September 2019 - mathfun@mailman.xmission.com

Re: [math-fun] Who has written a clear summary of what research mathematicians do?
by Steve Witham 01 Oct '19

01 Oct '19

Scott Kim wrote-- > I think it's an occupational hazard of being an > expert in any field, not just math, that it can be hard to explain what you > do to an outsider. Nonetheless I think it's important to do, and not just > for outsiders, but also for people inside the field. At least it's > important to me Â— the dearth of mathematicians talking about what > mathematics is I find intolerable, and makes me distrustful of mathematics > as a field. Ted Nelson's _Computer Lib_ (1974) for me was a taste of what the good thing can be: An enthusiastic insider in a field who is also a skeptical outsider without missing a beat. There's the public-facing importance of that (especially with something commercial like computers), but also I like the way I oriented myself after reading that book. Stewart Brand, Martin Gardner, Aubrey de Grey, Vi Hart. https://youtu.be/v-pyuaThp-c "Doodling in Math Class: Connecting Dots" (talk about not missing beats) --Steve

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[math-fun] NxN generalization of Hilbert / Wunderlich Curves
by Brad Klee 30 Sep '19

30 Sep '19

AltFWord[let_, n_] := Riffle[Table[let, n], Table["F", n - 1]] MoveWord[2, "L"] = "+RF-LFL-FR+"; MoveWord[2, "R"] = "-LF+RFR+FL-"; MoveWord[n_, "L"] := If[OddQ[n], "", "+"] <> StringTake[ MoveWord[n - 1, "R"] <> "F-" <> AltFWord["L", n] <> "-" <> AltFWord["R", n][[2 ;; -1]] <> "+", {If[OddQ[n], 2, 1], -1}]; MoveWord[n_, "R"] := If[OddQ[n], "", "-"] <> StringTake[ MoveWord[n - 1, "L"] <> "F+" <> AltFWord["R", n] <> "+" <> AltFWord["L", n][[2 ;; -1]] <> "-", {If[OddQ[n], 2, 1], -1}]; HilbertGen[n_] := {"L" -> MoveWord[n, "L"], "R" -> MoveWord[n, "R"]} ToVec[str_] := FoldList[Switch[#2, "+", #1*{1, I}, "-", #1*{1, -I}, "F", {Total[#1], #1[[2]]}, _, #1] &, {0, 1}, Characters[str]] Grid[Transpose@{Function[{n}, Labeled[ Map[Graphics[{Line[ReIm[ToVec[#][[All, 1]]]]}, ImageSize -> 500] &, NestList[StringReplace[#, HilbertGen[n]] &, "L", 2]][[-1]], n] ] /@ Range[2, 10]}, Frame -> All] Out[]:= https://0x0.st/zvPI.png

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[math-fun] Proof on Catalan Numbers
by Brad Klee 30 Sep '19

30 Sep '19

Here is an excerpt from one of my unpublished manuscripts: https://github.com/bradklee/Docs/blob/master/CatalanExcerpt.pdf It describes a derivation of the Catalan numbers, which I have not seen elsewhere. Does this proof work for everyone? Has anyone seen it written down elsewhere? How do you think it compares to using André's reflection method? Thanks --Brad

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[math-fun] Wunderhilbertlich Curves
by Brad Klee 30 Sep '19

30 Sep '19

HilbertC = {"L" -> "+RF-LFL-FR+", "R" -> "-LF+RFR+FL-"}; WunderlichF5 = { "L" -> "+RFRF-LFLFL-F-LF+RFR+FL", "R" -> "-LFLF+RFRFR+F+RF-LFL-FR"}; AsymWunderhilbertlich = { "L" -> "+YF-XFX-FY+", "R" -> "-XF+YFY+FX-", "X" -> "+RFRF-LFLFL-F-LF+RFR+FL", "Y" -> "-LFLF+RFRFR+F+RF-LFL-FR"}; SymWunderhilbertlich ={ "L" -> "+YF-XFW-FZ+", "R" -> "-WF+ZFY+FX-", "X" -> "+RFRF-LFLFL-F-LF+RFR+FL", "Y" -> "-LFLF+RFRFR+F+RF-LFL-FR", "W" -> "LF+RFR+FL-F-LFLFL-FRFR+", "Z" -> "RF-LFL-FR+F+RFRFR+FLFL-"}; ToVec[str_] := FoldList[Switch[#2, "+", #1*{1, I}, "-", #1*{1, -I}, "F", {Total[#1], #1[[2]]}, _, #1] &, {0, 1}, Characters[str]] Grid[Partition[Map[Graphics[{Line[ReIm[ToVec[#][[All, 1]]]]} , ImageSize -> 250] &, NestList[StringReplace[#, AsymWunderhilbertlich] &, "L", 5]], 3], Frame -> All] Grid[Partition[Map[Graphics[{Line[ReIm[ToVec[#][[All, 1]]]]} , ImageSize -> 250] &, NestList[StringReplace[#, SymWunderhilbertlich] &, "L", 5]], 3], Frame -> All] Out[]: https://0x0.st/zt76.png https://0x0.st/zt7Y.png

