October 2010 - mathfun@mailman.xmission.com

[math-fun] Negative yield?
by Dan Asimov 25 Oct '10

25 Oct '10

Apparently some treasury bonds have just sold, at government auction, with negative yields: "Inflation Bonds Are Sold With Negative Yield for First Time" at < http://www.nytimes.com/2010/10/26/business/26markets.html >. But what does "negative yield" mean? I feel confident that it does *not* mean that the purchaser of the bond must make periodic coupon payments to the seller. But perhaps it means that the redemption value of the bond at maturation is *less* than the price it is purchased for? (If so, it's hard for me to imagine why it's not better to just keep that money in a safe all those years. Maybe a good safe is too expensive?) --Dan Those who sleep faster are sooner rested.

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Re: [math-fun] Negative yield?
by Henry Baker 25 Oct '10

25 Oct '10

The redemption value is less, unless there has been inflation. NB: I believe that TIPS are indexed to "core" inflation, which does not include some important things like energy prices, so TIPS won't protect against rapid increases in the price of oil. Bernanke hasn't ruled out negative coupons (or equivalent). See his 2002 speech here: http://www.federalreserve.gov/BOARDDOCS/SPEECHES/2002/20021121/default.htm --- Countries with a habit of too much inflation also have a habit of declaring old currency worthless, from time to time, just to keep the amount stuffed in mattresses under control. Due to the popularity of U.S. currency in the international drug trade, other countries have occasionally called upon the U.S. to "update" our currency to new, crisp, version 2.0 bills, after which the old currency would become worthless. Drug dealers & money launderers would then have to scramble to exchange their millions of bills into new ones, producing lots of chances for them to be noticed. Despite our willingness to spend enormous amounts of money to stop the drug trade, the U.S. hasn't (yet) taken this relatively cheap step. The U.S. government did sever the relationship between gold & dollars back in the 1930's, and severed the relationship between silver & dollars in the 1960's. Harvard&MIT-educated Bernanke would simply take this sequence to its limit, severing the relationship between anything of value and dollars. At 03:43 PM 10/25/2010, Dan Asimov wrote: >Apparently some treasury bonds have just sold, at government auction, with negative yields: > >"Inflation Bonds Are Sold With Negative Yield for First Time" at >< http://www.nytimes.com/2010/10/26/business/26markets.html >. > >But what does "negative yield" mean? > >I feel confident that it does *not* mean that the purchaser of the bond must make periodic coupon payments to the seller. > >But perhaps it means that the redemption value of the bond at maturation is *less* than the price it is purchased for? (If so, it's hard for me to imagine why it's not better to just keep that money in a safe all those years. Maybe a good safe is too expensive?) > >--Dan

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[math-fun] What is my cell-number
by Eric Angelini 25 Oct '10

25 Oct '10

Hello Math-Fun, ... slightly adapted from a puzzle found here (might be old hat): "This Book Does Not Exist: Adventures In The Paradoxical", by Gary Hayden and Michael Picard; +---+---+---+---+---+ | 0 | 0 | 0 | 0 | 0 | +---+---+---+---+---+ | 0 | 1 | 2 | 3 | 0 | +---+---+---+---+---+ | 0 | 4 | 5 | 6 | 0 | +---+---+---+---+---+ | 0 | 7 | 8 | 9 | 0 | +---+---+---+---+---+ | 0 | 0 | 0 | 0 | 0 | +---+---+---+---+---+ You wake up tighten on a chair in one of the cubic cells with number 1 to 9 -- but in which one? You are surrounded by four similar walls -- you don't know which one you are facing: for instance, should you be in cell #4, wouldn't you know if you are facing the wall between your cell and the cell #7, or #5, or #1 or #0. You may ask two questions to the warden -- one after the other; he will give you the correct answers, one after the other. The questions you may ask are only those: - What cell number has the cell in front of me? - What cell number has the cell on my right? - What cell number has the cell behind me? - What cell number has the cell on my left? If you then guess the cell number you are occupying, you will be free -- if not you'll be killed. What are my survival odds if I'm a good math-puzzler? Best, É.

