July 2014 - mathfun@mailman.xmission.com

[math-fun] Fwd: Thue-Morse sums
by Bill Gosper 10 Jul '14

10 Jul '14

---------- Forwarded message ---------- From: Julian Ziegler Hunts <julianj.zh(a)gmail.com> Date: Tue, Jul 8, 2014 at 9:37 AM Subject: Re: [math-fun] Thue-Morse sums To: Bill Gosper <billgosper(a)gmail.com> >then S0 = -3 S1 and S3 = (S1 - 3 S2)/2. Nope. The convergence is great; 2^18 terms gets you >30 digits and these are only accurate to about 10^-5. Julian YIKES! I must've been had by FindIntegerNullVector. Or just general Mma numerics. Note here spontaneous reduction to machine precision: In[172]:= Precision /@ Take[%168, -4] Out[172]= {26.7284, MachinePrecision, MachinePrecision, MachinePrecision} In[168]:= %152 =.; Table[ Sum[morse[n]/(1`69 - 1 + n), {n, 2^(k - 1), 2^k - 1}], {k, 23}] Out[168]= \ {-1.000000000000000000000000000000000000000000000000000000000000000000\ 000, -0.16666666666666666666666666666666666666666666666666666666666666\ 6666667, -0.\ 026190476190476190476190476190476190476190476190476190476190476190476,\ -0.003149628149628149628149628149628149628149628149628149628149628149\ 6281, -0.0002619192183025316892818376272195153227023190373683308580086\ 696571249, \ -0.0000140961597127378916942716179595142349011573826289517744899239499\ 297, -4.68535396826140722399442374299172711851286747074109655227506937\ 39*10^-7, \ -9.29696753332452313450490608814161365712389608058600043630547620*10^-\ 9, -1.0739453689565117144058603338727124796070485046345754188182264*\ 10^-10, -7.\ 08627180236246614574305619137829547332797328422000406663275*10^-13, \ -2.631884639466089306668070775152803160494886428701003929236*10^-15, \ -5.438339487510447306781452686083673103623574531677729791*10^-18, \ -6.192989936483548192430918684852680607679431272493048*10^-21, \ -3.8561058372588959715408154984974065907955402095482*10^-24, \ -1.3041154300564990475170368702167108513355356894*10^-27, \ -2.381829871124105427498092722592385259369002*10^-31, \ -2.337561278904883240397854728219244957183*10^-35, \ -1.22733321882602548663709666820289370*10^-39, \ -3.434100513325279736346293287953*10^-44, \ -5.10267262676374843629844283*10^-49, -4.01376*10^-54, \ -1.66664*10^-59, -3.64371*10^-65} Even weirder: spontaneous *creation* of "precision": In[167]:= Table[ Sum[morse[n]/(1`69 + n), {n, 2^(k - 1), 2^k - 1}], {k, 25}] Out[167]= \ {-0.500000000000000000000000000000000000000000000000000000000000000000\ 000, -0.08333333333333333333333333333333333333333333333333333333333333\ 3333333, -0.\ 015476190476190476190476190476190476190476190476190476190476190476190,\ -0.002207514707514707514707514707514707514707514707514707514707514707\ 5147, -0.0002091375027541893674392190938372057340187376836883901987123\ 870639319, \ -0.0000122946980614332692270376487316405594406332942110185551582173466\ 668, -4.32195428826170511217542239637665018410757461892500010625794694\ 03*10^-7, \ -8.87301646666058245644516124261223306314074651021882186455064548*10^-\ 9, -1.0458099643631916758928638937741904903086993472013174399559272*\ 10^-10, -6.\ 98143049429755311075516385788269917584660543240898536451685*10^-13, \ -2.610180763779562442726545899627011713573506161849931176559*10^-15, \ -5.413598355752984663980985076812050357066558893858646547*10^-18, \ -6.177575942247773207802885950958692670828387637048574*10^-21, \ -3.8508912806285963424755514484965618347262775126888*10^-24, \ -1.3031628850933154851149881302056671832958127403*10^-27, \ -2.380894944793706582745607332625954939245743*10^-31, \ -2.337070373089340522029095103932355659111*10^-35, \ -1.22719585888756635716102476624359666*10^-39, \ -3.433896409631194067248804676899*10^-44, \ -5.10251207751970643813210988*10^-49, \ -4.0136977684427557028670456513860079925490740717925430787290285399384\ 8*10^-54, \ -1.6666230725159058889654514475037587573071924772089658199531293682217\ 2*10^-59, \ -3.6436950646124655027220392184731237917460692833546825636890121781519\ 5*10^-65, \ -4.1844261203426874910214038879337676295476794340326942203659352222110\ 4*10^-71, \ -2.5187326323755771625453269689899173317142745156407402958430970838190\ 2*10^-77} I should have regarded these results with greater suspicion. No wonder people believed that I believed that ISC delusion. At any rate, the convergence seems to *accelerate*, like a confluent hypergeometric series. --rwg On Mon, Jul 7, 2014 at 10:33 PM, Bill Gosper <billgosper(a)gmail.com> wrote: > On 2014-07-07 00:39, Bill Gosper wrote: > > [Adding subject; fixing bug.] > > > > > Whoa, how are you getting these? Convergence is absent or useless. > > Holy crapoli, I completely missed the point of this Flanelle business. > > (Wasn't he the guy who helped Dumbledore make a Philosopher's Stone?) > > > WDS> > > > > h(n) = (-1)^(parity of bit-sum of binary representation of n) > > > > > f(x) S=SUM[ f(n) * h(n), for n=0..2^k-1 with k large] > > > ln(x+1) S=-log(2)/2 > > ln(x+2) S = -0.1379330125 > > ln(x+3) -0.07070756527 > > x^j 0 for any fixed integer j>=0 > > 1/(x+1) 0.398761088108 > > > > [http://isc.carma.newcastle.edu.au/advancedCalc identifies this as > > > > 3^(1/6)/Zeta[3]^(5/237)/3 . I'm surprised you didn't notice.] > http://arewomenhuman.me/wp-content/uploads/2012/10/trolls.jpeg > > > > > > 1/(x+2) 0.1049709156499 > > sqrt(x) -0.63407426 > > 1/sqrt(x+1) 0.1983140804979 > > > It appears that the cases f(x):=1/(x+k) (various k) are simply related. If > > > S0:=SUM[h(n)/n, for n=1..2^k-1 with k large] ~ -1.19628326432525643722229, > > S1:=SUM[h(n)/(n+1), for n=0..2^k-1 with k large] ~ 0.398761088108 > > S2:=SUM[h(n)/(n+2), for n=0..2^k-1 with k large] ~ 0.1049709156499 > > > S3:=SUM[h(n)/(n+3), for n=0..2^k-1 with k large] ~ 0.0419241705 > > then S0 = -3 S1 and > > S3 = (S1 - 3 S2)/2 . > > --rwg > > -- > You received this message because you are subscribed to the Google Groups > "mathfuneavesdroppers" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to mathfuneavesdroppers+unsubscribe(a)googlegroups.com. > For more options, visit https://groups.google.com/d/optout. >

