Category Archives: Analysis

Real and complex analysis, functional analysis, PDE.

Noncommutative probability I: motivations and examples

Posted on 21/05/2018 by lamwk

In many areas of mathematics, one can usually assign a commutative algebraic structure to the “space” that one is studying. For example, instead of studying an (affine) algebraic variety, one can study algebraic functions on that variety, which gives rise … Continue reading →

Posted in Functional analysis, Probability | Leave a comment

A simple proof of the Gauss-Bonnet theorem for geodesic ball

Posted on 26/02/2018 by KKK

In this short note, we will give a simple proof of the Gauss-Bonnet theorem for a geodesic ball on a surface. The only prerequisite is the first variation formula and some knowledge of Jacobi field (second variation formula), in particular … Continue reading →

Posted in Calculus, Differential equations, Differential geometry, Geometry | Leave a comment

Zeros of random polynomials

Posted on 21/11/2017 by lamwk

Given a polynomial , where the coefficients are random, what can we say about the distribution of the roots (on )? Of course, it would depend on what “random” means. Here, “random” means that the sequence is an i.i.d. sequence … Continue reading →

Posted in Algebra, Complex analysis, Potential theory, Probability | Leave a comment

Euler’s formula e^ix = cos x + i sin x: a geometric approach

Posted on 08/11/2017 by KKK

Today I mentioned the famous Euler’s formula briefly in my calculus class (when discussing hyperbolic functions, lecture notes here): where is a solution to (usually denoted by “”, but indeed there is no single-valued square root for complex numbers, or … Continue reading →

Posted in Analysis, Calculus, Complex analysis, Geometry | Leave a comment

An inequality for functions on the plane

Posted on 29/09/2017 by KKK

I accidentally came across a curious inequality for functions of two variables. I would like to know if this inequality is a special case of a more general result but I was unable to find a reference. It would also … Continue reading →

Posted in Analysis, Geometry, Inequalities | Leave a comment

Weighted isoperimetric inequalities in warped product manifolds

Posted on 14/09/2017 by KKK

1. Introduction The classical isoperimetric inequality on the plane states that for a simple closed curve on , we have , where is the length of the curve and is the area of the region enclosed by it. The equality … Continue reading →

Posted in Analysis, Calculus, Differential equations, Functional analysis, Geometry, Inequalities | 2 Comments

Faber-Krahn inequality

Posted on 09/09/2017 by KKK

I record a proof of the Faber-Krahn inequality here, mainly for my own benefit. Let be one of the standard space forms: the Euclidean space , the unit sphere , or the hyperbolic space . Suppose is a bounded domain … Continue reading →

Posted in Analysis, Differential equations, Functional analysis, Geometry, Inequalities | Leave a comment

Why is a² + b² ≥ 2ab ?

Posted on 14/06/2017 by KKK

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 | 1 Comment

The Cauchy-Schwarz inequality and the Lagrange identity

Posted on 20/05/2017 by KKK

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 | 1 Comment

On the existence of a metric compatible with a given connection

Posted on 05/12/2016 by KKK

Question: Suppose we are given a torsion-free (i.e. the torsion tensor vanishes) affine connection on a smooth connected manifold . Does there exist a Riemannian metric such that its Levi-Civita connection is ? If so, is it unique if we … Continue reading →

Posted in Differential equations, Geometry | 2 Comments

Weighted Hsiung-Minkowski formulas and rigidity of umbilic hypersurfaces

Posted on 04/09/2016 by KKK

1. Motivation and Main Results A. D. Alexandrov [Ale1956], [Ale1962] proved that the only closed hypersurfaces of constant (higher order) mean curvature embedded in are round hyperspheres. The embeddedness assumption is essential. For instance, admits immersed tori with constant mean … Continue reading →

Posted in Calculus, General Relativity, Geometry, Inequalities | Leave a comment