July 2009 - mathfun@mailman.xmission.com

Re: [math-fun] factoring almost completely monic polynomials
by Dan Asimov 02 Aug '09

02 Aug '09

What are the cases you know that don't work? --Dan Erich wrote: << for integers m and n, i've been factoring (over the integers) the polynomials: P = (x^n-1) / (x-1) - m obviously when m = (k^n-1)/(k-1) for some integer k, there is a linear factor. i'm interested in the cases when P factors without a linear factor. i've only found 6 cases: [details re 5, 6, 8, 9, 11, 14 omitted] is there some rhyme or reason for these, or are they just random? >> _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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Re: [math-fun] L versus 2L
by Dan Asimov 01 Aug '09

01 Aug '09

Is this the "Webster's Third New International" ? In an interesting etymological note, RWG wrote: << . . . The off-line (and far superior) MW gives . . . >>> _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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Re: [math-fun] L versus 2L
by Dan Asimov 30 Jul '09

30 Jul '09

Interesting that "pace" means 2 steps. It certainly also means just 1 step, in common parlance. Just one of those little contradictions of language. Here is def. 3 for pace, according to < www.m-w.com >: 3 a : step 2a(1) b : any of various units of distance based on the length of a human step and this is def. 2a(1) for step: 2 a (1) : an advance or movement made by raising the foot and bringing it down elsewhere --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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[math-fun] L versus 2L
by James Propp 30 Jul '09

30 Jul '09

Is there a words that means the operation of taking a pair of steps (one with the left foot and one with the right), or the distance travelled during such an operation? I had hoped that the term "stride" meant this. And indeed, according to at least one website, the stride of a horse is defined as the distance the horse has advanced when each of its legs has returned to its initial position (that is, the distance the horse travels during a full period). But that same site says that a person's stride is defined as the distance the person travels when taking a single step (that is, the distance the person travels during a half-period). For that matter, is a "lap" equal to one pool-length, or two? Jim

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Re: [math-fun] dumb question about general relativity
by Mike Stay 28 Jul '09

28 Jul '09

On Mon, Jul 27, 2009 at 10:59 PM, Steve Witham<sw(a)tiac.net> wrote: > it looks like the only kind of symmetries it applies to are > transformations that you can have continuous versions of, like > translation and rotation. Is this true? It makes me sad! No. > I would have liked a conservation law to pop out of > *every* kind of symmetry! James Gilliam did his thesis on "discrete mechanics", a generalization of classical mechanics in which both the phase space and time are discrete. There's a lot of work on mechanics in which time takes integer values, so the real novelty here was adapting the use of calculus in physics to situations where the phase space is also discrete. It turns out that the Euler-Lagrange equation, Noether's theorem, and the symplectic structure on phase space all generalize to this context. http://www.math.ucr.edu/home/baez/thesis_gilliam.pdf -- Mike Stay - metaweta(a)gmail.com http://math.ucr.edu/~mike http://reperiendi.wordpress.com

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Re: [math-fun] Fwd: HAKMEM 32
by Dan Asimov 27 Jul '09

27 Jul '09

What does "have a cluster point at 0" mean here? E.g., it might mean that the underlying density is infinite, or has a local max, at 0. --Dan << ITEM 32 (Schroeppel) Take a random real number and raise it to large powers; we expect the fraction part to be uniformly distributed. . . . . . . Now, 3 + sqrt2 is suspicious; it looks non-uniform, and seems to have a cluster point at zero. PROBLEM: Is it non-uniform? >> _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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Re: [math-fun] dumb question about general relativity
by Dan Asimov 27 Jul '09

27 Jul '09

I once did speculate about this very possibility, back in the '70s and came up with a paradoxical scenario: Suppose by prearrangement -- or just by luck (?) -- two oppositely time-oriented humanoids met in the same room in some habitable place for one hour -- A of them arriving at noon, planning to stay until 1 p.m., and B arriving at 1 p.m., planning to stay until noon (in some fixed time zone calibration). They sit quietly in separate chair for 1/2 hour "after" their arrivals, where this refers to noon to 12:30 for A, but refers to 1 p.m. to 12:30 for B. Unfortunately, B runs amok at 12:30 p.m., whips out his pistol and shoots A, resulting in what would normally be an immediately fatal injury to A. --Dan << I will not attempt to speculate about what one might observe if a visitor from a faraway part of the universe arrived here with time reversed orientation. _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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Re: [math-fun] dumb question about general relativity
by Dan Asimov 27 Jul '09

