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In mathematics, the Picard group of a ringed space X, denoted by Pic(X), is the group of isomorphism classes of invertible sheaves (or line bundles) on X, with the group operation being tensor product. This construction is a global version of the construction of the divisor class group, or ideal class group, and is much used in algebraic geometry and the theory of complex manifolds. Alternatively, the Picard group can be defined as the sheaf cohomology group H 1 ( X , O X ∗ ) .
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https://en.wikipedia.org/wiki/Picard_scheme
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{\displaystyle H^{1}(X,{\mathcal {O}}_{X}^{*}).\,} For integral schemes the Picard group is isomorphic to the class group of Cartier divisors. For complex manifolds the exponential sheaf sequence gives basic information on the Picard group. The name is in honour of Émile Picard's theories, in particular of divisors on algebraic surfaces.
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https://en.wikipedia.org/wiki/Picard_scheme
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In mathematics, the Picard–Fuchs equation, named after Émile Picard and Lazarus Fuchs, is a linear ordinary differential equation whose solutions describe the periods of elliptic curves.
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https://en.wikipedia.org/wiki/Picard–Fuchs_equation
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In mathematics, the Pidduck polynomials sn(x) are polynomials introduced by Pidduck (1910, 1912) given by the generating function ∑ n s n ( x ) n ! t n = ( 1 + t 1 − t ) x ( 1 − t ) − 1 {\displaystyle \displaystyle \sum _{n}{\frac {s_{n}(x)}{n! }}t^{n}=\left({\frac {1+t}{1-t}}\right)^{x}(1-t)^{-1}} (Roman 1984, 4.4.3), (Boas & Buck 1958, p.38)
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https://en.wikipedia.org/wiki/Pidduck_polynomials
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In mathematics, the Pincherle derivative T ′ {\displaystyle T'} of a linear operator T: K → K {\displaystyle T:\mathbb {K} \to \mathbb {K} } on the vector space of polynomials in the variable x over a field K {\displaystyle \mathbb {K} } is the commutator of T {\displaystyle T} with the multiplication by x in the algebra of endomorphisms End ( K ) {\displaystyle \operatorname {End} (\mathbb {K} )} . That is, T ′ {\displaystyle T'} is another linear operator T ′: K → K {\displaystyle T':\mathbb {K} \to \mathbb {K} } T ′ := = T x − x T = − ad ( x ) T , {\displaystyle T':==Tx-xT=-\operatorname {ad} (x)T,\,} (for the origin of the ad {\displaystyle \operatorname {ad} } notation, see the article on the adjoint representation) so that T ′ { p ( x ) } = T { x p ( x ) } − x T { p ( x ) } ∀ p ( x ) ∈ K . {\displaystyle T'\{p(x)\}=T\{xp(x)\}-xT\{p(x)\}\qquad \forall p(x)\in \mathbb {K} .} This concept is named after the Italian mathematician Salvatore Pincherle (1853–1936).
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https://en.wikipedia.org/wiki/Pincherle_derivative
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In mathematics, the Pincherle polynomials Pn(x) are polynomials introduced by S. Pincherle (1891) given by the generating function ( 1 − 3 x t + t 3 ) − 1 / 2 = ∑ n = 0 ∞ P n ( x ) t n {\displaystyle \displaystyle (1-3xt+t^{3})^{-1/2}=\sum _{n=0}^{\infty }P_{n}(x)t^{n}} Humbert polynomials are a generalization of Pincherle polynomials
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https://en.wikipedia.org/wiki/Pincherle_polynomials
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In mathematics, the Pinsky phenomenon is a result in Fourier analysis. This phenomenon was discovered by Mark Pinsky of Northwestern University. It involves the spherical inversion of the Fourier transform.
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https://en.wikipedia.org/wiki/Pinsky_phenomenon
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The phenomenon involves a lack of convergence at a point due to a discontinuity at boundary. This lack of convergence in the Pinsky phenomenon happens far away from the boundary of the discontinuity, rather than at the discontinuity itself seen in the Gibbs phenomenon. This non-local phenomenon is caused by a lensing effect.
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https://en.wikipedia.org/wiki/Pinsky_phenomenon
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In mathematics, the Plancherel theorem (sometimes called the Parseval–Plancherel identity) is a result in harmonic analysis, proven by Michel Plancherel in 1910. It states that the integral of a function's squared modulus is equal to the integral of the squared modulus of its frequency spectrum. That is, if f ( x ) {\displaystyle f(x)} is a function on the real line, and f ^ ( ξ ) {\displaystyle {\widehat {f}}(\xi )} is its frequency spectrum, then A more precise formulation is that if a function is in both Lp spaces L 1 ( R ) {\displaystyle L^{1}(\mathbb {R} )} and L 2 ( R ) {\displaystyle L^{2}(\mathbb {R} )} , then its Fourier transform is in L 2 ( R ) {\displaystyle L^{2}(\mathbb {R} )} , and the Fourier transform map is an isometry with respect to the L2 norm. This implies that the Fourier transform map restricted to L 1 ( R ) ∩ L 2 ( R ) {\displaystyle L^{1}(\mathbb {R} )\cap L^{2}(\mathbb {R} )} has a unique extension to a linear isometric map L 2 ( R ) ↦ L 2 ( R ) {\displaystyle L^{2}(\mathbb {R} )\mapsto L^{2}(\mathbb {R} )} , sometimes called the Plancherel transform.
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https://en.wikipedia.org/wiki/Plancherel_theorem
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This isometry is actually a unitary map. In effect, this makes it possible to speak of Fourier transforms of quadratically integrable functions. Plancherel's theorem remains valid as stated on n-dimensional Euclidean space R n {\displaystyle \mathbb {R} ^{n}} .
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https://en.wikipedia.org/wiki/Plancherel_theorem
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The theorem also holds more generally in locally compact abelian groups. There is also a version of the Plancherel theorem which makes sense for non-commutative locally compact groups satisfying certain technical assumptions. This is the subject of non-commutative harmonic analysis.
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https://en.wikipedia.org/wiki/Plancherel_theorem
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The unitarity of the Fourier transform is often called Parseval's theorem in science and engineering fields, based on an earlier (but less general) result that was used to prove the unitarity of the Fourier series. Due to the polarization identity, one can also apply Plancherel's theorem to the L 2 ( R ) {\displaystyle L^{2}(\mathbb {R} )} inner product of two functions. That is, if f ( x ) {\displaystyle f(x)} and g ( x ) {\displaystyle g(x)} are two L 2 ( R ) {\displaystyle L^{2}(\mathbb {R} )} functions, and P {\displaystyle {\mathcal {P}}} denotes the Plancherel transform, then and if f ( x ) {\displaystyle f(x)} and g ( x ) {\displaystyle g(x)} are furthermore L 1 ( R ) {\displaystyle L^{1}(\mathbb {R} )} functions, then and so
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https://en.wikipedia.org/wiki/Plancherel_theorem
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In mathematics, the Plancherel theorem for spherical functions is an important result in the representation theory of semisimple Lie groups, due in its final form to Harish-Chandra. It is a natural generalisation in non-commutative harmonic analysis of the Plancherel formula and Fourier inversion formula in the representation theory of the group of real numbers in classical harmonic analysis and has a similarly close interconnection with the theory of differential equations. It is the special case for zonal spherical functions of the general Plancherel theorem for semisimple Lie groups, also proved by Harish-Chandra.
