May 2017 - mathfun@mailman.xmission.com

[math-fun] Directions to a map without knowing the starting position
by Seb Perez-D 18 May '17

18 May '17

Hello everyone I am looking for the name and/or more information about a class of problems, which consist of providing directions to reach a specific point on a map, without knowing where one is starting from. If I remember well, it wasn't even clear whether such maps existed - but some non-trivial examples have been found. The paths were coloured red or blue and directions given "take the red path, take the blue path, etc." Searching for the suspected keywords sends me to Google Maps etc., so I turn to the wisdom and knowledge of math-fun! Thanks in advance Seb

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[math-fun] 45º pessimal?
by Bill Gosper 18 May '17

18 May '17

Everybody knows that, in vacuo, 45º elevation maximizes projectile range. Trick question: Given our atmosphere, is the range maximized by > or < 45º? SPOILER. Baseball: <. Cannonball: >. Well, long long distance artillery anyway. The idea is to fly above the denser, low level atmosphere. This raises the interesting question: Are there projectile densities and energies for which the range improves for elevations both > and < 45º? --rwg

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[math-fun] HPE unveils The Machine with 160 TBytes (2^48 ?)
by Henry Baker 17 May '17

17 May '17

FYI -- I could swear that someone at HP was reading math-fun's discussion of table-driven computing a month or two ago. 'HPE' = 'HyPE' ? The Machine scales to ~ 2^92 bytes, so pointers are > 64 bits long, even though the current prototype has only ~ 2^48 bytes of memory. The article doesn't say, but my guess of the location of the prototype is a suburb of Salt Lake City, where the first application is find more hay in haystacks and storing discrete log tables. https://venturebeat.com/2017/05/16/hp-enterprise-unveils-single-memory-160-… HP Enterprise unveils The Machine, a single-memory computer capable of addressing 160 terabytes Dean Takahashi @deantak May 16, 2017 6:01 AM Hewlett Packard Enterprise announced what it is calling a big breakthrough Â creating a prototype of a computer with a single bank of memory that can process enormous amounts of information. The computer, known as The Machine, is a custom-built device made for the era of big data. https://news.hpe.com/content-hub/memory-driven-computing/ HPE said it has created the worldÂs largest single-memory computer. The R&D program is the largest in the history of HPE, the former enterprise division of HP that split apart from the consumer-focused division. ... Based on the current prototype, HPE expects the architecture could easily scale to an exabyte-scale single-memory system and, beyond that, to a nearly limitless pool of memory Â 4,096 yottabytes [2^92 bytes]. For context, that is 250,000 times the entire digital universe today. ... The new prototype has 160 TB of shared memory spread across 40 physical nodes, interconnected using a high-performance fabric protocol. It has an optimized Linux-based operating system (OS) running on ThunderX2, CaviumÂs flagship second generation dual socket capable ARMv8-A workload optimized System on a Chip. It also has photonics and optical communication links, including the new X1 photonics module. And HPE has built software programming tools designed to take advantage of abundant persistent memory.

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Re: [math-fun] Egan's Orthogonal trilogy
by Keith F. Lynch 17 May '17

