### A Pluto.jl notebook ###
# v0.18.1

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

# ╔═╡ 06d2666a-8723-11eb-1395-0febdf3dc2a4
begin
	import ImageMagick
	using Plots, PlutoUI, Colors, Images, StatsBase, Distributions
	using Statistics
end


# ╔═╡ 3a4957ec-8723-11eb-22a0-8b35322596e2
html"""
<div style="
position: absolute;
width: calc(100% - 30px);
border: 50vw solid #282936;
border-top: 500px solid #282936;
border-bottom: none;
box-sizing: content-box;
left: calc(-50vw + 15px);
top: -500px;
height: 500px;
pointer-events: none;
"></div>

<div style="
height: 500px;
width: 100%;
background: #282936;
color: #fff;
padding-top: 68px;
">
<span style="
font-family: Vollkorn, serif;
font-weight: 700;
font-feature-settings: 'lnum', 'pnum';
"> <p style="
font-size: 1.5rem;
opacity: .8;
"><em>Section 2.2</em></p>
<p style="text-align: center; font-size: 2rem;">
<em> Sampling and Random Variables </em>
</p>

<p style="
font-size: 1.5rem;
text-align: center;
opacity: .8;
"><em>Lecture Video</em></p>
<div style="display: flex; justify-content: center;">
<div  notthestyle="position: relative; right: 0; top: 0; z-index: 300;">
<iframe src="https://www.youtube.com/embed/7HrpoFZzITI" width=400 height=250  frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe></div>
</div>
</div>

<style>
body {
overflow-x: hidden;
}
</style>"""

# ╔═╡ 0a70bca4-8723-11eb-1bcf-e9abb9b1ab75
PlutoUI.TableOfContents(aside=true)

# ╔═╡ 472a41d2-8724-11eb-31b3-0b81612f0083
md"""
## Julia: Useful tidbits 
"""

# ╔═╡ aeb99f72-8725-11eb-2efd-d3e44686be03
md"""
The package that each function comes from is shown in brackets, unless it comes from `Base`.
"""

# ╔═╡ 4f9bd326-8724-11eb-2c9b-db1ac9464f1e
md"""

- Julia `Base` library (no `using` required):

  - `if...else...end`
  - `Dict`: Julia's dictionary type
  - `÷` or `div`:  integer division (type `\div` + <tab>)

  - `sum(S)`: sum of elements in the collection `S`, e.g. an array
  - `count(S)`: count the number of true elements of a Boolean collection

  - `rand(S)`: random sampling from a collection `S` 


- `Statistics.jl` (pre-loaded standard library; just needs `using`)

  - `mean(S)`: calculate the mean of a collection `S`
  - `std(S)`:  calculate the standard deviation of a collection `S`



- `StatsBase.jl`:
  - `countmap`


- `Plots.jl`:

  - `histogram(x)`: Plot a histogram of data vector `x` [Plots]

  - `bar(d)`: Plot a bar graph of categorical data [Plots]


- `Colors.jl`:
  - `distinguishable_colors(n)`: Make `n` distinguishable colours [Colors]

"""

# ╔═╡ db2d25de-86b1-11eb-0c78-d1ee52e019ca
md"""
## Random sampling with `rand`
"""

# ╔═╡ e33fe4c8-86b1-11eb-1031-cf45717a3dc9
md"""
The `rand` function in Julia is quite versatile: it tries to generate, or **sample**, a random object from the argument that you pass in:
"""

# ╔═╡ f49191a2-86b1-11eb-3eab-b392ba058415
rand(1:6)

# ╔═╡ 1abda6c4-86b2-11eb-2aa3-4d1148bb52b7
rand([2, 3, 5, 7, 11])

# ╔═╡ 30b12f28-86b2-11eb-087b-8d50ec429b89
rand("MIT")

# ╔═╡ 4ce946c6-86b2-11eb-1820-0728798665ab
rand('a':'z')

# ╔═╡ fae3d138-8743-11eb-1014-b3a2a9b49aba
typeof('a':'z')

# ╔═╡ 6cdea3ae-86b2-11eb-107a-17bea3f54bc9
rand()   # random number between 0 and 1

# ╔═╡ 1c769d58-8744-11eb-3bd3-ab11ea1503ed
rand

# ╔═╡ 297fdfa0-8744-11eb-1934-9fe31e8be534
methods(rand);

# ╔═╡ 776ec3f2-86b3-11eb-0216-9b71d07e99f3
md"""
We can take random objects from a collection of objects of *any* type, for example:
"""

# ╔═╡ 5fcf8d4e-8744-11eb-080e-cba749004b08
distinguishable_colors(10)

# ╔═╡ 4898106a-8744-11eb-128a-35fec741e6b8
typeof(distinguishable_colors(10))

# ╔═╡ 0926366a-86b2-11eb-0f6d-31ae6981598c
rand(distinguishable_colors(3))   # from Colors.jl package

# ╔═╡ 7c8d7b72-86b2-11eb-2dd5-4f77bc5fb8ff
md"""
### Several random objects
"""

# ╔═╡ 2090b7f2-86b3-11eb-2a99-ed98800e1d63
md"""
To sample several random objects from the same collection, we could write an array comprehension:
"""

# ╔═╡ a7dff55c-86b2-11eb-330f-3d6279347095
[rand(1:6) for i in 1:10]

# ╔═╡ 2db33022-86b3-11eb-17dd-13c534ac9892
md"""
But in fact, just adding another argument to `rand` does the trick:
"""

# ╔═╡ 0de6f23e-86b2-11eb-39ff-318bbc4ecbcf
rand(1:6, 10)

# ╔═╡ 36c3da4a-86b3-11eb-0b2f-fffdde06fcd2
md"""
In fact, you can also generate not only random vectors, but also random matrices:
"""

# ╔═╡ 940c2bf6-86b2-11eb-0a5e-011abdd6352b
rand(1:6, 10, 12)

# ╔═╡ 5a4e7fc4-86b3-11eb-3376-0941b79574aa
rand(distinguishable_colors(5), 10, 10)

# ╔═╡ c433104e-86b3-11eb-20bb-af608bb281cc
md"""
We can also use random images:
"""

# ╔═╡ 78dc94e2-8723-11eb-1ff2-bb7104b62033
penny_image = load(download("https://www.usacoinbook.com/us-coins/lincoln-memorial-cent.jpg"))

# ╔═╡ bb1465c4-8723-11eb-1abc-bdb5a7028cf2
begin
	head = penny_image[:, 1:end÷2] # end -> last index of a collection
	tail = penny_image[:, end÷2:end]
end;

# ╔═╡ e04f3828-8723-11eb-3452-09f821391ad0
rand( [head, tail], 5, 5)

# ╔═╡ b7793f7a-8726-11eb-11d8-cd928f1a3645
md"""
## Uniform sampling
"""

# ╔═╡ ba80cc78-8726-11eb-2f33-e364f19295d8
md"""
So far, each use of `rand` has done **uniform sampling**, i.e. each possible object as the *same* probability.

