibson Lake (Greater Sudbury)
Gillies Lake
Gloucester Pool
Go Home Lake
Golden Lake
Gordon Lake
Ghost Lake
Gould Lake (disambiguation), several lakes
Green Lake
Grundy Lake
Guelph Lake
Gull Lake (Ontario)
Gullrock Lake
Gunter Lake
H
Halet Lake
Halls Lake (Haliburton County)
Hammer Lake
Head Lake (Kawartha Lakes)
Head Lake (Haliburton County)
Heart Lake
Herbert Lake
Holden Lake
Lake Huron
Horseshoe Lake multiple lakes
I
Inn Lake
Indian Lake
Innis Lake
Irwin Lake
Ivanhoe Lake
J
Jack Lake
Jeff Lake
Lake Joseph
Jules Lake
Jumping Cariboo Lake
K
Kabinakagami Lake
Lake Kagawong
Kahshe Lake
Kamaniskeg Lake
Kashagawigamog Lake
Kashwakamak Lake
Kasshabog Lake
Kawagama Lake
Kawartha Lakes
Lake Kelso
Kennisis Lake
Kesagami Lake
Kimber Lake
Kushog Lake
Lake Kairiskons
Lake Kishkatina
L
Lake Bernard (Parry Sound District)
Lac des Mille Lacs
Lady Evelyn Lake
Lake Madawaska
Lake of Bays (Kenora District)
Lake of Bays (Muskoka lake)
Lake of the Woods
Lake of Two Islands
Larder Lake
Little Branch Lake
Little Lake (Peterborough)
Little Moose Lake
Little Papineau Lake
Little Sachigo Lake
Little Sucker Lake
Little Yirkie Lake
Limerick Lake
Long Lake
Loughborough Lake
Lower Beverley Lake
Lower Buckhorn Lake
M
Mabel Lake
MacDowell Lake
Madawaska Lake
Mameigwess Lake (north Kenora District)
Mameigwess Lake (south Kenora District)
Lake Manitou
Lake Manitouwabing
Maple Lake
Marmion Lake
Mary Lake
Lake Matinenda
Maul Lake
Maynard Lake
Mazinaw Lake
McArthur Lake
McKay Lake (Ottawa)
McKay Lake (Pic River)
McLaren Lake
Mirror Lake
Mississauga Lake
Mississippi Lake
Lake Mindemoya
Minnitaki Lake
Missisa Lake
Mojikit Lake
Mong Lake
Morrison Lake
Mountain Lake
Mozhabong Lake
Mud Lake
Muldrew Lake
Lake Muskoka
Muskrat Lake
N
Nameless Lake (Sudbury District)
Net Lake
Nettleton Lake
Nicholls Lake
Night Hawk Lake
Lake Nipigon
Lake Nipissing
North Caribou Lake
Lake Nosbonsing
Nungesser Lake
Nishin Lake
O
Oak Lake
Oba Lake - North
Oba Lake - South
Octopus Lake
Lake Ogoki
Old Man's Lake
Onaman Lake
Onigam Lake
Onion Lake
Lake Ontario
Opeongo Lake
Opinicon Lake
Otter Tail Lake
Ottertooth Lake
Otty Lake
Ozhiski Lake
P
Packsack Lake
Lake Panache
Paint Lake
Pakeshkag Lake
Papineau Lake
Paudash Lake
Peninsula Lake
Perch Lake
Pelican Lake
Percy Lake
Peters Lake (Sudbury District)
Pierce Lake
Pierre Lake
Pike Lake
Pigeon Lake
Pog Lake
Pokei Lake
Pot Lake
Priamo Lake
Professor's Lake
Pumphouse Lake
Puslinch Lake
Pine Lake
R
Rainbow Lake
Rainy Lake
Lake Ramsey
Rebecca Lake, multiple lakes
Redbridge Lake
Red Cedar Lake
Red Squirrel Lake
Redstone Lake (Haliburton County)
Redstone Lake (Sudbury District)
Restoul
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https://en.wikipedia.org/wiki/Method%20of%20undetermined%20coefficients
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In mathematics, the method of undetermined coefficients is an approach to finding a particular solution to certain nonhomogeneous ordinary differential equations and recurrence relations. It is closely related to the annihilator method, but instead of using a particular kind of differential operator (the annihilator) in order to find the best possible form of the particular solution, an ansatz or 'guess' is made as to the appropriate form, which is then tested by differentiating the resulting equation. For complex equations, the annihilator method or variation of parameters is less time-consuming to perform.
Undetermined coefficients is not as general a method as variation of parameters, since it only works for differential equations that follow certain forms.
Description of the method
Consider a linear non-homogeneous ordinary differential equation of the form
where denotes the i-th derivative of , and denotes a function of .
The method of undetermined coefficients provides a straightforward method of obtaining the solution to this ODE when two criteria are met:
are constants.
g(x) is a constant, a polynomial function, exponential function , sine or cosine functions or , or finite sums and products of these functions (, constants).
The method consists of finding the general homogeneous solution for the complementary linear homogeneous differential equation
and a particular integral of the linear non-homogeneous ordinary differential equation based on . Then the general solution to the linear non-homogeneous ordinary differential equation would be
If consists of the sum of two functions and we say that is the solution based on and the solution based on . Then, using a superposition principle, we can say that the particular integral is
Typical forms of the particular integral
In order to find the particular integral, we need to 'guess' its form, with some coefficients left as variables to be solved for. This takes the form of the first derivative of the complementary function. Below is a table of some typical functions and the solution to guess for them.
If a term in the above particular integral for y appears in the homogeneous solution, it is necessary to multiply by a sufficiently large power of x in order to make the solution independent. If the function of x is a sum of terms in the above table, the particular integral can be guessed using a sum of the corresponding terms for y.
Examples
Example 1
Find a particular integral of the equation
The right side t cos t has the form
with n = 2, α = 0, and β = 1.
Since α + iβ = i is a simple root of the characteristic equation
we should try a particular integral of the form
Substituting yp into the differential equation, we have the identity
Comparing both sides, we have
which has the solution
We then have a particular integral
Example 2
Consider the following linear nonhomogeneous differential equation:
This is like the first example above, except that
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https://en.wikipedia.org/wiki/Harald%20Ludvig%20Westergaard
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Harald Ludvig Westergaard (April 19, 1853 in Copenhagen – December 13, 1936 in Copenhagen) was a Danish statistician and economist known for his work in demography and the history of statistics.
Harald Westergaard was born in Copenhagen and apart from a period studying in England and Germany in 1877-78 he lived there all his life. His subject at the University of Copenhagen was mathematics but he became interested in economics and, while he was in England, he seems to have met William Stanley Jevons. In the preface to the second edition (1879) of the Theory of Political Economy Jevons refers to Westergaard's mathematical suggestions. However, after this spectacular debut Westergaard seems not to have contributed further to mathematical economics.
In 1880-1882, Westergaard worked for the Danish Insurance Office and he developed an interest in demography. His international reputation was made by the publication of Die Lehre von der Mortalität und Morbilität (1881). This work won him a gold medal from the University and led to his appointment as a lecturer in 1883. In 1886, he became a Professor at the early age of 33. He retired in 1924.
Westergaard's late work Contributions to the History of Statistics (1932) described the history of vital and economic statistics up to the end of the nineteenth century. Statistical theory, whether of the Laplace or Pearson variety, is discussed but given a subordinate place. In the Introduction, Westergaard remarks, "For a long while ... the calculus of probabilities had less influence on statistics than might have been expected, the authors confining themselves to abstract theories which had little or nothing to do with reality."
Westergaard was well-known and respected internationally. The obituary in the Journal of the Royal Statistical Society of London, begins, "By [his] death Europe has lost her senior statistician" and ends, "This is not the place to write at length about his personal charm, marked by simplicity, helpfulness and friendliness; but it was this as much as his intellectual eminence that gave him a unique place in the society of economists and statisticians."
Books by Harald Westergaard
Die Lehre von der Mortalität und Morbilität: Anthropologisch-statistische Untersuchungen. 1881
Grundzüge der Theorie der Statistik. 1890
Economic Development in Denmark before and during the World War. 1922
Contributions to the History of Statistics, 1932, reprinted 1969 New York: Kelley
References
“ Westergaard, Harald Ludvig”, pp. 319–320 in Leading Personalities in Statistical Sciences from the Seventeenth Century to the Present, (ed. N. L. Johnson and S. Kotz) 1997. New York: Wiley. Originally published in Encyclopedia of Statistical Science.
Obituary: Harald Westergaard, Journal of the Royal Statistical Society, Vol. 100, No. 1. (1937), pp. 149–150.
External links
There is a photograph at
Harald Ludvig Westergaard on the Portraits of Statisticians page.
1853 births
1936 deaths
D
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https://en.wikipedia.org/wiki/Philip%20Hall
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Philip Hall FRS (11 April 1904 – 30 December 1982), was an English mathematician. His major work was on group theory, notably on finite groups and solvable groups.
Biography
He was educated first at Christ's Hospital, where he won the Thompson Gold Medal for mathematics, and later at King's College, Cambridge. He was elected a Fellow of the Royal Society in 1951 and awarded its Sylvester Medal in 1961. He was President of the London Mathematical Society in 1955–1957, and awarded its Berwick Prize in 1958 and De Morgan Medal in 1965.
Publications
See also
Abstract clone
Commutator collecting process
Isoclinism of groups
Regular p-group
Three subgroups lemma
Hall algebra, and Hall polynomials
Hall subgroup
Hall–Higman theorem
Hall–Littlewood polynomial
Hall's universal group
Hall's marriage theorem
Hall word
Hall–Witt identity
Irwin–Hall distribution
Zappa–Szép product
References
1904 births
1982 deaths
20th-century English mathematicians
Algebraists
Group theorists
People educated at Christ's Hospital
Alumni of King's College, Cambridge
Fellows of the Royal Society
Bletchley Park people
Presidents of the London Mathematical Society
Sadleirian Professors of Pure Mathematics
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https://en.wikipedia.org/wiki/Heinz%20Pr%C3%BCfer
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Ernst Paul Heinz Prüfer (10 November 1896 – 7 April 1934) was a German Jewish mathematician born in Wilhelmshaven. His major contributions were on abelian groups, graph theory, algebraic numbers, knot theory and Sturm–Liouville theory.
In 1915 he began his university studies in mathematics, Physics and Chemistry in Berlin. After that he started his doctorate degree with Issai Schur as his advisor at Friedrich Wilhelm University, Berlin. In 1921 he obtained his doctorate degree. His thesis was named Unendliche Abelsche Gruppen von Elementen endlicher Ordnung (Infinite abelian groups of elements of finite order). This thesis set the road for his contributions on abelian groups. In 1922 he worked with mathematician Paul Koebe in the University of Jena, and in 1923 he obtained tenure and was at this university until 1927. In that year he moved to Münster University where he worked until the end of his life. His final work was about projective geometry, but it was posthumously completed by his students Gustav Fleddermann and Gottfried Köthe.
Heinz Prüfer was married, but never had children. He died prematurely at 37 years of age in 1934 in Münster Germany, due to lung cancer.
Mathematical contributions
Heinz Prüfer created the following mathematical notions that were later named after him:
Prüfer sequence (also known as a Prüfer code; it has broad applications in graph theory and network theory).
Prüfer domain. Also see Bézout domain, which is a Prüfer domain
Prüfer rank
Prüfer manifold also known as Prüfer surface or Prüfer analytical manifold
Prüfer group
Prüfer theorems
References
Jürgen Elstrodt and Norbert Schmitz: History of Münster University (2013). Chapter 52. page 111, Heinz Prüfer Biographie. Chapter 52. page 111
External links
1896 births
1934 deaths
20th-century German mathematicians
20th-century German Jews
Academic staff of the University of Münster
People from Wilhelmshaven
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https://en.wikipedia.org/wiki/Zerah%20Colburn
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Zerah Colburn may refer to:
Zerah Colburn (mental calculator) (1804–1840), American mathematics prodigy
Zerah Colburn (locomotive designer) (1832–1870), American steam locomotive designer and railroad author
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https://en.wikipedia.org/wiki/1023%20%28number%29
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1023 (one thousand [and] twenty-three) is the natural number following 1022 and preceding 1024.
In mathematics
1023 is the tenth Mersenne number of the form .
In binary, it is also the tenth repdigit 11111111112 as all Mersenne numbers in decimal are repdigits in binary.
It is equal to the sum of five consecutive prime numbers 193 + 197 + 199 + 211 + 223.
It is the number of three-dimensional polycubes with 7 cells.
1023 is the number of elements in the 9-simplex, as well as the number of uniform polytopes in the tenth-dimensional hypercubic family , and the number of noncompact solutions in the family of paracompact honeycombs that shares symmetries with .
In other fields
Computing
Floating-point units in computers often run a IEEE 754 64-bit, floating-point excess-1023 format in 11-bit binary. In this format, also called binary64, the exponent of a floating-point number (e.g. 1.009001 E1031) appears as an unsigned binary integer from 0 to 2047, where subtracting 1023 from it gives the actual signed value.
1023 is the number of dimensions or length of messages of an error-correcting Reed-Muller code made of 64 block codes.
Technology
The Global Positioning System (GPS) works on a ten-digit binary counter that runs for 1023 weeks, at which point an integer overflow causes its internal value to roll over to zero again.
1023 being , is the maximum number that a 10-bit ADC converter can return when measuring the highest voltage in range.
See also
The year AD 1023
References
Integers
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https://en.wikipedia.org/wiki/Feit%E2%80%93Thompson%20theorem
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In mathematics, the Feit–Thompson theorem, or odd order theorem, states that every finite group of odd order is solvable. It was proved by .
History
conjectured that every nonabelian finite simple group has even order. suggested using the centralizers of involutions of simple groups as the basis for the classification of finite simple groups, as the Brauer–Fowler theorem shows that there are only a finite number of finite simple groups with given centralizer of an involution. A group of odd order has no involutions, so to carry out Brauer's program it is first necessary to show that non-cyclic finite simple groups never have odd order. This is equivalent to showing that odd order groups are solvable, which is what Feit and Thompson proved.
