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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 example, if A is the C*-algebra of compact operators on a separable Hilbert space, M(A) is B(H), the C*-algebra of all bounded operators on H.
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https://en.wikipedia.org/wiki/Multiplier_algebra
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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 as the following integral over the p × p {\displaystyle p\times p} positive-definite real matrices: Γ p ( a ) = ∫ S > 0 exp ( − t r ( S ) ) | S | a − p + 1 2 d S , {\displaystyle \Gamma _{p}(a)=\int _{S>0}\exp \left(-{\rm {tr}}(S)\right)\,\left|S\right|^{a-{\frac {p+1}{2}}}dS,} where | S | {\displaystyle |S|} denotes the determinant of S {\displaystyle S} .
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https://en.wikipedia.org/wiki/Multivariate_gamma_function
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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 {\displaystyle \Re (a)>(p-1)/2} .
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https://en.wikipedia.org/wiki/Multivariate_gamma_function
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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 ) / 2 ) . {\displaystyle \Gamma _{p}(a)=\pi ^{(p-1)/2}\Gamma (a)\Gamma _{p-1}(a-{\tfrac {1}{2}})=\pi ^{(p-1)/2}\Gamma _{p-1}(a)\Gamma (a+(1-p)/2).}
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https://en.wikipedia.org/wiki/Multivariate_gamma_function
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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 {\displaystyle p} with the expression: Γ p ( a ) = π p ( p − 1 ) / 4 G ( a + 1 2 ) G ( a + 1 ) G ( a + 1 − p 2 ) G ( a + 1 − p 2 ) {\displaystyle \Gamma _{p}(a)=\pi ^{p(p-1)/4}{\frac {G(a+{\frac {1}{2}})G(a+1)}{G(a+{\frac {1-p}{2}})G(a+1-{\frac {p}{2}})}}} Where G is the Barnes G-function, the indefinite product of the Gamma function. The function is derived by Anderson from first principles who also cites earlier work by Wishart, Mahalanobis and others. There also exists a version of the multivariate gamma function which instead of a single complex number takes a p {\displaystyle p} -dimensional vector of complex numbers as its argument. It generalizes the above defined multivariate gamma function insofar as the latter is obtained by a particular choice of multivariate argument of the former.
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https://en.wikipedia.org/wiki/Multivariate_gamma_function
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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.
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https://en.wikipedia.org/wiki/N!_conjecture
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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 z i {\displaystyle z_{i}} are the (complex) coordinates on the complex n-space. Complex coordinate space is a vector space over the complex numbers, with componentwise addition and scalar multiplication.
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https://en.wikipedia.org/wiki/Complex_coordinate_space
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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 numbers.
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https://en.wikipedia.org/wiki/Complex_coordinate_space
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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.
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https://en.wikipedia.org/wiki/Complex_coordinate_space
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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 lattice. Z n {\displaystyle \mathbb {Z} ^{n}} is the simplest example of a root lattice. The integer lattice is an odd unimodular lattice.
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https://en.wikipedia.org/wiki/Integer_point
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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 is therefore a complex manifold.
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https://en.wikipedia.org/wiki/Symmetric_product_of_an_algebraic_curve
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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 product ΣnC can be identified with complex projective space C P n {\displaystyle \mathbb {CP} ^{n}} of dimension n. If G has genus g ≥ 1 then the ΣnC are closely related to the Jacobian variety J of C. More accurately for n taking values up to g they form a sequence of approximations to J from below: their images in J under addition on J (see theta-divisor) have dimension n and fill up J, with some identifications caused by special divisors. For g = n we have ΣgC actually birationally equivalent to J; the Jacobian is a blowing down of the symmetric product.
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https://en.wikipedia.org/wiki/Symmetric_product_of_an_algebraic_curve
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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 'birational data'. Other ways of constructing J, for example as a Picard variety, are preferred now but this does mean that for any rational function F on C F(x1) + ... + F(xg)makes sense as a rational function on J, for the xi staying away from the poles of F. For n > g the mapping from ΣnC to J by addition fibers it over J; when n is large enough (around twice g) this becomes a projective space bundle (the Picard bundle). It has been studied in detail, for example by Kempf and Mukai.
