January 2015 - mathfun@mailman.xmission.com

[math-fun] NAND flash memory
by Henry Baker 06 Jan '15

06 Jan '15

>From what I understand, NAND flash memory consists of rather large "eraseblocks"/"gcblocks" of perhaps 64KBytes, which can be read in 512byte blocks (?) and written in 512byte blocks (?), but the "write" operation here is really an AND operation, since an erase operation on such a block resets the block to all 1's. Obviously, the usual "FTL" and file system layers cover all of this up -- for good reasons, including bad block handling and "wear leveling", to avoid erasing a block too many times. The question here is whether the ability to perform a simple logic operation -- AND in this case -- can be put to good use. (Let's ignore for the moment that these devices are slower than RAM memories.) The most obvious use which comes to mind is that of a bitvector for keeping track of which "eraseblocks" are free; the most obvious convention would be that a "1" bit indicates that the block is free! Thus, a single 64KB eraseblock could keep track of the free/allocated status of 32GB of flash memory. Note that simply _reading_ or _AND'ing to memory_ don't cause any "wear"; the wear is caused by _erasing_ a complete "eraseblock". So in order to make good use of these blocks, we have to arrange data structures to put stuff together in the same "eraseblock" that all gets erased at the same time, and keep stuff in separate eraseblocks that needs to be independently erased. Next, we notice that algorithms which live in a lattice and which compute LUB's or GLB's can be mapped to such a memory -- e.g., lattice elements the are long bitvectors. So if each bitvector occupied a separate eraseblock, then an optimizing algorithm that computed an LUB, for example, could be mapped to such a flash memory, and only cause "wear" at the end of each computation when the bitvectors are reset. We could also do Connection Machine-style bit-serial processing on 2^19 bit-processors, but for wear-levelling, we would assume a WORM (write-once, read-many) memory, and then recycle the eraseblocks only after all of them had been used at least once. (Note that in CM-style processing, each bitvector i indicates the state of the i'th memory bit in all 2^19 processors in an SIMD array of processors. Also, any ALU ops are "free" because they can be performed by the processor accessing the NAND flash memory.) Can we do faster bit-matrix multiplication? Vaughan Pratt wrote a paper about long bitvector machines, but I'm not sure how much of this would carry (!) over to this model. Can any other lattices other than Boolean lattices be emulated with these long AND operations? Any other ideas?

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[math-fun] Fwd: Countdown
by Bill Gosper 05 Jan '15

05 Jan '15

This is starting to make me want one of those "Ten fingers are overrated" T shirts. ---------- Forwarded message ---------- From: Julian Ziegler Hunts <julianj.zh(a)gmail.com> Date: Wed, Dec 31, 2014 at 5:36 AM Subject: Re: Countdown To: Bill Gosper <billgosper(a)gmail.com> Hans Havermann> > > If we insist on only one the four operators (+-*/) as inbetweens, 2016 is > already impossible. > The cheat is to allow concatenation as well. This gives us up to (but not > including) 2102, > even if the concatenation is disallowed after the 10 (to force the saying > of 'ten'). > Except for 1992: the only way to get it is 1098+765+4+32+1. Julian

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[math-fun] Architectural Pseudo-domes ?
by Henry Baker 05 Jan '15

05 Jan '15

The dome of the Roman Pantheon is a hemisphere, whose Gaussian curvature is positive. Has anyone built a "dome" using a pseudosphere (tractrix of revolution) ? http://en.wikipedia.org/wiki/Pseudosphere BTW, for what architectural constraints would a pseudo-dome be the optimum answer? (I don't know an answer to this.)

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Re: [math-fun] Double moves
by rcs＠xmission.com 04 Jan '15

04 Jan '15

[This is from Scott Kim, at some alternate email address. --Rich] Date: Tue, 30 Dec 2014 12:45:05 -0800 Subject: Re: [math-fun] Double moves From: Scott Kim <scottekim1(a)gmail.com> The board game "Robot Turtles", which teaches programming to kids, includes the idea of each player taking some fixed number of moves in succession. I play it this way because it is both more fun and requires more thinking ahead.

