December 2015 - mathfun@mailman.xmission.com

[math-fun] convert vector into sum of permutation vectors?
by Warren D Smith 29 Dec '15

29 Dec '15

>Andy Latto: >Even assuming the coordinates of V are non-negative, this is not in >general possible. >What does your algorithm yield when asked to write (1, 10) >as a positive linear combination of (1, 2), (2, 1), and (1,1)? --First of all, it's (0,1), (1,0) and (1,1); second, I only claimed the coefficients of the perm vectors would be positive with the coefficient of (1,1,1,...,1) allowed to be negative; and third, (1,10) = 9*(0,1) + (1,1) is the answer. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] Gosper island double surprise
by Joerg Arndt 29 Dec '15

29 Dec '15

Two renderings of Gosper's island, one traverses all points of the (3.4.6.4)-tiling once, the other (was awfully hard to obtain and) traverses all edges once: http://jjj.de/tmp-xmas/ The edge version is to my best knowledge the first such curve for (3.4.6.4), versions for (3^6) and (4^4) are known for quite some time. Ventrella gave the first for (3.6.3.6) in 2012 in his book "Brain-Filling Curves". All other uniform tilings have odd incidences on some points, hence no such curve exists on them. The curve I use is not Gosper's (I need a curve that turns by 120 degrees after every edge). Enjoy! Now that is my chance to ask about terminology (again). Do the following appear OK? I call... ... the arrangement of points and edges of some tiling a "grid", as in the "square grid" for what is (4^4), the (uniform) tiling into unit squares in Gruenbaum and Shephard. ... specifically, the grids for (3^6) the "triangular grid", for (6^3) the "hexagonal grid", for (4^4) the "square grid" (as said), and for (3.6.3.6) the "tri-hexagonal grid". ... curves that traverse all points once "point-covering" (and could in analogy call those that traverse all edges once "edge-covering", but indeed call them "grid-filling", should I prefer "edge-covering"?). The (pdf) images above are examples of each. These are two corner cases of "plane-filling" on a grid. Also I define Eisenstein integers as numbers of the form x + \omega_6 * y while the rest of the world seems to use x + \omega_3 * y (my norm is x^2 + x*y + y^2, the other is x^2 - x*y + y^2). Does anybody want to kill me for that? Yes, I am writing something up and would like to avoid annoying the readers with bad terminology (and nobody in my personal reach could possibly answer the questions above). Best regards, jj

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[math-fun] Real division algebras
by Warren D Smith 29 Dec '15

29 Dec '15

Well (to Dan Asimov) First: you & Gerver, you said, were using real division algebras. You did not say how. If you did not need the full strength of the distributive law, e.g. only need right-distributivity, then you could use my 16-ons. My question to you, was: can you? Second: you said the definition of "division algebra" is well established and my answer is: that may be true in your mind... but if you actually look in the literature, I have found verbatim definitions under which my 16-ons are one. Even though they are proven not to be one. By a famous impossibility theorem. The reason is the literature definitions tend to be careless about the distributive law, they sort of assume it without explicitly saying so, and so forth. I suspect they had in mind what you do, but the fact is, their verbatim words say otherwise in at least some cases. Basically, it has occurred to very few people, ever, to think about (what I call) "nonlinear algebras." That is understandable since they are hard to deal with. But in the specific case of my "2^N-on" things, I was able to do it since they have a nice structure. There might be other nonlinear algebras with nice structure waiting to be found -- who knows. Certainly not me. But I was able to prove some stuff, like the only normed ones have to be power-of-2 dimensional.

