### A Pluto.jl notebook ### # v0.16.0 using Markdown using InteractiveUtils # ╔═╡ 9a7fa844-0e94-11eb-0850-31e34452fd9b md"""# Notebook 8 -- Math 2121, Fall 2021 There is a bounded region of space in $\mathbb{R}^n$ that we naturally associate to an $n \times n$ matrix: namely, consider the *$n$-dimensional parallelogram* (called a *parallelepiped* when $n=3$) whose edges are the columns of your matrix. """ # ╔═╡ 6ce856dc-0e95-11eb-072e-6b1a7cf12830 # ╔═╡ 51eb2cec-0e95-11eb-0920-79c8815b98da md"""Our goal today is to uncover a formula for the volume for this region. But first we need to better understand what the $n$-dimensional parallelogram associated to an $n \times n$ matrix looks like and what "volume" means for a region in $\mathbb{R}^n$.""" # ╔═╡ 867f0302-0e95-11eb-2633-8be05070104d # ╔═╡ 87892b74-0e95-11eb-187f-256d157b7d21 md"""A $1\times 1$ matrix is just a scalar. However our definitions work, they should associate a volume of $|a|$ to $A = [a]$ when $n=1$.""" # ╔═╡ c78e4e3e-0e95-11eb-27d8-8fdca3bc353a # ╔═╡ c8e33a4c-0e95-11eb-2e77-e96a550786e3 md"""The $n=2$ case is more informative. In $2$ dimensions, we usually refer to volume as *area*. Consider the following $2\times 2$ matrix: """ # ╔═╡ 3e8ba024-0e96-11eb-36a5-61db9665d7d0 A = rand(-5:0.01:5, 2, 2) # ╔═╡ 2dc9055e-0e96-11eb-04a8-cf3d9d9683a7 begin using LinearAlgebra using Plots plotly() function matrix_plot(A) p = scatter([0], [0], color = [:red], legend = false, label = "origin", aspect_ratio=:equal, title="Parallelogram of a 2-by-2 matrix") scatter!([A[1, 1]], [A[2, 1]], label = "first column", color = [:red]) scatter!([A[1, 2]], [A[2, 2]], label = "second column", color = [:red]) scatter!( [A[1, 1] + A[1, 2]], [A[2, 1] + A[2, 2]], label = "sum of columns", color = [:red]) quiver!(quiver = ([A[1, 1]], [A[2, 1]]), [0], [0], color = [:blue],) quiver!(quiver = ([A[1, 2]], [A[2, 2]]), [0], [0], color = [:blue],) plot!(Shape( [0, A[1, 1], A[1, 1] + A[1, 2], A[1, 2]], [0, A[2, 1], A[2, 1] + A[2, 2], A[2, 2]],), opacity=.25, color = [:blue]) return p end matrix_plot(A) end # ╔═╡ d7d4030a-0ea0-11eb-050f-e9e4e781a13c md"Here is a picture of the parallelogram spanned by the columns of $M$:" # ╔═╡ 87aa9f88-0ea0-11eb-04da-99c18a359c09 md"How can we characterize this parallelogram? Well, every point inside the parallelogram has the form $Av$ for some vector $v = \left[\begin{array}{c} v_1 \\ v_2\end{array}\right] \in \mathbb{R}^2$ with $0\leq v_1 \leq 1$ and $0\leq v_2 \leq 1$. This suggests a general definition: The *$n$-dimensional parallelogram* of an $n\times n$ matrix $A$ is the set of all points $\left\{ Av : v= \left[\begin{array}{c} v_1 \\ v_2 \\ \vdots \\ v_n \end{array}\right]\in\mathbb{R}^n\text{ with } 0\leq v_i \leq 1 \text{ for }i=1,2,\dots,n\right\}.$ Then we define the *volume* of an $n\times n$ matrix to be the volume of this region. " # ╔═╡ 368edece-0ea1-11eb-3d62-4934f3ed5484 # ╔═╡ 32b5167e-0ea1-11eb-3db0-cdc1d26df891 md"Next issue: how is volume defined in $\mathbb{R}^n$? The definition should obey our intuition that the volume of a rectangular region is $\text{length} \times \text{width} \times \text{height} \times \dots$ and volume should be *invariant under translation*. That is, if $L_1,L_2,\dots,L_n$ are positive numbers then the region $\mathrm{rect}(L_1,L_2,\dots,L_n) = \{ v \in \mathbb{R}^n : 0\leq v_i \leq L_i\text{ for }i=1,2,\dots,n\}$ should have volume $L_1\cdot L_2\cdots L_n$. Moreover, any translate of this region $u + \mathrm{rect}(L_1,L_2,\dots,L_n) = \{ u + v \in \mathbb{R}^n : 0\leq v_i \leq L_i\text{ for }i=1,2,\dots,n\}$ where $u \in \mathbb{R}^n$ is some given element should have the same volume. Finally, if we have two regions in $\mathbb{R}^n$ that don't overlap, then the volume of their union should be the sum of their individual volumes. " # ╔═╡ 8d0c461e-0ea2-11eb-3ecc-3765095f7bef # ╔═╡ 8dfe61c4-0ea2-11eb-0a03-3b0fbeaed045 md"Here is one definition of the volume of a region $S$ in $\mathbb{R}^n$ that satisfies these properties: choose small positive numbers $L_1,L_2,\dots,L_n > 0$ and fill up as much of $S$ as possible with nonoverlapping translates of the $n$-dimensional rectangle $\mathrm{rect}(L_1,L_2,\dots,L_n)$. If you can fit $N$ such regions completely inside $S$ without any overlap, then you say your *estimated* volume is $N \cdot L_1 \cdot L_2\cdots L_n.