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List Of Things Named After Adrien-Marie Legendre
Adrien-Marie Legendre (1752–1833) is the eponym of all of the things listed below. *26950 Legendre *Associated Legendre polynomials * Fourier–Legendre series *Gauss–Legendre algorithm *Gauss–Legendre method *Gauss–Legendre quadrature *Legendre (crater) *Legendre chi function * Legendre duplication formula * Legendre–Papoulis filter *Legendre form *Legendre function * Legendre moment *Legendre polynomials *Legendre pseudospectral method *Legendre rational functions * Legendre relation *Legendre sieve *Legendre symbol * Legendre transformation **Legendre transform (integral transform) **Finite Legendre transform *Legendre wavelet *Legendre–Clebsch condition *Legendre–Fenchel transformation * Legendre's conjecture *Legendre's constant * Legendre's differential equation *Legendre's equation * Legendre's formula *Legendrian knot *Legendrian submanifold * Saccheri–Legendre theorem * Legendre's theorem on spherical triangles *Legendre's three-square theorem In mathe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adrien-Marie Legendre
Adrien-Marie Legendre (; ; 18 September 1752 – 9 January 1833) was a French mathematician who made numerous contributions to mathematics. Well-known and important concepts such as the Legendre polynomials and Legendre transformation are named after him. Life Adrien-Marie Legendre was born in Paris on 18 September 1752 to a wealthy family. He received his education at the Collège Mazarin in Paris, and defended his thesis in physics and mathematics in 1770. He taught at the École Militaire in Paris from 1775 to 1780 and at the École Normale from 1795. At the same time, he was associated with the Bureau des Longitudes. In 1782, the Berlin Academy awarded Legendre a prize for his treatise on projectiles in resistant media. This treatise also brought him to the attention of Lagrange. The '' Académie des sciences'' made Legendre an adjoint member in 1783 and an associate in 1785. In 1789, he was elected a Fellow of the Royal Society. He assisted with the Anglo-French ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Legendre Relation
In mathematics, Legendre's relation can be expressed in either of two forms: as a relation between complete elliptic integrals, or as a relation between periods and quasiperiods of elliptic functions. The two forms are equivalent as the periods and quasiperiods can be expressed in terms of complete elliptic integrals. It was introduced (for complete elliptic integrals) by . Complete elliptic integrals Legendre's relation stated using complete elliptic integrals is : K'E + KE' - KK' = \frac \pi 2 where ''K'' and ''K''′ are the complete elliptic integrals of the first kind for values satisfying , and ''E'' and ''E''′ are the complete elliptic integrals of the second kind. This form of Legendre's relation expresses the fact that the Wronskian of the complete elliptic integrals (considered as solutions of a differential equation) is a constant. Elliptic functions Legendre's relation stated using elliptic functions is : \omega_2 \eta_1 - \omega_1 \eta_2 = 2\pi i \, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Legendre's Formula
In mathematics, Legendre's formula gives an expression for the exponent of the largest power of a prime ''p'' that divides the factorial ''n''!. It is named after Adrien-Marie Legendre. It is also sometimes known as de Polignac's formula, after Alphonse de Polignac. Statement For any prime number ''p'' and any positive integer ''n'', let \nu_p(n) be the exponent of the largest power of ''p'' that divides ''n'' (that is, the ''p''-adic valuation of ''n''). Then :\nu_p(n!) = \sum_^ \left\lfloor \frac \right\rfloor, where \lfloor x \rfloor is the floor function. While the sum on the right side is an infinite sum, for any particular values of ''n'' and ''p'' it has only finitely many nonzero terms: for every ''i'' large enough that p^i > n, one has \textstyle \left\lfloor \frac \right\rfloor = 0. This reduces the infinite sum above to :\nu_p(n!) = \sum_^ \left\lfloor \frac \right\rfloor \, , where L = \lfloor \log_ n \rfloor. Example For ''n'' = 6, one has 6! = 720 = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Legendre's Equation
