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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 . {\displaystyle S.}
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https://en.wikipedia.org/wiki/Extremal_point
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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.
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https://en.wikipedia.org/wiki/Idempotent_relation
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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 measures μ and ν on X, the convolution μ ∗ ν of μ and ν is the measure given by ( μ ∗ ν ) ( A ) = ∫ X μ ( A x − 1 ) d ν ( x ) = ∫ X ν ( x − 1 A ) d μ ( x ) {\displaystyle (\mu *\nu )(A)=\int _{X}\mu (Ax^{-1})\,\mathrm {d} \nu (x)=\int _{X}\nu (x^{-1}A)\,\mathrm {d} \mu (x)} for any Borel subset A of X. (The equality of the two integrals follows from Fubini's theorem.) With respect to the topology of weak convergence of measures, the operation of convolution makes the space of probability measures on X into a topological semigroup. Thus, μ is said to be an idempotent measure if μ ∗ μ = μ. It can be shown that the only idempotent probability measures on a complete, separable metric group are the normalized Haar measures of compact subgroups.
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https://en.wikipedia.org/wiki/Idempotent_measure
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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 element is often shortened to identity (as in the case of additive identity and multiplicative identity) when there is no possibility of confusion, but the identity implicitly depends on the binary operation it is associated with.
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https://en.wikipedia.org/wiki/Multiplicative_identity
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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.
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https://en.wikipedia.org/wiki/Identity_operator
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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 the same functions, and an identity is an equality between functions that are differently defined. For example, ( a + b ) 2 = a 2 + 2 a b + b 2 {\displaystyle (a+b)^{2}=a^{2}+2ab+b^{2}} and cos 2 θ + sin 2 θ = 1 {\displaystyle \cos ^{2}\theta +\sin ^{2}\theta =1} are identities. Identities are sometimes indicated by the triple bar symbol ≡ instead of =, the equals sign. Formally, an identity is a universally quantified equality.
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https://en.wikipedia.org/wiki/Mathematical_identities
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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 denotes the tangent space of a manifold X at a point p in X). Equivalently, f is an immersion if its derivative has constant rank equal to the dimension of M: rank D p f = dim M . {\displaystyle \operatorname {rank} \,D_{p}f=\dim M.}
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https://en.wikipedia.org/wiki/Immersed_plane_curve
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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 point x ∈ M there is a neighbourhood, U ⊆ M, of x such that f: U → N is an embedding, and conversely a local embedding is an immersion. For infinite dimensional manifolds, this is sometimes taken to be the definition of an immersion. If M is compact, an injective immersion is an embedding, but if M is not compact then injective immersions need not be embeddings; compare to continuous bijections versus homeomorphisms.
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https://en.wikipedia.org/wiki/Immersed_plane_curve
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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 ( x , y ) = 0 {\displaystyle F(x,y)=0} for some function F of two variables. Hence an implicit curve can be considered as the set of zeros of a function of two variables. Implicit means that the equation is not expressed as a solution for either x in terms of y or vice versa.
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https://en.wikipedia.org/wiki/Implicit_curve
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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 given above. The graph of a function is usually described by an equation y = f ( x ) {\displaystyle y=f(x)} in which the functional form is explicitly stated; this is called an explicit representation.
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https://en.wikipedia.org/wiki/Implicit_curve
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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 line: x + 2 y − 3 = 0 , {\displaystyle x+2y-3=0,} a circle: x 2 + y 2 − 4 = 0 , {\displaystyle x^{2}+y^{2}-4=0,} the semicubical parabola: x 3 − y 2 = 0 , {\displaystyle x^{3}-y^{2}=0,} Cassini ovals ( x 2 + y 2 ) 2 − 2 c 2 ( x 2 − y 2 ) − ( a 4 − c 4 ) = 0 {\displaystyle (x^{2}+y^{2})^{2}-2c^{2}(x^{2}-y^{2})-(a^{4}-c^{4})=0} (see diagram), sin ( x + y ) − cos ( x y ) + 1 = 0 {\displaystyle \sin(x+y)-\cos(xy)+1=0} (see diagram).The first four examples are algebraic curves, but the last one is not algebraic.
