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	Ergodic theory - Wikipedia, the free encyclopedia
		This is the celebrated ergodic theorem, in an abstract form due to George David Birkhoff.
		The equidistribution theorem is a special case of the ergodic theorem, dealing specifically with the distribution of probabilities on the unit interval.
		The ergodicity of the geodesic flow on manifolds of constant negative curvature was discovered by E.
	en.wikipedia.org /wiki/Ergodic_theory (783 words)

	Birkhoff's theorem (relativity) - Wikipedia, the free encyclopedia
		Then Birkhoff's theorem says that the exterior geometry must be Schwarzschild; the only effect of the pulsation is to change the location of the stellar surface.
		Another interesting consequence of Birkhoff's theorem is that for a spherically symmetric thin shell, the interior solution must be given by the Minkowski metric; in other words, the gravitational field must vanish inside a spherically symmetric shell.
		Birkhoff's theorem can be generalized: any spherically symmetric solution of the Einstein/Maxwell field equations must be static and asymptotically flat, so the exterior geometry of a spherically symmetric charged star must be given by the Reissner-Nordström electrovacuum.
	en.wikipedia.org /wiki/Birkhoff's_theorem_(relativity) (421 words)

	PlanetMath: ergodic theorem
		Birkhoff's ergodic theorem (often called the pointwise or strong ergodic theorem) states that there exists
		This is often interpreted in the following way: for an ergodic transformation, the time average equals the space average almost surely.
		This is version 8 of ergodic theorem, born on 2002-02-16, modified 2006-06-08.
	planetmath.org /encyclopedia/ErgodicTheorem.html (122 words)

	Read This: Ergodic Theory of Numbers
		The ergodic theorem is then applied to, as stated in the preface, "obtain old and new results in an elegant and straightforward manner".
		The Ergodic Theorem, coupled with the natural extension machinery applied to the Gauss map, is used in chapter 5 to obtain arithmetical properties of the approximation coefficients for continued fraction expansions.
		Ergodicity and the Ergodic Theorem are used as tools to arrive at results of interest to the authors.
	www.maa.org /reviews/ergodicnt.html (3054 words)

	Harmonic Analysis and Ergodic Theory
		The traditional classes: ergodic, weak mixing, strong mixing, are easily seen to be Borel sets (when the probability space is separable and one uses the strong topology of operators on Hilbert space.) For mild mixing this is no longer so.
		Abstract: We use the Koosis theorem to find the asymptotics of orthogonal polynomials for the case when a Szego measure on the unit circumference is perturbed by an arbitrary finite measure inside the unit disk and an arbitrary Blaschke sequence of point masses outside the unit disk.
		Ergodic theorems are an extremely broad and rich field with wide application.
	condor.depaul.edu /~haet/speakers.htm (1443 words)

	Search Results for theorem
		The theorem is then a sort of topological form of the particle-wave equivalence of quantum mechanics, and the quest for 'truly' understanding these and analogous dualities has been one of the great motivating forces in the mathematics of the last fifty years.
		It is nevertheless certain that the theorem on the sum of the three angles of the triangle should be considered one of those fundamental truths that are impossible to contest and that are an enduring example of mathematical certitude.
		Pick's theorem states that the area of a reticular polygon is L + B/2 - 1 where L is the number of reticular points inside the polygon and B is the number of reticular points on the edges of the polygon.
	www-groups.dcs.st-and.ac.uk /history/Search/historysearch.cgi?SUGGESTION=theorem&CONTEXT=1 (15691 words)

	Search Results for theorem*
		Theorem 2 of Euclid's Phaenomena consists of four propositions with proofs for only three of them while the missing one is replaced by the remark "that this is the case has been shown elsewhere"; indeed theorem and proof are found as Theorem 10 in Autolycus's 'Rotating Sphere'.
		This means that a theorem in Autolycus's work has first a general statement, then a construction related to a particular figure with points in the figure denoted by letters, next comes the demonstration of the theorem, and finally a conclusion relating to the general statement is sometimes drawn.
		The theorem relating convergence almost everywhere and uniform convergence by D F Egorov, one of Bugaev's pupils, in 1911 is seen as marking the beginning of the Moscow school of the theory of functions of a real variable.
	www-groups.dcs.st-and.ac.uk /history/Search/historysearch.cgi?SUGGESTION=theorem*&CONTEXT=1 (16791 words)

