## 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

## Progressive Waves

*Yvonne Choquet-Bruhat*

### in General Relativity and the Einstein Equations

- Published in print:
- 2008
- Published Online:
- May 2009
- ISBN:
- 9780199230723
- eISBN:
- 9780191710872
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199230723.003.0011
- Subject:
- Mathematics, Applied Mathematics

This chapter draws on the treatment of progressive waves for non-linear equations used in Sections III.12 and III.13 to construct weak gravitational and electromagnetic waves on a given electrovac ... More

## ADVECTION AND ADVECTION DIFFUSION

*George Em Karniadakis and Spencer J. Sherwin*

### in Spectral/hp Element Methods for Computational Fluid Dynamics

- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198528692
- eISBN:
- 9780191713491
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198528692.003.0006
- Subject:
- Mathematics, Numerical Analysis

This chapter focuses on the scalar advection equation and develops a Galerkin discretization using the techniques described in Chapter 4. It includes an extended presentation of the discontinuous ... More

##
Understanding Polynomial Equations in *n* Unknowns

*Victor J. Katz and Karen Hunger Parshall*

### in Taming the Unknown: A History of Algebra from Antiquity to the Early Twentieth Century

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691149059
- eISBN:
- 9781400850525
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691149059.003.0012
- Subject:
- Mathematics, History of Mathematics

This chapter examines the search for simultaneous solutions which eventually led to the creation of whole new algebraic constructs over the course of the eighteenth and nineteenth centuries. The ... More

## Linear Stability Analysis

*Gary A. Glatzmaier*

### in Introduction to Modeling Convection in Planets and Stars: Magnetic Field, Density Stratification, Rotation

- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691141725
- eISBN:
- 9781400848904
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691141725.003.0003
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

This chapter describes a linear stability analysis (that is, solving for the critical Rayleigh number Ra and mode) that allows readers to check their linear codes against the analytic solution. For ... More

## INTRODUCTION

*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/9780198569039.003.0001
- Subject:
- Mathematics, Mathematical Physics

This chapter introduces the mathematical theory of the nonlinear heat equation referred to as the porous medium equation (PME). This equation is one of the simplest examples of a nonlinear evolution ... More

## Linear Stochastic Difference Equations

*Lars Peter Hansen and Thomas J. Sargent*

### in Recursive Models of Dynamic Linear Economies

- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691042770
- eISBN:
- 9781400848188
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691042770.003.0002
- Subject:
- Economics and Finance, History of Economic Thought

This chapter describes the vector first-order linear stochastic difference equation. It is first used to represent information flowing to economic agents, then again to represent competitive ... More

## Applications of the Pilot ACE and the DEUCE

*Tom Vickers*

### in Alan Turing's Automatic Computing Engine: The Master Codebreaker's Struggle to build the Modern Computer

- Published in print:
- 2005
- Published Online:
- January 2008
- ISBN:
- 9780198565932
- eISBN:
- 9780191714016
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198565932.003.0013
- Subject:
- Mathematics, History of Mathematics

This chapter discusses applications of the Pilot ACE. These applications include linear equations, photogrammetry, traffic simulation, matrix programmes, demonstration programmes, and ... More

## Programming the Pilot ACE

*J. G. Hayes*

### in Alan Turing's Automatic Computing Engine: The Master Codebreaker's Struggle to build the Modern Computer

- Published in print:
- 2005
- Published Online:
- January 2008
- ISBN:
- 9780198565932
- eISBN:
- 9780191714016
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198565932.003.0011
- Subject:
- Mathematics, History of Mathematics

This chapter presents an overview of the programming process for the Pilot ACE. This programme was for the back-substitution phase of solving a set of linear algebraic equations, with multiple ... More

## Why Growth Rates Differ, I

*Maurice FitzGerald Scott*

### in A New View of Economic Growth

- Published in print:
- 1991
- Published Online:
- November 2003
- ISBN:
- 9780198287421
- eISBN:
- 9780191596872
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198287429.003.0010
- Subject:
- Economics and Finance, Development, Growth, and Environmental

A linear equation, in which growth of output is determined by the share of investment and the growth of quality‐adjusted employment, is derived from the model, and fitted by ordinary least squares ... More

## Optimal Fiscal Policy in a Linear Stochastic Economy

*Thomas J. Sargent and François R. Velde*

### 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:
- 10.1093/0199248273.003.0009
- Subject:
- Economics and Finance, Macro- and Monetary Economics

The Lucas and Stokey (1983) economy without capital is used to exhibit features of the Lucas and Stokey model of optimal taxation, and show how they compare with Barro's (1979) tax‐smoothing model. ... More

