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[math-fun] Why is the Tits group not a "proper" sporadic group?
by Colin Wright 12 Oct '18

12 Oct '18

Reading around for other things I came across this: Re:://en.wikipedia.org/wiki/Classification_of_finite_simple_groups#Statemen… """ Theorem: Every finite simple group is isomorphic to one of the following groups: * a member of one of three infinite classes of such, namely: * the cyclic groups of prime order, * the alternating groups of degree at least 5, * the groups of Lie type * one of 26 groups called the "sporadic groups" * the Tits group (which is sometimes considered a 27th sporadic group). """ So the Tits group is not one of the infinite families, and it's a finite simple group, so why isn't it fully and properly classed as one of the sporadic groups? What is a sporadic group if not just a finite simple group that's not one of the infinite families? Colour me puzzled. Colin -- Mathematicians are like Frenchmen: whatever you say to them they translate into their own language and forthwith it is something entirely different. -- Johann Wolfgang von Goethe 1749-1842

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[math-fun] Dice rotation games
by James Propp 11 Oct '18

11 Oct '18

Colm Mulcahy's Hamilternion Card Games page http://www.mathsireland.ie/hamilternion made me wonder if there are any games that (a) hinge on the fact that 3D rotation is noncommutative and (b) require only ordinary dice. One idea I like is that we can have one die act on another: if the first die has its m-face facing upward (for m between 1 and 6), we rotate the second die ninety degrees clockwise around the face that shows an m. Note that the operations of the form "rotate this die ninety degrees clockwise around the face that shows an m" generate a group. It's not the usual way we think about rotation groups, since the specified axes are carried along by the object. A fun way to see that this is still associative is to imagine that the operation is "hold the object still and rotate the rest of the universe ninety degrees counterclockwise around the face that shows an m". To make a game that's fun to play, we'd probably want to incorporate a random element. But hey, we have dice! To make a game that's fun to play, we'd probably want players to be able to score "points". But hey, we have dice! Anyone want to suggest a game along these lines that's got elegantly simple rules and is fun to play? Maybe I'll try out a few such games at the upcoming Celebration of Mind. Jim Propp

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Re: [math-fun] Simple model for branching network flows
by Stuart Anderson 11 Oct '18

