Moving-average model: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
← Older edit
en>Jack Greenmaven
m Reverted 1 edit by 128.239.152.36 identified as test/vandalism using STiki
en>Frederik159
No edit summary
 
Line 1: Line 1:
In [[mathematics]] the '''Rudin–Shapiro sequence''', also known as the '''Golay–Rudin–Shapiro sequence''' is an infinite [[automatic sequence]] named after [[Marcel J.E. Golay|Marcel Golay]], [[Walter Rudin]] and [[Harold S. Shapiro]], who independently investigated its properties.<ref name=bm>[http://www.joma.org/images/upload_library/22/Ford/Brillhart-Morton854-869.pdf A Case Study in Mathematical Research: The Golay–Rudin–Shapiro Sequence], John Brillhart and Patrick Morton</ref>
It involves expertise and knowledge of various tools and technologies used for creating websites. Also, you may want to opt for a more professioanl theme if you are planning on showing your site off to a high volume of potential customers each day. Should you go with simple HTML or use a platform like Wordpress. In the recent years, there has been a notable rise in the number of companies hiring Indian Word - Press developers. Over a million people are using Wordpress to blog and the number of Wordpress users is increasing every day. <br><br>


==Definition==
Always remember that an effective linkwheel strategy strives to answer all the demands of popular  search engines while reacting to the latest marketing number trends. Some of the Wordpress development services offered by us are:. Some plugins ask users to match pictures or add numbers, and although effective, they appear unprofessional and unnecessary. t need to use the back button or the URL to get to your home page. Aided by the completely foolproof j - Query color selector, you're able to change the colors of factors of your theme a the click on the screen, with very little previous web site design experience. <br><br>Here are a few reasons as to why people prefer Word - Press over other software's. Now if we talk about them one by one then -wordpress blog customization means customization of your blog such as installation of wordpress on your server by wordpress developer which will help you to acquire the SEO friendly blog application integrated with your site design as well as separate blog administration panel for starting up your own business blog,which demands a experienced wordpress designer. Possibly the most downloaded Word - Press plugin, the Google XML Sitemaps plugin but not only automatically creates a site map linking to everyone your pages and posts, it also notifies Google, Bing, Yahoo, and Ask. Nonetheless, with stylish Facebook themes obtainable on the Globe Broad Internet, half of your enterprise is done previously. Have you heard about niche marketing and advertising. <br><br>The next thing I did after installing Wordpress was to find myself a free good-looking Wordpress-theme offering the functionality I was after. And, make no mistake,India's Fertility Clinics and IVF specialists are amongst the best in the world,and have been for some time. Next you'll go by way of to your simple Word - Press site. 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. Word - Press offers constant updated services and products, that too, absolutely free of cost. <br><br>Website security has become a major concern among individuals all over the world. If you treasured this article and you also would like to be given more info with regards to [http://ll.my/wordpressbackupplugin841096 wordpress backup] kindly visit the page. An ease of use which pertains to both internet site back-end and front-end users alike. However, you must also manually approve or reject comments so that your website does not promote parasitic behavior. Page speed is an important factor in ranking, especially with Google. I have never seen a plugin with such a massive array of features, this does everything that platinum SEO and All In One SEO, also throws in the functionality found within SEO Smart Links and a number of other plugins it is essentially the swiss army knife of Word - Press plugins.
Each term of the Rudin–Shapiro sequence is either +1 or &minus;1. The ''n''<sup>th</sup> term of the sequence, ''b''<sub>''n''</sub>, is defined by the rules:
 
:<math>a_n=\textstyle\sum \varepsilon_i \varepsilon_{i+1}</math>
 
:<math>b_n=(-1)^{a_n}</math>
 
where the ε<sub>''i''</sub> are the digits in the binary expansion of ''n''. Thus ''a''<sub>''n''</sub> counts the number of (possibly overlapping) occurrences of the sub-string 11 in the binary expansion of ''n'', and ''b''<sub>''n''</sub> is +1 if ''a''<sub>''n''</sub> is even and &minus;1 if ''a''<sub>''n''</sub> is odd.<ref>{{MathWorld|urlname=Rudin-ShapiroSequence|title=Rudin-Shapiro Sequence}}</ref><ref name=PF42/><ref>Everest et al (2003) p.234</ref>
 
For example, ''a''<sub>6</sub> = 1 and ''b''<sub>6</sub> = &minus;1 because the binary representation of 6 is 110, which contains one occurrence of 11; whereas ''a''<sub>7</sub> = 2  and ''b''<sub>7</sub> = +1 because the binary representation of 7 is 111, which contains two (overlapping) occurrences of 11.
 
