Motivation: (See the attatchments.) I am not sure where else I can get help.
I want to convert the mathematics in the attatchments into code. This might be the only way to apply the equations in the articles.
Here is the abstract:
Suppose $n\in\mathbb{N}$, $f:\mathbb{R}^{n}\to\mathbb{R}$ is a Borel function and any Borel measure is defined on the Borel $\sigma$-algebra. For all real numbers $a_1<b_1,a_2<b_2,\cdots,a_n<b_n$, suppose that $\mathcal{B}_{n}:=(a_1,b_1)\times(a_2,b_2)\times\cdots\times(a_n,b_n)\subset\mathbb{R}^{n}$ is an $n$-dimensional interval.
We want the mean of explicit, unbounded $f$ (defined without axiom of choice) that is non-Lebesgue integrable on any $\mathcal{B}_n\subset\mathbb{R}^{n}$. The problem is the mean of $f$, w.r.t. the $n$-dimensional Lebesgue measure, is always undefined. Thus, take the average w.r.t. the Hausdorff measure in its dimension of a family of bounded functions with different domains converging to $f$. Nevertheless, depending on the chosen family of bounded functions converging to $f$, the new mean can be one of several values; that is, the new mean is non-unique. Hence, we define a modeling question (its answer, a mathematical model of a real-life phenomenon) that chooses a unique, "satisfying", and finite average of $f$.
The applications in the modeling question include a reference point $\mathbf{R}\in\mathbb{R}^{n+1}$ (i.e., the chosen "center" of the observable universe), the actual rate versus expected rate of expansion of a family of each bounded function's graph (i.e., the rate of expansion of the observable universe), and the "measure" of the chosen families of each bounded function's graph (i.e., the "entropic density" of the observable universe). The "measure" partitions each graph into equal measure sets which take a sample point from each partition, pathways of line segments between sample points, lengths of line segments in each pathway, removed lengths which are outliers, remaining lengths which are converted into a probability distribution, and the entropy of the distribution.
Therefore, assuming $n\in\mathbb{N}$, we want to average the special case of $f:\mathbb{R}^{n}\to\mathbb{R}$ (i.e., $\mathcal{G}:\mathbb{R}^{n}\to\mathbb{R}$) where for all real numbers $a_1<b_1,a_2<b_2,\cdots,a_n<b_n$, supposing that $\mathcal{B}_{n}:=(a_1,b_1)\times(a_2,b_2)\times\cdots\times(a_n,b_n)\subset\mathbb{R}^{n}$ is an $n$-dimensional interval:
We want explicit, unbounded $\mathcal{G}$ (defined without the axiom of choice) such that:
- The restriction of $\mathcal{G}$ to any $n$-dimensional interval has infinite $(n+1)$-dimensional volume both above and below $v=0$, where $v:\mathbb{R}^{n}\to\mathbb{R}$ is a function.
- For all $n$-dimensional intervals $\mathcal{B}_n\subset\mathbb{R}^{n}$ (look above this quote) and $c<d$ real numbers, $\{(x_1,\cdots, x_n)\in\mathcal{B}_n:\mathcal{G}(x_1,\cdots,x_n) \in (c,d)\}$ is a set with positive $n$-dimensional Lebesgue measure.
Here is an example, when $n=1$:
(Example of $\mathcal{G}$) Let $(q_t)_{t\in\mathbb{N}}$ be a numbering of the rational numbers and $k_t=2^{2^t}$, and define functions $s_t$ as follows:
If $s_0=0$ everywhere and: $$\small{ s_{t+1}(x)=\begin{cases} q_{t/2} & x\in\bigcup_{j\in\{1,\cdots,t\}}(q_j-1/k_t,q_j+1/k_t), \, t \text{ is even}\\ k_t^2 & x \in\bigcup_{j\in\{1,\cdots,t\}}(q_j-1/k_t,q_j+1/k_t), \, t \text{ is odd}\\ s_t(x) & \text{otherwise} \end{cases}}$$ each $s_{t+1}$ agrees with $s_t$ at all real numbers except a set of measure $<1/2^t$ for big $t$, so one could consider $\mathcal{G}$ the pointwise limit of the functions $s_t$, which is defined everywhere except measure $0$ (in those bad points just define $\mathcal{G}=0$).
Here is the partial code to define the example:
Clear["Global`*"]
enumerateRationals[n_Integer?Positive] :=
Module[{posRationals,
fullList},(*Generate enough positive rationals using the Calkin-
Wilf step*)
posRationals = NestList[1/(2 Floor[#] - # + 1) &, 1, Ceiling[n/2]];
(*Interleave:0,q1,-q1,q2,-q2...*)
fullList = Riffle[posRationals, -posRationals];
Prepend[fullList, 0][[1 ;; n]]];
enumerateRationals[0] = {};
q[t_] := q[t] = enumerateRationals[t][[t]]
(*Takes the t-value in an enumeration*)
k[t_] := k[t] = 2^(2^t)
s[0, _] = 0;
s[t_Integer, x_] :=
s[t, x] =
Piecewise[{{q[(t - 1)/2],
EvenQ[t - 1] &&
0 <= Min[Abs[x - enumerateRationals[t - 1]]] < 1/k[t - 1]}, {k[
t - 1]^2,
OddQ[t - 1] &&
0 <= Min[Abs[x - enumerateRationals[t - 1]]] < 1/k[t - 1]}},
s[t - 1, x]]
FullSimplify[PiecewiseExpand[s[4, x]]]
I need to compute $\mathcal{G}(x)$ by taking s[t,x] as t->Infinity, but I'm unable to do this with the current code.
Notes on the Remaining Paper:
To obtain a unique, "satisfying", and finite average from $\mathcal{G}(x)$, we use an answer to the modeling question (see abstract or introduction) in Section 3.1 (pg. 28-30). In the longer attatchment, we answer the recent modeling question in Section 4 (pg. 31-43) and represent it with code in Section 4.4.2.1 (pg. 123-158).
Since the explanation of the answer in Section 4 [pg. 44-122] is extremely long, you don't have to read it.
Therefore, after reading most of the papers (excluding certain parts of pg. 44-122) I have the following question:
Question:
When $n=1$, using $\mathcal{G}(x)$ in Section 2.1.1 (pg. 6), Definition 13 (pg. 13-14), and the answer to Section 3.1 (pg. 28-30):
What is $M_{\mathcal{G},\mathbb{R}}((f_r^{\star}),(A_r^{\star});1)$ depending on (see the abstract or introduction) the reference point $\mathbf{R}\in\mathbb{R}^{2}$, the fixed expected rate of expansion $E$, and the actual rate of expansion $\mathcal{E}(\mathbf{R},\mathrm{graph}(f_r^{\star}))$?
(If you want to acknowledge that you've read the attatchments, see this online link.)
Attachments:
