June 2015 - mathfun@mailman.xmission.com

[math-fun] Brief "golden era" aka "flowering period" for universe. Enjoy it while can.
by Warren D Smith 13 Jun '15

13 Jun '15

It is well known that the entropy of the universe increases. However, the entropy of the observable portion of the universe (within the de Sitter horizon) can and will decrease. Indeed, in about 1000 gigayears, the part of the universe observable from point X will, for all but an extremely small-measure-fraction of X, contain only a few bits of entropy and be almost empty, containing zero atoms and only a few very cold photons. This is due to the exponential expansion of the (Lambda-dominated) future universe with e-folding time of about 14 gigayears. There will however remain a few "islands" e.g. gravitationally bound agglomerations such as our galaxy (unaffected by the overall expansion), that will continue to be interesting for well beyond that time, e.g. the smallest red dwarf stars are forecast to continue shining for 10000 gigayears. But these places will be exponentially rare. So if we plot on a horizontal axis, time, and on the vertical axis, entropy of the observable part of the universe viewed from typical point, the curve will increase at first, reach a max, then decrease. Let me call the max-info point, the "golden era" when the universe is "most interesting." I estimate that this golden era is happening approximately now. More precisely, somewhere between 5 Gyr and 30 Gyr of age (present age is 13.8 Gyr). I have not seen this whole remark before. Why now? Basically because Lambda-domination began at age 9 Gyr and the entropy-increasing processes such as star formation & shining, black hole formation & mergers, etc, are increasing entropy at approximately a linear rate, which cannot compete with the exponential decrease that sets in once Lambda-domination gets rolling. (The observable portion of the universe has a fixed radius of about 14 giga light years, which will remain approximately constant forevermore, say the mathematics of de Sitter space.) It indeed has been estimated that over 90% of the stars that will ever form, have already, and presumably the same is true for black holes (since they arise only from very short-lived stars). -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] Don't cry for me, Pentagonia.
by Bill Gosper 13 Jun '15

13 Jun '15

Adam Goucher has made a startling observation (quoted without permission): ---- I was reading the third edition of 'Minskys and Trinskys' by Gosper, Holloway and (Ziegler Hunts)^2: http://www.blurb.com/books/2172660-minskys-trinskys-3rd-edition The pattern of disjoint D10-symmetric snowflakes (or 'Rastermen', as they're referred to in the book) shown in Figure 31 appears to be affinely isomorphic to the pattern obtained by marking Penrose tilings: http://condellpark.com/kd/penrmark24bg.gif This suggests that it should be possible to create a very integer-esque method of drawing an affined Penrose tiling on a square grid: -- Run the Minsky algorithm on every square and mark any squares which participate in sufficiently long cycles. -- Exploit the fact that the Penrose tiling and its associated markings are mutually locally derivable. This could possibly be useful for running cellular automata on Penrose tilings in Golly -- just embed the tiling on a square grid, similar to Tim Hutton's 'slanted hexagon' approach. ------------ "P5d1" is the Minsky recurrence with d multiplier 1 and e chosen to make the unfloored period = 5. It makes those fractal pentagrams that fascinate me. It assigns to each gridpoint (x,y) the period of the minsky loop containing (x,y). Those periods are 5, 20, 30, 70, 130, 270, 530, 1070, ... Here <http://gosper.org/p5d1fun.pdf> is a raw sandbox of plotting the (x,y) with those periods. For some periods, the points form little figures (e.g., rhombi, hexagons, minipentagrams,...) whose centers are vertices of a rhombic Penrose tiling. Other periods produce patterns strongly resembling quasicrystal diffractograms. According to Wikipedia, Penrose began with gosper.org/6pents.bmp (minus the two Müller-Lyer segments). Hey kids: Animate this. Start with one chorded pentagon. Unfold it into two sharing the edge parallel to the chord. Then, with a Manipulate, slide the four blank pentagons in from the sides. Pause for effect. Slide them away. Finally, these same analyses led to gosper.org/pentfill.pdf and the earlier www.tweedledum.com/rwg/pentagonfill.pdf Sorry you have to click these yourselves. Even sorrier that every single gosper.org would-be attachment that I tried tonight was deleted, and needed to be restored. Just like 1970s quota enforcement. --rwg PS, I never got a copy of ---------- Forwarded message ---------- From: Bill Gosper <billgosper(a)gmail.com> Date: Sun, Jun 7, 2015 at 6:45 AM Subject: Fwd: What is the period of this integer sequence? To: math-fun(a)mailman.xmission.com I just noticed a typo in this item dated 2014-05-08 12:56: "Thus the only "restoring force" capable of pulling a[2n],a[2n+1] back to 0,1 ..." should say "back to (0,8)" Anyone who was confused by this is paying way too much attention. The item was primarily about the "Minsky Stock Index" restated toward the end. Last I heard, Tom Rokicki had chased it about 13 quadrillion steps without returning to (1,0). --rwg ----------------------------------------- [...] By way of couragement, here is a similar sequence we did track. ----- More xmission chupandose?

