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In mathematics, the 57-cell (pentacontakaiheptachoron) is a self-dual abstract regular 4-polytope (four-dimensional polytope). Its 57 cells are hemi-dodecahedra. It also has 57 vertices, 171 edges and 171 two-dimensional faces. The symmetry order is 3420, from the product of the number of cells (57) and the symmetry of each cell (60).
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https://en.wikipedia.org/wiki/57-cell
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The symmetry abstract structure is the projective special linear group, L2(19). It has Schläfli type {5,3,5} with 5 hemi-dodecahedral cells around each edge. It was discovered by H. S. M. Coxeter (1982).
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https://en.wikipedia.org/wiki/57-cell
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In mathematics, the ADE classification (originally A-D-E classifications) is a situation where certain kinds of objects are in correspondence with simply laced Dynkin diagrams. The question of giving a common origin to these classifications, rather than a posteriori verification of a parallelism, was posed in (Arnold 1976). The complete list of simply laced Dynkin diagrams comprises A n , D n , E 6 , E 7 , E 8 . {\displaystyle A_{n},\,D_{n},\,E_{6},\,E_{7},\,E_{8}.}
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https://en.wikipedia.org/wiki/Buckyball_surface
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Here "simply laced" means that there are no multiple edges, which corresponds to all simple roots in the root system forming angles of π / 2 = 90 ∘ {\displaystyle \pi /2=90^{\circ }} (no edge between the vertices) or 2 π / 3 = 120 ∘ {\displaystyle 2\pi /3=120^{\circ }} (single edge between the vertices). These are two of the four families of Dynkin diagrams (omitting B n {\displaystyle B_{n}} and C n {\displaystyle C_{n}} ), and three of the five exceptional Dynkin diagrams (omitting F 4 {\displaystyle F_{4}} and G 2 {\displaystyle G_{2}} ). This list is non-redundant if one takes n ≥ 4 {\displaystyle n\geq 4} for D n .
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https://en.wikipedia.org/wiki/Buckyball_surface
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{\displaystyle D_{n}.} If one extends the families to include redundant terms, one obtains the exceptional isomorphisms D 3 ≅ A 3 , E 4 ≅ A 4 , E 5 ≅ D 5 , {\displaystyle D_{3}\cong A_{3},E_{4}\cong A_{4},E_{5}\cong D_{5},} and corresponding isomorphisms of classified objects. The A, D, E nomenclature also yields the simply laced finite Coxeter groups, by the same diagrams: in this case the Dynkin diagrams exactly coincide with the Coxeter diagrams, as there are no multiple edges.
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https://en.wikipedia.org/wiki/Buckyball_surface
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In mathematics, the AKNS system is an integrable system of partial differential equations, introduced by and named after Mark J. Ablowitz, David J. Kaup, Alan C. Newell, and Harvey Segur from their publication in Studies in Applied Mathematics: Ablowitz, Kaup, and Newell et al. (1974).
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https://en.wikipedia.org/wiki/AKNS_system
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In mathematics, the ATS theorem is the theorem on the approximation of a trigonometric sum by a shorter one. The application of the ATS theorem in certain problems of mathematical and theoretical physics can be very helpful.
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https://en.wikipedia.org/wiki/ATS_theorem
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In mathematics, the Abel transform, named for Niels Henrik Abel, is an integral transform often used in the analysis of spherically symmetric or axially symmetric functions. The Abel transform of a function f(r) is given by F ( y ) = 2 ∫ y ∞ f ( r ) r r 2 − y 2 d r . {\displaystyle F(y)=2\int _{y}^{\infty }{\frac {f(r)r}{\sqrt {r^{2}-y^{2}}}}\,dr.} Assuming that f(r) drops to zero more quickly than 1/r, the inverse Abel transform is given by f ( r ) = − 1 π ∫ r ∞ d F d y d y y 2 − r 2 .
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https://en.wikipedia.org/wiki/Abel_transform
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{\displaystyle f(r)=-{\frac {1}{\pi }}\int _{r}^{\infty }{\frac {dF}{dy}}\,{\frac {dy}{\sqrt {y^{2}-r^{2}}}}.} In image analysis, the forward Abel transform is used to project an optically thin, axially symmetric emission function onto a plane, and the inverse Abel transform is used to calculate the emission function given a projection (i.e. a scan or a photograph) of that emission function.
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https://en.wikipedia.org/wiki/Abel_transform
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In absorption spectroscopy of cylindrical flames or plumes, the forward Abel transform is the integrated absorbance along a ray with closest distance y from the center of the flame, while the inverse Abel transform gives the local absorption coefficient at a distance r from the center. Abel transform is limited to applications with axially symmetric geometries. For more general asymmetrical cases, more general-oriented reconstruction algorithms such as algebraic reconstruction technique (ART), maximum likelihood expectation maximization (MLEM), filtered back-projection (FBP) algorithms should be employed. In recent years, the inverse Abel transform (and its variants) has become the cornerstone of data analysis in photofragment-ion imaging and photoelectron imaging. Among recent most notable extensions of inverse Abel transform are the "onion peeling" and "basis set expansion" (BASEX) methods of photoelectron and photoion image analysis.
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https://en.wikipedia.org/wiki/Abel_transform
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In mathematics, the Abel–Jacobi map is a construction of algebraic geometry which relates an algebraic curve to its Jacobian variety. In Riemannian geometry, it is a more general construction mapping a manifold to its Jacobi torus. The name derives from the theorem of Abel and Jacobi that two effective divisors are linearly equivalent if and only if they are indistinguishable under the Abel–Jacobi map.
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https://en.wikipedia.org/wiki/Abel–Jacobi_theorem
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In mathematics, the Abel–Plana formula is a summation formula discovered independently by Niels Henrik Abel (1823) and Giovanni Antonio Amedeo Plana (1820). It states that ∑ n = 0 ∞ f ( a + n ) = ∫ a ∞ f ( x ) d x + f ( a ) 2 + ∫ 0 ∞ f ( a − i x ) − f ( a + i x ) i ( e 2 π x − 1 ) d x {\displaystyle \sum _{n=0}^{\infty }f\left(a+n\right)=\int _{a}^{\infty }f\left(x\right)dx+{\frac {f\left(a\right)}{2}}+\int _{0}^{\infty }{\frac {f\left(a-ix\right)-f\left(a+ix\right)}{i\left(e^{2\pi x}-1\right)}}dx} For the case a = 0 {\displaystyle a=0} we have ∑ n = 0 ∞ f ( n ) = 1 2 f ( 0 ) + ∫ 0 ∞ f ( x ) d x + i ∫ 0 ∞ f ( i t ) − f ( − i t ) e 2 π t − 1 d t . {\displaystyle \sum _{n=0}^{\infty }f(n)={\frac {1}{2}}f(0)+\int _{0}^{\infty }f(x)\,dx+i\int _{0}^{\infty }{\frac {f(it)-f(-it)}{e^{2\pi t}-1}}\,dt.} It holds for functions ƒ that are holomorphic in the region Re(z) ≥ 0, and satisfy a suitable growth condition in this region; for example it is enough to assume that |ƒ| is bounded by C/|z|1+ε in this region for some constants C, ε > 0, though the formula also holds under much weaker bounds.
