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Discrete time

S. G. Rajeev

in Advanced Mechanics: From Euler's Determinism to Arnold's Chaos

Published in print:
2013
Published Online:
December 2013
ISBN:
9780199670857
eISBN:
9780191775154
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199670857.003.0015
Subject:
Physics, Atomic, Laser, and Optical Physics

Whether time is discrete at the fundamental level is still unknown. But for computational purposes, it is very useful to have a form of mechanics in which time increases in finite steps. Each step ... More


Symmetric Markovian Semigroups and Dirichlet Forms

Zhen-Qing Chen and Masatoshi Fukushima

in Symmetric Markov Processes, Time Change, and Boundary Theory (LMS-35)

Published in print:
2011
Published Online:
October 2017
ISBN:
9780691136059
eISBN:
9781400840564
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691136059.003.0001
Subject:
Mathematics, Probability / Statistics

This chapter studies the concepts of Dirichlet form and Dirichlet space by first working with a σ‎-finite measure space (E,B(E),m) without any topological assumption on E and establish the ... More


Geometry on my Mind

David D. Nolte

in Galileo Unbound: A Path Across Life, the Universe and Everything

Published in print:
2018
Published Online:
August 2018
ISBN:
9780198805847
eISBN:
9780191843808
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198805847.003.0005
Subject:
Physics, History of Physics

This chapter reviews the history of modern geometry with a focus on the topics that provided the foundation for the new visualization of physics. It begins with Carl Gauss and Bernhard Riemann, who ... More


Complex Numbers and Grassmann Numbers

Bryan J. Dalton, John Jeffers, and Stephen M. Barnett

in Phase Space Methods for Degenerate Quantum Gases

Published in print:
2014
Published Online:
April 2015
ISBN:
9780199562749
eISBN:
9780191747311
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199562749.003.0003
Subject:
Physics, Condensed Matter Physics / Materials, Atomic, Laser, and Optical Physics

This chapter sets out the basic algebraic properties of both complex numbers (c-numbers) and Grassmann numbers (g-numbers). The formal rules of addition and multiplication together with the ... More


Separation Properties, Convergence, and Extensions

Tim Maudlin

in New Foundations for Physical Geometry: The Theory of Linear Structures

Published in print:
2014
Published Online:
April 2014
ISBN:
9780198701309
eISBN:
9780191771613
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198701309.003.0005
Subject:
Philosophy, Logic/Philosophy of Mathematics

Standard topological spaces are characterized by various sorts of separation properties among points. For example, in a Hausdorff space, any two points belong to non-overlapping open sets. This ... More


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