- mathfun@mailman.xmission.com - mailman.xmission.com

[math-fun] Fixed Point Question
by Tom Karzes 19 Feb '20

19 Feb '20

Does a function which is a scaled proper rotation plus an offset in R^n always have a fixed point? More precisely, let R be a proper rotation matrix in R^n. Reflections are excluded. So R is an orthogonal matrix with determinant 1. Let s be a scale factor, 0 < s < 1. Let A be the matrix s*R, combining the scaling and the rotation. Let b be a vector in R^n. Then: f(v) = A*v + b It's clear that f can have at most one fixed point, since f changes the distances between points by a factor of s. Since the distance between two fixed points must remain constant, there can be at most one. I want to know if f always has a fixed point, or if there are examples where it does not. I.e., can I always find a fixed point vf that satisfies: vf = A*vf + b So we need vf to satisfy: (I - A)*vf = b where I is the identity matrix. If the determinant of (I - A) is zero, then there should be either no solution or multiple solutions. But we already know that there can be at most one solution, so I believe the question is equivalent to asking whether: det(I - A) = 0 If so, then I believe there is no fixed point. Otherwise we have: vf = inv(I - A)*b Now consider the following. If b is the zero vector, then vf exists and is also the zero vector, for any A that satisfies the problem definition. This implies that det(I - A) cannot be zero for any A that satisfies the problem definition. This is independent of the value of b. So (I - A) always has an inverse, and f always has exactly one fixed point. Note that this argument works if R has determinant -1 as well, i.e. if R includes a reflection, so R can be any orthogonal matrix. The argument also works for s > 1. Does this reasoning sound correct? Or did I miss something? Thanks, Tom

3 8

[math-fun] Palindromes multiplied
by Éric Angelini 18 Feb '20

18 Feb '20

Hello Math-Fun, is this seq finite (and correct)? S = 1,2,3,11,4,22,101,5,111,6,1001,7,88,77,8,1111,9,... (my knowledges about repunits and palindromes are close to the palindrome 0 themselves) Definition: Lexicographically earliest seq of distinct palindromes such that the product of two successive palindromes is a palindrome. Best, É.

3 5

[math-fun] Fibonacci
by Brent Meeker 17 Feb '20

17 Feb '20

My son just told me that his new Fibonacci joke is as bad as the previous two Fibonacci jokes put together. Brent

8 9

[math-fun] Morley's Theorem
by James Propp 16 Feb '20

16 Feb '20

Is Donald J. Newman's proof of Morley's Theorem the same as Conway's? The article https://www.researchgate.net/publication/267921593_Is_John_Conway's_Proof_of_Morley's_Theorem_the_Simplest_and_Free_of_A_Deus_Ex_Machina references Newman's article (reference [9]) but never discusses it, and from what I can see at https://www.cut-the-knot.org/triangle/Morley/newman.shtml it seems to me that the key insight is the same in both cases. Have any of you studied Morley's Theorem and its proofs in detail, and feel qualified to weigh in on this? There's more good stuff at https://www.cut-the-knot.org/ctk/MorleysRedux.shtml than I can process... Thanks, Jim Propp

1 0

Re: [math-fun] a long series of primes
by Keith F. Lynch 15 Feb '20

15 Feb '20

Mike Stay <metaweta(a)gmail.com> wrote: > Simon Plouffe <simon.plouffe(a)gmail.com> wrote: >> Is there simpler method to generate any number of primes ? > "Simplicity" depends entirely on the programming language. In > Mathematica, the simplest way is Prime[n]. One might think that > there are "natural" universal Turing machines, in the sense that > unnatural ones can simulate natural ones with short interpreters, > but not vice-versa. Somewhat surprisingly, that turns out to > be false. Cite? Also, from context I think Simon really means a faster way, not a simpler way. There's often a trade-off between the speed and the complexity of an algorithm. And how difficult it is to write a program often depends on how well the programming language hides the complexity. Just because it's easy to write Factor(1+2^3^4^5^6^7^8^9) doesn't mean you'll get an answer before the heat death of the universe. Also, physically building Turing-equivalent machines requires physical components, such as transistors, gears, or electromechanical relays. Some tasks inherently require either more such components, or that the fixed number of such components change state more times, than other tasks do.

