https://community.wolfram.com RSS Feed for Wolfram Community showing any discussions tagged with Computer-Based Maths sorted by active. https://community.wolfram.com/groups/-/m/t/3780998 &[Wolfram Notebook][1] [1]: https://www.wolframcloud.com/obj/db87a296-2bf6-4bfd-a245-c299a8ee06e8 Alejandro Latorre Chirot 2026-08-14T20:03:30Z https://community.wolfram.com/groups/-/m/t/3781059 &[Wolfram Notebook][1] [1]: https://www.wolframcloud.com/obj/a4214910-c607-425d-b63c-990572539210 Alejandro Latorre Chirot 2026-08-14T14:13:30Z https://community.wolfram.com/groups/-/m/t/3778606 ![Visualize OEIS A001333 Pell-Lucas numbers with NestGraph and NestTree][1] &[Wolfram Notebook][2] [1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=vs.png&userId=23928 [2]: https://www.wolframcloud.com/obj/1c36401b-0aa2-4a40-b331-f17ba4510ee4 Shenghui Yang 2026-08-10T22:26:44Z https://community.wolfram.com/groups/-/m/t/3778637 This is a restatement of [this post][1], since it is difficult to read and [this post][2] since the latter might later be closed. I don't know where else to share. **Motivation:** Consider the function $\mathcal{G}:\mathbb{R}\to\mathbb{R}$, > (**Example of** $\mathcal{G}$) Let $(q_t)_{t\in\mathbb{N}}$ be a [numbering][3] 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 of the example using Mathematica: 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[5, x]]] (*Example of s[t,x]*) (I want to compute `s[t,x]` as `t->Infinity` to get $\mathcal{G}(x)$ or `bigG[x]`.) In the usual sense, the mean of $\mathcal{G}$ with respect to the Lebesgue measure over a family of bounded sets converging to $\mathrm{dom}(\mathcal{G})=\mathbb{R}$ is always undefined, since $\mathcal{G}$ satisfies two properties: 1. The restriction of $\mathcal{G}$ to any interval has infinite area both above and below $y=0$. 2. For all $a<b$ and $c<d$ real numbers, $\{x\in (a,b):\mathcal{G}(x) \in (c,d)\}$ is a set with positive Lebesgue measure. Furthermore, the mean w.r.t. the Hausdorff measure in its dimension of a family of bounded functions (with different bounded domains) converging to $\mathcal{G}$ is defined but has different values depending on the family of bounded functions chosen. Hence, the former mean is non-unique. **Question:** In my concept paper (see the following summary and attatchments), assuming $\mathbf{R}$ is the origin and $E=1/2$, what is the *average* of the example of $\mathcal{G}$ using Mathematica? Is the "average value" non-zero and finite? If not, explain why? > Every reference in the next section refers to the Sections, Definitions, and pages of the concept paper. ## Summary of The Concept Paper## My attempt to average of $\mathcal{G}(x)$ on $\small{(-\infty,+\infty)}$ is defined w.r.t. to four definitions: - The [reference point][6] $\mathbf{R}\in\mathbb{R}^{2}$: > For every *chosen* reference point, there should be a unique, "satisfying" (i.e., see the section "**Modeling Question Summary**" at the bottom), and either: > > 1. a finite mean of $\mathcal{G}$ > 2. if not a finite mean of $\mathcal{G}$, then an infinite mean > 3. if not a finite or infinite mean, then an undefined mean of $\mathcal{G}$. - The "measure" (Definitions 20-22 (pg. 20-25) of a family of each bounded function’s graph, where $\{G_r^{\star}:r\in\mathbb{R}^{+}\}=\{\mathrm{graph}(f_r^{\star}):r\in\mathbb{R}^{+}\}$ and $\{f_r^{\star}:r\in\mathbb{R}^{+}\}$ is a family of bounded functions with different bounded domains converging to $\mathcal{G}$): > The preliminary step to defining the "measure" of a family of each bounded function's graph (Definition 20, pg. 20-22) is to partition each of their graphs 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, the entropy of the distribution, and the maximum entropy w.r.t all pathways. > > ----------------- > > 'The "measure" (Definition 