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https://en.wikipedia.org/wiki/Plancherel_theorem_for_spherical_functions
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The Plancherel theorem gives the eigenfunction expansion of radial functions for the Laplacian operator on the associated symmetric space X; it also gives the direct integral decomposition into irreducible representations of the regular representation on L2(X). In the case of hyperbolic space, these expansions were known from prior results of Mehler, Weyl and Fock. The main reference for almost all this material is the encyclopedic text of Helgason (1984).
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https://en.wikipedia.org/wiki/Plancherel_theorem_for_spherical_functions
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In mathematics, the Plücker map embeds the Grassmannian G r ( k , V ) {\displaystyle \mathbf {Gr} (k,V)} , whose elements are k-dimensional subspaces of an n-dimensional vector space V, either real or complex, in a projective space, thereby realizing it as an algebraic variety. More precisely, the Plücker map embeds G r ( k , V ) {\displaystyle \mathbf {Gr} (k,V)} into the projectivization P ( Λ k V ) {\displaystyle \mathbf {P} (\Lambda ^{k}V)} of the k {\displaystyle k} -th exterior power of V {\displaystyle V} . The image is algebraic, consisting of the intersection of a number of quadrics defined by the Plücker relations (see below). The Plücker embedding was first defined by Julius Plücker in the case k = 2 , n = 4 {\displaystyle k=2,n=4} as a way of describing the lines in three-dimensional space (which, as projective lines in real projective space, correspond to two-dimensional subspaces of a four-dimensional vector space). The image of that embedding is the Klein quadric in RP5. Hermann Grassmann generalized Plücker's embedding to arbitrary k and n. The homogeneous coordinates of the image of the Grassmannian G r ( k , V ) {\displaystyle \mathbf {Gr} (k,V)} under the Plücker embedding, relative to the basis in the exterior space Λ k V {\displaystyle \Lambda ^{k}V} corresponding to the natural basis in V = K n {\displaystyle V=K^{n}} (where K {\displaystyle K} is the base field) are called Plücker coordinates.
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https://en.wikipedia.org/wiki/Plucker_relations
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In mathematics, the Pochhammer contour, introduced by Camille Jordan (1887) and Leo Pochhammer (1890), is a contour in the complex plane with two points removed, used for contour integration. If A and B are loops around the two points, both starting at some fixed point P, then the Pochhammer contour is the commutator ABA−1B−1, where the superscript −1 denotes a path taken in the opposite direction. With the two points taken as 0 and 1, the fixed basepoint P being on the real axis between them, an example is the path that starts at P, encircles the point 1 in the counter-clockwise direction and returns to P, then encircles 0 counter-clockwise and returns to P, after that circling 1 and then 0 clockwise, before coming back to P. The class of the contour is an actual commutator when it is considered in the fundamental group with basepoint P of the complement in the complex plane (or Riemann sphere) of the two points looped. When it comes to taking contour integrals, moving basepoint from P to another choice Q makes no difference to the result, since there will be cancellation of integrals from P to Q and back.
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https://en.wikipedia.org/wiki/Pochhammer_contour
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In mathematics, the Pocklington–Lehmer primality test is a primality test devised by Henry Cabourn Pocklington and Derrick Henry Lehmer. The test uses a partial factorization of N − 1 {\displaystyle N-1} to prove that an integer N {\displaystyle N} is prime. It produces a primality certificate to be found with less effort than the Lucas primality test, which requires the full factorization of N − 1 {\displaystyle N-1} .
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https://en.wikipedia.org/wiki/Pocklington_primality_test
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In mathematics, the Poincaré conjecture is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space. The conjecture states that: Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere. An equivalent form of the conjecture involves a coarser form of equivalence than homeomorphism called homotopy equivalence: if a 3-manifold is homotopy equivalent to the 3-sphere, then it is necessarily homeomorphic to it. Originally conjectured by Henri Poincaré in 1904, the theorem concerns a space that locally looks like ordinary three-dimensional space but is connected, finite in size, and lacks any boundary (a closed 3-manifold).
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https://en.wikipedia.org/wiki/Mathematical_conjecture
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The Poincaré conjecture claims that if such a space has the additional property that each loop in the space can be continuously tightened to a point, then it is necessarily a three-dimensional sphere. An analogous result has been known in higher dimensions for some time. After nearly a century of effort by mathematicians, Grigori Perelman presented a proof of the conjecture in three papers made available in 2002 and 2003 on arXiv.
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https://en.wikipedia.org/wiki/Mathematical_conjecture
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The proof followed on from the program of Richard S. Hamilton to use the Ricci flow to attempt to solve the problem. Hamilton later introduced a modification of the standard Ricci flow, called Ricci flow with surgery to systematically excise singular regions as they develop, in a controlled way, but was unable to prove this method "converged" in three dimensions. Perelman completed this portion of the proof. Several teams of mathematicians have verified that Perelman's proof is correct. The Poincaré conjecture, before being proven, was one of the most important open questions in topology.
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https://en.wikipedia.org/wiki/Mathematical_conjecture
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In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds. It states that if M is an n-dimensional oriented closed manifold (compact and without boundary), then the kth cohomology group of M is isomorphic to the ( n − k {\displaystyle n-k} )th homology group of M, for all integers k H k ( M ) ≅ H n − k ( M ) . {\displaystyle H^{k}(M)\cong H_{n-k}(M).} Poincaré duality holds for any coefficient ring, so long as one has taken an orientation with respect to that coefficient ring; in particular, since every manifold has a unique orientation mod 2, Poincaré duality holds mod 2 without any assumption of orientation.
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https://en.wikipedia.org/wiki/Poincaré_duality_theorem
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In mathematics, the Poincaré inequality is a result in the theory of Sobolev spaces, named after the French mathematician Henri Poincaré. The inequality allows one to obtain bounds on a function using bounds on its derivatives and the geometry of its domain of definition. Such bounds are of great importance in the modern, direct methods of the calculus of variations. A very closely related result is Friedrichs' inequality.
