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| | <br><br>Code your site well when trying to use SEO. Is going to a messy Javascript code for content, you may find your site getting much less traffic, because the spiders are incapable of indexing things from a site. Text-free Flash will be entirely overlooked by bugs.<br><br>Web space and bandwidth are increased metabolism important issues. When you choose a hosting provider, you should certainly consider web space and bandwidth certain. Most of the hosting providers offer unlimited bandwidth and unlimited web space. You should not select a hosting companies which gives limited web space or limited information.<br><br>While web traffic is all fine and dandy, the traffic is yield you the most results is targeted web web traffic visitors. Why is this? Well one of the many rules of advertising is to obtain traffic can be targeted in your specific product or corporation. If you don't find targeted traffic, you need to have a difficult time converting your traffic into sales and profit. Internet site that way to try to to this is simply by using a targeted [http://www.trafficfaze.com/ website traffic] spurning campaign.<br><br>Many unethical website owners try to gain search engine visibility through the help of keywords have got nothing in to do with their websites. They place unrelated keywords in a page (such as "sex", the naming of a known celebrity, the hot search topic of the day, thus.) inside a meta tag for a website. The keyword doesn't have everything to do more than page round. However, since the keyword is popular, they think this will boost their visibility. Particular is considered spam via search engines and might result in the page (or sometimes the whole site) to become removed via the search engine listing.<br><br>Different ezines and publications prefer different formats, and various other ways of submitting your work. You may want to write your article in Word format first but then save your work in ASCII (standard) text format too. In addition, you may wish to embed some HTML formatting into your text version as most sites accept some Html page. If the publication requires you to email your work to an editor, commonly prefer in order to email it to them in MS Word framework. If the ezine is self-service (you submit it using a web form), often they require text (with some HTML) or sometimes just plain text sole.<br><br>Your purchase of the internet marketing field might not be as high as the opposite marketing farms. But if you opt for SEO for leads then probably the rate of return on investment will be much better. This is a very cheap option for web marketing and so that the return are usually much greater. This is why opting for SEO for the exact purpose of getting leads will any day be a more rewarding option in order to.<br><br>Let us say, you perform your research and at dusk. The next morning if at all possible do a lot more writing.You will save a considerable time by applying system-tested system.<br><br>As a result, has got more get by sharing information than by hoarding understand it. The most effective means to disseminate education is create articles and publish them on the actual. In return, you can get a higher search engine rank, leads, and new company. If you diligently put your knowledge out in the world, rewards will flow back to you in due time. |
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| In [[mathematics]], a [[Diophantine equation]] is an equation of the form ''P''(''x''<sub>1</sub>, ..., ''x''<sub>''j''</sub>, ''y''<sub>1</sub>, ..., ''y''<sub>''k''</sub>)=0 (usually abbreviated ''P''(''{{overline|x}}'',''{{overline|y}}'')=0 ) where ''P''(''{{overline|x}}'',''{{overline|y}}'') is a polynomial with integer [[coefficient]]s. A '''Diophantine set''' is a [[set (mathematics)|subset]] ''S'' of '''N'''<sup>j</sup> <ref> [http://planetmath.org/encyclopedia/DiophantineSet.html Planet Math Definition]</ref> so that for some [[Diophantine equation]] ''P''(''{{overline|x}}'',''{{overline|y}}'')=0.
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| :<math>\bar{n} \in S \iff (\exists \bar{m} \in \mathbb{N}^{k})(P(\bar{n},\bar{m})=0) </math>
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| That is, a parameter value is in the Diophantine set S [[if and only if]] the associated Diophantine equation is [[Satisfiability|satisfiable]] under that parameter value. Note that the use of natural numbers both in ''S'' and the existential quantification merely reflects the usual applications in computability and model theory. We can equally well speak of Diophantine sets of integers and freely replace quantification over natural numbers with quantification over the integers.<ref> The two definitions are equivalent. This can be proved using [[Lagrange's four-square theorem]]. </ref> Also it is sufficient to assume ''P'' is a polynomial over <math>\mathbb{Q}</math> and multiply ''P'' by the appropriate denominators to yield integer coefficients. However, whether quantification over rationals can also be substituted for quantification over the integers it is a notoriously hard open problem.