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https://en.wikipedia.org/wiki/Symmetric_product_of_an_algebraic_curve
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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.
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https://en.wikipedia.org/wiki/Cabtaxi_number
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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 positive integers. Harmonic numbers have been studied since antiquity and are important in various branches of number theory. They are sometimes loosely termed harmonic series, are closely related to the Riemann zeta function, and appear in the expressions of various special functions.
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https://en.wikipedia.org/wiki/Harmonic_number
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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 plane by Bernhard Riemann in 1859, leading directly to the celebrated Riemann hypothesis about the distribution of prime numbers.
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https://en.wikipedia.org/wiki/Harmonic_number
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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, except for the case n = 1, the harmonic numbers are never integers.
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https://en.wikipedia.org/wiki/Harmonic_number
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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)}\quad (r>0).} In particular, H n = H n ( 1 ) {\displaystyle H_{n}=H_{n}^{(1)}} is the n-th harmonic number. The hyperharmonic numbers were discussed by J. H. Conway and R. K. Guy in their 1995 book The Book of Numbers. : 258
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https://en.wikipedia.org/wiki/Hyperharmonic_number
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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: In linear algebra, the n-th symmetric power of a vector space V is the vector subspace of the symmetric algebra of V consisting of degree-n elements (here the product is a tensor product). In algebraic topology, the n-th symmetric power of a topological space X is the quotient space X n / S n {\displaystyle X^{n}/{\mathfrak {S}}_{n}} , as in the beginning of this article. In algebraic geometry, a symmetric power is defined in a way similar to that in algebraic topology. For example, if X = Spec ( A ) {\displaystyle X=\operatorname {Spec} (A)} is an affine variety, then the GIT quotient Spec ( ( A ⊗ k ⋯ ⊗ k A ) S n ) {\displaystyle \operatorname {Spec} ((A\otimes _{k}\dots \otimes _{k}A)^{{\mathfrak {S}}_{n}})} is the n-th symmetric power of X.
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https://en.wikipedia.org/wiki/Symmetric_power
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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} (n)} . Many authors prefer slightly different notations, usually differing by factors of 2. The notation used here is consistent with the size of the most common matrices which represent the groups.
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https://en.wikipedia.org/wiki/Symplectic_group
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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 by their dimension n. The name "symplectic group" is due to Hermann Weyl as a replacement for the previous confusing names (line) complex group and Abelian linear group, and is the Greek analog of "complex". The metaplectic group is a double cover of the symplectic group over R; it has analogues over other local fields, finite fields, and adele rings.
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https://en.wikipedia.org/wiki/Symplectic_group
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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, ... Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, while in other writings, that term is used instead for the integers (including negative integers). In common language, particularly in primary school education, natural numbers may be called counting numbers to intuitively exclude the negative integers and zero, and also to contrast the discreteness of counting to the continuity of measurement—a hallmark characteristic of real numbers. The natural numbers can be used for counting (as in "there are six coins on the table"), in which case they serve as cardinal numbers.
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https://en.wikipedia.org/wiki/Counting_numbers
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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 numbers form a set, often symbolized as N {\textstyle \mathbb {N} } . Many other number sets are built by successively extending the set of natural numbers: the integers, by including an additive identity 0 (if not yet in) and an additive inverse −n for each nonzero natural number n; the rational numbers, by including a multiplicative inverse 1 / n {\displaystyle 1/n} for each nonzero integer n (and also the product of these inverses by integers); the real numbers by including the limits of Cauchy sequences of rationals; the complex numbers, by adjoining to the real numbers a square root of −1 (and also the sums and products thereof); and so on.
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https://en.wikipedia.org/wiki/Counting_numbers
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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 combinatorics.
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https://en.wikipedia.org/wiki/Counting_numbers
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In mathematics, the necklace ring is a ring introduced by Metropolis and Rota (1983) to elucidate the multiplicative properties of necklace polynomials.
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https://en.wikipedia.org/wiki/Necklace_ring
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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.
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https://en.wikipedia.org/wiki/Nil-Coxeter_algebra
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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 representations ρ: g → End ( V ) } . {\displaystyle {\mathcal {N}}=\{a\in {\mathfrak {g}}:\rho (a){\mbox{ is nilpotent for all representations }}\rho :{\mathfrak {g}}\to \operatorname {End} (V)\}.} The nilpotent cone is an irreducible subvariety of g {\displaystyle {\mathfrak {g}}} (considered as a vector space).
