Appendix 1. Minor Hypotheses
Let octavor0(Q)=0.5( vor0(Q)+ vor0( hat Q)), and
octavor(Q) = 0.5( vor(Q)+ vor( hat Q)).
kappa(S) is defined in Section 3.4.
We let taunu, tauV, tau0, and tauGamma be
f- sol zeta pt, where f= nu, vor, vor0, and Gamma,
respectively.
betapsi is defined in Section 2.8.
Each calculation is accompanied by one or more reference numbers.
These identification numbers are needed to find further details
about the calculation in cite{H}.
The numbering that follows reflects the sections from which the hypotheses are drawn. Edge lengths whose bounds are not specified are assumed
to be between 2 and 2.51.
Section 2
-
{ 2.8.1} betapsi(y1,y3,y5) < dih3(S),
if y2,y3 in[2,2.23], y4 in[2.77,2 sqrt{2}], cos psi=y1/2.77.
{757995764}
-
{ 2.10.1} vor(Q,1.385)<0.00005, if y4 in[2.77,2 sqrt{2}],
and eta456 => sqrt2.
{275706375}
-
{ 2.10.2} vor(Q,1.385)<0.00005, if y4 in[2.77,2 sqrt{2}],
and eta234 => sqrt2.
{324536936}
-
{ 2.10.3} tauV(Q,1.385)>1.232 , pt, if y4 in[2.77,2 sqrt{2}],
and eta456 => sqrt2.
{983547118}
-
{ 2.10.4} tauV(Q,1.385)>1.232 , pt, if y4 in[2.77,2 sqrt{2}],
and eta234 => sqrt2.
{206278009}
Section 3
In Inequalities 3.23.1 -- 3.23.12, the domain is the set of upright quarters.
The dihedral angle is measured along the diagonal.
In Inequalities 3.23.13-- 3.23.25, y1 in[2.51,2 sqrt{2}] and
y4 in[2.51,2 sqrt{2}] and dih<2.46.
Let vorx = vor if the simplex has type C and
vorx= vor0 otherwise. Set taux = sol zeta pt - vorx.
In the Inequalities 3.23.26 -- 3.23.38, we assume y1 in[2.51,2 sqrt{2}],
y4 in[2 sqrt{2},3.2], and dih<2.46.
-
{ 3.6.1} Gamma< octavor0, if
y1 in[2.696,2 sqrt{2}].
{214637273}
-
{ 3.6.2} octavor< octavor0, if y1 in[2.696,2 sqrt{2}],
y2,y3 in[2,2.45].
{378432183}
-
{ 3.6.3} octavor< octavor0, if y1 in[2.696,2 sqrt{2}],
y2,y5 in[2.45,2.51].
{572206659}
-
{ 3.6.4} vor< vor0 + 0.003521, if y1 in[2.51,2 sqrt{2}].
{310679005}
-
{ 3.6.5} vor < vor0 - 0.003521, if
y1 in[2.696,2 sqrt{2}], y2,y6 in[2.45,2.51],
y4 in[2.77,2 sqrt{2}], eta234, eta456 <= sqrt{2}.
{122573656}
-
{ 3.6.6} vor < vor0 - 0.003521, if
y1 in[2.696,2 sqrt{2}], y2,y6 in[2.45,2.51],
y4 in[2.51,2.77].
{284970880}
-
{ 3.6.7} kappa< -0.003521,
y1 in[2.696,2 sqrt{2}], y2,y6 in[2.45,2.51],
y4 => 2.77,
eta234 => sqrt{2}.
{956875054}
-
{ 3.6.8} kappa< -0.003521,
y1 in[2.696,2 sqrt{2}], y2,y6 in[2.45,2.51],
y4 => 2.77,
eta456 => sqrt{2}.
{597997492}
-
{ 3.6.9} Gamma < octavor0 + 0.01561,
if y1 in[2.51,2 sqrt{2}].
