This page contains information on the interval verifications
for Sphere Packings IV. This was adapted from Sphere Packings IV,
Appendix 1 on May 6, 1998.
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% RETIRED
% retired 5/6/98: refno{597997492}, refno{279477953}
% retired 5/6/98: refno{357223197} refno{19971603}
% retired 5/10/98:$\dih<1.63$, if $y_6\in[2.51,3.02]$. \refno{614013885}
% ret. 5/10/98: $\dih<1.51$, if $y_5,y_6\in[2.51,3.02]$. \refno{734934433}
%\refno{247814184-5/11/98-A1}
%\refno{909431001-5/11/98-A1}
%\refno{471064815-5/11/98-A1}
%\refno{122573656-5/11/98-A11}
Let octavor0(Q)=0.5( vor0(Q)+ vor0(hat Q)), and
octavor(Q) = 0.5( vor(Q)+ vor(hat Q)).
We let taunu, tauV, tau0, and tauGamma be
f- sol zeta pt, where f=nu, vor, vor0, and Gamma,
respectively.
Edge lengths whose bounds are not specified are assumed
to be between 2 and 2.51.
Section A1
betapsi is defined in Section 2.8.
-
betapsi(y1,y3,y5)
betapsi(y1,y3,y5) < dih3(S),
provided y4=3.2, y5=2.51, y6=2, cos psi=y1/2.51.
refno{735258244}
-
betapsi(y1,y2,y6)
betapsi(y1,y2,y6)
dih(R(y2/2,eta126,y1/(2cospsi)))
vor(Q,1.385)<0.00005, if y4 in[2.77,2 sqrt{2}],
and eta{456} ge sqrt2.
refno{275706375}
-
vor(Q,1.385)<0.00005, if y4 in[2.77,2 sqrt{2}],
and eta{234} ge sqrt2.
refno{324536936}
-
tauV(Q,1.385)>0.0682, if y4 in[2.77,2 sqrt{2}],
and eta{456} ge sqrt2.
refno{983547118}
-
tauV(Q,1.385)>0.0682, if y4 in[2.77,2 sqrt{2}],
and eta{234} ge sqrt2.
refno{206278009}
Section A2
In Inequalities A2 and A3,
the domain is the set of upright quarters.
The dihedral angle is measured along the diagonal.
-
nu < -4.3223 + 4.10113 dih.
refno{413688580}
-
nu < -0.9871 + 0.80449 dih
refno{805296510}
-
nu < -0.8756 + 0.70186 dih
refno{136610219}
-
nu < -0.3404 + 0.24573 dih
refno{379204810}
-
nu < -0.0024 + 0.00154 dih
refno{878731435}
-
nu < 0.1196 - 0.07611 dih
refno{891740103}
Section A3
-
- taunu < -4.42873 + 4.16523 dih
refno{334002329}
-
- taunu < -1.01104 + 0.78701 dih
refno{883139937}
-
- taunu < -0.99937 + 0.77627 dih
refno{507989176}
-
- taunu < -0.34877 + 0.21916 dih
refno{244435805}
-
- taunu < -0.11434 + 0.05107 dih
refno{930176500}
-
- taunu < 0.07749 - 0.07106 dih
refno{815681339}
Section A4
In A4 and A5, 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.
-
vorx < -3.421 + 2.28501 dih
refno{649592321}
-
vorx < -2.616 + 1.67382 dih
refno{600996944}
-
vorx < -1.4486 + 0.8285 dih
refno{70667639}
-
vorx < -0.79 + 0.390925 dih
refno{99182343}
-
vorx < -0.3088 + 0.12012 dih
refno{578762805}
-
vorx < -0.1558 + 0.0501 dih
refno{557125557}
Section A5
-
- taux < -3.3407 + 2.1747 dih
refno{719735900}
-
- taux < -2.945 + 1.87427 dih
refno{359616783}
-
- taux < -1.5035 + 0.83046 dih
refno{440833181}
-
- taux < -1.0009 + 0.48263 dih
refno{578578364}
-
- taux < -0.7787 + 0.34833 dih
refno{327398152}
-
- taux < -0.4475 + 0.1694 dih
refno{314861952}
-
- taux < -0.2568 + 0.0822 dih
refno{234753056}
-
Section A6
In the Inequalities A6 and A7, we assume y1 in[2.51,2 sqrt{2}],
y4 in[2 sqrt{2},3.2], and dih<2.46.
