February 2014 - mathfun@mailman.xmission.com

[math-fun] Projection of a dodecahedron onto an unrolled cylinder
by Hilarie Orman 05 Feb '14

05 Feb '14

I'm looking for a diagram of a dodecahedron projected onto a 1x1 cylinder. svg file would be delightful. Google is perversely biased towards projections of the earth onto dodecahedra. Hilarie

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Re: [math-fun] More 4th grade "math"--The field with 1 element.
by W. Edwin Clark 04 Feb '14

04 Feb '14

If F is a finite field of order q then the number of subspace of dimension k of an n-dimensional vector space over F is the q-binomial coefficient<http://en.wikipedia.org/wiki/Gaussian_binomial_coefficient>: binomial_q(n,k). The limit as q-->0 of binomial_q(n,k) is binomial(n,k) --the usual binomial coefficient - the number of k element subsets of an n element set. For this and similar reasons people that study q-analogues <http://en.wikipedia.org/wiki/Q-analog>sometimes refer to sets as vector spaces over the field with 1 element and subsets as being subspaces of such a vector space. See http://en.wikipedia.org/wiki/Field_with_one_element or Peter Cameron's blog entry http://cameroncounts.wordpress.com/2011/07/20/the-field-with-one-element/ --Edwin On Tue, Feb 4, 2014 at 11:11 PM, Dan Asimov <dasimov(a)earthlink.net> wrote: > I think (0,+,*) ought to be considered a field. But apparently > algebraists ignore it, I think by requiring that > 1 != 0, probably to avoid having a special case. > > --Dan > > > On 2014-02-04, at 8:00 PM, Henry Baker wrote: > > > Is {0,+,*} a field or not? > > > > All of its non-zero elements have multiplicative inverses (vacuously) ! > > > > At 05:39 PM 2/4/2014, James Buddenhagen wrote: > >> Back in 1914 the number 1 was a prime. At least according to D. N. > >> Lehmer. See D. N. Lehmer, List of primes numbers from 1 to > >> 10,006,721, Carnegie Institution Washington, D.C., 1914 > >> > >> On Tue, Feb 4, 2014 at 1:38 PM, Andy Latto <andy.latto(a)pobox.com> > wrote: > >>> On Tue, Feb 4, 2014 at 2:12 PM, Eugene Salamin <gene_salamin(a)yahoo.com> > wrote: > >>>> According to a friend who volunteers in the Santa Cruz CA Public > Schools, the official view is that 1 is a prime, because its only divisors > are 1 and itself. However, a teacher did mention that in more advanced > mathematics, 1 is not a prime. > >>> > >>> Do they also teach that unique factorization is false? Or is it stated > >>> as "numbers > 1 have a unique factorization into primes other than 1"? > >>> > >>> Andy > > > > > > _______________________________________________ > > math-fun mailing list > > math-fun(a)mailman.xmission.com > > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >

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Re: [math-fun] obfuscated programs, moron detection interlude
by Steve Witham 04 Feb '14

04 Feb '14

>Date: Tue, 4 Feb 2014 14:57:18 -0500 >From: Warren D Smith <warren.wds(a)gmail.com> > >Date: Wed, 5 Feb 2014 02:34:39 +0000 >From: Fred Lunnon <fred.lunnon(a)gmail.com> >From: Warren D Smith <warren.wds(a)gmail.com> > >--I can write an interpreter WI for interpreting any program >written in Warren's Programming Language (WPL). >WI works as follows. First, if decrypts the WPL program by >public key cryptography with the decryption key >4103943687648904689346189789676897642975685988043769349707438017463179432. >Second, it views the result as a BASIC program and runs it. >Now. Doesn't the net program (WI+its input) at seem pretty darn obfuscated >to you? Here's the definition of indistinguishability obfuscation: >A program-garbling procedure qualifies as an indistinguishability >obfuscator if, whenever two programs that do exactly the same thing >pass through the obfuscator, no one is able to tell which garbled >program came from which original. That is, no one can tell which is which by any (tractable) means, including watching and deeply studying the sequences of internal states as the programs run. Watching WI run, we would see the plaintext BASIC program pop out, so that's not very obfuscated. - - - >From: Fred Lunnon <fred.lunnon(a)gmail.com> > > Most intriguing notion of "homomorphic encryption", of which I knew not: > http://en.wikipedia.org/wiki/Homomorphic_encryption > >But I'm unconvinced about some the claims apparently being made for it. Do you mean you've read the proofs? Not that I have. >For example, if I encrypt a large integer and send it off to an insecure >factorising server, am I going to receive in return its encrypted >factorisation, >despite the server being unable to decrypt the input? Hmm ... That's exactly what I read the wikipedia article to be claiming. -> for certain operations <- I don't know whether factoring can be constructed from those operations. But if the hard problem inside indistinguishability obfuscation turns out to be truly hard, the answer is a definite "Yes, but." You write a straightforward: input an encrypted message decrypt it with this secret-key cypher factor it encrypt the result with this secret-key cypher output the encrypted result ...which you would compile into the obfuscator's input language and run thru the obfuscator, producing a circuit diagram which you would send to the server in advance. Then you could send it an encrypted number, and get back the encrypted factorization. BUT, with the current state of the art, the run time is much larger than that of the original, and the circuit size is proportional to that big run time, which makes it impractical to send or store. --Steve

