id stringlengths 14 14 | text stringlengths 9 3.55k | source stringlengths 1 250 |
|---|---|---|
c_2gbfw5391rt0 | 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 algebras were introduced by Busby (1968). For ... | Multiplier algebra |
c_90nvnc6a494u | 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 beta distribution.It has two equivalent definitions. One is given ... | Multivariate gamma function |
c_i4yzax20nep7 | 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 complex number whose real part satisfies ℜ ( a ) > ( p − 1 ) / 2 {... | Multivariate gamma function |
c_jlvbu5oxuyuw | 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 − 1 ( a − 1 2 ) = π ( p − 1 ) / 2 Γ p − 1 ( a ) Γ ( a + ( 1 − p ... | Multivariate gamma function |
c_q476x3chwj4g | 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 on. This can also be extended to non-integer values of p {\displ... | Multivariate gamma function |
c_chuzgs2hr5bm | 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. | N! conjecture |
c_m0f39k5ipsac | 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 \mathbb {C} } with itself. Symbolically, or The variables ... | Complex coordinate space |
c_pnumoqu4t5vb | 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 {C} ^{n}} is a topological vector space over the complex nu... | Complex coordinate space |
c_yuzu1p4si0xr | 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 space for holomorphic coordinate systems on complex manifolds. | Complex coordinate space |
c_rzebhy2glbi2 | 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 called the square lattice, or grid la... | Integer point |
c_f9xx3hehas91 | 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 exists as a smooth algebraic variety denoted by ΣnC. If C is a compact Riemann surface, ΣnC... | Symmetric product of an algebraic curve |
c_h8lml7lohuoa | 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 the Riemann sphere C {\displaystyle \mathbb {C} } ∪ {∞} ≈ S2), its nth symmetric produc... | Symmetric product of an algebraic curve |
c_hveqzjbfhpz4 | 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 source of André Weil's technique of constructing J as an abstract variety from 'birationa... | Symmetric product of an algebraic curve |
c_y70ra0c5jwo2 | 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 numbers. | Cabtaxi number |
c_o1mdh6ba01au | 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 of the harmonic mean of the first n posit... | Harmonic number |
c_qx4ng94zzini | 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 work was extended into the complex plan... | Harmonic number |
c_h6ykxip02y8x | 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 Bertrand-Chebyshev theorem implies that, exc... | Harmonic number |
c_5w6er8a9xqwi | 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 ) . {\displaystyle H_{n}^{(r)}=\sum _{k=1}^{n}H_{k}^{(r-1)}\... | Hyperharmonic number |
c_cpfc8gm9abb9 | 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 exists at least in the following three areas... | Symmetric power |
c_tbhaac0p6tc8 | 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 by U S p ( n ) {\displaystyle \mathrm {USp... | Symplectic group |
c_7p9f9bfo7ya6 | 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 of (compact) symplectic groups, indexed b... | Symplectic group |
c_ksbd5qfbmcmc | 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, corresponding to the positive integers 1, 2, 3... | Counting numbers |
c_bw1tq0aajh81 | 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. sports jersey numbers).The natural numbe... | Counting numbers |
c_7v5miu0yfcke | 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 partitioning and enumerations, are studied in combi... | Counting numbers |
c_7zt3i4vy0y5d | In mathematics, the necklace ring is a ring introduced by Metropolis and Rota (1983) to elucidate the multiplicative properties of necklace polynomials. | Necklace ring |
c_epnve9b219gh | 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. | Nil-Coxeter algebra |
c_ice3gse4w5fo | 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 ∈ g: ρ ( a ) is nilpotent for all repres... | Nilpotent cone |
c_a3xqwfvi0w7v | 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 and ordinal multiplica... | Nimber |
c_ex9z8frekhhy | 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. | Nimber |
c_s9q9f0q0k5ij | 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, then the top row is exact as wel... | Nine lemma |
c_kyhwi5cvzq3h | 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. | Nine lemma |
c_8t72a76xpf6s | 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 of the horizontal and vertical ... | Nine lemma |
c_p53bi0j4s6a8 | 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 precisely, for every component U in the Fatou set of f, the sequ... | No-wandering-domain theorem |
c_fp0abccit2zc | {\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 result can be generalized to many situations where the functions ... | No-wandering-domain theorem |
