Problem l112o23
Unknowns | Inequalities |
Equations |
Solution 1 |
Solution 2 |
Solution 3 |
Solution 4 |
Solution 5 |
Solution 6 |
Solution 7 |
Solution 8 |
Solution 9 |
Solution 10 |
Solution 11 |
Solution 12 |
Solution 13 |
Solution 14 |
Solution 15 |
Solution 16 |
Solution 17 |
Solution 18 |
Solution 19 |
Solution 20 |
Solution 21 |
Solution 22 |
Solution 23 |
Solution 24 |
Solution 25 |
Solution 26 |
Solution 27 |
Solution 28 |
Solution 29 |
Solution 30 |
Solution 31 |
Solution 32 |
Solution 33 |
Solution 34 |
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Unknowns
All solutions for the following 61 unknowns have to be determined:
a1, ..., a19, b1, ..., b42
Inequalities
Each of the following lists represents one inequality which states
that not all unknowns in this list may vanish. These inequalities
filter out solutions which are trivial for the application.
{a1,a5,a12},
{a6,a4,a5,a9,a7,a11,a10,a8},
{a13,a14,a18,a15},
{b1,b7,b24},
{b1,b2,b3,b4,b5,b6,b7,b8,b9,b10,b11,b12,b13,
b14,b15,b16,b17,b18,b19,b20,b21,b22,b23},
{b24,b25,b26,b27,b28,b29,b30,b31,b32,b33,
b34,b35,b36,b37,b38,b39,b40,b41,b42}
Equations
All comma separated 162 expressions involving 1678 terms have to vanish.
All terms are products of one a- and one b-unknown.
a3*b27,
a3*b2,
a3*b5,
2*a12*b27 - 3*a14*b24,
a14*b5 - 2*a3*b29,
2*a1*b2 - 3*a2*b1,
2*a1*b3 - 3*a2*b1,
a2*b5 - 2*a3*b4,
2*a15*b5 - 3*a3*b30,
2*a12*b31 - 3*a16*b24,
2*a15*b31 + a8*b27,
a13*b7 - a5*b25,
a10*b27 + 2*a18*b31,
a11*b27 + 2*a19*b31,
2*a1*b4 - a2*b2 - 9*a3*b1,
2*a1*b4 - a2*b3 - 6*a3*b1,
4*a1*b5 - 2*a3*b2 - a3*b3,
a15*b14 - a8*b30 + a8*b5,
a11*b41 - a18*b23 + 2*a19*b42,
4*a12*b29 - a14*b27 - 6*a15*b24 - a2*b27,
a1*b25 - a12*b25 - a13*b1 + a13*b24,
2*a12*b32 - 6*a16*b24 - 3*a17*b24 - a5*b27,
4*a12*b36 - 4*a16*b27 - 6*a18*b24 - a7*b27,
2*a12*b34 + a14*b7 - 3*a17*b24 - a5*b26,
a13*b11 + 2*a15*b7 - a5*b28 - a7*b25,
a1*b7 - a12*b7 - a5*b1 + a5*b24,
a13*b16 + a18*b7 - a5*b35 - a9*b25,
a11*b36 - a14*b23 - 2*a16*b42 + 4*a19*b39,
a10*b23 - a11*b22 + 6*a11*b42 - 8*a19*b23,
a1*b27 - 2*a12*b26 - a12*b27 + 3*a13*b24 + 3*a14*b24,
2*a12*b31 + 2*a12*b32 - 6*a16*b24 - 3*a17*b24 - a4*b27,
a14*b14 - 2*a15*b36 + 2*a16*b30 - a3*b36 - 2*a8*b29,
2*a12*b33 + a13*b6 - 3*a17*b24 - 2*a4*b25 - a5*b25,
8*a12*b39 - 4*a16*b31 - 2*a17*b31 - 12*a19*b24 - a9*b27,
a10*b41 + 2*a11*b38 - 2*a15*b23 - a18*b22 + 4*a18*b42,
a1*b26 - 2*a12*b25 - a12*b26 + 3*a13*b24 - a14*b1 + a14*b24,
2*a12*b28 + 2*a12*b29 - a14*b26 + a14*b3 - 6*a15*b24 - a2*b26,
6*a12*b30 + a14*b29 + a14*b4 - 4*a15*b27 - 2*a2*b29 - 3*a3*b26,