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Re: [math-fun] How many spokes does a bicycle need?
by Dan Asimov 25 Oct '10

25 Oct '10

This seems to be the extent of Amy C. Edmondson's discussion of the minimum number of spokes needed to fix the hub of a bicycle wheel relative to its rim -- sans 4 illustrations -- after Buckminster Fuller. The idea of reducing the degrees of freedom to 0 makes sense. But I can't say I follow her argument, which seems thoroughly non-rigorous, and especially mentions of "negative" degrees of freedom. (In fact, there are indeed 6 degrees of freedom (in the usual sense) for the position and orientation of the hub relative to the rim, which would suggest the possibility that only 6 spokes might work -- if their constraints were independent. Which may explain why Coxeter is cited as claiming that 7 is the correct minimum. What happens if -- as in one of Edmondson's illustrations -- 3 spokes are on one side of the hub at 120 degrees apart, and another three in the same configuration on the other side of the hub . . . but rotated 60 degrees with respect to the first 3 spokes? I don't see intuitively why that won't work, though both books agree that it won't. --Dan << Degrees of Freedom The subject of twelve fundamental directions of symmetry, with their six natural positive-negative pairs, leads directly to a discussion of the "twelve degrees of freedom" inherent in space. The term is almost self-descriptive, but can be best explained in reverse. That is, we explore the number of degrees of freedom inherent in space (and thus affecting every system) in terms of how many restraining forces are necessary to completely inhibit a system's motion. What is the minimum number of applied forces necessary to anchor a body in space? Fig. 7-9 Click on thumbnail for larger image. Again, we can start with a planar analogy. Imagine a fiat circular disk, such as a coaster, lying on a table and held in place by two taut strings pulling in opposite directions. The disk looks stable, but actually is free to move back and forth, at 90 degrees to the line of the two restraints (Fig. 7-9a). So, we try applying three tension forces, 120 degrees apart (Fig. 7-9a), and observe that the circle's position is fixed. Actually, it turns out that only the location of the exact center of the circle is fixed, for the disk is free to rotate slightly in place. Because rotation involves motion directed at 90 degrees to all three strings, there is nothing to restrain the circle from twisting back and forth, as shown in Figure 7-9b. Three additional strings to counteract each of the original restraints would have to be added to prevent all motion, for a total of three positive and three negative vectors. Fig. 7-10. Minimum of twelve spokes needed for stability (a) wheel with six spokes (b) wheel with twelve spokes. Click on thumbnail for larger image. In space, a similar procedure involves a bicycle wheel. Suppose that our goal is to anchor the hub with a minimum of spokes. At first glance this may appear to be the same problem as the previous planar example; however, in this case, the hub has both width and length. Both circular ends of the narrow hub—typically about a half inch wide and 3 inches long—must be stabilized. With only six spokes attaching the hub to the rim (three fixing the position of each end, as shown in Figure 7-l0a), the system feels quite rigid; force can be applied to the hub from any direction—up, down, back or forth—without budging it. However, the hub has no resistence to an applied torque, the effect of which occurs at ninety degrees to the spokes, and is therefore able to twist slightly about its long axis. Three more spokes at each end, to counterbalance the original six, remove the remaining flexibility. All twelve degrees of freedom are finally accounted for, with a minimum of twelve spokes (Fig. 7-l0b). This experiment is quite rewarding to experience—well worth trying for yourself. You don't need to go as far as dismantling a bicycle wheel; just find a hoop of any material and size and a short dowel segment, and then connect the two with radial strings added one at a time until the hub suddenly becomes rigidly restrained. It is enormously satisfying to feel the hub become absolutely immobile (all "freedom" taken away), with the surprisingly low number of twelve spokes. (4) What both the planar and spatial procedures indicate is that degrees of freedom are both positive and negative. In anchoring the hub of the bicycle wheel, there at first appear to be six degrees of freedom; however, each has a positive and negative direction. In conclusion, degrees of freedom measure the extent of a system's mobility: how many alternative directions of motion must be impeded before the body in space is completely restrained. (5) >> Those who sleep faster are more rested sooner.

