July 2015 - mathfun@mailman.xmission.com

[math-fun] Somsky gears, aka unsymmetric off-center planetary gears. Don't really work.
by Warren D Smith 09 Jul '15

09 Jul '15

https://www.youtube.com/watch?v=M_BUn4TDns8 demonstrates. And there is a proof, which appears momentarily in the video at 3:35, if you freeze the video and view at full screen magnification you can actually try to read the proof. The first line in the proof is the necessary condition about tooth count sum which makes it work. This is quite a beautiful thing, which is highly non-obvious, but has a very simple pre-college ancient-Greek sort of proof. However, if the radii b and b' are unequal then Somsky's proof indicates the two gears with centers B and B' will rotate at constant angular velocity hence necessarily eventually collide. Therefore, this kind of planetary gear is IMPOSSIBLE if all gears coplanar and you want to be able to rotate the central gear (center C) an infinite number of times without being killed by a collision. By making the gears have suitable nonzero thicknesses and with the gears with centers B' and B being thinner and in sufficiently offset parallel planes, then they can "pass through" each other rather than colliding... but that will not save us because gear with center C will be unable to mesh with them both. So, I conclude this kind of planetary gear is not possible unless the travel distance is restricted; you cannot do an infinite number of rotations. So it seems practically useless, unfortunately, except in the well known symmetric case where all planets same size. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] Monotonicity of wobble
by James Propp 08 Jul '15

08 Jul '15

I'm nearly finished with the square-wheeled tricycle essay, especially the annoyingly technical Extras section (http://mathenchant.org/wheel-extras.pdf) in which I use high school math to work out how much wobble you get from various non-catenary roads (to make up for the fact that most of the relevant math is college level). One nicety I neglected is that in computing the wobble, it's not obvious that it suffices to compare how high the center of mass of the wheel is when it's symmetrically balanced atop a hump, versus how high the center of mass of the wheel is when it's symmetrically wedged between two humps. That's because it's not a priori clear that the height of the center of mass of the wheel will change in a monotone fashion as one goes between the two positions ("atop" vs. "wedged"). Assuming I did the work correctly, for the case of the flat road and the quarter-circle road, the center of mass is slightly higher for the " wedged" position, while for the case of the half-circle road, the center of mass is slightly higher for the "atop" position. The monotonicity property I desire is clear for the case of a flat road, but I don't see why it should be true for the half-circle road or the quarter-circle road (and maybe it's not even true!). Can anyone help me out? As always, the essay will acknowledge everyone who provided assistance along the way. The current draft is posted at http://mathenchant.org/wheel-draft.rtf . Thanks, Jim Propp

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[math-fun] 11,373,076:1 gear train
by Henry Baker 08 Jul '15

08 Jul '15

FYI -- Modern day antikythera device? https://en.wikipedia.org/wiki/Antikythera_mechanism http://3dprint.com/79049/3d-printed-extreme-reduction/ 3D Printed Extreme Reduction Gearing Device Offers Exponential Force of 1 by 11 Million "The multi-colored, 3D printed device is compact, with separate gears designated by the multi-colored design skipping a tooth with each turn. With a crank and planetary gears built into the first gear or ring, it works to drive the next ring, which moves in the opposite direction. This means that the green ring is moving forward, and the red ring is moving backward, almost at the exactly the same speed, with the net effect being a one to eleven million gearing ratio due to an algorithm developed by van Deventer."

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[math-fun] dark matter cross sections, a few simple calculations
by Warren D Smith 08 Jul '15

08 Jul '15

According to my calculations, white dwarfs are highly transparent to dark matter particles -- proven by experiments setting upper limits on dark matter cross sections with nucleons of 10^(-46) to 10^(-50) meter^2. Assuming the dark matter particles have masses 1-10000 GeV/c^2. Certainly over 99% transmission through a white dwarf. [However white dwarfs should be quite opaque to neutrinos with energies >=1 GeV, based on cross section data & formulas here: http://cupp.oulu.fi/neutrino/nd-cross.html .] However, neutron stars could be anywhere from highly transparent to highly opaque to dark matter, we cannot tell which from detector experiments up to year 2015. It has been alleged based on astronomical observations that dark matter has a cross section for interaction with other dark matter particles (1.7 +- 0.7) * 10^(-4) cm^2/gram. That for a dark matter particle mass of 1-10000 GeV/c^2 would be 3*10^(-32) to 3*10^(-28) meter^2 cross section for interaction per particle. This is INSANELY LARGER than the known upper limits from detectors. How can it be that dark matter is much more interactive with other dark matter, than it is with normal matter? This is simply not possible with the known physics forces. The astronomers blissfully never did this simple calculation, thus happily remaining ignorant of what would seem to be a huge contradiction. Seems to me the only escapes are either (a) the observers are wrong, (b) dark matter particles are tremendously more massive than previously suspected, (c) my calculations are wrong, (d) there are new "dark forces" in physics that do not affect normal matter, but do affect dark matter.

