Fatigue (material): Difference between revisions

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In [[abstract algebra]] a '''linearly ordered''' or '''totally ordered group''' is a [[group (mathematics)|group]] ''G'' equipped with a [[total order]] "≤", that is ''translation-invariant''. This may have different meanings. Let ''a'', ''b'', ''c'' ∈ ''G'', we say that ''(G, ≤)'' is a
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* '''left-ordered group''' if ''a''&nbsp;≤&nbsp;''b'' implies ''c a''&nbsp;≤&nbsp;''c b''
* '''bi-ordered group''' if ''a''&nbsp;≤&nbsp;''b'' implies ''c a''&nbsp;≤&nbsp;''c b'' and ''a c''&nbsp;≤&nbsp;''b c''
 
In analogy with ordinary numbers, we call an element ''c'' of an ordered group '''positive''' if 0&nbsp;≤&nbsp;''c'' and ''c''&nbsp;≠&nbsp;0, where "0" here denotes the [[identity element]] of the group (not necessarily the familiar zero of the real numbers). The set of positive elements in a group is often denoted with ''G''<sub>+</sub>.<ref>Note that the + is written as a subscript, to distinguish from ''G''<sup>+</sup> which includes the identity element. See e.g. [http://www.nongnu.org/isarmathlib/IsarMathLib/document.pdf IsarMathLib], p. 344.</ref>
 
For every element ''a'' of a linearly ordered group ''G'' either ''a''&nbsp;∈&nbsp;''G''<sub>+</sub>, or '' -a'' &nbsp;∈&nbsp;''G''<sub>+</sub>, or ''a''&nbsp;=&nbsp;0. If a linearly ordered group ''G'' is not trivial (i.e. 0 is not its only element), then ''G''<sub>+</sub> is infinite. Therefore, every nontrivial linearly ordered group is infinite.
 
If ''a'' is an element of a linearly ordered group ''G'', then the [[absolute value]] of ''a'', denoted by |''a''|, is defined to be:
 
:<math>|a|:=\begin{cases}a, & \text{if }a\geqslant0,\\ -a, & \text{otherwise}.\end{cases}</math>
 
If in addition the group ''G'' is [[abelian group|abelian]], then for any ''a'',&nbsp;''b''&nbsp;∈&nbsp;''G'' the [[triangle inequality]] is satisfied: |''a''&nbsp;+&nbsp;''b''| ≤&nbsp;|''a''|&nbsp;+&nbsp;|''b''|.
 
==Examples==
 
 
Any totally ordered group is [[Torsion (algebra)|torsion-free]]. Conversely, [[F. W. Levi]] showed that an [[abelian group]] admits a linear order if and only if it is torsion free {{harv|Levi|1942}}.  
 
[[Otto Hölder]] showed that every bi-ordered group satisfying an [[Archimedean property]] is [[isomorphism|isomorphic]] to a subgroup of the additive group of [[real number]]s, {{harv|Fuchs|Salce|2001|p=61}}.
If we write the archimedean l.o. group multiplicatively, this may be shown by considering the dedekind completion, <math>\widehat{G}</math> of the closure of an  l.o. group under <math>n</math>th roots. We endow this space with the usual topology of a linear order, and then it can be shown that for each <math>g\in\widehat{G}</math> the exponential maps <math>g^{\cdot}:(\mathbb{R},+)\to(\widehat{G},\cdot) :\lim_{i}q_{i}\in\mathbb{Q}\mapsto \lim_{i}g^{q_{i}}</math> are well defined order preserving/reversing, topological group isomorphisms. Completing an l.o. group can be difficult in the non-archimedean case. In these cases, one may classify a group by its rank: which is related to the order type of the largest sequence of convex subgroups.
 
A large source of examples of left-orderable groups comes from groups acting on the real line by order preserving homeomorphisms. Actually, for countable groups, this is known to be a characterization of left-orderability, see for instance {{harv|Ghys|2001}}.
 
==See also==
*[[Cyclically ordered group]]
 
==Notes==
<references />
 
==References==
 
*{{Citation | last1=Levi | first1=F.W. | title=Ordered groups. | journal=Proc. Indian Acad. Sci. | year=1942 | volume=A16 | pages=256–263}}
 
*{{Citation | last1=Fuchs | first1=László | last2=Salce | first2=Luigi | title=Modules over non-Noetherian domains | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Mathematical Surveys and Monographs | isbn=978-0-8218-1963-0 | id={{MathSciNet | id = 1794715}} | year=2001 | volume=84}}
 
*{{Citation | last1=Ghys | first1=É. | title=Groups acting on the circle. | journal=L´Eins. Math. | year=2001 | volume=47 | pages=329–407}}
 
[[Category:Ordered groups]]

Latest revision as of 17:36, 10 January 2015

Msvcr71.dll is an significant file which assists support Windows process different components of the program including important files. Specifically, the file is chosen to aid run corresponding files inside the "Virtual C Runtime Library". These files are important inside accessing any settings which support the different applications and programs inside the system. The msvcr71.dll file fulfills many significant functions; though it's not spared from getting damaged or corrupted. Once the file gets corrupted or damaged, the computer can have a difficult time processing and reading components of the program. However, users need not panic because this problem might be solved by following many procedures. And I usually show you certain tips about Msvcr71.dll.

Carry out window's program restore. It is pretty important to do this considering it removes incorrect changes that have happened in the program. Some of the errors outcome from inability of the program to create restore point regularly.

So what must you look for whenever we compare registry cleaners. Many of the registry cleaners accessible today, have really synonymous qualities. The principal ones which you should be searching for are these.

If which does not work you need to try plus repair the problem with a 'registry cleaner'. What arises on several computers is the fact that their registry database becomes damaged plus unable to show a computer where the DLL files which it requirements are. Every Windows PC has a central 'registry' database that stores information regarding all of the DLL files on a computer.

So to fix this, we just have to be able to make all of the registry files non-corrupted again. This may dramatically accelerate the loading time of your computer and usually allow we to a large amount of aspects on it again. And fixing these files couldn't be easier - we just have to employ a tool called a tuneup utilities 2014.

You must furthermore see with it that it is quite easy to download plus install. We could avoid those products that can require you a extremely complicated set of instructions. Additionally, you should no longer want any alternative program requirements.

When the registry is corrupt or full of mistakes, the signs is felt by the computer owner. The slow performance, the frequent program crashes and the nightmare of all computer owners, the blue screen of death.

You are able to click here to locate out how to accelerate Windows and grow PC perfomance. And you can click here to download a registry cleaner to aid you clean up registry.

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