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## Statistical Shape Theory

*Wilfrid S. Kendall and Huiling Le*

### in New Perspectives in Stochastic Geometry

- Published in print:
- 2009
- Published Online:
- February 2010
- ISBN:
- 9780199232574
- eISBN:
- 9780191716393
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199232574.003.0010
- Subject:
- Mathematics, Geometry / Topology

There are many variations on what one may regard as statistical shape, depending on the application in mind. The focus of this chapter is the statistical analysis of the shapes determined by finite ... More

## Complex-valued harmonic morphisms on three-dimensional Euclidean space

*Paul Baird and John C. Wood*

### in Harmonic Morphisms Between Riemannian Manifolds

- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198503620
- eISBN:
- 9780191708435
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198503620.003.0001
- Subject:
- Mathematics, Pure Mathematics

This chapter presents an introduction to the theory of harmonic morphisms for the case of maps from open subsets of Euclidean 3-space to the complex plane. Harmonic morphisms are characterized by ... More

## THE CONSTRUCTIVE UNIQUENESS OF THE LOCALLY CONVEX TOPOLOGY ON ℝ N

*Douglas Bridges and Luminita Vîtă*

### in From Sets and Types to Topology and Analysis: Towards practicable foundations for constructive mathematics

- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198566519
- eISBN:
- 9780191713927
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198566519.003.0020
- Subject:
- Mathematics, Logic / Computer Science / Mathematical Philosophy

This chapter proves constructively that every n-dimensional real locally convex space is homeomorphic to ℝn. Although essentially nonconstructive theorems of real analysis are used in the customary ... More

## Metric Spaces

*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:
- 10.1093/0198774036.003.0005
- Subject:
- Economics and Finance, Econometrics

This chapter introduces and illustrates the concept of a metric (distance measure), and the definition of a metric space. Open, closed, and compact sets are discussed in a general context, and the ... More

## Preliminaries to Main Results

*Joram Lindenstrauss, David Preiss, and Tiˇser Jaroslav*

### in Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179)

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153551
- eISBN:
- 9781400842698
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153551.003.0009
- Subject:
- Mathematics, Analysis

This chapter presents a number of results and notions that will be used in subsequent chapters. In particular, it considers the concept of regular differentiability and the lemma on deformation of ... More

## Coxeter Groups

*Adam Piggott*

### in Office Hours with a Geometric Group Theorist

- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691158662
- eISBN:
- 9781400885398
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691158662.003.0013
- Subject:
- Mathematics, Geometry / Topology

This chapter considers Coxeter groups and how to find a space on which a group acts by building a space using combinatorics from the group. It first describes groups generated by reflections, ... More

## The Standard Metric

*Bernhard M¨uhlherr, Holger P. Petersson, and Richard M. Weiss*

### in Descent in Buildings (AM-190)

- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691166902
- eISBN:
- 9781400874019
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691166902.003.0026
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter presents some results about groups generated by reflections and the standard metric on a Bruhat-Tits building. It begins with definitions relating to an affine subspace, an affine ... More

## The Euclidean Functional Integrals

*Laurent Baulieu, John Iliopoulos, and Roland Sénéor*

### in From Classical to Quantum Fields

- Published in print:
- 2017
- Published Online:
- May 2017
- ISBN:
- 9780198788393
- eISBN:
- 9780191830310
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198788393.003.0010
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology, Theoretical, Computational, and Statistical Physics

The Wick rotation and the functional integral in Euclidean space. Some mathematical theorems. Perturbation theory and Feynman rules in Euclidean space.

## The action functional and the harmonic oscillator

*Jean-Michel Bismut*

### in Hypoelliptic Laplacian and Orbital Integrals (AM-177)

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691151298
- eISBN:
- 9781400840571
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691151298.003.0011
- Subject:
- Mathematics, Geometry / Topology

This chapter solves explicitly certain natural variational problems associated with a scalar hypoelliptic Laplacian, in the case where the underlying Riemannian manifold is an Euclidean vector space. ... More

## Minkowski spacetime

*Nathalie Deruelle and Jean-Philippe Uzan*

### in Relativity in Modern Physics

- Published in print:
- 2018
- Published Online:
- October 2018
- ISBN:
- 9780198786399
- eISBN:
- 9780191828669
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198786399.003.0019
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

This chapter presents the main features of the Minkowski spacetime, which is the geometrical framework in which the laws of relativistic dynamics are formulated. It is a very simple mathematical ... More

## A review: The Laplacian and the d’Alembertian

*Christopher D. Sogge*

### in Hangzhou Lectures on Eigenfunctions of the Laplacian (AM-188)

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691160757
- eISBN:
- 9781400850549
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691160757.003.0001
- Subject:
- Mathematics, Numerical Analysis

This chapter reviews the Laplacian and the d'Alembertian. It begins with a brief discussion on the solution of wave equation both in Euclidean space and on manifolds and how this knowledge can be ... More

## Wright-Fisher Geometry

*Charles L. Epstein and Rafe1 Mazzeo*

### in Degenerate Diffusion Operators Arising in Population Biology (AM-185)

- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691157122
- eISBN:
- 9781400846108
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691157122.003.0002
- Subject:
- Mathematics, Probability / Statistics

This chapter introduces the geometric preliminaries needed to analyze generalized Kimura diffusions, with particular emphasis on Wright–Fisher geometry. It begins with a discussion of the natural ... More

## Second Order and Gaussian Fields

*Ulf Grenander and Michael I. Miller*

### in Pattern Theory: From representation to inference

- Published in print:
- 2006
- Published Online:
- November 2020
- ISBN:
- 9780198505709
- eISBN:
- 9780191916564
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198505709.003.0010
- Subject:
- Computer Science, Programming Languages

This chapter studies second order and Gaussian fields on the background spaces which are the continuum limits of the finite graphs. For this random processes in ... More

## Intensionality

*E. A. Ashcroft, A. A. Faustini, R. Jaggannathan, and W. W. Wadge*

### in Multidimensional Programming

- Published in print:
- 1995
- Published Online:
- November 2020
- ISBN:
- 9780195075977
- eISBN:
- 9780197560327
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780195075977.003.0005
- Subject:
- Computer Science, Software Engineering

The intensional programming language, Lucid, described in Chapter 1 is based directly on intensional logic, a family of mathematical formal systems that permit ... More

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