Let's *count* heads and tails using the `countmap` function from the `StatsBase.jl` package:
"""

# ╔═╡ 8fe715e4-8727-11eb-2e7f-15b723bb8d9d
tosses = rand( ["head", "tail"], 10000)

# ╔═╡ 9da4e6f4-8727-11eb-08cb-d55e3bbff0e4
toss_counts = countmap(tosses)

# ╔═╡ 9647840c-8745-11eb-33fc-898ae351ddfe
"tail" => 3

# ╔═╡ 9d782aa6-8745-11eb-034e-0f8b01be987b
typeof("tail" => 3)

# ╔═╡ a693582c-8745-11eb-261b-ef79e503420e
toss_counts["tail"]

# ╔═╡ c3255d7a-8727-11eb-0421-d99faf27c7b0
md"""

We see that `countmap` returns a **dictionary** (`Dict`), which maps **keys** to **values**; we will say more about dictionaries in another chapter.
"""

# ╔═╡ e2037806-8727-11eb-3c36-1500f1dd545b
prob_tail = toss_counts["tail"] / length(tosses)

# ╔═╡ 28ff621c-8728-11eb-2ec0-2579cb313315
md"""
As we increase the number of tosses, we "expect" the probability to get closer to $1/2$.
"""

# ╔═╡ 57125768-8728-11eb-3c3b-1dd37e1ac189
md"""
## Tossing a weighted coin
"""

# ╔═╡ 6a439c78-8728-11eb-1969-27a19d254704
md"""
How could we model a coin that is **weighted**, so that it is more likely to come up heads? We want to assign a probability $p = 0.7$ to heads, and $q = 0.3$ to tails.
"""

# ╔═╡ 9f8ac08c-8728-11eb-10ad-f93ca225ce38
md"""
One way would be to generate random integers between 1 and 10 and assign heads to a subset of the possible results with the desired probability, e.g. 1:7 get heads, and 8:10 get tails:
"""

# ╔═╡ 062b400a-8729-11eb-16c5-235cef648edb
function simple_weighted_coin()
	outcome = if rand(1:10) ≤ 7
		"heads"
	else      # could have elseif
		"tails"
	end
	
	return outcome
end

# ╔═╡ 7c099606-8746-11eb-37fe-c3befde06e9d
function simple_weighted_coin2()
	if rand(1:10) ≤ 7
		"heads"
	else      # could have elseif
		"tails"
	end
end

# ╔═╡ 97cb3dde-8746-11eb-0b00-690f20cb26dc
result = for i in 1:10
	
end

# ╔═╡ 9d8cdc8e-8746-11eb-2b9a-b30a52026f09
result == nothing

# ╔═╡ 81a30c9e-8746-11eb-38c8-9be4f6ba2e80
simple_weighted_coin2()

# ╔═╡ 2a81ba88-8729-11eb-3dcb-1db26e468066
md"""
(Note that `if` statements have a return value in Julia.)
"""

# ╔═╡ 5ea5838a-8729-11eb-1749-030533fb0656
md"""
How could we generalise this to an arbitrary probability $p ∈ [0, 1]$?

We can generate a uniform floating-point number between 0 and 1, and check if it is less than $p$. This is called a **Bernoulli trial**.
"""

# ╔═╡ c9f21046-8746-11eb-27c6-910807240fd1
rand()

# ╔═╡ 806e6aae-8729-11eb-19ea-33722c60edf0
rand() < 0.314159

# ╔═╡ 90642564-8729-11eb-3cd7-b3d41a1553b4
md"""
Note that comparisons also return a value in Julia. Here we have switched from heads/tails to true/false as the output.
"""

# ╔═╡ 9a426a20-8729-11eb-0c0f-31e1d4dc91bc
md"""
Let's make that into a function:
"""

# ╔═╡ a4870b14-8729-11eb-20ee-e531d4a7108d
bernoulli(p) = rand() < p

# ╔═╡ 081e3796-8747-11eb-32ec-dfd998605737
bernoulli(0.7)

# ╔═╡ 008a40d2-872a-11eb-224d-5b3331f29c99
md"""
p = $(@bind p Slider(0.0:0.01:1.0, show_value=true, default=0.7))
"""

# ╔═╡ baed5908-8729-11eb-00e0-9f749406c30c
countmap( [bernoulli(p) for i in 1:1000] )

# ╔═╡ ed94eae8-86b3-11eb-3f1b-15c7a54903f5
md"""
## Bar charts and histograms
"""

# ╔═╡ f20504de-86b3-11eb-3125-3140e0e060b0
md"""
Once we have generated several random objects, it is natural to want to count **how many times** each one occurred.
"""

# ╔═╡ 7f1a3766-86b4-11eb-353f-b13acaf1503e
md"""
Let's roll a (fair) die 1000 times:
"""

# ╔═╡ 371838f8-86b4-11eb-1633-8d282e42a085
md"""
An obvious way to find the counts would be to run through the data looking for 1s, then run through again looking for 2s, etc.:
"""

# ╔═╡ 9e9d3556-86b5-11eb-3dfb-916e625da235
md"""
Note that this is *not* the most efficient algorithm!
"""

# ╔═╡ 90844738-8738-11eb-0604-3d23662152d9
md"""
We can plot **categorical data** using a **bar chart**, `bar` in Plots.jl. This counts each discrete item.
"""

# ╔═╡ 02d03642-86b4-11eb-365a-63ff61ddd3b5
rolls = rand(1:6, 100)   # try modifying 100 by adding more zeros

# ╔═╡ 94688c1a-8747-11eb-13a3-eb36f731674c
rolls .== 1

# ╔═╡ ad701cdc-8747-11eb-3804-63a0fc881547
count(rolls .== 1)

# ╔═╡ 2405eb68-86b4-11eb-31b0-dff8e355d88e
counts = [count(rolls .== i) for i in 1:6]

# ╔═╡ 2d71fa88-86b5-11eb-0e55-35566c2246d7
begin
	bar(counts, alpha=0.5, leg=false, size=(500, 300))
	hline!([length(rolls) / 6], ls=:dash)
	title!("number of die rolls = $(length(rolls))")
	ylims!(0, length(rolls) / 3)
end

# ╔═╡ cb8a9762-86b1-11eb-0484-6b6cc8b1b14c
md"""
## Probability densities 

### Rolling multiple dice
"""

# ╔═╡ d0c9814e-86b1-11eb-2f29-1d041bccc649
roll_dice(n) = sum( rand(1:6, n) )

# ╔═╡ 7a16b674-86b7-11eb-3aa5-83712cdc8580
trials = 10^6

# ╔═╡ 2bfa712a-8738-11eb-3248-6f9bb93154e8
md"""
### Converging shape
"""

# ╔═╡ 6c133ab6-86b7-11eb-15f6-7780da5afc31
md"""
n = $(@bind n Slider(1:50, show_value=true))
"""

# ╔═╡ b81b1090-8735-11eb-3a52-2dca4d4ed472
experiment() = roll_dice(n) 

# experiment() = sum([randn()^2 for i in 1:n])

# ╔═╡ e8e811de-86b6-11eb-1cbf-6d4aeaee510a
data = [experiment() for t in 1:trials]

# ╔═╡ e4abcbf4-86b8-11eb-167a-d97c61e07837
data

# ╔═╡ 514f6be0-86b8-11eb-30c9-d1020f783afe
histogram(data, alpha=0.5, legend=false, bins=200, c=:lightsalmon1, title="n = $n")  
# c = RGB(0.1, 0.2, 0.3))

# ╔═╡ a15fc456-8738-11eb-25bd-b15c2b16d461
md"""
Here we have switched from a bar chart to a **histogram**, which counts the number of items falling into a given range or **bin**. When $n$ is small this tends to look like a bar chart, but it looks like a "smooth bar chart" as $n$ gets larger, due to the **aggregation**.
"""

# ╔═╡ dd753568-8736-11eb-1f20-1b81110ae807
md"""
Does the above histogram look like a bell to you?
"""

# ╔═╡ 8ab9001a-8737-11eb-1009-5717fbe83af7
begin
	bell = load(download("https://encrypted-tbn0.gstatic.com/images?q=tbn:ANd9GcRmxRAIQt_L-X99A_4FoP3vsC-l_WHlC3TtAw&usqp=CAU"))
	
	bell[1:end*9÷10, :]
end

# ╔═╡ f8b2dd20-8737-11eb-1593-43659c693109
md"""
### Normalising the $y$-axis
"""

# ╔═╡ 89bb366a-8737-11eb-10a6-e754ee817f9a
md"""
Notice that the *shape* of the curve seems to converge to a bell-shaped curve, but the axes do not. What can we do about this?

