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In mathematics, the Mohr–Mascheroni theorem states that any geometric construction that can be performed by a compass and straightedge can be performed by a compass alone. It must be understood that "any geometric construction" refers to figures that contain no straight lines, as it is clearly impossible to draw a stra... | https://en.wikipedia.org/wiki/Mohr–Mascheroni_theorem |
In mathematics, the Moore determinant is a determinant defined for Hermitian matrices over a quaternion algebra, introduced by Moore (1922). | https://en.wikipedia.org/wiki/Moore_determinant_of_a_Hermitian_matrix |
In mathematics, the Moore plane, also sometimes called Niemytzki plane (or Nemytskii plane, Nemytskii's tangent disk topology), is a topological space. It is a completely regular Hausdorff space (also called Tychonoff space) that is not normal. It is named after Robert Lee Moore and Viktor Vladimirovich Nemytskii. | https://en.wikipedia.org/wiki/Moore_plane |
In mathematics, the Mordell–Weil theorem states that for an abelian variety A {\displaystyle A} over a number field K {\displaystyle K} , the group A ( K ) {\displaystyle A(K)} of K-rational points of A {\displaystyle A} is a finitely-generated abelian group, called the Mordell–Weil group. The case with A {\displaystyl... | https://en.wikipedia.org/wiki/Mordell–Weil_theorem |
In mathematics, the Morlet wavelet (or Gabor wavelet) is a wavelet composed of a complex exponential (carrier) multiplied by a Gaussian window (envelope). This wavelet is closely related to human perception, both hearing and vision. | https://en.wikipedia.org/wiki/Morlet_wavelet |
In mathematics, the Morrey–Campanato spaces (named after Charles B. Morrey, Jr. and Sergio Campanato) L λ , p ( Ω ) {\displaystyle L^{\lambda ,p}(\Omega )} are Banach spaces which extend the notion of functions of bounded mean oscillation, describing situations where the oscillation of the function in a ball is proport... | https://en.wikipedia.org/wiki/Morrey–Campanato_space |
The seminorm of the Morrey spaces is given by ( λ , p ) p = sup 0 < r < diam ( Ω ) , x 0 ∈ Ω 1 r λ ∫ B r ( x 0 ) ∩ Ω | u ( y ) | p d y . {\displaystyle {\bigl (}_{\lambda ,p}{\bigr )}^{p}=\sup _{0 n {\displaystyle \lambda >n} , the space contains only the 0 function. Note that this is a norm for p ≥ 1 {\displaystyle... | https://en.wikipedia.org/wiki/Morrey–Campanato_space |
The seminorm of the Campanato space is given by ( λ , p ) p = sup 0 < r < diam ( Ω ) , x 0 ∈ Ω 1 r λ ∫ B r ( x 0 ) ∩ Ω | u ( y ) − u r , x 0 | p d y {\displaystyle {\bigl (}_{\lambda ,p}{\bigr )}^{p}=\sup _{0 A r n {\displaystyle |\Omega \cap B_{r}(x_{0})|>Ar^{n}} for every x 0 ∈ Ω {\displaystyle x_{0}\in \Omega } a... | https://en.wikipedia.org/wiki/Morrey–Campanato_space |
In mathematics, the Morse–Palais lemma is a result in the calculus of variations and theory of Hilbert spaces. Roughly speaking, it states that a smooth enough function near a critical point can be expressed as a quadratic form after a suitable change of coordinates. The Morse–Palais lemma was originally proved in the ... | https://en.wikipedia.org/wiki/Morse–Palais_lemma |
In mathematics, the Mostow–Palais theorem is an equivariant version of the Whitney embedding theorem. It states that if a manifold is acted on by a compact Lie group with finitely many orbit types, then it can be embedded into some finite-dimensional orthogonal representation. It was introduced by Mostow (1957) and Pal... | https://en.wikipedia.org/wiki/Mostow–Palais_theorem |
In mathematics, the Moyal product (after José Enrique Moyal; also called the star product or Weyl–Groenewold product, after Hermann Weyl and Hilbrand J. Groenewold) is an example of a phase-space star product. It is an associative, non-commutative product, ★, on the functions on ℝ2n, equipped with its Poisson bracket (... | https://en.wikipedia.org/wiki/Star_product_(quantization) |
In mathematics, the Muller–Schupp theorem states that a finitely generated group G has context-free word problem if and only if G is virtually free. The theorem was proved by David Muller and Paul Schupp in 1983. | https://en.wikipedia.org/wiki/Muller–Schupp_theorem |
In mathematics, the Munn semigroup is the inverse semigroup of isomorphisms between principal ideals of a semilattice (a commutative semigroup of idempotents). Munn semigroups are named for the Scottish mathematician Walter Douglas Munn (1929–2008). | https://en.wikipedia.org/wiki/Munn_semigroup |
