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notebook.jl
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notebook.jl
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### A Pluto.jl notebook ###
# v0.19.47
using Markdown
using InteractiveUtils
# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error).
macro bind(def, element)
quote
local iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end
local el = $(esc(element))
global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el)
el
end
end
# ╔═╡ bafcc8b1-2519-4dc3-837b-e65fa36cbfef
using Plots # main plotting package
# ╔═╡ 74266685-f5c4-4b19-ab58-1dc890f885c5
using PlotlyJS: PlotlyJS # provides an interactive backend for web browsers
# ╔═╡ 48cc1cbf-91b0-424a-8645-035cd40c194a
using Unitful # package for adding physical units to numbers
# ╔═╡ 15b99748-69eb-4789-b9e8-243a97f3a36e
using Optim
# ╔═╡ dc860b9f-bceb-4c19-8675-aaa43ccf3536
using DataFrames
# ╔═╡ 4cb5a162-2f49-4569-9093-bb1e1d3024a4
using LsqFit
# ╔═╡ 0b299610-118d-4bcb-8b74-b4bc4f2ebe46
using PlutoUI
# ╔═╡ eb48686d-86a9-48fb-9543-b809d5d025be
html"<button onclick='present()'>Toggle presentation mode</button>"
# ╔═╡ b956f723-9404-4a2f-bda2-1ea984332064
md"""
# Scientific computing with Julia
"""
# ╔═╡ f37832b0-a362-4df8-9046-74d54828032d
md"""
## Pluto: reproducibility by default
$(Resource("https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4544406/bin/f1000research-4-7374-g0000.jpg"))
_Image by Paul Blow in article ["Reproducibility: The risks of replication drive"](https://doi.org/10.1038/503333a) by Mina Bissel, Nature, 2013._
Julia is an interactive programming language, much like Python and R, and can be used in notebooks for exploratory work or data science workflows.
It can be used in Jupyter notebooks (the "Ju" in the name stands for "Julia"!) via the [`IJulia.jl`](https://github.com/JuliaLang/IJulia.jl) kernel, but here we want to demonstrate [`Pluto.jl`](https://github.com/fonsp/Pluto.jl) 🎈, an alternative type of notebook with focus on reproducibility.
Pluto is _reactive_, which means that interdependent cells are automatically updated, much like a spreadsheet.
You can also use widgets to set the value of variables.
"""
# ╔═╡ 64a8c492-f6cd-4b31-b200-f15210fdc6af
a = 2
## to make `a` into a slider, uncomment the following line and comment the above one.
## Comments in Julia use the # syntax.
# @bind a Slider(0:10; show_value=true, default=2)
# ╔═╡ ef5fe3c7-d01d-48de-a9c5-1010d8c75f5e
b = a ^ 2
# ╔═╡ 4f425f14-e2ec-4b5c-a2d2-f5614da38c7a
md"""
Pluto has also a tight integration with the Julia package manager, to make sure the notebook you run locally will use the same set packages on someone else's machine.
To achieve this, Pluto creates a virtual environment under the hood, but whenever you use Julia outside of Pluto do remember to create a [local environment](https://pkgdocs.julialang.org/v1/environments/) for your projects!
"""
# ╔═╡ 9ff517e2-d978-4b52-ac09-3dc16b222d11
md"""
## What's Julia?
Julia is a modern, dynamic, general-purpose, compiled programming language.
It's interactive ("like Python"), can be used in a REPL or notebooks, like Jupyter (it's the "Ju") or Pluto (this one🎈).
Julia has a runtime which includes a just-in-time (JIT) compiler and a garbage collector (GC), for automatic memory management.
Julia is mainly used for numerical computing; the differential equations solvers suite is particularly popular.
The main paradigm of Julia is multiple dispatch: the ability of functions or methods to depend on the type and number of _all_ its arguments.
Fun fact: version 1.0 was released in London in 2018 during JuliaCon hosted at UCL!
### Why Julia?
From "[My Target Audience](https://scientificcoder.com/my-target-audience)" by Matthijs Cox:
$(Resource("https://cdn.hashnode.com/res/hashnode/image/upload/v1681735971356/91b6e886-7ce1-41a3-9d9f-29b7b096e7f2.png"))
$(Resource("https://cdn.hashnode.com/res/hashnode/image/upload/v1681735992315/62fdd58f-4630-4120-8eb4-5238740543e8.png"))
### Two-language problem
This is another way to look at two-community problem discussed above:
* one language for the prototype, a different one for production, or
* one language for the front-end, a different one for the backend.
