Gilbreath's conjecture: Difference between revisions

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In [[mathematics]], the '''Bourbaki–Witt theorem''' in [[order theory]], named after [[Nicolas Bourbaki]] and [[Ernst Witt]], is a basic [[fixed point theorem]] for [[partially ordered set]]s. It states that if ''X'' is a non-empty [[chain complete]] [[poset]], and
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: <math>f : X \to X</math>
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such that
  <li>[http://savantgaming.com/index.php?option=com_kunena&view=topic&catid=5&id=86037&Itemid=0#86066 http://savantgaming.com/index.php?option=com_kunena&view=topic&catid=5&id=86037&Itemid=0#86066]</li>
 
 
: <math>f (x) \geq x</math> for all <math>x,</math>
  <li>[http://www.siksik-immobilier.com/spip.php?article303 http://www.siksik-immobilier.com/spip.php?article303]</li>
 
 
then ''f'' has a [[fixed point (mathematics)|fixed point]]. Such a function ''f'' is called ''inflationary'' or ''progressive''.
  <li>[http://passerelle.ethiopie.free.fr/spip.php?article81/ http://passerelle.ethiopie.free.fr/spip.php?article81/]</li>
 
 
== Special case of a finite poset ==
  <li>[http://affairs.kyu.edu.tw:8080/bbs/upload/viewthread.php?tid=310277&extra= http://affairs.kyu.edu.tw:8080/bbs/upload/viewthread.php?tid=310277&extra=]</li>
 
 
If the poset ''X'' is finite then the statement of the theorem has a clear interpretation that leads to the proof. The sequence of successive iterates,
</ul>
 
: <math> x_{n+1}=f(x_n), n=0,1,2,\ldots, </math>
 
where ''x''<sub>0</sub> is any element of ''X'', is monotone increasing. By the finiteness of ''X'', it stabilizes:
 
: <math> x_n=x_{\infty},\quad</math> for ''n'' sufficiently large.  
 
It follows that ''x''<sub>∞</sub> is a fixed point of ''f''.
 
== Proof of the theorem ==
 
Pick some <math>y \in X</math>. Define a function ''K'' recursively on the ordinals as follows:
 
:<math>\,K(0) = y</math>
 
:<math>\,K( \alpha+1 ) = f( K( \alpha ) ).</math>
 
If <math> \beta </math> is a [[limit ordinal]], then by construction
 
:<math>\{ K( \alpha ) \ : \ \alpha < \beta \}</math>
 
is a chain in ''X''. Define
 
:<math>K( \beta ) = \sup \{ K( \alpha ) \ : \ \alpha < \beta \}.</math>
 
This is now an increasing function from the ordinals into ''X''. It cannot be strictly increasing, as if it were we would have an [[injective function]] from the ordinals into a set, violating [[Hartogs number|Hartogs' lemma]]. Therefore the function must be eventually constant, so for some
 
:<math> \alpha , \ \ K( \alpha+1 ) = K ( \alpha ); </math>
 
that is,
 
:<math>\,f( K( \alpha ) ) = K ( \alpha ).</math>
 
So letting
 
:<math>\,x = K ( \alpha ),</math>
 
we have our desired fixed point. [[Q.E.D.]]
 
==Applications==
The Bourbaki–Witt theorem has various important applications. One of the most common is in the proof that the [[axiom of choice]] implies [[Zorn's lemma]]. We first prove it for the case where ''X'' is chain complete and has no maximal element. Let ''g'' be a choice function on
 
:<math>P(X) - \{ \varnothing \}.</math>
 
Define a function
 
:<math>f : X \to X</math>  
 
by
 
:<math>f(x) = g( \{ y \ : \ y > x \} ).</math>
 
This is allowed as, by assumption, the set is non-empty. Then ''f''(''x'') > ''x'', so ''f'' is an inflationary function with no fixed point, contradicting the theorem.
 
This special case of Zorn's lemma is then used to prove the [[Hausdorff maximality principle]], that every poset has a maximal chain, which is easily seen to be equivalent to Zorn's Lemma.
 
Bourbaki–Witt has other applications. In particular in [[computer science]], it is used in the theory of [[computable function]]s.
It is also used to define recursive data types, e.g. linked lists, in [[domain theory]].
 
==References==
*{{cite journal | author=Nicolas Bourbaki | title=Sur le théorème de Zorn
| journal = Archiv der Mathematik | volume=2:6 | year=1949 | pages=434&ndash;437}}
*{{cite journal | author=Ernst Witt | title=Beweisstudien zum Satz von M. Zorn | journal = [[Mathematische Nachrichten]] | volume=4 | year=1951 | pages=434&ndash;438}}
 
{{DEFAULTSORT:Bourbaki-Witt theorem}}
[[Category:Order theory]]
[[Category:Fixed-point theorems]]
[[Category:Theorems in the foundations of mathematics]]
[[Category:Articles containing proofs]]

Latest revision as of 21:14, 2 October 2014

Abercrombie & Fitch Perth Australia In women

Coffee tastes good nevertheless is it good for you? More than half of Americans are java junkies, yet the average joe doesn know beans about the health effects of our each day brew. In fact, scientists declare there are grounds to claim of which coffee has many Abercrombie & Fitch Perth Australia effects in health some good, some awful. Keep clicking as main coffee researcher Abercrombie And Fitch Perth Wa Dr. The fact is, scientists say there are grounds to claim that coffee has numerous effects on health some good, some bad. Keep hitting as Abercrombie Fitch Australian Stockists leading coffee science tecnistions Dr. Rob van Dam, adjunct asst professor of nutrition at Harvard School of Public Health and an associate professor at Yong Loo Lin School of Medicine in Singapore, filter systems coffee truth from espresso fallacy and be prepared for quite a few jolts.

TRUE OR Fake: Coffee can raise cholesterol TRUE. Coffee beans contain a strong cholesterol raising compound identified as cafestol. If you drink instant java or coffee that's passed through a paper filter, your current brew will contain only a negligible amount of cafestol.

But traditional Turkish, Traditional, Scandinavian coffee and coffee that's well prepared via the French press process Abercrombie Australia Sydney can contain high numbers of cafestol. Studies have shown that drinking a great deal of these kinds of coffee can increase cholesterol levels and the risk of heart attack. In contrast, a large body involving research suggests that regular consumption of filtered coffee does not increase risk for heart disease as well as stroke.

TRUE OR Untrue: Coffee causes Parkinson disease Fake: Studies have shown that men who beverage lots of coffee have a substandard risk for Parkinson's, a mind condition marked by shaking and difficulty to put together movements. Animal studies claim that caffeine prevents the dying of nerve cells that leave the neurotransmitter dopamine, which is a main problem in Parkinson's.

In women, the relationship concerning caffeine intake and Parkinson's may be a bit more complicated. Recent studies suggest that caffeine has a advantageous effect on women not utilizing estrogen replacement therapy, but is not in those who do bring hormones.

TRUE OR Bogus: Coffee causes liver hurt FALSE: Recent evidence suggests that coffee lowers the risk for the liver cirrhosis and liver cancer, though there's no clear outline of the apparent protective outcome. Similarly, concerns that heavy coffee consumption can increase the risk for other forms of cancer are not confirmed in recent, far more rigorous studies.

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