January 2019 - mathfun@mailman.xmission.com

Re: [math-fun] Could change of base (binary ?> ternary) speed up computation?
by Keith F. Lynch 12 Jan '19

12 Jan '19

From: Richard Howard <rich(a)richardehoward.com> > A cold, slow universe is an interesting concept. > I note that no one has mentioned "The Last Question" by Isaac Asimov > (1956) where he explored the idea of long times and dying universes. > http://www.multivax.com/last_question.html Perhaps because few people find its main premise, that a sufficiently powerful computer would somehow get the ability to create things, plausible. It could simulate things, but it would need a substrate to run on. Unless you accept Tegmark's idea, that every self-consistent system automatically exists. Tegmark says that if software could exist that would contain conscious sapient beings and a world for them to interact with, then nobody needs to run it, or even write it, unless they want to see what the beings do. The beings would perceive and act the same regardless. Just as pi definitely has a specific googolplexth decimal digit whether anyone ever calculates it or not. But then Multivax was redundant; its program would have run anyway, as would all other possible programs. As for a cold, slow, vast, inconceivably ancient, run down, and worn out universe, Stephen Baxter has played with that concept in several of his novels. In one of them, the big reveal at the end was that the viewpoint characters weren't in our distant future, but in our past, and that the setting was less than a trillionth of a nanosecond after the Big Bang. One person's hot and fast is another's cold and slow. As someone who prefers much warmer temperatures than most people, I can sympathize. > I wonder when quantum zero point energy will be large enough > relative to the remaining energy in the cold universe to make > organized life impossible. We may not have infinite time to > do the calculations. We should be okay for at least the next googol (10^100) years or so. Most of the known energy reserves of the universe are gravitational, i.e. we can power our civilization by dropping things into other things. As I implied in my mention of Tipler's scenario, if things can get arbitrarily dense and close to each other, you can get unlimited energy. Thanks to event horizons, there are limits, but you can come pretty close to total conversion of mass into energy. That does require that we have continued access to stuff. If everything beyond our solar system eventually retreats beyond the Hubble horizon, the future looks a lot more bleak. If we have our galaxy to play with, along with Andromeda (which is projected to collide and merge with our galaxy fairly soon (i.e. about 4 billion years from next Tuesday)), we should do pretty well for a while. > I was thinking of neutronium in the Robert Forward sense--on the > surface of a neutron star. Yes, the Cheela lived on the surface. And reasonable rates of cooling would require access to the surface in any case. (As does the plot of that novel, which involves them noticing and communicating with humans.) But then we're not talking about neutronium (material consisting almost entirely of neutrons), but about degenerate matter, or even ordinary matter. Neutronium requires lots of pressure from overlaying material. Whether degenerate matter can be solid at such temperatures, I don't think anyone knows. The pressure of the atmosphere may suffice to increase its melting point, and it's anyone's guess what an extremely strong magnetic field would do. > Gravitational energy is comparable to the nuclear binding forces--I > don't see why it would be unpredictable or threaten to blow up the > computer. When a nucleon decays or otherwise changes, is random, as is the direction and fate of its decay products. If you think otherwise, please explain how, in principle, it would be possible to generate a coherent beam of neutrinos. > After all, all of our matter is held together by quantum forces > (everything scaled down) and our computers don't often blow up. > Aside from being a bit toasty, reactions should be well behaved. Could you design a computer that would operate in the center of the sun? Could life evolve inside the sun?

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[math-fun] Math is hard, but Mattel steps up
by Allan Wechsler 12 Jan '19

12 Jan '19

Many of us remember the cringeworthy episode in 1992, in which one of Teen Talk Barbie's canned phrases was "Math class is tough!" Mattel is now manufacturing an "Inspiring Women" series of Barbies, based on real-life role models. Here is their Katherine Johnson doll: https://barbie.mattel.com/shop/en-us/ba/inspiring-women-series/barbie-inspi… Johnson was a trajectory analyst at NASA during the early manned space program, and her life was dramatized in the film "Hidden Figures". (It's out of focus, but I really want to hear more about the scale model 1960's engineering office in which the doll is posed.) Johnson turned 100 last fall.

