January 2017 - mathfun@mailman.xmission.com

Re: [math-fun] ITS (PDP-10) Lives! -- How about the PDP-1?
by rcs＠xmission.com 30 Jan '17

30 Jan '17

I don't know of anyplace that reads old tapes. I got a tape from Bill Plummer long ago, and I have a bunch of my own too. I may still have some old paper tapes, but no system stuff. I did do a PDP1X simulator in Lisp c. 1985, but it's also on large tape somewhere. --Rich ------- [forwarded from WBA. --Rich] Date: Mon, 30 Jan 2017 23:03:36 -0500 Subject: Re: [math-fun] ITS (PDP-10) Lives! -- How about the PDP-1? From: wba <wbackerman(a)gmail.com> The discussion of the PDP-10 reminds me of another problem that's been sitting in my living room for decades. The MIT RLE PDP-1 timesharing system was the world's first, and for a long time the oldest timesharing system still running. It was the original "spacewar" machine. Some time ago, with the assistance of David Moon, we saved all the system stuff onto tape ("macrotapes" we called them then). Both on the ITS "MC" machine, which had "modern" 9-track tape drives, as opposed to the 7-track drives of AI, ML, and DM, and onto TOPS-20 drives from machine mit-xx. You're supposed to copy stuff over onto modern technology every 10 years or so to prevent disasters like this. But we managed to skip 8-inch floppies, 5-inch floppies, 10-cm floppies, various tape cassettes, CD's, DVD's, and a whole lot of other stuff. Of course I should have attended to this stuff decades ago. Does anyone know of any place on the planet where these tapes could be read? Rich: I believe you got some copies of this when you came to visit a long time ago. You didn't do anything with it by any chance? Bill Ackerman

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[math-fun] social networking cellular computations
by Henry Baker 30 Jan '17

30 Jan '17

After attending CES2017 (Consumer Electronics Show) in Las Vegas this month, I concluded that the following device is inevitable, so I started thinking about what sorts of computations & patterns that might be obtained. One of the interesting things is the progress of "wearable computers", in which computation is embedded in socks, gloves, shirts, glasses, etc. I haven't seen a fully-electronic T-shirt yet, but it's inevitable. By "electronic T-shirt", I mean one with a (bit-mapped?) display on the front & back which can display any stupid thing that your heart desires -- e.g., "I'm with stupid", etc. If there is a display on your sleeve, you could literally wear year heart on your sleeve. Now such a stand-alone display is interesting, but would get boring after perhaps 5 minutes. Far more interesting is a display *that senses & reacts to other people's displays* -- probably by using Bluetooth LE, ANT+, or some other short-range wireless technology. You've probably heard the phrase "he was swayed by the last person he talked with"; an old joke recognizes this tendency and suggests that such information simply be displayed on his forehead. Well, an electronic T-shirt could do exactly that: indicate some amalgam of the electronic T-shirts that our target T-shirt had been recently close to. So, what if each such T-shirt had the same finite set of states, and all the T-shirts had the same rule for combining the states of the N closest T-shirts in order to compute this T-shirt's next state ? What kinds of computations & patterns could be obtained from such a cellular model? One could implement such a device today using Bluetooth smartwatches or smartrings (yes, they were at CES2017) communicating amongst themselves; of course, displaying the results on a T-shirt would be a lot more dramatic. BTW, this sort of model isn't 100% novel; I referred to a similar model here last November: SCHELLING, T. 1971. "Dynamic Models of Segregation". Journal of Mathematical Sociology 1: 143-186. https://www.stat.berkeley.edu/~aldous/157/Papers/Schelling_Seg_Models.pdf

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[math-fun] This is a test - please ignore only if somebody has already replied to this message Date: 2017-01-30 08:36
by Bill Gosper 30 Jan '17

30 Jan '17

From: Seb Perez-D <sbprzd+mathfun(a)gmail.com> To: math-fun <math-fun(a)mailman.xmission.com> Reply-To: math-fun <math-fun(a)mailman.xmission.com> I fear my messages are not reaching the mailing list - could at least one person reply? many thanks in advance Sébastien --------- You're supposed to ask, "Can you hear me?" and then when we say yes, you slap us with a huge and unwelcome telemarketing charge. --rwg

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[math-fun] A question about infinite words
by James Propp 30 Jan '17

