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"<html>\\(\\displaystyle \\left(\\frac{\\kappa}{\\Omega} < \\frac{q^{2}}{2 \\, r}, \\frac{\\kappa}{\\Omega} < -\\frac{{\\left(\\psi - q\\right)}^{2}}{2 \\, {\\left(R - r\\right)}}\\right)\\)</html>"
],
"text/latex": [
"$\\displaystyle \\left(\\frac{\\kappa}{\\Omega} < \\frac{q^{2}}{2 \\, r}, \\frac{\\kappa}{\\Omega} < -\\frac{{\\left(\\psi - q\\right)}^{2}}{2 \\, {\\left(R - r\\right)}}\\right)$"
],
"text/plain": [
"(kappa/Omega < 1/2*q^2/r, kappa/Omega < -1/2*(psi - q)^2/(R - r))"
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"execution_count": 26,
"metadata": {},
"output_type": "execute_result"
"var(\"a_v b_q n Omega\") # Define symbols introduce for values of beta and q\n",
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"lcm_m_2n2 = Omega # more semantic variable name\n",
"\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",
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"assymptote_gap_condition1 = (kappa/lcm_m_2n2 < bgmlv2_d_upperbound_terms.hyperbolic)\n",
"assymptote_gap_condition2 = (kappa/lcm_m_2n2 < bgmlv3_d_upperbound_terms.hyperbolic)\n",
"assymptote_gap_condition1, assymptote_gap_condition2"
]
},
{
"cell_type": "markdown",
"id": "ce8bc94f",
"metadata": {},
"source": [
"Rearrange these two conditions into bounds for $r$:"
]
},
{
"cell_type": "code",
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"execution_count": 27,
"id": "553bba31",
"metadata": {
"scrolled": true
},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\left(r < \\frac{\\Omega q^{2}}{2 \\, \\kappa}, r < \\frac{\\Omega {\\left(\\psi - q\\right)}^{2}}{2 \\, \\kappa} + R\\right)\\)</html>"
],
"text/latex": [
"$\\displaystyle \\left(r < \\frac{\\Omega q^{2}}{2 \\, \\kappa}, r < \\frac{\\Omega {\\left(\\psi - q\\right)}^{2}}{2 \\, \\kappa} + R\\right)$"
],
"text/plain": [
"(r < 1/2*Omega*q^2/kappa, r < 1/2*Omega*(psi - q)^2/kappa + R)"
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"execution_count": 27,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"r_upper_bound1 = (\n",
" assymptote_gap_condition1\n",
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" * r * lcm_m_2n2 / kappa\n",
")\n",
"\n",
"assert r_upper_bound1.lhs() == r\n",
"\n",
"r_upper_bound2 = (\n",
" assymptote_gap_condition2\n",
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" * (r-R) * lcm_m_2n2 / kappa + R\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",
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"execution_count": 28,
"id": "602840cc",
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle r < \\min\\left( \\frac{1}{2} \\, \\Omega q^{2} , \\frac{1}{2} \\, \\Omega {\\left(\\psi - q\\right)}^{2} + R \\right)\\)</html>"
],
"text/latex": [
"$\\displaystyle r < \\min\\left( \\frac{1}{2} \\, \\Omega q^{2} , \\frac{1}{2} \\, \\Omega {\\left(\\psi - q\\right)}^{2} + R \\right)$"
],
"text/plain": [
"r < \\min\\left( \\frac{1}{2} \\, \\Omega q^{2} , \\frac{1}{2} \\, \\Omega {\\left(\\psi - q\\right)}^{2} + R \\right)"
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"execution_count": 28,
"metadata": {},
"output_type": "execute_result"
"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",
"r\"r < \\min\\left(\" + latex(main_theorem1.r_upper_bound1) + \",\" + latex(main_theorem1.r_upper_bound2) + r\"\\right)\""
"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",
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"execution_count": 29,
"metadata": {
"scrolled": true
},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\frac{\\Omega \\psi^{2} + 2 \\, R}{2 \\, \\Omega \\psi}\\)</html>"
],
"text/latex": [
"$\\displaystyle \\frac{\\Omega \\psi^{2} + 2 \\, R}{2 \\, \\Omega \\psi}$"
],
"text/plain": [
"1/2*(Omega*psi^2 + 2*R)/(Omega*psi)"
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"execution_count": 29,
"metadata": {},
"output_type": "execute_result"
}
],
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"var(\"Delta m\", domain=\"real\")\n",
"# Delta to represent bogomolov(v)\n",
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"# m to represent \\ell^2\n",
"\n",
"q_sol = solve(\n",
" main_theorem1.r_upper_bound1\n",
" == main_theorem1.r_upper_bound2\n",
" , q\n",
")[0].rhs()\n",
"\n",
"q_sol"
]
},
{
"cell_type": "code",
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"execution_count": 30,
"id": "c89865ff",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\frac{1}{8} \\, \\Omega \\psi^{2} + \\frac{1}{2} \\, R + \\frac{R^{2}}{2 \\, \\Omega \\psi^{2}}\\)</html>"
],
"text/latex": [
"$\\displaystyle \\frac{1}{8} \\, \\Omega \\psi^{2} + \\frac{1}{2} \\, R + \\frac{R^{2}}{2 \\, \\Omega \\psi^{2}}$"
],
"text/plain": [
"1/8*Omega*psi^2 + 1/2*R + 1/2*R^2/(Omega*psi^2)"
