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I have a hash and print that in the following Pattern :
Input 1 :
{
a: [1,2,3,4,5,6,7,8,9],
b: [1,2,3,4,5,6],
c: [2,3,4,5,6,7]
}
Output 1 :
A1 A2 A3 A4 A5 A6 A7 A8 A9
B1 B2 B3 B4 B5 B6
C2 C3 C4 C5 C6 C7
Input 2 :
{
a: [1,2,3,4,5,6,7],
b: [2,3,4,5,6],
c: [1,2,3,4,5,6,7]
}
Output 2 :
A1 A2 A3 A4 A5 A6 A7
B2 B3 B4 B5 B6
C1 C2 C3 C4 C5 C6 C7
For e.g.
If we consider Input 1 then the expectation is, the solution should add blank spaces at the missing number position.
=> It should return :
[
["A1","A2","A3","A4","A5","A6","A7","A8","A9"],
["B1","B2","B3","B4","B5","B6","","",""],
["","C2","C3","C4","C5","C6","C7","",""]
]
You can use collection, map or each for this
data = {a: [1,2,3,4,5,6,7,8,9],b: [1,2,3,4,5,6],c: [2,3,4,5,6,7]}
data.map{|k,v| (1..9).map{|a| data[k].include?(a) ? k.to_s.upcase() +a.to_s : ' '}}
Lets say I 3 different matrix on 3 different threads:
a1 a2 a3 a4 a5 a6
b1 b2 b3 b4 b5 b6
c1 c2 c3 c4 c5 c6
d1 d2 d3 d4 d5 d6
e1 e2 e3 e4 e5 e6
f1 f2 f3 f4 f5 f6
I want to gather them by using MPI_Gather into this one matrix
a1 a2 a3 a4 a5 a6
b1 b2 b3 b4 b5 b6
c1 c2 c3 c4 c5 c6
d1 d2 d3 d4 d5 d6
e1 e2 e3 e4 e5 e6
f1 f2 f3 f4 f5 f6
I am using
MPI_Gather(&oldMatrix,N/size,columnType,&newMatrix,N/size,gatherColType,0,MPI_COMM_WORLD);
Where coltype is:
MPI_Type_vector(N,
1,
N/size,
MPI_FLOAT,
&column);
MPI_Type_commit(&column);
MPI_Type_create_resized(column, 0, 1*sizeof(float), &columnType);
MPI_Type_commit(&columnType);
However, I am not sure how should i create gatherColType (if it is needed). Can you help me about that?
note: N is matrix size and size is thread count (i.e. 3 in this question)
I have written this code:
digraph G {
A254 -> A10[style=invis];
A10 -> A9[style=invis];
A9 -> A8[style=invis];
A8 -> A7[style=invis];
A7 -> A6[style=invis];
A6 -> A5[style=invis];
A5 -> A4[style=invis];
A4 -> A3[style=invis];
A3 -> A2[style=invis];
A2 -> A1[style=invis];
A254 -> A8 [label="t"];
A8 -> A10 [label="t", style=dotted];
A8 -> A9 [label="t", style=dotted];
A8 -> A7 [label="t", style=dotted];
A8 -> A6 [label="t", style=dotted];
A8 -> A3 [constraint = false, label="t"];
A3 -> A5 [label="t", style=dotted];
A3 -> A4 [label="t", style=dotted];
A3 -> A2 [label="t", style=dotted];
A3 -> A1 [label="t", style=dotted];
A254[style=filled]
A3[style=filled]
A8[style=filled]
{rank=same; A254,A10,A9,A8,A7,A6,A5,A4,A3,A2,A1}
}
It produces the following graph:
Actually I have three questions:
1.How can I make the edge A8 -> A3 neater? It looks very bad.
2.How can I make the edges A254 -> A8 + A8 -> A3 rectangular?
3.How can I make this graph vertical?
EDIT:
It is essential that the nodes are lined up and in the same order shown in the graph above.
