Friday, October 28, 2016

[fyspvpzy] Chess Turing Test

Consider a chess tournament with both human and computer participants, and each player cannot tell who they are playing.  Along with the main tournament incentive to do well, there is also a Turing Test-like sub competition: the humans try to identify which of their opponents are computers, and the computers try to disguise their play to appear to be human.

Of course, the computers will need to decrease its playing strength, but they will also need to make human-like mistakes and avoid computer-like moves, things that computers don't do well yet.

[qjjzjrrw] Another note about the Goodstein Theorem

Another important part of the proof (elided in the current Wikipedia article) is showing that the "minus 1" operation, after mapping to omegas, results in an ordinal number that is less than the previous one.  Note that, after subtracting one from the finite number, writing the resulting finite number in hereditary base notation may result in a very different "shape" than the previous number, compare by analogy 420000 and 419999.  We'll need to invoke exactly how hereditary base notation is constructed and its resulting properties, as well as the definition of "less than" for transfinite ordinal numbers.  Previous note.

[vgoghfhr] Swype

Swype and similar gesture typing are elegant input interfaces well adapted to touchscreens.  Can they improved?

Add "a" "an" and "the" as single buttons that can be swiped like prefixes to a word, without having to lift the finger.  Possibly other common words, too: prepositions.  Punctuation as suffixes.

Even without dedicated buttons, common word sequences (as above) could be swiped in one continuous gesture, maybe without the space, or with it.

Rearrange the letters to avoid ambiguity between common words.  However, this is difficult because rapid gesture typing critically relies on the user having memorized the qwerty keyboard layout.  Memorization is necessary because one cannot easily hunt for the next letter mid-gesture because the finger covers up a large portion of the screen where the next letter might be hiding.  (This is not an issue for virtual keyboards in which keys are tapped: you can see the whole keyboard in between letters.)  Perhaps display a second reference copy of the keyboard above the gesture keyboard while learning a new layout.

Add extra copies of certain letters carefully placed so that the user can learn alternate gestures which avoid misparsing.

Shrink the size of the keyboard to give more space to the app, and rely on stronger NLP to guess the word.  Possibly decrease the number of rows of letters, rearranging.

Use as context, setting prior probabilities, all the words previously typed.  Harder: also use as context later words. This is probably best presented as a variation of spell checking, underlining words likely misparsed, rather than changing words long after they have been typed.  Later context is also available when editing the middle of text, though sentences are often not grammatical in the midst of being edited.

[yryrofrq] Combining music and pictures

Consider an art form creating video accompanied by sound, with categories of various limits of the average bitrate (after compression) of the video.  (Perhaps also categories for limiting the bitrate of the sound.)

Original idea was to strictly limit to static images (slideshow), but the desire to allow Ken Burns style pan and zoom, as well quickly shuffling images suggest expressing the limitation purely information theoretically.

Maybe a contest with prizes.  At the very lowest average bitrate, it might be a contest to come up with the best single image to accompany the music.

Very similar to demoscene, though we imagine standard video compression.  Also similar to music videos, if the soundtrack is taken as fixed.

[qerzxorn] All the superheavy elements

If I understand correctly, the intense neutron flux inside a supernova will produce, via the r-process, all the superheavy elements, not just the ones typically labeled "naturally occurring" up to uranium or so.  Those beyond uranium do get produced, but decay quick enough that there is none left by the time the supernova remnant has recongealed to become (say) planet Earth.  The supernova also produces all the superheavy elements that have not even been discovered yet, so it's not the case that when scientists create a new element on Earth that it might be the first time that element has ever existed in the universe.  In particular, because of the high neutron flux, a supernova produces the neutron rich isotopes of the superheavy elements, which tend to be more stable than the ones produced by fusion of heavy nuclei on earth.  If a nucleus is unstable because it has too many neutrons, it will beta decay until the balance is just right.

If there are any more stable, or long lived, elements, a supernova would have produced them.  This means the hypothesized "island of stability" cannot really be that stable.  But this is at odds with some scientists predicting that the island of stability might contain elements with half lives of a billion years.  Are there isotopes with a nice ratio of protons and neutrons which cannot be reached by the r-process and beta decay?

Of course, there is an upper limit to the r-process, where perhaps cluster decay or spontaneous fission occur before the next neutron can reach the nucleus.

For the core part of the supernova undergoing gravitational collapse into a neutron star, the neutron flux could be even higher, and even heavier elements might very briefly form (still bound by the strong force).  Later, a neutron star itself might be considered a giant nucleus, albeit bound by the gravitational force not strong force.  (Tangentially, "nuclear pasta")

Design a spacecraft, or some other method, to capture and confirm heavy elements produced soon after a supernova.  Maybe go in a few minutes or a day after the event.  This will be pretty difficult: maybe a shield the size of a planet accelerated to relativistic speeds toward the Wolf-Rayet star or white dwarf, having guessed when the supernova will go off, or triggering it, so that the craft arrives at just the right moment.

[xhuasskf] Leaking Clinton campaign emails

Russia the state leaking Hilary Clinton campaign emails in hopes of influencing the U.S. election seems very out of character for how a state intelligence agency normally works.

(If the attack merely originated in Russia the geographic region or been perpetrated by someone from Russia unaffiliated with state intelligence, then it would not be remarkable.)

Leaking the entirety, or a huge number, of the emails reveals how much was leaked ("they know you know").  Leaking them to the public prevents their use privately for extortion or precision tactics, for example leaking selected emails directly to the Trump campaign.  Leaking the emails reveals compromised security, which would induce the Clinton campaign to plug up holes, holes which might have been better exploited for more valuable information in the future.  Leaking them to the public prevents selling the documents to the highest bidder.

The Russian state intelligence agency, assuming it has discovered the hacker on Russian soil, is probably punishing him/her/them for doing all these things wrong.

Thursday, October 27, 2016

[calatnpl] Rth

The polar coordinates, r and theta (sometimes rho and theta), when written as symbols and placed together then pronounced as letters: "earth".

Wednesday, October 26, 2016

[vhrnytmf] Packing dice pips

Create dice marked with pips corresponding to the centers of the densest packing of circles in a square.  3 and 6 will be different from traditional.

Could optimize distance between dots on adjacent faces to choose the among the 4 possible orientations of a each face.

Instead of dots, compute the Voronoi partition of the face around the circle centers, then color the regions with as few colors as possible.

Apply to other dice used in gaming -- non-square faces, d10.  The pattern could become difficult to recognize for large numbers.

Dominoes.  7 has a rattler: make the packing circle of the rattler as large as possible.

[vsratisj] Tenor and piano

Piano players, vocal tenors, and maybe a few others learn to read both treble and bass clef (octave treble clef in the case of the tenor part in some choral music).  It is a subtle bilingualism that they might not consider remarkable until they meet everyone else, people who sing or play any other instrument or part, who have difficulty reading the clef they are not used to.

There's probably something interesting going on inside the brain wherein the notes on the page are mapped to an intermediate representation first.

Tuesday, October 25, 2016

[nmegekrf] Modified minuscule

Place a macron over some lowercase letters to distinguish them from uppercase versions.  Perhaps useful for base 52 or 62 or 64.

c o p s v w x z

Lowercase L should be drawn with a loop as in cursive, and capital i should have serifs to keep them looking different.  The macron over lowercase w is optional if it has curved bottoms like two U's.  (Incidentally, we use an apostrophe in the pluralization of u to avoid confusion with the word us.)  Lowercase k is somewhat close to capital K, perhaps draw lowercase with a loop as in cursive.

None of the lowercase letters above have ascenders.  However, if added, lowercase thorn does, and looks like its uppercase, as does theta.

[xtgdugcl] Mini galaxy

Put the sun and solar system out to Neptune in orbit around a more massive object, perhaps a black hole.  How far from the central object would the sun have to orbit, and how massive must the central object be, in order for the whole arrangement to be stable?  Place multiple solar systems in orbit around the central object.  How far apart do they need to be spaced for things to be stable?  Travel between star systems, each with their own habitable zone, might not be too difficult.

Can such a system naturally form?  (Probably not.)  Consider astroengineering to construct one.

Stability needs a time scale defined: maybe long enough for life to evolve and set the stage for science fiction.

Inspired by the Firefly system, with multiple stars.

[jiqyawnz] Unexpected no-limit hold 'em

No-limit Texas Hold'em poker, because all the money can be bet at once, might have too high a variance that it might be impossible to calculate expected values.  The central limit theorem might not hold.

