https://community.wolfram.com RSS Feed for Wolfram Community showing any discussions tagged with Discrete Mathematics with no replies sorted by active. https://community.wolfram.com/groups/-/m/t/3740868 &[Wolfram Notebook][1] [1]: https://www.wolframcloud.com/obj/32e11114-d41a-43eb-b18e-db0907d77579 James Butterfield 2026-06-29T14:31:23Z https://community.wolfram.com/groups/-/m/t/3716598 I would like to share a compact deterministic arithmetic program that produces four numerical values lying within the CODATA 2022 one-standard-uncertainty windows for: - the inverse fine-structure constant - the proton/electron mass ratio - the muon/electron mass ratio - the tau/electron mass ratio The code was motivated by a broader physical framework, but the point of this post is not to argue for that framework. The narrower question is this: Can this compact computational object be explained as ordinary numerical fitting, and if so, where is the fitted information stored? The program contains: - no empirical target constants as inputs - no decimal fit coefficients - no continuous optimization loop - no output-specific adjustable parameters In addition, the three mass-ratio outputs share the same denominator ladder. Wolfram Language code: ```wolfram u = 1; r = 3; A = r*(r - u); B = 2; n = A + B - u; rel = A + B + r; ph = n + B; h = (rel + u) + n; nVal = r + rel + ph + h; cVal = 2*nVal; atten = 1 - A/n^2; Phi = nVal*r + rel + r/cVal + (rel - ph)/cVal^2 + (18/(n*Pi))*atten/cVal^3 + (4*Pi)*atten/cVal^4; d1 = r + ph + u; d2 = h*(nVal - (r + u)); d3 = d1*d2; d4 = d2*(nVal - (r + u))*A*(rel - r); Psi1 = cVal*(h + r) - (ph + r) + (rel - ph)/d1 - u/d2 + (rel - ph)/d3 - u/d4; Psi2 = h*rel - r + (rel - u)/d1 - u/d2 + (r + u)/d3 + (h - r - u)/d4; Psi3 = nVal*(cVal - u) - ph + (rel - ph)/d1 - (rel - ph)/d2 + (r + u)/d3 - (r*h)/d4; NumberForm[N[{Phi, Psi1, Psi2, Psi3}, 15], 15] Output: {137.035999165800, 1836.15267343627, 206.768283284455, 3477.15145895483} For comparison, the CODATA 2022 reference values are approximately: alpha^-1 = 137.035999177(21) proton/electron = 1836.152673426(32) muon/electron = 206.7682827(46) tau/electron = 3477.23(23) All four outputs lie within the corresponding one-standard-uncertainty windows. Under a conservative unit-range normalization, the four simultaneous window hits roughly correspond to ~66 bits of target-window information. So my question is simple: **If this is ordinary numerical fitting, what is the information channel?** For completeness, I include a short note containing the description-length analysis: [https://doi.org/10.5281/zenodo.20158423][1] [1]: https://doi.org/10.5281/zenodo.20158423 Tetsuya Momose 2026-05-15T11:09:21Z https://community.wolfram.com/groups/-/m/t/3696660 A chromatic cellular automaton on {0,1}³ generates a cruciform pattern from a single seed (established in Post 3). At every timestep, the null-space complement of the pattern — obtained by mapping each state s → 7−s — is computed and compared to the positive space using five independent complexity measures: Shannon entropy, joint entropy of adjacent pairs, spatial mutual information, 2×2 block entropy, and Kolmogorov complexity approximation via compression. Result: the first four measures show exact equality C(P,t) = C(N,t) at every step. The fifth (Kolmogorov complexity approximated via compression) shows near-equality with fractional difference under 3%, consistent with encoder-dependent variation in a computable upper bound rather than genuine structural asymmetry. The structural complexity of the complement matches the complexity of the pattern — not approximately, but identically — across all information-theoretic measures tested. The complement operator also reveals a vertex absent from the automaton's lifecycle. The lifecycle visits Red, Green, Blue, Yellow, and Magenta. Cyan (0,1,1) never appears. In the complement, Red maps to Cyan. Since Red is