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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


Some Mathematics

Norma van Surdam Graham

in Visual Pattern Analyzers

Published in print:
1989
Published Online:
January 2008
ISBN:
9780195051544
eISBN:
9780199872183
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780195051544.003.0002
Subject:
Psychology, Cognitive Neuroscience

This chapter provides the mathematical background for using Fourier Analysis (Linear Systems Analysis) to study visual pattern analyzers. It introduces sinusoidal, Gabor, and delta functions as ... More


Random Processes in Physics and Finance

Melvin Lax, Wei Cai, and Min Xu

Published in print:
2006
Published Online:
January 2010
ISBN:
9780198567769
eISBN:
9780191718359
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198567769.001.0001
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This book discusses random processes in physics, from basic probability theory to the Fokker-Planck process and Langevin equation for the Fokker Planck process and diffusion. It includes diverse ... More


Formal means

Helmut Hofmann

in The Physics of Warm Nuclei: with Analogies to Mesoscopic Systems

Published in print:
2008
Published Online:
September 2008
ISBN:
9780198504016
eISBN:
9780191708480
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198504016.003.0026
Subject:
Physics, Nuclear and Plasma Physics

This chapter elucidates various mathematical formulas. Based on expressions for Gaussian integrals in one and many dimensions, the methods of stationary phase and steepest descent are deduced, ... More


Fourier transforms and convolutions

Dennis Sherwood and Jon Cooper

in Crystals, X-rays and Proteins: Comprehensive Protein Crystallography

Published in print:
2010
Published Online:
January 2011
ISBN:
9780199559046
eISBN:
9780191595028
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199559046.003.0005
Subject:
Physics, Crystallography: Physics

This chapter revisits the essential mathematics of integral calculus and introduces the important concept of the Fourier transform, the properties of which are elegantly demonstrated by sketching the ... More


Review of probability

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/9780198567769.003.0001
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter reviews probability theory, limiting the discussion to the study of random events as opposed to random processes, the latter being a sequence of random events extended over a period of ... More


Mathematics and Physics: Strategies of Assimilation

Alasdair Urquhart

in The Philosophy of Mathematical Practice

Published in print:
2008
Published Online:
February 2010
ISBN:
9780199296453
eISBN:
9780191711961
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199296453.003.0017
Subject:
Philosophy, Logic/Philosophy of Mathematics

The main topic of this chapter is the problem posed to the mathematical community by the non-rigorous methods used by physicists in investigating mathematical problems. These methods are often highly ... More


Applying Problematic Mathematics, Interpreting Successful Structures: From the Delta Function to the Positron

Otávio Bueno and Steven French

in Applying Mathematics: Immersion, Inference, Interpretation

Published in print:
2018
Published Online:
June 2018
ISBN:
9780198815044
eISBN:
9780191852862
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198815044.003.0007
Subject:
Philosophy, Logic/Philosophy of Mathematics

In this chapter we consider whether, in applying mathematics in the ways described in previous chapters, there are grounds for taking the relevant mathematical entities to be indispensable. Our ... More


Continuous Random Numbers

Robert H. Swendsen

in An Introduction to Statistical Mechanics and Thermodynamics

Published in print:
2012
Published Online:
December 2013
ISBN:
9780199646944
eISBN:
9780191775123
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199646944.003.0005
Subject:
Physics, Condensed Matter Physics / Materials

This chapter extends the development of probability theory to include continuous random variables. Dirac delta functions are used to simplify the transformation of continuous random variables, as ... More


Imaging

B. D. Guenther

in Modern Optics Simplified

Published in print:
2019
Published Online:
January 2020
ISBN:
9780198842859
eISBN:
9780191878770
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198842859.003.0011
Subject:
Physics, Atomic, Laser, and Optical Physics

Treating an imaging system as a linear system and use llinear system properties to d iscuss both coherent and incoherent imaging. Use a one dimensional pin hole camera to study the theory of ... More


Bound states in one dimension

Efstratios Manousakis

in Practical Quantum Mechanics: Modern Tools and Applications

Published in print:
2015
Published Online:
December 2015
ISBN:
9780198749349
eISBN:
9780191813474
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198749349.003.0008
Subject:
Physics, Atomic, Laser, and Optical Physics

