February 2014 - mathfun@mailman.xmission.com

Re: [math-fun] Expansion of pi in factorial base
by Keith F. Lynch 19 Feb '14

19 Feb '14

Simon Plouffe <simon.plouffe(a)gmail.com> wrote: > In the factorial base , the number E is 1, of course ... Wouldn't it be 1.1111111... repeating? After all, 1/e is the harmonic mean of the factorials. (I've never seen it phrased that way before; I'm just being different.)

1 0

[math-fun] The universe as simulation
by Whitfield Diffie 18 Feb '14

18 Feb '14

I sent the following paragraph to Dave Fraber's IP list in response to a discussion of the universe as simulation. Back in the sixties someone (it may have been Minsky but it may equally well have been Silver, or McCarthy, or one of the other members of that circle) told me the notion that the universe might be a simulation of Newtonian mechanics on a fixed-wordlength machine, with relativity as the overflow errors and quantum mechanics as the underflow errors. If you took this seriously, you might be able to construct experiments that would give you insight into the wordlength and instruction set of the machine. Bob Taylor said that it was probably Fredkin but I think he was confusing it with Fredkin's pre-mature Wolframism. Does anyone remember who thought this up? Does anyone know Fredkin's email address? Whit

2 1

[math-fun] Possibility puzzle
by Bill Gosper 18 Feb '14

18 Feb '14

Apparently, possibility is the intersection of probability with Boolean algebra. A certain 4th grade teacher decided to enrich Saxon Math with a few days of re-education from a private copy of something called "Logic Liftoff", followed by a test supposedly copied therefrom, on which a certain confused 4th grader apparently was reduced to bluffing with ambigrams: gosper.org/logic.png <http://gosper.org/logic.png>. I don't understand how he got credit on the first five answers, since their correct answers are clearly "the sun will rise tomorrow." Or how he didn't lose 5 more points for "Febuary". I actually had to Google for reassurance that they weren't dumbing down the spelling. --rwg

6 6

[math-fun] Cryptocurrencies based on prime numbers
by Henry Baker 17 Feb '14

17 Feb '14

I've been reading about "crypto-currencies" like Bitcoin. Here's one based on prime numbers. http://cryptomining-blog.com/tag/prime-numbers/ http://en.wikipedia.org/wiki/Primecoin Many of the existing cryptocoins are based on SHA256 or SCRYPT: http://dustcoin.com/ The largest supercomputer in existence today is the entire might of the Bitcoin mining industry. (Hmmm... I wonder what fraction of those Bluffdale computer cycles are mining bitcoins right now...) The MIT Tidbit crew seems to have replaced "captchas" with coin-miners (as in, "we don't care if you're a bot or not, so long as your computer is willing to mine bitcoins for us"). I wonder if there aren't some other hard problems that would also advance the world instead of simply burning computer cycles. Can't we base a crypto currency on the Riemann hypothesis or perhaps protein folding?

1 0

Re: [math-fun] Possibility puzzle
by Adam P. Goucher 17 Feb '14

17 Feb '14

> Bill Gosper wrote: > > Apparently, possibility is the intersection of probability with Boolean > algebra. A certain > 4th grade teacher decided to enrich Saxon Math with a few days of What's Saxon Math? Does it involve the Angle bisector theorem? > reduced to > bluffing with ambigrams: gosper.org/logic.png I can remember an anecdote that in some Cambridge lecture on propositional logic, the lecturer wrote a few axioms such as this: (S ^ H) v N <==> (S v N) ^ (H v N) And then rotated the blackboard through 180° to give the remaining axioms. By the way, has anyone else realised that Soddy's hexlet generalises by adding extra layers of spheres? http://cp4space.wordpress.com/2014/02/15/soddys-hexlet/ Sincerely, Adam P. Goucher

3 3

[math-fun] Simple model of Earth magnetic field, which sounded promising for a little while
by Warren D Smith 14 Feb '14

