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Topic: Aleksandr Lyapunov

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	Aleksandr Lyapunov
		'''Aleksandr Mikhailovich Lyapunov''' (&1040;&1083;&1077;&1082;&1089;&1072;&1085;&1076;&1088; &1052;&1080;&1093;&1072;&1081;&1083;&1086;&1074;&1080;&1095; &1051;&1103;&1087;&1091;&1085;&1086;&1074;) (June 6 1857 – November 3 1918, all new style) was a Russian mathematician, mechanician and physicist.
		Lyapunov had already begun to study this stability in His previous two-years attempts at solving the task.
		Lyapunov lectured at the university on themes from theoretical mechanics, integrals of differential equations and the theory of probability.
	aleksandr-lyapunov.mindbit.com (1635 words)

	Aleksandr Lyapunov - Wikipedia, the free encyclopedia
		Aleksandr Mikhailovich Lyapunov (Александр Михайлович Ляпунов) (June 6, 1857 – November 3, 1918, all new style) was a Russian mathematician, mechanician and physicist.
		Lyapunov had lectured already from 1880 at the Department of Mechanics and this had taken him a lot of time.
		Lyapunov lectured at the university on themes from theoretical mechanics, integrals of differential equations and the theory of probability.
	en.wikipedia.org /wiki/Aleksandr_Mikhailovich_Lyapunov (1642 words)

	Lyapunov biography
		Lyapunov was not the only one being coached by Sechenov who was teaching his own daughter Natalia Rafailovna Sechenov at the same time.
		Lyapunov graduated in 1880 and remained at St Petersburg to undertake research.
		Lyapunov established that with variation in the angular velocity of revolution Maclaurin ellipsoids pass into Jacobi ellipsoids.
	www-history.mcs.st-and.ac.uk /history/Biographies/Lyapunov.html (1229 words)

	Knowledge King - Aleksandr Lyapunov (Site not responding. Last check: 2007-10-02)
		Aleksandr Mikhailovich Lyapunov (Александр Михайлович Ляпунов) (June 6 1857 - November 3 1918) was a Russian mathematician, mechanician and physicist.
		Lyapunov wrote his first independent scientific works under the guidance of professor of mechanics, D. Bobilev.
		Lyapunov had lectured already from 1880 at the faculty for mechanics and this had taken him a lot of time.
	www.knowledgeking.net /encyclopedia/a/al/aleksandr_lyapunov.html (1621 words)

	Lyapunov stability - Wikipedia, the free encyclopedia
		Lyapunov stability is applicable to only unforced (no control input) dynamical systems.
		Then V(x) is called a Lyapunov function candidate and the system is asymptotically stable in the sense of Lyapunov.
		Lyapunov's realisation was that stability can be proven without requiring knowledge of the true physical energy, providing a Lyapunov function can be found to satisfy the above constraints.
	www.wikipedia.org /wiki/Lyapunov_stability (237 words)

	Encyclopedia: Aleksandr Lyapunov
		Aleksandr Lyapunov File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version.
		In probability theory, Lyapunovs central limit theorem is one of the variants of the central limit theorems.
		Standard Lyapunov logistic fractal with iteration sequence AB Generalized Lyapunov logistic fractal with iteration sequence AABAB In mathematics Lyapunov fractals (also known as Markus-Lyapunov fractals) are bifurcational fractals derived from an extension of the logistic map in which the degree of the growth of the population, r, periodically switches...
	www.nationmaster.com /encyclopedia/Aleksandr-Lyapunov (3697 words)

	Lyapunov exponent - Encyclopedia, History, Geography and Biography
		The Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a measure that characterizes rate of separation of infinitesimally close trajectories.
		Lyapunov spectrum can be used as a tool in analysis of a system.
		The commonly used numerical procedures calculate the largest Lyapunov exponents and recover the others by means of the Gram-Schmidt process.
	www.arikah.net /encyclopedia/Lyapunov_test (560 words)

	Aleksandr Lyapunov -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-02)
		Lyapunov was born in (Click link for more info and facts about Yaroslavl) Yaroslavl, (Click link for more info and facts about Imperial Russia) Imperial Russia.
		Lyapunov had already begun to study this (The quality or attribute of being firm and steadfast) stability in his previous two-years attempts at solving the task.
		Lyapunov lectured at the university on themes from theoretical mechanics, (The result of a mathematical integration; F(x) is the integral of f(x) if dF/dx = f(x)) integrals of (An equation containing differentials of a function) differential equations and the (Click link for more info and facts about theory of probability) theory of probability.
	www.absoluteastronomy.com /encyclopedia/A/Al/Aleksandr_Lyapunov.htm (1992 words)

