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c_ibtwmglj9q1j | In mathematics, an extreme point of a convex set S {\displaystyle S} in a real or complex vector space is a point in S {\displaystyle S} which does not lie in any open line segment joining two points of S . {\displaystyle S.} In linear programming problems, an extreme point is also called vertex or corner point of S . ... | Extremal point |
c_famlctff6hqy | In mathematics, an idempotent binary relation is a binary relation R on a set X (a subset of Cartesian product X × X) for which the composition of relations R ∘ R is the same as R. This notion generalizes that of an idempotent function to relations. | Idempotent relation |
c_k7db9437zga5 | In mathematics, an idempotent measure on a metric group is a probability measure that equals its convolution with itself; in other words, an idempotent measure is an idempotent element in the topological semigroup of probability measures on the given metric group. Explicitly, given a metric group X and two probability ... | Idempotent measure |
c_lscrizhsvrzv | In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is used in algebraic structures such as groups and rings. The term identity el... | Multiplicative identity |
c_mb1dmzwceyma | In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unchanged. That is, when f is the identity function, the equality f(X) = X is true for all values of X to which f can be applied. | Identity operator |
c_xc9pjnj6t5ec | In mathematics, an identity is an equality relating one mathematical expression A to another mathematical expression B, such that A and B (which might contain some variables) produce the same value for all values of the variables within a certain range of validity. In other words, A = B is an identity if A and B define... | Mathematical identities |
c_4tfit2fh68o6 | In mathematics, an immersion is a differentiable function between differentiable manifolds whose differential pushforward is everywhere injective. Explicitly, f: M → N is an immersion if D p f: T p M → T f ( p ) N {\displaystyle D_{p}f:T_{p}M\to T_{f(p)}N\,} is an injective function at every point p of M (where TpX den... | Immersed plane curve |
c_873m5hc3s6bb | The function f itself need not be injective, only its derivative must be. A related concept is that of an embedding. A smooth embedding is an injective immersion f: M → N that is also a topological embedding, so that M is diffeomorphic to its image in N. An immersion is precisely a local embedding – that is, for any po... | Immersed plane curve |
c_cxbncq9jjc4x | In mathematics, an implicit curve is a plane curve defined by an implicit equation relating two coordinate variables, commonly x and y. For example, the unit circle is defined by the implicit equation x 2 + y 2 = 1 {\displaystyle x^{2}+y^{2}=1} . In general, every implicit curve is defined by an equation of the form F ... | Implicit curve |
c_fer5j1o0md5y | If F ( x , y ) {\displaystyle F(x,y)} is a polynomial in two variables, the corresponding curve is called an algebraic curve, and specific methods are available for studying it. Plane curves can be represented in Cartesian coordinates (x, y coordinates) by any of three methods, one of which is the implicit equation giv... | Implicit curve |
c_ydgkx95zokbl | The third essential description of a curve is the parametric one, where the x- and y-coordinates of curve points are represented by two functions x(t), y(t) both of whose functional forms are explicitly stated, and which are dependent on a common parameter t . {\displaystyle t.} Examples of implicit curves include: a l... | Implicit curve |
c_pgxaw0d2378v | The first three examples possess simple parametric representations, which is not true for the fourth and fifth examples. The fifth example shows the possibly complicated geometric structure of an implicit curve. The implicit function theorem describes conditions under which an equation F ( x , y ) = 0 {\displaystyle F(... | Implicit curve |
c_8w48ksho3wjw | This theorem is the key for the computation of essential geometric features of the curve: tangents, normals, and curvature. In practice implicit curves have an essential drawback: their visualization is difficult. But there are computer programs enabling one to display an implicit curve. Special properties of implicit ... | Implicit curve |