30 Jan '17

The following question has arisen from my research on one-dimensional packing; I'm wondering if someone can help me prove it. Say we have an infinite word W = (w_1, w_2, w_3, ...) consisting of letters from some finite alphabet each of which occurs infinitely often in W. Suppose further that the finite alphabet consists of two kinds of letters which I'll call "consonants" and "vowels". We are also given a rational number r, and our infinite word satisfies a "proportionality property" (P): (P) If w_i = w_j (with i < j), then the number of vowels in (w_i, w_{i+1}, ..., w_{j-1}) (counted with multiplicity) divided by the number of consonants in (w_i, w_{i+1}, ..., w_{j-1}) (counted with multiplicity) equals r. Then I would like to be able to conclude that the consonant-vowel pattern of W is eventually periodic. Example: W = (a, b, c, e, d, c, a, b, c, a, b, c, e, d, c, a, b, c, a, b, c, a, b, c, e, d, c, ....) where a and e are vowels and b, c, and d are consonants. (If the pattern isn't clear, write the word schematically as XY XXY XXXY XXXXY ... where X = (a, b, c) and Y = (e, d, c).) Let r = 1/2. With i = 1 and j = 7, we see that the number of vowels in positions 1 through 6 divided by the number of consonants in positions 1 through 6 equals 2/4, or 1/2. More generally, given any two positions i and j that both have the same letter in W, if we look at all the letters that occur in positions i through j-1, we see that the number of vowels divided by the number of consonants is 1/2, so property (P) is satisfied. And if we look at the consonant-vowel pattern, it's just vowel-consonant-consonant over and over again, repeating periodically. My gut tells me that this is at the level of a hard IMO problem, and that it has a pretty solution. But that's no consolation to me, given that I don't see how to solve it. A solution to this puzzle would advance me towards a seemingly unrelated goal in the theory of sphere-packing. Jim

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[math-fun] This is a test - please ignore only if somebody has already replied to this message
by Seb Perez-D 30 Jan '17

30 Jan '17

I fear my messages are not reaching the mailing list - could at least one person reply? many thanks in advance Sébastien

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Re: [math-fun] Packing question
by Dan Asimov 30 Jan '17

30 Jan '17

Is the problem that the densest packings are not unique? I don't see this as a problem per se. For example: In the Kepler problem of packing spheres in 3D, the packings obtained by any sequence of what are sometimes called the "a", "b", and "c" packings in successive layers — as long as two successive layers are not given by the same letter — is equally densest. Which gives uncountably many ties. This may be surprising, but it's just the way things are and I see no need to arrange things so this not be the case. On the other hand, if asymptotic density is the only thing to maximize, I find it somewhat unpleasant that by starting with say the fcc packing but removing all (unit) spheres lying closer to the origin than a distance of 10^10^10^10^10^10^10^10^10^10 (exponents grouped from the top down) results in yet another packing tied for densest. One could avoid such "cheater" configurations by requiring that all sufficiently large configurations cannot be made denser by adding more spheres. I'm not sure if this condition is known to always allow the densest of all configurations to satisfy it. Certainly there are densest configurations in dimensions 1, 2, 8, and 24 that *do* satisfy it, however, and I believe no dimensions known (i.e., not dimension 3, either) where some densest configuration does not satisfy it. But in any case this cuts down on some of the unpleasantness. Jim: Before we get to how your theory might be constructed, I would like to understand much better what it is you are aiming to achieve or avoid. —Dan ----- Here's some background on where my recent question (so quickly resolved by Andy Latto) came from: The itch I'm trying to scratch is the fact that even though the sphere-packing problems in dimensions 2, 8, and 24 have beautiful canonical solutions, these solutions are only known to be "optimal" in ways I find unsatisfactory. We say for instance that the six-around-one packing in 2D "achieves the maximal packing density" --- but then so do lots of other packings (for instance, any perturbed version of the six-around-one packing). Or, we say that the six-around-one packing is, up to isometry, the "unique lattice packing", but the lattice condition seems extraneous (and indeed the lattice condition becomes problematic in higher dimensions, where it's not even known if densest lattice packings exist). My diagnosis of the problem is that density is too coarse a way to measure packings. We need a non-Archimedean measure (or rather valuation) that can "see" both the density of a lattice packing and the finite perturbations we can apply to it, even though they are separated by infinite orders of magnitude. I have various ideas for how such a theory could be constructed, but none of them feel quite right, and there are significant technical challenges to applying them. So, rather than attack the sphere-packings problem head-on, I'm taking a detour through one-dimensional packing problems. And, to make things even simpler, I'm starting by focussing on discrete packing problems. Furthermore, instead of packing Z, I'm just packing the right half of it. So I want a way to measure subsets of {0,1,2,...} that includes cardinality and density as special cases. One approach is to measure a set S by the limiting behavior of its generating function \sum_{n \in S} q^n as q approaches 1; you can see some low-hanging fruits of this idea at http://mathoverflow.net/questions/248994/comparing-sizes-of-sets-of-natural… . I was trying to apply this approach to packing problems and related sorts of combinatorial optimization problems. For instance: Consider subsets of {0,1,2,...} having the property that no two elements of S differ by 3 or 5. In what sense is S_0 := {0,2,4,...} the unique "biggest" set of this kind? It has density 1/2, but so does S_1 := {1,3,5,...}. And the set S_2 := {0,1,2, 8,9,10, 16,17,19, ...} starts out looking denser than S_0. Examine at the associated q-series as q approaches 1: using the notation of the MathOverflow article, we have |S_0|_q > |S_1|_q > |S_2|_q for all q<1 sufficiently close to 1. So in this sense S_0 is "bigger" than the other two sets. I'd like to prove that for all finite sets D (of positive integers), not just the finite set {3,5}, there is a unique biggest set of natural numbers, S, such that no two elements of S differ by an element of D, where "biggest" denotes comparison of generating functions as q goes to 1. Naturally, I expect that this unique maximal S is an eventually periodic set. (Where I call a set eventually periodic, I mean of course that its indicator function is eventually periodic.) If I could prove this, a simple coding would give results about packings of the set of natural numbers by translates of a finite set. Anyone have any thoughts about this? -----

