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List Of Things Named After Joseph Liouville
{{refimprove, date=February 2016 Several concepts from mathematics and physics are named after the French mathematician Joseph Liouville. *Euler–Liouville equation *Liouville–Arnold theorem *Liouville–Bratu–Gelfand equation * Liouville–Green method *Liouville's equation *Liouville's formula *Liouville function * Liouville dynamical system *Liouville field theory * Liouville gravity * Liouville integrability *Liouville measure *Liouville number * Liouville one-form *Liouville operator *Liouville space * Liouville surface *Liouville–Neumann series *Liouvillian function *Riemann–Liouville integral * Quantum Liouville equation *Sturm–Liouville theory Liouville's theorem *Liouville's theorem (complex analysis) * Liouville's theorem (harmonic functions) *Liouville's theorem (conformal mappings) *Liouville's theorem (differential algebra) * Liouville's theorem (diophantine approximation) *Liouville's theorem (Hamiltonian) In physics, Liouville's theorem, named after th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Liouville Operator
In physics, Liouville's theorem, named after the French mathematician Joseph Liouville, is a key theorem in classical statistical and Hamiltonian mechanics. It asserts that ''the phase-space distribution function is constant along the 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. There are related mathematical results in symplectic topology and ergodic theory; systems obeying Liouville's theorem are examples of incompressible dynamical systems. There are extensions of Liouville's theorem to stochastic systems. Liouville equations The Liouville equation describes the time evolution of the ''phase space distribution function''. Although the equation is usually referred to as the "Liouville equation", Josiah Willard Gibbs was the first to recognize the impor ... [...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 1 + \log_2(d) ~) no pair of integers ~(\,p,\,q\,)~ exists that simultaneously satisfies the pair of bracketing inequalities :0 0 ~, then, since c\,q - d\,p is an integer, we can assert the sharper inequality \left, c\,q - d\,p \ \ge 1 ~. From this it follows that :\left, x - \frac\= \frac \ge \frac Now for any integer ~n > 1 + \log_2(d)~, the last inequality above implies :\left, x - \frac \ \ge \frac > \frac \ge \frac ~. Therefore, in the case ~ \left, c\,q - d\,p \ > 0 ~ such pair of integers ~(\,p,\,q\,)~ would violate the ''second'' inequality in the definition of a Liouville number, for some positive integer . We conclude that there is no pair of integers ~(\,p,\,q\,)~, with ~ q > 1 ~, that would qualify such an ~ x = c / d ~, as a Liouville number. Hence a Liouville number, if ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Liouville's Theorem (differential Algebra)
In mathematics, Liouville's theorem, originally formulated by 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, must be in the same differential field as the function, plus possibly a finite number of logarithms. 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) for some transcendental t) such that D t = ... [...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 theorem about conformal mappings in Euclidean space. It states that any smooth conformal mapping on a domain of R''n'', where ''n'' > 2, can be expressed as a composition of translations, similarities, orthogonal transformations and inversions: they are 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 R3 and higher-dimensional spaces. By contrast, conformal mappings in R2 can be much more complicated – for example, all simply connected planar domains are conformally equivalent, by the Riemann mapping theorem. Generalizations of the theorem hold for transformations that are only weakly differentiable . The focus of such a study is the non-linear Cauchy–Ri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harmonic Function
In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f: 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 "harmonic" ... [...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. Proof This important theorem has several proofs. A standard analytical proof uses the fact that holomorphic functions are analytic. Another proof uses the mean value property of harmonic functions. The proof can be adapted to the case where the harmonic function f is merely bounded above or below. See Harmonic function#Liouville's theorem. Corollaries Fundamental theorem of algebra There is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sturm–Liouville Theory
