· b T Ii j Jij Kk II JJ K KJK II J JI J KK
15
x y θ A B x A y A A x ¨ A y θ © Reuven Segev, 2002
Transcript of · b T Ii j Jij Kk II JJ K KJK II J JI J KK
x
y
θAB
xA yA
A
x¨ A
yθ
© Reuven Segev, 2002
rB
000 )Z,Y,X(
)z,y,x(
I0
J0
K0
i
j
k
k,j,iz,y,xI J K0 0 0, ,
X Y Z0 0 0, ,
R I J K0 0 0 0 0 0 0= + +X Y Z
r r R= ( )0
© Reuven Segev, 2002
r r R= ( , )0 t
ρ
m dVV
= ∫ ρ
r r1 2( ) , ( )t tt
| ( ) ( ) | | ( ) ( ) |r r r r1 2 1 2 0t td
dtt t− = − =constant ,
t
0 2 1 2 1= − ⋅ −( ) ( )v v r r
( , , )X Y Z0 0 0
tAt
Ar0XAr
ArA
© Reuven Segev, 2002
0Y0Z
Z,Y,XI J K( ), ( ), ( )t t t
Z,Y,X
B(X0,Y0, Z0)
I0 J0
K0
ij
k
ΑI(t)
J(t)
K(t)
rA
X0
Y0
Z0
x
y
z
R0
R
(x,y,z)
r
B Bt X
YZ
0R BR R= ( )tBtAr r= ( )tBi j k, ,
z,y,x
Ri j k, ,I J K( ), ( ), ( )t t t
BZ,Y,X
X Y Z0 0 0, ,RI J K( ), ( ), ( )t t t
R I J K= + +X Y Z0 0 0
r r R r I J K= + = + + +A A X Y Z0 0 0
r I J KA , , ,X Y Z0 0 0, ,
ArI J K, ,
© Reuven Segev, 2002
ArAI J K, ,
I J K, ,
X Y Z0 0 0, ,z,y,xz,y,x
X Y Z0 0 0, ,z,y,x
Z,Y,X
I J K, ,
AI J K, ,
xXAx IyZAyK
A
A
A
xX
yX
xZ
= ⋅ =
= ⋅ =
= ⋅ =
I i I i
I j I j
K i K i
cos( , )
cos( , )
cos( , )
M M M
cos( , )I jIj
[ ]A =
A A A
A A A
A A A
xX xY xZ
yX yY yZ
zX zY zZ
I i j k
J i j k
K i j k
= + +
= + +
= + +
A A A
A A A
A A A
xX yX zX
xY yY zY
xZ yZ zZ.
IJK
© Reuven Segev, 2002
]A[
I0 J0
K0
ij
k
X0
Y0
Z0
x
y
z
X
Y
Z I
J
KA
B
A
B
I k i j k
J j i j k
K i i j k
= − = + +
= = + +
= = + +
A A A
A A A
A A A
xX yX zX
xY yY zY
xZ yZ zZ.
A A AxX yX zX= = = −0 0 1, ,
[ ]A =
=
−
A A A
A A A
A A A
xX xY xZ
yX yY yZ
zX zY zZ
0 0 1
0 1 0
1 0 0
ψk'K', J', Iθ'I"K", J", I
φ"KK, J, I
© Reuven Segev, 2002
I0 = i
J0 = j
K0 = k
θ'I ')X(
ψ kz
i
j
I'
J '
K '= k
ψ
ψ
ψ
i
j
J '
K '= k
ψ
ψ
K"J"
θ
I"= I'
θ
θ
φ"K"Z
i
j
J '
K '= k
J"
I"= I'
θK = K"
I
J
φ
φ
φ
φ, θ, ψψθφ
© Reuven Segev, 2002
i I j J, ' , , 'J J K k K K' , " , ' , "= ='I
I I I J J" ' , , " ,=K K= "
i
j
J '
K '= k
J"
I"= I'
θ
K = K"
I
J
φ
φ
ψ
θ
i
j
k
I
J
K = K"
ψ
θ
φ I'
© Reuven Segev, 2002
I i j
J i j
K k
I I
J J K
K J K
I I J
J I J
K K
' cos sin
' sin cos
'
" '
" cos ' sin '
" sin ' cos '
cos " sin "
sin " cos "
, "
= +
= − +
=
=
= +
= − +
= +
= − +
=
ψ ψ
ψ ψ
θ θ
θ θ
φ φ
φ φ
y,x'Z', Y
I i j
J i j k
K i j k
I i j i j k
J i j i j k
" cos sin
" cos ( sin cos ) sin
" sin ( sin cos ) cos
cos (cos sin ) sin [cos ( sin cos ) sin ]
sin (cos sin ) cos [cos ( sin cos ) sin ]
= +
= − + +
= − − + +
= + + − + +
= − + + − + +
ψ ψ
θ ψ ψ θ
θ ψ ψ θ
φ ψ ψ φ θ ψ ψ θ
φ ψ ψ φ θ ψ ψ θ
.. sin ( sin cos ) cosK i j k= − − + +θ ψ ψ θ
I i j k
J i j k
K i j k
= − + + +
= − − + − + +
= − +
(cos cos sin cos sin ) (cos sin sin cos cos ) sin sin
