I am, however, going to try to understand Bryant's paper on Chern's discoveries regarding $G_2$ and the non-existence of complex structures on $S^6$. That's differential systems stuff that I can follow :)
for intervals is infinity commonly a ( or [ ? I see a lot of (-infinity,infinity) but surely you would want to include the "infinitely last number" so it should be [-infinity,infinity]. Does it make much difference when we are talking about infinity?
@WDUK As a definition, $[a,b]$ is the set of numbers from $a$ to $b$ inclusive, $[a,\infty)$ is the set of numbers greater than or equal to $a$, etc. It's a lot of casework, which is annoying, but it's the easiest way to get to a definition.
I am proving mu is an inner product. Mu has the following condition: If vāVvāV satisfies Ī¼(v,w)=0Ī¼(v,w)=0 for all wāVwāV, then v=0vv=0v. Someone answered that this condition "tells you that the map has a trivial kernel. However, an inner product must be positive-definite, which may fail even if the kernel is reduced to the null vector." Can someone explain this?
Give an example of an open $U$ in $\Bbb R^2$ and a smooth function $w$ from $U$ to $S^1$ that is not the form of $\exp(i t)$ for a smooth $t$ from $U$ to $\Bbb R$
currently i have begun with analysis(I) and linear algebra(I), seems infested with proofs. I guess my real question is, can those proof techniques be applied to another field? (e.g. proofs by computers, or pointing to fallacies in textes)
Well, in Europe there's much more emphasis on theoretical mathematics than, say, in the US. There is, nevertheless, a lot of good applied mathematics. Take some courses in numerical analysis, cryptography, etc.
@Ali: If you can prove something about continuity/smoothness on a connected open set, you can prove it on any open set by doing it component by component. It does not need to be connected, but he should have said connected, yes.
NOOO ... @Ali. See, this is why you confuzled me. Part (a) is for intervals in $\Bbb R$. Part (b) and (c) move to $\Bbb R^2$.
@TedShifrin I mean, for me this is really alot, not necessarily hard to understand. I wonder if later you just keep handwaving your way through with lemmas you dont know anymore.
@shaihorowitz I don't think there is a nicer reason than saying that if you have two bodies and you calculate the effective potential a third body would have due to gravity and centrifugal force then there are 5 stationary points
If you want to think of the symmetry of it they appear in a nice way. L L Body1 L Body 2 L L
Those L's above and below are supposed to be above the middle L
@arctictern I really don't know how I'm supposed to do that here. if I assume that $x\in\text{The \bigcup stuff}$ I have the same problem as before with the "finiteness". It's still infinitely many $X_1\cup X_2\cup\cdots$.
@NaCl $x\in \bigcup_{n=1}^\infty \bigcap_{m=n}^\infty X_m$ if and only if there exists an $n$ for which $x\in \bigcap_{m=n}^\infty X_m$ if and only if $x$ is in all but finitely man $X_m$s
@Ted I looked at a couple questions after thinking they had interesting titles on this site. They were both random philosophy about definitions. So it's a better use of my time than that.
@Jeff also, I tried to contradict, see $4^{2n-1}-1=p\cdot q$ for any $p,q$. Then $4^{2n-1}-1=\frac{4^{2n}}{4}-\frac{4}{4}=\frac{4^{2n}-4}{4}=pq$ which clearly holds, no contradiction
@NaCl If $x\in \bigcap_{m=n}^\infty X_m$ then $x$ is in every $X_m$ except for possibly $X_1,\cdots,X_{n-1}$, which is finitely many $X_m$s. Conversely, if $x$ is in all but finitely man $X_m$, then let $X_{n-1}$ be the one it's not in with highest subscript, then $x$ is in $\bigcap_{m=n}^\infty X_m$.
@robjohn Hello Sir. Can you tell me a good book to study theta functions and related concepts such as elliptic functions, elliptic integrals, dedekind eta functions etc. in depth?
Hello could someone help me with this word problem: The capacity of a mobile telephone battery is the difference betweeen full charge and the charge at a given time x. the rate of charging is proportional to minus the capacity. Suppose that it takes one hour to go from 10% charge to half charge. How long does it take to achieve 90% charge?
The area of triangle formed by the lines
$$a_1x + b_1y + c_1z=0$$
$$a_2x + b_2y + c_2z=0$$
$$a_3x + b_3y + c_3z=0$$
Is $$\frac {\Delta^2}{2\lambda_1 \lambda_2 \lambda_3}$$
Where
$$\Delta =
\begin{vmatrix}
a_1 & b_1 & c_1 \\
a_2 & a_2 & c_2 \\
a_3 & b_3 & c_3...
