### A Pluto.jl notebook ### # v0.16.4 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 # ╔═╡ f2118adc-1ebb-11eb-080a-0b61e26ca1a0 using Plots # ╔═╡ 76d253ae-1ec0-11eb-1d04-cbbf959c3984 using PlutoUI # ╔═╡ b8779288-1ec2-11eb-2381-7db4b2380595 using LinearAlgebra # ╔═╡ 624a8396-1f1e-11eb-3ddf-aff41274881c md"# Notebook 12 -- Math 2121, Fall 2021 In this notebook we will explore how to draw complex numbers and then discuss a visual proof of the fundamental theorem of algebra. The way we can visualize complex numbers is very similar to how we draw vectors in $\mathbb{R}^2$." # ╔═╡ 9b354e40-1ecc-11eb-1fbd-c9d33504aaaa md"In lecture, we *defined* the complex number $a+bi$ to be the $2\times 2$ matrix $\left[\begin{array}{rr} a & -b \\ b & a\end{array}\right].$ A very useful way of visualizing the number $a+bi$ is by drawing the vector $\left[\begin{array}{c} a \\ b \end{array}\right] \in \mathbb{R}^2$ which is the first column of the matrix above. This first column determines the second column, so there is no loss of information." # ╔═╡ f88a7884-1ecc-11eb-25f3-27782e93a793 function plot_vector(v, title="") xlim = (-4, 4) ylim = (-4, 4) scatter([0], [0], legend=false; aspect_ratio=:equal, title=title) quiver!(quiver = ([v[1]],[v[2]]), [0], [0], xlimit=xlim, ylimit=ylim, color=:blue) end # ╔═╡ 8fd1de9a-1ece-11eb-1c83-0d036a825aad function plot_sum(u, v, title="") xlim = (-4, 4) ylim = (-4, 4) scatter([0], [0], legend=false; aspect_ratio=:equal, title=title) quiver!( quiver = ([u[1]],[u[2]]), [0], [0], xlimit=xlim, ylimit=ylim, linestyle = :dashdot, color = :grey, opacity = 0.5) quiver!( quiver = ([v[1]],[v[2]]), [u[1]], [u[2]], xlimit=xlim, ylimit=ylim, linestyle = :dashdot, color = :grey, opacity = 0.5) quiver!(quiver = ([u[1]+ v[1]],[u[2] + v[2]]), [0], [0], color=:blue) end # ╔═╡ 9c770878-1ed3-11eb-1838-2d7e3d1aad90 function complex_product(u, v) return [u[1] * v[1] - u[2] * v[2], u[1] * v[2] + u[2] * v[1]] end # ╔═╡ 625e8dac-1ecf-11eb-1246-7b2b9657f8a2 function plot_multiple(u, v, title="") q = norm(u) * norm(v) + 1 xlim = (-q, q) ylim = (-q, q) scatter([0], [0], legend=false; aspect_ratio=:equal, title=title) quiver!( quiver = ([u[1]],[u[2]]), [0], [0], xlimit=xlim, ylimit=ylim, linestyle = :dashdot, color = :grey, opacity = 0.5) quiver!( quiver = ([v[1]],[v[2]]), [0], [0], xlimit=xlim, ylimit=ylim, linestyle = :dashdot, color = :grey, opacity = 0.5) uv = complex_product(u, v) quiver!(quiver = ([uv[1] ],[uv[2]]), [0], [0], color=:blue) end # ╔═╡ 0c3f4810-1ece-11eb-2283-c556fcd8030e md"We draw such vectors as arrows in the $xy$-plane in the usual way." # ╔═╡ 825b9e6e-1ecd-11eb-0c7d-374c815d571c u = [-2;2] # ╔═╡ 17f95c72-1ece-11eb-1697-9dbc93d4d92f md"This represents the complex number $u$ = $(u[1]) +$(u[2])*i*." # ╔═╡ 865778e4-1ecd-11eb-1396-b18dbe8ddb2b v = [4; 2] # ╔═╡ 304a50f6-1ece-11eb-2b75-45047654a0da md"This represents the complex number $v$ = $(v[1]) +$(v[2])*i*." # ╔═╡ 36973140-1ece-11eb-02b3-d70eeedd08a7 md"The sum of these complex numbers is $u + v$ = $(u[1] + v[1]) +$(u[2] + v[2])*i*. In terms of arrows, the sum $u + v$ corresponds to translating the start of $v$ to the end of $u$ and the following the origin to the endpoint of the translated copy of $v$. " # ╔═╡ 0c4b7f14-1ecd-11eb-0186-f5af4fffe54e plot(plot_vector(u, "u"), plot_vector(v, "v"), plot_sum(u, v, "u + v"), layout=(1,3)) # ╔═╡ 0a4640a2-1ed0-11eb-22e1-cdcc2964b14e md"There is also a simple of way of visualizing the product of two complex numbers. The **length** or **norm** of $z = a + bi$ is $|z| = \sqrt{a^2 + b^2}$. This is also how we define the **length** of the vector $\left[\begin{array}{r} a \\ b\end{array}\right]$. The **angle** of the vector $\left[\begin{array}{r} a \\ b\end{array}\right]$ is the angle that the vector makes with the positive $x$-axis. The **angle** of $z = a + bi$ is defined to be the angle of $\left[\begin{array}{r} a \\ b\end{array}\right]$ The complex number $z$ is uniquely determined by its length and angle: * If these are $r$ and $\theta$, respectively, then $z = r \cos \theta + i r \sin \theta$. Here is a method to compute the angle of $z$: " # ╔═╡ 3d468af6-1ec5-11eb-2169-59305780810d function angle(z) # returns angle between 0.0 and 2 * pi u1, u2 = z[1], z[2] if u1 == 0 return u2 > 0 ? pi/2 : 3 * pi/2 end if u2 == 0 return u1 > 0 ? 