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Re: [math-fun] 11,373,076:1 gear train
by rwg 07 Jul '15

07 Jul '15

On 2015-07-07 14:26, Richard Fateman wrote: > There is a gear train in the SF exploratorium with a motor going > 200rpm on one end and the > far end has a gear embedded in concrete. The last gear will rotate > once in, what, a trillion years... > RJF How many megatons if it actually stored all the energy from that motor? http://www.exploratorium.edu/arts/works/machine-concrete emanating from http://www.chadweisshaar.com/blog/2013/03/26/trip-to-boston/ --rwg

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[math-fun] Modern Egyptian mathematicians
by James Propp 07 Jul '15

07 Jul '15

Do any of you have connections with any contemporary Egyptian mathematicians? I'm asking because I'm curious about the wooden "quarter circles" (not really quarter circles but that's apparently what they're called) that have been found near the pyramids and that may have played a role in their construction. Knowing the precise dimensions of these objects might enable one to assess some theories of their purpose (such as Gerard Fonte's theory that the bounding arcs were approximate catenaries, and that large blocks of limestone were rolled end over end along tracks made of the catenaries). Fonte tells me that these relics are not on display, but are held by the Egyptian government. I am quite politically naive, but I'm guessing that an Egyptian mathematician with an interest in understanding ancient Egyptian technology would stand a chance of gaining access to the quarter-circles and taking some photographs (or at least recording their dimensions). Jim Propp

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[math-fun] Blancmange is a dessert
by James Buddenhagen 07 Jul '15

07 Jul '15

But APG was probably testing us. http://mathworld.wolfram.com/BlancmangeFunction.html

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[math-fun] Discriminant of cubic eqn
by Henry Baker 06 Jul '15

06 Jul '15

If D is the discriminant of (z-r1)(z-r2)(z-r3), then D = (r1-r2)^2 (r1-r3)^2 (r2-r3)^2, r1,r2,r3 complex. |D|=|r1-r2|^2 |r1-r3|^2 |r2-r3|^2 =(abc)^2 (a,b,c are the side-lengths of the triangle of roots) =(4 R A)^2 (R = circumradius of triangle of roots; A = area of triangle) = 16 R^2 A^2 Of course, this formula doesn't work when the triangle flattens out -- e.g., all roots are on the real axis -- because A->0 and R-> oo. Better off to use _curvature_ k=1/R instead of radius, so at least we have |D| = 16 A^2 / k^2 = 16 (A/k)^2 where A->0 and k->0 simultaneously in the 3-real-roots case.

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[math-fun] Definition of mathematics
by Warren D Smith 06 Jul '15

