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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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