### A Pluto.jl notebook ### # v0.16.3 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 # ╔═╡ fa46ec78-17af-11eb-36de-4318924fc007 using PlutoUI # ╔═╡ fcf6a148-17af-11eb-18dd-156fef488646 using Plots # ╔═╡ ff00f51a-17af-11eb-1292-677833269e20 using LinearAlgebra # ╔═╡ c57bd6d4-1809-11eb-2125-535e3ef2acf6 using QuadGK # package to do numerical integration # ╔═╡ d3aebe38-17af-11eb-2f81-332a33974a71 md"# Notebook 10 -- Math 2121, Fall 2021 In practice, (abstract) vector spaces that are not readily identified with subspaces of $\mathbb{R}^n$ often arise as *function spaces*, like the vector space of functions $\mathbb{R} \to \mathbb{R}$. Let's examine how we can think of *vectors*, *vector addition*, and *scalar multiplication* in such spaces." # ╔═╡ f227162c-1808-11eb-23f2-45f28382c51b md"Recall from a few weeks ago how we can visualize vector operations in $\mathbb{R}^2$." # ╔═╡ 8e844750-17b0-11eb-1075-c1d57b6b6cb3 md"First, create some vectors." # ╔═╡ 10a8e0fc-17b0-11eb-2a48-29cd82bffd61 u = rand(-10:10, 2, 1) # ╔═╡ 619dbea6-17b0-11eb-0cd2-a53415f07008 v = rand(-10:10, 2, 1) # ╔═╡ ad34ea6a-17b0-11eb-06c0-e576aa1f3941 begin a_slider = @bind a Slider(-10:0.1:10, default=1, show_value=true) b_slider = @bind b Slider(-10:0.1:10, default=1, show_value=true) md"""Now choose some scalars. a = $(a_slider) b =$(b_slider) """ end # ╔═╡ e689bebc-17b0-11eb-0c1d-814a992495f4 begin lims = (-20, 20) u1, u2 = u[1], u[2] v1, v2 = v[1], v[2] p1 = scatter([0], [0], xlims = lims, ylims = lims, legend = false, label = "origin", aspect_ratio=:equal, title = "a•u and b•v") quiver!(quiver = ([a*u1],[a*u2]), [0], [0], color = [:blue],) quiver!(quiver = ([b*v1],[b*v2]), [0], [0], color = [:orange],) p2 = scatter([0], [0], xlims = lims, ylims = lims, legend = false, label = "origin", aspect_ratio=:equal, title = "a•u + b•v") quiver!( quiver = ( [a*u1 + b*v1], [a*u2 + b*v2] ), [0], [0], color = [:black], ) quiver!( quiver = ([a*u1],[a*u2]), [0], [0], color = [:blue], linestyle = :dashdot, ) quiver!( quiver = ([b*v1],[b*v2]), [a*u1], [a*u2], color = [:orange], linestyle = :dashdot, ) plot(p1, p2, layout=2) end # ╔═╡ 5113b4e2-1809-11eb-0004-d3643fbc3b50 md"Visualizing the vector space operations for $\mathsf{Fun}(\mathbb{R},\mathbb{R}) = \{\text{functions }f:\mathbb{R} \to \mathbb{R}\}$ is not so different." # ╔═╡ 9c13a860-17b1-11eb-3414-b1f003ab07e8 function plot_functions(f, g) plot(f, label="f", title="functions f and g") plot!(g, label="g") end # ╔═╡ 116aa720-17bd-11eb-1042-c5cc7806b979 function plot_combinations(f, g, c1, c2) # plot scalar multiples of two functions on the left, # plot their linear combination on the right p3 = plot( x -> c1 * f(x), ylims = (-5, 5), label="$(c1)•f", title="functions$(c1)•f and $(c2)•g") plot!