Community RSS Feed https://community.wolfram.com RSS Feed for Wolfram Community showing any discussions in tag Wolfram Language sorted by active Clear all variables defined in a given cell. https://community.wolfram.com/groups/-/m/t/2623005 Based on the suggested method [here](https://mathematica.stackexchange.com/questions/155643/is-it-possible-to-clear-all-variables-defined-in-a-given-cell), I try to clear all variables defined in a given cell as follows: In:= SelectionMove[EvaluationNotebook[],All,EvaluationCell];Options[NotebookSelection[EvaluationNotebook[]],CellLabel] {cell} = Cells@@% ClearAll[#] &amp; /@ Cases[NotebookRead[cell], HoldPattern[RowBox[{sym_, &#034;=&#034;, _}]] :&gt; sym, Infinity]; gen2=SGGenSet229me[[2,1;;3,1;;3]]; ubasSG229me//FactorMatrix//Last//Inverse//HermiteDecomposition//Last (* tmInv is one of the bases of SG229me *) tmInv=TMSG229meToITA[[1;;3,1;;3]]//Inverse; %//HermiteDecomposition//Last G=Transpose[tmInv] . tmInv %==Transpose[gen2] . G . gen2 (* tm is the transformation matrix which convert the representation of this group to an orthogonal basis *) tm=tmInv//Inverse conjgen2=noTranSGGenSet229meToITA[[2,1;;3,1;;3]] Inverse[tm] . gen2 . tm==conjgen2 conj//OrthogonalMatrixQ (* The metric change under change-of-basis *) Transpose[tm] . G . tm == IdentityMatrix {valgen2,vecgen2}=Eigensystem[gen2] {valconjgen2,vecconjgen2}=Eigensystem[conjgen2] Out= {CellLabel -&gt; &#034;In:=&#034;} Out= { CellObject[ &#034;09eef945-c108-482c-a081-f7198678f646&#034;, &#034;7e18a8bb-14fb-460e-8275-756c0388fd0c&#034;]} During evaluation of In:= ClearAll::ssym: RowBox[{{,RowBox[{valgen2,,,vecgen2}],}}] is not a symbol or a string. During evaluation of In:= ClearAll::ssym: RowBox[{{,RowBox[{valconjgen2,,,vecconjgen2}],}}] is not a symbol or a string. Out= {{1/2, 1/2, 1/2}, {0, 2, 0}, {0, 0, 2}} Out= {{1/2, 1/2, 1/2}, {0, 2, 0}, {0, 0, 2}} Out= {{11/4, -(5/4), -(5/4)}, {-(5/4), 11/ 4, -(5/4)}, {-(5/4), -(5/4), 11/4}} Out= True Out= {{1, 1/2, 1/2}, {1/2, 1, 1/2}, {1/2, 1/2, 1}} Out= {{0, 1, 0}, {-1, 0, 0}, {0, 0, -1}} Out= True Out= True Out= True Out= {{-1, I, -I}, {{1/2, 1/2, 1}, {3/2 - I/2, 3/2 + I/2, 1}, {3/2 + I/2, 3/2 - I/2, 1}}} Out= {{-1, I, -I}, {{0, 0, 1}, {-I, 1, 0}, {I, 1, 0}}} As you can see, this method will trigger the following warning messages: During evaluation of In:= ClearAll::ssym: RowBox[{{,RowBox[{valgen2,,,vecgen2}],}}] is not a symbol or a string. During evaluation of In:= ClearAll::ssym: RowBox[{{,RowBox[{valconjgen2,,,vecconjgen2}],}}] is not a symbol or a string. Any tips for fixing this problem? Regards, Zhao Hongyi Zhao 2022-09-24T23:56:17Z Convert a complex number into the exponent form. https://community.wolfram.com/groups/-/m/t/2622820 See my following code snippet: In:= (* https://mathematica.stackexchange.com/questions/16414/how-can-i-convert-a-complex-number-into-an-exponent-form *) polarForm = Expand[# /. z_?NumericQ :&gt; Abs[z] Exp[I Arg[z]]] &amp;; polarForm1 = ComplexExpand[# /. z_?NumericQ :&gt; Abs[z] Exp[I Arg[z]], TargetFunctions -&gt; {Re, Im}] &amp;; x1=Eigenvectors[gen1] polarForm/@#&amp;/@x1 polarForm1/@#&amp;/@x1 Out= {{1, 1, 1}, {1/2 (-1 - I Sqrt), 1/2 (-1 + I Sqrt), 1}, {1/2 (-1 + I Sqrt), 1/2 (-1 - I Sqrt), 1}} Out= {{1, 1, 1}, {E^(-((2 I \[Pi])/3)), E^((2 I \[Pi])/3), 1}, {E^((2 I \[Pi])/3), E^(-((2 I \[Pi])/3)), 1}} Out= {{1, 1, 1}, {-(1/2) - (I Sqrt)/2, -(1/2) + (I Sqrt)/2, 1}, {-(1/2) + (I Sqrt)/2, -(1/2) - (I Sqrt)/2, 1}} Why doesn&#039;t the second method work? Regards, Zhao Hongyi Zhao 2022-09-25T07:18:10Z About the norm determination algorithm used by Eigenvectors command. https://community.wolfram.com/groups/-/m/t/2623014 See the following example: In:= gen1 Eigenvectors[gen1] Norm/@% Out= {{0, -1, 0}, {0, 0, -1}, {-1, 0, 0}} Out= {{1, 1, 1}, {1/2 (-1 - I Sqrt), 1/2 (-1 + I Sqrt), 1}, {1/2 (-1 + I Sqrt), 1/2 (-1 - I Sqrt), 1}} Out= {Sqrt, Sqrt, Sqrt} I want to know how the algorithm of **Eigenvectors** command determines the length/norm of the eigenvectors. Regards, Zhao Hongyi Zhao 2022-09-25T02:28:10Z Voting communities in the UN general assembly https://community.wolfram.com/groups/-/m/t/2622627 ![enter image description here] &amp;[Wolfram Notebook] *Also available [here].* : https://community.wolfram.com//c/portal/getImageAttachment?filename=lead_geo.gif&amp;userId=20103 : https://www.wolframcloud.com/obj/ad9b91d8-e0e9-49ad-8327-2d864c274d92 : https://christopherwolfram.com/projects/un-voting-modularity/ Christopher Wolfram 2022-09-24T07:22:46Z Can the element indices in a Table-Map-Manipulate structure be determined? https://community.wolfram.com/groups/-/m/t/2622909 The attached notebook is a table populated by solving the equations inside a FindRoot structure. I&#039;m trying to come up with a Map process that detects negative numbers in column 3 and replaces them with revised numbers I calculate. I can acquire the element coordinates outside of the manipulate structure but if I try to wrap the table inside a Map structure I get no output. Is there a way to keep track of the element position as the table is filled in Manipulate? Thanks in advance, G &amp;[Wolfram Notebook] : https://www.wolframcloud.com/obj/03fe398d-82a8-4489-bd79-bf07abb12eb2 Gerald Proteau 2022-09-24T17:58:36Z How to display errors in cells that are wrapped? https://community.wolfram.com/groups/-/m/t/2622380 For a large notebook, it is convenient to add sections and wrap codes for easy display. However, I find that when I evaluate a piece of wrapped code by clicking on the vertical bar on the right hand side and press Shift+Enter, if the wrapped code contains error such as false assertion, then, this piece of code is still wrapped, so I cannot see whether there is errors when evaluating the wrapped code. Hence, is there a way to unwrap and display cells with errors when evaluating wrapped code? Or is there a way to fast locate wrapped cells with errors? Or, at the least, is there a way to know if errors happen when evaluating wrapped code? Thank you. &amp;[Wolfram Notebook] : https://www.wolframcloud.com/obj/d5287f64-f105-4fd0-b37e-3245203919eb Francis Cong 2022-09-23T18:33:03Z Data collector using SQL operators package https://community.wolfram.com/groups/-/m/t/2619525 &amp;[Wolfram Notebook] : https://www.wolframcloud.com/obj/aa72f721-ee9e-4730-a813-f1803af5215c Damian Calin 2022-09-19T19:25:41Z Change file extension https://community.wolfram.com/groups/-/m/t/2621297 Dear All, I have to open two files that have the same name, but different extension. In the same directory I have one binary file (i.e test.bin) and one text file (i.e. test.txt). I use this string to open the bin file NAME = SystemDialogInput[&#034;FileOpen&#034;]; Now Name is C://path//test.bin Is there the possibility to define in an automatic way NAME2=C://path//test.txt Thank you very much and best regards Bruno Vusini 2022-09-23T06:03:12Z Concatenating a group of csv files https://community.wolfram.com/groups/-/m/t/2621190 I don&#039;t know why I seem to be stuck on something so. seemingly, simple. I have a set of Excel csv files. Call them f1.csv, f2.csv.... fn.csv. They are all in the same format meaning that they all have the same header). What I want to do is to read them all and concatenate them as I read them into one big file with a single header. I have tried OpenAppend, Write, Import and Export and nothing seems to be working. Could someone please provide me with a function such that I could use it as follows: contatenate[singleFilename,#]&amp; /@ {f1.csv,f2.csv,....fn.csv} The concatemate function would create a CSV file that would have the common header and all the rows of all the fx.csv files I apologize in advance if this is trivial (which I know it must be) but that I somehow seem to be stock on. Henrick Jeanty 2022-09-22T02:37:09Z Notebook won't save to directory https://community.wolfram.com/groups/-/m/t/2622392 re: Mathematica 13.0 (home edition) running on macOS Monterey Version 12.5 **Issue** I worked on an existing notebook yesterday morning and saved the notebook at 11:05 AM without incident. After further work, some 20 minutes later, I attempted to save the notebook again and the following popped up: ![enter image description here] Between saving the document at 11:05 AM and attempting to save it again, I made no changes to permissions of directories or anything else. I only edited some simple code (which worked and continues to work). **Of note**: I have no problems saving other kinds of files. I could (at the time) save the notebook to a different directory (file folder). **Today**: This issue has now spread to other directories and other notebooks. I do not have this issue when saving other types of files; it seems, at least for now, a Mathematica issue. Thoughts appreciated. : https://community.wolfram.com//c/portal/getImageAttachment?filename=ScreenShot2022-09-23at3.08.15PM.png&amp;userId=50524 Andreas Agas 2022-09-23T19:15:08Z [GiF] Shifting Lands (Level sets of x*y*z) https://community.wolfram.com/groups/-/m/t/817678 ![Level sets of x*y*z] **Shifting Lands** Same idea [as before]: this shows the stereographic image of the level