|
|
| Line 1: |
Line 1: |
| {{cleanup|date=April 2011}}
| | There are some methods in which you buy traffic to match your website. To reach more visitors to your site very rapidly there is actually option. [http://www.trafficfaze.com/ Buy website traffic] through the pay per click (buy web site traffic) applications. The top services from and you'll discover more traffic are Bing search, google and yahoo search placement. The main advantage to make use of the top PPC ( pay per click) search engines are that individuals can buy website traffic from most used motors like google.<br><br>You also require to take life lightly by finding things that you can relate with them on and attempt to build on those materials. If you enjoy working together with people and having to know them better, this part can be a regarding fun. Nonetheless it can even be demanding when you've to make sure that you set the time aside regular to do the updates and posting of content.<br><br>There is a lot of hosting companies which offer good service but want to consider following three features while selecting an outstanding service vendor.<br><br><br><br>Build rapport with internet users. Before it is possible to get visitors to trust you, you need to get in order to like you first, valid? Use conversational and friendly tone whenever you're writing your copies to easily put customers at no difficulty. Don't forget to ask questions whenever appropriate and empathize specifically when you are writing about the pressing issues of your prospects. Performing so, you will make these people feel which do understand where tend to be coming from and a person are genuinely interested in assisting them through.<br><br>Set up a Community on your website with your banner ad attached at the top. And invite others to link going without and in order to for pretty own sites. Certain you keep that you will maintain the discussion. Purchase also post on other boards with a link to be able to your board of directors.<br><br>People accustomed to add AdSense on the software. But sometime it's the competition level and also the failure of the banner ads will not allow in order to definitely get the amount of hike in web traffic and business. Here email marketing comes into play money-back guarantee effective yet inexpensive marketing technique can deliver tremendous result for your effort.<br><br>Having a website does not guarantee outcomes. Much is to be done after placing a web presence. After all, the World Wide Web is actually a large expanse that without SEO, customers will not be able to locate stores. Without SEO a questionable income scheme will reside in anonymity.<br><br>As a result, we've got more obtain by sharing information compared to hoarding the item. The most effective means to disseminate know-how is to write articles and publish them on the actual. In return, you may have a higher search engine rank, leads, and new business. If you diligently put your knowledge out into the world, rewards will regurgitate to you in due time. |
| | |
| 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.
| |
| | |
| :<math>\bar{n} \in S \iff (\exists \bar{m} \in \mathbb{N}^{k})(P(\bar{n},\bar{m})=0) </math>
| |
| | |
| 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.
| |
| | |
| [[#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.
| |
| | |
| 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.
| |
| | |
| ==Examples==
| |
| The well known [[Pell equation]]
| |
| | |
| :<math>x^2-d(y+1)^2= 1</math>
| |
| | |
| 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
| |
| | |
| :{0,2,3,5,6,7,8,10,...}
| |
| | |
| consisting of 0 and the natural numbers that are not perfect squares. Other examples of Diophantine definitions are as follows:
| |
| | |
| * The equation <math>a =(2x+3)y</math> only has solutions in <math>\mathbb{N}</math> when a is not a power of 2.
| |
| | |
| * 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]].
| |
| | |
| * The equation <math>a+x=b</math> defines the set of pairs <math>(a\,,\,b)</math> such that <math>a\le b.\,</math>
| |
| | |
| ==Matiyasevich's theorem==
| |
| | |
| Matiyasevich's theorem says:
| |
| | |
| :Every [[recursively enumerable set|computably enumerable set]] is Diophantine.
| |
| | |
| 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>)
| |
| such that an integer ''n'' is in ''S'' if and only if there exist some integers
| |
| ''x''<sub>1</sub>, ..., ''x''<sub>''k''</sub>
| |
| such that ''f''(''n'', ''x''<sub>1</sub>, ..., ''x''<sub>''k''</sub>) = 0.
| |
| | |
| It is not hard to see that every Diophantine set is recursively enumerable:
| |
| consider a Diophantine equation ''f''(''n'', ''x''<sub>1</sub>, ..., ''x''<sub>''k''</sub>) = 0.