This
is generated from
digraph G {
A254 -> A10[style=invis];
A10 -> A9[style=invis];
A9 -> A8[style=invis];
A8 -> A7[style=invis];
A7 -> A6[style=invis];
A6 -> A5[style=invis];
A5 -> A4[style=invis];
A4 -> A3[style=invis];
A3 -> A2[style=invis];
A2 -> A1[style=invis];
A254 -> A8 [label="t"];
A8 -> A10 [label="t", style=dotted];
A8 -> A9 [label="t", style=dotted];
A8 -> A7 [label="t", style=dotted];
A8 -> A6 [label="t", style=dotted];
A8 -> A3 [label="t"];
A3 -> A5 [label="t", style=dotted];
A3 -> A4 [label="t", style=dotted];
A3 -> A2 [label="t", style=dotted];
A3 -> A1 [label="t", style=dotted];
A254[style=filled]
A3[style=filled]
A8[style=filled]
{rank=same; A254,A8,A3}
}
The best thing to do is read the documentation and experiment with the ideas in there
I have found the solution, here it is
You can make the edge neater by making a suitable adjustments to the distance between nodes and to the size of the nodes. And the most important thing is to force the A8->A3 edge to be longer by setting the minlen attribute of the corrosponding edge.
The following code
digraph G {
nodesep=0.5
node[fixedsize=true, shape="circle", width=0.5]
A254 -> A10[style=invis];
A10 -> A9[style=invis];
A9 -> A8[style=invis];
A8 -> A7[style=invis];
A7 -> A6[style=invis];
A6 -> A5[style=invis];
A5 -> A4[style=invis];
A4 -> A3[style=invis];
A3 -> A2[style=invis];
A2 -> A1[style=invis];
A254 -> A8 [label="t"];
A8 -> A10 [label="t", style=dotted];
A8 -> A9 [label="t", style=dotted];
A8 -> A7 [label="t", style=dotted];
A8 -> A6 [label="t", style=dotted];
A8 -> A3 [minlen = 3, constraint = false, label="t"];
A3 -> A5 [label="t", style=dotted];
A3 -> A4 [label="t", style=dotted];
A3 -> A2 [label="t", style=dotted];
A3 -> A1 [label="t", style=dotted];
A254[style=filled]
A3[style=filled]
A8[style=filled]
{rank=same; A254,A10,A9,A8,A7,A6,A5,A4,A3,A2,A1}
}
generates the following graph:
Edges can not be rectangular.
You can make the graph vertical by using the command rotate=90
#include <stdio.h>
int main()
{
int alpha,numeric;
for(alpha='A';alpha<'K';alpha++)
{
for(numeric=0;numeric<10;numeric++)
{
printf("%c%d\t",alpha,numeric);
}
}
return 0;
}
The output I got is as follows.
A0 A1 A2 A3 A4 A5 A6 A7 A8 A9
B0 B1 B2 B3 B4 B5 B6 B7 B8 B9
C0 C1 C2 C3 C4 C5 C6 C7 C8 C9
D0 D1 D2 D3 D4 D5 D6 D7 D8 D9
E0 E1 E2 E3 E4 E5 E6 E7 E8 E9
F0 F1 F2 F3 F4 F5 F6 F7 F8 F9
G0 G1 G2 G3 G4 G5 G6 G7 G8 G9
H0 H1 H2 H3 H4 H5 H6 H7 H8 H9
I0 I1 I2 I3 I4 I5 I6 I7 I8 I9
J0 J1 J2 J3 J4 J5 J6 J7 J8 J9
But, more neat and in form of a matrix, without any space between lines.
Someone please tell me how the printing is entering to new line even though there is no newline(\n) character in the program.
I have been trying to integrate a function AND get an output that is sufficiently simple as to be usable. Simply using the Simplify and FullSimplify commands has not done nearly enough, and it takes me over 2 hours to get a result from this integration. Are there any further tricks I can use to force further simplifications? Any assumptions I can put in to make it simpler or faster?
Be warned, the example output is stupidly long, so much so that it is impossible to read, much less manually simplify.
EDIT: Updated input and output to reflect help from commentators. Still nowhere near being usable.
