### A Pluto.jl notebook ###
# v0.16.1

using Markdown
using InteractiveUtils

# ╔═╡ dcb245de-bc64-4892-a542-faaf9b6d7303
# setup the Julia environment -
begin
	using ImageIO
	using FileIO
	using Images
	using Plots
	using ImageMagick
	using LinearAlgebra
	using Optim
	using PlutoUI
end

# ╔═╡ ce580d07-e462-4c11-8cd6-b314f3ea13af
PlutoUI.LocalResource("./figs/Reactor.png")

# ╔═╡ c38c856c-153d-11ec-0945-f7587b6f2699
md"""
#### Example: Steady-state concentration balances

Consider a reactor in which enzyme $E$ catalyzes the conversion of some substrate $S$ (starting material) to a product $P$ at a rate $\hat{r}_{1}$ (units: mmol/L-time). Enzyme $E$, which is not stable, degrades in the reactor at rate $r_{2}$ (units: mmol/L-time). Susbtrate $S$ is introduced into the reactor in stream 1, while enzyme $E$ is introduced in stream 2. Lastly, unreacted subrate $S$, enzyme $E$ and product $P$ leave the reactor in stream 3.

Let the concentrations of $S$ be given by $C_{1}$ (units: mmol/L), the enzyme $E$ by $C_{2}$ (units: mmol/L) and the product $P$ by $C_{3}$ (units: mmol/L). Then, the kinetic expression for reaction $\hat{r}_{1}$ is given by:

$$\hat{r}_{1} = k_{cat}E\left(\frac{C_{1}}{K+C_{1}}\right)$$

where $k_{cat}$ denotes the turnover number (catalytic rate constant with units 1/time) for the enzyme $E$ and substate $S$, and $K$ denotes the saturation constant for enzyme $E$ and substrate $S$ (units: concentration). The rate of degradation of enzyme $E$ is assumed to be first-order with rate:

$$\hat{r}_{2} = k_{d}E$$

where $k_{d}$ denotes the degradation constant for enzyme $E$ (units: 1/time). 

__Estimate the composition of stream 3.__

###### Assumptions

* Reactor is at steady-state and well-mixed
* Reactor is at a constant T, P and V
* Density of stream 1, 2 and 3 is constant and equal to water at T and P. 

###### Problem setup
The volume of the reactor $V$ = 30$\mu$L. Stream 1 has a volumetric flow rate $F_{1}$ = 10$\mu$L/h and Stream 2 has a volumetric flow rate $F_{2}$ = 5$\mu$L/h. The concentration(s) of susbstrate in the stream 1 is $C_{1,1}$ = 10 mmol/L, enzyme $E$ in stream 2 is $C_{2,2}$ = 2.0 mmol/L. The turnover number of enzyme $E$ is give by $k_{cat}$ = 16.2 s$^{-1}$, the saturation coefficient $K$ = 5 mmol/L and degradation constant $k_{d}$ = 0.1 h$^{-1}$. 
"""

# ╔═╡ 2baefe3c-f9d1-4851-85e8-2f81d49f60c6


# ╔═╡ 61f56a22-d722-41a7-b371-06058bfd5da5
md"""

##### Steady-state concentration balance equations

Let's start with the steady-state concetration balances that we developed last time from your notes, and model the composition of the output stream (concentration of $S$, $E$ and $P$). 

Concentration balances:

$$\sum_{s~=~1}^{\mathcal{S}}v_{s}D_{s}C_{k,s} + \sum_{r~=~1}^{\mathcal{R}}\sigma_{kr}\hat{r}_{r} + \sum_{j~=~1}^{\mathcal{T}}\tau_{kj}q_{j}X = 0\qquad{k=1,2,\cdots, \mathcal{M}}$$

We have three streams so $\mathcal{S}$=3, three chemical components $\mathcal{M}$=3, two _bulk phase_ reactions so $\mathcal{R}$ = 2, and no cells ($\mathcal{T}$ = 0). Thus, we have three equations of the form:

$$\sum_{s~=~1}^{3}v_{s}D_{s}C_{k,s} + \sum_{r~=~1}^{2}\sigma_{kr}\hat{r}_{r} = 0\qquad{k=1,2, 3}$$

__Hmmm.__ We can't solve this problem now. 

* Issue 1: We don't have equations for $C_{1}$, $C_{2}$ and $C_{3}$. These terms appear in the kinetics, but nowhere else. What do we do?
* Issue 2: How we calculate $D_{3}$? 
* Issue 3: The kinetics are non-linear. How do we handle that?
"""

