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[math-fun] Topology of the Lorenz System
by Brad Klee 22 Jan '21

22 Jan '21

A question was asked whether or not the "template" of the Lorenz strange attractor could have the same topology of the YGB surface, and I said, for typical parameters, the answer is No. See Figure 3 of: https://arxiv.org/pdf/1610.07079.pdf Cut along the midline and glue together the two loose ends, then you have a mobius strip. Repeat: When self-intersecting a mobius strip on an edge, there is a choice between two alternatives. The Lorenz template features an intersection with a point-like join between two cycles. The YGB surface involves a line-like join between two cycles (Try drawing graphs with two T-nodes and two cycles). Is there a chaotic system of ODE's whose attractor template is topologically equivalent to the YGB surface? The other issue is almost-periodic analysis of the Lorenz system, when considered as two joined oscillation disks. Has anything been written on Period Functions? Does this question sound interesting to anyone else? --Brad

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Re: [math-fun] September 10000, 1993
by Keith F. Lynch 21 Jan '21

21 Jan '21

M F Hasler <mhasler(a)dsi972.fr> wrote: > Except for the fact that there was certainly no month more that > 10^20 (or so) years ago, because the universe didn't exist yet > and/or more specifically, time as we define it doesn't reach that > far into the past. Yes, as I mentioned. The Big Bang was a mere 10^10.14 years ago. Whether time existed before that, nobody knows. But this is math, not physics or history. Numbers go on forever. Both Earth and the known universe were around 10^12 days ago, but not 10^13. 33.0% of all days in a Gregorian period (400 years or 146097 days) are the 10^Nth day of some month in the past 10^12 days. That's just under a third. The Gregorian calendar was in use 10^5 days ago (albeit only in Catholic countries), but not 10^6. 18.8% of all days in a Gregorian period are the 10^Nth day of some month in the past 10^5 days. > But yes, the answer to the original question relies only on residues > (mod 7) of powers of 10 , so it is essentially the same for any > calender that has weeks of seven days and no exceptions to that. It > is quite obvious that one particular value will never appear as such > a residue, Right, since 10 and 7 are relatively prime. > slightly less obvious that all others will (and do so for N < 7 ? > incidentally corresponding to roughly the age of mankind). Only because the decimal expansion of 1/7 repeats after 6 digits. Had I tried a different base, the expansion could repeat after any of 1, 2, 3, or 6 digits. If not 6, then several days of the week would never appear. And it's not obvious to me that if I were to re-run my second question for other bases that I'd get a solution for every day, e.g. that every day is the 11^Nth day of some prior month (possibly in the absurdly distant past) for some N. Here's how many of the 146097 days in a Gregorian period are the 10^Nth day of some month for each N, not counting days for which a smaller N applies. There are 4800 for N=0 since every month has a first day, and there's obviously never any previous month with a smaller N. There are 4800 for N=1 since every month has a tenth day, which is of course never also its first day. But there are only 4097 for N=2 since some 100th days of a month land on the 10th day of subsequent month. The smallest N that doesn't appear is 75. The largest N that does appear is 306, for just 9 out of every 146097 days. The largest N with 4800 is 9; the billionth day of each month somehow manages to never land on any 1st, 10th, 100th, 1000th, ... or hundred millionth day of any month. 4800 0 4800 1 4097 2 4800 3 4800 4 4198 5 3234 6 3247 7 4466 8 4800 9 1316 10 2322 11 1354 12 4264 13 4632 14 2834 15 2906 16 3565 17 3246 18 2807 19 2169 20 1768 21 1881 22 2949 23 3752 24 799 25 551 26 259 27 4466 28 989 29 1896 30 753 31 1861 32 1144 33 1827 34 2505 35 1505 36 1579 37 1482 38 2345 39 982 40 721 41 1882 42 856 43 1440 44 1622 45 1546 46 560 47 1641 48 843 49 405 50 1169 51 975 52 619 53 714 54 1099 55 1003 56 460 57 137 58 777 59 257 60 1149 61 905 62 748 63 287 64 124 65 39 66 632 67 74 68 397 69 1222 70 58 71 141 72 341 73 620 74 181 76 110 77 153 78 10 79 446 80 346 81 647 82 371 83 63 84 258 85 173 86 175 87 243 88 243 89 44 90 44 91 17 92 12 93 195 94 22 95 101 96 539 97 219 98 87 99 93 100 134 101 94 102 266 103 304 104 173 105 107 106 376 107 404 108 22 109 102 110 77 111 280 112 37 114 58 115 121 116 12 118 187 119 66 120 17 121 44 122 35 123 240 124 106 125 80 126 121 127 16 128 114 130 114 131 99 132 12 133 6 134 21 135 18 136 140 137 14 138 411 140 87 141 9 142 9 143 54 145 33 146 48 147 20 148 28 149 15 150 5 151 10 152 7 154 2 155 6 156 307 159 2 161 189 162 7 163 12 164 15 165 48 166 10 169 9 170 9 172 81 176 82 177 23 178 25 181 2 182 28 183 26 184 9 186 5 187 16 188 33 190 17 197 16 199 4 202 2 204 35 206 23 213 12 216 17 219 3 222 2 223 13 225 33 227 3 229 1 238 1 261 2 280 9 306 A hundred days ago was last October. A thousand days ago was in April of 2018. Ten thousand days ago was in September 1993. A hundred thousand days ago was in April 1747, less than six years before Britain and its colonies adopted the Gregorian calendar. A million days ago was in February of 718 BCE, 35 years after Rome's founding. Ten million days ago was in the middle of the last glacial period, long before recorded history. A hundred million days ago was early in the third from last glacial period. A billion days ago was during the Pliocene epoch, before our species existed. Ten billion days ago was during the Oligocene epoch, early in the age of mammals. A hundred billion days ago was during the Permian period, the last part of the Paleozoic era, before mammals, or even dinosaurs, existed, but after the great coal forests. A trillion days ago was during the late Archean or early Proterozoic eon, when the only life was single cells, and there was little or no oxygen in the atmosphere. Ten trillion days ago was twice as long ago as the Big Bang. But there are isotopes that will last much longer than that, stars that will still be shining longer than that, and computer programs that would take much longer than that to finish.

