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spherical harmonic function | These functions form an orthogonal system, and are thus basic to the expansion of a general function on the sphere as alluded to above. Spherical harmonics are important in many theoretical and practical applications, including the representation of multipole electrostatic and electromagnetic fields, electron configurations, gravitational fields, geoids, the magnetic fields of planetary bodies and stars, and the cosmic microwave background radiation. In 3D computer graphics, spherical harmonics play a role in a wide variety of topics including indirect lighting (ambient occlusion, global illumination, precomputed radiance transfer, etc.) and modelling of 3D shapes. | wikipedia |
canonical commutation relation algebra | In mathematics and physics CCR algebras (after canonical commutation relations) and CAR algebras (after canonical anticommutation relations) arise from the quantum mechanical study of bosons and fermions respectively. They play a prominent role in quantum statistical mechanics and quantum field theory. | wikipedia |
quantum graph | In mathematics and physics, a quantum graph is a linear, network-shaped structure of vertices connected on edges (i.e., a graph) in which each edge is given a length and where a differential (or pseudo-differential) equation is posed on each edge. An example would be a power network consisting of power lines (edges) connected at transformer stations (vertices); the differential equations would then describe the voltage along each of the lines, with boundary conditions for each edge provided at the adjacent vertices ensuring that the current added over all edges adds to zero at each vertex. Quantum graphs were first studied by Linus Pauling as models of free electrons in organic molecules in the 1930s. They also arise in a variety of mathematical contexts, e.g. as model systems in quantum chaos, in the study of waveguides, in photonic crystals and in Anderson localization, or as limit on shrinking thin wires. Quantum graphs have become prominent models in mesoscopic physics used to obtain a theoretical understanding of nanotechnology. Another, more simple notion of quantum graphs was introduced by Freedman et al.Aside from actually solving the differential equations posed on a quantum graph for purposes of concrete applications, typical questions that arise are those of controllability (what inputs have to be provided to bring the system into a desired state, for example providing sufficient power to all houses on a power network) and identifiability (how and where one has to measure something to obtain a complete picture of the state of the system, for example measuring the pressure of a water pipe network to determine whether or not there is a leaking pipe). | wikipedia |
scalar fields | In mathematics and physics, a scalar field is a function associating a single number to every point in a space – possibly physical space. The scalar may either be a pure mathematical number (dimensionless) or a scalar physical quantity (with units). In a physical context, scalar fields are required to be independent of the choice of reference frame. That is, any two observers using the same units will agree on the value of the scalar field at the same absolute point in space (or spacetime) regardless of their respective points of origin. Examples used in physics include the temperature distribution throughout space, the pressure distribution in a fluid, and spin-zero quantum fields, such as the Higgs field. These fields are the subject of scalar field theory. | wikipedia |
acceleration vector | In mathematics and physics, acceleration is the rate of change of velocity of a curve with respect to a given linear connection. This operation provides us with a measure of the rate and direction of the "bend". | wikipedia |
geometric center | In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any object in n-dimensional Euclidean space.In geometry, one often assumes uniform mass density, in which case the barycenter or center of mass coincides with the centroid. Informally, it can be understood as the point at which a cutout of the shape (with uniformly distributed mass) could be perfectly balanced on the tip of a pin.In physics, if variations in gravity are considered, then a center of gravity can be defined as the weighted mean of all points weighted by their specific weight. In geography, the centroid of a radial projection of a region of the Earth's surface to sea level is the region's geographical center. | wikipedia |
generator (mathematics) | In mathematics and physics, the term generator or generating set may refer to any of a number of related concepts. The underlying concept in each case is that of a smaller set of objects, together with a set of operations that can be applied to it, that result in the creation of a larger collection of objects, called the generated set. The larger set is then said to be generated by the smaller set. It is commonly the case that the generating set has a simpler set of properties than the generated set, thus making it easier to discuss and examine. It is usually the case that properties of the generating set are in some way preserved by the act of generation; likewise, the properties of the generated set are often reflected in the generating set. | wikipedia |
continuum percolation theory | In mathematics and probability theory, continuum percolation theory is a branch of mathematics that extends discrete percolation theory to continuous space (often Euclidean space ℝn). More specifically, the underlying points of discrete percolation form types of lattices whereas the underlying points of continuum percolation are often randomly positioned in some continuous space and form a type of point process. For each point, a random shape is frequently placed on it and the shapes overlap each with other to form clumps or components. | wikipedia |
continuum percolation theory | As in discrete percolation, a common research focus of continuum percolation is studying the conditions of occurrence for infinite or giant components. Other shared concepts and analysis techniques exist in these two types of percolation theory as well as the study of random graphs and random geometric graphs. Continuum percolation arose from an early mathematical model for wireless networks, which, with the rise of several wireless network technologies in recent years, has been generalized and studied in order to determine the theoretical bounds of information capacity and performance in wireless networks. In addition to this setting, continuum percolation has gained application in other disciplines including biology, geology, and physics, such as the study of porous material and semiconductors, while becoming a subject of mathematical interest in its own right. | wikipedia |
dimension of an algebraic variety | In mathematics and specifically in algebraic geometry, the dimension of an algebraic variety may be defined in various equivalent ways. Some of these definitions are of geometric nature, while some other are purely algebraic and rely on commutative algebra. Some are restricted to algebraic varieties while others apply also to any algebraic set. Some are intrinsic, as independent of any embedding of the variety into an affine or projective space, while other are related to such an embedding. | wikipedia |
