Proof of fundamental theorem of arithmetic

I read it a couple days ago and actually remembered it this time in a way that I will never forget it. It invokes Euclid’s lemma, which states that if p | ab for p prime, then p | a or p | b, which can be proved using Bezout’s lemma. For existence, it does induction on the number of factors, with 1 as the trivial base case. For the non base case, wherein our number is composite, apply the inductive hypothesis on the factors. For uniqueness, assume two distinct factorizations: p_1p_2\ldots p_n = q_1q_2\ldots q_n. By Euclid’s lemma, each of the p_is divides and is thus equal to one of the q_is. Keep invoking Euclid’s lemma, canceling out a prime factor on each iteration and eventually we must end with 1 = 1 in order for the two sides to be equal.



今天又听了诸多苏联歌曲,甚悦耳,大多来自于薛范的YouTube account。音乐与镜头配的好的不能再好了。记得曾经薛范在采访中说过:做他的工作需要音乐,俄文,和中文三项通,具备此条件的人并不多。

所听的,我能想到的有:москва майская, москва-пекин, прощание славянки, подмосковные вечера, марш защитников москвы, в путь, 等等。这些无论是其音乐还是其歌词艺术质量都很高。我认为一个人可以不喜欢这些歌代表的苏联的政治那一套,但是很难从一个纯粹艺术角度上低评它,就像一个天才可能是个asshole,但是这并不改变他是天才的现实。在这里边,对我印象比较深的是在《莫斯科-北京》里,还有,若我没判断错,郭沫若的镜头。另一位出现的我记得的好像是胡乔木,但这我不能太肯定。

A mechanics problem

Today, I solved one from the Cracking the AP Physics C Exam 2008 Edition, which I had bought in high school. It is


Questions (c), (d), and (e) are not included. They are to determine the relationship between the magnitude of the acceleration of the block and the linear acceleration of the cylinder, to determine the acceleration of the cylinder, and to determine the acceleration of the block respectively. It is obvious that from (b) that the magnitude of the acceleration of the block is twice the linear acceleration of the cylinder, and the answer to (e) can be derived from that of (d) via this. As for (d), I first solved it with a free body diagram and Newton’s laws. Then, my ultra gifted friend Brian Bi suggested using the Euler-Lagrange equations, which simplifies things greatly, to the extent that he characterized himself as having forgotten how to solve mechanics problems the way I had done (obviously not true, but you get what he’s saying). I was able to work it out this way shortly, getting the same answer.


Note: I left overlooked the 4 coefficient initially and only filled it in at the end. It is still omitted in some of the intermediate equations.

First blog post

I plan, hope to be writing out here a miscellany of thoughts and musings pertaining to my intellectual interests, which can be roughly divided into a primary one, which is pure math and other topics of a mathematical nature, and secondary ones, which, of the ones I can think of now, are Chinese language, language (in general), psychometrics, neuroscience, and stories of scientists and of unusual people of one form or another.