Input (ai, bi, ci,and di are unit vectors, and so will be replaced with the appropriate cosine later. ki has a value but is still kept track of so the proper cosine can be used later. the cosines are independent of the integration, so it makes it marginally shorter to do it this way):
r = {r1, r2, r3};
a = {a1, a2, a3}/\[Sigma]a;
b = {b1, b2, b3}/\[Sigma]b;
c = {c1, c2, c3}/\[Sigma]c;
d = {d1, d2, d3}/\[Sigma]d;
k = {k1, k2, k3};
S = {{S11, S12, S13}, {S21, S22, S23}, {S31, S32, S33}};
FullSimplify[Integrate[(1/(2*Pi*\[Sigma]a*\[Sigma]b))*Exp[-(1/2)*((a.r)^2 + (b.r)^2)]*(1/(2*Pi*\[Sigma]c*\[Sigma]d))*Exp[-(1/2)*((c.r)^2 + (d.r)^2)], {r1, -Infinity, Infinity}, {r2, -Infinity, Infinity}, {r3, -Infinity, Infinity}, Assumptions -> Element[{r1, r2, r3, a1, a2, a3, b1, b2, b3, c1, c2, c3, d1, d2, d3, k1, k2, k3, S11, S12, S13, S21, S22, S23, S31, S32, S33, \[Sigma]a, \[Sigma]b, \[Sigma]c, \[Sigma]d, \[Tau]}, Reals] && \[Sigma]a > 0 && \[Sigma]b > 0 && \[Sigma]c > 0 && \[Sigma]d > 0 && \[Tau] >= 0 && 1 >= a1 >= -1 && 1 >= a2 >= -1 && 1 >= a3 >= -1 && 1 >= b1 >= -1 && 1 >= b2 >= -1 && 1 >= b3 >= -1 && 1 >= c1 >= -1 && 1 >= c2 >= -1 && 1 >= c3 >= -1 && 1 >= d1 >= -1 && 1 >= d2 >= -1 && 1 >= d3 >= -1]]]]
Output:
ConditionalExpression[
1/(Sqrt[2 \[Pi]] \[Sqrt]((b3^2 c2^2 d1^2 \[Sigma]a^2 -
2 b2 b3 c2 c3 d1^2 \[Sigma]a^2 +
b2^2 c3^2 d1^2 \[Sigma]a^2 -
2 b3^2 c1 c2 d1 d2 \[Sigma]a^2 +
2 b2 b3 c1 c3 d1 d2 \[Sigma]a^2 +
2 b1 b3 c2 c3 d1 d2 \[Sigma]a^2 -
2 b1 b2 c3^2 d1 d2 \[Sigma]a^2 +
b3^2 c1^2 d2^2 \[Sigma]a^2 - 2 b1 b3 c1 c3 d2^2 \[Sigma]a^2 +
b1^2 c3^2 d2^2 \[Sigma]a^2 +
2 b2 b3 c1 c2 d1 d3 \[Sigma]a^2 -
2 b1 b3 c2^2 d1 d3 \[Sigma]a^2 -
2 b2^2 c1 c3 d1 d3 \[Sigma]a^2 +
2 b1 b2 c2 c3 d1 d3 \[Sigma]a^2 -
2 b2 b3 c1^2 d2 d3 \[Sigma]a^2 +
2 b1 b3 c1 c2 d2 d3 \[Sigma]a^2 +
2 b1 b2 c1 c3 d2 d3 \[Sigma]a^2 -
2 b1^2 c2 c3 d2 d3 \[Sigma]a^2 +
