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https://en.wikipedia.org/wiki/Postfix
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Postfix may refer to:
Postfix (linguistics), an affix which is placed after the stem of a word
Postfix notation, a way of writing algebraic and other expressions
Postfix (software), a mail transfer agent
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https://en.wikipedia.org/wiki/Acyclic
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Acyclic may refer to:
In chemistry, a compound which is an open-chain compound, e.g. alkanes and acyclic aliphatic compounds
In mathematics:
A graph without a cycle, especially
A directed acyclic graph
An acyclic complex is a chain complex all of whose homology groups are zero
In economics, an economic indicator with little or no correlation to the business cycle
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https://en.wikipedia.org/wiki/P-adic%20analysis
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In mathematics, p-adic analysis is a branch of number theory that deals with the mathematical analysis of functions of p-adic numbers.
The theory of complex-valued numerical functions on the p-adic numbers is part of the theory of locally compact groups. The usual meaning taken for p-adic analysis is the theory of p-adic-valued functions on spaces of interest.
Applications of p-adic analysis have mainly been in number theory, where it has a significant role in diophantine geometry and diophantine approximation. Some applications have required the development of p-adic functional analysis and spectral theory. In many ways p-adic analysis is less subtle than classical analysis, since the ultrametric inequality means, for example, that convergence of infinite series of p-adic numbers is much simpler. Topological vector spaces over p-adic fields show distinctive features; for example aspects relating to convexity and the Hahn–Banach theorem are different.
Important results
Ostrowski's theorem
Ostrowski's theorem, due to Alexander Ostrowski (1916), states that every non-trivial absolute value on the rational numbers Q is equivalent to either the usual real absolute value or a -adic absolute value.
Mahler's theorem
Mahler's theorem, introduced by Kurt Mahler, expresses continuous p-adic functions in terms of polynomials.
In any field of characteristic 0, one has the following result. Let
be the forward difference operator. Then for polynomial functions f we have the Newton series:
where
is the kth binomial coefficient polynomial.
Over the field of real numbers, the assumption that the function f is a polynomial can be weakened, but it cannot be weakened all the way down to mere continuity.
Mahler proved the following result:
Mahler's theorem: If f is a continuous p-adic-valued function on the p-adic integers then the same identity holds.
Hensel's lemma
Hensel's lemma, also known as Hensel's lifting lemma, named after Kurt Hensel, is a result in modular arithmetic, stating that if a polynomial equation has a simple root modulo a prime number , then this root corresponds to a unique root of the same equation modulo any higher power of , which can be found by iteratively "lifting" the solution modulo successive powers of . More generally it is used as a generic name for analogues for complete commutative rings (including p-adic fields in particular) of the Newton method for solving equations. Since p-adic analysis is in some ways simpler than real analysis, there are relatively easy criteria guaranteeing a root of a polynomial.
To state the result, let be a polynomial with integer (or p-adic integer) coefficients, and let m,k be positive integers such that m ≤ k. If r is an integer such that
and
then there exists an integer s such that
and
Furthermore, this s is unique modulo pk+m, and can be computed explicitly as
where
Applications
P-adic quantum mechanics
P-adic quantum mechanics is a relatively recent approach to unde
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https://en.wikipedia.org/wiki/Sadleirian%20Professor%20of%20Pure%20Mathematics
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The Sadleirian Professorship of Pure Mathematics, originally spelled in the statutes and for the first two professors as Sadlerian, is a professorship in pure mathematics within the DPMMS at the University of Cambridge. It was founded on a bequest from Lady Mary Sadleir for lectureships "for the full and clear explication and teaching that part of mathematical knowledge commonly called algebra". She died in 1706 and lectures began in 1710 but eventually these failed to attract undergraduates. In 1860 the foundation was used to establish the professorship. On 10 June 1863 Arthur Cayley was elected with the statutory duty "to explain and teach the principles of pure mathematics, and to apply himself to the advancement of that science." The stipend attached to the professorship was modest although it improved in the course of subsequent legislation.
List of Sadlerian Lecturers of Pure Mathematics
1746–1769 William Ludlam
1826–1835 Lawrence Stephenson
List of Sadleirian Lecturers of Pure Mathematics
1845–1847 Arthur Scratchley
1847–1857 George Ferns Reyner
1851 Stephen Hanson
1855–1858 William Charles Green
1857–1864 John Robert Lunn
List of Sadleirian Professors of Pure Mathematics
1863–1895 Arthur Cayley
1895–1910 Andrew Russell Forsyth
1910–1931 E. W. Hobson
1931–1942 G. H. Hardy
1945–1953 Louis Mordell
1953–1967 Philip Hall
1967–1986 J. W. S. Cassels
1986–2012 John H. Coates
2013–2014 Vladimir Markovic
2017–2021 Emmanuel Breuillard
References
Sources
Obituary Notices of Fellows Deceased. (1895). Proceedings of the Royal Society of London, 58, I-Lx. Retrieved from https://www.jstor.org/stable/115800 (Obituary of Arthur Cayley written by Andrew Forsyth).
University of Cambridge DPMMS https://web.archive.org/web/20160624155328/http://www.admin.cam.ac.uk/offices/academic/secretary/professorships/sadleirian.pdf
Pure Mathematics, Sadleirian
Faculty of Mathematics, University of Cambridge
Pure Mathematics, Sadleirian, Cambridge
Mathematics education in the United Kingdom
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https://en.wikipedia.org/wiki/Fatou%27s%20lemma
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In mathematics, Fatou's lemma establishes an inequality relating the Lebesgue integral of the limit inferior of a sequence of functions to the limit inferior of integrals of these functions. The lemma is named after Pierre Fatou.
Fatou's lemma can be used to prove the Fatou–Lebesgue theorem and Lebesgue's dominated convergence theorem.
Standard statement
In what follows, denotes the -algebra of Borel sets on .
Fatou's lemma remains true if its assumptions hold -almost everywhere. In other words, it is enough that there is a null set such that the values are non-negative for every To see this, note that the integrals appearing in Fatou's lemma are unchanged if we change each function on .
Proof
Fatou's lemma does not require the monotone convergence theorem, but the latter can be used to provide a quick proof. A proof directly from the definitions of integrals is given further below.
In each case, the proof begins by analyzing the properties of . These satisfy:
the sequence is pointwise non-decreasing at any and
, .
Since , we immediately see that is measurable.
Via the Monotone Convergence Theorem
Moreover,
By the Monotone Convergence Theorem and property (1), the limit and integral may be interchanged:
where the last step used property (2).
From "first principles"
To demonstrate that the monotone convergence theorem is not "hidden", the proof below does not use any properties of Lebesgue integral except those established here.
Denote by the set of simple -measurable functions such that on .
Now we turn to the main theorem
The proof is complete.
Examples for strict inequality
Equip the space with the Borel σ-algebra and the Lebesgue measure.
Example for a probability space: Let denote the unit interval. For every natural number define
Example with uniform convergence: Let denote the set of all real numbers. Define
These sequences converge on pointwise (respectively uniformly) to the zero function (with zero integral), but every has integral one.
The role of non-negativity
A suitable assumption concerning the negative parts of the sequence f1, f2, . . . of functions is necessary for Fatou's lemma, as the following example shows. Let S denote the half line [0,∞) with the Borel σ-algebra and the Lebesgue measure. For every natural number n define
This sequence converges uniformly on S to the zero function and the limit, 0, is reached in a finite number of steps: for every x ≥ 0, if , then fn(x) = 0. However, every function fn has integral −1. Contrary to Fatou's lemma, this value is strictly less than the integral of the limit (0).
As discussed in below, the problem is that there is no uniform integrable bound on the sequence from below, while 0 is the uniform bound from above.
Reverse Fatou lemma
Let f1, f2, . . . be a sequence of extended real-valued measurable functions defined on a measure space (S,Σ,μ). If there exists a non-negative integrable function g on S such that fn ≤ g for all n,
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https://en.wikipedia.org/wiki/Glossary%20of%20field%20theory
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Field theory is the branch of mathematics in which fields are studied. This is a glossary of some terms of the subject. (See field theory (physics) for the unrelated field theories in physics.)
Definition of a field
A field is a commutative ring in which and every nonzero element has a multiplicative inverse. In a field we thus can perform the operations addition, subtraction, multiplication, and division.
The non-zero elements of a field F form an abelian group under multiplication; this group is typically denoted by F×;
The ring of polynomials in the variable x with coefficients in F is denoted by F[x].
Basic definitions
Characteristic The characteristic of the field F is the smallest positive integer n such that ; here n·1 stands for n summands . If no such n exists, we say the characteristic is zero. Every non-zero characteristic is a prime number. For example, the rational numbers, the real numbers and the p-adic numbers have characteristic 0, while the finite field Zp with p being prime has characteristic p.
Subfield A subfield of a field F is a subset of F which is closed under the field operation + and * of F and which, with these operations, forms itself a field.
Prime field The prime field of the field F is the unique smallest subfield of F.
Extension field If F is a subfield of E then E is an extension field of F. We then also say that E/F is a field extension.
Degree of an extension Given an extension E/F, the field E can be considered as a vector space over the field F, and the dimension of this vector space is the degree of the extension, denoted by [E : F].
Finite extension A finite extension is a field extension whose degree is finite.
Algebraic extension If an element α of an extension field E over F is the root of a non-zero polynomial in F[x], then α is algebraic over F. If every element of E is algebraic over F, then E/F is an algebraic extension.
Generating set Given a field extension E/F and a subset S of E, we write F(S) for the smallest subfield of E that contains both F and S. It consists of all the elements of E that can be obtained by repeatedly using the operations +, −, *, / on the elements of F and S. If , we say that E is generated by S over F.
Primitive element An element α of an extension field E over a field F is called a primitive element if E=F(α), the smallest extension field containing α. Such an extension is called a simple extension.
Splitting field A field extension generated by the complete factorisation of a polynomial.
Normal extension A field extension generated by the complete factorisation of a set of polynomials.
Separable extension An extension generated by roots of separable polynomials.
Perfect field A field such that every finite extension is separable. All fields of characteristic zero, and all finite fields, are perfect.
Imperfect degree Let F be a field of characteristic ; then Fp is a subfield. The degree is called the imperfect degree of F. The field F
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https://en.wikipedia.org/wiki/Seminorm
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In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk and, conversely, the Minkowski functional of any such set is a seminorm.
A topological vector space is locally convex if and only if its topology is induced by a family of seminorms.
Definition
Let be a vector space over either the real numbers or the complex numbers
A real-valued function is called a if it satisfies the following two conditions:
Subadditivity/Triangle inequality: for all
Absolute homogeneity: for all and all scalars
These two conditions imply that and that every seminorm also has the following property:
Nonnegativity: for all
Some authors include non-negativity as part of the definition of "seminorm" (and also sometimes of "norm"), although this is not necessary since it follows from the other two properties.
By definition, a norm on is a seminorm that also separates points, meaning that it has the following additional property:
Positive definite/Positive/: whenever satisfies then
A is a pair consisting of a vector space and a seminorm on If the seminorm is also a norm then the seminormed space is called a .
Since absolute homogeneity implies positive homogeneity, every seminorm is a type of function called a sublinear function. A map is called a if it is subadditive and positive homogeneous. Unlike a seminorm, a sublinear function is necessarily nonnegative. Sublinear functions are often encountered in the context of the Hahn–Banach theorem.
A real-valued function is a seminorm if and only if it is a sublinear and balanced function.
Examples
The on which refers to the constant map on induces the indiscrete topology on
Let be a measure on a space . For an arbitrary constant , let be the set of all functions for which
exists and is finite. It can be shown that is a vector space, and the functional is a seminorm on . However, it is not always a norm (e.g. if and is the Lebesgue measure) because does not always imply . To make a norm, quotient by the closed subspace of functions with . The resulting space, , has a norm induced by .
If is any linear form on a vector space then its absolute value defined by is a seminorm.
A sublinear function on a real vector space is a seminorm if and only if it is a , meaning that for all
Every real-valued sublinear function on a real vector space induces a seminorm defined by
Any finite sum of seminorms is a seminorm. The restriction of a seminorm (respectively, norm) to a vector subspace is once again a seminorm (respectively, norm).
If and are seminorms (respectively, norms) on and then the map defined by is a seminorm (respectively, a norm) on In particular, the maps on defined by and are both seminorms on
If and are seminorms on then so are
and
where and
The space
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https://en.wikipedia.org/wiki/Sarvadaman%20Chowla
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Sarvadaman D. S. Chowla (22 October 1907 – 10 December 1995) was an Indian American mathematician, specializing in number theory.
Early life
He was born in London, since his father, Gopal Chowla, a professor of mathematics in Lahore, was then studying in Cambridge. His family returned to India, where he received his master's degree in 1928 from the Government College in Lahore. In 1931 he received his doctorate from the University of Cambridge, where he studied under J. E. Littlewood.
Career and awards
Chowla then returned to India, where he taught at several universities, becoming head of mathematics at Government College, Lahore in 1936. During the difficulties arising from the partition of India in 1947, he left for the United States. There he visited the Institute for Advanced Study until the fall of 1949, then taught at the University of Kansas in Lawrence until moving to the University of Colorado in 1952. He moved to Penn State in 1963 as a research professor, where he remained until his retirement in 1976. He was a member of the Indian National Science Academy.
Among his contributions are a number of results which bear his name. These include the Bruck–Ryser–Chowla theorem, the Ankeny–Artin–Chowla congruence, the Chowla–Mordell theorem, and the Chowla–Selberg formula, and the Mian–Chowla sequence.
Works
Notes
External links
American Hindus
1907 births
1995 deaths
20th-century Indian mathematicians
Indian number theorists
Alumni of Trinity College, Cambridge
British emigrants to the United States
University of Colorado faculty
University of Kansas faculty
Pennsylvania State University faculty
Government College University, Lahore alumni
English Hindus
British people of Indian descent
American academics of Indian descent
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https://en.wikipedia.org/wiki/Self-selection%20bias
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In statistics, self-selection bias arises in any situation in which individuals select themselves into a group, causing a biased sample with nonprobability sampling. It is commonly used to describe situations where the characteristics of the people which cause them to select themselves in the group create abnormal or undesirable conditions in the group. It is closely related to the non-response bias, describing when the group of people responding has different responses than the group of people not responding.
Self-selection bias is a major problem in research in sociology, psychology, economics and many other social sciences. In such fields, a poll suffering from such bias is termed a self-selected listener opinion poll or "SLOP".
The term is also used in criminology to describe the process by which specific predispositions may lead an offender to choose a criminal career and lifestyle.
While the effects of self-selection bias are closely related to those of selection bias, the problem arises for rather different reasons; thus there may be a purposeful intent on the part of respondents leading to self-selection bias whereas other types of selection bias may arise more inadvertently, possibly as the result of mistakes by those designing any given study.
Explanation
Self-selection makes determination of causation more difficult. For example, when attempting to assess the effect of a test preparation course in increasing participant's test scores, significantly higher test scores might be observed among students who choose to participate in the preparation course itself. Due to self-selection, there may be a number of differences between the people who choose to take the course and those who choose not to, such as motivation, socioeconomic status, or prior test-taking experience. Due to self-selection according to such factors, a significant difference in mean test scores could be observed between the two populations independent of any ability of the course to affect test scores. An outcome might be that those who elect to do the preparation course would have achieved higher scores in the actual test anyway. If the study measures an improvement in absolute test scores due to participation in the preparation course, they may be skewed to show a higher effect. A relative measure of 'improvement' might improve the reliability of the study somewhat, but only partially.
Self-selection bias causes problems for research about programs or products. In particular, self-selection affects evaluation of whether or not a given program has some effect, and complicates interpretation of market research.
The Roy model provides one of the earliest academic illustrations of the self-selection problem.
See also
Convenience sampling
Sampling bias
Selection bias
References
Jacobs, B., Hartog, J., Vijverberg, W. (2009) "Self-selection bias in estimated wage premiums for earnings risk", Empirical Economics, 37 (2), 271–286.
External links
Self-selection bias
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https://en.wikipedia.org/wiki/Richard%20Borcherds
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Richard Ewen Borcherds (; born 29 November 1959) is a British mathematician currently working in quantum field theory. He is known for his work in lattices, group theory, and infinite-dimensional algebras, for which he was awarded the Fields Medal in 1998.
Early life
Borcherds was born in Cape Town, South Africa, but the family moved to Birmingham in the United Kingdom when he was six months old.
Education
Borcherds was educated at King Edward's School, Birmingham, and Trinity College, Cambridge, where he studied under John Horton Conway.
Career
After receiving his doctorate in 1985, Borcherds has held various alternating positions at Cambridge and the University of California, Berkeley, serving as Morrey Assistant Professor of Mathematics at Berkeley from 1987 to 1988. He was a Royal Society University Research Fellow. From 1996 he held a Royal Society Research Professorship at Cambridge before returning to Berkeley in 1999 as Professor of Mathematics.
An interview with Simon Singh for The Guardian, in which Borcherds suggested he might have some traits associated with Asperger syndrome, subsequently led to a chapter about him in a book on autism by Simon Baron-Cohen. Baron-Cohen concluded that while Borcherds had many autistic traits, he did not merit a formal diagnosis of Asperger syndrome.
Awards and honours
In 1992 Borcherds was one of the first recipients of the EMS prizes awarded at the first European Congress of Mathematics in Paris, and in 1994 he was an invited speaker at the International Congress of Mathematicians in Zurich. In 1994, he was elected to be a Fellow of the Royal Society. In 1998 at the 23rd International Congress of Mathematicians in Berlin, Germany he received the Fields Medal together with Maxim Kontsevich, William Timothy Gowers and Curtis T. McMullen. The award cited him "for his contributions to algebra, the theory of automorphic forms, and mathematical physics, including the introduction of vertex algebras and Borcherds' Lie algebras, the proof of the Conway-Norton moonshine conjecture and the discovery of a new class of automorphic infinite products." In 2012 he became a fellow of the American Mathematical Society, and in 2014 he was elected to the National Academy of Sciences.
References
Further reading
Conway and Sloane, Sphere Packings, Lattices, and Groups, Third Edition, Springer, 1998 .
Frenkel, Lepowsky and Meurman, Vertex Operator Algebras and the Monster, Academic Press, 1988 .
Kac, Victor, Vertex Algebras for Beginners, Second Edition, AMS 1997 .
External links
20th-century British mathematicians
21st-century British mathematicians
Fields Medalists
Group theorists
Fellows of the Royal Society
Fellows of the American Mathematical Society
Members of the United States National Academy of Sciences
University of California, Berkeley College of Letters and Science faculty
1959 births
Living people
People educated at King Edward's School, Birmingham
Alumni of Trinity College, Cambridge
Wh
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https://en.wikipedia.org/wiki/Partial
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Partial may refer to:
Mathematics
Partial derivative, derivative with respect to one of several variables of a function, with the other variables held constant
∂, a symbol that can denote a partial derivative, sometimes pronounced "partial dee"
Partial differential equation, a differential equation that contains unknown multivariable functions and their partial derivatives
Other uses
Partial application, in computer science the process of fixing a number of arguments to a function, producing another function
Partial charge or net atomic charge, in chemistry a charge value that is not an integer or whole number
Partial fingerprint, impression of human fingers used in criminology or forensic science
Partial seizure or focal seizure, a seizure that initially affects only one hemisphere of the brain
Partial or Part score, in contract bridge a trick score less than 100, as well as other meanings
Partial or Partial wave, one sound wave of which a complex tone is composed in a harmonic series
Showing partiality, favor, or bias
Arts and entertainment
Partial (music)
Partial (website)
Partials (novel)
See also
Part (disambiguation)
Partial function in mathematics, a function for some subset of a total function
Partially ordered set in mathematics, an ordering, sequencing, or arrangement of the elements of a set
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https://en.wikipedia.org/wiki/Stochastic
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Stochastic (; ) refers to the property of being well-described by a random probability distribution. Although stochasticity and randomness are distinct in that the former refers to a modeling approach and the latter refers to phenomena themselves, these two terms are often used synonymously. Furthermore, in probability theory, the formal concept of a stochastic process is also referred to as a random process.
Stochasticity is used in many different fields, including the natural sciences such as biology, chemistry, ecology, neuroscience, and physics, as well as technology and engineering fields such as image processing, signal processing, information theory, computer science, cryptography, and telecommunications. It is also used in finance, due to seemingly random changes in financial markets as well as in medicine, linguistics, music, media, colour theory, botany, manufacturing, and geomorphology.
Etymology
The word stochastic in English was originally used as an adjective with the definition "pertaining to conjecturing", and stemming from a Greek word meaning "to aim at a mark, guess", and the Oxford English Dictionary gives the year 1662 as its earliest occurrence. In his work on probability Ars Conjectandi, originally published in Latin in 1713, Jakob Bernoulli used the phrase "Ars Conjectandi sive Stochastice", which has been translated to "the art of conjecturing or stochastics". This phrase was used, with reference to Bernoulli, by Ladislaus Bortkiewicz, who in 1917 wrote in German the word Stochastik with a sense meaning random. The term stochastic process first appeared in English in a 1934 paper by Joseph Doob. For the term and a specific mathematical definition, Doob cited another 1934 paper, where the term stochastischer Prozeß was used in German by Aleksandr Khinchin, though the German term had been used earlier in 1931 by Andrey Kolmogorov.
Mathematics
In the early 1930s, Aleksandr Khinchin gave the first mathematical definition of a stochastic process as a family of random variables indexed by the real line. Further fundamental work on probability theory and stochastic processes was done by Khinchin as well as other mathematicians such as Andrey Kolmogorov, Joseph Doob, William Feller, Maurice Fréchet, Paul Lévy, Wolfgang Doeblin, and Harald Cramér. Decades later Cramér referred to the 1930s as the "heroic period of mathematical probability theory".
In mathematics, the theory of stochastic processes is an important contribution to probability theory, and continues to be an active topic of research for both theory and applications.
The word stochastic is used to describe other terms and objects in mathematics. Examples include a stochastic matrix, which describes a stochastic process known as a Markov process, and stochastic calculus, which involves differential equations and integrals based on stochastic processes such as the Wiener process, also called the Brownian motion process.
Natural science
One of the simplest
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https://en.wikipedia.org/wiki/Topologist%27s%20sine%20curve
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In the branch of mathematics known as topology, the topologist's sine curve or Warsaw sine curve is a topological space with several interesting properties that make it an important textbook example.
It can be defined as the graph of the function sin(1/x) on the half-open interval (0, 1], together with the origin, under the topology induced from the Euclidean plane:
Properties
The topologist's sine curve T is connected but neither locally connected nor path connected. This is because it includes the point (0,0) but there is no way to link the function to the origin so as to make a path.
The space T is the continuous image of a locally compact space (namely, let V be the space {−1} ∪ (0, 1], and use the map f from V to T defined by f(−1) = (0,0) and f(x) = (x, sin(1/x)) for x > 0), but T is not locally compact itself.
The topological dimension of T is 1.
Variants
Two variants of the topologist's sine curve have other interesting properties.
The closed topologist's sine curve can be defined by taking the topologist's sine curve and adding its set of limit points, ; some texts define the topologist's sine curve itself as this closed version, as they prefer to use the term 'closed topologist's sine curve' to refer to another curve. This space is closed and bounded and so compact by the Heine–Borel theorem, but has similar properties to the topologist's sine curve—it too is connected but neither locally connected nor path-connected.
The extended topologist's sine curve can be defined by taking the closed topologist's sine curve and adding to it the set . It is arc connected but not locally connected.
See also
List of topologies
Warsaw circle
References
Topological spaces
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https://en.wikipedia.org/wiki/Combinatorial%20game%20theory
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Combinatorial game theory is a branch of mathematics and theoretical computer science that typically studies sequential games with perfect information. Study has been largely confined to two-player games that have a position that the players take turns changing in defined ways or moves to achieve a defined winning condition. Combinatorial game theory has not traditionally studied games of chance or those that use imperfect or incomplete information, favoring games that offer perfect information in which the state of the game and the set of available moves is always known by both players. However, as mathematical techniques advance, the types of game that can be mathematically analyzed expands, thus the boundaries of the field are ever changing. Scholars will generally define what they mean by a "game" at the beginning of a paper, and these definitions often vary as they are specific to the game being analyzed and are not meant to represent the entire scope of the field.
Combinatorial games include well-known games such as chess, checkers, and Go, which are regarded as non-trivial, and tic-tac-toe, which is considered trivial, in the sense of being "easy to solve". Some combinatorial games may also have an unbounded playing area, such as infinite chess. In combinatorial game theory, the moves in these and other games are represented as a game tree.
Combinatorial games also include one-player combinatorial puzzles such as Sudoku, and no-player automata, such as Conway's Game of Life, (although in the strictest definition, "games" can be said to require more than one participant, thus the designations of "puzzle" and "automata".)
Game theory in general includes games of chance, games of imperfect knowledge, and games in which players can move simultaneously, and they tend to represent real-life decision making situations.
Combinatorial game theory has a different emphasis than "traditional" or "economic" game theory, which was initially developed to study games with simple combinatorial structure, but with elements of chance (although it also considers sequential moves, see extensive-form game). Essentially, combinatorial game theory has contributed new methods for analyzing game trees, for example using surreal numbers, which are a subclass of all two-player perfect-information games. The type of games studied by combinatorial game theory is also of interest in artificial intelligence, particularly for automated planning and scheduling. In combinatorial game theory there has been less emphasis on refining practical search algorithms (such as the alpha–beta pruning heuristic included in most artificial intelligence textbooks), but more emphasis on descriptive theoretical results (such as measures of game complexity or proofs of optimal solution existence without necessarily specifying an algorithm, such as the strategy-stealing argument).
An important notion in combinatorial game theory is that of the solved game. For example, tic-tac-toe is co
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https://en.wikipedia.org/wiki/Transitive%20closure
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In mathematics, the transitive closure of a homogeneous binary relation on a set is the smallest relation on that contains and is transitive. For finite sets, "smallest" can be taken in its usual sense, of having the fewest related pairs; for infinite sets is the unique minimal transitive superset of .
For example, if is a set of airports and means "there is a direct flight from airport to airport " (for and in ), then the transitive closure of on is the relation such that means "it is possible to fly from to in one or more flights".
More formally, the transitive closure of a binary relation on a set is the smallest (w.r.t. ⊆) transitive relation on such that ⊆ ; see . We have = if, and only if, itself is transitive.
Conversely, transitive reduction adduces a minimal relation from a given relation such that they have the same closure, that is, ; however, many different with this property may exist.
Both transitive closure and transitive reduction are also used in the closely related area of graph theory.
Transitive relations and examples
A relation R on a set X is transitive if, for all x, y, z in X, whenever and then . Examples of transitive relations include the equality relation on any set, the "less than or equal" relation on any linearly ordered set, and the relation "x was born before y" on the set of all people. Symbolically, this can be denoted as: if and then .
