АЛЬТЕРНАТИВНЫЕ ДЕЙСТВИТЕЛЬНЫЕ ЛИНЕЙНЫЕ ПРОСТРАНСТВА РАЗМЕРНОСТЕЙ 2, 3 И 4

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<ul><li><p> 1 (17), 2011 - . </p><p> 3</p><p> 512+514.126 </p><p>. . , . . </p><p> 2, 3 4 </p><p> . - , 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. </p><p> 2 [1], : ( , ) ( , ) ( , )x y a b x a y b , ( , ) ( , )t x y xt yt , tR ; (1) </p><p> ( , ) ( , ) ( , )x y a b x a y b ax , 2 ( 1)( , ) ,2</p><p>t tt x y xt yt x </p><p>, tR . (2) </p><p> 2L , - 2a L . 2L : </p><p>, , ..., , ...o a x ; -</p><p> . (2) 2a L (0,0) ; , ( , )x y , </p><p>2( , ) ( , )x y x y x . </p></li><li><p> . </p><p> 4</p><p> 2a L [1]. - 2a L - ( : ( , ) ( , ) ( , )x y u v x u y v xu ; 2. (2) 2a L ). 2a L </p><p>( , ) , (1,0), (0,1) . ( , )x y - : </p><p> ( , )x y = ( 1)2</p><p>x xx y </p><p>. (3) </p><p> ,x y ( 1),</p><p>2x xx y </p><p> . : </p><p> = ( 1), (1,0), 2 (2,1), ..., , , ...2</p><p>t tt t </p><p>; </p><p> = , (0,1), 2 (0,2), ..., (0, ), ...t t . </p><p> (3), x y 2a L , -</p><p> , 2a L 1- : 2a L = + . = ( , ) 2a L </p><p>( , )a p , (0, )b , </p><p>2</p><p>,( 1)( )</p><p>2</p><p>x axx xy a b px by</p><p> , . [1]. . . 2L -. 2a L -. 2L 2a L [1]. . , - 2L 2a L . [1]. </p><p> [2] 2a A - 2a L . ( , , )O B : ( , )A a h , ( , )M x y , </p><p>( , )m p . ,A , A , - </p><p>2 ( 1), ( ) .2</p><p>t tx mt a y am p m h </p></li><li><p> 1 (17), 2011 - . </p><p> 5</p><p> . - . ( , , )O B -</p><p> 2a A , ( , )O c d , ( , )a p , (0, )b , ( , )M x y , - B , ( , )M x y , B , - B M ( , )x y . - </p><p>2</p><p>,( 1)( ) ( ) .</p><p>2</p><p>x ax cx xy a b p ac x by d</p><p> . . [3, c. 237280]. [4, c. 3536]. </p><p> 2A 2L . -</p><p> 2a A . 2a A - 2A 2L . - 2A 2a A , , . . </p><p> 3 4, . </p><p>1. (3)UT </p><p>1.1. 3- </p><p> 3- [5]. 3R - : </p><p> ( , , ) ( , , ) ( , , )x y z a b c x a y b z c ay ; (4) </p><p> ( 1)( , , ) , ,2</p><p>t tt x y z xt yt zt xy </p><p>, tR , (5) </p><p>c. [5, c. 107]. (4) . 3 , - . [5, c. 166215] , . (0,0,0) ; - ( , , )x y z </p><p>( , , )x y z = ( , , )x y z xy . </p></li><li><p> . </p><p> 6</p><p> 3 2. [6] - 2 . </p><p> s 3 ( , , )a b c . : </p><p> , </p><p> ( , , )x y z ; ( , , )x y z s . </p><p> (4) -: </p><p>,</p><p>;</p><p>x x ay y bz z ay c</p><p>1 0 0 01 0 00 1 00 1</p><p>am</p><p>bc a</p><p>. </p><p> ( , , )x y z </p><p>1 0 01 0</p><p>1xz y</p><p> (3)UT . : </p><p> 1 0 0</p><p>1 01</p><p>xz y</p><p>1 0 01 0</p><p>1ac b</p><p> = 1 0 0</p><p>1 01</p><p>x az ay c y b</p><p>. (6) </p><p> [7, c. 