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https://en.wikipedia.org/wiki/Nilpotent_cone
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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 multiplication. Because of the Sprague–Grundy theorem which states that every impartial game is equivalent to a Nim heap of a certain size, nimbers arise in a much larger class of impartial games. They may also occur in partisan games like Domineering.
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https://en.wikipedia.org/wiki/Nimber
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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.
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https://en.wikipedia.org/wiki/Nimber
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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 well. Likewise, if all columns as well as the two top rows are exact, then the bottom row is exact as well. Similarly, because the diagram is symmetric about its diagonal, rows and columns may be interchanged in the above as well.
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https://en.wikipedia.org/wiki/Nine_lemma
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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.
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https://en.wikipedia.org/wiki/Nine_lemma
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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 lines. Make faces. You have now proved: (a) the Nine Lemma (b) the Sixteen Lemma (c) the Twenty-five Lemma..."There are two variants of nine lemma: sharp nine lemma and symmetric nine lemma (see Lemmas 3.3, 3.4 in Chapter XII of ).
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https://en.wikipedia.org/wiki/Nine_lemma
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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 sequence U , f ( U ) , f ( f ( U ) ) , … , f n ( U ) , … {\displaystyle U,f(U),f(f(U)),\dots ,f^{n}(U),\dots } will eventually become periodic. Here, f n denotes the n-fold iteration of f, that is, f n = f ∘ f ∘ ⋯ ∘ f ⏟ n .
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https://en.wikipedia.org/wiki/No-wandering-domain_theorem
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{\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 naturally belong to a finite-dimensional parameter space, most notably to transcendental entire and meromorphic functions with a finite number of singular values.
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https://en.wikipedia.org/wiki/No-wandering-domain_theorem
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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 graded Hopf algebra. It has the Hopf algebra of symmetric functions as a quotient, and is a subalgebra of the Hopf algebra of permutations, and is the graded dual of the Hopf algebra of quasisymmetric function. Over the rational numbers it is isomorphic as a Hopf algebra to the universal enveloping algebra of the free Lie algebra on countably many variables.
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https://en.wikipedia.org/wiki/Noncommutative_symmetric_function
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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.
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https://en.wikipedia.org/wiki/Nonmetricity_tensor
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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 forms, algebraic K-theory and the theory of motives. The theorem asserts that a certain statement holds true for any prime ℓ {\displaystyle \ell } and any natural number n {\displaystyle n} .
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https://en.wikipedia.org/wiki/Merkurjev–Suslin_theorem
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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 conjecture to distinguish it from the Bloch–Kato conjecture on values of L-functions. The norm residue isomorphism theorem was proved by Vladimir Voevodsky using a number of highly innovative results of Markus Rost.
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https://en.wikipedia.org/wiki/Merkurjev–Suslin_theorem
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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 equivalent to the normal form of the bifurcation. For example, the normal form of a saddle-node bifurcation is d x d t = μ + x 2 {\displaystyle {\frac {\mathrm {d} x}{\mathrm {d} t}}=\mu +x^{2}} where μ {\displaystyle \mu } is the bifurcation parameter. The transcritical bifurcation d x d t = r ln x + x − 1 {\displaystyle {\frac {\mathrm {d} x}{\mathrm {d} t}}=r\ln x+x-1} near x = 1 {\displaystyle x=1} can be converted to the normal form d u d t = μ u − u 2 + O ( u 3 ) {\displaystyle {\frac {\mathrm {d} u}{\mathrm {d} t}}=\mu u-u^{2}+O(u^{3})} with the transformation u = x − 1 , μ = r + 1 {\displaystyle u=x-1,\mu =r+1} .See also canonical form for use of the terms canonical form, normal form, or standard form more generally in mathematics.
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https://en.wikipedia.org/wiki/Normal_form_(dynamical_systems)
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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) construction.
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https://en.wikipedia.org/wiki/Cyclic_and_separating_vector
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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 commutative rings because of its relationship with the ideal structure of such rings.
