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<p class="MsoNormal"><span>Calculating
information content, example DNA</span></p>
<p class="MsoNormal"><span> </span></p>
<p class="MsoNormal" style="margin-left:36pt"><span><span>1)<span style="font:7pt "times new roman"">
</span></span></span><span>Discussion
about beer cans and living objects</span></p>
<p class="MsoNormal" style="margin-left:18pt"><span>Information is an abstract concept which can only be entertained in the
minds of entities that can deal with abstract concepts, usually humans. The
present discussion about whether inanimate objects can exchange information is
comparable to a discussion about whether plants and animals can master
trigonometry. We see that bees build and snails carry geometrical forms and
ferns form fractals, although no one could entertain the motion that they know
how to calculate, using the appropriate algorithms. </span></p>
<p class="MsoNormal" style="margin-left:18pt"><span>The existence of natural laws and the ability to speak about the
existence of natural laws are two different pairs of shoes. In the following I
shall show how information is a part of Nature, regardless of whether we
recognise it or not, like the laws of trigonometry are a part of Nature.</span></p>
<p class="MsoNormal" style="margin-left:36pt"><span><span><br></span></span></p><p class="MsoNormal" style="margin-left:36pt"><span><span>2)<span style="font:7pt "times new roman"">
</span></span></span><span>Necessary
half-steps</span></p>
<p class="MsoNormal" style="margin-left:18pt"><span>In order to understand the explication that follows, it is necessary
that the reader looks up and understands the following entries in the OEIS
(Online Encyclopaedia of Integer Sequences, <a href="http://www.oeis.org">www.oeis.org</a>)</span></p>
<p class="MsoNormal" style="margin-left:36pt"><span><span>a)<span style="font:7pt "times new roman"">
</span></span></span><span>OEIS/A000142:
the number of permutations of n</span></p>
<p class="MsoNormal" style="margin-left:36pt"><span><span>b)<span style="font:7pt "times new roman"">
</span></span></span><span>OEIS/A000254:
the number of cycles in permutations of n</span></p>
<p class="MsoNormal" style="margin-left:36pt"><span><span>c)<span style="font:7pt "times new roman"">
</span></span></span><span>OEIS/A235647:
the number of cycles when reordering summands (a,b) into summands (b,a)</span></p>
<p class="MsoNormal" style="margin-left:36pt"><span><span>d)<span style="font:7pt "times new roman"">
</span></span></span><span>OEIS/A242615:
the upper limits for the numbers of commutative and sequenced assemblies of n
objects.</span></p>
<p class="MsoNormal" style="margin-left:18pt"><span>It may be helpful to work through the treatise „Natural Orders – de ordinibus
naturalibus” (ISBN <span class="gmail-m-1498043997744764680gmail-st">9783990571378</span>)
to have a general overview of the subject.</span></p>
<p class="MsoNormal" style="margin-left:36pt"><span><span><br></span></span></p><p class="MsoNormal" style="margin-left:36pt"><span><span>3)<span style="font:7pt "times new roman"">
</span></span></span><span>Definition</span></p>
<p class="MsoNormal" style="margin-left:18pt"><span>From “Natural Orders”:</span></p>
<p class="MsoNormal" style="margin-left:18pt"><span>8.3.3.3 <span style="background:yellow none repeat scroll 0% 0%">Information is a description of
what is not the case</span></span>. [Let <i>x = a</i><i><span style="font-size:8pt;line-height:107%">k</span></i><span>. This is a statement, no information
contained. Let <i>x = a</i></span><i><span style="font-size:8pt;line-height:107%">k</span></i><span>
and <i>k </i></span><i><span style="font-size:12pt;font-family:symbol" lang="FR-CA"><span><symbol for is_included_in> </span></span></i><span><i><span style="font-family:symbol" lang="DE-AT"><span></span></span></i></span><i><span> {1,2,...,k,...,n}</span></i><span>. This statement contains the information <i>k </i></span><i><span style="font-size:12pt;font-family:symbol" lang="FR-CA"><span><symbol for is_not_included_in></span></span></i><i><span lang="DE-AT"> </span></i><i><span>{1,2,...,k-<wbr>1,k+1,...,n}</span></i><span>.]</span></p><p class="MsoNormal" style="margin-left:18pt"><span style="font-size:12pt;font-family:"liberation serif",serif">(Sorry for the included & not-included symbols not making it thru the simplified text editor in use here.)</span></p><div class="gmail-yj6qo gmail-ajU"><div id="gmail-:2a3" class="gmail-ajR" tabindex="0"><img class="gmail-ajT" src="https://ssl.gstatic.com/ui/v1/icons/mail/images/cleardot.gif"></div></div>
