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

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).

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 $$\delta C_j(\vec{r}, t) = C_j(\vec{r}, t)-\bar{C}_j$$.

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).

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).

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

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 ($$x=h(x')$$).

## 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

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

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。