In this note we publish those symbolic and numerical calculations which need to the paper in title. We used Mathematica 10 and its translating program into ...
COMPUTATIONS WITH MATHEMATICA 10, AN ADDENDUM TO THE PAPER ´ ´ FEJES-TOTH”. ´ ”ON THE ICOSAHEDRON INEQUALITY OF LASZL O ´ ´ AKOS G.HORVATH
In this note we publish those symbolic and numerical calculations which need to the paper in title. We used Mathematica 10 and its translating program into Latex. In[1]: D[1/6Sin[c](Cos[(x-c)/2]-Cos[x/2]Cos[c/2])/(1-Cos[x/2]Cos[c/2]), x,c] Out[1]:
−
1 x Cos[ 2c ]Cos[c](−Cos[ 2c ]Cos[ x 2 ]+Cos[ 2 (−c+x)])Sin[ 2 ] 2
12(1−Cos[ 2c ]Cos[ x 2 ])
+
(−Cos[ 2c ]Cos[ x2 ]+Cos[ 12 (−c+x)])Sin[ 2c ]Sin[c]Sin[ x2 ] + 2 24(1−Cos[ 2c ]Cos[ x 2 ]) 1 1 1 c x 1 c x 1 1 Sin[c]( 4 Cos[ 2 (−c+x)]− 4 Sin[ 2 ]Sin[ 2 ]) Cos[c]( 2 Cos[ 2 ]Sin[ 2 ]− 2 Sin[ 2 (−c+x)]) + − 6(1−Cos[ 2c ]Cos[ x 6(1−Cos[ 2c ]Cos[ x 2 ]) 2 ]) 1 c 1 c x 1 c x 1 x Cos[ x Sin Sin[c] Cos Sin − Sin (−c+x) Cos Sin[c]Sin Cos [2] (2 [2] [2] 2 [2 ]) [2] [ 2 ]( 2 [ 2 ]Sin[ 2c ]+ 12 Sin[ 12 (−c+x)]) 2] − 2 2 12(1−Cos[ 2c ]Cos[ x 12(1−Cos[ 2c ]Cos[ x 2 ]) 2 ]) c x 1 c x Cos[ 2c ]Cos[ x 2 ](−Cos[ 2 ]Cos[ 2 ]+Cos[ 2 (−c+x)])Sin[ 2 ]Sin[c]Sin[ 2 ] 3
12(1−Cos[ 2c ]Cos[ x 2 ])
+
In[2]: Simplify[%] (( [ ] [x] [ c ]3 Out[2]: 9Cos 3c Cos[x]+ 2 − 2(5 + 10Cos[c] + Cos[2c])Cos 2 + Cos 2 ( ( [ ] ) [ ]) [ ] [ x ])3 ) 1 c c c Cos (30 + (11 + 3Cos[c])Cos[x]) Sin / 96 −1 + Cos Cos 2 2 2 2 2 In[3]: D[1/6Sin[c](Cos[(x-c)/2]-Cos[x/2]Cos[c/2])/(1-Cos[x/2]Cos[c/2]), {x,2}] [ ] [ ] [ ]) ( 41 Cos[ 2c ]Cos[ x2 ]− 14 Cos[ 12 (−c+x)])Sin[c] 1 ( Out[3]: + 6 −Cos 2c Cos x2 + Cos 12 (−c + x) · 6(1−Cos[ 2c ]Cos[ x ]) 2 ( ) 2 2 1 c x 1 1 Cos[ 2c ]Cos[ x Cos[ 2c ] Sin[ x Cos[ 2c ]Sin[c]Sin[ x 2] 2] 2 ]( 2 Cos[ 2 ]Sin[ 2 ]− 2 Sin[ 2 (−c+x)]) Sin[c] − − 2 + 3 2 c x c x 4(1−Cos[ 2c ]Cos[ x 2 1−Cos Cos 6 1−Cos Cos ]) ( [ ] [ ]) ( [ ] [ ]) 2 2 2 2 2 In[4]: Simplify[%] Out[4]:
c Sin[ 2c ]Sin[c](−2Cos[c]Sin[ x 2 ]+Cos[ 2 ]Sin[x]) 3
48(−1+Cos[ 2c ]Cos[ x 2 ])
In[5]: D[1/6Sin[c](Cos[(x-c)/2]-Cos[x/2]Cos[c/2])/(1-Cos[x/2]Cos[c/2]), {c,2}] ( ) 2 c 2 ( [c] [x] [1 ]) Cos[ 2c ]Cos[ x Cos[ x 1 2] 2 ] Sin[ 2 ] Out[5]: − Sin[c] + 2 + 3 6 −Cos 2 Cos 2 + Cos 2 (−c + x) 4(1−Cos[ 2c ]Cos[ x 2(1−Cos[ 2c ]Cos[ x 2 ]) 2 ]) (( [ ] [ ] [ ]) c x 1 1 1 1 4 Cos 2 Cos 2 − 4 Cos 2 (−c + x) Sin[c]− 6(1−Cos[ 2c ]Cos[ x 2 ]) ( [ ] [ ] [ ]) ( [ ] [ ] [ ])) − −Cos 2c Cos x2 + Cos 21 (−c + x) Sin[c] + 2Cos[c] 12 Cos x2 Sin 2c + 21 Sin 12 (−c + x) − [ ] [ ] ( ( [ ] [ ] x c 1 Cos[c] −Cos 2c Cos x2 + 2 Cos 2 Sin 2 6(1−Cos[ 2c ]Cos[ x 2 ]) [ ]) ( [ ] [ ])) [ ] Cos 12 (−c + x) + Sin[c] 21 Cos x2 Sin 2c + 12 Sin 12 (−c + x) In[6]: Simplify[%] [ ] ( ] [ ] [ − 11Sin[2c − x]+ Out[6]: 4Sin 12 (c − 3x) + Sin 21 (5c − 3x) − 24Sin[c − x] + 3Sin 3(c−x) 2 [ ] [1 ] [1 ] 39Sin 2 (3c − x) + Sin 2 (5c − x) − 6Sin[x] + 24Sin[c + x] − 3Sin 3(c+x) + 11Sin[2c + x]− 2 [1 ] [1 ] [1 ] [1 ]) − 39Sin 2 (3c + x) − Sin 2 (5c + x) − 4Sin 2 (c + 3x) − Sin 2 (5c + 3x) / (384 (−1+ [ ] [ ])3 ) +Cos 2c Cos x2 Date: 2014 Nov. 1
´ G.HORVATH ´ A.
2
In[7]: D[1/6Sin[c](Cos[(x − c)/2] − Cos[x/2]Cos[c/2])/(1 − Cos[x/2]Cos[c/2]), {x, 2}] · · D[1/6Sin[c](Cos[(x − c)/2] − Cos[x/2]Cos[c/2])/(1 − Cos[x/2]Cos[c/2]), {c, 2}] − D[1/6Sin[c](Cos[(x − c)/2] − Cos[x/2]Cos[c/2])/(1 − Cos[x/2]Cos[c/2]), x, c]∧ 2 (
( ) 2 c 2 ]) Cos[ 2c ]Cos[ x Cos[ x 2] 2 ] Sin[ 2 ] Out[7]: −Cos 2 Cos 2 + Cos 2 (−c + x) − Sin[c]+ 2 + 3 4(1−Cos[ 2c ]Cos[ x 2(1−Cos[ 2c ]Cos[ x 2 ]) 2 ]) (( [ ] [ ] [ ]) ( [ ] [ ] [ ]) 1 1 c x 1 1 c x 1 Cos − Sin[c] − −Cos Cos + Cos Sin[c]+ Cos Cos (−c + x) (−c + x) c x 4 2 2 4 2 2 2 2 6(1−Cos[ 2 ]Cos[ 2 ]) (1 [x] [c] 1 [1 ])) ( [x] [c]( ( [c] [x] [1 ]) 2Cos[c] 2 Cos 2 Sin 2 + 2 Sin 