## Effective Theories for Ostwald Ripening

*Barbara Niethammer*

### in Analysis and Stochastics of Growth Processes and Interface Models

- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780199239252
- eISBN:
- 9780191716911
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199239252.003.0010
- Subject:
- Mathematics, Probability / Statistics, Analysis

This chapter discusses the derivation and analysis of reduced models for a coarsening process known as Ostwald ripening, which is a paradigm for statistical self-similarity in coarsening systems. The ... More

## FULL ANALYSIS OF SELF-SIMILARITY

*Juan Luis Vázquez*

### in The Porous Medium Equation: Mathematical Theory

- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780198569039
- eISBN:
- 9780191717468
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198569039.003.0016
- Subject:
- Mathematics, Mathematical Physics

This chapter examines the class of solutions of the PME that are invariant under the scaling group in the variables (x, t, u), and take therefore the so-called self-similar form. After a detailed ... More

## Extinction rates and asymptotics for 0 < m < m c

*Juan Luis Vázquez*

### in Smoothing and Decay Estimates for Nonlinear Diffusion Equations: Equations of Porous Medium Type

- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780199202973
- eISBN:
- 9780191707919
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199202973.003.0008
- Subject:
- Mathematics, Applied Mathematics

This chapter deals with the actual asymptotic behaviour of the solutions of the FDE in the exponent range 0 < m < mc . This behaviour depends on the class of initial data. The ... More

## The Subtle Mathematics of Music

*Peter van der Merwe*

### in Roots of the Classical: The Popular Origins of Western Music

- Published in print:
- 2004
- Published Online:
- May 2008
- ISBN:
- 9780198166474
- eISBN:
- 9780191713880
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198166474.003.0002
- Subject:
- Music, History, Western

What distinguishes music from other sounds is symmetry. Only in music are sounds related to one another by precise, quasi-mathematical ratios, and even there only in melodic pitch and rhythmic ... 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

## The Universality of Chaos

*Robert C. Hilborn*

### in Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers

- Published in print:
- 2000
- Published Online:
- January 2010
- ISBN:
- 9780198507239
- eISBN:
- 9780191709340
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198507239.003.0002
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

The many universal features of nonlinear behaviour took the scientific and mathematics worlds by surprise. For the period-doubling route to chaotic behaviour, the famous Feigenbaum numbers provide ... More

## Mechanisms in dynamically complex systems

*Meinard Kuhlmann*

### in Causality in the Sciences

- Published in print:
- 2011
- Published Online:
- September 2011
- ISBN:
- 9780199574131
- eISBN:
- 9780191728921
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199574131.003.0041
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy

In recent debates mechanisms are often discussed in the context of ‘complex systems’ which are understood as having a complicated compositional structure. The chapter wants to draw attention to ... More

## Aggregation

*Paul Charbonneau*

### in Natural Complexity: A Modeling Handbook

- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691176840
- eISBN:
- 9781400885497
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691176840.003.0003
- Subject:
- Computer Science, Programming Languages

This chapter explores how naturally occurring inanimate structures grow by accretion of smaller-sized components, focusing on one specific accretion process: diffusion-limited aggregation (DLA). In ... 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

## Like Me: A Homophily-Based Account of Human Culture

*Daniel B. M. Haun and Harriet Over*

### in Cultural Evolution: Society, Technology, Language, and Religion

- Published in print:
- 2013
- Published Online:
- May 2015
- ISBN:
- 9780262019750
- eISBN:
- 9780262318297
- Item type:
- chapter

- Publisher:
- The MIT Press
- DOI:
- 10.7551/mitpress/9780262019750.003.0005
- Subject:
- Biology, Evolutionary Biology / Genetics

This chapter presents a homophily-based account of human social structure and cultural transmission, wherein a tendency to favor similar others (homophily) is a key driving force in creating ... More

## Painting By Numbers

*Nicholas Mee*

### in Celestial Tapestry: The Warp and Weft of Art and Mathematics

- Published in print:
- 2020
- Published Online:
- September 2020
- ISBN:
- 9780198851950
- eISBN:
- 9780191886690
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198851950.003.0025
- Subject:
- Physics, History of Physics

Fractals have a property of self-similarity. They are similar to themselves in different regions, and similar to themselves on different length scales, so an enlargement of part of a fractal looks ... More

## Global network invariants

*Ernesto Estrada*

### in The Structure of Complex Networks: Theory and Applications

- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780199591756
- eISBN:
- 9780191774959
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199591756.003.0008
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter analyses several global structural invariants for networks. The discussions cover degree-based global invariants, such as degree centralisation and variance; graph irregularity and ... 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

## 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

## 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

## Gaia and Complexity

*Lee F. Klinger*

### in Scientists Debate Gaia: The Next Century

- Published in print:
- 2004
- Published Online:
- August 2013
- ISBN:
- 9780262194983
- eISBN:
- 9780262283182
- Item type:
- chapter

- Publisher:
- The MIT Press
- DOI:
- 10.7551/mitpress/9780262194983.003.0018
- Subject:
- Environmental Science, Climate

This chapter considers complexity, the body of theory on self-organization and self-regulation, and Gaia, the view of the Earth as a living system, in a study of the Earth and its ecosystems for ... More

## Nonlinear systems∗

*Chun Wa Wong*

### in Introduction to Mathematical Physics: Methods & Concepts

- Published in print:
- 2013
- Published Online:
- May 2013
- ISBN:
- 9780199641390
- eISBN:
- 9780191747786
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199641390.003.0006
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

Nonlinear systems show two very different faces: unpredictability and unusual coherence. Unpredictability is shown to arise from instabilities and bifurcations that are highly sensitive to small ... More

## Scaling

*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/9780198821939.003.0003
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

Scaling appears practically everywhere in science; it basically quantifies how the properties or shapes of an object change with the scale of the object. Scaling laws are always associated with power ... More