123] (3)UT </p><p> 3R : </p><p> 1 0 0</p><p>1 01</p><p>xz y</p><p>( , , )x y z . (7) </p><p> (6) (3)UT (4) </p><p> 3 . (7) : </p><p>1 0 01 0</p><p>1</p><p>t</p><p>xz y</p><p> = 1 0 0</p><p>1 01</p><p>xth yt</p><p>, ( 1)2</p><p>t th zt xy . </p><p> 3m . (3)UT , -</p><p> : </p></li><li><p> 1 (17), 2011 - . </p><p> 7</p><p>21 0 0</p><p>1 00 1</p><p>vm xz</p><p>, 21 0 0</p><p>1 01</p><p>am xz x</p><p>. </p><p>1 0 01 00 1</p><p>xz</p><p>1 0 01 00 1</p><p>ac</p><p> = 1 0 0</p><p>1 00 1</p><p>x az c</p><p>, </p><p>1 0 01 0</p><p>1xz x</p><p>1 0 01 0</p><p>1ac a</p><p> = 1 0 0</p><p>1 01</p><p>x az ax c x a</p><p>, . (7) </p><p> ( ,0, ) ( ,0, ) ( ,0, )x z a c x a z c , ( ,0, ) ( ,0, )t x z xt zt ; (8) </p><p>( , , ) ( , , ) ( , , )x x z a a c x a x a z c ax , 2 ( 1)( , , ) , ,2</p><p>t tt x x z xt xt zt x </p><p>. (9) </p><p> (7) 2 2( , ), ( , )v am x z m x z ; </p><p> 2R (1) (2): ( , ) ( , ) ( , )x z a c x a z c , ( , ) ( , )t x z xt zt , tR ; </p><p>( , ) ( , ) ( , )x z a c x a z c ax , 2 ( 1)( , ) ,2</p><p>t tt x z xt zt x </p><p>, tR . </p><p> ( ,0, )x z ( , , )x x z 3 , 2- -. 2L 2a L (1) (2) 2- - (8) (9) . , - , . </p><p>1.2. 2- </p><p> (1) 2L : ps : r p r , </p><p> ( , )r x y ( , )r x y ps ( , )p a b . (1) ,x x a y y b ; </p><p> 2vm : </p><p>21 0 0</p><p>1 00 1</p><p>vm ab</p><p>. </p></li><li><p> . </p><p> 8</p><p> ps 2L -</p><p>. 2a L </p><p>(3) s : </p><p>s : , </p><p> ( , )x y , ( , )x y , ( , )a b . s 2a L </p><p>,;</p><p>x x ay y ax b </p><p> 2</p><p>1 0 01 0</p><p>1am a</p><p>b a</p><p>. </p><p> s 2a L -</p><p> : </p><p>,;</p><p>x x ay vx y b </p><p>1 0 01 0</p><p>1ab v</p><p>. </p><p> 2L - (1) ps . -</p><p> 2a L (2) s . -</p><p> 2AutL 2aAut L , , -. . </p><p>2. 3 </p><p>2.1. (4)UT </p><p> (4)UT 4 : </p><p>3vm =</p><p>1 0 0 01 0 00 1 00 0 1</p><p>abc</p><p>; 31am =</p><p>1 0 0 01 0 0</p><p>1 00 0 1</p><p>ab ac</p><p>; 32am =</p><p>1 0 0 01 0 0</p><p>1 00 1</p><p>ab ac a</p><p>; </p><p>33am =</p><p>1 0 0 01 0 00 1 0</p><p>1</p><p>abc b a</p><p>; 34am =</p><p>1 0 0 01 0 0</p><p>1 01</p><p>ab ac b a</p><p>; </p></li><li><p> 1 (17), 2011 - . </p><p> 9</p><p>32vm =</p><p>1 0 0 01 0 0</p><p>0 0 1 00 0 1</p><p>a</p><p>b</p><p>; 32m = </p><p>1 0 0 01 0 00 1 0</p><p>1</p><p>aab a a</p><p>; </p><p> (4)UT 2vm (3)UT 32vm . </p><p> : </p><p>3vm : </p><p>1 0 0 01 0 00 1 00 0 1</p><p>xyz</p><p>1 0 0 01 0 00 1 00 0 1</p><p>abc</p><p>=</p><p>1 0 0 01 0 00 1 00 0 1</p><p>x ay bz c</p><p>; </p><p>31am : </p><p>1 0 0 01 0 