December 2012 - mathfun@mailman.xmission.com

Re: [math-fun] Black hole lifetimes
by Henry Baker 27 Dec '12

27 Dec '12

Cool page! Thanks! So, if I have a black hole of 1,000 metric tons = 1,000,000 kg, the lifetime is only 84 seconds! A black hole whose lifetime is equal to that of the current age of the universe is approx. 1.7x10^8 metric tons = 1.7x10^11 kg. which isn't very big: about the mass of a small asteroid. At 02:47 PM 12/23/2012, Warren Smith wrote: >On 12/23/12, Henry Baker <hbaker1(a)pipeline.com> wrote: >> Interesting calculation. >> >> What's the situation with small black holes? >> >> I understand that the really tiny ones evaporate pretty quickly due to >> Hawking radiation, but what about a Moon mass black hole? > >--use http://xaonon.dyndns.org/hawking/

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[math-fun] glasses are liquids?
by Warren Smith 27 Dec '12

27 Dec '12

Well,ok (to reply to Meeker) first of all, even a single grain perfect crystal metal could deform via the creation of a new grain and stretching of old one (via thermal fluctuations) , to a longer state. So it could stretch. It might be interesting to see actual systematic experiments on cable stretch for different kinds of cables, including monofilament/not, crystal/polycrystal/amorphous, etc. Nothing difficult about doing such experiments, it just requires time. It also is possible (re my nylon line that stretched ) that the rain and/or sun had something to do with it. Second, re the claim "there is no tensile strength" would be news to structural engineers -- well yes. They deserve to be edified. (Also the claim glass is a liquid might also be news to them.) I still think tensile strength is a myth,but the reason it is a persistent myth is that this kind of stretching can be very very slow. In particular I have seen estimates that window pane deformation via glass "flow" should take way longer than age of universe before it gets large enough to observe. Finally re Meeker's question about "exponential in what" well, for example, many phenomena obey a temperature dependence like exp(-E/(k*T)) where T=temperature and E,K are constant. This might be expected if defoming a solid requires overcoming an energy barrier E to do a rearrangement. Get 10 times colder and these phenomena get WAY slower. However, with E's that are distributed according to a power law, say, one might well get rate laws that depend on T in a way much less powerfully cut off when you get 10X colder. So this would provide a way to claim amorphous stuff and crystal stuff both stretch, but the cutoff as temp changes might exhibit different behaviors, thus allowing a distinction to be drawn, but not as silly as a Yes/no 100-0 distinction.

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[math-fun] glasses are liquids?
by Warren Smith 27 Dec '12

27 Dec '12

>> But insofar as glass is amorphous there are >> arbitrarily >> small energy barriers against changes of configuration and so it is >> technically a liquid, >> even though in practical terms it's a solid. >> >> Brent Meeker > > --I deny this. I claim, it is at least in principle possible for an > amorphous solid to exist > in which any change requires at least a fixed energy change. One simple example > would be usual periodic packing of bricks, but each brick is in one of > 2 orientations > ("upside down" or "rightside up") selected randomly, and any switch > takes 1 electron volt to get you over the barrier. Meeker: How is that amorphous? It seems to have a lot of periodic structure. --true. It is however, aperiodic. If you want an example apparently without periodic structure, then put point "atoms" in space, linked by "bonds" whenever the distance between any atom pair is in a certain subset of the positive real axis (e.g.: between 1 and 2). Each bond has energy 1. I suspect by choosing the distance-set right this will have amorphous ground state. > Also, re the claim glass is a "liquid with very high viscosity" how about solid > crystalline metals? They can be deformed. In fact, they are a lot > more deformable than glass. So are they liquids with high viscosity? Meeker: The criterion isn't whether they can be deformed, it's whether they can be permanently deformed under arbitrarily small stresses. --that sounds like a good criterion for a "high viscosity liquid/." But a metal might obey this criterion. Thermal randomness (and quantum tunneling) will occasionally rearrange the metal atoms in funny ways that allow it to stretch. Due to this, I expect a chunk of metal under any permanent small tension, will gradually stretch. I actually think the whole notion that a metal rod has a "tensile strength" is a myth, there really is no such thing. I can also tell you it is an empirical fact that cables stretch, even though made of allegedly-crystalline materials. I had a plastic (nylon I think; nylon is regarded as crystalline) monofilament line hanging for a few years, and it stretched quite a lot, I had to retighten it several times. So... perhaps you could save your idea by arguing somehow that the stretching of crystals is somehow far more suppressed than for amorphous solids, in some mathematical sense like e.g. some exponential-style cutoff with temperature vs a power-law cutoff.