fuzzy concept | In mathematics and statistics, a fuzzy variable (such as "the temperature", "hot" or "cold") is a value which could lie in a probable range defined by some quantitative limits or parameters, and which can be usefully described with imprecise categories (such as "high", "medium" or "low") using some kind of scale or conceptual hierarchy. | wikipedia |
piecewise-linear function | In mathematics and statistics, a piecewise linear, PL or segmented function is a real-valued function of a real variable, whose graph is composed of straight-line segments. | wikipedia |
absolute deviation | In mathematics and statistics, deviation is a measure of difference between the observed value of a variable and some other value, often that variable's mean. The sign of the deviation reports the direction of that difference (the deviation is positive when the observed value exceeds the reference value). The magnitude of the value indicates the size of the difference. | wikipedia |
statistical mean | In mathematics and statistics, the arithmetic mean ( arr-ith-MET-ik), arithmetic average, or just the mean or average (when the context is clear) is the sum of a collection of numbers divided by the count of numbers in the collection. The collection is often a set of results from an experiment, an observational study, or a survey. The term "arithmetic mean" is preferred in some mathematics and statistics contexts because it helps distinguish it from other types of means, such as geometric and harmonic. In addition to mathematics and statistics, the arithmetic mean is frequently used in economics, anthropology, history, and almost every academic field to some extent. | wikipedia |
statistical mean | For example, per capita income is the arithmetic average income of a nation's population. While the arithmetic mean is often used to report central tendencies, it is not a robust statistic: it is greatly influenced by outliers (values much larger or smaller than most others). For skewed distributions, such as the distribution of income for which a few people's incomes are substantially higher than most people's, the arithmetic mean may not coincide with one's notion of "middle". In that case, robust statistics, such as the median, may provide a better description of central tendency. | wikipedia |
spectral network | In mathematics and supersymmetric gauge theory, spectral networks are "networks of trajectories on Riemann surfaces obeying certain local rules. Spectral networks arise naturally in four-dimensional N = 2 theories coupled to surface defects, particularly the theories of class S." == References == | wikipedia |
stochastic geometry models of wireless networks | In mathematics and telecommunications, stochastic geometry models of wireless networks refer to mathematical models based on stochastic geometry that are designed to represent aspects of wireless networks. The related research consists of analyzing these models with the aim of better understanding wireless communication networks in order to predict and control various network performance metrics. The models require using techniques from stochastic geometry and related fields including point processes, spatial statistics, geometric probability, percolation theory, as well as methods from more general mathematical disciplines such as geometry, probability theory, stochastic processes, queueing theory, information theory, and Fourier analysis.In the early 1960s a stochastic geometry model was developed to study wireless networks. | wikipedia |
stochastic geometry models of wireless networks | This model is considered to be pioneering and the origin of continuum percolation. Network models based on geometric probability were later proposed and used in the late 1970s and continued throughout the 1980s for examining packet radio networks. Later their use increased significantly for studying a number of wireless network technologies including mobile ad hoc networks, sensor networks, vehicular ad hoc networks, cognitive radio networks and several types of cellular networks, such as heterogeneous cellular networks. Key performance and quality of service quantities are often based on concepts from information theory such as the signal-to-interference-plus-noise ratio, which forms the mathematical basis for defining network connectivity and coverage.The principal idea underlying the research of these stochastic geometry models, also known as random spatial models, is that it is best to assume that the locations of nodes or the network structure and the aforementioned quantities are random in nature due to the size and unpredictability of users in wireless networks. The use of stochastic geometry can then allow for the derivation of closed-form or semi-closed-form expressions for these quantities without resorting to simulation methods or (possibly intractable or inaccurate) deterministic models. | wikipedia |
transport function | In mathematics and the field of transportation theory, the transport functions J(n,x) are defined by J ( n , x ) = ∫ 0 x t n e t ( e t − 1 ) 2 d t . {\displaystyle J(n,x)=\int _{0}^{x}t^{n}{\frac {e^{t}}{(e^{t}-1)^{2}}}\,dt.} Note that e t ( e t − 1 ) 2 = ∑ k = 0 ∞ k e k t . {\displaystyle {\frac {e^{t}}{(e^{t}-1)^{2}}}=\sum _{k=0}^{\infty }k\,e^{kt}.} | wikipedia |
ray (quantum theory) | In mathematics and the foundations of quantum mechanics, the projective Hilbert space P ( H ) {\displaystyle P(H)} of a complex Hilbert space H {\displaystyle H} is the set of equivalence classes of non-zero vectors v {\displaystyle v} in H {\displaystyle H} , for the relation ∼ {\displaystyle \sim } on H {\displaystyle H} given by w ∼ v {\displaystyle w\sim v} if and only if v = λ w {\displaystyle v=\lambda w} for some non-zero complex number λ {\displaystyle \lambda } .The equivalence classes of v {\displaystyle v} for the relation ∼ {\displaystyle \sim } are also called rays or projective rays. This is the usual construction of projectivization, applied to a complex Hilbert space. | wikipedia |
analysis of boolean functions | In mathematics and theoretical computer science, analysis of Boolean functions is the study of real-valued functions on { 0 , 1 } n {\displaystyle \{0,1\}^{n}} or { − 1 , 1 } n {\displaystyle \{-1,1\}^{n}} (such functions are sometimes known as pseudo-Boolean functions) from a spectral perspective. The functions studied are often, but not always, Boolean-valued, making them Boolean functions. The area has found many applications in combinatorics, social choice theory, random graphs, and theoretical computer science, especially in hardness of approximation, property testing, and PAC learning. | wikipedia |
gerstenhaber algebra | In mathematics and theoretical physics, a Gerstenhaber algebra (sometimes called an antibracket algebra or braid algebra) is an algebraic structure discovered by Murray Gerstenhaber (1963) that combines the structures of a supercommutative ring and a graded Lie superalgebra. It is used in the Batalin–Vilkovisky formalism. It appears also in the generalization of Hamiltonian formalism known as the De Donder–Weyl theory as the algebra of generalized Poisson brackets defined on differential forms. | wikipedia |