c_5bw7vpk80sg4 | In mathematics, an implicit equation is a relation of the form R ( x 1 , … , x n ) = 0 , {\displaystyle R(x_{1},\dots ,x_{n})=0,} where R is a function of several variables (often a polynomial). For example, the implicit equation of the unit circle is x 2 + y 2 − 1 = 0. {\displaystyle x^{2}+y^{2}-1=0.} | Implicit differentiation |
c_cwkw1m4wcuvw | An implicit function is a function that is defined by an implicit equation, that relates one of the variables, considered as the value of the function, with the others considered as the arguments. : 204–206 For example, the equation x 2 + y 2 − 1 = 0 {\displaystyle x^{2}+y^{2}-1=0} of the unit circle defines y as an im... | Implicit differentiation |
c_pin4s7h1u36u | In mathematics, an implicit surface is a surface in Euclidean space defined by an equation F ( x , y , z ) = 0. {\displaystyle F(x,y,z)=0.} An implicit surface is the set of zeros of a function of three variables. Implicit means that the equation is not solved for x or y or z. The graph of a function is usually describ... | Implicit surface |
c_u7tdx1u1w1zv | The third essential description of a surface is the parametric one: ( x ( s , t ) , y ( s , t ) , z ( s , t ) ) {\displaystyle (x(s,t),y(s,t),z(s,t))} , where the x-, y- and z-coordinates of surface points are represented by three functions x ( s , t ) , y ( s , t ) , z ( s , t ) {\displaystyle x(s,t)\,,y(s,t)\,,z(s,t)... | Implicit surface |
c_o6rwxegc2m0i | {\displaystyle x+2y-3z+1=0.} The sphere x 2 + y 2 + z 2 − 4 = 0. {\displaystyle x^{2}+y^{2}+z^{2}-4=0.} | Implicit surface |
c_88esngtd3x4n | The torus ( x 2 + y 2 + z 2 + R 2 − a 2 ) 2 − 4 R 2 ( x 2 + y 2 ) = 0. {\displaystyle (x^{2}+y^{2}+z^{2}+R^{2}-a^{2})^{2}-4R^{2}(x^{2}+y^{2})=0.} | Implicit surface |
c_phlu4mpsp78h | A surface of genus 2: 2 y ( y 2 − 3 x 2 ) ( 1 − z 2 ) + ( x 2 + y 2 ) 2 − ( 9 z 2 − 1 ) ( 1 − z 2 ) = 0 {\displaystyle 2y(y^{2}-3x^{2})(1-z^{2})+(x^{2}+y^{2})^{2}-(9z^{2}-1)(1-z^{2})=0} (see diagram). The surface of revolution x 2 + y 2 − ( ln ( z + 3.2 ) ) 2 − 0.02 = 0 {\displaystyle x^{2}+y^{2}-(\ln(z+3.2))^{2}-0.0... | Implicit surface |
c_gjkwp2frtg6b | The implicit function theorem describes conditions under which an equation F ( x , y , z ) = 0 {\displaystyle F(x,y,z)=0} can be solved (at least implicitly) for x, y or z. But in general the solution may not be made explicit. This theorem is the key to the computation of essential geometric features of a surface: tang... | Implicit surface |
c_udj2s64uh8bh | If F ( x , y , z ) {\displaystyle F(x,y,z)} is polynomial in x, y and z, the surface is called algebraic. Example 5 is non-algebraic. Despite difficulty of visualization, implicit surfaces provide relatively simple techniques to generate theoretically (e.g. Steiner surface) and practically (see below) interesting surfa... | Implicit surface |
c_72zkuxz5l9c9 | In mathematics, an incidence matrix is a logical matrix that shows the relationship between two classes of objects, usually called an incidence relation. If the first class is X and the second is Y, the matrix has one row for each element of X and one column for each element of Y. The entry in row x and column y is 1 i... | Incidence matrix |
c_a6z0bfnn3lb3 | In mathematics, an incidence poset or incidence order is a type of partially ordered set that represents the incidence relation between vertices and edges of an undirected graph. The incidence poset of a graph G has an element for each vertex or edge in G; in this poset, there is an order relation x ≤ y if and only if ... | Incidence poset |
c_qwp54gf71kd2 | In mathematics, an incidence structure is an abstract system consisting of two types of objects and a single relationship between these types of objects. Consider the points and lines of the Euclidean plane as the two types of objects and ignore all the properties of this geometry except for the relation of which point... | Incidence structure |
