October 2013 - mathfun@mailman.xmission.com

Re: [math-fun] the real reason for twisted ropes
by Henry Baker 26 Oct '13

26 Oct '13

It's too bad that the Wikipedia editors are such jerks; these would be good points to add to the article on Rope. At 10:16 AM 10/26/2013, you wrote: >if ropes were simply untwisted parallel strands, each a cylinder, and each strand >assumed pre-cut at a random location, then pull on rope ==> it comes apart. >Well, actually, the different strands will stick due to van der Waals forces, >but it'll be very small tensile strength. > >Contrast: with suitable twisted construction: >apply tension, the outer strands compress the inner ones >radially. This causes the strands to stick together due to >friction =coeff*(normal force), and we now have positive normal force. > >Now note, the normal force actually is *proportional* to applied tension >which means no matter how much tension you apply, the rope will not break >(assuming unbreakable strands) even though topologically speaking it already is >broken by assumption. > >So that is a profound theorem. I claim there are related theorems (none proven, but they >should be!) about a lot of knots. That is, knots are mechanisms, >sometimes very ingenious ones. And for many knots/links, even though topologically the unknot or >unlinked, they will not come apart no matter how hard you pull on them >(short of breaking) because the design similarly amplifies friction, the harder you pull >the more frictional force, so it will not come undone no matter how hard you pull. >For example, known rope "bends" (the term of art for tying two ropes together to get an >effectively longer rope) will not fail no matter how hard you pull, short of >actual breakage. But naive bend designs will fail, like if my mother tied them. > >Brilliant knots-as-mechanisms have been designed over the centuries by sailors, truckers, farmers, etc, >but mathematicians and physicists never seem to have designed any, nor >am I aware of proofs of any of these no-fail theorems.

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[math-fun] Another multi-D numerical integration idea -- orthogonal arrays and product rules
by Warren D Smith 26 Oct '13