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[math-fun] No such thing as a self-avoiding spacefill
by Bill Gosper 29 Dec '15

29 Dec '15

On 2015-12-23 12:00, Warren D Smith wrote: >> J?rg, these recursive arcs are beautiful and ingenious, but in the >> limit, they all describe the same area-filling function, where in >> this case, the area filled is 1/3 of a "France Flake" island. All such >> area-fills map closed intervals onto closed sets, hitting *all* the >> points at least once, uncountably many at least twice, and at least >> countably many at least thrice. > > --it seems to me, there must be some value in an area-filling curve > that is the "limit" of a sequence curve1, curve2, ..., in which every curveN > never self-intersects and "stays away from intersecting itself" by at least > some natural distance f(N). Since in any real application, N will be finite. > > Possible way to make that precise: > Each point of curve N is distance >= f(N) away from any other point of > curveN that is not at ArcDistance <= 2*f(N). > > So then the question is: which area-filling curves can be manufactured > in this way, and which cannot? > One might argue they all can be done (proof: local surgery as needed) > but perhaps not in a "nice" way (simple nice definition). > Sequences of curves are beguiling and misleading, by concealing that something singular happens as D reaches 2. The textures and designs that seem to distinguish these curves are just superficial treatments applied to the bottommost recursion. Recall that traditional pictures of Heighway's Dragon show whole grids of self-contacts. But simply joining the midpoints of its segments produces a completely self-avoiding "median curve": gosper.org/twindrag.png . Alternatively, sampling the limiting "curve" at a frequency slightly off from a power of 2 produces beat frequencies: gosper.org/insuck.png showing whole intervals of phases where the polygonal approximations self-avoid. Finally, recall the plain old square grid four-around-one (base i+2, digits 0, i^(0..3)): gosper.org/erez2.png and how different it looks when quarter-circles replace the line segments at the bottom level: gosper.org/erez7.PNG Bottom line: There is no bottom level. --rwg An interesting rendition of a spacefill might be to use Julian's inverter on a big bunch of double and triple points, darkening each one in proportion to the separation of its inverse images. I expect it would resemble leaf veins.

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[math-fun] convert vector into sum of permutation vectors?
by Warren D Smith 28 Dec '15

28 Dec '15

A "permutation vector" shall mean an N-vector whose entries are some permutation of (0,1,2,3,...,N-1). For example (4,2,1,3,0). Let V be an N-dimensional real vector. I want "the natural method" for converting V into a weighted sum of permutation vectors plus some multiple of the all-1 vector. What is "natural"? Well, you may have your own ideas about that, and I'm open to revisions, but here are mine: 1. few nonzero weights. And they tend to be positive. 2. The weights are minimal in some norm (e.g. L1 or L2). 3. Fast algorithm to perform the conversion, i.e. compute the weights and the used permutations. E.g: I have a simple way to convert V into a weighted sum of at most 2N-2 perm-vectors plus some multiple of the all-1-vector, which takes O(N) operations if V is pre-sorted, and all weights automatically are nonnegative. But this is almost certainly improvable. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] Real division algebras
by Warren D Smith 28 Dec '15

28 Dec '15

Dan, I already knew what you said about real division algebras and topology. However, I do not understand those topological proofs. I in fact asked Milnor for help at one point. I have a conjecture that we ought to be able to use "generalized smoothness" (a new topological notion I introduced in this paper) to prove that the only real division algebras, (wide sense) were 1,2,34,8,16 dimensional. Milnor and another Topologist, Denis Sullivan;, agreed this was interesting problem. However, Milnor said he no longer was able to do this stuff and no longer remembered how the topological proofs he was so famous for, work. Sullivan claimed he was going to work on it, but if so, he didn't accomplish anything, at least not that he told me about. I suggest reading my paper, especially the parts about division, e.g. sections 19-23. Here are some quotes: nnacd = not necessarily associative, commutative, or distributive. 2^n-ons: for n=1,2,3,4 these are the reals, complexes, quaternions, octonions, and my 16-ons. 19. What is a “division algebra” and are the 2^n-ons one? One (boring) definition of “division algebra” is “an algebra in which xy = 0 implies x = 0 or y = 0.” By that definition, all of our 2^n-ons are nnacd “division algebras” simply because of the multiplicativity of Euclidean norm. A stronger definition of “division algebra” is “an algebra in which, given b and given a not= 0, 1. there exists z so that az = b, and 2. there exists q so that qa = b.” Finally, a still stronger definition would require, not only that the solutions z and q of these two division problems must exist, but that, furthermore, they must be unique. The first of these division problems, for our 2^n-ons, trivially does have a unique solution z... The second division problem, however, is not so trivial because there is no right-cancellation law in the 16-ons... Theorem 70 (16-on division). There exists a unique solution to any generic 16-on division problem. -- Note also, my 16-ons do NOT satisfy the distributive law ("nonlinear algebra") although they do satisfy right-distributivity and weakened versions of left-distributivity. The 16-ons are very interesting and I would like to see some use found for them. In any event, they do not deserve to be lightly dismissed as though I were ignorant. Had you bothered to look at my paper, you would have found I'd already said everything you just told me, plus considerably more, as part of my review of that area. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] space filling curves
by Warren D Smith 28 Dec '15

