1.
2.
3.
, - - .
1) () , - , ;
2) , .. ;
3) (, ) -.
- - : . ( ) .
1. y 0
11 (. 11-1). , A B, . - . , , , . , , -. , , . , , , ( ? ? ?....). , - , .. .
, 1 m A 1 n B. A m ´ n, - . - i Î{1,..., m } j Î{1,..., n }, A aij - , - i - j - , a bij - - B, i - j - .
1. . . , . :
1. ;
2. .
. :
= , B = .
4 . 1- , $ 100, 100 . A 1- , - 2- , A , -$ 101 ( a 12 = 0, b 12 = 101). A 2- , - 1- , a 21 = 101, b 21 = 0. , 2- , $ 1, 1 .
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, , i, j, 1- (2-) A () i j. :
■ (1)
1.1. . 11 , B = - . . , . - , - , . , (. 1 1 1.2).
( ) - . á i *, j *ñ ,
()( j) (( ≤ ) ( ≤ )). (2)
, A i * i,
j *, ≤ A - , i *. - ( ≤ ). , , , - .
2. , -. , 4 : á1, 1ñ, á1, 2ñ, á2, 1ñ, á2, 2ñ. - 1 2 (2 1), (2) i ≠ i * j ≠ j * :
á1, 1ñ: ( ≤ ) ( ≤ ); (2.11)
á1, 2ñ: ( ≤ ) ( ≤ ); (2.12)
á2, 1ñ: ( ≤ ) ( ≤ ); (2.21)
á2, 2ñ: ( ≤ ) ( ≤ ). (2.22)
, , , - .
1, (1).
, á1, 1ñ , (2.11). (1) a 21 = 101, a 11 = 100, .. 1- (2.11) . á1, 1ñ .
, á1, 2ñ , (2.12). (1) a 22 = 1, a 12 = 0, .. 1- (2.12) . á1, 2ñ .
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, á2, 1ñ , (2.21). (1) a 11 = 100, a 21 = 101, .. 1- (2.21) -. (1) b 21 = 0, b 22 = 1, .. 2- (2.21) . á2, 1ñ .
, á2, 2ñ , (2.22). (1) a 22 = 1, a 12 = 0, .. 1- (2.22) . (1) b 22 = 1, b 21 = 0, .. 2- (2.22) . á2, 2ñ .
, 1- 100 - 100 , á2, 2ñ. 2- 1 , 1- . , , , . , , , . ■
, .
A i *, - i
. (3a)
B j *, - j
. (3b)
1. á i *, j *ñ . - i * j * ■
, , -, .
3. : Q = . , 2, 4- á1, 1ñ, á1, 2ñ, á2, 1ñ, á2, 2ñ (2.11) - (2.22), , - ■
1. - . . 2 3.
■
1.2. . - , . A i c xi (i = 1,..., m), B - j c yj (j = 1,..., n), , A B x = (x 1,..., xm) y = (y 1,..., yn). , , , 0, 1.
A B x y. A
g A(x, y) = , (4a)
g B(x, y) = .(4b)
-, . á x *, y *ñ ,
()( y) ((g A(x, y *) ≤ g A(x *, y *) (g B(x *, y) ≤ g B(x *, y *)). (5)
A x *, - x
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. (6a)
B y *, - y
. (6b)
2. á x *, y *ñ . - x * y * ■
(. 1.6):
3. - ■
, , - 3.
1.3. 2×2. - , - , . -, ( 2×2). , 1 2. 2×2, , .
A = B = . - , 1- A, 2- B. i j A aij, B bij (i, j = 1, 2). - (4) :
g A(x, y) = (a 11 a 12 a 21 + a 22) xy + (a 12 a 22) x + (a 21 a 22) y + a 22; (7a)
g B(x, y) = (b 11 b 12 b 21 + b 22) xy + (b 12 b 22) x + (b 21 b 22) y + b 22, (7b)
x y 1- A B (. (1.20) (1.23)).
1.5.
a 1 = (a 11 a 12 a 21 + a 22), (8a)
a 2 = a 22 a 12; (9)
b 1 = (b 11 b 12 b 21 + b 22), (8b)
b 2 = b 22 b 21. (9b)
(7) :
g A(x, y) = a 1 xy a 2 x + (a 21 a 22) y + a 22, (10a)
g B(x, y) = b 1 xy b 2 y + (b 12 b 22) x + b 22. (10b)
u = a 1 y a 2, v = (a 21 a 22) y + a 22, (11a)
s = b 1 x b 2, t = (b 12 b 22) x + b 22. (11b)
(10)
g A(x, y) = ux + v, (12a)
g B(x, y) = sy+t, (12b)
u v x, s t y. yg A(x, y) - x, xg (x, y) - y. u, (11a), y, - s, (11b), x.
y 0 = a 2 a 1, (13a)
x 0 = b 2 b 1. (13b)
(11a), (13a) (11b), (13b) , a 1≠ 0 u = a 1 y 0 a 2 = 0, b 1≠ 0 s = b 1 x 0 b 2 = 0. u = a 1 y a 2 a 1 a 2, - , .1. - s = b 1 x b 2 , .