We need to **normalise** so that the total shaded *area* is 1. We can use the histogram option `norm=true` to do so.
"""

# ╔═╡ 14f0a090-8737-11eb-0ccf-391249267401
histogram(data, alpha=0.5, legend=false, bins=50, norm=true,
			c=:lightsalmon1, title="n = $n", ylims=(0, 0.05))  


# ╔═╡ e305467e-8738-11eb-1213-eb11aaebe151
md"""
### Normalising the $x$ axis
"""

# ╔═╡ e8341288-8738-11eb-27ae-0795fa7e4a7e
md"""
As we changed $n$ above, the range of values on the $x$ axis also changed. We want to find a way to rescale $x$ at the same time so that both axes and shape stop changing as we increase $n$.

We need to make sure that the data is centred in the same place -- we will choose 0. And we need to make sure that the width is the same -- we will divide by the standard deviation.
"""

# ╔═╡ 07e8ae34-873b-11eb-1df2-175392ac4678
σ = std(data)

# ╔═╡ 77616afe-873a-11eb-3f11-1bc417f53138
mean(data), std(data)

# ╔═╡ 16023bea-8739-11eb-1e32-79f2d006b093
normalised_data = ( data .- mean(data) ) ./ std(data)   

# ╔═╡ bfc52118-873b-11eb-0fbb-952c60bc7cc2
histogram(normalised_data, bins=-5 - (1/(2σ)):(1/σ):5, norm=true,
	alpha=0.5, leg=false, ylim=(0, 0.41), size=(500, 300))

# ╔═╡ aa6126f8-873b-11eb-3b4a-0f96fe07b7fb
plot!(x -> exp(-x^2 / 2) / √(2π), lw=2, c=:red, alpha=0.5)

# ╔═╡ 308547c6-873d-11eb-3a42-833f8bf496ae
md"""
Note that in the limit, the data becomes *continuous*, no longer discrete. The probability of any particular value is 0 ! We then talk about a **probability density function**, $f(x)$; the integral of this function over the interval $[a, b]$ gives the probability of being in that interval.
"""

# ╔═╡ cb2fb68e-8749-11eb-29ea-9729ac0c63b4
md"""
### Options for plotting functions
"""

# ╔═╡ e0a1863e-8735-11eb-1182-1b3c59b1e05a
md"""
Other options for the `histogram` function options:
- `legend=false` turns off the legend (key), i.e. the plot labels
- `bins=50` specifies the number of **bins**; can also be a vector of bin edges
- `linetype=:stephist`: use steps instead of bars


- `alpha`: a general plot option specifying transparency (0=invisible; 1=opaque)
- `c` or `color`: a general plot option for the colour
- `lw`: line width (default=1)


There are several different ways to specify colours. [Here](http://juliagraphics.github.io/Colors.jl/stable/namedcolors/) is a list of named colours, but you can also specify `RGB(0.1, 0.2, 0.3)`.
"""

# ╔═╡ 900af0c8-86ba-11eb-2270-71b1869b9a1a
md"""
Note that `linetype=:stephist` will give a stepped version of the histogram:
"""

# ╔═╡ be6e4c00-873c-11eb-1413-5326aba54216
md"""
## Sampling from other distributions
"""

# ╔═╡ 9a1136c2-873c-11eb-124f-c3939972ce4a
md"""
dof = $(@bind dof Slider(1:50, show_value=true))  
"""

# ╔═╡ e01b6f70-873c-11eb-04a1-ad8e86578982
chisq_data = rand( Chisq(dof), 100000 )

# ╔═╡ b5251f76-873c-11eb-38cb-7db300c8fe3c
histogram( chisq_data, norm=true, bins=100, size=(500, 300), leg=false, alpha=0.5,
	xlims=(0, 10*√(dof)))

# ╔═╡ da62fd1c-873c-11eb-0758-e7cb48e964f1
histogram( [ sum( randn().^2 for _=1:dof )  for _ = 1:100000], norm=true,
	alpha=0.5, leg=false)

# ╔═╡ 00000000-0000-0000-0000-000000000001
PLUTO_PROJECT_TOML_CONTENTS = """
[deps]
Colors = "5ae59095-9a9b-59fe-a467-6f913c188581"
Distributions = "31c24e10-a181-5473-b8eb-7969acd0382f"
ImageMagick = "6218d12a-5da1-5696-b52f-db25d2ecc6d1"
Images = "916415d5-f1e6-5110-898d-aaa5f9f070e0"
Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8"
Statistics = "10745b16-79ce-11e8-11f9-7d13ad32a3b2"
StatsBase = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91"

[compat]
Colors = "~0.12.8"
Distributions = "~0.25.16"
ImageMagick = "~1.2.1"
Images = "~0.24.1"
Plots = "~1.21.3"
PlutoUI = "~0.7.9"
StatsBase = "~0.33.10"
"""

# ╔═╡ 00000000-0000-0000-0000-000000000002
PLUTO_MANIFEST_TOML_CONTENTS = """
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[[Base64]]
uuid = "2a0f44e3-6c83-55bd-87e4-b1978d98bd5f"

[[Bzip2_jll]]
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git-tree-sha1 = "19a35467a82e236ff51bc17a3a44b69ef35185a2"
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[[CEnum]]
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[[Cairo_jll]]
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git-tree-sha1 = "4b859a208b2397a7a623a03449e4636bdb17bcf2"
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[[CatIndices]]
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[[ChainRulesCore]]
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[[ColorSchemes]]
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git-tree-sha1 = "9995eb3977fbf67b86d0a0a0508e83017ded03f2"
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[[ColorTypes]]
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[[ColorVectorSpace]]
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[[Colors]]
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[[Compat]]
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git-tree-sha1 = "4866e381721b30fac8dda4c8cb1d9db45c8d2994"
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[[CompilerSupportLibraries_jll]]
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[[ComputationalResources]]
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[[Contour]]
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[[CoordinateTransformations]]
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[[CustomUnitRanges]]
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[[DataAPI]]
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[[DataStructures]]
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[[DataValueInterfaces]]
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[[Dates]]
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[[DelimitedFiles]]
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[[Distances]]
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[[Distributed]]
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uuid = "8ba89e20-285c-5b6f-9357-94700520ee1b"

[[Distributions]]
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git-tree-sha1 = "f4efaa4b5157e0cdb8283ae0b5428bc9208436ed"
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[[DocStringExtensions]]
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[[Downloads]]
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[[EarCut_jll]]
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git-tree-sha1 = "3f3a2501fa7236e9b911e0f7a588c657e822bb6d"
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[[EllipsisNotation]]
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[[Expat_jll]]
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git-tree-sha1 = "b3bfd02e98aedfa5cf885665493c5598c350cd2f"
uuid = "2e619515-83b5-522b-bb60-26c02a35a201"
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[[FFMPEG]]
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[[FFMPEG_jll]]
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git-tree-sha1 = "d8a578692e3077ac998b50c0217dfd67f21d1e5f"
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[[FFTViews]]
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[[FFTW]]
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[[FFTW_jll]]
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git-tree-sha1 = "3676abafff7e4ff07bbd2c42b3d8201f31653dcc"
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[[FileIO]]
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[[FillArrays]]
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[[FixedPointNumbers]]
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[[Fontconfig_jll]]
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git-tree-sha1 = "21efd19106a55620a188615da6d3d06cd7f6ee03"
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[[Formatting]]
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[[FreeType2_jll]]
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git-tree-sha1 = "87eb71354d8ec1a96d4a7636bd57a7347dde3ef9"
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[[FriBidi_jll]]
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[[GLFW_jll]]
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[[GR]]
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git-tree-sha1 = "182da592436e287758ded5be6e32c406de3a2e47"
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version = "0.58.1"

