December 2019 - mathfun@mailman.xmission.com

Re: [math-fun] Boustrophedon primes
by Walter Trump 20 Dec '19

20 Dec '19

Sorry for the error in my last contribution. Of course A282178 does not start with number 1. I wanted to say that the elements of A282178 are in the column with number 1. The primes of A330339 are equivalent to 1 modulo 4. The primes of A282178 are equivalent to 3 modulo 4. If n is odd p = prime(n) belongs to A282178 if 2*(s1(n) - s2(n)) + p = 1 p = prime(n) does not belong to A330339 Where s1(n) = sum(prime(k) for k=1,3,5,...,n-2) and s2(n) = sum(prime(k) for k=2,4,6,...,n-1) If n is even p = prime(n) belongs to A282178 if 2*(s1(n) - s2(n)) - p = 1 p = prime(n) belongs to A330339 if 2*(s1(n) - s2(n)) - p = -1 Where s1(n) = sum(prime(k) for k=1,3,5,...,n-1) and s2(n) = sum(prime(k) for k=2,4,6,...,n-2) Walter

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Re: [math-fun] walking every road in a city - math-fun Digest, Vol 202, Issue 28
by Stuart Anderson 18 Dec '19

18 Dec '19

One method to walk every road in a city, especially if, like a postman you must walk both sides of every street, is to treat the road network as an electrical network with unit resistors in each street , then designating one street as the battery, calculate the currents and voltages using node analysis. If the voltage is chosen as the number of spanning trees in the network, the currents and voltages should come out as integers. The network of streets then becomes a squared rectangle in the squared rectangle theory of Brooks, Stone, Smith and Tutte. To walk both sides of every street exactly once, starting at an outside corner of the rectangle, walk the diagonal to the opposite corner of that square, then at the T-intersection 'bounce' at 45 degrees like a reflected laser beam into the adjacent square, heading for it's opposite corner, and so on. The diagonal walk will eventually exit at one of the other corners. Then enter at one of the other unused corners and diagonally walk until you exit the final corner. It may happen that these two walks exhaust all the squares, and hence traverse all the streets in both directions, but it may not, there may be intranversed squares. If there are, enter one the intraversed squares at the boundary of the rectangle, and diagonal walk from there until you return to where you started that particular walk. Keep doing this until all squares are traversed. Some complications, the network may have bridges and overpasses so it is not planar. Partition the network into planar and K5 or K2-3 components and traverse those separately. One way streets and dead ends shouldn't be a problem as the can be taken out of the network and added back to the network later. Another complication is that I stead of a T-intersection in the interior of the squares rectangle you may encounter a cross, in which you need to decide a convention on which way to turn at the corner. > > Message: 4 > Date: Tue, 17 Dec 2019 22:30:17 -0500 (EST) > From: "Keith F. Lynch" <kfl(a)KeithLynch.net> > To: math-fun <math-fun(a)mailman.xmission.com> > Subject: [math-fun] Walking every road in a city > Message-ID: <47d0rT2pbFz1ZVx(a)panix2.panix.com> > > With a map in hand and an app on his phone, Jarad Schofer is taking > the concept of a jog through the city to a whole new level: He plans > to run every road and alley in the District. > > "You know I've lived here a long time and I kind of thought I'd run > everywhere and now I can see that's really nowhere near the case," > he said. > > He's a teacher with a doctorate in mathematics and only recently got > into running. He started with 5Ks, then moved to 10K races. > > Now with a few big races under his belt, Schofer has knocked out > 860 miles on his mission to tackle the streets of D.C. In all, he > intends to run around 2,500 miles. > > . . . > > He said the biggest challenge has been overlapping certain streets > but according to his calculations, it's impossible to avoid. > > > https://wtop.com/dc/2019/12/dc-runner-is-on-a-mission-to-cover-every-inch-o… > > What's the best algorithm for walking (or running) every road in a > city while minimizing duplication? > > Only a small part of DC is on a regular grid. Mostly its road network > is pretty much random. > > > > ------------------------------ > > Message: 5 > Date: Tue, 17 Dec 2019 23:21:13 -0500 > From: Neil Sloane <njasloane(a)gmail.com> > To: math-fun <math-fun(a)mailman.xmission.com> > Subject: Re: [math-fun] Boustrophedon primes > Message-ID: > < > CAAOnSgTeWCo3smnjuCWL94xwj-TYX+z0Qi_1+f63rzUiKrhQYQ(a)mail.gmail.com> > Content-Type: text/plain; charset="UTF-8" > > Allan, I had exactly the same idea, and this afternoon I created A330545! > It is indeed the key to the whole problem - and of course the recurrence > makes it a lot easier to compute. > > > > > On Tue, Dec 17, 2019 at 5:27 PM Allan Wechsler <acwacw(a)gmail.com> wrote: > > > There is an underlying sequence, which tells what columns successive > prime > > numbers land in. This is the alternating sum of the decremented first > > differences of the primes, with the sequence starting at 2. Just reading > > off Eric's picture, I get > > > > 2,2,3,2,5,4,7,6,9,4,5,0,3,2,5,0,.... > > > > A330339 is then the primes indexed by occurrences of zero in this > > alternating sum. > > > > On Tue, Dec 17, 2019 at 5:14 PM Neil Sloane <njasloane(a)gmail.com> wrote: > > > > > See https://oeis.org/A330339 > > > > > > > > > > > > On Tue, Dec 17, 2019 at 4:24 PM ?ric Angelini <eric.angelini(a)skynet.be > > > > > wrote: > > > > > > > Thank you Walter! This is an incentive > > > > for ski slopes! > > > > Best, > > > > ?. > > > > > > > > > > > > > Le 17 d?c. 2019 ? 20:05, Walter Trump <w(a)trump.de> a ?crit : > > > > > > > > > > The next Boustrophedon primes are 3821 and 3989. > > > > > Here is a continuation of Eric's arrangement of numbers (you cannot > > > read > > > > the > > > > > digits, they are too small): > > > > > https//www.trump.de/Boustrophedon-Primes.png > > > > > > > > > > Walter > > > > > > > > > > > > > > > > > > > > _______________________________________________ > > > > > 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 > > > > > _______________________________________________ > > math-fun mailing list > > math-fun(a)mailman.xmission.com > > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > > > > ------------------------------ > > 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 202, Issue 28 > ***************************************** >