4*a13*b5 + 3*a14*b30 - 2*a15*b29 + 2*a15*b4 - 3*a2*b30 - 4*a3*b28,
2*a12*b36 - 2*a13*b31 - 2*a14*b31 + a16*b27 - 3*a18*b24 - a6*b27,
2*a12*b37 + a14*b11 - 2*a17*b27 - 6*a18*b24 - 2*a5*b29 - a7*b26,
4*a12*b38 + a14*b13 - 2*a15*b32 - 4*a18*b27 - 2*a7*b29 - 2*a8*b26,
2*a12*b34 + a13*b6 + a13*b7 - 3*a17*b24 - 2*a4*b25 - 2*a5*b25,
a13*b13 + 2*a15*b11 - 2*a15*b34 - 3*a5*b30 - a7*b28 - 2*a8*b25,
2*a15*b14 - 2*a15*b38 + 2*a18*b30 + a18*b5 - 2*a3*b38 - 3*a8*b30,
a1*b6 + 2*a1*b7 - a12*b6 - 2*a4*b1 + 2*a4*b24 - 3*a5*b1,
2*a10*b29 - a14*b20 + 4*a15*b39 - 2*a16*b38 + 2*a18*b36 + a8*b36,
a10*b36 + 2*a11*b29 - a14*b22 - 2*a16*b41 + 4*a18*b39 + 2*a19*b36,
6*a11*b25 - a13*b21 - a18*b16 + 2*a19*b34 + a5*b41 + a9*b35,
2*a10*b42 - a11*b20 + 3*a11*b41 - 4*a18*b23 - 3*a19*b22 + a8*b23,
a1*b28 - 3*a12*b28 - a13*b2 + a13*b26 - 2*a13*b3 + 6*a15*b24 + 3*a2*b25,
2*a13*b14 + 2*a15*b13 - 2*a15*b37 + 2*a17*b30 - a3*b37 - 3*a7*b30 - 2*a8*b28,
a13*b13 + 2*a15*b11 + 2*a15*b8 - 3*a5*b30 - a7*b28 + a8*b2 - 2*a8*b25,
3*a1*b6 + a1*b7 - a12*b6 - a12*b7 - 6*a4*b1 - 4*a5*b1 + a5*b24,
8*a12*b39 + 4*a12*b40 + a14*b17 - 2*a16*b32 - 2*a17*b32 - 24*a19*b24 - 2*a9*b26
,
3*a10*b30 - 2*a15*b20 + 4*a15*b41 - a18*b14 - 4*a19*b30 + a3*b41 + 2*a8*b38,
4*a10*b30 - 3*a10*b5 - 4*a15*b20 - 2*a18*b14 + a3*b20 - a8*b14 + 2*a8*b38,
a11*b2 - 6*a11*b25 + a13*b21 + a18*b16 + 2*a19*b8 - a5*b41 - a9*b35,
6*a11*b35 + a11*b37 - 8*a13*b23 - 2*a17*b42 - a18*b21 + 4*a19*b40 + a9*b41,
a1*b28 - a12*b28 - a13*b2 + a13*b27 - a14*b25 - 2*a15*b1 + 2*a15*b24 + a2*b25,
6*a12*b30 + 2*a13*b4 + a14*b28 - 2*a15*b26 - 2*a15*b27 + 2*a15*b3 - 2*a2*b28 -
6*a3*b25,
2*a12*b32 + 4*a12*b33 + 2*a12*b34 + a14*b6 - 6*a16*b24 - 9*a17*b24 - 2*a4*b26 -
a5*b26,
2*a12*b36 + 2*a12*b37 + a14*b10 - 2*a16*b27 - 2*a17*b27 - 6*a18*b24 - 2*a4*b29
- a7*b26,
2*a12*b38 + a13*b13 + 2*a15*b10 - 2*a15*b33 - 2*a18*b27 - 3*a4*b30 - a7*b28 - 2
*a8*b25,
4*a1*b8 - a13*b6 - 3*a13*b7 - a14*b7 + 2*a4*b25 + 2*a5*b25 + a5*b26 - 6*a6*b1,
2*a1*b12 - 2*a13*b8 - a13*b9 - 6*a15*b7 - 2*a2*b8 + 3*a5*b28 + 2*a6*b25 - 6*a8*
b1,
a1*b11 - a12*b11 - a14*b7 + a2*b7 - a5*b2 + a5*b27 - a7*b1 + a7*b24,
2*a14*b14 + 2*a15*b13 - 2*a2*b14 - a3*b13 - 2*a7*b30 + 3*a7*b5 - 2*a8*b29 + 2*
a8*b4,
2*a12*b40 + a14*b16 + 2*a16*b34 - 2*a17*b31 - a17*b32 - 12*a19*b24 - a5*b36 -
a9*b26,
4*a10*b25 - a13*b19 - 2*a15*b16 - a18*b11 + 2*a18*b34 + 2*a5*b38 + a7*b35 + a9*
b28,
2*a1*b15 + 2*a1*b17 - 2*a16*b6 - 2*a17*b6 - a4*b10 + 2*a4*b32 + 2*a4*b33 - 6*a9
*b1,
6*a11*b26 - 12*a12*b42 - a14*b21 - 2*a16*b40 + 4*a17*b39 + 8*a19*b31 + 6*a19*
b32 + a9*b36,
