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{
"cell_type": "markdown",
"id": "fdf7befd",
"metadata": {},
"source": [
"Express the two conditions corresponding to the upper bounds on $d$ from $\\Delta(u) \\geq 0$ or $\\Delta(v-u) \\geq 0$ being more than $\\frac{\\kappa}{2n^2}$ higher than the lowerbound given by $\\chern_2^{\\beta}(u) > 0$"
]
},
{
"cell_type": "code",
"execution_count": 26,
"id": "1377923b",
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\left(\\frac{\\kappa}{2 \\, n^{2}} < \\frac{q^{2}}{2 \\, r}, \\frac{\\kappa}{2 \\, n^{2}} < -\\frac{{\\left(\\psi - q\\right)}^{2}}{2 \\, {\\left(R - r\\right)}}\\right)\\)</html>"
],
"text/latex": [
"$\\displaystyle \\left(\\frac{\\kappa}{2 \\, n^{2}} < \\frac{q^{2}}{2 \\, r}, \\frac{\\kappa}{2 \\, n^{2}} < -\\frac{{\\left(\\psi - q\\right)}^{2}}{2 \\, {\\left(R - r\\right)}}\\right)$"
],
"text/plain": [
"(1/2*kappa/n^2 < 1/2*q^2/r, 1/2*kappa/n^2 < -1/2*(psi - q)^2/(R - r))"
]
},
"execution_count": 26,
"metadata": {},
"output_type": "execute_result"
"source": [
"var(\"a_v b_q n\") # Define symbols introduce for values of beta and q\n",
"beta_value_expr = (beta == a_v/n)\n",
"q_value_expr = (q == b_q/n)\n",
"# placeholder for the specific values of k (start with 1):\n",
"var(\"kappa\", domain=\"real\")\n",
"\n",
"assymptote_gap_condition1 = (kappa/(2*n^2) < bgmlv2_d_upperbound_terms.hyperbolic)\n",
"assymptote_gap_condition2 = (kappa/(2*n^2) < bgmlv3_d_upperbound_terms.hyperbolic)\n",
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"assymptote_gap_condition1, assymptote_gap_condition2"
]
},
{
"cell_type": "markdown",
"id": "ce8bc94f",
"metadata": {},
"source": [
"Rearrange these two conditions into bounds for $r$:"
]
},
{
"cell_type": "code",
"execution_count": 27,
"id": "553bba31",
"metadata": {
"scrolled": true
},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\left(r < \\frac{n^{2} q^{2}}{\\kappa}, r < \\frac{n^{2} {\\left(\\psi - q\\right)}^{2}}{\\kappa} + R\\right)\\)</html>"
],
"text/latex": [
"$\\displaystyle \\left(r < \\frac{n^{2} q^{2}}{\\kappa}, r < \\frac{n^{2} {\\left(\\psi - q\\right)}^{2}}{\\kappa} + R\\right)$"
],
"text/plain": [
"(r < n^2*q^2/kappa, r < n^2*(psi - q)^2/kappa + R)"
]
},
"execution_count": 27,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"r_upper_bound1 = (\n",
" assymptote_gap_condition1\n",
" * r * 2*n^2 / kappa\n",
")\n",
"\n",
"assert r_upper_bound1.lhs() == r\n",
"\n",
"r_upper_bound2 = (\n",
" assymptote_gap_condition2\n",
" * (r-R) * 2*n^2 / kappa + R\n",
")\n",
"\n",
"assert r_upper_bound2.lhs() == r\n",
"\n",
"(r_upper_bound1, r_upper_bound2)"
"cell_type": "markdown",
"id": "7f4476b5",
"metadata": {},
"source": [
"### Main Theorem 1"
"cell_type": "markdown",
"id": "f6f4b131",
"metadata": {},
"source": [
"The first main theorem is about these two upper bounds on $r$ needing to be satisfied for $\\kappa = 1$ (weakest form)"
]
},
{
"cell_type": "code",
"execution_count": 28,
"id": "602840cc",
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\verb|r|\\verb| |\\verb|<|\\verb| |\\verb|min| \\left(n^{2} q^{2}, n^{2} {\\left(\\psi - q\\right)}^{2} + R\\right)\\)</html>"
],
"text/latex": [
"$\\displaystyle \\verb|r|\\verb| |\\verb|<|\\verb| |\\verb|min| \\left(n^{2} q^{2}, n^{2} {\\left(\\psi - q\\right)}^{2} + R\\right)$"
],
"text/plain": [