0 : pi end a = atan(abs(u2) / abs(u1)) if u1 > 0 && u2 > 0 return a elseif u1 > 0 && u1 > 0 return 2 * pi - a elseif u1 < 0 && u2 > 0 return pi - a else return pi + a end end # ╔═╡ 0620fd88-1ed4-11eb-06f6-07ff4fbc570a md"Here is how to interpret the product of two complex numbers $y$ and $z$ in terms of arrows: * The vector in $\mathbb{R}^2$ representing $yz$ is the vector whose angle is the **sum** of the angles of $y$ and $z$, and whose length is the **product** of the lengths of $y$ and $z$." # ╔═╡ e17dd5a0-1f1e-11eb-29a7-9fb15de2f63a md"Recall that u = $(u[1]) +$(u[2])*i* and v = $(v[1]) +$(v[2])*i*." # ╔═╡ 45b83704-1f1f-11eb-0692-dff71b862998 norm(u), angle(u) * 180 / pi # ╔═╡ 49cdc0fa-1f1f-11eb-0257-8bd931a28896 norm(v), angle(v) * 180 / pi # ╔═╡ 3fae670c-1f1f-11eb-2703-5f51d71908ce uv = complex_product(u, v) # ╔═╡ 5d14cf2a-1f1f-11eb-1534-6dce533e7d47 [norm(uv) angle(uv) / pi * 180; norm(u) * norm(v) (angle(u) + angle(v)) * 180 /pi] # ╔═╡ 99138700-1f1f-11eb-33ae-039fd141ae38 plot_multiple(u, v, "uv = ($(u[1]) +$(u[2])i)($(v[1]) +$(v[2])i)") # ╔═╡ ab3bc9b0-1f1f-11eb-2990-f7195fa1101b md"Some more examples of products of complex numbers:" # ╔═╡ ddcbaf3a-1ecf-11eb-31fd-b3384790a096 y = [0.707 0.707] * 2 # ╔═╡ e44cde2e-1ecf-11eb-0099-33b6f29cbed1 z = [-0.707 0.707] / 2 # ╔═╡ c871f996-1f1f-11eb-320b-27b5e28f10a6 md"These vectors represent y = $(y[1]) +$(y[2])*i* and z = $(z[1]) +$(z[2])*i*" # ╔═╡ 0edd2c54-1ed1-11eb-137a-798796c1bfc4 md"These numbers have length $2$ and $1/2$, so the length of their product is $1$. The angle of $y$ is $\pi/4$ and the angle of $z$ is $3\pi/4$, so the angle of $yz$ is $\pi/4 + 3\pi/4 = \pi$ radians." # ╔═╡ dae46768-1ed3-11eb-17f7-97158c4168f8 norm(y), norm(z), norm(y) * norm(z), norm(complex_product(y, z)) # ╔═╡ 71065f0e-1ed3-11eb-19ed-e3123cd9ee34 angle(y), angle(z), angle(y) + angle(z), angle(complex_product(y, z)) # ╔═╡ 59ec8b1c-1ecf-11eb-25ae-eb3febbd005e plot_multiple(y, z, "yz = ($(y[1]) +$(y[2])i)($(z[1]) +$(z[2])i) = -1") # ╔═╡ cd56145e-1ed1-11eb-31cb-a5b58bbd444b plot_multiple(y, y, "y^2 = ($(y[1]) +$(y[2])i)^2") # ╔═╡ 939ed0c0-1ed6-11eb-3dc6-531ca37f777e plot_multiple(z, z, "z^2 = ($(z[1]) +$(z[2])i)^2") # ╔═╡ 5d249e30-1ed1-11eb-1b13-f7863ba96193 md"Multiplying by $i$ corresponds to rotating counterclockwise by $\pi/2$ radians." # ╔═╡ 6bd6b3ac-1ed1-11eb-3e42-9bd16cc4cc68 plot_multiple([0, 1], [3, 4], "i(3 + 4i) = -4 + 3i") # ╔═╡ 2b0cd1be-1ed2-11eb-3b36-f50d87bbdea0 md"These pictures give us some visual intuition for adding, multiplying, and taking powers $z \mapsto z^n$ of complex numbers. The last operation corresponds to multiplying the angle of the arrow representing $z$ by $n$ and exponentiating the length. Using these abilities together lets us imagine the output of a *polynomial function* $f(x) = z_0 + z_1 x + z_2 x^2 + \dots + z_n x^n$ where $x$ is a variable and $z_0,z_1,\dots,z_n \in \mathbb{C}$. We can encode such a function in Julia as a $2\times (n+1)$ real matrix, whose columns record the coefficients $z_0, z_1,\dots,z_n$. Here are some methods to create polynomials encoded like this:" # ╔═╡ 08e519f6-1ebc-11eb-05ee-c58d68673a3c function random_real_polynomial(degree) return rand(-10:10, 1, degree + 1) end # ╔═╡ 3b32a9ee-1ebc-11eb-3cae-63d5e57edd52 function random_complex_polynomial(degree) return rand(-10:10, 2, degree + 1) end # ╔═╡ 7beb1758-1ebe-11eb-3a4d-e1937b4bd74b function monomial(degree) ans = zeros(1, degree + 1) ans[1, end] = 1 return ans end # ╔═╡ 2bd19ea8-1ed3-11eb-31b6-5f8937810389 md"Here is a very simple method to print a 2-row matrix as a polynomial in the usual way." # ╔═╡ 51b018aa-1ebc-11eb-38e9-6b85dbdd4aa7 function print_polynomial(f) m, n = size(f) s = "" for i = 1:n if f[1, i] != 0.0 || (m == 2 && f[2, i] != 0.0) coeff = m == 1 ? f[i] : string(f[1, i], " + ", f[2, i], "i") var = i == 1 ? "" : i == 2 ? " * x" : string(" * x^", i - 1) s *= string(i == 1 ? "" : " + ", "(", coeff, ")", var) end end if s[1] == ' ' s = s[4:end] end return s end # ╔═╡ 448a300e-1ed3-11eb-16ef-c7bb6f9c611c f = random_real_polynomial(5) # ╔═╡ 3adfa720-1ed4-11eb-0ee8-99436bfc313f Text(print_polynomial(f)) # ╔═╡ 3fdde72a-1ed4-11eb-1c22-77be0a9afbd4 g = random_complex_polynomial(4) # ╔═╡ 478141c0-1ed4-11eb-00c8-a374788df3ed Text(print_polynomial(g)) # ╔═╡ 4b80b3aa-1ed4-11eb-3847-d1561b2d0052 h = monomial(7) # ╔═╡ 4f50774a-1ed4-11eb-2652-419c7e831266 Text(print_polynomial(h)) # ╔═╡ abf1a0e8-1ed4-11eb-14de-0bc276e5a74b md"The following method evaluates our polynomial $f$ at a complex number $z = a + bi$, assuming that $f$ is inputted as a 1- or 2-row array and $z$ is inputted as the 2-row vector $\left[\begin{array}{c} a \\ b \end{array}\right]$. " # ╔═╡ 50363304-1ebe-11eb-2f9d-0b09757ba4f0 function evaluate_polynomial(f, z) ans = zeros(2, 2) # convert z back to 2-by-2 matrix z = [z[1] -z[2]; z[2] z[1]] m, n = size(f) degree = n - 1 for i=0:degree a = f[1, i + 1] b = m == 1 ? 0 : f[2, i + 1] # convert coefficient of f back to 2-by-2 matrix coefficient = [a -b; b a] # multiply together and add ans += coefficient * z^i end # return just first columne of 2-by-2 matrix return ans[:, 1] end # ╔═╡ 6b9345b0-1ed7-11eb-185e-7d58cfdfb766 # evaluates f(x) at x = 1 = 1 + 0i evaluate_polynomial(f, [1, 0]), sum(f) # ╔═╡ 87f6bc82-1ed7-11eb-1e84-df55e602a9b3 # evalutes f(x) at x = i = 0 + i evaluate_polynomial(f, [0, 1]), [f[1] - f[3] + f[5], f[2] - f[4] + f[6]] # ╔═╡ e32f61a4-1ed6-11eb-354f-17282dcb5fff md"To visualize a set of complex numbers $z=a+bi$, we just draw the endpoints of the vectors $\left[\begin{array}{c} a \\ b \end{array}\right]$. Below is a picture of the set $\{ z \in \mathbb{C} : |z| = 2 \}$, which forms a circle:" # ╔═╡ fd7b492a-1ebc-11eb-2df6-89f5b82300cb function circle_path(r, N=1000) path = zeros(2, N) for n = 0:N - 1 path[1, n + 1] = r * cos(2 * pi / N * n) path[2, n + 1] = r * sin(2 * pi / N * n) end return path end # ╔═╡ a2a6692a-1ebd-11eb-0adc-eb0a68681a98 function plot_path(path, title="") plot(path[1,:], path[2,:], aspect_ratio=:equal, title=title, legend=false) scatter!([0], [0]) end # ╔═╡ 21238f4e-1ed7-11eb-1edc-c7ea159b7d80 plot_path(circle_path(2), "{ z : |z| = 2 }") # ╔═╡ 55b249b2-1ed7-11eb-28db-d35a383166cb md"Below are the analogous pictures $\{ f(z) : |z| = 2\}$, $\{ g(z) : |z| = 2\}$, and $\{ h(z) : |z| = 2\}$." # ╔═╡ b7b2f7ba-1ebe-11eb-385c-8342199f5c7d function compute_path(f, domain) # evaluates polynomial f at all columns in domain _, N = size(domain) path = zeros(2, N) for n = 1:N path[:, n] = evaluate_polynomial(f, domain[:, n]) end return path end # ╔═╡ e819c960-1ed7-11eb-3830-bb1a74cc489d plot_path(compute_path(f, circle_path(2)), "{ f(z) : |z| = 2 }") # ╔═╡ f712ca5c-1ed7-11eb-397f-f98f0e2e137b plot_path(compute_path(g, circle_path(2)), "{ g(z) : |z| = 2 }") # ╔═╡ fc861c64-1ed7-11eb-3007-1d0b2472e928 plot_path(compute_path(h, circle_path(2)), "{ h(z) : |z| = 2 }") # ╔═╡ 0030678e-1ed8-11eb-04c8-231ce86cc088 md"The picture of $\{h(z) : |z| = 2\}$ is also a circle, but of radius $(2^(size(h)[2] - 1)). This makes sense because$h(x) = x^n$for$n$=$(size(h)[2] - 1). (Can you explain this using our visual interpretation of complex multiplication?)" # ╔═╡ 6c327e18-1ed8-11eb-3209-9b57a26050c0 md"## Fundamental theorem of algebra The fundamental theorem of algebra result says that if $n>0$ then any polynomial $p(x) = z_0 + z_1 x + z_2 x^2 + \dots + z_n x^n$ with $z_{n} \neq 0$ can be factored as $p(x) = z_n ( x - \alpha_1 )(x - \alpha_2)\cdots (x - \alpha_n)$ for some complex numbers $\alpha_1,\alpha_2,\dots,\alpha_n \in \mathbb{C}$ which do not need to be distinct. We can sketch a visual proof of this fact, extending the discussion above. " # ╔═╡ 6bee0f08-7b9a-411c-9713-1e8b784cdad4 # ╔═╡ cf9f64e2-1ed9-11eb-05fb-1dbac3e72537 md"__Some preliminaries__: **Fact.** If $p(0) = 0$ then $p(x) = x q(x)$ for some polynomial $q(x)$. *Proof:* This is obvious since $p(0) = z_0$. **Fact.** If $p(\alpha) = 0$ then $p(x) = (x - \alpha) q(x)$ for some polynomial $q(x)$. *Proof:* If $p(\alpha) = 0$ then the polynomial $f(x) = p(x + \alpha)$ has $f(0) = 0$, so $f(x) = x g(x)$. But then $p(x) = f(x - \alpha) = (x-\alpha) q(x)$ for the polynomial $q(x)= g(x-\alpha)$. **Conclusion.