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https://en.wikipedia.org/wiki/Poincaré_inequality
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In mathematics, the Poincaré lemma gives a sufficient condition for a closed differential form to be exact (while an exact form is necessarily closed). Precisely, it states that every closed p-form on an open ball in Rn is exact for p with 1 ≤ p ≤ n. The lemma was introduced by Henri Poincaré in 1886.Especially in calculus, the Poincaré lemma also says that every closed 1-form on a simply connected open subset in R n {\displaystyle \mathbb {R} ^{n}} is exact. In the language of cohomology, the Poincaré lemma says that the k-th de Rham cohomology group of a contractible open subset of a manifold M (e.g., M = R n {\displaystyle M=\mathbb {R} ^{n}} ) vanishes for k ≥ 1 {\displaystyle k\geq 1} . In particular, it implies that the de Rham complex yields a resolution of the constant sheaf R M {\displaystyle \mathbb {R} _{M}} on M. The singular cohomology of a contractible space vanishes in positive degree, but the Poincaré lemma does not follow from this, since the fact that the singular cohomology of a manifold can be computed as the de Rham cohomology of it, that is, the de Rham theorem, relies on the Poincaré lemma. It does, however, mean that it is enough to prove the Poincaré lemma for open balls; the version for contractible manifolds then follows from the topological consideration. The Poincaré lemma is also a special case of the homotopy invariance of de Rham cohomology; in fact, it is common to establish the lemma by showing the homotopy invariance or at least a version of it.
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https://en.wikipedia.org/wiki/Poincare_lemma
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In mathematics, the Poincaré metric, named after Henri Poincaré, is the metric tensor describing a two-dimensional surface of constant negative curvature. It is the natural metric commonly used in a variety of calculations in hyperbolic geometry or Riemann surfaces. There are three equivalent representations commonly used in two-dimensional hyperbolic geometry. One is the Poincaré half-plane model, defining a model of hyperbolic space on the upper half-plane.
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https://en.wikipedia.org/wiki/Constant_negative_curvature
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The Poincaré disk model defines a model for hyperbolic space on the unit disk. The disk and the upper half plane are related by a conformal map, and isometries are given by Möbius transformations. A third representation is on the punctured disk, where relations for q-analogues are sometimes expressed. These various forms are reviewed below.
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https://en.wikipedia.org/wiki/Constant_negative_curvature
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In mathematics, the Poincaré residue is a generalization, to several complex variables and complex manifold theory, of the residue at a pole of complex function theory. It is just one of a number of such possible extensions. Given a hypersurface X ⊂ P n {\displaystyle X\subset \mathbb {P} ^{n}} defined by a degree d {\displaystyle d} polynomial F {\displaystyle F} and a rational n {\displaystyle n} -form ω {\displaystyle \omega } on P n {\displaystyle \mathbb {P} ^{n}} with a pole of order k > 0 {\displaystyle k>0} on X {\displaystyle X} , then we can construct a cohomology class Res ( ω ) ∈ H n − 1 ( X ; C ) {\displaystyle \operatorname {Res} (\omega )\in H^{n-1}(X;\mathbb {C} )} . If n = 1 {\displaystyle n=1} we recover the classical residue construction.
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https://en.wikipedia.org/wiki/Poincare_residue
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In mathematics, the Poincaré separation theorem, also known as the Cauchy interlacing theorem, gives some upper and lower bounds of eigenvalues of a real symmetric matrix B'AB that can be considered as the orthogonal projection of a larger real symmetric matrix A onto a linear subspace spanned by the columns of B. The theorem is named after Henri Poincaré. More specifically, let A be an n × n real symmetric matrix and B an n × r semi-orthogonal matrix such that B'B = Ir. Denote by λ i {\displaystyle \lambda _{i}} , i = 1, 2, ..., n and μ i {\displaystyle \mu _{i}} , i = 1, 2, ..., r the eigenvalues of A and B'AB, respectively (in descending order). We have λ i ≥ μ i ≥ λ n − r + i , {\displaystyle \lambda _{i}\geq \mu _{i}\geq \lambda _{n-r+i},}
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https://en.wikipedia.org/wiki/Poincaré_separation_theorem
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In mathematics, the Poincaré–Bendixson theorem is a statement about the long-term behaviour of orbits of continuous dynamical systems on the plane, cylinder, or two-sphere.
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https://en.wikipedia.org/wiki/Poincaré–Bendixson_theorem
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In mathematics, the Poincaré–Hopf theorem (also known as the Poincaré–Hopf index formula, Poincaré–Hopf index theorem, or Hopf index theorem) is an important theorem that is used in differential topology. It is named after Henri Poincaré and Heinz Hopf. The Poincaré–Hopf theorem is often illustrated by the special case of the hairy ball theorem, which simply states that there is no smooth vector field on an even-dimensional n-sphere having no sources or sinks.
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https://en.wikipedia.org/wiki/Poincaré–Hopf_theorem
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In mathematics, the Poincaré–Lelong equation, studied by Lelong (1964), is the partial differential equation i ∂ ∂ ¯ u = ρ {\displaystyle i\partial {\overline {\partial }}u=\rho } on a Kähler manifold, where ρ is a positive (1,1)-form.
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https://en.wikipedia.org/wiki/Poincaré–Lelong_equation
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In mathematics, the Poincaré–Miranda theorem is a generalization of intermediate value theorem, from a single function in a single dimension, to n functions in n dimensions. It says as follows: Consider n {\displaystyle n} continuous functions of n {\displaystyle n} variables, f 1 , … , f n {\displaystyle f_{1},\ldots ,f_{n}} . Assume that for each variable x i {\displaystyle x_{i}} , the function f i {\displaystyle f_{i}} is nonpositive when x i = − 1 {\displaystyle x_{i}=-1} and nonnegative when x i = 1 {\displaystyle x_{i}=1} . Then there is a point in the n {\displaystyle n} -dimensional cube n {\displaystyle ^{n}} in which all functions are simultaneously equal to 0 {\displaystyle 0} .The theorem is named after Henri Poincaré - who conjectured it in 1883, and Carlo Miranda - who in 1940 showed that it is equivalent to the Brouwer fixed-point theorem. : 545 It is sometimes called the Miranda theorem or the Bolzano-Poincare-Miranda theorem.
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https://en.wikipedia.org/wiki/Poincaré-Miranda_theorem
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In mathematics, the Poisson boundary is a measure space associated to a random walk. It is an object designed to encode the asymptotic behaviour of the random walk, i.e. how trajectories diverge when the number of steps goes to infinity. Despite being called a boundary it is in general a purely measure-theoretical object and not a boundary in the topological sense. However, in the case where the random walk is on a topological space the Poisson boundary can be related to the Martin boundary, which is an analytic construction yielding a genuine topological boundary. Both boundaries are related to harmonic functions on the space via generalisations of the Poisson formula.
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https://en.wikipedia.org/wiki/Poisson_boundary
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In mathematics, the Poisson summation formula is an equation that relates the Fourier series coefficients of the periodic summation of a function to values of the function's continuous Fourier transform. Consequently, the periodic summation of a function is completely defined by discrete samples of the original function's Fourier transform. And conversely, the periodic summation of a function's Fourier transform is completely defined by discrete samples of the original function. The Poisson summation formula was discovered by Siméon Denis Poisson and is sometimes called Poisson resummation.
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https://en.wikipedia.org/wiki/Poisson_summation_formula
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In mathematics, the Pompeiu problem is a conjecture in integral geometry, named for Dimitrie Pompeiu, who posed the problem in 1929, as follows. Suppose f is a nonzero continuous function defined on a Euclidean space, and K is a simply connected Lipschitz domain, so that the integral of f vanishes on every congruent copy of K. Then the domain is a ball. A special case is Schiffer's conjecture.
