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matrix coefficient | They arise naturally from finite-dimensional representations of G as the matrix-entry functions of the corresponding matrix representations. The Peter–Weyl theorem says that the matrix coefficients on G are dense in the Hilbert space of square-integrable functions on G. Matrix coefficients of representations of Lie gro... | wikipedia |
m-theory | In mathematics, a matrix is a rectangular array of numbers or other data. In physics, a matrix model is a particular kind of physical theory whose mathematical formulation involves the notion of a matrix in an important way. A matrix model describes the behavior of a set of matrices within the framework of quantum mech... | wikipedia |
m-theory | In their original paper, these authors showed, among other things, that the low energy limit of this matrix model is described by eleven-dimensional supergravity. These calculations led them to propose that the BFSS matrix model is exactly equivalent to M-theory. The BFSS matrix model can therefore be used as a prototy... | wikipedia |
gauge–gravity duality | In mathematics, a matrix is a rectangular array of numbers or other data. In physics, a matrix model is a particular kind of physical theory whose mathematical formulation involves the notion of a matrix in an important way. A matrix model describes the behavior of a set of matrices within the framework of quantum mech... | wikipedia |
gauge–gravity duality | In their original paper, these authors showed, among other things, that the low energy limit of this matrix model is described by eleven-dimensional supergravity. These calculations led them to propose that the BFSS matrix model is exactly equivalent to M-theory. The BFSS matrix model can therefore be used as a prototy... | wikipedia |
gauge–gravity duality | This subject is a generalization of ordinary geometry in which mathematicians define new geometric notions using tools from noncommutative algebra. In a paper from 1998, Alain Connes, Michael R. Douglas, and Albert Schwarz showed that some aspects of matrix models and M-theory are described by a noncommutative quantum ... | wikipedia |
meander (mathematics) | In mathematics, a meander or closed meander is a self-avoiding closed curve which crosses a given line a number of times, meaning that it intersects the line while passing from one side to the other. Intuitively, a meander can be viewed as a meandering river with a straight road crossing the river over a number of brid... | wikipedia |
measure algebra | In mathematics, a measure algebra is a Boolean algebra with a countably additive positive measure. A probability measure on a measure space gives a measure algebra on the Boolean algebra of measurable sets modulo null sets. | wikipedia |
mock theta functions | In mathematics, a mock modular form is the holomorphic part of a harmonic weak Maass form, and a mock theta function is essentially a mock modular form of weight 1/2. The first examples of mock theta functions were described by Srinivasa Ramanujan in his last 1920 letter to G. H. Hardy and in his lost notebook. Sander ... | wikipedia |
modular lie algebra | In mathematics, a modular Lie algebra is a Lie algebra over a field of positive characteristic. The theory of modular Lie algebras is significantly different from the theory of real and complex Lie algebras. This difference can be traced to the properties of Frobenius automorphism and to the failure of the exponential ... | wikipedia |
elliptic modular form | In mathematics, a modular form is a (complex) analytic function on the upper half-plane that satisfies: a kind of functional equation with respect to the group action of the modular group, and a growth condition.The theory of modular forms therefore belongs to complex analysis. The main importance of the theory is its ... | wikipedia |
elliptic modular form | Modular form theory is a special case of the more general theory of automorphic forms, which are functions defined on Lie groups that transform nicely with respect to the action of certain discrete subgroups, generalizing the example of the modular group S L 2 ( Z ) ⊂ S L 2 ( R ) {\displaystyle \mathrm {SL} _{2}(\mathb... | wikipedia |
