## Representing Response Times as Random Variables

*R. Duncan Luce*

### in Response Times: Their Role in Inferring Elementary Mental Organization

- Published in print:
- 1991
- Published Online:
- January 2008
- ISBN:
- 9780195070019
- eISBN:
- 9780199869879
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195070019.003.0001
- Subject:
- Psychology, Cognitive Models and Architectures

This chapter begins with a discussion of the study of response times. Response times are treated as observations of a random variable. The mathematics of stochastic processes is then used to ... More

## RANDOM PLANAR GRAPHS AND THE NUMBER OF PLANAR GRAPHS

*Marc Noy*

### 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/9780198571278.003.0014
- Subject:
- Mathematics, Probability / Statistics

This chapter surveys recent progress on the enumeration of labelled planar graphs and on the distribution of several parameters in random planar graphs. It shows how to use generating functions to ... More

## Random graphs with general degree distributions: This chapter describes more sophisticated random graph models that mimic networks with arbitrary degree distributions

*M. E. J. Newman*

### in Networks: An Introduction

- Published in print:
- 2010
- Published Online:
- September 2010
- ISBN:
- 9780199206650
- eISBN:
- 9780191594175
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199206650.003.0013
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

The previous chapter looked at the classic random graph model, in which pairs of vertices are connected at random with uniform probabilities. Although this model has proved tremendously useful as a ... More

## Generating Functions

*Oliver Johns*

### in Analytical Mechanics for Relativity and Quantum Mechanics

- Published in print:
- 2005
- Published Online:
- January 2010
- ISBN:
- 9780198567264
- eISBN:
- 9780191717987
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198567264.003.0018
- Subject:
- Physics, Atomic, Laser, and Optical Physics

In the previous chapter, several conditions were given that could be used for testing transformations to see whether they were canonical or not. But no methods, other than trial and error, were given ... More

## Sylvester denumerant

*J. L. Ramírez Alfonsín*

### in The Diophantine Frobenius Problem

- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198568209
- eISBN:
- 9780191718229
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198568209.003.0004
- Subject:
- Mathematics, Algebra, Combinatorics / Graph Theory / Discrete Mathematics

In 1857, while investigating the partition number function, J. J. Sylvester defined the function d(m; a1, . . . , an), called the denumerant, as the number of nonnegative integer representations of m ... More

## NEW CONCEPTS AND METHODS—PRE-BAYESIAN ERA

*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/9780198525318.003.0006
- Subject:
- Mathematics, Probability / Statistics

After James Bernoulli, the main contributors to probability theory and its applications in the 18th century worked either to solve more intricate problems of games of chance (De Moivre), or to build ... More

## Single Spherical and Cylindrical Boundaries

*M. Bordag, G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko*

### in Advances in the Casimir Effect

- Published in print:
- 2009
- Published Online:
- September 2009
- ISBN:
- 9780199238743
- eISBN:
- 9780191716461
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199238743.003.0009
- Subject:
- Physics, Condensed Matter Physics / Materials, Atomic, Laser, and Optical Physics

This chapter presents important results obtained for the Casimir effect in the presence of spherical and cylindrical shells with various boundary conditions. It also includes the Casimir effect for a ... More

## Blast Theory

*M. Vidyasagar*

### in Hidden Markov Processes: Theory and Applications to Biology

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691133157
- eISBN:
- 9781400850518
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691133157.003.0009
- Subject:
- Mathematics, Probability / Statistics

This chapter deals with BLAST theory. BLAST (Basic Local Alignment Search Tool) is a widely used statistical method for finding similarities between sequences of symbols from finite alphabets. While ... More

## Quantum Statistics of Fields

*STEPHEN M. BARNETT and PAUL M. RADMORE*

### in Methods in Theoretical Quantum Optics

- Published in print:
- 2002
- Published Online:
- February 2010
- ISBN:
- 9780198563617
- eISBN:
- 9780191714245
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198563617.003.0004
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter considers methods for describing the quantum statistics of a single mode of the electromagnetic field. Some of these methods may be extended to multimode fields, and this is discussed in ... More

