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New Perspectives in Stochastic Geometry

Wilfrid S. Kendall and Ilya Molchanov (eds)

Published in print:
2009
Published Online:
February 2010
ISBN:
9780199232574
eISBN:
9780191716393
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199232574.001.0001
Subject:
Mathematics, Geometry / Topology

Stochastic geometry is a subject with roots stretching back at least 300 years, but one which has only been formed as an academic area in the last 50 years. It covers the study of random patterns, ... More


Modeling Reality: How Computers Mirror Life

Iwo Bialynicki-Birula and Iwona Bialynicka-Birula

Published in print:
2004
Published Online:
January 2010
ISBN:
9780198531005
eISBN:
9780191713033
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198531005.001.0001
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This book covers a wide range of subjects concerning the use of computer modeling to solve a diverse set of problems. The book covers some advanced topics (cellular automata, Shannon measure of ... More


Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers

Robert C. Hilborn

Published in print:
2000
Published Online:
January 2010
ISBN:
9780198507239
eISBN:
9780191709340
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198507239.001.0001
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This book introduces the full range of activity in the rapidly growing field of nonlinear dynamics. Using a step-by-step introduction to dynamics and geometry in state space as the central focus of ... More


Brownian Motion: Fluctuations, Dynamics, and Applications

Robert M. Mazo

Published in print:
2008
Published Online:
January 2010
ISBN:
9780199556441
eISBN:
9780191705625
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199556441.001.0001
Subject:
Physics, Condensed Matter Physics / Materials

Brownian motion is the incessant motion of small particles immersed in an ambient medium. It is due to fluctuations in the motion of the medium particles on the molecular scale. The name has been ... More


Random Fractals

Peter Mörters

in New Perspectives in Stochastic Geometry

Published in print:
2009
Published Online:
February 2010
ISBN:
9780199232574
eISBN:
9780191716393
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199232574.003.0008
Subject:
Mathematics, Geometry / Topology

This chapter expounds the theory of random fractals, using tree representation as a unifying principle. Applications to the fine structure of Brownian motion are discussed.


From Cantor to Mandelbrot: Self-similarity and fractals

Iwo Białynicki-Birula and Iwona Białynicka-Birula

in Modeling Reality: How Computers Mirror Life

Published in print:
2004
Published Online:
January 2010
ISBN:
9780198531005
eISBN:
9780191713033
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198531005.003.0008
Subject:
Physics, Theoretical, Computational, and Statistical Physics

Objects that look similar after magnification are said to have fractal structure. Perfect fractals can exist only in mathematics but in the real world we have plenty of approximate fractals, from ... More


Chaos and Fractals: An Elementary Introduction

David P. Feldman

Published in print:
2012
Published Online:
December 2013
ISBN:
9780199566433
eISBN:
9780191774966
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199566433.001.0001
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This book provides an elementary introduction to chaos and fractals. It introduces the key phenomena of chaos — aperiodicity, sensitive dependence on initial conditions, bifurcations — via simple ... More


Introducing Fractals

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/9780199566433.003.0016
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter focuses on fractals and the role of iteration in their generation. It first considers three familiar shapes from geometry: a circle, a line segment, and a rectangle. It then describes a ... More


Random walks on percolation structures

J. Klafter and I.M. Sokolov

in First Steps in Random Walks: From Tools to Applications

Published in print:
2011
Published Online:
December 2013
ISBN:
9780199234868
eISBN:
9780191775024
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199234868.003.0010
Subject:
Physics, Soft Matter / Biological Physics

In this chapter another, experimentally widespread, situation is considered. The random walk takes place not on a homogeneous lattice, where each site in principle accessible to the walker, but on a ... More


Hierarchies, Complex Fractal Dimensions, and Log-Periodicity

Didier Sornette

in Why Stock Markets Crash: Critical Events in Complex Financial Systems

Published in print:
2017
Published Online:
May 2018
ISBN:
9780691175959
eISBN:
9781400885091
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691175959.003.0006
Subject:
Business and Management, Finance, Accounting, and Banking

This chapter describes the concept of fractals and their self-similarity, including fractals with complex dimensions. It shows how these geometric and mathematical objects enable one to codify the ... More


The geometry of a single fracture

Pierre M. Adler, Jean-François Thovert, and Valeri V. Mourzenko

in Fractured Porous Media

Published in print:
2012
Published Online:
January 2013
ISBN:
9780199666515
eISBN:
9780191748639
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199666515.003.0002
Subject:
Physics, Condensed Matter Physics / Materials

This chapter addresses the geometry of a single fracture with a double objective which is to characterize it and to reproduce it. The geometrical quantities which characterize the structure of a ... More


Extreme Outcomes, Connectivity, and Power Laws: Towards an Econophysics of Organization

Max Boisot and Bill McKelvey

in Knowledge, Organization, and Management: Building on the Work of Max Boisot

Published in print:
2013
Published Online:
September 2013
ISBN:
9780199669165
eISBN:
9780191749346
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199669165.003.0004
Subject:
Business and Management, Organization Studies, Knowledge Management

Organization science to study extremes more rigorously. For a new perspective, we turn to an emerging new physics expressly aimed at dealing with complexity dynamics, Econophysics. It incorporates ... More


Opening Remarks

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/9780199566433.003.0001
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter starts by introducing the terms ‘chaos’ and ‘fractal’ and talks briefly about their history. Since the 1970s, these two concepts — chaos and fractals — have become an important element ... More


Dimensions

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/9780199566433.003.0017
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter explains how to characterise fractals by means of the dimension. It defines dimension in terms of the scaling properties of a shape in order to describe fractals quantitatively. The ... More


Random Fractals

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/9780199566433.003.0018
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter explores ways of generating fractals other than a deterministic procedure. In particular, it considers fractal-generating mechanisms that involve randomness or irregularity. The ... More


The Box‐Counting Dimension

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/9780199566433.003.0019
Subject:
Physics, Theoretical, Computational, and Statistical Physics

There are several examples of fractals that are not exactly self-similar, as is the case with small parts of the random Koch curve which exhibit statistical self-similarity but not identicality. The ... More


When do Averages Exist?

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/9780199566433.003.0020
Subject:
Physics, Theoretical, Computational, and Statistical Physics

Since fractals are self-similar, it is often not useful to describe them in terms of an average size. Stating an average size does not capture what is interesting or noteworthy about the shape of a ... More


Power Laws and Long Tails

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/9780199566433.003.0021
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter examines distributions known as power laws, which are fractal because they are scale-free. It first considers normal or Gaussian distributions to provide a useful contrast to power-law ... More


Infinities, Big and Small

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/9780199566433.003.0022
Subject:
Physics, Theoretical, Computational, and Statistical Physics

The Cantor set and the Koch curve are two examples of fractals may be considered not only as geometrical objects, but also as a means to explore apparent paradoxes and difficulties in set theory and ... More


Conclusion

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/9780199566433.003.0033
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This concluding chapter highlights some of the key themes and lessons of chaos and fractals and reflects on how their impact can be characterised. It discusses how deterministic dynamical systems can ... More


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