October 2018 - mathfun@mailman.xmission.com

[math-fun] A (non-)proof of the Collatz 3n+1 problem.
by Joerg Arndt 17 Oct '18

17 Oct '18

Dear funsters, Georg Fischer has an attempt of a proof for the Collatz 3n+1 problem online here: http://www.teherba.org/index.php/OEIS/3x%2B1_Problem He suspects that it contains a flaw, hence refrains from contacting any big-wigs. With his permission I'd like to ask on this list what the problem may be. Any takers? Georg will certainly add the reason for the invalidity of his proof as soon as someone points it out. Best regards, jj

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[math-fun] (OT) Animal behavior
by Dan Asimov 15 Oct '18

15 Oct '18

The recent fascinating revelations in this forum about red blood cells remind me of something I discovered only in the past couple of years. I didn't think that animal behavior was a subject that interested me that much; I still have no desire to *study* it. But (and I recommend a high-resolution TV or monitor) now that they can plant a 24/7 camera where animals live and then filter through years of footage to find the most interesting ... well, I never would have guessed how interesting this could be. (I can hardly overemphasize this.) So if you haven't already, you might like this. —Dan

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Re: [math-fun] Why we have red blood cells: not turtles all the way down
by Henry Baker 14 Oct '18

14 Oct '18

After additional reading, it appears that RBC's are far from "passive" hemoglobin carriers. * RBC's appear to emit NO (a vasodilator) when mechanically squeezed, as if to say "open up, so I can get through!" (RBC's can squeeze through capillaries slightly smaller than their diameter by folding over.) * RBC's appear to also emit NO when pH is low, as if to say "open up, because I'm bringing you a present of fresh O2, which I can tell you need due to the carbonic acid and lactic acid that I'm sensing !" The reason why capillaries & arterioles don't simply dilate on their own, is that this would lower the blood pressure and divert flow *without bringing in any corresponding oxygen-carrying hemoglobin*. Thus, RBC's can help direct oxygen-rich hemoglobin towards arterioles and capillaries where the pH is lower -- e.g., where the muscles have been working hard & producing carbonic acid and lactic acid. At 05:21 PM 10/13/2018, Warren D Smith wrote: >--that seems bullshit since other animals have quite different red blood cells. >Also, it is quite obvious to anybody who knows anything that >hemoglobin in RBCs is superior to just molecules floating in the blood >because the conditions are maintainable more precisely, and also >more amenable to recycling and less problems with e.g precipitation >and bacteria eating you for lunch, to have to worry about >(easily life threatening). > >> "Also, the biconcave shape allows RBC's to undergo extreme deformations >> while maintaining a constant surface area for gas exchange." >> >> Canham, P.B. "The minimum energy of bending as a possible explanation of >> the biconcave shape of the human red blood cell. J. Theoretical Biology, >> 1970. > >-- >Warren D. Smith >http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)

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Re: [math-fun] Why we have red blood cells: not turtles all the way down
by Dan Asimov 14 Oct '18

14 Oct '18

No. The film they showed my 8th grade science class (c. 1960) talked about the optimal shape for some kind of absorbency condition. However, that's very interesting. I find that optimal shapes with respect to interesting conditions can often be quite fascinating. I wonder how one might define the problem in a very simple way. It is striking that, for the actual shape, a) the usual shape given for an erythrocyte is a *solid of revolution*; b) the cross-section that is rotated is not a smooth curve, but is a continuous curve made up of several smooth arcs whose endpoints, but not derivatives, agree. —Dan ----- Perhaps this is what you recall? "Also, the biconcave shape allows RBC's to undergo extreme deformations while maintaining a constant surface area for gas exchange." Canham, P.B. "The minimum energy of bending as a possible explanation of the biconcave shape of the human red blood cell. J. Theoretical Biology, 1970. At 05:17 PM 10/12/2018, Dan Asimov wrote: >On this topic, I remember c. 8th grade being shown a film addressing this topic, that concluded the lenticular shape of the red blood cell i.e., its hourglass cross==section was exactly what some computer simulation shoowed was ideal (for exactly what conditions, I don't recall). > >I later learned that the film was funded by a possibly biased organization, and even so I don't know if those claims stand the test of time (c. 58 years since 8th grade). > >Dan >----- >... why we had red blood cells ... ... >----- -----

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Re: [math-fun] Why we have red blood cells: not turtles all the way down
by Henry Baker 14 Oct '18

