Adjacency matrix: Difference between revisions

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In [[mathematics]], the '''rank''', '''Prüfer rank''', or '''torsion-free rank''' of an [[abelian group]] ''A'' is the [[cardinality]] of a maximal [[linearly independent]] subset. The rank of ''A'' determines the size of the largest [[free abelian group]] contained in ''A''. If ''A'' is [[Torsion (algebra)|torsion-free]] then it embeds into a [[vector space]] over the [[rational numbers]] of dimension rank ''A''. For [[finitely generated abelian group]]s, rank is a strong invariant and every such group is determined up to isomorphism by its rank and [[torsion subgroup]]. [[Torsion-free abelian groups of rank 1]] have been completely classified. However, the theory of abelian groups of higher rank is more involved.
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The term rank has a different meaning in the context of [[elementary abelian group]]s.
 
== Definition ==
 
A subset {''a''<sub>''α''</sub>} of an abelian group is '''[[linearly independent]]''' (over '''Z''') if the only linear combination of these elements that is equal to zero is trivial: if
 
: <math>\sum_\alpha n_\alpha a_\alpha = 0, \quad n_\alpha\in\mathbb{Z},</math>
 
where all but finitely many coefficients ''n''<sub>''α''</sub> are zero (so that the sum is, in effect, finite), then all coefficients are 0. Any two maximal linearly independent sets in ''A'' have the same [[cardinality]], which is called the '''rank''' of ''A''.
 
Rank of an abelian group is analogous to the [[vector space dimension|dimension]] of a [[vector space]]. The main difference with the case of vector space is a presence of [[torsion (algebra)|torsion]]. An element of an abelian group ''A'' is classified as torsion if its [[order (group theory)|order]] is finite. The set of all torsion elements is a subgroup, called the [[torsion subgroup]] and denoted ''T''(''A''). A group is called torsion-free if it has no non-trivial torsion elements. The factor-group ''A''/''T''(''A'') is the unique maximal torsion-free quotient of ''A'' and its rank coincides with the rank of ''A''.
 
The notion of rank with analogous properties can be defined for [[module (mathematics)|modules]] over any [[integral domain]], the case of abelian groups corresponding to modules over '''Z'''.
 
== Properties ==
 
* The rank of an abelian group ''A'' coincides with the dimension of the '''Q'''-vector space ''A'' ⊗ '''Q'''. If ''A'' is torsion-free then the canonical map ''A'' → ''A'' ⊗ '''Q''' is [[injective]] and the rank of ''A'' is the minimum dimension of '''Q'''-vector space containing ''A'' as an abelian subgroup. In particular, any intermediate group '''Z'''<sup>''n''</sup> < ''A'' < '''Q'''<sup>''n''</sup> has rank ''n''.
 
* Abelian groups of rank 0 are exactly the [[periodic group|periodic abelian groups]].  
 
* The group '''Q''' of rational numbers has rank 1. [[Torsion-free abelian groups of rank 1]] are realized as subgroups of '''Q''' and there is a satisfactory classification of them up to isomorphism. By contrast, there is no satisfactory classification of torsion-free abelian groups of rank 2.{{Citation needed|date=July 2010}}
 
* Rank is additive over [[short exact sequence]]s: if
 
::<math>0\to A\to B\to C\to 0\;</math>
 
:is a short exact sequence of abelian groups then rk ''B'' = rk ''A'' + rk ''C''. This follows from the [[flat module|flatness]] of '''Q''' and the corresponding fact for vector spaces.
 
* Rank is additive over arbitrary [[direct sum]]s:  
 
::<math>\operatorname{rank}\left(\bigoplus_{j\in J}A_j\right) = \sum_{j\in J}\operatorname{rank}(A_j),</math>
 
: where the sum in the right hand side uses [[cardinal arithmetic]].
 
== Groups of higher rank ==
 
Abelian groups of rank greater than 1 are sources of interesting examples. For instance, for every cardinal ''d'' there exist torsion-free abelian groups of rank ''d'' that are [[indecomposable module|indecomposable]], i.e. cannot be expressed as a direct sum of a pair of their proper subgroups. These examples demonstrate that torsion-free abelian group of rank greater than 1 cannot be simply built by direct sums from torsion-free abelian groups of rank 1, whose theory is well-understood. Moreover, for every integer ''n'' ≥ 3, there is a torsion-free abelian group of rank 2''n'' &minus; 2 that is simultaneously a sum of two indecomposable groups, and a sum of ''n'' indecomposable groups.{{Citation needed|date=July 2010}} Hence even the number of indecomposable summands of a group of an even rank greater or equal than 4 is not well-defined.
 