06 Jul '15

I still like mine better than everybody else's: it is the study of thinking & how to be a better thinker. Is it the study of structure, patterns & form? No, because it also encompasses the study of phenomena that seem structureless and formless. How about chaos, fluid turbulence, and random number generators? Are they to be excluded from mathematics? Is it algebraic & symbolic manipulations? No, that would exclude much ancient greek geometry-based mathematics. Is it the study of "numbers" and their descendants? How about set theory, topology, and logic as counterexamples? Is it physics? Well, plenty of maths has nothing to do with physics, in fact sometimes intentionally so, such as the study of laws intentionally designed to be wrong-physics. Is it the study of tautologies and obviously-true statements? Dubious, counterexamples have already been posted. Is it the study of rule-sets we make up (Vi Hart)? Not a bad try, that one. I think all those definitions were trying to be my definition but falling a little short, except for maybe Vi Hart's definition which arguably is equivalent to mine. So there :) On 7/6/15, 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 (meekerdb) > 2. Re: Definition of mathematics (James Propp) > 3. Re: Definition of mathematics (James Propp) > 4. Re: Definition of mathematics (meekerdb) > 5. Re: New(?) way to tabulate pFq identities (Bill Gosper) > > > ---------------------------------------------------------------------- > > Message: 1 > Date: Sun, 05 Jul 2015 18:28:11 -0700 > From: meekerdb <meekerdb(a)verizon.net> > To: math-fun <math-fun(a)mailman.xmission.com> > Subject: Re: [math-fun] Definition of mathematics > Message-ID: <5599D9AB.9080209(a)verizon.net> > Content-Type: text/plain; charset=utf-8; format=flowed > > I can add a few: > > "The duty of abstract mathematics, as I see it, is precisely to > expand our capacity for hypothesizing possible ontologies." > --- Norm Levitt > > A physicist goes off to a conference. After a week his suit?s gotten soiled > and crumpled, > so he goes out to look for a dry cleaner. Walking down the main street of > town, he comes > upon a store with a lot of signs out front. One of them says ?Dry Cleaning.? > So he goes in > with his dirty suit and asks when he can come back to pick it up. The > mathematician who > owns the shop replies, ?I?m terribly sorry, but we don?t do dry cleaning.? > ?What?? > exclaims the puzzled physicist. ?The sign outside says ?Dry Cleaning?!? ?We > do not do > anything here,? replies the mathematician. ?We only sell signs!? > --- Alain Connes, in Changeux > > "A mathematician is like a mad tailor: he is making "all possible clothes" > and hopes to > make also something suitable for dressing" > --- Stanislaw Lem, Summa Techologiae > > "Mathematics is part of physics. Physics is an experimental science, a part > of natural > science. Mathematics is the part of physics where experiments are cheap." > --- Vladimir Arnold. > > "Math is a cybervirus that lives in human minds, evolves therein and > reproduces itself via > language." > --- Stephen Paul King > > Brent Meeker > > On 7/5/2015 2:29 PM, Thane Plambeck wrote: >> i just noticed there are several proposed definitions (some of them >> familiar to me, others not), on the wikipedia page for bertrand russell, >> and dropped in below >> >> Bertrand Russell wrote this famous tongue-in-cheek definition, describing >> the way all terms in mathematics are ultimately defined by reference to >> undefined terms: >> >> The subject in which we never know what we are talking about, nor whether >> what we are saying is true.[12] >> <https://en.m.wikipedia.org/wiki/Definitions_of_mathematics#cite_note-12> >> Bertrand >> Russell <https://en.m.wikipedia.org/wiki/Bertrand_Russell> 1901 >> >> Many other attempts to characterize mathematics have led to humor or >> poetic >> prose: >> >> "Mathematics is about making up rules and seeing what happens."[13] >> <https://en.m.wikipedia.org/wiki/Definitions_of_mathematics#cite_note-13> >> Vi >> Hart <https://en.m.wikipedia.org/wiki/Vi_Hart> >> >> A mathematician is a blind man in a dark room looking for a black cat >> which >> isn't there.[14] >> <https://en.m.wikipedia.org/wiki/Definitions_of_mathematics#cite_note-14> >> Charles >> Darwin <https://en.m.wikipedia.org/wiki/Charles_Darwin> >> >> A mathematician, like a painter or poet, is a maker of patterns. If his >> patterns are more permanent than theirs, it is because they are made with >> ideas. G. H. Hardy <https://en.m.wikipedia.org/wiki/G._H._Hardy>, 1940 >> >> Mathematics is the art of giving the same name to different things.