26 Oct '13

It turns out the same idea I recently thought of and will describe here, was invented earlier by Greg Kuperberg, and also he found out a version of his idea had been invented even earlier by Nicolas Victoir: Asymmetric cubature formulae with few points in high dimension for symmetric measures, SIAM J. Numer. Anal. 42,1 (2004) 209-227 http://www.math.hu-berlin.de/~finance/papers/victoir2.pdf Greg Kuperberg: Numerical cubature using error-correcting codes, http://arxiv.org/abs/math/0402047 My idea was the following. A "product rule" for integrating a function over an D-dimensional hypercube, is simply to integrate in each dimension one at a time, causing a K-node rule to become a K^D-node rule, with the same degree of exactness t as the 1-dimensonal rule had to start with (e.g. t=2*K-1 for a Gauss rule). View each of the K^D points as indexed by a D-letter word in a K-letter alphabet. Now discard points. If the words corresponding to the points you keep form, combinatorially, an "orthogonal array of strength>=t" (in radix K), then the new slimmed down rule will still have degree of exactness t. (Of course it needs to have all weights rescaled by an overall constant factor to compensate for the discardings.) There are various further games one can play, but that is the basic construction. As a result: THEOREM: the D-cube enjoys (for any fixed t>0) an exactness-degree-t quadrature formula with all weights EQUAL and O(D^floor(t/2)) nodes, all interior to the cube. The constant (which depends upon t) hidden in the O may be large, but for any particular fixed t this result is best possible. The same result is also true, but with a better constant inside the O, if all weights are positive but not necessarily equal. PROOF SKETCH: I'd actually originally shown this for O(D^(c*t)) for various constants c with 0<c. That comes by combining known results on 1-dimensional equal-weight quadrature rules (specifically Gauss rules for unequal weights, and Chebyshev and related rules for equal weights) and known results from the theory of orthogonal arrays as duals of linear error correcting codes within prime fields (specifically BCH codes), and finally known results about gaps between consecutive primes. The fact one can get c down to 1/2, which is minimum possible, was shown by Kuperberg via his own extensions of Chebyshev rules (which I had not known about but which are elegantly suited for this purpose) which enable always regarding the radix as 2 or a power of 2 and thus enable the use of BINARY BCH codes (or any binary codes better than BCH). Some results I have that were not mentioned by Kuperberg include: * You can product up, not 1-dimensional rules, but rather 2-diml or 3-diml basic rules. For example there is a 7-point degree-5 rule in the 2-dimensional square by Albrecht & Collatz (beating the naive 9 point count), yielding after D-fold producting a 2*D-dimensional rule of degree 5 with 7^D points (before slimming). Using an wordsize=D orthogonal array of radix=7 and strength=5 we slim this to a number of points bounded by a QuadraticPolynomial(D). J.Albecht & L.Collatz: Z.Angew.Math.Mech. 38 (1958) 1-15 (German). There also is a 32-point degree-5 rule for the 3-dimensional cube with all point-weights equal, by L.N. Dobrodeev: Cubature rules with equal coefficients for integrating functions with respect to symmetric domains, U.S.S.R. Comput. Math. and Math. Phys. 18,4 (1978) 27-34 (in English) I'm unsure how much this idea can pay off. * You can use the product+slimming method to create a rule "almost" of degree of exactness t, but not quite. That is, it will be exact for all degree-t polynomials in D variables as integrand, except for (say) the univariate monomials x^t or bivariate monomials x^A * y^B for A+B=t. Then you can handle these few exceptions by adjoining a comparatively small number of additional integration nodes (with careful design of their locations and weights). Warning: the weights of these extra nodes can sometimes be negative. For example, Genz & Malik devised a rule in D dimensions with 2^D + 2*D^2 + 2*D + 1 points (centrosymmetric) and degree of exactness 7, where the 2^D points in the leftmost term form the vertices of a hypercube. The 2^D may safely be slimmed down to only the number of words in a radix=2 orthogonal array with strength=6. (Septics are integrated exactly to 0 automatically just by centrosymmetry, so strength=7 not needed.) For example a "Kerdock code" accomplishes that. The 2^D points alone will then integrate every degree<=7 monomial exactly except for x^6 and x^4 * y^2, and Genz & Malik's extra points deal with correcting things in those cases. These extra points can have negative weights but the total sum of |weights| is bounded by 0.082*D*D + O(D). A.C. Genz and A.A. Malik. Remarks on algorithm 006: An adaptive algorithm for numerical integration over an n-dimensional rectangular region, J. of Computational and Applied Mathematics 6,4 (1980) 295-302 A.C. Genz and A.A. Malik: An imbedded family of fully symmetric numerical integration rules, SIAM Journal on Numerical Analysis 20,3 (June 1983) 580-588. The Genz and Malik extra points and weights are: 1 point at (0,0,0...,0) with weight= V*(12824 - 9129*D + 400*D^2 ) / 19683 each 2*D points (+-A, 0, 0, ..., 0) with weight = 980*V/6561 each 2*D points (+-B, 0, 0, ..., 0) with weight = V*(1820-400*D)/19683 each [negative if D>=5] D*(D-1) points +-(C,C,0,0,...,0) with weight = V*200/19683 each A=sqrt(9/70) B=sqrt(9/10) C=sqrt(9/19) V=2^D=volume of hypercube [-1,1]^D. The Genz-Malik "extra points" actually form a degree-5 integration rule by themselves (albeit with different weights needed) and the single point (0,0,0...0) is a degree-1 integration rule by itself, which is very convenient for software error-estimation purposes. I believe, however, that the "slimmed" Genz-Malik rule will be far superior for the purposes they used it for, efficiency-wise, than their original "fat" rule. For example, in D=256 dimensions, there is a binary orthogonal array of strength 6 with 2^23 (about 8 million) words arising from a linear code [256,233,6] over GF(2), which I looked up at http://codetables.de . Hence 2^23 + 2*256^2 + 2*256 + 1 = 8520193 points suffice to get an integration rule in a 256-dimensional hypercube exact for polynomials of degree<=7. (Could we do a bit better with a nonlinear code? Have not checked.) This blows away all previous practical 256-dimensional cubature formulas, far as I can tell (at least if you are willing to live with some negative weights). Apparently the numerical integration software community has not used these ideas (Genz & Malik's unslimmed rule is used in some widely used software, though, e.g. CERN library), which is a pity because they've thereby lost a lot of efficiency. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] Moore-Schulman base-3/2 conjecture and OEIS
by Keith F. Lynch 26 Oct '13

