Postgrad Exam Diary 001

WARNING

Preface

For various reasons, I have no choice now but to prepare for the postgraduate entrance exam.
This diary is both a review of each day’s study and a place where I can talk into the void a little. I first thought of naming it Day 1, but then I realized I would not have time to write every day, nor would I necessarily have something to write every day. There will always be days of slackness and days too busy for words. So I kept it as 001.

To be honest, I never truly believed I could start now and get in within three months. Speaking without much ambition, someone like me, who does not plan to marry, does not plan to buy a house, and supposedly has low material desire, can probably afford to keep trying for several years. This year is more of a trial run. I do not feel much pressure myself, but my parents are very anxious because my attempt at direct PhD admission failed.

Today was only my third time going to the university library. The first time was when I had just entered XDU and came to look around, and the second was for a meeting.
Ever since that first visit to the so-called rat-slaying library, I have felt it was not a place that suited me for studying. The atmosphere is too serious and oppressive. I sit stiffly in the chair, worrying that I might make a sound and disturb others, and I keep wondering whether the people around me are watching to see if I am actually studying. It feels as if even glancing at my phone is a crime. I know that is only in my head and in fact nobody cares, but a place full of strangers that is this quiet really makes a socially anxious person tense up. It feels nothing like self-study sessions in high school.
Thankfully the library has neither Big Brother’s eyes nor Miss Yang recording videos with her phone. If only 0721’s 凌地宁宁 were here.

I started with Math I. This year I plan to follow Wu Zhongxiang and jump straight into the intensive course. I watched two chapters in one day. Maybe because it has been too long since I last used my brain properly for study, I get sleepy as soon as I start learning. Fortunately, the first two chapters on functions and limits were not difficult, and only today did I realize how much my old Mathematical Analysis course had helped me. Back then I thought that course, so theoretical and science-oriented, was useless for my engineering major. I fell behind, then wanted to study it even less, because it was simply too hard. In the end the teacher helped me through the makeup exam and gave me a passing score in the sixties. But Mathematical Analysis did not only give me proofs of basic theorems. It was not just a toy for mathematicians. It was also training and enlightenment for the mind. Shame on me, truly.

I have never been the type to take notes. Anything with internal logic I usually keep only in my head. But memory fades with time. I am no longer living the kind of life where I review everything every day like in high school, so I should write down some knowledge now and then for later review.

Review

Functions

• Basic elements of a function: domain, mapping rule

• Properties of functions: monotonicity, parity, periodicity, boundedness

• Odd functions

  $$ ln{\frac{1-x}{1+x}},ln(x+\sqrt[2]{1+x^2}),\frac{e^x-1}{e^x+1},f(x)-f(-x) $$

• Even functions

  $$ f(x)+f(-x) $$

• If $f(x)$ is odd, then $\int ^x_0 f(x)dx+C$ is even

• If $f(x)$ is even, then $\int ^x_0 f(x)dx+C$ is odd only when $C=0$

• The derivative of a differentiable periodic function is periodic

• If the derivative is periodic, the original function is not necessarily periodic

• If the derivative is periodic and its integral over one period is $0$, then the original function is periodic, and this is a necessary and sufficient condition

Limits

• Local boundedness: if a limit exists in a deleted neighborhood, then the function is locally bounded; local boundedness alone does not imply the limit exists
• Sign preservation
• Order preservation

Criteria for the existence of a sequence limit

• Squeeze theorem
• Monotone bounded theorem

Relation between divergence to infinity and unbounded variables:

• A sequence diverges to infinity if its terms eventually become arbitrarily large in absolute value $$ \forall M \gt 0 ,\exist N \gt 0,当 n \gt N时,恒有\left\vert x_n \right\vert \gt M $$
• An unbounded variable only requires that for any given number, there exists some term whose absolute value exceeds it $$ \forall M \gt 0 ,\exist N \gt 0,使得\left\vert x_N \right\vert \gt M $$

Postscript

I do not understand, I do not understand, how high and far the stars are.
When will I finally be able to stand among them?