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https://en.wikipedia.org/wiki/Pompeiu_problem
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In mathematics, the Pontryagin classes, named after Lev Pontryagin, are certain characteristic classes of real vector bundles. The Pontryagin classes lie in cohomology groups with degrees a multiple of four.
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https://en.wikipedia.org/wiki/Pontryagin_number
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In mathematics, the Pontryagin product, introduced by Lev Pontryagin (1939), is a product on the homology of a topological space induced by a product on the topological space. Special cases include the Pontryagin product on the homology of an abelian group, the Pontryagin product on an H-space, and the Pontryagin product on a loop space.
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https://en.wikipedia.org/wiki/Pontryagin_product
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In mathematics, the Porteous formula, or Thom–Porteous formula, or Giambelli–Thom–Porteous formula, is an expression for the fundamental class of a degeneracy locus (or determinantal variety) of a morphism of vector bundles in terms of Chern classes. Giambelli's formula is roughly the special case when the vector bundles are sums of line bundles over projective space. Thom (1957) pointed out that the fundamental class must be a polynomial in the Chern classes and found this polynomial in a few special cases, and Porteous (1971) found the polynomial in general. Kempf & Laksov (1974) proved a more general version, and Fulton (1992) generalized it further.
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https://en.wikipedia.org/wiki/Degeneracy_loci
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In mathematics, the Property P conjecture is a statement about 3-manifolds obtained by Dehn surgery on a knot in the 3-sphere. A knot in the 3-sphere is said to have Property P if every 3-manifold obtained by performing (non-trivial) Dehn surgery on the knot is not simply-connected. The conjecture states that all knots, except the unknot, have Property P. Research on Property P was started by R. H. Bing, who popularized the name and conjecture.
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https://en.wikipedia.org/wiki/Property_P_conjecture
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This conjecture can be thought of as a first step to resolving the Poincaré conjecture, since the Lickorish–Wallace theorem says any closed, orientable 3-manifold results from Dehn surgery on a link. If a knot K ⊂ S 3 {\displaystyle K\subset \mathbb {S} ^{3}} has Property P, then one cannot construct a counterexample to the Poincaré conjecture by surgery along K {\displaystyle K} . A proof was announced in 2004, as the combined result of efforts of mathematicians working in several different fields.
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https://en.wikipedia.org/wiki/Property_P_conjecture
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In mathematics, the Prouhet–Tarry–Escott problem asks for two disjoint multisets A and B of n integers each, whose first k power sum symmetric polynomials are all equal. That is, the two multisets should satisfy the equations ∑ a ∈ A a i = ∑ b ∈ B b i {\displaystyle \sum _{a\in A}a^{i}=\sum _{b\in B}b^{i}} for each integer i from 1 to a given k. It has been shown that n must be strictly greater than k. Solutions with k = n − 1 {\displaystyle k=n-1} are called ideal solutions. Ideal solutions are known for 3 ≤ n ≤ 10 {\displaystyle 3\leq n\leq 10} and for n = 12 {\displaystyle n=12} . No ideal solution is known for n = 11 {\displaystyle n=11} or for n ≥ 13 {\displaystyle n\geq 13} .This problem was named after Eugène Prouhet, who studied it in the early 1850s, and Gaston Tarry and Edward B. Escott, who studied it in the early 1910s. The problem originates from letters of Christian Goldbach and Leonhard Euler (1750/1751).
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https://en.wikipedia.org/wiki/Prouhet–Tarry–Escott_problem
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In mathematics, the Prouhet–Thue–Morse constant, named for Eugène Prouhet, Axel Thue, and Marston Morse, is the number—denoted by τ—whose binary expansion 0.01101001100101101001011001101001... is given by the Prouhet–Thue–Morse sequence. That is, τ = ∑ n = 0 ∞ t n 2 n + 1 = 0.412454033640 … {\displaystyle \tau =\sum _{n=0}^{\infty }{\frac {t_{n}}{2^{n+1}}}=0.412454033640\ldots } where tn is the nth element of the Prouhet–Thue–Morse sequence.
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https://en.wikipedia.org/wiki/Prouhet–Thue–Morse_constant
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In mathematics, the Prym variety construction (named for Friedrich Prym) is a method in algebraic geometry of making an abelian variety from a morphism of algebraic curves. In its original form, it was applied to an unramified double covering of a Riemann surface, and was used by F. Schottky and H. W. E. Jung in relation with the Schottky problem, as it is now called, of characterising Jacobian varieties among abelian varieties. It is said to have appeared first in the late work of Riemann, and was extensively studied by Wirtinger in 1895, including degenerate cases. Given a non-constant morphism φ: C1 → C2of algebraic curves, write Ji for the Jacobian variety of Ci.
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https://en.wikipedia.org/wiki/Prym_variety
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Then from φ construct the corresponding morphism ψ: J1 → J2,which can be defined on a divisor class D of degree zero by applying φ to each point of the divisor. This is a well-defined morphism, often called the norm homomorphism. Then the Prym variety of φ is the kernel of ψ. To qualify that somewhat, to get an abelian variety, the connected component of the identity of the reduced scheme underlying the kernel may be intended.
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https://en.wikipedia.org/wiki/Prym_variety
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Or in other words take the largest abelian subvariety of J1 on which ψ is trivial. The theory of Prym varieties was dormant for a long time, until revived by David Mumford around 1970. It now plays a substantial role in some contemporary theories, for example of the Kadomtsev–Petviashvili equation.
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https://en.wikipedia.org/wiki/Prym_variety
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One advantage of the method is that it allows one to apply the theory of curves to the study of a wider class of abelian varieties than Jacobians. For example, principally polarized abelian varieties (p.p.a.v. 's) of dimension > 3 are not generally Jacobians, but all p.p.a.v.
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https://en.wikipedia.org/wiki/Prym_variety
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's of dimension 5 or less are Prym varieties. It is for this reason that p.p.a.v. 's are fairly well understood up to dimension 5.
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https://en.wikipedia.org/wiki/Prym_variety
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In mathematics, the Prékopa–Leindler inequality is an integral inequality closely related to the reverse Young's inequality, the Brunn–Minkowski inequality and a number of other important and classical inequalities in analysis. The result is named after the Hungarian mathematicians András Prékopa and László Leindler.
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https://en.wikipedia.org/wiki/Prékopa–Leindler_inequality
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In mathematics, the Prüfer manifold or Prüfer surface is a 2-dimensional Hausdorff real analytic manifold that is not paracompact. It was introduced by Radó (1925) and named after Heinz Prüfer.
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https://en.wikipedia.org/wiki/Prüfer_surface
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In mathematics, the Puppe sequence is a construction of homotopy theory, so named after Dieter Puppe. It comes in two forms: a long exact sequence, built from the mapping fibre (a fibration), and a long coexact sequence, built from the mapping cone (which is a cofibration). Intuitively, the Puppe sequence allows us to think of homology theory as a functor that takes spaces to long-exact sequences of groups. It is also useful as a tool to build long exact sequences of relative homotopy groups.