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https://en.wikipedia.org/wiki/Divisibility_(ring_theory)
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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 functions or subsets in question will have some property, such as being analytic or smooth, but in general this is not needed (the functions in question need not even be continuous); it is however necessary that the space on/in which the object is defined is a topological space, in order that the word local has some meaning.
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https://en.wikipedia.org/wiki/Set_germ
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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.
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https://en.wikipedia.org/wiki/Quasi-continuous_function
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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 {C} .} The notion of a real form can also be defined for complex Lie groups. Real forms of complex semisimple Lie groups and Lie algebras have been completely classified by Élie Cartan.
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https://en.wikipedia.org/wiki/Split_form
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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.
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https://en.wikipedia.org/wiki/Dimension_function
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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.
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https://en.wikipedia.org/wiki/Compactly_embedded
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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 (or is right-cancellative) if for all b and c in M, b ∗ a = c ∗ a always implies that b = c. An element a in a magma (M, ∗) has the two-sided cancellation property (or is cancellative) if it is both left- and right-cancellative. A magma (M, ∗) has the left cancellation property (or is left-cancellative) if all a in the magma are left cancellative, and similar definitions apply for the right cancellative or two-sided cancellative properties. A left-invertible element is left-cancellative, and analogously for right and two-sided. If a⁻¹ is the inverse of a, then a ∗ b = a ∗ c implies a⁻¹ ∗ a ∗ b = a⁻¹ ∗ a ∗ c which implies b = c. For example, every quasigroup, and thus every group, is cancellative.
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https://en.wikipedia.org/wiki/Cancellation_property
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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 operations that model quantification and equality. They differ from polyadic algebras in that the latter do not model equality.
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https://en.wikipedia.org/wiki/Cylindric_algebra
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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.
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https://en.wikipedia.org/wiki/Expansivity_constant
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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 notion of end point, to study the divergence of nets in terms of convergence to end points and it is a useful tool for the study and classification of some families of non compact topological spaces. It can also be used to approach a topological space as the limit of other topological spaces: the externologies are very useful when a compact metric space embedded in a Hilbert space is approached by its open neighbourhoods.
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https://en.wikipedia.org/wiki/Exterior_space
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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 numbers.
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https://en.wikipedia.org/wiki/Automorphic_function
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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 under the action of G {\displaystyle G} .
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https://en.wikipedia.org/wiki/Automorphic_function
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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} on the multiplicative group of everywhere nonzero holomorphic functions.
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https://en.wikipedia.org/wiki/Automorphic_function
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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 factors of automorphy.Relation between factors of automorphy and other notions: Let Γ {\displaystyle \Gamma } be a lattice in a Lie group G {\displaystyle G} . Then, a factor of automorphy for Γ {\displaystyle \Gamma } corresponds to a line bundle on the quotient group G / Γ {\displaystyle G/\Gamma } . Further, the automorphic forms for a given factor of automorphy correspond to sections of the corresponding line bundle.The specific case of Γ {\displaystyle \Gamma } a subgroup of SL(2, R), acting on the upper half-plane, is treated in the article on automorphic factors.
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https://en.wikipedia.org/wiki/Automorphic_function
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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 numbers. 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\cdot x)=j_{g}(x)f(x)} where j g ( x ) {\displaystyle j_{g}(x)} is an everywhere nonzero holomorphic function.
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https://en.wikipedia.org/wiki/Automorphic_cuspidal_representation
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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.
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https://en.wikipedia.org/wiki/Automorphic_cuspidal_representation
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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 eigenfunction of certain Casimir operators on G; and to satisfy a "moderate growth" asymptotic condition a height function.It is the first of these that makes F automorphic, that is, satisfy an interesting functional equation relating F(g) with F(γg) for γ ∈ Γ {\displaystyle \gamma \in \Gamma } . In the vector-valued case the specification can involve a finite-dimensional group representation ρ acting on the components to 'twist' them. The Casimir operator condition says that some Laplacians have F as eigenfunction; this ensures that F has excellent analytic properties, but whether it is actually a complex-analytic function depends on the particular case.
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https://en.wikipedia.org/wiki/Automorphic_cuspidal_representation
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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 the possibility of vector-valued automorphic forms.