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[math-fun] Length of a _hollow_ stalactite
by Henry Baker 03 Jan '15

03 Jan '15

It appears that the optimum shape for a _hollow_ stalactite of uniform thickness d (d small wrt all other dimensions) is also exponentially tapering, except that instead of a function proportional to exp(-h/B), we have a function proportional to exp(-h/(B/2))=exp(-2h/B). Suppose the thickness of the stalactite surface is the radius of 1 iron atom = 126 picometers = 126*10-12 m. We keep a base of radius 1/sqrt(pi) = 1.834'. With this radius and this thickness, the load-bearing area of our base is 2*pi*R*d = 4.4666x10-10 m^2. If we solve for h where R(h)=the radius of an iron atom, we get 44.7 miles -- exactly 1/2 of the height for the solid stalactite case. If we still want to reach a height of 89 miles, we double the size of the base, for a radius of 2/sqrt(pi) = 3.667'. But the entire weight of the structure is still only 0.1259878322205 kgf ~= 126 gf ~= 4.5 oz. The solid structure weighs about 600,000 times as much as the hollow structure. We have only checked the vertical forces on this thin membrane, but given the extremely small amount of taper, the horizontal forces aren't going to be very large relative to the vertical forces. Notice that if we flip this structure upside down, so that tension becomes compression and vice versa, then the compressive strength of steel is a good deal less than the tensile strength of steel. Furthermore, with compressive loads, we now have to deal with Euler buckling in addition to simple compressive crush strength, so that the structure would have a somewhat different shape and could not reach as high. Finally, the structure would have to be further strengthened in order to withstand wind loads. The result would be something not much different from the Eiffel Tower in Paris.

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[math-fun] "Experiencing Meanings in Geometry"
by Henry Baker 03 Jan '15

03 Jan '15

Very cool paper & pix: http://www.math.cornell.edu/~dwh/papers/Aesthetics/Aesthetics.pdf https://dspace.library.cornell.edu/bitstream/1813/2714/1/2003-4.pdf

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Re: [math-fun] Is Algebra Necessary?
by Henry Baker 03 Jan '15

