Community RSS Feed https://community.wolfram.com RSS Feed for Wolfram Community showing any discussions in tag Mathematics sorted by active How can this Euler Method be implemented in Mathematica? https://community.wolfram.com/groups/-/m/t/2824249 Hello, Has anyone already implemented the Euler method in Mathematica and can he/she post the implementation? ![enter image description here] ![enter image description here] ![enter image description here] [https://math.libretexts.org/Courses/Monroe_Community_College/MTH_225_Differential_Equations/3%3A_Numerical_Methods/3.1%3A_Euler&#039;s_Method] I would like something like this, using a For loop (if possible, but can be as well even other types of implementation): https://community.wolfram.com/groups/-/m/t/2823888 Thank you. : https://community.wolfram.com//c/portal/getImageAttachment?filename=3666Capture1.JPG&amp;userId=2803344 : https://community.wolfram.com//c/portal/getImageAttachment?filename=6106Capture2.JPG&amp;userId=2803344 : https://community.wolfram.com//c/portal/getImageAttachment?filename=4387Capture3.JPG&amp;userId=2803344 : https://math.libretexts.org/Courses/Monroe_Community_College/MTH_225_Differential_Equations/3:_Numerical_Methods/3.1:_Euler%27s_Method Cornel B. 2023-02-05T20:17:22Z No output from Solve[ ]? https://community.wolfram.com/groups/-/m/t/2241678 Could someone please look at this and tell me what&#039;s wrong? It&#039;s probably a simple mistake. I think this system of equations should return {{pd -&gt; -3}, {pd -&gt; 10}}. At least, that&#039;s what I get when I use a pencil and paper. But my input below doesn&#039;t return anything. Not even an error message. &amp;[Wolfram Notebook] : https://www.wolframcloud.com/obj/6d7028ba-c944-483e-8c18-3b422736dce5 Jay Gourley 2021-04-12T05:45:56Z Wolfram Mathematica vs Standard Maple vs Maple Flow vs Mathcad vs Matlab https://community.wolfram.com/groups/-/m/t/2827048 Hello, I have at least some questions that I would like to address to you: 1. What would be the advantages of using Wolfram Mathematica compared to the following software: Standard Maple, Maple Flow, Mathcad, or Matlab? 2. What would be the disadvantages of using Wolfram Mathematica compared to the following software: Standard Maple, Maple Flow, Mathcad, and Matlab? 3. Why do you prefer Wolfram Mathematica instead of the other software: Standard Maple, Maple Flow, Mathcad, or Matlab? 4. Or do you use the other software listed above besides Wolfram Mathematica? 5. Is it really better to use Wolfram Mathematica compared to the other software above, at least from the point of view of say the following fields: Engineering/Electrical Engineering, Control Systems, Signal Processing, Calculus, Matrices, Algorithm (convergence, accuracy, precision), working with units, programming? 6. Is it worth messing around with Wolfram Mathematica to learn this software, or would it be better if you turned to one of the other software above? Thank you. Cornel B. 2023-02-09T07:09:09Z How to obtain partial transpose of a 5*5 matrix? https://community.wolfram.com/groups/-/m/t/2826630 I have a 5*5 matrix as follow &amp;[Wolfram Notebook] its a density matrix of a quantum state. suppose the quantum state is as follow: |w&gt; = |10000&gt;+|01000&gt;+|00100&gt;+|00010&gt;+|00001&gt; I want to obtain the partial transpose of the matrix (THE PERES PARTIAL TRANSPOSITION MATRIX) note: there are 5 partial transpose matrix for a 5*5 matrix. I want all of them. I don&#039;t know if there is a command in Mathematica for it? : https://www.wolframcloud.com/obj/717fc1e8-ca15-4861-9332-6e8f3e1388c2 Reza Hamzeh 2023-02-08T20:25:24Z Shading a region with a changing boundary? https://community.wolfram.com/groups/-/m/t/2826615 I&#039;d like to shade the region marked &#034;fill this region&#034;. I&#039;d prefer a method that doesn&#039;t require me to break the region up into smaller regions. Maybe using one application of RegionPlot? &amp;[Wolfram Notebook] : https://www.wolframcloud.com/obj/ecbb062f-4c2b-4add-af1e-ef3bea8cc994 Jon Joseph 2023-02-08T18:37:45Z Gaussian Integration didn't give correct result https://community.wolfram.com/groups/-/m/t/2825872 As shown in the noteboook, I was calculating the variance of a Gaussian distribution centered at x=0. The correct result should be just w^2, which is a property of Gaussian distribution. The result here is not only wrong, but even still has x in the formula, which is not what I would expect after an integration. Can anyone see what the problem here is? Did I type the formula in a correct way? I&#039;m new to Mathematica. &amp;[Wolfram Notebook] : https://www.wolframcloud.com/obj/40b5d9f1-72d8-42db-8519-87d961a03766 Wei Liang 2023-02-08T10:45:08Z Try to beat these MRB constant records! https://community.wolfram.com/groups/-/m/t/366628 ![If you see this instead of an image, reload the page.] §0. (from §6 below to whet your appetite!) Q&amp;A: ==== Q: What can you expect from reading about **C**&lt;sub&gt;*MRB*&lt;/sub&gt; and its record computations? A: &gt; As you see, the war treated me kindly enough, in spite of the heavy &gt; gunfire, to allow me to get away from it all and take this walk in the &gt; land of your ideas. &#x2014; Karl Schwarzschild (1915), “Letter to Einstein”, Dec 22 Q: Can you calculate more digits of **C**&lt;sub&gt;*MRB*&lt;/sub&gt;? A: &gt; With the availability of high-speed electronic computers, it is now &gt; quite convenient to devise statistical experiments for the purpose of &gt; estimating certain mathematical constants and functions. Copyright © 1966 ACM (Association for Computing Machinery) New York, NY, United States Q: How can you compute them? A: &gt; The value of $\pi$ has engaged the attention of many mathematicians and &gt; calculators from the time of Archimedes to the present day, and has &gt; been computed from so many different formulae, that a complete account &gt; of its calculation would almost amount to a history of mathematics. - James Glaisher (1848-1928) Q: Why should you do it? A: &gt; While it is never safe to affirm that the future of Physical Science &gt; has no marvels in store even more astonishing than those of the past, &gt; it seems probable that most of the grand underlying principles have &gt; been firmly established and that further advances are to be sought &gt; chiefly in the rigorous application of these principles to all the &gt; phenomena which come under our notice. It is here that the science of &gt; measurement shows its importance &#x2014; where quantitative work is more to &gt; be desired than qualitative work. An eminent physicist remarked that &gt; the future truths of physical science are to be looked for in the &gt; sixth place of decimals. Albert A. Michelson (1894) Q: Why are those digits there? A: &gt; [The principle, &#034;nothing is without reason (nihil est sine ratione), or there is no effect without a cause&#034;] must be considered one of the greatest and most fruitful of all human knowledge, for upon it is built a great part of metaphysics, physics, and moral science. (G VII 301/L 227). Gottfried Wilhelm Leibniz (1646&#x2013;1716) ---------- POSTED BY: Marvin Ray Burns. ======== ![If you see this text, the images are not showing. Refresh the page.] ![The first 100 partial sums of] {![the CMRB series.]} ---------- **After receiving Wikipedia&#039;s and MathWorld&#039;s articles on the MRB constant (the upper limit point of the sequence of the above partial sums), Google OpenAI Chat CPT described the constant as follows.** &gt; ![Chat AI 1] &gt; ![Chat AI 2] References: &gt; &gt; - Burns, M. R. &#034;An Alternating Series Involving n^(th) Roots.&#034; Unpublished note, 1999. &gt; - Burns, M. R. &#034;Try to Beat These MRB Constant Records!&#034; &gt; - http://community.wolfram.com/groups/-/m/t/366628. &gt; - Crandall, R. E. &#034;Unified Algorithms for Polylogarithm, L-Series, and Zeta Variants.&#034; 2012a. &gt; - http://www.marvinrayburns.com/UniversalTOC25.pdf. &gt; - Crandall, R. E. &#034;The MRB Constant.&#034; §7.5 in Algorithmic Reflections: Selected Works. PSI Press, pp. 28-29, 2012b. &gt; - Finch, S. R. Mathematical Constants. Cambridge, England: Cambridge University Press, p. 450, 2003. &gt; - Plouffe, S. &#034;MRB Constant.&#034; http://pi.lacim.uqam.ca/piDATA/mrburns.txt. &gt; - Sloane, N. J. A. Sequences A037077 in &#034;The On-Line Encyclopedia of Integer Sequences.&#034; &gt; With the Mathematica toolbox, I&#039;m doing just that by finding patterns in its numeric expansions, performing basic numeric, real and complex analysis from original viewpoints, and tying together basic concepts from every branch of mathematics. Join me in doing so below. For the best viewing, wait a minute until the word LaTeX in the LaTex script is centered below. $$\LaTeX$$ If the phrase [Math Processing Error] is shown, or the LATEX script has vanished from the center of the above line, some of the math below might be missing or appear in the LaTex code instead of the script. For easy navigation, use the ![CTRL+f] keys on your keyboard. Cues in the forms of §&#039;s and keywords in quotes are provided in the ![CTRL+f] &#034;Index&#034;. If you can see the header and the words Reply | Flag at the same time, in any of the following replies, you&#039;ll need to refresh the page to see them. ---------- ---------- ![CMRB funnel] That **C**&lt;sub&gt;*MRB*&lt;/sub&gt; nomenclature for the MRB constant was devised by Wolfram Alpha, as seen next. ------------ &gt; ![W\A] ---------- ---------- ---------- ---------- &#039; **An Easter egg for you to find below: ** &gt; ![In another reality, I invented CMRB and then discovered many of its qualities.] Thus, here are three open questions about &#034;constructing rather than finding&#034; math. - If we assume the MRB constant exists and that it was of my making did its qualities exist before I invented it? - If they didn&#039;t, does that mean I invented them too? - If so, does the same principle hold that we invented all of the unintended consequences, of all other mathematical constructs, such as constants, numbers, theorems, shapes, etc.? ---------- ---------- ---------- Index ===== The first post -------------- We first analyze the prototypical series for the MRB constant, ![Sn^(1/n)-1] (Select § with the given number or the keywords in quotes, and then press the ![CTRL+f] keys on your keyboard to move to that section.) §1. Is that series convergent? §2. Is -1 the only term that series is convergent for? §3. Is that series absolutely convergent? §4. Is that series &#034;efficient?