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

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 el = $(esc(element))
        global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : missing
        el
    end
end

# ╔═╡ 942d22f0-fdd2-11ea-067f-ffddf84d1c39
begin
	using Plots 
	using PlutoUI
	md"##### Helper methods"
end

# ╔═╡ 500b0a2e-fdd2-11ea-32d0-d702f3410357
md"# Notebook 5 -- Math 2121, Fall 2021
In today's notebook we'll first explore some plots of linear transformations. Then we'll take a look at the frequency that a random linear transformation is one-to-one or onto.

##### Running *this* notebook (optional)
If you have Pluto up and running, you can access the notebook we are currently viewing by entering [this link](https://www.math.hkust.edu.hk/~emarberg/teaching/2021/Math2121/julia/05_Math2121_Fall2021.jl) (right click -> Copy Link)  in the *Open from file* menu in Pluto."


# ╔═╡ 333a2fce-fdcd-11ea-1ab5-7d88e0217376
function pprint(tagx, tagy, rx, ry)
	s = [string(rx[1]), string(rx[2]), string(ry[1]), string(ry[2])]
	pad = maximum([4, maximum(map(length,s))])
	for i=1:length(s)
		s[i] = " "^(pad - length(s[i])) * s[i]
	end

	top = join([            tagx * " = [$(s[1])]", 
							tagy * " = [$(s[3])]"], "     ")
	bot = join([" "^length(tagx) * "   [$(s[2])]", 
				" "^length(tagy) * "   [$(s[4])]"], "     ")
	return Text(join([top, bot], "\n"))	
end

# ╔═╡ 8a23eb6e-fdc2-11ea-0603-8b191ff11d0e
function input_plot(x, y, title, lims=(-2,2))

	p = scatter([0], [0], xlims = lims, ylims = lims, 
		legend = false, label = "origin",  aspect_ratio=:equal,
		title = title)
	
	quiver!(quiver = ([x[1]],[x[2]]), [0], [0], color = [:blue],)	
	quiver!(
		quiver = ([x[1]],[x[2]]), [y[1]], [y[2]], 
		color = [:lightgrey], linestyle = :dashdot, linecolor=:lightgrey)
	
	quiver!(quiver = ([y[1]],[y[2]]), [0], [0], color = [:red],)
	quiver!(
		quiver = ([y[1]],[y[2]]), [x[1]], [x[2]], 
		color = [:lightgrey], linestyle = :dashdot, linecolor=:lightgrey)
	
	return p
end

# ╔═╡ e41e382c-fdcc-11ea-3d1c-17127b462f9e
function transformation_plot(A, x, y, title, lims=(-2,2))
	p = scatter([0], [0], xlim=lims, ylim=lims,
	legend = false, label = "origin",  aspect_ratio=:equal,
	title = title)
	
	Ux = A * x
	Uy = A * y
	Uxy = A * (x + y)
	
	quiver!(quiver = ([Ux[1]], [Ux[2]]), [0], [0], color = [:blue],)
	quiver!(
		quiver = ([Ux[1]], [Ux[2]]), [Uy[1]], [Uy[2]], 
		color = [:lightgrey], linestyle = :dashdot, linecolor=:lightgrey)
	
	quiver!(quiver = ([Uy[1]], [Uy[2]]), [0], [0], color = [:red],)
	quiver!(
		quiver = ([Uy[1]], [Uy[2]]), [Ux[1]], [Ux[2]], 
		color = [:lightgrey], linestyle = :dashdot, linecolor=:lightgrey)
	
	return p
end

# ╔═╡ 4c6f5762-fdcd-11ea-364e-d70107e4bc0b
function compare_plots(x, y; A)
	p1 = input_plot(x, y, "x and y")
	p2 = transformation_plot(A, x, y, "Ax and Ay")
	plot(p1, p2, layout=2)
end

# ╔═╡ 421e0dbc-fdcd-11ea-0ef8-83890a2328e5
md"##### Linear transformations from $2\times 2$ matrices"

# ╔═╡ 019217d6-fdcd-11ea-1e36-b5cbc03cf1d5
begin
	r_slider = @bind r Slider(0:0.1:2 * pi, default=1, show_value=false)
	theta_slider = @bind theta Slider(0:0.1:2 * pi, default=1, show_value=true)
	
	x1_slider = @bind x1 Slider(-2:0.1:2, default=1, show_value=false)
	x2_slider = @bind x2 Slider(-2:0.1:2, default=0, show_value=false)

	y1_slider = @bind y1 Slider(-2:0.1:2, default=0, show_value=false)
	y2_slider = @bind y2 Slider(-2:0.1:2, default=1, show_value=false)
	
	k_slider = @bind k Slider(-2:0.1:2, default=1.5, show_value=true)
	
	md"""**input parameters**
	
	`x1` = $(x1_slider) `y1` = $(y1_slider) rotate = $(r_slider)
	
