October 2013 - mathfun@mailman.xmission.com

[math-fun] An idea about teaching calculus
by Michael Collins 28 Oct '13

28 Oct '13

Here is an idea about teaching calculus that I have been pondering for a while, wondering what others might think of it (or if I have reinvented the square wheel?) The concept of "lim F(x) as x approaches a" is (pedagogicaly) problematic since we need artificial-looking examples to get this limit to be something other than just F(a). I'm thinking the first time students encounter limits, it should be the limit of a sequence {a_n} as n approaches infinity. Then we can define the derivative F'(a) as the limit of the sequence (F(a + d_n) - F(a))/d_n, with F being differentiable at a iff this limit exists and is the same for all sequences d_n that approach zero. Limits at infinity seem like the right place to start for a couple of reasons. First, it is possibly more clear that we really do need a new concept to talk about what happens "at infinity". Also, the two variables over which we are quantifying are more clearly playing distinct roles, since one is a large integer and the other is small. And the proposed definition of the derivative is closer to the way mathematicians actually think about such things -- the limit of an ever-improving sequence of approximations.

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[math-fun] The Simpsons ...
by Guy Haworth 27 Oct '13

27 Oct '13

One of favourite script-bites is the self-referential: "Why is there no sign for 'Imminent Meltdown: run for your lives?' " "Don't be silly: if there was a meltdown, there'd be no power to light the sign." Either Russell or Klein, or both, would be proud. Guy Haworth

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[math-fun] ropes...
by Warren D Smith 27 Oct '13

27 Oct '13

http://www.industrialhemp.net/ gives a bill in US congress in 2007, apparently never passed, which would have legalized industrial hemp. Discussion follows, claiming basically it is legal to grow it in Canada+Europe, but in the USA it is illegal unless have special permit which in practice is essentially impossible to obtain (they give evidence of this impossibility). It is not clear to me synthetic rope is better than hemp rope. I mean, you'd naively think growing it is cheaper than synthesizing it. About "WD Smith rope model" it is ok if the inner rope core is untwisted straight fibers, if outer sheath is twisted and/or braided so that tension on rope compresses the rope radially, thus increasing inter-fiber friction, thus preventing break even if all fibers are cut in random locations.

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Re: [math-fun] Earth-sized object orbiting black hole at 5 x Schwarzschild radius:
by Keith F. Lynch 27 Oct '13

27 Oct '13

Henry Baker <hbaker1(a)pipeline.com> wrote: > Keith F. Lynch wrote: >> Henry Baker <hbaker1(a)pipeline.com> wrote: >>> Light would orbit this black hole in a few minutes; I would >>> presume that an Earth-sized object would have to orbit at nearly >>> the speed of light even at 5 AU (Jupiter's distance from the Sun). >> That orbit wouldn't be stable. The planet would lose energy by >> emitting gravitational radiation, and within a few orbits would >> spiral into the black hole. > Are you sure that such a small object like the Earth would lose that > much energy in a few orbits? > All of the simulations that I've seen show _similar-sized_ objects > -- e.g., binary black holes -- converging in a few orbits. > But the Earth is such a small mass relative an Earth-orbit-sized > black hole that I wouldn't think that gravitational radiation would > cause the orbit to collapse so quickly. Gravitational radiation is proportional to mass. So, of course, is the object's inertia which resists spiraling in. The two effects cancel out. So objects will spiral in at a rate independent of their masses. It would contradict the uniqueness of free fall and the equivalence principle if this wasn't the case. > Tidal forces would likely rip the Earth into pieces & possibly > schmear it out like Saturn's rings, but I'm skeptical about it > collapsing so quickly from gravitational radiation. The tidal effect at the event horizon depends on the size of the black hole. A sufficiently large one would have no more tidal effect on a planet passing through its event horizon than the moon has on our planet. Whether a black hole that large exists, I don't know. There's the claimed "firewall effect," which would incinerate anything passing through the event horizon. Whether it exists, I don't know. I suspect not, since there are event horizons that have nothing to do with black holes, and those definitely don't have a firewall effect. Why should black hole event horizons behave differently? One example of such an event horizon is the edge of the observable universe, beyond which objects are receding from us faster than light. (The speed limit only applies relative to a local observer.) And we're on that cosmological event horizon to distant (hypothetical) observers. Another example is that there's an event horizon behind anyone who is accelerating. It's about 10/A light years behind you, where A is your acceleration in meters per second per second. As long as you maintain that acceleration, light, matter, signals, etc., from behind that horizon will *never* catch up to you. Once again, there are (hypothetical) observers to whom we are right on the event horizon. And yet it's still cold outdoors, at least where I live.

