text
stringlengths 9
3.55k
| source
stringlengths 31
280
|
|---|---|
In mathematics, the Riemann zeta function is a function in complex analysis, which is also important in number theory. It is often denoted ζ(s) and is named after the mathematician Bernhard Riemann. When the argument s is a real number greater than one, the zeta function satisfies the equation It can therefore provide the sum of various convergent infinite series, such as ζ ( 2 ) = 1 1 2 + {\textstyle \zeta (2)={\frac {1}{1^{2}}}+} 1 2 2 + {\textstyle {\frac {1}{2^{2}}}+} 1 3 2 + … . {\textstyle {\frac {1}{3^{2}}}+\ldots \,.}
|
https://en.wikipedia.org/wiki/Particular_values_of_the_Riemann_zeta_function
|
Explicit or numerically efficient formulae exist for ζ(s) at integer arguments, all of which have real values, including this example. This article lists these formulae, together with tables of values. It also includes derivatives and some series composed of the zeta function at integer arguments.
|
https://en.wikipedia.org/wiki/Particular_values_of_the_Riemann_zeta_function
|
The same equation in s above also holds when s is a complex number whose real part is greater than one, ensuring that the infinite sum still converges. The zeta function can then be extended to the whole of the complex plane by analytic continuation, except for a simple pole at s = 1. The complex derivative exists in this more general region, making the zeta function a meromorphic function.
|
https://en.wikipedia.org/wiki/Particular_values_of_the_Riemann_zeta_function
|
The above equation no longer applies for these extended values of s, for which the corresponding summation would diverge. For example, the full zeta function exists at s = −1 (and is therefore finite there), but the corresponding series would be 1 + 2 + 3 + … , {\textstyle 1+2+3+\ldots \,,} whose partial sums would grow indefinitely large. The zeta function values listed below include function values at the negative even numbers (s = −2, −4, etc.), for which ζ(s) = 0 and which make up the so-called trivial zeros. The Riemann zeta function article includes a colour plot illustrating how the function varies over a continuous rectangular region of the complex plane. The successful characterisation of its non-trivial zeros in the wider plane is important in number theory, because of the Riemann hypothesis.
|
https://en.wikipedia.org/wiki/Particular_values_of_the_Riemann_zeta_function
|
In mathematics, the Riemannian connection on a surface or Riemannian 2-manifold refers to several intrinsic geometric structures discovered by Tullio Levi-Civita, Élie Cartan and Hermann Weyl in the early part of the twentieth century: parallel transport, covariant derivative and connection form. These concepts were put in their current form with principal bundles only in the 1950s. The classical nineteenth century approach to the differential geometry of surfaces, due in large part to Carl Friedrich Gauss, has been reworked in this modern framework, which provides the natural setting for the classical theory of the moving frame as well as the Riemannian geometry of higher-dimensional Riemannian manifolds. This account is intended as an introduction to the theory of connections.
|
https://en.wikipedia.org/wiki/Riemannian_connection_on_a_surface
|
In mathematics, the Riemann–Hurwitz formula, named after Bernhard Riemann and Adolf Hurwitz, describes the relationship of the Euler characteristics of two surfaces when one is a ramified covering of the other. It therefore connects ramification with algebraic topology, in this case. It is a prototype result for many others, and is often applied in the theory of Riemann surfaces (which is its origin) and algebraic curves.
|
https://en.wikipedia.org/wiki/Riemann–Hurwitz_formula
|
In mathematics, the Riemann–Lebesgue lemma, named after Bernhard Riemann and Henri Lebesgue, states that the Fourier transform or Laplace transform of an L1 function vanishes at infinity. It is of importance in harmonic analysis and asymptotic analysis.
|
https://en.wikipedia.org/wiki/Riemann–Lebesgue_lemma
|
In mathematics, the Riemann–Liouville integral associates with a real function f: R → R {\displaystyle f:\mathbb {R} \rightarrow \mathbb {R} } another function Iα f of the same kind for each value of the parameter α > 0. The integral is a manner of generalization of the repeated antiderivative of f in the sense that for positive integer values of α, Iα f is an iterated antiderivative of f of order α. The Riemann–Liouville integral is named for Bernhard Riemann and Joseph Liouville, the latter of whom was the first to consider the possibility of fractional calculus in 1832. The operator agrees with the Euler transform, after Leonhard Euler, when applied to analytic functions. It was generalized to arbitrary dimensions by Marcel Riesz, who introduced the Riesz potential.
|
https://en.wikipedia.org/wiki/Riemann–Liouville_differintegral
|
In mathematics, the Riemann–Roch theorem for surfaces describes the dimension of linear systems on an algebraic surface. The classical form of it was first given by Castelnuovo (1896, 1897), after preliminary versions of it were found by Max Noether (1886) and Enriques (1894). The sheaf-theoretic version is due to Hirzebruch.
|
https://en.wikipedia.org/wiki/Riemann–Roch_theorem_for_surfaces
|
In mathematics, the Riemann–Siegel formula is an asymptotic formula for the error of the approximate functional equation of the Riemann zeta function, an approximation of the zeta function by a sum of two finite Dirichlet series. It was found by Siegel (1932) in unpublished manuscripts of Bernhard Riemann dating from the 1850s. Siegel derived it from the Riemann–Siegel integral formula, an expression for the zeta function involving contour integrals. It is often used to compute values of the Riemann–Siegel formula, sometimes in combination with the Odlyzko–Schönhage algorithm which speeds it up considerably.
|
https://en.wikipedia.org/wiki/Riemann–Siegel_formula
|
When used along the critical line, it is often useful to use it in a form where it becomes a formula for the Z function. If M and N are non-negative integers, then the zeta function is equal to ζ ( s ) = ∑ n = 1 N 1 n s + γ ( 1 − s ) ∑ n = 1 M 1 n 1 − s + R ( s ) {\displaystyle \zeta (s)=\sum _{n=1}^{N}{\frac {1}{n^{s}}}+\gamma (1-s)\sum _{n=1}^{M}{\frac {1}{n^{1-s}}}+R(s)} where γ ( s ) = π 1 2 − s Γ ( s 2 ) Γ ( 1 2 ( 1 − s ) ) {\displaystyle \gamma (s)=\pi ^{{\tfrac {1}{2}}-s}{\frac {\Gamma \left({\tfrac {s}{2}}\right)}{\Gamma \left({\tfrac {1}{2}}(1-s)\right)}}} is the factor appearing in the functional equation ζ(s) = γ(1 − s) ζ(1 − s), and R ( s ) = − Γ ( 1 − s ) 2 π i ∫ ( − x ) s − 1 e − N x e x − 1 d x {\displaystyle R(s)={\frac {-\Gamma (1-s)}{2\pi i}}\int {\frac {(-x)^{s-1}e^{-Nx}}{e^{x}-1}}dx} is a contour integral whose contour starts and ends at +∞ and circles the singularities of absolute value at most 2πM.
|
https://en.wikipedia.org/wiki/Riemann–Siegel_formula
|
The approximate functional equation gives an estimate for the size of the error term. Siegel (1932) and Edwards (1974) derive the Riemann–Siegel formula from this by applying the method of steepest descent to this integral to give an asymptotic expansion for the error term R(s) as a series of negative powers of Im(s). In applications s is usually on the critical line, and the positive integers M and N are chosen to be about (2πIm(s))1/2. Gabcke (1979) found good bounds for the error of the Riemann–Siegel formula.