	`x2` = $(x2_slider) `y2` = $(y2_slider) 
	
	**angle parameter**
	
	`θ` = $(theta_slider) 
	
	"""
end

# ╔═╡ 1d86c41c-fdcd-11ea-002b-69c19dd80a3f
begin
	rmat = [cos(r) -sin(r); sin(r) cos(r)]
	x = rmat * [x1; x2]
	y = rmat * [y1; y2]
	pprint("x", "y", x, y)
end

# ╔═╡ fa05c5ff-0163-46e8-a755-fb962d46bb81
compare_plots(x, y; A=[cos(theta) -sin(theta)
					   sin(theta)  cos(theta)]
) # rotates counter clockwise by angle theta

# ╔═╡ dda79521-46b0-4a52-bd32-63d96bfe7459
compare_plots(x, y; A=[1 0
					   0 -1]
) # flips input across horizontal axis

# ╔═╡ 0105937c-f57c-4429-a32d-c226c815ec79
compare_plots(x, y; A=[-1 0
					    0 1]
) # flips input across the vertical axis

# ╔═╡ 001a77d5-f56a-4f34-83d2-b022c6966a41
compare_plots(x, y; A=[0 1
					   1 0]
) # reflects across the line x_1 = x_2

# ╔═╡ bc5852b4-f78f-45b7-b312-b2ea6b1ba809
begin
	md"""
	**another parameter**
	
	`k` = $(k_slider)
	
	"""
end

# ╔═╡ 0df14d8c-11aa-4d52-82ff-f359e1fdc1b0
compare_plots(x, y; A=[k 0
					   0 1]
) # scales the horizontal components by factor k

# ╔═╡ 7f4aa4d0-f2b2-4ee6-92ae-7541a17b367b
compare_plots(x, y; A=[1 0
					   0 k]
) # scales the vertical components by factor k

# ╔═╡ 935fddb6-7931-4805-becd-4e6049b6e9c6
compare_plots(x, y; A=[1 k
					   0 1]
) # shearing/skewing operation in horizontal direction

# ╔═╡ 133108da-045e-41f8-998b-9772ead84b32
compare_plots(x, y; A=[1 0 
		               k 1]
) # shearing/skewing operation in vertical direction

# ╔═╡ 3a9bc594-fe26-11ea-3767-5bdf350fb6fb
md"##### Random matrices, onto and one-to-one linear transformations

We can generate a random linear transformation $T: \mathbb{R}^n \to \mathbb{R}^m$
by generating a random $m\times n$ matrix $A$ and setting $T(x) = Ax$.

If $n > m$ then $T$ is never one-to-one, and if $n < m$ then $T$ is never onto.

If $n \leq m$, however, then a random linear transformation $T$ is injective with a high probability. 

If $n \geq m$, similarly, then a random linear transformation $T$ is surjective with a high probability.

In other words, a sufficiently random linear transformation is almost always one-to-one if this is possible (that is, if $n \leq m$) and almost always onto if this is possible (that is, if $n \geq m$).

If $n=m$ then a random linear transformation is almost always both one-to-one and onto (which will be our definition of an *invertible function* next lecture).

To explore this phenomenon, we'll consider our usual model of random matrix which is not *that* random: namely, $01$-matrices whose entries are independently $0$ or $1$ with some probability $p$. 

For such matrices, we can see some interesting behavior while also observing convergence to the ''almost always'' properties described above.
"

# ╔═╡ b16e53c8-fe26-11ea-206b-4b8acae80e60
md"##### Helper methods"

# ╔═╡ e5ee9a5a-fe1e-11ea-11e9-95bc915a0ee6
begin
	# Returns true if row has all zeros.
	function is_zero_row(A, row, tolerance=10e-16)
		(m, n) = size(A)
		return all([abs(A[row,col]) < tolerance for col=1:n])
	end
	