The attack on Burnside's conjecture was started by , who studied CA groups; these are groups such that the Centralizer of every non-trivial element is Abelian. In a pioneering paper he showed that all CA groups of odd order are solvable. (He later classified all the simple CA groups, and more generally all simple groups such that the centralizer of any involution has a normal 2-Sylow subgroup, finding an overlooked family of simple groups of Lie type in the process, that are now called Suzuki groups.)
extended Suzuki's work to the family of CN groups; these are groups such that the Centralizer of every non-trivial element is Nilpotent. They showed that every CN group of odd order is solvable. Their proof is similar to Suzuki's proof. It was about 17 pages long, which at the time was thought to be very long for a proof in group theory.
The Feit–Thompson theorem can be thought of as the next step in this process: they show that there is no non-cyclic simple group of odd order such that every proper subgroup is solvable. This proves that every finite group of odd order is solvable, as a minimal counterexample must be a simple group such that every proper subgroup is solvable. Although the proof follows the same general outline as the CA theorem and the CN theorem, the details are vastly more complicated. The final paper is 255 pages long.
Significance of the proof
The Feit–Thompson theorem showed that the classification of finite simple groups using centralizers of involutions might be possible, as every nonabelian simple group has an involution. Many of the techniques they introduced in their proof, especially the idea of local analysis, were developed further into tools used in the classification. Perhaps the most revolutionary aspect of the proof was its length: before the Feit–Thompson paper, few arguments in group theory were more than a few pages long and most could be read in a day. Once group theorists realized that such long arguments could work, a series of papers that were several hundred pages long started to appear. Some of these dwarfed even the Feit–Thompson paper; the paper by Michael Aschbacher and Stephen D. Smith on quasithin groups was 1,221 pages long.
Revision of the proof
M
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https://en.wikipedia.org/wiki/Edge-of-the-wedge%20theorem
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In mathematics, Bogoliubov's edge-of-the-wedge theorem implies that holomorphic functions on two "wedges" with an "edge" in common are analytic continuations of each other provided they both give the same continuous function on the edge. It is used in quantum field theory to construct the analytic continuation of Wightman functions. The formulation and the first proof of the theorem were presented by Nikolay Bogoliubov at the International Conference on Theoretical Physics, Seattle, USA (September, 1956) and also published in the book Problems in the Theory of Dispersion Relations. Further proofs and generalizations of the theorem were given by R. Jost and H. Lehmann (1957), F. Dyson (1958), H. Epstein (1960), and by other researchers.
The one-dimensional case
Continuous boundary values
In one dimension, a simple case of the edge-of-the-wedge theorem can be stated as follows.
Suppose that f is a continuous complex-valued function on the complex plane that is holomorphic on the upper half-plane, and on the lower half-plane. Then it is holomorphic everywhere.
In this example, the two wedges are the upper half-plane and the lower half plane, and their common edge is the real axis. This result can be proved from Morera's theorem. Indeed, a function is holomorphic provided its integral round any contour vanishes; a contour which crosses the real axis can be broken up into contours in the upper and lower half-planes and the integral round these vanishes by hypothesis.
Distributional boundary values on a circle
The more general case is phrased in terms of distributions. This is technically simplest in the case where the common boundary is the unit circle in the complex plane. In that case holomorphic functions f, g in the regions and have Laurent expansions
absolutely convergent in the same regions and have distributional boundary values given by the formal Fourier series
Their distributional boundary values are equal if for all n. It is then elementary that the common Laurent series converges absolutely in the whole region .
Distributional boundary values on an interval
In general given an open interval on the real axis and holomorphic functions defined in and satisfying
for some non-negative integer N, the boundary values of can be defined as distributions on the real axis by the formulas
Existence can be proved by noting that, under the hypothesis, is the -th complex derivative of a holomorphic function which extends to a continuous function on the boundary. If f is defined as above and below the real axis and F is the distribution defined on the rectangle
by the formula
then F equals off the real axis and the distribution is induced by the distribution on the real axis.
In particular if the hypotheses of the edge-of-the-wedge theorem apply, i.e. , then
By elliptic regularity it then follows that the function F is holomorphic in .
In this case elliptic regularity can be deduced directly from the fact that is known to
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https://en.wikipedia.org/wiki/Integral%20%28disambiguation%29
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Integral is a concept in calculus.
Integral may also refer to:
in mathematics
Integer, a number
Integral symbol
Integral (measure theory), or Lebesgue integration
Integral element
in computer science
Integral data type, a data type that represents some range of mathematical integers
in philosophy and spirituality
Integral humanism (India), political philosophy in Hindu nationalism
Integral theory, an area of discourse emanating from Ken Wilber's thought on spiritual evolution, methodology and ontology. Also known under other names, including integral philosophy, integral worldview, etc.
Integral Culture, transmodern subculture referred to by sociologist Paul H. Ray
as a proper name
INTEGRAL, the International Gamma-Ray Astrophysics Laboratory
Intégral: The Journal of Applied Musical Thought, a music-theory journal
"Integral (song)", a Pet Shop Boys song from Fundamental
The Integral, a glass spaceship in Yevgeny Zamyatin's novel We
Integral (horse), a British Thoroughbred racehorse
Integral, an extended play by The Sixth Lie
Integral (album)
Integral (train), diesel multiple unit train type
See also
Integralism, ideology according to which a nation is an organic unity
Integrality, in commutative algebra, the notions of an element integral over a ring
Integration (disambiguation)
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https://en.wikipedia.org/wiki/RKWard
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RKWard is a transparent front-end to the R programming language, a scripting-language with a strong focus on statistics functions. RKWard tries to combine the power of the R language with the ease of use of commercial statistical packages.
RKWard is written in C++ and although it can run in numerous environments, it was designed for and integrates the KDE desktop environment with the Qt (software) libraries.
Features
RKWard's features include
Spreadsheet-like data editor
Syntax highlighting, code folding and code completion
Data import (e.g. SPSS, Stata and CSV)
Plot preview and browsable history
R package management
Workspace browser
GUI dialogs for all kinds of statistics and plots
Interface
RKWard aims to be easy to use, both for people with deep knowledge of R, and for users who, although they have experience in statistics, are not familiar with the language. The application design offers the possibility of using the graphic tools as well as ignoring many of them and using the program as integrated development environment.
It includes a workspace viewer, which gives access to packages, functions and variables loaded by R or imported from other sources. It also has a file viewer, and data set editing windows, display of the contents of the variables, help, command log and HTML output.
It also offers components that help in code editing and direct order execution, such as the script window and the R console, where you can enter complete commands or programs as you would in the original R text interface. It provides additional help such as syntax coloring documentation of functions while writing, and includes the feature of capturing graphs or emerging dialogs produced by offering additional options for handling, saving and exporting them.
Package Management
The R package management is carried out through a configuration dialog that allows one to, either automatically (because a plug-in requires it) or manually, install new packages from the repository's official project, update existing ones, delete them or upload / download them from the workspace.
Add-ons system
Thanks to its add-ons system RKWard constantly expands the number of functions that can be accessed without writing the code directly. These components allow, from a graphical user interface, instructions to be generated in R for the most common or complex statistical operations. In this way, even without having deep knowledge about the language it is possible to perform advanced data analysis or elaborated graphs. The results of the computations are formatted and presented as HTML, making it possible, with a single click and drag, to export tables and graphs to, for example, office suites.
rk.Teaching
RKTeaching (stylized as rk.Teaching) is a package specially designed for use in teaching and learning statistics, integrating modern packages (such as R2HTML, plyr and ggplot2 among others) as RKWard native outputs. As of 2020, RKTeaching is in version 1.3.0.
See also
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https://en.wikipedia.org/wiki/Boris%20Tsirelson
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Boris Semyonovich Tsirelson (May 4, 1950 – January 21, 2020) (, ) was a Russian–Israeli mathematician and Professor of Mathematics at Tel Aviv University in Israel, as well as a Wikipedia editor.
Biography
Tsirelson was born in Leningrad to a Russian Jewish family. From his father Simeon's side, he was the great-nephew of rabbi Yehuda Leib Tsirelson, chief rabbi of Bessarabia from 1918 to 1941, and a prominent posek and Jewish leader. He obtained his Master of Science from the University of Leningrad and remained there to pursue graduate studies. He obtained his Ph.D. in 1975, with thesis "General properties of bounded Gaussian processes and related questions" written under the direction of Ildar Abdulovich Ibragimov.
Later, he participated in the refusenik movement, but only received permission to emigrate to Israel in 1991. From then until 2017, he was a professor at Tel-Aviv University.
In 1998 he was an Invited Speaker at the International Congress of Mathematicians in Berlin.
Contributions to mathematics
Tsirelson made notable contributions to probability theory and functional analysis. These include:
Tsirelson's bound, in quantum mechanics, is an inequality, related to the issue of quantum nonlocality.
Tsirelson space is an example of a reflexive Banach space in which neither a l p space nor a c0 space can be embedded.
The Tsirelson's drift, a counterexample in the theory of stochastic differential equations, it's a SDE which has a weak solution but no strong solution.
The Gaussian isoperimetric inequality (proved by Vladimir Sudakov and Tsirelson, and independently by Christer Borell), stating that affine halfspaces are the isoperimetric sets for the Gaussian measure.
References
External links
Tsirelson's homepage, at Tel Aviv University
Mourning page, at Tel Aviv University
Mathematicians from Saint Petersburg
Israeli mathematicians
Israeli Jews
Israeli people of Russian-Jewish descent
Soviet emigrants to Israel
Academic staff of Tel Aviv University
1950 births
2020 deaths
Probability theorists
Wikipedia people
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https://en.wikipedia.org/wiki/Mapping%20cylinder
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In mathematics, specifically algebraic topology, the mapping cylinder of a continuous function between topological spaces and is the quotient
where the denotes the disjoint union, and ∼ is the equivalence relation generated by
That is, the mapping cylinder is obtained by gluing one end of to via the map . Notice that the "top" of the cylinder is homeomorphic to , while the "bottom" is the space . It is common to write for , and to use the notation or for the mapping cylinder construction. That is, one writes
with the subscripted cup symbol denoting the equivalence. The mapping cylinder is commonly used to construct the mapping cone , obtained by collapsing one end of the cylinder to a point. Mapping cylinders are central to the definition of cofibrations.
Basic properties
The bottom Y is a deformation retract of .
The projection splits (via ), and the deformation retraction is given by:
(where points in stay fixed because for all ).
The map is a homotopy equivalence if and only if the "top" is a strong deformation retract of . An explicit formula for the strong deformation retraction can be worked out.
Examples
Mapping cylinder of a fiber bundle
For a fiber bundle with fiber , the mapping cylinder
has the equivalence relation
for . Then, there is a canonical map sending a point
to the point , giving a fiber bundle
whose fiber is the cone . To see this, notice the fiber over a point is the quotient space
where every point in is equivalent.
Interpretation
The mapping cylinder may be viewed as a way to replace an arbitrary map by an equivalent cofibration, in the following sense:
Given a map , the mapping cylinder is a space , together with a cofibration and a surjective homotopy equivalence (indeed, Y is a deformation retract of ), such that the composition equals f.
Thus the space Y gets replaced with a homotopy equivalent space , and the map f with a lifted map . Equivalently, the diagram
gets replaced with a diagram
together with a homotopy equivalence between them.
The construction serves to replace any map of topological spaces by a homotopy equivalent cofibration.
Note that pointwise, a cofibration is a closed inclusion.
Applications
Mapping cylinders are quite common homotopical tools. One use of mapping cylinders is to apply theorems concerning inclusions of spaces to general maps, which might not be injective.
Consequently, theorems or techniques (such as homology, cohomology or homotopy theory) which are only dependent on the homotopy class of spaces and maps involved may be applied to with the assumption that and that is actually the inclusion of a subspace.
Another, more intuitive appeal of the construction is that it accords with the usual mental image of a function as "sending" points of to points of and hence of embedding within despite the fact that the function need not be one-to-one.
Categorical application and interpretation
One can use the mapping cylinder to construct ho
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https://en.wikipedia.org/wiki/Geometry%20of%20Complex%20Numbers
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Geometry of Complex Numbers: Circle Geometry, Moebius Transformation, Non-Euclidean Geometry is an undergraduate textbook on geometry, whose topics include circles, the complex plane, inversive geometry, and non-Euclidean geometry. It was written by Hans Schwerdtfeger, and originally published in 1962 as Volume 13 of the Mathematical Expositions series of the University of Toronto Press. A corrected edition was published in 1979 in the Dover Books on Advanced Mathematics series of Dover Publications (). The Basic Library List Committee of the Mathematical Association of America has suggested its inclusion in undergraduate mathematics libraries.
Topics
The book is divided into three chapters, corresponding to the three parts of its subtitle: circle geometry, Möbius transformations, and non-Euclidean geometry. Each of these is further divided into sections (which in other books would be called chapters) and sub-sections. An underlying theme of the book is the representation of the Euclidean plane as the plane of complex numbers, and the use of complex numbers as coordinates to describe geometric objects and their transformations.
The chapter on circles covers the analytic geometry of circles in the complex plane. It describes the representation of circles by Hermitian matrices, the inversion of circles, stereographic projection, pencils of circles (certain one-parameter families of circles) and their two-parameter analogue, bundles of circles, and the cross-ratio of four complex numbers.
The chapter on Möbius transformations is the central part of the book, and defines these transformations as the fractional linear transformations of the complex plane (one of several standard ways of defining them). It includes material on the classification of these transformations, on the characteristic parallelograms of these transformations, on the subgroups of the group of transformations, on iterated transformations that either return to the identity (forming a periodic sequence) or produce an infinite sequence of transformations, and a geometric characterization of these transformations as the circle-preserving transformations of the complex plane. This chapter also briefly discusses applications of Möbius transformations in understanding the projectivities and perspectivities of projective geometry.