( x -> c2 * g(x), label="$(c2)•g") p4 = plot( x -> c1 * f(x) + c2 * g(x), ylims = (-5, 5), legend=false, title="$(c1)•f +$(c2)•g", color=:black) plot(p3, p4, layout=2) end # ╔═╡ af27fd2c-1809-11eb-229b-6f80e7c887c0 md"Create some functions $\mathbb{R}\to\mathbb{R}$." # ╔═╡ 3785e62e-17b1-11eb-212b-7111d6649452 f(x) = cos(x) # ╔═╡ afd6ad20-17b1-11eb-14b4-03ffbe732afd g(x) = sinc(x) # ╔═╡ dddf29ec-1809-11eb-38b8-635e101b761b md"The *sinc* function is defined by $\mathrm{sinc}(x) = \frac{\sin(\pi x)}{\pi x}$. You may remember from calculus that $\lim_{x\to 0} \frac{\sin(x)}{x} = 1$. The sinc function is therefore defined at all $x \in \mathbb{R}$ and has $\mathrm{sinc}(0) = 1$." # ╔═╡ 05a0b948-17bd-11eb-3801-b3b2c4a5e807 plot_functions(f, g) # ╔═╡ 45a17b14-17b2-11eb-37b5-4fbde4394139 begin aa_slider = @bind aa Slider(-10:0.1:10, default=1, show_value=true) bb_slider = @bind bb Slider(-10:0.1:10, default=1, show_value=true) md"""Choose some scalars in $$\mathbb{R}$$: a = $(aa_slider) b =$(bb_slider) """ end # ╔═╡ 564d242e-17b2-11eb-1daa-d988a077f421 plot_combinations(f, g, aa, bb) # ╔═╡ 91ca8f28-180a-11eb-05c0-3da484bd3ea7 md"The function space $\mathsf{Fun}(\mathbb{R},\mathbb{R}) = \{\text{functions }f:\mathbb{R} \to \mathbb{R}\}$ is really big, too big for many natural operations like differentiation and integration to make sense for all elements. The space $\mathsf{Fun}(\mathbb{R},\mathbb{R})$ contains many natural subspaces that may be more suitable to applications: * The space $L^1(\mathbb{R}) = \{ f \in \mathsf{Fun}(\mathbb{R},\mathbb{R}) : \int_{-\infty}^\infty |f(x)| dx < \infty\}$ of *integrable functions*. * The space $L^2(\mathbb{R}) = \{ f \in \mathsf{Fun}(\mathbb{R},\mathbb{R}) : f^2 \in L^1(\mathbb{R})\}$ of *square-integrable functions*. * The space $L^\infty(\mathbb{R}) = \{ f \in \mathsf{Fun}(\mathbb{R},\mathbb{R}) : \exists N\text{ with }|f(x)| quadgk(y -> f(y) * cos(2 * pi * x * y), a, b, rtol=tol)[1] end # ╔═╡ 2e6dd04c-17b6-11eb-3c56-3145c312987b function plot_fourier_transform(fn, tol=1e-3; a=-100, b=100) plt1 = plot( x -> fn(x), legend=false, title="function") plt2 = plot( x -> FT(fn, tol; a=a, b=b)(x), legend=false, title="Fourier transform") plot(plt1, plt2, layout=2) end # ╔═╡ 22cc50d6-180d-11eb-1217-e33548bb2e55 md"Our definition of the Fourer transform does not make sense for$\cos(x)$, since this function is not square-integrable:$\int_{-\infty}^\infty |\cos(x)|^2 dx = \infty$. But we can apply$\mathsf{FT}$to things like the truncated cosine function$\cos(x) 1_{[-2\pi,2\pi]}(x)$where$1_{[a,b]}(x)$is the function that takes values$1$for$a\leq b\leq x$and$0$for all other$x$." # ╔═╡ 95590622-17d0-11eb-36b7-a7013327ea5f plot_fourier_transform(x -> abs(x) < 2 * pi ? cos(x) : 0.0) # ╔═╡ 8ca56aba-180d-11eb-170d-23352fdc3b21 md"The result is the sum of two 'pulses' at$x=\pm \frac{1}{2\pi}$." # ╔═╡ 0f4c7886-180f-11eb-1ec6-9d635fc1dea2 # ╔═╡ e8f5c12e-180e-11eb-3e22-85d4d140461c md"What about the