|
https://en.wikipedia.org/wiki/Riemann–Siegel_formula
|
In mathematics, the Riemann–Siegel theta function is defined in terms of the gamma function as θ ( t ) = arg ( Γ ( 1 4 + i t 2 ) ) − log π 2 t {\displaystyle \theta (t)=\arg \left(\Gamma \left({\frac {1}{4}}+{\frac {it}{2}}\right)\right)-{\frac {\log \pi }{2}}t} for real values of t. Here the argument is chosen in such a way that a continuous function is obtained and θ ( 0 ) = 0 {\displaystyle \theta (0)=0} holds, i.e., in the same way that the principal branch of the log-gamma function is defined. It has an asymptotic expansion θ ( t ) ∼ t 2 log t 2 π − t 2 − π 8 + 1 48 t + 7 5760 t 3 + ⋯ {\displaystyle \theta (t)\sim {\frac {t}{2}}\log {\frac {t}{2\pi }}-{\frac {t}{2}}-{\frac {\pi }{8}}+{\frac {1}{48t}}+{\frac {7}{5760t^{3}}}+\cdots } which is not convergent, but whose first few terms give a good approximation for t ≫ 1 {\displaystyle t\gg 1} . Its Taylor-series at 0 which converges for | t | < 1 / 2 {\displaystyle |t|<1/2} is θ ( t ) = − t 2 log π + ∑ k = 0 ∞ ( − 1 ) k ψ ( 2 k ) ( 1 4 ) ( 2 k + 1 ) !
|
https://en.wikipedia.org/wiki/Riemann–Siegel_theta_function
|
( t 2 ) 2 k + 1 {\displaystyle \theta (t)=-{\frac {t}{2}}\log \pi +\sum _{k=0}^{\infty }{\frac {(-1)^{k}\psi ^{(2k)}\left({\frac {1}{4}}\right)}{(2k+1)! }}\left({\frac {t}{2}}\right)^{2k+1}} where ψ ( 2 k ) {\displaystyle \psi ^{(2k)}} denotes the polygamma function of order 2 k {\displaystyle 2k} . The Riemann–Siegel theta function is of interest in studying the Riemann zeta function, since it can rotate the Riemann zeta function such that it becomes the totally real valued Z function on the critical line s = 1 / 2 + i t {\displaystyle s=1/2+it} .
|
https://en.wikipedia.org/wiki/Riemann–Siegel_theta_function
|
In mathematics, the Riemann–Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes. The definition of this integral was first published in 1894 by Stieltjes. It serves as an instructive and useful precursor of the Lebesgue integral, and an invaluable tool in unifying equivalent forms of statistical theorems that apply to discrete and continuous probability.
|
https://en.wikipedia.org/wiki/Riemann–Stieltjes_Integral
|
In mathematics, the Riemann–von Mangoldt formula, named for Bernhard Riemann and Hans Carl Friedrich von Mangoldt, describes the distribution of the zeros of the Riemann zeta function. The formula states that the number N(T) of zeros of the zeta function with imaginary part greater than 0 and less than or equal to T satisfies N ( T ) = T 2 π log T 2 π − T 2 π + O ( log T ) . {\displaystyle N(T)={\frac {T}{2\pi }}\log {\frac {T}{2\pi }}-{\frac {T}{2\pi }}+O(\log {T}).} The formula was stated by Riemann in his notable paper "On the Number of Primes Less Than a Given Magnitude" (1859) and was finally proved by Mangoldt in 1905.
|
https://en.wikipedia.org/wiki/Riemann–von_Mangoldt_formula
|
Backlund gives an explicit form of the error for all T > 2: | N ( T ) − ( T 2 π log T 2 π − T 2 π − 7 8 ) | < 0.137 log T + 0.443 log log T + 4.350 . {\displaystyle \left\vert {N(T)-\left({{\frac {T}{2\pi }}\log {\frac {T}{2\pi }}-{\frac {T}{2\pi }}}-{\frac {7}{8}}\right)}\right\vert <0.137\log T+0.443\log \log T+4.350\ .} Under the Lindelöf and Riemann hypotheses the error term can be improved to o ( log T ) {\displaystyle o(\log {T})} and O ( log T / log log T ) {\displaystyle O(\log {T}/\log {\log {T}})} respectively.Similarly, for any primitive Dirichlet character χ modulo q, we have N ( T , χ ) = T π log q T 2 π e + O ( log q T ) , {\displaystyle N(T,\chi )={\frac {T}{\pi }}\log {\frac {qT}{2\pi e}}+O(\log {qT}),} where N(T,χ) denotes the number of zeros of L(s,χ) with imaginary part between -T and T.
|
https://en.wikipedia.org/wiki/Riemann–von_Mangoldt_formula
|
In mathematics, the Riesz function is an entire function defined by Marcel Riesz in connection with the Riemann hypothesis, by means of the power series R i e s z ( x ) = − ∑ k = 1 ∞ ( − x ) k ( k − 1 ) ! ζ ( 2 k ) = x ∑ n = 1 ∞ μ ( n ) n 2 exp ( − x n 2 ) . {\displaystyle {\rm {Riesz}}(x)=-\sum _{k=1}^{\infty }{\frac {(-x)^{k}}{(k-1)!\zeta (2k)}}=x\sum _{n=1}^{\infty }{\frac {\mu (n)}{n^{2}}}\exp \left({\frac {-x}{n^{2}}}\right).} If we set F ( x ) = 1 2 R i e s z ( 4 π 2 x ) {\displaystyle F(x)={\frac {1}{2}}{\rm {Riesz}}(4\pi ^{2}x)} we may define it in terms of the coefficients of the Laurent series development of the hyperbolic (or equivalently, the ordinary) cotangent around zero.
|
https://en.wikipedia.org/wiki/Riesz_function
|
If x 2 coth x 2 = ∑ n = 0 ∞ c n x n = 1 + 1 12 x 2 − 1 720 x 4 + ⋯ {\displaystyle {\frac {x}{2}}\coth {\frac {x}{2}}=\sum _{n=0}^{\infty }c_{n}x^{n}=1+{\frac {1}{12}}x^{2}-{\frac {1}{720}}x^{4}+\cdots } then F may be defined as F ( x ) = ∑ k = 1 ∞ x k c 2 k ( k − 1 ) ! = 12 x − 720 x 2 + 15120 x 3 − ⋯ {\displaystyle F(x)=\sum _{k=1}^{\infty }{\frac {x^{k}}{c_{2k}(k-1)! }}=12x-720x^{2}+15120x^{3}-\cdots } The values of ζ(2k) approach one for increasing k, and comparing the series for the Riesz function with that for x exp ( − x ) {\displaystyle x\ \exp(-x)} shows that it defines an entire function.
|
https://en.wikipedia.org/wiki/Riesz_function
|
Alternatively, F may be defined as F ( x ) = ∑ k = 1 ∞ k k + 1 ¯ x k B 2 k . {\displaystyle F(x)=\sum _{k=1}^{\infty }{\frac {k^{\overline {k+1}}x^{k}}{B_{2k}}}.\ } n k ¯ {\displaystyle n^{\overline {k}}} denotes the rising factorial power in the notation of D. E. Knuth and the number Bn are the Bernoulli number. The series is one of alternating terms and the function quickly tends to minus infinity for increasingly negative values of x. Positive values of x are more interesting and delicate.
|
https://en.wikipedia.org/wiki/Riesz_function
|
In mathematics, the Riesz mean is a certain mean of the terms in a series. They were introduced by Marcel Riesz in 1911 as an improvement over the Cesàro mean. The Riesz mean should not be confused with the Bochner–Riesz mean or the Strong–Riesz mean.
