АЛЬТЕРНАТИВНЫЕ ДЕЙСТВИТЕЛЬНЫЕ ЛИНЕЙНЫЕ ПРОСТРАНСТВА РАЗМЕРНОСТЕЙ 2, 3 И 4

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  • 1 (17), 2011 - .

    3

    512+514.126

    . . , . .

    2, 3 4

    . - , 2, 3, 4 . , . - -. - . - . 3, 6, 10. : -, . Abstrat. The article considers Abelian subgroups of real unitriangular groups of the third, fourth and fifth orders and isomorphic to them tuple length groups of 2, 3, 4 real numbers. The authors receive linear spaces alternative to arithmetical space on the basis of tuple length groups. Operations over alternative space vectors are set by nonlinear formulas. Groups of automorphism spaces of one dimension are set by nonlinear formulas of a various kind. All considered linear spaces are subsibsons. The article defines sibsons of dimensions 3, 6, 10. Key words: real linear spaces with nonlinear operations, sibsons.

    2 [1], : ( , ) ( , ) ( , )x y a b x a y b , ( , ) ( , )t x y xt yt , tR ; (1)

    ( , ) ( , ) ( , )x y a b x a y b ax , 2 ( 1)( , ) ,2

    t tt x y xt yt x

    , tR . (2)

    2L , - 2a L . 2L :

    , , ..., , ...o a x ; -

    . (2) 2a L (0,0) ; , ( , )x y ,

    2( , ) ( , )x y x y x .

  • .

    4

    2a L [1]. - 2a L - ( : ( , ) ( , ) ( , )x y u v x u y v xu ; 2. (2) 2a L ). 2a L

    ( , ) , (1,0), (0,1) . ( , )x y - :

    ( , )x y = ( 1)2

    x xx y

    . (3)

    ,x y ( 1),

    2x xx y

    . :

    = ( 1), (1,0), 2 (2,1), ..., , , ...2

    t tt t

    ;

    = , (0,1), 2 (0,2), ..., (0, ), ...t t .

    (3), x y 2a L , -

    , 2a L 1- : 2a L = + . = ( , ) 2a L

    ( , )a p , (0, )b ,

    2

    ,( 1)( )

    2

    x axx xy a b px by

    , . [1]. . . 2L -. 2a L -. 2L 2a L [1]. . , - 2L 2a L . [1].

    [2] 2a A - 2a L . ( , , )O B : ( , )A a h , ( , )M x y ,

    ( , )m p . ,A , A , -

    2 ( 1), ( ) .2

    t tx mt a y am p m h

  • 1 (17), 2011 - .

    5

    . - . ( , , )O B -

    2a A , ( , )O c d , ( , )a p , (0, )b , ( , )M x y , - B , ( , )M x y , B , - B M ( , )x y . -

    2

    ,( 1)( ) ( ) .

    2

    x ax cx xy a b p ac x by d

    . . [3, c. 237280]. [4, c. 3536].

    2A 2L . -

    2a A . 2a A - 2A 2L . - 2A 2a A , , . .

    3 4, .

    1. (3)UT

    1.1. 3-

    3- [5]. 3R - :

    ( , , ) ( , , ) ( , , )x y z a b c x a y b z c ay ; (4)

    ( 1)( , , ) , ,2

    t tt x y z xt yt zt xy

    , tR , (5)

    c. [5, c. 107]. (4) . 3 , - . [5, c. 166215] , . (0,0,0) ; - ( , , )x y z

    ( , , )x y z = ( , , )x y z xy .

  • .

    6

    3 2. [6] - 2 .

    s 3 ( , , )a b c . :

    ,

    ( , , )x y z ; ( , , )x y z s .

    (4) -:

    ,

    ;

    x x ay y bz z ay c

    1 0 0 01 0 00 1 00 1

    am

    bc a

    .

