## 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

## 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

## 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

## 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

## 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

## 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

## Finite‐Difference Methods for Continuous‐Time Dynamic Programming

*Graham V. Candler*

### in Computational Methods for the Study of Dynamic Economies

- Published in print:
- 2001
- Published Online:
- November 2003
- ISBN:
- 9780199248278
- eISBN:
- 9780191596605
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0199248273.003.0008
- Subject:
- Economics and Finance, Macro- and Monetary Economics

Introduces some of the methods and underlying ideas behind computational fluid dynamics—in particular, the use is discussed of finite‐difference methods for the simulation of dynamic economies. A ... More

## ELASTIC 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/9780198528418.003.0009
- Subject:
- Mathematics, Applied Mathematics

This chapter introduces elastic registration using the general framework introduced in Chapter 8. A physical motivation for this particular regularizer is given. The two images are viewed as two ... More

## Mathematical and computational modelling tools

*Baltazar D. Aguda and Avner Friedman*

### in Models of Cellular Regulation

- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780198570912
- eISBN:
- 9780191718717
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198570912.003.0003
- Subject:
- Physics, Soft Matter / Biological Physics

This chapter reviews chemical kinetics to illustrate the formulation of model equations for a given reaction mechanism. For spatially uniform systems, these model equations are usually ordinary ... More

## Differential equations in physics

*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/9780199641390.003.0005
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

Ordinary and partial differential equations appear in physics as equations of motion or of state. They are often linear differential equations for which a sum of solutions remains a solution. The ... More

## Appell Hypergeometric Functions

*Paula Tretkoff*

### in Complex Ball Quotients and Line Arrangements in the Projective Plane (MN-51)

- Published in print:
- 2016
- Published Online:
- October 2017
- ISBN:
- 9780691144771
- eISBN:
- 9781400881253
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691144771.003.0008
- Subject:
- Mathematics, Geometry / Topology

This chapter discusses the complete quadrilateral line arrangement, and especially its relationship with the space of regular points of the system of partial differential equations defining the ... More

## Generalized Fokker–Planck equation

*Melvin Lax, Wei Cai, and Min Xu*

### in Random Processes in Physics and Finance

- Published in print:
- 2006
- Published Online:
- January 2010
- ISBN:
- 9780198567769
- eISBN:
- 9780191718359
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198567769.003.0008
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

Thermal noise and Gaussian noise sources combine to create a category of Markovian processes known as Fokker–Planck processes. This chapter discusses Fokker–Planck processes such as ... More

## BIFURCATIONS AND INSTABILITIES

*Nasr M. Ghoniem and Daniel D. Walgraef*

### in Instabilities and Self-Organization in Materials: Volume I: Fundamentals of Nanoscience, Volume II: Applications in Materials Design and Nanotechnology

- Published in print:
- 2008
- Published Online:
- May 2008
- ISBN:
- 9780199298686
- eISBN:
- 9780191720222
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199298686.003.0007
- Subject:
- Physics, Condensed Matter Physics / Materials

This chapter focuses on the mathematical structures underlying the notions of stability, bifurcation, and instabilities in complex nonlinear dynamical systems described by sets of ordinary or partial ... 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

## Introduction

*Karsten Urban*

### in Wavelet Methods for Elliptic Partial Differential Equations

- Published in print:
- 2008
- Published Online:
- May 2009
- ISBN:
- 9780198526056
- eISBN:
- 9780191712340
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198526056.003.0001
- Subject:
- Mathematics, Applied Mathematics, Mathematical Finance

This chapter gives an outline of the history of wavelets, in particular with respect to their use to numerically solve partial differential equations. The scope of the book and its outline is ... More

## Differential Equations

*Bard Ermentrout and John Rinzel*

### in Computational Modeling Methods for Neuroscientists

- Published in print:
- 2009
- Published Online:
- August 2013
- ISBN:
- 9780262013277
- eISBN:
- 9780262258722
- Item type:
- chapter

- Publisher:
- The MIT Press
- DOI:
- 10.7551/mitpress/9780262013277.003.0002
- Subject:
- Neuroscience, Techniques

This chapter, which addresses the partial differential equations (PDEs) with the example of finding the speed and profile of a propagating impulse for a Hodgkin-Huxley-like cable equation, highlights ... More

## Stochastic Homogenization and Energy of Infinite Sets of Points

*Xavier Blanc*

### 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/9780199239252.003.0004
- Subject:
- Mathematics, Probability / Statistics, Analysis

This chapter presents, in a synthetic way, a series of recent works by X. Blanc, C. Le Bris, and P-L. Lions on homogenization of an elliptic partial differential equation under certain periodic or ... More

## Looking at the World Sideways

*Craig Callender*

### in What Makes Time Special?

- Published in print:
- 2017
- Published Online:
- July 2017
- ISBN:
- 9780198797302
- eISBN:
- 9780191839603
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198797302.003.0008
- Subject:
- Philosophy, Metaphysics/Epistemology

When physics tells its story of the world, it writes on spatial pages and we flip pages in the temporal directions. The present moment contains the seeds of what happens next. Relativity challenges ... More

## Fundamentals of fluid dynamics with an introduction to the importance of interfaces

*H. A. Stone*

### in Soft Interfaces: Lecture Notes of the Les Houches Summer School: Volume 98, July 2012

- Published in print:
- 2017
- Published Online:
- November 2017
- ISBN:
- 9780198789352
- eISBN:
- 9780191831201
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198789352.003.0001
- Subject:
- Physics, Soft Matter / Biological Physics

The topics discussed are all related to basic fluid mechanics. In these introductory notes I highlight some of the main features of fluid flows and their mathematical characterization. There is much ... More

## A Review of General Asset Pricing Theory

*Claus Munk*

### in Fixed Income Modelling

- Published in print:
- 2011
- Published Online:
- September 2011
- ISBN:
- 9780199575084
- eISBN:
- 9780191728648
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199575084.003.0004
- Subject:
- Economics and Finance, Financial Economics

The pricing of fixed income securities follows the same general principles as the pricing of all other financial assets. This chapter explains some important general concepts and results in asset ... More