## Equivalence Criteria for Fourier–Mukai Transforms

*D. Huybrechts*

### in Fourier-Mukai Transforms in Algebraic Geometry

- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780199296866
- eISBN:
- 9780191711329
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199296866.003.0007
- Subject:
- Mathematics, Geometry / Topology

Criteria must be developed to determine whether a given Fourier-Mukai transform is in fact an equivalence. Applying the techniques from Chapter 1, this chapter explains the fully faithfulness of a ... More

## NILPOTENT GROUPS

*John C. Lennox and Derek J. S. Robinson*

### in The Theory of Infinite Soluble Groups

- Published in print:
- 2004
- Published Online:
- September 2007
- ISBN:
- 9780198507284
- eISBN:
- 9780191709326
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198507284.003.0002
- Subject:
- Mathematics, Pure Mathematics

This chapter discusses extraction of roots in nilpotent groups. Topics covered include the Mal’cev completion for a torsion-free nilpotent group, the commutator collection process, and isolators.

## Some Problems of Unlikely Intersections in Arithmetic and Geometry (AM-181)

*Umberto Zannier*

- Published in print:
- 2012
- Published Online:
- October 2017
- ISBN:
- 9780691153704
- eISBN:
- 9781400842711
- Item type:
- book

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691153704.001.0001
- Subject:
- Mathematics, Geometry / Topology

This book considers the so-called unlikely intersections, a topic that embraces well-known issues, such as Lang's and Manin–Mumford's, concerning torsion points in subvarieties of tori or abelian ... More

## Circular and Thin-Wall Torsion

*Sanjay Govindjee*

### in Engineering Mechanics of Deformable Solids: A Presentation with Exercises

- Published in print:
- 2012
- Published Online:
- December 2013
- ISBN:
- 9780199651641
- eISBN:
- 9780191775017
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199651641.003.0007
- Subject:
- Physics, Condensed Matter Physics / Materials

This chapter introduces the fundamental kinematic assumption associated with the torsion of circular bars. It treats the equilibrium of twisted rods and examines the behaviour of elastic as well as ... More

## Measurements of the Casimir Force Between Metals

*M. Bordag, G. L. Klimchitskaya, U. Mohideen, and V. M. Mostepanenko*

### in Advances in the Casimir Effect

- Published in print:
- 2009
- Published Online:
- September 2009
- ISBN:
- 9780199238743
- eISBN:
- 9780191716461
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199238743.003.0019
- Subject:
- Physics, Condensed Matter Physics / Materials, Atomic, Laser, and Optical Physics

This chapter starts chronologically with the first measurement, by means of a torsion pendulum, in the recent phase of Casimir force experiments. Then the main breakthroughs in the measurement of the ... More

## Saccades and fixations

*Nicholas J. Wade and Benjamin W. Tatler*

### in The Moving Tablet of the Eye: The origins of modern eye movement research

- Published in print:
- 2005
- Published Online:
- April 2010
- ISBN:
- 9780198566175
- eISBN:
- 9780191584954
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198566175.003.0004
- Subject:
- Psychology, Cognitive Psychology

Fixations were examined before saccades were measured. Concerns were the stability of fixation and torsion of the eyes. The latter resulted in Donders' law and Listing's law, both of which were ... More

## Engineering Mechanics of Deformable Solids: A Presentation with Exercises

*Sanjay Govindjee*

- Published in print:
- 2012
- Published Online:
- December 2013
- ISBN:
- 9780199651641
- eISBN:
- 9780191775017
- Item type:
- book

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199651641.001.0001
- Subject:
- Physics, Condensed Matter Physics / Materials

This book covers the essential elements of engineering mechanics for mechanical elements in tension-compression, torsion, and bending. Its approach emphasizes a fundamental bottom-up approach to the ... More

## BROWNIAN MOTION AND NOISE

*Robert M. Mazo*

### in Brownian Motion: Fluctuations, Dynamics, and Applications

- Published in print:
- 2008
- Published Online:
- January 2010
- ISBN:
- 9780199556441
- eISBN:
- 9780191705625
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199556441.003.0013
- Subject:
- Physics, Condensed Matter Physics / Materials

This chapter treats Brownian motion as a fundamental limit on the precision of experimental measurements. As mechanical examples, the oscillations of a small mirror mounted on a torsion fibre are ... More

## Solutions to some classical problems in linear elastostatics

*Lallit Anand and Sanjay Govindjee*

### in Continuum Mechanics of Solids

- Published in print:
- 2020
- Published Online:
- September 2020
- ISBN:
- 9780198864721
- eISBN:
- 9780191896767
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198864721.003.0009
- Subject:
- Physics, Condensed Matter Physics / Materials

This chapter presents and discusses the solution of several classical problems in linear elastostatics, including thick-walled spheres and cylinders under external and internal pressure; bending and ... More

## The Cartan structure equations

*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/9780198786399.003.0065
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology

This chapter focuses on Cartan structure equations. It first introduces a 1-form and its exterior derivative, before turning to a study of the connection and torsion forms, thereby expressing the ... More