[[GR_jll]]
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git-tree-sha1 = "ef49a187604f865f4708c90e3f431890724e9012"
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[[GeometryBasics]]
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[[Gettext_jll]]
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git-tree-sha1 = "9b02998aba7bf074d14de89f9d37ca24a1a0b046"
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[[Glib_jll]]
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git-tree-sha1 = "a32d672ac2c967f3deb8a81d828afc739c838a06"
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[[Graphics]]
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[[Graphite2_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "344bf40dcab1073aca04aa0df4fb092f920e4011"
uuid = "3b182d85-2403-5c21-9c21-1e1f0cc25472"
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[[Grisu]]
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uuid = "42e2da0e-8278-4e71-bc24-59509adca0fe"
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[[HTTP]]
deps = ["Base64", "Dates", "IniFile", "Logging", "MbedTLS", "NetworkOptions", "Sockets", "URIs"]
git-tree-sha1 = "60ed5f1643927479f845b0135bb369b031b541fa"
uuid = "cd3eb016-35fb-5094-929b-558a96fad6f3"
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[[HarfBuzz_jll]]
deps = ["Artifacts", "Cairo_jll", "Fontconfig_jll", "FreeType2_jll", "Glib_jll", "Graphite2_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Pkg"]
git-tree-sha1 = "129acf094d168394e80ee1dc4bc06ec835e510a3"
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[[IdentityRanges]]
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[[IfElse]]
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[[ImageAxes]]
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[[ImageContrastAdjustment]]
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[[ImageCore]]
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[[ImageDistances]]
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[[ImageFiltering]]
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[[ImageIO]]
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[[ImageMagick]]
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[[ImageMagick_jll]]
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uuid = "c73af94c-d91f-53ed-93a7-00f77d67a9d7"
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[[ImageMetadata]]
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[[ImageMorphology]]
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[[ImageQualityIndexes]]
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[[ImageShow]]
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[[ImageTransformations]]
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[[Images]]
deps = ["AxisArrays", "Base64", "ColorVectorSpace", "FileIO", "Graphics", "ImageAxes", "ImageContrastAdjustment", "ImageCore", "ImageDistances", "ImageFiltering", "ImageIO", "ImageMagick", "ImageMetadata", "ImageMorphology", "ImageQualityIndexes", "ImageShow", "ImageTransformations", "IndirectArrays", "OffsetArrays", "Random", "Reexport", "SparseArrays", "StaticArrays", "Statistics", "StatsBase", "TiledIteration"]
git-tree-sha1 = "8b714d5e11c91a0d945717430ec20f9251af4bd2"
uuid = "916415d5-f1e6-5110-898d-aaa5f9f070e0"
version = "0.24.1"

[[Imath_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "87f7662e03a649cffa2e05bf19c303e168732d3e"
uuid = "905a6f67-0a94-5f89-b386-d35d92009cd1"
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[[IndirectArrays]]
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uuid = "9b13fd28-a010-5f03-acff-a1bbcff69959"
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[[Inflate]]
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uuid = "d25df0c9-e2be-5dd7-82c8-3ad0b3e990b9"
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[[IniFile]]
deps = ["Test"]
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uuid = "83e8ac13-25f8-5344-8a64-a9f2b223428f"
version = "0.5.0"

[[IntelOpenMP_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "d979e54b71da82f3a65b62553da4fc3d18c9004c"
uuid = "1d5cc7b8-4909-519e-a0f8-d0f5ad9712d0"
version = "2018.0.3+2"

[[InteractiveUtils]]
deps = ["Markdown"]
uuid = "b77e0a4c-d291-57a0-90e8-8db25a27a240"

[[Interpolations]]
deps = ["AxisAlgorithms", "ChainRulesCore", "LinearAlgebra", "OffsetArrays", "Random", "Ratios", "Requires", "SharedArrays", "SparseArrays", "StaticArrays", "WoodburyMatrices"]
git-tree-sha1 = "61aa005707ea2cebf47c8d780da8dc9bc4e0c512"
uuid = "a98d9a8b-a2ab-59e6-89dd-64a1c18fca59"
version = "0.13.4"

[[IntervalSets]]
deps = ["Dates", "EllipsisNotation", "Statistics"]
git-tree-sha1 = "3cc368af3f110a767ac786560045dceddfc16758"
uuid = "8197267c-284f-5f27-9208-e0e47529a953"
version = "0.5.3"

[[IrrationalConstants]]
git-tree-sha1 = "f76424439413893a832026ca355fe273e93bce94"
uuid = "92d709cd-6900-40b7-9082-c6be49f344b6"
version = "0.1.0"

[[IterTools]]
git-tree-sha1 = "05110a2ab1fc5f932622ffea2a003221f4782c18"
uuid = "c8e1da08-722c-5040-9ed9-7db0dc04731e"
version = "1.3.0"

[[IteratorInterfaceExtensions]]
git-tree-sha1 = "a3f24677c21f5bbe9d2a714f95dcd58337fb2856"
uuid = "82899510-4779-5014-852e-03e436cf321d"
version = "1.0.0"

[[JLLWrappers]]
deps = ["Preferences"]
git-tree-sha1 = "642a199af8b68253517b80bd3bfd17eb4e84df6e"
uuid = "692b3bcd-3c85-4b1f-b108-f13ce0eb3210"
version = "1.3.0"

[[JSON]]
deps = ["Dates", "Mmap", "Parsers", "Unicode"]
git-tree-sha1 = "8076680b162ada2a031f707ac7b4953e30667a37"
uuid = "682c06a0-de6a-54ab-a142-c8b1cf79cde6"
version = "0.21.2"

[[JpegTurbo_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "d735490ac75c5cb9f1b00d8b5509c11984dc6943"
uuid = "aacddb02-875f-59d6-b918-886e6ef4fbf8"
version = "2.1.0+0"

[[LAME_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "f6250b16881adf048549549fba48b1161acdac8c"
uuid = "c1c5ebd0-6772-5130-a774-d5fcae4a789d"
version = "3.100.1+0"

[[LERC_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "bf36f528eec6634efc60d7ec062008f171071434"
uuid = "88015f11-f218-50d7-93a8-a6af411a945d"
version = "3.0.0+1"

[[LZO_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "e5b909bcf985c5e2605737d2ce278ed791b89be6"
uuid = "dd4b983a-f0e5-5f8d-a1b7-129d4a5fb1ac"
version = "2.10.1+0"

[[LaTeXStrings]]
git-tree-sha1 = "c7f1c695e06c01b95a67f0cd1d34994f3e7db104"
uuid = "b964fa9f-0449-5b57-a5c2-d3ea65f4040f"
version = "1.2.1"

[[Latexify]]
deps = ["Formatting", "InteractiveUtils", "LaTeXStrings", "MacroTools", "Markdown", "Printf", "Requires"]
git-tree-sha1 = "a4b12a1bd2ebade87891ab7e36fdbce582301a92"
uuid = "23fbe1c1-3f47-55db-b15f-69d7ec21a316"
version = "0.15.6"

[[LazyArtifacts]]
deps = ["Artifacts", "Pkg"]
uuid = "4af54fe1-eca0-43a8-85a7-787d91b784e3"

[[LibCURL]]
deps = ["LibCURL_jll", "MozillaCACerts_jll"]
uuid = "b27032c2-a3e7-50c8-80cd-2d36dbcbfd21"

[[LibCURL_jll]]
deps = ["Artifacts", "LibSSH2_jll", "Libdl", "MbedTLS_jll", "Zlib_jll", "nghttp2_jll"]
uuid = "deac9b47-8bc7-5906-a0fe-35ac56dc84c0"

