How do we change the five sign functions, in each criteria of the final code, to get what I want?

bharathk98 · April 3, 2026, 10:22pm

Suppose, we have the following code (i.e., f[r] and g[r] can be any function, where f[r+c]<V[r]<g[r+d] or g[r+d]<V[r]<f[r+c] and c, d are small constants; e.g., between -10 and 10.)

V[r_]:=V[r]=r!+1
f[r]:=f[r]=
g[r_]:=g[r]=
LengthS[r_] := LengthS[r] = {f[r],g[r]}
LengthS1[r_, x_] := LengthS1[r, x] = LengthS[r][[x]]
LengthS2[j_, y_] := LengthS2[j, y] = LengthS[j][[y]]

We approximate the constants, in the code below, using this equation:

If F:\mathbb{N}\to\mathbb{R} and G:\mathbb{N}\to\mathbb{R} are
arbitrary functions, we want to calculate this equation with
Mathematica:

\begin{equation} c=\inf\left\{|1-\mathbf{c_1}|:\forall(\varepsilon>0)\exists(\mathbf{c_1}>0)\forall(r\in\mathbb{N})\exists(v\in\mathbb{N})\left(\left|\frac{F(r)}{G(v)}-\mathbf{c_1}\right|<\varepsilon\right)\right\} \tag{1} \end{equation}

cV1[r_, 1, 2] := cV1[r, 1, 2] = N[Min[Table[
 RealAbs[1 - V[r]/LengthS1[v, x]], {v, 1, 30000}]]]

cV1[r_, 2, 1 ] := cV1[r, 2, 1] = N[Min[Table[
 RealAbs[1 - LengthS1[r, x]]/V[v], {v, 1, 30000}]]] 

cV2[r_, 1, 2] := cV2[r, 1, 2] = N[Min[Table[
 RealAbs[1 - V[r]/LengthS2[v, y]], {v, 1, 30000}]]] 

cV2[r_, 2, 1] := cV2[r, 2, 1] = N[Min[Table[
 RealAbs[1 - LengthS2[r, y]/V[v]], {v, 1, 30000}]]] 

c[r_, 1, 2] := c[r, 1, 2] = N[Min[Table[
 RealAbs[1 - LengthS1[r, x]/LengthS2[v, y]], {v, 1, 30000}]]] 

c[r_, 2, 1] := c[r, 2, 1] = N[Min[Table[
 RealAbs[1 - LengthS2[r, y]/LengthS1[v, x]], {v, 1, 30000}]]]

In case the outputs are incorrect, use Equation \eqref{eq:1} to solve the exact value.

In addition, w.r.t. the inequality f[r+c]<V[r]<g[r+d], we adjust r+c and r+d into r+c1 and r+d1, using P1 (i.e., c1==c+1||c1==c||c1==c-1 and d1==d+1||d1==d||d1==d-1). If this makes no sense, then analyze the code.

P1 = 3 (*P1 can be any constant positive integer*)

Min11[r_, x_] := 
 Min11[r, x] = 
  Max[r1 /. 
    FindInstance[LengthS1[r1, x] <= V[r] < LengthS1[r1 + 1, x], {r1}, 
     PositiveIntegers]]

Min12[r_, x_] := 
 Min12[r, x] = 
  ArgMin[{RealAbs[LengthS1[r2, x] - V[r]], r - P1 <= r2 <= r + P1}, 
   r2, PositiveIntegers]

Min21[r_, y_] := 
 Min21[r, y] = 
  Max[r3 /. 
    FindInstance[LengthS2[r3, y] <= V[r] < LengthS2[r3 + 1, y], {r3}, 
     PositiveIntegers]]

Min22[r_, y_] := 
 Min22[r, y] = 
  ArgMin[{RealAbs[LengthS1[r4, y] - V[r]], r - P1 <= r4 <= r + P1}, 
   r4, PositiveIntegers]

rMin1[r_, x_] := 
 rMin1[r, x] = 
  Min12[r, x] + Sign[Floor[RealAbs[2 r - Min11[r, x] - Min12[r, x]]/2]]

rMin2[r_, y_] := 
 rMin2[r, y] = 
  Min22[r, y] + Sign[Floor[RealAbs[2 r - Min21[r, y] - Min22[r, y]]/2]]

Putting it together, here is what I want:

For any LengthS1[r,x]=f[r+c1] and LengthS2[r,y]==g[r+d1] such that for any function f and g, where f[r+c1]<V[r]<g[r+d1] or g[r+d1]<V[r]<f[r+c1], consider four out of sixteen cases. (I figured out the other cases, so I will not include them):

Case 11:

 LengthS1[rMin1[r, x], x] > V[r] && V[r] > LengthS2[rMin2[r, y], y] && 
 RealAbs[LengthS1[rMin1[r, x], x] - V[r]] >
  RealAbs[LengthS2[rMin2[r, y], y] - V[r]]

and extra criteria including cV1, cV2, and c (see Equation \ref{eq:1}).

