Bounds of Step 4, bounds:
Section 2.
- 2.2.1. Gamma< vor0A, if
y1 in [2.696,2sqrt2].
- 2.2.2. vorA< vor0A, if y1 in [2.696,2sqrt2],
y2,y3 in [2,2.45].
- 2.2.3. vorA< vor0A, if y1 in [2.696,2sqrt2],
y2,y5 in [2.45,2.51].
- 2.2.4. vor< vor0 + 0.003521, if y1 in [2.51,2sqrt2].
- 2.2.5. vor< vor0 - 0.003521, if
y1 in [2.696,2sqrt2], y2,y6 in [2.45,2.51],
y4 in [2.51,2sqrt2].
- 2.2.6. kappa < -0.003521,
y1 in [2.696,2sqrt2], y4 => 2sqrt2, y2,y6 => 2.45.
(Note anc(y1,y2,y6)=0.)
- 2.2.7.
Gamma< vor0A + 0.017, if y1 in [2.51,2.696].
- 2.2.8. vor< vor0 - 0.009, if y4 in [2.51,2sqrt2],
y1 in [2.51,2.696].
- 2.2.9. kappa< -0.017,
if y1 in [2.51,2.696], y4 => 2sqrt2.
- 2.4.1.
dih> 0.956 if y1 in [2.51,2sqrt2], y4 => 2.
- 2.6.1.
dih> 1.65 if y1 in [2.51,2sqrt2], y4 => 3.2.
Note: By Lemma of Cross Diag Stuff: 5/10/97, we have
D[delta,xi]>=0, i = 2,3,5,6, so wlog x2=x3=x5=x6=2.51^2.
- 2.6.2.
tau0> 1.01104 - 0.78701 dih, if y1 in [2.51,2sqrt2].
- 2.6.3.
tau> 1.01104 - 0.78701 dih, if y1 in [2.51,2sqrt2],
y4 in [2.51,2sqrt2] and dih<1.51.
- 2.6.4.
tauV,0 > 1.01104 - 0.78701 dih, if y1 in [2.51,2sqrt2],
y4 in [2sqrt2,3.2], and dih<1.51.
- 2.7.1. kappa< -0.029, if
if y1 in [2.51,2sqrt2] and y4 => 3.2.
- 2.7.2.
Gamma < vor0A + 0.01561, if y1 in [2.51,2sqrt2].
Section 2.10:
- 2.10.1.
dih> 1.714, if y1 in [2.7,2sqrt2], y4 => 3.2.
- 2.10.2.
tau0 > 0.99937 - 0.77627 dih,
if y1 in [2.51,2sqrt2].
- 2.10.3.
tauV > 0.99937 - 0.77627 dih,
if y1 in [2.51,2sqrt2], y4 in [2.51,2sqrt2],
and dih < 2pi-3(0.956)-1.65.
- 2.10.4.
tauV,0 > 0.99937 - 0.77627 dih,
if y1 in [2.51,2sqrt2], y4 in [2sqrt2,3.2],
and dih < 2pi-3(0.956)-1.65.
- 2.10.5.
dih> 1.714, if y1 in [2.51,2.7], y4 => 3.2,
y2 in [2,2.25].
- 2.11.1.
tauV> 1.039 - 0.77627 dih, if
y1 in [2.51,2.7], y4 in [2.51,2sqrt2],
and dih < 2pi-3(0.956)-1.65.
- 2.11.2.
tauV,0 > 1.039 - 0.77627 dih, if
y1 in [2.51,2.7], y4 in [2sqrt2,3.2],
and dih < 2pi-3(0.956)-1.65.
- 2.11.3.
tauG> 1.03869 - 0.77627 dih, if
y1 in [2.51,2.7], y2,y6 in [2.25,2.51].
- 2.11.4.
tauG> 1.03869 - 0.77627 dih, if
y1 in [2.51,2.7], y2,y6 in [2,2.25].
- 2.11.5. tauV> 1.039 - 0.77627 dih, if
y1 in [2.51,2.7], y2,y6 in [2.25,2.51],
and eta135 => sqrt2.