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https://en.wikipedia.org/wiki/Fiber_sequence
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In mathematics, the Pythagoras number or reduced height of a field describes the structure of the set of squares in the field. The Pythagoras number p(K) of a field K is the smallest positive integer p such that every sum of squares in K is a sum of p squares. A Pythagorean field is a field with Pythagoras number 1: that is, every sum of squares is already a square.
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https://en.wikipedia.org/wiki/Pythagoras_number
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In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other two sides. The theorem can be written as an equation relating the lengths of the sides a, b and the hypotenuse c, sometimes called the Pythagorean equation: a 2 + b 2 = c 2 . {\displaystyle a^{2}+b^{2}=c^{2}.}
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https://en.wikipedia.org/wiki/The_Pythagorean_theorem
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The theorem is named for the Greek philosopher Pythagoras, born around 570 BC. The theorem has been proved numerous times by many different methods – possibly the most for any mathematical theorem. The proofs are diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years.
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https://en.wikipedia.org/wiki/The_Pythagorean_theorem
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When Euclidean space is represented by a Cartesian coordinate system in analytic geometry, Euclidean distance satisfies the Pythagorean relation: the squared distance between two points equals the sum of squares of the difference in each coordinate between the points. The theorem can be generalized in various ways: to higher-dimensional spaces, to spaces that are not Euclidean, to objects that are not right triangles, and to objects that are not triangles at all but n-dimensional solids. The Pythagorean theorem has attracted interest outside mathematics as a symbol of mathematical abstruseness, mystique, or intellectual power; popular references in literature, plays, musicals, songs, stamps, and cartoons abound.
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https://en.wikipedia.org/wiki/The_Pythagorean_theorem
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In mathematics, the Quillen determinant line bundle is a line bundle over the space of Cauchy–Riemann operators of a vector bundle over a Riemann surface, introduced by Quillen (1985). Quillen proved the existence of the Quillen metric on the determinant line bundle, a Hermitian metric defined using the analytic torsion of a family of differential operators.
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https://en.wikipedia.org/wiki/Quillen_determinant_line_bundle
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In mathematics, the Quillen–Lichtenbaum conjecture is a conjecture relating étale cohomology to algebraic K-theory introduced by Quillen (1975, p. 175), who was inspired by earlier conjectures of Lichtenbaum (1973). Kahn (1997) and Rognes & Weibel (2000) proved the Quillen–Lichtenbaum conjecture at the prime 2 for some number fields. Voevodsky, using some important results of Markus Rost, has proved the Bloch–Kato conjecture, which implies the Quillen–Lichtenbaum conjecture for all primes.
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https://en.wikipedia.org/wiki/Lichtenbaum_conjecture
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In mathematics, the RSA numbers are a set of large semiprimes (numbers with exactly two prime factors) that were part of the RSA Factoring Challenge. The challenge was to find the prime factors of each number. It was created by RSA Laboratories in March 1991 to encourage research into computational number theory and the practical difficulty of factoring large integers. The challenge was ended in 2007.RSA Laboratories (which is an acronym of the creators of the technique; Rivest, Shamir and Adleman) published a number of semiprimes with 100 to 617 decimal digits.
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https://en.wikipedia.org/wiki/RSA_number
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Cash prizes of varying size, up to US$200,000 (and prizes up to $20,000 awarded), were offered for factorization of some of them. The smallest RSA number was factored in a few days. Most of the numbers have still not been factored and many of them are expected to remain unfactored for many years to come.
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https://en.wikipedia.org/wiki/RSA_number
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As of February 2020, the smallest 23 of the 54 listed numbers have been factored. While the RSA challenge officially ended in 2007, people are still attempting to find the factorizations. According to RSA Laboratories, "Now that the industry has a considerably more advanced understanding of the cryptanalytic strength of common symmetric-key and public-key algorithms, these challenges are no longer active."
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https://en.wikipedia.org/wiki/RSA_number
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Some of the smaller prizes had been awarded at the time. The remaining prizes were retracted.
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https://en.wikipedia.org/wiki/RSA_number
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The first RSA numbers generated, from RSA-100 to RSA-500, were labeled according to their number of decimal digits. Later, beginning with RSA-576, binary digits are counted instead. An exception to this is RSA-617, which was created before the change in the numbering scheme. The numbers are listed in increasing order below.
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https://en.wikipedia.org/wiki/RSA_number
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In mathematics, the Rabinowitsch trick, introduced by J.L. Rabinowitsch (1929), is a short way of proving the general case of the Hilbert Nullstellensatz from an easier special case (the so-called weak Nullstellensatz), by introducing an extra variable. The Rabinowitsch trick goes as follows.
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https://en.wikipedia.org/wiki/Rabinowitsch_trick
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Let K be an algebraically closed field. Suppose the polynomial f in K vanishes whenever all polynomials f1,....,fm vanish. Then the polynomials f1,....,fm, 1 − x0f have no common zeros (where we have introduced a new variable x0), so by the weak Nullstellensatz for K they generate the unit ideal of K. Spelt out, this means there are polynomials g 0 , g 1 , … , g m ∈ K {\displaystyle g_{0},g_{1},\dots ,g_{m}\in K} such that 1 = g 0 ( x 0 , x 1 , … , x n ) ( 1 − x 0 f ( x 1 , … , x n ) ) + ∑ i = 1 m g i ( x 0 , x 1 , … , x n ) f i ( x 1 , … , x n ) {\displaystyle 1=g_{0}(x_{0},x_{1},\dots ,x_{n})(1-x_{0}f(x_{1},\dots ,x_{n}))+\sum _{i=1}^{m}g_{i}(x_{0},x_{1},\dots ,x_{n})f_{i}(x_{1},\dots ,x_{n})} as an equality of elements of the polynomial ring K {\displaystyle K} .
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https://en.wikipedia.org/wiki/Rabinowitsch_trick
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In mathematics, the Radon transform is the integral transform which takes a function f defined on the plane to a function Rf defined on the (two-dimensional) space of lines in the plane, whose value at a particular line is equal to the line integral of the function over that line. The transform was introduced in 1917 by Johann Radon, who also provided a formula for the inverse transform. Radon further included formulas for the transform in three dimensions, in which the integral is taken over planes (integrating over lines is known as the X-ray transform).
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https://en.wikipedia.org/wiki/Filtered_backprojection
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It was later generalized to higher-dimensional Euclidean spaces and more broadly in the context of integral geometry. The complex analogue of the Radon transform is known as the Penrose transform. The Radon transform is widely applicable to tomography, the creation of an image from the projection data associated with cross-sectional scans of an object.
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https://en.wikipedia.org/wiki/Filtered_backprojection
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In mathematics, the Radon–Nikodym theorem is a result in measure theory that expresses the relationship between two measures defined on the same measurable space. A measure is a set function that assigns a consistent magnitude to the measurable subsets of a measurable space. Examples of a measure include area and volume, where the subsets are sets of points; or the probability of an event, which is a subset of possible outcomes within a wider probability space. One way to derive a new measure from one already given is to assign a density to each point of the space, then integrate over the measurable subset of interest.
