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Commit 0ac0465d authored by Luke Naylor's avatar Luke Naylor
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Correct the references in example notebook references to more general expressions

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%% Cell type:markdown id:63724666 tags: %% Cell type:markdown id:63724666 tags:
# Utilities # Utilities
%% Cell type:code id:abe149f0 tags: %% Cell type:code id:abe149f0 tags:
``` sage ``` sage
# Requires extra package: # Requires extra package:
#! sage -pip install "pseudowalls==0.0.3" --extra-index-url https://gitlab.com/api/v4/projects/43962374/packages/pypi/simple #! sage -pip install "pseudowalls==0.0.3" --extra-index-url https://gitlab.com/api/v4/projects/43962374/packages/pypi/simple
%display latex %display latex
from pseudowalls import * from pseudowalls import *
Δ = lambda v: v.Q_tilt() Δ = lambda v: v.Q_tilt()
mu = stability.Mumford().slope mu = stability.Mumford().slope
ts = stability.Tilt ts = stability.Tilt
var("beta", domain="real") var("beta", domain="real")
def beta_minus(v): def beta_minus(v):
beta = stability.Tilt().beta beta = stability.Tilt().beta
solutions = solve( solutions = solve(
stability.Tilt(alpha=0).degree(v)==0, stability.Tilt(alpha=0).degree(v)==0,
beta) beta)
return min(map(lambda s: s.rhs(), solutions)) return min(map(lambda s: s.rhs(), solutions))
class Object(object): class Object(object):
pass pass
``` ```
%% Cell type:markdown id:0b574b63 tags: %% Cell type:markdown id:0b574b63 tags:
# Define Cherns in Examples # Define Cherns in Examples
%% Cell type:markdown id:f1c3a606 tags: %% Cell type:markdown id:f1c3a606 tags:
Define two recurring examples used to illustrate performance of the different theorems to find semistabilizer bounds: Define two recurring examples used to illustrate performance of the different theorems to find semistabilizer bounds:
%% Cell type:code id:30d3738a tags: %% Cell type:code id:30d3738a tags:
``` sage ``` sage
recurring = Object() recurring = Object()
recurring.chern = Chern_Char(3, 2, -2) recurring.chern = Chern_Char(3, 2, -2)
recurring.chern recurring.chern
``` ```
%% Output %% Output
$\displaystyle \text{Chern Character:} \\ \begin{array}{l} \mathrm{ch}_{0} = 3 \\ \mathrm{ch}_{1} = 2 \ell^{1} \\ \mathrm{ch}_{2} = -2 \ell^{2} \end{array}$ $\displaystyle \text{Chern Character:} \\ \begin{array}{l} \mathrm{ch}_{0} = 3 \\ \mathrm{ch}_{1} = 2 \ell^{1} \\ \mathrm{ch}_{2} = -2 \ell^{2} \end{array}$
<pseudowalls.chern_character.Chern_Char object at 0x7f25db3a9990> <pseudowalls.chern_character.Chern_Char object at 0x7ff9d8e4c110>
%% Cell type:code id:fe394909 tags: %% Cell type:code id:fe394909 tags:
``` sage ``` sage
extravagant = Object() extravagant = Object()
extravagant.chern = Chern_Char(29, 13, -3/2) extravagant.chern = Chern_Char(29, 13, -3/2)
extravagant.chern extravagant.chern
``` ```
%% Output %% Output
$\displaystyle \text{Chern Character:} \\ \begin{array}{l} \mathrm{ch}_{0} = 29 \\ \mathrm{ch}_{1} = 13 \ell^{1} \\ \mathrm{ch}_{2} = -\frac{3}{2} \ell^{2} \end{array}$ $\displaystyle \text{Chern Character:} \\ \begin{array}{l} \mathrm{ch}_{0} = 29 \\ \mathrm{ch}_{1} = 13 \ell^{1} \\ \mathrm{ch}_{2} = -\frac{3}{2} \ell^{2} \end{array}$
<pseudowalls.chern_character.Chern_Char object at 0x7f25d1953e20> <pseudowalls.chern_character.Chern_Char object at 0x7ff9ce32e490>
%% Cell type:markdown id:09692ba5 tags: %% Cell type:markdown id:09692ba5 tags:
# Calculate Preliminary Quantities # Calculate Preliminary Quantities
%% Cell type:code id:7576ebe4 tags: %% Cell type:code id:7576ebe4 tags:
``` sage ``` sage
m = 1 # value of $\ell^2$
# add attributes for ... # add attributes for ...
for example in [recurring, extravagant]: for example in [recurring, extravagant]:
example.betaminus = beta_minus(example.chern) example.betaminus = beta_minus(example.chern)
example.twisted = example.chern.twist(example.betaminus) example.twisted = example.chern.twist(example.betaminus)
example.n = example.betaminus.denominator() example.n = example.betaminus.denominator()
example.m = 1 # \ell^2 = 1 on P^1
example.bgmlv = example.chern.Q_tilt() example.bgmlv = example.chern.Q_tilt()
# Actual maximal rank of Pseudo-Semistabilizers # Actual maximal rank of Pseudo-Semistabilizers
# (needs to be calculated elsewhere) # (needs to be calculated elsewhere)
recurring.actual_rmax = 25 recurring.actual_rmax = 25
extravagant.actual_rmax = 49313 extravagant.actual_rmax = 49313
``` ```
%% Cell type:markdown id:500abba3 tags: %% Cell type:markdown id:500abba3 tags:
# First Loose Bound # First Loose Bound
%% Cell type:markdown id:f3e61899 tags: %% Cell type:markdown id:f3e61899 tags:
Formula for loose bound: Formula for loose bound:
%% Cell type:code id:77fff07c tags: %% Cell type:code id:77fff07c tags:
``` sage ``` sage
def loose_bound(example): def loose_bound(example):
n = example.n n = example.n
twisted = example.twisted twisted = example.twisted
return ( m*n^2*twisted.ch[1]^2 return ( example.m*n^2*twisted.ch[1]^2
) / gcd(m, 2*n^2) ) / gcd(example.m, 2*n^2)
for example in [recurring, extravagant]: for example in [recurring, extravagant]:
example.loose_bound = loose_bound(example) example.loose_bound = loose_bound(example)
``` ```
%% Cell type:markdown id:218fa169 tags: %% Cell type:markdown id:218fa169 tags:
Loose bounds for the two examples: Loose bounds for the two examples:
%% Cell type:code id:9edbb954 tags: %% Cell type:code id:9edbb954 tags:
``` sage ``` sage
recurring.loose_bound recurring.loose_bound
``` ```
%% Output %% Output
$\displaystyle 144$ $\displaystyle 144$
144 144
%% Cell type:code id:21b52c4d tags: %% Cell type:code id:21b52c4d tags:
``` sage ``` sage
extravagant.loose_bound extravagant.loose_bound
``` ```
%% Output %% Output
$\displaystyle 215296$ $\displaystyle 215296$
215296 215296
%% Cell type:markdown id:22bc4b5f tags: %% Cell type:markdown id:22bc4b5f tags:
# Stronger Convenient Bound # Stronger Convenient Bound
%% Cell type:markdown id:ee140573 tags: %% Cell type:markdown id:ee140573 tags:
Use expression for bound used in main document Use expression for bound used in main document
%% Cell type:markdown id:8de156d1 tags: %% Cell type:markdown id:8de156d1 tags:
$\renewcommand\nu\ell$ $\renewcommand\nu\ell$
Redefine \nu to $\nu$ in latex Redefine \nu to $\nu$ in latex
$\let\originalDelta\Delta$ $\let\originalDelta\Delta$
$\renewcommand\Delta{\originalDelta(v)}$ $\renewcommand\Delta{\originalDelta(v)}$
Redefine \Delta in latex to be $\Delta$ $\renewcommand\Omega{\operatorname{lcm}(m,2n^2)}$
Redefine \Delta in latex to be $\Delta$ and \Omega to be $\Omega$
%% Cell type:code id:712b7324 tags: %% Cell type:code id:712b7324 tags:
``` sage ``` sage
from plots_and_expressions import main_theorem1, Delta, nu, R, n from plots_and_expressions import main_theorem1, Delta, m, R, n, lcm_m_2n2
