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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 straight line without a straightedge. It is understood that a line is determined provided that two distinct points on that line are given or constructed, even though no visual representation of the line will be present. The theorem can be stated more precisely as: Any Euclidean construction, insofar as the given and required elements are points (or circles), may be completed with the compass alone if it can be completed with both the compass and the straightedge together.Though the use of a straightedge can make a construction significantly easier, the theorem shows that any set of points that fully defines a constructed figure can be determined with compass alone, and the only reason to use a straightedge is for the aesthetics of seeing straight lines, which for the purposes of construction is functionally unnecessary.
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https://en.wikipedia.org/wiki/Mohr–Mascheroni_theorem
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In mathematics, the Moore determinant is a determinant defined for Hermitian matrices over a quaternion algebra, introduced by Moore (1922).
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https://en.wikipedia.org/wiki/Moore_determinant_of_a_Hermitian_matrix
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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.
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https://en.wikipedia.org/wiki/Moore_plane
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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 {\displaystyle A} an elliptic curve E {\displaystyle E} and K {\displaystyle K} the field of rational numbers is Mordell's theorem, answering a question apparently posed by Henri Poincaré around 1901; it was proved by Louis Mordell in 1922. It is a foundational theorem of Diophantine geometry and the arithmetic of abelian varieties.
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https://en.wikipedia.org/wiki/Mordell–Weil_theorem
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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.
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https://en.wikipedia.org/wiki/Morlet_wavelet
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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 proportional to some power of the radius other than the dimension. They are used in the theory of elliptic partial differential equations, since for certain values of λ {\displaystyle \lambda } , elements of the space L λ , p ( Ω ) {\displaystyle L^{\lambda ,p}(\Omega )} are Hölder continuous functions over the domain Ω {\displaystyle \Omega } .
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https://en.wikipedia.org/wiki/Morrey–Campanato_space
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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 p\geq 1} .
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https://en.wikipedia.org/wiki/Morrey–Campanato_space
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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 } and r < diam ( Ω ) {\displaystyle r<\operatorname {diam} (\Omega )} . When n = λ {\displaystyle n=\lambda } , the Campanato space is the space of functions of bounded mean oscillation. When n < λ ≤ n + p {\displaystyle n<\lambda \leq n+p} , the Campanato space is the space of Hölder continuous functions C α ( Ω ) {\displaystyle C^{\alpha }(\Omega )} with α = λ − n p {\displaystyle \alpha ={\frac {\lambda -n}{p}}} . For λ > n + p {\displaystyle \lambda >n+p} , the space contains only constant functions.
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https://en.wikipedia.org/wiki/Morrey–Campanato_space
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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 finite-dimensional case by the American mathematician Marston Morse, using the Gram–Schmidt orthogonalization process. This result plays a crucial role in Morse theory. The generalization to Hilbert spaces is due to Richard Palais and Stephen Smale.
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https://en.wikipedia.org/wiki/Morse–Palais_lemma
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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 Palais (1957).
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https://en.wikipedia.org/wiki/Mostow–Palais_theorem
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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 (with a generalization to symplectic manifolds, described below). It is a special case of the ★-product of the "algebra of symbols" of a universal enveloping algebra.
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https://en.wikipedia.org/wiki/Star_product_(quantization)
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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.
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https://en.wikipedia.org/wiki/Muller–Schupp_theorem
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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).
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https://en.wikipedia.org/wiki/Munn_semigroup
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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 result under gradient descent is of the same knot type. Invariance of Möbius energy under Möbius transformations was demonstrated by Michael Freedman, Zheng-Xu He, and Zhenghan Wang (1994) who used it to show the existence of a C 1 , 1 {\displaystyle C^{1,1}} energy minimizer in each isotopy class of a prime knot.
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https://en.wikipedia.org/wiki/Freedman–He–Wang_conjecture
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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 minimizer for the knot sum of two trefoils (although this is not a proof). Recall that the Möbius transformations of the 3-sphere S 3 = R 3 ∪ ∞ {\displaystyle S^{3}=\mathbf {R} ^{3}\cup \infty } are the ten-dimensional group of angle-preserving diffeomorphisms generated by inversion in 2-spheres.
