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Boas–Buck polynomials
Summary
Boas–Buck_polynomials
In mathematics, Boas–Buck polynomials are sequences of polynomials Φ n ( r ) ( z ) {\displaystyle \Phi _{n}^{(r)}(z)} defined from analytic functions B {\displaystyle B} and C {\displaystyle C} by generating functions of the form C ( z t r B ( t ) ) = ∑ n ≥ 0 Φ n ( r ) ( z ) t n {\displaystyle \displaystyle C(zt^{r}B(t))=\sum _{n\geq 0}\Phi _{n}^{(r)}(z)t^{n}} .The case r = 1 {\displaystyle r=1} , sometimes called generalized Appell polynomials, was studied by Boas and Buck (1958).
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Bochner space
Summary
Bochner_space
In mathematics, Bochner spaces are a generalization of the concept of L p {\displaystyle L^{p}} spaces to functions whose values lie in a Banach space which is not necessarily the space R {\displaystyle \mathbb {R} } or C {\displaystyle \mathbb {C} } of real or complex numbers. The space L p ( X ) {\displaystyle L^{p}(X)} consists of (equivalence classes of) all Bochner measurable functions f {\displaystyle f} with values in the Banach space X {\displaystyle X} whose norm ‖ f ‖ X {\displaystyle \|f\|_{X}} lies in the standard L p {\displaystyle L^{p}} space. Thus, if X {\displaystyle X} is the set of complex numbers, it is the standard Lebesgue L p {\displaystyle L^{p}} space.
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Bochner space
Summary
Bochner_space
Almost all standard results on L p {\displaystyle L^{p}} spaces do hold on Bochner spaces too; in particular, the Bochner spaces L p ( X ) {\displaystyle L^{p}(X)} are Banach spaces for 1 ≤ p ≤ ∞ . {\displaystyle 1\leq p\leq \infty .} Bochner spaces are named for the mathematician Salomon Bochner.
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Bochner formula
Summary
Bochner_formula
In mathematics, Bochner's formula is a statement relating harmonic functions on a Riemannian manifold ( M , g ) {\displaystyle (M,g)} to the Ricci curvature. The formula is named after the American mathematician Salomon Bochner.
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Bochner theorem
Summary
Bochner's_theorem
In mathematics, Bochner's theorem (named for Salomon Bochner) characterizes the Fourier transform of a positive finite Borel measure on the real line. More generally in harmonic analysis, Bochner's theorem asserts that under Fourier transform a continuous positive-definite function on a locally compact abelian group corresponds to a finite positive measure on the Pontryagin dual group. The case of sequences was first established by Gustav Herglotz (see also the related Herglotz representation theorem.)
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Bochner's tube theorem
Summary
Bochner's_tube_theorem
In mathematics, Bochner's tube theorem (named for Salomon Bochner) shows that every function holomorphic on a tube domain in C n {\displaystyle \mathbb {C} ^{n}} can be extended to the convex hull of this domain. Theorem Let ω ⊂ R n {\displaystyle \omega \subset \mathbb {R} ^{n}} be a connected open set. Then every function f ( z ) {\displaystyle f(z)} holomorphic on the tube domain Ω = ω + i R n {\displaystyle \Omega =\omega +i\mathbb {R} ^{n}} can be extended to a function holomorphic on the convex hull ch ⁡ ( Ω ) {\displaystyle \operatorname {ch} (\Omega )} . A classic reference is (Theorem 9). See also for other proofs.
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Edge-of-the-wedge theorem
Summary
Edge-of-the-wedge_theorem
In mathematics, Bogoliubov's edge-of-the-wedge theorem implies that holomorphic functions on two "wedges" with an "edge" in common are analytic continuations of each other provided they both give the same continuous function on the edge. It is used in quantum field theory to construct the analytic continuation of Wightman functions. The formulation and the first proof of the theorem were presented by Nikolay Bogoliubov at the International Conference on Theoretical Physics, Seattle, USA (September, 1956) and also published in the book Problems in the Theory of Dispersion Relations. Further proofs and generalizations of the theorem were given by R. Jost and H. Lehmann (1957), F. Dyson (1958), H. Epstein (1960), and by other researchers.
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Bondy's theorem
Summary
Bondy's_theorem
In mathematics, Bondy's theorem is a bound on the number of elements needed to distinguish the sets in a family of sets from each other. It belongs to the field of combinatorics, and is named after John Adrian Bondy, who published it in 1972.
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Boole's rule
Summary
Boole's_rule
In mathematics, Boole's rule, named after George Boole, is a method of numerical integration.
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Borel summation
Summary
Borel_summability
In mathematics, Borel summation is a summation method for divergent series, introduced by Émile Borel (1899). It is particularly useful for summing divergent asymptotic series, and in some sense gives the best possible sum for such series. There are several variations of this method that are also called Borel summation, and a generalization of it called Mittag-Leffler summation.
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Borel's lemma
Summary
Borel's_lemma
In mathematics, Borel's lemma, named after Émile Borel, is an important result used in the theory of asymptotic expansions and partial differential equations.
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Borel–de Siebenthal theory
Summary
Borel–de_Siebenthal_theory
In mathematics, Borel–de Siebenthal theory describes the closed connected subgroups of a compact Lie group that have maximal rank, i.e. contain a maximal torus. It is named after the Swiss mathematicians Armand Borel and Jean de Siebenthal who developed the theory in 1949. Each such subgroup is the identity component of the centralizer of its center. They can be described recursively in terms of the associated root system of the group. The subgroups for which the corresponding homogeneous space has an invariant complex structure correspond to parabolic subgroups in the complexification of the compact Lie group, a reductive algebraic group.
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Borwein's algorithm
Summary
Borwein's_algorithm
In mathematics, Borwein's algorithm is an algorithm devised by Jonathan and Peter Borwein to calculate the value of 1/π. They devised several other algorithms. They published the book Pi and the AGM – A Study in Analytic Number Theory and Computational Complexity.
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Bour's minimal surface
Summary
Bour's_minimal_surface
In mathematics, Bour's minimal surface is a two-dimensional minimal surface, embedded with self-crossings into three-dimensional Euclidean space. It is named after Edmond Bour, whose work on minimal surfaces won him the 1861 mathematics prize of the French Academy of Sciences.
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Brandt matrix
Summary
Brandt_matrix
In mathematics, Brandt matrices are matrices, introduced by Brandt (1943), that are related to the number of ideals of given norm in an ideal class of a definite quaternion algebra over the rationals, and that give a representation of the Hecke algebra. Eichler (1955) calculated the traces of the Brandt matrices. Let O be an order in a quaternion algebra with class number H, and Ii,...,IH invertible left O-ideals representing the classes.
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Brandt matrix
Summary
Brandt_matrix
Fix an integer m. Let ej denote the number of units in the right order of Ij and let Bij denote the number of α in Ij−1Ii with reduced norm N(α) equal to mN(Ii)/N(Ij). The Brandt matrix B(m) is the H×H matrix with entries Bij. Up to conjugation by a permutation matrix it is independent of the choice of representatives Ij; it is dependent only on the level of the order O.