In mathematics, Legendre's equation is the Diophantine equation ax^2+by^2+cz^2=0. The equation is named for Adrien-Marie Legendre who proved in 1785 that it is solvable in integers ''x'', ''y'', ''z'', not all zero, if and only if −''bc'', −''ca'' and −''ab'' are quadratic residues modulo ''a'', ''b'' and ''c'', respectively, where ''a'', ''b'', ''c'' are nonzero, square-free, pairwise relatively prime integers, not all positive or all negative . References * L. E. Dickson Leonard Eugene Dickson (January 22, 1874 – January 17, 1954) was an American mathematician. He was one of the first American researchers in abstract algebra, in particular the theory of finite fields and classical groups, and is also remem ..., '' History of the Theory of Numbers. Vol.II: Diophantine Analysis'', Chelsea Publishing, 1971, . Chap.XIII, p. 422. * J.E. Cremona and D. Rusin, "Efficient solution of rational conics", Math. Comp., 72 (2003) pp. 1417-1441. D ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Legendre's Differential Equation
In physical science and mathematics, Legendre polynomials (named after Adrien-Marie Legendre, who discovered them in 1782) are a system of complete and orthogonal polynomials, with a vast number of mathematical properties, and numerous applications. They can be defined in many ways, and the various definitions highlight different aspects as well as suggest generalizations and connections to different mathematical structures and physical and numerical applications. Closely related to the Legendre polynomials are associated Legendre polynomials, Legendre functions, Legendre functions of the second kind, and associated Legendre functions. Definition by construction as an orthogonal system In this approach, the polynomials are defined as an orthogonal system with respect to the weight function w(x) = 1 over the interval 1,1/math>. That is, P_n(x) is a polynomial of degree n, such that \int_^1 P_m(x) P_n(x) \,dx = 0 \quad \text n \ne m. With the additional standardization condi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Legendre's Constant
Legendre's constant is a mathematical constant occurring in a formula conjectured by Adrien-Marie Legendre to capture the asymptotic behavior of the prime-counting function \pi(x). Its value is now known to be 1. Examination of available numerical evidence for known primes led Legendre to suspect that \pi(x) satisfies an approximate formula. Legendre conjectured in 1808 that : \pi(x) = \frac where \lim_ B(x) = 1.08366 .... Or similarly, :\lim_ \left( \ln(n) - \right)= B where ''B'' is Legendre's constant. He guessed ''B'' to be about 1.08366, but regardless of its exact value, the existence of ''B'' implies the prime number theorem. Pafnuty Chebyshev proved in 1849 that if the limit ''B'' exists, it must be equal to 1. An easier proof was given by Pintz in 1980. It is an immediate consequence of the prime number theorem, under the precise form with an explicit estimate of the error term : \pi(x)= (x) + O \left(x e^\right) \quad\text x \to \infty (for some posit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Legendre's Conjecture
Legendre's conjecture, proposed by Adrien-Marie Legendre, states that there is a prime number between n^2 and (n+1)^2 for every positive integer n. The conjecture is one of Landau's problems (1912) on prime numbers; , the conjecture has neither been proved nor disproved. Prime gaps If Legendre's conjecture is true, the gap between any prime ''p'' and the next largest prime would be O(\sqrt p), as expressed in big O notation. It is one of a family of results and conjectures related to prime gaps, that is, to the spacing between prime numbers. Others include Bertrand's postulate, on the existence of a prime between n and 2n, Oppermann's conjecture on the existence of primes between n^2, n(n+1), and (n+1)^2, Andrica's conjecture and Brocard's conjecture on the existence of primes between squares of consecutive primes, and Cramér's conjecture that the gaps are always much smaller, of the order (\log p)^2. If Cramér's conjecture is true, Legendre's conjecture would follow for all ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Legendre–Fenchel Transformation
In mathematics and mathematical optimization, the convex conjugate of a function is a generalization of the Legendre transformation which applies to non-convex functions. It is also known as Legendre–Fenchel transformation, Fenchel transformation, or Fenchel conjugate (after Adrien-Marie Legendre and Werner Fenchel). It allows in particular for a far reaching generalization of Lagrangian duality. Definition Let X be a real topological vector space and let X^ be the dual space to X. Denote by :\langle \cdot , \cdot \rangle : X^ \times X \to \mathbb the canonical dual pairing, which is defined by \left( x^*, x \right) \mapsto x^* (x). For a function f : X \to \mathbb \cup \ taking values on the extended real number line, its is the function :f^ : X^ \to \mathbb \cup \ whose value at x^* \in X^ is defined to be the supremum: :f^ \left( x^ \right) := \sup \left\, or, equivalently, in terms of the infimum: :f^ \left( x^ \right) := - \inf \left\. This definition can b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Legendre–Clebsch Condition
