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formula | In all cases, however, formulas form the basis for calculations. Expressions are distinct from formulas in that they cannot contain an equals sign (=). Expressions can be likened to phrases the same way formulas can be likened to grammatical sentences. | wikipedia |
fractal geometry | In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illustrated in successive magnifications of the Mandelbrot set. This exhibition of... | wikipedia |
fractal geometry | One way that fractals are different from finite geometric figures is how they scale. Doubling the edge lengths of a filled polygon multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the conventional dimension of the filled polygon). Likewise, if the radiu... | wikipedia |
fractal geometry | However, if a fractal's one-dimensional lengths are all doubled, the spatial content of the fractal scales by a power that is not necessarily an integer and is in general greater than its conventional dimension. This power is called the fractal dimension of the geometric object, to distinguish it from the conventional ... | wikipedia |
fractal geometry | Starting in the 17th century with notions of recursion, fractals have moved through increasingly rigorous mathematical treatment to the study of continuous but not differentiable functions in the 19th century by the seminal work of Bernard Bolzano, Bernhard Riemann, and Karl Weierstrass, and on to the coining of the wo... | wikipedia |
fractal geometry | That's fractals." More formally, in 1982 Mandelbrot defined fractal as follows: "A fractal is by definition a set for which the Hausdorff–Besicovitch dimension strictly exceeds the topological dimension." Later, seeing this as too restrictive, he simplified and expanded the definition to this: "A fractal is a rough or ... | wikipedia |
fractal geometry | Still later, Mandelbrot proposed "to use fractal without a pedantic definition, to use fractal dimension as a generic term applicable to all the variants".The consensus among mathematicians is that theoretical fractals are infinitely self-similar iterated and detailed mathematical constructs, of which many examples hav... | wikipedia |
free boolean algebra | In mathematics, a free Boolean algebra is a Boolean algebra with a distinguished set of elements, called generators, such that: Each element of the Boolean algebra can be expressed as a finite combination of generators, using the Boolean operations, and The generators are as independent as possible, in the sense that t... | wikipedia |
free lie algebra | In mathematics, a free Lie algebra over a field K is a Lie algebra generated by a set X, without any imposed relations other than the defining relations of alternating K-bilinearity and the Jacobi identity. | wikipedia |
bounded type (mathematics) | In mathematics, a function defined on a region of the complex plane is said to be of bounded type if it is equal to the ratio of two analytic functions bounded in that region. But more generally, a function is of bounded type in a region Ω {\displaystyle \Omega } if and only if f {\displaystyle f} is analytic on Ω {\di... | wikipedia |
bounded sequences | In mathematics, a function f defined on some set X with real or complex values is called bounded if the set of its values is bounded. In other words, there exists a real number M such that | f ( x ) | ≤ M {\displaystyle |f(x)|\leq M} for all x in X. A function that is not bounded is said to be unbounded.If f is real-va... | wikipedia |
cofunction | In mathematics, a function f is cofunction of a function g if f(A) = g(B) whenever A and B are complementary angles. This definition typically applies to trigonometric functions. The prefix "co-" can be found already in Edmund Gunter's Canon triangulorum (1620).For example, sine (Latin: sinus) and cosine (Latin: cosinu... | wikipedia |
logarithmically convex function | In mathematics, a function f is logarithmically convex or superconvex if log ∘ f {\displaystyle {\log }\circ f} , the composition of the logarithm with f, is itself a convex function. | wikipedia |
supermodular function | In mathematics, a function f: R k → R {\displaystyle f\colon \mathbb {R} ^{k}\to \mathbb {R} } is supermodular if f ( x ↑ y ) + f ( x ↓ y ) ≥ f ( x ) + f ( y ) {\displaystyle f(x\uparrow y)+f(x\downarrow y)\geq f(x)+f(y)} for all x {\displaystyle x} , y ∈ R k {\displaystyle y\in \mathbb {R} ^{k}} , where x ↑ y {\displa... | wikipedia |
closed convex function | In mathematics, a function f: R n → R {\displaystyle f:\mathbb {R} ^{n}\rightarrow \mathbb {R} } is said to be closed if for each α ∈ R {\displaystyle \alpha \in \mathbb {R} } , the sublevel set { x ∈ dom f | f ( x ) ≤ α } {\displaystyle \{x\in {\mbox{dom}}f\vert f(x)\leq \alpha \}} is a closed set. Equivalently, if th... | wikipedia |
