January 2019 - mathfun@mailman.xmission.com

[math-fun] (no subject)
by James Buddenhagen 16 Jan '19

16 Jan '19

Not sure if this appeared here already. https://www.theguardian.com/science/2019/jan/15/sir-michael-atiyah-obituary

1 0

[math-fun] Draft of January 2019 blog post
by James Propp 16 Jan '19

16 Jan '19

I'm cutting things awfully close this month, but if any of you happen to have time to read my new essay on Wednesday morning or afternoon and send me suggestions for how to improve the essay before the evening, I'll do my best to incorporate (and, as always, acknowledge) them when I post it in the evening. http://mathenchant.org/044-draft.pdf Suggestions after Wednesday night are welcome too, since I can edit the essay even after I publish it. And Comments (in the Comments section of the blog) will be welcome at that time as well. Thanks, Jim Propp

3 2

[math-fun] pseudo-homomorphic encryption ??
by Henry Baker 14 Jan '19

14 Jan '19

Here's a real-world problem: You have some sort of IoT (Internet of Things) sensor -- perhaps a camera -- and you want to train an AI/machine learning algorithm to recognize something that is exposed to the camera. But perhaps you don't trust the AI/ML developer. So you send him/her only an encrypted dataset along with the classification data (yes/no or perhaps a finite set of possibilities); this classification data isn't encrypted, and there isn't any easy way to figure out from the sequence of classifications any useful info about the encrypted dataset. So far as I know, homomorphic encryption hasn't matured to the point where the entire training process could operate on homomorphically encrypted data. But we're not talking here about completely generic calculations -- we're talking about quite limited calculations, just in enormous quantities (10^18 calculations). Perhaps there are "homomorphic" encryption systems that do *just enough* and AI/ML systems that are dumbed down *just enough* that the two constraints can meet in the middle. After all, AI/ML systems don't seem to care about most kinds of image distortions, so perhaps they could still be capable of characterizing certain pictures even after encryption ? Obviously, if such things are possible, then there are clearly information leaks, but these might even be useful.

3 2

[math-fun] 3D practal reactal fractals
by Henry Baker 14 Jan '19

14 Jan '19

We talked several months ago about the 3D fractal patterns in the vertebrate *lung* which enables *incoming blood*, *outgoing blood* and air to interact. The problem with lungs is that they have to expand and contract to bring in new air and expel old air. So perhaps we need *four* interacting fractals: * incoming fluid (blood ?) * outgoing fluid (blood ?) * incoming gas (air/oxygen ?) * outgoing gas (air/CO2/whatever ?) So here's the problem: Is there a * fractal pattern that is * 3D printable * that can be made out of a 3D printable material (plastic, metal) * that can be built out of small number of 3D "tiles" (or whatever they're called in 3D) * that can be put together (perhaps with glue) * With an input plenum for air, an output plenum for air, an input plenum for fluid, an output plenum for fluid The idea is to bring the air & fluid together in microscopic lowest level fractal "tubing" (probably not circular, but perhaps) where the gas & fluid can "react". An artificial lung might be one application, but there's lots of applications where gasses & liquids need to come into intimate contact.

2 1

Re: [math-fun] Some serious algebraic weirdness for higher integers
by Dan Asimov 14 Jan '19

14 Jan '19

The paper is "Integral Euler characteristic of Out(F_11)" by Shigeyuki Morita, Takuya Sakasai, and Masaaki Suzuki, available at <https://arxiv.org/pdf/1405.4063.pdf>. —Dan Paul Palmer wrote: ----- More details on the Morita paper? I couldn't find it. Thanks On Sat, Jan 12, 2019, 8:34 PM Dan Asimov <dasimov(a)earthlink.net> wrote: > Just something amusing I'm learning about — For each integer let F_n > be the free group on n generators, nonabelian for n >= 2.° > > These groups are utterly natural yet quite intriguing. Like the fact that > F_2 contains an isomorphic copy of F_n for any n, even if n = aleph_0. > > So let's ask about its *automorphism* group, the group of all > self-homomorphisms > a : F_n —> F_n that are invertible and have inverse that is also a > homomorphism. > (Too much in a rush at the moment to check if the last condition is > redundant.) > > Anyhow, all groups act on themselves by inner automorphisms. For any g in > a group g > there is the inner automorphism x —> g x g^(-1) of G for any x in G. > > Definition: > ----------- > Let Inn(G) be the group of all inner automorphisms of G. (It's sometimes > isomorphic > to G, other times not.) > > So, the interesting automorphism thing is the *outer* automorphisms, the > quotient group > > Out(G) = Aut(G) / Inn(G) > > Anyhow, people have compared Out(F_n) for n = 2, 3, 4, .... One way to > compare them > is to consider the INTEGER X(Out(F_n)) where X(G), for any group G, > denotes the Euler > characteristic of the topological space K(G,1). (The space is uniquely > defined only > up to homotopy type, but that's enough to uniquely determine its Euler > characteristic. > > Anyhoo, the point of the above is to convince you that the table below is > a natural > thing to consider: > > n 2 3 4 5 6 7 8 9 10 11 > ————————————————————————————————————————————————————————————————————————— > X(Out(F_n)) 1 1 2 1 2 1 1 -24 -121 -1202 > > (From a 2014 paper by Shigeyuki Morita et al.) This is seriously weird. > > —Dan > —————————————————————————————————————————————————————————————————————————— > * If you don't know about the free group F_n: This is the smallest group > having n distinct generators and no relations (which defines it up to > isomorphism. F_1 = Z but for n >= 2 they're nonabelian. F_n if made abelian > becomes the direct sum of Z n times. -----

1 0

[math-fun] Some serious algebraic weirdness for higher integers
by Dan Asimov 14 Jan '19