Case 12:

 LengthS1[rMin1[r, y], y] > V[r] && V[r] > LengthS2[rMin2[r, x], x] && 
 RealAbs[LengthS1[rMin1[r, y], y] - V[r]] > 
 RealAbs[LengthS2[rMin2[r, x], x] - V[r]]

and extra criteria including cV1, cV2 and c (see Equation \ref{eq:1})

Case 15:

 LengthS2[rMin2[r, y], y] > V[r] && V[r] > LengthS1[rMin1[r, x], x] &&
  RealAbs[LengthS2[rMin2[r, y], y] - V[r]] >
  RealAbs[LengthS1[rMin1[r, x], x] - V[r]]

and extra criteria including cV1, cV2 and c (see Equation \ref{eq:1})

Case 16:

 LengthS2[rMin2[r, x], x] > V[r] && V[r] > LengthS1[rMin1[r, y], y] &&
  RealAbs[LengthS2[rMin1[r, x], x] - V[r]] >
  RealAbs[LengthS1[rMin2[r, y], y] - V[r]]

and extra criteria including cV1, cV2 and c (see Equation \ref{eq:1}).

In the code below (I assume) there should be five sign functions involved with cV1, cV2, and c in Equation \eqref{eq:1} that are multiplied by c[r,x,y] (in Q11), c[r,y,x] (in Q12), c[r,y,x] (in Q15), and c[r,x,y] (in Q16).

Note: In my research paper, I want the sign functions in (Q11 and Q12) and (Q15 and Q16) to be the same, except that the x and y are swapped. (See the code, for an example of such functions.)

Question: How do I change the five sign functions (in each criteria) to match any example that satisfies the former criteria? (Keep reading for specific examples.)

Here is what I got:

Q11[r_, x_, y_] := 
 Q11[r, x, y] =
       Sign[cV2[r, y, x] - cV1[r, y, x]] Sign[cV1[r, y, x] - cV2[r, x, y]] 
       Sign[cV2[r, x, y] - cV1[r, x, y]] Sign[c[r, x, y] - c[r, y, x]] 
       Sign[cV2[r, x, y] - cV2[r, y, x]] c[r, x, y] 
       >= c[r, y, x] && c[r, y, x] >= cV2[r, y, x] && LengthS1[rMin1[r, x], x] > 
       V[r] && V[r] > LengthS2[rMin2[r, y], y] && RealAbs[LengthS1[rMin1[r, x], x] - V[r]] 
       > RealAbs[LengthS2[rMin2[r, y], y] - V[r]]

Q12[r_, x_, y_] := 
 Q12[r, x, y] =
        Sign[cV2[r, x, y] - cV1[r, x, y]] Sign[cV1[r, x, y] - cV2[r, y, x]] 
        Sign[cV2[r, y, x] - cV1[r, y, x]] Sign[c[r, y, x] - c[r, x, y]] 
        Sign[cV2[r, y, x] - cV2[r, x, y]] c[r, y, x]
         >= c[r, x, y] && c[r, x, y] >= cV2[r, x, y] && LengthS1[rMin1[r, y], y] >
        V[r] && V[r] > LengthS2[rMin2[r, x], x] && RealAbs[LengthS1[rMin1[r, y], y] -V[r]] 
        > RealAbs[LengthS2[rMin2[r, x], x] - V[r]]

Q15[r_, x_, y_] := 
 Q15[r, x, y] = 
       Sign[cV2[r, x, y] - cV1[r, x, y]] Sign[cV1[r, x, y] - cV2[r, y, x]] 
       Sign[cV1[r, y, x] - cV2[r, y, x]] Sign[c[r, y, x] - c[r, x, y]] 
       Sign[cV1[r, x, y] - cV1[r, y, x]] c[r, y, x] 
       >= c[r, x, y] && c[r, x, y] >= cV1[r, y, x] && LengthS2[rMin2[r, y], y] > V[r] && V[r] > 
       LengthS1[rMin1[r, x], x] && RealAbs[LengthS2[rMin2[r, y], y] - V[r]]
       > RealAbs[LengthS1[rMin1[r, x], x] - V[r]]

