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Dec
12
awarded  general-topology
Dec
10
awarded  Caucus
Nov
22
awarded  Enlightened
Nov
22
awarded  Nice Answer
Nov
22
awarded  Enlightened
Nov
22
awarded  Nice Answer
Nov
21
comment Does Nash's theorem allow an embedded representation of the Riemann tensor without loss of generality?
Yes and no. Yes: it is crutch that can be leaned on if all else fails. No: the great discovery of Gauss was that curvature can be described intrinsically, so philosophically extrinsic descriptions of Riemann curvature should be in subservient roles. // The Riemann tensor measures non-commutativity of covariant derivatives. So yes, it manifests itself as unavailability of "flat" embeddings, and this picture can be an auxiliary guide. But emphasis is better placed on angle defects and things like that.
Nov
19
comment How to draw greek letters on paper / blackboard?
mathoverflow.net/a/5968/3948
Nov
18
comment Convolution of convex polygons and a Gaussian
As a final comment though: if you are doing numerical convolution, you don't do "hundreds of lookups and operators" for "each pixel". You store the graph of the original function $f$ as a giant vector, and convolution is simply achieved by matrix multiplication against a good kernel. Sure the dimensions of your image may be large, but the last I checked, vector addition and multiplication is not that slow. (Your kernel is essentially fixed from the get-go and should be prepopulated...)
Nov
18
comment Convolution of convex polygons and a Gaussian
Well, good luck with that. For the rectangles you are using heavily that the sides are parallel to the axes. The Fourier transform for a general triangle is a complete mess. You are unlikely to get good closed form formulas that apply universally. What you probably want is to divide into cases where the polygon is small/big compared to the Gaussian function. In the small case carry a table of precomputed values for a large collection of polygons. In the big case build your approximations based on pre-computed values for infinite pie-wedges.
Nov
18
comment Convolution of convex polygons and a Gaussian
If this is for computer graphics, why do you actually need the explicit solution? Why not just convolve in physical space numerically? If you just need the numerical value that would be much, much quicker.
Nov
18
revised Find point after Kth turn
deleted 51 characters in body
Nov
17
revised Ahlfors regular
added 17 characters in body
Nov
17
answered Does Nash's theorem allow an embedded representation of the Riemann tensor without loss of generality?
Nov
14
comment Besov spaces---concrete description of spatial inhomogeneity
@Michael: no. I am pretty sure not. I want the Littlewood Paley pieces in $L^2$, measured with $\alpha$ derivatives, summed in small $\ell_q$ to converge. So $q$ goes on the outside.
Nov
14
revised Besov spaces---concrete description of spatial inhomogeneity
edited body
Nov
12
revised “sup” in an equation
edited tags
Nov
11
answered Sectional Curvature of Paraboloid
Nov
11
comment Sectional Curvature of Paraboloid
And looking at it slightly more carefully, the expression for $R_{1212}$ is also wrong. So is his computation of $g$. Manifestly based on the $g_{11}$ and $g_{22}$ he showed, $g \neq r^2$. I think you should just avoid using that particular website altogether, especially considering that the correct answer can be found on Wikipedia.
Nov
11
comment Sectional Curvature of Paraboloid
There's also a typo in the linked "solution" for $\Gamma^1_{11}$ and $\Gamma^1_{22}$. The denominator should have $1+a^2 r^2$ and not $1 + a^2 r$ as written.