### A Pluto.jl notebook ###
# v0.16.0

using Markdown
using InteractiveUtils

# ╔═╡ 9a7fa844-0e94-11eb-0850-31e34452fd9b
md"""# Notebook 8 -- Math 2121, Fall 2021

There is a bounded region of space in $\mathbb{R}^n$ that we naturally associate to an $n \times n$ matrix: namely, consider the *$n$-dimensional parallelogram* (called a  *parallelepiped* when $n=3$) whose edges are the columns of your matrix.
"""

# ╔═╡ 6ce856dc-0e95-11eb-072e-6b1a7cf12830


# ╔═╡ 51eb2cec-0e95-11eb-0920-79c8815b98da
md"""Our goal today is to uncover a formula for the volume for this region. 

But first we need to better understand what the $n$-dimensional parallelogram associated to an $n \times n$ matrix looks like and what "volume" means for a region in $\mathbb{R}^n$."""

# ╔═╡ 867f0302-0e95-11eb-2633-8be05070104d


# ╔═╡ 87892b74-0e95-11eb-187f-256d157b7d21
md"""A $1\times 1$ matrix is just a scalar.

However our definitions work, they should associate a volume of $|a|$ to $A = [a]$ when $n=1$."""

# ╔═╡ c78e4e3e-0e95-11eb-27d8-8fdca3bc353a


# ╔═╡ c8e33a4c-0e95-11eb-2e77-e96a550786e3
md"""The $n=2$ case is more informative. 

In $2$ dimensions, we usually refer to volume as *area*. 

Consider the following $2\times 2$ matrix:
"""

# ╔═╡ 3e8ba024-0e96-11eb-36a5-61db9665d7d0
A = rand(-5:0.01:5, 2, 2)

# ╔═╡ 2dc9055e-0e96-11eb-04a8-cf3d9d9683a7
begin
	using LinearAlgebra
	using Plots
	plotly()
	
	function matrix_plot(A)
		p = scatter([0], [0], color = [:red], legend = false, label = "origin",
			aspect_ratio=:equal,
			title="Parallelogram of a 2-by-2 matrix")
		scatter!([A[1, 1]], [A[2, 1]], label = "first column", color = [:red])
		scatter!([A[1, 2]], [A[2, 2]], label = "second column", color = [:red])
		scatter!(
			[A[1, 1] + A[1, 2]], [A[2, 1] + A[2, 2]], 
			label = "sum of columns", color = [:red])

		quiver!(quiver = ([A[1, 1]], [A[2, 1]]), [0], [0], color = [:blue],)
		quiver!(quiver = ([A[1, 2]], [A[2, 2]]), [0], [0], color = [:blue],)
		plot!(Shape(
				[0, A[1, 1], A[1, 1] + A[1, 2], A[1, 2]],
				[0, A[2, 1], A[2, 1] + A[2, 2], A[2, 2]],), opacity=.25, color = [:blue])

		return p
	end
	
	matrix_plot(A)
end

# ╔═╡ d7d4030a-0ea0-11eb-050f-e9e4e781a13c
md"Here is a picture of the parallelogram spanned by the columns of $M$:"

# ╔═╡ 87aa9f88-0ea0-11eb-04da-99c18a359c09
md"How can we characterize this parallelogram? 

Well, every point inside the parallelogram has the form $Av$ for some vector 

$v = \left[\begin{array}{c} v_1 \\ v_2\end{array}\right] \in \mathbb{R}^2$ 

with $0\leq v_1 \leq 1$ and $0\leq v_2 \leq 1$. This suggests a general definition:

The *$n$-dimensional parallelogram* of an $n\times n$ matrix $A$ is the set of all points

$\left\{ Av : v= \left[\begin{array}{c} v_1 \\ v_2 \\ \vdots \\ v_n \end{array}\right]\in\mathbb{R}^n\text{ with } 0\leq v_i \leq 1 \text{ for }i=1,2,\dots,n\right\}.$

Then we define the *volume* of an $n\times n$ matrix to be the volume of this region.
"

# ╔═╡ 368edece-0ea1-11eb-3d62-4934f3ed5484


# ╔═╡ 32b5167e-0ea1-11eb-3db0-cdc1d26df891
md"Next issue: how is volume defined in $\mathbb{R}^n$?

The definition should obey our intuition that the volume of a rectangular region is 

$\text{length} \times \text{width} \times \text{height} \times \dots$

and volume should be *invariant under translation*.