2 (−c + x) − Cos 2 Sin 2 Cos[c] −Cos 2 Cos 2 + Cos 2 (−c + x) + ( ( [ ] [ ] [ ]))) ( ( [ ] [ ])2 )) ( 41 Cos[ 2c ]Cos[ x2 ]− 14 Cos[ 12 (−c+x)])Sin[c] + Sin[c] 21 Cos x2 Sin 2c + 12 Sin 12 (−c + x) / 6 1 − Cos 2c Cos x2 6(1−Cos[ 2c ]Cos[ x 2 ]) ( ) 2 2 ( [c] [x] [1 ]) Cos[ 2c ]Cos[ x Cos[ 2c ] Sin[ x 1 2] 2] −Cos Cos + Cos (−c + x) Sin[c] − + − 2 3 x x c c 6 2 2 2 4 1−Cos[ 2 ]Cos[ 2 ]) 2(1−Cos[ 2 ]Cos[ 2 ]) ) ( ( 1 1 c x 1 1 x Cos[ 2c ]Sin[c]Sin[ x Cos[ 2c ]Cos[c](−Cos[ 2c ]Cos[ x 2 ]( 2 Cos[ 2 ]Sin[ 2 ]− 2 Sin[ 2 (−c+x)]) 2 ]+Cos[ 2 (−c+x)])Sin[ 2 ] − − + 2 2 c x 6(1−Cos[ 2c ]Cos[ x 12 1−Cos Cos ( [ 2 ] [ 2 ]) 2 ]) c x 1 c x Cos[ 2c ]Cos[ x (−Cos[ 2c ]Cos[ x2 ]+Cos[ 12 (−c+x)])Sin[ 2c ]Sin[c]Sin[ x2 ] 2 ](−Cos[ 2 ]Cos[ 2 ]+Cos[ 2 (−c+x)])Sin[ 2 ]Sin[c]Sin[ 2 ] + + 3 2 12(1−Cos[ 2c ]Cos[ x 24(1−Cos[ 2c ]Cos[ x ]) 2 2 ]) 1 1 1 c x 1 c x 1 1 Sin[c]( 4 Cos[ 2 (−c+x)]− 4 Sin[ 2 ]Sin[ 2 ]) Cos[c]( 2 Cos[ 2 ]Sin[ 2 ]− 2 Sin[ 2 (−c+x)]) + − 6(1−Cos[ 2c ]Cos[ x 6(1−Cos[ 2c ]Cos[ x 2 ]) 2 ]) )2 c 1 c x 1 1 1 x c 1 1 Cos[ x Cos[ 2c ]Sin[c]Sin[ x 2 ]Sin[ 2 ]Sin[c]( 2 Cos[ 2 ]Sin[ 2 ]− 2 Sin[ 2 (−c+x)]) 2 ]( 2 Cos[ 2 ]Sin[ 2 ]+ 2 Sin[ 2 (−c+x)]) − 2 2 12(1−Cos[ 2c ]Cos[ x 12(1−Cos[ 2c ]Cos[ x 2 ]) 2 ]) 1 6
(
[c]
[x]
[1
In[8]: Simplify[%] (( [ ] [ ] Out[8]: 336 + 502Cos[c] + 256Cos[2c] + 26Cos[3c] − 66Cos 21 (c − 3x) − 14Cos 21 (5c − 3x) − [ ] [ ] [ ] 3(c−x) 2Cos 21 (7c − 3x) + 7Cos[c − 2x] + Cos[3c − 2x] − 498Cos c−x + 190Cos[c − x] − 46Cos + 2 2 [1 ] [1 ] 4Cos[2(c − x)] + 100Cos[2c − x] − 298Cos 2 (3c − x) + 18Cos[3c − x] − 98Cos 2 (5c − x) − [ ] [ ] [ ] 3(c+x) 2Cos 12 (7c − x) + 280Cos[x] + 8Cos[2x] − 498Cos c+x + 190Cos[c + x] − 46Cos + 2 2 [1 ] [1 ] 4Cos[2(c + x)] + 100Cos[2c + x] − 298Cos 2 (3c + x) + 18Cos[3c + x] − 98Cos 2 (5c + x) − ] [ ] [ ] [ 2Cos 12 (7c + x) + 7Cos[c + 2x] + Cos[3c + 2x] − 66Cos 12 (c + 3x) − 14Cos 12 (5c + 3x) − [ ]) [ ]2 ) ( ( [ ] [ ])5 ) 2Cos 12 (7c + 3x) Sin 2c / 576 −2 + 2Cos 2c Cos x2 In[9]: Plot3D[1/6Sin[c](Cos[(x-c)/2]-Cos[x/2]Cos[c/2])/(1-Cos[x/2]Cos[c/2]), {x,0,Pi/2}, {c,x,2ArcSin[Sqrt[2/3]]}] Out[9]:
3
In[10]: D[1/6Sin[c](Cos[(x-c)/2]-Cos[x/2]Cos[c/2])/(1-Cos[x/2]Cos[c/2]), c]
Out[10]: +
Sin[c](
1 Cos[c](−Cos[ 2c ]Cos[ x 2 ]+Cos[ 2 (−c+x)])
6(1−Cos[
[ ]Sin[ ] ]
1 x 2 Cos 2
c 2 c 2
6(1−Cos[
[ [ ])
c 2
]Cos[ ]) ]) x 2
−
c x 1 c Cos[ x 2 ](−Cos[ 2 ]Cos[ 2 ]+Cos[ 2 (−c+x)])Sin[ 2 ]Sin[c]
12(1−Cos[ 2c ]Cos[ x 2 ])
+ 12 Sin 12 (−c+x) Cos x 2
In[11]: Simplify[%] Out[11]:
( ) c 3 Sin[ 2c ] (1+3Cos[c])Sin[ x 2 ]−2Cos[ 2 ] Sin[x] 2
12(−1+Cos[ 2c ]Cos[ x 2 ])
In[12]: Plot3D[Out[11], {x,0,Pi/2}, {c,x,2ArcSin[Sqrt[2/3]]}] Out[12]:
In[13]: RegionPlot[Out[11]>=0 ,{x,0,Pi/2},{c,0,2ArcSin[Sqrt[2/3]]}] Out[13]:
2
+
4
´ G.HORVATH ´ A.
In[14]: Plot3D[Out[8], {x,0,Pi/2}, {c,x,2ArcSin[Sqrt[2/3]]}] Out[14]:
In[15]:RegionPlot[Out[8]>=0 ,{x,0,Pi/2},{c,0,2ArcSin[Sqrt[2/3]]}] Out[15]:
In[16]: Reduce[ Out[8]==0 && x==Pi/2 && 0 < c < 2ArcSin[Sqrt[2/3]], {x,c}] ] [√ [ ] 2 3 4 π Out[16] x == 2 &&c == 4ArcTan Root 1 − 24#1 + 78#1 − 24#1 + #1 &, 1 In[17]: N[%] Out[17]: x == 1.5708&&c == 0.875793 In[18]: Solve[z ∧ 4 − 24z ∧ 3 + 78z ∧ 2 − 24z + 1==0] {{ √ √ √ √ } { √ √ } Out[18]: z → 6 − 34 − 35 − 6 34 , z → 6 − 34 + 35 − 6 34 , { √ √ √ √ } { √ √ }} z → 6 + 34 − 35 + 6 34 , z → 6 + 34 + 35 + 6 34
5
In[19]: N[%] Out[19]:
{{z → 0.049513}, {z → 0.288583}, {z → 3.46521}, {z → 20.1967}}
In[20]: Reduce[ Out[8]==0 && c==Pi/2 && 0 < x < Pi/2, {x,c}] [√ ] √ √ 2 5 3 2√ √ Out[20]: x == 4ArcTan − 10+7 + &&c == π2 2 10+7 2 In[21] N[%] Out[21]:
x == 0.427922&&c == 1.5708
