### A Pluto.jl notebook ### # v0.16.0 using Markdown using InteractiveUtils # This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error). macro bind(def, element) quote local el = $(esc(element)) global$(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : missing el end end # ╔═╡ 942d22f0-fdd2-11ea-067f-ffddf84d1c39 begin using Plots using PlutoUI md"##### Helper methods" end # ╔═╡ 500b0a2e-fdd2-11ea-32d0-d702f3410357 md"# Notebook 5 -- Math 2121, Fall 2021 In today's notebook we'll first explore some plots of linear transformations. Then we'll take a look at the frequency that a random linear transformation is one-to-one or onto. ##### Running *this* notebook (optional) If you have Pluto up and running, you can access the notebook we are currently viewing by entering [this link](https://www.math.hkust.edu.hk/~emarberg/teaching/2021/Math2121/julia/05_Math2121_Fall2021.jl) (right click -> Copy Link) in the *Open from file* menu in Pluto." # ╔═╡ 333a2fce-fdcd-11ea-1ab5-7d88e0217376 function pprint(tagx, tagy, rx, ry) s = [string(rx[1]), string(rx[2]), string(ry[1]), string(ry[2])] pad = maximum([4, maximum(map(length,s))]) for i=1:length(s) s[i] = " "^(pad - length(s[i])) * s[i] end top = join([ tagx * " = [$(s[1])]", tagy * " = [$(s[3])]"], " ") bot = join([" "^length(tagx) * " [$(s[2])]", " "^length(tagy) * " [$(s[4])]"], " ") return Text(join([top, bot], "\n")) end # ╔═╡ 8a23eb6e-fdc2-11ea-0603-8b191ff11d0e function input_plot(x, y, title, lims=(-2,2)) p = scatter([0], [0], xlims = lims, ylims = lims, legend = false, label = "origin", aspect_ratio=:equal, title = title) quiver!(quiver = ([x[1]],[x[2]]), [0], [0], color = [:blue],) quiver!( quiver = ([x[1]],[x[2]]), [y[1]], [y[2]], color = [:lightgrey], linestyle = :dashdot, linecolor=:lightgrey) quiver!(quiver = ([y[1]],[y[2]]), [0], [0], color = [:red],) quiver!( quiver = ([y[1]],[y[2]]), [x[1]], [x[2]], color = [:lightgrey], linestyle = :dashdot, linecolor=:lightgrey) return p end # ╔═╡ e41e382c-fdcc-11ea-3d1c-17127b462f9e function transformation_plot(A, x, y, title, lims=(-2,2)) p = scatter([0], [0], xlim=lims, ylim=lims, legend = false, label = "origin", aspect_ratio=:equal, title = title) Ux = A * x Uy = A * y Uxy = A * (x + y) quiver!(quiver = ([Ux[1]], [Ux[2]]), [0], [0], color = [:blue],) quiver!( quiver = ([Ux[1]], [Ux[2]]), [Uy[1]], [Uy[2]], color = [:lightgrey], linestyle = :dashdot, linecolor=:lightgrey) quiver!(quiver = ([Uy[1]], [Uy[2]]), [0], [0], color = [:red],) quiver!( quiver = ([Uy[1]], [Uy[2]]), [Ux[1]], [Ux[2]], color = [:lightgrey], linestyle = :dashdot, linecolor=:lightgrey) return p end # ╔═╡ 4c6f5762-fdcd-11ea-364e-d70107e4bc0b function compare_plots(x, y; A) p1 = input_plot(x, y, "x and y") p2 = transformation_plot(A, x, y, "Ax and Ay") plot(p1, p2, layout=2) end # ╔═╡ 421e0dbc-fdcd-11ea-0ef8-83890a2328e5 md"##### Linear transformations from $2\times 2$ matrices" # ╔═╡ 019217d6-fdcd-11ea-1e36-b5cbc03cf1d5 begin r_slider = @bind r Slider(0:0.1:2 * pi, default=1, show_value=false) theta_slider = @bind theta Slider(0:0.1:2 * pi, default=1, show_value=true) x1_slider = @bind x1 Slider(-2:0.1:2, default=1, show_value=false) x2_slider = @bind x2 Slider(-2:0.1:2, default=0, show_value=false) y1_slider = @bind