December 2020 - mathfun@mailman.xmission.com

[math-fun] Is this entire function surjective?
by Dan Asimov 22 Dec '20

22 Dec '20

A complex-valued function f : C —> C on the complex plane C is called "entire" if it's analytic for all z ∊ C. Or in other words it can be defined by a power series f(z) = ∑ c_n z^n with an infinite radius of convergence. The "little theorem" of Picard states that an entire function is surjective or omits just one complex value, or else it is a constant. Clearly the function f(z) = z exp(-z) is not a constant. Does it omit a value? In more generality, what about entire functions of the form g(z) = P(z) exp(Q(z)) where P and Q are polynomials? —Dan

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[math-fun] Tapered peaks
by Éric Angelini 22 Dec '20

22 Dec '20

Hello Math-Fun, How high will you climb? Read one digit d at the time, starting from the first one, and extend S with the product d*(last term of S). When a zero is read, replace this product by the smallest integer not yet present in S and go on. S = 2,4,16,16,96,96,576,5184,31104,279936,1679616,8398080,58786560,352719360,1763596800,1763596800,14108774400,56435097600,169305292800,169305292800,1,4,8,56,... Integers with two or more consecutive zeros must be read carefully. I hope I did. P.-S. How I read and multiplied is visible here, on my blog: http://cinquantesignes.blogspot.com/2020/12/tapered-peaks.html Best, É.

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Re: [math-fun] The True Tau Day(s)
by Keith F. Lynch 21 Dec '20

21 Dec '20

Greg Huber <greg.huber(a)czbiohub.org> wrote: > c'mon people, seriously, grow up -- there's no need to throw away > another Greek letter, 'cause they ain't making any new ones!) There are lots of other alphabets. I suggest using the whole of Unicode for the names of mathematical constants. Some upgrades to Mathematica may be necessary. Last month Randall Munroe discovered that raising an emoji to an emoji power will crash Mathematica.

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[math-fun] Question about perfect matchings
by Dan Asimov 20 Dec '20

20 Dec '20

For a finite graph G (assume G is connected and without loops or multiple edges), a "perfect matching" is a set of disjoint edges all of whose endpoints form the vertex set V(G) of G. (Fun exercise: For each of the five Platonic solids, the graph of its edges has a perfect matching.) Obviously, a *necessary* condition for G to have a perfect matching is that |V(G)| be *even*. Question: --------- Is a simple sufficient condition for G to have a perfect matching known? (A vertex v contained in only one edge e of G would force e to be in any perfect matching, and this leads to easy counterexamples, like the Y-shaped graph on 4 vertices. (So let's also assume there are no vertices contained in only one edge.) —Dan

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[math-fun] Braced Heptagon
by ed pegg 20 Dec '20

20 Dec '20

In Mathematica GraphData[{"UnitDistance", {21, 2}}] Is that a rigid graph? If so, it uses just 42 unit rods, beating the record at https://erich-friedman.github.io/mathmagic/0100.html

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[math-fun] Super-weak Skolem-Langford numbers
by Éric Angelini 18 Dec '20

18 Dec '20

Hello Math-fun, find the first 5 consecutive numbers that are also Super-weak Skolem-Langford. (I guess 6 is impossible). http://oeis.org/A339803 à+ É. Catapulté de mon aPhone

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[math-fun] The True Tau Day(s)
by Greg Huber 17 Dec '20

17 Dec '20

Dear Bill - I agree with you 100% that the Greek letter *tau *should honor Gamma(1/4) = 3.625... (though I confess to having a fondness for *tau*^4, maybe because it's easier -- for me -- to remember that it's near 173). Should some enlightened individuals start a campaign to celebrate a true *Tau Day *in honor of Gamma(1/4)? (2π and June 28 are OK, but NOT *tau* = 2π -- c'mon people, seriously, grow up -- there's no need to throw away another Greek letter, 'cause they ain't making any new ones!) But when to kick off the new, true Tau Day?? Perhaps on 3/6/25, which could mean either March 6, 2025 or 3 June 2025 depending on where you are located? That would give two different target dates to aim for, in case only 3.6... people show up for the party on March 6. Or should it be 10 days after Star Wars Day, with the slogan "May One-Four ^th Be With You"? *Or, perhaps the funniest idea so far, should the annual celebrations commence in Israel, in conjunction with 1 April, since "Gam ma 1/4" means "Also from April 1" *(Heb./Eng. translation: G.Huber)*.* This latter possibility could help to alleviate the worries of those stereotypical Jewish parents who spent undue time and effort sending their kids money in foreign currency, all because of some joker named Rothberg, according to "The Schmooze" section of *The Forward:* https://forward.com/schmooze/154158/israeli-april-fools-jokes-went-too-far/ . Instead, those anxious parents would know that on April Fools' Day their children are studying the properties of the Gamma function, not to mention the many shekels said parents and kids would save because Artin's book is available as a Dover reprint. Furthermore, to the list of leg-pulling utterances such as "I saw your car being towed" and "A beautiful lady asked me to give this phone number to you" (both suggested here https://www.hebrewpod101.com/blog/2018/03/19/how-to-celebrate-april-fools-d… ) one could add remarks especially designed for a dual-purpose April-Fools'/Tau-Day, such as: "I saw your keys around the first zero of the Gamma function" (oh boy! imagine the belly laughs while the remarkee looks around for that), or "Let's do a Zoom call around Gamma of 0 'clock" (they wait centuries in the Zoom room, as you barely suppress guffaws) So, Bill, it seems a true *tau*-inspired Tau Day would truly be a multifaceted laughfest, whenever it happens to happen. -- Greg. ==================== G Huber Theory, Biohub San Francisco, CA On Wed, Dec 16, 2020 at 2:22 AM Bill Gosper <billgosper(a)gmail.com> wrote: > The arclength of one period of sin x = the circumference of an ellipse > with semiaxes 1 and √2. > > In[112]:= #1 == #2 == FunctionExpand@#1== N@# &[ArcLength[Circle[{0, 0}, > {√2, 1}]], ArcLength[Sin@x, {x, 0, 2 π]}]] > > Out[112]= 4 EllipticE[-1] == 4 √2 EllipticE[1/2] == 4 √2 > π^(3/2)/Gamma[1/4]^2 + Gamma[1/4]^2/√(2π) == 7.64039557805542 > > 𝚪(¼) is the rightful value of the symbol 𝛕. And someone should write > Beckmann II: A History of 𝛕. > —rwg >

