April 2015 - mathfun@mailman.xmission.com

Re: [math-fun] Prime ladders (Tom Karzes)
by Warren D Smith 30 Apr '15

30 Apr '15

Re the suggestion by RCS that "the stats become much more favorable if one considers changing TWO digits"... we ask... QUESTION: Does there exist an integer K>0 such that all primes (except perhaps for a finite set of exceptions) are "connected" where two primes are called connected if they differ in K or fewer digit locations? HEURISTIC ANSWER: The number of D-digit primes is about 10^D / (D*ln(10)) and the number of "neighbor numbers" connected to a given such prime, is 9^K * (D choose K) if all K digits change (plus some lower order terms if fewer than all K change). The "expected number" of those neighbors that are prime is about 9^K * (D choose K) / (D*ln10). The chance that all neighbors are nonprime is about exp( -9^K * (D choose K) / (D*ln10) ). The expected number of D-digit "isolanis" (primes lacking prime neighbors) is thus about exp( -9^K * (D choose K) / (D*ln10) ) * 9^K * (D choose K) / (D*ln10). We now sum this over all D=1,2,3... and observe that the result is finite if K>=2. If K=1 then the sum is infinite similarly to SUM(1), If K=2 then the sum is finite similarly to SUM(D*exp(-D)). Of course the heuristic reasoning above should be altered by inserting various constant fudge factors in various places, but it will not be enough to alter these finiteness/not results. Now you can also consider, not merely "isolanis" which are singleton connected components, but also connected componets containing C primes, for C=1,2,3,4,... and sum over C. This does not alter the conclusion. So: I would conjecture that if we call two primes 'connected' when they differ in 2 or fewer digits, then: all besides a finite set of exceptional primes are connected. (And this is also true in any radix>=2.)

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[math-fun] transfinite epistemic logic puzzle
by Warren D Smith 29 Apr '15

29 Apr '15

>From: Jason Holt <credentiality(a)gmail.com> >Subject: [math-fun] transfinite epistemic logic puzzle >Apologies if this has already been posted: > http://jdh.hamkins.org/transfinite-epistemic-logic-puzzle-challenge/ --wow. That was spectacular. I also found on the internet this related puzzle by evan(a)scd.hp.com (Evan Whitney) dated January 1998, (which I have not solved): Here's a variation of Mr. P and Mr. S that I just made up. I believe the solution is unique and the problem completely defined. One logician, Mr. S, is told the sum of two integers. The other logician, Mr. P, is told the product of the same two integers. Mr. P knows that Mr. S knows the sum, and Mr. S knows that Mr. P knows the product. The following conversation ensues, you decide who is who: Mr. X: "I don't know what the two integers are." Mr. Y: "I knew you didn't know what they are." Mr. X: "I knew you knew that." Mr. Y: "This conversation could go on forever." Mr. X: "I still don't know what the integers are." Mr. Y: "Now I know what they are." Mr. X: "Now I know what they are, too." What are the two integers?

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[math-fun] transfinite epistemic logic puzzle
by Jason Holt 29 Apr '15

29 Apr '15

Apologies if this has already been posted: http://jdh.hamkins.org/transfinite-epistemic-logic-puzzle-challenge/

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[math-fun] The guy who coined the words scientist, physicist, ion, dielectric, anode, cathode, ...
by Bill Gosper 29 Apr '15

29 Apr '15

http://en.wikipedia.org/wiki/William_Whewell famously wrote an accidental poem which I've never seen correctly quoted: "Hence no force however great can stretch a cord however fine into a horizontal line which is accurately straight" http://babel.hathitrust.org/cgi/pt?id=hvd.hxj4g9;view=1up;seq=74 He hastily deleted it from future editions. --rwg

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[math-fun] Dupes in OEIS?
by Leo Broukhis 28 Apr '15

28 Apr '15

Is there supposed to be a difference between https://oeis.org/A206398 and https://oeis.org/A000793 ? (This came to my attention when I remembered a puzzle I invented as a kid: how many times a text might need to be typed on a typewriter with hammers re-soldered in a random order to get the desired result, given the number of the typewriter keys.) Thanks, Leo

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Re: [math-fun] exotic R4 relevant to physics?
by Warren D Smith 27 Apr '15

27 Apr '15

You might want to check my papers on "shape of the universe". Topology can say things about this question. I, however, had no training in topology. Therefore, I probably did not figure out everything that can be figured out, and a "real topologist" ought to be able to do better. Perhaps even pretty easily. Essentially, there are (a) certain experimental physical facts, and (b) there also are certain assumptions you could make, about the universe. If you choose your favorite set from columns (a) and (b), then the question is (c) what can be deduced about the topology of the universe? Quite a lot can be said, and I did, but I believe quite a lot more could be said and this whole topic has been not been studied very much by very good people, so there's "low hanging fruit" to be had. (Indeed the very fact I managed to get as far as I did with no training in topology, kind of proves that.) Anyhow, the papers I am speaking of are #55, #63, #75 here: http://rangevoting.org/WarrenSmithPages/homepage/works.html

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[math-fun] Nonstandard smooth structure on R^4
by Dan Asimov 27 Apr '15