17 May '17

Henry Baker <hbaker1(a)pipeline.com> wrote: > We can't tell "from inside" whether we're in a left-handed universe > or a right-handed universe. Thus, there's no way "from inside" to > distinguish +t from -t. > Yes, there's entropy, which is the usual way to tell a normal movie > from a time-reversed movie, but all the usual mechanical laws are > time-symmetric. In the -t direction entropy is lower and there are events which we remember. In the +t direction entropy is higher and we have free will. (You can choose today what to have for breakfast tomorrow. You can't choose today what to have had for breakfast yesterday.) It's exactly the same in the Orthogonal universe -- until the events of the third book. > "Keith F. Lynch" <kfl(a)KeithLynch.net> wrote: >> (Disclaimer: That's special relativity. In general relativity, >> reverse causality appears to be possible under extreme conditions.) > With GR, gravitational waves from coalescing black holes seem to > distinguish +t from -t; Certainly. That's an instance of entropy increasing. In fact the first announced LIGO observation of gravitational waves was the most rapid entropy increase ever observed -- tne energy equivalent of 200 solar masses radiated per second, albeit only for a few milliseconds. (It's ironic that one of the LIGOs is at Hanford. Once that town was known for studying the most dangerous known radiation. Now they're studying the least dangerous known radiation. I estimate it would take at least 10^30 watts of gravitational waves hitting you to kill you.) But I was thinking of subtler applications of GR. Tipler's rotating cylinder. Stabilized wormholes. Flying through the center of a donut-shaped black hole. > it might be possible to focus gravitational waves onto a black hole > to get it to split in two, but that might be asking a little too > much from our current technology. To focus gravitational waves all you need is a gravitational lens. Observing continuous smooth point-source gravitational waves through a massive object would be a useful way to probe the mass distribution inside the object. If everything is reversible in principle, why can't black holes be turned into stars? You wouldn't even need to focus any gravitational waves, since a radially symmetric collapse doesn't generate any. At least in Egan's universe they don't have any black holes or event horizons to worry about. Note that if you want to play around with an event horizon (in our universe) you don't need a black hole. SImply accelerate, and there will be an event horizon behind you. Its distance in light years is roughly equal to the reciprocal of your acceleration in Gs. Although you can never travel faster than light, light from behind that event horizon can never reach you (as long as you continue with the same acceleration). In your frame, time is going backwards there. > Perhaps I should read these books; I'm having a hard time (!) > understanding how one could consider one of the space coordinates > as a time coordinate. The characters have a hard time with it too. Time doesn't *seem* like a spacial dimension to them any more than to us. They finally just have to accept that that's how the universe is. Under ordinary conditions time behaves exactly like in our universe. Their version of Newton's laws are identical to ours. > As you point out, how do you set up a differential equation in > "quaternion space" such that a wave can propagate in the other > space coordinates? Answered in the books and on Egan's website. Hint: In their universe the speed of light varies with wavelength, and can be anything from zero to infinite. Also, light has three polarizations, each at right angles to the other two. In addition to the two polarizations we all know and love, there are longitudinal waves, in which the fields are in the direction of motion.

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Re: [math-fun] Egan's Orthogonal trilogy
by Keith F. Lynch 16 May '17

16 May '17

Henry Baker <hbaker1(a)pipeline.com> wrote: > This is the *quaternion universe*. Egan mentions quaternions by name. And has one of the characters discover them, only with a different notation. But I think it would make more sense to view *our* universe as the quaternion universe. As soon as I learned about quaternions, about 40 years ago, I made the connection. Three dimensions that are just alike, and one that is different and has a direction associated with it. There is no way to define the units i, j, k, -i, -j, or -k except by reference to the other units. Your i could be my -j and our moderator's (j-k)/sqrt(2). Exactly as with space's x, y, and z axes. But there's no way to confuse 1 with -1 or with any of the imaginary units. Exactly like our t axis. There's a unique "arrow of time." Relativity can mix space and time to an extent, but every interval between two events is unabiguously either spacelike or timelike, depending on whether the invariant, x^2 + y^2 + z^2 - ct^2, is positive or negative. And if it's timelike, it's unambiguous which event is earlier and which is later. There's no frame of reference in which Macron's inauguration came before Trump's. (Disclaimer: That's special relativity. In general relativity, reverse causality appears to be possible under extreme conditions.) By contrast, in Egan's universe the four coordinates are completely interchangable. Once their spaceship reaches infinite speed, one of their planet's space axes is their spaceships's time axis, and one of their spaceship's space axes is their planet's time axis. There's no unique arrow of time. Each object carries its own arrow of time along with it. Hence it's possible to land on a time-reversed planet. But you had better bring a flashlight, since you won't be able to see by the light of its time-reversed sun. Also, watch out for the time-reversed dust that was in your spaceship before you landed, and you weren't able to remove it then. The good news is that it was all gone by the time you took off, as you had, without trying, tracked it all out. You also removed (by walking in them) all the footprints that were around your landing site when you landed.

