R. Duncan Luce
- Published in print:
- 1991
- Published Online:
- January 2008
- ISBN:
- 9780195070019
- eISBN:
- 9780199869879
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195070019.003.0008
- Subject:
- Psychology, Cognitive Models and Architectures
This chapter examines closely-related models for possible processes underlying choice reaction time. Topics discussed include accumulator models; random walks with boundaries; restrictions on the ...
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This chapter examines closely-related models for possible processes underlying choice reaction time. Topics discussed include accumulator models; random walks with boundaries; restrictions on the random walk model; and modifications of the random walk.Less
This chapter examines closely-related models for possible processes underlying choice reaction time. Topics discussed include accumulator models; random walks with boundaries; restrictions on the random walk model; and modifications of the random walk.
Sergey N. Dorogovtsev
- Published in print:
- 2010
- Published Online:
- May 2010
- ISBN:
- 9780199548927
- eISBN:
- 9780191720574
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199548927.003.0011
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This chapter considers the specifics of walks, navigation, and search processes in networks of various architectures and geometries. It starts with discussion of random walks on complex networks, ...
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This chapter considers the specifics of walks, navigation, and search processes in networks of various architectures and geometries. It starts with discussion of random walks on complex networks, then biased random walks and in application to packet routing in the Internet. It discusses the famous Kleinberg's problem and searchability of networks with underlining metric spaces. Finally, the Google PageRank is explained.Less
This chapter considers the specifics of walks, navigation, and search processes in networks of various architectures and geometries. It starts with discussion of random walks on complex networks, then biased random walks and in application to packet routing in the Internet. It discusses the famous Kleinberg's problem and searchability of networks with underlining metric spaces. Finally, the Google PageRank is explained.
Robert M. Mazo
- Published in print:
- 2008
- Published Online:
- January 2010
- ISBN:
- 9780199556441
- eISBN:
- 9780191705625
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199556441.003.0009
- Subject:
- Physics, Condensed Matter Physics / Materials
This chapter introduces the idea of random walk and studies a number of simple cases to illustrate the concepts, mostly in one dimension. The symmetric Pearson walk, the biased walk, and the ...
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This chapter introduces the idea of random walk and studies a number of simple cases to illustrate the concepts, mostly in one dimension. The symmetric Pearson walk, the biased walk, and the persistent walk are treated for lattice walks in discrete time, and the results generalized to continuum walks. Asymptotic results for large number of steps or long times are emphasized. The problems of random walk with boundaries and first passage times are introduced. Remarks on further generalizations and problems are briefly made.Less
This chapter introduces the idea of random walk and studies a number of simple cases to illustrate the concepts, mostly in one dimension. The symmetric Pearson walk, the biased walk, and the persistent walk are treated for lattice walks in discrete time, and the results generalized to continuum walks. Asymptotic results for large number of steps or long times are emphasized. The problems of random walk with boundaries and first passage times are introduced. Remarks on further generalizations and problems are briefly made.
Eric Renshaw
- Published in print:
- 2011
- Published Online:
- September 2011
- ISBN:
- 9780199575312
- eISBN:
- 9780191728778
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199575312.003.0004
- Subject:
- Mathematics, Applied Mathematics, Mathematical Biology
This chapter considers path analyses for random walks, which leads on to first passage and return probabilities and the Arc Sine Law. Both absorbing and reflecting barriers are considered in some ...
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This chapter considers path analyses for random walks, which leads on to first passage and return probabilities and the Arc Sine Law. Both absorbing and reflecting barriers are considered in some depth. Since the assumption of independent steps is not universally true, a classic case being the movement of share prices driven by market sentiment, the correlated random walk is introduced.Less
This chapter considers path analyses for random walks, which leads on to first passage and return probabilities and the Arc Sine Law. Both absorbing and reflecting barriers are considered in some depth. Since the assumption of independent steps is not universally true, a classic case being the movement of share prices driven by market sentiment, the correlated random walk is introduced.
Christian Gollier
- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691148762
- eISBN:
- 9781400845408
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691148762.003.0004
- Subject:
- Economics and Finance, Development, Growth, and Environmental
This chapter illustrates that the shape of the term structure of discount rates is determined by the way the wealth effect and the precautionary effects evolve with the time horizon. When the growth ...