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[math-fun] one way window for light
by Warren D Smith 12 Jun '15

12 Jun '15

"One way mirrors" seen in police interview rooms on TV, aren't really. Light goes both ways through them, it is just that there is a lot less light coming from one side, since it is a dark place. The other side is brightly lit; and then if you partially-silver the window the scattering and partial reflection tends to overwhelm the transmission from the dark side, making it *appear* to be a mirror from the bright side. But our recent discussion re the "Faraday effect" being a "loophole" in the time-reversibility claims for Maxwell equations, made it occur to me: we really can make a one-way window for light. ---------------------xxx----------------------- ----light->>----A---F----B----light->>---- ---------------------xxx----------------------- let A be a polarizer, F is a faraday rotatory cell containing magnetic field (generated by cylindrical solenoid xxx surrounding F) pointed rightward, and B is another polarizer whose angle is 45 degrees to A, selected to transmit the 45-rotated linearly-polarized light from A as rotated by F (length of F and magnetic field strength selected to make the rotation angle 45). Result: light transmits thru left-to-right, albeit halved in power since only one polarization is transmitted. But if you try to go backward, then the Faraday F will rotate 45 degrees causing the plane-polarized light from B now to be orthogonal to A, and transmission is blocked. Note this does not work with sugar solution doing the rotating instead of Faraday. Then the backward rotation would undo the forward rotation and transmits equally both ways. Nor would any set of dielectric, partially silvered, polarizing, rotating, etc slabs work -- by time reversibility of Maxwell equations. Faraday effect based on EXTERNALLY applied magnetic field, is crucial to make it work. On 6/12/15, math-fun-request(a)mailman.xmission.com <math-fun-request(a)mailman.xmission.com> wrote: > Send math-fun mailing list submissions to > math-fun(a)mailman.xmission.com > > To subscribe or unsubscribe via the World Wide Web, visit > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > or, via email, send a message with subject or body 'help' to > math-fun-request(a)mailman.xmission.com > > You can reach the person managing the list at > math-fun-owner(a)mailman.xmission.com > > When replying, please edit your Subject line so it is more specific > than "Re: Contents of math-fun digest..." > > > Today's Topics: > > 1. Re: Definition of mathematics (Adam P. Goucher) > 2. Re: DIY penrose shower floor (Joerg Arndt) > 3. Zimbabweans will get $5 for 175 quadrillion ? (Henry Baker) > 4. Re: Zimbabweans will get $5 for 175 quadrillion ? > (Charles Greathouse) > 5. Re: Zimbabweans will get $5 for 175 quadrillion ? (Andy Latto) > > > ---------------------------------------------------------------------- > > Message: 1 > Date: Fri, 12 Jun 2015 04:37:16 +0200 > From: "Adam P. Goucher" <apgoucher(a)gmx.com> > To: math-fun(a)mailman.xmission.com > Subject: Re: [math-fun] Definition of mathematics > Message-ID: > <trinity-4c2799e5-0665-4dc3-9ed1-6e896eca9c48-1434076635968@3capp-mailcom-bs07> > > Content-Type: text/plain; charset=UTF-8 > > Gauss once said: > > "Mathematics is the Queen of the Sciences" > > Assuming we've defined `queen' and `science', and established uniqueness, > we have a concrete definition of mathematics. > > > Sincerely, > > > Adam P. Goucher > >> Sent: Friday, June 12, 2015 at 2:34 AM >> From: "Dan Asimov" <asimov(a)msri.org> >> To: math-fun <math-fun(a)mailman.xmission.com> >> Subject: Re: [math-fun] Definition of mathematics >> >> As a grad student, ca. 1971, I came up with >> >> "Mathematics is the science of