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Abstract
B. V. Novikov, Semigroup cohomologies: a survey, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 1, pp. 1{18.
A survey of research in semigroup cohomologies and their applications is given.
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.
1] . . . .: , 1987. 84:1A352]
2] !" #. . $%"& // (). * +$. | 1964. | .. 33,
/ 2. | . 263{269. 65:1A207]
3] !" #. . $%"& "&23 // (). * +$. |
1965. | .. 39, / 1. | . 3{10. 65:11A225]
4] !" #. . $%"& " // (). * +$. |
1966. | .. 41, / 3. | . 513{520. 66:7A240]
5] !" #. . $%"& 44523 5"23 // ().
* +$. | 1967. | .. 46, / 1. | . 11{18. 67:10A134]
6] !" #. . "42" 52 6"7 46"7 "(2 3 ""& // .. .(. 4". -4 * +$. | 1975. | .. 48.
76:8A504]
7] 4 *., 97"(" . +6"& "(. | .: :, 1960. 61:2A238]
8] 4 !" $. ;. < &3 5 // (). * +$. | 1985. |
.. 117, / 3. | . 465{468. 86:1A492]
9] == *., >"4 +. *"(6"& 4"& . .. 1, 2. | .:
, 1972. 64:10A191, 68:10A123]
10] >. ;. >2 & // ?6. !. :+> . +"@". |
1971. | .. 404. | . 275{284. 71:12A163]
11] >. ;. <( !=!" 7 // 5""& "(.
B2. V. | :., 1976. | . 71{74. 77:10A249]
12] >. ;. +6"& 6"& 34"4 (()"
" 6"3 // !5. 5!5. 4". | 1982. | / 5. | . 30{34.
82:9A340]
13] " .". < %" 5 // (). * +$. |
1976. | .. 83, / 1. | . 25{28. 77:5A279]
14] " .". < &3 5 // (). * +$. | 1977. |
.. 85, / 3. | . 545{548. 77:12A424]
15] " .". < "423 =423 57453 5 // (). * +$. | 1977. | .. 87, / 2. | . 281{284. 78:6A416]
16] 55 . B. < 0-&3 // .". . 5. ==. -7
"(. | "5: 5 , 1978. | . 185{188. 79:6A368]
17] 55 . B. < "4523 " 45"&3 // ;. * ?$,
". *. | 1979. | / 6. | . 474{478. 79:11A161]
18] 55 . B. 0- 5" 0-423 // B"4 #C.
. -4. | 1981. | B2. 46, / 221. | . 80{85. 82:6A356]
19] 55 . B. < 526" 7 "423 // B"4
#C. . -4. | 1981. | B2. 46, / 221. | . 96. 82:6A357]
: 15
20] 55 . B. 4" 7 4"!" 46" // .. .(. 4. -4
* +$. | 1982. | .. 70. | . 52{55. 83:5A341]
21] 55 . B. <" "&D)" 4%"& 0- 7 // ."& "" E. > 6. 2. >2 "(!57. | 45: ! -5 45. -4, 1983. | . 94{99. 83:11A213]
22] 55 . B. F462" 3 E"& // !5. 5!5. 4". | 1988. | / 11. | . 25{32. 89:6A317]
23] 55 . B. 4452" 2 )"" !"4 1 //
4". !"4. | 1990. | .. 48, / 1. | . 148{149. 90:12A363]
24] 55 . B. $"="452" 4" // ;5i i * ?GH. | 1994. | / 8. | . 10{12.
25] 55 . B. < " I // 4". !"4. | 1995. | .. 57, / 4. |
. 633{636. MR 96f:12005]
26] 55 . B. < 3 6"7 !"4 1 // ;5i i *
?GH. | 1996. | / 8. | . 6{8.
27] 55 . B. <( ("7 4"!" 46" // ;5i i * ?GH. |
1998. | / 3. | . 26{27.
28] <) :. *. "42" &4& ""& 4" "(""523 7 // .. . 4. (-5. | 1967. | .. 17. | . 45{88. 68:12A270]
29] >6& *. . < %"&3 "7 // (). * +$. | 1976. |
.. 84, / 3. | . 545{548. 77:10A248]
30] >6& *. . $%"& "7 ) 5 3 6"& 34"4 // (). * +$. | 1977. | .. 86, / 1. |
. 21{24. 78:1A372]
31] >6& *. . & 5 I==@"4 5 &3 //
(). * +$. | 1977. | .. 86, / 3. | . 546{548. 78:3A277]
32] >6& *. . C2" %7"52 %"& "7 // .. .(.
4". -4 * +$. | 1979. | .. 62. | . 76{90. 80:1A430]
33] >6& *. . < !5 23 =43 =45 !6"& 5 4" "7 // .. .(. 4". -4 * +$. | 1986. | .. 83. |
. 60{75. 87:4A437]
34] >6& *. . < 3 // .. .(. 4". -4 *
+$. | 1988. | .. 91. | . 36{43. 89:7A308]
35] J" . K., KC"7=" L. +. <52 4" 4"7. | .: ,
1974. 75:3A379]
36] K&"5 B. . <( 5"23 3 2 G- // ;. *
$. | 1970. | .. 14, / 9. | . 782{785. 71:4A142]
37] K&"5 B. . <( %" 5 " 5"23 // B"4
+?, cep. 1. | 1971. | / 1 (7). | . 15{21. 71:7A185]
38] Adams W. W., RiePel M. A. Adjoint functors and derived functors with an application to the cohomology of semigroups // J. Algebra. | 1967. | Vol. 7, no. 1. |
P. 25{34. 68:4A266]
39] AguadTe J. Cohomology of binary systems // Arch. math. | 1981. | Vol. 36, no. 5. |
P. 434{444. 82:1A486]
16
. . 40] Barr M., Beck J. Homology and standard construction // Lect. Notes in Math. |
1969. | Vol. 80. | P. 245{335. 72:7A312]
41] Barrat M. G. A note on the cohomology of semigroups // J. London Math. Soc. |
1961. | Vol. 36. | P. 496{498. 62:8A228]
42] Bernstein N. On the cohomology of semigroups // Dissert. Abstrs. | 1965. | Vol. 25,
no. 1. | P. 6644{6645. 66:7A309]
43] Bernstein N. Standard cohomology of semigroups // Portugal. Math. | 1973. |
Vol. 32, P. 21{23. MR 1974, vol. 47, N6813.]
44] Carbonne P. Cohomologie des semi-groupes d'entiers // J. Algebra. | 1983. |
Vol. 84, no. 1. | P. 1{13. 84:4A396]
45] Cheng Charles Ching-an, Shapiro J. Cohomological dimension of an abelian
monoid // Proc. Amer. Math. Soc. | 1980. | Vol. 80, no. 4. | P. 547{551.
81:7A377]
46] Clark W. E. Cohomology of semigroups via topology with an application to semigroup algebras // Commun. Algebra. | 1976. | Vol. 4. | P. 979{997. 77:7A387]
47] Cohen D. E. A monoid which is right FP1 but not left FP1 // Bull. London Math.
Soc. | 1992. | Vol. 24, no. 4. | P. 340{342. 93:4A137]
48] Davidson T. M. K., Ebanks B. R. Cocycles on cancellative semigroups // Publ. Math.
Debrecen. | 1995. | Vol. 46, no. 1{2. | P. 137{147. MR 96c:20116]
49] Eilenberg S., MacLane S. Homology theories for multiplicative systems // Trans.
Amer. Math. Soc. | 1951. | Vol. 71, no. 2. | P. 294{330. MR 1952, vol. 13,
p. 314.]
50] Grassmann H. On Schreier extention of |nite semigroups // Algebraic theory of
semigroups. | Amsterdam e.a., 1979. | P. 219{224. 80:4A142]
51] Grillet P. A. Building semigroups from groups (and reduced semigroups) // Semigroup Forum. | 1972. | Vol. 4, no. 4. | P. 327{334. 73:2A151]
52] Grillet P. A. Left coset extensions // Semigroup Forum. | 1974. | Vol. 7. |
P. 200{263. 74:9A182]
53] Grillet P. A. Commutative semigroup cohomology // Semigroup Forum. | 1991. |
Vol. 43, no. 2. | P. 247{252. 92:2A313]
54] Haile D. E. On crossed product algebras arising from weak cocycles // J. Algebra. |
1982. | Vol. 74. | P. 270{279. 82:9A334]
55] Haile D. E. The Brauer monoid of a |eld // J. Algebra. | 1983. | Vol. 81, no. 2. |
P. 521{539. 83:11A474]
56] Haile D. E., Larson R. G., Sweedler M. E. A new invariant for C over R: almost
invertible cohomology theory and the classi|cation of idempotent cohomology classes
and algebras by partially ordered sets with Galois group action // Amer. J. Math. |
1983. | Vol. 105, no. 3. | P. 689{814. 84:1A350]
57] Hancock V. R. Commutative Schreier semigroup extensions of a group // Acta Sci.
Math. | 1964. | Vol. 25, no. 2. | P. 129{134. 65:2A312]
58] Hurwitz C. M. On the homotopy theory of monoids // J. Austral. Math. Soc.,
ser. A. | 1989. | Vol. 47, no. 2. | P. 171{185. 91:1A627]
59] Kobayashi Y. Complete rewriting systems and homology of monoid algebras //
J. Pure and Appl. Algebra. | 1990. | Vol. 65, no. 3. | P. 263{275. 91:4A450]
: 17
60] Lafon Y. A new |nitness condition for monoids presented by complete rewriting
systems (after C. C. Squier) // J. Pure and Appl. Algebra. | 1995. | Vol. 98,
no. 3. | P. 229{244. MR 96c:20107]
61] Lausch H. Relative cohomology of groups // Lect. Notes Math. | 1977. |
Vol. 573. | P. 66{72. 77:12A423]
62] Lausch H. Cohomology of inverse semigroups // J. Algebra. | 1975. | Vol. 35,
no. 1{3. | P. 273{303. 76:1A430]
63] Leech J. E. Two papers: H -coextensions of monoids and The structure of a band of
groups // Mem. Amer. Math. Soc. | 1975. | Vol. 157. 75:10A172]
64] Leech J. E. Extending group by monoids // J. Algebra. | 1982. | Vol. 74, no. 1. |
P. 1{19. 82:9A335]
65] Leech J. E. Cohomology theory for monoid congruences // Houston J. Math. |
1985. | Vol. 11, no. 2. | P. 207{223. 86:1A491]
66] Loganathan M. Cohomology of inverse semigroups // J. Algebra. | 1981. | Vol. 70,
no. 2. | P. 375{393. 82:1A483]
67] Loganathan M. Idempotent-separating extensions of regular semigroups with Abelian
kernel // J. Austral. Math. Soc., ser. A. | 1982. | Vol. 32. | P. 104{113.
82:10A132]
68] Loganathan M. Cohomology and extensions of regular semigroups // J. Austral.
Math. Soc., ser. A. | 1983. | Vol. 35, no. 2. | P. 178{193. 84:3A484]
69] MacLane S. Origins of the cohomology of groups // Monogr. Enseign. Math. |
1978. | No. 26. | P. 191{219. 79:4A439]
70] McDuP D. On the classifying spaces of discrete monoids // Topology. | 1979. |
Vol. 18, no. 4. | P. 313{320. 80:6A572]
71] Mitchell B. On the dimension of objects and categories. I. Monoids // J. Algebra. |
1968. | Vol. 9, no. 3. | P. 314{340. 69:5A311]
72] Nguyen Huu Khang. Schreiersche Erweiterungen von Halbgruppen // Seminarber.
Humboldt-Univ. Berlin: Sek. Math. | 1981. | No. 34. 82:3A139]
73] Nguyen Huu Khang. Schreier extensions of semigroups // Math. Nachr. | 1983. |
Vol. 113. | P. 191{207. 84:6A125]
74] Nguyen Xuan Tuyen. On some exact sequences of cohomology of monoids // Bull.
Acad. pol. sci. Ser. sci. math. | 1979{1980. | Vol. 27, no.. 7{8. | P. 521{523.
80:11A404]
75] Nico W. R. On the cohomology of |nite semigroups // Dissert. Abstrs. | 1967. |
Vol. B28, no. 1. | P. 265. 68:6A341]
76] Nico W. R. On the cohomology of |nite semigroups // J. Algebra. | 1969. | Vol. 11,
no. 4. | P. 598{612. 69:9A279]
77] Nico W. R. Homological dimension in semigroup algebras // J. Algebra. | 1971. |
Vol. 18, no. 3. | P. 404{413. 72:4A440]
78] Nico W. R. A counterexample in the cohomology of monoids // Semigroup Forum. |
1972. | Vol. 4. | P. 93{94. 72:8A459]
79] Nico W. R. An improved upper bound for global dimension in semigroup algebras //
Proc. Amer. Math. Soc. | 1972. | Vol. 35. | P. 34{36. 73:5A404]
80] Nico W. R. A property of projective ideals in semigroup algebras // Proc. Amer.
Math. Soc. | 1973. | Vol. 37, no. 2. | P. 407{410. 73:10A244]
18
. . 81] Nico W. R. Wreath products and extension // Houston J. Math. | 1983. | Vol. 9. |
P. 71{99. 83:11A299]
82] Novikov B. V. On partial cohomologies of semigroups // Semigroup Forum. |
1984. | Vol. 28, no. 1{3. | P. 355{364. 84:8A347]
83] Novikov B. V. On modi|cation of the Galois group // Filomat (Ni„s). | 1995. |
Vol. 9, no. 3. | P. 867{872.
84] Redei L. Die Verallgemeinerung der Schreierschen Erweiterungstheorie // Acta Sci.
Math. Szeged. | 1952. | Vol. 14. | P. 252{273. MR 1953, vol. 14, p. 614.]
85] RiePel M. A. A characterization of the group algebra of |nite groups // Pacif. J.
Math. | 1966. | Vol. 16, no. 2. | P. 347{363. 67:3A149]
86] Rodrigez G. Sull'ampliamento di Schreier dei semigruppi // Rend. Ist. Lombardo.
Acad. sci. e lett. | 1971. | Vol. A105, no. 5. | P. 859{883. 72:11A112]
87] Sribala S. Cohomology and extension of inverse semigroup // J. Algebra. | 1977. |
Vol. 47, no. 1. | P. 1{17. 78:3A275]
88] Stallings J. R. The cohomology of pregroups // Lect. Notes Math. | 1973. |
Vol. 319. | P. 169{182. 74:1A396]
89] Strecker R. Verallgemeinerte Schreiersche Halbgruppenerweiterungen // Monnats.
Dtsch. Akad. Wiss. Berlin. | 1969. | B. 11, N. 5{6. | S. 325{328. 70:4A201]
90] Strecker R. UG ber kommutative Schreiersche Halbgruppenerweiterungen // Acta
Math. Acad. Sci. Hung. | 1972. | B. 28, N. 1{2. | S. 33{44. 73:6A187]
91] Sweedler M. E. Weak cohomology // Contemp. Math. | 1982. | Vol. 13. |
P. 109{119. 83:7A359]
92] Szendrei M. B. On an extension of semigroups // Acta Sci. Math. | 1977. | Vol. 39,
no. 3{4. | P. 367{389. 78:7A218]
93] Takahashi M. Extensions of semimodules. I // Math. Semin. Notes, Kobe Univ. |
1982. | Vol. 10. | P. 563{592. 83:12A321]
94] Takahashi M. Extensions of semimodules. II // Math. Semin. Notes, Kobe Univ. |
1983. | Vol. 11. | P. 83{118. 84:9A250]
95] Wallace A. D. The structure of topological semigroups // Bull. Amer. Math. Soc. |
1955. | Vol. 61, no. 2. | P. 95{112. 57:7672]
96] Wells Ch. Extension theories for monoids // Semigroup Forum. | 1978. | Vol. 16,
no. 1. | P. 13{35. 79:1A169]
& ' 1997 .
. . 515.123
: , .
! , # . #
!
$
! ! % # . , &' ' (
' !, $
)
# !. * ) ' !
.
Abstract
B. G. Averbuh, On transformations of families of pseudometrics, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 1, pp. 19{32.
Arbitrary transformations of a family of pseudometrics are studied which result
in uniform continuous pseudometrics again. Then their properties are applied for
a descriptionof a quotient uniformstructure by means of pseudometrics on the initial
space. It is proved that if a uniform structure on this space is subinvariant with
respect to some given set of its transformations then the quotient uniform structure
is subinvariant with respect to the induced transformations. The minimization for
a given family of pseudometrics is considered in the last section.
1. S = f g 2A | X . A Q R+, ! R+ | ! !S X X , ! !S (x1 x2) = !(f (x1 x2)g ), $ x1 x2 2 X . % &' ( &( , !, &' !S &' X , '
, !
S . ) !,
! !, '! &. *$ &' & +1] ,
$ S .
, 2001, 7, 1 1, . 19{32.
c 2001 ,
!" #
20
. . 0 ! ( &
, ' '( !: | !& $ X , ' , & !, $ ! X X
!S > .
, X 2 R.
% &' $ , ! (
S Y = X=R, ! !
$ UNIF ! . 3 &'.
! ' ! .
F | &
X X , (!( 2 R,
X &' , &'( F . 4$ Y & 2! &
Y Y , F .
5 ( &
, !$ ! .
% ' ! : !& S S 0 , ! ! , S 0 $ & '( $
.
1. % 6 ! (
&
.
A | . 2 A R+ R+, $ ''
! ,
jx1 ; x2j. 7& RA+ Q
R+ , &6 ((
) 2A
. 0 r80 = fr0 g r800 = fr00 g RA+ & r80 6 r800 ,
r0 6 r00 ( 9 ' r81 r82 r83 2 RA+ & ' &! $, '
( ( ' ( $( (
). 0 $ &( , &' $ R+
&' ( !( .
1. %! ! !, ! RA
+, & ' & A, ! :
: 1) !(8r ) > 0 !&$ r8 2 RA+, !(80) = 0, $ 80 | RA+ ' 9
: 2) ! ' 809
: 3) ' r81 r82 r83 &! $ RA+, !(8ri ) 6
6 !(8rj )+ !(8rk ), $ i j k 2 f19 29 3g, . . ! ( (
21
&! $ R+.
: & 6 !, !&'(
r81 r82 2 RA+ r81 6 r82 !(8r1 ) 6 !(8r2 ). 0 !( ( &
: 3) &!,
&' !(8r1 + r82) 6 !(8r1 ) + !(8r2) &' !&'(
r81 r82 2 RA+. 4 '! &'.
. 1. A $ . 4$ & !( :
(
!c(x) = x 0 6 x 6 c $ c 2 R+9
c x > c
!(x) = xp $ x 2 R+ 0 < p 6 19
8
x 0 6 x 6 2
>
>
<4 ; x 2 < x 6 3
!(x) = >
>x ; 2 3 < x 6 4
:
2 x > 4:
', & & & '. % '
R+ &, : 1) : 2).
2. A f g 2A | '( '( . 4$ P!&$ c > 0
& !(r) = !c (r ), $
r8 = fr g 2 RA+.
%' ( &
.
: 4) A RA+.
. 0 $ " > 0 & U ' RA+, $ (80 8h) 2 U !(8h) < "2 , $ h8 2 RA+. : , U 6 ' jr0 ; r00 j < , $
> 0. (8r0 r800) 2 U . 7& r8 r = min(r0 r00), 2 A. 4$ r8 6 r80 r800 ! ' 8h0 8h00 2 RA+, r8 + 8h0 = r80 , r8 + h8 00 = r800, 6 (8r 8h0 r80) (8r 8h00 r800) & $. 7, (80 8h0) (80 8h00) 2 U , !(8h0 ) !(8h00) < "2 . > : 3) !(8r ) ; !(8h0 ) 6
6 !(8r0 ) 6 !(8r) + !(8h0 ), !(8r ) ; !(8h00) 6 !(8r00 ) 6 !(8r) + !(8h00 ), j!(8r0) ; !(8r00 )j < ".
: 5) A0 | A. 0 : RA+ ! R+A i0 : RA+ ! RA+ . !
! | 0
0
22
. . R
A, !0 = !ji ( A+ ) | A0 . !
!0 | A0 , !0 0 | A.
@ $, & ! A 2 A, A0 = A n fg ! = !0 0. > : 4) , & & 6$ A.
! .
#
! A $ 2 A, !ji ( +) %, i | R+ R+.
. A ! , !ji( +) 0, & i (R+ ) 0 : RA+ ! RA+ , $ A0 = A nfg,
80. 0 & . r8 2 RA+ | '
, r80 = (i0 0 )(8r) r80 = (i )(8r), $ | RA+ R+ .
4$ r8 = r80 + r80 r80 r80 6 r8, (8r r80 r80) & $.
B, !(8r0 ) ; !(8r0 ) 6 !(8r) 6 !(8r0 ) + !(8r0 ) !(8r) = !(8r0 ).
: 6) &
. !
$ %, .
. ' !( '( fA g2B , f! g2B ( &
,
6 !& 2 B ! A , S
& ! B . A A = A ,
'
''(
Q
RA+ = RA+ . &
! : RA+ ! R+ Q
& RB+, !^ = ! ! , &!
RA+ R+. , !^ & A. %' : 1) : 2) .
5 (8r1 r82 r83) RA+, &!! $.
6 '
RA+ , 2 B , (8r1 r82 r83 ), &! $. '(
(! (8r1 ) ! (8r2 ) ! (8r3 )) (f! (8r1 )g f! (8r2 )g f! (8r3 )g )
RB+ & . & ! $ : 3), !^ .
r81 r82 2 RA+ r81 6 r82. A r8i = fr8i g , $ i = 1 2, r8i 2 RA+ , r81 6 r82 , , ! (8r1 ) 6 ! (8r2 ) ( 2 B .
0
0
R
R
0
23
B, f! (8r1 )g 6 f! (8r2 )g RB+, !^ (8r1) = !(f! (8r1 )g ) 6
6 !(f! (8r2 )g ) = !^ (8r2 ).
: 7) r8 | $ RA+, r80 | $,
r80 6 28r. % ! A !(8r0 ) 6 2!(8r).
. 4
(8r0 r8 r8) & $. 76 : 3).
: 8) 5 ' ( &
R+.
! | &, " > 0 h > 0 | , !(h) < ".
4$ !(x) < 2" x 2 +0 2h]. 7! : h > 0, !(h) = 0, !(x) 0 R+. A, , !(x) !, "0 > 0, ( &2( x !(x) > "0 . 0 < " < "0 fx : !(x) < "g $. ' ! &2 x, $ !(x) = "9 &
$ ("). ) (") ! ", !.
: 9) !
! A ' r8 2 RA+, %' , %.
. ( ' '2 &, r8 =
= (i )(8r ). r8 6= 0 r8 6 r8, : 5) : 7)
!, !ji( +) !, . . ! $ A. J : 1).
X | S = f g | '( X , ' A.
$ $ & A ( ( , . . S - . K
! RA+ ! !S X X ,
! !S (x1 x2) = !(f (x1 x2)g ), $ x1 x2 2 X . 5
' $ .
1. (
!S X , )
! : 1), : 2),
: 3).
. L&( '( ', ' !&$ ( X S , '( !S .
. , ! | &, !S | X . 7 ' & ! S . % ,
&
9 ,
R
24
. . ' $ : 3). , !S . U | RA+, !(8r ) < " r8 = fr g 2 U , $ " | 6 . : , U 6 r < ,
$ &$ A0 A, > 0 !&$ 2 A0 . 7& V ' X ,
(x1 x2) 2 V ( 2 A0 ! (x1 x2) < .
4$, , (x1 x2) 2 V !S (x1 x2) < ".
( &(, : 3). (8r1 r82 r83) | RA+, &! $. 2 A R2 $ P1 P2 P3 P1 P2 = r1 , P1 P3 = Qr2 , P2 P3 = r3 . X = R2 P1 = fP1 g,
P2 = fP2 g, P3 = fP3 g | X . &' '( ( X R2 , 2 A, & . 4$ (P1 P2) =
= r1 , (P1 P3) = r2 , (P2 P3) = r3 ( 2 A, S = f g, : !S (P1 P2) = !(8r1 ), !S (P1 P3) = !(8r2),
!S (P2 P3) = !(8r3), &( : 3). L&( '( & $ . X = RA+. % R+ !, (0 r ) = r . A, '2, | & X S = f g, f (80 r8)g = r8 !S (80 r8) = !(8r ), ' &.
5 ' X , '(
' &.
5 1) B fA g2B f! g2B ! | , )
: 6), S = f g 2A , 2 B , | S
X , A , S = S . ; Q ! ! S !, M = f(! )S g2B , %.
.
S = f g 2A 2B &O
$ &
A ( X 9 S , A = A . M , '
B .
. >:
Y Y ! ! (x1 x2) = ! ! (f (x1 x2)g
S
2A 2B ) =
= !(f! (f (x1 x2)g 2A )g2B ) = !(f(! )S (x1 x2)g2B ) = ! (x1 x2)
!&'( x1 x2 2 X .
25
5 2) * % % ! % S , ' , !S .
. x1 x2 2 X x1 6= x2. 4$ $ !S (x1 x2) = !(f (x1 x2)g) RA+ ' . !S (x1 x2) 6= 0 : 9).
5 3) * % % ! , X !S , , S , $
A, !.
. A0 | A, & , '( !, S 0 | S , A0 . 7&, '2, i0 RA+
RA+ !0 | $ !ji ( A+ ) . 4$ $ , !S0 !S X !. > ' 1 , , !S , , 2, , S 0 .
0 & 2. 5 $ U ', S 0 . : , 6 < , $ &$
A00 A0 > 0. L
6 " > 0,
$ ! f(x1 x2) 2 X X : !S (x1 x2) < "g U .
7& i R+ RA+ , $ 2 A00 . : & !0 ji( +) : 5) , : 8) " > 0, $ !0 ji( +) (x) < "
" , (x1 x2) | , x < . " = 21 min
2A
0
0
X X , !S (x1 x2) = ! (f (x1 x2)g 2A ) < ". 7& r8 f (x1 x2)g 2A RA+ , r8 | $ ! !
i (R+ ), $ 2 A00. J $ $ ' '
!, ' , (x1 x2). 7, r8 6 r8,
!0 (8r ) 6 2!0(8r) < 2" 6 " (x1 x2) < A00.
5 4) &(' !. P&'
$, & S , ((
( S .
5 4) S1 S2 | X , S1 = S2 . % , %' ( , %' ) $
, .
. 5 '
. 0 S = f g 2A $ !O$ & f : A ! S . %6 A 2 , 1 2 ,
f (1 ) = f (2 ), & A .
0
0
0
R
0
0
R
R
00
0
0
0
0
26
. . A & & A ! S S ,
A . 0, S S .
$, '& ! , A A, S | S . 4$ & ! A &
& ! A, . 5 !S = !S ', '
$ & S &' S .
P&' & !,
& Q$ a 2 A Q
Qa $ & R+a R+ . 4$
Qa 2a
a2A
''
(!
Q Q
RA+ RA+, '
R+ . 7&
a2A 2a
$ & D. 7$ D !& & ! A, & ! A . 0 !&'( x1 x2 2 X ! f (x1 x2)g 2A 2 D, ', !S = !S . 3
& $ .
7&
, S1 S2 ( &
! ' &.
5 5) | X , , S .
' %' !, , !S ,
% .
. %'& S = f g 2A & 6
S 0 , &' $ ! S 0 . %6 $ (! ! '( f"n g $ "n 6 '
& An A &T '( Qn = f g 2An ,
'( !
f(x1 x2) 2 X X : (x1 x2) < g 2An
f(x1 x2) 2 X X : (x1 x2) < "n g. 4$ S 0 S
0
, A = An .
n
! | ! & A, $ A0 . 5 3) '. 4 & , , ! & 2, > 0, 2 A0 .
5 6) ' , ,
, ' %' 27
!, % X X !S > .
. ' 5 1) !, $ S , &
. @ , , 6 1. 5 !
R+, ! !^ (s) = sup (x1 x2)
(x1 x2 )6s
, &' ! &, .
7, $, !^ & ! :
1) 0 6 !^ (s) 6 1 !&$ s 2 R+9
2) !^ (0) = 09
3) !^ (s) 9
4) !^ (s) ' .
% ! $, , , , ! .
4 , ' ( !, $ $ ( '! & f(s t) 2 R2 : s 2 R+ 0 6 t 6 !^ (s)g ( ( ( $ '( ''( ). % , ! '
! $, !&$ " > 0 k > 0, $ ' f(s t) 2 R2 : s 2 R+ 0 6 t 6 " + ksg, ! | !^ , & ! | 6 '. 3 2 .
% , !!$ $ ,
! :
X Y | , 6 Y $
, f : X ! Y | ' &. 4$ &
, X , $ & f '.
7, 5 6) &(. 6 6
$
$ . 7 $ RA+ !
^(8r) = sup (x1 x2):
f (x1 x2 )g6r
4$ ! .
1. +
!S , )
^ % .
2. X | R | 2 . % ' 28
. . Y = X=R 6 ! ! (, '( '
& p : X ! Y . 7& p ' $ ( & '() '( ( ''( &
. Y
$ $ , $ 6 & X .
:' !, (
$ , !$ (! ! S , , ! !
Y .
L6 ' '( , $ $ 6 . A , ' !& , .
6 ' (
&
, & !&$
$ &, $ . *$ !( ( &
. 5 4) , $ 6 .
5 !!, - & !,
!. | X , y1 y2 | '
Y . R
k , !
' X , & '
!&! 2k-! Z = fx000 x01 x001 : : : x0k;1 x00k;1 x0kg X , !! ! : p(x000 ) = y1 , p(x0k ) = y2 ,
p(x0i) = p(x00i ) i = 1 : : : k ;1. 0
Z 6
Pk
(Z ) = (x00i;1 x0i).
1
7 ^ Y , ^ (y1 y2 ) = Z(yinfy ) (Z ),
1 2
$ Z (y1 y2) & ( , !( y1 y2 .
Y , ' & X ,
& ' ( (. 7 :
1) & & p Y , 9
2) ( , '( X , $
, $' 2 ' ( 9
3) X & .
5 $ S X & ' S^ Y , , (( S .
29
A Y ! , 6! ^ X , & p & ''. , Y 2,
( $ , 6'( S , . 3 !! ! Y &
' & , !
S .
, S = f g 2A $
X ( , !( ( &
), ' ' Y !. , , | $ ( Y ) , | 6 & X . 4$ S ,
! . ( , S^, !! .
4 &, ! .
2. X | , R | ,
$
X S | X ,
%'
. ) Y = X=R , %' S .
K ' , $
X &
, $'( R. 7& F = ff g2B &
X &. X 6 &
$
, !&'( x1 x2 2 X , 2 B (f (x1) f (x2)) 6 (x1 x2). 5! X 6 &
F , &' , &!( .
, $ !$ $ & ! S = f g &'( F
&
. % , : !S (f (x1) f (x2 )) = !(f (f (x1) f (x2 ))g ) 6 !(f (x1 x2)g) =
= !S (x1 x2). 4 &, & &' ' ' !( ( &
&'( .
, F $ 2 R, . . (f f )(R) R !&$ 2 B . 4$ - Y &
F^ . ,
' 2 ' ! .
2. !
X F , ) Y F^ .
30
. . . y1 y2 | ' ' Y , Z |
, ! ( X . A f^ | & F^ , f | ! $ & F , &' f , &!(
Z , ! Z , !! f^ (y1 ) f^ (y2 ). S |
!( ( &
&'( F X . 4$ &' &' 2 S , Z &2 ' Z . B, ^ ^ (f^ (y1 ) f^ (y2 )) 6 ^ (y1 y2 ), . . & F^ .
76 , 2 ! Y .
3. 5 ( &
$ $ .
6 ' (
&
, ( $
& '( $ .
3. .
. S , A. % S !! ! S = f0 1 : : :g, & . ! + 1 T = fT0 T1 : : :g S , ! '
6 T ! :
T1) 2 T > 9
T2) T | 9
T3) T \f : < g $ & '( T .
7, T ' .
, $, T0 = S , T ' ( < 6 .
A 2, , = + 1 $ & T n f g, T = T . A $ &, T = T n f g. %' T1), T2), T3)
.
T
| '
. 7& $ P = T .
<
P & , $' T2) T3):
P1) 2 P > 9
P2) P \ f : < g $ & '( P .
31
A P , T = P . A P ' , & Q S , , !( X ! , !
, !
P . Q < . ,
Q R , $ T = P R .
$, Q S . 5 ' Q ! !: Q = f0 1 : : :g, $ < 9 & . !!
! +1 fR0 R1 : : :g Q ,
!! ! :
R 1) !&$ 6 ! R f : < g, 6
1 < 2 6 R2 \ f : < 1 g = R1 9
R 2) , X !&
,
, P R , < 9 2 R , ( , P f : < g9
R 3) (P R ) \f : < g $ & '( P R
!& 6 .
% R0 6 . , , R ' < 6 , 2,
, = + 1. 4$ R = R , ,
X , ,
P R 9 R 1), R 2), R 3) ''. A $ , $ R = R f g. 4$ $
' R 1) R 2). 7& ! R3), , &' $
& P R, $ $ &, $ . % , , $ , X , &' , .
7' ' (P R ) \ f : < g $ &' ( &
! .
5, S , , $ | '
. 7& $
R R . %' R1) R2) < $, R 3). - (P R ) \ f : < g $ & $( P R . 4$ P , ,
-'(, $ $ & $
&' &' Q , -'(, &' $ $' 32
. . P , &' P2). , ,
2 R . 4$ $' $ $ & $ P ( ! Q ). K $,
$ &' 0 Q , !( , &' ! R2), ', ! , $ R , < , $ 0. $
&' $
&, $' $ P & 2! Q .
4 ' fR : 6 g . R = R , T = P R 9 T1), T2), T3) T
! P1), R2), R3) P R , ' .
1] ., . , // Math. Slovaca. | 1981. | Vol. 31, no. 1. | P. 3{12.
$ % & 1997 .
. . , . . . . . 517.987.1+517.518.1+517.982.3
: , , {.
" #$% , &'( 1909{1914 ', : , ! .
. 1952{1953 % / 0 0% 0 &/, &#, 1 . 20% % % 3( 0 / 1956 .
5. .. "%0.
. 1996{1997 % 0 '2 3 360% /, 0
$ , 3' , 7.
" 8 00 , "{"
? . , 7 60, 8 .
Abstract
V. K. Zakharov, A. V. Mikhalev, Connections between the integral Radonean
representations for locally compact and Hausdor% spaces, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 1, pp. 33{46.
After the fundamental papers of Riesz, Radon and Hausdor; in 1909{1914
the problem of general Radonean representation became actual: &nd for Hausdor% topological spaces a class of linear functionals isomorphically integrally representable by all Radon measures. In 1952{1953 the bijective solution of the problem
of Radonean representation for locally compact spaces was obtained by Halmos,
Hewitt, Edwards, etc. For bounded Radon measures on a Tychono; space the problem of isomorphic Radonean representationwas solved in 1956 by Yu. V. Prokhorov.
In 1996{1997 the authors obtained one of possible solutions of the problem of
general Radonean representation using the family of metasemicontinuous functions
with compact supports and the class of thin functionals on it.
'
! , 2001, 7, < 1, . 33{46.
c 2001 ( )*,
+ ! ,- .
34
. . , . . After this the question if the theorem about general Radonean representation
covers the Riesz{Radon theorem was still left open. In this paper the positive
answer to this question is given.
1], 2] 3]
1909{1914 !"# .
%" & '. ( ' )!' (T G ) '- '# RMw0(T G )0 ."/ 0! !-! ' 1 B(T G ) 0 1
R ]. 2.- ' . i d !' MI(T B(T G ) ) B(T G )-1''/ -'/
!3# f 1 T R.
4 ' ! !3/ i '. '" ' !'
T A(T ), .' !' UI(T ) (MI(T B(T G ) ) j 2 RMw0(T G )0 ) B(T G )- " !1/ !
', !' ) 7! i jA(T) 1 '. RMw0 (T G )0
! A(T ) #/ !3 A(T ) --- 5!/'.
4 . - 1 1! (/) ''- ): ! - ! A(T ) !' A(T ) #/ !3 &' /" !/ (. . "1 '/) ' ! ! 0&)
A(T)4 , ! ) ."- )" (A(T)4 )+ ) '#' A(T)rf0 fi jA(T ) j 2 RMw0(T G )0 g ."/ ! !3 A(T).
0 '/ ! - - )- !" !'
! / '0' 7], "' 8] 9' 9]
1952{1953 1 !! {
('., ', 11] 12]).
. (T G ) | Cc (T G ) |
(T G ) . :
1) ! ! 7! i jCc(T G ) " RMw0 (T G )0 Cc (T G ) $%
2) Cc (T G )rf0 ! ! & (Cc (T G ) )+ 35
-! Cc (T G ).
; 1 1'./ 0# '/ ! - )- 1" / '
# " 1997 . ('. 14,15,39,40]).
( ). (T G ) |
, S(T G ) | (
) (T G ) Sc (T G ) |
&( . 1) ! ! 7! i jSc (T G ) " RMw0 (T G )0 Sc (T G ) $%
2) Sc (T G )rf0 ! ! & (Sc (T G )4 )+ -!
Sc (T G ).
/ !1/ # 14], 15], 39] 40] - !//'
, {
? 4 # " &- ."/# < : ' { 1 '/ ' !' ('. ' ).
