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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


SEPARATION PROPERTIES IN CONSTRUCTIVE TOPOLOGY

Peter Aczel and Christopher Fox

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/9780198566519.003.0011
Subject:
Mathematics, Logic / Computer Science / Mathematical Philosophy

This chapter studies separation properties in topology as done on the basis of the formal system CZF. Until the 1970s, there was only a limited focus on the general notion of a topological space in ... More


APPROXIMATING INTEGRABLE SETS BY COMPACTS CONSTRUCTIVELY

Bas Spitters

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/9780198566519.003.0017
Subject:
Mathematics, Logic / Computer Science / Mathematical Philosophy

The interplay between intuitionistic mathematics and Bishop-style constructive mathematics is crucial for this chapter's contribution. In locally compact spaces, (Borel-)measurable sets can be ... More


Weak Convergence in 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:
10.1093/0198774036.003.0026
Subject:
Economics and Finance, Econometrics

This chapter lays the foundations for functional limit theory by considering the case of general metric spaces, from a topological standpoint. It develops tools to replace the method of ... More


The Metric Topology

Adel N. Boules

in Fundamentals of Mathematical Analysis

Published in print:
2021
Published Online:
May 2021
ISBN:
9780198868781
eISBN:
9780191905254
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198868781.003.0004
Subject:
Mathematics, Analysis, Applied Mathematics

The chapter is an extensive account of the metric topology and is a prerequisite for all the subsequent chapters. The leading sections develop the basic metric properties such as closure and ... More


Introduction and Preliminaries

Fon-Che Liu

in Real Analysis

Published in print:
2016
Published Online:
January 2017
ISBN:
9780198790426
eISBN:
9780191831676
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198790426.003.0001
Subject:
Mathematics, Analysis

This chapter on introduction and preliminaries is to prepare the reader for a more systematic development of the methods of real analysis through a brief account of the summability of systems of ... More


and Spaces

Matt Clay and Dan Margalit

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/9780691158662.003.0002
Subject:
Mathematics, Geometry / Topology

This chapter discusses the notion of space, first by explaining what it means for a group to be a group of symmetries of a geometric object. This is the idea of group action, and some examples are ... More


Asymptotic Dimension

Greg Bell

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/9780691158662.003.0011
Subject:
Mathematics, Geometry / Topology

This chapter considers the notion of asymptotic dimension. It first provides an overview of asymptotic dimension before explaining the asymptotic dimensions of free abelian and nonabelian groups, ... More


Metric Spaces and Geodesic Motion

David D. Nolte

in Introduction to Modern Dynamics: Chaos, Networks, Space, and Time

Published in print:
2019
Published Online:
November 2019
ISBN:
9780198844624
eISBN:
9780191880216
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198844624.003.0011
Subject:
Physics, Theoretical, Computational, and Statistical Physics

The metric tensor uniquely defines the geometric properties of a metric space, while differential geometry is concerned with the derivatives of vectors and tensors within the metric space. ... More


Metric Spaces

James Davidson

in Stochastic Limit Theory: An Introduction for Econometricians

Published in print:
2021
Published Online:
November 2021
ISBN:
9780192844507
eISBN:
9780191927201
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780192844507.003.0005
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


Vector spaces and matrices: Basic theory

Leiba Rodman

in Topics in Quaternion Linear Algebra

Published in print:
2014
Published Online:
October 2017
ISBN:
9780691161853
eISBN:
9781400852741
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691161853.003.0003
Subject:
Mathematics, Algebra

This chapter covers the basics on the vector space of columns with quaternion components, matrix algebra, and various matrix decompositions. The real and complex representations of quaternions are ... More


Hierarchical Temporal Gestalt Perception in Music: (1978)

James Tenney

Larry Polansky, Lauren Pratt, Robert Wannamaker, and Michael Winter (eds)

in From Scratch: Writings in Music Theory

Published in print:
2015
Published Online:
April 2017
ISBN:
9780252038723
eISBN:
9780252096679
Item type:
chapter
Publisher:
University of Illinois Press
DOI:
10.5406/illinois/9780252038723.003.0009
Subject:
Music, Theory, Analysis, Composition

James Tenney talks about hierarchical temporal gestalt perception in music based on a metric space model, which he applied to compositions by Anton Webern, Edgard Varèse, and Claude Debussy. He ... More


Affine Fixed Point Buildings

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/9780691166902.003.0027
Subject:
Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter shows that if Ξ‎ is an affine building and Γ‎ is a finite descent group of Ξ‎, then Γ‎ is a descent group of Ξ‎∞ and (Ξ‎∞) is congruent to (Ξ‎∞). Ξ‎Γ‎ and Ξ‎ can be viewed as metric ... More


Measures on Metric Spaces

James Davidson

in Stochastic Limit Theory: An Introduction for Econometricians

Published in print:
2021
Published Online:
November 2021
ISBN:
9780192844507
eISBN:
9780191927201
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780192844507.003.0027
Subject:
Economics and Finance, Econometrics

This chapter lays the foundations for functional limit theory, considering the case of general metric spaces from a topological standpoint. The issues of separability and measurability and techniques ... More


Matching with Transfers: Basic Notions

Pierre-André Chiappori

in Matching with Transfers: The Economics of Love and Marriage

Published in print:
2017
Published Online:
May 2018
ISBN:
9780691171739
eISBN:
9781400885732
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691171739.003.0002
Subject:
Economics and Finance, Macro- and Monetary Economics

This chapter describes the basic notions of matching with transfers. It first introduces the notations that will be used throughout the book, including two compact, separable metric spaces: the space ... More


Fundamentals of Mathematical Analysis

Adel N. Boules

Published in print:
2021
Published Online:
May 2021
ISBN:
9780198868781
eISBN:
9780191905254
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198868781.001.0001
Subject:
Mathematics, Analysis, Applied Mathematics

Fundamentals of Mathematical Analysis is a beginning graduate textbook on real and functional analysis, with a substantial component on topology. The three leading chapters furnish background ... More


Timespan Intervals

Jason Yust

in Organized Time: Rhythm, Tonality, and Form

Published in print:
2018
Published Online:
May 2018
ISBN:
9780190696481
eISBN:
9780190696511
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780190696481.003.0006
Subject:
Music, Theory, Analysis, Composition

David Lewin’s concept of timespan interval is applied to theories of hypermeter and rhythmic structure. The special property of timespan intervals is that the distances between them are measured ... More


Back to the Twins

Alvaro De Rújula

in Enjoy Our Universe: You Have No Other Choice

Published in print:
2018
Published Online:
February 2020
ISBN:
9780198817802
eISBN:
9780191859366
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198817802.003.0016
Subject:
Physics, Particle Physics / Astrophysics / Cosmology

A very detailed discussion on the Twin Paradox and its resolution(s). How science may be difficult for the (pioneering) scientists. Muons as the optimal clocks. Is the relativistic time dilation due ... More


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