    ( , , )x y z

    1 0 01 0

    1xz y

    (3)UT . :

    1 0 0

    1 01

    xz y

    1 0 01 0

    1ac b

    = 1 0 0

    1 01

    x az ay c y b

    . (6)

    [7, c. 123] (3)UT

    3R :

    1 0 0

    1 01

    xz y

    ( , , )x y z . (7)

    (6) (3)UT (4)

    3 . (7) :

    1 0 01 0

    1

    t

    xz y

    = 1 0 0

    1 01

    xth yt

    , ( 1)2

    t th zt xy .

    3m . (3)UT , -

    :

  • 1 (17), 2011 - .

    7

    21 0 0

    1 00 1

    vm xz

    , 21 0 0

    1 01

    am xz x

    .

    1 0 01 00 1

    xz

    1 0 01 00 1

    ac

    = 1 0 0

    1 00 1

    x az c

    ,

    1 0 01 0

    1xz x

    1 0 01 0

    1ac a

    = 1 0 0

    1 01

    x az ax c x a

    , . (7)

    ( ,0, ) ( ,0, ) ( ,0, )x z a c x a z c , ( ,0, ) ( ,0, )t x z xt zt ; (8)

    ( , , ) ( , , ) ( , , )x x z a a c x a x a z c ax , 2 ( 1)( , , ) , ,2

    t tt x x z xt xt zt x

    . (9)

    (7) 2 2( , ), ( , )v am x z m x z ;

    2R (1) (2): ( , ) ( , ) ( , )x z a c x a z c , ( , ) ( , )t x z xt zt , tR ;

    ( , ) ( , ) ( , )x z a c x a z c ax , 2 ( 1)( , ) ,2

    t tt x z xt zt x

    , tR .

    ( ,0, )x z ( , , )x x z 3 , 2- -. 2L 2a L (1) (2) 2- - (8) (9) . , - , .

    1.2. 2-

    (1) 2L : ps : r p r ,

    ( , )r x y ( , )r x y ps ( , )p a b . (1) ,x x a y y b ;

    2vm :

    21 0 0

    1 00 1

    vm ab

    .

  • .

    8

    ps 2L -

    . 2a L

    (3) s :

    s : ,

    ( , )x y , ( , )x y , ( , )a b . s 2a L

    ,;

    x x ay y ax b

    2

    1 0 01 0

    1am a

    b a

    .

    s 2a L -

    :

    ,;

    x x ay vx y b

    1 0 01 0

    1ab v

    .

    2L - (1) ps . -

    2a L (2) s . -

    2AutL 2aAut L , , -. .

    2. 3

    2.1. (4)UT

    (4)UT 4 :

    3vm =

    1 0 0 01 0 00 1 00 0 1

    abc

    ; 31am =

    1 0 0 01 0 0

    1 00 0 1

    ab ac

    ; 32am =

    1 0 0 01 0 0

    1 00 1

    ab ac a

    ;

    33am =

    1 0 0 01 0 00 1 0

    1

    abc b a

    ; 34am =

    1 0 0 01 0 0

    1 01

    ab ac b a

    ;

  • 1 (17), 2011 - .

    9

    32vm =

    1 0 0 01 0 0

    0 0 1 00 0 1

    a

    b

    ; 32m =

    1 0 0 01 0 00 1 0

    1

    aab a a

    ;

    (4)UT 2vm (3)UT 32vm .

    :

    3vm :

    1 0 0 01 0 00 1 00 0 1

    xyz

    1 0 0 01 0 00 1 00 0 1

    abc

    =

    1 0 0 01 0 00 1 00 0 1

    x ay bz c

    ;

    31am :

    1 0 0 01 0 0

    1 00 0 1

    xy xz

    1 0 0 01 0 0

    1 00 0 1

    ab ac

    =

    1 0 0 01 0 0

    1 00 0 1

    x ay ax b x az c

    ;

    32am :

    1 0 0 01 0 0

    1 00 1

    xy xz x

    1 0 0 01 0 0

    1 00 1

    ab ac a

    =

    1 0 0 01 0 0

    1 00 1

    x ay ax b x az ax c x a

    ;

    33am :