=- "- --- /' .' " 39]. <' 1 ' '/' -' /!' '/ /' ! <# " ('. !. 40]). > , '/ .' '3
! " 39] !. '3 ! 3# /.
?/ /. " ?. 4. @! 1 ! ., ' 1 1" ! -"
- .
6. 4 <' ! '/ !.', ) ' { - !" !'
! 1 '/ ' !' - .
" T | !" !'
! ' | #/# ."/# !3 Cc (T), . . ' 2 (Cc (T) )+ . !.', ) !3 ' .- /' 1' # ."
! !3 Sc (T ), . . ' 2 (Sc (T )4 )+ .
" A(T) | 1" '. F(T ). (
'', )
!3 1 A(T ) R 1/- !!
, f = p-lim(fm j m 2 M) )& f = lim(fm j m 2 M) - !.# (fm 2 A(T ) j m 2 M) !.# !3 f 2 A(T). D A(T ) | 0&)
36
. . , . . ! , < # " !.' 1 #: 1) (fm 2 A(T)+ j m 2 M) # 0 F(T) )& (fm j m 2 M) # 0 RG 2) - f 2 A(T) (fm 2 A(T )+ j m 2 M) " f
F(T ) ! (fm j m 2 M) " f R. ?)- "" - !!
-
"-'
(fm 2 A(T) j m 2 M !).
%- '' --- !)# - "#0 .#. <' '/ ' & !1"', - --- 0 1#
('. 12, 73D]).
1. )
' Cc (T) .
. " (fm 2 Cc (T)+ j m 2 M) # 0 F(T). J!' m0 2 M '' !3 g fm0 !'
! '.
C clft 2 T j g(t) > 0g. =- C !3- h 2 Cc (T )+ , !- )
h > (C).
41"'&' " > 0. K #&- a > 0, ! ) a'(h) < ". ''
!// '. Gm ft 2 T j fm (t) < ag. K! !! (fm j m 2 M) # 0, (Gm j m 2 M) " T . <' !) !/ (Gn j n 2 N M)
'. C. L1 M , ) m 2 M, !
) n 6 m - n 2 N. <' Gm Gn - n. J),
C Gm . 41"'&' l 2 M, ! ) l > m0 l > m. K fl 6 g fl 6 fm . L1 , ) H coz fl C Gm , 1
, ) fl (t) 6 fm (t) < a = a(C)(t) 6 ah(t) # )! t 2 H. J), fl 6 ah )& 'fl 6 a'(h) < ". D k < l, 'fk 6 'fl < ". K!' 1', ('fm j m 2 M) # 0. M
L
"1- # ))# / !3 ' ' " . !/' &/' ' ' =-,
/' 21].
'' '. Ybc )/ !3# !'
!/'
-' f 2 Fbc (T), ! ) f = sup(fm 2 Cc (T) j m 2 M) F(T) - !# (fm 2 Cc (T) j m 2 M) ". ?)/' 1' '' '. Zbc !3# f 2 Fbc (T ), ! ) f = inf(fm 2 Cc (T) j m 2 M)
F(T ) - !# (fm 2 Cc (T ) j m 2 M) #. @, )
Cc (T) = Ybc \ Zbc . = ' ' " Y Z.
2. *!
Y Z F(T ) & &(
P : P
1) (ai gi j i 2 I) 2 Y (ai hi j i 2 I) 2 Z & (ai 2 R+ j i 2 I), (gi 2 Y j i 2 I) (hi 2 Z j i 2 I)%
2) inf(gi j i 2 I) 2 Y sup(hi j i 2 I) 2 Z & (gi 2 Y j i 2 I) (hi 2 Z j i 2 I)%
3) sup(gi j i 2 I) 2 Y inf(hi j i 2 I) 2 Z & (gi 2 Y j i 2 I) (hi 2 Z j i 2 I) & u v 2 Fbc (T), gi 6 u hi > v i%
37
4) g ^ 1 2 Y, g _ (;1) 2 Y, h ^ 1 2 Z h _ (;1) 2 Z & +
g 2 Y h 2 Z%
5) ;Y Z ;Z Y, . . ;g 2 Z ;h 2 Y & g 2 Y h 2 Z%
6) Y = Cc (T ) + Y+ , . . & g 2 Y (& f 2 Cc (T) g0 2 Y+ fg 2 Y j g > 0g, g = f + g0 %
7) Z = Cc (T ) ; Y+ , . . & h 2 Z (& f 2 Cc(T ) h0 2 Y+ ,
h = f ; h0 .
. %# 1){5) --- /' -' 1 # / ). ' # 6). " g 2 Y, . . g = sup(fm j m 2 M) - !# (fm 2 Cc (T) j m 2 M) ". 41"'&' f fm0 '' gm (fm _ f) ; f 2
2 Cc (T)+ g0 g ; f. K! !! (gm j m 2 M) " g0 , g0 2 Y+ . J),
g = f + g0 . ; . 1 1). %# 7) -). M
;
' Y /# !3 ', - 'g supf'f j
f 2 Cc (T ) ^ f 6 gg - !. g 2 Y. 9 !!. =#", ! !! !3- g ) ' !'
!/# ", !'
! '. C ) a > 0, ! ) g 6 a(C).
<' g 6 a(C) 6 f0 - !# !3 f0 2 Cc (T ). 4 1"
'g 6 'f0 < 1.
?)/' 1' ' Z /# !3 ', - 'h inf f'f j f 2 Cc(T ) ^ f > hg - !. h 2 Z. @, ) ' '
--- .-' '.
% '" ''/ 2 !1/ 3. , g 2 Y (fm j m 2 M) " g (fm 2 Cc(T )+ j m 2 M), ('fm j m 2 M) " 'g.
. K! !! fm 6 g, 'fm 6 'g b - m. '' ) a sup('fm j m 2 M) 6 b. 41"'&' 1" !3
f 2 Cc (T )+ , ! ) f 6 g. K ((f ; fm )+ j m 2 M) # (f ; g)+ = 0 F(T).
4 ''/ 1 '/ )' inf('((f ; fm )+ ) j m 2 M) = 0. K! !! ' 1, '((f ; fm )+ ) > '(f ; fm ). <' 'f = 'f ; inf('((f ; fm )+ ) j
m 2 M) = sup('f ; '((f ; fm )+ ) j m 2 M) 6 sup('f ; '(f ; fm ) j m 2 M) =
= sup('fm j m 2 M) = a. ; , ) b 6 a. M
4. )
' ! Y+ &( :
1) ' "
%
2) '(ag) = a'g & a 2 R+ g 2 Y+ %
3) '(g0 + g00) = 'g0 + 'g00 & g0 g00 2 Y+ .
. %# 1) 2) )/.
3) '' / / '. M 0 ff 2 Cc(T )+ j
f 6 g0 g M 00 ff 2 Cc (T)+ j f 6 g00g. 'g0 = sup('f 0 j
38
. . , . . f 0 2 M 0) 'g00 = sup('f 00 j f 00 2 M 00). '' '. M M 0 M 00. K! !! g0 = sup(f 0 j f 0 2 M 0 ) g00 = sup(f 00 j
f 00 2 M 00), g0 + g00 = sup(f 0 + f 00 j (f 0 f 00) 2 M). %",
(f 0 + f 00 j (f 0 f 00 ) 2 M) " (g0 + g00 ). ?)/' 1' ('f 0 + 'f 00 j
(f 0 f 00 ) 2 M) " ('g0 + 'g00 ). '' 3 ('(f 0 + f 00 ) j (f 0 f 00 ) 2 M) " '(g0 + g00).
L1 '' '(g0 + g00 ) = 'g0 + 'g00 . M
5. )
' ! Y ' ! Z
& &( :
1) ' ' "
&%
2) '(f + g) = 'f + 'g '(f ; g) = 'f ; 'g & f 2 Cc (T)
g 2 Y+P
%
P
P
'(
(aigi j i 2 I)) = (ai 'gi j i 2 I) '( (ai hi j i 2 I)) =
3)
P
= (ai 'hi j i 2 I) & (ai 2 R+ j i 2 I),
(gi 2 Y j i 2 I) (hi 2 Z j i 2 I)%
4) ('(gm ) j m 2 M) " 'g & (gm 2 Y j m 2 M) & g 2 Y, (gm j m 2 M) " g F(T )%
5) '(;g) = ;'(g) '(;h) = ;'(h) & g 2 Y h 2 Z%
6) 'v 6 'u 'u 6 'w & u 2 Y v w 2 Z, v 6 u 6 w.
. %# 1) 1 #.
2) '(f + g) = supf'u j u 2 Cc (T ) ^ u 6 f + gg 'g =
= supf'v j v 2 Cc (T )+ ^ v 6 gg. K! !! u 6 f + g, u ; f 6 g )&
'u ; 'f 6 'g, . . 'u 6 'f + 'g. %", '(f + g) 6 'f + 'g. K! !!
v 6 g, f + v 6 f + g )& 'f + 'v 6 '(f + g), . . 'v 6 '(f + g) ; 'f.
%", 'g 6 '(f + g) ; 'f, . . 'g + 'f 6 '(f +g). 4 -- ).
%# 3) " 1 # 2), # 6) 7) ''/ 2 #
2) 3) ''/ 4.
4) '' /
S '. Ll fu 2 Cc(T ) j u 6 gl g.
'' '. P (fmg Lm j m 2 M). O
-)' , (l u) 6 (m v), l 6 m u 6 v. D p (l u) q (m v) | 1"/
<'/ 1 P , n 2 M, ! ) l 6 n m 6 n. K! !!
u 6 gl 6 gn v 6 gm 6 gn , w u _ v 2 Ln . <' p 6 (n w) q 6 (n w).
%", '. P --- /' .
'' " (fp 2 Cc (T)+ j p 2 P), ! ) fp u - !.
p (l u), u 2 Ll . " p = (l u), q = (m v) p 6 q. K u 6 v )&
fp u 6 v fq . J), (fp j p 2 P ) ".
" t 2 T " > 0. K g(t) ; "=2 < gl (t) - ! l 2 M. =,
gl (t) ; "=2 < u(t) = fp (t) - ! u 2 Ll p (l u). %",
g(t) ; " < fp (t). K! !! " 1", g(t) = sup(fp (t) j t 2 P ). J),
(fp j p 2 P ) " g. '' 3 ('fp j p 2 P) " 'g.
" " > 0. K 'g ; " < 'fp - ! p (l u). K! !! fp = u 2
2 Ll , 'fp = 'u 6 'gl )& 'g ; " < 'gl . J), 'g = sup('gm j m 2 M).
39
%# 5) ).
6) #' 6) 7) ''/ 2 u = f 0 + g0 v = f 00 ; g00 - !/
0
f f 00 2 Cc (T) g0 g00 2 Y+ . K! !! f 00 ; f 0 6 g0 + g00 2 Y, # 3)
'f 00 ; 'f 0 6 '(g0 + g00 ) = 'g0 + 'g00. K
" # 2) 'v = 'f 00 ; 'g00 6
6 'f 0 + 'g0 = 'u.
=, 'u = supf'f j f 2 Cc (T ) ^ f 6 ug 'w = inf f'f j f 2 Cc (T) ^ f > wg
)& f 6 u 6 w 6 f. %", 'f 6 'f. ; 'u 6 'f 'u 6
6 'w. M
K
" ' Fbc (T ) / !3/ 'e ': , 'ef inf f'g j g 2 Y ^ g > f g ': f supf'h j h 2 Z ^ h 6 f g - !.#
!3 f 2 Fbc (T ). @, ) 'e --- .' ', ': --
.' '.
1. )
'e ': Fbc (T) & &(
:
1) 'e ': "
&%
2) ': f 6 'ef & f 2 Fbc (T)%
3) 'e(;f) = ;': f ': (;f) = ;'ef & f 2 Fbc (T)%
4) 'e(af) = a'ef ': (af) = a': f & a 2 R+ f 2 Fbc (T)%
5) 'e(f 0 +f 00 ) 6 'ef 0 + 'ef 00 ': (f 0 +f 00 ) > ': f 0 +': f 00 & f 0 f 00 2 Fbc (T)%
6) ('e(fn ) j n 2 N) " 'ef & (fn 2 Fbc (T ) j n 2 N)
& f 2 Fbc (T), (fn j n 2 N) " f F(T ).
. %# 1) ).
2) ': f supf'h j h 2 Z ^ h 6 f g 'ef inf f'g j g 2 Y ^
^ g > f g. K! !! h 6 f 6 g, # 6) ''/ 5 'h 6 'g. ;
': f 6 'g ': f 6 'ef.
3) K! !! 'e(;f) inf f'g j g 2 Y ^ g > ;f g, f > ;g 2 Z )&
': f > '(;g). # 5) ''/ 5 '(;g) = ;'g. <' ;': f 6 'g )&
;': f 6 'e(;f).
% # /, ! !! ': f = supf'h j h 2 Z ^ h 6 f g, ;f 6 ;h 2 Y
)& 'e(;f) 6 '(;h) = ;'h. <' ;'e(;f) > 'h )& ;'e(;f) > ': f.
%# 4) 1 # 3) ''/ 5.
5) 'e(f 0 + f 00 ) inf f'g j g 2 Y ^ g > (f 0 + f 00 )g,
0
'ef inf f'g0 j g0 2 Y ^ g0 > f 0 g 'ef 00 inf f'g00 j g00 2 Y ^ g00 > f 00 g. K! !!
g0 +g00 > f 0 +f 00 g0 +g00 2 Y, # 3) ''/ 5 'e(f 0 +f 00 ) 6 '(g0 +g00) =
= 'g0 + 'g00. ; 'e(f 0 + f 00 ) ; 'g0 6 'g00 )& 'e(f 0 + f 00 ) ; 'g0 6 'ef 00 , 'e(f 0 + f 00 ) ; 'ef 00 6 'g0 )& 'e(f 0 + f 00 ) ; 'ef 00 6 'ef 0 .
6) K! !! ('efn j n 2 N) " 'efn 6 'ef b - n, a sup('efn j n 2 N) 6 b. P!' " > 0. K - !. n gn0 2 Y, ! ) gn0 > fn 'efn + "=2n > 'gn0 > 'efn . K! !! f 2 Fbc (T), f 6 u - ! u 2 Cc (T). '' gn00 gn0 ^ u 2 Y. # 3)
40
. . , . . ''/ 2 g 2 Y, ! ) g = sup(gn00 j n 2 N). K! !! gn00 6 gn0 , # 1) ''/ 5 'gn00 6 'gn0 < 'efn +"=2n . 2' , gn00 > fn ^ f = fn .
'' gn sup(gi00 j i = 1 : : : n) 2 Y. K (gn j n 2 N) " g. # 4) ''/ 5 ('gn j n 2 N) " 'g.
' !3, ) 'gn 6 'efn + " ; "=2n . =- n = 1 'g1 = 'g100 6 'ef1 + "=2 = 'ef1 + " ; "=21. 1' #
/ ) gn + gn00+1 = gn ^ gn00+1 + gn _ gn00+1 . K! !! gn ^ gn00+1 >
> gn00 ^ gn00+1 > fn ^ fn+1 = fn , '(gn ^ gn00+1 ) > 'efn . 2' , gn _ gn00+1 =
= gn+1. 4 1" 'gn + 'gn00+1 = '(gn ^ gn00+1) + 'gn+1 > 'efn + 'gn+1 .
%", 'gn+1 6 ('gn ; 'efn )+'gn00+1 6 " ; "=2n +'gn00+1 < " ; 2"=2n+1 +
+ 'efn+1 + "=2n+1 = 'efn+1 + " ; "=2n+1.
- ! !1' , )' 'g = lim('gn j
n 2 N) 6 lim('efn +" ; "=2n j n 2 N) = lim('efn j n 2 N)+" ; lim("=2n j n 2 N) =
= a + ". 2' , f = p-lim(fn j n 2 N) 6 p-lim(gn00 j n 2 N) 6
6 p-lim(gn j n 2 N) = g 2 Y )& b 'ef 6 'g. 4 1" b 6 a + ". K!
!! " 1", b 6 a 6 b. M
'' Fbc (T ) '. Xbc' ff 2 Fbc (T) j 'ef = ': f g. ;
' /# !3 '^ Xbc' , . 'f
^ 'ef = ': f. =
' ' " X' .
% '" / .# !1/- - '.
1. T | ' 2
2 (Cc (T ) )+ . 1) X' = ff 2 Fbc (T) j 8" > 0 9g 2 Y 9h 2 Z (h 6 f 6 g ^ 'g ; 'h < ")g%
2) X' | %
3) Y Z X' %
4) '^ ! ! '%
5) '^ - X' .
. 1) ;1)' )" 1) )1 X.
'ef inf f'g j g 2 Y ^ g > f g ': f supf'h j h 2 Z ^ h 6 f g.
=- f 2 Fbc (T ) " > 0 g 2 Y h 2 Z, ! ) h 6 f 6 g 'ef + "=2 > 'g ': f ; "=2 < 'h. D f 2 X' , 'g ; 'h < ('ef + "=2) ;
; (': f ; "=2) = " )&
f 2 X. ;, f 2 X " > 0, 'ef 6 'g ': f > 'h ! 'ef 6 'g < 'h + " 6 ': f + ". J), 'ef 6 ': f. L
"1# 2) 1 .- 1, 1!)', ) 'ef = ': f, . . f 2 X' .
2) " f 2 X' a 2 R. K # 3) 1 .- 1
'e(;f) = ;': f = ;'ef = ': (;f). J), ;f 2 X' .
D a > 0, 'e(af) = a'ef = a': f = ': (af). D a < 0, 'e(af) =
= (;a)'e(;f) = (;a)': (;f) = ': ((;a)(;f)) = ': (af). J), af 2 X.
" f 0 f 00 2 X' . K #' 2) 5) 1 .- 1 ': (f 0 +f 00 ) 6
6 'e(f 0 +f 00 ) 6 'ef 0 + 'ef 00 = ': f 0 +': f 00 6 ': (f 0 +f 00). %", f 0 +f 00 2 X' .
41
3) " g 2 Y. K 1 'g supf'f j f 2 Cc (T) ^ f 6 gg , ) - " > 0 f 2 Cc (T ) Z, ! ) f 6 g 6 g
'g ; " < 'f = 'f. J), g 2 X' . ?), h 2 Z, 1
'h = inf f'f j f 2 Cc(T) ^ f 6 hg , ) - " f 2 Cc (T) Y,
! ) h 6 h 6 f 'h + " > 'f = 'f. J), h 2 X' .
4) " f 2 Cc (T) Y X' . K 'f
^ = 'ef = 'f = 'f.
D f 2 X' f > 0, '^f 'ef > 'e0 = 0.
" a 2 R f 2 X' . D a > 0, '(af)
^
= ': (af) = a': f = a'f.
^ D
a < 0, '^(af) = 'e((;a)(;f)) = (;a)'e(;f) = a': f = a'f.
^
" f 0 f 00 2 X' . K 1 !1" # 2) , )
'(f
^ 0 + f 00 ) = ': (f 0 + f 00 ) = ': f 0 + ': f 00 = 'f
^ 0 + 'f
^ 00 .
5) K! !! '^ = 'ejX' , # 6) 1 .- 1 (fn 2 X' j n 2 N) "
" f 2 X' )& ('f
^ n j n 2 N) " '^f. K! !! X' --- !/'
0&)/' ', < # " ))# -
/. M
=!.' ", ) Sc(T) Xbc'.
= )1 Gc ' 1)" '. !// !'
!/ '. 1 T .
6. T | A 2 Ac (T G ).
A = S(Ci \ Di j i 2 I) (Ci 2 C j i 2 I)
(Di 2 Gc j i 2 I).
. A = S(Fi \ Gi j i 2 I) C - !/ !)/ !!3# (Fi 2 F j i 2 I) (Gi 2 G j i 2 I) !
!'
! '. C. K! !! T !" !'
!, !/ !'
! '. H, ! ) C H. <'
Fi \ Gi = Fi \ Gi \ C \ H = (Fi \ C)S\ (Gi \ H). '' '.
Ci Fi \ C Di Gi \ H. K A = (Ci \ Di j i 2 I). M
7. A 2 Ac (T G ). (A) 2 Xbc'.
. D D | !/ !'
! '., g (D) 2 Y X' . D C | !'
! '., h (C) 2 Z X' .
K! !! X' | 0&S!, (C \ D) = g ^ h 2 X' .
'' 6 A = (Ci \ Di j i 2 I). <' (A) = sup((Ci \ Di) j i 2 I) 2
2 X' . M
2. Sc (T) Xbc' .
. " f 2 Sc (T). 41"'&' " > 0. !1" '/ 1 (II.4) 1 " 39] / , ) - " )- !3- u 2 St(T A(T G )), !- ) jf(t) ; u(t)j < "=4
- t 2 T. 41"'&' g0 u ; ("=4)1 h0 u + ("=4)1. K
g0 (t) = (u(t) ; f(t)) + f(t) ; "=4 6 f(t) h0 (t) > (u(t) ; f(t)) + f(t) + "=4 > f(t).
42
. . , . . % # /, g0(t) = (u(t) ; f(t)) + f(t) ; "=4 > f(t) ; "=2 0
h (t) = (u(t) ; f(t)) + f(t) + "=4 < f(t) + "=2. K! !! !3- f -
) !'
!/' ', !'
! '. C ) a b, ! ) a(C) 6 f 6 b(C). ;1)' (C) )1 x '' )/ !3 !'
!/' -' v (ax _ g0 ) ^ bx w (ax _ h0 ) ^ bx. K f(t) ; "=2 < v(t) 6 f(t) 6 w(t) < f(t)+"=2 - t 2 T 0 6 w ; v 6 "x.
P
K! !! v 2 St(T A(T G )), v = (ak (Ak ) j k 2 K) - !/
!)/ !!3# (ak 2 R j k 2 K) (Ak 2 A(T G ) j k 2 K). 4 , ) A(T G ) | , '. )", ) S '. Ak !- ak )/ -. K (Ak j k 2 K) = coz v C.
%", !. '. Ak --- !'
!/'. '' 7
(Ak ) 2 X' . J), v 2 X' . ?)/' 1' w 2 X' .
41"'&' 1" > 0. !1' /0 - " =(3'(x))
^
&' v w 1 X' . 4 .- 1) 1 '/ 1
g0 g00 2 Y h0 h00 2 Z, ! ) h0 6 v 6 g0 , h00 6 w 6 g00 ,
'g0 ; 'h0 < =3 'g00 ; 'h00 < =3. @, ) h0 6 f 6 g00 . 2' ,
'g00 ; 'h0 = ('g00 ; 'h00 ) + ('h00 ; 'g0 ) + ('g0 ; 'h0 ) < 2=3 + 'h
^ 00 ; 'g
^06
6 2=3 + 'w
^ ; 'v
^ = 2=3 + '(w
^ ; v) 6 2=3 + "'x
^ = . %",
f 2 X' . M
'' " #/# ."/# !3 '^ '. Sc (T) !.', ) <' '. --- !'.
8. - & A 2 A(T G ) & " > 0 ( K 2 C ,
K A '^((A n K)) < ".
. " D | !/ !'
! '. g (D). K! !! g = supff 2 Cc (T )+ j f 6 gg, - " > 0 f 2 Cc (T)+ , ! ) f 6 g 'g ; "=2 < 'f. '' !/
'. D0 coz f !3 g0 (D0 ). K! !! 'f = 'f 6 'g0 , 'g ; "=2 < 'g0. '' 7 g g0 2 X' . <' 'g
^ ; '^g0 = 'eg ; 'eg0 =
0
= 'g ; 'g < "=2.
'' 1'!/ '. Kn ft 2 T j f(t) > 1=ng D0 !3 hn (Kn ). @, ) !'
!/. K! !! (Kn j n 2 N) " D0 ,
(hn j n 2 N) " g0 . 4 ))# -
/ !3 '^ '/ )' ('h
^ n j n 2 N) " 'g
^ 0 . <' n, ! )
0
0
^
n Kn )) =
'g
^ ; "=2 6 'h
^ n 6 'g
^ . 4 1" g ; hn = (D n Kn ) )& '((D
= '^g ; 'h
^ n 6 'g
^ ; 'g
^ 0 + "=2 < ".
" " C | !'
! '.. =- '. D " > 0
' /0 !'
! '. L, ! ) L D '((D
^
n L)) < ". '' !'
! '.
K C \ L C \ D. K )' '((C
^
\ D n K)) = '^((C \ (D n L))) 6
6 '^((D n L)) < ".
S
(!3, " A 2 Ac (T G ). '' 6 A = (Ci \ Di j i = 1 : : : n).
=- " > 0 !'
!/ '. Ki , ! ) Ki Ci \ Di
S
43
'((C
^ i \ Di n Ki )) < "=n. ''
'.
S Di n!'
!
S K (Ki j
i = 1 : : : n). K! !! A n K = (Ci \P
K j i = 1 : : : n) (Ci \ Di n Ki j
i =P 1 : : : n), '((A
^
n K)) 6 '(
^ ((Ci \ Di n Ki ) j i = 1 : : : n)) =
= ('((C
^ i \ Di n Ki )) j i = 1 : : : n) < ". M
2. T | ' 2
2 (Cc (T ) )+ . '^ Sc (T) ! . . , ! ! ' Sc (T).
. " (An 2 Ac (T G ) j n 2 N) # 0 " > 0.
'' 8 - An !'
! '. Ln, ! )
Ln TAn '((A
^ n n Ln )) < "=2n+1 . ''S !'
!/ '.
KnS (Li j i = 1 : : : n) An . K! !! An n KnP= (An n Li j i = 1 : : : n) P(Ai n Li j i = 1 : : : n), '^((AnPn Kn )) 6 '(
^ ((Ai n Li ) j i = 1 : : : n)) =
= ('((A
^ i n Li )) j i = 1 : : : n) < " (1=2i+1 j iT= 1 : : : n) 6 "=2.
'' !'
! '. K (Kn j n 2 N) !3
h (K) hn (Kn ). K! !! (Kn j n 2 N) # K, (hn j n 2 N) # h. 4 ))# -
/ !3 '^ )' ('h
^ n j n 2 N) # 'h.
^
%", n, ! ) '^h + "=2 > '^hn. 4 1"
'(A
^ n n K) = '(A
^ n ) ; 'h
^ < '(A
^ n ) ; '^hn + "=2 = '(A
^ n n Kn ) + "=2 6 ".
J), !3 '^ --- ."/' !'
!/'.
=, - # )! t 2 T !'
!- !/!" D. '' !3 g (D) 2 X' . D f 2 X' ^ 6 '^g )& j'f
^ j 6 'g
^ < 1. J), '^
jf j 6 g, ;'g
^ = '(
^ ;g) 6 'f
--- !" )/'.
(!3, # ))# -
/ 1 '/ 1. K!' 1', !3 '^ --- !'.
" " --- #/' ."/' !' !3'
Sc(T ), .' !3 '. =!.', ) = '.
^
%) !.', ) (D) = '(D)
^
- !'
! !/ '. D. K! !! --- ."/' !'
!/', - " > 0 !'
! '. C D,
! ) (D n C) < ". ;1)' (D) (C) )1 g h . %", g < h + ". '' 1'! '. F T n D. K! !! T | ! '. C | !'
!, /- !3- f T, !) 0 6 f 6 1, f(t) = 0 - t 2 F f(t) = 1 - t 2 C.
@, ) g 6 f 6 h. <' g < h + " 6 f + " = 'f + ". J),
g = supf'f j f 2 Cc (T ) ^ f 6 gg 'g = '^g.
" " K | 1" !'
! '.. K !'
! !/ '. D K. '' !'
! !/ '. E D n K. K! !! (K) = (D) ; (E), (K) = (D) ; (E) = '(D)
^
; '(E)
^
= '(K).
^
44
. . , . . " " A 2 Ac (T G ). K! ., !! /0, - " > 0 !'
! '. K A, ! ) (A n K) < "=2. ?)/' 1', !'
! '. L A, ! )
'(A
^ n L) < "=2. '' !'
! '. C K L A.
K! !! '^ 1, (A n C) < "=2 '^(A n C) < "=2. %", j(A) ; '(A)
^ j 6 j(A) ; (C)j + j'(C)
^
; '(A)
^ j < ". K! !!
" 1", (A) = '(A).
^
(!3, " f 2 Sc (T). 41"'&' 1" > 0 " =(x),
x | !3- 1 !1" .- 2. =P " 1"'&' !3 v w 1 !1" .- 2. K v = (ak (Ak ) j k 2 K)
G ) j k 2 K). %",
- !#
!)# !!3
P
P ^(Ak k2) Aj kc(T
v = (ak (Ak ) j k 2 K) = (ak '(A
2 K) = 'v.
^ ?)/' 1', w = 'w.
^ K! !! v 6 f 6 w 0 6 w ; v 6 "x, f 6 w 6 v + "x =
= 'v
^ + 6 '^f + 'f
^ 6 '^w 6 'v
^ + "'x
^ = v + 6 f + . %",
jf ; 'f
^ j 6 . K! !! 1", f = 'f.
^ M
% <# ' '/ '.' !! " . P0
1 (Cc (T) )+ (Sc (T )4 )+ , - P0' '^.
# 1. T | . ! P0 " (Cc (T) )+ (Sc (T)4 )+ .
. D '0 6= '00, , ), '^0 =6 '^00. J), P0 5!. D 2 (Sc (T )4 )+ , !3 ' jCc (T) --- #/'
."/'. '' . '.
^ ' 2 = '.
^
J), P0 !. M
K
" '/ '.' !1" ' < !.
3 ( ). /
" ( " .
. " T | !" !'
! . ' 2 !3- P0 1 (Cc (T) )+ (Sc (T )4 )+ . ' 1
(II.5) 1 39] ' !' !3- V
1 (Sc (T)4 )+ RMw0(T)0 . >!3- V P0 & ' !
. M
% 1 ! ' 2 '. 1)" )".
4. 0 ! P0 ""
P " Cc (T) Sc (T )4 , P ' = P0('+ ) ; P0 (;'; ) !
' 2 Cc (T) .
. ' ), ) P0(a') = aP0' P0 ('0 + '00) =
= P0'0 + P0 '00 - / a 2 R+ '0 '00 2 (Cc (T ) )+ .
45
R. )", ) a > 0. ;1)' a' )1 . 41"'&' / g 2 Y
" > 0. '' '. Lg ff 2 Cc (T ) j f 6 gg. K 1
'g = sup('f j f 2 Lg ) g = sup(f j f 2 Lg ) , ) u v 2 Lg , ! ) 'g ; "=(2a) < 'u g ; "=2 < v. '' !3
w u _ v 2 Lg . K j(a')g ; gj 6 ja'g ; a'wj + jw ; gj < aj'g ; 'wj +
+ "=2 < ". L1 1" " , ) (a')g = g.
%0 )/' 1' !1/-, ) (a')h = h - h 2 Z.
" " f 2 Sc (T ) " > 0. '' '. Zf fh 2 Z j
^ = sup(h j Zf ) ! ., !!
h 6 f g. K 1 'f
^ = ': f = sup('h j h 2 Zf ) f
^
/0, , ) (a')f
^ = f.
;1)' '0 + '00 )1 . " g 2 Y " > 0. K 1 ' 0 g =
= sup('0f j f 2 Lg ), ' 00g = sup('00 f j f 2 Lg ) g = sup(f j f 2 Lg ) ,
) u v w 2 Lg , ! ) ' 0g ; "=3 < '0 u, ' 00 g ; "=3 < '00v
g ; "=3 < w. '' !3 x u _ v _ w 2 Lg . K
j(' 0 + ' 00)g ; gj 6 j' 0g ; '0xj + j' 00g ; '00 xj + jx ; gj < ". %", (' 0 + ' 00 )g = g.
?)/' 1' !1/-, ) ('0 +'00)h = h - h 2 Z.
" " f 2 Sc (T ) " > 0. K '^0f = ': 0 f = sup('0h j h 2 Zf ),
^ = sup(h j h 2 Zf ). K! ., !! /0, '^00f = sup('00h j h 2 Zf ) f
0
00
^
, ) ('^ + '^ )f = f.
' ", ) . P0 1 - / -!/ 3/. " ' 6 . 41"'&' g 2 Y ''
'. Lg ff 2 Cc(T) j f 6 gg. K 1 'g = sup('f j f 2 Lg ) g = sup(f j f 2 Lg ) , ) 'g 6 g.
41"'&' " f 2 Sc (T ) '' '. Yf fg 2 Y j
^ = inf(g j g 2 Yf ) ,
g > f g. K 1 'f
^ = 'ef = inf('g j g 2 Yf ) f
^ K!' 1', P0 --- '/'. L1 ! P0
) 'f
^ 6 f.
" , ) P0 --- 1/'. %", P0 -
/ -!/ 3/.
4 .- 3.6.1 1 16] . P0 ' 0 5! # P 1 A Cc (T)
B Sc (T )4 , ! ) P' = P0('+ ) ; P0(;'; ).
K! !! P0 - / -!/ 3/, '1 ^ '2 = 0 )&
P'1 ^ P'2 = P0 '1 ^ P0'2 = P0('1 ^ '2 ) = 0. # 14E(b) 1 12]
P --- 0&)/' #/' '.
D 2 B, = + + ; . % 1 1 '/ 2 + =
= P0'0 ;; = P0'00 - !/ '0 '00 2 A+ . '' !3
' '0 ; '00 2 A. K = P0'0 ; P0 '00 = P '0 ; P'00 = P '. 9 1), )
P --- 5!/'. K!' 1', P " !/# 0&)/#
#/# . <' P " '/# 1'1'. M
46
. . , . . ()
15] . ., . . ! "#
// %!& '(. | 1998. | ,. 360,
0 1. | 1. 13{15.
39] . ., . . 3 ! 4
"#
// 5" ! !. | 1997. | ,. 3, 0 4. | 1. 1135{1172.
40] . ., . . 9 ! 4
"#
// 34
'(. 1. . |
1999. | ,. 63, 0 5. | 1. 37{82.
/! 0 2001 .
. . . . . 510.643
: , , .
!! r , #! $% % & !!
r (F ) = (F ), (p) | & + + p. , L(r) r + L #! % ! L
& $ , r, # &! (,!) L1 ! L2
-. : L1 ,! L2 , L1 = L2 (r) + r.
.+ ! 0! , 0 % ! + &!+ . 10 ! 2 !
+! !% % : !% +, % K, K4, T, S4, S5, GL, Grz, . 1!
# !, -.% ! 0+ % 2 +! ! %.
Abstract
E. E. Zolin, Relative interpretability of modal logics, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 1, pp. 47{69.
This paper introduces the notion of modality as an operator r , de7ned on
the set of propositional modal formulas by the equality r (F ) = (F ), where (p)
is a formula of one variable p. De7ning the logic L(r) of modality r over logic L as
the set of all provable in L formulas of the propositional language extended by the
operator r, the notion of exact interpretability (,!) of a logic L1 in a logic L2 can be
formalized as follows: L1 ,! L2 i8 L1 = L2 (r) for some modality r. The question
about the number of logics, which are exactly interpretable in some 7xed logic, is
considered in this paper. Answers to this question are obtained for the following
family of known modal logics: logics of boolean modalities, normal logics K, K4, T,
S4, S5, GL, Grz, logics of provability. A number of results concerning the absence
of exact interpretability of some logics of this family in others are o8ered as well.
1. . (p) , 2001, 7, 9 1, . 47{69.
c 2001 !,
"#
$% &
48
. . p # r , r (F) = (F ):
& 3, : :p. ( ) * ) - L:
1) -
. / L ), * , 0 L. 1*
) * 0) L. 2 ) , , 31, . 11], 32{4].
2) -/
. ), *
, * , L -.. 7 *,
r tr : tr (A) A )
B r(B). 1 L(r) r L # ) , r- )
L. - /
. ) * , L ( # L). & ) : * *) L , M
( r, *
L(r) = M) L.
* , 0
), 31, . 12]: 8{9# GL ;, ;p = p ^ p, 8* Grz, GL(;) = Grz.
: * 310] L, , * * n > 1 L( n ) = L.
, ). : *) ) K, K4, T, S4, KB, S5, GL,
Grz 31], , #) 35], * *) . = 0) .
r
r
2. 7 L, *# ) ) p0 p1 : : :, ? ! -
49
. ? , > : ^ _ $ 3. ( Fm 0 * . & (p) |
L p.
. , (p), r ( r : Fm ! Fm), #
: r (F ) = (F).
: , ) ? p, *
? () . 1 , * )
:
(i) ? | ( ) A
(ii) () | () A
(iii) r1 r2 | , r1 ! r2 | A
(iv) r | , r | .
/ () ! ? () : : . * ( , ) -
. ; = () ^ , n , n > 1,
/ _ :.
C r tr : Fm ! Fm : tr (A) * A ) r. 7 *: tr (?) = ?A tr (p) = p pA
tr (A ! B) = tr (A) ! tr (B)A tr ( A) = r(tr (A)).
( ) L,
* L ) :
MP (modus ponens)
A A ! B ` BA
Sub ( )
A ` A3B=p]A
RE ( 0
) A $ B ` A $ B:
D A3B=p] | , * A ) )
p B. H L ` A A 2 L. : *) ( ) ?) ).
& L | * ( ), , X | . 1 L + X * *, * * L X * ,
LX | *, * L X, * , | MP.
r
r
r
r
r
r
r
r
r
50
. . . 9
L(r) r L # , r- ) L:
L(r) = fA 2 Fm j L ` tr (A)g:
J , * L(r) | .