2*a10*b38 + 3*a11*b30 - 2*a15*b22 + 6*a15*b42 - a18*b20 + 2*a18*b41 - 2*a19*b38
+ a8*b41,
a11*b12 - 6*a11*b35 + 8*a13*b23 + a18*b21 + 4*a19*b18 - a2*b23 - 2*a6*b42 - a9*
b41,
2*a1*b29 - 2*a12*b28 - 2*a12*b29 + 2*a13*b27 - a14*b2 - a14*b26 + a14*b27 + 6*
a15*b24 + a2*b26,
3*a1*b30 - 3*a12*b30 + a13*b29 - a13*b4 - a14*b28 - 2*a15*b2 + 2*a15*b27 + a2*
b28 + 3*a3*b25,
4*a12*b35 + 4*a12*b36 + 2*a12*b37 - 2*a13*b32 - 2*a14*b32 + a14*b9 - 12*a18*b24
- 2*a6*b26 - a7*b26,
2*a12*b37 + 2*a13*b10 + a13*b11 + 2*a15*b6 - 2*a17*b27 - 6*a18*b24 - 2*a4*b28 -
a5*b28 - 3*a7*b25,
2*a1*b8 + 2*a1*b9 - 2*a13*b6 - a14*b6 - a4*b2 + 2*a4*b25 + 2*a4*b26 - 6*a6*b1 -
3*a7*b1,
2*a13*b14 + 2*a15*b12 + 2*a15*b13 - a2*b14 + a3*b12 - 2*a6*b30 - 3*a7*b30 - 2*
a8*b28 + 2*a8*b4,
2*a1*b10 + a1*b11 - a12*b11 - a14*b6 + a2*b6 - 2*a4*b2 - a5*b2 + a5*b27 - 3*a7*
b1,
a10*b2 - 4*a10*b25 + a13*b19 + 2*a15*b16 + a18*b11 + 2*a18*b8 - 2*a5*b38 - a7*
b35 - a9*b28,
2*a1*b20 - a10*b3 - 4*a13*b18 - 2*a15*b16 - a18*b9 + 2*a5*b38 + 2*a6*b35 - 2*a8
*b8 + a9*b28,
6*a11*b25 - 6*a12*b42 - a13*b21 - a18*b15 + 4*a19*b31 - 2*a19*b33 + 4*a19*b34 +
a4*b41 + a9*b35,
a10*b14 - 2*a10*b38 - 3*a11*b30 + 2*a11*b5 + 3*a15*b22 + 2*a18*b20 + a19*b14 -
a3*b22 - a8*b41,
a1*b35 - a12*b35 + a13*b31 + a13*b34 - a13*b8 - a16*b25 - a17*b25 - a18*b1 +
a18*b24 + a6*b25,
2*a12*b35 + 2*a12*b37 + a13*b10 - 2*a13*b33 + a13*b9 - 2*a14*b33 - 6*a18*b24 -
a4*b28 - 2*a6*b25 - 2*a7*b25,
2*a12*b37 + a13*b11 - 2*a13*b34 + a13*b9 - 2*a14*b34 + a17*b27 - 6*a18*b24 - a5
*b28 - 2*a6*b25 - 2*a7*b25,
6*a1*b14 - 2*a13*b12 - 2*a15*b11 - 2*a15*b9 - 6*a3*b8 + 6*a5*b30 + 2*a6*b28 +
a7*b28 - 2*a8*b2 - 2*a8*b3,
2*a1*b14 - a14*b13 - 2*a15*b10 + a2*b13 + 2*a3*b10 + 2*a4*b30 - 4*a4*b5 + a7*
b29 - a7*b4 - 2*a8*b2,
a10*b26 - 2*a12*b41 + 4*a13*b39 - a14*b18 + a14*b39 - 2*a16*b35 - a16*b36 + 2*
a18*b31 + a18*b32 + a6*b36,
4*a12*b39 + 2*a12*b40 + a14*b15 - 2*a16*b31 + 2*a16*b33 - 2*a17*b31 - a17*b32 -
12*a19*b24 - a4*b36 - a9*b26,
4*a10*b26 - 8*a12*b41 - a14*b19 - 2*a16*b37 + 2*a17*b36 + 4*a18*b31 + 4*a18*b32
+ 8*a19*b27 + a7*b36 + 2*a9*b29,
a1*b16 - a12*b16 - a16*b7 - a17*b7 + a5*b31 + a5*b34 - a5*b8 + a6*b7 - a9*b1 +
a9*b24,
a10*b20 - 4*a10*b41 + 3*a11*b14 - 6*a11*b38 + 8*a15*b23 + 6*a18*b22 + 4*a19*b20
- 3*a3*b23 - a8*b22 - 2*a8*b42,
a1*b36 - 2*a12*b35 - a12*b36 + 2*a13*b31 + a13*b32 + a14*b31 - a14*b8 - 2*a16*
b25 - a16*b26 + 3*a18*b24 + a6*b26,
4*a12*b38 - 2*a13*b36 + a14*b12 - 4*a15*b31 - 2*a15*b32 + 2*a16*b28 + 2*a16*b29
- 2*a18*b27 - a2*b36 - 2*a6*b29 - 2*a8*b26,
2*a12*b40 + 2*a13*b15 + a13*b16 - 2*a17*b31 + 2*a17*b33 - 2*a17*b34 + a18*b6 -