[[LibGit2]]
deps = ["Base64", "NetworkOptions", "Printf", "SHA"]
uuid = "76f85450-5226-5b5a-8eaa-529ad045b433"

[[LibSSH2_jll]]
deps = ["Artifacts", "Libdl", "MbedTLS_jll"]
uuid = "29816b5a-b9ab-546f-933c-edad1886dfa8"

[[Libdl]]
uuid = "8f399da3-3557-5675-b5ff-fb832c97cbdb"

[[Libffi_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "0b4a5d71f3e5200a7dff793393e09dfc2d874290"
uuid = "e9f186c6-92d2-5b65-8a66-fee21dc1b490"
version = "3.2.2+1"

[[Libgcrypt_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libgpg_error_jll", "Pkg"]
git-tree-sha1 = "64613c82a59c120435c067c2b809fc61cf5166ae"
uuid = "d4300ac3-e22c-5743-9152-c294e39db1e4"
version = "1.8.7+0"

[[Libglvnd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll", "Xorg_libXext_jll"]
git-tree-sha1 = "7739f837d6447403596a75d19ed01fd08d6f56bf"
uuid = "7e76a0d4-f3c7-5321-8279-8d96eeed0f29"
version = "1.3.0+3"

[[Libgpg_error_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "c333716e46366857753e273ce6a69ee0945a6db9"
uuid = "7add5ba3-2f88-524e-9cd5-f83b8a55f7b8"
version = "1.42.0+0"

[[Libiconv_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "42b62845d70a619f063a7da093d995ec8e15e778"
uuid = "94ce4f54-9a6c-5748-9c1c-f9c7231a4531"
version = "1.16.1+1"

[[Libmount_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "9c30530bf0effd46e15e0fdcf2b8636e78cbbd73"
uuid = "4b2f31a3-9ecc-558c-b454-b3730dcb73e9"
version = "2.35.0+0"

[[Libtiff_jll]]
deps = ["Artifacts", "JLLWrappers", "JpegTurbo_jll", "LERC_jll", "Libdl", "Pkg", "Zlib_jll", "Zstd_jll"]
git-tree-sha1 = "c9551dd26e31ab17b86cbd00c2ede019c08758eb"
uuid = "89763e89-9b03-5906-acba-b20f662cd828"
version = "4.3.0+1"

[[Libuuid_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "7f3efec06033682db852f8b3bc3c1d2b0a0ab066"
uuid = "38a345b3-de98-5d2b-a5d3-14cd9215e700"
version = "2.36.0+0"

[[LinearAlgebra]]
deps = ["Libdl", "libblastrampoline_jll"]
uuid = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"

[[LogExpFunctions]]
deps = ["ChainRulesCore", "DocStringExtensions", "IrrationalConstants", "LinearAlgebra"]
git-tree-sha1 = "34dc30f868e368f8a17b728a1238f3fcda43931a"
uuid = "2ab3a3ac-af41-5b50-aa03-7779005ae688"
version = "0.3.3"

[[Logging]]
uuid = "56ddb016-857b-54e1-b83d-db4d58db5568"

[[MKL_jll]]
deps = ["Artifacts", "IntelOpenMP_jll", "JLLWrappers", "LazyArtifacts", "Libdl", "Pkg"]
git-tree-sha1 = "5455aef09b40e5020e1520f551fa3135040d4ed0"
uuid = "856f044c-d86e-5d09-b602-aeab76dc8ba7"
version = "2021.1.1+2"

[[MacroTools]]
deps = ["Markdown", "Random"]
git-tree-sha1 = "5a5bc6bf062f0f95e62d0fe0a2d99699fed82dd9"
uuid = "1914dd2f-81c6-5fcd-8719-6d5c9610ff09"
version = "0.5.8"

[[MappedArrays]]
git-tree-sha1 = "e8b359ef06ec72e8c030463fe02efe5527ee5142"
uuid = "dbb5928d-eab1-5f90-85c2-b9b0edb7c900"
version = "0.4.1"

[[Markdown]]
deps = ["Base64"]
uuid = "d6f4376e-aef5-505a-96c1-9c027394607a"

[[MbedTLS]]
deps = ["Dates", "MbedTLS_jll", "Random", "Sockets"]
git-tree-sha1 = "1c38e51c3d08ef2278062ebceade0e46cefc96fe"
uuid = "739be429-bea8-5141-9913-cc70e7f3736d"
version = "1.0.3"

[[MbedTLS_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "c8ffd9c3-330d-5841-b78e-0817d7145fa1"

[[Measures]]
git-tree-sha1 = "e498ddeee6f9fdb4551ce855a46f54dbd900245f"
uuid = "442fdcdd-2543-5da2-b0f3-8c86c306513e"
version = "0.3.1"

[[Missings]]
deps = ["DataAPI"]
git-tree-sha1 = "bf210ce90b6c9eed32d25dbcae1ebc565df2687f"
uuid = "e1d29d7a-bbdc-5cf2-9ac0-f12de2c33e28"
version = "1.0.2"

[[Mmap]]
uuid = "a63ad114-7e13-5084-954f-fe012c677804"

[[MosaicViews]]
deps = ["MappedArrays", "OffsetArrays", "PaddedViews", "StackViews"]
git-tree-sha1 = "b34e3bc3ca7c94914418637cb10cc4d1d80d877d"
uuid = "e94cdb99-869f-56ef-bcf0-1ae2bcbe0389"
version = "0.3.3"

[[MozillaCACerts_jll]]
uuid = "14a3606d-f60d-562e-9121-12d972cd8159"

[[NaNMath]]
git-tree-sha1 = "bfe47e760d60b82b66b61d2d44128b62e3a369fb"
uuid = "77ba4419-2d1f-58cd-9bb1-8ffee604a2e3"
version = "0.3.5"

[[Netpbm]]
deps = ["ColorVectorSpace", "FileIO", "ImageCore"]
git-tree-sha1 = "09589171688f0039f13ebe0fdcc7288f50228b52"
uuid = "f09324ee-3d7c-5217-9330-fc30815ba969"
version = "1.0.1"

[[NetworkOptions]]
uuid = "ca575930-c2e3-43a9-ace4-1e988b2c1908"

[[OffsetArrays]]
deps = ["Adapt"]
git-tree-sha1 = "c870a0d713b51e4b49be6432eff0e26a4325afee"
uuid = "6fe1bfb0-de20-5000-8ca7-80f57d26f881"
version = "1.10.6"

[[Ogg_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "887579a3eb005446d514ab7aeac5d1d027658b8f"
uuid = "e7412a2a-1a6e-54c0-be00-318e2571c051"
version = "1.3.5+1"

[[OpenBLAS_jll]]
deps = ["Artifacts", "CompilerSupportLibraries_jll", "Libdl"]
uuid = "4536629a-c528-5b80-bd46-f80d51c5b363"

[[OpenEXR]]
deps = ["Colors", "FileIO", "OpenEXR_jll"]
git-tree-sha1 = "327f53360fdb54df7ecd01e96ef1983536d1e633"
uuid = "52e1d378-f018-4a11-a4be-720524705ac7"
version = "0.3.2"

[[OpenEXR_jll]]
deps = ["Artifacts", "Imath_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "923319661e9a22712f24596ce81c54fc0366f304"
uuid = "18a262bb-aa17-5467-a713-aee519bc75cb"
version = "3.1.1+0"

[[OpenSSL_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "15003dcb7d8db3c6c857fda14891a539a8f2705a"
uuid = "458c3c95-2e84-50aa-8efc-19380b2a3a95"
version = "1.1.10+0"