|
https://en.wikipedia.org/wiki/Riesz_mean
|
In mathematics, the Riesz potential is a potential named after its discoverer, the Hungarian mathematician Marcel Riesz. In a sense, the Riesz potential defines an inverse for a power of the Laplace operator on Euclidean space. They generalize to several variables the Riemann–Liouville integrals of one variable.
|
https://en.wikipedia.org/wiki/Riesz_potential
|
In mathematics, the Riesz rearrangement inequality, sometimes called Riesz–Sobolev inequality, states that any three non-negative functions f: R n → R + {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{+}} , g: R n → R + {\displaystyle g:\mathbb {R} ^{n}\to \mathbb {R} ^{+}} and h: R n → R + {\displaystyle h:\mathbb {R} ^{n}\to \mathbb {R} ^{+}} satisfy the inequality ∬ R n × R n f ( x ) g ( x − y ) h ( y ) d x d y ≤ ∬ R n × R n f ∗ ( x ) g ∗ ( x − y ) h ∗ ( y ) d x d y , {\displaystyle \iint _{\mathbb {R} ^{n}\times \mathbb {R} ^{n}}f(x)g(x-y)h(y)\,dx\,dy\leq \iint _{\mathbb {R} ^{n}\times \mathbb {R} ^{n}}f^{*}(x)g^{*}(x-y)h^{*}(y)\,dx\,dy,} where f ∗: R n → R + {\displaystyle f^{*}:\mathbb {R} ^{n}\to \mathbb {R} ^{+}} , g ∗: R n → R + {\displaystyle g^{*}:\mathbb {R} ^{n}\to \mathbb {R} ^{+}} and h ∗: R n → R + {\displaystyle h^{*}:\mathbb {R} ^{n}\to \mathbb {R} ^{+}} are the symmetric decreasing rearrangements of the functions f {\displaystyle f} , g {\displaystyle g} and h {\displaystyle h} respectively.
|
https://en.wikipedia.org/wiki/Riesz_rearrangement_inequality
|
In mathematics, the Riesz–Fischer theorem in real analysis is any of a number of closely related results concerning the properties of the space L2 of square integrable functions. The theorem was proven independently in 1907 by Frigyes Riesz and Ernst Sigismund Fischer. For many authors, the Riesz–Fischer theorem refers to the fact that the Lp spaces L p {\displaystyle L^{p}} from Lebesgue integration theory are complete.
|
https://en.wikipedia.org/wiki/Riesz-Fischer_theorem
|
In mathematics, the Riesz–Markov–Kakutani representation theorem relates linear functionals on spaces of continuous functions on a locally compact space to measures in measure theory. The theorem is named for Frigyes Riesz (1909) who introduced it for continuous functions on the unit interval, Andrey Markov (1938) who extended the result to some non-compact spaces, and Shizuo Kakutani (1941) who extended the result to compact Hausdorff spaces. There are many closely related variations of the theorem, as the linear functionals can be complex, real, or positive, the space they are defined on may be the unit interval or a compact space or a locally compact space, the continuous functions may be vanishing at infinity or have compact support, and the measures can be Baire measures or regular Borel measures or Radon measures or signed measures or complex measures.
|
https://en.wikipedia.org/wiki/Riesz-Markov-Kakutani_representation_theorem
|
In mathematics, the Riesz–Thorin theorem, often referred to as the Riesz–Thorin interpolation theorem or the Riesz–Thorin convexity theorem, is a result about interpolation of operators. It is named after Marcel Riesz and his student G. Olof Thorin. This theorem bounds the norms of linear maps acting between Lp spaces.
|
https://en.wikipedia.org/wiki/Riesz–Thorin_theorem
|
Its usefulness stems from the fact that some of these spaces have rather simpler structure than others. Usually that refers to L2 which is a Hilbert space, or to L1 and L∞. Therefore one may prove theorems about the more complicated cases by proving them in two simple cases and then using the Riesz–Thorin theorem to pass from the simple cases to the complicated cases. The Marcinkiewicz theorem is similar but applies also to a class of non-linear maps.
|
https://en.wikipedia.org/wiki/Riesz–Thorin_theorem
|
In mathematics, the Robin boundary condition (; properly French: ), or third type boundary condition, is a type of boundary condition, named after Victor Gustave Robin (1855–1897). When imposed on an ordinary or a partial differential equation, it is a specification of a linear combination of the values of a function and the values of its derivative on the boundary of the domain. Other equivalent names in use are Fourier-type condition and radiation condition.
|
https://en.wikipedia.org/wiki/Robin_boundary_conditions
|
In mathematics, the Robinson–Schensted correspondence is a bijective correspondence between permutations and pairs of standard Young tableaux of the same shape. It has various descriptions, all of which are of algorithmic nature, it has many remarkable properties, and it has applications in combinatorics and other areas such as representation theory. The correspondence has been generalized in numerous ways, notably by Knuth to what is known as the Robinson–Schensted–Knuth correspondence, and a further generalization to pictures by Zelevinsky. The simplest description of the correspondence is using the Schensted algorithm (Schensted 1961), a procedure that constructs one tableau by successively inserting the values of the permutation according to a specific rule, while the other tableau records the evolution of the shape during construction.
|
https://en.wikipedia.org/wiki/Schensted_algorithm
|
The correspondence had been described, in a rather different form, much earlier by Robinson (Robinson 1938), in an attempt to prove the Littlewood–Richardson rule. The correspondence is often referred to as the Robinson–Schensted algorithm, although the procedure used by Robinson is radically different from the Schensted algorithm, and almost entirely forgotten.
|
https://en.wikipedia.org/wiki/Schensted_algorithm
|
Other methods of defining the correspondence include a nondeterministic algorithm in terms of jeu de taquin. The bijective nature of the correspondence relates it to the enumerative identity ∑ λ ∈ P n ( t λ ) 2 = n ! {\displaystyle \sum _{\lambda \in {\mathcal {P}}_{n}}(t_{\lambda })^{2}=n!} where P n {\displaystyle {\mathcal {P}}_{n}} denotes the set of partitions of n (or of Young diagrams with n squares), and tλ denotes the number of standard Young tableaux of shape λ.
|
https://en.wikipedia.org/wiki/Schensted_algorithm
|
In mathematics, the Robinson–Schensted–Knuth correspondence, also referred to as the RSK correspondence or RSK algorithm, is a combinatorial bijection between matrices A with non-negative integer entries and pairs (P,Q) of semistandard Young tableaux of equal shape, whose size equals the sum of the entries of A. More precisely the weight of P is given by the column sums of A, and the weight of Q by its row sums. It is a generalization of the Robinson–Schensted correspondence, in the sense that taking A to be a permutation matrix, the pair (P,Q) will be the pair of standard tableaux associated to the permutation under the Robinson–Schensted correspondence. The Robinson–Schensted–Knuth correspondence extends many of the remarkable properties of the Robinson–Schensted correspondence, notably its symmetry: transposition of the matrix A results in interchange of the tableaux P,Q.
|
https://en.wikipedia.org/wiki/Robinson–Schensted–Knuth_correspondence
|
In mathematics, the Rogers polynomials, also called Rogers–Askey–Ismail polynomials and continuous q-ultraspherical polynomials, are a family of orthogonal polynomials introduced by Rogers (1892, 1893, 1894) in the course of his work on the Rogers–Ramanujan identities. They are q-analogs of ultraspherical polynomials, and are the Macdonald polynomials for the special case of the A1 affine root system (Macdonald 2003, p.156). Askey & Ismail (1983) and Gasper & Rahman (2004, 7.4) discuss the properties of Rogers polynomials in detail.