Possibly tricky to state questions so that expectation is even invoked.

[ynmfnabs] Peddling influence

Business model, step 1. Gain subscribers, followers, readers, access, etc., by any means possible, perhaps even by doing good.  2. Monetize the trust and influence by selling it off to the highest bidder.

The highest bidder is pretty much guaranteed to be someone seeking to manipulate, deceive, and defraud.  The highest bidder might be yourself.

The cycle continues: people learn to no longer trust the thing previously believed trustworthy, so newcomers appear to fill the vacuum.

Does the world have to work this way?

Humans lack the intellectual capacity to constantly evaluate from first principles whether something is good or not, even if the information is available.  Therefore, they place their trust about such evaluations in other people or entities.  Computers may have or soon have the desired kind of intellectual capacity.

[zsennlvq] Future nothing up my sleeve

Propose a cryptography function incorporating "nothing up my sleeve" parameters, traditionally things like square roots of prime numbers.

Improve on tradition: the proposal specifies parameters will be generated from an unpredictable value to be measured in the future, after the proposal has been made.  For example, PRNG seeded from the stock market.

[vwwsnaes] Nearly square rectangles

Rectangles whose dimensions differ by at most 2.  27 items, or 44 if allowing rotations.

1x1=1 2x1=2 3x1=3 2x2=4 3x2=6 4x2=8 3x3=9 4x3=12 5x3=15 4x4=16 5x4=20 6x4=24 5x5=25 6x5=30 7x5=35 6x6=36 7x6=42 8x6=48 7x7=49 8x7=56 9x7=63 8x8=64 9x8=72 10x8=80 9x9=81 10x9=90 10x10=100

Rectangles whose aspect ratio is between 1 and 2.  17 items, or 33 if allowing rotations.

1x1=1 10x9=90 9x8=72 8x7=56 7x6=42 6x5=30 5x4=20 9x7=63 4x3=12 7x5=35 10x7=70 3x2=6 8x5=40 5x3=15 7x4=28 9x5=45 2x1=2

[rlmmcnvh] Fastest everyday objects

What are the fastest massive (not massless) objects a regular person (i.e., someone not working at a particle accelerator) will encounter?  Of course, massless objects -- photons, gravitons -- move at the speed of light (299792458 m/s).

Neutrinos -- recently proven to have mass -- stream through your body constantly.  They travel very very close to the speed of light, though their precise actual speed has yet to be determined.

Dark matter also streams through your body.  Unknown speed; some probably close to the speed of light.

The decay products of cosmic rays hitting the atmosphere also hit or pass through your body.  I am unable to look up their speed, but many are close to the speed of light: the muons must be traveling at that speed or else they would have decayed before they got to you.

Obsolete technology, part 1: The electrons in a cathode ray tube (CRT) in a television or monitor travel about 90000000 m/s or 0.3c, impressively fast.  CRTs are technically particle accelerators.

Next are a few items of thermal velocity.  Velocity is proportional to (only) the square root of the temperature.

The air molecules (or ions) heated within lightning stroke are probably traveling pretty fast.

Does a small static electricity discharge cause high speed air?

Hydrogen molecules are light, so move faster at a given temperature compared to more massive molecules.  An unburnt hydrogen molecule in the vicinity of a hydrogen flame might be traveling 5800 m/s or 0.00002c.

Obsolete technology, part 2: An argon atom inside an incandescent light, having just bounced off the 5800K hot filament might be traveling 1000 m/s.

Gas molecules at room temperature range from 300 - 2000 m/s.

Next, physical objects:

A typical bullet travels 350 m/s.

A cracked whip exceeds the speed of sound, 343 m/s.

A commercial jet plane cruising at 600 mph travels 268 m/s.

A car going 60 mph travels 26.8 m/s.

Monday, October 24, 2016

[ldkoddtj] Liquid labor

Does society benefit by having liquidity in the labor market, i.e., people being easily able to quit their jobs and work for someone else?  Or is it purely an internal calculation: each worker should be permitted the freedom of entering into labor contracts that are easier or harder to quit with the market setting different wage premiums for different types of contracts?

Probably an illiquid labor market benefits established businesses and discourages innovation.

Consider regulation which penalizes businesses for practices discouraging their workers from quitting.  Or at least, such practices must be transparent and declared at the outset.

[oudyzksb] 2x2 checkerboard alphabet

Take a rectangle, pick a point inside, probably a lattice point, and divide the rectangle into 4 rectangles.  Color the four pieces black or white in a checkerboard pattern (two ways).

Probably consider starting rectangles of a few different heights.

Instead of coloring a quarter straight black, fill it in with a half circle (ellipse), quarter circle, or triangle.  Kerning possibilities.

Sunday, October 23, 2016

[vsbujnmz] Drug dogs

If there is anything a drug-sniffing dog really wants to do, it is to please its Master.  Thus the biases of its handler get reflected in the dog's discoveries or omissions.

Devise an experiment to measure this bias, then use it as basis to legally challenge, under violation Equal Protection, convictions based on evidence discovered by drug dogs.

There's probably plenty of bias in the decision to call in the drug dog.

Friday, October 21, 2016

[lczglumn] Gomboc notes

A gomboc is interesting only if it is accompanied by some proof that is homogeneous inside: no cavities or hidden weights.

The most common way to do this is to make it out of a transparent material.  Can this be cheated?  Create an object, perhaps a sphere, out of a transparent material which changes in density inside so exhibits self-righting properties like a gomboc.  It might need to have constant index of refraction throughout, though with a sufficiently weird shape, people might not notice the index of refraction smoothly changing inside.

Another way is to make it out of an extremely dense material.  The density can be confirmed by measurements of volume and mass.  If there is a cavity inside, then there would also have to be a plug of even denser material.  Such a plug would be impossible if the gomboc is made out of osmium or iridium (or an alloy of them both), the two densest materials, probably.  (Are their any compounds that are stable at standard temperature and pressure which are denser?  Is there proof that there aren't?)  Osmium, despite being a noble metal, oxidizes in air to form very toxic osmium tetroxide, so go with iridium.

Alternatively, make it out of platinum, then use the price as proof that it does not contain a hidden plug of osmium or iridium, which are exorbitantly expensive (compared to platinum).  Platinum is kind of soft, so we worry that the shape might be easily altered in an accident.

Similarly alternatively, make it out of tungsten.  Every material denser than tungsten is very expensive.  Unfortunately, regular pure tungsten is very brittle, and damage to a gomboc will destroy its balance properties.  Single-crystal tungsten is supposedly less brittle, and maybe extremely pure tungsten also, but those versions of tungsten probably approach prices similar to the more expensive, denser materials.  Tungsten is very hard to machine, so difficult to construct things out of.

Just how much would one have to cheat to create an object which practically behaves like a gomboc but which isn't one?  Create a sphere with a very small cavity or dense plug on one side and show how it rolls.

A still unsolved problem is designing a gomboc which has high tolerance for error in manufacturing.  (Tangentially, the AK-47 is notable as a machine gun with high tolerance for error in manufacturing yet still functioning well.)  Alternatively, some way to cheaply make objects of very precise dimensions.  Another unsolved problem is designing a gomboc that functions on as rough a surface as possible.

[hlxggeky] Zeta contours around the pole

What is the shape of the Riemann zeta function in the vicinity of its pole at s=1?  Circularly symmetric?  It shoots up to infinity, but how quickly?  This should be easy to investigate.  The questions might be slightly tricky because it is complex-valued.

Magnitude might be an interesting shape to 3D print, a funnel.

Invert the function centered at the pole, transforming it to a pole at infinity.

[mtwyzxzv] Vader correctly choking

Darth Vader seems to capriciously Force choke to death poorly performing subordinates.  Tell stories that these acts were not as capricious as they seemed: Vader had used his precognition to see that had these officers been allowed to live, perhaps demotion or other traditional military punishment, then the results would have been disastrous.

[ngatfdwd] No arrow of time in the far future

Sean Carroll https://youtu.be/9VFGuupXwng makes the neat point that we can distinguish between up and down in space because we are spatially near a large object in space, namely the Earth.  In deep space, we would not be able to distinguish between directions.  Analogously we can distinguish between forward and backward in time because we are temporally near a large object in time, namely the Big Bang, when there was very low entropy, and entropy has been noticeably increasing since then.

In the far future, 10^{at least 3 digits} years (though it is fun to write 10^10^10 just because), entropy will have increased almost to the maximum, and analogous to deep space, "we" will not be able to distinguish forward versus backward in time.  Predicting the future will be just as easy as remembering the past: things aren't changing.  It is hard to imagine life existing under such conditions, but who knows, life, uh, finds a way.

