Mike Henderson

I may have the counting wrong - it's been a while. Maybe it's three parameters and that it can't be embedded in 4d. I dunno.

A complicated surface with portions colored differently. Symmetry planes are shown.

A complicated surface with portions colored differently. Symmetry planes are shown. Mostly the same as the previous picture, but not all symmetries are shown and there are black lines showing the parameter value contours.

These are out-takes from
www.worldscientific.com/doi/10.1142/...
A very weird surface. Points on it satisfy 3d ODE with bcs and four parameters. It can be embedded in 4d, but projected into 3d it crosses itself.

The colors have to do with a classification of the solutions - # loops and twists.

This is the same algorithm (and code) as for the torus. The continuation is limited to the inside of a box, so you only see the yellow spheres until they cross that. The polyhedra around the spheres are more visible. They're used to find the boundary of the union of spherical balls.

This is an animation of an algorithm for covering a torus with the projection of disks tangent to the surface. Red is the boundary, and the yellow contribute to the boundary. Blue ones are interior.

Each step picks a point on the boundary for a new disk. The boundary is a simple list.

I have always had a blind spot for people who say things with confidence. I need to think about things, and can't seem to avoid reacting as if confidence means that they've thought about what they're saying. I know I'm not alone in this.

So not surprised that LLMs come across as intelligent.

Ugh. I moved my stuff to a new machine, and when I ran configure on my code I goofed and told it to use lapack64 instead of lapack. Strange messages about subroutine arguments having bad values.

I hate debugging. I hate installing and configuring code.

Or in my case, I worked on computing manifolds. I can do it for any dimension (it can be expensive). Now all I hear is -- "Well, we can't draw anything higher than k=2, so what's it good for?"

Thanks,

On the question about what counts as a solution, I've seen people spend a lot of time and doing some brilliant things to get a closed form solution. It's nevertheless 10 lines long and the response is usually -- "Oh, good".

Its like "what's your favorite song?". Well it was Bowie, "Young Americans" until Cohen's "Alexandra Leaving" played on iTunes. Now it's that. What do you mean?

I'm great at small talk. The IBM execs loved me too. An answer, any answer, as long as it's confidently stated.

Something we used to talk about all the time is what counts as "solving" a differential equation? Does it require a (simple) closed form? An asymptotic expansion? A numerical solution?

And I've yet to discover what "solving" the 3-body problem means. Stop that.

I do have my opinions.

There's not even a best case where there is any positive benefit.

The coach gets paid several million a year. For what?

So you spend all these millions and they do shit (watching PSU game). But even if they win, what exactly is the benefit????

Even a tenth.

I'm just trying to imagine if the amount of attention (and money) that goes into college football went instead into Applied Math (just a random alternative).

Point Cloud Continuation: Extracting Manifolds from Observations of a Dynamical System | SIAM Journal on Applied Dynamical Systems Abstract. In the numerical study of dynamical systems, there are applications where equations or good initial guesses are not available, and a data-driven approach based on a set of observations of th...

Our paper on point cloud continuation
has appeared in SIADSL

epubs.siam.org/eprint/N2VHD...

I should say these differential forms are vectors. All are tensors.

Ok. I have a blind spot for the whole "differential forms" thing. They're vectors. Rate of work is F.v, so applied force over a displacement in a direction. Heat is a flow. Sheesh.

BTW, (1/2 m v.v)'=ma.v=F.v

Robert Hermann 1931-2020 I was sorry to hear today of the recent death of Robert Hermann, at the age of 88. While I unfortunately never got to meet him, his writing had a lot of influence on me, as it likely did for many o…

Robert Hermann is apparently quite well known in math physics.

www.math.columbia.edu/~woit/wordpr...

Geometry, Physics, and Systems Robert Hermann Department of Mathematics Rutgers University New Brunswick, New Jersey Marcel Dekker, Inc. New York 1973

Returning to the sumanifold phi given in the form (2.6),i.e., three of the thermodynamic variables expressed as functions of the other two, we see that conditions (21) and (2.3) lead to equations of the following dorrt: dU/dV - T dS/dV = P dU/dP = T dS/dP Similar relations are obtained bking the other pairs of thermodynamic variables independently Unfortunately, physicists and chemists have invented an awkward notation to hide this possibility of change of variable. For example, if (P,V) are independent variables, the partial derivatives of U(P,V) are denoted by some such symbol as (dU/dP)_V, which is read "the change in U with respect to P at constant V".

This geometric framework is adequate to set up all of the "formal" material of thermodynamics, to be found in the textbooks, concerning the algebraic and differential relations between thermodynamic variables in equilibrium situations. This material is, in fact, rather limited in extent. Apparently, one of the emotional excesses of physicists in the 19th and early 20th centuries, was to overemphasize and make unduly mystical this material, particularly that part connected with "entropy." A reasonable approach towards entropy is to consider it simply the function that occurs in (3.1), in the role "conjugate" to T, relative to the 1-form theta.

I 'm reading Hermann's "Geometry, Physics, and Systems", 1973. It's got a chapter on Thermo. I love this guy.

He doesn't like the (dU/dP)_V notation or mysticism (his word) about entropy.

But what's with the bibliography? All references are labelled 1? Later there are some 2's.

Click the wrong box so that my retirement is invested in the wrong place, decide to withdraw too much every month, ... x/x

I guess it's good to know that if I don't press the wrong button I'm safe. But I'm concerned. Is there someone enforcing this? Can I get it in writing? Same for real life? What if I sign the wrong document, say the wrong thing, am in the wrong place, or born to the wrong parents? 2/x

Since grad school I never let myself play computer games. Novels were bad enough. But now I play solitaire. It's as much about dodging the ads. One claims I'm out of date (maybe true). If I press the wrong button I have to uninstall all sorts of games and find how to get the screen manager back 1/x

(subscript)

A question: this thermo stuff uses the notation
(\frac{\partial T}{\partial V})_S
Why the subscr\frac{\partial T}{\partial V}ipt? Isn't that what the partial means? If S was a function of V it'd be
\partial T/\partial V+ \partial T/\partial S \partial S/\partial V
No?

And then I read chemical perspectives that say things like each phase has a Gibbs free energy and the state with the lowest G is the stable one. Is this empirical? G=H-TS, so why?

In other applications I'd expect the existence of an energy (U,F,G,H) to define a surface on which the system can move. Fix the energies and there's a range of accessible states. Since there's no dynamics maybe fixed energies defines a single state?

The phase diagram, or constraint, is part of a bifurcation diagram. You have a dynamical system and let it come to an equilibrium, then change the parameters. At a set of parameter values it tells you which configuration of the system is stable.

Then there are these rules for how you can move

So we (USA) live in a democracy and are free. Except if you work for a company, which is kind of hard to avoid in one way or another. And the company is pretty much a dictatorship. Who says what goes? The shareholders? (Not in little things at least). The CEO? Anyone in "Management"? Not right.

If you use Stoke's theorem (path integral of tangential component equals area integral of curl) this would say that the curl is zero, but the phase manifold is one constraint on T, P, V, S? So do paths have to lie on a 2d submanifold of the 2d phase manifold?