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location Palo Alto, CA
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May
25
accepted Representations of a cyclic group of order p over a field of characteristic p?
May
25
comment Qualms about the axioms of probability
When I saw, for example, the "group axioms", I didn't really think of them as "axioms", but more as the definition of a group. I guess in this case, it makes sense to think of the "probability axioms" more as the definition of a probability.
May
24
comment Qualms about the axioms of probability
Pete, that's a great question. Maybe "discrete-ly"? I'm taking an intro probability course intended for computer science undergrads, and I got the axioms from Ross's text, "A First Course in Probability".
May
24
asked Qualms about the axioms of probability
May
24
asked Flip a coin until a head comes up. Why is “infinitely many tails” an event we need to consider?
Apr
20
comment Representations of a cyclic group of order p over a field of characteristic p?
This is really cool! I think I see how to make this work when $K = \mathbb{F}_p$, but what if $K$ is not a finite field (so that $V$ contains infinitely many vectors)?
Apr
19
asked Representations of a cyclic group of order p over a field of characteristic p?
Mar
2
accepted Why can we think of the second fundamental form as a Hessian matrix?
Feb
28
answered Why can we think of the second fundamental form as a Hessian matrix?
Feb
28
comment Why can we think of the second fundamental form as a Hessian matrix?
Oh thanks, that makes sense: when we have a parametrization of the form $f(u, v) = (u, v, h(u, v))$, the Hessian evaluated at h(0, 0) is precisely the second fundamental form at f(0, 0). I'm not sure how to generalize this to an arbitrary point of an arbitrary parametrization, but that's my own fault for not knowing the Implicit Function Theorem...
Feb
27
comment Why can we think of the second fundamental form as a Hessian matrix?
I think the interpretation is that their matrices are the same, i.e. we think of the matrix element $(h_{ij})$ of the fundamental form as the second derivative $\partial^2 h / \partial u_i \partial u_j$ of some function $h$. Also, I'm not sure if there are different ways to define the second fundamental form, but in Kuhnel's text, it's defined as a bilinear form on the tangent space: $II(X, Y) = \langle LX, Y\rangle$ where $L: T_pM \rightarrow T_pM$ is the shape operator.
Feb
27
asked Why can we think of the second fundamental form as a Hessian matrix?
Feb
26
comment Smooth curve with no Frenet frame
Are you sure that this can be done with a smooth curve? Also, I think it might be important to mention that $X_1, \ldots, X_n$ are smooth vector fields along $\gamma$...
Feb
26
comment Understanding the intuition behind math
I think this is a hard question to answer in general (though I also think it's definitely worth trying to answer). On the other hand, are there any particular concepts for which you're seeking intuition?
Feb
26
accepted Eigenvectors of real symmetric matrices?
Feb
26
asked Eigenvectors of real symmetric matrices?
Feb
22
awarded  Popular Question
Feb
10
awarded  Citizen Patrol
Jan
29
awarded  Nice Question
Jan
29
comment Why should we care about groups at all?
So if I'm not convinced that groups are important, the way to convince myself would be to learn more math!