the most frequently renewed state (Dead → Idea), Cyan becomes the dominant active state in the null space. The vertex absent from the dynamics is the most prominent vertex of the shadow. Four questions for the community: 1. Continuous extension. The information symmetry C(P,t) = C(N,t) holds by construction for any bijective complement on a finite vertex set. Does it extend to continuous state spaces on [0,1]³? 2. Cyan dominance. The complement reveals Cyan (0,1,1) as the dominant active state of the null space. If Cyan were introduced into the lifecycle, would the complement become impoverished at a corresponding vertex? 3. Simultaneous threshold crossing. The null-space complement achieves identical structural complexity without its own dynamics. In any system where the positive space crosses a self-referential complexity threshold, does the complement cross simultaneously? 4. Universality challenge. Five complexity measures all show C(P,t) = C(N,t). Is there ANY computable complexity measure for which the equality fails? Notebook Attached. https://www.wolframcloud.com/obj/f17551bc-aec2-415b-9323-ad8865b5d46d Dustin Sprenger 2026-04-20T02:45:27Z https://community.wolfram.com/groups/-/m/t/3634252 Hi everyone, My name is Tigran. I am working with the Wolfram Language on cellular automata and would like to ask whether a particular larger-neighborhood extension of Rule 110, defined in algebraic normal form (ANF) over GF(2), is already known, and how best to validate it rigorously. ⸻ Rule family (ANF over GF(2)) For each neighborhood size m >= 3, I define a Boolean local rule f_m(x1, …, xm) = x_{m-1} + x_m + x_{m-1}x_m + (x1x2*…*xm) (mod 2) where: • addition is XOR (mod 2), • multiplication is AND. For m = 3, this becomes f_3(x1, x2, x3) = x2 + x3 + x2x3 + x1x2*x3 (mod 2) which corresponds exactly to Wolfram Rule 110. Converting the ANF to a lookup table and then to a Wolfram rule index gives, for example: m = 3 -> 110 m = 4 -> 28398 m = 6 -> 7993589098607472366 ⸻ Evolution setting I evolve a one-sided (causal) wedge with boundary zeros and a single-seed initial condition: a(0,0) = 1 a(0,t > 0) = 0 Out-of-range indices are treated as zero. The update rule is: a(n+1, t) = f_m( a(n, t-m+1), …, a(n, t) ) This is a triangular / wedge evolution rather than a standard bi-infinite CA. ⸻ Observation Empirically, for certain neighborhood sizes — most notably m = 2^j + 2 the resulting spacetime diagrams show persistent activity and localized structures strongly reminiscent of Rule 110, including glider-like behavior. This does not appear to be a trivial left-right symmetry or a simple shift. ⸻ Questions 1. Prior art Is this specific ANF extension x_{m-1} + x_m + x_{m-1}x_m + (x1x2*…*xm) already known or studied (in Wolfram’s work, the CA literature, or related discussions of Rule-110 generalizations)? 2. Validation / factorization What are the standard techniques to show that a larger-neighborhood Boolean CA factors onto Rule 110, for example via: • a sliding block code, • coarse-graining or renormalization, • substitution or rescaling, • or a proof of a semiconjugacy to Rule 110? Does the use of a wedge evolution change what would count as a valid factor or simulation? 3. Practical workflow If the right goal is to exhibit an explicit map pi such that pi( evolution under f_m ) = evolution under Rule 110 applied to pi(state), what would be a reasonable computational strategy (for example, searching small block maps) to test this in Wolfram Language? If useful, I can share a minimal notebook or GitHub repository Thank you very much for your time and for any pointers or references. Tigran ClearAll[NumberCellularAutomatonFastC]; NumberCellularAutomatonFastC[nMin_Integer, nMax_Integer, kMin_Integer, kMax_Integer, ruleIndex_Integer, M_Integer /; M > 0] := Module[{bits, tMin = nMin, tMax = nMax, nMaxIter = kMax, outAll, slice, cf}, bits = Reverse@IntegerDigits[ruleIndex, 