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MATHEMATICAL MODELING, SYSTEMS ANALYSIS
??? 512.54
DOI 10.17150/1993-3541.2014.24(6).152-158
?. ?. ??????
???????? ???????? ?????? ? ?????? ?????????? ?? ???,
?. ???????, ?????????? ?????????
?. ?. ????????
??????????? ??????????????? ??????????? ????????? ? ?????,
?. ???????, ?????????? ?????????
? ??????? ?????????? ??????????????? ?????????? ?????
? ???????? ?????? ????????????
?????????. ??????????????? ?????? ? ???????? ?????? ???????? ???????????? ????????? ??????
?????? ????? ????????????. ?????? ?? ??? ??????? ??????? ????? n = 2k, k ? ? ???????? ???????
????? ? n ??????????????. ? 1973 ?. ?. ?. ????????? ? ?????? Ф? ??????? ????????????? ?????????? ???????╗ ???? ????????? ??????? ??????????? ?????????? ????? ????????? ?????, ??????????? 4k ???????? ???????????? ??? ??????? k ? ?, ? ???????? ??????? ? ???, ??? ? ??????????
?????????? ??????????????? ??????? ? ???????? ?????? ???????? ???????????? ??? ????? ??????
???????. ? ?? ?? ????? ??? s-???????? ?????????? ????? ??? s > 2 ?????????? ?? ?????? ???????
??????????????? ?????? ? ???????? ?????? ???????? ????????????. ???? ?????? ???????????? ?
?????? ??????, ??? ????? ?. ?. ???????? ???????????????? ?? ???????????? ?????????? ??????
????????? ????? ? ??????????? ????????????? ???????????. ? ???? ?????? ????? ???????? ???????????? ?????? ??????, ? ??? k = 1, 2, 3, 4, 9 ?????????? ??????? ???????????? ?????????? ?????,
??????????? n = 8k ???????? ????????????. ??? ????????? k ?????? ? ????????????? ????????????
?????????? ?????, ??????????? n = 8k ???????? ????????????, ???????? ????????. ???????, ???
????? ?????? ?? ???????? ? ????????????? s-???????? ?????????? ????? ??? s > 3, ???????????
???????? ????? ???????? ????????????.
???????? ?????. ??????????????? ??????; ?????????? ??????; ???????? ??????? ?? ??????.
?????????? ? ??????. ???? ??????????? 1 ??????? 2014 ?.; ???? ???????? ? ?????? 29 ??????? 2014 ?.;
???? ??????-?????????? 29 ??????? 2014 ?.
V. V. BLUDOV
Institute for System Dynamics and Control Theory
of Siberian Branch of Russian Academy of Sciences,
Irkutsk, Russian Federation
L. E. BADMAEVA
Baikal State University of Economics and Law,
Irkutsk, Russian Federation
ON THE METHOD OF CONSTRUCTING ORDERABLE SOLUBLE GROUPS
WITH FINITELY MANY ORDERINGS
Abstract. The ordered groups with a finite number of total orderings permit only an even number of orderings.
However, we do not know the examples of groups with n orderings for every even number n = 2k, k ? ?.
In 1973, V. Kopytov in the article ФOn totally ordered soluble groups╗ gave examples of soluble class two
groups of finite rank admitting 4k total orderings for each k ? ? and proved the theorem according to which
in soluble non-Abelian orderable groups with a finite number of total orderings that number is divisible by
four. At the same time, there are no examples of ordered groups with a finite number of total orderings for
soluble class s groups with s > 2. This gap is filled in this article where Kopytov?s method is applied to soluble
class three groups of finite rank with a nilpotent class two commutatant. In this case, the number of finite
orderings is divisible by eight, and for k = 1, 2, 3, 4, 9 the examples of soluble class three groups admitting
n = 8k of total orderings are given. For the rest k, the issue of the existence of soluble class three groups
admitting nа= 8k of total orderings remains open. It is important to stress that nothing is known about the
existence of soluble class s groups with s > 3 admitting a finite number of total orderings.
Keywords. Ordered group; soluble group; total order.
Article info. Received October 1, 2014; accepted October 29, 2014; available online December 29, 2014.
Е ?. ?. ??????, ?. ?. ????????, 2014
152
?. ?. ??????, ?. ?. ????????
Izvestiya of Irkutsk State Economics Academy,
2014, no. 6 (98), pp. 152?158. ISSN 1993-3541
???????? ????????? ??????????????? ????????????? ????????.
2014. ? 6 (98). ?. 152?158. ISSN 1993-3541
V. V. BLUDOV, L. E. BADMAEVA
???? ?????? ????????? ?????? ???????? ????? ????????????, ?? ??? ????? ??????, ?????????