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Brandt semigroup
Summary
Brandt_semigroup
In mathematics, Brandt semigroups are completely 0-simple inverse semigroups. In other words, they are semigroups without proper ideals and which are also inverse semigroups. They are built in the same way as completely 0-simple semigroups: Let G be a group and I , J {\displaystyle I,J} be non-empty sets.
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Brandt semigroup
Summary
Brandt_semigroup
Define a matrix P {\displaystyle P} of dimension | I | × | J | {\displaystyle |I|\times |J|} with entries in G 0 = G ∪ { 0 } . {\displaystyle G^{0}=G\cup \{0\}.} Then, it can be shown that every 0-simple semigroup is of the form S = ( I × G 0 × J ) {\displaystyle S=(I\times G^{0}\times J)} with the operation ( i , a , j ) ∗ ( k , b , n ) = ( i , a p j k b , n ) {\displaystyle (i,a,j)*(k,b,n)=(i,ap_{jk}b,n)} .
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Brandt semigroup
Summary
Brandt_semigroup
As Brandt semigroups are also inverse semigroups, the construction is more specialized and in fact, I = J (Howie 1995). Thus, a Brandt semigroup has the form S = ( I × G 0 × I ) {\displaystyle S=(I\times G^{0}\times I)} with the operation ( i , a , j ) ∗ ( k , b , n ) = ( i , a p j k b , n ) {\displaystyle (i,a,j)*(k,b,n)=(i,ap_{jk}b,n)} . Moreover, the matrix P {\displaystyle P} is diagonal with only the identity element e of the group G in its diagonal.
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Brenke polynomials
Summary
Brenke_polynomials
In mathematics, Brenke polynomials are special cases of generalized Appell polynomials, and Brenke–Chihara polynomials are the Brenke polynomials that are also orthogonal polynomials. Brenke (1945) introduced sequences of Brenke polynomials Pn, which are special cases of generalized Appell polynomials with generating function of the form A ( w ) B ( x w ) = ∑ n = 0 ∞ P n ( x ) w n . {\displaystyle A(w)B(xw)=\sum _{n=0}^{\infty }P_{n}(x)w^{n}.}
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Brenke polynomials
Summary
Brenke_polynomials
Brenke observed that Hermite polynomials and Laguerre polynomials are examples of Brenke polynomials, and asked if there are any other sequences of orthogonal polynomials of this form. Geronimus (1947) found some further examples of orthogonal Brenke polynomials. Chihara (1968, 1971) completely classified all Brenke polynomials that form orthogonal sequences, which are now called Brenke–Chihara polynomials, and found their orthogonality relations.
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Brewer sum
Summary
Brewer_sum
In mathematics, Brewer sums are finite character sum introduced by Brewer (1961, 1966) related to Jacobsthal sums.
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Bring's curve
Summary
Bring's_curve
In mathematics, Bring's curve (also called Bring's surface and, by analogy with the Klein quartic, the Bring sextic) is the curve in P 4 {\displaystyle \mathbb {P} ^{4}} cut out by the homogeneous equations v + w + x + y + z = v 2 + w 2 + x 2 + y 2 + z 2 = v 3 + w 3 + x 3 + y 3 + z 3 = 0. {\displaystyle v+w+x+y+z=v^{2}+w^{2}+x^{2}+y^{2}+z^{2}=v^{3}+w^{3}+x^{3}+y^{3}+z^{3}=0.} It was named by Klein (2003, p.157) after Erland Samuel Bring who studied a similar construction in 1786 in a Promotionschrift submitted to the University of Lund. Note that the roots xi of the Bring quintic x 5 + a x + b = 0 {\displaystyle x^{5}+ax+b=0} satisfies Bring's curve since ∑ i = 1 5 x i k = 0 {\displaystyle \sum _{i=1}^{5}x_{i}^{k}=0} for k = 1 , 2 , 3.
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Bring's curve
Summary
Bring's_curve
{\displaystyle k=1,2,3.} The automorphism group of the curve is the symmetric group S5 of order 120, given by permutations of the 5 coordinates.
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Bring's curve
Summary
Bring's_curve
This is the largest possible automorphism group of a genus 4 complex curve. The curve can be realized as a triple cover of the sphere branched in 12 points, and is the Riemann surface associated to the small stellated dodecahedron. It has genus 4. The full group of symmetries (including reflections) is the direct product S 5 × Z 2 {\displaystyle S_{5}\times \mathbb {Z} _{2}} , which has order 240.
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Brown's representability theorem
Summary
Brown_representability
In mathematics, Brown's representability theorem in homotopy theory gives necessary and sufficient conditions for a contravariant functor F on the homotopy category Hotc of pointed connected CW complexes, to the category of sets Set, to be a representable functor. More specifically, we are given F: Hotcop → Set,and there are certain obviously necessary conditions for F to be of type Hom(—, C), with C a pointed connected CW-complex that can be deduced from category theory alone. The statement of the substantive part of the theorem is that these necessary conditions are then sufficient. For technical reasons, the theorem is often stated for functors to the category of pointed sets; in other words the sets are also given a base point.
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Brownian Motion
Mathematics
Brownian_diffusion > Mathematics
In mathematics, Brownian motion is described by the Wiener process, a continuous-time stochastic process named in honor of Norbert Wiener. It is one of the best known Lévy processes (càdlàg stochastic processes with stationary independent increments) and occurs frequently in pure and applied mathematics, economics and physics. The Wiener process Wt is characterized by four facts: W0 = 0 Wt is almost surely continuous Wt has independent increments W t − W s ∼ N ( 0 , t − s ) {\displaystyle W_{t}-W_{s}\sim {\mathcal {N}}(0,t-s)} (for 0 ≤ s ≤ t {\displaystyle 0\leq s\leq t} ). N ( μ , σ 2 ) {\displaystyle {\mathcal {N}}(\mu ,\sigma ^{2})} denotes the normal distribution with expected value μ and variance σ2. The condition that it has independent increments means that if 0 ≤ s 1 < t 1 ≤ s 2 < t 2 {\displaystyle 0\leq s_{1}
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Brown–Peterson cohomology
Summary
Brown-Peterson_cohomology
In mathematics, Brown–Peterson cohomology is a generalized cohomology theory introduced by Edgar H. Brown and Franklin P. Peterson (1966), depending on a choice of prime p. It is described in detail by Douglas Ravenel (2003, Chapter 4). Its representing spectrum is denoted by BP.
