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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


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