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https://en.wikipedia.org/wiki/Density_function_(measure_theory)
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This can be expressed as ν ( A ) = ∫ A f d μ , {\displaystyle \nu (A)=\int _{A}f\,d\mu ,} where ν is the new measure being defined for any measurable subset A and the function f is the density at a given point. The integral is with respect to an existing measure μ, which may often be the canonical Lebesgue measure on the real line R or the n-dimensional Euclidean space Rn (corresponding to our standard notions of length, area and volume). For example, if f represented mass density and μ was the Lebesgue measure in three-dimensional space R3, then ν(A) would equal the total mass in a spatial region A. The Radon–Nikodym theorem essentially states that, under certain conditions, any measure ν can be expressed in this way with respect to another measure μ on the same space.
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https://en.wikipedia.org/wiki/Density_function_(measure_theory)
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The function f is then called the Radon–Nikodym derivative and is denoted by d ν d μ {\displaystyle {\tfrac {d\nu }{d\mu }}} . An important application is in probability theory, leading to the probability density function of a random variable. The theorem is named after Johann Radon, who proved the theorem for the special case where the underlying space is Rn in 1913, and for Otto Nikodym who proved the general case in 1930. In 1936 Hans Freudenthal generalized the Radon–Nikodym theorem by proving the Freudenthal spectral theorem, a result in Riesz space theory; this contains the Radon–Nikodym theorem as a special case.A Banach space Y is said to have the Radon–Nikodym property if the generalization of the Radon–Nikodym theorem also holds, mutatis mutandis, for functions with values in Y. All Hilbert spaces have the Radon–Nikodym property.
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https://en.wikipedia.org/wiki/Density_function_(measure_theory)
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In mathematics, the Radó–Kneser–Choquet theorem, named after Tibor Radó, Hellmuth Kneser and Gustave Choquet, states that the Poisson integral of a homeomorphism of the unit circle is a harmonic diffeomorphism of the open unit disk. The result was stated as a problem by Radó and solved shortly afterwards by Kneser in 1926. Choquet, unaware of the work of Radó and Kneser, rediscovered the result with a different proof in 1945. Choquet also generalized the result to the Poisson integral of a homeomorphism from the unit circle to a simple Jordan curve bounding a convex region.
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https://en.wikipedia.org/wiki/Radó–Kneser–Choquet_theorem
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In mathematics, the Rainville polynomials pn(z) are polynomials introduced by Rainville (1945) given by the generating function e w I 0 ( z w ) = ∑ n p n ( z ) w n {\displaystyle \displaystyle e^{w}I_{0}(zw)=\sum _{n}p_{n}(z)w^{n}} Boas & Buck (1958, p.46).
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https://en.wikipedia.org/wiki/Rainville_polynomials
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In mathematics, the Ramanujan conjecture, due to Srinivasa Ramanujan (1916, p. 176), states that Ramanujan's tau function given by the Fourier coefficients τ(n) of the cusp form Δ(z) of weight 12 Δ ( z ) = ∑ n > 0 τ ( n ) q n = q ∏ n > 0 ( 1 − q n ) 24 = q − 24 q 2 + 252 q 3 − 1472 q 4 + 4830 q 5 − ⋯ , {\displaystyle \Delta (z)=\sum _{n>0}\tau (n)q^{n}=q\prod _{n>0}\left(1-q^{n}\right)^{24}=q-24q^{2}+252q^{3}-1472q^{4}+4830q^{5}-\cdots ,} where q = e 2 π i z {\displaystyle q=e^{2\pi iz}} , satisfies | τ ( p ) | ≤ 2 p 11 / 2 , {\displaystyle |\tau (p)|\leq 2p^{11/2},} when p is a prime number. The generalized Ramanujan conjecture or Ramanujan–Petersson conjecture, introduced by Petersson (1930), is a generalization to other modular forms or automorphic forms.
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https://en.wikipedia.org/wiki/Ramanujan_conjecture
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In mathematics, the Ramanujan–Soldner constant (also called the Soldner constant) is a mathematical constant defined as the unique positive zero of the logarithmic integral function. It is named after Srinivasa Ramanujan and Johann Georg von Soldner. Its value is approximately μ ≈ 1.45136923488338105028396848589202744949303228… (sequence A070769 in the OEIS) Since the logarithmic integral is defined by l i ( x ) = ∫ 0 x d t ln t , {\displaystyle \mathrm {li} (x)=\int _{0}^{x}{\frac {dt}{\ln t}},} then using l i ( μ ) = 0 , {\displaystyle \mathrm {li} (\mu )=0,} we have l i ( x ) = l i ( x ) − l i ( μ ) = ∫ 0 x d t ln t − ∫ 0 μ d t ln t = ∫ μ x d t ln t , {\displaystyle \mathrm {li} (x)\;=\;\mathrm {li} (x)-\mathrm {li} (\mu )=\int _{0}^{x}{\frac {dt}{\ln t}}-\int _{0}^{\mu }{\frac {dt}{\ln t}}=\int _{\mu }^{x}{\frac {dt}{\ln t}},} thus easing calculation for numbers greater than μ. Also, since the exponential integral function satisfies the equation l i ( x ) = E i ( ln x ) , {\displaystyle \mathrm {li} (x)\;=\;\mathrm {Ei} (\ln {x}),} the only positive zero of the exponential integral occurs at the natural logarithm of the Ramanujan–Soldner constant, whose value is approximately ln(μ) ≈ 0.372507410781366634461991866… (sequence A091723 in the OEIS)
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https://en.wikipedia.org/wiki/Ramanujan–Soldner_constant
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In mathematics, the Ran space (or Ran's space) of a topological space X is a topological space Ran ( X ) {\displaystyle \operatorname {Ran} (X)} whose underlying set is the set of all nonempty finite subsets of X: for a metric space X the topology is induced by the Hausdorff distance. The notion is named after Ziv Ran.
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https://en.wikipedia.org/wiki/Ran_space
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In mathematics, the Rankin–Cohen bracket of two modular forms is another modular form, generalizing the product of two modular forms. Rankin (1956, 1957) gave some general conditions for polynomials in derivatives of modular forms to be modular forms, and Cohen (1975) found the explicit examples of such polynomials that give Rankin–Cohen brackets. They were named by Zagier (1994), who introduced Rankin–Cohen algebras as an abstract setting for Rankin–Cohen brackets.
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https://en.wikipedia.org/wiki/Rankin–Cohen_bracket
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In mathematics, the Rauzy fractal is a fractal set associated with the Tribonacci substitution s ( 1 ) = 12 , s ( 2 ) = 13 , s ( 3 ) = 1 . {\displaystyle s(1)=12,\ s(2)=13,\ s(3)=1\,.} It was studied in 1981 by Gérard Rauzy, with the idea of generalizing the dynamic properties of the Fibonacci morphism. That fractal set can be generalized to other maps over a 3-letter alphabet, generating other fractal sets with interesting properties, such as periodic tiling of the plane and self-similarity in three homothetic parts.