0</p><p>1 00 0 1</p><p>xy xz</p><p>1 0 0 01 0 0</p><p>1 00 0 1</p><p>ab ac</p><p>=</p><p>1 0 0 01 0 0</p><p>1 00 0 1</p><p>x ay ax b x az c</p><p>; </p><p>32am : </p><p>1 0 0 01 0 0</p><p>1 00 1</p><p>xy xz x</p><p>1 0 0 01 0 0</p><p>1 00 1</p><p>ab ac a</p><p>=</p><p>1 0 0 01 0 0</p><p>1 00 1</p><p>x ay ax b x az ax c x a</p><p>; </p><p>33am : </p><p>1 0 0 01 0 00 1 0</p><p>1</p><p>xyz y x</p><p>1 0 0 01 0 00 1 0</p><p>1</p><p>abc b a</p><p>=</p><p>1 0 0 01 0 00 1 0</p><p>1</p><p>x ay bz ay bx c y b x a</p><p>; </p><p>34am : </p><p>1 0 0 01 0 0</p><p>1 01</p><p>xy xz y x</p><p>1 0 0 01 0 0</p><p>1 01</p><p>ab ac b a</p><p>=</p><p>1 0 0 01 0 0</p><p>1 01</p><p>x ay ax b x az ay bx c y ax b x a</p><p>; </p><p>32vm : </p><p>1 0 0 01 0 0</p><p>0 0 1 00 0 1</p><p>x</p><p>y</p><p>1 0 0 01 0 0</p><p>0 0 1 00 0 1</p><p>a</p><p>b</p><p>=</p><p>1 0 0 01 0 0</p><p>0 0 1 00 0 1</p><p>x a</p><p>y b</p><p>; </p><p>32m : </p><p>1 0 0 01 0 00 1 0</p><p>1</p><p>xxy x x</p><p>1 0 0 01 0 00 1 0</p><p>1</p><p>aab a a</p><p>=</p><p>1 0 0 01 0 00 1 0</p><p>2 1</p><p>x ax ay ax c x a x a</p><p>. </p></li><li><p> . </p><p> 10</p><p> 3R : 3vm ( , , )a b c , </p><p>3iam ( , , )a b c , 1,4i . </p><p> 32vm , 2vm , -</p><p> . 1.1. 3iam (4)UT </p><p>3R : </p><p> ( , , ) ( , , )vx y z a b c = ( , , )x a y b z c 3vm ; (10) </p><p> 1( , , ) ( , , )ax y z a b c = ( , , )x a y ax b z c 31am ; (11) </p><p> 2( , , ) ( , , )ax y z a b c = ( , , )x a y ax b z ax c 32am ; (12) </p><p> 3( , , ) ( , , )ax y z a b c = ( , , )x a y b z ay bx c 33am ; (13) </p><p> 4( , , ) ( , , )ax y z a b c = ( , , )x a y ax b z ay bx c 34am . (14) </p><p> (10)(14) 3R 3( , )vR , </p><p>3( , )iaR , 1,4i . 3R (4)UT </p><p> (4)UT , </p><p>(10)(14) . 32m 32m ( , )a b , </p><p>2R : </p><p> 2( , ) ( , )x y a b = ( , 2 )x a y ax b 32m . (15) </p><p> - (10)(14). , (10)(14) ( , , )a b c vs , ias : </p><p>( , , ) ( , , ) ( , , )x y z a b c x y z . </p><p>3vm , </p><p>32vm , </p><p>3iam , 1,4i , </p><p>22m = </p><p>1 0 01 0</p><p>2 1ab a</p><p>. </p><p> ( m ) 3- [8, c. 301]: </p><p>,,</p><p>;</p><p>x x ay hx y bz dx fy z c</p><p> m =</p><p>1 0 0 01 0 0</p><p>1 01</p><p>ab hc d f</p><p>. </p></li><li><p> 1 (17), 2011 - . </p><p> 11</p><p> - , . </p><p>2.2. 