That is, if $L_1,L_2,\dots,L_n$ are positive numbers then the region 

$\mathrm{rect}(L_1,L_2,\dots,L_n) = \{ v \in \mathbb{R}^n : 0\leq v_i \leq L_i\text{ for }i=1,2,\dots,n\}$

should have volume $L_1\cdot L_2\cdots L_n$. Moreover, any translate of this region

$u + \mathrm{rect}(L_1,L_2,\dots,L_n) = \{ u + v \in \mathbb{R}^n : 0\leq v_i \leq L_i\text{ for }i=1,2,\dots,n\}$

where $u \in \mathbb{R}^n$ is some given element should have the same volume.

Finally, if we have two regions in $\mathbb{R}^n$ that don't overlap, then the volume of their union should be the sum of their individual volumes. "

# ╔═╡ 8d0c461e-0ea2-11eb-3ecc-3765095f7bef


# ╔═╡ 8dfe61c4-0ea2-11eb-0a03-3b0fbeaed045
md"Here is one definition of the volume of a region $S$ in $\mathbb{R}^n$ that satisfies these properties: choose small positive numbers $L_1,L_2,\dots,L_n > 0$ and fill up as much of $S$ as possible with nonoverlapping translates of the $n$-dimensional rectangle $\mathrm{rect}(L_1,L_2,\dots,L_n)$. If you can fit $N$ such regions completely inside $S$ without any overlap, then you say your *estimated* volume is 

$N \cdot L_1 \cdot L_2\cdots L_n.$

The true *volume* of $S$ is then the *limit superior* of all possible estimated volumes. 

To define *limit superior*, we need calculus; you can think of this as just the *maximum*."

# ╔═╡ 925334a2-0ea2-11eb-3112-2f59c8ce51d4


# ╔═╡ eef14ab4-0ea2-11eb-0133-b3a60328c86f
md"This conforms with our usual definiton of area in two dimensions.

But in $\mathbb{R}^2$ we can also calculate area more directly:"

# ╔═╡ ac657e72-0e98-11eb-1a8c-b70f72979c6b
begin
	function enclosing_rectangle(A)
		u = A[1:2, 1]
		v = A[1:2, 2]
		w = u + v
	
		x1 = minimum([0 u[1] v[1] w[1]])
		x2 = maximum([0 u[1] v[1] w[1]])
	
		y1 = minimum([0 u[2] v[2] w[2]])
		y2 = maximum([0 u[2] v[2] w[2]])
	
		return x1, x2, y1, y2
	end

	x1, x2, y1, y2 = enclosing_rectangle(A)
	
	mplot = matrix_plot(A)
	plot!(Shape([x1, x2, x2, x1], [y1, y1, y2, y2]), color = [:blue], opacity =.1)
end

# ╔═╡ 18b799c1-435e-427b-9be8-2e712d3c662c
# x-coordinates of points in picture
xcoords = 0, A[1,1], A[1,2], A[1,1] + A[1,2]

# ╔═╡ 9fe4f768-9d41-416c-9c1b-d8737c1a9a69
# y-coordinates of points in picture
ycoords = 0, A[2,1], A[2,2], A[2,1] + A[2,2]

# ╔═╡ 0139bd8b-57da-43df-9c51-9e81bbed4111
# outer rectangle area
(maximum(xcoords) - minimum(xcoords)) * (maximum(ycoords) - minimum(ycoords))

# ╔═╡ 744c9460-db89-4bef-9edc-9681c1d98d21
# outer triangle areas
0.88 * 2.05 + 3.06 * 2.38

# ╔═╡ b89360fe-614a-4e02-bbac-98f788b6b4d8
# remaining area
parallelogram_area = 17.4542 - 9.0868

# ╔═╡ e078d7c6-0ea3-11eb-307b-d1c904fb5539


# ╔═╡ e1faae26-0ea3-11eb-318d-77713a78456a
md"Here is an alternate, probabilistic definition of volume that is more amenable to estimation.

Consider a region $S$ in $\mathbb{R}^n$. Define the volume of 

$R = u + \mathrm{rect}(L_1,L_2,\dots,L_n) = \{ u + v \in \mathbb{R}^n : 0\leq v_i \leq L_i\text{ for }i=1,2,\dots,n\}$

to be $L_1\cdot L_2\cdots L_n$ as usual. Choose $u$ and $L_1,L_2,\dots,L_n$ such that

$S \subseteq R.$

Now sample many points uniformly at random from the region $R$. The *volume* of $S$ should then be (close to) the volume of $R$ times the fraction of these points that are contained in $S$. 

