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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 configura... | 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) co... | 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... | 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 unifor... | 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 t... | 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 perco... | 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. Continuu... | 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 ... | 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 con... | 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... | 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, ... | 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 distr... | 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 communicatio... | 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 wireles... | 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}}... | 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 {\displaystyl... | 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 studie... | 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 formali... | 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 unit... | 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 ... | 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 phy... | 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, ... | 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 exchang... | 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*... | 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 a... | 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 proble... | 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 attr... | 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 fin... | 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 br... | 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... | 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 count... | 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... | 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,... | 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 c... | 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 c... | 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 ... | 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 divis... | 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 {\di... | 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 {\displa... | 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 \Ga... | 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... | 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 o... | 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 (... | 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... | 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 sim... | 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 ... | 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 modifie... | 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 a... | 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\righta... | 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... | 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 exponentia... | 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... | 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 sim... | 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 ... | 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 ... | 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()), ... | 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 ari... | 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 state... | 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 on... | 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 ... | 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 modul... | 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 c... | 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, ... | 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 simpl... | 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... | 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 ... | 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... | 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 fu... | 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, functiona... | 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 t... | 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 ann... | 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 isom... | 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)... | 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... | 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 automorphis... | 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... | wikipedia |
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