03 Jan '15

FYI -- Perhaps writers at the NYTimes or The New Yorker would appreciate algebra better if the word problems had more intrigue, more suspense, and of course more diversity in names ... [Thanks to Robert Dodier <robert.dodier(a)gmail.com> for bringing this essay to my attention.] http://www.online-literature.com/stephen-leacock/literary-lapses/40/ THE HUMAN ELEMENT IN MATHEMATICS Stephen Leacock 1869-1944 The student of arithmetic who has mastered the first four rules of his art, and successfully striven with money sums and fractions, finds himself confronted by an unbroken expanse of questions known as problems. These are short stories of adventure and industry with the end omitted, and though betraying a strong family resemblance, are not without a certain element of romance. The characters in the plot of a problem are three people called A, B, and C. The form of the question is generally of this sort: "A, B, and C do a certain piece of work. A can do as much work in one hour as B in two, or C in four. Find how long they work at it." Or thus: "A, B, and C are employed to dig a ditch. A can dig as much in one hour as B can dig in two, and B can dig twice as fast as C. Find how long, etc. etc." Or after this wise: "A lays a wager that he can walk faster than B or C. A can walk half as fast again as B, and C is only an indifferent walker. Find how far, and so forth." The occupations of A, B, and C are many and varied. In the older arithmetics they contented themselves with doing "a certain piece of work." This statement of the case however, was found too sly and mysterious, or possibly lacking in romantic charm. It became the fashion to define the job more clearly and to set them at walking matches, ditch-digging, regattas, and piling cord wood. At times, they became commercial and entered into partnership, having with their old mystery a "certain" capital. Above all they revel in motion. When they tire of walking-matches--A rides on horseback, or borrows a bicycle and competes with his weaker-minded associates on foot. Now they race on locomotives; now they row; or again they become historical and engage stage-coaches; or at times they are aquatic and swim. If their occupation is actual work they prefer to pump water into cisterns, two of which leak through holes in the bottom and one of which is water-tight. A, of course, has the good one; he also takes the bicycle, and the best locomotive, and the right of swimming with the current. Whatever they do they put money on it, being all three sports. A always wins. In the early chapters of the arithmetic, their identity is concealed under the names John, William, and Henry, and they wrangle over the division of marbles. In algebra they are often called X, Y, Z. But these are only their Christian names, and they are really the same people. Now to one who has followed the history of these men through countless pages of problems, watched them in their leisure hours dallying with cord wood, and seen their panting sides heave in the full frenzy of filling a cistern with a leak in it, they become something more than mere symbols. They appear as creatures of flesh and blood, living men with their own passions, ambitions, and aspirations like the rest of us. Let us view them in turn. A is a full-blooded blustering fellow, of energetic temperament, hot-headed and strong-willed. It is he who proposes everything, challenges B to work, makes the bets, and bends the others to his will. He is a man of great physical strength and phenomenal endurance. He has been known to walk forty-eight hours at a stretch, and to pump ninety-six. His life is arduous and full of peril. A mistake in the working of a sum may keep him digging a fortnight without sleep. A repeating decimal in the answer might kill him. B is a quiet, easy-going fellow, afraid of A and bullied by him, but very gentle and brotherly to little C, the weakling. He is quite in A's power, having lost all his money in bets. Poor C is an undersized, frail man, with a plaintive face. Constant walking, digging, and pumping has broken his health and ruined his nervous system. His joyless life has driven him to drink and smoke more than is good for him, and his hand often shakes as he digs ditches. He has not the strength to work as the others can, in fact, as Hamlin Smith has said, "A can do more work in one hour than C in four." The first time that ever I saw these men was one evening after a regatta. They had all been rowing in it, and it had transpired that A could row as much in one hour as B in two, or C in four. B and C had come in dead fagged and C was coughing badly. "Never mind, old fellow," I heard B say, "I'll fix you up on the sofa and get you some hot tea." Just then A came blustering in and shouted, "I say, you fellows, Hamlin Smith has shown me three cisterns in his garden and he says we can pump them until to-morrow night. I bet I can beat you both. Come on. You can pump in your rowing things, you know. Your cistern leaks a little, I think, C." I heard B growl that it was a dirty shame and that C was used up now, but they went, and presently I could tell from the sound of the water that A was pumping four times as fast as C. For years after that I used to see them constantly about town and always busy. I never heard of any of them eating or sleeping. Then owing to a long absence from home, I lost sight of them. On my return I was surprised to no longer find A, B, and C at their accustomed tasks; on inquiry I heard that work in this line was now done by N, M, and O, and that some people were employing for algebraica jobs four foreigners called Alpha, Beta, Gamma, and Delta. Now it chanced one day that I stumbled upon old D, in the little garden in front of his cottage, hoeing in the sun. D is an aged labouring man who used occasionally to be called in to help A, B, and C. "Did I know 'em, sir?" he answered, "why, I knowed 'em ever since they was little fellows in brackets. Master A, he were a fine lad, sir, though I always said, give me Master B for kind-heartedness-like. Many's the job as we've been on together, sir, though I never did no racing nor aught of that, but just the plain labour, as you might say. I'm getting a bit too old and stiff for it nowadays, sir--just scratch about in the garden here and grow a bit of a logarithm, or raise a common denominator or two. But Mr. Euclid he use me still for them propositions, he do." >From the garrulous old man I learned the melancholy end of my former acquaintances. Soon after I left town, he told me, C had been taken ill. It seems that A and B had been rowing on the river for a wager, and C had been running on the bank and then sat in a draught. Of course the bank had refused the draught and C was taken ill. A and B came home and found C lying helpless in bed. A shook him roughly and said, "Get up, C, we're going to pile wood." C looked so worn and pitiful that B said, "Look here, A, I won't stand this, he isn't fit to pile wood to-night." C smiled feebly and said, "Perhaps I might pile a little if I sat up in bed." Then B, thoroughly alarmed, said, "See here, A, I'm going to fetch a doctor; he's dying." A flared up and answered, "You've no money to fetch a doctor." "I'll reduce him to his lowest terms," B said firmly, "that'll fetch him." C's life might even then have been saved but they made a mistake about the medicine. It stood at the head of the bed on a bracket, and the nurse accidentally removed it from the bracket without changing the sign. After the fatal blunder C seems to have sunk rapidly. On the evening of the next day, as the shadows deepened in the little room, it was clear to all that the end was near. I think that even A was affected at the last as he stood with bowed head, aimlessly offering to bet with the doctor on C's laboured breathing. "A," whispered C, "I think I'm going fast." "How fast do you think you'll go, old man?" murmured A. "I don't know," said C, "but I'm going at any rate."--The end came soon after that. C rallied for a moment and asked for a certain piece of work that he had left downstairs. A put it in his arms and he expired. As his soul sped heavenward A watched its flight with melancholy admiration. B burst into a passionate flood of tears and sobbed, "Put away his little cistern and the rowing clothes he used to wear, I feel as if I could hardly ever dig again."--The funeral was plain and unostentatious. It differed in nothing from the ordinary, except that out of deference to sporting men and mathematicians, A engaged two hearses. Both vehicles started at the same time, B driving the one which bore the sable parallelopiped containing the last remains of his ill-fated friend. A on the box of the empty hearse generously consented to a handicap of a hundred yards, but arrived first at the cemetery by driving four times as fast as B. (Find the distance to the cemetery.) As the sarcophagus was lowered, the grave was surrounded by the broken figures of the first book of Euclid.--It was noticed that after the death of C, A became a changed man. He lost interest in racing with B, and dug but languidly. He finally gave up his work and settled down to live on the interest of his bets.--B never recovered from the shock of C's death; his grief preyed upon his intellect and it became deranged. He grew moody and spoke only in monosyllables. His disease became rapidly aggravated, and he presently spoke only in words whose spelling was regular and which presented no difficulty to the beginner. Realizing his precarious condition he voluntarily submitted to be incarcerated in an asylum, where he abjured mathematics and devoted himself to writing the History of the Swiss Family Robinson in words of one syllable.