14 Oct '18

Perhaps this is what you recall? "Also, the biconcave shape allows RBC's to undergo extreme deformations while maintaining a constant surface area for gas exchange." Canham, P.B. "The minimum energy of bending as a possible explanation of the biconcave shape of the human red blood cell. J. Theoretical Biology, 1970. At 05:17 PM 10/12/2018, Dan Asimov wrote: >On this topic, I remember c. 8th grade being shown a film addressing this topic, that concluded the lenticular shape of the red blood cell Â i.e., its hourglass cross==section Â was exactly what some computer simulation shoowed was ideal (for exactly what conditions, I don't recall). > >I later learned that the film was funded by a possibly biased organization, and even so I don't know if those claims stand the test of time (c. 58 years since 8th grade). > >ÂDan >----- >... why we had red blood cells ... ... >-----

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Re: [math-fun] Why we have red blood cells: not turtles all the way down
by Dan Asimov 13 Oct '18

13 Oct '18

On this topic, I remember c. 8th grade being shown a film addressing this topic, that concluded the lenticular shape of the red blood cell — i.e., its hourglass cross=section — was exactly what some computer simulation showed was ideal (for exactly what conditions, I don't recall). I later learned that the film was funded by a possibly biased organization, and even so I don't know if those claims stand the test of time (c. 58 years since 8th grade). —Dan ----- ... why we had red blood cells ... ... -----

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[math-fun] geometric pies
by Cris Moore 13 Oct '18

13 Oct '18

https://www.washingtonpost.com/news/in-sight/wp/2018/10/11/lauren-ko-finds-… <https://www.washingtonpost.com/news/in-sight/wp/2018/10/11/lauren-ko-finds-…>

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Re: [math-fun] Lie group terminology
by Dan Asimov 13 Oct '18

13 Oct '18

The example of this kind of thing mentioned most often seems to be SL(2,R)^, which means the unique simply connected Lie group with the same Lie algebra as SL(2,R). Where SL(2,R) is defined as the matrix group SL(2,R) = {n x n real matrices with determinant = 1}. Then the simply connected Lie group SL(2,R)^ is not a subgroup of the group of invertible m x m matrices for any m. —Dan Andy Latto wrote: ----- The prototypical examples of Lie groups are matrix groups, subgroups of GL(n), the group of invertible n x n matrices. I think it's almost, but not quite, true, that all Lie groups are subgroups of matrix groups. (the "almost" is that I think there are a few (families of?) Lie groups that are double-covers of matrix groups, but aren't matrix groups themselves. ... ... -----

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[math-fun] Why we have red blood cells: not turtles all the way down
by Henry Baker 12 Oct '18

12 Oct '18

As you know, I've been looking at fractal networks -- e.g., bifurcating trees like the air pathways in the lung, the arterial & venous systems for the blood. Ever since I was a kid, I always wondered why we had red blood cells flowing through our arteries & veins instead of having hemoglobin as just another component of blood plasma. I think I finally found the answer. The human arterial system is a fractal binary tree approximately 30 levels deep, with the aorta at the root, and the capillaries at the leaves. At every level there are arterioles where the blood flow is governed by the "Poiseuille" equation Q = pi*R^4*dP/(8*mu*L) where Q is the quantity of flow, R is the radius of the arteriole, dP is the pressure differential, L is the length of the arteriole, and mu is the "effective" viscosity. https://en.wikipedia.org/wiki/Hagen%E2%80%93Poiseuille_equation That Q is proportional to R^4 has been tested experimentally by putting blood through glass pipes of varying sizes, and holds experimentally over 4 orders of magnitude in radius. Here's the problem: Poiseuille tells us that the flow through the capillaries should be too small to support our body's requirements for oxygen. The solution (!) is that *mu* -- the "effective" viscosity -- varies quite a bit over the size range from aorta-sized (cross sectional area of ~ 2.5cm^2) to capillary-sized (cross sectional area of ~5-7 micrometers), due to the "FÃ¥hrÃ¦usÂLindqvist" effect. In particular, the situation at the capillary level is considerably different from the situation at the aortic level of the arterial tree. At the aortic level, blood is quite viscous, and flows at 40-50 cm/sec (~ 1 mph), while at the capillary level, the red blood cells (RBC's) travel in *single file* (ballistically?) at ~1 mm/sec (~ 1/500 mph), as the capillaries are too small for the RBC's to pass one another within the capillary. As a result of this orderly flow through the capillaries, the resistance of the capillaries is smaller than one would expect from Poiseuille, and we have the paradoxical effect that the RBC's travel through the capillaries *faster* than the plasma itself travels through the capillaries. At least at scales approaching those of the capillaries (where the FÃ¥hrÃ¦usÂLindqvist effect operates), the plasma becomes essentially a *lubricant* to enable the quicker passage of the RBC's. (Perhaps the FÃ¥hrÃ¦usÂLindqvist effect should be called a kind of "superconducting" effect for RBC's through capillaries?) The net result is that the oxygen is carried through the capillaries faster via RBC's then it could have been carried by the plasma itself. This is a cool evolutionary hack: similar in coolness to that of a high jumper or pole vaulter whose *center of gravity* NEVER gets over the bar, even though the athlete himself/herself does. https://en.wikipedia.org/wiki/F%C3%A5hr%C3%A6us%E2%80%93Lindqvist_effect "The FÃ¥hrÃ¦usÂLindqvist effect ... is an effect where the viscosity of a fluid, in this case blood, changes with the diameter of the tube it travels through; in particular there's a decrease of viscosity as the tube's diameter decreases (only if the vessel diameter is between 10 and 300 micrometers). This is because erythrocytes [RBC's] move over the center of the vessel, leaving plasma at the wall of the vessel." "First, mu decreased with decreasing capillary radius, R. This decrease was most pronounced for capillary diameters < 0.5mm." "These initially confusing results can be explained by the concept of a plasma cell-free layer, a thin layer adjacent to the capillary wall that is depleted of red blood cells. Because the cell-free layer is red cell-poor, its effective viscosity is lower than that of whole blood. This layer therefore acts to reduce flow resistance within the capillary, with the net effect that the effective viscosity is less than that for whole blood. Because the cell-free layer is very thin (approximately 3 micrometers) this effect is insignificant in capillaries whose diameter is large." Lipowsky, Herbert H. "Microvascular Rheology and Hemodynamics", Microcirculation 12 (2005), 5-15.

