January 2018 - mathfun@mailman.xmission.com

[math-fun] Polygon discriminant sequence
by Ed Pegg Jr 01 Feb '18

01 Feb '18

I've been looking at discriminants today. Specifically, sequence https://oeis.org/A193681 Discriminant of minimal polynomial of 2*cos(Pi/n) I've noticed that all these numbers are almost-powers of n. For n with the following values 1, 5, 7, 9, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 45, 47, 49, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179 The power is exactly 1, 1, 2, 2, 4, 5, 7, 8, 10, 13, 14, 17, 19, 20, 9, 22, 19, 25, 28, 29, 32, 34, 35, 38, 40, 43, 47, 49, 50, 52, 53, 55, 52, 62, 64, 67, 68, 73, 74, 77, 80, 82, 85, 88 What about the rest? Here are values {n,p,a,b} so that discriminant of minimal polynomial of 2*cos(Pi/n) is (n^p a^b) discriminant( 2*cos(Pi/17) ) = 17^7 (Gauss's 17-gon) discriminant( 2*cos(Pi/168) ) = 168^48 189^(-8) discriminant( 2*cos(Pi/179) ) = 179^88 Is there any rhyme or reason to this sequence? --Ed Pegg Jr {{1,1,1,1}, {2,1,2,-1}, {3,1,3,-1}, {4,1,2,1}, {5,1,1,1}, {6,1,2,1}, {7,2,1,1}, {8,4,2,-1}, {9,2,1,1}, {10,3,2,1}, {11,4,1,1}, {12,4,3,-2}, {13,5,1,1}, {14,5,2,1}, {15,3,3,-1}, {16,8,2,-1}, {17,7,1,1}, {18,4,12,1}, {19,8,1,1}, {20,6,2,4}, {21,5,3,-2}, {22,9,2,1}, {23,10,1,1}, {24,8,3,-4}, {25,8,5,1}, {26,11,2,1}, {27,7,3,1}, {28,10,2,4}, {29,13,1,1}, {30,4,20,2}, {31,14,1,1}, {32,16,2,-1}, {33,9,3,-4}, {34,15,2,1}, {35,10,5,-1}, {36,9,2,6}, {37,17,1,1}, {38,17,2,1}, {39,11,3,-5}, {40,16,5,-4}, {41,19,1,1}, {42,12,189,-2}, {43,20,1,1}, {44,18,2,4}, {45,9,1,1}, {46,21,2,1}, {47,22,1,1}, {48,16,3,-8}, {49,19,1,1}, {50,17,40,1}, {51,15,3,-7}, {52,22,2,4}, {53,25,1,1}, {54,15,2,3}, {55,18,5,-3}, {56,24,7,-4}, {57,17,3,-8}, {58,27,2,1}, {59,28,1,1}, {60,16,45,-4}, {61,29,1,1}, {62,29,2,1}, {63,15,3,-3}, {64,32,2,-1}, {65,22,5,-4}, {66,20,2673,-2}, {67,32,1,1}, {68,30,2,4}, {69,21,3,-10}, {70,18,56,2}, {71,34,1,1}, {72,18,2,18}, {73,35,1,1}, {74,35,2,1}, {75,18,32805,-1}, {76,34,2,4}, {77,27,7,-2}, {78,24,9477,-2}, {79,38,1,1}, {80,32,5,-8}, {81,23,3,2}, {82,39,2,1}, {83,40,1,1}, {84,24,189,-4}, {85,30,5,-6}, {86,41,2,1}, {87,27,3,-13}, {88,36,2,12}, {89,43,1,1}, {90,18,2,6}, {91,33,7,-3}, {92,42,2,4}, {93,29,3,-14}, {94,45,2,1}, {95,34,5,-7}, {96,32,3,-16}, {97,47,1,1}, {98,38,112,1}, {99,27,3,-9}, {100,35,2,10}, {101,49,1,1}, {102,32,111537,-2}, {103,50,1,1}, {104,44,2,12}, {105,20,405,-2}, {106,51,2,1}, {107,52,1,1}, {108,30,2,12}, {109,53,1,1}, {110,30,42592,2}, {111,35,3,-17}, {112,48,7,-8}, {113,55,1,1}, {114,36,373977,-2}, {115,42,5,-9}, {116,54,2,4}, {117,33,3,-12}, {118,57,2,1}, {119,45,7,-5}, {120,32,45,-8}, {121,52,1,1}, {122,59,2,1}, {123,39,3,-19}, {124,58,2,4}, {125,46,5,-1}, {126,27,56,3}, {127,62,1,1}, {128,64,2,-1}, {129,41,3,-20}, {130,36,1352,4}, {131,64,1,1}, {132,40,2673,-4}, {133,51,7,-6}, {134,65,2,1}, {135,30,5,-3}, {136,60,2,12}, {137,67,1,1}, {138,44,4074381,-2}, {139,68,1,1}, {140,36,448,4}, {141,45,3,-22}, {142,69,2,1}, {143,55,11,-1}, {144,48,3,-24}, {145,54,5,-12}, {146,71,2,1}, {147,39,2711943423,-1}, {148,70,2,4}, {149,73,1,1}, {150,40,45,-10}, {151,74,1,1}, {152,68,2,12}, {153,45,3,-18}, {154,50,3872,2}, {155,58,5,-13}, {156,48,9477,-4}, {157,77,1,1}, {158,77,2,1}, {159,51,3,-25}, {160,64,5,-16}, {161,63,7,-8}, {162,47,384,1}, {163,80,1,1}, {164,78,2,4}, {165,36,820125,-2}, {166,81,2,1}, {167,82,1,1}, {168,48,189,-8}, {169,74,13,1}, {170,64,10625,-4}, {171,51,3,-21}, {172,82,2,4}, {173,85,1,1}, {174,56,138706101,-2}, {175,53,1715,-1}, {176,80,11,-8}, {177,57,3,-28}, {178,87,2,1}, {179,88,1,1}, {180,36,2,24}}