Starting at ''n'' = 0, the first few terms of the ''a''<sub>''n''</sub> sequence are:
 
:0, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 1, 1, 1, 2, 3, ... {{OEIS|A014081}}
 
and the corresponding terms ''b''<sub>''n''</sub> of the Rudin–Shapiro sequence are:
 
:+1, +1, +1, &minus;1, +1, +1, &minus;1, +1, +1, +1, +1, &minus;1, &minus;1, &minus;1, +1, &minus;1, ... {{OEIS|A020985}}
 
==Properties==
The Rudin–Shapiro sequence can be generated by a four state [[finite state machine|automaton]].<ref>[http://www.emis.de/journals/SLC/opapers/s30allouche.pdf Finite automata and arithmetic], Jean-Paul Allouche</ref>
 
There is a recursive definition<ref name=PF42>Pytheas Fogg (2002) p.42</ref>
:<math>
\begin{cases}
b_{2n} & = b_n  \\
b_{2n+1} & = (-1)^n b_n
\end{cases}</math>
 
The values of the terms ''a''<sub>''n''</sub> and ''b''<sub>''n''</sub> in the Rudin–Shapiro sequence can be found recursively as follows. If ''n'' = ''m''.2<sup>''k''</sup> where ''m'' is odd then
 
:<math>a_n =
\begin{cases}
a_{(m-1)/4} & \text{if } m = 1 \mod 4 \\
a_{(m-1)/2} + 1 & \text{if } m = 3 \mod 4
\end{cases}</math>
 
:<math>b_n =
\begin{cases}
b_{(m-1)/4} & \text{if } m = 1 \mod 4 \\
-b_{(m-1)/2} & \text{if } m = 3 \mod 4
\end{cases}</math>
 
Thus ''a''<sub>108</sub> = ''a''<sub>13</sub> + 1 = ''a''<sub>3</sub> + 1 = ''a''<sub>1</sub> + 2 = ''a''<sub>0</sub> + 2 = 2, which can be verified by observing that the binary representation of 108, which is 1101100, contains two sub-strings 11. And so ''b''<sub>108</sub> = (&minus;1)<sup>2</sup> = +1.
 
The Rudin-Shapiro word +1 +1 +1 &minus;1 +1 +1 &minus;1 +1 +1 +1 +1 &minus;1 &minus;1 &minus;1 +1 &minus;1 ..., which is created by concatenating the terms of the Rudin–Shapiro sequence, is a fixed point of the morphism or [[string substitution]] rules
 
:+1 +1  '''→'''  +1 +1 +1 &minus;1
:+1 &minus;1  '''→'''  +1 +1 &minus;1 +1
:&minus;1 +1  '''→'''  &minus;1 &minus;1 +1 &minus;1
:&minus;1 &minus;1  '''→'''  &minus;1 &minus;1 &minus;1 +1
 
as follows:
 
:+1 +1 '''→''' +1 +1 +1 &minus;1 '''→''' +1 +1 +1 &minus;1 +1 +1 &minus;1 +1 '''→''' +1 +1 +1 &minus;1 +1 +1 &minus;1 +1 +1 +1 +1 &minus;1 &minus;1 &minus;1 +1 &minus;1 ...
 
It can be seen from the morphism rules that the Rudin–Shapiro string contains at most four consecutive +1s and at most four consecutive &minus;1s.
 
The sequence of partial sums of the Rudin–Shapiro sequence, defined by
 
:<math>s_n = \sum_{k=0}^n b_k \, ,</math>
 
with values
 
:1, 2, 3, 2, 3, 4, 3, 4, 5, 6, 7, 6, 5, 4, 5, 4, ... {{OEIS|A020986}}
 
can be shown to satisfy the inequality
 
:<math>\sqrt{\frac{3n}{5}} < s_n < \sqrt{6n} \text{ for } n \ge 1 \, .</math><ref name=bm />
 
==See also==
*[[Shapiro polynomials]]
 