|
https://en.wikipedia.org/wiki/Rogers_polynomials
|
In mathematics, the Rogers–Ramanujan identities are two identities related to basic hypergeometric series and integer partitions. The identities were first discovered and proved by Leonard James Rogers (1894), and were subsequently rediscovered (without a proof) by Srinivasa Ramanujan some time before 1913. Ramanujan had no proof, but rediscovered Rogers's paper in 1917, and they then published a joint new proof (Rogers & Ramanujan 1919). Issai Schur (1917) independently rediscovered and proved the identities.
|
https://en.wikipedia.org/wiki/Rogers–Ramanujan_identities
|
In mathematics, the Rogers–Szegő polynomials are a family of polynomials orthogonal on the unit circle introduced by Szegő (1926), who was inspired by the continuous q-Hermite polynomials studied by Leonard James Rogers. They are given by h n ( x ; q ) = ∑ k = 0 n ( q ; q ) n ( q ; q ) k ( q ; q ) n − k x k {\displaystyle h_{n}(x;q)=\sum _{k=0}^{n}{\frac {(q;q)_{n}}{(q;q)_{k}(q;q)_{n-k}}}x^{k}} where (q;q)n is the descending q-Pochhammer symbol. Furthermore, the h n ( x ; q ) {\displaystyle h_{n}(x;q)} satisfy (for n ≥ 1 {\displaystyle n\geq 1} ) the recurrence relation h n + 1 ( x ; q ) = ( 1 + x ) h n ( x ; q ) + x ( q n − 1 ) h n − 1 ( x ; q ) {\displaystyle h_{n+1}(x;q)=(1+x)h_{n}(x;q)+x(q^{n}-1)h_{n-1}(x;q)} with h 0 ( x ; q ) = 1 {\displaystyle h_{0}(x;q)=1} and h 1 ( x ; q ) = 1 + x {\displaystyle h_{1}(x;q)=1+x} .
|
https://en.wikipedia.org/wiki/Rogers–Szegő_polynomials
|
In mathematics, the Rokhlin lemma, or Kakutani–Rokhlin lemma is an important result in ergodic theory. It states that an aperiodic measure preserving dynamical system can be decomposed to an arbitrary high tower of measurable sets and a remainder of arbitrarily small measure. It was proven by Vladimir Abramovich Rokhlin and independently by Shizuo Kakutani. The lemma is used extensively in ergodic theory, for example in Ornstein theory and has many generalizations.
|
https://en.wikipedia.org/wiki/Rokhlin_lemma
|
In mathematics, the Roman surface or Steiner surface is a self-intersecting mapping of the real projective plane into three-dimensional space, with an unusually high degree of symmetry. This mapping is not an immersion of the projective plane; however, the figure resulting from removing six singular points is one. Its name arises because it was discovered by Jakob Steiner when he was in Rome in 1844.The simplest construction is as the image of a sphere centered at the origin under the map f ( x , y , z ) = ( y z , x z , x y ) . {\displaystyle f(x,y,z)=(yz,xz,xy).}
|
https://en.wikipedia.org/wiki/Roman_surface
|
This gives an implicit formula of x 2 y 2 + y 2 z 2 + z 2 x 2 − r 2 x y z = 0. {\displaystyle x^{2}y^{2}+y^{2}z^{2}+z^{2}x^{2}-r^{2}xyz=0.\,} Also, taking a parametrization of the sphere in terms of longitude (θ) and latitude (φ), gives parametric equations for the Roman surface as follows: x = r 2 cos θ cos φ sin φ {\displaystyle x=r^{2}\cos \theta \cos \varphi \sin \varphi } y = r 2 sin θ cos φ sin φ {\displaystyle y=r^{2}\sin \theta \cos \varphi \sin \varphi } z = r 2 cos θ sin θ cos 2 φ {\displaystyle z=r^{2}\cos \theta \sin \theta \cos ^{2}\varphi } The origin is a triple point, and each of the xy-, yz-, and xz-planes are tangential to the surface there. The other places of self-intersection are double points, defining segments along each coordinate axis which terminate in six pinch points. The entire surface has tetrahedral symmetry. It is a particular type (called type 1) of Steiner surface, that is, a 3-dimensional linear projection of the Veronese surface.
|
https://en.wikipedia.org/wiki/Roman_surface
|
In mathematics, the Romanovski polynomials are one of three finite subsets of real orthogonal polynomials discovered by Vsevolod Romanovsky (Romanovski in French transcription) within the context of probability distribution functions in statistics. They form an orthogonal subset of a more general family of little-known Routh polynomials introduced by Edward John Routh in 1884. The term Romanovski polynomials was put forward by Raposo, with reference to the so-called 'pseudo-Jacobi polynomials in Lesky's classification scheme. It seems more consistent to refer to them as Romanovski–Routh polynomials, by analogy with the terms Romanovski–Bessel and Romanovski–Jacobi used by Lesky for two other sets of orthogonal polynomials. In some contrast to the standard classical orthogonal polynomials, the polynomials under consideration differ, in so far as for arbitrary parameters only a finite number of them are orthogonal, as discussed in more detail below.
|
https://en.wikipedia.org/wiki/Romanovski_polynomials
|
In mathematics, the Rost invariant is a cohomological invariant of an absolutely simple simply connected algebraic group G over a field k, which associates an element of the Galois cohomology group H3(k, Q/Z(2)) to a principal homogeneous space for G. Here the coefficient group Q/Z(2) is the tensor product of the group of roots of unity of an algebraic closure of k with itself. Markus Rost (1991) first introduced the invariant for groups of type F4 and later extended it to more general groups in unpublished work that was summarized by Serre (1995). The Rost invariant is a generalization of the Arason invariant.
|
https://en.wikipedia.org/wiki/Rost_invariant
|
In mathematics, the Rothe–Hagen identity is a mathematical identity valid for all complex numbers ( x , y , z {\displaystyle x,y,z} ) except where its denominators vanish: ∑ k = 0 n x x + k z ( x + k z k ) y y + ( n − k ) z ( y + ( n − k ) z n − k ) = x + y x + y + n z ( x + y + n z n ) . {\displaystyle \sum _{k=0}^{n}{\frac {x}{x+kz}}{x+kz \choose k}{\frac {y}{y+(n-k)z}}{y+(n-k)z \choose n-k}={\frac {x+y}{x+y+nz}}{x+y+nz \choose n}.} It is a generalization of Vandermonde's identity, and is named after Heinrich August Rothe and Johann Georg Hagen.
|
https://en.wikipedia.org/wiki/Rothe–Hagen_identity
|
In mathematics, the Routh–Hurwitz theorem gives a test to determine whether all roots of a given polynomial lie in the left half-plane. Polynomials with this property are called Hurwitz stable polynomials. The Routh–Hurwitz theorem is important in dynamical systems and control theory, because the characteristic polynomial of the differential equations of a stable linear system has roots limited to the left half plane (negative eigenvalues). Thus the theorem provides a mathematical test, the Routh-Hurwitz stability criterion, to determine whether a linear dynamical system is stable without solving the system. The Routh–Hurwitz theorem was proved in 1895, and it was named after Edward John Routh and Adolf Hurwitz.