In[22]: Reduce[ Out[8]==0 && c==2ArcSin[Sqrt[2/3]] && 0¡x¡Pi/2, {x,c}] [ [√ ]] [√ [{ [ [√ ]] √ 2 2 + 10368Cos 4ArcSin + Out[22]: x == 4ArcTan Root 14712 − 256 3 + 20331Cos 2ArcSin 3 3 [ [√ ]] ( [ [√ ]] [ [√ ]] √ 2 2 2 +1053Cos 6ArcSin + 51168 − 3072 3 + 81324Cos 2ArcSin + 41472Cos 4ArcSin + 3 3 3 [ [√ ]]) ( [ [√ ]] [ [√ ]] 2 2 2 + 4212Cos 6ArcSin #1 + 66768 + 121986Cos 2ArcSin + 62208Cos 4ArcSin + 3 3 3 [ [√ ]]) ( [ [√ ]] √ 2 2 + 6318Cos 6ArcSin #12 + + 51168 + 3072 3 + 81324Cos 2ArcSin + 3 3 [ [√ ]] [ [√ ]]) ( [ [√ ]] √ 2 2 2 +41472Cos 4ArcSin + 4212Cos 6ArcSin #13 + 14712 + 256 3 + 20331Cos 2ArcSin + 3 3 3 [ [√ ]] [ [√ ]]) }]] [√ ] 2 2 2 +10368Cos 4ArcSin + +1053Cos 6ArcSin #14 &, 0.03105 &&c == 2ArcSin 3 3 3 In[23]: N[%] Out[23]: x == 0.697715&&c == 1.91063 In[24]:Reduce[ Out[8]==0 && x==-2Pi/3+2ArcSin[Sqrt[14]/4] && x¡c¡2ArcSin[Sqrt[2/3]], {x,c}] ( [ √ ]) 7 √ √ √ [√ [{ 2 Out[24]: x == − 3 π − 3ArcSin 2 2 &&c == 4ArcTan Root 2699 − 768 2 + 163 21 − 448 42+ [ ( [ √ ])] [ ( [ √ ])] 7 7 2 4 2 +560Cos 3 π − 3ArcSin 2 + 16Cos 3 π − 3ArcSin 2 2 + [ √ ])] ( [ ( 7 √ √ √ 2 + + −18227 + 4992 2 − 2059 21 + 2592 42 + 3920Cos 3 π − 3ArcSin 2 2 [ ( [ √ ])]) 7 √ √ ( +112Cos 43 π − 3ArcSin 2 2 #1 + 16167 − 9600 2 + 1695 21− [ √ ])] [ ( [ √ ])]) [ ( 7 7 √ 2 4 2 + 336Cos 3 π − 3ArcSin 2 2 #12 + −3360 42 + 11760Cos 3 π − 3ArcSin 2 ( [ ( [ √ ])] 7 √ √ √ 2 + −2751 + 9216 2 − 3447 21 − 2304 42 + 19600Cos 3 π − 3ArcSin 2 2 + [ ( [ √ ])]) 7 √ √ √ ( +560Cos 43 π − 3ArcSin 2 2 #13 + −2751 − 9216 2 − 3447 21 + 2304 42+ [ ( [ √ ])] [ ( [ √ ])]) 7 7 2 4 2 +19600Cos 3 π − 3ArcSin 2 + 560Cos 3 π − 3ArcSin 2 2 #14 + [ √ ])] ( [ ( 7 √ √ √ 2 + 16167 + 9600 2 + 1695 21 + 3360 42 + 11760Cos 3 π − 3ArcSin 2 2 [ √ ])]) [ ( 7 √ √ √ ( #15 + −18227 − 4992 2 − 2059 21 − 2592 42+ +336Cos 43 π − 3ArcSin 2 2 [ √ ])] [ ( [ √ ])]) [ ( 7 7 √ ( 4 2 2 + 112Cos 3 π − 3ArcSin 2 2 #16 + 2699 + 768 2+ +3920Cos 3 π − 3ArcSin 2 [ √ ])] [ ( 7 √ √ + +163 21 + 448 42 + 560Cos 23 π − 3ArcSin 2 2
´ G.HORVATH ´ A.