patterns." >> >> ?Dan >> >> >> > On Jun 11, 2015, at 5:13 PM, Allan Wechsler <acwacw(a)gmail.com> wrote: >> > >> > Mathematics is the study of the formal consequences of formal rules. >> > Even >> > more succinctly, mathematics is the study of form. >> > >> > On Thu, Jun 11, 2015 at 6:23 PM, James Propp <jamespropp(a)gmail.com> >> > wrote: >> > >> >> A couple of weeks ago, in a gathering of mathematicians throwing out >> >> candidate definitions of mathematics, I half-seriously ventured the >> >> opinion that mathematics is the subset of philosophy consisting of >> >> those >> >> philosophical questions that actually have answers, along with the >> >> answers >> >> to those questions. >> >> >> >> But I don't think this is original. Whom am I quoting (or >> >> paraphrasing) here? >> >> >> >> Jim Propp >> >> _______________________________________________ >> >> math-fun mailing list >> >> math-fun(a)mailman.xmission.com >> >> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >> >> >> > _______________________________________________ >> > math-fun mailing list >> > math-fun(a)mailman.xmission.com >> > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >> >> >> _______________________________________________ >> math-fun mailing list >> math-fun(a)mailman.xmission.com >> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >> > > > > ------------------------------ > > Message: 2 > Date: Fri, 12 Jun 2015 15:46:51 +0200 > From: Joerg Arndt <arndt(a)jjj.de> > To: math-fun <math-fun(a)mailman.xmission.com> > Subject: Re: [math-fun] DIY penrose shower floor > Message-ID: <20150612134651.GA2990(a)jjj.de> > Content-Type: text/plain; charset=us-ascii > > Fantastic work! > > * Dirk Lattermann <dlatt(a)alqualonde.de> [Jun 12. 2015 14:42]: >> [...] >> >> Yes, I already considered this. Maybe sometime, it'd be nice! >> >> It was a major effort finding a round drain. Only square ones seem to be >> readily available (and affordable). >> When my plumber finally came around with this circular drain, it >> reminded me at once of the metal samples a water jet cutting company >> had shown me when I inquired them about cutting the tiles (very >> expensive, but would have saved much time and grief glueing). >> >> For this round drain, I paid about 10 times the price of a square drain. >> A custom designed penrose drain plate will be expensive, and I have not >> figured out how it will look like when the rhombs are rather small to >> leave big enough bridges for stability. >> > > We should have (laser-)cutters powerful enough for metal > at our school. I'd need to know the dimensions, mostly > the thickness, to be able to ask around. > >> [...] >> _______________________________________________ >> math-fun mailing list >> math-fun(a)mailman.xmission.com >> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > > > ------------------------------ > > Message: 3 > Date: Fri, 12 Jun 2015 08:05:48 -0700 > From: Henry Baker <hbaker1(a)pipeline.com> > To: math-fun(a)mailman.xmission.com > Subject: [math-fun] Zimbabweans will get $5 for 175 quadrillion ? > Message-ID: <E1Z3QXM-0004M0-5z(a)elasmtp-banded.atl.sa.earthlink.net> > Content-Type: text/plain; charset="us-ascii" > > FYI -- I can't tell whether "quadrillion" means 10^15 or 10^24, or whether > "trillion" means 10^12 or 10^18. > > Also, I couldn't tell whether the exchange was forced by an exponent > overflow problem or a precision problem. > > 175 quadrillion requires at least 58 bits ("short scale"), so it is already > beyond IEEE754 double. However, the Intel 80-bit x86 extended precision can > handle up to 63 bits of precision, so the govt computers are probably > running Windoze. > > Now we know why there is such interest in Bitcoins. > > https://en.wikipedia.org/wiki/Quadrillion > > https://en.wikipedia.org/wiki/Trillion > > https://news.yahoo.com/currency-dies-zimbabweans-5-175-quadrillion-local-do… > > As currency dies, Zimbabweans will get $5 for 175 quadrillion local dollars > > Reuters > > By MacDonald Dzirutwe 23 hours ago > > A man holds up newly issued 200 million and 500 million Zimbabwe dollar > notes in the capital Harare > > By MacDonald Dzirutwe > > HARARE (Reuters) - Zimbabweans will start exchanging 'quadrillions' of local > dollars for a few U.S. dollars next week, as President Robert Mugabe's > government discards its virtually worthless national currency, the central > bank said on Thursday. > > The southern African country started using foreign currencies like the U.S. > dollar and South African rand in 2009 after the Zimbabwean dollar was ruined > by hyper-inflation, which hit 500 billion percent in 2008. > > At the height of Zimbabwe's economic crisis in 2008, Zimbabweans had to > carry plastic bags bulging with bank notes to buy basic goods like bread and > milk. Prices were rising at least twice a day. > > >From Monday, customers who held Zimbabwean dollar accounts before March > 2009 can approach their banks to convert their Zimbabwean dollar balance > into dollars, Reserve Bank of Zimbabwe (RBZ) Governor John Mangudya said in > a statement. > > The process will legally end the local currency. Zimbabweans have until > September to turn in their old bank notes, which some people sell as > souvenirs to tourists. > > Bank accounts with balances of up to 175 quadrillion Zimbabwean dollars will > be paid $5. Those with balances above 175 quadrillion dollars will be paid > at an exchange rate of $1 to 35 quadrillion Zimbabwean dollars. > > The highest - and last - bank note to be printed by the RBZ in 2008 was 100 > trillion Zimbabwean dollars. It was not enough to ride a public bus to work > for a week. > > The RBZ said customers who still have stashes of old Zimbabwean dollar notes > can walk into any bank and get $1 for every 250 trillion they hold. > > That means a holder of a 100 trillion bank note will on Monday get 40 cents. > The RBZ has set aside $20 million to pay Zimbabwean dollar currency > holders. > > > > > ------------------------------ > > Message: 4 > Date: Fri, 12 Jun 2015 11:18:22 -0400 > From: Charles Greathouse <charles.greathouse(a)case.edu> > To: math-fun <math-fun(a)mailman.xmission.com> > Subject: Re: [math-fun] Zimbabweans will get $5 for 175 quadrillion ? > Message-ID: > <CAAkfSGJH-+u7nWr_2C+BYX3Ka_G+fUiQaUXFNj3zMdzoj8FoPw(a)mail.gmail.com> > Content-Type: text/plain; charset=UTF-8 > > Here "quadrillion" is 10^15 (the "short scale"). I don't think the change > was driven by computer issues at all, rather the need for currency > stability. (It's easy enough to use "trillions of dollars" as your unit to > avoid precision/overflow issues.) > > Charles Greathouse > Analyst/Programmer > Case Western Reserve University > > On Fri, Jun 12, 2015 at 11:05 AM, Henry Baker <hbaker1(a)pipeline.com> wrote: > >> FYI -- I can't tell whether "quadrillion" means 10^15 or 10^24, or >> whether >> "trillion" means 10^12 or 10^18. >> >> Also, I couldn't tell whether the exchange was forced by an exponent >> overflow problem or a precision problem. >> >> 175 quadrillion requires at least 58 bits ("short scale"), so it is >> already beyond IEEE754 double. However, the Intel 80-bit x86 extended >> precision can handle up to 63 bits of precision, so the govt computers >> are >> probably running Windoze. >> >> Now we know why there is such interest in Bitcoins. >> >> https://en.wikipedia.org/wiki/Quadrillion >> >> https://en.wikipedia.org/wiki/Trillion >> >> >> https://news.yahoo.com/currency-dies-zimbabweans-5-175-quadrillion-local-do… >> >> As currency dies, Zimbabweans will get $5 for 175 quadrillion local >> dollars >> >> Reuters >> >> By MacDonald Dzirutwe 23 hours ago >> >> A man holds up newly issued 200 million and 500 million Zimbabwe dollar >> notes in the capital Harare >> >> By MacDonald Dzirutwe >> >> HARARE (Reuters) - Zimbabweans will start exchanging 'quadrillions' of >> local dollars for a few U.S. dollars next week, as President Robert >> Mugabe's government discards its virtually worthless national currency, >> the >> central bank said on Thursday. >> >> The southern African country started using foreign currencies like the >> U.S. dollar and South African rand in 2009 after the Zimbabwean dollar >> was >> ruined by hyper-inflation, which hit 500 billion percent in 2008. >> >> At the height of Zimbabwe's economic crisis in 2008, Zimbabweans had to >> carry plastic bags bulging with bank notes to buy basic goods like bread >> and milk. Prices were rising at least twice a day. >> >> From Monday, customers who held Zimbabwean dollar accounts before March >> 2009 can approach their banks to convert their Zimbabwean dollar balance >> into dollars, Reserve Bank of Zimbabwe (RBZ) Governor John Mangudya said >> in >> a statement. >> >> The process will legally end the local currency. Zimbabweans have until >> September to turn in their old bank notes, which some people sell as >> souvenirs to tourists. >> >> Bank accounts with balances of up to 175 quadrillion Zimbabwean dollars >> will be paid $5. Those with balances above 175 quadrillion dollars will >> be >> paid at an exchange rate of $1 to 35 quadrillion Zimbabwean dollars. >> >> The highest - and last - bank note to be printed by the RBZ in 2008 was >> 100 trillion Zimbabwean dollars. It was not enough to ride a public bus >> to >> work for a week. >> >> The RBZ said customers who still have stashes of old Zimbabwean dollar >> notes can walk into any bank and get $1 for every 250 trillion they hold. >> >> That means a holder of a 100 trillion bank note will on Monday get 40 >> cents. The RBZ has set aside $20 million to pay Zimbabwean dollar >> currency >> holders. >> >> >> _______________________________________________ >> math-fun mailing list >> math-fun(a)mailman.xmission.com >> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >> > > > ------------------------------ > > Message: 5 > Date: Fri, 12 Jun 2015 12:11:23 -0400 > From: Andy Latto <andy.latto(a)pobox.com> > To: math-fun <math-fun(a)mailman.xmission.com> > Subject: Re: [math-fun] Zimbabweans will get $5 for 175 quadrillion ? > Message-ID: > <CAKqg3U0ChRRbvnPiHZv8N-qXBHi0CQK6o47vy_ShQi_d1zL_+g(a)mail.gmail.com> > Content-Type: text/plain; charset=UTF-8 > > On Fri, Jun 12, 2015 at 11:05 AM, Henry Baker <hbaker1(a)pipeline.com> wrote: > > >> >> Bank accounts with balances of up to 175 quadrillion Zimbabwean dollars >> will be paid $5. Those with balances above 175 quadrillion dollars will >> be paid at an exchange rate of $1 to 35 quadrillion Zimbabwean dollars. > >> >> The RBZ said customers who still have stashes of old Zimbabwean dollar >> notes can walk into any bank and get $1 for every 250 trillion they hold. > > Wonder why the physical currency is so much more valuable (less > worthless?) than the bank accounts. These two exchange rates differ by > a factor of 140. If you had substantial amounts of money in a > Zimbabwean bank account, withdrawing it as currency (which I assume is > no longer possible) would make a big difference in how many dollars > you ended up with. > > Andy > > > > ------------------------------ > > Subject: Digest Footer > > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > > ------------------------------ > > End of math-fun Digest, Vol 148, Issue 28 > ***************************************** > -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] The maximums and the minimums
by Warren D Smith 12 Jun '15