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https://en.wikipedia.org/wiki/Abel–Plana_formula
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In mathematics, the Abel–Ruffini theorem (also known as Abel's impossibility theorem) states that there is no solution in radicals to general polynomial equations of degree five or higher with arbitrary coefficients. Here, general means that the coefficients of the equation are viewed and manipulated as indeterminates. The theorem is named after Paolo Ruffini, who made an incomplete proof in 1799, (which was refined and completed in 1813 and accepted by Cauchy) and Niels Henrik Abel, who provided a proof in 1824.Abel–Ruffini theorem refers also to the slightly stronger result that there are equations of degree five and higher that cannot be solved by radicals. This does not follow from Abel's statement of the theorem, but is a corollary of his proof, as his proof is based on the fact that some polynomials in the coefficients of the equation are not the zero polynomial.
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https://en.wikipedia.org/wiki/Abel–Ruffini_theorem
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This improved statement follows directly from Galois theory § A non-solvable quintic example. Galois theory implies also that x 5 − x − 1 = 0 {\displaystyle x^{5}-x-1=0} is the simplest equation that cannot be solved in radicals, and that almost all polynomials of degree five or higher cannot be solved in radicals. The impossibility of solving in degree five or higher contrasts with the case of lower degree: one has the quadratic formula, the cubic formula, and the quartic formula for degrees two, three, and four, respectively.
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https://en.wikipedia.org/wiki/Abel–Ruffini_theorem
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In mathematics, the Abhyankar–Moh theorem states that if L {\displaystyle L} is a complex line in the complex affine plane C 2 {\displaystyle \mathbb {C} ^{2}} , then every embedding of L {\displaystyle L} into C 2 {\displaystyle \mathbb {C} ^{2}} extends to an automorphism of the plane. It is named after Shreeram Shankar Abhyankar and Tzuong-Tsieng Moh, who published it in 1975. More generally, the same theorem applies to lines and planes over any algebraically closed field of characteristic zero, and to certain well-behaved subsets of higher-dimensional complex affine spaces.
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https://en.wikipedia.org/wiki/Abhyankar–Moh_theorem
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In mathematics, the Ackermann ordinal is a certain large countable ordinal, named after Wilhelm Ackermann. The term "Ackermann ordinal" is also occasionally used for the small Veblen ordinal, a somewhat larger ordinal. Unfortunately there is no standard notation for ordinals beyond the Feferman–Schütte ordinal Γ0.
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https://en.wikipedia.org/wiki/Ackermann_ordinal
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Most systems of notation use symbols such as ψ(α), θ(α), ψα(β), some of which are modifications of the Veblen functions to produce countable ordinals even for uncountable arguments, and some of which are "collapsing functions". The last one is an extension of the Veblen functions for more than 2 arguments. The smaller Ackermann ordinal is the limit of a system of ordinal notations invented by Ackermann (1951), and is sometimes denoted by φ Ω 2 ( 0 ) {\displaystyle \varphi _{\Omega ^{2}}(0)} or θ ( Ω 2 ) {\displaystyle \theta (\Omega ^{2})} , ψ ( Ω Ω 2 ) {\displaystyle \psi (\Omega ^{\Omega ^{2}})} , or φ ( 1 , 0 , 0 , 0 ) {\displaystyle \varphi (1,0,0,0)} , where Ω is the smallest uncountable ordinal. Ackermann's system of notation is weaker than the system introduced much earlier by Veblen (1908), which he seems to have been unaware of.
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https://en.wikipedia.org/wiki/Ackermann_ordinal
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In mathematics, the Adams spectral sequence is a spectral sequence introduced by J. Frank Adams (1958) which computes the stable homotopy groups of topological spaces. Like all spectral sequences, it is a computational tool; it relates homology theory to what is now called stable homotopy theory. It is a reformulation using homological algebra, and an extension, of a technique called 'killing homotopy groups' applied by the French school of Henri Cartan and Jean-Pierre Serre.
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https://en.wikipedia.org/wiki/Adams–Novikov_spectral_sequence
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In mathematics, the Ahlfors conjecture, now a theorem, states that the limit set of a finitely-generated Kleinian group is either the whole Riemann sphere, or has measure 0. The conjecture was introduced by Ahlfors (1966), who proved it in the case that the Kleinian group has a fundamental domain with a finite number of sides. Canary (1993) proved the Ahlfors conjecture for topologically tame groups, by showing that a topologically tame Kleinian group is geometrically tame, so the Ahlfors conjecture follows from Marden's tameness conjecture that hyperbolic 3-manifolds with finitely generated fundamental groups are topologically tame (homeomorphic to the interior of compact 3-manifolds). This latter conjecture was proved, independently, by Agol (2004) and by Calegari & Gabai (2006). Canary (1993) also showed that in the case when the limit set is the whole sphere, the action of the Kleinian group on the limit set is ergodic.
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https://en.wikipedia.org/wiki/Ahlfors_measure_conjecture
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In mathematics, the Al-Salam–Ismail polynomials are a family of orthogonal polynomials introduced by Al-Salam and Ismail (1983).
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https://en.wikipedia.org/wiki/Al-Salam–Ismail_polynomials
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In mathematics, the Albanese variety A ( V ) {\displaystyle A(V)} , named for Giacomo Albanese, is a generalization of the Jacobian variety of a curve.
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https://en.wikipedia.org/wiki/Albanese_variety
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In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander II discovered this, the first knot polynomial, in 1923. In 1969, John Conway showed a version of this polynomial, now called the Alexander–Conway polynomial, could be computed using a skein relation, although its significance was not realized until the discovery of the Jones polynomial in 1984. Soon after Conway's reworking of the Alexander polynomial, it was realized that a similar skein relation was exhibited in Alexander's paper on his polynomial.
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https://en.wikipedia.org/wiki/Skein_module
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In mathematics, the Almgren–Pitts min-max theory (named after Frederick J. Almgren, Jr. and his student Jon T. Pitts) is an analogue of Morse theory for hypersurfaces. The theory started with the efforts for generalizing George David Birkhoff's method for the construction of simple closed geodesics on the sphere, to allow the construction of embedded minimal surfaces in arbitrary 3-manifolds.It has played roles in the solutions to a number of conjectures in geometry and topology found by Almgren and Pitts themselves and also by other mathematicians, such as Mikhail Gromov, Richard Schoen, Shing-Tung Yau, Fernando Codá Marques, André Neves, Ian Agol, among others.
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https://en.wikipedia.org/wiki/Almgren–Pitts_min-max_theory
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In mathematics, the Alperin–Brauer–Gorenstein theorem characterizes the finite simple groups with quasidihedral or wreathed Sylow 2-subgroups. These are isomorphic either to three-dimensional projective special linear groups or projective special unitary groups over a finite field of odd order, depending on a certain congruence, or to the Mathieu group M 11 {\displaystyle M_{11}} . Alperin, Brauer & Gorenstein (1970) proved this in the course of 261 pages. The subdivision by 2-fusion is sketched there, given as an exercise in Gorenstein (1968, Ch. 7), and presented in some detail in Kwon et al. (1980).
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https://en.wikipedia.org/wiki/Alperin–Brauer–Gorenstein_theorem
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In mathematics, the Alvis–Curtis duality is a duality operation on the characters of a reductive group over a finite field, introduced by Charles W. Curtis (1980) and studied by his student Dean Alvis (1979). Kawanaka (1981, 1982) introduced a similar duality operation for Lie algebras. Alvis–Curtis duality has order 2 and is an isometry on generalized characters. Carter (1985, 8.2) discusses Alvis–Curtis duality in detail.
