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Multiplier algebra Summary Multiplier_algebra In mathematics, the multiplier algebra, denoted by M(A), of a C*-algebra A is a unital C*-algebra that is the largest unital C*-algebra that contains A as an ideal in a "non-degenerate" way. It is the noncommutative generalization of Stone–Čech compactification. Multiplier ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Multivariate gamma function Summary Multivariate_gamma_function In mathematics, the multivariate gamma function Γp is a generalization of the gamma function. It is useful in multivariate statistics, appearing in the probability density function of the Wishart and inverse Wishart distributions, and the matrix variate be... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Multivariate gamma function Summary Multivariate_gamma_function The other one, more useful to obtain a numerical result is: Γ p ( a ) = π p ( p − 1 ) / 4 ∏ j = 1 p Γ ( a + ( 1 − j ) / 2 ) . {\displaystyle \Gamma _{p}(a)=\pi ^{p(p-1)/4}\prod _{j=1}^{p}\Gamma (a+(1-j)/2).} In both definitions, a {\displaystyle a} is a co... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Multivariate gamma function Summary Multivariate_gamma_function Note that Γ 1 ( a ) {\displaystyle \Gamma _{1}(a)} reduces to the ordinary gamma function. The second of the above definitions allows to directly obtain the recursive relationships for p ≥ 2 {\displaystyle p\geq 2}: Γ p ( a ) = π ( p − 1 ) / 2 Γ ( a ) Γ p ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Multivariate gamma function Summary Multivariate_gamma_function Thus Γ 2 ( a ) = π 1 / 2 Γ ( a ) Γ ( a − 1 / 2 ) {\displaystyle \Gamma _{2}(a)=\pi ^{1/2}\Gamma (a)\Gamma (a-1/2)} Γ 3 ( a ) = π 3 / 2 Γ ( a ) Γ ( a − 1 / 2 ) Γ ( a − 1 ) {\displaystyle \Gamma _{3}(a)=\pi ^{3/2}\Gamma (a)\Gamma (a-1/2)\Gamma (a-1)} and so ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
N! conjecture Summary N!_conjecture In mathematics, the n! conjecture is the conjecture that the dimension of a certain bi-graded module of diagonal harmonics is n!. It was made by A. M. Garsia and M. Haiman and later proved by M. Haiman. It implies Macdonald's positivity conjecture about the Macdonald polynomials. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Complex coordinate space Summary Complex_coordinate_space In mathematics, the n-dimensional complex coordinate space (or complex n-space) is the set of all ordered n-tuples of complex numbers. It is denoted C n {\displaystyle \mathbb {C} ^{n}} , and is the n-fold Cartesian product of the complex plane C {\displaystyle ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Complex coordinate space Summary Complex_coordinate_space The real and imaginary parts of the coordinates set up a bijection of C n {\displaystyle \mathbb {C} ^{n}} with the 2n-dimensional real coordinate space, R 2 n {\displaystyle \mathbb {R} ^{2n}} . With the standard Euclidean topology, C n {\displaystyle \mathbb {... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Complex coordinate space Summary Complex_coordinate_space A function on an open subset of complex n-space is holomorphic if it is holomorphic in each complex coordinate separately. Several complex variables is the study of such holomorphic functions in n variables. More generally, the complex n-space is the target spac... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Integer point Summary Integer_lattice In mathematics, the n-dimensional integer lattice (or cubic lattice), denoted Z n {\displaystyle \mathbb {Z} ^{n}} , is the lattice in the Euclidean space R n {\displaystyle \mathbb {R} ^{n}} whose lattice points are n-tuples of integers. The two-dimensional integer lattice is also... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Symmetric product of an algebraic curve Summary Symmetric_product_of_an_algebraic_curve In mathematics, the n-fold symmetric product of an algebraic curve C is the quotient space of the n-fold cartesian product C × C × ... × Cor Cn by the group action of the symmetric group Sn on n letters permuting the factors. It exi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Symmetric product of an algebraic curve Summary Symmetric_product_of_an_algebraic_curve Its interest in relation to the classical geometry of curves is that its points correspond to effective divisors on C of degree n, that is, formal sums of points with non-negative integer coefficients. For C the projective line (say... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Symmetric product of an algebraic curve Summary Symmetric_product_of_an_algebraic_curve That means that at the level of function fields it is possible to construct J by taking linearly disjoint copies of the function field of C, and within their compositum taking the fixed subfield of the symmetric group. This is the s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cabtaxi number Summary Cabtaxi_number In mathematics, the n-th cabtaxi number, typically denoted Cabtaxi(n), is defined as the smallest positive integer that can be written as the sum of two positive or negative or 0 cubes in n ways. Such numbers exist for all n, which follows from the analogous result for taxicab numb... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Harmonic number Summary Harmonic_number In mathematics, the n-th harmonic number is the sum of the reciprocals of the first n natural numbers: Starting from n = 1, the sequence of harmonic numbers begins: Harmonic numbers are related to the harmonic mean in that the n-th harmonic number is also n times the reciprocal o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Harmonic number Summary Harmonic_number The harmonic numbers roughly approximate the natural logarithm function: 143 and thus the associated harmonic series grows without limit, albeit slowly. In 1737, Leonhard Euler used the divergence of the harmonic series to provide a new proof of the infinity of prime numbers. His... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Harmonic number Summary Harmonic_number When the value of a large quantity of items has a Zipf's law distribution, the total value of the n most-valuable items is proportional to the n-th harmonic number. This leads to a variety of surprising conclusions regarding the long tail and the theory of network value. The Bert... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hyperharmonic number Summary Hyperharmonic_number In mathematics, the n-th hyperharmonic number of order r, denoted by H n ( r ) {\displaystyle H_{n}^{(r)}} , is recursively defined by the relations: H n ( 0 ) = 1 n , {\displaystyle H_{n}^{(0)}={\frac {1}{n}},} and H n ( r ) = ∑ k = 1 n H k ( r − 1 ) ( r > 0 ) . {\disp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Symmetric power Summary Symmetric_power In mathematics, the n-th symmetric power of an object X is the quotient of the n-fold product X n := X × ⋯ × X {\displaystyle X^{n}:=X\times \cdots \times X} by the permutation action of the symmetric group S n {\displaystyle {\mathfrak {S}}_{n}} . More precisely, the notion exis... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Symplectic group Summary Symplectic_group In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted Sp(2n, F) and Sp(n) for positive integer n and field F (usually C or R). The latter is called the compact symplectic group and is also denoted ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Symplectic group Summary Symplectic_group In Cartan's classification of the simple Lie algebras, the Lie algebra of the complex group Sp(2n, C) is denoted Cn, and Sp(n) is the compact real form of Sp(2n, C). Note that when we refer to the (compact) symplectic group it is implied that we are talking about the collection... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Counting numbers Summary Zermelo_ordinals In mathematics, the natural numbers are the numbers 1, 2, 3, etc., possibly including 0 as well. Some definitions, including the standard ISO 80000-2, begin the natural numbers with 0, corresponding to the non-negative integers 0, 1, 2, 3, ..., whereas others start with 1, corr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Counting numbers Summary Zermelo_ordinals They may also be used for ordering (as in "this is the third largest city in the country"), in which case they serve as ordinal numbers. Natural numbers are sometimes used as labels, known as nominal numbers, having none of the properties of numbers in a mathematical sense (e.g... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Counting numbers Summary Zermelo_ordinals This chain of extensions canonically embeds the natural numbers in the other number systems. Properties of the natural numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partition... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Necklace ring Summary Necklace_ring In mathematics, the necklace ring is a ring introduced by Metropolis and Rota (1983) to elucidate the multiplicative properties of necklace polynomials. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Nil-Coxeter algebra Summary Nil-Coxeter_algebra In mathematics, the nil-Coxeter algebra, introduced by Fomin & Stanley (1994), is an algebra similar to the group algebra of a Coxeter group except that the generators are nilpotent. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Nilpotent cone Summary Nilpotent_cone In mathematics, the nilpotent cone N {\displaystyle {\mathcal {N}}} of a finite-dimensional semisimple Lie algebra g {\displaystyle {\mathfrak {g}}} is the set of elements that act nilpotently in all representations of g . {\displaystyle {\mathfrak {g}}.} In other words, N = { a ∈ ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Nimber Summary Nimber In mathematics, the nimbers, also called Grundy numbers, are introduced in combinatorial game theory, where they are defined as the values of heaps in the game Nim. The nimbers are the ordinal numbers endowed with nimber addition and nimber multiplication, which are distinct from ordinal addition ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Nimber Summary Nimber The nimber addition and multiplication operations are associative and commutative. Each nimber is its own negative. In particular for some pairs of ordinals, their nimber sum is smaller than either addend. The minimum excludant operation is applied to sets of nimbers. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Nine lemma Summary Nine_lemma In mathematics, the nine lemma (or 3×3 lemma) is a statement about commutative diagrams and exact sequences valid in the category of groups and any abelian category. It states: if the diagram to the right is a commutative diagram and all columns as well as the two bottom rows are exact, th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Nine lemma Summary Nine_lemma The nine lemma can be proved by direct diagram chasing, or by applying the snake lemma (to the two bottom rows in the first case, and to the two top rows in the second case). Linderholm (p. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Nine lemma Summary Nine_lemma 201) offers a satirical view of the nine lemma: "Draw a noughts-and-crosses board... Do not fill it in with noughts and crosses... Instead, use curved arrows... Wave your hands about in complicated patterns over this board. Make some noughts, but not in the squares; put them at both ends o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
No-wandering-domain theorem Summary No_wandering_domain_theorem In mathematics, the no-wandering-domain theorem is a result on dynamical systems, proven by Dennis Sullivan in 1985. The theorem states that a rational map f: Ĉ → Ĉ with deg(f) ≥ 2 does not have a wandering domain, where Ĉ denotes the Riemann sphere. More ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
No-wandering-domain theorem Summary No_wandering_domain_theorem {\displaystyle f^{n}=\underbrace {f\circ f\circ \cdots \circ f} _{n}.} The theorem does not hold for arbitrary maps; for example, the transcendental map f ( z ) = z + 2 π sin ( z ) {\displaystyle f(z)=z+2\pi \sin(z)} has wandering domains. However, the r... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Noncommutative symmetric function Summary Noncommutative_symmetric_function In mathematics, the noncommutative symmetric functions form a Hopf algebra NSymm analogous to the Hopf algebra of symmetric functions. The Hopf algebra NSymm was introduced by Israel M. Gelfand, Daniel Krob, Alain Lascoux, Bernard Leclerc, Vlad... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Nonmetricity tensor Summary Nonmetricity_tensor In mathematics, the nonmetricity tensor in differential geometry is the covariant derivative of the metric tensor. It is therefore a tensor field of order three. It vanishes for the case of Riemannian geometry and can be used to study non-Riemannian spacetimes. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Merkurjev–Suslin theorem Summary Galois_symbol In mathematics, the norm residue isomorphism theorem is a long-sought result relating Milnor K-theory and Galois cohomology. The result has a relatively elementary formulation and at the same time represents the key juncture in the proofs of many seemingly unrelated theore... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Merkurjev–Suslin theorem Summary Galois_symbol John Milnor speculated that this theorem might be true for ℓ = 2 {\displaystyle \ell =2} and all n {\displaystyle n} , and this question became known as Milnor's conjecture. The general case was conjectured by Spencer Bloch and Kazuya Kato and became known as the Bloch–Kat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Normal form (dynamical systems) Summary Normal_form_(dynamical_systems) In mathematics, the normal form of a dynamical system is a simplified form that can be useful in determining the system's behavior. Normal forms are often used for determining local bifurcations in a system. All systems exhibiting a certain type of... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cyclic and separating vector Summary Cyclic_and_separating_vector In mathematics, the notion of a cyclic and separating vector is important in the theory of von Neumann algebras, and in particular in Tomita–Takesaki theory. A related notion is that of a vector which is cyclic for a given operator. The existence of cycl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Divisibility (ring theory) Summary Divisor_(ring_theory) In mathematics, the notion of a divisor originally arose within the context of arithmetic of whole numbers. With the development of abstract rings, of which the integers are the archetype, the original notion of divisor found a natural extension. Divisibility is ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Set germ Summary Germ_(mathematics) In mathematics, the notion of a germ of an object in/on a topological space is an equivalence class of that object and others of the same kind that captures their shared local properties. In particular, the objects in question are mostly functions (or maps) and subsets. In specific i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Quasi-continuous function Summary Quasi-continuous_function In mathematics, the notion of a quasi-continuous function is similar to, but weaker than, the notion of a continuous function. All continuous functions are quasi-continuous but the converse is not true in general. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Split form Summary Real_form In mathematics, the notion of a real form relates objects defined over the field of real and complex numbers. A real Lie algebra g0 is called a real form of a complex Lie algebra g if g is the complexification of g0: g ≃ g 0 ⊗ R C . {\displaystyle {\mathfrak {g}}\simeq {\mathfrak {g}}_{0}\o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Dimension function Summary Dimension_function In mathematics, the notion of an (exact) dimension function (also known as a gauge function) is a tool in the study of fractals and other subsets of metric spaces. Dimension functions are a generalisation of the simple "diameter to the dimension" power law used in the const... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Compactly embedded Summary Compact_embedding In mathematics, the notion of being compactly embedded expresses the idea that one set or space is "well contained" inside another. There are versions of this concept appropriate to general topology and functional analysis. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cancellation property Summary Cancellation_property In mathematics, the notion of cancellativity (or cancellability) is a generalization of the notion of invertibility. An element a in a magma (M, ∗) has the left cancellation property (or is left-cancellative) if for all b and c in M, a ∗ b = a ∗ c always implies that ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cylindric algebra Summary Cylindric_algebra In mathematics, the notion of cylindric algebra, invented by Alfred Tarski, arises naturally in the algebraization of first-order logic with equality. This is comparable to the role Boolean algebras play for propositional logic. Cylindric algebras are Boolean algebras equippe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Expansivity constant Summary Expansivity_constant In mathematics, the notion of expansivity formalizes the notion of points moving away from one another under the action of an iterated function. The idea of expansivity is fairly rigid, as the definition of positive expansivity, below, as well as the Schwarz–Ahlfors–Pic... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Exterior space Summary Exterior_space In mathematics, the notion of externology in a topological space X generalizes the basic properties of the family εXcc = {E ⊆ X: X\E is a closed compact subset of X}of complements of the closed compact subspaces of X, which are used to construct its Alexandroff compactification. An... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Automorphic function Factor of automorphy Factor_of_automorphy > Factor of automorphy In mathematics, the notion of factor of automorphy arises for a group acting on a complex-analytic manifold. Suppose a group G {\displaystyle G} acts on a complex-analytic manifold X {\displaystyle X} . Then, G {\displaystyle G} also ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Automorphic function Factor of automorphy Factor_of_automorphy > Factor of automorphy A function f {\displaystyle f} is termed an automorphic form if the following holds: f ( g . x ) = j g ( x ) f ( x ) {\displaystyle f(g.x)=j_{g}(x)f(x)} where j g ( x ) {\displaystyle j_{g}(x)} is an everywhere nonzero holomorphic fun... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Automorphic function Factor of automorphy Factor_of_automorphy > Factor of automorphy The factor of automorphy for the automorphic form f {\displaystyle f} is the function j {\displaystyle j} . An automorphic function is an automorphic form for which j {\displaystyle j} is the identity. Some facts about factors of auto... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Automorphic function Factor of automorphy Factor_of_automorphy > Factor of automorphy The factor of automorphy is a coboundary if and only if it arises from an everywhere nonzero automorphic form. For a given factor of automorphy, the space of automorphic forms is a vector space. The pointwise product of two automorphi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Automorphic cuspidal representation Definition Automorphic_forms > Definition In mathematics, the notion of factor of automorphy arises for a group acting on a complex-analytic manifold. Suppose a group G {\displaystyle G} acts on a complex-analytic manifold X {\displaystyle X} . Then, G {\displaystyle G} also acts on ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Automorphic cuspidal representation Definition Automorphic_forms > Definition Equivalently, an automorphic form is a function whose divisor is invariant under the action of G {\displaystyle G} . The factor of automorphy for the automorphic form f {\displaystyle f} is the function j {\displaystyle j} . An automorphic fu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Automorphic cuspidal representation Definition Automorphic_forms > Definition An automorphic form is a function F on G (with values in some fixed finite-dimensional vector space V, in the vector-valued case), subject to three kinds of conditions: to transform under translation by elements γ ∈ Γ {\displaystyle \gamma \i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Automorphic cuspidal representation Definition Automorphic_forms > Definition The third condition is to handle the case where G/Γ is not compact but has cusps. The formulation requires the general notion of factor of automorphy j for Γ, which is a type of 1-cocycle in the language of group cohomology. The values of j m... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Automorphic cuspidal representation Definition Automorphic_forms > Definition The cocycle condition imposed on the factor of automorphy is something that can be routinely checked, when j is derived from a Jacobian matrix, by means of the chain rule. A more straightforward but technically advanced definition using class... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Automorphic cuspidal representation Definition Automorphic_forms > Definition Herein, the analytical structure of its L-function allows for generalizations with various algebro-geometric properties; and the resultant Langlands program. To oversimplify, automorphic forms in this general perspective, are analytic functio... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Automorphic cuspidal representation Definition Automorphic_forms > Definition Allowing a powerful mathematical tool for analyzing the invariant constructs of virtually any numerical structure. Examples of automorphic forms in an explicit unabstracted state are difficult to obtain, though some have directly analytical p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Automorphic cuspidal representation Definition Automorphic_forms > Definition - Generally any harmonic analytic object as a functor over Galois groups which is invariant on its ideal class group (or idele). As a general principle, automorphic forms can be thought of as analytic functions on abstract structures, which a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Polyconvex function Summary Polyconvex_function In mathematics, the notion of polyconvexity is a generalization of the notion of convexity for functions defined on spaces of matrices. Let Mm×n(K) denote the space of all m × n matrices over the field K, which may be either the real numbers R, or the complex numbers C. A... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hemicontinuity Summary Upper_hemicontinuous In mathematics, the notion of the continuity of functions is not immediately extensible to set-valued functions between two sets A and B. The dual concepts of upper hemicontinuity and lower hemicontinuity facilitate such an extension. A set-valued function that has both prope... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Prevalent and shy sets Summary Prevalent_and_shy_sets In mathematics, the notions of prevalence and shyness are notions of "almost everywhere" and "measure zero" that are well-suited to the study of infinite-dimensional spaces and make use of the translation-invariant Lebesgue measure on finite-dimensional real spaces.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Motzkin number Summary Motzkin_number In mathematics, the nth Motzkin number is the number of different ways of drawing non-intersecting chords between n points on a circle (not necessarily touching every point by a chord). The Motzkin numbers are named after Theodore Motzkin and have diverse applications in geometry, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cyclotomic polynomial Summary Cyclotomic_polynomial In mathematics, the nth cyclotomic polynomial, for any positive integer n, is the unique irreducible polynomial with integer coefficients that is a divisor of x n − 1 {\displaystyle x^{n}-1} and is not a divisor of x k − 1 {\displaystyle x^{k}-1} for any k < n. Its ro... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Taxicab number Summary Taxicab_number In mathematics, the nth taxicab number, typically denoted Ta(n) or Taxicab(n), also called the nth Ramanujan–Hardy number, is defined as the smallest integer that can be expressed as a sum of two positive integer cubes in n distinct ways. The most famous taxicab number is 1729 = Ta... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Taxicab number Summary Taxicab_number As told by Hardy: I remember once going to see him when he was lying ill at Putney. I had ridden in taxi-cab No. 1729, and remarked that the number seemed to be rather a dull one, and that I hoped it was not an unfavourable omen. "No," he replied, "it is a very interesting number;... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Nth-term test Summary Nth-term_test In mathematics, the nth-term test for divergence is a simple test for the divergence of an infinite series:If lim n → ∞ a n ≠ 0 {\displaystyle \lim _{n\to \infty }a_{n}\neq 0} or if the limit does not exist, then ∑ n = 1 ∞ a n {\displaystyle \sum _{n=1}^{\infty }a_{n}} diverges.Many ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Null sign Use in mathematics Null_sign > Use in mathematics In mathematics, the null sign (∅) denotes the empty set. Note that a null set is not necessarily an empty set. Common notations for the empty set include "{}", "∅", and " ∅ {\displaystyle \emptyset } ". | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Null sign Use in mathematics Null_sign > Use in mathematics The latter two symbols were introduced by the Bourbaki group (specifically André Weil) in 1939, inspired by the letter Ø in the Danish and Norwegian alphabets (and not related in any way to the Greek letter Φ).Empty sets are used in set operations. For example... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cayley numbers Summary Octonion_multiplication In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system. The octonions are usually represented by the capital letter O, using boldface O or blackboard bold O {\displaystyle \mathbb {O} } . Octonions have eight... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cayley numbers Summary Octonion_multiplication They are also power associative. Octonions are not as well known as the quaternions and complex numbers, which are much more widely studied and used. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cayley numbers Summary Octonion_multiplication Octonions are related to exceptional structures in mathematics, among them the exceptional Lie groups. Octonions have applications in fields such as string theory, special relativity and quantum logic. Applying the Cayley–Dickson construction to the octonions produces the ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Open unit disc Summary Open_unit_disk In mathematics, the open unit disk (or disc) around P (where P is a given point in the plane), is the set of points whose distance from P is less than 1: D 1 ( P ) = { Q: | P − Q | < 1 } . {\displaystyle D_{1}(P)=\{Q:\vert P-Q\vert <1\}.\,} The closed unit disk around P is the set ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Open unit disc Summary Open_unit_disk It is the interior of a circle of radius 1, centered at the origin. This set can be identified with the set of all complex numbers of absolute value less than one. When viewed as a subset of the complex plane (C), the unit disk is often denoted D {\displaystyle \mathbb {D} } . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Norm topology Summary Operator_norm In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its operator norm. Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces. Informally, the operator norm ‖ ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Kirillov orbit theory Summary Kirillov_orbit_theory In mathematics, the orbit method (also known as the Kirillov theory, the method of coadjoint orbits and by a few similar names) establishes a correspondence between irreducible unitary representations of a Lie group and its coadjoint orbits: orbits of the action of th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Finite order Summary Order_(group_theory) In mathematics, the order of a finite group is the number of its elements. If a group is not finite, one says that its order is infinite. The order of an element of a group (also called period length or period) is the order of the subgroup generated by the element. If the group... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Finite order Summary Order_(group_theory) The order of a group G is denoted by ord(G) or |G|, and the order of an element a is denoted by ord(a) or |a|, instead of ord ( ⟨ a ⟩ ) , {\displaystyle \operatorname {ord} (\langle a\rangle ),} where the brackets denote the generated group. Lagrange's theorem states that for... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Order polytope Summary Order_polytope In mathematics, the order polytope of a finite partially ordered set is a convex polytope defined from the set. The points of the order polytope are the monotonic functions from the given set to the unit interval, its vertices correspond to the upper sets of the partial order, and ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cobordism ring Summary Cobordism_ring In mathematics, the oriented cobordism ring is a ring where elements are oriented cobordism classes of manifolds, the multiplication is given by the Cartesian product of manifolds and the addition is given as the disjoint union of manifolds. The ring is graded by dimensions of mani... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Origin (mathematics) Summary Origin_(mathematics) In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space. In physical problems, the choice of origin is often arbitrary, meaning any choice of origin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Complex orthogonal group Summary Special_orthogonal_group In mathematics, the orthogonal group in dimension n {\displaystyle n} , denoted O ( n ) {\displaystyle \operatorname {O} (n)} , is the group of distance-preserving transformations of a Euclidean space of dimension n {\displaystyle n} that preserve a fixed poin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Complex orthogonal group Summary Special_orthogonal_group It is compact. The orthogonal group in dimension n {\displaystyle n} has two connected components. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Complex orthogonal group Summary Special_orthogonal_group The one that contains the identity element is a normal subgroup, called the special orthogonal group, and denoted SO ( n ) {\displaystyle \operatorname {SO} (n)} . It consists of all orthogonal matrices of determinant 1. This group is also called the rotation ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Complex orthogonal group Summary Special_orthogonal_group In low dimension, these groups have been widely studied, see SO(2), SO(3) and SO(4). The other component consists of all orthogonal matrices of determinant –1. This component does not form a group, as the product of any two of its elements is of determinant 1, a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Complex orthogonal group Summary Special_orthogonal_group By extension, for any field F {\displaystyle F} , an n × n {\displaystyle n\times n} matrix with entries in F {\displaystyle F} such that its inverse equals its transpose is called an orthogonal matrix over F {\displaystyle F} . The n × n {\displaystyle n\times ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Oscillation (mathematics) Summary Oscillation_of_a_function_at_a_point In mathematics, the oscillation of a function or a sequence is a number that quantifies how much that sequence or function varies between its extreme values as it approaches infinity or a point. As is the case with limits, there are several definiti... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Weyl calculus Summary Metaplectic_representation In mathematics, the oscillator representation is a projective unitary representation of the symplectic group, first investigated by Irving Segal, David Shale, and André Weil. A natural extension of the representation leads to a semigroup of contraction operators, introdu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Weyl calculus Summary Metaplectic_representation It acts as Möbius transformations on the extended complex plane, leaving the unit circle invariant. In that case the oscillator representation is a unitary representation of a double cover of SU(1,1) and the oscillator semigroup corresponds to a representation by contrac... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Weyl calculus Summary Metaplectic_representation On an infinitesimal level the semigroup is described by a cone in the Lie algebra of SU(1,1) that can be identified with a light cone. The same framework generalizes to the symplectic group in higher dimensions, including its analogue in infinite dimensions. This article... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Outer automorphism group Summary Outer_automorphism In mathematics, the outer automorphism group of a group, G, is the quotient, Aut(G) / Inn(G), where Aut(G) is the automorphism group of G and Inn(G) is the subgroup consisting of inner automorphisms. The outer automorphism group is usually denoted Out(G). If Out(G) is... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Outer automorphism group Summary Outer_automorphism The cosets of Inn(G) with respect to outer automorphisms are then the elements of Out(G); this is an instance of the fact that quotients of groups are not, in general, (isomorphic to) subgroups. If the inner automorphism group is trivial (when a group is abelian), the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Overlapping interval topology Summary Overlapping_interval_topology In mathematics, the overlapping interval topology is a topology which is used to illustrate various topological principles. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
P-Laplacian Summary P-Laplacian In mathematics, the p-Laplacian, or the p-Laplace operator, is a quasilinear elliptic partial differential operator of 2nd order. It is a nonlinear generalization of the Laplace operator, where p {\displaystyle p} is allowed to range over 1 < p < ∞ {\displaystyle 1 | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
P-adic gamma function Summary P-adic_gamma_function In mathematics, the p-adic gamma function Γp is a function of a p-adic variable analogous to the gamma function. It was first explicitly defined by Morita (1975), though Boyarsky (1980) pointed out that Dwork (1964) implicitly used the same function. Diamond (1977) de... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Packing dimension Summary Packing_dimension In mathematics, the packing dimension is one of a number of concepts that can be used to define the dimension of a subset of a metric space. Packing dimension is in some sense dual to Hausdorff dimension, since packing dimension is constructed by "packing" small open balls in... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Parabolic cylinder function Summary Parabolic_cylinder_function In mathematics, the parabolic cylinder functions are special functions defined as solutions to the differential equation This equation is found when the technique of separation of variables is used on Laplace's equation when expressed in parabolic cylindri... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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