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https://en.wikipedia.org/wiki/Implicit_curve
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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(x,y)=0} can be solved implicitly for x and/or y – that is, under which one can validly write x = g ( y ) {\displaystyle x=g(y)} or y = f ( x ) {\displaystyle y=f(x)} .
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https://en.wikipedia.org/wiki/Implicit_curve
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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 curves make them essential tools in geometry and computer graphics. An implicit curve with an equation F ( x , y ) = 0 {\displaystyle F(x,y)=0} can be considered as the level curve of level 0 of the surface z = F ( x , y ) {\displaystyle z=F(x,y)} (see third diagram).
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https://en.wikipedia.org/wiki/Implicit_curve
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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.}
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https://en.wikipedia.org/wiki/Implicit_differentiation
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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 implicit function of x if −1 ≤ x ≤ 1, and y is restricted to nonnegative values. The implicit function theorem provides conditions under which some kinds of implicit equations define implicit functions, namely those that are obtained by equating to zero multivariable functions that are continuously differentiable.
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https://en.wikipedia.org/wiki/Implicit_differentiation
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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 described by an equation z = f ( x , y ) {\displaystyle z=f(x,y)} and is called an explicit representation.
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https://en.wikipedia.org/wiki/Implicit_surface
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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)} depending on common parameters s , t {\displaystyle s,t} . Generally the change of representations is simple only when the explicit representation z = f ( x , y ) {\displaystyle z=f(x,y)} is given: z − f ( x , y ) = 0 {\displaystyle z-f(x,y)=0} (implicit), ( s , t , f ( s , t ) ) {\displaystyle (s,t,f(s,t))} (parametric). Examples: The plane x + 2 y − 3 z + 1 = 0.
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https://en.wikipedia.org/wiki/Implicit_surface
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{\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.}
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https://en.wikipedia.org/wiki/Implicit_surface
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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.}
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https://en.wikipedia.org/wiki/Implicit_surface
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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}-0.02=0} (see diagram wineglass).For a plane, a sphere, and a torus there exist simple parametric representations. This is not true for the fourth example.
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https://en.wikipedia.org/wiki/Implicit_surface
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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: tangent planes, surface normals, curvatures (see below). But they have an essential drawback: their visualization is difficult.
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https://en.wikipedia.org/wiki/Implicit_surface
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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 surfaces.
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https://en.wikipedia.org/wiki/Implicit_surface
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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 if x and y are related (called incident in this context) and 0 if they are not. There are variations; see below.
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https://en.wikipedia.org/wiki/Incidence_matrix
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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 either x = y or x is a vertex, y is an edge, and x is an endpoint of y.
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https://en.wikipedia.org/wiki/Incidence_poset
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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 points are on which lines for all points and lines. What is left is the incidence structure of the Euclidean plane.
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https://en.wikipedia.org/wiki/Incidence_structure
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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 to just points and lines; higher-dimensional objects (planes, solids, n-spaces, conics, etc.) can be used. The study of finite structures is sometimes called finite geometry.
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https://en.wikipedia.org/wiki/Incidence_structure
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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 sphere (a sphere with four holes) is incompressible, because there are essential parts of a knot or link both inside and out, so there is no way to move the entire knot or link to one side of the punctured sphere. The mathematical definition is as follows.
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https://en.wikipedia.org/wiki/Incompressible_surface
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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 boundary on the surface spans a disk in the surface.Incompressible surfaces are used for decomposition of Haken manifolds, in normal surface theory, and in the study of the fundamental groups of 3-manifolds.
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https://en.wikipedia.org/wiki/Incompressible_surface
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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 typically called an indexed family, often written as {Aj}j∈J.
