@TedShifrin, do you know anything about uniquely geodesic spaces? John Voight has some use for them, so far I sent him some pages from Cheeger and Ebin that tell me that an oriented compact surface cannot be a UG space. See mathoverflow.net/questions/97523/… for the middle of the discussion
@WillJagy: No, I was never much of a Riemannian expert, sorry. Still remember a bit of standard graduate material and complex geometry, but my brain is dissolving :)
Oh, I'm more worried about random variables. If they're as dumb as I am, they'll have been very confused by random variables, what with they're not variables.
@TedShifrin The usual! Just waiting for the semester to start now and trying to rush through some writing I'd like to have done by the time my advisor gets back to town
but I keep getting distracted by potentially impossible problems
I've been hiding because of all of this Donald Trump nonsense. I keep telling myself no one actually likes him and I sequester myself from the world so as to never be proven wrong
@TedShifrin Yea it seems totally hopeless so far. If one tries to consider a small region, you end up with all the coefficients of your taylor series being expressed interms of the integral of the unknown function of the whole domain. Not really helpful.
My recollection is that one likes to turn diff eqns into integral eqns to apply Banach functional analysis, but one loses the local nature of some diff eqns.
@MikeMiller I'm familiar with the record and how he's cloest to Ted Cruz in voting record. But he's also basically the only GOP candidate who isn't a hawk and isn't for NSA spying.
@TedShifrin Yea we started with a diff eq. than an integral eq. then a transformed version on that integral eq. I'd LOVE to turn it back into a diff eq. to do some local analysis, but I havent figured out how yet
Well, that's not 100% true. I think I can turn it into a linear diff eq of INFINITE order, but I dont know anything about those either
@MikeMiller He's also not obviously in the pocket of big business. He sort of spurned the Koch brothers and has lost out on some high profile donors because he refuses to schmooze them a lot
@Kevin: I think I probably shouldn't have started this debate here, since a math chatroom is not a proper place for political debates. I'd be glad to continue talking if you email me.
@Ted: I'm wondering how to present random variables in my recap. The data one really has is a random variable $\Omega \to \mathbb R$, where $\Omega$ has a probability measure to start with. Books tend to avoid this and never say this out loud, then talking about probability mass functions as if they're part of the data of the random variable. Should I try to talk about $\Omega$ as if it has a "probability density" on it to start with, and the probability mass function of an RV comes from this?
I don't know what your course assumes or where it's headed, so I'm not in a good position to give advice (not that I'm an experienced teacher of this material, anyhow).
Illustrative examples are particularly good in this subject. What text are y'all using?
I know this. That's why I'm asking for advice. Where does the PDF come from if I don't give them a piece of data on the sample space that looks like, smells like, tastes like, a probability measure?
Okay, so 1 I know basically 0 about how this would work if you had a continuous distribution. And what I'm syaing clearly doesn't generalize well to that case. But does one need a mapping from ALL the measurable subsets to a probability? Isn't just a mapping from each 'individual' event plus the additivity property sufficient?
Ah, I covered most of that, I guess, but don't expect me to be an expert. BTW, @MikeM, do they emphasize indicator functions? I realized too late teaching out of Ross's book that he doesn't emphasize/use them nearly enough. Very powerful tool. Our friend @Studentmath was very good with them.
the idea that there is some variable whose value we don't know but we attribute various probabilities to its being in certain intervals is pretty intuitive. how we formalize an intuitive idea sometimes requires some technical sophistication.
@Ted: You act as if when I took probability I was a grad student. I was a sophomore in undergrad. I was not a wunderkind. Fellow classmates struggled with the same things I did.
Your math brain is different from most of your students'. You are well along in your Ph.D. and advanced for a first-year student, by far. Most of your undergraduate majors will go nowhere near graduate school.
I won't keep arguing with you. In a few years, you'll understand what I was saying.
Well lets say if you get bad marks in one of your math exams(when you are ahead of what the rest of the class is on) does that imply you are bad at math? @Ted
Many of you whiz kids are too impatient to learn computations. I personally believe that they are important and eventually help you understand theory better.
@Rememberme I don't have @Ted's experience, but my advice is to as much as possible give your full effort. Usually that means doing more practice and going over things you already know more than is fun. When I was an undergraduate I knew, even at the time, that I wasn't giving 100% and I regret that lack of focus now.
So I was thinking more about my post about what conjugation "is.". For one thing, I've simplified my explanation of matrices - using the categorical imperative to think with arrows instead of elements. (An ordered basis for V is essentially an isomorphism V->F^n or F^n->V. This is even how Wikipedia presents change-of-bases.)
