This is an R Markdown document. R Markdown is a simple formatting syntax for authoring HTML, PDF, and MS Word documents (among others!). I recommend using it to create HTML documents, and to instead use LaTeX or knitr (LaTeX with embedded R code) documents to make PDFs (For more on LateX and/or knitr see https://pauljhurtado.com/latex/).

For further information on using R Markdown see http://rmarkdown.rstudio.com, especially the section Get Started. There, you can find various resources including an R Markdown Cheatsheet – I like the older version at https://rstudio.com/wp-content/uploads/2015/02/rmarkdown-cheatsheet.pdf – and the R Markdown Reference Guide https://rstudio.com/wp-content/uploads/2015/03/rmarkdown-reference.pdf.

In R Studio (https://www.rstudio.com/), you can start a new R Markdown document by going to **File > New File > R Markdown**. When you click the **Knit** button a document will be generated that includes both content as well as the output of any embedded R code chunks within the document. We’ll get to that shortly, but first, a little more on basic formatting and equations. To get R and R Studio installed (along side the free software LaTeX) see http://www.pauljhurtado.com/teaching/software.html.

View the source file for this document to view examples. To make text **bold** place a pair asterisks (or underscore characters) before and after the text. To make text *italic*, put a single asterisk (or underscore character) before and after the text. To format inline (R) code, such as names of `funcions`

and `packages`

, place grave accents (`, not apostrophes like ’) on either end of the text. Use multiple # at the start of a line to define a section (*## Section Title*) and subsection headers (*### Subsection Title*). If compiling to produce an HTML document, then HTML and CSS can also be included to format text __like this__ or like this.

For additional resources, see the links above.

In short, LaTeX works!

```
$$\frac{dx_1}{dt} = f(x_1)$$
```

\[\frac{dx_1}{dt} = f(x_1)\]

Similarly,

```
\begin{equation}
\frac{dy}{dt}=g(y)
\label{eq:dydt}
\end{equation}
```

\[\begin{equation} \frac{dy}{dt}=g(y) \label{eq:dydt} \end{equation}\]

Note that to get the equation numbering for eq. \eqref{eq:dydt} above to work, I had to add the following to the top of the R markdown source file (https://stackoverflow.com/questions/35026405/auto-number-equations-in-r-markdown-documents-in-rstudio):

```
<script type="text/x-mathjax-config">
MathJax.Hub.Config({
TeX: {
equationNumbers: {
autoNumber: "AMS",
formatNumber: function (n) {return '1.'+n}
}
}
});
</script>
```

The amsmath package is mostly functional (it’s all run through mathjax by pandoc), for example

\[A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\]

Some environments are limited however. For example, wrapping the following in a subequations environment fails.

\[\begin{align} \frac{dN}{dt} =& \; r\,N\bigg(1-\frac{N}{K}\bigg) - a\,N\,P \\ \frac{dP}{dt} =& \; \chi\,a\,N\,P - \mu\,P \end{align}\]

- Mathematical expression examples: https://www.calvin.edu/~rpruim/courses/s341/S17/from-class/MathinRmd.html
- LaTeX resources: https://www.pauljhurtado.com/latex/

You can embed an R code chunk like this:

```
```{r cars}
summary(cars)
```
```

This yields the output

`summary(cars)`

```
## speed dist
## Min. : 4.0 Min. : 2.00
## 1st Qu.:12.0 1st Qu.: 26.00
## Median :15.0 Median : 36.00
## Mean :15.4 Mean : 42.98
## 3rd Qu.:19.0 3rd Qu.: 56.00
## Max. :25.0 Max. :120.00
```

You can also embed plots, for example:

```
```{r pressure, echo=FALSE}
plot(pressure)
```
```

yields…

Note that, above, the optional `echo = FALSE`

parameter was added to the code chunk to prevent printing of the R code that generated the plot.

For more on Code Chunk options, see https://rmarkdown.rstudio.com/lesson-3.html and the R Markdown Reference Guide at https://rstudio.com/wp-content/uploads/2015/03/rmarkdown-reference.pdf.

You can also call other languages, e.g., if you have python installed (Anaconda users need to add the additional lines 4-5 per https://stackoverflow.com/questions/50352614/r-markdown-how-can-i-make-rstudio-display-python-plots-inline-instead-of-in-new/50711837#50711837):

```
import matplotlib.pyplot as plt
plt.style.use('seaborn-whitegrid')
import numpy as np
import os
os.environ['QT_QPA_PLATFORM_PLUGIN_PATH'] = '/Users/MrStandardUser/Anaconda3/Library/plugins/platforms'
x = np.linspace(0, 10, 30)
y = np.sin(x)
plt.plot(x, y, 'o', color='black')
plt.show()
```

For more on using python in conjunction with R, see https://rstudio.github.io/reticulate/articles/r_markdown.html.

For more on the different “engines” available beyond R and Python, run the following in the R console: `names(knitr::knit_engines$get())`

.

- Google, Stack Exchange, anything by Yihui Xie
- https://rmarkdown.rstudio.com/
- R Bootcamp materials by Kevin Shoemaker: http://naes.unr.edu/shoemaker/teaching/R-Bootcamp/index.html