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https://en.wikipedia.org/wiki/Freedman–He–Wang_conjecture
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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 {x} -\mathbf {a} ).} Consider a rectifiable simple curve γ ( u ) {\displaystyle \gamma (u)} in the Euclidean 3-space R 3 {\displaystyle \mathbf {R} ^{3}} , where u {\displaystyle u} belongs to R 1 {\displaystyle \mathbf {R} ^{1}} or S 1 {\displaystyle S^{1}} .
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https://en.wikipedia.org/wiki/Freedman–He–Wang_conjecture
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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 ( γ ( u ) , γ ( v ) ) {\displaystyle D(\gamma (u),\gamma (v))} is the shortest arc distance between γ ( u ) {\displaystyle \gamma (u)} and γ ( v ) {\displaystyle \gamma (v)} on the curve. The second term of the integrand is called a regularization. It is easy to see that E ( γ ) {\displaystyle E(\gamma )} is independent of parametrization and is unchanged if γ {\displaystyle \gamma } is changed by a similarity of R 3 {\displaystyle \mathbf {R} ^{3}} .
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https://en.wikipedia.org/wiki/Freedman–He–Wang_conjecture
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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.
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https://en.wikipedia.org/wiki/Freedman–He–Wang_conjecture
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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)=(\cos t,\sin t,0)} denote a unit circle. We have | γ 0 ( x ) − γ 0 ( y ) | 2 = ( 2 sin 1 2 ( x − y ) ) 2 {\displaystyle |\gamma _{0}(x)-\gamma _{0}(y)|^{2}={\left(2\sin {\tfrac {1}{2}}(x-y)\right)^{2}}} and consequently, E ( γ 0 ) = ∫ − π π d x ∫ x − π x + π d y = 4 π ∫ 0 π d y = 2 π ∫ 0 π / 2 d y = 2 π u = 0 π / 2 = 4 {\displaystyle {\begin{aligned}E(\gamma _{0})&=\int _{-\pi }^{\pi }{}dx\int _{x-\pi }^{x+\pi }\leftdy\\&=4\pi \int _{0}^{\pi }\leftdy\\&=2\pi \int _{0}^{\pi /2}\leftdy\\&=2\pi \left_{u=0}^{\pi /2}=4\end{aligned}}} since 1 u − cot u = u 3 − ⋯ {\displaystyle {\frac {1}{u}}-\cot u={\frac {u}{3}}-\cdots } .
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https://en.wikipedia.org/wiki/Freedman–He–Wang_conjecture
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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.
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https://en.wikipedia.org/wiki/Nagata's_conjecture_on_algebraic_curves
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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.
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https://en.wikipedia.org/wiki/Nagata–Biran_conjecture
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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.
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https://en.wikipedia.org/wiki/Nagell–Lutz_theorem
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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 smooth variety. The converse statement, that smooth algebraic varieties have rings of differential operators that are generated by their derivations, is a result of Alexander Grothendieck.The Nakai conjecture is known to be true for algebraic curves and Stanley–Reisner rings. A proof of the conjecture would also establish the Zariski–Lipman conjecture, for a complex variety V with coordinate ring R. This conjecture states that if the derivations of R are a free module over R, then V is smooth. == References ==
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https://en.wikipedia.org/wiki/Nakai_conjecture
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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 trivial bundles, i.e. those of degree zero (and the other cases are a minor technical extension of this case). This case of the Narasimhan–Seshadri theorem says that a degree zero holomorphic vector bundle over a Riemann surface is stable if and only if it comes from an irreducible unitary representation of the fundamental group of the Riemann surface.
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https://en.wikipedia.org/wiki/Narasimhan–Seshadri_theorem
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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 stable if and only if it admits a flat unitary connection compatible with its holomorphic structure. Then the fundamental group representation appearing in the original statement is just the monodromy representation of this flat unitary connection.