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https://en.wikipedia.org/wiki/Automorphic_cuspidal_representation
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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 as embeddings of Galois groups to their underlying global field extensions. In this formulation, automorphic forms are certain finite invariants, mapping from the idele class group under the Artin reciprocity law.
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https://en.wikipedia.org/wiki/Automorphic_cuspidal_representation
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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, therefore indicating the 'primitivity' of their fundamental structure.
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https://en.wikipedia.org/wiki/Automorphic_cuspidal_representation
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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 certain field extensions as Abelian groups. - Specific generalizations of Dirichlet L-functions as class field-theoretic objects.
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https://en.wikipedia.org/wiki/Automorphic_cuspidal_representation
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- 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 an abstracted irreducible fundamental representation). As mentioned, automorphic functions can be seen as generalizations of modular forms (as therefore elliptic curves), constructed by some zeta function analogue on an automorphic structure. In the simplest sense, automorphic forms are modular forms defined on general Lie groups; because of their symmetry properties. Therefore in simpler terms, a general function which analyzes the invariance of a structure with respect to its prime 'morphology'.
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https://en.wikipedia.org/wiki/Automorphic_cuspidal_representation
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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 polyconvex if A ↦ f ( A ) {\displaystyle A\mapsto f(A)} can be written as a convex function of the p × p subdeterminants of A, for 1 ≤ p ≤ min{m, n}. Polyconvexity is a weaker property than convexity. For example, the function f given by f ( A ) = { 1 det ( A ) , det ( A ) > 0 ; + ∞ , det ( A ) ≤ 0 ; {\displaystyle f(A)={\begin{cases}{\frac {1}{\det(A)}},&\det(A)>0;\\+\infty ,&\det(A)\leq 0;\end{cases}}} is polyconvex but not convex.
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https://en.wikipedia.org/wiki/Polyconvex_function
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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 to the property of the same name for single-valued functions. Roughly speaking, a function is upper hemicontinuous if when (1) a convergent sequence of points in the domain maps to a sequence of sets in the range which (2) contain another convergent sequence, then the image of the limiting point in the domain must contain the limit of the sequence in the range. Lower hemicontinuity essentially reverses this, saying if a sequence in the domain converges, given a point in the range of the limit, then you can find a sub-sequence whose image contains a convergent sequence to the given point.
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https://en.wikipedia.org/wiki/Hemicontinuity
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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 mathematician John Milnor.
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https://en.wikipedia.org/wiki/Prevalent_and_shy_sets
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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 Motzkin numbers M n {\displaystyle M_{n}} for n = 0 , 1 , … {\displaystyle n=0,1,\dots } form the sequence: 1, 1, 2, 4, 9, 21, 51, 127, 323, 835, ... (sequence A001006 in the OEIS)
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https://en.wikipedia.org/wiki/Motzkin_number
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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 {\displaystyle e^{2i\pi {\frac {k}{n}}}} , where k runs over the positive integers not greater than n and coprime to n (and i is the imaginary unit). In other words, the nth cyclotomic polynomial is equal to Φ n ( x ) = ∏ gcd ( k , n ) = 1 1 ≤ k ≤ n ( x − e 2 i π k n ) . {\displaystyle \Phi _{n}(x)=\prod _{\stackrel {1\leq k\leq n}{\gcd(k,n)=1}}\left(x-e^{2i\pi {\frac {k}{n}}}\right).} It may also be defined as the monic polynomial with integer coefficients that is the minimal polynomial over the field of the rational numbers of any primitive nth-root of unity ( e 2 i π / n {\displaystyle e^{2i\pi /n}} is an example of such a root). An important relation linking cyclotomic polynomials and primitive roots of unity is ∏ d ∣ n Φ d ( x ) = x n − 1 , {\displaystyle \prod _{d\mid n}\Phi _{d}(x)=x^{n}-1,} showing that x is a root of x n − 1 {\displaystyle x^{n}-1} if and only if it is a d th primitive root of unity for some d that divides n.
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https://en.wikipedia.org/wiki/Cyclotomic_polynomial
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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 derived from a conversation in about 1919 involving mathematicians G. H. Hardy and Srinivasa Ramanujan.
