(tgx)
csc x
(sec x)
(csc x)
(a )
(arccos x)
sec x tgx
csc x ctgx
tgxdx
ln cos x
ctgxdx ln sin x
1
1 x2
1
(arcctgx )
1 x2
dx
cos 2 x
dx
sin 2 x
C
C
sec xdx ln sec x tgx C
x
1
arctg C
2
2
a
a x
a
dx
x a
1
C
ln
2
2
x a
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dx
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ln
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dx
x
C
arcsin
a
a2 x2
In
sin xdx
0
x2
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a2
cos x
x 2 dx
1 u2
1 u2
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chx C
chxdx
shx C
x
cos n xdx
0
x 2 a 2 dx
2u
1 u2
2
ax
C
ln a
dx
x 2
x a2
2
x 2
x a2
2
x 2
a x2
2
u tg
x
2
a
n 1
In
n
2
ctgx C
sec x C
a x dx
2
tgx C
csc 2 xdx
csc x ctgxdx
dx
n
sec 2 xdx
sec x tgxdx
csc xdx ln csc x ctgx C
2
1 x2
(arctgx)
x
a ln a
1
(log a x)
x ln a
sin x
1 x2
1
2
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x
1
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sec 2 x
ln( x
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x2
2
a2
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ln( x
2
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ln x
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2
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x
arcsin
C
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dx
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e x
: shx
2
x
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: chx
2
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: thx
chx e x
sin x
1
x
1
lim(1 ) x e
x
x
ex
arthx
0
x
2.718281828459045...
x
e
e
x
x2 1
arshx ln( x
archx
lim
x 2 1)
ln( x
1 1 x
ln
2 1 x
A
sin(
cos(
) sin cos
) cos cos
tg (
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ctg (
sin
cos
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cos
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90°-
cos
sin
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90°+
cos
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sin
cos
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cos sin
sin sin
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tg
1 tg tg
ctg ctg 1
)
ctg
ctg
sin
sin
2 sin
sin
sin
2 cos
cos
cos
2 cos
cos
cos
2 sin
2
2
2
2
cos
sin
cos
sin
2
2
2
2
sin 2
2 sin cos
cos 2
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tg 2
tg
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ctg 2 1
2ctg
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1 tg 2
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sin
1 1 2 sin 2
cos
sin
1 cos
1 cos
sin
1 cos
1 cos
2
a
sin A
b
sin B
c
sin C
arcsin x
n
(uv) ( n )
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cos 3
4 cos3
2
ctg
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2
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arctgx
3 cos
3
1 cos
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1 cos
1 cos
c2
2R
4 sin 3
tg
3tg
1 3tg 2
tg 3
1 cos
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2
sin 3
2
sin
1 cos
1 cos
sin
a2 b2
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arcctgx
C nk u ( n k ) v ( k )
k 0
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u
v
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u ( n ) v nu ( n 1) v
n(n 1)
(n k 1) ( n k ) ( k )
u
v
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f (b) f (a ) f ( )(b a)
f (b) f (a) f ( )
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x
1 y 2 dx,
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K
s
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M
K
a
.
:
lim
s
0
0;
K
1
.
a
M
s
y
tg
M
d
ds
s
y
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.
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f ( x)
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n
f ( x)
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b
a
b
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a
W
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p A
mm
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y
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pt
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1
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1
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z
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p ( x, y )
x
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f
l
y y0
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l
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z
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2
x
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,
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F ( r , , ) r 2 sin dr
d
0
0
1
M
z dv
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t
(
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M
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),
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dv
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(t )
x
y
L
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L
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Pdx Qdy
L
Q cos )ds
L
L
(
D
y, Q
P
Q
x
P
)dxdy
y
Q
x
x
(
Pdx Qdy
L
P
y
D
2
D
Q
x
P
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y
A
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D
Pdx Qdy
L
1
xdy ydx
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·
1 G
Q
x
2 P ( x, y ) Q ( x, y ) G
P
y
(0,0)
·
Q
x
P
y
u ( x, y )
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( x, y)
u ( x, y )
P ( x, y )dx Q( x, y )dy
x0
y0
0
( x0 , y0 )
f [ x, y, z ( x, y )] 1 z x2 ( x, y ) z y2 ( x, y )dxdy
f ( x, y, z )ds
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P( x, y, z )dydz Q( x, y, z )dzdx R( x, y, z )dxdy
R( x, y, z )dxdy
R[ x, y, z ( x, y )]dxdy
Dxy
P( x, y, z )dydz
P[ x( y, z ), y, z ]dydz
D yz
Q ( x, y, z )dzdx
Q[ x, y ( z, x), z ]dzdx
Dzx
Pdydz Qdzdx Rdxdy
( P cos
Q cos
R cos )ds
P
x
(
Q
y
R
) dv
z
P
x
div
Pdydz
Q
y
A n ds
Qdzdx
R
y
An ds
Q
P
)dydz (
z
z
(P cos
Q cos
y
Q
P
)dxdy
y
z
R
R
y
i
j
k
x
P
y
Q
z
R
A
R cos ) ds
An ds
R
Q
)dzdx (
x
x
x
P
Q
z
1 q q2
1
1
2
1
3
qn
n
1
n
1 qn
1 q
( n 1)n
2
1
Pdx Qdy Rdz
cos
cos
cos
x
P
y
Q
z
R
P
z
Pdx Qdy Rdz
1 2 3
Q cos
R cos ) ds
div
dydz dzdx dxdy
rotA
( P cos
R
,
z
div Adv
(
Rdxdy
R
x
Q
x
A t ds
P
y
0,
...