sets of a function on the unit sphere; in this case, the function $f(x,y,z)=xyz$. Here&#039;s the code (you can make the Manipulate more responsive but worse-looking by reducing PlotPoints): Manipulate[ ContourPlot[( 2 (x - y) (x + y) (-1 + x^2 + y^2))/(1 + x^2 + y^2)^3, {x, -1.2, 1.2}, {y, -1.2, 1.2}, Axes -&gt; None, Frame -&gt; False, ImageSize -&gt; 540, Background -&gt; GrayLevel[.97], ContourStyle -&gt; Thickness[.004], PlotPoints -&gt; 100, PlotRangePadding -&gt; -0.01, ColorFunctionScaling -&gt; False, ColorFunction -&gt; (Blend[ RGBColor /@ {&#034;#1B435D&#034;, &#034;#78BBE6&#034;, &#034;#D5EEFF&#034;, &#034;#FF895D&#034;}, 1/8 + 7/8 * 3 Sqrt/2 (# + 1/(3 Sqrt))] &amp;), Contours -&gt; Table[r + t, {r, -1/(3 Sqrt) - 2/(12 Sqrt), 1/(3 Sqrt) + 2/(12 Sqrt), 1/(12 Sqrt)}]], {t, 0., 1/(12 Sqrt)}] : http://community.wolfram.com//c/portal/getImageAttachment?filename=cubes12.gif&amp;userId=610054 : http://community.wolfram.com/groups/-/m/t/811841 Clayton Shonkwiler 2016-03-05T03:35:16Z Is this a typo in the "Elementary Introduction to WL" book? https://community.wolfram.com/groups/-/m/t/2617889 I&#039;m reading the notebook version of the book on my desktop copy of Mathematica. In chapter 28 &#034;Tests and conditionals&#034;, the text (as rendered) shows &gt; You can also test for less than or equal using ≤, which is typed as ≤. &gt; You can test whether two things are not equal using ≠, which is typed as ≠ Clearly the &#034;which is typed as&#034; is not so useful here, and the second instances of those symbols were not supposed to be &#034;pretty-printed&#034; like that: Instead, the intent was for the sentences to end as &gt; ... using ≤, which is typed as &lt;=. &gt; ... using ≠, which is typed as !=. David Liu 2022-09-17T13:15:34Z Orthogonalize a matrix through similarity transformation. https://community.wolfram.com/groups/-/m/t/2621861 I want to orthogonalize a matrix through similarity transformation, as shown below: In:= gen tm=TMSG229meToITA[[1;;3,1;;3]] conjgen=noTranSGGenSet229meToITA[[2,1;;3,1;;3]] Inverse[tm] . gen . tm==conjgen conjgen//OrthogonalMatrixQ Out= {{-1, 2, -1}, {-(3/2), 3/2, -(1/2)}, {-(1/2), 3/2, -(3/2)}} Out= {{1, 1/2, 1/2}, {1/2, 1, 1/2}, {1/2, 1/2, 1}} Out= {{0, 1, 0}, {-1, 0, 0}, {0, 0, -1}} Out= True Out= True As you can see, **tm** is the desired matrix, but I don&#039;t know how to find such matrices. Any tips will be appreciated. Regards, Zhao Hongyi Zhao 2022-09-23T12:58:15Z How could I deal with and mark point of singularity or non-existance? https://community.wolfram.com/groups/-/m/t/2621755 ClearAll[&#034;Global*&#034;]; ode1 = y&#039;[t] == -Sin[x[t]]/y[t]; ode2 = x&#039;[t] == -Cos[x[t]] (6 Sin[x[t]] Cos[x[t]] + y[t] (b - c (1 + 3*y[t]^2)))/(2*y[t]^3*(b + c (y[t]^2 - 1))); ode3 = v&#039;[t] == -(b + c*(y[t]^2 - 1))/(4*y[t]*Cos[x[t]]) + Sin[x[t]]/(2*y[t]^2); bc = {x[t0] == 0, y[t0] == Br, v[t0] == Log[Dr]}; Do[tstar = -3 + i/2; sols[i] = ParametricNDSolve[{ode1, ode2, ode3, bc}, {x, y, v}, {t, tstar, 0}, {b, c, Br,Dr}]; data[i] = Table[{Br, Dr} /. FindRoot[{(y[2, c, Br, Dr] - 1) /. sols[i], v[2, c, Br, Dr] /. sols[i]}, {{Br, 1}, {Dr, 1}}] // Quiet, {c, 0.3, 2.2,.01}]; lst[i] = Thread[{data[i][[All, 2]], data[i][[All, 1]]}], {i, 0, 4, 1}] ListLinePlot[Table[lst[i], {i, 0, 4, 1}], Frame -&gt; True, FrameLabel -&gt; {{&#034;Br&#034;, &#034;&#034;}, {&#034;Dr&#034;, &#034;&#034;}}, PlotLegends -&gt; Table[Row[{&#034;tstar =&#034;, -3. + i/2}], {i, 0, 4, 1}]] For the above problem, I want to run two more loops for the parameter Br =1 to 3 with any step size and Dr =20 to 40 with any step size and plot them for several points of t0 like above. Also, for &#039;FindRoot&#039; ,we will now determine parameters b and c rather than Br and Dr. We will have certaintly many points of Br and Dr where solutions will not exist(singularity) and I need to mark those points where I don&#039;t have solutions(mark some rectangle or circle or something else or could fill in some boxes if solution exist vs leave them blank if not exist). Dibbo 123 2022-09-23T02:46:45Z Where are the palletes of Wolfram|One on the cloud? https://community.wolfram.com/groups/-/m/t/2621564 Are there no palettes installed, e.g. Basic Math Assistant? Athanasios Paraskevopoulos 2022-09-22T23:06:34Z Quantum tic-tac-toe: a case study on multicomputation and quantum https://community.wolfram.com/groups/-/m/t/2620886 ![enter image description here] &amp;[Wolfram Notebook] : https://community.wolfram.com//c/portal/getImageAttachment?filename=ScreenShot2022-09-22at12.54.59PM.jpg&amp;userId=11733 : https://www.wolframcloud.com/obj/727ba202-5221-4d66-8453-735d1a952d4f Mads Bahrami 2022-09-22T17:30:30Z Polynomial fit for recursion https://community.wolfram.com/groups/-/m/t/2620494 If I&#039;m not posting this in the correct place, someone, please tell me where to post it. I have a linear recursion: f(n + 2) = (1/n) + (1 + 1/n) f(n), that I am trying to estimate the behavior of for large values of n. The solution is hideous (which is why I&#039;m using Mathematica), and I don&#039;t know if I&#039;m using the right tool. I solved the recursion using RSolve. Then I tried to use Series[(...),{n, Infinity, 2}] to get the large n behavior, but the program flatly refused to do it. I fiddled with it for a while but couldn&#039;t get anywhere. Too complicated, I suspect. Is there another tool I can approach this with? (I can, of course, just calculate the series and do a polynomial fit, but I&#039;m trying to find a way to generate the solution from the recursion itself.) Thanks! -Dan Daniel Boyce 2022-09-21T09:46:19Z [GIF] Moving bars illusion https://community.wolfram.com/groups/-/m/t/1127522 ![enter image description here] These bars are moving at a constant speed together, but it looks as if they move to the right in an alternating fashion Code: size={sizex,sizey}={600,400}; n=16; bars=sizex Subdivide[-1/2,1/2,2n-1]; width=N[bars[]-bars[]]; bars=Rectangle[{#1,-sizey/2},{#2,sizey/2}]&amp;@@@Partition[bars,2]; cols={Lighter[Yellow,1/5],Darker[Blue,1/5]}; heights=N[sizey Subdivide[-1/2,1/2,Length[cols]+1][[2;;-2]]]; ClearAll[CreateScene] CreateScene[\[Alpha]_]:=Module[{recs}, recs=MapThread[{#1,Rectangle[{\[Alpha] sizex-2 width,#2-width/2},{\[Alpha] sizex+2 width,#2+width/2}]}&amp;,{cols,heights}]; Rasterize[Graphics[{bars,recs},PlotRange-&gt;({{-1,1},{-1,1}}size/2),PlotRangePadding-&gt;0],&#034;Image&#034;,RasterSize-&gt;size,ImageSize-&gt;size] ] Manipulate[CreateScene[a], {a, -0.6, 0.6}] enjoy! : http://community.wolfram.com//c/portal/getImageAttachment?filename=movingbars.gif&amp;userId=73716 Sander Huisman 2017-06-24T15:45:59Z Factoring out the factor of a matrix with the results elements Abs <=1 https://community.wolfram.com/groups/-/m/t/2621310 I have some matrices, which are all composed of numbers with absolute value less than or equal to 1. I want to find the common factor for each of them, so that the final elements in a matrix still have the absolute values ≤ 1 and is as close to 1 as possible. The following is an example: In:= FactorMatrix//ClearAll; FactorMatrix[m_?MatrixQ]:=Block[{gcd}, gcd = PolynomialGCD @@ Flatten[m]; {gcd, m/gcd} ] ubasSGBC141 %//FactorMatrix ubasSG229me %//FactorMatrix Out= {{1/2, 0, 0}, {0, 1/2, 0}, {0, 0, 1/2}} Out= {1/2, {{1, 0, 0}, {0, 1, 0}, {0, 0, 1}}} Out= {{-(1/16), -(1/16), 1/8}, {1/16, 0, 1/16}, {0, 1/16, 1/16}} Out= {1/16, {{-1, -1, 2}, {1, 0, 1}, {0, 1, 1}}} In the above results, the first one meets the requirements, but the second one fails. Specifically, the result should be as follows: {1/8, {{-(1/2), -(1/2), 1}, {1/2, 0, 1/2}, {0, 1/2, 1/2}}} Are there any hints for achieving this goal? Regards, Zhao Hongyi Zhao 2022-09-22T06:55:43Z How to define piecewise functions of two variables for NMinimize? https://community.wolfram.com/groups/-/m/t/2621015 Hey, I&#039;m new, so sorry if I somehow messed up by posting in the wrong place. Anyway, onto my question: I defined two functions \[Beta][w_, x_] := Piecewise[{{0, w &lt; 150 &amp;&amp; x &lt; 150}, {1/ 3, (w &lt; 150 &amp;&amp; 150 &lt;= x &lt; 300) || (150 &lt;= w &lt; 300 &amp;&amp; x &lt; 150)}, {1/ 2, (w &lt; 150 &amp;&amp; x &gt;= 300) || (x &lt; 150 &amp;&amp; w &gt;= 300)}, {2/ 3, (150 &lt;= w &lt; 300 &amp;&amp; 150 &lt;= x &lt; 300)}, {5/ 6, (150 &lt;= w &lt; 300 &amp;&amp; x &gt;= 300 ) || (150 &lt;= x &lt; 300 &amp;&amp; w &gt;= 300 )}, {1, w &gt;= 300 &amp;&amp; x &gt;= 300}}] and \[Gamma][w_, x_, y_] := Piecewise[{{0, w &lt; 150 &amp;&amp; x &lt; 150 &amp;&amp; y &lt; 150}, {1/ 3, (150 &lt;= w &lt; 300 &amp;&amp; x &lt; 150 &amp;&amp; y &lt; 150) || (150 &lt;= x &lt; 300 &amp;&amp; w &lt; 150 &amp;&amp; y &lt; 150) || (150 &lt;= y &lt; 300 &amp;&amp; w &lt; 150 &amp;&amp; x &lt; 150)}, {1/ 2, (w &gt;= 300 &amp;&amp; x &lt; 150 &amp;&amp; y &lt; 150 ) || (x &gt;= 300 &amp;&amp; w &lt; 150 &amp;&amp; y &lt; 150 ) || (y &gt;= 300 &amp;&amp; x &lt; 150 &amp;&amp; w &lt; 150 )}, {2/ 3, (150 &lt;= w &lt; 300 &amp;&amp; 150 &lt;= x &lt; 300 &amp;&amp; y &lt; 150) || (150 &lt;= w &lt; 300 &amp;&amp; 150 &lt;= y &lt; 300 &amp;&amp; x &lt; 150) || (150 &lt;= y &lt; 300 &amp;&amp; 150 &lt;= x &lt; 300 &amp;&amp; w &lt; 150)}, {5/ 6, (w &gt;= 300 &amp;&amp; 150 &lt;= x &lt; 300 &amp;&amp; y &lt; 150) || (w &gt;= 300 &amp;&amp; 150 &lt;= y &lt; 300 &amp;&amp; x &lt; 150) || (x &gt;= 300 &amp;&amp; 150 &lt;= w &lt; 300 &amp;&amp; y &lt; 150) || (x &gt;= 300 &amp;&amp; 150 &lt;= y &lt; 300 &amp;&amp; w &lt; 150) || (y &gt;= 300 &amp;&amp; 150 &lt;= w &lt; 300 &amp;&amp; x &lt; 150) || (y &gt;= 300 &amp;&amp; 150 &lt;= x &lt; 300 &amp;&amp; w &lt; 150)}, {1, (w &gt;= 150 &amp;&amp; x &gt;= 150 &amp;&amp; y &gt;= 150) || (w &lt; 150 &amp;&amp; x &gt;= 300 &amp;&amp; y &gt;= 300) || (x &lt; 150 &amp;&amp; w &gt;= 300 &amp;&amp; y &gt;= 300) || (y &lt; 150 &amp;&amp; x &gt;= 300 &amp;&amp; w &gt;= 300)}}] and then use them along with a third piecewise function in one variable \[Alpha][w_] := Piecewise[{{0, w &lt; 150}, {1/3, 150 &lt;= w &lt; 300}, {1/2, w &gt;= 300}}] The piecewise functions themselves all work, but I get an error when evaluating: NMinimize[{w + x + y + z, w + \[Alpha][w] x + \[Beta][w, x] y + \[Gamma][w, x, y] z &gt;= 1/3 &amp;&amp; w \[Element] PositiveIntegers &amp;&amp; x \[Element] PositiveIntegers &amp;&amp; y \[Element] PositiveIntegers &amp;&amp; z \[Element] PositiveIntegers }, {w, x, y, z}] Can anyone tell me why this doesn&#039;t work and if there is some way to do this correctly? Thank you very much! Jesco Nevihosteny 2022-09-21T16:31:19Z ⭐ [R&DL] Wolfram R&D Developers on LIVE Stream https://community.wolfram.com/groups/-/m/t/2593151 **Join us for the unique Wolfram R&amp;D livestreams on [Twitch] and [YouTube] led by our developers! Share this with your friends and colleagues:** https://wolfr.am/RDlive You will see **LIVE** stream indicators on these channels on the dates listed below. The live streams provide tutorials and behind the scenes look at Mathematica and the Wolfram Language directly from developers. Join our livestreams every Wednesday at 11 AM CST and interact with developers who work on data science, machine learning, image processing, visualization, geometry, and other areas. ---------- ⭕ **UPCOMING** EVENTS - Sept. 28th -- Q&amp;A with Visualization &amp; Graphics Developers ---------- ✅ **PAST** EVENTS - Sept. 14th -- [Paclet Development] - Sept. 7th -- [Overview of Chemistry] - Aug. 24th -- [Dive into Visualization] - Aug. 17th -- [Latest in Graphics &amp; Shaders] - Aug. 10th -- [What&#039;s new in Calculus &amp; Algebra] &gt; **What are your interests? Leave a comment here on this post to share your favorite topic suggestions for our livestreams.** Follow us on out live broadcasting channels [Twitch] and [YouTube] and for the up-to-date announcements on our social media: [Facebook] and [Twitter]. [![enter image description here]] : https://www.twitch.tv/wolfram : https://www.youtube.com/channel/UCJekgf6k62CQHdENWf2NgAQ : https://community.wolfram.com/web/connorg : https://community.wolfram.com/web/giulioa : https://community.wolfram.com/groups/-/m/t/2616863 : https://community.wolfram.com/groups/-/m/t/2613617 : https://community.wolfram.com/groups/-/m/t/2605432 : https://community.wolfram.com/groups/-/m/t/2600997 : https://community.wolfram.com/groups/-/m/t/2596451 : https://www.twitch.tv/wolfram : https://www.youtube.com/channel/UCJekgf6k62CQHdENWf2NgAQ : https://www.facebook.com/wolframresearch : https://twitter.com/WolframResearch : https://community.wolfram.com//c/portal/getImageAttachment?filename=RDAnno_image.png&amp;userId=20103 : https://www.wolfram.com/mathematica/core-areas/ Charles Pooh 2022-08-05T21:37:19Z Change the order of digits and symbols in the output expression https://community.wolfram.com/groups/-/m/t/2619841 In the following example: In:= SGGenSet229me[[1,1;;3]] % . {x,y,z,1} Out= {{0, -1, 0, 1/4}, {0, 0, -1, 1/4}, {-1, 0, 0, 1/4}} Out= {1/4 - y, 1/4 - z, 1/4 - x} The desired result is as follows: **{-y + 1/4, -z + 1/4, -x + 1/4}**. How can I achieve this goal? Regards, Zhao Hongyi Zhao 2022-09-20T09:34:12Z Reconstructing Escher's cubic space division https://community.wolfram.com/groups/-/m/t/2603029 ![Cubic Space Division in action] &amp;[Wolfram Notebook] : https://community.wolfram.com//c/portal/getImageAttachment?filename=out.gif&amp;userId=93201 : https://www.wolframcloud.com/obj/3d2d6a50-db87-40a8-af86-11e27b959ae1 Silvia Hao 2022-08-22T21:12:13Z Default values for arguments of functions https://community.wolfram.com/groups/-/m/t/2620033 Hi! &amp;[Wolfram Notebook] : https://www.wolframcloud.com/obj/bf7c1b48-0a6d-4d24-bfa8-2f279450dd80 Vladimir Ivanov 2022-09-20T13:42:11Z 32 rhombic hexecontahedra (RH) at vertices of rhombic triacontahedra https://community.wolfram.com/groups/-/m/t/2611323 ![enter image description here] &amp;[Wolfram Notebook] : https://community.wolfram.com//c/portal/getImageAttachment?filename=Lead32rhombic.png&amp;userId=20103 : https://www.wolframcloud.com/obj/ac2a6dff-5cbd-4a94-8f2c-d890db939fd0 Sandor Kabai 2022-09-05T03:56:01Z Kernel freezes in some "Elementary Introduction to WL" exercises? https://community.wolfram.com/groups/-/m/t/2617877 In [chapter 7 &#034;colors and styles&#034; ] section of the book, there are some exercises that seem to freeze the kernel when I click &#034;check answer&#034;. For example, exercise 2.4 says &gt; &#034;Make a list of colors with hues varying from 0 to 1 in steps of 0.02.&#034; When I put the correct answer Table[Hue[x], {x, 0, 1, 0.02}], I can evaluate and it compare to the &#034;expected output&#034; and it matches and works fine. However, if I click &#034;Check my solution&#034;, it will run indefinitely and prevent me from evaluating anything else until I restart Mathematica. Even clicking &#034;abort evaluation&#034; in the menu or &#034;remove from evaluation queue&#034; does nothing, only a restart of Mathematica works. This is in the notebook version of the book. I&#039;m using Mathematica 13.1 on Debian 11. Also, this exact same behavior happens on the Wolfram cloud version of the book, and there it hung my notebooks for longer since there&#039;s no easy analog of reset; I think I had to do some random things that I no longer remember in order to get my web notebooks to work again. : https://www.wolfram.com/language/elementary-introduction/2nd-ed/07-colors-and-styles.html David Liu 2022-09-17T13:09:46Z How to Plot several functions on the same axis? https://community.wolfram.com/groups/-/m/t/2619990 ![enter image description here] : https://community.wolfram.com//c/portal/getImageAttachment?filename=ScreenShot2022-09-20at10.35.48PM.png&amp;userId=2614227 Abeer Ahamed 2022-09-20T16:39:02Z Custom documentation not rendering in 13.1 https://community.wolfram.com/groups/-/m/t/2564451 I use PacletBuild in 13.1 to build my documentation, then I PacletInstall it. This was working fine in 13.0. After installing, I could type my function name and press F1 to bring up the documentation. Now it looks like I have to type the full path to my function in the documentation address bar e.g. myContext/ref/myFuncName. Can someone walk me through some troubleshooting tips to try? Eric Smith 2022-07-05T21:48:53Z [R&DL] Wolfram R&D LIVE: Wolfram Paclets--Extending the Wolfram System https://community.wolfram.com/groups/-/m/t/2616863 *MODERATOR NOTE: This is the notebook used in the livestream &#034;Paclet Development&#034; on Wednesday, September 14 -- a part of Wolfram R&amp;D livestream series announced and scheduled here: https://wolfr.am/RDlive For questions about this livestream, please leave a comment below.* &amp;[Wolfram Notebook] : https://www.wolframcloud.com/obj/5243a81f-c772-4f75-96fa-9288e0b98303 Connor Gray 2022-09-14T21:30:20Z IGraph/M: graph theory and network analysis with Mathematica https://community.wolfram.com/groups/-/m/t/560469 *WOLFRAM MATERIALS for the ARTICLE:* &gt; Szabolcs Horvát, Jakub Podkalicki, Gábor Csárdi, Tamás Nepusz, Vincent Traag, Fabio Zanini, Daniel Noom, (2022). &gt; *IGraph/M: graph theory and network analysis for Mathematica*. &gt; arXiv:2209.09145 **[physics.soc-ph]**. &gt; https://doi.org/10.48550/arXiv.2209.09145 [![Discourse topics](https://img.shields.io/discourse/topics?color=limegreen&amp;server=https%3A%2F%2Figraph.discourse.group)](https://igraph.discourse.group) [![GitHub (pre-)release](https://img.shields.io/github/release/szhorvat/IGraphM/all.svg)](https://github.com/szhorvat/IGraphM/releases) [![Contributions welcome](https://img.shields.io/badge/contributions-welcome-brightgreen.svg)](https://github.com/szhorvat/IGraphM#contributions) [![DOI](https://zenodo.org/badge/DOI/10.5281/zenodo.1134932.svg)](https://doi.org/10.5281/zenodo.1134932) ---- ##Article abstract IGraph/M is an efficient general purpose graph theory and network analysis package for Mathematica. IGraph/M serves as the Wolfram Language interfaces to the igraph C library, and also provides several unique pieces of functionality not yet present in igraph, but made possible by combining its capabilities with Mathematica&#039;s. The package is designed to support both graph theoretical research as well as the analysis of large-scale empirical networks. ---- ##Introduction The post below was written for the original