So if we generate $N$ points within $R$ and $m$ of them are inside $S$, then we expect 

$\mathrm{volume}(S) \approx \frac{m}{N} \cdot L_1\cdot L_2\cdots L_n.$

With enough random points $N$ this approximate volume will converge to the true volume.

"

# ╔═╡ fe034b28-0ea3-11eb-1b1a-1f474ab12faf


# ╔═╡ f921e1cc-0ea4-11eb-21cd-2305c9faf2d9
md"When our matrix $A$ is invertible and $S$ is the $n$-dimensional parallelogram associated to $A$, then there is an easy way for us to test whether a random point $w \in R$ belongs to $S$: 

We just check if $A^{-1} w$ has all coordinates between $0.0$ and $1.0$.

Thus, we can try to implement the probabilistic definition to estimate the matrix volume."

# ╔═╡ 5e0bb100-0e49-11eb-272e-db4dce3e1696
function volume_estimates(A, lower_bound, upper_bound, trials)
	# we are trying to estimate the volume of the (n-dimension) parallelogram
	# whose edges include the vectors that are the columns of the matrix A
	#
	# we will sample points from a rectangular region R
	#
	# this outer region R is u + rect(L1, L2, ..., Ln) where
	#
	#   L1 = L2 = ... = Ln = upper_bound - lower_bound
	#   u = [lower_bound; lower_bound; ...; lower_bound]
	#
	
	(n, p) = size(A)
	@assert n == p
	
	estimates = []
	samples = []
	
	# let's only record the cumulative estimate every 1000 trials
	interval = maximum([Int(floor(trials/1000)) 1])
	
	# size of region R
	sample_region_size = (upper_bound - lower_bound)^n
	
	# rather than generate a single vector v many times,
	# we construct one random matrix with many columns
	v = rand(lower_bound:0.00001:upper_bound, n, trials)
	# now we can just do one matrix multiplication with all of our samples 
	w = A^-1 * v

	count = 0
	for j=1:trials
		# column j of v belongs to the matrix volume if and only if 
		# column j of w = A^-1 * v has all entries between 0.0 and 1.0
		if 0.0 <= minimum(w[:,j]) && maximum(w[:,j]) <= 1.0
			count += 1
		end
		
		if j % interval == 0
			fraction = count / j
			append!(estimates, fraction * sample_region_size)
			append!(samples, j)
		end
	end
	
	return estimates, samples
end

# ╔═╡ 9398ea44-0e4f-11eb-2412-23acd564dc41
function estimate_bounds(A, trials=1000)
	# we get more accurate volume estimates by using a smaller region R
	#
	# this method estimates optimal choices for lower_bound, upper_bound
	# so that R is as small as possible while containing the paralllelogram of A
	w = A * rand(0:1, size(A)[1], trials)
	return (minimum(w), maximum(w))
end

# ╔═╡ 7bdd6824-0e4a-11eb-063c-530768c6a7fe
function plot_estimates(matrix, lower, upper, trials)
	estimates, samples = volume_estimates(matrix, lower, upper, trials)
	plot(
		samples,
		estimates,
		xlabel="samples N",
		ylabel="volume estimate after N samples",
		ylims = (0.0, 2 * estimates[end]),
		label="first estimate",
		title="Volume estimates of $(size(matrix)[1])-dimensional parallelogram",
		legend=:bottomright
	)
	
	estimates, samples = volume_estimates(matrix, lower, upper, trials)
	plot!(samples, estimates, label="second estimate")
	
	estimates, samples = volume_estimates(matrix, lower, upper, trials)
	plot!(samples, estimates, label="third estimate")
end

# ╔═╡ 7fb874de-0ea6-11eb-0449-39b8d954819a
trials = 10000

# ╔═╡ 843b9826-0ea6-11eb-3042-af3bcedae2ec
matrix = A

# ╔═╡ c5e50efa-0eb2-11eb-28dd-fdf089198c2d
lower, upper = estimate_bounds(matrix)

# ╔═╡ e21e9614-0ea6-11eb-2aba-97dd7bda343b
md"The following graph shows what happens when we sample a large number $N$ of points from a rectangular region $R$ containing the $n$-dimensional parallelogram $S$ of our matrix $A$, and then multiply the faction of the points that are in $S$ by the volume of $R$.

The resulting plots should estimate the volume of the parallelogram spanned by the columns $A$."

