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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 *Generalized_Fourier_series#Example_(Fourier–Legendre_series), Fourier–Legendre series *Gauss–Legendre algorithm *Gauss–Legendre method *Gauss–Legendre quadrature *Legendre (crater) *Legendre chi function *Legendre duplication formula *Optimum "L" filter, 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 t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adrien-Marie Legendre
Adrien-Marie Legendre (; ; 18 September 1752 – 9 January 1833) was a French people, 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. He is also known for his contributions to the Least squares, method of least squares, and was the first to officially publish on it, though Carl Friedrich Gauss had discovered it before 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 Supérieure, École Normale from 1795. At the same time, he was associated with the Bureau des Longitudes. In 1782, the Prussian Academy of Sciences, Berlin Academy awarded Legendre a prize for his treatise on projectiles in resistant m ... [...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 a Diophantine equation of the form: ax^2+by^2+cz^2=0. The equation is named for Adrien-Marie Legendre who proved it 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 and also not all positive or all negative. References * L. E. Dickson, ''History of the Theory of Numbers. Vol.II: Diophantine Analysis'', Chelsea Publishing The Chelsea Publishing Company was a publisher of mathematical books, based in New York City New York, often called New York City (NYC), is the most populous city in the United States, located at the southern tip of New York State on on ..., 1971, . Chap.XIII, p. 422. * J.E. Cremona and D. Rusin, "Efficient solution of rational conics", Math. Comp., 72 (2003) pp.&n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Legendre's Differential Equation
In mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a wide 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, big q-Legendre polynomials, and associated Legendre functions. Definition and representation 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 stand ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Legendre's Constant
Legendre's constant is a mathematical constant occurring in a formula constructed by Adrien-Marie Legendre to approximate the behavior of the prime-counting function \pi(x). The value that corresponds precisely to its asymptotic behavior is now known to be 1. Examination of available numerical data for known values of \pi(x) led Legendre to an approximating formula. Legendre proposed in 1808 the formula y=\frac, (), as giving an approximation of y=\pi(x) with a "very satisfying precision". However, if one defines the real function B(x) by \pi(x)=\frac, and if B(x) converges to a real constant B as x tends to infinity, then this constant satisfies B = \lim_ \left( \log(x) - \right). Not only is it now known that the limit exists, but also that its value is equal to 1, somewhat less than Legendre's . Regardless of its exact value, the existence of the limit B implies the prime number theorem. Pafnuty Chebyshev proved in 1849 that if the limit ''B'' exists, it must be equ ... [...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, and is one of many open problems on the spacing of prime numbers. Prime gaps If Legendre's conjecture is true, the gap between any prime ''p'' and the next largest prime would be O(\sqrt\,), 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 ... [...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). The convex conjugate is widely used for constructing the dual problem in optimization theory, thus generalizing 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\langle x^*, x \right\rangle \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: ... [...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 (hysteresis, 2 step or onâ ... [...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 ou ... [...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 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 Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse, the inverse of a number that, when added to the ... 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] |