14 Feb '14

> What sets the magnetic field value? > > If there is a fixed velocity field, then induced voltage V is > proportional to magnetic field, which via Ohm's law and the law of > inductance, means we should approach a current I set by V=I*R where > R=resistance, which in turn generates the magnetic field. > > These equations are linear, so if multiply input magnetic field by > X... we multiply output magnetic field by X. There is nothing to set > any particular level. And if the output > exceeds the input, we would get growth, and what would stop that growth? > > So something else needs to be put into this picture, some nonlinear > effect, --The problem with my last post's idea, that the solar wind was this "nonlinear effect", is: Suppose the Earth were just a boring permanent bar magnet (no radioactivity, liquid flow, or anything complicated). Then the same effects on the solar wind would happen, but this Earth's field is not being generated by any kind of power expenditure, so something is silly re the idea this is a power loss & etc. Despite the numerical coincidence that this "loss" and "heat input" were close. Revert to the more-boring idea that the Joule heat from electrical resistance, is emitted in a different place (anyhow different distribution) than the radioactive heat. For example, the solid inner core, presumably has 2 or more times less electrical resistivity than the molten-metal outer core, and has no fluid convection to induce voltages, hence would be expected to be warmed much less by I*I*R ohmic heating, than the liquid outer core. So the dynamo's joule heat is emitted higher up than the radioactive input-heat, so the whole kaboodle serves as another heat-transfer mechanism. Another reason this should be true is increasing pressure decreases resistivity R, so more R at higher altitudes, so more I*I*R joule heating happens there if electrical current densities do not depend much on location. If we crudely approximate the heat-output and heat-input as being in totally disjoint locations, then it is impossible for the I*I*R ohm-joule-heat power to exceed the total radioactive heat input. This sets an upper limit on the Earth's magnetic field. What is that upper bound? And is is relevant? The current I required to generate the observed Earth magnetic dipole moment of 8*10^22 amp*meter^2 is (assuming it flows equatorially at radius=3000 km) is I=2.8*10^9 amps. The resistivity of the molten metal is thought to be about 0.7*10^(-6) ohm-meters. Suggesting R ought to be about 2*10^(-13) ohms. Thus the present joule heat ought to be about I*I*R = 1.6 megawatts. Which is tiny compared to the many terawatt heat source. So, either this effect is irrelevant, or the fluid flow is complicated and turbulent and the electrical current is complicated too, but time-averages to a simple ring current while the fluid flow time-averages to a simple "rise to poles sink at equator" flow. With this postulate, the true complicated current is much larger and yields much greater joule heating. With this explanation, if we postulated that the true fluctuating current densities are perhaps 3000 times larger than the time-averaged current, then the upper bound would yield about the actual magnetic field. Bah. Fluid turbulence. Uncalculatable crud.

1 0

[math-fun] Simple model of Earth magnetic field, which sounded promising for a little while
by Warren D Smith 14 Feb '14