c_ofola0xzr2jh | 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, Vladimir Retakh, and Jean-Yves Thibon. It is noncommutative but cocommutative gr... | Noncommutative symmetric function |
c_u90z345lru3u | 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. | Nonmetricity tensor |
c_om9by747zcb3 | 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 theorems from abstract algebra, theory of quadratic f... | Merkurjev–Suslin theorem |
c_lvxv9z9emd8q | 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–Kato conjecture or the motivic Bloch–Kato conjectu... | Merkurjev–Suslin theorem |
c_uih0tzha64xo | 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 bifurcation are locally (around the equilibrium) topologically equivale... | Normal form (dynamical systems) |
c_23oif2le0rl9 | 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 cyclic vectors is guaranteed by the Gelfand–Naimark–Segal (GNS) constr... | Cyclic and separating vector |
c_77p1qjl5mmxu | 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 a useful concept for the analysis of the structure of com... | Divisibility (ring theory) |
c_un6jt9kgywxs | 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 implementations of this idea, the fun... | Set germ |
c_c4pug3cb6w9z | 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. | Quasi-continuous function |
c_srtq53snfups | 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}\otimes _{\mathbb {R} }\mathbb ... | Split form |
c_4yu8i5j3t5z5 | 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 construction of s-dimensional Hausdorff measure. | Dimension function |
c_uxo9evfvqmmd | 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. | Compactly embedded |
c_2n7wf5qqrmzl | 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 b = c. An element a in a magma (M, ∗) has the right ... | Cancellation property |
c_le47y8vuilfb | 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 equipped with additional cylindrification operation... | Cylindric algebra |
c_61ajzg5b1823 | 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–Pick theorem demonstrate. | Expansivity constant |
c_f3xr2xxzvym5 | 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 externology permits to introduce a no... | Exterior space |
c_h8kdkye7ajmv | 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 the space of holomorphic functions from X {\displaystyle X} to the complex num... | Automorphic function |
c_jn56ibbkumyf | 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 function. Equivalently, an automorphic form is a function whose divisor is invariant unde... | Automorphic function |
c_bquhzux6lak9 | 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 automorphy: Every factor of automorphy is a cocycle for the action of G {\displaystyle G} ... | Automorphic function |
c_tbvy2q6sqf7n | 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 automorphic forms is an automorphic form corresponding to the product of the corresponding facto... | Automorphic function |
c_1366ye7ltksf | 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 the space of holomorphic functions from X {\displaystyle X} to the complex num... | Automorphic cuspidal representation |
c_ga98b49hmuoc | 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 function is an automorphic form for which j {\displaystyle j} is the identity. | Automorphic cuspidal representation |
c_orj54gypap8w | 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 \in \Gamma } according to the given factor of automorphy j; to be an eigenfuncti... | Automorphic cuspidal representation |
c_bbb1nhx4e4sv | 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 may be complex numbers, or in fact complex square matrices, corresponding to th... | Automorphic cuspidal representation |
c_sn3wx3daw9vf | 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 field theory, constructs automorphic forms and their correspondent functions ... | Automorphic cuspidal representation |
c_0592on4qnyco | 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 functionals quantifying the invariance of number fields in a most abstract sense, the... | Automorphic cuspidal representation |
c_2bgszsjcpqe7 | 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 properties: - The Eisenstein series (which is a prototypical modular form) over... | Automorphic cuspidal representation |
c_or63kege68ti | - 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 are invariant with respect to a generalized analogue of their prime ideal (or a... | Automorphic cuspidal representation |
c_rw1s13pccz6m | 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 function f: Mm×n(K) → R ∪ {±∞} is said to be po... | Polyconvex function |
c_pc1ml39vfi0k | 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 properties is said to be continuous in an analogy... | Hemicontinuity |
c_veunh27yoqf2 | 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. The term "shy" was suggested by the American mathemat... | Prevalent and shy sets |