In mathematics, the Möbius energy of a knot is a particular knot energy, i.e., a functional on the space of knots. It was discovered by Jun O'Hara, who demonstrated that the energy blows up as the knot's strands get close to one another. This is a useful property because it prevents self-intersection and ensures the re... | https://en.wikipedia.org/wiki/Freedman–He–Wang_conjecture |
They also showed the minimum energy of any knot conformation is achieved by a round circle.Conjecturally, there is no energy minimizer for composite knots. Robert B. Kusner and John M. Sullivan have done computer experiments with a discretized version of the Möbius energy and concluded that there should be no energy mi... | https://en.wikipedia.org/wiki/Freedman–He–Wang_conjecture |
For example, the inversion in the sphere { v ∈ R 3: | v − a | = ρ } {\displaystyle \{\mathbf {v} \in \mathbf {R} ^{3}\colon |\mathbf {v} -\mathbf {a} |=\rho \}} is defined by x → a + ρ 2 | x − a | 2 ⋅ ( x − a ) . {\displaystyle \mathbf {x} \to \mathbf {a} +{\rho ^{2} \over |\mathbf {x} -\mathbf {a} |^{2}}\cdot (\mathbf... | https://en.wikipedia.org/wiki/Freedman–He–Wang_conjecture |
Define its energy by E ( γ ) = ∬ { 1 | γ ( u ) − γ ( v ) | 2 − 1 D ( γ ( u ) , γ ( v ) ) 2 } | γ ˙ ( u ) | | γ ˙ ( v ) | d u d v , {\displaystyle E(\gamma )=\iint \left\{{\frac {1}{|\gamma (u)-\gamma (v)|^{2}}}-{\frac {1}{D(\gamma (u),\gamma (v))^{2}}}\right\}|{\dot {\gamma }}(u)||{\dot {\gamma }}(v)|\,du\,dv,} where D... | https://en.wikipedia.org/wiki/Freedman–He–Wang_conjecture |
Moreover, the energy of any line is 0, the energy of any circle is 4 {\displaystyle 4} . In fact, let us use the arc-length parameterization. | https://en.wikipedia.org/wiki/Freedman–He–Wang_conjecture |
Denote by ℓ {\displaystyle \ell } the length of the curve γ {\displaystyle \gamma } . Then E ( γ ) = ∫ − ℓ / 2 ℓ / 2 d x ∫ x − ℓ / 2 x + ℓ / 2 d y . {\displaystyle E(\gamma )=\int _{-\ell /2}^{\ell /2}{}dx\int _{x-\ell /2}^{x+\ell /2}\leftdy.} Let γ 0 ( t ) = ( cos t , sin t , 0 ) {\displaystyle \gamma _{0}(t)=(\c... | https://en.wikipedia.org/wiki/Freedman–He–Wang_conjecture |
In mathematics, the Nagata conjecture on curves, named after Masayoshi Nagata, governs the minimal degree required for a plane algebraic curve to pass through a collection of very general points with prescribed multiplicities. | https://en.wikipedia.org/wiki/Nagata's_conjecture_on_algebraic_curves |
In mathematics, the Nagata–Biran conjecture, named after Masayoshi Nagata and Paul Biran, is a generalisation of Nagata's conjecture on curves to arbitrary polarised surfaces. | https://en.wikipedia.org/wiki/Nagata–Biran_conjecture |
In mathematics, the Nagell–Lutz theorem is a result in the diophantine geometry of elliptic curves, which describes rational torsion points on elliptic curves over the integers. It is named for Trygve Nagell and Élisabeth Lutz. | https://en.wikipedia.org/wiki/Nagell–Lutz_theorem |
In mathematics, the Nakai conjecture is an unproven characterization of smooth algebraic varieties, conjectured by Japanese mathematician Yoshikazu Nakai in 1961. It states that if V is a complex algebraic variety, such that its ring of differential operators is generated by the derivations it contains, then V is a smo... | https://en.wikipedia.org/wiki/Nakai_conjecture |
In mathematics, the Narasimhan–Seshadri theorem, proved by Narasimhan and Seshadri (1965), says that a holomorphic vector bundle over a Riemann surface is stable if and only if it comes from an irreducible projective unitary representation of the fundamental group. The main case to understand is that of topologically t... | https://en.wikipedia.org/wiki/Narasimhan–Seshadri_theorem |
Donaldson (1983) gave another proof using differential geometry, and showed that the stable vector bundles have an essentially unique unitary connection of constant (scalar) curvature. In the degree zero case, Donaldson's version of the theorem says that a degree zero holomorphic vector bundle over a Riemann surface is... | https://en.wikipedia.org/wiki/Narasimhan–Seshadri_theorem |
In mathematics, the Narumi polynomials sn(x) are polynomials introduced by Narumi (1929) given by the generating function ∑ s n ( x ) t n / n ! = ( t log ( 1 + t ) ) a ( 1 + t ) x {\displaystyle \displaystyle \sum s_{n}(x)t^{n}/n!=\left({\frac {t}{\log(1+t)}}\right)^{a}(1+t)^{x}} (Roman 1984, 4.4), (Boas & Buck 1958,... | https://en.wikipedia.org/wiki/Narumi_polynomials |