2, 2^M]; cf = Compile[{{b, _Integer, 1}, {tmax, _Integer}, {nmax, _Integer}, {m, _Integer}}, Module[{prev, curr, out, n, t, j, idx, pattern, val}, prev = ConstantArray[0, tmax + 1]; prev[[1]] = 1; out = ConstantArray[0, {nmax + 1, tmax + 1}]; out[[1]] = prev; For[n = 1, n <= nmax, n++, curr = ConstantArray[0, tmax + 1]; For[t = 0, t <= Min[tmax, (m - 1) n], t++, pattern = 0; (* pattern built from prev[t-(m-1)]..prev[t] *) For[j = 1, j <= m, j++, idx = t - m + j; val = If[idx >= 0, prev[[idx + 1]], 0]; pattern = 2 pattern + val; ]; curr[[t + 1]] = b[[pattern + 1]]; ]; prev = curr; out[[n + 1]] = prev; ]; out ], CompilationTarget -> "C", RuntimeOptions -> "Speed" ]; outAll = cf[bits, tMax, nMaxIter, M]; slice = outAll[[kMin + 1 ;; kMax + 1, tMin + 1 ;; tMax + 1]]; Transpose[slice] ] XX[n_] := Symbol["x" <> ToString[n]] + Symbol["x" <> ToString[n - 1]] + Symbol["x" <> ToString[n - 1]]*Symbol["x" <> ToString[n]] + Product[Symbol["x" <> ToString[j]], {j, 1, n}]; Table[ {anfToRule[XX[(2^j + 2)], (2^j + 2)], XX[(2^j + 2)], ArrayPlot[ NumberCellularAutomatonFastC[ 0, 10*(2^j + 2)^2 + 100, 0, 10*(2^j + 2)^2 + 100, anfToRule[XX[(2^j + 2)], (2^j + 2)], (2^j + 2)], ColorRules -> {0 -> Black, 1 -> LightBlue}] }, {j, 0, 3} ] Tigran Nersissian 2026-02-04T13:27:33Z https://community.wolfram.com/groups/-/m/t/3580921 &[Wolfram Notebook][1] [1]: https://www.wolframcloud.com/obj/c6388a9d-cdfc-4d41-9d79-caccd5ba5c68 Adael Reyes 2025-11-23T22:34:43Z https://community.wolfram.com/groups/-/m/t/3567195 Hi I'm an independent researcher exploring a potential link between the shortcut Collatz function (`T(n)`) and ECA Rule 54 (single cell evolution). I need help verifying a formula from the MathWorld page for Rule 54. The page presents this piecewise formula: $$ a(n) = \begin{cases} \frac{1}{15}(4^{n+2} - 1) & \text{if } n \text{ is even} \newline \frac{7}{15}(4^{n+1} - 1) & \text{if } n \text{ is odd} \end{cases} \quad (n \ge 1) $$ [https://mathworld.wolfram.com/Rule54.html][1] My work proves that `T(n) = wH(a(n)) - 1` (where `wH` is binary Hamming weight), assuming `a(n)` represents the normalized decimal value (odd-part) of the n'th row pattern. Could someone please help clarify: 1. Is the interpretation of `a(n)` (as normalized value) correct for Rule 54 (single cell) ? 2. What is the source or derivation for this specific `a(n)` formula? 3. Is the formula original to MathWorld? 4. If a primary source isn't readily available, is citing the MathWorld page directly considered acceptable practice for this formula in a formal paper? Verifying this is crucial for my paper connecting Collatz to Rule 54. While Dr. Weisstein might be the most direct source, perhaps he isn't very active here. Any insights, references, or verification from the community would be greatly appreciated. Thanks so much! Rudi B. Stranden [1]: https://mathworld.wolfram.com/Rule54.html Rudi Bjørnsen Stranden 2025-10-29T17:38:56Z https://community.wolfram.com/groups/-/m/t/3514752 A computational essay that serves as 'Code as Proof,' formally modeling the foundational flaws of classical networking and demonstrating the superiority of the Dædælus transactional fabric. The notebook begins by deconstructing the shared-media contention of the 1976 Metcalfe-Boggs model and the latency-bound throughput of TCP (Mathis equation), proving how traditional bandwidth-multiplexing over unreliable links inevitably leads to efficiency collapse, the loss of epistemic certainty, and unbounded tail latency. It then contrasts this with the Dædælus architecture, simulating how a reliable N2N Lattice that uses Transaction-Multiplexing and the Reversible Snake protocol eliminates contention, is immune to packet loss, and provides the deterministic performance necessary for a Truncated Tail Latency. &[Wolfram Notebook][1] [1]: https://www.wolframcloud.com/obj/5940dabf-c983-4d15-aac3-59b831fde427 Dean Gladish 2025-07-18T09:33:17Z