In the chapter on non-Euclidean geometry, the topics include the Poincaré disk model of the hyperbolic plane, elliptic geometry, spherical geometry, and (in line with Felix Klein's Erlangen program) the transformation groups of these geometries as subgroups of Möbious transformations.
This work brings together multiple areas of mathematics, with the intent of broadening the connections between abstract algebra, the theory of complex numbers, the theory of matrices, and geometry.
Reviewer Howard Eves writes that, in its selection of material and its formulation of geometry, the book "largely reflects work of C. Caratheodory and E. Cartan".
Audience and rec
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https://en.wikipedia.org/wiki/Prime%20model
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In mathematics, and in particular model theory, a prime model is a model that is as simple as possible. Specifically, a model is prime if it admits an elementary embedding into any model to which it is elementarily equivalent (that is, into any model satisfying the same complete theory as ).
Cardinality
In contrast with the notion of saturated model, prime models are restricted to very specific cardinalities by the Löwenheim–Skolem theorem. If is a first-order language with cardinality and is a complete theory over then this theorem guarantees a model for of cardinality Therefore no prime model of can have larger cardinality since at the very least it must be elementarily embedded in such a model. This still leaves much ambiguity in the actual cardinality. In the case of countable languages, all prime models are at most countably infinite.
Relationship with saturated models
There is a duality between the definitions of prime and saturated models. Half of this duality is discussed in the article on saturated models, while the other half is as follows. While a saturated model realizes as many types as possible, a prime model realizes as few as possible: it is an atomic model, realizing only the types that cannot be omitted and omitting the remainder. This may be interpreted in the sense that a prime model admits "no frills": any characteristic of a model that is optional is ignored in it.
For example, the model is a prime model of the theory of the natural numbers N with a successor operation S; a non-prime model might be meaning that there is a copy of the full integers that lies disjoint from the original copy of the natural numbers within this model; in this add-on, arithmetic works as usual. These models are elementarily equivalent; their theory admits the following axiomatization (verbally):
There is a unique element that is not the successor of any element;
No two distinct elements have the same successor;
No element satisfies Sn(x) = x with n > 0.
These are, in fact, two of Peano's axioms, while the third follows from the first by induction (another of Peano's axioms). Any model of this theory consists of disjoint copies of the full integers in addition to the natural numbers, since once one generates a submodel from 0 all remaining points admit both predecessors and successors indefinitely. This is the outline of a proof that is a prime model.
References
Mathematical logic
Model theory
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https://en.wikipedia.org/wiki/Mereotopology
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In formal ontology, a branch of metaphysics, and in ontological computer science, mereotopology is a first-order theory, embodying mereological and topological concepts, of the relations among wholes, parts, parts of parts, and the boundaries between parts.
History and motivation
Mereotopology begins in philosophy with theories articulated by A. N. Whitehead in several books and articles he published between 1916 and 1929, drawing in part on the mereogeometry of De Laguna (1922). The first to have proposed the idea of a point-free definition of the concept of topological space in mathematics was Karl Menger in his book Dimensionstheorie (1928) -- see also his (1940). The early historical background of mereotopology is documented in Bélanger and Marquis (2013) and Whitehead's early work is discussed in Kneebone (1963: ch. 13.5) and Simons (1987: 2.9.1). The theory of Whitehead's 1929 Process and Reality augmented the part-whole relation with topological notions such as contiguity and connection. Despite Whitehead's acumen as a mathematician, his theories were insufficiently formal, even flawed. By showing how Whitehead's theories could be fully formalized and repaired, Clarke (1981, 1985) founded contemporary mereotopology. The theories of Clarke and Whitehead are discussed in Simons (1987: 2.10.2), and Lucas (2000: ch. 10). The entry Whitehead's point-free geometry includes two contemporary treatments of Whitehead's theories, due to Giangiacomo Gerla, each different from the theory set out in the next section.
Although mereotopology is a mathematical theory, we owe its subsequent development to logicians and theoretical computer scientists. Lucas (2000: ch. 10) and Casati and Varzi (1999: ch. 4,5) are introductions to mereotopology that can be read by anyone having done a course in first-order logic. More advanced treatments of mereotopology include Cohn and Varzi (2003) and, for the mathematically sophisticated, Roeper (1997). For a mathematical treatment of point-free geometry, see Gerla (1995). Lattice-theoretic (algebraic) treatments of mereotopology as contact algebras have been applied to separate the topological from the mereological structure, see Stell (2000), Düntsch and Winter (2004).
Applications
Barry Smith, Anthony Cohn, Achille Varzi and their co-authors have shown that mereotopology can be useful in formal ontology and computer science, by allowing the formalization of relations such as contact, connection, boundaries, interiors, holes, and so on. Mereotopology has been applied also as a tool for qualitative spatial-temporal reasoning, with constraint calculi such as the Region Connection Calculus (RCC). It provides the starting point for the theory of fiat boundaries developed by Smith and Varzi, which grew out of the attempt to distinguish formally between
boundaries (in geography, geopolitics, and other domains) which reflect more or less arbitrary human demarcations and
boundaries which reflect bona fide physical disconti
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https://en.wikipedia.org/wiki/Pullback%20bundle
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In mathematics, a pullback bundle or induced bundle is the fiber bundle that is induced by a map of its base-space. Given a fiber bundle and a continuous map one can define a "pullback" of by as a bundle over . The fiber of over a point in is just the fiber of over . Thus is the disjoint union of all these fibers equipped with a suitable topology.
Formal definition
Let be a fiber bundle with abstract fiber and let be a continuous map. Define the pullback bundle by
and equip it with the subspace topology and the projection map given by the projection onto the first factor, i.e.,
The projection onto the second factor gives a map
such that the following diagram commutes:
If is a local trivialization of then is a local trivialization of where
It then follows that is a fiber bundle over with fiber . The bundle is called the pullback of E by or the bundle induced by . The map is then a bundle morphism covering .
Properties
Any section of over induces a section of , called the pullback section , simply by defining
for all .
If the bundle has structure group with transition functions (with respect to a family of local trivializations ) then the pullback bundle also has structure group . The transition functions in are given by
If is a vector bundle or principal bundle then so is the pullback . In the case of a principal bundle the right action of on is given by
It then follows that the map covering is equivariant and so defines a morphism of principal bundles.
In the language of category theory, the pullback bundle construction is an example of the more general categorical pullback. As such it satisfies the corresponding universal property.
The construction of the pullback bundle can be carried out in subcategories of the category of topological spaces, such as the category of smooth manifolds. The latter construction is useful in differential geometry and topology.
Bundles and sheaves
Bundles may also be described by their sheaves of sections. The pullback of bundles then corresponds to the inverse image of sheaves, which is a contravariant functor. A sheaf, however, is more naturally a covariant object, since it has a pushforward, called the direct image of a sheaf. The tension and interplay between bundles and sheaves, or inverse and direct image, can be advantageous in many areas of geometry. However, the direct image of a sheaf of sections of a bundle is not in general the sheaf of sections of some direct image bundle, so that although the notion of a 'pushforward of a bundle' is defined in some contexts (for example, the pushforward by a diffeomorphism), in general it is better understood in the category of sheaves, because the objects it creates cannot in general be bundles.
References
Sources
Further reading
Fiber bundles
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https://en.wikipedia.org/wiki/Euclid%20number
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In mathematics, Euclid numbers are integers of the form , where pn # is the nth primorial, i.e. the product of the first n prime numbers. They are named after the ancient Greek mathematician Euclid, in connection with Euclid's theorem that there are infinitely many prime numbers.
Examples
For example, the first three primes are 2, 3, 5; their product is 30, and the corresponding Euclid number is 31.
The first few Euclid numbers are 3, 7, 31, 211, 2311, 30031, 510511, 9699691, 223092871, 6469693231, 200560490131, ... .
History
It is sometimes falsely stated that Euclid's celebrated proof of the infinitude of prime numbers relied on these numbers. Euclid did not begin with the assumption that the set of all primes is finite. Rather, he said: consider any finite set of primes (he did not assume that it contained only the first n primes, e.g. it could have been ) and reasoned from there to the conclusion that at least one prime exists that is not in that set.
Nevertheless, Euclid's argument, applied to the set of the first n primes, shows that the nth Euclid number has a prime factor that is not in this set.
Properties
Not all Euclid numbers are prime.
E6 = 13# + 1 = 30031 = 59 × 509 is the first composite Euclid number.
Every Euclid number is congruent to 3 modulo 4 since the primorial of which it is composed is twice the product of only odd primes and thus congruent to 2 modulo 4. This property implies that no Euclid number can be a square.
For all the last digit of En is 1, since is divisible by 2 and 5. In other words, since all primorial numbers greater than E2 have 2 and 5 as prime factors, they are divisible by 10, thus all En ≥ 3 + 1 have a final digit of 1.
Unsolved problems
It is not known whether there is an infinite number of prime Euclid numbers (primorial primes).
It is also unknown whether every Euclid number is a squarefree number.
Generalization
A Euclid number of the second kind (also called Kummer number) is an integer of the form En = pn # − 1, where pn # is the nth primorial. The first few such numbers are:
1, 5, 29, 209, 2309, 30029, 510509, 9699689, 223092869, 6469693229, 200560490129, ...
As with the Euclid numbers, it is not known whether there are infinitely many prime Kummer numbers. The first of these numbers to be composite is 209.
See also
Euclid–Mullin sequence
Proof of the infinitude of the primes (Euclid's theorem)
References
Eponymous numbers in mathematics
Integer sequences
Unsolved problems in number theory
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https://en.wikipedia.org/wiki/Diffusion%20process
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In probability theory and statistics, diffusion processes are a class of continuous-time Markov process with almost surely continuous sample paths. Diffusion process is stochastic in nature and hence is used to model many real-life stochastic systems. Brownian motion, reflected Brownian motion and Ornstein–Uhlenbeck processes are examples of diffusion processes. It is used heavily in statistical physics, statistical analysis, information theory, data science, neural networks, finance and marketing.
A sample path of a diffusion process models the trajectory of a particle embedded in a flowing fluid and subjected to random displacements due to collisions with other particles, which is called Brownian motion. The position of the particle is then random; its probability density function as a function of space and time is governed by a convection–diffusion equation.
Mathematical definition
A diffusion process is a Markov process with continuous sample paths for which the Kolmogorov forward equation is the Fokker–Planck equation.
See also
Diffusion
Itô diffusion
Jump diffusion
Sample-continuous process
References
Markov processes
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https://en.wikipedia.org/wiki/Semi-Hilbert%20space
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In mathematics, a semi-Hilbert space is a generalization of a Hilbert space in functional analysis, in which, roughly speaking, the inner product is required only to be positive semi-definite rather than positive definite, so that it gives rise to a seminorm rather than a vector space norm.
The quotient of this space by the kernel of this seminorm is also required to be a Hilbert space in the usual sense.
References
Optimal Interpolation in Semi-Hilbert Spaces
Topological vector spaces
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https://en.wikipedia.org/wiki/Fundamental%20polygon
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In mathematics, a fundamental polygon can be defined for every compact Riemann surface of genus greater than 0. It encodes not only the topology of the surface through its fundamental group but also determines the Riemann surface up to conformal equivalence. By the uniformization theorem, every compact Riemann surface has simply connected universal covering surface given by exactly one of the following:
the Riemann sphere,
the complex plane,
the unit disk D or equivalently the upper half-plane H.
In the first case of genus zero, the surface is conformally equivalent to the Riemann sphere.
In the second case of genus one, the surface is conformally equivalent to a torus C/Λ for some lattice Λ in C. The fundamental polygon of Λ, if assumed convex, may be taken to be either a period parallelogram or a centrally symmetric hexagon, a result first proved by Fedorov in 1891.
In the last case of genus g > 1, the Riemann surface is conformally equivalent to H/Γ where Γ is a Fuchsian group of Möbius transformations. A fundamental domain for Γ is given by a convex polygon for the hyperbolic metric on H. These can be defined by Dirichlet polygons and have an even number of sides. The structure of the fundamental group Γ can be read off from such a polygon. Using the theory of quasiconformal mappings and the Beltrami equation, it can be shown there is a canonical convex Dirichlet polygon with 4g sides, first defined by Fricke, which corresponds to the standard presentation of Γ as the group with 2g generators a1, b1, a2, b2, ..., ag, bg and the single relation [a1,b1][a2,b2] ⋅⋅⋅ [ag,bg] = 1, where [a,b] = a b a−1b−1.
Any Riemannian metric on an oriented closed 2-manifold M defines a complex structure on M, making M a compact Riemann surface. Through the use of fundamental polygons, it follows that two oriented closed 2-manifolds are classified by their genus, that is half the rank of the Abelian group Γ/[Γ,Γ], where Γ = 1(M). Moreover, it also follows from the theory of quasiconformal mappings that two compact Riemann surfaces are diffeomorphic if and only if they are homeomorphic. Consequently, two closed oriented 2-manifolds are
homeomorphic if and only if they are diffeomorphic. Such a result can also be proved using the methods of differential topology.
Fundamental polygons in genus one
Parallelograms and centrally symmetric hexagons
In the case of genus one, a fundamental convex polygon is sought for the action by translation of Λ = Z a ⊕ Z b on R2 = C where a and b are linearly independent over R. (After performing a real linear transformation on R2, it can be assumed if necessary that Λ = Z2 = Z + Z i; for a genus one Riemann surface it can be taken to have the form Λ = Z2 = Z + Z ω, with Im ω > 0.) A fundamental domain is given by the parallelogram for where and are generators of Λ.
If C is the interior of a fundamental convex polygon, then the translates + x cover R2 as x runs over Λ. It follows that the boundary points of C are f
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https://en.wikipedia.org/wiki/Severi%E2%80%93Brauer%20variety
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In mathematics, a Severi–Brauer variety over a field K is an algebraic variety V which becomes isomorphic to a projective space over an algebraic closure of K. The varieties are associated to central simple algebras in such a way that the algebra splits over K if and only if the variety has a rational point over K. studied these varieties, and they are also named after Richard Brauer because of their close relation to the Brauer group.