release of IGraph/M. The package has come a long way since then and now contains ~300 functions. See http://szhorvat.net/mathematica/IGraphM for more details on the current release. Compatibility: 64-it Windows/macOS/Linux or Raspberry Pi; Mathematica &lt;del&gt;10.0&lt;/del&gt; 11.0 or later. &lt;a href=&#034;http://szhorvat.net/mathematica/IGraphM&#034;&gt;&lt;img src=&#034;https://community.wolfram.com//c/portal/getImageAttachment?filename=IGraphM-ad-3.png&amp;userId=38370&#034; width=&#034;300&#034;&gt;&lt;/a&gt; ---- I would like to announce IGraph/M, a new igraph interface for Mathematica: http://szhorvat.net/mathematica/IGraphM [igraph](http://igraph.org/) is a graph manipulation and analysis package. IGraph/M makes its functionality available from Mathematica. This initial release, version 0.1, covers only some igraph functions, as I focused on the things that I need personally. However the main framework is complete, and new functions can be added quickly. If anyone would like to contribute, please contact me. Binary packages for OS X (10.9 or later) and Linux can be downloaded [from GitHub](https://github.com/szhorvat/IGraphM/releases). Unfortunately, I was unable to compile the development version of igraph for Windows, so I cannot provide a Windows version. If you can help with compiling igraph itself (not IGraph/M) on Windows, please let me know! Functionality in this release that is not built into Mathematica: * Vertex betweenness centrality for weighted graphs * Estimates of vertex betweenness, edge betweenness and closeness centrality; for large graphs * Minimum feedback arc set for weighted and unweighted graphs * Find all cliques (not just maximal ones) * Count 3- and 4-motifs * Rewire edges, keeping either the density or the degree sequence * Alternative algorithms for isomorphism testing: Bliss, VF2 * Subgraph isomorphism * Test if a degree sequence is graphical * Alternative algorithms for generating random graphs with given degree sequence * Layout algorithms that take weights into account Note that IGraph/M is *not a replacement* for Mathematica&#039;s graphs and networks functionality. It is meant to complement what is already available in Mathematica, thus it primarily focuses on adding functionality that is not already present. Why did I release the package before covering most of the igraph functionality? I do not have time to work on things I do not personally need or use, so I am unlikely to extend it further unless the need comes up. I do think that the functions that are included in v0.1 can already be useful to others too. I would also like to give the opportunity for people to contribute to the project if they wish to. The groundwork has been laid, so further extensions should be quick and relatively easy. Also check out a related project, [IGraphR](https://github.com/szhorvat/IGraphR), which makes igraph available for Mathematica users through RLink. I wrote IGraph/M because I needed higher performance and greater reliability (especially for parallel computing) than what RLink could provide. ---- **A request:** If any of you have used IGraphR in the past to access igraph from Mathematica, please post a response to this thread and let me know which specific functions you were using. &amp;[Wolfram Notebook] : https://www.wolframcloud.com/obj/94639221-60b4-47e3-8862-d996caa40388 Szabolcs Horvát 2015-09-06T12:55:14Z HermiteDecomposit of the lattices generated by column or row matrices https://community.wolfram.com/groups/-/m/t/2619424 I noticed the following description given in the document of [HermiteDecomposition](https://reference.wolfram.com/language/ref/HermiteDecomposition.html?q=HermiteDecomposition): &gt; HermiteDecomposition can be used to determine if two integer lattices &gt; are equivalent. If the generators of the lattices are put in row &gt; matrices, the lattices are equal if and only if the matrices of the &gt; Hermite decompositions of the corresponding matrices are the same. As you can see, this command needs **&#034;the generators of the lattices are put in row matrices&#034;**. But in practice, I do not know whether the generators of different lattices are stored in column or row matrices. In this case, how should I use this command? Regards, Zhao Hongyi Zhao 2022-09-19T14:48:33Z Function hover button failure in generated paclet https://community.wolfram.com/groups/-/m/t/2198795 I have a fairly sizable Paclet built and generated in the latest (albeit outdated) Eclipse/Workbench. Everything is generating acceptably in Mathematica 12.1 (Other posts have warned me not to upgrade to 12.2, but that&#039;s for another day). The problem is that when I hover over my functions (symbols) in a notebook, I get the hover buttons and the left one (double chevron) brings up the tool-tops nicely from my package symbol::usage statements. The right one (info circle), however, does not bring up my symbol reference page for that function; in fact, it is grayed out and can&#039;t be selected. Any idea why this is happening? Am I missing some switch that would link the right symbol hover button to the respective documentation page? Is this a known issue? Thanks! Jeff Jeffery Henning 2021-02-19T12:24:07Z Enhanced documentation description of SmithDecomposition may be required https://community.wolfram.com/groups/-/m/t/2619401 I noticed the following description [here](https://reference.wolfram.com/language/ref/SmithDecomposition.html): &gt; Details &gt; &gt; The result is given in the form {u,r,v}, where u and v are unimodular matrices, r is a diagonal matrix with each diagonal entry &gt; dividing the next one, and u.m.v==r. &gt; The unimodular matrices u and v are integer matrices with Abs[Det[u]]==1, and their inverses are also integer matrices. As you can see, there are two unimodular integer matrices, namely, u and v. Therefore, I think the above description should be strengthened as follows: The unimodular matrices u and v are integer matrices with Abs[Det[u]]==1 **and Abs[Det[v]]==1**, and their inverses are also integer matrices. Regards, Zhao Hongyi Zhao 2022-09-19T13:55:15Z How to assign a keyboard shortcut to a style? https://community.wolfram.com/groups/-/m/t/2619955 I want to assign a keyboard shortcut, say &#034;Shift+Ctrl+I&#034;, to apply the style &#034;InlineFormula&#034; to the current selection. (I have changed this style, but this does not matter here). I know that this is possible by modifying the &#034;KeyEventTranslations.tr&#034;- file of my Wolfram installation or copy and modify it and store it in my $UserBaseDirectory. I have even found a complete package &#034;[Shortcuts]&#034; for many additional shortcuts. But this seems to be overkill for my simple purpose. Is there some simple WL-expression, a KeyEventHandler or the like, which I can use for my purpose? : https://github.com/rolfmertig/Shortcuts Werner Geiger 2022-09-20T12:52:00Z Understanding the syntax of Flatten https://community.wolfram.com/groups/-/m/t/2619105 I humbly ask ... Is there a post somewhere that describes how the following syntax of Flatten works? I&#039;m having trouble understanding this from what is described on the Flatten documentation page. Flatten[list, { {s11,s12,...},{s21,s22, ...}, ...}] flattens list by combining all levels sij to make each level i in the result The related examples didn&#039;t help my understanding. Perhaps I&#039;m also not understanding the meaning of Level 1, level 2, ... and the concept of outermost, innermost. John Burgers 2022-09-18T20:19:33Z How to control the Image Assistant Toolbar? https://community.wolfram.com/groups/-/m/t/2618380 &lt;sup&gt;(Cross-posted from [MMa.SE].)&lt;/sup&gt; When we select an Image in the FrontEnd, the [Image Assistant Toolbar] appears attached under it: [![toolbar]] How can we temporarily prevent the toolbar from appearing when an Image is selected? : https://mathematica.stackexchange.com/q/273612/280 : http://reference.wolfram.com/language/workflow/GetCoordinatesFromAnImage.html : https://i.stack.imgur.com/42a31.png : https://i.stack.imgur.com/42a31.png Alexey Popkov 2022-09-18T09:27:15Z [WSG23] Daily Study Group: Quantum Computation Framework (Oct 24, 2022) https://community.wolfram.com/groups/-/m/t/2619353 A new study group devoted to [Quantum Computation Framework] begins **Monday, October 24th!