&#034; (defined as how it compares to other series and integrals that compute CMRB in speed and computational cost.) §5. Is there a geometric [isomorphism] between that series and the edges of hypercubes? §6. Q and A, §7. My claim to the MRB constant (CMRB), or a case of calculus déjà vu? §8. Where is it found? §9. What exactly is it? §10. How it all began, §11. Scholarly works §12. The why and what of the **C**&lt;sub&gt;*MRB*&lt;/sub&gt; Records, Second post: ------------ ![CTRL+f] &#034;The following might help anyone serious about breaking my record.&#034; Third post ---------- ![CTRL+f] &#034;The following email Crandall sent me before he died might be helpful for anyone checking their results.&#034; Fourth post ----------- ![CTRL+f] &#034;Perhaps some of these speed records will be easier to beat.&#034; Many more interesting posts --------------------------- ... including, not to omit, Real-World and beyond, Applications which have been moved to [this discussion] to save on loading times. ...including the ![CTRL+f] &#034;MRB constant supercomputer&#034;s 1 and 2. ...including records of computing the MRB constant from ![CTRL+f] &#034;Crandall&#039;s eta derivative formulas&#034;. ...including all the methods used to compute **C**&lt;sub&gt;*MRB*&lt;/sub&gt; ![CTRL+f] &#034;and their efficiency&#034;. ...including the dispersion of the 0-9th decimals in **C**&lt;sub&gt;*MRB*&lt;/sub&gt; ![CTRL+f] &#034;decimal expansions&#034;. ...including the ![CTRL+f] &#034;convergence rate&#034; of 3 primary forms of **C**&lt;sub&gt;*MRB*&lt;/sub&gt;. ...including complete documentation of all multimillion-digit records with many highlights. ...including ![CTRL+f] &#034;arbitrarily close approximation formulas&#034; for **C**&lt;sub&gt;*MRB*&lt;/sub&gt;. ...including !![CTRL+f] &#034;efficient programs&#034; to compute the integrated analog (MKB) of **C**&lt;sub&gt;*MRB*&lt;/sub&gt;. ...including a recent discovery that could ![CTRL+f] &#034;help in verifying&#034; digital expansions of the integrated analog (MKB) of **C**&lt;sub&gt;*MRB*&lt;/sub&gt;. ...including an inquiry for a ![CTRL+f] &#034;closed form&#034; for CMRB. ...including a question about ![CTRL+f] &#034;how normal&#034; CMRB is, and what Google OpenAI Chat CPT says. ...including a few attempts at a ![CTRL+f] &#034;A cool 7 million digits?&#034; ... including an overview of all **C**&lt;sub&gt;*MRB*&lt;/sub&gt; ![CTRL+f] &#034;speed records of MRB constant&#034;, by platform. ... including ![CTRL+f] &#034;yet another attempt at 7 million digits&#034;. ---------- ---------- ---------- The MRB constant relates to the divergent series: ![divegrent series] =![DNE] The sequence of its partial sums has two limit points with an upper limit point known as the MRB constant (CMRB). So, out of the many series for CMRB, we first analyze the sum prototype, i.e., the series ![{CMRB}] =![Sn^(1/n)-1] ---------- Concerning the sum prototype for CMRB §1. Is the series convergent? After being programmed with the rules of convergence and this series, Google Open AI answered: &gt; ![enter image description here] So, I proved its convergence by that test, as shown next. PROOF ===== Below, we show by the Squeeze theorem (sandwich theorem) and by plotting the following series, the qualifications for the Leibniz criterion (alternating series test) are satisfied for the MRB constant (CMRB) in CMRB ![enter image description here] by showing a(n)=(n^{1/n}-1)&gt;0 is monotonically decreasing for $n≥3$ and has a limit as n goes to infinity of zero. Of course, $\sum_1^3(-1)^n(n^{1/n}-1)$, converges, and the sum of two convergent series converges. &amp;[Wolfram Notebook] We have shown by plotting and the Squeeze theorem (sandwich theorem) that the Leibniz criterion (alternating series test) holds. As we have seen, for n&gt;1, the derivative is 0 only at e; there are no more critical points for the plot to cease to decrease. Thus, a(n)=(n^{1/n}-1)&gt;0 is monotonically decreasing for $n≥3$ and has a limit, as n goes to infinity, of zero. Finally, $\sum_1^3(-1)^n(n^{1/n}-1)$, converges, and the sum of two convergent series converges. Therefore the series is convergent.∎ The Leibniz criterion, summoned above, is defined and proven [here]: ---------- ---------- §2. Next, we ask and observe, ![enter image description here] The short explanation is that $z_0$ must be real for the limit to be 0, and since $lim_{n-&gt;\infty}n^{1/n}=1,$ $z_0=1.$ Using the same limit, for a non-constant term f(n) in $\sum_{n=1}^\infty(-1)^n(n^{1/n}-f(n)), lim_{n-&gt;\infty}f(n)=1,$ As we will see soon (![CTRL+f] &#034;As for efficiency&#034; ). ---------- ---------- §3. ![divergent proof] Plot[{n^(1/n) - 1, 1/n}, {n, 1, Infinity}, PlotLegends -&gt; LineLegend[&#034;Expressions&#034;]] ![plot] ...showing its terms are larger than those of the divergent [Harmonic Series]. Surprisingly, both sides of all three inequalities meet at the first Foias Constant **instead of e**: In:= N[FindRoot[n^(1/n) - 1 - 1/n == 0, {n, .1}, WorkingPrecision -&gt; 38], 32] Out= {n -&gt; 2.2931662874118610315080282912508} In:= N[ FindRoot[n == (1 + 1/n)^n, {n, 1.2}, WorkingPrecision -&gt; 38], 32] Out= {n -&gt; 2.2931662874118610315080282912508} In:= N[FindRoot[n^(1/n) == (1 + 1/n), {n, 1.2}, WorkingPrecision -&gt; 38], 32] Out= {n -&gt; 2.2931662874118610315080282912508} ![Foias 1] ![Foias 2] ---------- For the series of absolute values, I noticed $$-1&lt;\sum_{n=1}^x\left(n^{\frac{1}{n}}-1\right)-\sqrt{x}-1&lt;0.5$$ for $$11\leq x\leq 286$$ ---------- Analogously to the [Riemann Zeta Function] , ![Riemann Zeta] , we see another analogy to the [harmonic series] on how n below, at, and above 1 affects the convergence of the series of absolute values, Similarly to the Riemann Zeta Function: For n&lt;1, ![fraction] (the sum of absolute values) diverges. For n=1, the series also diverges. Only when n&gt;1 does the series converge: ![white backgrond] ![ yellow, green, and bluebackgrounds] Assuming[n &gt; 1, SumConvergence[{(-1)^x (x^(1/x) - 1)/x, (x^(1/x) - 1)/ x, (-1)^x x^(1/x)/x^1, x^(1/x)/ x^1, (-1)^x x^(1/x)/x^(n) and x^(1/x)/x^(n)}, x]] ---------- ---------- §4. As for efficiency, we will look at several much faster-converging series for CMRB throughout this discussion. Here is how the &#034;regular&#034; one (dr) mentioned above compares to the two related ones (dr1) for &#034;d one direction&#034; and (dn) for &#034;d new.&#034; So, below we have the expressions involving a sum followed by how close to zero of a result they give after the given number of partial summations. &amp;[Wolfram Notebook] That increase in efficiency is the Cesàro method in dn: ![enter image description here] As Wikipedia says, &gt; ![enter image description here] For how more efficient forms compare, ![CTRL+f] &#034;the rate of convergence&#034; of 3 major forms. ---------- ---------- §5. Before I deeply considered the full ramifications of the word Isomorphism, I called the Geometry of the MRB constant from that sum CMRB=$\sum_{n=1}^\infty(-1)^n(n^{1/n}-1)$, &gt; the process that plots values from constructions arising from a &gt; peculiar non-Euclidean geometric [isomorphism] between its partial &gt; sums and hypercubes of many dimensions, where we have the following. &gt; ![special hypercubes] &gt; &gt; &gt; Then find a ![sum to that series.] There in Diagram 3, M, at the point of the segment is where the z=MRB constant would be, and the base of that segment is the MRB constant -1. However, in mathematics, an isomorphism is a &#034;structure-preserving mapping between two structures of the same type that can be reversed by inverse mapping.&#034; Hence, I&#039;m unsure if the term applies here. ---------- §6. Q&amp;A: [Moved to the beginning.] ---------- §7. This discussion is not crass bragging; it is an attempt by this amateur to share his discoveries with the greatest audience possible. Amateurs have made a few significant discoveries, as discussed in ![enter image description here] [here.] This amateur has tried his best to prove his discoveries and has often asked for help. Great thanks to all of those who offered a hand! If I&#039;ve failed to give you credit for any of your suggestions, let me know, and I will correct that issue! As I went more and more public with my discoveries, I made several attempts to see what portions were original. I concluded from these investigations that the only original thought I had was the obstinacy to think anything meaningful could be found in the infinite sum shown next. ![CMRB sum] Nonetheless, someone might have a claim to this thought to whom I have not given proper credit. If that is you, I apologize. The last thing we need is another calculus war, this time for a constant. However, if your thought was published after mine, as Newton said concerning Leibniz&#039;s claim to calculus, “To take away the Right of the first inventor, and divide it between him and that other would be an Act of Injustice.” [Sir Isaac Newton, The Correspondence of Isaac Newton, 7 v., edited by H. W. Turnbull, J. F. Scott, A. Rupert Hall, and Laura Tilling, Cambridge University Press, 1959&#x2013;1977: VI, p. 455] Here is what Google says about the MRB constant as of August 8, 2022, at https://www.google.com/search?q=who+discovered+the+%22MRB+constant%22 ![enter image description here] ![If you see this instead of an image, reload the page.]