The laws of the universe won't have changed: time travel into the past will remain impossible.  However, if will be impossible to tell that you haven't broken the laws of the universe, that you haven't traveled backwards in time.

[eshzohcu] Understanding physics through aliens

If something is physically possible to do, then some alien civilization -- many -- will have already done it: look to the heavens for evidence of it being done.  If you don't see evidence of it, then it can't be done, perhaps due to difficulties we have not discovered yet.

Vaguely reminiscent of Rule 34 of the internet.

Problems with this reasoning: If it is physically possible for intelligent life to exist in this universe, then there should be plenty of astronomical evidence of its existence.  Some things may be possible, but might take billions of years or longer for an alien civilization to accomplish, but the universe isn't yet old enough for us to see evidence of their accomplishments.

Previously, tapping quasars.

[upwngaul] Game controllers as keyboards

The basic shape of game controllers, disregarding radical experiments by Nintendo, has not changed much in many years, suggesting it might be a near optimal shape for a handheld input device.

The thumbs do most of the work, plus a few "shoulder" buttons for the other fingers.  Even the radical Wii controllers follow this pattern.

They are obviously very different in shape from a typical keyboard, having fewer total buttons, but people may be able to learn to type on a controller with fewer buttons with combinations or chords.

Thursday, October 20, 2016

[sunifufr] Function from zero crossings

Given a list of zero-crossings on the X axis, construct a wavy function which is zero at only those points.  (Need to say "only", or else we can simply construct the zero function.  Or perhaps those are the only zeros inside the interval defined by the list; there may be further zeros outside the interval.)  This is an underconstrained problem.  Construct a "nice" function, for some definition of nice.

Splines are the obvious first idea.  An analytic function (infinitely differentiable) would be nicer.  Low order polynomial that has "large" absolute value between zeros but low maximum absolute value on the interval.

What if the list of zero-crossings is infinite, e.g., Riemann-Siegel Z function, or sine, or cosine?  Construct a sum of sines and cosines which, when truncated, get the zeros approximately correct, for some definition of approximate.

Original inspiration, the zero crossings of sin(x) sin(m*x) where m is an irrational number, have no periodic pattern.  The zeros are where a line of irrational slope intersect a regular grid of orthogonal lines.

Wednesday, October 19, 2016

[iyuepsrr] Ultimate Lorem Ipsum

Provide a random excerpt of the original Cicero text (De Finibus), looping back to the start if necessary.  This should be easy.

[esvcmzuy] Hemispherical jigsaw puzzle

Start with a hemisphere with a picture on it.  Partition it into regions, then make pyramids (with bases with spherical caps) by extruding (intruding) the regions to the center.  If the pieces are designed to fit together in only one way, then it can be a jigsaw puzzle, even having shapes that stay together if correctly matched (e.g., dovetail).

Two hemispherical puzzles (the pieces of both jumbled together) could, after both being solved, be joined together into a sphere.

All the pieces could be shaped the same, but this is much harder for a hemisphere than for a sphere.  Maybe a special base, though that will give away most of the challenge of figuring out how the pieces are supposed to fit.

Trim off or round off the tips of the pyramids to make the pieces less dangerous to accidentally step on.

[yidzlgbu] Isohedral pyramids

Any convex polyhedron can be decomposed into a collection of pyramids (pyramid base = polygon face, pyramid apex = polygon interior point).  For an isohedron with apex at center, all the pyramids are congruent.

Might make for a nice puzzle, easy to construct because all the pieces are identical.  Maybe round off the tips to make it less dangerous to accidentally step on.

[cjvvgoqf] Music player filesystem

Simplest is 1 "program", a bunch of songs in order.  Possible UI actions: play program in order from the beginning, go forward/backward one song, repeat program, repeat one song, shuffle program.

This was the model for the iPod shuffle that did not have a display.  It is similar to physical media formats: tape, records, CDs (though the latter 2 do have rudimentary random access).

Next is multiple programs: play the default (first) program, go forward/backward one program, repeat all programs, repeat one program, repeat one song.  The same song could be included on multiple programs; the device should deduplicate.  A program could be an album or playlist.

Is two levels or hierarchy too complicated to navigate without a display?  More levels?

Filesystem features required: symbolic links for deduplication, ordering of files and directories, metadata.

Inspiration was the lack of a standardized interface to play a music library stored on an external device on a car stereo, using an interface built into the car.

Tuesday, October 18, 2016

[umqmxbar] Numbers whose factorial is just less than a power of 2

3 5 22 28 48 63 64 90 536 548 557 2689 6627 35053 49692 55139 2418263 9519742 18582322 19015320 43430008 43578373 45601897 103481702 455189228 760006976 1855572486

Pari/gp:
s=0; m=0; for(i=1, 10^100, s+=log(i)/log(2); q=s; f=q-floor(q); if(f>m, m=f; printf(" %d",i))); print("")

64 was artificially inserted into the sequence; it is the same as 63.  64! = 0.9966 * 2^296 = 0.9966 * 256^37.

If one only has a uniform random bit generator, permutations of this many items can be generated easily, with high probability (low probability of rejection with rejection sampling).  In fact, looking at the high order bits when they come out are enough to (usually) quickly know to reject.

Going the other direction: 3 6 7 11 15 20 24 62 65. Stopping at 81 because it's hard to imagine someone with a good random permutation generator larger than shuffling a deck of 81 Set cards.

How can the above record sequences be computed (extended) quickly?  There seem to be many algorithmic tricks possible.

Monday, October 17, 2016

[seuezdxy] My Little Inferno

Create an alternate ending to the My Little Inferno video game which makes the point that the player just burnt away a chunk of their life playing the game.  Inspired by the Yogscast playthrough commentary.

[piurrvti] Zeta calculator

In this video, the interviewer asks "If I were to enter into a calculator, 1 plus 2 plus 3 plus 4 and so on out to infinity, and then hit equals, would I get the answer minus 1 twelfth?"  The answer is of course, what does it means to do that "out to infinity?"

An interesting alternative valid question is, "If I were to enter -1 into a calculator, and then hit the zeta function button, would I get the answer -1/12?"  The answer is, what kind of calculator provides the zeta function?  The zeta function is a little bit more complicated to compute than, say, sine, but not too bad.  It's not considered useful enough to provide, though perhaps if it were, it would stimulate more interest in analytic number theory.

[esojjvln] Single elimination blitz

To make a chess tournament attractive to many people entering (as an alternative to segregating by class or rating), do things that increase variance -- increase the probability that a low-rated player can win or do well in the tournament.  Single games of blitz has higher variance.  Single-elimination tournaments have high variance.  Chess has draws, so not exactly suited for single-elimination; perhaps Swiss permitting repeated encounters until a decisive result occurs.  Tournaments with only a small number of rounds have high variance.

Inspired by poker and bridge, which have variance built into the game by being a game of incomplete information.

No matter what the structure, monetary prizes have to be carefully structured, or else there will be incentives for people to pay people off to throw games.  Every game must be zero-sum in terms of prizes or expected prizes (though this will induce many draws due to risk averseness).  For a single large grand prize, those still in contention for the grand prize must only play each other: there might be some byes due to draws causing an odd number of people to be in the money.

[ndvgnkvu] Porn blocker as history and cache blacklist

Repurpose the list of sites in a porn blocker or other parental control software as a blacklist of sites not to cache or store in browser history.  Because the list of sites is public -- included in the publicly available porn blocker -- the user can plausibly deny even knowing about the sites, which could not be done if the user had to hand-create a blacklist.

[wbcjkvuu] Look to zoom

How pleasant a UI is it to show a large image in a virtual reality environment, then use head tracking to choose where to zoom?  The zoom center probably slowly drifts toward the center so the user does not have to keep head cocked constantly, but that means the head has to track this drift.

Anything that requires pointing is equivalent to mouse on a regular display.

[houaffqc] Chess960 hiding 1 move

Modify Chess960 very slightly so that the initial position is not revealed to black until after white's first move.

White's clock starts running as soon as white sees the initial position.

[rmosscuf] Human CA

Create a demonstration with a large number of people, each doing a very simple task, resulting in very complicated global behavior.  Famous is cellular automata, though it would be nicer to have something more robust to people making mistakes.

[bdqrodel] Crazyhouse limited drops

Variations on the drop rule for crazyhouse chess (or shogi):

Pieces may only be dropped on the original position (two possible places for rook, knight, bishop, 1 place for queen, 8 for pawn).