14 Feb '14

The Cambridge Encyclopedia of Earth Science claims what generates the Earth's magnetic field is not understood. Here I suggest a simple model. FACTS: 1. The field is roughly a dipole pointed approximately North, and at previous times in Earth history has pointed approximately South instead. North and South (with respect to rotation axis) appear to be stable states, but pointing in other directions (or nonexistent field) appears to be an unstable state which quickly vanishes and is replaced by one of the two stable states. This historical record is revealed by magnetism trapped in volcanic rocks at time of formation, versus that time. You can see a graphic in http://en.wikipedia.org/wiki/Geomagnetic_reversal which shows 9 N-pointing and 9 S-pointing periods occurred during the last 5 Million years, each with apparently-random durations. There may also have been additional switchovers with short durations. The short ones are hard to detect. The best known such short-duration event was the "Laschamp event" in about the year 39000 BC, a reversal lasting only about 440 years with the actual change of polarity lasting around 250 years. The reversed field was 75% weaker and the strength dropped to only 5% of the current strength during the transition. This resulted in greater radiation reaching the Earth, causing greater production of beryllium 10 and higher levels of carbon 14. Perhaps these short events are best regarded as failed attempts to reverse -- we drop to a low field state, and then that state is attracted to the two N- or S-pointing stable states. If attracted back to the original one then we get a "failed reversal" instead of a genuine reversal. The switchover-times are each <=50,000 years. There was at least one 40 million year span containing no reversal, but overall average has been about 2.2 reversals per million years. 2. The Moon has only tiny magnetic field, mostly a few nanoTesla. (Versus: Earth's is 25-65 microTesla.) Comparably-tiny fields: Mars, Venus. Intermediate strength: Mercury. Jupiter's field is 10X Earth's. Saturn also has a strong field. Uranus and Neptune have large magnetic fields, but they are not well axis aligned. MY MODEL: The field is generated by a "dynamo" from convection currents of hot electrically conductive fluid inside the planet. Moon and Mars have no field since they have no such internal convection currents, they are geo-tectonically dead in the sense they have no volcanos (although Mars at an earlier time did have volcanos) and no plate tectonics, indicating internal convection currents are much weaker and less-well defined (i.e. tend to cancel out over time averaging). Venus still has active volcanos, but its lack of magnetic field will be explained below. Mercury's, Saturn's, and Jupiter's fields are presumed to be generated by same mechanism as Earth's. The more gravity and more internal heat a (non-dead) planet has, the more powerful its convection and hence more powerful its fields -- also axis alignment more-perpendicular to planet-sun direction helps generate more field for reason explained below -- all explaining why Mercury<Earth<Saturn<Jupiter. Now my claim is that the convection currents flow outward from core toward equator. The fluid then cools, and drops back to the core from the poles. This requires that (and is predicted by) the poles tend to exert more cooling effect than the equator. If so, then a N- or S-pointing magnetic field, will induce (via Lorentz force) an equator-circling electrical current, This in turn will reinforce the field. Result is a stable "dynamo" powered ultimately by internal heat (radioactively generated and/or left over from planet formation). For Earth, situation is complicated by the presence of oceans over thin crust, and continents over thick crust. Presumably there is more cooling in the ocean regions and/or volcanic regions. Also there are "hot spots" such as Hawaii. Also the Earth's rotation axis precesses and is not exactly perpendicular to the Earth-sun line. All this complexity, plus fluid "turbulence," presumably has something to do with why we get occasional reversals. I would presume the field continually fluctuates and whenever (rarely) a fluctuation is comparable to the field itself, then we are ready for either a reversal or failed-reversal. In the case of Venus, there are no plate tectonics (although there is volcanism) and there is a huge atmosphere which redistributes heat efficiently, meaning the equator may not have a much greater cooling effect than the poles. For Uranus and Neptune the sun is so far away that its thermal effects are pretty negligible, explaining why their fields are not axis aligned (and Uranus has a huge axial tilt too, so even if the sun's heat non-negligible, that'd still mostly cancel out its effects). SO??? That seemed pretty easy. Why wasn't this thought of ages ago? Too shed light on that, let's do some crude numerical calculations. Assume convection speed of v = 1 cm/year -- same order as continent drift speeds. Total voltage induced circling equator (25000 mile circumf.) then is of order v * (50 microTesla) * (25000 miles) = 6.4 * 10^(-7) volts. Assuming 40 ohm*meter electrical resistivity for lava (this was a Hawaii measurement) the earth going round the equator is about a 10^(-5) ohm resistor. [However it actually is found that high pressures and high temperatures both act to decrease resistivity for molten rock apparently like Arhennius law exp(const/T), plus there are stairsteps caused by temperature surpassing ionization-energy thresholds.] So this voltage would induce a current circling equator of order 0.06 Amps. That in turn would cause a magnetic dipole moment of about 8*10^12 * meter^2 * ampere which is 10^10 times smaller than the actual Earth dipole moment 8*10^22 * meter^2 * Ampere. Oops. It seems still within the reasonable realm that the resistivity might be only 1 ohm*meter and the flow speed might be 10 cm/year. This would increase our predicted dipole moment by a factor of 40*10=400. The flow speeds necessarily will be larger still nearer the center of the Earth due to volumes being smaller there, and the magnetic fields there might be much larger due to larger currents at smaller distances. So assuming both 10X greater flow speed and magnetic field down there, we could get another factor of a few hundred. But this is still well short of the required 10^10 factor. CONCLUSION: The model sounded great for a little while, but it is predicting too small fields compared to observation. What went wrong?? -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

