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Liouville's Theorem (transcendence Theory)
Liouville's theorem has various meanings, all mathematical results named after Joseph Liouville: * In complex analysis, see Liouville's theorem (complex analysis) ** There is also a related theorem on harmonic functions * In conformal mappings, see Liouville's theorem (conformal mappings) * In Hamiltonian mechanics, see Liouville's theorem (Hamiltonian) and Liouville–Arnold theorem * In linear differential equations, see Liouville's formula * In transcendence theory and diophantine approximations, the theorem that any Liouville number is transcendental * In differential algebra, see Liouville's theorem (differential algebra) * In differential geometry, see Liouville's equation * In coarse-grained modelling, see Liouville's equation : ''For Liouville's equation in dynamical systems, see Liouville's theorem (Hamiltonian).'' : ''For Liouville's equation in quantum mechanics, see Von Neumann equation.'' : ''For Liouville's equation in Euclidean space, see Liouville–Bratu� ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Joseph Liouville
Joseph Liouville ( ; ; 24 March 1809 – 8 September 1882) was a French mathematician and engineer. Life and work He was born in Saint-Omer in France on 24 March 1809. His parents were Claude-Joseph Liouville (an army officer) and Thérèse Liouville (née Balland). Liouville gained admission to the École Polytechnique in 1825 and graduated in 1827. Just like Augustin-Louis Cauchy before him, Liouville studied engineering at École des Ponts et Chaussées after graduating from the Polytechnique, but opted instead for a career in mathematics. After some years as an assistant at various institutions including the École Centrale Paris, he was appointed as professor at the École Polytechnique in 1838. He obtained a chair in mathematics at the Collège de France in 1850 and a chair in mechanics at the Faculté des Sciences in 1857. Besides his academic achievements, he was very talented in organisational matters. Liouville founded the ''Journal de Mathématiques Pures et A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Liouville's Theorem (complex Analysis)
In complex analysis, Liouville's theorem, named after Joseph Liouville (although the theorem was first proven by Cauchy in 1844), states that every bounded entire function must be constant. That is, every holomorphic function f for which there exists a positive number M such that , f(z), \leq M for all z\in\Complex is constant. Equivalently, non-constant holomorphic functions on \Complex have unbounded images. The theorem is considerably improved by Picard's little theorem, which says that every entire function whose image omits two or more complex numbers must be constant. Statement Liouville's theorem: Every holomorphic function f:\mathbb C \to \mathbb C for which there exists a positive number M such that , f(z), \leq M for all z\in\Complex is constant. More succinctly, Liouville's theorem states that every bounded entire function must be constant. Proof This important theorem has several proofs. A standard analytical proof uses the fact that holomorphic functi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harmonic Functions
In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f\colon U \to \mathbb R, where is an open subset of that satisfies Laplace's equation, that is, \frac + \frac + \cdots + \frac = 0 everywhere on . This is usually written as \nabla^2 f = 0 or \Delta f = 0 Etymology of the term "harmonic" The descriptor "harmonic" in the name "harmonic function" originates from a point on a taut string which is undergoing harmonic motion. The solution to the differential equation for this type of motion can be written in terms of sines and cosines, functions which are thus referred to as "harmonics." Fourier analysis involves expanding functions on the unit circle in terms of a series of these harmonics. Considering higher dimensional analogues of the harmonics on the unit ''n''-sphere, one arrives at the spherical harmonics. These functions satisfy Laplace's equation and, over time, "harmo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Liouville's Theorem (conformal Mappings)
In mathematics, Liouville's theorem, proved by Joseph Liouville in 1850, is a rigidity (mathematics), rigidity theorem about conformal mappings in Euclidean space. It states that every smooth function, smooth conformal mapping on a domain of R, where ''n'' > 2, can be expressed as a composition of translation (geometry), translations, similarity (geometry), similarities, orthogonal matrix, orthogonal transformations and inversive geometry#In higher dimensions, inversions: they are Möbius transformation#Higher dimensions, Möbius transformations (in ''n'' dimensions).Philip Hartman (1947Systems of Total Differential Equations and Liouville's theorem on Conformal MappingAmerican Journal of Mathematics 69(2);329–332. This theorem severely limits the variety of possible conformal mappings in R and higher-dimensional spaces. By contrast, conformal mappings in R can be much more complicated – for example, all simply connected planar domains are conformally equivalent, by the Riemann ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Liouville's Theorem (Hamiltonian)
In physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical mechanics, statistical and Hamiltonian mechanics. It asserts that ''the phase space, phase-space distribution function is constant along the Trajectory, trajectories of the system''—that is that the density of system points in the vicinity of a given system point traveling through phase-space is constant with time. This time-independent density is in statistical mechanics known as the classical a priori probability. Liouville's theorem applies to conservative systems, that is, systems in which the effects of friction are absent or can be ignored. The general mathematical formulation for such systems is the measure-preserving dynamical system. Liouville's theorem applies when there are degrees of freedom that can be interpreted as positions and momenta; not all measure-preserving dynamical systems have these, but Hamiltonian systems do. The general se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Liouville–Arnold Theorem