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| [[#Matiyasevich.27s theorem| The MRDP theorem]] states that a set of integers is Diophantine if and only if it is [[recursively enumerable set|computably enumerable]]. <ref>The final piece of this result was published in 1970 by Matiyasevich and is thus also known as Matiyasevich's theorem but pedantically speaking Matiyasevich's theorem refers to the representability of exponentiation in Diophantine sets and the mathematical community has moved to calling the equivalence result the MRDP theorem or the Davis-Putnam-Robinson-Matiyasevich theorem after the mathematicians providing key pieces of the theorem.</ref> A set ''S'' is recursively enumerable precisely if there is an algorithm that, when given an integer, eventually halts if that input is a member of ''S'' and otherwise runs forever. This means that the concept of general Diophantine set, apparently belonging to [[number theory]], can be taken rather in logical or recursion-theoretic terms. This is far from obvious, however, and represented the culmination of some decades of work.
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| Matiyasevich's completion of the MRDP theorem settled [[Hilbert's tenth problem]]. [[David Hilbert|Hilbert's]] tenth problem<ref>[[David Hilbert]] posed the problem in his celebrated list, from his 1900 address to the [[International Congress of Mathematicians]].</ref> was to find a general [[algorithm]] which can decide whether a given Diophantine equation has a solution among the integers. While Hilbert's tenth problem is not a formal mathematical statement as such the nearly universal acceptance of the (philosophical) identification of a decision [[algorithm]] with a [[recursive set|total computable predicate]] allows us to use the MRDP theorem to conclude the tenth problem is unsolvable.
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| ==Examples==
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| The well known [[Pell equation]]
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| :<math>x^2-d(y+1)^2= 1</math>
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| is an example of a Diophantine equation with a parameter. As has long been known, the equation has a solution in the unknowns <math>x,y</math> precisely when the parameter <math>d</math> is 0 or not a [[square number|perfect square]]. In the present context, one says that this equation provides a ''Diophantine definition'' of the set
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| :{0,2,3,5,6,7,8,10,...}
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| consisting of 0 and the natural numbers that are not perfect squares. Other examples of Diophantine definitions are as follows:
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| * The equation <math>a =(2x+3)y</math> only has solutions in <math>\mathbb{N}</math> when a is not a power of 2.
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| * The equation <math>a=(x+2)(y+2)</math> only has solutions in <math>\mathbb{N}</math> when a is greater than 1 and is not a [[prime number]].
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| * The equation <math>a+x=b</math> defines the set of pairs <math>(a\,,\,b)</math> such that <math>a\le b.\,</math>
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| ==Matiyasevich's theorem==
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| Matiyasevich's theorem says:
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| :Every [[recursively enumerable set|computably enumerable set]] is Diophantine.
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| A set ''S'' of integers is '''[[recursively enumerable set|computably enumerable]]''' if there is an algorithm that behaves as follows: When given as input an integer ''n'', if ''n'' is a member of ''S'', then the algorithm eventually halts; otherwise it runs forever. That is equivalent to saying there is an algorithm that runs forever and lists the members of ''S''. A set ''S'' is '''Diophantine''' precisely if there is some [[polynomial]] with integer coefficients ''f''(''n'', ''x''<sub>1</sub>, ..., ''x''<sub>''k''</sub>)
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| such that an integer ''n'' is in ''S'' if and only if there exist some integers
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| ''x''<sub>1</sub>, ..., ''x''<sub>''k''</sub>
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| such that ''f''(''n'', ''x''<sub>1</sub>, ..., ''x''<sub>''k''</sub>) = 0.
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| It is not hard to see that every Diophantine set is recursively enumerable:
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| consider a Diophantine equation ''f''(''n'', ''x''<sub>1</sub>, ..., ''x''<sub>''k''</sub>) = 0.
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| Now we make an algorithm which simply tries all possible values for
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| ''n'', ''x''<sub>1</sub>, ..., ''x''<sub>''k''</sub>, in the increasing order of the sum of their absolute values,
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| and prints ''n'' every time ''f''(''n'', ''x''<sub>1</sub>, ..., ''x''<sub>''k''</sub>) = 0.