6
[ ( [ √ ])]) }]] 7 7 4 2 +16Cos 3 π − 3ArcSin 2 #1 &, 0.1324047495312671 In[25]: N[%]
Out[25]: x == 0.324463&&c == 1.39593 In[26]: Reduce[ Out[8]==0 && x==-4ArcSin[Sqrt[14]/4]+5Pi/3 && x¡c¡Pi/2, {x,c}] ( [ √ ]) 7 √ √ √ [√ [{ Out[26]: x == 13 5π − 12ArcSin 2 2 &&c == 4ArcTan Root 9361 − 5376 3 − 2432 7 + 933 21+ [ ( [ √ ])] [ ( [ √ ])] 7 7 √ √ ( +2240Cos 13 5π − 12ArcSin 2 2 +64Cos 23 5π − 12ArcSin 2 2 + −55113 + 31104 3 + 15168 7− [ ( [ √ ])] [ ( [ √ ])]) 7 7 √ −11709 21 + 15680Cos 13 5π − 12ArcSin 2 2 + 448Cos 23 5π − 12ArcSin 2 2 #1+ ( [ ( [ √ ])] 7 √ √ √ + 51813 − 40320 3 − 25920 7 + 9225 21 + 47040Cos 13 5π − 12ArcSin 2 2 + [ ( [ √ ])]) 7 √ √ √ ( +1344Cos 23 5π − 12ArcSin 2 2 #12 + 17811 − 27648 3 + 13824 7 − 19377 21+ [ ( [ √ ])] [ ( [ √ ])]) 7 7 +78400Cos 13 5π − 12ArcSin 2 2 + 2240Cos 23 5π − 12ArcSin 2 2 #13 + ( [ ( [ √ ])] 7 √ √ √ + 17811 + 27648 3 − 13824 7 − 19377 21 + 78400Cos 13 5π − 12ArcSin 2 2 + [ ( [ √ ])]) 7 √ √ √ ( +2240Cos 23 5π − 12ArcSin 2 2 #14 + 51813 + 40320 3 + 25920 7 + 9225 21+ [ ( [ √ ])] [ ( [ √ ])]) 7 7 +47040Cos 13 5π − 12ArcSin 2 2 + 1344Cos 23 5π − 12ArcSin 2 2 #15 + ( [ ( [ √ ])] 7 √ √ √ + −55113 − 31104 3 − 15168 7 − 11709 21 + 15680Cos 13 5π − 12ArcSin 2 2 + [ ( [ √ ])]) 7 +448Cos 23 5π − 12ArcSin 2 2 #16 + ( [ ( [ √ ])] 7 √ √ √ + 9361 + 5376 3 + 2432 7 + 933 21 + 2240Cos 13 5π − 12ArcSin 2 2 + [ ( [ √ ])]) }]] 7 +64Cos 23 5π − 12ArcSin 2 2 #17 &, 0.1604829573029123 In[27]: N[%] Out[27]: x == 0.398271&&c == 1.52411 In[28]:Reduce[Out[8]==0&&x==Pi/5&&0 < c < 2ArcSin[Sqrt[2/3]], {x, c}] ( (( [{ x = π5 ∧ c = 4 tan−1 Root #12 − 5&, 2#1 + #22 − 10&, −2#1 + #32 − 10&, 29#1#47 − 189#1#46 +393#1#45 −105#1#44 −105#1#43 +393#1#42 −189#1#4+29#1+8#2#47 −60#2#46 + 156#2#45 − 288#2#44 + 288#2#43 − 156#2#42 + 60#2#4 − 8#2 + 56#3#47 − #3#46 + 420#3#45 + 288#3#44 − 288#3#43 − 420#3#42 + 324#3#4 − 56#3 + 167#47 − 951#46 + 1083#45 + } ])1/2 ) 4 3 2 ∨c=4 341#4 + 341#4 + 1083#4 − 951#4 + 167& , {2, 2, 2, 4} (( [{ 2 2 2 −1 tan Root #1 − 5&, 2#1 + #2 − 10&, −2#1 + #3 − 10&, 29#1#47 − 189#1#46 + 393#1#45 − 105#1#44 − 105#1#43 + 393#1#42 − 189#1#4 + 29#1 + 8#2#47 − 60#2#46 + 156#2#45 − 288#2#44 + 288#2#43 − 156#2#42 + 60#2#4 − 8#2 + 56#3#47 − 324#3#46 + 420#3#45 + 288#3#44 − 288#3#43 − 420#3#42 + 324#3#4 − 56#3 + )) 167#47 − } ]) 1/2 951#46 + 1083#45 + 341#44 + 341#43 + 1083#42 − 951#4 + 167& , {2, 2, 2, 5} In[29]:N [%]
7
Out[29]: x = 0.628319 ∧ (c = 0.36077 ∨ c = 1.83487) [{ sin(u) sin(v) sin(x) sin(y) sin(z) In[30]: NMaximize √3−cos(u) + √3−cos(v) + √3−cos(x) + √3−cos(y) + √3−cos(z) , } ] π u + v + x + y + z ≤ 2 , 0 < x, 0 < y, 0 < z, 0 < u, 0 < v , {x, y, z, u, v} Out[30]: {1.97836, {x → 0.314159, y → 0.314159, z → 0.314159, u → 0.314159, v → 0.314159}}