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Re: [math-fun] Moore-Schulman base-3/2 conjecture and OEIS
by Keith F. Lynch 27 Oct '13

27 Oct '13

Yesterday I wrote, in part: > If not, has anyone tried summing (2/3)^j for every appropriate > sequence in OEIS? I did this this afternoon, for all sequences in OEIS, skipping negative, zero, oversized, and duplicate terms in each sequence. I was briefly excited to see that A077468 was equal to 1 for at least 17 places, which would be extremely unlikely to happen by chance. But it turns out that that entry is "Greedy powers of (2/3): sum_{n=1..inf} (2/3)^a(n) = 1. 25" At least I know my program works. ("Oversized" because (2/3)^j for j>100 is too small to make a difference to a sum close to 1 with 64-bit precision.) I wrote the program in C, using my DEC VT420 terminal, and ran it directly on my shell-account ISP, Panix. It took less than three seconds to run even though more than 50 other users were logged in. The program is just 28 lines long.

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[math-fun] "The Simpsons and Their Mathematical Secrets"
by Henry Baker 27 Oct '13

27 Oct '13

FYI -- I think that this qualifies as both "math" & "fun", whether you like The Simpsons or not: http://www.theguardian.com/tv-and-radio/2013/sep/22/the-simpsons-secret-for… Guide to Mathematics and Mathematicians on The Simpsons http://homepage.smc.edu/nestler_andrew/SimpsonsMath.htm http://mathsci2.appstate.edu/~sjg/simpsonsmath/

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Re: [math-fun] the real reason for twisted ropes
by Henry Baker 27 Oct '13

27 Oct '13

Pretty good 14-minute WWII film about growing hemp in the USA (because the US lost access to sisal & manilla from the Phillipines) & making rope. Many of these techniques could have been used by the ancients, albeit with somewhat less automation. It says in this film that you have to have a special marihuana permit to grow hemp. The leaves of this plant look exactly like you would expect them to look. If anyone wants materials to experiment with ancient rope-making techniques, there's plenty of these materials available (many on US govt national parklands -- e.g., back country of Santa Barbara, back country of Big Sur, etc.). Unfortunately, many of the people engaging in rope-growing are starting to carry automatic weapons. It's also easy to see from this film why it's a bit difficult to grow rope in your back yard, unless you have really, really tall fences. Hemp for Victory - Entire Film - US Government asks farmers to grow hemp for twine & rope. Uploaded on Aug 15, 2010 Produced by the US Government in 1942... http://www.youtube.com/watch?v=W0xHCkOnn-A At 05:42 PM 10/26/2013, Whitfield Diffie wrote: >> The USA actually banned hemp because it is related to marijuana >> plant and hence plant contains some tiny amount of intoxicating >> substances, ... > > Are you sure of this. The anti-marijuana act dates from the >thirties and I am sure there was lots of hemp rope in my youth in the >forties and fifties. I thought is was just superseded by more >cost-effective materials.

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[math-fun] WDSmith-type rope model
by Henry Baker 27 Oct '13

27 Oct '13

In order to "prove" that rope "works"--i.e., it doesn't fall apart under normal operating conditions--it would be nice to have a simple mathematical model. Here's my first attempt at a "Smith-type" model of rope, in which the tension on the rope itself acts to make sure that the parts of the rope stick together ever more tightly. I've seen cheap clothesline rope that is constructed in the following fashion. Take a yarn made up of a large number N of _straight_ fibers, but the fibers have a particular _mean length_ L, so that within any particular 1-inch length of this yarn, there will be some number of fibers that begin, and an approximately equal number of fibers that end. Surrounding this yarn is a woven outer _sleeve_, which is very compliant in the longitudinal direction. The key point about this sleeve is that this sleeve _carries no load_ itself, but when it is lengthened, it _compresses_ the yarn inside. This lateral compression presses the fibers of the inner yarn together so that the sliding _friction_ of these yarn fibers against one another carries the load of the rope. We now take this model to its simplest logical conclusion: The inner "yarn" now consists of exactly N=2 _straight_ fibers at any point along the rope; each fiber segment is of length exactly L, and the breaks between fiber segments occur at exactly the mid-point of the other fiber segment. The 2 yard fibers are thus arranged to look like this: ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- ----- Obviously, there is nothing whatsoever to hold this 2-fiber yarn arrangement together until the outer sleeve is place around this yarn. Consider now that the sleeve is in place. The sleeve carries no longitudinal force, but any longitudinal force on it is immediately translated into a linearly proportional lateral--e.g., _normal_--force pushing the 2 yarn fibers together. We now have a classic situation in high-school physics (see Wikipedia): "The elementary properties of sliding (kinetic) friction were discovered by experiment in the 15th to 18th centuries and were expressed as ... empirical laws: "Amontons' First Law: The force of friction is directly proportional to the applied load. "Amontons' Second Law: The force of friction is independent of the apparent area of contact." http://en.wikipedia.org/wiki/Friction There is a coefficient of friction which characterizes the friction between our 2 yarn fibers parameterized by the numerical value of the normal force. When the 2 fibers aren't slipping against one another, this friction will tend to be higher--i.e., it is "stiction". http://en.wikipedia.org/wiki/Stiction We can now put together the various pieces of the model. When a longitudinal load is applied to the rope, it lengthens the sleeve while pulling apart the various fibers. Until the lateral force reaches a certain level, the rope will lengthen _without resistance_. However, beyond a certain stretch of the sleeve, enough lateral force is produced on the 2 fibers in the yarn for the friction to start opposing the longitudinal load. Eventually, the sleeve will produce enough lateral force for the friction between the yarn fibers to stop any movement, at which point the fibers themselves will be stretched according to their Young's modulus. The rope then stretches as if it were a single long fiber until it breaks at one of the transition points between one fiber segment and the next fiber segment. The problem with having only 2 fibers/yarn is that the effective strength is halved. However, if we cleverly arrange N fibers so that transitions between fiber segments never occur simultaneously, we can get an effective strength of (N-1)/N of the strength of a single fiber, at a weight of N fibers. Thus, a yarn of 100 fibers will have the strength of 99 fibers at a weight of 100 -- an efficiency of 99%. (This assumes that the sleeve is "free".) At 05:12 PM 10/26/2013, Warren D Smith wrote: >--Actually, the correct formulation of these "theorems" (for now conjectures) >would involve the friction coefficients K for their rope surfaces >and the "knot does not fail" claim would only be true if >K > threshold (and threshold varies depending on the knot type). >(The elastic constants of the rope material and the rope radius both seem >irrelevant, kind of cancel out of the question.) Albeit the usual knots found >in knot manuals evidently have quite safe thresholds versus realistic K values >for actual rope materials. I think there are a few knot types >which are not considered safe using slippery m(modern synthetic) rope materials >but were considered safe using older, less slippery, rope materials like hemp.

