User Fargle | chat.stackexchange.com

Fargle

Apr 13 23:58

Thanks!

Fargle

Apr 13 23:58

Demanding that on the transition maps translates to it being forced on the chart maps, so it really is equivalent. That was the lightbulb I needed (though I should go do that verification again just to be sure I've got it under my belt).

Fargle

Apr 13 23:57

Ahh! Of course.

Fargle

Apr 13 23:54

Like, I do understand that you can't place that demand on the chart maps themselves in the atlas definition, because you can't talk about smoothness before having a smooth structure; I suppose what I mean is, can you "weaken" the embedded definition to say simply that it has local homeomorphisms, but with transition maps that are diffeomorphisms? I can see that it would be inconvenient to add such bells and whistles, but does that fall out as equivalent?

Fargle

Apr 13 23:52

This might be a nonsense question, but I'll ask it anyway:
One way to define a smooth manifold is to say it's a subspace of $\Bbb R^M$ which is everywhere locally diffeomorphic to $\Bbb R^d$ for some fixed $d$. Another way is to let it be a topological manifold whose chart transition maps are smooth for any chart in its atlas.
So my question is: it's not intuitive to me that the demand that the *chart maps* be smooth in the embedded definition isn't, somehow, *stronger* than the same demand on just the *transition maps* in the general definition. Does this fall out of, say, the Whitney embe…

Fargle

Mar 9, 2024 19:04

@Jakobian You were the one in the beginning who commented that the people you've seen in your school et cetera.

Fargle

Mar 9, 2024 18:53

Also, hi folks.

Fargle

Mar 9, 2024 18:49

Here, "$x(1) = \pi$" is just a snooty and pedantic way to say $x_1 = \pi$. The nice thing about this definition is that you can still work with tuples in the usual ways you're used to, but now the idea can be extended to more than just finite index sets.

Fargle

Mar 9, 2024 18:48

Word

Fargle

Mar 9, 2024 18:48

So if you want to pick the point with coordinates $(\pi, e, \sqrt{2})$, you'd map $1 \mapsto \pi$, etc.

Fargle

Mar 9, 2024 18:47

A 3-tuple of real numbers is a function $\{1,2,3\} \to \Bbb R$

Fargle

Mar 9, 2024 18:47

Well, it'd be mapping $1 \mapsto 1$, $2 \mapsto 3$, $3 \mapsto 2$.

Fargle

Mar 9, 2024 18:45

In a way, I suppose. You're using some exterior, well-known set (like the numbers from $1$ to $m$, or the natural numbers as a whole, or whatever) to index into the set in some ordered way.

Fargle

Mar 9, 2024 18:43

You don't want to try to define the tuple as a function from x-values to x-values --- you want a tuple to be something where, if you give it a number $i$ from $1$ to $m$, it obligingly spits out $x_i$.

Fargle

Mar 9, 2024 18:41

The $x_i$ are what get pointed to by the indices $i$, so you want the function to be from $\{1,\dots,m\}$

Fargle

Mar 9, 2024 18:41

@Obliv Right --- effectively, in this framing, $x_1$ is thought of as just shorthand for the value $x(1)$, and the ordered tuple $(x_1,\dots,x_m)$ is a stand-in for the whole function.

Fargle

Mar 8, 2024 19:04

Indeed.

Fargle

Mar 8, 2024 19:02

The construction isn't exactly simple and it could detour you for some days

Fargle

Mar 8, 2024 19:02

@Obliv Learning to construct the real numbers is an edifying exercise but I'd recommend you go along with Munkres here if your goal is to progress in the book

Fargle

Mar 8, 2024 18:46

How goes it Ted?

Fargle

Mar 8, 2024 18:45

Good afternoon (or whatever it is where you are), everyone.

Fargle

Mar 6, 2024 20:16

@EE18 That would be a partial order --- total orders are nonreflexive.

Fargle

Feb 19, 2024 18:38

(Sort of a goofy model though. The set $\{x, y\}$ would work just as well >_>)

Fargle

Feb 19, 2024 18:37

It does work as a model for unordered pairs at least

Fargle

Feb 19, 2024 18:35

@EE18 Yeah --- with this definition, where you just define $p = \{\{x\},\{y\}\}$, the sets $X \times Y$ and $Y \times X$ are literally identical, but you would like them not to be the same set.