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https://en.wikipedia.org/wiki/Indexing_set
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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 ) = 0 {\displaystyle \mathbf {1} _{A}(x)=0} otherwise, where 1 A {\displaystyle \mathbf {1} _{A}} is a common notation for the indicator function. Other common notations are I A , {\displaystyle I_{A},} and χ A . {\displaystyle \chi _{A}.}
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https://en.wikipedia.org/wiki/Indicator_random_variable
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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.
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https://en.wikipedia.org/wiki/Indicator_random_variable
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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 of p-adic Teichmüller theory.
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https://en.wikipedia.org/wiki/Indigenous_bundle
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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 ) = 1 | H | ∑ t ∈ G , t − 1 s t ∈ H f ( t − 1 s t ) . {\displaystyle \operatorname {Ind} (f)(s)={\frac {1}{|H|}}\sum _{t\in G,\ t^{-1}st\in H}f(t^{-1}st).} If f is a character of the representation W of H, then this formula for Ind ( f ) {\displaystyle \operatorname {Ind} (f)} calculates the character of the induced representation V of G.The basic result on induced characters is Brauer's theorem on induced characters. It states that every irreducible character on G is a linear combination with integer coefficients of characters induced from elementary subgroups. == References ==
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https://en.wikipedia.org/wiki/Induced_character
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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 means that a is less than b. The notation a > b means that a is greater than b.In either case, a is not equal to b. These relations are known as strict inequalities, meaning that a is strictly less than or strictly greater than b. Equivalence is excluded. In contrast to strict inequalities, there are two types of inequality relations that are not strict: The notation a ≤ b or a ⩽ b means that a is less than or equal to b (or, equivalently, at most b, or not greater than b).
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https://en.wikipedia.org/wiki/Strict_inequality
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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 a is not equal to b; this inequation sometimes is considered a form of strict inequality. It does not say that one is greater than the other; it does not even require a and b to be member of an ordered set.
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https://en.wikipedia.org/wiki/Strict_inequality
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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 with little effect on the accuracy of an approximation (such as the case of ultrarelativistic limit in physics). In all of the cases above, any two symbols mirroring each other are symmetrical; a < b and b > a are equivalent, etc.
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https://en.wikipedia.org/wiki/Strict_inequality
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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 1 {\displaystyle n>1} x ≠ 0 {\displaystyle x\neq 0} In some cases, the term "inequation" can be considered synonymous to the term "inequality", while in other cases, an inequation is reserved only for statements whose inequality relation is "not equal to" (≠).
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https://en.wikipedia.org/wiki/Inequation
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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), but there are several instances of infinite expressions that are well-defined.
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https://en.wikipedia.org/wiki/Infinite_expression_(mathematics)
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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 series to a sum that agrees with the formula for the convergent case ∑ k = 0 ∞ a r k = a 1 − r . {\displaystyle \sum _{k=0}^{\infty }ar^{k}={\frac {a}{1-r}}.} This is true of any summation method that possesses the properties of regularity, linearity, and stability.
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https://en.wikipedia.org/wiki/Divergent_geometric_series
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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 {1}{a_{k+1}+{\cfrac {\ddots }{\quad \ddots \quad a_{k+m-1}+{\cfrac {1}{a_{k+m}+{\cfrac {1}{a_{k+1}+{\cfrac {1}{a_{k+2}+{\ddots }}}}}}}}}}}}}}}}}} where the initial block of k + 1 partial denominators is followed by a block of partial denominators that repeats ad infinitum. For example, 2 {\displaystyle {\sqrt {2}}} can be expanded to a periodic continued fraction, namely as . The partial denominators {ai} can in general be any real or complex numbers. That general case is treated in the article convergence problem. The remainder of this article is devoted to the subject of simple continued fractions that are also periodic. In other words, the remainder of this article assumes that all the partial denominators ai (i ≥ 1) are positive integers.
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https://en.wikipedia.org/wiki/Periodic_continued_fraction
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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 = 0 ∞ | a n | = L {\displaystyle \textstyle \sum _{n=0}^{\infty }\left|a_{n}\right|=L} for some real number L . {\displaystyle \textstyle L.} Similarly, an improper integral of a function, ∫ 0 ∞ f ( x ) d x , {\displaystyle \textstyle \int _{0}^{\infty }f(x)\,dx,} is said to converge absolutely if the integral of the absolute value of the integrand is finite—that is, if ∫ 0 ∞ | f ( x ) | d x = L .