One example of a non-transitive relation is "city x can be reached via a direct flight from city y" on the set of all cities. Simply because there is a direct flight from one city to a second city, and a direct flight from the second city to the third, does not imply there is a direct flight from the first city to the third. The transitive closure of this relation is a different relation, namely "there is a sequence of direct flights that begins at city x and ends at city y". Every relation can be extended in a similar way to a transitive relation.
An example of a non-transitive relation with a less meaningful transitive closure is "x is the day of the week after y". The transitive closure of this relation is "some day x comes after a day y on the calendar", which is trivially true for all days of the week x and y (and thus equivalent to the Cartesian square, which is "x and y are both days of the week").
Existence and description
For any relation R, the transitive closure of R always exists. To see this, note that the intersection of any family of transitive relations is again transitive. Furthermore, there exists at least one transitive relation containing R, namely the trivial one: X × X. The transitive closure of R is then given by the intersection of all transitive relations containing R.
For finite sets, we can construct the transitive closure step by step, starting from R and adding transitive edges.
This gives the intuition for a general construction. For any set X, we
can prove that transitive closure is given by the following e
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https://en.wikipedia.org/wiki/One-to-one
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One-to-one or one to one may refer to:
Mathematics and communication
One-to-one function, also called an injective function
One-to-one correspondence, also called a bijective function
One-to-one (communication), the act of an individual communicating with another
One-to-one (data model), a relationship in a data model
One to one computing (education), an initiative for a computer for every student
One-to-one marketing or personalized marketing, an attempt to make a unique product offering for each customer
Music
One to One (band), a 1980s Canadian pop music group
One to One (Carole King album), 1982
One to One (Christine Fan album), 2005
One to One (Howard Jones album), 1986
One to One (Syreeta album), 1977
One to One (Ed Bruce album), 1981
"One to One" (Freeez song)
"One to One" (Joe Jackson Band song)
Other uses
One to One (Apple), Apple's personal training service
One 2 One, a defunct British mobile telecommunications company, which became T-Mobile UK
One to One (TV series), an Irish TV series
See also
1-1 (disambiguation)
One-to-many (disambiguation)
Many-to-many
One on One (disambiguation)
One-way (disambiguation)
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https://en.wikipedia.org/wiki/Cube%20root
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In mathematics, a cube root of a number is a number such that . All nonzero real numbers have exactly one real cube root and a pair of complex conjugate cube roots, and all nonzero complex numbers have three distinct complex cube roots. For example, the real cube root of , denoted , is , because , while the other cube roots of are and . The three cube roots of are
In some contexts, particularly when the number whose cube root is to be taken is a real number, one of the cube roots (in this particular case the real one) is referred to as the principal cube root, denoted with the radical sign The cube root is the inverse function of the cube function if considering only real numbers, but not if considering also complex numbers: although one has always the cube of a nonzero number has more than one complex cube root and its principal cube root may not be the number that was cubed. For example, , but
Formal definition
The cube roots of a number x are the numbers y which satisfy the equation
Properties
Real numbers
For any real number x, there is one real number y such that y3 = x. The cube function is increasing, so does not give the same result for two different inputs, and it covers all real numbers. In other words, it is a bijection, or one-to-one. Then we can define an inverse function that is also one-to-one. For real numbers, we can define a unique cube root of all real numbers. If this definition is used, the cube root of a negative number is a negative number.
If x and y are allowed to be complex, then there are three solutions (if x is non-zero) and so x has three cube roots. A real number has one real cube root and two further cube roots which form a complex conjugate pair. For instance, the cube roots of 1 are:
The last two of these roots lead to a relationship between all roots of any real or complex number. If a number is one cube root of a particular real or complex number, the other two cube roots can be found by multiplying that cube root by one or the other of the two complex cube roots of 1.
Complex numbers
For complex numbers, the principal cube root is usually defined as the cube root that has the greatest real part, or, equivalently, the cube root whose argument has the least absolute value. It is related to the principal value of the natural logarithm by the formula
If we write x as
where r is a non-negative real number and θ lies in the range
,
then the principal complex cube root is
This means that in polar coordinates, we are taking the cube root of the radius and dividing the polar angle by three in order to define a cube root. With this definition, the principal cube root of a negative number is a complex number, and for instance will not be −2, but rather .
This difficulty can also be solved by considering the cube root as a multivalued function: if we write the original complex number x in three equivalent forms, namely
The principal complex cube roots of these three forms are then respectively
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https://en.wikipedia.org/wiki/Representation%20of%20a%20Lie%20group
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In mathematics and theoretical physics, a representation of a Lie group is a linear action of a Lie group on a vector space. Equivalently, a representation is a smooth homomorphism of the group into the group of invertible operators on the vector space. Representations play an important role in the study of continuous symmetry. A great deal is known about such representations, a basic tool in their study being the use of the corresponding 'infinitesimal' representations of Lie algebras.
Finite-dimensional representations
Representations
A complex representation of a group is an action by a group on a finite-dimensional vector space over the field . A representation of the Lie group G, acting on an n-dimensional vector space V over is then a smooth group homomorphism
,
where is the general linear group of all invertible linear transformations of under their composition. Since all n-dimensional spaces are isomorphic, the group can be identified with the group of the invertible, complex generally Smoothness of the map can be regarded as a technicality, in that any continuous homomorphism will automatically be smooth.
We can alternatively describe a representation of a Lie group as a linear action of on a vector space . Notationally, we would then write in place of for the way a group element acts on the vector .
A typical example in which representations arise in physics would be the study of a linear partial differential equation having symmetry group . Although the individual solutions of the equation may not be invariant under the action of , the space of all solutions is invariant under the action of . Thus, constitutes a representation of . See the example of SO(3), discussed below.
Basic definitions
If the homomorphism is injective (i.e., a monomorphism), the representation is said to be faithful.
If a basis for the complex vector space V is chosen, the representation can be expressed as a homomorphism into general linear group . This is known as a matrix representation. Two representations of G on vector spaces V, W are equivalent if they have the same matrix representations with respect to some choices of bases
for V and W.
Given a representation , we say that a subspace W of V is an invariant subspace if for all and . The representation is said to be irreducible if the only invariant subspaces of V are the zero space and V itself. For certain types of Lie groups, namely compact and semisimple groups, every finite-dimensional representation decomposes as a direct sum of irreducible representations, a property known as complete reducibility. For such groups, a typical goal of representation theory is to classify all finite-dimensional irreducible representations of the given group, up to isomorphism. (See the Classification section below.)
A unitary representation on a finite-dimensional inner product space is defined in the same way, except that is required to map into the group of unitary operators. If G is a c
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https://en.wikipedia.org/wiki/Unitary%20representation
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In mathematics, a unitary representation of a group G is a linear representation π of G on a complex Hilbert space V such that π(g) is a unitary operator for every g ∈ G. The general theory is well-developed in the case that G is a locally compact (Hausdorff) topological group and the representations are strongly continuous.
The theory has been widely applied in quantum mechanics since the 1920s, particularly influenced by Hermann Weyl's 1928 book Gruppentheorie and Quantenmechanik. One of the pioneers in constructing a general theory of unitary representations, for any group G rather than just for particular groups useful in applications, was George Mackey.
Context in harmonic analysis
The theory of unitary representations of topological groups is closely connected with harmonic analysis. In the case of an abelian group G, a fairly complete picture of the representation theory of G is given by Pontryagin duality. In general, the unitary equivalence classes (see below) of irreducible unitary representations of G make up its unitary dual. This set can be identified with the spectrum of the C*-algebra associated with G by the group C*-algebra construction. This is a topological space.
The general form of the Plancherel theorem tries to describe the regular representation of G on L2(G) using a measure on the unitary dual. For G abelian this is given by the Pontryagin duality theory. For G compact, this is done by the Peter–Weyl theorem; in that case, the unitary dual is a discrete space, and the measure attaches an atom to each point of mass equal to its degree.
Formal definitions
Let G be a topological group. A strongly continuous unitary representation of G on a Hilbert space H is a group homomorphism from G into the unitary group of H,
such that g → π(g) ξ is a norm continuous function for every ξ ∈ H.
Note that if G is a Lie group, the Hilbert space also admits underlying smooth and analytic structures. A vector ξ in H is said to be smooth or analytic if the map g → π(g) ξ is smooth or analytic (in the norm or weak topologies on H). Smooth vectors are dense in H by a classical argument of Lars Gårding, since convolution by smooth functions of compact support yields smooth vectors. Analytic vectors are dense by a classical argument of Edward Nelson, amplified by Roe Goodman, since vectors in the image of a heat operator e–tD, corresponding to an elliptic differential operator D in the universal enveloping algebra of G, are analytic. Not only do smooth or analytic vectors form dense subspaces; but they also form common cores for the unbounded skew-adjoint operators corresponding to the elements of the Lie algebra, in the sense of spectral theory.
Two unitary representations π1: G → U(H1), π2: G → U(H2) are said to be unitarily equivalent if there is a unitary transformation A:H1 → H2 such that π1(g) = A* ∘ π2(g) ∘ A for all g in G. When this holds, A is said to be an intertwining operator for the representations .
If is a representa
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https://en.wikipedia.org/wiki/Simple%20Lie%20group
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In mathematics, a simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian symmetric spaces.
Together with the commutative Lie group of the real numbers, , and that of the unit-magnitude complex numbers, U(1) (the unit circle), simple Lie groups give the atomic "blocks" that make up all (finite-dimensional) connected Lie groups via the operation of group extension. Many commonly encountered Lie groups are either simple or 'close' to being simple: for example, the so-called "special linear group" SL(n) of n by n matrices with determinant equal to 1 is simple for all n > 1.
The first classification of simple Lie groups was by Wilhelm Killing, and this work was later perfected by Élie Cartan. The final classification is often referred to as Killing-Cartan classification.
Definition
Unfortunately, there is no universally accepted definition of a simple Lie group. In particular, it is not always defined as a Lie group that is simple as an abstract group. Authors differ on whether a simple Lie group has to be connected, or on whether it is allowed to have a non-trivial center, or on whether is a simple Lie group.
The most common definition is that a Lie group is simple if it is connected, non-abelian, and every closed connected normal subgroup is either the identity or the whole group. In particular, simple groups are allowed to have a non-trivial center, but is not simple.
In this article the connected simple Lie groups with trivial center are listed. Once these are known, the ones with non-trivial center are easy to list as follows. Any simple Lie group with trivial center has a universal cover, whose center is the fundamental group of the simple Lie group. The corresponding simple Lie groups with non-trivial center can be obtained as quotients of this universal cover by a subgroup of the center.
Alternatives
An equivalent definition of a simple Lie group follows from the Lie correspondence: A connected Lie group is simple if its Lie algebra is simple. An important technical point is that a simple Lie group may contain discrete normal subgroups. For this reason, the definition of a simple Lie group is not equivalent to the definition of a Lie group that is simple as an abstract group.
Simple Lie groups include many classical Lie groups, which provide a group-theoretic underpinning for spherical geometry, projective geometry and related geometries in the sense of Felix Klein's Erlangen program. It emerged in the course of classification of simple Lie groups that there exist also several exceptional possibilities not corresponding to any familiar geometry. These exceptional groups account for many special examples and configurations in other branches of mathematics, as well as contemporary theoretical physics.
As a counterexample, the general linear group is neither simple, nor semisim
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https://en.wikipedia.org/wiki/Symplectic%20vector%20space
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In mathematics, a symplectic vector space is a vector space V over a field F (for example the real numbers R) equipped with a symplectic bilinear form.
A symplectic bilinear form is a mapping that is
Bilinear Linear in each argument separately;
Alternating holds for all ; and
Non-degenerate for all implies that .
If the underlying field has characteristic not 2, alternation is equivalent to skew-symmetry. If the characteristic is 2, the skew-symmetry is implied by, but does not imply alternation. In this case every symplectic form is a symmetric form, but not vice versa.
Working in a fixed basis, ω can be represented by a matrix. The conditions above are equivalent to this matrix being skew-symmetric, nonsingular, and hollow (all diagonal entries are zero). This should not be confused with a symplectic matrix, which represents a symplectic transformation of the space. If V is finite-dimensional, then its dimension must necessarily be even since every skew-symmetric, hollow matrix of odd size has determinant zero. Notice that the condition that the matrix be hollow is not redundant if the characteristic of the field is 2. A symplectic form behaves quite differently from a symmetric form, for example, the scalar product on Euclidean vector spaces.
Standard symplectic space
The standard symplectic space is R2n with the symplectic form given by a nonsingular, skew-symmetric matrix. Typically ω is chosen to be the block matrix
where In is the identity matrix. In terms of basis vectors :
A modified version of the Gram–Schmidt process shows that any finite-dimensional symplectic vector space has a basis such that ω takes this form, often called a Darboux basis or symplectic basis.
Sketch of process:
Start with an arbitrary basis , and represent the dual of each basis vector by the dual basis: . This gives us a matrix with entries . Solve for its null space. Now for any in the null space, we have , so the null space gives us the degenerate subspace .
Now arbitrarily pick a complementary such that , and let be a basis of . Since , and , WLOG . Now scale so that . Then define for each of . Iterate.
Notice that this method applies for symplectic vector space over any field, not just the field of real numbers.
Case of real or complex field:
When the space is over the field of real numbers, then we can modify the modified Gram-Schmidt process as follows: Start the same way. Let be an orthonormal basis (with respect to the usual inner product on ) of . Since , and , WLOG . Now multiply by a sign, so that . Then define for each of , then scale each so that it has norm one. Iterate.
Similarly, for the field of complex numbers, we may choose a unitary basis. This proves the spectral theory of antisymmetric matrices.
Lagrangian form
There is another way to interpret this standard symplectic form. Since the model space R2n used above carries much canonical structure which might easily lead to misinterpretation, we will use "an
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https://en.wikipedia.org/wiki/G2%20%28mathematics%29
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{{DISPLAYTITLE:G2 (mathematics)}}
In mathematics, G2 is the name of three simple Lie groups (a complex form, a compact real form and a split real form), their Lie algebras as well as some algebraic groups. They are the smallest of the five exceptional simple Lie groups. G2 has rank 2 and dimension 14. It has two fundamental representations, with dimension 7 and 14.
The compact form of G2 can be described as the automorphism group of the octonion algebra or, equivalently, as the subgroup of SO(7) that preserves any chosen particular vector in its 8-dimensional real spinor representation (a spin representation).
History
The Lie algebra , being the smallest exceptional simple Lie algebra, was the first of these to be discovered in the attempt to classify simple Lie algebras. On May 23, 1887, Wilhelm Killing wrote a letter to Friedrich Engel saying that he had found a 14-dimensional simple Lie algebra, which we now call .
In 1893, Élie Cartan published a note describing an open set in equipped with a 2-dimensional distribution—that is, a smoothly varying field of 2-dimensional subspaces of the tangent space—for which the Lie algebra appears as the infinitesimal symmetries. In the same year, in the same journal, Engel noticed the same thing. Later it was discovered that the 2-dimensional distribution is closely related to a ball rolling on another ball. The space of configurations of the rolling ball is 5-dimensional, with a 2-dimensional distribution that describes motions of the ball where it rolls without slipping or twisting.
In 1900, Engel discovered that a generic antisymmetric trilinear form (or 3-form) on a 7-dimensional complex vector space is preserved by a group isomorphic to the complex form of G2.
In 1908 Cartan mentioned that the automorphism group of the octonions is a 14-dimensional simple Lie group. In 1914 he stated that this is the compact real form of G2.
In older books and papers, G2 is sometimes denoted by E2.
Real forms
There are 3 simple real Lie algebras associated with this root system:
The underlying real Lie algebra of the complex Lie algebra G2 has dimension 28. It has complex conjugation as an outer automorphism and is simply connected. The maximal compact subgroup of its associated group is the compact form of G2.
The Lie algebra of the compact form is 14-dimensional. The associated Lie group has no outer automorphisms, no center, and is simply connected and compact.
The Lie algebra of the non-compact (split) form has dimension 14. The associated simple Lie group has fundamental group of order 2 and its outer automorphism group is the trivial group. Its maximal compact subgroup is . It has a non-algebraic double cover that is simply connected.
Algebra
Dynkin diagram and Cartan matrix
The Dynkin diagram for G2 is given by .
Its Cartan matrix is:
Roots of G2
Although they span a 2-dimensional space, as drawn, it is much more symmetric to consider them as vectors in a 2-dimensional subspace of a
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https://en.wikipedia.org/wiki/F4%20%28mathematics%29
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{{DISPLAYTITLE:F4 (mathematics)}}
In mathematics, F4 is the name of a Lie group and also its Lie algebra f4. It is one of the five exceptional simple Lie groups. F4 has rank 4 and dimension 52. The compact form is simply connected and its outer automorphism group is the trivial group. Its fundamental representation is 26-dimensional.
The compact real form of F4 is the isometry group of a 16-dimensional Riemannian manifold known as the octonionic projective plane OP2. This can be seen systematically using a construction known as the magic square, due to Hans Freudenthal and Jacques Tits.
There are 3 real forms: a compact one, a split one, and a third one. They are the isometry groups of the three real Albert algebras.
The F4 Lie algebra may be constructed by adding 16 generators transforming as a spinor to the 36-dimensional Lie algebra so(9), in analogy with the construction of E8.
In older books and papers, F4 is sometimes denoted by E4.
Algebra
Dynkin diagram
The Dynkin diagram for F4 is: .
Weyl/Coxeter group
Its Weyl/Coxeter group is the symmetry group of the 24-cell: it is a solvable group of order 1152. It has minimal faithful degree , which is realized by the action on the 24-cell.
Cartan matrix
F4 lattice
The F4 lattice is a four-dimensional body-centered cubic lattice (i.e. the union of two hypercubic lattices, each lying in the center of the other). They form a ring called the Hurwitz quaternion ring. The 24 Hurwitz quaternions of norm 1 form the vertices of a 24-cell centered at the origin.
Roots of F4
The 48 root vectors of F4 can be found as the vertices of the 24-cell in two dual configurations, representing the vertices of a disphenoidal 288-cell if the edge lengths of the 24-cells are equal:
24-cell vertices:
24 roots by (±1, ±1, 0, 0), permuting coordinate positions
Dual 24-cell vertices:
8 roots by (±1, 0, 0, 0), permuting coordinate positions
16 roots by (±1/2, ±1/2, ±1/2, ±1/2).
Simple roots
One choice of simple roots for F4, , is given by the rows of the following matrix:
The Hasse diagram for the F4 root poset is shown below right.
F4 polynomial invariant
Just as O(n) is the group of automorphisms which keep the quadratic polynomials invariant, F4 is the group of automorphisms of the following set of 3 polynomials in 27 variables. (The first can easily be substituted into other two making 26 variables).
Where x, y, z are real-valued and X, Y, Z are octonion valued. Another way of writing these invariants is as (combinations of) Tr(M), Tr(M2) and Tr(M3) of the hermitian octonion matrix:
The set of polynomials defines a 24-dimensional compact surface.
Representations
The characters of finite dimensional representations of the real and complex Lie algebras and Lie groups are all given by the Weyl character formula. The dimensions of the smallest irreducible representations are :
1, 26, 52, 273, 324, 1053 (twice), 1274, 2652, 4096, 8424, 10829, 12376, 16302, 17901, 19278, 19448, 29172, 34749,
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https://en.wikipedia.org/wiki/Suprematism
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Suprematism () is an early twentieth-century art movement focused on the fundamentals of geometry (circles, squares, rectangles), painted in a limited range of colors. The term suprematism refers to an abstract art based upon "the supremacy of pure artistic feeling" rather than on visual depiction of objects.
Founded by Russian artist Kazimir Malevich in 1913, Supremus (Russian: Супремус) conceived of the artist as liberated from everything that pre-determined the ideal structure of life and art. Projecting that vision onto Cubism, which Malevich admired for its ability to deconstruct art, and in the process change its reference points of art, he led a group of Russian avant-garde artists — including Aleksandra Ekster, Liubov Popova, Olga Rozanova, Ivan Kliun, Ivan Puni, Nadezhda Udaltsova, Nina Genke-Meller, Ksenia Boguslavskaya and others — in what's been described as the first attempt to independently found a Russian avant-garde movement, seceding from the trajectory of prior Russian art history.
To support the movement, Malevich established the journal Supremus (initially titled Nul or Nothing), which received contributions from artists and philosophers. The publication, however, never took off and its first issue was never distributed due to the Russian Revolution. The movement itself, however, was announced in Malevich's 1915 Last Futurist Exhibition of Paintings 0,10, in St. Petersburg, where he, and several others in his group, exhibited 36 works in a similar style.
Birth of the movement
Kazimir Malevich developed the concept of Suprematism when he was already an established painter, having exhibited in the Donkey's Tail and the Der Blaue Reiter (The Blue Rider) exhibitions of 1912 with cubo-futurist works. The proliferation of new artistic forms in painting, poetry and theatre as well as a revival of interest in the traditional folk art of Russia provided a rich environment in which a Modernist culture was born.
In "Suprematism" (Part II of his book The Non-Objective World, which was published 1927 in Munich as Bauhaus Book No. 11), Malevich clearly stated the core concept of Suprematism:
He created a suprematist "grammar" based on fundamental geometric forms; in particular, the square and the circle. In the 0.10 Exhibition in 1915, Malevich exhibited his early experiments in suprematist painting. The centerpiece of his show was the Black Square, placed in what is called the red/beautiful corner in Russian Orthodox tradition; the place of the main icon in a house. "Black Square" was painted in 1915 and was presented as a breakthrough in his career and in art in general. Malevich also painted White on White which was also heralded as a milestone. White on White marked a shift from polychrome to monochrome Suprematism.
Distinct from Constructivism
Malevich's Suprematism is fundamentally opposed to the postrevolutionary positions of Constructivism and materialism. Constructivism, with its cult of the object, is concerned with utili
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https://en.wikipedia.org/wiki/Character%20%28mathematics%29
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In mathematics, a character is (most commonly) a special kind of function from a group to a field (such as the complex numbers). There are at least two distinct, but overlapping meanings. Other uses of the word "character" are almost always qualified.
Multiplicative character
A multiplicative character (or linear character, or simply character) on a group G is a group homomorphism from G to the multiplicative group of a field , usually the field of complex numbers. If G is any group, then the set Ch(G) of these morphisms forms an abelian group under pointwise multiplication.
This group is referred to as the character group of G. Sometimes only unitary characters are considered (thus the image is in the unit circle); other such homomorphisms are then called quasi-characters. Dirichlet characters can be seen as a special case of this definition.
Multiplicative characters are linearly independent, i.e. if are different characters on a group G then from it follows that .
Character of a representation
The character of a representation of a group G on a finite-dimensional vector space V over a field F is the trace of the representation , i.e.
for
In general, the trace is not a group homomorphism, nor does the set of traces form a group. The characters of one-dimensional representations are identical to one-dimensional representations, so the above notion of multiplicative character can be seen as a special case of higher-dimensional characters. The study of representations using characters is called "character theory" and one-dimensional characters are also called "linear characters" within this context.
Alternative definition
If restricted to finite abelian group with representation in (i.e. ), the following alternative definition would be equivalent to the above (For abelian groups, every matrix representation decomposes into a direct sum of representations. For non-abelian groups, the original definition would be more general than this one):
A character of group is a group homomorphism i.e. for all
If is a finite abelian group, the characters play the role of harmonics. For infinite abelian groups, the above would be replaced by where is the circle group.
See also
Character group
Dirichlet character
Harish-Chandra character
Hecke character
Infinitesimal character
Alternating character
Characterization (mathematics)
Pontryagin duality
References
Lectures Delivered at the University of Notre Dame
External links
Representation theory
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https://en.wikipedia.org/wiki/%2A-algebra
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In mathematics, and more specifically in abstract algebra, a *-algebra (or involutive algebra) is a mathematical structure consisting of two involutive rings and , where is commutative and has the structure of an associative algebra over . Involutive algebras generalize the idea of a number system equipped with conjugation, for example the complex numbers and complex conjugation, matrices over the complex numbers and conjugate transpose, and linear operators over a Hilbert space and Hermitian adjoints.
However, it may happen that an algebra admits no involution.
Definitions
*-ring
In mathematics, a *-ring is a ring with a map that is an antiautomorphism and an involution.
More precisely, is required to satisfy the following properties:
for all in .
This is also called an involutive ring, involutory ring, and ring with involution. The third axiom is implied by the second and fourth axioms, making it redundant.
Elements such that are called self-adjoint.
Archetypical examples of a *-ring are fields of complex numbers and algebraic numbers with complex conjugation as the involution. One can define a sesquilinear form over any *-ring.
Also, one can define *-versions of algebraic objects, such as ideal and subring, with the requirement to be *-invariant: and so on.
*-rings are unrelated to star semirings in the theory of computation.
*-algebra
A *-algebra is a *-ring, with involution * that is an associative algebra over a commutative *-ring with involution , such that .
The base *-ring is often the complex numbers (with acting as complex conjugation).
It follows from the axioms that * on is conjugate-linear in , meaning
for .
A *-homomorphism is an algebra homomorphism that is compatible with the involutions of and , i.e.,
for all in .
Philosophy of the *-operation
The *-operation on a *-ring is analogous to complex conjugation on the complex numbers. The *-operation on a *-algebra is analogous to taking adjoints in complex matrix algebras.
Notation
The * involution is a unary operation written with a postfixed star glyph centered above or near the mean line:
, or
(TeX: x^*),
but not as ""; see the asterisk article for details.
Examples
Any commutative ring becomes a *-ring with the trivial (identical) involution.
The most familiar example of a *-ring and a *-algebra over reals is the field of complex numbers where * is just complex conjugation.
More generally, a field extension made by adjunction of a square root (such as the imaginary unit ) is a *-algebra over the original field, considered as a trivially-*-ring. The * flips the sign of that square root.
A quadratic integer ring (for some ) is a commutative *-ring with the * defined in the similar way; quadratic fields are *-algebras over appropriate quadratic integer rings.
Quaternions, split-complex numbers, dual numbers, and possibly other hypercomplex number systems form *-rings (with their built-in conjugation operation) and *-algebr
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https://en.wikipedia.org/wiki/Antilinear%20map
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In mathematics, a function between two complex vector spaces is said to be antilinear or conjugate-linear if
hold for all vectors and every complex number where denotes the complex conjugate of
Antilinear maps stand in contrast to linear maps, which are additive maps that are homogeneous rather than conjugate homogeneous. If the vector spaces are real then antilinearity is the same as linearity.
Antilinear maps occur in quantum mechanics in the study of time reversal and in spinor calculus, where it is customary to replace the bars over the basis vectors and the components of geometric objects by dots put above the indices. Scalar-valued antilinear maps often arise when dealing with complex inner products and Hilbert spaces.
Definitions and characterizations
A function is called or if it is additive and conjugate homogeneous.