![enter image description here] (the calculus war for CMRB) ---------- CREDIT https://soundcloud.com/cmrb/homer-simpson-vs-peter-griffin-cmrb &#039; Wikipedia, the free encyclopedia &#039; The calculus controversy (German: Prioritätsstreit, &#034;priority dispute&#034;) was an argument between the mathematicians Isaac Newton and Gottfried Wilhelm Leibniz over who had first invented calculus. ![enter image description here] (Newton&#039;s notation as published in PRINCIPIS MATHEMATICA [PRINCIPLES OF MATHEMATICS]) ![enter image description here] ( Leibniz&#039;s notation as published in the scholarly journal Acta Eruditorum [Reports of Scholars]) Whether or not we divide the credit between the two pioneers, ![Wikipedia] said one thing that distinguishes their finds from the work of their antecedents: &gt; Newton came to calculus as part of his investigations in physics and &gt; geometry. He viewed calculus as the scientific description of the &gt; generation of motion and magnitudes. In comparison, Leibniz focused on &gt; the tangent problem and came to believe that calculus was a &gt; metaphysical explanation of the change. Importantly, the core of their &gt; insight was the formalization of the inverse properties between the &gt; integral and the differential of a function. This insight had been &gt; anticipated by their predecessors, but they were the first to conceive &gt; calculus as a system in which new rhetoric and descriptive terms were &gt; created. Their unique discoveries lay not only in their &gt; imagination but also in their ability to synthesize the insights &gt; around them into a universal algorithmic process, thereby forming a &gt; new mathematical system. Like as Newton and Leibniz created a *new system* from the elaborate, confusing structure designed and built by their predecessors, my forerunners studied series for centuries leading to a labyrinth of sums, and then, I created a &#034;new scheme&#034; for the CMRB &#034;realities&#034; to escape it! ---------- ---------- §8. ![it] is defined in the following places, most of which attribute it to my curiosity. - [ค่าคงที่ลุ่มแม่น้ำโขง] (in Thai); - [ar.wikipedia.org/wiki/] (In Arabic); - [Constante MRB] (in French); - [Constanta MRB - MRB constant] (in Romanian); - http://constant.one/ ; - Crandall, R. E. &#034;The MRB Constant.&#034; §7.5 in [Algorithmic Reflections: Selected Works]. PSI Press, pp. 28-29, 2012,ISBN-10 : ‎193563819X ISBN-13: ‎978-1935638193; - Crandall, R. E. &#034;[Unified Algorithms for Polylogarithm, L-Series, and Zeta Variants].&#034; 2012; - [https://en-academic.com/], Wikipedia, Mathematical constant; - Encyclopedia of Mathematics (Series #94); - [Engineering Tools] of the Iran Civil Center (translated from Persian), an international community dedicated to the construction industry, ISSN: 1735&#x2013;2614; - Etymologie CA Kanada Zahlen&#034; (in German). [etymologie.info]; - Finch, S. R. [Mathematical Constants], Cambridge, England: Cambridge University Press, p. 450, 2003, ISBN-13: 978-0521818056, ISBN-10: 0521818052; - Finch&#039;s original essay on [Iterated Exponential Constants]; - Finch, Steven &amp; Wimp, Jet. (2004). Mathematical constants. The Mathematical Intelligencer. 26. 70-74. 10.1007/BF02985660; - Journal of Mathematics Research; [Vol. 11, No. 6; December 2019] ISSN 1916-9795 E-ISSN 1916-9809 Published by Canadian Center of Science and Education; - [Library of General Functions (LGF) for SIMATIC S7-1200] - Mauro Fiorentina’s [math notes] (in Italian); - MATHAR, RICHARD J. &#034;NUMERICAL EVALUATION OF THE OSCILLATORY INTEGRAL OVER exp(iπx) x^(1/x) BETWEEN 1 AND INFINITY&#034; [(PDF)]. arxiv. Cornell University; - [Mathematical Constants and Sequences] a selection compiled by Stanislav Sýkora, Extra Byte, Castano Primo, Italy. Stan’s Library, ISSN 2421-1230, Vol.II; - [&#034;Matematıksel Sabıtler&#034;] (in Turkish). Türk Biyofizik Derneği; - [MathWorld Encyclopedia]; - [MRB常数] (in Chinese); - [mrb constantとは] 意味・読み方・使い方 ( in Japanese); - [MRB константа] (in Bulgarian); - [OEIS Encyclopedia (The MRB constant);] - Patuloy ang MRB - [MRB constant] (in Filipino) - [Plouffe&#039;s Inverter;] - the LACM [Inverse Symbolic Calculator;] - The On-Line Encyclopedia of Integer Sequences® (OEIS®) as [A037077], Notices Am. Math. Soc. 50 (2003), no. 8, 912&#x2013;915, MR 1992789 (2004f:11151); - [Wikipedia Encyclopedia]. ![enter image description here] ---------- ---------- §9. ![CMRB] ## = B =## ![enter image description here] and from Richard Crandall in 2012 courtesy of Apple Computer&#039;s advanced computational group, we have the following computational scheme using equivalent sums of the zeta variant, Dirichlet eta: &gt; ![enter image description here] &gt; ![enter image description here] The expressions ![Etam] and ![eta0] denote the mth derivative of the Dirichlet eta function of m and 0, respectively. The c&lt;sub&gt;j&lt;/sub&gt;&#039;s are found by the code, N[ Table[Sum[(-1)^j Binomial[k, j] j^(k - j), {j, 1, k}], {k, 1, 10}]] (* {-1., -1., 2., 9., 4., -95., -414., 49., 10088., 55521.}*) Crandall&#039;s first &#034;B&#034; is proven below by Gottfried Helms, and it is proven more rigorously, considering the conditionally convergent sum,![CMRB sum] afterward. Then the formula (44) is a Taylor expansion of eta(s) around s = 0. &gt; ![n^(1/n)-1] At ![enter image description here] [here,] we have the following explanation. ![enter image description here] ![enter image description here]![enter image description here] ---------- ![enter image description here] **The integral forms for CMRB and MKB differ by only a trigonometric multiplicand to that of its analog.** ![enter image description here] In:= CMRB = Re[NIntegrate[(I*E^(r*t))/Sin[Pi*t] /. r -&gt; Log[t^(1/t) - 1]/t, {t, 1, I*Infinity}, WorkingPrecision -&gt; 30]] Out= 0.187859642462067120248517934054 In:= CMRB - N[NSum[(E^( r*t))/Cos[Pi*t] /. r -&gt; Log[t^(1/t) - 1]/t, {t, 1, Infinity}, Method -&gt; &#034;AlternatingSigns&#034;, WorkingPrecision -&gt; 37], 30] Out= 5.*10^-30 In:= CMRB - Quiet[N[NSum[ E^(r*t)*(Cos[Pi*t] + I*Sin[Pi*t]) /. r -&gt; Log[t^(1/t) - 1]/t, {t, 1, Infinity}, Method -&gt; &#034;AlternatingSigns&#034;, WorkingPrecision -&gt; 37], 30]] Out= 5.*10^-30 In:= Quiet[ MKB = NIntegrate[ E^(r*t)*(Cos[Pi*t] + I*Sin[Pi*t]) /. r -&gt; Log[t^(1/t) - 1]/t, {t, 1, I*Infinity}, WorkingPrecision -&gt; 30, Method -&gt; &#034;Trapezoidal&#034;]] Out= 0.0707760393115292541357595979381 - 0.0473806170703505012595927346527 I ---------- \&amp;[MRB equations] ---------- ---------- ---------- ---------- ---------- ---------- §10. How it all began ================== **From these meager beginnings: ** My life has proven that one&#039;s grades in school are not necessarily a prognostication of achievement in mathematics. See [my report cards] for evidence of my poor grades. The eldest child, raised by my sixth grade-educated mother, I was a D and F student through 6th grade the second time, but in Jr high, in 1976, we were given a self-paced program. Then I noticed there was more to math than rote multiplication and division of 3 and 4-digit numbers! Instead of repetition, I was able to explore what was out there. The more I researched, the better my grades got! It was amazing!! So, having become proficient in mathematics during my high school years, on my birthday in 1994, I decided to put down the TV remote control and pick up a pencil. I began by writing out the powers of 2, like 2 times 2, 2 times 2 times 2, etc. I started making up algebra problems to work at solving and even started buying books on introductory calculus. Then came my first opportunity to attend university. I cared for my mother, who suffered from Alzheimer&#039;s, so instead of working my usual 60+ hours a week. I started taking a class or two a semester. After my mom passed away, I went back to working my long number of hours but always kept up with my math hobby! Occasionally, I make a point of attending school and taking a class or two to enrich myself and my math hobby. This has become such a successful routine that some strangers listed me on Wikipedia as an amateur mathematician alphabetically following Jost Bürgi, who constructed a table of progressions that is now understood as antilogarithms independently of John Napier at the behest of Johannes Kepler. I&#039;ve even studied a few graduate-level topics in Mathematics. Why I started so slow and am now a pioneer is a mystery! Could it say something about the educational system? Can the reason be found deep in psychology? (After all, I never made any progress in math or discoveries without first assuming I could, even when others told me I couldn&#039;t!) Or could it be that the truth is a little of both and more? ---------- **From these meager beginnings: ** On January 11 and 23,1999, I wrote, &gt; I have started a search for a new mathematical constant! Does anyone &gt; want to help me? Consider, 1^(1/1)-2^(1/2)+3^(1/3)...I will take it &gt; apart and examine it &#034;bit by bit.&#034; I hope to find connections to all &gt; kinds of arithmetical manipulations. I realize I am in &#034;no man&#039;s &gt; land,&#034; but I work best there! If anyone else is foolhardy enough to &gt; come along and offer advice, I welcome you. The point is that I *found* the MRB constant (**C**&lt;sub&gt;*MRB*&lt;/sub&gt;), meaning after all the giants were through roaming the forest of numbers and founding all they found, one virgin mustard seedling caught my eye. So, I carefully &#034;brought it up&#034; to a level of maturity and my understanding of math along with it! (In another reality, I invented **C**&lt;sub&gt;*MRB*&lt;/sub&gt; and then discovered many of its qualities.) In doing so, I came to find out that this constant (**C**&lt;sub&gt;*MRB*&lt;/sub&gt;) &gt; ![MRB math world snippit] (from https://mathworld.wolfram.com/MRBConstant.html) was more closely related to other constants than I could have imagined. As the apprentice of all, building upon the foundation of Chebyshev (1854&#x2013;1859) on the best uniform approximation of functions, as vowed on January 23, 1999. &#034;I took **C**&lt;sub&gt;*MRB*&lt;/sub&gt; apart and examined it &#039;bit by bit,&#039; finding connections to all kinds of arithmetical manipulations.&#034; Not satisfied with conveniently construed constructions (half-hazardously put together formulas) that a naïve use of numeric search engines like Wolfram Alpha or the OEIS might give, I set out to determine the most interesting (by being the most improbable but true) approximations for each constant in relation to it. For example, consider α=the fine structure constant, arguably the most fundamental constant of all, with a value that nearly equals 1/137. Or 1/137.03599913= 0.00729735257..., to be precise, and essentially and **metaphorically** equals (-133+60*10^(2/5))CMRB/456 . It is denoted by the Greek letter alpha &#x2013; α. Let m be the MRB constant. Then we have &amp;[Wolfram Notebook] ---------- We have a strong arithmetic relationship with the Lemniscate Constant: According to Wikipedia In mathematics, the lemniscate constant ϖ is a transcendental mathematical constant that is the ratio of the perimeter of Bernoulli&#039;s lemniscate to its diameter, analogous to the definition of π for the circle. &amp;[Wolfram Notebook] ---------- Consider its relationship to Viswanath&#039;s constant (VC) &gt; ![Viswanath&#039; math world snippit] (from https://mathworld.wolfram.com/RandomFibonacciSequence.html) **With both being functions of x&lt;sup&gt;1/x&lt;/sup&gt; alone, ** we have these near-zeors of VC using **C**&lt;sub&gt;*MRB*&lt;/sub&gt;, which have a ratio of [Gelfond&#039;s constant] $=e^\pi.