|
https://en.wikipedia.org/wiki/Routh–Hurwitz_theorem
|
In mathematics, the Rudin–Shapiro sequence, also known as the Golay–Rudin–Shapiro sequence, is an infinite 2-automatic sequence named after Marcel Golay, Walter Rudin, and Harold S. Shapiro, who independently investigated its properties.
|
https://en.wikipedia.org/wiki/Rudin–Shapiro_sequence
|
In mathematics, the Ruelle zeta function is a zeta function associated with a dynamical system. It is named after mathematical physicist David Ruelle.
|
https://en.wikipedia.org/wiki/Ruelle_zeta-function
|
In mathematics, the Runge–Kutta–Fehlberg method (or Fehlberg method) is an algorithm in numerical analysis for the numerical solution of ordinary differential equations. It was developed by the German mathematician Erwin Fehlberg and is based on the large class of Runge–Kutta methods. The novelty of Fehlberg's method is that it is an embedded method from the Runge–Kutta family, meaning that identical function evaluations are used in conjunction with each other to create methods of varying order and similar error constants. The method presented in Fehlberg's 1969 paper has been dubbed the RKF45 method, and is a method of order O(h4) with an error estimator of order O(h5). By performing one extra calculation, the error in the solution can be estimated and controlled by using the higher-order embedded method that allows for an adaptive stepsize to be determined automatically.
|
https://en.wikipedia.org/wiki/Runge–Kutta–Fehlberg_method
|
In mathematics, the Russo–Dye theorem is a result in the field of functional analysis. It states that in a unital C*-algebra, the closure of the convex hull of the unitary elements is the closed unit ball. : 44 The theorem was published by B. Russo and H. A. Dye in 1966.
|
https://en.wikipedia.org/wiki/Russo–Dye_theorem
|
In mathematics, the Samuelson–Berkowitz algorithm efficiently computes the characteristic polynomial of an n × n {\displaystyle n\times n} matrix whose entries may be elements of any unital commutative ring. Unlike the Faddeev–LeVerrier algorithm, it performs no divisions, so may be applied to a wider range of algebraic structures.
|
https://en.wikipedia.org/wiki/Samuelson–Berkowitz_algorithm
|
In mathematics, the Satake isomorphism, introduced by Ichirō Satake (1963), identifies the Hecke algebra of a reductive group over a local field with a ring of invariants of the Weyl group. The geometric Satake equivalence is a geometric version of the Satake isomorphism, proved by Ivan Mirković and Kari Vilonen (2007).
|
https://en.wikipedia.org/wiki/Geometric_Satake_correspondence
|
In mathematics, the Sato–Tate conjecture is a statistical statement about the family of elliptic curves Ep obtained from an elliptic curve E over the rational numbers by reduction modulo almost all prime numbers p. Mikio Sato and John Tate independently posed the conjecture around 1960. If Np denotes the number of points on the elliptic curve Ep defined over the finite field with p elements, the conjecture gives an answer to the distribution of the second-order term for Np. By Hasse's theorem on elliptic curves, N p / p = 1 + O ( 1 / p ) {\displaystyle N_{p}/p=1+\mathrm {O} (1/\!
|
https://en.wikipedia.org/wiki/Lang–Trotter_conjecture
|
{\sqrt {p}})\ } as p → ∞ {\displaystyle p\to \infty } , and the point of the conjecture is to predict how the O-term varies. The original conjecture and its generalization to all totally real fields was proved by Laurent Clozel, Michael Harris, Nicholas Shepherd-Barron, and Richard Taylor under mild assumptions in 2008, and completed by Thomas Barnet-Lamb, David Geraghty, Harris, and Taylor in 2011. Several generalizations to other algebraic varieties and fields are open.
|
https://en.wikipedia.org/wiki/Lang–Trotter_conjecture
|
In mathematics, the Schneider–Lang theorem is a refinement by Lang (1966) of a theorem of Schneider (1949) about the transcendence of values of meromorphic functions. The theorem implies both the Hermite–Lindemann and Gelfond–Schneider theorems, and implies the transcendence of some values of elliptic functions and elliptic modular functions.
|
https://en.wikipedia.org/wiki/Schneider–Lang_theorem
|
In mathematics, the Schoenflies problem or Schoenflies theorem, of geometric topology is a sharpening of the Jordan curve theorem by Arthur Schoenflies. For Jordan curves in the plane it is often referred to as the Jordan–Schoenflies theorem.
|
https://en.wikipedia.org/wiki/Schönflies_problem
|
In mathematics, the Schoen–Yau conjecture is a disproved conjecture in hyperbolic geometry, named after the mathematicians Richard Schoen and Shing-Tung Yau. It was inspired by a theorem of Erhard Heinz (1952). One method of disproof is the use of Scherk surfaces, as used by Harold Rosenberg and Pascal Collin (2006).
|
https://en.wikipedia.org/wiki/Schoen–Yau_conjecture
|
In mathematics, the Scholz conjecture is a conjecture on the length of certain addition chains. It is sometimes also called the Scholz–Brauer conjecture or the Brauer–Scholz conjecture, after Arnold Scholz who formulated it in 1937 and Alfred Brauer who studied it soon afterward and proved a weaker bound.
|
https://en.wikipedia.org/wiki/Scholz_conjecture
|
In mathematics, the Schottky form or Schottky's invariant is a Siegel cusp form J of degree 4 and weight 8, introduced by Friedrich Schottky (1888, 1903) as a degree 16 polynomial in the Thetanullwerte of genus 4. He showed that it vanished at all Jacobian points (the points of the degree 4 Siegel upper half-space corresponding to 4-dimensional abelian varieties that are the Jacobian varieties of genus 4 curves). Igusa (1981) showed that it is a multiple of the difference θ4(E8 ⊕ E8) − θ4(E16) of the two genus 4 theta functions of the two 16-dimensional even unimodular lattices and that its divisor of zeros is irreducible. Poor & Yuen (1996) showed that it generates the 1-dimensional space of level 1 genus 4 weight 8 Siegel cusp forms. Ikeda showed that the Schottky form is the image of the Dedekind Delta function under the Ikeda lift.
|
https://en.wikipedia.org/wiki/Schottky_form
|
In mathematics, the Schottky problem, named after Friedrich Schottky, is a classical question of algebraic geometry, asking for a characterisation of Jacobian varieties amongst abelian varieties.
|
https://en.wikipedia.org/wiki/Schottky_problem
|
In mathematics, the Schreier refinement theorem of group theory states that any two subnormal series of subgroups of a given group have equivalent refinements, where two series are equivalent if there is a bijection between their factor groups that sends each factor group to an isomorphic one. The theorem is named after the Austrian mathematician Otto Schreier who proved it in 1928. It provides an elegant proof of the Jordan–Hölder theorem. It is often proved using the Zassenhaus lemma. Baumslag (2006) gives a short proof by intersecting the terms in one subnormal series with those in the other series.
|
https://en.wikipedia.org/wiki/Schreier_refinement_theorem
|
In mathematics, the Schröder number S n , {\displaystyle S_{n},} also called a large Schröder number or big Schröder number, describes the number of lattice paths from the southwest corner ( 0 , 0 ) {\displaystyle (0,0)} of an n × n {\displaystyle n\times n} grid to the northeast corner ( n , n ) , {\displaystyle (n,n),} using only single steps north, ( 0 , 1 ) ; {\displaystyle (0,1);} northeast, ( 1 , 1 ) ; {\displaystyle (1,1);} or east, ( 1 , 0 ) , {\displaystyle (1,0),} that do not rise above the SW–NE diagonal.The first few Schröder numbers are 1, 2, 6, 22, 90, 394, 1806, 8558, ... (sequence A006318 in the OEIS).where S 0 = 1 {\displaystyle S_{0}=1} and S 1 = 2. {\displaystyle S_{1}=2.} They were named after the German mathematician Ernst Schröder.