Fargle

Feb 19, 2024 18:13

@TedShifrin Great question. @Stan? ;)

Fargle

Feb 19, 2024 18:09

@TedShifrin I don't mean that it's incredible, just that, whereas the example I always held in my head of such a homeo was arctan for some reason, $\frac{d}{1+d}$ is a very much simpler one, and it falls out of a common real-world interaction.

Fargle

Feb 19, 2024 18:02

And they furthermore are exactly related by that formula rather than the min function or arctan, or whichever other you like

Fargle

Feb 19, 2024 18:01

@Jakobian I guess what I mean is that the interest-discount case is just a special case of demanding such a bound --- an arbitrary nonnegative interest rate makes sense, but discount rates are only sensible below 100%

Fargle

Feb 19, 2024 17:57

But I dunno, sometimes I find myself infodumping upon unsuspecting bystanders in my life. I am sure they love that.

Fargle

Feb 19, 2024 17:56

@TedShifrin You've got me there.

Fargle

Feb 19, 2024 17:50

I don't disagree. Just had never seen it from this angle before, and it strikes me that it could be used as a "concrete" way to introduce that map as a motivating example of a homeomorphism, for those inclined to think in terms of money.

Fargle

Feb 19, 2024 17:46

I figure --- I just didn't realize that it showed up in the context of interest/discount.

Fargle

Feb 19, 2024 17:33

Just noticed something neat while reading some stuff --- I saw $re^{i\theta} \mapsto \frac{r}{1+r}e^{i\theta}$ given as an example of a homeomorphism $\Bbb C \to $ the open unit disk, and I realized that, restricted to the positive real line, this is just the relationship between interest rate and rate of discount in compound interest!

Fargle

Feb 17, 2024 01:15

Howdy folks

Fargle

Oct 21, 2023 17:33

I dunno. Ask three math nerds, get five answers.

Fargle

Oct 21, 2023 17:30

@Jakobian Maybe it's just that this boundary is fuzzy.

Fargle

Oct 21, 2023 17:29

Thanks! It feels really, really natural. Secondary education has its problems (God only knows...) but I feel very fulfilled and energized by my job.

Fargle

Oct 21, 2023 17:28

Perhaps we'll have to find a third language in which to say "to each their own". ;)

Fargle

Oct 21, 2023 17:27

To be clear I mean only to speak to my cognitive biases. Probably I'm misusing some language as well. Being a newly minted high school teacher means I haven't had nearly the time to digest the math that I like as I have in the past...

Fargle

Oct 21, 2023 17:26

The Cauchy bit feels more topological. Having terms eventually be within neighborhoods of one another.

Fargle

Oct 21, 2023 17:22

@Jakobian This is a definite weakness of the Cauchy idea, that I grant. Then again, analysis was never really my "bag", as the kids say.

Fargle

Oct 21, 2023 17:19

Granted! But it feels less gross to my poor primate brain than the Dedekind construction at least. A case where aesthetics is a serious (and perhaps unacceptable) bias in my thinking.

Fargle

Oct 21, 2023 17:18

I personally always felt like the Cauchy sequence construction made more intuitive sense to me for that "gap-filling", but suum cuique.

Fargle

Oct 21, 2023 17:08

@Serilena To do it rigorously, it turns out, it does in fact have to get quite messy. The informal understanding works for some applications but, if we want to be sure we're doing things right, the real numbers are a very complicated object with very unintuitive properties to begin with.

Fargle

Oct 21, 2023 17:06

Heya @Ted! How goes it?

Fargle

Oct 21, 2023 17:04

And even after that many working mathematicians were uncomfortable with these notions --- but I say that to emphasize that our modern understanding is so rock solid precisely because so many of the questions are asked and answered in the literature.

Fargle

Oct 21, 2023 17:03

Well, the ensuing time after that makes things really interesting. The debate to which I refer raged mostly until the work of "rigorizing" calculus and analysis was undertaken in the 19th century by figures like Cauchy, Bolzano, Weierstrass, Riemann, and many others.

Fargle

Oct 21, 2023 16:59

@Serilena The references in the Wikipedia article would be a good place to start; it's hard to say where to get into this discussion though, because again, it ranges over two centuries. Something like Bishop Berkeley's "The Analyst" could be an example of early criticisms of notions of infinity and infinitesimals as they appeared in early calculus, but the terms of that two-century debate might be hard to understand coming from a less math-oriented background.

Fargle

Oct 21, 2023 16:52

We've, in some sense, "had this argument before".

Fargle

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