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Buchsbaum ring
Summary
Weak_sequence
In mathematics, Buchsbaum rings are Noetherian local rings such that every system of parameters is a weak sequence. A sequence ( a 1 , ⋯ , a r ) {\displaystyle (a_{1},\cdots ,a_{r})} of the maximal ideal m {\displaystyle m} is called a weak sequence if m ⋅ ( ( a 1 , ⋯ , a i − 1 ): a i ) ⊂ ( a 1 , ⋯ , a i − 1 ) {\displaystyle m\cdot ((a_{1},\cdots ,a_{i-1})\colon a_{i})\subset (a_{1},\cdots ,a_{i-1})} for all i {\displaystyle i} . They were introduced by Jürgen Stückrad and Wolfgang Vogel (1973) and are named after David Buchsbaum. Every Cohen–Macaulay local ring is a Buchsbaum ring. Every Buchsbaum ring is a generalized Cohen–Macaulay ring.
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Budan's Theorem
Summary
Budan's_Theorem
In mathematics, Budan's theorem is a theorem for bounding the number of real roots of a polynomial in an interval, and computing the parity of this number. It was published in 1807 by François Budan de Boislaurent. A similar theorem was published independently by Joseph Fourier in 1820. Each of these theorems is a corollary of the other. Fourier's statement appears more often in the literature of 19th century and has been referred to as Fourier's, Budan–Fourier, Fourier–Budan, and even Budan's theorem Budan's original formulation is used in fast modern algorithms for real-root isolation of polynomials.
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Burnside theorem
Summary
Burnside's_theorem
In mathematics, Burnside's theorem in group theory states that if G is a finite group of order p a q b {\displaystyle p^{a}q^{b}} where p and q are prime numbers, and a and b are non-negative integers, then G is solvable. Hence each non-Abelian finite simple group has order divisible by at least three distinct primes.
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Busemann's theorem
Summary
Busemann's_theorem
In mathematics, Busemann's theorem is a theorem in Euclidean geometry and geometric tomography. It was first proved by Herbert Busemann in 1949 and was motivated by his theory of area in Finsler spaces.
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Bäcklund transform
Summary
Bäcklund_transform
In mathematics, Bäcklund transforms or Bäcklund transformations (named after the Swedish mathematician Albert Victor Bäcklund) relate partial differential equations and their solutions. They are an important tool in soliton theory and integrable systems. A Bäcklund transform is typically a system of first order partial differential equations relating two functions, and often depending on an additional parameter. It implies that the two functions separately satisfy partial differential equations, and each of the two functions is then said to be a Bäcklund transformation of the other.
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Bäcklund transform
Summary
Bäcklund_transform
A Bäcklund transform which relates solutions of the same equation is called an invariant Bäcklund transform or auto-Bäcklund transform. If such a transform can be found, much can be deduced about the solutions of the equation especially if the Bäcklund transform contains a parameter. However, no systematic way of finding Bäcklund transforms is known.
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Bézout's lemma
Summary
Bézout's_lemma
In mathematics, Bézout's identity (also called Bézout's lemma), named after Étienne Bézout, is the following theorem: Here the greatest common divisor of 0 and 0 is taken to be 0. The integers x and y are called Bézout coefficients for (a, b); they are not unique. A pair of Bézout coefficients can be computed by the extended Euclidean algorithm, and this pair is, in the case of integers one of the two pairs such that | x | ≤ | b / d | {\displaystyle |x|\leq |b/d|} and | y | ≤ | a / d | ; {\displaystyle |y|\leq |a/d|;} equality occurs only if one of a and b is a multiple of the other.
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Bézout's lemma
Summary
Bézout's_lemma
As an example, the greatest common divisor of 15 and 69 is 3, and 3 can be written as a combination of 15 and 69 as 3 = 15 × (−9) + 69 × 2, with Bézout coefficients −9 and 2. Many other theorems in elementary number theory, such as Euclid's lemma or the Chinese remainder theorem, result from Bézout's identity. A Bézout domain is an integral domain in which Bézout's identity holds. In particular, Bézout's identity holds in principal ideal domains. Every theorem that results from Bézout's identity is thus true in all principal ideal domains.
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Bôcher's theorem
Summary
Bôcher's_theorem
In mathematics, Bôcher's theorem is either of two theorems named after the American mathematician Maxime Bôcher.
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Capelli's identity
Summary
Capelli_identity
In mathematics, Capelli's identity, named after Alfredo Capelli (1887), is an analogue of the formula det(AB) = det(A) det(B), for certain matrices with noncommuting entries, related to the representation theory of the Lie algebra g l n {\displaystyle {\mathfrak {gl}}_{n}} . It can be used to relate an invariant ƒ to the invariant Ωƒ, where Ω is Cayley's Ω process.
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Carathéodory existence theorem
Summary
Carathéodory's_existence_theorem
In mathematics, Carathéodory's existence theorem says that an ordinary differential equation has a solution under relatively mild conditions. It is a generalization of Peano's existence theorem. Peano's theorem requires that the right-hand side of the differential equation be continuous, while Carathéodory's theorem shows existence of solutions (in a more general sense) for some discontinuous equations. The theorem is named after Constantin Carathéodory.
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Carathéodory's theorem (conformal mapping)
Summary
Carathéodory's_theorem_(conformal_mapping)
In mathematics, Carathéodory's theorem is a theorem in complex analysis, named after Constantin Carathéodory, which extends the Riemann mapping theorem. The theorem, first proved in 1913, states that any conformal mapping sending the unit disk to some region in the complex plane bounded by a Jordan curve extends continuously to a homeomorphism from the unit circle onto the Jordan curve. The result is one of Carathéodory's results on prime ends and the boundary behaviour of univalent holomorphic functions.
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Carleman linearization
Summary
Carleman_linearization
In mathematics, Carleman linearization (or Carleman embedding) is a technique to transform a finite-dimensional nonlinear dynamical system into an infinite-dimensional linear system. It was introduced by the Swedish mathematician Torsten Carleman in 1932. Carleman linearization is related to composition operator and has been widely used in the study of dynamical systems. It also been used in many applied fields, such as in control theory and in quantum computing.
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Carleman's equation
Summary
Carleman's_equation
In mathematics, Carleman's equation is a Fredholm integral equation of the first kind with a logarithmic kernel. Its solution was first given by Torsten Carleman in 1922. The equation is ∫ a b ln ⁡ | x − t | y ( t ) d t = f ( x ) {\displaystyle \int _{a}^{b}\ln |x-t|\,y(t)\,dt=f(x)} The solution for b − a ≠ 4 is y ( x ) = 1 π 2 ( x − a ) ( b − x ) ∫ a b f ( t ) d t ( t − a ) ( b − t ) ] {\displaystyle y(x)={\frac {1}{\pi ^{2}{\sqrt {(x-a)(b-x)}}}}\left}}\int _{a}^{b}{\frac {f(t)\,dt}{\sqrt {(t-a)(b-t)}}}\right]} If b − a = 4 then the equation is solvable only if the following condition is satisfied ∫ a b f ( t ) d t ( t − a ) ( b − t ) = 0 {\displaystyle \int _{a}^{b}{\frac {f(t)\,dt}{\sqrt {(t-a)(b-t)}}}=0} In this case the solution has the form y ( x ) = 1 π 2 ( x − a ) ( b − x ) {\displaystyle y(x)={\frac {1}{\pi ^{2}{\sqrt {(x-a)(b-x)}}}}\left} where C is an arbitrary constant. For the special case f(t) = 1 (in which case it is necessary to have b − a ≠ 4), useful in some applications, we get y ( x ) = 1 π ln ⁡ 1 ( x − a ) ( b − x ) {\displaystyle y(x)={\frac {1}{\pi \ln \left}}{\frac {1}{\sqrt {(x-a)(b-x)}}}}
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Carmichael's totient function conjecture
Summary
Carmichael's_totient_function_conjecture
In mathematics, Carmichael's totient function conjecture concerns the multiplicity of values of Euler's totient function φ(n), which counts the number of integers less than and coprime to n. It states that, for every n there is at least one other integer m ≠ n such that φ(m) = φ(n). Robert Carmichael first stated this conjecture in 1907, but as a theorem rather than as a conjecture. However, his proof was faulty, and in 1922, he retracted his claim and stated the conjecture as an open problem.