In dimension one, the Severi–Brauer varieties are conics. The corresponding central simple algebras are the quaternion algebras. The algebra (a,b)K corresponds to the conic C(a,b) with equation
and the algebra (a,b)K splits, that is, (a,b)K is isomorphic to a matrix algebra over K, if and only if C(a,b) has a point defined over K: this is in turn equivalent to C(a,b) being isomorphic to the projective line over K.
Such varieties are of interest not only in diophantine geometry, but also in Galois cohomology. They represent (at least if K is a perfect field) Galois cohomology classes in
H1(PGLn),
where PGLn
is the projective linear group, and n is the dimension of
the variety V. There is a short exact sequence
1 → GL1 → GLn → PGLn → 1
of algebraic groups. This implies a connecting homomorphism
H1(PGLn) → H2(GL1)
at the level of cohomology. Here H2(GL1) is identified with the Brauer group of K, while the kernel is trivial because
H1(GLn) = {1}
by an extension of Hilbert's Theorem 90. Therefore, Severi–Brauer varieties can be faithfully represented by Brauer group elements, i.e. classes of central simple algebras.
Lichtenbaum showed that if X is a Severi–Brauer variety over K then there is an exact sequence
Here the map δ sends 1 to the Brauer class corresponding to X.
As a consequence, we see that if the class of X has order d in the Brauer group then there is a divisor class of degree d on X. The associated linear system defines the d-dimensional embedding of X over a splitting field L.
See also
projective bundle
Note
References
Further reading
External links
Expository paper on Galois descent (PDF)
Algebraic varieties
Diophantine geometry
Homological algebra
Algebraic groups
Ring theory
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https://en.wikipedia.org/wiki/England%20national%20football%20team%20records%20and%20statistics
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The history of the England national football team, also known as the Three Lions, begins with the first representative international match in 1870 and the first officially-recognised match two years later. England primarily competed in the British Home Championship over the following decades. Although the FA had joined the international governing body of association football FIFA in 1906, the relationship with the British associations was fraught. In 1928, the British nations withdrew from FIFA, in a dispute over payments to amateur players. This meant that England did not enter the first three World Cups.
The Three Lions first entered the World Cup in 1950 and have since qualified for 16 of the 19 finals tournaments to 2022. They won the 1966 World Cup on home soil making them one of only eight nations to have won a FIFA World Cup. They have reached the semi-finals on two other occasions, in 1990 and 2018. The Three Lions have been eliminated from the World Cup quarter-final stage on seven occasions – more often than any other nation. England failed to qualify for the finals in 1974, 1978, and 1994.
England also compete in the UEFA European Championship. During UEFA Euro 2020, they reached the final of the competition for the first time, finishing as runners-up. England have also reached the semi-final of the competition in 1968 and 1996 with the latter held on home soil. England's most capped player is Peter Shilton with 125 caps and its top goalscorer is Harry Kane with 61 goals. England compete in the FIFA World Cup, UEFA European Championship, and UEFA Nations League. However, as a constituent country of the United Kingdom, England are not a member of the International Olympic Committee so are not eligible to compete in the Olympic games.
This list encompasses honours won by the England national team, and records set by both players and managers including appearance and goal records. It also records England's record victories.
Honours and achievements
Source:
Major
FIFA World Cup
Champions: 1966
UEFA European Championship
Runners-up: 2020
Third place: 1968, 1996
UEFA Nations League Finals
Third place: 2019
Regional
British Home Championship
Champions outright (40): 1887–88, 1889–90, 1890–91, 1891–92, 1892–93, 1894–95, 1897–98, 1898–99, 1900–01, 1902–03, 1903–04, 1904–05, 1908–09, 1910–11, 1912–13, 1929–30, 1930–31, 1931–32, 1934–35, 1937–38, 1946–47, 1947–48, 1949–50, 1953–54, 1954–55, 1956–57, 1960–61, 1964–65, 1965–66, 1967–68, 1968–69, 1970–71, 1972–73, 1974–75, 1977–78, 1978–79, 1981–82, 1982–83
Shared (14): 1885–86, 1905–06, 1907–08, 1911–12, 1938–39, 1951–52, 1952–53, 1955–56, 1957–58, 1958–59, 1959–60, 1963–64, 1969–70, 1973–74
Rous Cup
Champions: 1986, 1988, 1989
Minor
England Challenge Cup
Champions: 1991
Tournoi de France
Champions: 1997
FA Summer Tournament
Champions: 2004
Awards
FIFA World Cup:
FIFA Fair Play Trophy: 1990, 1998 (shared), 2022
BBC Sports Personality of the Year:
BBC Sports Team of the Year Award:
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https://en.wikipedia.org/wiki/CoCoA
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CoCoA (Computations in Commutative Algebra)
is a free computer algebra system developed by the University of Genova, Italy, used to compute with numbers and polynomials. The CoCoA Library (CoCoALib)
is available under GNU General Public License. CoCoA has been ported to many operating systems including Macintosh on PPC and x86, Linux on x86, Unix x86-64 & PPC, Solaris on SPARC and Windows on x86.
CoCoA is mainly used by researchers (see citations at
and),
but can be useful even for "simple" computations.
CoCoA's features include:
Very big integers and rational numbers using the GNU Multi-Precision Library
Multivariate Polynomials
Gröbner basis
User interfaces: text; Emacs-based; Qt-based
It is able to perform simple and sophisticated operations on multivariate polynomials and on various data related to them (ideals, modules, matrices, rational functions). For example, it can readily compute Gröbner basis, syzygies and minimal free resolutions, intersection, division, the radical of an ideal, the ideal of zero-dimensional schemes, Poincaré series and Hilbert functions, factorization of polynomials, and toric ideals. The capabilities of CoCoA and the flexibility of its use are further enhanced by the dedicated high-level programming language.
Its mathematical core, CoCoALib, has been designed as an open source C++ library, focussing on ease of use and flexibility.
CoCoALib is based on GNU Multi-Precision Library.
CoCoALib is used by
ApCoCoA
and
NmzIntegrate
See also
List of computer algebra systems
Standard Template Library
References
External links
ApCoCoA, an extension of CoCoA
Computer algebra system software for Linux
Free computer algebra systems
Science software that uses Qt
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https://en.wikipedia.org/wiki/Walter%20Feit
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Walter Feit (October 26, 1930 – July 29, 2004) was an Austrian-born American mathematician who worked in finite group theory and representation theory. His contributions provided elementary infrastructure used in algebra, geometry, topology, number theory, and logic. His work helped the development and utilization of sectors like cryptography, chemistry, and physics.
He was born to a Jewish family in Vienna and escaped for England in 1939 via the Kindertransport. He moved to the United States in 1946 where he became an undergraduate at the University of Chicago. He did his Ph.D. at the University of Michigan, and became a professor at Cornell University in 1952, and at Yale University in 1964.
His most famous result is his proof, joint with John G. Thompson, of the Feit–Thompson theorem that all finite groups of odd order are solvable. At the time it was written, it was probably the most complicated and difficult mathematical proof ever completed.He wrote almost a hundred other papers, mostly on finite group theory, character theory (in particular introducing the concept of a coherent set of characters), and modular representation theory. Another regular theme in his research was the study of linear groups of small degree, that is, finite groups of matrices in low dimensions. It was often the case that, while the conclusions concerned groups of complex matrices, the techniques employed were from modular representation theory.
He also wrote the books:The representation theory of finite groups and Characters of finite groups, which are now standard references on character theory, including treatments of modular representations
and modular characters.
Feit was an invited speaker at the International Congress of Mathematicians (ICM) in Nice in 1970.
He was awarded the Cole Prize by the American Mathematical Society in 1965, and was elected to the United States National Academy of Sciences and the American Academy of Arts and Sciences. He also served as Vice-President of the International Mathematical Union.
"In October 2003, on the eve of Professor Feit's retirement, colleagues and former students gathered at Yale for a special four-day "Conference on Groups, Representations and Galois Theory" to honor him and his contributions. Nearly 80 researchers from around the world met to exchange ideas in the fields he had helped to create."
He died in Branford, Connecticut in 2004 and was survived by his wife, Dr. Sidnie Feit, and a son and daughter.
"A memorial service was held on Sunday, October 10, 2004, at the New Haven Lawn Club, 193 Whitney Avenue, New Haven, CT."
Selected publications
References
External links
Yale obituary
Walter Feit (1930–2004), Notices of the American Mathematical Society; vol. 52, no. 7 (August 2005).
1930 births
2004 deaths
20th-century American mathematicians
Mathematicians from Vienna
Group theorists
Institute for Advanced Study visiting scholars
Cornell University faculty
Yale University faculty
Universi
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https://en.wikipedia.org/wiki/Space%20partitioning
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In geometry, space partitioning is the process of dividing an entire space (usually a Euclidean space) into two or more disjoint subsets (see also partition of a set). In other words, space partitioning divides a space into non-overlapping regions. Any point in the space can then be identified to lie in exactly one of the regions.
Overview
Space-partitioning systems are often hierarchical, meaning that a space (or a region of space) is divided into several regions, and then the same space-partitioning system is recursively applied to each of the regions thus created. The regions can be organized into a tree, called a space-partitioning tree.
Most space-partitioning systems use planes (or, in higher dimensions, hyperplanes) to divide space: points on one side of the plane form one region, and points on the other side form another. Points exactly on the plane are usually arbitrarily assigned to one or the other side. Recursively partitioning space using planes in this way produces a BSP tree, one of the most common forms of space partitioning.
Uses
In computer graphics
Space partitioning is particularly important in computer graphics, especially heavily used in ray tracing, where it is frequently used to organize the objects in a virtual scene. A typical scene may contain millions of polygons. Performing a ray/polygon intersection test with each would be a very computationally expensive task.
Storing objects in a space-partitioning data structure (k-d tree or BSP tree for example) makes it easy and fast to perform certain kinds of geometry queries—for example in determining whether a ray intersects an object, space partitioning can reduce the number of intersection test to just a few per primary ray, yielding a logarithmic time complexity with respect to the number of polygons.
Space partitioning is also often used in scanline algorithms to eliminate the polygons out of the camera's viewing frustum, limiting the number of polygons processed by the pipeline. There is also a usage in collision detection: determining whether two objects are close to each other can be much faster using space partitioning.
In integrated circuit design
In integrated circuit design, an important step is design rule check. This step ensures that the completed design is manufacturable. The check involves rules that specify widths and spacings and other geometry patterns. A modern design can have billions of polygons that represent wires and transistors. Efficient checking relies heavily on geometry query. For example, a rule may specify that any polygon must be at least n nanometers from any other polygon. This is converted into a geometry query by enlarging a polygon by n/2 at all sides and query to find all intersecting polygons.
In probability and statistical learning theory
The number of components in a space partition plays a central role in some results in probability theory. See Growth function for more details.
In Geography and GIS
There are m
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https://en.wikipedia.org/wiki/Hyperfunction
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In mathematics, hyperfunctions are generalizations of functions, as a 'jump' from one holomorphic function to another at a boundary, and can be thought of informally as distributions of infinite order. Hyperfunctions were introduced by Mikio Sato in 1958 in Japanese, (1959, 1960 in English), building upon earlier work by Laurent Schwartz, Grothendieck and others.
Formulation
A hyperfunction on the real line can be conceived of as the 'difference' between one holomorphic function defined on the upper half-plane and another on the lower half-plane. That is, a hyperfunction is specified by a pair (f, g), where f is a holomorphic function on the upper half-plane and g is a holomorphic function on the lower half-plane.
Informally, the hyperfunction is what the difference would be at the real line itself. This difference is not affected by adding the same holomorphic function to both f and g, so if h is a holomorphic function on the whole complex plane, the hyperfunctions (f, g) and (f + h, g + h) are defined to be equivalent.
Definition in one dimension
The motivation can be concretely implemented using ideas from sheaf cohomology. Let be the sheaf of holomorphic functions on Define the hyperfunctions on the real line as the first local cohomology group:
Concretely, let and be the upper half-plane and lower half-plane respectively. Then so
Since the zeroth cohomology group of any sheaf is simply the global sections of that sheaf, we see that a hyperfunction is a pair of holomorphic functions one each on the upper and lower complex halfplane modulo entire holomorphic functions.
More generally one can define for any open set as the quotient where is any open set with . One can show that this definition does not depend on the choice of giving another reason to think of hyperfunctions as "boundary values" of holomorphic functions.
Examples
If f is any holomorphic function on the whole complex plane, then the restriction of f to the real axis is a hyperfunction, represented by either (f, 0) or (0, −f).
The Heaviside step function can be represented as where is the principal value of the complex logarithm of .
The Dirac delta "function" is represented by This is really a restatement of Cauchy's integral formula. To verify it one can calculate the integration of f just below the real line, and subtract integration of g just above the real line - both from left to right. Note that the hyperfunction can be non-trivial, even if the components are analytic continuation of the same function. Also this can be easily checked by differentiating the Heaviside function.
If g is a continuous function (or more generally a distribution) on the real line with support contained in a bounded interval I, then g corresponds to the hyperfunction (f, −f), where f is a holomorphic function on the complement of I defined by This function f jumps in value by g(x) when crossing the real axis at the point x. The formula for f follows from the previous exam
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https://en.wikipedia.org/wiki/Fodor%27s%20lemma
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In mathematics, particularly in set theory, Fodor's lemma states the following:
If is a regular, uncountable cardinal, is a stationary subset of , and is regressive (that is, for any , ) then there is some and some stationary such that for any . In modern parlance, the nonstationary ideal is normal.
The lemma was first proved by the Hungarian set theorist, Géza Fodor in 1956. It is sometimes also called "The Pressing Down Lemma".
Proof
We can assume that (by removing 0, if necessary).
If Fodor's lemma is false, for every there is some club set such that . Let . The club sets are closed under diagonal intersection, so is also club and therefore there is some . Then for each , and so there can be no such that , so , a contradiction.
