August 2020 - mathfun@mailman.xmission.com

[math-fun] Never Go Down
by Éric Angelini 14 Aug '20

14 Aug '20

Hello Math-fun If someone could compute more terms of http://oeis.org/A336611 ... the authors would appreciate a lot (because they are stuck around the 73rd term of the sequence — with no pattern in view). Best, É.

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[math-fun] I punted the messy realparts and imagparts
by Bill Gosper 14 Aug '20

14 Aug '20

from three hilberts <http://gosper.org/3hilbthets.png> and three draguns <http://gosper.org/3dragthets.png> at theta constants. Such identities seem rather abundant, but probably not reliably enough to find a formula with a parameter. —rwg There's a pretty good chance I can find a closed form for a terDragon[𝜗[]] or two. —rwg I think these formulas can be proved with finite-state machines, but I'll just settle for a hundred digits.

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[math-fun] EE: optimum power transfer from multiple sources?
by Henry Baker 13 Aug '20

13 Aug '20

This may be off-topic for math-fun, but here goes. The standard 'maximum power transfer' theorem says that the maximum power is transferred from a source to a sink when the source and the sink have conjugate impedances. For resistive loads, this means equal (real) impedances. My question is what happens when one has a single sink, but *multiple* sources with a multiplicity of power capacities, possibly time-varying ? This problem is prompted by an array of solar panels, each with different characteristics -- e.g., orientation -- and shading, so the output power is time-varying among the different panels in the array. In the modern world, everything can be digitized, or at least sampled. Solar panels with 'microinverters' produce alternating current (AC), so it is useful to consider each 1/60 sec *cycle period* from a collection of solar panels separately. Consider using microinverters with enough power storage to store a few seconds worth of power; each microinverter will have the same optimal impedance for the time during which it is connected to the load. Then it is conceivable to use a *scheduling* algorithm to enable optimum power transfer from all of the micro- inverters. Thus, a 'round-robin' scheduling of the microinverters would have each microinverter dump a 'unit' amount of power for 1/60 sec at the optimum source impedance to the load, in sequence. If a micro-inverter doesn't have enough power to contribute when its 'turn' comes up, it 'passes', and zero power transfer takes place during that 1/60 sec. Does this scheme provide optimal power transfer from the sources to the sinks?? I considered the idea of simply tying the outputs of the sources together and skipping the round-robin idea, but if the sources are different, then such a scheme would end up transferring power among the sources instead of transferring it to the load.

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[math-fun] Contract-tracing virus ('CTV') (was Re: Retrospective contact tracing)
by Henry Baker 12 Aug '20

12 Aug '20

I've been following TWiV (This Week in Virology) ever since March, so I've been musing about various things that could be done with viruses and infections. Also, '60 Minutes' recently had a segment about a very cool technology which uses a CRISPR-like technology to correct the DNA 'SNIP' which causes sickle cell disease. Here are some ideas (please save this email to inoculate the world against someone filing for an invalid patent on these thoughts): 1. Construct a *non-disease* virus that is extremely contagious, and replicates, but doesn't cause any signs, symptoms or disease. However, it includes a certain 'identification sequence' which incorporates a huge amount of redundancy to guard against information loss. This non-disease infection ('NDI') *automatically* does contract tracing, because it is easy to test someone for the 'NDI' and its ID sequence. Possible application: infect everyone who gets on a particular airplane with an flight#-distinguishable NDI, so that one can trace not only everyone who was on the airplane, but also everyone who contacted them in the subsequent weeks. If the NDI is suitably engineered, it shouldn't last in the body longer than a month or so, which provides enough time to do 'contact tracing', but not enough time to be a permanent privacy problem. Risk: what if the NDI mutates into a DI -- i.e., it produces an actual disease? 2. Ditto the above, except that the NDI is actually a non-disease *vaccine*! Getting the NDI inoculates the victim against a real disease. This is a variant on the Salk/Sabin dead/weakened vaccine idea, except that this NDI 'vaccine' can *propagate all by itself*. This idea is analogous to releasing sterile males into a population of bad actor animals so that they no longer reproduce. 3. Ditto the above, but incorporate CRISPR technology to insert data into a host cell's DNA sequence. This version allows for permanent storage. 4. Ditto the above, but with the sickle cell repairer CRISPR-like technology. Everyone can get & transmit this NDI, but only those with sickle cell disease will see an effect -- the elimination of the sickle cell disease. 5. Combine this idea with a Turing-universal set of RNA/DNA CRISPR-like 'transformers'. This enables an arbitrary distributed computation while the virus propagates. At 11:21 AM 8/12/2020, Michael Kleber wrote: >Following up on our discussion in April/May about the non-uniformity in >COVID transmission event sizes -- i.e. the average reproduction number R_e >doing a bad job of explaining dynamics because the distribution is highly >skewed. > >Japan's COVID contact tracing (and now Massachusetts's also) are using >*retrospective* rather than prospective tracing. In brief, when someone >tests positive, the standard prospective approach is to try to track down >the people they might have given it to -- their children in the >transmission tree. In retrospective tracing, you instead look for their >*siblings* in the transmission tree: other people who might have become >infected the same time they did. > >In a regime where lots of transmission happens at "superspreading events", >you get a lot of benefit from this change -- even after you pay the latency >cost of trying to find people one infection-generation later in their >disease cycle. Here's some analysis of the dynamics: > >Twitter thread: >https://twitter.com/AdamJKucharski/status/1293472353758871552 >Preprint: >https://www.medrxiv.org/content/10.1101/2020.08.01.20166595v1.full.pdf > >--Michael

