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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: Linear Algebra
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 CauchySchwarz inequality and the Lagrange identity
The classical Lagrange identity is the following: This can be proven by expanding and separating the terms into the crossterms part and the non crossterms part. The Lagrange identity implies the CauchySchwarz inequality in . And when , this can … Continue reading
Posted in Algebra, Group theory, Inequalities, Linear Algebra
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Why a vector field rotates about its curl?
Let be a smooth vector field on . We define its curl to be the vector field It is often claimed in advanced calculus that measures how fast “rotates”. More specifically, suppose , then locally around , the vector field … Continue reading
Posted in Calculus, Linear Algebra
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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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Three by three skewsymmetric matrix
This is just a short remark about 3 x 3 skewsymmetricmatrix. Theorem. If is nonzero and skewsymmetric, then two of its singular values are nonzero and equal, while the other one is zero. Proof. … 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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SLemma
This post aims at proving a standard result in quadratic programming called SLemma. The use of this result will be obvious in a future post. We begin with a lemma. Lemma 1. Let be two … Continue reading
Posted in Applied mathematics, Linear Algebra, Optimization
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