Fodor's lemma also holds for Thomas Jech's notion of stationary sets as well as for the general notion of stationary set.
Fodor's lemma for trees
Another related statement, also known as Fodor's lemma (or Pressing-Down-lemma), is the following:
For every non-special tree and regressive mapping (that is, , with respect to the order on , for every ), there is a non-special subtree on which is constant.
References
G. Fodor, Eine Bemerkung zur Theorie der regressiven Funktionen, Acta Sci. Math. Szeged, 17(1956), 139-142 .
Karel Hrbacek & Thomas Jech, Introduction to Set Theory, 3rd edition, Chapter 11, Section 3.
Mark Howard, Applications of Fodor's Lemma to Vaught's Conjecture. Ann. Pure and Appl. Logic 42(1): 1-19 (1989).
Simon Thomas, The Automorphism Tower Problem. PostScript file at
S. Todorcevic, Combinatorial dichotomies in set theory. pdf at
Articles containing proofs
Lemmas in set theory
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https://en.wikipedia.org/wiki/Stationary%20set
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In mathematics, specifically set theory and model theory, a stationary set is a set that is not too small in the sense that it intersects all club sets and is analogous to a set of non-zero measure in measure theory. There are at least three closely related notions of stationary set, depending on whether one is looking at subsets of an ordinal, or subsets of something of given cardinality, or a powerset.
Classical notion
If is a cardinal of uncountable cofinality, and intersects every club set in then is called a stationary set. If a set is not stationary, then it is called a thin set. This notion should not be confused with the notion of a thin set in number theory.
If is a stationary set and is a club set, then their intersection is also stationary. This is because if is any club set, then is a club set, thus is nonempty. Therefore, must be stationary.
See also: Fodor's lemma
The restriction to uncountable cofinality is in order to avoid trivialities: Suppose has countable cofinality. Then is stationary in if and only if is bounded in . In particular, if the cofinality of is , then any two stationary subsets of have stationary intersection.
This is no longer the case if the cofinality of is uncountable. In fact, suppose is moreover regular and is stationary. Then can be partitioned into many disjoint stationary sets. This result is due to Solovay. If is a successor cardinal, this result is due to Ulam and is easily shown by means of what is called an Ulam matrix.
H. Friedman has shown that for every countable successor ordinal , every stationary subset of contains a closed subset of order type .
Jech's notion
There is also a notion of stationary subset of , for a cardinal and a set such that , where is the set of subsets of of cardinality : . This notion is due to Thomas Jech. As before, is stationary if and only if it meets every club, where a club subset of is a set unbounded under and closed under union of chains of length at most . These notions are in general different, although for and they coincide in the sense that is stationary if and only if is stationary in .
The appropriate version of Fodor's lemma also holds for this notion.
Generalized notion
There is yet a third notion, model theoretic in nature and sometimes referred to as generalized stationarity. This notion is probably due to Magidor, Foreman and Shelah and has also been used prominently by Woodin.
Now let be a nonempty set. A set is club (closed and unbounded) if and only if there is a function such that . Here, is the collection of finite subsets of .
is stationary in if and only if it meets every club subset of .
To see the connection with model theory, notice that if is a structure with universe in a countable language and is a Skolem function for , then a stationary must contain an elementary substructure of . In fact, is stationary if and only if for any such structure there is an elementary substructure of t
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https://en.wikipedia.org/wiki/Thin%20set
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In mathematics, thin set may refer to:
Thin set (analysis) in analysis of several complex variables
Thin set (Serre) in algebraic geometry
In set theory, a set that is not a stationary set
Thin set can also refer to thin set mortar.
See also
Meagre set
Shrinking space
Slender group
Small set
Thin category
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https://en.wikipedia.org/wiki/Diagonal%20intersection
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Diagonal intersection is a term used in mathematics, especially in set theory.
If is an ordinal number and
is a sequence of subsets of , then the diagonal intersection, denoted by
is defined to be
That is, an ordinal is in the diagonal intersection if and only if it is contained in the first members of the sequence. This is the same as
where the closed interval from 0 to is used to
avoid restricting the range of the intersection.
See also
Club filter
Club set
Fodor's lemma
References
Thomas Jech, Set Theory, The Third Millennium Edition, Springer-Verlag Berlin Heidelberg New York, 2003, page 92.
Akihiro Kanamori, The Higher Infinite, Second Edition, Springer-Verlag Berlin Heidelberg, 2009, page 2.
Ordinal numbers
Set theory
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https://en.wikipedia.org/wiki/Club%20filter
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In mathematics, particularly in set theory, if is a regular uncountable cardinal then the filter of all sets containing a club subset of is a -complete filter closed under diagonal intersection called the club filter.
To see that this is a filter, note that since it is thus both closed and unbounded (see club set). If then any subset of containing is also in since and therefore anything containing it, contains a club set.
It is a -complete filter because the intersection of fewer than club sets is a club set. To see this, suppose is a sequence of club sets where Obviously is closed, since any sequence which appears in appears in every and therefore its limit is also in every To show that it is unbounded, take some Let be an increasing sequence with and for every Such a sequence can be constructed, since every is unbounded. Since and is regular, the limit of this sequence is less than We call it and define a new sequence similar to the previous sequence. We can repeat this process, getting a sequence of sequences where each element of a sequence is greater than every member of the previous sequences. Then for each is an increasing sequence contained in and all these sequences have the same limit (the limit of ). This limit is then contained in every and therefore and is greater than
To see that is closed under diagonal intersection, let be a sequence of club sets, and let To show is closed, suppose and Then for each for all Since each is closed, for all so To show is unbounded, let and define a sequence as follows: and is the minimal element of such that Such an element exists since by the above, the intersection of club sets is club. Then and since it is in each with
See also
References
Jech, Thomas, 2003. Set Theory: The Third Millennium Edition, Revised and Expanded. Springer. .
Set theory
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https://en.wikipedia.org/wiki/Triangulated%20category
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In mathematics, a triangulated category is a category with the additional structure of a "translation functor" and a class of "exact triangles". Prominent examples are the derived category of an abelian category, as well as the stable homotopy category. The exact triangles generalize the short exact sequences in an abelian category, as well as fiber sequences and cofiber sequences in topology.
Much of homological algebra is clarified and extended by the language of triangulated categories, an important example being the theory of sheaf cohomology. In the 1960s, a typical use of triangulated categories was to extend properties of sheaves on a space X to complexes of sheaves, viewed as objects of the derived category of sheaves on X. More recently, triangulated categories have become objects of interest in their own right. Many equivalences between triangulated categories of different origins have been proved or conjectured. For example, the homological mirror symmetry conjecture predicts that the derived category of a Calabi–Yau manifold is equivalent to the Fukaya category of its "mirror" symplectic manifold. Shift operator is a decategorified analogue of triangulated category.
History
Triangulated categories were introduced independently by Dieter Puppe (1962) and Jean-Louis Verdier (1963), although Puppe's axioms were less complete (lacking the octahedral axiom (TR 4)). Puppe was motivated by the stable homotopy category. Verdier's key example was the derived category of an abelian category, which he also defined, developing ideas of Alexander Grothendieck. The early applications of derived categories included coherent duality and Verdier duality, which extends Poincaré duality to singular spaces.
Definition
A shift or translation functor on a category D is an additive automorphism (or for some authors, an auto-equivalence) from D to D. It is common to write for integers n.
A triangle (X, Y, Z, u, v, w) consists of three objects X, Y, and Z, together with morphisms , and . Triangles are generally written in the unravelled form:
or
for short.
A triangulated category is an additive category D with a translation functor and a class of triangles, called exact triangles (or distinguished triangles), satisfying the following properties (TR 1), (TR 2), (TR 3) and (TR 4). (These axioms are not entirely independent, since (TR 3) can be derived from the others.)
TR 1
For every object X, the following triangle is exact:
For every morphism , there is an object Z (called a cone or cofiber of the morphism u) fitting into an exact triangle
The name "cone" comes from the cone of a map of chain complexes, which in turn was inspired by the mapping cone in topology. It follows from the other axioms that an exact triangle (and in particular the object Z) is determined up to isomorphism by the morphism , although not always up to a unique isomorphism.
Every triangle isomorphic to an exact triangle is exact. This means that if
is an exact triangle
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https://en.wikipedia.org/wiki/All%20India%20Trade%20Union%20Congress
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The All India Trade Union Congress (AITUC) is the oldest trade union federation in India. It is associated with the Communist Party of India. According to provisional statistics from the Ministry of Labour, AITUC had a membership of 14.2 million in 2013. It was founded on 31 October 1920 with Lala Lajpat Rai as its first president.
In Bombay by Lala Lajpat Rai, Joseph Baptista, N. M. Joshi, Diwan Chaman Lall and a few others and, until 1945 when unions became organised on party lines, it was the primary trade union organisation in India. Since then, it has been associated with the Communist Party of India.
AITUC is governed by a body headed by National President Ramendra Kumar and General Secretary Amarjeet Kaur, both the politician affiliated with Communist Party of India. "Trade Union Record" is the fortnightly journal of the AITUC.
AITUC is a founder member of the World Federation of Trade Unions. Today, its institutional records are part of the Archives at the Nehru Memorial Museum & Library, at Teen Murti House, Delhi.
History
Background
The beginning of the labour upsurge against oppression and exploitation goes back to the second half of 19th century, with the emergence of class of casual general labour during British Raj in India. The self-sufficient Village economy was shattered with no new structures in place, creating impoverished peasantry and landless labour force.
The dumping of cheap industrial goods resulting in millions of artisans, spinners, weavers, craftsmen, smelters, smiths, potters, etc., who could no more live on agriculture also turned into landless labourers. This led to widespread famines in India through the period from 1850 to 1890 resulting in deaths of several lakhs and also reducing millions as beggars.
The anguish of impoverished masses, ruined peasantry was up in revolt which resulted in several movements even though crushed by the rulers. This background did help the 1857 revolt by princely states and the common masses against the disempowering policies of British rule.
Till this time trade unionism was not known to workers, they were reacting to extreme exploitative working conditions and very low wages. They formed themselves as 'jamaats' which were based more on social caste basis in order to fight back oppression of employers. This was beginning of organization by the workers even though not the trade unions in essence.
From 1905 onwards there was notable advance in the working class actions and it was more and more closing its ranks with the advance of freedom struggle in the country.
A strike took place in Bombay against extension of working hours. The printing press workers in Calcutta also struck work. Another great event of the period was strike by industrial workers of Bombay from July 24 to 28, 1908, in protest against the pronouncement of judgment sentencing six years imprisonment to freedom fighter Bal Gangadhar Tilak. There were street fights between workers and police and military of Br
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https://en.wikipedia.org/wiki/Cycle%20graph%20%28algebra%29
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In group theory, a subfield of abstract algebra, a group cycle graph illustrates the various cycles of a group and is particularly useful in visualizing the structure of small finite groups.
A cycle is the set of powers of a given group element a, where an, the n-th power of an element a is defined as the product of a multiplied by itself n times. The element a is said to generate the cycle. In a finite group, some non-zero power of a must be the group identity, e; the lowest such power is the order of the cycle, the number of distinct elements in it. In a cycle graph, the cycle is represented as a polygon, with the vertices representing the group elements, and the connecting lines indicating that all elements in that polygon are members of the same cycle.
Cycles
Cycles can overlap, or they can have no element in common but the identity. The cycle graph displays each interesting cycle as a polygon.
If a generates a cycle of order 6 (or, more shortly, has order 6), then a6 = e. Then the set of powers of a2, {a2, a4, e} is a cycle, but this is really no new information. Similarly, a5 generates the same cycle as a itself.
So, only the primitive cycles need be considered, namely those that are not subsets of another cycle. Each of these is generated by some primitive element, a. Take one point for each element of the original group. For each primitive element, connect e to a, a to a2, ..., an−1 to an, etc., until e is reached. The result is the cycle graph.
When a2 = e, a has order 2 (is an involution), and is connected to e by two edges. Except when the intent is to emphasize the two edges of the cycle, it is typically drawn as a single line between the two elements.
Properties
As an example of a group cycle graph, consider the dihedral group Dih4. The multiplication table for this group is shown on the left, and the cycle graph is shown on the right with e specifying the identity element.
Notice the cycle {e, a, a2, a3} in the multiplication table, with a4 = e. The inverse a−1 = a3 is also a generator of this cycle: (, , and . Similarly, any cycle in any group has at least two generators, and may be traversed in either direction. More generally, the number of generators of a cycle with n elements is given by the Euler φ function of n, and any of these generators may be written as the first node in the cycle (next to the identity e); or more commonly the nodes are left unmarked. Two distinct cycles cannot intersect in a generator.
Cycles that contain a non-prime number of elements have cyclic subgroups that are not shown in the graph. For the group Dih4 above, we could draw a line between a2 and e since , but since a2 is part of a larger cycle, this is not an edge of the cycle graph.
There can be ambiguity when two cycles share a non-identity element. For example, the 8-element quaternion group has cycle graph shown at right. Each of the elements in the middle row when multiplied by itself gives −1 (where 1 is the identity element).
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https://en.wikipedia.org/wiki/Static%20spacetime
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In general relativity, a spacetime is said to be static if it does not change over time and is also irrotational. It is a special case of a stationary spacetime, which is the geometry of a stationary spacetime that does not change in time but can rotate. Thus, the Kerr solution provides an example of a stationary spacetime that is not static; the non-rotating Schwarzschild solution is an example that is static.
Formally, a spacetime is static if it admits a global, non-vanishing, timelike Killing vector field which is irrotational, i.e., whose orthogonal distribution is involutive. (Note that the leaves of the associated foliation are necessarily space-like hypersurfaces.) Thus, a static spacetime is a stationary spacetime satisfying this additional integrability condition. These spacetimes form one of the simplest classes of Lorentzian manifolds.
Locally, every static spacetime looks like a standard static spacetime which is a Lorentzian warped product R S with a metric of the form
,
where R is the real line, is a (positive definite) metric and is a positive function on the Riemannian manifold S.