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[math-fun] Corey's denester just did a quotient,
by Bill Gosper 11 Jan '19

11 Jan '19

(2 - Sqrt[2] 3^(1/4))^(1/4)/(-1 + Sqrt[3])^(7/12) = = (23 - 12 Sqrt[2] 3^(1/4) + 13 Sqrt[3] - 7 Sqrt[2] 3^(3/4))^(1/6)/2^(1/3) but not the numerator nor denominator separately. Corey's RationalizeDenominator further simplifies to (2 - Sqrt[2] 3^(1/4))^(1/4)/(-1 + Sqrt[3])^(7/12) = = 1/(5 - Sqrt[3] + Sqrt[2] 3^(3/4))^(1/6) https://community.wolfram.com/groups/-/m/t/980264 —rwg Given the reciprocal of the above quotient, the denester prints out the 6th root of the trinomial, but then hangs up for hours.

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Re: [math-fun] Could change of base (binary ?> ternary) speed up computation?
by Keith F. Lynch 10 Jan '19

10 Jan '19

Richard Howard <rich(a)richardehoward.com> wrote: > It seems like going cold is the wrong way to go. It leaves you cold, huh? > Radiative cooling rate drops as T^4, while energy per erasure > drops as T. Nitpick: In optically thin media, it drops as T^3. That's relevant especially when speaking of computation in the extremely far future. As I said, you'd need very low density material to minimize self- gravitation and the resulting Unruh effect. Also, you'd need to build the computers out of something other than matter as we know it after all the protons decay. (You can make new protons, but it would require energy you don't have.) > Lose big trying to dump energy to the CMB at 3K. In the distant future, the CMB will asymptotically approach 0K. (That reminds me, one of the things on my to-do list is to work out what the present-day universe would look like from an Earth-like planet deep in intergalactic space. Very cold and dark, you say? Sure. But what if the planet is moving fast enough that the CMB in the direction of travel is blue-shifted to be as bright as the sun is from Earth? Would it look anything like the sun? How wide would it be, and what would its color temperature be? How long, in the planet's frame of reference, until the CMB's temperature dropped significantly? Or would the drag from the radiation slow down the planet before that happened?) To be fair, I've been talking about two very different things: Computation in the extremely distant future and computation in perhaps a century or two once we approach the kT log(2) limit, which we're nowhere near yet. > On the other hand, Robert Forward may have had it right in Dragon's > Egg. A good computer would be built out of neutronium matter and > run at the rate of nuclear reactions. Maybe. As far as anyone knows, neutronium is a liquid even at cryogenic temperatures, let alone at megakelvins, albeit one with an absurdly high surface tension. Also, nuclear reactions are rather unpredictable due to quantum effects, release enough energy to blow the computer apart, and mostly aren't much faster than Drexlerian rod logic. > It is easy to dump waste heat at 10^6 K or higher--t^4 becomes > pretty big relative to kTln2/erasure. Yes, if you want *fast* computations, hot and compact is the way to do it. If neutronium computers are possible at all, they would be a good choice if time is expensive and energy is cheap. Even without nuclear energy, you can get plenty of energy from the neutron star's own heat, using space as the heat sink. Neutronium also has an awesome specific heat, so neutron stars would probably take a few trillion years to cool down appreciably. If you're a real speed freak, you might prefer Tipler's future to Dyson's. As I mentioned, Dyson claimed it would be possible to do infinite computation in infinite time with finite total energy in an expanding and cooling universe. The computation would get colder and slower without limit, but so what? What's the hurry? Tipler, by contrast, postulated a contracting and heating universe, one that would end with a Big Crunch. He claimed it would be possible in such a scenario to do infinite computation in a finite time with infinite total energy, by running hotter, faster, and more compactly, without limit. It now appears that the universe will expand and cool forever. But a local Tipler future might be possible by tossing your computer into a large black hole. Too bad it can't convey its results to you unless you travel with it. A third possible ultimate fate of the universe is the Big Rip, in which the Hubble distance approaches zero, perhaps in just a few billion years, and everything is torn apart. As far as I know nobody has yet claimed to have found a way to live forever in such a system.