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Cartan criterion
Summary
Cartan's_criterion
In mathematics, Cartan's criterion gives conditions for a Lie algebra in characteristic 0 to be solvable, which implies a related criterion for the Lie algebra to be semisimple. It is based on the notion of the Killing form, a symmetric bilinear form on g {\displaystyle {\mathfrak {g}}} defined by the formula B ( u , v ) = tr ⁡ ( ad ⁡ ( u ) ad ⁡ ( v ) ) , {\displaystyle B(u,v)=\operatorname {tr} (\operatorname {ad} (u)\operatorname {ad} (v)),} where tr denotes the trace of a linear operator. The criterion was introduced by Élie Cartan (1894).
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Theory of equivalence
Summary
Cartan's_equivalence_method
In mathematics, Cartan's equivalence method is a technique in differential geometry for determining whether two geometrical structures are the same up to a diffeomorphism. For example, if M and N are two Riemannian manifolds with metrics g and h, respectively, when is there a diffeomorphism ϕ: M → N {\displaystyle \phi :M\rightarrow N} such that ϕ ∗ h = g {\displaystyle \phi ^{*}h=g} ?Although the answer to this particular question was known in dimension 2 to Gauss and in higher dimensions to Christoffel and perhaps Riemann as well, Élie Cartan and his intellectual heirs developed a technique for answering similar questions for radically different geometric structures. (For example see the Cartan–Karlhede algorithm.) Cartan successfully applied his equivalence method to many such structures, including projective structures, CR structures, and complex structures, as well as ostensibly non-geometrical structures such as the equivalence of Lagrangians and ordinary differential equations.
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Theory of equivalence
Summary
Cartan's_equivalence_method
(His techniques were later developed more fully by many others, such as D. C. Spencer and Shiing-Shen Chern.) The equivalence method is an essentially algorithmic procedure for determining when two geometric structures are identical. For Cartan, the primary geometrical information was expressed in a coframe or collection of coframes on a differentiable manifold. See method of moving frames.
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Cartan's lemma
Summary
Cartan's_lemma
In mathematics, Cartan's lemma refers to a number of results named after either Élie Cartan or his son Henri Cartan: In exterior algebra: Suppose that v1, ..., vp are linearly independent elements of a vector space V and w1, ..., wp are such that v 1 ∧ w 1 + ⋯ + v p ∧ w p = 0 {\displaystyle v_{1}\wedge w_{1}+\cdots +v_{p}\wedge w_{p}=0} in ΛV. Then there are scalars hij = hji such that w i = ∑ j = 1 p h i j v j . {\displaystyle w_{i}=\sum _{j=1}^{p}h_{ij}v_{j}.} In several complex variables: Let a1 < a2 < a3 < a4 and b1 < b2 and define rectangles in the complex plane C by K 1 = { z 1 = x 1 + i y 1 | a 2 < x 1 < a 3 , b 1 < y 1 < b 2 } K 1 ′ = { z 1 = x 1 + i y 1 | a 1 < x 1 < a 3 , b 1 < y 1 < b 2 } K 1 ″ = { z 1 = x 1 + i y 1 | a 2 < x 1 < a 4 , b 1 < y 1 < b 2 } {\displaystyle {\begin{aligned}K_{1}&=\{z_{1}=x_{1}+iy_{1}|a_{2}
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Cartan's theorems A and B
Summary
Cartan's_theorems_A_and_B
In mathematics, Cartan's theorems A and B are two results proved by Henri Cartan around 1951, concerning a coherent sheaf F on a Stein manifold X. They are significant both as applied to several complex variables, and in the general development of sheaf cohomology. Theorem B is stated in cohomological terms (a formulation that Cartan (1953, p. 51) attributes to J.-P. Serre): Analogous properties were established by Serre (1957) for coherent sheaves in algebraic geometry, when X is an affine scheme.
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Cartan's theorems A and B
Summary
Cartan's_theorems_A_and_B
The analogue of Theorem B in this context is as follows (Hartshorne 1977, Theorem III.3.7): These theorems have many important applications. For instance, they imply that a holomorphic function on a closed complex submanifold, Z, of a Stein manifold X can be extended to a holomorphic function on all of X. At a deeper level, these theorems were used by Jean-Pierre Serre to prove the GAGA theorem. Theorem B is sharp in the sense that if H1(X, F) = 0 for all coherent sheaves F on a complex manifold X (resp.
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Cartan's theorems A and B
Summary
Cartan's_theorems_A_and_B
quasi-coherent sheaves F on a noetherian scheme X), then X is Stein (resp. affine); see (Serre 1956) (resp. (Serre 1957) and (Hartshorne 1977, Theorem III.3.7)).
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Cartier duality
Summary
Cartier_dual
In mathematics, Cartier duality is an analogue of Pontryagin duality for commutative group schemes. It was introduced by Pierre Cartier (1962).
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Casey's theorem
Summary
Casey's_theorem
In mathematics, Casey's theorem, also known as the generalized Ptolemy's theorem, is a theorem in Euclidean geometry named after the Irish mathematician John Casey.
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Castelnuovo's contraction theorem
Summary
Castelnuovo's_contraction_theorem
In mathematics, Castelnuovo's contraction theorem is used in the classification theory of algebraic surfaces to construct the minimal model of a given smooth algebraic surface. More precisely, let X {\displaystyle X} be a smooth projective surface over C {\displaystyle \mathbb {C} } and C {\displaystyle C} a (−1)-curve on X {\displaystyle X} (which means a smooth rational curve of self-intersection number −1), then there exists a morphism from X {\displaystyle X} to another smooth projective surface Y {\displaystyle Y} such that the curve C {\displaystyle C} has been contracted to one point P {\displaystyle P} , and moreover this morphism is an isomorphism outside C {\displaystyle C} (i.e., X ∖ C {\displaystyle X\setminus C} is isomorphic with Y ∖ P {\displaystyle Y\setminus P} ). This contraction morphism is sometimes called a blowdown, which is the inverse operation of blowup. The curve C {\displaystyle C} is also called an exceptional curve of the first kind.
