July 2018 - mathfun@mailman.xmission.com

Re: [math-fun] Free surface effect
by Dan Asimov 30 Jul '18

30 Jul '18

That's much clearer to me, thanks! The phrase "doesn't move" is presumably with respect to some inertial frame of reference? —Dan Brent Meeker wrote: ----- ... I was imagining a ship that was just a cylindrical tube, that it partly full of some fluid, say oil, and it has a heavy keel. If it rolls, say counter-clockwise looking forward, the oil will stay exactly where it was (neglecting boundary layer friction) while the ship keel will move to the right and upward. Hence the ship structure CG will move to the right, while the oil CG doesn't move. Did that help? -----

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Re: [math-fun] Free surface effect
by Dan Asimov 30 Jul '18

30 Jul '18

If CG means center of gravity, then I don't understand this. Otherwise, I don't understand it because I don't know what CG means. —Dan ----- Sure. Imagine a circular cylindrical tank. When partially full roll of the ship would leave the tankage CG unchanged while the CG of the rest of the ship (assuming it's stabilized by a keel) would move opposite the roll. So the overall CG would change. But you can't tell the stability of a floating object in general just by considering the CG location. You have to consider the center of buoyancy too and their relative motion. Think of a catamaran. The CG doesn't change when it rolls, but the center of bouyancy moves to provide a righting moment. -----

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Re: [math-fun] Free surface effect
by Henry Baker 30 Jul '18

30 Jul '18

It boils down to the changed shape of the *displaced water* under rotation of the hull. Ideally, any rotation about the COG should *dramatically increase* the displacement, thus raising the COG enough to supply a force counter to that twisting the hull in the first place. For example, a fast sailing ship leans over & increases its displacement, until the force trying to right the ship equals the force of the wind. Fast sailing yachts take advantage of this principle to dramatically *increase the length of the waterline* because the longer hulls can go faster for the same total displacement. At 10:30 PM 7/29/2018, Jason Holt wrote: >I just learned about free surface effect: >https://en.m.wikipedia.org/wiki/Free_surface_effect > >And it makes me wonder: imagine a cross section blueprint of a ship >perpendicular to the keel. We want to design the cross section of an oil >compartment that will run stem to stern and minimizes free surface effect. > >To keep it simple, let's just consider a static case where the ship has >listed to port or starboard and the fluid in our tank has settled. The >tank is, say, 50% full. Can tank shape affect the ship's change in center >of gravity?

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[math-fun] Free surface effect
by Jason Holt 30 Jul '18

30 Jul '18

I just learned about free surface effect: https://en.m.wikipedia.org/wiki/Free_surface_effect And it makes me wonder: imagine a cross section blueprint of a ship perpendicular to the keel. We want to design the cross section of an oil compartment that will run stem to stern and minimizes free surface effect. To keep it simple, let's just consider a static case where the ship has listed to port or starboard and the fluid in our tank has settled. The tank is, say, 50% full. Can tank shape affect the ship's change in center of gravity?

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Re: [math-fun] Free surface effect
by Dan Asimov 30 Jul '18

30 Jul '18

Here's the basic definition from the linked article: ----- The free surface effect is a mechanism which can cause a watercraft to become unstable and capsize. It refers to the tendency of liquids — and of unbound aggregates of small solid objects, like seeds, gravel, or crushed ore, whose behavior approximates that of liquids — to move in response to changes in the attitude of a craft's cargo holds, decks, or liquid tanks in reaction to operator-induced motions (or sea states caused by waves and wind acting upon the craft). ----- That certainly sounds like the description of the phenomenon in *practical* terms. But for anyone interested in what is the basic thing going on in math or physics, it's *so vague* that it's not much help: "The free surface effect is the tendency for [lots of things] to move in response to [lots of other things] [caused by motion of a vessel] [which itself may be caused by ... lots of things]" —Dan ----- ... free surface effect: https://en.m.wikipedia.org/wiki/Free_surface_effect ..... -----