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https://en.wikipedia.org/wiki/Narasimhan–Seshadri_theorem
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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, p.37)
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https://en.wikipedia.org/wiki/Narumi_polynomials
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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 all the applied aspects of the both transforms. Most recently, F. B. M. Belgacem has renamed it the natural transform and has proposed a detail theory and applications.
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https://en.wikipedia.org/wiki/N-transform
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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 thought of as a continuum mean instead of a collection of particles) using continuum mechanics. The equations are a statement of Newton's second law, with the forces modeled according to those in a viscous Newtonian fluid—as the sum of contributions by pressure, viscous stress and an external body force. Since the setting of the problem proposed by the Clay Mathematics Institute is in three dimensions, for an incompressible and homogeneous fluid, only that case is considered below.
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https://en.wikipedia.org/wiki/Navier–Stokes_existence_and_smoothness
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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 {\frac {\partial \mathbf {v} }{\partial t}}+(\mathbf {v} \cdot \nabla )\mathbf {v} =-{\frac {1}{\rho }}\nabla p+\nu \Delta \mathbf {v} +\mathbf {f} ({\boldsymbol {x}},t)} where ν > 0 {\displaystyle \nu >0} is the kinematic viscosity, f ( x , t ) {\displaystyle \mathbf {f} ({\boldsymbol {x}},t)} the external volumetric force, ∇ {\displaystyle \nabla } is the gradient operator and Δ {\displaystyle \displaystyle \Delta } is the Laplacian operator, which is also denoted by ∇ ⋅ ∇ {\displaystyle \nabla \cdot \nabla } or ∇ 2 {\displaystyle \nabla ^{2}} . Note that this is a vector equation, i.e. it has three scalar equations.
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https://en.wikipedia.org/wiki/Navier–Stokes_existence_and_smoothness
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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.
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https://en.wikipedia.org/wiki/Navier–Stokes_existence_and_smoothness
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{\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.
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https://en.wikipedia.org/wiki/Navier–Stokes_existence_and_smoothness
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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.
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https://en.wikipedia.org/wiki/Navier–Stokes_existence_and_smoothness
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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.
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https://en.wikipedia.org/wiki/Navier–Stokes_existence_and_smoothness
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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 other complex flow patterns. However, the nonlinearity of the Navier–Stokes equations also makes them more difficult to solve, as traditional linear methods may not work.
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https://en.wikipedia.org/wiki/Navier–Stokes_existence_and_smoothness
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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 nonlinear in the velocity vector v. This means that the acceleration of the fluid depends on the magnitude and direction of the velocity, as well as the spatial distribution of the velocity within the fluid.
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https://en.wikipedia.org/wiki/Navier–Stokes_existence_and_smoothness
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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 acceleration is therefore dependent on the magnitude and direction of both vectors. Another source of nonlinearity in the Navier–Stokes equations is the pressure term − 1 ρ ∇ p {\displaystyle -{\frac {1}{\rho }}\nabla p} .
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https://en.wikipedia.org/wiki/Navier–Stokes_existence_and_smoothness
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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 the obstacle will be higher than the velocity of the fluid farther away from the obstacle.
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https://en.wikipedia.org/wiki/Navier–Stokes_existence_and_smoothness
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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 } . Let v ( x , t ) {\displaystyle \mathbf {v} (\mathbf {x} ,t)} be the velocity of the fluid at position x {\displaystyle \mathbf {x} } and time t {\displaystyle t} , and let p ( x , t ) {\displaystyle p(\mathbf {x} ,t)} be the pressure at the same position and time.
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https://en.wikipedia.org/wiki/Navier–Stokes_existence_and_smoothness
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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 } is the kinematic viscosity of the fluid. Assuming that the flow is steady (meaning that the velocity and pressure do not vary with time), we can set the time derivative terms equal to zero: ( v ⋅ ∇ ) v = − 1 ρ ∇ p + ν Δ v {\displaystyle (\mathbf {v} \cdot \nabla )\mathbf {v} =-{\frac {1}{\rho }}\nabla p+\nu \Delta \mathbf {v} } ∇ ⋅ v = 0 {\displaystyle \nabla \cdot \mathbf {v} =0} We can now consider the flow near the circular obstacle. In this region, the velocity of the fluid will be higher than the uniform flow velocity v 0 {\displaystyle \mathbf {v_{0}} } due to the presence of the obstacle.