monopole (mathematics) | In mathematics, a monopole is a connection over a principal bundle G with a section of the associated adjoint bundle. | wikipedia |
monotonically non-decreasing | In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory. | wikipedia |
many-valued function | In mathematics, a multivalued function, also called multifunction and many-valued function, is a set-valued function with continuity properties that allow considering it locally as an ordinary function. Multivalued functions arise commonly in applications of the implicit function theorem, since this theorem can be view... | wikipedia |
many-valued function | It cannot be considered as an ordinary function, since, when one follows one value of the logarithm along a circle centered at 0, one gets another value than the starting one after a complete turn. This phenomenon is called monodromy. Another common way for defining a multivalued function is analytic continuation, whic... | wikipedia |
isotope (jordan algebra) | In mathematics, a mutation, also called a homotope, of a unital Jordan algebra is a new Jordan algebra defined by a given element of the Jordan algebra. The mutation has a unit if and only if the given element is invertible, in which case the mutation is called a proper mutation or an isotope. Mutations were first intr... | wikipedia |
isotope (jordan algebra) | Their functorial properties allow an explicit construction of the corresponding Hermitian symmetric space of compact type as a compactification of a finite-dimensional complex semisimple Jordan algebra. The automorphism group of the compactification becomes a complex subgroup, the complexification of its maximal compac... | wikipedia |
near-field (mathematics) | In mathematics, a near-field is an algebraic structure similar to a division ring, except that it has only one of the two distributive laws. Alternatively, a near-field is a near-ring in which there is a multiplicative identity and every non-zero element has a multiplicative inverse. | wikipedia |
relativistic system (mathematics) | In mathematics, a non-autonomous system of ordinary differential equations is defined to be a dynamic equation on a smooth fiber bundle Q → R {\displaystyle Q\to \mathbb {R} } over R {\displaystyle \mathbb {R} } . For instance, this is the case of non-relativistic non-autonomous mechanics, but not relativistic mechanic... | wikipedia |
relativistic system (mathematics) | Therefore, it is called the relativistic system. In particular, Special Relativity on the Minkowski space Q = R 4 {\displaystyle Q=\mathbb {R} ^{4}} is of this type. Since a configuration space Q {\displaystyle Q} of a relativistic system has no preferable fibration over R {\displaystyle \mathbb {R} } , a velocity spac... | wikipedia |
relativistic system (mathematics) | The notion of jets of submanifolds generalizes that of jets of sections of fiber bundles which are utilized in covariant classical field theory and non-autonomous mechanics. A first order jet bundle J 1 1 Q → Q {\displaystyle J_{1}^{1}Q\to Q} is projective and, following the terminology of Special Relativity, one can t... | wikipedia |
relativistic system (mathematics) | Given coordinates ( q 0 , q i ) {\displaystyle (q^{0},q^{i})} on Q {\displaystyle Q} , a first order jet manifold J 1 1 Q {\displaystyle J_{1}^{1}Q} is provided with the adapted coordinates ( q 0 , q i , q 0 i ) {\displaystyle (q^{0},q^{i},q_{0}^{i})} possessing transition functions q ′ 0 = q ′ 0 ( q 0 , q k ) , q ′ i ... | wikipedia |
relativistic system (mathematics) | Then a generic equation of motion of a relativistic system in terms of relativistic velocities reads ( ∂ λ G μ α 2 … α 2 N 2 N − ∂ μ G λ α 2 … α 2 N ) q τ μ q τ α 2 ⋯ q τ α 2 N − ( 2 N − 1 ) G λ μ α 3 … α 2 N q τ τ μ q τ α 3 ⋯ q τ α 2 N + F λ μ q τ μ = 0 , {\displaystyle \left({\frac {\partial _{\lambda }G_{\mu \alpha ... | wikipedia |
normed algebra | In mathematics, a normed algebra A is an algebra over a field which has a sub-multiplicative norm: ∀ x , y ∈ A ‖ x y ‖ ≤ ‖ x ‖ ‖ y ‖ . {\displaystyle \forall x,y\in A\qquad \|xy\|\leq \|x\|\|y\|.} Some authors require it to have a multiplicative identity 1A such that ║1A║ = 1. | wikipedia |