{751772680}
-
{ 3.6.10} vor < vor0 -0.009,
if eta234, eta456 <= sqrt{2},
y1 in[2.51,2.696], y4 in[2.51,2 sqrt{2}].
{7444003}
-
{ 3.6.11} vor < vor0 -0.009.
if y1 in[2.51,2.696], y4 in[2.51,2.77].
{139787058}
-
{ 3.6.12} kappa < -0.017,
if y1 in[2.51,2.696], y4 in[2.77,2 sqrt{2}], eta234 => sqrt{2}.
{664200787}
-
{ 3.6.13} kappa < -0.017,
if y1 in[2.51,2.696], y4 in[2.77,2 sqrt{2}],
eta456 => sqrt{2}.
{390273147}
-
{ 3.8.1} dih> 0.956 if y1 in[2.51,2 sqrt{2}], y4 => 2.
{125103581}
-
{ 3.12.1} dih > 0.28, if y1 in[2.51,2 sqrt{2}], y4 ge2,
y5 in[2,2 sqrt{2}].
{504968542}
-
{ 3.14.1} tauV(S) > 0.13 + 0.2 ( dih(S)- pi/2),
if y1,y2 in[2.51,2 sqrt{2}], and rad(S) <= sqrt{2}.
{630927837}
-
{ 3.14.2} tauV(S, sqrt{2}) > 0.13 + 0.2 ( dih(S)- pi/2),
if y1,y2 in[2.51,2 sqrt{2}], and rad(S) => sqrt{2}.
{825907374}
-
{ 3.17.1} dih> 1.65 if y1 in[2.51,2 sqrt{2}], y4 ge3.2.
[Verification relies on Cross Diag. 5/10/97 lemma.]
{410091263}
-
{ 3.17.2} taunu> 1.01104 - 0.78701 dih, if y1 in[2.51,2 sqrt{2}].
{503100724}
-
{ 3.17.3} tauV > 1.01104 - 0.78701 dih, if
dih<1.51, y1 in[2.51,2 sqrt{2}], y4 in[2.51,2 sqrt{2}],
eta234, eta456 <= sqrt{2}.
{543902992}
-
{ 3.17.4} tauV > 1.01104 - 0.78701 dih, if
dih<1.51, y1 in[2.51,2 sqrt{2}], y4 in[2.51,2.77].
{324222121}
-
{ 3.17.5} tau0 > 1.01104 - 0.78701 dih, if
dih<1.51, y1 in[2.51,2 sqrt{2}], y4 in[2.77,3.2],
eta456 => sqrt{2}.
{988906740}
-
{ 3.17.6} tau0 > 1.01104 - 0.78701 dih, if
dih<1.51, y1 in[2.51,2 sqrt{2}], y4 in[2.77,3.2],
eta234 => sqrt{2}.
{58687038}
-
{ 3.17.7}
dih>1.23 if y1 in [2.51,2 sqrt{2}], and y4 => 2.51.
The verification uses monotonicity.
{499014780}
-
{ 3.17.8}
dih > 1.4167, if y1 in[2.51,2 sqrt{2}], and y4 ge2 sqrt{2}.
{901845849}
-
{ 3.18.1} kappa< -0.029, if
if y1 in[2.51,2 sqrt{2}], y4 => 3.2.
{19971603}
-
{ 3.21.1} dih> 1.714, if y1 in[2.7,2 sqrt{2}], y4 ge3.2.
The verification uses monotonicity as in 3.17.1 to
reduce to y2=y3=y5=y6=2.51.
{770716154}
-
{ 3.21.2} taunu> 0.99937 - 0.77627 dih,
if y1 in[2.51,2 sqrt{2}].
{555573208}
-
{ 3.21.3} tauV> 0.99937 - 0.77627 dih,
if y1 in[2.51,2 sqrt{2}], y4 in[2.51,2 sqrt{2}],
and dih < 2 pi-3(0.956)-1.65,
eta234, eta456 <= sqrt{2}.
