Page 35 - Undergraduate Research Opportunities Program Proceedings 2017-18

School of Science
Department of Mathematics

Potential effects of meteorological change, transport of pollutants, or primary pollutants on PM2.5 concentrations on
PM2.5 levels in Hong Kong are investigated based on the Kolmogorov-Zurbenko filter (KZ filter). Meteorological impacts
on PM2.5 was studied using a joint analysis of KZ filter and linear regression model. Influences of meteorological variables
on PM2.5 are high except in Central. The short-term variations of PM2.5 were studied with wind direction. It reveals that
PM2.5 showed strong increasing tendency with northern or northwestern wind direction. The long-term variations were
explored with normalized long-term trend of NO2, SO2 and CO. PM2.5 generally shows similar pattern with primary
pollutants. This paper will be useful for Hong Kong Government regarding PM2.5 policies.

Integro-differential Equations: Theory and Applications

Supervisor: JIN Tianling / MATH
Student: HUNG Chun Kit / MATH Course: UROP1100, Summer

s,p
s,2
In this report, we first study the fractional Sobolev Space W (Ω), in particular, W (Ω) for any s ∈ (0,1), which is a
Banach space tightly related to the fraction Laplacian (−∆) . Two equivalent definitions of it will be introduced, one is
s
through the Fourier multiplier, the other is through a singular integral. Afterward, we investigate the spectral properties
of it and turns out to be similar to classical symmetric elliptic operators. Finally, we prove that if s = 1/k for some k ∈N,
then the first eigenfunction is superharmonic.

Supervisor: JIN Tianling / MATH
Student: TAM Tsz Yan / PHYS Course: UROP1100, Summer

The report mainly studies the results from the paper “On the superhar monicity of the first eigenfunction of the
fractional Laplacian for certain exponents”, and discussed some technical details supporting the original proof.
Properties of the eigenvalues and first eigenfunction of the fractional Laplacian were studied, followed by discussing the
main theorem of the selected paper. It is shown that eigenvalues of fractional Laplacian are real and the limit diverges,
2
and their corresponding eigenfunctions forms an orthonormal basis of L (Ω). Several properties of the first
eigenfunction of the fractional Laplacian were also studied. With these results, the main theorem on the selected paper
were discussed.

Supervisor: JIN Tianling / MATH
Student: YIP Chi Hoi / MATH-PMA Course: UROP1100, Fall

We mainly study the existence, regularity of weak solution of boundary value problem. For the existence, we first
introduce Lax-Milgram Theorem and energy estimate, and then use them to get the existence theorem. For the
regularity, we consider various kinds of space embeddings, and we should divide them into 3 subclasses to study. We
first introduce Gagliado-Nirenberg-Sobolev inequality and Morrey inequality, which deal with the embedding of W (Ω),
1,p
and then we introduce General Sobolev inequality, which is a generalization to the embedding to W (Ω). Finally, we
k,p
study the interior regularity. The amazing conclusion is that weak solutions are in fact smooth.

Efficient Algorithms for Visualizing Dynamical Systems

Supervisor: LEUNG Shing Yu / MATH
Student: LEUNG Kai Lok / MAE Course: UROP1100, Summer

25