### A Pluto.jl notebook ### # v0.16.4 using Markdown using InteractiveUtils # ╔═╡ dd969a86-22a6-11eb-1a42-1d1b5868f305 using LinearAlgebra # ╔═╡ bfd67682-22a7-11eb-3721-c92d93a61153 using Plots # ╔═╡ 86ab217a-2377-11eb-1f82-9dfb61bd42a2 md"# Notebook 13 -- Math 2121, Fall 2021 In this notebook we will explore an application of eigenvalues and eigenvectors to drawing graphs (in the sense of discrete mathematics, rather than calculus)." # ╔═╡ 0c90dbe4-2312-11eb-2c76-b5cdbba77632 md"A *(simple, undirected) graph* is a set of *vertices* $V = \{v_1,v_2,\dots,v_n\}$ along with a set of *edges* $v_i \longleftrightarrow v_j$ between distinct vertices. We can draw a graph by plotting points that correspond to the vertices and drawing lines between vertices that are connected by an edge." # ╔═╡ 146e2a4a-22b8-11eb-08d1-251c5464596a function plot_graph(edges, index, xy) # helper method for plotting graphs, which uses the columns of xy # (which should be a 2-by-n matrix where n = number of vertices in graph) # as the vertex locations p = plot(grid=nothing, axis=nothing, foreground_color_subplot="white", aspect_ratio=:equal) for (v, adjacent) in edges if haskey(index, v) i = index[v] x, y = xy[1, i], xy[2, i] for w in adjacent if haskey(index, w) j = index[w] dx, dy = xy[1, j] - x, xy[2, j] - y quiver!(quiver = ([dx],[dy]), [x], [y], color=:blue) end end end end scatter!(xy[1,:], xy[2,:], markersize = 5, legend=false) return p end # ╔═╡ 79176da8-2312-11eb-2f2b-675dfba186a5 function plot_graph_random(edges, index) # this method just passes a random 2-row vector xy to the plot_graph function xy = rand(2, length(index)) plot_graph(edges, index, xy) end # ╔═╡ a1acca14-282a-11eb-19c9-b7cbaa0440ae function graph_index(edges) vertices = [v for (v, k)=edges] degrees = [(-length(k), v) for (v, k)=edges] perm = sortperm(degrees) return Dict([degrees[perm[i]][2] => i for i=1:length(perm)]) end # ╔═╡ e84d076c-2312-11eb-2a84-5142f49ba93c md"For example, here is a graph with 6 vertices and 7 edges:" # ╔═╡ 8928cb74-2312-11eb-0af5-d173a317a75d begin example_edges = Dict( 1 => [2, 5], 2 => [1, 3, 5], 3 => [2, 4], 4 => [3, 5, 6], 5 => [1, 2, 4], 6 => [4]) example_index = Dict([i => i for i=1:6]) plot_graph_random(example_edges, example_index) end # ╔═╡ 44308286-2313-11eb-0ec7-e9a341fca047 md"The *adjacency matrix* of a graph with vertices $V=\{v_1,v_2,\dots,v_n\}$ is the $n\times n$ matrix that has $1$ in position $(i,j)$ if $v_i \longleftrightarrow v_j$ is an edge and $0$ in position $(i,j)$ if $v_i \longleftrightarrow v_j$ is not an edge. The *degree* of a vertex $v_i$ is the number of edges that involve $v_i$. This is the sum of the entries in column $i$ (or row $i$) of the adjacency matrix." # ╔═╡ 91fd9026-2313-11eb-2f22-b306cd9623f4 function graph_adjacency_matrix(edges, index) n = length(index) A = zeros(Int64, n, n) for (v, adjacent) in edges if haskey(index, v) for w in adjacent if haskey(index, w) i = index[v] j = index[w] A[i, j] = 1 end end end end return A end # ╔═╡ 