Oh lol, well, it definitely gets you thinking about different types of things
The idea is that you take a function and iterate it (there's also a continuous version where you have a one-parameter family of functions that form a semigroup)
Now, of course you could do that to anything ever, so w/e m8 who cares?
Well, first off you're often looking at certain special cases, like measure spaces and measurable functions, or, in my current case, topological spaces and continuous functions
And there are some cool things you can say about stuff. For example, the proof I just figured out for real this time is that if given a homeomorphism of a metrizable compact set with no isolated points, if there is some point $x$ such that $\{f^n(x) : n\in\mathbb{Z}\}$ is dense, then there's a point $y$ such that $\{f^n(y) : n\in\mathbb{N}\}$ is dense.
So let's say you have $X$ and $Y$ are locally compact, second countable, and Hausdorff
If for any non-empty open sets $U$ and $V$, we have that there's some $n$ such that $f^n(U) \cap V \ne \emptyset$, then there's $f$ is topologically transitive
(Topologically transitive means there's a point with a forward dense orbit)
Anyway reminder that our endgame theorem is that if you have a homeomorphism of a compact, perfect metric space $f$ such that there's a point $x$ with a dense full orbit, then there's a point $y$ with a dense forward orbit
Note that the $n_k$ are not necessarily in order, this could be a negative sequence, the point is only that $|n_k| \to \infty$
Anyway, so either infinitely many $n_k$ are positive, in which case the orbit is a subset of $O(x) \subset \overline{O^+(x)}$, so we win
Or infinitely many $n_k$ are negative. Now, given $U,V$ open subsets of $X$, you can find $i < j < 0$ such that $f^i(x)\in U$ and $f^j(x) \in V$, then $f^{j-i}(U) \cap V \ne \emptyset$, but then the above theorem rekks you
Right now, I'm working on a problem (don't spoil if you figure it out first) which is looking at the map $f(x,y) = (x+\alpha,x+y)$ on the torus (which is a homeomorphism) when $\alpha$ is irrational
Okay so there's a stronger condition called topological mixing, which is when if you gimme $U,V$ open in $X$, there's some $N$ such that $n\ge N \implies f^n(U) \cap V \ne \emptyset$
We'll get to that next section but for now I gotta figure out the torus
I hadn't gotten the chance to review them well since it was the day before the lecture and we we spent the whole time trying to figure out the $\sum \frac{1}{n^4}$ problem
It's a shame, his comments were nice, much as I was like "m8 w0t?" once he started talking physics, though I guess it's good to take this time off now that he actually has it
Because at this point I almost don't really know the subject much anymore, so I he wrote down some equation about conservation of energy and I'm like "K I'll just roll with that"
Perhaps, though I dunno, I tried physics and burned really fast. In high school the subject was just words and putting equations together when nothing was rotating and everything was going in a straight line, or at least could be approximated as such
@Eric I don't necessarily agree that historical motivation needs to be understood. In the cases it gives a better motivation than the existing in-subject motivations, I would agree.
Like, sure it's cool to just sorta know the fact that there are some physical systems moving around work the way your equations say they do, but getting a higher grip than that seems to involve some kinda physical intuition
you're missing my point with terminological arguments :p i'm just saying, there are cases where (a) there exists completely mathematical intuitions which covers up the motivations very well (b) the historical physical motivations are better than the in-math intuitions
like you shouldnt know it because it'll make your intuition better or something but knowing the history of your subject will generally be helpful in other ways
I see, so whatever you said so far does not have much to do with the mathematical point of view of the subjects. Can you elaborate why knowing the history of the subject might be helpful?
(Just to close off my point, as an example of (a) I would say eg harmonic functions - I think Laplace equation originated from physical issues (I know how it comes up in electromagnetism but surely they come up in SHM and whatnot) - but I think thinking of them as real part of holomorphic functions is a perfectly good intuition)
I dunno how much the overlap is but they're definitely not synonymous or something
@Eric I mean sure, but those connections came more after the fact, the algebra wasn't really motivated by the physics, I think more of the motivation came from number theory
At least as far as I know
But anyway, yeah I remember there was this one point when I was like
I primarily don't agree with it, like it's dank to know for people who are into it and all but knowing it deeply is not something I could justify putting as an imperative on anyone
But yeah this one time I was on Reddit and was like, I know Fourier analysis is a thing that physicists and signal people do because it makes life better or something, and I've got a vague idea that my friend is turning it into rep theory, but really tho what's the deal with it and why should I care if I'm not a physicist etc?
Eh, still don't buy it, working in the fields is different from knowing the physics, and I mean, I don't want to chase funding to the expense of doing what I'm into, which isn't really physics at this point
By sometimes are you saying in more specific things or like, a life direction? And I know you're not saying that, I'm responding to an argument that's floating around
That's fair I guess, for some reason this stuff is taking quite a lot of time. This week and whenever I'll be lecturing it'll likely take maybe nearly as long as a Soug pset
it's also written about 98% dialogue with occasional scene transitions; the dialogue is unquoted (written in --line style, like Joyce) and so you have to interpret who the characters are from context and their voices
Okay so, I'm having an iffy time figuring out why the map $(x,y) \mapsto (x+\alpha,x+y)$ on the torus (thought of as a square) has every point with a dense orbit
Like, its restriction to either coordinate is dense
But getting it on the whole square is... not going as hoped
But yeah so the way I'm thinking about this is that if you shift $y$, you shouldn't kill density, due to its dependence it's just a vertical shift by $y-y_0$, taken mod 1
Meaning it should suffice to prove that $(x_0,0)$ has a dense orbit
Oh I meant all of this in terms of forward orbit for reference, no need to worry about inverses
(I may go to sleep in the meantime since it's possible that doing this at 4AM just kinda isn't the best idea around, so if I do, see you tomorrow)
(Sorta one of those things where, you're too braindead to work, but also a bit too braindead to realize that you're too braindead to work. Such is the struggle.)
Let $x=\sqrt{2}+\sqrt{3}$. Note that $x^2=2+2\sqrt{6}+3$ and therefore $x^2-5=2\sqrt{6}$. We can square both sides of this equation and obtain $(x^2-5)^2=24$. You should now be able to show that $x$ is an algebraic number (over $\mathbb{Q}$). (In fact, it is instructive to expand this equation an...
In this answer what should be the solution for exercise2? Will the least degree be $2^n$?
user84215
10:27
From where can I learn easily MathJax to type math formulas in this chat room?