In such a local coordinate representation the Killing field may be identified with and S, the manifold of -trajectories, may be regarded as the instantaneous 3-space of stationary observers. If is the square of the norm of the Killing vector field, , both and are independent of time (in fact ). It is from the latter fact that a static spacetime obtains its name, as the geometry of the space-like slice S does not change over time.
Examples of static spacetimes
The (exterior) Schwarzschild solution.
de Sitter space (the portion of it covered by the static patch).
Reissner–Nordström space.
The Weyl solution, a static axisymmetric solution of the Einstein vacuum field equations discovered by Hermann Weyl.
Examples of non-static spacetimes
In general, "almost all" spacetimes will not be static. Some explicit examples include:
Spherically symmetric spacetimes, which are irrotational, but not static.
The Kerr solution, since it describes a rotating black hole, is a stationary spacetime that is not static.
Spacetimes with gravitational waves in them are not even stationary.
References
Lorentzian manifolds
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https://en.wikipedia.org/wiki/Asymptotically%20flat%20spacetime
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An asymptotically flat spacetime is a Lorentzian manifold in which, roughly speaking, the curvature vanishes at large distances from some region, so that at large distances, the geometry becomes indistinguishable from that of Minkowski spacetime.
While this notion makes sense for any Lorentzian manifold, it is most often applied to a spacetime standing as a solution to the field equations of some metric theory of gravitation, particularly general relativity. In this case, we can say that an asymptotically flat spacetime is one in which the gravitational field, as well as any matter or other fields which may be present, become negligible in magnitude at large distances from some region. In particular, in an asymptotically flat vacuum solution, the gravitational field (curvature) becomes negligible at large distances from the source of the field (typically some isolated massive object such as a star).
Intuitive significance
The condition of asymptotic flatness is analogous to similar conditions in mathematics and in other physical theories. Such conditions say that some physical field or mathematical function is asymptotically vanishing in a suitable sense.
In general relativity, an asymptotically flat vacuum solution models the exterior gravitational field of an isolated massive object. Therefore, such a spacetime can be considered as an isolated system: a system in which exterior influences can be neglected. Indeed, physicists rarely imagine a universe containing a single star and nothing else when they construct an asymptotically flat model of a star. Rather, they are interested in modeling the interior of the star together with an exterior region in which gravitational effects due to the presence of other objects can be neglected. Since typical distances between astrophysical bodies tend to be much larger than the diameter of each body, we often can get away with this idealization, which usually helps to greatly simplify the construction and analysis of solutions.
Formal definitions
A manifold is asymptotically simple if it admits a conformal compactification such that every null geodesic in has future and past endpoints on the boundary of .
Since the latter excludes black holes, one defines a weakly asymptotically simple manifold as a manifold with an open set isometric to a neighbourhood of the boundary of , where is the conformal compactification of some asymptotically simple manifold.
A manifold is asymptotically flat if it is weakly asymptotically simple and asymptotically empty in the sense that its Ricci tensor vanishes in a neighbourhood of the boundary of .
Some examples and nonexamples
Only spacetimes which model an isolated object are asymptotically flat. Many other familiar exact solutions, such as the FRW models, are not.
A simple example of an asymptotically flat spacetime is the Schwarzschild metric solution. More generally, the Kerr metric is also asymptotically flat. But another well known generalization
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https://en.wikipedia.org/wiki/Monster%20Lie%20algebra
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In mathematics, the monster Lie algebra is an infinite-dimensional generalized Kac–Moody algebra acted on by the monster group, which was used to prove the monstrous moonshine conjectures.
Structure
The monster Lie algebra m is a Z2-graded Lie algebra. The piece of degree (m, n) has dimension cmn if (m, n) ≠ (0, 0) and dimension 2 if (m, n) = (0, 0).
The integers cn are the coefficients of qn of the j-invariant as elliptic modular function
The Cartan subalgebra is the 2-dimensional subspace of degree (0, 0), so the monster Lie algebra has rank 2.
The monster Lie algebra has just one real simple root, given by the vector (1, −1), and the Weyl group has order 2, and acts by mapping (m, n) to (n, m). The imaginary simple roots are the vectors (1, n) for n = 1, 2, 3, ..., and they have multiplicities cn.
The denominator formula for the monster Lie algebra is the product formula for the j-invariant:
The denominator formula (sometimes called the Koike-Norton-Zagier infinite product identity) was discovered in the 1980s. Several mathematicians, including Masao Koike, Simon P. Norton, and Don Zagier, independently made the discovery.
Construction
There are two ways to construct the monster Lie algebra. As it is a generalized Kac–Moody algebra whose simple roots are known, it can be defined by explicit generators and relations; however, this presentation does not give an action of the monster group on it.
It can also be constructed from the monster vertex algebra by using the Goddard–Thorn theorem of string theory. This construction is much harder, but also proves that the monster group acts naturally on it.
References
;
(Introductory study text with a brief account of Borcherds algebra in Ch. 21)
Lie algebras
Moonshine theory
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https://en.wikipedia.org/wiki/Forcing%20function
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Forcing function can mean:
In differential calculus, a function that appears in the equations and is only a function of time, and not of any of the other variables.
In interaction design, a behavior-shaping constraint, a means of preventing undesirable user input usually made by mistake.
A forcing function is any task, activity or event that forces one to take action and produce a result.
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https://en.wikipedia.org/wiki/Nielsen%E2%80%93Thurston%20classification
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In mathematics, Thurston's classification theorem characterizes homeomorphisms of a compact orientable surface. William Thurston's theorem completes the work initiated by .
Given a homeomorphism f : S → S, there is a map g isotopic to f such that at least one of the following holds:
g is periodic, i.e. some power of g is the identity;
g preserves some finite union of disjoint simple closed curves on S (in this case, g is called reducible); or
g is pseudo-Anosov.
The case where S is a torus (i.e., a surface whose genus is one) is handled separately (see torus bundle) and was known before Thurston's work. If the genus of S is two or greater, then S is naturally hyperbolic, and the tools of Teichmüller theory become useful. In what follows, we assume S has genus at least two, as this is the case Thurston considered. (Note, however, that the cases where S has boundary or is not orientable are definitely still of interest.)
The three types in this classification are not mutually exclusive, though a pseudo-Anosov homeomorphism is never periodic or reducible. A reducible homeomorphism g can be further analyzed by cutting the surface along the preserved union of simple closed curves Γ. Each of the resulting compact surfaces with boundary is acted upon by some power (i.e. iterated composition) of g, and the classification can again be applied to this homeomorphism.
The mapping class group for surfaces of higher genus
Thurston's classification applies to homeomorphisms of orientable surfaces of genus ≥ 2, but the type of a homeomorphism only depends on its associated element of the mapping class group Mod(S). In fact, the proof of the classification theorem leads to a canonical representative of each mapping class with good geometric properties. For example:
When g is periodic, there is an element of its mapping class that is an isometry of a hyperbolic structure on S.
When g is pseudo-Anosov, there is an element of its mapping class that preserves a pair of transverse singular foliations of S, stretching the leaves of one (the unstable foliation) while contracting the leaves of the other (the stable foliation).
Mapping tori
Thurston's original motivation for developing this classification was to find geometric structures on mapping tori of the type predicted by the Geometrization conjecture. The mapping torus Mg of a homeomorphism g of a surface S is the 3-manifold obtained from S × [0,1] by gluing S × {0} to S × {1} using g. If S has genus at least two, the geometric structure of Mg is related to the type of g in the classification as follows:
If g is periodic, then Mg has an H2 × R structure;
If g is reducible, then Mg has incompressible tori, and should be cut along these tori to yield pieces that each have geometric structures (the JSJ decomposition);
If g is pseudo-Anosov, then Mg has a hyperbolic (i.e. H3) structure.
The first two cases are comparatively easy, while the existence of a hyperbolic structure on the mapping torus of
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https://en.wikipedia.org/wiki/Olga%20Taussky-Todd
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Olga Taussky-Todd (August 30, 1906 – October 7, 1995) was an Austrian and later Czech-American mathematician. She published more than 300 research papers on algebraic number theory, integral matrices, and matrices in algebra and analysis.
Early life
Olga Taussky was born into a Jewish family in what is now Olomouc, Czech Republic, on August 30, 1906. Her father, Julius David Taussky, was an industrial chemist and her mother, Ida Pollach, was a housewife. She was the second of three children. Her father preferred that, if his daughters had careers, they be in the arts, but they all went into the sciences. Ilona, three years older than Olga, became a consulting chemist in the glyceride industry, and Hertha, three years younger than Olga, became a pharmacist and later a clinical chemist at Cornell University Medical College in New York City.
At the age of three, her family moved to Vienna and lived there until the middle of World War I. Later Taussky's father accepted a position as director of a vinegar factory at Linz in Upper Austria. At a young age, Taussky displayed a keen interest in mathematics. After her father died during her last year at school, she worked through the summer at her father's vinegar factory and was pressured by her family to study chemistry in order to take over her father's work. Her elder sister, however, qualified in chemistry and took over her father's work. In "Red Vienna" of the day, the Social Democratic Party of Austria encouraged woman to pursue higher education, and Taussky enrolled at the University of Vienna in the fall of 1925 to study mathematics.
Career
Taussky worked first in algebraic number theory, with a doctorate at the University of Vienna supervised by Philipp Furtwängler, a number theorist from Germany. During that time, she attended meetings of the so-called Vienna Circle, the group of philosophers and logicians developing the philosophy of logical positivism. Taussky, like Olga Hahn-Neurath and Rose Rand, was one of the first women to join the group, which included Otto Neurath, Rudolf Carnap, and Kurt Gödel and which was strongly influenced by Ludwig Wittgenstein.
Taussky is best known for her work in matrix theory (in particular the computational stability of complex matrices), algebraic number theory, group theory, and numerical analysis.
According to Gian-Carlo Rota, as a young mathematician she was hired by a group of German mathematicians to find and correct the many mathematical errors in the works of David Hilbert, so that they could be collected into a volume to be presented to him on his birthday. There was only one paper, on the continuum hypothesis, that she was unable to repair.
In 1935, she moved to England and became a Fellow at Girton College, Cambridge University, as well as at Bryn Mawr College. Soon after, in 1938, she married the Irish mathematician Jack Todd, a colleague at the University of London.
Later, she started to use matrices to analyze vibrations of airplanes dur
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https://en.wikipedia.org/wiki/GEANT-3
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GEANT is the name of a series of simulation software designed to describe the passage of elementary particles through matter, using Monte Carlo methods. The name is an acronym formed from "GEometry ANd Tracking". Originally developed at CERN for high energy physics experiments, GEANT-3 has been used in many other fields.
History
The very first version of GEANT dates back to 1974, while the first version of GEANT-3 dates back to 1982. Versions of GEANT through 3.21 were written in FORTRAN and eventually maintained as part of CERNLIB. Since about 2000, the last FORTRAN release has been essentially in stasis and receives only occasional bug fixes. GEANT3 was, however, still in use by some experiments for some time thereafter. Most of GEANT-3 is available under the GNU General Public License, with the exception of some hadronic interaction code contributed by the FLUKA collaboration.
GEANT-3 was used by a majority of high energy physics experiments from the late 1980s to the early 2000s. The largest experiments using were three of the experiments at the Large Electron-Positron collider, including ALEPH, L3 and OPAL. It was also a key tool in the design and optimization of the detectors of all experiments at the Large Hadron Collider (LHC) – see e.g. the ATLAS Technical Design Report. GEANT-3.21 based programs remained main simulation engine of ATLAS, CMS and LHCb at LHC until 2004, when these experiments moved to Geant4-based simulations. Even in 2019 it remains the primary simulation tool for the ALICE experiment at the LHC.
A related (but separate) product is Geant4 (when referring to this version, the name is typically no longer capitalized). It is a complete rewrite in C++ with a modern object-oriented design. Geant4 was developed by the RD44 collaboration in 1994–1998 and is being maintained and improved now by the Geant4 international collaboration. For quite some time Geant4 did not have a clearly defined software license. As of version 8.1 (released June 30, 2006) this omission has been remedied. Geant4 is now available under the Geant4 Software License.
See also
EGS (program)
CLHEP and FreeHEP, libraries for high energy physics
References
External links
Geant4 publicly accessible webpage
GEANT webpage at CERN (only available to CERN users)
Free software programmed in Fortran
Monte Carlo particle physics software
Physics software
CERN software
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https://en.wikipedia.org/wiki/Infinity%20plus%20one
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In mathematics, infinity plus one is a concept which has a well-defined formal meaning in some number systems, and may refer to:
Transfinite numbers, numbers that are larger than all the finite numbers.
Cardinal numbers, representations of sizes (cardinalities) of abstract sets, which may be infinite.
Ordinal numbers, representations of order types of well-ordered sets, which may also be infinite.
Hyperreal numbers, an extension of the real number system that contains infinite and infinitesimal numbers.
Surreal numbers, another extension of the real numbers, contain the hyperreal and all the transfinite ordinal numbers.
English phrases
Infinity
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https://en.wikipedia.org/wiki/Hyperrectangle
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In geometry, an hyperrectangle (also called a box, hyperbox, or orthotope), is the generalization of a rectangle (a plane figure) and the rectangular cuboid (a solid figure) to higher dimensions.
A necessary and sufficient condition is that it is congruent to the Cartesian product of finite intervals. If all of the edges are equal length, it is a hypercube.
A hyperrectangle is a special case of a parallelotope.
Types
A four-dimensional orthotope is likely a hypercuboid.
The special case of an n-dimensional orthotope where all edges have equal length is the n-cube or hypercube.
By analogy, the term "hyperrectangle" can refer to Cartesian products of orthogonal intervals of other kinds, such as ranges of keys in database theory or ranges of integers, rather than real numbers.
Dual polytope
The dual polytope of an n-orthotope has been variously called a rectangular n-orthoplex, rhombic n-fusil, or n-lozenge. It is constructed by 2n points located in the center of the orthotope rectangular faces.