I am thinking a fourth example could be orbits and stabilizers. Any orbit Gx is iso as a G-set to G/Stab(x). The "x" amounts to picking an "origin" for this orbit. One could explain vector spaces vs. affine spaces to illustrate "origins." Then Stab(gx)=gStab(x)g^-1. Any ideas on how to present this as an example of conjugation "changing our perspective"?
That's how I've taught it in group theory -- precisely like the change of basis theorem in linear algebra. You're moving a symmetry that stabilizes one element to a symmetry stabilizing another in the orbit of the first.
heya @Stan. I now live in California, so this is early :P
@TedShifrin Have any idea what physics students need to take to get into mathematical physics? I'm guessing analysis and algebra, and maybe some things that depend on the field.
On the whole, everything went smoothly, @Stan. I'm almost entirely moved in. Now have to spend a week shopping for food, staples, some furniture ... get a driver's license, renew my passport, etc., etc.
@TedShifrin Okay, let me be more specific. Suppose I wanted to get into studying theory of approximation, like pade approximants, asymptotic analysis, etc.
Ah, read Bender & Orzsag for starters. You need a bunch of analysis. No algebra in that. Algebra is relevant to some extent if you want to be on the representation theory side.
@TedShifrin Yea I've been thinking a lot about what I REALLY enjoy. And mostly it's not exactly the physics so much as having a HARD problem and wringing every piece of information out of it to understand what's going on, even if you can't fully solve the problem
Yes, it's hard for me to feel the difference. But for me personally, it feels like I'm more interested in the process than the answers. So, I've been considering whether I shouldn't be studying the process itself.
@TedShifrin At this level, I'm not sure I understand the difference anymore. Are techniques for finding approximate solutions to integral equations that arise from physics problems a math question or a physics question?
@StanShunpike Well his father left him $200 million, so that helps
If he had merely invested that money in a market index fund he'd have significantly more money today than he actually has. His investments havent done as well as the market
@PVAL That's not shocking. You would kind of need to do stuff like that to make that money if you had such a low level of intelligence as he seems to have.
@TedShifrin I have not talked to those people in years. I have seen recruitment vids from "Vector Marketing" (and probably other incarnations of this) and Trump is prominently featured. These can probably be found by a google search.
@MikeMiller I'm not sure what you mean. There are various ways graduate students can be replaced by alternatives. It seems like a market like any other to me. Also, my comment wasn't meant ot be pejorative
As a member of a discipline that is sometimes sloppy with math, I try and cut them some slack. Macroeconomics problems are VERY hard. But maybe I'm too generous
Business schools on the other hand, I've seen some downright criminal use of statistics there
it can occur either due to some object obstructing your airway, this is called obstructive, or it can occur due to a malfunction in your neurological process of breathing, this is called central sleep apnea
Well I was thinking of this question: Are there nice examples of connected metric spaces in which every open ball can be covered with disjoint compact sets
@StanShunpike Ya its quite bad. Because you're periodically suffocating you never enter the later stages of sleep. As a result your body never really rests and so you're sleepy constantly.
@StanShunpike As you weight increases the fat around your throat and lungs makes it harder to breathe. So it easier for parts of your body to obstruct your airway during sleep. There's more stuff there to get in the way and its harder to move.
@StanShunpike the good news is the treatment is harmless and effective, although its annoying. You have t sleep wearing a mask that basically hooked ot a pump that forces air into your lungs should you stop breathing.
@StanShunpike You get used ot it. It doesn't actually take that much pressure to keep your airway open. For example, people generally do it without help while awake, so you just have to simulate the state of your airway while youre awake
@Rememberme For any topological space, the singleton set of a single point is always a compact subset. So any subset of a topological space can be written as a disjoint union of singletons, hence you have your condition. You probably want to restrict something in your question.
@StanShunpike Yup, quite. Just to give you some numbers i looked up, most CPAP machines have to provide a pressure of ~ 10cmH20 to keep your ariway open. Standard air pressure is about 1000 cmH20, so its not a crazy increase in pressure. Wasnt developed until the 70s.
So can there be a connected metric space where the open balls can be covered disjoint open compact metric spaces which are not homeomorphic to $\Bbb{R}$
I guess made a mistake writing the question above . The revised question is : Can there be a connected metric space where the open balls can be covered by disjoint open compact sets which are not homeomorphic to $\Bbb{R}$
A proof for that ^^ I guess it will be my homework
@anon I don't get this notation {X} of yours ... Do you mean that if I take a $S^1\times S^1$ then an open covering for it will be $\{S^1\times S^1\}$ ? Is that what you mean to say?