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https://en.wikipedia.org/wiki/Navier–Stokes_existence_and_smoothness
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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 pressure farther away. This can be seen by considering the continuity equation, which states that the mass flow rate through any surface must be constant.
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https://en.wikipedia.org/wiki/Navier–Stokes_existence_and_smoothness
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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 result of these nonlinear effects, the Navier–Stokes equations in this case become difficult to solve, and approximations or numerical methods must be used to find the velocity and pressure fields in the flow.
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https://en.wikipedia.org/wiki/Navier–Stokes_existence_and_smoothness
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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 involves approximating the derivative terms in the equation using finite differences.
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https://en.wikipedia.org/wiki/Navier–Stokes_existence_and_smoothness
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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 proving Neukirch's conjecture that automorphisms of the algebraic number field correspond to outer automorphisms of its absolute Galois group. Florian Pop (1990, 1994) extended the result to infinite fields that are finitely generated over prime fields. The Neukirch–Uchida theorem is one of the foundational results of anabelian geometry, whose main theme is to reduce properties of geometric objects to properties of their fundamental groups, provided these fundamental groups are sufficiently non-abelian.
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https://en.wikipedia.org/wiki/Neukirch–Uchida_theorem
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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 using other boundary conditions: a Dirichlet boundary condition specifies the values of the solution itself (as opposed to its derivative) on the boundary, whereas the Cauchy boundary condition, mixed boundary condition and Robin boundary condition are all different types of combinations of the Neumann and Dirichlet boundary conditions.
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https://en.wikipedia.org/wiki/Neumann_boundary_condition
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( − 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 ) J α + n ( z ) , {\displaystyle {\frac {\left({\frac {z}{2}}\right)^{\alpha }}{\Gamma (\alpha +1)}}{\frac {1}{t-z}}=\sum _{n=0}O_{n}^{(\alpha )}(t)J_{\alpha +n}(z),} where J are Bessel functions. To expand a function f in the form f ( z ) = ∑ n = 0 a n J α + n ( z ) {\displaystyle f(z)=\sum _{n=0}a_{n}J_{\alpha +n}(z)\,} for | z | < c {\displaystyle |z|
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https://en.wikipedia.org/wiki/Neumann_polynomial
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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 mesh, the subdomains are assigned Neumann or Dirichlet problems in a checkerboard fashion.
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https://en.wikipedia.org/wiki/Neumann–Dirichlet_method
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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 reduces the partial differential equation to an integral equation on the boundary to which the theory of Fredholm operators can be applied. The theory is particularly simple in two dimensions—the case treated in detail in this article—where it is related to complex function theory, the conjugate Beurling transform or complex Hilbert transform and the Fredholm eigenvalues of bounded planar domains.
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https://en.wikipedia.org/wiki/Neumann–Poincaré_operator
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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.
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https://en.wikipedia.org/wiki/Nevanlinna_invariant
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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 {n}{k}}},} satisfy the inequality S k − 1 S k + 1 ≤ S k 2 . {\displaystyle S_{k-1}S_{k+1}\leq S_{k}^{2}.} If all the numbers ai are non-zero, then equality holds if and only if all the numbers ai are equal. It can be seen that S1 is the arithmetic mean, and Sn is the n-th power of the geometric mean.
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https://en.wikipedia.org/wiki/Elementary_symmetric_mean
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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 formal power series ring K ] {\displaystyle K]} , over K {\displaystyle K} , where K {\displaystyle K} was the real number or complex number field. This is still of considerable utility with respect to Puiseux expansions. The Newton polygon is an effective device for understanding the leading terms a X r {\displaystyle aX^{r}} of the power series expansion solutions to equations P ( F ( X ) ) = 0 {\displaystyle P(F(X))=0} where P {\displaystyle P} is a polynomial with coefficients in K {\displaystyle K} , the polynomial ring; that is, implicitly defined algebraic functions.