An on a vector space is a scalar-valued antilinear map.
A function is called if
while it is called if
In contrast, a linear map is a function that is additive and homogeneous, where is called if
An antilinear map may be equivalently described in terms of the linear map from to the complex conjugate vector space
Examples
Anti-linear dual map
Given a complex vector space of rank 1, we can construct an anti-linear dual map which is an anti-linear map sending an element for to for some fixed real numbers We can extend this to any finite dimensional complex vector space, where if we write out the standard basis and each standard basis element as then an anti-linear complex map to will be of the form for
Isomorphism of anti-linear dual with real dual
The anti-linear dualpg 36 of a complex vector space is a special example because it is isomorphic to the real dual of the underlying real vector space of This is given by the map sending an anti-linear map to In the other direction, there is the inverse map sending a real dual vector to giving the desired map.
Properties
The composite of two antilinear maps is a linear map. The class of semilinear maps generalizes the class of antilinear maps.
Anti-dual space
The vector space of all antilinear forms on a vector space is called the of If is a topological vector space, then the vector space of all antilinear functionals on denoted by is called the or simply the of if no confusion can arise.
When is a normed space then the canonical norm on the (continuous) anti-dual space denoted by is defined by using this same equation:
This formula is identical to the formula for the on the continuous dual space of which is defined by
Canonical isometry between the dual and anti-dual
The complex conjugate of a functional is defined by sending to It satisfies
for every and every
This says exactly that the canonical antilinear bijection defined by
as well as its inverse are antilinear isometries and consequently also homeomorphisms.
If then and this canonical map reduces down to the identity map.
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https://en.wikipedia.org/wiki/Involution%20%28mathematics%29
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In mathematics, an involution, involutory function, or self-inverse function is a function that is its own inverse,
for all in the domain of . Equivalently, applying twice produces the original value.
General properties
Any involution is a bijection.
The identity map is a trivial example of an involution. Examples of nontrivial involutions include negation (), reciprocation (), and complex conjugation () in arithmetic; reflection, half-turn rotation, and circle inversion in geometry; complementation in set theory; and reciprocal ciphers such as the ROT13 transformation and the Beaufort polyalphabetic cipher.
The composition of two involutions f and g is an involution if and only if they commute: .
Involutions on finite sets
The number of involutions, including the identity involution, on a set with elements is given by a recurrence relation found by Heinrich August Rothe in 1800:
and for
The first few terms of this sequence are 1, 1, 2, 4, 10, 26, 76, 232 ; these numbers are called the telephone numbers, and they also count the number of Young tableaux with a given number of cells.
The number can also be expressed by non-recursive formulas, such as the sum
The number of fixed points of an involution on a finite set and its number of elements have the same parity. Thus the number of fixed points of all the involutions on a given finite set have the same parity. In particular, every involution on an odd number of elements has at least one fixed point. This can be used to prove Fermat's two squares theorem.
Involution throughout the fields of mathematics
Real-valued functions
Some basic examples of involutions include the functions
the composition
and more generally the function
is an involution for constants and that satisfy
Another one is
The graph of an involution (on the real numbers) is symmetric across the line . This is due to the fact that the inverse of any general function will be its reflection over the line . This can be seen by "swapping" with . If, in particular, the function is an involution, then its graph is its own reflection.
Other elementary involutions are useful in solving functional equations.
Euclidean geometry
A simple example of an involution of the three-dimensional Euclidean space is reflection through a plane. Performing a reflection twice brings a point back to its original coordinates.
Another involution is reflection through the origin; not a reflection in the above sense, and so, a distinct example.
These transformations are examples of affine involutions.
Projective geometry
An involution is a projectivity of period 2, that is, a projectivity that interchanges pairs of points.
Any projectivity that interchanges two points is an involution.
The three pairs of opposite sides of a complete quadrangle meet any line (not through a vertex) in three pairs of an involution. This theorem has been called Desargues's Involution Theorem. Its origins can be seen in Lemma IV of the lemmas to
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https://en.wikipedia.org/wiki/24-cell
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In geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,4,3}. It is also called C24, or the icositetrachoron, octaplex (short for "octahedral complex"), icosatetrahedroid, octacube, hyper-diamond or polyoctahedron, being constructed of octahedral cells.
The boundary of the 24-cell is composed of 24 octahedral cells with six meeting at each vertex, and three at each edge. Together they have 96 triangular faces, 96 edges, and 24 vertices. The vertex figure is a cube. The 24-cell is self-dual. The 24-cell and the tesseract are the only convex regular 4-polytopes in which the edge length equals the radius.
The 24-cell does not have a regular analogue in 3 dimensions. It is the only one of the six convex regular 4-polytopes which is not the four-dimensional analogue of one of the five regular Platonic solids. It is the unique regular polytope, in any number of dimensions, which has no regular analogue in the adjacent dimension, either below or above. However, it can be seen as the analogue of a pair of irregular solids: the cuboctahedron and its dual the rhombic dodecahedron.
Translated copies of the 24-cell can tile four-dimensional space face-to-face, forming the 24-cell honeycomb. As a polytope that can tile by translation, the 24-cell is an example of a parallelotope, the simplest one that is not also a zonotope.
Geometry
The 24-cell incorporates the geometries of every convex regular polytope in the first four dimensions, except the 5-cell, those with a 5 in their Schlӓfli symbol, and the polygons {7} and above. It is especially useful to explore the 24-cell, because one can see the geometric relationships among all of these regular polytopes in a single 24-cell or its honeycomb.
The 24-cell is the fourth in the sequence of 6 convex regular 4-polytopes (in order of size and complexity). It can be deconstructed into 3 overlapping instances of its predecessor the tesseract (8-cell), as the 8-cell can be deconstructed into 2 overlapping instances of its predecessor the 16-cell. The reverse procedure to construct each of these from an instance of its predecessor preserves the radius of the predecessor, but generally produces a successor with a smaller edge length.
Coordinates
Squares
The 24-cell is the convex hull of its vertices which can be described as the 24 coordinate permutations of:
Those coordinates can be constructed as , rectifying the 16-cell with 8 vertices permutations of (±2,0,0,0). The vertex figure of a 16-cell is the octahedron; thus, cutting the vertices of the 16-cell at the midpoint of its incident edges produces 8 octahedral cells. This process also rectifies the tetrahedral cells of the 16-cell which become 16 octahedra, giving the 24-cell 24 octahedral cells.
In this frame of reference the 24-cell has edges of length and is inscribed in a 3-sphere of radius . Remarkably, the edge length equals the circumradius, as in the hexagon, or the cuboct
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https://en.wikipedia.org/wiki/Orbifold
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In the mathematical disciplines of topology and geometry, an orbifold (for "orbit-manifold") is a generalization of a manifold. Roughly speaking, an orbifold is a topological space which is locally a finite group quotient of a Euclidean space.
Definitions of orbifold have been given several times: by Ichirô Satake in the context of automorphic forms in the 1950s under the name V-manifold; by William Thurston in the context of the geometry of 3-manifolds in the 1970s when he coined the name orbifold, after a vote by his students; and by André Haefliger in the 1980s in the context of Mikhail Gromov's programme on CAT(k) spaces under the name orbihedron.
Historically, orbifolds arose first as surfaces with singular points long before they were formally defined. One of the first classical examples arose in the theory of modular forms with the action of the modular group on the upper half-plane: a version of the Riemann–Roch theorem holds after the quotient is compactified by the addition of two orbifold cusp points. In 3-manifold theory, the theory of Seifert fiber spaces, initiated by Herbert Seifert, can be phrased in terms of 2-dimensional orbifolds. In geometric group theory, post-Gromov, discrete groups have been studied in terms of the local curvature properties of orbihedra and their covering spaces.
In string theory, the word "orbifold" has a slightly different meaning, discussed in detail below. In two-dimensional conformal field theory, it refers to the theory attached to the fixed point subalgebra of a vertex algebra under the action of a finite group of automorphisms.
The main example of underlying space is a quotient space of a manifold under the properly discontinuous action of a possibly infinite group of diffeomorphisms with finite isotropy subgroups. In particular this applies to any action of a finite group; thus a manifold with boundary carries a natural orbifold structure, since it is the quotient of its double by an action of .
One topological space can carry different orbifold structures. For example, consider the orbifold O associated with a quotient space of the 2-sphere along a rotation by ; it is homeomorphic to the 2-sphere, but the natural orbifold structure is different. It is possible to adopt most of the characteristics of manifolds to orbifolds and these characteristics are usually different from correspondent characteristics of underlying space. In the above example, the orbifold fundamental group of O is and its orbifold Euler characteristic is 1.
Formal definitions
Definition using orbifold atlas
Like a manifold, an orbifold is specified by local conditions; however, instead of being locally modelled on open subsets of , an orbifold is locally modelled on quotients of open subsets of by finite group actions. The structure of an orbifold encodes not only that of the underlying quotient space, which need not be a manifold, but also that of the isotropy subgroups.
An n-dimensional orbifold is a Hausdorff to
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https://en.wikipedia.org/wiki/Indefinite%20orthogonal%20group
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In mathematics, the indefinite orthogonal group, is the Lie group of all linear transformations of an n-dimensional real vector space that leave invariant a nondegenerate, symmetric bilinear form of signature , where . It is also called the pseudo-orthogonal group or generalized orthogonal group. The dimension of the group is .
The indefinite special orthogonal group, is the subgroup of consisting of all elements with determinant 1. Unlike in the definite case, is not connected – it has 2 components – and there are two additional finite index subgroups, namely the connected and , which has 2 components – see for definition and discussion.
The signature of the form determines the group up to isomorphism; interchanging p with q amounts to replacing the metric by its negative, and so gives the same group. If either p or q equals zero, then the group is isomorphic to the ordinary orthogonal group O(n). We assume in what follows that both p and q are positive.
The group is defined for vector spaces over the reals. For complex spaces, all groups are isomorphic to the usual orthogonal group , since the transform changes the signature of a form. This should not be confused with the indefinite unitary group which preserves a sesquilinear form of signature .
In even dimension , is known as the split orthogonal group.
Examples
The basic example is the squeeze mappings, which is the group of (the identity component of) linear transforms preserving the unit hyperbola. Concretely, these are the matrices and can be interpreted as hyperbolic rotations, just as the group SO(2) can be interpreted as circular rotations.
In physics, the Lorentz group is of central importance, being the setting for electromagnetism and special relativity. (Some texts use for the Lorentz group; however, is prevalent in quantum field theory because the geometric properties of the Dirac equation are more natural in .)
Matrix definition
One can define as a group of matrices, just as for the classical orthogonal group O(n). Consider the diagonal matrix given by
Then we may define a symmetric bilinear form on by the formula
,
where is the standard inner product on .
We then define to be the group of matrices that preserve this bilinear form:
.
More explicitly, consists of matrices such that
,
where is the transpose of .
One obtains an isomorphic group (indeed, a conjugate subgroup of ) by replacing g with any symmetric matrix with p positive eigenvalues and q negative ones. Diagonalizing this matrix gives a conjugation of this group with the standard group .
Subgroups
The group and related subgroups of can be described algebraically. Partition a matrix L in as a block matrix:
where A, B, C, and D are p×p, p×q, q×p, and q×q blocks, respectively. It can be shown that the set of matrices in whose upper-left p×p block A has positive determinant is a subgroup. Or, to put it another way, if
are in , then
The analogous result for the bottom-right q
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https://en.wikipedia.org/wiki/Eddington%E2%80%93Finkelstein%20coordinates
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In general relativity, Eddington–Finkelstein coordinates are a pair of coordinate systems for a Schwarzschild geometry (e.g. a spherically symmetric black hole) which are adapted to radial null geodesics. Null geodesics are the worldlines of photons; radial ones are those that are moving directly towards or away from the central mass. They are named for Arthur Stanley Eddington and David Finkelstein. Although they appear to have inspired the idea, neither ever wrote down these coordinates or the metric in these coordinates. Roger Penrose seems to have been the first to write down the null form but credits it to the above paper by Finkelstein, and, in his Adams Prize essay later that year, to Eddington and Finkelstein. Most influentially, Misner, Thorne and Wheeler, in their book Gravitation, refer to the null coordinates by that name.
In these coordinate systems, outward (inward) traveling radial light rays (which each follow a null geodesic) define the surfaces of constant "time", while the radial coordinate is the usual area coordinate so that the surfaces of rotation symmetry have an area of . One advantage of this coordinate system is that it shows that the apparent singularity at the Schwarzschild radius is only a coordinate singularity and is not a true physical singularity. While this fact was recognized by Finkelstein, it was not recognized (or at least not commented on) by Eddington, whose primary purpose was to compare and contrast the spherically symmetric solutions in Whitehead's theory of gravitation and Einstein's version of the theory of relativity.
Schwarzschild metric
The Schwarzschild coordinates are , and in these coordinates the Schwarzschild metric is well known:
where
is the standard Riemannian metric of the unit 2-sphere.
Note the conventions being used here are the metric signature of (− + + +) and the natural units where c = 1 is the dimensionless speed of light, G the gravitational constant, and M is the characteristic mass of the Schwarzschild geometry.
Tortoise coordinate
Eddington–Finkelstein coordinates are founded upon the tortoise coordinate – a name that comes from one of Zeno of Elea's paradoxes on an imaginary footrace between "swift-footed" Achilles and a tortoise.
The tortoise coordinate is defined:
so as to satisfy:
The tortoise coordinate approaches as approaches the Schwarzschild radius .
When some probe (such as a light ray or an observer) approaches a black hole event horizon, its Schwarzschild time coordinate grows infinite. The outgoing null rays in this coordinate system have an infinite change in t on travelling out from the horizon. The tortoise coordinate is intended to grow infinite at the appropriate rate such as to cancel out this singular behaviour in coordinate systems constructed from it.
The increase in the time coordinate to infinity as one approaches the event horizon is why information could never be received back from any probe that is sent through such an event horizon.
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https://en.wikipedia.org/wiki/E6%20%28mathematics%29
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{{DISPLAYTITLE:E6 (mathematics)}}
In mathematics, E6 is the name of some closely related Lie groups, linear algebraic groups or their Lie algebras , all of which have dimension 78; the same notation E6 is used for the corresponding root lattice, which has rank 6. The designation E6 comes from the Cartan–Killing classification of the complex simple Lie algebras (see ). This classifies Lie algebras into four infinite series labeled An, Bn, Cn, Dn, and five exceptional cases labeled E6, E7, E8, F4, and G2. The E6 algebra is thus one of the five exceptional cases.
The fundamental group of the complex form, compact real form, or any algebraic version of E6 is the cyclic group Z/3Z, and its outer automorphism group is the cyclic group Z/2Z. Its fundamental representation is 27-dimensional (complex), and a basis is given by the 27 lines on a cubic surface. The dual representation, which is inequivalent, is also 27-dimensional.
In particle physics, E6 plays a role in some grand unified theories.
Real and complex forms
There is a unique complex Lie algebra of type E6, corresponding to a complex group of complex dimension 78. The complex adjoint Lie group E6 of complex dimension 78 can be considered as a simple real Lie group of real dimension 156. This has fundamental group Z/3Z, has maximal compact subgroup the compact form (see below) of E6, and has an outer automorphism group non-cyclic of order 4 generated by complex conjugation and by the outer automorphism which already exists as a complex automorphism.
As well as the complex Lie group of type E6, there are five real forms of the Lie algebra, and correspondingly five real forms of the group with trivial center (all of which have an algebraic double cover, and three of which have further non-algebraic covers, giving further real forms), all of real dimension 78, as follows:
The compact form (which is usually the one meant if no other information is given), which has fundamental group Z/3Z and outer automorphism group Z/2Z.
The split form, EI (or E6(6)), which has maximal compact subgroup Sp(4)/(±1), fundamental group of order 2 and outer automorphism group of order 2.
The quasi-split form EII (or E6(2)), which has maximal compact subgroup SU(2) × SU(6)/(center), fundamental group cyclic of order 6 and outer automorphism group of order 2.
EIII (or E6(-14)), which has maximal compact subgroup SO(2) × Spin(10)/(center), fundamental group Z and trivial outer automorphism group.
EIV (or E6(-26)), which has maximal compact subgroup F4, trivial fundamental group cyclic and outer automorphism group of order 2.
The EIV form of E6 is the group of collineations (line-preserving transformations) of the octonionic projective plane OP2. It is also the group of determinant-preserving linear transformations of the exceptional Jordan algebra. The exceptional Jordan algebra is 27-dimensional, which explains why the compact real form of E6 has a 27-dimensional complex representation. The compact real for
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https://en.wikipedia.org/wiki/E6
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E6, E06, E.VI or E-6 can mean:
Science, mathematics and engineering
The E6 series (number series) of preferred numbers for electronic components
E6 (mathematics), a Lie group in mathematics
E6 polytope in geometry
E06, Thyroiditis ICD-10 code
E-6 process, a common photographic process for developing transparency film
E6 protein, a protein encoded by Human papillomavirus
Honda E6, one of the predecessors of Honda's ASIMO robot
Transport
E-6 Mercury, a US Navy derivative of the Boeing 707
E6 Series Shinkansen, a Japanese high-speed train
BYD e6, an electric car by BYD Auto
EMD E6, a diesel locomotive
E6 European long distance path, a long-distance hiking trail
Eggenfellner E6, an American aircraft engine design
European route E6, a European highway route
LB&SCR E6 class, a British steam locomotive
LNER Class E6, a class of British steam locomotives
London Buses route E6, a Transport for London contracted bus route
Pfalz E.VI, a World War I German aircraft
PRR E6, an American steam locomotive
Jōban Expressway, Sendai-Tōbu Road, Sanriku Expressway (between Sendaiko-kita IC and Rifu JCT) and Sendai-Hokubu Road, the E6 expressway in Japan
North–South Expressway Central Link, route E6 in Malaysia
NAIA Expressway, route E6 in the Philippines
Other uses
Motorola ROKR E6, a 2006 multimedia phone model
Nokia E6, a smartphone
E6, a London postcode district in the E postcode area
An error by the Shortstop in baseball
E-6 (rank), the sixth rank of enlisted soldier in the US armed services
The Elephant 6 Recording Company, a collective of independent American musicians
Electric Six, a Detroit rock band
E6, the first note of the whistle register
The E6 grade of difficulty in rock climbing
E6, a song by Norwegian band D.D.E. The song title is a reference to the European route E6.
E6 (short for Epic 6), a variant of Dungeons & Dragons v3.5
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https://en.wikipedia.org/wiki/Closure%20%28mathematics%29
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In mathematics, a subset of a given set is closed under an operation of the larger set if performing that operation on members of the subset always produces a member of that subset. For example, the natural numbers are closed under addition, but not under subtraction: is not a natural number, although both 1 and 2 are.
Similarly, a subset is said to be closed under a collection of operations if it is closed under each of the operations individually.
The closure of a subset is the result of a closure operator applied to the subset. The closure of a subset under some operations is the smallest superset that is closed under these operations. It is often called the span (for example linear span) or the generated set.
Definitions
Let be a set equipped with one or several methods for producing elements of from other elements of .
A subset of is said to be closed under these methods, if, when all input elements are in , then all possible results are also in . Sometimes, one may also say that has the .
The main property of closed sets, which results immediately from the definition, is that every intersection of closed sets is a closed set. It follows that for every subset of , there is a smallest closed subset of such that (it is the intersection of all closed subsets that contain ). Depending on the context, is called the closure of or the set generated or spanned by .
The concepts of closed sets and closure are often extended to any property of subsets that are stable under intersection; that is, every intersection of subsets that have the property has also the property. For example, in a Zariski-closed set, also known as an algebraic set, is the set of the common zeros of a family of polynomials, and the Zariski closure of a set of points is the smallest algebraic set that contains .
In algebraic structures
An algebraic structure is a set equipped with operations that satisfy some axioms. These axioms may be identities. Some axioms may contain existential quantifiers in this case it is worth to add some auxiliary operations in order that all axioms become identities or purely universally quantified formulas. See Algebraic structure for details.
In this context, given an algebraic structure , a substructure of is a subset that is closed under all operations of , including the auxiliary operations that are needed for avoiding existential quantifiers. A substructure is an algebraic structure of the same type as . It follows that, in a specific example, when closeness is proved, there is no need to check the axioms for proving that a substructure is a structure of the same type.
Given a subset of an algebraic structure , the closure of is the smallest substructure of that is closed under all operations of . In the context of algebraic structures, this closure is generally called the substructure generated or spanned by , and one says that is a generating set of the substructure.
For example, a group is a set with an associat
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https://en.wikipedia.org/wiki/Aleph%20number
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In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets that can be well-ordered. They were introduced by the mathematician Georg Cantor and are named after the symbol he used to denote them, the Hebrew letter aleph ().
The cardinality of the natural numbers is (read aleph-nought or aleph-zero; the term aleph-null is also sometimes used), the next larger cardinality of a well-ordered set is aleph-one then and so on. Continuing in this manner, it is possible to define a cardinal number for every ordinal number as described below.
The concept and notation are due to Georg Cantor,
who defined the notion of cardinality and realized that infinite sets can have different cardinalities.
The aleph numbers differ from the infinity () commonly found in algebra and calculus, in that the alephs measure the sizes of sets, while infinity is commonly defined either as an extreme limit of the real number line (applied to a function or sequence that "diverges to infinity" or "increases without bound"), or as an extreme point of the extended real number line.
Aleph-zero
(aleph-zero, also aleph-nought or aleph-null) is the cardinality of the set of all natural numbers, and is an infinite cardinal. The set of all finite ordinals, called or (where is the lowercase Greek letter omega), has cardinality . A set has cardinality if and only if it is countably infinite, that is, there is a bijection (one-to-one correspondence) between it and the natural numbers. Examples of such sets are
the set of all integers,
any infinite subset of the integers, such as the set of all square numbers or the set of all prime numbers,
the set of all rational numbers,
the set of all constructible numbers (in the geometric sense),
the set of all algebraic numbers,
the set of all computable numbers,
the set of all computable functions,
the set of all binary strings of finite length, and
the set of all finite subsets of any given countably infinite set.
These infinite ordinals: and are among the countably infinite sets. For example, the sequence (with ordinality ) of all positive odd integers followed by all positive even integers
is an ordering of the set (with cardinality ) of positive integers.
If the axiom of countable choice (a weaker version of the axiom of choice) holds, then is smaller than any other infinite cardinal.
Aleph-one
is the cardinality of the set of all countable ordinal numbers, called or sometimes . This is itself an ordinal number larger than all countable ones, so it is an uncountable set. Therefore, is distinct from . The definition of implies (in ZF, Zermelo–Fraenkel set theory without the axiom of choice) that no cardinal number is between and . If the axiom of choice is used, it can be further proved that the class of cardinal numbers is totally ordered, and thus is the second-smallest infinite cardinal number. One can show o
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https://en.wikipedia.org/wiki/Arithmetization%20of%20analysis
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The arithmetization of analysis was a research program in the foundations of mathematics carried out in the second half of the 19th century.
History
Kronecker originally introduced the term arithmetization of analysis, by which he meant its constructivization in the context of the natural numbers (see quotation at bottom of page). The meaning of the term later shifted to signify the set-theoretic construction of the real line. Its main proponent was Weierstrass, who argued the geometric foundations of calculus were not solid enough for rigorous work.
Research program
The highlights of this research program are:
the various (but equivalent) constructions of the real numbers by Dedekind and Cantor resulting in the modern axiomatic definition of the real number field;
the epsilon-delta definition of limit; and
the naïve set-theoretic definition of function.
Legacy
An important spinoff of the arithmetization of analysis is set theory. Naive set theory was created by Cantor and others after arithmetization was completed as a way to study the singularities of functions appearing in calculus.
The arithmetization of analysis had several important consequences:
the widely held belief in the banishment of infinitesimals from mathematics until the creation of non-standard analysis by Abraham Robinson in the 1960s, whereas in reality the work on non-Archimedean systems continued unabated, as documented by P. Ehrlich;
the shift of the emphasis from geometric to algebraic reasoning: this has had important consequences in the way mathematics is taught today;
it made possible the development of modern measure theory by Lebesgue and the rudiments of functional analysis by Hilbert;
it motivated the currently prevalent philosophical position that all of mathematics should be derivable from logic and set theory, ultimately leading to Hilbert's program, Gödel's theorems and non-standard analysis.
Quotations
"God created the natural numbers, all else is the work of man." — Kronecker
References
Torina Dechaune Lewis (2006) The Arithmetization of Analysis: From Eudoxus to Dedekind, Southern University.
Carl B. Boyer, Uta C. Merzbach (2011) A History of Mathematics John Wiley & Sons.
Arithmetization of analysis at Encyclopedia of Mathematics.
History of mathematics
Philosophy of mathematics
Mathematical analysis
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https://en.wikipedia.org/wiki/Von%20Neumann%20algebra
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In mathematics, a von Neumann algebra or W*-algebra is a *-algebra of bounded operators on a Hilbert space that is closed in the weak operator topology and contains the identity operator. It is a special type of C*-algebra.
Von Neumann algebras were originally introduced by John von Neumann, motivated by his study of single operators, group representations, ergodic theory and quantum mechanics. His double commutant theorem shows that the analytic definition is equivalent to a purely algebraic definition as an algebra of symmetries.
Two basic examples of von Neumann algebras are as follows:
The ring of essentially bounded measurable functions on the real line is a commutative von Neumann algebra, whose elements act as multiplication operators by pointwise multiplication on the Hilbert space of square-integrable functions.
The algebra of all bounded operators on a Hilbert space is a von Neumann algebra, non-commutative if the Hilbert space has dimension at least .
Von Neumann algebras were first studied by in 1929; he and Francis Murray developed the basic theory, under the original name of rings of operators, in a series of papers written in the 1930s and 1940s (; ), reprinted in the collected works of .
Introductory accounts of von Neumann algebras are given in the online notes of and and the books by , , and . The three volume work by gives an encyclopedic account of the theory. The book by discusses more advanced topics.
Definitions
There are three common ways to define von Neumann algebras.
The first and most common way is to define them as weakly closed *-algebras of bounded operators (on a Hilbert space) containing the identity. In this definition the weak (operator) topology can be replaced by many other common topologies including the strong, ultrastrong or ultraweak operator topologies. The *-algebras of bounded operators that are closed in the norm topology are C*-algebras, so in particular any von Neumann algebra is a C*-algebra.
The second definition is that a von Neumann algebra is a subalgebra of the bounded operators closed under involution (the *-operation) and equal to its double commutant, or equivalently the commutant of some subalgebra closed under *. The von Neumann double commutant theorem says that the first two definitions are equivalent.