locally compact quantum group | In mathematics and theoretical physics, a locally compact quantum group is a relatively new C*-algebraic approach toward quantum groups that generalizes the Kac algebra, compact-quantum-group and Hopf-algebra approaches. Earlier attempts at a unifying definition of quantum groups using, for example, multiplicative unitaries have enjoyed some success but have also encountered several technical problems. One of the main features distinguishing this new approach from its predecessors is the axiomatic existence of left and right invariant weights. This gives a noncommutative analogue of left and right Haar measures on a locally compact Hausdorff group. | wikipedia |
representation theory of lie groups | 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. | wikipedia |
associative superalgebra | In mathematics and theoretical physics, a superalgebra is a Z2-graded algebra. That is, it is an algebra over a commutative ring or field with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading. The prefix super- comes from the theory of supersymmetry in theoretical physics. | wikipedia |
associative superalgebra | Superalgebras and their representations, supermodules, provide an algebraic framework for formulating supersymmetry. The study of such objects is sometimes called super linear algebra. Superalgebras also play an important role in related field of supergeometry where they enter into the definitions of graded manifolds, supermanifolds and superschemes. | wikipedia |
braid statistics | In mathematics and theoretical physics, braid statistics is a generalization of the spin statistics of bosons and fermions based on the concept of braid group. While for fermions (Bosons) the corresponding statistics is associated to a phase gain of π {\displaystyle \pi } ( 2 π {\displaystyle 2\pi } ) under the exchange of identical particles, a particle with braid statistics leads to a rational fraction of π {\displaystyle \pi } under such exchange or even a non-trivial unitary transformation in the Hilbert space (see non-Abelian anyons). A similar notion exists using a loop braid group. Braid statistics are applicable to theoretical particles such as the two-dimensional anyons and their higher-dimensional analogues known as plektons. | wikipedia |
quantum group | In mathematics and theoretical physics, the term quantum group denotes one of a few different kinds of noncommutative algebras with additional structure. These include Drinfeld–Jimbo type quantum groups (which are quasitriangular Hopf algebras), compact matrix quantum groups (which are structures on unital separable C*-algebras), and bicrossproduct quantum groups. Despite their name, they do not themselves have a natural group structure, though they are in some sense 'close' to a group. The term "quantum group" first appeared in the theory of quantum integrable systems, which was then formalized by Vladimir Drinfeld and Michio Jimbo as a particular class of Hopf algebra. | wikipedia |
quantum group | The same term is also used for other Hopf algebras that deform or are close to classical Lie groups or Lie algebras, such as a "bicrossproduct" class of quantum groups introduced by Shahn Majid a little after the work of Drinfeld and Jimbo. In Drinfeld's approach, quantum groups arise as Hopf algebras depending on an auxiliary parameter q or h, which become universal enveloping algebras of a certain Lie algebra, frequently semisimple or affine, when q = 1 or h = 0. Closely related are certain dual objects, also Hopf algebras and also called quantum groups, deforming the algebra of functions on the corresponding semisimple algebraic group or a compact Lie group. | wikipedia |
zeta function regularization | In mathematics and theoretical physics, zeta function regularization is a type of regularization or summability method that assigns finite values to divergent sums or products, and in particular can be used to define determinants and traces of some self-adjoint operators. The technique is now commonly applied to problems in physics, but has its origins in attempts to give precise meanings to ill-conditioned sums appearing in number theory. | wikipedia |
umbral calculus | In mathematics before the 1970s, the term umbral calculus referred to the surprising similarity between seemingly unrelated polynomial equations and certain shadowy techniques used to "prove" them. These techniques were introduced by John Blissard and are sometimes called Blissard's symbolic method. They are often attributed to Édouard Lucas (or James Joseph Sylvester), who used the technique extensively. | wikipedia |
finite mathematics | In mathematics education, Finite Mathematics is a syllabus in college and university mathematics that is independent of calculus. A course in precalculus may be a prerequisite for Finite Mathematics. Contents of the course include an eclectic selection of topics often applied in social science and business, such as finite probability spaces, matrix multiplication, Markov processes, finite graphs, or mathematical models. | wikipedia |
finite mathematics | These topics were used in Finite Mathematics courses at Dartmouth College as developed by John G. Kemeny, Gerald L. Thompson, and J. Laurie Snell and published by Prentice-Hall. Other publishers followed with their own topics. With the arrival of software to facilitate computations, teaching and usage shifted from a broad-spectrum Finite Mathematics with paper and pen, into development and usage of software. | wikipedia |
mathematical manipulative | In mathematics education, a manipulative is an object which is designed so that a learner can perceive some mathematical concept by manipulating it, hence its name. The use of manipulatives provides a way for children to learn concepts through developmentally appropriate hands-on experience. The use of manipulatives in mathematics classrooms throughout the world grew considerably in popularity throughout the second half of the 20th century. Mathematical manipulatives are frequently used in the first step of teaching mathematical concepts, that of concrete representation. | wikipedia |
mathematical manipulative | The second and third steps are representational and abstract, respectively. Mathematical manipulatives can be purchased or constructed by the teacher. | wikipedia |
mathematical manipulative | Examples of common manipulatives include number lines, Cuisenaire rods; fraction strips, blocks, or stacks; base ten blocks (also known as Dienes or multibase blocks); interlocking linking cubes (such as Unifix); construction sets (such as Polydron and Zometool); colored tiles or tangrams; pattern blocks; colored counting chips; numicon tiles; chainable links; abaci such as "rekenreks", and geoboards. Improvised teacher-made manipulatives used in teaching place value include beans and bean sticks, or single popsicle sticks and bundles of ten popsicle sticks. Virtual manipulatives for mathematics are computer models of these objects. | wikipedia |