$ ![=e^\pi] ---------- Then there is the Rogers - Ramanujan Continued Fraction, R(q), of **C**&lt;sub&gt;*MRB*&lt;/sub&gt; that is well-linearly approximated by terms of other terms of **C**&lt;sub&gt;*MRB*&lt;/sub&gt; alone: ![enter image description here] ---------- ---------- What about &#034;?&#034; for m= the MRB constant? ![enter image description here] --------------- &amp;[Wolfram Notebook] ---------- ---------- ---------- **From these meager beginnings: ** On Feb 22, 2009, I wrote, &gt; It appears that the absolute value, minus 1/2, of the limit(integral of (-1)^x*x^(1/x) from 1 to 2N as N-&gt;infinity) would equal the partial sum of (-1)^x*x^(1/x) from 1 to where the upper summation is even and growing without bound. Is anyone interested in improving or disproving this conjecture? &gt; &gt; ![enter image description here] I came to find out my discovery, a very slow-to-converge [oscillatory integral,] would later be further defined by [Google Scholar.] Here is proof of a faster converging integral for its integrated analog (The MKB constant) by Ariel Gershon. g(x)=x^(1/x), M1=![hypothesis] Which is the same as ![enter image description here] because changing the upper limit to 2N + 1 increases MI by $2i/\pi.$ MKB constant calculations have been moved to their discussion at [http://community.wolfram.com/groups/-/m/t/1323951?p_p_auth=W3TxvEwH] . ![Iimofg-&gt;1] ![Cauchy&#039;s Integral Theorem] ![Lim surface h gamma r=0] ![Lim surface h beta r=0] ![limit to 2n-1] ![limit to 2n-] Plugging in equations  and  into equation  gives us: ![left]![right] Now take the limit as N?? and apply equations  and : ![QED] He went on to note that, ![enter image description here] After I mentioned it to him, Richard Mathar published his meaningful work on it [here in arxiv], where M is the MRB constant and M1 is MKB: &gt; ![enter image description here] M1 has a convergent series, ![enter image description here] which has lines of symmetry across whole-and-half number points on the x-axis, and **half-periods of exactly 1**, for both real and imaginary parts as in the following plots. ReImPlot[(-1)^x (x^(1/x) - 1), {x, 1, Infinity}, PlotStyle -&gt; Blue, Filling -&gt; Axis, FillingStyle -&gt; {Green, Red}] ![big plot] ![small plot] Also where &amp;[Wolfram Notebook] Then f[x_] = Exp[I Pi x] (x^(1/x) - 1); Assuming[ x \[Element] Integers &amp;&amp; x &gt; 1, FullSimplify[Re[f[x + 1/2]] - Im[f[x]]]] gives 0 ---------- M2 and CMRB are connected: ---- &amp;[Wolfram Notebook] ---------- In the complex plane, they even converge at the same rate. ---- Every 75 *i* of the upper value of the partial integration yields 100 additional digits of M2=![enter image description here] and of CMRB=![enter image description here]=![enter image description here] &amp;[Wolfram Notebook] Here is a heuristic explanation for the observed behavior. Write the integral as an infinite series, $m= \sum_{k = 1}^\infty a_k$ with $a_k = \int_{i kM}^{i (k+1)M} \frac{t^{1/t}-1}{\sin (\pi t)} \, dt$ for $k \ge 2$ and the obvious modification for $k = 1$. we are computing the partial sums of these series with $M = 75$ and the question is why the series remainders decrease by a factor of $10^{-100}$ for each additional term. The integrand is a quotient with numerator $t^{1/t} - 1 \approx \log t\, / t$ and denominator $1/\sin \pi t \approx e^{i \pi t}$ for large imaginary $t$. The absolute values of these terms therefore are $|a_k| \approx \log |kM|/|kM| \cdot e^{-\pi kM}$. This implies: ![o$] as$k \to \infty$. Consequently, the remainders$\sum_{k = N}^\infty$behave like$e^{- \pi N M}$. They decrease by a factor of$e^{-\pi M}$for each additional term. And for$M = 75$, this is approximately$10^{-100}$, predicting an increase in accuracy of 100 digits whenever the upper integration bound increased by$75i$. I used the fact that ![enter image description here] The following &#034;partial proof of it&#034; is from Quora. While![enter image description here] ![enter image description here] ![enter image description here] ---------- **I developed a lot more theory behind it and ways of computing many more digits in [this linked] Wolfram Community post.** ---------- ---------- **From these meager beginnings:** In October 2016, I wrote the following [here in researchgate]: First, we will follow the path the author took to find out that, for ![ratio of a-1 to a], the limit of the ratio of a to a - 1, as a goes to infinity, is Gelfond&#039;s Constant, (e ^ pi). We will consider the hypothesis and provide hints for proof using L’ Hospital’s Rule (since we have indeterminate forms as a goes to infinity): The following should help in proving the hypothesis: Cos[Pi*I*x] == Cosh[Pi*x], Sin[Pi*I*x] == I Sin-h[Pi*x], and Limit[x^(1/x),x-&gt;Infinity]==1. Using L’Hospital’s Rule, we have the following: ![L’ Hospital’s a&#039;s] (17) (PDF) Gelfond&#039;s Constant using MKB-constant-like integrals. Available from: [https://www.researchgate.net/publication/309187705_Gelfond%27_s_Constant_using_MKB_constant_like_integrals] [accessed Aug 16 2022]. We find no limit &#034;a&#034; goes to infinity of the ratio of the previous forms of integrals when the &#034;I&#034; is left out, and we give a small proof for their divergence. That was responsible for the integral-equation-discovery mentioned in one of the following posts, where it is written, &#034;Using those ratios, it looks like&#034; (There ![m] is the MRB constant.) &gt; ![enter image description here] ---------- **From these meager beginnings:** In November 2013, I wrote:$C$MRB is approximately 0.1878596424620671202485179340542732. See [this](http://www.wolframalpha.com/input/?i=0.1878596424620671202485179340542732300559030949001387&amp;lk=1&amp;a=ClashPrefs_*Math-) and [this.](http://www.wolframalpha.com/input/?i=mrb+constant&amp;t=elga01)$\sum_{n=1}^\infty (-1)^n\times(n^{1/n}-a)$is formally convergent only when$a =1$. However, if you extend the meaning of$\sum$through &#034;summation methods&#034;, whereby series that diverge in one sense converge in another sense (e.g., Cesaro, etc.), you get results for other$a$. A few years ago, it came to me to ask what value of$a$gives $$\sum_{n=1}^\infty (-1)^n\times(n^{1/n}-a)=0\text{ ?}$$(For what value of a is the Levin&#039;s u-transform&#039;s and Cesàro&#039;s sum result 0 considering weak convergence?) The solution I got surprised me: it was$a=1-2\times C\mathrm{MRB}=0.6242807150758657595029641318914535398881938101997224\ldots$. ![enter image description here] I asked, &#034;If that&#039;s correct can you explain why?&#034; and got the following comment. ![enter image description here] To see this for yourself in Mathematica enter FindRoot[NSum[(-1)^n*(n^(1/n) - x), {n, 1, Infinity}], {x, 1}] where regularization is used so that the sum that formally diverges returns a result that can be interpreted as evaluation of the analytic extension of the series. ![enter image description here] See [here.](http://oeis.org/A173273) Also, ![enter image description here] ---------- ---------- ---------- §11. Scholarly works about **C**&lt;sub&gt;*MRB*&lt;/sub&gt;. ------------------ **From these meager beginnings:** In 2015 I wrote: &gt; Mathematica makes some attempts to project hyper-dimensions onto &gt; 2-space with the Hypercube command. Likewise, some attempts at tying &gt; them to our universe are mentioned at &gt; [https://bctp.berkeley.edu/extraD.html] . The MRB constant from &gt; infinite-dimensional space is described at &gt; http://marvinrayburns.com/ThegeometryV12.pdf. &gt; It is my theory that like the MRB constant, the universe, under inflation, started in &gt; an infinite number of space dimensions. They almost all &gt; instantly collapsed, leaving all but the few we enjoy today. I&#039;m not the first to think the universe consists of an infinitude of dimensions. Some string theories and theorists propose it too. Michele Nardelli added a vast amount of string theory analysis and its connection to dimensions and the MRB constant. He said, &gt; In the following links, there are several works concerning various &gt; sectors of Theoretical Physics, Cosmology, and Applied Mathematics, in &gt; which MRB Constant is used as a &#034;regularizer&#034; or &#034;normalizer&#034;. This &gt; constant allows to obtain a better approximation to the solutions &gt; obtained, developing the various equations that are analyzed. The &gt; solutions in turn, lead to four numbers that are called &#034;recurring &gt; numbers&#034;. They are zeta (2) = Pi^2/6, 1729 (Hardy-Ramanujan number), &gt; 4096 (which multiplied by 2 gives the gauge group SO (8192)) and the &gt; Golden Ratio 1.61803398 ... HE HAS PUBLISHED HUNDREDS OF PAPERS ON STRING THEORY AND THE MRB CONSTANT! [https://www.academia.edu/search?q=MRB%20constant] [https://www.researchgate.net/profile/Michele-Nardelli] **[Dr. Richard Crandall] called the MRB constant a [key fundamental constant]** &gt; ![enter image description here] **in [this linked] well-sourced and equally greatly cited Google Scholar promoted paper. Also [here].** **[Dr. Richard J. Mathar] wrote on the MRB constant [here.]** **Xun Zhou, School of Water Resources and Environment, China University of Geosciences (Beijing), [wrote] the following in &#034;on Some Series and Mathematic Constants Arising in Radioactive Decay&#034; for the Journal of Mathematics Research, 2019.