Gauss's Theorema Egregium
Gauss's theorema egregium states that the
Gaussian curvature of a surface embedded in three-space may be understood
intrinsically to that surface. "Residents" of the surface may observe the
Gaussian curvature of the surface without ever venturing into full
three-dimensional space; they can observe the curvature of the surface they
live in without even knowing about the three-dimensional space in which they
are embedded.
In particular, Gaussian curvature can be measured by checking how closely
the arc lengths of circles of small radii correspond to what they should be
in Euclidean space, . If the arc length of circles tends to be smaller than
what is expected in Euclidean space, then the space is positively curved; if
larger, negatively; if the same, 0 Gaussian curvature.
Gauss (effectively) expressed the theorema egregium by saying that the
Gaussian curvature at a point is given by -R(v,w)v,w where R is the Riemann
tensor, and v and w are an orthonormal basis for the tangent space.
Excerpt from MathWorld
Absolute Minimum
Well, so I'm back here to this abandoned place. I am uncertain if I should post this. But maybe just for the record. I guess I have reached the lowest point that I can remember in recent memory. Glad to have friends that helped me through the night. I guess things just happened so quickly that I cannot follow. Looking back though, I think I did the right thing. Perhaps its the most noble thing I could do. I certainly don't like it. Nonetheless, it was a rational decision. As much as I have lost, I hope I still upheld my principles.
I hope one day, when I visit this abandoned place again. I could look back to this whole experience not as a minimum point in my life, but as an inflection point towards greater good.
State of Disorder
It seems to be an unusual time for me to revisit this abandon place. I guess I just need a moment to write a couple of lines. Haha without going into the detail, I really need to find method in this chaos. Blah. =|
3 * 7 = 21
So i'm 21 now. Yeah.
March Madness!
Haha well it's march madness time. Really have been spending too much time watching NCAA and playing computer games. =|
With 3 weeks left only, I better gather myself and stay focus again. Another school year already... =)
A monotonically decreasing function
Haha, well that's probably the best description of my current state. I don't know if I am putting things in the right perspective, it seems to be quite a difficult time. I guess it's probably good to face some difficulties. But whatever the case, I can't dwell on this too much, there are still much to be done. Hopefully I can turn things around soon. =)
As epsilon goes to zero
Well, about time to conclude my two weeks break and perhaps the semester that just passed. I actually know not of what to write. I wonder if I have really put myself to the test or the test is flawed itself. Perhaps I will never know. It takes a lot of confidence to stay in my condition after so long, but perhaps I have too much of it. Like the irony of larger stars burns faster for example.
I do not know of what the next step will be, but there are always choices to be made. There are times where I have doubts of my choice of path thus far. But maybe I have moved to0 far, or maybe I just deep down wants to stay on course. Whatever the case, I offer no resolution.
It seems I have reached the point where this entry shall end meaninglessly. Well I hope I can add some meaning to it in the near future. For now, as long as I know epsilon is not zero, that's all it matters.
Theorem of the Day -- 13/11/2005
Roth's Theorem
For algebraic a
|a - p/q| < 1/p^(2+epsilon)
with epsilon > 0, has finitely many solutions. Klaus Roth received a Fields medal for this result.
(Excerpt from Mathworld)
Comments:
Well, so that's what it takes to win a Fields Medal. Haha, first I have to understand what the above really means. Blah =|
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