$(Resource("https://pde-on-gpu.vaw.ethz.ch/assets/literate_figures/l1_two_lang.png"))
$(Resource("https://pde-on-gpu.vaw.ethz.ch/assets/literate_figures/l1_flux-vs-tensorflow.png"))
### Performance
!!! note "About performance"
Julia _can_ be fast, but it is also easy to write slow code.
Writing efficient code can take some practice, but you can remain in the same language, without being forced to resort to rewrite your program in a different language to achieve better performance.
_Plots in this section are from the presentation of the training course [Julia for HPC @ UCL 2024](https://github.com/carstenbauer/JuliaUCL24)_
#### Julia can be fast
$(Resource("https://i.imgur.com/vQjzPEF.png"))
$(Resource("https://i.imgur.com/7HLD6Ta.png"))
Gradual performance improvement, no disruptive
language change!
#### Julia can be parallel
$(Resource("https://i.imgur.com/FvDZF1Z.png"))
$(Resource("https://i.imgur.com/iHh7KsQ.png"))
### Productivity
$(Resource("https://i.imgur.com/Ym5H0Pz.jpeg"))
$(Resource("https://i.imgur.com/KZMZSru.jpeg"))
from <https://twitter.com/ChapelLanguage/status/1623389242822111232>
### Summary
* Explorable & Understandable
* Composability thanks to multiple dispatch
* User-defined types are as fast and compact as built-ins
* Code that is close to the mathematics
* No need to switch languages for performance...
* ...but you can still call C-like shared libraries with simple Foreign Function Interface (FFI) if you want to
* Built-in package manager
* Most packages are written in Julia: end users and software developers talk the same language
* MIT licensed: free and open source
"""
# ╔═╡ 7e38a429-1d1d-4981-97a9-e3d6e530ab4f
md"""
## What does Julia code look like
_This section is partly inspired by [ETH's course 101-0250-00L on solving partial differential equations (PDEs) in parallel on graphical processing units (GPUs) with the Julia language](https://pde-on-gpu.vaw.ethz.ch/lecture1/)._
"""
# ╔═╡ 54f48b85-33d5-4284-b906-6062085c3c18
function lorenz(x)
σ = 10
β = 8/3
ρ = 28
return [
σ * (x[2] - x[1]),
x[1] * (ρ - x[3]) - x[2],
x[1] * x[2] - β * x[3]
]
end
# ╔═╡ cfc6b81c-69f9-4a64-8c48-659b6b7be51f
function integrate()
# integrate dx/dt = lorenz(t,x) numerically for 500 steps
dt = 0.01
x₀ = [2.0, 0.0, 0.0]
out = zeros(3, 500) # increase second dimension for more steps
out[:,1] = x₀
for i=2:size(out,2)
out[:,i] = out[:,i-1] + lorenz(out[:,i-1]) * dt
end
return out
end
# ╔═╡ 1d8cf13d-dbf3-4d10-9387-830fb80e6fca
attractor = integrate()
# ╔═╡ 08be6ab3-9ef3-442c-bd23-97aa68efad92
plotlyjs(); # enable the PlotlyJS backend
# ╔═╡ 3c4c6b49-fadc-4833-a886-c8c2edcc5f7b
plot(attractor[1,:], attractor[2,:], attractor[3,:])
# ╔═╡ 4a7ba915-ef5a-4c2e-a0c0-b044017f0316
md"""
## Crash course on multiple dispatch
From "[SIAM CSE19: Solving the Two Language Problem in Scientific Computing and Machine Learning with Julia](https://www.youtube.com/watch?v=OfMP5PTFQk0)" (acceptance speech for J. H. Wilkinson Prize 2019 for Numerical Software)
$(Resource("https://i.imgur.com/QqUPohw.png"))
_Based on the blogpost "[Rock–paper–scissors game in less than 10 lines of code](https://giordano.github.io/blog/2017-11-03-rock-paper-scissors)"._
"""
# ╔═╡ 2fc1cab7-d477-4487-a101-1a1aae214865
begin
abstract type Shape end
struct Rock <: Shape end
struct Paper <: Shape end
