Reputation
4,110
Next privilege 5,000 Rep.
Approve tag wiki edits
Badges
2 11 30
Impact
~54k people reached

Jun
8
comment Every path has a simple “subpath”
Can the length of $f$ be infinite? If not, then such $g$ is found with very simple reparameterization argument.
Jun
8
comment Set {1,1} = Set {1}, origin of this convention
If $A=\{1,1,2,3\}$ and $B=\{1,2,3\}$, then $A\subset B$ (any element you take from $A$ is in $B$) and $B\subset A$ implies $A=B$. A good book that I could recommend for reading for related topics is Paul Halmos; Naive Set Theory. It starts building the axioms of set theory very nicely.
Jun
8
comment Is it generally accepted that if you throw a dart at a number line you will NEVER hit a rational number?
@StefanWalter: This doesn't really change anything. Either $C$ is an event of neither tails nor heads (e.g. wind blew the coin away) in which case, again, if $C$ is zero measurable it does not mean that it can not happen. It only says that tails or heads are gotten almost surely. On the hand, if you consider no other options than tail or heads, then $C$ is empty. For the r.v. that takes two values, any set not containing them has empty preimage. In fair coin tossing there are no non-empty zero measurable sets separating $A$ and $B$.
Jun
7
comment Is it generally accepted that if you throw a dart at a number line you will NEVER hit a rational number?
@StefanWalter: It really depends on how you define the word impossible. If impossible means that it can not happen, then clearly this does not coincide with the concept of probability zero and it would indeed be wrong/misguided to say that it would.
Jun
7
comment Is $\lim\limits_{k\to\infty}\sum\limits_{n=k+1}^{2k}{\frac{1}{n}} = 0$?
@QiaochuYuan: what sum?
Jun
7
comment Weak a.s. convergence VS a.s.weak convergence
@NateEldredge. thanks for clearing this:)
Jun
7
comment Weak a.s. convergence VS a.s.weak convergence
What measure is $\mathbb{P}$?
Jun
7
comment Is it generally accepted that if you throw a dart at a number line you will NEVER hit a rational number?
Hitting any individual number has probability zero. So with this logic we will never hit any point when throwing a dart. Can you see the flaw in this? Eventhough rationals cumulate zero mass on the real line, this doesn't mean that they don't exist.
Jun
7
comment Unit speed geodesics
@WillieWong: Thanks.
Jun
7
comment Condition on function $f:\mathbb{R}\rightarrow \mathbb{R}$ so that $(a,b)\mapsto | f(a) - f(b)|$ generates a metric on $\mathbb{R}$
@copper.hat: You're probably assuming that $f$ is in addition continuous?
Jun
6
comment Unit speed geodesics
Thanks. I knew there existed a simple way to see this!:)
Jun
6
comment Unit speed geodesics
@Thomas: You're probably right. If $d(x,y)=\inf_{\gamma\in\mathscr{D}(x,y)}L_{d}(\gamma)$ for all $x,y\in X$: where $L_{d}$ is the length-function induced by $d$ and $\mathscr{D}(x,y)$ is the collection of those $\gamma\in X^{[0,1]}$ that are continuous, $\gamma(0)=x$ and $\gamma(1)=y$ and $L_{d}(\gamma)<\infty$ - we say that $(X,d)$ is a length space. In other words, if the distance of any pair of points is obtained as infimum over the lengths of rectifiable paths that join the points.
Jun
6
accepted Unit speed geodesics
Jun
6
revised Unit speed geodesics
added tag
Jun
6
accepted Compactly supported function?
Jun
6
asked Unit speed geodesics
Jun
5
awarded  Civic Duty
Jun
5
comment Cross product of paths is a path with length?
By saying that $a(t)$ is a path with length in $[c,d]$, do you mean that the length of $a(t)$ is a real between $c$ and $d$?
Jun
5
comment Is Koch snowflake a continuous curve?
If the convergence is uniform, then a limit of continuous functions is continuous.
Jun
5
comment (Un-)Countable union of open sets
@DaveL.Renfro. True, I didn't somehow even think about that :-) So $T_{1}$ would be the least requirement for this construction.