c_gaidvjmud4nh | 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, combinatorics and number theory. The M... | Motzkin number |
c_uvu2u8edj37t | 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 roots are all nth primitive roots of unity e 2 i π k n... | Cyclotomic polynomial |
c_ht21brmxbedc | 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(2) = 13 + 123 = 93 + 103. The name is... | Taxicab number |
c_p16y3ztdjs3c | 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; it is the smallest number expressible... | Taxicab number |
c_r6bcttq3wijg | 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 authors do not name this test or giv... | Nth-term test |
c_wky04wbf4sev | 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 } ". | Null sign |
c_pln3rblz5hrf | 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: A = { 2 , 3 , 5 , 7 , 11 } {\displaystyle A=\{2,3,5,7,11\}... | Null sign |
c_28sqr0wldm3c | 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 dimensions; twice the number of dimensions of ... | Cayley numbers |
c_n22nlzntdkfj | 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. | Cayley numbers |
c_n1p1z86yujz7 | 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 sedenions. | Cayley numbers |
c_kudgzz00swzz | 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 of points whose distance from P is les... | Open unit disc |
c_b8mf95bbbdw7 | 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} } . | Open unit disc |
c_yjoz3ibjtwc7 | 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 ‖ T ‖ {\displaystyle \|T\|} of a linea... | Norm topology |
c_ppdv67102xbt | 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 the group on the dual space of its Lie algebra. The th... | Kirillov orbit theory |
c_akhz898wsxn1 | 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 operation is denoted as a multiplication,... | Finite order |
c_70ciok3f5bp4 | 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 any subgroup H of a finite group G, the o... | Finite order |
c_0dj97zuuma6g | 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 its dimension is the number of element... | Order polytope |
c_4bprdd8f6d06 | 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 manifolds and is denoted by Ω ∗ S O = ⊕ 0 ... | Cobordism ring |
c_ocfxxixpsec4 | 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 will ultimately give the same answer. This allows... | Origin (mathematics) |
c_r3e4zdrcdqw7 | 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 point, where the group operation is given by composing transfo... | Complex orthogonal group |
c_m6b7ospe6hsq | It is compact. The orthogonal group in dimension n {\displaystyle n} has two connected components. | Complex orthogonal group |
c_nubc9akvjnqi | 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 group, generalizing the fact that in dimensions 2 and 3, i... | Complex orthogonal group |
c_w5uo0vgizl8g | 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, and therefore not an element of the component. | Complex orthogonal group |
c_cjcw782ibx7g | 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 n} orthogonal matrices form a subgroup, denoted O ( n , ... | Complex orthogonal group |
c_gnndhjp9gt18 | 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 definitions that put the intuitive concept into a form suitable for a mathemati... | Oscillation (mathematics) |
c_flu249rrfvjd | 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, introduced as the oscillator semigroup by Roger Howe in ... | Weyl calculus |
c_r7d1x2uwmpb4 | 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 contraction operators of the semigroup in SL(2,C) corres... | Weyl calculus |
c_hlfpugpy2f91 | 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 explains the theory for SU(1,1) in detail and su... | Weyl calculus |
c_j5fu40jxr9nz | 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 trivial and G has a trivial center, then G is said ... | Outer automorphism group |
c_szifagzfy7rv | 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 automorphism group and outer automorphism group are... | Outer automorphism group |
c_utm8gwx9g4rv | In mathematics, the overlapping interval topology is a topology which is used to illustrate various topological principles. | Overlapping interval topology |
c_c68txn4xa9bs | 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 | P-Laplacian |
c_y2tki826uhf7 | 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) defined a p-adic analog Gp of log Γ. Overholtzer (1952... | P-adic gamma function |
c_4iw6wse18o48 | 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 inside the given subset, whereas Hausdorff dim... | Packing dimension |
c_w1qontqzu7su | 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 cylindrical coordinates. The above equation may be brought into two dist... | Parabolic cylinder function |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.