struct Scissors <: Shape end
play(::Type{Paper}, ::Type{Rock}) = "Paper wins"
play(::Type{Paper}, ::Type{Scissors}) = "Scissors wins"
play(::Type{Rock}, ::Type{Scissors}) = "Rock wins"
play(::Type{T}, ::Type{T}) where {T<: Shape} = "Tie, try again"
play(a::Type{<:Shape}, b::Type{<:Shape}) = play(b, a) # Commutativity
end
# ╔═╡ a8a6ab01-1c32-45c5-80f9-1ba0f49cdfcb
md"""
You can also extend to more data structures
"""
# ╔═╡ 6074a761-3264-42cd-9a67-fb8fcea557ca
begin
struct Well <: Shape end
play(::Type{Well}, ::Type{Rock}) = "Well wins"
play(::Type{Well}, ::Type{Scissors}) = "Well wins"
play(::Type{Well}, ::Type{Paper}) = "Paper wins"
end
# ╔═╡ c75a74b1-651b-4d26-b8e3-0b6c4c3d3a59
play(Paper, Scissors)
# ╔═╡ f04b4438-3bea-4d7f-bbd8-6c7ea800cc9f
play(Rock, Rock)
# ╔═╡ cde75794-0f43-40e7-b2e4-9a748bf67431
play(Rock, Paper)
# ╔═╡ 8f875943-61d1-45f8-9b37-a75fff536da9
play(Paper, Well)
# ╔═╡ 2730f18e-dc79-448d-97cd-b32da0221abc
play(Well, Rock)
# ╔═╡ 534d35aa-e5cc-4507-b440-cbf53764a4dd
play(Well, Well)
# ╔═╡ 87a54b0d-2e42-4fe9-a40e-4567ae9fcb87
md"""
Multiple dispatch is the key ingredient to make Julia packages very composable!
"""
# ╔═╡ 1ab78041-e834-4fac-a1f7-1fb3b14fc394
md"""
## Play with Julia code
### Basic datatypes
#### Numbers
"""
# ╔═╡ 0d2421e4-4a9f-405d-831b-9146056110aa
my_integer = 42
# ╔═╡ 506a56cf-675f-42f4-99c6-45ca025ec7f4
my_float = 3.14
# ╔═╡ d614d6c7-49b4-4b38-822a-6fbc295e675d
my_integer + my_float
# ╔═╡ f50d27fd-f101-489c-b240-ee14c8c1dd33
my_float / my_integer
# ╔═╡ 1a1e2a98-2338-4673-b599-a435f83a1fbe
md"""
#### Strings
"""
# ╔═╡ 8ae49a36-670b-493d-8957-29436ff9519c
my_string = "hello world"
# ╔═╡ 4266fc68-5fbf-4c2f-85a6-f2064b1ca52d
println(my_string)
# ╔═╡ df57efba-663f-4408-914a-e4ff00902d97
greet(name::String) = println("Hello ", name)
# ╔═╡ b8901b06-fa08-4a6c-ad44-0965a55f8ca8
greet("UCL ARC")
# ╔═╡ de339ded-7430-4d87-aa00-edbf3eee0f1a
md"""
#### Vectors and multi-dimensional arrays
"""
# ╔═╡ c8afa835-47c7-488b-8827-e72c3df3a43f
my_vector = [1, 2, 3, 5, 7, 11, 13]
# ╔═╡ 2a13f83a-cfa7-48d7-9c42-651698428b6d
my_vector[1]
# ╔═╡ 936fb03d-ea61-42a3-b389-21cbb351fd35
my_vector[3:5]
# ╔═╡ 3e60fa18-e8e7-44c2-9c46-45ab1fca1537
new_vector = [1, 1, 2, 3, 5, 8, 13]
# ╔═╡ 2f32386b-eed2-4f42-847a-4629d9522701
my_vector + new_vector
# ╔═╡ 61e6ab7f-e16e-444c-b73a-df3ac15cb7e7
my_vector * new_vector'
# ╔═╡ 020d657b-f8fd-44d5-aea9-dd579e4af180
my_matrix = randn(3, 3)
# ╔═╡ 7ffe9fd0-d724-4567-86da-bf57bd215215
my_matrix[2, 2]
# ╔═╡ 9bb42815-39ad-4cc2-a3c0-30b2fd6c2152
my_matrix[1:3, 2]
# ╔═╡ 0d71f10f-f753-48cd-baff-da0460e3a419
my_matrix ^ 2
# ╔═╡ 132aa1da-7051-4da7-8643-2c5497fa6b3d
my_matrix * my_vector[1:3]
# ╔═╡ c6163ccc-3a4e-409c-b21d-347858e7a134
md"""
#### Dictionaries
"""
# ╔═╡ d0e2fbe1-fc53-42dd-a984-08607c097b7d
my_dictionary = Dict("a" => 97, "b" => 98, "c" => 99)
# ╔═╡ 84469f69-f90f-4a6e-bb21-de399bcf16a5
my_dictionary["a"]
# ╔═╡ 0057156f-9a23-4ec7-bd2f-f5d1fa28b3fd
my_dictionary["d"] = 100
# ╔═╡ 4d6cecf8-273d-42e6-a405-2fbf45ec8a21
my_dictionary
# ╔═╡ b2f7edec-14d9-4011-8d84-8fc65fd0d266
md"""
## Hands-on tutorial: ideal projectile (no air resistance)
For this hands-on we will study the [motion of an ideal projectile](https://en.wikipedia.org/wiki/Projectile_motion), without air resistance or any other complication.