# ╔═╡ cb4478c2-ca0a-40bb-bb9f-cd4e0437381e
md"""
###### Issue 1: $C_{1}$, $C_{2}$ and $C_{3}$. What can we do?

The well mixed assumption allows us to say that the concentration inside the reactor i.e., $C_{1}$, $C_{2}$ and $C_{3}$ is approximately the same as the concentration in the exit streams. Thus:

$$C_{i}\simeq C_{i3}\qquad{i=1,2,3}$$
"""

# ╔═╡ e78f1041-4259-491f-86d4-1f7d6ced4f15
md"""

###### Issue 2: How do we calculate $D_{3}$?

Let's start from the steady-state total mass balance:

$$\sum_{s=1}^{\mathcal{S}}v_{s}\dot{m}_{s} = 0$$

However, from the definition of density $\rho$ (units: mass/volume) we know that: $\dot{m}_{s} = \rho_{s}\dot{V}_{s}$
where $\dot{V}_{s}$ is another representation of the volumetric flow rate of stream $s$ (units: volume/time). 
Plugging $\dot{m}_{s}$ (in terms of the density and volumetric flow rate) into the total mass balance gives:

$$\sum_{s=1}^{\mathcal{S}}v_{s}\rho_{s}\dot{V}_{s} = 0$$

However, from the problem statement we know that $\rho_{1}\simeq\rho_{2}\simeq\rho_{3}\equiv\rho$. 
Thus, we can divide out the density to give:

$$\sum_{s=1}^{\mathcal{S}}v_{s}\dot{V}_{s} = 0$$

This equation is referred to as a _volume balance_ but it's just another form of the total mass balance (given the assumption of constant density). Dividing the steady-state volume balance by the reactor volume $V$ gives:

$$\sum_{s=1}^{\mathcal{S}}v_{s}D_{s} = 0$$

which we can use to estimate $D_{3}$:

$$D_{1}+D_{2} = D_{3}$$

###### Aside: How good is the constant density assumption?
In liquids, the constant density assumption is sometimes reasonable, however, for gasses the validity of this assumption needs to rigorously confirmed. 

"""

# ╔═╡ f954ba83-d8a5-4ef0-b84e-e513e95dbb0f
md"""
###### Issue 3: We have a set of non-linear algebraic equations. How can we solve these?

How can we turn a currently unsolvable problem into a problem that we can solve?
"""

# ╔═╡ 777bae38-20e6-4956-a0c9-6245aae429b4
md"""
#### Turn that frown upside down: approximate methods to solve for the exit compostion
"""

# ╔═╡ 5852e1da-8b57-40a4-9f93-2c2b40dd101a
md"""

###### Method 1: The $C_{1}\gg{K}$ case

If $C_{1}\gg{K}$, then $\hat{r}_{1}$ becomes linear:

$$\hat{r}_{1}\simeq k_{cat}C_{2}$$

which means we can write the concetration balances (in matrix vector form) as:

$\begin{pmatrix}
-D_{3} & -k_{cat} & 0 \\
0 & -\left(D_{3}+k_{d}\right) & 0 \\
0 & k_{cat} & -D_{3} 
\end{pmatrix}
\begin{pmatrix}
C_{1} \\
C_{2} \\
C_{3}
\end{pmatrix} = -
\begin{pmatrix}
D_{1}C_{11} \\
D_{2}C_{22} \\
0
\end{pmatrix}$

This linear system of equations is in the form: $\mathbf{A}\mathbf{x} = \mathbf{b}$. Thus, we can solve for the unknown composition vector by computing the inverse of the $\mathbf{A}$ matrix or:

$$\mathbf{x} = \mathbf{A}^{-1}\mathbf{b}$$
"""

# ╔═╡ e3fd51c5-f8c0-4c64-af7e-a0b6943a62a3
begin
	
	# initialize -
	parameter_dict = Dict{String,Any}()
	
	# Problem parameters (make sure we have the correct units!)
	F1 = 10.0 			# units: μL/hr
	F2 = 5.0 			# units: μL/hr
	V = 30.0 			# units: μL
	kd = 0.10   		# units: hr^-1
	kcat = 16.2*(3600) 	# units: hr^-1
	K = 5.0 			# units: mmol/L
	
