@HenningMakholm I thought that everyone can delete his own question. Is some number of reputation posts needed for that? (Cyril is asking in comments to that question that moderator should do that for him.)
@WillihamTotland If I is the n-by-n identity matrix and A is an n-by-p matrix, then IA is equal to A. If A is p-by-n, then AI is equal to A. In general, as long as IA or AI is defined, it will be equal to A.
@HenningMakholm We should celebrate. Open the champagne! =)
@HenningMakholm Well, if it is a synonym for elementary NT, then it cannot cover questions from logic or set theory, no? In particular, questions pertaining to the definition of natural numbers?
"Questions on congruences, linear diophantine equations, greatest common divisor, divisibility, etc." -- At least the tag wiki summary doesn't cover this definition of natural numbers part...
@HenningMakholm Oh, I thought Algol (but not sure), but didn't know about mathematical physics used it. Thanks.
No, but I'd say that the definition is even more elementary than those things.
@Srivatsan It's not that I have actually seen any pre-Algol math/physics text use it. But I have seen physics texts that follow a long expression with " =: M" and thereby give a name to it.
iirc, one of my Intro to Modern Number Theory pdfs says that there is "formal number theory" (which relates NT, and diophantine equs in particular, to computation and logic). That might be relevant but I haven't been following the discussion there.
the point is you want to go in between the group perspective and the freer algebra perspective but keep clear which you're in and when. I suppose it gets more important for derivation and conceptual understanding depending on the specific application.
It has one group literally over another instead of side-by-side, as I've always seen it. Also, \to and \cong mean two distinct things: the first is referring to some (canonical) morphism map and the lateral says they're isomorphic.
I can tell you right now it's not "just ASCII" (did you make that up on the spot?), but really my only qualm is that it's not as aesthetic when describing groups to put them in anything but inline except in the case of diagramming.
And I wouldn't say common, I would say near-universal, because I've literally never seen anything else till now.
@QED: So are you aware of any instances of writing group quotients the other way? Do you have historical knowledge that mathematicians wrote it the other way pre-ASCII or at all commonly in history?
hmm. I guess there is are real instances then. I'm still not sure about the conjecture that ASCII or print-type is the reason we have the "/" notation so overwhelmingly.
It's a first. Usually Didier comes up with the simple answer using The Law of Total Variance, but his answer looks more like mine usually do and I answered using the Law of Total Variance
@anon That is pretty amazing...
Is that native in Mma, or is that something that someone is selling?
I don't know. I just copied it over from the blog Enthusiasms (which likewise has no explanation). You could email Simen (the author) or Tineye it or something.
Oh. Well good then because I couldn't make any sense of it. This means I messed up somewhere while doing the sums. I'll just post the whole thing as a question, that'll be easier.
The canonical name for G, while not being a member of the ground model, is simply the collection of names of conditions in P. So when you interpret it by G you only take those p's which are in G, and you have G.
@ZhenLin No, the answer is different, no? In a discrete metric, it is the case that the only open balls are \emptyset and X. -- or am I confusing something?
@Srivatsan If you have the discrete metric, i.e. distance either 0 or 1 then every singleton is open because you can take an open ball of radius 1/2 and it will contain just the point (because every other point has distance 1)
But part of the "Oh, he's a logician, let's bash him." approach some folks here have, I was ridiculed for saying "There are no empty metric spaces because it makes FOL stiffer and less comfortable."
And if folks agreed to accept FOL as a basis for mathematics, perhaps they would be kind enough to accept its rules.
@AsafKaragila Can you explain? Why does FOL conflict with empty metric spaces? Is it something to do with empty metric spaces or is it do with any empty structure?
Of course you can change the way you handle things, but then you always have to require that the structure is nonempty and so and so. It's just easier to do it this way.
I think the only problem with allowing the empty model is that you no longer have soundness for the usual rules for \forall elimination and \exists introduction
@ZhenLin But for all elimination would say, you can substitute x in that sentence where x is from the structure. But I can still do it, no? Just that I cannot find any x to put in the sentence...
Ok, first take the k outside the integral: it's a harmless constant. You are left with integral of exp(-3x) between the limits ... and ... (let's come to the limits later).
What's the anti-derivative (or primitive or indefinite integral - I don't know which term you're most comfortable with) of exp(-3x)?