3 </p><p> 3( , )vR , 3( , )iaR , 1,4i , </p><p> : (10): </p><p> ( , , )t x y z = ( , , )xt yt zt , tR ; (16) </p><p> (11): </p><p> ( , , )t x y z = 2 ( 1), ,2</p><p>t txt yt x zt </p><p>, tR ; (17) </p><p> (12): </p><p> ( , , )t x y z = 2 2( 1) ( 1), ,2 2</p><p>t t t txt yt x zt x </p><p>, tR ; (18) </p><p> (13): </p><p> ( , , )t x y z = ( , , ( 1) )xt yt zt xy t t , tR ; (19) </p><p> (14): </p><p> ( , , )t x y z = 2 ( 1), , ( 1)2</p><p>t txt yt x zt xy t t </p><p>, tR ; (20) </p><p> (15): </p><p> ( , )t x y = 2, ( 1)xt yt x t t , tR . (21) -</p><p> 3: </p><p>3L = 3, , ( )v R v R , 3aiL = 3, , ( )ia R ia R , 1,4i . </p><p> 3L , 3a iL </p><p> . 22a L 2 -</p><p> (15), (21), 2L , 2a L . (4)UT 6R : </p><p>1 0 0 01 0 0</p><p>1 01</p><p>ap br q c</p><p> ( , , , , , )a b c p q r . </p></li><li><p> . </p><p> 12</p><p>1 0 0 01 0 0</p><p>1 01</p><p>xu yw v z</p><p>1 0 0 01 0 0</p><p>1 01</p><p>ap br q c</p><p> = </p><p>1 0 0 01 0 0</p><p>1 01</p><p>x au ay p y bw av pz r v bz q z c</p><p> : ( , , , , , )x y z u v w + ( , , , , , )a b c p q r = </p><p>= ( , , , , , )x a y b z c u ay p v bz q w av pz r . </p><p>, 6R - 6( , )R . 6( , )R </p><p>( )R : </p><p>( , , , , , )t x y z u v w = </p><p>= ( 1) ( 1) ( 1) ( 2)( 1), , , , , ( )2 2 2 6</p><p>t t t t t t t t txt yt zt ut xy vt yz wt xv zu xyz </p><p>. </p><p> 6 = 3( , , ( ))R R , 6- </p><p>, 3. 6 , , - 3L , 3a iL , 1,4i , </p><p>22</p><p>a L . -</p><p> 3vm , 3iam , </p><p>32m . -</p><p> 3L , 3a iL , 22</p><p>a L 6 . </p><p>3. 4 </p><p>3.1. (5)UT </p><p> (5)UT </p><p>4vm = </p><p>1 0 0 0 01 0 0 00 1 0 00 0 1 00 0 0 1</p><p>abcd</p><p>; 41am = </p><p>1 0 0 0 01 0 0 0</p><p>1 0 00 0 1 00 0 0 1</p><p>ab acd</p><p>; </p><p>42am = </p><p>1 0 0 0 01 0 0 0</p><p>1 0 00 1 0</p><p>0 0 0 1</p><p>ab ac ad</p><p>; 43am = </p><p>1 0 0 0 01 0 0 0</p><p>1 0 00 1 00 0 1</p><p>ab ac ad a</p><p>; </p></li><li><p> 1 (17), 2011 - . </p><p> 13</p><p>44am = </p><p>1 0 0 0 01 0 0 0</p><p>1 0 01 0</p><p>0 0 0 1</p><p>ab ac b ad</p><p>; 45am = </p><p>1 0 0 0 01 0 0 00 1 0 00 0 1 0</p><p>1</p><p>abcd c b a</p><p>; </p><p>46am = </p><p>1 0 0 0 01 0 0 00 1 0 0</p><p>0 1 01</p><p>abc ad c b a</p><p>; 47am = </p><p>1 0 0 0 01 0 0 0</p><p>1 0 01 0</p><p>1</p><p>ab ac b ad c b a</p><p>; </p><p>48am = </p><p>1 0 0 0 01 0 0 0</p><p>1 0 00 0 0 1 00 0 0 1</p><p>ab a</p><p>c</p><p>; 43m = </p><p>1 0 0 0 01 0 0 00 1 0 00 0 1 0</p><p>1</p><p>aaab a a a</p><p>; 42,1m = </p><p>1 0 0 0 01 0 0 00 1 0 00 0 1 0</p><p>1</p><p>aabc b a a</p><p>. </p><p> (5)UT ; . : </p><p> 4vm : </p><p>1 0 0 0 01 0 0 00 1 0 00 0 1 00 0 0 1</p><p>xyzw</p><p>1 0 0 0 01 0 0 00 1 0 00 0 1 00 0 0 1</p><p>abcd</p><p> = </p><p>1 0 0 0 01 0 0 00 1 0 00 0 1 00 0 0 1</p><p>x