Q16[r_, x_, y_] := 
 Q16[r, x, y] = 
       Sign[cV2[r, y, x] - cV1[r, y, x]] Sign[cV2[r, y, x] - cV1[r, x, y]] 
       Sign[cV1[r, x, y] - cV2[r, x, y]] Sign[c[r, x, y] - c[r, y, x]] 
       Sign[cV1[r, x, y] - cV1[r, y, x]] c[r,x, y] 
       >= c[r, y, x] && c[r, y, x] >= cV1[r, x, y] && LengthS2[rMin2[r, x], x] > V[r] && V[r] > 
       LengthS1[rMin1[r, y], y] && RealAbs[LengthS2[rMin2[r, x], x] - V[r]] > 
       RealAbs[LengthS1[rMin1[r, y], y] - V[r]]

Examples:

Here are the cases I worked with (they should satisfy the known criteria in the post). Note that r can be any natural number (e.g., r==10). I have not gotten the output I wanted for all the specific cases.

LengthS[r_]:=LengthS[r]={11r!/8+1, 2r!/3+1} (it should return {Q11[r,1,2],Q12[r,1,2],Q15[r,1,2],Q16[r,1,2]}=={True,False,False,False} and {Q11[r,2,1],Q12[r,2,1],Q15[r,2,1],Q16[r,2,1]}=={False,True,False,False})
LengthS[r_]:=LengthS[r]={2r!/3+1, 11r!/8+1} (it should return {Q11[r,1,2],Q12[r,1,2],Q15[r,1,2],Q16[r,1,2]}=={False,False,True,False} and {Q11[r,2,1],Q12[r,2,1],Q15[r,2,1],Q16[r,2,1]}=={False,False,False,True})
LengthS[r_]:=LengthS[r]={4r!/3+1, 7r!/8+1} (it should return {Q11[r,1,2],Q12[r,1,2],Q15[r,1,2],Q16[r,1,2]}=={True,False,False,False} and {Q11[r,2,1],Q12[r,2,1],Q13[r,2,1],Q14[r,2,1]}=={False,True,False,False})
LengthS[r_]:=LengthS[r]={7r!/8+1, 4r!/3+1} (it should return {Q11[r,1,2],Q12[r,1,2],Q15[r,1,2],Q16[r,1,2]}=={False,False,True,False} and {Q11[r,2,1],Q12[r,2,1],Q15[r,2,1],Q16[r,2,1]}=={False,False,False,True})
LengthS[r_]:=LengthS[r]={11r!/8+1, 33r!/50+1} (it should return {Q11[r,1,2],Q12[r,1,2],Q15[r,1,2],Q16[r,1,2]}=={True,False,False,False} and {Q11[r,2,1],Q12[r,2,1],Q15[r,2,1],Q16[r,2,1]}=={False,True,False,False})
LengthS[r_]:=LengthS[r]={33r!/50+1, 11r!/8+1} (it should return {Q11[r,1,2],Q12[r,1,2],Q15[r,1,2],Q16[r,1,2]}=={False,False,True,False} and {Q11[r,2,1],Q12[r,2,1],Q15[r,2,1],Q16[r,2,1]}=={False,False,False,True})
LengthS[r_]:=LengthS[r]={7r!/9+1, 11r!/8+1} (it should return {Q11[r,1,2],Q12[r,1,2],Q15[r,1,2],Q16[r,1,2]}=={True,False,False,False} and {Q11[r,2,1],Q12[r,2,1],Q15[r,2,1],Q16[r,2,1]}=={False,True,False,False})
LengthS[r_]:=LengthS[r]={11r!/8+1, 7r!/9+1} (it should return {Q11[r,1,2],Q12[r,1,2],Q15[r,1,2],Q16[r,1,2]}=={False,False,True,False} and {Q11[r,2,1],Q12[r,2,1],Q15[r,2,1],Q16[r,2,1]}=={False,False,False,True})

gianlucagorni · April 4, 2026, 1:53pm

Reading your first block of code, it seems that the functions LengthS1 and LengthS2 are exactly the same function, with just different names for the variables.

bharathk98 · April 4, 2026, 2:02pm

Yes, that is true, but the variables x or y can be 1 or 2. As long as the numbers in the variables are different, the functions are different.

I define LengthS1 and LengthS2 this way for a special reason. (This would take a long time to explain.)

bharathk98 · April 10, 2026, 4:24pm

Cross-posted: https://mathematica.stackexchange.com/questions/319150/how-do-we-change-the-five-sign-functions-in-each-criteria-of-the-final-code-to

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How do we change the five sign functions, in each criteria of the final code, to get what I want?

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