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Extremal point Summary Extreme_point 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Idempotent relation Summary Idempotent_relation 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Idempotent measure Summary Idempotent_measure 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. Explicitl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Multiplicative identity Summary Left_identity 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Identity operator Summary Identity_operator 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 a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Mathematical identities Summary Identity_(mathematics) 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.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Immersed plane curve Summary Immersed_plane_curve 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 injec... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Immersed plane curve Summary Immersed_plane_curve 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Implicit curve Summary Implicit_curve 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Implicit curve Summary Implicit_curve 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Implicit curve Summary Implicit_curve 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.} Ex... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Implicit curve Summary Implicit_curve 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 equa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Implicit curve Summary Implicit_curve 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Implicit differentiation Summary Explicit_function 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Implicit differentiation Summary Explicit_function 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^{... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Implicit surface Summary Implicit_surface 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. ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Implicit surface Summary Implicit_surface 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Implicit surface Summary Implicit_surface {\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.} | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Implicit surface Summary Implicit_surface 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.} | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Implicit surface Summary Implicit_surface 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 {\dis... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Implicit surface Summary Implicit_surface 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 essen... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Implicit surface Summary Implicit_surface 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Incidence matrix Summary Incidence_relation 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Incidence poset Summary Incidence_poset 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Incidence structure Summary Incidence_structure 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Incidence structure Summary Incidence_structure 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Incompressible surface Summary Incompressible_surface 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 h... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Incompressible surface Summary Incompressible_surface 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Indexing set Summary Indexing_set 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Indicator random variable Summary Representing_function 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} _{... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Indicator random variable Summary Representing_function 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 number... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Indigenous bundle Summary Indigenous_bundle 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Induced character Summary Induced_character 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Strict inequality Summary Strict_inequality 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 differe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Strict inequality Summary Strict_inequality 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 th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Strict inequality Summary Strict_inequality 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 imp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Inequation Summary Inequation 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 inequation... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Infinite expression (mathematics) Summary Infinite_expression 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 o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Divergent geometric series Summary Divergent_geometric_series 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Periodic continued fraction Summary Periodic_continued_fraction 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}+... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Absolute convergence Summary Absolutely_convergent_series 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Absolute convergence Summary Absolutely_convergent_series {\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 th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Absolute convergence Summary Absolutely_convergent_series 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}}-{... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Infinitesimal Summary Infinitesimal 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. In... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Infinitesimal Summary Infinitesimal 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 ana... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Infinitesimal Summary Infinitesimal 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Infinitesimal Summary Infinitesimal 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 s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Infinitesimal Summary Infinitesimal 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 sm... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Infinitesimal Summary Infinitesimal 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 hi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Infinitesimal Summary Infinitesimal 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 th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Infinitesimal Summary Infinitesimal 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 wou... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Infinitesimal Summary Infinitesimal 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Infinitesimal Summary Infinitesimal 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Infinitesimal Summary Infinitesimal 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Infinitesimal Summary Infinitesimal 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 continuu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Infinitesimal operator Summary Infinitesimal_operator 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Information source (mathematics) Summary Information_source_(mathematics) 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Infrastructure (number theory) Summary Infrastructure_(number_theory) In mathematics, an infrastructure is a group-like structure appearing in global fields. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Initial algebra Summary Initial_algebra 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
One-to-one mapping Summary Injective_mapping 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 cont... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
One-to-one mapping Summary Injective_mapping 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
One-to-one mapping Summary Injective_mapping 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Inner form Summary Inner_form 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 ¯ {\displ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Inner form Summary Inner_form 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}})... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Inner product spaces Summary Inner_product_spaces 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 su... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Inner product spaces Summary Inner_product_spaces 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 Giusep... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Inner product spaces Summary Inner_product_spaces 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 tha... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Inner regular Summary Inner_regular_measure In mathematics, an inner regular measure is one for which the measure of a set can be approximated from within by compact subsets. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Integer matrix Summary Integer_matrix 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Integer sequences Summary Integer_sequences 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Integer sequences Summary Integer_sequences 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 deter... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Integer-valued function Summary Integer-valued_function 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, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Numerical polynomial Summary Integral-valued_polynomial 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-value... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Integral curve Summary Integral_curve In mathematics, an integral curve is a parametric curve that represents a specific solution to an ordinary differential equation or system of equations. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Definite Integrals Summary Integral_over_time 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. Int... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Definite Integrals Summary Integral_over_time 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 r... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Definite Integrals Summary Integral_over_time 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Definite Integrals Summary Integral_over_time 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 R... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Ehrhart polynomial Summary Ehrhart_polynomials 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 generaliza... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Kernel (integral operator) Summary Kernel_(integral_operator) 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 manip... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Integrating factor technique Summary Integration_factor 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Integration by parts operator Summary Integration_by_parts_operator 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 an... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Integrodifferential equation Summary Integrodifferential_equation In mathematics, an integro-differential equation is an equation that involves both integrals and derivatives of a function. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Integrodifference equation Summary Integrodifference_equation 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 } {\displa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Integrodifference equation Summary Integrodifference_equation 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}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Interleave sequence Summary Interleave_sequence 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 seque... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Interleave sequence Summary Interleave_sequence 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{\... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Interprime Summary Interprime 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 i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Interval contractor Summary Interval_contractor 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(... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Interval exchange transformation Summary Interval_exchange_map 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 permut... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Invariant convex cone Summary Invariant_convex_cone 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Invariant convex cone Summary Invariant_convex_cone 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 gr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Invariant convex cone Summary Invariant_convex_cone 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Invariant convex cone Summary Invariant_convex_cone 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Invariant convex cone Summary Invariant_convex_cone 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Invariant convex cone Summary Invariant_convex_cone 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 infini... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Invariant set Summary Invariant_(mathematics) 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Invariant set Summary Invariant_(mathematics) 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 transfor... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Invariant measure Summary Invariant_measure 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 i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Invariant polynomial Summary Invariant_polynomial 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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