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[math-fun] Geometry of Number Walls Update
by Fred Lunnon 16 May '17

16 May '17

A brief intro & summary of recent number wall activity has been uploaded at https://www.dropbox.com/s/rocryne1zhp98of/geom_wall_update.txt , avoiding the dreaded Gmail spurious line breaks (why?, why?), and permitting covert correction of my inevitable catastrophic typos. But Dropbox descends steadily into inutility, a recent `improvement' being automatic updates which cannot be disabled: the resulting potential for chaos could surely have been foreseen! Can anyone suggest acceptable alternatives? Fred Lunnon

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[math-fun] Scientists Achieve Counterfactual Quantum Communication For The First Time
by Henry Baker 15 May '17

15 May '17

FYI -- I assume that "counterfactual" here doesn't mean "fake news"... https://www.sciencealert.com/scientists-have-achieved-direct-counterfactual… Scientists Achieve Direct Counterfactual Quantum Communication For The First Time Communication without particle transmission. FIONA MACDONALD 10 MAY 2017 Theoretical physicists have long proposed that such a form of communication would be possible, but now, for the first time, researchers have been able to experimentally achieve it - transferring a black and white bitmap image from one location to another without sending any physical particles. https://www.ncbi.nlm.nih.gov/pubmed/23679694 Direct counterfactual quantum communication on the other hands relies on something other than quantum entanglement. Instead, it uses a phenomenon called the quantum Zeno effect. Very simply, the quantum Zeno effect occurs when an unstable quantum system is repeatedly measured. In the quantum world, whenever you look at a system, or measure it, the system changes. And in this case, unstable particles can never decay while they're being measured (just like the proverbial watched kettle that will never boil), so the quantum Zeno effect creates a system that's effectively frozen with a very high probability. If you want to delve a little deeper, the video below gives a great explanation: http://www.youtube.com/watch?v=dOY0NbR1ppw Counterfactual quantum communication is based on this quantum Zeno effect, and is defined as the transfer of a quantum state from one site to another without any quantum or classical particle being transmitted between them. This requires a quantum channel to run between two sites, which means there's always a small probability that a quantum particle will cross the channel. If that happens, the system is discarded and a new one is set up. To set up such a complex system, researchers from the University of Science and Technology of China placed two single-photon detectors in the output ports of the last of an array of beam splitters. Because of the quantum Zeno effect, the system is frozen in a certain state, so it's possible to predict which of the detectors would 'click' whenever photons passed through. A series of nested interferometers measure the state of the system to make sure it doesn't change. It works based on the fact that, in the quantum world, all light particles can be fully described by wave functions, rather than as particles. So by embedding messages in light the researchers were able to transmit this message without ever directly sending a particle. The team explains that the basic idea for this set up came from holography technology. "In the 1940s, a new imaging technique - holography - was developed to record not only light intensity but also the phase of light," the researchers write in the journal Proceedings of the National Academy of Sciences. http://www.pnas.org/content/early/2017/04/19/1614560114 "One may then pose the question: Can the phase of light itself be used for imaging? The answer is yes." The basic idea is this - someone wants to send an image to Alice using only light (which acts as a wave, not a particle, in the quantum realm). Alice transfers a single photon to the nested interferometer, where it can be detected by three single-photon detectors: D0, D1, and Df. If D0 or D1 'click', Alice can conclude a logic result of one or zero. If Df clicks, the result is considered inconclusive. As Christopher Packham explains for Phys.org: https://phys.org/news/2017-05-counterfactual-quantum.html "After the communication of all bits, the researchers were able to reassemble the image - a monochrome bitmap of a Chinese knot. Black pixels were defined as logic 0, while white pixels were defined as logic 1 ... In the experiment, the phase of light itself became the carrier of information, and the intensity of the light was irrelevant to the experiment." Not only is this a big step forward for quantum communication, the team explains it's technology that could also be used for imaging sensitive ancient artefacts that couldn't surprise direct light shined on them. The results will now need to be verified by external researchers to make sure what the researchers saw was a true example of counterfactual quantum communication. Either way, it's a pretty cool demonstration of just how bizarre and unexplored the quantum world is. The research has been published in the journal Proceedings of the National Academy of Sciences. http://www.pnas.org/content/early/2017/04/19/1614560114