Fourier transform of$\cos(x^2) 1_{[-4\pi^2,4\pi^2]}(x)$?" # ╔═╡ 85daa168-180e-11eb-2d6b-bd507896f0e3 plot_fourier_transform(x -> abs(x) < 4 * pi^2 ? cos(x^2) : 0.0) # ╔═╡ 150576ec-180f-11eb-0354-dbd84048269e md"Ignoring error, the Fourier transform of the above function is exactly$\sqrt{\pi} \cos(\pi^2 x^2 - \pi/4)$as we can confirm by comparing the plots below:" # ╔═╡ af1bc278-180e-11eb-3e7d-d7b145cd318c begin pl1 = plot(title="Fourier transform plot", FT(x -> abs(x) < 4 * pi^2 ? cos(x^2) : 0.0), ylims=(-5,5), legend=false) pl2 = plot(title="Exact formula plot", x -> sqrt(pi) * cos(pi^2 * x^2 - pi/4), ylims=(-5,5), legend=false) plot(pl1, pl2, layout=2) end # ╔═╡ 6a251658-180f-11eb-3a5e-6f962c2ce71b md"An interesting example comes$\mathrm{sinc}(x)$, whose Fourier transform looks exactly like$1_{[-\frac{1}{2},\frac{1}{2}]}(x)$." # ╔═╡ bbbc29a8-17c0-11eb-094e-9bc91a6f0a35 plot_fourier_transform(x -> sinc(x)) # ╔═╡ b3be8544-180f-11eb-35b1-696e52fafb83 md"The Fourier transform of$1_{[-\frac{1}{2},\frac{1}{2}]}(x)$, in turn, looks just like$\mathrm{sinc}(x)$." # ╔═╡ 1b9c8106-17c1-11eb-0f57-2d05107850c3 plot_fourier_transform(x -> (abs(x) < 0.5 ? 1.0 : 0)) # ╔═╡ c93a6c6a-180f-11eb-2bb8-89dc98712240 md"We can use these examples to construct *eigenvectors* of$\mathsf{FT}$. If$T : V \to V$is a linear transformation from a vector space to itself, then an *eigenvector* is a nonzero element$v \in V$such that$T(v) = \lambda v$for some scalar$\lambda \in \mathbb{R}$. In this case$\lambda$is the corresponding *eigenvalue*." # ╔═╡ 892df760-1811-11eb-0d87-ed458ddc6115 # ╔═╡ 8a63824c-1811-11eb-27bc-ad1f40a75ce7 md"Now, if$\mathsf{FT}(f) = g$and$\mathsf{FT}(g) = f$where$f\neq g$, then by linearity$\mathsf{FT}(f+g) = \mathsf{FT}(f) + \mathsf{FT}(g) = g +f = f+g$so$f +g $is an eigenvector with eigenvalue$1$. Similarly,$\mathsf{FT}(f-g) = \mathsf{FT}(f) - \mathsf{FT}(g) = g -f = -(f-g)$so$f-g$is an eigenvector with eigenvalue$-1$. We can apply this to$f(x) = \mathrm{sinc}(x)$and$g(x) = 1_{[-\frac{1}{2},\frac{1}{2}]}(x)$." # ╔═╡ 6a104b4c-1807-11eb-0d90-e59edf914c21 plot_fourier_transform(x -> sinc(x) + (abs(x) < 0.5 ? 1.0 : 0)) # ╔═╡ 7ea997fc-1807-11eb-381d-397358972310 plot_fourier_transform(x -> sinc(x) - (abs(x) < 0.5 ? 1.0 : 0)) # ╔═╡ eb3d47ec-1811-11eb-0446-d75261e5f002 md"This shows that$\lambda=-1$and$\lambda=1$are both eigenvalues of$\mathsf{FT}$. Here are some other interesting eigenvectors of$\mathsf{FT}$for these eigenvalues:" # ╔═╡ 2bfe4e5c-1812-11eb-23a1-0163b06f87ff plot_fourier_transform(x -> exp(-pi * x^2)) # ╔═╡ 57e339a6-1812-11eb-25d8-1388cdb4d35c plot_fourier_transform(x -> exp(-pi * x^2) * (8 * pi * x^2 - 2)) # ╔═╡ 61fe4dc2-1812-11eb-1ab5-73e4d56ae352 md"Are there are any other eigenvalues$\lambda$for$\mathsf{FT}$besides$\lambda=1$and$\lambda=-1$? Yes,$\lambda=0$:" # ╔═╡ 77f62500-1812-11eb-1303-b744fd7f9a91 