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[math-fun] Lie group terminology
by Allan Wechsler 12 Oct '18

12 Oct '18

I've been tantalized by various aspects of group theory for a long time, but never had formal instruction beyond introductory abstract algebra. When I try to improve myself, I almost always fetch up against some difficulty that makes me throw up my hands and walk away. The same thing just happened, and I would like to look at the difficulty carefully to see what it is that's stopping me. I wanted to know what kind of thing E8 was. I understand that E8 is actually a bunch of different things: it's a group, it's a polyhedron with that group as its symmetry, it's a root system, it's a lattice ... OK, none of that is too challenging in principle, so I thought I would tackle the group aspect first. What kind of thing, then, is the group E8? I knew it was a big finite group, but didn't know anything about how it was defined. So (please don't laugh) I looked it up in Wikipedia. The article "E8 (mathematics)" starts out by saying "any of several closely related exceptional simple Lie groups". I don't mind that the definition isn't precise; that sort of thing happens all the time, and it doesn't bother me that there is are several different groups that sloppily get called E8. But I don't know what exceptional simple Lie groups are, and the phrase was hotlinked, so I clicked through to learn what an exceptional simple Lie group was. The link took me to a section of a more general article on simple Lie groups. OK, so obviously I need to know what a simple Lie group is before I can learn about the exceptional ones. This article said that a simple Lie group was a "connected non-Abelian Lie group" with some additional constraints to make it qualify as "simple", which looked analogous to the constraints that make an ordinary simple group simple (no normal substructures). If I really wanted to know what they were talking about, though, I obviously had to understand what a Lie group was. So I clicked through to the article about Lie groups in general. The first thing it said was that a Lie group is "a group that is also a differentiable manifold". Wait just a ding danged minute. Do they mean that the points of the manifold form a group? Apparently so, and the group operation has to be continuous and smooth. OK, that makes sense, and the examples they give (the real line under addition, for instance) are consistent with this definition. But: E8 is a finite group. And it's an exceptional simple Lie group, which is a kind of simple Lie group, which is a kind of Lie group, which (can be viewed as) a differentiable manifold. How can any finite group be a differentiable manifold? Differentiable manifolds all have at least the cardinality of the continuum. Then I noticed that the article "Lie group" began with a caveat, "Not to be confused with 'Group of Lie type'." Looking more closely, it seems like the article "Simple Lie group" is actually discussing two separate concepts: differentiable-manifold Lie groups that have a concept of simplicity appropriate to that category; and groups of Lie type that are simple in the ordinary group-theoretical sense. It seems like it bops back and forth between the two concepts indiscriminately, and it is confusing the heck out of me. Help.

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