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Mathematical Preliminaries

Partha P. Mitra and Hemant Bokil

in Observed Brain Dynamics

Published in print:
2007
Published Online:
May 2009
ISBN:
9780195178081
eISBN:
9780199864829
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780195178081.003.0005
Subject:
Neuroscience, Techniques, Molecular and Cellular Systems

This chapter reviews a broad range of mathematical topics relevant to the rest of the book. It begins with a brief discussion of real and complex numbers and elementary real and complex functions, ... More


Linear Systems Analysis

Russell L. De Valois and Karen K. De Valois

in Spatial Vision

Published in print:
1991
Published Online:
January 2008
ISBN:
9780195066579
eISBN:
9780199872220
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780195066579.003.0001
Subject:
Psychology, Cognitive Neuroscience

This chapter provides a background on linear systems analysis to allow the reader to follow the applications to visual problems discussed in this book. Specifically, it discusses the Fourier theorem ... More


FOURIER ANALYSIS ON FINITE ABELIAN GROUPS: SOME GRAPHICAL APPLICATIONS

Andrew Goodall

in Combinatorics, Complexity, and Chance: A Tribute to Dominic Welsh

Published in print:
2007
Published Online:
September 2007
ISBN:
9780198571278
eISBN:
9780191718885
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198571278.003.0007
Subject:
Mathematics, Probability / Statistics

This article reviews basic techniques of Fourier analysis on a finite abelian group Q, with subsequent applications in graph theory. These include evaluations of the Tutte polynomial of a graph G in ... More


Visual Pattern Analyzers

Norma Van Surdam Graham

Published in print:
1989
Published Online:
January 2008
ISBN:
9780195051544
eISBN:
9780199872183
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780195051544.001.0001
Subject:
Psychology, Cognitive Neuroscience

From the light that falls on the retina, the visual system must extract meaningful information about what is where in our environment. At an early stage in this process, it analyzes the incoming ... More


LINEAR CAUCHY PROBLEM WITH CONSTANT COEFFICIENTS

Sylvie Benzoni-Gavage and Denis Serre

in Multi-dimensional hyperbolic partial differential equations: First-order systems and applications

Published in print:
2006
Published Online:
September 2007
ISBN:
9780199211234
eISBN:
9780191705700
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199211234.003.0001
Subject:
Mathematics, Applied Mathematics

For first-order systems, the well-posedness of the Cauchy problem is equivalent to an algebraic condition on the symbol of the operator: the hyperbolicity. The hyperbolic Cauchy problem has the ... More


Historical Prelude

Roger D. Roger and Miles A. Whittington

in Cortical Oscillations in Health and Disease

Published in print:
2010
Published Online:
May 2010
ISBN:
9780195342796
eISBN:
9780199776276
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780195342796.003.0002
Subject:
Neuroscience, Molecular and Cellular Systems, Development

One of the pioneers in the mathematical and conceptual study of oscillations was Norbert Wiener, whose book Cybernetics first appeared in 1948. The authors review the type of data, experimental ... More


Fourier Analysis and Benfordʼs Law

Steven J. Miller

in Benford's Law: Theory and Applications

Published in print:
2015
Published Online:
October 2017
ISBN:
9780691147611
eISBN:
9781400866595
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691147611.003.0003
Subject:
Mathematics, Probability / Statistics

This chapter continues the development of the theory of Benford's law. It uses Fourier analysis (in particular, Poisson Summation) to prove many systems either satisfy or almost satisfy the ... More


Selected Theorems by Eli Stein

Charles Fefferman

Charles Fefferman, Alexandru D. Ionescu, D. H. Phong, Stephen Wainger, Charles Fefferman, Alexandru D. Ionescu, D. H. Phong, and Stephen Wainger (eds)

in Advances in Analysis: The Legacy of Elias M. Stein

Published in print:
2014
Published Online:
October 2017
ISBN:
9780691159416
eISBN:
9781400848935
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691159416.003.0001
Subject:
Mathematics, Numerical Analysis

This chapter is an overview of the scope and originality of Eli Stein's contributions to analysis. His work deals with representation theory, classical Fourier analysis, and partial differential ... More


Fourier analysis in diffraction and image formation

Christopher Hammond

in The Basics of Crystallography and Diffraction

Published in print:
2015
Published Online:
August 2015
ISBN:
9780198738671
eISBN:
9780191801938
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198738671.003.0013
Subject:
Physics, Crystallography, Condensed Matter Physics / Materials

This chapter starts with the Fourier series and Fourier transforms, and the representation of periodic functions. It then looks at Fourier analysis in crystallography. It details electron density, ... More


Advances in Analysis: The Legacy of Elias M. Stein

Charles Fefferman, Alexandru D. Ionescu, D.H. Phong, and Stephen Wainger

Published in print:
2014
Published Online:
October 2017
ISBN:
9780691159416
eISBN:
9781400848935
Item type:
book
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691159416.001.0001
Subject:
Mathematics, Numerical Analysis

Princeton University's Elias Stein was the first mathematician to see the profound interconnections that tie classical Fourier analysis to several complex variables and representation theory. His ... More


The Basics of Crystallography and Diffraction

Christopher Hammond

Published in print:
2015
Published Online:
August 2015
ISBN:
9780198738671
eISBN:
9780191801938
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198738671.001.0001
Subject:
Physics, Crystallography, Condensed Matter Physics / Materials

This book provides an introduction to crystallography, light, X-ray, and electron diffraction. The book also shows, by historical and biographical references, how the subject has developed from the ... More


Pattern Perception

Peter H. Schiller and Edward J. Tehovnik

in Vision and the Visual System

Published in print:
2015
Published Online:
September 2015
ISBN:
9780199936533
eISBN:
9780190258054
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199936533.003.0013
Subject:
Psychology, Vision

This chapter has seven subdivisions. Section A delineates the major theories that have been formulated about how living organisms process pattern and shape information. Section B describes ... More


Waves

William J. Mullin

in Quantum Weirdness

Published in print:
2017
Published Online:
March 2017
ISBN:
9780198795131
eISBN:
9780191836480
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198795131.003.0001
Subject:
Physics, Particle Physics / Astrophysics / Cosmology

Quantum mechanics is introduced here as a fundamental theory that must treat particles equally often as waves. Here, classical waves are introduced in some detail, with discussions of the important ... More


Heat flow and thermal relaxation

Andrew M. Steane

in Thermodynamics: A complete undergraduate course

Published in print:
2016
Published Online:
January 2017
ISBN:
9780198788560
eISBN:
9780191830426
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198788560.003.0010
Subject:
Physics, Theoretical, Computational, and Statistical Physics

The solution of heat flow problems involving the thermal diffusion equation is explained. Conduction through a flat wall and into a sphere is described. A worked example using Fourier analysis is ... More


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