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"assymptote_gap_condition2 = (k/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",
Luke Naylor
committed
"execution_count": 27,
"id": "553bba31",
"metadata": {
"scrolled": true
},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\left(r < \\frac{{\\operatorname{lcm}(m,2n^2)} q^{2}}{2 \\, {k_{v,q}}}, r < \\frac{{\\left({\\operatorname{ch}_1^{\\beta}(v)} - q\\right)}^{2} {\\operatorname{lcm}(m,2n^2)}}{2 \\, {k_{v,q}}} + R\\right)\\)</html>"
],
"text/latex": [
"$\\displaystyle \\left(r < \\frac{{\\operatorname{lcm}(m,2n^2)} q^{2}}{2 \\, {k_{v,q}}}, r < \\frac{{\\left({\\operatorname{ch}_1^{\\beta}(v)} - q\\right)}^{2} {\\operatorname{lcm}(m,2n^2)}}{2 \\, {k_{v,q}}} + R\\right)$"
],
"text/plain": [
"(r < 1/2*lcm_m_2n2*q^2/k, r < 1/2*(ch1bv - q)^2*lcm_m_2n2/k + R)"
Luke Naylor
committed
"execution_count": 27,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"r_upper_bound1 = (\n",
" assymptote_gap_condition1\n",
")\n",
"\n",
"assert r_upper_bound1.lhs() == r\n",
"\n",
"r_upper_bound2 = (\n",
" assymptote_gap_condition2\n",
" * (r-R) * lcm_m_2n2 / k + 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",
Luke Naylor
committed
"execution_count": 28,
"id": "602840cc",
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle r < \\min\\left( \\frac{1}{2} \\, {\\operatorname{lcm}(m,2n^2)} q^{2} , \\frac{1}{2} \\, {\\left({\\operatorname{ch}_1^{\\beta}(v)} - q\\right)}^{2} {\\operatorname{lcm}(m,2n^2)} + R \\right)\\)</html>"
],
"text/latex": [
"$\\displaystyle r < \\min\\left( \\frac{1}{2} \\, {\\operatorname{lcm}(m,2n^2)} q^{2} , \\frac{1}{2} \\, {\\left({\\operatorname{ch}_1^{\\beta}(v)} - q\\right)}^{2} {\\operatorname{lcm}(m,2n^2)} + R \\right)$"
],
"text/plain": [
"r < \\min\\left( \\frac{1}{2} \\, {\\operatorname{lcm}(m,2n^2)} q^{2} , \\frac{1}{2} \\, {\\left({\\operatorname{ch}_1^{\\beta}(v)} - q\\right)}^{2} {\\operatorname{lcm}(m,2n^2)} + R \\right)"
Luke Naylor
committed
"execution_count": 28,
"metadata": {},
"output_type": "execute_result"
"main_theorem1 = Object()\n",
"main_theorem1.r_upper_bound1 = r_upper_bound1.subs(k == 1).rhs()\n",
"main_theorem1.r_upper_bound2 = r_upper_bound2.subs(k == 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": "code",
Luke Naylor
committed
"execution_count": 29,
"metadata": {
"scrolled": true
},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\frac{{\\operatorname{ch}_1^{\\beta}(v)}^{2} {\\operatorname{lcm}(m,2n^2)} + 2 \\, R}{2 \\, {\\operatorname{ch}_1^{\\beta}(v)} {\\operatorname{lcm}(m,2n^2)}}\\)</html>"
],
"text/latex": [
"$\\displaystyle \\frac{{\\operatorname{ch}_1^{\\beta}(v)}^{2} {\\operatorname{lcm}(m,2n^2)} + 2 \\, R}{2 \\, {\\operatorname{ch}_1^{\\beta}(v)} {\\operatorname{lcm}(m,2n^2)}}$"
],
"text/plain": [
"1/2*(ch1bv^2*lcm_m_2n2 + 2*R)/(ch1bv*lcm_m_2n2)"
Luke Naylor
committed
"execution_count": 29,
"metadata": {},
"output_type": "execute_result"
}
],
"var(\"m\", domain=\"real\")\n",
"var(\"bgmlv_v\", domain=\"real\", latex_name=r\"\\Delta(v)\")\n",
"var(\"l\", latex_name=r\"\\ell\")\n",
"# Delta to represent bogomolov(v)\n",
Luke Naylor
committed