In mathematics and its applications, classical Sturm–Liouville theory is the theory of ''real'' second-order ''linear'' ordinary differential equations of the form: for given coefficient functions , , and , an unknown function ''y = y''(''x'') of the free variable , and an unknown constant λ. All homogeneous (i.e. with the right-hand side equal to zero) second-order linear ordinary differential equations can be reduced to this form. In addition, the solution is typically required to satisfy some boundary conditions at extreme values of ''x''. Each such equation () together with its boundary conditions constitutes a Sturm–Liouville problem. In the simplest case where all coefficients are continuous on the finite closed interval and has continuous derivative, a function ''y = y''(''x'') is called a ''solution'' if it is continuously differentiable and satisfies the equation () at every x\in (a,b). In the case of more general , , , the solutions must be understood in a weak ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Method Of Quantum Characteristics
Quantum characteristics are phase-space trajectories that arise in the phase space formulation of quantum mechanics through the Wigner transform of Heisenberg operators of canonical coordinates and momenta. These trajectories obey the Hamilton equations in quantum form and play the role of characteristics in terms of which time-dependent Weyl's symbols of quantum operators can be expressed. In the classical limit, quantum characteristics reduce to classical trajectories. The knowledge of quantum characteristics is equivalent to the knowledge of quantum dynamics. Weyl–Wigner association rule In Hamiltonian dynamics, classical systems with n degrees of freedom are described by 2n canonical coordinates and momenta :\xi^ = (x^1, \ldots , x^n, p_1, \ldots , p_n) \in \R^, that form a coordinate system in the phase space. These variables satisfy the Poisson bracket relations :\=-I^. The skew-symmetric matrix I^, :\left\, I\right\, = \begin 0 & -E_ \\ E_ & 0 \end, where E_n i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann–Liouville Integral
In mathematics, the Riemann–Liouville integral associates with a real function f: \mathbb \rightarrow \mathbb another function of the same kind for each value of the parameter . The integral is a manner of generalization of the repeated antiderivative of in the sense that for positive integer values of , is an iterated antiderivative of of order . The Riemann–Liouville integral is named for Bernhard Riemann and Joseph Liouville, the latter of whom was the first to consider the possibility of fractional calculus in 1832. The operator agrees with the Euler transform, after Leonhard Euler, when applied to analytic functions. It was generalized to arbitrary dimensions by Marcel Riesz, who introduced the Riesz potential. Definition The Riemann–Liouville integral is defined by :I^\alpha f(x) = \frac\int_a^xf(t)(x-t)^\,dt where is the gamma function and is an arbitrary but fixed base point. The integral is well-defined provided is a locally integrable function, and is a co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Liouvillian Function
In mathematics, the Liouvillian functions comprise a set of functions including the elementary functions and their repeated integrals. Liouvillian functions can be recursively defined as integrals of other Liouvillian functions. More explicitly, a Liouvillian function is a function of one variable which is the composition of a finite number of arithmetic operations , exponentials, constants, solutions of algebraic equations (a generalization of ''n''th roots), and antiderivatives. The logarithm function does not need to be explicitly included since it is the integral of 1/x. It follows directly from the definition that the set of Liouvillian functions is closed under arithmetic operations, composition, and integration. It is also closed under differentiation. It is not closed under limits and infinite sums. Liouvillian functions were introduced by Joseph Liouville in a series of papers from 1833 to 1841. Examples All elementary functions are Liouvillian. Examples of we ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Liouville–Neumann Series
In mathematics, the Liouville–Neumann series is an infinite series that corresponds to the resolvent formalism technique of solving the Fredholm integral equations in Fredholm theory. Definition The Liouville–Neumann (iterative) series is defined as :\phi\left(x\right) = \sum^\infty_ \lambda^n \phi_n \left(x\right) which, provided that \lambda is small enough so that the series converges, is the unique continuous solution of the Fredholm integral equation of the second kind, If the ''n''th iterated kernel is defined as ''n''−1 nested integrals of ''n'' operators , :K_n\left(x,z\right) = \int\int\cdots\int K\left(x,y_1\right)K\left(y_1,y_2\right) \cdots K\left(y_, z\right) dy_1 dy_2 \cdots dy_ then :\phi_n\left(x\right) = \int K_n\left(x,z\right)f\left(z\right)dz with :\phi_0\left(x\right) = f\left(x\right)~, so ''K''0 may be taken to be . The resolvent (or solving kernel for the integral operator) is then given by a schematic analog "geometric series", :R\left(x, z;\lambd ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