{195592955}
-
{ 3.21.4} tauV> 0.99937 - 0.77627 dih,
if y1 in[2.51,2 sqrt{2}], y4 in[2.51,2.77],
and dih < 2 pi-3(0.956)-1.65.
{782587164}
-
{ 3.21.5} tau0> 0.99937 - 0.77627 dih,
if y1 in[2.51,2 sqrt{2}], y4 in[2.77,2 sqrt{2}],
and dih < 2 pi-3(0.956)-1.65,
eta456 => sqrt{2}.
{505147860}
-
{ 3.21.6} tau0> 0.99937 - 0.77627 dih,
if y1 in[2.51,2 sqrt{2}], y4 in[2.77,2 sqrt{2}],
and dih < 2 pi-3(0.956)-1.65,
eta234 => sqrt{2}.
{766168323}
-
{ 3.21.7} tau0> 0.99937 - 0.77627 dih,
if y1 in[2.51,2 sqrt{2}], y4 in[2 sqrt{2},3.2],
and dih < 2 pi-3(0.956)-1.65.
{535396206}
-
{ 3.21.8} dih> 1.714, if y1 in[2.51,2.7], y4 ge3.2,
y2 in[2,2.25].
Dimension reductions as in 3.17.1, reduce us to
y3=y5=y6=2.51, y2=2.25, y4=3.2.
{666090270}
-
{ 3.21.9} tauG> 1.03869 - 0.77627 dih, if
eta126, eta135 <= sqrt{2}.
y1 in[2.51,2.7], y2,y6 in[2.25,2.51],
{873528047}
-
{ 3.21.10} tauG> 1.03869 - 0.77627 dih, if
y1 in[2.51,2.7], y2,y6 in[2,2.25],
and eta126, eta135 <= sqrt{2}.
{608557364}
-
{ 3.21.11} tauV> 1.039 - 0.77627 dih, if
y1 in[2.51,2.7], y2,y6 in[2.25,2.51],
and eta135 => sqrt{2}.
{735941538}
-
{ 3.21.12} tauV> 1.039 - 0.77627 dih, if
y1 in[2.51,2.7], y2,y6,y3 in[2.25,2.51].
{660620561}
-
{ 3.21.13} tauV> 1.039 - 0.77627 dih, if
y1 in[2.51,2.7], y2,y6,y5 in[2.25,2.51].
{298303482}
-
{ 3.21.14} tauV> 1.039 - 0.77627 dih, if
y1 in[2.51,2.7], y3 in[2,2.25],
eta135 => sqrt{2}, eta126 <= sqrt{2}.
{768640004}
-
{ 3.21.15} tau0 > 1.039 - 0.77627 dih + (1.189+0.307) , pt,
if y1 in[2.51,2 sqrt{2}], y4 => 2 sqrt{2},
and dih < 1.77.
A monotonicity argument gives y4 <= 3.365.
%This is the
%monotonicity argument that is embedded in the verification function
%{ tt x4upperfromdihupper}.
{653493427}
-
{ 3.23.1} nu < -4.3223 + 4.10113 dih.