The first two definitions describe a von Neumann algebra concretely as a set of operators acting on some given Hilbert space. showed that von Neumann algebras can also be defined abstractly as C*-algebras that have a predual; in other words the von Neumann algebra, considered as a Banach space, is the dual of some other Banach space called the predual. The predual of a von Neumann algebra is in fact unique up to isomorphism. Some authors use "von Neumann algebra" for the algebras together with a Hilbert space action, and "W*-algebra" for the abstract concept, so a von Neumann algebra is a W*-algebra together with a Hilbert space and a suitable faithful unital action o
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https://en.wikipedia.org/wiki/Commute
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Commute, commutation or commutative may refer to:
Commuting, the process of travelling between a place of residence and a place of work
Mathematics
Commutative property, a property of a mathematical operation whose result is insensitive to the order of its arguments
Equivariant map, a function whose composition with another function has the commutative property
Commutative diagram, a graphical description of commuting compositions of arrows in a mathematical category
Commutative semigroup, commutative monoid, abelian group, and commutative ring, algebraic structures with the commutative property
Commuting matrices, sets of matrices whose products do not depend on the order of multiplication
Commutator, a measure of the failure of two elements to be commutative in a group or ring
Science and technology
Commutator (electric), a rotary switch on the shaft of an electric motor or generator
Commutation (neurophysiology), how certain neural circuits in the brain exhibit noncommutativity
Commutation (telemetry), a form of time-division multiplexing
Commutation, a synonym for packet switching in computer networking and telecommunications
Other uses
Commutation (law) (of sentence), a reduction in severity of punishment
Commutation (finance) (law) to lessen periodic dues (usually rents, fares or tithes) by paying a lump sum
Commutation (finance) under a cash option, encashment. The act of a pension member/annuitant who gives up part or all in exchange for a lump sum payment.
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https://en.wikipedia.org/wiki/Symplectic%20geometry
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Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry has its origins in the Hamiltonian formulation of classical mechanics where the phase space of certain classical systems takes on the structure of a symplectic manifold.
The term "symplectic", introduced by Weyl, is a calque of "complex"; previously, the "symplectic group" had been called the "line complex group". "Complex" comes from the Latin com-plexus, meaning "braided together" (co- + plexus), while symplectic comes from the corresponding Greek sym-plektikos (συμπλεκτικός); in both cases the stem comes from the Indo-European root *pleḱ- The name reflects the deep connections between complex and symplectic structures.
By Darboux's Theorem, symplectic manifolds are isomorphic to the standard symplectic vector space locally, hence only have global (topological) invariants. "Symplectic topology," which studies global properties of symplectic manifolds, is often used interchangeably with "symplectic geometry."
Introduction
A symplectic geometry is defined on a smooth even-dimensional space that is a differentiable manifold. On this space is defined a geometric object, the symplectic 2-form, that allows for the measurement of sizes of two-dimensional objects in the space. The symplectic form in symplectic geometry plays a role analogous to that of the metric tensor in Riemannian geometry. Where the metric tensor measures lengths and angles, the symplectic form measures oriented areas.
Symplectic geometry arose from the study of classical mechanics and an example of a symplectic structure is the motion of an object in one dimension. To specify the trajectory of the object, one requires both the position q and the momentum p, which form a point (p,q) in the Euclidean plane ℝ2. In this case, the symplectic form is
and is an area form that measures the area A of a region S in the plane through integration:
The area is important because as conservative dynamical systems evolve in time, this area is invariant.
Higher dimensional symplectic geometries are defined analogously. A 2n-dimensional symplectic geometry is formed of pairs of directions
in a 2n-dimensional manifold along with a symplectic form
This symplectic form yields the size of a 2n-dimensional region V in the space as the sum of the areas of the projections of V onto each of the planes formed by the pairs of directions
Comparison with Riemannian geometry
Symplectic geometry has a number of similarities with and differences from Riemannian geometry, which is the study of differentiable manifolds equipped with nondegenerate, symmetric 2-tensors (called metric tensors). Unlike in the Riemannian case, symplectic manifolds have no local invariants such as curvature. This is a consequence of Darboux's theorem which states that a neighborhood of any point of a 2n-dimensio
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https://en.wikipedia.org/wiki/Poisson%20bracket
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In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. The Poisson bracket also distinguishes a certain class of coordinate transformations, called canonical transformations, which map canonical coordinate systems into canonical coordinate systems. A "canonical coordinate system" consists of canonical position and momentum variables (below symbolized by and , respectively) that satisfy canonical Poisson bracket relations. The set of possible canonical transformations is always very rich. For instance, it is often possible to choose the Hamiltonian itself as one of the new canonical momentum coordinates.
In a more general sense, the Poisson bracket is used to define a Poisson algebra, of which the algebra of functions on a Poisson manifold is a special case. There are other general examples, as well: it occurs in the theory of Lie algebras, where the tensor algebra of a Lie algebra forms a Poisson algebra; a detailed construction of how this comes about is given in the universal enveloping algebra article. Quantum deformations of the universal enveloping algebra lead to the notion of quantum groups.
All of these objects are named in honor of Siméon Denis Poisson.
Properties
Given two functions and that depend on phase space and time, their Poisson bracket is another function that depends on phase space and time. The following rules hold for any three functions of phase space and time:
Anticommutativity
Bilinearity
Leibniz's rule
Jacobi identity
Also, if a function is constant over phase space (but may depend on time), then for any .
Definition in canonical coordinates
In canonical coordinates (also known as Darboux coordinates) on the phase space, given two functions and , the Poisson bracket takes the form
The Poisson brackets of the canonical coordinates are
where is the Kronecker delta.
Hamilton's equations of motion
Hamilton's equations of motion have an equivalent expression in terms of the Poisson bracket. This may be most directly demonstrated in an explicit coordinate frame. Suppose that is a function on the solution's trajectory-manifold. Then from the multivariable chain rule,
Further, one may take and to be solutions to Hamilton's equations; that is,
Then
Thus, the time evolution of a function on a symplectic manifold can be given as a one-parameter family of symplectomorphisms (i.e., canonical transformations, area-preserving diffeomorphisms), with the time being the parameter: Hamiltonian motion is a canonical transformation generated by the Hamiltonian. That is, Poisson brackets are preserved in it, so that any time in the solution to Hamilton's equations,
can serve as the bracket coordinates. Poisson brackets are canonical invariants.
Dropping the coordinates,
The operator in the convective part of the derivat
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https://en.wikipedia.org/wiki/Chern%20class
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In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since become fundamental concepts in many branches of mathematics and physics, such as string theory, Chern–Simons theory, knot theory, Gromov–Witten invariants.
Chern classes were introduced by .
Geometric approach
Basic idea and motivation
Chern classes are characteristic classes. They are topological invariants associated with vector bundles on a smooth manifold. The question of whether two ostensibly different vector bundles are the same can be quite hard to answer. The Chern classes provide a simple test: if the Chern classes of a pair of vector bundles do not agree, then the vector bundles are different. The converse, however, is not true.
In topology, differential geometry, and algebraic geometry, it is often important to count how many linearly independent sections a vector bundle has. The Chern classes offer some information about this through, for instance, the Riemann–Roch theorem and the Atiyah–Singer index theorem.
Chern classes are also feasible to calculate in practice. In differential geometry (and some types of algebraic geometry), the Chern classes can be expressed as polynomials in the coefficients of the curvature form.
Construction
There are various ways of approaching the subject, each of which focuses on a slightly different flavor of Chern class.
The original approach to Chern classes was via algebraic topology: the Chern classes arise via homotopy theory which provides a mapping associated with a vector bundle to a classifying space (an infinite Grassmannian in this case). For any complex vector bundle V over a manifold M, there exists a map f from M to the classifying space such that the bundle V is equal to the pullback, by f, of a universal bundle over the classifying space, and the Chern classes of V can therefore be defined as the pullback of the Chern classes of the universal bundle. In turn, these universal Chern classes can be explicitly written down in terms of Schubert cycles.
It can be shown that for any two maps f, g from M to the classifying space whose pullbacks are the same bundle V, the maps must be homotopic. Therefore, the pullback by either f or g of any universal Chern class to a cohomology class of M must be the same class. This shows that the Chern classes of V are well-defined.
Chern's approach used differential geometry, via the curvature approach described predominantly in this article. He showed that the earlier definition was in fact equivalent to his. The resulting theory is known as the Chern–Weil theory.
There is also an approach of Alexander Grothendieck showing that axiomatically one need only define the line bundle case.
Chern classes arise naturally in algebraic geometry. The generalized Chern classes in algebraic geometry can be defined for vector bundles (or more precisely, local
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https://en.wikipedia.org/wiki/Anticommutative%20property
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In mathematics, anticommutativity is a specific property of some non-commutative mathematical operations. Swapping the position of two arguments of an antisymmetric operation yields a result which is the inverse of the result with unswapped arguments. The notion inverse refers to a group structure on the operation's codomain, possibly with another operation. Subtraction is an anticommutative operation because commuting the operands of gives for example, Another prominent example of an anticommutative operation is the Lie bracket.
In mathematical physics, where symmetry is of central importance, these operations are mostly called antisymmetric operations, and are extended in an associative setting to cover more than two arguments.
Definition
If are two abelian groups, a bilinear map is anticommutative if for all we have
More generally, a multilinear map is anticommutative if for all we have
where is the sign of the permutation .
Properties
If the abelian group has no 2-torsion, implying that if then , then any anticommutative bilinear map satisfies
More generally, by transposing two elements, any anticommutative multilinear map satisfies
if any of the are equal; such a map is said to be alternating. Conversely, using multilinearity, any alternating map is anticommutative. In the binary case this works as follows: if is alternating then by bilinearity we have
and the proof in the multilinear case is the same but in only two of the inputs.
Examples
Examples of anticommutative binary operations include:
Cross product
Lie bracket of a Lie algebra
Lie bracket of a Lie ring
Subtraction
See also
Commutativity
Commutator
Exterior algebra
Graded-commutative ring
Operation (mathematics)
Symmetry in mathematics
Particle statistics (for anticommutativity in physics).
References
.
External links
. Which references the Original Russian work
Properties of binary operations
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https://en.wikipedia.org/wiki/Jacobi%20identity
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In mathematics, the Jacobi identity is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operation. By contrast, for operations with the associative property, any order of evaluation gives the same result (parentheses in a multiple product are not needed). The identity is named after the German mathematician Carl Gustav Jacob Jacobi.
The cross product and the Lie bracket operation both satisfy the Jacobi identity. In analytical mechanics, the Jacobi identity is satisfied by the Poisson brackets. In quantum mechanics, it is satisfied by operator commutators on a Hilbert space and equivalently in the phase space formulation of quantum mechanics by the Moyal bracket.
Definition
Let and be two binary operations, and let be the neutral element for . The is
Notice the pattern in the variables on the left side of this identity. In each subsequent expression of the form , the variables , and are permuted according to the cycle . Alternatively, we may observe that the ordered triples , and , are the even permutations of the ordered triple .
Commutator bracket form
The simplest informative example of a Lie algebra is constructed from the (associative) ring of matrices, which may be thought of as infinitesimal motions of an n-dimensional vector space. The × operation is the commutator, which measures the failure of commutativity in matrix multiplication. Instead of , the Lie bracket notation is used:
In that notation, the Jacobi identity is:
That is easily checked by computation.
More generally, if is an associative algebra and is a subspace of that is closed under the bracket operation: belongs to for all , the Jacobi identity continues to hold on . Thus, if a binary operation satisfies the Jacobi identity, it may be said that it behaves as if it were given by in some associative algebra even if it is not actually defined that way.
Using the antisymmetry property , the Jacobi identity may be rewritten as a modification of the associative property:
If is the action of the infinitesimal motion on , that can be stated as:
There is also a plethora of graded Jacobi identities involving anticommutators , such as:
Adjoint form
Most common examples of the Jacobi identity come from the bracket multiplication on Lie algebras and Lie rings. The Jacobi identity is written as:
Because the bracket multiplication is antisymmetric, the Jacobi identity admits two equivalent reformulations. Defining the adjoint operator , the identity becomes:
Thus, the Jacobi identity for Lie algebras states that the action of any element on the algebra is a derivation. That form of the Jacobi identity is also used to define the notion of Leibniz algebra.
Another rearrangement shows that the Jacobi identity is equivalent to the following identity between the operators of the adjoint representation:
There, the bracket on the left side is the operat
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https://en.wikipedia.org/wiki/Chern%E2%80%93Simons%20form
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In mathematics, the Chern–Simons forms are certain secondary characteristic classes. The theory is named for Shiing-Shen Chern and James Harris Simons, co-authors of a 1974 paper entitled "Characteristic Forms and Geometric Invariants," from which the theory arose.
Definition
Given a manifold and a Lie algebra valued 1-form over it, we can define a family of p-forms:
In one dimension, the Chern–Simons 1-form is given by
In three dimensions, the Chern–Simons 3-form is given by
In five dimensions, the Chern–Simons 5-form is given by
where the curvature F is defined as
The general Chern–Simons form is defined in such a way that
where the wedge product is used to define Fk. The right-hand side of this equation is proportional to the k-th Chern character of the connection .
In general, the Chern–Simons p-form is defined for any odd p.
Application to physics
In 1978, Albert Schwarz formulated Chern–Simons theory, early topological quantum field theory, using Chern-Simons forms.
In the gauge theory, the integral of Chern-Simons form is a global geometric invariant, and is typically gauge invariant modulo addition of an integer.
See also
Chern–Weil homomorphism
Chiral anomaly
Topological quantum field theory
Jones polynomial
References
Further reading
Homology theory
Algebraic topology
Differential geometry
String theory
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https://en.wikipedia.org/wiki/Commutative%20property
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In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of the property that says something like or , the property can also be used in more advanced settings. The name is needed because there are operations, such as division and subtraction, that do not have it (for example, ); such operations are not commutative, and so are referred to as noncommutative operations. The idea that simple operations, such as the multiplication and addition of numbers, are commutative was for many years implicitly assumed. Thus, this property was not named until the 19th century, when mathematics started to become formalized. A similar property exists for binary relations; a binary relation is said to be symmetric if the relation applies regardless of the order of its operands; for example, equality is symmetric as two equal mathematical objects are equal regardless of their order.
Mathematical definitions
A binary operation on a set S is called commutative if
In other words, an operation is commutative if every two elements commute. An operation that does not satisfy the above property is called noncommutative.
One says that commutes with or that and commute under if
That is, a specific pair of elements may commute even if the operation is (strictly) noncommutative.
Examples
Commutative operations
Addition and multiplication are commutative in most number systems, and, in particular, between natural numbers, integers, rational numbers, real numbers and complex numbers. This is also true in every field.
Addition is commutative in every vector space and in every algebra.
Union and intersection are commutative operations on sets.
"And" and "or" are commutative logical operations.
Noncommutative operations
Some noncommutative binary operations:
Division, subtraction, and exponentiation
Division is noncommutative, since .
Subtraction is noncommutative, since . However it is classified more precisely as anti-commutative, since .
Exponentiation is noncommutative, since . This property leads to two different "inverse" operations of exponentiation (namely, the nth-root operation and the logarithm operation), which is unlike the multiplication.
Truth functions
Some truth functions are noncommutative, since the truth tables for the functions are different when one changes the order of the operands. For example, the truth tables for and are
Function composition of linear functions
Function composition of linear functions from the real numbers to the real numbers is almost always noncommutative. For example, let and . Then
and
This also applies more generally for linear and affine transformations from a vector space to itself (see below for the Matrix representation).
Matrix multiplication
Matrix multiplication of square matrices is almost always noncom
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https://en.wikipedia.org/wiki/Lie%20derivative
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In differential geometry, the Lie derivative ( ), named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector field. This change is coordinate invariant and therefore the Lie derivative is defined on any differentiable manifold.
Functions, tensor fields and forms can be differentiated with respect to a vector field. If T is a tensor field and X is a vector field, then the Lie derivative of T with respect to X is denoted . The differential operator is a derivation of the algebra of tensor fields of the underlying manifold.
The Lie derivative commutes with contraction and the exterior derivative on differential forms.
Although there are many concepts of taking a derivative in differential geometry, they all agree when the expression being differentiated is a function or scalar field. Thus in this case the word "Lie" is dropped, and one simply speaks of the derivative of a function.
The Lie derivative of a vector field Y with respect to another vector field X is known as the "Lie bracket" of X and Y, and is often denoted [X,Y] instead of . The space of vector fields forms a Lie algebra with respect to this Lie bracket. The Lie derivative constitutes an infinite-dimensional Lie algebra representation of this Lie algebra, due to the identity
valid for any vector fields X and Y and any tensor field T.
Considering vector fields as infinitesimal generators of flows (i.e. one-dimensional groups of diffeomorphisms) on M, the Lie derivative is the differential of the representation of the diffeomorphism group on tensor fields, analogous to Lie algebra representations as infinitesimal representations associated to group representation in Lie group theory.
Generalisations exist for spinor fields, fibre bundles with a connection and vector-valued differential forms.
Motivation
A 'naïve' attempt to define the derivative of a tensor field with respect to a vector field would be to take the components of the tensor field and take the directional derivative of each component with respect to the vector field. However, this definition is undesirable because it is not invariant under changes of coordinate system, e.g. the naive derivative expressed in polar or spherical coordinates differs from the naive derivative of the components in Cartesian coordinates. On an abstract manifold such a definition is meaningless and ill defined. In differential geometry, there are three main coordinate independent notions of differentiation of tensor fields: Lie derivatives, derivatives with respect to connections, and the exterior derivative of totally antisymmetric covariant tensors, i.e. differential forms. The main difference between the Lie derivative and a derivative with respect to a connection is that the latter derivative of a tensor field with respect to a tangent vector is well-defined even if it is not specified how to extend that
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https://en.wikipedia.org/wiki/Peter%E2%80%93Weyl%20theorem
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In mathematics, the Peter–Weyl theorem is a basic result in the theory of harmonic analysis, applying to topological groups that are compact, but are not necessarily abelian. It was initially proved by Hermann Weyl, with his student Fritz Peter, in the setting of a compact topological group G . The theorem is a collection of results generalizing the significant facts about the decomposition of the regular representation of any finite group, as discovered by Ferdinand Georg Frobenius and Issai Schur.
Let G be a compact group. The theorem has three parts. The first part states that the matrix coefficients of irreducible representations of G are dense in the space C(G) of continuous complex-valued functions on G, and thus also in the space L2(G) of square-integrable functions. The second part asserts the complete reducibility of unitary representations of G. The third part then asserts that the regular representation of G on L2(G) decomposes as the direct sum of all irreducible unitary representations. Moreover, the matrix coefficients of the irreducible unitary representations form an orthonormal basis of L2(G). In the case that G is the group of unit complex numbers, this last result is simply a standard result from Fourier series.
Matrix coefficients
A matrix coefficient of the group G is a complex-valued function on G given as the composition
where π : G → GL(V) is a finite-dimensional (continuous) group representation of G, and L is a linear functional on the vector space of endomorphisms of V (e.g. trace), which contains GL(V) as an open subset. Matrix coefficients are continuous, since representations are by definition continuous, and linear functionals on finite-dimensional spaces are also continuous.
The first part of the Peter–Weyl theorem asserts (; ):
Peter–Weyl Theorem (Part I). The set of matrix coefficients of G is dense in the space of continuous complex functions C(G) on G, equipped with the uniform norm.
This first result resembles the Stone–Weierstrass theorem in that it indicates the density of a set of functions in the space of all continuous functions, subject only to an algebraic characterization. In fact, the matrix coefficients form a unital algebra invariant under complex conjugation because the product of two matrix coefficients is a matrix coefficient of the tensor product representation, and the complex conjugate is a matrix coefficient of the dual representation. Hence the theorem follows directly from the Stone–Weierstrass theorem if the matrix coefficients separate points, which is obvious if G is a matrix group . Conversely, it is a consequence of the theorem that any compact Lie group is isomorphic to a matrix group .
A corollary of this result is that the matrix coefficients of G are dense in L2(G).
Decomposition of a unitary representation
The second part of the theorem gives the existence of a decomposition of a unitary representation of G into finite-dimensional representations. Now, intuitively g
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https://en.wikipedia.org/wiki/Smooth
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Smooth may refer to:
Mathematics
Smooth function, a function that is infinitely differentiable; used in calculus and topology
Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions
Smooth algebraic variety, an algebraic variety with no singular points
Smooth number, a number whose prime factors are all less than a certain value; used in applications of number theory
Smoothsort, a sorting algorithm
"Analysis of the Jane Curve", an applied mathematics article by Norbert Schappacher referring to a surface that is infinitely smooth.
Arts and entertainment
Music
Smooth (singer), Juanita Stokes, American singer, rapper and actress
Smooth (Smooth album), 1995
Smooth (Gerald Albright album), 1994
"Smooth" (Florida Georgia Line song), 2017
"Smooth" (iiO song), 2004
"Smooth" (Santana song), featuring Rob Thomas, 1999
"Smooth", a mashup by Neil Cicierega from Mouth Moods, 2017
Other media
Smooth (magazine), an American publication for young black men
Smooth Radio (disambiguation), UK radio station networks
smoothfm, an Australian radio network
Foxtel Smooth, a defunct Australian pay-television music channel
See also
Smooth Island (disambiguation)
Smoother (disambiguation)
Smoothing (disambiguation)
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https://en.wikipedia.org/wiki/Euler%E2%80%93Lagrange%20equation
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In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered in the 1750s by Swiss mathematician Leonhard Euler and Italian mathematician Joseph-Louis Lagrange.
Because a differentiable functional is stationary at its local extrema, the Euler–Lagrange equation is useful for solving optimization problems in which, given some functional, one seeks the function minimizing or maximizing it. This is analogous to Fermat's theorem in calculus, stating that at any point where a differentiable function attains a local extremum its derivative is zero.
In Lagrangian mechanics, according to Hamilton's principle of stationary action, the evolution of a physical system is described by the solutions to the Euler equation for the action of the system. In this context Euler equations are usually called Lagrange equations. In classical mechanics, it is equivalent to Newton's laws of motion; indeed, the Euler-Lagrange equations will produce the same equations as Newton's Laws. This is particularly useful when analyzing systems whose force vectors are particularly complicated. It has the advantage that it takes the same form in any system of generalized coordinates, and it is better suited to generalizations. In classical field theory there is an analogous equation to calculate the dynamics of a field.
History
The Euler–Lagrange equation was developed in the 1750s by Euler and Lagrange in connection with their studies of the tautochrone problem. This is the problem of determining a curve on which a weighted particle will fall to a fixed point in a fixed amount of time, independent of the starting point.
Lagrange solved this problem in 1755 and sent the solution to Euler. Both further developed Lagrange's method and applied it to mechanics, which led to the formulation of Lagrangian mechanics. Their correspondence ultimately led to the calculus of variations, a term coined by Euler himself in 1766.
Statement
Let be a real dynamical system with degrees of freedom. Here is the configuration space and the Lagrangian, i.e. a smooth real-valued function such that and is an -dimensional "vector of speed". (For those familiar with differential geometry, is a smooth manifold, and where is the tangent bundle of
Let be the set of smooth paths for which and The action functional is defined via
A path is a stationary point of if and only if
Here, is the time derivative of When we say stationary point, we mean a stationary point of with respect to any small perturbation in . See proofs below for more rigorous detail.
Example
A standard example is finding the real-valued function y(x) on the interval [a, b], such that y(a) = c and y(b) = d, for which the path length along the curve traced by y is as short as possible.
the integrand function being .
The partial
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https://en.wikipedia.org/wiki/Locally%20constant%20function
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In mathematics, a locally constant function is a function from a topological space into a set with the property that around every point of its domain, there exists some neighborhood of that point on which it restricts to a constant function.
Definition
Let be a function from a topological space into a set
If then is said to locally constant at if there exists a neighborhood of such that is constant on which by definition means that for all
The function is called locally constant if it is locally constant at every point in its domain.
Examples
Every constant function is locally constant. The converse will hold if its domain is a connected space.
Every locally constant function from the real numbers to is constant, by the connectedness of But the function from the rationals to defined by and is locally constant (this uses the fact that is irrational and that therefore the two sets and are both open in ).
If is locally constant, then it is constant on any connected component of The converse is true for locally connected spaces, which are spaces whose connected components are open subsets.
Further examples include the following:
Given a covering map then to each point we can assign the cardinality of the fiber over ; this assignment is locally constant.
A map from a topological space to a discrete space is continuous if and only if it is locally constant.
Connection with sheaf theory
There are of locally constant functions on To be more definite, the locally constant integer-valued functions on form a sheaf in the sense that for each open set of we can form the functions of this kind; and then verify that the sheaf hold for this construction, giving us a sheaf of abelian groups (even commutative rings). This sheaf could be written ; described by means of we have stalk a copy of at for each This can be referred to a , meaning exactly taking their values in the (same) group. The typical sheaf of course is not constant in this way; but the construction is useful in linking up sheaf cohomology with homology theory, and in logical applications of sheaves. The idea of local coefficient system is that we can have a theory of sheaves that look like such 'harmless' sheaves (near any ), but from a global point of view exhibit some 'twisting'.
See also
Locally constant sheaf
Sheaf theory
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https://en.wikipedia.org/wiki/Reflection%20%28mathematics%29
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In mathematics, a reflection (also spelled reflexion) is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as a set of fixed points; this set is called the axis (in dimension 2) or plane (in dimension 3) of reflection. The image of a figure by a reflection is its mirror image in the axis or plane of reflection. For example the mirror image of the small Latin letter p for a reflection with respect to a vertical axis (a vertical reflection) would look like q. Its image by reflection in a horizontal axis (a horizontal reflection) would look like b. A reflection is an involution: when applied twice in succession, every point returns to its original location, and every geometrical object is restored to its original state.
The term reflection is sometimes used for a larger class of mappings from a Euclidean space to itself, namely the non-identity isometries that are involutions. Such isometries have a set of fixed points (the "mirror") that is an affine subspace, but is possibly smaller than a hyperplane. For instance a reflection through a point is an involutive isometry with just one fixed point; the image of the letter p under it
would look like a d. This operation is also known as a central inversion , and exhibits Euclidean space as a symmetric space. In a Euclidean vector space, the reflection in the point situated at the origin is the same as vector negation. Other examples include reflections in a line in three-dimensional space. Typically, however, unqualified use of the term "reflection" means reflection in a hyperplane.
Some mathematicians use "flip" as a synonym for "reflection".
Construction
In a plane (or, respectively, 3-dimensional) geometry, to find the reflection of a point drop a perpendicular from the point to the line (plane) used for reflection, and extend it the same distance on the other side. To find the reflection of a figure, reflect each point in the figure.
To reflect point through the line using compass and straightedge, proceed as follows (see figure):
Step 1 (red): construct a circle with center at and some fixed radius to create points and on the line , which will be equidistant from .
Step 2 (green): construct circles centered at and having radius . and will be the points of intersection of these two circles.
Point is then the reflection of point through line .
