## 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