Re: [math-fun] how many decimal places exist in the mass of an electron (& other such things)?
ok, to reply to the commenters 1. particle physicists are experts on error analysis. --well, methinks they overrate themselves, but anyhow, this does not answer my original question which pertains to the FUNDAMENTAL limitations on knowability of electron mass. Not the limits of measurements so far, which are almost irrelevant. See, you might say, the electron mass is a real number, defined to infinite #decimals. Due to human inadequacies, we do not know them all, but they exist. But instead you might say, NO, the electron mass really is not, and cannot ever be, defined to more than 50 decimal places. (Or something.) 2. What means 5667565+-11 in the physics literature? --yes, normally this denotes "+-1 standard error" meaning RMS deviation. If other meaning is intended, the author should say so. 3. Million electrons... they repel... try measuring energy of electron/positron annihilation? --yes, electrons do repel and that would be an issue. The annihilation plan will encounter a limit from the momentum uncertainty of starting ingredients, which due to Heisenberg & the finite size of the observable universe sets a fundamental limit on the accuracy of this measurement. However, by measuring not 1, but 1000000 events, then you can decrease the error by factor 1000. Hence my point is, this "fundamental limit" aint necessarily so. Which leads to the question, again, what is the fundamental limit, or maybe there is none? On 11/23/13, math-fun-request@mailman.xmission.com <math-fun-request@mailman.xmission.com> wrote:
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Today's Topics:
1. Re: Newman = -1/Oldman (Bill Gosper) 2. Fwd: [LifeCA] Geminoid replicator (rcs@xmission.com) 3. how many decimal places exist in the mass of an electron (& other such things)? (Warren D Smith) 4. Re: "Happy Valentinukkah?s Day!" (Mike Speciner) 5. Re: how many decimal places exist in the mass of an electron (& other such things)? (Henry Baker) 6. Re: how many decimal places exist in the mass of an electron (& other such things)? (Rowan Hamilton)
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Message: 1 Date: Sat, 23 Nov 2013 11:54:21 -0800 From: Bill Gosper <billgosper@gmail.com> To: math-fun@mailman.xmission.com Subject: Re: [math-fun] Newman = -1/Oldman Message-ID: <CAA-4O0EsQAE9Y4o9eTXrHpo45dRSWpSU+=bJaNfNwVmKNntmYg@mail.gmail.com> Content-Type: text/plain; charset=ISO-8859-1
On Thu, Nov 21, 2013 at 5:01 AM, Bill Gosper <billgosper@gmail.com> wrote:
rwg>Maybe later today NeilB and I can find a way to speed this up, or maybe discover that it's all a bad dream. ------ Sure enough, Neil quickly realized that Adam's page http://cp4space.wordpress.com/2013/10/24/enumerating-the-rationals/ contains logarithmic-time recipes for both the forward map pos2newman(index) -> rational and newman2pos(rational)->index. Using these, the scrozzlement of the 11 term of coth 1 = 1 3 5 7 9 11 ... occurs at oldman step 16712177, which could be found by log search, if nothing else, even if Neil never writes, e.g., newmanposinterval that works on things like Coth[1], since we can work with finite CFs and generalize afterward:
Recall
In[68]:= newmanstep[n_] := 1/(2*Floor[n] + 1 - n)
In[74]:= tim[newman2pos[FromContinuedFraction[{1, 3, 5, 7, 9, 11, 2}]]]
0.000284 (seconds)
Out[74]= 206191919601
The terminal 2 could be any rational > 1. Using our log search value, minus a couple: In[75]:= tim[Divide @@ pos2newman[% - 16712179]]
0.000157 seconds
Out[75]= 577/602
Here's the magic moment, stepping forward (opposite of oldman): In[85]:= ColumnForm[ContinuedFraction /@ NestList[newmanstep, %75, 4]]
{0, 1, 23, 12, 2} {24, 12, 2} {0, 24, 1, 11, 2} {1, 23, 1, 11, 2} {0, 1, 1, 22, 1, 11, 2}
We can now simulate the effect of 206191919601-16712178 newman steps on Coth[1]:
In[79]:= {Numerator[#], Denominator[#]} &@ Convergents[{1, 3, 5, 7, 9, 11, 2}][[-2 ;; -1]]
Out[79]= {{15331, 32042}, {11676, 24403}}
In[80]:= {Numerator[#], Denominator[#]} &@ Convergents[{24, 12, 2}][[-2 ;; -1]]
Out[80]= {{289, 602}, {12, 25}}
In[81]:= %.Inverse[%%].{{Coth[1]}, {1}}
Out[81]= {{-30876 + 23515 Coth[1]}, {-1229 + 936 Coth[1]}}
In[82]:= ContinuedFraction[Divide @@ %, 6]
Out[82]= {{24, 12, 13, 15, 17, 19}}
I.e., we've back up past the 11. Step once forward.