ay bz cw d</p><p>; </p><p>41am : </p><p>1 0 0 0 01 0 0 0</p><p>1 0 00 0 1 00 0 0 1</p><p>xy xzw</p><p>1 0 0 0 01 0 0 0</p><p>1 0 00 0 1 00 0 0 1</p><p>ab acd</p><p> = </p><p>1 0 0 0 01 0 0 0</p><p>1 0 00 0 1 00 0 0 1</p><p>x ay ax b x a</p><p>z cw d</p><p>; </p><p>42am : </p><p>1 0 0 0 01 0 0 0</p><p>1 0 00 1 0</p><p>0 0 0 1</p><p>xy xz xw</p><p>1 0 0 0 01 0 0 0</p><p>1 0 00 1 0</p><p>0 0 0 1</p><p>ab ac ad</p><p> = </p><p>1 0 0 0 01 0 0 0</p><p>1 0 00 1 0</p><p>0 0 0 1</p><p>x ay ax b x az ax c x aw d</p><p>; </p></li><li><p> . </p><p> 14</p><p>43am : </p><p>1 0 0 0 01 0 0 0</p><p>1 0 00 1 00 0 1</p><p>xy xz xw x</p><p>1 0 0 0 01 0 0 0</p><p>1 0 00 1 00 0 1</p><p>ab ac ad a</p><p> = </p><p>1 0 0 0 01 0 0 0</p><p>1 0 00 1 00 0 1</p><p>x ay ax b x az ax c x aw ax d x a</p><p>; </p><p>44am : </p><p>1 0 0 0 01 0 0 0</p><p>1 0 01 0</p><p>0 0 0 1</p><p>xy xz y xw</p><p>1 0 0 0 01 0 0 0</p><p>1 0 01 0</p><p>0 0 0 1</p><p>ab ac b ad</p><p> = </p><p>= </p><p>1 0 0 0 01 0 0 0</p><p>1 0 01 0</p><p>0 0 0 1</p><p>x ay ax b x az ay bx c y ax b x aw d</p><p>; </p><p>45am : </p><p>1 0 0 0 01 0 0 00 1 0 00 0 1 0</p><p>1</p><p>xyzw z y x</p><p>1 0 0 0 01 0 0 00 1 0 00 0 1 0</p><p>1</p><p>abcd c b a</p><p> = </p><p>= </p><p>1 0 0 0 01 0 0 00 1 0 00 0 1 0</p><p>1</p><p>x ay bz cw az by cx d z c y b x a</p><p>; </p><p>46am :</p><p>1 0 0 0 01 0 0 00 1 0 0</p><p>0 1 01</p><p>xyz xw z y x</p><p>1 0 0 0 01 0 0 00 1 0 0</p><p>0 1 01</p><p>abc ad c b a</p><p> = </p><p>= </p><p>1 0 0 0 01 0 0 00 1 0 0</p><p>0 1 01</p><p>x ay bz ax c x aw az by cx d z ax c y b x a</p><p>; </p></li><li><p> 1 (17), 2011 - . </p><p> 15</p><p>47am : </p><p>1 0 0 0 01 0 0 0</p><p>1 0 01 0</p><p>1</p><p>xy xz y xw z y x</p><p>1 0 0 0 01 0 0 0</p><p>1 0 01 0</p><p>1</p><p>ab ac b ad c b a</p><p> = </p><p>= </p><p>1 0 0 0 01 0 0 0</p><p>1 0 01 0</p><p>1</p><p>x ay ax b x az ay bx c y ax b x aw az by cx d z ay bx c y ax b x a</p><p>; </p><p>48am : </p><p>1 0 0 0 01 0 0 0</p><p>1 0 00 0 0 1 00 0 0 1</p><p>xy x</p><p>z</p><p>1 0 0 0 01 0 0 0</p><p>1 0 00 0 0 1 00 0 0 1</p><p>ab a</p><p>c</p><p> = </p><p>1 0 0 0 01 0 0 0</p><p>1 0 00 0 0 1 00 0 0 1</p><p>x ay ax b x a</p><p>z c</p><p>; </p><p>43m : </p><p>1 0 0 0 01 0 0 00 1 0 00 0 1 0</p><p>1</p><p>xxxy x x x</p><p>1 0 0 0 01 0 0 00 1 0 00 0 1 0</p><p>1</p><p>aaab a a a</p><p> = </p><p>= </p><p>1 0 0 0 01 0 0 00 1 0 00 0 1 0</p><p>3 1</p><p>x ax ax ay ax b x a x a x a</p><p>; </p><p>42,1m : </p><p>1 0 0 0 01 0 0 00 1 0 00 0 1 0</p><p>1</p><p>xxyz y x x</p><p>1 0 0 0 01 0 0 00 1 0 00 0 1 0</p><p>1</p><p>aabc b a a</p><p> = </p><p>= </p><p>1 0 0 0 01 0 0 00 1 0 00 0 1 0</p><p>( ) 1</p><p>x ax ay bz bx a b y c y b x a x a</p><p>. </p></li><li><p> . </p><p> 16</p><p> - (5)UT . 