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https://en.wikipedia.org/wiki/Alvis–Curtis_dual
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In mathematics, the Andreotti–Frankel theorem, introduced by Aldo Andreotti and Theodore Frankel (1959), states that if V {\displaystyle V} is a smooth, complex affine variety of complex dimension n {\displaystyle n} or, more generally, if V {\displaystyle V} is any Stein manifold of dimension n {\displaystyle n} , then V {\displaystyle V} admits a Morse function with critical points of index at most n, and so V {\displaystyle V} is homotopy equivalent to a CW complex of real dimension at most n. Consequently, if V ⊆ C r {\displaystyle V\subseteq \mathbb {C} ^{r}} is a closed connected complex submanifold of complex dimension n {\displaystyle n} , then V {\displaystyle V} has the homotopy type of a CW complex of real dimension ≤ n {\displaystyle \leq n} . Therefore H i ( V ; Z ) = 0 , for i > n {\displaystyle H^{i}(V;\mathbb {Z} )=0,{\text{ for }}i>n} and H i ( V ; Z ) = 0 , for i > n . {\displaystyle H_{i}(V;\mathbb {Z} )=0,{\text{ for }}i>n.} This theorem applies in particular to any smooth, complex affine variety of dimension n {\displaystyle n} .
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https://en.wikipedia.org/wiki/Andreotti–Frankel_theorem
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In mathematics, the Andreotti–Grauert theorem, introduced by Andreotti and Grauert (1962), gives conditions for cohomology groups of coherent sheaves over complex manifolds to vanish or to be finite-dimensional.
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https://en.wikipedia.org/wiki/Andreotti–Grauert_theorem
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In mathematics, the Andreotti–Vesentini separation theorem, introduced by Aldo Andreotti and Edoardo Vesentini (1965, 1965b) states that certain cohomology groups of coherent sheaves are separated.
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https://en.wikipedia.org/wiki/Andreotti–Vesentini_theorem
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In mathematics, the Andrews–Curtis conjecture states that every balanced presentation of the trivial group can be transformed into a trivial presentation by a sequence of Nielsen transformations on the relators together with conjugations of relators, named after James J. Andrews and Morton L. Curtis who proposed it in 1965. It is difficult to verify whether the conjecture holds for a given balanced presentation or not. It is widely believed that the Andrews–Curtis conjecture is false. While there are no counterexamples known, there are numerous potential counterexamples. It is known that the Zeeman conjecture on collapsibility implies the Andrews–Curtis conjecture.
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https://en.wikipedia.org/wiki/Andrews–Curtis_conjecture
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In mathematics, the André–Oort conjecture is a problem in Diophantine geometry, a branch of number theory, that can be seen as a non-abelian analogue of the Manin–Mumford conjecture, which is now a theorem (proven in several different ways). The conjecture concerns itself with a characterization of the Zariski closure of sets of special points in Shimura varieties. A special case of the conjecture was stated by Yves André in 1989 and a more general statement (albeit with a restriction on the type of the Shimura variety) was conjectured by Frans Oort in 1995. The modern version is a natural generalization of these two conjectures.
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https://en.wikipedia.org/wiki/André–Oort_conjecture
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In mathematics, the Angelescu polynomials πn(x) are a series of polynomials generalizing the Laguerre polynomials introduced by Angelescu (1938). The polynomials can be given by the generating functionBoas & Buck (1958, p.41) They can also be defined by the equation where A n ( x ) n ! {\displaystyle {\frac {A_{n}(x)}{n!}}} is an Appell set of polynomials (see Shukla (1981)).
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https://en.wikipedia.org/wiki/Angelescu_polynomials
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In mathematics, the Anger function, introduced by C. T. Anger (1855), is a function defined as J ν ( z ) = 1 π ∫ 0 π cos ( ν θ − z sin θ ) d θ {\displaystyle \mathbf {J} _{\nu }(z)={\frac {1}{\pi }}\int _{0}^{\pi }\cos(\nu \theta -z\sin \theta )\,d\theta } with complex parameter v and complex variable x. It is closely related to the Bessel functions. The Weber function (also known as Lommel–Weber function), introduced by H. F. Weber (1879), is a closely related function defined by E ν ( z ) = 1 π ∫ 0 π sin ( ν θ − z sin θ ) d θ {\displaystyle \mathbf {E} _{\nu }(z)={\frac {1}{\pi }}\int _{0}^{\pi }\sin(\nu \theta -z\sin \theta )\,d\theta } and is closely related to Bessel functions of the second kind.
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https://en.wikipedia.org/wiki/Anger_function
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In mathematics, the Appell–Humbert theorem describes the line bundles on a complex torus or complex abelian variety. It was proved for 2-dimensional tori by Appell (1891) and Humbert (1893), and in general by Lefschetz (1921)
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https://en.wikipedia.org/wiki/Appell–Humbert_theorem
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In mathematics, the Arason invariant is a cohomological invariant associated to a quadratic form of even rank and trivial discriminant and Clifford invariant over a field k of characteristic not 2, taking values in H3(k,Z/2Z). It was introduced by (Arason 1975, Theorem 5.7). The Rost invariant is a generalization of the Arason invariant to other algebraic groups.
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https://en.wikipedia.org/wiki/Arason_invariant
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In mathematics, the Arens square is a topological space, named for Richard Friederich Arens. Its role is mainly to serve as a counterexample.
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https://en.wikipedia.org/wiki/Arens_square
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In mathematics, the Arens–Fort space is a special example in the theory of topological spaces, named for Richard Friederich Arens and M. K. Fort, Jr.
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https://en.wikipedia.org/wiki/Arens–Fort_space
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In mathematics, the Arf invariant of a nonsingular quadratic form over a field of characteristic 2 was defined by Turkish mathematician Cahit Arf (1941) when he started the systematic study of quadratic forms over arbitrary fields of characteristic 2. The Arf invariant is the substitute, in characteristic 2, for the discriminant for quadratic forms in characteristic not 2. Arf used his invariant, among others, in his endeavor to classify quadratic forms in characteristic 2. In the special case of the 2-element field F2 the Arf invariant can be described as the element of F2 that occurs most often among the values of the form.
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https://en.wikipedia.org/wiki/Arf_invariant
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Two nonsingular quadratic forms over F2 are isomorphic if and only if they have the same dimension and the same Arf invariant. This fact was essentially known to Leonard Dickson (1901), even for any finite field of characteristic 2, and Arf proved it for an arbitrary perfect field. The Arf invariant is particularly applied in geometric topology, where it is primarily used to define an invariant of (4k + 2)-dimensional manifolds (singly even-dimensional manifolds: surfaces (2-manifolds), 6-manifolds, 10-manifolds, etc.) with certain additional structure called a framing, and thus the Arf–Kervaire invariant and the Arf invariant of a knot. The Arf invariant is analogous to the signature of a manifold, which is defined for 4k-dimensional manifolds (doubly even-dimensional); this 4-fold periodicity corresponds to the 4-fold periodicity of L-theory. The Arf invariant can also be defined more generally for certain 2k-dimensional manifolds.
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https://en.wikipedia.org/wiki/Arf_invariant
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In mathematics, the Arthur conjectures are some conjectures about automorphic representations of reductive groups over the adeles and unitary representations of reductive groups over local fields made by James Arthur (1989), motivated by the Arthur–Selberg trace formula. Arthur's conjectures imply the generalized Ramanujan conjectures for cusp forms on general linear groups.
