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[math-fun] a nice day
by françois mendzina essomba2 14 Mar '18

14 Mar '18

Nice day in Yaoundé today... log(2)=sum((-1)^k*(1-1/2^k)*zeta(k), k= 2..infinity); An improvement in convergence. S(n) = sum((-1)^k*(1-1/2^k)*zeta(k), k= 2..n); log(2) = (S(n+1) + S(n))/2 ; n ----- infinity... FME...

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[math-fun] Piday
by françois mendzina essomba2 14 Mar '18

14 Mar '18

In memory of Pi... a(n+1)=2^(n+3/2)*sqrt(1/r-((1-2^(-2*n-1)*(a(n))^2*r)^(1/r)+1)^(r/2)/(r*2^(r/2))); a1=2^(3/2)/sqrt(r) ; r in R ; n----infinity ; a(n) ---- Pi.

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Re: [math-fun] History of swap/exch computer instructions ?
by rcs＠xmission.com 13 Mar '18

13 Mar '18

[The list software objected to this message; I don't know why. Retrying. -- Rich] Date: Tue, 13 Mar 2018 21:13:52 -0400 Subject: Re: [math-fun] History of swap/exch computer instructions ? From: wba <wbackerman(a)gmail.com> I the early days, many computers used the read-clear and write-to-previously-cleared properties of core memory. To recapitulate how that worked, a memory cycle began by sending a pulse to the core that clears it, while sense amplifiers sense the weak signal telling whether or not the core "turned over". Then the new data was written back by selectively sending pulses to the core in the other direction, setting bits to 1. Even if you only wanted to read, or only wanted to write, you had to do both halves of the cycle, to rewrite what you read, or to pre-clear what you were about to write. These properties were used for "add to memory" types of instructions, as well as various atomic synchronization primitives, such as semaphores. And, of course, for reg<->mem swaps. The atomic synchronization primitives were harder to implement when core memory got replaced by plain RAM. The DEC PDP-1 at the Research Laboratory of Electronics at MIT (a number of correspondents on this list remember that machine) had an interesting application of swapping something with memory. The time-sharing system needed to swap whole memory fields (4096 words!) onto and off of the drum when switching processes. A given field on the drum (one revolution, 4096 bits, 18 fixed tracks) had to be read into core, while the previous contents of core were written to a different 18 tracks. This required a sort of "supraluminal" information transfer. Faster than light? No. Just faster than the drum's rotation. Writing bits onto a track required blasting a powerful pulse from the write amplifiers onto the write heads. The electrical noise from this blast completely overwhelmed the sensitive read amplifiers trying to listen to another track. So they took advantage of the fact that whatever pulse was delivered to the write heads would have its magnetic field sort of smeared out in space, because the heads rode about .001 inches above the surface. Similarly, the read heads would "hear" an average of the smeared out magnetic field of the drum bit going by. So the mechanism would read data from memory, clearing same and holding that memory word open, and then blast that onto the desired write track. Just before the blast occurred, the read heads on the other track could sort of hear the other bit coming. The read amplifiers sensed the yet-to-arrive data just before the blast, and write that back into core.

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Re: [math-fun] History of swap/exch computer instructions ?
by Henry Baker 13 Mar '18

13 Mar '18

Thanks, Rich! I looked at the 7094 instruction set manual online, but didn't see anything like an exchange/swap instruction. I looked at an online GE645 instruction set manual, but didn't find anything there. Exchange/swap were considered back then for two reasons: 1. memory & registers were extremely expensive, so a single instruction exchange/swap would save both data/register memory and instruction memory. 2. main memory was made from ferrite cores, so reading was *destructive*. Since you had to write *something* back to memory anyway, you might as well write something *else* just as easily as writing back what you just read. (The PDP-8 curiously did this backwards: its "DCA" (Deposit and Clear Accumulator) stored the accumulator and set it to zero, but the "LDA" (LoaD Accumulator) instruction did *not* clear the memory location from which it was loaded! It would be interesting to know if there were any computers who "utilized" this *destructive read* "feature" of core memory.) I do recall that some computers back then had the ability to exchange a memory location with the *program counter*! This enabled fast Fortran I subroutining (no stack; no recursion; no service) and "co-routining". Perhaps the PDP-6/10 did this sort of thing as well. At 03:59 PM 3/13/2018, rcs(a)xmission.com wrote: >The PDP6 had EXCH A,M which swapped an Accumulator with a Memory location. >The manual is dated 1964. > >At that time, EXCH as an atomic instruction was believed useful for certain >safely-interruptible algorithms --- futzing with linked lists &c. > >Maybe check out an old IBM 7094 manual? > >Rich >---- >Quoting Michael Greenwald <mbgreen(a)seas.upenn.edu>: >>On 2018-03-08 08:10, Henry Baker wrote: >>>I'm curious as to the origin of the exch/swap instruction in computer >>>instruction sets. >>> >>>Does anyone know the first computer having such an instruction? >>> >>>I'm not interested in the modern "compare-and-swap" type of >>>instruction, which seems to have originated with Maurice Herlihy, but >>>the original, non-conditional type. >> >>I don't think Maurice originated the instruction; I am pretty sure >>some versions of it predated him by decades (IBM 370). He was just >>the one who pointed out and proved that it was universal (in the >>context of non-blocking synchronization), as was the pair LL SC. >> >>But I have no answer to your main question, sorry. >> >>>Yes, early computer scientists (before they were called that!) loved >>>these instructions because they allowed for clever algorithms like a >>>stackless GC algorithm.