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[math-fun] glasses are liquids?
by Warren Smith 26 Dec '12

26 Dec '12

> But insofar as glass is amorphous there are > arbitrarily > small energy barriers against changes of configuration and so it is > technically a liquid, > even though in practical terms it's a solid. > > Brent Meeker --I deny this. I claim, it is at least in principle possible for an amorphous solid to exist in which any change requires at least a fixed energy change. One simple example would be usual periodic packing of bricks, but each brick is in one of 2 orientations ("upside down" or "rightside up") selected randomly, and any switch takes 1 electron volt to get you over the barrier. Also, re the claim glass is a "liquid with very high viscosity" how about solid crystalline metals? They can be deformed. In fact, they are a lot more deformable than glass. So are they liquids with high viscosity?

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Re: [math-fun] glasses are liquids?
by Henry Baker 26 Dec '12

26 Dec '12

You all might enjoy the "Pitch drop experiment" "The most famous version of the experiment was started in 1927 by Professor Thomas Parnell of the University of Queensland in Brisbane, Australia, to demonstrate to students that some substances that appear to be solid are in fact very-high-viscosity fluids. Parnell poured a heated sample of pitch into a sealed funnel and allowed it to settle for three years. In 1930, the seal at the neck of the funnel was cut, allowing the pitch to start flowing. Large droplets form and fall over the period of about a decade. The eighth drop fell on 28 November 2000, allowing experimenters to calculate that the pitch has a viscosity approximately 230 billion (2.3Ã1011) times that of water.[1] The ninth drop is expected to fall in 2012 or 2013. [2]" http://en.wikipedia.org/wiki/Pitch_drop_experiment The _live_ "pitch drop webcam" is located here: http://smp.uq.edu.au/content/pitch-drop-experiment At 02:57 PM 12/26/2012, Warren Smith wrote: >> But insofar as glass is amorphous there are arbitrarily >> small energy barriers against changes of configuration and so it is >> technically a liquid, even though in practical terms it's a solid. >> >> Brent Meeker > >--I deny this. I claim, it is at least in principle possible for an amorphous solid to exist >in which any change requires at least a fixed energy change. One simple example >would be usual periodic packing of bricks, but each brick is in one of 2 orientations >("upside down" or "rightside up") selected randomly, and any switch >takes 1 electron volt to get you over the barrier. > >Also, re the claim glass is a "liquid with very high viscosity" how about solid >crystalline metals? They can be deformed. In fact, they are a lot >more deformable than glass. So are they liquids with high viscosity?

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[math-fun] Entering a New Phase
by Warren Smith 26 Dec '12

26 Dec '12

http://www.lassp.cornell.edu/sethna/OrderParameters/BrokenSymmetry.html Interesting view: phase changes are about symmetry changes. One counterexample to this is glasses (amorphous solids). They can be solid or liquid but no symmetry changes that I know of. The fact glasses tend to have non-sharp melting points, as opposed to a lot of other phase changes with more sharply-defined changeovers, could be argued to support his view, though(?). Another way to look at it, which also supports his view, if the "freedom of motion" in a liquid IS a symmetry. Also, for many glasses, there is a crystalline ordered phase which can form, but takes ages to do so. If so we could argue this not really a counterexample. Must such an ordered phase exist with lower energy than the disordered phase? I claim "no" and as proof I point out that certain 2D tiling problems have min-energy state (i.e. tiling) corresponding to Turing machine valid computations... which can be arbitrarily infinitely messy.