93a22518-2313-11eb-2694-f300179f03c1 A = graph_adjacency_matrix(example_edges, example_index) # ╔═╡ d80ec394-24aa-11eb-3d3a-6d30db1cd8b0 example_edges # ╔═╡ 9c2f77f8-2313-11eb-1fc8-1fb7a3c00df0 md"The *Laplacian matrix* of a graph is the matrix $L = D - A$ where $A$ is the adjacency matrix and $D$ is the diagonal matrix whose entry in position $(i,i)$ is the degree of vertex $v_i$." # ╔═╡ c1737492-2313-11eb-2b8a-dd47ede4e43e function graph_laplacian_matrix(edges, index) A = graph_adjacency_matrix(edges, index) D = Diagonal(A * ones(Int64, length(index))) return D - A end # ╔═╡ c2de6148-2313-11eb-267d-97b73b8b3a6b L = graph_laplacian_matrix(example_edges, example_index) # ╔═╡ df41ac78-2313-11eb-04e0-710dfa13f2d0 md"Some interesting facts: * The eigenvalues of the adjacency matrix $A$ are always all real. * The eigenvalues of the Laplacian matrix $L$ are all real and nonnegative. * The matrix $L$ always has $0$ as an eigenvalue. * The multiplicity of the $0$-eigenvalue of $L$ is the number of *connected components* of the graph. " # ╔═╡ 0fabb8d6-2314-11eb-101c-2b35f746714a eigen(A).values # ╔═╡ 296c0c50-2314-11eb-2a17-a9f7c731f65b eigen(L).values # ╔═╡ bc44837c-2312-11eb-0b0a-11ca6bebcf2f function random_graph(n, p) # creates a random graph with n vertices edges = Dict(i => [] for i=1:n) for i=1:n for j=i+1:n if rand() < p append!(edges[i], j) append!(edges[j], i) end end end return edges, graph_index(edges) end # ╔═╡ bda02576-2314-11eb-0cc0-55ee6c9cdcbc function connected_components(edges, index) # computes connected components using graph Laplacian matrix L = graph_laplacian_matrix(edges, index) return length([v for v=eigen(L).values if abs(v) < 1e-14]) end # ╔═╡ 70c5ae8a-2314-11eb-10a6-394f037729b5 begin redges, rindex = random_graph(8, 0.12) plot_graph_random(redges, rindex) end # ╔═╡ c0159040-2314-11eb-063a-39fa40158634 connected_components(redges, rindex) # ╔═╡ cccefba8-2314-11eb-26ea-e926eea9e7e2 md"Here is an interesting algorithm for *drawing* a complicated graph: * Compute the adjacency matrix $A$ or Laplacian matrix $L$. * Take two distinct eigenvectors $u$ and $v$ for this matrix. * Rescale these eigenvectors to have the same length. * Use the coordinates $(u_1,v_1)$, $(u_2,v_2)$, $\dots$, $(u_n,v_n)$ for the vertices. * Often what works well is to use the two eigenvectors for $L$ with smallest nonzero eigenvalue. * Another choice is the use the two eigenvectors for $L$ with the largest eigenvalues." # ╔═╡ d776525c-2307-11eb-3e91-9d3fc5122bca function path_graph(n) # creates path graph with n vertices index = Dict(i => i for i=1:n) edges = Dict(i => (i == n ? [n - 1] : i == 1 ? [2] : [i - 1, i + 1]) for i=1:n) return edges, index end # ╔═╡ 94372020-2304-11eb-144d-d7eba810a5d3 function circle_graph(n) # creates circle graph with n vertices index = Dict(i => i for i=1:n) edges = Dict(i => [i == n ? 