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This chapter illustrates that the shape of the term structure of discount rates is determined by the way the wealth effect and the precautionary effects evolve with the time horizon. When the growth rate of consumption is constant, then consumption increases exponentially, and the intertemporal rate of substitution, which is the discount factor, decreases exponentially. This requires that the discount rate is constant. The simplest extension of this to uncertainty is to assume that the growth rate of the economy follows a random walk. In that case, the variance of log consumption increases linearly, which yields an exponentially increasing precautionary effect for the discount factor. This justifies a constant precautionary effect on the discount rate, yielding a crucial result for the theory of efficient discount rates.Less
This chapter illustrates that the shape of the term structure of discount rates is determined by the way the wealth effect and the precautionary effects evolve with the time horizon. When the growth rate of consumption is constant, then consumption increases exponentially, and the intertemporal rate of substitution, which is the discount factor, decreases exponentially. This requires that the discount rate is constant. The simplest extension of this to uncertainty is to assume that the growth rate of the economy follows a random walk. In that case, the variance of log consumption increases linearly, which yields an exponentially increasing precautionary effect for the discount factor. This justifies a constant precautionary effect on the discount rate, yielding a crucial result for the theory of efficient discount rates.
Max A. Alekseyev and Toby Berger
- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691164038
- eISBN:
- 9781400881338
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691164038.003.0005
- Subject:
- Mathematics, History of Mathematics
This chapter studies solutions of the Tower of Hanoi puzzle and some of its variants with random moves, where each move is chosen uniformly from the set of the valid moves in the current state. The ...
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This chapter studies solutions of the Tower of Hanoi puzzle and some of its variants with random moves, where each move is chosen uniformly from the set of the valid moves in the current state. The Tower of Hanoi puzzle consists of n disks of distinct sizes distributed across three pegs. At a single move it is permitted to transfer a disk from the top of one peg to the top of another peg, if this results in a valid state, i.e. a particular distribution of the disks across the pegs. The chapter proves the exact formulas for the expected number of random moves to solve the puzzles. It also presents an alternative proof for one of the formulas that couples a theorem about expected commute times of random walks on graphs with the delta-to-wye transformation used in the analysis of three-phase AC systems for electrical power distribution.Less
This chapter studies solutions of the Tower of Hanoi puzzle and some of its variants with random moves, where each move is chosen uniformly from the set of the valid moves in the current state. The Tower of Hanoi puzzle consists of n disks of distinct sizes distributed across three pegs. At a single move it is permitted to transfer a disk from the top of one peg to the top of another peg, if this results in a valid state, i.e. a particular distribution of the disks across the pegs. The chapter proves the exact formulas for the expected number of random moves to solve the puzzles. It also presents an alternative proof for one of the formulas that couples a theorem about expected commute times of random walks on graphs with the delta-to-wye transformation used in the analysis of three-phase AC systems for electrical power distribution.
Hendrik S. Houthakker and Peter J. Williamson
- Published in print:
- 1996
- Published Online:
- November 2003
- ISBN:
- 9780195044072
- eISBN:
- 9780199832958
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/019504407X.003.0005
- Subject:
- Economics and Finance, Financial Economics
The discussion in this chapter begins with an analysis of central trading places, which looks at the economics of securities trading and the rationale for brokers and central trading places (with ...
