diff --git a/main.tex b/main.tex
index e66912c0606cc00408e002c939ad99907dcf6f2a..0065588d181069e9c5a6b3ea9861c56916ca646d 100644
--- a/main.tex
+++ b/main.tex
@@ -167,7 +167,11 @@ $\chern(F)$, and so we can write:
\begin{sagesilent}
ts = stability.Tilt
beta_min = var("beta", domain="real")
- c_lower_bound = -(ts(beta=beta_min).rank(u)/ts().alpha).expand() + c
+
+ c_lower_bound = -(
+ ts(beta=beta_min).rank(u)
+ /ts().alpha
+ ).expand() + c
var("q", domain="real")
c_in_terms_of_q = c_lower_bound + q
@@ -205,7 +209,11 @@ Expressing $c$ in terms of $q$ as defined in (eqn \ref{eqn-cintermsofm})
we get the following:
\begin{sagesilent}
- bgmlv1_with_q = bgmlv1.expand().subs(c == c_in_terms_of_q)
+ bgmlv1_with_q = (
+ bgmlv1
+ .expand()
+ .subs(c == c_in_terms_of_q)
+ )
\end{sagesilent}
\begin{equation}
@@ -220,6 +228,7 @@ This can be rearranged to express a bound on $d$ as follows:
var("r_alt",domain="real") # r_alt = r - R/2 temporary substitution
bgmlv1_with_q_reparam = (bgmlv1_with_q.subs(r == r_alt + R/2)/r_alt).expand()
+
bgmlv1_d_ineq = (
((0 >= -bgmlv1_with_q_reparam)/4 + d) # Rearrange for d
.subs(r_alt == r - R/2) # Resubstitute r back in
@@ -236,45 +245,47 @@ This can be rearranged to express a bound on $d$ as follows:
\begin{sagesilent}
# Seperate out the terms of the lower bound for d
- bgmlv1_d_lowerbound_exp_term = (bgmlv1_d_lowerbound*(R-2*r)).expand().subs(r==2*R)/(R-2*r)
- bgmlv1_d_lowerbound_const_term = bgmlv1_d_lowerbound.subs(1/(R-2*r) == 0).subs(r==0)
- bgmlv1_d_lowerbound_linear_term = bgmlv1_d_lowerbound.subs(1/(R-2*r) == 0).subs(r==1)*r
+ bgmlv1_d_lowerbound_without_hyp = bgmlv1_d_lowerbound.subs(1/(R-2*r) == 0)
- # Verify the simplified forms of the terms that will be mentioned in text
- assert bgmlv1_d_lowerbound_const_term == v.twist(beta_min).ch[2]/2 + beta_min*q, "fail"
- assert bgmlv1_d_lowerbound_linear_term == (v.twist(beta_min).ch[2]/2 + beta_min^2/2 + beta_min*q)*r, "fail"
+ bgmlv1_d_lowerbound_exp_term = (
+ bgmlv1_d_lowerbound
+ - bgmlv1_d_lowerbound_without_hyp
+ ).expand()
- assert bgmlv1_d_lowerbound_exp_term == (
- - 2*R*v.twist(beta_min).ch[2]
- - 3*R^2*beta_min^2
- - 4*R*beta_min*q
- + C*q
- - q^2
- ).expand()/(R-2*r), "fail"
+ bgmlv1_d_lowerbound_const_term = bgmlv1_d_lowerbound_without_hyp.subs(r==0)
- assert bgmlv1_d_lowerbound_exp_term == (
- - 2*D*R + (C^2)/4
- - ((C - 4*R*beta_min)/2 - q)^2
- ).expand()/(R-2*r), "fail"
+ bgmlv1_d_lowerbound_linear_term = (
+ bgmlv1_d_lowerbound_without_hyp
+ - bgmlv1_d_lowerbound_const_term
+ ).expand()
+
+ # Verify the simplified forms of the terms that will be mentioned in text
+ assert bgmlv1_d_lowerbound_const_term == (
+ v.twist(beta_min).ch[2]/2
+ + beta_min*q
+ )
assert bgmlv1_d_lowerbound_exp_term == (
- (2*R*beta_min + q)
- *(2*R*beta_min + q - C)
- + 2*D*R
- ).expand()/(2*r - R), "fail"
+ (
+ - R*v.twist(beta_min).ch[2]/2
+ - R*beta_min*q
+ + C*q
+ - q^2
+ )/(R-2*r)
+ ).expand()
\end{sagesilent}
+
\noindent
Viewing equation \ref{eqn-bgmlv1_d_lowerbound} as a lower bound for $d$ given
as a function of $r$, the terms can be rewritten as follows.
The constant term in $r$ is
-$\chern^{\beta}_2(F) + \beta q$.
+$\chern^{\beta}_2(F)/2 + \beta q$.
The linear term in $r$ is
-$(\chern^{\beta}_2(F)/2 + \beta^2/2 + \beta q)r$.
+$\sage{bgmlv1_d_lowerbound_linear_term}$.
Finally, there's an hyperbolic term in $r$ which tends to 0 as $r \to \infty$,
and can be written:
-$\frac{-2R\chern^{\beta}_2(F) - 3R^2\beta^2 - 4Rq\beta + Cq - q^2}{R-2r}$ or
-$\frac{(2R\beta + q)(2R\beta + q - C) + 2DR}{2r-R}$.