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Catalan's constant
Summary
Catalan's_constant
In mathematics, Catalan's constant G, is defined by G = β ( 2 ) = ∑ n = 0 ∞ ( − 1 ) n ( 2 n + 1 ) 2 = 1 1 2 − 1 3 2 + 1 5 2 − 1 7 2 + 1 9 2 − ⋯ , {\displaystyle G=\beta (2)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(2n+1)^{2}}}={\frac {1}{1^{2}}}-{\frac {1}{3^{2}}}+{\frac {1}{5^{2}}}-{\frac {1}{7^{2}}}+{\frac {1}{9^{2}}}-\cdots ,} where β is the Dirichlet beta function. Its numerical value is approximately (sequence A006752 in the OEIS) G = 0.915965594177219015054603514932384110774… It is not known whether G is irrational, let alone transcendental. G has been called "arguably the most basic constant whose irrationality and transcendence (though strongly suspected) remain unproven".Catalan's constant was named after Eugène Charles Catalan, who found quickly-converging series for its calculation and published a memoir on it in 1865.
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Cauchy integral formula
Summary
Cauchy's_integral_formula
In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary of the disk, and it provides integral formulas for all derivatives of a holomorphic function. Cauchy's formula shows that, in complex analysis, "differentiation is equivalent to integration": complex differentiation, like integration, behaves well under uniform limits – a result that does not hold in real analysis.
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Cayley's Ω process
Summary
Cayley's_Ω_process
In mathematics, Cayley's Ω process, introduced by Arthur Cayley (1846), is a relatively invariant differential operator on the general linear group, that is used to construct invariants of a group action. As a partial differential operator acting on functions of n2 variables xij, the omega operator is given by the determinant Ω = | ∂ ∂ x 11 ⋯ ∂ ∂ x 1 n ⋮ ⋱ ⋮ ∂ ∂ x n 1 ⋯ ∂ ∂ x n n | . {\displaystyle \Omega ={\begin{vmatrix}{\frac {\partial }{\partial x_{11}}}&\cdots &{\frac {\partial }{\partial x_{1n}}}\\\vdots &\ddots &\vdots \\{\frac {\partial }{\partial x_{n1}}}&\cdots &{\frac {\partial }{\partial x_{nn}}}\end{vmatrix}}.} For binary forms f in x1, y1 and g in x2, y2 the Ω operator is ∂ 2 f g ∂ x 1 ∂ y 2 − ∂ 2 f g ∂ x 2 ∂ y 1 {\displaystyle {\frac {\partial ^{2}fg}{\partial x_{1}\partial y_{2}}}-{\frac {\partial ^{2}fg}{\partial x_{2}\partial y_{1}}}} . The r-fold Ω process Ωr(f, g) on two forms f and g in the variables x and y is then Convert f to a form in x1, y1 and g to a form in x2, y2 Apply the Ω operator r times to the function fg, that is, f times g in these four variables Substitute x for x1 and x2, y for y1 and y2 in the resultThe result of the r-fold Ω process Ωr(f, g) on the two forms f and g is also called the r-th transvectant and is commonly written (f, g)r.
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Charlier polynomials
Summary
Charlier_polynomials
In mathematics, Charlier polynomials (also called Poisson–Charlier polynomials) are a family of orthogonal polynomials introduced by Carl Charlier. They are given in terms of the generalized hypergeometric function by C n ( x ; μ ) = 2 F 0 ( − n , − x ; − ; − 1 / μ ) = ( − 1 ) n n ! L n ( − 1 − x ) ( − 1 μ ) , {\displaystyle C_{n}(x;\mu )={}_{2}F_{0}(-n,-x;-;-1/\mu )=(-1)^{n}n!L_{n}^{(-1-x)}\left(-{\frac {1}{\mu }}\right),} where L {\displaystyle L} are generalized Laguerre polynomials.
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Charlier polynomials
Summary
Charlier_polynomials
They satisfy the orthogonality relation ∑ x = 0 ∞ μ x x ! C n ( x ; μ ) C m ( x ; μ ) = μ − n e μ n ! δ n m , μ > 0.
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Charlier polynomials
Summary
Charlier_polynomials
{\displaystyle \sum _{x=0}^{\infty }{\frac {\mu ^{x}}{x! }}C_{n}(x;\mu )C_{m}(x;\mu )=\mu ^{-n}e^{\mu }n!\delta _{nm},\quad \mu >0.} They form a Sheffer sequence related to the Poisson process, similar to how Hermite polynomials relate to the Brownian motion.
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Chebyshev distance
Summary
Maximum_metric
In mathematics, Chebyshev distance (or Tchebychev distance), maximum metric, or L∞ metric is a metric defined on a vector space where the distance between two vectors is the greatest of their differences along any coordinate dimension. It is named after Pafnuty Chebyshev. It is also known as chessboard distance, since in the game of chess the minimum number of moves needed by a king to go from one square on a chessboard to another equals the Chebyshev distance between the centers of the squares, if the squares have side length one, as represented in 2-D spatial coordinates with axes aligned to the edges of the board. For example, the Chebyshev distance between f6 and e2 equals 4.
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Chebyshev's sum inequality
Summary
Chebyshev's_sum_inequality
In mathematics, Chebyshev's sum inequality, named after Pafnuty Chebyshev, states that if a 1 ≥ a 2 ≥ ⋯ ≥ a n {\displaystyle a_{1}\geq a_{2}\geq \cdots \geq a_{n}\quad } and b 1 ≥ b 2 ≥ ⋯ ≥ b n , {\displaystyle \quad b_{1}\geq b_{2}\geq \cdots \geq b_{n},} then 1 n ∑ k = 1 n a k b k ≥ ( 1 n ∑ k = 1 n a k ) ( 1 n ∑ k = 1 n b k ) . {\displaystyle {1 \over n}\sum _{k=1}^{n}a_{k}b_{k}\geq \left({1 \over n}\sum _{k=1}^{n}a_{k}\right)\!\!\left({1 \over n}\sum _{k=1}^{n}b_{k}\right)\!.} Similarly, if a 1 ≤ a 2 ≤ ⋯ ≤ a n {\displaystyle a_{1}\leq a_{2}\leq \cdots \leq a_{n}\quad } and b 1 ≥ b 2 ≥ ⋯ ≥ b n , {\displaystyle \quad b_{1}\geq b_{2}\geq \cdots \geq b_{n},} then 1 n ∑ k = 1 n a k b k ≤ ( 1 n ∑ k = 1 n a k ) ( 1 n ∑ k = 1 n b k ) . {\displaystyle {1 \over n}\sum _{k=1}^{n}a_{k}b_{k}\leq \left({1 \over n}\sum _{k=1}^{n}a_{k}\right)\!\!\left({1 \over n}\sum _{k=1}^{n}b_{k}\right)\!.}
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Choi's theorem on completely positive maps
Summary
Choi's_theorem_on_completely_positive_maps
In mathematics, Choi's theorem on completely positive maps is a result that classifies completely positive maps between finite-dimensional (matrix) C*-algebras. An infinite-dimensional algebraic generalization of Choi's theorem is known as Belavkin's "Radon–Nikodym" theorem for completely positive maps.