. : r1 r2 L, L ` r1 p $ r2 pA L, L(r1) = L(r2 ).
? 0 ) *: , L ` r1 p $ r2 p, L ` tr 1 (A) $ tr 2 (A) A.
, # , ). # , (p), p ) .
r
r
r
3. 0 15 , * * ) , * *
L r L(r), 0 . ,
* , 0 L(r), * r r
L, r L.
. , ) ) . f(x1 : : : xn): f? >gn ! f? >g , 0 , C:JN 0
.
J ) ) / ** :
f 6 g , f(x y) ! g(x y) .
. !
()..) r L # ) ) (x y), *
(i) L ` (r? r>)A
(ii) f(x y), *
L ` f(r? r>) 6 f:
. : r )..
r /
0 FLr = ff(x y) j L ` f(r? r>)g
51
** 6, # /. 2 ,
* FLr O, ,
/ 0 FLr O ) FLr.
& r | , rp p : rp b(p Q1 : : : Qn), Qi | .
N b p:
L ` b(p Q1 : : : Qn) $ 3p ^ b(> Q1 : : : Qn) _ :p ^ b(? Q1 : : : Qn)]
L ` rp $ (p ^ r> _ :p ^ r?). 1 , p $ ((p ^ >) _ (:p ^ ?)):
()
3.1. L1 L2 | , r1 r2 | , ri Li .. i , i = 1 2. L1 (r1) L2 (r2 ) , 2 6 1 :
. U* Fi = FLiri . ( Fi f, ) f > i . &0 F1 F2 , 2 6 1. , * L1 (r1) L2 (r2) ) F1 F2 . U# .
& F1 F2. # A(p) 2 L1 (r1), p =
= (p1 : : : pn) | # ). () A 0
L1 (r1) b(p ? >) ) p ? >,
*, b(p ? >) 2 L1 (r1). = 0
p
#
_ ;p ^ b( ? >) 2 L1(r1)
2f?>gn
p p1 1 ^ : : : ^ pnn , pi ? :pi , pi > pi .
&0 b( ? >) 2 L1 (r1) 2 f? >gn, *,
b( x y) F1 , F2 . & , * b(p ? >) 2 L2 (r2) () * A 2 L2 (r2).
3.2. 3.1 :
L1 (r1) = L2 (r2 ) , 1 = 2 :
J , * * 0) L
/, 3.2 , * *) L ) * 15. :
, 15 ) ),
) ?, ).. ?
*
.
J L 4 0)
: r0 = ?, r1 = (), r2 = :, r3 = >A *, * )
52
. . ).. L:
? = :x ^ :y () = :x ^ y : = x ^ :y > = x ^ y:
&0 0) LA * )
* L? L() L: L> .
( ).. ) )A , ) / ) Lrj , j 2 f0 1 2 3g, 0 *#) ? () : >. U .
3.3. ! " (x y) 6 ? # , ! $ " .
.
& ? 6= J f0 1 2 3g. &, * T
LJ = Lrj ).. C:JN:
j 2J
J (x y) =
_ ;xrj ? ^ yrj >:
j 2J
C , Lrj ` J ( ? >) j 2 J. :
:
Lrj ` p $ rj p, 0 j-
O
* J ( ? >)
0 > Lrj .
C , f < J , C:JN f(x y) j- O * j 2 J. J O * f( ? >), *, 0 ? Lrj .
C, f( ? >) Lrj , * LJ .
1 , J /
)
O) *, J | ).. LJ . U# , * (x y) 6 ? J J, *
? 6= J f0 1 2 3g.
& 3.3 * *
Lfrj jj 2J g A , J = f0 3g, LJ = L? \ L> = L?> .
&# ) . U* * E .
3.4. % # :
I: L? = E f p $ ?gA L() = E f p $ pgA
L: = E f p $ :pgA L> = E f p $ >g:
II: L?() = E f p $ (p ^ >)gA
L?: = E f p $ (:p ^ ?)gA
L?> = E f p $ ?gA
L(): = E f p $ (p $ >)gA
53
L()> = E f p $ (p _ ?)gA
L:> = E f p $ (:p _ >)g:
III: L?(): = E f p $ ((p $ >) ^ (:p $ ?))gA
L?()> = E f p $ ((p ^ >) _ ?)gA
L?:> = E f p $ ((:p ^ ?) _ >)gA
L():> = E f p $ ((p $ >) _ (:p $ ?))g:
IV: L?():> = E f p $ ((p ^ >) _ (:p ^ ?))g:
3.4.1. p $ rp,
r | , 0 * ) *) RE.
3.4.2. U* * kf k * f? >g2,
) f(x y) >. 1 3.3
, * * / , * N (N = I, II, III, IV) ).. , k k = N.
. * () *.
() I. N j 2 f0 1 2 3g * Ej = E f p $ rj pg.
?
F , * F $ tr j (F) Ej , * Lrj . &0 F 2 Lrj ,
tr j (F ) 2 Lrj , , * | , , E Ej A , * F 2 Ej .
II, III, IV. : F G
) ), E fF g \ E fGg = E fF _ Gg.
1, * *, , L?() = L? \ L() = E f( p $ ?) _ ( q $ q)g, * , *
A A ^ >, * . Y* ) .
& L | . ? I ,
* r * L rj , j 2 f0 1 2 3g, r 0 rj L.
I II LJ = E f p $ (p r)g, (p q) | , r | " ( 0
? > 0
) ? >. 7 , *
(p q) LJ .
3.5. L r, (p q) LJ I II, L( (() r)) = LJ . $
, L?> | ! ! L: L(r) = L?> .
r
r
54
. . #, , (p q) = p ^ q.
0 * ).. Q = (() r) L , *,
(x y) = :x = J , J = f0 1g, *, L(Q) = LJ 3.3.
: ) (p q) *.
3.6. '
! "
!
(!.
. & L | , L ` A(p q) ! C(q r), p = (p1 : : : pm ), q = (q1 : : : qn), r = (r1 : : : rk).
() *, * A C | ) ? >, L ` A(p q ? >) ! C(q r ? >).
&, * B(q ? >) W;
_
2f?>gm
A( q ? >):
? L ` A $ p ^ A( q ? >) L ` A ! B. 1
L ` A( q ? >) ! C , L ` B ! C.
4. ! & * - L * 0) LA * * "(L). 33] 0 /# ) ) D D, ) / (! 6) (! 6) ( 33] -
.). 31, . 11], 32, . 96, 106, 115, 138], 34] - ., :.
( *
)
)
, * *) L, * * (L). D (L), "(L) | * 1. D, * *
L
"(L) > 4 (L) > 4A , L
RE (L) 6 "(L).
0 * ) "(L) (L) ) A , * *) ) L = K, K4, T, S4, KB, GL, Grz 31] "(L) = 1
, * (L) = 1, , , "(S5) = (S5) = 16. 7 * (L) ) (. 35]).
55
4.1. "(L) (L)
0 * * "(L) (L) ) , ) 3. J # 3.4 ) .
. 9 M " L (*: M ,! L), # r, * M = L(r).
U*, * * (L) * M, * ) L.
4.1.1. () N (N = I, II, III) $
! ! ! ) .
. : N = I 0 *.
N = II. L II r
( , ) * 0
), 0
3.5 L0 II # (p q), * L( (() r)) = L0 .
N = III. :, , * L?()> ,! L():> . J L():> ).. x _ y, 0
` ?_ >.
1 r, ).. :x _ y. J 0
, *, r , (p) (:p ^ : ?) _ (p ^ >), L():> ` r? $ : ?,
` r> $ >, *, ` :r? _ r>. &0 L():> (r) = L?()>.
Y* III ).. , * kk = 3.
4.1.2. () N (N = I, II, III) $
! N 0 > N .
. C*
N = I *. 4.1.1 / ,! * N = II, III N, * - N 0 = N + 1.
N = II. r = > L?(): , *,
L?(): (r) = L?> 3.5.
N = III. & L | III. & 3.4 :
L = E f p $ rpg, *# r | . 9 , *
).. r r L?():> ).. L.
& 3.2 * L?():> (r) = L.
. Q1 Q2 L, ) , L, A , L )* r ,
(p) (p ^ Q1) _ (:p ^ Q2), >.
56
. . 4.1.3. * N (N = II, III, IV) $
! N 0 < N .
. / ,! * N 0 = N ; 1 * / ) . & N = II *.
N = III. L?> 0 () >, *, 3.5 L?> *
/ I II.
N = IV. L?(): ) , *, , ).. ) >, 0 L?():> 6,! L?(): .
4.1.4. + L | N = I, II, III, IV,
(L) = 4 10 14 15 "(L) = 4 16 64 256 .
(L) ).
L 0 b(() ? >)A 256 ) . # b1 b2 b (b1 $ b2). U*, *
L ` b(p ? >) , b( x y) > (x y) 2 f? >g ) ) (. 3).
D*, L ` b1 $ b2 , b1(p x y) b2 (p x y) * /
) ) f( 0 1) 2 f? >g3 j (0 1) = ?g
2 (4 ; k k), *# k k = N (. * 3.4.2). U
"(L) = 256=22(4;N ) = 22N .
4.2. 0 * ) "(L) (L) ) *) ) .
. # (. 31, c. 4]) , (A1) * L
(A2) (A ! B) ! ( A ! B) ()
MP, Sub Nec ( ) A ` A:
( * * K. J / , * .
4.2.1. + )
L ) K MP Sub, L | ( L RE) ,
, L | ( L Nec).
57
0 * , K ) :
(A3) p ! p
()
(A4) p ! p
()
(A5) p ! 3p
(*)
(A6) 3p ! 3p
()
(A7) ( p ! p) ! p
() 9#)
(A8) ( (p ! p) ! p) ! p () 8*):
J 31, . 5]: T = K + (A3)A
K4 = K + (A4)A S4 = T + (A4)A KB = K + (A5)A S5 = S4 + (A5) = T + (A6)A
GL = K + (A7) | 8{9#A Grz = K + (A8) | 8*.
& 0) 31, . 5, 12], ,
* L 0 , S5, "(L) = 1, "(S5) = 16. 0 0 * (L).
4.2.2. + L | K L GL, "(L) = (L) = 1.
. & "(L) > (L), * , *
(L) = 1. U :
r1 = ()A rn+1 = () ^ 3rn n > 1:
(1)
&, * rn L *. &
N > m > 1 | *, p1 : : : pN | * ,
N = f1 : : : N g. = :
ANm _ j 2N
pj !
_
J N jJ j=m
_ j 2J
pj :
4.2.3. + m > n, ANm 2 K(rn) N > m.
. 31, . 5] K:
K ` A , A * ) *) /) .
& (W R j=) |; , x1 2 W. &W pj A 0 *
, * x1 j= rn
: (
j 2N
*) 0 x2 : : : xn 2 W, * x1 R x2 R : : : R xn, 8i = 1 : : : n 9j = j(i) 2 N xi j= pj . 1,
;
W pj . J N , jJ j = m, * J fj(1) : : : j(n)g, * x1 j= rn
j 2J
4.2.4. + m < n, ANm 2= GL(rn) N > n.
. 9 GL *) -
) ) / 31, . 5]. = W = f1 : : : ng * < ) *)A 58
. . i j= pj , i = j, 1 6 i 6 n, j 2 N . 1 (W < j=) GL 1 =
6j tr n (ANm ), * .
? 4.2.3 4.2.4 , * ANm 2 L(rn ) , m > n N > maxfm ng, *, L(rn) 6= L(rr ) n 6= r.
* K K4 GL 31, . 1] , * (K4) = 1.
= - . (. 31, . 10]), :, ::.
4.2.5. * ! GL $ ,
-, )# : .
.
9 r m 3Q
m
: 3Q, m > 1, Q | . 1 GL ` m 3A $ m ?, r 0 GL m ? : m ?, *, GL(r) = L?> 3.5.
310], GL : GL( n ) = GL n > 1. C, GL
7 *) - .:
() : : : 3 : .
= , (A3). 2 ) 4.2.2A , )
(1) 0 (). U .
4.2.6. + L | K L Grz, "(L) = (L) = 1.
. :* , * (L) = 1. # :
r01 = ()A r0n+1 = () ^ 3(:() ^ 3r0n) n > 1:
(2)
&, * L(r0n ) *. 7 ANm , 4.2.2.
4.2.7. + m > n, ANm 2 K(r0n) N > m.
4.2.3 *# ,
; W p
* x1 j= r0n
j *, * ( j 2N
*) 0 y2 x2 : : : yn xn 2 W , *
x1 R y2 R x2 R : : : R yn R xn
8i = 1 : : : n 8j 2 N yi =
6j pj 8i = 1 : : : n 9j = j(i) 2 N xi j= pj :
r
4.2.8. + m < n, ANm 2= Grz(r0n) N > n.
59
. , * Grz *) ) ) *) / 31,
. 12]. J W = f1 : : : 2n ; 1g * 6 / j= : i j= pj , i = 2j ; 1, i 2 W,
j 2 N . 1 (W 6 j=) | Grz 1 =6j tr n (ANm ).
? ) .
1 * K T S4 Grz (. 31]), 4.2.6 (T) = (S4) = 1.
KB rn (1) n > 1 0 r2 A * (2).
1 .
4.2.9. + L | K L KB, "(L) = (L) = 1.
. 1 "(KB) = 1, "(L) = 1. = r00n, ) 00
00
00
1 (p) = p ^ 3pA
n+1 (p) = p ^ 3(p ^ 3(:p ^ 3(:p ^ 3 n(p)))) n > 1 (3)
ANm , 4.2.2. 9 ,
* m > 2n, ANm 2 K(r00n) N > m. & KB *) / , , * m < 2n, ANm 2= KB(r00n) N > 2n. C, L(r00n )
* (L) = 1.
J, S5. 2 / (W R) / R = W W.
4.2.10. "(S5) = (S5) = 16.
. 9 S5 0 ) ( 0)) :
(i) r, :r, r:, :r:, r | 0 = () ! A
(ii) r, :r, r | 0 () = ^ 3A
(iii) r, :r, r | 0 ? = _ :.
: * , * 0 ! . D /, *
0 0 () , 0 0 ! *.
& , * (S5) = 16. ( (iii) $
, ) ( S5) :p $ p.
( (ii) $
: ) :p $ : p. 9 ? > () : *
) .
r0
60
. . :, ( p $ p) 2 S5(:) n S5()A p 2 S5() n S5(:). D*, (ii) (iii) *. J * (i).
4.3.
!
D 35] .
& T U | * , T *. = * , )
* , T .
1 ) , ) U , .
( 0) RE , *,
/ . &0 4.3 RE.
J ) ) :
D = GLf: ? ( p _ q) ! ( p _ q)g | :A
S = GLf p ! pg | C.
# * Fn n+1 ? ! n ?, n 2 ! = f0 1 : : :g. 36] , * * :
GL = GLfFn j n 2 g D = D \ GL GL = GL
_
;
n=2
;
:Fn S = S \ GL ;
!, ! n *. _ ! n *, GL D S GL A , D! = D, S! = S, GL! = Fm.
= * C S, ) * ) . U/
S, 37].
. !
M = (W j=)
( W | , | **
W ), 0 r %""
b, * (1) fx 2 W j r xg | * ( | /, / )A
(2) fx 2 W j x rg * (! + 1) A
(3) 0 W rA
(4) b | /
0 WA
;
;
61
(5) ) *) fx 2 W j x rg.
N A M, *: M b j= A.
4.3.1 (%7]). S ` A , A ! ! .
& S, , * (2) 0 SA , "(S) = 1.
4.3.2. (S) = 1.
. = (2) ANm , 4.2.2. 4.2.7, m > n, ANm 2 K(r0n ) S(r0n) N > m.
& m < n. & ) M = (W j=), tr n (ANm ) N > n.
& W = fbg V , V = f: : : x;2 x;1 x0 x1 y2 x2 : : : yn xng, W * / (! +1) ,
0 b | *, 0) V : : : : x;2 x;1 x0 x1 y2 x2 : : : yn xn .
U / : b j= p1A x;i j= p1 ) i > 0A xi j= pi
i = 1 : : : n. 1 M | ) ) 0 r = x1 ,
b =
6j tr n (ANm ).
4.3.3. + L | ( RE), $
K L S, "(L) = (L) = 1.
1 , L " 35], L S, "(L) = (L) = 1. ? 36] , *
, S ( " 35]), * GL . &, * L "(L) < 1 (L) < 1.
& k = 3k 1) = fn 2 ! j n > kg, k > 0. = * GLk = GLk . U*, * GLk = GLf k ?g.
. & ) F
# * , ) ) 0 ) F .
H F (k) * , * F ) )
k ?. # * (r )(k) = r (k) . '
deg(r ) r # (p), deg(?) = deg( ()) = 0A
deg(r1 ! r2) = maxfdeg(r1 ) deg(r2)gA
deg( r) = 1 + deg(r):
r
r0
;
;
;
;
62
. . 4.3.4. GLk ` F $ F (k) ! F .
.
:* k , * GL `
k
(k )
;
`
? ! (F $ F ).
4.3.5. "(GLk ) < 1, (GLk ) < 1.
. J GLk r 0 r(k),
;
;
;
/ * * 0) *
, *, "(GLk ) < 1. :,
35] , * GLk Nec,
* 4.2.1 RE, , *
(GLk ) 6 "(GLk ) < 1.
# ! * .
C k > 0, * (! n ) fn 2 ! j n < kg, 0 3k 1) GLk GL . C, "(GL ) 6 "(GLk ) < 1.
U, , *, * (GL ) <
< 1, GL , , RE. 1 , , * L = GL r * r(k).
& A 2 L(r), L ` tr (A). 1 L GLk , 4.3.4 L ` (tr (A))(k) . 9 (
A r), * (tr (A))(k) (tr (k) (A))(k) , L ` (tr (k) (A))(k) , 4.3.4 L ` tr (k) (A), *
A 2 L(r(k)). 2 , 0 L(r) = L(r(k)). 1
, (GL ) < 1.
;
;
;
;
;
;
;
;
;
;
;
;
r
r
r
r
r
r
;
5. # 0 , MP Sub. 4.1 (,!)
) A * 0 / ) . D 0 / ) ) ), * ) ) ) ( . ) 4.2 4.3). 7 )) *
) , *
) ) ). # * .
. 9 M " L, r, * M L(r).
U/ , , 38] ) ) 9 ( * , *
M L, 38] - L
M-.).
63
5.1. () !
.
. & L | . ? , *) 39], , * L ` : ?, L L() , *
L L> . J L() L> * .
C 5.2 5.4 ) , , *, ) *
.
5.2. M L(r) L RE, 3M] L(r), 3M] | , )# M RE.
5.3. + 0 2 !, GL GL.
. & 4.2.1 3GL] Nec, 0 : ? ( ? ! ?), GL 0 ?. D*, 3GL ] * GL.
5.4. + M L L ) !-
L? L() L: L> , M ) !
.
. _ L Lrj , j 2 f0 1 2 3g (* .
3), L(r) Ltr j (r) .
: * # 35] t(L) L. & (W ) | * r.
(%5]). _ x | (W ), d(x) = 0A
* d(x) = maxfd(y) j x yg. (
M = (W j=) # . ' A t(A) = fn 2 ! j
M n r, * M r =6j Ag. '
t(L) S L O ) , ) L:
t(L) = t(A).
A2L
= , ) GL. : L 0 : L L> , L
0 , 0 2= t(L)A , L L? L() L: . U .
5.5. + L M ) GL, 0 2= t(L) 0 2 t(M), M L.
. & L L> , M L? L() L: L> .
: *
.
r
64
. . 5.6. % # , $
M ,! L:
(i) "(M) 6 "(L) (M) 6 (L).
(ii) $ M $/ $ L.
(iii) const(M) 6 const(L), const(L) $
0
L.
(iv) L RE, 0
) M .
5.7.
(i) * $
! .
(ii) + L RE ) > : ?, GL 6,! L.
(iii) + L ) GL ) ! $ , GL 6,! L.
(iv) GL $
, ! GL, ) , ! 3k 1), k > 0.
.
(i) C 5.6,(i) 4.3.
(ii) L ) .
(iii) ? 31, . 7], * GL 0 n ?, n 2 GL `
` n ? $ n+1 ?, 0 const(GL ) < 1. D*, const(L) < 1, const(GL) = 1, 0 GL 6,! L 5.6,(iii).
(iv) 35] , * Nec ( RE, * 4.2.1) GL GL , ) 3k 1) k > 0.
. N A p, ) 0
A ) . N A # , A , A B.
& p = (p1 : : : pn) | , A , ( * ) 0 :
_ (p ^ B )
A$
(])
;
2f?>gn
B | .
. 7 , * L , A L ` A L ` B 2 f? >gn,
65
B | (]). : , L , ) ) .
5.8. L | , M | , )# $
(A5) p ! 3p, M L(r)
! r. L(r) ! L() ,
L> , L()>.
. :* , * L()> L(r). =
rp. _# (]) :
rp $ ((p ^ Qp) _ (:p ^ Q0 p)):
9 M , *, L ` r(p ! q) ! (rp ! rq). & (]) 0
L, *:
(a) L ` Q(p ! q) ! (Q0 p ! Q0q)A
(b) L ` Q(p ! q) ! (Q0 p ! Qq)A
(c) L ` Q0 (p ! q) ! (Qp ! Q0q)A
(d) L ` Q(p ! q) ! (Qp ! Qq):
Y* (p ! 3p) L ` (:Q0:p ^ Q:r:p) _ (Q0:p ^ Q0:r:p), 0 O
) ) ( :p p):
(e) L ` Q0 p ! Q0:rpA
(f) L ` Q0 p _ Q:rp:
, M Nec, 0 r- M L (g) L ` A, L ` QA.
1 L ` Q(p ! p), , p q (b), *
(h) L ` Q0 p ! Qp:
(e) (f) (i) L ` Q0 :rp _ Q:rp
(h) (j) L ` Q:rpA
r 0 *
(k) L ` Q((Qp ! :p) ^ (Q0p ! p)):
66
. . ? (d) (g) , * L(Q) , 0 :
(l) L ` Q(A ^ B) ! QA
, (k) *
(m) L ` Q(Qp ! :p)
(d) (n) L ` QQp ! Q:p:
? (m) (g) (o) L ` QQ(Qp ! :p)A
0
(n), *
(p) L ` Q(p ^ Qp)
(q) L ` Qp:
1 ,
(r) L ` rp $ (p _ Q0p)
*
(s) L ` r? $ Q0?:
1 (a) * L ` Q0p ! Q0 q, *,
(t) L ` Q0 p $ Q0?:
` (s) (t) (r) (u) L ` rp $ (p _ r?)
L()> (. 3.4) *: L()> L(r).
H * ) ,
.
5.9. '
GL .
. = (]) - A.
: GL 6 ` B 2 f? >gn, 67
GL 31, . 84] | * r, * r =6j B . ` r =6j B ) * r, 0 / j= 0
*, r j= pi , i = >. 1 r =
6j A, * GL 6 ` A.
5.10. + K 6 ` A, !/ ( (W R j=) 0 r 2 W , $
r =6j A # x 2 W , $
x R r.
. & K 31, . 5] M = (W R j=), * r =
6j A r 2 W. &
# M0 = (W 0 R0 j=0). & W 0 = W fr0g, r0 2= W.
U/ R0 :
0) W n frg / R0 R, x y 2 W n frg, x R0 y , x R yA
M0 r r0 * W n frg, * r )
, r R0 x , r0 R0 x , r R x x 2 W n frgA
M0 0 r0 ) 0 W n frg, ) r )
, x R0 r0 , x R r x 2 W n frgA
r R r, r R0 r0 R0 r0.
U*, * x 2 W 0 x R0 r.
& / j=0 / j= W 0 : r0 j=0 p , r j= p p. 1 , * F :
r0 j=0 F , r j= F 8x 2 W (x j=0 F , x j= F):
C, r =6j 0 A.
5.11. '
K K4 .
( *# 5.10, K4) 5.9.
5.12. + L | X | )
, LX | .
.
V : A LX ` A 0 , * L ` ; ! A * ; X.
? (]) A, * ^
_
p ^
; ! B :
;!A $
2f?>gn
V
V
? LX ` A L ` ; ! A, L ` ; ! B ^
2 f? >gn, *, LX ` B .
68
. . 5.13. 1
GL , D , GL , !, ! n $
.
J * ; = () ^ .
5.14. L )
, L(;) L L(;) L.
L(;) = L + f p ! pg.
. * () *. : *: L(;) L1 , L1 = L + f p ! pg. & L1 ` p $ ;p, L1 ` A $ tr; (A) A. & A 2 L(;), tr; (A) 2 L,
tr; (A) 2 L1 , *, A 2 L1 .
31, . 12], GL(;) = Grz. ? ,
* K(;) = T, K4(;) = S4. J B: B = T + (A5),
(A5) | *: p ! 3p.
5.15. '
KB, B, S5, ) 2 S5, )# L()> , $
K, T, K4,
S4, Grz, GL , D , GL , !, ! n $
.
. ? 5.8 , * ) , ) *, / , * ) ): 0
L() , L> L()> . :, 311] , * / S5
. &0 K, K4, GL , D GL ) . : T, S4 Grz / ,! , * T ,! K,
S4 ,! K4 Grz ,! GL.
&# , ) GL
* .
5.16. GL ,! GL $
) !.
. 310] GL:
GL( n ) = GL n > 1. # k > 1, * 30 k) = fn 2 ! j n < kg. = *.
0 2= . &, * GL( k ) = GL( k ). * () *.
& A 2 GL ( k ), GL ` tr k (A). 1 t(tr k (A)) 30 k). U*, * GL- *) ,
/
* k, tr k (A) tr> (A) . J * * GL L> tr> (A) 0
( , tr k (A)) . U
GL ` tr k (A), *, A 2 GL( k ).
0 2 . 0 * , * GL ( k+1 ) = GL( k+1 ). D * V* ().
U* F = Fn. U*, * GL ` B , GL ` F ! B.
;
;
;
n2
69
& A 2= GL( k+1 ), (W j=) r, * r =
6j tr k+1 (A). _ d(r) < k, / (W ) )
0 xk xk;1 : : : xd(r) = r ( * d(xk ) = k) / j= , * *) xk r .
1 *) xk r tr k+1 (F ),
* xk =6j tr k+1 (A).
1 , *, * d(r) > k, r j= F ,
t(F ) = 30 k). C, r =
6j F ! tr k+1 (A), *,
GL 6 ` F ! tr k+1 (A) A 2= GL ( k+1 ).
Y _. y. J
1. 9. {
) *
.
1] Boolos G. Logic of Provability. | Cambridge: Cambridge University Press, 1993.
2] . . | ., 1974.
3] Makinson D. There are innitely many Diodorean modal functions // Journal of
Symbolic Logic. | 1966. | Vol. 31, no. 4. | P. 406{408.
4] Sugihara T. The number of modalities in T supplemented by the axiom CL2 pL3 p //
Journal of Symbolic Logic. | 1962. | Vol. 27, no. 4. | P. 407{408.
5] $%&'( ). *. + ',- -, '&.%/01- ./'& // 2.(.
$* ))). )% '&'. | 1985. | 3. 49, 4 6. | ). 1123{1155.
6] 5'6( 7. 8. + 9: ;%;.:,- ./'& //
2.(. $* ))). )% '&'. | 1989. | 3. 53, 4 5. | ). 915{943.
7] Visser A. The provability logics of recursively enumerable theories // Journal of
Philosophical Logic. | 1984. | Vol. 13. | P. 97{113.
8] Zeman J. Modal systems in which necessity is <factorable= // Notre Dame Journal
of Formal Logic. | 1969. | Vol. 10. | P. 247{256.
9] Makinson D. Some embedding theorems for modal logic // Notre Dame Journal of
Formal Logic. | 1971. | Vol. 12. | P. 252{254.
10] $>6. . $. )(&( &%&(& ( - ./'& // 3.,
8 ?0.. 9. ; '&. . (, &. 1986. | ). 4.
11] Scroggs S. J. Extensions of the Lewis system S 5 // Journal of Symbolic Logic. |
1951. | Vol. 16, no. 2. | P. 407{408.
' ( ) 1997 .
. . . . . 517.588+519.68
: , , !" " "#, $% "&.
'( ) * * ) ) ("
"!$ ( ( ( &" &+ ( !" " "# * (, %+* !() ! " "$ . , !" "$ &-% (
!&) &"$. " * ( "! * !"$(,
% (%+) & .) " "%&* * (, " ( !(! !. /, !("$$
"$ (* .), "!* ( ) !+* &* (, &(% (#!% " "#) | ( "
.! !! ) ) * |
(#$ (( $% " "*
(* &() &-") " ! .
Abstract
A. W. Niukkanen, Analytical continuation formulas for multiple hypergeometric
series, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 1, pp. 71{86.
Applying canonical forms of multiple hypergeometric series along with the use
of the operator factorization method makes it possible to obtain, in explicit and
most general form, the analytical continuation formulas directly applicable to arbitrary series having the Gaussian type (2==1) with respect to one or several arguments. The formulas help us to unify a great number of particular formulas scattered
throughout the literature. Moreover they give us a complete set of relations for any
non-standard series provided that it pertains to the Gaussian type with respect to
at least one of its variables. Due to simplicity and universality of the basic relations
/"# "(), (* &) () * (, ("% !"$( ) &, # , (") "$). !"!&"6 ( "#), "!" #! 7) !"$* "() ( 97{01{00317 01{01{00380).
, 2001, 7, 9 1, . 71{86.
c 2001 !
"#
$%&,
' () *
72
. . there arises an importantpossibilityto implementcomputer-aided analysis of numerous repeated transformations with respect to di:erent arguments of the series and
to join these transformations with other important types of transformations. This
possibility may have signi;cance for mathematical analysis, mathematical physics,
computer algebra and theoretical chemistry.
1. 1] 2{5] !!"$ , & !'$$$ (2==1) ' . N F(x1 : : : xN ) ' xn $'$ "$ (p==q) ! Fqp, '"! N F ' $ N F $&" xn .
+ $ & ' "$! !!"$ . , !
$$"$ $$ ' -'$$ $$ $ ' ' $ $$
$'$ "$. $'" ! $$'& ' !& '& $! '& .
.'$, ' "$! & $&" $$ ". / ', ' "$! $
!! $ , &$ ! 6,7]
($'" ' ! !) 8] ($'" ' $ ). ! 8] &$ !"$ $$ ! , $ $ ' "$! " 1.
2" "$! ' -'$$ F12 $ & 3$ ', $& '! $ '! 24 3 +' !!"$! ' -'$$ 6, 7]. 5$ ' "$!
F12 6,7]
1
6
z
1
+
6
1
2
0
2
;2
1
2
2
0
z
F = ; (;z) F 1 + +
0
1 02
21 1
1
+
+ ; 02 21 (;z);1 F 1 +10 16 z (1)
01
12
1 < (, .. " $, &" !&"( &, %+* . "! ) $, ! (. =9{15], # ( $). /"$!
& ", "$! (, "$ "# ( &* =9{11], " ( ", ($ " ".$ ? 6.
...
73
!
j arg(;z)j < ;
a b==c d] = ;(a);(b)=;(c);(d)
i| = i ; j :
(2)
;' "$! !!"$ ! '" ! .
< $ $$ , " $$ ' "$! !!"$ ( &! "$ ), & !'$$$ (2==1) ' . , $ $$ ! $$ ($ ! ''&$ $ ) '$ , " '
"$! , ! , $ , G -. / '$ " ' $$ 3
N +1F N + 1 (x0 x1 : : : xN ) (x0 x), ! x0 ! '& . $$ N +1F '
$ $ h jq mi, $ $$ = ( qi0 + m i) ( qi0 + m1 i1 + : : : + mN iN ), ! ( i) = ;( + 1)=;() |
$ ,!, (i0 i1 : : : iN ) (i0 i) | $' N +1F (q m1 : : : mN ) (q m) | $ $ "$
. , "$ ' " , $$ , ". $
L " = L(i1 : : : iN ), $ !
$' i0 , "$'& ' G ! !!"$! N +1F, &! !'$$$ (2==1) ' x0, $ G = N +1F h1j1 m1i h2j1 m2i==h0j1 m0i L 6 x0 x]:
(3)
,$ ' $ !!"$ , & (2==1) ' $ , &$ "$ $'" (3), & ! "$! , "$ &
' "$! , $ "$ $'" $$'&! "$ G . ? $ & '$ $$& ! $$ ' "$! "$ G $ $ & .
/ &"$ $ ! , "
@ ''& "$'&
$' ! "$ ! !!"$ , $ = $$ $$ | =$"$ ' | , $ $ "$
!!"$ ', $"$ . .
74
. . 2. A$ ' "$ "$ ' $ ' "$! G , = (N + 1)- $
! , ' $' $
' i0 , $$'&' !'' x0, 3 | $
$ i1 : : : iN . $' (3) ' $' '" ' -'$$ F12, $
i1 : : : iN . , = ' '' (1) ' ' '& BC
$ B!C, ' '" "$
G $' ' , & ' (1) $'" ! "$ ($. '' (15)).
D$ '! $$ , " $'$ $
'' (1), ' ! $, $ & ! ! $ E{F$ 6]. , = " $ ' @ ' "$! G $! =$"$ , $ . ,'$ ! $$
, " @ $ $3 (1) $
$ ' G , '" $3 (1) "$ "$! $'" $ '.
G' $ , ' $
$ ' $3 $ H-=
$3 ( & $3 f f1 = f f2 , ! | H-' 4], H-
$3& f1 = f2 ). $$
H-= $3 ' $3 $3, ' ! $ . / ' "$! G $ ' $3,
& $$ ' "$! $! b = N +1F h2j1 m2i L 6 x0 x]. ? (1==0) ' x0 .
1. b b
(;x0 );2 N +1F
h2j1 m2i L 6 x;0 1 (;x0);m2 x]:
(4)
. , ' b '' b = F01
26 ;x0d(s)] exp(;s) NF
L6 (;x0);m2 xsm2 ]js=0
$ ' "$ (5) $3
(5)
...
75
F01 26 ;x0d(s)] exp(;s) , F01
26 ;x0d(s) + x0]
(6)
;1
1
;2 1
F0 26 ;x0d(s) + x0] (;x0 ) F0 26 d(s) + x0 ]
(7)
;1
;1
1
1
F0 26 d(s) + x0 ] , F0 26 d(s)] exp(x0 s)
(8)
! ' (7) $ ! ' "$! F01
a6 x] (;x);a F01
a6 x;1] ! F01
a6 x] = (1 ; x);a , jxj < 1,
'! & $$ $3 H-=$ (
& F1(s d(s)) F2(s d(s)) H-= '! '!',
$ F1(s d(s))J(s)js=0 = F2 (s d(s))J(s)js=0 & J(s)6 $
' $ F1 , F2). ' '& b '' (5) ( ! ! !"$'), '" $3 (4), " 3 $ 1.
1. G (4) (7) $ !, " "' " = ' B""!C $ ' '. / " x0 2 m2, b "$ (4) $$, b "$ (4) $$. D, $ $$ b "$ (4),
b "$ (4) $$ $ BC () " b $, ! @ " " $$.
2. b -
G
G
2
;
0 12==1 02]C1G ;
0 21==2 01]C2G (9)
(9)
3
1
x
m
1
2
0
C1G = (;x0 );2 F 4h1 + 20j1 m20i h2j1 m2i L 6 x0 (;1) xm
0 2 5 (10)
h1 + 21j1 m21i
2
3
1
x
m
2
1
0
C2G = (;x0 );1 F 4h1j1 m1i h1 + 10j1 m10i L 6 x0 (;1) xm
0 1 5 (11)
h1 + 12j1 m12i
;- (2).
. , (4) x0 ! x0st, x ! xsm1 tm0 ,
"$ (4) Q F01
16 d(s)] F10
==06 d(t)]js=t=0. G & '! ' ($. 4]) ' $ Q '& "$ (4) $ "$ G . / "$ "$ $ '
! F
d6 d(s)]s $3 H-=$
F d6 d(s)]s , (d )F
d + 6 d(s)]
(12)
76
. . !" ! ' "$' $'"& = ', $$'&' " = n = 0 1 2 : ::. G & '! ' (12) G , H-=
' (4), '" G (;x0 );2 (1 ; 2)(0 ; 2);1 F01
1 ; 26 d(s)]F10
==0 ; 26 d(t)]
N +1F
h2j1 m2i L 6 x0;1s;1 t;1 (;x0 );m2 xsm1 ;m2 tm0 ;m2 ]js=t=0: (13)
, (13) '' !
!"$', L1 m12i h2j1 m2i L 6 x;0 1 (;xx0 )m2 (
h
j
1
; 2) N +1
1
2
;
2
G (;x0 ) ( ; ) F
(14)
h02jL1 m02i
0 2
!
( ) ;( + )=;() 12 = 1 ; 2 m12 = m1 ; m2
. . /' "$ (14) $
( ;i) = (;1)i (1 ; i);1 (14) ' ' (9) 3
$ $ 2, $3, '"$'& ' ($. (9)), "$ ! $ 1 $ 2, m1 $ m2.
2. ;' (9) (9) !" ' (4) $$,
" &$ B""!C $ ' N +1F , = $$ " $ " ! x0 . , ! $ ' (4) $3 (9), (9) $ "$! G $ , " ' G $$.