__NOTOC__ In the calculus of variations the Legendre–Clebsch condition is a second-order condition which a solution of the Euler–Lagrange equation must satisfy in order to be a minimum. For the problem of minimizing : \int_^ L(t,x,x')\, dt . \, the condition is :L_(t,x(t),x'(t)) \ge 0, \, \forall t \in ,b/math> Generalized Legendre–Clebsch In optimal control, the situation is more complicated because of the possibility of a singular solution. The generalized Legendre–Clebsch condition, also known as convexity, is a sufficient condition for local optimality such that when the linear sensitivity of the Hamiltonian to changes in u is zero, i.e., : \frac = 0 The Hessian of the Hamiltonian is positive definite along the trajectory of the solution: : \frac > 0 In words, the generalized LC condition guarantees that over a singular arc, the Hamiltonian is minimized. See also * Bang–bang control In control theory, a bang–bang controller (2 step or on–off controlle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Legendre Wavelet
In functional analysis, compactly supported wavelets derived from Legendre polynomials are termed Legendre wavelets or spherical harmonic wavelets. Legendre functions have widespread applications in which spherical coordinate system is appropriate.Colomer and Colomer As with many wavelets there is no nice analytical formula for describing these harmonic spherical wavelets. The low-pass filter associated to Legendre multiresolution analysis is a finite impulse response (FIR) filter. Wavelets associated to FIR filters are commonly preferred in most applications. An extra appealing feature is that the Legendre filters are ''linear phase'' FIR (i.e. multiresolution analysis associated with linear phase filters). These wavelets have been implemented on MATLAB (wavelet toolbox). Although being compactly supported wavelet, legdN are not orthogonal (but for ''N'' = 1). Legendre multiresolution filters Associated Legendre polynomials are the colatitudinal part of the spherical harmonics w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Legendre Transform
The finite Legendre transform (fLT) transforms a mathematical function defined on the finite interval into its Legendre spectrum. Conversely, the inverse fLT (ifLT) reconstructs the original function from the components of the Legendre spectrum and the Legendre polynomials, which are orthogonal on the interval ˆ’1,1 Specifically, assume a function ''x''(''t'') to be defined on an interval ˆ’1,1and discretized into ''N'' equidistant points on this interval. The fLT then yields the decomposition of ''x''(''t'') into its spectral Legendre components, :L_x (k) = \frac\sum_^x(t)P_k(t), where the factor (2''k'' + 1)/''N'' serves as normalization factor and ''L''''x''(''k'') gives the contribution of the ''k''-th Legendre polynomial to ''x''(''t'') such that (ifLT) :x(t) = \sum_k L_x(k) P_k(t). The fLT should not be confused with the Legendre transform or Legendre transformation used in thermodynamics and quantum physics. Legendre filter The fLT of a noisy experimental o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Legendre Transform (integral Transform)
In mathematics, Legendre transform is an integral transform named after the mathematician Adrien-Marie Legendre, which uses Legendre polynomials In physical science and mathematics, Legendre polynomials (named after Adrien-Marie Legendre, who discovered them in 1782) are a system of complete and orthogonal polynomials, with a vast number of mathematical properties, and numerous applicat ... P_n(x) as kernels of the transform. Legendre transform is a special case of Jacobi transform. The Legendre transform of a function f(x) isChurchill, R. V., and C. L. Dolph. "Inverse transforms of products of Legendre transforms." Proceedings of the American Mathematical Society 5.1 (1954): 93–100. :\mathcal_n\ = \tilde f(n) = \int_^1 P_n(x)\ f(x) \ dx The inverse Legendre transform is given by :\mathcal_n^\ = f(x) = \sum_^\infty \frac \tilde f(n) P_n(x) Associated Legendre transform Associated Legendre transform is defined as :\mathcal_\ = \tilde f(n,m) = \int_^1 (1-x^2)^P_n^m(x) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