[8] >> <https://en.m.wikipedia.org/wiki/Definitions_of_mathematics#cite_note-eves-8> >> Henri Poincar? <https://en.m.wikipedia.org/wiki/Henri_Poincar%C3%A9> >> >> Mathematics is the science of skilful operations with concepts and rules >> invented just for this purpose. [this purpose being the skilful operation >> ....][15] >> <https://en.m.wikipedia.org/wiki/Definitions_of_mathematics#cite_note-15> >> Eugene >> Wigner <https://en.m.wikipedia.org/wiki/Eugene_Wigner> >> >> Mathematics is not a book confined within a cover and bound between >> brazen >> clasps, whose contents it needs only patience to ransack; it is not a >> mine, >> whose treasures may take long to reduce into possession, but which fill >> only a limited number of veins and lodes; it is not a soil, whose >> fertility >> can be exhausted by the yield of successive harvests; it is not a >> continent >> or an ocean, whose area can be mapped out and its contour defined: it is >> limitless as that space which it finds too narrow for its aspirations; >> its >> possibilities are as infinite as the worlds which are forever crowding in >> and multiplying upon the astronomer's gaze; it is as incapable of being >> restricted within assigned boundaries or being reduced to definitions of >> permanent validity, as the consciousness of life, which seems to slumber >> in >> each monad, in every atom of matter, in each leaf and bud cell, and is >> forever ready to burst forth into new forms of vegetable and animal >> existence.[16] >> <https://en.m.wikipedia.org/wiki/Definitions_of_mathematics#cite_note-Stewar…> >> James Joseph Sylvester >> <https://en.m.wikipedia.org/wiki/James_Joseph_Sylvester> >> >> What is mathematics? What is it for? What are mathematicians doing >> nowadays? Wasn't it all finished long ago? How many new numbers can you >> invent anyway? Is today's mathematics just a matter of huge calculations, >> with the mathematician as a kind of zookeeper, making sure the precious >> computers are fed and watered? If it's not, what is it other than the >> incomprehensible outpourings of superpowered brainboxes with their heads >> in >> the clouds and their feet dangling from the lofty balconies of their >> ivory >> towers? Mathematics is all of these, and none. Mostly, it's just >> different. >> It's not what you expect it to be, you turn your back for a moment and >> it's >> changed. It's certainly not just a fixed body of knowledge, its growth is >> not confined to inventing new numbers, and its hidden tendrils pervade >> every aspect of modern life.[16] >> <https://en.m.wikipedia.org/wiki/Definitions_of_mathematics#cite_note-Stewar…> >> Ian Stewart >> <https://en.m.wikipedia.org/wiki/Ian_Stewart_(mathematician)> >> >> On Sunday, July 5, 2015, Andy Latto <andy.latto(a)pobox.com> wrote: >> >>> While you and I may both agree that the set of things that we couldn't >>> imagine were otherwise roughly corresponds to mathematics, there are a >>> lot of theological arguments that porport to prove that there is a >>> God, and that he has certain properties, and that this couldn't >>> possibly be otherwise. So I don't see a good way to patch this >>> definition to exclude theology. >>> >>> Andy >>> >>> >>> On Sat, Jul 4, 2015 at 3:30 PM, James Propp <jamespropp(a)gmail.com >>> <javascript:;>> wrote: >>>> Here's a start of a different sort of approach to the question of what >>> math >>>> is, taking off from a remark in Jordan Ellenberg's book "How Not To Be >>>> Wrong" about how a times b equals b times a, for all a and b, "because >>>> it >>>> couldn't be otherwise". >>>> >>>> Let's make this more modest and say that we can't IMAGINE how it could >>>> be >>>> otherwise. >>>> >>>> That is: I can just barely imagine (with a mental squint, and with an >>> inner >>>> acknowledgments of my limitations as a reasoner) *that* ordinary >>>> multiplication of ordinary natural numbers might not be commutative. >>>> But >>> I >>>> cannot imagine in any kind of detail *how* it might fail to be >>> commutative. >>>> There are probably lots of things that humans aren't able to doubt >>>> (such >>> as >>>> "I exist") that don't count as mathematics, so this definition will >>>> need >>> to >>>> be modified before it