	# Setup the dilution D1, D2 and D3 -
	D1 = (F1/V) 		# units: hr^-1
	D2 = (F2/V) 		# units: hr^-1
	D3 = D1 + D2 		# units: hr^-1
	
	# What is C11 and C12?
	C11 = 10.0 			# units: mmol/L
	C22 = 2.0 			# units: mmol/L
	
	# package the parameters -
	parameter_dict["F1"] = F1
	parameter_dict["F2"] = F2
	parameter_dict["V"] = V
	parameter_dict["kd"] = kd
	parameter_dict["kcat"] = kcat
	parameter_dict["K"] = K
	parameter_dict["D1"] = D1
	parameter_dict["D2"] = D2
	parameter_dict["D3"] = D3
	parameter_dict["C11"] = C11
	parameter_dict["C22"] = C22
	
	# return -
	nothing
end

# ╔═╡ 204fa395-1288-4dcd-90f9-16eaca2c7f07
begin
	
	# setup A1 -
	A1 = [-D3 -kcat 0 ; 0 -(D3+kd) 0 ; 0 kcat -D3]
	
	# setup bV1 -
	bV1 = [-D1*C11 ; -D2*C22 ; 0]

	# return -
	nothing
end

# ╔═╡ efacdd42-3564-4788-82a8-eec6d1dee154
# compute the output composition?
c_out_1 = inv(A1)*bV1

# ╔═╡ fc2dc16e-495a-4fd9-a9f7-859583d959d4
# check:
I = inv(A1)*A1

# ╔═╡ bf5521d9-192f-4378-9cbc-270142ab689b
A1

# ╔═╡ e58faa37-938c-4475-9a73-fd5922da721f
md"""

##### Method 2: Taylor expansion of $\hat{r}_{1}$ around some point $x^{\star}$ = ($C^{\star}_{1}$, $C^{\star}_{2}$) 

The Taylor expansion of $\hat{r}_{1}$ is given by:

$$\hat{r}_{1}\simeq \hat{r}_{1}\left(x^{\star}\right)+
\left(\frac{\partial{\hat{r}_{1}}}{\partial C_{1}}\right)\Bigg\vert_{x^{\star}}\left(C_{1} - C^{\star}_{1} \right) + 
\left(\frac{\partial{\hat{r}_{1}}}{\partial C_{2}}\right)\Bigg\vert_{x^{\star}}\left(C_{2} - C^{\star}_{2} \right) + H.O.T $$

where the derivative(s) are evaluated at $x^{\circ}$:

$$a = \left(\frac{\partial{\hat{r}_{1}}}{\partial C_{1}}\right)\Bigg\vert_{x^{\star}} = \frac{k_{cat}C^{\star}_{2}K}{\left(K+C^{\star}_{1}\right)^2}$$

and:

$$b = \left(\frac{\partial{\hat{r}_{1}}}{\partial C_{2}}\right)\Bigg\vert_{x^{\star}} = \frac{k_{cat}C^{\star}_{1}}{\left(K+C^{\star}_{1}\right)}$$

Substituting the Taylor expansion of $\hat{r}_{1}$ into the concentration balance and re-writing the system in matrix-vector form
gives:

$\begin{pmatrix}
-(D_{3} + a) & -b & 0 \\
0 & -\left(D_{3}+k_{d}\right) & 0 \\
a & b & -D_{3} 
\end{pmatrix}
\begin{pmatrix}
C_{1} \\
C_{2} \\
C_{3}
\end{pmatrix} = 
\begin{pmatrix}
-D_{1}C_{11} + \hat{r}^{\star}_{1} - aC^{\star}_{1} - bC^{\star}_{2} \\
-D_{2}C_{22} \\
-\hat{r}^{\star}_{1} + aC^{\star}_{1} + bC^{\star}_{2}
\end{pmatrix}$
"""