In dynamical systems theory, the Liouville–Arnold theorem states that if, in a Hamiltonian dynamical system with ''n'' degrees of freedom, there are also ''n'' independent, Poisson commuting first integrals of motion, and the level sets of all first integrals are compact, then there exists a canonical transformation to action-angle coordinates in which the transformed Hamiltonian is dependent only upon the action coordinates and the angle coordinates evolve linearly in time. Thus the equations of motion for the system can be solved in quadratures if the level simultaneous set conditions can be separated. The theorem is named after Joseph Liouville and Vladimir Arnold.J. Liouville, « Note sur l'intégration des équations différentielles de la Dynamique, présentée au Bureau des Longitudes le 29 juin 1853 », '' JMPA'', 1855, pdf/ref> History The theorem was proven in its original form by Liouville in 1853 for functions on \mathbb^ with canonical symplectic structure. It wa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Liouville's Formula
In mathematics, Liouville's formula, also known as the Abel–Jacobi–Liouville identity, is an equation that expresses the determinant of a square-matrix solution of a first-order system of homogeneous linear differential equations in terms of the sum of the diagonal coefficients of the system. The formula is named after the French mathematician Joseph Liouville. Jacobi's formula provides another representation of the same mathematical relationship. Liouville's formula is a generalization of Abel's identity and can be used to prove it. Since Liouville's formula relates the different linearly independent solutions of the system of differential equations, it can help to find one solution from the other(s), see the example application below. Statement of Liouville's formula Consider the -dimensional first-order homogeneous linear differential equation :y'=A(t)y on an interval of the real line, where for denotes a square matrix of dimension with real or complex entries. Le ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diophantine Approximations
In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria. The first problem was to know how well a real number can be approximated by rational numbers. For this problem, a rational number ''p''/''q'' is a "good" approximation of a real number ''α'' if the absolute value of the difference between ''p''/''q'' and ''α'' may not decrease if ''p''/''q'' is replaced by another rational number with a smaller denominator. This problem was solved during the 18th century by means of simple continued fractions. Knowing the "best" approximations of a given number, the main problem of the field is to find sharp upper and lower bounds of the above difference, expressed as a function of the denominator. It appears that these bounds depend on the nature of the real numbers to be approximated: the lower bound for the approximation of a rational number by another rational number is larg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Liouville Number
In number theory, a Liouville number is a real number x with the property that, for every positive integer n, there exists a pair of integers (p,q) with q>1 such that :0<\left, x-\frac\<\frac. The inequality implies that Liouville numbers possess an excellent sequence of approximations. In 1844, Joseph Liouville proved a bound showing that there is a limit to how well s can be approximated by rational numbers, and he defined Liouville numbers specifically so that they would have rational approximations better than the ones allowed by this bound. Liouville also exhibited examples of Liouville nu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Liouville's Theorem (differential Algebra)
In mathematics, Liouville's theorem, originally formulated by French mathematician Joseph Liouville in 1833 to 1841, places an important restriction on antiderivatives that can be expressed as elementary functions. The antiderivatives of certain elementary functions cannot themselves be expressed as elementary functions. These are called nonelementary antiderivatives. A standard example of such a function is e^, whose antiderivative is (with a multiplier of a constant) the error function, familiar from statistics. Other examples include the functions \frac and x^x. Liouville's theorem states that elementary antiderivatives, if they exist, are in the same differential field as the function, plus possibly a finite number of applications of the logarithm function. Definitions For any differential field F, the of F is the subfield \operatorname(F) = \. Given two differential fields F and G, G is called a of F if G is a simple transcendental extension of F (that is, G = F(t) fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Liouville's Equation
: ''For Liouville's equation in dynamical systems, see Liouville's theorem (Hamiltonian).'' : ''For Liouville's equation in quantum mechanics, see Von Neumann equation.'' : ''For Liouville's equation in Euclidean space, see Liouville–Bratu–Gelfand equation.'' In differential geometry, Liouville's equation, named after Joseph Liouville, is the nonlinear partial differential equation satisfied by the conformal factor of a metric on a surface of constant Gaussian curvature : :\Delta_0\log f = -K f^2, where is the flat Laplace operator :\Delta_0 = \frac +\frac = 4 \frac \frac. Liouville's equation appears in the study of isothermal coordinates in differential geometry: the independent variables are the coordinates, while can be described as the conformal factor with respect to the flat metric. Occasionally it is the square that is referred to as the conformal factor, instead of itself. Liouville's equation was also taken as an example by David Hilbert in the formulati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