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[math-fun] History of swap/exch computer instructions ?
by Henry Baker 13 Mar '18

13 Mar '18

I'm curious as to the origin of the exch/swap instruction in computer instruction sets. Does anyone know the first computer having such an instruction? I'm not interested in the modern "compare-and-swap" type of instruction, which seems to have originated with Maurice Herlihy, but the original, non-conditional type. Yes, early computer scientists (before they were called that!) loved these instructions because they allowed for clever algorithms like a stackless GC algorithm.

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[math-fun] Wieferich #s
by Bill Gosper 13 Mar '18

13 Mar '18

https://oeis.org/A077816 , e.g., In[19]:= Select[Range@9999, Mod[2^EulerPhi@#, #^2] == 1 &] Out[19]= {1093, 3279, 3511, 7651} A different definition might be In[18]:= Select[Range@999999, Mod[2^#, #^2] == 2 &] Out[18]= {1093, 3511} No mention of primes! https://oeis.org/A001220 Arthur Wieferich showed that if p is not a term of this sequence, then the First Case of Fermat's Last Theorem has no solution in x, y and z for prime exponent p (cf. Wieferich, 1909). - Felix Fröhlich <https://oeis.org/wiki/User:Felix_Fröhlich>, May 27 2016 --rwg

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[math-fun] Common subsequence length
by David Wilson 11 Mar '18

11 Mar '18

Let S be a finite set with |S| = s. Let A be a finite random sequence of elements from S with |A| = a. Let B be a finite random sequence of elements from S with |B| = b. What is the expected length f(s, a, b) of a longest common subsequence of A and B? As an example, let A and B be finite sequences of 10 decimal digits with |A| = |B| = 10. Then S = set of decimal digits with |S| = 10. Two possible values of A and B are A = (3,1,4,1,5,9,2,6,5,3) B = (1,4,1,4,2,1,3,5,6,2) A longest common subsequence of A and B is (1,4,1,2,5) of length 5.

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[math-fun] Just two more to go ...
by Hilarie Orman 11 Mar '18

11 Mar '18

Another global disaster looms as we approach Vonnegut crystalization boundary: http://science.sciencemag.org/content/359/6380/1136 Hilarie

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[math-fun] Series puzzle
by Dan Asimov 10 Mar '18

10 Mar '18

A series puzzle that appeared a long time ago in a puzzle section of a math magazine is this nice one, that some recent math-fun posts reminded me of: PUZZLE: Given positive integers K and L, express in closed form the Sum_{0 <= n < oo} 1 / (Kn + L)! I've never tried to evaluate this if Kn + L is replaced by a quadratic or higher-degree integer polynomial in n. —Dan

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[math-fun] Bale problem's
by françois mendzina essomba2 10 Mar '18

10 Mar '18

Hello In an unfortunate attempt to solve zeta (2n + 1), I became interested in certain functions. I was able to find: cos (x / sqrt (2)) * cosh (x / sqrt (2)) = sum ((- 1) ^ n * x ^ (4 * n) / (4 * n)!,n=0..infinity) but I am lost to the next sum. sum ((- 1) ^ n * x ^ (3 * n) / (3 * n)!,n=0..infinity) = ??? Do you know the function returned by this sum? I just noticed that the solutions of the equation sum ((- 1) ^ n * x ^ (3 * n) / (3 * n)!,n=0..infinity) = 0 are substantially close to: (2 * n + 1) * Pi / sqrt (3) Is there more information on this?

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