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Re: [math-fun] Solving some logarithmic equations exactly
by Bill Gosper 14 Aug '19

14 Aug '19

We need a similar function to solve Kepler's equation! *M* = *E* - e sin *E .* http://www.tweedledum.com/rwg/pizza.htm Will this suffice to describe motion in an elliptical orbit? E.g., {A cos(u(t)), B sin(u(t))}. What is u(t)? Is it the same, modulo scaling t, for all orbits of the shape "A by B"? Does scaling the "sun"'s gravity just scale t somehow? —rwg Date: 2019-08-12 13:11 From: Andres Valloud <avalloud(a)smalltalk.comcastbiz.net> To: math-fun <math-fun(a)mailman.xmission.com> Reply-To: "andres.valloud(a)gmail.com" <andres.valloud(a)gmail.com>, math-fun < math-fun(a)mailman.xmission.com> Thanks! The Wikipedia page notes W cannot be expressed in terms of elementary functions. No wonder trying to clear out x from the original equation was rather difficult! On 8/12/19 2:37 , Neil Bickford wrote: > I think Example 1 in > https://en.wikipedia.org/wiki/Lambert_W_function#Example_1 might be > applicable to this! > > For instance, for a x + b = ln x, set y = ln x - b to get (a e^b) e^y = > y. Then -a e^b = -y e^(-y), and so y = -W(-a e^b). Substituting back > then gives x = e^(b-W(-a e^b)), assuming I’ve done my math right! > > Mathematica informs me that since a x + b = ln x, this can also be > written as x = -W(-a e^b)/a. > > --Neil Bickford > > On Mon, Aug 12, 2019 at 2:11 AM Andres Valloud > <avalloud(a)smalltalk.comcastbiz.net > <mailto:avalloud@smalltalk.comcastbiz.net>> wrote: > > What's a neat way to solve equations such as the below exactly? > > ax + b = ln x > > In particular, what to do when b = 0? > > Andres.

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Re: [math-fun] Random nonempty intersection of cube and plane, revisited
by Keith F. Lynch 13 Aug '19

13 Aug '19

Tomas Rokicki <rokicki(a)gmail.com> wrote: > A quick way to debug this. Keep track of the highest area seen so > far and every time you see a new high area print out the points. > The highest area I'm seeing is 4 sqrt(2) or about 5.656, but > you report a high area of 6.9. Done. For a quadrilateral, with ten million (pseudo-)random planes, the largest area I get is 6.894347. The equation of the plane is 0.002074x - 0.706580y - 0.707630z + 0.005191 = 0. The x,y,z coordinates of the four points, in order, are: -1.000000,-1.000005, 0.995590 1.000000,-0.994135, 0.995590 1.000000, 0.997070,-0.992659 -1.000000, 0.997070,-0.998520 I'm a little puzzled by these numbers. If I understand my code of two years ago correctly, two of the three coordinates of each point should always be exactly -1 or +1, and the third should always be strictly between -1 and +1. What area do you get? By what method? I use the version of the shoelace formula found at https://en.wikipedia.org/wiki/Shoelace_formula immediately after the word "Alternatively." I also rely on Tom Duff's claim that "In 3 dimensions you can ... compute the projected areas in the xy, yz and zx planes and take the square root of the sum of the squares."

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[math-fun] Solving some logarithmic equations exactly
by Andres Valloud 12 Aug '19

12 Aug '19

What's a neat way to solve equations such as the below exactly? ax + b = ln x In particular, what to do when b = 0? Andres.

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[math-fun] Papilio philenor
by Brad Klee 12 Aug '19

12 Aug '19

"Underside of hind wing brilliant metallic blue with orange and black spots" [1]. More precisely, the spots, usually seven of them, have orange centers with predominantly black circumference. However, the surroundings of an orange spot rarely amount to a fully black annulus, and are typically dappled with a white obstruction. On the fifth spot from anterior to posterior, you will often find not just one, but two dabs of white. This description can be checked against numerous digital photographs, and even against video evidence [2-3], but the most fun is to go out in nature and find one yourself [4]! Or if you haven't any in your region, maybe check the history books going back as far as the late 1700s [5]. Most of the more recent articles I could find were more about chemical defence, than chemical morphogenesis. The later was an interest of Alan Turing in the years before his untimely death. These days, not only can we take more and more data integrals, it should also be easier to work out the theory using all sorts of computer algorithms. Any thoughts? Cheers --Brad [1] https://scholarworks.uark.edu/cgi/viewcontent.cgi?article=3106&context=jaas [2] https://www.butterfliesandmoths.org/species/Battus-philenor [3] https://www.youtube.com/watch?v=u2cE86AA1q0&app=desktop [4] https://0x0.st/zO7w.JPG [5] http://entnemdept.ufl.edu/creatures/bfly/pipevine_swallowtail.htm

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Re: [math-fun] Math Jobs?
by Brad Klee 12 Aug '19

12 Aug '19

DELIRIUM: 1. ( Blue and Green ) Look everybody! I just made a t-H-E-r-O-e-M! 2. ( Yellow ) Then the big fat nice man gave me a cracker, and told me to go home. see also: https://literature.stackexchange.com/questions/2459/whats-up-with-the-color…

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[math-fun] The big fix
by Hans Havermann 11 Aug '19

11 Aug '19

In April 2017 I wrote an article on counting t-free ordinals that mentioned (at the end) an issue with Mathematica's then-just-released version 11.1 that incorrectly pluralized newly introduced number names for 10^102 (one "trestrigintillions") all the way up to 10^303 (one "centillions"). A stupid mistake (how did this get past the vetting process?) but an easy (say, ten-minute) fix. "I've alerted Wolfram to the bug", I wrote. Versions 11.1.1, 11.2, and 11.3 came and went without the fix so I complained again about it (just short of a year after my first alert). Somebody at the Wolfram Technology Group thought it helpful to offer this workaround: removePluralS[str_String] := StringReplace[str, "illions" -> "illion"] I did not think to check version 12 when it came out last April. An email from the Wolfram technical support team on August 8 alerted me to the fact that the issue had in fact been resolved. So, two years!