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https://en.wikipedia.org/wiki/Arthur's_conjectures
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In mathematics, the Arthur–Selberg trace formula is a generalization of the Selberg trace formula from the group SL2 to arbitrary reductive groups over global fields, developed by James Arthur in a long series of papers from 1974 to 2003. It describes the character of the representation of G(A) on the discrete part L20(G(F)\G(A)) of L2(G(F)\G(A)) in terms of geometric data, where G is a reductive algebraic group defined over a global field F and A is the ring of adeles of F. There are several different versions of the trace formula. The first version was the unrefined trace formula, whose terms depend on truncation operators and have the disadvantage that they are not invariant. Arthur later found the invariant trace formula and the stable trace formula which are more suitable for applications.
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https://en.wikipedia.org/wiki/Relative_trace_formula
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The simple trace formula (Flicker & Kazhdan 1988) is less general but easier to prove. The local trace formula is an analogue over local fields. Jacquet's relative trace formula is a generalization where one integrates the kernel function over non-diagonal subgroups.
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https://en.wikipedia.org/wiki/Relative_trace_formula
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In mathematics, the Artin approximation theorem is a fundamental result of Michael Artin (1969) in deformation theory which implies that formal power series with coefficients in a field k are well-approximated by the algebraic functions on k. More precisely, Artin proved two such theorems: one, in 1968, on approximation of complex analytic solutions by formal solutions (in the case k = C {\displaystyle k=\mathbb {C} } ); and an algebraic version of this theorem in 1969.
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https://en.wikipedia.org/wiki/Artin_approximation_theorem
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In mathematics, the Artin conductor is a number or ideal associated to a character of a Galois group of a local or global field, introduced by Emil Artin (1930, 1931) as an expression appearing in the functional equation of an Artin L-function.
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https://en.wikipedia.org/wiki/Artin_conductor
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In mathematics, the Artin–Hasse exponential, introduced by Artin and Hasse (1928), is the power series given by E p ( x ) = exp ( x + x p p + x p 2 p 2 + x p 3 p 3 + ⋯ ) . {\displaystyle E_{p}(x)=\exp \left(x+{\frac {x^{p}}{p}}+{\frac {x^{p^{2}}}{p^{2}}}+{\frac {x^{p^{3}}}{p^{3}}}+\cdots \right).}
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https://en.wikipedia.org/wiki/Dwork's_lemma
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In mathematics, the Artin–Mazur zeta function, named after Michael Artin and Barry Mazur, is a function that is used for studying the iterated functions that occur in dynamical systems and fractals. It is defined from a given function f {\displaystyle f} as the formal power series ζ f ( z ) = exp ( ∑ n = 1 ∞ | Fix ( f n ) | z n n ) , {\displaystyle \zeta _{f}(z)=\exp \left(\sum _{n=1}^{\infty }{\bigl |}\operatorname {Fix} (f^{n}){\bigr |}{\frac {z^{n}}{n}}\right),} where Fix ( f n ) {\displaystyle \operatorname {Fix} (f^{n})} is the set of fixed points of the n {\displaystyle n} th iterate of the function f {\displaystyle f} , and | Fix ( f n ) | {\displaystyle |\operatorname {Fix} (f^{n})|} is the number of fixed points (i.e. the cardinality of that set). Note that the zeta function is defined only if the set of fixed points is finite for each n {\displaystyle n} . This definition is formal in that the series does not always have a positive radius of convergence. The Artin–Mazur zeta function is invariant under topological conjugation. The Milnor–Thurston theorem states that the Artin–Mazur zeta function of an interval map f {\displaystyle f} is the inverse of the kneading determinant of f {\displaystyle f} .
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https://en.wikipedia.org/wiki/Artin–Mazur_zeta_function
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In mathematics, the Artin–Rees lemma is a basic result about modules over a Noetherian ring, along with results such as the Hilbert basis theorem. It was proved in the 1950s in independent works by the mathematicians Emil Artin and David Rees; a special case was known to Oscar Zariski prior to their work. An intuitive characterization of the lemma involves the notion that a submodule N of a module M over some ring A with specified ideal I holds a priori two topologies: one induced by the I-adic topology on M, and the other when considered as a module in its own right over A. Then Artin-Rees dictates that these topologies actually coincide, at least when A is Noetherian and M finitely-generated.
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https://en.wikipedia.org/wiki/Krull's_intersection_theorem
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One consequence of the lemma is the Krull intersection theorem. The result is also used to prove the exactness property of completion. The lemma also plays a key role in the study of ℓ-adic sheaves.
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https://en.wikipedia.org/wiki/Krull's_intersection_theorem
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In mathematics, the Artin–Zorn theorem, named after Emil Artin and Max Zorn, states that any finite alternative division ring is necessarily a finite field. It was first published in 1930 by Zorn, but in his publication Zorn credited it to Artin.The Artin–Zorn theorem is a generalization of the Wedderburn theorem, which states that finite associative division rings are fields. As a geometric consequence, every finite Moufang plane is the classical projective plane over a finite field. == References ==
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https://en.wikipedia.org/wiki/Artin–Zorn_theorem
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In mathematics, the Askey scheme is a way of organizing orthogonal polynomials of hypergeometric or basic hypergeometric type into a hierarchy. For the classical orthogonal polynomials discussed in Andrews & Askey (1985), the Askey scheme was first drawn by Labelle (1985) and by Askey and Wilson (1985), and has since been extended by Koekoek & Swarttouw (1998) and Koekoek, Lesky & Swarttouw (2010) to cover basic orthogonal polynomials.
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https://en.wikipedia.org/wiki/Askey_scheme
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In mathematics, the Askey–Gasper inequality is an inequality for Jacobi polynomials proved by Richard Askey and George Gasper (1976) and used in the proof of the Bieberbach conjecture.
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https://en.wikipedia.org/wiki/Askey–Gasper_inequality
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In mathematics, the Askey–Wilson polynomials (or q-Wilson polynomials) are a family of orthogonal polynomials introduced by Askey and Wilson (1985) as q-analogs of the Wilson polynomials. They include many of the other orthogonal polynomials in 1 variable as special or limiting cases, described in the Askey scheme. Askey–Wilson polynomials are the special case of Macdonald polynomials (or Koornwinder polynomials) for the non-reduced affine root system of type (C∨1, C1), and their 4 parameters a, b, c, d correspond to the 4 orbits of roots of this root system. They are defined by p n ( x ) = p n ( x ; a , b , c , d ∣ q ) := a − n ( a b , a c , a d ; q ) n 4 ϕ 3 {\displaystyle p_{n}(x)=p_{n}(x;a,b,c,d\mid q):=a^{-n}(ab,ac,ad;q)_{n}\;_{4}\phi _{3}\left} where φ is a basic hypergeometric function, x = cos θ, and (,,,)n is the q-Pochhammer symbol. Askey–Wilson functions are a generalization to non-integral values of n.
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https://en.wikipedia.org/wiki/Askey–Wilson_polynomial
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In mathematics, the Assouad–Nagata dimension (sometimes simply Nagata dimension) is a notion of dimension for metric spaces, introduced by Jun-iti Nagata in 1958 and reformulated by Patrice Assouad in 1982, who introduced the now-usual definition.
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https://en.wikipedia.org/wiki/Assouad–Nagata_dimension
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In mathematics, the Atiyah conjecture is a collective term for a number of statements about restrictions on possible values of l 2 {\displaystyle l^{2}} -Betti numbers.