$(Resource("https://upload.wikimedia.org/wikipedia/commons/8/8f/Moto_parabolico.png"))
The coordinates $$x$$ and $$y$$ of the projectile at time $$t$$, launched at time $$t = 0$$ from position $$(0,0)$$ with launch angle $$\theta$$ and initial speed with magnitude $$v_0$$ are given by the formulas:
```math
\begin{aligned}
x &= v_0 t \cos(\theta), \\
y &= v_0 t \sin(\theta) - \frac{1}{2}gt^2.
\end{aligned}
```
Furthermore, the [time of flight](https://en.wikipedia.org/wiki/Projectile_motion#Time_of_flight_or_total_time_of_the_whole_journey) of the projectile (the time after which the projectile reaches the ground again, i.e. $$y = 0$$) is given by
```math
t = \frac{2v_0\sin(\theta)}{g}.
```
"""
# ╔═╡ 4e81e65a-810e-4aa7-9fc2-2d8b3d0049d6
md"""
where $$g$$ is the constant of the [acceleration of gravity](https://en.wikipedia.org/wiki/Standard_gravity). Let's set a Julia variable with value $$9.81~\mathrm{m}/\mathrm{s}^2$$:
"""
# ╔═╡ 33367bb4-c7d7-4011-9195-1c7b0719ec26
g = 9.81 * u"m/s^2"
# ╔═╡ bb56b280-ba25-43f7-9cc0-c9e3e66570df
md"""
Let's now define the functions for the coordinates $$x$$ and $$y$$, and the time of flight:
"""
# ╔═╡ 2068fa89-94d4-4b8d-802b-aa06258e1f90
displacement_x(t, v₀, θ, g) = v₀ * t * cos(θ)
# ╔═╡ 06de99b9-46fa-4f2a-bdf3-4bcc12f89d0c
displacement_y(t, v₀, θ, g) = v₀ * t * sin(θ) - (g * t ^ 2) / 2
# ╔═╡ 171d9bc1-6d38-4064-93a2-7e2224ab6c72
time_of_flight(v₀, θ, g) = 2 * v₀ * sin(θ) / g
# ╔═╡ cfddaa20-b7d7-4d96-9c89-9f3ade90b9a0
md"""
Finally, let's define the variables which represent the initial speed and launch angle of the projectile.
We can use Pluto sliders for this, to more easily change their values
"""
# ╔═╡ 4f22a3c0-bdb6-415b-87bf-f8775eafa49a
@bind tmp_v₀ Slider(0:0.1:10; show_value=true, default=5)
# ╔═╡ ebf792fa-6cfe-4e43-b50c-800afa835e3f
v₀ = tmp_v₀ * u"m/s"
# ╔═╡ ab591e9d-c2f0-430a-a06d-c86d67e54ec9
@bind θ Slider(range(; start=0, stop=pi/2, length=101); show_value=true, default=pi/4)
# ╔═╡ 16f4e440-a59c-4267-8d9f-58f8c25cf7b2
total_t_range = range(; start=0.0 * u"s", stop=time_of_flight(v₀, θ, g), length=100)
# ╔═╡ 8ad79456-9bfb-4e4a-89f0-0df46775432d
md"""
Move the following slider to see fewer or more points in the plot below.
