March 2017 - mathfun@mailman.xmission.com

Re: [math-fun] probability
by Dan Asimov 25 Mar '17

25 Mar '17

My favorite probability paradox is intransitive dice. In one of its simplest forms, three 6-sided dice A, B, C with A beating B, B beating C, and C beating A — each more than half the time. E.g., A: 2, 2, 4, 4, 9, 9 B: 1, 1, 6, 6, 8, 8 C: 3, 3, 5, 5, 7, 7. where the common probability of A>B, or of B>C, or of C>A, is 5/9. —Dan >From: Cris Moore <moore(a)santafe.edu> >Is there a good source for intro-level probability “paradoxes” that would give me an opportunity to pit cognitive biases against mathematical level-headedness? I have in mind things suitable for e.g. 6th-graders, things like: “I flip 8 coins. Which is more likely, that they come up HTTHHTTT or HHHHHHHH?”

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Re: [math-fun] probability
by Dan Asimov 25 Mar '17

25 Mar '17

----- . . . that something can be true of every subset but not of the whole set. ----- That's not quite how Simpson's paradox is usually stated. The Wikipedia article at https://en.wikipedia.org/wiki/Simpson's_paradox has some nice examples. —Dan -----Original Message----- >From: "Keith F. Lynch" <kfl(a)KeithLynch.net> >My favorite probability paradox is Simpson's paradox, i.e. that >something can be true of every subset but not of the whole set. >A surprising real-world example is that until recently US life >expectancy was going up, but that there were two groups for which >it was going down: Smokers and non-smokers. > >The explanation is that, roughly speaking, life expectancy was only >increasing because of people quitting smoking, and that all else was >a net loss, presumably because improvements in medical care were more >than offset by the increasing unaffordability of that care. > >Of course it's not really a paradox, just an example of how >mathematical intuition is often wrong.

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Re: [math-fun] probability
by Keith F. Lynch 25 Mar '17

25 Mar '17

My favorite probability paradox is Simpson's paradox, i.e. that something can be true of every subset but not of the whole set. A surprising real-world example is that until recently US life expectancy was going up, but that there were two groups for which it was going down: Smokers and non-smokers. The explanation is that, roughly speaking, life expectancy was only increasing because of people quitting smoking, and that all else was a net loss, presumably because improvements in medical care were more than offset by the increasing unaffordability of that care. Of course it's not really a paradox, just an example of how mathematical intuition is often wrong.

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Re: [math-fun] Beyond Floats: Next Gen Computer Arithmetic
by Leo Broukhis 25 Mar '17

25 Mar '17

On Sat, 25 Mar 2017 06:42:50 -0700, Henry Baker <hbaker1(a)pipeline.com> wrote > > > Were you able to figure out how the "regime" bits in posits work? > The "regime" term is a misnomer. The exponent uses a mixed unary-binary representation of integers with a fixed-sized binary tail (say, 3); 0 - 10 000 1 - 10 001 ... 7 - 10 111 8 - 110 000 9 - 110 001 ... 16 - 1110 000 ... 25 - 11110 001 ... 32 - 111110 000 ... -1 - 01 111 ... -8 - 01 000 -9 - 001 111 In other words, if the number starts with N 1s, its value is (N-1)*2^3 + tail, otherwise -N*2^3 + tail. Leo >

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Re: [math-fun] Beyond Floats: Next Gen Computer Arithmetic
by Leo Broukhis 25 Mar '17