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[math-fun] The enclosed area in
by Bill Gosper 12 Aug '20

12 Aug '20

frame 83 <http://gosper.org/drankleframe83.png> of the flexing twindragon <http://gosper.org/drankle2304.gif> strongly resembles the fully flexed (mono)dragon <http://gosper.org/dragtrip!.png>, except that the nth largest (unsymmetrical, bisectable?) blob area is 1/2^(n+1) of the total, instead of 4/5/2^n. (Somebody should check me.) —rwg (Note that the fully flexed Dragon encloses no area. It's all boundary. And it was never a closed curve!) (Irony would be if I wind up having to write a book on the bleeping Dragon Curve, of which I was never very fond. Just very curious. )

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[math-fun] Retrospective contact tracing
by Michael Kleber 12 Aug '20

12 Aug '20

Following up on our discussion in April/May about the non-uniformity in COVID transmission event sizes -- i.e. the average reproduction number R_e doing a bad job of explaining dynamics because the distribution is highly skewed. Japan's COVID contact tracing (and now Massachusetts's also) are using *retrospective* rather than prospective tracing. In brief, when someone tests positive, the standard prospective approach is to try to track down the people they might have given it to -- their children in the transmission tree. In retrospective tracing, you instead look for their *siblings* in the transmission tree: other people who might have become infected the same time they did. In a regime where lots of transmission happens at "superspreading events", you get a lot of benefit from this change -- even after you pay the latency cost of trying to find people one infection-generation later in their disease cycle. Here's some analysis of the dynamics: Twitter thread: https://twitter.com/AdamJKucharski/status/1293472353758871552 Preprint: https://www.medrxiv.org/content/10.1101/2020.08.01.20166595v1.full.pdf --Michael -- Forewarned is worth an octopus in the bush.

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[math-fun] Fwd: Erich: No more https://www2.stetson.edu/~efriedma/packing.html??
by Bill Gosper 12 Aug '20

12 Aug '20

---------- Forwarded message --------- From: Erich Friedman <efriedma(a)stetson.edu> Date: Fri, Aug 7, 2020 at 5:37 AM Subject: Re: Erich: No more https://www2.stetson.edu/~efriedma/packing.html ?? To: Bill Gosper <billgosper(a)gmail.com> my university, in its infinite wisdom, decided without warning that it would no longer host personal webpages. sorry. perhaps this material will appear somewhere else someday. erich ------------------------------ *From:* Bill Gosper <billgosper(a)gmail.com> *Sent:* Friday, July 31, 2020 3:18 PM *To:* Erich Friedman <efriedma(a)stetson.edu> *Subject:* Erich: No more https://www2.stetson.edu/~efriedma/packing.html?? CAUTION - External Email - —Bill

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Re: [math-fun] math-fun Digest, Vol 210, Issue 12
by Stuart Anderson 11 Aug '20

