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	Joseph Liouville - Wikipedia, the free encyclopedia
		Liouville graduated from the École Polytechnique in 1827.
		Liouville was also involved in politics for some time, and he became member of the Constituting Assembly in 1848.
		Liouville worked in a number of different fields in mathematics, including number theory, complex analysis, differential geometry and topology, but also mathematical physics and even astronomy.
	en.wikipedia.org /wiki/Joseph_Liouville (285 words)

	Liouville number - Wikipedia, the free encyclopedia
		A Liouville number can then be approximated "quite closely" by a sequence of rational numbers.
		In 1844, Joseph Liouville showed that numbers with this property are not just irrational, but are always transcendental (see proof below).
		A Liouville number is irrational but does not have this property, so it can't be algebraic and must be transcendental.
	en.wikipedia.org /wiki/Liouville_number (725 words)

	Liouville (Site not responding. Last check: 2007-10-21)
		Liouville had already gained an international reputation with papers published in Crelle's Journal but at the same time the quality of Crelle's Journal made him aware of deficiencies in the avenues for mathematical publications which there were in France.
		In 1837 Liouville was appointed to lecture at the Collège de France as a substitute for Biot.
		Liouville investigated criteria for integrals of algebraic functions to be algebraic during the period 1832-33.
	www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Liouville.html (1814 words)

	Joseph Liouville (Site not responding. Last check: 2007-10-21)
		Liouville worked in a number of different in mathematics including number theory complex analysis differential geometry but also mathematical physics and even astronomy.
		He is remembered particularly for Liouville's theorem a nowadays rather basic result in analysis.
		In number theory he was the to prove the existence of transcendental numbers by a construction using continued fractions (Liouville numbers).
	www.freeglossary.com /Liouville (477 words)

	Liouville (Site not responding. Last check: 2007-10-21)
		Liouville was also involved in politics forsome time, and he became member of the Constituting Assembly in1848.
		In numbertheory, he was the first to prove the existence of transcendental numbers by a construction using continued fractions (Liouville numbers).In mathematical physics, the Sturm-Liouville theory which was joint work with Sturm is now a standard procedure to solve certain types of integral equations.
		Moreover, there is a second " Liouville's theorem " in Hamiltoniandynamics.
	www.therfcc.org /liouville-32809.html (257 words)

	Liouville, Joseph - Hutchinson encyclopedia article about Liouville, Joseph (Site not responding. Last check: 2007-10-21)
		Liouville was born in St Omer, Pas-de-Calais, and studied in Paris at the Ecole Polytechnique and the Ecole des Ponts et Chaussées.
		In collaboration with Jacques Charles-François Sturm, Liouville published papers in 1836 on vibration, which laid the foundations of the theory of linear differential equations.
		This information should not be considered complete, up to date, and is not intended to be used in place of a visit, consultation, or advice of a legal, medical, or any other professional.
	encyclopedia.farlex.com /Liouville%2c+Joseph (233 words)

	Search Results for Liouville (Site not responding. Last check: 2007-10-21)
		In 1837 Liouville was appointed to lecture at the College de France as a substitute for Biot.
		Liouville was in many ways someone who Libri should not have competed with, for he was an outstanding mathematician who could usually come up with a more elegant proof of Libri's results than he could himself.
		Liouville made a number of very important mathematical discoveries while working on the theory of perturbations including the discovery of Liouville's theorem "when a bounded domain in phase space evolves according to Hamilton's equations its volume is conserved".
	www-groups.dcs.st-and.ac.uk /~history/Search/historysearch.cgi?SUGGESTION=Liouville&CONTEXT=1 (3224 words)

	Liouville function - Wikipedia, the free encyclopedia
		The Liouville function, denoted by λ(n) and named after Joseph Liouville, is an important function in number theory.
		The Liouville function is related to the Riemann zeta function by the formula
		Haselgrove, C.B. A disproof of a conjecture of Polya.
	www.wikipedia.org /wiki/Liouville_function (198 words)