26 Oct '13

Has it been proven that there's no *one-sided* counterexample to the Moore-Schulman base-3/2 conjecture? In other words, sum (2/3)^j = 1 for j in some unending integer sequence with an asymptotic density of zero such as the squares, primes, fibonaccis, or factorials? (It's trivial that there's no such *terminating* sequence, for either a one-sides or two-sided counterexample. Any finite sum (or difference) of (2/3)^j for positive integers j would be an even number over an odd number.) If not, has anyone tried summing (2/3)^j for every appropriate sequence in OEIS? (By "appropriate" I mean all terms are positive integers, they increase monotonically, there are infinitely many terms, and the asymptotic density of terms is zero. (A small number of duplicated terms at the beginning of a sequence, as in the factorials and fibonaccis, can be "forgiven.")) If the result is suspiciously close to 1, you could then compute more terms of the series and see if it gets closer and closer to 1. Similarly if the result is very close to some integer power of 2/3, since that's equivalent to the sum over the series with an offset (i.e. adding N to each term). If it does, you could then try to find a proof or disproof that this is not just a coincidence. If there's no such proof and if nobody has tried this, perhaps I will. There are only about 200,000 OEIS entries. And I see that there's an easy way to download all of them, and no rule against doing so for non-commercial purposes. If it *has* been proven that there's no one-sided counterexample, you could still do the above, but then look the the differences between each pair of sums, regarding one sum as representing the +1 digits and the other sum as representing the -1 digits. (Any collisions would be regarded as 0 digits. For instance sum (2/3)^squares minus sum (2/3)^cubes would have (2/3)^sixth-powers in both sequences.) This would be a lot harder, as there would be about 20 billion such differences. Far more if you allow different offsets for the plus and the minus sequences. It's certain that many of these 20 billion differences would be very close to 1 just by chance. But it might be worth doing with just the most popular OEIS sequences, i.e. those that occur most often in math, in the same sense as pi and e are more popular than the twin primes constant or Khinchin's constant. This popularity probably correlates pretty closely with the more ordinary sort of popularity -- web hit count. Am I correct that the most popular OEIS sequences in both senses of the word are the lowest-numbered ones? I think there's also the possibility that the Moore-Schulman base-3/2 conjecture is false, but that no counteraxmple can ever be found because it would require infinite computation, i.e. more and more backtracking without limit. Any thoughts on this dismal possibility?

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Re: [math-fun] the real reason for twisted ropes
by Henry Baker 26 Oct '13

26 Oct '13

Excellent point! Thanks. At 10:16 AM 10/26/2013, Warren D Smith wrote: >if ropes were simply untwisted parallel strands, each a cylinder, and each strand >assumed pre-cut at a random location, then pull on rope ==> it comes apart. >Well, actually, the different strands will stick due to van der Waals forces, >but it'll be very small tensile strength. > >Contrast: with suitable twisted construction: >apply tension, the outer strands compress the inner ones >radially. This causes the strands to stick together due to >friction =coeff*(normal force), and we now have positive normal force. > >Now note, the normal force actually is *proportional* to applied tension >which means no matter how much tension you apply, the rope will not break >(assuming unbreakable strands) even though topologically speaking it already is >broken by assumption.

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[math-fun] the real reason for twisted ropes
by Warren D Smith 26 Oct '13

26 Oct '13

if ropes were simply untwisted parallel strands, each a cylinder, and each strand assumed pre-cut at a random location, then pull on rope ==> it comes apart. Well, actually, the different strands will stick due to van der Waals forces, but it'll be very small tensile strength. Contrast: with suitable twisted construction: apply tension, the outer strands compress the inner ones radially. This causes the strands to stick together due to friction =coeff*(normal force), and we now have positive normal force. Now note, the normal force actually is *proportional* to applied tension which means no matter how much tension you apply, the rope will not break (assuming unbreakable strands) even though topologically speaking it already is broken by assumption. So that is a profound theorem. I claim there are related theorems (none proven, but they should be!) about a lot of knots. That is, knots are mechanisms, sometimes very ingenious ones. And for many knots/links, even though topologically the unknot or unlinked, they will not come apart no matter how hard you pull on them (short of breaking) because the design similarly amplifies friction, the harder you pull the more frictional force, so it will not come undone no matter how hard you pull. For example, known rope "bends" (the term of art for tying two ropes together to get an effectively longer rope) will not fail no matter how hard you pull, short of actual breakage. But naive bend designs will fail, like if my mother tied them. Brilliant knots-as-mechanisms have been designed over the centuries by sailors, truckers, farmers, etc, but mathematicians and physicists never seem to have designed any, nor am I aware of proofs of any of these no-fail theorems.