12*a19*b24 - 2*a4*b35 - a5*b35 - 3*a9*b25,
6*a12*b40 + 2*a13*b15 + 2*a13*b17 - 4*a16*b33 - 2*a16*b34 + a17*b32 - 2*a17*b33
- 2*a17*b34 - 24*a19*b24 - a4*b37 - 6*a9*b25,
4*a10*b25 - 4*a12*b41 - a13*b19 - 2*a15*b15 - a18*b10 + 2*a18*b31 + 2*a18*b34 +
4*a19*b27 + 2*a4*b38 + a7*b35 + a9*b28,
4*a12*b40 + 2*a13*b16 + 2*a13*b17 - 6*a16*b34 + 2*a17*b31 + a17*b32 - 2*a17*b33
- 2*a17*b34 - 24*a19*b24 - a5*b37 - 6*a9*b25,
8*a10*b25 - 8*a12*b41 - 2*a13*b19 - 2*a15*b17 + 2*a16*b37 - 2*a17*b36 + 4*a18*
b33 + 4*a18*b34 + 8*a19*b27 + a7*b37 + 2*a9*b28,
4*a10*b28 - 4*a13*b20 - 2*a15*b19 + 4*a15*b40 - 2*a17*b38 - a18*b13 + 2*a18*b37
+ 2*a7*b38 + 2*a8*b35 + a8*b37 + 3*a9*b30,
2*a1*b16 + 2*a1*b17 - 2*a12*b16 - 6*a16*b7 - a17*b6 - 3*a17*b7 - a5*b11 + 2*a5*
b31 + 2*a5*b32 + 2*a5*b34 - 6*a9*b1,
a10*b35 + a11*b28 - a13*b22 + 6*a13*b42 - a16*b41 - a17*b41 - a18*b18 + a18*b39
+ a18*b40 - 2*a19*b35 + a6*b41,
24*a11*b25 - 24*a12*b42 - 4*a13*b21 + 2*a16*b40 - 4*a17*b39 - a18*b17 + 8*a19*
b31 + 8*a19*b33 + 16*a19*b34 + 2*a9*b35 + a9*b37,
4*a10*b35 + a10*b37 + 6*a11*b28 - 6*a13*b22 - 2*a15*b21 - 2*a17*b41 - a18*b19 +
4*a18*b40 + 2*a19*b37 + a7*b41 + 2*a9*b38,
2*a12*b38 + a13*b12 - 2*a13*b35 - 2*a15*b31 - 2*a15*b34 + a16*b28 + a17*b28 -
a18*b26 + a18*b3 - a2*b35 - a6*b28 - 2*a8*b25,
2*a1*b8 + 2*a1*b9 - 2*a12*b8 - a13*b6 - 3*a13*b7 - 3*a14*b7 - a5*b2 + 2*a5*b25
+ 2*a5*b26 + a5*b27 - 6*a6*b1 - 3*a7*b1,
2*a1*b11 + a1*b9 - a12*b9 - 2*a13*b7 - 3*a14*b7 + a2*b7 + a5*b26 + 2*a5*b27 - 2
*a5*b3 - 2*a6*b1 + 2*a6*b24 - 3*a7*b1,
2*a1*b10 + 2*a1*b11 - 2*a12*b10 - a14*b6 - a14*b7 + a2*b6 + a2*b7 - 2*a4*b2 + 2
*a4*b27 - 2*a5*b2 - 4*a7*b1 + a7*b24,
a1*b13 - a12*b13 - a14*b11 - 2*a15*b7 + a2*b11 + 3*a3*b7 + a5*b29 - a5*b4 - a7*
b2 + a7*b27 - 2*a8*b1 + 2*a8*b24,
2*a1*b14 - 2*a12*b14 - a14*b13 - 2*a15*b11 + a2*b13 + 2*a3*b11 + 2*a5*b30 - 4*
a5*b5 + a7*b29 - a7*b4 - 2*a8*b2 + 2*a8*b27,
2*a1*b15 + a1*b16 - a12*b16 - a16*b6 - 3*a17*b7 - 2*a4*b8 + a5*b31 + 2*a5*b33 +
a5*b34 - a5*b8 + a6*b6 - 3*a9*b1,
8*a1*b21 - 48*a11*b1 - 4*a16*b17 - 4*a17*b17 - 8*a19*b6 - 24*a19*b7 + 4*a4*b40
+ 8*a5*b39 + 4*a5*b40 + a7*b17 + 2*a9*b32 - 2*a9*b9,
a10*b21 - 6*a11*b18 - a11*b19 + 6*a11*b39 + 6*a11*b40 - 8*a16*b23 - 8*a17*b23 -
10*a19*b21 + 8*a6*b23 + a7*b23 - a9*b22 + 10*a9*b42,
4*a1*b12 + 2*a1*b13 - 2*a13*b9 - a14*b9 - 2*a15*b6 - 6*a15*b7 + 2*a4*b28 + 2*a5
*b28 + 2*a5*b29 + 2*a6*b26 - 4*a6*b3 - a7*b3 - 12*a8*b1,
a1*b41 + a10*b25 - a12*b41 - a13*b18 + a13*b39 + a13*b40 - a16*b35 - a17*b35 +
a18*b31 + a18*b34 - a18*b8 - 4*a19*b25 + a6*b35,