+$\frac{R\chern^{\beta}_2(F)/2 + R\beta q - Cq + q^2 }{2r-R}$.
In the case $\beta = \beta_{-}$ (or $\beta_{+}$) we have
$\chern^{\beta}_2(F) = 0$,
so some of these expressions simplify.
@@ -298,7 +309,11 @@ Expressing $c$ in terms of $q$ as defined in (eqn \ref{eqn-cintermsofm})
we get the following:
\begin{sagesilent}
- bgmlv2_with_q = bgmlv2.expand().subs(c == c_in_terms_of_q)
+ bgmlv2_with_q = (
+ bgmlv2
+ .expand()
+ .subs(c == c_in_terms_of_q)
+ )
\end{sagesilent}
\begin{equation}
@@ -311,11 +326,12 @@ This can be rearranged to express a bound on $d$ as follows:
\begin{sagesilent}
bgmlv2_d_ineq = (
- ((0 <= bgmlv2_with_q)/2/r + d) # Rearrange for d
- .expand()
- )
+ (0 <= bgmlv2_with_q)/2/r # rescale assuming r > 0
+ + d # Rearrange for d
+ ).expand()
- bgmlv2_d_lowerbound = bgmlv2_d_ineq.rhs() # Keep hold of lower bound for d
+ # Keep hold of lower bound for d
+ bgmlv2_d_lowerbound = bgmlv2_d_ineq.rhs()
\end{sagesilent}
\begin{equation}
@@ -325,9 +341,16 @@ This can be rearranged to express a bound on $d$ as follows:
\begin{sagesilent}
# Seperate out the terms of the lower bound for d
- bgmlv2_d_lowerbound_exp_term = (bgmlv2_d_lowerbound*r).expand().subs(r==0)/r
- bgmlv2_d_lowerbound_const_term = bgmlv2_d_lowerbound.subs(1/r == 0).subs(r==0)
- bgmlv2_d_lowerbound_linear_term = bgmlv2_d_lowerbound.subs(1/r == 0).subs(r==1)*r
+ bgmlv2_d_lowerbound_without_hyp = bgmlv2_d_lowerbound.subs(1/r == 0)
+ bgmlv2_d_lowerbound_const_term = bgmlv2_d_lowerbound_without_hyp.subs(r==0)
+ bgmlv2_d_lowerbound_linear_term = (
+ bgmlv2_d_lowerbound_without_hyp
+ - bgmlv2_d_lowerbound_const_term
+ ).expand()
+ bgmlv2_d_lowerbound_exp_term = (
+ bgmlv2_d_lowerbound
+ - bgmlv2_d_lowerbound_without_hyp
+ ).expand()
\end{sagesilent}
Viewing equation \ref{eqn-bgmlv2_d_lowerbound} as a lower bound for $d$ in term
@@ -361,7 +384,11 @@ Expressing $c$ in terms of $q$ as defined in (eqn \ref{eqn-cintermsofm})
we get the following:
\begin{sagesilent}
- bgmlv3_with_q = bgmlv3.expand().subs(c == c_in_terms_of_q)
+ bgmlv3_with_q = (
+ bgmlv3
+ .expand()
+ .subs(c == c_in_terms_of_q)
+ )
\end{sagesilent}
\begin{equation}
@@ -376,14 +403,17 @@ This can be rearranged to express a bound on $d$ as follows:
var("r_alt",domain="real") # r_alt = r - R temporary substitution
bgmlv3_with_q_reparam = (
- bgmlv3_with_q.subs(r == r_alt + R)
+ bgmlv3_with_q
+ .subs(r == r_alt + R)
/r_alt # This operation assumes r_alt > 0
).expand()
+
bgmlv3_d_ineq = (
((0 <= bgmlv3_with_q_reparam)/2 + d) # Rearrange for d
.subs(r_alt == r - R) # Resubstitute r back in
.expand()
)
+
# Check that this equation represents a bound for d
assert bgmlv3_d_ineq.lhs() == d, f"Inequality is of the form: {bgmlv3_d_ineq}"
bgmlv3_d_upperbound = bgmlv3_d_ineq.rhs() # Keep hold of lower bound for d
@@ -398,21 +428,26 @@ This can be rearranged to express a bound on $d$ as follows:
# Seperate out the terms of the lower bound for d
bgmlv3_d_upperbound_without_hyp = bgmlv3_d_upperbound.subs(1/(R-r) == 0)
+
bgmlv3_d_upperbound_const_term = bgmlv3_d_upperbound_without_hyp.subs(r==0)
+
bgmlv3_d_upperbound_linear_term = (
bgmlv3_d_upperbound_without_hyp
- bgmlv3_d_upperbound_const_term
).expand()
+
bgmlv3_d_upperbound_exp_term = (
bgmlv3_d_upperbound
- bgmlv3_d_upperbound_without_hyp
).expand()
# Verify the simplified forms of the terms that will be mentioned in text
+
assert bgmlv3_d_upperbound_const_term == (
v.twist(beta_min).ch[2]
+ beta_min*q
).expand()
+
assert bgmlv3_d_upperbound_exp_term == (
R*v.twist(beta_min).ch[2]
+ (C - q)^2/2