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Doing math.


Oct
20
comment The Complex projective space is homeomorphic to the n-sphere
What definition of the complex projective space do you use?
Oct
15
comment Question about the closure of a set
@Luis: That sequence does not have a limit.
Oct
2
comment The set of rational numbers in the interval $(0,1)$ cannot be expressed as the intersection of a countable collection of open sets
@YiorgosS.Smyrlis. Got it. Thanks.
Oct
2
comment The set of rational numbers in the interval $(0,1)$ cannot be expressed as the intersection of a countable collection of open sets
@YiorgosS.Smyrlis: Why do $U_{n}$ have to be dense? The statement only says that rationals in $(0,1)$ can not be expressed as a countable intersection of open sets, not necessarily dense ones.
Sep
12
comment Why is differential geometry called differential geometry?
Usually the Riemannian metric of a smooth manifold captures it's geometric properties such as curvatures and angles. One doesn't need to integrate on the manifold to capture these notions.
May
28
comment Research Area Choice: PDE vs Optimization
What type of industry would you like to consider outside of academia? Bear in mind that PDEs are used in finance too.
May
15
comment How to prove or disprove $P(\overline A) = P(U) - P(A)$
Could you clarify what $P(A)$ stands for? And what exactly do you mean by the statement "$U-A$ implies $A\cap \overline{U}$"?
May
15
comment Show that if sup{∑|f(a)|}<∞, then {a∈A:f(a) is not zer0} is countable.
@user140794: Assume that there are infinitely many $a_{1},a_{2},...\in A$ with $f(a_{k})>\frac{1}{n}$ for all $k\in\mathbb{N}$. Then $\sup_{N\in\mathbb{N}}\sum_{k=1}^{N}f(a_{k})=\infty$, which is a contradiction to your assumption. And note for the last part: for absolutely convergent series the order of terms doesn't affect the sum.
May
13
comment Geometric meaning of symmetric connection
Have you computed both sides of $(2)$ for the Euclidean connection on $\mathbb{R}^{n}$?
May
11
comment Equivalent definitions of vector field
@soporhs. You're welcome. I'll gladly answer any questions you might have, but I advice that you try to work the details on your own at first.
May
11
comment Equivalent definitions of vector field
What definition of $TM$ are you working with? It seems that this answer only opens the usual definition of the tangent space but I can't see how you identify those two definitions of a vector field.
May
11
comment Equivalent definitions of vector field
What is your definition of $TM$?
May
6
comment $\mathbb R$ has the same cardinality of any interval
$g$ is not defined at $x=-1$.
May
2
comment $L^p$ not sequentially compact
Didn't you just show there that $\lim_{n\to\infty}\sqrt{sin(n\pi x)}= 0$ in the $L^{2}$ topology? Every subsequence of this sequence converges in $L^{2}$, so how is this a counter example?
May
1
comment Show a bijection between sets
What are your thoughts so far? Have you tried some candidates?
Apr
28
comment Is $A = f\mathbb{R^2}$ complete? $f:\mathbb{R^2}\to\mathbb{R^3}$, $f(x,y) = (x,y,x^2-y)$
@souf: Note that $f\mathbb{R}^{2}=\mathrm{graph}(g)$, where $g:\mathbb{R}^{2}\to \mathbb{R}$ as Goos defined above. So you only need to show that the graph of $g$ is a closed subset of $\mathbb{R}^{3}$ because closed subsets of complete spaces are complete.
Apr
13
comment Why do some series converge and others diverge?
@Ethan: Because in the sum you add those numbers together. If you add numbers that are large then obviously the end result will be large. If you add negligent numbers together, then end result will stay small.
Apr
13
comment Why do some series converge and others diverge?
@Ethan: I mean that the sum is always less than $1$, for any finitely many terms that you sum together. By converging fast enough to zero I mean the following. By looking at the sequences $(\frac{1}{n})_{n=1}^{\infty}$ and $(\frac{1}{2^{n}})_{n=1}^{\infty}$ you will see that both of them converge to zero, but e.g. the third term in the first sequence is $\frac{1}{3}$, whereas in the second sequence it is $\frac{1}{8}$: much less. And for any given $n\in\mathbb{N}$‚ the other sequence will always have small values than the other. The terms go faster to zero so to say.
Apr
13
comment Why do some series converge and others diverge?
@Ethan: By a simple observation you can see that whenever you add finitely many of those terms $\frac{1}{2},\frac{1}{4},\frac{1}{8},...$ the sum is always bounded above by $1$. So the infinite sum must also be bounded above by $1$. The infinite sum is defined as the limit of the partial sums. they might be diverging or converging to a specific number, that all depends on the numbers that you're adding. Usually you can think that if the terms of the sum converge "fast enough" to zero, then the corresponding sum is finite.
Apr
10
comment Using Cantor's intersection theorem
@Mitsos: It's false in general for continuous functions as well. It's true for example, always when $f$ is injective. You have to argue with what you know about the sets $F_{n}$ relationship to each other.