. . - y (, y 1- , 0 ≤ y ≤ 1)
j A(y) = { x | () (g A(x', y) ≤ g A(x, y))}. (14a)
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, j A(y) x , g A(x, y) y.
.1a. a 1= 0, a 2 = 0
.1b. a 1= 0, a 2< 0
.1c. a 1> 0, a 2< 0
.1d. a 2< a 1< 0
.1e. a 1= 0, a 2> 0
.1f. a 2> a 1> 0
.1g. a 1< 0, a 2> 0
.1h. a 1< 0, a 2 = 0 .1i. a 1> 0, a 2 = 0
.1j. a 1< 0, a 2 = a 1.1k. a 1> 0, a 2 = a 1
.1 l. a 1 < a 2< 0 .1m. a 1 > a 2> 0
4. j A(y)
j A(y) = (15)
. y x g A(x, y) = ux + v (. (12)), [0, 1] x = 0, u < 0, x = 1, u > 0. u = 0 g A(x, y) ≡ v, .. x. - [0, 1], ■
Φ A = {(x, y) | y Î[0, 1], x Î j A(y)}. (16)
Φ A . u = a 1 y a 2 a 1 a 2 .1, (15) (16) - a 1 a 2. . a 1 = a 2 = 0 u ≡ 0 (. . 1), , , j A(y) = [0, 1] y, .2 ( - ). 3- (.1b, 1c 1d) 0 ≤ y ≤ 1 u (y) > 0, , (15), j A(y) ≡ {1}, .2b. 3- (.1e, 1f 1g) 0 ≤ y ≤ 1 u (y) < 0, , (15), j A(y) ≡ {0}, - .2c.
, u = a 1 y a 2 , .1h, u (0) = 0, u (y) < 0 y > 0. (15) j A(0) = [0, 1], j A(y) = 0 y > 0, .2d. , , - .1i, j A(0) = [0, 1], j A(y) = 1 y > 0, .2e. , - .1j, u (1) = 0, u (y) > 0 y < 1, j A(1) = [0, 1], j A(y) = 1 y < 1, .2f. , , .1k, .2g.
.2a. a 1= 0, a 2 = 0 .2b. a 1= 0, a 2< 0; a 1> 0, .2c. a 1= 0, a 2> 0; a 2 > a 1> 0;
a 2< 0; a 2< a 1< 0 a 1< 0, a 2> 0
.2d. a 1< 0, a 2 = 0 .2e. a 1> 0, a 2 = 0 .2f. a 1< 0, a 2 = a 1
.2g. a 1> 0, a 2 = a 1 .2h. a 1 < a 2< 0 .2i. a 1 > a 2> 0
, , .1 l 1m , .2h 2i, .1.1.
, .2 - a 1 a 2.
. x
y B(x) = { y | () (g B(x, y') ≤ g B(x, y))}. (14b)
, y B(x) y B, g B(x, y) x.
, (16):
Y = {(x, y) | x Î[0, 1], y Î y B(x)}. (16b)
, , - , 9 , .3.
.3a. b 1= 0, b 2 = 0 .3b. b 1= 0, b 2< 0; b 1> 0, .3c. b 1= 0, b 2> 0; b 2 > b 1> 0;
b 2< 0; b 2< b 1< 0 b 1< 0, b 2> 0
.3d. b 1< 0, b 2 = 0 .3e. b 1> 0, b 2 = 0 .3f. b 1< 0, b 2 = b 1
.3g. b 1> 0, b 2 = b 1 .3h. b 1 < b 2< 0 .3i. b 1 > b 2> 0
(14) (14b)
5. (x *, y *), Φ A Y , - ■
, :
6. a 1, a 2, b 1 b 2 Φ A Y -.
, 1- 9 .2, 2- 9 .3. -, 2- . , 3- , . , , - . 2b, 2c, 2h, 2i 3b, 3c, 3h, 3i. 16- , : á2h, 3iñ á2i, 3hñ -. , .11-1.3 ■
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.
1. 2×2.
1. B , (8) (9), - a 1, a 2, b 1, b 2.
2. a 1 a 2, , .2.
3. b 1 b 2, B, .3.
4. .
5. (x *, y *) , x = (x *, 1 x *), y = (y *, 1 y *). .