Properties
The matrix for a reflection is orthogonal with determinant −1 and eigenvalues −1, 1, 1, ..., 1. The product of two such matrices is a special orthogonal matrix that represents a rotation. Every rotation is the result of reflecting in an even number of reflections in hyperplanes through the origin, and every improper rotation is the result of reflecting in an odd number. Thus reflections generate the orthogonal group, and this result is known as the Cartan–Dieudonné theorem.
Similarly the Euclidean group, which consists of all isometries of Euclidean space, is generated by reflections in affine hy
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https://en.wikipedia.org/wiki/Inversive%20geometry
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In geometry, inversive geometry is the study of inversion, a transformation of the Euclidean plane that maps circles or lines to other circles or lines and that preserves the angles between crossing curves. Many difficult problems in geometry become much more tractable when an inversion is applied. Inversion seems to have been discovered by a number of people contemporaneously, including Steiner (1824), Quetelet (1825), Bellavitis (1836), Stubbs and Ingram (1842-3) and Kelvin (1845).
The concept of inversion can be generalized to higher-dimensional spaces.
Inversion in a circle
Inverse of a point
To invert a number in arithmetic usually means to take its reciprocal. A closely related idea in geometry is that of "inverting" a point. In the plane, the inverse of a point P with respect to a reference circle (Ø) with center O and radius r is a point P, lying on the ray from O through P such that
This is called circle inversion or plane inversion. The inversion taking any point P (other than O) to its image P also takes P back to P, so the result of applying the same inversion twice is the identity transformation on all the points of the plane other than O (self-inversion). To make inversion an involution it is necessary to introduce a point at infinity, a single point placed on all the lines, and extend the inversion, by definition, to interchange the center O and this point at infinity.
It follows from the definition that the inversion of any point inside the reference circle must lie outside it, and vice versa, with the center and the point at infinity changing positions, whilst any point on the circle is unaffected (is invariant under inversion). In summary, the nearer a point to the center, the further away its transformation, and vice versa.
Compass and straightedge construction
Point outside circle
To construct the inverse P of a point P outside a circle Ø:
Draw the segment from O (center of circle Ø) to P.
Let M be the midpoint of OP. (Not shown)
Draw the circle c with center M going through P. (Not labeled. It's the blue circle)
Let N and N be the points where Ø and c intersect.
Draw segment NN.
P is where OP and NN intersect.
Point inside circle
To construct the inverse P of a point P inside a circle Ø:
Draw ray r from O (center of circle Ø) through P. (Not labeled, it's the horizontal line)
Draw line s through P perpendicular to r. (Not labeled. It's the vertical line)
Let N be one of the points where Ø and s intersect.
Draw the segment ON.
Draw line t through N perpendicular to ON.
P is where ray r and line t intersect.
Dutta's construction
There is a construction of the inverse point to A with respect to a circle P that is independent of whether A is inside or outside P.
Consider a circle P with center O and a point A which may lie inside or outside the circle P.
Take the intersection point C of the ray OA with the circle P.
Connect the point C with an arbitrary point B on the circle P (different from C)
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https://en.wikipedia.org/wiki/Conformal%20geometry
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In mathematics, conformal geometry is the study of the set of angle-preserving (conformal) transformations on a space.
In a real two dimensional space, conformal geometry is precisely the geometry of Riemann surfaces. In space higher than two dimensions, conformal geometry may refer either to the study of conformal transformations of what are called "flat spaces" (such as Euclidean spaces or spheres), or to the study of conformal manifolds which are Riemannian or pseudo-Riemannian manifolds with a class of metrics that are defined up to scale. Study of the flat structures is sometimes termed Möbius geometry, and is a type of Klein geometry.
Conformal manifolds
A conformal manifold is a pseudo-Riemannian manifold equipped with an equivalence class of metric tensors, in which two metrics g and h are equivalent if and only if
where λ is a real-valued smooth function defined on the manifold and is called the conformal factor. An equivalence class of such metrics is known as a conformal metric or conformal class. Thus, a conformal metric may be regarded as a metric that is only defined "up to scale". Often conformal metrics are treated by selecting a metric in the conformal class, and applying only "conformally invariant" constructions to the chosen metric.
A conformal metric is conformally flat if there is a metric representing it that is flat, in the usual sense that the Riemann curvature tensor vanishes. It may only be possible to find a metric in the conformal class that is flat in an open neighborhood of each point. When it is necessary to distinguish these cases, the latter is called locally conformally flat, although often in the literature no distinction is maintained. The n-sphere is a locally conformally flat manifold that is not globally conformally flat in this sense, whereas a Euclidean space, a torus, or any conformal manifold that is covered by an open subset of Euclidean space is (globally) conformally flat in this sense. A locally conformally flat manifold is locally conformal to a Möbius geometry, meaning that there exists an angle preserving local diffeomorphism from the manifold into a Möbius geometry. In two dimensions, every conformal metric is locally conformally flat. In dimension a conformal metric is locally conformally flat if and only if its Weyl tensor vanishes; in dimension , if and only if the Cotton tensor vanishes.
Conformal geometry has a number of features which distinguish it from (pseudo-)Riemannian geometry. The first is that although in (pseudo-)Riemannian geometry one has a well-defined metric at each point, in conformal geometry one only has a class of metrics. Thus the length of a tangent vector cannot be defined, but the angle between two vectors still can. Another feature is that there is no Levi-Civita connection because if g and λ2g are two representatives of the conformal structure, then the Christoffel symbols of g and λ2g would not agree. Those associated with λ2g would involve derivativ
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https://en.wikipedia.org/wiki/Noncommutative%20geometry
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Noncommutative geometry (NCG) is a branch of mathematics concerned with a geometric approach to noncommutative algebras, and with the construction of spaces that are locally presented by noncommutative algebras of functions (possibly in some generalized sense). A noncommutative algebra is an associative algebra in which the multiplication is not commutative, that is, for which does not always equal ; or more generally an algebraic structure in which one of the principal binary operations is not commutative; one also allows additional structures, e.g. topology or norm, to be possibly carried by the noncommutative algebra of functions.
An approach giving deep insight about noncommutative spaces is through operator algebras (i.e. algebras of bounded linear operators on a Hilbert space). Perhaps one of the typical examples of a noncommutative space is the "noncommutative tori", which played a key role in the early development of this field in 1980s and lead to noncommutative versions of vector bundles, connections, curvature, etc.
Motivation
The main motivation is to extend the commutative duality between spaces and functions to the noncommutative setting. In mathematics, spaces, which are geometric in nature, can be related to numerical functions on them. In general, such functions will form a commutative ring. For instance, one may take the ring C(X) of continuous complex-valued functions on a topological space X. In many cases (e.g., if X is a compact Hausdorff space), we can recover X from C(X), and therefore it makes some sense to say that X has commutative topology.
More specifically, in topology, compact Hausdorff topological spaces can be reconstructed from the Banach algebra of functions on the space (Gelfand–Naimark). In commutative algebraic geometry, algebraic schemes are locally prime spectra of commutative unital rings (A. Grothendieck), and every quasi-separated scheme can be reconstructed up to isomorphism of schemes from the category of quasicoherent sheaves of -modules (P. Gabriel–A. Rosenberg). For Grothendieck topologies, the cohomological properties of a site are invariants of the corresponding category of sheaves of sets viewed abstractly as a topos (A. Grothendieck). In all these cases, a space is reconstructed from the algebra of functions or its categorified version—some category of sheaves on that space.
Functions on a topological space can be multiplied and added pointwise hence they form a commutative algebra; in fact these operations are local in the topology of the base space, hence the functions form a sheaf of commutative rings over the base space.
The dream of noncommutative geometry is to generalize this duality to the duality between noncommutative algebras, or sheaves of noncommutative algebras, or sheaf-like noncommutative algebraic or operator-algebraic structures, and geometric entities of certain kinds, and give an interaction between the algebraic and geometric description of those via this duality.
R
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https://en.wikipedia.org/wiki/Field%20theory
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Field theory may refer to:
Science
Field (mathematics), the theory of the algebraic concept of field
Field theory (physics), a physical theory which employs fields in the physical sense, consisting of three types:
Classical field theory, the theory and dynamics of classical fields
Quantum field theory, the theory of quantum mechanical fields
Statistical field theory, the theory of critical phase transitions
Grand unified theory
Social science
Field theory (psychology), a psychological theory which examines patterns of interaction between the individual and his or her environment
Field theory (sociology), a sociological theory concerning the relationship between social actors and local social orders
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https://en.wikipedia.org/wiki/BPS
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BPS, Bps or bps may refer to:
Science and mathematics
Plural of bp, base pair, a measure of length of DNA
Plural of bp, basis point, one one-hundredth of a percentage point - ‱
Battered person syndrome, a physical and psychological condition found in victims of abuse
Best practice statement, a qualification of a method used in guidelines documents
Bisphenol S, an organic chemical compound
Bladder pain syndrome, a disorder characterised by pain associated with urination
Bogomol'nyi–Prasad–Sommerfield bound, a mathematical concept in field and string theory
Bogomol'nyi–Prasad–Sommerfield state, solutions saturating the BPS bound
BPS domain, a protein domain
Bronchopulmonary sequestration, where a section of lung tissue has a decreased blood supply
Bovine papular stomatitis, a zoonotic farmyard pox
Computing
IBM Basic Programming Support, BPS/360
Bits per second (bps), a data rate unit
Bytes per second (Bps), a data rate unit
Bits per sample (bps), referring to color depth
Organizations
Badan Pusat Statistik, Indonesian statistical survey institute
Banco de Previsión Social, Uruguayan state-owned social security institute
Barbados Postal Service, national postal agency of Barbados
Biophysical Society, scientific society
Bosnian-Herzegovinian Patriotic Party-Sefer Halilović, a mayor Bosnian political party
Botswana Prison Service, corrections agency of Botswana
British Psychological Society, the representative body for psychologists and psychology in the United Kingdom
British Pteridological Society, the focal point for fern enthusiasts throughout the United Kingdom
Buddhist Publication Society, a charity aiming to spread the teachings of Buddha
Board of Pharmacy Specialties, a certification agency for specialized pharmacists
Education
Belilios Public School, a government secondary school in Hong Kong
Bismarck Public Schools, North Dakota, US school district
Boston Public Schools, a school district in Boston, Massachusetts, United States
British Parachute Schools, a parachuting drop zone in Nottinghamshire, England
Brockton Public Schools, the school district serving Brockton, Massachusetts, US
Birmingham City School District or Birmingham Public Schools, Michigan, US
Birla Public School, Pilani, Rajasthan, India
Brainerd Public Schools, Brainerd, Minnesota, US
Other uses
Bachelor of Professional Studies, an undergraduate degree
Battle Programmer Shirase, an anime television series
Baltic Pipeline System, an oil transport system in Western Europe
Biopsychosocial model, an interdisciplinary model that examines the interconnection between biology, psychology, and socio-environmental factors
Porto Seguro Airport, Brazil, IATA code
Basic Pay Scale, public sector term for the grade of an official in Pakistan
BPS grade tea
BLAST Pro Series, a Counter-Strike: Global Offensive tournament
See also
BP (disambiguation) (for BPs)
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https://en.wikipedia.org/wiki/Erd%C5%91s%E2%80%93Ko%E2%80%93Rado%20theorem
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In mathematics, the Erdős–Ko–Rado theorem limits the number of sets in a family of sets for which every two sets have at least one element in common. Paul Erdős, Chao Ko, and Richard Rado proved the theorem in 1938, but did not publish it until 1961. It is part of the field of combinatorics, and one of the central results of
The theorem applies to families of sets that all have the same and are all subsets of some larger set of size One way to construct a family of sets with these parameters, each two sharing an element, is to choose a single element to belong to all the subsets, and then form all of the subsets that contain the chosen element. The Erdős–Ko–Rado theorem states that when is large enough for the problem to be nontrivial this construction produces the largest possible intersecting families. When there are other equally-large families, but for larger values of only the families constructed in this way can be largest.
The Erdős–Ko–Rado theorem can also be described in terms of hypergraphs or independent sets in Kneser graphs. Several analogous theorems apply to other kinds of mathematical object than sets, including linear subspaces, permutations, and strings. They again describe the largest possible intersecting families as being formed by choosing an element and forming the family of all objects that contain the chosen element.
Statement
Suppose that is a family of distinct subsets of an set and that each two subsets share at least one element. Then the theorem states that the number of sets in is at most the binomial coefficient
The requirement that is necessary for the problem to be nontrivial: all sets intersect, and the largest intersecting family consists of all sets, with
The same result can be formulated as part of the theory of hypergraphs. A family of sets may also be called a hypergraph, and when all the sets (which are called "hyperedges" in this context) are the same it is called an hypergraph. The theorem thus gives an upper bound for the number of pairwise overlapping hyperedges in an hypergraph with
The theorem may also be formulated in terms of graph theory: the independence number of the Kneser graph for is
This is a graph with a vertex for each subset of an set, and an edge between every pair of disjoint sets. An independent set is a collection of vertices that has no edges between its pairs, and the independence number is the size of the largest Because Kneser graphs have symmetries taking any vertex to any other vertex (they are vertex-transitive graphs), their fractional chromatic number equals the ratio of their number of vertices to their independence number, so another way of expressing the Erdős–Ko–Rado theorem is that these graphs have fractional chromatic number
History
Paul Erdős, Chao Ko, and Richard Rado proved this theorem in 1938 after working together on it in England. Rado had moved from Berlin to the University of Cambridge and Erdős from Hungary to the Universi
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https://en.wikipedia.org/wiki/Institut%20national%20de%20la%20statistique%20et%20des%20%C3%A9tudes%20%C3%A9conomiques
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The National Institute of Statistics and Economic Studies (), abbreviated INSEE or Insee ( , ), is the national statistics bureau of France. It collects and publishes information about the French economy and people and carries out the periodic national census. Headquartered in Montrouge, a commune in the southern Parisian suburbs, it is the French branch of Eurostat. The INSEE was created in 1946 as a successor to the Vichy regime's National Statistics Service (SNS). It works in close cooperation with the Institut national d'études démographiques (INED).
Purpose
The INSEE is responsible for the production and analysis of official statistics in France. Its best known responsibilities include:
Organising and publishing the national census.
Producing various indices – which are widely recognised as being of excellent quality – including an inflation index used for determining the rates of rents and the costs associated with construction.
Eurostat uses INSEE statistics in combination with those of other national statistical agencies to compile comparable statistics for the European Union as a whole. It is also widely recognized as representing France on international questions of statistics.
Organisation
The INSEE is the responsibility of MINEFI, the French Ministry of Finance. The current director is Jean-Luc Tavernier. However, Eurostat considers INSEE as an independent body, although its independence is not written in the law.
Teaching and research
Research and teaching for the INSEE is undertaken by GENES or Group of the National Schools of the Economy and Statistics (French: Groupe des Écoles Nationales d'Économie et Statistique) which includes:
ENSAE (École nationale de la statistique et de l'administration économique), a grande école which trains INSEE administrators and engineers specialized in statistics, the economy, and finance.
ENSAI, (École nationale de la statistique et de l'analyse de l'information), an engineering school.
Codes and numbering system
INSEE gives numerical indexing codes (French: les Codes INSEE) to various entities in France:
INSEE codes (known as COG) are given to various administrative units, notably the French communes (they do not coincide with postcodes). The 'complete' code has 8 digits and 3 spaces within, but there is a popular 'simplified' code with 5 digits and no space within:
2 digits (département) and 3 digits (commune) for the 96 départements of Metropolitan France.
3 digits (département or collectivity) and 2 digits (commune) for the Overseas departments, Overseas Territorial Collectivities and Overseas Countries and Territories.
INSEE numbers (13 digits + a two-digit key) are national identification numbers given to people. The format is as follows: syymmlllllooo kk, where
s is 1 for a male, 2 for a female for a permanent number; it is 7 for a male, 8 for a female for a temporary number,
yy are the last two digits of the year of birth,
mm is the month of birth or a number abov
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https://en.wikipedia.org/wiki/Weyl%20group
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In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections through the hyperplanes orthogonal to the roots, and as such is a finite reflection group. In fact it turns out that most finite reflection groups are Weyl groups. Abstractly, Weyl groups are finite Coxeter groups, and are important examples of these.
The Weyl group of a semisimple Lie group, a semisimple Lie algebra, a semisimple linear algebraic group, etc. is the Weyl group of the root system of that group or algebra.
Definition and examples
Let be a root system in a Euclidean space . For each root , let denote the reflection about the hyperplane perpendicular to , which is given explicitly as
,
where is the inner product on . The Weyl group of is the subgroup of the orthogonal group generated by all the 's. By the definition of a root system, each preserves , from which it follows that is a finite group.
In the case of the root system, for example, the hyperplanes perpendicular to the roots are just lines, and the Weyl group is the symmetry group of an equilateral triangle, as indicated in the figure. As a group, is isomorphic to the permutation group on three elements, which we may think of as the vertices of the triangle. Note that in this case, is not the full symmetry group of the root system; a 60-degree rotation preserves but is not an element of .
We may consider also the root system. In this case, is the space of all vectors in whose entries sum to zero. The roots consist of the vectors of the form , where is the th standard basis element for . The reflection associated to such a root is the transformation of obtained by interchanging the th and th entries of each vector. The Weyl group for is then the permutation group on elements.
Weyl chambers
If is a root system, we may consider the hyperplane perpendicular to each root . Recall that denotes the reflection about the hyperplane and that the Weyl group is the group of transformations of generated by all the 's. The complement of the set of hyperplanes is disconnected, and each connected component is called a Weyl chamber. If we have fixed a particular set Δ of simple roots, we may define the fundamental Weyl chamber associated to Δ as the set of points such that for all .
Since the reflections preserve , they also preserve the set of hyperplanes perpendicular to the roots. Thus, each Weyl group element permutes the Weyl chambers.
The figure illustrates the case of the A2 root system. The "hyperplanes" (in this case, one dimensional) orthogonal to the roots are indicated by dashed lines. The six 60-degree sectors are the Weyl chambers and the shaded region is the fundamental Weyl chamber associated to the indicated base.
A basic general theorem about Weyl chambers is this:
Theorem: The Weyl group acts freely an
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https://en.wikipedia.org/wiki/Schild%27s%20Ladder
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Schild's Ladder is a 2002 science fiction novel by Australian author Greg Egan. The book derives its name from Schild's ladder, a construction in differential geometry, devised by the mathematician and physicist Alfred Schild.
Plot summary
Twenty-thousand years in the future, Cass, a humanoid physicist from Earth, travels to an orbital station in the vicinity of the star Mimosa, and begins a series of experiments to test the extremities of the fictitious Sarumpaet rules – a set of fundamental equations in "Quantum Graph Theory", which holds that physical existence is a manifestation of complex constructions of mathematical graphs. However, the experiments unexpectedly create a bubble of something more stable than ordinary vacuum, dubbed "novo-vacuum", that expands outward at half the speed of light as ordinary vacuum collapses to this new state at the border, hinting at more general laws beyond the Sarumpaet rules. The local population is forced to flee to ever more distant star systems to escape the steadily approaching border, but since the expansion never slows, it is just a matter of time before the novo-vacuum encompasses any given region within the Local Group.
Two factions develop as the expanding bubble swallows star after star: the Preservationists, who wish to stop the expansion and preserve the Milky Way at any cost; and the Yielders, who consider the novo-vacuum to be too important a discovery to destroy without understanding.
Six hundred years after the initial experiment, a vessel called the Rindler has matched velocities with an ever-expanding novo-vacuum region at the border, powered by multispectral light emitted as the ordinary vacuum collapses into its lower energy-state. A variety of refugees are probing the novo-vacuum in order to understand the physics that makes it possible. The novo-vacuum turns out to be more complicated than anyone suspects, however, and Egan's usual topics of simulation and quantum ontology are taken to the extreme when we learn that a whole ordered universe exists within this zone of apparent chaos, existing as direct elaborations of the quantum graph's lattice structure, of which elementary particles, fundamental interactions, and our spacetime itself are only special cases.
It is ultimately revealed that the novo-vacuum's exotic geometry contains living organisms and even civilizations. This ecosystem is based on "vendeks," microbe-like complexes of quantum graph structures only 10−33 meters across. Agglomerations of vendeks form "xennobes," analogous to multicellular organisms but only 10−27 meters across. This discovery greatly increases the importance of the Yielders' mission, since destroying the novo-vacuum would be tantamount to genocide, and a solution must be found to the metastability of the novo-vacuum's border region within our spacetime.
See also
Quantum graph
External links
Official Website
2002 Australian novels
Transhumanist books
Novels by Greg Egan
Australian science fic
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https://en.wikipedia.org/wiki/Construction%20of%20the%20real%20numbers
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In mathematics, there are several equivalent ways of defining the real numbers. One of them is that they form a complete ordered field that does not contain any smaller complete ordered field. Such a definition does not prove that such a complete ordered field exists, and the existence proof consists of constructing a mathematical structure that satisfies the definition.
The article presents several such constructions. They are equivalent in the sense that, given the result of any two such constructions, there is a unique isomorphism of ordered field between them. This results from the above definition and is independent of particular constructions. These isomorphisms allow identifying the results of the constructions, and, in practice, to forget which construction has been chosen.
Axiomatic definitions
An axiomatic definition of the real numbers consists of defining them as the elements of a complete ordered field. This means the following. The real numbers form a set, commonly denoted , containing two distinguished elements denoted 0 and 1, and on which are defined two binary operations and one binary relation; the operations are called addition and multiplication of real numbers and denoted respectively with and ; the binary relation is inequality, denoted Moreover, the following properties called axioms must be satisfied.
The existence of such a structure is a theorem, which is proved by constructing such a structure. A consequence of the axioms is that this structure is unique up to an isomorphism, and thus, the real numbers can be used and manipulated, without referring to the method of construction.
Axioms
is a field under addition and multiplication. In other words,
For all x, y, and z in , x + (y + z) = (x + y) + z and x × (y × z) = (x × y) × z. (associativity of addition and multiplication)
For all x and y in , x + y = y + x and x × y = y × x. (commutativity of addition and multiplication)
For all x, y, and z in , x × (y + z) = (x × y) + (x × z). (distributivity of multiplication over addition)
For all x in , x + 0 = x. (existence of additive identity)
0 is not equal to 1, and for all x in , x × 1 = x. (existence of multiplicative identity)
For every x in , there exists an element −x in , such that x + (−x) = 0. (existence of additive inverses)
For every x ≠ 0 in , there exists an element x−1 in , such that x × x−1 = 1. (existence of multiplicative inverses)
is totally ordered for . In other words,
For all x in , x ≤ x. (reflexivity)
For all x and y in , if x ≤ y and y ≤ x, then x = y. (antisymmetry)
For all x, y, and z in , if x ≤ y and y ≤ z, then x ≤ z. (transitivity)
For all x and y in , x ≤ y or y ≤ x. (totality)
Addition and multiplication are compatible with the order. In other words,
For all x, y and z in , if x ≤ y, then x + z ≤ y + z. (preservation of order under addition)
For all x and y in , if 0 ≤ x and 0 ≤ y, then 0 ≤ x × y (preservation of order under multiplication)
The order ≤ is complete in
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https://en.wikipedia.org/wiki/Universe%20%28mathematics%29
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In mathematics, and particularly in set theory, category theory, type theory, and the foundations of mathematics, a universe is a collection that contains all the entities one wishes to consider in a given situation.
In set theory, universes are often classes that contain (as elements) all sets for which one hopes to prove a particular theorem. These classes can serve as inner models for various axiomatic systems such as ZFC or Morse–Kelley set theory. Universes are of critical importance to formalizing concepts in category theory inside set-theoretical foundations. For instance, the canonical motivating example of a category is Set, the category of all sets, which cannot be formalized in a set theory without some notion of a universe.
In type theory, a universe is a type whose elements are types.
In a specific context
Perhaps the simplest version is that any set can be a universe, so long as the object of study is confined to that particular set. If the object of study is formed by the real numbers, then the real line R, which is the real number set, could be the universe under consideration. Implicitly, this is the universe that Georg Cantor was using when he first developed modern naive set theory and cardinality in the 1870s and 1880s in applications to real analysis. The only sets that Cantor was originally interested in were subsets of R.
This concept of a universe is reflected in the use of Venn diagrams. In a Venn diagram, the action traditionally takes place inside a large rectangle that represents the universe U. One generally says that sets are represented by circles; but these sets can only be subsets of U. The complement of a set A is then given by that portion of the rectangle outside of As circle. Strictly speaking, this is the relative complement U \ A of A relative to U; but in a context where U is the universe, it can be regarded as the absolute complement AC of A. Similarly, there is a notion of the nullary intersection, that is the intersection of zero sets (meaning no sets, not null sets).
Without a universe, the nullary intersection would be the set of absolutely everything, which is generally regarded as impossible; but with the universe in mind, the nullary intersection can be treated as the set of everything under consideration, which is simply U. These conventions are quite useful in the algebraic approach to basic set theory, based on Boolean lattices. Except in some non-standard forms of axiomatic set theory (such as New Foundations), the class of all sets is not a Boolean lattice (it is only a relatively complemented lattice).
In contrast, the class of all subsets of U, called the power set of U, is a Boolean lattice. The absolute complement described above is the complement operation in the Boolean lattice; and U, as the nullary intersection, serves as the top element (or nullary meet) in the Boolean lattice. Then De Morgan's laws, which deal with complements of meets and joins (which are unions in set theory)
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https://en.wikipedia.org/wiki/Necklace%20problem
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The necklace problem is a problem in recreational mathematics concerning the reconstruction of necklaces (cyclic arrangements of binary values) from partial information.
Formulation
The necklace problem involves the reconstruction of a necklace of beads, each of which is either black or white, from partial information. The information specifies how many copies the necklace contains of each possible arrangement of black beads. For instance, for , the specified information gives the number of pairs of black beads that are separated by positions, for .
This can be made formal by defining a -configuration to be a necklace of black beads and white beads, and counting the number of ways of rotating a -configuration so that each of its black beads coincides with one of the black beads of the given necklace.
The necklace problem asks: if is given, and the numbers of copies of each -configuration are known up to some threshold , how large does the threshold need to be before this information completely determines the necklace that it describes? Equivalently, if the information about -configurations is provided in stages, where the th stage provides the numbers of copies of each -configuration, how many stages are needed (in the worst case) in order to reconstruct the precise pattern of black and white beads in the original necklace?
Upper bounds
Alon, Caro, Krasikov and Roditty showed that 1 + log2(n) is sufficient, using a cleverly enhanced inclusion–exclusion principle.
Radcliffe and Scott showed that if n is prime, 3 is sufficient, and for any n, 9 times the number of prime factors of n is sufficient.
Pebody showed that for any n, 6 is sufficient and, in a followup paper, that for odd n, 4 is sufficient. He conjectured that 4 is again sufficient for even n greater than 10, but this remains unproven.