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The discussion in this chapter begins with an analysis of central trading places, which looks at the economics of securities trading and the rationale for brokers and central trading places (with reference to US stock exchanges), the types of orders buyers or sellers may place in the market and the way these are executed, and the system of ‘specialists’ commonly found in stock exchanges (which is designed to provide a smooth and continuous market for individual stocks). Next, it briefly examines financial markets without central trading places. This is followed by a look at the mechanics of securities trading in the stock exchanges in London (UK) and Tokyo (Japan). The last section of the chapter discusses the operational efficiency of the stock market and the efficient market hypothesis (EFM), looking at the implications of central exchanges in which information flows rapidly between participants for the efficiency of the stock market as a whole and, in particular, at the ‘random walk’ behavior of share prices.Less
The discussion in this chapter begins with an analysis of central trading places, which looks at the economics of securities trading and the rationale for brokers and central trading places (with reference to US stock exchanges), the types of orders buyers or sellers may place in the market and the way these are executed, and the system of ‘specialists’ commonly found in stock exchanges (which is designed to provide a smooth and continuous market for individual stocks). Next, it briefly examines financial markets without central trading places. This is followed by a look at the mechanics of securities trading in the stock exchanges in London (UK) and Tokyo (Japan). The last section of the chapter discusses the operational efficiency of the stock market and the efficient market hypothesis (EFM), looking at the implications of central exchanges in which information flows rapidly between participants for the efficiency of the stock market as a whole and, in particular, at the ‘random walk’ behavior of share prices.
M. E. J. Newman
- Published in print:
- 2010
- Published Online:
- September 2010
- ISBN:
- 9780199206650
- eISBN:
- 9780191594175
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199206650.003.0006
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This chapter introduces the basic theoretical tools used to describe and analyze networks, most of which come from graph theory, the branch of mathematics that deals with networks. Topics covered ...
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This chapter introduces the basic theoretical tools used to describe and analyze networks, most of which come from graph theory, the branch of mathematics that deals with networks. Topics covered include the adjacency matrix, weighted networks, directed networks, hypergraphs, bipartite networks, trees, planar networks, the graph Laplacian, and random walks. Exercises are provided at the end of the chapter.Less
This chapter introduces the basic theoretical tools used to describe and analyze networks, most of which come from graph theory, the branch of mathematics that deals with networks. Topics covered include the adjacency matrix, weighted networks, directed networks, hypergraphs, bipartite networks, trees, planar networks, the graph Laplacian, and random walks. Exercises are provided at the end of the chapter.
R. Duncan Luce
- Published in print:
- 1991
- Published Online:
- January 2008
- ISBN:
- 9780195070019
- eISBN:
- 9780199869879
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195070019.003.0009
- Subject:
- Psychology, Cognitive Models and Architectures
This chapter examines closely-related models for possible processes underlying choice reaction time. The models are similar to those discussed in the preceding chapter, except that the accumulation ...
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This chapter examines closely-related models for possible processes underlying choice reaction time. The models are similar to those discussed in the preceding chapter, except that the accumulation of information is assumed here to occur continuously. Two main types of model are considered: continuous analogues of the random walk model and models based on the idea that information accumulates in a punctate manner according to a renewal counting process.Less
This chapter examines closely-related models for possible processes underlying choice reaction time. The models are similar to those discussed in the preceding chapter, except that the accumulation of information is assumed here to occur continuously. Two main types of model are considered: continuous analogues of the random walk model and models based on the idea that information accumulates in a punctate manner according to a renewal counting process.
Ralph Skomski
- Published in print:
- 2008
- Published Online:
- January 2010
- ISBN:
- 9780198570752
- eISBN:
- 9780191718816
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198570752.003.0007
- Subject:
- Physics, Condensed Matter Physics / Materials
Atomic disorder has far-reaching consequences for the behavior of magnetic materials. It modifies the electronic structure but does not necessarily destroy ferromagnetism, as exemplified by amorphous ...