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[math-fun] Walking every road in a city
by Keith F. Lynch 18 Dec '19

18 Dec '19

With a map in hand and an app on his phone, Jarad Schofer is taking the concept of a jog through the city to a whole new level: He plans to run every road and alley in the District. "You know I've lived here a long time and I kind of thought I'd run everywhere and now I can see that's really nowhere near the case," he said. He's a teacher with a doctorate in mathematics and only recently got into running. He started with 5Ks, then moved to 10K races. Now with a few big races under his belt, Schofer has knocked out 860 miles on his mission to tackle the streets of D.C. In all, he intends to run around 2,500 miles. . . . He said the biggest challenge has been overlapping certain streets but according to his calculations, it's impossible to avoid. https://wtop.com/dc/2019/12/dc-runner-is-on-a-mission-to-cover-every-inch-o… What's the best algorithm for walking (or running) every road in a city while minimizing duplication? Only a small part of DC is on a regular grid. Mostly its road network is pretty much random.

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[math-fun] Tetranacci Tetrahedron
by ed pegg 18 Dec '19

18 Dec '19

Let t^2 =1.92756 be the tetranacci constant. Build a tetrahedron with edgelengths t^0 opposite t^3 t^1 opposite t^1 t^2 opposite t^2 This has 4 similar triangles of 2 sizes. If this tetrahedron is rolled on its edges, it only reaches 12 different orientations. -Ed Pegg Jr

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[math-fun] More On Ellipse Area Integrals
by Brad Klee 17 Dec '19

17 Dec '19

Define an ellipse of eccentricity "e" as the real points {x,y} satisfying: (1-e^2)*(x+e)^2 + y^2 = (1-e^2), y[x] = sqrt((1-e^2)*(1-(x+e)^2)). Notice that (e^2-1)*d/de*y[x-e] - e*y[x-e] = 0, and that: (e^2-1)*d/de*y[x] - e*y[x] = d/dx*((e^2-1)*y[x]) = d/dx*Cy[x] . Action angle coordinates {p,q} satisfy {x,y}=sqrt(2*p)*{cos[q],sin[q]}, with y[x] replaced by p[q]=(1/2)*((1-e^2)/(1+e*cos[q]))^2, and then: (e^2-1)*d/de*p[q] - e*p[q] = d/dq*((2*sin[q]+e*cos[q]*sin[q])*p[q]) = d/dq*Cp[q] . Subtracting both equations, and integrating between points on the ellipse circumference, obtains Int_{P1}^{P2} e*( p[q]*dq - y[x]*dx ) = e*(Apq - Axy) = Cy[x_P2] - Cy[x_P1] - Cp[q_P2] + Cp[q_P1], which says that the difference of sectorial area Apq and Cartesian area Axy is a difference of certificates evaluated at P1 and P2. ( I tested this equation numerically and it works, no problem. ) The difference Apq - Axy is also the area difference of a triangle and a quadrilateral, so can alternatively be calculated by trigonometry. When this is done separately, it should then be possible to reverse the derivation and conclude that p[q] is the desired function "1/2*r^2". It satisfies a certain differential equation, with a known, unique certificate. --Brad PS. In case there are any typos in the above, here are the computes: Factor@Dot[ D[Sqrt[(1 - e^2) (1 - x^2)], {e, #}] & /@ {0, 1}, {-e, e^2 - 1}] With[{ye = Sqrt[(1 - e^2) (1 - (x + e)^2)]}, Factor@Plus[Factor@Dot[D[ye, {e, #}] & /@ {0, 1}, {-e, e^2 - 1}], D[(1 - e^2) ye, x]]] With[{r2e = (1/2)*((1 - e^2)/(1 + e Cos[q]))^2}, ReplaceAll[ Factor@Subtract[Dot[D[r2e, {e, #}] & /@ {0, 1}, {-e, e^2 - 1}], D[(2*Sin[q] + e*Cos[q] Sin[q]) r2e, q]], Sin[q]^2 -> 1 - Cos[q]^2]]