25 Mar '17

On Mon, 20 Mar 2017 14:18:16 -0700, Henry Baker <hbaker1(a)pipeline.com> wrote: FYI -- > > https://www.youtube.com/watch?v=aP0Y1uAA-2Y > > Stanford Seminar: Beyond Floating Point: Next Generation Computer > Arithmetic > > John L. Gustafson, Natl Univ of Singapore > > https://web.stanford.edu/class/ee380/Abstracts/170201.html > > A new data type called a "posit" is designed for direct drop-in > replacement for IEEE Standard 754 floats. > I whipped up my own C++ implementation and verified that the formula on page 40 is indeed evaluated with better precision using 32-bit posits than 32-bit floats. However, using "posits" will require more care to avoid losing precision due to the order of magnitude of the intermediate results. I've plugged my Posit class into enquire.c ( http://homepages.cwi.nl/~steven/enquire.html) and paranoia.c ( http://www.netlib.org/paranoia/paranoia.c) The former gets very confused, claiming that some bits aren't used, and the latter finds several flaws, defects, and failures. Leo

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Re: [math-fun] Beyond Floats: Next Gen Computer Arithmetic
by Henry Baker 25 Mar '17

25 Mar '17

Hi Leo: Were you able to figure out how the "regime" bits in posits work? At 11:21 PM 3/24/2017, Leo Broukhis wrote: >On Mon, 20 Mar 2017 14:18:16 -0700, Henry Baker <hbaker1(a)pipeline.com> wrote: > >FYI -- > >> https://www.youtube.com/watch?v=aP0Y1uAA-2Y >> >> Stanford Seminar: Beyond Floating Point: Next Generation Computer Arithmetic >> >> John L. Gustafson, Natl Univ of Singapore >> >> https://web.stanford.edu/class/ee380/Abstracts/170201.html >> >> A new data type called a "posit" is designed for direct drop-in >> replacement for IEEE Standard 754 floats. > >I whipped up my own C++ implementation and verified that the formula on >page 40 is indeed evaluated with better precision using 32-bit posits than >32-bit floats. > >However, using "posits" will require more care to avoid losing precision >due to the order of magnitude of the intermediate results. > >I've plugged my Posit class into enquire.c ( >http://homepages.cwi.nl/~steven/enquire.html) and paranoia.c ( >http://www.netlib.org/paranoia/paranoia.c) The former gets very confused, >claiming that some bits aren't used, and the latter finds several flaws, >defects, and failures. > >Leo

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Re: [math-fun] Help with some math
by Bill Gosper 25 Mar '17

25 Mar '17

gosper.org/xx-2.png ries (@ -l6) failed to recognize c2(-2-) ~ -4.95208960443219142648286540385 - 1.88898604639119460912777031678 I --rwg On 2017-03-24 17:07, David Wilson wrote: > I presume you are talking about evaluating > > c2(x) = c(c(x)) - x^2 + 2 > > with > > c(x) = ((x + √(x^2 - 4))/2)^√2 + ((x - √(x^2 - 4))/2)^√2 > > If you start with |x| >= 2, you end up with > > c(x) = y^√2 + z^√2 > > where y and z are real, so it is reasonable to take y^√2 and z^√2 real as well. > > If |x| < 2, however, we end up with > > c(x) = y^√2 + z^√2 > > Where y and z are complex numbers. It then seems that y^√2 and z^√2 > take on an infinitude of non-real complex values, none better than > another. So how do you compute y^√2 or z^√2 with y or z complex? Is > there some principle value? > > At any rate, rwg and J. Buddenhagen seem to agree on an evaluation of > c2(x) on |x| < 2. > I would like to see a plot of c2(x) on the real line. > >> -----Original Message----- [Chopped]

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Re: [math-fun] Help with some math
by Bill Gosper 25 Mar '17