Pieces may only be dropped on their original rank (1st or 2nd rank).  Further possibility: some (maybe all) pieces start in hand, to be dropped on their original rank whenever the player feels it fit to do so.

Pieces may only be dropped on squares not attacked by the opponent.

Pieces may only be dropped on squares controlled (attacked) by the player.  (Previously explored.)  Further possibility: pieces may be dropped directly onto an opponent's piece, capturing it.

[uitadwod] Stackage

Stackage for Haskell packages has the curious behavior that packages can disappear from it even though they were perfectly fine.  The cause of such a disappearance of say a package B is as follows: package B was originally pulled in as a dependency of another package A, and the maintainer of package A quit, so package A and all its dependencies, including package B, are candidates to be removed from Stackage.  Package B survives only if it has a direct maintainer in Stackage or is a dependency of another maintained package.

Inspired by the many packages that got dropped when lambdabot got removed from Stackage nightly, e.g., brainfuck.

Although the stated goal of Stackage is a curated collection of Haskell packages, each with an explicit maintainer willing to fix bugs and compilation problems (e.g., with new versions of GHC), I have found that a side feature is more useful: the identification of a large mutually compatible collection of packages without version dependency problems.  Such a side feature -- such a collection -- could be computed automatically without having to have a direct or indirect maintainer for each package in the collection.  I wish such a larger collection existed.

Start with, say, Stackage Nightly and expand it to include every package in Hackage that compiles cleanly and is compatible with Stackage Nightly and with every other package in the expanded collection.  There may be tricky cases of mutually incompatible packages in a potential expanded set which will need to be resolved, e.g., the newest version of A requires an old version of B, and the newest version of B requires an old version of A.  Perhaps resolve such conflicts in favor of the choice which causes the expanded set to be as large as possible.

Tangentially, how can one safely build a package (to test whether it compiles cleanly) if one is not sure whether a package's build script is evil?  Probably some kind of operating system container or sandbox.  Identify packages which use simple, presumably safe, build mechanisms, probably pure Haskell, versus packages which do something unusual, e.g., call a Makefile, which ought to be scrutinized before building.  (Inspired by a build script of software, I think maxima computer algebra, which creepily attempted to send email back to the author every time it was compiled.)

Can compiling a carefully crafted source file with GHC allow the author of the source file to perform arbitrary user-level actions within the operating system?

Friday, October 14, 2016

[cqvfuwdi] Decisive Swiss

Modify Swiss pairing for chess tournaments so that two players who have played already can meet again if all their previous meetings were draws.  Helps later when using head-to-head result as a tiebreaker.

Thursday, October 13, 2016

[cozkvurf] Unabomber math

Tell a story of the Unabomber sitting in prison working on proving the Riemann Hypothesis.  Not too unrealistic: his Ph.D. was in analysis.  The story is, this was always his endgame: he knew he might be spending the last few decades of his life in prison, and he knew he could do something productive and satisfying with his time there.

Tell a story of someone else, not yet in prison but considering embarking on a project very likely to result in a long prison sentence, studying up on math beforehand -- maybe the Clay Millennium problems -- to prepare for eventually doing time.

It would be politically uneasy if a prisoner, especially an especially bad one such as the Unabomber, were to achieve positive renown for doing something like proving the Riemann Hypothesis while imprisoned.  Therefore, we expect prisons to be set up to make such work difficult: what are those things?  Ends up being a roundabout restriction on speech.  Study whether such roundabout restrictions on speech also occur outside of prisons.

A vindictive society wants to make sure people in prison suffer, not be able to do anything pleasant, even if thinking and writing about math should be pleasant.  Should vindictive society get its way?

[dmvmqfvo] Change from within and without

Attempting to cause change from within: working within the system.

Attempting to cause change from without: disrupting the system, e.g., protests.

Which succeeds when (if ever)?  Why?

When does both being done simultaneously synergistically help each other?  When does both being done simultaneously impede each other?  What causes the difference?

Probably matters: zero-sum versus positive-sum.

Pessimistic hypothesis: change from without never succeeds, always impedes.  We almost suspect conspiracy-theory-style social manipulation for the social forces that cause it to be attempted (often) despite it being counterproductive.  Inspired by identity politics driving wedges, causing groups to want to or be willing to hurt each other more.

[pgmjypdp] Lens Flare The Force Awakens

Create versions of JJ Abrams Star Wars movies adding egregious lens flares everywhere.

Wednesday, October 12, 2016

[yvxbdfoi] VR flashcards

An evolution of technology, for example, for learning foreign language vocubulary.

Foreign word - English word
Foreign word - picture
Foreign word - animation or movie
Foreign word - 3D still scene or 3D movie

E.g., learning names of vegetables: get a sense of scale, turn it around to see it from different angles.

Maybe gross anatomy medical education.

[gwpxocih] More apostrophes

Take a string of letters with apostrophes, substitute the regular expression [a-z]+ for each apostrophe.  If there is a unique English word that matches that pattern, then let the contracted form with apostrophes be a valid contraction of that word.  Currently existing contractions are grandfathered in, despite violating this rule.  Maybe we need an alternate character that is not an apostrophe.

(Tangentially, there seem to be very few current cases where leaving out an apostrophe leaves a valid sentence or fragment with a different meaning.  Maybe apostrophe-s versus s-apostrophe versus s-alone for possessive singular versus possessive plural versus adjective (maybe name) ending with an s.  It is ironic that apostrophes are what people are sticklers about; they are almost purely decorative -- vestigial.)

(Another tangent: there is ambiguity between apostrophe-s meaning "is" or possession: Bob's back.)

Perhaps forbid apostrophes in the first or last position.

What contractions will exist?  List them in order of word frequency (and probably stop before the words become too obscure: trying to figure out an obscure word with missing letters is too hard.)

Finding these contractions seems computationally difficult.

[gxpgqixq] Notes on higher dimensional mazes

Consider two different types of 2D mazes on a grid of squares:

Some of the squares are black (indicating blocked); other squares are white (indicating passage).  Call this type of maze a "maze of blocks".

All the squares are white, but there are infinitesimally thin walls preventing passage between certain adjacent squares.  Call this type of maze a "maze of walls".

Depicting a 3D maze of blocks in 2D is fairly easy: just show slices.

A 4D maze of blocks is similarly easy.  A MxNxPxQ maze could be depicted as a PxQ array of MxN slices.  It could also be depicted as a MxN array of PxQ slices.  These different depictions are interesting when two of the dimensions are small, 2 or 3.

Consider a MxNx3x3 maze of blocks.  First, depict it as a 3x3 array of MxN mazes.  One can determine whether one can move along the 3rd or 4th dimension by looking at the corresponding square on an adjacent MxN maze.  This can be annoying or time consuming if the MxN maze is large.  Add annotations to each square in the MxN maze showing which 3rd and 4th dimension steps are possible: thin colored rectangles.  The ultimate goal is to provide enough information so that a person can form a model of the maze (or local parts of it) in their heads (interestingly, a 4-dimensional model) and solve chunks of it in their heads.

I have seen a 3D maze of walls depicted in slices, with certain squares in each slice marked with a small black square indicating a hole down or a larger thin ring indicating passageway up the ceiling.  Could be generalized to 4D with the same per-square annotations described above.

***

The simplest maze structure is a tree: there exists a unique path between any pair of nodes.  Make the maze a little bit harder (or easier, depending on the point of view) by adding loops.  The obvious starting idea is just one loop, and lots of trees hanging off of it.  What next?

In 3 dimensions and more, there are no constraints on graphs that can be embedded in a maze: there are no problems with paths needing to cross like in 2D.  Other loopy graphs: tetrahedron and higher simplices (equivalently complete graphs), torus grid of 2 or more dimensions, hypercubes, cross polytopes.  Cubic graphs, including snarks, are neat.  Cross polytopes and simplices have very small diameter.  Which loopy graphs are interesting for mazes?

***

Consider a maze of blocks of N dimensions, where N is quite large, but the maze has width 2 in all dimensions.  (Maze total volume is 2^N.)  Is it difficult?  How can one depict it to aid solving?

For a GUI, N switches denoting the coordinates, a light above each switch signifying whether changing that coordinate is permitted (not a solid block).  Maybe not just a light but the name of the room (block).

What interesting loopy graphs above can be embedded in a 2^N maze of blocks?  If not 2^N, then maybe 3^N offers enough space.