22, pg. 23-25) involves the supremum of the sample size (i.e., when applying the preliminary step to each bounded function's graph in a chosen family) such that the entropy in the preliminary step is less than or equal to the entropy (i.e., when applying the preliminary step to each bounded function's graph in a "non-equivalent" family [Definition 18-19, pg. 19-20]), then dividing the former by the original sample size (i.e., when applying the preliminary step to each bounded function's graph in a chosen family) and taking its "supremum" and "infimum" w.r.t. to all possible partitions and samples. (The "supremum" and "infimum" should be equal.) To better understand the definitions, consider the following examples (Section A.7.1-A.7.3, pg. 179-197). - The expected rate of expansion: > The expected rate of expansion is an arbitrary fixed constant and is > written as the function $E:\mathcal{A}(A)\to\mathbb{R}$ (i.e., > $\mathcal{A}(A):=\mathbb{R}^{+}$ is the chosen index set of a family of functions or > sets). - The actual rate of expansion (Definition 23, pg. 25-26): > The actual rate of expansion of a family of each bounded function's > graph is a function of the "average" $2$-dimensional Euclidean > distance between every point in each bounded function's graph and the > reference point $\mathbf{R}\in\mathbb{R}^{2}$ ------- ### Modeling Question Summary: ### > Combining "the measure" and the actual rate of expansion, we get a general notion of the choice function in the modeling question (Section 3.1, pg. 27-30). The choice function should pick "equivelant" families (Definition 15-17, pg. 17-18) of bounded functions converging to $\mathcal{G}$ which satisfy all the criteria in the modeling question, such that: > - the "measure" (Definition 20-22, pg. 20-25) of each chosen family of each bounded function's graph increases at a rate linear or superlinear (Definition 22, pg. 23-25) to that of each "non-equivelant" family (Definition 18-19, pg. 19-20) of each bounded function's graph (Section 3.1 criterion 2) - the absolute difference between the $y$-coordinate of the reference point $\mathbf{R}\in\mathbb{R}^{2}$ and the mean of each chosen family of bounded functions converging to $\mathcal{G}$ is minimized (Definition 13 [pg. 13-14], Section 3.1 criterion 4) - the absolute difference between the expected rate of expansion and the actual rate of expansion of each chosen family of each bounded function's graph is minimized (Definition 23 [pg. 25-26], Section 3.1 criterion 4), > so that it's more likely the choice function, which answers the modeling question, chooses the desired families. Hence, we take the mean of the chosen "equivelant" (Definition 15-17, pg. 17-18) families as the new mean (i.e., the "equivelant" families should have the same mean). [1]: https://community.wolfram.com/groups/-/m/t/3769559?p_p_auth=EIl3sQDy [2]: https://scicomp.stackexchange.com/questions/45512/can-we-average-pathological-function-mathcalg-mathbbr-to-mathbbr-in-a [3]: https://en.wikipedia.org/wiki/Calkin%E2%80%93Wilf_tree [4]: https://mathematica.stackexchange.com/a/319780/34171 [5]: https://www.researchgate.net/publication/410022187_Reformatted_Averaging_an_Explicit_Non-Lebesgue_Integrable_and_Unbounded_Function_That_Is_Defined_Without_The_Axiom_of_Choice [6]: https://en.wikipedia.org/wiki/Frame_of_reference Bharath Krishnan 2026-08-11T00:50:32Z https://community.wolfram.com/groups/-/m/t/3776323 &[Wolfram Notebook][1] [1]: https://www.wolframcloud.com/obj/879a9153-44d4-403e-829c-9850ddbd49ea Alejandro Latorre Chirot 2026-08-08T22:49:43Z https://community.wolfram.com/groups/-/m/t/3761691 &[Wolfram Notebook][1] [1]: https://www.wolframcloud.com/obj/e71d5af0-8a32-4917-88a2-a42a5fbf2416 Robert Cowen 2026-07-16T17:09:06Z 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