"""
# ╔═╡ 6ec5e4ea-7b18-400f-b417-fffbb74415cd
@bind t Slider(1:1:length(total_t_range); default=90, show_value=true)
# ╔═╡ 48d63c64-dee4-47a8-87df-8db849cdc18d
let
p = plot(displacement_x.(total_t_range[1:t], v₀, θ, g), displacement_y.(total_t_range[1:t], v₀, θ, g);
ylims=(0, Inf),
label="",
xlabel="x",
ylabel="y",
title="Projectile trajectory", marker=:o,
)
end
# ╔═╡ a34cdc3d-3d38-44ca-9835-7db8b451a468
md"""
### Find condition for maximum range of projectile
[It can be derived](https://en.wikipedia.org/wiki/Range_of_a_projectile#Flat_ground) that the range of the projectile $d$ is given by the formula:
```math
d = \frac{v_0^2 \sin(2\theta)}{\lvert{}g\rvert{}}
```
We want to find the condition under which the range of the projectile is maximum.
We can observe $d$ increases monotonically with the initial speed $v_0$, but the relation with $\theta$ is (slightly!) more complicated, so we can write some code to find the value of $\theta$ which maximizes the range.
To do this, we'll use a package for [numerical optimisation](https://en.wikipedia.org/wiki/Mathematical_optimization) called [`Optim.jl`](https://github.com/JuliaNLSolvers/Optim.jl).
"""
# ╔═╡ d2c5fbe4-a364-4f48-919c-bc192cc01af4
md"""
Let's define the function `total_distance` which represents the range of the projectile as a function of initial speed $v_0$, launch angle $\theta$ and acceleration of gravity $g$
!!! warning "Units in Optim.jl"
Due to the complexity of the operations performed internally, `Optim.jl` doesn't guarantee dimensional consistency of its operations, and as such it isn't compatible with `Unitful.jl` or other unit-checking packages.
We will have to strip the units, in meters, from the return value of `total_distance`.
"""
# ╔═╡ 8cc58d50-40e7-4ac5-81ed-6ce2a02d9d3c
total_distance(v₀, θ, g) = ustrip(u"m", v₀ ^ 2 * sin(2 * θ) / abs(g))
# ╔═╡ 8dfeac4b-47f5-4e7a-a34b-0976ae60304a
md"""
We will use the `optimize` function from `Optmi.jl`, which tries to _minimize_ the value of the objective function passed as input.
However, in our case we want to find when `total_distance` is _maximum_, so to do this we'll try to minimize the function `-total_distance`.
Additionally, we want to vary only the launch angle, $\theta$, while keeping the other parameters fixed.
To do this, we can write an [anonymous function](https://docs.julialang.org/en/v1/manual/functions/#man-anonymous-functions) with the `->` syntax.
Finally, note that the [`Optim.jl` API](https://julianlsolvers.github.io/Optim.jl/stable/user/minimization/) expects the objective function to take a _vector_ of parameters as the only input argument, even if it is only one, so we'll write the anonymous function keeping this in mind (always read the documentation!).
With the drop-down menu below you can choose different minimization algorithims.
"""
# ╔═╡ 4775cb36-5991-4ef5-a150-cb614caf8326
@bind minim_algorithm Select([NelderMead, SimulatedAnnealing, BFGS, LBFGS, ConjugateGradient, GradientDescent, MomentumGradientDescent, AcceleratedGradientDescent, Newton, NewtonTrustRegion]; default=BFGS)
# ╔═╡ c58d3cfb-a234-44cd-89b8-1cb072b84c26
result = optimize(
θ -> -total_distance(v₀, θ[1], g), # Target function to minimize: remember we want to maximize the total distance!
[1.0], # Initial guess for the free parameter(s)
minim_algorithm() # Minimization algorithm (change it with the drop-drown menu)
)
# ╔═╡ 48ce8d19-e01d-4894-8441-5eaca0c675a2
md"""
The value of the initial $\theta$ which maximizes the range of the projectile is thus
"""
# ╔═╡ 36cd06ae-f7fc-4775-ba15-57bf1fcdc709
mod(Optim.minimizer(result)[1], pi/2) # We want angles in the range [0, pi/2]
# ╔═╡ 1722c037-2231-4d0a-ad16-8a63ec374046
md"""
or $\theta = \pi/4 = 45\degree$.