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[math-fun] High-dimensional numerical integration idea: space-filling curves
by Warren D Smith 26 Oct '13

26 Oct '13

In high dimensions (>20, say) the best known numerical integration method still usually seems to be the monte-carlo method "VEGAS" by G.Peter Lepage. This is monte carlo, enhanced with a crude (but often quite effective) attempt to add "importance-based sampling biases." (The effectiveness of that depends very heavily on the structure of the function being integrated.) G. Peter Lepage: A New Algorithm for Adaptive Multidimensional Integration, J. Computational Physics 27,2 (1978) 192-203. http://www-zeus.desy.de/~tassi/Lepage-slac-pub-1839.pdf Here's a new idea which might be better. View multi-D integration as 1D integration along a uniform space filling curve filling the unit D-dimensional hypercube (assuming that is the region of integration). Accomplish that 1D integration, by employing a crude differential-equation solver to solve for F(1)-F(0) in the equation F'(t) = G(t) where G is the integrand and t is the space filling curve parameter, 0<t<1. Make this solver employ "adaptive step sizes," i.e. if the integrand within the current step t-interval has too great a mean-square deviation from linearity, then redo the step with smaller size; if too small a variance, then increase step size for the next step. Versus VEGAS, this scheme seems on the surface to enjoy several advantages. 1. very simple algorithm [once you know how to convert t to locations (x,y,z...) within the cube] 2. essentially zero memory required 3. low "waste" i.e. function evaluations that are not directly used 4. seems to be more flexible and fine grained in its "adaption" i.e. importance sampling, unlike VEGAS which has a very procrustean importance sampling notion; also the error estimate which the integrator could output, also could be very fine grained. 5. random-point monte-carlo is bad in the sense that points which happen to be close together, are partially "wasted." So-called "quasi monte carlo" methods use "low discrepancy point sets" which avoid such defects and in practice, as well as theoretically asymptotically, perform better than plain monte carlo (i.e. random uniform points). Points equispaced along space filling curve would (I would expect) be somewhat "better than random" points, thus gaining the advantages of quasi monte carlo to some extent. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] Fractal ropes ?
by Henry Baker 25 Oct '13

25 Oct '13

Ropes have been made by mankind since at least 18,000 years ago, yet I have had a difficult time finding much mathematical treatment of them, with the following exception: J. Bohr and K. Olsen 2011 EPL 93 60004 doi:10.1209/0295-5075/93/60004 "The ancient art of laying rope". http://iopscience.iop.org/0295-5075/93/6/60004 Ropes are fractal-like, consisting of several levels: fibers, yarns, strands, rope. Fibers are "spun" into yarns, which are "spun" into strands, which are spun into rope. Each level is spun in the direction opposite to the previous level. The numbers of fibers, yarns, strands affect the geometry of the rope. Question #1: Why have any complexity in ropes at all? The tensile strength of a material isn't improved or even affected by subdividing it (to a first approximation), so why subdivide it? A1. Flexibility. Subdivided material can more easily bend without breaking (taking Euler buckling to its opposite limit). This flexibility allows ropes to be stored in coils. A2. Stress relief. Subdivided material automatically rearranges the stress among the various fibers more uniformly. A3. Ease of manufacture from natural materials. Natural fibers have lengths limited by the size of the plant or animal, so making very long ropes requires some amount of splicing. Splicing involves interweaving of fibers in a manner similar to building a rope in the first place, so there is a smooth transition between the concept of a splice and a rope. Question #2: Why have a twist? A. You need to keep the strands together for basic rope operation (e.g., for pulleys, coiling, etc.), so twisted strands stay together better than untwisted strands. Question #3: How come ropes don't untwist under stress? A. That's the cool part of multiple levels of counter-twisted strands: a properly designed rope won't twist in either direction under load. Question #4: What are the best numbers for fibers, yarns, strands ? A. Bohr & Olsen (link above) talk about this to some extent. The final level of 3 strands seems to have been the most popular rope design since antiquity. The numbers for the other levels are not typically special or even specified. ---- Now that we have 3D printers, we could conceivably build ropes of any design that we please. But what designs would we want to build? I would imagine that there are more interesting relationships between integer numbers of fibers/yarns/strands that would have interesting properties. One might investigate the properties of non-round fibers/yarns/strands; e.g., the ancient Egyptians used a kind of "flat rope" ("tape" ?) to hold their boats together. Various kinds of braided patterns come to mind. Carbon nanotubes could form the fiber-level structure, but perhaps there are more interesting designs possible at the molecular level. If one had a 3D printer that could position individual carbon atoms, perhaps structures more interesting than simple nanotubes could be built. These would presumably have features with some combination of nanotube and diamond-like properties. Biological systems, both plants & animals, have developed extremely sophisticated fibers/strands/ropes. For example, muscle tissue itself is a kind of rope that can contract & perform work when properly instructed. There are lots of excellent ideas from these biological systems that could be re-used for more hi-tech ropes. --- Probably the most heavily investigated "high tech" rope is that of the use of carbon nanotubes to construct a "space elevator" -- probably in the form of a flat tape rather than a round rope. At least one elevator cable company is already in production on carbon fiber elevator cables, although these cables are still nowhere close to the strength/weight requirements for an Earth space elevator. But perhaps we're not being clever enough. If we end up building such a structure atom-by-atom anyway, then perhaps there are better designs that can do more interesting things while we're at it: conduct electricity for powering the elevator, micro-mechanical "muscles" that could perform some kind of "peristalsis" that could provide the motive power for moving stuff up/down the elevator, etc. --- One of the hottest topics in architecture is "tensegrity": the integration of strong-but-limp ropes which are strong under tensile loads with stiff-but-not-so-strong members which are strong under compressive loads. Biological systems are particularly adept at tensegrity structures -- e.g., the human hand with its muscles & tendons going all the way up the forearm. In addition to building fractal compression members of the Farr-type, we need to also consider fractal tensile ropes. http://www.london-institute.org/people/farr/fractals.shtml Future 3D printers will have to have the ability to "print" both compressive and tensile members at the same time, and so will require the ability to "print" at least 2 materials. (Such printers may actually require at least 3 materials, because some structures will require the "printing" of scaffolding, which can later be dissolved away, leaving the compression & tension members free to do their thing(s).)