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https://en.wikipedia.org/wiki/Atiyah_conjecture
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In mathematics, the Atiyah conjecture on configurations is a conjecture introduced by Atiyah (2000, 2001) stating that a certain n by n matrix depending on n points in R3 is always non-singular.
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https://en.wikipedia.org/wiki/Atiyah_conjecture_on_configurations
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In mathematics, the Atiyah–Bott fixed-point theorem, proven by Michael Atiyah and Raoul Bott in the 1960s, is a general form of the Lefschetz fixed-point theorem for smooth manifolds M, which uses an elliptic complex on M. This is a system of elliptic differential operators on vector bundles, generalizing the de Rham complex constructed from smooth differential forms which appears in the original Lefschetz fixed-point theorem.
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https://en.wikipedia.org/wiki/Atiyah-Bott_fixed_point_theorem
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In mathematics, the Atiyah–Hirzebruch spectral sequence is a spectral sequence for calculating generalized cohomology, introduced by Michael Atiyah and Friedrich Hirzebruch (1961) in the special case of topological K-theory. For a CW complex X {\displaystyle X} and a generalized cohomology theory E ∙ {\displaystyle E^{\bullet }} , it relates the generalized cohomology groups E i ( X ) {\displaystyle E^{i}(X)} with 'ordinary' cohomology groups H j {\displaystyle H^{j}} with coefficients in the generalized cohomology of a point. More precisely, the E 2 {\displaystyle E_{2}} term of the spectral sequence is H p ( X ; E q ( p t ) ) {\displaystyle H^{p}(X;E^{q}(pt))} , and the spectral sequence converges conditionally to E p + q ( X ) {\displaystyle E^{p+q}(X)} . Atiyah and Hirzebruch pointed out a generalization of their spectral sequence that also generalizes the Serre spectral sequence, and reduces to it in the case where E = H Sing {\displaystyle E=H_{\text{Sing}}} .
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https://en.wikipedia.org/wiki/Atiyah-Hirzebruch_spectral_sequence
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It can be derived from an exact couple that gives the E 1 {\displaystyle E_{1}} page of the Serre spectral sequence, except with the ordinary cohomology groups replaced with E {\displaystyle E} . In detail, assume X {\displaystyle X} to be the total space of a Serre fibration with fibre F {\displaystyle F} and base space B {\displaystyle B} . The filtration of B {\displaystyle B} by its n {\displaystyle n} -skeletons B n {\displaystyle B_{n}} gives rise to a filtration of X {\displaystyle X} . There is a corresponding spectral sequence with E 2 {\displaystyle E_{2}} term H p ( B ; E q ( F ) ) {\displaystyle H^{p}(B;E^{q}(F))} and converging to the associated graded ring of the filtered ring E ∞ p , q = E p + q ( X ) {\displaystyle E_{\infty }^{p,q}=E^{p+q}(X)} .This is the Atiyah–Hirzebruch spectral sequence in the case where the fibre F {\displaystyle F} is a point.
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https://en.wikipedia.org/wiki/Atiyah-Hirzebruch_spectral_sequence
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In mathematics, the Atiyah–Jones conjecture is a conjecture about the homology of the moduli spaces of instantons. The original form of the conjecture considered instantons over a 4-dimensional sphere. It was introduced by Michael Francis Atiyah and John D. S. Jones (1978) and proved by Charles P. Boyer, Jacques C. Hurtubise, and Benjamin M. Mann et al. (1992, 1993).
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https://en.wikipedia.org/wiki/Atiyah–Jones_conjecture
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The more general version of the Atiyah–Jones conjecture is a question about the homology of the moduli spaces of instantons on any 4-dimensional real manifold, or on a complex surface. The Atiyah–Jones conjecture has been proved for ruled surfaces by R. J. Milgram and J. Hurtubise, and for rational surfaces by Elizabeth Gasparim. The conjecture remains unproved for other types of 4 manifolds.
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https://en.wikipedia.org/wiki/Atiyah–Jones_conjecture
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In mathematics, the Aubin–Lions lemma (or theorem) is the result in the theory of Sobolev spaces of Banach space-valued functions, which provides a compactness criterion that is useful in the study of nonlinear evolutionary partial differential equations. Typically, to prove the existence of solutions one first constructs approximate solutions (for example, by a Galerkin method or by mollification of the equation), then uses the compactness lemma to show that there is a convergent subsequence of approximate solutions whose limit is a solution. The result is named after the French mathematicians Jean-Pierre Aubin and Jacques-Louis Lions. In the original proof by Aubin, the spaces X0 and X1 in the statement of the lemma were assumed to be reflexive, but this assumption was removed by Simon, so the result is also referred to as the Aubin–Lions–Simon lemma.
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https://en.wikipedia.org/wiki/Aubin–Lions_lemma
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In mathematics, the Auslander algebra of an algebra A is the endomorphism ring of the sum of the indecomposable modules of A. It was introduced by Auslander (1974). An Artin algebra Γ is called an Auslander algebra if gl dim Γ ≤ 2 and if 0→Γ→I→J→K→0 is a minimal injective resolution of Γ then I and J are projective Γ-modules.
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https://en.wikipedia.org/wiki/Auslander_algebra
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In mathematics, the Ax–Grothendieck theorem is a result about injectivity and surjectivity of polynomials that was proved independently by James Ax and Alexander Grothendieck.The theorem is often given as this special case: If P is an injective polynomial function from an n-dimensional complex vector space to itself then P is bijective. That is, if P always maps distinct arguments to distinct values, then the values of P cover all of Cn.The full theorem generalizes to any algebraic variety over an algebraically closed field.
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https://en.wikipedia.org/wiki/Ax-Grothendieck_theorem
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In mathematics, the Babenko–Beckner inequality (after K. Ivan Babenko and William E. Beckner) is a sharpened form of the Hausdorff–Young inequality having applications to uncertainty principles in the Fourier analysis of Lp spaces. The (q, p)-norm of the n-dimensional Fourier transform is defined to be ‖ F ‖ q , p = sup f ∈ L p ( R n ) ‖ F f ‖ q ‖ f ‖ p , where 1 < p ≤ 2 , and 1 p + 1 q = 1. {\displaystyle \|{\mathcal {F}}\|_{q,p}=\sup _{f\in L^{p}(\mathbb {R} ^{n})}{\frac {\|{\mathcal {F}}f\|_{q}}{\|f\|_{p}}},{\text{ where }}1
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https://en.wikipedia.org/wiki/Babenko–Beckner_inequality
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In mathematics, the Babuška–Lax–Milgram theorem is a generalization of the famous Lax–Milgram theorem, which gives conditions under which a bilinear form can be "inverted" to show the existence and uniqueness of a weak solution to a given boundary value problem. The result is named after the mathematicians Ivo Babuška, Peter Lax and Arthur Milgram.
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https://en.wikipedia.org/wiki/Babuška–Lax–Milgram_theorem
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In mathematics, the Bachmann–Howard ordinal (also known as the Howard ordinal, or Howard-Bachmann ordinal) is a large countable ordinal. It is the proof-theoretic ordinal of several mathematical theories, such as Kripke–Platek set theory (with the axiom of infinity) and the system CZF of constructive set theory. It was introduced by Heinz Bachmann (1950) and William Alvin Howard (1972).