|
https://en.wikipedia.org/wiki/Schröder_number
|
In mathematics, the Schuette–Nesbitt formula is a generalization of the inclusion–exclusion principle. It is named after Donald R. Schuette and Cecil J. Nesbitt. The probabilistic version of the Schuette–Nesbitt formula has practical applications in actuarial science, where it is used to calculate the net single premium for life annuities and life insurances based on the general symmetric status.
|
https://en.wikipedia.org/wiki/Schuette–Nesbitt_formula
|
In mathematics, the Schur orthogonality relations, which were proven by Issai Schur through Schur's lemma, express a central fact about representations of finite groups. They admit a generalization to the case of compact groups in general, and in particular compact Lie groups, such as the rotation group SO(3).
|
https://en.wikipedia.org/wiki/Schur_orthogonality_relations
|
In mathematics, the Schwartz kernel theorem is a foundational result in the theory of generalized functions, published by Laurent Schwartz in 1952. It states, in broad terms, that the generalized functions introduced by Schwartz (Schwartz distributions) have a two-variable theory that includes all reasonable bilinear forms on the space D {\displaystyle {\mathcal {D}}} of test functions. The space D {\displaystyle {\mathcal {D}}} itself consists of smooth functions of compact support.
|
https://en.wikipedia.org/wiki/Schwartz_kernel_theorem
|
In mathematics, the Schwartz–Zippel lemma (also called the DeMillo–Lipton–Schwartz–Zippel lemma) is a tool commonly used in probabilistic polynomial identity testing, i.e. in the problem of determining whether a given multivariate polynomial is the 0-polynomial (or identically equal to 0). It was discovered independently by Jack Schwartz, Richard Zippel, and Richard DeMillo and Richard J. Lipton, although DeMillo and Lipton's version was shown a year prior to Schwartz and Zippel's result. The finite field version of this bound was proved by Øystein Ore in 1922.
|
https://en.wikipedia.org/wiki/Schwartz–Zippel_lemma
|
In mathematics, the Schwarz alternating method or alternating process is an iterative method introduced in 1869–1870 by Hermann Schwarz in the theory of conformal mapping. Given two overlapping regions in the complex plane in each of which the Dirichlet problem could be solved, Schwarz described an iterative method for solving the Dirichlet problem in their union, provided their intersection was suitably well behaved. This was one of several constructive techniques of conformal mapping developed by Schwarz as a contribution to the problem of uniformization, posed by Riemann in the 1850s and first resolved rigorously by Koebe and Poincaré in 1907. It furnished a scheme for uniformizing the union of two regions knowing how to uniformize each of them separately, provided their intersection was topologically a disk or an annulus.
|
https://en.wikipedia.org/wiki/Schwarz_alternating_method
|
From 1870 onwards Carl Neumann also contributed to this theory. In the 1950s Schwarz's method was generalized in the theory of partial differential equations to an iterative method for finding the solution of an elliptic boundary value problem on a domain which is the union of two overlapping subdomains. It involves solving the boundary value problem on each of the two subdomains in turn, taking always the last values of the approximate solution as the next boundary conditions. It is used in numerical analysis, under the name multiplicative Schwarz method (in opposition to additive Schwarz method) as a domain decomposition method.
|
https://en.wikipedia.org/wiki/Schwarz_alternating_method
|
In mathematics, the Schwarz lantern is a polyhedral approximation to a cylinder, used as a pathological example of the difficulty of defining the area of a smooth (curved) surface as the limit of the areas of polyhedra. It is formed by stacked rings of isosceles triangles, arranged within each ring in the same pattern as an antiprism. The resulting shape can be folded from paper, and is named after mathematician Hermann Schwarz and for its resemblance to a cylindrical paper lantern.
|
https://en.wikipedia.org/wiki/Schwarz_lantern
|
It is also known as Schwarz's boot, Schwarz's polyhedron, or the Chinese lantern.As Schwarz showed, for the surface area of a polyhedron to converge to the surface area of a curved surface, it is not sufficient to simply increase the number of rings and the number of isosceles triangles per ring. Depending on the relation of the number of rings to the number of triangles per ring, the area of the lantern can converge to the area of the cylinder, to a limit arbitrarily larger than the area of the cylinder, or to infinity—in other words, the area can diverge. The Schwarz lantern demonstrates that sampling a curved surface by close-together points and connecting them by small triangles is inadequate to ensure an accurate approximation of area, in contrast to the accurate approximation of arc length by inscribed polygonal chains. The phenomenon that closely sampled points can lead to inaccurate approximations of area has been called the Schwarz paradox. The Schwarz lantern is an instructive example in calculus and highlights the need for care when choosing a triangulation for applications in computer graphics and the finite element method.
|
https://en.wikipedia.org/wiki/Schwarz_lantern
|
In mathematics, the Schwarz lemma, named after Hermann Amandus Schwarz, is a result in complex analysis about holomorphic functions from the open unit disk to itself. The lemma is less celebrated than deeper theorems, such as the Riemann mapping theorem, which it helps to prove. It is, however, one of the simplest results capturing the rigidity of holomorphic functions.
|
https://en.wikipedia.org/wiki/Schwarz's_lemma
|
In mathematics, the Schwarz reflection principle is a way to extend the domain of definition of a complex analytic function, i.e., it is a form of analytic continuation. It states that if an analytic function is defined on the upper half-plane, and has well-defined (non-singular) real values on the real axis, then it can be extended to the conjugate function on the lower half-plane. In notation, if F ( z ) {\displaystyle F(z)} is a function that satisfies the above requirements, then its extension to the rest of the complex plane is given by the formula, That is, we make the definition that agrees along the real axis. The result proved by Hermann Schwarz is as follows.
|
https://en.wikipedia.org/wiki/Schwarz_reflection_principle
|
Suppose that F is a continuous function on the closed upper half plane { z ∈ C ∣ Im ( z ) ≥ 0 } {\displaystyle \left\{z\in \mathbb {C} \mid \operatorname {Im} (z)\geq 0\right\}} , holomorphic on the upper half plane { z ∈ C ∣ Im ( z ) > 0 } {\displaystyle \left\{z\in \mathbb {C} \mid \operatorname {Im} (z)>0\right\}} , which takes real values on the real axis. Then the extension formula given above is an analytic continuation to the whole complex plane.In practice it would be better to have a theorem that allows F certain singularities, for example F a meromorphic function. To understand such extensions, one needs a proof method that can be weakened.
|
https://en.wikipedia.org/wiki/Schwarz_reflection_principle
|
In fact Morera's theorem is well adapted to proving such statements. Contour integrals involving the extension of F clearly split into two, using part of the real axis. So, given that the principle is rather easy to prove in the special case from Morera's theorem, understanding the proof is enough to generate other results. The principle also adapts to apply to harmonic functions.
|
https://en.wikipedia.org/wiki/Schwarz_reflection_principle
|
In mathematics, the Schwarzian derivative is an operator similar to the derivative which is invariant under Möbius transformations. Thus, it occurs in the theory of the complex projective line, and in particular, in the theory of modular forms and hypergeometric functions. It plays an important role in the theory of univalent functions, conformal mapping and Teichmüller spaces. It is named after the German mathematician Hermann Schwarz.
