KJW

Clever. I’m sure the Clay Mathematics Institute would like to know. 😂 My understanding is that this may have been tried but not yet arrived upon a successful solution (and the word Yet is doing a lot of heavy lifting there) And if after spending US$1 million worth of tokens, the problem has not been solved, should one continue? It seems to me that this sort of thing could be something new on which to gamble.

LOL

Has anyone given it The Riemann Hypothesis to solve? It would be interesting to see if it can be solved for less than US$1 million worth of tokens.

I don't know about "one has to account for the self-interaction of gravity", but general relativity has a somewhat peculiar form of energy conservation that is the zero covariant divergence of the energy-momentum tensor field, a form that is true in all coordinate systems of an arbitrarily curved spacetime. However, due to the restriction of covariance and the nature of the energy-momentum tensor field, unlike a conserved vector field, conservation of the energy-momentum tensor field as it is normally understood only works locally. That is, there is no integral form of the conservation of the energy-momentum tensor field that would provide a global notion of the conservation of energy-momentum. By contrast, for a conserved vector field, the generalised Stokes theorem provides the integral form from the differential form that allows the global notion of conservation.

Not all renewables, but certainly solar panels, are typically made from semi-conducting materials, so they fall into the category that Swansont has mentioned in the OP. That was my first thought, but I subsequently realised that there is a fundamental difference in that advancement of computer technology is largely based on miniaturisation whereas advancement of solar energy technology is largely based on solar cell efficiency.

IIRC, Karl Marx claimed that the slaves in the USA were better off than the workers in England. I understand his point was that the freedom that the workers in England had was a mere illusion, and that the working conditions of the workers in England were much poorer than the working conditions of the slaves in the USA. Nowadays in well-functioning democracies, government regulations curtail the excesses of capitalism, but even in these places, there are still very uneven distributions of wealth, unless a government has made a point of adopting strongly socialist policies within the otherwise capitalist system.

It is also worth noting that a wavefunction obeying the Klein-Gordon equation (a relativistic form of the Schrödinger equation) for a massive particle has a phase velocity that is greater than c. But the group velocity is the classical velocity of the particle and is less than c.

Depending on the particular music being discussed, I am a viewer of "The Daily Doug". I've watched enough of his videos that this video took me by surprise. It pleases me that he was able to escape the clutches of such a controlling religion, but it is sad that it was at the cost of a relationship with his family. I agree with what he said about capitalism - that for it to work, it requires both winners and losers. And yes, I am a fan of progressive rock.

Although I have not contributed to this thread until now, I have been following it since the beginning. From the title of the thread: "The difference between Truth and Fact.", it seems to me that the category of a statement is more significant than its truth-value. Both truths and facts are true and therefore are not distinguished by this. What it means for a statement to be true is itself an interesting question, and it's not clear to me that it can be rigorously defined, particularly from a mathematical standpoint. In this thread, some have spoken of the conditional nature of a truth or fact. I tend to think that if a qualifying condition is not explicitly stated, then the currency condition becomes implicitly assumed. I tend to regard facts as the atoms of truths. Facts tend to be specific, truths tend to be general. Facts tend not to be open to interpretation, truths tend to be open to interpretation. Facts tend to say what is or what happened, truths tend to say how or why. Nevertheless, I haven't been able to come up with a precise distinction between the two even though I have identified particular statements I have made as "facts", for example that the spacetime surrounding the earth is curved. In this particular case, I was distinguishing a statement that is the result of an observation from something that is falsifiable.

I don't recall where I saw it or when, but I do remember seeing a quote saying that a neural network is the second-best way to solve any problem. The above diagram kind of reminds me of what a second-best solution would look like.

Regardless of the technology, the laws of thermodynamics still apply.

You may also be interested in the concept of the fractional Fourier transform. Actually, there is a whole bunch of discrete concepts that may be able to be made fractional. Recently, I posted the following in another thread that may be of interest to you:

Today, I learned that one can place LaTeX in the title of a thread. Well done, @Markus Hanke ! 👍

What's a "multifractal"?

Actually, that's a very good point, and fits in with a pet peeve of mine concerning [math]\sqrt{-\mathfrak{g}}[/math] in relativity texts. I think it should be [math]\sqrt{\mathfrak{g}}[/math]. That is, four-dimensional volumes in spacetime are imaginary-valued.