Hi all, I'd been struggling with motivation for a bit with the subject, as a first year the stuff I've seen so far hasn't really satisfied me but even just being on this site and scrolling past all the questions I can't answer has inspired me and given me back the motivation. Wanted to give thanks for that somewhere, also, if you don't mind answering, what are your favourite, more specific areas of physics? :^ )
@0celouvsky (A) That hasn't been true for years (but was for a long time), and (B) part of the reason for that is that "Converting an integer to a string" represent the operation in the wrong way for C++.
@BenNiehoff your nomination for guest chat spkr has now been 2ndd by @skullpetrol, what would it take to convince you? =D physics.meta.stackexchange.com/questions/7783/… ... plz consider it/ let me know; otherwise may have to resort to more extreme measures :P
Usually seen in the company of tk. Especially on the cover of books. As in tcl/tk. It was popular in the early and mid 1990 and made the easy things easy, the moderately difficult things slightly more than moderately difficult and really complicated things collapse in a tangle of unresolvable questions about which level of indirection would be used to evaluate the value of this variable in that context.
@Secret Suppose you have a metric space $X$ with two different metrics, $d_1$ and $d_2$. Suppose $d_1$ and $d_2$ have the same convergent sequences. Are the topologies induced by these metrics equal?
I think I need more information here, so by same convergent sequences, you mean given the set $S_i$ of all possible sequences in $X$ where $i=1,2$, then $S_1=S_2$ and that for every matching pair taken from each set, (e.g. any sequences $u_1 \in S_1$ and its corresponding pair $u_2 \in S_2$), they converge to the same limit?
(It seems my metric knowledge is still rather preliminary, result in so much background googling...) Since $d_1$ and $d_2$ has the same set of convergent sequences and these have the same limit, and all metric spaces are first countable, therefore $d_1=d_2$. Therefore their open balls are the same and therefore they must induce the same topology
Community!
I'm working on the following problem:
Let $X$ be a non-empty set and $d_1,d_2$ metrics on X. Show that the following conditions are equivalent:
1) $d_1$ and $d_2$ are equivalent, i.e.
$\forall x \in X, \epsilon < 0 \exists \delta >0: U_\delta^1 \subset U_\epsilon^2 \land U_\delta^2 ...
@Secret We want to show that if $U\subset X$ is open in $d_1$, then it is open in $d_2$. But it's enough to show that $C=X-U$ is closed for both.
We know it's closed for $d_1$ by definition of closed.
Now $C$ being closed means it contains all of its limit points. In a metric space, this means that any sequence $\subset C$ that converges in $X$, converges in $C$.
Quasiregular singularities are "$R_{abcd}$ OK in parallel frame" because the Riemann tensor can be easily extended along a geodesic curve in that direction, but that might not mesh together along different geodesic curves
Since they are locally extendible but not globally
@0celouvsky If the open set is the interior of a unit square, it seems naively that can be a result of countable union of open 2-balls. Similarly a unit 2-ball can be made with a countable union of open sets that are interior of squares. The topologies by these two metrics seemed to contain each other, therefore the topologies are equivalent?
Community!
I'm working on the following problem:
Let $X$ be a non-empty set and $d_1,d_2$ metrics on X. Show that the following conditions are equivalent:
1) $d_1$ and $d_2$ are equivalent, i.e.
$\forall x \in X, \epsilon < 0 \exists \delta >0: U_\delta^1 \subset U_\epsilon^2 \land U_\delta^2 ...
