SC0x_01_Mathematical_Functions.jl

### A Pluto.jl notebook ###
# v0.19.5

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)
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# ╔═╡ 89497e80-afd6-11ec-39d1-ffadb6c24524
begin
	using HypertextLiteral,PlutoUI,LaTeXStrings, Formatting
	
	using Logging
	Logging.disable_logging(Logging.Info) # Disables printing on the (pluto) terminal
	
	using Plots,Plots.PlotMeasures # To be able to use cm, mm, etc.
	default(legend=false,
			size=(250,225),
			tickfontsize=8,
			line=(:2, :blue),
			framestyle=:origin
	)
	
	html"""
	Notebook Packages & Settings
	<style> main {max-width: 925px; } </style>
	<style> .rich_output {color: black;}</style>
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end

# ╔═╡ e84c21e3-1b08-4a59-99e6-579dc2f4fef1
function center_div(something)
	@htl "<div align='center'>$something</div>"
end;

# ╔═╡ 630dcdb5-457e-426e-aa7c-c5423c5b882b
md"""$\require{physics}$"""

# ╔═╡ 600220f8-60e2-4c6a-84cd-af5c06bfb943
TableOfContents(title="Mathematical Functions")

# ╔═╡ 46e43464-09a3-4e32-9537-a0dd0df3da5b
md"""
# Simple Functions
These are some simple mathematical models used in SCM.
## Linear

$\begin{align}
f(x) = mx + b
\end{align}$ 

**Cost functions:**\
Of the form: $f(\text{Units}) = \text{Fixed Cost + Unitary Cost(Units)}$

$\begin{aligned}
&\begin{array}{left}
\hline
\text{Truckload Transportation costs} & f(d) = € 500 + \frac{€ 14.5}{d}(d) & d=\text{distance}\\
\text{Warehousing costs} &  f(\text{cases}) = € 800 + \frac{€ 3.5}{\text{case}}(\text{cases}) \\
\hline
\end{array}
\end{aligned}$
"""

# ╔═╡ 14c91a8a-edf6-4a91-b625-1d90d2c28809
let
	T(d) = 500 + 14.5d
	d = Vector(0:20)
	W(c) = 800 + 3.5c
	c = Vector(0:50)
	fig = plot(layout=grid(1,2), size=(600, 250), framestyle=:origin, ylims=[0,1000], bottom_margin=3mm)
	plot!(fig[1], d, T.(d), line=(2, :blue), right_margin=5mm,
		xlabel="distance traveled", ylabel="Transportation Costs")
	plot!(fig[2], c, W.(c), line=(2, :red), left_margin=5mm,
		xlabel="cases stored", ylabel="Warehousing Costs")
	center_div(fig)
end

# ╔═╡ 7d582600-904d-486c-a9e6-8b24510807e7
md"""
## Quadratic

$\begin{align}
f(x)=ax^2+bx+c
\end{align}$

**Profit function**\
Assumes sales volume, and therefore revenue, are based on price, and that production levels/costs are also determined by the sales volume.

$\begin{aligned}
&\begin{array}{left}
\hline
\text{Unit Sales} & S(p) = 20,000 - 80p & p=\text{price} \\
\text{Revenue} & R(p) = (20,000 - 80p)p \\
\text{Production Costs} &  C(p) = 500,000 + 75(20,000 - 80p) \\
\text{Profit} &  P(p) = 20,000p - 80p^2 - 500,000 - 1,500,000 + 6000p \\
	& P(p) = -80p^2 + 26,000p - 2,000,000 \\
\hline
\end{array}
\end{aligned}$
"""

# ╔═╡ 8d6ffe41-48e0-4b13-b98e-d46da3eec4f4
let
	P(p) = -80p^2 + 26_000p - 2_000_000
	p = Vector(115:210)
	fig = plot(p, P.(p), framestyle=:origin, size=(400,250), line=(:2, :blue),
		ylabel="Profit", xlabel="price", yformatter=y->round(y, digits=0))
	p_max = 26000/160 # Derivation
	scatter!([p_max],[P(p_max)], color=:black, label="")
	annotate!(p_max, P(p_max), text("\$ ($p_max, $(P(p_max))) \$", 11, :bottom, :right))
	center_div(fig)
end

# ╔═╡ e440e65f-61d3-4507-b13e-ecbd67aa45ab
md"""
**Parcel Trucking**\
Parcel carriers combine many orders into a single shipment, so this function represents the cost of shipment for an individual order based on its weight.