** &gt; A divergent infinite series may also lead to mathematical constants if &gt; its partial sum is bounded. The Marvin Ray Burns’ (MRB) constant is &gt; the upper bounded value of the partial sum of the divergent and &gt; alternating infinite series: &gt; -1&lt;sup&gt;1/1&lt;/sup&gt;+2&lt;sup&gt;1/2&lt;/sup&gt;-3&lt;sup&gt;1/3&lt;/sup&gt;+4&lt;sup&gt;1/4&lt;/sup&gt;-5&lt;sup&gt;1/5&lt;/sup&gt;+6&lt;sup&gt;1/6&lt;/sup&gt;-···=0.187859···(M. Chen, &amp; S. Chen, 2016). Thus, construction of new infinite series has the possibility &gt; of leading to new mathematical constants. ---------- ---------- ---------- ---------- §12. MRB Constant Records, ==================== **Google OpenAI Chat CPT gave the following introduction to the MRB constant records:** &gt; It is not uncommon for researchers and mathematicians to compute large &gt; numbers of digits for mathematical constants or other mathematical &gt; quantities for various reasons. One reason might be to test and &gt; improve numerical algorithms for computing the value of the constant. &gt; Another reason might be to use the constant as a benchmark to test the &gt; performance of a computer or to compare the performance of different &gt; computers. Some people may also be interested in the mathematical &gt; properties of the constant, and computing a large number of digits can &gt; help to reveal patterns or other features of the constant that may not &gt; be apparent with fewer digits. Additionally, some people may simply &gt; find the process of calculating a large number of digits to be a &gt; challenging and rewarding intellectual pursuit. &gt; It seems that the MRB constant, &gt; also known as the CMRB constant, is a mathematical constant defined as &gt; the sum of the infinite series Sum[(-1)^k(k^(1/k)-1),{k,1,n}] and has &gt; a decimal expansion of 0.187859... The MRB constant has been &gt; calculated to various digits using various methods and computers. The &gt; previous record number of digits was 40,000, calculated in 2006 using a &gt; program written in Mathematica and run on a Windows desktop computer. &gt; It is not known whether the MRB constant is algebraic, transcendental, &gt; or irrational. The process of computing the MRB constant using various &gt; methods and devices, including computers and software such as Mathcad, &gt; Mathematica, and Sigma. You have calculated 6,500,000 digits of the &gt; MRB constant using a supercomputer, with a computation time of 72.526 &gt; days and an absolute time of 185.491 days. You have mentioned that the &gt; error from a 6,000,000 or more digit calculation that used a different &gt; method is 0.*10^-6029992. **When asked how, in detail, to calculate the digits of the MRB constant:** &gt; To calculate the digits of the MRB constant in more detail, you can &gt; use the following steps: &gt; &gt; Write a program or use an existing software tool that can calculate &gt; the partial sums of the series (-1)^k k^(1/k) for a given value of n. &gt; &gt; Use this program or tool to calculate the partial sums for a large &gt; value of n, such as n = 10,000 or n = 100,000. &gt; &gt; Repeat step 2 for increasing values of n until the partial sums &gt; converge to a constant value. The more digits of accuracy you want, &gt; the larger n you will need to use. &gt; &gt; The digits of the MRB constant are the digits of the converged value &gt; obtained in step 3. &gt; &gt; It is important to note that calculating the digits of the MRB &gt; constant is a computationally intensive task that requires a fast &gt; computer and a good implementation of the algorithms for calculating &gt; the partial sums of the series (-1)^k k^(1/k). In addition, it is not &gt; known whether the MRB constant is algebraic, transcendental, or even &gt; irrational, so it may not be possible to calculate all of its digits. &gt; &gt; I hope this helps. Please let me know if you have any other questions. My inspiration to compute a lot of digits of **C**&lt;sub&gt;*MRB*&lt;/sub&gt; came from [the following website by Simon Plouffe]. There, computer mathematicians calculate millions, then billions of digits of constants like pi, when with only 65 decimal places of pi, we could determine the size of the observable universe to within a Planck length (where the uncertainty of our measure of the universe would be greater than the universe itself)! In contrast, 65 digits of the MRB constant &#034;measures&#034; the value of -1+ssqrt(2)-3^(1/3) up to n^(1/n) where n is 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000, which can be called 1 unvigintillion or just 10^66. And why compute 65 digits of the MRB constant? Because having that much precision is the only way to solve such a problem as &gt; 1465528573348167959709563453947173222018952610559967812891154^ m-m, &gt; where m is the MRB constant, which gives the near integer &#034;to beat &gt; all,&#034; &gt; 200799291330.9999999999999999999999999999999999999999999999999999999999999900450... And why compute millions of digits of it? uhhhhhhhhhh.... &#034;Because it&#039;s there!&#034; (...Yeah, thanks George Mallory!) And why?? (c&#039;est ma raison d&#039;être!!!) &gt; ![enter image description here] &gt; ![enter image description here] &gt; ![enter image description here] &gt;![enter image description here] So, below you find reproducible results with methods. The utmost care has been taken to assure the accuracy of the record number of digits calculations. These records represent the advancement of consumer-level computers, 21st-century Iterative methods, and clever programming over the past 23 years. Here are some record computations of **C**&lt;sub&gt;*MRB*&lt;/sub&gt;. If you know of any others, let me know, and I will probably add them! 1 digit of the **C**&lt;sub&gt;*MRB*&lt;/sub&gt; with my TI-92s, by adding -1+sqrt(2)-3^(1/3)+4^(1/4)-5^(1/5)+6^(1/6)... as far as I could, was computed. That first digit, by the way, was just 0. Then by using the sum key, to compute$\sum _{n=1}^{1000 } (-1)^n \left(n^{1/n}\right),$the first correct decimal i.e. (.1). It gave (.1_91323989714) which is close to what Mathematica gives for summing to only an upper limit of 1000. ![Ti-92&#039;s] ---------- 4 decimals(.1878) of CMRB were computed on Jan 11, 1999, with the Inverse Symbolic Calculator, applying the command evalf( 0.1879019633921476926565342538468+sum((-1)^n* (n^(1/n)-1),n=140001..150000)); where 0.1879019633921476926565342538468 was the running total of t=sum((-1)^n* (n^(1/n)-1),n=1..10000), then t= t+the sum from (10001.. 20000), then t=t+the sum from (20001..30000) ... up to t=t+the sum from (130001..140000). ---------- 5 correct decimals (rounded to .18786), in Jan of 1999, were drawn from CMRB using Mathcad 3.1 on a 50 MHz 80486 IBM 486 personal computer operating on Windows 95. ---------- 9 digits of CMRB shortly afterward using Mathcad 7 professional on the Pentium II mentioned below, by summing (-1)^x x^(1/x) for x=1 to 10,000,000, 20,000,000, and many more, then linearly approximating the sum to a what a few billion terms would have given. ---------- 500 digits of CMRB with an online tool called Sigma on Jan 23, 1999. See [http://marvinrayburns.com/Original_MRB_Post.html] if you can read the printed and scanned copy there. Sigma still can be found [here.] ---------- 5,000 digits of CMRB in September of 1999 in 2 hours on a 350 MHz PentiumII,133 MHz 64 MB of RAM using the simple PARI commands \p 5000;sumalt(n=1,((-1)^n*(n^(1/n)-1))), after allocating enough memory. To beat that, I did it on July 4, 2022, in 1 second on the 5.5 GHz CMRBSC 3 with 4800MHz 64 GB of RAM by Newton&#039;s method using Convergence acceleration of alternating series. Henri Cohen, Fernando Rodriguez Villegas, Don Zagier acceleration &#034;Algorithm 1&#034; to at least 5000 decimals. (* Newer loop with Newton interior. *) ![PII] documentation [here] ---------- 6,995 accurate digits of CMRB were computed on June 10-11, 2003, over a period, of 10 hours, on a 450 MHz P3 with an available 512 MB RAM, To beat that, I did it in &lt;2.5 seconds on the MRBCSC 3 on July 7, 2022 (more than 14,400 times as fast!) ![PIII] documentation [here] ---------- 8000 digits of CMRB completed, using a Sony Vaio P4 2.66 GHz laptop computer with 960 MB of available RAM, at 2:04 PM 3/25/2004, ---------- 11,000 digits of CMRB&gt; on March 01, 2006, with a 3 GHz PD with 2 GB RAM available calculated. ---------- 40 000 digits of CMRB in 33 hours and 26 min via my program written in Mathematica 5.2 on Nov 24, 2006. The computation was run on a 32-bit Windows 3 GHz PD desktop computer using 3.25 GB of Ram. The program was Block[{a, b = -1, c = -1 - d, d = (3 + Sqrt)^n, n = 131 Ceiling[40000/100], s = 0}, a = 1; d = (d + 1/d)/2; For[m = 1, m &lt; n, a[m] = (1 + m)^(1/(1 + m)); m++]; For[k = 0, k &lt; n, c = b - c; b = b (k + n) (k - n)/((k + 1/2) (k + 1)); s = s + c*a[k]; k++]; N[1/2 - s/d, 40000]] ---------- 60,000 digits of CMRB on July 29, 2007, at 11:57 PM EST in 50.51 hours on a 2.6 GHz AMD Athlon with 64-bit Windows XP. The max memory used was 4.0 GB of RAM. ---------- 65,000 digits of CMRB in only 50.50 hours on a 2.66 GHz Core 2 Duo using 64-bit Windows XP on Aug 3, 2007, at 12:40 AM EST, The max memory used was 5.0 GB of RAM. ---------- 100,000 digits of CMRB on Aug 12, 2007, at 8:00 PM EST, were computed in 170 hours on a 2.66 GHz Core 2 Duo using 64-bit Windows XP. The max memory used was 11.3 GB of RAM. The typical daily record of memory used was 8.5 GB of RAM. To beat that, on the 4th of July, 2022, I computed the same digits in 1/4 of an hour. ![CTRL+f] &#034;4th of July, 2022&#034; for documentation. To beat that, on the 7th of July, 2022, I computed the same digits in 1/5 of an hour. ![CTRL+f] &#034;7th of July, 2022&#034; for documentation (850 times as fast as the first 100,000 run!) ---------- 150,000 digits of CMRB on Sep 23, 2007, at 11:00 AM EST. Computed in 330 hours on a 2.66 GHz Core 2 Duo using 64-bit Windows XP. The max memory used was 22 GB of RAM. The typical daily record of memory used was 17 GB of RAM. ---------- 200,000 digits of CMRB using Mathematica 5.2 on March 16, 2008, at 3:00 PM EST,. Found in 845 hours, on a 2.66 GHz Core 2 Duo using 64-bit Windows XP. The max memory used was 47 GB of RAM. The typical daily record of memory used was 28 GB of RAM. ---------- 300,000 digits of CMRB were destroyed (washed away by Hurricane Ike ) on September 13, 2008 sometime between 2:00 PM - 8:00 PM EST. Computed for a long 4015. Hours (23.899 weeks or 1.4454*10^7 seconds) on a 2.66 GHz Core 2 Duo using 64-bit Windows XP. The max memory used was 91 GB of RAM. The Mathematica 6.0 code is used as follows: Block[{$MaxExtraPrecision = 300000 + 8, a, b = -1, c = -1 - d, d = (3 + Sqrt)^n, n = 131 Ceiling[300000/100], s = 0}, a = 1; d = (d + 1/d)/2; For[m = 1, m &lt; n, a[m] = (1 + m)^(1/(1 + m)); m++]; For[k = 0, k &lt; n, c = b - c; b = b (k + n) (k - n)/((k + 1/2) (k + 1)); s = s + c*a[k]; k++]; N[1/2 - s/d, 300000]] ---------- 225,000 digits of CMRB were started with a 2.66 GHz Core 2 Duo using 64-bit Windows XP on September 18, 2008. It was completed in 1072 hours. ---------- 250,000 digits were attempted but failed to be completed to a serious internal error that restarted the machine. The error occurred sometime on December 24, 2008, between 9:00 AM and 9:00 PM. The computation began on November 16, 2008, at 10:03 PM EST. The Max memory used was 60.5 GB. ---------- 250,000 digits of CMRB on Jan 29, 2009, 1:26:19 pm (UTC-0500) EST, with a multiple-step Mathematica command running on a dedicated 64-bit XP using 4 GB DDR2 RAM onboard and 36 GB virtual. The computation took only 333.102 hours. The digits are at http://marvinrayburns.com/250KMRB.txt. The computation is completely documented. ---------- 300000 digit search of CMRB was initiated using an i7 with 8.0 GB of DDR3 RAM onboard on Sun 28 Mar 2010 at 21:44:50 (UTC-0500) EST, but it failed due to hardware problems. ---------- 299,998 Digits of CMRB: The computation began Fri 13 Aug 2010 10:16:20 pm EDT and ended 2.23199*10^6 seconds later | Wednesday, September 8, 2010. I used Mathematica 6.0 for Microsoft Windows (64-bit) (June 19, 2007), which averages 7.44 seconds per digit. I used my Dell Studio XPS 8100 i7 860 @ 2.80 GHz with 8GB physical DDR3 RAM. Windows 7 reserved an additional 48.929 GB of virtual Ram. ---------- 300,000 digits to the right of the decimal point of CMRB from Sat 8 Oct 2011 23:50:40 to Sat 5 Nov 2011 19:53:42 (2.405*10^6 seconds later). This run was 0.5766 seconds per digit slower than the 299,998 digit computation even though it used 16 GB physical DDR3 RAM on the same machine. The working precision and accuracy goal combination were maximized for exactly 300,000 digits, and the result was automatically saved as a file instead of just being displayed on the front end. Windows reserved a total of 63 GB of working memory, of which 52 GB were recorded as being used. The 300,000 digits came from the Mathematica 7.0 command Quit; DateString[] digits = 300000; str = OpenWrite[]; SetOptions[str, PageWidth -&gt; 1000]; time = SessionTime[]; Write[str, NSum[(-1)^n*(n^(1/n) - 1), {n, \[Infinity]}, WorkingPrecision -&gt; digits + 3, AccuracyGoal -&gt; digits, Method -&gt; &#034;AlternatingSigns&#034;]]; timeused = SessionTime[] - time; here = Close[str] DateString[] ---------- 314159 digits of the constant took 3 tries due to hardware failure. Finishing on September 18, 2012, I computed 314159 digits, taking 59 GB of RAM. The digits came from the Mathematica 8.0.4 code DateString[] NSum[(-1)^n*(n^(1/n) - 1), {n, \[Infinity]}, WorkingPrecision -&gt; 314169, Method -&gt; &#034;AlternatingSigns&#034;] // Timing DateString[] ---------- 1,000,000 digits of CMRB for the first time in history in 18 days, 9 hours 11 minutes, 34.253417 seconds by Sam Noble of the Apple Advanced Computation Group. ---------- 1,048,576 digits of CMRB in a lightning-fast 76.4 hours, finishing on Dec 11, 2012, were scored by Dr. Richard Crandall, an Apple scientist and head of its advanced computational group. That&#039;s on a 2.93 GHz 8-core Nehalem, 1066 MHz, PC3-8500 DDR3 ECC RAM. To beat that, in Aug of 2018, I computed 1,004,993 digits in 53.5 hours 34 hours computation time (from the timing command) with 10 DDR4 RAM (of up to 3000 MHz) supported processor cores overclocked up to 4.7 GHz! Search this post for &#034;53.5&#034; for documentation. To beat that, on Sept 21, 2018: I computed 1,004,993 digits in 50.37 hours of absolute time and 35.4 hours of computation time (from the timing command) with 18 (DDR3 and DDR4) processor cores! Search this post for &#034;50.37 hours&#034; for documentation.** To beat that, on May 11, 2019, I computed over 1,004,993 digits in 45.5 hours of absolute time and only 32.5 hours of computation time, using 28 kernels on 18 DDR4 RAM (of up to 3200 MHz) supported cores overclocked up to 5.1 GHz Search &#039;Documented in the attached &#034;:3 fastest computers together 3.nb.&#034; &#039; for the post that has the attached documenting notebook. To beat that, I accumulated over 1,004,993 correct digits in 44 hours of absolute time and 35.4206 hours of computation time on 10/19/20, using 3/4 of the MRB constant supercomputer 2 -- see https://www.wolframcloud.com/obj/bmmmburns/Published/44%20hour%20million.nb for documentation. To beat that, I did a 1,004,993 correct digits computation in 36.7 hours of absolute time and only 26.4 hours of computation time on Sun 15 May 2022 at 06:10:50, using 3/4 of the MRB constant supercomputer 3. Ram Speed was 4800MHz, and all 30 cores were clocked at up to 5.2 GHz. To beat that, I did a 1,004,993 correct digits computation in 30.5 hours of absolute time and 15.7 hours of computation time from the Timing[] command using 3/4 of the MRB constant supercomputer 4, finishing Dec 5, 2022. Ram Speed was 5200MHz, and all of the 24 performance cores were clocked at up to 5.95 GHz, plus 32 efficiency cores running slower. I used 24 kernels on the Wolfram Lightweight grid over an i-12900k, 12900KS, and 13900K. [36.7 hours million notebook] [30.5 hours million] ---------- A little over 1,200,000 digits, previously, of CMRB in 11 days, 21 hours, 17 minutes, and 41 seconds (I finished on March 31, 2013, using a six-core Intel(R) Core(TM) i7-3930K CPU @ 3.20 GHz. see https://www.wolframcloud.com/obj/bmmmburns/Published/36%20hour%20million.nb for details. ---------- 2,000,000 or more digit computation of CMRB on May 17, 2013, using only around 10GB of RAM. It took 37 days 5 hours, 6 minutes 47.1870579 seconds. I used my six-core Intel(R) Core(TM) i7-3930K CPU @ 3.20 GHz. ---------- 3,014,991 digits of CMRB, world record computation of **C**&lt;sub&gt;*MRB*&lt;/sub&gt; was finished on Sun 21 Sep 2014 at 18:35:06. It took 1 month 27 days 2 hours 45 minutes 15 seconds. The processor time from the 3,000,000+ digit computation was 22 days. I computed the 3,014,991 digits of **C**&lt;sub&gt;*MRB*&lt;/sub&gt; with Mathematica 10.0. I Used my new version of Richard Crandall&#039;s code in the attached 3M.nb, optimized for my platform and large computations. I also used a six-core Intel(R) Core(TM) i7-3930K CPU @ 3.20 GHz with 64 GB of RAM, of which only 16 GB was used. Can you beat it (in more digits, less memory used, or less time taken)? This confirms that my previous &#034;2,000,000 or more digit computation&#034; was accurate to 2,009,993 digits. they were used to check the first several digits of this computation. See attached 3M.nb for the full code and digits. ---------- Over 4 million digits of CMRB were finished on Wed 16 Jan 2019, 19:55:20. It took 4 years of continuous tries. This successful run took 65.13 days absolute time, with a processor time of 25.17 days, on a 3.7 GHz overclocked up to 4.7 GHz on all cores Intel 6 core computer with 3000 MHz RAM. According to this computation, the previous record, 3,000,000+ digit computation, was accurate to 3,014,871 decimals, as this computation used my algorithm for computing n^(1/n) as found in chapter 3 in the paper at https://www.sciencedirect.com/science/article/pii/0898122189900242 and the 3 million+ computation used Crandall&#039;s algorithm. Both algorithms outperform Newton&#039;s method per calculation and iteration. Example use of M R Burns&#039; algorithm to compute 123456789^(1/123456789) 10,000,000 digits: ClearSystemCache[]; n = 123456789; (*n is the n in n^(1/n)*) x = N[n^(1/n),100]; (*x starts out as a relatively small precision approximation to n^(1/n)*) pc = Precision[x]; pr = 10000000; (*pr is the desired precision of your n^(1/n)*) Print[t0 = Timing[While[pc &lt; pr, pc = Min[4 pc, pr]; x = SetPrecision[x, pc]; y = x^n; z = (n - y)/y; t = 2 n - 1; t2 = t^2; x = x*(1 + SetPrecision[4.5, pc] (n - 1)/t2 + (n + 1) z/(2 n t) - SetPrecision[13.5, pc] n (n - 1)/(3 n t2 + t^3 z))]; (*You get a much faster version of N[n^(1/n),pr]*) N[n - x^n, 10]](*The error*)]; ClearSystemCache[]; n = 123456789; Print[t1 = Timing[N[n - N[n^(1/n), pr]^n, 10]]] Gives {25.5469,0.*10^-9999984} {101.359,0.*10^-9999984} More information is available upon request. ---------- More than 5 million digits of CMRB were found on Fri 19 Jul 2019, 18:49:02; methods are described in the reply below, which begins with &#034;Attempts at a 5,000,000 digit calculation .&#034; For this 5 million digit calculation of **C**&lt;sub&gt;*MRB*&lt;/sub&gt; using the 3 node MRB supercomputer: processor time was 40 days. And the actual time was 64 days. That is in less absolute time than the 4-million-digit computation, which used just one node. ---------- 6,000,000 digits of CMRB after 8 tries in 19 months. (Search &#034;8/24/2019 It&#039;s time for more digits!&#034; below.) finishing on Tue, 30 Mar 2021, at 22:02:49 in 160 days. The MRB constant supercomputer 2 said the following: Finished on Tue 30 Mar 2021, 22:02:49. computation and absolute time were 5.28815859375*10^6 and 1.38935720536301*10^7 s. respectively Enter MRB1 to print 6029991 digits. The error from a 5,000,000 or more-digit calculation that used a different method is 0.*10^-5024993. That means that the 5,000,000-digit computation Was accurate to 5024993 decimals!!! ---------- 5,609,880, verified by 2 distinct algorithms for x^(1/x), digits of CMRB on Thu 4 Mar 2021 at 08:03:45. The 5,500,000+ digit computation using a totally different method showed that many decimals are in common with the 6,000,000+ digit computation in 160.805 days. ---------- 6,500,000 digits of CMRB on my second try, The MRB constant supercomputer said, Finished on Wed 16 Mar 2022 02: 02: 10. computation and absolute time were 6.26628*10^6 and 1.60264035419592*10^7s respectively Enter MRB1 to print 6532491 digits. The error from a 6, 000, 000 or more digit the calculation that used a different method is 0.