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[math-fun] The Origin of Membrane Bioenergetics
by Warren Smith 26 Dec '12

26 Dec '12

>Not necessarily math fun but, for our origin-of-life followers, this paper suggests insight: "Harnessing energy as ion gradients across membranes is as universal as the genetic code. We leverage new insights into anaerobe metabolism to propose geochemical origins that account for the ubiquity of chemiosmotic coupling, and Na+/H+ transporters in particular. Natural proton gradients acting across thin FeS walls within alkaline hydrothermal vents could drive carbon assimilation, leading to the emergence of protocells within vent pores. Protocell membranes that were initially leaky would eventually become less permeable, forcing cells dependent on natural H+ gradients to pump Na+ ions. Our hypothesis accounts for the Na+/H+ promiscuity of bioenergetic proteins, as well as the deep divergence between bacteria and archaea." http://download.cell.com/pdf/PIIS0092867412014389.pdf --Here is something an amateur scientist might be able to figure out (if not already known). Consider a metal. It corrodes, especially in water. It produces electric potential in nature of a battery as it does so. (Similarly, certain semiconductors like iron pyrites also can do this.) Now meanwhile, bacteria use chemcial energy (ATP) to pump protons across membrane, also creating a battery with two "terminals" -- inside the cell & outside. This battery is used to fuel various cellular processes, for example the "motor" that spins flagellum, is electrically-powered. QUESTION: can bacteria "cheat" by sticking themselves to a metal surface and pirating potential difference? I have noticed bacteria forming films on metal surfaces which make me suspect this might happen.

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[math-fun] The Origin of Membrane Bioenergetics
by Hans Havermann 26 Dec '12

26 Dec '12

Not necessarily math fun but, for our origin-of-life followers, this paper suggests insight: "Harnessing energy as ion gradients across membranes is as universal as the genetic code. We leverage new insights into anaerobe metabolism to propose geochemical origins that account for the ubiquity of chemiosmotic coupling, and Na+/H+ transporters in particular. Natural proton gradients acting across thin FeS walls within alkaline hydrothermal vents could drive carbon assimilation, leading to the emergence of protocells within vent pores. Protocell membranes that were initially leaky would eventually become less permeable, forcing cells dependent on natural H+ gradients to pump Na+ ions. Our hypothesis accounts for the Na+/H+ promiscuity of bioenergetic proteins, as well as the deep divergence between bacteria and archaea." http://download.cell.com/pdf/PIIS0092867412014389.pdf

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[math-fun] 2013 jumps
by Eric Angelini 26 Dec '12

26 Dec '12

Hello Math-Fun, here is a puzzle for the new year -- hope it is sound (the puzzle); 1) Let's say we need 2 "jumps" to rearrange ABCD into CABD, like this: A B C D Jump #1: A C B D Jump #2: C A B D 2) Let's say we always look for the smallest quantity of jumps to rearrange a word "a" into one of its anagrams "b". We'll then need 3 "jumps" to rearrange ABCD into CBAD: A B C D Jump #1: B A C D Jump #2: B C A D Jump #3: C B A D Note: - to transform the word ABCD into CABD you need 2 jumps, BUT to transform the word ABAD into AABD you only need 1 jump. 3) Instead of letter-words "a" and "b", let's consider integers. We will deal only with integers "a" and "b" which obey these simple rules: - a is not equal to b - neither a nor b have a leading zero - both a and b > 9 ----------------------------------------------------------------- Let's start a sequence S of b(n)s where b is the smallest integer which can be produced in n jumps. I guess S starts like this: S = 12, 102, 102, ... b(1) = 12 because "12" is the smallest integer obtainable in 1 jump [the jump 21 -> 12] b(2) = 102 because "102" is the smallest integer obtainable in 2 jumps [210-120-102] b(3) = 102 because "102" is the smallest integer obtainable in 3 jumps [201-210-120-102] etc. Remark 1: we do NOT accept intermediate "integers" with leading zero(es). That is, you cannot produce 102 in 3 jumps like this: 201-021-012-102 Remark 2: we do NOT accept two intermediate "integers" being the same. What is the smallest integer obtainable in 2013 jumps? Best, É.

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[math-fun] Entering a New Phase
by Hans Havermann 26 Dec '12

26 Dec '12

21 Dec 2012: Forget solid, liquid, and gas: there are in fact more than 500 phases of matter. In a major paper in today's issue of Science, Perimeter Faculty member Xiao-Gang Wen reveals a modern reclassification of all of them. http://perimeterinstitute.ca/news/entering-new-phase

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