14 Jan '19

Just something amusing I'm learning about — For each integer let F_n be the free group on n generators, nonabelian for n >= 2.° These groups are utterly natural yet quite intriguing. Like the fact that F_2 contains an isomorphic copy of F_n for any n, even if n = aleph_0. So let's ask about its *automorphism* group, the group of all self-homomorphisms a : F_n —> F_n that are invertible and have inverse that is also a homomorphism. (Too much in a rush at the moment to check if the last condition is redundant.) Anyhow, all groups act on themselves by inner automorphisms. For any g in a group g there is the inner automorphism x —> g x g^(-1) of G for any x in G. Definition: ----------- Let Inn(G) be the group of all inner automorphisms of G. (It's sometimes isomorphic to G, other times not.) So, the interesting automorphism thing is the *outer* automorphisms, the quotient group Out(G) = Aut(G) / Inn(G) Anyhow, people have compared Out(F_n) for n = 2, 3, 4, .... One way to compare them is to consider the INTEGER X(Out(F_n)) where X(G), for any group G, denotes the Euler characteristic of the topological space K(G,1). (The space is uniquely defined only up to homotopy type, but that's enough to uniquely determine its Euler characteristic. Anyhoo, the point of the above is to convince you that the table below is a natural thing to consider: n 2 3 4 5 6 7 8 9 10 11 ————————————————————————————————————————————————————————————————————————— X(Out(F_n)) 1 1 2 1 2 1 1 -24 -121 -1202 (From a 2014 paper by Shigeyuki Morita et al.) This is seriously weird. —Dan —————————————————————————————————————————————————————————————————————————— * If you don't know about the free group F_n: This is the smallest group having n distinct generators and no relations (which defines it up to isomorphism. F_1 = Z but for n >= 2 they're nonabelian. F_n if made abelian becomes the direct sum of Z n times.

3 2

[math-fun] four fours with floor
by Cris Moore 13 Jan '19

13 Jan '19

Some variants of the classic “four fours” problem allows the floor function, which rounds down to the nearest integer. But I have the impression that by starting with a single 4, and nesting uses of factorial!, sqrt(), and floor(), we can get any integer. Do you think that’s true? Cris

3 3

Re: [math-fun] four fours with floor
by Dan Asimov 13 Jan '19

13 Jan '19

Say we are given a finite F set of unary and binary functions Z —> Z (like +, -, x, integer division, factorial) and a finite set J of allowable initial inputs (like {4}). I wonder if there may be mild conditions under which we can conclude that one or another of two possibilities holds: Maybe, the set of all obtainable numbers has a) density = 0 in Z, or else b) density = 1 in Z. (Or some other dramatic dichotomy if not that one.) The mild conditions might avoid the situations where all numbers obtained avoid some remainder mod p for some prime p. —Dan

1 0

[math-fun] Is there a name for the (presumably 100% of) algebraic numbers inexpressible in radicals?
by Bill Gosper 12 Jan '19

12 Jan '19

(E.g., the four duodecics @ http://mathworld.wolfram.com/SquareDissection.html) If not, I propose irradical. —rwg

4 5

Re: [math-fun] Could change of base (binary ?> ternary) speed up computation?
by Steve Witham 12 Jan '19

12 Jan '19

> From: Richard Howard <rich(a)richardehoward.com> > Date: 1/10/19, 5:16 PM > > A cold, slow universe is an interesting concept. Everyone should know or remember Freeman Dyson's lecture series / paper, "TIME WITHOUT END: PHYSICS AND BIOLOGY IN AN OPEN UNIVERSE" from 1979. https://www.aleph.se/Trans/Global/Omega/dyson.txt http://scilib-physics.narod.ru/Dyson/dyson.pdf I shall discuss three principal questions within the framework of the open universe with the metric (6). (1) Does the universe freeze into a state of permanent physical quiescence as it expands and cools? (2) Is it possible for life and intelligence to survive indefinitely? (3) Is it possible to maintain communication and transmit information across the constantly expanding distances between galaxies? Skimming just now, I remember bits that made me jump the first time. I'll switch to all quotes: T = 10^65 yr. (36) Even the most rigid materials cannot preserve their shapes or their chemical structures for times long compared with (36). On a time scale of 10^65 yr, every piece of rock behaves like a liquid, flowing into a spherical shape under the influence of gravity. Its atoms and molecules will be ceaselessly diffusing around like the molecules in a drop of water. --- For two nuclei combining to form iron, Z = 26, A = 56, S = 3500, and T = 10^1500 yr. (41) On the time scale (41), ordinary matter is radioactive and is constantly generating nuclear energy. --- The lifetime is then by (30) T = exp(120N^(4/3))T_0. (46) For a typical low-mass star we have N = 10^56, S = 10^77, T = 10^(10^76) yr. (47) In (46) it is completely immaterial whether T_0 [sw: the start time] is a small fraction of a second or a large number of years. We do not know whether every collapse of an iron star into a neutron star will produce a supernova explosion. At the very least, it will produce a huge outburst of energy in the form of neutrinos and a modest burst of energy in the form of x rays and visible light. The universe will still be producing occasional fireworks after times as long as (47). --- TABLE I. Summary of time scales. Closed Universe Total duration 10^11 yr Open Universe Low-mass stars cool off 10^14 yr Planets detached from stars 10^15 yr Stars detached from galaxies 10^19 yr Decay of orbits by gravitational radiation 10^20 yr Decay of black holes by Hawking process 10^64 yr Matter liquid at zero temperature 10^65 yr All matter decays to iron 10^1500 yr Collapse of ordinary matter to black hole [alternative (ii)] 10^(10^26) yr Collapse of stars to neutron stars or black holes [alternative (iv)] 10^(10^76) yr --Steve

1 0