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https://en.wikipedia.org/wiki/Bachmann–Howard_ordinal
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In mathematics, the Backus–Gilbert method, also known as the optimally localized average (OLA) method is named for its discoverers, geophysicists George E. Backus and James Freeman Gilbert. It is a regularization method for obtaining meaningful solutions to ill-posed inverse problems. Where other regularization methods, such as the frequently used Tikhonov regularization method, seek to impose smoothness constraints on the solution, Backus–Gilbert instead seeks to impose stability constraints, so that the solution would vary as little as possible if the input data were resampled multiple times. In practice, and to the extent that is justified by the data, smoothness results from this.
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https://en.wikipedia.org/wiki/Backus–Gilbert_method
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Given a data array X, the basic Backus-Gilbert inverse is: H θ = C − 1 G θ G θ T C − 1 G θ {\displaystyle \mathbf {H} _{\theta }={\frac {\mathbf {C} ^{-1}\mathbf {G} _{\theta }}{\mathbf {G} _{\theta }^{T}\mathbf {C} ^{-1}\mathbf {G} _{\theta }}}} where C is the covariance matrix of the data, and Gθ is an a priori constraint representing the source θ for which a solution is sought. Regularization is implemented by "whitening" the covariance matrix: C ′ = C + λ I {\displaystyle \mathbf {C} '=\mathbf {C} +\lambda \mathbf {I} } with C′ replacing C in the equation for Hθ. Then, H θ T X {\displaystyle \mathbf {H} _{\theta }^{T}\mathbf {X} } is an estimate of the activity of the source θ.
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https://en.wikipedia.org/wiki/Backus–Gilbert_method
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In mathematics, the Baily–Borel compactification is a compactification of a quotient of a Hermitian symmetric space by an arithmetic group, introduced by Walter L. Baily and Armand Borel (1964, 1966).
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https://en.wikipedia.org/wiki/Baily–Borel_compactification
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In mathematics, the Baker–Campbell–Hausdorff formula is the solution for Z {\displaystyle Z} to the equation for possibly noncommutative X and Y in the Lie algebra of a Lie group. There are various ways of writing the formula, but all ultimately yield an expression for Z {\displaystyle Z} in Lie algebraic terms, that is, as a formal series (not necessarily convergent) in X {\displaystyle X} and Y {\displaystyle Y} and iterated commutators thereof. The first few terms of this series are: where " ⋯ {\displaystyle \cdots } " indicates terms involving higher commutators of X {\displaystyle X} and Y {\displaystyle Y} . If X {\displaystyle X} and Y {\displaystyle Y} are sufficiently small elements of the Lie algebra g {\displaystyle {\mathfrak {g}}} of a Lie group G {\displaystyle G} , the series is convergent.
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https://en.wikipedia.org/wiki/Baker-Campbell-Hausdorff_formula
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Meanwhile, every element g {\displaystyle g} sufficiently close to the identity in G {\displaystyle G} can be expressed as g = e X {\displaystyle g=e^{X}} for a small X {\displaystyle X} in g {\displaystyle {\mathfrak {g}}} . Thus, we can say that near the identity the group multiplication in G {\displaystyle G} —written as e X e Y = e Z {\displaystyle e^{X}e^{Y}=e^{Z}} —can be expressed in purely Lie algebraic terms. The Baker–Campbell–Hausdorff formula can be used to give comparatively simple proofs of deep results in the Lie group–Lie algebra correspondence.
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https://en.wikipedia.org/wiki/Baker-Campbell-Hausdorff_formula
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If X {\displaystyle X} and Y {\displaystyle Y} are sufficiently small n × n {\displaystyle n\times n} matrices, then Z {\displaystyle Z} can be computed as the logarithm of e X e Y {\displaystyle e^{X}e^{Y}} , where the exponentials and the logarithm can be computed as power series. The point of the Baker–Campbell–Hausdorff formula is then the highly nonobvious claim that Z := log ( e X e Y ) {\displaystyle Z:=\log \left(e^{X}e^{Y}\right)} can be expressed as a series in repeated commutators of X {\displaystyle X} and Y {\displaystyle Y} . Modern expositions of the formula can be found in, among other places, the books of Rossmann and Hall.
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https://en.wikipedia.org/wiki/Baker-Campbell-Hausdorff_formula
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In mathematics, the Balian–Low theorem in Fourier analysis is named for Roger Balian and Francis E. Low. The theorem states that there is no well-localized window function (or Gabor atom) g either in time or frequency for an exact Gabor frame (Riesz Basis).
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https://en.wikipedia.org/wiki/Balian–Low_theorem
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In mathematics, the Banach fixed-point theorem (also known as the contraction mapping theorem or contractive mapping theorem or Banach-Caccioppoli theorem) is an important tool in the theory of metric spaces; it guarantees the existence and uniqueness of fixed points of certain self-maps of metric spaces, and provides a constructive method to find those fixed points. It can be understood as an abstract formulation of Picard's method of successive approximations. The theorem is named after Stefan Banach (1892–1945) who first stated it in 1922.
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https://en.wikipedia.org/wiki/Banach_fixed_point_theorem
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In mathematics, the Banach game is a topological game introduced by Stefan Banach in 1935 in the second addendum to problem 43 of the Scottish book as a variation of the Banach–Mazur game.Given a subset X {\displaystyle X} of real numbers, two players alternatively write down arbitrary (not necessarily in X {\displaystyle X} ) positive real numbers x 0 , x 1 , x 2 , … {\displaystyle x_{0},x_{1},x_{2},\ldots } such that x 0 > x 1 > x 2 > ⋯ {\displaystyle x_{0}>x_{1}>x_{2}>\cdots } Player one wins if and only if ∑ i = 0 ∞ x i {\displaystyle \sum _{i=0}^{\infty }x_{i}} exists and is in X {\displaystyle X} .One observation about the game is that if X {\displaystyle X} is a countable set, then either of the players can cause the final sum to avoid the set. Thus in this situation the second player can win.
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https://en.wikipedia.org/wiki/Banach_game
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In mathematics, the Banach–Stone theorem is a classical result in the theory of continuous functions on topological spaces, named after the mathematicians Stefan Banach and Marshall Stone. In brief, the Banach–Stone theorem allows one to recover a compact Hausdorff space X from the Banach space structure of the space C(X) of continuous real- or complex-valued functions on X. If one is allowed to invoke the algebra structure of C(X) this is easy – we can identify X with the spectrum of C(X), the set of algebra homomorphisms into the scalar field, equipped with the weak*-topology inherited from the dual space C(X)*. The Banach-Stone theorem avoids reference to multiplicative structure by recovering X from the extreme points of the unit ball of C(X)*.
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https://en.wikipedia.org/wiki/Banach–Stone_theorem
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In mathematics, the Bareiss algorithm, named after Erwin Bareiss, is an algorithm to calculate the determinant or the echelon form of a matrix with integer entries using only integer arithmetic; any divisions that are performed are guaranteed to be exact (there is no remainder). The method can also be used to compute the determinant of matrices with (approximated) real entries, avoiding the introduction of any round-off errors beyond those already present in the input.
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https://en.wikipedia.org/wiki/Bareiss_Algorithm
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In mathematics, the Barnes G-function G(z) is a function that is an extension of superfactorials to the complex numbers. It is related to the gamma function, the K-function and the Glaisher–Kinkelin constant, and was named after mathematician Ernest William Barnes. It can be written in terms of the double gamma function.