** A list of daily topics can be found on our Daily Study Groups page. This group will be led by me and [@Nikolay Murzin][at0], from Wolfram Quantum Team. We will meet daily, Monday to Friday. Study group sessions include time for exercises, discussion and Q&amp;A. This study group will help you acquire an intro knowledge of quantum computation and learn how to implement quantum algorithms in the [Wolfram Quantum Computation Framework]. Feel free to explore our framework. &gt; [REGISTER HERE] ![enter image description here] ![enter image description here] [at0]: https://community.wolfram.com/web/murzinnikolay : https://community.wolfram.com//c/portal/getImageAttachment?filename=ScreenShot2022-01-17at3.53.39PM.png&amp;userId=1539902 : https://community.wolfram.com//c/portal/getImageAttachment?filename=circuit.png&amp;userId=1539902 : http://wolfr.am/wolfram-quantum : http://wolfr.am/wolfram-quantum : https://www.bigmarker.com/series/dsg-wolfram-quantum-framework/series_details?utm_bmcr_source=events Mads Bahrami 2022-09-19T18:43:04Z Limit the scope of graphics directives specified for PlotStyle? https://community.wolfram.com/groups/-/m/t/2618917 Is there an easy way to limit the scope of graphics directives specified for the PlotStyle option of functions such as ListPlot and List LinePlot to just the lines joining the points? The reason I ask is that I&#039;m seeing directives such as Dotted and Dashed applied to unfilled plot markers. Whilst I understand from the documentation that this is the intended behaviour, it&#039;s probably not the best default behaviour in most situations --- I can&#039;t think of a single occasion on which I&#039;ve wanted to dash- or dot-style the lines used to draw the plot markers. The issue is highlighted in the code snippet below. ListLinePlot[ Table[Table[{i, i*RandomReal@{1, 4}}, {i, 3}], 10] , PlotLegends -&gt; Automatic , PlotMarkers -&gt; {Automatic, 10} , PlotStyle -&gt; Directive[Dotted] ] This code generates a plot like the one shown below. In this plot, the Dotted PlotStyle, which I really only want to be applied to the lines joining the points, is also being applied to the lines used to draw the unfilled plot markers for plot numbers 6 to 10. ![enter image description here] One obvious solution that occurs to me is to explicitly specify the plot markers such that the issue doesn&#039;t arise. But I&#039;d rather not have to do this if there&#039;s a simple way of stipulating that the PlotStyle should be restricted to the lines joining the points or should not be applied to the lines used to draw the plot markers. Or alternatively that only filled plot markers should be used. : https://community.wolfram.com//c/portal/getImageAttachment?filename=9074PlotStyleDotted.jpg&amp;userId=842054 Ian Williams 2022-09-18T12:27:33Z Rough & ready routines for styling an ePub more like a book https://community.wolfram.com/groups/-/m/t/2618219 &amp;[Wolfram Notebook] : https://www.wolframcloud.com/obj/bc381caf-406e-4684-bf6a-71a938b474da Thomas Colignatus 2022-09-17T08:10:30Z Use Which command more concisely https://community.wolfram.com/groups/-/m/t/2618072 I defined the following function in my package: gensFINDSSG//ClearAll; gensFINDSSG[{m__?MatrixQ} | m__?MatrixQ]:=Block[{gens,dim,vec,x,y,z,t,u,v}, gens={m}; dim=Dimensions[gens][]-1; Which[ dim==3,vec={x, y, z, 1}, dim==4,vec={x, y, z, t, 1}, dim==5,vec={x, y, z, t, u, 1}, dim==6,vec={x, y, z, t, u, v, 1} ]; Map[Dot[#, vec] &amp;, gens[[All, 1 ;; dim]], {-2}]// InputForm // ToString //StringReplace[#,{&#034;(&#034;-&gt;&#034;&#034;,&#034;)&#034;-&gt;&#034;&#034;}]&amp; ] But I want to know if there are more concise ways to rewrite the following part: Which[ dim==3,vec={x, y, z, 1}, dim==4,vec={x, y, z, t, 1}, dim==5,vec={x, y, z, t, u, 1}, dim==6,vec={x, y, z, t, u, v, 1} ]; Regards, Zhao Hongyi Zhao 2022-09-17T01:00:48Z [WSG22] Daily Study Group: Writing a Wolfram Language Function https://community.wolfram.com/groups/-/m/t/2596110 **A Wolfram U Daily Study Group on &#034;Writing a Wolfram Language Function&#034;** begins on September 12, 2022. Join me and a group of fellow learners in an exploration of what it takes to make a &#034;good&#034; function in the Wolfram Language. We&#039;ll begin with standard function structure and then cover arguments, patterns, options and how to extend function capability with overloading, recursion and iteration, memoization and up-values. Later sessions will discuss error handling and extra tips. Sessions include short lessons, poll questions to review key concepts, practice problems and live Q&amp;A. A certificate of program completion will be awarded to participants who attend online sessions and pass a quiz. Some prior Wolfram Language experience or knowledge is recommended for this Study Group. [**REGISTER HERE**] ![Wolfram U Banner] : https://www.bigmarker.com/series/dsg-writing-a-wolfram-language-function/series_details?utm_bmcr_source=community : https://community.wolfram.com//c/portal/getImageAttachment?filename=WolframUBanner.jpeg&amp;userId=1861308 Daniel Robinson 2022-08-11T12:04:15Z Announcing the Wolfram Student Podcast https://community.wolfram.com/groups/-/m/t/2618765 Are you interested in seeing how high schoolers are using the Wolfram Language? Excited to learn about the concepts behind their innovative projects? Join Sam Natarajan and Rushank Goyal every two weeks for a new episode of the Wolfram Student Podcast! Every episode, the two of us interview a high schooler who has created a cool project&#x2014;from 4D cellular automata to computationally-generated music to neural networks with Boolean functions. We spend 25-30 minutes discussing the ideas behind their work and exploring important pieces of code. Occasionally, we&#039;ll do a special episode featuring developers and specialists working at Wolfram. The first two episodes are already up on the Wolfram YouTube channel, with more to come every other Wednesday. Access our playlist [here]. You can also listen to the audio version on [Spotify]. If you&#039;re a high schooler who would like to be featured, feel free to fill out [this form]! ![Thumbnail of Episode 1] : https://youtube.com/playlist?list=PLxn-kpJHbPx08CzJeEJsVtv_Igzga4wv6 &#034;here&#034; : https://open.spotify.com/show/1saGtQW9a1NNvFTzX5ra0o : https://community.wolfram.com//c/portal/getImageAttachment?filename=hqdefault.jpg&amp;userId=1851088 : https://wolfr.am/wsp-interest-form Rushank Goyal 2022-09-18T04:17:26Z Try to beat these MRB constant records! https://community.wolfram.com/groups/-/m/t/366628 CMRB is the MRB constant. POSTED BY: Marvin Ray Burns, ![enter image description here]![enter image description here] &#039; WITH COAUTHORS: Giuseppe Peano in the ![enter image description here] (FORMULATION of CMRB MATHEMATICS) ------------------------------------------------------------------------ &#039; Issac Newton in the ![enter image description here] (PRINCIPLES OF CMRB MATHEMATICS) ------------------------------ &#039; Gottfried Wilhelm Leibniz in the ![enter image description here] fandom.com/ (REPORTS OF CMRB SCHOLARS) --------------------------------------------- &#039; and most significantly, Euclid in the ![enter image description here] important7.com/ ELEMENTS OF CMRB GEOMETRY ========================= (assuming a different form of the parallel postulate) . When asked for an image of that matches everything contained in this discussion, Google&#039;s AI gave the following cartoon credited to V.J Motto. ![enter image description here] **An Easter egg for you to find below:** &gt; (In another reality, I invented CMRB and then discovered many of its &gt; qualities.) Content of the first post, as of September 2, 2022 ========================== 1. Q and A, 2. My claim to the MRB constant (CMRB), or a case of calculus déjà vu? 3. What exactly is it? 4. Where is it found? 5. How it all began, 6. The why and what of the **C**&lt;sub&gt;*MRB*&lt;/sub&gt; Records, 7. **C**&lt;sub&gt;*MRB*&lt;/sub&gt; and its applications. The following contents of the first post have been moved. Use CNRL+F to locate. 1. CNRTL+F &#034;Real-World, and beyond, Applications&#034;. 2. CNRL+F &#034;MeijerG Representation for&#034; **C**&lt;sub&gt;*MRB*&lt;/sub&gt;, 3. CNRL+F &#034;the Laplace transform analogy to&#034; CMRB. 4. **C**&lt;sub&gt;*MRB*&lt;/sub&gt; CNRL+F &#034;formulas and identities&#034;, 5. CNRL+F &#034;Primary Proof 1&#034;, 6. CNRL+F &#034;Primary Proof 2&#034;, 7. CNRL+F &#034;Primary Proof 3&#034;, 8. CNRL+F &#034;The relationship between&#034; **C**&lt;sub&gt;*MRB*&lt;/sub&gt; and its integrated analog, 9. The MRB constant supercomputer 0 Second post: ------------ The following might help anyone serious about breaking my record. Third post ---------- The following email Crandall sent me before he died might be helpful for anyone checking their results. Fourth post ----------- Perhaps some of these speed records will be easier to beat. Many more interesting posts --------------------------- ...including the MRB constant supercomputers 1 and 2. ...including records of computing the MRB constant from Crandall&#039;s eta derivative formulas. ...including all the methods used to compute **C**&lt;sub&gt;*MRB*&lt;/sub&gt; and their efficiency. ...including the dispersion of the 0-9th decimals in **C**&lt;sub&gt;*MRB*&lt;/sub&gt; decimal expansions. ...including the convergence rate of 3 major different forms of **C**&lt;sub&gt;*MRB*&lt;/sub&gt;. ...including complete documentation of all multimillion-digit records with many highlights. ...including arbitrarily close approximation formulas for **C**&lt;sub&gt;*MRB*&lt;/sub&gt;. ...including efficient programs to compute the integrated analog (MKB) of **C**&lt;sub&gt;*MRB*&lt;/sub&gt;. ...including a recent discovery that could help in verifying digital expansions of the integrated analog (MKB) of **C**&lt;sub&gt;*MRB*&lt;/sub&gt;. ...including CNRL+F &#034;the Laplace transform analogy to&#034; **C**&lt;sub&gt;*MRB*&lt;/sub&gt;. ...including CNRTL+F &#034;Real-World, and beyond, Applications&#034;. ... including an overview of all **C**&lt;sub&gt;*MRB*&lt;/sub&gt; speed records, by platform. ...including a few attempts at a spectacular 7 million digits using Mathematica. ...including an investigation into whether ![CMRB sum] converges only for z&lt;sub&gt;0&lt;/sub&gt;=1? Also as to whether it is absolutely convergent. ... including a similarly rigorous investigation that proves whether ![enter image description here] ---------- ---------- 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 the MRB constant? 