Category Archives: Algebra

The Cauchy-Schwarz inequality and the Lagrange identity

The classical Lagrange identity is the following: This can be proven by expanding and separating the terms into the cross-terms part and the non cross-terms part. The Lagrange identity implies the Cauchy-Schwarz inequality in . And when , this can … Continue reading

Posted in Algebra, Group theory, Inequalities, Linear Algebra | Leave a comment

27 lines on a smooth cubic surface

Here describes two different proofs of a general smooth cubic surface containing exactly 27 lines. One approach uses blow-ups and the other one uses the Grassmannian. If I have time I will elaborate on the discussion.  

Posted in Algebraic geometry | Leave a comment

Toric perspective 1

This series of posts will be about toric varieties. The author is not sure if there is a part 2, but he still calls this part 1. This post is about computing the dimensions and degrees of popular toric varieties. … Continue reading

Posted in Algebraic geometry, Combinatorics | 1 Comment

A Fibonacci-like sequence

What’s the pattern of the following sequence? Can you guess the next term? The answer is

Posted in Algebra, Combinatorics | Leave a comment

High degree polynomial with few real roots

There are univariate polyomials that the number of non-real roots is significantly larger than that of real roots. The simplest example is where is odd. It has one real root and non-real roots. In this post we show an example … Continue reading

Posted in Algebra, Calculus, Miscellaneous | Leave a comment

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 | Leave a 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

Posted in Algebra, Applied mathematics, Optimization | Leave a comment