27 Apr '15

The following will rely heavily on Alexandru Scorpan's book, The Wild World of 4-Manifolds. Part I. (There are a number of parts to this.) We'll talk about compact 4-manifolds. This will mean a compact topological space every point of which has a neighborhood homeomorphic to 4-space. (Or equivalently, to the open unit ball in R^4).* 0. We'll also assume all manifolds are simply connected (every loop can be shrunk to a point). 1. The integer homology group H_2[M] represents a way of thinking of [all the closed surfaces inside of M] as a group. It is always a finitely-generated free abelian group. Given a basis, we can view it as a direct sum Z^n of n copies of the integers Z. 2. Given any two homology classes u, v in H_2[M], there is an integer associated to them. First think of each as a surface in M. Then <u,v> := the number of intersection points of these surfaces. The points need to be counted with algebraic sign depending on the orientation of M. This may require perturbing them slightly to be in general position. It turns out to always be the same integer no matter how this is done. (As an analogy, think of the generic intersection of two closed curves in a closed surface — which also give an integer.) In fact, this mapping H_2[M] x H_2[M] -> Z is a symmetric bilinear form over the integers. Of special interest are the self-intersection numbers <u,u>. Restricting the bilinear form to self-intersections, it becomes a quadratic form Q: H_2[M] -> Z via Q(u) := <u,u>. This is called the INTERSECTION FORM of M. (Two such quadratic forms are "equivalent" if there is a Z-basis for each one such that the resulting symmetric matrices are the same.) FACT: Because of "Poincaré duality", Q is always UNIMODULAR, which means that det(Q) = ±1. Another property of quadratic forms is DEFINITENESS: If all values taken by Q(u) are postive, or else all values are negative, the form is respectively POSITIVE or NEGATIVE DEFINITE. Otherwise the form is INDEFINITE. 3. MAJOR RESULT: Theorem of Serre: ---------------- "Two indefinite unimodular integer quadratic forms are equivalent if and only if they have the same rank (the n here), signature, and parity." ----- The (symmetric integer) matrix of a quadratic form can always be diagonalized over the REALS. When this has been done, its SIGNATURE is the number of positive terms minus the number of negative terms on the diagonal. The parity is EVEN if the quadratic form takes only even values. Otherwise it is ODD. Unfortunately, any classification of *definite* quadratic forms over Z is currently out of reach. 4. MAJOR RESULT: Theorem of Michael Freedman, ca. 1981: ---------------- "The topological classification of simply connected compact 4-manifolds is as follows: There is at least one for each inequivalent quadratic form Q, having that form as its intersection form. "If Q is even: -------------- There is exactly one such manifold up to topological equivalence. "If Q is odd: ------------- There are two such manifolds, at least one of which cannot be smoothed." ----- Example: The simplest nonzero quadratic form has the matrix [1]. An example of this is the complex projective plane CP^2. This is not only smooth, but the nicest kind of complex analytic manifold: a Kähler manifold. But since [1] is an odd form, there must be another topological manifold with the same quadratic form that is not smoothable at all. -----END OF PART I----- --Dan ——Dan __________________________________________________________________ * To avoid pathologies, we also assume manifolds are Hausdorff and separable (have a countable dense subset).

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[math-fun] California water crisis
by Henry Baker 27 Apr '15

27 Apr '15

Now that Captain Kirk is using Kickstarter to build a water pipeline to California from someplace with "too much" water, I was thinking about what would be cheap ways to get large amounts of desalinated water. One suggestion (Popular Mechanics??) from my childhood: towing an iceberg (assuming we can still find one) to San Diego/Long Beach/San Francisco harbor and harvest it into the water supply. So long as you don't care much about speed, moving stuff through water can be quite energy-efficient. So even if it took a number of weeks to move a large iceberg, the percent lost to melting would be relatively small. Furthermore, if you wanted to reduce the melting, you could wrap the iceberg in bubble-wrap to better insulate it. Another option would be to wrap the iceberg in plastic wrap to catch the melted (desalinated) water & pipe it into a nearby oil tanker. You could presumably use an oil tanker itself to bring fresh water from someplace with "too much", and this would likely be energetically cheaper than using a pipeline, but such a tanker could carry only perhaps 80% (by volume) of the amount of oil it would normally carry. If you were forced to desalinate the seawater locally, it would be best if you could use solar power to do it. Solar panels are becoming cheap enough that they might allow relatively inexpensive desalinators. Even better, solar panels are _distributed power_, so it makes sense to build small, modular desalinators -- perhaps the size of a standard shipping crate. If you made these modules so that they were waterproof & floated, you could hook them together into a floating desalinization plant which required no expensive space on the California coast. Furthermore, if you provided a long enough hose, you could put these (very low profile) modular solar desalinization plants over the horizon so that they wouldn't mess up the view from expensive California real estate (this is a real problem with offshore wind farms). Putting the desal plant far out in the ocean also solves another problem: what to do with the discharge water with higher salt content. If you discharge this close to shore, it messes up the local sea life, whereas if you do it far out to sea, it can get diluted much more in the deeper waters.

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[math-fun] heptagons in a triangular grid
by Bill Gosper 26 Apr '15

26 Apr '15

gosper.org/gridheptagons.png --rwg

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[math-fun] Observation on Collatz
by David Wilson 26 Apr '15

26 Apr '15

I did a more direct test. Using iterate c(n) = n/2 if n even; (3n+1)/2 if n odd I looked at small b >= 1, counting the trajectories of n on [0, 2^b), that first reached a value < n on the b-th step. It turns out that for certain b, the trajectory of n never reaches a value < n for the first time on the b-th step. My program says these b are (3, 6, 9, 11, 14, 17, 19, 22, 25, 28, 30, .). If my interpretation of A182137 is correct, these are the b for which A182137(b) = 2*A182137(b-1). These b are (3, 6, 9, 11, 14, 17, 19, 22, 25, 28, 30, 33, 36, 37, .). I might have guessed that the b fall into a Beatty-like sequence b(n) = ceil(c*n) with 2.7 <= c < 3. However, in such a sequence, first differences must be either 2 or 3, inconsistent with adjacent elements 36 and 37 implied by A182137. This means either my intuition is wrong (the set of b is more interesting and mysterious than a Beatty sequence), or A182137 is wrong.

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