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[math-fun] How can they call them twin primes
by Bill Gosper 31 Jan '18

31 Jan '18

In[1]:= PrimeQ[5280 + {1, -1}] Out[1]= {True, True} if there's a mile between them? --rwg

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[math-fun] Fourier Transform video
by Henry Baker 31 Jan '18

31 Jan '18

I'm curious as to what others think about this Fourier Transform video (1/26/2018). http://www.3blue1brown.com/ https://www.youtube.com/c/3blue1brown

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Re: [math-fun] Non-electronic analogue computers
by Henry Baker 30 Jan '18

30 Jan '18

There's the famous 1949 "MONIAC" ("Money AC"; get it?), which is one of the few computers not influenced by Al Gore, who was but 1 year old at the time. Rube Goldberg and Heath Robinson would both have been proud of the MONIAC. "Other [MONIAC] computers went to Harvard Business School", where it presumably educated an entire generation of corporate and political leaders about how macroeconomics and tax policy work. :-) I understand that some members of Congress still think that this is the way computers and the Internet are built -- out of a series of tubes and tanks, leaking/dripping with sarcasm. :-) I'm sure that someone is working on a blockchain model of the economy as we speak. :-) https://en.wikipedia.org/wiki/MONIAC_Computer The MONIAC was approximately 2 m high, 1.2 m wide and almost 1 m deep, and consisted of a series of transparent plastic tanks and pipes which were fastened to a wooden board. Each tank represented some aspect of the UK national economy and the flow of money around the economy was illustrated by coloured water. At the top of the board was a large tank called the treasury. Water (representing money) flowed from the treasury to other tanks representing the various ways in which a country could spend its money. For example, there were tanks for health and education. To increase spending on health care a tap could be opened to drain water from the treasury to the tank which represented health spending. Water then ran further down the model to other tanks, representing other interactions in the economy. Water could be pumped back to the treasury from some of the tanks to represent taxation. Changes in tax rates were modeled by increasing or decreasing pumping speeds. ... The actual flow of the water was automatically controlled through a series of floats, counterweights, electrodes and cords. When the level of water reached a certain level in a tank, pumps and drains would be activated. To their surprise, Phillips and his associate Walter Newlyn found that MONIAC could be calibrated to an accuracy of Â±2%. [Perhaps the success of this analogue computer explains the anemic British post-WWII economy?] The first MONIAC was created in his landlady's garage in Croydon at a cost of Â£400 (equivalent to Â£13,000 in 2016). Phillips first demonstrated the MONIAC to a number of leading economists at the LSE in 1949. It was very well received and Phillips was soon offered a teaching position at the LSE. At 11:58 AM 1/29/2018, Mike Stay wrote: >Gitton's water clock implemented a counter using tanks and siphons.

