MathGroup Archive 2004

[Date Index] [Thread Index] [Author Index]

Search the Archive

An entropy measure of rational number fractal dimension: d=0.3732201657487591656832916543359

  • To: mathgroup at smc.vnet.net
  • Subject: [mg50315] An entropy measure of rational number fractal dimension: d=0.3732201657487591656832916543359
  • From: Roger Bagula <tftn at earthlink.net>
  • Date: Thu, 26 Aug 2004 06:51:19 -0400 (EDT)
  • Reply-to: tftn at earthlink.net
  • Sender: owner-wri-mathgroup at wolfram.com

  I've done some other work using the Farey tree functions on the interval

[0,1].
In this case because of the singularity at Log[0], I limit
myself to single repetitions of fractions in the interval (0,1], zero
excluded.
As the number on points gets larger this average gets smaller but very 
slowly
so this is an upper bound for the dimension by an entropy measure of 
this type
( Ulam base two information entropy).
It is in the right range for the 3 set, 1/3 sets model:
1/3 rational numbers
1/3 irrational algebraic numbers
1/3 transcendental numbers
where rational numbers never touch transcendental numbers.
This program is a start of 49000 where duplicates are deleted and zero 
is deleted
to deplete that number in the end average.
It is an honest effort to make an estimate of the true fractal dimension 
of the rational numbers.


Mathematica notebook:

(***********************************************************************

                    Mathematica-Compatible Notebook

This notebook can be used on any computer system with Mathematica 3.0,
MathReader 3.0, or any compatible application. The data for the notebook 
starts with the line of stars above.

To get the notebook into a Mathematica-compatible application, do one of 
the following:

* Save the data starting with the line of stars above into a file
  with a name ending in .nb, then open the file inside the application;

* Copy the data starting with the line of stars above to the
  clipboard, then use the Paste menu command inside the application.

Data for notebooks contains only printable 7-bit ASCII and can be
sent directly in email or through ftp in text mode.  Newlines can be
CR, LF or CRLF (Unix, Macintosh or MS-DOS style).

NOTE: If you modify the data for this notebook not in a Mathematica-
compatible application, you must delete the line below containing the 
word CacheID, otherwise Mathematica-compatible applications may try to 
use invalid cache data.

For more information on notebooks and Mathematica-compatible 
applications, contact Wolfram Research:
  web: http://www.wolfram.com
  email: info at wolfram.com
  phone: +1-217-398-0700 (U.S.)

Notebook reader applications are available free of charge from 
Wolfram Research.
***********************************************************************)

(*CacheID: 232*)


(*NotebookFileLineBreakTest
NotebookFileLineBreakTest*)
(*NotebookOptionsPosition[      2931,         92]*)
(*NotebookOutlinePosition[      3774,        119]*)
(*  CellTagsIndexPosition[      3730,        115]*)
(*WindowFrame->Normal*)



Notebook[{
Cell[BoxData[
    \(Clear\ [f, a, b]\)], "Input"],

Cell[BoxData[{
    \(f[a_, b_] := \((\((a/b)\)/\((1 - a/b)\))\) /; 0 <= a/b <= 1/2\), 
    \(f[a_, b_] := \((\((1 - a/b)\)/\((a/b)\))\) /; 1/2 < a/b <= 1\)}], 
  "Input"],

Cell[BoxData[
    \( (*\ symmetrical\ table\ of\ Farey\ rational\ numbers*) \)], "Input"],

Cell[BoxData[
    \(\(a = Table[If[n < m, f[n, m], f[m, n]], {n, 1, 700}, {m, 1, \ 700}]; 
    \)\)], "Input"],

Cell[BoxData[
    \( (*\ members\ only\ once\ with\ zero\ excluded\ interval\ \((0, 1\)]
        \ *) \)], "Input"],

Cell[BoxData[
    \(\(a2 = Delete[Union[Flatten[a]], 1]; \)\)], "Input"],

Cell[BoxData[
    \( (*\ 
      average\ binary\ entropy\ taken\ as\ H0 = 
        \(-Log\) \((base2)\)[1/2]\ to\ give\ a\ Shannon\ information/\ Ulam\ 
          entropy\ based\ dimension*) \)], "Input"],

Cell[BoxData[
    \( (*\ H = \(H0 + Sum[\(-p\)*Log[p]/Log[2], n]/\(n\  : \ d\) = H/H0\)*) 
      \)], "Input"],

Cell[CellGroupData[{

Cell[BoxData[
    \(Dim_entropy = 
      N[\((Apply[Plus, \(-Log[a2]\)*a2/Log[2]]/\(Dimensions[a2]\)[\([1]\)])\)/
          \((\(-Log[1/2]\)/Log[2])\), 20]\)], "Input"],

Cell[BoxData[
    \(0.3732201657487591656832916543359`20\)], "Output"]
}, Open  ]]
},
FrontEndVersion->"Macintosh 3.0",
ScreenRectangle->{{0, 1920}, {0, 1060}},
WindowSize->{1730, 963},
WindowMargins->{{Automatic, 76}, {17, Automatic}},
MacintoshSystemPageSetup->"\<\
00/0001804P000000_@2@?olonh35@9B7`<5:@?l0040004/0B`000003509H04/
02d5X5k/02H20@4101P00BL?00400@0000000000000000010000000000000000
0000000000000002000000@210D00000\>"
]


(***********************************************************************
Cached data follows.  If you edit this Notebook file directly, not using
Mathematica, you must remove the line containing CacheID at the top of 
the file.  The cache data will then be recreated when you save this file 
from within Mathematica.
***********************************************************************)

(*CellTagsOutline
CellTagsIndex->{}
*)

(*CellTagsIndex
CellTagsIndex->{}
*)

(*NotebookFileOutline
Notebook[{
Cell[1709, 49, 49, 1, 27, "Input"],
Cell[1761, 52, 170, 3, 43, "Input"],
Cell[1934, 57, 89, 1, 27, "Input"],
Cell[2026, 60, 110, 2, 27, "Input"],
Cell[2139, 64, 115, 2, 27, "Input"],
Cell[2257, 68, 72, 1, 27, "Input"],
Cell[2332, 71, 204, 4, 27, "Input"],
Cell[2539, 77, 110, 2, 27, "Input"],

Cell[CellGroupData[{
Cell[2674, 83, 168, 3, 27, "Input"],
Cell[2845, 88, 70, 1, 26, "Output"]
}, Open  ]]
}
]
*)




(***********************************************************************
End of Mathematica Notebook file.
***********************************************************************)



Respectfully, Roger L. Bagula

tftn at earthlink.net, 11759Waterhill Road, Lakeside,Ca 92040-2905,tel: 619-5610814 :
URL :  http://home.earthlink.net/~tftn
URL :  http://victorian.fortunecity.com/carmelita/435/ 





  • Prev by Date: Re: Publicon problems converting sample document to LaTeX
  • Next by Date: Re: FindMinimum and the minimum-radius circle
  • Previous by thread: Re: Opposite of ReadList[ ... ,Byte]?
  • Next by thread: Re: An entropy measure of rational number fractal dimension: d=0.3732201657487591656832916543359