	ALEKSANDR LYAPUNOV FACTS AND INFORMATION (Site not responding. Last check: 2007-10-02)
		Aleksandr Mikhailovich Lyapunov (ÐÐ»ÐµÐºÑÐ°Ð½Ð´Ñ ÐÐ¸ÑÐ°Ð¹Ð»Ð¾Ð²Ð¸Ñ ÐÑÐ¿ÑÐ½Ð¾Ð²) (June_6 1857 – November_3 1918, all new_style) was a Russian mathematician, mechanician and physicist.
		As was said by Vladimir_Andreevich_Steklov, "Chebyshev saw in the young man such an immense research power, that he had dared to lay on him such a toilsome task".
		In 1885 he became private reader of the University at Kharkov in the chair of mechanics, where he replaced V. Imshenecky, who had been chosen as a member of the Russian_Academy_of_Sciences.
	www.amysflowershop.com /Aleksandr_Lyapunov (1595 words)

	Lyapunov (Site not responding. Last check: 2007-10-02)
		Aleksandr Mikhailovich began his education at home, then later one of his uncles R M Sechenov prepared him for entering the Gymnasium.
		Lyapunov was not the only one being coached by Sechenov who was teaching his own daughter Natalia Rafailovna Sechenov at the same time.
		Lyapunov established that with variation in the angular velocity of revolution Maclaurin ellipsoids pass into Jacobi ellipsoids.
	www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Lyapunov.html (1165 words)

	Aleksandr Lyapunov - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-02)
		Aleksandr Mikhailovich Lyapunov (Александр Михайлович Ляпунов) (June 6 1857 – November 3 1918, all new style) was a Russian mathematician, mechanician and physicist.
		After the sudden death of his father Aleksandr was educated by his uncle R. Sechenov, brother of the famous philosopher.
		Mathematics was taught at that time by Chebyshev and his students and.
	www.newlenox.us /project/wikipedia/index.php/Aleksandr_Lyapunov (1671 words)

	Read about Talk:Aleksandr Lyapunov at WorldVillage Encyclopedia. Research Talk:Aleksandr Lyapunov and learn about ... (Site not responding. Last check: 2007-10-02)
		To make 1894 even more doubtful: It would mean that Aleksandr's father would have devoted himself to his son's education at a time when the latter already had his PhD.
		So I do believe it was a mistake in an x-reference and I suspect it was really a year 1864, when Aleksandr was 7 years old.
		Lyapunov had also begun to study this stability in his previous two-years attempts at solving the task.
	encyclopedia.worldvillage.com /s/b/Talk:Aleksandr_Lyapunov (926 words)

	References for Lyapunov (Site not responding. Last check: 2007-10-02)
		V A Dobrovol'skii, A M Lyapunov's manuscripts on the theory of ordinary differential equations (Russian), Istor.-Mat.
		J Mawhin, The centennial legacy of Poincaré and Lyapunov in ordinary differential equations, Rend.
		V V Pavlovskaya, A M Lyapunov's investigations on stability of ellipsoidal figures of equilibrium of a rotating fluid mass (Russian), Analytic methods for investigating nonlinear oscillations (Kiev, 1980), 129-138.
	www.mathsoc.spb.ru /pantheon/lyapunov/ref.html (321 words)

	Aleksandr Lyapunov Summary
		After earning his degree at the university in 1880, Lyapunov decided to heed the encouragement of his professors and stayed on as a member of the school's Mechanics Department in preparation for a professorial career.
		Aleksandr Mikhailovich Lyapunov (Александр Михайлович Ляпунов) (June 6 1857 – November 3 1918, all new style) was a Russian mathematician, mechanician and physicist.
		After the death of Chebyshev in 1894, Lyapunov became in 1901 a head of applied mathematics at the University at Saint Petersburg, where he was entirely devoted to tutorage and research work.
	www.bookrags.com /Aleksandr_Lyapunov (2044 words)

	Lyapunov Page
		When the Lyapunov formula is applied to Markus' model, it produces an exponent for each pair of parameters.
		Readers who demand precision can more accurately estimate the lyapunov exponent by increasing the number of iterations and at the end of the procedure, by dividing the sum logarithms by the number of iterations.
		If the Lyapunov exponent is plotted for a succession of initial x values, it may take on a specific value, say 0.015 for a number of these initial values.
	rawilson.net /chaos/lyapunov (1745 words)

	Lyapunov time - Wikipedia, the free encyclopedia
		In mathematics, the Lyapunov time is the length of time for a dynamical system to become chaotic.
		The Lyapunov time reflects the limits of the predictability of the system.
		By convention, it is measured as the time for nearby trajectories of the system to diverge by e.
	www.wikipedia.org /wiki/Lyapunov%20time (73 words)

