e

It isn't doable for equations of higher degree.

What would you do in physics if you're God at the beginning of the universe and you could tweak all the constants around the Big Bang? What would you do to change that fact? What would you do to change the digits of E? There's nothing you can do, right? These facts, the fact that the prime numbers have the certain properties and certain symmetry groups, these things, you can't just fire all the mathematicians and hope the physics department discovers these things.

You can't tweak anything in the universe, in the physical universe to make these things different.

So I would claim we already know that there's a space of truth that is not physics, and yet it affects what happens in the physical world in terms of biology, in terms of physics.

If you want to know something about particles, you say, well, why do the fermions do this or that? Eventually you get the answer, oh, it's because this symmetry group has this and this properties.

Okay.

If you want to know, hey, why do the cicadas come out at 13 and 17 years? Oh, so the predators don't time their emergence cycles.

Okay, but why 13 and 17? Ah, because they're prime.

But the actual numerical value of the dark energy necessary to explain the observations is bizarre.

We're used to numbers in physics like one, zero, square root of two, E, pi, things like that.

The number that popped out from the observations to explain the accelerated expansion is 10 to like the minus 120, a decimal point with 120 zeros and a one.

And we're like, how can we ever explain a number like that? Well, Lenny comes along and says, different shapes for the extra dimensions give rise to different values for the dark energy.

So this branching of universe is something that I kept thinking, maybe in the middle of night I wake up and say that, okay, what is happening here? Is there any other universe?

Yes.

And that's closest in spirit to the quantum mechanical version of the multiverse story, the one that does go directly back to Hugh Everett in 1957, because...

In quantum mechanics, as I mentioned, the best we can do is predict the probabilities of various outcomes.

Then S has a branch down here to T and T has a branch down here to you.

And another one to V call is D, E and F.

And then, you know, it's going to going to be done in a minute.

But as you know, I'm running out of letters.

So each each Q.R.S.T.

So on through Z. There's 10 of them.

Each each can be either zero or one.

So I have two to the tenth of states thousand and twenty four states of the system.

The one constraint is that these functions have to have derivatives, but this will be true for most of the functions you think to write down.

In this case, there are really only two more functions that you need to understand, e to the z and the natural log.

I want to settle in and spend some meaningful time understanding both of these, and I think you'll agree that it's time well spent.

Everybody deserves, in my opinion, at least one time in their life to experience the joy of understanding a complex logarithm.

First, though, a necessary prerequisite is to understand the complex exponential.

Regular viewers will be familiar with what it looks like to raise e to the power of a complex number, but a little review just never hurts.

We can start scaffolding the transformation just by focusing on the more familiar examples of real number inputs and the real number outputs they correspond to.

For example, e to the 0 is 1, so this point at 0 maps over to this point at 1, and then every time you let that input increase by 1, the output grows by a factor of e. meaning it runs away from us actually quite quickly.

That's just a superposition principle, so that's why there's a s- there's a sum here.

And then the signal from each target, it has what we call a spatial phase term, which is this E to the J big omega PI.

P for the index, again, for the virtual antenna, I for the target.

That's why there's double indexing here.

Better at small angles, meaning down boresight, worse at larger angles.

And also, uh, to improve your DOA resolution, you need arrays with bigger apertures.

Either more elements or space them further apart, bigger aperture, better resolution.

Okay?

So all of that is lumped into just one, uh, complex factor Alpha.

And then, um, you have basically, uh, uh, uh, a, uh, time varying term, uh, with frequency delta F, E to the J, two pi delta FT. Um, so, so, so, so that's the frequency of the IF signal delta F.

And then you have a phase term, uh, which doesn't change over time, um, but it depends on the round trip delay tau.

So we just call that E to the J, uh, phi of tau.

So this is our phase term essentially, and that's our frequency Um, or time varying term.

Okay? So what is, uh, specifically delta F is, uh, uh, we showed that there is, uh, uh, basically two, the two things contributing to the frequency of the, of the IF signal.

Okay? So we, we, we checked with some, um, you know, um, um, numbers that are actually, um, um, kind of, uh, uh, pretty conservative estimates.

So we assume a target going at 100 miles per hour that was just a few meters in front of the receiver, and receiver working at seventy-seven gigahertz.

You are going to completely miss all the nuanced human headlines that say things like, "Seismic event devastates region," or, "Tremors collapse local infrastructure."

Oh, right, because those headlines do not contain the literal string of characters E-A-R-T-H-Q-U-A-K-E.

Exactly.

Furthermore, you create a completely non-representative mathematical sample.

The machine learning model does not learn the complex linguistic structures of how disasters are reported.

The remaining pool of ninety thousand headlines has been artificially drained of all its confusing items.

You literally skimmed all the hard stuff off the top.

Exactly.