LaTeX editing mathematical formulas basic syntax elements
LaTeX The mathematical model has two forms:
- inline and display.
- The former refers to a line between the text insert mathematical formulas, which are arranged independently, may or may not numbered.
- Interline formula (inline)
- $ Enclosed by the formula.
- Interblock formula (displayed)
- $$ enclosed by the formula is a number of free form
- There are [.....] unnumbered independent formula Markdown form but does not seem to support.
- Block element is centered between the default display.
Greek alphabet edit various types of table
Subscripts and superscripts, root, ellipsis
- Subscript: x_i: \ (x_i \)
- Superscript: X ^ 2: \ (X ^ 2 \)
- NOTE: If more than one vertical mark letters or symbols, requires a pair of enclosed {} {I1} X_: \ (X_ {I1} \) \ (AT X ^ {} \)
- Root: \ sqrt [n-] {}. 5: \ (\ sqrt [n-]. 5} {\)
- Omitted issue: \ Cdots: \ (\ Cdots \)
Operators
Basic operators + - * ÷
Summation:
- \sum_1^n: \(\sum_1^n\)
- \ sum_ {x, y}: \ (\ sum_ {x, y} \)
integral:
- \int_1^n: \(\int_1^n\)
limit
- lim_ {x \ a \ IFN?} \ (lim \ _ {x \ to \ infty} \)
Determinant
\[ X=\left| \begin{matrix} x_{11} & x_{12} & \cdots & x_{1d}\\ x_{21} & x_{22} & \cdots & x_{2d}\\ \vdots & \vdots & \ddots & \vdots \\ x_{11} & x_{12} & \cdots & x_{1d}\\ \end{matrix} \right| \]$$ X=\left| \begin{matrix} x_{11} & x_{12} & \cdots & x_{1d}\\ x_{21} & x_{22} & \cdots & x_{2d}\\ \vdots & \vdots & \ddots & \vdots \\ x_{11} & x_{12} & \cdots & x_{1d}\\ \end{matrix} \right| $$
matrix
\[ \begin{matrix} 1 & x & x^2\\ 1 & y & y^2\\ 1 & z & z^2\\ \end{matrix} \]$$ \begin{matrix} 1 & x & x^2\\ 1 & y & y^2\\ 1 & z & z^2\\ \end{matrix} $$
arrow
Piecewise functions
\[ f(n)= \begin{cases} n/2, & \text{if $n$ is even}\\ 3n+1,& \text{if $n$ is odd} \end{cases} \]$$ f(n)= \begin{cases} n/2, & \text{if $n$ is even}\\ 3n+1,& \text{if $n$ is odd} \end{cases} $$
equation set
\[ \left\{ \begin{array}{c} a_1x+b_1y+c_1z=d_1\\ a_2x+b_2y+c_2z=d_2\\ a_3x+b_3y+c_3z=d_3 \end{array} \right. \]$$ \left\{ \begin{array}{c} a_1x+b_1y+c_1z=d_1\\ a_2x+b_2y+c_2z=d_2\\ a_3x+b_3y+c_3z=d_3 \end{array} \right. $$
Common formula
Linear model
\[ h(\theta) = \sum_{j=0} ^n \theta_j x_j \]$$ h(\theta) = \sum_{j=0} ^n \theta_j x_j $$
Mean square error
\[ J(\theta) = \frac{1}{2m}\sum_{i=0}^m(y^i - h_\theta(x^i))^2 \]$$ J(\theta) = \frac{1}{2m}\sum_{i=0}^m(y^i - h_\theta(x^i))^2 $$
I asked 积公 formula
$$ H_c=\sum_{l_1+\dots +l_p}\prod^p_{i=1} \binom{n_i}{l_i} $$\$$ H_c=\sum_{l_1+\dots +l_p}\prod^p_{i=1} \binom{n_i}{l_i} \$$
Batch gradient descent
\[ \frac{\partial J(\theta)}{\partial\theta_j} = -\frac1m\sum_{i=0}^m(y^i - h_\theta(x^i))x^i_j \]$$ \frac{\partial J(\theta)}{\partial\theta_j} = -\frac1m\sum_{i=0}^m(y^i - h_\theta(x^i))x^i_j $$
The derivation process
\[ \begin{align} \frac{\partial J(\theta)}{\partial\theta_j} & = -\frac1m\sum_{i=0}^m(y^i - h_\theta(x^i)) \frac{\partial}{\partial\theta_j}(y^i-h_\theta(x^i))\\ & = -\frac1m\sum_{i=0}^m(y^i-h_\theta(x^i)) \frac{\partial}{\partial\theta_j}(\sum_{j=0}^n\theta_j x^i_j-y^i)\\ &=-\frac1m\sum_{i=0}^m(y^i -h_\theta(x^i)) x^i_j \end{align} \]$$ \begin{align} \frac{\partial J(\theta)}{\partial\theta_j} & = -\frac1m\sum_{i=0}^m(y^i - h_\theta(x^i)) \frac{\partial}{\partial\theta_j}(y^i-h_\theta(x^i))\\ & = -\frac1m\sum_{i=0}^m(y^i-h_\theta(x^i)) \frac{\partial}{\partial\theta_j}(\sum_{j=0}^n\theta_j x^i_j-y^i)\\ &=-\frac1m\sum_{i=0}^m(y^i -h_\theta(x^i)) x^i_j \end{align} $$
Subscript characters
\[ \max \limits_{a<x<b}\{f(x)\} \]$$ \max \limits_{a<x<b}\{f(x)\} $$
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