Play with the slider of `θ` above to verify this result.
### Dealing with data
In this section we want to have a look at the tools available in the Julia ecosystem that may be useful for data science.
The reference package in this domain is [`DataFrames.jl`](https://github.com/JuliaData/DataFrames.jl), which has similar functionalities to analogous packages in other packages like `data.frame` in R, or Pandas and Polars in Python.
Let's generate a dataframe which contains the data of a simulated projectile trajectory.
Columns represent the time `t`, and the corresponding coordinates `x` and `y`:
"""
# ╔═╡ 82228247-05a0-40be-b42a-4b1338c18b8b
data = let
v₀ = rand() * 10 * u"m/s"
θ = rand() * pi/2
t_range = range(; start=0u"s", stop=time_of_flight(v₀, θ, g), length=25)
# Some quantities to modulate the random noise
max_height = v₀ ^ 2 * sin(θ) ^ 2 / (2 * abs(g))
d = total_distance(v₀, θ, g) * u"m"
# Generate the data for x and y, and add random noise
x = displacement_x.(t_range, v₀, θ, g) .+ randn.() .* (max_height / 100)
y = displacement_y.(t_range, v₀, θ, g) .+ randn.() .* (d / 100)
# Save in a dataframe
DataFrame(:t => t_range, :x => x, :y => y)
end
# ╔═╡ ee419b81-f381-41d3-bbf1-37b16a4bc0cc
scatter(data.x, data.y; label="", xlabel="x", ylabel="y")
# ╔═╡ 812119b0-6915-4975-9c16-465d00912870
md"""
Then we can use the package [`LsqFit.jl`](https://github.com/JuliaNLSolvers/LsqFit.jl) for doing a simple curve fitting.
!!! warning "Caveats"
For simplicity we'll only fit the data for the vertical displacement, discarding the horizontal one.
Also, for obtaining the best fit, we will have to strip the units.
"""
# ╔═╡ 5368ecc6-d37b-4d88-8fd6-c5fa78da92d1
best_fit = curve_fit(
(t, p) -> ustrip(displacement_y.(t, p[1], p[2], ustrip(g))), # Function to fit
ustrip.(data.t), # data on the "x axis"
ustrip.(data.y), # data on the "y axis"
[0.5, 0.5], # Initial guesses for parameters `v₀` and `θ`: tweak them for better models
)
# ╔═╡ 4a50cea7-f0eb-4a16-bcf2-522753f5712b
let
scatter(ustrip.(data.y); label="data")
plot!(ustrip.(displacement_y.(ustrip.(data.t), best_fit.param[1], best_fit.param[2], ustrip.(g))); label="best model")
end
# ╔═╡ b5dfea75-9910-45db-bb69-67a956b9851b
md"""
#### More about data wrangling
You can use `DataFrames.jl` to do all the classical data wrangling operations .
Check out [the documentation](https://dataframes.juliadata.org/stable/) and play with the data below.
"""
# ╔═╡ 128fa05d-89a9-44fc-96a3-5e0fa54d27db
filter(:x => <(1u"m"), data)
# ╔═╡ 1eb5ec9f-2304-4e91-9e9a-ab57f335434d
describe(data)
# ╔═╡ 7a647923-f4a6-4cbe-ae13-546e55760b36
md"""
## Do you need help with Julia?
* Get in touch with the [ARC Collaborations group](https://www.ucl.ac.uk/advanced-research-computing/collaborations-and-consultancy) for collaboration on funded project
* Engage with the [Julia community](https://julialang.org/community/), e.g. [Discourse web forum](https://discourse.julialang.org/)
* Learning resource:
* [Official documentation](https://docs.julialang.org/)
* [Other learning resources](https://julialang.org/learning/)
* [doggo dot jl YouTube channel](https://www.youtube.com/@doggodotjl)
* [Julia for HPC @ UCL 2024](https://github.com/carstenbauer/JuliaUCL24)
"""
# ╔═╡ 490efc11-a562-4042-bec6-60b6256c2002
md"""
## Notebook setup
This section is not very interesting, it contains only some setup code for this notebook.
"""
# ╔═╡ 67d426e1-5985-4e9a-99cd-b9fae9e0abf1
PlutoUI.TableOfContents(; include_definitions=false, depth=4)
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