?????? ? ????????? ???????? ????????, ?????
????? ? ??????????????? ???????. ??????????, ??? ??????? ?? ??????? ????? 2n ??????????
??????????????? ?????? ? 2n ??????????????.
??????? ?????? ? ???????? ?????? ???????????? ????????? ?????? ??? ???????? ????????????, ??? ????? ???????? ??? ????????? ?????????? ?????? ???????????? ?????. ???????? ???
??????? ?????????? ??????????????? ?????,
??????????? ??? ???????? ???????????? [1;а13]
? ????? ???????? ?????, ??????????? 8nа + 6,
nа? ? ???????????? [10], ?? ??? ??? ?????? ???????????. ??? ???????? ?????????? ??????????????? ????? ? ???????? ?????? ????????????, ?? ????? ???????????? ????? ????? ?????
4n, n ? ? (??.:а[5; 6, ???????а5.3.2]). ?????? ???
??????? ????? ? 4n ??????????????, ??????????? ? ?????? [5], ????????? ? ?????? ???????????
?????????? ?????, ? ??? ?????????? ?????
??????? ???????????? sа> 2 ?? ?????? ???????
????? ? ???????? ?????? ???????????? ?? ???
??? ?? ?????????.
? ?????? ????????? ???????????? ?????????? ??????, ? ??????? ??????????? ??????????? ????????????? ??????. ?? ???? ????????
????????? ?????????? ? ?????? ? ???????
???????? ??? ?????????, ?????????????, ???????? ??????????? ???????? ????????? H.
? ???? ?????? H ????????? ? ?????? ? ? ??????????? ???????? H ????????? ?? ?????? ???????????, ??? ???? ??? ?????????? ????????
????????? ? ????? ???????? ????????????
?????? ?????????? ??????? ??????. ??? ? ?
?????? [5], ?? ??????? ??????? ?????, ? ??????? ??????????? ???????????? ??????????
?? ????? ???????????? ???????????. ?????????? ?????????? ????? ????? ??????????????
??? ?????????? ????? ? ???????? ?????? ????????????, ?? ?? ????? ?????????? ?? ???????????, ????????? ?????? ???????, ?????????
? ??????????? ????????????, ????????????
?? ???????????? ????????? ?????????? ????
??????????? ? ???????? ? ???????? ? ????????????? ????????????.
??? ????? ???????? ????????? ?????? ??????
????? ?????????????? ???????????? ? ???????????, ???????? ? ?????? ? ??????? [1?7; 11]:
?а[g, h] = g?1h?1gh?? ?????????? ?????????
g ? h;
?а ?S? ? ?????????, ??????????? ?????????? S;
?аG? = [G, G]?? ????????? ?????? G;
?а ????????? H ?????? G ?????????, ????
g?1Hgа= H ??? ???? g ? G;
?а ???????????? ????????? H?? ??? ????????? NG(F) = ?g ? G|g?1Hg = H?.
???????? ??????????? ????????????? ? ?????????? ??????. ????? G?? ??????.
??????????? 1. ?????????? ???
1 = G0 < G1 < иии < Gs = G ?????????? ???????????, ???? ??????-?????? ??? ??????? ??????????, ?. ?. Gi + 1/Gi ? Z(G/Gi),
???, ??? ???????????, [Giа +а 1, G] ? Gi ??? ????
0 ? i < s. ??????, ?????????? ???????????
?????, ?????????? ?????????????, ????? ????
s?? ??????? ???????????????.
??????????? 2. ?????????? ???
1 = G(n) < G(n ? 1) < и и и < G(1) < G(0) = G ??????????
??????????, ???? ??????-?????? ??? ???????
???????, ???, ??? ???????????, [G(i), G(i)] ? G(i + 1)
??? ???? 0 ? i < n ? 1. ??????, ?????????? ?????????? ?????, ?????????? ??????????,
????? ???? n?? ??????? ????????????.
??????????? 3. ?????????????? ???????
?G, и, ??, ? ??????? ?G, и? ???????? ??????, ?
?G,а ???? ??????? ????????????? ?????????,
?????????? ??????? ????????????? ???????,
???? ????????? ? ???????? ??????? ?? ????????? G ??????????? ?? ???????: a ? b ??????
gahа? gbh ??? ???? a, b, g, h ? G.
??????????? 4. ????????? H ??????? ??????????????? ?????? G ?????????? ????????,
???? ??? ????? x ? G ? h ? H ?? 1 < x < h ???????
x ? H. ??????? ???? ???????? ???????? ??????? ????????????? ?????? G ?????????? ?????