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[math-fun] Inverse function of error function
by Warren D Smith 09 Jul '14

09 Jul '14

If x=erf(y), then say y=erfinv(x). SMP jocks: What is the Maclaurin series of erfinv(x)? Obviously the erfinv function is useful for statistics. It also occurs to me: This series should have infinite radius of convergence, i.e. erfinv(x) should be a very well behaved "entire" function. Because: the derivative of erf is never zero anywhere in the complex plane, and the value of erf is always finite everywhere in the plane, so its inverse function should exist everywhere.

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[math-fun] Quaternion/vector algebra
by Henry Baker 09 Jul '14

09 Jul '14

The recent discussions of quaternions on math-fun reminded me that the algebraic manipulations of quaternions are quite alien and inscrutable to those familiar with commutative and/or matrix algebra. In particular, the appearance of a _quaternion square root_ in one standard quaternion formula heavily used in computer graphics seems completely contrived/deus ex machina. Notationally, I will use lowercase letters to represent quaternion variables and uppercase letters to represent vector variables, where vectors are a subset of quaternions which have zero scalar parts. Quaternion multiplication is represented by juxtaposition. It is a simple exercise in algebra to convince oneself that the function taking U->qUconj(q) takes vectors to vectors; i.e., the scalar part of the image is also zero, because the contributions from q and conj(q) cancel. It so happens that this transformation rotates U about the axis V(q), where V(q) is the vector part of q, and also scales the vector U by q conj(q)=N(q). Suppose that we are given a vector A which will be mapped into the vector A' by an unknown quaternion q; i.e., we want to find q such that A' = q A conj(q). How can we find q using algebra? The simplest rotation that maps A->A' has the axis AxA', where "x" here denotes the traditional Gibbs vector cross product. For our purposes, the most interesting fact about this axis is that it is perpendicular to both A and A'; i.e., A.(AxA')=A'.(AxA')=0, where "." here denotes the traditional Gibbs vector dot product. We now need a little lemma to show how algebraic manipulations of quaternions with perpendicular vector parts can be performed. Lemma. If the vector part of q is perpendicular to vector U, then Uq=conj(q)U. Proof: We express q in terms of its scalar and vector parts: q = S(q) + V(q) The perpendicularity condition is U.V(q)=0. Uq = U(S(q)+V(q)) = U S(q) + U V(q) = S(q)U + UxV(q) - U.V(q) = S(q)U + UxV(q) = S(q)U - V(q)xU = S(q)U - (V(q)xU - V(q).U) = S(q)U - V(q)U = conj(q)U QED Let us now go back to our equation A'=qAconj(q), which we would still like to solve for q. q A conj(q) = A' q q A = A' q^2 = A'/A q = +- sqrt(A'/A) Not only does algebra work, it works here even if |A'|/=|A|, i.e., even if the lengths of the vectors A,A' aren't equal. Note that A'/A = A'conj(A)/N(A) = A'(-A)/N(A) = -A'A/N(A) so q = +- sqrt(-A'A)/sqrt(N(A)) = +- sqrt(-A'A)/|A| which is the more traditional form of the quaternion solution.