    1 0 0 01 0 00 1 0

    1

    xyz y x

    1 0 0 01 0 00 1 0

    1

    abc b a

    =

    1 0 0 01 0 00 1 0

    1

    x ay bz ay bx c y b x a

    ;

    34am :

    1 0 0 01 0 0

    1 01

    xy xz y x

    1 0 0 01 0 0

    1 01

    ab ac b a

    =

    1 0 0 01 0 0

    1 01

    x ay ax b x az ay bx c y ax b x a

    ;

    32vm :

    1 0 0 01 0 0

    0 0 1 00 0 1

    x

    y

    1 0 0 01 0 0

    0 0 1 00 0 1

    a

    b

    =

    1 0 0 01 0 0

    0 0 1 00 0 1

    x a

    y b

    ;

    32m :

    1 0 0 01 0 00 1 0

    1

    xxy x x

    1 0 0 01 0 00 1 0

    1

    aab a a

    =

    1 0 0 01 0 00 1 0

    2 1

    x ax ay ax c x a x a

    .

  • .

    10

    3R : 3vm ( , , )a b c ,

    3iam ( , , )a b c , 1,4i .

    32vm , 2vm , -

    . 1.1. 3iam (4)UT

    3R :

    ( , , ) ( , , )vx y z a b c = ( , , )x a y b z c 3vm ; (10)

    1( , , ) ( , , )ax y z a b c = ( , , )x a y ax b z c 31am ; (11)

    2( , , ) ( , , )ax y z a b c = ( , , )x a y ax b z ax c 32am ; (12)

    3( , , ) ( , , )ax y z a b c = ( , , )x a y b z ay bx c 33am ; (13)

    4( , , ) ( , , )ax y z a b c = ( , , )x a y ax b z ay bx c 34am . (14)

    (10)(14) 3R 3( , )vR ,

    3( , )iaR , 1,4i . 3R (4)UT

    (4)UT ,

    (10)(14) . 32m 32m ( , )a b ,

    2R :

    2( , ) ( , )x y a b = ( , 2 )x a y ax b 32m . (15)

    - (10)(14). , (10)(14) ( , , )a b c vs , ias :

    ( , , ) ( , , ) ( , , )x y z a b c x y z .

    3vm ,

    32vm ,

    3iam , 1,4i ,

    22m =

    1 0 01 0

    2 1ab a

    .

    ( m ) 3- [8, c. 301]:

    ,,

    ;

    x x ay hx y bz dx fy z c

    m =

    1 0 0 01 0 0

    1 01

    ab hc d f

    .

  • 1 (17), 2011 - .

    11

    - , .

    2.2. 3

    3( , )vR , 3( , )iaR , 1,4i ,

    : (10):

    ( , , )t x y z = ( , , )xt yt zt , tR ; (16)

    (11):

    ( , , )t x y z = 2 ( 1), ,2

    t txt yt x zt

    , tR ; (17)

    (12):

    ( , , )t x y z = 2 2( 1) ( 1), ,2 2

    t t t txt yt x zt x

    , tR ; (18)

    (13):

    ( , , )t x y z = ( , , ( 1) )xt yt zt xy t t , tR ; (19)

    (14):

    ( , , )t x y z = 2 ( 1), , ( 1)2

    t txt yt x zt xy t t

    , tR ; (20)

    (15):

    ( , )t x y = 2, ( 1)xt yt x t t , tR . (21) -

    3:

    3L = 3, , ( )v R v R , 3aiL = 3, , ( )ia R ia R , 1,4i .

    3L , 3a iL

    . 22a L 2 -

    (15), (21), 2L , 2a L . (4)UT 6R :

    1 0 0 01 0 0

    1 01

    ap br q c

    ( , , , , , )a b c p q r .

  • .

    12

    1 0 0 01 0 0

    1 01

    xu yw v z

    1 0 0 01 0 0

    1 01

    ap br q c

    =

    1 0 0 01 0 0

    1 01

    x au ay p y bw av pz r v bz q z c

    : ( , , , , , )x y z u v w + ( , , , , , )a b c p q r =

    =