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[math-fun] 'in-band' signaling in voting systems
by Henry Baker 15 Aug '20

15 Aug '20

A voting system is a communications system with some extremely weird characteristics. Nevertheless, this communication system still shares problems with more traditional communications systems. The simplest problem is ensuring that the system is 'live' -- i.e., that it actually is carrying information. Also, voting systems are highly targeted by hackers, so standard problems like 'replay attacks' must be solved. When Akamai (or any other large-scale communications company) wants to map out the internet, it sends packets, sometimes in pairs, in order to detect connectivity, latency, throughput, error rates. These packets aren't 'payload', in the sense that Akamai doesn't get *paid* for transmitting these packets, but it needs to ascertain the characteristics of the various communications links in order to better plan for actual paying traffic. The voting system also desperately needs to map out its network, and make sure that it is working, even in the presence of 'noise', out of order delivery, random delays and/or active *jamming*. Due to the weird nature of the voting network, however, these 'signaling' packets need to be distinguished from 'regular' packets (payloads). But due to the nature of detecting hacking, jamming and other 'active attacks', it may be necessary to make the signaling packets indistinguishable from payload packets as they are crossing the network, and the only knowledge of which packets are which is found in the testing logic itself. I.e., in order to properly test the voting network, it may be necessary to inject a *known* number of 'false' votes into the network, and subsequently subtract these false votes from the totals. Such 'false' votes can also be used to better protect voters' privacy in precincts where there are only a handful of voters. In these precincts, the 'false' votes can temporarily drown out the real votes, but these 'false' votes will later be subtracted from the totals for the 'real' vote. Clearly, there are great dangers in a poorly implemented voting system with 'in-band' signaling. However, there are also great dangers in *not testing* a voting system for link failures and active hacking attacks. Has anyone else considered these kinds of problems?

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[math-fun] More spirals and 2 x 2 square squares
by Éric Angelini 15 Aug '20

15 Aug '20

Hello Math-fun, We've started a spiral of distinct integers with 0 in the middle of the plane, then 1, then 2. At this point the next integer must be 6 as we want the content of EVERY 2 x 2 SQUARE of the plane to sum up to a square (0+1+2+6=9). When we have a choice, we always extend the spiral with the smallest unused term. Thus here, after 6, the term 3. This 3 forces us to fill the next cell with 7 (indeed 0+6+3+7=16) and not 8 (the sum 17 is not a square) or 40 (though 0+6+3+40=49 is also a square) -- remember, we must always extend the spiral with the smallest unused integer. This means that the only constraint on the next integer (after 7) is to be the smallest unused one, thus 4. Now to complete the next 2 x 2 square we must use 5 (as 0+7+4+5=16), etc. +---+---+---+---+---+---+---+---+---+---+ | | | | | | | | | | | +---+---+---+---+---+---+---+---+---+---+ | | | | | | | | | | | +---+---+---+---+---+---+---+---+---+---+ | | |13 |38 |34 |15 |31 | A | | | +---+---+---+---+---+---+---+---+---+---+ | | |26 | 4 | 5 |10 | 8 | B | | | +---+---+---+---+---+---+---+---+---+---+ | | |12 | 7 | 0 | 1 |17 | C | | | +---+---+---+---+---+---+---+---+---+---+ | | |14 | 3 | 6 | 2 |16 | . | | | +---+---+---+---+---+---+---+---+---+---+ | | |11 |21 |19 |22 | 9 | . | | | +---+---+---+---+---+---+---+---+---+---+ | | | | | | | . | . | | | +---+---+---+---+---+---+---+---+---+---+ | | | | | | | | | | | +---+---+---+---+---+---+---+---+---+---+ This spiral leads to the lexicographically earliest sequence of distinct positive terms 0, 1, 2, 6, 3, 7, 4, 5, 10, 8, 17, 16, 9, 22, 19, 21, 11, 14, 12, 26, 13, 38, 34, 15, 31, A, B, C, ... which is not in the OEIS and perhaps of general interest (caveat: handmade, as usual). (Our next move after dinner, soccer on TV, sleep, breakfast? Each 2 x 2 square sums up to a prime, of course). Best, É.

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[math-fun] Rearranging digits
by James Propp 15 Aug '20

15 Aug '20

This tweet from @AlgebraFact gave me a good way to get my brain working this morning: “Suppose n is a power of 2. Then you cannot rearrange the base 10 digits of n to form a different power of 2.” What about Fibonacci numbers? Jim Propp

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[math-fun] Sum of consecutive terms = shortcut
by Éric Angelini 15 Aug '20

15 Aug '20

Hello Math-fun, could I recommend to our members who like erratic behaviors to have a look at http://oeis.org/A336897 Best, É. Catapulté de mon aPhone

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[math-fun] intriguing partial result
by Bill Gosper 14 Aug '20

14 Aug '20

4 (Re[(1-I) hilbert[EllipticTheta[4, 1/2]]]) | (2 + √2) EllipticTheta[4,1/√2] - (√2 - 2) EllipticTheta[3,1/√2] /.R : EllipticTheta[__] :> Round[R, 2^-691] // N[#, 69] & During evaluation of In[679]:= EllipticTheta::argtu: EllipticTheta called with 1 argument; 2 or 3 arguments are expected. {1.78030371851856017082096982647118260720864907614581765107300156501703} | 1.78030371851856017082096982647118260720864907614581765107300156501703 (The extra braces are to allow hilbert to be multivalued.) Kids: The pattern EllipticTheta[___] means EllipticTheta of any number of arguments. So why is it complaining? And how do I shut it up? (I won't say if nobody asks!) Kids: True or false: Im[x]==Re[I*x] —rwg

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[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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