- 2.11.6.
tauV> 1.039 - 0.77627 dih, if
y1 in [2.51,2.7], y2,y6,y3 in [2.25,2.51].
- 2.11.7.
tauV> 1.039 - 0.77627 dih, if
y1 in [2.51,2.7], y2,y6,y5 in [2.25,2.51].
- 2.11.8. tauV> 1.039 - 0.77627 dih, if
y1 in [2.51,2.7], y3 in [2,2.25],
eta135 => sqrt2, eta126 <= sqrt2.
- 2.11.9.
tauV,0 > 1.039 - 0.77627 dih + (1.189+0.307) pt,
if y1 in [2.51,2sqrt2], y4 => 2sqrt2,
and dih < 1.77.
- 2.14.1.
Gamma < vor0A + 0.00935, if y1 in [2.57,2sqrt2].
- 2.14.2. vorA < vor0A,
if y1 in [2.51,2.57],
eta126 => sqrt2.
- 2.14.3. kappa(S)< -0.039,
if y1 in [2.51,2.57],
y2,y3,y5,y6 in [2,2.25],
y4 => 3.2.
- 2.14.4. kappa(S)< -0.033,
if y1 in [2.51,2.57],
y2,y3,y5 in [2,2.25],
y4 => 3.2.
- 2.14.5.
Gamma< vor0A+0.0927,
if y1 in [2.51,2.57],
y2 => 2.25.
- 2.14.6.
Gamma< vor0A, if y1 in [2.51,2.57], y2,y6 => 2.25.
- 2.14.7. vorA< vorA0 - 0.01,
if y1 in [2.51,2sqrt2],
eta126 <= sqrt2,
eta135 => sqrt2,
y3<=2.2.
In Inequalities 2.16.1 -- 2.16.12, the domain is all upright quarters.
The dihedral angle is measured along the diagonal.
- 2.16.1.
sigma0 < -4.3223 + 4.10113 dih.
- 2.16.2.
sigma0 < -0.9871 + 0.80449 dih
- 2.16.3.
sigma0 < -0.8756 + 0.70186 dih
- 2.16.4.
sigma0 < -0.3429 + 0.24573 dih
- 2.16.5.
sigma0 < -0.0024 + 0.00154 dih
- 2.16.6.
sigma0 < 0.1196 - 0.07611 dih
- 2.16.7.
- tau0 < -4.42873 + 4.16523 dih
- 2.16.8.
- tau0 < -1.01104 + 0.78701 dih
- 2.16.9.
- tau0 < -0.99937 + 0.77627 dih
- 2.16.10.
- tau0 < -0.34877 + 0.21916 dih
- 2.16.11.
- tau0 < -0.11434 + 0.05107 dih
- 2.16.12.
- tau0 < 0.07749 - 0.07106 dih
In Inequalities 2.16.13--2.16.25, y1 in [2.51,2sqrt2] and
y4 in [2.51,2sqrt2] and dih<2.46.
In Inequalities 2.16.26--2.16.38, y1 in [2.51,2sqrt(2)], y4 in [2sqrt(2),3.2],
and dih < 2.46.
- 2.16.26. vor0 < -3.58 + 2.28501 dih
- 2.16.27. vor0 < -2.715+ 1.67382 dih
- 2.16.28. vor0 < -1.517 + 0.8285 dih
- 2.16.29. vor0 < -0.858 + 0.390925 dih
- 2.16.30. vor0 < -0.358 + 0.12012 dih
- 2.16.31. vor0 < -0.186 + 0.05007 dih
- 2.16.32.
- tauV0 < -3.48 + 2.1747 dih
- 2.16.33.
- tauV0 < -3.06 + 1.87427 dih
- 2.16.34.
- tauV0 < -1.58 + 0.83046 dih
- 2.16.35.
- tauV0 < -1.06 + 0.48263 dih
- 2.16.36.
- tauV0 < -0.83 + 0.34833 dih
- 2.16.37.
- tauV0 < -0.50 + 0.1694 dih
- 2.16.38.
- tauV0 < -0.29 + 0.0822 dih
- 2.17.1.
dih < 2.184, y1 in [2.51,2sqrt2].
