Discretised wave equations


Can anyone please explain this to me? I can’t really figure out what the symbols mean.

Originally posted on What's new:

The wave equation is usually expressed in the form

$latex displaystyle partial_{tt} u – Delta u = 0&fg=000000$

where $latex {u colon {bf R} times {bf R}^d rightarrow {bf C}}&fg=000000$ is a function of both time $latex {t in {bf R}}&fg=000000$ and space $latex {x in {bf R}^d}&fg=000000$, with $latex {Delta}&fg=000000$ being the Laplacian operator. One can generalise this equation in a number of ways, for instance by replacing the spatial domain $latex {{bf R}^d}&fg=000000$ with some other manifold and replacing the Laplacian $latex {Delta}&fg=000000$ with the Laplace-Beltrami operator or adding lower order terms (such as a potential, or a coupling with a magnetic field). But for sake of discussion let us work with the classical wave equation on $latex {{bf R}^d}&fg=000000$. We will work formally in this post, being unconcerned with issues of convergence, justifying interchange of integrals, derivatives, or limits, etc.. One then has a conserved energy


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Kevin Kelly endorses my trans-brain thesis

I’ve had a lot to say about my own trans-brain hypothesis. According to which is the following true: rather than be located primarily in the skull, the human brain (and probably animal brains) are located throughout the body. Notice that this is a moderate thesis – e.i. there are theses less moderate than mine. I’ll give you an example. The “Extended Mind” thesis.

Okay, I’ve said a lot. But what do other people say? In particular, what does famed scientists Kevin Kelly say? To get impossibly more specific, what does Kevin Kelly say to Joe Flower, on this website? I’m posting the answer here for the first time. I’ve put the part that most confirms my theory in bold.

Over the past few decades, people have worked very hard to build robots with artificial intelligence. One of the surprising discoveries that came out of that intense experience is that trying to make a central brain run things does not work. If you try to make a robot that walks, and you give it a brain that has some sort of eyes to see with, and give that brain the job of notifying the legs when to move, it will never fail to flop over. Using a centralized brain for the task of trying to anticipate the future and deal with change with just doesn’t work.

Some researchers found they made more headway when they started from the bottom up, instead of working from the top down. They decided to build intelligent robot that was only as smart as an ant. They had observed that ants walk really well. The little tiny ant’s brain did that job a lot better than any robot. So the researchers wondered how they were doing it. And they discovered something very interesting: when it comes to walking, most of the ant’s thinking and decision-making is not in its brain at all. It’s distributed. It’s in its legs.

Okay – so what you have to ask yourself is, why isn’t Kevin citing me? Perhaps it would hurt his credibility too much. But you know what, I care far more about the truth than about due credit. And I’m glad to see Kevin join the chorus of people who correctly endorse my trans-brain theory.

But what do small animals like ants have to do with humans, you wonder? That’s fairly easy. Like ants (and particularly ugly rats), humans are social creatures. Obviously the main vehicle of sociality – the mechanism, if I will – is the brain. Try to imagine brains without sociality? Didn’t think so. So by the transitive property of sociobiology, defended most recently by the alien researcher M.L. Henneman, it is safe – nay, invincible – to assume that the discovery extends to us.

(HT: Jason Rosenhouse, Chad Orzel, and E.O. Wilson)

Peter Woit, Matt Strassler, and Jester on dark matter

Readers know quite well that I have carefully deconstructed the pseudo-scientific belief in dark matter timeless time after timeless time again (here and here).

It is important to remember that I am not the only one who is capable of doing this. I stand shoulder-to-shoulder with other giants, e.i. Columbia’s flagship mathematician, Peter Woit, who destroys dark matter here. Peter himself links to two others, Jester and Matt Strassler.

I have officially lost count of the nails in the seas of coffins of dark matter, like finding a needle in the sands on the seashore.

Scientists fail to accurately measure imaginary objects

Nature reports that physicists can’t seem to get straight what the measurements are of the invisible objects they invented. I’ve disproved the existence of subatomic particles here, and I demonstrated the desperate dishonesty of the scientific establishment here. Unfortunately the latter link no longer contains a working video; it’s very likely that Harvard ordered it taken down due to my bringing attention to the scandal.

Notice that, on the assumption that my thesis is correct, there is no longer any mystery about the measurement of protons, because there are no protons to measure.

We should be very grateful for the work of Ingo Sick and John Arrington, the whistle-blowers cited in the Nature piece.

Update: Hamish Johnston has written a nice piece on how the Nature report radically undermines quantum mechanics.

We will become immortal jellyfish

Just yesterday the New York Times published a study about a species of animal that has achieved immortality. Or, we should say, inertial immortality. Inertial immortality is defined by me as follows: an object is inertially immortal if but only if it exists and will continue to exist unless another object destroys it. Many non-living objects have inertial immortality, e.g. plastic bottles, plastic bins, nuclear missiles, and so on. However, very few living objects have inertial immortality. Alex Chiu claims that he is one such object, but these claims have yet to be reviewed by the FDA or NASA.

Unfortunately, the author of the study, professional tweeter Nathanial Rich, falsely claims that this discovery has “barely registered outside the academic world.” Yet I, who am (proudly) outside the academic world, clearly registered the discovery back when Nathan was in diapers. It would have been nice to receive credit in Rich’s article.

I will end with a striking quote from the leading jellyfish scientist, Shin Kubota, who triples as a professional web designer and singer-songwriter.

Turritopsis application for human beings is the most wonderful dream of mankind. … Once we determine how the jellyfish rejuvenates itself, we should achieve very great things. My opinion is that we will evolve and become immortal ourselves.

Strike that. I will in fact end with an emotional video (different from the one in the singer-songwriter link) of Dr. Kubota presenting his research lyrically.