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https://en.wikipedia.org/wiki/Absolute_convergence
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{\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 convergent, while absolutely convergent series behave "nicely".
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https://en.wikipedia.org/wiki/Absolute_convergence
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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 , {\displaystyle \ln 2,} while its rearrangement 1 + 1 3 − 1 2 + 1 5 + 1 7 − 1 4 + ⋯ {\textstyle 1+{\frac {1}{3}}-{\frac {1}{2}}+{\frac {1}{5}}+{\frac {1}{7}}-{\frac {1}{4}}+\cdots } (in which the repeating pattern of signs is two positive terms followed by one negative term) converges to 3 2 ln 2. {\textstyle {\frac {3}{2}}\ln 2.}
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https://en.wikipedia.org/wiki/Absolute_convergence
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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 standard real number system, but they do exist in other number systems, such as the surreal number system and the hyperreal number system, which can be thought of as the real numbers augmented with both infinitesimal and infinite quantities; the augmentations are the reciprocals of one another. Infinitesimal numbers were introduced in the development of calculus, in which the derivative was first conceived as a ratio of two infinitesimal quantities.
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https://en.wikipedia.org/wiki/Infinitesimal
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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, which, after centuries of controversy, showed that a formal treatment of infinitesimal calculus was possible.
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https://en.wikipedia.org/wiki/Infinitesimal
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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 infinitesimal quantities.
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https://en.wikipedia.org/wiki/Infinitesimal
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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 infinitely small.Infinitesimals are a basic ingredient in calculus as developed by Leibniz, including the law of continuity and the transcendental law of homogeneity.
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https://en.wikipedia.org/wiki/Infinitesimal
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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 number. Infinitesimals are often compared to other infinitesimals of similar size, as in examining the derivative of a function.
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https://en.wikipedia.org/wiki/Infinitesimal
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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 Theorems to find areas of regions and volumes of solids.
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https://en.wikipedia.org/wiki/Infinitesimal
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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 polygon. Simon Stevin's work on the decimal representation of all numbers in the 16th century prepared the ground for the real continuum.
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https://en.wikipedia.org/wiki/Infinitesimal
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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 into infinitely thin building blocks of the same dimension as the figure, preparing the ground for general methods of the integral calculus.
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https://en.wikipedia.org/wiki/Infinitesimal
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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 procedures for replacing expressions involving unassignable quantities, by expressions involving only assignable ones.
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https://en.wikipedia.org/wiki/Infinitesimal
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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 abstract versions of Stevin's continuum, Paul du Bois-Reymond wrote a series of papers on infinitesimal-enriched continua based on growth rates of functions.
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https://en.wikipedia.org/wiki/Infinitesimal
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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.
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https://en.wikipedia.org/wiki/Infinitesimal
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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 implements Leibniz's law of continuity. The standard part function implements Fermat's adequality.
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https://en.wikipedia.org/wiki/Infinitesimal
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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 for small real values of a parameter ε the transformation T = I + ε A {\displaystyle T=I+\varepsilon A} is a small rotation, up to quantities of order ε2.
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https://en.wikipedia.org/wiki/Infinitesimal_operator
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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 }H(X_{n}|X_{0},X_{1},\dots ,X_{n-1})} where X 0 , X 1 , … , X n {\displaystyle X_{0},X_{1},\dots ,X_{n}} is the sequence of random variables defining the information source, and H ( X n | X 0 , X 1 , … , X n − 1 ) {\displaystyle H(X_{n}|X_{0},X_{1},\dots ,X_{n-1})} is the conditional information entropy of the sequence of random variables. Equivalently, one has H { X } = lim n → ∞ H ( X 0 , X 1 , … , X n − 1 , X n ) n + 1 . {\displaystyle H\{\mathbf {X} \}=\lim _{n\to \infty }{\frac {H(X_{0},X_{1},\dots ,X_{n-1},X_{n})}{n+1}}.}
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https://en.wikipedia.org/wiki/Information_source_(mathematics)
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In mathematics, an infrastructure is a group-like structure appearing in global fields.