y1 Slider(-2:0.1:2, default=0, show_value=false) y2_slider = @bind y2 Slider(-2:0.1:2, default=1, show_value=false) k_slider = @bind k Slider(-2:0.1:2, default=1.5, show_value=true) md"""**input parameters** x1 = $(x1_slider) y1 =$(y1_slider) rotate = $(r_slider) x2 =$(x2_slider) y2 = $(y2_slider) **angle parameter** θ =$(theta_slider) """ end # ╔═╡ 1d86c41c-fdcd-11ea-002b-69c19dd80a3f begin rmat = [cos(r) -sin(r); sin(r) cos(r)] x = rmat * [x1; x2] y = rmat * [y1; y2] pprint("x", "y", x, y) end # ╔═╡ fa05c5ff-0163-46e8-a755-fb962d46bb81 compare_plots(x, y; A=[cos(theta) -sin(theta) sin(theta) cos(theta)] ) # rotates counter clockwise by angle theta # ╔═╡ dda79521-46b0-4a52-bd32-63d96bfe7459 compare_plots(x, y; A=[1 0 0 -1] ) # flips input across horizontal axis # ╔═╡ 0105937c-f57c-4429-a32d-c226c815ec79 compare_plots(x, y; A=[-1 0 0 1] ) # flips input across the vertical axis # ╔═╡ 001a77d5-f56a-4f34-83d2-b022c6966a41 compare_plots(x, y; A=[0 1 1 0] ) # reflects across the line x_1 = x_2 # ╔═╡ bc5852b4-f78f-45b7-b312-b2ea6b1ba809 begin md""" **another parameter** k = $(k_slider) """ end # ╔═╡ 0df14d8c-11aa-4d52-82ff-f359e1fdc1b0 compare_plots(x, y; A=[k 0 0 1] ) # scales the horizontal components by factor k # ╔═╡ 7f4aa4d0-f2b2-4ee6-92ae-7541a17b367b compare_plots(x, y; A=[1 0 0 k] ) # scales the vertical components by factor k # ╔═╡ 935fddb6-7931-4805-becd-4e6049b6e9c6 compare_plots(x, y; A=[1 k 0 1] ) # shearing/skewing operation in horizontal direction # ╔═╡ 133108da-045e-41f8-998b-9772ead84b32 compare_plots(x, y; A=[1 0 k 1] ) # shearing/skewing operation in vertical direction # ╔═╡ 3a9bc594-fe26-11ea-3767-5bdf350fb6fb md"##### Random matrices, onto and one-to-one linear transformations We can generate a random linear transformation$T: \mathbb{R}^n \to \mathbb{R}^m$by generating a random$m\times n$matrix$A$and setting$T(x) = Ax$. If$n > m$then$T$is never one-to-one, and if$n < m$then$T$is never onto. If$n \leq m$, however, then a random linear transformation$T$is injective with a high probability. If$n \geq m$, similarly, then a random linear transformation$T$is surjective with a high probability. In other words, a sufficiently random linear transformation is almost always one-to-one if this is possible (that is, if$n \leq m$) and almost always onto if this is possible (that is, if$n \geq m$). If$n=m$then a random linear transformation is almost always both one-to-one and onto (which will be our definition of an *invertible function* next lecture). To explore this phenomenon, we'll consider our usual model of random matrix which is not *that* random: namely,$01$-matrices whose entries are independently$0$or$1$with some probability$p$. For such matrices, we can see some interesting behavior while also observing convergence to the ''almost always'' properties described above. " # ╔═╡ b16e53c8-fe26-11ea-206b-4b8acae80e60 md"##### Helper methods" # ╔═╡ e5ee9a5a-fe1e-11ea-11e9-95bc915a0ee6 begin # Returns true if row has all zeros. function is_zero_row(A, row, tolerance=10e-16) (m, n) = size(A) return all([abs(A[row,col]) < tolerance for col=1:n]) end # Returns first index of nonzero entry in row, or 0 if none exists. function find_leading(A, row=1, tolerance=10e-16) (m, n) = size(A) for