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 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
 A curious identity on the median triangle
 27 lines on a smooth cubic surface
 Weighted HsiungMinkowski formulas and rigidity of umbilic hypersurfaces
 The discrete GaussBonnet theorem
 Why a vector field rotates about its curl?
 A functional inequality on the boundary of static manifolds
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Category Archives: Algebra
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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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 blowups and the other one uses the Grassmannian. If I have time I will elaborate on the discussion.
Posted in Algebraic geometry
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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
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A Fibonaccilike sequence
What’s the pattern of the following sequence? Can you guess the next term? The answer is
Posted in Algebra, Combinatorics
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High degree polynomial with few real roots
There are univariate polyomials that the number of nonreal roots is significantly larger than that of real roots. The simplest example is where is odd. It has one real root and nonreal roots. In this post we show an example … Continue reading
Posted in Algebra, Calculus, Miscellaneous
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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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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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