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In [[mathematics]], a '''Følner sequence''' for a [[group (mathematics)|group]] is a [[sequence]] of [[Set (mathematics)|sets]] satisfying a particular condition. If a group has a Følner sequence with respect to its action on itself, the group is [[amenable group|amenable]]. A more general notion of Følner [[net (mathematics)|nets]] can be defined analogously, and is suited for the study of [[uncountable]] groups. Følner sequences are named for [[Erling Følner]].
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== Definition ==
 
Given a group <math>G</math> that [[group action|acts]] on a countable set <math>X</math>, a Følner sequence for the action is a sequence of finite [[subset]]s <math>F_1, F_2, \dots</math> of <math>X</math> which exhaust <math>X</math> and which "don't move too much" when acted on by any group element. Precisely,
:For every <math>x\in X</math>, there exists some <math>i</math> such that <math>x \in F_j</math> for all <math> j > i</math>, and
:<math>\lim_{i\to\infty}\frac{|gF_i\,\triangle\,F_i|}{|F_i|} = 0</math> for all group elements <math>g</math> in <math>G</math>.
Explanation of the notation used above:
*<math>gF_i\ </math> is the result of the set <math>F_i\ </math> being acted on the left by <math>g</math>.  It consists of elements of the form <math>gf</math> for all <math>f</math> in <math>F_i</math>.
*<math>\triangle</math> is the [[symmetric difference]] operator, i.e., <math>A\triangle B</math> is the set of elements in exactly one of the sets <math>A</math> and <math>B</math>.
*<math>|A|</math> is the [[cardinality]] of a set <math>A</math>.
Thus, what this definition says is that for any group element <math>g</math>, the proportion of elements of <math>F_i\ </math> that are moved away by <math>g</math> goes to 0 as <math>i</math> gets large.
 
In the setting of a [[locally compact group]] acting on a measure space <math>(X,\mu)</math> there is a more general definition. Instead of being finite, the sets are required to have finite, non-zero measure, and so the Følner requirement will be that
* <math>\lim_{i\to\infty}\frac{\mu(gF_i\,\triangle\,F_i)}{\mu(F_i)} = 0</math>,
analogously to the discrete case. The standard case is that of the group acting on itself by left translation, in which case the measure in question is normally assumed to be the Haar measure.
 
== Examples ==
 
* Any finite group <math>G</math> trivially has a Følner sequence <math>F_i=G</math> for each <math>i</math>.
 
* Consider the group of [[integer]]s, acting on itself by addition.  Let <math>F_i</math> consist of the integers between <math>-i</math> and <math>i</math>.  Then <math>gF_i</math> consists of integers between <math>g-i</math> and <math>g+i</math>. For large <math>i</math>, the symmetric difference has size <math>2g</math>, while <math>F_i</math> has size <math>2i+1</math>.  The resulting ratio is <math>2g/(2i+1)</math>, which goes to 0 as <math>i</math> gets large.
 
* With the original definition of Følner sequence, a countable group has a Følner sequence [[if and only if]] it is amenable.  An [[amenable group]] has a Følner sequence if and only if it is countable.  A group which has a Følner sequence is countable if and only if it is amenable.
 
* A locally compact group has a Følner sequence (with the generalized definition) if and only if it is amenable and [[second countable]].
 
== Proof of amenability {{Citation needed|date=May 2013}}==
We have a group <math>G</math> and a Følner sequence <math>F_i</math>, and we need to define a measure <math>\mu</math> on <math>G</math>, which philosophically speaking says how much of <math>G</math> any subset <math>A</math> takes up. The natural definition that uses our Følner sequence would be
:<math>\mu(A)=\lim_{i\to\infty}{|A\cap F_i|\over|F_i|}.</math>
Of course, this limit doesn't necessarily exist.  To overcome this technicality, we take an [[ultrafilter]] <math>U</math> on the natural numbers that contains intervals <math>[n,\infty)</math>. Then we use an [[ultralimit]] instead of the regular [[limit (mathematics)|limit]]:
:<math>\mu(A)=U{\textrm-}\lim{|A\cap F_i|\over|F_i|}.</math>
It turns out ultralimits have all the properties we need.  Namely,
#<math>\mu</math> is a ''probability measure''. That is, <math>\mu(G)=U\textrm{-}\lim1=1</math>, since the ultralimit coincides with the regular limit when it exists.
#<math>\mu</math> is ''finitely additive''. This is since ultralimits commute with addition just as regular limits do.
#<math>\mu</math> is ''left invariant''.  This is since
#:<math>\left|{|gA\cap F_i|\over|F_i|}-{|A\cap F_i|\over|F_i|}\right| = \left|{|A\cap g^{-1}F_i|\over|F_i|}-{|A\cap F_i|\over|F_i|}\right|</math>
#:::<math>\leq{|A\cap(g^{-1}F_i\,\triangle\,F_i)|\over|F_i|}\to0</math>
:by the Følner sequence definition.
 
== References ==
 
* {{Cite journal|author=Erling Følner|title=On groups with full Banach mean value|journal=[[Mathematica Scandinavica]]|volume=3|pages=243&ndash;254|year=1955|url=http://resolver.sub.uni-goettingen.de/purl?GDZPPN00234405X}}
 
{{DEFAULTSORT:Folner Sequence}}
[[Category:Geometric group theory]]

Revision as of 23:32, 21 February 2014

The best part of the Tadalis is that men suffering with impotence can even enjoy the multiple orgasms during the sexual copulation. Another thing you should try to do is to take a quick power nap when your lunch break comes around. Almonds, pistachios, musli (safed musli), dates, raisins are used in aphrodisiac preparations of Unani Medicines. Oral impotence drugs cannot be consumed by impotent individuals with life threatening diseases as it produces host of side effects from head ache to heart attacks. In this way, husbands and wives are notorious in forcing these stereotyped roles on one another.

Today, thanks to new surgical techniques that spare the nerves and vessels feeding the penis, erectile dysfunction is far from inevitable. Who think that even though we're writing about the internet, we should still make an effort to be entertaining and erudite. The Sex-related Guidance Organization has useful factsheets on all of this. The difference between the two types of impotence is in the frequency of occurrence of the impotency episodes. Lots of scientific findings confirm it helps increase immune system as well as improve level of energy.

Did you know that the way you breathe also has a major impact upon your impotency problems. It is ideal for food cooking, baking allowing it to do wonders to your own wellness, hair and skin. In such cases, there will be no physical factors accompanying erectile dysfunction. To begin with, whenever you requests something via Pharmaplanet they're not required to be supplied with a doctor's prescription for any product. Nausea and constipation are common symptoms of Paxil.

This process involves an intraocular implant which is even popularly known as the iris implants and with the help of this you can easily change one's eye colour instead of using the color contacts or the lenses. It will help your stylist get an idea of what type of cut or color you may have in mind on a future visit. In fact, hops may be part of the reason why heavy beer drinkers often experience hormonal shifts and develop enlarged bellies and breasts. PETA claims that meat eaters have higher rates of impotency due to artery blockage. Also, if you keep sitting in one position throughout the day, the chances are rife that you could suffer from impotency too.

4T Plus capsule is a best recommended herbal supplement to treat male impotence in old age. The insertion process is painless and does not have any side effects on your body. However, without proper knowledge, they can also be used improperly. An ability to maintain a penile erection requires a healthy & sound nervous system, healthy arteries in and near corpora cavernosa, smooth muscles & fibrous tissues in corpora cavernosa and abundant levels of nitric oxide. The blood is kept within the penis to maintain erection.

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