mathematical manipulative | Notable collections of virtual manipulatives include The National Library of Virtual Manipulatives and the Ubersketch. Multiple experiences with manipulatives provide children with the conceptual foundation to understand mathematics at a conceptual level and are recommended by the NCTM.Some of the manipulatives are now used in other subjects in addition to mathematics. For example, Cuisenaire rods are now used in language arts and grammar, and pattern blocks are used in fine arts. | wikipedia |
multiple representations (mathematics education) | In mathematics education, a representation is a way of encoding an idea or a relationship, and can be both internal (e.g., mental construct) and external (e.g., graph). Thus multiple representations are ways to symbolize, to describe and to refer to the same mathematical entity. They are used to understand, to develop, and to communicate different mathematical features of the same object or operation, as well as connections between different properties. Multiple representations include graphs and diagrams, tables and grids, formulas, symbols, words, gestures, software code, videos, concrete models, physical and virtual manipulatives, pictures, and sounds. Representations are thinking tools for doing mathematics. | wikipedia |
differential and integral calculus | In mathematics education, calculus denotes courses of elementary mathematical analysis, which are mainly devoted to the study of functions and limits. The word calculus is Latin for "small pebble" (the diminutive of calx, meaning "stone"), a meaning which still persists in medicine. Because such pebbles were used for counting out distances, tallying votes, and doing abacus arithmetic, the word came to mean a method of computation. In this sense, it was used in English at least as early as 1672, several years prior to the publications of Leibniz and Newton.In addition to the differential calculus and integral calculus, the term is also used for naming specific methods of calculation and related theories which seek to model a particular concept in terms of mathematics. Examples of this convention include propositional calculus, Ricci calculus, calculus of variations, lambda calculus, and process calculus. Furthermore, the term "calculus" has variously been applied in ethics and philosophy, for such systems as Bentham's felicific calculus, and the ethical calculus. | wikipedia |
history of calculus | In mathematics education, calculus denotes courses of elementary mathematical analysis, which are mainly devoted to the study of functions and limits. The word calculus is Latin for "small pebble" (the diminutive of calx, meaning "stone"), a meaning which still persists in medicine. Because such pebbles were used for counting out distances, tallying votes, and doing abacus arithmetic, the word came to mean a method of computation. In this sense, it was used in English at least as early as 1672, several years prior to the publications of Leibniz and Newton.In addition to the differential calculus and integral calculus, the term is also used widely for naming specific methods of calculation. Examples of this include propositional calculus in logic, the calculus of variations in mathematics, process calculus in computing, and the felicific calculus in philosophy. | wikipedia |
ethnomathematics | In mathematics education, ethnomathematics is the study of the relationship between mathematics and culture. Often associated with "cultures without written expression", it may also be defined as "the mathematics which is practised among identifiable cultural groups". It refers to a broad cluster of ideas ranging from distinct numerical and mathematical systems to multicultural mathematics education. The goal of ethnomathematics is to contribute both to the understanding of culture and the understanding of mathematics, and mainly to lead to an appreciation of the connections between the two. | wikipedia |
precalculus | In mathematics education, precalculus is a course, or a set of courses, that includes algebra and trigonometry at a level which is designed to prepare students for the study of calculus, thus the name precalculus. Schools often distinguish between algebra and trigonometry as two separate parts of the coursework. | wikipedia |
unit fraction | In mathematics education, unit fractions are often introduced earlier than other kinds of fractions, because of the ease of explaining them visually as equal parts of a whole. A common practical use of unit fractions is to divide food equally among a number of people, and exercises in performing this sort of fair division are a standard classroom example in teaching students to work with unit fractions. | wikipedia |
function field sieve | In mathematics the Function Field Sieve is one of the most efficient algorithms to solve the Discrete Logarithm Problem (DLP) in a finite field. It has heuristic subexponential complexity. Leonard Adleman developed it in 1994 and then elaborated it together with M. D. Huang in 1999. | wikipedia |
function field sieve | Previous work includes the work of D. Coppersmith about the DLP in fields of characteristic two. The discrete logarithm problem in a finite field consists of solving the equation a x = b {\displaystyle a^{x}=b} for a , b ∈ F p n {\displaystyle a,b\in \mathbb {F} _{p^{n}}} , p {\displaystyle p} a prime number and n {\displaystyle n} an integer. The function f: F p n → F p n , x ↦ a x {\displaystyle f:\mathbb {F} _{p^{n}}\to \mathbb {F} _{p^{n}},x\mapsto a^{x}} for a fixed a ∈ F p n {\displaystyle a\in \mathbb {F} _{p^{n}}} is a one-way function used in cryptography. Several cryptographic methods are based on the DLP such as the Diffie-Hellman key exchange, the El Gamal cryptosystem and the Digital Signature Algorithm. | wikipedia |
differential calculus over commutative algebras | In mathematics the differential calculus over commutative algebras is a part of commutative algebra based on the observation that most concepts known from classical differential calculus can be formulated in purely algebraic terms. Instances of this are: The whole topological information of a smooth manifold M {\displaystyle M} is encoded in the algebraic properties of its R {\displaystyle \mathbb {R} } -algebra of smooth functions A = C ∞ ( M ) , {\displaystyle A=C^{\infty }(M),} as in the Banach–Stone theorem. Vector bundles over M {\displaystyle M} correspond to projective finitely generated modules over A , {\displaystyle A,} via the functor Γ {\displaystyle \Gamma } which associates to a vector bundle its module of sections. Vector fields on M {\displaystyle M} are naturally identified with derivations of the algebra A {\displaystyle A} . | wikipedia |
