Quasiperiodicity - Revision history

BrianMcMahon: Tidied translations and added German and Spanish (U. Mueller)

2017-11-17T10:16:36Z

Tidied translations and added German and Spanish (U. Mueller)

BrianMcMahon: Style edits to align with printed edition

2017-05-16T13:49:10Z

Style edits to align with printed edition

MassimoNespolo: cat

2015-02-26T07:57:31Z

cat

MassimoNespolo: Lang.

2015-02-17T04:48:29Z

Lang.

BrianMcMahon at 17:28, 7 February 2012

2012-02-07T17:28:56Z

TedJanssen at 18:28, 18 May 2009

2009-05-18T18:28:20Z

TedJanssen at 15:49, 18 May 2009

2009-05-18T15:49:09Z

TedJanssen at 14:50, 18 May 2009

2009-05-18T14:50:00Z

New page

[[Quasiperiodicity]]

Quasi-periodicité (Fr.)

'''Definition'''

A function is called ''quasiperiodic'' if its Fourier transform consists of δ-peaks
on positions

<math>{\bf k}~=~\sum_{i=1}^n h_i {\bf a}_i^*,~~({\rm integers ~}h_i) </math>

for basis vectors '''a'''<sub>i</sub><sup>*</sup> in a space of dimension ''m''. If the basis vectors
form a basis for the space (''n'' equal to the space dimension, and linearly
independent basis vectors over the real numbers) then the function is lattice periodic.
If ''n'' is larger than the space dimension, then the function is ''aperiodic''.

'''Comment'''

Sometimes the definition includes that the function is not lattice periodic.

A quasiperiodic function may be expressed in a convergent trigonometric series.

<math>f({\bf r})~=~\sum_{{\bf k}} A({\bf k}) \cos \left( 2\pi {\bf k}.{\bf r}+\phi ({\bf k}) \right). </math>

It is a special case of an almost periodic function. An ''almost periodic function''
is a function ''f''('''r''') such that for every small number ε there is
a translation '''a''' such that the difference between the function and the function shifted over
'''a''' is smaller than the chosen quantity:

| <math>f({\bf r}+{\bf a})-f({\bf r}) |~<~ \epsilon~~{\rm for ~all~{\bf r}} .</math>

A quasiperiodic function is always an almost periodic function, but the converse
is not true.

The theory of almost-periodic functions goes back to the work by H. Bohr.

← Older revision		Revision as of 10:16, 17 November 2017
Line 1:		Line 1:
−	<~~Font~~ color="blue">Quasi-periodicité </font>(''Fr''). <~~Font~~ color="black">Quasi-periodicità </font>(''It''). <~~Font~~ color="purple">準周期性 </font>(''Ja'').	+	<font color="blue">Quasi-periodicité</font> (''Fr''). <font color="red">Quasiperiodizität</font> (''Ge''). <font color="black">Quasi-periodicità</font> (''It''). <font color="purple">準周期性</font> (''Ja''). <font color="green">Cuasiperiodicidad</font> (''Sp'').

	== Definition ==		== Definition ==

@@ Line 1: / Line 1: @@
-<Font color="blue">Quasi-periodicit&eacute; </font>(''Fr'');　<Font color="black">Quasi-periodicit&agrave; </font>(''It'');　<Font color="purple">準周期性 </font>(''Ja'').
+<Font color="blue">Quasi-periodicit&eacute; </font>(''Fr''). <Font color="black">Quasi-periodicit&agrave; </font>(''It''). <Font color="purple">準周期性 </font>(''Ja'').
 == Definition ==
@@ Line 14: / Line 14: @@
 Sometimes the definition includes that the function is not lattice periodic.
-A quasiperiodic function may be expressed in a convergent trigonometric series.
+A quasiperiodic function may be expressed in a convergent trigonometric series:
-<math>f( r)~=~\sum_k A(k) \cos \left( 2\pi  k. r+\phi ( k) \right). </math>
+<math>f(r)~=~\sum_k A(k) \cos [ 2\pi  k. r+\varphi (k) ]. </math>
 It is a special case of an almost periodic function. An  ''almost periodic function''
@@ Line 23: / Line 23: @@
 '''a''' is smaller than the chosen quantity:
-<math>| f(r+ a)-f( r) |~<~ \epsilon~~{\rm for ~all~ r} .</math>
+<math>| f(r+ a)-f( r) |~<~ \varepsilon~~{\rm for ~all~ r} .</math>
 A quasiperiodic function is always an almost periodic function, but the converse is not true.

← Older revision		Revision as of 07:57, 26 February 2015
Line 28:		Line 28:

	The theory of almost-periodic functions goes back to the work by H. Bohr.		The theory of almost-periodic functions goes back to the work by H. Bohr.
		+
		+	[[Category: Fundamental crystallography]]

← Older revision		Revision as of 04:48, 17 February 2015
Line 1:		Line 1:
−	<Font color="blue">Quasi-periodicité </font>(Fr.)	+	<Font color="blue">Quasi-periodicité </font>(''Fr'');　<Font color="black">Quasi-periodicità </font>(''It'');　<Font color="purple">準周期性 </font>(''Ja'').

	== Definition ==		== Definition ==

@@ Line 1: / Line 1: @@
 <Font color="blue">Quasi-periodicit&eacute; </font>(Fr.)
- '''Definition'''
+== Definition ==
-A function is called  ''quasiperiodic'' if its Fourier transform consists of &delta;-peaks
+A function is called  ''quasiperiodic'' if its Fourier transform consists of &delta;-peaks on positions
- <math> k~=~\sum_{i=1}^n h_i  a_i^*,~~({\rm integers ~}h_i)  </math>
+<math> k~=~\sum_{i=1}^n h_i  a_i^*,~~({\rm integers ~}h_i)  </math>
-for basis vectors  '''a'''<sub>i</sub><sup>*</sup> in a space of dimension ''m''. If the basis vectors
+for basis vectors  '''a'''<sub>i</sub><sup>*</sup> in a space of dimension ''m''. If the basis vectors form a basis for the space (''n'' equal to the space dimension, and linearly
-form a basis for the space (''n'' equal to the space dimension, and linearly
+independent basis vectors over the real numbers) then the function is lattice periodic. If ''n'' is larger than the space dimension, then the function is  ''aperiodic''.
- '''Comment'''
+== Comment ==
 Sometimes the definition includes that the function is not lattice periodic.
@@ Line 22: / Line 16: @@
 A quasiperiodic function may be expressed in a convergent trigonometric series.
-  <math>f( r)~=~\sum_k A(k) \cos \left( 2\pi  k. r+\phi ( k) \right). </math>
+<math>f( r)~=~\sum_k A(k) \cos \left( 2\pi  k. r+\phi ( k) \right). </math>
 It is a special case of an almost periodic function. An  ''almost periodic function''
@@ Line 29: / Line 23: @@
 '''a''' is smaller than the chosen quantity:
- | <math>f(r+ a)-f( r) |~<~ \epsilon~~{\rm for ~all~ r} .</math>
+<math>| f(r+ a)-f( r) |~<~ \epsilon~~{\rm for ~all~ r} .</math>
-A quasiperiodic function is always an almost periodic function, but the converse
+A quasiperiodic function is always an almost periodic function, but the converse is not true.
 The theory of almost-periodic functions goes back to the work by H. Bohr.