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;There is nothing without a reason. Read more at: https://minimalistquotes.com/gottfried-leibniz-quote-229585/ ---------- ---------- ---------- This discussion is not crass bragging; it is an attempt by this amateur to share his discoveries with the greatest audience possible. Amateurs have been known to make a few significant discoveries as discussed in ![enter image description here] [here.] This amateur has made his best attempts at proving his discoveries and has often asked for help in doing so. 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 portion of them was original. What I concluded from these investigations was that the only original thought I had was the obstinacy to think anything meaningful can be found in the infinite sum shown next. ![CMRB sum] Nonetheless, it is possible that 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 abut 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; From 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! ---------- ![it] is defined in all of the following places, the majority of which attribute 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; - 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] ---------- ---------- ![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] Eta denotes the kth 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.}*) &gt; ... &gt; ![enter image description here]![enter image description here] 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. &gt; Even though one has cause to be a little bit wary around formal &gt; rearrangements of conditionally convergent sums (see the [Riemann series theorem]), it&#039;s not very difficult to validate the formal &gt; manipulation of Helms. The idea is to cordon off a big chunk of the &gt; infinite double summation (all the terms from the second column on) &gt; that we know is *absolutely* convergent, which we are then free to &gt; rearrange with impunity. (Most relevantly for our purposes here, see &gt; pages 80-85 of this [document.] &gt; culminating with the Fubini theorem which is essentially the &gt; manipulation Helms is using.) &gt; &gt; So, by definition the MRB constant$B$is the conditionally convergent &gt; sum$\sum_{n \geq 1} (-1)^n (n^{1/n} - 1)$. Put$a_n = (-1)^n (n^{1/n}= - 1)$, &gt; so$B = \sum_{n \geq 1} a_n.$&gt; &gt; Looking at the first column, put$b_n = (-1)^n \frac{\log(n)}{n}.$&gt; &gt; so$\eta^{(1)}(1) = \sum_{n \geq 1}&gt; b_n$&gt; &gt; as a conditionally convergent series. &gt; &gt; We have &gt; &gt; $$B - \eta^{(1)} = \sum_{n \geq 1} a_n - b_n = \sum_{n \geq 1} \sum_{m &gt; \geq 2} (-1)^n \frac{(\log n)^m}{n^m m!}$$ &gt; &gt; (The first equation is an elementary limit statement that says if &gt;$\sum_{n \geq 1} a_n$converges and$\sum_{n \geq 1} b_n$converges, &gt; then also$\sum_{n \geq 1} a_n - b_n$converges and$\sum_{n \geq 1} &gt; a_n - \sum_{n \geq 1} b_n = \sum_{n \geq 1} a_n - b_n$. It doesn&#039;t at &gt; all matter whether the convergence of either series is conditional or &gt; absolute.) &gt; &gt; So now we check the *absolute* convergence of the right-hand side, &gt; i.e., that$\sum_{n \geq 1} \sum_{m \geq 2} \frac{(\log n)^m}{n^m m!}$&gt; converges. (Remember what this means in the case of infinite sums of &gt; *positive* terms: it means that there is a number$K$such that every finite partial sum$S$is bounded above by$K$; the least such upper &gt; bound will be the number that the infinite sum converges to.) So take &gt; any such finite partial sum$S$, and rearrange its terms so that the &gt; terms in the$m = 2$column come first, then the terms in the$m = 3$&gt; column, and so on. An upper bound for the terms of$S$in the$m = 2$&gt; column is$\frac{\zeta^{(2)}(2)}{2!}$. Put that one aside. &gt; &gt; For the$m = 3$column, an upper bound is$\sum_{n \geq 2} \frac{(\log &gt; n)^3}{n^3 3!}$(we drop the$n=1$term which is$0$). By calculus we &gt; have$\log n \leq n^{1/2}$for all$n \geq 2$, so this has upper bound &gt;$\frac1{3!} \sum_{n \geq 2} \frac1{n^{3/2}} \leq \frac1{3!} &gt; \int_1^\infty \frac{dx}{x^{3/2}}$by an integral test, which yields &gt;$\frac{2}{3!}$as an upper bound. Applying the same reasoning for the &gt;$m$column from$m = 4$on, an upper bound for that column would be &gt;$\frac1{m!} \int_1^\infty \frac{dx}{x^{m/2}} = \frac{2}{m!(m-2)}$. &gt; Adding all those upper bounds together, an upper bound for the entire &gt; doubly infinite sum would be &gt; &gt; $$\frac{\zeta^{(2)}(2)}{2!} + \sum_{m \geq 3} \frac{2}{m!(m-2)}$$ &gt; &gt; which certainly converges. So we have absolute convergence of the &gt; doubly infinite sum. &gt; &gt; Thus we are in a position to apply the Fubini theorem, which justifies &gt; the rearrangement expressed in the first of the following equations &gt; &gt; $$\sum_{n \geq 1} \sum_{m \geq 2} (-1)^n \frac{(\log n)^m}{n^m m!} = &gt; \sum_{m \geq 2} \sum_{n \geq 1} (-1)^n \frac{(\log n)^m}{n^m m!} = &gt; \sum_{m \geq 2} (-1)^{m+1} \frac{\eta^{(m)}(m)}{m!}$$ &gt; &gt; giving us what we wanted. ---------- ![enter image description here] **The integral forms for CMRB 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:= 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 ![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 ---------- We derive the preceding and following integral forms of CMRB from the Abel-Plana Formula, considering the following result. ------------------------------------------------------------------------ ExpToTrig[Re[Exp[2 Pi z - 1]]] ![r] ![APformula] ![APproof] \&amp;[MRB equations] ---------- ---------- ---------- ---------- ---------- ---------- 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. For evidence of my poor grades see [my report cards]. 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*2, 2*2*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 took care of 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 on my math hobby! Occasionally, I make a point of going to 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 were telling 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, I have started a search for a new mathematical constant! Does anyone want to help me? Consider, 1^(1/1)-2^(1/2)+3^(1/3)...I will take it apart and examine it &#034;bit by bit.&#034; I hope to find connections to all kinds of arithmetical manipulations. I realize I am in &#034;no man&#039;s land,&#034; but I work best there! If anyone else is foolhardy enough to 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 own 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 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] Notice, by &#034;near-zeros,&#034; I mean we have the following. ![near-zeros] VC/(6*(11/7 - ProductLog)) - CMRB CMRB - (5*VC^6)/56 3.4164*10^-8 1.47*10^-9 Out= 3.4164*^-8 Out= 1.47*^-9 See [cloud notebook]. The near zero, CMRB - (5*VC^6)/56, is so small that Wolfram Alpha yields a rational power of VC for the nth root of 56/6 CMRB. ![enter image description here] ---------- ---------- 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 itself alone: &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. And where 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 ReImPlot[(-1)^x (x^(1/x) - 1), {x, 1, Infinity}, PlotStyle -&gt; Blue, Filling -&gt; Axis, FillingStyle -&gt; {Green, Red}] ![big plot] ![small plot] ---------- 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 post.** Here is how my analysis (along with improvements to Mathematica) has improved the speed of calculating that constant&#039;s digits: (digits and seconds) ![enter image description here] Better 2022 results are expected soon! 2022 results documentation: &amp;[Wolfram Notebook] ---------- ---------- **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 a proof of 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 there is no limit a goes to infinity, of the ratio of the previous forms of integrals when the &#034;I&#034; is left out, and 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 $a=1-2\times C\mathrm{MRB}=0.6242807150758657595029641318914535398881938101997224\ldots$. Where $C\mathrm{MRB}$ is $\sum_{n=1}^\infty (-1)^n\times(n^{1/n}-1)$. 