12 Jun '15

https://dl.dropboxusercontent.com/u/3507527/GodsUnits.html explains the maximum and minimum possible values of things in the world. Any comments or further suggestions of stuff to bound? Also, inside is my claimed resolution of the magnetic pair-creation problem. Insofar as I resolved it. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] Zimbabweans will get $5 for 175 quadrillion ?
by Henry Baker 12 Jun '15

12 Jun '15

FYI -- I can't tell whether "quadrillion" means 10^15 or 10^24, or whether "trillion" means 10^12 or 10^18. Also, I couldn't tell whether the exchange was forced by an exponent overflow problem or a precision problem. 175 quadrillion requires at least 58 bits ("short scale"), so it is already beyond IEEE754 double. However, the Intel 80-bit x86 extended precision can handle up to 63 bits of precision, so the govt computers are probably running Windoze. Now we know why there is such interest in Bitcoins. https://en.wikipedia.org/wiki/Quadrillion https://en.wikipedia.org/wiki/Trillion https://news.yahoo.com/currency-dies-zimbabweans-5-175-quadrillion-local-do… As currency dies, Zimbabweans will get $5 for 175 quadrillion local dollars Reuters By MacDonald Dzirutwe 23 hours ago A man holds up newly issued 200 million and 500 million Zimbabwe dollar notes in the capital Harare By MacDonald Dzirutwe HARARE (Reuters) - Zimbabweans will start exchanging 'quadrillions' of local dollars for a few U.S. dollars next week, as President Robert Mugabe's government discards its virtually worthless national currency, the central bank said on Thursday. The southern African country started using foreign currencies like the U.S. dollar and South African rand in 2009 after the Zimbabwean dollar was ruined by hyper-inflation, which hit 500 billion percent in 2008. At the height of Zimbabwe's economic crisis in 2008, Zimbabweans had to carry plastic bags bulging with bank notes to buy basic goods like bread and milk. Prices were rising at least twice a day. >From Monday, customers who held Zimbabwean dollar accounts before March 2009 can approach their banks to convert their Zimbabwean dollar balance into dollars, Reserve Bank of Zimbabwe (RBZ) Governor John Mangudya said in a statement. The process will legally end the local currency. Zimbabweans have until September to turn in their old bank notes, which some people sell as souvenirs to tourists. Bank accounts with balances of up to 175 quadrillion Zimbabwean dollars will be paid $5. Those with balances above 175 quadrillion dollars will be paid at an exchange rate of $1 to 35 quadrillion Zimbabwean dollars. The highest - and last - bank note to be printed by the RBZ in 2008 was 100 trillion Zimbabwean dollars. It was not enough to ride a public bus to work for a week. The RBZ said customers who still have stashes of old Zimbabwean dollar notes can walk into any bank and get $1 for every 250 trillion they hold. That means a holder of a 100 trillion bank note will on Monday get 40 cents. The RBZ has set aside $20 million to pay Zimbabwean dollar currency holders.

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[math-fun] DIY penrose shower floor
by Dirk Lattermann 12 Jun '15

12 Jun '15

Finally, a year after we moved into our new (old) house, I managed to finish the penrose mosaic for the shower in our ground floor bathroom. I put two quick pictures, before the glass wall was installed, at http://folgenschwer.de/mosaik/ Thought some of you might enjoy, Dirk

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[math-fun] Similar permutations problem
by Dave Dyer 11 Jun '15

11 Jun '15

Here's a real life problem. I have a lot of strings (~20,000) which are permutations of 14 letter "words" starting with ABCDEFGHIJKLMN I suspect a lot of these words are similar, meaning they have a large number of letters in the same position. Checking for identical words is trivial - just sort and compare adjacents. Is there an approach better than the brute force 20,000! pairwise comparisons to find similar permutations? (For those who like a less abstract descriptions, the 20,000 are solutions to a set of puzzles, and I suspect a lot of the solutions are similar even though the puzzles are distinct.)