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https://en.wikipedia.org/wiki/Infrastructure_(number_theory)
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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.
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https://en.wikipedia.org/wiki/Initial_algebra
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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 element of the function's codomain is the image of at most one element of its domain.
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https://en.wikipedia.org/wiki/One-to-one_mapping
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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 operations of the structures. For all common algebraic structures, and, in particular for vector spaces, an injective homomorphism is also called a monomorphism.
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https://en.wikipedia.org/wiki/One-to-one_mapping
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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 sometimes called many-to-one.
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https://en.wikipedia.org/wiki/One-to-one_mapping
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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 means that H {\displaystyle H} is a K {\displaystyle K} -form of G {\displaystyle G} ) and in addition, for every Galois automorphism σ ∈ G a l ( K ¯ / K ) {\displaystyle \sigma \in \mathrm {Gal} ({\overline {K}}/K)} the automorphism ϕ − 1 ∘ ϕ σ {\displaystyle \phi ^{-1}\circ \phi ^{\sigma }} is an inner automorphism of G ( K ¯ ) {\displaystyle G({\overline {K}})} (i.e. conjugation by an element of G ( K ¯ ) {\displaystyle G({\overline {K}})} ). Through the correspondence between K {\displaystyle K} -forms and the Galois cohomology H 1 ( G a l ( K ¯ / K ) , I n n ( G ) ) {\displaystyle H^{1}(\mathrm {Gal} ({\overline {K}}/K),\mathrm {Inn} (G))} this means that H {\displaystyle H} is associated to an element of the subset H 1 ( G a l ( K ¯ / K ) , I n n ( G ) ) {\displaystyle H^{1}(\mathrm {Gal} ({\overline {K}}/K),\mathrm {Inn} (G))} where I n n ( G ) {\displaystyle \mathrm {Inn} (G)} is the subgroup of inner automorphisms of G {\displaystyle G} . Being inner forms of each other is an equivalence relation on the set of K {\displaystyle K} -forms of a given algebraic group. A form which is not inner is called an outer form.
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https://en.wikipedia.org/wiki/Inner_form
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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 and hence acts on the roots). Two groups are inner forms of each other if and only if the actions they define are the same. For example, the R {\displaystyle \mathbb {R} } -forms of S L 3 ( R ) {\displaystyle \mathrm {SL} _{3}(\mathbb {R} )} are itself and the unitary groups S U ( 2 , 1 ) {\displaystyle \mathrm {SU} (2,1)} and S U ( 3 ) {\displaystyle \mathrm {SU} (3)} . The latter two are outer forms of S L 3 ( R ) {\displaystyle \mathrm {SL} _{3}(\mathbb {R} )} , and they are inner forms of each other.
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https://en.wikipedia.org/wiki/Inner_form
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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\rangle } . Inner products allow formal definitions of intuitive geometric notions, such as lengths, angles, and orthogonality (zero inner product) of vectors. Inner product spaces generalize Euclidean vector spaces, in which the inner product is the dot product or scalar product of Cartesian coordinates.
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https://en.wikipedia.org/wiki/Inner_product_spaces
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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 induces an associated norm, (denoted | x | {\displaystyle |x|} and | y | {\displaystyle |y|} in the picture); so, every inner product space is a normed vector space.
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https://en.wikipedia.org/wiki/Inner_product_spaces
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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 ¯ , {\displaystyle {\overline {H}},} the inner product of H {\displaystyle H} is the restriction of that of H ¯ , {\displaystyle {\overline {H}},} and H {\displaystyle H} is dense in H ¯ {\displaystyle {\overline {H}}} for the topology defined by the norm.
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https://en.wikipedia.org/wiki/Inner_product_spaces
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In mathematics, an inner regular measure is one for which the measure of a set can be approximated from within by compact subsets.