,$ ' ' G & $ !" $
(9) (9), "&$ $ , $ ' $
G ($. (3)) $" $ $, , " ($) ' "$! G $' ' , (9) (9),
$ $"$ '& " $$'& $ $ $$ x0, '! $$ $"$ .
A @ 3 =$"$ " 1. G
C G = ;
0 12==1 02]C1G + ;
0 21==2 01]C2G (15)
C1G C2G (10) (11) -
.
G
$'" x1 = x2 = : : : = xN = 0 ' (15) "$ $ $
' (1). + ' "$ 3, '' (15) $ ...
77
" =$"$ & $ & $ (1)
$ $$'& G ($. "
2).
3. 24 ! "
3. M , @ '
(9) (9), $" $ x0 x1 : : : xN , ' C1G C2 G '& $$ ' L(F ) = 0, ' $ '
F = G.
, ! $ 3 C1 F12 C2F12 $$"$! ' -'$$ 6, 7] &" , " ' C1G C2G &$ ' 3 $$ L(F) = 0 $$ " x0 = 1.
,$ ' & 3 $$ L(F) = 0 $ & ' 3 = $$, ' (15) $$ ' $, & '& G , $ ' 3 $$ L(F) = 0 $$ " x0 = 0,
" ' 3 $$ x0 = 1.
/ $ ' $ $" ! 3 $$ L(F) = 0, ' G , B C =' '& $ & C1 C2, =$! L1 L2 ($. 1]) $ =
. , ', & ' ! BC, $'& '' (15), '" $ ' $, ! ' ', $$& $$' L(F) = 0.
, ' ' G @ 24 " ,
&$ 6 !' ', $$'& @ ' 3 ' -'$$ $$ $ "
0 1 1 8]. N 24 ! &$ !
24 3 +' !!"$! ' -'$$ ($. . 6.4 8]
. 2.9 7]). $'" x1 = x2 = : : : = xN = 0 = "$ $&
$ 3 +'.
M ' 3 ' -'$$ $$
" 0 1 1, " $ u 6] w 7], & (u1 w1 u5 w2 ), (u2 w3 u6 w4) (u3 w5 u4 w6 ). / "
! ! ' un wn ' $ $ $ Uni Wni , $ ', 6] 7], " " = & 6 ', $ '!
$ '! L1 L2 L0 L1 L2 $ & !
! $ i = 1 2 3 4.
78
. . 2. 24 ! " , $ % 1. & %, % G % L1 L2 C1 C2, ! % (1.1){(1.24), '. ( , %, $! % 1, '
% L1 L2 C1 C2 ! )
! %.
@&" 1
, "
#$ % # G
=G =F
"
#
h1 j1 m1 i h2 j1 m2 i L x0 x
h0 j1 m0 i
U11
=
U12
= W12 = L0 G = (1 ; x0 ) 012 "
#
x
h
01 j1 m01 i h02 j1 m02 i h1 j0 m1 i h2 j0 m2 i L x0 m
12
0
(1 ; x0 )
F
h0 j1 m0 i h01 j0 m01 i h02 j0 m02 i
U13
U14
U51
U52
W11
= W13 = L2 G = (1 ; x0 ); 1 #
"
x0
;m1
F h1 j1 m1 i h02 j1 m02 i h2 j0 m2 i L x0 ; 1 x(1 ; x0 )
h0 j1 m0 i h02 j0 m02 i
= W14 = L1 G = (1 ; x0 ); 2 "
#
x0
h
01 j1 m01 i h2 j1 m2 i h1 j0 m1 i L x(1 ; x0 );m2
x0 ; 1
F
h0 j1 m0 i h01 j0 m01 i
= W21 = L2 C0 L1 G = L1 C0 L2 G = x10; 0 "
#
h
1 + 20 j1 m20 i h1 + 10 j1 m10 i L x0 xx;0 m0
F
h2 ; 0 j1 ;m0 i h01 j0 m01 i
(1.1)
(1.2)
(1.3)
(1.4)
(1.5)
= W22 = L0 U51 = L1 C0 L1 G = L2 C0 L2 G = x10; 0 (1 ; x0 ) 012 2
m0 3
x
x;
0
F 4h1 ; 2 j1 ;m2 i h1 ; 1 j1 ;m1 i h1 j0 m1 i h2 j0 m2 i L x0 (1 ; x0 )m120 5
h2 ; 0 j1 ;m0 i h01 j0 m01 i h02 j0 m02 i
(1.6)
79
...
U53
U54
= W23 = L1 U51 = L0 C0 L1 G = C0 L2 G = x10; 0 (1 ; x0 ) 01 ;1 2
x0
x(;x0 );m0 3
h
1
;
j
1
;
m
i
h
1
+
j
1
m
i
h
j
0
m
i
L
2
2
2
2
1
0
1
0
F4
x0 ; 1 (1 ; x0 )m10 5
h2 ; 0 j1 ;m0 i h02 j0 m02 i
(1.7)
= W24 = L2 U51 = C0 L1 G = L0 C0 L2 G = x10; 0 (1 ; x0 ) 02 ;1 3
x
(
;
x0 );m0
x0
h
1
+
j
1
m
i
h
1
;
j
1
;
m
i
h
j
0
m
i
L
1
1
1
1
20
20
F4
x0 ; 1 (1 ; x0 )m20 5
h2 ; 0 j1 ;m0 i h01 j0 m01 i
2
= L1 C1 L1 G = F
"
h1 j1 m1 i h2 j1 m2 i L 1 ; x0 (;1);m120 x
h1 + 120 j1 m120 i h01 j0 m01 i h02 j0 m02 i
#
(1.8)
U21
=
U22
= W32 = L0 U21 = L2 C1 L1 G = x10; 0 "
#
h
1 + 20 j1 m20 i h1 + 10 j1 m10 i h1 j0 m1 i h2 j0 m2 i L 1 ; x0 (;xx)m0
0
F
h1 + 120 j1 m120 i
(1.10)
U23
= W33 = L2 U21 = L0 C1 L1 G = C2 L2 G = x;0 1 "
1 (;1);m2 xx;m1 #
h
1 j1 m1 i h1 + 10 j1 m10 i h2 j0 m2 i L 1 ;
0
F
x0
h1 + 120 j1 m120 i h02 j0 m02 i
U24
U61
U62
W31
= W34 = L1 U21 = C1 L1 G = L0 C2 L2 G = x;0 2 "
#
1
;m1
;m2
F h1 + 20 j1 m20 i h2 j1 m2 i h1 j0 m1 i L 1 ; x0 (;1) xx0
h1 + 120 j1 m120 i h01 j0 m01 i
= W41 = L2 C2 L1 G = (1 ; x0 ) 012 "
#
h
01 j1 m01 i h02 j1 m02 i L 1 ; x0 x(x0 ; 1)m012
F h1 + j1 m i h j0 m i h j0 m i
01 2
012
01
0
1
0
2
0
2
(1.9)
(1.11)
(1.12)
(1.13)
= W42 = L0 U61 = L1 C2 L1 G = x1; 0 (1 ; x0 ) 012 3
x
(;x0 );m0
h
1
;
j
1
;
m
i
h
1
;
j
1
;
m
i
h
j
0
m
i
h
j
0
m
i
L
1
;
x
2
2
1
1
1
1
2
2
0
F4
(1 ; x0 )m120 5
h1 + 012 j1 m012 i
2
(1.14)
80
U63
. . = W43 = L2 U61 = C2 L1 G = L0 C1 L2 G = x010 (1 ; x0 ) 012 2
h j1 m i h1 ; j1 ;m i h j0 m i L 1 ; 1 (;1)m02
F4
U64
01
01
1
1
1
1
h1 + 012 j1 m012 i h01 j0 m01 i
x0
(1.15)
= W44 = L1 U61 = L0 C2 L1 G = C1 L2 G = x020 (1 ; x0 ) 012 2
h1 ; j1 ;m i h j1 m i h j0 m i L 1 ; 1 (;1)m01
2
2
02
02
F4
h1 + 012 j1 m012 i h02 j0 m02 i
2
2
xxm0 10 3
(1 ; x0 )m120 5
x0
xxm0 20 3
(1 ; x0 )m120 5
2
U31
=
W51
= C2 G = (; )
x0 ; 1 F
(1.16)
1 (;1)m210 x 3
h
j
1
m
i
h
1
+
j
1
m
i
L
1
1
1
0
1
0
15
4
x0
xm
0
h1 + 12 j1 m12 i
(1.17)
U32
U33
U34
= W52 = L0 U31 = L0 C2 G = (;x0 ) 20 (1 ; x0 ) 012 2
3
2
0
1 xxm
0
h
1
;
j
1
;
m
i
h
j
1
m
i
h
j
0
m
i
h
j
0
m
i
L
2
2
1
1
2
2
02
02
F4
x0 (x0 ; 1)m120 5
h1 + 12 j1 m12 i h01 j0 m01 i h02 j0 m02 i
= W53 = L2 U31 = L2 C2 G = (1 ; x0 ); 1 "
x #
;1
m12
F h1 j1 m1 i h02 j1 m02 i L (1 ; x0 ) (;1) (1 ; x0 )m1
h1 + 12 j1 m12 i h01 j0 m01 i h02 j0 m02 i
= W54 = L1 U31 = L1 C2 G = (;x0 )1; 0 (1 ; x0 ) 01 ;1 2
(1.18)
(1.19)
3
1 xx;0 m0
h
1
;
2 j1 ;m2 i h1 + 10 j1 m10 i h1 j0 m1 i h2 j0 m2 i L 4
F
1 ; x0 (1 ; x0 )m10 5
h1 + 12 j1 m12 i
(1.20)
U41
U42
= W61 = C1 G = (;x0 ); 2 2
1 (;1)m120 x 3
h
1
+
j
1
m
i
h
j
1
m
i
L
2
2
2
0
2
0
25
F4
x0
xm
0
h1 + 21 j1 m21 i
(1.21)
= W62 = L0 U41 = L0 C1 G = (;x0 ) 012 3
1
0
1 xxm
0
i
L
h
j
1
m
i
h
1
;
j
1
;
m
i
h
j
0
m
i
h
j
0
m
1
1
1
1
2
2
F 4 01 01
x0 (x0 ; 1)m120 5
h1 + 21 j1 m21 i h01 j0 m01 i h02 j0 m02 i
2
(1.22)
...
U43
U44
= W63 = L1 U41 = L1 C1 G = (1 ; x0 ); 2 "
x #
h
01 j1 m01 i h2 j1 m2 i L (1 ; x0 );1 (1
;
x0 )m2
F
h1 + 21 j1 m21 i h01 j0 m01 i h02 j0 m02 i
= W64 = L2 U41 = L2 C1 G = (;x0 )1; 0 2
F 4h1 + 20 j1 m20 i h1 ; 1 j1 ;m1 i h1 j0 m1 i h2 j0 m2 i L
h1 + 21 j1 m21 i
81
(1.23)
3
;m0
1 ;1x (1x;xx0 )m20 5
0
0
(1.24)
2 $'$ " L1 L2 C1 C2. L1 L2 C1 C2 ' G "@ " '
U14 U13 U41 U31 $$. O$ B$3C Li Cj CiLj BC Li Lj CiCj , ! i j = 1 2. $ $3 ' U43 U44 U33 U23 U24 U63 U64 U34 ($. ' 1). A $ " = $' & I:
L21 = L22 = C12 = C22 = I
(16)
$ " & " L0 C0, ''& '! $ '!:
L0 L1 L2 = L2 L1 C0 C1 C2 = C2 C1 L0 C0 = C0 L0 :
(17)
, L0 $ G " ' U12 ,
C0 | ', "&$ "! U51
A(0 ; 1):
C0 G = A(0 ; 1)U51 A(a) xa0 (;x0 );a :
(18)
!': () $ L0 C0 6 () $ . A 8 (a)
" C0L2 C0L1 L0C2 L0 C1 " ' U53 U54 U32 U42 $$, " '! $$ "$
3 ' $ & $" $3
L1 C0 = A(0 ; 1)C0L2 L2 C0 = A(0 ; 1)C0L1 (19)
C1L0 = L0 C2 C2L0 = L0 C1:
(20)
O$ (), & Li Cj Lk CiLj Ck
(i j k = 1 2), '& $3
82
. . B(0 i )CiLi Ci = Lj Cj Lj B(0 0 ; i)Ci Li Ci = Lj Cj Lj B(1 ; 0 1 + i ; 0 )CiLi Ci = Lj Cj Lj B(1 ; 0 1 ; i )CiLi Ci = Lj Cj Lj 0
0
0
0
0
0
0
(21)
(22)
(23)
(24)
! i j = 1 2, i0 = 3 ; i, j 0 = 3 ; j, B(b c) | B(b c) = xb0(;x0 );b (1 ; x0 )c (x0 ; 1);c :
(25)
"$ "@ $ () , , L1 C1L1 , L2 C2L1 , L2 C1L1 L1 C2L1 , " ' U21 , U61, U22 U62 $$. Q& '" , $ ' ' & $ Li Ci (i = 1 2). ,
" , $ () (), $ ' $! $3 (16){(22),
' , " L0 C0 L1 L2 C1C2 $ $ " , '$& ' & 3! "$ $. ,& L0 C0 $$'
" ' U52. D , $$ 5 $, $'$', $' $3 L0 C0 = C0 L0 " $
Li L0 = L0 Li = Li Ci C0 = C0 Ci = Ci
0
0
(i = 1 26 i0 = 3 ; i)
(26)
&! $ Li Ci & L0C0 ' &. / 23 '' 3 ' Uni
$'& '& U11 G , '" 24 ', @ 1,
" 3 $ 2.
4. #! $
!,
& ! ! G-
3. 12 , *! % (U1 U5), (U2 U6) (U3 U4),
, $ i % 2. Un C1Uni , |
12 C2Uni .
83
...
@&" 2
&$ , % #% G
0 012
U 4 + ; 0 120 U63
01 02 2
1 2
U14
=;
U54
= ; 1 +2 ; 0 1+120 20
10
U24
120 1 ; 0
= ; 11 +
+ 20 1 + 10
U63
= ; 1 + 012 0 ; 1
U11
=;
U51
= ; 1 +2 ; 0 1;21
1
20
U31
= ; 1 + 12 0 ; 1
U41
21 1 ; 0
= ; 11 +
+ 20 1 ; 1
U22
= ; 1+ 112+0 21
2
20
U62
= ; 11+;012 12
2 02
U34
= ; 11+;12 012
2 02
U44
= ; 1+ 12+1 120
2
20
01
02
U63 + ;
U14 + ;
U54 + ;
(2.1)
2 ; 0 012
1 ; 1 1 ; 2
1 + 120 0 ; 1
1 2
(2.2)
U24
1 + 012 1 ; 0
1 ; 1 1 ; 2
U54
(2.3)
U14
(2.4)
0 12
U 1 + ; 0 21 U31
1 02 4
2 01
1
02
(2.5)
0
ei ( 0 ;1) U31 + ;
ei (1;
0
U11 + ;
ei
0
ei
0
2 ; 0 12
1 + 10 1 ; 2
1 + 21 0 ; 1
ei
1 + 120 12
3 +;
1 1 + 10
1U4
1 + 012 21
4 +;
1 ; 1 01
0
1U4
ei
1
0U2
1U2
(2.6)
U11
(2.7)
1 + 21 012
6 +;
1 ; 01
1
0
( 0 ;1) 1
U5
(2.8)
ei
1 + 12 120
2 +;
1 1 + 10
0
0
ei ( 0 ;1) U41
1 + 12 1 ; 0
5 +;
1 + 10 1 ; 2
0 )U 1
2 01
e;i
0
ei
(2.9)
2U4
0
4
0
2U4
0
ei
3
0
e;i
0
(2.10)
2
0U2
6
(2.11)
2U2
(2.12)
2
84
. . . + 12 ', @ 2, $ ' $ ' (15) ' Uni , ' "$ '. R $! "$ '$$ $ & $! $, @ $
2, ', @ 1. / '" A B, 12 " ',
, ! $ x0
;x0. ? $ $$ , &! !'& !" ' xa0 . $'" 0 1),
! = 1, ;x0 = exp(i0 )x0 0 = sgn
Imx0] A (a) = exp(;i0 a): (27)
2 0 1) ', $ B-$, $ = 1, ;x0 = exp(i 0 )x0 0 = sgn
Imx0] B (b c) = exp
i 0 (c ; b)]: (28)
D 3! $ ' (15) $ "@ U1i (i = 1 2 3 4) ' U1 ($. ' 1). $' $! $3 (20,) (15) $'" U11 U12 @ ' '
'' (2.5), $'" U13 U14 | '' (2.1). "$
(2.1) (2.5) & 4 ' U2 U6 U3 U4 , '$ 4 " $ ($. ' 1) @, ! $ $'" U1 , ' "$! . .' '& 8 $3 ($. (2.3), (2.4), (2.7){(2.12)) $ $'&
'& '& U5 . , ' (15) "@ $ = ' $ ' U2 U6 U3 U4 ($.
(2.2) (2.6)) 3 $ $ 3.
$'" x1 = x2 = : : : = xN = 0, = = ;1 $3 (2.1){(2.10) 2 $& $ $$'& ', @ . 2.9
'$$! ! 6], ' (2.11), (2.12) "&$ $$'& 3" ' (38), (40) ! 6] $! "$.
5. )*
+ &
O3 +$ '& ($. 8, $. 297]) $ $ , $@ ' & "$ ' "$!
, "& = $, " B&$ $" $ $$"$ $$ $$'& $ ! !!"$ C. $ $$
...
85
' "& $' &. E! '" ' "$! ( "$$, $ ', @ 8]), &$ $$ "$ $'" $ ' (15), ' ! =
' $ . ,$ , $ $ ' (15), "$ $&" 3. D, !
"$ ' ($. , ' $. 295, 297 ! 8]).
,'" 1] $ $ $ ', -', $"& $ $ $3 ' !!"$ , !' $
- . ' $, '$ $ !"$$ $ $3 & ''& $' $$ ! ?E, ! $" !"$ "$ $$ = $3.
. % &'(, *+'+ * * ,,
' ( '& (- *&. .(
, */( 0 12{5], ( 4( 19] /- 40 &*&/0 ' *,( ( ( ,
*6,() *,( /. 7 *' /, +8 * /(0 * /- ( G , / /& *'6. 9( '
',(- (0 *- (/,(0 *: ' /( **( * * ;& ' 110].
<,8 *', *'*6 6 * ' ' /- & +(0 * * * *' &(0 &*&/0 ' *4,8 *&
/0 *- ' 7=> 111]. = '* 40
-(0 *- 11, 11] -'( *,( -( *,
(*84 ' ?** F4 - @ H1 G2 112]. A 0 (/ ;
,(0 6- '( *( : * ' (' 6 * B+{@' 113]. A &
/- & & *&
* &*&/0 ', 840 '(- * * 6' *(0 114]. * 4 /- /- '/ 9 *' ', ' '( *'0' * 6 , * C-;, *84& ,
(, &( 6' ', '84 4 - /- -, ;
*,, ; ' (/,(0
- 115].
,
11] ?. =. . Transformation theory of multiple hypergeometric series and com-
86
12]
13]
14]
15]
16]
17]
18]
19]
110]
111]
112]
113]
114]
115]
. . puter aided symbolic calculations // Proceedings of the 9th International Conference
FComputational ModelingG. | Dubna: JINR, 1997. | P. 219{223.
?. =. . Generalized hypergeometric series NF (x1 : : : xN ) arising in physical and quantum chemical applications // J. Phys. A: Math. Gen. | 1983. |
Vol. 16. | P. 1813{1825.
?. =. . Generalized operator reduction formulae for multiple hypergeometric series NF (x1 : : : xN ) // J. Phys. A: Math. Gen. | 1984. | Vol. 17. |
L731{L736.
?. =. . (- ' &*&/0 ' *,(0 - /- // U*0 . . | 1988. |
W. 43, (*. 3. | X. 191{192.
?. =. . (- *'0' &*&/0 ' *,(0 - /- // >. . | 1991. | W. 50,
(*. 1. | X. 65{73.
@. 9-, ?. 7'-. =(+ '( . W. 1. | >.: ,
1973.
Y. Z8. X*,( / 0 **. >.: >,
1980.
H. M. Srivastava, P. W. Karlsson. Multiple hypergeometric series. | Chichester:
Ellis Hoorwood, 1985.
?. =. . A* ** &*&/ '( 4& ' // >. . | 2000. | W. 67, (*. 4. |
X. 573{581.
?. =. . &*&/0 ' **( *, 0 *,8- &( // .'. *. . |
1999. | W. 5, (*. 3. | X. 717{745.
A. W. Niukkanen, O. S. Paramonova. Computer generation of complicated transformations and reduction formulas for multiple hypergeometric series // Computer
Physics Communications. | 2000. | Vol. 126. | P. 141{148.
?. =. . >' ( * ?** F4 - @ H1 G2 // U*0 . . | 1999. | W. 54. |
X. 169{170.
A. W. Niukkanen. Operator factorization method and addition formulas for hypergeometric functions // Integral transforms and special functions. | 2001. | Vol. 11. |
P. 25{48.
?. =. , <. ?. . j& 0 &*&/0 ' &0 *(0 // <&. | 2001. |
=(*. 1. | X. 41{48.
A. W. Niukkanen, V. I. Perevozchikov, V. A. Lurie. A generalization of a classical
relation between J +n (z ) J (z ) and J ;1 (z ) with comments on the modern state
and trends in the theory of special functions // Fractional calculus and applied
analysis. | 2000. | Vol. 3, no. 2. | P. 119{132.
+ ! # ! 1996 .
. . . . . 512.533.8
: , , , , , !" , #$ , #$# .
&'( : Sub S ;! Sub T $ " * , ], ! " ,' A S A "$" " *$ , $ ] $ T ! , A ' $ $,- $ $. , !# # #
# $ #" " " # $ $ '! . .#" " ( $ $ ! " $
#$# . &
'(", /,-
! ( $ $ $! $!) $. 2 , ,' $!
$! $! $ / " .
Abstract
A. Ja. Ovsyannikov, Ideal lattice isomorphisms of semigroups, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 1, pp. 87{103.
A lattice isomorphism of a semigroup S upon a semigroup T is called an ideal
lattice isomorphism if it induces a bijection of the set of ideals of S onto the corresponding set of T . Left and right ideal lattice isomorphisms are de6ned in a similar
way. The order on idempotents and the property of being a subgroup are proved
to retain under lattice isomorphisms of these kinds. The property of a semigroup
of being decomposable in a semilattice of Archimedean semigroups is retained as
well. Mappings that induce ideal lattice isomorphisms of idempotent semigroups
are described. In particular, each left ideal or right ideal lattice isomorphism of
an idempotent semigroup is induced by an isomorphism.
7' $
8 $ 7 9.
, 2001, 7, : 1, . 87{103.
c 2001 ! " #" "
,
$ %& '
88
x
1. . . ! " #
$1] (
) $2]. +
, S T Sub S S
Sub T . -
" " # S # T . .
/ , , ! , 0 1 # !
. 2 S T | . +
#
: Sub S ;! Sub T $
, ], A S A $ , ] T , A ! . 5 ! $1, x 23] (. # $2, x 30]): , , S T , S . . -
, ! , .
+! !
! , ! $
] , $
] . 8
1{5. 2 , $
] . ; 3 , 4 (
) . +
", 5 $
] .
- $3] $4]. , # $5] $6].
x
2. +
! . > , ! $
-
89
]: "
/
#
. ;# , !, . - S S S@. A ", , , ! .
2 , ! , . 2 S T ! ".
a e f 0
a0 e0 f 0 00
a 0 0 a 0
a0 00 00 a0 00
S: e 0 e 0 0 T : e0 a0 e0 00 00 :
f 0 0 f 0
f 0 00 00 f 0 00
0 0 0 0 0
00 00 00 00 00
2 , S T S 1 a = fa 0g S 1 e = fe 0g S 1 f = fa f 0g S 1 0 = f0g
(1)
1
0
0 0 0
1
0
0
1
0
0 0
1
0
0 0
T a = fa 0 g T e = fe 0 g T f = fa f 0 g T 0 = f0 g: (2)
8#
' S T , x x0 x 2 S , " S T , ( ) 1
'(xy) 2 h'(x) '(y)i ';1 ('(x)'(y)) 2 hx yi:
(3)
C
S T (1) (2) . D , ' " " # S ! # T , # " # S ! # T , # E
. ; ,
' . 8
' , , fe0 00g fe 0g S ,
T .
F
, , . ; , .
G, " ' S T " , 1 ; $" $ ! , " ,'# A S B T ( $ '(A) ';1 (B ) "$", " $ T S $ $.
90
. . /
x y 2 S 1 (3). +# ", "! , , " 0 x y z 2 S x = yz
'(x) 2 h'(x) '(y)i1 0 2 T = ';1 () 2 h';1 ( ) ';1 ( )i1.
8 , ! . H #
.
I J (A) $ L(A)] $ ] , #
# A. . A /
a, J (fag) L(fag) , , J (a) L(a).
1. S T :
() | $
] () !
A S J (A) = J (A) $L(A) = L(A)]
() !" x y 2 S hxi J (y) $hxi L(y)] hxi J (hyi) $hxi L(hyi)].
( ).
() ) (). 2 | S T , A 2
2 Sub S , A 6= ?. ; A J (A), A J (A). 2 J (A) |
, J (A) J (A). 2
#
;1 , ;1 J (A) J (A). J, S , #! A,
, / ,
J (A) = J (A).
() ) (). 2 J (A) = J (A) A 2 Sub S , A 6= ?. H I S . ; J (I ) = J (I ) = I , . . I
T . K
, ;1 K S K T .
" () ) () () ) () #
#
.
+
, J -
$L-
] ... J - $L-]. #
. $ $
] S T % !% '. &
' " ' J $
L], . . !" a b 2 S
91
aJ b , '(a)J '(b) $aLb , '(a)L'(b)]:
) , ' " J - $L-] % J -
$L-
] S J -
$L-
] T.
L (. 24.1 24.2 $1, 2] 31.1 31.2 $2]), S T " " ' # B /
, !
, /
S
! # T , x 2 B n h'(x)i = hxi '(xn ) = '(x)n .
5 # ' ", .
F J - ,
J - / 2. $ S | | S T. &
T + , +
" S "
S T. ) ,
! (. . ) " .
x
3. ! "# ,
%&#% 2 $ ], # xy = x $xy = y]. 2 ,
.
1. $ | $
] -
S T , ' | !%, %
. &
!" e f 2 ES e 6 f , '(e) 6 '(f ):
. - . H S T , e 6 f '(e) 6 '(f ). 2 e < f , . . ef = fe = e 6= f . 2#,
'(e) <6 '(f ), / #
. M
#, '(e) J '(f ):
(4)
5 /
, h'(e) '(f )i = f'(e) '(f )g , 92
. . /
, #
'(e) > '(f ). H (4) .
H , , '(f ) 2 J ('(e)). H #
e < f e 2 J (f ), #
1 '(e) 2 J ('(f )).
; '(f ) 2 J ('(e)), , (4) / .
- F = J (e)=I (e) S . 8
#
e f 2.38 $5] 0- .
2 e < f , F # 0-
. 2#, F ! " B = ha bi, f " ab = f , ba 6 e. ; eJ f ,
f = set s t 2 S . 2 e < f , !
#
s t 2 fSf . f = se et, f 6= et se, f = ef = fe, #
e < f .
2# a = se b = et. D , a b # "
B , ab = f , ba = etse 6 e. H $7] (. # $2, 41.8]) ' B "
, /
e 2= B
(5)
'(e) < '(f ). M , ba < e.
F, '(f ) " '(B ).
J, ea 2= B:
(6)
n
m
n
m
F , ea = b a e = ef = eab = b a b 2 B ,
(5). 2 , ea /
. F : n m (ea)n = (ea)n+m :
(7)
G, ba < e ae = (aba)e = a(ba)e = aba = a, . .
ae = a:
(8)
n
n
+
m
H (8) (7) ea = ea . F
# a (8), an = an+m , #
" B . ; , ea /
, /
'(ea). '(ea) 2 h'(e) '(a)i. H
'(e) '(f ), : f'(e) '(f )g | '(e) < '(f ).
2 f'(e) '(f )g | , , ,
( ). ; '(f ) | " '(B ) '(f ) = '(f )'(e), '(a) = '(a)'(f ) = '(a)'(f )'(e) = '(a)'(e), . . '(a) = '(a)'(e). ;
h'(a) '(e)i = h'(a)i h'(e)i '(e)h'(a)i. ; '(ea) 2= h'(a)i h'(e)i,
'(ea) 2 '(e)h'(a)i, . . '(ea) = '(e)'(a)n n. F, '(f )'(ea) = '(f )'(e)'(a)n = '(f )'(a)n = '(a)n = '(an ), . .
93
an 2 hf eai. 2 e < f , , hf eai = hf i heai. 8 an 2 heai
f = anbn 2 eS , . . ef = f , #
e < f .
2 '(f ) < '(e). ; '(f ) | " '(B ), '(e)'(a) =
= '(a)'(e) = '(a). ; , h'(e) '(a)i = h'(e)i h'(a)i '(ea) 2
2 h'(a)i, . . ea 2 hai B , (6). 2
.
2 | . L , , e f 2 ES e < f '(e) < '(f ). H , ef = e e 2 L(f ), #
() #
1 '(e) 2 L('(f )),
. . '(e)'(f ) = '(e). . '(e) <6 '(f ), , f'(e) '(f )g , '(f )'(e) = '(f ) '(f ) 2 L('(e)) #
() #
1 f 2 L(e), . . fe = f , #
e < f . , '(e) < '(f ), .
F .
2#, .
2. $ | S T . &
G S , G
T.
. H S T , G S , G T . 2 G | S .
- . +
, /
, /
, , .
H /
2 G #, . F . ; /
#
27.3 $1, 2], . # $2, #
34.3]. M , h i ! /
Sub T , ;1 h i " S 3.1 $1, 1] $2]. 2 ;1 h i = hgi g 2 G. ;
g | /
, L(g) = L(H ) H hgi.
H #
() #
1 L( ) = L( ), | h i. ; = 2 T , . .
= . M , | /
. 2
.
-
! .
1. $ A + a b g , G = hgi , B = ha bi !% %
94
. . e = ab, % G, ga 2 hgn ai n. &
gn + G.
. 8
, e " A. 2# C = hg ai. F#, G \ Ca = ?:
(9)
m
H , g = ca m !
c 2 C , gm ba = caba = ca = gm , . . gm ba = gm , , # g;m , ba = e, #
. ; ga 2 hgn ai, ga = cgn ak gnr (10)
c 2 hgn ai, k | , r | " " .
- #
r.
2 r = 0. ; (10) ga = cgn ak . . k > 1, g = ge =
= gab = cgn ak b = cgn ak;1, . . g = cgn ak;1 2 Ca, (9).
M , k = 1, . . ga = cgn a g = gab = cgn ab = cgn, . . g = cgn
, , g 2 hc gni. c = g g;n 2 G, c 2 hgn ai, (9) , c 2 hgn i. M , g 2 hgn i gn # G.
2 r > 0. 2#, / k = 1. 2#,
k > 1. 2 (10), : ga =
= cgn;1(ga)ak;1 gnr = cgn;1 (cgn ak gnr )ak;1gnr = cgn;1cgn ak gnr;1(ga)ak;2 gnr =
= cgn;1 cgn ak gnr;1 (cgn ak gnr )ak;2gnr . 2# , ga = cgn;1(cgn ak gnr;1 )k;2cgn ak gnr agnr. 2# d = cgn;1(cgn ak gnr;1 )k;2cgn .
; d 2 C ga = dak gnr agnr . F, ga = dak gnr;1(ga)gnr , ga = (dak gnr;1)m (ga)(gnr )m m. 2 jgj /
g. 2 m = jgj ga = (dak gnr;1 )m ga = hgnr a,
h = (dak gnr;1 )m;1 dak . 2 k > 1, , h 2 Ca. ga = hgnr a g = hgnr , h 2 hgi. J, h 2 Ca, (9). A , k = 1 (10).
-
(10) ga = cgn agnr :
(11)
n
;
1
nr
n
;
1
m
2# m = jgj. (11) ga = cg (ga)g = (cg ) ga, . .
ga = (cgn;1)m ga. F
# bg;1, e = (cgn;1 )m . ;
c 2 hgn ai, (9) c 2 hgn i.
(11) , ga = gnl agnr ! l, a = gnl;1 agnr :
(12)
F#, n jgj / . 2 ! t, tn jgj gt 6= e.
(12) a = (gnl;1 )ta(gnr )t = g;ta, g;t = e. 2
.
G 2. 2 | S T . G 95
G S . 8 , , G .
; G ! /
. 2# hgi = ;1 h i.
-, g 2 G. - J - Jg S J T Fg F , ! / J -. ; G Jg , G J . 2 F # /
, # ", h i F ( " 2 h i J ), , F # , # 0- . ; , F 0-, 0- # . 2 1 Fg # # . D
, Fg # 0-, 0- .
2 e " G, h"i = hei. 2 2.54 $5] Fg
" B = ha bi " e, e = ab.
2# A = hg a bi #, A 1. ; h i,
, ! /
Sub T , , hgi ! /
Sub S . 2 # hgi ,
g 2 G, , # ".
M , ;1 h"i = hgn i n. 2#,
ga 2 hgn ai. H $7] B "
. 2# = '(a). 2 hgi \ hai = ?, ga 2= hgi hai
ga 2 hg ain(hgihai). M , hgai hg ain(h ihi), hgai h" in(h"ihi). ; , ga 2 hgn ai. H 1
A . , hgn i = hgi . 2
, 2.
8 .
1. , !
$
] S T !% ' +
S n Gr S +
T n Gr T, !
x 2 S hxi = h'(x)i. $ !" x y 2 S n Gr S xJ y , '(x)J '(y)
$xLy , '(x)L'(y)]:
-
, x 2 S n Gr S n | , xn 2= Gr S,
'(xn ) = '(x)n +
" :
) | ) S | ) !
x | !
, !
xk 2 Gr S, xk;1 2= Gr S
, k 6= 4.
. . x 2= Gr S, hxi ! /
Sub S , . . hxi | ! /
Sub T . 2 3.1 ) $1] $2] hxi = h i 2 T . H 2
96
. . , 2= Gr T . > , #
x 7! ". ; #
1.
F #
, x |
/
, '(xn ) = '(x)n 24.2 $1, 2], .
# $2, 31.2]. . x | /
, hxi\Gr S 6= ?.
2 k | , xk 2 Gr S . . k 6= 4, '(xn) = '(x)n $8]. 2 k = 4. ; $8] '(xn ) = '(x)n , '(x2 ) = '(x)3 '(x3) = '(x)2 .
2#, #
# ) ).
A . , '(x2) = '(x)3
'(x3) = '(x)2 .
2#, | . 8
, '(x)3 2 L('(x)2 ). 8 x2 2 L(x3 ), . . x2 = yx3 !
y 2 S . 2 e | hxi xm = e. ; x2 = yx3 = y x2 x =
= ym x2 xm = ym x2e x2 = x2e, #
x3 2= Gr S .
2
).
2 S | | . ; '(x)3 2 J ('(x)2 ), x2 2 J (x3 ), . . x2 = yx3 z ! y z 2 S . H s , /
z s
, # z s = f . ; x2 = yx x2 z = (yx)s x2 z s = y1 x3f
! y1 2 S , , x2 = y1 x3 f , x2 f = x2 .
8 x3 f = x3 , . . x2 = y1 x3. L ! ", , .
2. , !
$
] S T !
, !
| .
x
4. , (
! - , # . H $6, . 105]
AS-. I S /
" S .
3. $ | AS -
S T. &
T AS -
, -
!
"
A S A "
T +
"
" S S=S T=T .
97
. L (., , $6, . 105]), AS- , /
a b
, a b , a2 bn ! n. 2 / T . G 2 T , , . .
2 J ():
(13)
. 2 J (2), J () = J (2 ), : 2 J (2).
2#, 2= J (2):
(14)
;, , hi , / hi ! /
Sub T ;1 hi " S ,
# C = ;1 hi = hai.
2#, a 2= J (a2):
(15)
2# : a 2 J (a2 ). ; J (a) = J (a2 ), #
1
J () = J (ha2 i):
(16)
J, 2 ha2 i. F , 2= ha2 i, ha2 i hi,
. . ha2 i J (2), (16) J () J (2 ), (14). F
C = ha2i, , C " . 8
, D C J (C ) = J (D). H D = ;1 ha2 i, J (a) = J (C ) = J (;1 h2i), #
1 J () = J (2), #
(14).
2# D = ;1 h i. (13) #
1 D J (C ). ; S AS-, , , D \ J (C 2 ) 6= ?.
.! #
1, , h i\ J (C 2 ) 6= ?. 2 C 2 J (a2 ), (15) J (a2 ) J (2), h i \ J (2) 6= ?, . .
2 n ! n. , T AS-.
2 A | S B = A. 2#,
B T . 2 2 B . J, T ,
. . # . ; ;1 hi ;1h i A,
A ;1 h i \ J (;1 hi) 6= ?. 8 ! #
1 h i\ J () 6= ?. M , AC (x) AS-, #! /
x,
AC ( ) 6 AC (). 2 , AC () 6 AC ( ). M , AC () = AC ( ) B # C T . 2
#
98
. . ;1 C , , #
# S . ; , ;1 C A, , B = A, B = C .
2
#
#
2.