{9052168}
-
{ 3.23.2} nu < -0.9871 + 0.80449 dih
{746202672}
-
{ 3.23.3} nu < -0.8756 + 0.70186 dih
{664841332}
-
{ 3.23.4} nu < -0.3429 + 0.24573 dih
{124343130}
-
{ 3.23.5} nu < -0.0024 + 0.00154 dih
{169912374}
-
{ 3.23.6} nu < 0.1196 - 0.07611 dih
{649751800}
-
{ 3.23.7} - taunu < -4.42873 + 4.16523 dih
{26167284}
-
{ 3.23.8} - taunu < -1.01104 + 0.78701 dih
{934011796}
-
{ 3.23.9} - taunu < -0.99937 + 0.77627 dih
{718142034}
-
{ 3.23.10} - taunu < -0.34877 + 0.21916 dih
{929512070}
-
{ 3.23.11} - taunu < -0.11434 + 0.05107 dih
{312204748}
-
{ 3.23.12} - taunu < 0.07749 - 0.07106 dih
{705959985}
-
{ 3.23.13} vorx < -3.421 + 2.28501 dih
{649592321}
-
{ 3.23.14} vorx < -2.616 + 1.67382 dih
{600996944}
-
{ 3.23.15} vorx < -1.4486 + 0.8285 dih
{70667639}
-
{ 3.23.16} vorx < -0.79 + 0.390925 dih
{99182343}
-
{ 3.23.17} vorx < -0.3088 + 0.12012 dih
{578762805}
-
{ 3.23.18} vorx < -0.1558 + 0.0501 dih
{557125557}
-
{ 3.23.19} - taux < -3.3407 + 2.1747 dih
{719735900}
-
{ 3.23.20} - taux < -2.945 + 1.87427 dih
{359616783}
-
{ 3.23.21} - taux < -1.5035 + 0.83046 dih
{440833181}
-
{ 3.23.22} - taux < -1.0009 + 0.48263 dih
{578578364}
-
{ 3.23.23} - taux < -0.7787 + 0.34833 dih
{327398152}
-
{ 3.23.24} - taux < -0.4475 + 0.1694 dih
{314861952}
-
{ 3.23.25} - taux < -0.2568 + 0.0822 dih
{234753056}
-
{ 3.23.26} vor0 < -3.58 + 2.28501 dih
{555481748}
-
{ 3.23.27} vor0 < -2.715 + 1.67382 dih
{615152889}
-
{ 3.23.28} vor0 < -1.517 + 0.8285 dih
{647971645}
-
{ 3.23.29} vor0 < -0.858 + 0.390925 dih
{516606403}
-
{ 3.23.30} vor0 < -0.358 + 0.12012 dih
{690552204}
-
{ 3.23.31} vor0 < -0.186 + 0.0501 dih
{852763473}
-
{ 3.23.32} - tau0 < -3.48 + 2.1747 dih
{679673664}
-
{ 3.23.33} - tau0 < -3.06 + 1.87427 dih
{926514235}
-
{ 3.23.34} - tau0 < -1.58 + 0.83046 dih
{459744700}
-
{ 3.23.35} - tau0 < -1.06 + 0.48263 dih
{79400832}
-
{ 3.23.36} - tau0 < -0.83 + 0.34833 dih
{277388353}
-
{ 3.23.37} - tau0 < -0.50 + 0.1694 dih
{839852751}
-
{ 3.23.38} - tau0 < -0.29 + 0.0822 dih
{787458652}
-
{ 3.26.1} Gamma < octavor0 + 0.00935, if
y1 in[2.57,2 sqrt{2}].
{366146051}
-
{ 3.26.2} octavor < octavor0,
if y1 in[2.51,2.57],
eta126 => sqrt2.
{875762896}
-
{ 3.26.3} kappa(S)< -0.039,
if y1 in[2.51,2.57],
y2,y3,y5,y6 in[2,2.25],
y4 ge3.2, Delta ge0.
{357223197}
-
{ 3.26.4} kappa(S)< -0.033,
if y1 in[2.51,2.57],
y2,y3,y5 in[2,2.25],
y4 ge3.2.
{279477953}
-
{ 3.26.5} Gamma< octavor0+0.00928,
if y1 in[2.51,2.57],
y2 in [2.25,2.51].
{675766140}
-
{ 3.26.6} Gamma< octavor0,
if y1 in[2.51,2.57],
y2,y6 in [2.25,2.51].
{520734758}
-
{ 3.26.7} octavor< octavor0 - 0.004131,
if y1 in[2.51,2 sqrt{2}],
eta126 <= sqrt2,
eta135 => sqrt2,
y3 le2.2.