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https://en.wikipedia.org/wiki/Newton_polygon
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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 effective, algorithmic approach to calculating d {\displaystyle d} . After the introduction of the p-adic numbers, it was shown that the Newton polygon is just as useful in questions of ramification for local fields, and hence in algebraic number theory. Newton polygons have also been useful in the study of elliptic curves.
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https://en.wikipedia.org/wiki/Newton_polygon
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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},\ldots ,x_{n})} of variables and a finite family ( a k ) k {\displaystyle (\mathbf {a} _{k})_{k}} of pairwise distinct vectors from N n {\displaystyle \mathbb {N} ^{n}} each encoding the exponents within a monomial, consider the multivariate polynomial f ( x ) = ∑ k c k x a k {\displaystyle f(\mathbf {x} )=\sum _{k}c_{k}\mathbf {x} ^{\mathbf {a} _{k}}} where we use the shorthand notation ( x 1 , … , x n ) ( y 1 , … , y n ) {\displaystyle (x_{1},\ldots ,x_{n})^{(y_{1},\ldots ,y_{n})}} for the monomial x 1 y 1 x 2 y 2 ⋯ x n y n {\displaystyle x_{1}^{y_{1}}x_{2}^{y_{2}}\cdots x_{n}^{y_{n}}} . Then the Newton polytope associated to f {\displaystyle f} is the convex hull of the vectors a k {\displaystyle \mathbf {a} _{k}} ; that is Newt ( f ) = { ∑ k α k a k: ∑ k α k = 1 & ∀ j α j ≥ 0 } .
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https://en.wikipedia.org/wiki/Newton_polytope
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{\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)=\operatorname {Newt} (f)+\operatorname {Newt} (g)} where the addition is in the sense of Minkowski. Newton polytopes are the central object of study in tropical geometry and characterize the Gröbner bases for an ideal.
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https://en.wikipedia.org/wiki/Newton_polytope
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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 integral operator, defined by convolution with a function having a mathematical singularity at the origin, the Newtonian kernel Γ which is the fundamental solution of the Laplace equation. It is named for Isaac Newton, who first discovered it and proved that it was a harmonic function in the special case of three variables, where it served as the fundamental gravitational potential in Newton's law of universal gravitation.
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https://en.wikipedia.org/wiki/Single_layer_potential
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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 conventions may vary; compare (Evans 1998) and (Gilbarg & Trudinger 1983)). For example, for d = 3 {\displaystyle d=3} we have Γ ( x ) = − 1 / ( 4 π | x | ) .
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https://en.wikipedia.org/wiki/Single_layer_potential
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{\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.
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https://en.wikipedia.org/wiki/Single_layer_potential
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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 differentiable.
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https://en.wikipedia.org/wiki/Single_layer_potential
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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 Newtonian potential to obtain a solution, and then adjusts by adding a harmonic function to get the correct boundary data. The Newtonian potential is defined more broadly as the convolution when μ is a compactly supported Radon measure.
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https://en.wikipedia.org/wiki/Single_layer_potential
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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 outside the support of f In dimension d = 3, this reduces to Newton's theorem that the potential energy of a small mass outside a much larger spherically symmetric mass distribution is the same as if all of the mass of the larger object were concentrated at its center.
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https://en.wikipedia.org/wiki/Single_layer_potential
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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 equation except on S. They appear naturally in the study of electrostatics in the context of the electrostatic potential associated to a charge distribution on a closed surface. If dμ = f dH is the product of a continuous function on S with the (d − 1)-dimensional Hausdorff measure, then at a point y of S, the normal derivative undergoes a jump discontinuity f(y) when crossing the layer. Furthermore, the normal derivative of w is a well-defined continuous function on S. This makes simple layers particularly suited to the study of the Neumann problem for the Laplace equation.