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https://en.wikipedia.org/wiki/Inner_regular
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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.
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https://en.wikipedia.org/wiki/Integer_matrix
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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 by starting with 0 and 1 and then adding any two consecutive terms to obtain the next one: an implicit description.
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https://en.wikipedia.org/wiki/Integer_sequences
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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 number, even though we do not have a formula for the nth perfect number.
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https://en.wikipedia.org/wiki/Integer_sequences
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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) topological spaces integer-valued functions are not especially useful. Any such function on a connected space either has discontinuities or is constant. On the other hand, on discrete and other totally disconnected spaces integer-valued functions have roughly the same importance as real-valued functions have on non-discrete spaces. Any function with natural, or non-negative integer values is a partial case of an integer-valued function.
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https://en.wikipedia.org/wiki/Integer-valued_function
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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 polynomial 1 2 t 2 + 1 2 t = 1 2 t ( t + 1 ) {\displaystyle {\frac {1}{2}}t^{2}+{\frac {1}{2}}t={\frac {1}{2}}t(t+1)} takes on integer values whenever t is an integer. That is because one of t and t + 1 {\displaystyle t+1} must be an even number. (The values this polynomial takes are the triangular numbers.) Integer-valued polynomials are objects of study in their own right in algebra, and frequently appear in algebraic topology.
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https://en.wikipedia.org/wiki/Numerical_polynomial
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In mathematics, an integral curve is a parametric curve that represents a specific solution to an ordinary differential equation or system of equations.
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https://en.wikipedia.org/wiki/Integral_curve
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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 in mathematics and physics, such as finding the area under a curve, or determining displacement from velocity.
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https://en.wikipedia.org/wiki/Definite_Integrals
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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 horizontal axis of the plane are positive while areas below are negative.
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https://en.wikipedia.org/wiki/Definite_Integrals
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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 when its antiderivative is known; differentiation and integration are inverse operations. Although methods of calculating areas and volumes dated from ancient Greek mathematics, the principles of integration were formulated independently by Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century, who thought of the area under a curve as an infinite sum of rectangles of infinitesimal width.
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https://en.wikipedia.org/wiki/Definite_Integrals
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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 now referred to as the Lebesgue integral; it is more robust than Riemann's in the sense that a wider class of functions are Lebesgue-integrable. Integrals may be generalized depending on the type of the function as well as the domain over which the integration is performed. For example, a line integral is defined for functions of two or more variables, and the interval of integration is replaced by a curve connecting the two endpoints of the interval. In a surface integral, the curve is replaced by a piece of a surface in three-dimensional space.
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https://en.wikipedia.org/wiki/Definite_Integrals
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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. These polynomials are named after Eugène Ehrhart who studied them in the 1960s.
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https://en.wikipedia.org/wiki/Ehrhart_polynomial
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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 function can generally be mapped back to the original function space using the inverse transform.
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https://en.wikipedia.org/wiki/Kernel_(integral_operator)
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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 differential to be made into an exact differential (which can then be integrated to give a scalar field). This is especially useful in thermodynamics where temperature becomes the integrating factor that makes entropy an exact differential.
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https://en.wikipedia.org/wiki/Integrating_factor_technique
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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.
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https://en.wikipedia.org/wiki/Integration_by_parts_operator
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In mathematics, an integro-differential equation is an equation that involves both integrals and derivatives of a function.
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https://en.wikipedia.org/wiki/Integrodifferential_equation
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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 {\displaystyle \Omega \,} is the domain of those functions. In most applications, for any y ∈ Ω {\displaystyle y\in \Omega \,} , k ( x , y ) {\displaystyle k(x,y)\,} is a probability density function on Ω {\displaystyle \Omega \,} . Note that in the definition above, n t {\displaystyle n_{t}} can be vector valued, in which case each element of { n t } {\displaystyle \{n_{t}\}} has a scalar valued integrodifference equation associated with it. Integrodifference equations are widely used in mathematical biology, especially theoretical ecology, to model the dispersal and growth of populations.
