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## 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. A simple illustration of the algorithm is the following expression:

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

The next expression defines the general case better. A real number r is converted into two input parameters n (an integer) and s (a real number) for the $$J_0$$ function.

$$J_1(r) = J_0(n,s)$$ 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. Examples and definitions of the the $$J_0$$ function are available here

## Calculated Examples up to $$J_1(r)$$

$$J_1(0) = f_{\omega}(1)$$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$$J_1(13) = f_{\omega^{\omega^{\omega^{2}.2 + \omega.3 + 3}.2 + \omega^{2}.2}}(4)$$

$$J_1(14) = f_{\omega^{\omega^{\omega^{3} + \omega^{2} + \omega + 1}}}(4)$$

$$J_1(15) = f_{\omega^{\omega^{\omega^{3}.2 + \omega^{2}.2 + \omega.2 + 1}}}(4)$$

$$J_1(16) = f_{\omega^{\omega^{\omega^{2} + 4}.4 + 1}}(5)$$

$$J_1(17) = f_{\omega^{\omega^{\omega^{3}.2 + \omega^{2}.4 + \omega.2 + 4}.4 + \omega^{3}.4 + \omega^{2}.4 + \omega.4 + 4}.4 + \omega^{3}.4 + \omega^{2}.4 + \omega.4 + 4}^{4}(5)$$

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

$$J_1(20) = f_{\omega^{\omega^{\omega^{\omega.3 + 1} + 2}.3 + \omega.4 + 1}.4 + \omega^{3}.4 + \omega^{2}.4 + \omega.4 + 4}^{4}(5)$$

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.
• jLog Function - is defined here
• jString Function - is defined here
• Work In Progress