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[math-fun] for the fun
by françois mendzina essomba2 30 Sep '19

30 Sep '19

k*Pi/2=-(sum(((-sqrt(-4*cos((c+a)*x)*cos((d+b)*x)+cos((d+a)*x)^2+2*cos((c+b)*x)*cos((d+a)*x)+cos((c+b)*x)^2)+cos((d+a)*x)+cos((c+b)*x))^n*(sin((d-c)*n*x)+sin((b-a)*n*x)))/(n*2^n*cos((d+b)*x)^n),n=1..infinity))+(c+a)*x; k in Z a,b,c,d in R when sqrt(delta) exist in R

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[math-fun] a multiple of pi
by françois mendzina essomba2 30 Sep '19

30 Sep '19

Hi, k*Pi/2=a*x+sum((1/n)*(cos(a*x)/cos(b*x))^n*sin((a-b)*n*x),n=1..infinity); k in Z a,b in R with the exceptions

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[math-fun] For natural log
by françois mendzina essomba2 29 Sep '19

29 Sep '19

Hi, log(t)/2=-sum(((cosh(x)-sqrt(cosh(x)^2+t-1))^n*cosh(n*x))/n,n=1..infinity); log(t)/2=-sum(((cos(x)-sqrt(cos(x)^2+t-1))^n*cos(n*x))/n,n=1..infinity); The exceptions for the values of x are obvious for my previous message. Best regards

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[math-fun] arctan function
by françois mendzina essomba2 29 Sep '19

29 Sep '19

Hi, atan(t)/2=sum((t^(-2*n-1)*(sqrt(sin(x)^2+t^2)-sin(x))^(2*n+1)*sin((2*n+1)*x))/(2*n+1),n=0..infinity); Best regards

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[math-fun] A046693 and sparse rulers. Simple proof that smallest covering subset of {0, ..., n} has a size less than sqrt(3 n)+4
by ed pegg 28 Sep '19

28 Sep '19

A Wichmann ruler has the form 1^r, r+1, (2r+1)^r, (4r+3)^s, (2r+2)^(r+1), 1^r. I find it more convenient to code this as {{1,1+r,1+2 r,3+4 r,2+2 r,1},{r,1,r,s,1+r,r}} This ruler can be extended with a mark at the end. 1^r, r+1, (2r+1)^r, (4r+3)^s, (2r+2)^(r+1), 1^r, (r+1)^1. The last mark could be of size less than r+1. The ruler could also be extended multiple times. The following Mathematica code gives complete rulers of length 123 to 1256, each having less than sqrt(3 n)+4 marks. I arbitrarily cut this off at r=9, but a much larger r could be used to extend the complete rulers to infinity. xx=9;upto9=Flatten[Table[Transpose[Select[Transpose[Flatten/@{{1,1+r,1+2 r,3+4 r,2+2 r,1,If[t<r+1,{t},{r+1,Last[IntegerDigits[t, r+1]]}]},{r,1,r,s,1+r,r,If[t<r+1,{1},{First[IntegerDigits[t, r+1]],1}]}}], First[#]>0&]],{r,3,xx},{s,2 r-2,2r+3},{t,0,4 r+2}],2]; I think sqrt(3 n)+4 is a better upper bound than the John Leech bound of sqrt(3.348 n), and it builds the subset. What do people think of this demonstrative proof? --Ed Pegg Jr

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[math-fun] "Transformer model" machine learning tool outperforms Mathematica on integrals and diffeqs
by Mike Stay 28 Sep '19

28 Sep '19

This is the same kind of machine learning system that's been in the news lately for doing so much better at writing comprehensible English. https://openreview.net/pdf?id=S1eZYeHFDS -- Mike Stay - metaweta(a)gmail.com http://math.ucr.edu/~mike https://reperiendi.wordpress.com

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