4*a10*b28 - 2*a10*b4 - 4*a13*b20 - 4*a15*b18 - 2*a15*b19 - 2*a18*b12 - a18*b13
+ a2*b20 + 2*a6*b38 + 2*a7*b38 - a8*b12 + 2*a8*b35 + 3*a9*b30,
a10*b12 - 4*a10*b35 - 6*a11*b28 + 2*a11*b4 + 6*a13*b22 + 2*a15*b21 + 4*a18*b18
+ a18*b19 + 2*a19*b12 - a2*b22 - 2*a6*b41 - a7*b41 - 2*a9*b38,
a1*b37 - 4*a12*b35 - a12*b37 + a13*b32 + 2*a13*b33 + 2*a13*b34 - 2*a13*b8 - a13
*b9 + a14*b34 - 2*a17*b25 - a17*b26 + 6*a18*b24 + 4*a6*b25 + a7*b25,
4*a1*b10 + 2*a1*b11 + 3*a1*b9 - a12*b9 - 2*a13*b6 - 3*a14*b6 + a2*b6 + 2*a4*b26
+ 2*a4*b27 - 4*a4*b3 + a5*b26 - 2*a5*b3 - 6*a6*b1 - 9*a7*b1,
4*a1*b16 + 2*a1*b17 - 2*a12*b17 - 8*a16*b7 - a17*b6 - 5*a17*b7 + 2*a4*b34 + 4*
a5*b31 + 2*a5*b32 + 2*a5*b34 - a5*b9 + a7*b7 - 8*a9*b1 + 2*a9*b24,
8*a12*b38 + 2*a13*b12 + 2*a13*b13 - 2*a13*b37 - 2*a15*b32 - 4*a15*b33 - 4*a15*
b34 + 2*a15*b9 + 2*a17*b28 + 2*a17*b29 - 4*a18*b27 - a2*b37 - 2*a6*b28 - 2*a7*
b28 - 8*a8*b25,
3*a1*b13 - a12*b13 - 2*a14*b10 - a14*b11 - 2*a15*b6 + 2*a2*b10 + a2*b11 + 3*a3*
b6 + 2*a4*b29 - 2*a4*b4 + a5*b29 - a5*b4 - 3*a7*b2 + a7*b27 - 6*a8*b1,
2*a1*b15 + 2*a1*b16 - 2*a12*b15 - a16*b6 - a16*b7 - 4*a17*b7 + 2*a4*b31 - 2*a4*
b8 + 2*a5*b33 + 2*a5*b34 - 2*a5*b8 + a6*b6 + a6*b7 - 4*a9*b1 + a9*b24,
a1*b21 + a10*b7 - 6*a11*b1 + 6*a11*b24 - a12*b21 - a16*b16 - a17*b16 - 6*a19*b7
- a5*b18 + a5*b39 + a5*b40 + a6*b16 + a9*b31 + a9*b34 - a9*b8,
3*a13*b14 - 2*a13*b38 + 2*a14*b38 + 2*a15*b12 - 2*a15*b35 - 2*a15*b36 - 2*a15*
b37 + 3*a16*b30 + 3*a17*b30 + 2*a18*b28 - a18*b29 + a18*b4 - 2*a2*b38 - 3*a3*
b35 - 3*a6*b30 - 3*a8*b28,
4*a1*b12 - 2*a13*b10 - 2*a13*b11 - a14*b11 - 2*a14*b8 - 2*a15*b6 - 6*a15*b7 + 2
*a2*b8 + 2*a4*b28 + 2*a5*b28 + 2*a5*b29 - 4*a6*b2 - a7*b2 + 2*a7*b25 + a7*b26 -
12*a8*b1,
2*a1*b12 + 4*a1*b13 - 2*a13*b10 - 2*a14*b10 - a14*b9 - 4*a15*b6 + a2*b10 + a2*
b9 + 2*a4*b28 + 4*a4*b29 - 2*a4*b4 - 2*a6*b2 - 2*a7*b2 + a7*b26 - 2*a7*b3 - 12*
a8*b1,
2*a10*b28 - 2*a13*b20 + 4*a13*b41 - a14*b41 - 2*a15*b18 + 2*a15*b39 + 2*a15*b40
- 2*a16*b38 - 2*a17*b38 - a18*b12 + a18*b36 + a18*b37 - 4*a19*b28 + a2*b41 + 2
*a6*b38 + 2*a8*b35,
2*a1*b15 + 2*a1*b16 + 4*a1*b17 - 2*a12*b15 - 2*a16*b6 - 6*a16*b7 - 3*a17*b6 - 3
*a17*b7 - a4*b11 + 2*a4*b32 + 2*a4*b34 - a5*b10 + 2*a5*b31 + 2*a5*b32 + 2*a5*
b33 - 12*a9*b1,
8*a1*b23 + a10*b15 - a11*b10 + 6*a11*b33 - 6*a11*b8 - a16*b21 - a17*b21 - 4*a19
*b15 - 4*a19*b16 - a4*b22 + 2*a4*b42 + 6*a5*b42 + a6*b21 - a9*b18 + a9*b39 + a9
*b40,
8*a1*b23 + a10*b16 - a11*b11 + 6*a11*b31 + 6*a11*b34 - 6*a11*b8 - 8*a12*b23 -
a16*b21 - a17*b21 - 8*a19*b16 - a5*b22 + 8*a5*b42 + a6*b21 - a9*b18 + a9*b39 +
a9*b40,
2*a1*b38 - 2*a12*b38 - a13*b12 + a13*b36 + a13*b37 - a14*b35 + 2*a15*b31 + 2*