# Delta: symbol for Δ(v) # Delta: symbol for Δ(v)
# n: symbol for denominator for β_(v) # n: symbol for denominator for β_(v)
# R : symbol for chern_0(v) # R : symbol for chern_0(v)
# nu : ... # nu : ...
main_theorem1.corollary_r_bound main_theorem1.corollary_r_bound
``` ```
%% Output %% Output
$\displaystyle \frac{1}{2} \, R + \frac{\Delta n^{2}}{4 \, \nu^{2}} + \frac{R^{2} \nu^{2}}{4 \, \Delta n^{2}}$ $\displaystyle \frac{1}{2} \, R + \frac{\Delta \Omega}{8 \, m} + \frac{R^{2} m}{2 \, \Delta \Omega}$
1/2*R + 1/4*Delta*n^2/nu^2 + 1/4*R^2*nu^2/(Delta*n^2) 1/2*R + 1/8*Delta*Omega/m + 1/2*R^2*m/(Delta*Omega)
%% Cell type:code id:43aa5f56 tags: %% Cell type:code id:43aa5f56 tags:
``` sage ``` sage
def corrolary_bound(example): def corrolary_bound(example):
return ( return (
main_theorem1.corollary_r_bound main_theorem1.corollary_r_bound
.subs(Delta==example.bgmlv) .subs(Delta==example.bgmlv)
.subs(nu==1) .subs(m==example.m)
.subs(R==example.chern.ch[0]) .subs(R==example.chern.ch[0])
.subs(n==example.n) .subs(n==example.n)
.subs(lcm_m_2n2==lcm(example.m,2*example.n^2))
) )
for example in [recurring, extravagant]: for example in [recurring, extravagant]:
example.corrolary_bound = corrolary_bound(example) example.corrolary_bound = corrolary_bound(example)
``` ```
%% Cell type:code id:5baed51c tags: %% Cell type:code id:5baed51c tags:
``` sage ``` sage
float(extravagant.corrolary_bound) extravagant.corrolary_bound
``` ```
%% Output %% Output
$\displaystyle 53838.5009765625$ $\displaystyle \frac{55130625}{1024}$
53838.5009765625 55130625/1024
%% Cell type:code id:fcc15f60 tags: %% Cell type:code id:fcc15f60 tags:
``` sage ``` sage
float(recurring.corrolary_bound) float(recurring.corrolary_bound)
``` ```
%% Output %% Output
$\displaystyle 37.515625$ $\displaystyle 37.515625$
37.515625 37.515625
%% Cell type:markdown id:e678a488 tags: %% Cell type:markdown id:e678a488 tags:
# Stronger Complicated Bounds # Stronger Complicated Bounds
%% Cell type:code id:ef8f09ff tags: %% Cell type:code id:ef8f09ff tags:
``` sage ``` sage
import numpy as np import numpy as np
def bound_comparisons(example): def bound_comparisons(example):
n = example.n n = example.n
a_v = example.betaminus.numerator() a_v = example.betaminus.numerator()
def theorem_bound(v_twisted, q_val, k): def theorem_bound(v_twisted, q_val, k):
return int(min( return int(min(
n^2*q_val^2/k, n^2*q_val^2/k,
v_twisted.ch[0] v_twisted.ch[0]
+ n^2*(v_twisted.ch[1] - q_val)^2/k + n^2*(v_twisted.ch[1] - q_val)^2/k
)) ))
def k(n, a_v, b_q): def k(n, a_v, b_q):
n = int(n) n = int(n)
a_v = int(a_v) a_v = int(a_v)
b_q = int(b_q) b_q = int(b_q)
k = -a_v*b_q % n k = -a_v*b_q % n
return k if k > 0 else k + n return k if k > 0 else k + n
b_qs = list(range(example.twisted.ch[1]*n+1)) b_qs = list(range(example.twisted.ch[1]*n+1))
qs = list(map(lambda x: x/n,b_qs)) qs = list(map(lambda x: x/n,b_qs))
ks = list(map(lambda b_q: k(n, a_v, b_q), b_qs)) ks = list(map(lambda b_q: k(n, a_v, b_q), b_qs))
theorem2_bounds = [ theorem2_bounds = [
