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 Zeros of random polynomials
 Hopf fibration double covers circle bundle of sphere
 Euler’s formula e^ix = cos x + i sin x: a geometric approach
 An inequality for functions on the plane
 Weighted isoperimetric inequalities in warped product manifolds
 FaberKrahn inequality
 Why is a² + b² ≥ 2ab ?
 A remark on the divergence theorem
 The CauchySchwarz inequality and the Lagrange identity
 On the existence of a metric compatible with a given connection
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Category Archives: Discrete Mathematics
Why is a² + b² ≥ 2ab ?
This post can be regarded as a sequel to my previous (and very ancient) post on 1+2+3+…. Though these two posts are not quite logically related, they share the same spirit (I’m asking a dumb question again). How can one … Continue reading
Posted in Calculus, Discrete Mathematics, Geometry, Inequalities, Linear Algebra, Probability
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The discrete GaussBonnet theorem
This is a slight extension of my previous note on discrete GaussBonnet theorem. As mentioned in that note, this is a generalization of the wellknown fact that the sum of the exterior angles of a polygon is always , which … Continue reading
Posted in Calculus, Combinatorics, Discrete Mathematics, Geometry, Topology
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“Useless” Circle Properties: The Green Flash In Mathematics
Since F.4 I’ve been thinking long and hard about reallife applications of circle properties. Why do we study them in secondary school anyway? Someone told me they’re used in designing cylindrical structures, but I couldn’t find a satisfactory book or website that … Continue reading
Water puzzles
Here is a typical water puzzle: You have two cups. Their capacities are 5 units and 3 units respectively. You may get water, pour water way or to another cup, but there are no marks on the cups. If you … Continue reading
Posted in Discrete Mathematics
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Stokes’ theorem on a simplicial complex
This is a note on a Stokes’ theorem on a simplicial complex. Originally I wanted to establish some formulas on a graph, it turns out that it’s better to work on a simplicial complex. After discussing with Raymond, we arrived … Continue reading
Posted in Analysis, Discrete Mathematics
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