nowhere continuous function | In mathematics, a nowhere continuous function, also called an everywhere discontinuous function, is a function that is not continuous at any point of its domain. If f {\displaystyle f} is a function from real numbers to real numbers, then f {\displaystyle f} is nowhere continuous if for each point x {\displaystyle x} t... | wikipedia |
harmonic (mathematics) | In mathematics, a number of concepts employ the word harmonic. The similarity of this terminology to that of music is not accidental: the equations of motion of vibrating strings, drums and columns of air are given by formulas involving Laplacians; the solutions to which are given by eigenvalues corresponding to their ... | wikipedia |
data cube | In mathematics, a one-dimensional array corresponds to a vector, a two-dimensional array resembles a matrix; more generally, a tensor may be represented as an n-dimensional data cube. | wikipedia |
p-adic zeta function | In mathematics, a p-adic zeta function, or more generally a p-adic L-function, is a function analogous to the Riemann zeta function, or more general L-functions, but whose domain and target are p-adic (where p is a prime number). For example, the domain could be the p-adic integers Zp, a profinite p-group, or a p-adic ... | wikipedia |
p-adic zeta function | For example, Kubota–Leopoldt used Kummer's congruences for Bernoulli numbers to construct a p-adic L-function, the p-adic Riemann zeta function ζp(s), whose values at negative odd integers are those of the Riemann zeta function at negative odd integers (up to an explicit correction factor). p-adic L-functions arising i... | wikipedia |
p-adic zeta function | A p-adic L-function arising in this way is typically called an arithmetic p-adic L-function as it encodes arithmetic data of the Galois module involved. The main conjecture of Iwasawa theory (now a theorem due to Barry Mazur and Andrew Wiles) is the statement that the Kubota–Leopoldt p-adic L-function and an arithmetic... | wikipedia |
pair of pants (mathematics) | In mathematics, a pair of pants is a surface which is homeomorphic to the three-holed sphere. The name comes from considering one of the removed disks as the waist and the two others as the cuffs of a pair of pants. Pairs of pants are used as building blocks for compact surfaces in various theories. Two important appli... | wikipedia |
cantor's pairing function | In mathematics, a pairing function is a process to uniquely encode two natural numbers into a single natural number.Any pairing function can be used in set theory to prove that integers and rational numbers have the same cardinality as natural numbers. | wikipedia |
parallelization (mathematics) | In mathematics, a parallelization of a manifold M {\displaystyle M\,} of dimension n is a set of n global smooth linearly independent vector fields. | wikipedia |
paratopological group | In mathematics, a paratopological group is a topological semigroup that is algebraically a group. In other words, it is a group G with a topology such that the group's product operation is a continuous function from G × G to G. This differs from the definition of a topological group in that the group inverse is not req... | wikipedia |
parent function | In mathematics, a parent function is the core representation of a function type without manipulations such as translation and dilation. For example, for the family of quadratic functions having the general form y = a x 2 + b x + c , {\displaystyle y=ax^{2}+bx+c\,,} the simplest function is y = x 2 {\displaystyle y=x^{2... | wikipedia |
parent function | For example, the graph of y = x2 − 4x + 7 can be obtained from the graph of y = x2 by translating +2 units along the X axis and +3 units along Y axis. This is because the equation can also be written as y − 3 = (x − 2)2. For many trigonometric functions, the parent function is usually a basic sin(x), cos(x), or tan(x). | wikipedia |
parent function | For example, the graph of y = A sin(x) + B cos(x) can be obtained from the graph of y = sin(x) by translating it through an angle α along the positive X axis (where tan(α) = A⁄B), then stretching it parallel to the Y axis using a stretch factor R, where R2 = A2 + B2. This is because A sin(x) + B cos(x) can be written a... | wikipedia |