c_944e7nzzka4h | Incidence structures are most often considered in the geometrical context where they are abstracted from, and hence generalize, planes (such as affine, projective, and Möbius planes), but the concept is very broad and not limited to geometric settings. Even in a geometric setting, incidence structures are not limited t... | Incidence structure |
c_z0myxmjc5ae0 | In mathematics, an incompressible surface is a surface properly embedded in a 3-manifold, which, in intuitive terms, is a "nontrivial" surface that cannot be simplified. In non-mathematical terms, the surface of a suitcase is compressible, because we could cut the handle and shrink it into the surface. But a Conway sph... | Incompressible surface |
c_oawd9xidq517 | There are two cases to consider. A sphere is incompressible if both inside and outside the sphere there are some obstructions that prevent the sphere from shrinking to a point and also prevent the sphere from expanding to encompass all of space. A surface other than a sphere is incompressible if any disk with its bound... | Incompressible surface |
c_pwb5okoyjmhi | In mathematics, an index set is a set whose members label (or index) members of another set. For instance, if the elements of a set A may be indexed or labeled by means of the elements of a set J, then J is an index set. The indexing consists of a surjective function from J onto A, and the indexed collection is typical... | Indexing set |
c_2y1p37se1sus | In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if A is a subset of some set X, then 1 A ( x ) = 1 {\displaystyle \mathbf {1} _{A}(x)=1} if x ∈ A , {\displaystyle x\in A,} and 1 A ( x ... | Indicator random variable |
c_mx8xq63qbewj | The indicator function of A is the Iverson bracket of the property of belonging to A; that is, 1 A ( x ) = . {\displaystyle \mathbf {1} _{A}(x)=.} For example, the Dirichlet function is the indicator function of the rational numbers as a subset of the real numbers. | Indicator random variable |
c_1a3z67ftgxue | In mathematics, an indigenous bundle on a Riemann surface is a fiber bundle with a flat connection associated to some complex projective structure. Indigenous bundles were introduced by Robert C. Gunning (1967). Indigenous bundles for curves over p-adic fields were introduced by Shinichi Mochizuki (1996) in his study o... | Indigenous bundle |
c_56hqt5zr8qbs | In mathematics, an induced character is the character of the representation V of a finite group G induced from a representation W of a subgroup H ≤ G. More generally, there is also a notion of induction Ind ( f ) {\displaystyle \operatorname {Ind} (f)} of a class function f on H given by the formula Ind ( f ) ( s )... | Induced character |
c_ntqr2aig80al | In mathematics, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. It is used most often to compare two numbers on the number line by their size. There are several different notations used to represent different kinds of inequalities: The notation a < b... | Strict inequality |
c_q7mgvj9pnugu | The notation a ≥ b or a ⩾ b means that a is greater than or equal to b (or, equivalently, at least b, or not less than b).The relation not greater than can also be represented by a ≯ b, the symbol for "greater than" bisected by a slash, "not". The same is true for not less than and a ≮ b. The notation a ≠ b means that ... | Strict inequality |
c_v4ol6ggov3ze | In engineering sciences, less formal use of the notation is to state that one quantity is "much greater" than another, normally by several orders of magnitude. The notation a ≪ b means that a is much less than b. The notation a ≫ b means that a is much greater than b.This implies that the lesser value can be neglected ... | Strict inequality |
c_ztb8r7n3mhrx | In mathematics, an inequation is a statement that an inequality holds between two values. It is usually written in the form of a pair of expressions denoting the values in question, with a relational sign between them indicating the specific inequality relation. Some examples of inequations are: a < b {\displaystyle a ... | Inequation |
c_tsqpmu763x5m | In mathematics, an infinite expression is an expression in which some operators take an infinite number of arguments, or in which the nesting of the operators continues to an infinite depth. A generic concept for infinite expression can lead to ill-defined or self-inconsistent constructions (much like a set of all sets... | Infinite expression (mathematics) |
