Announcing Pebble Hound

Hello world (and Googlebot), here are the links to our new store:

Pebble Hound

Pebble Hound Store

Susan wants me to unload the garage, so this is my world wide garage sale store!

A simple nonlinear PDE wave equation

In searching for solitons in free space, consider the scalar wave equation:

Uxx + Uyy + Uzz = (1/c^2)*Utt

but suppose the wave velocity c is not constant, but rather equal to the field value itself:

c=U

then

Uxx + Uyy + Uzz - Utt/U^2 = 0

Solutions to this nonlinear PDE might yield semi-stable reflexive standing waves with a unique scale. The mean value of c would emerge rather than being specified apriori as a constant of nature.

Questions:
1. Are there closed form solutions to this nonlinear PDE with 3 space and 1 time dimension?
2. Do additional space or time dimensions yield sensible solutions closer to our world?
3. Are there stable symmetric solutions in space or time?

These questions could shed light on light and how it self mixes into what we call matter waves through a nonlinear action upon itself! It would be satisfying to see particular soliton condensates that correspond to the zoo of elementary particles.

Time for bed . . .

Simplest Field Equation For Six Dimensions

The simplest field equation in six dimensions that makes any sense when projected to fewer dimensions is the second order PDE:

Uxx + Uyy + Uzz - Urr - Uss - Utt = 0

Here U(x,y,z,r,s,t) is a six dimensional scalar field, and the equation is a simple extension of Laplaces Equation.

I wonder if solutions to this equation in any way correspond with our apparent 4D space-time world. No nonlinearities, but two more time-like dimensions! Just a thought . . .

Iron Compounds Super Conduct!

I read with some surprise that iron compounds super conduct (NewScientist, 16Aug.2008, p.31). What will next turn out to super conduct?

Why do we keep finding these things by experiment if quantum mechanics is such a great model of the physical world? What could guide us toward the ultimate materials for high temperature superconductors, magnets, etc. ?

Mankind needs a workable mathematical model that predicts material properties. I'd be happy to see the current version of quantum mechanics fall by the wayside if a new model could at least harness all the available computer gigaflops we have in the world to help us design better materials without resorting to what looks like alchemy!

Science claims quantum mechanics explains all chemical properties, and yet relies so much on accident and experiment to develop new materials! Quantum mechanics as currently expressed might turn out to be as useful to material science as epicycles were to astronomy! Yes, it works, but where does it lead? How can it guide experimental intuition?

Perhaps the blame is the current state of numerical computation. I don't really believe this to be the case. My intuition is that our current models use too many computer flops because they miss the key dynamics of nature's inner workings. As Copernicus showed, Kepler computed, and Newton demonstrated; get the model right and the calculations collapse to what is necessary and sufficient!

Why spend so many research dollars on subatomic theory when we have relatively poor theoretical tools to make the materials that future generations depend on? Our theoreticians must get to work assisting experimentalists, rather than vice versa! The earth is in a world of hurt. We might even find by solving the more mundane problems of material science that we also solve some of the remaining mysteries of the subatomic world!

End of rant!

P.S. I've had one of those miserable two week colds, so this is how I react to news!

Wave Particle Duality - Playing with Alice!

The Fourier transform in 3D converts a cube to a triple product of sinc functions in the three dimensions. A cubic particle becomes an infinite wave. It has the appearence of a Coulomb potetential without the infinity. No information is lost or gained. Local becomes global,

Performing the same transform operation a second time returns to the cube, but with a sign change.

Performing the same transform operation yet a third time yields the triple sinc product with a sign change.

Performing the same transform operation yet a fourth time returns to the original cube. As in all iterations, no information is lost or gained. You are back where you started, but you traveled with Alice through the looking glass four times!

The "world operator" employs such an iteration. When we declare that a local measurement has been made, we are viewing the particle domain. When we see non-local effects, we are viewing the wave domain. The standing waves of a persistent iterating operator create our eigen world of the observable.

The world appears local on one side, and global on the other side of the looking glass! It is the same world, but stepping through an iterating operation.

What exactly is the mathematical form of the "world operator" and the object it operates upon? This mysterious operation iterates ad infinitum, taking the world from the past to the present to the future. It plays the frames of the world manifold like a well oiled machine. It is lossless, reversable, and deterministic. In what space-time manifold does it play?

More nonlinear first order scalar partial differential equations!

I wouldn't mind playing with this nonlinear first order scalar partial differential equation:

(h*Ut)^2 - (h*c*Ux)^2 - (h*c*Uy)^2 - (h*c*Uz) ^2 . . . = ((m*c^2)*U)^2

or even this nonlinear first order scalar partial differential equation:

(h*Ut)^2 - (h*c*Ux)^2 - (h*c*Uy)^2 - (h*c*Uz) ^2 . . . = ((m*c^2)^2)*U

Hmmm, first order quadratic terms remind me of something. Perhaps the field equations of general relativity that are quadratic in the first derivatives and linear in the second derivatives.

Linear fields superpose; they don't couple. Nonlinear fields couple. What kind of world do we live in? Nonlinear, . . . digital?

If I were a betting man, I'd bet that the elementary particle mass ratios have something to do with stationary states of nonlinear fields. What do you think?

Time for bed.

Happy Birthday (whoopee!)

Another day older and slightly less deeper in debt!

Consider this nonlinear first order scalar partial differential equation:

(h*Ut)^2 - (h*c*Ux + h*c*Uy + h*c*Uz + . . . )^2 = ((m*c^2)U)^2

Solutions welcomed!