	# Returns first index of nonzero entry in row, or 0 if none exists.
	function find_leading(A, row=1, tolerance=10e-16)
		(m, n) = size(A)
		for col=1:n
			if abs(A[row, col]) > tolerance
				return col
			end
		end
		return 0
	end

	function find_nonzeros_in_column(A, row_start, col, tol=10e-12)
		(m, n) = size(A)
		return [i for i=row_start:m if abs(A[i, col]) > tol]
	end

	function _rowop_replace(A, source_row, target_row, scalar_factor)
		@assert source_row != target_row
		A[target_row,:] += A[source_row,:] * scalar_factor
		return A
	end

	function _rowop_scale(A, target_row, scalar_factor)
		@assert scalar_factor != 0
		A[target_row,:] *= scalar_factor
		A
	end
	
	function _rowop_swap(A, s, t)
		A[t,:], A[s,:] = A[s,:], A[t,:]
		A
	end

	function RREF(A, tol=10e-12)
		A = float(copy(A))
		(m, n) = size(A)
		
		row = 1
		for col=1:n
			nonzeros = find_nonzeros_in_column(A, row, col, tol)
			if length(nonzeros) == 0
				continue
			end
			
			i = nonzeros[1]
			
			if abs(A[i, col] - 1) < tol
				A[i, col] = 1
			else
				A = _rowop_scale(A, i, 1 / A[i, col])
			end
			
			for j=nonzeros[2:end]
				A = _rowop_replace(A, i, j, -A[j, col])
			end

			if i != row
				A = _rowop_swap(A, row, i)
			end
			
			row += 1
		end
		
		for row=m:-1:1
			l = find_leading(A, row, tol)
			if l > 0
				for k=1:row - 1
					if abs(A[k,l]) > tol
						_rowop_replace(A, row, k, -A[k,l])
					end
				end
			end
		end
		
		# round entries to give nicer output; could omit this step
		for row=1:m
			for col=1:n
				if abs(A[row,col] - round(A[row,col])) < tol
					A[row,col] = round(A[row,col])
				end
			end
		end
		
		return A
	end
end

# ╔═╡ 110a5bd4-fe1f-11ea-262e-730e1ff90e74
begin
	function pivot_positions(A, tol=10e-12)
		rref = RREF(A, tol)
		return [
			(i, find_leading(rref, i)) 
			for i=1:size(A)[1] if find_leading(rref, i) > 0
		]
	end

	function npivots(A, tol=10e-12)
		return length(pivot_positions(A, tol))
	end
end

# ╔═╡ ec4f8f1c-fe1e-11ea-2dfa-05f83ed0f00c
function is_onto(A)
	(m, n) = size(A)
	return npivots(A) == m
end

# ╔═╡ 29b54ba8-fe1f-11ea-2dd4-2baa0503defd
function is_one_to_one(A)
	(m, n) = size(A)
	return npivots(A) == n
end

# ╔═╡ bbf682e6-fe1f-11ea-1f7d-4f55724ebd9b
function random_boolean_matrix(m, n, zero_probability)
	A = rand(m, n)
	for i=1:m
		for j=1:n
			A[i, j] = Int(A[i, j] > zero_probability)
		end
	end
	return A
end

# ╔═╡ a0d257ee-fe1f-11ea-16d9-a7ecc379d56e
function accumulate_mean(returns_boolean, m, n, zero_prob, trials)
	s = 0
	for i=1:trials
		s += Int(returns_boolean(random_boolean_matrix(m, n, zero_prob)))
	end
	mean = s / trials
	return mean
end

# ╔═╡ a65f1cf6-fe1f-11ea-0d70-1f2f1b0ce2dd
function accumulate_std(returns_boolean, m, n, zero_prob, trials, mean)
	s = 0
	for i=1:trials
		s += (Int(returns_boolean(random_boolean_matrix(m, n, zero_prob))) - mean)^2
	end
	std = sqrt(s / (trials - 1))
	return std
end

# ╔═╡ 92d46126-fe21-11ea-3b1d-0be2fce36f72
function accumulate_plot(returns_boolean, title, trials, M, N)
	prob = 0:0.01:1
	mu = [accumulate_mean(returns_boolean, M, N, p, trials) for p in prob]
	std = [accumulate_std(returns_boolean, M, N, prob[i], trials, mu[i]) for i=1:length(prob)]

	return plot(
		prob, 
		hcat(mu, std), 
		xlabel="Probablity p that individual matrix entry is zero",
		ylabel="Proportion",
		title=title,
		label=["mean" "std"]
	)
end