comes close to drawing the line between math and >>>> non-math in approximately the right place. >>>> >>>> A variant of this approach would be to define pure mathematics as the >>> study >>>> of fantasies that possess a certain kind of coherence. >>>> >>>> Jim Propp >>>> _______________________________________________ >>>> math-fun mailing list >>>> math-fun(a)mailman.xmission.com <javascript:;> >>>> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >>> >>> >>> -- >>> Andy.Latto(a)pobox.com <javascript:;> >>> >>> _______________________________________________ >>> math-fun mailing list >>> math-fun(a)mailman.xmission.com <javascript:;> >>> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >>> >> > > > > > ------------------------------ > > Message: 2 > Date: Sun, 5 Jul 2015 22:02:13 -0400 > From: James Propp <jamespropp(a)gmail.com> > To: math-fun <math-fun(a)mailman.xmission.com> > Subject: Re: [math-fun] Definition of mathematics > Message-ID: > <CA+G9J-cAFDyFBFzVBxMH0r=Ao_hNtF_08knNd88evmPAjWmoFA(a)mail.gmail.com> > Content-Type: text/plain; charset=UTF-8 > > Thanks, Brent. > > I'll comment on a couple of these: > > "The duty of abstract mathematics, as I see it, is precisely to >> expand our capacity for hypothesizing possible ontologies." >> > > I'll combine this with quotes from J. B. S. Haldane ("*Now my own suspicion > is that the Universe is not only queerer than we suppose, but queerer than > we can suppose*") and Francis Bacon (the end goal of science is "the > effecting of all things possible") and assert that the job of the pure > mathematician is the imagining of all things possible, however queer. > >> > "Math is a cybervirus that lives in human minds, evolves therein and >> reproduces itself via language." >> > > Here's my more specific analysis of the memetic ecology of math (this time > based on a quote that is often credited to Samuel Butler): A theorem is > only a question's way of making more questions. > > Jim Propp > > > ------------------------------ > > Message: 3 > Date: Sun, 5 Jul 2015 22:26:25 -0400 > From: James Propp <jamespropp(a)gmail.com> > To: math-fun <math-fun(a)mailman.xmission.com> > Subject: Re: [math-fun] Definition of mathematics > Message-ID: > <CA+G9J-fqhhuOEzohQFzw3Jds5ssyTEv_=DR=Kfb2AqtANZho6A(a)mail.gmail.com> > Content-Type: text/plain; charset=UTF-8 > > Is the quote from Levitt part of an essay that elaborates on this point, > and if so, where can I find it? > > "The duty of abstract mathematics, as I see it, is precisely to >> expand our capacity for hypothesizing possible ontologies." >> --- Norm Levitt >> > > Jim Propp > > > ------------------------------ > > Message: 4 > Date: Sun, 05 Jul 2015 22:11:00 -0700 > From: meekerdb <meekerdb(a)verizon.net> > To: math-fun <math-fun(a)mailman.xmission.com> > Subject: Re: [math-fun] Definition of mathematics > Message-ID: <559A0DE4.4070201(a)verizon.net> > Content-Type: text/plain; charset=windows-1252; format=flowed > > No, it's "private communication" as he and I both posted on Vic Stenger's > email list. > Sadly both Vic and Norm have died. > > Brent Meeker > > On 7/5/2015 7:26 PM, James Propp wrote: >> Is the quote from Levitt part of an essay that elaborates on this point, >> and if so, where can I find it? >> >> "The duty of abstract mathematics, as I see it, is precisely to >>> expand our capacity for hypothesizing possible ontologies." >>> --- Norm Levitt >>> >> Jim Propp >> _______________________________________________ >> math-fun mailing list >> math-fun(a)mailman.xmission.com >> https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun >> > > > > > ------------------------------ > > Message: 5 > Date: Sun, 5 Jul 2015 22:27:04 -0700 > From: Bill Gosper <billgosper(a)gmail.com> > To: math-fun(a)mailman.xmission.com > Subject: Re: [math-fun] New(?) way to tabulate pFq identities > Message-ID: > <CAA-4O0H6Uj=yzcWJ+hZc8ZGNuCsuLmP1Xp6QCZe5xnAg1BRSmg(a)mail.gmail.com> > Content-Type: text/plain; charset=UTF-8 > > On Tue, Jun 30, 2015 at 12:25 PM, Bill Gosper <billgosper(a)gmail.com> wrote: > >> By a process not obvious how to automate, >> > > In very rough and somewhat gory form: gosper.org/hyper3by3.pdf > (Pleasant introduction: gosper.org/pathi.pdf) > Included: a triple 4F3[+1/4] = double 5F4 = single 6f5 (with conjugate > quadratic > surd parameters) all with three freedoms, as a subproblem. Probably nobody > wants quadratic surd parameters, but there are