partial functions | In mathematics, a partial function f from a set X to a set Y is a function from a subset S of X (possibly the whole X itself) to Y. The subset S, that is, the domain of f viewed as a function, is called the domain of definition or natural domain of f. If S equals X, that is, if f is defined on every element in X, then ... | wikipedia |
partial functions | This is the case in calculus, where, for example, the quotient of two functions is a partial function whose domain of definition cannot contain the zeros of the denominator. For this reason, in calculus, and more generally in mathematical analysis, a partial function is generally called simply a function. | wikipedia |
partial functions | In computability theory, a general recursive function is a partial function from the integers to the integers; no algorithm can exist for deciding whether an arbitrary such function is in fact total. When arrow notation is used for functions, a partial function f {\displaystyle f} from X {\displaystyle X} to Y {\displa... | wikipedia |
partial group algebra | In mathematics, a partial group algebra is an associative algebra related to the partial representations of a group. | wikipedia |
phase line (mathematics) | In mathematics, a phase line is a diagram that shows the qualitative behaviour of an autonomous ordinary differential equation in a single variable, d y d x = f ( y ) {\displaystyle {\tfrac {dy}{dx}}=f(y)} . The phase line is the 1-dimensional form of the general n {\displaystyle n} -dimensional phase space, and can be... | wikipedia |
piecewise algebraic space | In mathematics, a piecewise algebraic space is a generalization of a semialgebraic set, introduced by Maxim Kontsevich and Yan Soibelman. The motivation was for the proof of Deligne's conjecture on Hochschild cohomology. Robert Hardt, Pascal Lambrechts, Victor Turchin, and Ismar Volić later developed the theory. | wikipedia |
point process | In mathematics, a point process is a random element whose values are "point patterns" on a set S. While in the exact mathematical definition a point pattern is specified as a locally finite counting measure, it is sufficient for more applied purposes to think of a point pattern as a countable subset of S that has no li... | wikipedia |
point source | In mathematics, a point source is a singularity from which flux or flow is emanating. Although singularities such as this do not exist in the observable universe, mathematical point sources are often used as approximations to reality in physics and other fields. | wikipedia |
polylogarithmic function | In mathematics, a polylogarithmic function in n is a polynomial in the logarithm of n, a k ( log n ) k + a k − 1 ( log n ) k − 1 + ⋯ + a 1 ( log n ) + a 0 . {\displaystyle a_{k}(\log n)^{k}+a_{k-1}(\log n)^{k-1}+\cdots +a_{1}(\log n)+a_{0}.} The notation logkn is often used as a shorthand for (log n)k, analogous ... | wikipedia |
polylogarithmic function | In computer science, polylogarithmic functions occur as the order of time or memory used by some algorithms (e.g., "it has polylogarithmic order"), such as in the definition of QPTAS (see PTAS). All polylogarithmic functions of n are o(nε) for every exponent ε > 0 (for the meaning of this symbol, see small o notation),... | wikipedia |
polynomial multiplication | In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example of a polynomial of a single indeterminate x is x2 − 4x + 7. An example wit... | wikipedia |
polynomial multiplication | Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemist... | wikipedia |
positive-semidefinite function | In mathematics, a positive-definite function is, depending on the context, either of two types of function. | wikipedia |
pre-lie algebra | In mathematics, a pre-Lie algebra is an algebraic structure on a vector space that describes some properties of objects such as rooted trees and vector fields on affine space. The notion of pre-Lie algebra has been introduced by Murray Gerstenhaber in his work on deformations of algebras. Pre-Lie algebras have been con... | wikipedia |