6. (10), B - (10b), x y ( - 1- ). , , : - . , - á1, 1ñ, a 11, B b 11, ..■
1 .
4. 1. Ÿ - (1). 2 , á2, 2ñ . x = (0, 1), y = (0, 1). , .
a 11 = 100, a 12 = 0, a 21 = 101, a 22 = 1;
b 11 = 100, b 12 = 101, b 21 = 0, b 22 = 1.
(8) (9)
a 1 = a 11 a 12 a 21 + a 22 = 100 0 101 + 1 = 0;
a 2 = a 22 a 12 = 1 0 = 1;
b 1 = b 11 b 12 b 21 + b 22 = 100 101 0 + 1 = 0;
b 2 = b 22 b 21 = 1 0 = 1.
a 1 = 0, a 2> 0 .2c; B b 1 = 0, b 2> 0 .3c. .4. - (0, 0), x = (0, 1), y = (0, 1).
. 6 1 g A(0, 0) = a 22 = 1; g (0, 0) = b 22 = 1■
5. 3 Q = , . a 1, a 2, b 1, b 2 (8), (9):
a 1 = a 11 a 12 a 21 + a 22 = 0 2 3 + 1 = 4;
a 2 = a 22 a 12 = 1 2 = 1;
b 1 = b 11 b 12 b 21 + b 22 = 8 6 5 + 7 = 4;
b 2 = b 22 b 21 = 7 5 = 2.
a 1 = 4, a 2 = 1 (a 1 < a 2< 0) .2h; B b 1 = 4, b 2 = 2 (b 1 > b 2> 0) .3i. y 0 = 0,25; x 0 = 0,5. .5. (0,5; 0,25), x = (0,5; 0,5), y = (0,25; 0,75).
(10):
g A(0,5; 0,25) = 40,50,25 +10,5+ (3 1) 0,25+ 1 = 0,5 + 0,5 + 0,5 + 1 = 1,5;
g B(0,5; 0,25) = 40,50,25 20,25 + (6 7) 0,5 + 7 = 0,5 0,5 0,5 + 7 = 6,5 ■
.4. . | .5. . |
6. . - Q = . a 1, a 2, b 1, b 2 (8), (9):
a 1 = a 11 a 12 a 21 + a 22 = 1 0 0 + 2 = 3;
a 2 = a 22 a 12 = 2 0 = 2;
b 1 = b 11 b 12 b 21 + b 22 = 2 0 0 + 1 = 3;
b 2 = b 22 b 21 = 1 0 = 1.
a 1 = 3, a 2 = 2 (a 1 > a 2> 0) .2i; B b 1 = 3, b 2 = 1 (b 1 > b 2> 0) .3i. y 0 = ⅔; x 0 = ⅓. .6. ,
.6. .
. á1, 1ñ, (1, 1), 1 2. - á2, 2ñ, (0, 0), 2 1. -, , (⅓,⅔), (10):
g A(⅓,⅔) = 3⅓⅔ 2⅓ + (0 2)⅔ + 2 = ⅔ ⅔ 2⅔ + 2 = ⅔;
g B(⅓,⅔) = 3⅓⅔ 1⅔ + (0 1)⅓ + 1 = ⅔ ⅔ ⅓ + 1 = ⅔.
4 5, 3 . - - (), : - ? , 2, 1. , á2, 2ñ, 2, 1. á1, 1ñ, 2, 1. x = (⅓, ⅔), y = (⅔,⅓) ⅔,.. , . , .. ( - ), . , ó ( ). ( ) . ■
, - 2×2. , - ( ). - 1 ( ) .
2. 1 - . . 4, 5 6.
■
, , , ( ) .
7. . - . . , 6 5 1 4 2, 3 3. . 1- , 2- 3 , .. , - , . .7 7 . 7 . . á5-1-1ñ , : 1- 5 , 1- . - . á5-1-1ñ : 5 3- 2- 4- 1- . , - á5-1-1ñ á3-2-1-1ñ á4-1-1-1ñ. á2-2-2-1ñ, á2-2-1-1-1ñ á2-1-1-1-1-1ñ , . á7ñ - .
.7. 7 ■
8. . , 4 . :
1. 1- .
2. 2- .
3. .
. , . , , - : , .
, .8. - , , , .. . 1- 2- . (4,4) (0,0). ( 1- ), ( 2- ) ( ).
.8. ■
. - (.. ) . . ( ) . , - , . , , 1- - . - , -. , , . (. 7 8) , .
, . , . , - , , -. , (0,0), , . , . .
1. . z z. z . , -. , -, -. , , , , . , -.
2. , , . z, . , , -, , .. . , - , .. .
, , , .
, . , (.. ) 0 1. , , .