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[math-fun] Interstellar's Black Hole
by Eric Angelini 02 Jan '15

02 Jan '15

Are those computer graphics of any help to understand what is going on there? I have my doubts... http://youtu.be/7yxaUi8LIzE Best, É. Catapulté de mon aPhone

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Re: [math-fun] math-fun Digest, Vol 143, Issue 1
by Stuart Anderson 02 Jan '15

02 Jan '15

Can you give an example, I don't see how this is possible? > > Every even perfect number is of the form a(n) + 1 for some n. > >>

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[math-fun] Length of a stalactite
by Bill Gosper 02 Jan '15

02 Jan '15

An oldie: http://www.frc.ri.cmu.edu/~hpm/project.archive/1976.skyhook/papers/scasci.t… (via http://en.wikipedia.org/wiki/Space_elevator) On 2015-01-01 13:21, Mike Speciner wrote: > I don't think it's quite the same calculation. For a space elevator, > r**-2 gravity, and centrifugal force are significant. > > --ms > On 01-Jan-15 15:55, Tom Knight wrote: >> This is the same calculation as for erecting space elevators, so it might be worth it to look up their take on the situation. >> >>> On Jan 1, 2015, at 3:33 PM, Henry Baker <hbaker1(a)pipeline.com> wrote: >>> >>> The recent discussions of Gaudi prompted me to think about stalactites. >>> >>> I wanted to understand how long they could get. >>> >>> Suppose a stalactite is made out of structural steel. >>> >>> Suppose that the stalactite is a solid body of revolution, and that the only forces are the vertical tensile forces that oppose gravity, and that the weight of the stalactite is uniformly distributed over the circular face of the base of the stalactite. We also assume that the steel doesn't stretch at all (bad assumption, but it probably doesn't change the weights very much, even though the lengths will grow somewhat). >>> >>> Structural steel has a tensile strength of approx. 250MPa = 250x10^6 kg m/sec^2. >>> >>> Structural steel has a density of approx. 7800 kg/m^3, and g = 9.81 m/sec^2. >>> >>> The optimum shape (with these constraints) for a stalactite appears to be an exponential function R(h)=exp(-h/B)/A, for suitable constants A,B>0. >>> >>> Let's consider a stalactite whose base area is 1 m^2. Then A=sqrt(pi). >>> >>> A steel base with area 1 m^2 can support 250*10^6 Newtons. We need to compute the volume of steel which weighs 250*10^6 Newtons. >>> >>> A 1x1x1 m cube of steel weighs 7800x9.81 = 76518 Newtons. So our base can support hanging 3267 m^3 of steel. >>> >>> So 3267 = integrate(pi*(exp(-h/B)/sqrt(pi))^2,h,0,inf) = integrate(exp(-2h/B),0,inf) = B/2, so B=6534. >>> >>> But our stalactite is infinitely long, which can't be right. Suppose that it ends with a single iron atom of radius 126 picometers = 126*10^-12 m. Then h = 145201 m = 89 miles. >>> >>> And this is ordinary structural steel; specialty wire steels are 10x stronger for the same density. >>> >>> http://en.wikipedia.org/wiki/Ultimate_tensile_strength >>>

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