b2^2 c1^2 d3^2 \[Sigma]a^2 - 2 b1 b2 c1 c2 d3^2 \[Sigma]a^2 +
b1^2 c2^2 d3^2 \[Sigma]a^2 + a3^2 c2^2 d1^2 \[Sigma]b^2 -
2 a2 a3 c2 c3 d1^2 \[Sigma]b^2 +
a2^2 c3^2 d1^2 \[Sigma]b^2 -
2 a3^2 c1 c2 d1 d2 \[Sigma]b^2 +
2 a2 a3 c1 c3 d1 d2 \[Sigma]b^2 +
2 a1 a3 c2 c3 d1 d2 \[Sigma]b^2 -
2 a1 a2 c3^2 d1 d2 \[Sigma]b^2 +
a3^2 c1^2 d2^2 \[Sigma]b^2 - 2 a1 a3 c1 c3 d2^2 \[Sigma]b^2 +
a1^2 c3^2 d2^2 \[Sigma]b^2 +
2 a2 a3 c1 c2 d1 d3 \[Sigma]b^2 -
2 a1 a3 c2^2 d1 d3 \[Sigma]b^2 -
2 a2^2 c1 c3 d1 d3 \[Sigma]b^2 +
2 a1 a2 c2 c3 d1 d3 \[Sigma]b^2 -
2 a2 a3 c1^2 d2 d3 \[Sigma]b^2 +
2 a1 a3 c1 c2 d2 d3 \[Sigma]b^2 +
2 a1 a2 c1 c3 d2 d3 \[Sigma]b^2 -
2 a1^2 c2 c3 d2 d3 \[Sigma]b^2 +
a2^2 c1^2 d3^2 \[Sigma]b^2 - 2 a1 a2 c1 c2 d3^2 \[Sigma]b^2 +
a1^2 c2^2 d3^2 \[Sigma]b^2 + a3^2 b2^2 d1^2 \[Sigma]c^2 -
2 a2 a3 b2 b3 d1^2 \[Sigma]c^2 +
a2^2 b3^2 d1^2 \[Sigma]c^2 -
2 a3^2 b1 b2 d1 d2 \[Sigma]c^2 +
2 a2 a3 b1 b3 d1 d2 \[Sigma]c^2 +
2 a1 a3 b2 b3 d1 d2 \[Sigma]c^2 -
2 a1 a2 b3^2 d1 d2 \[Sigma]c^2 +
a3^2 b1^2 d2^2 \[Sigma]c^2 - 2 a1 a3 b1 b3 d2^2 \[Sigma]c^2 +
a1^2 b3^2 d2^2 \[Sigma]c^2 +
2 a2 a3 b1 b2 d1 d3 \[Sigma]c^2 -
2 a1 a3 b2^2 d1 d3 \[Sigma]c^2 -
2 a2^2 b1 b3 d1 d3 \[Sigma]c^2 +
2 a1 a2 b2 b3 d1 d3 \[Sigma]c^2 -
2 a2 a3 b1^2 d2 d3 \[Sigma]c^2 +
2 a1 a3 b1 b2 d2 d3 \[Sigma]c^2 +
2 a1 a2 b1 b3 d2 d3 \[Sigma]c^2 -
2 a1^2 b2 b3 d2 d3 \[Sigma]c^2 +
a2^2 b1^2 d3^2 \[Sigma]c^2 - 2 a1 a2 b1 b2 d3^2 \[Sigma]c^2 +
a1^2 b2^2 d3^2 \[Sigma]c^2 + (a3 b2 c1 - a2 b3 c1 -
a3 b1 c2 + a1 b3 c2 + a2 b1 c3 -
a1 b2 c3)^2 \[Sigma]d^2)/(c3^2 (d2^2 \[Sigma]a^2 \
\[Sigma]b^2 + (b2^2 \[Sigma]a^2 + a2^2 \[Sigma]b^2) \[Sigma]d^2) -
2 c2 c3 (d2 d3 \[Sigma]a^2 \[Sigma]b^2 + (b2 b3 \[Sigma]a^2 +
a2 a3 \[Sigma]b^2) \[Sigma]d^2) +
c2^2 (d3^2 \[Sigma]a^2 \[Sigma]b^2 + (b3^2 \[Sigma]a^2 +
a3^2 \[Sigma]b^2) \[Sigma]d^2) + \[Sigma]c^2 ((a3 d2 -