col=1:n if abs(A[row, col]) > tolerance return col end end return 0 end function find_nonzeros_in_column(A, row_start, col, tol=10e-12) (m, n) = size(A) return [i for i=row_start:m if abs(A[i, col]) > tol] end function _rowop_replace(A, source_row, target_row, scalar_factor) @assert source_row != target_row A[target_row,:] += A[source_row,:] * scalar_factor return A end function _rowop_scale(A, target_row, scalar_factor) @assert scalar_factor != 0 A[target_row,:] *= scalar_factor A end function _rowop_swap(A, s, t) A[t,:], A[s,:] = A[s,:], A[t,:] A end function RREF(A, tol=10e-12) A = float(copy(A)) (m, n) = size(A) row = 1 for col=1:n nonzeros = find_nonzeros_in_column(A, row, col, tol) if length(nonzeros) == 0 continue end i = nonzeros[1] if abs(A[i, col] - 1) < tol A[i, col] = 1 else A = _rowop_scale(A, i, 1 / A[i, col]) end for j=nonzeros[2:end] A = _rowop_replace(A, i, j, -A[j, col]) end if i != row A = _rowop_swap(A, row, i) end row += 1 end for row=m:-1:1 l = find_leading(A, row, tol) if l > 0 for k=1:row - 1 if abs(A[k,l]) > tol _rowop_replace(A, row, k, -A[k,l]) end end end end # round entries to give nicer output; could omit this step for row=1:m for col=1:n if abs(A[row,col] - round(A[row,col])) < tol A[row,col] = round(A[row,col]) end end end return A end end # ╔═╡ 110a5bd4-fe1f-11ea-262e-730e1ff90e74 begin function pivot_positions(A, tol=10e-12) rref = RREF(A, tol) return [ (i, find_leading(rref, i)) for i=1:size(A)[1] if find_leading(rref, i) > 0 ] end function npivots(A, tol=10e-12) return length(pivot_positions(A, tol)) end end # ╔═╡ ec4f8f1c-fe1e-11ea-2dfa-05f83ed0f00c function is_onto(A) (m, n) = size(A) return npivots(A) == m end # ╔═╡ 29b54ba8-fe1f-11ea-2dd4-2baa0503defd function is_one_to_one(A) (m, n) = size(A) return npivots(A) == n end # ╔═╡ bbf682e6-fe1f-11ea-1f7d-4f55724ebd9b function random_boolean_matrix(m, n, zero_probability) A = rand(m, n) for i=1:m for j=1:n A[i, j] = Int(A[i, j] > zero_probability) end end return A end # ╔═╡ a0d257ee-fe1f-11ea-16d9-a7ecc379d56e function accumulate_mean(returns_boolean, m, n, zero_prob, trials) s = 0 for i=1:trials s += Int(returns_boolean(random_boolean_matrix(m, n, zero_prob))) end mean = s / trials return mean end # ╔═╡ a65f1cf6-fe1f-11ea-0d70-1f2f1b0ce2dd function accumulate_std(returns_boolean, m, n, zero_prob, trials, mean) s = 0 for i=1:trials s += (Int(returns_boolean(random_boolean_matrix(m, n, zero_prob))) - mean)^2 end std = sqrt(s / (trials - 1)) return std end # ╔═╡ 92d46126-fe21-11ea-3b1d-0be2fce36f72 function accumulate_plot(returns_boolean, title, trials, M, N) prob = 0:0.01:1 mu = [accumulate_mean(returns_boolean, M, N, p, trials) for p in prob] std = [accumulate_std(returns_boolean, M, N, prob[i], trials, mu[i]) for i=1:length(prob)] return plot( prob, hcat(mu, std), xlabel="Probablity p that individual matrix entry is zero", ylabel="Proportion", title=title, label=["mean" "std"] ) end # ╔═╡ b676d206-fe37-11ea-238d-a9efe71dad02 plotly() # ╔═╡ bcc2b6b8-fe26-11ea-16d8-b9391dc06fe6 md"##### Parameters" # ╔═╡ c29c6fae-fe1f-11ea-2524-4b9c4588b70d trials = 1000 # ╔═╡ c5190434-a974-4124-8765-65574de60494 nrows, ncols = 6, 10 # ╔═╡ 1b8f276c-fe3b-11ea-375d-574e523dbc3b md"To make the following plot, for various