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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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