differential calculus over commutative algebras | More generally, a linear differential operator of order k, sending sections of a vector bundle E → M {\displaystyle E\rightarrow M} to sections of another bundle F → M {\displaystyle F\rightarrow M} is seen to be an R {\displaystyle \mathbb {R} } -linear map Δ: Γ ( E ) → Γ ( F ) {\displaystyle \Delta :\Gamma (E)\to \Gamma (F)} between the associated modules, such that for any k + 1 {\displaystyle k+1} elements f 0 , … , f k ∈ A {\displaystyle f_{0},\ldots ,f_{k}\in A}: where the bracket : Γ ( E ) → Γ ( F ) {\displaystyle :\Gamma (E)\to \Gamma (F)} is defined as the commutator Denoting the set of k {\displaystyle k} th order linear differential operators from an A {\displaystyle A} -module P {\displaystyle P} to an A {\displaystyle A} -module Q {\displaystyle Q} with D i f f k ( P , Q ) {\displaystyle \mathrm {Diff} _{k}(P,Q)} we obtain a bi-functor with values in the category of A {\displaystyle A} -modules. Other natural concepts of calculus such as jet spaces, differential forms are then obtained as representing objects of the functors D i f f k {\displaystyle \mathrm {Diff} _{k}} and related functors. Seen from this point of view calculus may in fact be understood as the theory of these functors and their representing objects. | wikipedia |
differential calculus over commutative algebras | Replacing the real numbers R {\displaystyle \mathbb {R} } with any commutative ring, and the algebra C ∞ ( M ) {\displaystyle C^{\infty }(M)} with any commutative algebra the above said remains meaningful, hence differential calculus can be developed for arbitrary commutative algebras. Many of these concepts are widely used in algebraic geometry, differential geometry and secondary calculus. Moreover, the theory generalizes naturally to the setting of graded commutative algebra, allowing for a natural foundation of calculus on supermanifolds, graded manifolds and associated concepts like the Berezin integral. | wikipedia |
elliptic rational function | In mathematics the elliptic rational functions are a sequence of rational functions with real coefficients. Elliptic rational functions are extensively used in the design of elliptic electronic filters. (These functions are sometimes called Chebyshev rational functions, not to be confused with certain other functions of the same name). | wikipedia |
elliptic rational function | Rational elliptic functions are identified by a positive integer order n and include a parameter ξ ≥ 1 called the selectivity factor. A rational elliptic function of degree n in x with selectivity factor ξ is generally defined as: R n ( ξ , x ) ≡ c d ( n K ( 1 / L n ( ξ ) ) K ( 1 / ξ ) c d − 1 ( x , 1 / ξ ) , 1 / L n ( ξ ) ) {\displaystyle R_{n}(\xi ,x)\equiv \mathrm {cd} \left(n{\frac {K(1/L_{n}(\xi ))}{K(1/\xi )}}\,\mathrm {cd} ^{-1}(x,1/\xi ),1/L_{n}(\xi )\right)} where cd(u,k) is the Jacobi elliptic cosine function. K() is a complete elliptic integral of the first kind. L n ( ξ ) = R n ( ξ , ξ ) {\displaystyle L_{n}(\xi )=R_{n}(\xi ,\xi )} is the discrimination factor, equal to the minimum value of the magnitude of R n ( ξ , x ) {\displaystyle R_{n}(\xi ,x)} for | x | ≥ ξ {\displaystyle |x|\geq \xi } .For many cases, in particular for orders of the form n = 2a3b where a and b are integers, the elliptic rational functions can be expressed using algebraic functions alone. Elliptic rational functions are closely related to the Chebyshev polynomials: Just as the circular trigonometric functions are special cases of the Jacobi elliptic functions, so the Chebyshev polynomials are special cases of the elliptic rational functions. | wikipedia |
let expression | In mathematics the let expression is described as the conjunction of expressions. In functional languages the let expression is also used to limit scope. In mathematics scope is described by quantifiers. | wikipedia |
let expression | The let expression is a conjunction within an existential quantifier. ( ∃ x E ∧ F ) ⟺ let x: E in F {\displaystyle (\exists xE\land F)\iff \operatorname {let} x:E\operatorname {in} F} where E and F are of type Boolean. The let expression allows the substitution to be applied to another expression. | wikipedia |
let expression | This substitution may be applied within a restricted scope, to a sub expression. The natural use of the let expression is in application to a restricted scope (called lambda dropping). These rules define how the scope may be restricted; { x ∉ FV ( E ) ∧ x ∈ FV ( F ) ⟹ let x: G in E F = E ( let x: G in F ) x ∈ FV ( E ) ∧ x ∉ FV ( F ) ⟹ let x: G in E F = ( let x: G in E ) F x ∉ FV ( E ) ∧ x ∉ FV ( F ) ⟹ let x: G in E F = E F {\displaystyle {\begin{cases}x\not \in \operatorname {FV} (E)\land x\in \operatorname {FV} (F)\implies \operatorname {let} x:G\operatorname {in} E\ F=E\ (\operatorname {let} x:G\operatorname {in} F)\\x\in \operatorname {FV} (E)\land x\not \in \operatorname {FV} (F)\implies \operatorname {let} x:G\operatorname {in} E\ F=(\operatorname {let} x:G\operatorname {in} E)\ F\\x\not \in \operatorname {FV} (E)\land x\not \in \operatorname {FV} (F)\implies \operatorname {let} x:G\operatorname {in} E\ F=E\ F\end{cases}}} where F is not of type Boolean. | wikipedia |
let expression | From this definition the following standard definition of a let expression, as used in a functional language may be derived. x ∉ FV ( y ) ⟹ ( let x: x = y in z ) = z = ( λ x . z ) y {\displaystyle x\not \in \operatorname {FV} (y)\implies (\operatorname {let} x:x=y\operatorname {in} z)=z=(\lambda x.z)\ y} For simplicity the marker specifying the existential variable, x: {\displaystyle x:} , will be omitted from the expression where it is clear from the context. x ∉ FV ( y ) ⟹ ( let x = y in z ) = z = ( λ x . z ) y {\displaystyle x\not \in \operatorname {FV} (y)\implies (\operatorname {let} x=y\operatorname {in} z)=z=(\lambda x.z)\ y} | wikipedia |
negative sign | In mathematics the one-sided limit x → a+ means x approaches a from the right (i.e., right-sided limit), and x → a− means x approaches a from the left (i.e., left-sided limit). For example, 1/x → + ∞ {\displaystyle \infty } as x → 0+ but 1/x → − ∞ {\displaystyle \infty } as x → 0−. | wikipedia |
signal-to-noise statistic | In mathematics the signal-to-noise statistic distance between two vectors a and b with mean values μ a {\displaystyle \mu _{a}} and μ b {\displaystyle \mu _{b}} and standard deviation σ a {\displaystyle \sigma _{a}} and σ b {\displaystyle \sigma _{b}} respectively is: D s n = ( μ a − μ b ) ( σ a + σ b ) {\displaystyle D_{sn}={(\mu _{a}-\mu _{b}) \over (\sigma _{a}+\sigma _{b})}} In the case of Gaussian-distributed data and unbiased class distributions, this statistic can be related to classification accuracy given an ideal linear discrimination, and a decision boundary can be derived.This distance is frequently used to identify vectors that have significant difference. One usage is in bioinformatics to locate genes that are differential expressed on microarray experiments. | wikipedia |
synchrotron function | In mathematics the synchrotron functions are defined as follows (for x ≥ 0): First synchrotron function F ( x ) = x ∫ x ∞ K 5 3 ( t ) d t {\displaystyle F(x)=x\int _{x}^{\infty }K_{\frac {5}{3}}(t)\,dt} Second synchrotron function G ( x ) = x K 2 3 ( x ) {\displaystyle G(x)=xK_{\frac {2}{3}}(x)} where Kj is the modified Bessel function of the second kind. | wikipedia |