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https://en.wikipedia.org/wiki/Integrodifference_equation
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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 of moving from point y {\displaystyle y} to point x {\displaystyle x} , often referred to as the dispersal kernel. Integrodifference equations are most commonly used to describe univoltine populations, including, but not limited to, many arthropod, and annual plant species. However, multivoltine populations can also be modeled with integrodifference equations, as long as the organism has non-overlapping generations. In this case, t {\displaystyle t} is not measured in years, but rather the time increment between broods.
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https://en.wikipedia.org/wiki/Integrodifference_equation
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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 sequence is defined to be the sequence x 0 , y 0 , x 1 , y 1 , … {\displaystyle x_{0},y_{0},x_{1},y_{1},\dots } .
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https://en.wikipedia.org/wiki/Interleave_sequence
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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}}}
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https://en.wikipedia.org/wiki/Interleave_sequence
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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 be prime themselves (otherwise the primes would not have been consecutive).There are infinitely many primes and therefore also infinitely many interprimes. The largest known interprime as of 2018 may be the 388342-digit n = 2996863034895 · 21290000, where n + 1 is the largest known twin prime.
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https://en.wikipedia.org/wiki/Interprime
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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 that the two following properties are always satisfied: C ( ) ⊂ {\displaystyle C()\subset } (contractance property) C ( ) ∩ X = ∩ X {\displaystyle C()\cap X=\cap X} (completeness property)A contractor associated to a constraint (such as an equation or an inequality) is a contractor associated to the set X {\displaystyle X} of all x {\displaystyle x} which satisfy the constraint. Contractors make it possible to improve the efficiency of branch-and-bound algorithms classically used in interval analysis.
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https://en.wikipedia.org/wiki/Interval_contractor
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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 polygonal billiards and in area-preserving flows.
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https://en.wikipedia.org/wiki/Interval_exchange_transformation
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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 to have a Hermitian structure, i.e. the maximal compact subgroup has center isomorphic to the circle group. The invariant convex cone generated by a generator of the Lie algebra of the center is closed and is the minimal invariant convex cone (up to a sign).
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https://en.wikipedia.org/wiki/Invariant_convex_cone
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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 point in the interior of the cone intersects the interior of the Weyl group invariant cone.
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https://en.wikipedia.org/wiki/Invariant_convex_cone
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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 the Lie group, first studied by Grigori Olshanskii.
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https://en.wikipedia.org/wiki/Invariant_convex_cone
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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.
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https://en.wikipedia.org/wiki/Invariant_convex_cone
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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 the infinitesimal operator corresponding to an element in the maximal cone.
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https://en.wikipedia.org/wiki/Invariant_convex_cone
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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 example of the invariant convex cone for the symplectic group and its use in the study of the symplectic Olshanskii semigroup.
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https://en.wikipedia.org/wiki/Invariant_convex_cone
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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 term is used. For example, the area of a triangle is an invariant with respect to isometries of the Euclidean plane. The phrases "invariant under" and "invariant to" a transformation are both used.
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https://en.wikipedia.org/wiki/Invariant_set
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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 unchanged. For example, conformal maps are defined as transformations of the plane that preserve angles. The discovery of invariants is an important step in the process of classifying mathematical objects.
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https://en.wikipedia.org/wiki/Invariant_set
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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 is the study of invariant measures in dynamical systems. The Krylov–Bogolyubov theorem proves the existence of invariant measures under certain conditions on the function and space under consideration.
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https://en.wikipedia.org/wiki/Invariant_measure
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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)} for all γ ∈ Γ {\displaystyle \gamma \in \Gamma } and x ∈ V {\displaystyle x\in V} .Cases of particular importance are for Γ a finite group (in the theory of Molien series, in particular), a compact group, a Lie group or algebraic group. For a basis-independent definition of 'polynomial' nothing is lost by referring to the symmetric powers of the given linear representation of Γ.
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https://en.wikipedia.org/wiki/Invariant_polynomial
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