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Re: [math-fun] Estimating the aspect ratio of a rectangle
by Dan Asimov 29 Jul '18

29 Jul '18

That's pretty much how I decide if I'm happy with the equilateral triangle I just tried to draw. —Dan ----- ... To decide by eye whether octagon ABCDEFGH is regular, hold it so that segment AB is horizontal, and see if the octagon appears to have bilateral symmetry. Then hold it so that segment AE is vertical, and see if it still appears to have bilateral symmetry. ... -----

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Re: [math-fun] Cohl Furey & the Octonians
by Henry Baker 29 Jul '18

29 Jul '18

1. I don't see how "division" relates to "time-reversal/unitarity" at all. I seem to recall that time-reversal is more like complex conjugation than division, so that when a wave travelling in the forward direction hits a wave travelling in the reverse direction, the result is abs()^2 -- i.e., (X)(X*) = |X|^2. See https://en.wikipedia.org/wiki/Transactional_interpretation 2. This still leaves us with the problem of division. So what *is* the physical interpretation of the map x -> 1/x ? At 10:15 PM 7/28/2018, Dan Asimov wrote: >Now I'm even completely loster. > >Does "x Â> 1/x" have anything to do with a) "Time reversal / unitarity" or with b) " > >Henry Baker wrote: >----- >"Time reversal / unitarity." > >???? > >Ok, I'm completely lost. > >What is the physical meaning of x -> 1/x (assuming that it's unique in a non-associative algebra. If not, choose a left or right inverse or.) > >At 12:31 PM 7/28/2018, Mike Stay wrote: >>>> "Where does she use *division*?" Time reversal / unitarity.

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[math-fun] octonionic physics
by Olivier Gerard 29 Jul '18

29 Jul '18

I love octonions as well as other algebraic objects in the large spectrum of non-associative mathematical laws, I investigate some of their properties and use for other parts of mathematics but I don't take for granted that the mathematics I like ought to be incorporated into the fabric of space-time. This is the contrary of the very ancient habit documented by platonic dialogs were each regular polyhedron is associated with a given "element". The current thread on the theories of one colorful Cohl Furey brings the more general problem of looking into non-associativity in mainstream mathematical physics, and by that I don't mean very very high energy physics happening hypothetically one femtosecond after the Big Bang or inside a black hole or at the Planck scale. Those are valuable pursuits, but to me three centuries of modern physics have given us remarquable inter-reactions between mathematical speculations and physics investigations at the observatory, laboratory or industrial plant scale. Take a few families of examples - complex numbers for electromagnetism (notably electric signals, filters), wave mechanics, - functional analysis, non commutativity and quantum mechanics, atomic structure, - path integrals and high energy particles collisions (QCD, quarks and gluons, etc.). Is there at least one case in this kind of current or future "core" physics where an octonionic or sedenionic model or interpretation brings an added value to our knowledge of phenomena or our prediction ability or reveal an internal structure ? (I propose not to include classical Lie algebras in this thread, although these are partially non-associative) As an example, I have heard several times of octonionic or generally 2^n-dimensional algebra based fluid mechanics but never saw something very definite or a report about a real-world problem computed in this way or an insight gleaned about the equations of fluids or the structure of space. Olivier

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Re: [math-fun] Is zero “weakly positive”?
by Dan Asimov 29 Jul '18

29 Jul '18

What? I find this hard to believe. I spent a year at a French math institute near Paris and never heard that "positif" includes 0. —Dan ----- In French, "positive" includes 0, whereas in English, it excludes zero. -----

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Re: [math-fun] Cohl Furey & the Octonians
by Dan Asimov 29 Jul '18

29 Jul '18

Sorry for the accidental send. In the meantime I've decided that I'm so lost, I withdraw the comment that I didn't mean to send. —Dan ----- Now I'm even completely loster. -----

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