### A Pluto.jl notebook ###
# v1.0.3

using Markdown
using InteractiveUtils

# ╔═╡ c2911073-a379-4e31-ae99-a3253faa2012
md"# **MATH 2121 - Fall 2026**
Course demonstration written using the `Julia` programming language and shown using the interactive `Pluto` environment."

# ╔═╡ cf7739e0-020a-11eb-10a5-2d8802f52f4c
md"Today's notebook will investigate how we can use matrix-vector multiplication to analyze a random walk.
"

# ╔═╡ a832924a-0209-11eb-06e0-0b066e883ca0
md"### Random walks with matrix-vector multiplication

Suppose you start with $x$ dollars and every minute you play a game, where you have a chance to either win 1 dollar or lose one dollar.

Let's suppose the probability of $p(x)$ of winning is a function of how much money you currently have. The process can stop in two ways: either **you lose all of your money**, or you win enough times that you reach some pre-determined amount where **you decide to quit**.

You could think of this as a gambling process, or some kind of competition.

We can graph your current state (the amount of money you have) as a function of time."

# ╔═╡ a6ceb284-01a6-11eb-0ace-9f348322f51d
starting_amount = 20

# ╔═╡ 3d5c7556-01a7-11eb-137a-0190746a61e1
maximum_amount = 2 * starting_amount - 1

# ╔═╡ 42eb6482-01a7-11eb-2c15-91426c26cf69
begin
	using SparseArrays
	
	mid = starting_amount
	n = maximum_amount
	
	function elementary_vector(n, i)
    	ans = sparse(zeros(n, 1))
	    ans[i, 1] = 1
    	return ans
	end

	start = elementary_vector(n, mid)
	probs = [0.5 for i=1:n]
end

# ╔═╡ aa50a5ae-01a9-11eb-113f-eddb0212e75d
begin
	using Plots 
	bar(probs, ylim=(0,1), legend=false, title="Upstep probabilites")
end

# ╔═╡ 0f48f422-0211-11eb-3a38-7fe02ed182e5
md"We need to assign the upstep probabilities $p(x)$. 

The simplest choice is just to set $p(x) = 0.5$  for all $x$.

Here is the vector showing this choice of probabilities."

# ╔═╡ 378646ec-0211-11eb-2331-c36c104c2373
md"Below is an animation of our random walk after some number of steps."

# ╔═╡ 74751370-01a9-11eb-31ed-9bd4067dcd81
steps = 100

# ╔═╡ 6405c0ae-01a7-11eb-14fe-dbb524d516fe
begin
	function transform(probs)
		n = length(probs)
		A = sparse(zeros(n, n))
		for i=1:n
			j = rand(Float64) < probs[i] ? i + 1 : i - 1
			if 1 <= j <= n
				A[j,i] = 1
			end
		end
		return A
	end
	
	function nz_position(sparse_matrix, j=1)
		I, V = findnz(sparse_matrix[1:end, j])
		return length(I) > 0 ? I[1] : 0
	end
	
    Base.@kwdef mutable struct RandomWalk
        x::Int64 = mid
    end
    
    function step!(w::RandomWalk, probs)
		if w.x != 0 && w.x != n + 1
			T = transform(probs)
			curr_vector = elementary_vector(n, w.x)
			next_vector = T * curr_vector
			x = nz_position(next_vector)
			w.x = (w.x == n && x == 0) ? n + 1 : x
		end
    end
	
    plt = plot(
        1,
        xlim = (1, steps),
        ylim = (0, n + 1),
        title = "Random walk",
        marker = 1,
        legend = false,
		markerstrokewidth = 0,
    )
    
	walk = RandomWalk()

   @gif for j=1:steps
        push!(plt, 1, walk.x)
		step!(walk, probs) 
	end every 1
	# plot(plt)
end

# ╔═╡ 4a0f0a42-0211-11eb-13c6-e79a45d5ddce
md"An essential question to ask is: after $(steps) steps, how likely are we to have a given amount of money? 

We can approximate the answer by simulating our random walk for $(steps) steps some large number of trials and making a histogram of the possible outcomes."