** To prove the fundamental theorem of algebra, it suffices to show that if $n > 0$ then $p(\alpha) = 0$ for *some* complex number $\alpha \in \mathbb{C}$. A proof of this last property is what we will outline below. " # ╔═╡ e7132e88-1ed9-11eb-1fcf-0bc4d97ab97f # ╔═╡ 7c34744a-1eda-11eb-15ae-ed6cedf51067 md"Our proof involves the notion of the **winding number** of a curve in $\mathbb{R}^2$. Consider a circle $\{ z \in \mathbb{R}^2 : |z| = r\}$. We order the points on this circle to start at the unique point the positive $x$-axis and travel counterclockwise. Passing the points on the circle in order as inputs to $p(x)$ traces a curve in $\mathbb{R}^2$. The **winding number** of this curve is the number of times the curves goes completely around the origin counter clockwise. " # ╔═╡ 6804a480-1ec2-11eb-3393-23be1df93255 function winding_number(p, r) # computes winding number of curve tracing { p(z) : |z| = r } # here is the simple idea to compute this: # # * take two successive points z_0, z_1 on the circle # * compute the small change in angles between these points # * add up these angle changes over the whole circle, then divide by 2 pi # # "adding" in this context is secretly some kind of integration domain = circle_path(r) path = compute_path(p, domain) _, N = size(path) ans = zeros(1, N) for i=1:N - 1 z_0 = path[:, i] z_1 = path[:, i + 1] angle_0 = angle(z_0) angle_1 = angle(z_1) dtheta = angle_1 - angle_0 # we have to adjust dtheta in the case when our angle function returns # something close to 0 for z_0 and close to 2 pi for z_1 (or vice versa) if dtheta > pi dtheta -= 2 * pi elseif dtheta < -pi dtheta += 2 * pi end ans[i + 1] = ans[i] + dtheta / 2 / pi end return ans end # ╔═╡ 5c8ff45c-1ebf-11eb-3a9a-a1ebe9f355b0 function plot_winding(f, r) deg = size(f)[2] - 1 domain = circle_path(r) path = compute_path(f, domain) winding = winding_number(f, r) p1 = plot_path(domain, "{ z: |z| = $(r) }") p2 = plot_path(path, "{ p(z): |z| =$(r) }") p3 = plot(transpose(winding), legend=false, ylimit=(-1, deg + 1), title="winding") plot(p1, p2, p3, layout=(1,3)) end # ╔═╡ 4b41654a-1edb-11eb-3921-2570403c5194 md"For example, $p(x) = -1 + x + x^2$ has winding number 1 for $r=1$:" # ╔═╡ 4f6bee38-1edb-11eb-2976-efb3da72cee1 plot_winding([-1 1 1; 0 0 0], 1) # ╔═╡ e53f1ee4-1edb-11eb-31ce-fb56c6e2c13c md"However, for $r=2$ the polynomial $p(x) = -1 + x + x^2$ has winding number $2$:" # ╔═╡ a4c4ef24-1edb-11eb-385b-cda09270681b plot_winding([-1 1 1; 0 0 0], 2) # ╔═╡ 0f130bd6-1edc-11eb-286e-8505ba498a3e md"Whereas for $r=0.5$ the same polynomial has winding number $0$:" # ╔═╡ 0221ac34-1edc-11eb-0303-e731f883d9f9 plot_winding([-1 1 1; 0 0 0], 0.5) # ╔═╡ 5d29226e-1edd-11eb-13e4-09d39766a6f0 md"__Key observations:__ * When $p(0) \neq 0$ and $r$ is very small, the winding number will be zero. * If $p(x)=x^n$ then the winding number will always be $n$, for any $r$. * If $p(x)$ has degree $n$ and $r$ is very large, then $p(x)$ has the same winding number as $x^n$. " # ╔═╡ 20a1b52a-1eda-11eb-0694-af37314d3479 p = random_real_polynomial(8) # ╔═╡ 28415d94-1eda-11eb-143d-73c3a168fa4d print_polynomial(p) # ╔═╡ 907ecb0e-1ec0-11eb-337f-f16c8b49db19 begin slider = @bind parameter Slider(0.01:0.0001:0.9999, default=0.01, show_value=false) md"parameter = $(slider)" end # ╔═╡ fa058db4-1ec1-11eb-140c-312802809fe9 r = Int(round(10000 / (1 - parameter) - 1)) / 100000 # ╔═╡ d768f702-1edd-11eb-38dd-4182f741ffe1 winding = Int(round(winding_number(p, r)[end])) # ╔═╡ ab20e9d4-1ebf-11eb-0c0d-c773646dd160 plot_winding(p, r) # ╔═╡ 13575dbc-1ede-11eb-2bb9-454f48210788 md"**Why does this prove the fundamental theorem of algebra?** * Winding number can only change as we vary$r$if$\{ p(z): |z| = r\}$passes through the origin. * This must happen if$p(x)$has degree$n>0$and$p(0) \neq 0$. * (Since the winding number is zero if$r$is small and the winding number is$n$if$r$is large.) * But this means for some real$r > 0$there is an$\alpha \in \mathbb{C}$with$|\alpha| = r$and$p(\alpha) =0\$." # ╔═╡ 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" [compat] Plots = "~1.23.2" PlutoUI = "~0.7.17" """ # ╔═╡ 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" [[ArgTools]] uuid = "0dad84c5-d112-42e6-8d28-ef12dabb789f" [[Artifacts]] uuid = "56f22d72-fd6d-98f1-02f0-08ddc0907c33" [[Base64]] uuid = "2a0f44e3-6c83-55bd-87e4-b1978d98bd5f" [[Bzip2_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] git-tree-sha1 = "19a35467a82e236ff51bc17a3a44b69ef35185a2" uuid = "6e34b625-4abd-537c-b88f-471c36dfa7a0" version = "1.0.8+0" [[Cairo_jll]] deps = ["Artifacts", "Bzip2_jll", "Fontconfig_jll", "FreeType2_jll", "Glib_jll", "JLLWrappers", "LZO_jll", "Libdl", "Pixman_jll", "Pkg", "Xorg_libXext_jll", "Xorg_libXrender_jll", "Zlib_jll", "libpng_jll"] git-tree-sha1 = "f2202b55d816427cd385a9a4f3ffb226bee80f99" uuid = "83423d85-b0ee-5818-9007-b63ccbeb887a" version = "1.16.1+0" [[ChainRulesCore]] deps = ["Compat", "LinearAlgebra", "SparseArrays"] git-tree-sha1 = "3533f5a691e60601fe60c90d8bc47a27aa2907ec" uuid = "d360d2e6-b24c-11e9-a2a3-2a2ae2dbcce4" version = "1.11.0" [[ColorSchemes]] deps = ["ColorTypes", "Colors", "FixedPointNumbers", "Random"] git-tree-sha1 = "a851fec56cb73cfdf43762999ec72eff5b86882a" uuid = "35d6a980-a343-548e-a6ea-1d62b119f2f4" version = "3.15.0" [[ColorTypes]] deps = ["FixedPointNumbers", "Random"] git-tree-sha1 = "024fe24d83e4a5bf5fc80501a314ce0d1aa35597" uuid = "3da002f7-5984-5a60-b8a6-cbb66c0b333f" version = "0.11.0" [[Colors]] deps = ["ColorTypes", "FixedPointNumbers", "Reexport"] git-tree-sha1 = "417b0ed7b8b838aa6ca0a87aadf1bb9eb111ce40" uuid = "5ae59095-9a9b-59fe-a467-6f913c188581" version = "0.12.8" [[Compat]] deps = ["Base64", "Dates", "DelimitedFiles", "Distributed", "InteractiveUtils", "LibGit2", "Libdl", "LinearAlgebra", "Markdown", "Mmap", "Pkg", "Printf", "REPL", "Random", "SHA", "Serialization", "SharedArrays", "Sockets", "SparseArrays", "Statistics", "Test", "UUIDs", "Unicode"] git-tree-sha1 = "dce3e3fea680869eaa0b774b2e8343e9ff442313" uuid = "34da2185-b29b-5c13-b0c7-acf172513d20" version = "3.40.0" [[CompilerSupportLibraries_jll]] deps = ["Artifacts", "Libdl"] uuid = "e66e0078-7015-5450-92f7-15fbd957f2ae" [[Contour]] deps = ["StaticArrays"] git-tree-sha1 = "9f02045d934dc030edad45944ea80dbd1f0ebea7" uuid = "d38c429a-6771-53c6-b99e-75d170b6e991" version = "0.5.7" [[DataAPI]] git-tree-sha1 = "cc70b17275652eb47bc9e5f81635981f13cea5c8" uuid = "9a962f9c-6df0-11e9-0e5d-c546b8b5ee8a" version = "1.9.0" [[DataStructures]] deps = ["Compat", "InteractiveUtils", "OrderedCollections"] git-tree-sha1 = "7d9d316f04214f7efdbb6398d545446e246eff02" uuid = "864edb3b-99cc-5e75-8d2d-829cb0a9cfe8" version = "0.18.10" [[DataValueInterfaces]] git-tree-sha1 = "bfc1187b79289637fa0ef6d4436ebdfe6905cbd6" uuid = "e2d170a0-9d28-54be-80f0-106bbe20a464" version = "1.0.0" [[Dates]] deps = ["Printf"] uuid = "ade2ca70-3891-5945-98fb-dc099432e06a" [[DelimitedFiles]] deps = ["Mmap"] uuid = "8bb1440f-4735-579b-a4ab-409b98df4dab" [[Distributed]] deps = ["Random", "Serialization", "Sockets"] uuid = "8ba89e20-285c-5b6f-9357-94700520ee1b" [[DocStringExtensions]] deps = ["LibGit2"] git-tree-sha1 = "b19534d1895d702889b219c382a6e18010797f0b" uuid = "ffbed154-4ef7-542d-bbb7-c09d3a79fcae" version = "0.8.6" [[Downloads]] deps = ["ArgTools", "LibCURL", "NetworkOptions"] uuid = "f43a241f-c20a-4ad4-852c-f6b1247861c6" [[EarCut_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] git-tree-sha1 = "3f3a2501fa7236e9b911e0f7a588c657e822bb6d" uuid = "5ae413db-bbd1-5e63-b57d-d24a61df00f5" version = "2.2.3+0" [[Expat_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] git-tree-sha1 = "b3bfd02e98aedfa5cf885665493c5598c350cd2f" uuid = "2e619515-83b5-522b-bb60-26c02a35a201" version = "2.2.10+0" [[FFMPEG]] deps = ["FFMPEG_jll"] git-tree-sha1 = "b57e3acbe22f8484b4b5ff66a7499717fe1a9cc8" uuid = "c87230d0-a227-11e9-1b43-d7ebe4e7570a" version = "0.4.1" [[FFMPEG_jll]] deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "JLLWrappers", "LAME_jll", "Libdl", "Ogg_jll", "OpenSSL_jll", "Opus_jll", "Pkg", "Zlib_jll", "libass_jll", "libfdk_aac_jll", "libvorbis_jll", "x264_jll", "x265_jll"] git-tree-sha1 = "d8a578692e3077ac998b50c0217dfd67f21d1e5f" uuid = "b22a6f82-2f65-5046-a5b2-351ab43fb4e5" version = "4.4.0+0" [[FixedPointNumbers]] deps = ["Statistics"] git-tree-sha1 = "335bfdceacc84c5cdf16aadc768aa5ddfc5383cc" uuid = "53c48c17-4a7d-5ca2-90c5-79b7896eea93" version = "0.8.4" [[Fontconfig_jll]] deps = ["Artifacts", "Bzip2_jll", "Expat_jll", "FreeType2_jll", "JLLWrappers", "Libdl", "Libuuid_jll", "Pkg", "Zlib_jll"] git-tree-sha1 = "21efd19106a55620a188615da6d3d06cd7f6ee03" uuid = "a3f928ae-7b40-5064-980b-68af3947d34b" version = "2.13.93+0" [[Formatting]] deps = ["Printf"] git-tree-sha1 = "8339d61043228fdd3eb658d86c926cb282ae72a8" uuid = "59287772-0a20-5a39-b81b-1366585eb4c0" version = "0.4.2" [[FreeType2_jll]] deps = ["Artifacts", "Bzip2_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"] git-tree-sha1 = "87eb71354d8ec1a96d4a7636bd57a7347dde3ef9" uuid = "d7e528f0-a631-5988-bf34-fe36492bcfd7" version = "2.10.4+0" [[FriBidi_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] git-tree-sha1 = "aa31987c2ba8704e23c6c8ba8a4f769d5d7e4f91" uuid = "559328eb-81f9-559d-9380-de523a88c83c" version = "1.0.10+0" [[GLFW_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Libglvnd_jll", "Pkg", "Xorg_libXcursor_jll", "Xorg_libXi_jll", "Xorg_libXinerama_jll", "Xorg_libXrandr_jll"] git-tree-sha1 = "0c603255764a1fa0b61752d2bec14cfbd18f7fe8" uuid = "0656b61e-2033-5cc2-a64a-77c0f6c09b89" version = "3.3.5+1" [[GR]] deps = ["Base64", "DelimitedFiles", "GR_jll", "HTTP", "JSON", "Libdl", "LinearAlgebra", "Pkg", "Printf", "Random", "Serialization", "Sockets", "Test", "UUIDs"] git-tree-sha1 = "d189c6d2004f63fd3c91748c458b09f26de0efaa" uuid = "28b8d3ca-fb5f-59d9-8090-bfdbd6d07a71" version = "0.61.0" [[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 = "fd75fa3a2080109a2c0ec9864a6e14c60cca3866" uuid = "d2c73de3-f751-5644-a686-071e5b155ba9" version = "0.62.0+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 = "5efcf53d798efede8fee5b2c8b09284be359bf24" uuid = "ac1192a8-f4b3-4bfe-ba22-af5b92cd3ab2" version = "0.9.2" [[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 = "a8f4f279b6fa3c3c4f1adadd78a621b13a506bce" uuid = "23fbe1c1-3f47-55db-b15f-69d7ec21a316" version = "0.15.9" [[LibCURL]] deps = ["LibCURL_jll", "MozillaCACerts_jll"] uuid = "b27032c2-a3e7-50c8-80cd-2d36dbcbfd21" [[LibCURL_jll]] deps = ["Artifacts", "LibSSH2_jll", "Libdl", "MbedTLS_jll", "Zlib_jll", "nghttp2_jll"] uuid = "deac9b47-8bc7-5906-a0fe-35ac56dc84c0" [[LibGit2]] deps = ["Base64", "NetworkOptions", "Printf", "SHA"] uuid = "76f85450-5226-5b5a-8eaa-529ad045b433" [[LibSSH2_jll]] deps = ["Artifacts", "Libdl", "MbedTLS_jll"] uuid = "29816b5a-b9ab-546f-933c-edad1886dfa8" [[Libdl]] uuid = "8f399da3-3557-5675-b5ff-fb832c97cbdb" [[Libffi_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] git-tree-sha1 = "761a393aeccd6aa92ec3515e428c26bf99575b3b" uuid = "e9f186c6-92d2-5b65-8a66-fee21dc1b490" version = "3.2.2+0" [[Libgcrypt_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Libgpg_error_jll", "Pkg"] git-tree-sha1 = "64613c82a59c120435c067c2b809fc61cf5166ae" uuid = "d4300ac3-e22c-5743-9152-c294e39db1e4" version = "1.8.7+0" [[Libglvnd_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll", "Xorg_libXext_jll"] git-tree-sha1 = "7739f837d6447403596a75d19ed01fd08d6f56bf" uuid = "7e76a0d4-f3c7-5321-8279-8d96eeed0f29" version = "1.3.0+3" [[Libgpg_error_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] git-tree-sha1 = "c333716e46366857753e273ce6a69ee0945a6db9" uuid = "7add5ba3-2f88-524e-9cd5-f83b8a55f7b8" version = "1.42.0+0" [[Libiconv_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] git-tree-sha1 = "42b62845d70a619f063a7da093d995ec8e15e778" uuid = "94ce4f54-9a6c-5748-9c1c-f9c7231a4531" version = "1.16.1+1" [[Libmount_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] git-tree-sha1 = "9c30530bf0effd46e15e0fdcf2b8636e78cbbd73" uuid = "4b2f31a3-9ecc-558c-b454-b3730dcb73e9" version = "2.35.0+0" [[Libtiff_jll]] deps = ["Artifacts", "JLLWrappers", "JpegTurbo_jll", "Libdl", "Pkg", "Zlib_jll", "Zstd_jll"] git-tree-sha1 = "340e257aada13f95f98ee352d316c3bed37c8ab9" uuid = "89763e89-9b03-5906-acba-b20f662cd828" version = "4.3.0+0" [[Libuuid_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] git-tree-sha1 = "7f3efec06033682db852f8b3bc3c1d2b0a0ab066" uuid = "38a345b3-de98-5d2b-a5d3-14cd9215e700" version = "2.36.0+0" [[LinearAlgebra]] deps = ["Libdl"] uuid = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e" [[LogExpFunctions]] deps = ["ChainRulesCore", "DocStringExtensions", "InverseFunctions", "IrrationalConstants", "LinearAlgebra"] git-tree-sha1 = "6193c3815f13ba1b78a51ce391db8be016ae9214" uuid = "2ab3a3ac-af41-5b50-aa03-7779005ae688" version = "0.3.4" [[Logging]] uuid = "56ddb016-857b-54e1-b83d-db4d58db5568" [[MacroTools]] deps = ["Markdown", "Random"] git-tree-sha1 = "3d3e902b31198a27340d0bf00d6ac452866021cf" uuid = "1914dd2f-81c6-5fcd-8719-6d5c9610ff09" version = "0.5.9" [[Markdown]] deps = ["Base64"] uuid = "d6f4376e-aef5-505a-96c1-9c027394607a" [[MbedTLS]] deps = ["Dates", "MbedTLS_jll", "Random", "Sockets"] git-tree-sha1 = "1c38e51c3d08ef2278062ebceade0e46cefc96fe" uuid = "739be429-bea8-5141-9913-cc70e7f3736d" version = "1.0.3" [[MbedTLS_jll]] deps = ["Artifacts", "Libdl"] uuid = "c8ffd9c3-330d-5841-b78e-0817d7145fa1" [[Measures]] git-tree-sha1 = "e498ddeee6f9fdb4551ce855a46f54dbd900245f" uuid = "442fdcdd-2543-5da2-b0f3-8c86c306513e" version = "0.3.1" [[Missings]] deps = ["DataAPI"] git-tree-sha1 = "bf210ce90b6c9eed32d25dbcae1ebc565df2687f" uuid = "e1d29d7a-bbdc-5cf2-9ac0-f12de2c33e28" version = "1.0.2" [[Mmap]] uuid = "a63ad114-7e13-5084-954f-fe012c677804" [[MozillaCACerts_jll]] uuid = "14a3606d-f60d-562e-9121-12d972cd8159" [[NaNMath]] git-tree-sha1 = "bfe47e760d60b82b66b61d2d44128b62e3a369fb" uuid = "77ba4419-2d1f-58cd-9bb1-8ffee604a2e3" version = "0.3.5" [[NetworkOptions]] uuid = "ca575930-c2e3-43a9-ace4-1e988b2c1908" [[Ogg_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] git-tree-sha1 = "7937eda4681660b4d6aeeecc2f7e1c81c8ee4e2f" uuid = "e7412a2a-1a6e-54c0-be00-318e2571c051" version = "1.3.5+0" [[OpenSSL_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] git-tree-sha1 = "15003dcb7d8db3c6c857fda14891a539a8f2705a" uuid = "458c3c95-2e84-50aa-8efc-19380b2a3a95" version = "1.1.10+0" [[Opus_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] git-tree-sha1 = "51a08fb14ec28da2ec7a927c4337e4332c2a4720" uuid = "91d4177d-7536-5919-b921-800302f37372" version = "1.3.2+0" [[OrderedCollections]] git-tree-sha1 = "85f8e6578bf1f9ee0d11e7bb1b1456435479d47c" uuid = "bac558e1-5e72-5ebc-8fee-abe8a469f55d" version = "1.4.1" [[PCRE_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] git-tree-sha1 = "b2a7af664e098055a7529ad1a900ded962bca488" uuid = "2f80f16e-611a-54ab-bc61-aa92de5b98fc" version = "8.44.0+0" [[Parsers]] deps = ["Dates"] git-tree-sha1 = "ae4bbcadb2906ccc085cf52ac286dc1377dceccc" uuid = "69de0a69-1ddd-5017-9359-2bf0b02dc9f0" version = "2.1.2" [[Pixman_jll]] deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"] git-tree-sha1 = "b4f5d02549a10e20780a24fce72bea96b6329e29" uuid = "30392449-352a-5448-841d-b1acce4e97dc" version = "0.40.1+0" [[Pkg]] deps = ["Artifacts", "Dates", "Downloads", "LibGit2", "Libdl", "Logging", "Markdown", "Printf", "REPL", "Random", "SHA", "Serialization", "TOML", "Tar", "UUIDs", "p7zip_jll"] uuid = "44cfe95a-1eb2-52ea-b672-e2afdf69b78f" [[PlotThemes]] deps = ["PlotUtils", "Requires", "Statistics"] git-tree-sha1 = "a3a964ce9dc7898193536002a6dd892b1b5a6f1d" uuid = "ccf2f8ad-2431-5c83-bf29-c5338b663b6a" version = "2.0.1" [[PlotUtils]] deps = ["ColorSchemes", "Colors", "Dates", "Printf", "Random", "Reexport", "Statistics"] git-tree-sha1 = "b084324b4af5a438cd63619fd006614b3b20b87b" uuid = "995b91a9-d308-5afd-9ec6-746e21dbc043" version = "1.0.15" [[Plots]] deps = ["Base64", "Contour", "Dates", "Downloads", "FFMPEG", "FixedPointNumbers", "GR", "GeometryBasics", "JSON", "Latexify", "LinearAlgebra", "Measures", "NaNMath", "PlotThemes", "PlotUtils", "Printf", "REPL", "Random", "RecipesBase", "RecipesPipeline", "Reexport", "Requires", "Scratch", "Showoff", "SparseArrays", "Statistics", "StatsBase", "UUIDs"] git-tree-sha1 = "ca7d534a27b1c279f05cd094196cb70c35e3d892" uuid = "91a5bcdd-55d7-5caf-9e0b-520d859cae80" version = "1.23.2" [[PlutoUI]] deps = ["Base64", "Dates", "Hyperscript", "HypertextLiteral", "IOCapture", "InteractiveUtils", "JSON", "Logging", "Markdown", "Random", "Reexport", "UUIDs"] git-tree-sha1 = "615f3a1eff94add4bca9476ded096de60b46443b" uuid = "7f904dfe-b85e-4ff6-b463-dae2292396a8" version = "0.7.17" [[Preferences]] deps = ["TOML"] git-tree-sha1 = "00cfd92944ca9c760982747e9a1d0d5d86ab1e5a" uuid = "21216c6a-2e73-6563-6e65-726566657250" version = "1.2.2" [[Printf]] deps = ["Unicode"] uuid = "de0858da-6303-5e67-8744-51eddeeeb8d7" [[Qt5Base_jll]] deps = ["Artifacts", "CompilerSupportLibraries_jll", "Fontconfig_jll", "Glib_jll", "JLLWrappers", "Libdl", "Libglvnd_jll", "OpenSSL_jll", "Pkg", "Xorg_libXext_jll", "Xorg_libxcb_jll", "Xorg_xcb_util_image_jll", "Xorg_xcb_util_keysyms_jll", "Xorg_xcb_util_renderutil_jll", "Xorg_xcb_util_wm_jll", "Zlib_jll", "xkbcommon_jll"] git-tree-sha1 = "ad368663a5e20dbb8d6dc2fddeefe4dae0781ae8" uuid = "ea2cea3b-5b76-57ae-a6ef-0a8af62496e1" version = "5.15.3+0" [[REPL]] deps = ["InteractiveUtils", "Markdown", "Sockets", "Unicode"] uuid = "3fa0cd96-eef1-5676-8a61-b3b8758bbffb" [[Random]] deps = ["Serialization"] uuid = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c" [[RecipesBase]] git-tree-sha1 = "44a75aa7a527910ee3d1751d1f0e4148698add9e" uuid = "3cdcf5f2-1ef4-517c-9805-6587b60abb01" version = "1.1.2" [[RecipesPipeline]] deps = ["Dates", "NaNMath", "PlotUtils", "RecipesBase"] git-tree-sha1 = "7ad0dfa8d03b7bcf8c597f59f5292801730c55b8" uuid = "01d81517-befc-4cb6-b9ec-a95719d0359c" version = "0.4.1" [[Reexport]] git-tree-sha1 = "45e428421666073eab6f2da5c9d310d99bb12f9b" uuid = "189a3867-3050-52da-a836-e630ba90ab69" version = "1.2.2" [[Requires]] deps = ["UUIDs"] git-tree-sha1 = "4036a3bd08ac7e968e27c203d45f5fff15020621" uuid = "ae029012-a4dd-5104-9daa-d747884805df" version = "1.1.3" [[SHA]] uuid = "ea8e919c-243c-51af-8825-aaa63cd721ce" [[Scratch]] deps = ["Dates"] git-tree-sha1 = "0b4b7f1393cff97c33891da2a0bf69c6ed241fda" uuid = "6c6a2e73-6563-6170-7368-637461726353" version = "1.1.0" [[Serialization]] uuid = "9e88b42a-f829-5b0c-bbe9-9e923198166b" [[SharedArrays]] deps = ["Distributed", "Mmap", "Random", "Serialization"] uuid = "1a1011a3-84de-559e-8e89-a11a2f7dc383" [[Showoff]] deps = ["Dates", "Grisu"] git-tree-sha1 = "91eddf657aca81df9ae6ceb20b959ae5653ad1de" uuid = "992d4aef-0814-514b-bc4d-f2e9a6c4116f" version = "1.0.3" [[Sockets]] uuid = "6462fe0b-24de-5631-8697-dd941f90decc" [[SortingAlgorithms]] deps = ["DataStructures"] git-tree-sha1 = "b3363d7460f7d098ca0912c69b082f75625d7508" uuid = "a2af1166-a08f-5f64-846c-94a0d3cef48c" version = "1.0.1" [[SparseArrays]] deps = ["LinearAlgebra", "Random"] uuid = "2f01184e-e22b-5df5-ae63-d93ebab69eaf" [[StaticArrays]] deps = ["LinearAlgebra", "Random", "Statistics"] git-tree-sha1 = "3c76dde64d03699e074ac02eb2e8ba8254d428da" uuid = "90137ffa-7385-5640-81b9-e52037218182" version = "1.2.13" [[Statistics]] deps = ["LinearAlgebra", "SparseArrays"] uuid = "10745b16-79ce-11e8-11f9-7d13ad32a3b2" [[StatsAPI]] git-tree-sha1 = "1958272568dc176a1d881acb797beb909c785510" uuid = "82ae8749-77ed-4fe6-ae5f-f523153014b0" version = "1.0.0" [[StatsBase]] deps = ["DataAPI", "DataStructures", "LinearAlgebra", "LogExpFunctions", "Missings", "Printf", "Random", "SortingAlgorithms", "SparseArrays", "Statistics", "StatsAPI"] git-tree-sha1 = "eb35dcc66558b2dda84079b9a1be17557d32091a" uuid = "2913bbd2-ae8a-5f71-8c99-4fb6c76f3a91" version = "0.33.12" [[StructArrays]] deps = ["Adapt", "DataAPI", "StaticArrays", "Tables"] git-tree-sha1 = "2ce41e0d042c60ecd131e9fb7154a3bfadbf50d3" uuid = "09ab397b-f2b6-538f-b94a-2f83cf4a842a" version = "0.6.3" [[TOML]] deps = ["Dates"] uuid = "fa267f1f-6049-4f14-aa54-33bafae1ed76" [[TableTraits]] deps = ["IteratorInterfaceExtensions"] git-tree-sha1 = "c06b2f539df1c6efa794486abfb6ed2022561a39" uuid = "3783bdb8-4a98-5b6b-af9a-565f29a5fe9c" version = "1.0.1" [[Tables]] deps = ["DataAPI", "DataValueInterfaces", "IteratorInterfaceExtensions", "LinearAlgebra", "TableTraits", "Test"] git-tree-sha1 = "fed34d0e71b91734bf0a7e10eb1bb05296ddbcd0" uuid = "bd369af6-aec1-5ad0-b16a-f7cc5008161c" version = "1.6.0" [[Tar]] deps = ["ArgTools", "SHA"] uuid = "a4e569a6-e804-4fa4-b0f3-eef7a1d5b13e" [[Test]] deps = ["InteractiveUtils", "Logging", "Random", "Serialization"] uuid = "8dfed614-e22c-5e08-85e1-65c5234f0b40" [[URIs]] git-tree-sha1 = "97bbe755a53fe859669cd907f2d96aee8d2c1355" uuid = "5c2747f8-b7ea-4ff2-ba2e-563bfd36b1d4" version = "1.3.0" [[UUIDs]] deps = ["Random", "SHA"] uuid = "cf7118a7-6976-5b1a-9a39-7adc72f591a4" [[Unicode]] uuid = "4ec0a83e-493e-50e2-b9ac-8f72acf5a8f5" [[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: # ╟─624a8396-1f1e-11eb-3ddf-aff41274881c # ╠═f2118adc-1ebb-11eb-080a-0b61e26ca1a0 # ╠═76d253ae-1ec0-11eb-1d04-cbbf959c3984 # ╠═b8779288-1ec2-11eb-2381-7db4b2380595 # ╟─9b354e40-1ecc-11eb-1fbd-c9d33504aaaa # ╟─f88a7884-1ecc-11eb-25f3-27782e93a793 # ╟─8fd1de9a-1ece-11eb-1c83-0d036a825aad # ╟─9c770878-1ed3-11eb-1838-2d7e3d1aad90 # ╟─625e8dac-1ecf-11eb-1246-7b2b9657f8a2 # 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