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Choquet theory
Summary
Choquet_theory
In mathematics, Choquet theory, named after Gustave Choquet, is an area of functional analysis and convex analysis concerned with measures which have support on the extreme points of a convex set C. Roughly speaking, every vector of C should appear as a weighted average of extreme points, a concept made more precise by generalizing the notion of weighted average from a convex combination to an integral taken over the set E of extreme points. Here C is a subset of a real vector space V, and the main thrust of the theory is to treat the cases where V is an infinite-dimensional (locally convex Hausdorff) topological vector space along lines similar to the finite-dimensional case. The main concerns of Gustave Choquet were in potential theory. Choquet theory has become a general paradigm, particularly for treating convex cones as determined by their extreme rays, and so for many different notions of positivity in mathematics.
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Choquet theory
Summary
Choquet_theory
The two ends of a line segment determine the points in between: in vector terms the segment from v to w consists of the λv + (1 − λ)w with 0 ≤ λ ≤ 1. The classical result of Hermann Minkowski says that in Euclidean space, a bounded, closed convex set C is the convex hull of its extreme point set E, so that any c in C is a (finite) convex combination of points e of E. Here E may be a finite or an infinite set. In vector terms, by assigning non-negative weights w(e) to the e in E, almost all 0, we can represent any c in C as with In any case the w(e) give a probability measure supported on a finite subset of E. For any affine function f on C, its value at the point c is In the infinite dimensional setting, one would like to make a similar statement.
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Chow's theorem
Summary
Chow's_theorem
In mathematics, Chow's theorem may refer to a number of theorems due to Wei-Liang Chow: Chow's theorem: The theorem that asserts that any analytic subvariety in projective space is actually algebraic. Chow–Rashevskii theorem: In sub-Riemannian geometry, the theorem that asserts that any two points are connected by a horizontal curve.
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Chrystal's equation
Summary
Chrystal's_equation
In mathematics, Chrystal's equation is a first order nonlinear ordinary differential equation, named after the mathematician George Chrystal, who discussed the singular solution of this equation in 1896. The equation reads as ( d y d x ) 2 + A x d y d x + B y + C x 2 = 0 {\displaystyle \left({\frac {dy}{dx}}\right)^{2}+Ax{\frac {dy}{dx}}+By+Cx^{2}=0} where A , B , C {\displaystyle A,\ B,\ C} are constants, which upon solving for d y / d x {\displaystyle dy/dx} , gives d y d x = − A 2 x ± 1 2 ( A 2 x 2 − 4 B y − 4 C x 2 ) 1 / 2 . {\displaystyle {\frac {dy}{dx}}=-{\frac {A}{2}}x\pm {\frac {1}{2}}(A^{2}x^{2}-4By-4Cx^{2})^{1/2}.} This equation is a generalization of Clairaut's equation since it reduces to Clairaut's equation under certain condition as given below.
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Church numerals
Summary
Church_encoding
In mathematics, Church encoding is a means of representing data and operators in the lambda calculus. The Church numerals are a representation of the natural numbers using lambda notation. The method is named for Alonzo Church, who first encoded data in the lambda calculus this way.
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Church numerals
Summary
Church_encoding
Terms that are usually considered primitive in other notations (such as integers, booleans, pairs, lists, and tagged unions) are mapped to higher-order functions under Church encoding. The Church-Turing thesis asserts that any computable operator (and its operands) can be represented under Church encoding. In the untyped lambda calculus the only primitive data type is the function.
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Clarke's generalized Jacobian
Summary
Clarke's_generalized_Jacobian
In mathematics, Clarke's generalized Jacobian is a generalization of the Jacobian matrix of a smooth function to non-smooth functions. It was introduced by Clarke (1983).
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Clarkson's inequalities
Summary
Clarkson's_inequalities
In mathematics, Clarkson's inequalities, named after James A. Clarkson, are results in the theory of Lp spaces. They give bounds for the Lp-norms of the sum and difference of two measurable functions in Lp in terms of the Lp-norms of those functions individually.
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Clausen's formula
Summary
Clausen's_formula
In mathematics, Clausen's formula, found by Thomas Clausen (1828), expresses the square of a Gaussian hypergeometric series as a generalized hypergeometric series. It states 2 F 1 2 = 3 F 2 {\displaystyle \;_{2}F_{1}\left^{2}=\;_{3}F_{2}\left} In particular it gives conditions for a hypergeometric series to be positive. This can be used to prove several inequalities, such as the Askey–Gasper inequality used in the proof of de Branges's theorem.
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Clifford theory
Summary
Clifford_theory
In mathematics, Clifford theory, introduced by Alfred H. Clifford (1937), describes the relation between representations of a group and those of a normal subgroup.
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Green's conjecture
Summary
Clifford's_theorem_on_special_divisors
In mathematics, Clifford's theorem on special divisors is a result of William K. Clifford (1878) on algebraic curves, showing the constraints on special linear systems on a curve C.
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Cohn's theorem
Summary
Cohn's_theorem
In mathematics, Cohn's theorem states that a nth-degree self-inversive polynomial p ( z ) {\displaystyle p(z)} has as many roots in the open unit disk D = { z ∈ C: | z | < 1 } {\displaystyle D=\{z\in \mathbb {C} :|z|<1\}} as the reciprocal polynomial of its derivative. Cohn's theorem is useful for studying the distribution of the roots of self-inversive and self-reciprocal polynomials in the complex plane.An nth-degree polynomial, p ( z ) = p 0 + p 1 z + ⋯ + p n z n {\displaystyle p(z)=p_{0}+p_{1}z+\cdots +p_{n}z^{n}} is called self-inversive if there exists a fixed complex number ( ω {\displaystyle \omega } ) of modulus 1 so that, p ( z ) = ω p ∗ ( z ) , ( | ω | = 1 ) , {\displaystyle p(z)=\omega p^{*}(z),\qquad \left(|\omega |=1\right),} where p ∗ ( z ) = z n p ¯ ( 1 / z ¯ ) = p ¯ n + p ¯ n − 1 z + ⋯ + p ¯ 0 z n {\displaystyle p^{*}(z)=z^{n}{\bar {p}}\left(1/{\bar {z}}\right)={\bar {p}}_{n}+{\bar {p}}_{n-1}z+\cdots +{\bar {p}}_{0}z^{n}} is the reciprocal polynomial associated with p ( z ) {\displaystyle p(z)} and the bar means complex conjugation. Self-inversive polynomials have many interesting properties. For instance, its roots are all symmetric with respect to the unit circle and a polynomial whose roots are all on the unit circle is necessarily self-inversive.
