Correct the references in example notebook references to more general expressions

0ac0465d · Luke Naylor · 65ae7d97 · 0ac0465d · 0ac0465d
Commit 0ac0465d authored 1 year ago by Luke Naylor
--- a/examples.ipynb
+++ b/examples.ipynb
@@ -69,7 +69,7 @@
       "$\\displaystyle \\text{Chern Character:} \\\\ \\begin{array}{l} \\mathrm{ch}_{0} = 3 \\\\ \\mathrm{ch}_{1} = 2 \\ell^{1} \\\\ \\mathrm{ch}_{2} = -2 \\ell^{2} \\end{array}$"
      ],
      "text/plain": [
-       "<pseudowalls.chern_character.Chern_Char object at 0x7f25db3a9990>"
+       "<pseudowalls.chern_character.Chern_Char object at 0x7ff9d8e4c110>"
      ]
     },
     "execution_count": 2,
@@ -100,7 +100,7 @@
       "$\\displaystyle \\text{Chern Character:} \\\\ \\begin{array}{l} \\mathrm{ch}_{0} = 29 \\\\ \\mathrm{ch}_{1} = 13 \\ell^{1} \\\\ \\mathrm{ch}_{2} = -\\frac{3}{2} \\ell^{2} \\end{array}$"
      ],
      "text/plain": [
-       "<pseudowalls.chern_character.Chern_Char object at 0x7f25d1953e20>"
+       "<pseudowalls.chern_character.Chern_Char object at 0x7ff9ce32e490>"
      ]
     },
     "execution_count": 3,
@@ -129,13 +129,12 @@
   "metadata": {},
   "outputs": [],
   "source": [
-    "m = 1 # value of $\\ell^2$\n",
-    "\n",
    "# add attributes for ...\n",
    "for example in [recurring, extravagant]:\n",
    "    example.betaminus = beta_minus(example.chern)\n",
    "    example.twisted = example.chern.twist(example.betaminus)\n",
    "    example.n = example.betaminus.denominator()\n",
+    "    example.m = 1 # \\ell^2 = 1 on P^1\n",
    "    example.bgmlv = example.chern.Q_tilt()\n",
    "\n",
    "# Actual maximal rank of Pseudo-Semistabilizers\n",
@@ -170,8 +169,8 @@
    "def loose_bound(example):\n",
    "    n = example.n\n",
    "    twisted = example.twisted\n",
-    "    return ( m*n^2*twisted.ch[1]^2\n",
-    "    ) / gcd(m, 2*n^2)\n",
+    "    return ( example.m*n^2*twisted.ch[1]^2\n",
+    "    ) / gcd(example.m, 2*n^2)\n",
    "\n",
    "for example in [recurring, extravagant]:\n",
    "    example.loose_bound = loose_bound(example)"
@@ -264,7 +263,8 @@
    "Redefine \\nu to $\\nu$ in latex\n",
    "$\\let\\originalDelta\\Delta$\n",
    "$\\renewcommand\\Delta{\\originalDelta(v)}$\n",
-    "Redefine \\Delta in latex to be $\\Delta$"
+    "$\\renewcommand\\Omega{\\operatorname{lcm}(m,2n^2)}$\n",
+    "Redefine \\Delta in latex to be $\\Delta$ and \\Omega to be $\\Omega$"
   ]
  },
  {
@@ -276,13 +276,13 @@
    {
     "data": {
      "text/html": [
-       "<html>\\(\\displaystyle \\frac{1}{2} \\, R + \\frac{\\Delta n^{2}}{4 \\, \\nu^{2}} + \\frac{R^{2} \\nu^{2}}{4 \\, \\Delta n^{2}}\\)</html>"
+       "<html>\\(\\displaystyle \\frac{1}{2} \\, R + \\frac{\\Delta \\Omega}{8 \\, m} + \\frac{R^{2} m}{2 \\, \\Delta \\Omega}\\)</html>"
      ],
      "text/latex": [
-       "$\\displaystyle \\frac{1}{2} \\, R + \\frac{\\Delta n^{2}}{4 \\, \\nu^{2}} + \\frac{R^{2} \\nu^{2}}{4 \\, \\Delta n^{2}}$"
+       "$\\displaystyle \\frac{1}{2} \\, R + \\frac{\\Delta \\Omega}{8 \\, m} + \\frac{R^{2} m}{2 \\, \\Delta \\Omega}$"
      ],
      "text/plain": [
-       "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)"
      ]
     },
     "execution_count": 8,
@@ -291,7 +291,7 @@
    }
   ],
   "source": [
-    "from plots_and_expressions import main_theorem1, Delta, nu, R, n\n",
+    "from plots_and_expressions import main_theorem1, Delta, m, R, n, lcm_m_2n2\n",
    "# Delta: symbol for Δ(v)\n",
    "# n: symbol for denominator for β_(v)\n",
    "# R : symbol for chern_0(v)\n",
@@ -310,9 +310,10 @@
    "    return (\n",
    "        main_theorem1.corollary_r_bound\n",
    "        .subs(Delta==example.bgmlv)\n",
-    "        .subs(nu==1)\n",
+    "        .subs(m==example.m)\n",
    "        .subs(R==example.chern.ch[0])\n",
    "        .subs(n==example.n)\n",
+    "        .subs(lcm_m_2n2==lcm(example.m,2*example.n^2))\n",
    "    )\n",
    "\n",
    "\n",
@@ -329,13 +330,13 @@
    {
     "data": {
      "text/html": [
-       "<html>\\(\\displaystyle 53838.5009765625\\)</html>"
+       "<html>\\(\\displaystyle \\frac{55130625}{1024}\\)</html>"
      ],
      "text/latex": [
-       "$\\displaystyle 53838.5009765625$"
+       "$\\displaystyle \\frac{55130625}{1024}$"
      ],
      "text/plain": [
-       "53838.5009765625"
+       "55130625/1024"
      ]
     },
     "execution_count": 10,
@@ -344,7 +345,7 @@
    }
   ],
   "source": [
-    "float(extravagant.corrolary_bound)"
+    "extravagant.corrolary_bound"
   ]
  },
  {
@@ -474,7 +475,7 @@
 ],
 "metadata": {
  "kernelspec": {
-   "display_name": "SageMath 9.7",
+   "display_name": "SageMath 9.8",
   "language": "sage",
   "name": "sagemath"
  },
@@ -488,7 +489,7 @@
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
-   "version": "3.10.8"
+   "version": "3.11.3"
  }
 },
 "nbformat": 4,