?(G) : E = G0 ? G1 ? . . . ? Gs = G.
??????????? 5. ?????? ?G, и? ??????????
???????????????, ???? ?? ????????? G
????? ?????????? ???????? ??????? ?, ???????????? ?G, и, ?? ? ??????? ?????????????
??????.
?????????? ????????? ???????????? ??????????????? ?????? ????? ???? ??? ????????,
??? ? ???????????.
??????? 1. ?????? G ?????????????? ????? ? ?????? ?????, ????? ?????????? ????????????????? ??????? ???????? ?(G), ?????, ???
???? A ?аB?? ?????? ? ?(G), ?? ??????-??????
B/A ????????? ????????? ????????? ??????? ????????????? ?????????? ?????? ???????????? ?????, ? ?????? ?????????????
??????-?????? B/A, ?????????????? ??????????? ?????????????? ????????????? NG(A),
????????? ????????? ?????????????????
?????? ????????????? ???????????? ?????
(??.: [6, ???????а2.3.1; 14]).
153
?????????????? ?????????????, ????????? ??????
Izvestiya of Irkutsk State Economics Academy,
2014, no. 6 (98), pp. 152?158. ISSN 1993-3541
???????? ????????? ??????????????? ????????????? ????????.
2014. ? 6 (98). ?. 152?158. ISSN 1993-3541
MATHEMATICAL MODELING, SYSTEMS ANALYSIS
gic = gi + 1, ??? 0 ? i ? m ? 2;
????? ?????????? ????? ? ??????????? ????????????? ???????????, ??????????? ???????? ????? ????????????:
1. ???? G?? ??????????????? ??????????
????????? ?????? ? ???????? ?????? ????????????, ?? ??? ??? ????? ???????? ???????
????? ????? ???????? ????? ???????? ?????????? ????????, ?????? ???? ?? ???? ????????
?????????? ????????? ? ?????? ?????? ???????
(????????? ??????? 5.3.1 [6]). ? ????? ??????
??? ????? ? ????????????? ??????????? ????????? ????? ???????? ?????????? ????????
?????? ???? ?? ??????? ???? ???. ????? G
????? ??? ?????????? ????????? N = [G, G] ?
D = [N, N], ???????? ??? ????????? ????????
??????? ?? ?????? G.
?????????? ??? ???????? ???????? 1 ? D ?
? N ? G. ??????-?????? G/N?? ??????? ??????,
??????? ?????? ????? ??? ???????? ????????????, ?. ?. ???? ??????? ????? 1.
??????-?????? G/D?? ??????????? ?????????? ?????? ? ???????? ?????? ???????? ????????????, ? ??? ??????? ??????? 1, ??????????
???????????? ?????? G ???????????? ?????????? ?? ????????????? ?????? ????????????? ?????????? ??? ?. ?? ? ???? ?????? ???
???????????? ???????????? ?? ????????????? ?????? D. ??????? ???????????? ??????
D ????? ?????? ?????????????? ??????????
?? ????????????? ?????? ?????????? ???????
??????????.
2. ????? Nm ?????????? ????????? ??????????? ????????????? ?????? ????? m ? ?????????? ????????? ??????????? {g0, ..., gm ? 1}.
????? Dm ????????? ????????? ?????? Nm.
????????? Nm?? ????????? ??????????? ????????????? ?????? ????? m, ?? Dm?? ?????????
??????? ?????? ????? m(m ? 1) 2 ? ??????????
????????? ??????????? {[gi, gj] | 0 ? i < j < m}.
??? ???? ??????????? ?????? ???????????:
[g0, g1] = d1, [g0, g2] = d2, ..., [g0, gm ? 1] = dm?? 1;
a
gmc ? 1 = g0a0 g1a1 , ..., gmm??11 . ???????, ??? ??????????? (2) ???????????? ??????? ?????????? ??????????? ?? ??????-?????? Nm/Dm, ?. ?. (gD)c = gcD.
????????? ??????-?????? Nm/Dm ????????????, ?? ?? ??? ????? ???????? ??? ??
?????? ??? ??????? ?. ? ???? ?????? ??????????? c ???????? ???????? Ac ? ??????
g0 = g0D, g1 = g1D, ..., gm ? 1 = gm ? 1D, ???
?0
?
?1
Ac = ? 0
?
??
?0
?
[gi , g j ]c = [gic , gcj ], 0 ? i < j < m.
?????? Dm ????????????, ?????????????
?? ??? ?????? ??? ?. ? ???? ?????? ????????
???????????? c ?? ????????? Dm ????? ??????