Let's assume that it has been modelled by the shipper based on historical data. As weight increases, it tends to taper down.

$\begin{aligned}
&\begin{array}{left}
\hline
\text{Cost per order} & f(w) = 33 + 0.067w - 0.00005w^2 & w=\text{weight} \\
\hline
\end{array}
\end{aligned}$

 Of course this function would be defined only for a particular interval, say $\forall w:0<w<1000$, so we cannot get too good or even negative costs.
"""

# ╔═╡ 25905a9a-360b-4ee2-975f-4fec94602756
let
	f(w) = 33 + 0.067w - 0.00005w^2
	w = Vector(0:1000)
	fig = plot(w, f.(w), framestyle=:origin, size=(400,250), line=(:2, :blue),
		ylabel="Cost of order", xlabel="weight", yformatter=y->round(y, digits=0))
	w_max = 0.067/(0.00005*2)
	scatter!([w_max], [f(w_max)])
	annotate!(w_max, f(w_max) - 3, text("\$ ($w_max, $(round(f(w_max), digits=2))) \$", 11, :top))
	center_div(fig)
end

# ╔═╡ 765cff9d-05da-4c86-8c1f-2d6585bac2ad
md"""
## Power
$f(x)=ax^b$

**Cost Absorption**\
Let's say we have a vehicle that has a monthly fixed cost of $€ 1000$ and has a capacity of $20$ full pallets. Then each pallet absorbs the fixed cost in the following manner

$\begin{aligned}
&\begin{array}{left}\hline
\text{Absorbed Transportation Cost} & f(p) = 1000*p^{-1} & p=\text{pallets} \\
\hline\end{array}
\end{aligned}$

"""

# ╔═╡ 0044dcd2-c7f1-4341-855e-c412e2253e5f
let
	f(x) = 1000x^-1
	x = Vector(1:0.1:20)
	fig = plot(x, f.(x), size=(400, 250), xlabel="pallets transported", ylabel="Fixed Cost 'per pallet'")
	center_div(fig)
end

# ╔═╡ 075efb44-57cc-4746-b78d-8433d531737b
md"""
## Exponential
$f(x)=ab^x$

**Compound interest**\
Behaviour of compound interest over time.

$\begin{aligned}
&\begin{array}{left}\hline
\text{Future Value} & f(t,i) = 1000(1+i)^t & i=\text{interest}, t=\text{periods} \\
\hline\end{array}
\end{aligned}$

"""

# ╔═╡ e4e4b914-217b-4ed5-bdd6-8d24030157d9
let
	f(x,y) = 1000(1+y)^x
	x = Vector(1:48)
	fig = plot(size=(400, 250), leg=true, legend=:topleft)
	colors = [:blue, :red, :green, :purple, :orange]
	for (j,i) in enumerate(0.01:0.01:0.05)
		plot!(x, f.(x,i), label="\$i=$i\$", line=colors[j], legendfont=10)
	end
	center_div(fig)
end

# ╔═╡ 5da33aeb-c250-4b0f-949d-cca5c96bb49b
md"""
## Logarithmic
$x=b^y \iff f(x)=\log_b(x)$

**Time needed for achieving a particular Future Value**\
Let's say we want to double in value with compound interest

$\begin{align}
F &= P(1 + i)^t \\
2P &= P(1 + i)^t \\
\ln(2) &= t \cdot \ln(1+i) \\
t &= \frac{\ln(2)}{\ln(1+i)}
\end{align}$

$\begin{aligned}
&\begin{array}{left}\hline
\text{Time to double initial value} & f(i) = \displaystyle\frac{\ln(2)}{\ln(1 + i)} & 
	i=\text{interest}, t=\text{periods} \\
\hline\end{array}
\end{aligned}$
"""

# ╔═╡ 13badf79-66a5-450f-9ef8-826aa90a23ac
let
	f(x) = log(2)/log(1+x)
	x = Vector(0.01:0.005:0.2)
	fig = plot(x, f.(x), size=(600, 250),
		xticks=(Vector(0.01:0.01:0.2), Vector(1:20)), bottom_margin=3mm,
		yticks=Vector(0:5:60), gridalpha=0.50,
		xlabel="Annual Interest Rate (%)", ylabel="Years to double")
	center_div(fig)
end