11 Aug '20

There's always the wayback machine http://web.archive.org/web/20200214235553/https://www2.stetson.edu/~efriedm… On Wed, Aug 12, 2020 at 4:00 AM <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: Fwd: Erich: No more > https://www2.stetson.edu/~efriedma/packing.html?? (smj) > > > ---------------------------------------------------------------------- > > Message: 1 > Date: Fri, 07 Aug 2020 13:44:22 -0700 > From: smj <smj(a)mab.sdf.org> > To: Bill Gosper <billgosper(a)gmail.com> > Cc: math-fun(a)mailman.xmission.com > Subject: Re: [math-fun] Fwd: Erich: No more > https://www2.stetson.edu/~efriedma/packing.html?? > Message-ID: <090bf51bf953ee880b0b05d6e40e002f(a)mab.sdf.org> > Content-Type: text/plain; charset=UTF-8; format=flowed > > On 2020-08-07 11:36, Bill Gosper wrote: > > ---------- Forwarded message --------- > > From: ERICH FRIEDMAN <efriedma(a)stetson.edu> > > Date: Fri, Aug 7, 2020 at 5:37 AM > > Subject: Re: Erich: No more > > https://www2.stetson.edu/~efriedma/packing.html?? > > To: Bill Gosper <billgosper(a)gmail.com> > > > > my university, in its infinite wisdom, decided without warning that > > it would no longer host personal webpages. sorry. > > > > perhaps this material will appear somewhere else someday. > > > > erich > > > > ------------------------- > > > > FROM: Bill Gosper <billgosper(a)gmail.com> > > SENT: Friday, July 31, 2020 3:18 PM > > TO: Erich Friedman <efriedma(a)stetson.edu> > > SUBJECT: Erich: No more > > https://www2.stetson.edu/~efriedma/packing.html?? > > > > CAUTION - External Email - > > > > ?Bill > > > This may be lost. > > Did you receive this message Bill? > FROM ME? > > > > ------------------------------ > > 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 210, Issue 12 > ***************************************** >

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Re: [math-fun] Shortest boundary of tiles in a tessellation of the plane
by Dan Asimov 10 Aug '20

10 Aug '20

Yep, you nailed it. I'm guessing Allan did, too, since his 4-digit figure of 3.8637+ agrees with what I got (√2 + √6 = (3.863703305156273+) for the boundary length, using the same non-regular hexagon you mention. However, I don't have a proof handy just now. —Dan William Somsky wrote: ----- Hexagons (not rotationally symmetric) with 120 degree interior angles and a-c = 1 unit, b-c = 1 unit, a-c perpendicular b-c a___ / \ / \c \ / \b____/ On Sun, Aug 9, 2020, 15:30 Allan Wechsler <acwacw(a)gmail.com> wrote: > A minimum must exist, very roughly, because each tile has unit area, and > the minimum circumference for a thingy of unit area is 2 * sqrt(pi), which > is 3.5449+. > > The circumference achieved by a square tiling is 4. I have a trick to get > as low as 3.8637+, and I have a strong feeling that it's the best trick, > but am nowhere near a proof. > > On Sun, Aug 9, 2020 at 6:03 PM Fred Lunnon <fred.lunnon(a)gmail.com> wrote: > > > Nit-picking: > > > > "each tile to a tile" perhaps? > > > > Unclear what deformations are implied by "adjust"! > > > > Finally, (why) must such a minimum exist? > > > > WFL > > > > > > > > On 8/9/20, Dan Asimov <dasimov(a)earthlink.net> wrote: > > > Suppose we tessellate the plane by the unit squares with vertices at > the > > > integer > > > points. We'd now like to adjust this tessellation so that > > > > > > 1) The boundary of each tile is a simple closed polygon; > > > > > > 2) The set of translations that carry each tile to another tile is > > exactly > > > the integer vectors (K,L) ∊ Z^2. > > > > > > Puzzle: Which tile shape satisfying 1) and 2) is such that each tile's > > > boundary > > > polygon has the shortest possible length? > > > > > > —Dan -----

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[math-fun] Shortest boundary of tiles in a tessellation of the plane
by Dan Asimov 10 Aug '20

10 Aug '20

Suppose we tessellate the plane by the unit squares with vertices at the integer points. We'd now like to adjust this tessellation so that 1) The boundary of each tile is a simple closed polygon; 2) The set of translations that carry each tile to another tile is exactly the integer vectors (K,L) ∊ Z^2. Puzzle: Which tile shape satisfying 1) and 2) is such that each tile's boundary polygon has the shortest possible length? —Dan

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