principal subalgebra | In mathematics, a principal subalgebra of a complex simple Lie algebra is a 3-dimensional simple subalgebra whose non-zero elements are regular. A finite-dimensional complex simple Lie algebra has a unique conjugacy class of principal subalgebras, each of which is the span of an sl2-triple. | wikipedia |
mathematical product | In mathematics, a product is the result of multiplication, or an expression that identifies objects (numbers or variables) to be multiplied, called factors. For example, 21 is the product of 3 and 7 (the result of multiplication), and x ⋅ ( 2 + x ) {\displaystyle x\cdot (2+x)} is the product of x {\displaystyle x} and ... | wikipedia |
mathematical product | When matrices or members of various other associative algebras are multiplied, the product usually depends on the order of the factors. Matrix multiplication, for example, is non-commutative, and so is multiplication in other algebras in general as well. There are many different kinds of products in mathematics: beside... | wikipedia |
progressive function | In mathematics, a progressive function ƒ ∈ L2(R) is a function whose Fourier transform is supported by positive frequencies only: s u p p f ^ ⊆ R + . {\displaystyle \mathop {\rm {supp}} {\hat {f}}\subseteq \mathbb {R} _{+}.} It is called super regressive if and only if the time reversed function f(−t) is progressive,... | wikipedia |
progressive function | The complex conjugate of a progressive function is regressive, and vice versa. The space of progressive functions is sometimes denoted H + 2 ( R ) {\displaystyle H_{+}^{2}(R)} , which is known as the Hardy space of the upper half-plane. This is because a progressive function has the Fourier inversion formula f ( t ) = ... | wikipedia |
progressive function | {\displaystyle f(t+iu)=\int _{0}^{\infty }e^{2\pi is(t+iu)}{\hat {f}}(s)\,ds=\int _{0}^{\infty }e^{2\pi ist}e^{-2\pi su}{\hat {f}}(s)\,ds.} Conversely, every holomorphic function on the upper half-plane which is uniformly square-integrable on every horizontal line will arise in this manner. Regressive functions are sim... | wikipedia |
projection (mathematics) | In mathematics, a projection is an idempotent mapping of a set (or other mathematical structure) into a subset (or sub-structure). In this case, idempotent means that projecting twice is the same as projecting once. The restriction to a subspace of a projection is also called a projection, even if the idempotence prope... | wikipedia |
projection (mathematics) | The shadow of a three-dimensional sphere is a closed disk. Originally, the notion of projection was introduced in Euclidean geometry to denote the projection of the three-dimensional Euclidean space onto a plane in it, like the shadow example. The two main projections of this kind are: The projection from a point onto ... | wikipedia |
projection (mathematics) | The points P such that the line CP is parallel to the plane does not have any image by the projection, but one often says that they project to a point at infinity of the plane (see Projective geometry for a formalization of this terminology). The projection of the point C itself is not defined. The projection parallel ... | wikipedia |
projection (mathematics) | This rudimentary idea was refined and abstracted, first in a geometric context and later in other branches of mathematics. Over time different versions of the concept developed, but today, in a sufficiently abstract setting, we can unify these variations.In cartography, a map projection is a map of a part of the surfac... | wikipedia |
projectionless c*-algebra | In mathematics, a projectionless C*-algebra is a C*-algebra with no nontrivial projections. For a unital C*-algebra, the projections 0 and 1 are trivial. While for a non-unital C*-algebra, only 0 is considered trivial. | wikipedia |
projectionless c*-algebra | The problem of whether simple infinite-dimensional C*-algebras with this property exist was posed in 1958 by Irving Kaplansky, and the first example of one was published in 1981 by Bruce Blackadar. For commutative C*-algebras, being projectionless is equivalent to its spectrum being connected. Due to this, being projec... | wikipedia |
