hm, well I'll leave it for now, because for some reason my book doesn't prove it anyways. I think they said they will give the proof in a later chapter
@TedShifrin I need to express the sum of products of the roots of an equation taken $i$ at a time compactly using the summation operator. I had seen them in an article and i don't think i understand them quite well.
Let $\{G_i \mid i \in I\}$ be a family of groups, then $\prod^w G_i$, the external weak direct product, is the internal weak direct product of the subgroups $\{i_k(G_k) \mid k \in I\}$, where $i_k : G_k \to \prod G_i$ is the canonical embedding.
I have already shown that the $i_k(G_k)$ are n...
Eric: I would argue that you should broaden yourself some at this point. So I would do low dim top, cuz that's stuff you might enjoy learning that's different from all the geometry/analysis.
@TedShifrin well that is true =p, ill try to figure things out more by myself , but you guys give me many exellent good ideas so thats y i want to copy them
Also, can we use the ridiculously weak bound of $\sum_{primes \; < n} log p < n log 4$ to prove Chebyshev's theorem to upper and lower obund $\pi(n)$ ?
@Balarka "This course will be an introduction to geometric and topological methods in the study of low-dimensional manifolds, especially in dimension 4." is the only thing up at this point
and minimal surfaces description reads: "Minimal surfaces have long been a central tool in geometry. I will present the general theory of existence and regularity for minimal surfaces. I will then focus on applications of the existence of minimal surfaces, namely its connection with positive scalar curvature, the Positive Mass Theorem, and classification of manifolds with positive isotropic curvature."
@TedShifrin No well basically the arguement to show $\sum_{p < n} log p < n log 4$ can be trivially adjusted to give upper bound, but the book says (a) Proving the lower bound can be also done with this method and (b) Doing (a) is trivial.
But how do you give the lower bound with this method ?
Let $\{G_i \mid i \in I\}$ be a family of groups, then $\prod^w G_i$, the external weak direct product, is the internal weak direct product of the subgroups $\{i_k(G_k) \mid k \in I\}$, where $i_k : G_k \to \prod G_i$ is the canonical embedding.
I have already shown that the $i_k(G_k)$ are n...
Well, Eric, my brain is disintegrating. I seem to be able to do precalculus ... we're getting to parametric equations and then complex numbers/algebra, and then linear algebra, so I imagine I'll be able to handle it :P
probably something to do with picking a frame that moves along the field line, and seeing that the electron mostly just does a circular orbit in this frame
@BalarkaSen Oh well actually if you clear the INMO, you can go there without giving any test (AFAIK, if you clear INMO, you get direct admission at CMI, and only interview at ISI)... but the problems given at INMO are excessively stupid and thoughtless now a days...
Well, no, then $e_1$ is tangent the curve, $e_3$ is normal to the surface, and $e_2 = e_3\times e_1$ is normal to the curve in the surface. @Semiclassic
@EricSilva: It's a very tricky, technical subject, so you need a great lecturer who knows what he's doing.
When I trained the grad students teaching calculus for the first time, I made sure they knew the fact, even if it wasn't going to be relevant in their Calc I course.
@Alessandro: Spivak has an exercise of which I'm very fond. If you have a function that has the intermediate value property and it takes on each value only finitely many times, then it must be continuous!
@EricSilva I guess my issue is this. the business of e^(-x^2) at infinity does show up if you do enough ODE stuff, and in particular it shows up in physics
but i guess it's kind of unnatural from a physical perspective but in the sense that it reveals this underlying principle in analysis that things are fucking weird it kind of is a natural thing to consider in that framework @Semi
Demonark, yeah, Eric's right. Abel, Liouville ... Differential algebra. Kaplansky wrote a book, so did a guy at Berkeley back in the 60s, but I'm blanking on his name.
Hmm, not sure why but I thought the AMS journals were already open access. It now seems that they just offer an open access option, but are subscription based. I wonder if anyone ever opts for the non-free publishing option.
@MaryStar I assume by 1,25 you mean 1 and 25 hundredths? Well, multiply 10 and 1 on the one hand, and multiply 0,0005 and 0,05 on the other, see how far apart the digits are.
Hi. I'm looking for (not very strong) sufficient conditions for $\lim_{K\to\infty}K^{-1}\sum_{i=1}^K a_{i,K}=0$, where $0\leq a_{i,K} \leq \ell < \infty$. Any suggestions?
In particular, for a start I'd like something weaker than $\sup_{i=1,\dots,K} a_{i,K}\to 0$ as $K\to\infty$
Let $\{G_i \mid i \in I\}$ be a family of groups, then $\prod^w G_i$, the external weak direct product, is the internal weak direct product of the subgroups $\{i_k(G_k) \mid k \in I\}$, where $i_k : G_k \to \prod G_i$ is the canonical embedding.
I have already shown that the $i_k(G_k)$ are n...
