##
Further *SL*(2) Examples, Based on the Legendre Family

*Nicholas M. Katz*

### in Convolution and Equidistribution: Sato-Tate Theorems for Finite-Field Mellin Transforms (AM-180)

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153308
- eISBN:
- 9781400842704
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153308.003.0016
- Subject:
- Mathematics, Number Theory

This chapter supposes that k has odd characteristic. It begins with the Legendre family of elliptic curves.

## PRO-SUBJECTIVE APPROACH LOSES AS SAMPLING THEORY GAINS GROUND

*Shoutir Kishore Chatterjee*

### in Statistical Thought: A Perspective and History

- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198525318
- eISBN:
- 9780191711657
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198525318.003.0008
- Subject:
- Mathematics, Probability / Statistics

In the first half of the 19th century, Laplace, Gauss, and a number of other contributors enriched statistical thought. The quest for a suitable model for the distribution of observational errors led ... More

## Mathematical Concepts and Definitions

*Jamie Tappenden*

### 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/9780199296453.003.0010
- Subject:
- Philosophy, Logic/Philosophy of Mathematics

This chapter lays some preliminary groundwork for answering the question: What kind of success have we attained when we introduce a ‘good’ mathematical concept or definition? What sort of ... More

## Algebraic Preliminaries

*JEAN ZINN-JUSTIN*

### in Quantum Field Theory and Critical Phenomena

- Published in print:
- 2002
- Published Online:
- January 2010
- ISBN:
- 9780198509233
- eISBN:
- 9780191708732
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198509233.003.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter reviews algebraic identities about gaussian integrals, in particular Wick's theorem, a result also relevant for gaussian probability distributions. It discusses the steepest descent ... More

## Ferromagnetic systems. Correlation functions

*Jean Zinn-Justin*

### in Phase Transitions and Renormalization Group

- Published in print:
- 2007
- Published Online:
- January 2010
- ISBN:
- 9780199227198
- eISBN:
- 9780191711107
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199227198.003.0006
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter defines more general statistical systems in an arbitrary number of space dimensions. For convenience, it uses the ferromagnetic language — even though, as a consequence of universality, ... More

## An Introduction to Statistical Mechanics and Thermodynamics

*Robert H. Swendsen*

- Published in print:
- 2019
- Published Online:
- February 2020
- ISBN:
- 9780198853237
- eISBN:
- 9780191887703
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198853237.001.0001
- Subject:
- Physics, Condensed Matter Physics / Materials, Theoretical, Computational, and Statistical Physics

This is a textbook on statistical mechanics and thermodynamics. It begins with the molecular nature of matter and the fact that we want to describe systems containing many (1020) particles. The first ... More

## Special functions

*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/9780199641390.003.0007
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

Special functions are advanced functions frequently used in science and engineering. The properties of Legendre, Hermite, Laguerre, Chebyshev and spherical harmonics polynomials and Bessel functions ... More

## Hamilton’s Principle in Phase Space

*Peter Mann*

### in Lagrangian and Hamiltonian Dynamics

- Published in print:
- 2018
- Published Online:
- August 2018
- ISBN:
- 9780198822370
- eISBN:
- 9780191861253
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198822370.003.0015
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter derives Hamilton’s equations using the Legendre transform and the definition of the Hamiltonian function. While, in the Newtonian formalism, conservation laws were rather difficult to ... More

## The Spectral-Element Method

*Heiner Igel*

### in Computational Seismology: A Practical Introduction

- Published in print:
- 2016
- Published Online:
- January 2017
- ISBN:
- 9780198717409
- eISBN:
- 9780191835070
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198717409.003.0007
- Subject:
- Physics, Geophysics, Atmospheric and Environmental Physics

The spectral-element method is introduced as a finite-element method with high-order Lagrange polynomials as interpolating functions. The concept of numerical integration is introduced and the ... More

## Angular momentum

*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/9780198749349.003.0015
- Subject:
- Physics, Atomic, Laser, and Optical Physics

This chapter discusses the algebra, the eigenvalues, and eigenstates of angular momentum operators.