|
https://en.wikipedia.org/wiki/Schwarzian_derivative
|
In mathematics, the Schwarz–Ahlfors–Pick theorem is an extension of the Schwarz lemma for hyperbolic geometry, such as the Poincaré half-plane model. The Schwarz–Pick lemma states that every holomorphic function from the unit disk U to itself, or from the upper half-plane H to itself, will not increase the Poincaré distance between points. The unit disk U with the Poincaré metric has negative Gaussian curvature −1. In 1938, Lars Ahlfors generalised the lemma to maps from the unit disk to other negatively curved surfaces: Theorem (Schwarz–Ahlfors–Pick).
|
https://en.wikipedia.org/wiki/Schwarz–Ahlfors–Pick_theorem
|
Let U be the unit disk with Poincaré metric ρ {\displaystyle \rho } ; let S be a Riemann surface endowed with a Hermitian metric σ {\displaystyle \sigma } whose Gaussian curvature is ≤ −1; let f: U → S {\displaystyle f:U\rightarrow S} be a holomorphic function. Then σ ( f ( z 1 ) , f ( z 2 ) ) ≤ ρ ( z 1 , z 2 ) {\displaystyle \sigma (f(z_{1}),f(z_{2}))\leq \rho (z_{1},z_{2})} for all z 1 , z 2 ∈ U .
|
https://en.wikipedia.org/wiki/Schwarz–Ahlfors–Pick_theorem
|
{\displaystyle z_{1},z_{2}\in U.} A generalization of this theorem was proved by Shing-Tung Yau in 1973. == References ==
|
https://en.wikipedia.org/wiki/Schwarz–Ahlfors–Pick_theorem
|
In mathematics, the Scott core theorem is a theorem about the finite presentability of fundamental groups of 3-manifolds due to G. Peter Scott, (Scott 1973). The precise statement is as follows: Given a 3-manifold (not necessarily compact) with finitely generated fundamental group, there is a compact three-dimensional submanifold, called the compact core or Scott core, such that its inclusion map induces an isomorphism on fundamental groups. In particular, this means a finitely generated 3-manifold group is finitely presentable. A simplified proof is given in (Rubinstein & Swarup 1990), and a stronger uniqueness statement is proven in (Harris & Scott 1996).
|
https://en.wikipedia.org/wiki/Scott_core_theorem
|
In mathematics, the Segal–Bargmann space (for Irving Segal and Valentine Bargmann), also known as the Bargmann space or Bargmann–Fock space, is the space of holomorphic functions F in n complex variables satisfying the square-integrability condition: ‖ F ‖ 2 := π − n ∫ C n | F ( z ) | 2 exp ( − | z | 2 ) d z < ∞ , {\displaystyle \|F\|^{2}:=\pi ^{-n}\int _{\mathbb {C} ^{n}}|F(z)|^{2}\exp(-|z|^{2})\,dz<\infty ,} where here dz denotes the 2n-dimensional Lebesgue measure on C n . {\displaystyle \mathbb {C} ^{n}.} It is a Hilbert space with respect to the associated inner product: ⟨ F ∣ G ⟩ = π − n ∫ C n F ( z ) ¯ G ( z ) exp ( − | z | 2 ) d z .
|
https://en.wikipedia.org/wiki/Segal–Bargmann_space
|
{\displaystyle \langle F\mid G\rangle =\pi ^{-n}\int _{\mathbb {C} ^{n}}{\overline {F(z)}}G(z)\exp(-|z|^{2})\,dz.} The space was introduced in the mathematical physics literature separately by Bargmann and Segal in the early 1960s; see Bargmann (1961) and Segal (1963). Basic information about the material in this section may be found in Folland (1989) and Hall (2000) . Segal worked from the beginning in the infinite-dimensional setting; see Baez, Segal & Zhou (1992) and Section 10 of Hall (2000) for more information on this aspect of the subject.
|
https://en.wikipedia.org/wiki/Segal–Bargmann_space
|
In mathematics, the Segre class is a characteristic class used in the study of cones, a generalization of vector bundles. For vector bundles the total Segre class is inverse to the total Chern class, and thus provides equivalent information; the advantage of the Segre class is that it generalizes to more general cones, while the Chern class does not. The Segre class was introduced in the non-singular case by Segre (1953).. In the modern treatment of intersection theory in algebraic geometry, as developed e.g. in the definitive book of Fulton (1998), Segre classes play a fundamental role.
|
https://en.wikipedia.org/wiki/Segre_class
|
In mathematics, the Segre embedding is used in projective geometry to consider the cartesian product (of sets) of two projective spaces as a projective variety. It is named after Corrado Segre.
|
https://en.wikipedia.org/wiki/Segre_mapping
|
In mathematics, the Seifert conjecture states that every nonsingular, continuous vector field on the 3-sphere has a closed orbit. It is named after Herbert Seifert. In a 1950 paper, Seifert asked if such a vector field exists, but did not phrase non-existence as a conjecture.
|
https://en.wikipedia.org/wiki/Seifert_conjecture
|
He also established the conjecture for perturbations of the Hopf fibration. The conjecture was disproven in 1974 by Paul Schweitzer, who exhibited a C 1 {\displaystyle C^{1}} counterexample. Schweitzer's construction was then modified by Jenny Harrison in 1988 to make a C 2 + δ {\displaystyle C^{2+\delta }} counterexample for some δ > 0 {\displaystyle \delta >0} . The existence of smoother counterexamples remained an open question until 1993 when Krystyna Kuperberg constructed a very different C ∞ {\displaystyle C^{\infty }} counterexample. Later this construction was shown to have real analytic and piecewise linear versions.
|
https://en.wikipedia.org/wiki/Seifert_conjecture
|
In mathematics, the Seifert–Van Kampen theorem of algebraic topology (named after Herbert Seifert and Egbert van Kampen), sometimes just called Van Kampen's theorem, expresses the structure of the fundamental group of a topological space X {\displaystyle X} in terms of the fundamental groups of two open, path-connected subspaces that cover X {\displaystyle X} . It can therefore be used for computations of the fundamental group of spaces that are constructed out of simpler ones.
|
https://en.wikipedia.org/wiki/Seifert–Van_Kampen_theorem
|
In mathematics, the Selberg conjecture, named after Atle Selberg, is a theorem about the density of zeros of the Riemann zeta function ζ(1/2 + it). It is known that the function has infinitely many zeroes on this line in the complex plane: the point at issue is how densely they are clustered. Results on this can be formulated in terms of N(T), the function counting zeroes on the line for which the value of t satisfies 0 ≤ t ≤ T.
|
https://en.wikipedia.org/wiki/Selberg's_zeta_function_conjecture
|
In mathematics, the Selberg integral is a generalization of Euler beta function to n dimensions introduced by Atle Selberg.
|
https://en.wikipedia.org/wiki/Selberg_integral
|
In mathematics, the Selberg trace formula, introduced by Selberg (1956), is an expression for the character of the unitary representation of a Lie group G on the space L2(Γ\G) of square-integrable functions, where Γ is a cofinite discrete group. The character is given by the trace of certain functions on G. The simplest case is when Γ is cocompact, when the representation breaks up into discrete summands. Here the trace formula is an extension of the Frobenius formula for the character of an induced representation of finite groups. When Γ is the cocompact subgroup Z of the real numbers G = R, the Selberg trace formula is essentially the Poisson summation formula.