8 . +
, # , / . 5 # A l S , A S .
I hhaii , #
/
a .
4. $ S | , | S T. &
T , !
X S X T, Sub X . .
X | S, Sub X
% .
/ " S
T .
. H 2 2 T . G X S #, A l X , A l X:
(17)
2 A l X . 8
M E
S , X , N | # M X . P
, M N | S , / M N l T .
2#, A l X , A M l S:
(18)
2 A l X , a 2 A M , s 2 S . J, sa 2 A M . . sa 2 M , . 2 sa 2= M . ;, a 2 N , # sa 2 N ,
. . sa 2 X . 8
, b 2 hhaii sb 2 X . 2
b | " hhaii, a = ba. sa = sba = (sb)a 2 A, A |
X a 2 A. , sa 2 A M .
2 (A M ) l S , a 2 A, x 2 X . ; xa 2 X \(A M ) = A. J (18)
.
2
(18) ,
(A M ) l T , 2#, A M l T:
99
(19)
B l X , B M l T:
(20)
2 B l X , 2 B M , 2 T . J, 2 B M . .
2 M , . 2 2= M . ;, 2 N
2 L( ), a ;1 L( ) N , , 2 N , . . 2 X . .
2 M , L( ) M 2 M . M , 2 X , . .
2 B . 8
, 2 hh ii 2 X . 2 " |
" hh ii, = " . = " = (") 2 B , B | X . , 2 B M .
8
" (20) #, (18).
(19) (20) , A l X . H S T #
(17) .
; , X .
. X | , ! , / #
31.1.2 $1], . # $2, 38.1.1]. . X |
, , X | , . . X | , X | . H 30.8 $1] (. # $2, 37.8]) X | , . . X /
. H , # # . > , , ,
#
2.7 $1] $2], , !
. , X | . ,
, Sub X / " . H X
30.8 $1] Sub X " . 2 , # " .
; X | T , # Y T . J, ;1 Y S T , , ;1 Y = X , . .
X T .
2
#
#
2 ,
S
/ ! T .
100
x
. . 5. +
, S T " ' # S T , x y 2 S '(xy) 2 f'(x)'(y) '(y)'(x)g:
(21)
+ ' , x y 2 S
f'(xy) '(yx)g = f'(x)'(y) '(y)'(x)g:
(22)
M! , ! . +, ! ha b j a2 = a b2 = b ab = ai
! . > , , #
# .
H / # ! #
.
5. -+ % .
, ! S T ! , !"
, ! S T.
F #
.
2. $ ' | !%, % S T, X Y | S,
'(X ) '(Y ) | T, X 6 Y '(X ) 6 '(Y ).
. ' X $
], !"
x 2 X, y 2 Y '(xy) = '(x)'(y), '(yx) = '(y)'(x)
$'(xy) = '(y)'(x), '(yx) = '(x)'(y)].
. F, ' X , #, '(xy) = '(x)'(y). H . P
, X < Y .
M
'(X ) < '(Y ). ; x xy 2 X , '(x) =
= '(x xy) = '(x)'(xy), . .
'(x) = '(x)'(xy):
(23)
2#, (23) '(xy) = '(x) '(xy) = '(x)'(y):
(24)
101
; '(xy) 2 h'(x) '(y)i '(x) 2 '(X ), '(y) 2 '(Y ), '(X ) < '(Y ),
, , #
/
'(x) '(y), (
) /
'(x) '(y) '(x)'(y) '(y)'(x) '(x)'(y)'(x) '(y)'(x)'(y)
, '(xy) 2 f'(x) '(x)'(y) '(y)'(x) '(x)'(y)'(x) '(y)'(x)'(y)g:
; '(Y ) | '(X ) < '(Y ), '(x)'(y)'(x) = '(x):
(25)
; , '(xy) '(x) '(y)
'(y), '(xy) = '(x)'(y), '(x), , # '(xy) = w('(x) '(y))'(x)
'(x) (23) (25), '(xy) = '(x).
J, (24) '(x)'(y) = '(x). H , '(xy) = '(x) xy = x, . . hx yi = fx y yxg.
. '(x)'(y) 6= '(x), '(X ) < '(Y ) '(x)'(y) = '(yx),
. . '(x)'(y) = '(yx x) = '(yx)'(x) '(x)'(y) = '(yx)'(x), '(x)'(y) = '(x)'(y)'(x) = '(x). 2
, '(xy) = '(x)'(y).
F
, ' X , '(xy) = '(y)'(x) '(yx) = '(x)'(y).
3. $ ' | !%, % S T , X Y | S, Z | " " S, '(X ) '(Y ) | T,
'(Z ) | " " T. . ' Z $
],
!" x 2 X, y 2 Y '(xy) = '(x)'(y) '(yx) = '(y)'(x) $'(xy) = '(y)'(x) '(yx) = '(x)'(y)].
. 2#, ' Z , #, '(xy) = '(x)'(y). H . 2
2, '(xy) = '(x)'(xy) =
= '(x)'(xy)'(y), . . '(xy) = '(x)'(xy)'(y). ; '(xy) 2 h'(x) '(y)i, .
F
, ' Z , '(xy) = '(y)'(x) '(yx) = '(x)'(y).
F #
5 . 2 S | , ' | #
, "! S T .
H 4 X S
'(X ) T , 'jX
102
. . ' S . 2
2 3, #, ' .
- #
5. F
! #
.
4. ) S
T % S T.
F# . 2 ' | ", "! S T . H #
2 # S T . M
30.8 $1] ' S . ;
, (22) # , x y # S , # X Y . H S T , # x y, (21).
- ! .
2 X Y . - #, X < Y ( Y < X ). ; #
2 '(X ) < '(Y ). F, ' X . ; '(xy) = '(x)'(y) 2.
K
2 , ' X , '(xy) = '(y)'(x).
2 X Y . 2 Z | X Y S .
2#, ' Z . ; #
2 '(X )'(Y ) '(Z ), 3 '(xy) = '(x)'(y). K
3 , '
X , '(xy) = '(y)'(x).
F #
5 . ; , 5 .
K # >. +. Q
"
.
*
1] . ., . . .
! 1. | #: %&- ( . - , 1990. ! 2. | #: %&-
( . - , 1991.
2] Shevrin L. N., Ovsyannikov A. J. Semigroups and their subsemigroup lattices. |
Dordrecht: Kluwer Academic Publishers, 1996.
103
3] . . % ! . &/0&/ // 2!
/3 04 5. #5. &. | 6 , 1993. | #. 242.
4] Ovsyannikov A. J. On ideal lattice isomorphisms of semigroups // Colloquium on
Semigroups. Szeged, 15{19 August 1994. Abstracts. | P. 28.
5] 600 ., ;. 5 . . 2. 1. | <.: <,
1972.
6] . . // 5> 5 . 2. 2. | <.: , 1991. |
#. 11{191.
7] Shevrin L. N. The bicyclic semigroup is determined by its subsemigroup lattice //
Simon Stevin. | 1993. | Vol. 67. | P. 49{53.
8] . . &/0&/ // // %&. &. < / . | 1966. | ? 1. | #. 153{160.
( #) * 1996 .
( | 1997 .).
,
,
. . . . . 517.51
: , , , , ! ", ! #{%!.
& '( '
(
)! !
#{%! '( ! ". (C,1) ,
!!
)! '( .
Abstract
A. N. Pavlikov, The Cesaro average for orthogonal-like decomposition systems with non-negative measure, Fundamentalnaya i prikladnaya matematika,
vol. 7 (2001), no. 1, pp. 105{119.
We prove that in case of orthogonal-like decomposition systems with non-negative measure the Abel{Poisson's method of summation is equivalent to positive
Cesaro's methods for convergence almost everywhere. A criterion for summability
of a sequence of partial integrals almost everywhere is given.
. . 1{3] !, # $
%
#, &%
# #.
. % H | $% R C ,
( | )
# &%
# # . * +
fe! g!2 H % (. . ) , .# +
y 2 H !! Z
y = (y e! )e! d(!)
1
' 1223 ( 4 96{01{00332).
, 2001, 7, 4 1, . 105{119.
c 2001 !,
"#
$% &
106
. . $ $ ! # # $ $ 2
& )
! H, ) ) % )
f(k g1
k=1 ( ( (k , (k (k+1 !
1
S
k 2 N, (k = (), ) 2
&! (y e! )e! $ $ (k k=1
Z
Z
!
!
y = (y e )e d(!) = klim
(y e! )e! d(!):
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k
3 !4# &%
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%
! ) .
% !! .4 .
1 ( ). c(!) | ( y=
Z
c(!)e! d(!)
R
kyk2 6 jc(!)j2 d(!).
*. 1], 2], 3].
2 (
!. ").
1
PR
E k k=1 E
E .
gk (x) > 0 gk (x) d < 1, gk (x) 1
P
k=1
*. 6, . 349{350].
x
1. (C,1)-
% ( | )
# &%
# # ,
f(k g1
(, ) (k ,
k=1 | 2
)
1
S
(0 = ?, (k (k+1 ! k 2 N, (k = (.
k=1
.R %!c(!) | )! 2
&! (. %
%
1
fsk gk=1 , $ sk = c(!)e d(!), $ $ ! k
( ),
R
c(!)e! d(!).
-
1
$ %
% fn g1
n=1, $ n = n (s1 + : : : + sn ) | %
% (C,1)-
(
2), ) 1 !)
! fsk g1
k=1.
107
. %
% fsk g1k=11 ! (C,1)-
S, %
% fngn=1 ! S
n ! 1.
% fe! g!2 | ! !, c(!) | )!
2
&! (.
" 1. n
fn g1
n=1 +1
1 < q 6 n , 1
2 Z
X
q
2
ksn ; n k 6 q2 ; 1 jc(!)j2 d(!):
(1)
n=1
#. ; ! n , )
Z
n Z
X
1
!
sn ; n = c(!)e d(!) ; n
c(!)e! d(!) =
k=1 k
n
Z
Z
n
n X
k
X
X
=
c(!)e! d(!) ; n1
c(!)e! d(!) =
k=1 m=1 m n m;1
k=1 k n k;1
Z
Z
n
n
X
X
c(!)e! d(!) =
=
c(!)e! d(!) ; n1 (n + 1 ; k)
k=1 k n k;1
k=1
kn k;1
Z
n
X
= n1 (k ; 1)
c(!)e! d(!)
k=1
kn k;1
X
2
Z
n
c(!)e! d(!) 6
ksn ; n k2 = 12 (k ; 1)
n k=1
k n k;1
Z
n
X
6 12 (k ; 1)2
jc(!)j2 d(!)
n k=1
k n k;1
1. $
Z
n
1
1
X
X
X
1
2
2
ksn ; n k 6 2 (k ; 1)
n=1
n=1 n k=1
6
X
1
jc(!)j2 d(!) 6
k n k;1
(k ; 1)2
k=1
Z
k n k; 1
jc(!)j2 d(!)
; ! n+1 > qn 1
X 1 1 X
1 =
q2 :
;2m = 1
q
6
2
2
k2 1 ; q12 k2(q2 ; 1)
n >k n k m=0
X 1
2:
n >k n
108
. . =.
1
1
2 X
X
ksn ; n k2 6 q2q; 1
n=1
Z
k=1 k n k;1
jc(!)j2 d(!) =
q2 Z jc(!)j2 d(!):
q2 ; 1
.
" 2. 1
X
n=1
nkn+1 ; nk2 6
1 Z jc(!)j2 d(!):
2
(2)
#. ; ! sk n sk =
Z
c(!)e! d(!) =
k
k
X
Z
m=1 m n m;1
c(!)e! d(!)
n
X
n+1 ; n = n +1 1 sn+1 ; n(n1+ 1) sk k=1
Z
Z
n
n
k
n
X
XX
X
sk =
c(!)e! d(!) = (n + 1 ; k)
c(!)e! d(!)
k=1 m=1 m n m;1
k=1
k n k;1
Z
Z
n
n
n
X
X
X
1
1
1
!
c(!)e d(!) ; n(n+1) k
c(!)e! d(!)
n(n+1) k=1 sk = n k=1
k=1 k n k;1
k n k;1
k=1
+
n 1 Z
X
1
n+1 ; n = n + 1 sn+1 ; n
c(!)e! d(!) ;
k=1 kn k;1
Z
Z
n
n
X
X
c(!)e! d(!) = n1
c(!)e! d(!) +
; n(n1+ 1) k
k=1 kn k;1
k=1 k n k;1
Z
n
nX
+1 Z
X
+ n(n1+ 1) k
c(!)e! d(!) + n +1 1
c(!)e! d(!) =
k=1 kn k;1
k=1 k n k;1
X
Z
n
c(!)e! d(!)+
= n +1 1 ; n1
k=1 k n k;1
Z
Z
n
X
1
1
!
+ n(n + 1) k
c(!)e! d(!) =
c(!)e d(!) + n + 1
k=1
kn k;1
n n n;1
109
= n +1 1
Z
n n n;1
= n(n1+ 1)
Z
n
X
c(!)e! d(!) + n(n1+ 1) (k ; 1)
nX
+1
Z
k=1
kn k;1
(k ; 1)
k=1
c(!)e! d(!) =
k n k;1
c(!)e! d(!):
kn+1 ; nk2 6 n2(n1+ 1)2
nX
+1
Z
k=1
k n k;1
(k ; 1)2
jc(!)j2 d(!)
1.
;%! )
. &
, Z
1
1
nX
+1
X
X
nkn+1 ; nk2 6 n2 (nn+ 1)2 (k ; 1)2
jc(!)j2 d(!) 6
n=1
n=1
k=1
k n k ;1
Z
n
+1
1
X X
jc(!)j2 d(!) =
6 n13 (k ; 1)2
n=1 k=1
k n k;1
Z
nX
+1
X
= (k ; 1)2
jc(!)j2 d(!) m13 6
k=1
m>k
k n k;1
Z
Z
1
X
6 12
jc(!)j2 d(!) = 12 jc(!)j2 d(!)
k=1
k n k;1
X 1 Z1 dt
1
3<
3 = 2(k ; 1)2 :
m
t
m>k
k;1
.
% p q | !
: p > q, % .4! %
% fn g1
n=1 ! (3)
1 < q 6 n+1 6 p:
n
% % H = L2 (X) | $ )
#
&%
# # , 2
&! c(!) , )
Z
jc(!)j2 d(!) < 1:
(4)
3. ! (4). "
! ! (C,1) X , , ! # fn g,
110
. . #
(3) ! fsn (x)g .
#.
. % %
% fn(x)g ! ) .
X. ;! &
(1), 1 Z
1
X
X
2
jsn (x) ; n (x)j d = ksn (x) ; n (x)k2 6
n=1 X
n=1
2 Z
6 q2q; 1 jc(!)j2 d(!) < 1:
1
P
!! 2, ) % ) . ! jsn (x) ;
n=1
; n (x)j2, , ) ) . X nlim
j
s
(x)
;
n (x)j = 0,
n
!1
. . %
% fsn (x)g ! ) ..
. % fsn (x)g ! ) .. 3 (1) %
% fn (x)g ! ) . X.
? % k (x) ; n (x) ! n < k < n+1 :
k
X
k (x) ; n (x) =
(j (x) ; j ;1(x))
j =n+1
X
2
k
jk(x) ; n (x)j2 = (j (x) ; j ;1(x)) 6
j =n +1
k p
k
2
X
X 1
pj
j j(j (x) ; j ;1(x))j2 6
6
6
j =n +1
X
n+1
X
n+1
j =n +1
1
j jj (x) ; j ;1(x)j2
j
j =n +1 j =n +1
@{A
!${B&.
D)!, ) n+1 6 pn , )
pn
X
pn
n+1
16 X
1 6 Z dt = lnp:
t
j =n +1 j j =n +1 j
n
=.
jk(x) ; n (x)j2 6 ln p
X
n+1
j =n +1
j jj (x) ; j ;1(x)j2:
(5)
111
;%! &
(2), ) 2 % ) . X
1
P
! njn(x) ; n;1(x)j2, ), (5) ! . )
n=1
. X, + %
% fn(x)g1
k=1 !4#! )
. X. .
1
% 1. P jckj2 < 1, f'k(x)g1k=1 | k=1
1
P
$ L2. "
! !% k 'k (x) k=1
X (C,1), , ! #
fng, #
(3), fsn (x)g .
#. 3 !
3 (= N, (k = f1 2 : : : kg,
1
R
P
(!) 1, jc(!)j2 d(!) = jck j2.
k=1
% 2 (. &4, . 127]).
1
P
jck j2 < 1, f'k (x)g1
k=1 | k=1
1
P
$ L2 . "
! !% k 'k (x)
k=1
X (C,1), ,
! # fn g, #
(3),
fsn (x)g .
#. E
! 1, )!, ) !!.! )
) &%
# #, ) 2.
x
2. (C )-
. F An = ;n+n ) n-# +22&
%-
1
P
$ ! Ant = (1;t1)1+ ( 6= ;1 ;2 : : :), An | ) F ! .
n=0
; $ ! %
+22&
! (.
n
n
P
P
4, . 75]) , ) An;;k1 = An G An++1 = Ak An;k G An = n+ An;1 G
k=0
k=1
An;1 = + n An .
, ) An n . ! .
; ! An 1
n
n X
1
X
X
log An = log 1 + k = k + O k2 :
k=1
k=1
k=1
+, )! ) C !
. H#, 112
. . X
1
j logAn ; log nj 6 C + o(1) + O k12 < M
k=1
$ M | .
! !
!G . 4
%
M1 M2 , ) n > 1
M1 < Ann < M2 :
(6)
=. ! An;;k1 =An ) &
(n ; M + 1);1 1
An;;k1
=
O
=
O
(7)
An
n
n :
. n() = sA(nn) , $ s(n) | )! ! , -
.
+,. I 1 > 2, (C 1) , ) (C 2), . .
(C 2)-# $ (C 1).
3)
n()(x) ! ))
$ % Z
n
X
1
(
)
n (x) = A An;k
c(!)e! (x) d(!):
n k=1
k n k;1
E
%, ) s(n) (x) !!! n- +22&
$ !
1
P
sn (x)tn
1
X
s(n) (x)tn = n=0
(1 ; t) :
n=0
% > ;1, >10. $ s(n+) (x) !!! n- +22&
-
P sn(x)tn
! ! (1;t) (1;1t) , , n
n
P
P
!, s(n+) (x) = An;;1k s(n) (x) = An;;1k Ak n()(x), n=0
k=0
k=0
n
n
X
X
s(n++1) (x) = s(k) (x)An;k n(+) (x) = 1+ An;;1k Akk() (x):
An k=0
k=1
n ( )
1 P
= n()(x) = n+1
jk (x) ; k(;1)(x)j2.
k=1
" 3. (4) > 21 lim () (x) = 0:
n!1 n
(8)
113
#. ;
k() (x) ; k(;1)(x) =
Z
Z
k
k
X
X
= A1 Ak;j
c(!)e! (x) d(!) ; 1;1 Ak;;j1
c(!)e! (x) d(!) =
A
k j =1
k j =1
j n j;1
j n j ;1
Z
k
X
= 1;1 Ak;j Ak ;1 ; Ak;;j1 Ak]
c(!)e! (x) d(!) =
Ak Ak j =1
j n j;1
Z
k
X
;1A;1 =
= 1;1
A
c(!)e! (x) d(!) +k ; j Ak;;j1Ak ;1 ; +k
k ;j k
Ak Ak j =1
n
j j ;1
Z
k
X
1
!
= ;1
c(!)e (x) d(!) ;j Ak;;j1Ak;1 =
Ak Ak j =1
j n j;1
Z
k
X
1
;1
= A (;j)Ak;j
c(!)e! (x) d(!)
k j =1
j n j;1
%
, 1
Z
Z
k
X
jk()(x) ; k(;1)(x)j2 d 6 2 (A1 )2 j 2 (Ak;;j1 )2
k j =1
X
Z
X
jc(!)j2 d(!)
j n j;1
2(n )(x) d 6
Z
2n
k
X
1 X
1
;1
2
jc(!)j2 d(!) =
()2 (2n + 1) k=1 (Ak )2 j =1 j Ak;j
j n j ;1
n
Z
2
2n A;1 !2
X 2
X
1
k;j :
2
= 2(2n + 1) j
jc(!)j d(!)
A
k
j =1
k=j
j n j;1
D)! ! (6) (7), )
;1 !2
2j A;1 !2
2n A;1 !2 X
1
X
X
Ak;j
k;j
k
;j
6
+
6
A
A
A
k
k=2j +1
j 1 X
C1 A 2 + X
M2 k;1 2 6 (A1 )2
k
j k=0 j
k=2j +1 M1 k
) . )%, > 12 , % )
k=j
k
k=j
k
114
. . 1 1
1 1 (j + 1) C1 M j 2 + M2 2 X
M12 j 2
j 2
M1 k=2j +1 k2 6
2
2j
1
1 = C2 :
2
2
2
2
;
2
6 M 2 j 2 C1 M2 j + M
M1 2j j
1
,
Z
Z
2n
X
C
2
(
)
jc(!)j2 d(!)
2n (x) d 6 22n j
j =1
X
j n j;1
1 Z
X
Z
1
2n
X
X
1
(
)
2n (x) d 6 C3 2n j
n=1 X
n=1 j =1 j n j;1
Z
1
X
2
6 C3 j j
j =1
jc(!)j2 d(!) 6
jc(!)j2 d(!) = 2C3
Z
jc(!)j2 d(!) < 1:
j n j;1
;, nlim
(n ) (x) = 0 ) ..
!1 2
J! 2n < k 6 2n+1 :
1
) (x) =
2(n+1
n
+1
2 +1
> 21 k +1 1
> 21 k +1 1
k() (x)
n+1
2X
m=1
k
X
n+1
2X
m=1
jm()(x) ; m(;1)(x)j2 >
jm()(x) ; m(;1)(x)j2 >
jm()(x) ; m(;1)(x)j2 = 12 k() (x)
m=1
(
)
22n+1 (x),
0 6
6
%
) . klim
() (x) = 0,
!1 k
) %. .
4. ! (4). R c(!)e! (x) d(!) E f(x) (C ) > 21 , E
n
1X
jf(x) ; k(;1) (x)j2 = 0
(9)
lim
n!1 n
k=1
n
1X
lim
jf(x) ; k(;1) (x)j = 0:
(10)
n!1 n
k=1
#. J ) (9). ;
115
n
X
k=1
jf(x) ; k(;1) (x)j2 =
n
X
jf(x) ; k() (x) + k() (x) ; k(;1) (x)j2 6
k=1
n
X
62
k=1
jf(x) ; k() (x)j2 + 2
n
X
k=1
jk() (x) ; k(;1) (x)j2:
!! 3 ! o(n), !! ! o(n) !. H (9).
n jf(x) ; (;1) (x)j
X
k
6
n
k=1
X
12 X
1
n
n
n
X
2 2
6
jf(x) ; k(;1) (x)j2 n12 = n1 jf(x) ; k(;1) (x)j
k=1
k=1
k=1
@{A
!${B&. =.
n jf(x) ; (;1) (x)j2 21
n jf(x) ; (;1) (x)j X
X
k
k
6 nlim
= 0:
lim
!1
n!1
n
n
k=1
k=1
%..
5. (4) ! (C 1) &
{ !
E .
#.
R
1. % c(!)e! (x) d(!) ! (C,1) 2
& f(x),
)
# ) . E.
3%! K! p
pX
;1
X
k
bk t = (b1 + : : : + bk )(tk ; tk+1 ) + (b1 + : : : + bp )tp =
k=1
k=1
pX
;1
= (1 ; t) (b1 + : : : + bk )tk + (b1 + : : : + bp )tp :
k=1
Z
pX
;1
!
k
c(!)e (x) d(!) = (1 ; t) sk (x)t + c(!)e! (x) d(!)tp =
k=1 k n k;1
k=1
p
p;2
X
k
2
= (1 ; t)
kk (x)t + (p ; 1)p;1 (x)tp;1 + sp (x)tp :
k=1
; jsp (x)j 6 pC1, jp;1(x)j 6 C2,
Z
p
X
k
Ft(x) = plim
t
c(!)e! (x) d(!) =
!1
k=1 k n k;1
p
X
tk
Z
116
. . pX
;2
2
k
p
;
1
p
= plim
(1 ; t)
kk (x)t + (p ; 1)p;1(x)t + sp (x)t =
!1
k=1
1
X
= (1 ; t)2 kk (x)tk :
k=1
1
P
@ $, ! t < 1 (1;1t)2 = ktk;1,
k=1
1
1
X
X
jFt(x) ; s(x)j = (1 ; t)2 kk (x)tk ; (1 ; t)2 ks(x)tk;1 =
k=1
k=1
1
X
= (1 ; t)2 ktk;1(k (x)t ; s(x))] =
k=1
N
1
X
X
2
k
;1
2
k
;1
(kt (k (x)t ; s(x))):
= (1 ; t) (kt (k (x)t ; s(x))) + (1 ; t)
k=1
k=N +1
3 N, 1 > t0 > 0, 1 > t1 > 0 ! %
$ " > 0 , ) jk(x) ; s(x)j < "4 ! k > N, jk (x)t ; s(x)j < "2 ! k > N, t > t0 .
N
N
P
P
(1 ; t)2 ktk;1jk (x)t ; s(x)j 6 (1 ; t)2 ktk;1C3 < (1 ; t)2 C3N 2 < "2 k=1
k=1
t > t1 (t1 )
1).
$ ! k > N, t > maxft0 t1g
1
1
X
X
ktk+1 2" < 2" + 2" (1 ; t)2 ktk+1 = 2" + 2" = ":
jFt(x) ; s(x)j 6 2" + (1 ; t)2
k=N +1
k=1
*%
, $ K!{
.
R
2. % c(!)e! (x) d(!) ! K!{
)
R
. EG jc(!)j2 d(!) < 1.
1
P
$ ! (n (x) ; n;1(x)) ) . E n=1
1
P
. K + )
%. ! njn(x) ; n;1(x)j2 (
1
n=1
P
(3)) % ! (n(x) ; n;1(x)). *%
, + !
n=1
R
! ) . E, $ c(!)e! (x) d(!) !
E ) . (C,1).
%..
6. ! (4). R c(!)e! (x) d(!) !
&
{ E
> 0, E (C ).
117
#. *
) , ) n
1X
(r) (x)j2 = 0 r > ; 1 lim
j
k
n!1 n
2
k=1
1
! " > 0 nlim
n(r+ 2 +") (x) = 0.
!1
3 , (6) , )
n
n
X
X
1
1
1
s(nr+ 2 +") (x) = s(kr) (x)An;;2 k+" = s(kr) (x)Ark A;n;2 k+" = r+ 12 = ; 12 +":
k=1
k=1
+
1
n
n;
X
(r+ 21 +") X
1 2
sn
(x) 6
jk(r) (x)j2 Ark An;;2 k+" 2 6
k=1
k=1
v
v
v
u
u
u
n
n
n
p
X
X
u
uX
u
(
r
)
;1+2"
2
r
2
t
t
jk (x)j K
Ak An;k = K t jk(r) (x)j2 A2nr;1+2"+1 =
6
k=1
k=1
k=1
v
u
n
p
uX
= K t jk(r) (x)j2 A2nr+2" = o(n1=2 )O(nr+" )
k=1
%.
; # 6, 5, $ ! (C,1) ) . E, ) 4 = 1 (9), . .
n (0)
1 P
jk (x) ; f(x)j2 = 0.
) . E nlim
!1 n
k=1
(0)
)
. !
; R?
%! k (x) ; f(x) R %
c(!)e (x) d(!) ; f(x) +
c(!)e! (x) d(!) + : : : % 1n 0
2n 1
, ) %, r = 0.
), )
h ( 1 +")
i
2 (x) ; f(x) = 0
lim
n
n!1
) . E.
;%! 4 4 = 12 + ", ) .4# %:
n 1
1X
n(; 2 +") (x) ; f(x)2 = 0
lim
n!1 n
k=1
) . E. =. 4%. # .)%,
)
lim (2")(x) ; f(x)] = 0
n!1 n
) . E.
118
. . A! % " = 2 , )%
.
.
7. (4)
R !
(C ) > 0 c(!)e! (x) d(!) ! % ! &
{ .
#.
1. J %
% (C,1) (C ) ! 0 < < 1. ; (C ) (C,1)-%G 5
% K!{
G 6 (C )-%.
2. J %
% (C,1) (C ) ! 1 < < 1. ; (C,1) (C )-%G % K!{
, %
.$ (C )G 5 % (C,1).
%..
% 3. "
! $ L2 ! (C ) > 0 ! % !
&
{ .
#. 3 !
7 (= N, (k = f1 2 : : : kg,
1
R
P
(!) 1, jc(!)j2 d(!) = jck j2.
k=1
% 4 (. &5, . 219]). "
! $ L2 !
(C ) > 0 ! % ! &
{ .
#. D
4$
! )
!, $ 2.
K $
2 . . , # .
"
#
1] . . , , " # # // %
. -
&. '. ()*. *. | ,*-
--
:
/-* 0 0-%, 1996. |
3. 117{118.
2] . . *#* , // '
. . 3*. )
"# . * . | '
: '89, 1997. | 3. 105.
3] . . " // '
%* -. 3 %., %. | 1997. | = 5.
4] ?@ 8. *. | %.:
119
/ *
# , 1963.
5] 0@ 3., A#
8. *. | %.: 8/B%, 1958.
6] 0* ?. &., B
3. '. C
)
"# )
"
. | %.: &, 1989.
' ( ) 1997 .
SV-
. . . . . 512.552
: , , V-, , , ,
, .
! " # $ SV-. % & SV- Soc (R) .
Abstract
V. N. Silaev, On right SV-rings, Fundamentalnaya i prikladnaya matematika,
vol. 7 (2001), no. 1, pp. 121{129.
In this paper we investigate the worst cases of SV-ring structure. We give two
constructions of SV-rings with strong restriction on all Soc (R) of Loewy chain.
. R, x 2 R y 2 R,
xyx = x. ! " #$ % & 1936 . ! + , ,, " ,
" , +, +$ -. . % /6].
1 ", V-
, , " 2. !%
/7] 1964 .
R, $ " R- .
5" ", ,. 6 +, "
() "$ 9. -, " , "
"+, :. ;
/5]. # ". 9+= "+ " R- M, " " Soc (M) "
" : "$ Soc0 (M) = 0 $ " Soc +1 (M) " " Soc +1 (M)= Soc (M) = Soc(M= Soc (M))
(Soc(M) | M)A | " ,
, 2001, 7, - 1, . 121{129.
c 2001 !,
"#
$% &
122
. . S
"$ Soc (M) =
Soc (M). 5 C <
6 jM j, Soc (M) = Soc+1 (M). D"
0 = Soc0(M) Soc1 (M) = Soc(M) Soc2 (M) : : : Soc (M)
+ (
) M. E M + , Soc (M) = MA +
M. - R + , RR ", " ", " ". F, "" %
-
, Soc (RR ) = Soc (R R), "
" Soc (R) .
1C "+ " 10{15 " SV-
, " " " V-
, ,
+
, " + . 6 ,
" /1,3,4], | /8].
9 #$. 2 /2], $ "= , "+ " ": 1) " u- "
SV-
+ 1A 2) "
" SV-
+ 2, V-
. -$ + , " , R , + %
R= Soc (R).
2 +
", %
J",K, "= " . 9 , " , ", SV-
"+ +1 ( " $ " , . . "
, ), < %
R= Soc (R) "
( " +1 %
-
R= Soc (R)
"" " 9={L "
"+ ", , " $ "
).
1 , " " .
L "+ , + %+
-
, , "% L. 9. E,= %+
-
, , "% #. 9. 5
+ "
+ "+ $.
&C " " %
" + #$. 2 /2]. M = , C +.
1 X | "+ $, D | . 5 + + CFMX (D) , X X D SV-
123
$ . QC +, + " UD = D(X ) += +%+ CFMX (D) ' End(UD )A
, Soc(CFMX (D)) + , , . 2 $ D "
, CFMX (D).
6 " , , + $ 8 6 "
To Q CFMX (D) ", X, :
1) Q ' CFMX (D)A
2) Q Q 8 : < 6 A
3) Q \ Soc(Q ) = 0 8 : < 6 A
4) D Q 8: 6 A
5) Soc(Q ) Soc(Q ) Soc(Q ), = min( ).
5 " " """ L Q1 :
L0 = 0A
L +1 =SL + Soc(Q +1 ) 8 < A
L = L " 6 ,
<
R = L + D " SV-
+ 1. 9 D " C $ "
Q1 , "= ". 5"
+ .
1. # "+ > 0 " SV-
R + 1, < %
-
R= Soc (R) "
.
. M " + 2.
&" + /2] + "= " ".
(2, lemma 4.1]). | X | , jX j = @ . < P X , 1) jY j = @ 8Y 2 P !
2) 6 , P P .
(2, proposition 4.2]). > 0 X | jX j = @ , D | Q = CFMX (D). 6 Q " # Q , $ Q 1) Q Q 8 6 6 !
2) Q \ Soc(Q ) = 0 8 < 6 !
3) D Q 8 6 .
. # 6 " P | + X + . # $ Y 2 P fY : @ ! Y + xYi = fY (i) $ i < @ .
124
. . # $ x 2 X 9! Y (x) 2 P , i (x) < @ , x = xY (x)i (x) .
L, < 6 , $ Z 2 P +
@ $, "$,
P , " S
gZ : @ ! P , Z = fgZ (j) j j < @ g, +
YZj = gZ (j) 8j < @ . # Y 2 P 9! Z (Y ) 2 P , j (Y ) < @ , Y = YZ (Y )j (Y ) .
# $ 6 " $:
' : CFMP (D) ! Q: (' (A)xy = i (x)i (y) AY (x)Y (y) A
' : CFMP (D) ! CFMP (D): (' (B)Y Y = j (Y )j (Y ) BZ (Y )Z (Y ) :
:
", ' ' | %+ 8 < 6 ' = ' ' .
# $ 6 Q = Im(' ).
1) 5
jP j = @ = jX j, Q ' Q.
2) 6+ ' = ' ' , Q Q 8 < 6 .
3) X 0 6= A 2 CFMP (D), , 9Z Z 2 P , AZZ 6= 0.
F Y , Z = F Y , ' (A)
5
Z =
Zj
Z j
YZj YZ j = AZZ 6= 0 8j < @ .
j< j< F, ' (A) $ @ , . 1 Im(' ) \ Soc(CFMP (D)) = 0, Q \ Soc(Q ) = 0.
Y, (Soc(Q )) 6 , """ Q Soc(Q ) \ Soc(Q ) = 0 " 6= A
Soc(Q ) Soc(Q ) Soc(Q ) = minf g:
# $ 6 " " """ L Q1 :
L0 = 0A
L +1 =SL + Soc(Q +1 ) 8 < A
L = L $ " 6 .
<
- R = L + D " SV-
+ 1. 9 D $ "
Q1 ,
"= ". M ", +.
!. 1 | , X | $ jX j = @0 .
!" "
DEMX (D) CFMX (D) = Q +:
0
0
0
0
0
0
@
0
0
0
@
DEMX (D) =
= fA 2 Q j 9n = n(A) > 1 8i j : i > n j > 1 Aij = Ai+1 j +1 g:
5 DEMX (D) Soc(Q) Soc(DEMX (D)) = Soc(Q). X + A B 2 DEMX (D) 0
0
SV-
(
(
125
j = i + 1 B = 1 i = j + 1
Aij = 01 A
ij
0 ,
A B = 1, B A 6= 1.
- , DEMX (D)= Soc(Q) = D(x) | , " x, x = A mod Soc(Q), x 1 = B mod Soc(Q).
[ " 2, P+1 , jP+1j = @0 . 9= + "$ +.
!" " R( + 1) +:
R( + 1) = L + '+1 (DEMP+1 (D)) jP+1 j = @0 :
5, '+1 (DEMX (D)) Q+1 Q 8 < + 1 | "
"
, Q Q 8 6 6 + 1, R( + 1) | "
Q1
(L ) 6 L+1 = L + Soc(Q+1 ) L+2 = R( + 1) | + "
R( + 1). # 6 L \ Q +1 =
= 0, L | Q +1 + L . 5
L Q +1 + L '+1 (DEMX (D)) Q +1 , ", R( + 1)=L "
(Q +1 +L )=L = Q +1 Soc(Q +1) = L +1 =L R( + 1)=L , ,
L +1 =L = Soc(R( + 1)=L ).
#, R( + 1)=L+1 = '+1 (DEMX (D))= Soc(Q+1 ) = D(x) "",
" (L ) 6+2 | "
R( + 1), " ",
+ 2.
#
$, R( + 1)=L ". # = + 1 + 2 ".
1 6 . M %
, $ X , a b 2 CFMX (D) + a Soc(CFMX (D)) b = 0 , a = 0
b = 0. #, b 6= 0 ) Im(b) 6= 0, a 6= 0 ) Ker(a) 6= UD = D(X ) ,
" x 2 Soc(CFMX (D)), " -
+ Im(b) + UD n Ker(a) , a x b 6= 0.
5", "
R( + 1)=L (Q +1 + L )=L = Q +1, x (R( + 1)=L ) y = 0 = x Soc(Q +1 ) y = 0, " + x = 0 y = 0, R( + 1)=L ".
