Adjust Examples notebook to main notebook changes'

bd6f7777 · Luke Naylor · 566cf53a · bd6f7777
Commit bd6f7777 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 0x7f679c692150>"
+       "<pseudowalls.chern_character.Chern_Char object at 0x7f25db3a9990>"
      ]
     },
     "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 0x7f67924d6810>"
+       "<pseudowalls.chern_character.Chern_Char object at 0x7f25d1953e20>"
      ]
     },
     "execution_count": 3,
@@ -256,59 +256,59 @@
   ]
  },
  {
-   "cell_type": "code",
-   "execution_count": 8,
-   "id": "712b7324",
+   "cell_type": "markdown",
+   "id": "8de156d1",
   "metadata": {},
-   "outputs": [],
   "source": [
-    "from plots_and_expressions import \\\n",
-    "r_upper_bound_all_q, Delta, nu, n, R\n",
-    "# Delta: symbol for Δ(v)\n",
-    "# n: symbol for denominator for β_(v)\n",
-    "# R : symbol for chern_0(v)\n",
-    "# nu : ..."
+    "$\\renewcommand\\nu\\ell$\n",
+    "Redefine \\nu to $\\nu$ in latex\n",
+    "$\\let\\originalDelta\\Delta$\n",
+    "$\\renewcommand\\Delta{\\originalDelta(v)}$\n",
+    "Redefine \\Delta in latex to be $\\Delta$"
   ]
  },
  {
   "cell_type": "code",
-   "execution_count": 9,
-   "id": "5814dd1d",
-   "metadata": {
-    "scrolled": true
-   },
+   "execution_count": 8,
+   "id": "712b7324",
+   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
-       "<html>\\(\\displaystyle \\frac{{\\left(R + \\frac{\\Delta n^{2}}{\\nu^{2}}\\right)}^{2} \\nu^{2}}{4 \\, \\Delta n^{2}}\\)</html>"
+       "<html>\\(\\displaystyle \\frac{1}{2} \\, R + \\frac{\\Delta n^{2}}{4 \\, \\nu^{2}} + \\frac{R^{2} \\nu^{2}}{4 \\, \\Delta n^{2}}\\)</html>"
      ],
      "text/latex": [
-       "$\\displaystyle \\frac{{\\left(R + \\frac{\\Delta n^{2}}{\\nu^{2}}\\right)}^{2} \\nu^{2}}{4 \\, \\Delta n^{2}}$"
+       "$\\displaystyle \\frac{1}{2} \\, R + \\frac{\\Delta n^{2}}{4 \\, \\nu^{2}} + \\frac{R^{2} \\nu^{2}}{4 \\, \\Delta n^{2}}$"
      ],
      "text/plain": [
-       "1/4*(R + Delta*n^2/nu^2)^2*nu^2/(Delta*n^2)"
+       "1/2*R + 1/4*Delta*n^2/nu^2 + 1/4*R^2*nu^2/(Delta*n^2)"
      ]
     },
-     "execution_count": 9,
+     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
-    "r_upper_bound_all_q"
+    "from plots_and_expressions import main_theorem1, Delta, nu, R, n\n",
+    "# Delta: symbol for Δ(v)\n",
+    "# n: symbol for denominator for β_(v)\n",
+    "# R : symbol for chern_0(v)\n",
+    "# nu : ...\n",
+    "main_theorem1.corollary_r_bound"
   ]
  },
  {
   "cell_type": "code",
-   "execution_count": 10,
+   "execution_count": 9,
   "id": "43aa5f56",
   "metadata": {},
   "outputs": [],
   "source": [
    "def corrolary_bound(example):\n",
    "    return (\n",
-    "        r_upper_bound_all_q.expand()\n",
+    "        main_theorem1.corollary_r_bound\n",
    "        .subs(Delta==example.bgmlv)\n",
    "        .subs(nu==1)\n",
    "        .subs(R==example.chern.ch[0])\n",
@@ -322,7 +322,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 11,
+   "execution_count": 10,
   "id": "5baed51c",
   "metadata": {},
   "outputs": [
@@ -338,7 +338,7 @@
       "53838.5009765625"
      ]
     },
-     "execution_count": 11,
+     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
@@ -349,7 +349,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 12,
+   "execution_count": 11,
   "id": "fcc15f60",
   "metadata": {},
   "outputs": [
@@ -365,7 +365,7 @@
       "37.515625"
      ]
     },
-     "execution_count": 12,
+     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
@@ -384,7 +384,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 13,
+   "execution_count": 12,
   "id": "ef8f09ff",
   "metadata": {},
   "outputs": [],
@@ -436,7 +436,7 @@
  },
  {
   "cell_type": "code",
-   "execution_count": 14,
+   "execution_count": 13,
   "id": "9cd102ac",
   "metadata": {},
   "outputs": [
@@ -462,7 +462,7 @@
       "       [0, 0, 4, 3, 8, 25, 12, 15, 19, 6, 5, 4, 3]], dtype=object)"
      ]
     },
-     "execution_count": 14,
+     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
@@ -474,7 +474,7 @@
 ],
 "metadata": {
  "kernelspec": {
-   "display_name": "SageMath 9.8",
+   "display_name": "SageMath 9.7",
   "language": "sage",
   "name": "sagemath"
  },
@@ -488,7 +488,7 @@
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
-   "version": "3.11.3"
+   "version": "3.10.8"
  }
 },
 "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 0x7f679c692150>
+    <pseudowalls.chern_character.Chern_Char object at 0x7f25db3a9990>

 %% 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 0x7f67924d6810>
+    <pseudowalls.chern_character.Chern_Char object at 0x7f25d1953e20>

 %% 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.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)

 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 $\Delta$
+
 %% Cell type:code id:712b7324 tags:

 ``` sage
-from plots_and_expressions import \
-r_upper_bound_all_q, Delta, nu, n, R
+from plots_and_expressions import main_theorem1, Delta, nu, R, n
 # Delta: symbol for Δ(v)
 # n: symbol for denominator for β_(v)
 # R : symbol for chern_0(v)
 # nu : ...
-```
-
-%% Cell type:code id:5814dd1d tags:
-
-``` sage
-r_upper_bound_all_q
+main_theorem1.corollary_r_bound
 ```

 %% Output

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

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

 ``` sage
 def corrolary_bound(example):
    return (
-        r_upper_bound_all_q.expand()
+        main_theorem1.corollary_r_bound
        .subs(Delta==example.bgmlv)
        .subs(nu==1)
        .subs(R==example.chern.ch[0])
        .subs(n==example.n)
    )


 for example in [recurring, extravagant]:
    example.corrolary_bound = corrolary_bound(example)
 ```

 %% Cell type:code id:5baed51c tags:

 ``` sage
 float(extravagant.corrolary_bound)
 ```

 %% Output

    $\displaystyle 53838.5009765625$
    53838.5009765625

 %% 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)