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[math-fun] Fwd: I just rediscovered Alan Adler's old Aerobie aerodynamics tutorial
by Bill Gosper 10 Aug '19

10 Aug '19

---------- Forwarded message --------- From: Alan Adler Date: Fri, Aug 9, 2019 at 10:40 PM Subject: Re: [math-fun] I just rediscovered Alan Adler's old Aerobie aerodynamics tutorial To: Bill Gosper <billgosper(a)gmail.com> https://www.youtube.com/watch?v=9Plt04KpoLc My aero lesson https://www.youtube.com/watch?v=6CLGnkM_5qo boomerangs to ballistics https://www.youtube.com/watch?v=c0YCY2Fd448 good morning America https://www.youtube.com/watch?v=dAfERVvGqco funny flying machines from the aero lesson https://www.youtube.com/watch?v=kK91CEc5TMc about Alan Adler

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[math-fun] Auditory ambiguity
by James Propp 09 Aug '19

09 Aug '19

Do any of you (all of you) know about this? https://youtu.be/5pRY3wlKwm8 It's a bit like Laurel-vs.-Yanni, but in my opinion it's much more striking. If you prepare your mind to hear "brainstorm" (e.g. by saying or audiating "brainstorm"), you'll hear "brainstorm"; if you prepare your mind to hear "green needle", you'll hear "green needle". It doesn't work for everybody, but it works for me and for many other people; the striking thing is that most people can hear both interpretations with equal ease, and can flip back and forth between them. (As opposed to Laurel vs. Yanni, where I think most people find one interpretation dominates the other.) The reason I'm asking is that I'd like to devise a magic trick in which I "convince" a sizeable subset of the audience that they have ESP, using an audio snippet that sounds like it's saying "heads" if the listener expects to hear "heads" and sound like it's saying "tails" if the listener expects to hear "tails". Thanks, Jim Propp

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[math-fun] Fwd: Re: I just rediscovered Alan Adler's old Aerobie aerodynamics tutorial
by Bill Gosper 09 Aug '19

09 Aug '19

The video begins by illustrating that the phenomenon of lift is only possible because air has mass. Somewhere ought to be interesting footage of frustrated birds and insects in an atmosphere with nitrogen replaced by balloon gas. —rwg -------- Original Message -------- Subject: Re: [math-fun] I just rediscovered Alan Adler's old Aerobie aerodynamics tutorial Date: 2017-07-08 03:24 From: Victor Miller <victorsmiller(a)gmail.com> To: math-fun <math-fun(a)mailman.xmission.com> Reply-To: math-fun <math-fun(a)mailman.xmission.com> Yes, I uploaded it. I don't remember where I found it. Victor On Sat, Jul 8, 2017 at 00:50 Bill Gosper <billgosper(a)gmail.com> wrote: > https://www.youtube.com/watch?v=9Plt04KpoLc > (Could the uploader be our own Victor Miller?) > Unfortunately, the video quality is low. The high resolution master > was > discarded by a fly-by-night cassette duplication outfit in Los Angeles. > If you find this video boring (as do most?), I think it's because you > think > you're understanding it, but aren't really. If you're young, come back > to > it > in a couple of years. If you really understand it, it's very > interesting. > --rwg > _________________________________________

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[math-fun] What the heck part is the 3rd factor of the denominator on the right?
by Bill Gosper 09 Aug '19

09 Aug '19

foo=DedekindEta[12 I] -> ((-1 + Sqrt[2])^(3/4) (-1 + Sqrt[3])^(1/6) Gamma[1/4])/( 2^(25/48) 3^(3/8) (2 + 3^(1/4) + Sqrt[3] (Sqrt[2] + 3^(1/4))) \[Pi]^(3/4)); Macsyma had a function named INPART almost identical to Mathematica's Part. Almost nobody used it, for the same reason that Part is a pain in the tush. Everybody used Macsyma's PART, which I really miss. I now realize it was brilliant: Part 1 of a quotient is the numerator, part 2 is the denominator. Clear@ndpart ndpart[xp_, 1, L___] := ndpart[Numerator@xp, L] /; Denominator@xp =!= 1 ndpart[xp_, 2, L___] := ndpart[Denominator@xp, L] /; Denominator@xp =!= 1 ndpart[xp_, p_, L___] := ndpart[xp[[p]], L] ndpart[xp_] := xp So the third factor of the denominator of the right side of foo is In[633]:= ndpart[foo, 2, 2, 3] Out[633]= 2 + 3^(1/4) + Sqrt[3] (Sqrt[2] + 3^(1/4)) In[634]:= Position[foo, %] Out[634]= {{2, 5, 1}} Was that your guess? —Bill

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