In[83]:= Simplify[newmanstep[(Divide @@ %%)[[1]]]]
Out[83]= (-1229 + 936 Coth[1])/(-29345 + 22349 Coth[1])
In[84]:= ContinuedFraction[%, 6]
Out[84]= {0, 24, 1, 11, 13, 15}
Now it should be easy to chronicle the scrozzlements of ... 7, 9, 13,15, etc. --rwg [...]
Not completely easy--the overshot predicate can give false negatives. Here are the scrozzlings of the first few terms of coth(1) = {1,3,5,7,9,...} : Out[275]= {1, {0, 4, 5, 69}} {241, {0, 8, 1, 7, 69}} {497, {0, 16, 9, 69}} {16712177, {0, 24, 1, 11, 69}} {33489393, {0, 36, 13, 69}} {281406290723313, {0, 48, 1, 15, 69}} {562881267433969, {0, 64, 17, 69}} {1208907373433436732588529, {0, 80, 1, 19, 69}} {2417833193048065907294705, {0, 100, 21, 69}} {1329226728136733477867453629484433905, {0, 120, 1, 23, 69}} {2658454723921649350771260689764778481, {0, 144, 25, 69}}
which means that after oldman step 1, {1,3,5,7,9,...} becomes {0,4,5,69} and after step 241, {0,8,1,7,9,11,...} and after step 497, {0,16,9,11,13,...}. Running each 1 more step (-1 newmans):
In[276]:= % /. {n_, cf_} :> {n + 1, cfzewm[cf, -1]}
Out[276]= {2, {4, 1, 4, 69}} {242, {8, 8, 69}} {498, {16, 1, 8, 69}} {16712178, {24, 12, 69}} {33489394, {36, 1, 12, 69}} {281406290723314, {48, 16, 69}} {562881267433970, {64, 1, 16, 69}} {1208907373433436732588530, {80, 20, 69}} {2417833193048065907294706, {100, 1, 20, 69}} {1329226728136733477867453629484433906, {120, 24, 69}} {2658454723921649350771260689764778482, {144, 1, 24, 69}}
we see the penultimate term alternately upscrozzled after a long run or downscrozzled after a (relatively short) doubling.
Instead of trying to guess the general sequence function(s) here, I switched to the simpler In[317]:= ContinuedFractionK[1, k, {k, \[Infinity]}]
Out[317]= BesselI[1, 2]/BesselI[0, 2]
i.e., {1,2,3,4,5,...}
Out[306]= {0, {1, 2, 69}} {1, {0, 3, 3, 69}} {25, {0, 5, 1, 4, 69}} {57, {0, 10, 5, 69}} {15417, {0, 14, 1, 6, 69}} {31801, {0, 21, 7, 69}} {132152377, {0, 27, 1, 8, 69}} {266370105, {0, 36, 9, 69}} {17523732937785, {0, 44, 1, 10, 69}} {35115918982201, {0, 55, 11, 69}} {36857494466319121465, {0, 65, 1, 12, 69}} {73750982613738224697, {0, 78, 13, 69}} {1237637881581459231343672377, {0, 90, 1, 14, 69}} {2475577920866839506242796601, {0, 105, 15, 69}} {664573435548828553977895141880396857, {0, 119, 1, 16, 69}} {1329187433441286490429798672020569145, {0, 136, 17, 69}} {5708903659867095197427783744328276684889422905, {0, 152, 1, 18, 69}} {11417894430690934721660927622126257230420409401, {0, 171, 19, 69}} {784634723779399736220988131476081826687296432438576708665, {0, 189, 1, 20, 69}} {1569272440702734831700461809377040128700090862996581022777, {0, 210, 21, 69}} {1725434941194652898093349545431774267571649300435431096073764288363577, {0, 230, 1, 22, 69}}
In[307]:= % /. {n_, cf_} :> {n + 1, cfzewm[cf, -1]}