In mathematics, the Natural transform is an integral transform similar to the Laplace transform and Sumudu transform, introduced by Zafar Hayat Khan in 2008. It converges to both Laplace and Sumudu transform just by changing variables. Given the convergence to the Laplace and Sumudu transforms, the N-transform inherits... | https://en.wikipedia.org/wiki/N-transform |
In mathematics, the Navier–Stokes equations are a system of nonlinear partial differential equations for abstract vector fields of any size. In physics and engineering, they are a system of equations that model the motion of liquids or non-rarefied gases (in which the mean free path is short enough so that it can be th... | https://en.wikipedia.org/wiki/Navier–Stokes_existence_and_smoothness |
Let v ( x , t ) {\displaystyle \mathbf {v} ({\boldsymbol {x}},t)} be a 3-dimensional vector field, the velocity of the fluid, and let p ( x , t ) {\displaystyle p({\boldsymbol {x}},t)} be the pressure of the fluid. The Navier–Stokes equations are: ∂ v ∂ t + ( v ⋅ ∇ ) v = − 1 ρ ∇ p + ν Δ v + f ( x , t ) {\displaystyle {... | https://en.wikipedia.org/wiki/Navier–Stokes_existence_and_smoothness |
Since in three dimensions, there are three equations and four unknowns (three scalar velocities and the pressure), then a supplementary equation is needed. This extra equation is the continuity equation for incompressible fluids that describes the conservation of mass of the fluid: ∇ ⋅ v = 0. | https://en.wikipedia.org/wiki/Navier–Stokes_existence_and_smoothness |
{\displaystyle \nabla \cdot \mathbf {v} =0.} Due to this last property, the solutions for the Navier–Stokes equations are searched in the set of solenoidal ("divergence-free") functions. | https://en.wikipedia.org/wiki/Navier–Stokes_existence_and_smoothness |
For this flow of a homogeneous medium, density and viscosity are constants. Since only its gradient appears, the pressure p can be eliminated by taking the curl of both sides of the Navier–Stokes equations. | https://en.wikipedia.org/wiki/Navier–Stokes_existence_and_smoothness |
In this case the Navier–Stokes equations reduce to the vorticity-transport equations. Now, we are going to look at nonlinearity. The Navier–Stokes equations are nonlinear because the terms in the equations do not have a simple linear relationship with each other. | https://en.wikipedia.org/wiki/Navier–Stokes_existence_and_smoothness |
This means that the equations cannot be solved using traditional linear techniques, and more advanced methods must be used instead. Nonlinearity is important in the Navier–Stokes equations because it allows the equations to describe a wide range of fluid dynamics phenomena, including the formation of shock waves and ot... | https://en.wikipedia.org/wiki/Navier–Stokes_existence_and_smoothness |
One way to understand the nonlinearity of the Navier–Stokes equations is to consider the term (v · ∇)v in the equations. This term represents the acceleration of the fluid, and it is a product of the velocity vector v and the gradient operator ∇. Because the gradient operator is a linear operator, the term (v · ∇)v is ... | https://en.wikipedia.org/wiki/Navier–Stokes_existence_and_smoothness |
The nonlinear nature of the Navier–Stokes equations can be seen in the term ( v ⋅ ∇ ) v {\displaystyle (\mathbf {v} \cdot \nabla )\mathbf {v} } , which represents the acceleration of the fluid due to its own velocity. This term is nonlinear because it involves the product of two velocity vectors, and the resulting acce... | https://en.wikipedia.org/wiki/Navier–Stokes_existence_and_smoothness |
The pressure in a fluid depends on the density and the gradient of the pressure, and this term is therefore nonlinear in the pressure. One example of the nonlinear nature of the Navier–Stokes equations can be seen in the case of a fluid flowing around a circular obstacle. In this case, the velocity of the fluid near th... | https://en.wikipedia.org/wiki/Navier–Stokes_existence_and_smoothness |
This results in a pressure gradient, with higher pressure near the obstacle and lower pressure farther away. To see this more explicitly, consider the case of a circular obstacle of radius R {\displaystyle R} placed in a uniform flow with velocity v 0 {\displaystyle \mathbf {v_{0}} } and density ρ {\displaystyle \rho }... | https://en.wikipedia.org/wiki/Navier–Stokes_existence_and_smoothness |