property (mathematics) | In mathematics, a property is any characteristic that applies to a given set. Rigorously, a property p defined for all elements of a set X is usually defined as a function p: X → {true, false}, that is true whenever the property holds; or equivalently, as the subset of X for which p holds; i.e. the set {x | p(x) = true... | wikipedia |
pseudo-reductive algebraic group | In mathematics, a pseudo-reductive group over a field k (sometimes called a k-reductive group) is a smooth connected affine algebraic group defined over k whose k-unipotent radical (i.e., largest smooth connected unipotent normal k-subgroup) is trivial. Over perfect fields these are the same as (connected) reductive gr... | wikipedia |
pseudo-reductive algebraic group | Pseudo-reductive groups arise naturally in the study of algebraic groups over function fields of positive-dimensional varieties in positive characteristic (even over a perfect field of constants). Springer (1998) gives an exposition of Tits' results on pseudo-reductive groups, while Conrad, Gabber & Prasad (2010) build... | wikipedia |
pseudogamma function | In mathematics, a pseudogamma function is a function that interpolates the factorial. The gamma function is the most famous solution to the problem of extending the notion of the factorial beyond the positive integers only. However, it is clearly not the only solution, as, for any set of points, an infinite number of c... | wikipedia |
pseudogamma function | Such a curve, namely one which interpolates the factorial but is not equal to the gamma function, is known as a pseudogamma function. The two most famous pseudogamma functions are Hadamard's gamma function: H ( x ) = ψ ( 1 − x 2 ) − ψ ( 1 2 − x 2 ) 2 Γ ( 1 − x ) {\displaystyle H(x)={\frac {\psi \left(1-{\frac {x}{2}}\r... | wikipedia |
quadratic algebra | In mathematics, a quadratic algebra is a filtered algebra generated by degree one elements, with defining relations of degree 2. It was pointed out by Yuri Manin that such algebras play an important role in the theory of quantum groups. The most important class of graded quadratic algebras is Koszul algebras. | wikipedia |
quadratic functions | In mathematics, a quadratic polynomial is a polynomial of degree two in one or more variables. A quadratic function is the polynomial function defined by a quadratic polynomial. Before the 20th century, the distinction was unclear between a polynomial and its associated polynomial function; so "quadratic polynomial" an... | wikipedia |
quadratic functions | For example, a univariate (single-variable) quadratic function has the form f ( x ) = a x 2 + b x + c , a ≠ 0 , {\displaystyle f(x)=ax^{2}+bx+c,\quad a\neq 0,} where x is its variable. The graph of a univariate quadratic function is a parabola, a curve that has an axis of symmetry parallel to the y-axis. If a quadratic... | wikipedia |
quadratic functions | The solutions of a quadratic equation are the zeros of the corresponding quadratic function. The bivariate case in terms of variables x and y has the form f ( x , y ) = a x 2 + b x y + c y 2 + d x + e y + f , {\displaystyle f(x,y)=ax^{2}+bxy+cy^{2}+dx+ey+f,} with at least one of a, b, c not equal to zero. The zeros of ... | wikipedia |
quadratic functions | A quadratic function in three variables x, y, and z contains exclusively terms x2, y2, z2, xy, xz, yz, x, y, z, and a constant: f ( x , y , z ) = a x 2 + b y 2 + c z 2 + d x y + e x z + f y z + g x + h y + i z + j , {\displaystyle f(x,y,z)=ax^{2}+by^{2}+cz^{2}+dxy+exz+fyz+gx+hy+iz+j,} where at least one of the coeffici... | wikipedia |
quadratic-linear algebra | In mathematics, a quadratic-linear algebra is an algebra over a field with a presentation such that all relations are sums of monomials of degrees 1 or 2 in the generators. They were introduced by Polishchuk and Positselski (2005, p.101). An example is the universal enveloping algebra of a Lie algebra, with generators ... | wikipedia |
quadric (algebraic geometry) | In mathematics, a quadric or quadric hypersurface is the subspace of N-dimensional space defined by a polynomial equation of degree 2 over a field. Quadrics are fundamental examples in algebraic geometry. The theory is simplified by working in projective space rather than affine space. | wikipedia |