| |
| Now we make an algorithm which simply tries all possible values for
| |
| ''n'', ''x''<sub>1</sub>, ..., ''x''<sub>''k''</sub>, in the increasing order of the sum of their absolute values,
| |
| and prints ''n'' every time ''f''(''n'', ''x''<sub>1</sub>, ..., ''x''<sub>''k''</sub>) = 0.
| |
| 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>.
| |
| | |
| ===Proof technique===
| |
| | |
| [[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.
| |
| | |
| ==Application to Hilbert's Tenth problem==
| |
| [[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.
| |
| | |
| ===Refinements===
| |
| | |
| 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).
| |
| | |
| ==Further applications==
| |
| | |
| Matiyasevich's theorem has since been used to prove that many problems from [[calculus]] and [[differential equation]]s are unsolvable.
| |
| | |
| One can also derive the following stronger form of [[Gödel's first incompleteness theorem]] from Matiyasevich's result:
| |
| :''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.''
| |
| | |
| == Notes ==
| |
| <references />
| |
| | |
| ==References==
| |
| | |
| * {{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.
| |
| * {{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 }}
| |
| * {{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 }}
| |
| * {{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 }}
| |
| * {{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 }}
| |
| | |
| == External links ==
| |
| * [http://www.scholarpedia.org/article/Matiyasevich_theorem Matiyasevich theorem] article on [[Scholarpedia]].
| |
| * [http://planetmath.org/encyclopedia/DiophantineSet.html Diophantine Set] article on [[PlanetMath]].
| |
| | |
| [[Category:Diophantine equations]]
| |
| [[Category:Hilbert's problems]]
| |
| | |
| [[fr:Diophantien]]
| |
| [[it:Teorema di Matiyasevich]]
| |
| [[he:הבעיה העשירית של הילברט]]
| |
| [[pt:Teorema de Matiyasevich]]
| |
| [[ru:Диофантово множество]]
| |
| [[zh:丟番圖集]]
| |
There are some methods in which you buy traffic to match your website. To reach more visitors to your site very rapidly there is actually option. Buy website traffic through the pay per click (buy web site traffic) applications. The top services from and you'll discover more traffic are Bing search, google and yahoo search placement. The main advantage to make use of the top PPC ( pay per click) search engines are that individuals can buy website traffic from most used motors like google.
You also require to take life lightly by finding things that you can relate with them on and attempt to build on those materials. If you enjoy working together with people and having to know them better, this part can be a regarding fun. Nonetheless it can even be demanding when you've to make sure that you set the time aside regular to do the updates and posting of content.
There is a lot of hosting companies which offer good service but want to consider following three features while selecting an outstanding service vendor.
Build rapport with internet users. Before it is possible to get visitors to trust you, you need to get in order to like you first, valid? Use conversational and friendly tone whenever you're writing your copies to easily put customers at no difficulty. Don't forget to ask questions whenever appropriate and empathize specifically when you are writing about the pressing issues of your prospects. Performing so, you will make these people feel which do understand where tend to be coming from and a person are genuinely interested in assisting them through.
Set up a Community on your website with your banner ad attached at the top. And invite others to link going without and in order to for pretty own sites. Certain you keep that you will maintain the discussion. Purchase also post on other boards with a link to be able to your board of directors.
People accustomed to add AdSense on the software. But sometime it's the competition level and also the failure of the banner ads will not allow in order to definitely get the amount of hike in web traffic and business. Here email marketing comes into play money-back guarantee effective yet inexpensive marketing technique can deliver tremendous result for your effort.
Having a website does not guarantee outcomes. Much is to be done after placing a web presence. After all, the World Wide Web is actually a large expanse that without SEO, customers will not be able to locate stores. Without SEO a questionable income scheme will reside in anonymity.
As a result, we've got more obtain by sharing information compared to hoarding the item. The most effective means to disseminate know-how is to write articles and publish them on the actual. In return, you may have a higher search engine rank, leads, and new business. If you diligently put your knowledge out into the world, rewards will regurgitate to you in due time.