c_fklw060a4hyg | In mathematics, an infinite geometric series of the form ∑ k = 0 ∞ a r k = a + a r + a r 2 + a r 3 + ⋯ {\displaystyle \sum _{k=0}^{\infty }ar^{k}=a+ar+ar^{2}+ar^{3}+\cdots } is divergent if and only if | r | ≥ 1. Methods for summation of divergent series are sometimes useful, and usually evaluate divergent geometric se... | Divergent geometric series |
c_8e9m1d8sx77a | In mathematics, an infinite periodic continued fraction is a continued fraction that can be placed in the form x = a 0 + 1 a 1 + 1 a 2 + 1 ⋱ a k + 1 a k + 1 + ⋱ ⋱ a k + m − 1 + 1 a k + m + 1 a k + 1 + 1 a k + 2 + ⋱ {\displaystyle x=a_{0}+{\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{\quad \ddots \quad a_{k}+{\cfrac {... | Periodic continued fraction |
c_smow38f7l7x8 | In mathematics, an infinite series of numbers is said to converge absolutely (or to be absolutely convergent) if the sum of the absolute values of the summands is finite. More precisely, a real or complex series ∑ n = 0 ∞ a n {\displaystyle \textstyle \sum _{n=0}^{\infty }a_{n}} is said to converge absolutely if ∑ n = ... | Absolute convergence |
c_qqojpdops3z0 | {\displaystyle \textstyle \int _{0}^{\infty }|f(x)|dx=L.} Absolute convergence is important for the study of infinite series because its definition is strong enough to have properties of finite sums that not all convergent series possess – a convergent series that is not absolutely convergent is called conditionally co... | Absolute convergence |
c_dlmrvl1wj6cp | For instance, rearrangements do not change the value of the sum. This is not true for conditionally convergent series: The alternating harmonic series 1 − 1 2 + 1 3 − 1 4 + 1 5 − 1 6 + ⋯ {\textstyle 1-{\frac {1}{2}}+{\frac {1}{3}}-{\frac {1}{4}}+{\frac {1}{5}}-{\frac {1}{6}}+\cdots } converges to ln 2 , {\displaystyl... | Absolute convergence |
c_56wjutktfg7k | In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word infinitesimal comes from a 17th-century Modern Latin coinage infinitesimus, which originally referred to the "infinity-th" item in a sequence. Infinitesimals do not exist in the sta... | Infinitesimal |
c_y0r1yxxbkiqv | This definition was not rigorously formalized. As calculus developed further, infinitesimals were replaced by limits, which can be calculated using the standard real numbers. Infinitesimals regained popularity in the 20th century with Abraham Robinson's development of nonstandard analysis and the hyperreal numbers, whi... | Infinitesimal |
c_ylzrigbx8hfg | Following this, mathematicians developed surreal numbers, a related formalization of infinite and infinitesimal numbers that include both hyperreal cardinal and ordinal numbers, which is the largest ordered field. Vladimir Arnold wrote in 1990: Nowadays, when teaching analysis, it is not very popular to talk about infi... | Infinitesimal |
c_2vfmqu209yfz | Consequently, present-day students are not fully in command of this language. Nevertheless, it is still necessary to have command of it. The crucial insight for making infinitesimals feasible mathematical entities was that they could still retain certain properties such as angle or slope, even if these entities were in... | Infinitesimal |
c_6oaq6nkxjcv8 | In common speech, an infinitesimal object is an object that is smaller than any feasible measurement, but not zero in size—or, so small that it cannot be distinguished from zero by any available means. Hence, when used as an adjective in mathematics, infinitesimal means infinitely small, smaller than any standard real ... | Infinitesimal |
c_i4leslpw2je4 | An infinite number of infinitesimals are summed to calculate an integral. The concept of infinitesimals was originally introduced around 1670 by either Nicolaus Mercator or Gottfried Wilhelm Leibniz. Archimedes used what eventually came to be known as the method of indivisibles in his work The Method of Mechanical Theo... | Infinitesimal |