## The Econometric Analysis of Economic Time Series

*David F. Hendry and Jean‐François Richard*

### in Econometrics: Alchemy or Science?: Essays in Econometric Methodology

- Published in print:
- 2000
- Published Online:
- November 2003
- ISBN:
- 9780198293545
- eISBN:
- 9780191596391
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198293542.003.0018
- Subject:
- Economics and Finance, Econometrics

This overview proposes a general framework linking economic analysis to a reduction‐based statistical methodology based on parameter change, expectations and contingent plans, conditioning and weak ... More

##
*n*-Point Correlation Functions: Descriptive Statistics

*P. J. E. Peebles*

### in The Large-Scale Structure of the Universe

- Published in print:
- 2020
- Published Online:
- May 2021
- ISBN:
- 9780691209838
- eISBN:
- 9780691206714
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691209838.003.0003
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

This chapter explores the statistical pattern of the galaxy distribution. It focuses on n-point correlation functions (analogs of the autocorrelation function and higher moments for a continuous ... More

## Linear Differential Polynomials

*Matthias Aschenbrenner, Lou van den Dries, and Joris van der Hoeven*

### in Asymptotic Differential Algebra and Model Theory of Transseries

- Published in print:
- 2017
- Published Online:
- October 2017
- ISBN:
- 9780691175423
- eISBN:
- 9781400885411
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691175423.003.0006
- Subject:
- Mathematics, Computational Mathematics / Optimization

This chapter introduces the reader to linear differential polynomials. It first considers homogeneous differential polynomials and the corresponding linear operators before proving various basic ... More

## Theoretical Underpinnings: Stellar Oscillations

*Sarbani Basu and William J. Chaplin*

### in Asteroseismic Data Analysis: Foundations and Techniques

- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691162928
- eISBN:
- 9781400888207
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691162928.003.0003
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

This chapter shows how the oscillations of a star are related to its structure. It also describes some of the important properties of stellar oscillations that allow us to use seismic data to ... More

## Parabolic Equations

*Rüdiger Verfürth*

### in A Posteriori Error Estimation Techniques for Finite Element Methods

- Published in print:
- 2013
- Published Online:
- May 2013
- ISBN:
- 9780199679423
- eISBN:
- 9780191758485
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199679423.003.0006
- Subject:
- Mathematics, Applied Mathematics, Numerical Analysis

We derive a posteriori error estimates for space–time finite element and certain finite volume discretisations of linear and nonlinear parabolic equations of second order. The starting point of our ... More

## Mathematical Tools

*Hans-Peter Eckle*

### in Models of Quantum Matter: A First Course on Integrability and the Bethe Ansatz

- Published in print:
- 2019
- Published Online:
- September 2019
- ISBN:
- 9780199678839
- eISBN:
- 9780191878589
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780199678839.003.0019
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics, Condensed Matter Physics / Materials

Chapter 19 introduces the mathematical techniques required to extract analytic infor- mation from the Bethe ansatz equations for a Heisenberg quantum spin chain of finite length. It discusses how the ... More

## Time-Differencing Schemes

*T. N. Krishnamurti, H. S. Bedi, and V. M. Hardiker*

### in An Introduction to Global Spectral Modeling

- Published in print:
- 1998
- Published Online:
- November 2020
- ISBN:
- 9780195094732
- eISBN:
- 9780197560761
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780195094732.003.0005
- Subject:
- Earth Sciences and Geography, Meteorology and Climatology

Atmospheric models generally require the solutions of partial differential equations. In spectral models, the governing partial differential equations reduce to a set of coupled ordinary nonlinear ... More

## Algebra

*Adrian Rice*

### in The Mathematical World of Charles L. Dodgson (Lewis Carroll)

- Published in print:
- 2019
- Published Online:
- April 2019
- ISBN:
- 9780198817000
- eISBN:
- 9780191858697
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198817000.003.0003
- Subject:
- Mathematics, History of Mathematics

This chapter is concerned with a single textbook written by Dodgson in 1867—An Elementary Treatise on Determinants. His book concerned the study of determinants, an algebraic construct that had come ... More

## The Survivor Problem: Simple Linear Regression with MCMC

*Therese M. Donovan and Ruth M. Mickey*

### in Bayesian Statistics for Beginners: a step-by-step approach

- Published in print:
- 2019
- Published Online:
- July 2019
- ISBN:
- 9780198841296
- eISBN:
- 9780191876820
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198841296.003.0017
- Subject:
- Biology, Biomathematics / Statistics and Data Analysis / Complexity Studies

While one of the most common uses of Bayes’ Theorem is in the statistical analysis of a dataset (i.e., statistical modeling), this chapter examines another application of Gibbs sampling: parameter ... More