# ╔═╡ eecc36b0-366f-469b-8743-f9f96a5f6e6c
# select a point to linearize around -
x = (0.001,0.5)

# ╔═╡ 543959b5-2646-4fa8-9b76-27d24adaa7ac
begin
	
	# compute r1star
	r1 = kcat*(x[2])*(x[1]/(K+x[1]))
	
	# compute a and b -
	a = (kcat*x[2]*K)/((K+x[1])^2)
	
	# compute b -
	b = (kcat*x[1])/(K+x[1])
	
	# Setup A2 -
	A2 = [-(D3+a) -b 0 ; 0 -(D3+kd) 0 ; a b -D3]
	
	# setup bV2 -
	bV2 = [-D1*C11+r1-a*x[1]-b*x[2] ; -D2*C22 ; -r1+a*x[1]+b*x[2]]
	
	# return -
	nothing
end

# ╔═╡ 453cc1b5-ab5a-4ed0-b126-9123601aaaa9
# compute the exit composition -
c_out_2 = inv(A2)*bV2

# ╔═╡ 60acab50-aa82-4622-ba02-db02282a2ed0
md"""

##### Method 3: Optimization approach: Nelder-Mead Method

The last approach (and the method most commonly used in practice) is to recast the problem of estimating the unknown exit composition as an _optimization problem_. In this case, we iteratively make a solution guess, test how good this guess is by computing the squared error using an _objective function_, and then use this error to inform our next guess of the solution. How the error is used to generate a new guess depends upon the _optimization algorithm_ (of which there is a huge number). 

In this case, let's use the $\texttt{Optim}$ package of Julia which encodes many optimization algorithm, and particularly the [Nelder-Mead](https://en.wikipedia.org/wiki/Nelder–Mead_method) algorithm which is a common direct (derivative free) approach for solving unconstrained optimization problems. 

"""

# ╔═╡ 45c480f6-2b41-4075-b764-d1f6a71fc0f9
begin
	
	"""
		compute_soln_error(soln_array::Array{Float64,1}, parameters::Dict{String,Any}) -> Array{Float64,1}
	"""
	function compute_soln_error(soln_array::Array{Float64,1}, parameters::Dict{String,Any})::Array{Float64,1}
		
		try
			
			# Get parameters from the Dict -
			F1 = parameters["F1"]
			F2 = parameters["F2"]
			V = parameters["V"]
			kd = parameters["kd"]
			kcat = parameters["kcat"]
			K = parameters["K"]
			D1 = parameters["D1"]
			D2 = parameters["D2"]
			D3 = parameters["D3"]
			C11 = parameters["C11"]
			C22 = parameters["C22"]

 			# Alias the soln -
			C1 = soln_array[1]
			C2 = soln_array[2]
			C3 = soln_array[3]

 			# compute r1 and r2 -
			r1 = kcat*C2*(C1/(K+C1))
			r2 = kd*C2

 			# setup A matrix and bV -
			AM = [-D3 0 0; 0 -D3 0; 0 0 -D3]
			bV = [D1*C11-r1; D2*C22-r2; r1]

 			# compute the error -
			error_array = AM*soln_array + bV

			# return -
			return error_array
		catch error
			rethrow()
		end
	end
	
	# return -
	nothing
end

# ╔═╡ cd7176c0-a8be-4982-b31d-2674fe20e84c
# compue the soln error -
soln_error_1 = compute_soln_error(c_out_1, parameter_dict)

# ╔═╡ 3fc4c269-2a2c-445e-8786-4b56fa60b80f
# compue the soln error -
soln_error = compute_soln_error(c_out_2, parameter_dict)