1
1
1
1
lim n u n
n
2
1
1
U
lim n 1
n
Un
1
3
sn
u1 u 2
u n ; lim sn
n
u1 u 2 u3 u 4
(
u1 u 2 u3
un un 1
lim u n 0
(2) u1
u2
un
u3
un
un
(2)
(1)
(2)
(1)
(1)
( 1) n
n
1
n
1
n2
p
1
np
p 1
0)
s u1 ,
n
(1)u1 u 2
, un
rn
rn
un
1
1 x x
2
x
3
x
n
x
1
x
1
a1 x a2 x 2
(3)a0
n
f ( x)
x
R
x
R
R
an an
1
(3)
f ( x)
1 mx
x3
3!
m(m 1) 2
x
2!
x5
5!
( 1) n
R
0
R
1
f ( n ) ( x0 )
( x x0 ) n
n!
f ( x0 )
( x x0 ) 2
2!
lim Rn
n
f ( 0)
m(m 1)
1
0
R 0
f ( x0 )( x x0 )
0
x
R
f ( n 1) ( )
( x x0 ) n 1 , f ( x )
(n 1)!
Rn
sin x
x
an 1
an
lim
(1 x) m
1 x
an x n
R
x0
1
f ( n ) ( 0) n
x
n!
f ( 0) 2
x
2!
f ( 0) x
(m n 1) n
x
n!
x 2n 1
(2n 1)!
(
x
0
( 1 x 1)
)
e ix
e ix
f (t )
e ix
cos x
2
ix
e e ix
sin x
2
cos x i sin x
An sin( n t
A0
n
)
n 1
a0
aA0 an
An sin
n
a0
2
bn
1, sin x, cos x, sin 2 x, cos 2 x
0
(an cos nx bn sin nx)
n 1
An cos
n
sin nx, cos nx
t
x
[
, ]
f ( x)
a0
2
an
bn
(an cos nx bn sin nx)
2
n 1
1
1
f ( x) cos nxdx
f ( x)sinnxdx
1 1
1 2
3 52
1
1
1
2
2
62
4
2
an
2
8
2
24
0 bn
(n 0,1,2 )
(n 1,2,3 )
1
1 2
2
1
1 2
2
2
1
32
1
32
1
42
1
42
2
6
2
12
f ( x) sin nxdx
n 1,2,3
f ( x)
f ( x) cos nxdx
n 0,1,2
f ( x)
bn sin nx
0
bn
0 an
2
0
2l
a0
2
an cos nx
f ( x)
a0
2
(an cos
n 1
n x
n x
)
bn sin
l
l
2l
l
an
1
n x
f ( x) cos
dx
l l
l
bn
1
n x
f ( x) sin
dx
l l
l
(n 0,1,2 )
l
y
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f ( x, y )
P ( x, y )dx Q( x, y )dy
0
g ( y )dy
g ( y )dy
f ( x)dx
G( y)
F ( x) C
dy
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y
x
u
dy
dx
u x
du
dx
u
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dx
P ( x ) y Q ( x)
1
Q( x) 0 ,
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y
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2
f ( x, y )
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dx
x
du
(u ) u
y
x
y
x
P ( x ) dx
( Q ( x)e
P ( x ) dx
dx C )e
P ( x ) dx
P( x) y Q( x) y n (n 0,1)
P( x, y )dx Q( x, y )dy
du ( x, y )
(u )
y Ce
Q( x) 0
f ( x)dx
0
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u
x
0
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u
y
Q ( x, y )
u ( x, y ) C
d2y
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py
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f ( x)
0
( )
f ( x) 0
f ( x) 0
p, q
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1
2
dy
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dx
pr q
r1 , r2
0
r2 r
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y ,y,y
u
3
r1 , r2
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y
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y
e x (c1 cos x c2 sin x)
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r1
i
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i
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y
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f ( x ) p, q
x
f ( x) e Pm ( x)
f ( x) e x [ Pl ( x) cos x Pn ( x) sin x]
1
A
A
A
A B
A
( AB) A
AB
A
A
A
A
A
(A
B)
A ( AB)
B
n
AB
n
Ai
AB
n
Ai
i 1
A
i 1
B
n
Ai
i 1
Ai
i 1
2
P( A ) 1 P( A)
A
B
P( B
A)
P( B) P( A)
A
c2 e r2 x
P( B A)
A, B,
P( B ) P( AB)
A, B,
P( A
B)
P( A) P ( B) P( AB)
P( A
B)
P( A) P( B)
n
P(
n
Ai )
P( Ai )
i 1
i 1
n
P( Ai A j )
1 i j n
P( Ai A j Ak )
1 i j k n
3
PB A
P ( AB)
P( A)
P ( AB)
P( A) P B A
P( A1 A2
An )
( P ( A) 0)
P( A1 ) P A2 A1
( P ( A1 A2
n
P ( A)
P ( ABi )
i 1
n
P An A1 A2
An 1 )
P( Bi ) P( A Bi )
i 1
Bayes
P( Bk A)
P( ABk )
P( A)
P( Bk ) P( A Bk )
n
P( Bi ) P( A Bi )
i 1
4
P (a
X
b)
P ( X b) P ( X
F (b) F (a)
a)
0)
An
1
( 1) n 1 P( A1 A2