==Notes==
{{reflist}}
{{refbegin}}
{{refend}}
 
==References==
* {{cite book | last1 = Allouche | first1 = Jean-Paul | last2 = Shallit | first2 = Jeffrey | author2-link = Jeffrey Shallit | isbn = 978-0-521-82332-6 | publisher = [[Cambridge University Press]] | title = Automatic Sequences: Theory, Applications, Generalizations | year = 2003 | zbl=1086.11015 }}
* {{cite book | last1=Everest | first1=Graham | last2=van der Poorten | first2=Alf | last3=Shparlinski | first3=Igor | last4=Ward | first4=Thomas | title=Recurrence sequences | series=Mathematical Surveys and Monographs | volume=104 | location=[[Providence, RI]] | publisher=[[American Mathematical Society]] | year=2003 | isbn=0-8218-3387-1 | zbl=1033.11006 }}
* {{cite book | last=Pytheas Fogg | first=N. | others=Editors Berthé, Valérie; Ferenczi, Sébastien; Mauduit, Christian; Siegel, A. | title=Substitutions in dynamics, arithmetics and combinatorics | series=Lecture Notes in Mathematics | volume=1794 | location=Berlin | publisher=[[Springer-Verlag]] | year=2002 | isbn=3-540-44141-7 | zbl=1014.11015 }}
* {{cite book | zbl=0724.11010 | last=Mendès France | first=Michel | chapter=The Rudin-Shapiro sequence, Ising chain, and paperfolding | pages=367–390 | editor1-last=Berndt | editor1-first=Bruce C. | editor1-link=Bruce C. Berndt | editor2-last=Diamond | editor2-first=Harold G. | editor3-last=Halberstam | editor3-first=Heini | editor3-link=Heini Halberstam | editor4-last=Hildebrand | editor4-first=Adolf | title=Analytic number theory. Proceedings of a conference in honor of Paul T. Bateman, held on April 25-27, 1989, at the University of Illinois, Urbana, IL (USA) | series=Progress in Mathematics | volume=85 | location=Boston | publisher=Birkhäuser | year=1990 | isbn=0-8176-3481-9 }}
 
{{DEFAULTSORT:Rudin-Shapiro sequence}}
[[Category:Binary sequences]]

Latest revision as of 11:59, 12 December 2014

It involves expertise and knowledge of various tools and technologies used for creating websites. Also, you may want to opt for a more professioanl theme if you are planning on showing your site off to a high volume of potential customers each day. Should you go with simple HTML or use a platform like Wordpress. In the recent years, there has been a notable rise in the number of companies hiring Indian Word - Press developers. Over a million people are using Wordpress to blog and the number of Wordpress users is increasing every day.

Always remember that an effective linkwheel strategy strives to answer all the demands of popular search engines while reacting to the latest marketing number trends. Some of the Wordpress development services offered by us are:. Some plugins ask users to match pictures or add numbers, and although effective, they appear unprofessional and unnecessary. t need to use the back button or the URL to get to your home page. Aided by the completely foolproof j - Query color selector, you're able to change the colors of factors of your theme a the click on the screen, with very little previous web site design experience.

Here are a few reasons as to why people prefer Word - Press over other software's. Now if we talk about them one by one then -wordpress blog customization means customization of your blog such as installation of wordpress on your server by wordpress developer which will help you to acquire the SEO friendly blog application integrated with your site design as well as separate blog administration panel for starting up your own business blog,which demands a experienced wordpress designer. Possibly the most downloaded Word - Press plugin, the Google XML Sitemaps plugin but not only automatically creates a site map linking to everyone your pages and posts, it also notifies Google, Bing, Yahoo, and Ask. Nonetheless, with stylish Facebook themes obtainable on the Globe Broad Internet, half of your enterprise is done previously. Have you heard about niche marketing and advertising.

The next thing I did after installing Wordpress was to find myself a free good-looking Wordpress-theme offering the functionality I was after. And, make no mistake,India's Fertility Clinics and IVF specialists are amongst the best in the world,and have been for some time. Next you'll go by way of to your simple Word - Press site. 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. Word - Press offers constant updated services and products, that too, absolutely free of cost.

Website security has become a major concern among individuals all over the world. If you treasured this article and you also would like to be given more info with regards to wordpress backup kindly visit the page. An ease of use which pertains to both internet site back-end and front-end users alike. However, you must also manually approve or reject comments so that your website does not promote parasitic behavior. Page speed is an important factor in ranking, especially with Google. I have never seen a plugin with such a massive array of features, this does everything that platinum SEO and All In One SEO, also throws in the functionality found within SEO Smart Links and a number of other plugins it is essentially the swiss army knife of Word - Press plugins.

Retrieved from "https://en.formulasearchengine.com/index.php?title=Moving-average_model&oldid=262762"