His exercise routine is somewhat variable but he usually goes for a short jog a few dozen times a day. Most often at 5 am, for whatever reason.

Since the article I also built and hooked up a treat dispenser for extra motivation.

These types of devices are fairly popular in Germany, and maybe other places. For someone like me, with very high peak daytime prices, even a small offset would be valuable, but the full price here seems to be $500, which is too much. No reason it couldn't be much cheaper, though.

If the determinant were zero, then (for basically the same reason as why you can't divide by zero) the matrix wouldn't be invertible. You wouldn't expect the polynomial to be invertible in that case, either.

If the determinant had free variables left over, then there is some combination of variables that will make it zero (e.g., if it came out to x^2-1, then it would be 0 for x=1 or -1). So maybe inversion won't fail everywhere, but it will fail in those spots, which would also lead one to think the polynomial is not invertible.

But if it's constant and non-zero, then there is no place where you can't invert the Jacobian. So it seems very plausible that this would apply to the polynomial as well. That's essentially what motivated the conjecture in the first place (leaving out some details, of course).

If so, you've probably heard of the determinant. It's a certain way of "summarizing" a matrix with one value.

The determinant in this case is of the Jacobian, which is a matrix you can construct from a multi-variable function. Each term is the partial derivative with respect to each variable (x, y, z, etc.), with one line per output variable (vector element).

The Jacobian of a polynomial function is, in general, going to be a matrix where every term is some polynomial expression. And the determinant of that will also be a complicated expression. But in some cases all the variable terms cancel out and you're left with a single constant (0 or some other value).

The conjecture says that if the Jacobian determinant is constant (i.e., all the terms cancel out), then there must be a polynomial inverse. And the key condition for an inverse is that there must not be two input points that evaluate to the same output. It's just like y=x^2. It's not invertible, because both +2 and -2 square to +4.

So if you can find a function where the Jacobian determinant is constant and also find two or more points that evaluate to the same output, then you've found a counterexample to the conjecture. And that's what's been done. And remarkably, the counterexample is pretty simple. It would be tedious but a bright high school student could verify it.

My desktop is like an actual desktop. I have my tools laid out exactly how I want them for the task I'm working on. I want to come back to them in exactly the same place when I pause for the night or weekend.

I have to clean it up occasionally, to close out things that I'm no longer working on or forgot about. But I'd rather do that incrementally than all at once. Forcing a reboot is like tipping over a physical workbench with a bunch of tools and in-progress projects on it. Just awful, and often takes hours to set back to its original position (at least browser tabs restore reliably these days).

Why not take the absolute value? Nature hates those, probably because the derivative is undefined at 0. So squaring it is.

The article mentions it already has a two-stage design, shifting bits and then bytes, so it can't be about shifting more than one bit at a time. Anyone know why?

Shouldn't that be an OR gate? Not only does the description above say "if and only if either of the original components had the state |1>", which is an OR, but the truth table listed above shows the same thing for the flag qubit.

Of course, one could say it's an AND on the |0> states, which is just De Morgan's law, but that's pretty awkward phrasing.

I'm not sure what fraction of devs actually fix things on their side, though. Once there's a driver workaround, and we've informed the devs, it's off our plate.

Performance is more of a gray area. We contact devs if there's something we can't work around, of course. And if there's something truly breaking. But for things that aren't exactly bugs, just things that could be improved, and we can improve on our own... well, we'll probably keep that for the competitive advantage.