	Springer Online Reference Works
		In "abstract" ergodic theory one studies various statistical properties of dynamical systems reflecting their behaviour over long periods of time (for example, ergodicity or mixing) as well as problems connected with the metric classification of systems (with respect to a metric isomorphism), and the two groups of problems turn out to be closely connected.
		a) The appearance of ergodic theory as an independent branch is connected with the von Neumann ergodic theorem and the Birkhoff ergodic theorem and the recognition of their metric nature.
		Applications of ergodic theory to number theory can be found in [11], and to the theory of lattices in semi-simple groups (work of Margulis) in [a4].
	eom.springer.de /e/e036150.htm (1285 words)

	VITAE
		An analog of the Marcinkiewicz integral in ergodic theory, Studia Mathematica, 68 (1981), 281-289.
		Ergodic theorems for convolutions of a measure on a group, (Co-authors: J. Rosenblatt, A. Tempelman), Illinois J.of Math., 38 (1994) 521-553.
		Convergence of ergodic averages, (Co-author: Mate Wierdl) Convergence in Ergodic Theory and Probability, Edited by Bergelson, March, and Rosenblatt, Walter de Gruyter \and Co., Berlin, 1996 229--247.
	condor.depaul.edu /~rjones/vitae.htm (1359 words)

	Ergodic theory
		An important fact that is worth being emphasized is that Theorem 3.2 relies on transferring the maximal inequality from the group to the measure space, this being possible due to the amenability of G. Theorem 3.3.
		The first one is the relevance of the ergodic theoretic approach to the number theoretic problem; this is about the fact that Theorem II implies Theorem I and it is discussed in the first part of section 1.
		A second issue is Furstenberg's theorem in the particular cases of weakly mixing and compact systems, which are the two opposite poles of the spectrum.
	www.math.ucla.edu /~thiele/workshop7/topics.html (1667 words)

	No title (Site not responding. Last check: 2007-11-01)
		This theorem is a generalization of Dunford and Schwartz's version of the Pointwise Ergodic Theorem, a result which extends Birkhoff's Ergodic Theorem.
		In the context of Bishop's constructive mathematics [3] we prove that for the Mean Ergodic Theorem to hold it is sufficient that the projection on the space of invariant functions exists.
		Ergodic Theorem can be proved classically for finite measure spaces using the pointwise ergodic theorem [12].
	www.cs.ru.nl /~spitters/ergodic.html (2637 words)

	Bulletin of the American Mathematical Society
		This is an introduction to some of the ideas in the proof, concentrating on the connections to ergodic theory.
		Polynomial extensions of van der Waerden's and Szemerédi's theorems.
		Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions.
	www.ams.org /bull/2006-43-01/S0273-0979-05-01086-4/home.html (324 words)

	[No title] (Site not responding. Last check: 2007-11-01)
		An interesting class of strictly ergodic subshifts is given by those subshifts that are generated by linearly recurrent sequences \cite{dur,lagple}: \begin{prop}\label{lrse} If $s$ is linearly recurrent, then $\Omega_s$ is strictly ergodic.
		Fix some strictly ergodic subshift $\Omega$ and define, for $w \in \mathcal{W}_\Omega$, the \textit{index of} $w$ to be $$ \mathrm{ind}(w) = \sup \{ r \in \Q : w^r \in \mathcal{W}_\Omega \}.
		The RAGE theorem establishes basic dynamical results in terms of the standard decomposition of the Hilbert space into pure point, singular continuous, and absolutely continuous subspaces.
	www.ma.utexas.edu /mp_arc/papers/05-309 (8765 words)