$ The true *volume* of $S$ is then the *limit superior* of all possible estimated volumes. To define *limit superior*, we need calculus; you can think of this as just the *maximum*." # ╔═╡ 925334a2-0ea2-11eb-3112-2f59c8ce51d4 # ╔═╡ eef14ab4-0ea2-11eb-0133-b3a60328c86f md"This conforms with our usual definiton of area in two dimensions. But in $\mathbb{R}^2$ we can also calculate area more directly:" # ╔═╡ ac657e72-0e98-11eb-1a8c-b70f72979c6b begin function enclosing_rectangle(A) u = A[1:2, 1] v = A[1:2, 2] w = u + v x1 = minimum([0 u[1] v[1] w[1]]) x2 = maximum([0 u[1] v[1] w[1]]) y1 = minimum([0 u[2] v[2] w[2]]) y2 = maximum([0 u[2] v[2] w[2]]) return x1, x2, y1, y2 end x1, x2, y1, y2 = enclosing_rectangle(A) mplot = matrix_plot(A) plot!(Shape([x1, x2, x2, x1], [y1, y1, y2, y2]), color = [:blue], opacity =.1) end # ╔═╡ 18b799c1-435e-427b-9be8-2e712d3c662c # x-coordinates of points in picture xcoords = 0, A[1,1], A[1,2], A[1,1] + A[1,2] # ╔═╡ 9fe4f768-9d41-416c-9c1b-d8737c1a9a69 # y-coordinates of points in picture ycoords = 0, A[2,1], A[2,2], A[2,1] + A[2,2] # ╔═╡ 0139bd8b-57da-43df-9c51-9e81bbed4111 # outer rectangle area (maximum(xcoords) - minimum(xcoords)) * (maximum(ycoords) - minimum(ycoords)) # ╔═╡ 744c9460-db89-4bef-9edc-9681c1d98d21 # outer triangle areas 0.88 * 2.05 + 3.06 * 2.38 # ╔═╡ b89360fe-614a-4e02-bbac-98f788b6b4d8 # remaining area parallelogram_area = 17.4542 - 9.0868 # ╔═╡ e078d7c6-0ea3-11eb-307b-d1c904fb5539 # ╔═╡ e1faae26-0ea3-11eb-318d-77713a78456a md"Here is an alternate, probabilistic definition of volume that is more amenable to estimation. Consider a region $S$ in $\mathbb{R}^n$. Define the volume of $R = u + \mathrm{rect}(L_1,L_2,\dots,L_n) = \{ u + v \in \mathbb{R}^n : 0\leq v_i \leq L_i\text{ for }i=1,2,\dots,n\}$ to be $L_1\cdot L_2\cdots L_n$ as usual. Choose $u$ and $L_1,L_2,\dots,L_n$ such that $S \subseteq R.$ Now sample many points uniformly at random from the region $R$. The *volume* of $S$ should then be (close to) the volume of $R$ times the fraction of these points that are contained in $S$. So if we generate $N$ points within $R$ and $m$ of them are inside $S$, then we expect $\mathrm{volume}(S) \approx \frac{m}{N} \cdot L_1\cdot L_2\cdots L_n.$ With enough random points $N$ this approximate volume will converge to the true volume. " # ╔═╡ fe034b28-0ea3-11eb-1b1a-1f474ab12faf # ╔═╡ f921e1cc-0ea4-11eb-21cd-2305c9faf2d9 md"When our matrix $A$ is invertible and $S$ is the $n$-dimensional parallelogram associated to $A$, then there is an easy way for us to test whether a random point $w \in R$ belongs to $S$: We just check if $A^{-1} w$ has all coordinates between $0.0$ and $1.0$. Thus, we can try to implement the probabilistic definition to estimate the matrix volume." # ╔═╡ 5e0bb100-0e49-11eb-272e-db4dce3e1696 function volume_estimates(A, lower_bound, upper_bound, trials) # we are trying to estimate the volume of the (n-dimension) parallelogram # whose edges include the vectors that are the columns of the matrix A # # we will sample points from a rectangular region R # # this outer region R is u + rect(L1, L2, ..., Ln) where # # L1 = L2 = ... = Ln = upper_bound - lower_bound # u = [lower_bound; lower_bound; ...; lower_bound] # (n, p) = size(A) @assert n == p estimates = [] samples = [] # let's only record the cumulative estimate every 1000 trials interval = maximum([Int(floor(trials/1000)) 1]) # size of region R sample_region_size = (upper_bound - lower_bound)^n # rather than generate a single vector v many times, # we construct one random matrix with many columns v = rand(lower_bound:0.00001:upper_bound, n, trials) # now we can just do one matrix multiplication with all of our samples w = A^-1 * v count = 0 for j=1:trials # column j of v belongs to the matrix volume if and only if # column j of w = A^-1 * v has all entries between 0.0 and 1.0 if 0.0 <= minimum(w[:,j]) && maximum(w[:,j]) <= 1.0 count += 1 end if j % interval == 0 fraction = count / j append!(estimates, fraction * sample_region_size) append!