# ╔═╡ 1cc8a0d8-27fb-4be8-8c1e-f1257af6e217
plot_estimates(matrix, lower, upper, trials)

# ╔═╡ 7c4e9c3b-5776-4f2d-90ea-bd8b75fbb098
# here is what we computed for the exact area:
parallelogram_area

# ╔═╡ 18a01504-0ea6-11eb-1b64-c33c95d4bcae
# another way to compute this number (up to +/- sign):
det(A)

# ╔═╡ 0908992e-0ea6-11eb-211a-51a9655c77b8
md"The punchline here: the volume of the parallelogram spanned by the columns of our matrix is *the absolute value of the determinant* of the matrix!

Thus the physical interpretation of $\det(A)$ is as the *signed volume of the parallelogram spanned by the columns of $A$*.

This works in any dimension, not just $n=2$.

However, in higher dimensions we need more samples to get a good estimate.
"

# ╔═╡ dd18e9c6-0eb3-11eb-31c4-296dbc11be15
M = rand(-5:0.001:5, 3, 3)

# ╔═╡ ece780c4-0eb3-11eb-1b24-0f804525f8ef
hlower, hupper = estimate_bounds(M)

# ╔═╡ c91999a2-0eb3-11eb-285a-85e605e2a1f6
htrials = 100000

# ╔═╡ c4dabe7a-0eb3-11eb-306b-b3001b0c4f73
plot_estimates(M, hlower, hupper, htrials)

# ╔═╡ 5645ac26-0eb4-11eb-3cfe-d13415915d87
# actual volume (up to +/- sign)
det(M)

# ╔═╡ bbcc1392-0eb3-11eb-1868-b39c9b4130d4
md"Next time: what is the physical interpretation of the sign of $\det(A)$?"

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uuid = "0656b61e-2033-5cc2-a64a-77c0f6c09b89"
version = "3.3.5+0"

[[GR]]
deps = ["Base64", "DelimitedFiles", "GR_jll", "HTTP", "JSON", "Libdl", "LinearAlgebra", "Pkg", "Printf", "Random", "Serialization", "Sockets", "Test", "UUIDs"]
git-tree-sha1 = "d189c6d2004f63fd3c91748c458b09f26de0efaa"
uuid = "28b8d3ca-fb5f-59d9-8090-bfdbd6d07a71"
version = "0.61.0"

[[GR_jll]]
deps = ["Artifacts", "Bzip2_jll", "Cairo_jll", "FFMPEG_jll", "Fontconfig_jll", "GLFW_jll", "JLLWrappers", "JpegTurbo_jll", "Libdl", "Libtiff_jll", "Pixman_jll", "Pkg", "Qt5Base_jll", "Zlib_jll", "libpng_jll"]
git-tree-sha1 = "cafe0823979a5c9bff86224b3b8de29ea5a44b2e"
uuid = "d2c73de3-f751-5644-a686-071e5b155ba9"
version = "0.61.0+0"

[[GeometryBasics]]
deps = ["EarCut_jll", "IterTools", "LinearAlgebra", "StaticArrays", "StructArrays", "Tables"]
git-tree-sha1 = "58bcdf5ebc057b085e58d95c138725628dd7453c"
uuid = "5c1252a2-5f33-56bf-86c9-59e7332b4326"
version = "0.4.1"

[[Gettext_jll]]
deps = ["Artifacts", "CompilerSupportLibraries_jll", "JLLWrappers", "Libdl", "Libiconv_jll", "Pkg", "XML2_jll"]
git-tree-sha1 = "9b02998aba7bf074d14de89f9d37ca24a1a0b046"
uuid = "78b55507-aeef-58d4-861c-77aaff3498b1"
version = "0.21.0+0"

[[Glib_jll]]
deps = ["Artifacts", "Gettext_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Libiconv_jll", "Libmount_jll", "PCRE_jll", "Pkg", "Zlib_jll"]
git-tree-sha1 = "7bf67e9a481712b3dbe9cb3dac852dc4b1162e02"
uuid = "7746bdde-850d-59dc-9ae8-88ece973131d"
version = "2.68.3+0"

[[Graphite2_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "344bf40dcab1073aca04aa0df4fb092f920e4011"
uuid = "3b182d85-2403-5c21-9c21-1e1f0cc25472"
version = "1.3.14+0"

[[Grisu]]
git-tree-sha1 = "53bb909d1151e57e2484c3d1b53e19552b887fb2"
uuid = "42e2da0e-8278-4e71-bc24-59509adca0fe"
version = "1.0.2"