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Cohn's theorem
Summary
Cohn's_theorem
The coefficients of self-inversive polynomials satisfy the relations. p k = ω p ¯ n − k , 0 ⩽ k ⩽ n . {\displaystyle p_{k}=\omega {\bar {p}}_{n-k},\qquad 0\leqslant k\leqslant n.}
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Cohn's theorem
Summary
Cohn's_theorem
In the case where ω = 1 , {\displaystyle \omega =1,} a self-inversive polynomial becomes a complex-reciprocal polynomial (also known as a self-conjugate polynomial). If its coefficients are real then it becomes a real self-reciprocal polynomial. The formal derivative of p ( z ) {\displaystyle p(z)} is a (n − 1)th-degree polynomial given by q ( z ) = p ′ ( z ) = p 1 + 2 p 2 z + ⋯ + n p n z n − 1 .
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Cohn's theorem
Summary
Cohn's_theorem
{\displaystyle q(z)=p'(z)=p_{1}+2p_{2}z+\cdots +np_{n}z^{n-1}.} Therefore, Cohn's theorem states that both p ( z ) {\displaystyle p(z)} and the polynomial q ∗ ( z ) = z n − 1 q ¯ n − 1 ( 1 / z ¯ ) = z n − 1 p ¯ ′ ( 1 / z ¯ ) = n p ¯ n + ( n − 1 ) p ¯ n − 1 z + ⋯ + p ¯ 1 z n − 1 {\displaystyle q^{*}(z)=z^{n-1}{\bar {q}}_{n-1}\left(1/{\bar {z}}\right)=z^{n-1}{\bar {p}}'\left(1/{\bar {z}}\right)=n{\bar {p}}_{n}+(n-1){\bar {p}}_{n-1}z+\cdots +{\bar {p}}_{1}z^{n-1}} have the same number of roots in | z | < 1. {\displaystyle |z|<1.}
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Costa's minimal surface
Summary
Costa's_minimal_surface
In mathematics, Costa's minimal surface, is an embedded minimal surface discovered in 1982 by the Brazilian mathematician Celso José da Costa. It is also a surface of finite topology, which means that it can be formed by puncturing a compact surface. Topologically, it is a thrice-punctured torus. Until its discovery, the plane, helicoid and the catenoid were believed to be the only embedded minimal surfaces that could be formed by puncturing a compact surface.
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Costa's minimal surface
Summary
Costa's_minimal_surface
The Costa surface evolves from a torus, which is deformed until the planar end becomes catenoidal. Defining these surfaces on rectangular tori of arbitrary dimensions yields the Costa surface. Its discovery triggered research and discovery into several new surfaces and open conjectures in topology. The Costa surface can be described using the Weierstrass zeta and the Weierstrass elliptic functions.
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Coxeter matroid
Summary
Coxeter_matroid
In mathematics, Coxeter matroids are generalization of matroids depending on a choice of a Coxeter group W and a parabolic subgroup P. Ordinary matroids correspond to the case when P is a maximal parabolic subgroup of a symmetric group W. They were introduced by Gelfand and Serganova (1987, 1987b), who named them after H. S. M. Coxeter. Borovik, Gelfand & White (2003) give a detailed account of Coxeter matroids.
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Cramer's paradox
Summary
Cramer's_paradox
In mathematics, Cramer's paradox or the Cramer–Euler paradox is the statement that the number of points of intersection of two higher-order curves in the plane can be greater than the number of arbitrary points that are usually needed to define one such curve. It is named after the Genevan mathematician Gabriel Cramer. This phenomenon appears paradoxical because the points of intersection fail to uniquely define any curve (they belong to at least two different curves) despite their large number.
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Cramer's paradox
Summary
Cramer's_paradox
It is the result of a naive understanding or a misapplication of two theorems: Bézout's theorem states that the number of points of intersection of two algebraic curves is equal to the product of their degrees, provided that certain necessary conditions are met. In particular, two curves of degree n {\displaystyle n} generally have n 2 {\displaystyle n^{2}} points of intersection. Cramer's theorem states that a curve of degree n {\displaystyle n} is determined by n ( n + 3 ) / 2 {\displaystyle n(n+3)/2} points, again assuming that certain conditions hold.For all n ≥ 3 {\displaystyle n\geq 3} , n 2 ≥ n ( n + 3 ) / 2 {\displaystyle n^{2}\geq n(n+3)/2} , so it would naively appear that for degree three or higher, the intersection of two curves would have enough points to define either of the curves uniquely.
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Cramer's paradox
Summary
Cramer's_paradox
However, because these points belong to both curves, they do not define a unique curve of this degree. The resolution of the paradox is that the n ( n + 3 ) / 2 {\displaystyle n(n+3)/2} bound on the number of points needed to define a curve only applies to points in general position. In certain degenerate cases, n ( n + 3 ) / 2 {\displaystyle n(n+3)/2} points are not enough to determine a curve uniquely.
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Cutler's bar notation
Summary
Cutler's_bar_notation
In mathematics, Cutler's bar notation is a notation system for large numbers, introduced by Mark Cutler in 2004. The idea is based on iterated exponentiation in much the same way that exponentiation is iterated multiplication.
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Dihedral group of order 6
Summary
Symmetric_group_of_degree_3
In mathematics, D3 (sometimes alternatively denoted by D6) is the dihedral group of degree 3 and order 6. It equals the symmetric group S3. It is also the smallest non-abelian group.This page illustrates many group concepts using this group as example.
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Danzer's configuration
Summary
Danzer's_configuration
In mathematics, Danzer's configuration is a self-dual configuration of 35 lines and 35 points, having 4 points on each line and 4 lines through each point. It is named after the German geometer Ludwig Danzer and was popularised by Branko Grünbaum. The Levi graph of the configuration is the Kronecker cover of the odd graph O4, and is isomorphic to the middle layer graph of the seven-dimensional hypercube graph Q7. The middle layer graph of an odd-dimensional hypercube graph Q2n+1(n,n+1) is a subgraph whose vertex set consists of all binary strings of length 2n + 1 that have exactly n or n + 1 entries equal to 1, with an edge between any two vertices for which the corresponding binary strings differ in exactly one bit.
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Danzer's configuration
Summary
Danzer's_configuration
Every middle layer graph is Hamiltonian.Danzer's configuration DCD(4) is the fourth term of an infinite series of ( ( 2 n − 1 n ) n ) {\displaystyle ({\tbinom {2n-1}{n}}_{n})} configurations DCD(n), where DCD(1) is the trivial configuration (11), DCD(2) is the trilateral (32) and DCD(3) is the Desargues configuration (103). In configurations DCD(n) were further generalized to the unbalanced ( ( n d ) d , ( n d − 1 ) n − d + 1 ) {\displaystyle ({\tbinom {n}{d}}_{d},{\tbinom {n}{d-1}}_{n-d+1})} configuration DCD(n,d) by introducing parameter d with connection DCD(n) = DCD(2n-1,n). DCD stands for Desargues-Cayley-Danzer. Each DCD(2n,d) configuration is a subconfiguration of the ( 2 2 n + 1 2 n ) {\displaystyle (2_{2n+1}^{2n})} Clifford configuration. While each DCD(n,d) admits a realisation as a geometric point-line configuration, the Clifford configuration can only be realised as a point-circle configuration and depicts the Clifford's circle theorems.