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https://en.wikipedia.org/wiki/Barnes_G-function
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Formally, the Barnes G-function is defined in the following Weierstrass product form: G ( 1 + z ) = ( 2 π ) z / 2 exp ( − z + z 2 ( 1 + γ ) 2 ) ∏ k = 1 ∞ { ( 1 + z k ) k exp ( z 2 2 k − z ) } {\displaystyle G(1+z)=(2\pi )^{z/2}\exp \left(-{\frac {z+z^{2}(1+\gamma )}{2}}\right)\,\prod _{k=1}^{\infty }\left\{\left(1+{\frac {z}{k}}\right)^{k}\exp \left({\frac {z^{2}}{2k}}-z\right)\right\}} where γ {\displaystyle \,\gamma } is the Euler–Mascheroni constant, exp(x) = ex is the exponential function, and Π denotes multiplication (capital pi notation). As an entire function, G is of order two, and of infinite type. This can be deduced from the asymptotic expansion given below.
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https://en.wikipedia.org/wiki/Barnes_G-function
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In mathematics, the Barnes–Wall lattice Λ16, discovered by Eric Stephen Barnes and G. E. (Tim) Wall (Barnes & Wall (1959)), is the 16-dimensional positive-definite even integral lattice of discriminant 28 with no norm-2 vectors. It is the sublattice of the Leech lattice fixed by a certain automorphism of order 2, and is analogous to the Coxeter–Todd lattice. The automorphism group of the Barnes–Wall lattice has order 89181388800 = 221 35 52 7 and has structure 21+8 PSO8+(F2).
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https://en.wikipedia.org/wiki/Barnes–Wall_lattice
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There are 4320 vectors of norm 4 in the Barnes–Wall lattice (the shortest nonzero vectors in this lattice). The genus of the Barnes–Wall lattice was described by Scharlau & Venkov (1994) and contains 24 lattices; all the elements other than the Barnes–Wall lattice have root system of maximal rank 16. The Barnes–Wall lattice is described in detail in (Conway & Sloane 1999, section 4.10).
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https://en.wikipedia.org/wiki/Barnes–Wall_lattice
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In mathematics, the Bass–Quillen conjecture relates vector bundles over a regular Noetherian ring A and over the polynomial ring A {\displaystyle A} . The conjecture is named for Hyman Bass and Daniel Quillen, who formulated the conjecture.
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https://en.wikipedia.org/wiki/Bass–Quillen_conjecture
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In mathematics, the Bateman function (or k-function) is a special case of the confluent hypergeometric function studied by Harry Bateman(1931). Bateman defined it by k n ( x ) = 2 π ∫ 0 π / 2 cos ( x tan θ − n θ ) d θ {\displaystyle \displaystyle k_{n}(x)={\frac {2}{\pi }}\int _{0}^{\pi /2}\cos(x\tan \theta -n\theta )\,d\theta } Bateman discovered this function, when Theodore von Kármán asked for the solution of the following differential equation which appeared in the theory of turbulence x d 2 u d x 2 = ( x − n ) u {\displaystyle x{\frac {d^{2}u}{dx^{2}}}=(x-n)u} and Bateman found this function as one of the solutions. Bateman denoted this function as "k" function in honor of Theodore von Kármán. This is not to be confused with another function of the same name which is used in Pharmacokinetics.
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https://en.wikipedia.org/wiki/Bateman_function
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In mathematics, the Bateman polynomials are a family Fn of orthogonal polynomials introduced by Bateman (1933). The Bateman–Pasternack polynomials are a generalization introduced by Pasternack (1939). Bateman polynomials can be defined by the relation F n ( d d x ) sech ( x ) = sech ( x ) P n ( tanh ( x ) ) . {\displaystyle F_{n}\left({\frac {d}{dx}}\right)\operatorname {sech} (x)=\operatorname {sech} (x)P_{n}(\tanh(x)).}
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https://en.wikipedia.org/wiki/Bateman_polynomials
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where Pn is a Legendre polynomial. In terms of generalized hypergeometric functions, they are given by F n ( x ) = 3 F 2 ( − n , n + 1 , 1 2 ( x + 1 ) 1 , 1 ; 1 ) . {\displaystyle F_{n}(x)={}_{3}F_{2}\left({\begin{array}{c}-n,~n+1,~{\tfrac {1}{2}}(x+1)\\1,~1\end{array}};1\right).}
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https://en.wikipedia.org/wiki/Bateman_polynomials
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Pasternack (1939) generalized the Bateman polynomials to polynomials Fmn with F n m ( d d x ) sech m + 1 ( x ) = sech m + 1 ( x ) P n ( tanh ( x ) ) {\displaystyle F_{n}^{m}\left({\frac {d}{dx}}\right)\operatorname {sech} ^{m+1}(x)=\operatorname {sech} ^{m+1}(x)P_{n}(\tanh(x))} These generalized polynomials also have a representation in terms of generalized hypergeometric functions, namely F n m ( x ) = 3 F 2 ( − n , n + 1 , 1 2 ( x + m + 1 ) 1 , m + 1 ; 1 ) . {\displaystyle F_{n}^{m}(x)={}_{3}F_{2}\left({\begin{array}{c}-n,~n+1,~{\tfrac {1}{2}}(x+m+1)\\1,~m+1\end{array}};1\right).} Carlitz (1957) showed that the polynomials Qn studied by Touchard (1956) , see Touchard polynomials, are the same as Bateman polynomials up to a change of variable: more precisely Q n ( x ) = ( − 1 ) n 2 n n ! ( 2 n n ) − 1 F n ( 2 x + 1 ) {\displaystyle Q_{n}(x)=(-1)^{n}2^{n}n! {\binom {2n}{n}}^{-1}F_{n}(2x+1)} Bateman and Pasternack's polynomials are special cases of the symmetric continuous Hahn polynomials.
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https://en.wikipedia.org/wiki/Bateman_polynomials
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In mathematics, the Bauer–Fike theorem is a standard result in the perturbation theory of the eigenvalue of a complex-valued diagonalizable matrix. In its substance, it states an absolute upper bound for the deviation of one perturbed matrix eigenvalue from a properly chosen eigenvalue of the exact matrix. Informally speaking, what it says is that the sensitivity of the eigenvalues is estimated by the condition number of the matrix of eigenvectors. The theorem was proved by Friedrich L. Bauer and C. T. Fike in 1960.
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https://en.wikipedia.org/wiki/Bauer–Fike_theorem
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In mathematics, the Beauville–Laszlo theorem is a result in commutative algebra and algebraic geometry that allows one to "glue" two sheaves over an infinitesimal neighborhood of a point on an algebraic curve. It was proved by Arnaud Beauville and Yves Laszlo (1995).
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https://en.wikipedia.org/wiki/Beauville–Laszlo_theorem
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In mathematics, the Beck–Fiala theorem is a major theorem in discrepancy theory due to József Beck and Tibor Fiala. Discrepancy is concerned with coloring elements of a ground set such that each set in a certain set system is as balanced as possible, i.e., has approximately the same number of elements of each color. The Beck–Fiala theorem is concerned with the case where each element doesn't appear many times across all sets. The theorem guarantees that if each element appears at most t times, then the elements can be colored so that the imbalance is at most 2t − 1.