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[math-fun] The Case Against Quantum Computing
by Hans Havermann 10 Jan '19

10 Jan '19

In the IEEE Spectrum last week, Mikhail Dyakonov presented his overview of the field: https://spectrum.ieee.org/computing/hardware/the-case-against-quantum-compu…

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[math-fun] Some questions from the 2018 Putnam Exam
by Dan Asimov 09 Jan '19

09 Jan '19

These are the questions from the recent Putnam Exam that seemed most interesting to me. I've changed some typography, and a word or two, for clarity. (I haven't tried to solve any yet. If anyone posts a solution, maybe include a spoiler warning for that problem number?) —Dan ************************************************************************* ----- A2 Let S_1, S_2, ..., S_(2n−1) be the nonempty subsets of {1,2,...,} in some order, and let M be the (2n−1)×(2n−1) matrix whose (i,j)th entry m(i,j) = 0 if S_i ∩ S_j is empty; m(i,j) = 1 otherwise. ----- ----- A5 Let f : R → R be an infinitely differentiable function satisfying f(0) = 0, f(1) = 1, and f(x) ≥ 0 for all x ∈ R. Show that there exist a positive integer n and a real number x such that the nth derivative of f is negative when evaluated at x: f^(n)(x) < 0. ----- ----- B2 Let n be a positive integer, and let f_n(z) = n +(n−1)z + (n−2)z^2 +··· + z^(n−1). Prove that f_n has no roots in the closed unit disk {z ∈ C: |z| ≤ 1}. ----- ----- B4 Given a real number a, we define a sequence by x_0 = 1, x_1 = x_2 = a, and x_(n+1) = 2 x_n x_(n−1) − x_(n−2) for n ≥ 2. Prove that if x_n = 0 for some n, then the sequence is periodic. ----- ----- B5 Let f = (f_1, f_2) be a function from R^2 to R^2 with continuous partial derivatives ∂f_i/∂x_j that are positive everywhere. Suppose that (∂f_1/∂x_1)(∂f_2/∂x_2) - (1/4)(∂f_1/∂x_2 + ∂f_2/∂x_1)^2 > 0 everywhere. Prove that f is one-to-one. ----- *************************************************************************

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Re: [math-fun] Could change of base (binary ?> ternary) speed up computation?
by Keith F. Lynch 09 Jan '19

09 Jan '19

Richard Howard <rich(a)richardehoward.com> wrote: > 2) Low temperature sounds great, but energy tends to go down with > temperature much slower than heat transfer. Low temperature > computers are really hard to cool. Right. That's why I said that running a computer in an air- conditioned room doesn't really gain you anything. The kT log(2) limit, i.e. that that's the minimum energy it takes to erase a bit, applies to the computer and air conditioner put together. That's why I suggested running the computer in the outer solar system instead. Or in nearby space behind a sun shield, like JWST. Tom Knight <tk(a)mit.edu> wrote: > I?ll let others worry about arbitrarily slow clocks. I like ones > running at multiple GHz. That's fine for next googol years or so, but when we approach the year googolplex, it's no longer realistic. For one thing, it takes a long time to radiate away waste heat at cryogenic temperatures. For another, the the components of the computer have to be widely separated so that they won't experience significant amounts of mutual gravitation, since an accelerated observer experiences a non-zero temperature (Unruh effect). For instance an observer experiencing 1 g (normal Earth surface gravity) will experience a temperature of at least 10^-20 K. Anyhow, there's not that much difference in speeds. A few gigahertz is one cycle every 10^33 Planck times or so. One cycle per eon (billion years) would be about one every 10^60 Planck times. Less than a factor of two difference (on a log scale).