- 2.20.1. vor(Q,1.385)<0.00005, if y4 in [2.77,2sqrt2],
and eta456 => sqrt2.
- 2.20.2. vor(Q,1.385)<0.00005, if y4 in [2.77,2sqrt2],
and eta234 => sqrt2.
- 2.20.3. tauV(Q,1.385)>1.232 pt, if y4 in [2.77,2sqrt2],
and eta456 => sqrt2.
- 2.20.4. tauV(Q,1.385)>1.232 pt, if y4 in [2.77,2sqrt2],
and eta234 => sqrt2.
Section 3.
- 3.1.1. Assume that the simplex is not a quasi-regular tetrahedron
or a simplex. Assume that the first three edges are between 2 and 2.51
and that the last three edges are between 2 and 3.2.
Then tauV,0 > 1.189 pt unless
y4 => 2sqrt2, y2,y3,y5,y6<=2.2, y1 => 2.29.
- 3.2.1. tauV,0 > -0.307 pt if
y1<= 2.3999, y4 in [2sqrt2,3.2]
(and Delta => 0 so it exists).
Section 4.
- 4.2.1. dih(y1,y2,t0,1.6,1.6,y6) <
dih(y1,y2,y3,y4,y5,y6),
if y4=3.2, y5,y6 in [2,3.2],
and Delta(y12,...,1.62,y62) => 0.
- 4.4.1.
beta< dih(S), where beta is as in the text, and
y1 in [2.51,2sqrt2], y4=2.1, y5=2.51.
- 4.4.2.
beta0(v,v4) + dih(0,v4,v3,v)> pi/2, where
|v4-v| in [2,2.1], |v3-v|=2.51, |v3| in [2.51,2sqrt2].
- 4.4.3.
beta0(v4,v) < dih(0,v4,v3,v), where
|v4-v| in [2,2.1], |v3-v|=2.51, |v3| in [2.51,2sqrt2].
- 4.6.1. If S1=S(y1,y2,y3,...,y6) is special
(so tauV,0 (S1)<1.189 pt and y4>2sqrt2, and so forth),
then dih2(S1) < dih(S(y2,y1,t0,1.6,1.6,y6)).
- 4.7.1.
If S1=S(y1,y2,y3,..,y6) is special
(so tauV,0 (S1)<1.189 pt and y4>2sqrt2, and so forth),
then dih(S1) > dih(S(y1,y2,t0,1.6,1.6,y6))
+ dih(S(y1,y3,t0,1.6,1.6,y5)).
- 4.8.1.
If P=(0,v,v1,v2) is special, with diagonal (v1,v2), then
vor0(VP(v,v1,v2))> 2.125 pt.
- 4.9.1.
vor0(VP(v,v1,v2))> 1.538, if S=(0,v,v1,v2)=S(y1,...,y6),
with y4 => 3.2, and y5,y6 in [2,3.2].
Section 5.
Define sigmamax as in (2.21.2).
- 5.1.1. zeta pt sol(Q) - sigmamax (Q) > 1.189 pt,
for all flat quarters Q.
- 5.1.2. tauV > 2.524 pt, if y5,y6 in [2.51,3.2].
- 5.1.3. tauV,0 > 3.032 pt,
if y5 in [2.77,3.2], y6 in [2.51,3.2].
- 5.1.4. tauV,0 => 2.85 pt, if y5,y6 in [2.51,2.77],
- 5.1.4. tauV,0 => 2.85 pt, if y5,y6 in [2.51,2.77],
and eta456 => sqrt2.
- 5.1.5. tauV,0 => 4.268 pt, if y4,y5,y6 in [2.51,3.2].
- 5.1.6. tauV,0 (VP(v,v1,v2)) => 1.53 if |v1-v2| => 3.2.
- 5.1.7. tauV,0 (VP(v,v1,v2)) => 1.63 if |v1-v2| => 3.2,
|v-v1| => 2.51.
- 5.1.8. tauV,0 (VP(v,v1,v2)) => 1.63 if |v1-v2| => 3.2,
|v-v1| => 2.51, |v-v2| => 2.51.