A 2^N maze of walls might lose most its geometric aspect; it might as well be an abstract graph.

[whsrvxdi] Spiral around a sphere

Draw a spiral from pole to pole on a sphere, then place points uniformly along the spiral.  This is one quick and dirty way to scatter points on a sphere avoiding points close to each other.  The best spiral probably has the latitude spacing between turns equal to the distance between adjacent points on the spiral.  Some messy details still need to be worked out.

[qgbcnfov] up dn ambigram

Lowercase up and dn (abbreviation for "down") can be flipped to read the other word.  Also arrow pointing up.

[ugawzrcp] Entropy loss in repeated hashing

How much entropy is lost when iterating a hash function?  We expand on this answer:

http://crypto.stackexchange.com/questions/15068/entropy-when-iterating-cryptographic-hash-functions

The probability a given bucket is occupied (gets hit), or equivalently the proportion of buckets occupied, when there are n buckets and f[i]*n trials is

f[i+1] = 1- (1 - 1/n)^(f[i] *n)
= 1 - z^f[i]
where z = (1 - 1/n)^n
n = 2^b where b is the width of the hash, e.g., b=128 for MD5.
f[0] = 1

The quantity z could be approximated as exp(-1) for large n, and the whole expression could be approximated as f[i+1] = 1-exp(-f[i]), but we do not have to make these approximations.  Evaluate z directly with arbitrary precision arithmetic to avoid round-off error.

Similarly, the recurrence could be approximately solved as a closed-form function of its index, but we do not have to approximate.  Instead, explicitly iterate the recurrence however many times you were planning to iterate the hash function.  This computation will take roughly the same amount of time that the iterated hash function iterations would have taken.  The recurrence computation can be done with double-precision floating point, or with arbitrary precision arithmetic if you are worried about round-off error.

(Aside: the first iteration loses approximately 0.6617 bits of entropy.  If we naively assume the amount of entropy lost each iteration is the same, then SHA-512 would converge to a fixed point after 774 iterations.  This intuition is false, because as the number of trials decreases, we are less likely to lose filled buckets due to collisions.)

Finally, evaluate the bits of entropy lost as -log(f[i])/log(2).  It may be instructive to choose the number of iterations to be a power of 2: i = 2^j.  Experimentally, I confirm that the answer about j-1 bits.

Note well, this is still an approximation because the it does not account for the hash function being the same each iteration, so that values can go in a loop.

Aside: run Floyd's cycle-finding algorithm on various hash functions and other cryptographic primitives to see if short cycles can be found.

Tuesday, October 11, 2016

[ssgyfepw] Tree digit separators

Insert different separators recursively (like a binary tree) between digits of a long digit string to make it easier to read.  Here we demonstrate one possible collection of separators on the SHA-512 hash of the empty string.

cf 83.e1 35:7e ef.b8 bd ** f1 54.28 50:d6 6d.80 07
d6 20.e4 05:0b 57.15 dc ** 83 f4.a9 21:d3 6c.e9 ce

47 d0.d1 3c:5d 85.f2 b0 ** ff 83.18 d2:87 7e.ec 2f
63 b9.31 bd:47 41.7a 81 ** a5 38.32 7a:f9 27.da 3e

The separators get fancier further up the tree. "nothing", space, period, colon, double-asterisk, line break, paragraph break, (to be used later) line of dashes, line of equal signs in its own paragraph.

Alternatively, we could repeat the smaller separators each time, building up a little pyramid.  This time we avoid newline separators but add some different ones: cf 83 . e1 35 .,. 7e ef . b8 bd .,:,. f1 54 . 28 50 .,. d6 6d . 80 07 .,:':,. d6 20 . e4 05 .,. 0b 57 . 15 dc .,:,. 83 f4 . a9 21 .,. d3 6c . e9 ce .,:'**':,. 47 d0 . d1 3c .,. 5d 85 . f2 b0 .,:,. ff 83 . 18 d2 .,. 87 7e . ec 2f .,:':,. 63 b9 . 31 bd .,. 47 41 . 7a 81 .,:,. a5 38 . 32 7a .,. f9 27 . da 3e

With groupings of 3 digits instead of 2, this is isomorphic to "extremely long scale".  Below is a 768-digit number between a "septillion" and an "octillion".

123 456.789 012:345 678.901 234 ** 567 890.123 456:789 012.345 678
901 234.567 890:123 456.789 012 ** 345 678.901 234:567 890.123 456

789 012.345 678:901 234.567 890 ** 123 456.789 012:345 678.901 234
567 890.123 456:789 012.345 678 ** 901 234.567 890:123 456.789 012
------------------------------------------------------------------
345 678.901 234:567 890.123 456 ** 789 012.345 678:901 234.567 890
123 456.789 012:345 678.901 234 ** 567 890.123 456:789 012.345 678

901 234.567 890:123 456.789 012 ** 345 678.901 234:567 890.123 456
789 012.345 678:901 234.567 890 ** 123 456.789 012:345 678.901 234

==================================================================

567 890.123 456:789 012.345 678 ** 901 234.567 890:123 456.789 012
345 678.901 234:567 890.123 456 ** 789 012.345 678:901 234.567 890

123 456.789 012:345 678.901 234 ** 567 890.123 456:789 012.345 678
901 234.567 890:123 456.789 012 ** 345 678.901 234:567 890.123 456
------------------------------------------------------------------
789 012.345 678:901 234.567 890 ** 123 456.789 012:345 678.901 234
567 890.123 456:789 012.345 678 ** 901 234.567 890:123 456.789 012

345 678.901 234:567 890.123 456 ** 789 012.345 678:901 234.567 890
123 456.789 012:345 678.901 234 ** 567 890.123 456:789 012.345 678

We can also experiment with a ternary tree where each separator, including "nothing", is used up to twice in a row.  (We again change up the sequence of separators.)  Here is a 243-digit number: 123 456 789 . 012 345 678 . 901 234 567 .:. 890 123 456 . 789 012 345 . 678 901 234 .:. 567 890 123 . 456 789 012 . 345 678 901 .:*:. 234 567 890 . 123 456 789 . 012 345 678 .:. 901 234 567 . 890 123 456 . 789 012 345 .:. 678 901 234 . 567 890 123 . 456 789 012 .:*:. 345 678 901 . 234 567 890 . 123 456 789 .:. 012 345 678 . 901 234 567 . 890 123 456 .:. 789 012 345 . 678 901 234 . 567 890 123

Not specified is what to do if the binary or ternary tree is not full.

Sunday, October 09, 2016

[hmnfenka] Dreams

Hypothesize that one of the reasons humans dream during sleep is to stabilize mental health: the brain generates happy dreams that seem real, for sad people.  The temporary happy "reality" causes hormonal and other beneficial physiological effects.

[fyvtfvvq] Some distributed technologies

Some of these are also sometimes labeled "federated".

Hashcash
Pump.io
Jabber
RSS, PubSubHubbub
Email
PGP / GnuPG (GPG)
Tor
Freenet
Bitcoin

[fliglzph] Making sex unenjoyable

Sexual assault and rape sometimes cause long-term trauma which makes having sex unpleasant.  People who find sex unpleasant are less likely to do it, and consequently less likely to have children.  Assuming controlling sex is one of the most important functions of society, we would expect to see sexual assault and rape institutionally deployed to accomplish this function (kind of horrifying).

Military victory is often followed by the invading army raping the conquered locals.  This might be purposeful to decrease the population growth rate of the conquered, allowing to conquerors to gain a population advantage, especially useful if the conflict lasts generations.

Are there other instances of this happening?

Sexual assault causes long-term trauma only sometimes, only under certain conditions (though we have not yet exactly identified those conditions).  A society or subculture might be engineered so that people deemed less desirable to breed encounter those trauma-inducing conditions more frequently, and then a background rate of sexual assault, causing trauma in some but not in others, causes the differential rate of having sex and consequently having children.  For example (perhaps this is more general than just an example): sexual assault might cause trauma only in people lacking a strong social support network.  A strong social support network is granted only to those who abide by the rules and membership requirements of the society or subculture; those who do not are shunned and live as lonely outcasts and suffer traumatic consequences.

[kuqjbyhj] Desirable features for a shell

Features of bash I use.  (Browsing through the manpage of bash.)  Motivation was thoughts of creating a better command-line shell, perhaps with a more regular syntax or more features, but a better one will have to retain old useful features.