# ╔═╡ 7aed3032-020f-11eb-0e2a-7d1e5ee7df20
trials = 1000

# ╔═╡ d054ed4e-0228-11eb-12ab-e5ae82ae2692
begin
	empirical_distribution = [0 for j=1:n + 2]
	for t=1:trials
		test = RandomWalk()
		for j=1:steps
    	    step!(test, probs) 
		end
		empirical_distribution[test.x + 1] += 1
	end
	empirical_distribution = empirical_distribution / trials
end

# ╔═╡ bd8fee00-01ad-11eb-2da9-ddd70baf04ae
begin
	bar(empirical_distribution, legend=false, title="Empirical distribution ($(trials) trials) after $(steps) steps")
end

# ╔═╡ 6f2e2478-0227-11eb-23a5-594cbd330381
if (steps + starting_amount) % 2 == 0
	md"Note the gaps in the above plot. Every other bar is zero because after $(steps) steps starting at $(starting_amount), your current amount of money must be even."
else
	md"Note the gaps in the above plot. Every other bar is zero because after $(steps) steps starting at $(starting_amount), your current amount of money must be odd."
end

# ╔═╡ 7a66eea0-0211-11eb-2418-51b9d54cc8eb
md"This is time consuming to calculate!

There is a better way.

It turns out that we can compute the **exact probability distribution** of states after $(steps) steps by construction a certain **transition matrix** T and multiplying this matrix $(steps) times with a vector that encodes our initial state."

# ╔═╡ ec17365c-0211-11eb-1e1c-1970de55ff31
md"Here is the transition matrix:"

# ╔═╡ 71fef19a-020a-11eb-15fc-616bf4464f72
begin
	function transition_matrix(probs)
		n = length(probs)
		A = zeros(n + 2, n + 2)
		for i=1:n
			A[i, i + 1] = 1 - probs[i]
			A[i + 2, i + 1] = probs[i]
		end
		A[1, 1] = 1
		A[n + 2, n + 2] = 1
		return A
	end
	
	T = transition_matrix(probs)
	spy(T)
end

# ╔═╡ 44986014-6fd7-481b-b690-aeb95e9bdb0d
md"The transition matrix $T$ in its usual form:"

# ╔═╡ ac3aaed7-fd34-4b89-9cbe-a0eb1f7b9a70
T

# ╔═╡ 1e0e3246-0212-11eb-3198-e751286833a2
md"Let $$v$$ be our starting state vector. 

This vector has $(2+maximum_amount) rows, representing the possible balances 0, 1, 2, $$\dots$$, $(1+maximum_amount). 

It has 1.0 in position $(starting_amount), and 0.0 everywhere else, since at the start of the game our balance is $(starting_amount) with 100% probability. 

Now, the theoretical distribution of states after $(steps) turns is computed by doing $(steps) matrix-vector multiplications $$T(T(T(T(\cdots (Tv)\cdots)))$$."

# ╔═╡ 129c5474-7120-4e95-948f-ae6d0d8f7e9b
# need to put start vector between two zero entries to have correct size
start_with_boundary = Matrix(vcat([0], start, [0]))

# ╔═╡ ca911331-02da-47bc-9b6c-e575493a99f6
md"Rather than computing $Tv$, then $T(Tv)$, then $T(T(Tv))$, and so on, there is a way we can multiply the matrix $T$ with itself many times to get a single matrix $$T$$^$(steps) that we can multiply with $$v$$ to get $$T(T(T(T(\cdots (Tv)\cdots)))$$. This is what is computed below.

The  matrix-matrix multiplication operation needed to define $$T$$^$(steps) will be covered next week."

# ╔═╡ b10842f8-0228-11eb-3a81-9b35e29288d5
theoretical_distribution = T^steps * start_with_boundary

# ╔═╡ 7525e902-0212-11eb-126a-09e3b8bd57aa
md"The graph below should roughly match the empirical distribution we calculated. 

When the number of trials is large enough it should be an exact match."

# ╔═╡ 84603420-01ae-11eb-275f-13937c7800f7
bar(theoretical_distribution, color=:green, legend=false, title="Theoretical distribution of states after $(steps) steps")

# ╔═╡ af553f42-0212-11eb-1827-596baabc3ec9
md"One benefit of the theoretical approach is we can actually figure out the limiting behavior of our random walk as the number of steps grows to infinity. This is not feasible to compute empirically."