	Joseph Liouville
		In 1838, Liouville was appointed Professor of Analysis and Mechanics at the Ecole Polytechnique.
		Sturm and Liouville examined general linear second order differential equations and examined properties of their eigenvalues, the behaviour of the eigenfunctions and the series expansion of arbitrary functions in terms of these eigenfunctions.
		Liouville was also a major influence in bringing Galois's work to general notice when he published Galois's notes in 1846.
	www.stetson.edu /~efriedma/periodictable/html/Lu.html (742 words)

	Liouville's theorem: Definition and Links by Encyclopedian.com - All about Liouville's theorem (Site not responding. Last check: 2007-10-21)
		Liouville's theorem in complex analysis states that every entire function (a holomorphic function f(z) defined on the whole complex plane C) which is bounded (i.e.
		Liouville's theorem can be used to give an elegant short proof for the fundamental theorem of algebra.
		The theorem is considerably improved by Picard's little theorem[?], which says that every entire function whose image omits at least two complex numbers must be constant.
	www.encyclopedian.com /li/Liouville%27s-theorem.html (189 words)

	Liouville's theorem (Site not responding. Last check: 2007-10-21)
		We now prove Liouville's theorem which is one of the fundamental theorems of statistical mechanics.
		Liouville's theorem states that the time derivative of this density as we move along the trajectory is zero, i.e.,
		To establish Liouville's theorem, we show that the RHS of this equation is zero.
	astron.berkeley.edu /~jrg/ay202/node27.html (794 words)

	Citebase - Structure Constants and Conformal Bootstrap in Liouville Field Theory (Site not responding. Last check: 2007-10-21)
		An analytic expression is proposed for the three-point function of the exponential fields in the Liouville field theory on a sphere.
		Using this function as the structure constant of the operator algebra we construct the four-point function of the exponential fields and verify numerically that it satisfies the conformal bootstrap equations, i.e., that the operator algebra thus defined is associative.
		Liouville field theory is considered with boundary conditions corresponding to a quantization of the classical Lobachevskiy plane (i.e.
	citebase.eprints.org /cgi-bin/citations?archiveID=oai:arXiv.org:hep-th/9506136 (1097 words)

	Liouville, Joseph (Site not responding. Last check: 2007-10-21)
		Liouville became professor at the École Polytechnique, Paris, in 1833.
		The Liouville theorem concerning the measure-preserving property of the Hamiltonian dynamics is basic to statistical mechanics and measure theory.
		In analysis Liouville was the first to deduce the theory of doubly periodic functions from general theorems (including his own) in the theory of analytic functions of a complex variable.
	www.phy.bg.ac.yu /web_projects/giants/liouvi~1.htm (309 words)

	Joseph Liouville - netlexikon (Site not responding. Last check: 2007-10-21)
		Liouville war auch zeitweise politisch aktiv und wurde 1848 in die Nationalversammlung gewählt.
		Liouville arbeitete in zahlreichen mathematischen Teilgebieten, darunter Zahlentheorie, Funktionentheorie und Differentialgeometrie, aber auch in mathematischer Physik und sogar in Astronomie.
		Liouville war auch der erste, dem ein Beweis für die Existenz transzendenter Zahlen gelang, indem er eine unendliche Klasse solcher Zahlen als Kettenbrüche konstruierte (Liouville-Zahlen).
	www.lexikon-definition.de /Joseph-Liouville.html (263 words)

	PlanetMath: proof of the fundamental theorem of algebra (Liouville's theorem)
		"proof of the fundamental theorem of algebra (Liouville's theorem)" is owned by Evandar.
		Cross-references: Liouville's theorem, compact set, bounded, continuous, entire, holomorphic, constant, root, polynomial
		This is version 3 of proof of the fundamental theorem of algebra (Liouville's theorem), born on 2002-02-13, modified 2004-09-16.
	planetmath.org /encyclopedia/ProofOfTheFundamentalTheoremOfAlgebra.html (110 words)