"'r < min' (n^2*q^2, n^2*(psi - q)^2 + R)"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"main_theorem1 = Object()\n",
"main_theorem1.r_upper_bound1 = r_upper_bound1.subs(kappa == 1).rhs()\n",
"main_theorem1.r_upper_bound2 = r_upper_bound2.subs(kappa == 1).rhs()\n",
"pretty_print(r\"r < min\",(main_theorem1.r_upper_bound1, main_theorem1.r_upper_bound2))"
"cell_type": "markdown",
"id": "8bf4b71c",
"metadata": {},
"source": [
"### Main Theorem 1 Corollary"
"cell_type": "markdown",
"id": "ddd87bcf",
"metadata": {},
"source": [
"$\\renewcommand\\nu\\ell$\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": 29,
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"metadata": {
"scrolled": true
},
"outputs": [
{
"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>"
],
"text/latex": [
"$\\displaystyle \\frac{1}{2} \\, R + \\frac{\\Delta n^{2}}{4 \\, \\nu^{2}} + \\frac{R^{2} \\nu^{2}}{4 \\, \\Delta n^{2}}$"
],
"text/plain": [
"1/2*R + 1/4*Delta*n^2/nu^2 + 1/4*R^2*nu^2/(Delta*n^2)"
]
},
"execution_count": 29,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"var(\"Delta nu\", domain=\"real\")\n",
"# Delta to represent bogomolov(v)\n",
"# nu to represent \\ell\n",
"\n",
"q_sol = solve(\n",
" main_theorem1.r_upper_bound1\n",
" == main_theorem1.r_upper_bound2\n",
" , q\n",
")[0].rhs()\n",
"\n",
"main_theorem1.corollary_r_bound = (main_theorem1.r_upper_bound1\n",
" .expand()\n",
" .subs(q==q_sol)\n",
" .subs(kappa==1)\n",
" .subs(psi**2 == Delta/nu^2)\n",
" .subs(1/psi**2 == nu^2/Delta)\n",
").expand()\n",
"\n",
"main_theorem1.corollary_r_bound"
{
"cell_type": "markdown",
"id": "6c1e0d68",
"metadata": {},
"source": [
"# Unsorted Extras"
]
},
{
"cell_type": "code",
"execution_count": 30,
"id": "896d26dd",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle c = \\frac{a_{v} r}{n} + \\frac{b_{q}}{n}\\)</html>"
],
"text/latex": [
"$\\displaystyle c = \\frac{a_{v} r}{n} + \\frac{b_{q}}{n}$"
],
"text/plain": [
"c == a_v*r/n + b_q/n"
]
},
"execution_count": 30,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"c_in_terms_of_q.subs([q_value_expr,beta_value_expr])"
]
},
{
"cell_type": "code",
"execution_count": 31,
"id": "51f22f7d",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle {\\left(a_{v} r + 2 \\, b_{q}\\right)} a_{v}\\)</html>"
],
"text/latex": [
"$\\displaystyle {\\left(a_{v} r + 2 \\, b_{q}\\right)} a_{v}$"
],
"text/plain": [
"(a_v*r + 2*b_q)*a_v"
]
},
"execution_count": 31,
"metadata": {},
"output_type": "execute_result"
}
],
"rhs_numerator = (positive_radius_condition_with_q\n",
" .rhs()\n",
" .subs([q_value_expr,beta_value_expr])\n",
" .factor()\n",
" .numerator()\n",
")\n",
"rhs_numerator"
]
},
{
"cell_type": "code",
"execution_count": 32,
"id": "8148f5cd",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle d > \\frac{{\\left(a_{v} r + 2 \\, b_{q}\\right)} a_{v}}{2 \\, n^{2}}\\)</html>"
],
"text/latex": [
"$\\displaystyle d > \\frac{{\\left(a_{v} r + 2 \\, b_{q}\\right)} a_{v}}{2 \\, n^{2}}$"
],
"text/plain": [
"d > 1/2*(a_v*r + 2*b_q)*a_v/n^2"
]
},
"execution_count": 32,
"metadata": {},
"output_type": "execute_result"
}
],
"(positive_radius_condition_with_q\n",
" .subs([q_value_expr,beta_value_expr])\n",
" .factor())"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "SageMath 9.8",
"language": "sage",
"name": "sagemath"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
}
},
"nbformat": 4,
"nbformat_minor": 5
}