Ken Crandall Blog

Announcing Pebble Hound

2009-04-26T11:27:00.000-07:00

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

2009-03-02T23:24:00.000-08:00

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

2008-12-13T22:02:00.000-08:00

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!

2008-08-17T20:20:00.000-07:00

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!

2008-05-20T23:22:00.000-07:00

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!

2008-03-11T23:05:00.000-07:00

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!)

2008-03-10T23:28:00.000-07:00

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!

Christmas Thoughts 2007

2007-12-25T16:54:00.000-08:00

Well, the year went by faster than expected! Here I sit here on Christmas day contemplating my navel and the stuff of the universe. I have lots to be grateful for, and am in a thoughtful state of mind.

As an electrical engineer in communications, I frequently use the function sinc(x) and its Fourier transform, the rectangular brick wall low pass filter in my work. It has always impressed me that a rectangle in one domain transforms to an infinite duration sinusoid with an envelope that roles off as 1/x.

If the dimensionality is increased from a one dimensional time domain signal to a four dimensional space-time signal, then could a hypercube in the frequency domain transform to a spherical sinc function in the space-time domain? This has all the appearance of a scalar potential standing wave and suggests the quantum mechanical amplitude potential since it falls off as 1/r. This potential has no problem with infinity at the center, since it limits to a finite value.

It strikes me that these infinite duration waves naturally occur in the time domain, but that our perception of a wave of infinite duration can only be detected by convolving with another similar wave. Our measuring instruments are doomed to be made of the same stuff that we are measuring! The net result is a "perception" of a particle of finite extent. We might call such a measurement a "photon", when we really just observed a wave exchanging energy with our measuring apparatus and appearing quite finite!

So, supposing some truth in the Fourier transform resolution of "wave-particle duality", then we can think in terms of cubes of matter, as long as we wink and take the Fourier transform to see what is "really" going on in the space-time domain. The cube starts in our model (our mind) in the frequency domain.

Is there a preferred number of spatial dimensions and time dimensions to accommodate our world?

This provides some motivation to play with n dimensional Fourier transforms and see if any combination of time and space dimensions result in our world when projected to ordinary four dimensional space-time. This is a non-trivial mathematical exercise! MatLab may have to come to the rescue.

Comments are invited!

A larger 6D hypercube set

2006-12-25T01:15:00.001-08:00

Perhaps just a bit tongue in cheek, there seems to be so many "fundamental particles" in the current zoo, that I had to shrink the size of the hypercube by half to make room for all of them! Pretty good alignment (or curve fitting?) with the hypercubes. Now it is left to describe the operation that describes the 6D evolution with time. What might it be? Also, what do you suppose the fundamental 6D hypercube is with a mass of just 125 electron volts? And for that matter, what is the size of the smallest hypercube?

n GeV Speculative Particle Association

1.0 1.24756e-007 fundamental 6D hypercube
2.0 7.98436e-006 electron neutrino
3.0 9.09468e-005 muon neutrino
4.0 0.000510999 electron
5.0 0.00194931 up quark
6.0 0.0058206 down quark
7.0 0.0146774 tau neutrino
8.0 0.0327039
9.0 0.0663002
10.0 0.124756 strange quark
11.0 0.221012
12.0 0.372518
13.0 0.602171
14.0 0.939352 neutron
15.0 1.42104 charmed quark
16.0 2.09305 tau lepton
17.0 3.0113
18.0 4.24322 bottom quark
19.0 5.86924
20.0 7.98436
21.0 10.6998
22.0 14.1448
23.0 18.4683
24.0 23.8412
25.0 30.4579
26.0 38.539
27.0 48.3329
28.0 60.1185
29.0 74.2075 W
30.0 90.9468 Z
31.0 110.721
32.0 133.955 Higgs boson
33.0 161.118 top quark
34.0 192.723
35.0 229.334
36.0 271.566
37.0 320.089
38.0 375.631
39.0 438.983
40.0 510.999
41.0 592.602
42.0 684.787
43.0 788.625
44.0 905.266
45.0 1035.94
46.0 1181.97
47.0 1344.77
48.0 1525.83
49.0 1726.78
50.0 1949.31

Speculative 6D Hypercube Particle Associations

2006-12-24T02:14:00.000-08:00

Ken Crandall Blog

n GeV
1 0.000008 neutrino
2 0.000511 electron
3 0.005821 down quark
4 0.032704
5 0.124756 strange quark
6 0.372518
7 0.939352 neutron
8 2.093052 tau
9 4.243215 bottom quark
10 7.984358
11 14.144778
12 23.841167
13 38.538973
14 60.118515
15 90.946833 Z
16 133.955307 Higgs
17 192.723004 top quark
18 271.565789

Where's the up quark?