[[OpenSpecFun_jll]]
deps = ["Artifacts", "CompilerSupportLibraries_jll", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "13652491f6856acfd2db29360e1bbcd4565d04f1"
uuid = "efe28fd5-8261-553b-a9e1-b2916fc3738e"
version = "0.5.5+0"

[[Opus_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "51a08fb14ec28da2ec7a927c4337e4332c2a4720"
uuid = "91d4177d-7536-5919-b921-800302f37372"
version = "1.3.2+0"

[[OrderedCollections]]
git-tree-sha1 = "85f8e6578bf1f9ee0d11e7bb1b1456435479d47c"
uuid = "bac558e1-5e72-5ebc-8fee-abe8a469f55d"
version = "1.4.1"

[[PCRE_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "b2a7af664e098055a7529ad1a900ded962bca488"
uuid = "2f80f16e-611a-54ab-bc61-aa92de5b98fc"
version = "8.44.0+0"

[[PDMats]]
deps = ["LinearAlgebra", "SparseArrays", "SuiteSparse"]
git-tree-sha1 = "4dd403333bcf0909341cfe57ec115152f937d7d8"
uuid = "90014a1f-27ba-587c-ab20-58faa44d9150"
version = "0.11.1"

[[PNGFiles]]
deps = ["Base64", "CEnum", "ImageCore", "IndirectArrays", "OffsetArrays", "libpng_jll"]
git-tree-sha1 = "e14c485f6beee0c7a8dcf6128bf70b85f1fe201e"
uuid = "f57f5aa1-a3ce-4bc8-8ab9-96f992907883"
version = "0.3.9"

[[PaddedViews]]
deps = ["OffsetArrays"]
git-tree-sha1 = "646eed6f6a5d8df6708f15ea7e02a7a2c4fe4800"
uuid = "5432bcbf-9aad-5242-b902-cca2824c8663"
version = "0.5.10"

[[Parameters]]
deps = ["OrderedCollections", "UnPack"]
git-tree-sha1 = "2276ac65f1e236e0a6ea70baff3f62ad4c625345"
uuid = "d96e819e-fc66-5662-9728-84c9c7592b0a"
version = "0.12.2"

[[Parsers]]
deps = ["Dates"]
git-tree-sha1 = "438d35d2d95ae2c5e8780b330592b6de8494e779"
uuid = "69de0a69-1ddd-5017-9359-2bf0b02dc9f0"
version = "2.0.3"

[[Pixman_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "b4f5d02549a10e20780a24fce72bea96b6329e29"
uuid = "30392449-352a-5448-841d-b1acce4e97dc"
version = "0.40.1+0"

[[Pkg]]
deps = ["Artifacts", "Dates", "Downloads", "LibGit2", "Libdl", "Logging", "Markdown", "Printf", "REPL", "Random", "SHA", "Serialization", "TOML", "Tar", "UUIDs", "p7zip_jll"]
uuid = "44cfe95a-1eb2-52ea-b672-e2afdf69b78f"

[[PkgVersion]]
deps = ["Pkg"]
git-tree-sha1 = "a7a7e1a88853564e551e4eba8650f8c38df79b37"
uuid = "eebad327-c553-4316-9ea0-9fa01ccd7688"
version = "0.1.1"

[[PlotThemes]]
deps = ["PlotUtils", "Requires", "Statistics"]
git-tree-sha1 = "a3a964ce9dc7898193536002a6dd892b1b5a6f1d"
uuid = "ccf2f8ad-2431-5c83-bf29-c5338b663b6a"
version = "2.0.1"

[[PlotUtils]]
deps = ["ColorSchemes", "Colors", "Dates", "Printf", "Random", "Reexport", "Statistics"]
git-tree-sha1 = "2537ed3c0ed5e03896927187f5f2ee6a4ab342db"
uuid = "995b91a9-d308-5afd-9ec6-746e21dbc043"
version = "1.0.14"

[[Plots]]
deps = ["Base64", "Contour", "Dates", "Downloads", "FFMPEG", "FixedPointNumbers", "GR", "GeometryBasics", "JSON", "Latexify", "LinearAlgebra", "Measures", "NaNMath", "PlotThemes", "PlotUtils", "Printf", "REPL", "Random", "RecipesBase", "RecipesPipeline", "Reexport", "Requires", "Scratch", "Showoff", "SparseArrays", "Statistics", "StatsBase", "UUIDs"]
git-tree-sha1 = "2dbafeadadcf7dadff20cd60046bba416b4912be"
uuid = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
version = "1.21.3"

[[PlutoUI]]
deps = ["Base64", "Dates", "InteractiveUtils", "JSON", "Logging", "Markdown", "Random", "Reexport", "Suppressor"]
git-tree-sha1 = "44e225d5837e2a2345e69a1d1e01ac2443ff9fcb"
uuid = "7f904dfe-b85e-4ff6-b463-dae2292396a8"
version = "0.7.9"

[[Preferences]]
deps = ["TOML"]
git-tree-sha1 = "00cfd92944ca9c760982747e9a1d0d5d86ab1e5a"
uuid = "21216c6a-2e73-6563-6e65-726566657250"
version = "1.2.2"

[[Printf]]
deps = ["Unicode"]
uuid = "de0858da-6303-5e67-8744-51eddeeeb8d7"

[[ProgressMeter]]
deps = ["Distributed", "Printf"]
git-tree-sha1 = "afadeba63d90ff223a6a48d2009434ecee2ec9e8"
uuid = "92933f4c-e287-5a05-a399-4b506db050ca"
version = "1.7.1"

[[Qt5Base_jll]]
deps = ["Artifacts", "CompilerSupportLibraries_jll", "Fontconfig_jll", "Glib_jll", "JLLWrappers", "Libdl", "Libglvnd_jll", "OpenSSL_jll", "Pkg", "Xorg_libXext_jll", "Xorg_libxcb_jll", "Xorg_xcb_util_image_jll", "Xorg_xcb_util_keysyms_jll", "Xorg_xcb_util_renderutil_jll", "Xorg_xcb_util_wm_jll", "Zlib_jll", "xkbcommon_jll"]
git-tree-sha1 = "ad368663a5e20dbb8d6dc2fddeefe4dae0781ae8"
uuid = "ea2cea3b-5b76-57ae-a6ef-0a8af62496e1"
version = "5.15.3+0"

[[QuadGK]]
deps = ["DataStructures", "LinearAlgebra"]
git-tree-sha1 = "12fbe86da16df6679be7521dfb39fbc861e1dc7b"
uuid = "1fd47b50-473d-5c70-9696-f719f8f3bcdc"
version = "2.4.1"

[[REPL]]
deps = ["InteractiveUtils", "Markdown", "Sockets", "Unicode"]
uuid = "3fa0cd96-eef1-5676-8a61-b3b8758bbffb"

[[Random]]
deps = ["SHA", "Serialization"]
uuid = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"

[[RangeArrays]]
git-tree-sha1 = "b9039e93773ddcfc828f12aadf7115b4b4d225f5"
uuid = "b3c3ace0-ae52-54e7-9d0b-2c1406fd6b9d"
version = "0.3.2"

[[Ratios]]
deps = ["Requires"]
git-tree-sha1 = "7dff99fbc740e2f8228c6878e2aad6d7c2678098"
uuid = "c84ed2f1-dad5-54f0-aa8e-dbefe2724439"
version = "0.4.1"

[[RecipesBase]]
git-tree-sha1 = "44a75aa7a527910ee3d1751d1f0e4148698add9e"
uuid = "3cdcf5f2-1ef4-517c-9805-6587b60abb01"
version = "1.1.2"

[[RecipesPipeline]]
deps = ["Dates", "NaNMath", "PlotUtils", "RecipesBase"]
git-tree-sha1 = "d4491becdc53580c6dadb0f6249f90caae888554"
uuid = "01d81517-befc-4cb6-b9ec-a95719d0359c"
version = "0.4.0"