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[math-fun] ITS (PDP-10) Lives!
by Henry Baker 30 Jan '17

30 Jan '17

FYI -- Revisit the most usable/least secure timesharing system ever made! https://github.com/PDP-10/its ITS, the Incompatible Timesharing System, is an operating system for the PDP-10 computer family. It was created by hackers at MIT in the 1960s. The MIT site was shut down in 1990, but enthusiasts continue to operate ITS systems to this day. About this project This repository contains source code, tools, and scripts to build an ITS system from scratch. The goals are: To provide an automated build from start to end. No user invervention is necessary. To check which programs have source code, and which programs are missing source code. To ensure that we know how to build all programs. In many cases we can just invoke MIDAS, but some require the use of DDT. To update programs with the latest bug fixes and enhancements. To be able to set configurable options across the whole system, such as host name, IP address, hardware devices. The build currently runs on the SIMH and KLH10 emulators. Of course, we'd be delighted to test this on a real KS10. Usage To build ITS with this repository, you need some tools installed: make, C compiler, and expect. Ensure all submodules are checked out, and then type make EMULATOR=simh or make EMULATOR=klh10. This will leave built files in the out directory, one of which is a disk image with ITS installed. To start ITS, type ./start. If you see KLH10#, type go and Enter. When you see the DSKDMP prompt, type its, press Enter, and then ESC g. Eventually, you will see SYSTEM JOB USING THIS CONSOLE. You are now ready to log in, so type Control-Z. See doc/DDT.md for a list of useful commands. When done, shut down orderly by typing :lock and then 5down. To install your personal user files, add a subdirectory under user with your files inside. They will be copied to ITS. The directory name is limited to six characters, and file names must have two six-character parts separated by a period. Here is an overview of the repository: bin - PDP-10 binary files necessary to bootstrap the system. build - build scripts. doc - documentation, most of which appear in the INFO system. src - source code for ITS and all programs. tools - build tools. out - build output. user - optional user files installed in ITS.

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[math-fun] Packing question
by James Propp 29 Jan '17