*10^-6029992. &#034;Computation time&#034; 72.526 days. &#034;Absolute time&#034; 185.491 days. ---------- ---------- ---------- ---------- ---------- ---------- ---------- : https://community.wolfram.com//c/portal/getImageAttachment?filename=5686test.gif&amp;userId=366611 : https://community.wolfram.com//c/portal/getImageAttachment?filename=Screenshot2022-12-15085733.jpg&amp;userId=366611 : https://community.wolfram.com//c/portal/getImageAttachment?filename=Screenshot2022-12-15084941.jpg&amp;userId=366611 : https://community.wolfram.com//c/portal/getImageAttachment?filename=Screenshot2023-02-04194312.jpg&amp;userId=366611 : https://community.wolfram.com//c/portal/getImageAttachment?filename=Screenshot2023-01-03030734.jpg&amp;userId=366611 : https://community.wolfram.com//c/portal/getImageAttachment?filename=0a.gif&amp;userId=366611 : 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https://community.wolfram.com//c/portal/getImageAttachment?filename=Screenshot2022-10-28054251.jpg&amp;userId=366611 : https://community.wolfram.com//c/portal/getImageAttachment?filename=Screenshot2023-01-18140654.jpg&amp;userId=366611 : https://www.wolframcloud.com/obj/965639ac-19d7-4dc8-ba67-4d47215e25c4 : https://community.wolfram.com//c/portal/getImageAttachment?filename=101513.PNG&amp;userId=366611 : https://en.wikipedia.org/wiki/Oscillatory_integral : https://scholar.google.com/scholar?hl=en&amp;as_sdt=0,15&amp;q=%22MKB%20constant%22&amp;btnG= : https://community.wolfram.com//c/portal/getImageAttachment?filename=46311.PNG&amp;userId=366611 : https://community.wolfram.com//c/portal/getImageAttachment?filename=580910.JPG&amp;userId=366611 : http://community.wolfram.com/groups/-/m/t/1323951?p_p_auth=W3TxvEwH : https://community.wolfram.com//c/portal/getImageAttachment?filename=28491.JPG&amp;userId=366611 : https://community.wolfram.com//c/portal/getImageAttachment?filename=76812.JPG&amp;userId=366611 : https://community.wolfram.com//c/portal/getImageAttachment?filename=100173.JPG&amp;userId=366611 : https://community.wolfram.com//c/portal/getImageAttachment?filename=57664.JPG&amp;userId=366611 : https://community.wolfram.com//c/portal/getImageAttachment?filename=74665.JPG&amp;userId=366611 : https://community.wolfram.com//c/portal/getImageAttachment?filename=49236.JPG&amp;userId=366611 : https://community.wolfram.com//c/portal/getImageAttachment?filename=15127.JPG&amp;userId=366611 : https://community.wolfram.com//c/portal/getImageAttachment?filename=92858.JPG&amp;userId=366611 : https://community.wolfram.com//c/portal/getImageAttachment?filename=49309.JPG&amp;userId=366611 : https://community.wolfram.com//c/portal/getImageAttachment?filename=3.PNG&amp;userId=366611 : https://arxiv.org/abs/0912.3844 : http://community.wolfram.com//c/portal/getImageAttachment?filename=Capturemkb2.JPG&amp;userId=366611 : https://community.wolfram.com//c/portal/getImageAttachment?filename=Screenshot2022-07-05020911.jpg&amp;userId=366611 : https://community.wolfram.com//c/portal/getImageAttachment?filename=Screenshot2022-07-05021130.jpg&amp;userId=366611 : https://community.wolfram.com//c/portal/getImageAttachment?filename=Screenshot2022-07-05022313.jpg&amp;userId=366611 : https://www.wolframcloud.com/obj/09dd1776-b25e-482e-9bd6-fcab9742a05c : https://www.wolframcloud.com/obj/9fea1f20-f12d-49a5-98f9-22a8e0026375 : https://community.wolfram.com//c/portal/getImageAttachment?filename=Screenshot2022-07-31235917.jpg&amp;userId=366611 : https://community.wolfram.com//c/portal/getImageAttachment?filename=Screenshot2022-08-01001529.jpg&amp;userId=366611 : https://community.wolfram.com//c/portal/getImageAttachment?filename=Screenshot2022-08-01001459.jpg&amp;userId=366611 : https://www.wolframcloud.com/obj/88e5ed59-3d0e-4a63-abdf-c48a053428cd : https://community.wolfram.com//c/portal/getImageAttachment?filename=Screenshot2022-08-14085504.jpg&amp;userId=366611 : https://community.wolfram.com//c/portal/getImageAttachment?filename=22421.JPG&amp;userId=366611 : https://community.wolfram.com//c/portal/getImageAttachment?filename=97362.JPG&amp;userId=366611 : https://community.wolfram.com//c/portal/getImageAttachment?filename=31093.JPG&amp;userId=366611 : https://community.wolfram.com//c/portal/getImageAttachment?filename=63064.JPG&amp;userId=366611 : https://community.wolfram.com/groups/-/m/t/1323951?p_p_auth=zHVSqCM8 : https://www.researchgate.net/publication/309187705_Gelfond%27_s_Constant_using_MKB_constant_like_integrals : https://community.wolfram.com//c/portal/getImageAttachment?filename=Screenshot2022-05-03083042.jpg&amp;userId=366611 : https://community.wolfram.com//c/portal/getImageAttachment?filename=Screenshot2022-08-16204240.jpg&amp;userId=366611 : https://www.researchgate.net/publication/309187705sConstantusingMKBconstantlikeintegrals : https://community.wolfram.com//c/portal/getImageAttachment?filename=Screenshot2022-06-08111905.jpg&amp;userId=366611 : https://community.wolfram.com//c/portal/getImageAttachment?filename=73892.JPG&amp;userId=366611 : https://community.wolfram.com//c/portal/getImageAttachment?filename=Screenshot2022-11-13083115.jpg&amp;userId=366611 : https://community.wolfram.com//c/portal/getImageAttachment?filename=Screenshot2022-08-16210900.jpg&amp;userId=366611 : https://community.wolfram.com//c/portal/getImageAttachment?filename=Screenshot2022-11-13083322.jpg&amp;userId=366611 : https://community.wolfram.com//c/portal/getImageAttachment?filename=b2.JPG&amp;userId=366611 : https://bctp.berkeley.edu/extraD.html : https://www.academia.edu/search?q=MRB%20constant : https://www.researchgate.net/profile/Michele-Nardelli : https://en.wikipedia.org/wiki/Richard_Crandall : https://scholar.google.com/scholar?hl=en&amp;as_sdt=0,15&amp;q=key%20fundamental%20constant%20zeta&amp;btnG= : https://community.wolfram.com//c/portal/getImageAttachment?filename=Screenshot2022-05-02113359.jpg&amp;userId=366611 : http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.695.5959&amp;rep=rep1&amp;type=pdf : https://www.marvinrayburns.com/UniversalTOC25.pdf : https://www2.mpia-hd.mpg.de/~mathar/ : https://arxiv.org/abs/0912.3844 : https://pdfs.semanticscholar.org/f3dc/9f828442123015c5d52535fdefc0ab450bf5.pdf : https://web.archive.org/web/20120310223600/http://pi.lacim.uqam.ca/eng/records_en.html : https://community.wolfram.com//c/portal/getImageAttachment?filename=P1.JPG&amp;userId=366611 : https://community.wolfram.com//c/portal/getImageAttachment?filename=P2.JPG&amp;userId=366611 : https://community.wolfram.com//c/portal/getImageAttachment?filename=P3.JPG&amp;userId=366611 : https://community.wolfram.com//c/portal/getImageAttachment?filename=P5.JPG&amp;userId=366611 : https://community.wolfram.com//c/portal/getImageAttachment?filename=Screenshot2023-01-31143800.jpg&amp;userId=366611 : http://marvinrayburns.com/Original_MRB_Post.html : https://wims.univ-cotedazur.fr/wims/wims.cgi?module=tool/analysis/sigma.en : https://community.wolfram.com//c/portal/getImageAttachment?filename=thumbnail%281%29.jpg&amp;userId=366611 : https://www.wolframcloud.com/obj/18c08d6b-abe9-4fbd-b33a-7e1167c9d243 : https://community.wolfram.com//c/portal/getImageAttachment?filename=thumbnail%282%29.jpg&amp;userId=366611 : https://www.wolframcloud.com/obj/0b304705-ed14-4c6b-bd27-fda9ff29536d : https://community.wolfram.com//c/portal/getImageAttachment?filename=0a.gif&amp;userId=366611 : https://community.wolfram.com//c/portal/getImageAttachment?filename=0a.gif&amp;userId=366611 : https://www.wolframcloud.com/obj/bmmmburns/Published/36%20hour%20million.nb : https://www.wolframcloud.com/obj/bmmmburns/Published/31_hour_million.nb Marvin Ray Burns 2014-10-09T18:08:49Z How to solve Sinh and Cosh functions summation over 0 to infinity https://community.wolfram.com/groups/-/m/t/2825897 Dear All, I am trying to calculate a non linear equation which consists of hyperbolic functions Sinh and Cosh. I would like to calculate the numerical value summation over 0 to infinity. by numerical minimization using Nminimize function. The notebook is attached here. When I am trying to minimise the function with respect to lambdaHybm, The calcultion does not work. However if I put some specific value (10, 100, 9999) insted of Infinite in equation, It gives some value. But, The value are different for different range of summation. How I can calculate from 0 to Infinite? Could you please give some suggestions. Thanks, Best Regards, Sukanta &amp;[Wolfram Notebook] : https://www.wolframcloud.com/obj/a822c7fa-5f24-4543-9701-586bb03dd27f Sukanta Kumar Jena 2023-02-08T11:02:23Z Compute definite integrals with tanh? no output https://community.wolfram.com/groups/-/m/t/2825265 I am trying to compute the following definite integral, Integrate[Sqrt[g*m]Divide[Tanh[Divide[Sqrt[g*k],Sqrt[m]]x],Sqrt[k]],{x,0,x}] (Paste this into Wolfram to see it) If you are too lazy, it is basically the definite integral from 0 to x of some constants multiplied by tanh(constants*x) dx For some reason, even though I have Wolfram Pro, it attempts to compute this integral for less than 10 seconds, then just stops, with no answer, no error message, nothing. What is going on? Is the integral simply too hard for Wolfram to compute? Please help. Ron Ellenbogen 2023-02-07T11:20:39Z How to compute the gradient and divergence of a 'Tensor Product' expression https://community.wolfram.com/groups/-/m/t/2825811 Dear Wolfram Community: I am facing the problem of computing the gradient and divergence of a general tensor field. In particular, expressions like the following ones (in Mathematica symbolic notation). Grad[TensorProduct[A[t,X], B[t,X]],X]; Div[TensorProduct[A[t,X], B[t,X]],X]; Where: A[t,X] and B[t,X] are two general tensor fields that are function of time &#034;t&#034; and the position vector &#034;X&#034;. Those expressions are the more general forms (at least I hope) of the cited operators, very useful in continuum mechanics (specially in fluid mechanics: the Reynolds stress tensor is a &#039;tensor product&#039; of turbulent velocity fields) Unfortunately, the related identities are difficult to find in the bibliography/references. I would like to know how to implement the product rule in symbolic notation. Thank you in advance, Alberto Silva Notes: 1) I tried to do it before in explicit, component-by component notation, and the result was a very big, messy and confusing expression... 2) I am leaving a small notebook written in the last version of Wolfram Mathematica as an example. 