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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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Author Archives: michaelngelo
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
Matrix multiplication as a convolution
1. The product of two power series/polynomials is The coefficients given by is sometimes called the Cauchy product. This is a convolution. 2. Let be a finite group and be the group algebra with complex coefficients. Let be two elements … Continue reading
Posted in Analysis, Fourier analysis, Linear Algebra
Tagged convolution, matrix multiplication
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From combinatorics to entropy
Let and . Then by Stirling’s formula. This is probably wellknown to people who have studied statistical physics, but some of us including myself may not be very aware of this kind of relation. I wonder if this was the … Continue reading
A meta post, and two articles
1. Is there any way to support MathJax here? It can translate directly the latex code to display math. It is supported on blogspot, I find it more convenient. 2. I believe many of us come across some very good articles … Continue reading
Posted in Dynamical system, Uncategorized
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Fourier coefficients as eigenvalues/spectrum
In this post I want to make a connection between Fourier coefficients and eigenvalues/spectrum. Let me put the claim up front: If , then is not invertible with respect to the convolution product. Please feel free to jump directly to … Continue reading
Posted in Fourier analysis, Linear Algebra
Tagged fourier coefficients, linear algebra
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