probabilities p in [0,1], we generate **$(trials)** random 01-matrices of size **$(nrows) by$(ncols)**. For each p, we compute the proportion of these matrices that correspond to **onto** linear transformations, as well as the standard deviation of this statistic. Then we plot both as a function of p." # ╔═╡ 633f693a-fe1f-11ea-1d89-1716c918cfb7 accumulate_plot(is_onto, "Random 01-matrices, size $(nrows)-by-$(ncols): onto transformations", trials, nrows, ncols) # ╔═╡ 87f1fc20-fe3d-11ea-0b45-f9825caf6913 md"Interesting properties of this graph: when $p$ is not too small or too large, our model of a random matrix is ''sufficiently'' random, so the measured proportion is 1.0: almost every random matrix corresponds to an onto linear transformation. The standard deviation is always bimodal, with maximum value 0.5 attained at two values $p_1$ and $p_2$ of the probability $p$, which are also where the mean and standard deviation graphs intersect. What are these two values of $p$? They are not symmetric, that is, $p_2 \neq 1-p_1$." # ╔═╡ e3c75cb8-fe3b-11ea-2970-89b816ef738e md"Why does the mean have value $0$ when $p=0$ and when $p=1$?" # ╔═╡ 2650306e-fe3c-11ea-00cf-5d7e15b2b8e9 md"Here is a random 01-matrix for $p=0$:" # ╔═╡ 180b696a-fe3c-11ea-18fe-37423a9b5827 R = random_boolean_matrix(nrows, ncols, 0) # ╔═╡ 2a064810-fe3c-11ea-141f-f71f37013d7a npivots(R) # ╔═╡ 4a125950-fe3c-11ea-3ae3-5546c0d4e11d md"Here is a random 01-matrix for $p=1$:" # ╔═╡ 292a5800-fe3c-11ea-3af5-f70d6dbc14a1 S = random_boolean_matrix(nrows, ncols, 1) # ╔═╡ 4fde7e68-fe3c-11ea-0c84-01b599a5fb56 npivots(S) # ╔═╡ 56266042-fe3c-11ea-23ca-d3f9240211c5 md"In both cases the number of pivots is much less than the number of rows or columns, so these matrices correspond to linear transformations that are neither onto nor one-to-one." # ╔═╡ b436be80-fe3b-11ea-25ef-c183d6181a2c md"To make the following plot, for various probabilities p in [0,1], we generate **$(trials)** random 01-matrices of size **$(ncols) by $(nrows)** (note: reversed from above). For each p, we compute the proportition of these matrices that correspond to **one-to-one** linear transformations, as well as the standard deviation of this statistic. Then we plot both as a function of p." # ╔═╡ 1ec4868e-fe22-11ea-295b-6929947c5b91 accumulate_plot(is_one_to_one, "Random 01-matrices, size$(ncols)-by-$(nrows): 1-to-1 transformations", trials, ncols, nrows) # ╔═╡ acf9c58a-fe3c-11ea-2ad9-fd57da314a66 md"Why are two graphs nearly the same? The **transpose** of a matrix$A$is then matrix$A^T$formed by interchanging the rows and columns: $$\left[\begin{array}{cc} 1 & 2 \\ 3 & 4 \\ 5 & 6 \end{array}\right]^T =\left[\begin{array}{ccc} 1 & 3 & 5 \\ 2 & 4 & 6 \end{array}\right].$$ We'll see later in the class, that the number of pivot positions in$A$is the same as in$A^T$." # ╔═╡ 27808bc2-fe3d-11ea-12fa-f35dd36e17a7 md"This means that the number of$(nrows) by $(ncols) matrices with pivots in every row (corresponding to **onto** linear transformations) is the same as the number of$(ncols) by \$(nrows) matrices with pivots in every column (corresponding to **one-to-one** linear transformations)." # ╔═╡ 00000000-0000-0000-0000-000000000001 PLUTO_PROJECT_TOML_CONTENTS = """ [deps] Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8" [compat] Plots = "~1.22.0" PlutoUI = "~0.7.9" """ # ╔═╡ 00000000-0000-0000-0000-000000000002 PLUTO_MANIFEST_TOML_CONTENTS = """ # This file is 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deps = ["Artifacts", "JLLWrappers", "Libdl", "Libglvnd_jll", "Pkg", "Xorg_libXcursor_jll", "Xorg_libXi_jll", "Xorg_libXinerama_jll", "Xorg_libXrandr_jll"] git-tree-sha1 = "dba1e8614e98949abfa60480b13653813d8f0157" uuid = "0656b61e-2033-5cc2-a64a-77c0f6c09b89" version = "3.3.5+0" [[GR]] deps = ["Base64", "DelimitedFiles", "GR_jll", "HTTP", "JSON", "Libdl", "LinearAlgebra", "Pkg", "Printf", "Random", "Serialization", "Sockets", "Test", "UUIDs"] git-tree-sha1 = "c2178cfbc0a5a552e16d097fae508f2024de61a3" uuid = "28b8d3ca-fb5f-59d9-8090-bfdbd6d07a71" version = "0.59.0" [[GR_jll]] deps = ["Artifacts", "Bzip2_jll", "Cairo_jll", "FFMPEG_jll", "Fontconfig_jll", "GLFW_jll", "JLLWrappers", "JpegTurbo_jll", "Libdl", "Libtiff_jll", "Pixman_jll", "Pkg", "Qt5Base_jll", "Zlib_jll", "libpng_jll"] git-tree-sha1 = "ef49a187604f865f4708c90e3f431890724e9012" uuid = "d2c73de3-f751-5644-a686-071e5b155ba9" version = "0.59.0+0" [[GeometryBasics]] deps = ["EarCut_jll", "IterTools", "LinearAlgebra", "StaticArrays", "StructArrays", "Tables"] git-tree-sha1 = "58bcdf5ebc057b085e58d95c138725628dd7453c" uuid = "5c1252a2-5f33-56bf-86c9-59e7332b4326" version = "0.4.1" [[Gettext_jll]] deps = ["Artifacts", "CompilerSupportLibraries_jll", "JLLWrappers", "Libdl", "Libiconv_jll", "Pkg", "XML2_jll"] git-tree-sha1 = "9b02998aba7bf074d14de89f9d37ca24a1a0b046" uuid = "78b55507-aeef-58d4-861c-77aaff3498b1" version = "0.21.0+0" [[Glib_jll]] deps = ["Artifacts", "Gettext_jll", "JLLWrappers", "Libdl", "Libffi_jll", "Libiconv_jll", "Libmount_jll", "PCRE_jll", "Pkg", "Zlib_jll"] git-tree-sha1 = "7bf67e9a481712b3dbe9cb3dac852dc4b1162e02" uuid = "7746bdde-850d-59dc-9ae8-88ece973131d" version = "2.68.3+0" [[Graphite2_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] git-tree-sha1 = "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 = "60ed5f1643927479f845b0135bb369b031b541fa" uuid = "cd3eb016-35fb-5094-929b-558a96fad6f3" version = "0.9.14" [[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" [[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" [[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 = "438d35d2d95ae2c5e8780b330592b6de8494e779" uuid = "69de0a69-1ddd-5017-9359-2bf0b02dc9f0" version = "2.0.3" [[Pixman_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] git-tree-sha1 = "b4f5d02549a10e20780a24fce72bea96b6329e29" uuid = "30392449-352a-5448-841d-b1acce4e97dc" version = "0.40.1+0" [[Pkg]] deps = ["Artifacts", "Dates", "Downloads", "LibGit2", "Libdl", "Logging", "Markdown", "Printf", "REPL", "Random", "SHA", "Serialization", "TOML", "Tar", "UUIDs", "p7zip_jll"] uuid = "44cfe95a-1eb2-52ea-b672-e2afdf69b78f" [[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 = "2537ed3c0ed5e03896927187f5f2ee6a4ab342db" uuid = "995b91a9-d308-5afd-9ec6-746e21dbc043" version = "1.0.14" [[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 = "b1a708d607125196ea1acf7264ee1118ce66931b" uuid = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" version = "1.22.0" [[PlutoUI]] deps = ["Base64", "Dates", "InteractiveUtils", "JSON", "Logging", "Markdown", "Random", "Reexport", "Suppressor"] git-tree-sha1 = "44e225d5837e2a2345e69a1d1e01ac2443ff9fcb" uuid = "7f904dfe-b85e-4ff6-b463-dae2292396a8" version = "0.7.9" [[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 = "3240808c6d463ac46f1c1cd7638375cd22abbccb" uuid = "90137ffa-7385-5640-81b9-e52037218182" version = "1.2.12" [[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", "Missings", "Printf", "Random", "SortingAlgorithms", "SparseArrays", "Statistics", "StatsAPI"] git-tree-sha1 = "8cbbc098554648c84f79a463c9ff0fd277144b6c" uuid = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91" version = "0.33.10" [[StructArrays]] deps = ["Adapt", "DataAPI", "StaticArrays", "Tables"] git-tree-sha1 = "f41020e84127781af49fc12b7e92becd7f5dd0ba" uuid = "09ab397b-f2b6-538f-b94a-2f83cf4a842a" version = "0.6.2" [[Suppressor]] git-tree-sha1 = "a819d77f31f83e5792a76081eee1ea6342ab8787" uuid = "fd094767-a336-5f1f-9728-57cf17d0bbfb" version = "0.2.0" [[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 = "1162ce4a6c4b7e31e0e6b14486a6986951c73be9" uuid = "bd369af6-aec1-5ad0-b16a-f7cc5008161c" version = "1.5.2" [[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: # ╟─500b0a2e-fdd2-11ea-32d0-d702f3410357 # ╟─942d22f0-fdd2-11ea-067f-ffddf84d1c39 # ╟─333a2fce-fdcd-11ea-1ab5-7d88e0217376 # ╟─8a23eb6e-fdc2-11ea-0603-8b191ff11d0e # ╟─e41e382c-fdcc-11ea-3d1c-17127b462f9e # ╟─4c6f5762-fdcd-11ea-364e-d70107e4bc0b # ╟─421e0dbc-fdcd-11ea-0ef8-83890a2328e5 # ╟─019217d6-fdcd-11ea-1e36-b5cbc03cf1d5 # ╟─1d86c41c-fdcd-11ea-002b-69c19dd80a3f # ╠═fa05c5ff-0163-46e8-a755-fb962d46bb81 # ╠═dda79521-46b0-4a52-bd32-63d96bfe7459 # ╠═0105937c-f57c-4429-a32d-c226c815ec79 # ╠═001a77d5-f56a-4f34-83d2-b022c6966a41 # ╟─bc5852b4-f78f-45b7-b312-b2ea6b1ba809 # ╠═0df14d8c-11aa-4d52-82ff-f359e1fdc1b0 # ╠═7f4aa4d0-f2b2-4ee6-92ae-7541a17b367b # ╠═935fddb6-7931-4805-becd-4e6049b6e9c6 # ╠═133108da-045e-41f8-998b-9772ead84b32 # ╟─3a9bc594-fe26-11ea-3767-5bdf350fb6fb # ╟─b16e53c8-fe26-11ea-206b-4b8acae80e60 # ╟─e5ee9a5a-fe1e-11ea-11e9-95bc915a0ee6 # ╟─110a5bd4-fe1f-11ea-262e-730e1ff90e74 # ╠═ec4f8f1c-fe1e-11ea-2dfa-05f83ed0f00c # ╠═29b54ba8-fe1f-11ea-2dd4-2baa0503defd # ╟─bbf682e6-fe1f-11ea-1f7d-4f55724ebd9b # ╟─a0d257ee-fe1f-11ea-16d9-a7ecc379d56e # ╟─a65f1cf6-fe1f-11ea-0d70-1f2f1b0ce2dd # ╟─92d46126-fe21-11ea-3b1d-0be2fce36f72 # ╠═b676d206-fe37-11ea-238d-a9efe71dad02 # ╟─bcc2b6b8-fe26-11ea-16d8-b9391dc06fe6 # ╠═c29c6fae-fe1f-11ea-2524-4b9c4588b70d # ╠═c5190434-a974-4124-8765-65574de60494 # ╟─1b8f276c-fe3b-11ea-375d-574e523dbc3b # ╠═633f693a-fe1f-11ea-1d89-1716c918cfb7 # ╟─87f1fc20-fe3d-11ea-0b45-f9825caf6913 # ╟─e3c75cb8-fe3b-11ea-2970-89b816ef738e # ╟─2650306e-fe3c-11ea-00cf-5d7e15b2b8e9 # ╠═180b696a-fe3c-11ea-18fe-37423a9b5827 # ╠═2a064810-fe3c-11ea-141f-f71f37013d7a # ╟─4a125950-fe3c-11ea-3ae3-5546c0d4e11d # ╠═292a5800-fe3c-11ea-3af5-f70d6dbc14a1 # ╠═4fde7e68-fe3c-11ea-0c84-01b599a5fb56 # ╟─56266042-fe3c-11ea-23ca-d3f9240211c5 # ╟─b436be80-fe3b-11ea-25ef-c183d6181a2c # ╠═1ec4868e-fe22-11ea-295b-6929947c5b91 # ╟─acf9c58a-fe3c-11ea-2ad9-fd57da314a66 # ╟─27808bc2-fe3d-11ea-12fa-f35dd36e17a7 # ╟─00000000-0000-0000-0000-000000000001 # ╟─00000000-0000-0000-0000-000000000002