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https://en.wikipedia.org/wiki/Single_layer_potential
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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.
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https://en.wikipedia.org/wiki/Noether_inequality
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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 the polynomial ring S = k . The integer d is equal to the Krull dimension of the ring A; and if A is an integral domain, d is also the transcendence degree of the field of fractions of A over k. The theorem has a geometric interpretation. Suppose A is the coordinate ring of an affine variety X, and consider S as the coordinate ring of a d-dimensional affine space A k d {\displaystyle \mathbb {A} _{k}^{d}} .
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https://en.wikipedia.org/wiki/Noether's_normalization_lemma
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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 a general projection from an affine space containing X to a d-dimensional subspace. More generally, in the language of schemes, the theorem can equivalently be stated as: every affine k-scheme (of finite type) X is finite over an affine n-dimensional space. The theorem can be refined to include a chain of ideals of R (equivalently, closed subsets of X) that are finite over the affine coordinate subspaces of the corresponding dimensions.The Noether normalization lemma can be used as an important step in proving Hilbert's Nullstellensatz, one of the most fundamental results of classical algebraic geometry. The normalization theorem is also an important tool in establishing the notions of Krull dimension for k-algebras.
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https://en.wikipedia.org/wiki/Noether's_normalization_lemma
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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 stationary Schrödinger equation. Similarly, the Korteweg–de Vries equation is integrable via the inverse scattering transform for the 1-dimensional Schrödinger equation.
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https://en.wikipedia.org/wiki/Novikov–Veselov_equation
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The equation is named after S.P. Novikov and A.P. Veselov who published it in Novikov & Veselov (1984).
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https://en.wikipedia.org/wiki/Novikov–Veselov_equation
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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 the criterion for elliptic curves. Serre and Tate (1968) used the results of André Néron (1964) to extend it to abelian varieties, and named the criterion after Ogg, Néron and Igor Shafarevich (commenting that Ogg's result seems to have been known to Shafarevich).
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https://en.wikipedia.org/wiki/Néron–Ogg–Shafarevich_criterion
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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 is named in honour of Niels Erik Nørlund and Stephen O. Rice. Nørlund's contribution was to define the integral; Rice's contribution was to demonstrate its utility by applying saddle-point techniques to its evaluation.
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https://en.wikipedia.org/wiki/Nørlund–Rice_integral
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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 for The Santa Cruz Conference on Finite Groups in 1979, with a footnote that Michael O'Nan had independently proved the same result. Michael Aschbacher and Scott later gave a corrected version of the statement of the theorem.The theorem states that a maximal subgroup of the symmetric group Sym(Ω), where |Ω| = n, is one of the following: Sk × Sn−k the stabilizer of a k-set (that is, intransitive) Sa wr Sb with n = ab, the stabilizer of a partition into b parts of size a (that is, imprimitive) primitive (that is, preserves no nontrivial partition) and of one of the following types:AGL(d,p) Sl wr Sk, the stabilizer of the product structure Ω = Δk a group of diagonal type an almost simple groupIn a survey paper written for the Bulletin of the London Mathematical Society, Peter J. Cameron seems to have been the first to recognize that the real power in the O'Nan–Scott theorem is in the ability to split the finite primitive groups into various types.
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https://en.wikipedia.org/wiki/O'Nan–Scott_theorem
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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.
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https://en.wikipedia.org/wiki/O'Nan–Scott_theorem
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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 from O(N2) to O(N1+ε) steps (at the cost of storing O(N1+ε) intermediate values). The Riemann–Siegel formula used for calculating the Riemann zeta function with imaginary part T uses a finite Dirichlet series with about N = T1/2 terms, so when finding about N values of the Riemann zeta function it is sped up by a factor of about T1/2. This reduces the time to find the zeros of the zeta function with imaginary part at most T from about T3/2+ε steps to about T1+ε steps. The algorithm can be used not just for the Riemann zeta function, but also for many other functions given by Dirichlet series. The algorithm was used by Gourdon (2004) to verify the Riemann hypothesis for the first 1013 zeros of the zeta function.