An )
5
(1)
0–1
P( X
p k (1 p)1 k , k
k)
0,1
B(n, p )
(2)
P(A)=p
k ) Cnk p k (1 p ) n k , k
P( X
0,1,
* Possion
lim npn
0
n
lim Cnk pnk (1 pn ) n
k
n
k
(3)
P( )
Poisson
k
P( X
k) e
k!
, k
0,1,2,
6
U ( a, b)
(1)
1
f ( x)
b a
, a
x b
0,
0,
F ( x)
x a
,
b a
1
E( )
(2)
f ( x)
e
0,
x
, x
0
k
e
k!
0,1,2,
,n
0,
x 0
x
1 e , x 0
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f ( x)
1
2
F ( x)
1
2
2)
)2
(x
e
x
2
2
x
)2
(t
e
2
2
dt
* N (0,1) —
( x)
1
2
( x)
1
2
x2
2
e
x
e
x
t2
2
dt
7.
( X ,Y )
x
F ( x, y )
FX ( x)
x
f X ( x)
FY ( y )
y
f (u , v)dvdu
f (u, v)dvdu
f ( x, v)dv
y
fY ( y )
f (u, v)dudv
f (u, y )du
8.
(1)
f ( x, y )
G
U(G)
1
, ( x, y ) G
A
0,
x
(2)
1
1
f ( x, y )
2
1
2
2
1
x
,
e
2 (1
(x
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f X ( x) f Y X ( y x)
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f ( x, y )dy
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f ( x, y )dx
fY X ( y x) f X ( x)dx
f X Y ( x y)
f ( x, y )
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f Y X ( y x) f X ( x)
f Y X ( y x)
f ( x, y )
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10.
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k 1
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A
I Ax
B
A
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min n, s
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1
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r A
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1
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1
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n
q
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x T Ax
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n
f x1 , x 2 ,
, xn
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x1 , x 2 ,
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c n1 y1
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c nn y n
f x1 , x 2 ,
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x
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A
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1
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f x1 , x 2 ,
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cs
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dim V
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dim L
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1
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k
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k
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c1 , c 2 ,
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1
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c c1 , c 2 ,
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c1i , c 2i ,
C
C
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2
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k
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,
2
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,
V
k
, c ki
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,
k
1
,
2
,
,
k
1
1
,
2
,
,
k
k
1
,
2
,
,
k
1
,
V
x
x
x1 , x 2 ,
, xk
1
,
2
,
,
k
1
,
2
,
,
k
2
,
1
,
T
,
2
,
y
C
k
,
1
k
y1 , y 2 ,
,
, yk
2
,
,
k
T
x
y
1
,
2
,
,
Cy
k
Cy
4
V
1
,
2
,
c1 , c 2 ,
c1 d1
,
5
AB
13
,
k
, ck
c2 d 2
E
a,0,
,0, a
T
a
0
A
E
T
1
BA
AB
E
AB
B
E
a
00
A*
1
0
1
0
0
1
0
3
0
0
1
0
0
0
0
8
=0
=BA
AB=
AB
20
, dk
ck d k
n
03
d1 , d 2 ,
n
A
B
AB
B
ABA
1
3E .
Ax=E
BA
aA bB
ab
0,
BA
BA
E
1
a
T