????????, ????????, ? ??????
d1, ..., d 1
,
2
, ..., [gm ? 2 , gm ? 1] = d 1
j ( j ? 1) + i + 1
2
m ( m ? 1)
m ( m ? 1)
???????????? ? ????????? (1).
? ????? ?????? ??? ???? ??????? ??????????. ?????? ? ??????????? ???????? ???????
????? ???????????? ? ????? ????, ? ?? ??????
????? ?? ?????????????????? ????????? k(?).
?????????? ????????????? ?????? ????? ?????????? ? ????????? ?????????? ???????? ???????????? ?? Dm.
???????? ?????????? ??????? E ? Dm ? Nm ? G
??? m = 2; 3; 4 ???????? ??????????:
? G/Nm?? ??????????? ?????? ?c?;
? Nm/Dm?? ??????? ?????? ????? m, ?????????? ?????????? ????????? ????????????
?????, ??????????? ????????? ?????? ?? ????????????? ?????? ?????????? f(x);
? Dm?? ??????? ?????? ????? w, ??????????
?????????? ????????? ???????????? ?????, ??????????? ????????? ?????? ?? ?????????????
?????? ?????????? k(?).
? ?????? ?????? ???????????? ?????? ?????????? ? ?????????????? ??????????????.
??? ??????? ????????????? ??? ???????? ?????????. ??????? ??????????? ? ?????????????
?????????????? ??????????????? ? ??? ??
?????.
(1)
??????????? ???????????? ??? ? ????????? f(x) = xm?? am?? 1xm?? 1?? ??? a1x?? a0 ? ???????? |a0| = 1, ??? ??? ? ??????, ???? ?????????
??????????, ?? ????????? ?????????????? ?????????, ????? ??????? ??????????? ? ????????.
???????? ?????????? ???????????? G ??????
Nm ?? ??????????? ?????? ?c? ???????????? ????????????, ????????? ?? ??????????? ????????? ????????? ????????:
2
a0 ?
?
a1 ?
1 ? 0 a2 ? .
?
? ? ?
? ?
0 ? 1 ai ? 1 ??
0 ? 0
0 ? 0
??????? ???????????? c ?? ????????? Dm
???????????? ?? ???????
[g0 , g2 ] = d m , ?, [gi , g j ] =
= d1
(2)
.
154
?. ?. ??????, ?. ?. ????????
Izvestiya of Irkutsk State Economics Academy,
2014, no. 6 (98), pp. 152?158. ISSN 1993-3541
???????? ????????? ??????????????? ????????????? ????????.
2014. ? 6 (98). ?. 152?158. ISSN 1993-3541
V. V. BLUDOV, L. E. BADMAEVA
???????? ?????????? f(x) ? ????????? ?????? ????????????? ??????, ???????? ?????? ?
????????? ?????? ????????????:
4.1. ???? ???????????? ????????? f(x) ???[g0 , g1]c = [g1, g0?1g1a1 ] = [g0 , g1]?1 = [g0 , g1].
?? ???? ????????????? ?????? ? ??? ??????????? ?????, ? ????????? k(?) ????????????, ??
??? a1 > 0 ????????? ????? ??? ??????????k(?) ????? ???? ????????????? ? ??? ?????????? ?????, ??? ???????? ? ????, ??? G/D2 ??????????? ??????????? ?????. ??? ??? ????????
????? 8 ???????????? (??.: [6, ???????а5.3.2]),
?? ??????? 1 ????????, ??? ????? ????????????
????????? D2?? ???????????, ????????? 2 ???????? 8.
????????? ?, ??????, ??? ?????? G ?????????
4.2. ???? ???????????? ????????? f(x) ???16а????????????.
??
????
????????????? ?????? ? ??? ??????????? ???????? ????????????? ?????????? ?
???
?????
? ????????? k(?) ????????????, ?? ??
????? ?????????????? ??????? ????? ?????,
?????
????
????????????? ? ??? ?????????????
2
????????, f(x) = x ?? 3x + 1.
?????.
?
????
?????? ????? ???????????? ?????
c
?1
???????, ??? ???? a0 = 1, ?? [g0, g1] = [g0, g1]
?????
8.
? ?????? ?? ????????? ???????? ????????????.
4.3. ???? ???????????? ????????? f(x) ???4. m = 3. ????????? f (x) = x3 ? a2x2 ? a1x ?
??
????? ? ???? ??????????a ???
a
?1
c
c
c
a0, a0 = ▒1, ?????????????? g0 = g1; g1 = g0 g2 ; g2 = g0 g1a g?????????????
2 .
a0 a1 a2
???
??????
?
?????????
k(?) ????????????, ??
?1
c
c
; g1 = g0 g2 ; g2 = g0 g1 g2 .