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https://en.wikipedia.org/wiki/Odlyzko–Schönhage_algorithm
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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.
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https://en.wikipedia.org/wiki/Oka's_coherence_theorem
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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 in fact isomorphic; these include all finite stationary stochastic processes, including Markov chains and subshifts of finite type, Anosov flows and Sinai's billiards, ergodic automorphisms of the n-torus, and the continued fraction transform.
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https://en.wikipedia.org/wiki/Ornstein_isomorphism_theorem
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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.
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https://en.wikipedia.org/wiki/Ornstein–Uhlenbeck_operator
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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 Eugene Uhlenbeck.
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https://en.wikipedia.org/wiki/Ornstein–Uhlenbeck_processes
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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 time, the process tends to drift towards its mean function: such a process is called mean-reverting. The process can be considered to be a modification of the random walk in continuous time, or Wiener process, in which the properties of the process have been changed so that there is a tendency of the walk to move back towards a central location, with a greater attraction when the process is further away from the center. The Ornstein–Uhlenbeck process can also be considered as the continuous-time analogue of the discrete-time AR(1) process.
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https://en.wikipedia.org/wiki/Ornstein–Uhlenbeck_processes
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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 boundary of its disc of convergence. The result is named after the mathematicians Alexander Ostrowski and Jacques Hadamard.
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https://en.wikipedia.org/wiki/Ostrowski–Hadamard_gap_theorem
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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 versions of the construction depending on whether q is congruent to 1 or 3 (mod 4).
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https://en.wikipedia.org/wiki/Paley_construction
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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.
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https://en.wikipedia.org/wiki/Paley–Wiener_integral
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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} , then P ( Z > θ E ) ≥ ( 1 − θ ) 2 E 2 E . {\displaystyle \operatorname {P} (Z>\theta \operatorname {E} )\geq (1-\theta )^{2}{\frac {\operatorname {E} ^{2}}{\operatorname {E} }}.}
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https://en.wikipedia.org/wiki/Paley–Zygmund_inequality
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Proof: First, E = E } ] + E } ] . {\displaystyle \operatorname {E} =\operatorname {E} \}}]+\operatorname {E} \}}].}
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https://en.wikipedia.org/wiki/Paley–Zygmund_inequality
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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. ∎
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https://en.wikipedia.org/wiki/Paley–Zygmund_inequality
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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, then the Pansu derivative of a function f: G 1 → G 2 {\displaystyle f\colon G_{1}\to G_{2}} at a point x ∈ G 1 {\displaystyle x\in G_{1}} is the function D f ( x ): G 1 → G 2 {\displaystyle Df(x)\colon G_{1}\to G_{2}} defined by D f ( x ) ( y ) = lim s → 0 δ 1 / s ( f ( x ) − 1 f ( x δ s y ) ) , {\displaystyle Df(x)(y)=\lim _{s\to 0}\delta _{1/s}(f(x)^{-1}f(x\delta _{s}y))\,,} provided that this limit exists. A key theorem in this area is the Pansu–Rademacher theorem, a generalization of Rademacher's theorem, which can be stated as follows: Lipschitz continuous functions between (measurable subsets of) Carnot groups are Pansu differentiable almost everywhere.
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https://en.wikipedia.org/wiki/Pansu_derivative
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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 coefficients in either algebraic or numerical form.
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https://en.wikipedia.org/wiki/Parker–Sochacki_method
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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 papers published in 1962.
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https://en.wikipedia.org/wiki/Parry–Daniels_map
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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 English mathematician Bill Parry and the American mathematician Dennis Sullivan, who introduced the invariant in a joint paper published in the journal Topology in 1975.