The Navier–Stokes equations in this case are: ∂ v ∂ t + ( v ⋅ ∇ ) v = − 1 ρ ∇ p + ν Δ v {\displaystyle {\frac {\partial \mathbf {v} }{\partial t}}+(\mathbf {v} \cdot \nabla )\mathbf {v} =-{\frac {1}{\rho }}\nabla p+\nu \Delta \mathbf {v} } ∇ ⋅ v = 0 {\displaystyle \nabla \cdot \mathbf {v} =0} where ν {\displaystyle \nu... | https://en.wikipedia.org/wiki/Navier–Stokes_existence_and_smoothness |
This results in a nonlinear term ( v ⋅ ∇ ) v {\displaystyle (\mathbf {v} \cdot \nabla )\mathbf {v} } in the Navier–Stokes equations that is proportional to the velocity of the fluid. At the same time, the presence of the obstacle will also result in a pressure gradient, with higher pressure near the obstacle and lower ... | https://en.wikipedia.org/wiki/Navier–Stokes_existence_and_smoothness |
Since the velocity is higher near the obstacle, the mass flow rate through a surface near the obstacle will be higher than the mass flow rate through a surface farther away from the obstacle. This can be compensated for by a pressure gradient, with higher pressure near the obstacle and lower pressure farther away. As a... | https://en.wikipedia.org/wiki/Navier–Stokes_existence_and_smoothness |
This system of ordinary differential equations can be solved using techniques such as the finite element method or spectral methods. Such method can be applied as: we can use a variety of techniques, such as the finite element method or spectral methods. One common approach is to use a finite difference method, which i... | https://en.wikipedia.org/wiki/Navier–Stokes_existence_and_smoothness |
In mathematics, the Neukirch–Uchida theorem shows that all problems about algebraic number fields can be reduced to problems about their absolute Galois groups. Jürgen Neukirch (1969) showed that two algebraic number fields with the same absolute Galois group are isomorphic, and Kôji Uchida (1976) strengthened this by ... | https://en.wikipedia.org/wiki/Neukirch–Uchida_theorem |
In mathematics, the Neumann (or second-type) boundary condition is a type of boundary condition, named after Carl Neumann. When imposed on an ordinary or a partial differential equation, the condition specifies the values of the derivative applied at the boundary of the domain. It is possible to describe the problem us... | https://en.wikipedia.org/wiki/Neumann_boundary_condition |
( − α n − k ) ( 2 t ) n + 1 − 2 k , {\displaystyle O_{n}^{(\alpha )}(t)={\frac {\alpha +n}{2\alpha }}\sum _{k=0}^{\lfloor n/2\rfloor }(-1)^{n-k}{\frac {(n-k)!}{k! }}{-\alpha \choose n-k}\left({\frac {2}{t}}\right)^{n+1-2k},} and they have the "generating function" ( z 2 ) α Γ ( α + 1 ) 1 t − z = ∑ n = 0 O n ( α ) ( t )... | https://en.wikipedia.org/wiki/Neumann_polynomial |
In mathematics, the Neumann–Dirichlet method is a domain decomposition preconditioner which involves solving Neumann boundary value problem on one subdomain and Dirichlet boundary value problem on another, adjacent across the interface between the subdomains. On a problem with many subdomains organized in a rectangular... | https://en.wikipedia.org/wiki/Neumann–Dirichlet_method |
In mathematics, the Neumann–Poincaré operator or Poincaré–Neumann operator, named after Carl Neumann and Henri Poincaré, is a non-self-adjoint compact operator introduced by Poincaré to solve boundary value problems for the Laplacian on bounded domains in Euclidean space. Within the language of potential theory it redu... | https://en.wikipedia.org/wiki/Neumann–Poincaré_operator |
In mathematics, the Nevanlinna invariant of an ample divisor D on a normal projective variety X is a real number connected with the rate of growth of the number of rational points on the variety with respect to the embedding defined by the divisor. The concept is named after Rolf Nevanlinna. | https://en.wikipedia.org/wiki/Nevanlinna_invariant |
In mathematics, the Newton inequalities are named after Isaac Newton. Suppose a1, a2, ..., an are real numbers and let e k {\displaystyle e_{k}} denote the kth elementary symmetric polynomial in a1, a2, ..., an. Then the elementary symmetric means, given by S k = e k ( n k ) , {\displaystyle S_{k}={\frac {e_{k}}{\binom... | https://en.wikipedia.org/wiki/Elementary_symmetric_mean |
In mathematics, the Newton polygon is a tool for understanding the behaviour of polynomials over local fields, or more generally, over ultrametric fields. In the original case, the local field of interest was essentially the field of formal Laurent series in the indeterminate X, i.e. the field of fractions of the forma... | https://en.wikipedia.org/wiki/Newton_polygon |