# ╔═╡ 08470b64-6f89-464d-b101-041c23ebbf13
md"""
## Multivariate

$y=f(x_1,x_2,...,x_n)$

These are functions with more than one independent variable or input.\
**Inventory Cost**\
One of the simplest models of total inventory cost includes
- Total cost of items or purchasing cost
- Ordering or set up cost
- [Inventory] holding cost
With the following parameters and variables
-  $c$ = unit cost
-  $D$ = total demand (mostly for a year)
-  $Q$ = order size (units/order)
-  $ct$ = cost of an individual order
-  $ce$ = cost of holding a unit over time

$\begin{aligned}
&\begin{array}{left}\hline
\text{Total Inventory Cost} & TC(c,ct,ce,Q,D)=cD + ct\frac{D}{Q} + ce\frac{Q}{2} & 
	 \\
\hline\end{array}
\end{aligned}$

"""

# ╔═╡ ac5bb82e-4667-4a53-85de-014c9009a886
md"""
c = $(@bind c Slider(5:0.1:10, show_value=true)) \
ct = $(@bind ct Slider(400:5:1000, show_value=true)) \
ce = $(@bind ce Slider(5:0.1:10, show_value=true)) \
D = $(@bind D Slider(3000:100:5000, show_value=true))
"""

# ╔═╡ d9e8bf3e-94c4-43e6-b83a-a55eb4b8c466
let
	f(Q) = c*D + ct*(D/Q) + ce*(Q/2)
	Q = Vector(100:5000)
	fig = plot([Q], [f.(Q)], size=(400, 250),
		xformatter=x->"\$ $(format(x, precision=0, commas=true)) \$",
		yformatter=y->"\$ $(format(y, precision=2, commas=true)) \$",
		tickfontsize=9,
		xlabel="Order Size (Q)", ylabel="Total Inventory Cost", labelfontsize=9)
	center_div(fig)
end

# ╔═╡ 64272bfa-a22d-4b6e-9a53-42cb446b6622
md"""
# Convexity
A function is convex in the interval $<x_1, x_2>$ if

$f\big[\lambda x_2 + (1-\lambda)x_1\big] \leq \lambda f(x_2) + (1 - \lambda)f(x_1)$

for all values $\lambda : 0 < \lambda < 1$

**Example**\
The function of the left is convex for the interval $<x_1, x_2>$, the function on the right is not.
"""

# ╔═╡ e31edee5-693b-419d-84cb-ae08b8b4b0ec
md"""
λ = $(@bind λ Slider(0:0.01:1, show_value=true, default=0.5))
"""

# ╔═╡ 8a27a019-38ae-4cc8-a978-f949b16ec246
let
	f(x) = 3x^2 - 4x + 2
	g(x) = (x-0.7)^5 + 4(x-0.7)^2 + 1
	functions = (f, g)
	x = Vector(-1.2:0.05:2.2)
	x1, x2 = -1, 2
	x_input = λ*x2 + (1 - λ)*x1
	fig = plot(layout=grid(1,2), size=(500, 250), yticks=[], tickfontsize=12, framestyle=:default)
	for i in 1:2
		plot!(fig[i], [x], [functions[i].(x)], xticks=([x1, x2], ["\$ x_1 \$","\$ x_2 \$"]))
		scatter!(fig[i], [x_input], [functions[i](x_input)], marker=(:red, 5))
		plot!(fig[i], [x1, x2], [functions[i](x1), functions[i](x2)], line=(:red, :dash, 1))
	end
	center_div(fig)
end

# ╔═╡ b403713d-7870-4a52-ac94-9a62e6e7d96a
md"""
# 🔨
"""

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git-tree-sha1 = "4f6ec5d99a28e1a749559ef7dd518663c5eca3d5"
uuid = "90137ffa-7385-5640-81b9-e52037218182"
version = "1.4.3"

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

[[deps.StatsAPI]]
deps = ["LinearAlgebra"]
git-tree-sha1 = "c3d8ba7f3fa0625b062b82853a7d5229cb728b6b"
uuid = "82ae8749-77ed-4fe6-ae5f-f523153014b0"
version = "1.2.1"