11 Oct '18

A version of Murray's Law was articulated by Leonardo Da Vinci https://phys.org/news/2012-01-leonardo-da-vinci-tree.html On Fri, Oct 12, 2018, 05:00 <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: Simple model for branching network flows (Henry Baker) > 2. Dice rotation games (James Propp) > 3. Re: Dice rotation games (Mike Stay) > > > ---------------------------------------------------------------------- > > Message: 1 > Date: Wed, 10 Oct 2018 13:12:56 -0700 > From: Henry Baker <hbaker1(a)pipeline.com> > To: math-fun(a)mailman.xmission.com > Subject: Re: [math-fun] Simple model for branching network flows > Message-ID: <E1gAKrS-000DlO-T7(a)elasmtp-mealy.atl.sa.earthlink.net> > Content-Type: text/plain; charset="us-ascii" > > As I thought, this is an extremely well-studied & modelled area. > The following is for incompressible fluid (blood) flow. > > For example, Murray's Law for a bifurcation: > > r^alpha = r1^alpha + r2^alpha > > where r is the radius of the parent node, and r1,r2 are the > radii of the child nodes. alpha is a constant >=2. > > alpha=2 => sum of cross sectional areas of the children = cross sectional > area of the parent. > > alpha=3 => minimizes work for laminar flow. > > alpha=7/3 => minimizes work for turbulent flow. > > https://en.wikipedia.org/wiki/Murray%27s_law > > Also, > > Adam, John A. "Blood Vessel Branching: Beyond the Standard Calculus > Problem". Math. Mag. 84 (2011), 196-207. > > Adam calculates the height of the binary tree as around k ~ 30, so we > don't need a 64-bit machine! > > Adam claims (sum area of children)/(area of parent) ~ 1.26 ~ 2^(1/3). > > https://en.wikipedia.org/wiki/Hemodynamics > > Cross-sectional area of the aorta (root) ~ 3-5 cm^2 velocity ~ 40 cm/s. > Total area of the capillaries ~ 4500-6000 cm^2 velocity ~ 0.03 cm/s. > i.e., total area ratio ~ 1000 = 10^3. > > --- > So if rc is the radius of each child (exactly 2 children), and if rp is > the radius of the parent, then > > (2*pi*rc^2)/(pi*rp^2) = 2^(1/3) > > So, rc/rp ~ 2^(-4/9) ~ 0.735 > > But Murray's Law gives > > rp^alpha = 2*rc^alpha, or > > 1/2 = (rc/rp)^alpha, or > > rc/rp = (1/2)^(1/alpha) > > Putting these equations together yields a Murray-type law with alpha = 9/4 > = 2.25, > i.e., rp^(9/4) = 2*rc^(9/4). > > The next thing to do is to estimate the flow resistance for each level > 1-30, which > are all in *series*. > > At 04:21 PM 10/9/2018, Henry Baker wrote: > >I'm interested in a balanced binary tree of air flow; > >the flow comes in at the root, and goes through k levels > >of binary branching, yielding 2^k outputs. > > > >I'm assuming that air flow resistance varies superlinearly > >with the flow -- i.e., trying to push 2x amount of air > >should face a resistance which is >2x -- perhaps 4x or 8x. > >Let's assume that whatever exponent is used, perhaps 2<alpha<3, > >the same alpha occurs at each level of the tree, so that the > >scaling is simple fractal. > > > >For the moment, let's assume that these pipes are cylindrical, > >and that the joint where one pipe becomes two is designed > >so that the primary resistance isn't the joint itself, but > >merely the sum of the resistances of the branches. There is > >also a slight "enlargement factor" at each branch, whereby > >the area of the branches at level k+1 is slightly larger than > >the area of the branches at level k. > > > >My intuition is that as the flow increases due to increased > >pressure at the root, the maximum resistance occurs at > >smaller and smaller k's -- i.e., closer to the root. > >Equivalently, as the pressure at the root decreases, the > >maximum resistance occurs at larger & larger k's -- i.e. > >further from the root. > > > >Is this intuition correct? > > > >Any references or links? > > > >Next question: what happens if the fluid is *incompressible* > >-- e.g., water? > > > > > ------------------------------ > > Message: 2 > Date: Thu, 11 Oct 2018 10:21:26 -0400 > From: James Propp <jamespropp(a)gmail.com> > To: math-fun <math-fun(a)mailman.xmission.com> > Subject: [math-fun] Dice rotation games > Message-ID: > <CA+G9J-fUiqQOtvnORWvxQ8Sa+Qsv=Fc5U0F1PJNitECu8= > o1Tg(a)mail.gmail.com> > Content-Type: text/plain; charset="UTF-8" > > Colm Mulcahy's Hamilternion Card Games page > > http://www.mathsireland.ie/hamilternion > > made me wonder if there are any games that (a) hinge on the fact that 3D > rotation is noncommutative and (b) require only ordinary dice. > > One idea I like is that we can have one die act on another: if the first > die has its m-face facing upward (for m between 1 and 6), we rotate the > second die ninety degrees clockwise around the face that shows an m. > > Note that the operations of the form "rotate this die ninety degrees > clockwise around the face that shows an m" generate a group. It's not the > usual way we think about rotation groups, since the specified axes are > carried along by the object. A fun way to see that this is still > associative is to imagine that the operation is "hold the object still and > rotate the rest of the universe ninety degrees counterclockwise around the > face that shows an m". > > To make a game that's fun to play, we'd probably want to incorporate a > random element. But hey, we have dice! > > To make a game that's fun to play, we'd probably want players to be able to > score "points". But hey, we have dice! > > Anyone want to suggest a game along these lines that's got elegantly simple > rules and is fun to play? Maybe I'll try out a few such games at the > upcoming Celebration of Mind. > > Jim Propp > > > ------------------------------ > > Message: 3 > Date: Thu, 11 Oct 2018 08:56:08 -0600 > From: Mike Stay <metaweta(a)gmail.com> > To: math-fun <math-fun(a)mailman.xmission.com> > Subject: Re: [math-fun] Dice rotation games > Message-ID: > <CAKQgqTb5EDX_sH++= > eHTeRQSYrCG9Tk16xQy3x522fjANT6F9g(a)mail.gmail.com> > Content-Type: text/plain; charset="UTF-8" > > I think such a game would be possible in the style of the various > games involving the orientation of rolled pig knuckles relative to > each other. The modern version my family has is called "Pass the > Pigs" (https://en.wikipedia.org/wiki/Pass_the_Pigs) In that game, > there's a score for each configuration, but one of the more common > configurations causes you to lose all your accumulated points for that > turn, so you're constantly faced with the decision of whether to roll > again or to pass the dice to the next player and keep the points > you've accumulated so far. > On Thu, Oct 11, 2018 at 8:22 AM James Propp <jamespropp(a)gmail.com> wrote: > > > > Colm Mulcahy's Hamilternion Card Games page > > > > http://www.mathsireland.ie/hamilternion > > > > made me wonder if there are any games that (a) hinge on the fact that 3D > > rotation is noncommutative and (b) require only ordinary dice. > > > > One idea I like is that we can have one die act on another: if the first > > die has its m-face facing upward (for m between 1 and 6), we rotate the > > second die ninety degrees clockwise around the face that shows an m. > > > > Note that the operations of the form "rotate this die ninety degrees > > clockwise around the face that shows an m" generate a group. It's not the > > usual way we think about rotation groups, since the specified axes are > > carried along by the object. A fun way to see that this is still > > associative is to imagine that the operation is "hold the object still > and > > rotate the rest of the universe ninety degrees counterclockwise around > the > > face that shows an m". > > > > To make a game that's fun to play, we'd probably want to incorporate a > > random element. But hey, we have dice! > > > > To make a game that's fun to play, we'd probably want players to be able > to > > score "points". But hey, we have dice! > > > > Anyone want to suggest a game along these lines that's got elegantly > simple > > rules and is fun to play? Maybe I'll try out a few such games at the > > upcoming Celebration of Mind. > > > > Jim Propp > > _______________________________________________ > > math-fun mailing list > > math-fun(a)mailman.xmission.com > > https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun > > > > -- > Mike Stay - metaweta(a)gmail.com > http://reperiendi.wordpress.com > > > > ------------------------------ > > 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 188, Issue 15 > ***************************************** >

