The tex2jax preprocessor defines the LaTex math delimiters, which are \\(...\\) for in-line math and \\[...\\] for displayed equations. It also defines the TeX delimiters $$...$$ for displayed equations, but it does not define $...$ as in-line math delimiters. We can change that, but I don’t want to, because they had a reason to do so, and I think it easy to understand what is it.

I like to use this tool to help me write LaTex expressions.

Very nice, let’s try this

\\[\frac{10^{120}}{10^{106}} = 10^{120-106} = 10^{14}\\]

\[\frac{10^{120}}{10^{106}} = 10^{120-106} = 10^{14}\]

Volume of a ball

$$\int_{\varphi=0}^{\pi}\int_{\theta = 0}^{2\pi}\int_{r=0}^{R}r^2\cdot\sin{\phi}\cdot d r \cdot d \phi \cdot d \theta = \frac{4}{3} \pi R^3$$
\[\int_{\varphi=0}^{\pi}\int_{\theta = 0}^{2\pi}\int_{r=0}^{R}r^2\cdot\sin{\phi}\cdot d r \cdot d \phi \cdot d \theta = \frac{4}{3} \pi R^3\]

Let’s say we want to add a little bit of inline math

Support Vector Machine

We try to separate positive from negative examples.

^
|\  \     +
| \  \
|  \  \ +  +
| - \  \   + +
|- - \  \ + 
| - - \  \ +
|______\__\_________

Let \(\mathbf{w}\) be a normal vector of the street, and \(\mathbf{x}\) a vector pointing to a sample. If we project \(\mathbf{x}\) on \(\mathbf{w}\), we obtain the scalar :

\[p = \frac{\mathbf{x} \cdot \mathbf{w}}{|| \mathbf{w} ||} = || \mathbf{x} || \cdot \cos(\mathbf{x},\mathbf{w})\]

\(\mathbf{w}\) is always the same, so its norm \(|| \mathbf{w} ||\) is constant. So our problem can be written as :

\[\mathbf{x} \cdot \mathbf{w} \geqslant C\]

If \(\mathbf{x} \cdot \mathbf{w} \geqslant C\) where is C is a constant then the sample x is a positive element, otherwise, it is a negative one. If we let b = -C, we can rewrite this :

\[\mathbf{x} \cdot \mathbf{w} + b \geqslant 0\]

This is what we call the DECISION RULE.

\[\begin{align*} \mathbf{w} \cdot \mathbf{x_+} + b \geqslant 1 \\ \mathbf{w} \cdot \mathbf{x_-} + b \leqslant -1 \end{align*}\]