# How do we define the function near the end of my post?

Suppose I have two sets $$F_{1,n}$$ and $$F_{2,n}$$ written as:

F1[t_] :=
F1[t] = Sort[
N[DeleteDuplicates[
Flatten[Sqrt[
Range[Ceiling[((t)!)], Floor[4 (t!)]]/((t!))]]]]]

F2[t_] :=
F2[t] = Sort[
N[DeleteDuplicates[
Flatten[Table[Sqrt[Range[s, 4 s]/s], {s, 1, t}]]]]]


And we have

\[CapitalDelta]F1[t_] := \[CapitalDelta]F1[t] = Differences[F1[t]]
\[CapitalDelta]F2[t_] := \[CapitalDelta]F2[t] = Differences[F2[t]]

H[t_]:=H[t]=t/Total[t]

E1[t_] :=
E1[t] = -Total[
H[\[CapitalDelta]F1[t]] Log[2, H[\[CapitalDelta]F1[t]]]]
E2[t_] :=
E2[t] = -Total[
H[\[CapitalDelta]F2[t]] Log[2, H[\[CapitalDelta]F2[t]]]]

T1[x_] :=
T1[x] = (2^(Mean[F1[x]]) (2^(E1[x + 1] - E1[x]))/(Length[F1[x + 1]]/
Length[F1[x]]))
T2[x_] :=
T2[x] = (2^
Mean[F2[x]] (2^(E2[x + 1] - E2[x]))/(Length[F2[x + 1]]/
Length[F2[x]]))


How do we define:

$$\sum\limits_{n=1}^{z}\sup\left\{|F_{2,j}|:j\in\mathbb{N},T2[j]\ge T1[n]\right\}$$

Attempt

I started with the code:

F2 \@ NSolve[T2[j] > T1[n], {j}, Integers]


but immedietely get:

Range::range: Range specification in Range[16 s,49 s] does not have appropriate bounds.


How fix this and complete the solution? Is there better code to be used?

• I think your syntax is off, what does \@ mean? Apr 17, 2022 at 22:15
• @CATrevillian I mistyped it. It should be @/. Apr 17, 2022 at 22:20
• @CATrevillian Either way, the NSolve part also returned an error. Apr 17, 2022 at 22:22
• What are the definitions for a, b,H? Apr 17, 2022 at 23:54
• @BobHanlon I made edits. Apr 17, 2022 at 23:58