Porosity: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
← Older edit
included example of testing method for porosity
en>Vsmith
→Porosity of rocks: remove non-image, wording
 
Line 1: Line 1:
Let ''k'' be a [[positive integer]]. In [[number theory]], '''Jordan's totient function''' <math>J_k(n)</math> of a positive integer ''n'' is the number of ''k''-tuples of positive integers all less than or equal to ''n'' that form a [[coprime]] (''k''&nbsp;+&nbsp;1)-tuple together with ''n''.  This is a generalisation of Euler's [[totient function]], which is ''J''<sub>1</sub>. The function is named after [[Camille Jordan]].
If you present photography effectively, it helps you look much more properly at the globe around you. It is thus, on these grounds that compel various web service provider companies to integrate the same in their packages too. SEO Ultimate - I think this plugin deserves more recognition than it's gotten up till now. 2- Ask for the designs and graphics that will be provided along with the Word - Press theme. You can easily customize the titles of the posts in Word - Press blog in a way that only title comes in the new post link and not the date or category of posts. <br><br>Most Word - Press web developers can provide quality CMS website solutions and they price their services at reasonable rates. WPTouch is among the more well known Word - Press smartphone plugins which is currently in use by thousands of users. With the free Word - Press blog, you have the liberty to come up with your own personalized domain name. From my very own experiences, I will let you know why you should choose WPZOOM Live journal templates. Moreover, many Word - Press themes need to be purchased and designing your own WP site can be boring. <br><br>Your Word - Press blog or site will also require a domain name which many hosting companies can also provide. The following piece of content is meant to make your choice easier and reassure you that the decision to go ahead with this conversion is requited with rich benefits:. After age 35, 18% of pregnancies will end in miscarriage. Storing write-ups in advance would have to be neccessary with the auto blogs.  If you loved this article and you would such as to get even more information concerning [http://emailsite.de/backup_plugin_6281524 wordpress dropbox backup] kindly browse through the page. Have you heard about niche marketing and advertising. <br><br>It has become a more prevalent cause of infertility and the fertility clinic are having more and more couples with infertility problems. Find more information about Design To Wordpress here. Websites that do rank highly, do so becaue they use keyword-heavy post titles. If you just want to share some picture and want to use it as a dairy, that you want to share with your friends and family members, then blogger would be an excellent choice. The popularity of Word - Press has increased the demand for Word - Press themes and these themes sells like hot cake on the internet. <br><br>Millions of individuals and organizations are now successfully using this tool throughout the world. If you operate a website that's been built on HTML then you might have to witness traffic losses because such a site isn't competent enough in grabbing the attention of potential consumers. However, there are a few other Wordpress plugins also for its development which requires adding files in your Wordpress setup. You should stay away from plugins that are full of flaws and bugs. You can check out the statistics of page of views for your web pages using free tools that are available on the internet.
 
==Definition==
 
Jordan's totient function is [[multiplicative function|multiplicative]] and may be evaluated as
:<math>J_k(n)=n^k \prod_{p|n}\left(1-\frac{1}{p^k}\right) .\,</math>
 
==Properties==
 
* <math>\sum_{d | n } J_k(d) = n^k. \, </math>
which may be written in the language of [[Dirichlet convolution]]s as<ref>Sándor & Crstici (2004) p.106</ref>
: <math>J_k(n) \star 1 = n^k\,</math>
and via [[Möbius inversion formula|Möbius inversion]] as
:<math>J_k(n) = \mu(n) \star n^k</math>.
Since the [[Dirichlet generating function]] of μ is 1/ζ(s) and the
Dirichlet generating function of n<sup>k</sup> is ζ(s-k), the series for
J<sub>k</sub> becomes
:<math>\sum_{n\ge 1}\frac{J_k(n)}{n^s} = \frac{\zeta(s-k)}{\zeta(s)}</math>.
 