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. Finally let a = M2 = $1-2\times C$MRB = 0.6242807150758... and the two limit-points of the series $\sum_{n=1}^\infty (-1)^n\times(n^{1/n}-M2)$ are +/- $C$MRB with its Levin&#039;s u-transform&#039;s result being 0. See [here.](http://oeis.org/A173273) Also, ![enter image description here] ---------- ---------- ---------- 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. It is my theory that &gt; like the MRB constant the universe, under inflation, started in an &gt; infinite number of space dimensions. And they almost all &gt; instantly collapsed, as far as our sensory realm is concerned, leaving &gt; all but the few we enjoy today. I&#039;m not the first person to think the universe consists of an infinitude of dimensions. Some string theories and theorists propose it too. Michele Nardelli added the following. &gt; In string theory, perturbation methods involve such a high degree of &gt; approximation that the theory is unable to identify which of the &gt; Calabi - Yau spaces are candidates for describing the universe. The &gt; consequence is that it does not describe a single universe, but &gt; something like 10^500 universes. In reality, admitting 10^500 &gt; different quantum voids would allow the only mechanism known at the &gt; moment to explain the present value of the cosmological constant &gt; following an idea by Steven Weinberg. Furthermore, a very large value &gt; of different voids is typical of any type of matter coupled to gravity &gt; and is also obtained when coupling the standard model. I believe that &gt; the multiverse is a &#034;space of infinite dimensions&#034; with infinite &gt; degrees of freedom and infinite possible potential wave functions that &gt; when they collapse, formalize a particle or a universe in a quantum &gt; state. The strings vibrating, like the strings of a musical &gt; instrument, emit frequencies that are not always precise numbers, &gt; indeed, very often they are decimal, irrational, and/or transcendent &gt; numbers. The MRB constant serves as a &#034;regularizer&#034; to obtain &gt; solutions as precise as possible and this in various sectors of string &gt; theory, black holes, and cosmology In [this physics.StackExchange question] his concept of the dimensions in string theory and a possible link with number theory is inquired about. Many MRB constant papers by Michele Nardelli are found [here in Google Scholar], which include previous versions of these. &gt; Hello. Here are the links of my more comprehensive articles describing &gt; the various applications of the CMRB in various fields of theoretical &gt; physics and cosmology. Thanks always for your availability, see you &gt; soon. &gt; -- &gt; &gt; [Analyzing several equations concerning Geometric Measure Theory, various Ramanujan parameters and the developments of the MRB Constant. New possible mathematical connections with some sectors of String Theory. XII] &gt; -- &gt; &gt; [On several equations concerning Geometric Measure Theory, various Ramanujan parameters and the developments of the MRB Constant. New possible mathematical connections with some sectors of Cosmology (Bubble universes) and String Theory] &gt; -- &gt; &gt; [Analyzing several equations concerning Geometric Measure Theory, various Ramanujan parameters and the developments of the MRB Constant. New possible mathematical connections with some sectors of Cosmology (Bubble universes) and String Theory. III] &gt; -- &gt; &gt; [Analyzing several equations concerning Geometric Measure Theory, various Ramanujan parameters and the developments of the MRB Constant. New possible mathematical connections with some equations concerning various aspects of Quantum Mechanics and String Theory. VI] &gt; -- &gt; &gt; [Analyzing several equations concerning various aspects of String Theory and one-loop graviton correction to the conformal scalar mode function. New possible mathematical connections with various Ramanujan parameters and some developments of the MRB Constant.] &gt; -- &gt; &gt; [Analyzing several equations concerning Geometric Measure Theory, various Ramanujan parameters and the developments of the MRB Constant. New possible mathematical connections with some equations concerning Multiverse models and the Lorentzian path integral for the vacuum decay process] &gt; -- &gt; &gt; [On the study of various equations concerning Primordial Gravitational Waves in Standard Cosmology and some sectors of String Theory. New possible mathematical connections with various Ramanujan formulas and various developments of the MRB Constant] &gt; -- &gt; &gt;[On the study of some equations concerning the mathematics of String Theory. New possible connections with some sectors of Number Theory and MRB Constant] &gt; -- &gt; &gt;[On the study of some equations concerning the mathematics of String Theory. New possible connections with some sectors of Number Theory and MRB Constant. II] &gt; -- &gt; &gt;[Analyzing some equations of Manuscript Book 2 of Srinivasa Ramanujan. New possible mathematical connections with several equations concerning the Geometric Measure Theory, the MRB Constant and various sectors of String Theory] &gt; -- &gt; &gt;[Analyzing the MRB Constant in Geometric Measure Theory and in a Ramanujan equation. New possible mathematical connections with ζ(2), ϕ , the Quantum Cosmological Constant and some sectors of String Theory] &gt; -- &gt; &gt; [Analyzing some equations of Manuscript Book 2 of Srinivasa Ramanujan. New possible mathematical connections with several equations concerning the Geometric Measure Theory, the MRB Constant, various sectors of Black Hole Physics and String Theory] &gt; -- &gt; &gt; [Analyzing further equations of Manuscript Book 2 of Srinivasa Ramanujan. New possible mathematical connections with the MRB Constant, the Ramanujan-Nardelli Mock General Formula and several equations concerning some sectors of String Theory III] His latest papers on the MRB constant, follow. &gt; Hi Marvin, for me the best links you could post are those related to &gt; the works concerning the Ramanujan continued fractions and &gt; mathematical connections with MRB Constant and various sectors of &gt; String Theory. &gt; &gt; Here are the links (in all there are 40): &gt; &gt; &gt; &gt; https://www.academia.edu/80247977/ https://www.academia.edu/80298701/ &gt; https://www.academia.edu/80376615/ https://www.academia.edu/80431963/ &gt; https://www.academia.edu/80508286/ https://www.academia.edu/80590932/ &gt; https://www.academia.edu/80660709/ https://www.academia.edu/80724379/ &gt; https://www.academia.edu/80799006/ https://www.academia.edu/80894850/ &gt; https://www.academia.edu/81033980/ https://www.academia.edu/81150262/ &gt; https://www.academia.edu/81231887/ https://www.academia.edu/81313294/ &gt; https://www.academia.edu/81536589/ https://www.academia.edu/81625054/ &gt; https://www.academia.edu/81705896/ https://www.academia.edu/81769347/ &gt; https://www.academia.edu/81812404/ https://www.academia.edu/81874954/ &gt; https://www.academia.edu/81959191/ https://www.academia.edu/82036273/ &gt; https://www.academia.edu/82080277/ https://www.academia.edu/82129372/ &gt; https://www.academia.edu/82155422/ https://www.academia.edu/82204999/ &gt; https://www.academia.edu/82231273/ https://www.academia.edu/82243774/ &gt; https://www.academia.edu/82347058/ https://www.academia.edu/82399680/ &gt; https://www.academia.edu/82441768/ https://www.academia.edu/82475969/ &gt; https://www.academia.edu/82516896/ https://www.academia.edu/82521506/ &gt; https://www.academia.edu/82532215/ https://www.academia.edu/82622577/ &gt; https://www.academia.edu/82679726/ https://www.academia.edu/82733681/ &gt; https://www.academia.edu/82777895/ https://www.academia.edu/82828901/ He recently added the following. &gt; Hi Marvin, &gt; &gt; The MRB Constant, also in the case of the Ramanujan&#039;s expressions that &gt; we are slowly analyzing, serves to &#034;normalize&#034;, therefore to rectify &gt; the approximations we obtain. For example, for the value of zeta (2), &gt; which is always approximate (1.64382 ....)&#034; [from the string theory equations, example below], &#034;adding an expression &gt; containing the MRB Constant gives a result much closer to the real &gt; value which is 1.644934 ... This procedure is carried out on all those &gt; we call &#034;recurring numbers&#034; (Pi, zeta (2), 4096, 1729 and the golden &gt; ratio), which, developing the expressions, are always approximations, &gt; from which, by inserting the CMRB in various ways, we obtain results &gt; much closer to the real values of the aforementioned recurring &gt; numbers. Finally, remember that Ramanujan&#039;s expressions and the &gt; recurring numbers that are obtained are connected to the frequencies &gt; of the strings, therefore to the vibrations of the same. One example of his procedure from [https://www.academia.edu/81812404/On_further_Ramanujans_continued_fractions_mathematical_connections_with_MRB_Constant_various_equations_concerning_some_sectors_of_String_Theory_XIX?] was to analyze some expressions from Ramanujan&#039;s notebooks. Finding other expressions from series of their anti-derivative and derivatives, in this case, dividing two previous expressions, after some calculations, he obtained this expression from it.