	Lyapunov exponent -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-02)
		Actually, the number of Lyapunov exponents is equal to the number of dimensions of the embedding phase space, but it is common to just refer to the largest one, because it determines the (The quality of being predictable) predictability of a dynamical system.
		The commonly used numerical procedures calculate the largest Lyapunov exponents and recover the others by means of the (Click link for more info and facts about Gram-Schmidt process) Gram-Schmidt process.
		The Lyapunov test, also known as the (Click link for more info and facts about Lyapunov exponent) Lyapunov exponent, the methods of approximation by (Click link for more info and facts about Aleksandr Lyapunov) Aleksandr Lyapunov which provide ways of determining the stability of sets of ordinary differential equations (determining the prediction horizon).
	www.absoluteastronomy.com /encyclopedia/L/Ly/Lyapunov_exponent.htm (384 words)

	aleksandr (Site not responding. Last check: 2007-10-02)
		Aleksandr Danilovic Aleksandrov's father was the headmaster of a secondary school in St Petersburg and his...
		Aleksandr Sergeevich Pushkin was born on May 26, 1799, in Moscow...
		Aleksandr Pushkin is, by common agreement -- at least among his own compatriots -- the greatest of all Russian writers...
	www.nieddu.biz /russia/aleksandr+.cgi (588 words)

	Russia
		2 Mar 1855 - 13 Mar 1881 Aleksandr II (b.
		Mar 1765 - 9 Mar 1775 Prince Aleksandr Mikhailovich (b.
		1991); Deputy chairman of the Defense Council Oleg Baklanov; Chairman of the Peasants' Union Vasiliy Starodubtsev; and Chairman of the Association of State Enterprises Aleksandr Tiziakov.
	www.worldstatesmen.org /Russia.htm (3695 words)

	Lyapunov function - Encyclopedia, History, Geography and Biography
		In the theory of dynamical systems, and control theory, Lyapunov functions, named after Aleksandr Mikhailovich Lyapunov, are a family of functions that can be used to demonstrate the stability of some state points of a system.
		The demonstration of stability require finding a Lyapunov function for that system.
		There is no direct way to obtain a Lyapunov function but there many tricks to simplify the task.
	www.arikah.net /encyclopedia/Lyapunov_function (136 words)

	Online Encyclopedia and Dictionary - Central limit theorem
		Several generalizations for finite variance exist which do not require identical distribution but incorporate some condition which guarantees that none of the variables exert a much larger influence than the others.
		Two such conditions are the Lindeberg condition and the Lyapunov condition.
		In the same setting and with the same notation as above, we can replace the Lyapunov condition with the following weaker one (from Lindeberg in 1920).
	fact-archive.com /encyclopedia/Central_limit_theorem (946 words)

	References for Lyapunov (Site not responding. Last check: 2007-10-02)
		V N Chinenova, On the history of the development of the extremal-energy criterion for stability (from Lagrange to Lyapunov) (Russian), in Mathematics and the mathematical sciences in Ukraine in the twentieth century (Ukrainian), Kam'yanets' -Podi l's'kii, 1999 (Kiev, 2001), 41-52.
		N A Pustovoitov, Theory of the stability of motion after Lyapunov.
		P S Shcherbakov, Alexander Mikhailovitch Lyapunov: on the centenary of his doctoral dissertation on stability of motion, Automatica J. O B Sheinin, Letters from A M Lyapunov to K A Andreev (Russian), Istor.-Mat.
	www-gap.dcs.st-and.ac.uk /~history/References/Lyapunov.html (411 words)

	Encyclopedia: Aleksei Lyapunov
		He was a distant relative of Aleksandr Lyapunov, and one of Petr Novikov's students.
		A mathematician is a person whose area of study and research is mathematics.
		Aleksandr Mikhailovich Lyapunov (Александр Михайлович Ляпунов) (June 6, 1857 - November 3, 1918, all new style) was a Russian mathematician, mechanician and physicist.
	www.nationmaster.com /encyclopedia/Aleksei-Lyapunov (188 words)

	Lyapunov exponents explained
		This Lyapunov exponent is a well known tool in the investigation of dynamic systems (elsewhere I explain how to use it to compute images of the mandelbrot set).
		He was born 6 June 1857 in Yaroslavl (Russia) and died 3 November 1918 in Odessa (he committed suicide three days after the death of his wive).
		The initial value of x usually plays no crucial role: if the Lyapunov exponent of a seed value x is positive, chances are good that it is positive for most other possible seed values.
	www.janthor.de /Lyapunov/explained.html (877 words)

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