January 2005 - mathfun@mailman.xmission.com

[math-fun] Lots of neat stuff about e^(pi sqrt(n))
by Mike Stay 24 Jan '05

24 Jan '05

http://www.geocities.com/titus_piezas/Ramanujan_a.htm -- Mike Stay staym(a)clear.net.nz http://www.cs.auckland.ac.nz/~msta039

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[math-fun] ~s mor(e)on units
by R. William Gosper 24 Jan '05

24 Jan '05

I hacked my units package to accept descending lists of target units, e.g., (c517) uconv(2*mph+inch/sec,[fps,furlong/fortnight]) 336 furlong (d517) ------------ + 3 fps 11 fortnight so you can have a pretty arbitrary mixed radix: (c520) uconv((furlong+inch)^2,[acres,feet,inch]) 2 2 (d520) inch + 110 feet + 10 acres The function anglicize contains a capricious (and negotiable) set of units lists: (c521) anglicize(13) (d521) bakersdozen (c523) anglicize(1000) (d523) thousand (c525) anglicize(1730) (d525) pair + greatgross (c560) anglicize(halfcrown) (d560) sixpence + 2 shilling (c526) anglicize(guinea-2*farthing) (d526) twopence + pound + ninepence + halfpenny (c535) anglicize(1.0*litre) (d535) 1.0 quart + 1.81388 fluidounce (c536) anglicize(mi^3-floz) 3 3 3 3 (d536) 166 yard + 63 rod + 3 quart + pint + 511 furlong + 10 foot 515 fluidounce 3 + -------------- + 999 chain 77 (c537) anglicize(d536+floz) 3 (d537) mile Now a question: my Webster's Unabridged gives sidereal day : the interval between two successive transits of the March equinox over the upper meridian of a place that is equal to 23 hours, 56 minutes, 4.09 seconds of mean solar time sidereal year : the time in which the earth completes one revolution in its orbit around the sun measured with respect to the fixed stars : 365 days, 6 hours, 9 minutes, and 9.54 seconds of solar time (c553) uconv(siderealyear,[days,hours,minutes,seconds]) 477 seconds (d553) ----------- + 9 minutes + 6 hours + 365 days 50 tropical year : the period occupied by the sun's center in passing from one equinox to the same again and having a mean length of 365 days, 5 hours, 48 minutes, 45.5 seconds (c551) anglicize(tropicalyear) 459747 second (d551) ------------- + 48 minute + 5 hour + 26 fortnight + day 10000 which differs by 10^-8 and is claimed to be better. I would think a sidereal day would be (c556) (1.d0/(1/siderealyear+1/day),%%=anglicize(%%)) 1.0d0 (d556) ------------------ = 4.09965986199637d0 second + 56.0d0 minute 1 1 ------------ + --- siderealyear day + 23.0d0 hour but (c555) (1.d0/(1/tropicalyear+1/day),%%=anglicize(%%)) 1.0d0 (d555) ------------------ = 4.09053820802342d0 second + 56.0d0 minute 1 1 ------------ + --- tropicalyear day + 23.0d0 hour agrees better with Webster. Is this because the March equinox is precessing? If so, what is the name of the interval required for the stars to complete one circle of Polaris? --rwg PS, the difference between the two notions of year (c557) anglicize(siderealyear-tropicalyear) 235653 second (d557) ------------- + 20 minute 10000 is called NECROPSIES of the equinoxes: tidal effects on Earth's nonsphericities hurry the seasons by 20+ minutes each orbit. Presently, the Northern Hemisphere is closer to the Sun in winter. In about (c558) canon(180.*days/d557) (d558) 12710.4 orbits, this ENCOPRESIS will switch this climatic moderation to the Southern Hemisphere, which also enjoys a higher %age of oceans. Go south, young carbophobe. (Sorry if I permuted a few letters, here.-) PPS, What a great language we have; "vulgar establishment" is a celestial term, and "celestial telescope" is a goldfish.