( sin cos cos cos sin ) ( sin sin cos cos cos ) cos sin
. sin sin sin cos cos
φ ψ φ θ ψ φ ψ φ θ ψ φ θ
φ ψ φ θ ψ φ ψ φ θ ψ φ θ
θ ψ θ ψ θ
I i j k
J i j k
K i j k
= + +
= + +
= + +
A A A
A A A
A A A
xX yX zX
xY yY zY
xZ yZ zZ,
© Reuven Segev, 2002
[ ]A =
A A A
A A A
A A A
xX xY xZ
yX yY yZ
zX zY zZ
[ ]
cos cos sin cos sin sin cos cos cos sin sin sin
cos sin sin cos cos sin sin cos cos cos sin cos
sin sin cos sin cos
A =
− − −
+ − + −
φ ψ φ θ ψ φ ψ φ θ ψ θ ψ
φ ψ φ θ ψ φ ψ φ θ ψ θ ψ
φ θ φ θ θ
x
y
z
x
y
z
x
y
z
90˚y 90˚z
x
y
z
90˚z
x
y
z
x
y
z
90˚y
© Reuven Segev, 2002
X Y Z0 0 0, ,
r r R= +A
R I J K= + +X Y Z0 0 0I J K, ,A
R i j k i j k i j k
R i j k
= + + + + + + + +
= + + + + + + + +
X Y Z
X Y Z X Y Z X Y Z
xX yX zX xY yY zY xZ yZ zZ
xX xY xZ yX yY yZ zX zY zZ
0 0 0
0 0 0 0 0 0 0 0 0
( ) ( ) ( ) ,
, ( ) ( ) ( )
A A A A A A A A A
A A A A A A A A A
R X Y Z
R X Y Z
R X Y Z
x xX xY xZ
y yX yY yZ
z zX zY zZ
= + +
= + +
= + +
A A A
A A A
A A A
0 0 0
0 0 0
0 0 0.
R
R
R
X
Y
Z
x
y
z
xX xY xZ
yX yY yZ
zX zY zZ
=
A A A
A A A
A A A
0
0
0
}0R]{A} = [R{
r
x
y
z
x
y
z
X
Y
Z
A
A
A
xX xY xZ
yX yY yZ
zX zY zZ
=
+
A A A
A A A
A A A
0
0
0
{ } { } [ ]{ }r r RA= + A 0
A
© Reuven Segev, 2002
c d e= = =0 5 0 2 0 1. , . . m m , m
A
r i j kA = + +0 5. m
B
ij
k
x
y
z
X
Y
Z I
J
K
A
B
I0 J0
K0
X0
Y0
Z0
AB
c
d
e
}0R{]A[}Ar{
{ }
.
.
.
R0
0 5
0 2
0 1
=
x
y
z
=
+
−
=
0 5
1
1
0 0 1
0 1 0
1 0 0
0 5
0 2
0 1
0 6
1 2
0 5
. .
.
.
.
.
.
r i j k= + +0 6 1 2 0 5. . . m
© Reuven Segev, 2002
]A[K, J, I
I I J J K K
I J J K K I
⋅ = ⋅ = ⋅ =
⋅ = ⋅ = ⋅ =
1 1 1
0 0 0
, ,
. , ,
]A[
A A A
A A A
A A A
A A A A A A
A A A A A A
A A A A A A
xX yX zX
xY yY zY
xZ yZ zZ
xX xY yX yY zX zY
xY xZ yY yZ zY zZ
xZ xX yZ yX zZ zX
2 2 2
2 2 2
2 2 2
1
1
1
0
0
0
+ + =
+ + =
+ + =
+ + =
+ + =
+ + =.
[ ] [ ] [ ]A AT = 1
[ ]A T]A[ ]A[[1]
J K I× =J K I× = −I
© Reuven Segev, 2002
I J K⋅ × =( ) 1
I J K⋅ × = −( ) 1
u v w⋅ ×( )
u v w⋅ × =( )
u u u
v v v
w w w
x y z
x y z
x y z
I J K⋅ × =
=
( )
.
A A A
A A A
A A A
A A A
A A A
A A A
xX yX zX
xY yY zY
xZ yZ zZ
xX xY xZ
yX yY yZ
zX zY zZ
I J K⋅ × =( ) A
A = 1 A = −1
y
© Reuven Segev, 2002
]A[AuA
]A[u [ ]{ˆ} {ˆ} {ˆ}A u u u= = 1u]A[
ArK, J, IAr
K, J, IAr
nn n nx y z
2 2 2 1+ + =IJI
I J⋅ = 0JIJ
K
© Reuven Segev, 2002
![BORIS S. MITYAGIN Abstract. arXiv:1407.4153v1 [math.SP] 15 ... · 8 BORISS.MITYAGIN But we can claim that U 1 is a trace-class operator as well, and (2.32) kU 1k 1 = kK(KWK)Kk 1 ≤](https://static.fdocument.org/doc/165x107/5ed5d69be7a2f75aeb1aecbd/boris-s-mityagin-abstract-arxiv14074153v1-mathsp-15-8-but-we-can-claim.jpg)