??
?????
????
?????????????
? ??? ??????? ?????? g0 , g1, g2 ???????
??????? ?????. ? ???? ?????? ????? ?????????? 0 0 a0 ?
??? ????? 16.
?
?
4.4. ???? ???????????? ????????? f(x) ?????
A = ? 1 0 a1 ? .
???
????????????? ????? ? ????????? k(?) ??????0 1 a ?
2?
?
???????, ?? ?? ????? ??? ????????????? ?????.
? ???? ?????? ????? ???????????? ????? 72.
? ????????? D3 ????????:
c
c
???????? ?????????? ???????, ????????d1 = [g0 , g1] = [g1, g2 ] = d 3 ;
???? ??? ??????????? ????????????? ??????.
? ?????? ?????? ? ???????? ????????d 2c = [g0 , g2 ]c = [g1, g0a0 g1a1g2a2 ] =
?? f(x) ????? ????? ????????? 3-? ???????
? a0
a2
? a0 a2
= [g0 , g1] [g1, g2 ] = d1 d 3 ;
f(x) =а x3 ? x ? 1. ?????? ????????? ????????
???????????? ??? ??????? ????? ?????, ??d 2c = [g2 , g0a0 g1a1g2a2 ] =
??????? ???????? x0 = ▒1 ?? ???????? ???????
= [g0 , g2 ]? a0 [g1, g2 ]? a1 = d 2? a0 d 3? a1 .
??????? ?????????.
?????? ????????? ?????????? ??????????? ?????? d1, d2, d3 ???????
??? ?????? ??????? ??????????. ?????????? ??0 ?
? 0 ?a0
??? ?????? (??.: [11; 12]). ????? f ?(x) = 3x 2 ? 1;
?
?
???? ??????????? ▒ 3 2. ?????????? ??????B = ? 0 0 ?a0 ? .
???
?????????? ???????????????? ???????
?
?
3. m = 2. ????????? f(x) = x2 ? a1x ? a0. ? ?????? a0 = ?1 ????????:
g0c = g1; g1c = g0?1g1a1 ;
0
?1
a2
?a1 ?
1
2
?
3?
?? ??; ?
?;
3 ??
?
?????? ?????????????????? ??????? C =
=а?E ? B ? ????????? ?? ????????????, ???????
?????????
?
3
3?
;
?? ?
?;
3 ??
? 3
? 3
?
; ? ?.
??
?
? 3
?
?? ?????????? (0; ?) ??????? f(x) ?????
???? ????????????? ??????, ?? ?????????
??????????? ?????????????? ?????? ???.
?а???? ????? ????????? ? ?? ??????? ???????
f(x)а=аx3 ? x ? 1 (???. 1).
?? ??????? (3) ??????? ?????????????????? ????????? ?????????????? ??????????
k(?) = ?3 + ?2?? 1, ??? k(?) ???????? ????????????
??????????? ? ????? 1 ????????????? ??????
?? ????????? (0; 1) (???. 2).
????? ????????? ???????????? ??????
2аиа2аиа2 = 8.
k(?) = ? 2 (? + a1) ? a02 + a0a2? =
(3)
????? ????????????? ?????? ??????? ?????????? ????? ?????????? ????? ?????????
???????? ???????????? ?????? D3 ??? ???????,
??? ????????? k(?) ????????????. ?????????
??????? ????? [11; 12], ????? ????????? ????????? ???????? ???????????? ?????? ??????????
f(x), ??????? ????? ??????? ??????????????
?????????? k(?).
= ? 3 + a1? 2 + a0a2? ? a02.
155
?????????????? ?????????????, ????????? ??????
Izvestiya of Irkutsk State Economics Academy,
2014, no. 6 (98), pp. 152?158. ISSN 1993-3541
???????? ????????? ??????????????? ????????????? ????????.
2014. ? 6 (98). ?. 152?158. ISSN 1993-3541
MATHEMATICAL MODELING, SYSTEMS ANALYSIS
d 2c = [g0 , g2 ]c = [g1, g3 ] = d 5 ;
Y
6
f(X)
4
d 3c = [g1, g2 ]c = [g2 , g3 ] = d 6 ;
2
d 4c = [g0 , g3 ]c = [g1, g0a0 g1a1g2a2 g3a3 ] =
?2 ?1,5 ?1 ?0,5 0
?2
0,5
1
1,5
= [g0 , g1]? a0 [g1, g2 ]a2 [g1, g3 ]a3 = d1? a0 d 3a2 d 5a3 ;
2 X
d 5c = [g1, g3 ]c = [g2 , g0a0 g1a1g2a2 g3a3 ] =
?4
= [g0 , g2 ]? a0 [g1, g2 ]? a1 [g2 , g3 ]a3 = d 2? a0 d 3? a1d 6a3 ;
?6
d 6c = [g2 , g3 ]c = [g3 , g0a0 g1a1g2a2 g3a3 ] =
?8
= [g0 , g3 ]? a0 [g1, g3 ]? a1 [g2 , g3 ]? a2 = d 4? a0 d 5? a1d 6? a2 .