See also
Necklace (combinatorics)
Bracelet (combinatorics)
Moreau's necklace-counting function
Necklace splitting problem
References
Combinatorics on words
Recreational mathematics
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https://en.wikipedia.org/wiki/Coxeter%20group
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In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example. However, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections. Coxeter groups were introduced in 1934 as abstractions of reflection groups , and finite Coxeter groups were classified in 1935 .
Coxeter groups find applications in many areas of mathematics. Examples of finite Coxeter groups include the symmetry groups of regular polytopes, and the Weyl groups of simple Lie algebras. Examples of infinite Coxeter groups include the triangle groups corresponding to regular tessellations of the Euclidean plane and the hyperbolic plane, and the Weyl groups of infinite-dimensional Kac–Moody algebras.
Standard references include
and .
Definition
Formally, a Coxeter group can be defined as a group with the presentation
where and for .
The condition means no relation of the form should be imposed.
The pair where is a Coxeter group with generators is called a Coxeter system. Note that in general is not uniquely determined by . For example, the Coxeter groups of type and are isomorphic but the Coxeter systems are not equivalent (see below for an explanation of this notation).
A number of conclusions can be drawn immediately from the above definition.
The relation means that for all ; as such the generators are involutions.
If , then the generators and commute. This follows by observing that
,
together with
implies that
.
Alternatively, since the generators are involutions, , so , and thus is equal to the commutator.
In order to avoid redundancy among the relations, it is necessary to assume that . This follows by observing that
,
together with
implies that
.
Alternatively, and are conjugate elements, as .
Coxeter matrix and Schläfli matrix
The Coxeter matrix is the symmetric matrix with entries . Indeed, every symmetric matrix with diagonal entries exclusively 1 and nondiagonal entries in the set is a Coxeter matrix.
The Coxeter matrix can be conveniently encoded by a Coxeter diagram, as per the following rules.
The vertices of the graph are labelled by generator subscripts.
Vertices and are adjacent if and only if .
An edge is labelled with the value of whenever the value is or greater.
In particular, two generators commute if and only if they are not connected by an edge.
Furthermore, if a Coxeter graph has two or more connected components, the associated group is the direct product of the groups associated to the individual components.
Thus the disjoint union of Coxeter graphs yields a direct product of Coxeter groups.
The Coxeter matrix, , is related to the Schläfli matrix with entries , but the elements are modified, being prop
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https://en.wikipedia.org/wiki/Word%20problem%20%28mathematics%20education%29
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In science education, a word problem is a mathematical exercise (such as in a textbook, worksheet, or exam) where significant background information on the problem is presented in ordinary language rather than in mathematical notation. As most word problems involve a narrative of some sort, they are sometimes referred to as story problems and may vary in the amount of technical language used.
Example
A typical word problem:
Tess paints two boards of a fence every four minutes, but Allie can paint three boards every two minutes. If there are 240 boards total, how many hours will it take them to paint the fence, working together?
Solution process
Word problems such as the above can be examined through five stages:
1. Problem Comprehension
2. Situational Solution Visualization
3. Mathematical Solution Planning
4. Solving for Solution
5. Situational Solution Visualization
The linguistic properties of a word problem need to be addressed first. To begin the solution process, one must first understand what the problem is asking and what type of solution the answer will be. In the problem above, the words "minutes", "total", "hours", and "together" need to be examined.
The next step is to visualize what the solution to this problem might mean. For our stated problem, the solution might be visualized by examining if the total number of hours will be greater or smaller than if it were stated in minutes. Also, it must be determined whether or not the two girls will finish at a faster or slower rate if they are working together.
After this, one must plan a solution method using mathematical terms. One scheme to analyze the mathematical properties is to classify the numerical quantities in the problem into known quantities (values given in the text), wanted quantities (values to be found), and auxiliary quantities (values found as intermediate stages of the problem). This is found in the "Variables" and "Equations" sections above.
Next, the mathematical processes must be applied to the formulated solution process. This is done solely in the mathematical context for now.
Finally, one must again visualize the proposed solution and determine if the solution seems to make sense for the realistic context of the problem. After visualizing if it is reasonable, one can then work to further analyze and draw connections between mathematical concepts and realistic problems.
The importance of these five steps in teacher education is discussed at the end of the following section.
Purpose and skill development
Word problems commonly include mathematical modelling questions, where data and information about a certain system is given and a student is required to develop a model. For example:
Jane had $5.00, then spent $2.00. How much does she have now?
In a cylindrical barrel with radius 2 m, the water is rising at a rate of 3 cm/s. What is the rate of increase of the volume of water?
As the developmental skills of students across grade levels varies, t
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https://en.wikipedia.org/wiki/Transcendental%20function
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In mathematics, a transcendental function is an analytic function that does not satisfy a polynomial equation, in contrast to an algebraic function.
In other words, a transcendental function "transcends" algebra in that it cannot be expressed algebraically using a finite amount of terms.
Examples of transcendental functions include the exponential function, the logarithm, and the trigonometric functions.
Definition
Formally, an analytic function of one real or complex variable is transcendental if it is algebraically independent of that variable. This can be extended to functions of several variables.
History
The transcendental functions sine and cosine were tabulated from physical measurements in antiquity, as evidenced in Greece (Hipparchus) and India (jya and koti-jya). In describing Ptolemy's table of chords, an equivalent to a table of sines, Olaf Pedersen wrote:
A revolutionary understanding of these circular functions occurred in the 17th century and was explicated by Leonhard Euler in 1748 in his Introduction to the Analysis of the Infinite. These ancient transcendental functions became known as continuous functions through quadrature of the rectangular hyperbola by Grégoire de Saint-Vincent in 1647, two millennia after Archimedes had produced The Quadrature of the Parabola.
The area under the hyperbola was shown to have the scaling property of constant area for a constant ratio of bounds. The hyperbolic logarithm function so described was of limited service until 1748 when Leonhard Euler related it to functions where a constant is raised to a variable exponent, such as the exponential function where the constant base is e. By introducing these transcendental functions and noting the bijection property that implies an inverse function, some facility was provided for algebraic manipulations of the natural logarithm even if it is not an algebraic function.
The exponential function is written Euler identified it with the infinite series where denotes the factorial of .
The even and odd terms of this series provide sums denoting and , so that These transcendental hyperbolic functions can be converted into circular functions sine and cosine by introducing into the series, resulting in alternating series. After Euler, mathematicians view the sine and cosine this way to relate the transcendence to logarithm and exponent functions, often through Euler's formula in complex number arithmetic.
Examples
Let be a positive constant. The following functions are transcendental:
For the second function , if we set equal to , the base of the natural logarithm, then we get that is a transcendental function. Similarly, if we set equal to in , then we get that (that is, the natural logarithm) is a transcendental function.
Algebraic and transcendental functions
The most familiar transcendental functions are the logarithm, the exponential (with any non-trivial base), the trigonometric, and the hyperbolic functions, and the inverses
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https://en.wikipedia.org/wiki/Aryabhata
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Aryabhata ( ISO: ) or Aryabhata I (476–550 CE) was the first of the major mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy. His works include the Āryabhaṭīya (which mentions that in 3600 Kali Yuga, 499 CE, he was 23 years old) and the Arya-siddhanta.
For his explicit mention of the relativity of motion, he also qualifies as a major early physicist.
Biography
Name
While there is a tendency to misspell his name as "Aryabhatta" by analogy with other names having the "bhatta" suffix, his name is properly spelled Aryabhata: every astronomical text spells his name thus, including Brahmagupta's references to him "in more than a hundred places by name". Furthermore, in most instances "Aryabhatta" would not fit the metre either.
Time and place of birth
Aryabhata mentions in the Aryabhatiya that he was 23 years old 3,600 years into the Kali Yuga, but this is not to mean that the text was composed at that time. This mentioned year corresponds to 499 CE, and implies that he was born in 476. Aryabhata called himself a native of Kusumapura or Pataliputra (present day Patna, Bihar).
Other hypothesis
Bhāskara I describes Aryabhata as āśmakīya, "one belonging to the Aśmaka country." During the Buddha's time, a branch of the Aśmaka people settled in the region between the Narmada and Godavari rivers in central India.
It has been claimed that the aśmaka (Sanskrit for "stone") where Aryabhata originated may be the present day Kodungallur which was the historical capital city of Thiruvanchikkulam of ancient Kerala. This is based on the belief that Koṭuṅṅallūr was earlier known as Koṭum-Kal-l-ūr ("city of hard stones"); however, old records show that the city was actually Koṭum-kol-ūr ("city of strict governance"). Similarly, the fact that several commentaries on the Aryabhatiya have come from Kerala has been used to suggest that it was Aryabhata's main place of life and activity; however, many commentaries have come from outside Kerala, and the Aryasiddhanta was completely unknown in Kerala. K. Chandra Hari has argued for the Kerala hypothesis on the basis of astronomical evidence.
Aryabhata mentions "Lanka" on several occasions in the Aryabhatiya, but his "Lanka" is an abstraction, standing for a point on the equator at the same longitude as his Ujjayini.
Education
It is fairly certain that, at some point, he went to Kusumapura for advanced studies and lived there for some time. Both Hindu and Buddhist tradition, as well as Bhāskara I (CE 629), identify Kusumapura as Pāṭaliputra, modern Patna. A verse mentions that Aryabhata was the head of an institution () at Kusumapura, and, because the university of Nalanda was in Pataliputra at the time and had an astronomical observatory, it is speculated that Aryabhata might have been the head of the Nalanda university as well. Aryabhata is also reputed to have set up an observatory at the Sun temple in Taregana, Bihar.
Works
Aryabhata is the author of several treatises on
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https://en.wikipedia.org/wiki/Brahmagupta
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Brahmagupta ( – ) was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the Brāhmasphuṭasiddhānta (BSS, "correctly established doctrine of Brahma", dated 628), a theoretical treatise, and the Khaṇḍakhādyaka ("edible bite", dated 665), a more practical text.
In 628 CE, Brahmagupta first described gravity as an attractive force, and used the term "gurutvākarṣaṇam (गुरुत्वाकर्षणम्)" in Sanskrit to describe it.
Brahmagupta is credited with first clear description of the quadratic formula (the solution of the quadratic equation) in his main work, the Brāhma-sphuṭa-siddhānta.
Life and career
Brahmagupta, according to his own statement, was born in 598 CE. Born in Bhillamāla in Gurjaradesa (modern Bhinmal in Rajasthan, India) during the reign of the Chavda dynasty ruler Vyagrahamukha, his ancestors were probably from Sindh. He was the son of Jishnugupta and was a Hindu by religion, in particular, a Shaivite. He lived and worked there for a good part of his life. Prithudaka Svamin, a later commentator, called him Bhillamalacharya, the teacher from Bhillamala.
Bhillamala was the capital of the Gurjaradesa, the second-largest kingdom of Western India, comprising southern Rajasthan and northern Gujarat in modern-day India. It was also a centre of learning for mathematics and astronomy. He became an astronomer of the Brahmapaksha school, one of the four major schools of Indian astronomy during this period. He studied the five traditional Siddhantas on Indian astronomy as well as the work of other astronomers including Aryabhata I, Latadeva, Pradyumna, Varahamihira, Simha, Srisena, Vijayanandin and Vishnuchandra.
In the year 628, at the age of 30, he composed the Brāhmasphuṭasiddhānta ("improved treatise of Brahma") which is believed to be a revised version of the received Siddhanta of the Brahmapaksha school of astronomy. Scholars state that he incorporated a great deal of originality into his revision, adding a considerable amount of new material. The book consists of 24 chapters with 1008 verses in the ārya metre. A good deal of it is astronomy, but it also contains key chapters on mathematics, including algebra, geometry, trigonometry and algorithmics, which are believed to contain new insights due to Brahmagupta himself.
Later, Brahmagupta moved to Ujjaini, Avanti, a major centre for astronomy in central India. At the age of 67, he composed his next well-known work Khanda-khādyaka, a practical manual of Indian astronomy in the karana category meant to be used by students.
Brahmagupta died in 668 CE, and he is presumed to have died in Ujjain.
Works
Brahmagupta composed the following treatises:
Brāhmasphuṭasiddhānta, composed in 628 CE.
Khaṇḍakhādyaka, composed in 665 CE.
Grahaṇārkajñāna, (ascribed in one manuscript)
Reception
Brahmagupta's mathematical advances were carried on further by Bhāskara II, a lineal descendant in Ujjain, who described Brahmagupta as the ganaka-chakra-chud
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https://en.wikipedia.org/wiki/Gelfand%E2%80%93Naimark%E2%80%93Segal%20construction
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In functional analysis, a discipline within mathematics, given a C*-algebra A, the Gelfand–Naimark–Segal construction establishes a correspondence between cyclic *-representations of A and certain linear functionals on A (called states). The correspondence is shown by an explicit construction of the *-representation from the state. It is named for Israel Gelfand, Mark Naimark, and Irving Segal.
States and representations
A *-representation of a C*-algebra A on a Hilbert space H is a mapping
π from A into the algebra of bounded operators on H such that
π is a ring homomorphism which carries involution on A into involution on operators
π is nondegenerate, that is the space of vectors π(x) ξ is dense as x ranges through A and ξ ranges through H. Note that if A has an identity, nondegeneracy means exactly π is unit-preserving, i.e. π maps the identity of A to the identity operator on H.
A state on a C*-algebra A is a positive linear functional f of norm 1. If A has a multiplicative unit element this condition is equivalent to f(1) = 1.
For a representation π of a C*-algebra A on a Hilbert space H, an element ξ is called a cyclic vector if the set of vectors
is norm dense in H, in which case π is called a cyclic representation. Any non-zero vector of an irreducible representation is cyclic. However, non-zero vectors in a general cyclic representation may fail to be cyclic.
The GNS construction
Let π be a *-representation of a C*-algebra A on the Hilbert space H and ξ be a unit norm cyclic vector for π. Then
is a state of A.
Conversely, every state of A may be viewed as a vector state as above, under a suitable canonical representation.
The method used to produce a *-representation from a state of A in the proof of the above theorem is called the GNS construction.
For a state of a C*-algebra A, the corresponding GNS representation is essentially uniquely determined by the condition, as seen in the theorem below.
Significance of the GNS construction
The GNS construction is at the heart of the proof of the Gelfand–Naimark theorem characterizing C*-algebras as algebras of operators. A C*-algebra has sufficiently many pure states (see below) so that the direct sum of corresponding irreducible GNS representations is faithful.
The direct sum of the corresponding GNS representations of all states is called the universal representation of A. The universal representation of A contains every cyclic representation. As every *-representation is a direct sum of cyclic representations, it follows that every *-representation of A is a direct summand of some sum of copies of the universal representation.
If Φ is the universal representation of a C*-algebra A, the closure of Φ(A) in the weak operator topology is called the enveloping von Neumann algebra of A. It can be identified with the double dual A**.
Irreducibility
Also of significance is the relation between irreducible *-representations and extreme points of the convex set of states. A r
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https://en.wikipedia.org/wiki/Composition%20series
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In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a module, into simple pieces. The need for considering composition series in the context of modules arises from the fact that many naturally occurring modules are not semisimple, hence cannot be decomposed into a direct sum of simple modules. A composition series of a module M is a finite increasing filtration of M by submodules such that the successive quotients are simple and serves as a replacement of the direct sum decomposition of M into its simple constituents.
A composition series may not exist, and when it does, it need not be unique. Nevertheless, a group of results known under the general name Jordan–Hölder theorem asserts that whenever composition series exist, the isomorphism classes of simple pieces (although, perhaps, not their location in the composition series in question) and their multiplicities are uniquely determined. Composition series may thus be used to define invariants of finite groups and Artinian modules.
A related but distinct concept is a chief series: a composition series is a maximal subnormal series, while a chief series is a maximal normal series.
For groups
If a group G has a normal subgroup N, then the factor group G/N may be formed, and some aspects of the study of the structure of G may be broken down by studying the "smaller" groups G/N and N. If G has no normal subgroup that is different from G and from the trivial group, then G is a simple group. Otherwise, the question naturally arises as to whether G can be reduced to simple "pieces", and if so, are there any unique features of the way this can be done?
More formally, a composition series of a group G is a subnormal series of finite length
with strict inclusions, such that each Hi is a maximal proper normal subgroup of Hi+1. Equivalently, a composition series is a subnormal series such that each factor group Hi+1 / Hi is simple. The factor groups are called composition factors.
A subnormal series is a composition series if and only if it is of maximal length. That is, there are no additional subgroups which can be "inserted" into a composition series. The length n of the series is called the composition length.
If a composition series exists for a group G, then any subnormal series of G can be refined to a composition series, informally, by inserting subgroups into the series up to maximality. Every finite group has a composition series, but not every infinite group has one. For example, has no composition series.
Uniqueness: Jordan–Hölder theorem
A group may have more than one composition series. However, the Jordan–Hölder theorem (named after Camille Jordan and Otto Hölder) states that any two composition series of a given group are equivalent. That is, they have the same composition length and the same composition factors, up to permutation and isomorphism. This theorem can be proved using the Schreier refinement theorem. The Jor
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https://en.wikipedia.org/wiki/Lie%20superalgebra
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In mathematics, a Lie superalgebra is a generalisation of a Lie algebra to include a Z2grading. Lie superalgebras are important in theoretical physics where they are used to describe the mathematics of supersymmetry. In most of these theories, the even elements of the superalgebra correspond to bosons and odd elements to fermions (but this is not always true; for example, the BRST supersymmetry is the other way around).
Definition
Formally, a Lie superalgebra is a nonassociative Z2-graded algebra, or superalgebra, over a commutative ring (typically R or C) whose product [·, ·], called the Lie superbracket or supercommutator, satisfies the two conditions (analogs of the usual Lie algebra axioms, with grading):
Super skew-symmetry:
The super Jacobi identity:
where x, y, and z are pure in the Z2-grading. Here, |x| denotes the degree of x (either 0 or 1). The degree of [x,y] is the sum of degree of x and y modulo 2.
One also sometimes adds the axioms for |x| = 0 (if 2 is invertible this follows automatically) and for |x| = 1 (if 3 is invertible this follows automatically). When the ground ring is the integers or the Lie superalgebra is a free module, these conditions are equivalent to the condition that the Poincaré–Birkhoff–Witt theorem holds (and, in general, they are necessary conditions for the theorem to hold).
Just as for Lie algebras, the universal enveloping algebra of the Lie superalgebra can be given a Hopf algebra structure.
A graded Lie algebra (say, graded by Z or N) that is anticommutative and Jacobi in the graded sense also has a grading (which is called "rolling up" the algebra into odd and even parts), but is not referred to as "super". See note at graded Lie algebra for discussion.
Properties
Let be a Lie superalgebra. By inspecting the Jacobi identity, one sees that there are eight cases depending on whether arguments are even or odd. These fall into four classes, indexed by the number of odd elements:
No odd elements. The statement is just that is an ordinary Lie algebra.
One odd element. Then is a -module for the action .
Two odd elements. The Jacobi identity says that the bracket is a symmetric -map.
Three odd elements. For all , .
Thus the even subalgebra of a Lie superalgebra forms a (normal) Lie algebra as all the signs disappear, and the superbracket becomes a normal Lie bracket, while is a linear representation of , and there exists a symmetric -equivariant linear map such that,
Conditions (1)–(3) are linear and can all be understood in terms of ordinary Lie algebras. Condition (4) is nonlinear, and is the most difficult one to verify when constructing a Lie superalgebra starting from an ordinary Lie algebra () and a representation ().
Involution
A ∗ Lie superalgebra is a complex Lie superalgebra equipped with an involutive antilinear map from itself to itself which respects the Z2 grading and satisfies
[x,y]* = [y*,x*] for all x and y in the Lie superalgebra. (Some authors prefer the convention [
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https://en.wikipedia.org/wiki/Homotopy%20group
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In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted which records information about loops in a space. Intuitively, homotopy groups record information about the basic shape, or holes, of a topological space.
To define the n-th homotopy group, the base-point-preserving maps from an n-dimensional sphere (with base point) into a given space (with base point) are collected into equivalence classes, called homotopy classes. Two mappings are homotopic if one can be continuously deformed into the other. These homotopy classes form a group, called the n-th homotopy group, of the given space X with base point. Topological spaces with differing homotopy groups are never equivalent (homeomorphic), but topological spaces that homeomorphic have the same homotopy groups.
The notion of homotopy of paths was introduced by Camille Jordan.
Introduction
In modern mathematics it is common to study a category by associating to every object of this category a simpler object that still retains sufficient information about the object of interest. Homotopy groups are such a way of associating groups to topological spaces.
That link between topology and groups lets mathematicians apply insights from group theory to topology. For example, if two topological objects have different homotopy groups, they cannot have the same topological structure—a fact that may be difficult to prove using only topological means. For example, the torus is different from the sphere: the torus has a "hole"; the sphere doesn't. However, since continuity (the basic notion of topology) only deals with the local structure, it can be difficult to formally define the obvious global difference. The homotopy groups, however, carry information about the global structure.
As for the example: the first homotopy group of the torus is
because the universal cover of the torus is the Euclidean plane mapping to the torus Here the quotient is in the category of topological spaces, rather than groups or rings. On the other hand, the sphere satisfies:
because every loop can be contracted to a constant map (see homotopy groups of spheres for this and more complicated examples of homotopy groups). Hence the torus is not homeomorphic to the sphere.
Definition
In the n-sphere we choose a base point a. For a space X with base point b, we define to be the set of homotopy classes of maps
that map the base point a to the base point b. In particular, the equivalence classes are given by homotopies that are constant on the basepoint of the sphere. Equivalently, define to be the group of homotopy classes of maps from the n-cube to X that take the boundary of the n-cube to b.
For the homotopy classes form a group. To define the group operation, recall that in the fundamental group, the product of two loops is defined by setting
The idea of composition in the fundamental group is that
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https://en.wikipedia.org/wiki/Jensen%27s%20inequality
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In mathematics, Jensen's inequality, named after the Danish mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function. It was proved by Jensen in 1906, building on an earlier proof of the same inequality for doubly-differentiable functions by Otto Hölder in 1889. Given its generality, the inequality appears in many forms depending on the context, some of which are presented below. In its simplest form the inequality states that the convex transformation of a mean is less than or equal to the mean applied after convex transformation; it is a simple corollary that the opposite is true of concave transformations.
Jensen's inequality generalizes the statement that the secant line of a convex function lies above the graph of the function, which is Jensen's inequality for two points: the secant line consists of weighted means of the convex function (for t ∈ [0,1]),
while the graph of the function is the convex function of the weighted means,
Thus, Jensen's inequality is
In the context of probability theory, it is generally stated in the following form: if X is a random variable and is a convex function, then
The difference between the two sides of the inequality, , is called the Jensen gap.
Statements
The classical form of Jensen's inequality involves several numbers and weights. The inequality can be stated quite generally using either the language of measure theory or (equivalently) probability. In the probabilistic setting, the inequality can be further generalized to its full strength.
Finite form
For a real convex function , numbers in its domain, and positive weights , Jensen's inequality can be stated as:
and the inequality is reversed if is concave, which is
Equality holds if and only if or is linear on a domain containing .
As a particular case, if the weights are all equal, then () and () become
For instance, the function is concave, so substituting in the previous formula () establishes the (logarithm of the) familiar arithmetic-mean/geometric-mean inequality:
A common application has as a function of another variable (or set of variables) , that is, . All of this carries directly over to the general continuous case: the weights are replaced by a non-negative integrable function , such as a probability distribution, and the summations are replaced by integrals.
Measure-theoretic form
Let be a probability space. Let be a -measurable function and be convex. Then:
In real analysis, we may require an estimate on
where , and is a non-negative Lebesgue-integrable function. In this case, the Lebesgue measure of need not be unity. However, by integration by substitution, the interval can be rescaled so that it has measure unity. Then Jensen's inequality can be applied to get
Probabilistic form
The same result can be equivalently stated in a probability theory setting, by a simple change of notation. Let be a probability space, X an integrable real-valued
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https://en.wikipedia.org/wiki/Chord%20%28geometry%29
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A chord (from the Latin chorda, meaning "bowstring") of a circle is a straight line segment whose endpoints both lie on a circular arc. If a chord were to be extended infinitely on both directions into a line, the object is a secant line. The perpendicular line passing through the chord's midpoint is called sagitta (Latin for "arrow").
More generally, a chord is a line segment joining two points on any curve, for instance, on an ellipse. A chord that passes through a circle's center point is the circle's diameter.
In circles
Among properties of chords of a circle are the following:
Chords are equidistant from the center if and only if their lengths are equal.
Equal chords are subtended by equal angles from the center of the circle.
A chord that passes through the center of a circle is called a diameter and is the longest chord of that specific circle.
If the line extensions (secant lines) of chords AB and CD intersect at a point P, then their lengths satisfy AP·PB = CP·PD (power of a point theorem).
In conics
The midpoints of a set of parallel chords of a conic are collinear (midpoint theorem for conics).
In trigonometry
Chords were used extensively in the early development of trigonometry. The first known trigonometric table, compiled by Hipparchus, tabulated the value of the chord function for every degrees. In the second century AD, Ptolemy of Alexandria compiled a more extensive table of chords in his book on astronomy, giving the value of the chord for angles ranging from to 180 degrees by increments of degree. The circle was of diameter 120, and the chord lengths are accurate to two base-60 digits after the integer part.
The chord function is defined geometrically as shown in the picture. The chord of an angle is the length of the chord between two points on a unit circle separated by that central angle. The angle θ is taken in the positive sense and must lie in the interval (radian measure). The chord function can be related to the modern sine function, by taking one of the points to be (1,0), and the other point to be (), and then using the Pythagorean theorem to calculate the chord length:
The last step uses the half-angle formula. Much as modern trigonometry is built on the sine function, ancient trigonometry was built on the chord function. Hipparchus is purported to have written a twelve-volume work on chords, all now lost, so presumably, a great deal was known about them. In the table below (where c is the chord length, and D the diameter of the circle) the chord function can be shown to satisfy many identities analogous to well-known modern ones:
The inverse function exists as well:
See also
Circular segment - the part of the sector that remains after removing the triangle formed by the center of the circle and the two endpoints of the circular arc on the boundary.
Scale of chords
Ptolemy's table of chords
Holditch's theorem, for a chord rotating in a convex closed curve
Circle graph
Exsecant and excosecant
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https://en.wikipedia.org/wiki/Brownian%20tree
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In probability theory, the Brownian tree, or Aldous tree, or Continuum Random Tree (CRT) is a random real tree that can be defined from a Brownian excursion. The Brownian tree was defined and studied by David Aldous in three articles published in 1991 and 1993. This tree has since then been generalized.
This random tree has several equivalent definitions and constructions: using sub-trees generated by finitely many leaves, using a Brownian excursion, Poisson separating a straight line or as a limit of Galton-Watson trees.