symmetrically continuous function | In mathematics, a function f: R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } is symmetrically continuous at a point x if lim h → 0 f ( x + h ) − f ( x − h ) = 0. {\displaystyle \lim _{h\to 0}f(x+h)-f(x-h)=0.} The usual definition of continuity implies symmetric continuity, but the converse is not true. | wikipedia |
symmetrically continuous function | For example, the function x − 2 {\displaystyle x^{-2}} is symmetrically continuous at x = 0 {\displaystyle x=0} , but not continuous. Also, symmetric differentiability implies symmetric continuity, but the converse is not true just like usual continuity does not imply differentiability. The set of the symmetrically con... | wikipedia |
function evaluation | In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the set Y is called the codomain of the function.Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the... | wikipedia |
function evaluation | A function is most often denoted by letters such as f, g and h, and the value of a function f at an element x of its domain is denoted by f(x); the numerical value resulting from the function evaluation at a particular input value is denoted by replacing x with this value; for example, the value of f at x = 4 is denote... | wikipedia |
function evaluation | (x+1)^{2}\right\vert _{x=4}} (which results in 25).A function is uniquely represented by the set of all pairs (x, f (x)), called the graph of the function, a popular means of illustrating the function. When the domain and the codomain are sets of real numbers, each such pair may be thought of as the Cartesian coordinat... | wikipedia |
response variable | In mathematics, a function is a rule for taking an input (in the simplest case, a number or set of numbers) and providing an output (which may also be a number). A symbol that stands for an arbitrary input is called an independent variable, while a symbol that stands for an arbitrary output is called a dependent variab... | wikipedia |
rapidly decreasing function | In mathematics, a function is said to vanish at infinity if its values approach 0 as the input grows without bounds. There are two different ways to define this with one definition applying to functions defined on normed vector spaces and the other applying to functions defined on locally compact spaces. Aside from thi... | wikipedia |
complete symmetric function | In mathematics, a function of n {\displaystyle n} variables is symmetric if its value is the same no matter the order of its arguments. For example, a function f ( x 1 , x 2 ) {\displaystyle f\left(x_{1},x_{2}\right)} of two arguments is a symmetric function if and only if f ( x 1 , x 2 ) = f ( x 2 , x 1 ) {\displaysty... | wikipedia |
complete symmetric function | A related notion is alternating polynomials, which change sign under an interchange of variables. Aside from polynomial functions, tensors that act as functions of several vectors can be symmetric, and in fact the space of symmetric k {\displaystyle k} -tensors on a vector space V {\displaystyle V} is isomorphic to the... | wikipedia |
step function | In mathematics, a function on the real numbers is called a step function if it can be written as a finite linear combination of indicator functions of intervals. Informally speaking, a step function is a piecewise constant function having only finitely many pieces. | wikipedia |
function space | In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set X into a vector space has a natural vector space structure given by pointwise addition ... | wikipedia |
functional (mathematics) | In mathematics, a functional (as a noun) is a certain type of function. The exact definition of the term varies depending on the subfield (and sometimes even the author). In linear algebra, it is synonymous with linear forms, which are linear mappings from a vector space V {\displaystyle V} into its field of scalars (t... | wikipedia |
functional (mathematics) | Depending on the author, such mappings may or may not be assumed to be linear, or to be defined on the whole space X . {\displaystyle X.} In computer science, it is synonymous with higher-order functions, that is, functions that take functions as arguments or return them.This article is mainly concerned with the second... | wikipedia |
functional (mathematics) | The first concept, which is more modern and abstract, is discussed in detail in a separate article, under the name linear form. The third concept is detailed in the computer science article on higher-order functions. In the case where the space X {\displaystyle X} is a space of functions, the functional is a "function ... | wikipedia |