	Preface: Discrete sample paths
		Of particular note in the discussion of process models is how ergodic theorists think of a stationary process, namely, as a measure-preserving transformation on a probability space, together with a partition of the space.
		The audiences included ergodic theorists, information theorists, and probabilists, as well as combinatorialists and people from engineering and other mathematics disciplines, ranging from undergraduate and graduate students through post-docs and junior faculty to senior professors and researchers.
		Many standard topics from ergodic theory are omitted or given only cursory treatment, in part because the book is already too long and in part because they are not close to the central focus of this book.
	www.math.utoledo.edu /~pshields/preface.html (1104 words)

	Encyclopedia :: encyclopedia : List of theorems (Site not responding. Last check: 2007-11-01)
		Most of the results below come from pure mathematics, but some are from theoretical physics, economics, and other applied fields.
		Haboush's theorem (algebraic groups, representation theory, invariant theory)
		Hilbert's Nullstellensatz (theorem of zeroes) (commutative algebra, algebraic geometry)
	www.hallencyclopedia.com /List_of_theorems (262 words)

	Amazon.com: Ergodic Theory (Cambridge Studies in Advanced Mathematics): Books: Karl E. Petersen (Site not responding. Last check: 2007-11-01)
		The author presents the fundamentals of the ergodic theory of point transformations and several advanced topics of intense research.
		Each of the basic aspects of ergodic theory--examples, convergence theorems, recurrence properties, and entropy--receives a basic and a specialized treatment.
		Without going into the details (to which the rest of the book is devoted), we mention some of the basic questions, examples, and constructions of ergodic theory, in order to provide an indication of the content and flavor of the subject as well as to establish reference points for terminology and notation.
	www.amazon.com /Ergodic-Cambridge-Studies-Advanced-Mathematics/dp/0521389976 (936 words)

	Some Ergodic Theory
		Let us first show that the Lebesgue measure is also ergodic.
		, is also an absolutely continuous invariant probability measure which is ergodic.
		The following theorem can be applied when one has an invariant measure.
	www.maths.warwick.ac.uk /~strien/MA424/HTMLversion/node14.html (103 words)

	HBA :: M.G. Nadkarni, University of Mumbai, Mumbai (Site not responding. Last check: 2007-11-01)
		The presentation has a slow pace and the book can be read by any person with a background in basic measure theory and metric topology.
		A new feature of the book is that the basic topics of Ergodic Theory such as the Poincaré recurrence lemma, induced automorphisms and Kakutani towers, compressibility and E. Hopf's theorem, the theorem of Ambrose on representation of flows are treated at the descriptive set-theoretic level before their measure-theoretic or topological versions are presented.
		In the second edition, a section on rank automorphisms and a brief discussion of the ergodic theorem due to Wiener and Wintner have been added.
	www.hindbook.com /Home.asp?P=15 (239 words)

	Birkhoff's Ergodic Theorem and its Applications (Site not responding. Last check: 2007-11-01)
		For ergodic systems their construction is a consequence of the Birkhoff Ergodic Theorem.
		We have focused on an alternative based on a method of S. Ulam and research on the relationship between these measures and the system's dynamics is underway.
		Another area of interest with applications to current NIST problems is the rate of convergence to ergodicity.
	math.nist.gov /reports/division/yearly/subsection2_2_1_13.html (151 words)

	Griffeath: An ergodic theorem for a class of spin systems
		Griffeath: An ergodic theorem for a class of spin systems
		An ergodic theorem for a class of spin systems.
		, Ergodic theorems for an infinite particle system with sinks and sources.
	math-doc.ujf-grenoble.fr /numdam-bin/item?id=AIHPB_1977__13_2_141_0 (140 words)