1 : i + 1, i == 1 ? n : i - 1] for i=1:n) return edges, index end # ╔═╡ c072d652-2307-11eb-3491-bd83d30d2316 function complete_graph(n) # creates complete graph with n vertices index = Dict(i => i for i=1:n) edges = Dict(i => [j for j=1:n if i != j] for i=1:n) return edges, index end # ╔═╡ aef6716c-2308-11eb-28fe-d70705e0d6fa function connected_random_graph(n, p, trials=1000) # attempts to create a random graph with one connected component for iter=1:trials e, i = random_graph(n, p) if connected_components(e, i) == 1 return e, i end end # gives up and returns graph with no edges return random_graph(n, 0) end # ╔═╡ 59bfed88-2315-11eb-013b-49e507c0f6f3 function plot_graph_adjacency_low(edges, index) # plots graph using lowest eigenvectors of adjacency matrix as xy-coordinates n = length(index) A = graph_adjacency_matrix(edges, index) xy = transpose(eigen(A).vectors[:,1:2]) plot_graph(edges, index, xy) end # ╔═╡ 4cbff80e-230a-11eb-1e49-f5ed69ed7c47 function plot_graph_adjacency_high(edges, index) # plots graph using highest eigenvectors of adjacency matrix as xy-coordinates n = length(index) A = graph_adjacency_matrix(edges, index) xy = transpose(eigen(A).vectors[:,end - 1:end]) plot_graph(edges, index, xy) end # ╔═╡ 64e9ea86-22a9-11eb-3cd2-a1aac2f66405 function plot_graph_laplacian_low(edges, index) # plots graph using lowest nonzero eigenvectors of Laplacian as xy-coordinates n = length(index) A = graph_laplacian_matrix(edges, index) c = 2 eig = eigen(A).values while c + 1 < n && abs(eig[c]) < 10e-14 c += 1 end xy = transpose(eigen(A).vectors[:, c:c+1]) plot_graph(edges, index, xy) end # ╔═╡ 300b3bf2-2309-11eb-387c-ef14710ee3e3 function plot_graph_laplacian_high(edges, index) # plots graph using highest eigenvectors of Laplacian as xy-coordinates n = length(index) A = graph_laplacian_matrix(edges, index) xy = transpose(eigen(A).vectors[:, end - 1:end]) plot_graph(edges, index, xy) end # ╔═╡ 38e58e1a-230b-11eb-2e2b-af306ffc966f function get_adjacency_plots(e, i) plot(plot_graph_adjacency_low(e, i), plot_graph_adjacency_high(e, i)) end # ╔═╡ aebfa540-230a-11eb-1b7e-f3f5868d4002 function get_laplacian_plots(e, i) plot(plot_graph_laplacian_low(e, i), plot_graph_laplacian_high(e, i)) end # ╔═╡ 73cebbb4-2379-11eb-132d-377db88d8802 md"Here is our starting graph from the example above, plotted randomly:" # ╔═╡ 23a2aa14-22a8-11eb-3621-95153be8bb6e begin test_edges = example_edges test_index = example_index plot_graph_random(test_edges, test_index) end # ╔═╡ 7dd38022-2379-11eb-2ff5-2d5a27c72d31 md"Here is the same graph, plotted using eigenvector methods based on the adjacency matrix:" # ╔═╡ 4911d9b0-230b-11eb-1769-a3a39d2d5e61 get_adjacency_plots(test_edges, test_index) # ╔═╡ 937fa7be-2379-11eb-1ecf-751d3483390d md"Here is the same graph, plotted using eigenvector methods based on the Laplacian matrix:" # ╔═╡ 4fd4b420-230b-11eb-3aff-27bdaead9528 get_laplacian_plots(test_edges, test_index) # ╔═╡ 98f78556-2379-11eb-2f8a-4d92cae0a949 md"In my opinion the plot using the eigenvectors of the Laplacian for the lowest nonzero eigenvalues as xy-coordinates gives the 'best' output. Here are a few more examples