https://community.wolfram.com/groups/-/m/t/3514743 A formal, executable model of the original 1976 Metcalfe-Boggs protocol for distributed packet switching. This notebook serves as 'Code as Proof,' deconstructing the dynamics of statistical arbitration on a shared, passive Ether. It mathematically models the interaction between the Transmission Interval, where a single station has acquired the Ether, and the Contention Interval, where multiple queued stations (Q) vie for access. By visualizing the best metrics of Acquisition Probability (A), Wait Time (W), and overall Efficiency (E), the model demonstrates the fundamental trade-offs and performance degradation inherent in a system reliant on simple packet collision and retransmission--setting the stage for the superior, deterministic model of Open Atomic Ethernet. &[Wolfram Notebook][1] [1]: https://www.wolframcloud.com/obj/e10b3b57-f841-4b2c-8999-734874b0d2f4 Dean Gladish 2025-07-18T09:22:46Z https://community.wolfram.com/groups/-/m/t/3481786 &[Wolfram Notebook][1] [1]: https://www.wolframcloud.com/obj/5c5cc7f5-cfb7-4fa1-ba79-bafb50a181d0 Jefferson Conza 2025-06-18T22:48:55Z https://community.wolfram.com/groups/-/m/t/3402381 I'm new to Mathematica, and going around with some Help from ChatGPT &[Wolfram Notebook][1] Day 2: &[Wolfram Notebook][2] Today: &[Wolfram Notebook][3] Thank You [1]: https://www.wolframcloud.com/obj/c8c55c46-e6ad-4ac5-8917-6b87229a61c2 [2]: https://www.wolframcloud.com/obj/06607baa-d76d-4006-a745-197ae5a33968 [3]: https://www.wolframcloud.com/obj/27473a5a-f3eb-4435-89a5-2126640b7cfb David Beteta 2025-02-28T14:21:33Z https://community.wolfram.com/groups/-/m/t/3379973 ![On functional iteration and roots][1] &[Wolfram Notebook][2] [1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=proj-6-cover.jpeg&userId=911151 [2]: https://www.wolframcloud.com/obj/930a433a-2ab4-478c-8542-f153add19e0f Wolfram Education Programs 2025-02-04T15:30:58Z https://community.wolfram.com/groups/-/m/t/3311103 &[Wolfram Notebook][1] [1]: https://www.wolframcloud.com/obj/ddceb7e2-1ad8-4a94-ab6b-ec11ed500a95 Jefferson Conza 2024-10-29T14:59:14Z https://community.wolfram.com/groups/-/m/t/3237111 &[Wolfram Notebook][1] [1]: https://www.wolframcloud.com/obj/9943b9c6-7bf8-4641-9399-ec43d18bc138 Richard Frost 2024-07-28T21:20:53Z https://community.wolfram.com/groups/-/m/t/3165799 Here is a quiz Mandelbrot gives. Of the difference graphs below, one is Brownian motion, one fractional Brownian motion, one a Levy process, two are real financial data, and three are multifractal forgeries. Can you tell which is which? ![Pick the Fake Mandelbrot Finance][1] [Answers here][2] I am fascinated by Mandelbrot's visual approach to science. In his book **Misbehavior of Markets** he comments, *Pictures are undervalued in science. A modern diagram or chart can be as precise as desired, and is no more costly than the computer that draws it. The picture can now aid, not mislead the scientist. It permits instant comparison, instant comprehension.* There was a famous scientific debate including the now late Daniel Kahneman over whether the "Hot Hand" effect was real or not. Current consensus is that it is real. I was curious if there was a "Hot Team" effect where teams were more likely to score after scoring. In my experience as a college basketball player and coach, it certainly feels like this exists. And basketball practitioners swear by it: "Basketball is a game of runs!", they say. I tried to tackle this problem using Mandelbrot's method, but found (at least on initial testing) that his eye test is not sensitive enough to pick up small changes of 2.5% dependence, which if the hot team effect exists it will be small. Here's what the Mandelbrot test looks like for 10,000 steps in a random walk with .525 probability dependence, meaning if you get a 1 you have a .525 chance of getting another 1 and same for -1. ![Mandelbrot