### A Pluto.jl notebook ### # v0.16.0 using Markdown using InteractiveUtils # ╔═╡ 606cd8fe-128e-11eb-0710-11dcb91feb8c using Images, FileIO, Plots, LinearAlgebra, Random # ╔═╡ c6106436-13d2-11eb-0045-d3143701e096 md"""# Notebook 9 -- Math 2121, Fall 2021 In this short demonstration we explore the physical meaning of the sign of a determinant. Some terminology: * An invertible matrix with positive determinant is called **orientation-preserving**. * An invertible matrix with negative determinant is called **orientation-reversing**. Whether an $n\times n$ matrix is orientation-preserving or orientation-reversing is something we can "just see," at least when $n=2$, simply by looking at how the matrix transforms a set of points. However, it's not so easy to make a rigorous definition of these properties without referencing the determinant. """ # ╔═╡ d54c0f16-1421-11eb-3797-a3014c4a090e md"A black and white image is encoded by a set of vectors $v = \left[\begin{array}{c} v_1 \\ v_2 \end{array}\right] \in \mathbb{R}^2$: one vector for each black pixel at horizontal position $v_1$ and vertical position $v_2$. Suppose we have a $2\times 2$ matrix $A$. Then we may use $A$ to transform our image to another image, in which each vector $v$ is replaced by $Av$. By comparing these images, we will be able to see whether $A$ is orientation-preserving or orientation-reversing." # ╔═╡ 78fc7c5c-128f-11eb-3205-edbf78e4c08d function extract_image(filename) # this function converts a png file to a list of vectors as we just described img = download(filename) |> load gx, gy = [], [] (m,n) = size(img) for x=1:m for y=1:n if img[x,y] != img[1,1] append!(gx, x) append!(gy, y) end end end # now resize list of coordinates into 2-by-N matrix return vcat(transpose(reverse(gx)), transpose(gy)) # note: we reverse order of x-coordinates because increasing the row # index goes down in a matrix, which is opposite of cartesian coordinates end # ╔═╡ 8c20450a-1421-11eb-1aeb-2542fa014cc1 g = extract_image("www.math.ust.hk/~emarberg/teaching/2020/Math2121/julia/base.png") # ╔═╡ da467cba-1424-11eb-2ccd-83b13013b77b md"Here is what this image looks like if we plot all of the points:" # ╔═╡ fc3eda98-12f2-11eb-0d68-a72b31994521 function make_displayable(g) # this function converts our image as list of vectors into something displayable padding = 3 gx, gy = g[1,:], g[2,:] minx, miny = Int(floor(minimum(gx))) - 1, Int(floor(minimum(gy))) - 1 maxx, maxy = Int(ceil(maximum(gx))), Int(ceil(maximum(gy))) ans = zeros(Int64, maxx - minx + padding, maxy - miny + padding) for i = 1:length(gx) x = Int(ceil(gx[i])) - minx y = Int(ceil(gy[i])) - miny for p=0:padding for q=0:padding ans[x + p, y + q] = 1 end end end return ans end # ╔═╡ c2987c9a-12f2-11eb-0e81-05c36bb8db55 heatmap( make_displayable(g), aspect_ratio=:equal, legend=false, #axis=([], false), color=:binary, title="Starting set in R^2" ) # ╔═╡ c2785f72-7e55-4196-8f76-08b65996dc7c md"If $A$ is an $n\times n$ matrix with $\det(A) \neq 0$, then the matrix $B = \frac{1}{|\det(A)|^{1/n}} A$ has $\det(B) = \pm 1$. If we transform an image using $B$ instead of $A$ (when $n=2$), then the resulting image will be the same, just rescaled to have the same size as the starting image. So to observe the qualitative difference between matrices with positive or negative determinant, we can just consider matrices with determinant $+1$ or $-1$." # ╔═╡ ea5c1812-1424-11eb-38f5-4d325f7dfb17 md"Below are some functions to create random $n$-by-$n$ matrices with determinant $+1$ or $-1$. We'll just need the $n=2$ case of these methods." # ╔═╡ 0c390026-1425-11eb-2f3e-0781966a13b3 function randmat(n) # returns n-by-n matrices with uniformly random entries from interval [-1,1] return 2 * rand(Float64, n, n) .