c_jqcxcov33f09 | In his formal published treatises, Archimedes solved the same problem using the method of exhaustion. The 15th century saw the work of Nicholas of Cusa, further developed in the 17th century by Johannes Kepler, in particular, the calculation of the area of a circle by representing the latter as an infinite-sided polygo... | Infinitesimal |
c_xcvv0lppmuga | Bonaventura Cavalieri's method of indivisibles led to an extension of the results of the classical authors. The method of indivisibles related to geometrical figures as being composed of entities of codimension 1. John Wallis's infinitesimals differed from indivisibles in that he would decompose geometrical figures int... | Infinitesimal |
c_iqfa0t1ghdq3 | He exploited an infinitesimal denoted 1/∞ in area calculations. The use of infinitesimals by Leibniz relied upon heuristic principles, such as the law of continuity: what succeeds for the finite numbers succeeds also for the infinite numbers and vice versa; and the transcendental law of homogeneity that specifies proce... | Infinitesimal |
c_maq2lfy2fvjd | The 18th century saw routine use of infinitesimals by mathematicians such as Leonhard Euler and Joseph-Louis Lagrange. Augustin-Louis Cauchy exploited infinitesimals both in defining continuity in his Cours d'Analyse, and in defining an early form of a Dirac delta function. As Cantor and Dedekind were developing more a... | Infinitesimal |
c_52p8qdy7ygis | Du Bois-Reymond's work inspired both Émile Borel and Thoralf Skolem. Borel explicitly linked du Bois-Reymond's work to Cauchy's work on rates of growth of infinitesimals. Skolem developed the first non-standard models of arithmetic in 1934. | Infinitesimal |
c_xnzsfl9m0bdc | A mathematical implementation of both the law of continuity and infinitesimals was achieved by Abraham Robinson in 1961, who developed nonstandard analysis based on earlier work by Edwin Hewitt in 1948 and Jerzy Łoś in 1955. The hyperreals implement an infinitesimal-enriched continuum and the transfer principle impleme... | Infinitesimal |
c_7h4mx0g9pl8f | In mathematics, an infinitesimal transformation is a limiting form of small transformation. For example one may talk about an infinitesimal rotation of a rigid body, in three-dimensional space. This is conventionally represented by a 3×3 skew-symmetric matrix A. It is not the matrix of an actual rotation in space; but ... | Infinitesimal operator |
c_9bltjnjd9u46 | In mathematics, an information source is a sequence of random variables ranging over a finite alphabet Γ, having a stationary distribution. The uncertainty, or entropy rate, of an information source is defined as H { X } = lim n → ∞ H ( X n | X 0 , X 1 , … , X n − 1 ) {\displaystyle H\{\mathbf {X} \}=\lim _{n\to \infty... | Information source (mathematics) |
c_5bwr8ybntomj | In mathematics, an infrastructure is a group-like structure appearing in global fields. | Infrastructure (number theory) |
c_zgm3bp0l3e42 | In mathematics, an initial algebra is an initial object in the category of F-algebras for a given endofunctor F. This initiality provides a general framework for induction and recursion. | Initial algebra |
c_et4lssawkpow | In mathematics, an injective function (also known as injection, or one-to-one function) is a function f that maps distinct elements of its domain to distinct elements; that is, x1 ≠ x2 implies f(x1) ≠ f(x2). (Equivalently, f(x1) = f(x2) implies x1 = x2 in the equivalent contrapositive statement.) In other words, every ... | One-to-one mapping |
c_adyt2kltpved | The term one-to-one function must not be confused with one-to-one correspondence that refers to bijective functions, which are functions such that each element in the codomain is an image of exactly one element in the domain. A homomorphism between algebraic structures is a function that is compatible with the operatio... | One-to-one mapping |
c_ixiosdg2y8vj | However, in the more general context of category theory, the definition of a monomorphism differs from that of an injective homomorphism. This is thus a theorem that they are equivalent for algebraic structures; see Homomorphism § Monomorphism for more details. A function f {\displaystyle f} that is not injective is so... | One-to-one mapping |