2 1

[math-fun] Quanta: "The Map of Mathematics"
by Brad Klee 14 Feb '20

14 Feb '20

See: https://www.quantamagazine.org/the-map-of-mathematics-20200213/ It's no surprise that scientists would eventually get to the idea of making a map of scientific ideas. Such a task could be useful for orienteering, and possibly an opportunity to explore Hofstadter-esque quine-making. Such creations can eventually be seen as most indicative of the psychology particular to the cartographer or cartographers. Often times you will get some sort of phylogenetic tree structure, which needs to be amended with extra connections. The necessity of amendments seems to be the only common feature between the various tree models, and this underscores the empirical fact that mathematical ideas tend to converge as often as they diverge (ideas are not ultimately constrained by sexual or genetic reproduction). Alexandru Buium did not make the same mistake with definite / indefinite articles when producing "a map of mathematics", see also: https://math.unm.edu/~buium/mapp.pdf These are both interesting takes, but not entirely similar to the map that I would probably prefer, because time is not included as a variable of the branching process. Is there any reason for "Function Phylogeny"? And what is the best way to limit subjectivity? Any thoughts in general? --Brad

2 2

[math-fun] Pennies in a Frying Pan
by Richard 14 Feb '20

14 Feb '20

Dear Funsters Imagine n pennies in a frying pan. They are all flipped randomly at once and those landing heads are removed and placed heads up on a flat surface. The process is repeated with the remaining pennies until all coins show heads. What is the expected number of flips, f(n), for n coins? f(1) = 2, f(2) = 8/3, f(3) = 66/3/7, f(4) = 1104/3/7/15, f(5) = 37050/3/7/15/31 etc. The numerators follow the sequence 2, 8, 66, 1104, 37050, 2482200, 31336530, 88081529760, 46640940221610.... 1. I don't find this sequence in the OEIS. Should it be there or is there a related sequence that is there? 2 f(64) = 7.34399... f(128) = 8.33837... f(256) = 9.33556... f(512) = 10.33415... f(1024) = 11.33345... f(2048) = 12.33309... f(4096) = 13.33292... f(8192) = 14.33283... f(16384) = 15.3327... Perhaps someone can verify these results and determine the limit of e(k) as k approaches infinity for f(2^k) = k+1+e(k). Any help is appreciated. Dick Hess

3 3

[math-fun] a long series of primes
by Simon Plouffe 13 Feb '20

13 Feb '20

Hello, I have been working on ways to generate primes with the simplest formula. Here is one : if c = 31622.7 ... then { c^n } is prime for n = 2 to 388 : 387 consecutive primes. { } is the nearest integer. The sequence begins like this : 1000000007 31622776952311 1000000014783746303 31622777186062677745609 1000000022175619536498921059 31622777419814234539614807614633 ... and not listed in the OEIS. Here is the complete list and more precise value of the constant. http://plouffe.fr/NEW/List%20of%20387%20consecutive%20primes.txt I developed a method based on Monte-Carlo and Simulated Annealing https://en.wikipedia.org/wiki/Simulated_annealing The method allows me to find other candidates which would give a longer series of primes. This formulation is the simplest I could imagine. Is there simpler method to generate any number of primes ? Best regards, Simon Plouffe

2 1

[math-fun] Sparse rulers proof
by ed pegg 12 Feb '20

12 Feb '20

TLDR version: ||||.......................|....|...|...|...|...|...|..|..| Sparse rulers allow the measuring of all distances up to length n (58) with a minimal number of marks M (13). The author believes B. Wichmann almost completely solved the problem in 1963. From Wichmann's sparse ruler recipe, M_n - ⌈√(3 n+9/4)⌋ can be shown to usually be 0 or 1 (the excess). The length 58 ruler at the start of this paragraph is conjectured to be the longest sparse ruler with fewer marks than a Wichmann construction, but no proof is known. Adding a single mark to the end of Wichmann's construction allows the building of excess-01 rulers for all lengths over 257992. The author's code found solutions for hundreds of difficult smaller lengths such as 1792, 5657 and 16617 to establish that excess-01 rulers always exist. The optimality of excess 1 solutions for lengths 474, 501, 582, 609, 669, 792, 793, ... may be doubtful since they break the excess pattern. The OEIS version is https://oeis.org/A326499 https://blog.wolfram.com/2020/02/12/hitting-all-the-marks-exploring-new-bou… --Ed Pegg Jr

1 0

[math-fun] Suggestion for next Iowa Basic Skills Test
by Brad Klee 11 Feb '20

11 Feb '20

Did anyone else have to take these tests as a kid? There was no real reward for doing well, but if you fail, you get labeled as "not-up-to-standard". I can't remember any of their problems, but thought of something else today: An election has five candidates finish with double-digit percentages, though, all less than 30%. After about 20 hours, the committee assigned to count votes decides at first to announce only 62% of all results. Define an "exclusion scenario" as a set C_1, C_2, ..., C_n of candidates who are announced to have zero percent support on a fair tally of the 62% subset. How many different exclusion scenarios are possible when allowing for any 62% subset? Is it possible to completely exclude the overall winner? I think this can be solved using standard combinatorics, but I am wondering if anyone would understand the implications. --Brad

1 3