## 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

## 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

## 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

## 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

## 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

## 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

## 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

## Population Density Methods in Large-Scale Neural Network Modelling

*Daniel Tranchina*

### 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/9780199235070.003.0007
- Subject:
- Mathematics, Biostatistics

Population density methods have a rich history in theoretical and computational neuroscience. In earlier years, these methods were used in large part to study the statistics of spike trains. Starting ... More