User blog:B1mb0w/J Function Sandpit J 1

J Function Sandpit \(J_1\)
The J Function is a work in progress. This sandpit defines a function called \(J_1\) which contains ideas that will be used in the final J Function. Click here for the J Function blog.

Definition
\(J_1\) uses an algorithm to generate \(J_0\) functions and ascending ordinal values. The simplest explanation of the algorithm is the following expression:

\(J_1(2^n) = J_0(n,n^n) = f_{\epsilon_0}(n)\)

or

\(J_1(r) = J_0(n,s)\) where n is an integer and r and s can be a real numbers

Example code for the J_1 function is provided at the end of this blog.

Calculated Examples up to \(J_1(r)\)
J_1(0) = f_{\omega}(1)

J_1(1) = f_{\omega}(1)

J_1(2) = f_{1}(2)

J_1(3) = f_{\omega + 1}(2)

J_1(4) = f_{\omega + 2}(3)

J_1(5) = f_{\omega^{\omega}.2}^{2}(3)

J_1(6) = f_{\omega^{\omega.2 + 1}.2 + 1}(3)

J_1(7) = f_{\omega^{\omega^{2} + \omega + 2}.2 + 2}^{2}(3)

J_1(7.5) = f_{\omega^{\omega^{2}.2 + \omega}}(3)

J_1(7.75) = f_{\omega^{\omega^{2}.2 + \omega.2}.2 + \omega.2 + 2}^{2}(3)

J_1(7.99) = f_{\omega^{\omega^{\omega}}}(3)

J_1(8) = f_{\omega^{\omega.3} + 3}(4)

J_1(9) = f_{\omega^{\omega^{3}.2 + \omega^{2}.2}}(4)

J_1(10) = f_{\omega^{\omega^{\omega + 3}.3 + \omega^{2}}}(4)

J_1(11) = f_{\omega^{\omega^{\omega.3 + 1} + 2} + \omega.3}(4)

J_1(12) = f_{\omega^{\omega^{\omega^{2} + \omega.2 + 2}.2 + \omega.2}}(4)

J_1(13) = f_{\omega^{\omega^{\omega^{2}.2 + \omega.2}}}(4)

J_1(14) = f_{\omega^{\omega^{\omega^{3}}}}(4)

J_1(15) = f_{\omega^{\omega^{\omega^{3}.2 + \omega^{2}.2}}}(4)

J_1(16) = 5

J_1(17) = f_{\omega^{\omega^{\omega.3 + 1} + 4}}(5)

J_1(18) = f_{\omega^{\omega^{\omega^{3} + \omega^{2}.2 + \omega.2 + 4}}}(5)

J_1(19) = f_{\omega^{\omega^{\omega^{4}.3 + \omega^{3}.3 + \omega^{2}.3 + \omega.4}}}(5)

J_1(20) = f_{\omega^{\omega^{\omega^{\omega + 4}.4 + \omega^{3}.4 + \omega^{2}.4 + 4}.4 + \omega^{3}.4 + \omega^{2}.4 + \omega.4 + 4}.4 + \omega^{3}.4 + \omega^{2}.4 + \omega.4 + 4}^{4}(5)

WORK IN PROGRESS

VBA Program Code
The following is VBA visual basic code and will run as a macro in Microsoft Excel. This function creates a string literal of a \(J_0\) Function. Example results are listed above.

Function j_1(r As Double) As String

Dim i As Integer, d As Double

i = jLog(2, r) + 1

d = i - 1 d = d ^ d d = jReal * (i ^ i - d) + d d = Int(d + 0.5)

j_1 = i & " - " & d j_1 = jString(i, d) End Function

How the Function Works

A description of how the code works will be provided here ... Work in Progress.


 * VBA Constants
 * VBA Data Structures
 * VBA Functions
 * j_1 Function -  This function extracts takes the Log Base 2 from a real number.  The integer part is used to set the \(n\) parameter of the \(J_0\) function.  The decimal part is then re-calibrated against a range up to the value of \(n^n\) and this becomes the s parameter of the \(J_0\) function.
 * Work In Progress