plot_fourier_transform(x -> sin(x) / exp(x^2)) # ╔═╡ 7c99f92e-1812-11eb-3427-1f21f30eda51 md"The numbers$-1, 0, 1$are the only eigenvalues of$\mathsf{FT}$. Here's why. A function$f: \mathbb{R} \to \mathbb{R}$is *even* if$f(-x) = f(x)$for all$x$and *odd* if$f(-x) = -f(x)$for all$x$. If$f \in L^2(\mathbb{R})$is odd then$\mathsf{FT}(f) = 0$, as in the example above. If$f \in L^2(\mathbb{R})$is even then$\mathsf{FT}(\mathsf{FT}(f)) = f$, as we saw with$f(x) = \mathrm{sinc}(x)$. Every$f \in L^2(\mathbb{R})$can be written as$f=f_{\mathrm{even}} + f_{\mathrm{odd}}$where$f_{\mathrm{even}}(x) = \frac{f(x) + f(-x)}{2}\qquad\text{and}\qquad f_{\mathrm{odd}}(x) = \frac{f(x) - f(-x)}{2}.$Also$\mathsf{FT}(f_{\mathrm{even}}) = \mathsf{FT}(f)_{\mathrm{even}}$. This means$\mathsf{FT}(f) = \mathsf{FT}(f_{\mathrm{even}}) = \mathsf{FT}(f)_{\mathrm{even}}$. So the only way we can have$\mathsf{FT}(f) = \lambda f$is if * the function$f$is odd and$\lambda=0$, or * the function$f$is even and$\lambda^2=1$, since then$\lambda^2 f = \lambda \mathsf{FT}(f) = \mathsf{FT}(\lambda f) = \mathsf{FT}(\mathsf{FT}(f)) = f.$" # ╔═╡ 52166baa-1822-11eb-29fe-410c310997eb # ╔═╡ ddc7b344-1821-11eb-0659-016e0381dbe5 md"Because the Fourier transform is a linear transformation$L^2(\mathbb{R}) \to L^2(\mathbb{R})$, which is infinite dimensional, there is no corresponding *standard matrix* as there would be for a linear transformation$\mathbb{R}^n \to \mathbb{R}^m$. As such, although we can figure out the eigenvalues of$\mathsf{FT}$, it is much less straightforward to find (a basis for) the eigenvectors: we cannot do this just by finding the null space of a certain matrix. Solving this probem is something covered in a more advanced course on Fourier/harmonic analysis." # ╔═╡ 56d4a494-1814-11eb-20a0-8dbb921ec932 plot_fourier_transform(x -> cos(2 * pi * x) * sinc(x), 1e-4) # ╔═╡ 5be9c976-1814-11eb-1660-d31bd88e1f9b plot_fourier_transform(x -> sinc(x)^2) # ╔═╡ a83c60fc-1813-11eb-2352-bdea9bac2ab0 md"Very briefly, why is the Fourier transform is an important/useful linear function: * The Fourier transform converts waves to bits. More precisely,$\mathsf{FT}$converts (decaying) sinusoidal signals to discrete signals. For this reason,$\mathsf{FT}$is the foundation of *signal processing.* * Not so much emphasized in our discussion, but$\mathsf{FT}$is also useful in algebra because it transforms differential equations to polynomial equations, which are easier to solve. * Finally, and signficantly: there are efficient algorithms to compute$\mathsf{FT}\$." # ╔═╡ 00000000-0000-0000-0000-000000000001 PLUTO_PROJECT_TOML_CONTENTS = """ [deps] LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e" Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" PlutoUI = "7f904dfe-b85e-4ff6-b463-dae2292396a8" QuadGK = "1fd47b50-473d-5c70-9696-f719f8f3bcdc" [compat] Plots = "~1.22.6" PlutoUI = "~0.7.16" QuadGK = "~2.4.2" """ # ╔═╡ 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 = "3.3.1" 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"cafe0823979a5c9bff86224b3b8de29ea5a44b2e" uuid = "d2c73de3-f751-5644-a686-071e5b155ba9" version = "0.61.0+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 = "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" [[Hyperscript]] deps = ["Test"] git-tree-sha1 = "8d511d5b81240fc8e6802386302675bdf47737b9" uuid = "47d2ed2b-36de-50cf-bf87-49c2cf4b8b91" version = "0.0.4" [[HypertextLiteral]] git-tree-sha1 = "f6532909bf3d40b308a0f360b6a0e626c0e263a8" uuid = "ac1192a8-f4b3-4bfe-ba22-af5b92cd3ab2" version = "0.9.1" [[IOCapture]] deps = ["Logging", "Random"] git-tree-sha1 = "f7be53659ab06ddc986428d3a9dcc95f6fa6705a" uuid = "b5f81e59-6552-4d32-b1f0-c071b021bf89" version = "0.2.2" [[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 = "c7f1c695e06c01b95a67f0cd1d34994f3e7db104" uuid = "b964fa9f-0449-5b57-a5c2-d3ea65f4040f" version = "1.2.1" [[Latexify]] deps = ["Formatting", "InteractiveUtils", "LaTeXStrings", "MacroTools", "Markdown", "Printf", "Requires"] git-tree-sha1 = "669315d963863322302137c4591ffce3cb5b8e68" uuid = "23fbe1c1-3f47-55db-b15f-69d7ec21a316" version = "0.15.8" [[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 = "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 = "f19e978f81eca5fd7620650d7dbea58f825802ee" uuid = "69de0a69-1ddd-5017-9359-2bf0b02dc9f0" version = "2.1.0" [[Pixman_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] git-tree-sha1 = "b4f5d02549a10e20780a24fce72bea96b6329e29" uuid = "30392449-352a-5448-841d-b1acce4e97dc" version = "0.40.1+0" [[Pkg]] deps = ["Artifacts", "Dates", "Downloads", "LibGit2", "Libdl", "Logging", "Markdown", "Printf", "REPL", "Random", "SHA", "Serialization", "TOML", "Tar", "UUIDs", "p7zip_jll"] uuid = "44cfe95a-1eb2-52ea-b672-e2afdf69b78f" [[PlotThemes]] deps = ["PlotUtils", "Requires", "Statistics"] git-tree-sha1 = "a3a964ce9dc7898193536002a6dd892b1b5a6f1d" uuid = "ccf2f8ad-2431-5c83-bf29-c5338b663b6a" version = "2.0.1" [[PlotUtils]] deps = ["ColorSchemes", "Colors", "Dates", "Printf", "Random", "Reexport", "Statistics"] git-tree-sha1 = "b084324b4af5a438cd63619fd006614b3b20b87b" uuid = "995b91a9-d308-5afd-9ec6-746e21dbc043" version = "1.0.15" [[Plots]] deps = ["Base64", "Contour", "Dates", "Downloads", "FFMPEG", "FixedPointNumbers", "GR", "GeometryBasics", "JSON", "Latexify", "LinearAlgebra", "Measures", "NaNMath", "PlotThemes", "PlotUtils", "Printf", "REPL", "Random", "RecipesBase", "RecipesPipeline", "Reexport", "Requires", "Scratch", "Showoff", "SparseArrays", "Statistics", "StatsBase", "UUIDs"] git-tree-sha1 = "ba43b248a1f04a9667ca4a9f782321d9211aa68e" uuid = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" version = "1.22.6" [[PlutoUI]] deps = ["Base64", "Dates", "Hyperscript", "HypertextLiteral", "IOCapture", "InteractiveUtils", "JSON", "Logging", "Markdown", "Random", "Reexport", "UUIDs"] git-tree-sha1 = "4c8a7d080daca18545c56f1cac28710c362478f3" uuid = "7f904dfe-b85e-4ff6-b463-dae2292396a8" version = "0.7.16" [[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" [[QuadGK]] deps = ["DataStructures", "LinearAlgebra"] git-tree-sha1 = "78aadffb3efd2155af139781b8a8df1ef279ea39" uuid = "1fd47b50-473d-5c70-9696-f719f8f3bcdc" version = "2.4.2" [[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" [[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: # ╟─d3aebe38-17af-11eb-2f81-332a33974a71 # ╠═fa46ec78-17af-11eb-36de-4318924fc007 # ╠═fcf6a148-17af-11eb-18dd-156fef488646 # ╠═ff00f51a-17af-11eb-1292-677833269e20 # ╟─f227162c-1808-11eb-23f2-45f28382c51b # ╟─8e844750-17b0-11eb-1075-c1d57b6b6cb3 # ╠═10a8e0fc-17b0-11eb-2a48-29cd82bffd61 # ╠═619dbea6-17b0-11eb-0cd2-a53415f07008 # ╟─ad34ea6a-17b0-11eb-06c0-e576aa1f3941 # ╟─e689bebc-17b0-11eb-0c1d-814a992495f4 # ╟─5113b4e2-1809-11eb-0004-d3643fbc3b50 # ╠═9c13a860-17b1-11eb-3414-b1f003ab07e8 # ╠═116aa720-17bd-11eb-1042-c5cc7806b979 # ╟─af27fd2c-1809-11eb-229b-6f80e7c887c0 # ╠═3785e62e-17b1-11eb-212b-7111d6649452 # ╠═afd6ad20-17b1-11eb-14b4-03ffbe732afd # ╟─dddf29ec-1809-11eb-38b8-635e101b761b # ╠═05a0b948-17bd-11eb-3801-b3b2c4a5e807 # ╟─45a17b14-17b2-11eb-37b5-4fbde4394139 # ╠═564d242e-17b2-11eb-1daa-d988a077f421 # ╟─91ca8f28-180a-11eb-05c0-3da484bd3ea7 # ╠═c57bd6d4-1809-11eb-2125-535e3ef2acf6 # ╟─98f0e97c-180b-11eb-301a-3f007d589bbd # ╠═14d5d0c6-17b6-11eb-0857-c7899b566855 # ╠═2e6dd04c-17b6-11eb-3c56-3145c312987b # ╟─22cc50d6-180d-11eb-1217-e33548bb2e55 # ╠═95590622-17d0-11eb-36b7-a7013327ea5f # ╟─8ca56aba-180d-11eb-170d-23352fdc3b21 # ╟─0f4c7886-180f-11eb-1ec6-9d635fc1dea2 # ╟─e8f5c12e-180e-11eb-3e22-85d4d140461c # ╠═85daa168-180e-11eb-2d6b-bd507896f0e3 # ╠═150576ec-180f-11eb-0354-dbd84048269e # ╠═af1bc278-180e-11eb-3e7d-d7b145cd318c # ╟─6a251658-180f-11eb-3a5e-6f962c2ce71b # ╠═bbbc29a8-17c0-11eb-094e-9bc91a6f0a35 # ╟─b3be8544-180f-11eb-35b1-696e52fafb83 # ╠═1b9c8106-17c1-11eb-0f57-2d05107850c3 # ╟─c93a6c6a-180f-11eb-2bb8-89dc98712240 # ╟─892df760-1811-11eb-0d87-ed458ddc6115 # ╟─8a63824c-1811-11eb-27bc-ad1f40a75ce7 # ╠═6a104b4c-1807-11eb-0d90-e59edf914c21 # ╠═7ea997fc-1807-11eb-381d-397358972310 # ╟─eb3d47ec-1811-11eb-0446-d75261e5f002 # ╠═2bfe4e5c-1812-11eb-23a1-0163b06f87ff # ╠═57e339a6-1812-11eb-25d8-1388cdb4d35c # ╟─61fe4dc2-1812-11eb-1ab5-73e4d56ae352 # ╠═77f62500-1812-11eb-1303-b744fd7f9a91 # ╟─7c99f92e-1812-11eb-3427-1f21f30eda51 # ╟─52166baa-1822-11eb-29fe-410c310997eb # ╟─ddc7b344-1821-11eb-0659-016e0381dbe5 # ╠═56d4a494-1814-11eb-20a0-8dbb921ec932 # ╠═5be9c976-1814-11eb-1660-d31bd88e1f9b # ╟─a83c60fc-1813-11eb-2352-bdea9bac2ab0 # ╟─00000000-0000-0000-0000-000000000001 # ╟─00000000-0000-0000-0000-000000000002