Time is valuable.

As my study plan proceeded, I began to realize that some of the previous decisions were immature, or using a better phrase, not fully considerate. The reason is that I found some particular materials I need to learn, and they have a higher priority. Anyway, I need to adjust my study plan and make some changes. I may not love to make these changes, but hey, you gotta do what you gotta do.

A mess of a night

Quantum Trajectory

Let’s start with a piece of good news: Progess in my research! I do think that now I understand the general setting of the problems. Once I got back to the problem to see the gap between two cases (finite dimension and infinite dimension), I immediately found a decisive issue in the transition: the definition of a measure. For k dimension, the measure is defined as \mathbb{P}^{\frac{1}{k}Id_{\mathbb{C}^k}} associated to the operator Id_{\mathbb{C}^k}/k, and trivially, it is not well defined when k goes to infinity. There are some other problems, but this one is the most obvious. The plan for next week is to begin to fix the problem from this breakthrough point, and I will continue from there.

Algorithms

I have started to learn the book Introduction to Algorithms. I have studied the first three sections of the first Chapter: I Foundations. It contains many materials that I learned before, like insert-sort, merge-sort, sigma-notation, etc. However, it was a long time ago, and the book indeed provided great insight and brought a new perspective for me. I also planned to do some of the exercises because some of them look really juicy and interesting.

Machine Learning

In Spring 2019, I audited a course called Mathematical Foundations of Data Science from a graduate seminar. The reason why I audited is that the professor didn’t allow me to register since he thought I did not have enough knowledge to understand the materials. Well, he did underestimate my understanding, but maybe he was also right because I didn’t pay much attention later in the course. Now I decided to review or learn the materials from today, and I think that this will be a great experience.

Stochastic Differential Equation

I have read Chapter 4: Stochastic integrals, Itˆo’s formula from Evans’ book An Introduction to Stochastic Differential Equations. However, the book only contains a few examples and no exercises. I was intended to read another book called Stochastic Differential Equation by Oksendal, which is a great book. However, it means that I will probably force myself to enjoy the book starting from the beginning, and I need to spend so much time on those dense and heavy materials. Therefore, given the situation, I decided to first continue reading the book from Evans and to understand everything in the book. After that, I will come back to Oksendal’s book and get a deeper understanding of the materials. The plan for the next week is to read Chapter 5, and I need to know how to correctly use the Ito’s Lemma.

Riemann-Hilbert Problem

This week I have learned the inverse scattering method both from a physics and mathematics point of view. The book also contains some examples and helps students to learn the method. Later that Chapter I finally arrived at the section where the Riemann-Hilbert Problems are introduced. The book provided a different approach to solve the Riemann-Hilbert Problem compared to the one that I read before. I will look into it tomorrow, and it looks really interesting.

I miss my girlfriend when she is not around. For real.

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