functional calculus | In mathematics, a functional calculus is a theory allowing one to apply mathematical functions to mathematical operators. It is now a branch (more accurately, several related areas) of the field of functional analysis, connected with spectral theory. (Historically, the term was also used synonymously with calculus of v... | wikipedia |
functional calculus | If f {\displaystyle f} is a function, say a numerical function of a real number, and M {\displaystyle M} is an operator, there is no particular reason why the expression f ( M ) {\displaystyle f(M)} should make sense. If it does, then we are no longer using f {\displaystyle f} on its original function domain. In the tr... | wikipedia |
functional calculus | This passes nearly unnoticed if we talk about 'squaring a matrix', though, which is the case of f ( x ) = x 2 {\displaystyle f(x)=x^{2}} and M {\displaystyle M} an n × n {\displaystyle n\times n} matrix. The idea of a functional calculus is to create a principled approach to this kind of overloading of the notation. Th... | wikipedia |
functional calculus | In the finite-dimensional case, the polynomial functional calculus yields quite a bit of information about the operator. For example, consider the family of polynomials which annihilates an operator T {\displaystyle T} . This family is an ideal in the ring of polynomials. | wikipedia |
functional calculus | Furthermore, it is a nontrivial ideal: let n {\displaystyle n} be the finite dimension of the algebra of matrices, then { I , T , T 2 , … , T n } {\displaystyle \{I,T,T^{2},\ldots ,T^{n}\}} is linearly dependent. So ∑ i = 0 n α i T i = 0 {\displaystyle \sum _{i=0}^{n}\alpha _{i}T^{i}=0} for some scalars α i {\displayst... | wikipedia |
functional calculus | Since the ring of polynomials is a principal ideal domain, this ideal is generated by some polynomial m {\displaystyle m} . Multiplying by a unit if necessary, we can choose m {\displaystyle m} to be monic. When this is done, the polynomial m {\displaystyle m} is precisely the minimal polynomial of T {\displaystyle T} ... | wikipedia |
functional calculus | This polynomial gives deep information about T {\displaystyle T} . For instance, a scalar α {\displaystyle \alpha } is an eigenvalue of T {\displaystyle T} if and only if α {\displaystyle \alpha } is a root of m {\displaystyle m} . | wikipedia |
functional calculus | Also, sometimes m {\displaystyle m} can be used to calculate the exponential of T {\displaystyle T} efficiently. The polynomial calculus is not as informative in the infinite-dimensional case. | wikipedia |
functional calculus | Consider the unilateral shift with the polynomials calculus; the ideal defined above is now trivial. Thus one is interested in functional calculi more general than polynomials. The subject is closely linked to spectral theory, since for a diagonal matrix or multiplication operator, it is rather clear what the definitio... | wikipedia |
abel's functional equation | In mathematics, a functional equation is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning is often used, where a functional equation is an equation that relates sever... | wikipedia |
abel's functional equation | {\displaystyle \log(xy)=\log(x)+\log(y).} If the domain of the unknown function is supposed to be the natural numbers, the function is generally viewed as a sequence, and, in this case, a functional equation (in the narrower meaning) is called a recurrence relation. | wikipedia |
abel's functional equation | Thus the term functional equation is used mainly for real functions and complex functions. Moreover a smoothness condition is often assumed for the solutions, since without such a condition, most functional equations have very irregular solutions. For example, the gamma function is a function that satisfies the functio... | wikipedia |
functional square root | In mathematics, a functional square root (sometimes called a half iterate) is a square root of a function with respect to the operation of function composition. In other words, a functional square root of a function g is a function f satisfying f(f(x)) = g(x) for all x. | wikipedia |
general hypergeometric function | In mathematics, a general hypergeometric function or Aomoto–Gelfand hypergeometric function is a generalization of the hypergeometric function that was introduced by Gelfand (1986). The general hypergeometric function is a function that is (more or less) defined on a Grassmannian, and depends on a choice of some comple... | wikipedia |
borcherds algebra | In mathematics, a generalized Kac–Moody algebra is a Lie algebra that is similar to a Kac–Moody algebra, except that it is allowed to have imaginary simple roots. Generalized Kac–Moody algebras are also sometimes called GKM algebras, Borcherds–Kac–Moody algebras, BKM algebras, or Borcherds algebras. The best known exam... | wikipedia |