Intuitively, the Brownian tree is a binary tree whose nodes (or branching points) are dense in the tree; which is to say that for any distinct two points of the tree, there will always exist a node between them. It is a fractal object which can be approximated with computers or by physical processes with dendritic structures.
Definitions
The following definitions are different characterisations of a Brownian tree, they are taken from Aldous's three articles. The notions of leaf, node, branch, root are the intuitive notions on a tree (for details, see real trees).
Finite-dimensional laws
This definition gives the finite-dimensional laws of the subtrees generated by finitely many leaves.
Let us consider the space of all binary trees with leaves numbered from to . These trees have edges with lengths . A tree is then defined by its shape (which is to say the order of the nodes) and the edge lengths. We define a probability law of a random variable on this space by:
where .
In other words, depends not on the shape of the tree but rather on the total sum of all the edge lengths.
In other words, the Brownian tree is defined from the laws of all the finite sub-trees one can generate from it.
Continuous tree
The Brownian tree is a real tree defined from a Brownian excursion (see characterisation 4 in Real tree).
Let be a Brownian excursion. Define a pseudometric on with
for any
We then define an equivalence relation, noted on which relates all points such that .
is then a distance on the quotient space .
It is customary to consider the excursion rather than .
Poisson line-breaking construction
This is also called stick-breaking construction.
Consider a non-homogeneous Poisson point process with intensity . In other words, for any , is a Poisson variable with parameter . Let be the points of . Then the lengths of the intervals are exponential variables with decreasing means. We then make the following construction:
(initialisation) The first step is to pick a random point uniformly on the interval . Then we glue the segment to (mathematically speaking, we define a new distance). We obtain a tree with a root (the point 0), two leaves ( and ), as well as one binary branching point (the point ).
(iteration) At step , the segment is similarly glued to the tree , on a uniformly random point of .
This algorithm may be used to simulate numerically Brownian trees.
Limit of Galton-Watson t
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https://en.wikipedia.org/wiki/Identity%20%28mathematics%29
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In mathematics, an identity is an equality relating one mathematical expression A to another mathematical expression B, such that A and B (which might contain some variables) produce the same value for all values of the variables within a certain range of validity. In other words, A = B is an identity if A and B define the same functions, and an identity is an equality between functions that are differently defined. For example, and are identities. Identities are sometimes indicated by the triple bar symbol instead of , the equals sign. Formally, an identity is a universally quantified equality.
Common identities
Algebraic identities
Certain identities, such as and , form the basis of algebra, while other identities, such as and , can be useful in simplifying algebraic expressions and expanding them.
Trigonometric identities
Geometrically, trigonometric identities are identities involving certain functions of one or more angles. They are distinct from triangle identities, which are identities involving both angles and side lengths of a triangle. Only the former are covered in this article.
These identities are useful whenever expressions involving trigonometric functions need to be simplified. Another important application is the integration of non-trigonometric functions: a common technique which involves first using the substitution rule with a trigonometric function, and then simplifying the resulting integral with a trigonometric identity.
One of the most prominent examples of trigonometric identities involves the equation which is true for all real values of . On the other hand, the equation
is only true for certain values of , not all. For example, this equation is true when but false when .
Another group of trigonometric identities concerns the so-called addition/subtraction formulas (e.g. the double-angle identity , the addition formula for ), which can be used to break down expressions of larger angles into those with smaller constituents.
Exponential identities
The following identities hold for all integer exponents, provided that the base is non-zero:
Unlike addition and multiplication, exponentiation is not commutative. For example, and , but whereas .
Also unlike addition and multiplication, exponentiation is not associative either. For example, and , but 23 to the 4 is 84 (or 4,096) whereas 2 to the 34 is 281 (or 2,417,851,639,229,258,349,412,352). When no parentheses are written, by convention the order is top-down, not bottom-up:
whereas
Logarithmic identities
Several important formulas, sometimes called logarithmic identities or log laws, relate logarithms to one another:
Product, quotient, power and root
The logarithm of a product is the sum of the logarithms of the numbers being multiplied; the logarithm of the ratio of two numbers is the difference of the logarithms. The logarithm of the th power of a number is times the logarithm of the number itself; the logarithm of a th root is the logar
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https://en.wikipedia.org/wiki/Affine%20group
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In mathematics, the affine group or general affine group of any affine space is the group of all invertible affine transformations from the space into itself. In the case of a Euclidean space (where the associated field of scalars is the real numbers), the affine group consists of those functions from the space to itself such that the image of every line is a line.
Over any field, the affine group may be viewed as a matrix group in a natural way. If the associated field of scalars the real or complex field, then the affine group is a Lie group.
Relation to general linear group
Construction from general linear group
Concretely, given a vector space , it has an underlying affine space obtained by "forgetting" the origin, with acting by translations, and the affine group of can be described concretely as the semidirect product of by , the general linear group of :
The action of on is the natural one (linear transformations are automorphisms), so this defines a semidirect product.
In terms of matrices, one writes:
where here the natural action of on is matrix multiplication of a vector.
Stabilizer of a point
Given the affine group of an affine space , the stabilizer of a point is isomorphic to the general linear group of the same dimension (so the stabilizer of a point in is isomorphic to ); formally, it is the general linear group of the vector space : recall that if one fixes a point, an affine space becomes a vector space.
All these subgroups are conjugate, where conjugation is given by translation from to (which is uniquely defined), however, no particular subgroup is a natural choice, since no point is special – this corresponds to the multiple choices of transverse subgroup, or splitting of the short exact sequence
In the case that the affine group was constructed by starting with a vector space, the subgroup that stabilizes the origin (of the vector space) is the original .
Matrix representation
Representing the affine group as a semidirect product of by , then by construction of the semidirect product, the elements are pairs , where is a vector in and is a linear transform in , and multiplication is given by
This can be represented as the block matrix
where is an matrix over , an column vector, 0 is a row of zeros, and 1 is the identity block matrix.
Formally, is naturally isomorphic to a subgroup of , with embedded as the affine plane , namely the stabilizer of this affine plane; the above matrix formulation is the (transpose of) the realization of this, with the and ) blocks corresponding to the direct sum decomposition .
A similar representation is any matrix in which the entries in each column sum to 1. The similarity for passing from the above kind to this kind is the identity matrix with the bottom row replaced by a row of all ones.
Each of these two classes of matrices is closed under matrix multiplication.
The simplest paradigm may well be the case , that is, the upper triangular matrice
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https://en.wikipedia.org/wiki/Affine%20space
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In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related to parallelism and ratio of lengths for parallel line segments. Affine space is the setting for affine geometry.
As in Euclidean space, the fundamental objects in an affine space are called points, which can be thought of as locations in the space without any size or shape: zero dimensional. Through any pair of points an infinite straight line can be drawn, a one-dimensional set of points; through any three points which are not collinear, a two-dimensional plane can be drawn; and in general through points in general position a -dimensional flat or affine subspace can be drawn. Affine space is characterized by a notion of pairs of parallel lines which lie within the same plane but never meet each-other (non-parallel lines within the same plane intersect in a point). Given any line, a line parallel to it can be drawn through any point in the space, and the equivalence class of parallel lines are said to share a direction.
Unlike for vectors in a vector space, in an affine space there is no distinguished point that serves as an origin. There is no predefined concept of adding or multiplying points together, or multiplying a point by a scalar number. However, for any affine space, an associated vector space can be constructed from the differences between start and end points, which are called displacement vectors, translation vectors or simply translations. Likewise, it makes sense to add a displacement vector to a point of an affine space, resulting in a new point translated from the starting point by that vector. While points cannot be arbitrarily added together, it is meaningful to take affine combinations of points: weighted sums with numerical coefficients summing to 1, resulting in another point. These coefficients define a barycentric coordinate system for the flat through the points.
Any vector space may be viewed as an affine space; this amounts to "forgetting" the special role played by the zero vector. In this case, elements of the vector space may be viewed either as points of the affine space or as displacement vectors or translations. When considered as a point, the zero vector is called the origin. Adding a fixed vector to the elements of a linear subspace (vector subspace) of a vector space produces an affine subspace of the vector space. One commonly says that this affine subspace has been obtained by translating (away from the origin) the linear subspace by the translation vector (the vector added to all the elements of the linear space). In finite dimensions, such an affine subspace is the solution set of an inhomogeneous linear system. The displacement vectors for that affine space are the solutions of the corresponding homogeneous linear system, which is a linear subspace. Linear subsp
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https://en.wikipedia.org/wiki/Affine%20combination
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In mathematics, an affine combination of is a linear combination
such that
Here, can be elements (vectors) of a vector space over a field , and the coefficients are elements of .
The elements can also be points of a Euclidean space, and, more generally, of an affine space over a field . In this case the are elements of (or for a Euclidean space), and the affine combination is also a point. See for the definition in this case.
This concept is fundamental in Euclidean geometry and affine geometry, because the set of all affine combinations of a set of points forms the smallest affine space containing the points, exactly as the linear combinations of a set of vectors form their linear span.
The affine combinations commute with any affine transformation in the sense that
In particular, any affine combination of the fixed points of a given affine transformation is also a fixed point of , so the set of fixed points of forms an affine space (in 3D: a line or a plane, and the trivial cases, a point or the whole space).
When a stochastic matrix, , acts on a column vector, , the result is a column vector whose entries are affine combinations of with coefficients from the rows in .
See also
Related combinations
Convex combination
Conical combination
Linear combination
Affine geometry
Affine space
Affine geometry
Affine hull
References
. See chapter 2.
External links
Notes on affine combinations.
Affine geometry
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https://en.wikipedia.org/wiki/List%20of%20things%20named%20after%20Carl%20Friedrich%20Gauss
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Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below.
There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymous adjective Gaussian is pronounced .
Mathematics
Algebra and linear algebra
Geometry and differential geometry
Number theory
Cyclotomic fields
Gaussian period
Gaussian rational
Gauss sum, an exponential sum over Dirichlet characters
Elliptic Gauss sum, an analog of a Gauss sum
Quadratic Gauss sum
Analysis, numerical analysis, vector calculus and calculus of variations
Complex analysis and convex analysis
Gauss–Lucas theorem
Gauss's continued fraction, an analytic continued fraction derived from the hypergeometric functions
Gauss's criterion – described on Encyclopedia of Mathematics
Gauss's hypergeometric theorem, an identity on hypergeometric series
Gauss plane
Statistics
Gauss–Kuzmin distribution, a discrete probability distribution
Gauss–Markov process
Gauss–Markov theorem
Gaussian copula
Gaussian measure
Gaussian correlation inequality
Gaussian isoperimetric inequality
Gauss's inequality
Gaussian function and topics named for it
Knot theory
Gauss code – described on website of University of Toronto
Gauss linking integral (knot theory)
Other mathematical areas
Gauss's algorithm for determination of the day of the week
Gauss's Easter algorithm
Gaussian brackets – described on WolframMathWorld
Gaussian's modular arithmetic
Gaussian integer, usually written as
Gaussian prime
Gaussian logarithms (also known as addition and subtraction logarithms)
Cartography
Gauss–Krüger coordinate system
Gaussian grid
Physics
Optics
Gauss lens
Double-Gauss lens
Gaussian optics
Classical mechanics
Gauss's principle of least constraint
For orbit determination in orbital mechanics:
Gauss's law for gravity
Gaussian gravitational constant
Gaussian year
Gauss's method
Quantum mechanics
Gaussian orbital
Electromagnetism
Gaussian units
gauss, the CGS unit for magnetic flux density
Degaussing, to demagnetize an object
Gauss rifle or coilgun
Gauss's law for magnetism
Gaussian surface
Gauss's law, giving the relationship between flux through a closed surface and the enclosed source
Awards and recognitions
Carl Friedrich Gauss Prize, a mathematics award
Gauss Lectureship, a mathematical distinction
The Gauss Mathematics Competition in Canadian junior high schools, an annual national mathematics competition administered by the Centre for Education in Mathematics and Computing
Other things named for him
Biology
Gaussia, a palm genus described by Hermann Wendland with the then new species Gaussia princeps, collected by Charles Wright in western Cuba. Named in "memoriam astronomi Caroli Friderici Gauss".
Gaussia, a genus of copepods
Informatics
Gaussian, a computational chemistry software program
GAUSS, a matrix programming language for mathematics and statistics
Place names and exped
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https://en.wikipedia.org/wiki/Double%20play
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In baseball and softball, a double play (denoted as DP in baseball statistics) is the act of making two outs during the same continuous play. Double plays can occur any time there is at least one baserunner and fewer than two outs.
In Major League Baseball (MLB), the double play is defined in the Official Rules in the Definitions of Terms, and for the official scorer in Rule 9.11. During the 2016 Major League Baseball season, teams completed an average 145 double plays per 162 games played during the regular season.
Examples
The simplest scenario for a double play is a runner on first base with less than two outs. In that context, five example double plays are:
The batter hits a ground ball
to an infielder or the pitcher, who throws the ball to one of the middle infielders, who steps on second base to force out the runner coming from first (first out), and then throws the ball to the first baseman in time to force out the batter (second out). As both outs are made by force plays, this is referred to as a "force double play". This is the most common double play. The neighborhood play is a source of controversy, as umpires sometimes call the runner at second base out despite the infielder not clearly touching that base, but merely being "in the neighborhood".
to the first baseman, who steps on first base to force out the batter (first out), and with the baserunner trying to advance from first base to second base, throws the ball to the shortstop who puts out the runner (second out). This is referred to as a "reverse force double play", although executing the first out removes the condition that forced the runner to take second base. The second out is not a force play and must be made with a tag.
The batter hits the ball in the air
a line drive to the first baseman, who catches it (first out), and then steps on first base before the baserunner can return to first to tag up (second out). This is also an example of an unassisted double play.
a deep fly ball to the right fielder, who catches it (first out), meanwhile the baserunner tags up and attempts to advance, and the outfielder throws the ball to the shortstop who tags the runner before he reaches second base (second out).
The batter strikes out (first out)
Meanwhile, the runner attempts to steal second base, and the catcher throws the ball to a middle infielder, who tags the runner before he reaches the base (second out). This is colloquially known as a "strike 'em out, throw 'em out" double play.
Double plays can occur in many ways in addition to these examples, and can involve many combinations of fielders. A double play can include an out resulting from a rare event, such as interference or an appeal play.
Recordkeeping
Per standard baseball positions, the examples given above are recorded, respectively, as:
4-6-3 (second baseman to shortstop to first baseman) or 6-4-3 (shortstop to second baseman to first baseman). Other combinations start with 1 (pitcher), 3 (first baseman), or
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https://en.wikipedia.org/wiki/Triple%20play
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In baseball, a triple play (denoted as TP in baseball statistics) is the act of making three outs during the same play. There have only been 735 triple plays in Major League Baseball (MLB) since 1876, an average of just over five per season.
They depend on a combination of two factors, which are themselves uncommon:
First, there must be at least two baserunners, and no outs. From analysis of all MLB games from 2011 to 2013, only 1.51% of at bats occur in such a scenario. By comparison, 27.06% of at bats occur with at least one baserunner and fewer than two outs, the scenario where a double play is possible.
Second, activity must occur during the play that enables the defense to make three outs. Common plays, such as the batter striking out or hitting a fly ball, do not normally provide an opportunity for a triple play. A ball hit sharply and directly to an infielder, who then takes very quick or unanticipated action, as well as confusion or mistakes by the baserunners is usually needed.
In baseball scorekeeping, the abbreviation GITP can be used if the batter grounded into a triple play.
Examples
The most likely scenario for a triple play is no outs with runners on first base and second base, which has been the case for the majority of MLB triple plays. In that context, two examples of triple plays are:
The batter hits a ground ball to the third baseman, who steps on third base to force out the runner coming from second (first out). The third baseman throws to the second baseman, who steps on second base to force out the runner coming from first (second out). The second baseman throws to the first baseman, with the throw arriving in time to force out the batter (third out). This is an example of grounding into a 5-4-3 triple play, also known as an "around the horn" triple play, per standard baseball positions.
During the 1973 season, Baltimore Orioles third baseman Brooks Robinson started two such 5-4-3 triple plays: one on July 7 against the Oakland Athletics, and one on September 20 against the Detroit Tigers. As a hitter, he is the all-time MLB leader for grounding into triple plays, with four in his career.
On July 17, 1990, the Minnesota Twins became the first (and to date, the only) team in MLB history to turn two triple plays in the same game. Both were 5-4-3 triple plays, executed by fielders Gary Gaetti, Al Newman, and Kent Hrbek in a game against the Boston Red Sox.
The baserunners start running in an attempt to steal or execute a hit and run play, and the batter hits a line drive to the second baseman, who catches it (first out). The second baseman throws to the shortstop, who steps on second base before the runner who started there can tag up (second out). The shortstop throws to the first baseman, who steps on first base before the runner who started there can tag up (third out). This is an example of lining out into a 4-6-3 triple play.
Most recent MLB triple play
The most recent triple play in MLB was turned by the Los A
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https://en.wikipedia.org/wiki/Error%20%28baseball%29
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In baseball and softball statistics, an error is an act, in the judgment of the official scorer, of a fielder misplaying a ball in a manner that allows a batter or baserunner to advance one or more bases or allows a plate appearance to continue after the batter should have been put out. The term error is sometimes used to refer to the play during which an error was committed.
Relationship to other statistical categories
An error does not count as a hit but still counts as an at bat for the batter unless, in the scorer's judgment, the batter would have reached first base safely but one or more of the additional bases reached was the result of the fielder's mistake. In that case, the play will be scored both as a hit (for the number of bases the fielders should have limited the batter to) and an error. However, if a batter is judged to have reached base solely because of a fielder's mistake, it is scored as a "reach on error (ROE)," and treated the same as if the batter had been put out, hence lowering his batting average.
Similarly, a batter does not receive credit for a run batted in (RBI) when runs score on an error, unless the scorer rules that a run would have scored even if the fielder had not made a mistake. For example, if a batter hits a ball to the outfield for what should be a sacrifice fly and the outfielder drops the ball for an error, the batter will still receive credit for the sacrifice fly and the run batted in.
If a play should have resulted in a fielder's choice with a runner being put out and the batter reaching base safely but the runner is safe due to an error, the play will be scored as a fielder's choice, with no hit being awarded to the batter and an error charged against the fielder.
Passed balls and wild pitches are separate statistical categories and are not scored as errors.
If a batted ball were hit on the fly into foul territory, with the batting team having no runners on base, and a fielder misplayed such ball for an error, it is possible for a team on the winning side of a perfect game to commit at least one error, yet still qualify as a perfect game.
There is a curious loophole in the rules on errors for catchers. If a catcher makes a "wild throw" in an attempt to prevent a stolen base and the runner is safe, the catcher is not charged with an error even if it could be argued that the runner would have been put out with "ordinary effort." There is therefore a "no fault" condition for the catcher attempting to prevent a steal. However, when considering that the majority of stolen base attempts are successful (around 2 successes per failure), this "no fault rule" is understandable due to the difficulty of throwing out runners. If the runner takes an additional base due to the wild throw, an error is charged for that advance. The other scenario where catchers may be given an error unrelated to fielding a ball in play is catcher’s interference, when the catcher's glove is hit by the bat during the swing. The ca
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https://en.wikipedia.org/wiki/Dy
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DY, D. Y., Dy, or dy may refer to:
In science and technology, and mathematics
Astronomy
DY Persei, a variable star in the Perseus constellation
DY Persei variable, a subclass of R Coronae Borealis variables
DY Eridani, a triple star system less than 16.5 light years away from Earth
Other sciences
, in calculus, Leibniz's notation for the differential of a variable y
Dysprosium, symbol Dy, a chemical element
1,4-Dioxane, a common solvent
Businesses
Norwegian Air Shuttle (IATA code DY)
Alyemda, a Yemeni airline (IATA code DY, until 1993)
DY, clothing brand of singer Daddy Yankee
People
Dy (surname), a surname in various cultures (including a list of people with the surname)
DY (rapper) (born 1984), Canadian rapper
DY, member of the American record production and songwriting team 808 Mafia
Lady Di, Diana, Princess of Wales
Places
DY, the official International vehicle registration code for Benin (formerly Dahomey)
DY postcode area in Britain
DY Patil Stadium, a cricket stadium in India
D. Y. Patil college of Engineering and Technology, Kolhapur in India
Dee Why, a suburb in Sydney, Australia
Other uses
dy (digraph), a digraph used in rendering the Xhosa and Shona languages, as well as some Australian Aboriginal languages such as Warlpiri
Deputy (disambiguation)
DY, a line of clothing from Puerto Rican singer Daddy Yankee
See also
Die (disambiguation)
Dye (disambiguation)
YD (disambiguation)
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https://en.wikipedia.org/wiki/James%20H.%20Wilkinson
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James Hardy Wilkinson FRS (27 September 1919 – 5 October 1986) was a prominent figure in the field of numerical analysis, a field at the boundary of applied mathematics and computer science particularly useful to physics and engineering.
Education
Born in Strood, England, he won a Foundation Scholarship to Sir Joseph Williamson's Mathematical School in Rochester. He studied the Cambridge Mathematical Tripos at Trinity College, Cambridge, where he graduated as Senior Wrangler.
Career
Taking up war work in 1940, he began working on ballistics but transferred to the National Physical Laboratory in 1946, where he worked with Alan Turing on the ACE computer project. Later, Wilkinson's interests took him into the numerical analysis field, where he discovered many significant algorithms.
Awards and honours
Wilkinson received the Turing Award in 1970 "for his research in numerical analysis to facilitate the use of the high-speed digital computer, having received special recognition for his work in computations in linear algebra and 'backward' error analysis." In the same year, he also gave the Society for Industrial and Applied Mathematics (SIAM) John von Neumann Lecture.
Wilkinson also received an Honorary Doctorate from Heriot-Watt University in 1973.
He was elected as a Distinguished Fellow of the British Computer Society in 1974 for his pioneering work in computer science.
The James H. Wilkinson Prize in Numerical Analysis and Scientific Computing, established in 1982 by SIAM, and J. H. Wilkinson Prize for Numerical Software, established in 1991, are named in his honour.
In 1987, Wilkinson won the Chauvenet Prize of the Mathematical Association of America, for his paper "The Perfidious Polynomial".
Personal life
Wilkinson married Heather Ware in 1945. He died at home of a heart attack on October 5, 1986. His wife and their son survived him, a daughter having predeceased him.
Selected works
(REAP)
Reprinted from SIAM in 2023, ISBN 978-1-61197-751-6.
(AEP)
with Christian Reinsch: Handbook for Automatic Computation, Volume II, Linear Algebra, Springer-Verlag, 1971
The Perfidious Polynomial. In: Studies in Numerical Analysis, pp. 1–28, MAA Stud. Math., 24, Math. Assoc. America, Washington, DC, 1984
References
External links
1919 births
1986 deaths
20th-century British mathematicians
British computer scientists
Turing Award laureates
Alumni of Trinity College, Cambridge
Fellows of the British Computer Society
Fellows of the Royal Society
People from Strood
People educated at Sir Joseph Williamson's Mathematical School
Senior Wranglers
Numerical analysts
Scientists of the National Physical Laboratory (United Kingdom)
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https://en.wikipedia.org/wiki/Functional%20derivative
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In the calculus of variations, a field of mathematical analysis, the functional derivative (or variational derivative) relates a change in a functional (a functional in this sense is a function that acts on functions) to a change in a function on which the functional depends.
In the calculus of variations, functionals are usually expressed in terms of an integral of functions, their arguments, and their derivatives. In an integrand of a functional, if a function is varied by adding to it another function that is arbitrarily small, and the resulting integrand is expanded in powers of , the coefficient of in the first order term is called the functional derivative.
For example, consider the functional
where . If is varied by adding to it a function , and the resulting integrand is expanded in powers of , then the change in the value of to first order in can be expressed as follows:
where the variation in the derivative, was rewritten as the derivative of the variation , and integration by parts was used in these derivatives.
Definition
In this section, the functional differential (or variation or first variation) Called first variation in , variation or first variation in , variation or differential in and differential in . is defined. Then the functional derivative is defined in terms of the functional differential.
Functional differential
Suppose is a Banach space and is a functional defined on .
The differential of at a point is the linear functional on defined by the condition that, for all ,
where is a real number that depends on in such a way that as . This means that is the Fréchet derivative of at .
However, this notion of functional differential is so strong it may not exist, and in those cases a weaker notion, like the Gateaux derivative is preferred. In many practical cases, the functional differential is defined as the directional derivative
Note that this notion of the functional differential can even be defined without a norm.
Functional derivative
In many applications, the domain of the functional is a space of differentiable functions defined on some space and is of the form
for some function that may depend on , the value and the derivative .
If this is the case and, moreover, can be written as the integral of times another function (denoted )
then this function is called the functional derivative of at . If is restricted to only certain functions (for example, if there are some boundary conditions imposed) then is restricted to functions such that continues to satisfy these conditions.
Heuristically, is the change in , so we 'formally' have , and then this is similar in form to the total differential of a function ,
where are independent variables.
Comparing the last two equations, the functional derivative has a role similar to that of the partial derivative , where the variable of integration is like a continuous version of the summation index . One thinks of as the gradi
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https://en.wikipedia.org/wiki/Functional%20integration
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Functional integration is a collection of results in mathematics and physics where the domain of an integral is no longer a region of space, but a space of functions. Functional integrals arise in probability, in the study of partial differential equations, and in the path integral approach to the quantum mechanics of particles and fields.
In an ordinary integral (in the sense of Lebesgue integration) there is a function to be integrated (the integrand) and a region of space over which to integrate the function (the domain of integration). The process of integration consists of adding up the values of the integrand for each point of the domain of integration. Making this procedure rigorous requires a limiting procedure, where the domain of integration is divided into smaller and smaller regions. For each small region, the value of the integrand cannot vary much, so it may be replaced by a single value. In a functional integral the domain of integration is a space of functions. For each function, the integrand returns a value to add up. Making this procedure rigorous poses challenges that continue to be topics of current research.
Functional integration was developed by Percy John Daniell in an article of 1919 and Norbert Wiener in a series of studies culminating in his articles of 1921 on Brownian motion. They developed a rigorous method (now known as the Wiener measure) for assigning a probability to a particle's random path. Richard Feynman developed another functional integral, the path integral, useful for computing the quantum properties of systems. In Feynman's path integral, the classical notion of a unique trajectory for a particle is replaced by an infinite sum of classical paths, each weighted differently according to its classical properties.
Functional integration is central to quantization techniques in theoretical physics. The algebraic properties of functional integrals are used to develop series used to calculate properties in quantum electrodynamics and the standard model of particle physics.
Functional integration
Whereas standard Riemann integration sums a function f(x) over a continuous range of values of x, functional integration sums a functional G[f], which can be thought of as a "function of a function" over a continuous range (or space) of functions f. Most functional integrals cannot be evaluated exactly but must be evaluated using perturbation methods. The formal definition of a functional integral is
However, in most cases the functions f(x) can be written in terms of an infinite series of orthogonal functions such as , and then the definition becomes
which is slightly more understandable. The integral is shown to be a functional integral with a capital . Sometimes the argument is written in square brackets , to indicate the functional dependence of the function in the functional integration measure.