quadric (algebraic geometry) | An example is the quadric surface x y = z w {\displaystyle xy=zw} in projective space P 3 {\displaystyle {\mathbf {P} }^{3}} over the complex numbers C. A quadric has a natural action of the orthogonal group, and so the study of quadrics can be considered as a descendant of Euclidean geometry. Many properties of quadri... | wikipedia |
quantum affine algebra | In mathematics, a quantum affine algebra (or affine quantum group) is a Hopf algebra that is a q-deformation of the universal enveloping algebra of an affine Lie algebra. They were introduced independently by Drinfeld (1985) and Jimbo (1985) as a special case of their general construction of a quantum group from a Cart... | wikipedia |
quantum groupoid | In mathematics, a quantum groupoid is any of a number of notions in noncommutative geometry analogous to the notion of groupoid. In usual geometry, the information of a groupoid can be contained in its monoidal category of representations (by a version of Tannaka–Krein duality), in its groupoid algebra or in the commut... | wikipedia |
quantized enveloping algebra | In mathematics, a quantum or quantized enveloping algebra is a q-analog of a universal enveloping algebra. Given a Lie algebra g {\displaystyle {\mathfrak {g}}} , the quantum enveloping algebra is typically denoted as U q ( g ) {\displaystyle U_{q}({\mathfrak {g}})} . The notation was introduced by Drinfeld and indepen... | wikipedia |
quasi-frobenius lie algebra | In mathematics, a quasi-Frobenius Lie algebra ( g , , β ) {\displaystyle ({\mathfrak {g}},,\beta )} over a field k {\displaystyle k} is a Lie algebra ( g , ) {\displaystyle ({\mathfrak {g}},)} equipped with a nondegenerate skew-symmetric bilinear form β: g × g → k {\displaystyle \beta :{\mathfrak {g}}\times {\mathfra... | wikipedia |
quasi-lie algebra | In mathematics, a quasi-Lie algebra in abstract algebra is just like a Lie algebra, but with the usual axiom = 0 {\displaystyle =0} replaced by = − {\displaystyle =-} (anti-symmetry).In characteristic other than 2, these are equivalent (in the presence of bilinearity), so this distinction doesn't arise when consider... | wikipedia |
denjoy–carleman theorem | In mathematics, a quasi-analytic class of functions is a generalization of the class of real analytic functions based upon the following fact: If f is an analytic function on an interval ⊂ R, and at some point f and all of its derivatives are zero, then f is identically zero on all of . Quasi-analytic classes are broa... | wikipedia |
quasitopological space | In mathematics, a quasi-topology on a set X is a function that associates to every compact Hausdorff space C a collection of mappings from C to X satisfying certain natural conditions. A set with a quasi-topology is called a quasitopological space. They were introduced by Spanier, who showed that there is a natural qua... | wikipedia |
quasiconcave function | In mathematics, a quasiconvex function is a real-valued function defined on an interval or on a convex subset of a real vector space such that the inverse image of any set of the form ( − ∞ , a ) {\displaystyle (-\infty ,a)} is a convex set. For a function of a single variable, along any stretch of the curve the highes... | wikipedia |
quasiperiodic function | In mathematics, a quasiperiodic function is a function that has a certain similarity to a periodic function. A function f {\displaystyle f} is quasiperiodic with quasiperiod ω {\displaystyle \omega } if f ( z + ω ) = g ( z , f ( z ) ) {\displaystyle f(z+\omega )=g(z,f(z))} , where g {\displaystyle g} is a "simpler" fun... | wikipedia |
quasiperiodic function | Another example is provided by the Weierstrass sigma function, which is quasiperiodic in two independent quasiperiods, the periods of the corresponding Weierstrass ℘ function. Functions with an additive functional equation f ( z + ω ) = f ( z ) + a z + b {\displaystyle f(z+\omega )=f(z)+az+b\ } are also called quasiper... | wikipedia |
quaternion algebra | In mathematics, a quaternion algebra over a field F is a central simple algebra A over F that has dimension 4 over F. Every quaternion algebra becomes a matrix algebra by extending scalars (equivalently, tensoring with a field extension), i.e. for a suitable field extension K of F, A ⊗ F K {\displaystyle A\otimes _{F}K... | wikipedia |