# ╔═╡ d810fc96-0212-11eb-23d4-19c189412fb3
begin
	tsteps= 3 * steps
	bar(T^tsteps * start_with_boundary, color=:green, legend=false, title="Theoretical distribution of states after $(tsteps) steps")
end

# ╔═╡ 53f57f98-0228-11eb-146f-4b37422ec0c4
md"In the constant probability 1/2 case, this last graph shows that if you continue long enough then you have a 1/2 chance of losing all your money and a 1/2 chance of making it to $(maximum_amount + 1). With sufficient time, every other possible outcome becomes vanishingly unlikely."

# ╔═╡ 86e088bc-0228-11eb-3db3-476cfd58d75a
md"Other choices of probabilities may lead to different long term outcomes."

# ╔═╡ 3fbfcd10-5547-4b16-b2c5-8cb780c620c1
begin
	function power(x, exp)
		return x < 0 ? -(abs(x)^exp) : abs(x)^exp
	end
	
	function compute_prob(i)
		x = (i - mid) / mid
		delta = -0.5 * power(x, 2)
		noise =  0 # 0.5 * (rand(Float64) - 0.5)
		p = 0.5 + delta + noise
		return maximum([0.00, minimum([1.0, p])])
	end

	newprobs = [compute_prob(i) for i=1:n]
end

# ╔═╡ 436148c0-5f17-497e-ae3c-b3bddb07b7af
bar(newprobs, ylim=(0,1), legend=false, title="Upstep probabilites")

# ╔═╡ 16c87eb4-940a-40f5-a3ae-e168a2005f09
md"For these upstep probabilities, which are not constant for all player balances, the game tends to last much longer. This because when our balance approaches 0 our winning probabilty goes up, and when our balance approaches $(1+maximum_amount) this probability goes down. Can you think of any scenarios where it would be useful to design this sort of gambling game?"

# ╔═╡ 1134ff84-f70f-4165-a293-b65a73957d9e
newT = transition_matrix(newprobs)

# ╔═╡ 01ee5d05-5cd2-4cf5-aa49-61964f9f4e01
begin
	newplt = plot(
        1,
        xlim = (1, steps),
        ylim = (0, n + 1),
        title = "Random walk",
        marker = 1,
        legend = false,
		markerstrokewidth = 0,
    )
    
	newwalk = RandomWalk()

   @gif for j=1:steps
        push!(newplt, 1, newwalk.x)
		step!(newwalk, newprobs) 
	end every 1
end

# ╔═╡ 8d17246c-cc44-4fc2-9e30-ec49f6e06b02
new_theoretical_distribution = newT^steps * start_with_boundary

# ╔═╡ 234cf2b8-b07f-40a4-8d9e-3c99cd116c0d
begin
	bar(new_theoretical_distribution, color=:green, legend=false, title="Theoretical distribution of states after $(steps) steps")
end

# ╔═╡ 4483d2e1-c918-4c8f-a280-a1acec75c8a9
begin
	newsteps= 100 * steps
	bar(newT^newsteps * start_with_boundary, color=:green, legend=false, title="Theoretical distribution of states after $(newsteps) steps")
end

# ╔═╡ 00000000-0000-0000-0000-000000000001
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git-tree-sha1 = "4f96c596b8c8258cc7d3b19797854d368f243ddc"
uuid = "860ef19b-820b-49d6-a774-d7a799459cd3"
version = "1.0.4"

[[deps.Statistics]]
deps = ["LinearAlgebra"]
git-tree-sha1 = "e2b53ce13a53367e96601081e33d34746b571bad"
uuid = "10745b16-79ce-11e8-11f9-7d13ad32a3b2"
version = "1.11.5"
weakdeps = ["SparseArrays"]

    [deps.Statistics.extensions]
    SparseArraysExt = ["SparseArrays"]

[[deps.StatsAPI]]
deps = ["LinearAlgebra"]
git-tree-sha1 = "178ed29fd5b2a2cfc3bd31c13375ae925623ff36"
uuid = "82ae8749-77ed-4fe6-ae5f-f523153014b0"
version = "1.8.0"