"# 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",
Luke Naylor
committed
"execution_count": 30,
"id": "c89865ff",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\frac{1}{8} \\, {\\operatorname{ch}_1^{\\beta}(v)}^{2} {\\operatorname{lcm}(m,2n^2)} + \\frac{1}{2} \\, R + \\frac{R^{2}}{2 \\, {\\operatorname{ch}_1^{\\beta}(v)}^{2} {\\operatorname{lcm}(m,2n^2)}}\\)</html>"
],
"text/latex": [
"$\\displaystyle \\frac{1}{8} \\, {\\operatorname{ch}_1^{\\beta}(v)}^{2} {\\operatorname{lcm}(m,2n^2)} + \\frac{1}{2} \\, R + \\frac{R^{2}}{2 \\, {\\operatorname{ch}_1^{\\beta}(v)}^{2} {\\operatorname{lcm}(m,2n^2)}}$"
],
"text/plain": [
"1/8*ch1bv^2*lcm_m_2n2 + 1/2*R + 1/2*R^2/(ch1bv^2*lcm_m_2n2)"
Luke Naylor
committed
"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",
").expand()\n",
"\n",
"main_theorem1.corollary_intermediate"
]
},
{
"cell_type": "code",
Luke Naylor
committed
"execution_count": 31,
"id": "9a56a088",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\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)}}\\)</html>"
],
"text/latex": [
"$\\displaystyle \\frac{1}{2} \\, R + \\frac{{\\Delta(v)} {\\operatorname{lcm}(m,2n^2)}}{8 \\, m} + \\frac{R^{2} m}{2 \\, {\\Delta(v)} {\\operatorname{lcm}(m,2n^2)}}$"
],
"text/plain": [
"1/2*R + 1/8*bgmlv_v*lcm_m_2n2/m + 1/2*R^2*m/(bgmlv_v*lcm_m_2n2)"
Luke Naylor
committed
"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(k==1)\n",
" .subs(ch1bv**2 == bgmlv_v/m)\n",
" .subs(1/ch1bv**2 == m/bgmlv_v)\n",
").expand()\n",
"\n",
"main_theorem1.corollary_r_bound"
{
"cell_type": "markdown",
"id": "aa54317e",
"metadata": {},
"source": [
"# Stronger Theorem Bound"
]
},
{
"cell_type": "code",
"execution_count": 32,
"id": "ec739837",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle r < \\min\\left( \\frac{{\\operatorname{lcm}(m,2n^2)} q^{2}}{2 \\, {k_{v,q}}} , \\frac{{\\left({\\operatorname{ch}_1^{\\beta}(v)} - q\\right)}^{2} {\\operatorname{lcm}(m,2n^2)}}{2 \\, {k_{v,q}}} + R \\right)\\)</html>"
],
"text/latex": [
"$\\displaystyle r < \\min\\left( \\frac{{\\operatorname{lcm}(m,2n^2)} q^{2}}{2 \\, {k_{v,q}}} , \\frac{{\\left({\\operatorname{ch}_1^{\\beta}(v)} - q\\right)}^{2} {\\operatorname{lcm}(m,2n^2)}}{2 \\, {k_{v,q}}} + R \\right)$"
],
"text/plain": [
"r < \\min\\left( \\frac{{\\operatorname{lcm}(m,2n^2)} q^{2}}{2 \\, {k_{v,q}}} , \\frac{{\\left({\\operatorname{ch}_1^{\\beta}(v)} - q\\right)}^{2} {\\operatorname{lcm}(m,2n^2)}}{2 \\, {k_{v,q}}} + R \\right)"
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]
},
"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}}{{k_{v,q}}} , \\frac{{\\left({\\operatorname{ch}_1^{\\beta}(v)} - q\\right)}^{2} n^{2}}{{k_{v,q}}} + R \\right)\\)</html>"
],
"text/latex": [
"$\\displaystyle r < \\min\\left( \\frac{n^{2} q^{2}}{{k_{v,q}}} , \\frac{{\\left({\\operatorname{ch}_1^{\\beta}(v)} - q\\right)}^{2} n^{2}}{{k_{v,q}}} + R \\right)$"
],
"text/plain": [
"r < \\min\\left( \\frac{n^{2} q^{2}}{{k_{v,q}}} , \\frac{{\\left({\\operatorname{ch}_1^{\\beta}(v)} - q\\right)}^{2} n^{2}}{{k_{v,q}}} + 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": {
"display_name": "SageMath 9.7",
"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
}