???. 1. ?????? ??????? f(x) = x3 ? x ? 1
? ?????? d1, d2, d3, d4, d5, d6 ???????
Y
?0
?
?0
?1
B=?
?0
?0
??
?0
k(?)
20
15
10
5
?1,5 ?1 ?0,5
0
0,5
1
1,5
2
2,5 X
0
0
0
0
1
0
0 ?a0
0 0
0 a2
0 0
0 a3
1 0
0
?a0
?a1
0
0
a3
0 ?
?
0 ?
0 ?
?.
?a0 ?
?a1 ?
?
?a2 ??
?????? ?????????????????? ???????
C =а?E ? B ? ????????? ?? ????????????, ? ??????? ?????????
?5
???. 2. ?????? ??????? k(?) = ?3 + ?2?? 1
k(?) = ? + a2? + (a1a3 + a0 )? +
?? ?????? ?????? ? ???????? ??????????
f(x) ??????? ????????? f(x) = x3 + 4x2?? x ? 1,
+ (a0a32 ? a12 + 2a0a2 )? 3 ?
(4)
??????? ???? ????????????? ?????? ? ??? ??2
2
2
??????????? ?????.
? a0 (a1a3 + a0 )? + a0 a2? ? a0 . ?? ??????? (3) ????????? k(?) = ?3 + ?2 +
????? ????????????? ?????? ??????? ???+а4? ? 1. ???????, ??? k(?) ????????????, ?????????? ????? ?????????? ????? ?????????
?? ???? ????????????? ??????. ????? ?????????
???????? ???????????? ?????? D4, ??? ???????,
???????????? ?????? 2 и 2 и 2 = 8.
??? ????????? k(?) ????????????.
? ??????? ?????? ? ???????? ?????????? f(x)
???????? ?????????? f(x) ? ?????? ??????
??????? ????????? f(x) = x3?? 5x2 + 2x + 1, ???????????????? ??????, ???????? ?????? ? ??????? ??? ????????????? ?????. ?? ???????
?????? ?????? ????????????:
(3) ????????? k(?) = ?3 ? 2?2 ? 5??? 1 ?????????5.1. ???? ???????????? ????????? f(x) ?????
???, ????? ???? ????????????? ??????. ?????
???? ????????????? ?????? ? ??? ?????????????
????????? ???????????? ?????? 2 и 4 и 2 = 16.
????? ? ????????? k(?) ????????????, ?? ?? ???? ????????? ?????? ? ???????? ??????????
??
??? ????????????? ? ??? ????????????? ?????.
f(x) ??????? ????????? f(x) = x3?? 5x2 + 6x ? 1,
? ???? ?????? ????? ???????????? ????? 24.
??????? ??? ????????????? ?????. ?? ???5.2. ???? ???????????? ????????? f(x) ??????? (3) ????????? k(?) = ?3 ? 6?2 + 5??? 1 ????? ??? ????????????? ????? ? ??? ???????????? ????????????, ????? ??? ?????????????
??? ????? ? ????????? k(?) ????????????, ?? ??
?????. ????? ????????? ???????????? ??????
????? ??? ????????????? ? ?????? ??????????6аиа6аиа2а=а72.
??? ?????. ? ???? ?????? ????? ????????????
5. m = 4. ????????? f(x) = x4 ? a3x3 ? a2x2 ?
????? 32.
? a1x? ? a0; a0 = ▒1, ?????????????? g0c = g1; g1c = g2 ; g2c = g3 ; g3c = g0a0 g1a1g2a2 g3a3 .
5.3. ???? ???????????? ????????? f(x) ???a0 a1 a2 a3
c
c
c
g1; g1 = g2 ; g2 = g3 ; g3 = g0 g1 g2 g3 .
?? ??? ????????????? ????? ? ??? ???????????
? ????????? D4 ????????:
????? ? ????????? k(?) ????????????, ?? ??
d1c = [g0 , g1]c = [g1, g2 ] = d 3 ;
????? ??? ????????????? ? ?????? ????????6
156
5
4
?. ?. ??????, ?. ?. ????????
Izvestiya of Irkutsk State Economics Academy,
2014, no. 6 (98), pp. 152?158. ISSN 1993-3541
???????? ????????? ??????????????? ????????????? ????????.