10 18

[math-fun] Connected components in random Cartesian mazes
by Thane Plambeck 12 Feb '14

12 Feb '14

Draw a maze on a two-dimensional n x n grid by erecting a wall between each pair of adjacent (ie, distance one) lattice points independently with probability p. Eyeballing these things in Mathematica, it looks to me like if p < 1/2, there tends to be one "large component" that connects almost all the cells that are not walled off into "locally small" (say 1x1 or 1x2) walled gardens. I'm sure I can't be the first to have considered something like this. I'd welcome information about prior work. -- Thane Plambeck tplambeck(a)gmail.com http://counterwave.com/

11 14

[math-fun] oldest stars?
by Warren D Smith 12 Feb '14

12 Feb '14

[math-fun] oldest stars? >You can read the methodology here > http://arxiv.org/abs/astro-ph/0703414 >but it looks like they rely on earlier work for relative abundance. See, >for example, > http://iopscience.iop.org/0004-637X/591/2/936/fulltext/ --the age estimates depend on knowing "initial" isotope ratios. These are then compared with "now." The initial ones seem to me to be a complete guess -- they say vaguely there are "theoretical models" of what they ought to be. Well, that seems to me to be dubious, especially if we are talking about 1 specific star's initial values, as opposed to say, the whole universe average at some time. So: I have trouble believing they can date single stars that old that accurately. However, on the bright side, they have about 5 different kinds of isotope "clocks" they were able to examine on this star, each checking on the others. That makes things sound better. On 2/12/14, 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 > http://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: Connected components in random Cartesian mazes (Dan Asimov) > 2. oldest stars? (Warren D Smith) > 3. Simple model of Earth magnetic field, which sounded promising > for a little while (Warren D Smith) > 4. Resistor network puzzle (Tom Karzes) > 5. Re: oldest stars? (Charles Greathouse) > > > ---------------------------------------------------------------------- > > Message: 1 > Date: Wed, 12 Feb 2014 11:28:05 -0800 > From: Dan Asimov <dasimov(a)earthlink.net> > To: math-fun <math-fun(a)mailman.xmission.com> > Subject: Re: [math-fun] Connected components in random Cartesian mazes > Message-ID: <2A761C13-86D5-462A-A60B-027D55622367(a)earthlink.net> > Content-Type: text/plain; charset=us-ascii > > < > http://www.amazon.com/s/ref=nb_sb_noss?url=search-alias%3Daps&field-keyword… >>. > > > On Feb 12, 2014, at 6:55 AM, Henry Baker <hbaker1(a)pipeline.com> wrote: > >> Perhaps it would be difficult to reproduce this experiment today. >> >> Is it even possible to purchase carbon paper anymore? >> >> I heard a program on NPR (Planet Money, perhaps?) where they said that the >> last >> bastion of carbon paper was India, where the govt still records lots of >> official >> stuff typed on paper with carbon duplicates/triplicates/... >> >> At 02:25 PM 2/11/2014, Richard E. Howard wrote: >>> The most fun paper in this field (in my opinion) is by Last and Thouless >>> in 1971. >>> >>> In this era of Higgs Boson-sized experimental budget, it is a study in >>> elegance. >>> They punched carefully randomized holes in carbon paper (2D conducting >>> sheet) and measured the resistivity as a function of hole density. >>> >>> The experiment trumped the theory at the time and the apparatus (for >>> once) >>> actually cost less than the pencil/paper theory... >> >> >> >> >> _______________________________________________ >> math-fun mailing list >> math-fun(a)mailman.xmission.com >> http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > > > > ------------------------------ > > Message: 2 > Date: Tue, 11 Feb 2014 19:48:50 -0500 > From: Warren D Smith <warren.wds(a)gmail.com> > To: math-fun <math-fun(a)mailman.xmission.com> > Subject: [math-fun] oldest stars? > Message-ID: > <CAAJP7Y3aK0UAr0A06PTEdmW+MNyo+5yqTHoBDRUrq-BU0zRJfA(a)mail.gmail.com> > Content-Type: text/plain; charset=ISO-8859-1 > > New York Times science section today discusses very old stars, in our > galaxy, found by searches for ones with very small iron content. > They then claim the oldest confirmed