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[math-fun] Hardin colourings & tri-colouring grids
by Fred Lunnon 04 Feb '14

04 Feb '14

Within the (exiguous) bowels of OEIS A078099 there lurks a throwaway comment to the effect that 3-colouring the nodes of a m x n grid such that edge-adjacent pairs of nodes are assigned distinct colours is equivalent to 2-colouring the nodes such that no square (4-circuit) has both diagonals assigned distinct colours. [Establishing this equivalence is not entirely trivial.] In default of any earlier attribution, I propose to dub the latter colourings "Hardin" colourings. Though I haven't sat down to prove it, it seems reasonable to assume that as n,m -> oo the density of diagonal pairs with equal colours approaches a limit d , for a `random' Hardin colouring. [We can just sum frequencies over distinct n x n colourings where n is large, then take means.] I can show d > 0.5382 , and extrapolation suggests d ~ 0.5395 +/- 0.00005 ; can anybody improve on these values, find an explicit expression, or show a non-trivial upper bound for d ? Fred Lunnon

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Re: [math-fun] More 4th grade "math"
by Henry Baker 04 Feb '14

04 Feb '14

Is {0,+,*} a field or not? All of its non-zero elements have multiplicative inverses (vacuously) ! At 05:39 PM 2/4/2014, James Buddenhagen wrote: >Back in 1914 the number 1 was a prime. At least according to D. N. >Lehmer. See D. N. Lehmer, List of primes numbers from 1 to >10,006,721, Carnegie Institution Washington, D.C., 1914 > >On Tue, Feb 4, 2014 at 1:38 PM, Andy Latto <andy.latto(a)pobox.com> wrote: >> On Tue, Feb 4, 2014 at 2:12 PM, Eugene Salamin <gene_salamin(a)yahoo.com> wrote: >>> According to a friend who volunteers in the Santa Cruz CA Public Schools, the official view is that 1 is a prime, because its only divisors are 1 and itself. However, a teacher did mention that in more advanced mathematics, 1 is not a prime. >> >> Do they also teach that unique factorization is false? Or is it stated >> as "numbers > 1 have a unique factorization into primes other than 1"? >> >> Andy

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Re: [math-fun] obfuscated programs, moron detection interlude
by Dave Dyer 04 Feb '14

04 Feb '14

>Doesn't the net program (WI+its input) at seem pretty darn obfuscated >to you? Whatever program would decrypt and run the obfuscated code would be hacked, and a separate program could be written to decrypt and publish the code. This kind of thing is done all the time to protect sensitive content. It does make the process of hacking it harder.

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[math-fun] Life started when universe only 10-17 MYr old?
by Warren D Smith 04 Feb '14

04 Feb '14

The Habitable Epoch of the Early Universe Abraham Loeb (Harvard) http://arxiv.org/abs/1312.0613 (Submitted on 2 Dec 2013 (v1), last revised 16 Jan 2014 (this version, v2)) In the redshift range 100<(1+z)<137, the cosmic microwave background (CMB) had a temperature of 273-373K (0-100 degrees Celsius), allowing early rocky planets (if any existed) to have liquid water chemistry on their surface and be habitable, irrespective of their distance from a star. In the standard LCDM cosmology, the first star-forming halos within our Hubble volume started collapsing at these redshifts, allowing the chemistry of life to possibly begin when the Universe was merely 10-17 million years old. The possibility of life starting when the average matter density was a million times bigger than it is today argues against the anthropic explanation for the low value of the cosmological constant. --WDS: but it seems a bit hard to believe there were enough atoms besides H, He, and Li at the time, for life to exist... but Loeb thinks by that time massive stars would have had time to form and then supernova, so there would have been heavy elements around...

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[math-fun] supernova
by rcs＠xmission.com 04 Feb '14

04 Feb '14

Has anyone observed the new supernova in M82? Utah has been cloudy recently. Rich

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[math-fun] Zero equals infinity?
by James Propp 04 Feb '14

04 Feb '14

My credit card company wrote to me saying "Your card has Zero Liability Protection", which scared me (what, I have no liability protection?!?), until I read the end of the sentence: "so you aren't responsible for fraudulent activity if you notify us." I would've called that "Infinite Liability Protection" myself... Jim Propp

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[math-fun] Zero equals infinity?
by Bill Gosper 04 Feb '14

04 Feb '14

JPropp> My credit card company wrote to me saying "Your card has Zero Liability Protection", which scared me (what, I have no liability protection?!?), until I read the end of the sentence: "so you aren't responsible for fraudulent activity if you notify us." I would've called that "Infinite Liability Protection" myself... <Jim Propp They must drive a stick. Neutral is 0th gear if you are accelerating, and ∞th gear if you are engine-braking. --rwg

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