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Re: [math-fun] Heat leaps through empty space -- quantum weirdness
by Henry Baker 17 Dec '19

17 Dec '19

Re: "Moving them apart makes the tunneling probability drop off exponentially" Hmmmm... At the vanishingly small absolute temperatures of current quantum computers, the Compton wavelength grows to become not very small; analogy to "slowing (i.e., 'cooling') neutrons to enhance probability of interaction in an atomic reactor". At 07:37 AM 12/17/2019, Mike Stay wrote: >On Tue, Dec 17, 2019 at 8:28 AM Henry Baker <hbaker1(a)pipeline.com> wrote: >> Remind me again how we're planning to insulate qubits from the environment ? > >In this particular case, the phonons tunnel from one resonator to the >other. Moving them apart makes the tunneling probability drop off >exponentially. > >For the more general case, topological qubits like nonabelian anyons >(think "smoke rings of charge that pass through each other") are >likely to be very resilient to environmental noise. >-- >Mike Stay - metaweta(a)gmail.com >http://math.ucr.edu/~mike >https://reperiendi.wordpress.com

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[math-fun] Prime modulo prime
by Éric Angelini 17 Dec '19

17 Dec '19

Hello Math-Fun, I was playing with a(n) being the remainder of the division by a(n+1) of a(n+2), with all terms prime. I've found this succession of primes, but I don't know if this is the lexico first such seq: S = 2,3,5,23,97,1187,14341,57073,... I found empirically by hand the last three terms because of this: the biggest factor that multiplies 5 to produce 3 as remainder is 4 (indeed, 23-4*5=3); the biggest factor that multiplies 23 to produce 97 as remainder is 4 (indeed, 97-4*23=5); the biggest factor that multiplies 97 to produce 23 as remainder is 12 (indeed, 1187-12*97=23); the biggest factor that multiplies 1187 to produce 97 as remainder is 12 (indeed, 14341-12*1187=97). Then, seing the pattern 4, 4, 12, 12,... I tried to multiply by 36... and got a hit: indeed 14341*36+1187 = 57073 which is prime. Any thoughts? à+ É. Catapulté de mon aPhone

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[math-fun] Heat leaps through empty space -- quantum weirdness
by Henry Baker 17 Dec '19