* An [[Average order of an arithmetic function|average order]] of ''J''<sub>''k''</sub>(''n'') is 
:<math>\frac{n^k}{\zeta(k+1)}</math>.
* The [[Dedekind psi function]] is
:<math>\psi(n) = \frac{J_2(n)}{J_1(n)}</math>,
and by inspection of the definition (recognizing that each factor in the product
over the primes is a cyclotomic polynomial of p<sup>-k</sup>), the arithmetic functions
defined by <math>\frac{J_k(n)}{J_1(n)}</math> or <math>\frac{J_{2k}(n)}{J_k(n)}</math>
can also be shown to be integer-valued multiplicative functions.
 
* <math>
\sum_{\delta\mid n}\delta^sJ_r(\delta)J_s\left(\frac{n}{\delta}\right) = J_{r+s}(n)
</math> &nbsp; &nbsp; &nbsp;<ref>Holden et al in external links The formula is Gegenbauer's</ref>
 
==Order of matrix groups==
 
The [[general linear group]] of matrices of order ''m'' over '''Z'''<sub>''n''</sub> has order<ref>All of these formulas are from Andrici and Priticari in [[#External links]]</ref>
:<math>
|\operatorname{GL}(m,\mathbf{Z}_n)|=n^{\frac{m(m-1)}{2}}\prod_{k=1}^m J_k(n).
</math>
 
The [[special linear group]] of matrices of order ''m'' over ''Z''<sub>''n''</sub> has order
:<math>
|\operatorname{SL}(m,\mathbf{Z}_n)|=n^{\frac{m(m-1)}{2}}\prod_{k=2}^m J_k(n).
</math>
 
The [[symplectic group]] of matrices of order ''m'' over ''Z''<sub>''n''</sub> has order
:<math>
|\operatorname{Sp}(2m,\mathbf{Z}_n)|=n^{m^2}\prod_{k=1}^m J_{2k}(n).
</math>
 
The first two formulas were discovered by Jordan.
 
==Examples==
 
Explicit lists in the [[OEIS]] are
J<sub>2</sub> in {{OEIS2C|A007434}},
J<sub>3</sub> in {{OEIS2C|A059376}},
J<sub>4</sub> in {{OEIS2C|A059377}},
J<sub>5</sub> in {{OEIS2C|A059378}},
J<sub>6</sub> up to J<sub>10</sub> in {{OEIS2C|A069091}}
up to {{OEIS2C|A069095}}.
       
                       
Multiplicative functions defined by ratios are
J<sub>2</sub>(n)/J<sub>1</sub>(n) in {{OEIS2C|A001615}},
J<sub>3</sub>(n)/J<sub>1</sub>(n) in {{OEIS2C|A160889}},
J<sub>4</sub>(n)/J<sub>1</sub>(n) in {{OEIS2C|A160891}},
J<sub>5</sub>(n)/J<sub>1</sub>(n) in {{OEIS2C|A160893}},
J<sub>6</sub>(n)/J<sub>1</sub>(n) in {{OEIS2C|A160895}},
J<sub>7</sub>(n)/J<sub>1</sub>(n) in {{OEIS2C|A160897}},
J<sub>8</sub>(n)/J<sub>1</sub>(n) in {{OEIS2C|A160908}},
J<sub>9</sub>(n)/J<sub>1</sub>(n) in {{OEIS2C|A160953}},
J<sub>10</sub>(n)/J<sub>1</sub>(n) in {{OEIS2C|A160957}},
J<sub>11</sub>(n)/J<sub>1</sub>(n) in {{OEIS2C|A160960}}.
       
                       
Examples of the ratios J<sub>2k</sub>(n)/J<sub>k</sub>(n) are
J<sub>4</sub>(n)/J<sub>2</sub>(n) in {{OEIS2C|A065958}},
J<sub>6</sub>(n)/J<sub>3</sub>(n) in {{OEIS2C|A065959}},
and
J<sub>8</sub>(n)/J<sub>4</sub>(n) in {{OEIS2C|A065960}}.
 