Examples
Most functional integrals are actually infinite, but often the limit of the quotient of two rel
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https://en.wikipedia.org/wiki/Anti-de%20Sitter%20space
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In mathematics and physics, n-dimensional anti-de Sitter space (AdSn) is a maximally symmetric Lorentzian manifold with constant negative scalar curvature. Anti-de Sitter space and de Sitter space are named after Willem de Sitter (1872–1934), professor of astronomy at Leiden University and director of the Leiden Observatory. Willem de Sitter and Albert Einstein worked together closely in Leiden in the 1920s on the spacetime structure of the universe.
Manifolds of constant curvature are most familiar in the case of two dimensions, where the elliptic plane or surface of a sphere is a surface of constant positive curvature, a flat (i.e., Euclidean) plane is a surface of constant zero curvature, and a hyperbolic plane is a surface of constant negative curvature.
Einstein's general theory of relativity places space and time on equal footing, so that one considers the geometry of a unified spacetime instead of considering space and time separately. The cases of spacetime of constant curvature are de Sitter space (positive), Minkowski space (zero), and anti-de Sitter space (negative). As such, they are exact solutions of the Einstein field equations for an empty universe with a positive, zero, or negative cosmological constant, respectively.
Anti-de Sitter space generalises to any number of space dimensions. In higher dimensions, it is best known for its role in the AdS/CFT correspondence, which suggests that it is possible to describe a force in quantum mechanics (like electromagnetism, the weak force or the strong force) in a certain number of dimensions (for example four) with a string theory where the strings exist in an anti-de Sitter space, with one additional (non-compact) dimension.
Non-technical explanation
Technical terms translated
A maximally symmetric Lorentzian manifold is a spacetime in which no point in space and time can be distinguished in any way from another, and (being Lorentzian) the only way in which a direction (or tangent to a path at a spacetime point) can be distinguished is whether it is spacelike, lightlike or timelike. The space of special relativity (Minkowski space) is an example.
A constant scalar curvature means a general relativity gravity-like bending of spacetime that has a curvature described by a single number that is the same everywhere in spacetime in the absence of matter or energy.
Negative curvature means curved hyperbolically, like a saddle surface or the Gabriel's Horn surface, similar to that of a trumpet bell. It might be described as being the "opposite" of the surface of a sphere, which has a positive curvature.
Spacetime in general relativity
General relativity is a theory of the nature of time, space and gravity in which gravity is a curvature of space and time that results from the presence of matter or energy. Energy and mass are equivalent (as expressed in the equation E = mc2). Space and time values can be converted into time or space units by multiplying or dividing the value by the spee
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https://en.wikipedia.org/wiki/Adjoint%20representation
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In mathematics, the adjoint representation (or adjoint action) of a Lie group G is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space. For example, if G is , the Lie group of real n-by-n invertible matrices, then the adjoint representation is the group homomorphism that sends an invertible n-by-n matrix to an endomorphism of the vector space of all linear transformations of defined by: .
For any Lie group, this natural representation is obtained by linearizing (i.e. taking the differential of) the action of G on itself by conjugation. The adjoint representation can be defined for linear algebraic groups over arbitrary fields.
Definition
Let G be a Lie group, and let
be the mapping ,
with Aut(G) the automorphism group of G and given by the inner automorphism (conjugation)
This Ψ is a Lie group homomorphism.
For each g in G, define to be the derivative of at the origin:
where is the differential and is the tangent space at the origin ( being the identity element of the group ). Since is a Lie group automorphism, Adg is a Lie algebra automorphism; i.e., an invertible linear transformation of to itself that preserves the Lie bracket. Moreover, since is a group homomorphism, too is a group homomorphism. Hence, the map
is a group representation called the adjoint representation of G.
If G is an immersed Lie subgroup of the general linear group (called immersely linear Lie group), then the Lie algebra consists of matrices and the exponential map is the matrix exponential for matrices X with small operator norms. We will compute the derivative of at . For g in G and small X in , the curve has derivative at t = 0, one then gets:
where on the right we have the products of matrices. If is a closed subgroup (that is, G is a matrix Lie group), then this formula is valid for all g in G and all X in .
Succinctly, an adjoint representation is an isotropy representation associated to the conjugation action of G around the identity element of G.
Derivative of Ad
One may always pass from a representation of a Lie group G to a representation of its Lie algebra by taking the derivative at the identity.
Taking the derivative of the adjoint map
at the identity element gives the adjoint representation of the Lie algebra of G:
where is the Lie algebra of which may be identified with the derivation algebra of . One can show that
for all , where the right hand side is given (induced) by the Lie bracket of vector fields. Indeed, recall that, viewing as the Lie algebra of left-invariant vector fields on G, the bracket on is given as: for left-invariant vector fields X, Y,
where denotes the flow generated by X. As it turns out, , roughly because both sides satisfy the same ODE defining the flow. That is, where denotes the right multiplication by . On the other hand, since , by chain rule,
as Y is left-invariant. Hence,
,
which is what was needed t
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https://en.wikipedia.org/wiki/Self-adjoint
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In mathematics, and more specifically in abstract algebra, an element x of a *-algebra is self-adjoint if . A self-adjoint element is also Hermitian, though the reverse doesn't necessarily hold.
A collection C of elements of a star-algebra is self-adjoint if it is closed under the involution operation. For example, if then since in a star-algebra, the set {x,y} is a self-adjoint set even though x and y need not be self-adjoint elements.
In functional analysis, a linear operator on a Hilbert space is called self-adjoint if it is equal to its own adjoint A. See self-adjoint operator for a detailed discussion. If the Hilbert space is finite-dimensional and an orthonormal basis has been chosen, then the operator A is self-adjoint if and only if the matrix describing A with respect to this basis is Hermitian, i.e. if it is equal to its own conjugate transpose. Hermitian matrices are also called self-adjoint.
In a dagger category, a morphism is called self-adjoint if ; this is possible only for an endomorphism .
See also
Hermitian matrix
Normal element
Symmetric matrix
Self-adjoint operator
Unitary element
References
Abstract algebra
Linear algebra
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https://en.wikipedia.org/wiki/Lies%2C%20damned%20lies%2C%20and%20statistics
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"Lies, damned lies, and statistics" is a phrase describing the persuasive power of statistics to bolster weak arguments, "one of the best, and best-known" critiques of applied statistics. It is also sometimes colloquially used to doubt statistics used to prove an opponent's point.
The phrase was popularized in the United States by Mark Twain (among others), who attributed it to the British prime minister Benjamin Disraeli. However, the phrase is not found in any of Disraeli's works and the earliest known appearances were years after his death. Several other people have been listed as originators of the quote, and it is often attributed to Twain himself.
History
Mark Twain popularized the saying in Chapters from My Autobiography, published in the North American Review in 1907. "Figures often beguile me," Twain wrote, "particularly when I have the arranging of them myself; in which case the remark attributed to Disraeli would often apply with justice and force: 'There are three kinds of lies: lies, damned lies, and statistics.'"
Colin White traces the origins to François Magendie (1783-1855). While arguing against using blood-letting to treat fever, and confronted with statistical numbers he believed to be manufactured, this French physiologist stated: "Thus the alteration of the truth which is already manifesting itself in the progressive form of lying and perjury, offers us, in the superlative, the statistics." In White's opinion, the world had a need of this phrase, many people "would have been proud" to coin it, and the origins are now obscured, as the phrase passed "from wit to wit".
Alternative attributions include, among many others (for example Walter Bagehot and Arthur James Balfour), the radical English journalist and politician Henry Du Pré Labouchère (1831–1912), Jervoise Athelstane Baines, and British politician and scholar Leonard H. Courtney, who used the phrase in 1895 and two years later became president of the Royal Statistical Society. Courtney is quoted by as attributing the phrase to a "wise statesman", but he may have been referring to a future statesman rather than a past one. The phrase has also been attributed to Arthur Wellesley, 1st Duke of Wellington.
The phrase is quoted frequently in 1895, but here is a 1894 example: "His less enthusiastic neighbor thinks of the proverbial kinds of falsehoods, “lies, damned lies, and statistics,” and replies: “Reports of large numbers of cases subjected to operation seldom fail to beget a suspicion of unjustifiable risk.”"
A Dictionary of English Folklore claims that the earliest instance resembling the phrase found in print is a letter written in the British newspaper National Observer on June 8, 1891, published June 13, 1891, p. 93(-94):
NATIONAL PENSIONS
[To the Editor of The National Observer]
London, 8 June 1891
"Sir, —It has been wittily remarked that there are three kinds of
falsehood: the first is a 'fib,' the second is a downright lie, and the third and most
aggravate
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https://en.wikipedia.org/wiki/Rhomboid
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Traditionally, in two-dimensional geometry, a rhomboid is a parallelogram in which adjacent sides are of unequal lengths and angles are non-right angled. The terms rhomboid and parallelogram are often erroneously conflated with each other (i.e, when most people refer to a "parallelogram" they almost always mean a rhomboid, a specific subtype of parallelogram), however while all rhomboids are parallelograms, not all parallelograms are rhomboids.
A parallelogram with sides of equal length (equilateral) is a rhombus but not a rhomboid.
A parallelogram with right angled corners is a rectangle but not a rhomboid.
The term rhomboid is now more often used for a rhombohedron or a more general parallelepiped, a solid figure with six faces in which each face is a parallelogram and pairs of opposite faces lie in parallel planes. Some crystals are formed in three-dimensional rhomboids. This solid is also sometimes called a rhombic prism. The term occurs frequently in science terminology referring to both its two- and three-dimensional meaning.
History
Euclid introduced the term in his Elements in Book I, Definition 22,
Euclid never used the definition of rhomboid again and introduced the word parallelogram in Proposition 34 of Book I; "In parallelogrammic areas the opposite sides and angles are equal to one another, and the diameter bisects the areas." Heath suggests that rhomboid was an older term already in use.
Symmetries
The rhomboid has no line of symmetry, but it has rotational symmetry of order 2.
In biology
In biology, rhomboid may describe a geometric rhomboid (e.g. the rhomboid muscles) or a bilaterally-symmetrical kite-shaped or diamond-shaped outline, as in leaves or cephalopod fins.
In medicine
In a type of arthritis called pseudogout, crystals of calcium pyrophosphate dihydrate accumulate in the joint, causing inflammation. Aspiration of the joint fluid reveals rhomboid-shaped crystals under a microscope.
References
External links
Types of quadrilaterals
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https://en.wikipedia.org/wiki/Wronskian
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In the mathematics of a square matrix, the Wronskian (or Wrońskian) is a determinant introduced by the Polish mathematician . It is used in the study of differential equations, where it can sometimes show linear independence of a set of solutions.
Definition
The Wronskian of two differentiable functions and is .
More generally, for real- or complex-valued functions , which are times differentiable on an interval , the Wronskian is a function on defined by
This is the determinant of the matrix constructed by placing the functions in the first row, the first derivatives of the functions in the second row, and so on through the derivative, thus forming a square matrix.
When the functions are solutions of a linear differential equation, the Wronskian can be found explicitly using Abel's identity, even if the functions are not known explicitly. (See below.)
The Wronskian and linear independence
If the functions are linearly dependent, then so are the columns of the Wronskian (since differentiation is a linear operation), and the Wronskian vanishes. Thus, one may show that a set of differentiable functions is linearly independent on an interval by showing that their Wronskian does not vanish identically. It may, however, vanish at isolated points.
A common misconception is that everywhere implies linear dependence, but pointed out that the functions and have continuous derivatives and their Wronskian vanishes everywhere, yet they are not linearly dependent in any neighborhood of . There are several extra conditions which combine with vanishing of the Wronskian in an interval to imply linear dependence.
Maxime Bôcher observed that if the functions are analytic, then the vanishing of the Wronskian in an interval implies that they are linearly dependent.
gave several other conditions for the vanishing of the Wronskian to imply linear dependence; for example, if the Wronskian of functions is identically zero and the Wronskians of of them do not all vanish at any point then the functions are linearly dependent.
gave a more general condition that together with the vanishing of the Wronskian implies linear dependence.
Over fields of positive characteristic the Wronskian may vanish even for linearly independent polynomials; for example, the Wronskian of and 1 is identically 0.
Application to linear differential equations
In general, for an th order linear differential equation, if solutions are known, the last one can be determined by using the Wronskian.
Consider the second order differential equation in Lagrange's notation:
where , are known, and y is the unknown function to be found. Let us call the two solutions of the equation and form their Wronskian
Then differentiating and using the fact that obey the above differential equation shows that
Therefore, the Wronskian obeys a simple first order differential equation and can be exactly solved:
where and is a constant.
Now suppose that we know one of the
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https://en.wikipedia.org/wiki/Crispin%20Wright
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Crispin James Garth Wright (; born 21 December 1942) is a British philosopher, who has written on neo-Fregean (neo-logicist) philosophy of mathematics, Wittgenstein's later philosophy, and on issues related to truth, realism, cognitivism, skepticism, knowledge, and objectivity. He is Professor of Philosophical Research at the University of Stirling, and taught previously at the University of St Andrews, University of Aberdeen, New York University, Princeton University and University of Michigan.
Life and career
Wright was born in Surrey and was educated at Birkenhead School (1950–61) and at Trinity College, Cambridge, graduating in Moral Sciences in 1964 and taking a PhD in 1968. He took an Oxford BPhil in 1969 and was elected Prize Fellow and then Research Fellow at All Souls College, Oxford, where he worked until 1978. He then moved to the University of St. Andrews, where he was appointed Professor of Logic and Metaphysics and then the first Bishop Wardlaw University Professorship in 1997. From fall 2008 to spring 2023, he was professor in the Department of Philosophy at New York University (NYU). He has also taught at the University of Michigan, Oxford University, Columbia University, and Princeton University. Crispin Wright was founder and director of Arché at the University of St. Andrews, which he left in September 2009 to take up leadership of the Northern Institute of Philosophy (NIP) at the University of Aberdeen. Crispin Wright's nephew, Stephen Wright, is also a philosopher at Oxford. Once NIP ceased operations in 2015, Wright moved to the University of Stirling.
Philosophical work
In the philosophy of mathematics, he is best known for his book Frege's Conception of Numbers as Objects (1983), where he argues that Frege's logicist project could be revived by removing the axiom schema of unrestricted comprehension (sometimes referred to as Basic Law V) from the formal system. Arithmetic is then derivable in second-order logic from Hume's principle. He gives informal arguments that (i) Hume's principle plus second-order logic is consistent, and (ii) from it one can produce the Dedekind–Peano axioms. Both results were proven informally by Gottlob Frege (Frege's Theorem), and would later be more rigorously proven by George Boolos and Richard Heck. Wright is one of the major proponents of neo-logicism, alongside his frequent collaborator Bob Hale. He has also written Wittgenstein and the Foundations of Mathematics (1980).
In general metaphysics, his most important work is Truth and Objectivity (Harvard University Press, 1992). He argues in this book that there need be no single, discourse-invariant thing in which truth consists, making an analogy with identity. There need only be some principles regarding how the truth predicate can be applied to a sentence, some 'platitudes' about true sentences. Wright also argues that in some contexts, probably including moral contexts, superassertibility will effectively function as a truth predic
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https://en.wikipedia.org/wiki/Gaussian%20process
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In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution, i.e. every finite linear combination of them is normally distributed. The distribution of a Gaussian process is the joint distribution of all those (infinitely many) random variables, and as such, it is a distribution over functions with a continuous domain, e.g. time or space.
The concept of Gaussian processes is named after Carl Friedrich Gauss because it is based on the notion of the Gaussian distribution (normal distribution). Gaussian processes can be seen as an infinite-dimensional generalization of multivariate normal distributions.
Gaussian processes are useful in statistical modelling, benefiting from properties inherited from the normal distribution. For example, if a random process is modelled as a Gaussian process, the distributions of various derived quantities can be obtained explicitly. Such quantities include the average value of the process over a range of times and the error in estimating the average using sample values at a small set of times. While exact models often scale poorly as the amount of data increases, multiple approximation methods have been developed which often retain good accuracy while drastically reducing computation time.
Definition
A time continuous stochastic process is Gaussian if and only if for every finite set of indices in the index set
is a multivariate Gaussian random variable. That is the same as saying every linear combination of has a univariate normal (or Gaussian) distribution.
Using characteristic functions of random variables, the Gaussian property can be formulated as follows: is Gaussian if and only if, for every finite set of indices , there are real-valued , with such that the following equality holds for all
where denotes the imaginary unit such that .
The numbers and can be shown to be the covariances and means of the variables in the process.
Variance
The variance of a Gaussian process is finite at any time , formally
Stationarity
For general stochastic processes strict-sense stationarity implies wide-sense stationarity but not every wide-sense stationary stochastic process is strict-sense stationary. However, for a Gaussian stochastic process the two concepts are equivalent.
A Gaussian stochastic process is strict-sense stationary if, and only if, it is wide-sense stationary.
Example
There is an explicit representation for stationary Gaussian processes. A simple example of this representation is
where and are independent random variables with the standard normal distribution.
Covariance functions
A key fact of Gaussian processes is that they can be completely defined by their second-order statistics. Thus, if a Gaussian process is assumed to have mean zero, defining the covariance function completely defines the process'
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https://en.wikipedia.org/wiki/Shing-Tung%20Yau
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Shing-Tung Yau (; ; born April 4, 1949) is a Chinese-American mathematician and the William Caspar Graustein Professor of Mathematics at Harvard University. In April 2022, Yau retired from Harvard to become a professor of mathematics at Tsinghua University.
Yau was born in Shantou, China, moved to Hong Kong at a young age, and to the United States in 1969. He was awarded the Fields Medal in 1982, in recognition of his contributions to partial differential equations, the Calabi conjecture, the positive energy theorem, and the Monge–Ampère equation. Yau is considered one of the major contributors to the development of modern differential geometry and geometric analysis.
The impact of Yau's work can be seen in the mathematical and physical fields of differential geometry, partial differential equations, convex geometry, algebraic geometry, enumerative geometry, mirror symmetry, general relativity, and string theory, while his work has also touched upon applied mathematics, engineering, and numerical analysis.
Biography
Yau was born in Shantou, China in 1949 to Hakka parents. Yau's ancestral hometown is Jiaoling county, China. His mother, Yeuk Lam Leung, was from Meixian District; his father, Chen Ying Chiu, was a Chinese scholar of philosophy, history, literature, and economics. He was the fifth of eight children, with Hakka ancestry.
During the Communist takeover of mainland China, when he was only a few months old, his family moved to Hong Kong where he was forced to learn to speak the Cantonese language as well as speak the Chinese dialect of Hakka. He was not able to revisit until 1979, at the invitation of Hua Luogeng, when mainland China entered the reform and opening era.. They had financial troubles from having lost all of their possessions, and his father and second-oldest sister died when he was thirteen. Yau began to read and appreciate his father's books, and became more devoted to schoolwork. After graduating from Pui Ching Middle School, he studied mathematics at the Chinese University of Hong Kong from 1966 to 1969, without receiving a degree due to graduating early. He left his textbooks with his younger brother, Stephen Shing-Toung Yau, who then decided to major in mathematics as well.
Yau left for the Ph.D. program in mathematics at University of California, Berkeley in the fall of 1969. Over the winter break, he read the first issues of the Journal of Differential Geometry, and was deeply inspired by John Milnor's papers on geometric group theory. Subsequently he formulated a generalization of Preissman's theorem, and developed his ideas further with Blaine Lawson over the next semester. Using this work, he received his Ph.D. the following year, in 1971, under the supervision of Shiing-Shen Chern.
He spent a year as a member of the Institute for Advanced Study at Princeton before joining Stony Brook University in 1972 as an assistant professor. In 1974, he became an associate professor at Stanford University. In 1976 he took
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https://en.wikipedia.org/wiki/Gudermannian%20function
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In mathematics, the Gudermannian function relates a hyperbolic angle measure to a circular angle measure called the gudermannian of and denoted . The Gudermannian function reveals a close relationship between the circular functions and hyperbolic functions. It was introduced in the 1760s by Johann Heinrich Lambert, and later named for Christoph Gudermann who also described the relationship between circular and hyperbolic functions in 1830. The gudermannian is sometimes called the hyperbolic amplitude as a limiting case of the Jacobi elliptic amplitude when parameter
The real Gudermannian function is typically defined for to be the integral of the hyperbolic secant
The real inverse Gudermannian function can be defined for as the integral of the secant
The hyperbolic angle measure is called the anti-gudermannian of or sometimes the lambertian of , denoted In the context of geodesy and navigation for latitude , (scaled by arbitrary constant ) was historically called the meridional part of (French: latitude croissante). It is the vertical coordinate of the Mercator projection.
The two angle measures and are related by a common stereographic projection
and this identity can serve as an alternative definition for and valid throughout the complex plane:
Circular–hyperbolic identities
We can evaluate the integral of the hyperbolic secant using the stereographic projection (hyperbolic half-tangent) as a change of variables:
Letting and we can derive a number of identities between hyperbolic functions of and circular functions of
These are commonly used as expressions for and for real values of and with For example, the numerically well-behaved formulas
(Note, for and for complex arguments, care must be taken choosing branches of the inverse functions.)
We can also express and in terms of
If we expand and in terms of the exponential, then we can see that and are all Möbius transformations of each-other (specifically, rotations of the Riemann sphere):
For real values of and with , these Möbius transformations can be written in terms of trigonometric functions in several ways,
These give further expressions for and for real arguments with For example,
Complex values
As a functions of a complex variable, conformally maps the infinite strip to the infinite strip while conformally maps the infinite strip to the infinite strip
Analytically continued by reflections to the whole complex plane, is a periodic function of period which sends any infinite strip of "height" onto the strip Likewise, extended to the whole complex plane, is a periodic function of period which sends any infinite strip of "width" onto the strip For all points in the complex plane, these functions can be correctly written as:
For the and functions to remain invertible with these extended domains, we might consider each to be a multivalued function (perhaps and , with and the principal branch) or consider their dom
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https://en.wikipedia.org/wiki/Totally%20disconnected%20space
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In topology and related branches of mathematics, a totally disconnected space is a topological space that has only singletons as connected subsets. In every topological space, the singletons (and, when it is considered connected, the empty set) are connected; in a totally disconnected space, these are the only connected subsets.
An important example of a totally disconnected space is the Cantor set, which is homeomorphic to the set of p-adic integers. Another example, playing a key role in algebraic number theory, is the field of p-adic numbers.
Definition
A topological space is totally disconnected if the connected components in are the one-point sets. Analogously, a topological space is totally path-disconnected if all path-components in are the one-point sets.
Another closely related notion is that of a totally separated space, i.e. a space where quasicomponents are singletons. That is, a topological space
is totally separated space if and only if for every , the intersection of all clopen neighborhoods of is the singleton . Equivalently, for each pair of distinct points , there is a pair of disjoint open neighborhoods of such that .
Every totally separated space is evidently totally disconnected but the converse is false even for metric spaces. For instance, take to be the Cantor's teepee, which is the Knaster–Kuratowski fan with the apex removed. Then is totally disconnected but its quasicomponents are not singletons. For locally compact Hausdorff spaces the two notions (totally disconnected and totally separated) are equivalent.
Unfortunately in the literature (for instance ), totally disconnected spaces are sometimes called hereditarily disconnected, while the terminology totally disconnected is used for totally separated spaces.
Examples
The following are examples of totally disconnected spaces:
Discrete spaces
The rational numbers
The irrational numbers
The p-adic numbers; more generally, all profinite groups are totally disconnected.
The Cantor set and the Cantor space
The Baire space
The Sorgenfrey line
Every Hausdorff space of small inductive dimension 0 is totally disconnected
The Erdős space ℓ2 is a totally disconnected Hausdorff space that does not have small inductive dimension 0.
Extremally disconnected Hausdorff spaces
Stone spaces
The Knaster–Kuratowski fan provides an example of a connected space, such that the removal of a single point produces a totally disconnected space.
Properties
Subspaces, products, and coproducts of totally disconnected spaces are totally disconnected.
Totally disconnected spaces are T1 spaces, since singletons are closed.
Continuous images of totally disconnected spaces are not necessarily totally disconnected, in fact, every compact metric space is a continuous image of the Cantor set.
A locally compact Hausdorff space has small inductive dimension 0 if and only if it is totally disconnected.
Every totally disconnected compact metric space is homeomorphic to a subset
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https://en.wikipedia.org/wiki/Quintic%20function
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In mathematics, a quintic function is a function of the form
where , , , , and are members of a field, typically the rational numbers, the real numbers or the complex numbers, and is nonzero. In other words, a quintic function is defined by a polynomial of degree five.
Because they have an odd degree, normal quintic functions appear similar to normal cubic functions when graphed, except they may possess one additional local maximum and one additional local minimum. The derivative of a quintic function is a quartic function.
Setting and assuming produces a quintic equation of the form:
Solving quintic equations in terms of radicals (nth roots) was a major problem in algebra from the 16th century, when cubic and quartic equations were solved, until the first half of the 19th century, when the impossibility of such a general solution was proved with the Abel–Ruffini theorem.
Finding roots of a quintic equation
Finding the roots (zeros) of a given polynomial has been a prominent mathematical problem.
Solving linear, quadratic, cubic and quartic equations by factorization into radicals can always be done, no matter whether the roots are rational or irrational, real or complex; there are formulae that yield the required solutions. However, there is no algebraic expression (that is, in terms of radicals) for the solutions of general quintic equations over the rationals; this statement is known as the Abel–Ruffini theorem, first asserted in 1799 and completely proved in 1824. This result also holds for equations of higher degree. An example of a quintic whose roots cannot be expressed in terms of radicals is .
Some quintics may be solved in terms of radicals. However, the solution is generally too complicated to be used in practice. Instead, numerical approximations are calculated using a root-finding algorithm for polynomials.
Solvable quintics
Some quintic equations can be solved in terms of radicals. These include the quintic equations defined by a polynomial that is reducible, such as . For example, it has been shown that
has solutions in radicals if and only if it has an integer solution or r is one of ±15, ±22440, or ±2759640, in which cases the polynomial is reducible.
As solving reducible quintic equations reduces immediately to solving polynomials of lower degree, only irreducible quintic equations are considered in the remainder of this section, and the term "quintic" will refer only to irreducible quintics. A solvable quintic is thus an irreducible quintic polynomial whose roots may be expressed in terms of radicals.
To characterize solvable quintics, and more generally solvable polynomials of higher degree, Évariste Galois developed techniques which gave rise to group theory and Galois theory. Applying these techniques, Arthur Cayley found a general criterion for determining whether any given quintic is solvable. This criterion is the following.
Given the equation
the Tschirnhaus transformation , which depresses the quint
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