^C
^L
^Z & bg fg wait "time wait" "wait -n" $! jobs
readline, history
pipes
redirection noclobber >|
interpolation (backquotes)
PATH
environment variables for child processes
subshells (parentheses) usually then piped
completion (tab)
glob expansion
cd pushd popd dirs pwd CDPATH
set -e -x "-o pipefail" -u
programming: control flow, local variables, functions
exit status && ||
prompt customization

[qmxwfkbd] Bullying the different

The common, perhaps universal, pattern of bullying is that "the different" are targeted.  Three possible evolutionary mechanisms, attempting to answer a previously asked question.

1. Bullying tests whether someone is in the tribe, to discover whether they can be trusted.  How would one pass the test?  This could be taught.

2. Someone has already been identified as not part of the tribe, someone who has allegiances elsewhere so cannot be trusted.  Bullying serves to expel them from activities and benefits of this tribe.

3. Bullying may serve as incentive to abandon their old tribe and maintain allegiance to the new one.

Attempts to regulate bullying seem doomed to fail, as they seem to be trying to regulate human nature.

People are different in many ways.  Which differences are selected for bullying, and which are not?  Why?  Assuming correct #2 above, it would be the differences, especially behavior, that signify tribe membership, whether someone can be trusted.  These behaviors might be subtle, and may only happen to correlate with the popularly believed reasons for bullying (e.g., race, gender, body shape, sexual orientation).

If someone is selected for bullying, how accurate are the bullies' determination that they are not part of the tribe, someone who cannot be trusted because they have allegiances elsewhere?  I suspect it is frighteningly accurate, but if not, this could be taught to be more accurate, though such teaching is politically incorrect.

[uakrhrvm] Questions about Rat Park

Why aren't there more replications?  Maybe negative results aren't getting published.

Are there political difficulties in getting funding to attempt a replication?  Political difficulties in publishing a confirmation (positive result)?

Is it a particularly difficult experiment to perform?  Perhaps needing to obtain and handle a controlled substance.

I suspect there are (or will be) some positive and some negative results in replications.  What determines the outcome?  Are some rat stimuli more effective than others at preventing or breaking addiction?  What neurological mechanisms are involved in the rat that cause the positive and negative results?

For which drugs is Rat Park effective for breaking addiction?  What dosages of the drugs, i.e., how addicted?  Which animals beyond just lab rats?

I suspect experimenter bias has a strong effect on the outcome.

Saturday, October 08, 2016

[myfnohif] Infinite cross bridges

Draw a square.  Divide it into thirds horizontally.  Take the middle rectangle and divide it into thirds vertically, yielding a small square in the center 1/9 the area of the original.  Recursively repeat infinitely on this center square.  Place electrical resistors on all segments and calculate the resistance between two corners of the original square.

Draw a square.  Draw a smaller square inside it.  Connect the corresponding vertices of the two squares.  Repeat.

Latter is equivalent to a prism with a square grid of resistors on its surface.  We can imagine other infinitely long thin geometric shapes: stack of octahedra.

[exgbexzz] Adversarial truth discovery

An adversarial framework seems like it should work well for discovering truth: each side challenges the other's version of the truth, causing lies and manipulation to be revealed.

We see this model in science, in the justice system, in political campaigns in democracies, in journalism.  However, these examples cynically or pessimistically demonstrate that reality is almost diametrically the opposite of theory.  It sometimes works in science (but often not: "cargo-cult science"), and both the justice and political systems are about who can bullshit better.  Journalism, motivated by advertising revenue, publishes the entertaining version of the story, not necessarily the true one.  In all these fields, effort seems better spent trying to take advantage of fallibility in human reasoning than to actually discover truth.

We imagine an alternate universe in which adversarial truth discovery does work well in all these fields.  It would have an interesting side effect: transcripts of jury trials and propaganda from a political campaign would be extremely good educational resources afterward.  Suppose unemployment were an important political issue between two parties.  As the sides debate and try to convince voters, both sides need voters to understand how the labor market functions and how domestic and macro economics work so that voters can evaluate proposed policy differences between the two parties.  Therefore, both sides will produce very good, easy to understand, educational material on these topics.  If the educational material is bad, confusing, the voter might come to the wrong conclusion and vote for the opponent, so there is tremendous incentive for the educational material to be good, and for the parties to attack opponent's bad educational material.  Compare this to the incentives (or disincentives) to create good educational material in the traditional education system.

Perhaps someday computers can help.

[odunhugu] Inclusive counting in music

Inclusive counting, not typically used in (American) English, is still present in musical terminology.  An interval of 7 notes higher than the base note of a scale is called an octave.  1 note higher is called a (major or minor) second.  6 octaves has nothing to do with the number 6*8 = 48; instead 6 octaves is 42 notes.

[ycnkrkrd] Replicate Riemann's zeta calculation

Replicate Riemann's 1859 hand calculation of the first few nontrivial zeros of the zeta function.  This seems formidable.  Feel free to use log tables and any other resources Riemann might have had at the time.

[jiekvsvp] Frobeniusly strong Lucas pseudoprimes

We compute composites (pseudoprimes) which pass the strong Lucas primality test with P and Q chosen by Selfridge Method A, and additionally satisfy the Frobenius-like criterion V[n+1] = 2Q (mod n) mentioned in Baillie's original paper, which can be tested almost for free.

There are 304 such pseudoprimes less than 10^9, and 757 less than 10^10.  The corresponding numbers for the strong Lucas pseudoprime test without the above Frobenius-like criterion are 1415 and 3622, so this almost-free test improves things significantly. (See OEIS A217255 and http://ntheory.org/pseudoprimes.html). These Frobeniusly strong Lucas pseudoprimes are a subset of the strong Lucas pseudoprimes.

Below are the 304 pseudoprimes less than 10^9.  Further pseudoprimes here.  Source code will be posted later.

5777 10877 75077 100127 113573 161027 162133 231703 430127 635627 851927 1033997 1106327 1256293 1388903 1697183 2263127 2435423 2662277 3175883 3399527 3452147 3774377 3900797 4109363 4226777 4403027 4828277 4870847 5208377 5942627 6003923 7353917 8518127 9401893 9713027 9793313 9922337 10054043 11637583 13277423 13455077 13695947 14015843 14985833 15754007 16485493 16685003 17497127 19168477 19347173 20018627 22361327 23307377 24157817 25948187 27854147 29395277 29604893 30299333 31622993 31673333 32702723 33816593 34134407 34175777 36061997 39240233 39247393 39850127 40928627 41177993 42389027 42525773 47297543 49219673 49476377 50075027 51931333 53697953 57464207 59268827 62133377 64610027 67081607 67237883 69244097 70894277 73295777 73780877 74580767 75239513 75245777 75983627 83241013 83963177 85015493 85903277 86023943 87471017 89746073 90686777 91418543 93400277 98385377 100981997 104943827 106728053 110734667 116853827 117772877 122879063 124477513 131017577 131990627 136579127 139904627 140782823 142593827 143221993 144967877 146278373 148472347 150204577 153256277 154308527 156715343 157132127 158197577 163578827 166850777 168018353 171579883 177991277 179295443 184135673 185504633 186003827 192227027 196754513 202368143 207023087 210089303 211099877 213361937 226525883 229206347 231437957 247030877 247882963 253755053 254194877 257339503 257815277 259179527 264250367 264689963 276795217 277932113 280075277 284828777 290256947 293485877 306219377 311387693 312189697 316701527 320234777 320297657 334046627 344107133 347522647 347547437 360783793 370020797 375578683 376682627 384646597 386628527 387009737 400091327 400657277 401790377 403675973 409245563 420717527 429992597 432988877 437118527 438894377 439744397 440964593 443146057 443969063 448504697 450825377 455039027 456780193 461700077 461807147 461819483 464407883 465523103 465964127 467486627 468245207 469721647 475167377 480053573 480891143 481033907 485326403 495101777 500337713 504097397 523827527 540136277 543893783 544339637 546540347 552576653 558030527 562046627 563601257 570122027 574181327 577647017 582182327 582934477 583031693 584238563 590901317 595402877 598147577 598364773 623709217 628919117 634888253 638227127 640151777 649354103 657333797 657665777 659936423 664939277 667595897 670042903 670786877 672810497 686258627 688773443 691455077 691593047 692726473 704907377 706840573 709706567 713971187 721652587 727615877 729645563 731349233 734498627 747587777 756310507 768614027 771325967 772719947 773515133 780421277 788342777 797102627 798790787 799500077 800484227 807775547 811541327 812957903 825393997 831065633 838472417 839350363 847053323 847887823 856901267 863097377 869420473 873933527 878330573 922483693 923962577 930039743 940801877 947756267 957697997 961095923 961931213 969210377 978920627 979805777 983720077 985125077 986079553 988367813