c_nfv9zaj4dbgv | In mathematics, an inner form of an algebraic group G {\displaystyle G} over a field K {\displaystyle K} is another algebraic group H {\displaystyle H} such that there exists an isomorphism ϕ {\displaystyle \phi } between G {\displaystyle G} and H {\displaystyle H} defined over K ¯ {\displaystyle {\overline {K}}} (this... | Inner form |
c_edg7vfndfbuo | In practice, to check whether a group is an inner or outer form one looks at the action of the Galois group G a l ( K ¯ / K ) {\displaystyle \mathrm {Gal} ({\overline {K}}/K)} on the Dynkin diagram of G {\displaystyle G} (induced by its action on G ( K ¯ ) {\displaystyle G({\overline {K}})} , which preserves any torus ... | Inner form |
c_4gwgaolqxwlu | In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often denoted with angle brackets such as in ⟨ a , b ⟩ {\displaystyle \langle a,b\rang... | Inner product spaces |
c_5j2nh2gb6yb0 | Inner product spaces of infinite dimension are widely used in functional analysis. Inner product spaces over the field of complex numbers are sometimes referred to as unitary spaces. The first usage of the concept of a vector space with an inner product is due to Giuseppe Peano, in 1898.An inner product naturally induc... | Inner product spaces |
c_pmgd00qejxp5 | If this normed space is also complete (that is, a Banach space) then the inner product space is a Hilbert space. If an inner product space H is not a Hilbert space, it can be extended by completion to a Hilbert space H ¯ . {\displaystyle {\overline {H}}.} This means that H {\displaystyle H} is a linear subspace of H ¯ ... | Inner product spaces |
c_pqhz214m9h6z | In mathematics, an inner regular measure is one for which the measure of a set can be approximated from within by compact subsets. | Inner regular |
c_hh1bniwfwzef | In mathematics, an integer matrix is a matrix whose entries are all integers. Examples include binary matrices, the zero matrix, the matrix of ones, the identity matrix, and the adjacency matrices used in graph theory, amongst many others. Integer matrices find frequent application in combinatorics. | Integer matrix |
c_h70m2je4nrr0 | In mathematics, an integer sequence is a sequence (i.e., an ordered list) of integers. An integer sequence may be specified explicitly by giving a formula for its nth term, or implicitly by giving a relationship between its terms. For example, the sequence 0, 1, 1, 2, 3, 5, 8, 13, ... (the Fibonacci sequence) is formed... | Integer sequences |
c_2n3j547n4dgr | The sequence 0, 3, 8, 15, ... is formed according to the formula n2 − 1 for the nth term: an explicit definition. Alternatively, an integer sequence may be defined by a property which members of the sequence possess and other integers do not possess. For example, we can determine whether a given integer is a perfect nu... | Integer sequences |
c_9hmb66ct05gm | In mathematics, an integer-valued function is a function whose values are integers. In other words, it is a function that assigns an integer to each member of its domain. The floor and ceiling functions are examples of integer-valued functions of a real variable, but on real numbers and, generally, on (non-disconnected... | Integer-valued function |
c_u7r339ti9r32 | In mathematics, an integer-valued polynomial (also known as a numerical polynomial) P ( t ) {\displaystyle P(t)} is a polynomial whose value P ( n ) {\displaystyle P(n)} is an integer for every integer n. Every polynomial with integer coefficients is integer-valued, but the converse is not true. For example, the polyno... | Numerical polynomial |
c_4fs3gwn9dm33 | In mathematics, an integral curve is a parametric curve that represents a specific solution to an ordinary differential equation or system of equations. | Integral curve |
c_bmr0gzqfjjkp | In mathematics, an integral is the continuous analog of a sum, which is used to calculate areas, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental operations of calculus, the other being differentiation. Integration started as a method to solve problems... | Definite Integrals |
c_n8t6ed4titd4 | Today integration is used in a wide variety of scientific fields. The integrals enumerated here are called definite integrals, which can be interpreted as the signed area of the region in the plane that is bounded by the graph of a given function between two points in the real line. Conventionally, areas above the hori... | Definite Integrals |