numerous ways of > rationalizing > them by surrendering one freedom. So this is a compact way of transmitting > them. A CAS could either exhaustively tabulate all the rational > specializations, > or just keep the general form, and check for a matching query by solving > algebraic equations on the fly. This might be pushing the envelope when we > get to cubics and beyond. --rwg > > I extracted >>> HypergeometricPFQ[{a, b, (2 a)/5 + (6 b)/5 + 1, 2 a + 2 b, >>> a + 3 b}, {(2 a)/5 + (6 b)/5, a + (3 b)/2 + 1/2, a + (3 b)/2 + 1, >>> 2 b + 1}, -(1/4)] == >>> (4^-a Gamma[b + 1/2] Gamma[2 a + 3 b + 1])/( >>> Gamma[a + b + 1/2] Gamma[a + 3 b + 1]) >>> >>> from a summation containing a messy, irreducible cubic factor, analogous >>> to the sort used to discover recurrence relations via "creative >>> telescopy". >>> Punting the cubic cost two degrees of freedom, but there were hundreds >>> of variants. The unpleasant choice seems to be between a useless messy >>> cubic, and a nearly useless identity with only two degrees of freedom. >>> Probably the identity you need was among the hundreds I threw away. >>> >>> Here's an interesting remedy. What you actually tabulate is, in this >>> case, >>> the fully general result in a somewhat intimidating >>> >> nauseating >> >>> form: >>> >>> ((-b - d) ((1/(-1 + b))(-1 + b - d) (-1 + a + b - d) (-1 + b + c - >>> d) HypergeometricPFQ[{a, -1 + b, c, >>> d, -1 - a + b + d, -1 + b - c + d}, {b/2 + d/2, >>> >> [This was so miscanonicalized that I'm vandalizing it here to make sure >> nobody wastes time trying to use it.] >> >>> >>> >> It's then fairly routine to determine whether your pFq[-1/4] is a special >>> case. >>> >> >> I.e., you check your z argument = -1/4 and your parameters are congruent >> mod1. >> >>> It's not pencil and paper, but many people now have access to computers. >>> --rwg >>> >> >> OK, here it is cleaned up. >> >> -(b - 2 d) (a + b - 2 d) (b + c - 2 d) HypergeometricPFQ[{a, -a + b, >> b - c, c, b - d, d}, {1/2 + b/2, 1 + b/2, 1 - a + b - c, >> 1 - a + d, 1 - c + d}, -(1/4)] + (b - d) (5 + 3 c + >> b (9 + 5 b + 4 c) + a (3 + 4 b + c - 7 d) - 18 d - >> 7 (3 b + c) d + 21 d^2) HypergeometricPFQ[{a, -a + b, b - c, c, >> 1 + b - d, d}, {1/2 + b/2, 1 + b/2, 1 - a + b - c, 1 - a + d, >> 1 - c + d}, -(1/4)] - 3 (5 + a + 3 b + c - 6 d) (b - d) (1 + b - >> d) HypergeometricPFQ[{a, -a + b, b - c, c, 2 + b - d, >> d}, {1/2 + b/2, 1 + b/2, 1 - a + b - c, 1 - a + d, 1 - c + d}, -( >> 1/4)] + 5 (b - d) (1 + b - d) (2 + b - >> d) HypergeometricPFQ[{a, -a + b, b - c, c, 3 + b - d, >> d}, {1/2 + b/2, 1 + b/2, 1 - a + b - c, 1 - a + d, 1 - c + d}, >> -(1/4)] == ( >> Gamma[1 + b] Gamma[1 - a + b - c] Gamma[1 - a + d] Gamma[1 - c + d])/ >> (Gamma[-a + b] Gamma[b - c] Gamma[d] Gamma[1 - a - c + d]) >> >> >> But maybe instead of a special case, you have a limiting case, >> say a -> oo : >> >> (b - 2 d) (b + c - 2 d) HypergeometricPFQ[{b - c, c, b - d, >> d}, {1/2 + b/2, 1 + b/2, 1 - c + d}, 1/ >> 4] - (3 + 4 b + c - 7 d) (b - d) HypergeometricPFQ[{b - c, c, >> 1 + b - d, d}, {1/2 + b/2, 1 + b/2, 1 - c + d}, 1/4] + >> 3 (b - d) (1 + b - d) HypergeometricPFQ[{b - c, c, 2 + b - d, >> d}, {1/2 + b/2, 1 + b/2, 1 - c + d}, 1/4] == ( >> Gamma[1 + b] Gamma[1 - c + d])/(Gamma[b - c] Gamma[d]) >> >> These are now 4F3[+1/4] instead of >> 6F5[-1/4] ! Also note the rhs of the a->oo limit is not >> the a->oo limit of the 6F5 rhs, but rather that limit transformed >> by Gamma[z]->1/Gamma[1-z] . I have only the vaguest >> notion why. Maybe there were illegal limit interchanges when >> I blew up my contour rectangles. >> >> I'm a big fan of handling all the contiguous cases, although it >> seems ambitious. I think for something like this you'd need a >> thirteen(?) dimensional system of 7x7 matrices, and seven-term >> recurrences. The chance that your 6F5[-1/4] will come out in >> Gammas seems mighty slim. Slightly better for certain >> 7F6[a+1,b,...g; a,h,...,i,j,k,L|-1/4] >> --rwg >> (Some of you may wonder about Mathematica's line-breaking. >> It's deliberately ghastly, to preclude premature end-of-expression >> on readback.) >> > > > ------------------------------ > > 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 149, Issue 17 > ***************************************** > -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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[math-fun] Definition of mathematics
by Warren D Smith 06 Jul '15

