You understand what the word “Full” means, right? It doesn’t mean 99%.

Don’t know what to call it but coast to coast seems close enough to full for me.

https://x.com/davidmoss/status/2006255297212358686

lol

> The assumption is that the SNR of logical (error-corrected) qubits is near infinite, and that such logical qubits can be constructed from noisey physical qubits.

This is an argument I've heard before and I don't really understand it[1]. I get that you can make a logical qubit out of physical qubits and build in error correction so the logical qubit has perfect SNR, but surely if (say the number of physical qubits you need to get the nth logical qubit is O(n^2) for example, then the SNR (of the whole system) isn't near infinite it's really bad.

[1] Which may well be because I don't understand quantum mechanics ...

The really important thing is that logical qbit error decreases exponentially with error correction amount. As such, for the ~1000 qbit regime needed for factoring, the amount of error correction ends up being essentially a constant factor (~1000x physical to logical). As long as you can build enough "decent" quality physical qbits and connect them, you can get near perfect logical qbits.

Having demonstrated error correction, some incremental improvements can now be made to make it more repeatable and with better characteristics.

The hard problem then remains how to connect those qubits at scale. Using a coaxial cable for each qubit is impractical; some form of multiplexing is needed. This, in turn, causes qubits to decohere while waiting for their control signal.

Everyone knows that it all hinges on why they’re being Gale. If they’re doing it so they can romance Shadowheart then it’s permissable.

You can romance Shadowheart as Laezel if you want and they hate each other at the start of the game. Don’t need Gale for that. You can “win” in act 1 with Gale though.

I have no idea what the normal process is but I have never been paid for any peer review I've ever done and none of those was for an open access publication.

Teams is straight up broken on web and in its native client. Not sure it’s fair to blame firefox for that.

This is completely untrue in my experience. I use firfox exclusively on my personal laptop and have done exclusively for years. I don’t even have chromium installed.

I can’t remember the last time a website was unusable on firefox. It’s certainly not common.

The definition of a subsequence is if you have a(n) as a sequence of real numbers and n_1 < n_2 <n_3 < ... is an increasing sequence of integers then

a(n_1), a(n_2), a(n_3), ... is a subsequence of a_n and is denoted a(n_k).

So the indexes don't need to be contiguous, just increasing.

So in your example 2, 1, 1/2, 1/3, ... is a decreasing subsequence.

edit: changed to using function-style notation because the nested subscript notation looks confusing in ascii

Thanks. I was thinking subsequence ~ substring but that’s a false analogy apparently!

Yeah it’s a bit confusing. It’s also confusing when you see them written because they’re actually written usually with a nested subscript. Like

   a
    n
     k

With the k smaller than the n which is in turn smaller than the a. Sequences of all kinds are just a function from the integers to the reals so I don’t know why we had to go and invent a whole new notation for them just to be extra obtuse.

To be fair he can't imagine many other aspects of what it is like to be a normal human being.

You don’t seem to appreciate: they paid for the ballroom. They have a right to set policy. That’s how an oligarchy works

That’s how campaign contributions have worked for a long time. Now it’s just a touch more blatant.

The rest of the world has always called it corruption.

> “…an abelian group is both associative and commutative…”

If something is not associative it is not a group. An abelian group is a group which is commutative.

So...an abelian group is both associative (because it's a group) and commutative (because it's abelian), which is exactly what the OP said? It sounds like you're disagreeing about something, but I'm not clear what your objection is.

I’m not disagreeing. I’m pointing out that in TFA it sounds as associativity is a property of abelian groups specifically whereas it as a property of all groups in general. In that sense it’s not wrong, just the emphasis is a bit misleading.

If you look in an abstract algebra textbook they all basically say the same definition for abelian groups (eg in Hien)

> “A group G is called abelian if its operation is commutative ie for all g, h in G, we have gh = hg”.

In an abstract algebra textbook, they define groups first and then abelian as a property that some groups have. Here, the author is defining abelian groups "from scratch" and doesn't have an earlier definition of groups to lean on.

In more advanced texts, they could simply say that a group is a moniod with inverses and could (by your reasoning, should) avoid specifying that groups are associative since this is a property of all monoids.

Well if I check such a book that takes a category-theoretic approach to teaching abstract algebra (Aluffi “Algebra Chapter 0”), he says the following:

   > “ A semigroup is a set endowed with an associative operation; a monoid is a semigroup with an identity element. Thus a group is a monoid in which every element has an inverse”.

So according to Aluffi at least, the operation of a monoid is also associative. As you can see he does in fact also remove the associativity criterion from the description of a group by defining it in terms of a monoid. So he’s consistent with me at least.

Right. And so is the article. When you are introducing an object you need to specify its properties, _including_those_it_inherits from objects you haven't defined.

If I haven't defined mammals, I say that bats are warm blooded animals that produce milk for their young, etc., but if I have (or expect my readers to know what a mammal is) I can just say they are mammals.