	Reports of the Mathematical Institute Leiden (Site not responding. Last check: 2007-10-21)
		Liouville's inequality gives an elementary lower bound for the difference of two algebraic numbers in terms of the heights of these numbers.
		The Thue-Siegel-Roth theorem may be viewed as a one-sided improvement of Liouville's inequality in which one fixes an algebraic number a and asserts that there are only finitely many numbers b in a given number field L which are close to a but not as close as in Liouville's inequality.
		In a symmetric improvement of Liouville's inequality one takes two number fields K and L and asserts that there are only finitely many pairs a in K, b in L which are close to each other but not as close as in Liouville's inequality.
	www.math.leidenuniv.nl /reports/1999-07.shtml (166 words)

	Geometry.Net - Scientists: Liouville Joseph
		Joseph Liouville's father was an army captain in Napoleon's armyso Joseph had to spend the first few years of his life with his uncle.
		In 1831 Liouville was appointed to his first academic post, as assistant to Claude Mathieu In 1836 Liouville founded a mathematics journal.
		Liouville had already gained an international reputation with papers published in Crelle 's Journal but at the same time the quality of Crelle 's Journal made him aware of deficiencies in the avenues for mathematical publications which there were in France.
	www.geometry.net /scientists/liouville_joseph.php (1685 words)

	Liouville equation - Information
		Liouville's theorem is also important in the study of symplectic topology, where it is formulated rather differently.
		The Liouville equation describes the time evolution of phase space distribution function (while density is the correct term from mathematics, physicists generally call it a distribution).
		The Liouville equation is integral to the proof of the fluctuation theorem from which the second law of thermodynamics can be derived.
	www.logicjungle.com /wiki/Liouville_equation (807 words)

	Extension of Liouville Formalism to Postinstability Dynamics (Site not responding. Last check: 2007-10-21)
		The means chosen to accomplish this is coupling of the governing dynamical equations with the corresponding Liouville equation that describes the evolution of the flow of error probability.
		Because the modified Liouville equation does not depend on the modified dynamical equations, the modified Liouville equation can be solved in advance, so that the stabilizing force becomes a known function.
		The modified Liouville equation is solved subject to a normalization constrain and to an initial condition (an initial probability distribution) that can be specified somewhat arbitrarily.
	www.nasatech.com /Briefs/Sept03/NPO30393.html (548 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		The first rigorous proofs that some differential equations are not solvable by quadrature were obtained in the 1830's by Liouville.
		Liouville was undoubtedly inspired by the results of Lagrange, Abel and Galois on the nonsolvability of algebraic equations by radicals.
		It is interesting that the Picard-Vessiot approach is close to the Liouville approach.
	www.math.toronto.edu /askold/obstr.txt (355 words)

	Le résultat de votre recherche
		Liouville, J. Théorème concernant la fonction numérique relative au nombre des représentations d'un entier sous la forme d'une somme de trois carrés.
		Liouville, J. Nombre des représentations du double d'un entier impair sous la forme d'une somme de douze carrés.
		Liouville, J. Théorème concernant les nombres premiers de l'une ou de l'autre des deux formes $40\mu+11,40\mu+19$.
	math-doc.ujf-grenoble.fr /cgi-bin/edbm_jmpa/JMPA?&srlgc=and&maxdocs=1000&format=jcomplete&type=html&py=1860 (526 words)

	Description
		Often, the equation of motion for the density matrix is written using superoperators in Liouville space rather than in terms of Hilbert space operators.
		We next consider the situation where the system is evolving under the effects of relaxation in the liquid state as treated by Redfield theory.
		When the value of ls is non-zero (it will be the Liouville space dimension) and the matrix siginf is empty, the acquisition is using a Liouville space treatment.
	www.gamma.ethz.ch /html/modules/level2/acquire5.htm (2977 words)

	Liouville's Theorem and Phase Space (Site not responding. Last check: 2007-10-21)
		Liouville's Theorem states that "the density of systems in the neighborhood of some given system in phase space remains constant in time" (Goldstein).
		The purpose of this project and presentation is to prove Liouville's Theorem and to explain what it means, specifically in terms of the time evolution of a phase space plot.
		The final topic to be covered in the project is a brief overview of basic applications of Liouville's Theorem in branches of physics such as statistical mechanics and quantum mechanics.
	filebox.vt.edu /users/mylinh/school/phys3356 (281 words)

