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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: Applied mathematics
“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
Polynomial Optimization 3: Why do we need generalize?
It has been a while since the last post. Let us recall what we have done. We study the unconstrained polynomial optimization problem () where is a real polynomial. This problem is equivalent to () … Continue reading
Posted in Algebra, Applied mathematics, Optimization
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AMGMHM Inequality: A Statistical Point of View
In this post we shall give another proof of the famous AMGMHM inequality: If are positive real numbers, then AM GM HM, precisely .
Posted in Applied mathematics, Calculus, Optimization, Statistics
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Polynomial Optimization 2: SOS and SDP
In this article we shall describe something called Grammatrix method which can decompose a polynomial into sum of squares. The notation means is a square symmetric positive semidefinite matrix. Proposition 1. Let , be a polynomial … Continue reading
Posted in Algebra, Applied mathematics, Optimization
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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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Polynomial Optimization 1: Motivation
(Nov 28, 2013) A (constrained) polynomial minimization problem is in the form () subject to where are polynomials in with real coefficients. If and , then the above problem becomes , an unconstrained polynomial minimization problem. … Continue reading
Posted in Algebra, Applied mathematics, Optimization
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The World of Complexity
(Sept 17, 2013) Our little plan is to write a series of posts discussing a sequence of optimization problems. Before we dive into them, it is cool to learn a rigorous treatment of standard complexity notions P, NP, NPhard and … Continue reading
Posted in Applied mathematics, Optimization
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