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Atomic disorder has far-reaching consequences for the behavior of magnetic materials. It modifies the electronic structure but does not necessarily destroy ferromagnetism, as exemplified by amorphous ferromagnets. Spin glasses combine disorder with competing exchange, and their ground state is neither ferromagnetic nor antiferromagnetic. The equilibrium and nonequilibrium properties of spin glasses have remained a complex problem, and several models have been developed, such as the Edwards–Anderson and Sherrington–Kirkpatrick models. On a mean-field level, the determination of ordering and spin-glass temperatures involves the diagonalisation of large random matrices. This chapter discusses disordered magnets and spin glasses, ferromagnetic order in inhomogeneous magnets, soft matter, transport, magnetism, random walks, polymers, diffusion, polymers and critical dimensionality, gases in magnetic metals, magnetoresistance, Bruggeman model, nanostructures, thin films, length scales in nanomagnetism, random anisotropy, and two-phase nanostructures. The use of models in disciplines other than magnetism is also considered, including metallurgy, biology and medicine, and social sciences.Less
Atomic disorder has far-reaching consequences for the behavior of magnetic materials. It modifies the electronic structure but does not necessarily destroy ferromagnetism, as exemplified by amorphous ferromagnets. Spin glasses combine disorder with competing exchange, and their ground state is neither ferromagnetic nor antiferromagnetic. The equilibrium and nonequilibrium properties of spin glasses have remained a complex problem, and several models have been developed, such as the Edwards–Anderson and Sherrington–Kirkpatrick models. On a mean-field level, the determination of ordering and spin-glass temperatures involves the diagonalisation of large random matrices. This chapter discusses disordered magnets and spin glasses, ferromagnetic order in inhomogeneous magnets, soft matter, transport, magnetism, random walks, polymers, diffusion, polymers and critical dimensionality, gases in magnetic metals, magnetoresistance, Bruggeman model, nanostructures, thin films, length scales in nanomagnetism, random anisotropy, and two-phase nanostructures. The use of models in disciplines other than magnetism is also considered, including metallurgy, biology and medicine, and social sciences.
Cristopher Moore and Stephan Mertens
- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780199233212
- eISBN:
- 9780191775079
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199233212.003.0012
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
Random sampling is a technique for dealing with possible states or solutions having an exponentially large space. The best method of random sampling generally involves a random walk or a Markov ...
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Random sampling is a technique for dealing with possible states or solutions having an exponentially large space. The best method of random sampling generally involves a random walk or a Markov chain. A Markov chain requires a number of steps to approach equilibrium, and thus provide a good random sample of the state space. This number of steps is called mixing time, which can be calculated by thinking about how quickly its choices overwhelm the system’s memory of its initial state, the extent to which one part of a system influences another, and how smoothly probability flows from one part of the state space to another. This chapter explores random walks and rapid mixing, first by considering a classic example from physics: a block of iron. It then discusses transition matrices, ergodicity, coupling, spectral gap, and expanders, as well as the role of conductance and the spectral gap in rapid mixing. It concludes by showing that temporal mixing is closely associated with spatial mixing.Less
Random sampling is a technique for dealing with possible states or solutions having an exponentially large space. The best method of random sampling generally involves a random walk or a Markov chain. A Markov chain requires a number of steps to approach equilibrium, and thus provide a good random sample of the state space. This number of steps is called mixing time, which can be calculated by thinking about how quickly its choices overwhelm the system’s memory of its initial state, the extent to which one part of a system influences another, and how smoothly probability flows from one part of the state space to another. This chapter explores random walks and rapid mixing, first by considering a classic example from physics: a block of iron. It then discusses transition matrices, ergodicity, coupling, spectral gap, and expanders, as well as the role of conductance and the spectral gap in rapid mixing. It concludes by showing that temporal mixing is closely associated with spatial mixing.
Anindya Banerjee, Juan J. Dolado, John W. Galbraith, and David F. Hendry
- Published in print:
- 1993
- Published Online:
- November 2003
- ISBN:
- 9780198288107
- eISBN:
- 9780191595899
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198288107.003.0003
- Subject:
- Economics and Finance, Econometrics
Presents the important properties of integrated variables and sets out some of the preliminary asymptotic theories essential for the consideration of such processes. It explores the concepts of unit ...
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Presents the important properties of integrated variables and sets out some of the preliminary asymptotic theories essential for the consideration of such processes. It explores the concepts of unit roots, non‐stationarity, orders of integration, and near integration, and demonstrates the use of the theory in understanding the behaviour of least‐squares estimators in spurious regressions and in models involving integrated data. The theoretical analysis is accompanied by evidence from Monte Carlo simulations. Several examples are also provided to illustrate the use of Wiener distribution theory in deriving asymptotic results for such models.Less
Presents the important properties of integrated variables and sets out some of the preliminary asymptotic theories essential for the consideration of such processes. It explores the concepts of unit roots, non‐stationarity, orders of integration, and near integration, and demonstrates the use of the theory in understanding the behaviour of least‐squares estimators in spurious regressions and in models involving integrated data. The theoretical analysis is accompanied by evidence from Monte Carlo simulations. Several examples are also provided to illustrate the use of Wiener distribution theory in deriving asymptotic results for such models.
