Category Archives: Applied mathematics

“Useless” Circle Properties: The Green Flash In Mathematics

Since F.4 I’ve been thinking long and hard about real-life 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

Posted in Applied mathematics, Discrete Mathematics, Geometry | Tagged , , , , , | 2 Comments

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

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AM-GM-HM Inequality: A Statistical Point of View

In this post we shall give another proof of the famous AM-GM-HM inequality: If are positive real numbers, then AM GM HM, precisely .

Posted in Applied mathematics, Calculus, Optimization, Statistics | 1 Comment

Polynomial Optimization 2: SOS and SDP

In this article we shall describe something called Gram-matrix 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

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S-Lemma

This post aims at proving a standard result in quadratic programming called S-Lemma. 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 | 4 Comments

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

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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, NP-hard and … Continue reading

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