May 2016 - mathfun@mailman.xmission.com

Re: [math-fun] bug lists non-well-ordered; roll over, Peano
by Henry Baker 21 May '16

21 May '16

This has actually happened to me enough times that I can't remember the first time... At 05:06 PM 5/21/2016, Fred Lunnon wrote: >You gotta laugh ... > >Is that an actual incident? What call-centre/conglomerate/bureacracy? > >WFL > >On 5/22/16, Henry Baker <hbaker1(a)pipeline.com> wrote: >> "Hi, I'd like to report a bug." >> >> "Could you please describe it?" >> >> [Describes bug.] >> >> [Clattering of keys.] >> >> "I'm sorry, but that bug has never been reported before." >> >> "I'm reporting it now." >> >> [With stronger accent] "I'm sorry, but you must be mistaken; that bug has >> never been reported before." >> >> {Sigh.] >> >> "But there has to be a first time report for every bug." >> >> [Clattering of keys on the Peano.] >> >> "You must be doing something wrong." >> >> [Hangs up.] >> >> ------ >> Next time, I think I'll try to get closure through open & closed sets.

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Re: [math-fun] annoyed replies to more totally confused gumf from FWL
by Warren D Smith 21 May '16

21 May '16

> A minor omission from the reduction algorithm concerns the final diagonal > matrix, which is not necessarily the identity matrix, but may incorporate > (an even number of) -1 entries along the main diagonal. It's unclear to > me whether these might somehow be absorbed into rotations earlier in the > sequence; otherwise they require an extra floor(n/2) rotations to > eliminate. --of course, they can be incorporated into rotations earlier. There of course are always exactly 2 choices for a Givens 2x2 each time, + or -. > WDS raised the question how many length m = n_C_2 sequences of Givens' > rotations R_ij are "valid", reducing a general orthonormal matrix to > diagonal. Denoting this number f(n) , for n = 1,...,5 , I compute > f(n) = 1, 1, 3, 50, 6821; > for example for n = 3 , there is a choice of > [[2, 1], [3, 1], [3, 2]] , > [[3, 1], [2, 1], [3, 2]] , > [[3, 2], [2, 1], [3, 1]] , > (restricted always to the lower triangle). --you're wrong again. You perhaps have solved some other problem however, and maybe someday you should figure out what it is. Rewriting your n=3 examples as orderings 1 2& 3* 2 1 3* 2 3&* 1 we see immediately that not a single one is valid, they each destroy previously created zeros at the * steps, and generically fail to zero that entry at all in the & steps. (And don't bother telling me your flawless computer program has shown the contrary. It is simply wrong. We already did that.) > I slipped up there --- thankyou. Fortunately for me, Maple gets it > right. WFL --wrong again, already pointed out your maple program was obviously wrong. > (1) Steps A0 to A2 operate on rows 3 and 4, not 1 and 2 . --obviously wrong since each step operates on two rows, not 1 row. You are just massively massively utterly wrong and confused about everything you are saying, over and over again, but say it with such a tone of confidence, and never admit there was a problem. Maybe you should run for US president? -- 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 159, Issue 58
by Warren D Smith 20 May '16

20 May '16

>> --ok, I immediately see from that last URL that you did it wrong. >> A0 to A1 looks reasonable. >> A1 to A2 is already wrong because note row #2 was >> left unaltered (should have changed) >> but row #1 changed (should have been unaltered). > > (1) Steps A0 to A2 operate on rows 3 and 4, not 1 and 2 . > > (2) My program executes the sequence you specified as follows --- > 7 > 4 8 > 2 5 9 > 1 3 6 ... > > I did previously point out this elementary malfunction, which is > the responsibility of the designer rather than the programmer. > Once an A_ij to be cleared is specified, Givens' R_ij alters > rows & cols i,j , intentionally or otherwise! --WRONG!! It alters rows i and j only, and in ALL columns simultaneously! > (3) I did however summon the initiative to investigate alternative > scanning sequences: NW instead of SE along diagonals, and inwards > rather than outwards. Unsurprisingly, they fail as well. > > [In fact, I still find it rather surprising that scanning columns does > work --- as you point out elsewhere, both N-ward and S-ward versions.] --you continue to be confused... -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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Re: [math-fun] Decomposing Rotation into set of sub-rotations
by Fred Lunnon 20 May '16