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[math-fun] Egan's Orthogonal trilogy
by Keith F. Lynch 15 May '17

15 May '17

In special relativity, sqrt(x^2 + y^2 + z^2 - ct^2) is invariant between any two events. Greg Egan wrote a science fiction trilogy in which it is instead sqrt(x^2 + y^2 + z^2 + ct^2) that is invariant between any two events. Note the single change of sign. He has worked out in great detail lots of consequences of this "tiny" change. He assumes that as much physics as possible remains unchanged. The series is of course set in a different universe. (That is surprisingly rare in science fiction.) The characters live on a planet that orbits a star, and are just starting to understand physics. The first novel, aptly named, is _The Clockwork Rocket_. The rocket is controlled by clockwork not because they're too primitive to have discovered electricity or steam, but because neither electric current nor liquids exist in their universe. Their planet is gravely threatened by a natural phenomenon. It would take far too long to develop the necessary science and engineering to solve the problem in time to save their planet. So they build an immense rocket designed to travel at an infinite (!) speed. People on board will live for generations and do research, and return to their home planet with the solution. By the planet's clocks this will be a very short time after they left. In our universe it would take an infinite amount of energy for a spaceship to travel at the speed of light. In their universe it would take an finite amount of energy for a spaceship to travel at an infinite speed. The novel also deals with their interpersonal relationships, and with political intrigue. There's even a saboteur aboard their giant rocket. But for me the most interesting part is watching them plan and conduct observations and experiments to figure out their physics (and their chemistry, and their biology). The second novel is _Eternal Flame_, the third is _The Arrows of Time_. The titles are minor spoilers for the physics. They can get unlimited energy by radiating equal amounts of positive and negative energy. And time travel is possible. There are some very interesting scenes set on a time-reversed planet they take a side trip to. Shortly after landing and taking a look around that lifeless planet, one of them sees something unexpected and wishes he'd been trained in time-reversed archeology. They were published in 2011, 2012, and 2013 respectively. For me, much of the fun was working out the physics for myself. For instance I discovered that if you were to accelerate at a uniform rate, you would eventually return to your starting point. Not just in space, but also in time and in velocity! For a real mathematical challenge, work out their version of Maxwell's equations. If you're curious about the physics, too lazy to work it out for yourself, and too cheap to buy the novels, there are spoilers for the physics (but not for the plot) at http://gregegan.net/ On this list, Egan is perhaps best known for the short story "Luminous" and its sequel "Dark Integers," based on the premise that no theorem is ever true of false until it makes a difference to some physical system (if only a mathematician's brain), that during the Big Bang, inconsistencies in math formed and got frozen in, and that by doing lots of proofs near the border between our math and the rival math it's possible to move the border. Bad guys want to get ahold of the secret location (in math-space) of the inconsistencies so that they can make a fortune, presumably by breaking arithmetic so that their bank balances add up to much larger numbers, heedless of the enormous risk to others. But have no fear, lots of things are very different in the Orthogonal universe, but their math is exactly like ours.

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[math-fun] Draft of my May 2017 blog post
by James Propp 12 May '17

12 May '17

I started writing a new draft titled "Minus Infinity" and would love to get your feedback. I plan on publishing it on May 17. Please leave your feedback here: https://mathenchant.wordpress.com?p=1645&shareadraft=59153fba18581 (keeping in mind that if you don't sign your comment I won't know who you, with the consequence that I won't be able to ask follow-up questions that your comments may elicit). Title: Minus Infinity Beginning: Today we'll talk about some paradoxical things, like the logarithm of zero and the maximum element of a set of real numbers that doesn't contain any real numbers at all. More importantly, we'll see how mathematicians try to wrap their heads around such things. To keep the discussion accessible, I'll... Read more: https://mathenchant.wordpress.com?p=1645&shareadraft= 59153fba18581 Thanks, Jim

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[math-fun] Double alphamagic squares
by Eric Angelini 09 May '17

09 May '17

This is insane! https://www.futilitycloset.com/2017/05/07/double-alphamagic-squares/ Best, É.

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