February 2014 - mathfun@mailman.xmission.com

[math-fun] Faultfree Perfect Squared Klein Bottles of orders 4-7
by Stuart Anderson 09 Feb '14

09 Feb '14

Geoffrey Morley has discovered more faultfree perfect squared Klein bottles of orders 4-7. A pdf showing the 91 known dissections is here ; http://www.squaring.net/downloads/FPSKBs4-7custom2.pdf Stuart

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[math-fun] permutations which fix an odd prefix -- solved
by Warren D Smith 08 Feb '14

08 Feb '14

I found the recurrence which had been eluding me. Let F(n) be the number of n-permutations which fix at least one odd-prefix. Start with F(0)=F(1)=0. Then I claim: F(n) = SUM(k=1,3,5,..., whichever is odd among{n-1, n-2})OF (k!-F(k))*(n-k)! is the magic recurrence from which all counts F(n), for every n>1, arise. Checking the first few cases manually: F(0)=0 F(1)=0 F(2)=1 F(3)=2 F(4)=6+4=10 F(5)=24+4*2=32 F(6)=120+4*6+88=232 F(7)=720+4*24+88*2=992 and those all indeed do agree with my computer counts. To see why this recurrence holds, enumerate all the F(n) permutations of {1,2,3,...,n} which fix an odd prefix. They are: perms of form their count 1... (n-1)! (123)... (3!-2)*(n-3)! where only counting the ones not of preceding form; that is, (3!-F(3))*(n-3)! (12345)... (5!-F(5))*(n-5)! where again only counting the ones not of preceding two forms and so on.

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[math-fun] Help with linear motion problem
by David Wilson 07 Feb '14

07 Feb '14

This is a linear motion problem (actually a work problem). At time 0, a particle is at distance 0 and known velocity v (d(0) = 0, v(0) = v). At known time t, the particle comes to a stop at known distance d (d(t) = d, v(t) = 0). Velocity decreases over the entire motion (a(x) <= 0 for 0 <= x <= t). The particle never reverse direction (v(x) >= 0 for 0 <= x <= t). I don't think this is soluble with constant acceleration (a(t) = c), perhaps with acceleration linear in time?

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Re: [math-fun] probability random permutation fixes at least one odd-cardinality prefix
by Warren D Smith 07 Feb '14

07 Feb '14

n n! n!*P(n) P(n) 1 1 0 0.000000 2 2 1 0.500000 3 6 2 0.333333 4 24 10 0.416667 5 120 32 0.266667 6 720 232 0.322222 7 5040 992 0.196825 8 40320 10096 0.250397 9 362880 53408 0.147178 10 3628800 727360 0.200441 11 39916800 4569536 0.114477 12 479001600 79501696 0.165974 13 6227020800 578101376 0.092838 14 87178291200 12337163008 0.141516 All numbers are exact except that the final column is rounded to 6 decimals. Arguably the odd-n and even-n subsequences should be separated. Considering even n only... Does the sequence 1, 10, 232, 10096, 727360, 79501696, 12337163008 obey any simple recurrence? I observe that 10-2*3*1 = 4 = 2^2 232-4*5*10 = 32 = 2^5 10096-6*7*232 = 352 = 11*2^5 727360-8*9*10096 = 448 = 7*2^6 79501696-10*11*727360 = -507904 = -31*2^14 12337163008-12*13*79501696 = -65101568 = -254303*2^8 These seem to reveal something interesting happening, but I do not know what it is.

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[math-fun] Home Plate doesn't exist.
by Bill Gosper 07 Feb '14

07 Feb '14

I just tried to age-minimize MathWorld's Home Plate article by using Conway's extreme √2 irrationality proof: http://gosper.org/homeplate.html --rwg

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[math-fun] probability random permutation fixes at least one odd-cardinality prefix
by Warren D Smith 06 Feb '14

06 Feb '14

The following problem recently arose (and was only approximately solved) in a paper in Notices AMS. Can mathfunners do better? GIVEN: a random permutation of {1,2,3,...,n} where n>2. QUESTION: what is the probability P(n) that the permutation fixes at least one odd-cardinality prefix? For example, if the prefix length is k, and k=odd, then this probability would be k! * (n-k)! / n!. However, the problem is asking not about just one prefix-length k, but ALL odd k: k=1,3,5,.. whichever is odd among {n-1, n-2}. The Notices paper claimed P(n) <= 2/n + O(n^(-2)). Can one say something more precise? -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] More 4th grade "math"
by Bill Gosper 06 Feb '14