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[math-fun] Alternatives to Gram-Schmidt
by Henry Baker 21 Jan '21

21 Jan '21

Gram-Schmidt takes a set of vectors, puts them into some arbitrary order, and then "orthogonalizes" them by subtracting off non-orthogonal components. The problem with G-S is that there seems to be little rhyme or reason for the initial sort order. I was curious about trying to *minimize* the *length* of the *description* of the final set of orthogonal vectors. By *description*, I mean the total length in bits of all of the numerators and denominators. My first attempt is a "greedy" algorithm that find the 2 vectors which are at the least orthogonal (greatest |angle-90|) from one another, replace them with the sum and difference of their normalizations, and then rinse and repeat. Note that the sum and difference of same-length vectors are orthogonal. Thus, if one starts with N independent vectors, then it appears that this "greedy" algorithm converges on a mutually orthogonal set. While the convergence isn't fast, it does appear to keep the total description lengths from exploding. The convergence rate appears to be similar to Jacobi/Givens. I would imagine that this algorithm would have interesting behavior when trying to mutually orthogonalize N+1 vectors in an N-dimensional space. It would attempt to find a distribute the vectors over the surface of the N-dimensional unit hypersphere with the arrow heads as far from one another as possible. Ditto for N+2, N+3, etc., vectors.

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Re: [math-fun] September 10000, 1993
by Keith F. Lynch 20 Jan '21

20 Jan '21

"Keith F. Lynch" <kfl(a)KeithLynch.net> wrote: > My full table, which shows just the most recent solution for each > day of a full Gregorian period (400 years = 146097 days), can be > downloaded from http://KeithLynch.net/146097.txt. The code that > generated it can be downloaded from http://KeithLynch.net/146097.c. If those URLs don't work for you, try leaving out the periods at the ends of the sentences. I wrongly thought that was obvious, but apparently not.

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Re: [math-fun] September 10000, 1993
by Keith F. Lynch 20 Jan '21

20 Jan '21

"Keith F. Lynch" <kfl(a)KeithLynch.net> wrote: > The solution to a related but more difficult calendar puzzle *does* > depend on the details of our calendar. What about power-of-ten dates > of *all* months? Is there any day which is *not* such a day? (Using > the proleptic Gregorian calendar.) The 1st and the 10th of each month > are of course the 0th and 1st powers of 10 of that month. But what > about the other 26, 27, 28, or 29 days of that month? . . . > Also, since the calendar does repeat, if we find solutions for > 146097 consecutive days, we'll have found a solution for all days. I have done exactly that, and confirmed that *every* day can indeed be expressed as day 10^N of some prior month in infinitely many ways, and that the most recent such N never exceeds 306. Here are the last few and next few days: W 1/13/2021 4/100000/1747 R 1/14/2021 12/10^55/1795 F 1/15/2021 6/10^64/1971 S 1/16/2021 9/10000/1993 N 1/17/2021 6/10^18/1913 M 1/18/2021 4/10^35/1954 T 1/19/2021 8/10^183/1833 W 1/20/2021 9/10^28/1963 R 1/21/2021 11/10^52/1790 F 1/22/2021 6/10^176/1713 S 1/23/2021 4/10^23/2087 N 1/24/2021 5/1000/2018 M 1/25/2021 3/1000000/2083 T 1/26/2021 2/10^12/1814 W 1/27/2021 1/10^16/1736 Note that the year on the right is usually wrong, sometimes by an immense number of years. But always an exact multple of 400 years. For instance the 24th of this month is of course not actually day million of March 2083, it's day million of March 718 BCE, exactly 2800 years earlier. > Since that number (400/146097) is rational, the decimal expansion > repeats, so there are only a finite number of distinct fractional > numbers of years ago, no matter how many powers of 10 you try. I > don't know the period of repetition of 400/146097, but since 146097 > is 3^3 * 7 * 773, I think it's at most 6*772=4632 decimal digits. It is exactly 1/4 of that. 1158 digits. Hence January 16 wasn't just September 10^4, 1993, it was also September 10^1162 of an enormously earlier year, September 10^2320 of an even earlier year, ad infinitum. Phrased another way, 10^1159 = 1 (mod 146097). My full table, which shows just the most recent solution for each day of a full Gregorian period (400 years = 146097 days), can be downloaded from http://KeithLynch.net/146097.txt. The code that generated it can be downloaded from http://KeithLynch.net/146097.c.