- 1.0 end # ╔═╡ c300769e-128c-11eb-348b-0502a48815ca function positive_determinant(n) while true A = randmat(n) if det(A) < 0.1 continue end return A / abs(det(A))^1/n end end # ╔═╡ df0afe86-13cc-11eb-283b-afc727e3ad62 function negative_determinant(n) while true A = randmat(n) if det(A) > -0.1 continue end return A / abs(det(A))^1/n end end # ╔═╡ 979446b6-128c-11eb-01be-bdea73e0d231 function illustrate(g, A, B) # function to plot side-by-side our image g transformed by A and B plot( heatmap( make_displayable(A * g), legend=false, color=:binary ), heatmap( make_displayable(B * g), legend=false, color=:binary ), layout = (1, 2), ) end # ╔═╡ 392285ee-1425-11eb-3561-85f1b6093337 md"""The plots below show our starting image transformed by random matrices $A$ with positive and negative determinant. If the matrix $A$ has positive determinant, then we say $A$ is **orientation-preserving** because the transformed image "looks the same (but rotated)." This is shown on the left. If the matrix $A$ has negative determinant, then we say $A$ is **orientation-reversing** because the transformed image is "the mirror image (but rotated)" of what we started with. This is on the right.""" # ╔═╡ 1d690c6c-12f2-11eb-21a8-8336324ab211 illustrate(g, positive_determinant(2), negative_determinant(2)) # ╔═╡ 00000000-0000-0000-0000-000000000001 PLUTO_PROJECT_TOML_CONTENTS = """ [deps] FileIO = "5789e2e9-d7fb-5bc7-8068-2c6fae9b9549" Images = "916415d5-f1e6-5110-898d-aaa5f9f070e0" LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e" Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c" [compat] FileIO = "~1.11.1" Images = "~0.24.1" Plots = "~1.22.6" """ # ╔═╡ 00000000-0000-0000-0000-000000000002 PLUTO_MANIFEST_TOML_CONTENTS = """ # This file is machine-generated - editing it directly is not advised [[AbstractFFTs]] deps = ["LinearAlgebra"] git-tree-sha1 = "485ee0867925449198280d4af84bdb46a2a404d0" uuid = "621f4979-c628-5d54-868e-fcf4e3e8185c" version = "1.0.1" [[Adapt]] deps = ["LinearAlgebra"] git-tree-sha1 = "84918055d15b3114ede17ac6a7182f68870c16f7" uuid = "79e6a3ab-5dfb-504d-930d-738a2a938a0e" version = 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"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" [[MKL_jll]] deps = ["Artifacts", "IntelOpenMP_jll", "JLLWrappers", "LazyArtifacts", "Libdl", "Pkg"] git-tree-sha1 = "5455aef09b40e5020e1520f551fa3135040d4ed0" uuid = "856f044c-d86e-5d09-b602-aeab76dc8ba7" version = "2021.1.1+2" [[MacroTools]] deps = ["Markdown", "Random"] git-tree-sha1 = "5a5bc6bf062f0f95e62d0fe0a2d99699fed82dd9" uuid = "1914dd2f-81c6-5fcd-8719-6d5c9610ff09" version = "0.5.8" [[MappedArrays]] git-tree-sha1 = "e8b359ef06ec72e8c030463fe02efe5527ee5142" uuid = "dbb5928d-eab1-5f90-85c2-b9b0edb7c900" version = "0.4.1" [[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" [[MosaicViews]] deps = ["MappedArrays", "OffsetArrays", "PaddedViews", "StackViews"] git-tree-sha1 = "b34e3bc3ca7c94914418637cb10cc4d1d80d877d" uuid = "e94cdb99-869f-56ef-bcf0-1ae2bcbe0389" version = "0.3.3" [[MozillaCACerts_jll]] uuid = "14a3606d-f60d-562e-9121-12d972cd8159" [[NaNMath]] git-tree-sha1 = "bfe47e760d60b82b66b61d2d44128b62e3a369fb" uuid = "77ba4419-2d1f-58cd-9bb1-8ffee604a2e3" version = "0.3.5" [[Netpbm]] deps = ["ColorVectorSpace", "FileIO", "ImageCore"] git-tree-sha1 = "09589171688f0039f13ebe0fdcc7288f50228b52" uuid = "f09324ee-3d7c-5217-9330-fc30815ba969" version = "1.0.1" [[NetworkOptions]] uuid = "ca575930-c2e3-43a9-ace4-1e988b2c1908" [[OffsetArrays]] deps = ["Adapt"] git-tree-sha1 = "c0e9e582987d36d5a61e650e6e543b9e44d9914b" uuid = 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"CompilerSupportLibraries_jll", "JLLWrappers", "Libdl", "Pkg"] git-tree-sha1 = "13652491f6856acfd2db29360e1bbcd4565d04f1" uuid = "efe28fd5-8261-553b-a9e1-b2916fc3738e" version = "0.5.5+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" [[PNGFiles]] deps = ["Base64", "CEnum", "ImageCore", "IndirectArrays", "OffsetArrays", "libpng_jll"] git-tree-sha1 = "85e3436b18980e47604dd0909e37e2f066f54398" uuid = "f57f5aa1-a3ce-4bc8-8ab9-96f992907883" version = "0.3.10" [[PaddedViews]] deps = ["OffsetArrays"] git-tree-sha1 = 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version = "0.1.1" [[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" [[ProgressMeter]] deps = ["Distributed", "Printf"] git-tree-sha1 = "afadeba63d90ff223a6a48d2009434ecee2ec9e8" uuid = "92933f4c-e287-5a05-a399-4b506db050ca" version = "1.7.1" [[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]] git-tree-sha1 = "b9039e93773ddcfc828f12aadf7115b4b4d225f5" uuid = "b3c3ace0-ae52-54e7-9d0b-2c1406fd6b9d" version = "0.3.2" [[Ratios]] deps = ["Requires"] git-tree-sha1 = "01d341f502250e81f6fec0afe662aa861392a3aa" uuid = "c84ed2f1-dad5-54f0-aa8e-dbefe2724439" version = "0.4.2" [[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" [[Rotations]] deps = ["LinearAlgebra", "StaticArrays", "Statistics"] git-tree-sha1 = "2ed8d8a16d703f900168822d83699b8c3c1a5cd8" uuid = "6038ab10-8711-5258-84ad-4b1120ba62dc" version = "1.0.2" [[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" [[SimpleTraits]] deps = ["InteractiveUtils", "MacroTools"] git-tree-sha1 = "5d7e3f4e11935503d3ecaf7186eac40602e7d231" uuid = "699a6c99-e7fa-54fc-8d76-47d257e15c1d" version = "0.9.4" [[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" [[SpecialFunctions]] deps = ["ChainRulesCore", "IrrationalConstants", "LogExpFunctions", "OpenLibm_jll", "OpenSpecFun_jll"] git-tree-sha1 = "2d57e14cd614083f132b6224874296287bfa3979" uuid = "276daf66-3868-5448-9aa4-cd146d93841b" version = "1.8.0" [[StackViews]] deps = ["OffsetArrays"] git-tree-sha1 = "46e589465204cd0c08b4bd97385e4fa79a0c770c" uuid = "cae243ae-269e-4f55-b966-ac2d0dc13c15" version = "0.1.1" [[Static]] deps = ["IfElse"] git-tree-sha1 = "a8f30abc7c64a39d389680b74e749cf33f872a70" uuid = "aedffcd0-7271-4cad-89d0-dc628f76c6d3" version = "0.3.3" [[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" [[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 = "945b8d87c5e8d5e34e6207ee15edb9d11ae44716" uuid = "731e570b-9d59-4bfa-96dc-6df516fadf69" version = "0.4.3" [[TiledIteration]] deps = ["OffsetArrays"] git-tree-sha1 = "5683455224ba92ef59db72d10690690f4a8dc297" uuid = "06e1c1a7-607b-532d-9fad-de7d9aa2abac" version = "0.3.1" [[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 = "9398e8fefd83bde121d5127114bd3b6762c764a6" uuid = "efce3f68-66dc-5838-9240-27a6d6f5f9b6" version = "0.5.4" [[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: # ╟─c6106436-13d2-11eb-0045-d3143701e096 # ╠═606cd8fe-128e-11eb-0710-11dcb91feb8c # ╟─d54c0f16-1421-11eb-3797-a3014c4a090e # ╠═78fc7c5c-128f-11eb-3205-edbf78e4c08d # ╠═8c20450a-1421-11eb-1aeb-2542fa014cc1 # ╟─da467cba-1424-11eb-2ccd-83b13013b77b # ╠═fc3eda98-12f2-11eb-0d68-a72b31994521 # ╟─c2987c9a-12f2-11eb-0e81-05c36bb8db55 # ╟─c2785f72-7e55-4196-8f76-08b65996dc7c # ╟─ea5c1812-1424-11eb-38f5-4d325f7dfb17 # ╠═0c390026-1425-11eb-2f3e-0781966a13b3 # ╠═c300769e-128c-11eb-348b-0502a48815ca # ╠═df0afe86-13cc-11eb-283b-afc727e3ad62 # ╠═979446b6-128c-11eb-01be-bdea73e0d231 # ╟─392285ee-1425-11eb-3561-85f1b6093337 # ╠═1d690c6c-12f2-11eb-21a8-8336324ab211 # ╟─00000000-0000-0000-0000-000000000001 # ╟─00000000-0000-0000-0000-000000000002