References
1. Kalashnikov, V. V. Consistent conjectures in mixed oligopoly / V. V. Kalashnikov, V. A. Bulavsky, N. I. Kalashnykova, F. J. Castillo Perez // European J. Oper. Res. 2011. 3(210). . 729-735.
2. Kalashnykova, N. I. Consistent conjectural variationse-quilibriu minamixed duopoly oligopoly / N. I. Kalashnykova, V. V. Kalashnikov, V. A. Bulavsky, F. J. Castillo Perez // Journal of Advanced Computational Intelligence and Intelligent Informatics. - 2011. - 15(4). - . 425-432.
3. , . . / . . // . - 1997. - 33(3). - . 112-134.
4. Liu, Y. F. Existence and uniqueness of consistent conjectural variation equilibrium in electricity markets / Y. F. Liu, Y. X. Ni, F. F. Wu, B. Cai // Int. J. Electrical Power Energy Sys. - 2007. - 29 (4). - . 455-461.
KALASHNIKOV Vyacheslav Ivanovich, doctor of physico-mathematical Sciences, Professor of the Department of Systems Engineering, Instituto Tecnologico y de Estudios Superiores de Monterrey (on leave from the Central Economics and Mathematics Institute (CEMI), Russian Academy of Sciences, Moscow). BULAVSKY Vladimir Aleksandrovich, doctor of physico-mathematical Sciences, Professor, Russia Professor of the Central Economics and Mathematics Institute (CEMI) of the Russian Academy of Sciences. KALASHNIKOVA Natalia Ivanovna, candidate of physico-mathematical Sciences, teacher of the Department of Mathematics, Universidad Autonoma de Nuevo Leon, San Nicolas de los Garza.
: kalash@itesm.mx
10.10.2012 .
V. I. Kalashnikov, V. A. Bulavsky, N. I. Kalashnikova
517.958:530.145 . .
. . . .



.
: , , , --, -, , . , 12-01-31400.
.
(,)() = '() , (1)
&^!1, ()(), .
(1) [1]. , (1) - , [2]. , [3]. , -
(, - ), . , .
, H(x, ) g : ^ (x)d:.
1 = 0, ['] = ,
= - g; a, b, c = 1, ..., dim g. (1) . (1) , , . , , -
1 (117) 2013 -
- 1 (117) 2013
36
.
, , dim g = n ^'() = . g G, - UcRn.
2. ^- . G , g , g* g , OX=G/GX - G, g*, GX X. nc gc ,
dimn = dimg-^dimC\, (,[,]) = 0,
X. [4] , X, .. . , 1 .
dim - dim g-indg, dimn = (dim g + ind g)/ 2 ,
ind g , . g
indg = inf dimgx, = Lie(Gx).
A.eg
^x=(Lg)*X, nx=-(^g)*X, - G, Xeg*. G g IX'-UX, IY]=I[XY], -. Q OX. {qa} Q - :
1=&-^+{). (0) = . dq
T 1- G.
( )() = { ( Mg '),
^() = ^<?>8(,')
-Dgg'(gr) ї TX, S(g,g') -, ^() Q, Xeg. Dqq'(g) :
, (. '1)={Dq,q ()()(),
(9)=|,'()(-')^()('^). (2)
d^(X) l- g.
[5]. (2) (-) ( l- . nx dim g , , IX(q; X), (dim g-ind g)/2. (2) , - , .
, H(x, dX), G, (), , .. , () , (2).
1. (), G, , , , , . gL(G)sg, . () . (dim g-ind g)/2 , , , ,
(dim g-ind g)/2<2. (3)
, .
3. - . G .
A=GAXV (4)
G g G, {eA} g, {eA} g*, nA=-(-Rg)* eA , GAB = G(eA, eB). , (4) : [A,y = 0.
G -:
(+2)() = 0, (5)
m "^). (5) (3).
ї , . FZ2(g; R) 2- g, R. 2- F :
F(\, ny) = F(X,Y), X,Yeg. F F=dA, A 1-, . 1- A g*- AeC(G) , A(nx)= A(X) Xeg. A 2- F :
nxA(Y)-nA(X)-A([X,Y])=F(X,Y). (6)
ї - (5) \ >|^ =\ _1(), :
( +m2)tp(g) = 0, (7)