	Citebase - The Shannon-McMillan Theorem for Ergodic Quantum Lattice Systems (Site not responding. Last check: 2007-11-01)
		The Shannon-McMillan Theorem for Ergodic Quantum Lattice Systems
		The theorem demonstrates the significance of the von Neumann entropy for translation invariant ergodic quantum spin systems on n-dimensional lattices: the entropy gives the logarithm of the essential number of eigenvectors of the system on large boxes.
		The one-dimensional case covers quantum information sources and is basic for coding theorems.
	www.citebase.org /cgi-bin/citations?id=oai:arXiv.org:math/0207121 (138 words)

	New Zealand Mathematical Societu Newsletter Number 83, December 2001
		David won the prize with a short speech giving a proof of the topological theorem that there must be at least one point in the ocean at which the mean tidal range is zero.
		These gems include Pythagoras' theorem, the golden rectangle and tilings with congruent tiles or just a few polygonal shapes but arranged in a pattern which is scalable (similar to a subdivision of itself into smaller tiles of the same shapes) but never repeats, featuring the results of Berger, Penrose and Conway.
		There are chapters on ergodic theory, spectral theory, entropy theory, general group actions, trajectory theory, smooth hyperbolic dynamical systems, billiards, one-dimensional mappings, dynamical systems on homogeneous spaces, and statistical mechanics.
	ifs.massey.ac.nz /mathnews/NZMS83/news83.htm (17750 words)

	A Quasi-Ergodic Theorem For Evanescent Processes (ResearchIndex)
		We prove a conditioned version of the ergodic theorem for Markov processes, which we call a quasi-ergodic theorem.
		We also prove a convergence result for conditioned processes as the conditioning event becomes rarer.
		1 A survey of limit theorems for Markov chains and processes o..
	citeseer.ist.psu.edu /393582.html (314 words)

	An Ergodic Theorem for Quantum Counting Processes (ResearchIndex)
		An Ergodic Theorem for Quantum Counting Processes (2003)
		If your firewall is blocking outgoing connections to port 3125, you can use these links to download local copies.
		Abstract: For a quantum-mechanical counting process we show ergodicity, under the condition that the underlying open quantum system approaches equilibrium in the time mean.
	citeseer.ist.psu.edu /592527.html (268 words)

	Ergodic theorem for contractive Markov systems (Site not responding. Last check: 2007-11-01)
		In this paper, we continue development of the theory of contractive Markov systems initiated in Werner (J.
		We prove an ergodic theorem for Markov chains associated with such systems using the coding map constructed in Werner (2003 Preprint).
		This is a generalization of the ergodic theorem of Elton (1987 Ergod.
	stacks.iop.org /0951-7715/17/2303 (214 words)

	subadditive ergodic theorem (Site not responding. Last check: 2007-11-01)
		Definition: If a stationary and ergodic process satisfies the subadditive inequality, it grows almost surely linearly in time.
		Note: From Algorithms and Theory of Computation Handbook, page 14-35, Copyright © 1999 by CRC Press LLC.
		Algorithms and Theory of Computation Handbook, CRC Press LLC, 1999, "subadditive ergodic theorem", in Dictionary of Algorithms and Data Structures [online], Paul E. Black, ed., U.S. National Institute of Standards and Technology.
	www.nist.gov /dads/HTML/subadditive.html (117 words)

	Homepage of Prof. Walter Trockel (Site not responding. Last check: 2007-11-01)
		An Alternative Proof for the Linear Utility Representation Theorem.
		Uniqueness of Mean Maximizers via an Ergodic Theorem.
		Border, K. Fixed Point Theorems with Applications to Economics and Game Theory.
	www.wiwi.uni-bielefeld.de /~imw/Members/WTrockel (477 words)

	Energy Citations Database (ECD) - Energy and Energy-Related Bibliographic Citations
		Energy Citations Database (ECD) Document #4413012 - ERGODIC THEOREM IN QUANTUM MECHANICS
		Availability information may be found in the Availability, Publisher, Research Organization, Resource Relation and/or Author (affiliation information) fields and/or via the "Full-text Availability" link.
		For a journal article, please see the Resource Relation field.
	www.osti.gov /energycitations/product.biblio.jsp?osti_id=4413012 (85 words)

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