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[math-fun] dice solitaire
by James Propp 24 Oct '10

24 Oct '10

Martin Gardner fan John Kemeny (no relation to the Dartmouth mathematician) tells me that his niece Allison plays a kind of dice-based solitaire game ("Apparently it is all the rage at Duke, where she was, but is now spending a year at the London School of Economics" writes John) and is curious what the optimal strategy is and what the odds of winning are under optimal play. The player's goal is to knock out numbers from the set {1,2,...,12} until none remain, where the player knocks out one or two numbers on each turn as follows: the player rolls two dice, and may knock out either the sum of the two numbers or may knock out any two distinct numbers that have the same sum as the sum of the numbers shown on the dice. (Note that all that matters is the sum of the numbers the player rolls, not the numbers shown by the two dice individually.) If a player is ever in a situation where no such move is possible, the game ends. This strikes me as do-able by brute force, since there are only 2^12 states of the game (based on which numbers have been crossed out), so, starting from endgame positions and working backwards, one should be able to work out, for each position, what the optimal plays are (for each die-roll) and what the chance of winning is under optimal play. A priori, I see no reason to believe that the optimal strategy has a nice description, but then again I see no reason to believe that it doesn't! Jim Propp

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Re: [math-fun] simple curves on punctured planes
by Dan Asimov 24 Oct '10

24 Oct '10

[Most of this was written before several posts that explain much of what I've described below, but for clarity I'm leaving this unedited.] Let C be a simple closed curve in R^3 - {p,q,r}, where #{p,q,r} = 3. If the winding numbers wind(C;p) = wind(C;q) = wind(C;r) = 0, then, as Bill points out, all of p, q, r must be external to C. The interior of any simple closed curve in R^2, union the curve itself, must be a closed 2-disk. Hence C continuously deforms to a point in R^2 - {p,q,r}. This means that the situation that Jim asks about cannot happen. >From this I infer that the long complicated word written on the wall next to the simple closed curve in the complement of 3 points in R^2, in the painting by "D.S. and B.T.", is *not* a word in the fundamental group of R^2 - {p,q,r} (which is the free group on 3 generators). Despite being 15 feet away when that painting was made, I don't recall anything I may have known about the math behind it. But I'd guess that the word is expressed in the two customary generators of the "isotopy group" G = Isot(R^2 - {p,q,r}) of R^2 - {p,q,r}. (G is the group of all self-homeomorphisms of R^2 - {p,q,r}, under composition, after identifying any two homeomorphisms that can be continuously deformed, one into the other, through homeomorphisms.) In case anyone is interested: Suppose p, q, r lie on the x-axis. Let A be a round annulus with p and q inside its inner circle, and r outside its outer circle; let B be an annulus with q and r inside its inner circle, and p outside its outer circle. Then the Dehn twist in A is the homeomorphism h_A: R^2 - {p,q,r} -> R^2 - {p,q,r} defined via h_A(x) = x for x not in A, but the circles between the inner and outer boundary components of A are each taken to themselves by a rotation by an angle that increases, with radius, from 0 to 2pi. Likewise for the Dehn twist in B, h_B: R^2 - {p,q,r} -> R^2 - {p,q,r}. Then the isotopy classes [h_A] and [h_B] are known to generate* the group Isot(R^2 - {p,q,r}). In the case of the simple closed curve of the painting, it's fun to let the exterior "ooze into" the curve, by shading that in. What remains is a very skinny and folded snake with one eye at each end (namely, the two punctures in its interior). Once this becomes clear, it's easy to imagine how a sequence of Dehn twists might result in unfolding the snake. The inverse of the corresponding word is, I'm guessing, what was painted on the wall. (But the resolution of the photo is too low for me to be able to read that word.) --Dan ___________________________________________________________________ * In general the isotopy classes of n-1 Dehn twists generate the group Isot(R^2 - {p_1,...,p_n}). Much as the n-1 transpositions (12), (23),..., (n-1 n) generate the symmetric group S_n. In fact there is an evident surjective homomorphism Isot(R^2 - {p_1,...,p_n}) -> S_n, according to the permutation of the holes {p_j}. ----------------------------------------------------------------------------- Bill T. wrote: << A simple curve in the plane is the boundary of a disk. If the disk has no points in its interior, the curve is contractible. The curve has winding number 1 or -1 about any point in the interior of the disk. The point is that in the plane with 3 or more punctures, there can be very complicated simple curves, even though they automatically have small winding numbers (0, 1 or -1) about all the punctures. [Jim P. wrote:] << What's the simplest example of a simple closed curve in the triply-punctured plane that has winding number 0 around each of the punctures but is not contractible? (Back when I was in grad school in Berkeley in the '80s, there was a painting of one such curve on the wall, along with the associated word in the fundamental group of the surface.) Also, what's the simplest way to prove that no analogous curve exists for the doubly-punctured plane? >> >> Those who sleep faster are more rested sooner.