{385332676}
-
{ 3.28.1} dih < 2.184, if y1 in[2.51,2 sqrt{2}].
{971555266}
Section 4
-
{ 4.2.1} tau0 > -0.307 , pt if
y1 <= 2.38, y4 in[2 sqrt2,3.2].
{596297497}
-
{ 4.2.2} vor0 < 0.832 , pt if
y1 <= 2.38, y4 in[2 sqrt2,3.2].
{738795425}
-
{ 4.2.4, 4.2.5}
( zeta pt - phi(t0,t0))s - sum alphai (1-hi/t0)
( phi(hi,t0)- phi(t0,t0))
&> -0.307 , pt,
phi(t0,t0)s + sum alphai (1-hi/t0)
( phi(hi,t0)- phi(t0,t0))
&< 0.832 , pt,
for y1 in[2.38,2.51], alphai= dihi(S), s = sol(S).
{932905720,422037052}
Section 5
Define sigmaMax as in Section 3.32.
-
{ 5.7.1} sol(Q) zeta , pt- sigmaMax(Q) > 1.189 , pt,
for all flat quarters Q.
{636885517}
-
{ 5.7.2} tau0 > 3.032 , pt,
if y5 in[2.77,3.2], y6 in[2.51,3.2].
{548033832}
-
{ 5.1.4} tau0 => 2.83 , pt, if y5,y6 in[2.51,2.77],
and eta456 => sqrt{2}.
{486752005}
-
{ 5.1.5} tau0> 4.268 , pt, if y4,y5,y6 in[2.51,3.2].
{718323672}
-
{ 5.10.1, 5.10.2} tau0(R) > 1.2 , pt and
vor0(R)< 0.5 , pt, where R is the
set formed by two adjacent simplices S(y1, ldots,y6), and
S(y1',y2,y3,y4,y5',y6'), if
y1 in[2.51,2 sqrt{2}], y4 in[2 sqrt{2},3.2], and
d => 2.51,
where d is the length of the diagonal extending
between the first vertices of the two simplices.
{554886232,273124689}
-
{ 5.13.1} sigmaMax(Q) <= 0, for all flat quarters
that are not of type B.
{515664917}
-
{ 5.13.2}
tau0 -2(0.008) > 2.518 , pt, if y5 in[2.51,3.2],
y6 in[2 sqrt{2},3.2].
{437034327}
-
{ 5.13.3}
tauV > 2.518 , pt, if y5,y6 in[2.51,2.77].
{697959796}
-
{ 5.13.4}
tau0 > 2.518 , pt, if y5,y6 in[2.51,2 sqrt{2}], and
the simplex does not have type A.
{146521567}
-
{ 5.13.5}
vor < -1.0319 , pt, if y5,y6 in[2.51,2.77].
{688766907}
-
{ 5.13.6}
vor0+2(0.008)< -1.0319 , pt, if y5 in[2.51,3.2],
y6 in[2 sqrt{2},3.2].
{181315466}
-
{ 5.13.7}
vor < -1.0319 , pt, if y5,y6 in[2.51,2.77], and
the simplex does not have type A.
{878617630}
-
{ 5.13.8}
tau0 > 4.0 , pt + 3(0.008), if y4 in[2.77,3.2],
y5,y6 in [2.51,3.2].
{294773427}
-
{ 5.13.9}
vor0< -2.063 , pt, if y4,y5,y6 in[2.51,3.2].
{640919742}
-
{ 5.13.10}
vor0+3(0.008)< -2.063 , pt, if y4 in[2.6,3.2],
y5,y6 in[2.51,3.2].
{555690672}
Section 6
-
{ 6.6.1}
dih<1.63, if y6 in[2.51,3.02].
{614013885}
-
{6.6.2}
dih<1.51, if y5,y6 in[2.51,3.02].
{734934433}