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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] Happy 103rd birthday!
by ed pegg 30 Sep '19

30 Sep '19

One of our subscribers turns 103 today.

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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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