The exponents r {\displaystyle r} here are certain rational numbers, depending on the branch chosen; and the solutions themselves are power series in K ] {\displaystyle K]} with Y = X 1 d {\displaystyle Y=X^{\frac {1}{d}}} for a denominator d {\displaystyle d} corresponding to the branch. The Newton polygon gives an e... | https://en.wikipedia.org/wiki/Newton_polygon |
In mathematics, the Newton polytope is an integral polytope associated with a multivariate polynomial. It can be used to analyze the polynomial's behavior when specific variables are considered negligible relative to the others. Specifically, given a vector x = ( x 1 , … , x n ) {\displaystyle \mathbf {x} =(x_{1},\ldot... | https://en.wikipedia.org/wiki/Newton_polytope |
{\displaystyle \operatorname {Newt} (f)=\left\{\sum _{k}\alpha _{k}\mathbf {a} _{k}:\sum _{k}\alpha _{k}=1\;\&\;\forall j\,\,\alpha _{j}\geq 0\right\}\!.} The Newton polytope satisfies the following homomorphism-type property: Newt ( f g ) = Newt ( f ) + Newt ( g ) {\displaystyle \operatorname {Newt} (fg)=\operat... | https://en.wikipedia.org/wiki/Newton_polytope |
In mathematics, the Newtonian potential or Newton potential is an operator in vector calculus that acts as the inverse to the negative Laplacian, on functions that are smooth and decay rapidly enough at infinity. As such, it is a fundamental object of study in potential theory. In its general nature, it is a singular i... | https://en.wikipedia.org/wiki/Single_layer_potential |
In modern potential theory, the Newtonian potential is instead thought of as an electrostatic potential. The Newtonian potential of a compactly supported integrable function f is defined as the convolution where the Newtonian kernel Γ in dimension d is defined by Here ωd is the volume of the unit d-ball (sometimes sign... | https://en.wikipedia.org/wiki/Single_layer_potential |
{\displaystyle \Gamma (x)=-1/(4\pi |x|).} The Newtonian potential w of f is a solution of the Poisson equation which is to say that the operation of taking the Newtonian potential of a function is a partial inverse to the Laplace operator. | https://en.wikipedia.org/wiki/Single_layer_potential |
Then w will be a classical solution, that is twice differentiable, if f is bounded and locally Hölder continuous as shown by Otto Hölder. It was an open question whether continuity alone is also sufficient. This was shown to be wrong by Henrik Petrini who gave an example of a continuous f for which w is not twice diffe... | https://en.wikipedia.org/wiki/Single_layer_potential |
The solution is not unique, since addition of any harmonic function to w will not affect the equation. This fact can be used to prove existence and uniqueness of solutions to the Dirichlet problem for the Poisson equation in suitably regular domains, and for suitably well-behaved functions f: one first applies a Newton... | https://en.wikipedia.org/wiki/Single_layer_potential |
It satisfies the Poisson equation in the sense of distributions. Moreover, when the measure is positive, the Newtonian potential is subharmonic on Rd. If f is a compactly supported continuous function (or, more generally, a finite measure) that is rotationally invariant, then the convolution of f with Γ satisfies for x... | https://en.wikipedia.org/wiki/Single_layer_potential |
When the measure μ is associated to a mass distribution on a sufficiently smooth hypersurface S (a Lyapunov surface of Hölder class C1,α) that divides Rd into two regions D+ and D−, then the Newtonian potential of μ is referred to as a simple layer potential. Simple layer potentials are continuous and solve the Laplace... | https://en.wikipedia.org/wiki/Single_layer_potential |
In mathematics, the Noether inequality, named after Max Noether, is a property of compact minimal complex surfaces that restricts the topological type of the underlying topological 4-manifold. It holds more generally for minimal projective surfaces of general type over an algebraically closed field. | https://en.wikipedia.org/wiki/Noether_inequality |
In mathematics, the Noether normalization lemma is a result of commutative algebra, introduced by Emmy Noether in 1926. It states that for any field k, and any finitely generated commutative k-algebra A, there exist algebraically independent elements y1, y2, ..., yd in A such that A is a finitely generated module over ... | https://en.wikipedia.org/wiki/Noether's_normalization_lemma |