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[math-fun] "smallest" degree n irreducible polynomial over GF(p)
by Mike Speciner 19 Jan '21

19 Jan '21

Let $n = 2^{e_0} q_1^{e_1} \cdots q_i^{e_i}$ be the prime factorization of $n$. Let $p$ be an odd prime. Is it true that $\exists k$ such that $x^n+k$ is irreducible over GF($p$) $\iff$ $(2^{min(e_0,2)} q_1 \cdots q_i) | (p-1)$ ? Are there similar criteria for "larger" polynomials such as $x^n + k_1 x + k_0$ ?

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[math-fun] Tictactoid game with arithmetic sequences.
by Allan Wechsler 18 Jan '21

18 Jan '21

I am sure this two-parameter family of games has been analyzed up the wazoo, so I am hoping somebody can just tell me what's known. X and O take turns claiming an integer from [1,n]. The winner is the first to have claimed an arithmetic sequence of length k. The classic strategy-stealing argument shows that this game is always either a win for the first player or a draw. What values of (n,k) have known outcomes? If k = 3, it's a draw for k < 6 and a win for k > 6; I haven't quite analyzed k = 6. Higher k's are more interesting. Is there a complete theory out there?

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[math-fun] Simpsonian conjecture
by Bill Gosper 18 Jan '21

18 Jan '21

If a plane P through the center of an ellipsoid E intersects it in area A, the volume of E is 2Ah/3, where h is the distance between the two planes parallel to P and tangent to E. —rwg

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[math-fun] Conjecture about the distribution of nontrivial powers.
by Allan Wechsler 17 Jan '21

17 Jan '21

A number b^e, where b is a positive integer, and e an integer at least 2, we will call a "nontrivial power". This term is from Neil Sloane, and I think it's superior to the term "perfect power" which seems to have some currency. The sequence of perfect powers is collected as A001597 at OEIS. We naturally turn our attention to the b's and e's of the entries in this sequence, but there is some ambiguity here, exemplified by 16 = 4^2 = 2^4. Let us, for the moment, resolve the ambiguity by preferring, for this purpose, the representation with the smallest exponent. With this choice, the sequence of b's is A072813, while the e's are at A072814. The exponent sequence A072814 is dominated by 2's, because cubes and other higher powers are much rarer than squares. Our attention is attracted by the rare cases when two higher powers are sandwiched between consecutive squares. The smallest example is the pair 3^3 = 27 and 2^5 = 32, sandwiched between the squares of 5 and 6. The next example occurs between the squares of 11 and 12, 5^3 = 125 and 2^7 = 128. The third example has 3^7 = 2187 and 13^3 = 2197, sandwiched between the squares of 46 and 47. Are there an infinite number of such examples? Are there any examples of *three* higher powers between consecutive squares? Hugo Pfoertner and I have expressed cautious skepticism; if I read Hugo's tone right, he and I both suspect there aren't any. Perhaps it is too ambitious to call this a conjecture, as I did in the subject of this message. Similar questions about the existence of primes in intervals of various sizes are notoriously hard, but I would expect this problem to be much easier.

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[math-fun] September 10000, 1993
by Keith F. Lynch 17 Jan '21

17 Jan '21

There's a bit of Internet folklore that says September 1993 never ended. (Look up "Eternal September" for details.) September 100, 1993 was otherwise known as Thursday, December 9th, 1993. September 1000, 1993 was otherwise known as Monday, May 27, 1996. September 10000, 1993 is today, otherwise known as Saturday, January 16, 2021. I thought about the pattern of the days of the week of the successive rollover days, and concluded that there's just one day of the week that they will never land on. Can you see which one, and why?

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