 %% Cell type:markdown id:63724666 tags:

 # Utilities

 %% 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 0x7f25db3a9990>
+    <pseudowalls.chern_character.Chern_Char object at 0x7ff9d8e4c110>

 %% 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 0x7f25d1953e20>
+    <pseudowalls.chern_character.Chern_Char object at 0x7ff9ce32e490>

 %% Cell type:markdown id:09692ba5 tags:

 # Calculate Preliminary Quantities

 %% Cell type:code id:7576ebe4 tags:

 ``` sage
-m = 1 # value of $\ell^2$
-
 # 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 ( m*n^2*twisted.ch[1]^2
-    ) / gcd(m, 2*n^2)
+    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:markdown id:8de156d1 tags:

 $\renewcommand\nu\ell$
 Redefine \nu to $\nu$ in latex
 $\let\originalDelta\Delta$
 $\renewcommand\Delta{\originalDelta(v)}$
-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:

 ``` 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)
 # n: symbol for denominator for β_(v)
 # R : symbol for chern_0(v)
 # nu : ...
 main_theorem1.corollary_r_bound
 ```

 %% Output

-    $\displaystyle \frac{1}{2} \, R + \frac{\Delta n^{2}}{4 \, \nu^{2}} + \frac{R^{2} \nu^{2}}{4 \, \Delta n^{2}}$
-    1/2*R + 1/4*Delta*n^2/nu^2 + 1/4*R^2*nu^2/(Delta*n^2)
+    $\displaystyle \frac{1}{2} \, R + \frac{\Delta \Omega}{8 \, m} + \frac{R^{2} m}{2 \, \Delta \Omega}$
+    1/2*R + 1/8*Delta*Omega/m + 1/2*R^2*m/(Delta*Omega)

 %% Cell type:code id:43aa5f56 tags:

 ``` sage
 def corrolary_bound(example):
    return (
        main_theorem1.corollary_r_bound
        .subs(Delta==example.bgmlv)
-        .subs(nu==1)
+        .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
-float(extravagant.corrolary_bound)
+extravagant.corrolary_bound
 ```

 %% Output

-    $\displaystyle 53838.5009765625$
-    53838.5009765625
+    $\displaystyle \frac{55130625}{1024}$
+    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)

--- a/plots_and_expressions.ipynb
+++ b/plots_and_expressions.ipynb