&#034; &gt; ![enter image description here] Then finally &#034;by inserting the CMRB, obtaining results much closer to the real values of the aforementioned recurring numbers:&#034; (referring to Ramanujan&#039;s equation, and the after more work..) &gt;, ![enter image description here] **You need to look at the paper entirely to see how he puts it all together.** He uses Wolfram Alpha for a lot of it. 7/7/2022 I just found a [video] he made concerning his work on string theory with its connection to Ramanujan and **C**&lt;sub&gt;*MRB*&lt;/sub&gt;. English subtitles are available on youtube. ---------- There are around 200 papers concerning the MRB contact [here] at acadeia.edu. ![enter image description here] More Google Scholar results on **C**&lt;sub&gt;*MRB*&lt;/sub&gt;t are [here,] which include the following. **[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. ---------- ---------- ---------- ---------- MRB Constant Records, ==================== 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 (additive inverse of ) **C**&lt;sub&gt;*MRB*&lt;/sub&gt; with my TI-92s, by adding 1-sqrt(2)+3^(1/3)-4^(1/4)+... 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 of $\text{CMRB}=\sum _{n=1}^{\infty } (-1)^n \left(n^{1/n}-1\right)$ i.e. (.1). It gave (.1_91323989714) which is close to what Mathematica gives for summing to only an upper limit of 1000. ---------- 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 a 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. ---------- 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. 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, 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. 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. CNTRL+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. CNTRL+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. 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. 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. Max memory used was 91 GB of RAM. The Mathematica 6.0 code is used 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 is an average of 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 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 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. To beat that, on 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 of the 30 cores were clocked at up to 5.2 GHz. To beat that I did a 1,004,993 correct digits computation in 35.6 hours of absolute time and only 25.3 hours of computation time, on Wed 3 Aug 2022 08:05:38, using the MRB constant supercomputer 3. Ram Speed was 4000MHz and all of the 40 cores were clocked at up to 5.5 GHz. [44 hours million notebook] [36.7 hours million notebook] [35.6 hours million notebook] ---------- 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 3.20 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 number of 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 was 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 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 actually 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 calculation that used a different method is 0.*10^-6029992. &#034;Computation time&#034; 72.526 days &#034;Absolute time&#034; 185.491 days ---------- ---------- ---------- ---------- ---------- ---------- ---------- **C**&lt;sub&gt;*MRB*&lt;/sub&gt; and its applications ===================================== **Definition 1** **C**&lt;sub&gt;*MRB*&lt;/sub&gt; is defined at [https://en.wikipedia.org/wiki/MRB_constant] . From [Wikipedia:] &gt; ![If you see this instead of an image, reload the page] &gt; ![If you see this instead of an image, reload the page] References Plouffe, Simon. &#034;mrburns&#034;. Retrieved 12 January 2015. Burns, Marvin R. (23 January 1999). &#034;RC&#034;. math2.org. Retrieved 5 May 2009. Plouffe, Simon (20 November 1999). &#034;Tables of Constants&#034; (PDF). Laboratoire de combinatoire et d&#039;informatique mathématique. Retrieved 5 May 2009. Weisstein, Eric W. &#034;MRB Constant&#034;. MathWorld. Mathar, Richard J. (2009). &#034;Numerical Evaluation of the Oscillatory Integral Over exp(iπx) x^*1/x) Between 1 and Infinity&#034;. arXiv:0912.3844 [math.CA]. Crandall, Richard. &#034;Unified algorithms for polylogarithm, L-series, and zeta variants&#034; (PDF). PSI Press. Archived from the original (PDF) on April 30, 2013. Retrieved 16 January 2015. (sequence A037077 in the OEIS) (sequence A160755 in the OEIS) (sequence A173273 in the OEIS) Fiorentini, Mauro. &#034;MRB (costante)&#034;. bitman.name (in Italian). Retrieved 14 January 2015. Finch, Steven R. (2003). Mathematical Constants. Cambridge, England: Cambridge University Press. p. 450. ISBN 0-521-81805-2.  The following equation that was shown in the Wikipedia definition shows how closely the MRB constant is related to root two. ![enter image description here] In:= N[Sum[Sqrt^(1/n)* Sqrt[n]^(1/n) - ((Sqrt^y*Sqrt^x)^(1/Sqrt^x))^Sqrt^(-y)/. x -&gt; 2*Log2[a^2 + b^2] /. y -&gt; 2*Log2[-ai^2 - bi^2] /. a -&gt; 1 - (2*n)^(1/4) /. b -&gt; 2^(5/8)*Sqrt[n^(1/4)] /. ai -&gt; 1 - I*(2*n)^(1/4) /. bi -&gt; 2^(5/8)*Sqrt[I*n^(1/4)], {n, 1, Infinity}], 7] Out= 0.1878596 + 0.*10^-8 I The complex roots and powers above are found to be well-defined because we get all either &#034;integer&#034; and &#034;rational&#034; the first of the following lists only, also by working from the bottom to the top of the above list of equations. ![enter image description here] Code: In:= Table[ Head[FullSimplify[ Expand[(Sqrt)^-y/(Sqrt)^x] //. x -&gt; 2 (Log[1 + Sqrt Sqrt[n]]/Log) /. y -&gt; 2 (Log[-1 + Sqrt Sqrt[n]]/Log)]], {n, 1, 10}] Out= {Integer, Rational, Rational, Rational, Rational, Rational, \ Rational, Rational, Rational, Rational} In:= Table[ Head[FullSimplify[ Expand[(Sqrt)^-y/(Sqrt)^x] //. x -&gt; 2 (Log[1 + Sqrt Sqrt[n]]/Log) /. y -&gt; 2 (Log[-1 + Sqrt Sqrt[n]]/Log)]], {n, 1, 10}] Out= {Times, Rational, Times, Times, Times, Times, Times, Times, \ Times, Times} ---------- **Definition 2** **C**&lt;sub&gt;*MRB*&lt;/sub&gt; is defined at [http://mathworld.wolfram.com/MRBConstant.html]. From MathWorld: &gt; ![MathWorld MRB] ![MathWorld MRB 2] &gt; &gt; SEE ALSO: &gt; Glaisher-Kinkelin Constant, Power Tower, Steiner&#039;s Problem &gt; REFERENCES: &gt; Burns, M. R. &#034;An Alternating Series Involving n^(th) Roots.&#034; Unpublished note, 1999. &gt; &gt; Burns, M. R. &#034;Try to Beat These MRB Constant Records!&#034; http://community.wolfram.com/groups/-/m/t/366628. &gt; &gt; Crandall, R. E. &#034;Unified Algorithms for Polylogarithm, L-Series, and Zeta Variants.&#034; 2012a. &gt; http://www.marvinrayburns.com/UniversalTOC25.pdf. &gt; &gt; Crandall, R. E. &#034;The MRB Constant.&#034; §7.5 in Algorithmic Reflections: Selected Works. PSI Press, pp. 28-29, 2012b. &gt; &gt; Finch, S. R. Mathematical Constants. Cambridge, England: Cambridge University Press, p. 450, 2003. &gt; &gt; Plouffe, S. &#034;MRB Constant.&#034; http://pi.lacim.uqam.ca/piDATA/mrburns.txt. &gt; &gt; Sloane, N. J. A. Sequences A037077 in &#034;The On-Line Encyclopedia of Integer Sequences.&#034; &gt; &gt; Referenced on Wolfram|Alpha: MRB Constant &gt; CITE THIS AS: &gt; Weisstein, Eric W. &#034;MRB Constant.&#034; From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MRBConstant.html **How would we show that the any of the series in the above MathWorld definition are convergent, or even absolutely convergent?** ---------- For &#034;a&#034;&lt;sub&gt;k&lt;/sub&gt;=k&lt;sup&gt;1/k&lt;/sup&gt;, given that the sequence is monotonically decreasing according to [Steiner&#039;s Problem], next, we would like to show (5) is the alternating sum of a sequence that converges to 0 monotonically and use the Alternating series test to see that it is conditionally convergent Here is proof that 1 is the limit of &#034;a&#034; as k goes to infinity: &gt; ![enter image description here] [Here] are many other proofs that 1 is the limit of &#034;a&#034; as k goes to infinity. Thus, (k&lt;sup&gt;1/k&lt;/sup&gt;-1) is a monotonically decreasing and bounded below by 0 **sequence.** If we want an absolutely convergent **series**, we can use (4). S&lt;sub&gt;k&lt;/sub&gt;![enter image description here] which, since the sum of the absolute values of the summands is finite, the sum converges absolutely! There is no closed-form for **C**&lt;sub&gt;*MRB*&lt;/sub&gt; in the MathWorld definition; this could be due to the following: in [Mathematical Constants],( Finch, S. R. Mathematical Constants, Cambridge, England: Cambridge University Press, p. 450), Steven Finch wrote that it is difficult to find an &#034;exact formula&#034; ([closed-form solution]) for it. &gt; ![enter image description here] ![enter image description here] ---------- ---------- ---------- ---------- Real-World, and beyond, Applications ==================================== This section and the rest of the content of this first post were moved below to improve loading times. 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https://community.wolfram.com//c/portal/getImageAttachment?filename=Screenshot2022-05-06204151.jpg&amp;userId=366611 Marvin Ray Burns 2014-10-09T18:08:49Z