exponential generating series | In mathematics, a generating function is a way of encoding an infinite sequence of numbers (an) by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary series, the formal power series is not required to converge: in fact, the gener... | wikipedia |
exponential generating series | One can generalize to formal power series in more than one indeterminate, to encode information about infinite multi-dimensional arrays of numbers. There are various types of generating functions, including ordinary generating functions, exponential generating functions, Lambert series, Bell series, and Dirichlet serie... | wikipedia |
exponential generating series | The particular generating function, if any, that is most useful in a given context will depend upon the nature of the sequence and the details of the problem being addressed. Generating functions are often expressed in closed form (rather than as a series), by some expression involving operations defined for formal ser... | wikipedia |
exponential generating series | However such interpretation is not required to be possible, because formal series are not required to give a convergent series when a nonzero numeric value is substituted for x. Also, not all expressions that are meaningful as functions of x are meaningful as expressions designating formal series; for example, negative... | wikipedia |
geometric algebra | In mathematics, a geometric algebra (also known as a real Clifford algebra) is an extension of elementary algebra to work with geometrical objects such as vectors. Geometric algebra is built out of two fundamental operations, addition and the geometric product. Multiplication of vectors results in higher-dimensional ob... | wikipedia |
geometric algebra | The geometric product was first briefly mentioned by Hermann Grassmann, who was chiefly interested in developing the closely related exterior algebra. In 1878, William Kingdon Clifford greatly expanded on Grassmann's work to form what are now usually called Clifford algebras in his honor (although Clifford himself chos... | wikipedia |
geometric algebra | Adding the dual of the Grassmann exterior product (the "meet") allows the use of the Grassmann–Cayley algebra, and a conformal version of the latter together with a conformal Clifford algebra yields a conformal geometric algebra (CGA) providing a framework for classical geometries. In practice, these and several derive... | wikipedia |
geometric algebra | The term "geometric algebra" was repopularized in the 1960s by Hestenes, who advocated its importance to relativistic physics.The scalars and vectors have their usual interpretation, and make up distinct subspaces of a geometric algebra. Bivectors provide a more natural representation of the pseudovector quantities in ... | wikipedia |
geometric algebra | A trivector can represent an oriented volume, and so on. An element called a blade may be used to represent a subspace of V {\displaystyle V} and orthogonal projections onto that subspace. | wikipedia |
geometric algebra | Rotations and reflections are represented as elements. Unlike a vector algebra, a geometric algebra naturally accommodates any number of dimensions and any quadratic form such as in relativity. Examples of geometric algebras applied in physics include the spacetime algebra (and the less common algebra of physical space... | wikipedia |
geometric algebra | Geometric calculus, an extension of GA that incorporates differentiation and integration, can be used to formulate other theories such as complex analysis and differential geometry, e.g. by using the Clifford algebra instead of differential forms. Geometric algebra has been advocated, most notably by David Hestenes and... | wikipedia |
geometrical progression | In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. For example, the sequence 2, 6, 18, 54, ... is a geometric progression with common ... | wikipedia |
geometrical progression | Examples of a geometric sequence are powers rk of a fixed non-zero number r, such as 2k and 3k. The general form of a geometric sequence is a , a r , a r 2 , a r 3 , a r 4 , … {\displaystyle a,\ ar,\ ar^{2},\ ar^{3},\ ar^{4},\ \ldots } where r ≠ 0 is the common ratio and a ≠ 0 is a scale factor, equal to the sequence's... | wikipedia |
geometric series | In mathematics, a geometric series is the sum of an infinite number of terms that have a constant ratio between successive terms. For example, the series 1 2 + 1 4 + 1 8 + 1 16 + ⋯ {\displaystyle {\frac {1}{2}}\,+\,{\frac {1}{4}}\,+\,{\frac {1}{8}}\,+\,{\frac {1}{16}}\,+\,\cdots } is geometric, because each successive ... | wikipedia |