quintic equation | In mathematics, a quintic function is a function of the form g ( x ) = a x 5 + b x 4 + c x 3 + d x 2 + e x + f , {\displaystyle g(x)=ax^{5}+bx^{4}+cx^{3}+dx^{2}+ex+f,\,} where a, b, c, d, e and f are members of a field, typically the rational numbers, the real numbers or the complex numbers, and a is nonzero. In other ... | wikipedia |
quintic equation | The derivative of a quintic function is a quartic function. Setting g(x) = 0 and assuming a ≠ 0 produces a quintic equation of the form: a x 5 + b x 4 + c x 3 + d x 2 + e x + f = 0. {\displaystyle ax^{5}+bx^{4}+cx^{3}+dx^{2}+ex+f=0.\,} Solving quintic equations in terms of radicals (nth roots) was a major problem in al... | wikipedia |
radial function | In mathematics, a radial function is a real-valued function defined on a Euclidean space Rn whose value at each point depends only on the distance between that point and the origin. The distance is usually the Euclidean distance. For example, a radial function Φ in two dimensions has the form Φ ( x , y ) = φ ( r ) , r ... | wikipedia |
radial function | A function is radial if and only if it is invariant under all rotations leaving the origin fixed. That is, ƒ is radial if and only if f ∘ ρ = f {\displaystyle f\circ \rho =f\,} for all ρ ∈ SO(n), the special orthogonal group in n dimensions. This characterization of radial functions makes it possible also to define rad... | wikipedia |
radial function | These are distributions S on Rn such that S = S {\displaystyle S=S} for every test function φ and rotation ρ. Given any (locally integrable) function ƒ, its radial part is given by averaging over spheres centered at the origin. To wit, ϕ ( x ) = 1 ω n − 1 ∫ S n − 1 f ( r x ′ ) d x ′ {\displaystyle \phi (x)={\frac {1}... | wikipedia |
radially unbounded function | In mathematics, a radially unbounded function is a function f: R n → R {\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} } for which Or equivalently, Such functions are applied in control theory and required in optimization for determination of compact spaces. Notice that the norm used in the definition can be a... | wikipedia |
proper rational function | In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rational numbers; they may be taken in any field K. In this case, one speaks of... | wikipedia |
read-once function | In mathematics, a read-once function is a special type of Boolean function that can be described by a Boolean expression in which each variable appears only once. More precisely, the expression is required to use only the operations of logical conjunction, logical disjunction, and negation. By applying De Morgan's laws... | wikipedia |
uniform continuity | In mathematics, a real function f {\displaystyle f} of real numbers is said to be uniformly continuous if there is a positive real number δ {\displaystyle \delta } such that function values over any function domain interval of the size δ {\displaystyle \delta } are as close to each other as we want. In other words, for... | wikipedia |
uniform continuity | The concepts of uniform continuity and continuity can be expanded to functions defined between metric spaces. Continuous functions can fail to be uniformly continuous if they are unbounded on a bounded domain, such as f ( x ) = 1 x {\displaystyle f(x)={\tfrac {1}{x}}} on ( 0 , 1 ) {\displaystyle (0,1)} , or if their sl... | wikipedia |
uniform continuity | However, any Lipschitz map between metric spaces is uniformly continuous, in particular any isometry (distance-preserving map). Although continuity can be defined for functions between general topological spaces, defining uniform continuity requires more structure. The concept relies on comparing the sizes of neighbour... | wikipedia |
singular function | In mathematics, a real-valued function f on the interval is said to be singular if it has the following properties: f is continuous on . (**) there exists a set N of measure 0 such that for all x outside of N the derivative f ′(x) exists and is zero, that is, the derivative of f vanishes almost everywhere. f is non-co... | wikipedia |
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