(samples, j) end end return estimates, samples end # ╔═╡ 9398ea44-0e4f-11eb-2412-23acd564dc41 function estimate_bounds(A, trials=1000) # we get more accurate volume estimates by using a smaller region R # # this method estimates optimal choices for lower_bound, upper_bound # so that R is as small as possible while containing the paralllelogram of A w = A * rand(0:1, size(A)[1], trials) return (minimum(w), maximum(w)) end # ╔═╡ 7bdd6824-0e4a-11eb-063c-530768c6a7fe function plot_estimates(matrix, lower, upper, trials) estimates, samples = volume_estimates(matrix, lower, upper, trials) plot( samples, estimates, xlabel="samples N", ylabel="volume estimate after N samples", ylims = (0.0, 2 * estimates[end]), label="first estimate", title="Volume estimates of $(size(matrix)[1])-dimensional parallelogram", legend=:bottomright ) estimates, samples = volume_estimates(matrix, lower, upper, trials) plot!(samples, estimates, label="second estimate") estimates, samples = volume_estimates(matrix, lower, upper, trials) plot!(samples, estimates, label="third estimate") end # ╔═╡ 7fb874de-0ea6-11eb-0449-39b8d954819a trials = 10000 # ╔═╡ 843b9826-0ea6-11eb-3042-af3bcedae2ec matrix = A # ╔═╡ c5e50efa-0eb2-11eb-28dd-fdf089198c2d lower, upper = estimate_bounds(matrix) # ╔═╡ e21e9614-0ea6-11eb-2aba-97dd7bda343b md"The following graph shows what happens when we sample a large number$N$of points from a rectangular region$R$containing the$n$-dimensional parallelogram$S$of our matrix$A$, and then multiply the faction of the points that are in$S$by the volume of$R$. The resulting plots should estimate the volume of the parallelogram spanned by the columns$A$." # ╔═╡ 1cc8a0d8-27fb-4be8-8c1e-f1257af6e217 plot_estimates(matrix, lower, upper, trials) # ╔═╡ 7c4e9c3b-5776-4f2d-90ea-bd8b75fbb098 # here is what we computed for the exact area: parallelogram_area # ╔═╡ 18a01504-0ea6-11eb-1b64-c33c95d4bcae # another way to compute this number (up to +/- sign): det(A) # ╔═╡ 0908992e-0ea6-11eb-211a-51a9655c77b8 md"The punchline here: the volume of the parallelogram spanned by the columns of our matrix is *the absolute value of the determinant* of the matrix! Thus the physical interpretation of$\det(A)$is as the *signed volume of the parallelogram spanned by the columns of$A$*. This works in any dimension, not just$n=2$. However, in higher dimensions we need more samples to get a good estimate. " # ╔═╡ dd18e9c6-0eb3-11eb-31c4-296dbc11be15 M = rand(-5:0.001:5, 3, 3) # ╔═╡ ece780c4-0eb3-11eb-1b24-0f804525f8ef hlower, hupper = estimate_bounds(M) # ╔═╡ c91999a2-0eb3-11eb-285a-85e605e2a1f6 htrials = 100000 # ╔═╡ c4dabe7a-0eb3-11eb-306b-b3001b0c4f73 plot_estimates(M, hlower, hupper, htrials) # ╔═╡ 5645ac26-0eb4-11eb-3cfe-d13415915d87 # actual volume (up to +/- sign) det(M) # ╔═╡ bbcc1392-0eb3-11eb-1868-b39c9b4130d4 md"Next time: what is the physical interpretation of the sign of$\det(A)\$?" # ╔═╡ 00000000-0000-0000-0000-000000000001 PLUTO_PROJECT_TOML_CONTENTS = """ [deps] LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e" Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" [compat] Plots = "~1.22.6" """ # ╔═╡ 00000000-0000-0000-0000-000000000002 PLUTO_MANIFEST_TOML_CONTENTS = """ # This file is machine-generated - editing it directly is not advised [[Adapt]] deps = ["LinearAlgebra"] git-tree-sha1 = "84918055d15b3114ede17ac6a7182f68870c16f7" uuid = 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"344bf40dcab1073aca04aa0df4fb092f920e4011" uuid = "3b182d85-2403-5c21-9c21-1e1f0cc25472" version = "1.3.14+0" [[Grisu]] git-tree-sha1 = "53bb909d1151e57e2484c3d1b53e19552b887fb2" uuid = "42e2da0e-8278-4e71-bc24-59509adca0fe" version = "1.0.2" [[HTTP]] deps = ["Base64", "Dates", "IniFile", "Logging", "MbedTLS", "NetworkOptions", "Sockets", "URIs"] git-tree-sha1 = "14eece7a3308b4d8be910e265c724a6ba51a9798" uuid = "cd3eb016-35fb-5094-929b-558a96fad6f3" version = "0.9.16" [[HarfBuzz_jll]] deps = ["Artifacts", "Cairo_jll", "Fontconfig_jll", "FreeType2_jll", "Glib_jll", "Graphite2_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Pkg"] git-tree-sha1 = "8a954fed8ac097d5be04921d595f741115c1b2ad" uuid = "2e76f6c2-a576-52d4-95c1-20adfe4de566" version = "2.8.1+0" [[IniFile]] deps = ["Test"] git-tree-sha1 = "098e4d2c533924c921f9f9847274f2ad89e018b8" uuid = "83e8ac13-25f8-5344-8a64-a9f2b223428f" version = "0.5.0" [[InteractiveUtils]] deps = ["Markdown"] uuid = "b77e0a4c-d291-57a0-90e8-8db25a27a240" [[IrrationalConstants]] git-tree-sha1 = "7fd44fd4ff43fc60815f8e764c0f352b83c49151" uuid = "92d709cd-6900-40b7-9082-c6be49f344b6" version = "0.1.1" [[IterTools]] git-tree-sha1 = "05110a2ab1fc5f932622ffea2a003221f4782c18" uuid = "c8e1da08-722c-5040-9ed9-7db0dc04731e" version = "1.3.0" [[IteratorInterfaceExtensions]] git-tree-sha1 = "a3f24677c21f5bbe9d2a714f95dcd58337fb2856" uuid = "82899510-4779-5014-852e-03e436cf321d" version = "1.0.0" [[JLLWrappers]] deps = ["Preferences"] git-tree-sha1 = "642a199af8b68253517b80bd3bfd17eb4e84df6e" uuid = "692b3bcd-3c85-4b1f-b108-f13ce0eb3210" version = "1.3.0" [[JSON]] deps = ["Dates", "Mmap", "Parsers", "Unicode"] git-tree-sha1 = "8076680b162ada2a031f707ac7b4953e30667a37" uuid = "682c06a0-de6a-54ab-a142-c8b1cf79cde6" version = "0.21.2" [[JpegTurbo_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] git-tree-sha1 = "d735490ac75c5cb9f1b00d8b5509c11984dc6943" uuid = "aacddb02-875f-59d6-b918-886e6ef4fbf8" version = "2.1.0+0" [[LAME_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] git-tree-sha1 = "f6250b16881adf048549549fba48b1161acdac8c" uuid = "c1c5ebd0-6772-5130-a774-d5fcae4a789d" version = "3.100.1+0" [[LZO_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] git-tree-sha1 = "e5b909bcf985c5e2605737d2ce278ed791b89be6" uuid = "dd4b983a-f0e5-5f8d-a1b7-129d4a5fb1ac" version = "2.10.1+0" [[LaTeXStrings]] git-tree-sha1 = "c7f1c695e06c01b95a67f0cd1d34994f3e7db104" uuid = "b964fa9f-0449-5b57-a5c2-d3ea65f4040f" version = "1.2.1" [[Latexify]] deps = ["Formatting", "InteractiveUtils", "LaTeXStrings", "MacroTools", "Markdown", "Printf", "Requires"] git-tree-sha1 = "a4b12a1bd2ebade87891ab7e36fdbce582301a92" uuid = "23fbe1c1-3f47-55db-b15f-69d7ec21a316" version = "0.15.6" [[LibCURL]] deps = ["LibCURL_jll", "MozillaCACerts_jll"] uuid = "b27032c2-a3e7-50c8-80cd-2d36dbcbfd21" [[LibCURL_jll]] deps = ["Artifacts", "LibSSH2_jll", "Libdl", "MbedTLS_jll", "Zlib_jll", "nghttp2_jll"] uuid = "deac9b47-8bc7-5906-a0fe-35ac56dc84c0" [[LibGit2]] deps = ["Base64", "NetworkOptions", "Printf", "SHA"] uuid = "76f85450-5226-5b5a-8eaa-529ad045b433" [[LibSSH2_jll]] deps = ["Artifacts", "Libdl", "MbedTLS_jll"] uuid = "29816b5a-b9ab-546f-933c-edad1886dfa8" [[Libdl]] uuid = "8f399da3-3557-5675-b5ff-fb832c97cbdb" [[Libffi_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] git-tree-sha1 = "761a393aeccd6aa92ec3515e428c26bf99575b3b" uuid = "e9f186c6-92d2-5b65-8a66-fee21dc1b490" version = "3.2.2+0" [[Libgcrypt_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Libgpg_error_jll", "Pkg"] git-tree-sha1 = "64613c82a59c120435c067c2b809fc61cf5166ae" uuid = "d4300ac3-e22c-5743-9152-c294e39db1e4" version = "1.8.7+0" [[Libglvnd_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll", "Xorg_libXext_jll"] git-tree-sha1 = "7739f837d6447403596a75d19ed01fd08d6f56bf" uuid = "7e76a0d4-f3c7-5321-8279-8d96eeed0f29" version = "1.3.0+3" [[Libgpg_error_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] git-tree-sha1 = "c333716e46366857753e273ce6a69ee0945a6db9" uuid = "7add5ba3-2f88-524e-9cd5-f83b8a55f7b8" version = "1.42.0+0" [[Libiconv_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] git-tree-sha1 = "42b62845d70a619f063a7da093d995ec8e15e778" uuid = "94ce4f54-9a6c-5748-9c1c-f9c7231a4531" version = "1.16.1+1" [[Libmount_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] git-tree-sha1 = "9c30530bf0effd46e15e0fdcf2b8636e78cbbd73" uuid = "4b2f31a3-9ecc-558c-b454-b3730dcb73e9" version = "2.35.0+0" [[Libtiff_jll]] deps = ["Artifacts", "JLLWrappers", "JpegTurbo_jll", "Libdl", "Pkg", "Zlib_jll", "Zstd_jll"] git-tree-sha1 = "340e257aada13f95f98ee352d316c3bed37c8ab9" uuid = "89763e89-9b03-5906-acba-b20f662cd828" version = "4.3.0+0" [[Libuuid_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] git-tree-sha1 = "7f3efec06033682db852f8b3bc3c1d2b0a0ab066" uuid = "38a345b3-de98-5d2b-a5d3-14cd9215e700" version = "2.36.0+0" [[LinearAlgebra]] deps = ["Libdl"] uuid = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e" [[LogExpFunctions]] deps = ["ChainRulesCore", "DocStringExtensions", "IrrationalConstants", "LinearAlgebra"] git-tree-sha1 = "34dc30f868e368f8a17b728a1238f3fcda43931a" uuid = "2ab3a3ac-af41-5b50-aa03-7779005ae688" version = "0.3.3" [[Logging]] uuid = "56ddb016-857b-54e1-b83d-db4d58db5568" [[MacroTools]] deps = ["Markdown", "Random"] git-tree-sha1 = "5a5bc6bf062f0f95e62d0fe0a2d99699fed82dd9" uuid = "1914dd2f-81c6-5fcd-8719-6d5c9610ff09" version = "0.5.8" [[Markdown]] deps = ["Base64"] uuid = "d6f4376e-aef5-505a-96c1-9c027394607a" [[MbedTLS]] deps = ["Dates", "MbedTLS_jll", "Random", "Sockets"] git-tree-sha1 = "1c38e51c3d08ef2278062ebceade0e46cefc96fe" uuid = "739be429-bea8-5141-9913-cc70e7f3736d" version = "1.0.3" [[MbedTLS_jll]] deps = ["Artifacts", "Libdl"] uuid = "c8ffd9c3-330d-5841-b78e-0817d7145fa1" [[Measures]] git-tree-sha1 = "e498ddeee6f9fdb4551ce855a46f54dbd900245f" uuid = "442fdcdd-2543-5da2-b0f3-8c86c306513e" version = "0.3.1" [[Missings]] deps = ["DataAPI"] git-tree-sha1 = "bf210ce90b6c9eed32d25dbcae1ebc565df2687f" uuid = "e1d29d7a-bbdc-5cf2-9ac0-f12de2c33e28" version = "1.0.2" [[Mmap]] uuid = "a63ad114-7e13-5084-954f-fe012c677804" [[MozillaCACerts_jll]] uuid = "14a3606d-f60d-562e-9121-12d972cd8159" [[NaNMath]] git-tree-sha1 = "bfe47e760d60b82b66b61d2d44128b62e3a369fb" uuid = "77ba4419-2d1f-58cd-9bb1-8ffee604a2e3" version = "0.3.5" [[NetworkOptions]] uuid = "ca575930-c2e3-43a9-ace4-1e988b2c1908" [[Ogg_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] git-tree-sha1 = "7937eda4681660b4d6aeeecc2f7e1c81c8ee4e2f" uuid = "e7412a2a-1a6e-54c0-be00-318e2571c051" version = "1.3.5+0" [[OpenSSL_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] git-tree-sha1 = "15003dcb7d8db3c6c857fda14891a539a8f2705a" uuid = "458c3c95-2e84-50aa-8efc-19380b2a3a95" version = "1.1.10+0" [[Opus_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] git-tree-sha1 = "51a08fb14ec28da2ec7a927c4337e4332c2a4720" uuid = "91d4177d-7536-5919-b921-800302f37372" version = "1.3.2+0" [[OrderedCollections]] git-tree-sha1 = "85f8e6578bf1f9ee0d11e7bb1b1456435479d47c" uuid = "bac558e1-5e72-5ebc-8fee-abe8a469f55d" version = "1.4.1" [[PCRE_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] git-tree-sha1 = "b2a7af664e098055a7529ad1a900ded962bca488" uuid = "2f80f16e-611a-54ab-bc61-aa92de5b98fc" version = "8.44.0+0" [[Parsers]] deps = ["Dates"] git-tree-sha1 = "98f59ff3639b3d9485a03a72f3ab35bab9465720" uuid = "69de0a69-1ddd-5017-9359-2bf0b02dc9f0" version = "2.0.6" [[Pixman_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] git-tree-sha1 = "b4f5d02549a10e20780a24fce72bea96b6329e29" uuid = "30392449-352a-5448-841d-b1acce4e97dc" version = "0.40.1+0" [[Pkg]] deps = ["Artifacts", "Dates", "Downloads", "LibGit2", "Libdl", "Logging", "Markdown", "Printf", "REPL", "Random", "SHA", "Serialization", "TOML", "Tar", "UUIDs", "p7zip_jll"] uuid = "44cfe95a-1eb2-52ea-b672-e2afdf69b78f" [[PlotThemes]] deps = ["PlotUtils", "Requires", "Statistics"] git-tree-sha1 = "a3a964ce9dc7898193536002a6dd892b1b5a6f1d" uuid = "ccf2f8ad-2431-5c83-bf29-c5338b663b6a" version = "2.0.1" [[PlotUtils]] deps = ["ColorSchemes", "Colors", "Dates", "Printf", "Random", "Reexport", "Statistics"] git-tree-sha1 = "b084324b4af5a438cd63619fd006614b3b20b87b" uuid = "995b91a9-d308-5afd-9ec6-746e21dbc043" version = "1.0.15" [[Plots]] deps = ["Base64", "Contour", "Dates", "Downloads", "FFMPEG", "FixedPointNumbers", "GR", "GeometryBasics", "JSON", "Latexify", "LinearAlgebra", "Measures", "NaNMath", "PlotThemes", "PlotUtils", "Printf", "REPL", "Random", "RecipesBase", "RecipesPipeline", "Reexport", "Requires", "Scratch", "Showoff", "SparseArrays", "Statistics", "StatsBase", "UUIDs"] git-tree-sha1 = "ba43b248a1f04a9667ca4a9f782321d9211aa68e" uuid = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" version = "1.22.6" [[Preferences]] deps = ["TOML"] git-tree-sha1 = "00cfd92944ca9c760982747e9a1d0d5d86ab1e5a" uuid = "21216c6a-2e73-6563-6e65-726566657250" version = "1.2.2" [[Printf]] deps = ["Unicode"] uuid = "de0858da-6303-5e67-8744-51eddeeeb8d7" [[Qt5Base_jll]] deps = ["Artifacts", "CompilerSupportLibraries_jll", "Fontconfig_jll", "Glib_jll", "JLLWrappers", "Libdl", "Libglvnd_jll", "OpenSSL_jll", "Pkg", "Xorg_libXext_jll", "Xorg_libxcb_jll", "Xorg_xcb_util_image_jll", "Xorg_xcb_util_keysyms_jll", "Xorg_xcb_util_renderutil_jll", "Xorg_xcb_util_wm_jll", "Zlib_jll", "xkbcommon_jll"] git-tree-sha1 = "ad368663a5e20dbb8d6dc2fddeefe4dae0781ae8" uuid = "ea2cea3b-5b76-57ae-a6ef-0a8af62496e1" version = "5.15.3+0" [[REPL]] deps = ["InteractiveUtils", "Markdown", "Sockets", "Unicode"] uuid = "3fa0cd96-eef1-5676-8a61-b3b8758bbffb" [[Random]] deps = ["Serialization"] uuid = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c" [[RecipesBase]] git-tree-sha1 = "44a75aa7a527910ee3d1751d1f0e4148698add9e" uuid = "3cdcf5f2-1ef4-517c-9805-6587b60abb01" version = "1.1.2" [[RecipesPipeline]] deps = ["Dates", "NaNMath", "PlotUtils", "RecipesBase"] git-tree-sha1 = "7ad0dfa8d03b7bcf8c597f59f5292801730c55b8" uuid = "01d81517-befc-4cb6-b9ec-a95719d0359c" version = "0.4.1" [[Reexport]] git-tree-sha1 = "45e428421666073eab6f2da5c9d310d99bb12f9b" uuid = "189a3867-3050-52da-a836-e630ba90ab69" version = "1.2.2" [[Requires]] deps = ["UUIDs"] git-tree-sha1 = "4036a3bd08ac7e968e27c203d45f5fff15020621" uuid = "ae029012-a4dd-5104-9daa-d747884805df" version = "1.1.3" [[SHA]] uuid = "ea8e919c-243c-51af-8825-aaa63cd721ce" [[Scratch]] deps = ["Dates"] git-tree-sha1 = "0b4b7f1393cff97c33891da2a0bf69c6ed241fda" uuid = "6c6a2e73-6563-6170-7368-637461726353" version = "1.1.0" [[Serialization]] uuid = "9e88b42a-f829-5b0c-bbe9-9e923198166b" [[SharedArrays]] deps = ["Distributed", "Mmap", "Random", "Serialization"] uuid = "1a1011a3-84de-559e-8e89-a11a2f7dc383" [[Showoff]] deps = ["Dates", "Grisu"] git-tree-sha1 = "91eddf657aca81df9ae6ceb20b959ae5653ad1de" uuid = "992d4aef-0814-514b-bc4d-f2e9a6c4116f" version = "1.0.3" [[Sockets]] uuid = "6462fe0b-24de-5631-8697-dd941f90decc" [[SortingAlgorithms]] deps = ["DataStructures"] git-tree-sha1 = "b3363d7460f7d098ca0912c69b082f75625d7508" uuid = "a2af1166-a08f-5f64-846c-94a0d3cef48c" version = "1.0.1" [[SparseArrays]] deps = ["LinearAlgebra", "Random"] uuid = "2f01184e-e22b-5df5-ae63-d93ebab69eaf" [[StaticArrays]] deps = ["LinearAlgebra", "Random", "Statistics"] git-tree-sha1 = "3c76dde64d03699e074ac02eb2e8ba8254d428da" uuid = "90137ffa-7385-5640-81b9-e52037218182" version = "1.2.13" [[Statistics]] deps = ["LinearAlgebra", "SparseArrays"] uuid = "10745b16-79ce-11e8-11f9-7d13ad32a3b2" [[StatsAPI]] git-tree-sha1 = "1958272568dc176a1d881acb797beb909c785510" uuid = "82ae8749-77ed-4fe6-ae5f-f523153014b0" version = "1.0.0" [[StatsBase]] deps = ["DataAPI", "DataStructures", "LinearAlgebra", "LogExpFunctions", "Missings", "Printf", "Random", "SortingAlgorithms", "SparseArrays", "Statistics", "StatsAPI"] git-tree-sha1 = "65fb73045d0e9aaa39ea9a29a5e7506d9ef6511f" uuid = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91" version = "0.33.11" [[StructArrays]] deps = ["Adapt", "DataAPI", "StaticArrays", "Tables"] git-tree-sha1 = "2ce41e0d042c60ecd131e9fb7154a3bfadbf50d3" uuid = "09ab397b-f2b6-538f-b94a-2f83cf4a842a" version = "0.6.3" [[TOML]] deps = ["Dates"] uuid = "fa267f1f-6049-4f14-aa54-33bafae1ed76" [[TableTraits]] deps = ["IteratorInterfaceExtensions"] git-tree-sha1 = "c06b2f539df1c6efa794486abfb6ed2022561a39" uuid = "3783bdb8-4a98-5b6b-af9a-565f29a5fe9c" version = "1.0.1" [[Tables]] deps = ["DataAPI", "DataValueInterfaces", "IteratorInterfaceExtensions", "LinearAlgebra", "TableTraits", "Test"] git-tree-sha1 = "fed34d0e71b91734bf0a7e10eb1bb05296ddbcd0" uuid = "bd369af6-aec1-5ad0-b16a-f7cc5008161c" version = "1.6.0" [[Tar]] deps = ["ArgTools", "SHA"] uuid = "a4e569a6-e804-4fa4-b0f3-eef7a1d5b13e" [[Test]] deps = ["InteractiveUtils", "Logging", "Random", "Serialization"] uuid = "8dfed614-e22c-5e08-85e1-65c5234f0b40" [[URIs]] git-tree-sha1 = "97bbe755a53fe859669cd907f2d96aee8d2c1355" uuid = "5c2747f8-b7ea-4ff2-ba2e-563bfd36b1d4" version = "1.3.0" [[UUIDs]] deps = ["Random", "SHA"] uuid = "cf7118a7-6976-5b1a-9a39-7adc72f591a4" [[Unicode]] uuid = "4ec0a83e-493e-50e2-b9ac-8f72acf5a8f5" [[Wayland_jll]] deps = ["Artifacts", "Expat_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Pkg", "XML2_jll"] git-tree-sha1 = "3e61f0b86f90dacb0bc0e73a0c5a83f6a8636e23" uuid = "a2964d1f-97da-50d4-b82a-358c7fce9d89" version = "1.19.0+0" [[Wayland_protocols_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll"] git-tree-sha1 = "2839f1c1296940218e35df0bbb220f2a79686670" uuid = "2381bf8a-dfd0-557d-9999-79630e7b1b91" version = "1.18.0+4" [[XML2_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Libiconv_jll", "Pkg", "Zlib_jll"] git-tree-sha1 = "1acf5bdf07aa0907e0a37d3718bb88d4b687b74a" uuid = "02c8fc9c-b97f-50b9-bbe4-9be30ff0a78a" version = "2.9.12+0" [[XSLT_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Libgcrypt_jll", "Libgpg_error_jll", "Libiconv_jll", "Pkg", "XML2_jll", "Zlib_jll"] git-tree-sha1 = "91844873c4085240b95e795f692c4cec4d805f8a" uuid = "aed1982a-8fda-507f-9586-7b0439959a61" version = "1.1.34+0" [[Xorg_libX11_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll", "Xorg_xtrans_jll"] git-tree-sha1 = "5be649d550f3f4b95308bf0183b82e2582876527" uuid = "4f6342f7-b3d2-589e-9d20-edeb45f2b2bc" version = "1.6.9+4" [[Xorg_libXau_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] git-tree-sha1 = "4e490d5c960c314f33885790ed410ff3a94ce67e" uuid = "0c0b7dd1-d40b-584c-a123-a41640f87eec" version = "1.0.9+4" [[Xorg_libXcursor_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXfixes_jll", "Xorg_libXrender_jll"] git-tree-sha1 = "12e0eb3bc634fa2080c1c37fccf56f7c22989afd" uuid = "935fb764-8cf2-53bf-bb30-45bb1f8bf724" version = "1.2.0+4" [[Xorg_libXdmcp_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] git-tree-sha1 = "4fe47bd2247248125c428978740e18a681372dd4" uuid = "a3789734-cfe1-5b06-b2d0-1dd0d9d62d05" version = "1.1.3+4" [[Xorg_libXext_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"] git-tree-sha1 = "b7c0aa8c376b31e4852b360222848637f481f8c3" uuid = "1082639a-0dae-5f34-9b06-72781eeb8cb3" version = "1.3.4+4" [[Xorg_libXfixes_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"] git-tree-sha1 = "0e0dc7431e7a0587559f9294aeec269471c991a4" uuid = "d091e8ba-531a-589c-9de9-94069b037ed8" version = "5.0.3+4" [[Xorg_libXi_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXfixes_jll"] git-tree-sha1 = "89b52bc2160aadc84d707093930ef0bffa641246" uuid = "a51aa0fd-4e3c-5386-b890-e753decda492" version = "1.7.10+4" [[Xorg_libXinerama_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll"] git-tree-sha1 = "26be8b1c342929259317d8b9f7b53bf2bb73b123" uuid = "d1454406-59df-5ea1-beac-c340f2130bc3" version = "1.1.4+4" [[Xorg_libXrandr_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXrender_jll"] git-tree-sha1 = "34cea83cb726fb58f325887bf0612c6b3fb17631" uuid = "ec84b674-ba8e-5d96-8ba1-2a689ba10484" version = "1.5.2+4" [[Xorg_libXrender_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"] git-tree-sha1 = "19560f30fd49f4d4efbe7002a1037f8c43d43b96" uuid = "ea2f1a96-1ddc-540d-b46f-429655e07cfa" version = "0.9.10+4" [[Xorg_libpthread_stubs_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] git-tree-sha1 = "6783737e45d3c59a4a4c4091f5f88cdcf0908cbb" uuid = "14d82f49-176c-5ed1-bb49-ad3f5cbd8c74" version = "0.1.0+3" [[Xorg_libxcb_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"] git-tree-sha1 = "daf17f441228e7a3833846cd048892861cff16d6" uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b" version = "1.13.0+3" [[Xorg_libxkbfile_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"] git-tree-sha1 = "926af861744212db0eb001d9e40b5d16292080b2" uuid = "cc61e674-0454-545c-8b26-ed2c68acab7a" version = "1.1.0+4" [[Xorg_xcb_util_image_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"] git-tree-sha1 = "0fab0a40349ba1cba2c1da699243396ff8e94b97" uuid = "12413925-8142-5f55-bb0e-6d7ca50bb09b" version = "0.4.0+1" [[Xorg_xcb_util_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll"] git-tree-sha1 = "e7fd7b2881fa2eaa72717420894d3938177862d1" uuid = "2def613f-5ad1-5310-b15b-b15d46f528f5" version = "0.4.0+1" [[Xorg_xcb_util_keysyms_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"] git-tree-sha1 = "d1151e2c45a544f32441a567d1690e701ec89b00" uuid = "975044d2-76e6-5fbe-bf08-97ce7c6574c7" version = "0.4.0+1" [[Xorg_xcb_util_renderutil_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"] git-tree-sha1 = "dfd7a8f38d4613b6a575253b3174dd991ca6183e" uuid = "0d47668e-0667-5a69-a72c-f761630bfb7e" version = "0.3.9+1" [[Xorg_xcb_util_wm_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xcb_util_jll"] git-tree-sha1 = "e78d10aab01a4a154142c5006ed44fd9e8e31b67" uuid = "c22f9ab0-d5fe-5066-847c-f4bb1cd4e361" version = "0.4.1+1" [[Xorg_xkbcomp_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxkbfile_jll"] git-tree-sha1 = "4bcbf660f6c2e714f87e960a171b119d06ee163b" uuid = "35661453-b289-5fab-8a00-3d9160c6a3a4" version = "1.4.2+4" [[Xorg_xkeyboard_config_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_xkbcomp_jll"] git-tree-sha1 = "5c8424f8a67c3f2209646d4425f3d415fee5931d" uuid = "33bec58e-1273-512f-9401-5d533626f822" version = "2.27.0+4" [[Xorg_xtrans_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] git-tree-sha1 = "79c31e7844f6ecf779705fbc12146eb190b7d845" uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10" version = "1.4.0+3" [[Zlib_jll]] deps = ["Libdl"] uuid = "83775a58-1f1d-513f-b197-d71354ab007a" [[Zstd_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] git-tree-sha1 = "cc4bf3fdde8b7e3e9fa0351bdeedba1cf3b7f6e6" uuid = "3161d3a3-bdf6-5164-811a-617609db77b4" version = "1.5.0+0" [[libass_jll]] deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"] git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47" uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0" version = "0.15.1+0" [[libfdk_aac_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55" uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280" version = "2.0.2+0" [[libpng_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"] git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c" uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f" version = "1.6.38+0" [[libvorbis_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"] git-tree-sha1 = "c45f4e40e7aafe9d086379e5578947ec8b95a8fb" uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a" version = "1.3.7+0" [[nghttp2_jll]] deps = ["Artifacts", "Libdl"] uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d" [[p7zip_jll]] deps = ["Artifacts", "Libdl"] uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0" [[x264_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2" uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a" version = "2021.5.5+0" [[x265_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9" uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76" version = "3.5.0+0" [[xkbcommon_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Wayland_jll", "Wayland_protocols_jll", "Xorg_libxcb_jll", "Xorg_xkeyboard_config_jll"] git-tree-sha1 = "ece2350174195bb31de1a63bea3a41ae1aa593b6" uuid = "d8fb68d0-12a3-5cfd-a85a-d49703b185fd" version = "0.9.1+5" """ # ╔═╡ Cell order: # ╟─9a7fa844-0e94-11eb-0850-31e34452fd9b # ╟─6ce856dc-0e95-11eb-072e-6b1a7cf12830 # ╟─51eb2cec-0e95-11eb-0920-79c8815b98da # ╟─867f0302-0e95-11eb-2633-8be05070104d # ╟─87892b74-0e95-11eb-187f-256d157b7d21 # ╟─c78e4e3e-0e95-11eb-27d8-8fdca3bc353a # ╟─c8e33a4c-0e95-11eb-2e77-e96a550786e3 # ╠═3e8ba024-0e96-11eb-36a5-61db9665d7d0 # ╟─d7d4030a-0ea0-11eb-050f-e9e4e781a13c # ╟─2dc9055e-0e96-11eb-04a8-cf3d9d9683a7 # ╟─87aa9f88-0ea0-11eb-04da-99c18a359c09 # ╟─368edece-0ea1-11eb-3d62-4934f3ed5484 # ╟─32b5167e-0ea1-11eb-3db0-cdc1d26df891 # ╟─8d0c461e-0ea2-11eb-3ecc-3765095f7bef # ╟─8dfe61c4-0ea2-11eb-0a03-3b0fbeaed045 # ╟─925334a2-0ea2-11eb-3112-2f59c8ce51d4 # ╟─eef14ab4-0ea2-11eb-0133-b3a60328c86f # ╟─ac657e72-0e98-11eb-1a8c-b70f72979c6b # ╠═18b799c1-435e-427b-9be8-2e712d3c662c # ╠═9fe4f768-9d41-416c-9c1b-d8737c1a9a69 # ╠═0139bd8b-57da-43df-9c51-9e81bbed4111 # ╠═744c9460-db89-4bef-9edc-9681c1d98d21 # ╠═b89360fe-614a-4e02-bbac-98f788b6b4d8 # ╟─e078d7c6-0ea3-11eb-307b-d1c904fb5539 # ╟─e1faae26-0ea3-11eb-318d-77713a78456a # ╟─fe034b28-0ea3-11eb-1b1a-1f474ab12faf # ╟─f921e1cc-0ea4-11eb-21cd-2305c9faf2d9 # ╠═5e0bb100-0e49-11eb-272e-db4dce3e1696 # ╠═9398ea44-0e4f-11eb-2412-23acd564dc41 # ╠═7bdd6824-0e4a-11eb-063c-530768c6a7fe # ╠═7fb874de-0ea6-11eb-0449-39b8d954819a # ╠═843b9826-0ea6-11eb-3042-af3bcedae2ec # ╠═c5e50efa-0eb2-11eb-28dd-fdf089198c2d # ╟─e21e9614-0ea6-11eb-2aba-97dd7bda343b # ╠═1cc8a0d8-27fb-4be8-8c1e-f1257af6e217 # ╠═7c4e9c3b-5776-4f2d-90ea-bd8b75fbb098 # ╠═18a01504-0ea6-11eb-1b64-c33c95d4bcae # ╟─0908992e-0ea6-11eb-211a-51a9655c77b8 # ╠═dd18e9c6-0eb3-11eb-31c4-296dbc11be15 # ╠═ece780c4-0eb3-11eb-1b24-0f804525f8ef # ╠═c91999a2-0eb3-11eb-285a-85e605e2a1f6 # ╠═c4dabe7a-0eb3-11eb-306b-b3001b0c4f73 # ╠═5645ac26-0eb4-11eb-3cfe-d13415915d87 # ╟─bbcc1392-0eb3-11eb-1868-b39c9b4130d4 # ╟─00000000-0000-0000-0000-000000000001 # ╟─00000000-0000-0000-0000-000000000002