geometric series | . {\displaystyle a+ar+ar^{2}+ar^{3}+...} , where a {\displaystyle a} is the coefficient of each term and r {\displaystyle r} is the common ratio between adjacent terms. The geometric series had an important role in the early development of calculus, is used throughout mathematics, and can serve as an introduction to fr... | wikipedia |
transformation (geometry) | In mathematics, a geometric transformation is any bijection of a set to itself (or to another such set) with some salient geometrical underpinning. More specifically, it is a function whose domain and range are sets of points — most often both R 2 {\displaystyle \mathbb {R} ^{2}} or both R 3 {\displaystyle \mathbb {R} ... | wikipedia |
graded lie algebra | In mathematics, a graded Lie algebra is a Lie algebra endowed with a gradation which is compatible with the Lie bracket. In other words, a graded Lie algebra is a Lie algebra which is also a nonassociative graded algebra under the bracket operation. A choice of Cartan decomposition endows any semisimple Lie algebra wit... | wikipedia |
graded lie algebra | A graded Lie superalgebra extends the notion of a graded Lie algebra in such a way that the Lie bracket is no longer assumed to be necessarily anticommutative. These arise in the study of derivations on graded algebras, in the deformation theory of Murray Gerstenhaber, Kunihiko Kodaira, and Donald C. Spencer, and in th... | wikipedia |
graded lie algebra | A supergraded Lie superalgebra is a further generalization of this notion to the category of superalgebras in which a graded Lie superalgebra is endowed with an additional super Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {Z} } -gradation. These arise when one forms a graded Lie superalgebra in a classical (non-supers... | wikipedia |
graph c*-algebra | In mathematics, a graph C*-algebra is a universal C*-algebra constructed from a directed graph. Graph C*-algebras are direct generalizations of the Cuntz algebras and Cuntz-Krieger algebras, but the class of graph C*-algebras has been shown to also include several other widely studied classes of C*-algebras. As a resul... | wikipedia |
graph c*-algebra | Although graph C*-algebras include numerous examples, they provide a class of C*-algebras that are surprisingly amenable to study and much more manageable than general C*-algebras. The graph not only determines the associated C*-algebra by specifying relations for generators, it also provides a useful tool for describi... | wikipedia |
extension (algebra) | In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q {\displaystyle Q} and N {\displaystyle N} are two groups, then G {\displaystyle G} is an extension of Q {\displaystyle Q} by N {\displaystyle N} if there is a short exact sequence... | wikipedia |
extension (algebra) | Group extensions arise in the context of the extension problem, where the groups Q {\displaystyle Q} and N {\displaystyle N} are known and the properties of G {\displaystyle G} are to be determined. Note that the phrasing " G {\displaystyle G} is an extension of N {\displaystyle N} by Q {\displaystyle Q} " is also used... | wikipedia |
half-exponential function | In mathematics, a half-exponential function is a functional square root of an exponential function. That is, a function f {\displaystyle f} such that f {\displaystyle f} composed with itself results in an exponential function: for some constants a {\displaystyle a} and b {\displaystyle b} . | wikipedia |
harmonic progression (mathematics) | In mathematics, a harmonic progression (or harmonic sequence) is a progression formed by taking the reciprocals of an arithmetic progression. Equivalently, a sequence is a harmonic progression when each term is the harmonic mean of the neighboring terms. As a third equivalent characterization, it is an infinite sequenc... | wikipedia |
helix | In mathematics, a helix is a curve in 3-dimensional space. The following parametrisation in Cartesian coordinates defines a particular helix; perhaps the simplest equations for one is x ( t ) = cos ( t ) , {\displaystyle x(t)=\cos(t),\,} y ( t ) = sin ( t ) , {\displaystyle y(t)=\sin(t),\,} z ( t ) = t . {\displays... | wikipedia |
helix | {\displaystyle h(t)=t.\,} A circular helix of radius a and slope a/b (or pitch 2πb) is described by the following parametrisation: x ( t ) = a cos ( t ) , {\displaystyle x(t)=a\cos(t),\,} y ( t ) = a sin ( t ) , {\displaystyle y(t)=a\sin(t),\,} z ( t ) = b t . {\displaystyle z(t)=bt.\,} Another way of mathematicall... | wikipedia |
heteroclinic network | In mathematics, a heteroclinic network is an invariant set in the phase space of a dynamical system. It can be thought of loosely as the union of more than one heteroclinic cycle. Heteroclinic networks arise naturally in a number of different types of applications, including fluid dynamics and populations dynamics. The... | wikipedia |