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https://en.wikipedia.org/wiki/Taxicab_number
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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 as the sum of two cubes in two different ways."
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https://en.wikipedia.org/wiki/Taxicab_number
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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 give it a shorter name.When testing if a series converges or diverges, this test is often checked first due to its ease of use. In the case of p-adic analysis the term test is a necessary and sufficient condition for convergence due to the non-archimedean triangle inequality.
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https://en.wikipedia.org/wiki/Nth-term_test
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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 } ".
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https://en.wikipedia.org/wiki/Null_sign
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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\}} B = { 4 , 6 , 8 , 9 } {\displaystyle B=\{4,6,8,9\}} A ∩ B = ? {\displaystyle A\cap B=?} There are no common elements in the solution; so it should be denoted as: A ∩ B = ∅ {\displaystyle A\cap B=\varnothing } or A ∩ B = { } {\displaystyle A\cap B=\{\}}
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https://en.wikipedia.org/wiki/Null_sign
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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 the quaternions, of which they are an extension. They are noncommutative and nonassociative, but satisfy a weaker form of associativity; namely, they are alternative.
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https://en.wikipedia.org/wiki/Cayley_numbers
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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.
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https://en.wikipedia.org/wiki/Cayley_numbers
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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.
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https://en.wikipedia.org/wiki/Cayley_numbers
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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 less than or equal to one: D ¯ 1 ( P ) = { Q: | P − Q | ≤ 1 } . {\displaystyle {\bar {D}}_{1}(P)=\{Q:|P-Q|\leq 1\}.\,} Unit disks are special cases of disks and unit balls; as such, they contain the interior of the unit circle and, in the case of the closed unit disk, the unit circle itself. Without further specifications, the term unit disk is used for the open unit disk about the origin, D 1 ( 0 ) {\displaystyle D_{1}(0)} , with respect to the standard Euclidean metric.
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https://en.wikipedia.org/wiki/Open_unit_disc
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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} } .
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https://en.wikipedia.org/wiki/Open_unit_disc
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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 linear map T: X → Y {\displaystyle T:X\to Y} is the maximum factor by which it "lengthens" vectors.
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https://en.wikipedia.org/wiki/Norm_topology
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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 theory was introduced by Kirillov (1961, 1962) for nilpotent groups and later extended by Bertram Kostant, Louis Auslander, Lajos Pukánszky and others to the case of solvable groups. Roger Howe found a version of the orbit method that applies to p-adic Lie groups.David Vogan proposed that the orbit method should serve as a unifying principle in the description of the unitary duals of real reductive Lie groups.
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https://en.wikipedia.org/wiki/Kirillov_orbit_theory
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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, the order of an element a of a group, is thus the smallest positive integer m such that am = e, where e denotes the identity element of the group, and am denotes the product of m copies of a. If no such m exists, the order of a is infinite.
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https://en.wikipedia.org/wiki/Finite_order
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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 order of the subgroup divides the order of the group; that is, |H| is a divisor of |G|. In particular, the order |a| of any element is a divisor of |G|.
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https://en.wikipedia.org/wiki/Finite_order
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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 elements in the partial order. The order polytope is a distributive polytope, meaning that coordinatewise minima and maxima of pairs of its points remain within the polytope. The order polytope of a partial order should be distinguished from the linear ordering polytope, a polytope defined from a number n {\displaystyle n} as the convex hull of indicator vectors of the sets of edges of n {\displaystyle n} -vertex transitive tournaments.
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https://en.wikipedia.org/wiki/Order_polytope
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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 ∞ Ω n S O {\displaystyle \Omega _{*}^{SO}=\oplus _{0}^{\infty }\Omega _{n}^{SO}} where Ω n S O {\displaystyle \Omega _{n}^{SO}} consists of oriented cobordism classes of manifolds of dimension n. One can also define an unoriented cobordism ring, denoted by Ω ∗ O {\displaystyle \Omega _{*}^{O}} . If O is replaced U, then one gets the complex cobordism ring, oriented or unoriented. In general, one writes Ω ∗ B {\displaystyle \Omega _{*}^{B}} for the cobordism ring of manifolds with structure B. A theorem of Thom says: Ω n O = π n ( M O ) {\displaystyle \Omega _{n}^{O}=\pi _{n}(MO)} where MO is the Thom spectrum.