25 Mar '17

c2(x) := c(c(x)) - x² + 2 is fascinating. Continuous and real for x > -2, where c2(-2+) = - 4 sin(π √2)² ~ -3.7164323713376353833, followed immediately by a minimum at x = -√(2 - 2 sin(π √2)) ~ -1.981869083895239 where c2(x) = 4 sin(π √2) ~ -3.8556101313995. c2(x) first reaches its permanent value of 0 at x = 2 cos(π/√2) ~ -1.2113997341576, where the slope is discontinuous. All these magic numbers are technically conjectural, from a spectacular performance by Munafo's ries. As David suggests, Mathematica's simplifiers were embarrassed at several points but excelled at others. Likewise the numerics. There are some bugs to report! --rwg (Note: David is using b^3, e.g., to mean b(b(b())).) On 2017-03-23 15:40, David Wilson wrote: > In the fortuitous case of > > b(x) = x^2 - 2 > > b is conjugate to > > a(x) = x^2 > > by the simple mapping > > f(x) = x + x^-1 > f^-1(x) = (x + sqrt(x^2 - 4))/2 > > with > > b^n = f o a^n o f^-1 > > Specifically, if > > c(x) = b^(1/2)(x) = ((x + sqrt(x^2 - 4))/2)^sqrt(2) + ((x - sqrt(x^2 - 4))/2)^sqrt(2) > > then > > c(c(x)) = x^2 - 2. > > It's a nice problem to give the c(x) formula and ask for c(c(x)). > It might be fun to blindside a math forum with this problem. > I can't get Wolfram Alpha to answer it. > > At any rate, the form of this f(x) led me to postulate the power series form > for the f mapping in the x -> x^2 + 1 case. > >> -----Original Message----- >> From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On >> Behalf Of rcs(a)xmission.com >> Sent: Thursday, March 23, 2017 12:42 AM >> To: math-fun(a)mailman.xmission.com >> Cc: rcs(a)xmission.com >> Subject: Re: [math-fun] Help with some math >> >> Dan's worked with flows quite a bit. >> >> Some of the rest of us have played around a little, >> looking for functional square-roots or real-indexed >> iterated functions. There are dragons lurking. >> >> Long ago, I wrote several flow programs in Lisp, >> using the nice feature of exact rational arithmetic. >> I tried to find a power series for the half flow of >> x+x^2, expanded around 0. I used f(x) = x+x^2, >> g(x) = x + x^2/2 + tbd, and the equation f(g(x)) = g(f(x)), >> matching the power series on each side. This led to >> a relatively simple formula for computing terms of g(). >> Things looked good at the beginning, but I got greedy and >> went out to a hundred terms. The numbers looked nice >> for a while, but soon the numerators & denominators >> ballooned. To make sense of things, I floated the >> rational coefficients, and took the nth root of |g_n| >> to estimate the reciprocal of the radius of convergence. >> A nasty surprise: the numerical evidence was that the >> radius of convergence seemed to be heading for 0. >> >> Some of our other experiments had odd results. >> >> Good luck with your approach! >> >> Rich >>

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[math-fun] Stable polyamorous marriage
by James Propp 24 Mar '17

24 Mar '17

A colleague wrote to me saying: *I was wondering if one of you might be able to help point me in the right direction to obtain a solution of a problem that I have created for myself. * *I teach a sophomore level design lab in the ME department and I want to arrange the students in each lab section into teams of three students. I asked each student to provide a ranked list of the five other students who they would like to work with. * *I thought that there would be a "simple" algorithm that would allow me to create the teams from this information. It would appear that I am trying to solve a variant of the stable roommate problem in which there are triples instead of pairs and the ranking of preferences does not include the entire set. * Do any of you know of relevant literature? Jim

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Re: [math-fun] Help with some math
by Dan Asimov 24 Mar '17

24 Mar '17

David, this (and replies thereto) is all very interesting. What led you to consider this particular functional equation? —Dan -----Original Message----- >From: David Wilson <davidwwilson(a)comcast.net> >Sent: Mar 21, 2017 3:01 PM >To: 'math-fun' <math-fun(a)mailman.xmission.com> >Subject: [math-fun] Help with some math > >Let > >f(x)^2 = f(x^2) - 1 > >and assume f is of the form > >f(x) = x + a x^-1 + b x^-3 + c x^-5 + ... > >What are the first few coefficients (say 20) as rational numbers? > >With the tools I have, it would take me a while to power through this. >I assume it would be easier with a symbolic math program. > > > > >_______________________________________________ >math-fun mailing list >math-fun(a)mailman.xmission.com >https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun

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