() =GABT1(e)11(e) =gab(t1a -1 -ieAB). (8)
(7) , , .
(7) , (8), :
~^+ ie(AdgA(X) + ()), (9)
g*- ^eC^(G) : X(X) = -\Fx ") [6], (9) g, 2- FeZ2(g; R).
(7) , 2, . . - G. 1- 1- g.
(7). [6]
indrF1g= inf dimkerF,
1 J Fe[F]
, . g [F]eH2(g; R).
2. . G 2- F, 2- FeZ2(g; R). (7) , . , , G , ,
(dim g-ind[F]g)/2<2.
, . ind[F] g = ind g, . , (3) (7).
4. .
- ( ) [7]. , , .
M = R^G - ds= dt2-a2(t)dI2, dI2 G, A = d? -a~2(t)Ac M, AG G. ,
Ac=GABrlAr]B=GAB^B,
Gab =tr(adeAad0B).
(^) M -
(+2)((,) = 0. (10)
(0)
(9.) = !'|(w-9v)dn(Sr),
d^(g) G.
(2), (0):
(Pa(ff) = &x(t)Dqq'(Sr)' G = [q,q',X) , (11)
X(t)
0x(f) + (a-2(f)K(X)+/n2)xW = O
: (_ )= 1. k(X)=- GAB IA(q; X) IB(q; X) ( q).
^F(t,g) (0), ['P(f,g)1
%.?) = /((^?)+(^?)^)(). (12)
a (), [aa,a++'] = 5(<j,a').
(2), :
( | %,<;)%',< |o)=J '-1 ),
= |j( 0x(f) I2 +2(01 &x(t) |2)xX(e)d^(X),

XX{g) = JDgq(g)d\i(q), = a_2(f)K(X) + m2.
, .. a = const, i(t) = exp(-iQ^t)^2- ,
1 (117) 2013 -
- 1 (117) 2013
38
0|%,)%',')|0
2,
xNffff-VuM, 8 = ^JnxNe)dn(X)
5. .
zp = tr(exp(-p ())),
(13)
A(x) ( M), p=1IkT . (1) A(x):
ZP=ZdnexP(-P^)-

dn En A(x). (heat kernel)
(, ) = () (- ()) () =
= ' (*) ' (-(3),

() (). , :
(*.*

+ ()(,') = 0,

(>) |=0- 5(,'),
zp=Jpp(x-xW-
. (14) . .
- G (14), = [-4 . [8]
VolG , G .
2
=^ = |(,;,()(,).
∞1
, , .. ,
= -&
2 S lnZo
2
-ip
2 <32lnzn
2
. . VolG 1
Zp=ln zp+ln VolG'

1. , . / . . - . : , 1981. - 336 .
2. , . . / . . // . . - 1980. - . 16. - 10. -. 1864- 1874.
3. Bagrov, V. G. Exact solutions of relativistic wave equations / V. G. Bagrov, D. M. Gitman. - Kluwer, 1990. - 331 p.
4. , . / . . - . : , 1978. - 407 .
5. , . . / . . , . . // . -1995. - . 104. - 2. - . 195-213.
6. , . . / . . , . . , . . // . -2008. - . 156. - . 2. - . 189-206.
7. , . . / . . , . . , . . // . - 2011. - . 167. - . 2. - . 78-95.
8. Mikheyev, V. V. Application of Coadjoint Orbits in the Thermodynamics of Non-Compact Manifolds / V. V. Mikheyev, I. V. Shirokov // Electronic Journal of Theoretical Physics. - 2005. -V. 7. - P. 1-10.
pptff.g'HjKptg-g'^Pgg'tg' V)
xdn(g)dn(qr')dn(X),
(15)
Rp(<7,g ;1)
fllfefog ;) +{_1.)^.) = 0|
(.,;*')1==5(<'')- (16)
, (14) (16) Q dim Q=(dim g-ind g) = 2.
(15),
zp = JPptff.S^Sr) = VolG Jflp(grg;X)d|j.(g)d|j.(a.).
, - , ї.
: magazev@mail.ru , , ї.
: vvm125@mail.ru , , (), ї. : iv_shirokov@mail.ru
14.09.2012 .
. . , . . , . .