# ╔═╡ b676d206-fe37-11ea-238d-a9efe71dad02
plotly()

# ╔═╡ bcc2b6b8-fe26-11ea-16d8-b9391dc06fe6
md"##### Parameters"

# ╔═╡ c29c6fae-fe1f-11ea-2524-4b9c4588b70d
trials = 1000

# ╔═╡ c5190434-a974-4124-8765-65574de60494
nrows, ncols = 6, 10

# ╔═╡ 1b8f276c-fe3b-11ea-375d-574e523dbc3b
md"To make the following plot, for various probabilities p in [0,1], we generate **$(trials)** random 01-matrices of size 
**$(nrows) by $(ncols)**. For each p, we compute the proportion of these matrices that correspond to **onto** linear transformations, as well as the standard deviation of this statistic. 

Then we plot both as a function of p."

# ╔═╡ 633f693a-fe1f-11ea-1d89-1716c918cfb7
accumulate_plot(is_onto, 
	"Random 01-matrices, size $(nrows)-by-$(ncols): onto transformations", 
	trials, nrows, ncols)

# ╔═╡ 87f1fc20-fe3d-11ea-0b45-f9825caf6913
md"Interesting properties of this graph: when $p$ is not too small or too large, our model of a random matrix is ''sufficiently'' random, so the measured proportion is 1.0: almost every random matrix corresponds to an onto linear transformation.

The standard deviation is always bimodal, with maximum value 0.5 attained at two values $p_1$ and $p_2$ of the probability $p$, which are also where the mean and standard deviation graphs intersect. 

What are these two values of $p$? They are not symmetric, that is, $p_2 \neq 1-p_1$."

# ╔═╡ e3c75cb8-fe3b-11ea-2970-89b816ef738e
md"Why does the mean have value $0$ when $p=0$ and when $p=1$?"

# ╔═╡ 2650306e-fe3c-11ea-00cf-5d7e15b2b8e9
md"Here is a random 01-matrix for $p=0$:"

# ╔═╡ 180b696a-fe3c-11ea-18fe-37423a9b5827
R = random_boolean_matrix(nrows, ncols, 0)

# ╔═╡ 2a064810-fe3c-11ea-141f-f71f37013d7a
npivots(R)

# ╔═╡ 4a125950-fe3c-11ea-3ae3-5546c0d4e11d
md"Here is a random 01-matrix for $p=1$:"

# ╔═╡ 292a5800-fe3c-11ea-3af5-f70d6dbc14a1
S = random_boolean_matrix(nrows, ncols, 1)

# ╔═╡ 4fde7e68-fe3c-11ea-0c84-01b599a5fb56
npivots(S)

# ╔═╡ 56266042-fe3c-11ea-23ca-d3f9240211c5
md"In both cases the number of pivots is much less than the number of rows or columns, so these matrices correspond to linear transformations that are neither onto nor one-to-one."

# ╔═╡ b436be80-fe3b-11ea-25ef-c183d6181a2c
md"To make the following plot, for various probabilities p in [0,1], we generate **$(trials)** random 01-matrices of size 
**$(ncols) by $(nrows)** (note: reversed from above). For each p, we compute the proportition of these matrices that correspond to **one-to-one** linear transformations, as well as the standard deviation of this statistic. 

Then we plot both as a function of p."

# ╔═╡ 1ec4868e-fe22-11ea-295b-6929947c5b91
accumulate_plot(is_one_to_one, 
	"Random 01-matrices, size $(ncols)-by-$(nrows): 1-to-1 transformations", 
	trials, ncols, nrows)

# ╔═╡ acf9c58a-fe3c-11ea-2ad9-fd57da314a66
md"Why are two graphs nearly the same?

The **transpose** of a matrix $A$ is then matrix $A^T$ formed by interchanging the rows and columns:

$$\left[\begin{array}{cc} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{array}\right]^T
=\left[\begin{array}{ccc} 1 & 3 & 5 \\ 2 & 4 & 6 \end{array}\right].$$

We'll see later in the class, that the number of pivot positions in $A$ is the same as in $A^T$."

# ╔═╡ 27808bc2-fe3d-11ea-12fa-f35dd36e17a7
md"This means that the number of $(nrows) by $(ncols) matrices with pivots in every row (corresponding to **onto** linear transformations) is the same as the number of $(ncols) by $(nrows) matrices with pivots in every column (corresponding to **one-to-one** linear transformations)."