48am </p><p> 31am . 43m </p><p>42,1m </p><p> (3)UT (4)UT : </p><p>23m = </p><p>1 0 01 0</p><p>2 1ab a</p><p> 32,1m = </p><p>1 0 0 01 0 00 1 0</p><p>1</p><p>abc b a b</p><p>. </p><p>3.2. 4R </p><p> (5)UT 10R : </p><p>1 0 0 0 01 0 0 0</p><p>1 0 01 0</p><p>1</p><p>ap bu q cw v r d</p><p>( , , , , , , , , , )a b c d p q r u v w . </p><p>, (4)UT 6R , . 2.2, , - 3vm , </p><p>3iam , 1,4i , -</p><p>, 3 ( , , )vm a b c , 3 ( , , )iam a b c </p><p>3vm , </p><p>3iam </p><p>3R . - (10)(14). . 4vm , </p><p>4iam , 1,7i , </p><p>4R . </p><p> 4R 4vm , 4iam . -</p><p> : </p><p> ( , , , ) ( , , , )vx y z w a b c d = ( , , , )x a y b z c w d ; (22) </p><p> 1( , , , ) ( , , , )ax y z w a b c d = ( , , , )x a y ax b z c w d ; (23) </p><p> 2( , , , ) ( , , , )ax y z w a b c d = ( , , , )x a y ax b z ax c w d ; (24) </p><p> 3( , , , ) ( , , , )ax y z w a b c d = ( , , , )x a y ax b z ax c w ax d ; (25) </p><p> 4( , , , ) ( , , , )ax y z w a b c d = ( , , , )x a y ax b z ay bx c w d ; (26) </p><p> 5( , , , ) ( , , , )ax y z w a b c d = ( , , , )x a y b z c w az by cx d ; (27) </p><p> 6( , , , ) ( , , , )ax y z w a b c d = ( , , , )x a y b z ax c w az by cx d ; (28) </p><p>7( , , , ) ( , , , )ax y z w a b c d = </p></li><li><p> 1 (17), 2011 - . </p><p> 17</p><p> = ( , , , )x a y ax b z ay bx c w az by cx d ; (29) </p><p> , : </p><p> 3( , ) ( , ) ( , 3 )x y a b x a y ax b , 23m ; (30) </p><p> 2,1( , , ) ( , , ) ( , , ( ) )x y z a b c x a y b z b a x ay c , 32,1m . (31) </p><p> 4( , )vR ; 4( , )iaR , </p><p>1,7i ; 2 3( , )R ; 3</p><p>2,1( , )R </p><p> 10 , : </p><p>( , , , , , , , , , ) ( , , , , , , , , , )x y z s p q r u v w a b c d e f g h k l = </p><p>= ( , , , , , , , ;x a y b z c s d p ay e q bz f r cs g u aq ez h </p><p>, )v br fs k w av er hs l ; </p><p>( , , , , , , , , , )t a b c d e f g h k l = </p><p>= ( 1) ( 1) ( 1), , , , , , ,2 2 2</p><p>t t t t t tat bt ct dt et ab ft bc gt cd </p><p>( 1) ( 2)( 1) ( 1) ( 2)( 1)( ) , ( ) ,2 6 2 6</p><p>t t t t t t t t t tht af ce abc kt bg df bcd </p><p>( 1) ( 2)( 1) ( 3)( 2)( 1)( ) ( )2 6 24</p><p>t t t t t t t t tlt ak dh eg abg adk cde abcd </p><p>. </p><p> 4R , 3R , 2R , tR : (22): ( , , , )t a b c d = ( , , , )at bt ct dt ; </p><p> (23): ( , , , )t a b c d = 2 ( 1), , ,2</p><p>t tat bt a ct dt </p><p>; </p><p> (24): ( , , , )t a b c d = 2 2( 1) ( 1), , ,2 2</p><p>t t t tat bt a ct a dt </p><p>; </p><p> (25): ( , , , )t a b c d = 2 2 2( 1) ( 1) ( 1), , ,2 2 2</p><p>t t t t t tat bt a ct a dt a </p><p>; </p><p> (26): ( , , , )t a b c d = 2 ( 1), , ( 1) ,2</p><p>t tat bt a ct ab t t dt </p><p>; </p><p> (27): ( , , , )t a b c d = 2 ( 1), , , ( 1)2</p><p>t tat bt ct dt ac t t b </p><p>; </p><p> (28): ( , , , )t a b c d = 2 2( 1) ( 1), , , ( 1)2 2</p><p>t t t tat bt ct a dt ac t t b </p><p>; </p></li><li><p> . </p><p> 18</p><p> (29): 2 3( 1) ( 2)( 1)( , , , ) , , ( 1) ,2 6</p><p>t t t t tt a b c d at bt a ct ab t t a </p><p> 2 2 4( 1) ( 2)( 1) ( 3)( 2)( 1)( 1) 32 6 24</p><p>t t t t t t t t tdt ac t t b a b a </p><p>; </p><p> (30): 2 ( 1)( , ) , 32</p><p>t tt a b at bt a </p><p>, 23m ; </p><p> (31): 2 ( 1)( , , ) , , ( 1)2</p><p>t tt a b c at bt ct ab t t a </p><p>, 32,1m . </p><p> 10 4( , , ( ))v R v R = </p><p>4L ; 4( , , ( ))ia R ia R = 4aiL , 1,7i ; </p><p>23 3( , , ( ))R R = </p><p>23L ; </p><p>32,1 2,1( , , ( ))R R = </p><p>32,1L </p><p> R . : 4L = a b c d </p><p> , 41a L = ,a b c d </p><p> , </p><p>44</p><p>a L = , ,a b c d . </p><p> 4L , 4a iL , 1,7i , 23L , </p><p>32,1L </p><p> , -, . 23L , </p><p>32,1L -</p><p>, , 2- 3- . </p><p> , . </p><p> , R : 2-, 3-, 4-. - . - . </p><p> 1. , . . / . . , </p><p>. . // . 2007. . 9, . 4. . 414. </p><p>2. , . . / . . , . . // . 2008. . 10, . 2. . 920. </p></li><li><p> 1 (17), 2011 - . </p><p> 19</p><p>3. Xy , . . / . . . (-) : , 2009. 476 . </p><p>4. , . . / . . , . . // , : . ., . 100- . . . (, 56 2010 .). : , 2010. . 3537. </p><p>5. , . . - / . . . : - , 2005. 306 . </p><p>6. , . . 2 - / . . // . . 2008. 12. . 1727. </p><p>7. , . / . . . : , 1986. 168 . </p><p>8. , . . / . . . . : , 1969. 548 . </p><p> - , , , </p><p>Dolgarev Ivan Arturovich Candidate of physical and mathematical sciences, associate professor, sub-department of mathematics and supercomputer modeling, Penza State University</p><p>E-mail: delivar@yandex.ru - , , , </p><p>Dolgarev Artur Ivanovich Candidate of physical and mathematical sciences, associate professor, sub-department of mathematics and supercomputer modeling, Penza State University</p><p>E-mail: delivar@yandex.ru </p><p> 512 + 514.126 </p><p>, . . -</p><p> 2, 3 4 / . . , . . // . . - . 2011. 1 (17). . 319. </p></li></ul>

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