	References for Liouville (Site not responding. Last check: 2007-10-21)
		J Lützen, Joseph Liouville's Contribution to the Theory of Integral Equations, Historia Mathematica 9 (1982), 371-391.
		J Lützen, Joseph Liouville's Work on the Figures of Equilibrium of a Rotating Mass of fluid, Rev.
		J Lützen, The birth of spectral theory - Joseph Liouville's contributions, in Proceedings of the International Congress of Mathematicians (Tokyo, 1991), 1651-1663.
	www-groups.dcs.st-and.ac.uk /~history/References/Liouville.html (230 words)

	Liouville's theorem for non-Hamiltonian systems
		Furthermore, a generalized Liouville equation for non-Hamiltonian systems can be derived which incorporates this metric factor.
		Thus, we have derived a modified version of Liouville's theorem and have shown that it leads to a conservation law for f equivalent to the Hamiltonian case.
		This, then, supports the generality of the Liouville equation for both Hamiltonian and non-Hamiltonian based ensembles, an important fact considering that this equation is the foundation of statistical mechanics.
	www.nyu.edu /classes/tuckerman/stat.mech/lectures/lecture_2/node3.html (323 words)

	The Hutchinson Dictionary of Scientific Biography: Liouville, Joseph (1809-1882)@ HighBeam Research (Site not responding. Last check: 2007-10-21)
		was a French mathematician who wrote prolifically on problems of analysis, but who is famous chiefly as the founder and first editor of the learned journal popularly known as the Journal de Liouville.
		Liouville was born at St Omer, Pas-de-Calais, on 24 March 1809 and studied at Commery and Toul before entering the Ecole Polytechnique in Paris in 1825.
		In 1827 he transferred to the Ecole des Ponts et Chausses, where he received his baccalaureate in 1830.
	www.highbeam.com /library/doc0.asp?DOCID=1P1:28910433&refid=ip_encyclopedia_hf (180 words)

	The Liouville operator and the Poisson bracket (Site not responding. Last check: 2007-10-21)
		The Liouville equation is the foundation on which statistical mechanics rests.
		It will now be cast in a form that will be suggestive of a more general structure that has a definite quantum analog (to be revisited when we treat the quantum Liouville equation).
		The Liouville equation is a partial differential equation for the phase space probability distribution function.
	www.nyu.edu /classes/tuckerman/stat.mech/lectures/lecture_2/node1.html (152 words)

	Liouville's Theorem (Site not responding. Last check: 2007-10-21)
		Establishment of a transport equation in a system of localised electron states...
		Liouville's theorem: a bounded analytic function is constant...
		Liouville's Theorem -- from Eric Weisstein's World of Physics...
	www.scienceoxygen.com /math/461.html (193 words)

	Liouville (crater) -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-21)
		Liouville is a small (Click link for more info and facts about lunar) lunar (Click link for more info and facts about impact crater) impact crater that is located near the eastern limb of the (Any natural satellite of a planet) Moon.
		It lies to the southeast of the larger (Click link for more info and facts about Dubyago crater) Dubyago crater, and was previously designated 'Dubyago S' before being given a name by the (Click link for more info and facts about IAU) IAU.
		Further to the west is the comparably-sized (Click link for more info and facts about Respighi crater) Respighi crater.
	www.absoluteastronomy.com /encyclopedia/L/Li/Liouville_(crater).htm (151 words)

	DicoMaths : Nombres de Liouville
		Il y a une infinité de nombres de Liouville, comme par exemple le nombre 1+1/2!+1/3!+1/4!+1/5!+....
		Liouville est le premier en 1844 à avoir exhibé des nombres transcendants, les nombres de Liouville.
		Hermite a prouvé ensuite en 1873 que e est transcendant, et Lindemann a fait de même pour pi en 1882.
	www.bibmath.net /dico/index.php3?action=affiche&quoi=./n/nbliouville.html (100 words)

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