[[Reexport]]
git-tree-sha1 = "45e428421666073eab6f2da5c9d310d99bb12f9b"
uuid = "189a3867-3050-52da-a836-e630ba90ab69"
version = "1.2.2"

[[Requires]]
deps = ["UUIDs"]
git-tree-sha1 = "4036a3bd08ac7e968e27c203d45f5fff15020621"
uuid = "ae029012-a4dd-5104-9daa-d747884805df"
version = "1.1.3"

[[Rmath]]
deps = ["Random", "Rmath_jll"]
git-tree-sha1 = "bf3188feca147ce108c76ad82c2792c57abe7b1f"
uuid = "79098fc4-a85e-5d69-aa6a-4863f24498fa"
version = "0.7.0"

[[Rmath_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "68db32dff12bb6127bac73c209881191bf0efbb7"
uuid = "f50d1b31-88e8-58de-be2c-1cc44531875f"
version = "0.3.0+0"

[[Rotations]]
deps = ["LinearAlgebra", "StaticArrays", "Statistics"]
git-tree-sha1 = "2ed8d8a16d703f900168822d83699b8c3c1a5cd8"
uuid = "6038ab10-8711-5258-84ad-4b1120ba62dc"
version = "1.0.2"

[[SHA]]
uuid = "ea8e919c-243c-51af-8825-aaa63cd721ce"

[[Scratch]]
deps = ["Dates"]
git-tree-sha1 = "0b4b7f1393cff97c33891da2a0bf69c6ed241fda"
uuid = "6c6a2e73-6563-6170-7368-637461726353"
version = "1.1.0"

[[Serialization]]
uuid = "9e88b42a-f829-5b0c-bbe9-9e923198166b"

[[SharedArrays]]
deps = ["Distributed", "Mmap", "Random", "Serialization"]
uuid = "1a1011a3-84de-559e-8e89-a11a2f7dc383"

[[Showoff]]
deps = ["Dates", "Grisu"]
git-tree-sha1 = "91eddf657aca81df9ae6ceb20b959ae5653ad1de"
uuid = "992d4aef-0814-514b-bc4d-f2e9a6c4116f"
version = "1.0.3"

[[SimpleTraits]]
deps = ["InteractiveUtils", "MacroTools"]
git-tree-sha1 = "5d7e3f4e11935503d3ecaf7186eac40602e7d231"
uuid = "699a6c99-e7fa-54fc-8d76-47d257e15c1d"
version = "0.9.4"

[[Sockets]]
uuid = "6462fe0b-24de-5631-8697-dd941f90decc"

[[SortingAlgorithms]]
deps = ["DataStructures"]
git-tree-sha1 = "b3363d7460f7d098ca0912c69b082f75625d7508"
uuid = "a2af1166-a08f-5f64-846c-94a0d3cef48c"
version = "1.0.1"

[[SparseArrays]]
deps = ["LinearAlgebra", "Random"]
uuid = "2f01184e-e22b-5df5-ae63-d93ebab69eaf"

[[SpecialFunctions]]
deps = ["ChainRulesCore", "LogExpFunctions", "OpenSpecFun_jll"]
git-tree-sha1 = "a322a9493e49c5f3a10b50df3aedaf1cdb3244b7"
uuid = "276daf66-3868-5448-9aa4-cd146d93841b"
version = "1.6.1"

[[StackViews]]
deps = ["OffsetArrays"]
git-tree-sha1 = "46e589465204cd0c08b4bd97385e4fa79a0c770c"
uuid = "cae243ae-269e-4f55-b966-ac2d0dc13c15"
version = "0.1.1"

[[Static]]
deps = ["IfElse"]
git-tree-sha1 = "a8f30abc7c64a39d389680b74e749cf33f872a70"
uuid = "aedffcd0-7271-4cad-89d0-dc628f76c6d3"
version = "0.3.3"

[[StaticArrays]]
deps = ["LinearAlgebra", "Random", "Statistics"]
git-tree-sha1 = "3240808c6d463ac46f1c1cd7638375cd22abbccb"
uuid = "90137ffa-7385-5640-81b9-e52037218182"
version = "1.2.12"

[[Statistics]]
deps = ["LinearAlgebra", "SparseArrays"]
uuid = "10745b16-79ce-11e8-11f9-7d13ad32a3b2"

[[StatsAPI]]
git-tree-sha1 = "1958272568dc176a1d881acb797beb909c785510"
uuid = "82ae8749-77ed-4fe6-ae5f-f523153014b0"
version = "1.0.0"

[[StatsBase]]
deps = ["DataAPI", "DataStructures", "LinearAlgebra", "Missings", "Printf", "Random", "SortingAlgorithms", "SparseArrays", "Statistics", "StatsAPI"]
git-tree-sha1 = "8cbbc098554648c84f79a463c9ff0fd277144b6c"
uuid = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91"
version = "0.33.10"

[[StatsFuns]]
deps = ["ChainRulesCore", "IrrationalConstants", "LogExpFunctions", "Reexport", "Rmath", "SpecialFunctions"]
git-tree-sha1 = "46d7ccc7104860c38b11966dd1f72ff042f382e4"
uuid = "4c63d2b9-4356-54db-8cca-17b64c39e42c"
version = "0.9.10"

[[StructArrays]]
deps = ["Adapt", "DataAPI", "StaticArrays", "Tables"]
git-tree-sha1 = "f41020e84127781af49fc12b7e92becd7f5dd0ba"
uuid = "09ab397b-f2b6-538f-b94a-2f83cf4a842a"
version = "0.6.2"

[[SuiteSparse]]
deps = ["Libdl", "LinearAlgebra", "Serialization", "SparseArrays"]
uuid = "4607b0f0-06f3-5cda-b6b1-a6196a1729e9"

[[Suppressor]]
git-tree-sha1 = "a819d77f31f83e5792a76081eee1ea6342ab8787"
uuid = "fd094767-a336-5f1f-9728-57cf17d0bbfb"
version = "0.2.0"

[[TOML]]
deps = ["Dates"]
uuid = "fa267f1f-6049-4f14-aa54-33bafae1ed76"

[[TableTraits]]
deps = ["IteratorInterfaceExtensions"]
git-tree-sha1 = "c06b2f539df1c6efa794486abfb6ed2022561a39"
uuid = "3783bdb8-4a98-5b6b-af9a-565f29a5fe9c"
version = "1.0.1"

[[Tables]]
deps = ["DataAPI", "DataValueInterfaces", "IteratorInterfaceExtensions", "LinearAlgebra", "TableTraits", "Test"]
git-tree-sha1 = "1162ce4a6c4b7e31e0e6b14486a6986951c73be9"
uuid = "bd369af6-aec1-5ad0-b16a-f7cc5008161c"
version = "1.5.2"

[[Tar]]
deps = ["ArgTools", "SHA"]
uuid = "a4e569a6-e804-4fa4-b0f3-eef7a1d5b13e"

[[TensorCore]]
deps = ["LinearAlgebra"]
git-tree-sha1 = "1feb45f88d133a655e001435632f019a9a1bcdb6"
uuid = "62fd8b95-f654-4bbd-a8a5-9c27f68ccd50"
version = "0.1.1"

[[Test]]
deps = ["InteractiveUtils", "Logging", "Random", "Serialization"]
uuid = "8dfed614-e22c-5e08-85e1-65c5234f0b40"

[[TiffImages]]
deps = ["ColorTypes", "DocStringExtensions", "FileIO", "FixedPointNumbers", "IndirectArrays", "Inflate", "OffsetArrays", "OrderedCollections", "PkgVersion", "ProgressMeter"]
git-tree-sha1 = "632a8d4dbbad6627a4d2d21b1c6ebcaeebb1e1ed"
uuid = "731e570b-9d59-4bfa-96dc-6df516fadf69"
version = "0.4.2"

