## 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

## 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

## 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

## Free Particle, Wave Packet and Dynamics, Quantum Dots, Defects and Traps

*Duncan G. Steel*

### in Introduction to Quantum Nanotechnology: A Problem Focused Approach

- Published in print:
- 2021
- Published Online:
- July 2021
- ISBN:
- 9780192895073
- eISBN:
- 9780191924149
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780192895073.003.0003
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics, Condensed Matter Physics / Materials

This chapter continues with a study of the time independent Schrödinger equation and seeks to contrast the quantum behavior of a free particle with that of a particle localized in a potential quantum ... More

## Ideal and Technical Filters

*Melle Jan Kromhout*

### in The Logic of Filtering: How Noise Shapes the Sound of Recorded Music

- Published in print:
- 2021
- Published Online:
- May 2021
- ISBN:
- 9780190070137
- eISBN:
- 9780190070175
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780190070137.003.0004
- Subject:
- Music, Popular

Chapter 3 shows how the logic of noise reduction is anchored in historical discourses on sound and technology, taking a closer look at the development, from the early nineteenth century onward, of ... 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

## Nano-Mechanical Oscillator and Basic Dynamics: Part I

*Duncan G. Steel*

### in Introduction to Quantum Nanotechnology: A Problem Focused Approach

- Published in print:
- 2021
- Published Online:
- July 2021
- ISBN:
- 9780192895073
- eISBN:
- 9780191924149
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780192895073.003.0002
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics, Condensed Matter Physics / Materials

This discussion introduces the student to the reality, in quantum technology, that analysis of any problem necessarily begins with the Hamiltonian representing the system. The quantum Hamiltonian ... More

## Explicit Sinc-Like Methods

*Marek A. Kowalski, Krzysztof 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:
- 10.1093/9780195080599.003.0007
- 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 possible the explicit and ... 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