@TedShifrin yeah, but when doing some substitutions, it is actually easier not to change names. At least, I find it so in substitutions where the limits of integration are being affected
@jcora originally, I did it in a couple of steps (in the comment above), but once I knew where I was going, I knew I wanted to map $-1$ to $a$ and $1$ to $b$. From there it is not too difficult to get $\frac{1-x}2a+\frac{1+x}2b$
I don't like how some books will show results and then prove they work without actually telling you how they thought of the particular substitution or Ansatz or whatever
just this past semester a professor taught me a perspective on how to prove things involving inequalities that really helped me
he said if you have a certain inequality, you try to "break it" by making the statement as strong as possible
which might, for example, turn a weak inequality into an equality
I never looked at it that way before
the context was something involving proving something was an inner product, i dont remember what it was exactly
"I don't like how some books will show results and then prove they work without actually telling you how they thought of the particular substitution or Ansatz or whatever" All rudin is like this
@TedShifrin I think the answer is yes, everything is linear. The best way to organize this is by fixed point sets. A cheap reduction is to use big technology: (I believe, but need to think about it) Smith theory says that $\text{rk} H^*(M^G) \leq \text{rk} H^*(M)$. Fixed point sets are submanifolds. So your fixed set is either empty, point, or $S^k$. You then organize by normal bundles of the fixed set, which have an action by the group of interest, and see if you can embed these in $S^4$.
yeah but this isn't just international, it is specifically India, and most books say "only for sale outside of the US" rather than explicitly saying "it is illegal to sell in the US"
lol the guy at proofwiki.org has been going through Knuth's exercises on binomial coefficients and putting them all up on the site, maybe you can find it there
@GFauxPas: I had a particularly annoying autistic student (presumably) in my class that got videoed. He was truly disruptive to the point where I had to yell at him to shut up. He was constantly asking off-topic questions, whether to show off or to pursue his pet topics. So I don't agree with you all the time.
But students really need to work out examples themselves. Too much time meant memorizing proofs and not wrestling with concrete examples/counterexamples.
@geocalc33: Your question makes no sense. Uniform convergence is about a sequence or series of functions.
No. He's thinking of a function build out of the Riemann zeta function $\zeta(s) = \sum n^{-s}$ ($s\in\Bbb C$, well, really, an open subset thereof), so it becomes $f(x,s)$. It simply is a two-variable function.
A parameter is a "letter" which is free to be set equal to various values. You know about parametric equations for curves or surfaces? Parameters are just variables.
You need to take a few courses like Intro to Higher Math, which begins to teach you to write proofs, and beginning abstract algebra. If you can't get solid A's in those, graduate school in math is not appropriate.
hi @amWhy. I'm just dispensing gloomy advice, as usual.
@GFauxPas: I generally advised most of my students to take algebra before analysis, because the proofs in algebra — for the most part — are not as laden with quantifiers and are therefore a little more straightforward.
@geocalc33: In a few words, I don't in this day and age cavalierly encourage people to go to grad school in math unless they have a real talent and passion. Perhaps a masters in applied math might be a reasonable choice. Regardless, your stat background and solid experience with programming would probably get you better — or comparable — jobs in the real world.
@GFauxPas: You can pffft all you want. Prospects for a career in academia grow dimmer by the year. More and more colleges and universities are hiring adjuncts at crap salaries rather than pay a somewhat reasonable salary to tenure-track faculty.
It's very hard to go into academia from industry (in mathematics). Unless one has a stellar publication record. Being removed from publications and teaching experience and the ridiculously competitive job market makes it very difficult. Far easier to try academia and leave in a huff for industry.
Physics is worse in that regard than math, Semiclassic. Math has so much service teaching to do. Everyone has to take some math. Physics, not so much. It's the premeds that keep you guys busy in the service arena.
Those of us who live and die to be dedicated teachers are in a different category, of course.
There are still small and liberal arts colleges that want dedicated teachers, but the research requirements have ramped up a bit there since my day because there's such huge supply ...
yeah. that's at least one route I should keep in mind
I feel like I'm a little better situated that some physics grads, since my research was stuff that involved paper/pencil/mathematica rather than some dedicated experimental lab or a huge colloboration
so a bit easier to sell that as being accessible to undergraduates. but there's a lot of speculation in there
@TedShifrin Absolutely, this is true. But research requirements allow for research in the teaching of mathematics, and/or mathematics education for undergrads.
@TedShifrin about teaching, one of my students who is aiming for a high school teaching degree (don't know how to say that in English), offered to give me some feedback, since she learns a lot about didactics of course in the specialized courses they take besides some bachelor classes. It was very helpful and encouraging
That's great, @Mathein. I've tried to give you some comments occasionally, too, but as you know I think you've improved and you definitely care about communicating and not being incomprehensible. :)
I am a big advocate of clever use of colored chalk. That and neat organization are both very helpful. (I got sneered at by a colleague when I spoke about this shortly after coming to UGA. Of course, he was about the worst mathematician and worst teacher on the faculty.)
my professor of topology told me about a professor he had that used to be a chemist but lost his hands in a lab accident, so he went into mathematics instead, after he learned to write on a chalkboard with artificial hands
@TedShifrin I that, too. When I write proofs, I tend to intersperse some commentary/justification for an equality/analogy/connection to other exercises etc. I'll try to use colored chalk to separate that stuff from the main proof more clearly
when my number theory professor thought a theorem was particularly important, he would say things like "this calls for colored chalk" and write the theorem in a color
So my problem with colored chalk at the moment is that it's kinda scratchy and is a pain to erase. Legend has it that Hagoromo colored chalk is better but I've never used it and also money
@quallenjäger a space of functions you can add, multiply and multiply by constants, I'd say. (Though you could have e.g. convolution instead of pointwise multiplication)