#
$, fL j < + 1g | $ , R( + 1), $, L+1 .
1 I | R( + 1), L+1 6 I. = minf 6 + 1 j L 6 I g, , + 1
< + 1 L I.
- R=L ", ""$ L 6= I, 0 6= (I=L ) (L+1 =L ) = (I \ L+1 )=L . M , Soc(R=L ) = L+1 =L |
R=L , , L+1 I | ".
#
$ ", R( + 1) " V-
. 9+=
" " R( + 1)- U. 5 UR(+1) = (R( + 1)=M)R(+1)
" M R( + 1). X 0
0
0
0
0
0
0
;
0
126
. . P = R( + 1) M = fr 2 R( + 1) j R( + 1) r M g $ L+1 , P = L+1 , L+1 | . X $ P 6 L+1 , "
+ P = L < + 1. 5
+, U " R( + 1)=L - (
" " P = Ann(U)).
1 U "
R( + 1)=L -.
1 /2, theorem 2.5], R "
" "+Q
Q , Q = End(U )D , U | " D -
",
;
L
(D ) ; | Soc(Q ) R, " "
R- . ;
9 C R( + 1)=L (
R( + 1)=L Q +1 ), " U R( + 1)=L -. &
UR(+1) " + /6].
(6, lemma 6.17]). ': R ! S | $
#, A |
%
S -. &
R S | , A %
R-.
$ (6, corollary 1.13]). '# R , ( ) R- .
5
+, " R( + 1) | " SV-
+ 2 "
(L ) 6+2 , R( + 1)=L "
6 + 1.
5" " | " . 9+= D "+ " F M
R = R( + 1) + F:
2
2
2
<
! " SV-
, , +1. 1 R= Soc (R) "
< . 9= +
"$.
(2, proposition 4.6]). (R ) | F -, F -
M
R = R + F
2
2
Q R. (:
2
L
(1) Soc (RR ) = Soc((R )R )!
(2) R , R sup + 1!
(3) R | , u-, , V-# , R .
2
2
127
SV-
1 + " " +
.
# $ " " ".
2. 1 n | , n > 2. 5 " SV-
R n "
0 = Soc0 (R) Soc1(R) : : : Socn (R)
%
-
R= Socm (R) m, 0 6 m 6 n;2, "
.
.
3. X | " , R | #. (-
$ h: CFMX (R) ! CFMX (CFMX (R)):
&. X | =, " X $ P = f(i j) j i j 2 Z 1 6 i < 1 1 6 j < 1g. !+ +
p1 p2 : P ! N "
" ".
5 p1 (P) p2(P ) $ =, + " +%+
h : CFMP (R) ! CFMp1 (P ) (CFMp2 (P ) (R)):
1 A 2 CFMP (R), " $ h h (A) = a 2 CFMp1 (P ) (CFMp2 (P ) (R))
ai1 i2 2 CFMp2 (P ) (R) (ai1 i2 )j1 j2 = A(i1 j1 )(i2j2 ) . !, h (1) = 1,
h (A + B) = h (A) + h (B). h (AB).
X
(h (AB)i1 i2 )j1 j2 = (AB)(i1 j1 )(i2 j2 ) =
A(i1 j1 )(kl) B(kl)(i2 j2 ) =
0
0
0
0
0
0
0
0
0
=
=
(kl) :
B(kl)(i2 j2 ) 6=0
X
(h (A)i1 k )j1 l (h (B)ki2 )lj2 =
0
kl :
(h0 (B)ki2 )lj2 6=0
X
0
X
(h (A)i1 k )j1 l (h (B)ki2 )lj2 = ((h (A)h (B))i1 i2 )j1 j2 0
l:
k:
h (B)ki2 6=0 (h (B)ki2 )lj2 6=0
0
0
0
0
0
, h (AB) = h (A)h (B). 1 h | %+ .
9+= A 6= 0. 9 (i1 j1) (i2 j2 ): A(i1 j1)(i2 j2 ) 6= 0,
(h (A)i1 i2 )j1 j2 6= 0, h (A) 6= 0. 1 h .
1 a 2 CFMp1 (P ) (CFMp2 (P ) (R)). 5 ((h ) 1 (a))(i1 j1 )(i2 j2 ) =(ai1 i2 )j1 j2 .
1 h , +.
1 F | ". # n > 1 + + Qn CFM
| X (CFM{zX (: : :CFMX }( F) : : :)):
0
0
0
0
0
0
0
0 ;
0
n 128
. . 9 Q0 += F.
$ 4. n $ hnn+1 : Qn ! Qn+1 :
. Qn = CFMX (Qn 1), Qn+1 = CFMX (CFMX (Qn 1)).
!= "+ 3. M +.
!+ 8i > 0 Qi + 1i . 5" " "
+ Si Qi 8i > 1 +:
Si = fa 2 Qi j 9N =N(a): 8l > N 8k > 1 alk = 0 8l 6 N 8k > 1 alk 2 F 1i 1g:
X= " "
DEMi Pi Qi 8i > 1 :
DEMi = fa 2 Qi j 9M = M(a): 8l > M 8k > 1 alk = al+1k+1 2 F 1i 1
8l 6 M 8k > 1 alk 2 F 1i 1g
Pi = fa 2 Qi j 8k l akl 2 F 1i 1 g:
5 Pi = CFMX (F), Si = Soc(Pi ) = Soc(DEMi ). !+ $ 8m n,
2 6 m < n, + hmn +%+ hn 1n : : :hm+1m+2 hmm+1 : Qm ! Qn .
5 hmn (Sm ) "
+ Qn, "= 8m m ,
2 6 m < m < n, hmn (Sm ) \ hm n (Sm ) = 0, hmn (Sm ) hm n(Sm ) hmn (Sm ).
" + , Sm = Soc(Pm ), hmm+1 (Sm ) \ Sm+1 = 0
hmm+1 (Pm ) Sm+1 " " hmm+1 + " Sm . L +, 8m n, 2 6 m < n, hmn (Sm ) \ DEMn = 0 hmn (Sm ) DEMn hmn (Sm ). 5" " "
Rn Qn:
Rn = h1n(S1 ) + h2n(S2 ) + : : : + hn 1n(Sn 1) + DEMn :
5 S1 | Q1 = CFMX (F ), , h1n(S1 ) | Qn A
" Soc(Rn ) = h1n(S1 ) Rn= Soc(Rn) = h2n(S2 ) + : : : + hn 1n(Sn 1 ) + DEMn = Rn 1
h2n(S2 ) + : : : + hn 1n(Sn 1 ) + DEMn
"
h2n(P2 ) = Qn 1 .
!, Soc(DEM1 ) = S1 , DEM1 = Soc(DEM1) = F(x) | "
, . 5
+, " ", Rn |
" n + 1 "
0 Soc1(Rn) = h1n(S1 ) Soc2 (Rn) = h1n(S1 ) + h2n(S2 ) : : : Socn (Rn) = h1n(S1 ) + : : : + hn 1n(Sn 1) + Sn Socn+1(Rn) = Rn, "=
Rn i, " "
.
8i, i 6 n ; 1, Rn= Soci (Rn) =
!= +, Rn | " V-
. !+ + R0 " F(x). 5 R0 | " V-
. E "= " + " " n.
;
;
;
;
;
;
;
0
0
0
0
0
;
;
;
;
;
;
;
;
;
;
0
;
SV-
129
1 Rn | " V-
. #
$, Rn+1 | $ "
V-
. 1 M | "+ " " Rn+1-.
X M Soc(Rn+1) = 0, M | " " Rn-, " ""$ M " Rn-. 5 " C /6, 1.13] /6, 6.17] M $ " Rn+1 -.
X M Soc(Rn+1 ) 6= 0, M Soc(Rn+1 ) = M , +, M % + Soc(Rn+1)Rn+1 . 5
Soc(Rn+1 )Rn+1 |
"" , M +% " Soc(Rn+1)Rn+1 .
5
+, $ , M | " Rn+1 .
Rn+1 Qn+1 = CFMX (F ), Qn+1 | " . 5
M Qn+1 = M Soc(Rn+1 ) Qn+1 = M Soc(Rn+1 ) = M (
Soc(Rn+1 ) = Soc(Qn+1 )), M | " Qn+1. 5 "
/6, 9.2] M | " Qn+1- " /6, 6.17]
M | " Rn+1-. [ +.
1] G. Baccella. Generalized V-rings and von Neumann regular rings // Rend. Sem. Mat.
Univ. Padova. | Vol. 72. | 1984. | P. 117{133.
2] G. Baccella. Semiartinian V-rings and semiartinian von Neumann regular rings.
3] G. Baccella. Von Neumann regularity of V-rings with Artinian primitive factor rings //
Proc. Amer. Math. Soc. | 1988. | Vol. 103, no. 3. | P. 747{749.
4] N. V. Dung, P. F. Smith. On semiartinian V-modules // J. Pure Appl. Algebra. |
1992. | Vol. 82, no. 1. | P. 27{37.
5] L. Fuchs. Torsion preradicals and ascending Loewy series of modules // J. Reine
Angew. Math. | 1969. | Vol. 239/240. | P. 169{179.
6] K. R. Goodearl. Von Neumann Regular Rings. | London: Pitman, 1979. | Monographs and Textbooks in Mathematics.
7] B. L. Osofsky. Rings all of whose nitely generated modules are injective // Pacic
J. Math. | 1964. | Vol. 14. | P. 645{650.
8] C. Nastasescu, N. Popescu. Anneaux semi-artiniens // Bull. Soc. Math. France. |
1968. | Vol. 96. | P. 357{368.
' ( ) 1997 .
. . . . 515.146.34+514.764.227
: , ", .
# $ %& ' (. & ) *+& ) ". ,-&& & " '
'&') " .
Abstract
S. Terzic, Cohomology with real coecients of generalized symmetric spaces,
Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 1, pp. 131{157.
In the article we consider generalized symmetric spaces of the compact simple
Lie groups. We give a classi2cation of these spaces and an explicit description of
their algebras of cohomology with real coe3cients. In the case of such spaces of
second category, the direct computation of their cohomology algebras is given.
.
1] !" # $ .
%& %$ . ' , !$" %% ), . *. +& 9].
- $ +, . (rank U = rank G) + , 2001, 7, 4 1, . 131{157.
c 2001 !,
"#
$% &
132
. (. 3]). -, , $ (rank U < rank G). 3 $, " & U
# %$ G ! G. $, 4 , +# $" %! + .
5 , ! $" !& H (BG ), +. 5, G=G $, ! $ t g $ $ t g.
- $! !" # . , . . An , Dn E6.
6 % 7! 8. -.
x
1. -! G | $ , ! H (G) | R. :# $ $%$
H (G) = ^(x1 : : : xl )
^(x1 : : : xl ) | # $
x1 : : : xl , (. 6]).
-! g | G t | g, v1 : : : vn. 3 t
!& R . * WG 4 !&,
! !& , ! WG . ' !& $ $ Rv1 : : : vn]WG .
-! R = rank G. <$, Rv1 : : : vn]WG R $". = , " .
1 (. . ).
(a) G | , WG t R (R = rank G) -
.
() k1 : : : kR | " , 2k1 ; 1 : : : 2kR ; 1 |
" , H (G) .
133
> ki , , " $ G " "
(. 10]):
g = An ki = 2 3 4 : : : n + 1C
g = Bn ki = 2 4 6 : : : 2nC
g = Cn ki = 2 4 6 : : : 2nC
g = Dn ki = 2 4 6 : : : 2n ; 2 nC
g = G2 ki = 2 6C
g = F4 ki = 2 6 8 12C
g = E6 ki = 2 5 6 8 912C
g = E7 ki = 2 6 8 10 12 14 18C
g = E8 ki = 2 8 12 14 18 202430:
F $" Rt]WG $
G.
2. # G | , $ Dl (l > 4), k1 : : : kR | % , 1 : : : n | % . &
$
n
Iki =
X
j =1
kj i (i = 1 2 : : : R)
Rt]WG.
- 4 G Al , Bl , Cl G2 6], G = F4 1], G = E6 | 10], G = E7 E8 |
7]. 7
Dn " !.
- $" !& Rt]WG G. 3
fxig $" g.
1. g = An (n > 1).
+!& Rt]WAn = S(x1 : : : xn+1)
S(x1 : : : xn+1) !& x1 : : : xn+1. 5 $" !& Rt]WAn
" fi (x) = i (x1 : : : xn+1) 2 6 i 6 n + 1g
i(x) $ i-" 4" " %&".
2. g = Bn (n > 2).
Rt]WBn = S(x21 : : : x2n)
fi(x2 ) = i (x21 : : : x2n) 1 6 i 6 ng:
134
. 3. g = Cn (n > 3).
Rt]WCn = S(x21 : : : x2n)
fi(x2 ) = i (x21 : : : x2n) 1 6 i 6 ng:
4. g = Dn (n > 4).
+!& Rt]WDn !&
!& t, !& S(x21 : : : x2n) x1 : : :xn.
<$" !& Rt]WDn " fi(x2 ) = i (x21 : : : x2n) 1 6 i 6 n ; 1 n(x) = x1 : : :xn g:
5. g = G2 .
<$" !& Rt]WG2 " P2 = 2
3
X
j =1
x2j P2 = 2
3
X
j =1
x6j :
6. g = F4 .
<$" !& Rt]WF4 " kl
X
X 1
Ikl = (xi )kl +
(
x
x
x
x
)
2 1 2 3 4 :
7. g = E6 .
E6 , G2 F4, !$!
2, $" !& Rt]WE6 . ) !
# $ $", $ 1]:
X
6
6
6
X
X
1
k
k
k
l
l
l
Ikl = 2
ai + b i +
cij i=1
i=1
ij =1
% ai , bi cij $" %
ai = xi + 12 (x1 + x2 + x3 + x4 + x5 + x6) (1 6 i 6 6)
bi = xi ; 32 (x1 + x2 + x3 + x4 + x5 + x6) (1 6 i 6 6)
cij = ;xi ; xj + 21 (x1 + x2 + x3 + x4 + x5 + x6 ) (1 6 i j 6 6):
8. g = E7 .
<$" Rt]WE7 " P2s = 2
X
2X
s;1
i<j
i=1
(xi + xj )2s = (16 ; 22s)p2s +
P8
s | &, s > 0, pi = xia , Cmn = n!(mm;! n)! .
a=1
C2i sp2s;ipi 9. g = E8 .
<$" Rt]WE8 " P2s = 18(5 + 32s;3 ; 22s;1)p2s +
+
pi =
x
P9 xi , p = 0.
a 1
a=1
2X
s;1
i=1
C2i sp2s;i
135
pi (9 + (;1)i ; 2i ) +
i;1
1X
j
3 j =1 Ci pi;j pj 2. -! ^(u1 : : : uR) | # ( )
4 u1 : : : uR deg u = 2k ; 1 ( = 1 2 : : : RC k |
& &! ).
-! Rv1 : : : vr ] | ( )
4 v1 : : : vr deg vi = 2li (i = 1 2 : : : rC li | &
&! ).
6 C = ^(u1 : : : uR) Rv1 : : : vr ]:
5 C deg(u v) = deg u + deg v (u 2 ^(u1 : : : uR ), v 2 Rv1 : : : vr ]):
X
C = C s C s = fc 2 C deg c = sg:
s>0
-! Rv1 : : : vr ] $ F1 : : : FR. = ! deg Fi = 2ki, i = 1 R ( 4 vi ).
6 C %%& d, d(ui 1) = 1 Fi
d(1 vj ) = 0 i = 1 R j = 1 r:
136
. <, d2 = 0.
)%%&! R- C $ .
6
C k Z k = Ker d \ C k , B k = Imd \ C k , H k = Z k =B k
L
H (C d) = H k | !& C #" %k>0
%& d.
<$, F1 F2 : : : FR , 4 " H (C d) ( !" $%$). 5, :
3. '
F1 F2 : : : FR F~1 F~2 : : : F~R ,
Fi F~i mod (F1 : : : Fi;1 Fi+1 : : : FR ) (i = 1 R). &
" (
),
. .
H (C dF1F2 :::FR ) = H (C dF~1F~2 :::F~R ):
)$! . 7].
3 " .
4. '
F1 F2 : : : FR *,
$ Fn+1 = Fn+2 = : : : = FR = 0. '
(C 0 d0) = ^(u1 u2 : : : un) Rv1 v2 : : : vr ]
&
d0 (ui 1) = 1 Fi i = 1 n d0(1 v) = 0 v 2 Rv1 : : : vr ]:
H (C dF1F2 :::FR ) = ^(un+1 : : : uR) H (C 0 d0):
)$! . 7].
. <, Fn+1 : : : FR hF1 : : : Fni, .
+ C " , " $
#
:
M
C = Ck Ck = fc 2 C : c = u1 ^ : : : ^ uk v v 2 Rv1 : : : vr ]g:
k>0
<! 4
d ! ;1. J ! Zk = Ker(dC
L k), Bk = Im(dCk+1) Hk = Zk=Bk . L H =
= H ((C d) R) = Hk $ +#. <,
k>0
H0 = Rv1 : : : vr ]=hF1 : : : FRi, $ hF1 : : : FRi $ ,
4 F1 : : : FR .
<$, ! + !"
C. 5, " .
137
5. H0(C d) | $ R = r, H (C d) H0(C d) ).
)$! . 7].
3# &! | ! %&" G=U. <$, j : Rt]WG ! Rs]WU , & j : U ! G, $ !" ! G=U " $ $. 3, ! 3],
% + $ " 4
% .
M!" ! + . ) ! ! 3]. J 4%%&. -!
BG | %&" G.
4%%& $ H (G) $ !" ! H (BG ).
5, " (. 3]).
6. # G | . #*, $
H (G) , "
$% 2k1 ; 1 : : : 2kR ; 1. &
(a) H (G) x1 : : : xR (Dxi = 2ki ; 1) +
() H (BG ) )
$ Kp y1 : : : yR ]. , E(n G) " yi , *
xi .
-! ! G=U E(n G) | ! G. 5 ! U. <$ $ (U G) %$
H (BG ) ! H (BU ), n "
$ n %$ H (B(n G)) ! H (B(n U)), & &
E(n G)=U
E(n G)=G.
5 + ! % " $
(. 3]).
7. # U | G x1 : : : xR | H (G), y1 : : : yR | H (BG ), x1 : : : xR
EG. &
H (G=U) $ )
H (H (BU ) H (G)), H (BU ) H (G)
*
))
, H (BU ) d(1 xi ) = (yi ) 1 (i = 1 : : : R).
5! , +. > H (BU ), "
138
. (. 3]).
1. T | G, )
(T G): H (BG ) ! H (BT ) ,.
' $, H (BG ) ! !
g, ! . > %$ (U G), "
3].
2. # U | G,
S T | U G, t | T s | %
, S . * H (BG ) H (BU )
$ t s , "))
, WG WU , ) (U G): H (BG ) ! H (BU ) * $ H (BG ) $ H (BU ).
- ! U G (. 7]) # " $! %! + .
8. # U | G (rank U = r), P1 : : : PR | Rt]WG.
j (Pr+1 ) : : : j (PR ) *
hj (P1 ) : : : j (Pr )i, j | *, , H (G=U) = fRs]WU =hj (P1 ) : : : j (Pr )ig ^(xr+1 : : : xR ):
!. N$ 7 , + G=U (L = H (BU ) H (G) d), d(u 1) = 0
d(1 xi ) = (U G)(yi ) 1 xi 2 H (G) u 2 H (BU ).
-!$! 1 2, , H (BG ) = Rt]WG = RP1 : : : PR]
H (BU ) = Rs]WU = RQ1 : : : Qr ]
(U G)Pi = j (Pi ) = Pi =s:
<" , L = Rs]WU ^(x1 : : : xR) = RQ1 : : : Qr ] ^(x1 : : : xR )
d(Qi 1) = 0 d(1 xi) = j (Pi) 1:
5 j (Pr+1 ) : : : j (PR ) hj (P1) : : : j (Pr )i, $
4 , H (L d) = H (L0 d0) ^(xr+1 : : : xR)
()
L0 = RQ1 : : : Qr ] ^(x1 : : : xr ) d0 (Qi 1) = 0 d0 (1 xi ) = j (Pi) 1:
139
5 ,
$% H (L d) . -4 $ ()
! H (L0 d0) , !, H0(L0 d0).
3 L0 R = r, 4 5
H (L0 d0) = H0(L0 d0) = RQ1 : : : Qr ]=hj (P1 ) : : : j (Pr )i:
N,
H (G=U) = H (L d) = fRs]WU =hj (P1 ) : : : j (Pr )ig ^(xr+1 : : : xR): 2
G=U, rank G = rank U, $
5 , + H (G=U) = Rt]WU =h(Rt]WG )+ i
(Rt]WG)+ $ !& !& Rt]WG, $ ! .
5 #
!" "
(. 1]).
9. # G | , U | % , H (G) = ^(x1 : : : xR ) | x1 : : : xR y1 : : : yR | H (BG ), x1 : : : xR . *
(U G)y1 : : : (U G)yr H (BU ) (U G)yr+1 = 0,. . . ,
(U G)yR = 0, H (G=U) = fH (BU )=h
(U G)y1 : : : (U G)yr ig ^(xr+1 : : : xR):
1. F , (U G)yr+1 = 0 : : : (U G)yR = 0
$! O
(U G)yr+1 : : : (U G)yR h
(U G)y1 : : : (U G)yr iP.
)
!, !$! 4, , H (G=U) = H (L0 d0) ^(xr+1 : : : xR)
L0 = H (BU ) ^(x1 : : : xr )
d0(1 xi) = (U G)yi 1 i = 1 r
d0(u 1) = 0 u 2 H (BU ):
5 (U G)yi , i = 1 r, " $ #
H (BU ), %
H (L0 d0) = H (BU )=h
(U G)y1 : : : (U G)yr i
%
() .
140
. 2. N$ G=U (rank G = R, rank U = r) ! H (BU )=h
(U G)y1 : : : (U G)yR i. <" , (U G)y1 : : : (U G)yR $! r %&! $
.
3. 7] $, , ! , - " :
R
Yr ; t2ki ) Y
P (G=UC t) = (1
(1 + t2kj ;1)
2li )
(1
;
t
i=1
j =r+1
ki (li ) " $ G ( U).
x
3. !
"#
$ , " , 4].
#$% 1. < | 4 (G U Q), G | $ , U | $ Q |
%$ G, G0 U G G = fg 2 G: Q(g) = gg G0 | & G .
*, (G U Q) m, Qm = id m ! !# & , " . 5 (G U Q) $ m ( ). J !
! .
- %& %& %$ !" Aut(G). -, $ %$ S1 S2 G, Aut(G), . .
S2 = S S1 S;1 S 2 Aut(G). 5 G(1 G(2 $ #
G(2 = S(G(1 ):
' $, (G G(1 S1) (G G(2 S2) $%.
141
* , !$ %&, ,
%$ Q g $! $ %$ 4
, & %$ ).
- %& !" # " 2].
10. # g | C | )
) k (k = 1 2 3), ) -
,
L ~ g. # g = gi | Zk -
i
. .* $ )
~
g0 . # Xi Yi Hi
(1 6 i 6 n) | /
g0 , 1 : : : n T(g0 ). # ~0 | L(g )
(0 1), X0 6= 0 g1 , $ xX0 2 L(g0 )~ 0 . #
(s0 : : : sn) | Pn
$+ * m = k aisi , ai 0
L(g ), 0 , ~ = (i 0) (1 6 i 6 n). &
(i) X0 X1 : : : Xn *
g 2isj
Xj (0 6 i 6 n)
$ ) m- g. 0 ) (s0 : : : snC k)+
(ii) i1 : : : it | , si1 = : : : = sit = 0. &
g0 (. . g0 ) )
(n ; t)- Q(Xj ) = e
m
, -
g(k) , i1 : : : it +
(iii) $ * ) g ) Q $
) " m.
11. , $ Q | ) (s0 : : : snC k). &
(i) Q ) , k = 1+
(ii) Q0 | ) (s00 : : : s0nC k), ) Q Q0
* ) g, r = r0 s * s0 )
g(k) .
5 $, %$ g $% +& g(k) $ k. - 4
%$ " +& g(1) , # %$ | +& g(2) g(3) .
142
. N$ (. 2]), ! An, Dn , E6 D4 " #
%$ ( !" Aut(g)). -4
Bn , Cn ,
G2, F4 , E7 E8 " .
- .
1. g = A2n .
Pk
n; n
g = t i=1 i Bn
2. g = A2n;1.
g
n; P ni
=t
3. g = Dn+1 .
g = t
4. g = E6 .
dim(g ) = 0.
dim(g ) = 1.
dim(g ) = 2.
k
i=1
n; P ni
k
i=1
1
An2 : : : Ank;1 Cnk :
Dn1 An2 : : : Ank;1 Cnk :
Bn1 An2 : : : Ank;1 Bnk :
g = F4
g = A1 B3 g = A2 A2 g = A3 A1 g = C4
m = 2
m = 4
m = 6
m = 4
m = 2:
g = t1 B3 g = t1 A1 A2 g = t1 C3
g = t1 A1 A1 A1 g = t1 C2 A1 g = t1 A3 m = 6
m = 8
m = 4
m = 8
m = 6
m = 6:
g = t2 A2
m = 8
2
g = t A1 A1 m = 10
g = t2 C2
m = 8:
dim(g ) = 3.
dim(g ) = 4.
5. g = D4 .
x
143
g = t3 A1 m = 12:
g = t4 m = 18:
g = G2
g = A2 g = A1 A1 g = t1 A1 g = t2
m = 3
m = 3
m = 6
m = 9
m = 12:
4. -! G=U | , &
%$ Q. U Q | %$, rank G = rank U,
, $ 2, ! 4 #
.
-, !, $ ! , & # %$. L
, t t | , " g g , $ $ $ $ t $ t , , +.
-, 4 ! An , Dn E6 .
-! Xi Yi Hi (1 6 i 6 R) | $"
V g
1 : : : R | 4
. 5 11 ) g +&{) g(k) (k = 2 k = 3), $ # % !
t ! , " g(k) .
1. g = A2n .
0 = n + n+1
1
i = 2 (i + 2n;i+1) (1 6 i 6 n ; 1) n = ; 12 (1 + : : : + 2n)
(1)
X
H0 = ;(H1 + : : : + H2n)
HX i = Hi + H2n;i+1 (1 6 i 6 n ; 1)
HX n = 2(Hn + Hn+1):
144
. 2. g = A2n;1.
0 = ; 21 (1 + 2n;1) + 2 + 2n;2 + : : : + n;1 + n+1 + n i = 12 (i + 2n;i) (1 6 i 6 n ; 1) n = n
(2)
HX 0 = ;(H1 + H2n;1 + 2H2 + : : : + 2H2n;2)
X
Hi = Hi + H2n;i (1 6 i 6 n ; 1) HX n = Hn :
3. g = Dn+1 .
0 = ; 1 + 2 + : : : + n;1 + 12 (n + n+1) i = i (1 6 i 6 n ; 1) n = 12 (n + n+1)
HX 0 = ;2(H1 + : : : + Hn;1) ; (Hn + Hn+1 )
X
Hi = Hi (1 6 i 6 n ; 1) HX n = Hn + Hn+1:
4. g = E6 .
5. g = D4 .
(3)
0 = ; 6 + 23 + 32 (2 + 4) + 1 + 5 1 = 6 2 = 3 3 = 21 (2 + 4) 4 = 12 (1 + 5 )
HX 0 = ;(2H1 + 3H2 + 4H3 + 3H4 + 2H5 + 2H6)
HX 1 = H6 HX 2 = H3 HX 3 = H2 + H4 HX 4 = H1 + H5 :
(4)
0 = ;2 ; 32 (1 + 3 + 4)
1 = 31 (1 + 3 + 4) 2 = 2 HX 0 = ;3H2 ; 2(H1 + H3 + H4)
HX 1 = H1 + H3 + H4 HX 2 = H2:
(5)
, ) #(2)
% ! +&{) A(2)
2n , A2n;1,
Dn(2)+1 , E6(2) D4(3) .
. Y, 4 HXi (i 6= 0) " + t g | & %$ g,
& %$ "
) (. 8]).
145
-! Q | %$ g t | ! g . -!$! # 4 HXi , , ! $" t !"
$" t.
6 $
t = (g ) + t0
(g ) | & g , t0 | ! g . 5 " .
12. # Q | ) g (m (s0 : : : sn ) k),
fi1 : : : it g i1 : : : it, $
si1 = : : : = sit = 0. #, , H1 : : : HR, X1 : : : XR , Y1 : : : YR | /
" , HX 1 : : : HX n | t g ,
$ H1 : : : HR )
(1){(5).
&
fHX i1 : : : HX it g t0 (g ) *
X
HX j ; cjkHX ik j 6= i1 : : : it t
k=1
cjk )
Xt
k=1
cjkail ik = ail j
")) ail ik 1 6 l k 6 t (
L(g ).
!. -! L(g ) | " g . +
! ~ L(g ) ! Pn
~ = ki ~i, f~0 : : : ~ng | 0
Pn
L(g ). <, deg ~ = kisi . 5 (. 2]) $0
%$
M
g0 =
L(g )~ :
deg ~=0
Pt
U ~ > 0 (;~ > 0), deg ~ = 0 ! , ~ = kir ~ ir .
r=1
N$ (. 5]), ~ > 0 L(g )~ ej1 : : : ejs ], ~ j1 + : : : + ~ js = ~ . 5 $,
, L L(g
HX i, eir , fir " $" )~ . -4
deg ~=0
HX ir = eir fir ], eir , fir " $" g0 .
-& (aik il )l6kl6t & + (aij ) L(g ) 146
. & +. X
HX j ; cjk HX ik (j 6= i1 : : : it)
t
k=1
" & g0. N$ &
, 4%%& cjk #
t
X
cjkail ik = ail j : 2
k=1
x
-
5. 1. g = A2n .
7 +&{) A(2)
2n :::
. <, $ 10 A(2)
2n , " & # %$ A2n n.
-! x1 : : : x2n+1 | g. -! HX i (0 6 i 6 n) | , (1). 5
xn+1 (HX i) = 0 0 6 i 6 n
xj (HX i) = ;x2n;j +2(HX i ) 0 6 i 6 n 1 6 j 6 n:
5 &
U G !
$ HX i (0 6 i 6 n), G " xn+1 = 0
xj = ;x2n;j +2:
-4
(U G)2j +1 = 0 1 6 j 6 n
(U G)2j = (;1)j (U G)j (x21 : : : x2n) 1 6 j 6 n:
5 $, , " & U #
%$ A2n ! A2n .
-4 $ 8 ($ ") +3
+3
147
13. 1
% $ A2n % )
H (A2n =U) = (H (BU )=h
(U A2n)2j i) ^(z3 : : : z2n+1) =
= (H (BU )=h
(U A2n)j (x21 : : : x2n)i) ^(z3 : : : z2n+1)
zi xi .
'. -, ! , Ck Bn;k . 7& t = L(HX 0 : : : HX k;1) L(HX k+1 : : : HX n):
-! y1 : : : yk | Ck , yk+1 : : : yn | Bn;k . + A2n t
$" " $:
x1 ! ;yk xk+1 ! yk+1 x2 ! ;yk;1 xk+2 ! yk+2 :: :: : :: :: :: :: :: : :: :: :: :: : : : :
xk ! ;y1 xn ! yn 4
(U G)2j (x1 : : : x2n+1) = (;1)j j (y12 : : : yn2 ) 1 6 j 6 n
H (A2n =Ck Bn;k ) =
= ((S(y12 : : : yk2 ) S(yk2+1 : : : yn2 ))=S + (y12 : : : yn2 )) ^(z3 : : : z2n+1):
2. g = A2n;1.
7 +&{) A(2)
2n;1 :::
. + A2n, " & # %$ A2n;1 n.
-! x1 : : : x2n | g. -! HX i (0 6 i 6 n) | , (2). 5 xj (HX i) = ;x2n;j +1(HX i ) 0 6 i 6 n 1 6 j 6 n:
5 &
U G !
$ HX i (0 6 i 6 n), xj = ;x2n;j +1:
ks
148
. -4 , (U G)2j +1 = 0 1 6 j 6 n ; 1
(U G)2j = (;1)j (U G)j (x21 : : : x2n) 1 6 j 6 n:
+ A2n, , " & U # %$ A2n;1 ! A2n;1. -4 " .
14. 1
% $ A2n;1 % )
H (A2n;1=U) = (H (BU )=h
(U A2n;1)2j i) ^(z3 : : : z2n;1) =
= (H (BU )=h
(U A2n;1)j (x21 : : : x2n)i) ^(z3 : : : z2n;1)
zi xi .
'. - Dk Cn;k . 7& t = L(HX 0 : : : HX k;1) L(HX k+1 : : : HX n):
-! y1 : : : yk | Dk , yk+1 : : : yn | Cn;k . + g $" " $:
x1 ! ;yk xk+1 ! yk+1 x2 ! ;yk;1 xk+2 ! yk+2 :: :: : :: :: :: :: :: : :: :: :: :: : : : :
xk ! ;y1 xn ! yn 4
(U G)2j (x1 : : : x2n) = (;1)j j (y12 : : : yn2 ) 1 6 i 6 n:
H (A2n;1=Dk Bn;k ) =
= (hS(y12 : : : yk2 ) y1 : : : yk i S(yk2+1 : : : yn2 )=S + (y12 : : : yn2 )) ^(z3 : : : z2n;1):
3. g = Dn+1 .
7 +&{) Dn(2)+1 :::
. " & # %$
Dn+1 n.
-! x1 : : : xn+1 | g. -! HX i (0 6 i 6 n) | , (3). 5
xn+1 (HX i) = 0 0 6 i 6 n:
ks
+3
149
5 $, , &
U # %$ G
(U G)n+1 = 0:
7!, & # %$ Dn+1 ! Dn+1 , A2n, " .
15. 1
% $ Dn+1 % )
H (Dn+1 =U) = (H (BU )=h
(U Dn+1 )1 : : : (U Dn+1 )ni) ^(zn+1 )
zi xi .
'. - Bk Bn;k
(0 6 k 6 n, B0 = ?, B1 = A1 ). 5 , t = L(HX 0 : : : HX k;1) L(HX k+1 : : : HX n):
-! y1 : : : yk | Bk , yk+1 : : : yn | Bn;k . + g $" " $:
x1 ! ;yk xk+1 ! yk+1 x2 ! ;yk;1 xk+2 ! yk+2 :: :: : :: :: :: :: :: : :: :: :: :: : : : :
xk ! ;y1 xn ! yn 4
(U G)i(x21 : : : x2n+1) = i (y12 : : : yn2 ) 1 6 i 6 n
H (n+1 =k Bn;k ) =
= (S(y12 : : : yk2 ) S(yk2+1 : : : yn2 ))=S + (y12 : : : yn2 ) ^(zn+1 ):
4. g = D4 .
7 +&{) D4(3) _
*4
. & D4 , &
# %$, 2.
J& + D4(3) 0 2 ;1 01
@;3 2 ;1A :
0 ;1 2
150
. -! x1 x2 x3 x4 | D4 HX i (0 6 i 6 3) |
, (4). 5 x1(HX 1 ) = 1 x1(HX 2 ) = 0 x1 (HX 0) = ;2
x2(HX 1 ) = ;1 x2(HX 2 ) = 1 x2 (HX 0) = ;1
x3(HX 1 ) = 2 x3(HX 2 ) = ;1 x3 (HX 0) = ;1
x4(HX 1 ) = 0 x4(HX 2 ) = 0 x4 (HX 0) = 0:
-! i(x21 x22 x23 x24), i 6 i 6 3, 4 = x1x2x3 x4 | $" Rt]WD4 .
<, &
U # %$ G x4 = 0
x1 = x2 + x3 :
5 $, , (U G)4 = 0
(U G)1 = (U G)(2(x2 + x3)2 ; x2x3])
(U G)2 = 41 (
(U G)1)2 (U G)3 = (U G)((x2 + x3)2 x22x23)
4 & # %$ D4
! D4 . - 8, "" .
16. 1
% $ D4 % )
H (D4 =U) = (H (BU )=h
(U G)1 (U G)3i) ^(z2 z4 ):
'(. - & U D4 .
1. g = A1 t1 .
t =; L(HX 1 ) L(HX 2 ; c21HX 1), c21a11 = a12 ) c21 = ; 12 , t = L(HX 1 ) L HX 2 + 21 HX 1 .
y1 , ;y1 | A1 , y2 | t1 ,
x2 ! ;y1 + 12 y2 x2 + x3 ! y1 + 12 y2
1
x3 ! 2y1 x2x3 ! 2y1 ;y1 + 2 y2 151
(A1 T 1 D4)1 = 3y12 + 41 y22 = J1
2
(A1 T 1 D4 )3 = 4y12 14 y22 ; y12 = J2 H (D4 =A1 T 1 ) = (Ry2] Sy12 ]=hJ1 J2i) ^(z2 z4):
2. g = A2 .
t = L(HX 1 HX 0).
y1 , y2 , y3 | t ,
x2 ! y3 ; y1 x2 + x3 ! y3 ; y2
x3 ! y1 ; y2 x2x3 ! (y3 ; y1 )(y1 ; y2 )
(A2 D4)1 = 2((y3 ; y2 )2 ; (y3 ; y1 )(y1 ; y2 )) = J1 (A2 D4)3 = (y3 ; y2 )2 (y3 ; y1 )2 (y1 ; y2 )2 = J2 H (D4 =A2) = (Sy1 y2 y3 ]=hJ1 J2i) ^(z2 z4):
3. g = g2.
y1 y2 y3 | g2 ,
x2 ! ;y3 x3 ! ;y2 (G2 D4 )1 = 2(y12 + y22 + y33 ) = P1
(G2 D4 )3 = y12 y22 y33 = P2 H (D4 =G2) = ^(z2 z4 ):
. & # %$ D4 g2 ( !
