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Commit 806b2648 authored by Luke Naylor's avatar Luke Naylor
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Annotate examples

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    %% Cell type:markdown id:63724666 tags:
    %% Cell type:markdown id:6879f29a tags:
    # Utilities
    Notebook for the Recurring examples that appear across Part $\text{\ref{part:fin-walls}}$
    %% Cell type:code id:abe149f0 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:0b574b63 tags:
    # Define Cherns in Examples
    %% Cell type:markdown id:f1c3a606 tags:
    Define two recurring examples used to illustrate performance of the different theorems to find semistabilizer bounds:
    %% Cell type:code id:30d3738a tags:
    ``` sage
    recurring = Object()
    recurring.chern = Chern_Char(3, 2, -2)
    recurring.chern
    ```
    %% Output
    $\displaystyle \text{Chern Character:} \\ \begin{array}{l} \mathrm{ch}_{0} = 3 \\ \mathrm{ch}_{1} = 2 \ell^{1} \\ \mathrm{ch}_{2} = -2 \ell^{2} \end{array}$
    <pseudowalls.chern_character.Chern_Char object at 0x7f1718d278e0>
    <pseudowalls.chern_character.Chern_Char object at 0x7f7ef1c08150>
    %% Cell type:code id:fe394909 tags:
    ``` sage
    extravagant = Object()
    extravagant.chern = Chern_Char(29, 13, -3/2)
    extravagant.chern
    ```
    %% Output
    $\displaystyle \text{Chern Character:} \\ \begin{array}{l} \mathrm{ch}_{0} = 29 \\ \mathrm{ch}_{1} = 13 \ell^{1} \\ \mathrm{ch}_{2} = -\frac{3}{2} \ell^{2} \end{array}$
    <pseudowalls.chern_character.Chern_Char object at 0x7f1718d263b0>
    <pseudowalls.chern_character.Chern_Char object at 0x7f7ee76c5850>
    %% Cell type:markdown id:09692ba5 tags:
    # Calculate Preliminary Quantities
    %% Cell type:code id:7576ebe4 tags:
    ``` sage
    # add attributes for ...
    for example in [recurring, extravagant]:
    example.betaminus = beta_minus(example.chern)
    example.twisted = example.chern.twist(example.betaminus)
    example.n = example.betaminus.denominator()
    example.m = 1 # \ell^2 = 1 on P^1
    example.bgmlv = example.chern.Q_tilt()
    # Actual maximal rank of Pseudo-Semistabilizers
    # (needs to be calculated elsewhere)
    recurring.actual_rmax = 25
    extravagant.actual_rmax = 49313
    ```
    %% Cell type:markdown id:500abba3 tags:
    # First Loose Bound
    %% Cell type:markdown id:f3e61899 tags:
    Formula for loose bound:
    %% Cell type:code id:77fff07c tags:
    ``` sage
    def loose_bound(example):
    n = example.n
    twisted = example.twisted
    return ( example.m*n^2*twisted.ch[1]^2
    ) / gcd(example.m, 2*n^2)
    for example in [recurring, extravagant]:
    example.loose_bound = loose_bound(example)
    ```
    %% Cell type:markdown id:218fa169 tags:
    Loose bounds for the two examples:
    %% Cell type:code id:9edbb954 tags:
    ``` sage
    recurring.loose_bound
    ```
    %% Output
    $\displaystyle 144$
    144
    %% Cell type:code id:21b52c4d tags:
    ``` sage
    extravagant.loose_bound
    ```
    %% Output
    $\displaystyle 215296$
    215296
    %% Cell type:markdown id:22bc4b5f tags:
    # Stronger Convenient Bound
    %% Cell type:markdown id:ee140573 tags:
    Use expression for bound used in main document
    %% Cell type:code id:712b7324 tags:
    ``` sage
    from plots_and_expressions import main_theorem1, bgmlv_v, m, R, n, lcm_m_2n2
    # Delta: symbol for Δ(v)
    # n: symbol for denominator for β_(v)
    # R : symbol for chern_0(v)
    # nu : ...
    main_theorem1.corollary_r_bound
    ```
    %% Output
    $\displaystyle \frac{1}{2} \, R + \frac{{\Delta(v)} {\operatorname{lcm}(m,2n^2)}}{8 \, m} + \frac{R^{2} m}{2 \, {\Delta(v)} {\operatorname{lcm}(m,2n^2)}}$
    1/2*R + 1/8*bgmlv_v*lcm_m_2n2/m + 1/2*R^2*m/(bgmlv_v*lcm_m_2n2)
    %% Cell type:code id:43aa5f56 tags:
    ``` sage
    def corrolary_bound(example):
    return (
    main_theorem1.corollary_r_bound
    .subs(bgmlv_v==example.bgmlv)
    .subs(m==example.m)
    .subs(R==example.chern.ch[0])
    .subs(n==example.n)
    .subs(lcm_m_2n2==lcm(example.m,2*example.n^2))
    )
    for example in [recurring, extravagant]:
    example.corrolary_bound = corrolary_bound(example)
    ```
    %% Cell type:code id:5baed51c tags:
    ``` sage
    extravagant.corrolary_bound
    ```
    %% Output
    $\displaystyle \frac{55130625}{1024}$
    55130625/1024
    %% Cell type:code id:fcc15f60 tags:
    ``` sage
    float(recurring.corrolary_bound)
    ```
    %% Output
    $\displaystyle 37.515625$
    37.515625
    %% Cell type:markdown id:e678a488 tags:
    # Stronger Complicated Bounds
    %% Cell type:code id:ef8f09ff tags:
    ``` sage
    import numpy as np
    def bound_comparisons(example):
    n = example.n
    a_v = example.betaminus.numerator()
    def theorem_bound(v_twisted, q_val, k):
    return int(min(
    n^2*q_val^2/k,
    v_twisted.ch[0]
    + n^2*(v_twisted.ch[1] - q_val)^2/k
    ))
    def k(n, a_v, b_q):
    n = int(n)
    a_v = int(a_v)
    b_q = int(b_q)
    k = -a_v*b_q % n
    return k if k > 0 else k + n
    b_qs = list(range(example.twisted.ch[1]*n+1))
    qs = list(map(lambda x: x/n,b_qs))
    ks = list(map(lambda b_q: k(n, a_v, b_q), b_qs))
    theorem2_bounds = [
    theorem_bound(example.twisted, q_val, 1)
    for q_val in qs
    ]
    theorem3_bounds = [
    theorem_bound(example.twisted, q_val, k)
    for q_val, k in zip(qs,ks)
    ]
    return qs, theorem2_bounds, theorem3_bounds
    ```
    %% Cell type:markdown id:b1c51cd8 tags:
    Array content:
    - 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 2 bounds on $ch_0(u)$ for u solutions to prob with $ch_1^{\beta}=q$
    %% Cell type:code id:9cd102ac tags:
    ``` sage
    np.array(bound_comparisons(recurring))
    ```
    %% Output
    $\displaystyle \begin{array}{l}
    \verb|[[0|\verb| |\verb|1/3|\verb| |\verb|2/3|\verb| |\verb|1|\verb| |\verb|4/3|\verb| |\verb|5/3|\verb| |\verb|2|\verb| |\verb|7/3|\verb| |\verb|8/3|\verb| |\verb|3|\verb| |\verb|10/3|\verb| |\verb|11/3|\verb| |\verb|4]|\\
    \verb| |\verb|[0|\verb| |\verb|1|\verb| |\verb|4|\verb| |\verb|9|\verb| |\verb|16|\verb| |\verb|25|\verb| |\verb|36|\verb| |\verb|28|\verb| |\verb|19|\verb| |\verb|12|\verb| |\verb|7|\verb| |\verb|4|\verb| |\verb|3]|\\
    \verb| |\verb|[0|\verb| |\verb|0|\verb| |\verb|4|\verb| |\verb|3|\verb| |\verb|8|\verb| |\verb|25|\verb| |\verb|12|\verb| |\verb|15|\verb| |\verb|19|\verb| |\verb|6|\verb| |\verb|5|\verb| |\verb|4|\verb| |\verb|3]]|
    \end{array}$
    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, 0, 4, 3, 8, 25, 12, 15, 19, 6, 5, 4, 3]], dtype=object)
    ......
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