Then the inclusion map S ↪ A {\displaystyle S\hookrightarrow A} induces a surjective finite morphism of affine varieties X → A k d {\displaystyle X\to \mathbb {A} _{k}^{d}}: that is, any affine variety is a branched covering of affine space. When k is infinite, such a branched covering map can be constructed by taking ... | https://en.wikipedia.org/wiki/Noether's_normalization_lemma |
In mathematics, the Novikov–Veselov equation (or Veselov–Novikov equation) is a natural (2+1)-dimensional analogue of the Korteweg–de Vries (KdV) equation. Unlike another (2+1)-dimensional analogue of KdV, the Kadomtsev–Petviashvili equation, it is integrable via the inverse scattering transform for the 2-dimensional s... | https://en.wikipedia.org/wiki/Novikov–Veselov_equation |
The equation is named after S.P. Novikov and A.P. Veselov who published it in Novikov & Veselov (1984). | https://en.wikipedia.org/wiki/Novikov–Veselov_equation |
In mathematics, the Néron–Ogg–Shafarevich criterion states that if A is an elliptic curve or abelian variety over a local field K and ℓ is a prime not dividing the characteristic of the residue field of K then A has good reduction if and only if the ℓ-adic Tate module Tℓ of A is unramified. Andrew Ogg (1967) introduced... | https://en.wikipedia.org/wiki/Néron–Ogg–Shafarevich_criterion |
In mathematics, the Nørlund–Rice integral, sometimes called Rice's method, relates the nth forward difference of a function to a line integral on the complex plane. It commonly appears in the theory of finite differences and has also been applied in computer science and graph theory to estimate binary tree lengths. It ... | https://en.wikipedia.org/wiki/Nørlund–Rice_integral |
In mathematics, the O'Nan–Scott theorem is one of the most influential theorems of permutation group theory; the classification of finite simple groups is what makes it so useful. Originally the theorem was about maximal subgroups of the symmetric group. It appeared as an appendix to a paper by Leonard Scott written fo... | https://en.wikipedia.org/wiki/O'Nan–Scott_theorem |
A complete version of the theorem with a self-contained proof was given by M.W. Liebeck, Cheryl Praeger and Jan Saxl. The theorem is now a standard part of textbooks on permutation groups. | https://en.wikipedia.org/wiki/O'Nan–Scott_theorem |
In mathematics, the Odlyzko–Schönhage algorithm is a fast algorithm for evaluating the Riemann zeta function at many points, introduced by (Odlyzko & Schönhage 1988). The main point is the use of the fast Fourier transform to speed up the evaluation of a finite Dirichlet series of length N at O(N) equally spaced values... | https://en.wikipedia.org/wiki/Odlyzko–Schönhage_algorithm |
In mathematics, the Oka coherence theorem, proved by Kiyoshi Oka (1950), states that the sheaf O := O C n {\displaystyle {\mathcal {O}}:={\mathcal {O}}_{\mathbb {C} _{n}}} of germs of holomorphic functions on C n {\displaystyle \mathbb {C} ^{n}} over a complex manifold is coherent. | https://en.wikipedia.org/wiki/Oka's_coherence_theorem |
In mathematics, the Ornstein isomorphism theorem is a deep result in ergodic theory. It states that if two Bernoulli schemes have the same Kolmogorov entropy, then they are isomorphic. The result, given by Donald Ornstein in 1970, is important because it states that many systems previously believed to be unrelated are ... | https://en.wikipedia.org/wiki/Ornstein_isomorphism_theorem |
In mathematics, the Ornstein–Uhlenbeck operator is a generalization of the Laplace operator to an infinite-dimensional setting. The Ornstein–Uhlenbeck operator plays a significant role in the Malliavin calculus. | https://en.wikipedia.org/wiki/Ornstein–Uhlenbeck_operator |
In mathematics, the Ornstein–Uhlenbeck process is a stochastic process with applications in financial mathematics and the physical sciences. Its original application in physics was as a model for the velocity of a massive Brownian particle under the influence of friction. It is named after Leonard Ornstein and George E... | https://en.wikipedia.org/wiki/Ornstein–Uhlenbeck_processes |
The Ornstein–Uhlenbeck process is a stationary Gauss–Markov process, which means that it is a Gaussian process, a Markov process, and is temporally homogeneous. In fact, it is the only nontrivial process that satisfies these three conditions, up to allowing linear transformations of the space and time variables. Over t... | https://en.wikipedia.org/wiki/Ornstein–Uhlenbeck_processes |