Out[307]= {1, {0, 3, 69}} {2, {3, 1, 2, 69}} {26, {5, 5, 69}} {58, {10, 1, 4, 69}} {15418, {14, 7, 69}} {31802, {21, 1, 6, 69}} {132152378, {27, 9, 69}} {266370106, {36, 1, 8, 69}} {17523732937786, {44, 11, 69}} {35115918982202, {55, 1, 10, 69}} {36857494466319121466, {65, 13, 69}} {73750982613738224698, {78, 1, 12, 69}} {1237637881581459231343672378, {90, 15, 69}} {2475577920866839506242796602, {105, 1, 14, 69}} {664573435548828553977895141880396858, {119, 17, 69}} {1329187433441286490429798672020569146, {136, 1, 16, 69}} {5708903659867095197427783744328276684889422906, {152, 19, 69}} {11417894430690934721660927622126257230420409402, {171, 1, 18, 69}} {784634723779399736220988131476081826687296432438576708666, {189, 21, 69}} {1569272440702734831700461809377040128700090862996581022778, {210, 1, 20, 69}} {1725434941194652898093349545431774267571649300435431096073764288363578, {230, 23, 69}}
Here the long steps are at {1, 26, 15418, 132152378, 17523732937786, 36857494466319121466, ...}
which has the empirical formula 2^(3*n + 2*n^2) - Sum[2^(k*(-1 + 2*k))*(-1 + 4^k), {k, 0, n}], n=0,1,2,... I'll yell if Koutschan's q-thing can do the sum. --rwg
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Message: 2 Date: Sat, 23 Nov 2013 14:43:32 -0700 From: rcs@xmission.com To: math-fun@mailman.xmission.com Cc: rcs@xmission.com Subject: [math-fun] Fwd: [LifeCA] Geminoid replicator Message-ID: <20131123144332.n1lvq95na8gss04w@webmail.xmission.com> Content-Type: text/plain; charset=ISO-8859-1; DelSp="Yes"; format="flowed"
Dave Green has put together a replicator for Conway Life. He has several versions. The smallest, fastest, has a bounding box of 3.7M^2, and takes 89M ticks to make a copy. If I'm reading his description correctly, the population of ON cells is 150K, most of which is information-carrying gliders.
Rich
----- Forwarded message from david.m.greene@gmail.com ----- Date: Sat, 23 Nov 2013 14:15:34 -0600 From: Dave Greene <david.m.greene@gmail.com> Reply-To: LifeCA@yahoogroups.com Subject: [LifeCA] Geminoid replicator To: LifeCA <LifeCA@yahoogroups.com>
The phase-shifted linear replicator is finally done -- several versions, actually, all using the same single-arm universal constructor and the same basic recipe. See
http://conwaylife.com/forums/viewtopic.php?f=2&t=1006&p=9908#p9901
The smallest pattern file is only 27K compressed, but the uncompressed macrocell file is almost 100K so I won't try quoting it here. There are links in the forum posting.
Keep the cheer,
Dave Greene
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Message: 3 Date: Sat, 23 Nov 2013 16:46:16 -0500 From: Warren D Smith <warren.wds@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: [math-fun] how many decimal places exist in the mass of an electron (& other such things)? Message-ID: <CAAJP7Y2HDV3+MVsoE3qX1zwp01gqsfrE9xznryhQsLqNJw-W3g@mail.gmail.com> Content-Type: text/plain; charset=ISO-8859-1
If you try to measure the mass m of an electron, some experimental error DELTAm, and the energy-time uncertainty principle combined with the finite lifetime of the universe (at least so far...) causes a limit on the accuracy of m.
So one could argue, the mass of the electron is inherently unknowable and undefined to more than a certain number of decimal places.
Except, somebody could measure the mass of a million-electron blob to try to dodge that limitation.