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Re: [math-fun] Simple formula for "Roth's constant"
by Bill Gosper 08 Jul '14

08 Jul '14

Jörg> I'll edit the sequence. If you do not see the edit within 10 days, remind me. Note the L-system I gave for the binary expansion, giving a divisionless method for computation. Best, jj I think it's been 10 days. There's also this glitch: "%t Nest[ Flatten[ # /. {0 -> {1, 1, 1}, 1 -> {1, 1, 0}}] &, {0}, 6] (* _Robert G. Wilson v_, Mar 09 2005 *)" always gets the last bit wrong: In[207]:= Nest[Flatten[# /. {0 -> {1, 1, 1}, 1 -> {1, 1, 0}}] &, {0}, 1] Out[207]= {1, 1, 1} In[208]:= Nest[Flatten[# /. {0 -> {1, 1, 1}, 1 -> {1, 1, 0}}] &, {0}, 2] Out[208]= {1, 1, 0, 1, 1, 0, 1, 1, 0} And Nest[Join[#, #, MapAt[1 - # &, #, -1]] &, {1}, 4] is shorter. --rwg On Tue, Apr 15, 2014 at 3:13 PM, Bill Gosper <billgosper(a)gmail.com> wrote: (A014578 <http://oeis.org/A014578>) Out[665]= k (-1) Sum[--------, {k, 0, ∞}] k 3 2 - 1 In[666]:= RealDigits[% /. ∞-> 4, 8] Out[666]= {{{6, 6, 7, 6, 6, 7, 6, 6, 6, 6, 6, 7, 6, 6, 7, 6, 6, 6, 6, 6, 7, 6, 6, 7, 6, 6, 7}}, 0} Note that this and the other partial sums are actual exemplars of Roth's criterion. In[685]:= FromDigits[%666, 8] Out[685]= 2077175852086666748456887/2417851639229258349412351 In[686]:= $MaxExtraPrecision = 9999; N[Round[1/(% - %665)], 239] N::meprec: Internal precision limit $MaxExtraPrecision = 9999.` reached while evaluating Round[1/((2077175852086666748456887/2417851639229258349412351)-\!$\*UnderoverscriptBox[\(\[Sum]$, $k = 0$, $\[Infinity]$]\*FractionBox[SuperscriptBox[$(\(-1$)\), $k$], $\(-1$ + Power[<<1>>]\)]\))]. >> (I don't understand this warning. 239 and 9999 are ridiculous overkill. Varying them (above a safe minimum) has no effect on the answer or the warning.) Out[686]= 14134776518227074636666380005943348126619871175004951664972849610340958207 In[697]:= Denominator[%685]^3 Out[697]= \ 14134776518227074636666362467923700155784852730794212624494301514198679551 where all that is needed is Denominator^(2+ϵ) (infinitely often). (As computed by the Life pattern x = 50, y = 50, rule = B3/S23 [...]) Tighter: #CXRLE #C Thue (really Roth) generator #C #C Four puffers produce gliders, whose collisions produce four #C lines of blinkers, each representing the transcendental number #C .110110111110110111110110110..., where the n'th bit is the #C the mod 2 of 1 + the largest power of 3 dividing n. #C Built by Bill Gosper, in or before 1987 x = 50, y = 50, rule = B3/S23 26b3o11b3o$26bo2bo10bo2bo$26bo6b3o4bo$26bo5bo2bo4bo$27bo7bo5bo2$bo8b2o $o8b4o$o3bo3b2ob2o28bo$4o5b2o29b3o$40bob2o$41b3o$41b2o2$2b3o$2bo$2bo$ 3bo$24b3o$24bo2bo$bo22bo$o23bo$o3bo14bo5bo$4o14bo$18bo3bo5b4o$18b4o5bo 3bo$31bo14b4o$24bo5bo14bo3bo$25bo23bo$25bo22bo$22bo2bo$23b3o$46bo$47bo $47bo$45b3o2$7b2o$6b3o$6b2obo$7b3o29b2o5b4o$8bo28b2ob2o3bo3bo$37b4o8bo $38b2o8bo2$8bo5bo7bo$9bo4bo2bo5bo$9bo4b3o6bo$6bo2bo10bo2bo$7b3o11b3o! --rwg

