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System Control and Rough Paths

Terry Lyons and Zhongmin Qian

Published in print:
2002
Published Online:
September 2007
ISBN:
9780198506485
eISBN:
9780191709395
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198506485.001.0001
Subject:
Mathematics, Probability / Statistics

Systems evolve and interact, and when they do this in a continuous way, differential equations can be used to provide accurate mathematical models for the interaction or reaction of the controlled ... More


The Porous Medium Equation: Mathematical Theory

Juan Luis Vazquez

Published in print:
2006
Published Online:
September 2007
ISBN:
9780198569039
eISBN:
9780191717468
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198569039.001.0001
Subject:
Mathematics, Mathematical Physics

The heat equation is one of the three classical linear partial differential equations of second order that form the basis of any elementary introduction to the area of PDEs, and only recently has it ... More


NON-PARAMETRIC IMAGE REGISTRATION

Jan Modersitzki

in Numerical Methods for Image Registration

Published in print:
2003
Published Online:
September 2007
ISBN:
9780198528418
eISBN:
9780191713583
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198528418.003.0008
Subject:
Mathematics, Applied Mathematics

This chapter introduces non-parametric registrations. The idea behind this type of registration is to come up with an appropriate measure both for the similarity as well as for the likelihood of a ... More


The Universality of the Radon Transform

Leon Ehrenpreis

Published in print:
2003
Published Online:
September 2007
ISBN:
9780198509783
eISBN:
9780191709166
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198509783.001.0001
Subject:
Mathematics, Mathematical Physics

Radon showed how to write arbitrary functions in Rn in terms of the characteristic functions (delta functions) of hyperplanes. This idea leads to various generalizations. For example, R can be ... More


Wavelet Methods for Elliptic Partial Differential Equations

Karsten Urban

Published in print:
2008
Published Online:
May 2009
ISBN:
9780198526056
eISBN:
9780191712340
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198526056.001.0001
Subject:
Mathematics, Applied Mathematics, Mathematical Finance

Wavelets have become a powerful tool in several applications by now. Their use for the numerical solution of operator equations has been investigated more recently. By now the theoretical ... More


EXTENSION OF SOLUTIONS OF DIFFERENTIAL EQUATIONS

Leon Ehrenpreis

in The Universality of the Radon Transform

Published in print:
2003
Published Online:
September 2007
ISBN:
9780198509783
eISBN:
9780191709166
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198509783.003.0009
Subject:
Mathematics, Mathematical Physics

This chapter shows how the book's main theory allows for the treatment of extension problems for partial differential equations. The culmination of this is a solution of the asymptotic edge of the ... More


Form of the Equations

W. Otto Friesen and Jonathon A. Friesen

in NeuroDynamix II: Concepts of Neurophysiology Illustrated by Computer Simulations

Published in print:
2009
Published Online:
February 2010
ISBN:
9780195371833
eISBN:
9780199865178
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780195371833.003.0023
Subject:
Psychology, Cognitive Neuroscience

This section describes the methods for numerical integration of the differential equations that underlie all of the dynamic models embodied in NeuroDynamix II. This chapter presents the general form ... More


Chaotic Economic Dynamics

Richard M. Goodwin

Published in print:
1990
Published Online:
November 2003
ISBN:
9780198283355
eISBN:
9780191596315
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/0198283350.001.0001
Subject:
Economics and Finance, Macro- and Monetary Economics

This collection of short essays provides an application of chaotic dynamics to economic systems. Each chapter presents several economic models incorporating differential (or difference) equations ... More


Adaptive Diversification and Speciation as Pattern Formation in Partial Differential Equation Models

Michael Doebeli

in Adaptive Diversification (MPB-48)

Published in print:
2011
Published Online:
October 2017
ISBN:
9780691128931
eISBN:
9781400838936
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691128931.003.0009
Subject:
Biology, Biodiversity / Conservation Biology

This chapter discusses partial differential equation models. Partial differential equations can describe the dynamics of phenotype distributions of polymorphic populations, and they allow for a ... More


HARMONIC FUNCTIONS IN R n

Leon Ehrenpreis

in The Universality of the Radon Transform

Published in print:
2003
Published Online:
September 2007
ISBN:
9780198509783
eISBN:
9780191709166
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198509783.003.0003
Subject:
Mathematics, Mathematical Physics

This chapter looks at a variation of “harmonic function” which was introduced by Chevalley. Ordinary harmonic functions are solutions of the Laplace equation; Chevalley harmonic functions are ... More


