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Aug
27
comment Why study Algebraic Geometry?
Dear Javier, it's "Stanford" not "Standford" (re: Ravi's notes).
May
19
awarded  Yearling
Dec
2
comment Tensors constructed out of metric other than the Riemann curvature tensor
I am pretty surprised to hear this. Do you know off the top of your head if the same thing is true for Lorentzian manifolds and manifolds with other signatures? (If not don't worry, I can work through the proof you cited if need be.)
Jul
31
awarded  Citizen Patrol
Jul
31
comment On analytic functions $f,g$ with the sum of their norms is constant
This has appeared a few times: math.stackexchange.com/questions/96257/… and math.stackexchange.com/questions/81030/…. The statements are not word-for-word the same as yours, but the accepted proofs go through basically verbatim for your problem.
Jul
27
comment Probability of a winning consecutive $k$-subset out of $n$ coin flips
Thanks for this! It's nice to see how slow the growth is in practice. In fact by using principles from the theory of large deviations you can see that, almost surely, the largest admissible $k$ is $\gg \log n$. By this I mean that if $C_n$ is the random variable denoting the length of the longest consecutive "> 50% winning" subsequence in $n$ tosses, then $\lim_n C_n/\log n = \infty$ almost surely.
Jul
5
comment Writing Integrals using Differential Forms
If you haven't seen this before then think about this: $dx$, $dy$, and $dz$ aren't special to your curve/surface/whatever, they come out of your ambient space. Sometimes it's nice not to have to assume anything about an ambient space and thus define and work with forms intrinsic to the manifold. I hope this makes sense, at least on some superficial level.
Jul
5
comment Writing Integrals using Differential Forms
Careful, in the wiki article $n$ is the dimension of the manifold (and not the ambient space). So indeed $n = 1$ and $n = 2$ in your two examples. Your confusion isn't unwarranted- there is a subtlety here. Very loosely speaking, in your entire post you've been working with differential forms of the ambient space ($dx$, $dy$, $dz$), whereas in slightly more "advanced" treatments of Riemannian geometry (such as the one in the wiki article) they work with differential forms of the manifold. It's a good exercise for you to figure out how one transforms into the other.
Jul
5
comment Writing Integrals using Differential Forms
Vincent, what you're looking for is the volume form: en.wikipedia.org/wiki/Volume_form
Jun
26
awarded  Commentator
Mar
12
comment Martingale inequality related to Kolmogorov's maximal inequality
It looks like your question got overlooked. Somebody else asked the same question a couple of weeks ago. You can find my answer over there: math.stackexchange.com/questions/313415/…
Mar
5
awarded  Critic
Mar
5
comment Definition of Ito process
Oh that's just the definition to take for $\mu_2$ to make your bullet points (1) and (2) be the same process.
Mar
5
comment Definition of Ito process
I just realized the article I pointed you to is ridiculously lacking in content, my bad. Supposing your time domain is, say, $[0,T]$, both $\mu, \sigma$ are functions of the form $f : \Omega \times [0, T] \to \mathbb{R}$ such that $f(\cdot, t)$ is measurable by your filtration. Here $\Omega$ is your sample space. This allows for position dependence. If $\mu_1$ is as in your first bullet point, the $\mu_2$ in your second bullet point would be $\mu_2(\omega, t) := \mu_1(X_1(\omega), t)$. Does this help? Edit: by position I meant past/present value.
Mar
5
comment Definition of Ito process
You're oversimplifying what you read in the Wiki article in your second bullet point. The formula you pasted is not quite the formula you can find in the article. If you look at the article carefully, it says that $\mu$, $\sigma$ are predictable processes, meaning they depend on time and position. en.wikipedia.org/wiki/Predictable_process
Mar
4
awarded  Informed
Mar
2
revised Martingale and bounded stopping time
added 2 characters in body
Mar
2
answered Martingale and bounded stopping time
Jan
7
comment Is the area of intersection of convex polygons always convex?
You are welcome my friend.
Jan
3
revised Is the area of intersection of convex polygons always convex?
describe algorithm to find maximal triangle + improved formatting