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https://en.wikipedia.org/wiki/Rauzy_fractal
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In mathematics, the Ravenel conjectures are a set of mathematical conjectures in the field of stable homotopy theory posed by Douglas Ravenel at the end of a paper published in 1984. It was earlier circulated in preprint. The problems involved have largely been resolved, with all but the "telescope conjecture" being proved in later papers by others.
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https://en.wikipedia.org/wiki/Ravenel's_conjectures
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The telescope conjecture is now generally believed not to be true, though there are some conflicting claims concerning it in the published literature, and is taken to be an open problem. Ravenel's conjectures exerted influence on the field through the founding of the approach of chromatic homotopy theory. The first of the seven conjectures, then the nilpotence conjecture, was proved in 1988 and is now known as the nilpotence theorem.
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https://en.wikipedia.org/wiki/Ravenel's_conjectures
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The telescope conjecture, which was #4 on the original list, remains of substantial interest because of its connection with the convergence of an Adams–Novikov spectral sequence. While opinion has been generally against the truth of the original statement, investigations of associated phenomena (for a triangulated category in general) have become a research area in its own right.On June 6, 2023, Robert Burklund, Jeremy Hahn, Ishan Levy, and Tomer Schlank announced a disproof of the telescope conjecture. Their preprint is forthcoming.
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https://en.wikipedia.org/wiki/Ravenel's_conjectures
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In mathematics, the Rayleigh quotient () for a given complex Hermitian matrix M {\displaystyle M} and nonzero vector x {\displaystyle x} is defined as:For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the conjugate transpose x ∗ {\displaystyle x^{*}} to the usual transpose x ′ {\displaystyle x'} . Note that R ( M , c x ) = R ( M , x ) {\displaystyle R(M,cx)=R(M,x)} for any non-zero scalar c {\displaystyle c} . Recall that a Hermitian (or real symmetric) matrix is diagonalizable with only real eigenvalues.
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https://en.wikipedia.org/wiki/Rayleigh_quotient
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It can be shown that, for a given matrix, the Rayleigh quotient reaches its minimum value λ min {\displaystyle \lambda _{\min }} (the smallest eigenvalue of M {\displaystyle M} ) when x {\displaystyle x} is v min {\displaystyle v_{\min }} (the corresponding eigenvector). Similarly, R ( M , x ) ≤ λ max {\displaystyle R(M,x)\leq \lambda _{\max }} and R ( M , v max ) = λ max {\displaystyle R(M,v_{\max })=\lambda _{\max }} . The Rayleigh quotient is used in the min-max theorem to get exact values of all eigenvalues.
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https://en.wikipedia.org/wiki/Rayleigh_quotient
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It is also used in eigenvalue algorithms (such as Rayleigh quotient iteration) to obtain an eigenvalue approximation from an eigenvector approximation. The range of the Rayleigh quotient (for any matrix, not necessarily Hermitian) is called a numerical range and contains its spectrum. When the matrix is Hermitian, the numerical radius is equal to the spectral norm.
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https://en.wikipedia.org/wiki/Rayleigh_quotient
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Still in functional analysis, λ max {\displaystyle \lambda _{\max }} is known as the spectral radius. In the context of C ⋆ {\displaystyle C^{\star }} -algebras or algebraic quantum mechanics, the function that to M {\displaystyle M} associates the Rayleigh–Ritz quotient R ( M , x ) {\displaystyle R(M,x)} for a fixed x {\displaystyle x} and M {\displaystyle M} varying through the algebra would be referred to as vector state of the algebra. In quantum mechanics, the Rayleigh quotient gives the expectation value of the observable corresponding to the operator M {\displaystyle M} for a system whose state is given by x {\displaystyle x} . If we fix the complex matrix M {\displaystyle M} , then the resulting Rayleigh quotient map (considered as a function of x {\displaystyle x} ) completely determines M {\displaystyle M} via the polarization identity; indeed, this remains true even if we allow M {\displaystyle M} to be non-Hermitian. (However, if we restrict the field of scalars to the real numbers, then the Rayleigh quotient only determines the symmetric part of M {\displaystyle M} .)
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https://en.wikipedia.org/wiki/Rayleigh_quotient
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In mathematics, the Rayleigh theorem for eigenvalues pertains to the behavior of the solutions of an eigenvalue equation as the number of basis functions employed in its resolution increases. Rayleigh, Lord Rayleigh, and 3rd Baron Rayleigh are the titles of John William Strutt, after the death of his father, the 2nd Baron Rayleigh. Lord Rayleigh made contributions not just to both theoretical and experimental physics, but also to applied mathematics. The Rayleigh theorem for eigenvalues, as discussed below, enables the energy minimization that is required in many self-consistent calculations of electronic and related properties of materials, from atoms, molecules, and nanostructures to semiconductors, insulators, and metals.
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https://en.wikipedia.org/wiki/Rayleigh_theorem_for_eigenvalues
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Except for metals, most of these other materials have an energy or a band gap, i.e., the difference between the lowest, unoccupied energy and the highest, occupied energy. For crystals, the energy spectrum is in bands and there is a band gap, if any, as opposed to energy gap. Given the diverse contributions of Lord Rayleigh, his name is associated with other theorems, including Parseval's theorem. For this reason, keeping the full name of "Rayleigh Theorem for Eigenvalues" avoids confusions.
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https://en.wikipedia.org/wiki/Rayleigh_theorem_for_eigenvalues
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In mathematics, the Redheffer star product is a binary operation on linear operators that arises in connection to solving coupled systems of linear equations. It was introduced by Raymond Redheffer in 1959, and has subsequently been widely adopted in computational methods for scattering matrices. Given two scattering matrices from different linear scatterers, the Redheffer star product yields the combined scattering matrix produced when some or all of the output channels of one scatterer are connected to inputs of another scatterer.
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https://en.wikipedia.org/wiki/Redheffer_star_product
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In mathematics, the Reeb foliation is a particular foliation of the 3-sphere, introduced by the French mathematician Georges Reeb (1920–1993). It is based on dividing the sphere into two solid tori, along a 2-torus: see Clifford torus. Each of the solid tori is then foliated internally, in codimension 1, and the dividing torus surface forms one more leaf. By Novikov's compact leaf theorem, every smooth foliation of the 3-sphere includes a compact torus leaf, bounding a solid torus foliated in the same way.
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https://en.wikipedia.org/wiki/Reeb_foliation
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In mathematics, the Reeb vector field, named after the French mathematician Georges Reeb, is a notion that appears in various domains of contact geometry including: in a contact manifold, given a contact 1-form α {\displaystyle \alpha } , the Reeb vector field satisfies R ∈ k e r d α , α ( R ) = 1 {\displaystyle R\in \mathrm {ker} \ d\alpha ,\ \alpha (R)=1} , in particular, in the context of Sasakian manifold#The Reeb vector field.