[[deps.StatsBase]]
deps = ["DataAPI", "DataStructures", "LinearAlgebra", "LogExpFunctions", "Missings", "Printf", "Random", "SortingAlgorithms", "SparseArrays", "Statistics", "StatsAPI"]
git-tree-sha1 = "8977b17906b0a1cc74ab2e3a05faa16cf08a8291"
uuid = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91"
version = "0.33.16"

[[deps.StructArrays]]
deps = ["Adapt", "DataAPI", "StaticArrays", "Tables"]
git-tree-sha1 = "57617b34fa34f91d536eb265df67c2d4519b8b98"
uuid = "09ab397b-f2b6-538f-b94a-2f83cf4a842a"
version = "0.6.5"

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

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

[[deps.Tables]]
deps = ["DataAPI", "DataValueInterfaces", "IteratorInterfaceExtensions", "LinearAlgebra", "OrderedCollections", "TableTraits", "Test"]
git-tree-sha1 = "5ce79ce186cc678bbb5c5681ca3379d1ddae11a1"
uuid = "bd369af6-aec1-5ad0-b16a-f7cc5008161c"
version = "1.7.0"

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

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

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

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

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

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

[[deps.Unzip]]
git-tree-sha1 = "34db80951901073501137bdbc3d5a8e7bbd06670"
uuid = "41fe7b60-77ed-43a1-b4f0-825fd5a5650d"
version = "0.1.2"

[[deps.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"

[[deps.Wayland_protocols_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4528479aa01ee1b3b4cd0e6faef0e04cf16466da"
uuid = "2381bf8a-dfd0-557d-9999-79630e7b1b91"
version = "1.25.0+0"

[[deps.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"

[[deps.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"

[[deps.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"

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

[[deps.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"

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

[[deps.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"

[[deps.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"

[[deps.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"

[[deps.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"

[[deps.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"

[[deps.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"

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

[[deps.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"

[[deps.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"

[[deps.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"

[[deps.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"

[[deps.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"

[[deps.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"

[[deps.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"

[[deps.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"

[[deps.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"

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

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

[[deps.Zstd_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "e45044cd873ded54b6a5bac0eb5c971392cf1927"
uuid = "3161d3a3-bdf6-5164-811a-617609db77b4"
version = "1.5.2+0"

[[deps.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"

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

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

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

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

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

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

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

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

[[deps.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:
# ╟─89497e80-afd6-11ec-39d1-ffadb6c24524
# ╟─e84c21e3-1b08-4a59-99e6-579dc2f4fef1
# ╟─630dcdb5-457e-426e-aa7c-c5423c5b882b
# ╠═600220f8-60e2-4c6a-84cd-af5c06bfb943
# ╟─46e43464-09a3-4e32-9537-a0dd0df3da5b
# ╟─14c91a8a-edf6-4a91-b625-1d90d2c28809
# ╟─7d582600-904d-486c-a9e6-8b24510807e7
# ╟─8d6ffe41-48e0-4b13-b98e-d46da3eec4f4
# ╟─e440e65f-61d3-4507-b13e-ecbd67aa45ab
# ╟─25905a9a-360b-4ee2-975f-4fec94602756
# ╟─765cff9d-05da-4c86-8c1f-2d6585bac2ad
# ╟─0044dcd2-c7f1-4341-855e-c412e2253e5f
# ╟─075efb44-57cc-4746-b78d-8433d531737b
# ╟─e4e4b914-217b-4ed5-bdd6-8d24030157d9
# ╟─5da33aeb-c250-4b0f-949d-cca5c96bb49b
# ╟─13badf79-66a5-450f-9ef8-826aa90a23ac
# ╟─08470b64-6f89-464d-b101-041c23ebbf13
# ╟─ac5bb82e-4667-4a53-85de-014c9009a886
# ╟─d9e8bf3e-94c4-43e6-b83a-a55eb4b8c466
# ╟─64272bfa-a22d-4b6e-9a53-42cb446b6622
# ╟─e31edee5-693b-419d-84cb-ae08b8b4b0ec
# ╟─8a27a019-38ae-4cc8-a978-f949b16ec246
# ╟─b403713d-7870-4a52-ac94-9a62e6e7d96a
# ╟─00000000-0000-0000-0000-000000000001
# ╟─00000000-0000-0000-0000-000000000002