a2 d3)^2 \[Sigma]b^2 +
b3^2 (d2^2 \[Sigma]a^2 + a2^2 \[Sigma]d^2) -
2 b2 b3 (d2 d3 \[Sigma]a^2 + a2 a3 \[Sigma]d^2) +
b2^2 (d3^2 \[Sigma]a^2 + a3^2 \[Sigma]d^2)))) Sqrt[
c3^2 (d2^2 \[Sigma]a^2 \[Sigma]b^2 + (b2^2 \[Sigma]a^2 +
a2^2 \[Sigma]b^2) \[Sigma]d^2) -
2 c2 c3 (d2 d3 \[Sigma]a^2 \[Sigma]b^2 + (b2 b3 \[Sigma]a^2 +
a2 a3 \[Sigma]b^2) \[Sigma]d^2) +
c2^2 (d3^2 \[Sigma]a^2 \[Sigma]b^2 + (b3^2 \[Sigma]a^2 +
a3^2 \[Sigma]b^2) \[Sigma]d^2) + \[Sigma]c^2 ((a3 d2 -
a2 d3)^2 \[Sigma]b^2 +
b3^2 (d2^2 \[Sigma]a^2 + a2^2 \[Sigma]d^2) -
2 b2 b3 (d2 d3 \[Sigma]a^2 + a2 a3 \[Sigma]d^2) +
b2^2 (d3^2 \[Sigma]a^2 +
a3^2 \[Sigma]d^2))]), (c3^2 (d2^2 \[Sigma]a^2 \[Sigma]b^2 \
+ (b2^2 \[Sigma]a^2 + a2^2 \[Sigma]b^2) \[Sigma]d^2) -
2 c2 c3 (d2 d3 \[Sigma]a^2 \[Sigma]b^2 + (b2 b3 \[Sigma]a^2 +
a2 a3 \[Sigma]b^2) \[Sigma]d^2) +
c2^2 (d3^2 \[Sigma]a^2 \[Sigma]b^2 + (b3^2 \[Sigma]a^2 +
a3^2 \[Sigma]b^2) \[Sigma]d^2) + \[Sigma]c^2 ((a3 d2 -
a2 d3)^2 \[Sigma]b^2 +
b3^2 (d2^2 \[Sigma]a^2 + a2^2 \[Sigma]d^2) -
2 b2 b3 (d2 d3 \[Sigma]a^2 + a2 a3 \[Sigma]d^2) +
b2^2 (d3^2 \[Sigma]a^2 +
a3^2 \[Sigma]d^2))) (b1^2 c3^2 d2^2 \[Sigma]a^2 -
2 b1^2 c2 c3 d2 d3 \[Sigma]a^2 + b1^2 c2^2 d3^2 \[Sigma]a^2 +
a3^2 c2^2 d1^2 \[Sigma]b^2 - 2 a2 a3 c2 c3 d1^2 \[Sigma]b^2 +
a2^2 c3^2 d1^2 \[Sigma]b^2 - 2 a3^2 c1 c2 d1 d2 \[Sigma]b^2 +
2 a2 a3 c1 c3 d1 d2 \[Sigma]b^2 +
2 a1 a3 c2 c3 d1 d2 \[Sigma]b^2 -
2 a1 a2 c3^2 d1 d2 \[Sigma]b^2 + a3^2 c1^2 d2^2 \[Sigma]b^2 -
2 a1 a3 c1 c3 d2^2 \[Sigma]b^2 + a1^2 c3^2 d2^2 \[Sigma]b^2 +
2 a2 a3 c1 c2 d1 d3 \[Sigma]b^2 -
2 a1 a3 c2^2 d1 d3 \[Sigma]b^2 -
2 a2^2 c1 c3 d1 d3 \[Sigma]b^2 +
2 a1 a2 c2 c3 d1 d3 \[Sigma]b^2 -
2 a2 a3 c1^2 d2 d3 \[Sigma]b^2 +
2 a1 a3 c1 c2 d2 d3 \[Sigma]b^2 +
2 a1 a2 c1 c3 d2 d3 \[Sigma]b^2 -
2 a1^2 c2 c3 d2 d3 \[Sigma]b^2 + a2^2 c1^2 d3^2 \[Sigma]b^2 -
2 a1 a2 c1 c2 d3^2 \[Sigma]b^2 + a1^2 c2^2 d3^2 \[Sigma]b^2 +