## Electron-density maps

*David Blow*

### in Outline of Crystallography for Biologists

- Published in print:
- 2002
- Published Online:
- November 2020
- ISBN:
- 9780198510512
- eISBN:
- 9780191919244
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198510512.003.0016
- Subject:
- Chemistry, Crystallography: Chemistry

When everything has been done to make the phases as good as possible, the time has come to examine the image of the structure in the form of an ... More

## Localization in Algebra

*Loring W. Tu*

### in Introductory Lectures on Equivariant Cohomology: (AMS-204)

- Published in print:
- 2020
- Published Online:
- January 2021
- ISBN:
- 9780691191751
- eISBN:
- 9780691197487
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691191751.003.0023
- Subject:
- Mathematics, Educational Mathematics

This chapter provides a digression concerning the all-important technique of localization in algebra. Localization generally means formally inverting a multiplicatively closed subset in a ring. ... More

## Free and Locally Free Actions

*Loring W. Tu*

### in Introductory Lectures on Equivariant Cohomology: (AMS-204)

- Published in print:
- 2020
- Published Online:
- January 2021
- ISBN:
- 9780691191751
- eISBN:
- 9780691197487
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691191751.003.0024
- Subject:
- Mathematics, Educational Mathematics

This chapter addresses free and locally free actions. It uses the Cartan model to compute the equivariant cohomology of a circle action, so equivariant cohomology is taken with real coefficients. An ... More

## Borel Localization for a Circle Action

*Loring W. Tu*

### in Introductory Lectures on Equivariant Cohomology: (AMS-204)

- Published in print:
- 2020
- Published Online:
- January 2021
- ISBN:
- 9780691191751
- eISBN:
- 9780691197487
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691191751.003.0026
- Subject:
- Mathematics, Educational Mathematics

This chapter explores Borel localization for a circle action. For a circle action, the Borel localization theorem says that up to torsion, the equivariant cohomology of an S1-manifold is concentrated ... More

## Some classical problems in rate-independent plasticity

*Lallit Anand and Sanjay Govindjee*

### in Continuum Mechanics of Solids

- Published in print:
- 2020
- Published Online:
- September 2020
- ISBN:
- 9780198864721
- eISBN:
- 9780191896767
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198864721.003.0024
- Subject:
- Physics, Condensed Matter Physics / Materials

This chapter presents analytical solutions to some classical problems in rate-independent plasticity. Solutions are presented for the elastic-plastic torsion of a round bar, including spring back; ... More

## Integral coefficients

*Claire Voisin*

### in Chow Rings, Decomposition of the Diagonal, and the Topology of Families (AM-187)

- Published in print:
- 2014
- Published Online:
- October 2017
- ISBN:
- 9780691160504
- eISBN:
- 9781400850532
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691160504.003.0006
- Subject:
- Mathematics, Geometry / Topology

This chapter turns to a number of birational invariants that can be defined using Z-coefficients. It is devoted in part to the study of the groups Z²ⁱ(X) measuring the failure of the Hodge conjecture ... More

## Modular curves, modular forms, lattices, Galois representations

*Bas Edixhoven*

### in Computational Aspects of Modular Forms and Galois Representations: How One Can Compute in Polynomial Time the Value of Ramanujan's Tau at a Prime (AM-176)

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691142012
- eISBN:
- 9781400839001
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691142012.003.0002
- Subject:
- Mathematics, Number Theory

This chapter provides the necessary background concerning modular curves and modular forms. It covers modular curves, modular forms, lattices and modular forms, Galois representations attached to ... More

##
Approximating *V*_{f} over the complex numbers

_{f}

*Jean-Marc Couveignes*

### in Computational Aspects of Modular Forms and Galois Representations: How One Can Compute in Polynomial Time the Value of Ramanujan's Tau at a Prime (AM-176)

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691142012
- eISBN:
- 9781400839001
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691142012.003.0012
- Subject:
- Mathematics, Number Theory

This chapter addresses the problem of computing torsion divisors on modular curves with an application to the explicit calculation of modular representations. The final result of the chapter is ... More

## Torsion

*Benson Farb and Dan Margalit*

### in A Primer on Mapping Class Groups (PMS-49)

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691147949
- eISBN:
- 9781400839049
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691147949.003.0008
- Subject:
- Mathematics, Geometry / Topology

This chapter deals with finite subgroups of the mapping class group. It first explains the distinction between finite-order mapping classes and finite-order homeomorphisms, focusing on the Nielsen ... More

## Some boundary-value problems for uniform isotropic incompressible materials

*David J. Steigmann*

### in Finite Elasticity Theory

- Published in print:
- 2017
- Published Online:
- December 2017
- ISBN:
- 9780198567783
- eISBN:
- 9780191746536
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198567783.003.0007
- Subject:
- Mathematics, Applied Mathematics

This chapter outlines the formulation and explicit solution of a number of simple boundary-value problems. Analysis is facilitated by the constraint of incompressibility. Examples include ... More