Jean Zinn-Justin
- Published in print:
- 2007
- Published Online:
- January 2010
- ISBN:
- 9780199227198
- eISBN:
- 9780191711107
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199227198.003.0003
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This chapter discusses the related questions of universality and macroscopic continuum limit in random systems with a large number of degrees of freedom. It first explains the notion of universality ...
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This chapter discusses the related questions of universality and macroscopic continuum limit in random systems with a large number of degrees of freedom. It first explains the notion of universality using the classical example of the central limit theorem in probability theory. It then discusses the properties of the random walk on a lattice, where universality is directly related to the continuum limit. In both examples, the chapter is interested in the collective properties of an infinite number of random variables in a situation where the probability of large deviations with respect to the mean value decreases fast enough. They differ in the sense that a random walk is based on a spatial structure that does not necessarily exist in the case of the central limit theorem. From the study of these first examples emerges the importance of Gaussian distributions, and this justifies the technical considerations of Chapter 2. The chapter introduces some transformations, acting on distributions, which decrease the number of random variables. It shows that Gaussian distributions are attractive fixed points for these transformations. This will provides the first, extremely simple, applications of the renormalization group (RG) ideas and allows the establishment of corresponding terminology. Finally, in this context of the random walk, a path integral representation is associated with the existence of a continuum limit. Exercises are provided at the end of the chapter.Less
This chapter discusses the related questions of universality and macroscopic continuum limit in random systems with a large number of degrees of freedom. It first explains the notion of universality using the classical example of the central limit theorem in probability theory. It then discusses the properties of the random walk on a lattice, where universality is directly related to the continuum limit. In both examples, the chapter is interested in the collective properties of an infinite number of random variables in a situation where the probability of large deviations with respect to the mean value decreases fast enough. They differ in the sense that a random walk is based on a spatial structure that does not necessarily exist in the case of the central limit theorem. From the study of these first examples emerges the importance of Gaussian distributions, and this justifies the technical considerations of Chapter 2. The chapter introduces some transformations, acting on distributions, which decrease the number of random variables. It shows that Gaussian distributions are attractive fixed points for these transformations. This will provides the first, extremely simple, applications of the renormalization group (RG) ideas and allows the establishment of corresponding terminology. Finally, in this context of the random walk, a path integral representation is associated with the existence of a continuum limit. Exercises are provided at the end of the chapter.
J. Klafter and I. M. Sokolov
- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780199234868
- eISBN:
- 9780191775024
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199234868.001.0001
- Subject:
- Physics, Soft Matter / Biological Physics
The name “random walk” for a problem of a displacement of a point in a sequence of independent random steps was coined by Karl Pearson in 1905 in a question posed to readers of “Nature”. The same ...
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The name “random walk” for a problem of a displacement of a point in a sequence of independent random steps was coined by Karl Pearson in 1905 in a question posed to readers of “Nature”. The same year, a similar problem was formulated by Albert Einstein in one of his Annus Mirabilis works. Even earlier problem was posed by Louis Bachelier in his thesis devoted to the theory of financial speculations in 1900. Nowadays theory of random walks was proved useful in physics and chemistry (diffusion, reactions, mixing in flows), economics, biology (from animal spread to motion of subcellular structures) and in many other disciplines. The random walk approach serves not only as a model of simple diffusion but of many complex sub‐ and superdiffusive transport processes as well. This book discusses main variants of the random walks and gives the most important mathematical tools for their theoretical description.Less
The name “random walk” for a problem of a displacement of a point in a sequence of independent random steps was coined by Karl Pearson in 1905 in a question posed to readers of “Nature”. The same year, a similar problem was formulated by Albert Einstein in one of his Annus Mirabilis works. Even earlier problem was posed by Louis Bachelier in his thesis devoted to the theory of financial speculations in 1900. Nowadays theory of random walks was proved useful in physics and chemistry (diffusion, reactions, mixing in flows), economics, biology (from animal spread to motion of subcellular structures) and in many other disciplines. The random walk approach serves not only as a model of simple diffusion but of many complex sub‐ and superdiffusive transport processes as well. This book discusses main variants of the random walks and gives the most important mathematical tools for their theoretical description.