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[math-fun] Gary Antonick is edging away from the following bonus puzzle
by Bill Gosper 08 Jul '14

08 Jul '14

because he thinks only Gene Salamin can solve it. Can any of you? (who didn't already know the answer) For a ceremonial match, a soccer ball is colored with a map of the Globe instead of a traditional pattern. At the start of the match, the ball rests on its south pole, with its lat 0, long 0 point aimed due east. At the end of the match, the ball lies forgotten on the pitch, in a truly random orientation. For any orientation, there will always be an axis through the center about which a single rotation will restore the original north up, 0,0 east orientation. What is the (surprising) expected magnitude of this required rotation? And yet last week Gary ran (with great satisfaction) Julian's tympanhedron, which, to my knowledge, has *never* been found when posed as a puzzle. --rwg

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Re: [math-fun] Tarski and Multiplication
by Henry Baker 08 Jul '14

08 Jul '14

Let me know if you find anything. You (& Tarski) now have me intrigued! At 12:22 PM 7/7/2014, Scott Fenton wrote: >Hi Henry, > >That citation's the one I found already. I'll try the editor though. Thanks >for the idea. > >-Scott > >On Mon, Jul 7, 2014 at 1:39 PM, Henry Baker <hbaker1(a)pipeline.com> wrote: > >> You might try the book that is referenced: >> >> Tarski, Alfred; Jan Tarski (24 March 1994). Introduction to Logic and to >> the Methodology of Deductive Sciences (4 ed.). USA: OUP. ISBN 978-0-19-504472-0 >> >> Or you might try to contact the wikipedia article editor. >> >> At 10:15 AM 7/7/2014, Scott Fenton wrote: >> >Hi all, >> > >> >In Tarski's "Introduction to Logic", he gave a nice little set of eight >> >axioms that he says uniquely define the real numbers ( >> >http://en.wikipedia.org/wiki/Tarski%27s_axiomatization_of_the_reals) He >> >further said that, using less than, cuts, and addition, you could define >> >multiplication in this system. Does anyone here know if he published >> >anything else on this system? I can't seem to dig anything up, and I'd like >> >to know how he defined things. >> > >> >Thanks, >> >-Scott

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[math-fun] Simons
by James Buddenhagen 08 Jul '14

08 Jul '14

perhaps this is of interest to some here: http://www.nytimes.com/2014/07/08/science/a-billionaire-mathematicians-life…

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Re: [math-fun] Thue-Morse sums
by Bill Gosper 08 Jul '14