G(1)
2 ).
5. g = E6 .
7 +&{) E6(2) ks
" & +:
0 2 ;1 0 0 01
B
;1 2 ;1 0 0C
B
C:
B
0 ;1 2 ;1 0C
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0 0 0 ;1 2
. & E6, &
# %$, 4.
152
. -! x1 : : : x6 | E6 Hi, 0 6 i 6 4, | , (5). F, (x1 + x6)(HX i ) = (x2 + x5 )(HX i) = (x3 + x4)(HX i ) i = 0X4
, !
t &
%$ Q
x1 + x6 = x2 + x5 = x3 + x4 :
5 $, % ai, bi cij , 1, t "
"
:
ai = xi + x1 + x6 bi = xi ; 2(x1 + x6)
bi = ;a7;i cij = ;ai ; bj = ;ai + a7;j :
-4 $ (U E6) " $:
(U E6)(Iki ) =
6
X
i=1
(U E6)(ai )ki +
5
X
j =2
(U E6)(c1j )ki +
X
j =34
(U E6)(c2j )ki :
- 4 , (U E6)(I5 ) = (U E6)(I9 ) = 0
4 & # %$ E6
! E6 . 5 $, 8 , 17. 1
% $ A2n % )
H (E6 =U) =
= (H (BU )=h
(U E6)I2 (U E6)I6 (U E6)I8 (U E6)I12 i) ^(z5 z9 ):
'(. - & U E6 .
1. dim((g )) = 3.
g = t3 A1 C i1 = 1, j = 2 3 4C t = t3 L(HX 1 )C t3 = L(HX j ; cj 1a11HX 1),
cj 1a11 = a1j .
t = L HX 2 + 12 HX 1 HX 3 HX 4 L(HX 1):
153
1 2 3 | t , y1 , ;y1 | A1 .
x1 ! 12 1 ; 13 3 + y1 x2 ! 12 1 ; 31 3 ; y1 x3 ! ;1 + 2 ; 13 3 x4 ! 1 ; 2 + 23 3 x5 ! ; 12 1 + 23 3 + y1 x6 ! ; 12 1 + 32 3 ; y1 x1 + x6 ! 13 3 a1 ! 21 1 c12 ! ;1 + 3 a2 ! 12 1 ; y1 c13 ! 12 1 ; 2 + 3 ; y1 a3 ! ;1 + 2 c14 ! ; 32 1 + 2 ; y1 a4 ! 1 ; 2 + 3 c15 ! ;2y1 1
a5 ! ; 2 1 + 3 + y1 c23 ! 12 1 ; 2 + 3 + y1 a6 ! ; 12 1 + 3 ; y1 c24 ! ; 32 1 + 2 + y1 :
H (E6 =T 3 SU(2)) =
= (R1 2 3 ] Ry12]=hq2 q6 q8 q12i) ^(z5 z9 ):
2. dim((g )) = 2.
g = t2 C2 C i1 = 2, i2 = 3, j = 1 4C t = t2 L(HX 2 HX 3)C
P2
P2
t2 = L HX j ; cjk HX ik , cjk ailik = ailj , 1 6 l 6 2.
k=1
k=1
t = L HX 1 + HX 2 + HX 3 HX 4 + 21 HX 2 + HX 3 L(HX 2 HX 3):
1 2 | t2 , y1 , y2 | C2.
x1 ! 1 ; 13 2 x2 ! 61 2 + y1 x3 ! 16 2 + y2 x4 ! 16 2 ; y2 x5 ! 16 2 ; y1 x6 ! ;1 + 23 2 x1 + x6 ! 13 2 154
. a1 ! 1 a2 ! 21 2 + y1 a3 ! 12 2 + y2 a4 ! 21 2 ; y2 a5 ! 21 2 + 3 ; y1 a6 ! ;1 + 2 c12 ! ;1 + 21 2 ; y1 c13 ! ;1 ; 2 + 21 2 ; y2 c14 ! ;1 + 12 2 + y2 c15 ! ;1 + 12 2 + y1 c23 ! ;y1 ; y2 c24 ! ;y1 + y2 :
H (E6 =T 2 SO(4)) =
= (R1 2 ] S(y12 y22 )=hq2 q6 q8 q12i) ^(z5 z9 ):
3. dim((g )) = 1.
g = t1 A3 C i1 = 0, i2 = 1, i3 = 2, j = 3C t = t1 L(HX 0 HX 1 HX 2)C
P3
P3
t1 = L HX 3 ; c3k HX ik , cjk ail ik = ail 3, 1 6 l 6 2.
k=1
k=1
t = L(HX 4 ) L(HX 0 HX 1 HX 2):
1 | t1 , y1 , y2 , y3 , y4 | A3 , " HX 0, HX 1, HX 2.
x2 ! ; 31 1 ; 13 y1 + y3 x1 ! ; 13 1 ; 13 y1 + y2 x3 ! ; 31 1 ; 43 y1 ; y2 ; y3 x4 ! 32 1 + 32 y1 + y2 + y3 x5 ! 23 1 ; 31 y1 ; y3 x6 ! 23 1 ; 31 y1 ; y2 x1 + x6 ! 13 1 ; 32 y1 a1 ! ;y1 + y2 c12 ! 1 ; y2 ; y3 a2 ! ;y1 + y3 c13 ! 1 + y1 + y3 a3 ! ;y1 + y4 c14 ! ;y2 + y4 a4 ! 1 + y2 + y3 c15 ! ;y2 + y3 a5 ! 1 ; y1 ; y3 c23 ! 1 + y1 + y2 a6 ! 1 ; y1 ; y2 c24 ! ;y3 + y4 :
H (E6 =T 1 SU(4)) =
= (R1] S(y1 y2 y3 y4)=hq2 q6 q8 q12i) ^(z5 z9 ):
155
4. dim((g )) = 0.
g = A1 B3 C i1 = 0, i2 = 2, i3 = 3, i4 = 4.
t = L(HX 0 HX 2 HX 3 HX 4):
y4 | A1 , y1 , y2 , y3 | B3 , " HX 2, HX 3, HX 4. 5
y3 (HX 4) = 0 y2 (HX 4) = ;1 y1 (HX 4) = 1
y3 (HX 3) = ;1 y2 (HX 3) = 1 y1 (HX 3) = 0
y3 (HX 2) = 2 y2 (HX 2) = 0 y1 (HX 3) = 0
x2 ! 16 y1 + 12 y2 + 21 y3 ; 31 y4 x1 ! ; 13 y1 ; 34 y4 x3 ! 16 y1 + 12 y2 ; 12 y3 ; 13 y4 x4 ! 16 y1 ; 12 y2 + 21 y3 ; 31 y4 x5 ! 16 y1 ; 12 y2 ; 12 y3 ; 13 y4 x6 ! 23 y1 + 23 y4 x1 + x6 ! 31 y1 ; 23 y4 a1 ! ;2y4 c12 ! 21 (y1 ; y2 ; y3 ) + y4 a2 ! 21 (y1 + y2 + y3 ) ; y4 c13 ! 12 (y1 ; y2 + y3 ) + y4 a3 ! 21 (y1 + y2 ; y3 ) ; y4 c14 ! 21 (y1 + y2 ; y3 ) + y4
a4 ! 12 (y1 ; y2 + y3 ) ; y4 c15 ! 21 (y1 + y2 + y3 ) + y4
a5 ! 12 (y1 ; y2 ; y3 ) ; y4 c23 ! ;y2 a6 ! y1 c24 ! ;y3 :
H (E6 =SU(2) SO(7)) =
= (Ry12] S(y12 y32 y42 )=hq2 q6 q8 q12i) ^(z5 z9 ):
g = F4C i1 = 1, i2 = 2, i3 = 3, i4 = 4.
t = L(HX 1 HX 2 HX 3 HX 4):
y1 , y2 , y3 , y4 | F4, "
HX 1, HX 2 , HX 3, HX 4 . 5
y4 (HX 1) = ;1 y3 (HX 1) = ;1 y2 (HX 1) = ;1 y1 (HX 1) = 1
y4 (HX 2) = 2 y3 (HX 2) = 0 y2 (HX 2) = 0 y1 (HX 2) = 0
y4 (HX 3) = 0 y3 (HX 3) = 1 y2 (HX 3) = 0 y1 (HX 3) = 0
y4 (HX 4) = 0 y3 (HX 4) = ;1 y2 (HX 4) = 1 y1 (HX 4) = 0
156
. x1 ! 32 y1 ; 13 y2 x2 ! 16 y1 + 16 y2 + 21 y3 + 21 y4 x3 ! 16 y1 + 16 y2 + 12 y3 ; 12 y4 x4 ! 16 y1 + 16 y2 ; 21 y3 + 21 y4 x5 ! 16 y1 + 16 y2 ; 12 y3 ; 12 y4 x6 ! ; 13 y1 + 32 y2 x1 + x6 ! 13 (y1 + y2 )
a1 ! y1 c12 ! 21 (;y1 + y2 ; y3 ; y4 )
a2 ! 21 (y1 + y2 + y3 ; y4 ) c13 ! 12 (;y1 + y2 ; y3 + y4 )
a3 ! 21 (y1 + y2 + y3 ; y4 ) c14 ! 12 (;y1 + y2 + y3 ; y4 )
a4 ! 21 (y1 + y2 ; y3 + y4 ) c15 ! 12 (;y1 + y2 + y3 + y4 )
a5 ! 21 (y1 + y2 ; y3 ; y4 ) c23 ! ;y3 a6 ! y2 c24 ! y4 :
1] $, qk = (Ik ) " $" H (BF4 ), 4
H (E6=F4) = ^(z5 z9 ):
. E6, & # %$ , F4 . ) C4 4 $
! 1], ! 4 , ! #
) F4 +&{) E6.
3, $ , - , $
$ 3.
&
1] Takeuci M. On Pontrjagin classes of compact symmetric spaces // J. Fac. Sci. Univ.
Tokyo Sec. I. | 1962. | Vol. 9. | P. 313{328.
2] Helgason S. Dierential Geometry, Lie groups and Symmetric spaces. | New York,
London: Acad. Press, 1978.
3] . !" ##!!" $#%!#% &!&!"
$#%!#% $%!" '$$ ( // )##!!" $#%!#% $+!. | ,.: -(, 1958. | /. 163{246.
4] 1 #2 . 334!!" #%5# $#%!#% . | ,.: ,, 1984.
157
5] 6!3 7. ., !4 . (. /! $ '$$ ( 35# '$$. | ,.: 8)//, 1995.
6] !4 . (. 9$ %!:% !" '$$ $3: !2. | ,., 1995.
7] ;! 1'!. <!" <'! $%!" &!&!" ! " $#%!#% # !$ &2 #%=!!2 '$$2 // 9. #. %. %!:. !. |
1968. | 6"$. 14. | /. 33{93.
8] 1= 6. #!5!!" 3" (. | ,.: ,, 1993.
9] 1= 6. %>:" !5! $& $'$#%" 3 ( // ?'!=.
!: $. | 1969. | 9. 3, "$. 3. | /. 94{96.
10] Coxeter H. S. M. The product of the generators of a Bnite group generated by reEections // Duke Math. J. | 1951. | Vol. 18. | P. 765{782.
' ( 1998 .
( = 3, D = 3)
. . , . e-mail: tishchen@kapella.gpi.ru
519.172.2+519.173+519.177
: , -, ! "#.
$%& '% (%)& 3 (%)&# &)* +& 3 % &,- & &- .#%# -((. /%" & 0%)& +& &# '% 1!+. /%2&
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Abstract
S. A. Tishchenko, Maximum size of a planar graph (4 = 3, D = 3), Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 1, pp. 159{171.
The problem of maximumsize of a graph of diameter 3 and maximumdegree 3 as
a function of its Euler characteristics is studied. The negative solution of an Erd!os
problem is obtained. A new approach to such problems is proposed which consists
in counting the paths between di:erent pairs of vertices in a graph.
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f3 2 F4 (G ). ; 4
A 3. $ A" x y (. . 5 $)), #
a1 a2 a3a4 2 P2(G), a1 a2 a3a4 f3 ,
a1a2 = a3 a4. B$ A
a1 a2 a4a3 $ a5 6 f3 .
C
, a1a2 a5 = a4 a3a5 . J
, a6 a1a2 = a6a4 a3, 6
3- 4-
.
F, 5, 6 3- pk ,
fi 2 F3(G ) F4(G ), A 3-, $ " !
. ; 3, 4 " 2, A pk 6 fj 2 F (G ). 6 $
$ #
$ , fi 2 F3 (G ) F4 (G ) !
$ a. ; 5, 6 $ 4-
, #
$ a " . ) $
! #$ $ ,
,# $ 4-
(
a) !
4-
, 7.
f4 2 F3(G ) f2 2 F5 (G ), !
$ a (. 1 )). 7 a3a4 a5 a2a7 a6 ,
$ $ , . ; 11 A p " 2, ,
p 6 fj 2 F(G ), 5, 6 8 #$ a4 a7 !
fi 2 F3(G ) F4 (G ) F5(G ), f2 , A 7 $ . " jE35j.
K 3- 4-
#$, !
3- 5-
, , 5{7 4jF4j + 6jF3j + jE35j pk 6 fj 2 F (G ). B$ pk fj 2 F(G ), (5). 4. #
! 6.
21 6 jV j(192; jV j) + jF5(max )j ; jF4(max )j ; jE35(max )j ; jV2 j: (6)
C
,
= jV j ; jE j + jF(max)j = (jV2 j + jV3j) ; jV2 j + 32 jV3j +
X
X
+ jFm (max )j =
jFm (max )j ; 21 jV3j (7)
m>3
m> 3
( = 3, D = 3)
X
m>3
mjFm j = 2jE j = 2jV2j + 3jV3j:
167
(8)
I$
(7) (8), "
X
15
3
(21 ; 3m)jFm (max )j = 21 + 21
2 jV3j; 6jV2j; 9jV3 j = 21 ; 2 jV2j + 2 jV j: (9)
m>3
), , (3) 4 (5) 5, X
X
(21 ; 3m)jFm j = (21 ; 3m)jFm j + 3jF6j + 6jF5j + 9jF4j + 12jF3j 6
m>3
m>7
6 jP~ j ; jF4j + jF5j ; jE35j 6 jV j(222; jV j) ; 17
2 jV2 j + jF5j ; jF4j ; jE35j: (10)
C$L
(9) (10), (6). M (6), , A " , jF5(max )j;jF4(max )j;jE35 (max )j;jV2 j,
" ". 6, ,
, !
.
7.
jV j2 ; 21jV j + 50 6 0:
(11)
I$
X
jF (max)j = jFm (max )j + jF5(max )j + jF4(max )j + jF3(max )j = 21 jV3j + m>6
2jE j =
X
m>6
mjFm j + 5jF5j + 4jF4j + 3jF3j = 3jV3j + 2jV2 j
(12)
(13)
"
X
(m ; 6)jFm (max )j;jF5(max )j; 2jF4(max )j; 3jF3(max )j 6 2jV2j; 6: (14)
m>6
E 5, 6 (8) 6, X
jE35j > 3jF3j ; mjFm j = 5jF5j + 4jF4j + 6jF3j ; 2jV2j ; 3jV3 j:
m>6
(15)
C$L
(14) (15), X
3jF5(max )j ; jE35(max )j + 2 (m ; 6)jFm (max )j 6 6jV2j + 3jV3j ; 12 !
m>6
! jF5(max )j ; jF4(max )j ; jE35(max )j ; jV2j 6 jV j ; 4: (16)
(16) (6) 6 (11). 168
. . 8.
jV j2 ; 20jV j + 40 6 0:
(17)
jF5(max )j 6 jF (max)j = 21 jV j ; 12 jV2 j + (18)
(6) (17). 1.
'# 3
-
"
3
-
"" (#
# #:
= 2 jV j 6 12 (
)N
= 1 jV j 6 16
) jV j > 12 # #
jV j (11) 7.
$) jV j > 16 # #
jV j (17) 8. %
1, " . B = 1 A , $ 2 " 14 . ; = 2 , .
12.
(G) = 2
f 2 F(G max)
a 2 E(G)
C , $ a , f " . C
, , . . " ,
" A $ 3. $ a " $ 17 " " (
1-, 2- !
3-). C
3(jV j ; 3) > 30 " , . 2.
3
3
12
C , # (11) jV j = 13. G 13. ; 1
f 2 F(G ) " $ 2. 2 " , "
2jV2j, 11 " $ 12jV2j. . (
"
$ " :
2jE j ; 12jV2j 6 3(jF j ; 2jV2j) ! 27 6 11jV2j
(19)
# #
jV2j jV2 j = 1. )
(6) )
$)
$ *
+
.
, %&#
.
# % , """ .
, -
( = 3, D = 3)
169
$ jF5j ; jF4j ; jE35j > 4:
(20)
O "," 2 $ 26, # (20) 11 5-
" 4-
. )
$, A , jF4j = 0N jF3j > 2:
(21)
A jF5j 6 5, (20) jE35j 6 1. (
, " $ g 2 F3(G ) $," #$ 5-
, . ., # 5, 6, # $ " fi 2= F3(G ) F4(G ) F5 (G ). ; (20){(21) " $ ", A " " #$ g. ; 3 A , A " #$ a 2 E(G), 12. 6 . E, $ 3 3, , 12, . A #
. 6 3-
3 12 . ( , 1 = 2 $ $ " $" 19].
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"
3 18 ( = ;1) 20 ( = ;2).
) . 8 " .
170
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(')). G& (&, "&(
5. ()*
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. C
,
. A $"
2 # (!
)
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( = 3, D = 3)
171
J $ ;. J. D, J. >. (
D. >. O$ $
$.
+ !
1] M. Fellows. Three for money: The degree/diameter problem. | http://www.c3.lanl.
gov/mega-math/workbk/graph/grthree.html.
2] J. C. Bermond, C. Delorme, G. Farhi. Large graphs with given degree and diameter II // J. Combin. Theory. | 1984. | B 36. | P. 32{48.
3] I. Alegre, M. A. Fiol, J. L. A. Yebra. Some large graphs with given degree and diameter // J. Graph Theory. | 1986. | V. 10. | P. 219{224.
4] F. R. K. Chung. Diameters of graphs: old problems and new results // Congressus
Numerantium. | 1987. | V. 60. | P. 295{317.
5] P. Hell, K. Seyarth. Largest planar graphs of diameter two and xed maximum
degree // Discrete Math. | 1993. | V. 111. | P. 313{332.
6] M. Fellows, P. Hell, K. Seyarth. Large planar graphs with given diameter and maximum degree // Discrete Appl. Math. | 1995. | V. 61. | P. 133{153.
7] F. Goebel, W. Kern. Planar regular graphs with prescribed diameter. | Univ. of
Twente (The Netherlands) Applied Math. Memorandum. No. 1183, December 1993.
8] M. Fellows, P. Hell, K. Seyarth. Constructions of large planar networks with given
degree and diameter // Networks. | 1998. | V. 32. | P. 275{281.
9] R. W. Pratt. The complete catalog of 3-regular, diameter-3 planar graphs. | http:
//www.unc.edu/~rpratt/graphtheory.html.
( ) 1999 .
. . . . . 515.16
: , , .
! " " #$ $ !%. " &
, . . '
"& Dk Dl ! Dk ,
&
" & ", &" $ ($
) (
@Dk Dl ! @Dk , Dk @Dl ! Dk @Dk @Dl ! @Dk )
&"
" "" $ $ . ,
" , ( !" &
$ $ ($ ) &" , . . ""
"
" . - ,'""
" ('. . ,&( ! (
&
!," & !,.
Abstract
S. M. Tuleshev, Spaces of immersions of elementary bre bundles, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 1, pp. 173{197.
In the paper we consider immersions of elementary 5bre objects. For 5bre disk,
i. e. a trivial 5bre bundle of type Dk Dl ! Dk , we give a de5nition of 5brewise immersion and for 5bre boundary spheres (5bre bundles @Dk Dl ! @Dk ,
Dk @Dl ! Dk and @Dk @Dl ! @Dk ) we de5ne the notion of framed 5brewise
immersions. It is proved that the natural maps from the space of immersions of 5bre
disk into spaces of framed immersions of 5bre boundary spheres satisfy the axiom
of the covering homotopy, i. e. are Serre bundles. This result is an initial step in
solving the problem on the homotopy description of space of immersions of one 5bre
manifold into another 5bre manifold.
1] . ! , # E # $ Dk Rn # B
- ! (
( && 6778 9 96{01{00287{.
, 2001, 7, 9 1, . 173{197.
c 2001 !,
"#
$% &
174
. . # $ $ ' S k;1 Rn, . . , ($ ( f : S k;1 ! Rn
( g: S k;1 ! Rn, $ # ' S k;1 f:
g(x) 2= f (Tx (S k;1 )) Rn. ,( : E ! B, - Dk S k;1 , # # (.
. , ( -
. / 0
-, , , ((( #
#- ' ( . 1(
$ 2, ( (, $ 3), $ ( $ $ .
30 # 4 0 ((( -
( E B, $ , $ 0$ 2 ( ' . 5 # , --( -.
3# E | $ ( $ Dk Dl $ Rm k Rn, m. . $ (F f) F : Dk Dl ! Rm Rn,
f : D ! R , ( $ -( :
Dk Dl
?
?
y
F
;;;;!
Rm ? Rn
?
y
:
f
Dk ;;;;!
Rm
. # 0$
$ # ( @Dk Dl , Dk @Dl @Dk @Dl ) ( - (,
-( '' # #$ .
- # B, C D. 7 ( # , ( E
$ E ! B, E ! C , E ! D (-
-
(. 2).
1. (( ( $ 0$ 2 (Dk Dl , @Dk Dl , Dk @Dl @Dk @Dl ) Rm+n = Rm Rn,
175
( ( $ 4(, ##( ( #.
3# k, l, m n | # , m > k, n > l. 3# Dr (((
r- Rr ( , . . Dr # (t x), t
# ( 0 , x ((( ' @Dr Dr . ($ Dk Dl , @Dk Dl Dk @Dl
(s x< t y), (x< t y) (s x< y) , s t 2 0 1],
x 2 @Dk , y 2 @Dl .
- 4: ( $ # #( ( .
1. 7 # (
Dk Dl ! Dk Rm Rn ! Rm (F f), F (((
C 1 - Dk Dl Rm+n, f | C 1 - Dk Rm, - Dk Dl
??
y
F
;;;;!
Rm+n =?Rm Rn
?
y
:
f
;;;;!
Rm
ln = E $ $ = Ekm
k
l
k
m
D D ! D R Rn ! Rm. > E C 1-- Dk
#- ((F 0 f 0 ) (F 00 f 00)) = maxf0 (F 0(X) F 00(X)) 0 (F 0 (V ) F 00(V )) j
X 2 Dk Dl V 2 TX (Dk Dl ) kV k = 1g
(F 0 f 0 ) (F 00 f 00) 2 E , 0 | Rm+n, Rm+n (( # TX (Dk Dl ) (((
# Dk Dl X.
2. 7 # ( @Dk Dl ! @Dk Rm Rn ! Rm G = ((G g) (G~ g~)), G g ((-( 1-( @Dk Dl Rm+n @Dk Rm , ( $ G Rm+n
@Dk Dl ;;;;!
?
?
?
?
y
y g
@Dk ;;;;!
Rm
~ ),
G~ ((( C 1 - @Dk Dl Rm+n n f0g, G(Y
Y 2 @Dk Dl , #
Y 176
. . G, . . G TY (@Dk Dl ), g~ | C 1- @Dk Rm nf0g, ( g~(y), y 2 @Dk , #
y g (=2 g Ty (@Dk )), G~
@Dk Dl ;;;;!
Rm+n n f0g Rm+n
??
?
?
y
y :
@Dk
g~
;;;;!
Rm n f0g Rm
ln = B $ $ A Bkm
# ( @Dk Dl ! @Dk Rm Rn ! Rm.
> B -- : ( G 0 = ((G0 g0 ) (G~ 0 g~0)), G 00 =
= ((G00 g00) (G~ 00 g~00)) 2 B #
~(G 0 G 00) = maxf0 (G0 (Y ) G00(Y )) 0 (G0 (V~ ) G00(V~ )) 0 (G~ 0(Y ) G~ 00(Y )) j
Y 2 @Dk Dl V~ 2 TY (@Dk Dl ) kV~ k = 1g
0 | , 4.
3. 7 # , ($ ( Dk @Dl ! Dk RmRn ! Rm, B H f ((-( C 1 -( Dk @Dl
H = ((H f) H),
m
+
n
k
m
R D R , Dk @Dl
?
?
y
H Rm+n
;;;;!
?
?
y
f
Dk ;;;;!
Rm
, HB ((( C 1- Dk @Dl Rm+n n f0g,
B
Z 2 Dk @Dl , #
Z
H(Z),
H (=2 H TZ (Dk @Dl )) H Rm+n n f0g Rm+n ;! Rm
Dk @Dl ;!
( ( Rm+n = Rm Rn #, Dk @Dl 0 2 Rm.
ln = C $ $ = Ckm
# (, ($ Dk @Dl ! Dk
Rm Rn ! Rm. .- C #- B(H0 H00) = maxf0 (H 0(Z) H 00 (Z)) 0 (H 0 (VB ) H 00(VB )) 0 (HB 0 (Z) HB 00(Z)) j
Z 2 Dk @Dl VB 2 TZ (Dk @Dl ) kVB k = 1g:
177
= : E ! B - . 1( (F f) 2 E
~ g~)), G g ((-( # (F f) = ((G g) (G
k
l
~ t y) = rsF (1 x< t y), g~(x) = rsf(1 x)
( F @D D f @Dk , G(x<
(# #
4 rs rt - '' s t
). C ^ : E ! C . 3# (
B H (((
(F f) 2 E ^ ( '
^ (F f) = ((H f) H),
k
l
B
F D @D , H(s x< y) = rtF(s x< 1 y). E #,
^ .
, , ( @Dk @Dl Rm+n.
4. 7 ( #$ 2- @Dk @Dl ! @Dk Rm Rn ! Rm ~ g~) G),
B G g ((-( 1 -( @Dk @Dl Rm+n
G = ((G
g) (G
k
m
@D R , ( $ G Rm+n
@Dk @Dl ;;;;!
?
?
?
?
y
y g
@Dk ;;;;!
Rm
1
k
l
~
G~ ((( C - @D @D Rm+n n f0g, ( G(Z),
Z 2 @Dk @Dl , #
Z G (=2 G TZ (@Dk @Dl )), g~ ((( C 1 - @Dk
Rmnf0g, g~(y), y 2 @Dk , #
y g (=2 g Ty (@Dk )), G~
@Dk @Dl ;;;;!
Rm+n n f0g Rm+n
??
?
?
y
y g~
@Dk ;;;;!
Rm n f0g Rm
, GB ((( C 1- @Dk @Dl Rm+n nf0g, B
G(Z),
Z 2 @Dk @Dl , #
Z G (=2 G TZ (@Dk @Dl )) G Rm+n n f0g Rm+n ;! Rm
@Dk @Dl ;!
@Dk @Dl 0 2 Rm. , ( , ~ G(Z),
B
G(Z)
Z 2 @Dk @Dl , .
ln
A Dkm = D $ $ ( ( #$ 2- @Dk @Dl ! @Dk
Rm Rn ! Rm. .- D #- -
: ( G 0 = ((G0 g0) (G~ 0 g~0) GB 0), G 00 = ((G00 g00) (G~ 00 g~00) GB 00) #
^(G 0 G 00) = maxf0 (G0 (Z) G00(Z)) 0 (G0 (V^ ) G00(V^ )) 0 (G~ 0(Z) G~ 00(Z))
0 (GB 0 (Z) GB 00(Z)) j Z 2 @Dk @Dl V^ 2 TZ (@Dk @Dl ) kV^ k = 1g:
178
. . = ~ : C ! D , 4 B 2 C #
: E ! B. 1( ((H f) H)
B
~
B
B
~ ((H f) H)) = ((G g) (G g~) G), G G ((-( ( H HB
~ y) = rsH(1 x< y),
@Dk @Dl , g ((( f @Dk , G(x<
g~(x) = rsf(1 x).
2. G(, ( -
, ((-( -
(. 1.1 1]) ( $ .
.
(i) : E ! B (m > k, n > l) ^ : E ! C (m > k, n > l)
.
(ii) ~ ^ : E ! D (m > k, n > l) .
. (i) 1# , ^ | (, # .
3# ( 0 P Gv : P ! B Hv : P ! C 0 6 v 6 1:
,$ # : ( p P
Gv(p) = ((Gv (p) gv(p)) (G~ v(p) g~v(p))) Hv(p) = ((Hv(p) fv (p)) HB v(p)):
H , # G0 H0 ( F F^ ,
. . - (
^ f(p))
^
F F^ : P ! E F (p) = (F(p) f(p)) F^(p) = (F(p)
p 2 P
^ = f0 (p) p 2 P):
G0 = F H0 = ^ F^ (f(p)
> - Fv F^v : P ! E 0 6 v 6 1 F0 = F F^0 = F^
( $ ( p P - Fv(p) = (Fv(p) fv (p)) F^v (p) = (F^v (p) f^v(p))
e Fv = Gv ^ F^v = Hv :
,$ # ( , 0 #, P |
.
179
3# "1 (v p< x< t y) # ( G~ v (p)(x< t y) #
Gv (p)] T(xty)(@Dk Dl ), #
"1 = minf"1 (v p< x< t y) j 0 6 v 6 1 p 2 P (x< t y) 2 @Dk Dl g
"2 = minfkrV Gv (p)(x< t y)k j
0 6 v 6 1 p 2 P (x< t y) 2 @Dk Dl V 2 T(xty) (@Dk Dl ) kV k = 1g
" = (1=10) minf"1 "2 1g. rV - # ( V .
C, # "^1 (v p< s x< y) (- HB v (p)(s x< y) Hv (p)] T(sxy) (Dk @Dl ) "^1 = minf"^1 (v p< s x< y) j 0 6 v 6 1 p 2 P (s x< y) 2 Dk @Dl g
"^2 = minfkrV Hv (p)(s x< y)k j
0 6 v 6 1 p 2 P (s x< y) 2 Dk @Dl VB 2 T(sxy) (Dk @Dl ) kVB k = 1g
"^ = (1=10) minf"^1 "^2 1g.
1( v v0 , $ # v ; v0 ( ( v ; v0 ), p 2 P Jvv0 (p)(x) x 2 @Dk Rm (
Kvv0 (p)(x< t y) Lvv0 (p)(s x< y) (x< t y) 2 @Dk Dl (s x< y) 2 Dk @Dl Rm+n - . 3# Mvv0 (p)(x) # (
# Rm, (( g~v0 (p)(x) g~v (p)(x), , vv0 (p)(x) | (, ( Mvv0 (p)(x) ( (~gv0 (p)(x) g~v (p)(x)) # v ; v0 # , 0 6 vv0 (p)(x) < ). 3 0 M~ vv0 (p)(x< t y) MB vv0 (p)(s x< y) Rm+n, ( (~gv0 (p)(x) 0) (~gv(p)(x) 0) HB v0 (p)(s x< y) HB v (p)(s x< y) (
-), vv0 (p)(x< t y) (= vv0 (p)(x)) vv0 (p)(s x< y) | - (, 0 6 vv0 (p)(s x< y) < ).
A
Jvv0 (p): @Dk ! SO(m R)
Kvv0 (p): @Dk Dl ! SO(m + n R)
Lvv0 (p): Dk @Dl ! SO(m + n R)
(, - x, (x< t y) (s x< y) (- ( $ m (
( m + n ( #, - Mvv0 (p)(x), M~ vv0 (p)(x< t y) MB vv0 (p)(s x< y) vv0 (p)(x), vv0 (p)(x< t y) vv0 (p)(s x< y) , | g~v0 (p)(x) g~v (p)(x),
180
. . | (~gv0 (p)(x) 0) (~gv (p)(x) 0),
| # HB v0 (p)(s x< y) HB v (p)(s x< y).
O Mvv0 (p)(x) , Jvv0 (p)(x)
# < , M~ vv0 (p)(x< t y) MB vv0 (p)(s x< y). >, (
Jvv0 (p): @Dk ! GL(m R)
Kvv0 (p): @Dk Dl ! GL(m + n R)
Lvv0 (p): Dk @Dl ! GL(m + n R)
'
k J (p)(x)
Jvv0 (p)(x) = kkg~g~v (p)(x)
x 2 @Dk <
v0 (p)(x)k vv0
k Kvv (p)(x< t y)
Kvv0 (p)(x< t y) = kkg~g~v (p)(x)
(x< t y) 2 @Dk Dl <
0
v0 (p)(x)k
B v (p)(s x< y)k
k
l
Lvv0 (p)(s x< y) = kkHH
B v0 (p)(s x< y)k Lvv0 (p)(s x< y) (s x< y) 2 D @D :
=, ( Jvv0 (p) Kvv0 (p) Lvv0 (p) ((-( C 1- # x (x< t y) (s x< y) .
H , # v ; v0 , Pvv0 (p)(x< t y) p 2 P (x< t y) 2 @Dk Dl Rm+n. 3# M^ vv0 (p)(x< t y) # ( # Rm+n,
(( Kvv0 (p)(x< t y)G~ v0 (p)(x< t y) G~ v (p)(x< t y), , vv0 (p)(x< t y) | (, 0 6 vv0 (p)(x< t y) < ). = Pvv0 (p): @Dk Dl ! SO(m + n R)
, (x< t y) @Dk Dl (- Rm+n, # M^ vv0 (p)(x< t y) vv0 (p)(x< t y), ( Kvv0 (p)(x< t y)G~ v0 (p)(x< t y) G~ v (p)(x< t y) ( M^ vv0 (p)(x< t y) , Pvv0 (p)(x< t y) # ). = Pvv0 (p): @Dk Dl ! GL(m + n R)
'
kG~v (p)(x< t y)k
Pvv0 (p)(x< t y) =
kKvv0 (p)(x< t y)G~ v0 (p)(x< t y)k Pvv0 (p)(x< t y)
(x< t y) 2 @Dk Dl . =, 0 ((( C 1 -
# (x< t y).
181
E #, Pvv0 (p)(x< t y)Kvv0 (p)(x< t y)G~ v0 (p)(x< t y) = G~ v (p)(x< t y)
Lvv0 (p)(s x< y)HB v0 (p)(s x< y) = HB v (p)(s x< y)
(1)
(10 )
vv (p)(xty) m+n
Rm?+n ;;;;;;;;;!
R?
0
?y
!vv (p)(xty) m+n
Rm?+n ;;;;;;;;;;
! R?
0
?
y ?
y
IdRm
vv (p)(x)
Rm ;;;;!
Rm ;;;;;;;!
Rm
"vv (p)(sxy) m+n
Rm?+n ;;;;;;;;;;
! R?
0
?
y <
(2)
Rm
0
?
y
Rm
IdRm
;;;;!
?
y :
(20 )
Rm
1# -
3]1.
1. m > k > 1, Gmk | k-
!" Rm S m;1 | # $
% ! Rm.
: Q ! Gmk , $ Q | !"
&$.
'$ ( }: Q ! S m;1 , ! % $ q 2 Q
! }(q) !
(q). & C 1 -$!, ( C 1 -$! }.
3 1, ( Q 0 I P @Dk (v p< x) #, (- g~v (p)(x) #- #
gv (p)] (Tx (@Dk )). 3# (g0 (p) g~0(p)) ( f(p) Dk Rm , #, ( -. 3 }: I P @Dk ! S m;1 C 1 - x 2 @Dk , }(v p< x) # (v p< x).
= R: I P @Dk Dl ! Gm+n k+l ( R(v p< x< t y) #, (- G~ v (p)(x< t y) #- # Gv (p)] (T(xty) (@Dk Dl )). . 1
. &,'
10 .
182
. . Gv (p)
@Dk Dl ;;;;!
Rm?+n
??
?y y
G~ v (p)
@Dk Dl ;;;;!
Rm+n n f0g Rm+n
?
?
y
?
?
y
g~v (p)
gv (p)
@Dk ;;;;!
Rm n f0g Rm
@Dk ;;;;!
Rm
# (v p< x) ((( R(v p< x< t y) Rm+n =
= Rm Rn ! Rm, <(v p< x< t y) def= (}(v p< x) 0) 2 Rm+n
( R(v p< x< t y). 3 (
< : I P @Dk Dl ! Sm+n;1 C 1 - (x< t y) 2 @Dk Dl .
1 . m > k > 1, n > l > 1, Gkm++l!nkk+l | $
Gm+n k+l , ( !
(k + l)- !", ! !
Rm+n = Rmk+l!Rkn ! Rm $ k- !
. S: Q ! Gm+n k+l 0
.
'$ ( = : Q ! S m+n;1 , ! % =(q) S(q) $ q 2 Q ! =
Q ;!