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[math-fun] Non-electronic analogue computers
by James Propp 29 Jan '18

29 Jan '18

Anyone know of any good designs for an easy-to-make analogue computer based on springs, masses, and dashpots? That is, we want a supply of easily-interconnectable components that we can combine in ways that correspond to a prescribed differential equation, so that the behavior of the system will be a solution of the equation. As I recall, there’s also a way to get hydraulic analogues of LRC networks (though I forget what the three sorts of components are called); is there a good design for a hydraulic computer? Jim Propp

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[math-fun] Non-electronic analogue computers
by Guy Haworth 29 Jan '18

29 Jan '18

Maybe I have come late to this conversation ... but I vaguely remember that Maurice Wilkes was initially put in charge of Cambridge's University's Mathematical Laboratory http://www.cl.cam.ac.uk/relics/history.html which: - had several mechanical machines for solving differential equations, https://en.wikipedia.org/wiki/Differential_analyser, and - maybe Meccano was involved too. There are several non-Babbage mechanical computers in the Science Museum in London (but maybe not now on show), and ... Jack Howlett wrote a book on early pre-electronic computers, maybe a translation from the original French. Sadly, I cannot immediately identify the book that Jack published. MIT Press might have been the publisher. Maybe it was 'The computer comes of age'. Guy

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Re: [math-fun] Non-electronic analogue computers
by Henry Baker 29 Jan '18

29 Jan '18

Jim: I just went through a similar search for pedagogical electronic design. Google "EDA" (Electronic Design and Analysis). There are a number of FOSS (Free and Open-Source Software) EDA tools for both Linux and Windows. The venerable program "Spice" can do time-stepped simulation/emulation of electronic circuits, but I don't think that Spice will do this symbolically. However, I think that there are some other programs can also read the same Spice files and -- at least for *linear time-invariant circuits* -- produce and solve symbolic equations. Spice files have a trivial format, wherein each line describes a link in the schematic graph structure: C23 2 3 0.1 would be a *capacitor* going from graph node labelled "2" to graph node labelled "3" and have a capacitive value of 0.1 Farads. The graph node labelled "0" is "ground", by convention. The process of going from the schematic circuit to the mathematical matrix format is called "Nodal Analysis" or more likely "Modified Nodal Analysis". Each "link" becomes *four* entries in the matrix -- at four corners of a rectangle -- called a "stamp" (?!?). I believe that the mechanical engineers utilize a similar -- if not identical -- format for their matrices. If you want to play with *digital signal processing*, I can also recommend "GnuRadio" and "GnuRadio Companion" (GRC). GRC allows you to lay out a digital signal processing system in block diagram form and then emulate it in real time. You can play with GRC using just your microphone and speakers -- you don't need any actual radio hardware. At 05:09 AM 1/29/2018, James Propp wrote: >Anyone know of any good designs for an easy-to-make analogue computer based on springs, masses, and dashpots? > >That is, we want a supply of easily-interconnectable components that we can combine in ways that correspond to a prescribed differential equation, so that the behavior of the system will be a solution of the equation. > >As I recall, there's also a way to get hydraulic analogues of LRC networks (though I forget what the three sorts of components are called); is there a good design for a hydraulic computer? > >Jim Propp

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Re: [math-fun] A binary sieve
by Dan Asimov 28 Jan '18

28 Jan '18

"... then delete every j for which ..." A priori it seems possible that the order in which such j are deleted could change the outcome. —Dan Fred Lunnon wrote: ----- Choose an integer k >= 2 , let q' = 2^k - 1 , initialise set S = {1, ..., q'-1} ; then delete every j for which i < j exists with j == i*2^h (mod q') for some h . For example, when k = 5 , q' = 31 , finally S = {1, 3, 5, 7, 11, 15} . I conjecture that max S = (q'-1)/2 for all k ; but why? -----

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[math-fun] A binary sieve
by Fred Lunnon 28 Jan '18

28 Jan '18

Choose an integer k >= 2 , let q' = 2^k - 1 , initialise set S = {1, ..., q'-1} ; then delete every j for which i < j exists with j == i*2^h (mod q') for some h . For example, when k = 5 , q' = 31 , finally S = {1, 3, 5, 7, 11, 15} . I conjecture that max S = (q'-1)/2 for all k ; but why? Fred Lunnon

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[math-fun] Nice numbers
by Keith F. Lynch 28 Jan '18

28 Jan '18

I've been exploring nice numbers. By "nice numbers" I of course refer to numbers that aren't mean. :-) I started by searching for sets of integers that don't contain the arithmetic mean of any pair of their elements. I learned that this is well explored territory. A greedy algorithm found a sequence which, when I looked it up in OEIS, was described as the numbers expressible in ternary without the digit 2. A005836. It's not obvious to me why that should be. Comments confirmed that the property I noticed is well known, but phrased in such a way that my search didn't find it. Next I modified my program to search for sets of integers that don't contain the *geometric* mean of any of their elements. It generated what turned out to be A000452, numbers whose prime factorization only contains exponents expressible in ternary without the digit 2. Again, comments confirmed that the property I noticed is well known, but, again, phrased in such a way that my search didn't find it. I'm seeking a set of totally nice numbers, a set of integers which doesn't contain any power mean of any pairs of elements. (I consider the geometric mean to be the zeroth power mean, which it technically is only in the limit.) Has anyone done this already? Thanks.

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