In mathematics, the Ostrowski–Hadamard gap theorem is a result about the analytic continuation of complex power series whose non-zero terms are of orders that have a suitable "gap" between them. Such a power series is "badly behaved" in the sense that it cannot be extended to be an analytic function anywhere on the bou... | https://en.wikipedia.org/wiki/Ostrowski–Hadamard_gap_theorem |
In mathematics, the Paley construction is a method for constructing Hadamard matrices using finite fields. The construction was described in 1933 by the English mathematician Raymond Paley. The Paley construction uses quadratic residues in a finite field GF(q) where q is a power of an odd prime number. There are two ve... | https://en.wikipedia.org/wiki/Paley_construction |
In mathematics, the Paley–Wiener integral is a simple stochastic integral. When applied to classical Wiener space, it is less general than the Itō integral, but the two agree when they are both defined. The integral is named after its discoverers, Raymond Paley and Norbert Wiener. | https://en.wikipedia.org/wiki/Paley–Wiener_integral |
In mathematics, the Paley–Zygmund inequality bounds the probability that a positive random variable is small, in terms of its first two moments. The inequality was proved by Raymond Paley and Antoni Zygmund. Theorem: If Z ≥ 0 is a random variable with finite variance, and if 0 ≤ θ ≤ 1 {\displaystyle 0\leq \theta \leq 1... | https://en.wikipedia.org/wiki/Paley–Zygmund_inequality |
Proof: First, E = E } ] + E } ] . {\displaystyle \operatorname {E} =\operatorname {E} \}}]+\operatorname {E} \}}].} | https://en.wikipedia.org/wiki/Paley–Zygmund_inequality |
The first addend is at most θ E {\displaystyle \theta \operatorname {E} } , while the second is at most E 1 / 2 P ( Z > θ E ) 1 / 2 {\displaystyle \operatorname {E} ^{1/2}\operatorname {P} (Z>\theta \operatorname {E} )^{1/2}} by the Cauchy–Schwarz inequality. The desired inequality then follows. ∎ | https://en.wikipedia.org/wiki/Paley–Zygmund_inequality |
In mathematics, the Pansu derivative is a derivative on a Carnot group, introduced by Pierre Pansu (1989). A Carnot group G {\displaystyle G} admits a one-parameter family of dilations, δ s: G → G {\displaystyle \delta _{s}\colon G\to G} . If G 1 {\displaystyle G_{1}} and G 2 {\displaystyle G_{2}} are Carnot groups, th... | https://en.wikipedia.org/wiki/Pansu_derivative |
In mathematics, the Parker–Sochacki method is an algorithm for solving systems of ordinary differential equations (ODEs), developed by G. Edgar Parker and James Sochacki, of the James Madison University Mathematics Department. The method produces Maclaurin series solutions to systems of differential equations, with the... | https://en.wikipedia.org/wiki/Parker–Sochacki_method |
In mathematics, the Parry–Daniels map is a function studied in the context of dynamical systems. Typical questions concern the existence of an invariant or ergodic measure for the map.It is named after the English mathematician Bill Parry and the British statistician Henry Daniels, who independently studied the map in ... | https://en.wikipedia.org/wiki/Parry–Daniels_map |
In mathematics, the Parry–Sullivan invariant (or Parry–Sullivan number) is a numerical quantity of interest in the study of incidence matrices in graph theory, and of certain one-dimensional dynamical systems. It provides a partial classification of non-trivial irreducible incidence matrices. It is named after the Engl... | https://en.wikipedia.org/wiki/Parry–Sullivan_invariant |
In mathematics, the Parseval–Gutzmer formula states that, if f {\displaystyle f} is an analytic function on a closed disk of radius r with Taylor series f ( z ) = ∑ k = 0 ∞ a k z k , {\displaystyle f(z)=\sum _{k=0}^{\infty }a_{k}z^{k},} then for z = reiθ on the boundary of the disk, ∫ 0 2 π | f ( r e i θ ) | 2 d θ = 2 ... | https://en.wikipedia.org/wiki/Parseval–Gutzmer_formula |
In mathematics, the Peano surface is the graph of the two-variable function f ( x , y ) = ( 2 x 2 − y ) ( y − x 2 ) . {\displaystyle f(x,y)=(2x^{2}-y)(y-x^{2}).} It was proposed by Giuseppe Peano in 1899 as a counterexample to a conjectured criterion for the existence of maxima and minima of functions of two variables.... | https://en.wikipedia.org/wiki/Peano_surface |
In mathematics, the Peano–Jordan measure (also known as the Jordan content) is an extension of the notion of size (length, area, volume) to shapes more complicated than, for example, a triangle, disk, or parallelepiped. It turns out that for a set to have Jordan measure it should be well-behaved in a certain restrictiv... | https://en.wikipedia.org/wiki/Jordan_measure |