06 Feb '14

I just failed to Google up any outrage over an infuriating TV quiz show I mistakenly watched several years ago. The contestant, (former astronaut Rusty Schweikart?) sailed right through to the million dollar question without even using his two "lifelines(?)", and then was sent home penniless for not listing 1 among the "factors" of 6. I wonder if, had he mentioned 1, they would have screwed him for confusing "factors" with "divisors". Anyway, ("drill and kill") Saxon Math is apparently using "factor" to mean divisor, presumably because it has fewer syllables: "Write the factors of 10 that are also factors of 15." Sounds like they are expecting a "factor" of 1. (I should've checked if they think 1 is a prime.) In the same assignment, "Divide 20/9. Write the quotient as a fraction." "Divide and write the quotient as a fraction: 25/6" It would probably be less harmful if they signaled their intent with gestures and whistles, as in canine obedience school. --rwg

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Re: [math-fun] obfuscated programs, moron detection interlude
by Fred Lunnon 05 Feb '14

05 Feb '14

I certainly have not read the proofs: just this thread, and the article I cited! But it looks as if your observation concerning resource inflation is probably the key --- it's all rather reminiscent of some claims about "universal" models of computation, which turn out to depend on some very dubious encoding scheme being required covertly to do most of the work. I simply don't believe that any practical scheme will encode --- for example --- my prime input into an obfuscated prime before forwarding it to the server; and if it fails to do so, the server cannot possibly factorise it correctly! So I'm afraid my hunch is to side with WDS: there's a gaping hole lurking somewhere. WFL On 2/5/14, Steve Witham <sw(a)tiac.net> wrote: >>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 > > _______________________________________________ > 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] Correction re Projection of a dodecahedron onto anunrolled cylinder
by Bill Gosper 05 Feb '14

05 Feb '14

Hilarie> Thank you, Neil. I think that Wikipedia may have what you are describing on their dodecahedron page (that page is marvelous, by the way). < Except that it still uses http://en.wikipedia.org/wiki/File:Pyritohedron_animation.gif instead of http://gosper.org/pyrominia.gif, and doesn't discuss the endododecahedron, nor Julian's equilateral, topologically regular one. Hilarie> Al[l]an, yes, I believe that dodecahedron edges project to pieces of sinusoids. The projection looks something like two lines of clenched teeth, 5 up, 5 down, with scalloped edges on the top and bottom. Hilarie Date: Tue, 4 Feb 2014 18:32:08 -0500 From: Neil Sloane <njasloane(a)gmail.com> Hilarie, This is not an answer to your question. But it might be helpful to start with nice coordinates. Let t = golden ratio (1+sqrt 5)/2. Then the 8 points (+-1, +-1, +-1) together with all twelve cyclic shifts of (0, +-1/t, +-t) form a dodecahedron (they lie on the sphere of radius sqrt 3). Given the points, you can do anything you want. Page 157 of my notebook Latt 22 has a sketch with lots more data - I could send you a pdf of a scan if you want. But that's a long way from an svg file of course Neil On Tue, Feb 4, 2014 at 6:05 PM, Hilarie Orman <ho(a)alum.mit.edu> wrote: > What I really want is the projection of a dodecahedron onto a sphere, and > that projection projected onto a cylinder (hat-box style), and that > projection > unrolled into a rectangle. As an svg file. > > Hilarie > > > Date: Tue, 04 Feb 2014 02:52:37 -0800 > > From: Henry Baker <hbaker1(a)pipeline.com> > > Do you mean a projection of the Archimedes "Hat Box Theorem" type? > > > At 12:55 AM 2/4/2014, Hilarie Orman wrote: > > >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 > --rwg When my brainscan didn't show anything it was like a great weight lifted from my shoulders.

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Re: [math-fun] More 4th grade "math"--The field with 1 element.
by Henry Baker 05 Feb '14

05 Feb '14

Thanks for the links! A field with 1 element would be more acceptable for typical algebra if a slightly different definition of _ring homomorphism_ were given, that didn't require f(1)=1. Things field-like things have been called "rngs" instead of "rings" (get the pun?). (Perhaps "felds" instead of "fields" ?) http://en.wikipedia.org/wiki/Ring_homomorphism http://en.wikipedia.org/wiki/Rng_(algebra) At 10:53 PM 2/4/2014, W. Edwin Clark wrote: >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

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