User blog:B1mb0w/The (new) J Function

The J Function
The J function is a reasonably fast growing function that will be progressively modified to simplify its presentation and increase its rate of growth. This blog will explain the structure of the function and will give various calculated values. The J Function will then be used by a brand new version of the Alpha function. Click here for more information about the Alpha Function. You can also Click here to see an out-dated version of this function.

Introduction
For an introduction, it will be useful to refer to my other blogs on the Strong D function. The following notation rules will also help to understand the behaviour of Strong D Functions and the basic structure of the J Function.

Notation:

\(D(m_{[x]}) = D(m_1,m_2,...,m_x)\)

\(D(m_{[x]},n_{[y]}) = D(m_1,m_2,...,m_x,n_1,n_2,...,n_y)\)

\(D(1,0_{[y]}) = D(D(1_{[y]})_{[y]})\)

Definition
The J Function is defined recursively as follows:

\(J(n) = D(n) = n+1\)

\(J(m,n) = D(n_{[D(m)]}) = D(n_1,n_2,...,n_{D(m)})\)

\(J(k,m,n) = D(n_{[D(m_{[D(k)]})]}) = D(n_1,n_2,...,n_{J(k,m)})\)

Basic J Function Examples:

\(J(0,n) = D(n_{[D(0)]}) = D(n_1) = D(n)\)

\(J(1,0) = J(m,0) = J(k,m,0) = D(0_{[D(m_{[D(k)]})]}) = D(0_1,0_2,...,0_{J(k,m)}) = D(0) = 1\) Because of the Leading Zero rule: L1

\(J(1,n) = D(n_{[D(1)]}) = D(n,n) >> f_{\omega}(n-1)\)

\(J(2,0) = D(4,0,0,0)\)

\(J(n,n-1,n) = D(n,0_{[n-1]},n) >> f_{\omega^2}(n)\)

\(J(1,2,3;4,5,6;7,8,9;...;...;...)\)

J Function behaviour follows similar rules to the Strong D Function. The rules are as follows:

\(J(k,m,n) = D(k,0_{[m]},n)\)

\(J(k,m,n) = J(k-1,m,J(k,m,n-1))\)

\(J(k,m,0) >> J(k-1,m,J(k-1,m,k))\) Explanation will be provided

\(J(1,m,0) >> J(J(1,m-1,2^m-1),m-1,J(1,m-1,2^m-1))\) Explanation will be provided

Note that:

\(J(0,m,n) = J(0,0,n) = J(n) = D(n)\) and

\(J(0,0,0) = J(0) = D(0) = 1\)

When more than one 3-tuple is used we get more complex behaviour. The general rule is:

\(J(f,g,h;k,m,n) = J(f,g,h;k-1,m,J(f,g,h;k,m,n-1))\)

\(J(h;0,0,n) = J(h;n) = J(h;J(h;n-1))\)

and

\(J(h;0,0,0) = J(h-1;J(h-1;h,h,h),J(h-1;h,h,h),J(h-1;h,h,h))\)

and

\(J(1;0,0,0) = J(0;J(0;1,1,1),J(0;1,1,1),J(0;1,1,1))\)

\(>> f_{\omega^2}(J(1,1,1)) = f_{\omega^2}(D(1,0,1)) >> f_{\omega^2}(f_3(6))\)

Calculated Examples up to J(1;0,0,n)
\(J(1;0,0,1) = J(1;1) = J(1;J(1;0)) >> f_{\omega^2}(f_{\omega^2}(f_3(6))) >> f_{\omega^2}^2(f_3(6))\)

and

\(J(1;0,0,n-1) >> f_{\omega^2}^{n}(f_3(6)) >> f_{\omega^2+1}(n)\) when \(n < f_3(6)\)

or

\(J(1;0,0,n) >> f_{\omega^2+1}(n)\) when \(n < f_{\omega^2}(f_3(6))\)

Calculated Examples up to J(1;n,0,n)
\(J(1;1,0,0) = J(1;0,0,J(1;0,0,1)) >> f_{\omega^2+1}(f_{\omega^2}^2(f_3(6)))\)

and

\(J(1;1,0,1) = J(1;0,0,J(1;1,0,0)) >> f_{\omega^2+1}^2(f_{\omega^2}^2(f_3(6)))\)

and

\(J(1;1,0,n-1) >> f_{\omega^2+1}^n(n) = f_{\omega^2+2}(n)\) when \(n < f_{\omega^2}^2(f_3(6))\)

or

\(J(1;1,0,n) >> f_{\omega^2+2}(n)\) when \(n < f_{\omega^2+1}(f_{\omega^2}^2(f_3(6)))\)

then

\(J(1;2,0,0) = J(1;1,0,J(1;1,0,2)) >> f_{\omega^2+2}(f_{\omega^2+1}^2(f_{\omega^2}^2(f_3(6))))\)

and

\(J(1;2,0,n) >> f_{\omega^2+3}(n)\)

and

\(J(1;3,0,n) >> f_{\omega^2+4}(n)\)

then

\(J(1;n,0,n) >> f_{\omega^2+\omega}(n)\)