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| This algorithm will obviously run forever and will list exactly the ''n'' | |
| for which ''f''(''n'', ''x''<sub>1</sub>, ..., ''x''<sub>''k''</sub>) = 0 has a solution | |
| in ''x''<sub>1</sub>, ..., ''x''<sub>''k''</sub>. | |
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| ===Proof technique===
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| [[Yuri Matiyasevich]] utilized a method involving [[Fibonacci number]]s in order to show that solutions to Diophantine equations may [[exponential growth|grow exponentially]]. Earlier work by [[Julia Robinson]], [[Martin Davis]] and [[Hilary Putnam]] had shown that this suffices to show that every [[recursively enumerable set|computably enumerable set]] is Diophantine.
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| ==Application to Hilbert's Tenth problem==
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| [[Hilbert's tenth problem]] asks for a general algorithm deciding the solvability of Diophantine equations. The conjunction of Matiyasevich's theorem with earlier results known collectively as the MRDP theorem implies that a solution to Hilbert's tenth problem is impossible.
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| ===Refinements===
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| Later work has shown that the question of solvability of a Diophantine equation is undecidable even if the equation only has 9 natural number variables (Matiyasevich, 1977) or 11 integer variables ([[Zhi Wei Sun]], 1992).
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| ==Further applications==
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| Matiyasevich's theorem has since been used to prove that many problems from [[calculus]] and [[differential equation]]s are unsolvable.
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| One can also derive the following stronger form of [[Gödel's first incompleteness theorem]] from Matiyasevich's result:
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| :''Corresponding to any given consistent axiomatization of number theory,<ref>More precisely, given a [[arithmetical hierarchy#The arithmetical hierarchy of formulas|<math>\Sigma^0_1</math>-formula]] representing the set of [[Gödel number]]s of [[sentence (mathematical logic)|sentences]] which recursively axiomatize a [[consistency|consistent]] [[theory (mathematical logic)|theory]] extending [[Robinson arithmetic]].</ref> one can explicitly construct a Diophantine equation which has no solutions, but such that this fact cannot be proved within the given axiomatization.''
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| == Notes ==
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| <references />
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| ==References==
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| * {{cite journal| last=Matiyasevich | first=Yuri V. | authorlink=Yuri Matiyasevich | year=1970 |title= Диофантовость перечислимых множеств|trans_title=Enumerable sets are Diophantine | journal=[[Doklady Akademii Nauk SSSR]] | volume=191 | pages=279–282 | language=Russian}} English translation in ''Soviet Mathematics'' '''11''' (2), pp. 354–357.
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| * {{cite journal | last=Davis | first=Martin | authorlink=Martin Davis | title=Hilbert's Tenth Problem is Unsolvable | journal=[[American Mathematical Monthly]] | volume=80 | pages=233–269 | year=1973 | issn=0002-9890 | zbl=0277.02008 }}
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| * {{cite book | first=Yuri V. | last=Matiyasevich | authorlink=Yuri Matiyasevich | title=Hilbert's 10th Problem | others=Foreword by Martin Davis and Hilary Putnam | publisher=MIT Press | isbn=0-262-13295-8 | series=MIT Press Series in the Foundations of Computing | location=Cambridge, MA | year=1993 | zbl=0790.03008 }}
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| * {{cite book | last=Shlapentokh | first=Alexandra | title=Hilbert's tenth problem. Diophantine classes and extensions to global fields | series=New Mathematical Monographs | volume=7 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2007 | isbn=0-521-83360-4 | zbl=1196.11166 }}
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| * {{cite journal | author=Sun Zhi-Wei | url=http://math.nju.edu.cn/~zwsun/12d.pdf | title=Reduction of unknowns in Diophantine representations | journal=Science China Mathematics | volume=35 | number=3 | year=1992 | pages=257–269 | zbl=0773.11077 }}
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| == External links ==
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| * [http://www.scholarpedia.org/article/Matiyasevich_theorem Matiyasevich theorem] article on [[Scholarpedia]].
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| * [http://planetmath.org/encyclopedia/DiophantineSet.html Diophantine Set] article on [[PlanetMath]].
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| [[Category:Diophantine equations]]
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| [[Category:Hilbert's problems]]
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| [[fr:Diophantien]]
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| [[it:Teorema di Matiyasevich]]
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| [[he:הבעיה העשירית של הילברט]]
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| [[pt:Teorema de Matiyasevich]]
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| [[ru:Диофантово множество]]
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| [[zh:丟番圖集]]
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