theory of | In mathematics the use of the term theory is different, necessarily so, since mathematics contains no explanations of natural phenomena, per se, even though it may help provide insight into natural systems or be inspired by them. In the general sense, a mathematical theory is a branch of or topic in mathematics, such as Set theory, Number theory, Group theory, Probability theory, Game theory, Control theory, Perturbation theory, etc., such as might be appropriate for a single textbook. In the same sense, but more specifically, the word theory is an extensive, structured collection of theorems, organized so that the proof of each theorem only requires the theorems and axioms that preceded it (no circular proofs), occurs as early as feasible in sequence (no postponed proofs), and the whole is as succinct as possible (no redundant proofs). Ideally, the sequence in which the theorems are presented is as easy to understand as possible, although illuminating a branch of mathematics is the purpose of textbooks, rather than the mathematical theory they might be written to cover. | wikipedia |
map coloring | In mathematics there is a close link between map coloring and graph coloring, since every map showing different areas has a corresponding graph. By far the most famous result in this area is the four color theorem, which states that any planar map can be colored with at most four colors. | wikipedia |
angled bracket | In mathematics they delimit sets and are often also used to denote the Poisson bracket between two quantities. In ring theory, braces denote the anticommutator where {a, b} is defined as a b + b a . | wikipedia |
bochner measurable function | In mathematics – specifically, in functional analysis – a Bochner-measurable function taking values in a Banach space is a function that equals almost everywhere the limit of a sequence of measurable countably-valued functions, i.e., f ( t ) = lim n → ∞ f n ( t ) for almost every t , {\displaystyle f(t)=\lim _{n\rightarrow \infty }f_{n}(t){\text{ for almost every }}t,\,} where the functions f n {\displaystyle f_{n}} each have a countable range and for which the pre-image f n − 1 ( { x } ) {\displaystyle f_{n}^{-1}(\{x\})} is measurable for each element x. The concept is named after Salomon Bochner. Bochner-measurable functions are sometimes called strongly measurable, μ {\displaystyle \mu } -measurable or just measurable (or uniformly measurable in case that the Banach space is the space of continuous linear operators between Banach spaces). | wikipedia |
cylindrical σ-algebra | In mathematics — specifically, in measure theory and functional analysis — the cylindrical σ-algebra or product σ-algebra is a type of σ-algebra which is often used when studying product measures or probability measures of random variables on Banach spaces. For a product space, the cylinder σ-algebra is the one that is generated by cylinder sets. In the context of a Banach space X , {\displaystyle X,} the cylindrical σ-algebra Cyl ( X ) {\displaystyle \operatorname {Cyl} (X)} is defined to be the coarsest σ-algebra (that is, the one with the fewest measurable sets) such that every continuous linear function on X {\displaystyle X} is a measurable function. In general, Cyl ( X ) {\displaystyle \operatorname {Cyl} (X)} is not the same as the Borel σ-algebra on X , {\displaystyle X,} which is the coarsest σ-algebra that contains all open subsets of X . {\displaystyle X.} | wikipedia |
question mark | In mathematics, "?" commonly denotes Minkowski's question mark function. In equations, it can mean "questioned" as opposed to "defined". U+225F ≟ QUESTIONED EQUAL TO U+2A7B ⩻ LESS-THAN WITH QUESTION MARK ABOVE U+2A7C ⩼ GREATER-THAN WITH QUESTION MARK ABOVEIn linear logic, the question mark denotes one of the exponential modalities that control weakening and contraction. | wikipedia |
2e6 (mathematics) | In mathematics, 2E6 is the name of a family of Steinberg or twisted Chevalley groups. It is a quasi-split form of E6, depending on a quadratic extension of fields K⊂L. Unfortunately the notation for the group is not standardized, as some authors write it as 2E6(K) (thinking of 2E6 as an algebraic group taking values in K) and some as 2E6(L) (thinking of the group as a subgroup of E6(L) fixed by an outer involution). Over finite fields these groups form one of the 18 infinite families of finite simple groups, and were introduced independently by Tits (1958) and Steinberg (1959). | wikipedia |
ibn al-haytham | In mathematics, Alhazen built on the mathematical works of Euclid and Thabit ibn Qurra and worked on "the beginnings of the link between algebra and geometry".He developed a formula for summing the first 100 natural numbers, using a geometric proof to prove the formula. | wikipedia |
arithmetic scheme | In mathematics, Arakelov theory (or Arakelov geometry) is an approach to Diophantine geometry, named for Suren Arakelov. It is used to study Diophantine equations in higher dimensions. | wikipedia |
bck algebra | In mathematics, BCI and BCK algebras are algebraic structures in universal algebra, which were introduced by Y. Imai, K. Iséki and S. Tanaka in 1966, that describe fragments of the propositional calculus involving implication known as BCI and BCK logics. | wikipedia |
bf-algebra | In mathematics, BF algebras are a class of algebraic structures arising out of a symmetric "Yin Yang" concept for Bipolar Fuzzy logic, the name was introduced by Andrzej Walendziak in 2007. The name covers discrete versions, but a canonical example arises in the BF space x of pairs of (false-ness, truth-ness). | wikipedia |
baire function | In mathematics, Baire functions are functions obtained from continuous functions by transfinite iteration of the operation of forming pointwise limits of sequences of functions. They were introduced by René-Louis Baire in 1899. A Baire set is a set whose characteristic function is a Baire function. (There are other similar, but inequivalent definitions of Baire sets.) | wikipedia |
banach algebra cohomology | In mathematics, Banach algebra cohomology of a Banach algebra with coefficients in a bimodule is a cohomology theory defined in a similar way to Hochschild cohomology of an abstract algebra, except that one takes the topology into account so that all cochains and so on are continuous. | wikipedia |
borwein's algorithm | In mathematics, Borwein's algorithm is an algorithm devised by Jonathan and Peter Borwein to calculate the value of 1/π. They devised several other algorithms. They published the book Pi and the AGM – A Study in Analytic Number Theory and Computational Complexity. | wikipedia |
brownian motion | In mathematics, Brownian motion is described by the Wiener process, a continuous-time stochastic process named in honor of Norbert Wiener. It is one of the best known Lévy processes (càdlàg stochastic processes with stationary independent increments) and occurs frequently in pure and applied mathematics, economics and physics. The Wiener process Wt is characterized by four facts: W0 = 0 Wt is almost surely continuous Wt has independent increments W t − W s ∼ N ( 0 , t − s ) {\displaystyle W_{t}-W_{s}\sim {\mathcal {N}}(0,t-s)} (for 0 ≤ s ≤ t {\displaystyle 0\leq s\leq t} ). N ( μ , σ 2 ) {\displaystyle {\mathcal {N}}(\mu ,\sigma ^{2})} denotes the normal distribution with expected value μ and variance σ2. The condition that it has independent increments means that if 0 ≤ s 1 < t 1 ≤ s 2 < t 2 {\displaystyle 0\leq s_{1} | wikipedia |