An n-fusil's Schläfli symbol can be represented by a sum of n orthogonal line segments: { } + { } + ... + { } or n{ }.
A 1-fusil is a line segment. A 2-fusil is a rhombus. Its plane cross selections in all pairs of axes are rhombi.
See also
Minimum bounding box
Cuboid
Notes
References
External links
Polytopes
Prismatoid polyhedra
Multi-dimensional geometry
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https://en.wikipedia.org/wiki/Probit
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In probability theory and statistics, the probit function is the quantile function associated with the standard normal distribution. It has applications in data analysis and machine learning, in particular exploratory statistical graphics and specialized regression modeling of binary response variables.
Mathematically, the probit is the inverse of the cumulative distribution function of the standard normal distribution, which is denoted as , so the probit is defined as
.
Largely because of the central limit theorem, the standard normal distribution plays a fundamental role in probability theory and statistics. If we consider the familiar fact that the standard normal distribution places 95% of probability between −1.96 and 1.96, and is symmetric around zero, it follows that
The probit function gives the 'inverse' computation, generating a value of a standard normal random variable, associated with specified cumulative probability. Continuing the example,
.
In general,
and
Conceptual development
The idea of the probit function was published by Chester Ittner Bliss in a 1934 article in Science on how to treat data such as the percentage of a pest killed by a pesticide. Bliss proposed transforming the percentage killed into a "probability unit" (or "probit") which was linearly related to the modern definition (he defined it arbitrarily as equal to 0 for 0.0001 and 1 for 0.9999):
He included a table to aid other researchers to convert their kill percentages to his probit, which they could then plot against the logarithm of the dose and thereby, it was hoped, obtain a more or less straight line. Such a so-called probit model is still important in toxicology, as well as other fields. The approach is justified in particular if response variation can be rationalized as a lognormal distribution of tolerances among subjects on test, where the tolerance of a particular subject is the dose just sufficient for the response of interest.
The method introduced by Bliss was carried forward in Probit Analysis, an important text on toxicological applications by D. J. Finney. Values tabled by Finney can be derived from probits as defined here by adding a value of 5. This distinction is summarized by Collett (p. 55): "The original definition of a probit [with 5 added] was primarily to avoid having to work with negative probits; ... This definition is still used in some quarters, but in the major statistical software packages for what is referred to as probit analysis, probits are defined without the addition of 5." It should be observed that probit methodology, including numerical optimization for fitting of probit functions, was introduced before widespread availability of electronic computing. When using tables, it was convenient to have probits uniformly positive. Common areas of application do not require positive probits.
Diagnosing deviation of a distribution from normality
In addition to providing a basis for important types of regression, the
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https://en.wikipedia.org/wiki/Group%20of%20Lie%20type
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In mathematics, specifically in group theory, the phrase group of Lie type usually refers to finite groups that are closely related to the group of rational points of a reductive linear algebraic group with values in a finite field. The phrase group of Lie type does not have a widely accepted precise definition, but the important collection of finite simple groups of Lie type does have a precise definition, and they make up most of the groups in the classification of finite simple groups.
The name "groups of Lie type" is due to the close relationship with the (infinite) Lie groups, since a compact Lie group may be viewed as the rational points of a reductive linear algebraic group over the field of real numbers. and are standard references for groups of Lie type.
Classical groups
An initial approach to this question was the definition and detailed study of the so-called classical groups over finite and other fields by . These groups were studied by L. E. Dickson and Jean Dieudonné. Emil Artin investigated the orders of such groups, with a view to classifying cases of coincidence.
A classical group is, roughly speaking, a special linear, orthogonal, symplectic, or unitary group. There are several minor variations of these, given by taking derived subgroups or central quotients, the latter yielding projective linear groups. They can be constructed over finite fields (or any other field) in much the same way that they are constructed over the real numbers. They correspond to the series An, Bn, Cn, Dn,2An, 2Dn of Chevalley and Steinberg groups.
Chevalley groups
Chevalley groups can be thought of as Lie groups over finite fields. The theory was clarified by the theory of algebraic groups, and the work of on Lie algebras, by means of which the Chevalley group concept was isolated. Chevalley constructed a Chevalley basis (a sort of integral form but over finite fields) for all the complex simple Lie algebras (or rather of their universal enveloping algebras), which can be used to define the corresponding algebraic groups over the integers. In particular, he could take their points with values in any finite field. For the Lie algebras An, Bn, Cn, Dn this gave well known classical groups, but his construction also gave groups associated to the exceptional Lie algebras E6, E7, E8, F4, and G2. The ones of type G2 (sometimes called Dickson groups) had already been constructed by , and the ones of type E6 by .
Steinberg groups
Chevalley's construction did not give all of the known classical groups: it omitted the unitary groups and the non-split orthogonal groups. found a modification of Chevalley's construction that gave these groups and two new families 3D4, 2E6, the second of which was discovered at about the same time from a different point of view by . This construction generalizes the usual construction of the unitary group from the general linear group.
The unitary group arises as follows: the general linear group over the complex numbers
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https://en.wikipedia.org/wiki/Oxygen%20difluoride
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Oxygen difluoride is a chemical compound with the formula . As predicted by VSEPR theory, the molecule adopts a bent molecular geometry. It is strong oxidizer and has attracted attention in rocketry for this reason. With a boiling point of −144.75 °C, OF2 is the most volatile (isolable) triatomic compound. The compound is one of many known oxygen fluorides.
Preparation
Oxygen difluoride was first reported in 1929; it was obtained by the electrolysis of molten potassium fluoride and hydrofluoric acid containing small quantities of water. The modern preparation entails the reaction of fluorine with a dilute aqueous solution of sodium hydroxide, with sodium fluoride as a side-product:
Structure and bonding
It is a covalently bonded molecule with a bent molecular geometry and a F-O-F bond angle of 103 degrees. Its powerful oxidizing properties are suggested by the oxidation number of +2 for the oxygen atom instead of its normal −2.
Reactions
Above 200 °C, decomposes to oxygen and fluorine by a radical mechanism.
reacts with many metals to yield oxides and fluorides. Nonmetals also react: phosphorus reacts with to form and ; sulfur gives and ; and unusually for a noble gas, xenon reacts (at elevated temperatures) yielding and xenon oxyfluorides.
Oxygen difluoride reacts very slowly with water to form hydrofluoric acid:
It can oxidize sulphur dioxide to sulfur trioxide and elemental fluorine:
However, in the presence of UV radiation, the products are sulfuryl fluoride () and pyrosulfuryl fluoride ():
Safety
Oxygen difluoride is considered an unsafe gas due to its oxidizing properties. Hydrofluoric acid produced by the hydrolysis of with water is highly corrosive and toxic, capable of causing necrosis, leaching calcium from the bones and causing cardiovascular damage, among a host of other insidious effects.
Popular culture
In Robert L. Forward's science fiction novel Camelot 30K, oxygen difluoride was used as a biochemical solvent by fictional life forms living in the solar system's Kuiper belt. While would be a solid at 30K, the fictional alien lifeforms were described as endothermic, maintaining elevated body temperatures and liquid blood by radiothermal heating.
Notes
References
External links
National Pollutant Inventory - Fluoride and compounds fact sheet
WebBook page for
CDC - NIOSH Pocket Guide to Chemical Hazards
Oxygen fluorides
Rocket oxidizers
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https://en.wikipedia.org/wiki/Yuri%20Manin
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Yuri Ivanovich Manin (; 16 February 1937 – 7 January 2023) was a Russian mathematician, known for work in algebraic geometry and diophantine geometry, and many expository works ranging from mathematical logic to theoretical physics.
Life and career
Manin was born on 16 February 1937 in Simferopol, Crimean ASSR, Soviet Union.
He received a doctorate in 1960 at the Steklov Mathematics Institute as a student of Igor Shafarevich. He became a professor at the Max-Planck-Institut für Mathematik in Bonn, where he was director from 1992 to 2005 and then director emeritus. He was also a professor emeritus at Northwestern University.
He had over the years more than 40 doctoral students, including Vladimir Berkovich, Mariusz Wodzicki, Alexander Beilinson, Ivan Cherednik, Alexei Skorobogatov, Vladimir Drinfeld, Mikhail Kapranov, Vyacheslav Shokurov, Ralph Kaufmann, Arend Bayer, Victor Kolyvagin and Hà Huy Khoái.
Manin died on 7 January 2023.
Research
Manin's early work included papers on the arithmetic and formal groups of abelian varieties, the Mordell conjecture in the function field case, and algebraic differential equations. The Gauss–Manin connection is a basic ingredient of the study of cohomology in families of algebraic varieties.
He developed the Manin obstruction, indicating the role of the Brauer group in accounting for obstructions to the Hasse principle via Grothendieck's theory of global Azumaya algebras, setting off a generation of further work.
Manin pioneered the field of arithmetic topology (along with John Tate, David Mumford, Michael Artin, and Barry Mazur). He also formulated the Manin conjecture, which predicts the asymptotic behaviour of the number of rational points of bounded height on algebraic varieties.
In mathematical physics, Manin wrote on Yang–Mills theory, quantum information, and mirror symmetry. He was one of the first to propose the idea of a quantum computer in 1980 with his book Computable and Uncomputable.
He wrote a book on cubic surfaces and cubic forms, showing how to apply both classical and contemporary methods of algebraic geometry, as well as nonassociative algebra.
Awards
He was awarded the Brouwer Medal in 1987, the first Nemmers Prize in Mathematics in 1994, the Schock Prize of the Royal Swedish Academy of Sciences in 1999, the Cantor Medal of the German Mathematical Society in 2002, the King Faisal International Prize in 2002, and the Bolyai Prize of the Hungarian Academy of Sciences in 2010.
In 1990, he became a foreign member of the Royal Netherlands Academy of Arts and Sciences. He was a member of eight other academies of science and was also an honorary member of the London Mathematical Society.
Selected works
, second expanded edition with new chapters by the author and Boris Zilber, Springer 2010.
See also
Arithmetic topology
Noncommutative residue
References
Further reading
External links
Manin's page at Max-Planck-Institut für Mathematik website
Good Proo
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https://en.wikipedia.org/wiki/Region%201
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Region 1 or Region I can refer to:
Region 1, a DVD region code
Region 1, Northwest Territories, a Statistics Canada census division
Northeastern United States, Region 1 for the US Census Bureau
Region 1, one of the health regions of Canada managed by Horizon Health Network
Former Region 1 (Johannesburg), an administrative district in the city of Johannesburg, South Africa, from 2000 to 2006
Region 1, an administrative region in Iran
Tarapacá Region, Chile
Ilocos Region, Philippines
Region name disambiguation pages
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https://en.wikipedia.org/wiki/Divisor%20%28algebraic%20geometry%29
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In algebraic geometry, divisors are a generalization of codimension-1 subvarieties of algebraic varieties. Two different generalizations are in common use, Cartier divisors and Weil divisors (named for Pierre Cartier and André Weil by David Mumford). Both are derived from the notion of divisibility in the integers and algebraic number fields.
Globally, every codimension-1 subvariety of projective space is defined by the vanishing of one homogeneous polynomial; by contrast, a codimension-r subvariety need not be definable by only r equations when r is greater than 1. (That is, not every subvariety of projective space is a complete intersection.) Locally, every codimension-1 subvariety of a smooth variety can be defined by one equation in a neighborhood of each point. Again, the analogous statement fails for higher-codimension subvarieties. As a result of this property, much of algebraic geometry studies an arbitrary variety by analysing its codimension-1 subvarieties and the corresponding line bundles.
On singular varieties, this property can also fail, and so one has to distinguish between codimension-1 subvarieties and varieties which can locally be defined by one equation. The former are Weil divisors while the latter are Cartier divisors.
Topologically, Weil divisors play the role of homology classes, while Cartier divisors represent cohomology classes. On a smooth variety (or more generally a regular scheme), a result analogous to Poincaré duality says that Weil and Cartier divisors are the same.
The name "divisor" goes back to the work of Dedekind and Weber, who showed the relevance of Dedekind domains to the study of algebraic curves. The group of divisors on a curve (the free abelian group generated by all divisors) is closely related to the group of fractional ideals for a Dedekind domain.
An algebraic cycle is a higher codimension generalization of a divisor; by definition, a Weil divisor is a cycle of codimension 1.
Divisors on a Riemann surface
A Riemann surface is a 1-dimensional complex manifold, and so its codimension-1 submanifolds have dimension 0. The group of divisors on a compact Riemann surface X is the free abelian group on the points of X.
Equivalently, a divisor on a compact Riemann surface X is a finite linear combination of points of X with integer coefficients. The degree of a divisor on X is the sum of its coefficients.
For any nonzero meromorphic function f on X, one can define the order of vanishing of f at a point p in X, ordp(f). It is an integer, negative if f has a pole at p. The divisor of a nonzero meromorphic function f on the compact Riemann surface X is defined as
which is a finite sum. Divisors of the form (f) are also called principal divisors. Since (fg) = (f) + (g), the set of principal divisors is a subgroup of the group of divisors. Two divisors that differ by a principal divisor are called linearly equivalent.
On a compact Riemann surface, the degree of a principal divisor is zero; that is, t
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https://en.wikipedia.org/wiki/Triple%20bar
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The triple bar or tribar, ≡, is a symbol with multiple, context-dependent meanings indicating equivalence of two different things. Its main uses are in mathematics and logic. It has the appearance of an equals sign with a third line.
Encoding
The triple bar character in Unicode is code point . The closely related code point is the same symbol with a slash through it, indicating the negation of its mathematical meaning.
In LaTeX mathematical formulas, the code \equiv produces the triple bar symbol and \not\equiv produces the negated triple bar symbol as output.
Uses
Mathematics and philosophy
In logic, it is used with two different but related meanings. It can refer to the if and only if connective, also called material equivalence. This is a binary operation whose value is true when its two arguments have the same value as each other. Alternatively, in some texts ⇔ is used with this meaning, while ≡ is used for the higher-level metalogical notion of logical equivalence, according to which two formulas are logically equivalent when all models give them the same value. Gottlob Frege used a triple bar for a more philosophical notion of identity, in which two statements (not necessarily in mathematics or formal logic) are identical if they can be freely substituted for each other without change of meaning.