[[HTTP]]
deps = ["Base64", "Dates", "IniFile", "Logging", "MbedTLS", "NetworkOptions", "Sockets", "URIs"]
git-tree-sha1 = "14eece7a3308b4d8be910e265c724a6ba51a9798"
uuid = "cd3eb016-35fb-5094-929b-558a96fad6f3"
version = "0.9.16"

[[HarfBuzz_jll]]
deps = ["Artifacts", "Cairo_jll", "Fontconfig_jll", "FreeType2_jll", "Glib_jll", "Graphite2_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Pkg"]
git-tree-sha1 = "8a954fed8ac097d5be04921d595f741115c1b2ad"
uuid = "2e76f6c2-a576-52d4-95c1-20adfe4de566"
version = "2.8.1+0"

[[IniFile]]
deps = ["Test"]
git-tree-sha1 = "098e4d2c533924c921f9f9847274f2ad89e018b8"
uuid = "83e8ac13-25f8-5344-8a64-a9f2b223428f"
version = "0.5.0"

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

[[IrrationalConstants]]
git-tree-sha1 = "7fd44fd4ff43fc60815f8e764c0f352b83c49151"
uuid = "92d709cd-6900-40b7-9082-c6be49f344b6"
version = "0.1.1"

[[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"

[[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"

[[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 = "761a393aeccd6aa92ec3515e428c26bf99575b3b"
uuid = "e9f186c6-92d2-5b65-8a66-fee21dc1b490"
version = "3.2.2+0"

[[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", "Libdl", "Pkg", "Zlib_jll", "Zstd_jll"]
git-tree-sha1 = "340e257aada13f95f98ee352d316c3bed37c8ab9"
uuid = "89763e89-9b03-5906-acba-b20f662cd828"
version = "4.3.0+0"

[[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"]
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"

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

[[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"

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

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

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

[[Ogg_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "7937eda4681660b4d6aeeecc2f7e1c81c8ee4e2f"
uuid = "e7412a2a-1a6e-54c0-be00-318e2571c051"
version = "1.3.5+0"

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

[[Parsers]]
deps = ["Dates"]
git-tree-sha1 = "98f59ff3639b3d9485a03a72f3ab35bab9465720"
uuid = "69de0a69-1ddd-5017-9359-2bf0b02dc9f0"
version = "2.0.6"

[[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"

[[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 = "b084324b4af5a438cd63619fd006614b3b20b87b"
uuid = "995b91a9-d308-5afd-9ec6-746e21dbc043"
version = "1.0.15"

[[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 = "ba43b248a1f04a9667ca4a9f782321d9211aa68e"
uuid = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
version = "1.22.6"

[[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"

[[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"

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

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

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

[[RecipesPipeline]]
deps = ["Dates", "NaNMath", "PlotUtils", "RecipesBase"]
git-tree-sha1 = "7ad0dfa8d03b7bcf8c597f59f5292801730c55b8"
uuid = "01d81517-befc-4cb6-b9ec-a95719d0359c"
version = "0.4.1"

[[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"

[[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"

[[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"

[[StaticArrays]]
deps = ["LinearAlgebra", "Random", "Statistics"]
git-tree-sha1 = "3c76dde64d03699e074ac02eb2e8ba8254d428da"
uuid = "90137ffa-7385-5640-81b9-e52037218182"
version = "1.2.13"

[[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", "LogExpFunctions", "Missings", "Printf", "Random", "SortingAlgorithms", "SparseArrays", "Statistics", "StatsAPI"]
git-tree-sha1 = "65fb73045d0e9aaa39ea9a29a5e7506d9ef6511f"
uuid = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91"
version = "0.33.11"

[[StructArrays]]
deps = ["Adapt", "DataAPI", "StaticArrays", "Tables"]
git-tree-sha1 = "2ce41e0d042c60ecd131e9fb7154a3bfadbf50d3"
uuid = "09ab397b-f2b6-538f-b94a-2f83cf4a842a"
version = "0.6.3"

[[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 = "fed34d0e71b91734bf0a7e10eb1bb05296ddbcd0"
uuid = "bd369af6-aec1-5ad0-b16a-f7cc5008161c"
version = "1.6.0"

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

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

[[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"

[[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"

[[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"

[[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 = "c45f4e40e7aafe9d086379e5578947ec8b95a8fb"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+0"

[[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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