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https://en.wikipedia.org/wiki/Parry–Sullivan_invariant
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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 π ∑ k = 0 ∞ | a k | 2 r 2 k , {\displaystyle \int _{0}^{2\pi }|f(re^{i\theta })|^{2}\,\mathrm {d} \theta =2\pi \sum _{k=0}^{\infty }|a_{k}|^{2}r^{2k},} which may also be written as 1 2 π ∫ 0 2 π | f ( r e i θ ) | 2 d θ = ∑ k = 0 ∞ | a k r k | 2 . {\displaystyle {\frac {1}{2\pi }}\int _{0}^{2\pi }|f(re^{i\theta })|^{2}\,\mathrm {d} \theta =\sum _{k=0}^{\infty }|a_{k}r^{k}|^{2}.}
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https://en.wikipedia.org/wiki/Parseval–Gutzmer_formula
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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.The surface was named the Peano surface (German: Peanosche Fläche) by Georg Scheffers in his 1920 book Lehrbuch der darstellenden Geometrie. It has also been called the Peano saddle.
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https://en.wikipedia.org/wiki/Peano_surface
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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 restrictive sense. For this reason, it is now more common to work with the Lebesgue measure, which is an extension of the Jordan measure to a larger class of sets. Historically speaking, the Jordan measure came first, towards the end of the nineteenth century.
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https://en.wikipedia.org/wiki/Jordan_measure
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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 {\displaystyle \mathbb {R} } each have a Jordan measure of 0, while Q ∩ {\displaystyle \mathbb {Q} \cap } , a countable union of them, is not Jordan-measurable. For this reason, some authors prefer to use the term Jordan content. The Peano–Jordan measure is named after its originators, the French mathematician Camille Jordan, and the Italian mathematician Giuseppe Peano.
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https://en.wikipedia.org/wiki/Jordan_measure
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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 optical diffraction problems The first numerical evaluation of this integral was performed by Trevor Pearcey using the quadrature formula. In optics, the Pearcey integral can be used to model diffraction effects at a cusp caustic, which corresponds to the boundary between two regions of geometric optics: on one side, each point is contained in three light rays; on the other side, each point is contained in one light ray. == References ==
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https://en.wikipedia.org/wiki/Pearcey_integral
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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, and 29. The numerators of the same sequence of approximations are half the companion Pell numbers or Pell–Lucas numbers; these numbers form a second infinite sequence that begins with 2, 6, 14, 34, and 82.
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https://en.wikipedia.org/wiki/Pell_numbers
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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 numbers can be used to find square triangular numbers, to construct integer approximations to the right isosceles triangle, and to solve certain combinatorial enumeration problems.As with Pell's equation, the name of the Pell numbers stems from Leonhard Euler's mistaken attribution of the equation and the numbers derived from it to John Pell. The Pell–Lucas numbers are also named after Édouard Lucas, who studied sequences defined by recurrences of this type; the Pell and companion Pell numbers are Lucas sequences.
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https://en.wikipedia.org/wiki/Pell_numbers
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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.
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https://en.wikipedia.org/wiki/Perkel_graph
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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 independent sets in an n-vertex cycle graph is counted by the nth Perrin number for n > 1.
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https://en.wikipedia.org/wiki/Perrin_pseudoprime
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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 }}}} (Roman 1984, 4.4.6), (Boas & Buck 1958, p.37). They are a generalization of the Boole polynomials.
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https://en.wikipedia.org/wiki/Peters_polynomials
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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.
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https://en.wikipedia.org/wiki/Peterson–Stein_formula
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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 theorem is a collection of results generalizing the significant facts about the decomposition of the regular representation of any finite group, as discovered by Ferdinand Georg Frobenius and Issai Schur.
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https://en.wikipedia.org/wiki/Peter–Weyl_theorem
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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.
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https://en.wikipedia.org/wiki/Peter–Weyl_theorem
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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 orthonormal basis of L2(G). In the case that G is the group of unit complex numbers, this last result is simply a standard result from Fourier series.
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https://en.wikipedia.org/wiki/Peter–Weyl_theorem
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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 interval with Lebesgue measure. The integral is also called the weak integral in contrast to the Bochner integral, which is the strong integral.
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https://en.wikipedia.org/wiki/Weak_integral
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