binary relations | In mathematics, a heterogeneous relation is a binary relation, a subset of a Cartesian product A × B , {\displaystyle A\times B,} where A and B are possibly distinct sets. The prefix hetero is from the Greek ἕτερος (heteros, "other, another, different"). A heterogeneous relation has been called a rectangular relation, ... | wikipedia |
hierarchy (mathematics) | In mathematics, a hierarchy is a set-theoretical object, consisting of a preorder defined on a set. This is often referred to as an ordered set, though that is an ambiguous term that many authors reserve for partially ordered sets or totally ordered sets. The term pre-ordered set is unambiguous, and is always synonymou... | wikipedia |
hierarchy (mathematics) | Sometimes, a set comes equipped with a natural hierarchical structure. For example, the set of natural numbers N is equipped with a natural pre-order structure, where n ≤ n ′ {\displaystyle n\leq n'} whenever we can find some other number m {\displaystyle m} so that n + m = n ′ {\displaystyle n+m=n'} . That is, n ′ {\d... | wikipedia |
hierarchy (mathematics) | This idea can be applied to any commutative monoid. On the other hand, the set of integers Z requires a more sophisticated argument for its hierarchical structure, since we can always solve the equation n + m = n ′ {\displaystyle n+m=n'} by writing m = ( n ′ − n ) {\displaystyle m=(n'-n)} .A mathematical hierarchy (a p... | wikipedia |
hierarchy (mathematics) | This is not just a pedantic claim; there are also mathematical hierarchies, in the general sense, that are not describable using set theory.Other natural hierarchies arise in computer science, where the word refers to partially ordered sets whose elements are classes of objects of increasing complexity. In that case, t... | wikipedia |
holomorphic map | In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space Cn. The existence of a complex derivative in a neighbourhood is a very strong condition: it implies that a holomorp... | wikipedia |
holomorphic map | That all holomorphic functions are complex analytic functions, and vice versa, is a major theorem in complex analysis.Holomorphic functions are also sometimes referred to as regular functions. A holomorphic function whose domain is the whole complex plane is called an entire function. The phrase "holomorphic at a point... | wikipedia |
homogenous function | In mathematics, a homogeneous function is a function of several variables such that, if all its arguments are multiplied by a scalar, then its value is multiplied by some power of this scalar, called the degree of homogeneity, or simply the degree; that is, if k is an integer, a function f of n variables is homogeneous... | wikipedia |
homogenous function | This definition is often further generalized to functions whose domain is not V, but a cone in V, that is, a subset C of V such that v ∈ C {\displaystyle \mathbf {v} \in C} implies s v ∈ C {\displaystyle s\mathbf {v} \in C} for every nonzero scalar s. In the case of functions of several real variables and real vector s... | wikipedia |
homogenous function | A norm over a real vector space is an example of a positively homogeneous function that is not homogeneous. A special case is the absolute value of real numbers. The quotient of two homogeneous polynomials of the same degree gives an example of a homogeneous function of degree zero. This example is fundamental in the d... | wikipedia |
inhomogeneous polynomial | In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x 5 + 2 x 3 y 2 + 9 x y 4 {\displaystyle x^{5}+2x^{3}y^{2}+9xy^{4}} is a homogeneous polynomial of degree 5, in two variables; the sum of the exponents in each te... | wikipedia |
inhomogeneous polynomial | An algebraic form, or simply form, is a function defined by a homogeneous polynomial. A binary form is a form in two variables. A form is also a function defined on a vector space, which may be expressed as a homogeneous function of the coordinates over any basis. | wikipedia |
inhomogeneous polynomial | A polynomial of degree 0 is always homogeneous; it is simply an element of the field or ring of the coefficients, usually called a constant or a scalar. A form of degree 1 is a linear form. A form of degree 2 is a quadratic form. | wikipedia |
inhomogeneous polynomial | In geometry, the Euclidean distance is the square root of a quadratic form. Homogeneous polynomials are ubiquitous in mathematics and physics. They play a fundamental role in algebraic geometry, as a projective algebraic variety is defined as the set of the common zeros of a set of homogeneous polynomials. | wikipedia |