##
Crystals and *p*-adic Integration

*Ben Brubaker, Daniel Bump, and Solomon Friedberg*

### in Weyl Group Multiple Dirichlet Series: Type A Combinatorial Theory (AM-175)

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691150659
- eISBN:
- 9781400838998
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691150659.003.0020
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter describes the properties of Kashiwara's crystal and its role in unipotent p-adic integrations related to Whittaker functions. In many cases, integrations of representation theoretic ... More

## GAUSSIAN INTEGRALS

*Jean Zinn-Justin*

### in Path Integrals in Quantum Mechanics

- Published in print:
- 2004
- Published Online:
- January 2010
- ISBN:
- 9780198566748
- eISBN:
- 9780191717994
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198566748.003.0001
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology, Theoretical, Computational, and Statistical Physics

This chapter introduces, in the case of ordinary integrals, concepts and methods that can be generalized to path integrals. The first part is devoted to the calculation of ordinary Gaussian ... More

## Gaussian expectation values. Steepest descent method

*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/9780199227198.003.0002
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

This chapter presents a number of technical results concerning generating functions, Gaussian measures, and the steepest descent method. It begins by introducing the notion of generating function of ... 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

## Characteristic Functions

*James Davidson*

### in Stochastic Limit Theory: An Introduction for Econometricians

- Published in print:
- 1994
- Published Online:
- November 2003
- ISBN:
- 9780198774037
- eISBN:
- 9780191596117
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/0198774036.003.0011
- Subject:
- Economics and Finance, Econometrics

This chapter begins with a look at convolutions and the distribution of sums of random variables. It briefly surveys complex number theory before defining the characteristic function and studying its ... More

## Generating Functions

*Oliver Davis Johns*

### in Analytical Mechanics for Relativity and Quantum Mechanics

- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780191001628
- eISBN:
- 9780191775161
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780191001628.003.0020
- Subject:
- Physics, Atomic, Laser, and Optical Physics

This chapter proposes methods for creating transformations that will automatically be canonical. Canonical transformations can be created by first choosing what are called generating functions. Using ... More

## The configuration model

*Mark Newman*

### in Networks

- Published in print:
- 2018
- Published Online:
- October 2018
- ISBN:
- 9780198805090
- eISBN:
- 9780191843235
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198805090.003.0012
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics

A discussion of the most fundamental of network models, the configuration model, which is a random graph model of a network with a specified degree sequence. Following a definition of the model a ... More

## Generating functions and applications

*J. Klafter and I.M. Sokolov*

### in First Steps in Random Walks: From Tools to Applications

- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780199234868
- eISBN:
- 9780191775024
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199234868.003.0002
- Subject:
- Physics, Soft Matter / Biological Physics

This chapter concentrates on the properties of the walks which do not immediately follow from the distribution of the particle's position after n steps, like the first passage and return ... More

## Generating functions

*Dusa McDuff and Dietmar Salamon*

### in Introduction to Symplectic Topology

- Published in print:
- 2017
- Published Online:
- June 2017
- ISBN:
- 9780198794899
- eISBN:
- 9780191836411
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198794899.003.0010
- Subject:
- Mathematics, Analysis, Geometry / Topology

This chapter discusses generating functions in more detail. It shows how generating functions give rise to discrete-time analogues of the symplectic action functional and hence lead to discrete ... 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

## Function: How Traits Interact with the Environment to Influence Reproduction and Survival

*Douglas W. Morris and Per Lundberg*

### in Pillars of Evolution: Fundamental principles of the eco-evolutionary process

- Published in print:
- 2011
- Published Online:
- December 2013
- ISBN:
- 9780198568797
- eISBN:
- 9780191774690
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198568797.003.0003
- Subject:
- Biology, Evolutionary Biology / Genetics

Interactions between the environment, traits, and their constituent genes demonstrate that it is impossible to predict evolutionary change in anything but the simplest environments. Even then, we can ... More