carmichael's totient function conjecture | In mathematics, Carmichael's totient function conjecture concerns the multiplicity of values of Euler's totient function φ(n), which counts the number of integers less than and coprime to n. It states that, for every n there is at least one other integer m ≠ n such that φ(m) = φ(n). Robert Carmichael first stated this conjecture in 1907, but as a theorem rather than as a conjecture. However, his proof was faulty, and in 1922, he retracted his claim and stated the conjecture as an open problem. | wikipedia |
darboux function | In mathematics, Darboux's theorem is a theorem in real analysis, named after Jean Gaston Darboux. It states that every function that results from the differentiation of another function has the intermediate value property: the image of an interval is also an interval. When ƒ is continuously differentiable (ƒ in C1()), this is a consequence of the intermediate value theorem. But even when ƒ′ is not continuous, Darboux's theorem places a severe restriction on what it can be. | wikipedia |
diophantine geometry | In mathematics, Diophantine geometry is the study of Diophantine equations by means of powerful methods in algebraic geometry. By the 20th century it became clear for some mathematicians that methods of algebraic geometry are ideal tools to study these equations. Diophantine geometry is part of the broader field of arithmetic geometry. Four theorems in Diophantine geometry which are of fundamental importance include: Mordell–Weil theorem Roth's theorem Siegel's theorem Faltings's theorem | wikipedia |
regulator of an algebraic number field | In mathematics, Dirichlet's unit theorem is a basic result in algebraic number theory due to Peter Gustav Lejeune Dirichlet. It determines the rank of the group of units in the ring OK of algebraic integers of a number field K. The regulator is a positive real number that determines how "dense" the units are. The statement is that the group of units is finitely generated and has rank (maximal number of multiplicatively independent elements) equal to where r1 is the number of real embeddings and r2 the number of conjugate pairs of complex embeddings of K. This characterisation of r1 and r2 is based on the idea that there will be as many ways to embed K in the complex number field as the degree n = {\displaystyle n=} ; these will either be into the real numbers, or pairs of embeddings related by complex conjugation, so that Note that if K is Galois over Q {\displaystyle \mathbb {Q} } then either r1 = 0 or r2 = 0. Other ways of determining r1 and r2 are use the primitive element theorem to write K = Q ( α ) {\displaystyle K=\mathbb {Q} (\alpha )} , and then r1 is the number of conjugates of α that are real, 2r2 the number that are complex; in other words, if f is the minimal polynomial of α over Q {\displaystyle \mathbb {Q} } , then r1 is the number of real roots and 2r2 is the number of non-real complex roots of f (which come in complex conjugate pairs); write the tensor product of fields K ⊗ Q R {\displaystyle K\otimes _{\mathbb {Q} }\mathbb {R} } as a product of fields, there being r1 copies of R {\displaystyle \mathbb {R} } and r2 copies of C {\displaystyle \mathbb {C} } .As an example, if K is a quadratic field, the rank is 1 if it is a real quadratic field, and 0 if an imaginary quadratic field. | wikipedia |
regulator of an algebraic number field | The theory for real quadratic fields is essentially the theory of Pell's equation. The rank is positive for all number fields besides Q {\displaystyle \mathbb {Q} } and imaginary quadratic fields, which have rank 0. The 'size' of the units is measured in general by a determinant called the regulator. | wikipedia |
regulator of an algebraic number field | In principle a basis for the units can be effectively computed; in practice the calculations are quite involved when n is large. The torsion in the group of units is the set of all roots of unity of K, which form a finite cyclic group. For a number field with at least one real embedding the torsion must therefore be only {1,−1}. | wikipedia |
regulator of an algebraic number field | There are number fields, for example most imaginary quadratic fields, having no real embeddings which also have {1,−1} for the torsion of its unit group. Totally real fields are special with respect to units. If L/K is a finite extension of number fields with degree greater than 1 and the units groups for the integers of L and K have the same rank then K is totally real and L is a totally complex quadratic extension. | wikipedia |
regulator of an algebraic number field | The converse holds too. (An example is K equal to the rationals and L equal to an imaginary quadratic field; both have unit rank 0.) | wikipedia |
regulator of an algebraic number field | The theorem not only applies to the maximal order OK but to any order O ⊂ OK. There is a generalisation of the unit theorem by Helmut Hasse (and later Claude Chevalley) to describe the structure of the group of S-units, determining the rank of the unit group in localizations of rings of integers. Also, the Galois module structure of Q ⊕ O K , S ⊗ Z Q {\displaystyle \mathbb {Q} \oplus O_{K,S}\otimes _{\mathbb {Z} }\mathbb {Q} } has been determined. | wikipedia |
e-function | In mathematics, E-functions are a type of power series that satisfy particular arithmetic conditions on the coefficients. They are of interest in transcendental number theory, and are more special than G-functions. | wikipedia |
e6 (mathematics) | In mathematics, E6 is the name of some closely related Lie groups, linear algebraic groups or their Lie algebras e 6 {\displaystyle {\mathfrak {e}}_{6}} , 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 Élie Cartan § Work). 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. | wikipedia |
e6 (mathematics) | 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. | wikipedia |
e7 (mathematics) | In mathematics, E7 is the name of several closely related Lie groups, linear algebraic groups or their Lie algebras e7, all of which have dimension 133; the same notation E7 is used for the corresponding root lattice, which has rank 7. The designation E7 comes from the Cartan–Killing classification of the complex simple Lie algebras, which fall into four infinite series labeled An, Bn, Cn, Dn, and five exceptional cases labeled E6, E7, E8, F4, and G2. The E7 algebra is thus one of the five exceptional cases. The fundamental group of the (adjoint) complex form, compact real form, or any algebraic version of E7 is the cyclic group Z/2Z, and its outer automorphism group is the trivial group. The dimension of its fundamental representation is 56. | wikipedia |
e8 lie algebra | In mathematics, E8 is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding root lattice, which has rank 8. The designation E8 comes from the Cartan–Killing classification of the complex simple Lie algebras, which fall into four infinite series labeled An, Bn, Cn, Dn, and five exceptional cases labeled G2, F4, E6, E7, and E8. The E8 algebra is the largest and most complicated of these exceptional cases. | wikipedia |
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. | wikipedia |
f4 (mathematics) | 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. | wikipedia |