For historical reasons, the term Jordan measure is now well-established for this set function, despite the fact that it is not a true measure in its modern definition, since Jordan-measurable sets do not form a σ-algebra. For example, singleton sets { x } x ∈ R {\displaystyle \{x\}_{x\in \mathbb {R} }} in R {\displayst... | https://en.wikipedia.org/wiki/Jordan_measure |
In mathematics, the Pearcey integral is defined as Pe ( x , y ) = ∫ − ∞ ∞ exp ( i ( t 4 + x t 2 + y t ) ) d t . {\displaystyle \operatorname {Pe} (x,y)=\int _{-\infty }^{\infty }\exp(i(t^{4}+xt^{2}+yt))\,dt.} The Pearcey integral is a class of canonical diffraction integrals, often used in wave propagation and opti... | https://en.wikipedia.org/wiki/Pearcey_integral |
In mathematics, the Pell numbers are an infinite sequence of integers, known since ancient times, that comprise the denominators of the closest rational approximations to the square root of 2. This sequence of approximations begins 1/1, 3/2, 7/5, 17/12, and 41/29, so the sequence of Pell numbers begins with 1, 2, 5, 12... | https://en.wikipedia.org/wiki/Pell_numbers |
Both the Pell numbers and the companion Pell numbers may be calculated by means of a recurrence relation similar to that for the Fibonacci numbers, and both sequences of numbers grow exponentially, proportionally to powers of the silver ratio 1 + √2. As well as being used to approximate the square root of two, Pell num... | https://en.wikipedia.org/wiki/Pell_numbers |
In mathematics, the Perkel graph, named after Manley Perkel, is a 6-regular graph with 57 vertices and 171 edges. It is the unique distance-regular graph with intersection array (6, 5, 2; 1, 1, 3). The Perkel graph is also distance-transitive. It is also the skeleton of an abstract regular polytope, the 57-cell. | https://en.wikipedia.org/wiki/Perkel_graph |
In mathematics, the Perrin numbers are defined by the recurrence relation P(n) = P(n − 2) + P(n − 3) for n > 2,with initial values P(0) = 3, P(1) = 0, P(2) = 2.The sequence of Perrin numbers starts with 3, 0, 2, 3, 2, 5, 5, 7, 10, 12, 17, 22, 29, 39, ... (sequence A001608 in the OEIS)The number of different maximal ind... | https://en.wikipedia.org/wiki/Perrin_pseudoprime |
In mathematics, the Peters polynomials sn(x) are polynomials studied by Peters (1956, 1956b) given by the generating function ∑ n = 0 + ∞ s n ( x ) t n n ! = ( 1 + t ) x ( 1 + ( 1 + t ) λ ) μ {\displaystyle \displaystyle \sum _{n=0}^{+\infty }s_{n}(x){\frac {t^{n}}{n! }}={\frac {(1+t)^{x}}{(1+(1+t)^{\lambda })^{\mu }}}... | https://en.wikipedia.org/wiki/Peters_polynomials |
In mathematics, the Peterson–Stein formula, introduced by Franklin P. Peterson and Norman Stein (1960), describes the Spanier–Whitehead dual of a secondary cohomology operation. | https://en.wikipedia.org/wiki/Peterson–Stein_formula |
In mathematics, the Peter–Weyl theorem is a basic result in the theory of harmonic analysis, applying to topological groups that are compact, but are not necessarily abelian. It was initially proved by Hermann Weyl, with his student Fritz Peter, in the setting of a compact topological group G (Peter & Weyl 1927). The t... | https://en.wikipedia.org/wiki/Peter–Weyl_theorem |
Let G be a compact group. The theorem has three parts. The first part states that the matrix coefficients of irreducible representations of G are dense in the space C(G) of continuous complex-valued functions on G, and thus also in the space L2(G) of square-integrable functions. | https://en.wikipedia.org/wiki/Peter–Weyl_theorem |
The second part asserts the complete reducibility of unitary representations of G. The third part then asserts that the regular representation of G on L2(G) decomposes as the direct sum of all irreducible unitary representations. Moreover, the matrix coefficients of the irreducible unitary representations form an ortho... | https://en.wikipedia.org/wiki/Peter–Weyl_theorem |
In mathematics, the Pettis integral or Gelfand–Pettis integral, named after Israel M. Gelfand and Billy James Pettis, extends the definition of the Lebesgue integral to vector-valued functions on a measure space, by exploiting duality. The integral was introduced by Gelfand for the case when the measure space is an int... | https://en.wikipedia.org/wiki/Weak_integral |
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