```Date: Mar 24, 2013 7:48 PM
Author: ross.finlayson@gmail.com
Subject: Re: Matheology § 224

On Mar 24, 4:16 pm, Virgil <vir...@ligriv.com> wrote:> In article> <ab63154d-ab7e-4a03-be1b-17ac93226...@hd10g2000pbc.googlegroups.com>,>  "Ross A. Finlayson" <ross.finlay...@gmail.com> wrote:>>>>>>>>>> > On Mar 24, 2:51 pm, fom <fomJ...@nyms.net> wrote:> > > On 3/24/2013 4:34 PM, WM wrote:>> > > > On 24 Mrz., 21:29, Virgil <vir...@ligriv.com> wrote:> > > >> In article> > > >> <729f073f-8948-4eb9-991a-2bd249ac5...@c6g2000yqh.googlegroups.com>,>> > > >> A binary tree that contains one path of each positive natural number> > > >> length will necessarily also contain exactly one path of infinite length.>> > > > Like the sequence> > > > 0.1> > > > 0.11> > > > 0.111> > > > ...> > > > that necessarily also contains its limit?>> > A binary tree that contains one path, of all zero-branches, of each> > finite length, will necessarily contain a path of 0-branches of> > infinite length.>> Better is>>     root>       |  \>       0   1>       |  \>       0   1>       |  \>       0   1>       |  \>       0   1>       |  \>       0   1>       |  \>       0   1>       |  \>       0   1>       |  \>>   And so on>> --A binary tree, of all 0's or 1's (0- or 1- branches) with paths ofeach finite length, has a path of infinite length: that is a subtreeof the intersection[*], of the finite paths.  A tree with paths ofeach length n having 0's except the last 1 shows a binary tree with aninfinite path:  that is a subtree of the union of the finite paths.As a space-complexity problem it's simpler to maintain theintersection[*] than the union.* [ To be sure, to well-define intersection (i.e. to keep it true), itis as the union where the existence of a disjoint leaves the resultundefined.  Basically this of the difference among unary andinfinitary, and binary and n-ary.]Basically in entropy coding and Huffman coding there are codes withthe prefix property that codes of arbitrary width can be put into asequence and demarcated courtesy the prefix property.  Then, an ideais that with paths as codes with the prefix property and distinctlength, there would be in the tree a path of each finite length, butthat paths of greater length would have a different prefix than eachpreceding path.https://en.wikipedia.org/wiki/Prefix_codehttps://en.wikipedia.org/wiki/Huffman_codingThen, where Huffman coding is to find the largest alphabet with theprefix property with the least average length of the path, here theidea for dense bounded tree coding is to find an alphabet with theprefix property with one code for each length: with the maximum suffixof the path that is disjoint the other code's paths, that each pair ismutually disjoint.Basically this is about codes with the prefix property (that therewould be a branch from lesser length codes, and the next), thepalindrome property, or otherwise with the prefix property, and thatthe suffix was not following the suffix of another code initialsegment, in reverse as it were.   Basically it is that the reverse ofthe code of length n as path, would have the prefix property, to theinitial segment of length n of any code of greater length.Then, saying that there would always be an infinite path in the unionof paths of each finite length of the binary tree is that there is no"dense bounded" coding as here.  Basically that is a question of whatare the most paths and most lengths (and as to complexity and entropy)there can be with the least amount of infinite paths.Here, again, the intersection[*] of the 0-paths is infinite, the unionof the 0*1-paths has a subtree that is infinite, a distinctdifference: the least number of finite paths, with an infinite path.There is a difference among representations, of unary, infinitary,binary, and n-ary (in spaces), structural difference withconcomitantly various results, as of their variety.Regards,Ross Finlayson
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