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:
$$\small{\begin{equation} c=\inf\left\{|1-\mathbf{c_1}|:\forall(\varepsilon>0)\exists(\mathbf{c_1}>0)\forall(r\in\mathbb{N})\exists(v\in\mathbb{N})\left(\left|\frac{F(r)}{G(v)}-\mathbf{c_1}\right|<\varepsilon\right)\right\} \tag{1}\label{eq:1} \end{equation}}$$
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[v, y]/V[r]], {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}).
Note, for the previous cases 1.-4. (see the specific examples at the bottom in case there is confusion):
- Case 11:
{Q11[r,1,2],Q12[r,1,2],Q15[r,1,2],Q16[r,1,2]}=={True,False,False,False}
- Case 12:
{Q11[r,2,1],Q12[r,2,1],Q15[r,2,1],Q16[r,2,1]}=={False,True,False,False}
- Case 15:
{Q11[r,1,2],Q12[r,1,2],Q15[r,1,2],Q16[r,1,2]}=={False,False,True,False}
- Case 16:
{Q11[r,2,1],Q12[r,2,1],Q15[r,2,1],Q16[r,2,1]}=={False,False,False,True}
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).
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})
I have not gotten the output I wanted for all the specific cases.
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