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https://en.wikipedia.org/wiki/Cobordism_ring
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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 one to pick an origin point that makes the mathematics as simple as possible, often by taking advantage of some kind of geometric symmetry.
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https://en.wikipedia.org/wiki/Origin_(mathematics)
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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 transformations. The orthogonal group is sometimes called the general orthogonal group, by analogy with the general linear group. Equivalently, it is the group of n × n {\displaystyle n\times n} orthogonal matrices, where the group operation is given by matrix multiplication (an orthogonal matrix is a real matrix whose inverse equals its transpose). The orthogonal group is an algebraic group and a Lie group.
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https://en.wikipedia.org/wiki/Complex_orthogonal_group
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It is compact. The orthogonal group in dimension n {\displaystyle n} has two connected components.
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https://en.wikipedia.org/wiki/Complex_orthogonal_group
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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, its elements are the usual rotations around a point (in dimension 2) or a line (in dimension 3).
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https://en.wikipedia.org/wiki/Complex_orthogonal_group
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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.
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https://en.wikipedia.org/wiki/Complex_orthogonal_group
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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 , F ) {\displaystyle \operatorname {O} (n,F)} , of the general linear group GL ( n , F ) {\displaystyle \operatorname {GL} (n,F)} ; that is More generally, given a non-degenerate symmetric bilinear form or quadratic form on a vector space over a field, the orthogonal group of the form is the group of invertible linear maps that preserve the form. The preceding orthogonal groups are the special case where, on some basis, the bilinear form is the dot product, or, equivalently, the quadratic form is the sum of the square of the coordinates. All orthogonal groups are algebraic groups, since the condition of preserving a form can be expressed as an equality of matrices.
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https://en.wikipedia.org/wiki/Complex_orthogonal_group
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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 mathematical treatment: oscillation of a sequence of real numbers, oscillation of a real-valued function at a point, and oscillation of a function on an interval (or open set).
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https://en.wikipedia.org/wiki/Oscillation_(mathematics)
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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 1988. The semigroup had previously been studied by other mathematicians and physicists, most notably Felix Berezin in the 1960s. The simplest example in one dimension is given by SU(1,1).
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https://en.wikipedia.org/wiki/Weyl_calculus
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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) corresponding to Möbius transformations that take the unit disk into itself. The contraction operators, determined only up to a sign, have kernels that are Gaussian functions.
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https://en.wikipedia.org/wiki/Weyl_calculus
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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 summarizes how the theory can be extended.
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https://en.wikipedia.org/wiki/Weyl_calculus
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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 to be complete. An automorphism of a group that is not inner is called an outer automorphism.
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https://en.wikipedia.org/wiki/Outer_automorphism_group
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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 naturally identified; that is, the outer automorphism group does act on the group. For example, for the alternating group, An, the outer automorphism group is usually the group of order 2, with exceptions noted below. Considering An as a subgroup of the symmetric group, Sn, conjugation by any odd permutation is an outer automorphism of An or more precisely "represents the class of the (non-trivial) outer automorphism of An", but the outer automorphism does not correspond to conjugation by any particular odd element, and all conjugations by odd elements are equivalent up to conjugation by an even element.
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https://en.wikipedia.org/wiki/Outer_automorphism_group
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In mathematics, the overlapping interval topology is a topology which is used to illustrate various topological principles.
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https://en.wikipedia.org/wiki/Overlapping_interval_topology
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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
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https://en.wikipedia.org/wiki/P-Laplacian
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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) had previously given a definition of a different p-adic analogue of the gamma function, but his function does not have satisfactory properties and is not used much.
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https://en.wikipedia.org/wiki/P-adic_gamma_function
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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 dimension is constructed by covering the given subset by such small open balls. The packing dimension was introduced by C. Tricot Jr. in 1982.
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https://en.wikipedia.org/wiki/Packing_dimension
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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 distinct forms (A) and (B) by completing the square and rescaling z, called H. F. Weber's equations: and If is a solution, then so are If is a solution of equation (A), then is a solution of (B), and, by symmetry, are also solutions of (B).
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https://en.wikipedia.org/wiki/Parabolic_cylinder_function
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