Vector module - Revision history

BrianMcMahon: Added German translation (U. Mueller)

2017-11-20T14:42:19Z

Added German translation (U. Mueller)

BrianMcMahon: small correction to sentence structure

2017-07-08T16:50:44Z

small correction to sentence structure

BrianMcMahon: Style edits to align with printed edition

2017-05-17T18:11:38Z

Style edits to align with printed edition

MassimoNespolo: It

2014-11-26T12:53:10Z

BrianMcMahon at 12:03, 8 February 2012

2012-02-08T12:03:42Z

TedJanssen at 15:05, 19 May 2009

2009-05-19T15:05:57Z

TedJanssen at 19:05, 18 May 2009

2009-05-18T19:05:32Z

TedJanssen at 19:04, 18 May 2009

2009-05-18T19:04:36Z

TedJanssen at 15:50, 18 May 2009

2009-05-18T15:50:21Z

TedJanssen at 15:39, 18 May 2009

2009-05-18T15:39:17Z

New page

[[Vector module]]

Module vectoriel (Fr.)

Synonymous: Z-module

'''Definition'''

A ''vector module'' is the set of vectors spanned by a number ''n'' of basis vectors
with integer coefficients. The basis vectors should be independent over the integers,
which means that any linear combination <math>\sum_i m_i {\bf a}_i</math> with ''m''<sub>i</sub> integers
is equal to zero if, and only if, all coefficients ''m''<sub>i</sub> are zero. The term Z-module
is sometimes used to underline the condition that the coefficients are integers.
The number of basis vectors is the ''rank'' of the vector module.

'''Comment'''

An ''n''-dimensional lattice in an ''n''-dimensional vector space is an example
of a vector module, with rank ''n''. In reciprocal space, the reciprocal lattice
corresponding to a crystallographic structure is a special case of a vector module.
The Bragg peaks for the crystal fall on the positions of the reciprocal lattice.
More generally, the Bragg peaks of an ''m''-dimensional aperiodic crystal structure
belong to a vector module of rank ''n'', larger than ''n''.

← Older revision		Revision as of 14:42, 20 November 2017
Line 1:		Line 1:
−	<~~Font~~ color="blue">Module vectoriel</font> (''Fr''). <~~Font~~ color="black">Modulo vettoriale</font> (''It'').	+	Synonyms: Z-module, Fourier module
		+
		+
		+	<font color="blue">Module vectoriel</font> (''Fr''). <font color="red">Vektormodul</font> (''Ge''). <font color="black">Modulo vettoriale</font> (''It'').

−	~~Synonyms: Z-module, Fourier module~~

	== Definition ==		== Definition ==

@@ Line 15: / Line 15: @@
 More generally, the Bragg peaks of an ''m''-dimensional aperiodic crystal structure
 belong to a vector module of rank ''n'', larger than ''m''. To indicate that this module
-exists in the reciprocal space, it is sometimes called ''Fourier module''.
+exists in reciprocal space, it is sometimes called the ''Fourier module''.
 [[Category:Fundamental crystallography]]

← Older revision		Revision as of 18:11, 17 May 2017
Line 1:		Line 1:
−	<Font color="blue">Module vectoriel</font> (''Fr''), <Font color="black">Modulo vettoriale</font> (''It'').	+	<Font color="blue">Module vectoriel</font> (''Fr''). <Font color="black">Modulo vettoriale</font> (''It'').

	Synonyms: Z-module, Fourier module		Synonyms: Z-module, Fourier module

← Older revision		Revision as of 12:53, 26 November 2014
Line 1:		Line 1:
−	<Font color="blue">Module vectoriel</font> (Fr.)	+	<Font color="blue">Module vectoriel</font> (''Fr''), <Font color="black">Modulo vettoriale</font> (''It'').

	Synonyms: Z-module, Fourier module		Synonyms: Z-module, Fourier module

@@ Line 1: / Line 1: @@
 <Font color="blue">Module vectoriel</font> (Fr.)
-Synonymous: Z-module, Fourier module
+Synonyms: Z-module, Fourier module
- '''Definition'''
+== Definition ==
-A  ''vector module'' is the set of vectors spanned by a number ''n'' of basis vectors
+A  ''vector module'' is the set of vectors spanned by a number ''n'' of basis vectors with integer coefficients. The basis vectors should be independent over the integers, which means that any linear combination <math>\sum_i m_i  a_i</math> with ''m''<sub>i</sub> integers is equal to zero if, and only if, all coefficients ''m''<sub>i</sub> are zero. The term Z-module is sometimes used to underline the condition that the coefficients are integers. The number of basis vectors is the  ''rank'' of the vector module.
- '''Comment'''
+== Comment ==
 An ''n''-dimensional lattice in an ''n''-dimensional vector space is an example
@@ Line 24: / Line 16: @@
 belong to a vector module of rank ''n'', larger than ''m''. To indicate that this module
 exists in the reciprocal space, it is sometimes called ''Fourier module''.

@@ Line 22: / Line 22: @@
 The Bragg peaks for the crystal fall on the positions of the reciprocal lattice.
 More generally, the Bragg peaks of an ''m''-dimensional aperiodic crystal structure
-belong to a vector module of rank ''n'', larger than ''n''.
+belong to a vector module of rank ''n'', larger than ''m''.