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[math-fun] Does someone have a q-extension of ζ(2) = π²/6 ?
by Bill Gosper 08 Jan '19

08 Jan '19

I could only get the odd terms: Out[368]= Sum[((1 - q)^2*q^k*(1 + q^(1 + 2*k)))/(1 - q^(1 + 2*k))^2, {k, 0, ∞}] = = ((1 + q)*QPochhammer[q^2, q^2]^3*QPochhammer[q^4, q^2])/ (QPochhammer[q, q^2]*QPochhammer[q^3, q^2]^3) Check q→1. Mathematica can't Limit the braindead QPochhammers. Punt them. In[369]:= depoch@% Out[369]= Sum[((1 - q)^2*q^k*(1 + q^(1 + 2*k)))/(1 - q^(1 + 2*k))^2, {k, 0, ∞}] = = ((1 + q)* Inactive[Product][1 - q^4*(q^2)^i, {i, 0, ∞}]* Inactive[Product][1 - (q^2)^(1 + i), {i, 0, ∞}]^3)/ (Inactive[Product][1 - q*(q^2)^i, {i, 0, ∞}]* Inactive[Product][1 - q^3*(q^2)^i, {i, 0, ∞}]^3) "Inactive" keeps them punted. In[370]:= Nest[prodcontract, %, 3] Out[370]= Sum[((1 - q)^2*q^k*(1 + q^(1 + 2*k)))/(1 - q^(1 + 2*k))^2, {k, 0, ∞}] = = (1 + q)*Inactive[Product][ ((1 - q^4*(q^2)^i)*(1 - (q^2)^(1 + i))^3)/((1 - q*(q^2)^i)*(1 - q^3*(q^2)^i)^3), {i, 0, ∞}] In[371]:= MapAt[Limit[#, q -> 1] &, %[[1]], 1] == MapAt[Limit[#, q -> 1] &, %[[2]], {2, 1}] Out[371]= Inactive[Sum][2/(1 + 2*k)^2, {k, 0, ∞}] == (1 + q)*Inactive[Product][(16*(1 + i)^3*(2 + i))/((1 + 2*i)*(3 + 2*i)^3), {i, 0, ∞}] In[372]:= Activate@% Out[372]= π^2/4 == (1 + q) π^2/8 (Left one q unsubstituted to avoid becoming True!) In[367]:= Series[%370 /. ∞ -> 8, {q, 0, 9}] Out[367] =1+2 q-q^2+3 q^4-6 q^5+3 q^6+8 q^7-16 q^8+7 q^9+O[q]^10= =1+2 q-q^2+3 q^4-6 q^5+3 q^6+8 q^7-16 q^8+8 q^9+O[q]^10 (Skimped on ∞ to avoid becoming True! But Macsyma could Taylor the infinite series.) Such a Lambertoid closed form Out[368,370] seems unfamiliar, both to me and Mathematica. I am too old and it is too young. —rwg Apologies for the "typesetting" imposed by the math-fun server.

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Re: [math-fun] Weisstein's compact CF notation
by rcs＠xmission.com 07 Jan '19

07 Jan '19

<note from RWG. I can't get his picture of the pi CF as a ticker tape into this message, but his URL works. It's hard to see the defects he's complaining about, but try squinting. The mathworld version doesn't have the glitches. -- rich> Date: Fri, 4 Jan 2019 03:22:41 -0800 Subject: Re: Weisstein's compact CF notation From: Bill Gosper <billgosper(a)gmail.com> Julian magically responded On Fri, Jan 4, 2019 at 12:44 AM Julian Ziegler Hunts <julianj.zh(a)gmail.com> wrote: I can't make it do what it *should* do and export a clear 256x9 image, but I can get a 256x13 with extra whitespace or a proper 512x18. The easiest way to do this is by adding Frame->False, PixelConstrained->True, ImageSize->256|512 to the ArrayPlot, followed by either Save Selection As? http://gosper.org/cfbits.png (<1K!) ? or Export (though Export adds even more padding for some reason, so it won't go below 256x15). Julian What's magic about this is that many of my ArrayPlots looked fine on the screen, but always crummy when I saved them! Stealth wysiNwyg! On Thu, Jan 3, 2019 at 7:03 PM Bill Gosper <billgosper(a)gmail.com> wrote: http://mathworld.wolfram.com/PiContinuedFraction.html renders the terms in strangely Arabesque binary, presumably with the pleasantly concise ArrayPlot[Transpose@PadLeft@IntegerDigits[ContinuedFraction[?, 256], 2]] ?cfbits.png from which you can read off the four Mersenne numbers 3 7 15 1, and then 4(1+8+64) = 292. But note the hypoplastic 4 bit, even after hours of experimentation. It is maddeningly difficult to prevent Mathematica, when you try to export a graphic, from blurring, rounding, stretching, shrinking, or outright amputating pixels, presumably due to premature quantization. Or incompetent DWIMmery. ?rwg Julian seems to be the only one who knows how to prevent it, and he just went back to UCLA. Eric seems to have achieved vertical fidelity, but just past the middle seems to be a double 57, which is what sucked me into this: ?rwg