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[math-fun] about arguments with the zeta function
by Simon Plouffe 27 Oct '13

27 Oct '13

Hello, I have foundsomething which could interest a few people, it is about the function 1/Pi*argument(Zeta(1/2+I*n)) and 1/Pi*GAMMA(1/4+I*n/2)), these functions appear in the context of the counting formula of Backlundwhich is (it does count the number of zeros of the zeta function). Z(n) = n/(2*Pi)*log(n/(2*Pi*exp(1))+ 11/8 + 1/Pi*argument(Zeta(1/2+I*n)). The unknown so far was that this particular Zeta with the argument is quite chaotic, well , it is still the same but I have found a general formula for it as well as one for the gamma function. The formula is quite simple, is uses [ ] and { }, the floor and fractional part of a number (or function)and Pi, log(Pi), log(2) and log(p) where p is a prime number. That kind of formula as far as I know was conjectured once by a certain Freeman Dyson, and some others as well like M.V. Berry, Connes, Hugh Montgomery , relating quasi-crystals to the zeta function. Well, that's one step for it. I think that mr Dyson was right. The article was submitted to certain friends and on the arxiv site, the abstract is in english and french and the article is in ... : french, YES, I know manyof you do NOT speak that language of MoliÃ¨re and the 'mot de Cambronne' will come to their mouth, I intend to think about it and make a serious effortfor it in the next few days, http://www.plouffe.fr/simon/On%20the%20values%20of%20the%20functions%20zeta… the article is also in Word, yes the vulgar Word : http://www.plouffe.fr/simon/On%20the%20values%20of%20the%20functions%20zeta… I have been working for this in the past 2 years. Here are the results, Best regards to all. Simon Plouffe/ /

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[math-fun] the real reason for twisted ropes
by Warren D Smith 27 Oct '13

27 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. --Actually, the correct formulation of these "theorems" (for now conjectures) would involve the friction coefficients K for their rope surfaces and the "knot does not fail" claim would only be true if K > threshold (and threshold varies depending on the knot type). (The elastic constants of the rope material and the rope radius both seem irrelevant, kind of cancel out of the question.) Albeit the usual knots found in knot manuals evidently have quite safe thresholds versus realistic K values for actual rope materials. I think there are a few knot types which are not considered safe using slippery m(modern synthetic) rope materials but were considered safe using older, less slippery, rope materials like hemp. The USA actually banned hemp because it is related to marijuana plant and hence plant contains some tiny amount of intoxicating substances, but it used to make excellent ropes in the old days, if it had been banned back then the British empire never would have been. Why do ribbon-shaped ropes (like shoelaces & seat belts) create knots harder to undo than ordinary cylindrical ropes? (If this percepton even is true?) Because have more surface area to be frictionally involved (isoperimetric theorem)? By the way, these thoughts on ropes+knots might be original with me, anyhow they are not discussed in a lot of places you'd think they would be. Which originality is pretty surprising, if true.

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