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M/G/1-TYPE MARKOV CHAINS

D. A. Bini, G. Latouche, and B. Meini

in Numerical Methods for Structured Markov Chains

Published in print:
2005
Published Online:
September 2007
ISBN:
9780198527688
eISBN:
9780191713286
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198527688.003.0004
Subject:
Mathematics, Numerical Analysis

This chapter is devoted to the theoretical analysis of M/G/1-type Markov chains. Conditions for the ergodicity are given in terms of the drift. The reduction to solving a nonlinear matrix equation is ... More


Markov Processes

M. Vidyasagar

in Hidden Markov Processes: Theory and Applications to Biology

Published in print:
2014
Published Online:
October 2017
ISBN:
9780691133157
eISBN:
9781400850518
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691133157.003.0004
Subject:
Mathematics, Probability / Statistics

This chapter deals with Markov processes. It first defines the “Markov property” and shows that all the relevant information about a Markov process assuming values in a finite set of cardinality n ... More


Dependence

James Davidson

in Stochastic Limit Theory: An Introduction for Econometricians

Published in print:
1994
Published Online:
November 2003
ISBN:
9780198774037
eISBN:
9780191596117
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/0198774036.003.0013
Subject:
Economics and Finance, Econometrics

A number of probabilistic approaches to the concept of dependence in stochastic sequences are contrasted. The fundamental idea is a shift transformation. The notions of stationarity and (serial) ... More


Econometric Tools and Techniques

David F. Hendry

in Dynamic Econometrics

Published in print:
1995
Published Online:
November 2003
ISBN:
9780198283164
eISBN:
9780191596384
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/0198283164.003.0003
Subject:
Economics and Finance, Econometrics

Least squares and recursive methods for estimating the values of unknown parameters and the logic of testing in empirical modelling, are discussed. The tools needed for investigating the properties ... More


INTRODUCTION TO THE QUANTUM MECHANICAL TIME-INDEPENDENT SCATTERING THEORY I: ONE-DIMENSIONAL SCATTERING

Pier A. Mello and Narendra Kumar

in Quantum Transport in Mesoscopic Systems: Complexity and Statistical Fluctuations. A Maximum Entropy Viewpoint

Published in print:
2004
Published Online:
September 2007
ISBN:
9780198525820
eISBN:
9780191712234
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198525820.003.0002
Subject:
Physics, Condensed Matter Physics / Materials

This chapter is devoted to basic potential scattering theory, focusing on the case of a one-dimensional conductor and an open cavity with a one-channel lead connected to it. The contents of this ... More


Hangzhou Lectures on Eigenfunctions of the Laplacian (AM-188)

Christopher D. Sogge

Published in print:
2014
Published Online:
October 2017
ISBN:
9780691160757
eISBN:
9781400850549
Item type:
book
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691160757.001.0001
Subject:
Mathematics, Numerical Analysis

Based on lectures given at Zhejiang University in Hangzhou, China, and Johns Hopkins University, this book introduces eigenfunctions on Riemannian manifolds. The book gives a proof of the sharp Weyl ... More


Ergodic Theory

Robert Alicki and Mark Fannes

in Quantum Dynamical Systems

Published in print:
2001
Published Online:
February 2010
ISBN:
9780198504009
eISBN:
9780191708503
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198504009.003.0007
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter defines and characterizes various degrees of ergodicity — both for classical and quantum systems — in terms of the Hilbert space formalism, Koopman formalism for classical systems, and ... More


A Matter of Degree: Making Sense of Phase Structure

Laura Ruetsche

in Interpreting Quantum Theories

Published in print:
2011
Published Online:
September 2011
ISBN:
9780199535408
eISBN:
9780191728525
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199535408.003.0012
Subject:
Philosophy, Philosophy of Science

This chapter presents three reasons to take quantum statistical mechanics to the thermodynamic limit, where the number of systems considered becomes infinite, in the form of three phenomena that ... More


Classical and quantum ergodicity

Christopher D. Sogge

in Hangzhou Lectures on Eigenfunctions of the Laplacian (AM-188)

Published in print:
2014
Published Online:
October 2017
ISBN:
9780691160757
eISBN:
9781400850549
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691160757.003.0006
Subject:
Mathematics, Numerical Analysis

This chapter proves results involving the quantum ergodicity of certain high-frequency eigenfunctions. Ergodic theory originally arose in the work of physicists studying statistical mechanics at the ... More


Measures and flows on Γ\𝔽 d

Jacques Franchi and Yves Le Jan

in Hyperbolic Dynamics and Brownian Motion: An Introduction

Published in print:
2012
Published Online:
January 2013
ISBN:
9780199654109
eISBN:
9780191745676
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199654109.003.0005
Subject:
Mathematics, Mathematical Physics

This chapter considers measures of Γ-invariant sets, and establish a mixing theorem for the action of the geodesic and horocyclic flows on square-integrable Γ-invariant functions. A Poincaré ... More