[[deps.StatsBase]]
deps = ["AliasTables", "DataAPI", "DataStructures", "IrrationalConstants", "LinearAlgebra", "LogExpFunctions", "Missings", "Printf", "Random", "SortingAlgorithms", "SparseArrays", "Statistics", "StatsAPI"]
git-tree-sha1 = "adb9da019510162e67a4493fc235c23203d8b09e"
uuid = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91"
version = "0.34.13"

[[deps.StructUtils]]
deps = ["Dates", "UUIDs"]
git-tree-sha1 = "2d0fc55c61321ba245c47be599570d11bac50303"
uuid = "ec057cc2-7a8d-4b58-b3b3-92acb9f63b42"
version = "2.8.5"

    [deps.StructUtils.extensions]
    StructUtilsMeasurementsExt = ["Measurements"]
    StructUtilsStaticArraysCoreExt = ["StaticArraysCore"]
    StructUtilsTablesExt = ["Tables"]

    [deps.StructUtils.weakdeps]
    Measurements = "eff96d63-e80a-5855-80a2-b1b0885c5ab7"
    StaticArraysCore = "1e83bf80-4336-4d27-bf5d-d5a4f845583c"
    Tables = "bd369af6-aec1-5ad0-b16a-f7cc5008161c"

[[deps.StyledStrings]]
uuid = "f489334b-da3d-4c2e-b8f0-e476e12c162b"
version = "1.11.0"

[[deps.SuiteSparse_jll]]
deps = ["Artifacts", "Libdl", "libblastrampoline_jll"]
uuid = "bea87d4a-7f5b-5778-9afe-8cc45184846c"
version = "7.8.3+2"

[[deps.TOML]]
deps = ["Dates"]
uuid = "fa267f1f-6049-4f14-aa54-33bafae1ed76"
version = "1.0.3"

[[deps.Tar]]
deps = ["ArgTools", "SHA"]
uuid = "a4e569a6-e804-4fa4-b0f3-eef7a1d5b13e"
version = "1.10.0"

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

[[deps.Test]]
deps = ["InteractiveUtils", "Logging", "Random", "Serialization"]
uuid = "8dfed614-e22c-5e08-85e1-65c5234f0b40"
version = "1.11.0"

[[deps.UUIDs]]
deps = ["Random", "SHA"]
uuid = "cf7118a7-6976-5b1a-9a39-7adc72f591a4"
version = "1.11.0"

[[deps.Unicode]]
uuid = "4ec0a83e-493e-50e2-b9ac-8f72acf5a8f5"
version = "1.11.0"

[[deps.UnicodeFun]]
deps = ["REPL"]
git-tree-sha1 = "53915e50200959667e78a92a418594b428dffddf"
uuid = "1cfade01-22cf-5700-b092-accc4b62d6e1"
version = "0.4.1"

[[deps.Unzip]]
git-tree-sha1 = "ca0969166a028236229f63514992fc073799bb78"
uuid = "41fe7b60-77ed-43a1-b4f0-825fd5a5650d"
version = "0.2.0"

[[deps.Vulkan_Loader_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Wayland_jll", "Xorg_libX11_jll", "Xorg_libXrandr_jll", "xkbcommon_jll"]
git-tree-sha1 = "2f0486047a07670caad3a81a075d2e518acc5c59"
uuid = "a44049a8-05dd-5a78-86c9-5fde0876e88c"
version = "1.3.243+0"

[[deps.Wayland_jll]]
deps = ["Artifacts", "EpollShim_jll", "Expat_jll", "JLLWrappers", "Libdl", "Libffi_jll"]
git-tree-sha1 = "96478df35bbc2f3e1e791bc7a3d0eeee559e60e9"
uuid = "a2964d1f-97da-50d4-b82a-358c7fce9d89"
version = "1.24.0+0"

[[deps.XZ_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "e52eca002a11c30a858185efdfb15311e1c7a6bf"
uuid = "ffd25f8a-64ca-5728-b0f7-c24cf3aae800"
version = "5.8.4+0"