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[math-fun] Multilingual numerical ambiguity
by James Propp 10 Jun '15

10 Jun '15

The word "billion" means different things in two different dialects of English (British and American). Can anyone give other examples where one number word means two different things into two languages or dialects? Jim Propp

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Re: [math-fun] math-fun Digest, Vol 148, Issue 23
by Leo Broukhis 10 Jun '15

10 Jun '15

On Wed, 10 Jun 2015 14:22:26 -0400, James Propp <jamespropp(a)gmail.com> wrote: > > The word "billion" means different things in two different dialects of > English (British and American). Can anyone give other examples where one > number word means two different things into two languages or dialects? If you allow other parts of speech with numerical meaning, then "once" (English/Spanish). Leo

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[math-fun] Pair production by electric field. How to compute exactly. And for magnetic.
by Warren D Smith 09 Jun '15

09 Jun '15

I found this paper, the most helpful paper I saw for trying to comprehend this. Barry Holstein seems to be good at explaining puzzling physics to the stupid (i.e. me) in mucho papers for the Amer.J.Physics, the journal put out by the Amer. Assoc of Physics Teachers which is big in trying for good writing and clarity. [At least one physics professor told me AJP is "not a real journal" and expressed contempt for it. But my experiences reading this journal have usually been good, in fact in the top class of physics journals, and I don't think his contempt was warranted. Just my opinion. I also like the J. Math'l Physics and Communications in Math'l Physics one hell of a lot more than most physicists I generally encounter; they ooze contempt for them too.] Barry R. Holstein: Strong field pair production, American Journal of Physics 67,6 (1999) 499-507. [Other AJP pieces by Holstein: Van der Waals interaction: AJP 69,4 (2001) 441-449; 69,12 (2001) 1283. Klein Paradox, AJP 66,6 (1998) 507- Second Born approximation and Coulomb scattering, AJP 75,6 (200-7) 537-539 Graviton physics, AJP 74,11 (2006) 1002-1011. How large is the "natural" magnetic moment? AJP 74,12 (2006) 1104-1111 Effective interactions and the hydrogen atom, AJP 72,3 (2004) 333-334. Neutrino physics AJP 68,1 (2000) 15-32: update, AJP 72,1 (2004) 18-24. Anomalies in quantum mechanics: The 1/r^2 potential, AJP 70,5 (2002) 513-. Gyroscope precession and general relativity, AJP 69,12 (2001) 1248-1256 Bound states of a uniform spherical charge distribution - revisited! AJP 68,7 (2000) 640- Blue skies and effective interactions, AJP 67,5 (1999) 422-427 Answer to question #76. Neutrino mass and helicity, AJP 66,12 (1998) 1045 The harmonic oscillator propagator, AJP 66,7 (1998) 583-589 Answer to question #4. Is there a physics application that is best analyzed in terms of continued fractions? AJP 65,12 (1997) 1133-1135. Diagonalization of the Dirac equation: An alternative approach, AJP 65,6 (1997) 519-522. The linear potential propagator, AJP 65,5 (1997) 414-418 Understanding alpha decay, AJP 64,8 (1996) 1061-1071. Quantum-Mechanics In Momentum-Space - The Coulomb System, AJP 63,8 (1995) 710-716. Quark Masses and Binding-Energy in a Proton - Answer to Question Number-2a, AJP 63,1 (1995) 14. Anomalies for Pedestrians, AJP 61,2 (1993) 142-147. Variations on the Aharonov-Bohm Effect, AJP 59,12 (1991) 1080-1085. Effective Lagrangians and Quantum-Mechanics - The Index of Refraction, AJP 57,2 (1989) 142-148. The Adiabatic Propagator, AJP 57,8 (1989) 714-720. The Adiabatic Theorem and Berry's Phase, AJP 57,12 (1989) 1079-1084 Gauge-Invariance and Quantization, AJP 56,5 (1988) 425-429 Temperature Measured by a Uniformly Accelerated Observer, AJP 52,8 (1984) 730. Bound-States, Virtual States, and Non-Exponential Decay via Path-Integrals, AJP 51,10 (1983) 897-901. Barrier Penetration via Path-Integrals, AJP 50,9 (1982) 833-839. Path-Integrals and the WKB Approximation, AJP 