mathematical limit | In mathematics, a limit is the value that a function (or sequence) approaches as the input (or index) approaches some value. Limits are essential to calculus and mathematical analysis, and are used to define continuity, derivatives, and integrals. The concept of a limit of a sequence is further generalized to the conce... | wikipedia |
limiting case (mathematics) | In mathematics, a limiting case of a mathematical object is a special case that arises when one or more components of the object take on their most extreme possible values. For example: In statistics, the limiting case of the binomial distribution is the Poisson distribution. As the number of events tends to infinity i... | wikipedia |
limiting case (mathematics) | Each type of figure is a circle for certain values of the defining parameters, and the generic figure appears more like a circle as the limiting values are approached. Archimedes calculated an approximate value of π by treating the circle as the limiting case of a regular polygon with 3 × 2n sides, as n gets large. In ... | wikipedia |
limiting case (mathematics) | In economics, two limiting cases of a demand curve or supply curve are those in which the elasticity is zero (the totally inelastic case) or infinity (the infinitely elastic case). In finance, continuous compounding is the limiting case of compound interest in which the compounding period becomes infinitesimally small,... | wikipedia |
limiting case (mathematics) | A parabola can degenerate into two distinct or coinciding parallel lines. An ellipse can degenerate into a single point or a line segment. A hyperbola can degenerate into two intersecting lines. | wikipedia |
linear algebraic group action | In mathematics, a linear algebraic group is a subgroup of the group of invertible n × n {\displaystyle n\times n} matrices (under matrix multiplication) that is defined by polynomial equations. An example is the orthogonal group, defined by the relation M T M = I n {\displaystyle M^{T}M=I_{n}} where M T {\displaystyle ... | wikipedia |
linear algebraic group action | The simple Lie groups were classified by Wilhelm Killing and Élie Cartan in the 1880s and 1890s. At that time, no special use was made of the fact that the group structure can be defined by polynomials, that is, that these are algebraic groups. The founders of the theory of algebraic groups include Maurer, Chevalley, a... | wikipedia |
linearity | In mathematics, a linear map or linear function f(x) is a function that satisfies the two properties: Additivity: f(x + y) = f(x) + f(y). Homogeneity of degree 1: f(αx) = α f(x) for all α.These properties are known as the superposition principle. In this definition, x is not necessarily a real number, but can in genera... | wikipedia |
linearity | Additivity alone implies homogeneity for rational α, since f ( x + x ) = f ( x ) + f ( x ) {\displaystyle f(x+x)=f(x)+f(x)} implies f ( n x ) = n f ( x ) {\displaystyle f(nx)=nf(x)} for any natural number n by mathematical induction, and then n f ( x ) = f ( n x ) = f ( m n m x ) = m f ( n m x ) {\displaystyle nf(x)=f(... | wikipedia |
locally constant function | In mathematics, a locally constant function is a function from a topological space into a set with the property that around every point of its domain, there exists some neighborhood of that point on which it restricts to a constant function. | wikipedia |
locally integrable function | In mathematics, a locally integrable function (sometimes also called locally summable function) is a function which is integrable (so its integral is finite) on every compact subset of its domain of definition. The importance of such functions lies in the fact that their function space is similar to Lp spaces, but its ... | wikipedia |
map (mathematics) | In mathematics, a map or mapping is a function in its general sense. These terms may have originated as from the process of making a geographical map: mapping the Earth surface to a sheet of paper.The term map may be used to distinguish some special types of functions, such as homomorphisms. For example, a linear map i... | wikipedia |
map (mathematics) | In category theory, a map may refer to a morphism. The term transformation can be used interchangeably, but transformation often refers to a function from a set to itself. There are also a few less common uses in logic and graph theory. | wikipedia |
matrix coefficient | In mathematics, a matrix coefficient (or matrix element) is a function on a group of a special form, which depends on a linear representation of the group and additional data. Precisely, it is a function on a compact topological group G obtained by composing a representation of G on a vector space V with a linear map f... | wikipedia |
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