c_so834yfblylp | Integrals also refer to the concept of an antiderivative, a function whose derivative is the given function; in this case, they are also called indefinite integrals. The fundamental theorem of calculus relates definite integrals with differentiation and provides a method to compute the definite integral of a function w... | Definite Integrals |
c_neadpf587map | Bernhard Riemann later gave a rigorous definition of integrals, which is based on a limiting procedure that approximates the area of a curvilinear region by breaking the region into infinitesimally thin vertical slabs. In the early 20th century, Henri Lebesgue generalized Riemann's formulation by introducing what is no... | Definite Integrals |
c_c2gkylomsonl | In mathematics, an integral polytope has an associated Ehrhart polynomial that encodes the relationship between the volume of a polytope and the number of integer points the polytope contains. The theory of Ehrhart polynomials can be seen as a higher-dimensional generalization of Pick's theorem in the Euclidean plane. ... | Ehrhart polynomial |
c_gakke60j84n1 | In mathematics, an integral transform is a type of transform that maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than in the original function space. The transformed fu... | Kernel (integral operator) |
c_2qwgoogcamqx | In mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given equation involving differentials. It is commonly used to solve ordinary differential equations, but is also used within multivariable calculus when multiplying through by an integrating factor allows an inexact diffe... | Integrating factor technique |
c_0s8b41k9kgzo | In mathematics, an integration by parts operator is a linear operator used to formulate integration by parts formulae; the most interesting examples of integration by parts operators occur in infinite-dimensional settings and find uses in stochastic analysis and its applications. | Integration by parts operator |
c_ayga2f524feh | In mathematics, an integro-differential equation is an equation that involves both integrals and derivatives of a function. | Integrodifferential equation |
c_9y3twgeiwk6c | In mathematics, an integrodifference equation is a recurrence relation on a function space, of the following form: n t + 1 ( x ) = ∫ Ω k ( x , y ) f ( n t ( y ) ) d y , {\displaystyle n_{t+1}(x)=\int _{\Omega }k(x,y)\,f(n_{t}(y))\,dy,} where { n t } {\displaystyle \{n_{t}\}\,} is a sequence in the function space and Ω ... | Integrodifference equation |
c_klv0nbapy09j | In this case, n t ( x ) {\displaystyle n_{t}(x)} is the population size or density at location x {\displaystyle x} at time t {\displaystyle t} , f ( n t ( x ) ) {\displaystyle f(n_{t}(x))} describes the local population growth at location x {\displaystyle x} and k ( x , y ) {\displaystyle k(x,y)} , is the probability o... | Integrodifference equation |
c_bpztbay877fs | In mathematics, an interleave sequence is obtained by merging two sequences via an in shuffle. Let S {\displaystyle S} be a set, and let ( x i ) {\displaystyle (x_{i})} and ( y i ) {\displaystyle (y_{i})} , i = 0 , 1 , 2 , … , {\displaystyle i=0,1,2,\ldots ,} be two sequences in S . {\displaystyle S.} The interleave se... | Interleave sequence |
c_a4oa4uzyaysr | Formally, it is the sequence ( z i ) , i = 0 , 1 , 2 , … {\displaystyle (z_{i}),i=0,1,2,\ldots } given by z i := { x i / 2 if i is even, y ( i − 1 ) / 2 if i is odd. {\displaystyle z_{i}:={\begin{cases}x_{i/2}&{\text{ if }}i{\text{ is even,}}\\y_{(i-1)/2}&{\text{ if }}i{\text{ is odd. }}\end{cases}}} | Interleave sequence |
c_avmaiipueur7 | In mathematics, an interprime is the average of two consecutive odd primes. For example, 9 is an interprime because it is the average of 7 and 11. The first interprimes are: 4, 6, 9, 12, 15, 18, 21, 26, 30, 34, 39, 42, 45, 50, 56, 60, 64, 69, 72, 76, 81, 86, 93, 99, ... (sequence A024675 in the OEIS)Interprimes cannot ... | Interprime |