-
vor0 < -3.58 + 2.28501 dih
refno{555481748}
-
vor0 < -2.715 + 1.67382 dih
refno{615152889}
-
vor0 < -1.517 + 0.8285 dih
refno{647971645}
-
vor0 < -0.858 + 0.390925 dih
refno{516606403}
-
vor0 < -0.358 + 0.12012 dih
refno{690552204}
-
vor0 < -0.186 + 0.0501 dih
refno{852763473}
Section A7
-
- tau0 < -3.48 + 2.1747 dih
refno{679673664}
-
- tau0 < -3.06 + 1.87427 dih
refno{926514235}
-
- tau0 < -1.58 + 0.83046 dih
refno{459744700}
-
- tau0 < -1.06 + 0.48263 dih
refno{79400832}
-
- tau0 < -0.83 + 0.34833 dih
refno{277388353}
-
- tau0 < -0.50 + 0.1694 dih
refno{839852751}
-
- tau0 < -0.29 + 0.0822 dih
refno{787458652}
Section A8
Some of these verifications rely on monotonicity arguments
established by hand and based on I.8.
-
dih>1.23 if y1 in [2.51,2 sqrt{2}], and y4 ge 2.51.
(uses monotonicity).
refno{499014780}
-
dih > 1.4167, if y1 in[2.51,2 sqrt{2}], and y4 ge2 sqrt{2}.
refno{901845849}
-
dih> 1.65 if y1 in[2.51,2 sqrt{2}], y4 ge3.2
(uses monotonicity).
refno{410091263}
-
dih> 0.956 if y1 in[2.51,2 sqrt{2}], y4 ge 2.
refno{125103581}
-
dih > 0.28, if y1 in[2.51,2 sqrt{2}], y4 ge2,
y5 in[2,2 sqrt{2}].
refno{504968542}
-
dih> 1.714, if y1 in[2.7,2 sqrt{2}], y4 ge3.2
(uses monotonicity).
refno{770716154}
-
dih> 1.714, if y1 in[2.51,2.7], y4 ge3.2,
y2 in[2,2.25]
(uses monotonicity).
refno{666090270}
-
dih < 2.184, if y1 in[2.51,2 sqrt{2}].
refno{971555266}
Section A9
kappa(S) is defined in Section 3.3.
-
kappa< -0.003521,
y1 in[2.696,2 sqrt{2}], y2,y6 in[2.45,2.51],
y4 ge 2.77,
refno{956875054}
-
kappa < -0.017,
if y1 in[2.51,2.696], y4 in[2.77,2 sqrt{2}], eta{234} ge sqrt{2}.
refno{664200787}
-
kappa < -0.017,
if y1 in[2.51,2.696], y4 in[2.77,2 sqrt{2}],
eta{456} ge sqrt{2}.
refno{390273147}
-
kappa< -0.02274 = xikG-xiG', if
y1 in[2.57,2 sqrt{2}], y4 ge 3.2,
Delta ge0.
By monotonicity we may assume y4=3.2.
refno{654422246}
-
kappa< xik = -0.029, if
y1 in[2.51,2.57], y4 ge 3.2,
Delta ge0.
By monotonicity we may assume y4=3.2.
refno{366536370}
-
kappa< -0.03883,
if y1 in[2.51,2.57],
y2,y3,y5,y6 in[2,2.25],
y4 ge3.2, Delta ge0.
By monotonicity we may assume y4=3.2.
refno{62532125}
-
kappa< -0.0325,
if y1 in[2.51,2.57],
y2,y3,y5 in[2,2.25],
y4 ge3.2,
Delta ge0.