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[math-fun] Prime-encoding constants/functions
by Leo Broukhis 14 Dec '20

14 Dec '20

Apropos https://www.youtube.com/watch?v=_gCKX6VMvmU There are many ways to produce the sequence of primes by iterating a function, e.g. starting with f_1 = ContinuedFraction(2, 3, 5, 7, 11, 13, ,,,) = 2.3130367364335829..., (http://oeis.org/A064442) and f_n+1 = 1/(f_n - Floor(f_n)) the sequence of Floor(f_n) will be the sequence of primes. Or, as in the clip, starting with the average of http://oeis.org/A053669 = 2.920050977316134712..., and f_n+1 = Floor(f_n) * (f_n - Floor(f_n) + 1) will do the same. Or the Blazys continued fraction, see http://oeis.org/A233588 The question is, which constant/function pair encodes primes in the most compact way. More precisely, given a denominator for a rational approximation, which function will correctly produce the most primes starting from the approximation of the corresponding constant? For example, with the IEEE double (64 bit floating point) approximations of the constants, the continued fraction starts diverging at 29, the Blazys continued fraction - at 47, the 2.92... constant - at 59. I was able to come up with the function f_n+1 = Floor(f) + 1/(f_n-Floor(f_n)); which is the delta-coded continued fraction of the sequence of primes, gives rise to f_1 = 2.7101020234300968374... and given its IEEE double approximation, starts diverging at 67. Improvements are welcome. Thanks, Leo

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[math-fun] Connecting simple closed curves of the same length
by Dan Asimov 10 Dec '20

10 Dec '20

Let C be a rectifiable simple closed curve* (RSCC) in the plane having length L(C) = 2π. Question: --------- Does there always exist a continuous family {C_t} of RSCC such that C_0 = C and C_1 is the unit circle in R^2, such that L(C_t) = 2π for all t in [0,1] ??? —Dan _____ * If C ⊂ R^2 is a RSCC, then C = A([0,1]) for some continuous function A : [0,1] —> R^2 where A(s) = A(t) for s ≠ t if and only if {s,t} = {0,1}, and such that L(C) = sup{∑ ‖A(x_(j+1)) - A(x_j)‖} < oo where the supremum is taken over all 0 = x_0 < x_1 < ... < x_n = 1. The length of C is then defined as L(C).

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[math-fun] Life at the Echigo House in the Shimmachi Quarter
by Brad Klee 10 Dec '20

10 Dec '20

<< Umegawa (to proprietress): Kiyo, that blockhead of a country bumpkin has been bothering me all day and my head is splitting. Hasn't Chubei showed up yet? I sneaked away from my customer, hoping to see you, my only connection with Chubei. Narrator: She slides open the /shoji/ door as she enters, even as she will slide it tomorrow at dawning. Kiyo: I'm glad you've come. There's a crowd of girls upstairs relaxing. They're drinking and playing /ken/ to pass the time until their customers call. Why don't you join them in a game of /ken/ and have a cup of sake? It'll cheer you up. Some of your friends are there. >> Major Plays of Chikamatsu, *Meido no hikyaku, Act 2, p. 173* Tr. Donald Keene (1922-2019, RIP). According to a footnote from Keene: /ken/ is "A game of chinese origin. Each player holds out none to five fingers; the one who guesses the total held out by both wins." See also: https://en.wiktionary.org/wiki/%E6%8B%B3 https://en.wikipedia.org/wiki/Sansukumi-ken https://en.wikipedia.org/wiki/Ken%27ichi In case anyone is curious or wanted to hear, I successfully defended my PhD dissertation on Nov. 20, and now I am expecting a permalink from proquest. Advance review copies of the dissertation can be made available on request to: bradklee(a)gmail.com . Until then, it looks like I'm locked in a combat situation with Julian Stephenson and / or one or more of his psychotic, possibly schizoidal minions "Chas", about whether or not google groups can help us decide if Nichiren Daishonin is the One True Shakyamuni Lotus Preacher, see also: http://ptlslzb87.blogspot.com/ . Hopefully the situation will improve by mid January or thereafter. Thanks again for your help and contributions earlier in time, --Brad

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