06 Jul '15

Jeez Louise, it appears no two mathematicians agree on the definition of mathematics. KM: In my view, mathematics is definitely *not* the study of thinking. There are huge areas in cognitive psychology for that and mathematics exists irrespective of one's level of thinking (yes, my philosophy is showing). I lean toward the view that math is the study of structure, patterns, and form that arise as a consequence of rule sets. The things that Warren listed that *seem* formless are not actually. Chaos has form; fluid turbulence has structure. Even truly random events & sets have structure that can be studied mathematically. --WDS: well then, is there *anything* you would call structureless? MS: Topology started with the bridges of Konigsberg, Euler's formula V-E+F=2, and Lhuilier's generalization to polyhedra with higher genus V-E+F=2-2g. --WDS: well, I would agree that numbers are very useful and almost unavoidable. But they are avoidable sometimes, and math sometimes proves statements that can be expressed without numbers. Thus proving: numbers are not the litmus test. JP: Graduate students in the humanities are learning how to think more intelligently in their chosen specialties. The same goes for lawyers. But they're not doing mathematics. --WDS: Well, as far as I've been able to tell, law does not involve thinking, it involves "lying" and "manipulating"; in particular, virtually everything you read about in a law book is false. Well, let me give you a few examples from my experiences in traffic court. I always fight traffic tickets. So far, I always lost the case. Anyhow, in court #1, the judge's first move was to tell me I wasn't allowed to wear what I was wearing, which was, jeans & a jacket. I mean, literally, his very first move, before trial even started. So, evidently, his personal fashion sensibilities trumped the very highest law of the land, the 1st amendment to the US constitution, which states that even if the US congress tried to make a law about what I was allowed to wear in court, that law would be invalid. Shortly thereafter, I presented two written motions to the judge, following the advice of a law book, which was to make your motions be written, citing verbatim law, etc etc. What happened then was not in my law book, which was: the judge refused to even read the motions. He refused even to look at them. Instead he told me to explain them orally, and immediately dismissed them after 1.5 sentences for motion #1, and 0.5 sentences for motion #2. In court #2, the judge opened ceremonies with a long long speech about the single most important principle, more important that any law in the world, even than the constitution (or so he said, for about 10 minutes on end) which was the principle that everybody is presumed innocent until proven guilty. So then after many many hours he got to my case. "How do you plead?" he said. "Innocent" I said. "You aren't allowed to plead innocent" said the judge. "If you wanted to do that, you had to do it in writing on a special form, not here at your trial." If there is any actual logic and fairness and thinking going on in the law, I do not believe it happens at a low budget level. I believe the written law is simply ignored until you pay at least $5000. It's basically an extortion scheme. But anyhow, re the humanities, the law, etc, I disagree with Propp in a more fundamental respect than that. Sure, people being educated in any field are learning how to think better. But, so what? That is not relevant: those fields do not help humanity as a whole to improve its thinking. I contend that humanity is presently much better at thinking than it was 1000 years ago. This is due to the development of math, science, technology. But is humanity any better at legal thinking than it was 1000 years ago? I would say: no. I think there has been zero improvement there. If you took a lawyer from today and time-transported him back 1000 years, would he be any better at it than a lawyer of that time? No. What about humanities generally? Well, we have more convenient access to information today, which helps, but if that factor is removed, then it seems to me that the only improvements in humanities thinking that happened during the last 1000 years, are (what I would call) mathematical. And as far as I can see, every mathematical contribution ever, has helped humanity to be better thinkers. (If not, then I claim it doesn't deserve to be called mathematics...) What does "better" mean? Means able to do more. Superset. And re JP's own suggestion that maths is the study of tautologies, how about if I do a computational study of approximation algorithm #1 vs #2, finding #1 seems to work better. I have no idea why, but 99% of the examples I try, it works better. Seems to me, that would be a perfectly legitimate and valuable mathematical contribution, and it would improve our thinking, but there is no theorem or obvious truth there at all. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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