By monotonicity we may assume y4=3.2.
refno{370631902}
Section A10
-
Gamma< octavor0, if
y1 in[2.696,2 sqrt{2}].
refno{214637273}
-
Gamma < octavor0 + 0.01561,
if y1 in[2.51,2 sqrt{2}].
refno{751772680}
-
Gamma < octavor0 + 0.00935, if
y1 in[2.57,2 sqrt{2}].
refno{366146051}
-
Gamma< octavor0+0.00928,
if y1 in[2.51,2.57],
y2 in [2.25,2.51].
refno{675766140}
-
Gamma< octavor0,
if y1 in[2.51,2.57],
y2,y6 in [2.25,2.51].
refno{520734758}
Section A11
-
octavor< octavor0, if y1 in[2.696,2 sqrt{2}],
y2,y3 in[2,2.45].
refno{378432183}
-
octavor< octavor0, if y1 in[2.696,2 sqrt{2}],
y2,y5 in[2.45,2.51].
refno{572206659}
-
vor< vor0 + 0.003521, if y1 in[2.51,2 sqrt{2}].
refno{310679005}
-
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].
refno{284970880}
-
vor < vor0 -0.009.
if y1 in[2.51,2.696], y4 in[2.51,2 sqrt2].
refno{972111620}
-
octavor < octavor0,
if y1 in[2.51,2.57],
eta{126} ge sqrt2.
refno{875762896}
-
octavor Section A12
-
tauV(S) > 0.13 + 0.2 ( dih(S)-pi/2),
if y1,y2 in[2.51,2 sqrt{2}], and rad(S) le sqrt{2}.
refno{630927837}
-
tauV(S, sqrt{2}) > 0.13 + 0.2 ( dih(S)-pi/2),
if y1,y2 in[2.51,2 sqrt{2}], and rad(S) ge sqrt{2}.
refno{825907374}
-
nu < -0.3429 + 0.24573 dih, for upright quarters with
y1 in[2.75,2 sqrt{2}].
refno{812894433}
-
vorx < -0.0571, for anchored simplices with
y4 in[2.51,2 sqrt2], y1 in[2.51,2.75], dih<2.2.
refno{404793781}
Section A13
-
taunu(S)>0.033, if S is an upright quarter.
refno{705592875}
-
tau0(S) > 0.06585- 0.0066,
if S is a flat quarter, and y4=2 sqrt{2}.
refno{747727191}
-
vor0(S) < 0.009, if S is a flat quarter, and y4=2 sqrt{2}.
refno{474496219}
-
vor0(S(2,2,y3,y4,2,2))<0.0461, if y4 in[2 sqrt{2},3.2].
refno{649551700}
-
vor0(S(2.51,2,y3,y4,2,2)) le0, if y4 in[2 sqrt{2},3.2].
refno{74657942}
-
vor0(S(y1,y2,2.51,y4,2,2))<0, if y4 in[2 sqrt{2},3.2].
refno{897129160}
-
tau0(S(2,2,y3,y4,2,2))>0.014, if y4 in[2 sqrt{2},3.2].
refno{760840103}
-
tau0(S(2.51,2,2,y4,2,2)) ge0, if y4 in[2 sqrt{2},3.2].
refno{675901554}
-
tau0(S(y1,y2,2.51,y4,2,2))>0.06585, if y4 in[2 sqrt{2},3.2].
refno{712696695}
Section A14
The verifications in this section were made by Samuel Ferguson.
Vi is defined in Section 4.9. The function f is defined
in Section 4.11.
-
V0 < 0,
if Delta ge0, y4 in[2,y2+y3], y5 in[2,3.2],
y6 in[y5,3.2].
refno{424011442}
-
V1 < 0,
if Delta ge0, y4 in[2,y2+y3], y5 in[2,3.2],
y6 in[y5,3.2].
refno{140881233}
-
Vj + 0.82 sqrt{421}<0, if y5 le2.189, y4 in[2 sqrt{2},3.2],
y5,y6 in[2,2.51], Delta ge0, j=0,1.
refno{601456709}
-
Vj + 0.82 sqrt{421}<0, if y5 le2.189, y4 in [3.2,y2+y3],
y5,y6 in[2,3.2], Delta ge0, j=0,1.