In this and in the following chapter, it reviews an important part of undergraduate material, namely solving for bound states and for scattering in one-dimension for the case where the potential is ... More


Retarded Green’s Functions

Norman J. Morgenstern Horing

in Quantum Statistical Field Theory: An Introduction to Schwinger's Variational Method with Green's Function Nanoapplications, Graphene and Superconductivity

Published in print:
2017
Published Online:
January 2018
ISBN:
9780198791942
eISBN:
9780191834165
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198791942.003.0005
Subject:
Physics, Theoretical, Computational, and Statistical Physics

Chapter 5 introduces single-particle retarded Green’s functions, which provide the probability amplitude that a particle created at (x, t) is later annihilated at (x′,t′). Partial Green’s functions, ... More


Molecular Scattering and Fourier Transforms

Peter B. Moore

in Visualizing the Invisible: Imaging Techniques for the Structural Biologist

Published in print:
2012
Published Online:
May 2015
ISBN:
9780199767090
eISBN:
9780190267841
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:osobl/9780199767090.003.0002
Subject:
Biology, Biochemistry / Molecular Biology

This chapter explores the mathematical relationships existing between known molecular structures from Chapter 1 and the structure factors, particularly dealing with the Fourier transforms and Fourier ... More


Fourier series and Fourier transforms

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/9780199641390.003.0004
Subject:
Physics, Theoretical, Computational, and Statistical Physics

The Fourier expansion of an arbitrary piecewise-smooth function in the finite interval —π 〈 x 〈 π as an infinite sum of terms (a 0; a n ... More


Soil Properties and Water Movement

J. -Y. Parlange and T. S. Steenhuis

in Vadose Zone Hydrology: Cutting Across Disciplines

Published in print:
1999
Published Online:
November 2020
ISBN:
9780195109900
eISBN:
9780197561058
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780195109900.003.0008
Subject:
Earth Sciences and Geography, Oceanography and Hydrology

For all spatial scales, from pore through local and field, to a watershed, interaction of the land surface with the atmosphere will be one of the crucial topics in hydrology and environmental ... 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


Explicit Sinc-Like Methods

Marek A. Kowalski, Krzystof A. Sikorski, and Frank Stenger

in Selected Topics in Approximation and Computation

Published in print:
1995
Published Online:
November 2020
ISBN:
9780195080599
eISBN:
9780197560402
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780195080599.003.0007
Subject:
Computer Science, Mathematical Theory of Computation

In this section we derive several methods of approximation using the function values {f(kh)}∞k=- ∞ . We present a family of simple rational functions, which make ... More


Some essential mathematics

J. Pierrus

in Solved Problems in Classical Electromagnetism: Analytical and Numerical Solutions with Comments

Published in print:
2018
Published Online:
October 2018
ISBN:
9780198821915
eISBN:
9780191861055
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198821915.003.0001
Subject:
Physics, Condensed Matter Physics / Materials

This chapter introduces most of the important mathematics that will be used repeatedly throughout this book. Whilst scalars like the electric potential Ф and the vector fields E and B are familiar ... More


Measurement of Scattering Coefficients

Dale Chimenti, Stanislav Rokhlin, and Peter Nagy

in Physical Ultrasonics of Composites

Published in print:
2011
Published Online:
November 2020
ISBN:
9780195079609
eISBN:
9780197560389
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780195079609.003.0012
Subject:
Chemistry, Industrial Chemistry

In the previous chapters, we saw how waves in composites behaved under various circumstances, depending on material anisotropy and wave propagation direction. The most ... More


Semantic Mimicry

Mark Wilson

in Physics Avoidance: and other essays in conceptual strategy

Published in print:
2017
Published Online:
December 2017
ISBN:
9780198803478
eISBN:
9780191841651
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198803478.003.0008
Subject:
Philosophy, Philosophy of Science, Philosophy of Language

Reasoning patterns can look superficially alike, yet differ significantly in their supportive underpinnings. Paradoxically, the progress of science is sometimes aided by these inferential pretenders, ... More


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