c_6t7wb8n86bil | In mathematics, an interval contractor (or contractor for short) associated to a set X {\displaystyle X} is an operator C {\displaystyle C} which associates to a hyperrectangle {\displaystyle } in R n {\displaystyle {\mathbf {R}}^{n}} another box C ( ) {\displaystyle C()} of R n {\displaystyle {\mathbf {R}}^{n}} such... | Interval contractor |
c_jsrmgr8fi6ck | In mathematics, an interval exchange transformation is a kind of dynamical system that generalises circle rotation. The phase space consists of the unit interval, and the transformation acts by cutting the interval into several subintervals, and then permuting these subintervals. They arise naturally in the study of po... | Interval exchange transformation |
c_o0xf36j6zk9e | In mathematics, an invariant convex cone is a closed convex cone in a Lie algebra of a connected Lie group that is invariant under inner automorphisms. The study of such cones was initiated by Ernest Vinberg and Bertram Kostant. For a simple Lie algebra, the existence of an invariant convex cone forces the Lie algebra ... | Invariant convex cone |
c_8v5lg4mvyn20 | The dual cone with respect to the Killing form is the maximal invariant convex cone. Any intermediate cone is uniquely determined by its intersection with the Lie algebra of a maximal torus in a maximal compact subgroup. The intersection is invariant under the Weyl group of the maximal torus and the orbit of every poin... | Invariant convex cone |
c_4t7neje5lac0 | For the real symplectic group, the maximal and minimal cone coincide, so there is only one invariant convex cone. When one is properly contained in the other, there is a continuum of intermediate invariant convex cones. Invariant convex cones arise in the analysis of holomorphic semigroups in the complexification of th... | Invariant convex cone |
c_7qida13k85vl | They are naturally associated with Hermitian symmetric spaces and their associated holomorphic discrete series. The semigroup is made up of those elements in the complexification which, when acting on the Hermitian symmetric space of compact type, leave invariant the bounded domain corresponding to the noncompact dual. | Invariant convex cone |
c_11iz35idnzlq | The semigroup acts by contraction operators on the holomorphic discrete series; its interior acts by Hilbert–Schmidt operators. The unitary part of their polar decomposition is the operator corresponding to an element in the original real Lie group, while the positive part is the exponential of an imaginary multiple of... | Invariant convex cone |
c_xknqiy2kfecz | A similar decomposition already occurs in the semigroup. The oscillator semigroup of Roger Howe concerns the special case of this theory for the real symplectic group. Historically this has been one of the most important applications and has been generalized to infinite dimensions. This article treats in detail the exa... | Invariant convex cone |
c_857zslii8xx1 | In mathematics, an invariant is a property of a mathematical object (or a class of mathematical objects) which remains unchanged after operations or transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the... | Invariant set |
c_c9uf4m4fw8ty | More generally, an invariant with respect to an equivalence relation is a property that is constant on each equivalence class.Invariants are used in diverse areas of mathematics such as geometry, topology, algebra and discrete mathematics. Some important classes of transformations are defined by an invariant they leave... | Invariant set |
c_zniu1i75xf52 | In mathematics, an invariant measure is a measure that is preserved by some function. The function may be a geometric transformation. For examples, circular angle is invariant under rotation, hyperbolic angle is invariant under squeeze mapping, and a difference of slopes is invariant under shear mapping.Ergodic theory ... | Invariant measure |
c_8eouc8b3cpop | In mathematics, an invariant polynomial is a polynomial P {\displaystyle P} that is invariant under a group Γ {\displaystyle \Gamma } acting on a vector space V {\displaystyle V} . Therefore, P {\displaystyle P} is a Γ {\displaystyle \Gamma } -invariant polynomial if P ( γ x ) = P ( x ) {\displaystyle P(\gamma x)=P(x)}... | Invariant polynomial |
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