Exploring the inverse mapping from a dynamics to a residue class - inputting a reduced dynamics or partial dynamics and outputting a residue class

Exploring the inverse mapping from a dynamics to a residue class - inputting a reduced dynamics or partial dynamics and outputting a residue class

Citation Author(s):
Wei Ren
Submitted by:
Wei Ren
Last updated:
Thu, 11/08/2018 - 10:34
DOI:
10.21227/qmzw-gn71
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Abstract: 

We study a reverse problem - given a reduced dynamics or partial dynamics, can we compute a residue class who presents that dynamics.

We design a dedicated algorithm that takes as input a dynamics with length t consists of “I” or “O” can output a residue class who present this dynamics in the first t transformations. We thus design computer program that can output a reside class by inputting a reduced dynamics or partial dynamics. That is, inputting c∈{I,O}^L,  CntO(s) ≤ ceil( CntI(s)*lambda), lambda=ln1.5/ln2=0.58469250, s = Substr(c,1,i), i=1,2,..., L. In other words, the dynamics is above of or cutting ratio line in our proposed Collatz graph.

Note that, the algorithm is quit lightweight and designed from our formal proof of Partition Theorem - We prove that all natural numbers are partitioned regularly corresponding to ongoing dynamics. Given any natural number x that is i module 2^t (i is an odd integer), the first t transformations in terms of I or O can be determined and identical with the first t transformations of of i. Once current value after t (t is greater or equal to 2) transformations of I or O, is less than x, then reduced dynamics of x is obtained. Otherwise, the residue class of x (namely, i module 2^t) can be partitioned into two halves (namely, i module 2^{t+1} and i+2^t module 2^{t+1}), and either half presents I or O in intermediately forthcoming (t+1)-th transformation.

 

Instructions: 

Step 1:  txpo25.c is source code in C that can be compiled into a executable program by any C compiler. It is ANSI C code, thus it can be complied for either Windows or Linux platform.

 

Step 2: Compile txpo25.c, generate txpo25.exe.

 

txpo25.exe is an executable program, and can be executed in any DOS command shell in Windows (if compiled for Windows platform) and computing results (as generated files) will be obtained.

 

Step 3: Run txpo25.exe by inputting argument that specifying a reduced dynamics or a partial dynamics.

 

tpxo25.exe IIOO > tpxo25-IIOO

tpxo25.exe II > tpxo25-II

tpxo25.exe O > tpxo25-O

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[1] Wei Ren, "Exploring the inverse mapping from a dynamics to a residue class - inputting a reduced dynamics or partial dynamics and outputting a residue class ", IEEE Dataport, 2018. [Online]. Available: http://dx.doi.org/10.21227/qmzw-gn71. Accessed: Jul. 09, 2020.
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doi = {10.21227/qmzw-gn71},
url = {http://dx.doi.org/10.21227/qmzw-gn71},
author = {Wei Ren },
publisher = {IEEE Dataport},
title = {Exploring the inverse mapping from a dynamics to a residue class - inputting a reduced dynamics or partial dynamics and outputting a residue class },
year = {2018} }
TY - DATA
T1 - Exploring the inverse mapping from a dynamics to a residue class - inputting a reduced dynamics or partial dynamics and outputting a residue class
AU - Wei Ren
PY - 2018
PB - IEEE Dataport
UR - 10.21227/qmzw-gn71
ER -
Wei Ren. (2018). Exploring the inverse mapping from a dynamics to a residue class - inputting a reduced dynamics or partial dynamics and outputting a residue class . IEEE Dataport. http://dx.doi.org/10.21227/qmzw-gn71
Wei Ren, 2018. Exploring the inverse mapping from a dynamics to a residue class - inputting a reduced dynamics or partial dynamics and outputting a residue class . Available at: http://dx.doi.org/10.21227/qmzw-gn71.
Wei Ren. (2018). "Exploring the inverse mapping from a dynamics to a residue class - inputting a reduced dynamics or partial dynamics and outputting a residue class ." Web.
1. Wei Ren. Exploring the inverse mapping from a dynamics to a residue class - inputting a reduced dynamics or partial dynamics and outputting a residue class [Internet]. IEEE Dataport; 2018. Available from : http://dx.doi.org/10.21227/qmzw-gn71
Wei Ren. "Exploring the inverse mapping from a dynamics to a residue class - inputting a reduced dynamics or partial dynamics and outputting a residue class ." doi: 10.21227/qmzw-gn71