## Lower-Order Spectral Model

*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/9780195094732.003.0007
- Subject:
- Earth Sciences and Geography, Meteorology and Climatology

This system was first developed by Lorenz (1960b). It is an elegant system that provides an introduction to the concepts of spectral modeling, based on the use of double Fourier series ... More

## Mathematical Aspects of Spectral Models

*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/9780195094732.003.0008
- Subject:
- Earth Sciences and Geography, Meteorology and Climatology

In this chapter we provide an introduction to the topic of spherical harmonics as basis functions for a global spectral model. The spherical harmonics are made up of trigonometric functions along ... More

## Thermodynamic Potentials

*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/9780199646944.003.0012
- Subject:
- Physics, Condensed Matter Physics / Materials

This chapter discusses the large variety of thermodynamic potentials that can be used to express a fundamental relation. It shows that the different thermodynamic potentials are related by Legendre ... More

## A Sturm-Liouville Equation: The Time-Independent One-Dimensional Schrödinger Equation

*John A. Adam*

### in Rays, Waves, and Scattering: Topics in Classical Mathematical Physics

- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691148373
- eISBN:
- 9781400885404
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691148373.003.0023
- Subject:
- Mathematics, Applied Mathematics

This chapter examines the mathematical properties of the time-independent one-dimensional Schrödinger equation as they relate to Sturm-Liouville problems. The regular Sturm-Liouville theory was ... More

## Resistance: Waiting for Power in Fuenteovejuna

*Philip Lorenz*

### in The Tears of Sovereignty: Perspectives of Power in Renaissance Drama

- Published in print:
- 2013
- Published Online:
- January 2014
- ISBN:
- 9780823251308
- eISBN:
- 9780823252633
- Item type:
- chapter

- Publisher:
- Fordham University Press
- DOI:
- 10.5422/fordham/9780823251308.003.0004
- Subject:
- Literature, Criticism/Theory

Lope de Vega's famous play about a popular uprising and revolt throws the “organ transplant” logic of sovereignty into crisis. Here, the conceptual and representational operations of transfer and ... More

## The Event of the Letter: Two Approaches to the Law and its Real

*Tracy McNulty*

### in Wrestling with the Angel: Experiments in Symbolic Life

- Published in print:
- 2014
- Published Online:
- November 2015
- ISBN:
- 9780231161190
- eISBN:
- 9780231537605
- Item type:
- chapter

- Publisher:
- Columbia University Press
- DOI:
- 10.7312/columbia/9780231161190.003.0005
- Subject:
- Political Science, Political Theory

This chapter examines Paul's polemic and its centrality to Alain Badiou's theory of the event, arguing that it overlooks the way in which the Mosaic law can itself be understood as an “event” by ... More

## Simple Systems

*Jochen Rau*

### in Statistical Physics and Thermodynamics: An Introduction to Key Concepts

- Published in print:
- 2017
- Published Online:
- November 2017
- ISBN:
- 9780199595068
- eISBN:
- 9780191844300
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780199595068.003.0004
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

Even though the general framework of statistical mechanics is ultimately targeted at the description of macroscopic systems, it is illustrative to apply it first to some simple systems: a harmonic ... More

## Temperature Fluctuations and Mixtures of Equilibrium States in the Canonical Ensemble

*Hugo Touchette*

### in Nonextensive Entropy: Interdisciplinary Applications

- Published in print:
- 2004
- Published Online:
- November 2020
- ISBN:
- 9780195159769
- eISBN:
- 9780197562024
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780195159769.003.0014
- Subject:
- Earth Sciences and Geography, Atmospheric Sciences

It has been suggested recently that "Q-exponential" distributions, which form the basis of Tsallis' nonextensive thermostatistical formalism, may be viewed as mixtures of exponential (Gibbs) ... More

## Nonextensive Entropies and Sensitivity to Initial Conditions of Complex Systems

*Marcelo L. Lyra*

### in Nonextensive Entropy: Interdisciplinary Applications

- Published in print:
- 2004
- Published Online:
- November 2020
- ISBN:
- 9780195159769
- eISBN:
- 9780197562024
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780195159769.003.0009
- Subject:
- Earth Sciences and Geography, Atmospheric Sciences

Tsallis generalized statistics has been successfully applied to describe some relevant features of several natural systems exhibiting a nonextensive character. It is based on an extended form for ... More

## Angular Momentum and Rotational Motion in Three Dimensions

*Jochen Autschbach*

### in Quantum Theory for Chemical Applications: From Basic Concepts to Advanced Topics

- Published in print:
- 2020
- Published Online:
- February 2021
- ISBN:
- 9780190920807
- eISBN:
- 9780197508350
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780190920807.003.0016
- Subject:
- Chemistry, Quantum and Theoretical Chemistry

This chapter treats rotations in three dimensions and the angular momentum quantum mechanically. The angular momentum is a 3-vector that plays the same role in the kinetic energy of rotating ... More