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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


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