refno{292977281}
-
Vj + 0.5 sqrt{421}<0, if y5 in[2.189,2.51], y4 in[2 sqrt{2},3.2],
y5,y6 in[2,2.51], Delta ge0, j=0,1.
refno{927286061}
-
Vj + 0.5 sqrt{421}<0, if y5 in[2.189,3.2], y4 in[3.2,y2+y3],
y5,y6 in[2,3.2], Delta ge0, j=0,1.
refno{340409511}
-
Delta<421, if y4 in[2 sqrt{2},y2+y3], y5,y6 in[2,3.2],
eta(x1,x3, x5) le t0.
refno{727498658}
-
-4doct u{135} partial/partial x5 (quo(R{135})+quo(R{315}))< 0.82.
refno{484314425}
-
-4doct u{135} partial/partial x5 ( quo(R{135})+ quo(R{315}))< 0.5,
if y5 in[2.189,2.51].
refno{440223030}
-
f(y1,y2) ge 0.887, lambda=1.945, y1,y2 in[2,2.51].
refno{115756648}
Section A15
Let Difj = partial^i/ partial x1^i fj(S),
f0= vor0, f1=- tau0, as in Section 5.1.
-
D2f0, D2f1>0, if Delta ge0, y4 ge2, y5=2, y6=2.
refno{329882546}
-
D2f0, D2f1>0, if Delta ge0, y4 ge2, y5=2, y6=2.51.
refno{427688691}
-
D2f0, D2f1>0, if Delta ge0, y4 ge2, y5=2, y6=2 sqrt{2}.
refno{562103670}
-
D2f0, D2f1>0, if Delta ge0, y4 ge2, y5=2.51, y6=2.51.
refno{564506426}
-
D2f0, D2f1>0, if Delta ge0, y4 ge2, y5=2.51, y6=2 sqrt{2}.
refno{288224597}
-
D2f0 >0, if D1f0 le0,
Delta ge0, y4 ge2, y5=2 sqrt{2}, y6=2 sqrt{2}.
refno{979916330}
-
D2f1 >0, if D1f1 le0,
Delta ge0, y4 ge2, y5=2 sqrt{2}, y6=2 sqrt{2}.
refno{749968927}
Section A16
-
tau(S) > 0.06585, if S is a flat quarter and tau(S) is
any of the functions for flat quarters in III.3.10.
refno{695180203}
-
sigma(S) le 0, if S is a flat quarter and tau(S) is
any of the functions for flat quarters in III.3.10.
refno{690626704}
-
sigma(S) < Z(3,2), for simplices S of type SA.
refno{807023313}
-
tau(S)> 0.13943, for simplices S of type SA.
refno{590577214}
-
vor0(S) < Z(3,2),
if y4,y5 in[2.51,2 sqrt2], and the
simplex S is not of type SA.
refno{949210508}
-
tau0(S) > 0.13943,
if y4,y5 in[2.51,2 sqrt2], and the
simplex S is not of type SA.
refno{671961774}
Section A17
Let x4,x5,x6 in[2,3.2]. Let k0, k1, k2 be the
number of variables in [2,2.51], [2.51,2 sqrt{2}], [2 sqrt{2},3.2],
respectively.(Make the intervals disjoint so that k0+k1+k2=3.)
Assume k1+2k2 >2. (k1+2k2=2 gives a flat treated in A16.)
Set
pitau = cases 0, & k2=0,
0.0254, & k0=k1=k2=1,
0.0463+(k0+2k2-3)0.008/3+k2(0.0066),& otherwise.
endcases
-
tau0(S) -pitau > D(3,k1+k2), for parameters (k_0,k_1,k_2)
satisfying k_0+k_1+k_2=3, k_1+2k_2>2.
refno{645264496}
-
tau0(S)-0.034052 >D(3,2), if y4 in [2.6,sqrt8], y5 in[2 sqrt2,3.2].
refno{910154674}
-
tau0(S)-0.034052-0.0066 >D(3,2), if y6=2, y4=2.51, y5=3.2.
refno{877743345}
Section A18
In the same context as A17,
set
pisigma = cases 0, & k2=0,
0.009, & k0=0,k2=1,
(k0+2k2)0.008/3 + 0.009k2,& otherwise.
endcases
vor0(S) + pisigma < Z(3,k1+k2).
refno{612259047}
-
Section A19
Let Q be a quadrilateral subcluster whose edges are described
by the vector
(a1,2,2,2,2,2,a4,b4). Assume both diagonals
have lengths in [2 sqrt2,3.2].
