Unitary transformation: Difference between revisions

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In [[mathematics]], a '''subsequence''' is a [[sequence]] that can be derived from another sequence by deleting some elements without changing the order of the remaining elements. For example, the sequence <math> \langle A,B,D \rangle </math> is a subsequence of <math> \langle A,B,C,D,E,F \rangle </math>. They should not be confused with [[substring]] which is a refinement of subsequence.
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Given two sequences ''X'' and ''Y'', a sequence ''G'' is said to be a ''common subsequence'' of ''X'' and ''Y'', if ''G'' is a subsequence of both ''X'' and ''Y''. For example, if
 
:<math> X = \langle A,C,B,D,E,G,C,E,D,B,G \rangle </math> and
 
:<math> Y = \langle B,E,G,C,F,E,U,B,K \rangle </math>
 
then a common subsequence of ''X'' and ''Y'' could be
 
:<math> G = \langle B,E,E \rangle. </math>
 
This would ''not'' be the ''[[Longest-common subsequence problem|longest common subsequence]]'', since ''G'' only has length 3, and the common subsequence <math> \langle B,E,E,B \rangle </math> has length&nbsp;4. The longest common subsequence of ''X'' and ''Y'' is <math> \langle B,E,G,C,E,B \rangle </math>.
 
==Applications==
Subsequences have applications to [[computer science]],<ref name="substrVsSubseq">In computer science, ''[[string (computer science)|string]]'' is often used as a synonym for ''sequence'', but it is important to note that ''[[substring]]'' and ''subsequence'' are not synonyms. Substrings are ''consecutive'' parts of a string, while subsequences need not be. This means that a substring of a string is always a subsequence of the string, but a subsequence of a string is not always a substring of the string, see: {{cite book | last = Gusfield | first = Dan | origyear = 1997 | year = 1999 | title = Algorithms on Strings, Trees and Sequences: Computer Science and Computational Biology | publisher = Cambridge University Press | location = USA | isbn = 0-521-58519-8 | pages = 4}}</ref> especially in the discipline of [[bioinformatics]], where computers are used to compare, analyze, and store [[DNA]] strands.
 
Take two strands of DNA, say:
 
:ORG<sub>1</sub> = <tt>ACGGTGTCGTGCTATGCTGATGCTGACTTATATGCTA</tt><BR> and
:ORG<sub>2</sub> = <tt>CGTTCGGCTATCGTACGTTCTATTCTATGATTTCTAA</tt>.
 
Subsequences are used to determine how similar the two strands of DNA are, using the DNA bases: [[adenine]], [[guanine]], [[cytosine]] and [[thymine]].
 
==See also==
*[[Subsequential limit]]
*[[Limit superior and limit inferior]]
*[[Longest increasing subsequence problem]]
*[[Erdős–Szekeres theorem]]
 
==Notes==
<references />
 
{{PlanetMath attribution|id=3300|title=subsequence}}
 
[[Category:Elementary mathematics]]
[[Category:Sequences and series]]

Latest revision as of 21:06, 11 January 2015

If you are reading this particular article then I can safely assume you are searching for certain hemorrhoid treatments we can do at house. If you are, then please read on. Let us initially define what hemorrhoids are plus what causes them, after knowing which, then we will discover effective house treatments we may use. Basically hemorrhoids are swollen veins in the anal canal. It is synonymous to varicose veins, however, instead of it being inside the legs it is found in our anal canal.

If you have decided which creams are the external hemorrhoid treatment we would choose then you'd find which they are found in many drug stores plus pharmacies, plus that are relatively inexpensive.

More fiber inside a diet is how to do away with hemorrhoids so which they may remain inactive. Eat plenty of fruits plus greens, and if important take a fiber supplement to be sure you're getting all of the fiber you want. Hemorrhoids thrive off of inactivity, and many persons that are bound to chairs or beds suffer from hemorrhoids. If it is possible for you to be less sedentary, then do so. Getting enough fiber inside the diet and getting normal exercise will greatly lower the hemorrhoids which are creating we suffer.

Venapro is really amazing considering there is not any need for awkward moments in the bathroom wherein you must put cream on the anus. Just spritz it and you're completed. In because brief because 3 days, positive effects are felt.

Another tip which would supply you with some immediate relief to the hemorrhoid problem is chatting a good warm bath. The bath usually sooth the pain we feel, and should you add a small salt into your shower, regarding a teaspoon or thus, and massage the hemorrhoid this may even further help with pain relief.

Witch Hazel, you are able to carry this with you and employ it whenever you're out. Pour a small of it on the bathroom paper and employ it to wash oneself. Not just usually you be cleaner, plus this really is important to heal the hemorrhoids, yet Witch Hazel will actively shrink the hemorrhoids.

It has been proven to be effective by tons of people through success stories inside an underground way of alternative medication. The program contains ingredient resources, charts, audio lessons and generally all which you should eliminate the hemorrhoids once and for all. I absolutely think which the H Miracle System is a awesome recommendation. There are many reviews from persons that have even gotten rid of especially serious hemorrhoid conditions.

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