28 Dec '15

just to make a rather boring point, the "D-dimensional Peano curve" is based on the fact that a hyperrectangle can be cut into 3 scaled copies of itself... if its edge lengths are 1, A, A^2, ..., A^(D-1) where A^D = 3. Isn't this meeting Dan Asimov's desires in any dimension? Then to make a rather more exciting (?) point, Dan said the only real division algebras are in dimensions 1,2,4,8. However, there actually is one in dimension 16 too, depending on the precise technical definition of "division algebra." That is explained in paper #73 here: http://rangevoting.org/WarrenSmithPages/homepage/works.html It would be rather nice if anybody figured out a use for it. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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Re: [math-fun] math-fun Digest, Vol 154, Issue 36
by Warren D Smith 24 Dec '15

24 Dec '15

> J?rg, these recursive arcs are beautiful and ingenious, but in the > limit, they all describe the same area-filling function, where in > this case, the area filled is 1/3 of a "France Flake" island. All such > area-fills map closed intervals onto closed sets, hitting *all* the > points at least once, uncountably many at least twice, and at least > countably many at least thrice. --it seems to me, there must be some value in an area-filling curve that is the "limit" of a sequence curve1, curve2, ..., in which every curveN never self-intersects and "stays away from intersecting itself" by at least some natural distance f(N). Since in any real application, N will be finite. Possible way to make that precise: Each point of curve N is distance >= f(N) away from any other point of curveN that is not at ArcDistance <= 2*f(N). So then the question is: which area-filling curves can be manufactured in this way, and which cannot? One might argue they all can be done (proof: local surgery as needed) but perhaps not in a "nice" way (simple nice definition).

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[math-fun] Happy Holidays
by David Wilson 22 Dec '15

22 Dec '15

Wishing all math-funsters and seqfans a happy holiday season. You've kept my brain alive for another year.

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[math-fun] black hole size according to Andrew King. Atrocious.
by Warren D Smith 22 Dec '15

22 Dec '15

http://arxiv.org/abs/1511.08502 The reason too-large black holes cannot accrete via an accretion disk allegedly is said disks become unstable to self-gravity-caused "clumping" perturbations. Then the gas either forms stars or is expelled by such stars, instead of spiraling into the hole. I'm finding this all extremely unconvincing. King is just multiplying a few letters like a monkey, and the reasons it all is valid (if it is) are very deep and due to other people not King; just tosses all that off in like 1 sentence, intermixing magic numbers like "0.001" and 3*10^16 that come out of the woodwork, as convenient. If this is correct you will be unable to tell it is correct by reading King; you'd also have to read a LOT of other stuff which is way deeper than King. Also, a lot of the letters he multiplies like monkey, he does not even define, he just says stuff like "alpha is the standard viscosity parameter" and eta is "the standard accretion efficiency" whatever those meant. So not only is it impossible to convince yourself King is correct by just reading King -- you cannot even understand what it is saying by just reading King. And although this ought to be possible to express in terms of fundamental constants, that is not possible by just reading King alone, and he did not try. In my opinion this author behavior is unacceptable. But since his paper was already accepted, evidently the journal does not share my notion of what should be "acceptable."

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