Ordinary differential equations (ODE) are important in physical modeling and deduction.

## Vector calculus and spherical average in SAXS

A vector is typically regarded as a geometric entity characterized by a magnitude and a direction. Here are some notes about vector calculus and its basic application in small-angle x-ray scattering (SAXS).

## Notes about fluorescence correlation spectroscopy formalism

Let's say the concentration of the component \(j\) in the solution is \(C_j(\vec{r}, t)\), which is a scalar function of coordinates \(\vec{r}\) and time \(t\). For a well-mixed solution, the average of concentration over coordinates and time should be a constant \(<C_j(\vec{r}, t)> = \bar{C}_j\). The local concentration \(C_j(\vec{r}, t)\), however, fluctuates with coordinates and time. The amount of local fluctuation is \(\delta C_j(\vec{r}, t) = C_j(\vec{r}, t)-\bar{C}_j\).

## Common probability density functions

Probability and statistics are one of the most common terminologies in scientific data interpretations. They provide us ways to find truths and make conclusions from this stochastic and noisy world. These different probability distributions are heavily coupled. In order to apply these distribution properly, we should understand their origin and derivation, which are the main purpose of this post (for continuous distributions).

## Common probability mass functions

Probability and statistics are one of the most common terminologies in scientific data interpretations. They provide us ways to find truths and make conclusions from this stochastic and noisy world. These different probability distributions are heavily coupled. In order to apply these distribution properly, we should understand their origin and derivation, which are the main purpose of this post (for discrete distributions).

## Generating functions of probability distributions

Probability mass/density distribution have defined generating functions which faciliate the derivation and the understanding of different probability distributions.

## Geometric interpretation about Jacobian determinant

In scientific computing, integration by substitution is a very common skill. For example, we measured a series of practical values of a variable (\(x'\)), of which the distribution (\(g(x')\)) was unknown. But we know the theoretic distribution of the variable (\(f(x)\)) and the correction relationship or mapping relationship between theoretic and practical values (\(x=h(x')\)).

## Combinations and particle statistics

## Permutation

Let's first consider a set of \(n\) objects, which are all different. The number of all possible arrangements (permutations) is

\[ n(n-1) \cdots 1 = n! \]

Generally, if we select \(k (<n)\) objects from \(n\), the number of permuations is

\[ n(n-1)\cdot\cdot\cdot(n-k+1) = \frac{n!}{(n-k)!} = P(n, k) \]

## python爬取大学及博士后信息

2020年开年不利，天启四骑士war, pestilence, famine, death已经降临了前3个，大家都宅在家里无法出门，于是我便利用这段时间学习一下爬虫的用法。

其实以前也用过爬虫，不过后来放了很久，很多包也更新了，所以值得重新学一遍。记得在实习的时候，经理曾经给了我一个报价网，该网将市场上几乎所以药品的中标价格都放在了网上，而且网页似乎并没有做任何反爬处理，可惜我已经忘记了网址T T

言归正传，这次用python的bs4、requests包来爬取网页信息，获得最近几年世界各地大学的变化趋势，以及postdoc的发布信息。

## 理想气体状态方程的统计力学推导

理想气体状态方程\(pV = nRT = Nk_BT\)想必大家已经非常熟悉了。下面记一下如何从统计力学的角度推出理想气体状态方程。

## 系综

一定条件下体系可能存在的状态的集合称之为ensemble（系综），是系统状态的概率分布，可与概率空间进行类比。