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Darboux function
Summary
Darboux's_theorem_(analysis)
In mathematics, Darboux's theorem is a theorem in real analysis, named after Jean Gaston Darboux. It states that every function that results from the differentiation of another function has the intermediate value property: the image of an interval is also an interval. When ƒ is continuously differentiable (ƒ in C1()), this is a consequence of the intermediate value theorem. But even when ƒ′ is not continuous, Darboux's theorem places a severe restriction on what it can be.
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De Gua's theorem
Summary
De_Gua's_theorem
In mathematics, De Gua's theorem is a three-dimensional analog of the Pythagorean theorem named after Jean Paul de Gua de Malves. It states that if a tetrahedron has a right-angle corner (like the corner of a cube), then the square of the area of the face opposite the right-angle corner is the sum of the squares of the areas of the other three faces: De Gua's theorem can be applied for proving a special case of Heron's formula.
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Completion (order theory)
Summary
Completion_(order_theory)
In mathematics, Dedekind cuts, named after German mathematician Richard Dedekind but previously considered by Joseph Bertrand, are а method of construction of the real numbers from the rational numbers. A Dedekind cut is a partition of the rational numbers into two sets A and B, such that all elements of A are less than all elements of B, and A contains no greatest element. The set B may or may not have a smallest element among the rationals. If B has a smallest element among the rationals, the cut corresponds to that rational.
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Completion (order theory)
Summary
Completion_(order_theory)
Otherwise, that cut defines a unique irrational number which, loosely speaking, fills the "gap" between A and B. In other words, A contains every rational number less than the cut, and B contains every rational number greater than or equal to the cut. An irrational cut is equated to an irrational number which is in neither set. Every real number, rational or not, is equated to one and only one cut of rationals.Dedekind cuts can be generalized from the rational numbers to any totally ordered set by defining a Dedekind cut as a partition of a totally ordered set into two non-empty parts A and B, such that A is closed downwards (meaning that for all a in A, x ≤ a implies that x is in A as well) and B is closed upwards, and A contains no greatest element.
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Completion (order theory)
Summary
Completion_(order_theory)
See also completeness (order theory). It is straightforward to show that a Dedekind cut among the real numbers is uniquely defined by the corresponding cut among the rational numbers. Similarly, every cut of reals is identical to the cut produced by a specific real number (which can be identified as the smallest element of the B set). In other words, the number line where every real number is defined as a Dedekind cut of rationals is a complete continuum without any further gaps.
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Dedekind sum
Summary
Dedekind_sum
In mathematics, Dedekind sums are certain sums of products of a sawtooth function, and are given by a function D of three integer variables. Dedekind introduced them to express the functional equation of the Dedekind eta function. They have subsequently been much studied in number theory, and have occurred in some problems of topology. Dedekind sums have a large number of functional equations; this article lists only a small fraction of these. Dedekind sums were introduced by Richard Dedekind in a commentary on fragment XXVIII of Bernhard Riemann's collected papers.
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Degen's eight-square identity
Summary
Degen's_eight-square_identity
In mathematics, Degen's eight-square identity establishes that the product of two numbers, each of which is a sum of eight squares, is itself the sum of eight squares. Namely: First discovered by Carl Ferdinand Degen around 1818, the identity was independently rediscovered by John Thomas Graves (1843) and Arthur Cayley (1845). The latter two derived it while working on an extension of quaternions called octonions.
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Degen's eight-square identity
Summary
Degen's_eight-square_identity
In algebraic terms the identity means that the norm of product of two octonions equals the product of their norms: ‖ a b ‖ = ‖ a ‖ ‖ b ‖ {\displaystyle \left\|ab\right\|=\left\|a\right\|\left\|b\right\|} . Similar statements are true for quaternions (Euler's four-square identity), complex numbers (the Brahmagupta–Fibonacci two-square identity) and real numbers. In 1898 Adolf Hurwitz proved that there is no similar bilinear identity for 16 squares (sedenions) or any other number of squares except for 1,2,4, and 8.
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Degen's eight-square identity
Summary
Degen's_eight-square_identity
However, in the 1960s, H. Zassenhaus, W. Eichhorn, and A. Pfister (independently) showed there can be a non-bilinear identity for 16 squares. Note that each quadrant reduces to a version of Euler's four-square identity: and similarly for the other three quadrants. Comment: The proof of the eight-square identity is by algebraic evaluation.
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Degen's eight-square identity
Summary
Degen's_eight-square_identity
The eight-square identity can be written in the form of a product of two inner products of 8-dimensional vectors, yielding again an inner product of 8-dimensional vectors: (a·a)(b·b) = (a×b)·(a×b). This defines the octonion multiplication rule a×b, which reflects Degen's 8-square identity and the mathematics of octonions. By Pfister's theorem, a different sort of eight-square identity can be given where the z i {\displaystyle z_{i}} , introduced below, are non-bilinear and merely rational functions of the x i , y i {\displaystyle x_{i},y_{i}} . Thus, where, and, with, Incidentally, the u i {\displaystyle u_{i}} obey the identity,
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Dehn's lemma
Summary
Dehn's_lemma
In mathematics, Dehn's lemma asserts that a piecewise-linear map of a disk into a 3-manifold, with the map's singularity set in the disk's interior, implies the existence of another piecewise-linear map of the disk which is an embedding and is identical to the original on the boundary of the disk. This theorem was thought to be proven by Max Dehn (1910), but Hellmuth Kneser (1929, page 260) found a gap in the proof. The status of Dehn's lemma remained in doubt until Christos Papakyriakopoulos (1957, 1957b) using work by Johansson (1938) proved it using his "tower construction". He also generalized the theorem to the loop theorem and sphere theorem.
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Deligne cohomology
Summary
Deligne_cohomology
In mathematics, Deligne cohomology is the hypercohomology of the Deligne complex of a complex manifold. It was introduced by Pierre Deligne in unpublished work in about 1972 as a cohomology theory for algebraic varieties that includes both ordinary cohomology and intermediate Jacobians. For introductory accounts of Deligne cohomology see Brylinski (2008, section 1.5), Esnault & Viehweg (1988), and Gomi (2009, section 2).
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Deligne–Lusztig theory
Summary
Deligne–Lusztig_theory
In mathematics, Deligne–Lusztig theory is a way of constructing linear representations of finite groups of Lie type using ℓ-adic cohomology with compact support, introduced by Pierre Deligne and George Lusztig (1976). Lusztig (1985) used these representations to find all representations of all finite simple groups of Lie type.