Method][3] I have to generate more samples to really double-check if it can pick up the difference but just from that image I can't really tell. So I devised a test that counts up the number of streaks of length 2 and plots their frequency. You can divide it up into chunks, to make it easier to see the difference between values. It looks like this: ![Dependence Eye Test][4] So here you can actually test yourself using the manipulate to see if you can tell which one is dependent and which one is independent. With 10 chunks of 1000 and a dependence of 0.525, I am about 95% accurate at telling the difference, which is an improvement on the initial Mandelbrot method which I couldn't really tell at all. With 10 chunks of 500 I performed at about 80%. Once you go lower than that it starts to become impossible because I think the information is not actually there. I encourage you to give it a try. I applied this method to the Portland Trail Blazers NBA Team data and found that it didn't look dependent, which is evidence for this paper [here][5]. Here's what it looked like: ![Blazers Dependence Test][6] On the left is fake independent data, in the middle is fake dependent data (p = 0.525) and on the right is the real data. I'm working on trying out my eye test on college basketball and tennis data. Overall I think it's a powerful tool for detecting dependence without knowing the mathematics behind it. It's science for dummies, like me, who can't get their head around probability formulas. On a bigger picture it's a method for finding effective visual models for random events. My plan is to build up a set of tests so that I can quickly understand data with minimal effort. Here's the notebook to use the manipulates for yourself: &[Wolfram Notebook][7] [1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=ScreenShot2024-03-18at6.30.28PM.png&userId=3163706 [2]: https://users.math.yale.edu/public_html/People/frame/Fractals/RandFrac/Market/Fake/FinanceQuizAns.html [3]: https://community.wolfram.com//c/portal/getImageAttachment?filename=ScreenShot2024-04-28at12.53.27AM.png&userId=3163706 [4]: https://community.wolfram.com//c/portal/getImageAttachment?filename=ScreenShot2024-04-28at12.03.30AM.png&userId=3163706 [5]: https://arxiv.org/pdf/1903.08716 [6]: https://community.wolfram.com//c/portal/getImageAttachment?filename=ScreenShot2024-04-28at1.24.28AM.png&userId=3163706 [7]: https://www.wolframcloud.com/obj/a0e02235-dc47-4d85-a2cc-a2d4488660a4 Willem Nielsen 2024-04-28T05:38:04Z https://community.wolfram.com/groups/-/m/t/2981522 &[Wolfram Notebook][1] [1]: https://www.wolframcloud.com/obj/ddf03176-0030-4109-8fe3-991ef29cd94f Peter Burbery 2023-08-02T02:03:55Z https://community.wolfram.com/groups/-/m/t/2840816 These are the so-called "recurring numbers", which result from analyzing various equations of that sector of mathematics studied by Srinivasa Ramanujan (Theory of Numbers) and various equations of various sectors of Theoretical Physics (String Theory) and of Theoretical Cosmology (Black Holes and cosmological inflation) ![enter image description here][1] ![enter image description here][2] ![enter image description here][3] ![enter image description here][4] ![enter image description here][5] ![enter image description here][6] [1]: https://community.wolfram.com//c/portal/getImageAttachment?filename=1270Immagine1.png&userId=2483773 [2]: https://community.wolfram.com//c/portal/getImageAttachment?filename=8411Immagine2.png&userId=2483773 [3]: https://community.wolfram.com//c/portal/getImageAttachment?filename=8715Immagine3.png&userId=2483773 [4]: https://community.wolfram.com//c/portal/getImageAttachment?filename=10443Immagine4.png&userId=2483773 [5]: https://community.wolfram.com//c/portal/getImageAttachment?filename=Immagine5.png&userId=2483773 [6]: https://community.wolfram.com//c/portal/getImageAttachment?filename=Immagine6.png&userId=2483773 Michele Nardelli 