29 Jan '17

Here's some background on where my recent question (so quickly resolved by Andy Latto) came from: The itch I'm trying to scratch is the fact that even though the sphere-packing problems in dimensions 2, 8, and 24 have beautiful canonical solutions, these solutions are only known to be "optimal" in ways I find unsatisfactory. We say for instance that the six-around-one packing in 2D "achieves the maximal packing density" --- but then so do lots of other packings (for instance, any perturbed version of the six-around-one packing). Or, we say that the six-around-one packing is, up to isometry, the "unique lattice packing", but the lattice condition seems extraneous (and indeed the lattice condition becomes problematic in higher dimensions, where it's not even known if densest lattice packings exist). My diagnosis of the problem is that density is too coarse a way to measure packings. We need a non-Archimedean measure (or rather valuation) that can "see" both the density of a lattice packing and the finite perturbations we can apply to it, even though they are separated by infinite orders of magnitude. I have various ideas for how such a theory could be constructed, but none of them feel quite right, and there are significant technical challenges to applying them. So, rather than attack the sphere-packings problem head-on, I'm taking a detour through one-dimensional packing problems. And, to make things even simpler, I'm starting by focussing on discrete packing problems. Furthermore, instead of packing Z, I'm just packing the right half of it. So I want a way to measure subsets of {0,1,2,...} that includes cardinality and density as special cases. One approach is to measure a set S by the limiting behavior of its generating function \sum_{n \in S} q^n as q approaches 1; you can see some low-hanging fruits of this idea at http://mathoverflow.net/questions/248994/comparing-sizes-of-sets-of-natural… . I was trying to apply this approach to packing problems and related sorts of combinatorial optimization problems. For instance: Consider subsets of {0,1,2,...} having the property that no two elements of S differ by 3 or 5. In what sense is S_0 := {0,2,4,...} the unique "biggest" set of this kind? It has density 1/2, but so does S_1 := {1,3,5,...}. And the set S_2 := {0,1,2, 8,9,10, 16,17,19, ...} starts out looking denser than S_0. Examine at the associated q-series as q approaches 1: using the notation of the MathOverflow article, we have |S_0|_q > |S_1|_q > |S_2|_q for all q<1 sufficiently close to 1. So in this sense S_0 is "bigger" than the other two sets. I'd like to prove that for all finite sets D (of positive integers), not just the finite set {3,5}, there is a unique biggest set of natural numbers, S, such that no two elements of S differ by an element of D, where "biggest" denotes comparison of generating functions as q goes to 1. Naturally, I expect that this unique maximal S is an eventually periodic set. (Where I call a set eventually periodic, I mean of course that its indicator function is eventually periodic.) If I could prove this, a simple coding would give results about packings of the set of natural numbers by translates of a finite set. Anyone have any thoughts about this? Jim Propp

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[math-fun] New form of matter: "time crystals"
by Henry Baker 29 Jan '17

29 Jan '17

FYI -- I wonder how "big" (both in space & time) these crystals can get? Perhaps a new type of cellular automaton can be built this way? New way to build a quantum computer? https://www.sciencealert.com/scientists-have-just-announced-a-brand-new-for… Scientists have confirmed a brand new form of matter: time crystals Perpetual motion without energy. FIONA MACDONALD 28 JAN 2017 For months now, there's been speculation that researchers might have finally created time crystals - strange crystals that have an atomic structure that repeats not just in space, but in time, putting them in perpetual motion without energy. Now it's official - researchers have just reported in detail how to make and measure these bizarre crystals. And two independent teams of scientists claim they've actually created time crystals in the lab based off this blueprint, confirming the existence of an entirely new form of matter. The discovery might sound pretty abstract, but it heralds in a whole new era in physics - for decades we've been studying matter that's defined as being 'in equilibrium', such as metals and insulators. But it's been predicted that there are many more strange types of matter out there in the Universe that aren't in equilibrium that we haven't even begun to look into, including time crystals. And now we know they're real. The fact that we now have the first example of non-equilibrium matter could lead to breakthroughs in our understanding of the world around us, as well as new technology such as quantum computing. "This is a new phase of matter, period, but it is also really cool because it is one of the first examples of non-equilibrium matter," said lead researcher Norman Yao from the University of California, Berkeley. "For the last half-century, we have been exploring equilibrium matter, like metals and insulators. We are just now starting to explore a whole new landscape of non-equilibrium matter." Let's take a step back for a second, because the concept of time crystals has been floating around for a few years now. First predicted by Nobel-Prize winning theoretical physicist Frank Wilczek back in 2012, time crystals are structures that appear to have movement even at their lowest energy state, known as a ground state. Usually when a material is in ground state, also known as the zero-point energy of a system, it means movement should theoretically be impossible, because that would require it to expend energy. But Wilczek predicted that this might not actually be the case for time crystals. Normal crystals have an atomic structure that repeats in space - just like the carbon lattice of a diamond. But, just like a ruby or a diamond, they're motionless because they're in equilibrium in their ground state. But time crystals have a structure that repeats in time, not just in space. And it keep oscillating in its ground state. Imagine it like jelly - when you tap it, it repeatedly jiggles. The same thing happens in time crystals, but the big difference here is that the motion occurs without any energy. A time crystal is like constantly oscillating jelly in its natural, ground state, and that's what makes it a whole new form of matter - non-equilibrium matter. It's incapable of sitting still. But it's one thing to predict these time crystals exist, it's another entirely to make them, which is where the new study comes in. Yao and his team have now come up with a detailed blueprint that describes exactly how to make and measure the properties of a time crystal, and even predict what the various phases surrounding the time crystals should be - which means they've mapped out the equivalent of the solid, liquid, and gas phases for the new form of matter. Published in Physical Review Letters, Yao calls the paper "the bridge between the theoretical idea and the experimental implementation". And it's not just speculation, either. Based on Yao's blueprint, two independent teams - one from the University of Maryland and one from Harvard - have now followed the instructions to create their own time crystals. Both of these developments were announced at the end of last year on the pre-print site arXiv.org (here and here), and have been submitted for publication in peer-reviewed journals. Yao is a co-author on both articles. While we're waiting for the papers to be published, we need to be skeptical about the two claims. But the fact that two separate teams have used the same blueprint to make time crystals out of vastly different systems is promising. The University of Maryland's time crystals were created by taking a conga line of 10 ytterbium ions, all with entangled electron spins. http://www.sciencealert.com/images/131711_web.jpg Chris Monroe, University of Maryland The key to turning that set-up into a time crystal was to keep the ions out of equilibrium, and to do that the researchers alternately hit them with two lasers. One laser created a magnetic field and the second laser partially flipped the spins of the atoms. Because the spins of all the atoms were entangled, the atoms settled into a stable, repetitive pattern of spin flipping that defines a crystal. That was normal enough, but to become a time crystal, the system had to break time symmetry. And observing the ytterbium atom conga line, the researchers noticed it was doing something odd. The two lasers that were periodically nudging the ytterbium atoms were producing a repetition in the system at twice the period of the nudges, something that couldn't occur in a normal system. "Wouldn't it be super weird if you jiggled the Jell-O and found that somehow it responded at a different period?" said Yao. "But that is the essence of the time crystal. You have some periodic driver that has a period 'T', but the system somehow synchronises so that you observe the system oscillating with a period that is larger than 'T'." Under different magnetic fields and laser pulsing, the time crystal would then change phase, just like an ice cube melting. http://www.sciencealert.com/images/131712_web.jpg Norman Yao, UC Berkeley The Harvard time crystal was different. The researchers set it up using densely packed nitrogen vacancy centres in diamonds, but with the same result. "Such similar results achieved in two wildly disparate systems underscore that time crystals are a broad new phase of matter, not simply a curiosity relegated to small or narrowly specific systems," explained Phil Richerme from Indiana University, who wasn't involved in the study, in a perspective piece accompanying the paper. "Observation of the discrete time crystal... confirms that symmetry breaking can occur in essentially all natural realms, and clears the way to several new avenues of research." Yao's blueprint has been published in Physical Review Letters, and you can see the Harvard time crystal paper here, and the University of Maryland paper here. https://arxiv.org/pdf/1610.08057v1.pdf https://arxiv.org/pdf/1609.08684v1.pdf