[wdgqdnby] Riemann zeta sound

In "Prime Number Races" by Andrew Granville and Greg Martin is an interesting formula involving the Riemann zeta function, reexpressed below in Haskell-style notation for mathematics:

\x -> 1 + 2 * (sum $ do { s <- rootOf zeta ; guard $ real_part s == 0.5 ; let { gamma = imag_part s} ; guard $ gamma > 0 ; return $ sin (gamma * log x) / gamma})

It is the sum of sine functions, like a Fourier transform.  This suggests synthesizing a sound using the same coefficients, the imaginary part of the roots of zeta along the critical line.  The first root is at 14.13, so we scale down every root by that factor yielding the sequence 1, 1.49, 1.77, 2.15,...  Then, take logarithms base 2^(1/12) to express the frequency ratios of the higher harmonics in musical semitones.  Here are the first 100 intervals in semitones:

0 6.87 9.88 13.27 14.64 16.93 18.40 19.39 21.17 21.79 22.87 23.97 24.84 25.27 26.44 26.96 27.58 28.20 29.05 29.38 29.87 30.63 31.00 31.55 31.82 32.52 32.92 33.14 33.67 34.10 34.51 34.79 35.07 35.68 35.81 36.19 36.48 36.85 37.23 37.45 37.63 38.08 38.36 38.56 38.87 39.04 39.46 39.66 39.84 40.08 40.42 40.59 40.90 41.00 41.24 41.58 41.75 41.88 42.14 42.33 42.60 42.77 42.97 43.05 43.40 43.54 43.70 43.89 44.04 44.26 44.51 44.58 44.73 44.93 45.17 45.26 45.46 45.60 45.70 45.98 46.09 46.23 46.33 46.54 46.68 46.86 46.99 47.09 47.22 47.45 47.58 47.63 47.83 47.91 48.10 48.24 48.39 48.44 48.57 48.78

Two potential issues:

The amplitude of each harmonic properly should be scaled by the reciprocal of gamma, as in the formula above.  However, the human ear is has variable sensitivity across the frequency spectrum, so it might be better to allow the listener to individually control the amplitude of each harmonic to explore different aspects of the music of the primes.  Let the listener also control the fundamental frequency and apply a band-pass filter, in order to hear different segments of harmonics.

The harmonics get more and more densely packed the higher one goes, in contrast to regular Fourier transforms, in which the harmonics are equally spaced in frequency.  For example, there are 5888 harmonics at 147 semitones above the fundamental (rounded to the nearest semitone), corresponding to the 88075th through 93962nd zeros.  (We used this table of Riemann zeta zeros.)

One idea is not to worry and synthesize the sound anyway.  Harmonics get somewhat densely packed even for regular Fourier transforms when viewed in the logarithmic semitone scale: there are 281 harmonics at 147 semitones above the fundamental, corresponding to the 4733rd through 5013th harmonics.

Another idea is to rescale the coefficients (roots) so that they are roughly equally spaced in frequency.  The spacing of the zeros along the critical line is related to Gram points and the Riemann-Siegel theta function.

Previous work on the sound of the Riemann zeta function.

Friday, October 07, 2016

[abymulnr] Primes in the neighborhood of powers of two

Prime numbers in the vicinity of "nice" powers of two.  The exponent is itself of the form 2^n, 3*2^n, or 5*2^n.  The larger primes were discovered with Pari/GP.  The gap between 2^65536-5627 and 2^65536-99267 took about 2.8 days.  2^65536 = 2^2^2^2^2.

Primes were verified additionally with 20 iterations of Miller-Rabin, with randomly chosen bases.

Search ranges: 2^2^15 -149000 +227000, 2^2^16 -154000 +163000.  For others, the search stopped at the last number.

Related OEIS: A058220 A058221 A129786.