Christian Gollier
- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691148762
- eISBN:
- 9781400845408
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691148762.003.0008
- Subject:
- Economics and Finance, Development, Growth, and Environmental
This chapter aims to provide a unified theoretical foundation to the term structure of discount rates. To do this the chapter develops a benchmark model based on two assumptions: individual ...
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This chapter aims to provide a unified theoretical foundation to the term structure of discount rates. To do this the chapter develops a benchmark model based on two assumptions: individual preferences toward risk, and the nature of the uncertainty over economic growth. Previously, it was shown that constant relative risk aversion, combined with a random walk for the growth of log consumption, yields a flat term structure for efficient discount rates. In this chapter, these two assumptions are relaxed by using a stochastic dominance approach. Stochastic models of economic growth with mean-reversion, Markov switches, and parametric uncertainty all exhibit some forms of positive statistical dependence of successive growth rates. Because this tends to magnify the long-term risk, it is the driving force of the decreasing nature of the term structure.Less
This chapter aims to provide a unified theoretical foundation to the term structure of discount rates. To do this the chapter develops a benchmark model based on two assumptions: individual preferences toward risk, and the nature of the uncertainty over economic growth. Previously, it was shown that constant relative risk aversion, combined with a random walk for the growth of log consumption, yields a flat term structure for efficient discount rates. In this chapter, these two assumptions are relaxed by using a stochastic dominance approach. Stochastic models of economic growth with mean-reversion, Markov switches, and parametric uncertainty all exhibit some forms of positive statistical dependence of successive growth rates. Because this tends to magnify the long-term risk, it is the driving force of the decreasing nature of the term structure.
Thomas Koshy
- Published in print:
- 2008
- Published Online:
- January 2009
- ISBN:
- 9780195334548
- eISBN:
- 9780199868766
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195334548.003.0002
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics
This chapter focuses on the ubiquitous central binomial coefficient and its properties. It confirms that Cn is a positive integer. The chapter covers Lagrange's identity.
This chapter focuses on the ubiquitous central binomial coefficient and its properties. It confirms that Cn is a positive integer. The chapter covers Lagrange's identity.
Carlos Gustavo Moreira and Daniel Smania
Araceli Bonifant, Mikhail Lyubich, and Scott Sutherland (eds)
- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691159294
- eISBN:
- 9781400851317
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691159294.003.0013
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics
This chapter considers the stability of metric (measure-theoretic) properties of dynamical systems. A well-known example is that of (C²) expanding maps on the circle; this class is structurally ...
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This chapter considers the stability of metric (measure-theoretic) properties of dynamical systems. A well-known example is that of (C²) expanding maps on the circle; this class is structurally stable, and all such maps have an absolutely continuous and ergodic invariant probability satisfying certain decay of correlations estimates. In particular, in the measure theoretic sense, most of the orbits are dense in the phase space. The chapter uses the idea of random walk, which describes transitions between various dynamical scales, to prove a surprising rigidity result: the conjugacy between two unimodal maps of the same degree with Feigenbaum or wild attractors is absolutely continuous.Less
This chapter considers the stability of metric (measure-theoretic) properties of dynamical systems. A well-known example is that of (C²) expanding maps on the circle; this class is structurally stable, and all such maps have an absolutely continuous and ergodic invariant probability satisfying certain decay of correlations estimates. In particular, in the measure theoretic sense, most of the orbits are dense in the phase space. The chapter uses the idea of random walk, which describes transitions between various dynamical scales, to prove a surprising rigidity result: the conjugacy between two unimodal maps of the same degree with Feigenbaum or wild attractors is absolutely continuous.