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https://en.wikipedia.org/wiki/Reeb_vector_field
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In mathematics, the Rees matrix semigroups are a special class of semigroups introduced by David Rees in 1940. They are of fundamental importance in semigroup theory because they are used to classify certain classes of simple semigroups.
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https://en.wikipedia.org/wiki/Rees_matrix_semigroup
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In mathematics, the Regiomontanus's angle maximization problem, is a famous optimization problem posed by the 15th-century German mathematician Johannes Müller (also known as Regiomontanus). The problem is as follows: A painting hangs from a wall. Given the heights of the top and bottom of the painting above the viewer's eye level, how far from the wall should the viewer stand in order to maximize the angle subtended by the painting and whose vertex is at the viewer's eye?If the viewer stands too close to the wall or too far from the wall, the angle is small; somewhere in between it is as large as possible. The same approach applies to finding the optimal place from which to kick a ball in rugby. For that matter, it is not necessary that the alignment of the picture be at right angles: we might be looking at a window of the Leaning Tower of Pisa or a realtor showing off the advantages of a sky-light in a sloping attic roof.
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https://en.wikipedia.org/wiki/Regiomontanus'_angle_maximization_problem
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In mathematics, the Rellich–Kondrachov theorem is a compact embedding theorem concerning Sobolev spaces. It is named after the Austrian-German mathematician Franz Rellich and the Russian mathematician Vladimir Iosifovich Kondrashov. Rellich proved the L2 theorem and Kondrashov the Lp theorem.
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https://en.wikipedia.org/wiki/Rellich–Kondrachov_theorem
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In mathematics, the Remez inequality, discovered by the Soviet mathematician Evgeny Yakovlevich Remez (Remez 1936), gives a bound on the sup norms of certain polynomials, the bound being attained by the Chebyshev polynomials.
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https://en.wikipedia.org/wiki/Remez_inequality
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In mathematics, the Riemann Xi function is a variant of the Riemann zeta function, and is defined so as to have a particularly simple functional equation. The function is named in honour of Bernhard Riemann.
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https://en.wikipedia.org/wiki/Riemann_Ξ_function
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In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2. Many consider it to be the most important unsolved problem in pure mathematics. It is of great interest in number theory because it implies results about the distribution of prime numbers. It was proposed by Bernhard Riemann (1859), after whom it is named.
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https://en.wikipedia.org/wiki/Critical_line_(mathematics)
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The Riemann hypothesis and some of its generalizations, along with Goldbach's conjecture and the twin prime conjecture, make up Hilbert's eighth problem in David Hilbert's list of twenty-three unsolved problems; it is also one of the Clay Mathematics Institute's Millennium Prize Problems, which offers a million dollars to anyone who solves any of them. The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields. The Riemann zeta function ζ(s) is a function whose argument s may be any complex number other than 1, and whose values are also complex.
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https://en.wikipedia.org/wiki/Critical_line_(mathematics)
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It has zeros at the negative even integers; that is, ζ(s) = 0 when s is one of −2, −4, −6, .... These are called its trivial zeros. The zeta function is also zero for other values of s, which are called nontrivial zeros. The Riemann hypothesis is concerned with the locations of these nontrivial zeros, and states that: The real part of every nontrivial zero of the Riemann zeta function is 1/2. Thus, if the hypothesis is correct, all the nontrivial zeros lie on the critical line consisting of the complex numbers 1/2 + i t, where t is a real number and i is the imaginary unit.
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https://en.wikipedia.org/wiki/Critical_line_(mathematics)
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In mathematics, the Riemann hypothesis, proposed by Bernhard Riemann (1859), is a conjecture that the non-trivial zeros of the Riemann zeta function all have real part 1/2. The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields. The Riemann hypothesis implies results about the distribution of prime numbers. Along with suitable generalizations, some mathematicians consider it the most important unresolved problem in pure mathematics. The Riemann hypothesis, along with the Goldbach conjecture, is part of Hilbert's eighth problem in David Hilbert's list of 23 unsolved problems; it is also one of the Clay Mathematics Institute Millennium Prize Problems.
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https://en.wikipedia.org/wiki/Conjecture
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In mathematics, the Riemann series theorem, also called the Riemann rearrangement theorem, named after 19th-century German mathematician Bernhard Riemann, says that if an infinite series of real numbers is conditionally convergent, then its terms can be arranged in a permutation so that the new series converges to an arbitrary real number, or diverges. This implies that a series of real numbers is absolutely convergent if and only if it is unconditionally convergent. As an example, the series 1 − 1 + 1/2 − 1/2 + 1/3 − 1/3 + ⋯ converges to 0 (for a sufficiently large number of terms, the partial sum gets arbitrarily near to 0); but replacing all terms with their absolute values gives 1 + 1 + 1/2 + 1/2 + 1/3 + 1/3 + ⋯, which sums to infinity.
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https://en.wikipedia.org/wiki/Riemann_rearrangement_theorem
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Thus the original series is conditionally convergent, and can be rearranged (by taking the first two positive terms followed by the first negative term, followed by the next two positive terms and then the next negative term, etc.) to give a series that converges to a different sum: 1 + 1/2 − 1 + 1/3 + 1/4 − 1/2 + ⋯ = ln 2. More generally, using this procedure with p positives followed by q negatives gives the sum ln(p/q). Other rearrangements give other finite sums or do not converge to any sum.
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https://en.wikipedia.org/wiki/Riemann_rearrangement_theorem
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In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers plus a value ∞ {\displaystyle \infty } for infinity. With the Riemann model, the point ∞ {\displaystyle \infty } is near to very large numbers, just as the point 0 {\displaystyle 0} is near to very small numbers. The extended complex numbers are useful in complex analysis because they allow for division by zero in some circumstances, in a way that makes expressions such as 1 / 0 = ∞ {\displaystyle 1/0=\infty } well-behaved.
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https://en.wikipedia.org/wiki/Extended_complex_numbers
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For example, any rational function on the complex plane can be extended to a holomorphic function on the Riemann sphere, with the poles of the rational function mapping to infinity. More generally, any meromorphic function can be thought of as a holomorphic function whose codomain is the Riemann sphere. In geometry, the Riemann sphere is the prototypical example of a Riemann surface, and is one of the simplest complex manifolds.
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https://en.wikipedia.org/wiki/Extended_complex_numbers
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In projective geometry, the sphere can be thought of as the complex projective line P 1 ( C ) {\displaystyle \mathbf {P} ^{1}(\mathbf {C} )} , the projective space of all complex lines in C 2 {\displaystyle \mathbf {C} ^{2}} . As with any compact Riemann surface, the sphere may also be viewed as a projective algebraic curve, making it a fundamental example in algebraic geometry. It also finds utility in other disciplines that depend on analysis and geometry, such as the Bloch sphere of quantum mechanics and in other branches of physics. The extended complex plane is also called the closed complex plane.
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https://en.wikipedia.org/wiki/Extended_complex_numbers
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