Author Archives: lamwk

Zeros of random polynomials

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

A brief introduction to Gamma-convergence

1. A motivating example Consider the -Laplace equation in , where is a nonempty bounded open subset of , and on . The energy functional associated with the PDE is . For , because of convexity, has a unique minimizer … Continue reading

Posted in Differential equations | Leave a comment

Weak L^1 is not locally convex

Let be the Lebesgue measure on . Consider , the space of Lebesgue measurable functions for which there exists some constant such that for every , . The purpose of this post is to show that this is not a … Continue reading

Posted in Functional analysis | Leave a comment

A dynamical proof of Fermat’s little theorem

If you have studied number theory (even just a little bit), you should know the Fermat’s little theorem: for any positive , and for any prime number , we have . Here I will give a proof by using a … Continue reading

Posted in Dynamical system, Number Theory | 3 Comments

Poincaré recurrence theorem and its friends

The Poincaré recurrence theorem states that a dynamical system (under suitable conditions) will eventually return to a condition that is very close to the original condition. Slightly more formally, for a compact set in , if is a volume preserving … Continue reading

Posted in Dynamical system, Number Theory | Leave a comment

Mathematics behind JPEG

Many of you should have this experience: when you save a picture with many words in Paint(小畫家) as .jpg file, some ugly artifacts appear. Then you may think why is such a low quality image format so common? At the … Continue reading

Posted in Miscellaneous | Leave a comment

A dynamical proof of some Diophantine approximations

Diophantine approximations is a field in number theory which asks how good a real number can be approximated by some rational number. Quite a number of classical results are proved by using combinatorics, which is not friendly to people who … Continue reading

Posted in Dynamical system, Number Theory | 2 Comments