Another result about non-uniqueness of direct sum decompositions is due to A.L.S. Corner: given integers ''n''&nbsp;≥&nbsp;''k''&nbsp;≥&nbsp;1, there exists a torsion-free abelian group ''A'' of rank ''n'' such that for any partition ''n'' = ''r''<sub>1</sub> + ... + ''r''<sub>''k''</sub> into ''k'' natural summands, the group ''A'' is the direct sum of ''k'' indecomposable subgroups of ranks ''r''<sub>1</sub>, ''r''<sub>2</sub>, ..., ''r''<sub>''k''</sub>.{{Citation needed|date=July 2010}} Thus the sequence of ranks of indecomposable summands in a certain direct sum decomposition of a torsion-free abelian group of finite rank is very far from being an invariant of ''A''.
 
Other surprising examples include torsion-free rank 2 groups ''A''<sub>''n'',''m''</sub> and ''B''<sub>''n'',''m''</sub> such that ''A''<sup>''n''</sup> is isomorphic to ''B''<sup>''n''</sup> if and only if ''n'' is divisible by ''m''.
 
For abelian groups of infinite rank, there is an example of a group ''K'' and a subgroup ''G'' such that
* ''K'' is indecomposable;
* ''K'' is generated by ''G'' and a single other element; and
* Every nonzero direct summand of ''G'' is decomposable.
 
==Generalization==
The notion of rank can be generalized for any module ''M'' over an [[integral domain]] ''R'', as the dimension over ''R''<sub>0</sub>, the [[quotient field]], of the [[tensor product]] of the module with the field:
::<math>\text{rank} (M)=\dim_{R_0} M\otimes_R R_0</math>
It makes sense, since ''R''<sub>0</sub> is a field, and thus any module (or, to be more specific, [[vector space]]) over it is free.
 
It is a generalization, since any abelian group is a module over the integers. It easily follows that the dimension of the product over '''Q''' is the cardinality of maximal linearly independent subset, since for any torsion element x and any rational q
::<math>x\otimes_{\mathbf Z} q = 0</math>
 
==See also==
*[[Rank of a group]]
 
==References==
{{Refimprove|date=September 2008}}
* Page 46 of {{Lang Algebra|edition=3}}
 
[[Category:Abelian group theory]]

Latest revision as of 10:57, 13 January 2015

It is time to address the slow computer issues whether or not we never learn how. Just considering the computer is working thus slow or keeps freezing up; does not signify to not address the issue and fix it. You may or can not be aware which any computer owner must know which there are certain points which the computer needs to keep the best performance. The sad truth is the fact that a lot of folks who own a program have no idea that it needs routine maintenance just like their vehicles.

But registry is conveniently corrupted and damaged whenever you may be using the computer. Overtime, without proper repair, it can be loaded with mistakes plus incorrect or missing information which may create a program unable to function properly or implement a certain task. And whenever your system cannot find the correct information, it may not understand what to do. Then it freezes up! That is the real cause of your trouble.

With the Internet, the risk to your registry is more and windows XP error messages may appear frequently. Why? The malicious wares like viruses, Trojans, spy-wares, ad wares, and the like gets recorded too. Cookies are best examples. You reach save passwords, and stuff, proper? That is a easy example of the register functioning.

Analysis a files plus clean it up regularly. Destroy all unwanted plus unused files because they just jam the computer program. It usually definitely improve the speed of the computer plus be careful that the computer do not infected by a virus. Remember always to update the antivirus software every time. If you never utilize your computer pretty frequently, you are able to take a free antivirus.

Use a registry reviver. This usually look your Windows registry for 3 types of keys that will really hurt PC performance. These are: duplicate, missing, plus corrupted.

S/w connected error handling - If the blue screen bodily memory dump happens following the installation of s/w application or perhaps a driver it may be which there is system incompatibility. By booting into secure mode and removing the software you are able to immediately fix this error. We may also try out a "program restore" to revert to an earlier state.

To accelerate a computer, we merely have to be able to do away with all these junk files, allowing a computer to locate what it wants, when it wants. Luckily, there's a tool which enables you to do this easily and rapidly. It's a tool called a 'registry cleaner'.

A registry cleaner is a system which cleans the registry. The Windows registry always gets flooded with junk data, info that has not been removed from uninstalled programs, erroneous file organization and different computer-misplaced entries. These clean little program software tools are quite well-known nowadays and you will find very a few good ones found on the Internet. The wise ones provide you choice to maintain, clean, update, backup, plus scan the System Registry. Whenever it finds supposedly unwanted ingredients inside it, the registry cleaner lists them plus recommends the user to delete or repair these orphaned entries and corrupt keys.

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