# ╔═╡ 5687b01f-2ace-4a80-9b3c-9e881f89eaf3
begin
	
	function objective_function(state_array, parameters)::Float64
		
		# compute the soln error -
		soln_error_array = compute_soln_error(state_array, parameters)
		
		# return -
		return transpose(soln_error_array)*soln_error_array
	end

	# return -
	nothing
end

# ╔═╡ 73a5e9c2-58c9-428c-8f03-17561d7630f6
begin
	# using optim -
	
	# Use the answer from Method 2 as a starting point 
	xinitial = [0.01, 0.1, 1.0]
	
	# setup the objective function -
	OF(p) = objective_function(p,parameter_dict)
    
    # call the optimizer -
    opt_result = optimize(OF,xinitial,NelderMead())
end

# ╔═╡ f5022387-72e1-4262-94ad-3bffd4010ae2
nm_soln = Optim.minimizer(opt_result)

# ╔═╡ 075683a3-3c19-4edd-9a21-a93f1d31e9bf
nm_error_check = compute_soln_error(nm_soln, parameter_dict)

# ╔═╡ 1e3fa40e-19f2-4c97-8353-1705a266ee5e
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main {
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a {
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</style>
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[[DataStructures]]
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[[DataValueInterfaces]]
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[[Dates]]
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[[DelimitedFiles]]
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[[DiffResults]]
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[[DiffRules]]
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[[Distances]]
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[[Distributed]]
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[[DocStringExtensions]]
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[[Downloads]]
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[[EarCut_jll]]
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[[Expat_jll]]
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[[FFTW_jll]]
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[[FileIO]]
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[[FillArrays]]
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[[FixedPointNumbers]]
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[[Fontconfig_jll]]
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[[ForwardDiff]]
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[[FreeType2_jll]]
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[[FriBidi_jll]]
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[[GLFW_jll]]
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[[GR]]
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[[GeometryBasics]]
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[[Gettext_jll]]
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[[Glib_jll]]
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[[Graphite2_jll]]
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[[Grisu]]
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[[HTTP]]
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[[HypertextLiteral]]
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[[IOCapture]]
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[[IfElse]]
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[[ImageIO]]
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[[ImageMagick]]
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[[ImageQualityIndexes]]
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[[Images]]
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[[Imath_jll]]
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[[IniFile]]
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[[IntelOpenMP_jll]]
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[[Interpolations]]
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[[IntervalSets]]
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[[IrrationalConstants]]
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[[JLLWrappers]]
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[[JSON]]
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[[JpegTurbo_jll]]
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[[LAME_jll]]
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[[LZO_jll]]
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[[LaTeXStrings]]
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[[Latexify]]
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uuid = "23fbe1c1-3f47-55db-b15f-69d7ec21a316"
version = "0.15.6"

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

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

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

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

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

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

[[Libffi_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "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"
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[[LineSearches]]
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version = "7.1.1"

[[LinearAlgebra]]
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[[LogExpFunctions]]
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[[Logging]]
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[[MKL_jll]]
deps = ["Artifacts", "IntelOpenMP_jll", "JLLWrappers", "LazyArtifacts", "Libdl", "Pkg"]
git-tree-sha1 = "5455aef09b40e5020e1520f551fa3135040d4ed0"
uuid = "856f044c-d86e-5d09-b602-aeab76dc8ba7"
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[[MacroTools]]
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[[MappedArrays]]
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[[Markdown]]
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[[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]]
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uuid = "442fdcdd-2543-5da2-b0f3-8c86c306513e"
version = "0.3.1"

[[Missings]]
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[[Mmap]]
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[[MosaicViews]]
deps = ["MappedArrays", "OffsetArrays", "PaddedViews", "StackViews"]
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uuid = "e94cdb99-869f-56ef-bcf0-1ae2bcbe0389"
version = "0.3.3"

[[MozillaCACerts_jll]]
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[[NLSolversBase]]
deps = ["DiffResults", "Distributed", "FiniteDiff", "ForwardDiff"]
git-tree-sha1 = "144bab5b1443545bc4e791536c9f1eacb4eed06a"
uuid = "d41bc354-129a-5804-8e4c-c37616107c6c"
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[[NaNMath]]
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[[Netpbm]]
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uuid = "f09324ee-3d7c-5217-9330-fc30815ba969"
version = "1.0.1"

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

[[Ogg_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "7937eda4681660b4d6aeeecc2f7e1c81c8ee4e2f"
uuid = "e7412a2a-1a6e-54c0-be00-318e2571c051"
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[[OpenEXR]]
deps = ["Colors", "FileIO", "OpenEXR_jll"]
git-tree-sha1 = "327f53360fdb54df7ecd01e96ef1983536d1e633"
uuid = "52e1d378-f018-4a11-a4be-720524705ac7"
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[[OpenEXR_jll]]
deps = ["Artifacts", "Imath_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "923319661e9a22712f24596ce81c54fc0366f304"
uuid = "18a262bb-aa17-5467-a713-aee519bc75cb"
version = "3.1.1+0"

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

[[OpenSpecFun_jll]]
deps = ["Artifacts", "CompilerSupportLibraries_jll", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "13652491f6856acfd2db29360e1bbcd4565d04f1"
uuid = "efe28fd5-8261-553b-a9e1-b2916fc3738e"
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[[Optim]]
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git-tree-sha1 = "7863df65dbb2a0fa8f85fcaf0a41167640d2ebed"
uuid = "429524aa-4258-5aef-a3af-852621145aeb"
version = "1.4.1"

[[Opus_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "51a08fb14ec28da2ec7a927c4337e4332c2a4720"
uuid = "91d4177d-7536-5919-b921-800302f37372"
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[[OrderedCollections]]
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uuid = "bac558e1-5e72-5ebc-8fee-abe8a469f55d"
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[[PCRE_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "b2a7af664e098055a7529ad1a900ded962bca488"
uuid = "2f80f16e-611a-54ab-bc61-aa92de5b98fc"
version = "8.44.0+0"