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[math-fun] Simple model for branching network flows
by Henry Baker 10 Oct '18

10 Oct '18

I'm interested in a balanced binary tree of air flow; the flow comes in at the root, and goes through k levels of binary branching, yielding 2^k outputs. I'm assuming that air flow resistance varies superlinearly with the flow -- i.e., trying to push 2x amount of air should face a resistance which is >2x -- perhaps 4x or 8x. Let's assume that whatever exponent is used, perhaps 2<alpha<3, the same alpha occurs at each level of the tree, so that the scaling is simple fractal. For the moment, let's assume that these pipes are cylindrical, and that the joint where one pipe becomes two is designed so that the primary resistance isn't the joint itself, but merely the sum of the resistances of the branches. There is also a slight "enlargement factor" at each branch, whereby the area of the branches at level k+1 is slightly larger than the area of the branches at level k. My intuition is that as the flow increases due to increased pressure at the root, the maximum resistance occurs at smaller and smaller k's -- i.e., closer to the root. Equivalently, as the pressure at the root decreases, the maximum resistance occurs at larger & larger k's -- i.e. further from the root. Is this intuition correct? Any references or links? Next question: what happens if the fluid is *incompressible* -- e.g., water?

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[math-fun] From an investment manager's newsletter
by Thane Plambeck 09 Oct '18

09 Oct '18

"as life expectancies continue to rise, it's not uncommon for parents to outlive their children..." on the other hand, i've seen articles making the point recently that the rapid increase in childhood obesity in recent years in the usa and other "advanced" nations makes it more likely that the kids will die *before* their parents. anyway, what is the history of this statistic and the expected trend here? googling hasn't helped me much, but maybe i'm missing something. -- Thane Plambeck tplambeck(a)gmail.com http://counterwave.com/

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[math-fun] Comparing two irrational numbers
by Dan Asimov 08 Oct '18

08 Oct '18

Suppose a, b are two irrational numbers that we want to compare somehow. Here's one idea: Consider A_n = {k*a mod 1 | 1 <= k <= n} and B_n = {k*b mod 1 | 1 <= k <= n} to be subsets of the unit-circumference circle R/Z, which is an abelian group. Let the distance d(x, Y) between a point x of R/Z and a finite subset Y of R/Z be defined as d(x, Y) = the shortest arc xy of R/Z with y in the set Y. Now for any two finite subsets X, Y of of R/Z, define their distance dist(X, Y) as dist(X, Y) = max {min_{x in X} d(x, Y), min_{y in Y} d(X, y)}. Now make the distance rotation-invariant by changing the definition to Dist(X, Y) = min_{g, h in R/Z} dist(g + X, h + Y) or equivalently Dist(X, Y) = min_{g in R/Z} dist(g + X, Y). Finally compute f(n) = D(A_n, B_n) = Dist(A_n, B_n). QUESTION: --------- What is the asymptotic behavior of f(n) as n —> oo (for various irrationals a, b) ??? —Dan

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[math-fun] Evolutionary bottlenecks and fat tails
by Henry Baker 08 Oct '18