Energy Levels of 6D Hypercubes

2006-12-19T01:37:00.000-08:00

Ken Crandall Blog

n GeV
1 0.000008
2 0.000511
3 0.005821
4 0.032704
5 0.124756
6 0.372518
7 0.939352
8 2.093052
9 4.243215
10 7.984358
11 14.144778
12 23.841167
13 38.538973
14 60.118515
15 90.946833
16 133.955307
17 192.723004
18 271.565789

Ken Crandall Blog

2006-12-15T22:14:00.000-08:00

Ken Crandall Blog

Well, now that we're on the subject of hypercubes, let's take a look at the hypercube triangle:
1
1 2
1 4 4
1 6 12 8
1 8 24 32 16
1 10 40 80 80 32
1 12 60 160 240 192 64
1 14 84 280 560 672 448 128
1 16 112 448 1120 1792 1792 1024 256
1 18 144 672 2016 4032 5376 4608 2304 512
1 20 180 960 3360 8064 13440 15360 11520 5120 1024
. . .

What is it? Well the flip answer is that they are the coefficients of the expanded form of (x+2)^n. The more interesting answer is that they describe the details of an n dimensional hypercube.

For example, if n=3, then we have 1 cube with 6 faces and 12 edges and 8 vertex points.

Since I am attracted to the n=6 hypercube, we can say:

One 6D hypercube with 12 5D hypercubes and 60 4D hypercubes and 160 cubes and 240 faces and 192 edges and 64 vertex points.

Try to visualize that! Well, there are several websites that do just that. Check out:

http://homepages.wmich.edu/~drichter/sixcube.htm

http://www.jpbowen.com/publications/ndcubes.html

So hypercubes are an interesting exercise in stretching your imagination. They also are where nature's inner workings go on. Not so far fetched to consider the lowly 6D hypercube as the building block of our universe! Huh?

Ken Crandall Blog

2006-11-17T00:16:00.000-08:00

Ken Crandall Blog

Assume, for fun, that fundamental particles are built out of cubes in n dimensions. Using MatLab, I did a quick search to see if any dimension yielded any interesting results for mass ratios, assuming that mass is proportional to the volume of the hypercube. To my surprise, six dimensions gives some pretty interesting results:

correspondence:
m1=1
m2=2^6 electron and positron
m3=3^6
m4=4^6
m5=5^6 muon
m6=6^6
m7=7^6 proton and neutron
m8=8^6 tau
m9=9^6
m10=10^6
m11=11^6
m12=12^6
m13=13^6
m14=14^6
m15=15^6 W and Z boson
m16=16^6 Higgs boson
m17=17^6
m18=18^6
. . .

Ratios of interest:

m7/m2 = 1838.26
m8/m2 = 4096.00
m15/m2 = 177978.51
m16/m2 = 262144.00

It would be interesting if the Higgs boson were found near 133.95 GeV to establish it as a rather heavy particle with a mass 262144 times greater than the electron!

I wonder why hypercubes in six dimensions give such a nice approximation to the masses of the zoo of subatomic particles.

Any thoughts out there in cyberspace?

Ken Crandall Blog

2006-11-05T23:56:00.000-08:00

Ken Crandall Blog
Being a big fan of MatLab, I dabbled a bit in "what if" games regarding the masses of subatomic particles. I have some interesting half-baked results, but would like to indulge in making the perhaps historic pre-announcement:

"CERN discovers new particle at 134 GEV."

Is this the Higgs Boson?

I'm interested in mathematical models, and would be interested in hearing from anyone else with ideas on why the constants of nature take on the values that they do, and what these values might be, and why!