NEIL F. JOHNSON, PAUL JEFFERIES, and PAK MING HUI
- Published in print:
- 2003
- Published Online:
- January 2010
- ISBN:
- 9780198526650
- eISBN:
- 9780191712104
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198526650.003.0002
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This chapter focuses on the basics of standard finance theory. It discusses the details of a random walk and quantifying risk using volatility σ. It then looks at the hedging of risk within standard ...
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This chapter focuses on the basics of standard finance theory. It discusses the details of a random walk and quantifying risk using volatility σ. It then looks at the hedging of risk within standard finance theory using derivatives beginning with a review of what derivatives are followed by a description of the Black–Scholes theory.Less
This chapter focuses on the basics of standard finance theory. It discusses the details of a random walk and quantifying risk using volatility σ. It then looks at the hedging of risk within standard finance theory using derivatives beginning with a review of what derivatives are followed by a description of the Black–Scholes theory.
David W. Stephens, Iain Couzin, and Luc-Alain Giraldeau
- Published in print:
- 2012
- Published Online:
- May 2016
- ISBN:
- 9780262018098
- eISBN:
- 9780262306003
- Item type:
- chapter
- Publisher:
- The MIT Press
- DOI:
- 10.7551/mitpress/9780262018098.003.0003
- Subject:
- Sociology, Social Psychology and Interaction
This chapter offers a selective review of behavioral and ecological perspectives on search behavior. Basic results from foraging theory are presented and their relationship to search is discussed. ...
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This chapter offers a selective review of behavioral and ecological perspectives on search behavior. Basic results from foraging theory are presented and their relationship to search is discussed. Techniques for the statistical description of searching motion are outlined, with a focus on the correlated random walk and the so-called Lévy flights—a technique that holds considerable promise. The problems of search in groups are reviewed at several levels. Both cooperative search (as conducted, e.g., by members of a social insect colony) and group movements of extremely selfish animals are considered. Finally, a review is provided of the producer-scrounger game, which considers the interactions within groups when some individuals parasitize the search behavior of others. The implications of these ideas are discussed and potential future directions for future enquiry are highlighted.Less
This chapter offers a selective review of behavioral and ecological perspectives on search behavior. Basic results from foraging theory are presented and their relationship to search is discussed. Techniques for the statistical description of searching motion are outlined, with a focus on the correlated random walk and the so-called Lévy flights—a technique that holds considerable promise. The problems of search in groups are reviewed at several levels. Both cooperative search (as conducted, e.g., by members of a social insect colony) and group movements of extremely selfish animals are considered. Finally, a review is provided of the producer-scrounger game, which considers the interactions within groups when some individuals parasitize the search behavior of others. The implications of these ideas are discussed and potential future directions for future enquiry are highlighted.
Melvin Lax, Wei Cai, and Min Xu
- Published in print:
- 2006
- Published Online:
- January 2010
- ISBN:
- 9780198567769
- eISBN:
- 9780191718359
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198567769.003.0003
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
Consider two physical problems describable by the same random process. The first process is the radioactive decay of a collection of nuclei. The second is the production of photoelectrons by a steady ...
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Consider two physical problems describable by the same random process. The first process is the radioactive decay of a collection of nuclei. The second is the production of photoelectrons by a steady beam of light on a photodetector. In both cases, we can let a discrete, positive, integer valued, variable n(t) represent the number of counts emitted in the time interval between 0 and t. This is the essence of the Poisson process, an example of Markovian process. Other examples of Markovian processes include the one dimensional random walk, gambler's ruin, diffusion processes and the Einstein relation, Brownian motion, Langevin theory of velocities in Brownian motion, Langevin theory of positions in Brownian motion, and chaos.Less
Consider two physical problems describable by the same random process. The first process is the radioactive decay of a collection of nuclei. The second is the production of photoelectrons by a steady beam of light on a photodetector. In both cases, we can let a discrete, positive, integer valued, variable n(t) represent the number of counts emitted in the time interval between 0 and t. This is the essence of the Poisson process, an example of Markovian process. Other examples of Markovian processes include the one dimensional random walk, gambler's ruin, diffusion processes and the Einstein relation, Brownian motion, Langevin theory of velocities in Brownian motion, Langevin theory of positions in Brownian motion, and chaos.