a3^2 b1^2 d2^2 \[Sigma]c^2 - 2 a2 a3 b1^2 d2 d3 \[Sigma]c^2 +
a2^2 b1^2 d3^2 \[Sigma]c^2 + a3^2 b1^2 c2^2 \[Sigma]d^2 -
2 a2 a3 b1^2 c2 c3 \[Sigma]d^2 + a2^2 b1^2 c3^2 \[Sigma]d^2 +
b3^2 ((a2 d1 - a1 d2)^2 \[Sigma]c^2 +
c2^2 (d1^2 \[Sigma]a^2 + a1^2 \[Sigma]d^2) -
2 c1 c2 (d1 d2 \[Sigma]a^2 + a1 a2 \[Sigma]d^2) +
c1^2 (d2^2 \[Sigma]a^2 + a2^2 \[Sigma]d^2)) +
b2^2 ((a3 d1 - a1 d3)^2 \[Sigma]c^2 +
c3^2 (d1^2 \[Sigma]a^2 + a1^2 \[Sigma]d^2) -
2 c1 c3 (d1 d3 \[Sigma]a^2 + a1 a3 \[Sigma]d^2) +
c1^2 (d3^2 \[Sigma]a^2 + a3^2 \[Sigma]d^2)) -
2 b1 b2 (-c3 (c2 d1 + c1 d2) d3 \[Sigma]a^2 + (a3 d1 -
a1 d3) (a3 d2 - a2 d3) \[Sigma]c^2 -
a3 (a2 c1 + a1 c2) c3 \[Sigma]d^2 +
c3^2 (d1 d2 \[Sigma]a^2 + a1 a2 \[Sigma]d^2) +
c1 c2 (d3^2 \[Sigma]a^2 + a3^2 \[Sigma]d^2)) -
2 b3 (b1 (-c2 d2 (c3 d1 + c1 d3) \[Sigma]a^2 + (a2 d1 -
a1 d2) (-a3 d2 + a2 d3) \[Sigma]c^2 -
a2 c2 (a3 c1 + a1 c3) \[Sigma]d^2 +
c1 c3 (d2^2 \[Sigma]a^2 + a2^2 \[Sigma]d^2) +
c2^2 (d1 d3 \[Sigma]a^2 + a1 a3 \[Sigma]d^2)) +
b2 (c2 d1 (c3 d1 - c1 d3) \[Sigma]a^2 + (a2 d1 -
a1 d2) (a3 d1 - a1 d3) \[Sigma]c^2 +
a1 c2 (-a3 c1 + a1 c3) \[Sigma]d^2 -
c1 c3 (d1 d2 \[Sigma]a^2 + a1 a2 \[Sigma]d^2) +
c1^2 (d2 d3 \[Sigma]a^2 + a2 a3 \[Sigma]d^2)))) > 0 ||
2 Abs[Arg[(b1^2 c3^2 d2^2 \[Sigma]a^2 -
2 b1^2 c2 c3 d2 d3 \[Sigma]a^2 + b1^2 c2^2 d3^2 \[Sigma]a^2 +
a3^2 c2^2 d1^2 \[Sigma]b^2 -
2 a2 a3 c2 c3 d1^2 \[Sigma]b^2 +
a2^2 c3^2 d1^2 \[Sigma]b^2 -
2 a3^2 c1 c2 d1 d2 \[Sigma]b^2 +
2 a2 a3 c1 c3 d1 d2 \[Sigma]b^2 +
2 a1 a3 c2 c3 d1 d2 \[Sigma]b^2 -
2 a1 a2 c3^2 d1 d2 \[Sigma]b^2 +
a3^2 c1^2 d2^2 \[Sigma]b^2 - 2 a1 a3 c1 c3 d2^2 \[Sigma]b^2 +
a1^2 c3^2 d2^2 \[Sigma]b^2 +
2 a2 a3 c1 c2 d1 d3 \[Sigma]b^2 -
2 a1 a3 c2^2 d1 d3 \[Sigma]b^2 -
2 a2^2 c1 c3 d1 d3 \[Sigma]b^2 +
2 a1 a2 c2 c3 d1 d3 \[Sigma]b^2 -
2 a2 a3 c1^2 d2 d3 \[Sigma]b^2 +
2 a1 a3 c1 c2 d2 d3 \[Sigma]b^2 +