Previously: Smaller numbers, Twin primes

2^1 +0 +1 +3 +5 +9 +11 +15 +17 +21 +27

2^2 -1 -2

2^2 +1 +3 +7 +9 +13 +15 +19 +25 +27 +33

2^3 -1 -3 -5 -6

2^3 +3 +5 +9 +11 +15 +21 +23 +29 +33 +35

2^4 -3 -5 -9 -11 -13 -14

2^4 +1 +3 +7 +13 +15 +21 +25 +27 +31 +37

2^5 -1 -3 -9 -13 -15 -19 -21 -25 -27 -29

2^5 +5 +9 +11 +15 +21 +27 +29 +35 +39 +41

2^6 -3 -5 -11 -17 -21 -23 -27 -33 -35 -41

2^6 +3 +7 +9 +15 +19 +25 +33 +37 +39 +43

2^8 -5 -15 -17 -23 -27 -29 -33 -45 -57 -59

2^8 +1 +7 +13 +15 +21 +25 +27 +37 +51 +55

2^10 -3 -5 -11 -15 -27 -33 -41 -47 -53 -57

2^10 +7 +9 +15 +25 +27 +37 +39 +45 +63 +67

2^12 -3 -5 -17 -23 -39 -45 -47 -69 -75 -77

2^12 +3 +15 +31 +33 +37 +43 +57 +61 +63 +81

2^16 -15 -17 -39 -57 -87 -89 -99 -113 -117 -123

2^16 +1 +3 +7 +15 +21 +27 +43 +45 +51 +63

2^20 -3 -5 -17 -27 -59 -69 -129 -143 -153 -185

2^20 +7 +13 +25 +33 +37 +51 +57 +85 +105 +127

2^24 -3 -17 -33 -63 -75 -77 -89 -95 -117 -167

2^24 +43 +73 +75 +115 +117 +121 +165 +205 +225 +231

2^32 -5 -17 -65 -99 -107 -135 -153 -185 -209 -267

2^32 +15 +61 +75 +81 +91 +93 +163 +181 +201 +217

2^40 -87 -167 -195 -203 -213 -285 -293 -299 -389 -437

2^40 +15 +27 +55 +97 +115 +141 +157 +177 +253 +277

2^48 -59 -65 -89 -93 -147 -165 -189 -233 -243 -257

2^48 +21 +61 +75 +91 +235 +241 +243 +253 +297 +315

2^64 -59 -83 -95 -179 -189 -257 -279 -323 -353 -363

2^64 +13 +37 +51 +81 +93 +141 +307 +331 +393 +493

2^80 -65 -93 -117 -143 -285 -317 -549 -645 -765 -933

2^80 +13 +85 +235 +253 +343 +435 +457 +555 +597 +753

2^96 -17 -87 -93 -147 -165 -189 -237 -243 -315 -347

2^96 +61 +81 +121 +151 +253 +403 +423 +627 +633 +765

2^128 -159 -173 -233 -237 -275 -357 -675 -713 -797 -1193

2^128 +51 +81 +165 +273 +385 +421 +463 +573 +625 +757

2^160 -47 -57 -75 -189 -285 -383 -465 -543 -659 -843

2^160 +7 +291 +357 +421 +471 +501 +643 +685 +861 +921

2^192 -237 -333 -399 -489 -527 -663 -915 -945 -1059 -1143

2^192 +133 +453 +511 +565 +813 +1005 +1045 +1113 +1131 +1423

2^256 -189 -357 -435 -587 -617 -923 -1053 -1299 -1539 -1883

2^256 +297 +301 +357 +487 +583 +757 +795 +807 +847 +931

2^320 -197 -743 -825 -843 -873 -1007 -1017 -1217 -1815 -2955

2^320 +27 +261 +391 +525 +561 +793 +931 +1413 +1857 +1981

2^384 -317 -1437 -1557 -1617 -2147 -2319 -2729 -3087 -3093 -3273

2^384 +231 +331 +417 +535 +735 +817 +823 +835 +933 +1015

2^512 -569 -629 -875 -975 -1695 -1827 -2529 -2807 -2967 -3143

2^512 +75 +145 +285 +727 +1105 +1147 +1273 +2743 +3177 +3913

2^640 -305 -503 -735 -995 -1019 -1215 -1593 -2015 -2033 -2805

2^640 +115 +303 +391 +757 +1287 +1485 +2943 +3627 +3711 +3861

2^768 -825 -1385 -1815 -1845 -2069 -2277 -2825 -3209 -6953 -9189

2^768 +183 +241 +427 +955 +1423 +1723 +1831 +2161 +2851 +3225

2^1024 -105 -179 -1397 -3177 -5025 -5409 -6083 -6369 -6615 -7137

2^1024 +643 +1081 +2113 +2715 +3711 +5335 +5793 +5947 +7015 +7447

2^1280 -1175 -1665 -3149 -5907 -7079 -7607 -7703 -8865 -10043 -10277

2^1280 +1815 +2251 +2553 +3217 +4075 +4405 +4435 +7081 +7405 +9087

2^1536 -3453 -4977 -7769 -8333 -9923 -10859 -11429 -11795 -11843 -11877

2^1536 +75 +1381 +1605 +2487 +4903 +6333 +6637 +7191 +7597 +7681

2^2048 -1557 -2543 -7437 -8507 -9443 -9509 -11339 -11837 -12459 -12855

2^2048 +981 +1617 +3063 +3211 +4143 +7405 +9843 +10665 +10725 +11097

2^2560 -75 -1347 -2745 -6599 -7245 -10205 -12659 -17417 -18413 -20159

2^2560 +903 +4077 +4125 +4363 +5025 +10593 +12153 +13581 +13891 +14767

2^3072 -47 -2165 -4737 -6117 -6489 -7625 -9617 -15269 -23993 -24279

2^3072 +813 +877 +3427 +5131 +9133 +12367 +12433 +13435 +16003 +17245

2^4096 -2549 -8067 -8627 -8799 -9443 -14477 -16859 -17555 -18365 -20655

2^4096 +1761 +7227 +7423 +10093 +10473 +13965 +17335 +17355 +19891 +22803

2^5120 -7097 -7779 -11619 -12689 -15435 -20799 -26745 -28727 -30527 -31955

2^5120 +5467 +6741 +20847 +25135 +26611 +29335 +30147 +33237 +33567 +40791

2^6144 -5157 -6369 -6639 -13773 -13785 -22437 -22923 -27029 -39879 -41895

2^6144 +375 +1275 +2853 +3901 +5545 +11877 +11953 +13171 +14875 +15481

2^8192 -2439 -5619 -9345 -9515 -19085 -19733 -21713 -45933 -46643 -51453

2^8192 +897 +9543 +10813 +13371 +14931 +14985 +15505 +15763 +16305 +19423

2^10240 -323 -10119 -27353 -34179 -40253 -44975 -47195 -66917 -76307 -82187

2^10240 +2601 +15295 +25011 +35671 +46123 +53697 +54387 +58843 +65215 +67963

2^12288 -27803 -30519 -39213 -41045 -43197 -60465 -65055 -72195 -73853 -78515

2^12288 +11293 +17437 +25951 +40413 +40803 +41965 +42775 +46555 +56623 +66175

2^16384 -13797 -29027 -62747 -99125 -101543 -107609 -115673 -118119 -134357 -134705

2^16384 +2775 +36333 +42027 +47025 +55933 +80391 +91395 +95665 +98581 +104565

2^20480 -479 -9159 -21203 -32885 -86435 -88767 -100073 -102509 -104097 -124599

2^20480 +8367 +24201 +28173 +32787 +36061 +46051 +52323 +70725 +72163 +81187

2^24576 -1875 -32507 -37217 -61373 -66905 -67475 -97023 -98079 -103815 -119235

2^24576 +241 +22461 +35193 +80967 +130677 +130965 +145611

2^32768 -25353 -38783 -53373 -61725 -67553 -79613 -112985

2^32768 +118113 +143905 +145027 +160771 +162873 +182005 +187861

2^40960 -48069 -70803

2^40960 +59815 +74031

2^49152 -34689 -37923

2^49152 +1605 +11923

2^65536 -5627 -99267 -123563

2^65536 +44061 +44181 +58227 +106417 +116193 +119031

Thursday, October 06, 2016

[oobhikls] Unsatisfying P != NP

Someday, it may be proven that P < NP; however, the proof might be disappointing or unsatisfying.  It might be proven that one (obscure) NP problem is not in P, but it unfortunately provides no insight as to whether another NP problem, a problem that you care about, perhaps integer factorization, is not in P.  It could be, or it could not be (assume the proof technique does not generalize).  Or it might be proven that in the worst-case, a problem cannot be solved in polynomial time, but this isn't helpful if your particular instance of the problem isn't in that worst-case class.  For example, it might be proven that factoring requires super-polynomial time for a set (infinite sequence) of very rare, very difficult to construct integers, but this doesn't help for cryptography because practically all RSA numbers will not be in that class.

If it is proven that P = NP by showing a polynomial time algorithm for an NP-complete problem, then things will be very exciting (or practically, perhaps not, if the exponent is huge): by reduction it will provide a polynomial-time algorithm for every problem in NP.  Is it possible that P = NP might be proven not going through the NP-complete route and yield a similarly disappointing or unsatisfying proof?  Perhaps a non-constructive proof: there exists some deterministic Turing machine equivalent to a nondeterministic Turing machine...

[sxbpxqpu] Splitting the primes by 3, 4, 6

All prime numbers except 2 are of the form 4k+1 or 4k+3.  (Actually, all odd numbers.)  This distinction is made famous in Fermat's Sum of Squares theorem and quadratic reciprocity.

All prime numbers except 3 are of the form 3k+1 or 3k+2.

All prime numbers except 2 and 3 are of the form 6k+1 or 6k+5.

Each of the residue classes contains roughly half the prime numbers.  (Nice article: "Prime Number Races" by Granville and Martin.)  I don't think there are other moduli which result in two classes of equal size.  (All prime numbers (all numbers) are of the form 2k+0 or 2k+1, but former contains only prime element, namely 2.)

The remainders for each of these can also be written +1 and -1.  Therefore, each prime can be annotated with 3 signs, moduli 3, 4, and 6.

The annotation for currently largest known Mersenne Prime M74207281 is +-+ (plus minus plus).  The annotation for the exponent is +++ (plus plus plus).

[vnluuail] 4D virtual reality

A good use of 3D virtual reality might be to depict 4-dimensional objects, kind of an analogy to how 2D displays are frequently used these days to depict 3-dimensional objects, especially in games.

There still remains the issue that what the eye sees is a 2-dimensional image projected on the retina.

[fuusnzvv] Prime enough

You come across a composite number that resists factorization, so you call it prime and use it accordingly.  Under what conditions or applications is this a bad idea?  Under what conditions is it fine (so long as no one discovers a factor)?

Probably a bad idea if you don't know the factorization but someone else (secretly) does, though I have no concrete examples of how things go wrong.  Any other cases?

Previously, if you can't factor N, then factor N-1.

Monday, October 03, 2016

[cmpuuxwa] Good teachers are bad for society

If school, not just business school and college, but even primary school, is intended to separate students destined to become high productivity workers from low, then good teachers, who help keep in school those students destined to become low productivity workers, and who help students destined to become low productivity workers pass tests intended to distinguish low productivity from high, these good teachers are interfering with the system, making it difficult for the rest of society to distinguish between low productivity people from high by looking at their education, a signaling mechanism in an imperfect (asymmetric) information game.

This model assumes that a person's productivity is exogenous: you don't actually learn anything in school, equivalently, nothing a teacher teaches ultimately affects productivity.  Productivity might be entirely determined by the social support structure around a person.

Does this model explain teacher salaries?  Society needs incompetent people, and only incompetent people, to be teachers.  Promotion and firing have to be geared toward retaining and rewarding the incompetent.  What mechanisms in the workplace and school system can accomplish this?  Kind of a kakistocracy.

Education as a signaling mechanism has most famously been analyzed for labor and employment, but it probably gets used in other fields, too, like courtship.  Good teachers might be indirectly affecting -- helping students subvert -- those fields, too.  Flaws in the courtship process cause at best a waste of time, at worst, sexual and domestic violence between people incompatible with each other.

[yxnisako] Base 26 checksum

Create an error correcting code or check digit for base 26, detecting mistakes that humans tend to do, like swapping adjacent letters, like the Verhoeff check sum for numbers.  Maybe adapt Verhoeff's idea for the dihedral group of order 5 to order 13.

Better might be taking input in base 27, allowing one escape character, and emitting a base 26 check letter or letters.

[huxoeaub] Bypassing SIM activation on Verizon Motorola Droid 4

The "touch 4 corners in succession" method works to bypass "A SIM card is needed to operate this phone" when initializing a factory-reset Droid 4.  It succeeded in portrait mode; I see other reports it also works in landscape mode.

What is the history of this circumvention?  Was it originally revealed by Motorola?  If not, how could anyone discover it?