-
tau0(Q) > 0.235, vor0(Q) < -0.075,
if b4 in[2.51,2 sqrt{2}],
-
tau0(Q) > 0.3109, vor0(Q) < -0.137,
if b4 in[2 sqrt{2},3.2],
refno{357477295}
Section A20
Let Q be a quadrilateral subcluster whose edges are described
by the vector (2,2,a2,2,2,b3,a4,b4). Assume
b4 ge b3, b4 in\{2.51,2 sqrt2\}, b3 in\{2,2.51,2 sqrt2\},
a2,a4 in\{2,2.51\}. Assume that the diagonal between corners
1 and 3 has length in [2 sqrt2,3.2], and that the other
diagonal has length ge 3.2. Let k0, k1, k2 be the
number of bi equal to 2, 2.51, 2 sqrt2, respectively.
If b4=2.51 and b3=2, no such subcluster exsits, so we exclude
this case.
-
vor0(Q)< Z(4,k1+k2) - 0.009 k2 -(k0+2k2)0.008/3.
refno{193776341}
-
tau0(Q) > D(4,k1+k2) + 0.04683 + (k0+2k2-3)0.008/3+0.0066k2.
refno{898647773}
-
vor0(Q) < Z(4,2) -0.0461 - 0.009 -2(0.008),
if a2 in {2,2.51}, a4 =2,
b4=2sqrt2, b3=2.51 or 2sqrt2.
refno{844634710}
-
tau0(Q) > D(5,1) +0.04683+0.008+2(0.0066),
if a2 in {2,2.51}, a4 =2,
b4=2sqrt2, b3=2.51 or 2sqrt2.
refno{328845176}
-
vor0(Q) < s5 - 0.0461-0.008,
if a2 in {2,2.51}, a4=2,
b3=2,
b4=2sqrt2.
refno{233273785}
-
tau0(Q) > t5 +0.008,
if a2 in {2,2.51},
a4=2,
b3=2,
b4=2sqrt2.
refno{966955550}
(The penalties used here are from Section 5.5.)
Section A21
-
vor0(S(2,2,2,y4,2,2))+ vor0(S(2,2,2,y4',2,2))
+ vor0(S(2,2,2,y4,y4',2))
< s5 -0.008, if y4,y4' in[2 sqrt2,3.2].
refno{275286804}
-
tau0(S(2,2,2,y4,2,2))+ tau0(S(2,2,2,y4',2,2))
+ tau0(S(2,2,2,y4,y4',2))
> t5+ 0.008, if y4,y4' in[2 sqrt2,3.2].
refno{627654828}
-
vor0(S(2,2,2,y4,y5,y6)) < -2(0.008) + s6-3(0.0461), if
y4,y5,y6 in[2 sqrt2,3.2].
refno{995177961}
-
tau0(S(2,2,2,y4,y5,y6)) > t6 + maxpi, if
y4,y5,y6 in[2 sqrt2,3.2].
refno{735892048}
Section A22
In A22 and A23,
y1 in [2.51,2 sqrt2], y4 in[2 sqrt2,3.2], and
dih<2.46.
vor0 denotes the truncated Voronoi function
on the union of an anchored simplex
and an adjacent special simplex S'. We apply deformation arguments
to the special simplex so that y5(S')=y6(S')=2. y1(S') in {2,2.51}.
Note 952096899: Let S' be the special simplex.
In the first group, and when y1(S')=2.51, S' might have
zero volume, which is numerically unstable.
But in this case vor0(S')<=0, so it can be dropped and the
result follows from the corresponding inequality in A6.
