## abelian group

Overview page. Subjects: Mathematics.

Suppose that *G* is a group with the operation ￮. Then *G* is abelian if the operation ￮ is commutative; that is, if, for all elements *a* and *b* in *G*, *a* ￮ *b*=*b* ￮ *a*.

## abelian group

in** The Concise Oxford Dictionary of Mathematics**

January 2009; p ublished online January 2009 .

Reference Entry. Subjects: Pure Mathematics. 38 words.

Suppose that *G* is a *group with the operation ￮. Then *G* is abelian if the operation ￮ is

## abelian group

in** The Concise Oxford Dictionary of Mathematics**

January 2014; p ublished online September 2014 .

Reference Entry. Subjects: Pure Mathematics. 41 words.

Suppose that *G* is a *group with the operation ￮. Then *G* is abelian if the operation ￮ is commutative; that is, if, for all elements ...

## abelian group

Overview page. Subjects: Mathematics.

Suppose that *G* is a group with the operation ￮. Then *G* is abelian if the operation ￮ is commutative; that is, if, for all elements *a* and *b* in *G*, *a* ￮ *b*=*b* ￮ *a*.

## Valued Abelian Groups

in** Asymptotic Differential Algebra and Model Theory of Transseries**

June 2017; p ublished online October 2017 .

Chapter. Subjects: Optimisation. 24508 words.

This chapter deals with valued abelian groups. It first introduces some terminology concerning ordered sets before discussing valued abelian groups and ordered abelian groups in more...

## abelian group

in** A Dictionary of Computer Science**

January 2016; p ublished online January 2016 .

Reference Entry. Subjects: Computing. 8 words.

## abelian group

in** A Dictionary of Computing**

January 2008; p ublished online January 2008 .

Reference Entry. Subjects: Computing. 6 words.

## Abelian group

in** A Dictionary of Physics**

January 2009; p ublished online January 2009 .

Reference Entry. Subjects: Physics. 5 words.

## Abelian group

in** A Dictionary of Physics**

January 2015; p ublished online September 2015 .

Reference Entry. Subjects: Physics. 5 words.

## Abelian group

in** A Dictionary of Physics**

March 2019; p ublished online March 2019 .

Reference Entry. Subjects: Physics. 5 words.

## Invisibility of Tate–Shafarevich Groups in Abelian Surfaces

in** International Mathematics Research Notices**

January 2014; p ublished online April 2013 .

Journal Article. Subjects: Mathematics. 3636 words.

We give some examples of elliptic curves over the rationals whose Tate–Shafarevich groups contain elements of order 6 or 7 that are not visible in an abelian surface. We then show that some...

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## FOURIER ANALYSIS ON FINITE ABELIAN GROUPS: SOME GRAPHICAL APPLICATIONS

in** Combinatorics, Complexity, and Chance**

January 2007; p ublished online September 2007 .

Chapter. Subjects: Probability and Statistics. 14520 words.

This article reviews basic techniques of Fourier analysis on a finite abelian group *Q*, with subsequent applications in graph theory. These include evaluations of the Tutte polynomial of a...

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** » ** abstract

## ZERO-SUM PROBLEMS IN FINITE ABELIAN GROUPS AND AFFINE CAPS

in** The Quarterly Journal of Mathematics**

June 2007; p ublished online May 2007 .

Journal Article. Subjects: Pure Mathematics. 0 words.

**Abstract**

For a finite abelian group *G*, let [math] (*G*) denote the smallest integer *l* such that every sequence *S* over *G* of length | *S*| ≥ *l* has a zero-sum...

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## COARSE CLASSIFICATION OF ABELIAN GROUPS AND AMENABLE SHIFT-HOMOGENEOUS METRIC SPACES

in** The Quarterly Journal of Mathematics**

December 2014; p ublished online March 2014 .

Journal Article. Subjects: Pure Mathematics. 0 words.

We classify countable locally finite-by-abelian groups up to coarse isomorphism. Our classification is derived from a coarse classification of amenable shift-homogeneous metric spaces.

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## Non-Abelian Stokes Theorem for Wilson Loops Associated with General Gauge Groups

in** Progress of Theoretical Physics**

January 2000; p ublished online January 2000 .

Journal Article. 0 words.

A formula constituting the non-Abelian Stokes theorem for general semi-simple compact gauge groups is presented. The formula involves a path integral over a group space and is applicable to...

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** » ** abstract

## Coding Techniques for Fault-Tolerant Parallel Prefix Computations in Abelian Groups

in** The Computer Journal**

January 2004; p ublished online January 2004 .

Journal Article. Subjects: Computer Science. 0 words.

This paper presents coding techniques that can be used to provide fault tolerance to a parallel prefix computation that is performed on a binary tree of processing nodes. More specifically,...

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## AF-Embeddings of the Crossed Products of AH-Algebras by Finitely Generated Abelian Groups

in** International Mathematics Research Papers**

January 2008; p ublished online January 2008 .

Journal Article. Subjects: Mathematics. 0 words.

Let *X* be a compact metric space and let Λ be a ℤ^{k} (*k ≥ 1*) action on *X*. We give a solution to a version of Voiculescu's problem of AF-embedding: The crossed product *C(X) ⋊ _{Λ} ℤ^{k}
* c...

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##
On a Family of Surfaces of General Type Attached to Abelian Four-Folds and the Weyl Group *W*(*E*
_{6})

in** International Mathematics Research Notices**

January 2015; p ublished online March 2015 .

Journal Article. Subjects: Mathematics; Pure Mathematics. 16116 words.

We study a family of surfaces of general type that arises from the intersections of two translates of the theta divisor on a principally polarized complex abelian four-fold. In particular,...

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## ENUMERATION OF FINITE SOLUBLE GROUPS WITH ABELIAN SYLOW SUBGROUPS

in** The Quarterly Journal of Mathematics**

March 1997; p ublished online March 1997 .

Journal Article. Subjects: Pure Mathematics. 0 words.

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##
Abelian subalgebras in ℤ_{2}-graded Lie algebras and affine Weyl groups

in** International Mathematics Research Notices**

January 2004; p ublished online January 2004 .

Journal Article. Subjects: Mathematics. 0 words.

Let [math] be a simple ℤ_{2}-graded Lie algebra and let [math] be a fixed Borel subalgebra of [math]. We describe and enumerate the abelian [math]-stable subalgebras of [math].

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## Abel, Niels Henrik (1802–29)

in** The New Oxford Dictionary for Scientific Writers and Editors**

January 2009; p ublished online January 2009 .

Reference Entry. Subjects: Language Reference. 11 words.

(1802–29)

Norwegian mathematician.

**Abelian group**

**Abel's theorem**