So anyhow... what are the inherent limits on how many decimal places can exist in such quantities (any further would have no meaning) and if so, estimate them.
-- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)
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Message: 4 Date: Sat, 23 Nov 2013 16:54:31 -0500 From: Mike Speciner <ms@alum.mit.edu> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] "Happy Valentinukkah?s Day!" Message-ID: <52912417.9010300@alum.mit.edu> Content-Type: text/plain; charset=windows-1252; format=flowed
At some point, the Gregorian calendar would add or subtract a February 29th out of the standard scheme in order to keep the calendar in sync with the seasons. It's a bit more interesting to contemplate what change should be made to the Jewish calendar in order to keep the seasons in sync. Two months have leap days and there is a leap month, so there are lots of possibilities, but not all combinations are allowed in any given year due to restrictions on the day of week.
I suppose another possibility is to just adjust the orbit of the earth, although one of the calendars would still have to be adjusted since their average year lengths are different.
--ms
On 22-Nov-13 23:32, Henry Baker wrote:
Cute discussion about the collisions of various calendars over the next 10's of thousands of years;
http://www.slate.com/articles/life/explainer/2013/11/thanksgivukkah_when_wil...
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Message: 5 Date: Sat, 23 Nov 2013 16:05:19 -0800 From: Henry Baker <hbaker1@pipeline.com> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] how many decimal places exist in the mass of an electron (& other such things)? Message-ID: <E1VkNDB-0004Gj-3Z@elasmtp-scoter.atl.sa.earthlink.net> Content-Type: text/plain; charset="us-ascii"
There's a bit of a problem in gathering together a million-electron blob: they repel one another like crazy. In fact, the amount of energy require to confine them is significant in the m=E/c^2 sense.
It might be easier to estimate the mass of an electron & an anti-electron by colliding them together & measuring the energy.
At 01:46 PM 11/23/2013, Warren D Smith wrote:
If you try to measure the mass m of an electron, some experimental error DELTAm, and the energy-time uncertainty principle combined with the finite lifetime of the universe (at least so far...) causes a limit on the accuracy of m.
So one could argue, the mass of the electron is inherently unknowable and undefined to more than a certain number of decimal places.
Except, somebody could measure the mass of a million-electron blob to try to dodge that limitation.
So anyhow... what are the inherent limits on how many decimal places can exist in such quantities (any further would have no meaning) and if so, estimate them. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)
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Message: 6 Date: Sat, 23 Nov 2013 19:57:45 -0800 From: Rowan Hamilton <rowanham@gmail.com> To: math-fun <math-fun@mailman.xmission.com> Subject: Re: [math-fun] how many decimal places exist in the mass of an electron (& other such things)? Message-ID: <CAM7qsEGM1HcQv_Zv=_dzPsoiM0ifb7rScnsju89ghfRL9O1f3A@mail.gmail.com> Content-Type: text/plain; charset=ISO-8859-1
Physicists spend a great deal of time and effort on understanding the errors in any physical measurement. A particle physics PhD involves about 2 years of classes, 2 years of slave labor, 1 year of measurement and then 3 years of error analysis. This is no joke. Since particle physics is an inherently statistical field, particle physicists are experts at error analysis.
http://pdg.lbl.gov/2013/listings/rpp2013-list-electron.pdf
On Sat, Nov 23, 2013 at 4:05 PM, Henry Baker <hbaker1@pipeline.com> wrote:
There's a bit of a problem in gathering together a million-electron blob: they repel one another like crazy. In fact, the amount of energy require to confine them is significant in the m=E/c^2 sense.
It might be easier to estimate the mass of an electron & an anti-electron by colliding them together & measuring the energy.
At 01:46 PM 11/23/2013, Warren D Smith wrote:
If you try to measure the mass m of an electron, some experimental error DELTAm, and the energy-time uncertainty principle combined with the finite lifetime of the universe (at least so far...) causes a limit on the accuracy of m.
So one could argue, the mass of the electron is inherently unknowable and undefined to more than a certain number of decimal places.
Except, somebody could measure the mass of a million-electron blob to try to dodge that limitation.
So anyhow... what are the inherent limits on how many decimal places can exist in such quantities (any further would have no meaning) and if so, estimate them. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)
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