LIPSCHITZ PATHS

Terry Lyons and Zhongmin Qian

in System Control and Rough Paths

Published in print:
2002
Published Online:
September 2007
ISBN:
9780198506485
eISBN:
9780191709395
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198506485.003.0002
Subject:
Mathematics, Probability / Statistics

The theory of controlled systems is worked out in detail for the case where the driving stimulus or noise is a Lipschitz function. The Itô functional, which is the map from stimulus to response, is ... More


Models in Finance

GREGORY C. CHOW

in Dynamic Economics: Optimization by the Lagrange Method

Published in print:
1997
Published Online:
October 2011
ISBN:
9780195101928
eISBN:
9780199855032
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780195101928.003.0007
Subject:
Economics and Finance, Financial Economics

Using stochastic differential equations instead of utilizing stochastic difference equations, most of the models involved in finance follow Merton’s work and are developed in continuous time. In this ... More


Dynamic Modeling of Core Processes

Domitilla Del Vecchio and Richard M. Murray

in Biomolecular Feedback Systems

Published in print:
2014
Published Online:
October 2017
ISBN:
9780691161532
eISBN:
9781400850501
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691161532.003.0002
Subject:
Biology, Biochemistry / Molecular Biology

This chapter describes basic biological mechanisms in a way that can be represented by simple dynamical models. It begins with a discussion of the basic modeling formalisms that will be utilized to ... More


Ecological Orbits: How Planets Move and Populations Grow

Lev Ginzburg and Mark Colyvan

Published in print:
2003
Published Online:
September 2007
ISBN:
9780195168167
eISBN:
9780199790159
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780195168167.001.0001
Subject:
Biology, Ecology

The main focus of the book is the presentation of the inertial view of population growth. This view provides a rather simple model for complex population dynamics, and is achieved at the level of the ... More


Towards a Non‐Linear Schwarz'S List

Philip Boalch

in The Many Facets of Geometry: A Tribute to Nigel Hitchin

Published in print:
2010
Published Online:
September 2010
ISBN:
9780199534920
eISBN:
9780191716010
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199534920.003.0011
Subject:
Mathematics, Geometry / Topology

This chapter focuses on ‘icosahedral’ solutions of (ordinary) differential equations. It considers various symmetry groups each of which could be thought of as the next in a sequence, following the ... More


UNIVERSAL LIMIT THEOREM

Terry Lyons and Zhongmin Qian

in System Control and Rough Paths

Published in print:
2002
Published Online:
September 2007
ISBN:
9780198506485
eISBN:
9780191709395
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198506485.003.0006
Subject:
Mathematics, Probability / Statistics

The key result presented in this book and the result which justifies the definition of a rough path is the universal limit theorem. The chapter proves in section 6.3 that a system of differential ... More


VECTOR FIELDS AND FLOW EQUATIONS

Terry Lyons and Zhongmin Qian

in System Control and Rough Paths

Published in print:
2002
Published Online:
September 2007
ISBN:
9780198506485
eISBN:
9780191709395
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198506485.003.0007
Subject:
Mathematics, Probability / Statistics

This chapter studies spaces of paths as geometric objects in their own right, and particularly looks at the possibility of constructing interesting and mathematically meaningful vector fields on ... More


Stochastic Dynamic Bifurcations and Excitability

Nils Berglund and Barbara Gentz

in Stochastic Methods in Neuroscience

Published in print:
2009
Published Online:
February 2010
ISBN:
9780199235070
eISBN:
9780191715778
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199235070.003.0003
Subject:
Mathematics, Biostatistics

Some models of action potential generation in neurons like the Fitzhugh–Nagumo and the Morris–Lecar model are given by slow–fast differential equations. We outline a general theory allowing us to ... More


Free systems

JESPER LÜTZEN

in Mechanistic Images in Geometric Form: Heinrich Hertz's 'Principles of Mechanics'

Published in print:
2005
Published Online:
January 2010
ISBN:
9780198567370
eISBN:
9780191717925
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198567370.003.0017
Subject:
Physics, History of Physics

Heinrich Hertz's strategy for studying the motion of mechanical systems was to deal first with free systems, to which the fundamental law applies, and then with unfree systems considered as ... More


One‐Dimensional Differential Equations

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/9780199566433.003.0030
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter deals with some one-dimensional differential equations and examines the types of dynamical behaviour that are (and are not) possible for such systems. It first considers the continuous ... More


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