2014. ? 6 (98). ?. 152?158. ISSN 1993-3541
V. V. BLUDOV, L. E. BADMAEVA
(mod 2) ? ?????? 3: p2(?) ? ?6 + 2?5 + 2?4 + ?3 +
+а?2 + 2? + 2 ? (?3 + 2? + 2)(?3 + 2?2 + 1) (modа3).
?????, ??? ????????? p1(?) ?????????????? ??
?????????? 1-? ? 4-? ??????? (? + 1), (?4 + ?3 +
+а?2 + ? + 1), ???????????? ??? ?2, ?????????
p2(?)?? ?? ??? ?????????? 3-? ??????? (?3 +
+а2? + 2), (?3 + 2?2 +1)?? ???????????? ??? ?3.
????????? p1(?) ? p2(?) ?????????????? ?? ?????????? ?????? ????????, ?? f(x) ?? ?????????????? ?? ?????????? ??? ?.
? ?????????? ????? ???????????? ?????
6аиа2аи 2 = 24.
?? ?????? ?????? ? ???????? ?????????? f(x)
??????? ????????? f(x) = x4 ? x3?? 11x2 + 5x + 5.
????????? ?????????? ??? ??????? ????? ?????, ??? ??? ??? ????????????? ????????????
????????? ? ?2: p(?) ? ?4 + ?3 + ?2 + ? + 1 (modа2),
??? ? ? ?????? 5.1. ???????, ??? f(x) ????? ???
????????????? ????? ? ??? ????????????? ?????. ??? ??????? ?? ????, ??? f(?4) > 0; f(?2) < 0;
f(0) > 0; f(2) < 0; f(4) > 0.
?? ??????? (4) ???????:
??? ?????. ? ???? ?????? ????? ????????????
????? ????? 32.
5.4. ???? ???????????? ????????? f(x) ????? ??? ????????????? ????? ? ???? ????????????? ?????? ? ????????? k(?) ????????????, ??
?? ????? ??? ????????????? ? ??? ????????????? ?????. ? ???? ?????? ????? ????????????
????? 72.
5.5. ???? ???????????? ????????? f(x) ????? ?????? ????????????? ????? ? ????????? k(?)
????????????, ?? ?? ????? ????? ????????????? ??????. ? ???? ?????? ????? ????????????
????? 192.
???????? ?????????? ??????? ??? ???????
5.1 ? 5.2.
? ?????? ?????? ? ???????? ?????????? f(x)
??????? ????????? f(x) = x4 + 11x3 + 7x2?? 7x ? 1.
????????? ?????????? ??? ??????? ????? ?????, ??? ??? ??? ????????????? ????????????
????????? ? ?2: p(?) ? ?4 + ?3 + ?2 + ? + 1 (modа2)
[8; 9, ????. ???????????? ???????????]. ???????, ??? f(x) ????? ???? ????????????? ? ???
????????????? ?????. ??? ??????? ?? ????, ???
f(?11) > 0; f(?2) < 0, f(?1) > 0; f(0) < 0; f(1) > 0.
?? ??????? (4) ??????? ?????????
k(?) =а?6 ? 7?5 ? 76?4 + 58?3 + 76?2 ? 7? ? 1.
?????? ????????? ????? ??? ?????????????
? ??? ????????????? ?????, ??? ??? k(?7) > 0;
k(?4) < 0; k(?0, 5) > 0; k(0) < 0; k(1) > 0; k(2)а<а0;
k(13) > 0. ???????? ??? ?? ?????????????? ???
??????? ????? ?????. ?????????? ?? ?????? 2:
p1(?) ? ?6 + ?5 + ? + 1 = (? + 1)2(?4а+ ?3 + ?2 + ? + 1)
k(?) = ?6 + 11?5 ? 10?4 ? 140?3 ? 50?2 + 275? + 125.
????????? k(?) ????? ??? ?????????????
? ?????? ????????????? ?????, ?????????
k(?11) > 0; k(?3) < 0; k(?2) > 0; k(?1) < 0;
k(0) > 0; k(2) < 0; k(4) > 0.
????????? k(?) ?????????? ??? ??????? ????? ?????, ??? ??? ??? ?????????? ?? ??????? 2
? 3 ????????? ? ???????? 5.1. ????? ?????????
???????????? 4 и 4 и 2 = 32.
?????? ?????????????? ??????????
1. ?????? ?. ?. ??????, ??????????????? ???????????? ???????? / ?. ?. ?????? // ??????? ? ??????.?? 1974.??
?. 13, ? 6.?? ?. 609?634.