<p class="MsoNormal" style="margin-left:36pt"><span><span>4)<span style="font:7pt "times new roman"">
</span></span></span><span>Discussion</span></p>
<p class="MsoNormal" style="margin-left:18pt"><span>4.1.) Comparison with the Shannon concept of information</span></p>
<p class="MsoNormal" style="margin-left:18pt"><span>If the alternatives are restricted to {0,1}, the above definition
becomes trivial, as non-0 = 1 and non-1 = 0. Therefore, the information concept
can only be demonstrated, if there are > 2 alternatives.</span></p>
<p class="MsoNormal" style="margin-left:18pt"><span>4.2.) Limited number of alternatives</span></p>
<p class="MsoNormal" style="margin-left:18pt"><span>If the number of alternatives is </span><span>∞</span><span>,
the information content can not be calculated. Therefore, the number of
alternatives must be limited. (n << </span><span>∞</span><span>)</span></p>
<p class="MsoNormal" style="margin-left:18pt"><span>4.3.) Interpretation of a number describing a non-infinite extent</span></p>
<p class="MsoNormal" style="margin-left:18pt"><span>A number is a permutation of a limited number of digits. Let us use for
demonstration purposes a decimal system, where the available digits are
{0,1,2,3,4,5,6,7,8,9}.</span></p>
<p class="MsoNormal" style="margin-left:18pt"><span>The number is traditionally seen as a permutation of digits with the
digits representing a sample with replacement. We traditionally assume, that
any of the available digits can repeatedly appear on any of the positions of
the permutation that yields the number.</span></p>
<p class="MsoNormal" style="margin-left:18pt"><span>4.4.) Samples with a limited number of replacements</span></p>
<p class="MsoNormal" style="margin-left:18pt"><span>We introduce the concept of a hybrid between samples without replacement
and samples with replacement. There are many, but not endlessly many, of the
digits that can appear on any of the positions of the permutation that
constitutes the number.</span></p>
<p class="MsoNormal" style="margin-left:18pt"><span>4.5) Keeping track of the available digits</span></p>
<p class="MsoNormal" style="margin-left:18pt"><span>Assuming a limited number of each of the digits, the total number of
different permutations that can be built from the universe of digits is
limited. We may naively believe that each number can be constructed by a
permutation of samples of digits, because we are used to the concept that the
digits come from samples with replacements, but the number of permutations of
digits that can actually be constructed is severely restricted by the limits on
the number of available digits.</span></p>
<p class="MsoNormal" style="margin-left:18pt"><span>4.6.) Using cycles as limiting factors on the number of digits</span></p>
<p class="MsoNormal" style="margin-left:18pt"><span>Each of the cycles assigns to each of the elements in its corpus one
specific identifying symbol (a digit or a combination of digits). These
identifying symbols are restricted in their number by the number of elements in
the cycle.</span></p>
<p class="MsoNormal" style="margin-left:18pt"><span>4.7.) Information content and degrees of freedom</span></p>
<p class="MsoNormal" style="margin-left:18pt"><span>A limited number of elements can not have an unlimited number of different
sequences, nor can they have an unlimited number of different commutative
relations among each other. The upper limits for both restrict the number of
actually realisable permutations. Those permutations that can not be actually
realised are the information content of the assembly of elements that already
have been sampled. This is analogous to the concept of degrees of freedom.</span></p><p class="MsoNormal" style="margin-left:18pt"><span><br></span></p><p class="MsoNormal" style="margin-left:18pt"><span>Karl<br></span></p>
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