[[TiledIteration]]
deps = ["OffsetArrays"]
git-tree-sha1 = "52c5f816857bfb3291c7d25420b1f4aca0a74d18"
uuid = "06e1c1a7-607b-532d-9fad-de7d9aa2abac"
version = "0.3.0"

[[URIs]]
git-tree-sha1 = "97bbe755a53fe859669cd907f2d96aee8d2c1355"
uuid = "5c2747f8-b7ea-4ff2-ba2e-563bfd36b1d4"
version = "1.3.0"

[[UUIDs]]
deps = ["Random", "SHA"]
uuid = "cf7118a7-6976-5b1a-9a39-7adc72f591a4"

[[UnPack]]
git-tree-sha1 = "387c1f73762231e86e0c9c5443ce3b4a0a9a0c2b"
uuid = "3a884ed6-31ef-47d7-9d2a-63182c4928ed"
version = "1.0.2"

[[Unicode]]
uuid = "4ec0a83e-493e-50e2-b9ac-8f72acf5a8f5"

[[Wayland_jll]]
deps = ["Artifacts", "Expat_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Pkg", "XML2_jll"]
git-tree-sha1 = "3e61f0b86f90dacb0bc0e73a0c5a83f6a8636e23"
uuid = "a2964d1f-97da-50d4-b82a-358c7fce9d89"
version = "1.19.0+0"

[[Wayland_protocols_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll"]
git-tree-sha1 = "2839f1c1296940218e35df0bbb220f2a79686670"
uuid = "2381bf8a-dfd0-557d-9999-79630e7b1b91"
version = "1.18.0+4"

[[WoodburyMatrices]]
deps = ["LinearAlgebra", "SparseArrays"]
git-tree-sha1 = "59e2ad8fd1591ea019a5259bd012d7aee15f995c"
uuid = "efce3f68-66dc-5838-9240-27a6d6f5f9b6"
version = "0.5.3"

[[XML2_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libiconv_jll", "Pkg", "Zlib_jll"]
git-tree-sha1 = "1acf5bdf07aa0907e0a37d3718bb88d4b687b74a"
uuid = "02c8fc9c-b97f-50b9-bbe4-9be30ff0a78a"
version = "2.9.12+0"

[[XSLT_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libgcrypt_jll", "Libgpg_error_jll", "Libiconv_jll", "Pkg", "XML2_jll", "Zlib_jll"]
git-tree-sha1 = "91844873c4085240b95e795f692c4cec4d805f8a"
uuid = "aed1982a-8fda-507f-9586-7b0439959a61"
version = "1.1.34+0"

[[Xorg_libX11_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll", "Xorg_xtrans_jll"]
git-tree-sha1 = "5be649d550f3f4b95308bf0183b82e2582876527"
uuid = "4f6342f7-b3d2-589e-9d20-edeb45f2b2bc"
version = "1.6.9+4"

[[Xorg_libXau_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4e490d5c960c314f33885790ed410ff3a94ce67e"
uuid = "0c0b7dd1-d40b-584c-a123-a41640f87eec"
version = "1.0.9+4"

[[Xorg_libXcursor_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXfixes_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "12e0eb3bc634fa2080c1c37fccf56f7c22989afd"
uuid = "935fb764-8cf2-53bf-bb30-45bb1f8bf724"
version = "1.2.0+4"

[[Xorg_libXdmcp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fe47bd2247248125c428978740e18a681372dd4"
uuid = "a3789734-cfe1-5b06-b2d0-1dd0d9d62d05"
version = "1.1.3+4"

[[Xorg_libXext_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "b7c0aa8c376b31e4852b360222848637f481f8c3"
uuid = "1082639a-0dae-5f34-9b06-72781eeb8cb3"
version = "1.3.4+4"

[[Xorg_libXfixes_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "0e0dc7431e7a0587559f9294aeec269471c991a4"
uuid = "d091e8ba-531a-589c-9de9-94069b037ed8"
version = "5.0.3+4"

[[Xorg_libXi_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXfixes_jll"]
git-tree-sha1 = "89b52bc2160aadc84d707093930ef0bffa641246"
uuid = "a51aa0fd-4e3c-5386-b890-e753decda492"
version = "1.7.10+4"

[[Xorg_libXinerama_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll"]
git-tree-sha1 = "26be8b1c342929259317d8b9f7b53bf2bb73b123"
uuid = "d1454406-59df-5ea1-beac-c340f2130bc3"
version = "1.1.4+4"

[[Xorg_libXrandr_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "34cea83cb726fb58f325887bf0612c6b3fb17631"
uuid = "ec84b674-ba8e-5d96-8ba1-2a689ba10484"
version = "1.5.2+4"

[[Xorg_libXrender_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "19560f30fd49f4d4efbe7002a1037f8c43d43b96"
uuid = "ea2f1a96-1ddc-540d-b46f-429655e07cfa"
version = "0.9.10+4"

[[Xorg_libpthread_stubs_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "6783737e45d3c59a4a4c4091f5f88cdcf0908cbb"
uuid = "14d82f49-176c-5ed1-bb49-ad3f5cbd8c74"
version = "0.1.0+3"

[[Xorg_libxcb_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"]
git-tree-sha1 = "daf17f441228e7a3833846cd048892861cff16d6"
uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b"
version = "1.13.0+3"

[[Xorg_libxkbfile_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "926af861744212db0eb001d9e40b5d16292080b2"
uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a"
version = "1.1.0+4"

[[Xorg_xcb_util_image_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "0fab0a40349ba1cba2c1da699243396ff8e94b97"
uuid = "12413925-8142-5f55-bb0e-6d7ca50bb09b"
version = "0.4.0+1"

[[Xorg_xcb_util_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll"]
git-tree-sha1 = "e7fd7b2881fa2eaa72717420894d3938177862d1"
uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5"
version = "0.4.0+1"

[[Xorg_xcb_util_keysyms_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "d1151e2c45a544f32441a567d1690e701ec89b00"
uuid = "975044d2-76e6-5fbe-bf08-97ce7c6574c7"
version = "0.4.0+1"

[[Xorg_xcb_util_renderutil_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "dfd7a8f38d4613b6a575253b3174dd991ca6183e"
uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e"
version = "0.3.9+1"

[[Xorg_xcb_util_wm_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"]
git-tree-sha1 = "e78d10aab01a4a154142c5006ed44fd9e8e31b67"
uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361"
version = "0.4.1+1"

[[Xorg_xkbcomp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxkbfile_jll"]
git-tree-sha1 = "4bcbf660f6c2e714f87e960a171b119d06ee163b"
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
version = "1.4.2+4"

[[Xorg_xkeyboard_config_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xkbcomp_jll"]
git-tree-sha1 = "5c8424f8a67c3f2209646d4425f3d415fee5931d"
uuid = "33bec58e-1273-512f-9401-5d533626f822"
version = "2.27.0+4"

[[Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "79c31e7844f6ecf779705fbc12146eb190b7d845"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.4.0+3"

[[Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"

[[Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "cc4bf3fdde8b7e3e9fa0351bdeedba1cf3b7f6e6"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.0+0"

[[libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.15.1+0"

[[libblastrampoline_jll]]
deps = ["Artifacts", "Libdl", "OpenBLAS_jll"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"

[[libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.2+0"

[[libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.38+0"

[[libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+1"

[[nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"

[[p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"

[[x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2021.5.5+0"

[[x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "3.5.0+0"

[[xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "ece2350174195bb31de1a63bea3a41ae1aa593b6"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "0.9.1+5"
"""

# ╔═╡ Cell order:
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