age of a star is 13.2 Gyr, confirmed > by use of uranium and thorium half-lives. > http://phys.org/news98033554.html > > QUESTION: > Huh? How can you date a star by measuring its U and Th isotope content? > It seems to me you'd have to know its isotope ratios when it started, > and you don't. > > Obviously, these stars are not the oldest, since they contain iron > that came from still-older stars -- but they come close to being first > generation. > > > -- > Warren D. Smith > http://RangeVoting.org <-- add your endorsement (by clicking > "endorse" as 1st step) > > > > ------------------------------ > > Message: 3 > Date: Wed, 12 Feb 2014 15:25:07 -0500 > From: Warren D Smith <warren.wds(a)gmail.com> > To: math-fun <math-fun(a)mailman.xmission.com> > Subject: [math-fun] Simple model of Earth magnetic field, which > sounded promising for a little while > Message-ID: > <CAAJP7Y06tu1dXAf8XM-ZWOKhc-soz9YRcXDGweP1g6p4atTmEQ(a)mail.gmail.com> > Content-Type: text/plain; charset=ISO-8859-1 > > The Cambridge Encyclopedia of Earth Science claims what generates the > Earth's magnetic field is not understood. Here I suggest a simple > model. > > FACTS: > 1. The field is roughly a dipole pointed approximately North, and at > previous times > in Earth history has pointed approximately South instead. North and South > (with respect to rotation axis) appear to be stable states, but > pointing in other > directions (or nonexistent field) appears to be an unstable state which > quickly > vanishes and is replaced by one of the two stable states. This > historical record is revealed by magnetism trapped in volcanic rocks > at time of formation, versus that time. > You can see a graphic in > http://en.wikipedia.org/wiki/Geomagnetic_reversal > which shows 9 N-pointing and 9 S-pointing periods occurred during the > last 5 Million years, each with apparently-random durations. > > There may also have been additional switchovers with short durations. > The short ones are hard to detect. The best known such > short-duration event was the "Laschamp event" in about the year 39000 > BC, a reversal lasting only about 440 years with the actual change of > polarity lasting around 250 years. The reversed field was 75% weaker > and the strength dropped to only 5% of the current strength during the > transition. This resulted in greater radiation reaching the Earth, > causing greater production of beryllium 10 and higher levels of carbon > 14. Perhaps these short events are best regarded as failed attempts > to reverse -- we drop to a low field state, and then that state is > attracted to the two N- or S-pointing stable states. If attracted back > to the original one then we get a "failed reversal" instead of a > genuine reversal. > > The switchover-times are each <=50,000 years. There was at least one > 40 million year span containing no reversal, but overall average has > been about 2.2 reversals per million years. > > 2. The Moon has only tiny magnetic field, mostly a few nanoTesla. > (Versus: Earth's is 25-65 microTesla.) > Comparably-tiny fields: Mars, Venus. > Intermediate strength: Mercury. > Jupiter's field is 10X Earth's. Saturn also has a strong field. > Uranus and Neptune have large magnetic fields, but they are not well > axis aligned. > > MY MODEL: > The field is generated by a "dynamo" from convection currents of hot > electrically conductive fluid inside the planet. Moon and Mars have > no field since they > have no such internal convection currents, they are geo-tectonically dead in > the > sense they have no volcanos (although Mars at an earlier time did have > volcanos) > and no plate tectonics, indicating internal convection currents are > much weaker and > less-well defined (i.e. tend to cancel out over time averaging). > Venus still has active volcanos, but its lack of magnetic field will > be explained below. > Mercury's, Saturn's, and Jupiter's fields are presumed to be generated > by same mechanism as Earth's. The more gravity and more internal heat > a (non-dead) planet has, the more powerful its convection and hence > more powerful its fields -- also axis > alignment more-perpendicular to planet-sun direction helps generate more > field > for reason explained below -- all explaining why > Mercury<Earth<Saturn<Jupiter. > > Now my claim is that the convection currents flow outward from core > toward equator. The fluid then cools, and drops back to the core > from the poles. > This requires that (and is predicted by) the poles tend to exert more > cooling effect than the equator. > > If so, then a N- or S-pointing magnetic field, will induce (via > Lorentz force) an equator-circling electrical current, This in turn > will reinforce the field. Result is a stable "dynamo" > powered ultimately by internal heat (radioactively generated and/or > left over from planet > formation). For Earth, situation is complicated by the presence of oceans > over > thin crust, and continents over thick crust. Presumably there is more > cooling in the ocean regions and/or volcanic regions. Also there are > "hot spots" such as Hawaii. Also > the Earth's rotation axis precesses and is not exactly perpendicular > to the Earth-sun line. > All this complexity, plus fluid "turbulence," presumably has something > to do with why we get occasional reversals. I would presume the field > continually fluctuates and whenever > (rarely) a fluctuation is comparable to the field itself, then we are > ready for either a reversal or failed-reversal. > > In the case of Venus, there are no plate tectonics (although there is > volcanism) and > there is a huge atmosphere which redistributes heat efficiently, > meaning the equator > may not have a much greater cooling effect than the poles. > > For Uranus and Neptune the sun is so far away that its thermal effects > are pretty negligible, explaining why their fields are not axis > aligned (and Uranus has a huge axial tilt too, so even if the sun's > heat non-negligible, that'd still mostly cancel out its effects). > > SO??? > > That seemed pretty easy. Why wasn't this thought of ages ago? > > Too shed light on that, let's do some crude numerical calculations. > Assume convection speed of v = 1 cm/year -- same order as continent > drift speeds. > Total voltage induced circling equator (25000 mile circumf.) then is of > order > v * (50 microTesla) * (25000 miles) = 6.4 * 10^(-7) volts. > Assuming 40 ohm*meter electrical resistivity for lava (this was a Hawaii > measurement) the earth going round the equator is about a 10^(-5) ohm > resistor. > [However it actually is found that high pressures and high > temperatures both act to decrease resistivity for molten rock > apparently like Arhennius law exp(const/T), > plus there are stairsteps caused by temperature surpassing > ionization-energy thresholds.] > > So this voltage would induce a current circling equator of order 0.06 Amps. > That in turn would cause a magnetic dipole moment of about > 8*10^12 * meter^2 * ampere > which is 10^10 times smaller than the actual Earth dipole moment > 8*10^22 * meter^2 * Ampere. > > Oops. It seems still within the reasonable realm that the > resistivity might be only > 1 ohm*meter and the flow speed might be 10 cm/year. This would increase > our > predicted dipole moment by a factor of 40*10=400. > The flow speeds necessarily will be larger still > nearer the center of the Earth due to volumes being smaller there, > and the magnetic fields there might be much larger due to larger > currents at smaller distances. So assuming both 10X greater > flow speed and magnetic field down there, we could get another factor > of a few hundred. > But this is still well short of the required 10^10 factor. > > CONCLUSION: > The model sounded great for a little while, but it is predicting too > small fields > compared to observation. What went wrong?? > > > > > > > > > > > -- > Warren D. Smith > http://RangeVoting.org <-- add your endorsement (by clicking > "endorse" as 1st step) > > > > ------------------------------ > > Message: 4 > Date: Wed, 12 Feb 2014 12:43:15 -0800 > From: Tom Karzes <karzes(a)sonic.net> > To: math-fun <math-fun(a)mailman.xmission.com> > Subject: [math-fun] Resistor network puzzle > Message-ID: <21243.56547.61304.994729(a)gargle.gargle.HOWL> > Content-Type: text/plain; charset=us-ascii > > This reminds me of an old puzzle: > > You have an infinite square grid of 1 ohm resistors. What is the > resistance between two adjacent nodes in the grid? > > More precisely, you have an infinte