a15*b34 - 2*a15*b8 - a16*b28 - a17*b28 - a18*b2 - 2*a18*b25 + a18*b27 + a2*b35
+ a6*b28 + 2*a8*b25,
2*a10*b29 - 2*a10*b4 - 2*a14*b20 - 2*a15*b19 - a16*b14 - a17*b14 - 2*a18*b13 +
2*a2*b20 + a3*b19 + a6*b14 - a7*b14 + 2*a7*b38 - a8*b12 + a8*b36 + a8*b37 + 2*
a9*b30 - 2*a9*b5,
8*a1*b18 - 12*a10*b1 - 2*a13*b15 - 2*a13*b16 - 2*a13*b17 - a14*b16 - 2*a16*b8 -
2*a17*b8 - a18*b6 - 9*a18*b7 + 2*a4*b35 + 6*a5*b35 + a5*b36 + a5*b37 - 2*a6*b8
- a9*b2 + 4*a9*b25 + a9*b26,
48*a1*b23 + a10*b17 + 6*a11*b32 - 24*a11*b8 - 6*a11*b9 - 6*a16*b21 - 6*a17*b21
- 8*a19*b15 - 16*a19*b16 - 12*a19*b17 + 12*a4*b42 + 36*a5*b42 + 4*a6*b21 + a7*
b21 - 2*a9*b18 - a9*b19 + 6*a9*b39 + 6*a9*b40,
2*a10*b18 - 2*a10*b39 - 2*a10*b40 + 3*a11*b12 + a11*b13 - 3*a11*b36 - 3*a11*b37
+ 4*a14*b23 + 3*a16*b22 + 3*a17*b22 + 4*a18*b21 + 4*a19*b19 - 4*a2*b23 - 3*a6*
b22 - 4*a7*b42 - a8*b21 + a9*b20 - 4*a9*b41,
12*a1*b14 - 2*a13*b13 - 2*a14*b12 - a14*b13 - 4*a15*b10 - 4*a15*b11 - 2*a15*b9
+ 2*a2*b12 + a2*b13 + 6*a4*b30 + 6*a5*b30 + 2*a6*b29 - 2*a6*b4 + 2*a7*b28 + 2*
a7*b29 - 2*a7*b4 - 8*a8*b2 + 2*a8*b26 - 4*a8*b3,
8*a10*b25 - 8*a12*b41 - 4*a13*b18 - a13*b19 + 4*a13*b40 + a14*b40 - a16*b37 - 2
*a17*b35 - a17*b36 - a17*b37 + 4*a18*b31 + a18*b32 + 2*a18*b33 + 6*a18*b34 -
a18*b9 + 2*a6*b35 + a6*b37 + a7*b35 + a9*b28,
8*a1*b18 + 4*a1*b19 - 24*a10*b1 - 4*a13*b17 - 2*a14*b17 - 2*a16*b9 - 2*a17*b9 -
4*a18*b6 - 12*a18*b7 + a2*b17 + 4*a4*b35 + 2*a4*b37 + 4*a5*b35 + 4*a5*b36 + 2*
a5*b37 + 2*a6*b32 - 2*a6*b9 + 2*a9*b26 - 4*a9*b3,
8*a1*b15 + 4*a1*b16 + 6*a1*b17 - 2*a12*b17 - 6*a16*b6 - 6*a16*b7 - 5*a17*b6 - 9
*a17*b7 + 4*a4*b31 + 2*a4*b32 + 4*a4*b33 + 2*a4*b34 - 2*a4*b9 + 4*a5*b32 + 4*a5
*b33 + 2*a5*b34 - a5*b9 + a7*b6 - 24*a9*b1,
a1*b19 - 4*a10*b1 + 4*a10*b24 - a12*b19 - a14*b16 - a16*b11 - a17*b11 - 4*a18*
b7 + a2*b16 - a5*b12 + a5*b36 + a5*b37 + a6*b11 + a7*b31 + a7*b34 - a7*b8 + 2*
a8*b7 - a9*b2 + a9*b27,
8*a1*b20 - 8*a10*b2 - 2*a14*b19 - 4*a15*b17 - 2*a16*b13 - 2*a17*b13 - 4*a18*b10
- 4*a18*b11 + 2*a2*b19 + 3*a3*b17 + 4*a4*b38 + 4*a5*b38 - a7*b13 + 2*a7*b36 +
2*a7*b37 + 2*a8*b32 - 2*a8*b9 + 2*a9*b29 - 2*a9*b4,
2*a1*b12 + 4*a1*b13 - 2*a12*b12 - 2*a13*b11 - 2*a14*b11 - a14*b9 - 2*a15*b6 - 6
*a15*b7 + a2*b11 + a2*b9 + 2*a5*b28 + 4*a5*b29 - 2*a5*b4 - 2*a6*b2 + 2*a6*b27 -
2*a7*b2 + a7*b26 + a7*b27 - 2*a7*b3 - 12*a8*b1,
3*a1*b21 + a10*b6 - 18*a11*b1 - a12*b21 - 2*a16*b15 - a16*b16 - 3*a17*b16 - 2*
a19*b6 - 12*a19*b7 - 2*a4*b18 + 2*a4*b39 - a5*b18 + a5*b39 + 3*a5*b40 + 2*a6*
b15 + a6*b16 + a9*b31 + 2*a9*b33 + a9*b34 - 3*a9*b8,