08 Oct '18

(Distributions having fat tails, that is.) Evolution, as usually presented -- e.g., by Darwin himself -- talks about *gradual* changes happening over hundreds/thousands of generations. As Moore's Law shows, even gradual change, if extended to enough generations, can produce dramatic change when viewed with enough historical perspective. Nevertheless, I contend that the most dramatic evolutionary changes occur at *bottlenecks*, where 80-90-95% of the population is wiped out, and the few remaining individuals have one or more features which might not have had any (or perhaps only a very modest) effect under "normal" times, but suddenly become the only thing that matters for survival. Assuming that the rest of the genes of these survivors are more-or-less independently distributed, then normal day-to-day evolution will continue to weed out "bugs" in the rest of these genes which are more common, while the one or two features that provided survival at the bottleneck are now shared by *all* of the remaining population. For example, essentially all humans have 5 fingers on each hand. This is quite bizarre, as under "normal" circumstances, having 3,4, or 6 fingers would seem to provide about the same survival advantage as having exactly 5 fingers. Suppose that there was a population bottleneck where survival depended upon have *at least* 5 fingers, and all of the survivors had 5,6,7,8, etc. fingers. More normal evolutionary pressures might then have eliminated those individuals having more than 5 fingers, but the gene pool as a whole would "remember" vividly the bottleneck event which wiped out everyone having fewer than 5 fingers. Note that these population bottlenecks don't have to happen very often. Indeed, it is estimated that the last great bottleneck for the human race was perhaps 70,000 years ago. Exactly which features enabled survival in that bottleneck isn't known, but whatever it was, this feature was clearly was a *rare* feature that suddenly became crucially important for survival. This means that really understanding evolution is going to exceedingly difficult, as some of the most important events are also the rarest events.

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[math-fun] Draft of October 2018 blog post
by James Propp 07 Oct '18

07 Oct '18

Usually I publish on the 17th, but this time I'm publishing on the 10th (in connection with Global Math Week). I'm not expecting many of you to spend your weekend hours perusing this, but even if only a couple of you do, I'm sure your comments will be helpful. http://mathenchant.org/040-second-draft.pdf Thanks, Jim Propp

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Re: [math-fun] Lie group terminology
by Dan Asimov 06 Oct '18

06 Oct '18

A Lie group G is a manifold that is also a group *such that* the group operations of multiplication : G x G —> G and inversion G —> G are each continuous. The solution of Hilbert's Fifth Problem showed that any such Lie group is in fact a *real-analytic* manifold on which those group operations are in fact real analytic. (Meaning: expressible locally as a tuple of power series.) A few sweeping things are known about Lie groups. For every dimension of Euclidean space R^n there is the Lie group SO(n) of all its isometries. (The rotation group SO(n) is the connected component of the identity element in O(n).) Every compact Lie group is a Lie subgroup of an O(n) for some n, so in that sense the O(n)'s are "universal". The rotation group SO(n) is abelian only for n = 1 or n = 2. The Lie groups that are abelian are precisely the cartesian products of Euclidean spaces and tori: R^n x T^p (where the p-torus T^p is the cartesian product of p circles S^1). The homotopy groups π_n(X) of a path-connected space X are the continuous maps of the n-sphere S^n into X, factored out by the equivalence relation of being continuously deformable into each other ("homotopy"). If we let each R^n sit in the next one as R^n x {0} in R^(n+1), then the orthogonal groups O(n) form an increasing union. We can then take the *union* of all the O(n)'s — to get a group (that isn't Lie) called O(oo). Here are the homotopy groups of O(oo), which striking depend only on n mod 8: n mod 8 0 1 2 3 4 5 6 7 π_n(O(oo)) Z/2 Z/2 0 Z 0 0 0 Z This was proved by Bott in 1957, and is called Bott periodicity. —Dan

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[math-fun] Law of Sines for Polyhedron Vertices
by Tom Karzes 06 Oct '18

06 Oct '18

Is anyone familiar with the following? Consider a polyhedron vertex with three faces (and three edges). Let the polygon angles at the vertex be A, B, and C. Let the dihedral angles at the vertex be a, b, and c, where a is opposite A, b is opposite B, and c is opposite C. Then: sin(A)/sin(a) = sin(B)/sin(b) = sin(C)/sin(c) This is just another Law of Sines, and seems pretty fundamental, but I can't find any reference to it (or proof of it). I can only find the version taught in high school trig (which uses the ratios of edge lengths to the sines of the opposite angles), and the spherical case (which uses the ratios of the sines of the surface triangle angles to the sines of the angles of the arcs from the center of the sphere). Is this well known? Does anyone have a reference for it? Or a simple proof? Tom

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