|
https://en.wikipedia.org/wiki/Selberg_trace_formula
|
The case when Γ\G is not compact is harder, because there is a continuous spectrum, described using Eisenstein series. Selberg worked out the non-compact case when G is the group SL(2, R); the extension to higher rank groups is the Arthur–Selberg trace formula.
|
https://en.wikipedia.org/wiki/Selberg_trace_formula
|
When Γ is the fundamental group of a Riemann surface, the Selberg trace formula describes the spectrum of differential operators such as the Laplacian in terms of geometric data involving the lengths of geodesics on the Riemann surface. In this case the Selberg trace formula is formally similar to the explicit formulas relating the zeros of the Riemann zeta function to prime numbers, with the zeta zeros corresponding to eigenvalues of the Laplacian, and the primes corresponding to geodesics. Motivated by the analogy, Selberg introduced the Selberg zeta function of a Riemann surface, whose analytic properties are encoded by the Selberg trace formula.
|
https://en.wikipedia.org/wiki/Selberg_trace_formula
|
In mathematics, the Serre spectral sequence (sometimes Leray–Serre spectral sequence to acknowledge earlier work of Jean Leray in the Leray spectral sequence) is an important tool in algebraic topology. It expresses, in the language of homological algebra, the singular (co)homology of the total space X of a (Serre) fibration in terms of the (co)homology of the base space B and the fiber F. The result is due to Jean-Pierre Serre in his doctoral dissertation.
|
https://en.wikipedia.org/wiki/Serre_spectral_sequence
|
In mathematics, the Shapiro inequality is an inequality proposed by Harold S. Shapiro in 1954.
|
https://en.wikipedia.org/wiki/Shapiro_inequality
|
In mathematics, the Shapiro polynomials are a sequence of polynomials which were first studied by Harold S. Shapiro in 1951 when considering the magnitude of specific trigonometric sums. In signal processing, the Shapiro polynomials have good autocorrelation properties and their values on the unit circle are small. The first few members of the sequence are: P 1 ( x ) = 1 + x P 2 ( x ) = 1 + x + x 2 − x 3 P 3 ( x ) = 1 + x + x 2 − x 3 + x 4 + x 5 − x 6 + x 7 . .
|
https://en.wikipedia.org/wiki/Shapiro_polynomials
|
. Q 1 ( x ) = 1 − x Q 2 ( x ) = 1 + x − x 2 + x 3 Q 3 ( x ) = 1 + x + x 2 − x 3 − x 4 − x 5 + x 6 − x 7 . . . {\displaystyle {\begin{aligned}P_{1}(x)&{}=1+x\\P_{2}(x)&{}=1+x+x^{2}-x^{3}\\P_{3}(x)&{}=1+x+x^{2}-x^{3}+x^{4}+x^{5}-x^{6}+x^{7}\\...\\Q_{1}(x)&{}=1-x\\Q_{2}(x)&{}=1+x-x^{2}+x^{3}\\Q_{3}(x)&{}=1+x+x^{2}-x^{3}-x^{4}-x^{5}+x^{6}-x^{7}\\...\\\end{aligned}}} where the second sequence, indicated by Q, is said to be complementary to the first sequence, indicated by P.
|
https://en.wikipedia.org/wiki/Shapiro_polynomials
|
In mathematics, the Sherman–Takeda theorem states that if A is a C*-algebra then its double dual is a W*-algebra, and is isomorphic to the weak closure of A in the universal representation of A. The theorem was announced by Sherman (1950) and proved by Takeda (1954). The double dual of A is called the universal enveloping W*-algebra of A.
|
https://en.wikipedia.org/wiki/Sherman–Takeda_theorem
|
In mathematics, the Shimizu L-function, introduced by Hideo Shimizu (1963), is a Dirichlet series associated to a totally real algebraic number field. Michael Francis Atiyah, H. Donnelly, and I. M. Singer (1983) defined the signature defect of the boundary of a manifold as the eta invariant, the value as s=0 of their eta function, and used this to show that Hirzebruch's signature defect of a cusp of a Hilbert modular surface can be expressed in terms of the value at s=0 or 1 of a Shimizu L-function.
|
https://en.wikipedia.org/wiki/Shimizu_L-function
|
In mathematics, the Shimura subgroup Σ(N) is a subgroup of the Jacobian of the modular curve X0(N) of level N, given by the kernel of the natural map to the Jacobian of X1(N). It is named after Goro Shimura. There is a similar subgroup Σ(N,D) associated to Shimura curves of quaternion algebras.
|
https://en.wikipedia.org/wiki/Shimura_subgroup
|
In mathematics, the Siegel G-functions are a class of functions in transcendental number theory introduced by C. L. Siegel. They satisfy a linear differential equation with polynomial coefficients, and the coefficients of their power series expansion lie in a fixed algebraic number field and have heights of at most exponential growth.
|
https://en.wikipedia.org/wiki/Siegel_G-function
|
In mathematics, the Siegel parabolic subgroup, named after Carl Ludwig Siegel, is the parabolic subgroup of the symplectic group with abelian radical, given by the matrices of the symplectic group whose lower left quadrant is 0 (for the standard symplectic form). == References ==
|
https://en.wikipedia.org/wiki/Siegel_parabolic_subgroup
|
In mathematics, the Siegel upper half-space of degree g (or genus g) (also called the Siegel upper half-plane) is the set of g × g symmetric matrices over the complex numbers whose imaginary part is positive definite. It was introduced by Siegel (1939). It is the symmetric space associated to the symplectic group Sp(2g, R). The Siegel upper half-space has properties as a complex manifold that generalize the properties of the upper half-plane, which is the Siegel upper half-space in the special case g = 1.
|
https://en.wikipedia.org/wiki/Siegel_upper_half-space
|
The group of automorphisms preserving the complex structure of the manifold is isomorphic to the symplectic group Sp(2g, R). Just as the two-dimensional hyperbolic metric is the unique (up to scaling) metric on the upper half-plane whose isometry group is the complex automorphism group SL(2, R) = Sp(2, R), the Siegel upper half-space has only one metric up to scaling whose isometry group is Sp(2g, R).
|
https://en.wikipedia.org/wiki/Siegel_upper_half-space
|
Writing a generic matrix Z in the Siegel upper half-space in terms of its real and imaginary parts as Z = X + iY, all metrics with isometry group Sp(2g, R) are proportional to d s 2 = tr ( Y − 1 d Z Y − 1 d Z ¯ ) . {\displaystyle ds^{2}={\text{tr}}(Y^{-1}dZY^{-1}d{\bar {Z}}).} The Siegel upper half-plane can be identified with the set of tame almost complex structures compatible with a symplectic structure ω {\displaystyle \omega } , on the underlying 2 n {\displaystyle 2n} dimensional real vector space V {\displaystyle V} , that is, the set of J ∈ H o m ( V ) {\displaystyle J\in Hom(V)} such that J 2 = − 1 {\displaystyle J^{2}=-1} and ω ( J v , v ) > 0 {\displaystyle \omega (Jv,v)>0} for all vectors v ≠ 0 {\displaystyle v\neq 0} .
|
https://en.wikipedia.org/wiki/Siegel_upper_half-space
|
In mathematics, the Siegel–Weil formula, introduced by Weil (1964, 1965) as an extension of the results of Siegel (1951, 1952), expresses an Eisenstein series as a weighted average of theta series of lattices in a genus, where the weights are proportional to the inverse of the order of the automorphism group of the lattice. For the constant terms this is essentially the Smith–Minkowski–Siegel mass formula.
|
https://en.wikipedia.org/wiki/Siegel–Weil_formula
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.