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[math-fun] distant galaxy
by Warren D Smith 24 Oct '13

24 Oct '13

What struck me as interesting, is they claimed this galaxy is 30 billion light years distant, a bit strange since the universe is only 13.8 billion years old. http://www.bbc.co.uk/news/science-environment-24637890 They then said they knew that, and the reason is "we see the galaxy as it appeared 13.1 Gyr ago, but the universe has expanded since then, so now it is 30 Glyr away." I actually think, this is true. So the universe is "expanding faster than light speed" in a way that actually affects galaxies we can see. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] Tiling question
by David Wilson 23 Oct '13

23 Oct '13

5x^2 + 5(x+1)^2 = (x+2)^2 + (3x+1)^2 Is there are nice geometric proof (e.g. tiling) of this equation?

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[math-fun] critical simplex in hypercube with super-symmetry
by Fred Lunnon 23 Oct '13

23 Oct '13

During a hunt for cyclic configurations improving the current lower bound 2.065457 on the side of a maximal simplex inscribed in the unit hypercube in (n = 9)-space, a specimen emerges equipped with intriguing symmetry. Its n+1 Cartesian vertices comprise n cyclic shifts of seed rows [a, a, a, a, a, a, a, a, a], [c, 0, 0, 1, b, 1, b, 0, 1] where a = 0.008275807801733629 or 0.9490527540582451 , b = 0.5549581320873712 , c = 0.1980622641951620 satisfy (horrible sextic), b^3 - 2*b^2 - b + 1 = 0 , c = b^2 - 2*b + 1 . Cyclic permutations of seed rows obviously leave the simplex invariant, and reversals yield another congruent simplex. But besides such trivial symmetries, the permuted second row seeds [c, 0, 0, 1, b, 1, b, 0, 1]; [c, 1, 0, b, 0, 0, 1, 1, b]; [c, 0, 1, 1, 0, 1, b, b, 0]; yield a triad of essentially distinct, congruent inscribed simplices; each has the same set of 4 formally distinct, numerically equal squared sides 4 b^2 - 2 b c - 2 b + 2 c^2 - 2 c + 4 , 2 b^2 - 2 b + 2 c^2 + 4 , 9 a^2 - 4 a b - 2 a c - 6 a + 2 b^2 + c^2 + 3 , 4 b^2 - 6 b + 2 c^2 - 2 c + 6 . Sadly, the side evaluates to a disappointing 1.893277 , which will hardly set any pulses racing (even mine). Still maybe something interesting is going on here --- while superficially it might appear that one should expect a shedload of locally maximal configurations, perhaps there is some deeper reason why in practice far fewer can actually turn up. Fred Lunnon

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