3) A year ago I got a clue about how to do this computation (inserting a &#039;custom operator&#039; using inactive expressions and replace rules) in the following community post: [How to use/ evaluate symbolic vector calculus identities?] The main problem that was left then was the true expression of the divergence was unknown and I just &#039;guessed&#039; it, so my guess could be wrong. : https://community.wolfram.com/groups/-/m/t/2303932 Alberto Silva Ariano 2023-02-07T23:25:12Z How do you create an array in the web-based Mathematica Student Edition? https://community.wolfram.com/groups/-/m/t/2825596 We are working with large matrices and it would help to be able to insert a blank matrix like you can in a notebook and quickly put in the numbers, rather than use the cumbersome {{a,b},{c,d}} notation. In a notebook, you can do Insert &gt; table/matrix, but this is not available in the Student edition. The basic math assistant palette is also missing. You also can&#039;t copy and paste a matrix in matrix form from the output to a new input to change the numbers. Any help would be greatly appreciated. Fumiko Futamura 2023-02-07T22:25:36Z Grassmann Algebra & Calculus Paclet Available https://community.wolfram.com/groups/-/m/t/2825558 A Mathematica paclet application for Grassmann Algebra and Calculus is now available at Dropbox. The link for the paclet and my email address are in my profile. The key attribute of Grassmann algebra is that it algebraicizes the notion and modeling of linear dependence and independence. It can be interpreted to many applications with contextual notation. The most important applications are mathematics, geometry, physics and engineering. Because of its basic antisymmetric nature, it is tensorial in nature being invariant to coordinate transformations. One can write coordinate free expressions applicable to finite dimensions. A metric construct and measure arises from its basic axioms. One can compute with or without a metric. It can distinguish between points and free vectors. It can treat basis vectors as derivative operators, and reciprocal basis vectors as differential forms. It is, as John Browne writes: &#034; a geometric calculus par excellence&#034;. The Grassmann algebra contains exterior, regressive, interior, generalized, hypercomplex, associative, and Clifford products. The Grassmann complement is a generalized Hodge star operator. The calculus routines include vector operators, the exterior derivative and the generalized vector calculus operators. All advances product expressions can be simplified to scalars and canonical exterior expressions. There are various built-in spaces with their associated coordinates, bases (vector, differential form and orthonormal), metrics and symbols. It&#039;s possible to define spaces and then switch between various spaces on the fly. Grassmann Calculus is thus a powerful application for education or work in multilinear algebra, geometry, differential geometry, physics and engineering. The GrassmannCalculus Palette, available from the Mathematica Palettes menu, is very useful when using the application. Another useful palette is the Common Grassmann Operations palette available from the GrassmannCalculus Palette, drop-down Palettes menu. John Browne&#039;s Foundations book is included (as notebooks) in the paclet. For the paclet to be useful one must learn the foundations, which is probably equivalent in time to a two semester college course or maybe learning tensor calculus. The extended Grassmann algebra theory and routines were developed by John Browne and follow closely Hermann Grassmann&#039;s work. The calculus routines were written by David Park who also designed the user interface. You can contact David Park at the email address in my profile. John Browne&#039;s (1942-2021) web site is at: [Grassmann Algebra] : https://sites.google.com/view/grassmann-algebra/home David J M Park Jr 2023-02-07T19:34:39Z How to fix Part::partw error https://community.wolfram.com/groups/-/m/t/2824909 How to correct the code which produces &gt; Part::partw &gt; Part::partd errors  Part::partw: Part 2 of Y[tau] does not exist. Part::partd: Part specification {Y-&gt;InterpolatingFunction[{{-2.11454,0.}},{5,3,1,{437},{4},0,0,0,0,Automatic,{},{},False},{{-2.11454,&lt;&lt;49&gt;&gt;,&lt;&lt;387&gt;&gt;}},{{{-34.6432,8.237},{3.35084*10^11,-6.32877*10^10}},&lt;&lt;49&gt;&gt;,&lt;&lt;387&gt;&gt;},{Automatic}]}[[1,All,1,2]] is longer than depth of object. &amp;[Code with error] : https://www.wolframcloud.com/obj/c367aef1-2586-4948-9440-17a34a2a6fb2 John Wick 2023-02-06T19:22:55Z How to integrate tanx/(x+sinx)? https://community.wolfram.com/groups/-/m/t/2824858 I use a website called integral of the day, all the previous ones have answers, but todays I cannot ind an answer for and no online calculators can do it either. Charlie Oke 2023-02-06T21:46:21Z [WSG23] Daily Study Group: Introduction to Probability https://community.wolfram.com/groups/-/m/t/2825207 A Wolfram U Daily Study Group on Introduction to Probability begins on **February 27th 2023**. Join me and a group of fellow learners to learn about the world of probability and statistics using the Wolfram Language. Our topics for the study group include the characterisation of randomness, random variable design and analysis, important random distributions and their applications, probability-based data science and advanced probability distributions. The idea behind this study group is to rapidly develop an intuitive understanding of probability for a college student, professional or interested hobbyist. A basic working knowledge of the Wolfram Language is recommended but not necessary. We are happy to help beginners get up to speed with Wolfram Language using resources already available on Wolfram U. Please feel free to use this thread to collaborate and share ideas, materials and links to other resources with fellow learners. [**REGISTER HERE**] ![Wolfram U Banner] : https://www.bigmarker.com/series/daily-study-group-probability-wsg36/series_details : https://community.wolfram.com//c/portal/getImageAttachment?filename=banner.jpg&amp;userId=2823613 Marc Vicuna 2023-02-07T01:15:38Z Same code with involving integer takes longer time https://community.wolfram.com/groups/-/m/t/2824228 Hello, Np = 500; a = 1; b = 4.0; h = (b - a)/Np; x0 = a; y = 1; Y = {{x0, y}}; f[x_, y_] = -y*(x + 1) - Cos[x]; For[m = 0, m &lt;= Np - 1, m++, xm = a + m*h; y[m + 1] = y[m] + h*f[xm, y[m]]; AppendTo[Y, {xm + h, y[m + 1]}]; ]; g2 = ListLinePlot[Y, PlotStyle -&gt; Green, PlotRange -&gt; All] ![enter image description here] Np = 500; a = 1; b = 4; h = (b - a)/Np; x0 = a; y = 1; Y = {{x0, y}}; f[x_, y_] = -y*(x + 1) - Cos[x]; For[m = 0, m &lt;= Np - 1, m++, xm = a + m*h; y[m + 1] = y[m] + h*f[xm, y[m]]; AppendTo[Y, {xm + h, y[m + 1]}]; ]; g2 = ListLinePlot[Y, PlotStyle -&gt; Green, PlotRange -&gt; All] ![enter image description here] : https://community.wolfram.com//c/portal/getImageAttachment?filename=5983Capture1.JPG&amp;userId=2803344 : https://community.wolfram.com//c/portal/getImageAttachment?filename=10109Capture2.JPG&amp;userId=2803344 So, the only difference is in b, in that in the first case b=4.0, while in the second case b=4. In the first case the code runs quickly and displays the result, while in the second case, the code runs endlessly and does not display the result. Can someone explain how exactly that .0 influences the running of the code so much? Thank you. Cornel B. 2023-02-05T18:18:47Z Improved Eulers Method Problem https://community.wolfram.com/groups/-/m/t/2815256 It is given the following problem: $$X&#039;=Z+(Y-\alpha)X$$ $$Y&#039;=1-\beta Y-X^2$$ $$Z&#039;=-X-\gamma Z$$ with initial conditions $(X(0),Y(0),Z(0)=(1,2,3)$. Where X: interest rate Υ:investment demand Z: price index $\alpha$: savings, $\beta$: cost per investment, $\gamma$: the absolute value of the elasticity of demand  \[Alpha]=0.9; \[Beta]=0.2; \[Gamma]=1.2; f[x_,y_,z_]:=z+(y-\[Alpha])*x g[x_,y_,z_]:=1-\[Beta]*y-x^2 h[x_,y_,z_]:=-x-\[Gamma]*z  I know how to define a system with two functions so I could use these - Method A: accuracy of order h S[a_, b_, h_, N_] := (u = a; u = a + h*b; Do[u[n + 1] = 2 u[n] - u[n - 1] + h*h*f[n*h, u[n], (u[n] - u[n - 1])/h], {n, 1, N}]) - Method B: accuracy of order h^2 Q[a_, b_, h_, N_] := (u = a; v = b; Do[{u[n + 1] = u[n] + h* F[u[n] + (h/2)*F[u[n], v[n]], v[n] + (h/2)* G[u[n], v[ n]]], \ v[n + 1] = v[n] + h* G[u[n] + (h/2)*F[u[n], v[n]], v[n] + (h/2)*G[u[n], v[n]]]}, {n, 0, N}]) Athanasios Paraskevopoulos 2023-01-26T12:03:14Z Formulas of interpolation/extrapolation method spline https://community.wolfram.com/groups/-/m/t/2824367 Hello, I would like to understand the formula behind the following interpolation and extrapolation done in mathematica in order to convert it in python: K is equal to {27.827, 18.53, -30.417} 0.205 {22.145, 13.687, -33.282} 0.197 {29.018, 18.841, -38.761} 0.204 {26.232, 22.327, -27.735} 0.197 {28.761, 21.565, -31.586} 0.212 {24.002, 17.759, -24.782} 0.208 {17.627, 18.224, -25.197} 0.204 {24.834, 20.538, -33.012} 0.205 {23.017, 23.037, -29.23} 0.211 {26.263, 23.686, -32.766} 0.215 K = ToExpression[Import[NotebookDirectory[] &lt;&gt; &#034;df.txt&#034;, &#034;TSV&#034;]]; func = Interpolation[Map[{#[], #[]} &amp;, K], InterpolationOrder -&gt; {1, 1, 1}, Method -&gt; &#034;Spline&#034;]; result= func[2, 2, -3] 0.0298707 result1 = func[20,20,-30] 0.168097 To have these results which is the formula applied using the variables above? Thanks Daniela Tortora 2023-02-06T12:10:34Z Symbolic formulas for Airy equations? https://community.wolfram.com/groups/-/m/t/2823543 I can find the Airy formula&#039;s in Mathematica under the author&#039;s name. Ai and Bi. I do not care to work with the formula&#039;s as named expressions. I would prefer to work with symbols, like Cosine and the Integral and so forth and so on. How do I convert these named expressions to symbolic relations? Alan White 2023-02-04T01:44:26Z Using vector variables and changing variables in ODEs https://community.wolfram.com/groups/-/m/t/2823806 &amp;[Wolfram Notebook] : https://www.wolframcloud.com/obj/ea736d97-ac0b-42f9-b87d-bc5902508eda Peter Burbery 2023-02-04T15:36:19Z Simplify function argument https://community.wolfram.com/groups/-/m/t/2823944 In a more complicated expression I would like Mathematica to simplify Integrate[f[x - Floor[x]], {x, 0, 1}] to Integrate[f[x], {x, 0,1 }] Simplify[...] doesn&#039;t seem to do it. Ideas? Thanks. julio kuplinsky 2023-02-04T14:48:53Z Great rhombic triacontahedron explaoned https://community.wolfram.com/groups/-/m/t/2823046 &amp;[Wolfram Notebook] : https://www.wolframcloud.com/obj/037b0049-c7c1-4ba4-92fa-1f00fa713279 Sandor Kabai 2023-02-03T17:49:46Z