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https://en.wikipedia.org/wiki/Beck–Fiala_theorem
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In mathematics, the Bell series is a formal power series used to study properties of arithmetical functions. Bell series were introduced and developed by Eric Temple Bell. Given an arithmetic function f {\displaystyle f} and a prime p {\displaystyle p} , define the formal power series f p ( x ) {\displaystyle f_{p}(x)} , called the Bell series of f {\displaystyle f} modulo p {\displaystyle p} as: f p ( x ) = ∑ n = 0 ∞ f ( p n ) x n . {\displaystyle f_{p}(x)=\sum _{n=0}^{\infty }f(p^{n})x^{n}.}
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https://en.wikipedia.org/wiki/Bell_series
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Two multiplicative functions can be shown to be identical if all of their Bell series are equal; this is sometimes called the uniqueness theorem: given multiplicative functions f {\displaystyle f} and g {\displaystyle g} , one has f = g {\displaystyle f=g} if and only if: f p ( x ) = g p ( x ) {\displaystyle f_{p}(x)=g_{p}(x)} for all primes p {\displaystyle p} .Two series may be multiplied (sometimes called the multiplication theorem): For any two arithmetic functions f {\displaystyle f} and g {\displaystyle g} , let h = f ∗ g {\displaystyle h=f*g} be their Dirichlet convolution. Then for every prime p {\displaystyle p} , one has: h p ( x ) = f p ( x ) g p ( x ) . {\displaystyle h_{p}(x)=f_{p}(x)g_{p}(x).\,} In particular, this makes it trivial to find the Bell series of a Dirichlet inverse. If f {\displaystyle f} is completely multiplicative, then formally: f p ( x ) = 1 1 − f ( p ) x . {\displaystyle f_{p}(x)={\frac {1}{1-f(p)x}}.}
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https://en.wikipedia.org/wiki/Bell_series
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In mathematics, the Bell triangle is a triangle of numbers analogous to Pascal's triangle, whose values count partitions of a set in which a given element is the largest singleton. It is named for its close connection to the Bell numbers, which may be found on both sides of the triangle, and which are in turn named after Eric Temple Bell. The Bell triangle has been discovered independently by multiple authors, beginning with Charles Sanders Peirce (1880) and including also Alexander Aitken (1933) and Cohn et al. (1962), and for that reason has also been called Aitken's array or the Peirce triangle.
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https://en.wikipedia.org/wiki/Aitken's_array
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In mathematics, the Beltrami equation, named after Eugenio Beltrami, is the partial differential equation ∂ w ∂ z ¯ = μ ∂ w ∂ z . {\displaystyle {\partial w \over \partial {\overline {z}}}=\mu {\partial w \over \partial z}.} for w a complex distribution of the complex variable z in some open set U, with derivatives that are locally L2, and where μ is a given complex function in L∞(U) of norm less than 1, called the Beltrami coefficient, and where ∂ / ∂ z {\displaystyle \partial /\partial z} and ∂ / ∂ z ¯ {\displaystyle \partial /\partial {\overline {z}}} are Wirtinger derivatives.
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https://en.wikipedia.org/wiki/Beltrami_differential_equation
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Classically this differential equation was used by Gauss to prove the existence locally of isothermal coordinates on a surface with analytic Riemannian metric. Various techniques have been developed for solving the equation. The most powerful, developed in the 1950s, provides global solutions of the equation on C and relies on the Lp theory of the Beurling transform, a singular integral operator defined on Lp(C) for all 1 < p < ∞.
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https://en.wikipedia.org/wiki/Beltrami_differential_equation
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The same method applies equally well on the unit disk and upper half plane and plays a fundamental role in Teichmüller theory and the theory of quasiconformal mappings. Various uniformization theorems can be proved using the equation, including the measurable Riemann mapping theorem and the simultaneous uniformization theorem. The existence of conformal weldings can also be derived using the Beltrami equation. One of the simplest applications is to the Riemann mapping theorem for simply connected bounded open domains in the complex plane. When the domain has smooth boundary, elliptic regularity for the equation can be used to show that the uniformizing map from the unit disk to the domain extends to a C∞ function from the closed disk to the closure of the domain.
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https://en.wikipedia.org/wiki/Beltrami_differential_equation
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In mathematics, the Bendixson–Dulac theorem on dynamical systems states that if there exists a C 1 {\displaystyle C^{1}} function φ ( x , y ) {\displaystyle \varphi (x,y)} (called the Dulac function) such that the expression ∂ ( φ f ) ∂ x + ∂ ( φ g ) ∂ y {\displaystyle {\frac {\partial (\varphi f)}{\partial x}}+{\frac {\partial (\varphi g)}{\partial y}}} has the same sign ( ≠ 0 {\displaystyle \neq 0} ) almost everywhere in a simply connected region of the plane, then the plane autonomous system d x d t = f ( x , y ) , {\displaystyle {\frac {dx}{dt}}=f(x,y),} d y d t = g ( x , y ) {\displaystyle {\frac {dy}{dt}}=g(x,y)} has no nonconstant periodic solutions lying entirely within the region. "Almost everywhere" means everywhere except possibly in a set of measure 0, such as a point or line. The theorem was first established by Swedish mathematician Ivar Bendixson in 1901 and further refined by French mathematician Henri Dulac in 1923 using Green's theorem.
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https://en.wikipedia.org/wiki/Bendixson–Dulac_theorem
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In mathematics, the Benjamin–Ono equation is a nonlinear partial integro-differential equation that describes one-dimensional internal waves in deep water. It was introduced by Benjamin (1967) and Ono (1975). The Benjamin–Ono equation is u t + u u x + H u x x = 0 {\displaystyle u_{t}+uu_{x}+Hu_{xx}=0} where H is the Hilbert transform. It possesses infinitely many conserved densities and symmetries; thus it is a completely integrable system.
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https://en.wikipedia.org/wiki/Benjamin–Ono_equation
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In mathematics, the Bergman–Weil formula is an integral representation for holomorphic functions of several variables generalizing the Cauchy integral formula. It was introduced by Bergmann (1936) and Weil (1935).
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https://en.wikipedia.org/wiki/Bergman–Weil_formula
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In mathematics, the Bernoulli numbers Bn are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of m-th powers of the first n positive integers, in the Euler–Maclaurin formula, and in expressions for certain values of the Riemann zeta function. The values of the first 20 Bernoulli numbers are given in the adjacent table. Two conventions are used in the literature, denoted here by B n − {\displaystyle B_{n}^{-{}}} and B n + {\displaystyle B_{n}^{+{}}} ; they differ only for n = 1, where B 1 − = − 1 / 2 {\displaystyle B_{1}^{-{}}=-1/2} and B 1 + = + 1 / 2 {\displaystyle B_{1}^{+{}}=+1/2} .
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https://en.wikipedia.org/wiki/Seidel_triangle
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For every odd n > 1, Bn = 0. For every even n > 0, Bn is negative if n is divisible by 4 and positive otherwise.
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https://en.wikipedia.org/wiki/Seidel_triangle
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The Bernoulli numbers are special values of the Bernoulli polynomials B n ( x ) {\displaystyle B_{n}(x)} , with B n − = B n ( 0 ) {\displaystyle B_{n}^{-{}}=B_{n}(0)} and B n + = B n ( 1 ) {\displaystyle B_{n}^{+}=B_{n}(1)} .The Bernoulli numbers were discovered around the same time by the Swiss mathematician Jacob Bernoulli, after whom they are named, and independently by Japanese mathematician Seki Takakazu. Seki's discovery was posthumously published in 1712 in his work Katsuyō Sanpō; Bernoulli's, also posthumously, in his Ars Conjectandi of 1713. Ada Lovelace's note G on the Analytical Engine from 1842 describes an algorithm for generating Bernoulli numbers with Babbage's machine. As a result, the Bernoulli numbers have the distinction of being the subject of the first published complex computer program.
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https://en.wikipedia.org/wiki/Seidel_triangle
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