[[PNGFiles]]
deps = ["Base64", "CEnum", "ImageCore", "IndirectArrays", "OffsetArrays", "libpng_jll"]
git-tree-sha1 = "e14c485f6beee0c7a8dcf6128bf70b85f1fe201e"
uuid = "f57f5aa1-a3ce-4bc8-8ab9-96f992907883"
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[[PaddedViews]]
deps = ["OffsetArrays"]
git-tree-sha1 = "646eed6f6a5d8df6708f15ea7e02a7a2c4fe4800"
uuid = "5432bcbf-9aad-5242-b902-cca2824c8663"
version = "0.5.10"

[[Parameters]]
deps = ["OrderedCollections", "UnPack"]
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[[Parsers]]
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uuid = "69de0a69-1ddd-5017-9359-2bf0b02dc9f0"
version = "2.0.3"

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

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

[[PkgVersion]]
deps = ["Pkg"]
git-tree-sha1 = "a7a7e1a88853564e551e4eba8650f8c38df79b37"
uuid = "eebad327-c553-4316-9ea0-9fa01ccd7688"
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[[PlotThemes]]
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[[PlotUtils]]
deps = ["ColorSchemes", "Colors", "Dates", "Printf", "Random", "Reexport", "Statistics"]
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uuid = "995b91a9-d308-5afd-9ec6-746e21dbc043"
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[[Plots]]
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git-tree-sha1 = "2dbafeadadcf7dadff20cd60046bba416b4912be"
uuid = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
version = "1.21.3"

[[PlutoUI]]
deps = ["Base64", "Dates", "HypertextLiteral", "IOCapture", "InteractiveUtils", "JSON", "Logging", "Markdown", "Random", "Reexport", "UUIDs"]
git-tree-sha1 = "f35ae11e070dbf123d5a6f54cbda45818d765ad2"
uuid = "7f904dfe-b85e-4ff6-b463-dae2292396a8"
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[[PositiveFactorizations]]
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uuid = "85a6dd25-e78a-55b7-8502-1745935b8125"
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[[Preferences]]
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[[Printf]]
deps = ["Unicode"]
uuid = "de0858da-6303-5e67-8744-51eddeeeb8d7"

[[ProgressMeter]]
deps = ["Distributed", "Printf"]
git-tree-sha1 = "afadeba63d90ff223a6a48d2009434ecee2ec9e8"
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[[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"

[[RangeArrays]]
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[[Ratios]]
deps = ["Requires"]
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[[RecipesBase]]
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[[RecipesPipeline]]
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[[Reexport]]
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[[Requires]]
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[[Rotations]]
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[[SHA]]
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[[Scratch]]
deps = ["Dates"]
git-tree-sha1 = "0b4b7f1393cff97c33891da2a0bf69c6ed241fda"
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[[Serialization]]
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[[SharedArrays]]
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[[Showoff]]
deps = ["Dates", "Grisu"]
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[[SimpleTraits]]
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[[Sockets]]
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[[SortingAlgorithms]]
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[[SparseArrays]]
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[[SpecialFunctions]]
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[[StackViews]]
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[[Static]]
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[[StaticArrays]]
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[[Statistics]]
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[[StatsAPI]]
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[[StatsBase]]
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[[StructArrays]]
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[[TOML]]
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[[TableTraits]]
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[[Tables]]
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git-tree-sha1 = "1162ce4a6c4b7e31e0e6b14486a6986951c73be9"
uuid = "bd369af6-aec1-5ad0-b16a-f7cc5008161c"
version = "1.5.2"

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[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:
# ╠═dcb245de-bc64-4892-a542-faaf9b6d7303
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# ╟─e78f1041-4259-491f-86d4-1f7d6ced4f15
# ╟─f954ba83-d8a5-4ef0-b84e-e513e95dbb0f
# ╟─777bae38-20e6-4956-a0c9-6245aae429b4
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# ╠═075683a3-3c19-4edd-9a21-a93f1d31e9bf
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# ╠═45c480f6-2b41-4075-b764-d1f6a71fc0f9
# ╟─1e3fa40e-19f2-4c97-8353-1705a266ee5e
# ╟─00000000-0000-0000-0000-000000000001
# ╟─00000000-0000-0000-0000-000000000002