4*a1*b18 + 2*a1*b19 - 12*a10*b1 - 4*a13*b15 - 2*a14*b15 - a16*b9 - a17*b9 - 2*
a18*b6 - 6*a18*b7 + a2*b15 - a4*b12 + 2*a4*b35 + 2*a4*b36 + 2*a5*b35 + 2*a5*b37
+ 2*a6*b33 - 2*a6*b8 + a6*b9 - 2*a7*b8 + a9*b26 - 2*a9*b3,
6*a1*b19 - 24*a10*b1 - 2*a14*b15 - 2*a14*b17 - 4*a16*b10 - 2*a16*b11 - 4*a17*
b10 - 2*a17*b11 - 6*a18*b6 - 6*a18*b7 + 2*a2*b15 + 2*a2*b17 - 2*a4*b13 + 4*a4*
b36 + 4*a4*b37 + 2*a5*b36 + 2*a5*b37 + 2*a7*b32 + 2*a7*b33 - a7*b9 - 6*a9*b2,
4*a1*b20 - 4*a10*b2 - a14*b19 - 4*a15*b15 - a16*b13 - a17*b13 - 2*a18*b10 - 2*
a18*b11 + a2*b19 + 3*a3*b15 - 3*a4*b14 + 2*a4*b38 + 2*a5*b38 + a6*b13 - a7*b12
+ a7*b36 + a7*b37 + a8*b10 + 2*a8*b33 - 2*a8*b8 + a9*b29 - a9*b4,
6*a1*b19 - 24*a10*b1 - 2*a12*b19 - 2*a14*b16 - 2*a14*b17 - 6*a16*b11 - 2*a17*
b10 - 4*a17*b11 - 4*a18*b6 - 12*a18*b7 + 2*a2*b16 + 2*a2*b17 + 2*a4*b37 - 2*a5*
b13 + 6*a5*b36 + 4*a5*b37 + 2*a7*b31 + 2*a7*b32 + 2*a7*b34 - a7*b9 - 6*a9*b2 +
2*a9*b27,
12*a1*b22 - 8*a10*b8 + 6*a11*b26 - 12*a11*b3 - 6*a13*b21 - a14*b21 - 4*a16*b18
- 4*a17*b18 - 2*a18*b15 - 6*a18*b16 - 2*a18*b17 - 6*a19*b9 + a2*b21 + 2*a4*b41
+ 10*a5*b41 + 2*a6*b18 + 2*a6*b39 + 2*a6*b40 - a9*b12 + 6*a9*b35 + a9*b36 + a9*
b37,
4*a1*b18 + 2*a1*b19 - 12*a10*b1 - 4*a12*b18 - 4*a13*b16 - 2*a14*b16 - a16*b9 -
a17*b9 - a18*b6 - 9*a18*b7 + a2*b16 - a5*b12 + 4*a5*b35 + 2*a5*b36 + 2*a5*b37 +
2*a6*b31 + 2*a6*b34 - 2*a6*b8 + a6*b9 - 2*a7*b8 + a9*b26 + a9*b27 - 2*a9*b3,
4*a1*b20 - 4*a10*b2 + 4*a10*b27 - 4*a12*b20 - a14*b19 - 4*a15*b16 - a16*b13 -
a17*b13 - 4*a18*b11 + a2*b19 + 3*a3*b16 - 3*a5*b14 + 4*a5*b38 + a6*b13 - a7*b12
+ a7*b36 + a7*b37 + a8*b11 + 2*a8*b31 + 2*a8*b34 - 2*a8*b8 + a9*b29 - a9*b4,
12*a1*b21 - 72*a11*b1 - 6*a16*b15 - 2*a16*b16 - 2*a16*b17 - 6*a17*b15 - 2*a17*
b16 - 2*a17*b17 - 12*a19*b6 - 36*a19*b7 - a4*b19 + 8*a4*b39 + 4*a4*b40 + 4*a5*
b39 + 8*a5*b40 + 2*a6*b15 + 2*a6*b17 + a7*b15 - a9*b10 + 2*a9*b32 + 4*a9*b33 -
6*a9*b8 - a9*b9,
6*a1*b22 + 4*a10*b33 - 4*a10*b8 - 6*a11*b2 - a14*b21 - a16*b19 - a17*b19 - 4*
a18*b15 - 2*a18*b16 - 2*a19*b10 - 4*a19*b11 + a2*b21 - 2*a4*b20 + 2*a4*b41 + 4*
a5*b41 + a6*b19 - a7*b18 + a7*b39 + a7*b40 + 2*a8*b15 - a9*b12 + a9*b36 + a9*
b37,
12*a1*b21 - 72*a11*b1 - 4*a12*b21 - 8*a16*b16 - 2*a16*b17 - 2*a17*b15 - 6*a17*
b16 - 2*a17*b17 - 8*a19*b6 - 48*a19*b7 + 2*a4*b40 - a5*b19 + 12*a5*b39 + 10*a5*
b40 + 2*a6*b16 + 2*a6*b17 + a7*b16 - a9*b11 + 4*a9*b31 + 2*a9*b32 + 4*a9*b34 -
6*a9*b8 - a9*b9,
6*a1*b22 + 4*a10*b31 + 4*a10*b34 - 4*a10*b8 - 6*a11*b2 + 6*a11*b27 - 6*a12*b22
- a14*b21 - a16*b19 - a17*b19 - 6*a18*b16 - 6*a19*b11 + a2*b21 - 2*a5*b20 + 6*