27 Jul '09

Lots of interesting things here. It does seem Rowan and Harold were discussing *classical* mechanics. (Can there be theoretical newtonian black holes? Actual ones?) Kind of ironic that using a black hole one could do all kinds of Fields medal-worthy research but be unable to report back about it. --Dan Mike wrote: << On Sat, Jul 25, 2009 at 10:21 PM, <mcintosh(a)servidor.unam.mx> wrote: > Quoting Rowan Hamilton <rowanham(a)gmail.com>: > >> Classical mechanics is *not* invariant under time reversal, as >> others have pointed out, since this violates the Second Law of >> Thermodynamics. > > But that is not true, at least for a time independent or symmetric > Hamiltonian. The Second Law is a statistical, or even empirical > law, and is not part of Classical Mechanics; look at Poincare's > recurrence theorem, for example. >From the perspective of an outside observer, the infalling matter never gets into the black hole. Time-reversing that gives a picture in which there's matter just outside the Schwarzschild radius that moves away from the star; that certainly happens all the time, so there's no contradiction with the second law. It's possible, though very difficult, to arrange things to end up in a low entropy state--for instance, Honda's Rube-Goldberg commercial used no computer graphics and required 606 takes. http://video.google.com/videoplay?docid=-4187430023476942057 For an observer falling into the star, the light outside gets more and more blue-shifted, because from his perspective time outside is passing more quickly. Crossing over the Schwarzschild radius, the light is infinitely blue shifted and infinite time passes. (So if you want to solve the halting problem, leave your computer in orbit around a black hole and jump in! It will do infinitely many calculations in a finite time from your perspective.) >> _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele

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[math-fun] Fwd: HAKMEM 32
by rcs＠xmission.com 27 Jul '09

27 Jul '09

Karl Heuer has done an experiment refuting a conjecture from Hakmem. I've appended the retyped item 32 after his message. I imagine that the original conjecture was based on a small sample. Wrt KHs conjecture, must the near-integer fraction p be rational? A unit fraction? (Think x = cbrt2.) It seems obvious that the set of x>1 with other-than uniform density must be of measure 0, but obvious isn't a proof. Also, imagine that x = 100+tiny. If the definition of being "near" an integer is within +-.05, then each additional power of x has quite a bit of wiggle room: The window grows from width .10 to 10, allowing us to tweak x to taste for each new power. So we could build practically any coarse distribution we want. What's possible here? Rich rcs(a)xmission.com ----- Forwarded message from kwzh(a)gnu.org ----- Date: Mon, 27 Jul 2009 06:44:12 -0400 From: Karl Heuer <kwzh(a)gnu.org> Reply-To: Karl Heuer <kwzh(a)gnu.org> Subject: HAKMEM 32 To: Richard Schroeppel <rcs(a)cs.arizona.edu> Hi there. Yesterday, on a whim, I decided to test your claim at the end of Item 32 of HAKMEM -- that { frac((3+sqrt(2))^n) | n in N } appears to have a non-uniform distribution and a cluster point at 0. Using exact Bignum arithmetic, I computed the first 250000 values of the sequence, each truncated to three decimal places; the histogram of those 1000 buckets looked consistent with a uniform distribution. The sample had a mean of 499.752 and standard deviation 288.651, quite close to what one should expect for the null hypothesis of uniformity. Can you shed any light on what led you to say otherwise, a few decades ago? Google search on [hakmem "item 32"] doesn't seem to have any hits other than the original text, which suggests that nobody in the Web-visible world has ever looked into it. (Conjecture: for any real x, frac(x^n) has near-integers with some density p, with the remaining 1-p being a uniform distribution on the unit interval.) Karl Heuer ----- End forwarded message ----- ITEM 32 (Schroeppel) Take a random real number and raise it to large powers; we expect the fraction part to be uniformly distributed. Some exceptions: 1 -- phi = (1 + sqrt5)/2 2 -- all -1 < X < 1 3 -- sqrt2 (half are integers, other half are probably uniformly distributed) 4 -- 1 + sqrt2 -- Proof: N N (1 + sqrt2) + (1 - sqrt2) = integer (by induction); N the (1 - sqrt2) goes to zero. 5 -- 2 + sqrt2 -- similar to 1 + sqrt2 6 -- any algebraic number whose conjugates are all inside the unit circle. Now, 3 + sqrt2 is suspicious; it looks non-uniform, and seems to have a cluster point at zero. PROBLEM: Is it non-uniform?

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[math-fun] Industry's First 1 TByte Mobile Hard Drive
by Henry Baker 27 Jul '09

27 Jul '09

FYI -- (nfi) Feynman was right! There's plenty of room at the bottom. ~ 2^40 bytes in your laptop! At USB 2.0 speeds, it takes nearly 5 hours to transfer all of these data. This drive can hold 2 years of mp3 stereo audio @ 1 MByte/min -- almost enough iPod music for a Mars trip. WD Ships Industry's First 1 TB Mobile Hard Drive LAKE FOREST, Calif., July 27 -- WD today announced ... new mobile hard drives... The highlight is a one terabyte model -- the industry's highest-capacity 2.5" drive available. Industry-leading 333 GB-per-platter technology enables the new WD SATA 2.5-inch hard drives to offer mobile storage device and notebook users an enormous 1 TB capacity. ...The manufacturer's suggested retail price for the WD 1 TB drive is $249.99 USD...

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