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"execution_count": 30,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"main_theorem1.corollary_intermediate = (main_theorem1.r_upper_bound1\n",
" .expand()\n",
" .subs(q==q_sol)\n",
" .subs(kappa==1)\n",
").expand()\n",
"\n",
"main_theorem1.corollary_intermediate"
]
},
{
"cell_type": "code",
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"execution_count": 31,
"id": "9a56a088",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
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"<html>\\(\\displaystyle \\frac{1}{2} \\, R + \\frac{\\Delta \\Omega}{8 \\, m} + \\frac{R^{2} m}{2 \\, \\Delta \\Omega}\\)</html>"
],
"text/latex": [
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"$\\displaystyle \\frac{1}{2} \\, R + \\frac{\\Delta \\Omega}{8 \\, m} + \\frac{R^{2} m}{2 \\, \\Delta \\Omega}$"
],
"text/plain": [
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"1/2*R + 1/8*Delta*Omega/m + 1/2*R^2*m/(Delta*Omega)"
Luke Naylor
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"execution_count": 31,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"main_theorem1.corollary_r_bound = (main_theorem1.r_upper_bound1\n",
" .expand()\n",
" .subs(q==q_sol)\n",
" .subs(kappa==1)\n",
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" .subs(ch1bv**2 == Delta/m)\n",
" .subs(1/ch1bv**2 == m/Delta)\n",
").expand()\n",
"\n",
"main_theorem1.corollary_r_bound"
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{
"cell_type": "markdown",
"id": "aa54317e",
"metadata": {},
"source": [
"# Stronger Theorem Bound"
]
},
{
"cell_type": "markdown",
"id": "8a736919",
"metadata": {},
"source": [
"$\\newcommand\\kappa{k_{v,q}}$\n",
"Redefine \\kappa to $\\kappa$ in latex"
]
},
{
"cell_type": "code",
"execution_count": 32,
"id": "ec739837",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle r < \\min\\left( \\frac{\\Omega q^{2}}{2 \\, \\kappa} , \\frac{\\Omega {\\left(\\psi - q\\right)}^{2}}{2 \\, \\kappa} + R \\right)\\)</html>"
],
"text/latex": [
"$\\displaystyle r < \\min\\left( \\frac{\\Omega q^{2}}{2 \\, \\kappa} , \\frac{\\Omega {\\left(\\psi - q\\right)}^{2}}{2 \\, \\kappa} + R \\right)$"
],
"text/plain": [
"r < \\min\\left( \\frac{\\Omega q^{2}}{2 \\, \\kappa} , \\frac{\\Omega {\\left(\\psi - q\\right)}^{2}}{2 \\, \\kappa} + R \\right)"
]
},
"execution_count": 32,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"main_theorem2 = Object()\n",
"main_theorem2.r_upper_bound1 = r_upper_bound1.rhs()\n",
"main_theorem2.r_upper_bound2 = r_upper_bound2.rhs()\n",
"\n",
"r\"r < \\min\\left(\" + latex(main_theorem2.r_upper_bound1) + \",\" + latex(main_theorem2.r_upper_bound2) + r\"\\right)\""
]
},
{
"cell_type": "markdown",
"id": "2fc47641",
"metadata": {},
"source": [
"# Stronger Theorem Corollary Bound"
]
},
{
"cell_type": "markdown",
"id": "51a6942e",
"metadata": {},
"source": [
"Specialize to $m=1$ or $2$"
]
},
{
"cell_type": "code",
"execution_count": 33,
"id": "78893533",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle r < \\min\\left( \\frac{n^{2} q^{2}}{\\kappa} , \\frac{n^{2} {\\left(\\psi - q\\right)}^{2}}{\\kappa} + R \\right)\\)</html>"
],
"text/latex": [
"$\\displaystyle r < \\min\\left( \\frac{n^{2} q^{2}}{\\kappa} , \\frac{n^{2} {\\left(\\psi - q\\right)}^{2}}{\\kappa} + R \\right)$"
],
"text/plain": [
"r < \\min\\left( \\frac{n^{2} q^{2}}{\\kappa} , \\frac{n^{2} {\\left(\\psi - q\\right)}^{2}}{\\kappa} + R \\right)"
]
},
"execution_count": 33,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"main_theorem2_corollary = Object()\n",
"main_theorem2_corollary.r_upper_bound1 = main_theorem2.r_upper_bound1.subs(lcm_m_2n2 == 2*n^2)\n",
"main_theorem2_corollary.r_upper_bound2 = main_theorem2.r_upper_bound2.subs(lcm_m_2n2 == 2*n^2)\n",
"\n",
"r\"r < \\min\\left(\" + latex(main_theorem2_corollary.r_upper_bound1) + \",\" + latex(main_theorem2_corollary.r_upper_bound2) + r\"\\right)\""
]
},
{
"cell_type": "markdown",
"id": "6c1e0d68",
"metadata": {},
"source": [
"# Unsorted Extras"
]
},
{
"cell_type": "code",
"execution_count": 34,
"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": 34,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"c_in_terms_of_q.subs([q_value_expr,beta_value_expr])"
]
},
{
"cell_type": "code",
"execution_count": 35,
"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": 35,
"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": 36,
"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": 36,
"metadata": {},
"output_type": "execute_result"
}
],
"(positive_radius_condition_with_q\n",
" .subs([q_value_expr,beta_value_expr])\n",
" .factor())"
]
}
],
"metadata": {
"kernelspec": {
Luke Naylor
committed
"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",
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"version": "3.11.3"
}
},
"nbformat": 4,
"nbformat_minor": 5
}