20 May '16

On 5/19/16, Warren D Smith <warren.wds(a)gmail.com> wrote: > it wouldn't let me post that to geometric-algebra, so you might > want to post it there for me. > > Or you could complain about how I'm wrong some more. > I have not programmed this or tried to... but > it still seems to me I'm obviously right plus was all along... > I'll take option 2, thanks. Screenshots of listing (sans comments) of program implementing algorithms under discussion at https://www.dropbox.com/s/3cr364xiw628i23/given_p0.tiff ; output of column-by-column algorithm test result at https://www.dropbox.com/s/uwsmq2habm4tfgp/given_p1.tiff ; output of triangular algorithm test result at https://www.dropbox.com/s/9v3t0nicq6huf2g/given_p2.tiff . Observe that the first result is finally diagonal, whereas the second has nonzero first column. Apologies for the excessive precision, courtesy of yet another "feature". Maple source code in attached text file, should you wish to experiment. > The procedures I gave will be stable in the sense numerical > errors will grow like polynomial(N), as opposed to many > procedures such as gaussian elimination with partial pivoting > where error can grow exponentially(N) for > unfriendly matrices. > > Wilkinson famously showed gauss with complete pivoting is backwards > stable in the sense of a certain bound on error growth, but I think > his bound was superpolynomial although subexponential. > Good points about stability, which I shall pass on to GA list in due course. Fred Lunnon

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[math-fun] Decomposing Rotation into set of sub-rotations - how many orderings?
by Warren D Smith 20 May '16

20 May '16

This combinatorics question may now have some interest: HOW MANY orderings of the (N-1)*N/2 entries in the lower triangle in an NxN matrix, are there, such that each one may be zeroed via an adjacent-coordinate 2x2 rotation, successively, without destroying the previously-created zeros? The going-up-columns raster scan (starting at leftmost column) and the diagonal raster (going southeast, on longer and long diagonals) ordering are two answers, but there evidently are many more possible orderings. (Here I am just considering multiplying by the 2x2 rotation from the left. I we also allow multiplications by 2x2s on the right, that'd be more.) The question is equivalent to the following. How many ways can you fill in that triangle with the integers 1,2,3,...,(N-1)*N/2 in such a way that: if some cell contains integer K, then each cell on the same row containing an integer<K, must be accompanied by a "mate" -- the cell immediately above it -- also with contents <K. ? A crude rough "probabilistic" heuristic estimate is as follows. Consider choosing one of the [(N-1)*N/2]! orderings at random. The chance it satisfies the "mate conditions" seems to be about C^(-N*N) for some constant C>1, perhaps C is about 2. So therefore I estimate the number of allowable orderings is about [(N-1)*N/2]! * C^(-N*N) for some constant C>1. Note this grows to infinity with N rather rapidly regardless of the value of C. But one might be able to get an exact not approximate formula. I would suggest determining the #orderings as a function of N for small N by computer enumeration. By hand: N #ord 1 1 2 1 3 1 4 ? -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] highly enriched uranium (HEU) improvised nuclear bomb
by Thane Plambeck 19 May '16

19 May '16

>From an email I just received that was signed by the governor of California and William J Perry "A quantity of HEU the size of a basketball would be sufficient to make an improvised nuclear bomb that had the explosive power of the Hiroshima bomb and was small enough to fit into a delivery van. Such a bomb, delivered by van (or fishing boat) and detonated in one of our cities, could essentially destroy that city, causing hundreds of thousands of casualties, as well as major social, political, and economic disruptions." My question: is this really true? I mean, supposing a small group of people accomplished getting that much HEU, is it really just a matter of "improvisation" to detonate it? I was under the perhaps mistaken impression that detonating it involved some serious engineering chops, especially if you dont want to kill or fatally irradiate yourself in the attempt. -- Thane Plambeck tplambeck(a)gmail.com http://counterwave.com/

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[math-fun] Attached Figure?
by Zak Seidov 19 May '16

19 May '16

Dear sir, I've just posted to m-f list but it looks like the attached figure is rejected, right? It's my first post... -- Zak Seidov

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Re: [math-fun] Orthogonal group generators
by Warren D Smith 19 May '16

