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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


Hydrodynamics

Stefano Atzeni and JÜrgen Meyer-Ter-Vehn

in The Physics of Inertial Fusion: BeamPlasma Interaction, Hydrodynamics, Hot Dense Matter

Published in print:
2004
Published Online:
January 2008
ISBN:
9780198562641
eISBN:
9780191714030
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198562641.003.0006
Subject:
Physics, Nuclear and Plasma Physics

This chapter is devoted to hydrodynamic concentration of energy, in particular to cold isentropic compression of matter by converging hydrodynamic flows. The discussion is restricted to ... More


Scale-invariance

Guido Caldarelli

in Scale-Free Networks: Complex Webs in Nature and Technology

Published in print:
2007
Published Online:
January 2010
ISBN:
9780199211517
eISBN:
9780191705984
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199211517.003.0004
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter presents an overview of self-similar phenomena in nature and why they are so important and how this applies to complex networks.


Scale-, Base-, and Sum-lnvariance

Arno Berger and Theodore P. Hill

in An Introduction to Benford's Law

Published in print:
2015
Published Online:
October 2017
ISBN:
9780691163062
eISBN:
9781400866588
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691163062.003.0005
Subject:
Mathematics, Probability / Statistics

This chapter establishes and illustrates three basic invariance properties of the Benford distribution that are instrumental in demonstrating whether or not certain datasets are Benford, and that ... More


Effective Field Theories for Nuclear and (Some) Atomic Physics

U. van Kolck

in Effective Field Theory in Particle Physics and Cosmology: Lecture Notes of the Les Houches Summer School: Volume 108, July 2017

Published in print:
2020
Published Online:
August 2020
ISBN:
9780198855743
eISBN:
9780191889370
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198855743.003.0006
Subject:
Physics, Particle Physics / Astrophysics / Cosmology

These lectures are a pedagogical—not comprehensive—introduction to the applications of effective field theory (EFT) in the context of nuclear and atomic physics. A common feature of these ... More


Kolmogorov’s (1941) theory revisited

W. David McComb

in Homogeneous, Isotropic Turbulence: Phenomenology, Renormalization and Statistical Closures

Published in print:
2014
Published Online:
April 2014
ISBN:
9780199689385
eISBN:
9780191768255
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199689385.003.0006
Subject:
Physics, Theoretical, Computational, and Statistical Physics

By restricting our attention in this book to HIT, we have ruled out effects due to mean shear, system rotation, density stratification; and so on. This leaves us with a stark choice: deviation from ... 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


Benfordʼs Law in the Natural Sciences

David Hoyle

in Benford's Law: Theory and Applications

Published in print:
2015
Published Online:
October 2017
ISBN:
9780691147611
eISBN:
9781400866595
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691147611.003.0016
Subject:
Mathematics, Probability / Statistics

This chapter focuses on the occurrence of Benford's law within the natural sciences, emphasizing that Benford's law is to be expected within many scientific data sets. This is a consequence of the ... More


A First Glance at Renormalisation and Symmetry

Laurent Baulieu, John Iliopoulos, and Roland Sénéor

in From Classical to Quantum Fields

Published in print:
2017
Published Online:
May 2017
ISBN:
9780198788393
eISBN:
9780191830310
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198788393.003.0017
Subject:
Physics, Particle Physics / Astrophysics / Cosmology, Theoretical, Computational, and Statistical Physics

The fate of the classical symmetries in a quantum field theory after renormalisation. Global symmetries versus gaugec symmetries. The Adler–Bell–Jackiw anomaly. The breaking of scale invariance and ... More


It Must Be Beautiful

Roger Arditi and Lev R. Ginzburg

in How Species Interact: Altering the Standard View on Trophic Ecology

Published in print:
2012
Published Online:
May 2015
ISBN:
9780199913831
eISBN:
9780190267902
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:osobl/9780199913831.003.0007
Subject:
Biology, Ecology

This chapter explains the role of ratio-dependence as an invariance in ecology. It analyzes the importance of using scaling invariance in mathematics and proposes that all models of interacting ... More


Hadrons, partons and QCD

Michael Kachelriess

in Quantum Fields: From the Hubble to the Planck Scale

Published in print:
2017
Published Online:
February 2018
ISBN:
9780198802877
eISBN:
9780191841330
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198802877.003.0018
Subject:
Physics, Particle Physics / Astrophysics / Cosmology, Theoretical, Computational, and Statistical Physics

This chapter first discusses the breaking of scale invariance of QCD with massless quarks by quantum corrections and explains that this effect is responsible for the bulk of hadron masses. Then the ... More


Percolation

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/9780691176840.003.0004
Subject:
Computer Science, Programming Languages

This chapter explores a lattice-based system where complex structures can arise from pure randomness: percolation, typically described as the passage of liquid through a porous or granular medium. In ... More


Epidemics

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/9780691176840.003.0009
Subject:
Computer Science, Programming Languages

This chapter examines the complex nature of the epidemic spread of contagious diseases. It first describes the model of epidemic spread constructed by adding random walks on a lattice to the ... More


Earthquakes

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/9780691176840.003.0008
Subject:
Computer Science, Programming Languages

This chapter examines the natural complexity of earthquakes. Earthquakes release energy that is quantified by their magnitude m, a logarithmic measure of seismic wave amplitudes. The number N of ... More


Complexity and Criticality

Helmut Satz

in The Rules of the Flock: Self-Organization and Swarm Structure in Animal Societies

Published in print:
2020
Published Online:
March 2020
ISBN:
9780198853398
eISBN:
9780191888052
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198853398.003.0007
Subject:
Physics, Soft Matter / Biological Physics, Particle Physics / Astrophysics / Cosmology

Complex systems and critical behavior in complex system are defined in terms of correlation between constituents in the medium, subject to screening by intermediate constituents. At a critical point, ... More


The Navier–Stokes Equations

S. G. Rajeev

in Fluid Mechanics: A Geometrical Point of View

Published in print:
2018
Published Online:
October 2018
ISBN:
9780198805021
eISBN:
9780191843136
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198805021.003.0003
Subject:
Physics, Soft Matter / Biological Physics, Condensed Matter Physics / Materials

When different layers of a fluid move at different velocities, there is some friction which results in loss of energy and momentum to molecular degrees of freedom. This dissipation is measured by a ... More


Singularities

S. G. Rajeev

in Fluid Mechanics: A Geometrical Point of View

Published in print:
2018
Published Online:
October 2018
ISBN:
9780198805021
eISBN:
9780191843136
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198805021.003.0012
Subject:
Physics, Soft Matter / Biological Physics, Condensed Matter Physics / Materials

The initial value problem of the incompressible Navier–Stokes equations is explained. Leray’s classic study of it (using Picard iteration) is simplified and described in the language of physics. The ... More


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