square grid of nodes. Each > horizontally adjacent pair of nodes is joined by a 1 ohm resistor, and > each vertically adjacent pair of nodes is joined by a 1 ohm resistor. > What is the resistance between two adjacent nodes (either horizontal > or vertical)? > > Tom > > Richard E. Howard writes: > > The most fun paper in this field (in my opinion) is by Last and Thouless > in > > 1971. > > > > In this era of Higgs Boson-sized experimental budget, it is a study in > > elegance. > > They punched carefully randomized holes in carbon paper (2D conducting > > sheet) and measured the resistivity as a function of hole density. > > > > The experiment trumped the theory at the time and the apparatus (for > once) > > actually cost less than the pencil/paper theory... > > > > --R > > > > ------------------------------ > > Message: 5 > Date: Wed, 12 Feb 2014 16:07:47 -0500 > From: Charles Greathouse <charles.greathouse(a)case.edu> > To: math-fun <math-fun(a)mailman.xmission.com> > Subject: Re: [math-fun] oldest stars? > Message-ID: > <CAAkfSGJx0aDsGx9vHyrriG8X7FRF0XendAUKsZP5e1EotocfSA(a)mail.gmail.com> > Content-Type: text/plain; charset=ISO-8859-1 > > You can read the methodology here > http://arxiv.org/abs/astro-ph/0703414 > but it looks like they rely on earlier work for relative abundance. See, > for example, > http://iopscience.iop.org/0004-637X/591/2/936/fulltext/ > > Charles Greathouse > Analyst/Programmer > Case Western Reserve University > > > On Tue, Feb 11, 2014 at 7:48 PM, Warren D Smith > <warren.wds(a)gmail.com>wrote: > >> New York Times science section today discusses very old stars, in our >> galaxy, found by searches for ones with very small iron content. >> They then claim the oldest confirmed age of a star is 13.2 Gyr, confirmed >> by use of uranium and thorium half-lives. >> http://phys.org/news98033554.html >> >> QUESTION: >> Huh? How can you date a star by measuring its U and Th isotope content? >> It seems to me you'd have to know its isotope ratios when it started, >> and you don't. >> >> Obviously, these stars are not the oldest, since they contain iron >> that came from still-older stars -- but they come close to being first >> generation. >> >> >> -- >> Warren D. Smith >> http://RangeVoting.org <-- add your endorsement (by clicking >> "endorse" as 1st step) >> >> _______________________________________________ >> math-fun mailing list >> math-fun(a)mailman.xmission.com >> http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >> > > > ------------------------------ > > _______________________________________________ > math-fun mailing list > math-fun(a)mailman.xmission.com > http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > > End of math-fun Digest, Vol 132, Issue 29 > ***************************************** > -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

1 0

Re: [math-fun] Simple model of Earth magnetic field, which sounded promising for a little while
by Warren D Smith 12 Feb '14

12 Feb '14

A few more notes. 1. Iron is non-magnetic at pressures above 10 Giga Pascals (where it enters an HCP-crystal-structure phase), so presumably the Earth's core is not ferromagnetic. (Sad, since if it were, that would help resolve the puzzle.) 2. The simple model I proposed could run in either (A) the direction where fluid flows up to equator & sinks at poles, or (B) the (opposite) rises-to-poles&sinks-at-equator direction. Only one of these directions yields a self-reinforcing dynamo. For the other direction, the "dynamo" effect should be of negative feedback! So, my hands got a workout applying "right hand rule" and suchlike to try to figure out which is the good direction -- I believe the answer is: Direction B yields self-reinforcing field! Contrary-sign to my original post - sorry. It is not obvious to me which of A,B the fluid inside the earth should prefer, although probably one of them is favored. 3. Higher pressure decreases electrical resistance in almost all materials. Higher temperature decreases it in semiconductive and insulating materials but increases it in metals. What is the electrical resistivity of the earth's interior? It might be that the resistivity varies with depth and is high at some particular particularly important depth(s).

1 0