Author Archives: lamwk

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

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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

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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

Arithmetic progression in some subsets of $\mathbb{Z}_N$ (Part 2)

Last time I have mentioned the idea of the proof (of theorem 2). Now I continue and give a full detail here.

Posted in Fourier analysis, Number Theory | 1 Comment