2. ?????? ?. ?. ? ?????????? ???????????? ???????? ?????????? ?????????? ??????????????? ????? / ?. ?. ??????, ?. ?. ???????, ?. ?. ??????? // ??????? ? ??????.?? 2009.?? ?. 48, ? 3.?? ?. 291?308.
3. ????? ?. ?. ?. ?????????????? ?-?????? / ?. ?. ?. ?????, ?. ?. ???????? // ??????? ? ??????.?? 2006.?? ?.а45,
? 1.?? ?. 20?27.
4. ????????? ?. ?. ? ???????? ???????????? ??????? ??????????? ????????????? ????? / ?. ?. ????????? //
???????, ?????? ? ?????????? : ??.?? ??????? : ???-?? ???, 1994.?? ?. 22?28.
5. ??????? ?. ?. ? ??????? ????????????? ?????????? ??????? / ?. ?. ??????? // ??????? ? ??????.?? 1973.??
?. 12, ? 6.?? ?. 655?666.
6. ??????? ?. ?. ?????????????????? ?????? / ?. ?. ???????, ?. ?. ????????.?? ??????????? : ????. ??.,
1996.?? 256 ?.
7. ????? ?. ?. ?????? ????? / ?. ?. ?????.?? ?. : ?????????, 2011.?? 805 ?.
8. ???? ?. ???????? ???? / ?. ????, ?. ???????????.?? ?. : ???, 1988.?? 430 c.
9. ???? ?. ?????????? ??????????? ??????? / ?. ????, ?. ?????.?? ???????????? : ???-?? ????. ??-??,
1996.?? 743 c.
10. ???????? ?. ?. ?????? ? ???????? ?????? ???????? ???????? / ?. ?. ???????? // ??????? ? ??????.??
1999.?? ?. 38, ? 2.?? ?. 176?200.
11. ???????? ?. ?. ?????????? / ?. ?. ????????.?? ?. : ?????, 2001.?? 336 c.
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13. Dlab V. On a family of simple ordered groups / V. Dlab // J. Aus. Math. ?1968.?? Soc. 8, ? 3.?? P. 591?608.
14. Iwasawa K. On linearly ordered groups / K. Iwasawa // J. Math. Soc. Japan.?? 1948.?? Vol. 1, ? 1.?? P. 1?9.
157
?????????????? ?????????????, ????????? ??????
Izvestiya of Irkutsk State Economics Academy,
2014, no. 6 (98), pp. 152?158. ISSN 1993-3541
???????? ????????? ??????????????? ????????????? ????????.
2014. ? 6 (98). ?. 152?158. ISSN 1993-3541
MATHEMATICAL MODELING, SYSTEMS ANALYSIS
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?????????? ?? ???????
Authors
?????? ??????? ???????????? ?????? ??????-?????????????? ????, ?????????, ???????? ???????? ?????? ? ?????? ?????????? ?? ???, 664054, ?. ???????,
??.а??????????, 134, e-mail: vasily-bludov@yandex.ru.
???????? ??????? ?????????? ? ????????,
??????? ??????????, ?????????? ? ????????????,
??????????? ??????????????? ??????????? ????????? ? ?????, 664003, ?. ???????, ??. ??????, 11, e-mail:
lb-lotus@rambler.ru.
Vasily V. Bludov ? Doctor habil. (Physical and Mathematical Sciences), Professor, Institute for System Dynamics
and Control Theory of Siberian Branch of Russian Academy
of Sciences, 134 Lermontov St., Irkutsk, 664054, Russian
Federation, e-mail: vasily-bludov@yandex.ru.
Luydmila E. Badmaeva ? PhD student, Department of
Mathematics, Econometrics and Statistics, Baikal State University of Economics and Law, 11 Lenin St., Irkutsk, 664003,
Russian Federation, e-mail: lb-lotus@rambler.ru.
????????????????? ???????? ??????
Reference to article
?????? ?. ?. ? ??????? ?????????? ??????????????? ?????????? ????? ? ???????? ?????? ????????????а / ?. ?. ??????, ?. ?. ???????? // ???????? ????????? ??????????????? ?????????????
????????.?? 2014.?? ? 6 (98).?? ?. 152?158.?? DOI:
10.17150/1993-3541.2014.24(6).152-158.
Bludov V. V., Badmaeva L. E. On the method constructing of orderable soluble groups with finitely many
orderings. Izvestiya Irkutskoy gosudarstvennoy ekonomicheskoy akademii?= Izvestiya of Irkutsk State Economics
Academy, 2014, no. 6 (98), pp. 152?158. (In Russian). DOI:
10.17150/1993-3541.2014.24(6).152-158.
158
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