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Re: [math-fun] They fit!
by Bill Gosper 27 Jan '17

27 Jan '17

On Fri, Jan 27, 2017 at 3:58 PM, Julian Ziegler Hunts <julianj.zh(a)gmail.com> wrote: > A little. Take six disks connected in an arc, with constant angles > between any consecutive three. Clearly, by varying that angle, we can make > two disks fit across as they do in the picture. What angle makes this > happen? The secant disks are symmetric with the two disks in the middle of > the arc (leftmost disks in the picture), so those angles are the same. And > the ends form equilateral triangles. So the total of the interior angles > of the beehive is 6*θ - 2π/3=4π, and θ=7π/9, the interior angle of a > nonagon. > > Nice! > Of course, that doesn't really tell you why the angle happens to be the > same as that of a nonagon. But it does prove by elementary geometry that > they fit. > > Julian > Re: [math-fun] They fit! Date: 2017-01-27 14:30 From: James Buddenhagen <jbuddenh(a)gmail.com> To: math-fun <math-fun(a)mailman.xmission.com> Reply-To: math-fun <math-fun(a)mailman.xmission.com> >Similar things for n=11 and n=13: >http://www.buddenbooks.com/jb/pack/circle/n11.htm Perhaps this should be named Melissen's Theorem. I missed the swap @ 3 o'clock. And that solution looks to me quite unremarkable by comparison. Presumably, the possibile swap should be just as surprising as the secant pair fitting perfectly. But only the latter jumps out at me. Can the swap finish Julian's proof? >http://www.buddenbooks.com/jb/pack/circle/n13.htm I don't remember ever seeing two optimal solutions, one loose, one tight. Did you really mean i^2 = 1? --rwg On Fri, Jan 27, 2017 at 4:16 PM, Bill Gosper <billgosper(a)gmail.com> wrote: > ccc11.gif <http://www2.stetson.edu/~efriedma/cirincir/ccc11.gif> seems to > assert that two unit disks precisely fit within a circle of nine. After > much bovoparturition, Mathematica agrees! Can anyone provide some insight? > --rwg

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