08 Jul '14

On 2014-07-07 00:39, Bill Gosper wrote: > [Adding subject; fixing bug.] > > Whoa, how are you getting these? Convergence is absent or useless. Holy crapoli, I completely missed the point of this Flanelle business. (Wasn't he the guy who helped Dumbledore make a Philosopher's Stone?) > WDS> > > h(n) = (-1)^(parity of bit-sum of binary representation of n) > > f(x) S=SUM[ f(n) * h(n), for n=0..2^k-1 with k large] > ln(x+1) S=-log(2)/2 > ln(x+2) S = -0.1379330125 > ln(x+3) -0.07070756527 > x^j 0 for any fixed integer j>=0 > 1/(x+1) 0.398761088108 > > [http://isc.carma.newcastle.edu.au/advancedCalc identifies this as > 3^(1/6)/Zeta[3]^(5/237)/3 . I'm surprised you didn't notice.] http://arewomenhuman.me/wp-content/uploads/2012/10/trolls.jpeg > > 1/(x+2) 0.1049709156499 > sqrt(x) -0.63407426 > 1/sqrt(x+1) 0.1983140804979 It appears that the cases f(x):=1/(x+k) (various k) are simply related. If S0:=SUM[h(n)/n, for n=1..2^k-1 with k large] ~ -1.19628326432525643722229, S1:=SUM[h(n)/(n+1), for n=0..2^k-1 with k large] ~ 0.398761088108 S2:=SUM[h(n)/(n+2), for n=0..2^k-1 with k large] ~ 0.1049709156499 S3:=SUM[h(n)/(n+3), for n=0..2^k-1 with k large] ~ 0.0419241705 then S0 = -3 S1 and S3 = (S1 - 3 S2)/2 . --rwg

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[math-fun] Consciousness (my guess)
by Warren D Smith 07 Jul '14

07 Jul '14

A possible interpretation of that discovery is: The human brain has a lot of parallel processing, but there is one "master" processor which runs sequentially. And this was it.

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[math-fun] Conscious? Push your reset button?
by Warren D Smith 07 Jul '14

07 Jul '14

Scientists Discovered Consciousness On-Off Switch In Human Brain [News story] Researchers at George Washington University have reported that while treating an epileptic patient they accidentally discovered what they referred as the human consciousness on-off switch, deep within the brain. The scientists claim that they have discovered that area of the brain that controls consciousness. The claustrum, , "a thin, sheet-like structure that lies hidden deep inside the brain," seems to bind together all of our senses, perceptions, and computations into single, cohesive experience. Scientists have been trying for long to figure out how exactly consciousness works and the concept behind it. And if proved right this discovery will open the path in certain areas of medicine. "This study is incredibly intriguing but it is one brick in a large edifice of consciousness that we're trying to build ... Ultimately, if we know how consciousness is created and which parts of the brain are involved then we can understand who has it and who doesn't," said American neuroscientist Cristof Koch. Led by Mohamad Koubeissi, of the George Washington University in Washington DC, the researchers were using deep brain electrodes to monitor brain signals of the patient and try to pinpoint the area of a patient's brain that was causing her seizures. One of the electrodes was placed on the claustrum with electrodes. This caused the woman to lose consciousness when the area was stimulated with high frequency electrical impulses. Her breathing slowed down and she stopped responding to visual and auditory stimuli. She regained her consciousness with no memory of what just happened. The experiment was repeated for two days, by zapping electrical impulses in the same area and the the same result was achieved every time. This test has been performed on one patient and a portion of her hippocampus was removed prior to the experiment as part of her epilepsy treatment. So the results, although consistent, are not representative of a "normal brain." The team will be investigating whether low frequency stimulation on the claustrum could also jolt back consciousness, which can help in treatment for people trapped in minimally conscious states. Koubeissi compared the human body to a car, where the ignition represents the claustrum of the brain. "There's only one spot where you turn the key and it all switches on and works together. So while consciousness is a complicated process created via many structures and networks -- we may have found the key," said Koubeissi. "Perhaps we could try to stimulate this region in an attempt to push them out of this state." The study, entitled Electrical stimulation of a small brain area reversibly disrupts consciousness, was published in the journal Epilepsy and Behavior.

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