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diag|
{}
I
{}
x
n
x
x
H
x
σ
x
2
Σ
X
x
()
j
ω
Xs
()
Xz
()
x
()
k
()
t
–diagonalmatrix,
D
=
diag
{
x
1
,...,
x
N
}
;
–(quadratic)identitymatrix,
I
=
diag1,...,1;
{
}
–sequenceofhomogenousscalarvariables
x(nEZ),
n
{}{
x
n
=
x
1,...,
x
N
}
;
–scalarrandomvariable;
–randomvector;
–expectedvalueofx
(
H
x
±
E
[]
x
)
;
–varianceofx
(
σ
x
2
±
Var
[]
x
)
;
–covariancematrixofx(
Σ
x±
Cov
{}
x);
–Fouriertransformofx(t);
–Laplacetransformofx(t);
–Laurenttransformof
{}
x
n
;
–k-thderivativeof
xt
()
,
k±
1,2,...
;equivalentto
xt
!
()
,
xt
!!
()
and
x
!!!
()
t
for
k±
1,2,3;
Thevariablesassociatedwithy,z,etc.aregeneratedintheanalogousway.
Basicoperators
arg
v
{}
|
Cov|
∆|
[]
[]
δ
[]
|
E|
[]
F
[]
|
inf|
{}
L
max|
{}
min|
{}
Pr|
()
sup|
{}
Var|
[]
Z
{}
|
|
|
2
|
®
||
|
*
O
–
operatorfordeterminingthevalueofavariable,satisfyingagivencondition,e.g.
v
ˆ
±
arg
v
{
5
v
+±
3
0
}
±–
0.6;
–
operatorfordeterminingthecovariancematrixofarandomvector,e.g.
Σ
x±
Cov
{}
x
;
–
operatorfordeterminingtheabsoluteerror,e.g.
∆±∆
x
~
[]
x
~
=–
x
~
x
-
;
–
operatorfordeterminingtherelativeerror,e.g.
δ
x
~
±
δ
[]
x
~
=
x
~
–
x
-
x
-
;
–
operatorfordeterminingtheexpectedvalueofarandomvariable,e.g.
H
x
±
E
[]
x
;
–
operatorfordeterminingtheFouriertransformofafunction,e.g.
X
()
j
ω
±
F
f
L
xt
()
1
J
;
–
operatorfordeterminingtheinfimum,e.g.
inf
{
–
1,0,2,3
}
±–
1;
–
operatorfordeterminingtheLaplacetransformofafunction,e.g.
Xs
()
±
L
f
L
xt
()
1
J
;
–
operatorfordeterminingthemaximum,e.g.
max
{
–
x
2
+
1
}
±
1;
–
operatorfordeterminingtheminimum,e.g.
min
{
x+
2
1
}
±
1;
–
operatorfordeterminingtheprobability,e.g.
Pr
(
x>
5
)
±
0.6;
–
operatorfordeterminingthesupremum,e.g.
sup
{
–
1,0,2,3
}
±
3;
–
operatorfordeterminingthevarianceofarandomvariable,e.g.
σ
x
2
±
Var
[]
x
;
–
operatorfordeterminingtheLaurenttransformofasequence,e.g.
Xz
()
=Z
{}
x
n
;
–
operatorfordeterminingtheabsolutevalueofanexpression,e.g.
–+
3
4j
±
5;
–
operatorfordeterminingtheEuclidiannormofavector,
x
2
±
x
1
2
+
+
x
2
N
;
...
–
operatorfordeterminingtheChebysheviannormofavector,
x
®
±
sup
{
x
1
,...,
x
N
}
;
–
operatorfordeterminingtheinnerproductoftwofunctions,e.g.
xt
()()
|
yt
±
5;
–
operatorofconvolution,e.g.
zt
()
±
xt
()
*
yt
()
;
–
operatorofcompositionoftwofunctionsoroperators,e.g.
(
f
O
g
)()
t
±
fgt
(
()
)
.
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