Inequality of arithmetic and geometric means: Difference between revisions

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{{Hatnote|For the theorem about the real analytic Eisenstein series see [[Kronecker limit formula]].}}
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In [[mathematics]], '''Kronecker's theorem''' is either of two theorems named after [[Leopold Kronecker]].
__NOTOC__
== The existence of extension fields ==
This is a theorem stating that a [[polynomial]] in a [[Field (mathematics)|field]], ''p''(''x'')&nbsp;&isin;&nbsp;''F''[''x''], has a root in an extension field <math>E \supset F</math>.<ref>[http://www.usna.edu/Users/math/wdj/book/node77.html Applied Abstract Algebra] by D. Joyner, R. Kreminski and J. Turisco.</ref>
 
For example, a polynomial in the [[Real number|reals]]  such as ''x''<sup>2</sup>&nbsp;+&nbsp;1&nbsp;=&nbsp;0 has two roots, both in the complex field.
 
This theorem is usually credited to Kronecker despite his original reluctance to accept the existence of numbers outside of the rationals;<ref>{{Cite book | last=Allenby | first=R. B. J. T.| title=Rings, fields and groups: an introduction to abstract algebra | date=1983 | publisher=E. Arnold | location=London  | isbn=0-7131-3476-3 | pages=140,141}}</ref> it provides a useful [[Constructivism (mathematics)|construction]] of many sets.
 
== A result in diophantine approximation ==
'''Kronecker's theorem''' may also refer to a result in [[diophantine approximation]]s applying to several [[real number]]s ''x<sub>i''</sub>, for 1 ≤ ''i'' ≤ ''N'', that generalises [[Dirichlet's approximation theorem]] to multiple variables. In terms of physical systems, it has the consequence that planets in circular orbits moving uniformly around a star will, over time, assume all alignments, unless there is an exact dependency between their orbital periods.
 
In the case of ''N'' numbers, taken as a single ''N''-[[tuple]] and point ''P'' of the [[torus]]
 
:''T'' = ''R<sup>N</sup>/Z<sup>N</sup>'',
 
the [[closure (mathematics)|closure]] of the subgroup <''P''> generated by ''P'' will be finite, or some torus ''T&prime;'' contained in ''T''. The original '''Kronecker's theorem''' ([[Leopold Kronecker]], 1884) stated that the [[necessary condition]] for
 
:''T&prime;'' = ''T'',
 
which is that the numbers ''x<sub>i''</sub> together with 1 should be [[linearly independent]] over the [[rational number]]s, is also [[sufficient condition|sufficient]]. Here it is easy to see that if some [[linear combination]] of the ''x<sub>i''</sub> and 1 with non-zero rational number coefficients is zero, then the coefficients may be taken as integers, and a [[character (mathematics)|character]] χ of the group ''T'' other than the [[trivial character]] takes the value 1 on ''P''. By [[Pontryagin duality]] we have ''T&prime;'' contained in the [[Kernel (group theory)|kernel]] of χ, and therefore not equal to ''T''.
 
In fact a thorough use of Pontryagin duality here shows that the whole Kronecker theorem describes the closure of <''P''> as the intersection of the kernels of the χ with
 
:χ(''P'') = 1.
 
This gives an ([[antitone]]) [[Galois connection]] between [[Monogenic semigroup|monogenic]] closed subgroups of ''T'' (those with a single generator, in the topological sense), and sets of characters with kernel containing a given point. Not all closed subgroups occur as monogenic; for example a subgroup that has a torus of dimension ≥ 1 as connected component of the identity element, and that is not connected, cannot be such a subgroup.
 
The theorem leaves open the question of how well (uniformly) the multiples ''mP'' of ''P'' fill up the closure. In the one-dimensional case, the distribution is uniform by the equidistribution theorem.
 
==See also==
* [[Kronecker set]]
* [[Weyl's criterion]]
 
== Notes and references ==
{{Springer|id=k/k055910|title=Kronecker's theorem}}
<references/>
 
[[Category:Diophantine approximation]]
[[Category:Topological groups]]
[[Category:Theorems in abstract algebra]]

Revision as of 10:06, 1 March 2014

A lagging computer is certainly annoying plus is very a headache. Almost every person that utilizes a computer faces this problem certain time or the other. If your computer also suffers from the same problem, you'll find it difficult to continue functioning because normal. In such a condition, the thought, "what must I do to make my PC run faster?" is recurring plus infuriating. There's a answer, still!

You are able to reformat the computer to make it run quicker. This usually reset a computer to when you initially used it. Always remember to back up all files plus programs before carrying this out because this usually remove your files from your database. Remember before we do this we want all of the motorists and installation files and this ought to be a last resort if you are trying to find slow computer tips.

The Windows registry is a system database of info. Windows and additional software store a great deal of settings plus alternative info inside it, and retrieve such info from the registry all time. The registry is furthermore a bottleneck inside that considering it is the heart of the operating program, any issues with it will cause errors and bring the running program down.

If that refuses to work you should try and repair the matter with a 'registry cleaner'. What happens on several computers is the fact that their registry database becomes damaged plus unable to show a computer where the DLL files that it requires are. Every Windows PC has a central 'registry' database that stores information regarding all DLL files on the computer.

After which, I equally purchased the Regtool tuneup utilities Software, and it further secure my computer having system crashes. All my registry problems are fixed, plus I could function peacefully.

Although I constantly use the latest variation of browser, sometimes different extensions and plugins become the cause of errors with my browser plus the system. The same is the story with my browser that was crashing frequently potentially due to the Flash player error.

The 'registry' is just the central database which stores all a settings plus options. It's a absolutely important part of the XP program, which means that Windows is consistently adding and updating the files inside it. The problems happen when Windows actually corrupts & loses a few of these files. This makes your computer run slow, because it tries hard to find them again.

Another significant system you'll wish To get is a registry cleaner. The registry is a huge list of everything installed on your computer, plus Windows references it whenever it opens a program or uses a device attached to your computer. When you delete a program, its registry entry could also be deleted, yet occasionally it's not. A registry cleaner could do away with these aged entries so Windows will search the registry faster. It moreover deletes or corrects any entries that viruses have corrupted.

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