17 Dec '19

FYI -- Remind me again how we're planning to insulate qubits from the environment ? This report doesn't even mention the Aharonov-Bohm effect, which also enables information transfer across "empty" space w/o utilizing electromagnetic radiation. No computation (classical or quantum) and proceed w/o heat insulation and "refrigerators" to protect the critical degrees of freedom of the computation itself. Quantum error correction could be a kind of "refrigeration" which rejects "heat"/"error" back to the environment, but it's going to require an enormous degree (!) of subtlety to pull this off. And even if we could build more reliable qubits, we won't be able to put them very close together. https://phys.org/news/2019-12-energy-space-quantum-weirdness.html Heat energy leaps through empty space, thanks to quantum weirdness by Kara Manke, University of California - Berkeley December 11, 2019 In a new study, University of California, Berkeley, researchers show that heat energy can travel through a complete vacuum thanks to invisible quantum fluctuations. In the experiment, the team placed two gold-coated silicon nitride membranes a few hundred nanometers apart inside a vacuum chamber. When they heated up one of the membranes, the other warmed up, too, even though there was nothing connecting the two membranes and negligible light energy passing between them. If you use a vacuum-insulated thermos to help keep your coffee hot, you may know it's a good insulator because heat energy has a hard time moving through empty space. Vibrations of atoms or molecules, which carry thermal energy, simply can't travel if there are no atoms or molecules around. But a new study by researchers at the University of California, Berkeley, shows how the weirdness of quantum mechanics can turn even this basic tenet of classical physics on its head. The study, appearing this week in the journal Nature, shows that heat energy can leap across a few hundred nanometers of a complete vacuum, thanks to a quantum mechanical phenomenon called the Casimir interaction. Though this interaction is only significant on very short length scales, it could have profound implications for the design of computer chips and other nanoscale electronic components where heat dissipation is key. It also upends what many of us learned about heat transfer in high school physics. "Heat is usually conducted in a solid through the vibrations of atoms or molecules, or so-called phonons--but in a vacuum, there is no physical medium. So, for many years, textbooks told us that phonons cannot travel through a vacuum," said Xiang Zhang, the professor of mechanical engineering at UC Berkeley who guided the study. "What we discovered, surprisingly, is that phonons can indeed be transferred across a vacuum by invisible quantum fluctuations." In the experiment, Zhang's team placed two gold-coated silicon nitride membranes a few hundred nanometers apart inside a vacuum chamber. When they heated up one of the membranes, the other warmed up, too--even though there was nothing connecting the two membranes and negligible light energy passing between them. "This discovery of a new mechanism of heat transfer opens up unprecedented opportunities for thermal management at the nanoscale, which is important for high-speed computation and data storage," said Hao-Kun Li, a former Ph.D. student in Zhang's group and co-first author of the study. "Now, we can engineer the quantum vacuum to extract heat in integrated circuits." No such thing as empty space The seemingly impossible feat of moving molecular vibrations across a vacuum can be accomplished because, according to quantum mechanics, there is no such thing as truly empty space, said King Yan Fong, a postdoctoral scholar at UC Berkeley and the study's other first author. "Even if you have empty space--no matter, no light--quantum mechanics says it cannot be truly empty. There are still some quantum field fluctuations in a vacuum," Fong said. "These fluctuations give rise to a force that connects two objects, which is called the Casimir interaction. So, when one object heats up and starts shaking and oscillating, that motion can actually be transmitted to the other object across the vacuum because of these quantum fluctuations." Though theorists have long speculated that the Casimir interaction could help molecular vibrations travel through empty space, proving it experimentally has been a major challenge. To do so, the team engineered extremely thin silicon nitride membranes, which they fabricated in a dust-free clean room, and then devised a way to precisely control and monitor their temperature. They found that, by carefully selecting the size and design of the membranes, they could transfer the heat energy over a few hundred nanometers of vacuum. This distance was far enough that other possible modes of heat transfer were negligible--such as energy carried by electromagnetic radiation, which is how energy from the sun heats up Earth. Because molecular vibrations are also the basis of the sounds that we hear, this discovery hints that sounds can also travel through a vacuum, Zhang said. "Twenty-five years ago, during my Ph.D. qualifying exam at Berkeley, one professor asked me 'Why can you hear my voice across this table?' I answered that, 'It is because your sound travels by vibrating molecules in the air.' He further asked, 'What if we suck all air molecules out of this room? Can you still hear me?' I said, 'No, because there is no medium to vibrate,'" Zhang said. "Today, what we discovered is a surprising new mode of heat conduction across a vacuum without a medium, which is achieved by the intriguing quantum vacuum fluctuations. So, I was wrong in my 1994 exam. Now, you can shout through a vacuum." More information: Phonon heat transfer across a vacuum through quantum fluctuations, Nature (2019). DOI: 10.1038/s41586-019-1800-4 , https://nature.com/articles/s41586-019-1800-4

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[math-fun] Identity transformation
by françois mendzina essomba2 16 Dec '19

16 Dec '19

Hi, Pi/2==sum((2*n)!*(1+x^(4*n+2))/(2^(2*n)*(n!)^2*(1+x^4)^(n+1/2)*(2*n+1)) ,n=0..inf); an identity transformation gives this result for any real number. I wonder what other process can lead to this formula Best regards.

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[math-fun] Another Ellipse Certificate
by Brad Klee 16 Dec '19

16 Dec '19

With[{r2x = 1/(1 - x*Cos[z])^2}, ReplaceAll[ Factor@Plus[ Dot[D[r2x, {x, #}] & /@ {0, 1}, {3 x, (x - 1) (x + 1)}], D[(2*Sin[z] - x*Cos[z] Sin[z]) r2x, z]], Sin[z]^2 -> 1 - Cos[z]^2]] This is no more fundamental a definition than the offering of yesterday; however, it may be easier to work with because r^2*dz is an area from, so A(x) = Pi/(1-x^2)^(3/2) , Should also follow from Cartesian definitions, i.e. from an x*dy integral of algebraic function. --Brad

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