f4 (mathematics) | 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. | wikipedia |
klein j-invariant | In mathematics, Felix Klein's j-invariant or j function, regarded as a function of a complex variable τ, is a modular function of weight zero for SL(2, Z) defined on the upper half-plane of complex numbers. It is the unique such function which is holomorphic away from a simple pole at the cusp such that j ( e 2 π i / 3 ) = 0 , j ( i ) = 1728 = 12 3 . {\displaystyle j\left(e^{2\pi i/3}\right)=0,\quad j(i)=1728=12^{3}.} Rational functions of j are modular, and in fact give all modular functions. Classically, the j-invariant was studied as a parameterization of elliptic curves over C, but it also has surprising connections to the symmetries of the Monster group (this connection is referred to as monstrous moonshine). | wikipedia |
ferrers function | In mathematics, Ferrers functions are certain special functions defined in terms of hypergeometric functions. They are named after Norman Macleod Ferrers. | wikipedia |
per enflo | In mathematics, Functional analysis is concerned with the study of vector spaces and operators acting upon them. It has its historical roots in the study of functional spaces, in particular transformations of functions, such as the Fourier transform, as well as in the study of differential and integral equations. In functional analysis, an important class of vector spaces consists of the complete normed vector spaces over the real or complex numbers, which are called Banach spaces. | wikipedia |
per enflo | An important example of a Banach space is a Hilbert space, where the norm arises from an inner product. Hilbert spaces are of fundamental importance in many areas, including the mathematical formulation of quantum mechanics, stochastic processes, and time-series analysis. Besides studying spaces of functions, functional analysis also studies the continuous linear operators on spaces of functions. | wikipedia |
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 g 2 , {\displaystyle {\mathfrak {g}}_{2},} 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). | wikipedia |
noncommutative algebra | In mathematics, Goldie's theorem is a basic structural result in ring theory, proved by Alfred Goldie during the 1950s. What is now termed a right Goldie ring is a ring R that has finite uniform dimension (also called "finite rank") as a right module over itself, and satisfies the ascending chain condition on right annihilators of subsets of R. Goldie's theorem states that the semiprime right Goldie rings are precisely those that have a semisimple Artinian right classical ring of quotients. The structure of this ring of quotients is then completely determined by the Artin–Wedderburn theorem. In particular, Goldie's theorem applies to semiprime right Noetherian rings, since by definition right Noetherian rings have the ascending chain condition on all right ideals. | wikipedia |
noncommutative algebra | This is sufficient to guarantee that a right-Noetherian ring is right Goldie. The converse does not hold: every right Ore domain is a right Goldie domain, and hence so is every commutative integral domain. A consequence of Goldie's theorem, again due to Goldie, is that every semiprime principal right ideal ring is isomorphic to a finite direct sum of prime principal right ideal rings. Every prime principal right ideal ring is isomorphic to a matrix ring over a right Ore domain. | wikipedia |
gosper's algorithm | In mathematics, Gosper's algorithm, due to Bill Gosper, is a procedure for finding sums of hypergeometric terms that are themselves hypergeometric terms. That is: suppose one has a(1) + ... + a(n) = S(n) − S(0), where S(n) is a hypergeometric term (i.e., S(n + 1)/S(n) is a rational function of n); then necessarily a(n) is itself a hypergeometric term, and given the formula for a(n) Gosper's algorithm finds that for S(n). | wikipedia |
grothendieck's galois theory | In mathematics, Grothendieck's Galois theory is an abstract approach to the Galois theory of fields, developed around 1960 to provide a way to study the fundamental group of algebraic topology in the setting of algebraic geometry. It provides, in the classical setting of field theory, an alternative perspective to that of Emil Artin based on linear algebra, which became standard from about the 1930s. The approach of Alexander Grothendieck is concerned with the category-theoretic properties that characterise the categories of finite G-sets for a fixed profinite group G. For example, G might be the group denoted Z ^ {\displaystyle {\hat {\mathbb {Z} }}} , which is the inverse limit of the cyclic additive groups Z/nZ — or equivalently the completion of the infinite cyclic group Z for the topology of subgroups of finite index. A finite G-set is then a finite set X on which G acts through a quotient finite cyclic group, so that it is specified by giving some permutation of X. In the above example, a connection with classical Galois theory can be seen by regarding Z ^ {\displaystyle {\hat {\mathbb {Z} }}} as the profinite Galois group Gal(F/F) of the algebraic closure F of any finite field F, over F. That is, the automorphisms of F fixing F are described by the inverse limit, as we take larger and larger finite splitting fields over F. The connection with geometry can be seen when we look at covering spaces of the unit disk in the complex plane with the origin removed: the finite covering realised by the zn map of the disk, thought of by means of a complex number variable z, corresponds to the subgroup n.Z of the fundamental group of the punctured disk. | wikipedia |
grothendieck's galois theory | The theory of Grothendieck, published in SGA1, shows how to reconstruct the category of G-sets from a fibre functor Φ, which in the geometric setting takes the fibre of a covering above a fixed base point (as a set). In fact there is an isomorphism proved of the type G ≅ Aut(Φ),the latter being the group of automorphisms (self-natural equivalences) of Φ. An abstract classification of categories with a functor to the category of sets is given, by means of which one can recognise categories of G-sets for G profinite. To see how this applies to the case of fields, one has to study the tensor product of fields. In topos theory this is a part of the study of atomic toposes. | wikipedia |
hadamard's gamma function | In mathematics, Hadamard's gamma function, named after Jacques Hadamard, is an extension of the factorial function, different from the classical gamma function (it is an instance of a pseudogamma function.) This function, with its argument shifted down by 1, interpolates the factorial and extends it to real and complex numbers in a different way than Euler's gamma function. It is defined as: H ( x ) = 1 Γ ( 1 − x ) d d x { ln ( Γ ( 1 2 − x 2 ) Γ ( 1 − x 2 ) ) } , {\displaystyle H(x)={\frac {1}{\Gamma (1-x)}}\,{\dfrac {d}{dx}}\left\{\ln \left({\frac {\Gamma ({\frac {1}{2}}-{\frac {x}{2}})}{\Gamma (1-{\frac {x}{2}})}}\right)\right\},} where Γ(x) denotes the classical gamma function. If n is a positive integer, then: H ( n ) = Γ ( n ) = ( n − 1 ) ! {\displaystyle H(n)=\Gamma (n)=(n-1)!} | wikipedia |
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