2 a1 a2 c1 c3 d2 d3 \[Sigma]b^2 -
2 a1^2 c2 c3 d2 d3 \[Sigma]b^2 +
a2^2 c1^2 d3^2 \[Sigma]b^2 - 2 a1 a2 c1 c2 d3^2 \[Sigma]b^2 +
a1^2 c2^2 d3^2 \[Sigma]b^2 + a3^2 b1^2 d2^2 \[Sigma]c^2 -
2 a2 a3 b1^2 d2 d3 \[Sigma]c^2 + a2^2 b1^2 d3^2 \[Sigma]c^2 +
a3^2 b1^2 c2^2 \[Sigma]d^2 -
2 a2 a3 b1^2 c2 c3 \[Sigma]d^2 + a2^2 b1^2 c3^2 \[Sigma]d^2 +
b3^2 ((a2 d1 - a1 d2)^2 \[Sigma]c^2 +
c2^2 (d1^2 \[Sigma]a^2 + a1^2 \[Sigma]d^2) -
2 c1 c2 (d1 d2 \[Sigma]a^2 + a1 a2 \[Sigma]d^2) +
c1^2 (d2^2 \[Sigma]a^2 + a2^2 \[Sigma]d^2)) +
b2^2 ((a3 d1 - a1 d3)^2 \[Sigma]c^2 +
c3^2 (d1^2 \[Sigma]a^2 + a1^2 \[Sigma]d^2) -
2 c1 c3 (d1 d3 \[Sigma]a^2 + a1 a3 \[Sigma]d^2) +
c1^2 (d3^2 \[Sigma]a^2 + a3^2 \[Sigma]d^2)) -
2 b1 b2 (-c3 (c2 d1 + c1 d2) d3 \[Sigma]a^2 + (a3 d1 -
a1 d3) (a3 d2 - a2 d3) \[Sigma]c^2 -
a3 (a2 c1 + a1 c2) c3 \[Sigma]d^2 +
c3^2 (d1 d2 \[Sigma]a^2 + a1 a2 \[Sigma]d^2) +
c1 c2 (d3^2 \[Sigma]a^2 + a3^2 \[Sigma]d^2)) -
2 b3 (b1 (-c2 d2 (c3 d1 + c1 d3) \[Sigma]a^2 + (a2 d1 -
a1 d2) (-a3 d2 + a2 d3) \[Sigma]c^2 -
a2 c2 (a3 c1 + a1 c3) \[Sigma]d^2 +
c1 c3 (d2^2 \[Sigma]a^2 + a2^2 \[Sigma]d^2) +
c2^2 (d1 d3 \[Sigma]a^2 + a1 a3 \[Sigma]d^2)) +
b2 (c2 d1 (c3 d1 - c1 d3) \[Sigma]a^2 + (a2 d1 -
a1 d2) (a3 d1 - a1 d3) \[Sigma]c^2 +
a1 c2 (-a3 c1 + a1 c3) \[Sigma]d^2 -
c1 c3 (d1 d2 \[Sigma]a^2 + a1 a2 \[Sigma]d^2) +
c1^2 (d2 d3 \[Sigma]a^2 +
a2 a3 \[Sigma]d^2))))/(c3^2 (d2^2 \[Sigma]a^2 \
\[Sigma]b^2 + (b2^2 \[Sigma]a^2 + a2^2 \[Sigma]b^2) \[Sigma]d^2) -
2 c2 c3 (d2 d3 \[Sigma]a^2 \[Sigma]b^2 + (b2 b3 \[Sigma]a^2 +
a2 a3 \[Sigma]b^2) \[Sigma]d^2) +
c2^2 (d3^2 \[Sigma]a^2 \[Sigma]b^2 + (b3^2 \[Sigma]a^2 +
a3^2 \[Sigma]b^2) \[Sigma]d^2) + \[Sigma]c^2 ((a3 d2 -
a2 d3)^2 \[Sigma]b^2 +
b3^2 (d2^2 \[Sigma]a^2 + a2^2 \[Sigma]d^2) -
2 b2 b3 (d2 d3 \[Sigma]a^2 + a2 a3 \[Sigma]d^2) +
b2^2 (d3^2 \[Sigma]a^2 + a3^2 \[Sigma]d^2)))]] < \[Pi]]