# ╔═╡ 00000000-0000-0000-0000-000000000001
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git-tree-sha1 = "b3bfd02e98aedfa5cf885665493c5598c350cd2f"
uuid = "2e619515-83b5-522b-bb60-26c02a35a201"
version = "2.2.10+0"

[[FFMPEG]]
deps = ["FFMPEG_jll"]
git-tree-sha1 = "b57e3acbe22f8484b4b5ff66a7499717fe1a9cc8"
uuid = "c87230d0-a227-11e9-1b43-d7ebe4e7570a"
version = "0.4.1"

[[FFMPEG_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "JLLWrappers", "LAME_jll", "Libdl", "Ogg_jll", "OpenSSL_jll", "Opus_jll", "Pkg", "Zlib_jll", "libass_jll", "libfdk_aac_jll", "libvorbis_jll", "x264_jll", "x265_jll"]
git-tree-sha1 = "d8a578692e3077ac998b50c0217dfd67f21d1e5f"
uuid = "b22a6f82-2f65-5046-a5b2-351ab43fb4e5"
version = "4.4.0+0"

[[FixedPointNumbers]]
deps = ["Statistics"]
git-tree-sha1 = "335bfdceacc84c5cdf16aadc768aa5ddfc5383cc"
uuid = "53c48c17-4a7d-5ca2-90c5-79b7896eea93"
version = "0.8.4"

[[Fontconfig_jll]]
deps = ["Artifacts", "Bzip2_jll", "Expat_jll", "FreeType2_jll", "JLLWrappers", "Libdl", "Libuuid_jll", "Pkg", "Zlib_jll"]
git-tree-sha1 = "21efd19106a55620a188615da6d3d06cd7f6ee03"
uuid = "a3f928ae-7b40-5064-980b-68af3947d34b"
version = "2.13.93+0"

[[Formatting]]
deps = ["Printf"]
git-tree-sha1 = "8339d61043228fdd3eb658d86c926cb282ae72a8"
uuid = "59287772-0a20-5a39-b81b-1366585eb4c0"
version = "0.4.2"

[[FreeType2_jll]]
deps = ["Artifacts", "Bzip2_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "87eb71354d8ec1a96d4a7636bd57a7347dde3ef9"
uuid = "d7e528f0-a631-5988-bf34-fe36492bcfd7"
version = "2.10.4+0"

[[FriBidi_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "aa31987c2ba8704e23c6c8ba8a4f769d5d7e4f91"
uuid = "559328eb-81f9-559d-9380-de523a88c83c"
version = "1.0.10+0"

[[GLFW_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libglvnd_jll", "Pkg", "Xorg_libXcursor_jll", "Xorg_libXi_jll", "Xorg_libXinerama_jll", "Xorg_libXrandr_jll"]
git-tree-sha1 = "dba1e8614e98949abfa60480b13653813d8f0157"
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 = "c2178cfbc0a5a552e16d097fae508f2024de61a3"
uuid = "28b8d3ca-fb5f-59d9-8090-bfdbd6d07a71"
version = "0.59.0"

[[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 = "ef49a187604f865f4708c90e3f431890724e9012"
uuid = "d2c73de3-f751-5644-a686-071e5b155ba9"
version = "0.59.0+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 = "60ed5f1643927479f845b0135bb369b031b541fa"
uuid = "cd3eb016-35fb-5094-929b-558a96fad6f3"
version = "0.9.14"

[[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"

[[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"

[[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 = "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"

[[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 = "b1a708d607125196ea1acf7264ee1118ce66931b"
uuid = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
version = "1.22.0"

[[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"

[[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 = "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"

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

[[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"

[[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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# ╠═133108da-045e-41f8-998b-9772ead84b32
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# ╠═b676d206-fe37-11ea-238d-a9efe71dad02
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# ╠═c5190434-a974-4124-8765-65574de60494
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# ╠═180b696a-fe3c-11ea-18fe-37423a9b5827
# ╠═2a064810-fe3c-11ea-141f-f71f37013d7a
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# ╠═4fde7e68-fe3c-11ea-0c84-01b599a5fb56
# ╟─56266042-fe3c-11ea-23ca-d3f9240211c5
# ╟─b436be80-fe3b-11ea-25ef-c183d6181a2c
# ╠═1ec4868e-fe22-11ea-295b-6929947c5b91
# ╟─acf9c58a-fe3c-11ea-2ad9-fd57da314a66
# ╟─27808bc2-fe3d-11ea-12fa-f35dd36e17a7
# ╟─00000000-0000-0000-0000-000000000001
# ╟─00000000-0000-0000-0000-000000000002