==Notes==
 
{{reflist}}
 
==References==
 
* {{cite book | author=L. E. Dickson | authorlink=Leonard Eugene Dickson | title=[[History of the Theory of Numbers]], Vol. I | origyear=1919 |year=1971 | publisher=[[Chelsea Publishing]] | isbn=0-8284-0086-5 | jfm=47.0100.04 | page=147 }}
*{{cite book | title=Problems in Analytic Number Theory | author=M. Ram Murty | authorlink=M. Ram Murty  | volume=206 | series=[[Graduate Texts in Mathematics]] | publisher=[[Springer-Verlag]] | year=2001 | isbn=0-387-95143-1 | zbl=0971.11001  | page=11 }}
* {{cite book | last1=Sándor | first1=Jozsef | last2=Crstici | first2=Borislav | title=Handbook of number theory II | location=Dordrecht | publisher=Kluwer Academic | year=2004 | isbn=1-4020-2546-7 | pages=32–36 | zbl=1079.11001 }}
 
==External links==
*{{cite journal|first1=Dorin |last1=Andrica |first2= Mihai |last2=Piticari
|url=http://www.emis.de/journals/AUA/acta7/Andrica%20.pdf
|journal=Acta universitatis Apulensis
|year=2004
|number=7
|mr=2157944
|title= On some Extensions of Jordan's arithmetical Functions
}}
* {{cite web |first1=Matthew |last1=Holden|first2=Michael |last2=Orrison
|first3=Michael |last3=Varble
|url=http://www.math.hmc.edu/~orrison/research/papers/totient.pdf
|title= Yet another Generalization of Euler's Totient Function
}}
 
{{Totient}}
 
[[Category:Number theory]]
[[Category:Modular arithmetic]]
[[Category:Multiplicative functions]]

Latest revision as of 16:20, 6 December 2014

If you present photography effectively, it helps you look much more properly at the globe around you. It is thus, on these grounds that compel various web service provider companies to integrate the same in their packages too. SEO Ultimate - I think this plugin deserves more recognition than it's gotten up till now. 2- Ask for the designs and graphics that will be provided along with the Word - Press theme. You can easily customize the titles of the posts in Word - Press blog in a way that only title comes in the new post link and not the date or category of posts.

Most Word - Press web developers can provide quality CMS website solutions and they price their services at reasonable rates. WPTouch is among the more well known Word - Press smartphone plugins which is currently in use by thousands of users. With the free Word - Press blog, you have the liberty to come up with your own personalized domain name. From my very own experiences, I will let you know why you should choose WPZOOM Live journal templates. Moreover, many Word - Press themes need to be purchased and designing your own WP site can be boring.

Your Word - Press blog or site will also require a domain name which many hosting companies can also provide. The following piece of content is meant to make your choice easier and reassure you that the decision to go ahead with this conversion is requited with rich benefits:. After age 35, 18% of pregnancies will end in miscarriage. Storing write-ups in advance would have to be neccessary with the auto blogs. If you loved this article and you would such as to get even more information concerning wordpress dropbox backup kindly browse through the page. Have you heard about niche marketing and advertising.

It has become a more prevalent cause of infertility and the fertility clinic are having more and more couples with infertility problems. Find more information about Design To Wordpress here. Websites that do rank highly, do so becaue they use keyword-heavy post titles. If you just want to share some picture and want to use it as a dairy, that you want to share with your friends and family members, then blogger would be an excellent choice. The popularity of Word - Press has increased the demand for Word - Press themes and these themes sells like hot cake on the internet.

Millions of individuals and organizations are now successfully using this tool throughout the world. If you operate a website that's been built on HTML then you might have to witness traffic losses because such a site isn't competent enough in grabbing the attention of potential consumers. However, there are a few other Wordpress plugins also for its development which requires adding files in your Wordpress setup. You should stay away from plugins that are full of flaws and bugs. You can check out the statistics of page of views for your web pages using free tools that are available on the internet.

Retrieved from "https://en.formulasearchengine.com/index.php?title=Porosity&oldid=262080"