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[math-fun] simple curves on punctured planes
by James Propp 24 Oct '10

24 Oct '10

What's the simplest example of a simple closed curve in the triply-punctured plane that has winding number 0 around each of the punctures but is not contractible? (Back when I was in grad school in Berkeley in the '80s, there was a painting of one such curve on the wall, along with the associated word in the fundamental group of the surface.) Also, what's the simplest way to prove that no analogous curve exists for the doubly-punctured plane? Jim Propp

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Re: [math-fun] simple curves on punctured planes
by James Propp 24 Oct '10

24 Oct '10

Thanks, everyone! Now I can get to my real question: The curve shown in http://math.berkeley.edu/publications/newsletter/2002/2002web%20photo%20sca… is a noncontractible simple closed curve in the triply-punctured plane with the property that if you fill in any one of the three punctures, the curve becomes contractible. For what n does there exist a noncontractible simple closed curve in the n-times punctured plane with the property that if you fill in any one of the n punctures, the curve becomes contractible? This question arose in connection with the work of Erik Demaine et al. on "picture-hanging puzzles". Jim Propp

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Re: [math-fun] How many spokes does a bicycle need?
by Gary Antonick 23 Oct '10

23 Oct '10

Amy C. Edmondson may be good to talk to. Wrote "A Fuller Explanation" referencing the 12-spoke rationale- pg 109 of the book. She teaches at the Harvard Business School. (617) 495-6732. On Fri, Oct 22, 2010 at 7:57 PM, Gary Antonick <gantonick(a)post.harvard.edu> wrote: > Amy C. Edmondson may be good to talk to. Wrote "A Fuller Explanation" > referencing the 12-spoke rationale (attached) > > She teaches at the Harvard Business School. (617) 495-6732. > > > On Fri, Oct 22, 2010 at 6:13 PM, Dan Asimov <dasimov(a)earthlink.net> wrote: >> In the biography of H.S.M. Coxeter "King of Infinite Space", >> one section describes his interaction with Buckminster Fuller. >> >> It states that Fuller erroneously claimed that a bicycle wheel >> needs 12 spokes "to hold it rigid", whereas the correct number >> is 7. This seems to be stated as a fact of geometry, rather than >> one of structural engineering. >> >> Anybody have any insight into this claim? >> >> --Dan >> >> P.S. In case anyone's interested, this is the only biography of >> Coxeter I know of and it's filled with interesting facts about >> his life. But it is sadly lacking in details about his mathematical >> accomplishments, other than in the vaguest terms. >> >> >> >> Those who sleep faster are more rested sooner. >> >> _______________________________________________ >> math-fun mailing list >> math-fun(a)mailman.xmission.com >> http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >> >

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[math-fun] How many spokes does a bicycle need?
by Dan Asimov 23 Oct '10

23 Oct '10

In the biography of H.S.M. Coxeter "King of Infinite Space", one section describes his interaction with Buckminster Fuller. It states that Fuller erroneously claimed that a bicycle wheel needs 12 spokes "to hold it rigid", whereas the correct number is 7. This seems to be stated as a fact of geometry, rather than one of structural engineering. Anybody have any insight into this claim? --Dan P.S. In case anyone's interested, this is the only biography of Coxeter I know of and it's filled with interesting facts about his life. But it is sadly lacking in details about his mathematical accomplishments, other than in the vaguest terms. Those who sleep faster are more rested sooner.

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