April 2015 - mathfun@mailman.xmission.com

[math-fun] Möbius (one-sided) disk gif
by Bill Gosper 16 Apr '15

16 Apr '15

http://gosper.org/mobber.gif (An old Macsyma bagatelle. It was somehow relevant here a couple of weeks ago, but it took me this long to rescue it from oblivion.) Source: /* -*- Mode: Macsyma -*- */ mob_animate(ribs):=block([plotnum1:2], plot_animate(v,0,1,eval(funmake('plotsurf,[mobber(ribs),t,0,%pi,u,v*%pi*2-%pi/2,v*%pi*2+%pi/2])),cyclic))$ mobber(n):=makelist(mob(t,(u/2+%pi*(k+1/4))/n-%pi/2),k,0,n-1)$ mob(t,u):=[cos(t)*cos(u)-2*sin(t)*sin(u)^4,sin(t)*sin(u)^2*sin(2*u)+cos(t)*sin(u),sin(t)*sin(2*u)]$ --rwg

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[math-fun] Help with a sum of a series
by Bill Gosper 16 Apr '15

16 Apr '15

On Tue, Apr 14, 2015 at 2:43 PM, Kerry Mitchell <lkmitch(a)gmail.com> wrote: Hi all, I'm working on a problem that involves the sum of an infinite series of fractions. The first few terms are: 1/2 + 1/8 + 2/32 + 5/128 + 14/512 + 42/2048 + ... I've not reduced them to lowest terms so that you could see the pattern: the numerators are Catalan numbers and the denominators are 2*4^n. Empirical results from the first 1000 terms and from projecting the asymptotic nature to the first million terms suggest that the limit of the sum is 1. However, neither my analytical skills nor my computational skills are sufficient to give me confidence in that. Can anyone give me some guidance on how to evaluate this sum? Thanks, Kerry Mathematica sees the pattern anyway: In[1]:= {1/2, 1/8, 2/32, 5/128, 14/512, 42/2048} Out[1]= {1/2, 1/8, 1/16, 5/128, 7/256, 21/1024} In[2]:= FindSequenceFunction[%, n] Out[2]= Pochhammer[1/2, -1 + n]/(2 Pochhammer[2, -1 + n]) In[3]:= Sum[%, {n, ∞}] Out[3]= 1 "Triangles, hell. Death to insight!" --rwg

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Re: [math-fun] What kind of structure are IEEE 754 doubles?
by Henry Baker 16 Apr '15

16 Apr '15

At 04:23 PM 4/14/2015, Mike Stay wrote: >Is addition commutative? Probably not, depending upon which computer you use. Leaving aside issues of NaN's, signaling and exponent issues, some computers have larger mantissas for "accumulation", so that very long iterated sums still "work". E.g., Intel x86 "extended precision" floats with 64-bit mantissa. http://en.wikipedia.org/wiki/Extended_precision Since you're now adding numbers of two different mantissa sizes, they are highly unlikely to be commutative. "satanism" is an anagram for "mantissa"; now do you see why floating point is so hard? http://en.wiktionary.org/wiki/mantissa

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[math-fun] Singapore exam question
by Warren D Smith 16 Apr '15

16 Apr '15

The whole internet apparently knows about this problem, so math-fun might as well too... http://cdn.theatlantic.com/assets/media/img/posts/2015/04/11155125_38575111…

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[math-fun] More dark matter news
by Warren D Smith 15 Apr '15

15 Apr '15

http://www.eso.org/public/archives/releases/sciencepapers/eso1514/eso1514a.… Richard Massey, Liliya Williams, 20 others: The behaviour of dark matter associated with 4 bright cluster galaxies in the 10 kpc core of Abell 3827, MNRAS (2014?) 15 pages now finds fairly direct evidence dark matter does interact with other dark matter particles (which, if so, I have previously argued here, but they do not argue, must be due to a new "dark force") with a cross-section (they claim) of (1.7 +- 0.7) * 10^(-4) cm^2/gram. The previous paper had claimed an upper bound of 0.47 cm^2/gram. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] To: math-fun <math-fun@mailman.xmission.com>
by Tanya Khovanova 15 Apr '15

15 Apr '15

There is a discussion of this puzzle here: http://blog.tanyakhovanova.com/2010/02/wise-men-without-hats/ and here http://blog.tanyakhovanova.com/2011/01/from-a-puzzle-to-a-magic-trick/ Tanya

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[math-fun] What kind of structure are IEEE 754 doubles?
by Mike Stay 15 Apr '15

15 Apr '15

Addition isn't even associative in floating point: http://en.wikipedia.org/wiki/Associative_property#Nonassociativity_of_float… so floating point numbers don't even form a monoid under addition. Is addition commutative? Is there some concise characterization of what kind of structure they do form, like "Addition and multiplication are magmas with an absorbing element NaN such that ..."? -- Mike Stay - metaweta(a)gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com

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[math-fun] Passwords
by James Propp 15 Apr '15

15 Apr '15

Changing gears slightly from the recent thread on protecting secrets from hackers and the NSA, I want to mention an approach to password security that I find very attractive: http://www.scientificamerican.com/article/memory-trick-increases-password-s… What I really like about this proposal is the way in which it makes use of things human brains are good at memorizing (namely images and stories) and builds them into the protocol. It also turns out that there's some combinatorial design theory involved with trading off ease-of-use against security, and trying to maximize the number of distinct passwords the system can create without burdening the user with memorizing too many Person-Action-Object sequences. For more information, check out https://www.youtube.com/watch?v=z4e5VImjc0Y . Jim Propp

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[math-fun] Help with a sum of a series
by Kerry Mitchell 15 Apr '15

15 Apr '15

Hi all, I'm working on a problem that involves the sum of an infinite series of fractions. The first few terms are: 1/2 + 1/8 + 2/32 + 5/128 + 14/512 + 42/2048 + ... I've not reduced them to lowest terms so that you could see the pattern: the numerators are Catalan numbers and the denominators are 2*4^n. Empirical results from the first 1000 terms and from projecting the asymptotic nature to the first million terms suggest that the limit of the sum is 1. However, neither my analytical skills nor my computational skills are sufficient to give me confidence in that. Can anyone give me some guidance on how to evaluate this sum? Thanks, Kerry

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Re: [math-fun] How to protect computer versus malware
by Dave Dyer 14 Apr '15

14 Apr '15

Code signing can be pretty good at certifying that code hasn't be altered since it was created, but the process of vetting signers and issuing certificates is a complete charade. Certificate authorities all the way up to ultimate trust have been compromised, and day-to-day issuing authorities' "due diligence" consists of little more than cashing your check. Once issued, there's no control over how certificates are stored, used, misused or stolen. On a more fundamental level, we know that code and data are interchangeable. If you allow a signed lisp interpreter to run, every piece of text on your computer is now a program. Many exploits start by breaking the code in some way - buffer overruns put unexpected data in play for example. All the trust in the world is useless if the trusted program has a bug. I'm not saying that code-centered security measures are useless, only that they are always going to be fallible.

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