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[math-fun] Numb3rs
by Daniel Asimov 24 Jan '05

24 Jan '05

I'm between TV sets, but would love to be able to (eventually) watch the premiere of this new show, Numb3rs, involving a detective, and his "math genius" brother who helps him solve cases by using math. It's on CBS tonight at 10 pm ET, 7pm PT; after tonight's premiere it will be at the same time on Fridays. If anyone's willing to tape it (or DVD it) and send me a copy, I'd be very grateful, and more than happy to overcompensate you for parts, labor, and postage. Among other things, I'd like to to see if the math is totally, completely bogus -- as in Good Will Hunting -- or not. Tx in advance. --Dan

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[math-fun] Re: A100832: What exactly are amenable numbers?
by Eric Angelini 24 Jan '05

24 Jan '05

Hello seqfans and math-fun, I've just published A101246 in Sloane's OEIS (see below). Could someone calculate for me the last term of the sequence (which, I'm sure, is finite) ? best, E. -------- 1,3,7,605,607,1446,1538,1608,2545,2618,3027,4616,4903, (...) « Insert the first differences between every pair of integers; this new sequence has it's first 10 digits all different one from another ; and so for the next 10 ; and for the next 10, etc. To construct the above sequence, start with 1 and add the smallest available integer which doesn't lead to a contradiction. » Example : 1 3 7 605 607 1446 1538 1608 2545 2618 (seq. A101246) 2 4 598 2 839 92 70 937 73 409 (first differ.) **10*dig**--10-dig--**10*dig**--10-dig--**10-dig** (10 digit boxes)

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[math-fun] Factors of Mersenne Numbers
by Guy Haworth 24 Jan '05

24 Jan '05

http://www.mersenne.org/ecmm.htm sheds some light. I sent my table of 'Mersenne Numbers: Consolidated Results' for M(p) with p < 130,000 or so to the Maths Comp. UMT via the late Dan Shanks. This has first factors, 2kp+1, indicating 'k' only. Guy

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[math-fun] Rotor-router article
by Michael Kleber 23 Jan '05

23 Jan '05

Hello Jim (and math-fun) : Mathematical Intelligencer 27#1 has hit finer newsstands near you, which means I can now post my rotor-router article on the web. At the moment it's linked to from my Brandeis home page -- I'll work on getting it up on the arXiv, but I think that will require human intervention, since it comes out as a 1 Meg pdf file (mostly due to its containing a picture of the 3 million bug blob), far larger than their usual size limit. My publications list, with "Goldbug Variations" at the top, is at http://people.brandeis.edu/~kleber/pubs.html There's also a link to just download/view the blob picture alone. Jim, you are of course welcome to pass the word on to your usual list of people interested in quasi-randomness. Enjoy! --Michael Kleber -- It is very dark and after 2000. If you continue you are likely to be eaten by a bleen.

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[math-fun] Mersenne divisors
by David Wilson 21 Jan '05

21 Jan '05

Where can I find a list of divisors for completely factored Mersenne numbers?

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[math-fun] most complicated variation on something like a rubic's cube?
by Ray Tayek 20 Jan '05

20 Jan '05

hi, one of the screen savers on my linux box caught my eye, since it was a box doing some strange and improbable rotations of parts of itself (the box looked like something out of some horror movie). so i was curious if anyone had figured out what limitations there were on some geometric solid that could actually be built with actual mechanical (rigid i guess) parts and what geometric shape if any would have the most states or some other measure of complexity. just a thought. thanks --- ray tayek <http://tayek.com/>, co-chair <http://www.ocjug.org/>, actively seeking telecommuting work.

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[math-fun] Spherical trig
by Don Reble 20 Jan '05

20 Jan '05

Given the three angles (A, B, C) of a spherical triangle, what's the radius (r) of the inscribed circle? My reference books are eager to tell me the radius of the circumcircle, or provide the inscribed radius if I know the three sides. Of course, one can get the sides from the angles, and do it that way. But Euclidean triangles have such a nice formula, area / r^2 = cot(A/2) + cot(B/2) + cot(C/2) you'd think there's a nice spherical formula, too. Anyway, I worked it out. For future reference: let u = sin^2 (A/2), v = sin^2 (B/2), w = sin^2 (C/2), p = uvw, q = uv+vw+wu, s = u+v+w -1 + 2s - s^2 + 4p Then sin^2 r = ------------------ 3 - 2s - s^2 + 4q That's satisfactorily symmetric, but still somewhat ugly. Is there a better formula? -- Don Reble djr(a)nk.ca

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[math-fun] Suggestion for Mathworld
by David Wilson 20 Jan '05

20 Jan '05

Eric once gave me the address for Mathworld suggestions/submissions but I have lost it. :-( Anyway, I wanted to suggest Voderberg Tile and Voderberg Spiral be added to Mathworld. The Wolfram site itself has examples.

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