2023-02-28T12:18:16Z https://community.wolfram.com/groups/-/m/t/2734106 I created a notebook where I decided to learn functions in the guide NumberDigits. I decided to do this for a few reasons. One I am interested in coming up with good names for functions and names like IntegerReverse which do one thing very well that is very specific are a good way to discuss how to name functions. The other reason is that I can understand these functions and there have been several times including when I was writing this notebook with CountDigits and [another post on Wolfram Community on visualizing Euler's number][1] I saw someone else make where someone pointed out that you could use RealDigits and I post I made where someone pointed out that I could solve my problem with RealDigits so I decided to work through the guide to not miss any simple digit functions. The function IntegerReverse is discussed in [Chapter 46 of an Elementary Introduction to the Wolfram Language by Stephen Wolfram: Writing Good Code][2]. Stephen Wolfram also writes about the addition of NumberDigit in the [blog post for 12.3][3] as a specific use case of RealDigits. There is one function that I don't have as strong an intuitive grasp on: MantissaExponent. &[Wolfram Notebook][4] [1]: https://community.wolfram.com/groups/-/m/t/1280901 [2]: https://www.wolfram.com/language/elementary-introduction/2nd-ed/46-writing-good-code.html [3]: https://writings.stephenwolfram.com/2021/05/launching-version-12-3-of-wolfram-language-mathematica/ [4]: https://www.wolframcloud.com/obj/6033c1a9-d7cf-41c8-9ed2-b4b40a93713e Peter Burbery 2022-12-17T02:00:37Z https://community.wolfram.com/groups/-/m/t/2707515 Have you heard? We are hosting another [Wolfram R&D Live][1] **Q&A session** on Twitch and YouTube featuring developers from the Calculus and Algebra team! Join us Wednesday November 30th at 11 AM CST. [Devendra Kapadia][2] and [Daniel Lichtblau][3] will be answering your questions. You can help us narrow down topics interesting to you by discussing them on this thread. Here are some examples of topics: - Continuous Calculus (integration, differential equations...) - Discrete Calculus (summation, transforms, ...) - Algebra (polynomials, algebraic equations, ...) - Special Functions (BesselJ, MeijerG, ...) Find out more about Calculus and Algebra [here][4]. Post your questions ahead of time below! [1]: https://wolfr.am/RDlive [2]: https://community.wolfram.com/web/dkapadia [3]: https://community.wolfram.com/web/danl [4]: https://wolfr.am/calculus Keren Garcia 2022-11-21T17:18:00Z https://community.wolfram.com/groups/-/m/t/2652917 I created a notebook where I designed questions with Question Generator and Question function. I have several examples of questions including prime numbers, matrices, and dates. &[Wolfram Notebook][1] [1]: https://www.wolframcloud.com/obj/67534e73-5e78-4be1-82b0-4402040d3372 Peter Burbery 2022-10-15T14:59:25Z https://community.wolfram.com/groups/-/m/t/2419451 Well, I have the following code: Clear["Global`*"]; a =; b =; \[Alpha] =; \[Beta] =; k =; ParallelTable[ If[TrueQ[Length[ Select[PowersRepresentations[n, a, b], DuplicateFreeQ[#] && ! MemberQ[#, 0] &]] >= k], n, Nothing], {n, \[Alpha], \[Beta]}] //. {} -> Nothing But for a given $a,b, k,\alpha$ and $\beta$ this calculation is really slow. Is there a way to speed things up a bit? I am a student in elementary number theory and looking for power representations of certain numbers. For relatively small sets, it is convenient to use this code but for larger sets, it becomes way too slow. >Thanks for any advice and help. ___ Example runnable code: Clear["Global`*"]; a = 3; b = 5; \[Alpha] = 500000; \[Beta] = 10000000; k = 6; ParallelTable[ If[TrueQ[Length[ Select[PowersRepresentations[n, a, b], DuplicateFreeQ[#] && ! MemberQ[#, 0] &]] >= k], n, Nothing], {n, \[Alpha], \[Beta]}] //. {} -> Nothing Jan Eerland 2021-12-06T10:34:40Z