State Preparation and Intramolecular Vibrational Energy Redistribution

Tomas Baer and William L. Hase

in Unimolecular Reaction Dynamics: Theory and Experiments

Published in print:
1996
Published Online:
November 2020
ISBN:
9780195074949
eISBN:
9780197560297
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780195074949.003.0006
Subject:
Chemistry, Theoretical Chemistry

The first step in a unimolecular reaction involves energizing the reactant molecule above its decomposition threshold. An accurate description of the ensuing ... More


Statistical Stability of Chaos

David P. Feldman

in Chaos and Fractals: An Elementary Introduction

Published in print:
2012
Published Online:
December 2013
ISBN:
9780199566433
eISBN:
9780191774966
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199566433.003.0014
Subject:
Physics, Theoretical, Computational, and Statistical Physics

Sensitive dependence on initial conditions makes it difficult to predict chaotic systems in the long term, as two initial conditions that begin close to each other rapidly get pushed apart. This ... More


Energetics, Energy Flow, and Scaling in Life

William H. Woodruff

in Protocells: Bridging Nonliving and Living Matter

Published in print:
2008
Published Online:
August 2013
ISBN:
9780262182683
eISBN:
9780262282093
Item type:
chapter
Publisher:
The MIT Press
DOI:
10.7551/mitpress/9780262182683.003.0021
Subject:
Biology, Microbiology

This chapter gives an overview of the origin of life, with particular attention to transition from ergodicity to contingency. It also considers physical insights from dynamical self-organization in ... More


Homogenisation

G. F. Roach, I. G. Stratis, and A. N. Yannacopoulos

in Mathematical Analysis of Deterministic and Stochastic Problems in Complex Media Electromagnetics

Published in print:
2012
Published Online:
October 2017
ISBN:
9780691142173
eISBN:
9781400842650
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691142173.003.0014
Subject:
Mathematics, Applied Mathematics

This chapter presents a theory of homogenisation for random bianisotropic media exhibiting an ergodic structure. It shows that for such a medium there exists a homogenised system of the Maxwell type. ... More


Minority game

František Slanina

in Essentials of Econophysics Modelling

Published in print:
2013
Published Online:
January 2014
ISBN:
9780199299683
eISBN:
9780191747038
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199299683.003.0006
Subject:
Physics, Theoretical, Computational, and Statistical Physics

In this chapter we present the minority game, which is one of the keystone models of econophysics. It is the mathematic implementation of the idea of inductive, rather than deductive, thinking. ... More


Mixing Flows on Moduli Spaces of Flat Bundles over Surfaces

Giovanni Forni and William M. Goldman

in Geometry and Physics: Volume II: A Festschrift in honour of Nigel Hitchin

Published in print:
2018
Published Online:
December 2018
ISBN:
9780198802020
eISBN:
9780191869068
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198802020.003.0021
Subject:
Mathematics, Geometry / Topology

This chapter extends Teichmüller dynamics to a flow on the total space of a flat bundle of deformation spaces of representations of the fundamental group of a fixed surface S in a Lie group G. The ... More


Introduction to Stochastic Processes

Florence Merlevède, Magda Peligrad, and Sergey Utev

in Functional Gaussian Approximation for Dependent Structures

Published in print:
2019
Published Online:
April 2019
ISBN:
9780198826941
eISBN:
9780191865961
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198826941.003.0001
Subject:
Mathematics, Probability / Statistics

We start by stating the need for a Gaussian approximation for dependent structures in the form of the central limit theorem (CLT) or of the functional CLT. To justify the need to quantify the ... More


Statistical Mechanics and Information Theory for Complex Systems

Stefan Thurner, Rudolf Hanel, and Peter Klimekl

in Introduction to the Theory of Complex Systems

Published in print:
2018
Published Online:
November 2018
ISBN:
9780198821939
eISBN:
9780191861062
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198821939.003.0006
Subject:
Physics, Theoretical, Computational, and Statistical Physics

Most complex systems are statistical systems. Statsitical mechanics and information theory usually do not apply to complex systems because the latter break the assumptions of ergodicity, ... More


Solutions of Mean Field Models

A. Campa, T. Dauxois, D. Fanelli, and S. Ruffo

in Physics of Long-Range Interacting Systems

Published in print:
2014
Published Online:
October 2014
ISBN:
9780199581931
eISBN:
9780191787140
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199581931.003.0004
Subject:
Physics, Theoretical, Computational, and Statistical Physics

In this chapter the theory and the tools described in the previous two chapters are applied to several simple mean-field models, computing both the microcanonical and the canonical partition ... More


Beyond Mean-Field Models

A. Campa, T. Dauxois, D. Fanelli, and S. Ruffo

in Physics of Long-Range Interacting Systems

Published in print:
2014
Published Online:
October 2014
ISBN:
9780199581931
eISBN:
9780191787140
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199581931.003.0005
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter aims at showing that the features occurring in mean-field models, described in the previous chapters, can be found also in the other long-range systems. The first four sections are ... More


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