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Commit 56aa72ae authored by Luke Naylor's avatar Luke Naylor
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Section and annotate characteristic curves notebook better

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characteristic_curves.ipynb
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%% Cell type:markdown id:39a3ff68 tags:
# Utilities
%% Cell type:code id:b87a49bc tags:
``` sage
# Requires extra package:
#! sage -pip install "pseudowalls==0.0.3" --extra-index-url https://gitlab.com/api/v4/projects/43962374/packages/pypi/simple
%display latex
from pseudowalls import *
Δ = lambda v: v.Q_tilt()
mu = stability.Mumford().slope
ts = stability.Tilt
var("beta", domain="real")
def beta_minus(v):
beta = stability.Tilt().beta
solutions = solve(
stability.Tilt(alpha=0).degree(v)==0,
beta)
return min(map(lambda s: s.rhs(), solutions))
class Object(object):
pass
```
%% Cell type:markdown id:5da7dba0 tags:
# Characteristic Curves Plots
%% Cell type:code id:5e309df3 tags:
``` sage
def charact_curves(v):
alpha = stability.Tilt().alpha
beta = stability.Tilt().beta
coords_range = (beta, -4, 5), (alpha, 0, 4)
text_args = {"fontsize":"xx-large", "clip":True}
black_text_args = {"rgbcolor": "black", **text_args}
p = (
implicit_plot(stability.Tilt().degree(v), *coords_range )
+ line([(mu(v),0),(mu(v),5)], linestyle = "dashed")
+ text(r"$\Theta_v^+$",[3.5, 2], rotation=45, **text_args)
+ text(r"$V_v$", [0.43, 1.5], rotation=90, **text_args)
+ text(r"$\Theta_v^-$", [-2.2, 2], rotation=-45, **text_args)
+ text(r"$\nu_{\alpha, \beta}(v)>0$", [-3, 1], **black_text_args)
+ text(r"$\nu_{\alpha, \beta}(v)<0$", [-1, 3], **black_text_args)
+ text(r"$\nu_{\alpha, \beta}(-v)>0$", [2, 3], **black_text_args)
+ text(r"$\nu_{\alpha, \beta}(-v)<0$", [4, 1], **black_text_args)
)
p.xmax(5)
p.xmin(-4)
p.ymax(4)
p.axes_labels([r"$\beta$", r"$\alpha$"])
return p
v1 = Chern_Char(3, 2, -2)
v2 = Chern_Char(3, 2, 2/3)
```
%% Cell type:code id:00428146 tags:
``` sage
typical_characteristic_curves = charact_curves(v1)
typical_characteristic_curves
```
%% Output
Graphics object consisting of 9 graphics primitives
%% Cell type:code id:51d49b55 tags:
``` sage
degenerate_characteristic_curves = charact_curves(v2)
degenerate_characteristic_curves
```
%% Output
Graphics object consisting of 9 graphics primitives
%% Cell type:markdown id:4f77e7d6 tags:
## Characteristic Curves for Pseudo-semistabilizers
%% Cell type:code id:87b4eeeb tags:
``` sage
def hyperbola_intersection_plot():
var("alpha beta", domain="real")
coords_range = (beta, -3, -1/2), (alpha, 0, 2.5)
delta1 = -sqrt(2)+1/100
delta2 = 1/2
pbeta=-1.5
text_args = {"fontsize":"large", "clip":True}
black_text_args = {"rgbcolor":"black", **text_args}
p = (
implicit_plot( beta^2 - alpha^2 == 2,
*coords_range , rgbcolor = "black", legend_label=r"a")
+ implicit_plot( (beta+4)^2 - (alpha)^2 == 2,
*coords_range , rgbcolor = "red")
+ implicit_plot( (beta+delta1)^2 - alpha^2 == (delta1-2)^2-2,
*coords_range , rgbcolor = "blue")
+ implicit_plot( (beta+delta2)^2 - alpha^2 == (delta2-2)^2-2,
*coords_range , rgbcolor = "green")
+ point([-2, sqrt(2)], size=50, rgbcolor="black", zorder=50)
+ text("Q",[-2, sqrt(2)+0.1], **black_text_args)
+ point([pbeta, sqrt(pbeta^2-2)], size=50, rgbcolor="black", zorder=50)
+ text("P",[pbeta+0.1, sqrt(pbeta^2-2)], **black_text_args)
+ circle((-2,0),sqrt(2), linestyle="dashed", rgbcolor="purple")