[[deps.Xorg_libICE_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "a3ea76ee3f4facd7a64684f9af25310825ee3668"
uuid = "f67eecfb-183a-506d-b269-f58e52b52d7c"
version = "1.1.2+0"

[[deps.Xorg_libSM_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libICE_jll"]
git-tree-sha1 = "9c7ad99c629a44f81e7799eb05ec2746abb5d588"
uuid = "c834827a-8449-5923-a945-d239c165b7dd"
version = "1.2.6+0"

[[deps.Xorg_libX11_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libxcb_jll", "Xorg_xtrans_jll"]
git-tree-sha1 = "808090ede1d41644447dd5cbafced4731c56bd2f"
uuid = "4f6342f7-b3d2-589e-9d20-edeb45f2b2bc"
version = "1.8.13+0"

[[deps.Xorg_libXau_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "aa1261ebbac3ccc8d16558ae6799524c450ed16b"
uuid = "0c0b7dd1-d40b-584c-a123-a41640f87eec"
version = "1.0.13+0"

[[deps.Xorg_libXcursor_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libXfixes_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "6c74ca84bbabc18c4547014765d194ff0b4dc9da"
uuid = "935fb764-8cf2-53bf-bb30-45bb1f8bf724"
version = "1.2.4+0"

[[deps.Xorg_libXdmcp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "52858d64353db33a56e13c341d7bf44cd0d7b309"
uuid = "a3789734-cfe1-5b06-b2d0-1dd0d9d62d05"
version = "1.1.6+0"

[[deps.Xorg_libXext_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libX11_jll"]
git-tree-sha1 = "1a4a26870bf1e5d26cd585e38038d399d7e65706"
uuid = "1082639a-0dae-5f34-9b06-72781eeb8cb3"
version = "1.3.8+0"

[[deps.Xorg_libXfixes_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libX11_jll"]
git-tree-sha1 = "75e00946e43621e09d431d9b95818ee751e6b2ef"
uuid = "d091e8ba-531a-589c-9de9-94069b037ed8"
version = "6.0.2+0"

[[deps.Xorg_libXi_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libXext_jll", "Xorg_libXfixes_jll"]
git-tree-sha1 = "dcb316b3ce0941f195537dda56bea4517fcd3ff5"
uuid = "a51aa0fd-4e3c-5386-b890-e753decda492"
version = "1.8.4+0"

[[deps.Xorg_libXinerama_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libXext_jll"]
git-tree-sha1 = "0ba01bc7396896a4ace8aab67db31403c71628f4"
uuid = "d1454406-59df-5ea1-beac-c340f2130bc3"
version = "1.1.7+0"

[[deps.Xorg_libXrandr_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libXext_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "6c174ef70c96c76f4c3f4d3cfbe09d018bcd1b53"
uuid = "ec84b674-ba8e-5d96-8ba1-2a689ba10484"
version = "1.5.6+0"

[[deps.Xorg_libXrender_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libX11_jll"]
git-tree-sha1 = "7ed9347888fac59a618302ee38216dd0379c480d"
uuid = "ea2f1a96-1ddc-540d-b46f-429655e07cfa"
version = "0.9.12+0"

[[deps.Xorg_libpciaccess_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Zlib_jll"]
git-tree-sha1 = "58972370b81423fc546c56a60ed1a009450177c3"
uuid = "a65dc6b1-eb27-53a1-bb3e-dea574b5389e"
version = "0.19.0+0"

[[deps.Xorg_libxcb_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libXau_jll", "Xorg_libXdmcp_jll"]
git-tree-sha1 = "bfcaf7ec088eaba362093393fe11aa141fa15422"
uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b"
version = "1.17.1+0"

[[deps.Xorg_libxkbfile_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libX11_jll"]
git-tree-sha1 = "ed756a03e95fff88d8f738ebc2849431bdd4fd1a"
uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a"
version = "1.2.0+0"

[[deps.Xorg_xcb_util_cursor_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xcb_util_image_jll", "Xorg_xcb_util_jll", "Xorg_xcb_util_renderutil_jll"]
git-tree-sha1 = "9750dc53819eba4e9a20be42349a6d3b86c7cdf8"
uuid = "e920d4aa-a673-5f3a-b3d7-f755a4d47c43"
version = "0.1.6+0"