Calculated Examples up to J(1;n,n,n)
\(J(1;1,1,0) = J(1;J(1;1,0,1),0,J(1;1,0,1)) >> f_{\omega^2+\omega}(J(1;1,0,1))\)

and

\(J(1;1,1,1) = J(1;1,1,J(1;1,1,0)) >> f_{\omega^2+\omega}^2(J(1;1,0,1))\)

then

\(J(1;1,1,n) >> f_{\omega^2+\omega}^n(n) = f_{\omega^2+\omega+1}(n)\)

and

\(J(1;n,1,n) >> f_{\omega^2+\omega+n}(n)\) = f_{\omega^2+\omega.2}(n)\)

then

\(J(1;1,2,n) >> f_{\omega^2+\omega.2}^n(n) = f_{\omega^2+\omega.2+1}(n)\)

and

\(J(1;n,2,n) >> f_{\omega^2+\omega.2+n}(n)\) = f_{\omega^2+\omega.3}(n)\)

then

\(J(1;n,3,n) >> f_{\omega^2+\omega.4}(n)\)

and

\(J(1;n,n,n) >> f_{\omega^2+\omega.n}(n) >> f_{\omega^2.2}(n)\)

Calculated Examples up to J(n;0,0,0)
\(J(2;0,0,0) = J(1;J(1;2,2,2),J(1;2,2,2),J(1;2,2,2)) >> f_{\omega^2.2}(J(1;2,2,2))\)

and

\(J(2;0,0,1) = J(2;0,0,J(2;0,0,0)) >> f_{\omega^2.2}^2(J(1;2,2,2))\)

then

\(J(2;0,0,n) >> f_{\omega^2.2+2}(n)\)

and

\(J(2;n,0,n) >> f_{\omega^2.2+\omega}(n)\)

and

\(J(2;n,n,n) >> f_{\omega^2.3}(n)\)

or

\(J(3;0,0,0) >> f_{\omega^2.3}(n)\)

then

\(J(n-1;n,n,n) >> f_{\omega^2.n}(n)\)

or

\(J(n;0,0,0) >> f_{\omega^3}(n)\)

Calculated Examples up to J(1,0,n;n,n,n)
If we return to the general rule:

\(J(f,g,h;k,m,n) = J(f,g,h;k-1,m,J(f,g,h;k,m,n-1))\)

then

\(J(1,0,0;k,m,n) = J(1,0,0;J(1,0,0;k,m,n-1),J(1,0,0;k,m,n-1),J(1,0,0;k,m,n-1))\)

and

\(J(1,0,0;0,0,0) = J(0,0,1;Z,Z,Z) = J(1;Z,Z,Z)\)

where \(Z = J(0,0,1;D(1,0),D(1,0),D(1,0))\) Explanation will be provided

or \(Z = J(1;D(1,0),D(1,0),D(1,0)) = J(1;3,3,3)\)

then

\(J(1,0,0;0,0,0) = J(1;J(1;3,3,3),J(1;3,3,3),J(1;3,3,3))\)

\(>> f_{\omega^2.2}(J(1;3,3,3))\)

and

\(J(1,0,0;0,0,1) >> f_{\omega^2.2}^2(J(1;3,3,3))\)

\(J(1,0,0;0,0,n) >> f_{\omega^2.2+1}(n)\)

\(J(1,0,0;n,0,n) >> f_{\omega^2+\omega}(n)\)

\(J(1,0,0;n,n,n) >> f_{\omega^3}(n)\)

then

\(J(1,0,1;0,0,0) >> J(1,0,0;J(1,0,0;4,4,4),J(1,0,0;4,4,4),J(1,0,0;4,4,4))\) Explanation will be provided

\(>> f_{\omega^3}(f_{\omega^3}(4))\)

and

\(J(1,0,1;0,0,n) >> f_{\omega^3+1}(n)\)

\(J(1,0,1;n,0,n) >> f_{\omega^3+\omega}(n)\)

\(J(1,0,1;n,n,n) >> f_{\omega^3+\omega^2}(n)\)

then

\(J(1,0,2;0,0,n) >> f_{\omega^3+\omega^2+1}(n)\)