to underscore this claim:" # ╔═╡ 4b0dd130-2309-11eb-07c0-9d6b2de9310b begin path_edges, path_index = path_graph(20) plot_graph_random(path_edges, path_index) end # ╔═╡ fdab2ef4-230a-11eb-3a07-cff509c68b58 get_adjacency_plots(path_edges, path_index) # ╔═╡ 65b6e718-230b-11eb-3565-a189238f7ce5 get_laplacian_plots(path_edges, path_index) # ╔═╡ a3ffaf30-2308-11eb-394c-0d1c79173cfb begin circle_edges, circle_index = circle_graph(20) plot_graph_random(circle_edges, circle_index) end # ╔═╡ 0a6796c8-230b-11eb-3204-118526952bf2 get_adjacency_plots(circle_edges, circle_index) # ╔═╡ 758b408a-230b-11eb-1dca-a90abb1a8eb8 get_laplacian_plots(circle_edges, circle_index) # ╔═╡ 845849dc-230b-11eb-3ba1-873b732ab100 begin complete_edges, complete_index = complete_graph(6) plot_graph_random(complete_edges, complete_index) end # ╔═╡ 9a5a0408-230b-11eb-31ff-fd6996bc1e0c get_adjacency_plots(complete_edges, complete_index) # ╔═╡ 9ca3598c-230b-11eb-2fe3-175d25cc1494 get_laplacian_plots(complete_edges, complete_index) # ╔═╡ f07b7cf6-2379-11eb-3a01-b992e29d89a4 md"**Remark:** our eigenvector methods do not give such great pictures of the complete graph on $n$ vertices." # ╔═╡ 94c4dea4-2437-11eb-1637-590d34bf930c begin connected_edges, connected_index = connected_random_graph(12, 0.1, 10000) plot_graph_random(connected_edges, connected_index) end # ╔═╡ a446a2f2-2437-11eb-2267-69844893bfb4 plot_graph_laplacian_low(connected_edges, connected_index) # ╔═╡ 00000000-0000-0000-0000-000000000001 PLUTO_PROJECT_TOML_CONTENTS = """ [deps] LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e" Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" [compat] Plots = "~1.23.5" """ # ╔═╡ 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 = "79e6a3ab-5dfb-504d-930d-738a2a938a0e" version 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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" [[InverseFunctions]] deps = ["Test"] git-tree-sha1 = "f0c6489b12d28fb4c2103073ec7452f3423bd308" uuid = "3587e190-3f89-42d0-90ee-14403ec27112" version = "0.1.1" [[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 = "f2355693d6778a178ade15952b7ac47a4ff97996" uuid = "b964fa9f-0449-5b57-a5c2-d3ea65f4040f" version = "1.3.0" [[Latexify]] deps = ["Formatting", "InteractiveUtils", "LaTeXStrings", "MacroTools", "Markdown", "Printf", "Requires"] git-tree-sha1 = "a8f4f279b6fa3c3c4f1adadd78a621b13a506bce" uuid = "23fbe1c1-3f47-55db-b15f-69d7ec21a316" version = "0.15.9" [[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", "InverseFunctions", "IrrationalConstants", "LinearAlgebra"] git-tree-sha1 = "6193c3815f13ba1b78a51ce391db8be016ae9214" uuid = "2ab3a3ac-af41-5b50-aa03-7779005ae688" version = "0.3.4" [[Logging]] uuid = "56ddb016-857b-54e1-b83d-db4d58db5568" [[MacroTools]] deps = ["Markdown", "Random"] git-tree-sha1 = "3d3e902b31198a27340d0bf00d6ac452866021cf" uuid = "1914dd2f-81c6-5fcd-8719-6d5c9610ff09" version = "0.5.9" [[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 = "ae4bbcadb2906ccc085cf52ac286dc1377dceccc" uuid = "69de0a69-1ddd-5017-9359-2bf0b02dc9f0" version = "2.1.2" [[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", "UnicodeFun"] git-tree-sha1 = "7dc03c2b145168f5854085a16d054429d612b637" uuid = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" version = "1.23.5" [[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 = "eb35dcc66558b2dda84079b9a1be17557d32091a" uuid = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91" version = "0.33.12" [[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" [[UnicodeFun]] deps = ["REPL"] git-tree-sha1 = "53915e50200959667e78a92a418594b428dffddf" uuid = "1cfade01-22cf-5700-b092-accc4b62d6e1" version = "0.4.1" [[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: # ╟─86ab217a-2377-11eb-1f82-9dfb61bd42a2 # ╠═dd969a86-22a6-11eb-1a42-1d1b5868f305 # ╠═bfd67682-22a7-11eb-3721-c92d93a61153 # ╟─0c90dbe4-2312-11eb-2c76-b5cdbba77632 # ╠═146e2a4a-22b8-11eb-08d1-251c5464596a # ╠═79176da8-2312-11eb-2f2b-675dfba186a5 # ╠═a1acca14-282a-11eb-19c9-b7cbaa0440ae # ╟─e84d076c-2312-11eb-2a84-5142f49ba93c # ╠═8928cb74-2312-11eb-0af5-d173a317a75d # ╟─44308286-2313-11eb-0ec7-e9a341fca047 # ╠═91fd9026-2313-11eb-2f22-b306cd9623f4 # ╠═93a22518-2313-11eb-2694-f300179f03c1 # ╠═d80ec394-24aa-11eb-3d3a-6d30db1cd8b0 # ╟─9c2f77f8-2313-11eb-1fc8-1fb7a3c00df0 # ╠═c1737492-2313-11eb-2b8a-dd47ede4e43e # ╠═c2de6148-2313-11eb-267d-97b73b8b3a6b # ╟─df41ac78-2313-11eb-04e0-710dfa13f2d0 # ╠═0fabb8d6-2314-11eb-101c-2b35f746714a # ╠═296c0c50-2314-11eb-2a17-a9f7c731f65b # ╠═bc44837c-2312-11eb-0b0a-11ca6bebcf2f # ╠═bda02576-2314-11eb-0cc0-55ee6c9cdcbc # ╠═70c5ae8a-2314-11eb-10a6-394f037729b5 # ╠═c0159040-2314-11eb-063a-39fa40158634 # ╟─cccefba8-2314-11eb-26ea-e926eea9e7e2 # ╠═d776525c-2307-11eb-3e91-9d3fc5122bca # ╠═94372020-2304-11eb-144d-d7eba810a5d3 # ╠═c072d652-2307-11eb-3491-bd83d30d2316 # ╠═aef6716c-2308-11eb-28fe-d70705e0d6fa # ╠═59bfed88-2315-11eb-013b-49e507c0f6f3 # ╠═4cbff80e-230a-11eb-1e49-f5ed69ed7c47 # ╠═64e9ea86-22a9-11eb-3cd2-a1aac2f66405 # ╠═300b3bf2-2309-11eb-387c-ef14710ee3e3 # ╠═38e58e1a-230b-11eb-2e2b-af306ffc966f # ╠═aebfa540-230a-11eb-1b7e-f3f5868d4002 # ╟─73cebbb4-2379-11eb-132d-377db88d8802 # ╠═23a2aa14-22a8-11eb-3621-95153be8bb6e # ╟─7dd38022-2379-11eb-2ff5-2d5a27c72d31 # ╠═4911d9b0-230b-11eb-1769-a3a39d2d5e61 # ╟─937fa7be-2379-11eb-1ecf-751d3483390d # ╠═4fd4b420-230b-11eb-3aff-27bdaead9528 # ╟─98f78556-2379-11eb-2f8a-4d92cae0a949 # ╠═4b0dd130-2309-11eb-07c0-9d6b2de9310b # ╠═fdab2ef4-230a-11eb-3a07-cff509c68b58 # ╠═65b6e718-230b-11eb-3565-a189238f7ce5 # ╠═a3ffaf30-2308-11eb-394c-0d1c79173cfb # ╠═0a6796c8-230b-11eb-3204-118526952bf2 # ╠═758b408a-230b-11eb-1dca-a90abb1a8eb8 # ╠═845849dc-230b-11eb-3ba1-873b732ab100 # ╠═9a5a0408-230b-11eb-31ff-fd6996bc1e0c # ╠═9ca3598c-230b-11eb-2fe3-175d25cc1494 # ╟─f07b7cf6-2379-11eb-3a01-b992e29d89a4 # ╠═94c4dea4-2437-11eb-1637-590d34bf930c # ╠═a446a2f2-2437-11eb-2267-69844893bfb4 # ╟─00000000-0000-0000-0000-000000000001 # ╟─00000000-0000-0000-0000-000000000002