[[deps.Xorg_xcb_util_image_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xcb_util_jll"]
git-tree-sha1 = "f4fc02e384b74418679983a97385644b67e1263b"
uuid = "12413925-8142-5f55-bb0e-6d7ca50bb09b"
version = "0.4.1+0"

[[deps.Xorg_xcb_util_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libxcb_jll"]
git-tree-sha1 = "68da27247e7d8d8dafd1fcf0c3654ad6506f5f97"
uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5"
version = "0.4.1+0"

[[deps.Xorg_xcb_util_keysyms_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xcb_util_jll"]
git-tree-sha1 = "44ec54b0e2acd408b0fb361e1e9244c60c9c3dd4"
uuid = "975044d2-76e6-5fbe-bf08-97ce7c6574c7"
version = "0.4.1+0"

[[deps.Xorg_xcb_util_renderutil_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xcb_util_jll"]
git-tree-sha1 = "5b0263b6d080716a02544c55fdff2c8d7f9a16a0"
uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e"
version = "0.3.10+0"

[[deps.Xorg_xcb_util_wm_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xcb_util_jll"]
git-tree-sha1 = "f233c83cad1fa0e70b7771e0e21b061a116f2763"
uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361"
version = "0.4.2+0"

[[deps.Xorg_xkbcomp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libxkbfile_jll"]
git-tree-sha1 = "801a858fc9fb90c11ffddee1801bb06a738bda9b"
uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4"
version = "1.4.7+0"

[[deps.Xorg_xkeyboard_config_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_xkbcomp_jll"]
git-tree-sha1 = "2e59214e017a55cb87474a00fa76035c82ac0e17"
uuid = "33bec58e-1273-512f-9401-5d533626f822"
version = "2.47.0+2"

[[deps.Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "a63799ff68005991f9d9491b6e95bd3478d783cb"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.6.0+0"

[[deps.Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"
version = "1.3.1+2"

[[deps.Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "446b23e73536f84e8037f5dce465e92275f6a308"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.7+1"

[[deps.eudev_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "c3b0e6196d50eab0c5ed34021aaa0bb463489510"
uuid = "35ca27e7-8b34-5b7f-bca9-bdc33f59eb06"
version = "3.2.14+0"

[[deps.fzf_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "b6a34e0e0960190ac2a4363a1bd003504772d631"
uuid = "214eeab7-80f7-51ab-84ad-2988db7cef09"
version = "0.61.1+0"

[[deps.libaom_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "ef17c47d22224aaecc76e597ab21a072e025cf7b"
uuid = "a4ae2306-e953-59d6-aa16-d00cac43593b"
version = "3.14.1+0"

[[deps.libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Zlib_jll"]
git-tree-sha1 = "cb007192783c56d8249db4cf0e3495001edfe414"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.17.5+0"

[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
version = "5.15.0+0"

[[deps.libdecor_jll]]
deps = ["Artifacts", "Dbus_jll", "JLLWrappers", "Libdl", "Libglvnd_jll", "Pango_jll", "Wayland_jll", "xkbcommon_jll"]
git-tree-sha1 = "9bf7903af251d2050b467f76bdbe57ce541f7f4f"
uuid = "1183f4f0-6f2a-5f1a-908b-139f9cdfea6f"
version = "0.2.2+0"

[[deps.libdrm_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libpciaccess_jll"]
git-tree-sha1 = "28e57478e8a160d346a19c28b3fffb9273bcc9c2"
uuid = "8e53e030-5e6c-5a89-a30b-be5b7263a166"
version = "2.4.134+0"

[[deps.libevdev_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "56d643b57b188d30cccc25e331d416d3d358e557"
uuid = "2db6ffa8-e38f-5e21-84af-90c45d0032cc"
version = "1.13.4+0"

[[deps.libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "646634dd19587a56ee2f1199563ec056c5f228df"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.4+0"