# dummy lines to add legends (circumvent bug in implicit_plot)
+ line([(2,0),(2,0)] , rgbcolor = "purple", linestyle="dotted",
legend_label=r"pseudo-wall")
+ line([(2,0),(2,0)] , rgbcolor = "black",
legend_label=r"$\Theta_v^-$")
+ line([(2,0),(2,0)] , rgbcolor = "red", legend_label=r"$\Theta_u$ case 1")
+ line([(2,0),(2,0)] , rgbcolor = "blue", legend_label=r"$\Theta_u$ case 2")
+ line([(2,0),(2,0)] , rgbcolor = "green", legend_label=r"$\Theta_u$ case 3")
)
p.set_legend_options(loc="upper right", font_size="x-large",
font_family="serif")
p.xmax(coords_range[0][2])
p.xmin(coords_range[0][1])
p.ymax(coords_range[1][2])
p.ymin(coords_range[1][1])
p.axes_labels([r"$\beta$", r"$\alpha$"])
return p
```
%% Cell type:markdown id:e5968de3 tags:
Considering the 3 different ways that $\Theta_u$ could intersect $\Theta_v^-$
%% Cell type:code id:ff33752c tags:
``` sage
hyperbola_intersection_plot()
```
%% Output
Graphics object consisting of 14 graphics primitives
%% Cell type:code id:0a940381 tags:
``` sage
def correct_hyperbola_intersection_plot():
var("alpha beta", domain="real")
coords_range = (beta, -2.5, 0.5), (alpha, 0, 3)
delta2 = 1/2
pbeta=-1.5
text_args = {"fontsize":"large", "clip":True}
black_text_args = {"rgbcolor":"black", **text_args}
p = (
implicit_plot( beta^2 - alpha^2 == 2,
*coords_range , rgbcolor = "black", legend_label=r"a")
+ implicit_plot((beta+delta2)^2 - alpha^2 == (delta2-2)^2-2,
*coords_range , rgbcolor = "green")
+ point([-2, sqrt(2)], size=50, rgbcolor="black", zorder=50)
+ text("Q",[-2, sqrt(2)+0.1], **black_text_args)
+ point([pbeta, sqrt(pbeta^2-2)], size=50, rgbcolor="black", zorder=50)
+ text("P",[pbeta+0.1, sqrt(pbeta^2-2)], **black_text_args)
+ circle((-2,0),sqrt(2), linestyle="dashed", rgbcolor="purple")
# dummy lines to add legends (circumvent bug in implicit_plot)
+ line([(2,0),(2,0)] , rgbcolor = "purple", linestyle="dotted",
legend_label=r"pseudo-wall")
+ line([(2,0),(2,0)] , rgbcolor = "black",
legend_label=r"$\Theta_v^-$")
+ line([(2,0),(2,0)] , rgbcolor = "green",
legend_label=r"$\Theta_u^-$")
# vertical characteristic lines
+ line([(0,0),(0,coords_range[1][2])],
rgbcolor="black", linestyle="dashed",
legend_label=r"$V_v$")
+ line([(-delta2,0),(-delta2,coords_range[1][2])],
rgbcolor="green", linestyle="dashed",
legend_label=r"$V_u$")
+ line([(-delta2,0),(-delta2-coords_range[1][2],coords_range[1][2])],
rgbcolor="green", linestyle="dotted",
legend_label=r"$\Theta_u^-$ assymptote")
+ line([(0,0),(-coords_range[1][2],coords_range[1][2])],
rgbcolor="black", linestyle="dotted",
legend_label=r"$\Theta_v^-$ assymptote")
)
p.set_legend_options(loc="upper right", font_size="x-large",
font_family="serif")
p.xmax(coords_range[0][2])
p.xmin(coords_range[0][1])
p.ymax(coords_range[1][2])
p.ymin(coords_range[1][1])
p.axes_labels([r"$\beta$", r"$\alpha$"])
return p
```
%% Cell type:code id:ff33752c tags:
``` sage
hyperbola_intersection_plot()
```
%% Output
%% Cell type:markdown id:fcf66820 tags:
Graphics object consisting of 14 graphics primitives
Full picture of characteristic curves, assymptotes, pseudo-walls for the scenario described in main problem (object $u$ destabilizing $v$ going down $\Theta_v^-$ before reaching some point $P$)
%% Cell type:code id:6068850f tags:
``` sage
correct_hyperbola_intersection_plot()
```
%% Output
Graphics object consisting of 14 graphics primitives
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