theorem_bound(example.twisted, q_val, 1) theorem_bound(example.twisted, q_val, 1)
for q_val in qs for q_val in qs
] ]
theorem3_bounds = [ theorem3_bounds = [
theorem_bound(example.twisted, q_val, k) theorem_bound(example.twisted, q_val, k)
for q_val, k in zip(qs,ks) for q_val, k in zip(qs,ks)
] ]
return qs, theorem2_bounds, theorem3_bounds return qs, theorem2_bounds, theorem3_bounds
``` ```
%% Cell type:markdown id:b1c51cd8 tags: %% Cell type:markdown id:b1c51cd8 tags:
Array content: Array content:
- First row: $q$-values - First row: $q$-values
- Second row: Theorem 1 bounds on $ch_0(u)$ for u solutions to prob with $ch_1^{\beta}=q$ - Second row: Theorem 1 bounds on $ch_0(u)$ for u solutions to prob with $ch_1^{\beta}=q$
- Second row: Theorem 2 bounds on $ch_0(u)$ for u solutions to prob with $ch_1^{\beta}=q$ - Second row: Theorem 2 bounds on $ch_0(u)$ for u solutions to prob with $ch_1^{\beta}=q$
%% Cell type:code id:9cd102ac tags: %% Cell type:code id:9cd102ac tags:
``` sage ``` sage
np.array(bound_comparisons(recurring)) np.array(bound_comparisons(recurring))
``` ```
%% Output %% Output
$\displaystyle \begin{array}{l} $\displaystyle \begin{array}{l}
\verb|[[0|\verb| |\verb|1/3|\verb| |\verb|2/3|\verb| |\verb|1|\verb| |\verb|4/3|\verb| |\verb|5/3|\verb| |\verb|2|\verb| |\verb|7/3|\verb| |\verb|8/3|\verb| |\verb|3|\verb| |\verb|10/3|\verb| |\verb|11/3|\verb| |\verb|4]|\\ \verb|[[0|\verb| |\verb|1/3|\verb| |\verb|2/3|\verb| |\verb|1|\verb| |\verb|4/3|\verb| |\verb|5/3|\verb| |\verb|2|\verb| |\verb|7/3|\verb| |\verb|8/3|\verb| |\verb|3|\verb| |\verb|10/3|\verb| |\verb|11/3|\verb| |\verb|4]|\\
\verb| |\verb|[0|\verb| |\verb|1|\verb| |\verb|4|\verb| |\verb|9|\verb| |\verb|16|\verb| |\verb|25|\verb| |\verb|36|\verb| |\verb|28|\verb| |\verb|19|\verb| |\verb|12|\verb| |\verb|7|\verb| |\verb|4|\verb| |\verb|3]|\\ \verb| |\verb|[0|\verb| |\verb|1|\verb| |\verb|4|\verb| |\verb|9|\verb| |\verb|16|\verb| |\verb|25|\verb| |\verb|36|\verb| |\verb|28|\verb| |\verb|19|\verb| |\verb|12|\verb| |\verb|7|\verb| |\verb|4|\verb| |\verb|3]|\\
\verb| |\verb|[0|\verb| |\verb|0|\verb| |\verb|4|\verb| |\verb|3|\verb| |\verb|8|\verb| |\verb|25|\verb| |\verb|12|\verb| |\verb|15|\verb| |\verb|19|\verb| |\verb|6|\verb| |\verb|5|\verb| |\verb|4|\verb| |\verb|3]]| \verb| |\verb|[0|\verb| |\verb|0|\verb| |\verb|4|\verb| |\verb|3|\verb| |\verb|8|\verb| |\verb|25|\verb| |\verb|12|\verb| |\verb|15|\verb| |\verb|19|\verb| |\verb|6|\verb| |\verb|5|\verb| |\verb|4|\verb| |\verb|3]]|
\end{array}$ \end{array}$
array([[0, 1/3, 2/3, 1, 4/3, 5/3, 2, 7/3, 8/3, 3, 10/3, 11/3, 4], array([[0, 1/3, 2/3, 1, 4/3, 5/3, 2, 7/3, 8/3, 3, 10/3, 11/3, 4],
[0, 1, 4, 9, 16, 25, 36, 28, 19, 12, 7, 4, 3], [0, 1, 4, 9, 16, 25, 36, 28, 19, 12, 7, 4, 3],
[0, 0, 4, 3, 8, 25, 12, 15, 19, 6, 5, 4, 3]], dtype=object) [0, 0, 4, 3, 8, 25, 12, 15, 19, 6, 5, 4, 3]], dtype=object)
......
plots_and_expressions.ipynb
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− 31
View file @ 0ac0465d
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