How to distinguish fixed from random effects in a model equation? When I'm reading model equations, I want to spot which factor is random or fixed. 
In this pdf they are showing an equation (see image) and saying that: 

residual effects are random


My question is is there a way to spot just by looking to an equation like this which factor is fixed or random? Also, is there a notation to make a difference between a fixed and a random factor? 
 A: There are common notations which can be used which make it very easy to know what is fixed and random, but the equation you posted does not adopt such a notation. In your case there is no way to tell, except for the residuals (which are very commonly denoted with $e$ and since it is indexed the same way as the response variable, this makes it rather likely that is it a subject-level residual). Even the description beneath the variables does not help, so you have to rely on the text that follows on from there, where you find that $HYS_l$ and $c_m$ are also random ! 
Here is one easy notation scheme: denote fixed effects with $X_1$, $X_2$ and $X_3 ...$ and random effects with lower case letters: $e$ (typically reserved for subject residuals), $u$ and $v...$.
So a random intercepts model with 2 $X$ variables and a response $Y$ could be simply written as 
$Y = \beta_0 + \beta_1X_1 + \beta_2X_2 + u + e$
where $u$ are the random intercepts and $e$ are the subject residuals.
To be more explicit, this type of model is often presented with indexing: Observations in each group are indexed with $i$ and $j$ referring to the $i$th subject in the $j$th group, where $i$ ranges from $1$ to $n_j$ and $j$ ranges from $1$ to $J$ where $J$ is the total number of groups.
Thus we have 


*

*$Y_{ij}$ is the response variable for the $i$th subject in the $j$th cluster.

*$X_{1ij}$ is the value of $X_1$ for the $i$th subject in the $j$th cluster.

*$X_{2ij}$ is the value of $X_2$ for the $i$th subject in the $j$th cluster.


Then, the above random intercepts model is
$$ Y_{ij}= (\beta_0 + u_{0j})+\beta_1 X_{1ij} +\beta_2 X_{2ij} +e_{ij},
$$
Since every subject has their own residual, $e$ are indexed by $e_{ij}$, and since every group has it's own residual, these are indexed by $u_{0j}$ (where the $0$ in the subscript corresponds to the zero in the $\beta$ subscript because $u_{0j}$ is the random intercept and $\beta_0$ is the global intercept).
Using this notation we can easily introduce a random slope for $X_1$ which will simply be another cluster-level residual, $u_{1j}$
$$ y_{ij}= (\beta_0 + u_{0j})+(\beta_1 + u_{1j}) X_{1ij} +\beta_2 X_{2ij} +e_{ij},
$$
and again for a random slope ($u_{2j}$) for $x_2$
$$ Y_{ij}= (\beta_0 + u_{0j})+(\beta_1 + u_{1j}) X_{1ij} +(\beta_2 +u_{2j}) X_{2ij} +e_{ij},
$$
Note how the subscripts on the beta-coefficient match those of the fixed effects $X_1$ and $X_2$ and the random effects $u_{0j}$ and $u_{1j}$
We can further extend this with interaction, further $X$ variables and further "levels". However it should be apparent that if we were to do that then this kind of notation quickly becomes difficult to work with, so there are 2 common ways to proceed.
The first is to write:
\begin{align}
\\
Y_{ij}&= \beta_{0j}+ \beta_{1j} X_{1ij} + \beta_{2j}X_{2ij} +e_{ij}
\\ \textrm{where}
\\ \beta_{0j}&=\beta_0 + u_{0j}
\\ \beta_{1j}&=\beta_1 + u_{1j}
\\ \beta_{2j}&=\beta_{2} +u_{2j}
\end{align}
and this is usually the approach taken by the multilevel and hierarchical linear modelling worlds (see for example the very well known book by Snjders and Bosker
Edit: To address the comment, in the case of an interaction we would write:
$$
Y_{ij}= \beta_{0j}+ \beta_{1j} X_{1ij} + \beta_{2j}X_{2ij}+ \beta_{3}X_{1ij}X_{2ij} +e_{ij}
$$
where in this case only the fixed effect of the interaction is modelled. We could easily include a random slope for the interaction too, in the same way that we did for $X_1$ and $X_2$.
A second way is to work in matrix form:
$$
\mathbf{y}=\mathbf{X\beta} + \mathbf{Z b} + \mathbf{e}
$$
where $\mathbf{y}$ is the response vector, $\mathbf{X}$ is a design matrix for the fixed effects ($\mathbf{\beta}$) and $\mathbf{Z}$ is a block-diagonal design matrix for the random effects ($\mathbf{b}$).
This is popular in the mixed effects world (see for example the book by Demidenko). To see how this notation works, we can partition the matrices for each group:
$$\begin{bmatrix}
 \mathbf{y_1}
\\ \mathbf{y_2}
\\ \vdots 
\\ \mathbf{y_J}
\end{bmatrix}= \begin{bmatrix}
X_1 \\ 
X_2 \\
\vdots \\ 
X_J 
\end{bmatrix} \begin{bmatrix}
\beta
\end{bmatrix}+\begin{bmatrix}
\mathbf{Z_1} & 0 & 0 & 0 \\ 
0 &  \mathbf{Z_2} & 0 & 0 \\ 
\vdots &  & \ddots  &  \\ 
0 & 0  & 0 & \mathbf{Z_J}
\end{bmatrix} +\begin{bmatrix}
b_1 \\ 
b_2  \\
\vdots \\ 
b_J 
\end{bmatrix}+\begin{bmatrix}
e_1 \\ 
e_2 \\ 
\vdots \\ 
e_J
\end{bmatrix}$$
where, in the case of the model without the interaction, but with random slopes for both $X_1$ and $X_2$,
$\mathbf{y_j} = \begin{bmatrix}
y_{1j}\\ 
y_{2j}\\ 
\vdots \\ 
y_{n_jj}
\end{bmatrix},$
$\mathbf{X_j}=\mathbf{Z_j}=\begin{bmatrix}
1 & X_{11j} & X_{21j}\\ 
1 & X_{12j} & X_{22j}\\ 
\vdots & \vdots & \vdots\\ 
1 & X_{1n_jj} & X_{2n_jj}
\end{bmatrix}, e_j = \begin{bmatrix}
e_{1j}\\ 
e_{2j}\\ 
\vdots \\ 
e_{n_jj}
\end{bmatrix},$
$\beta = \begin{pmatrix}
\beta_0 \\ 
\beta_1 \\
\beta_2
\end{pmatrix}$ and $ b_j=\begin{pmatrix}
u_{0j}\\ 
u_{1j} \\
u_{2j}
\end{pmatrix}$ 
Note that for this model we have that $\mathbf{X_j}=\mathbf{Z_J}$ because we have random effects for all 3 fixed effects (intercept, $X_1$ and $X_2$). If we had random slopes for only $X_1$ then we would have:
$\mathbf{Z_J}=\begin{bmatrix}
1 & X_{11j} \\ 
1 & X_{12j} \\ 
\vdots & \vdots \\ 
1 & X_{1n_jj}
\end{bmatrix}$
and if we had random intercepts only, we would have:
$\mathbf{Z_J}=\begin{bmatrix}
1  \\ 
1  \\ 
\vdots \\ 
1 
\end{bmatrix}$