a5*b41 + a6*b19 - a7*b18 + a7*b39 + a7*b40 + 2*a8*b16 - a9*b12 + a9*b36 + a9*
b37,
4*a10*b12 + a10*b13 - 4*a10*b36 - 4*a10*b37 - 6*a11*b29 + 6*a11*b4 + 6*a14*b22
+ 6*a15*b21 + 4*a16*b20 + 4*a17*b20 + 6*a18*b19 + 6*a19*b13 - 6*a2*b22 - 3*a3*
b21 - 4*a6*b20 + a7*b20 - 6*a7*b41 + 2*a8*b18 - a8*b19 - 2*a8*b39 - 2*a8*b40 +
3*a9*b14 - 6*a9*b38,
12*a1*b22 + 2*a10*b32 - 4*a10*b8 - 2*a10*b9 - 12*a11*b2 - 2*a14*b21 - 2*a16*b19
- 2*a17*b19 - 2*a18*b15 - 2*a18*b16 - 4*a18*b17 - 4*a19*b10 - 8*a19*b11 + 2*a2
*b21 + 4*a4*b41 + 8*a5*b41 + a6*b19 + 2*a7*b39 + 2*a7*b40 + a8*b17 - a9*b12 -
a9*b13 + 2*a9*b36 + 2*a9*b37,
3*a1*b19 - 12*a10*b1 - a12*b19 - 2*a14*b15 - a14*b16 - 2*a16*b10 - a16*b11 - 3*
a17*b11 - 2*a18*b6 - 6*a18*b7 + 2*a2*b15 + a2*b16 - 2*a4*b12 + 2*a4*b36 - a5*
b12 + a5*b36 + 3*a5*b37 + 2*a6*b10 + a6*b11 + a7*b31 + 2*a7*b33 + a7*b34 - 3*a7
*b8 + 2*a8*b6 - 3*a9*b2 + a9*b27,
16*a1*b20 - 8*a10*b2 + 4*a10*b26 - 8*a10*b3 - 4*a13*b19 - 4*a14*b18 - a14*b19 -
4*a15*b15 - 4*a15*b16 - 4*a15*b17 - 2*a16*b12 - 2*a17*b12 - 2*a18*b10 - 6*a18*
b11 - 4*a18*b9 + 4*a2*b18 + a2*b19 + 4*a4*b38 + 12*a5*b38 + 2*a6*b36 + 2*a6*b37
- a7*b12 + 4*a7*b35 + a7*b36 + a7*b37 - 8*a8*b8 + 4*a9*b28 + 2*a9*b29 - 2*a9*
b4
Relevance
In the following evolutionary system u=u(t,x) is a scalar function,
v=v(t,x) is a vector function and f(..,..) stands for the scalar
product of both vector arguments of f.
u = u *a1 + u *f(v,v)*a6 + u *a2*u + f(v ,v )*a4 + f(v,v )*a5
t 2x x x x x 2x
3 2
+ f(v,v )*f(v,v)*a9 + f(v,v )*a7*u + f(v,v) *a11 + f(v,v) *a10*u
x x
2 3
+ f(v,v)*a8*u + a3*u ,
v = u *a13*v + v *a12 + v *f(v,v)*a16 + v *a14*u + f(v,v )*a17*v
t x 2x x x x
2 2
+ f(v,v) *a19*v + f(v,v)*a18*u*v + a15*u *v}
For any solution of the above algebraic conditions for the
a's and b's this system has the following symmetry:
2
u = u *b1 + u *f(v,v)*b8 + u *b2*u + u *b3 + u *f(v,v )*b9
t 3x 2x 2x x x x
2 2
+ u *f(v,v) *b18 + u *f(v,v)*b12*u + u *b4*u + f(v ,v )*b6
x x x x 2x
+ f(v ,v )*f(v,v)*b15 + f(v ,v )*b10*u + f(v,v )*b7
x x x x 3x
2
+ f(v,v )*f(v,v)*b16 + f(v,v )*b11*u + f(v,v ) *b17
2x 2x x
2 2
+ f(v,v )*f(v,v) *b21 + f(v,v )*f(v,v)*b19*u + f(v,v )*b13*u
x x x
4 3 2 2 3 4
+ f(v,v) *b23 + f(v,v) *b22*u + f(v,v) *b20*u + f(v,v)*b14*u + b5*u ,
v = u *b25*v + u *v *b26 + u *f(v,v)*b35*v + u *b28*u*v + v *b24
t 2x x x x x 3x
2
+ v *f(v,v)*b31 + v *b27*u + v *f(v,v )*b32 + v *f(v,v) *b39
2x 2x x x x
2
+ v *f(v,v)*b36*u + v *b29*u + f(v ,v )*b33*v + f(v,v )*b34*v
x x x x 2x
3
+ f(v,v )*f(v,v)*b40*v + f(v,v )*b37*u*v + f(v,v) *b42*v
x x
2 2 3
+ f(v,v) *b41*u*v + f(v,v)*b38*u *v + b30*u *v