50,9 (1982) 829-832. Elementary Derivation Of The Radiation-Field From An Accelerated Charge, AJP 49,4 (1981) 346.] Holstein also follows: Fritz Sauter: U"ber das Verhalten eines Elektrons im Homogenen Elektrischen Feld Nach der Relativistischen Theorie Diracs, Zeitschrift Physik 69,11-12 (1931) 742-764 Zum Kleinschen Paradoxon, Zeitschrift Physik 73,7-8 (1932) 547-552 Sauter actually was probably the first person who understood the "Schwinger limit," and in my opinion better than Schwinger ever did 20 years later. The "Klein paradox" is the fact that, when you compute 1 dimensional plane wave transmission thru a "potential barrier" with Dirac's equation (the "barrier" actually might be a downstep, anyhow it is some localized fields) you can end up with reflection coefficient R and transmission coefficient T, with |R|>1. This historically was a big puzzle. Eventually interpreted as pair production. Sauter found exact solution of Dirac equation in an electrostatic potential of the form (V/2)*tanh(a*x/2) which steps V volts up, and avoids issues that might have worried somebody about non-smooth potentials with corners. The max electric field is a*V/4. Using this exact solution, we then can consider the "pre-filled Dirac sea" of electron states with negative energies in the field-free vacuum (all these states are exactly-known plane-wave states) and regard them as the input to Sauter. Sauter's formula for |R|^2+|T|^2-1, times an "attempt rate" aka "input current" for each state, can be summed over all Dirac-sea states, and regarded as a particle production rate. If the Dirac sea were restricted to a finite-volume box this actually effectively is a finite set of states to sum over (the ones with too-negative energies do not matter since they cannot be promoted to positive energies by the available step-size V). So it seems to me that, at least in principle, one can exactly obtain the pair production rate from the Sauter electric field, in this way. The details can get complicated. But that is, in principle, a way to do the calculation. Now. Suppose we try the same thing, but for a magnetic field. Incidentally: At least for *classical* point charges: if you have a magnetic field in the z-direction, which is a function of x, which is 0 when x<0, and positive when x>0, and bounded below by a positive constant for all large-enough x, then any such field is a "perfect mirror" in the sense that electrons flying into it from the x<0 region, get reflected back out, angle of incidence=angle of reflection. The same "perfect mirror" statement also would have classically been true for Sauter's pure-electric field provided the incident electron did not have enough energy to surmount the barrier. Anyhow, suppose we have magnetic field in the z-direction which is +B when x>0 and -B when x<0. In that case, a "Landau level" n in the right halfspace, with spin parameter -, automagically transitions to a Landau level n with spin parameter +, in the left halfspace, whenever cross the x=0 plane. I.e. the spin reverses with respect to the B-field, but aside from that the Landau states stay the same whenever cross the plane. This corresponds to an energy step up (or down) by 1 level. Nonrelativistically this step size is constant hbar*e*B / (m*c) but relativistically it is nonconstant i.e. depends upon n and decreases for n large, ultimately to 0. We regard the "Dirac sea" in this scenario as consisting entirely of Landau levels. In that case we have simply a potential step, from the view of the two Dirac seas. If because B is large, the size of this step exceeds 2*m*c^2, then it seems to me we ought to get pair production since a negative energy state in sea#1, can have positive energy in sea #2. Furthermore, the rate of particle production should be calculable since there are only a finite set of Landau levels for which the step size exceeds the threshold; we can sum over all of them therefore. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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