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Re: [math-fun] Could change of base (binary ?> ternary) speed up computation?
by Keith F. Lynch 07 Jan '19

07 Jan '19

Dan Asimov <dasimov(a)earthlink.net> wrote: > Since useful quanputers might be a long ways off, maybe some other > stunt could speed up current computation as we know it. > If trits could be stored, retrieved, and copied as easily as bits, > there would be less *space* needed by a factor of of log_c(2)/log_c(3) > = .6309+, where c = zeta(3). It's hard to tell with text whether someone is joking. I assume you know that log(a)/log(b) always has the same value regardless of the base of the log. For instance I was recently debating with myself whether to express a number as log2(10), 1/log10(2), or as log(10)/log(2). In the last case, I don't have to worry about the base. I'd think you'd also know that changing the radix of a computer may change its speed by some constant, but wouldn't change the big-O category of any algorithm. Replacing it with a quantum computer would, at least for a few special algorithms, such as Shor's. I've read that the Russians once had a computer that was based on balanced ternary. As for space, that's true of a trit takes the same space to store as a bit, but why should that be? > Perhaps someone knowledgeable can estimate how much trits could > actually speed up computation. Surely this has been much studied > by now. I can't see why it would speed it up at all, or increase the precision, so long as the word size in bit-equivalents remained the same. Computation is generally done one word at a time. A decimal computer would be slightly faster if you want the results in decimal. (I assume few people want results in ternary.) And it also avoids some rounding errors that binary doesn't, since it can exactly represent more fractions. But to maximize that, you'd want base 30, base 210, or some other primorial. Of course a binary (or ternary, etc.) computer can be programmed to emulate any other radix, at the cost of some speed and some space. Or to do exact rational calculations by storing everything as integer numerators and denominators. > But the practical question is, Is there a practical way to build > trit-based processors? Certainly. But fewer states are always easier, and two is the fewest you can have, since one state isn't good for anything. Similarly, life anywhere in the universe ought to evolve to have two sexes, not more, despite that novel your uncle wrote where they had three. > Would it be easier if they could be as large as a room, or larger? Easier for what? Easier for hand assembly of individual transistors? Sure, for that you'd want components that are a comfortable size to handle. And it would result in a computer with the power of an iPhone filling, not just a room, but a large building. And the speed-of- light delay means that it would run very slowly by today's standards. Tom Knight <tk(a)mit.edu> wrote: > Binary is optimal for information storage and transmission. This > is because for a fixed amount of noise (from either environment or > fabrication issues) binary requires only a single distance between > states, whereas higher radix storage requires n-1. That doesn't sound right to me. Only the most primitive modems used on-off keying or two-tone frequency shift keying. Information theory says that the most efficient way to use bandwidth involves having a vast number of subtly different states. Of course each state can encode multiple bits, which can be unpacked at the receiving end. > Energy dissipation is now the main issue for most computation. The > way forward is not higher radices, but reversible low-dissipation > computing, which for reasons I don?t understand, no one is taking > seriously. Maybe because we're still many orders of magnitude away from the kT log(2) limit. Reversible computation is only useful for breaking that limit. Also it generally involves computing very slowly. By the time we've approached that limit, it might be practical to do most of our computation far from the sun, where T is much lower. (Running a computer in an air conditioned room doesn't really gain you anything.) Dyson claims that we, or rather our cybernetic descendants or uploads, ought to be able to live forever in an expanding universe by running slower and slower, colder and colder, doing infinite computation in infinite time using finite total energy. Critics claim that that won't be possible since clocks can't be made arbitarily slow due to quantum limits, or because the background temperature of space isn't asymptotic to absolute zero due to Unruh radiation or other effects.

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