[[deps.libinput_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "eudev_jll", "libevdev_jll", "mtdev_jll"]
git-tree-sha1 = "91d05d7f4a9f67205bd6cf395e488009fe85b499"
uuid = "36db933b-70db-51c0-b978-0f229ee0e533"
version = "1.28.1+0"

[[deps.libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Zlib_jll"]
git-tree-sha1 = "e51150d5ab85cee6fc36726850f0e627ad2e4aba"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.58+0"

[[deps.libva_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libX11_jll", "Xorg_libXext_jll", "Xorg_libXfixes_jll", "libdrm_jll"]
git-tree-sha1 = "7dbf96baae3310fe2fa0df0ccbb3c6288d5816c9"
uuid = "9a156e7d-b971-5f62-b2c9-67348b8fb97c"
version = "2.23.0+0"

[[deps.libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll"]
git-tree-sha1 = "11e1772e7f3cc987e9d3de991dd4f6b2602663a5"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.8+0"

[[deps.mtdev_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "b4d631fd51f2e9cdd93724ae25b2efc198b059b1"
uuid = "009596ad-96f7-51b1-9f1b-5ce2d5e8a71e"
version = "1.1.7+0"

[[deps.nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"
version = "1.64.0+1"

[[deps.p7zip_jll]]
deps = ["Artifacts", "CompilerSupportLibraries_jll", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
version = "17.7.0+0"

[[deps.x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "14cc7083fc6dff3cc44f2bc435ee96d06ed79aa7"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "10164.0.1+0"

[[deps.x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl"]
git-tree-sha1 = "e7b67590c14d487e734dcb925924c5dc43ec85f3"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "4.1.0+0"

[[deps.xkbcommon_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"]
git-tree-sha1 = "a1fc6507a40bf504527d0d4067d718f8e179b2b8"
uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd"
version = "1.13.0+0"
"""

# ╔═╡ Cell order:
# ╟─c2911073-a379-4e31-ae99-a3253faa2012
# ╟─cf7739e0-020a-11eb-10a5-2d8802f52f4c
# ╟─a832924a-0209-11eb-06e0-0b066e883ca0
# ╠═a6ceb284-01a6-11eb-0ace-9f348322f51d
# ╠═3d5c7556-01a7-11eb-137a-0190746a61e1
# ╟─0f48f422-0211-11eb-3a38-7fe02ed182e5
# ╟─42eb6482-01a7-11eb-2c15-91426c26cf69
# ╟─aa50a5ae-01a9-11eb-113f-eddb0212e75d
# ╟─378646ec-0211-11eb-2331-c36c104c2373
# ╠═74751370-01a9-11eb-31ed-9bd4067dcd81
# ╟─6405c0ae-01a7-11eb-14fe-dbb524d516fe
# ╟─4a0f0a42-0211-11eb-13c6-e79a45d5ddce
# ╠═7aed3032-020f-11eb-0e2a-7d1e5ee7df20
# ╠═d054ed4e-0228-11eb-12ab-e5ae82ae2692
# ╟─bd8fee00-01ad-11eb-2da9-ddd70baf04ae
# ╟─6f2e2478-0227-11eb-23a5-594cbd330381
# ╟─7a66eea0-0211-11eb-2418-51b9d54cc8eb
# ╟─ec17365c-0211-11eb-1e1c-1970de55ff31
# ╟─71fef19a-020a-11eb-15fc-616bf4464f72
# ╟─44986014-6fd7-481b-b690-aeb95e9bdb0d
# ╠═ac3aaed7-fd34-4b89-9cbe-a0eb1f7b9a70
# ╟─1e0e3246-0212-11eb-3198-e751286833a2
# ╠═129c5474-7120-4e95-948f-ae6d0d8f7e9b
# ╟─ca911331-02da-47bc-9b6c-e575493a99f6
# ╠═b10842f8-0228-11eb-3a81-9b35e29288d5
# ╟─7525e902-0212-11eb-126a-09e3b8bd57aa
# ╟─84603420-01ae-11eb-275f-13937c7800f7
# ╟─af553f42-0212-11eb-1827-596baabc3ec9
# ╟─d810fc96-0212-11eb-23d4-19c189412fb3
# ╟─53f57f98-0228-11eb-146f-4b37422ec0c4
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