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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


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


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


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


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


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


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


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


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


KAM theory

S. G. Rajeev

in Advanced Mechanics: From Euler's Determinism to Arnold's Chaos

Published in print:
2013
Published Online:
December 2013
ISBN:
9780199670857
eISBN:
9780191775154
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199670857.003.0018
Subject:
Physics, Atomic, Laser, and Optical Physics

The Kolmogorov-Arnold-Moser (KAM) theorem presents limits to the notion of chaos. This is the deepest result to date on this subject. This chapter starts with a review of Newton's method for solving ... More


Multi-parameter Singular Integrals. (AM-189)

Brian Street

Published in print:
2014
Published Online:
October 2017
ISBN:
9780691162515
eISBN:
9781400852758
Item type:
book
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691162515.001.0001
Subject:
Mathematics, Analysis

This book develops a new theory of multi-parameter singular integrals associated with Carnot–Carathéodory balls. The book first details the classical theory of Calderón–Zygmund singular integrals and ... More


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