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 Why is a² + b² ≥ 2ab ?
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 On the existence of a metric compatible with a given connection
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 27 lines on a smooth cubic surface
 Weighted HsiungMinkowski formulas and rigidity of umbilic hypersurfaces
 The discrete GaussBonnet theorem
 Why a vector field rotates about its curl?
 A functional inequality on the boundary of static manifolds
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 Mean value properties for harmonic functions on Riemannian manifolds
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 Why does a mirror reverse left and right, but not top and bottom?
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Category Archives: Miscellaneous
A discussion forum
In order to facilitate discussion, I have set up a discussion forum here: http://cuhkmath.606h.net/phpBB3/viewforum.php?f=2 One of the most effective way of learning mathematics is by discussion. While there are excellent sites like MathOverflow or StackExchange where you may even find … Continue reading
Posted in Miscellaneous
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Why does a mirror reverse left and right, but not top and bottom?
If you stand in front of a mirror and raise your right hand, it appears in the mirror that you raise your left hand. Why does a mirror reverse left and right, but not top and bottom? For some of … Continue reading
Posted in Geometry, Linear Algebra, Miscellaneous
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The 10th Taiwan Geometry Symposium
This is not really a “mathematical” post. Since this is the first symposium I’ve ever coorganized, I am supposed to advertise it (a bit). So if you happen to be in Taiwan, and are interested in geometry (OK, I know … Continue reading
High degree polynomial with few real roots
There are univariate polyomials that the number of nonreal roots is significantly larger than that of real roots. The simplest example is where is odd. It has one real root and nonreal roots. In this post we show an example … Continue reading
Posted in Algebra, Calculus, Miscellaneous
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Two little remarks on sphere
1. “Take a sphere of radius in dimensions, large; then most points inside the sphere are in fact very close to the surface.” David Ruelle Let and . Let The fraction of the volume of to the volume of is … Continue reading
Mathematics behind JPEG
Many of you should have this experience: when you save a picture with many words in Paint(小畫家) as .jpg file, some ugly artifacts appear. Then you may think why is such a low quality image format so common? At the … Continue reading
Posted in Miscellaneous
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On integrating products of dot products
Today KKK showed me the following formula: There exists a constant such that for all . Here the integral is either over the unit sphere or over the ball (of course the corresponding constants are different). In this note we … Continue reading
Posted in Miscellaneous
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