\(J(1,0,2;n,0,n) >> f_{\omega^3+\omega^2+\omega}(n)\)

\(J(1,0,2;n,n,n) >> f_{\omega^3+\omega^2.2}(n)\)

then

\(J(1,0,3;n,n,n) >> f_{\omega^3+\omega^2.3}(n)\)

and

\(J(1,0,n;n,n,n) >> f_{\omega^3.2}(n)\)

Calculated Examples up to J(2,0,n;n,n,n)
Without proof, calculations should continue with:

\(J(2,0,0;0,0,n) >> f_{\omega^3.2+1}(n)\)

\(J(2,0,0;n,0,n) >> f_{\omega^3.2+\omega}(n)\)

\(J(2,0,0;n,n,n) >> f_{\omega^3.2+\omega^2}(n)\)

and

\(J(2,0,1;n,n,n) >> f_{\omega^3.2+\omega^2.2}(n)\)

\(J(2,0,2;n,n,n) >> f_{\omega^3.2+\omega^2.3}(n)\)

\(J(2,0,h-1;n,n,n) >> f_{\omega^3.2+\omega^2.h}(n)\)

and

\(J(2,0,n;n,n,n) >> f_{\omega^3.3}(n)\)

Calculated Examples up to J(n,n,n;n,n,n)
Without proof, calculations should continue with:

\(J(3,0,n;n,n,n) >> f_{\omega^3.4}(n)\)

\(J(f,0,n;n,n,n) >> f_{\omega^3.f}(n)\)

and

\(J(n,0,n;n,n,n) >> f_{\omega^4}(n)\)

then

\(J(1,1,0;0,0,n) >> f_{\omega^4+1}(n)\)

\(J(1,1,0;n,0,n) >> f_{\omega^4+\omega}(n)\)

\(J(1,1,0;n,n,n) >> f_{\omega^4+\omega^2}(n)\)

\(J(1,1,n;n,n,n) >> f_{\omega^4+\omega^3}(n)\)

\(J(n,1,n;n,n,n) >> f_{\omega^4.2}(n)\)

\(J(n,g,n;n,n,n) >> f_{\omega^4.g}(n)\)

and

\(J(n,n,n;n,n,n) >> f_{\omega^5}(n)\)

Growth Rate of the J Function
\(J(3;0,0,0) >> f_{\omega^3}(3) = f_{\omega^{\omega}}(3)\)

\(J(4,0,4;4,4,4) >> f_{\omega^4}(n) = f_{\omega^{\omega}}(4)\)

\(J(5,5,5;5,5,5) >> f_{\omega^5}(n) = f_{\omega^{\omega}}(5)\)

The growth rate of the J Function therefore appears to be:

\(J(n,n,n;n,n,n;...;n,n,n) >> f_{\omega^{\omega}}(n)\) with n+1 parameters in the J function

Each group of 3 parameters of value n represents one 3-tuple in the J Function. If we use the notation N to represent the 3-tuple n,n,n, then the growth rate of the J function may reach:

\(J(N_{[(n+1)/3]}) >> f_{\omega^{\omega}}(n)\) with n+1 parameters in the J function

\(J(N_{[n]}) = J(N;N;...;N) = J(n,n,n;n,n,n;...;n,n,n) >> f_{\omega^{\omega^2}}(n)\)

Some calculations for n=3
\(J(4,0,1) >> f_{\omega}(3)\)

\(J(3,1,1) >> f_{\omega.2}(3)\)

\(J(3,2,1) >> f_{\omega^2}(3)\)

\(J(3;0,0,0) >> f_{\omega^{\omega}}(3)\)

\(J(3;3,3,3;3,3,3) >> f_{\omega^{\omega.2}}(3)\)

If we use the notation T to represent the 3-tuple 3,3,3, then we can continue:

\(J(3;T;T;T) = J(3;3,3,3;3,3,3;3,3,3) >> f_{\omega^{\omega^2}}(3)\)

\(J(3;T_{[4]}) = J(3;T;T;T;T) >> f_{\omega^{\omega^2+\omega}}(3)\)

\(J(3;T_{[5]}) >> f_{\omega^{\omega^2+\omega.2}}(3)\)

\(J(3;T_{[6]}) >> f_{\omega^{\omega^2.2}}(3)\)

\(J(3;T_{[9]}) >> f_{\omega^{\omega^2.3}}(3) = f_{\omega^{\omega^3}}(3) = f_{\epsilon_0}(3)\)

Comments and Questions
Look forward to any comments and questions. If anybody is interested, the J Function was named by my wife. The full name is the Juki Function.

Cheers B1mb0w.