19 May '16

> From: Fred Lunnon <fred.lunnon(a)gmail.com> The following questions arose as a result of an enquiry raised on the list Geometric_Algebra <geometric_algebra(a)googlegroups.com> --- SO(n) denotes the proper isometries of an (n-1)-sphere, embedded in standard fashion in a Cartesian coordinate basis (x_1,...,x_n); R_i(t) denotes rotation through angle t about axis the coline meet of coordinate hyperplanes x_(i-1), x_i . [ Ambiguity in rotation sense need not concern us just now. ] Question (A) : Does the set G = { R_2, ..., R_n } generate SO(n) (irredundantly)? --you forgot to say which t. Question (B) : Given an arbitrary isometry, is its minimum length as a word over G necessarily at most n_C_2 = n(n-1)/2 (sharply)? --first of all, letting R_ij(t) denote rotation in plane generated by i and j coordinates, thru angle t: any member of SO(n) is generated by the R_ij(t)'s, in fact is a word with n_C_2 factors, each an R_ij(t). The t's will all differ in general. The (i,j) will also all differ. Standard Givens rotation matrix factoring/reduction algorithms show this. You seem interested not in general (i,j) but want only to allow j=i+1. So obviously, if we could express R_ab(t) in terms of the R_(a, a+1) we would be done. In fact it obviously suffices to express R_(1,3) in terms of R_(1,2) and R_(2,3). In quaternion terms it suffices to express k in terms of i and j. Well, uh, gee. k = j*i. Proof done? Well, not quite. More precisely, R_12(t) is i^(2*t/pi) in quaternion terms. So we need to express k^p in terms of i^q and j^r for arbitrary real p and some real q and r. But, well, look. Plainly, we can PERMUTE the coordinates arbitrarily using only the R_{a, a+1}(pi). Proof: Bubble sort. Therefore plainly we can permute to move any two coordinates adjacent, then use an R_{a,a+1} to rotate just those two coords, then unpermute. Therefore, the R_{a,a+1} generate the full set of R_{a,b} which in turn via standard Givens algorithms generate the full SO(n). Question (C) : Is there a practical algorithm expressing an isometry, given in the form (say) of a numerical matrix, as a word over G ? --yes.

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Re: [math-fun] solar cells using quantum dots?
by Henry Baker 19 May '16

19 May '16

Solar panel "efficiency" comes in several forms: * efficient use of space * efficient use of capital * efficient conversion of received energy etc. Except in the most intensive versions of solar farms, space efficiency isn't as important as capital cost & flexibility. Thus, if one could "paint" a 10% efficient coating cheaply onto an available surface, that would be far better than a 25% efficient solar array that required new construction to support the panel array. For rooftop use, weight & aesthetic issues may dominate efficiency. For example, if I could get good looking roof tiles that also functioned as solar collectors, that would be a big win, because I could cover 100% of my roof instead of only a small %. However, I've been told that nearly every reasonably efficient solar surface will end up looking *black*. So we need to find artists & architects who can make black roofs look good. BTW, the same "quantum dot" techniques used to improve solar cell efficiency can also be used to selectively *reflect* long infrared (aka "heat") while *absorbing & converting* shorter wavelengths, thus reducing the load on the air conditioning system & reducing the need for the electricity to run that A/C system. Here in California, I've noticed that farmers have been replacing *crop rows* with *solar panel rows*. This trend will continue, since solar panels don't have to be watered in order to produce profits! At 12:53 PM 5/19/2016, Warren D Smith wrote: >As far as I know some green plants and algae do >have several kinds of chlorophyll-containing macro-molecular >assembly, each tailored to different region of spectrum, but no single life >form employs a continuum. (If this was such a great idea, then why not?) >Perhaps however a community of 1000 algal species layered at different depths >in water could be argued to form such a continuum (not a realistic >model of nature). >And also, far as I know, the overall efficiency of green plants >at converting to sunlight to biomass=fuel is way worse than that 35% efficient >solar cell. What if the plants were living in a massively-CO2-rich atmosphere? >Would they be efficient then?

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[math-fun] solar cells using quantum dots?
by Warren D Smith 19 May '16

19 May '16

HB's idea sounds very promising, if you can develop theory and experiment and nanofab technologies to help with it... The 4-gap idea used in the record setting cell is a huge pain. If somehow we could manufacture (and do it well) quantum dots with controllable properties that varied spatially as desired, then we could presumably have energy gaps forming a continuum, allowing perfectly tailored access to entire solar spectrum, and without weird rare chemicals. That'd be the dream. Whether it merely is a pipe dream, that is the question. As far as I know some green plants and algae do have several kinds of chlorophyll-containing macro-molecular assembly, each tailored to different region of spectrum, but no single life form employs a continuum. (If this was such a great idea, then why not?) Perhaps however a community of 1000 algal species layered at different depths in water could be argued to form such a continuum (not a realistic model of nature). And also, far as I know, the overall efficiency of green plants at converting to sunlight to biomass=fuel is way worse than that 35% efficient solar cell. What if the plants were living in a massively-CO2-rich atmosphere? Would they be efficient then? -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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