-
vor0 < -3.58 + 2.28501 dih
refno{53502142}
-
vor0 < -2.715 + 1.67382 dih
refno{134398524}
-
vor0 < -1.517 + 0.8285 dih
refno{371491817}
-
vor0 < -0.858 + 0.390925 dih
refno{832922998}
-
vor0 < -0.358+0.009 + 0.12012 dih
refno{724796759}
-
vor0 < -0.186+0.009 + 0.0501 dih
refno{431940343}
Let S' be the special simplex. By deformations, we have
y5(S')=y6(S')=2, and y1(S') in {2,2.51}. If
y1(S')=2.51, and y4(S')le3, the inequalities listed
above follow from Section A7 and the inequality
tau0(S') > 0.06585, if y1=2.51, y4 in[2sqrt2,3],
y5=y6=2.
refno{66753311}
Similarly, the result follows if y2 or y3 ge2.2 from the
inequality
tau0(S') > 0.06585, if y4 in[3,3.2], y5=y6=2,
y1=2.51, y2 in[2.2,2.51].
refno{762922223}
Because of these reductions, we may assume in the first
batch of inequalities of A23 that when y1(S') ne2,
we have that y1(S')=2.51, y5(S')=y6(S')=2, y4 in[3,3.2],
y2(S'),y3(S') le2.2. In all but (371464244) and
(657011065), if y1(S')=2.51, we prove the inequality
with tau0(S') replaced with its lower bound 0.
Let S be an upright quarter with y5=2.51.
-
vor0 < -3.58/2 + 2.28501 dih
refno{980721294}
-
vor0 < -2.715/2 + 1.67382 dih
refno{989564937}
-
vor0 < -1.517/2 + 0.8285 dih
refno{263355808}
-
vor0 < -0.858/2 + 0.390925 dih
refno{445132132}
-
vor0 < (-0.358+0.009)/2 + 0.12012 dih +0.2(dih-1.23)
refno{806767374}
-
vor0 < (-0.186+0.009)/2 + 0.0501 dih +0.2(dih-1.23)
refno{511038592}
Section A23
tau0 denotes the truncated Voronoi function
on the union of an anchored simplex
(with y1 in[2.51,2 sqrt2], y4 in[2 sqrt2,3.2], dih<2.46)
and an adjacent special simplex S'. We apply deformation arguments
to the special simplex so that y5(S')=y6(S')=2. y1(S') in {2,2.51}.
Note 942752615: In the first group, and when y1(S')=2.51, S' might have
zero volume, which is numerically unstable.
it is enough to prove the
inequalities in two separate cases.
If y4(S')< 3., we show that tau0(S') > 0.6585, so that in this
case the inequality follows from corresponding inequality in A7.
If y4(S')>3., we use tau0(S')> 0 (there are no quoins
and the formula is geometric since y1(S')=2.51 gives Adih1=0).
-
- tau0+0.06585 < -3.48 + 2.1747 dih
refno{4591018}
-
- tau0+0.06585 < -3.06 + 1.87427 dih
refno{193728878}
-
- tau0+0.06585 < -1.58 + 0.83046 dih
refno{2724096}
-
- tau0+0.06585 < -1.06 + 0.48263 dih
refno{213514168}
-
- tau0+0.06585 < -0.83 + 0.34833 dih
refno{750768322}
-
- tau0+0.06585 < -0.50 + 0.1694 dih
refno{371464244}
-
- tau0+0.06585 < -0.29 +0.0014 + 0.0822 dih
refno{657011065}
Let S be an upright quarter with y5=2.51.
-
- tau0+0.06585/2 < -3.48/2 + 2.1747 dih
refno{953023504}
-
- tau0+0.06585/2 < -3.06/2 + 1.87427 dih
refno{887276655}
-
- tau0+0.06585/2 < -1.58/2 + 0.83046 dih
refno{246315515}
-
- tau0+0.06585/2 < -1.06/2 + 0.48263 dih
refno{784421604}
-
- tau0+0.06585/2 < -0.83/2 + 0.34833 dih
refno{258632246}
-
- tau0+0.06585/2 < -0.50/2 + 0.1694 dih +0.03(dih-1.23)
refno{404164527}
-
- tau0+0.06585/2 < -0.29/2 +0.0014/2 + 0.0822 dih +0.2(dih-1.23)
refno{163088471}