In mathematics, the triple bar is sometimes used as a symbol of identity or an equivalence relation (although not the only one; other common choices include ~ and ≈). Particularly, in geometry, it may be used either to show that two figures are congruent or that they are identical. In number theory, it has been used beginning with Carl Friedrich Gauss (who first used it with this meaning in 1801) to mean modular congruence: if N divides a − b.
In category theory, triple bars may be used to connect objects in a commutative diagram, indicating that they are actually the same object rather than being connected by an arrow of the category.
This symbol is also sometimes used in place of an equal sign for equations that define the symbol on the left-hand side of the equation, to contrast them with equations in which the terms on both sides of the equation were already defined. An alternative notation for this usage is to typeset the letters "def" above an ordinary equality sign, . Similarly, another alternative notation for this usage is to precede the equals sign with a colon, . The colon notation has the advantage that it reflects the inherent asymmetry in the definition of one object from already defined objects.
Science
In botanical nomenclature, the triple bar denotes homotypic synonyms (those based on the same type specimen), to distinguish them from heterotypic synonyms (those based on different type specimens), which are marked with an equals sign.
In chemistry, the triple bar can be used to represent a triple bond between atoms. For example, HC≡CH is a common shorthand for acetylene (systematic name: ethyne).
Application design
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https://en.wikipedia.org/wiki/Currency%20crisis
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A currency crisis is a type of financial crisis, and is often associated with a real economic crisis. A currency crisis raises the probability of a banking crisis or a default crisis. During a currency crisis the value of foreign denominated debt will rise drastically relative to the declining value of the home currency. Generally doubt exists as to whether a country's central bank has sufficient foreign exchange reserves to maintain the country's fixed exchange rate, if it has any. The crisis is often accompanied by a speculative attack in the foreign exchange market. A currency crisis results from chronic balance of payments deficits, and thus is also called a balance of payments crisis. Often such a crisis culminates in a devaluation of the currency. Financial institutions and the government will struggle to meet debt obligations and economic crisis may ensue. Causation also runs the other way. The probability of a currency crisis rises when a country is experiencing a banking or default crisis, while this probability is lower when an economy registers strong GDP growth and high levels of foreign exchange reserves. To offset the damage resulting from a banking or default crisis, a central bank will often increase currency issuance, which can decrease reserves to a point where a fixed exchange rate breaks. The linkage between currency, banking, and default crises increases the chance of twin crises or even triple crises, outcomes in which the economic cost of each individual crisis is enlarged.
Currency crises can be especially destructive to small open economies or bigger, but not sufficiently stable ones. Governments often take on the role of fending off such attacks by satisfying the excess demand for a given currency using the country's own currency reserves or its foreign reserves (usually in the United States dollar, Euro or Pound sterling). Currency crises have large, measurable costs on an economy, but the ability to predict the timing and magnitude of crises is limited by theoretical understanding of the complex interactions between macroeconomic fundamentals, investor expectations, and government policy. A currency crisis may also have political implications for those in power. Following a currency crisis a change in the head of government and a change in the finance minister and/or central bank governor are more likely to occur.
A currency crisis is normally considered as part of a financial crisis. Kaminsky et al. (1998), for instance, define currency crises as when a weighted average of monthly percentage depreciations in the exchange rate and monthly percentage declines in exchange reserves exceeds its mean by more than three standard deviations. Frankel and Rose (1996) define a currency crisis as a nominal depreciation of a currency of at least 25% but it is also defined at least 10% increase in the rate of depreciation. In general, a currency crisis can be defined as a situation when the participants in an exchange market come
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https://en.wikipedia.org/wiki/Planar%20algebra
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In mathematics, planar algebras first appeared in the work of Vaughan Jones on the standard invariant of a II1 subfactor. They also provide an appropriate algebraic framework for many knot invariants (in particular the Jones polynomial), and have been used in describing the properties of Khovanov homology with respect to tangle composition. Any subfactor planar algebra provides a family of unitary representations of Thompson groups.
Any finite group (and quantum generalization) can be encoded as a planar algebra.
Definition
The idea of the planar algebra is to be a diagrammatic axiomatization of the standard invariant.
Planar tangle
A (shaded) planar tangle is the data of finitely many input disks, one output disk, non-intersecting strings giving an even number, say , intervals per disk and one -marked interval per disk.
Here, the mark is shown as a -shape. On each input disk it is placed between two adjacent outgoing strings, and on the output disk it is placed between two adjacent incoming strings. A planar tangle is defined up to isotopy.
Composition
To compose two planar tangles, put the output disk of one into an input of the other, having as many intervals, same shading of marked intervals and such that the -marked intervals coincide. Finally we remove the coinciding circles. Note that two planar tangles can have zero, one or several possible compositions.
Planar operad
The planar operad is the set of all the planar tangles (up to isomorphism) with such compositions.
Planar algebra
A planar algebra is a representation of the planar operad; more precisely, it is a family of vector spaces , called -box spaces, on which acts the planar operad, i.e. for any tangle (with one output disk and input disks with and intervals respectively) there is a multilinear map
with according to the shading of the -marked intervals, and these maps (also called partition functions) respect the composition of tangle in such a way that all the diagrams as below commute.
Examples
Planar tangles
The family of vector spaces generated by the planar tangles having intervals on their output disk and a white (or black) -marked interval, admits a planar algebra structure.
Temperley–Lieb
The Temperley-Lieb planar algebra is generated by the planar tangles without input disk; its -box space is generated by
Moreover, a closed string is replaced by a multiplication by .
Note that the dimension of is the Catalan number .
This planar algebra encodes the notion of Temperley–Lieb algebra.
Hopf algebra
A semisimple and cosemisimple Hopf algebra over an algebraically closed field is encoded in a planar algebra defined by generators and relations, and "corresponds" (up to isomorphism) to a connected, irreducible, spherical, non degenerate planar algebra with non zero modulus and of depth two.
Note that connected means (as for evaluable below), irreducible means , spherical is defined below, and non-degenerate means that the traces (define
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https://en.wikipedia.org/wiki/Hilbert%27s%20ninth%20problem
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Hilbert's ninth problem, from the list of 23 Hilbert's problems (1900), asked to find the most general reciprocity law for the norm residues of k-th order in a general algebraic number field, where k is a power of a prime.
Progress made
The problem was partially solved by Emil Artin by establishing the Artin reciprocity law which deals with abelian extensions of algebraic number fields. Together with the work of Teiji Takagi and Helmut Hasse (who established the more general Hasse reciprocity law), this led to the development of the class field theory, realizing Hilbert's program in an abstract fashion. Certain explicit formulas for norm residues were later found by Igor Shafarevich (1948; 1949; 1950).
The non-abelian generalization, also connected with Hilbert's twelfth problem, is one of the long-standing challenges in number theory and is far from being complete.
See also
List of unsolved problems in mathematics
References
External links
English translation of Hilbert's original address
Algebraic number theory
Unsolved problems in number theory
09
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https://en.wikipedia.org/wiki/Hilbert%27s%20fourteenth%20problem
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In mathematics, Hilbert's fourteenth problem, that is, number 14 of Hilbert's problems proposed in 1900, asks whether certain algebras are finitely generated.
The setting is as follows: Assume that k is a field and let K be a subfield of the field of rational functions in n variables,
k(x1, ..., xn ) over k.
Consider now the k-algebra R defined as the intersection
Hilbert conjectured that all such algebras are finitely generated over k.
Some results were obtained confirming Hilbert's conjecture in special cases and for certain classes of rings (in particular the conjecture was proved unconditionally for n = 1 and n = 2 by Zariski in 1954). Then in 1959 Masayoshi Nagata found a counterexample to Hilbert's conjecture. The counterexample of Nagata is a suitably constructed ring of invariants for the action of a linear algebraic group.
History
The problem originally arose in algebraic invariant theory. Here the ring R is given as a (suitably defined) ring of polynomial invariants of a linear algebraic group over a field k acting algebraically on a polynomial ring k[x1, ..., xn] (or more generally, on a finitely generated algebra defined over a field). In this situation the field K is the field of rational functions (quotients of polynomials) in the variables xi which are invariant under the given action of the algebraic group, the ring R is the ring of polynomials which are invariant under the action. A classical example in nineteenth century was the extensive study (in particular by Cayley, Sylvester, Clebsch, Paul Gordan and also Hilbert) of invariants of binary forms in two variables with the natural action of the special linear group SL2(k) on it. Hilbert himself proved the finite generation of invariant rings in the case of the field of complex numbers for some classical semi-simple Lie groups (in particular the general linear group over the complex numbers) and specific linear actions on polynomial rings, i.e. actions coming from finite-dimensional representations of the Lie-group. This finiteness result was later extended by Hermann Weyl to the class of all semi-simple Lie-groups. A major ingredient in Hilbert's proof is the Hilbert basis theorem applied to the ideal inside the polynomial ring generated by the invariants.
Zariski's formulation
Zariski's formulation of Hilbert's fourteenth problem asks whether, for a quasi-affine algebraic variety X over a field k, possibly assuming X normal or smooth, the ring of regular functions on X is finitely generated over k.
Zariski's formulation was shown to be equivalent to the original problem, for X normal. (See also: Zariski's finiteness theorem.)
Éfendiev F.F. (Fuad Efendi) provided symmetric algorithm generating basis of invariants of n-ary forms of degree r.
Nagata's counterexample
gave the following counterexample to Hilbert's problem. The field k is a field containing 48 elements a1i, ...,a16i, for i=1, 2, 3 that are algebraically independent over the prime field. The ring R
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https://en.wikipedia.org/wiki/Cofibration
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In mathematics, in particular homotopy theory, a continuous mapping between topological spaces
,
is a cofibration if it has the homotopy extension property with respect to all topological spaces . That is, is a cofibration if for each topological space , and for any continuous maps and with , for any homotopy from to , there is a continuous map and a homotopy from to such that for all and . (Here, denotes the unit interval .)
This definition is formally dual to that of a fibration, which is required to satisfy the homotopy lifting property with respect to all spaces; this is one instance of the broader Eckmann–Hilton duality in topology.
Cofibrations are a fundamental concept of homotopy theory. Quillen has proposed the notion of model category as a formal framework for doing homotopy theory in more general categories; a model category is endowed with three distinguished classes of morphisms called fibrations, cofibrations and weak equivalences satisfying certain lifting and factorization axioms.
Definition
Homotopy theory
In what follows, let denote the unit interval.
A map of topological spaces is called a cofibrationpg 51 if for any map such that there is an extension to , meaning there is a map such that , we can extend a homotopy of maps to a homotopy of maps , whereWe can encode this condition in the following commutative diagramwhere is the path space of equipped with the compact-open topology.
For the notion of a cofibration in a model category, see model category.
Examples
In topology
Topologists have long studied notions of "good subspace embedding", many of which imply that the map is a cofibration, or the converse, or have similar formal properties with regards to homology. In 1937, Borsuk proved that if is a binormal space ( is normal, and its product with the unit interval is normal) then every closed subspace of has the homotopy extension property with respect to any absolute neighborhood retract. Likewise, if is a closed subspace of and the subspace inclusion is an absolute neighborhood retract, then the inclusion of into is a cofibration.
Hatcher's introductory textbook Algebraic Topology uses a technical notion of good pair which has the same long exact sequence in singular homology associated to a cofibration, but it is not equivalent. The notion of cofibration is distinguished from these because its homotopy-theoretic definition is more amenable to formal analysis and generalization.
If is a continuous map between topological spaces, there is an associated topological space called the mapping cylinder of . There is a canonical subspace embedding and a projection map such that as pictured in the commutative diagram below. Moreover, is a cofibration and is a homotopy equivalence. This result can be summarized by saying that "every map is equivalent in the homotopy category to a cofibration."
Arne Strøm has proved a strengthening of this result, that every map factors as the compo
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https://en.wikipedia.org/wiki/Abuse%20of%20notation
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In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not entirely formally correct, but which might help simplify the exposition or suggest the correct intuition (while possibly minimizing errors and confusion at the same time). However, since the concept of formal/syntactical correctness depends on both time and context, certain notations in mathematics that are flagged as abuse in one context could be formally correct in one or more other contexts. Time-dependent abuses of notation may occur when novel notations are introduced to a theory some time before the theory is first formalized; these may be formally corrected by solidifying and/or otherwise improving the theory. Abuse of notation should be contrasted with misuse of notation, which does not have the presentational benefits of the former and should be avoided (such as the misuse of constants of integration).
A related concept is abuse of language or abuse of terminology, where a term — rather than a notation — is misused. Abuse of language is an almost synonymous expression for abuses that are non-notational by nature. For example, while the word representation properly designates a group homomorphism from a group G to GL(V), where V is a vector space, it is common to call V "a representation of G". Another common abuse of language consists in identifying two mathematical objects that are different, but canonically isomorphic. Other examples include identifying a constant function with its value, identifying a group with a binary operation with the name of its underlying set, or identifying to the Euclidean space of dimension three equipped with a Cartesian coordinate system.
Examples
Structured mathematical objects
Many mathematical objects consist of a set, often called the underlying set, equipped with some additional structure, such as a mathematical operation or a topology. It is a common abuse of notation to use the same notation for the underlying set and the structured object (a phenomenon known as suppression of parameters). For example, may denote the set of the integers, the group of integers together with addition, or the ring of integers with addition and multiplication. In general, there is no problem with this if the object under reference is well understood, and avoiding such an abuse of notation might even make mathematical texts more pedantic and more difficult to read. When this abuse of notation may be confusing, one may distinguish between these structures by denoting the group of integers with addition, and the ring of integers.
Similarly, a topological space consists of a set (the underlying set) and a topology which is characterized by a set of subsets of (the open sets). Most frequently, one considers only one topology on , so there is usually no problem in referring as both the underlying set, and the pair consisting of and its topology — even though they are technically distinct mathematical object
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