S m+n;1 Rm+n ;! Rm
$
Q 0 2 Rm. )
S C 1 -$!, = C 1 -$!
.
!0
. 1# 10 . 3# Vmk++ln!kk+1l+1
| ( T'( Vm+n k+l+1 #$ $ (k +l +1)-
Rm+n, ( $ (R1 : : : Rk+l Rk+l+1 )
$ p(R1 ) : : : p(Rk+l), p: Rm+n ! Rm | (,
k $ p(Rk+l+1 ) = 0. = !0
p~: Vmk++ln!kk+1l+1
! Gkm++l!nkk+l
-, (- (R1 : : : Rk+l Rk+l+1)
# hR1 : : : Rk+l i, (- R1 : : : Rk+l. .
!0
k
Vmk++ln!kk+1l+1
, p~, ((( () Gkm++l!
n k+l
!0
Vk+l k+l S n;l;1 . 1( ( S (Vmk++ln!kk+1l+1
)
!0
S (Vmk++ln!kk+1l+1
)
??
yp~
0
#
!0
;;;;!
Vmk++ln!kk+1l+1
?
183
!
?
yp~
:
#
k
Q
;;;;!
Gkm++l!
n k+l
. S (, S (Vmk++ln!kk+1l!+10)
#, 0 !0
s: Q ! S (Vmk++ln!kk+1l+1
):
A
!0
p^: Vmk++ln!kk+1l+1
! Sm+n;1
-, (-- (R1 : : : Rk+l Rk+l+1)
Rk+l+1 . = = : Q ! S m+n;1 = = p^ S! s. . =(q), q 2 Q, # S(q), p(=(q)) = 0. 2
= k
S: I P Dk @Dl ! Gkm++l!
n k+l (( (v p< s x< y) # S(v p< s x< y), (- HB v (p)(s x< y) #- # Hv (p)] (T(sxy) (Dk @Dl )). = S (, (H0(p) HB 0(p)) (
F^ (p) Dk Dl Rm+n. 30 #
10. = : I P Dk @Dl ! Sm+n;1 C 1 - (s x< y) 2 Dk @Dl , =(v p< s x< y) S(v p< s x< y) =
I P Dk @Dl ;!
S m+n;1 Rm+n ;! Rm
I P Dk @Dl 0 2 Rm.
> 0 , ( v v0 , $ jv ; v0 j 6 , $
p 2 P (x< t y) 2 @Dk Dl V 2 T(xty) (@Dk Dl ), kV k = 1, - (.
G g~v0 (p)(x) g~v (p)(x) #4 (0 # ( Jvv0 (p)(x) (3)
Kvv0 (p)(x< t y)).
G Kvv0 (p)(x< t y)G~ v0 (p)(x< t y) G~ v (p)(x< t y)
#4 ( # ( - (4)
( Pvv0 (p)(x< t y)).
184
. . kKvv (p)(x< t y)G~v (p)(x< t y) ; G~v (p)(x< t y)k < 20" :
kG~ v(p)(x< t y) ; Kvv (p)(x< t y)G~v (p)(x< t y)k < " min 1 kKuu (p0 )(x0< t0 y0 )G~ u (p0 )(x0< t0 y0 )k < 20
kKuu (p0)(x0< t0 y0)k 0 6 u u0 6 1 p0 2 P (x0 < t0 y0 ) 2 @Dk Dl :
k(~gv (p)(x) 0) ; (~gv (p)(x) 0)k <
" minfk(~g (p0 )(x0) 0)k j 0 6 u 6 1 p0 2 P x0 2 @Dk g:
< 20
u
krV Gv (p)(x< t y) ; rV Gv (p)(x< t y)k < 10" :
kGv(p)(x< t y) ; Gv (p)(x< t y)k <
< ("=100)(1=(maxfkrV <(v0 p0< x0< t0 y0 )k j 0 6 v0 6 1 p0 2 P
(x0< t0 y0 ) 2 @Dk Dl V 0 2 T(x t y ) (@Dk Dl ) kV 0 k = 1 ))
0
0
0
0
(5)
0
0
0
0
(6)
0
0
(7)
(8)
0
0
(9)
( , 0 ().
=, , Gv (p) F (p).
1, ^ > 0 , ( v v0, $ jv ; v0 j 6 ^, $
p 2 P , (s x< y) 2 Dk @Dl VB 2 T(sxy) (Dk @Dl ), kVB k = 1, (#
- (.
G HB v0 (p)(s x< y) HB v (p)(s x< y) #4
( # ( ( (30 )
Lvv0 (p)(s x< y)).
kHB v(p)(s x< y) ; HB v0 (p)(s x< y)k < 10"^ minf1 kHB u(p0)(s0 x0< y0)k j
0 6 u 6 1 p0 2 P (s0 x0< y0 ) 2 Dk @Dl g:
(40 )
krV Hv(p)(s x< y) ; rV Hv0 (p)(s x< y)k < 10"^ :
(50 )
kHv(p)(s x< y) ; Hv0 (p)(s x< y)k <
< (^"=100)(1=(maxfkrV =(v0 p0< s0 x0< y0 )k j 0 6 v0 6 1 p0 2 P
(s0 x0< y0) 2 Dk @Dl VB 0 2 T(s x y ) (Dk @Dl ) kVB 0 k = 1g))
(60 )
( , ().
. ^ , Hv (p) F^ (p).
, (5), (6) (7) , ( v 6 kG~ v(p)(x< t y) ; G~ 0(p)(x< t y)k < 10" (10)
0
0
0
0
0
0
0
kKv0(p)(x< t y) ; IdRm n k k(~g0(p)(x) 0)k =
" k(~g (p)(x) 0)k
= k(~gv (p)(x) 0) ; (~g0(p)(x) 0)k < 20
0
"
kKv0(p)(x< t y) ; IdRm n k < 20 kPv0(p)(x< t y) ; IdRm n k kKv0(p)(x< t y)G~ 0(p)(x< t y)k =
= kG~ v (p)(x< t y) ; Kv0(p)(x< t y)G~ 0(p)(x< t y)k <
" kKv0(p)(x< t y)G~ 0(p)(x< t y)k < 20
kKv0(p)(x< t y)k
1
kPv0(p)(x< t y) ; IdRm n k < 20" kKv0(p)(x<
t y)k kPv0(p)(x< t y)Kv0(p)(x< t y) ; Kv0(p)(x< t y)k < 20" :
185
+
+
+
+
30 (11)
kPv0(p)(x< t y)Kv0(p)(x< t y) ; Id m+n k < 10" :
, (40 ) , ( v 6 ^
kLv0(p)(s x< y) ; Id m+n k kHB 0(p)(s x< y)k =
"^ kHB (p)(s x< y)k
= kHB v (p)(s x< y) ; HB 0(p)(s x< y)k < 10
0
kLv0(p)(s x< y) ; Id m+n k < 10"^ :
(70 )
-( ( 2].
2. ( s0, s0 < 1, ! % $ p 2 P s 2 s0 1],
(x< t y) 2 @Dk Dl F (p)(s x< t y) ; F(p)(1 x< t y) 6 4 krsF(p)(1 x< t y)k:
s0 ; 1
3
1( ( # # 0
.
. 1# 2. , ( F (p)(s x< t y) ; F(p)(1 x< t y)
rsF(p)(1 x< t y) = slim
!1
s;1
0 P , # s0 ,
s0 < 1, ( s 2 s0 1]
F (p)(s x< t y) ; F(p)(1 x< t y) ; rsF (p)(1 x< t y) 6 1 krsF (p)(1 x< t y)k:
s;1
3
3 #
F (p)(s x< t y) ; F(p)(1 x< t y) 6 4 krsF(p)(1 x< t y)k:
s;1
3
R
R
R
186
. . 1( s0 6 s 6 1
F(p)(s x< t y) ; F(p)(1 x< t y) F(p)(s x< t y) ; F (p)(1 x< t y) 6
:
s0 ; 1
s;1
G $ $ . 2
#
4 ( -
0
.
2 . ( t0, t0 < 1, ! % $ p 2 P t 2 t0 1],
(s x< y) 2 Dk @Dl
^
^
x< t y) ; F(p)(s
x< 1 y) 6 4 kr F(p)(s
F(p)(s
x< 1 y)k:
3 t ^
t0 ; 1
=(# 2, s0 , 1=2 < s0 < 1, , ( $
v 6 , p 2 P, s 2 s0 1], (x< t y) 2 @Dk Dl , V 2 T(xty) (@Dk Dl ), kV k = 1,
(# - (:
F(p)(s x< t y) ; F (p)(1 x< t y) < 2krsF(p)(1 x< t y)k<
(12)
s0 ; 1
krsF (p)(s x< t y) ; rsF (p)(1 x< t y)k < 10" <
(13)
(14)
krV F (p)(s x< t y) ; rV F(p)(1 x< t y)k < 10" <
kF(p)(s x< t y) ; F (p)(1 x< t y)k <
< ("=10)(1=(maxfkrV (Pv 0(p0 )(x0 < t0 y0 )Kv 0(p0 )(x0< t0 y0 ))k j 0 6 v0 6 1
p0 2 P (x0< t0 y0 ) 2 @Dk Dl V 0 2 T(x t y ) (@Dk Dl ) kV 0k = 1g)) (15)
( (15) , 0 ().
=, # s0 . 3# s1 = s0 + (1=3)(1 ; s0 ).
C, #( 20, # t0, 1=2 < t0 < 1, , ( $ v 6 ^, p 2 P , t 2 t0 1], (s x< y) 2 Dk @Dl , VB 2 T(sxy)(Dk @Dl ),
kVB k =1, ( F(p)(s
^
^
x< t y) ; F(p)(s
x< 1 y) < 2kr F(p)(s
x< 1 y)k<
(80 )
t^
t0 ; 1
"^ <
^
^
krtF(p)(s
x< t y) ; rtF(p)(s
x< 1 y)k < 10
(90 )
"^ <
^
^
krV F(p)(s
x< t y) ; rV F(p)(s
x< 1 y)k < 10
(100)
^
^
kF(p)(s
x< t y) ; F(p)(s
x< 1 y)k <
< (^"=10)(1=(maxfkrV Lv 0 (p0)(s0 x0< y0 )k j 0 6 v0 6 1 p0 2 P
(s0 x0< y0 ) 2 Dk @Dl VB 0 2 T(s x y ) (Dk @Dl ) kVB 0 k = 1g))
(110)
( , ().
0
0
0
0
0
0
0
0
0
0
0
0
187
=, t0 . 3# t1 = t0 + (1=3)(1 ; t0 ).
1
# C 1 - ' (s), (t), (s) (t) 0 1]
, (# - (:
(s) = 0 0 6 s 6 s1 <
(16)
(t) = 0 0 6 t 6 t1<
(120)
(1) = 1 0 (1) = 0<
(17)
(1) = 1 0 (1) = 0<
(130)
j(s)j 6 1 j 0(s)j < 1 ;2 s0 <
(18)
j(t)j 6 1 j0(t)j < 1 ;2 t0 <
(140)
(s) = 0 0 6 s 6 s0 <
(19)
(t) = 0 0 6 t 6 t0<
(150)
(1) = 0 (1) = 0<
(20)
(1) = 0(1) = 0<
(160)
j0(s)j > 10j 0(s)j s1 6 s 6 1<
(21)
0
0
j (t)j > 10j (t)j t1 6 t 6 1<
(170)
j(s)j 6 20<
(22)
j(t)j 6 20:
(180)
(
M(v) = maxfkGv (p)(x< t y) ; G0(p)(x< t y)k j p 2 P (x< t y) 2 @Dk Dl g
N(v) = maxfkHv (p)(s x< y) ; H0(p)(s x< y) j p 2 P (s x< y) 2 Dk @Dl g:
, - Fv (p) = (Fv (p) fv (p)) F^v (p) =
= (F^v (p) f^v (p)), p 2 P , ( v 6 v 6 ^ -
:
Fv (p)(s x< t y) = Id m+n +(s)(Pv0 (p)(x< t y)Kv0(p)(x< t y) ; Id m+n )] (F (p)(s x< t y) ; F(p)(1 x< t y)) + (s)(Gv (p)(x< t y) ; G0(p)(x< t y)) +
+ (s)M(v)<(v p< x< t y) + F(p)(1 x< t y)<
(23)
fv (p)(s x) = Id m +(s)(Jv0 (p)(x) ; Id m)](f(p)(s x) ; f(p)(1 x)) +
+ (s)(gv (p)(x) ; g0(p)(x)) + (s)M(v)}(v p< x) + f(p)(1 x)<
(24)
F^v (p)(s x< t y) = Id m+n +(t)(Lv0 (p)(s x< y) ; Id m+n )] ^
(F^(p)(s x< t y) ; F(p)(s
x< 1 y)) + (t)(Hv (p)(s x< y) ; H0(p)(s x< y)) +
^
+ (t)N(v)=(v p< s x< y) + F(p)(s
x< 1 y)<
(190)
f^v (p)(s x) = fv (p)(s x):
(200)
R
R
R
R
R
R
188
. . V, Fv (p)
Dk Dl ;;;;!
Rm+n
?
?
?
?
y
y (25)
f v ( p)
;;;;!
Rm
F^v (p)
Dk Dl ;;;;!
Rm?+n
?
Dk
?
y
?
y :
^
(210)
f v ( p)
Dk ;;;;!
Rm
4 ( - :
rsFv(p)(s x< t y) = 0(s)Pv0(p)(x< t y)Kv0(p)(x< t y) ; Id m+n] (F(p)(s x< t y) ; F(p)(1 x< t y)) +
+ Id m+n +(s)(Pv0 (p)(x< t y)Kv0(p)(x< t y) ; Id m+n )] rsF(p)(s x< t y) + 0(s)(Gv (p)(x< t y) ; G0(p)(x< t y)) +
+ 0 (s)M(v)<(v p< x< t y)<
rtF^v (p)(s x< t y) = 0(t)Lv0(p)(s x< y) ; Id m+n ] ^
(F^(p)(s x< t y) ; F(p)(s
x< 1 y)) +
+ Id m+n +(t)(Lv0 (p)(s x< y) ; Id m+n )]rtF^ (p)(s x< t y) +
+ 0 (t)(Hv (p)(s x< y) ; H0(p)(s x< y)) + 0(t)N(v)=(v p< s x< y):
1( V 2 T(xty)(@Dk Dl )
rV Fv(p)(s x< t y) = (s)rV (Pv0(p)(x< t y)Kv0(p)(x< t y)) (F(p)(s x< t y) ; F (p)(1 x< t y)) +
+ Id m+n +(s)(Pv0 (p)(x< t y)Kv0(p)(x< t y) ; Id m+n )] (rV F (p)(s x< t y) ; rV F(p)(1 x< t y)) +
+ (s)(rV Gv (p)(x< t y) ; rV G0(p)(x< t y)) +
+ (s)M(v)rV <(v p< x< t y) + rV F (p)(1 x< t y):
1( VB 2 T(sxy) (Dk @Dl )
rV F^v (p)(s x< t y) = (t)rV Lv0(p)(s x< y) ^
(F^(p)(s x< t y) ; F(p)(s
x< 1 y)) +
+ Id m+n +(t)(Lv0(p)(s x< y) ; Id m+n )] ^
(rV F(p)(s
x< t y) ; rV F^ (p)(s x< 1 y)) +
+ (t)(rV Hv (p)(s x< y) ; rV H0(p)(s x< y)) +
^
+ (t)N(v)rV =(v p< s x< y) + rV F(p)(s
x< 1 y):
R
R
R
R
R
R
R
(26)
(220)
R
R
(27)
R
(230)
189
1, Fv (p) - :
Fv (p)(s x< t y) ((( C 1 - (s x< t y)<
(28)
F0(p) = F(p)<
(29)
Fv (p)(1 x< t y) = Gv (p)(x< t y)<
(30)
rsFv (p)(1 x< t y) = G~v(p)(x< t y)<
(31)
Fv (p) ((( :
(32)
-$ , (28){(32) # (i) (-
, : E ! B | ). ,
(25) , fv (p) :
fv (p)(s x) ((( C 1 - (s x)<
(280)
f0 (p) = f(p)<
(290)
fv (p)(1 x) = gv (p)(x)<
(300)
rsfv (p)(1 x) = g~v(p)(x):
(310)
1] , fv (p) ((( .
(320)
(25), (28), (280 ), (32) (320 ) - # ( Fv : P ! E , 0 6 v 6 , ( ((( F (29), (290), (30), (300 ), (31) (310) -, Fv
Gv .
3
# (28){(32).
(28) , (, # Fv (p) (. ' (23)), ((-( C 1 -.
(29) (( #- (23).
(30) # (23), #( ( (17), (20), ' (t) (t), , G0(p)(x< t y) = F (p)(1 x< t y).
(31) (26) # (17), (1), (20) rsF (p)(1 x< t y) = G~0(p)(x< t y).
1( # , Fv (p) ((( , #, rV Fv (p)(s x< t y) 6= 0, V 2 T(sxty)(Dk Dl ). V
# V = Vs + V(xty) , V(xty) # ( V T(xty)(@Dk Dl ), Vs | ( V (-, # T(xty) (@Dk Dl ) T(sxty)(Dk Dl ). .
rV Fv (p)(s x< t y) = srsFv (p)(s x< t y) + (xty)rW Fv(p)(s x< t y) (33)
W = V(xty) =kV(xty)k, s (xty) | $( (. 3 0 s 6= 0, (xty) 6= 0.
3. ( ! B0 Rm+n, kB0k < " (" $ % $! ), !" %
rW Fv (p)(s x< t y) = rW Gv (p)(x< t y) + B0:
190
. . . 1# 3. , (18) (15) k(s)rW (Pv0(p)(x< t y)Kv0(p)(x< t y)) (F(p)(s x< t y) ; F(p)(1 x< t y))k < 10" :
,#( (18), (11) (14), kId m+n +(s)(Pv0 (p)(x< t y)Kv0(p)(x< t y) ; Id m+n)] (rW F (p)(s x< t y) ; rW F(p)(1 x< t y))k < 2"10 :
, (18) (8) , k(s)(rW Gv (p)(x< t y) ; rW G0(p)(x< t y))k < 10" :
1, #( (22) (9), k(s)M(v)rW <(v p< x< t y)k < 102" , , , F(p)(1 x< t y) = G0 (p)(x< t y) (8) ,
krW F(p)(1 x< t y) ; rW Gv (p)(x< t y)k < 10" :
G 3 (27) 0$ ( . 2
4. ( ! 0B, U U 0 Rm+n !
0
0
X, X , !
% kB k < ", X > 10X , kU k = kU k = 1, ! U !
R(v p< x< t y) rsFv (p)(s x< t y) = G~ v(p)(x< t y) + B + XU + X0U 0:
. 1# 4. , (26) , rsFv (p)(s x< t y) = 0(s)Pv0(p)(x< t y)Kv0(p)(x< t y) ; Id m+n] (F(p)(s x< t y) ; F(p)(1 x< t y)) + rsF(p)(1 x< t y) +
+ (s)Pv0 (p)(x< t y)Kv0(p)(x< t y) ; Id m+n ]rsF(p)(1 x< t y) +
+ Id m+n +(s)(Pv0 (p)(x< t y)Kv0(p)(x< t y) ; Id m+n )] (rsF(p)(s x< t y) ; rsF(p)(1 x< t y)) +
+ 0 (s)M~ (v p< x< t y)U 0 + 0 (s)M(v)U
~ p< x< t y) = kGv (p)(x< t y) ; G0 (p)(x< t y)k, U 0 = (Gv (p)(x< t y) ;
M(v
~ p< x< t y), U = <(v p< x< t y). V, 0 6 M(v
~ p< x< t y) 6
; G0(p)(x< t y))=M(v
6 M(v).
, (18), (12), (1), (10) rsF(p)(1 x< t y) = G~ 0(p)(x< t y) ,
k 0(s)Pv0(p)(x< t y)Kv0(p)(x< t y) ; Id m+n] (F(p)(s x< t y) ; F(p)(1 x< t y))k < 104" R
R
R
R
R
R
R
191
(10) , krsF (p)(1 x< t y) ; G~ v(p)(x< t y)k < 10" :
,#( (18), (1), (10) rs F(p)(1 x< t y) = G~ 0(p)(x< t y), k(s)Pv0(p)(x< t y)Kv0(p)(x< t y) ; Id m+n]rsF (p)(1 x< t y)k < 10" R
(18), (11) (13) kId m+n +(s)(Pv0 (p)(x< t y)Kv0(p)(x< t y) ; Id m+n)] (rsF(p)(s x< t y) ; rsF(p)(1 x< t y))k < 102" :
R
R
>, #( (21) 4 (, ( 4. 2
- (. 1]).
5. A, B, U , U 0, A0 B0 | ! R0m+n,
0
X, X | !, $(
$(
: kB k kB k <
< (1=10)0(A sA0 ) $ $"
s, kB 0 k < (1=10)kA0k, kU k = 1,
kU 0k = 1, X0 0> 10X0 0 0! U A A0. '$ ! A + B +
+ XU + X U A + B " .
.#, #( , #, Fv (p) ((( . H (33), #, rsFv (p)(s x< t y) rW Fv(p)(s x< t y) . 3, , 5, # 3 4
A = G~ v (p)(x< t y) A0 = rW Gv (p)(x< t y):
3 0 ( 5 (-(, #(, #( ".
3 4 ( #, < 1, #(
F (p) F (p). 5 -- - ( v 6 2. 3(
, -- - ( $ v 2 0 1]. 5
- # (i).
3
# .
1, F^v (p)(s x< t y) ((( C 1 - (s x< t y)<
(240)
^
F^0(p) = F(p)<
(250)
F^v (p)(s x< 1 y) = Hv (p)(s x< y)<
(260)
(270)
rtF^v (p)(s x< 1 y) = HB v (p)(s x< y)<
F^v (p) ((( .
(280)
192
. . , , (, # ' (190) ((-(
C -, (240).
(250) (( .
, (190), #( (130) (160), , H0(p)(s x< y) =
^
= F(p)(s x< 1 y), # (260).
^
(270) (220 ), (130), (10), (160) rtF(p)(s
x< 1 y) =
B
= H0(p)(s x< y).
(280) # , rV F^v (p)(s x< t y) 6= 0 ( $
BV 2 T(sxty)(Dk Dl ). /4 VB VB = VBt + VB(sxy) , VB(sxy) | ( VB T(sxy)(Dk @Dl ), VBt | ( VB (-,
#- T(sxy) (Dk @Dl ) T(sxty) (Dk Dl ). .
rV F^v(p)(s x< t y) = Bt rtF^v(p)(s x< t y) + B(sxy)rW F^v (p)(s x< t y) (290)
WB = VB(sxy)=kVB(sxy) k, Bt B(sxy) | $( (, $ -.
3 . ( ! BB0 Rm+n, kBB0k < "^, !"
%
rW F^v (p)(s x< t y) = rW Hv(p)(s x< y) + BB0 :
. 1# 30. , (140) (110) , "^ :
^
k(t)rW Lv0(p)(s x< y)(F^(p)(s x< t y) ; F(p)(s
x< 1 y))k < 10
=
kId m+n +(t)(Lv0(p)(s x< y) ; Id m+n)] 2^"
^
^
(rW F(p)(s
x< t y) ; rW F(p)(s
x< 1 y))k < 10
(140 ), (70 ) (100).
,#( (140) (50 ), k(t)(rW Hv(p)(s x< y) ; rW H0(p)(s x< y))k < 10"^ :
, (180) (60) k(t)N(v)rW =(v p< s x< y)k < 102^" , , (50 ) F^ (p)(s x< 1 y) = H0(p)(s x< y) , krW F^ (p)(s x< 1 y) ; rW Hv(p)(s x< y)k < 10"^ :
30 (230 ) ( 30 . 2
4 . ( ! BB , UB , UB 0 Rm+n, !
BX, XB 0, !
% kBB k < "^, XB > 10XB 0, kUB k = kUB 0k = 1, ! UB !
S(v p< s x< y) rtF^v(p)(s x< t y) = HB v (p)(s x< y) + BB + XB UB + XB 0UB 0:
1
0
R
R
0
193
. 1# 40. , (220) , rtF^v(p)(s x< t y) = 0(t)Lv0(p)(s x< y) ; Id m+n] ^
^
^
(F(p)(s
x< t y) ; F(p)(s
x< 1 y)) + rtF(p)(s
x< 1 y) +
+ (t)Lv0(p)(s x< y) ; Id m+n ]rtF^ (p)(s x< 1 y) +
+ Id m+n +(t)(Lv0 (p)(s x< y) ; Id m+n )] ^
^
(rtF(p)(s
x< t y) ; rtF(p)(s
x< 1 y)) +
0
0
0
~
B
B
+ (t)N(v p< s x< y)U + (t)N(v)U
R
R
R
R
~ p< s x< y) = kHv (p)(s x< y) ; H0(p)(s x< y)k
N(v
~ p< s x< y)
UB 0 = (Hv (p)(s x< y) ; H0(p)(s x< y))=N(v
UB = =(v p< s x< y):
~ p< s x< y) 6 N(v).
=, 0 6 N(v
^
,#( (140 ), (80 ), (10 ), (40) rt F(p)(s
x< 1 y) =
= HB 0(p)(s x< y), , 4^" ^
k0(t)Lv0(p)(s x< y) ; Id m+n ](F(p)(s
x< t y) ; F^ (p)(s x< 1 y))k < 10
(40) "^ :
^
krtF(p)(s
x< 1 y) ; HB v (p)(s x< y)k < 10
, (140), (10 ), (40 ) , "^ ^
k(t)Lv0(p)(s x< y) ; Id m+n]rtF(p)(s
x< 1 y)k < 10
(140), (70) (90 ) kId m+n +(t)(Lv0(p)(s x< y) ; Id m+n)] 2^" :
^
^
(rtF(p)(s
x< t y) ; rtF(p)(s
x< 1 y))k < 10
G (170) $ 4 . 2
1, F^v (p) ((( . 1( 0 #, rtF^v (p)(s x< t y) rW F^v (p)(s x< t y) (. (290)), 5, # 30 40 -
(
A = HB v (p)(s x< y) A0 = rW Hv (p)(s x< y)
B U 0 = UB 0 X = X
B X0 = XB 0:
B B 0 = BB 0 U = U
B = B
,#( "^, #(, ( 5 (-(.
R
R
R
R
194
. . >, #( (210), (240 ){(280 ), , ^ : E ! C ((( .
(210), (240), (280) - # ( F^v : P ! E , 0 6 v 6 ^, ( F^ ( (250)), (260) (270)
-, F^v Hv .
O ^ < 1, (, #( F^^(p) F^(p), -- - ( v 6 2^. 3( ,
-- - F^v (p) ( $ v 2 0 1].
,, (i) #- .
3 (ii) - (.
(ii)0. ~ : C ! D (m > k, n > l) .
1# 0 ( #- ( #
# (i), ( ( ( : E ! B. O , 0 C D # # (, ($, $ 0 B E . 30, (
#, 4# 0 .
3# ( Gv : P ! D, 0 6 v 6 1, P | #
0, Gv (p) - 4 Gv (p) = ((Gv (p) gv(p)) (G~ v(p) g~v (p)) GB v(p)):
1, # v = 0 Gv B
H : P ! C H(p) = ((H(p) f(p)) H(p))
p 2 P
. . G0 = ~ H. G, ~ ((( , ,
(
Hv : P ! C 0 6 v 6 1 Hv (p) = ((Hv(p) fv (p)) HB v(p)) p 2 P
-( Gv , . . ~ Hv = Gv , v = 0
H0 = H:
5 Hv (p) fv (p) Hv # ( Fv (p) fv (p) # (i).
3
- HB v (p).
7 -- $ $ , (- 0 Gv (p) Gv ( p P :
(qGv (p)) (T(Rn))
\
(Gv(p)) (T(Rm+n))
? l Id? l
k
k
@D @D
@D @D
T(@Dk
@Dl )
- q (T(Rn)) q- T(Rn)
\
(Gv (p))
T(Rm+n)
?+n q- ?n
Gv (p)
m
R
R
!
!
195
q: Rm+n = Rm Rn ! Rn ( # (, ( # $ ((-( (, ( ( ( (Gv (p))! '' (
Gv (p). 7# ( T(@Dk @Dl ) ! @Dk @Dl (qGv (p)) (T(Rn)) ! @Dk @Dl
@Dk @Dl ( l ; 1), 0 # 0$ , ((-( (qGv (p)) (T(Rn)) ! @Dk @Dl :
, Pv (p)(@Dk @Dl ) ! @Dk @Dl
# -. H , 0 . > , # #
.
C # Pv (p)(Dk @Dl ) ! Dk @Dl
# (qHv (p)) (T(Rn)) ! Dk @Dl
- T(Dk @Dl ) ! Dk @Dl (qHv (p)) (T(Rn)) ! Dk @Dl :
. Hv (p) Gv (p) ~ , (
Pv (p)(Dk @Dl ) ! Dk @Dl
@Dk @Dl Pv (p)(@Dk @Dl ) ! @Dk @Dl :
1, ( Gv (p) Hv (p) 0 6 v 6 1, v, # (
P(p)(I @Dk @Dl ) ! I @Dk @Dl
P(p)(I Dk @Dl ) ! I Dk @Dl ( $ fvg @Dk @Dl fvg Dk @Dl - Pv (p)(@Dk @Dl ) ! @Dk @Dl Pv (p)(Dk @Dl ) ! Dk @Dl
.
196
. . >, P(p)(Dk @Dl 0@Dk@Dl I @Dk @Dl ) ! Dk @Dl 0@Dk@Dl I @Dk @Dl (( (
P0 (p)(Dk @Dl ) ! Dk @Dl P(p)(I @Dk @Dl ) ! I @Dk @Dl
P0 (p)(@Dk @Dl ) ! @Dk @Dl :
=, ((( (
P(p)(I Dk @Dl ) ! I Dk @Dl
Dk @Dl f0g@Dk@Dl I @Dk @Dl .
3 - v (p) m+n
@Dk @Dl G;!
R n f0g Rm+n ;! Rm
@Dk @Dl 0 2 Rm, 0 #, GB v (p)(Z),
Z 2 @Dk @Dl , (
(qGv (p)) (T(Rn)) ! @Dk @Dl
Z. 1, GB v (p)(Z) #
Z Gv (p) (=2 (Gv (p)) TZ (@Dk @Dl )). 30, -- Pv (p)(@Dk @Dl ) ! @Dk @Dl GB v (p) # 0 (. =2B (( GB v (p), 0 6 v 6 1, v, G(p)
(
P(p)(I @Dk @Dl ) ! I @Dk @Dl :
C (, - - HB v (p), 0 6 v 6 1,
# (
Pv (p)(Dk @Dl ) ! Dk @Dl :
B (
=2 ( HB v (p) v, H(p)
P(p)(I Dk @Dl ) ! I Dk @Dl :
B GB 0 (p), HB 0(p) G 0 (p) G(p)
B (
HB 0(p) G(p)
P(p)(Dk @Dl 0@Dk@Dl I @Dk @Dl ) ! Dk @Dl 0@Dk@Dl I @Dk @Dl :
. I Dk @Dl '( Dk @Dl f0g@Dk@Dl I @Dk @Dl ,
B # ( H(p),
B HB 0(p) G 0 (p) G(p)
( - - HB v (p). > 0 4( # . 2
!
197
1] S. Smale. The classication of immersions of spheres in Euclidean spaces // Ann. of
Math. | 1959. | Vol. 69. | P. 327{344.
2] S. Smale. Regular curves on Riemannian manifolds // Trans. Amer. Math. Soc. |
1958. | Vol. 87. | P. 492{512.
3] S. Smale. A classication of immersions of two-sphere // Trans. Amer. Math. Soc. |
1958. | Vol. 90. | P. 281{290.
' ( ) 1997 .
. . . . 517.977
: , ,
, !" , #
.
$ "%&"'
"( '%( "( &**
( ", **( # "' " # * .
&
* " *( ( ' "&
#' & "' #. "& *
( "'% " # % !. %"" "&.
Abstract
R. Hildebrand, Classication of phase portraits of optimal syntheses, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 1, pp. 199{233.
The paper is devoted to the investigation of controllable oscillating systems of
ordinary di1erential equations a2ne in scalar nonsymmetric control in a neighborhood of a singular point of focus or center type. Integrands in value functionals are
quadratic in phase coordinates. We classify such systems in case of general position
by arising optimal syntheses. The existence of optimal synthesis is proved and its
structure is described.
1. 5]. !. H = H (u x ) | &
, ( u 2 U R. )
* ! @H
_
x_ = @H
@ = ; @x :
, 2001, 7, 3 1, . 199{233.
c 2001 !,
"#
$% &
200
. .
x | / & X , | 01
, !(
T X . u !( 2:
u(x ) = argmax H (u x ):
u2U
40
, & H u 2
0
U . 5 u .
6 /, &&/ !, & * && u: H (u x ) = H0(x ) + uH1(x ). 8 ,
H1 0 . !(
! 2 ! . 6 , 2/ 2. *2 2
0 q, 2 0 ! @ d 2q @H (u x ) 6= 0
@ d k @H (u x ) 0 8 k 6 2q ; 1:
@u dt
@u
@u dt
@u
5
:; 10] , 2 1
2 0
2 , !
. 5
, 2 0
/ 2 0 1
2 , !
. < &
14]. >
7].
5
:; , 2 0
u 0 0
/
U . 6 2
2
0, 0 0
U .
4 &&
! &&
/
!(
. 4
&
Z1
J = (F (x) + uG(x)) dt ! min
(1)
0
x(t) = x~:
(2)
x_ = A(x) + B (x)u@ x 2 U R2 u 2 0 1] t!lim
+1
5
2
0 & x~ 2 R2. U x~ . , u(t) | & t, A B & F G 0 C 3. A x~ 2
& : 2 @A
@x (~x) -01
201
2
A(~x) = 0. B U 0
.
* 0
5
& F x~ &, . . & F 1
x~ !. > G
0
x~. B
x(t) / U .
B ! !(/ ! &, Z1
x_ = y I = F (x) dt ! inf 0
y_ = ;x + u u 2 0 1]:
5
2
(0 0), 1
0
2
. .
, 2
0
x y 0, u 0, 0 /
.
1. C & 2
F (x) 1. D
, , 201
. 1 . : a 0 1
(;1 0). < !
1 0. : b
0 , 0
1, (1 0). <
2 0@ , & u = 1 (0 0). :
a b
! ! 2. 6 2, 0
/ a b, 1. 6 2 / 0.
y
a
6
y
u=0
u=0
M
u=1
6
a
x
u=1
b
$. 1
-
6
x
9
b
202
. 2. C & , . . F (x) = 12 x2 . < 3]. F , !( . 6 (0 0) . D a !
1
0. < / (0 0) . D 0
b !
0 1. !(
1
! a, !
0 (. . 1 ).
C
0 2
. 5
2, ! . 0
, / u = 0, 0
!. 5 2, ; , 2
. 5 !
! 2
.
6 2
0
, (1), (2) 2(
0
& A B F G !(/ 1
/
.
5 I. G U , (
!( . H
! !
1 0 2 0, !(
x~. (
,
x~. ! 2 0, ! x~.
5 II. G U , (
!( . H
! !
, 0 1
2, ! x~. 5
2! x~ 2
, ; 0 2
20 x~ ! .
4
!
! 2
.
5 III. (
. (
/ , / & J ;1. < 0, (1) 2
1
+1.
6 2
2
! &!
/ (1), (2). 4 & A B F G, / 0
5
!
x~. 5
& ! & 2(
0
. & 2! 0
2
, , 2
1
0
.
D U x~. U , 0
/ 5
!(/ & 0 2
, ! . 5
, x~ & A B F G.
2. 203
C 0
1
(A B F G) & C 3, 1
/ U 0 x~ 2 R2. ; A B |
, F G | & ! !(
:
i) A(~x) = 0,
ii) 2
A x~ | & , . .
1 tr @A (~x)2 ; det @A (~x) < 0
2
@x
@x
iii) B (~x) 6= 0,
iv) F (~x) = G(~x) = 0,
v) rF (~x) = 0,
tr | . 2 ; 0
1
S . 0
2
(
. :0 ((A B F G) U ), (
1
&
(A B F G), 1
/ U 0 !(/ i){v), U U 0 , (1), (2) U . B, (
U x~, & A B F G A0 B 0 F 0 G0 !.
5 ((A B F G) U ) ((A0 B 0 F 0 G0) U ) ! 0
(1), (2). , 0
U . 5 2, 0 1
x~ & A B F G, !(/ i){v), 1
(1), (2) , !(
!2/
/ !, !(
! U . 6 2
0
1
& (A B F G) 0
S !( (1), (2).
6 ; (1), (2). < S / ! 2
/ 2 . 6 , ;
2 2 ;
, 0 / !(
2 ; .
2 1
& (A B F G)
S , (
2 ;
.
:
11] 2 !(
2 ;
/ : x_ = A(x) + uB (x) x_ 0 = A0 (x0) + uB 0 (x0) u 2 U 204
. 1
/ V , V 0 , ! 2 ;
, (
&&
& D : V ! V 0 , ( !2/ / / u(t).
< , &&
& D A A0 B B 0 , . . A0 (D(x)) = D (A(x)) B 0 (D(x)) = D (B (x)).
.
D 2
&&
D. 6 , A, B , A0, B 0 ! i){iii), 2 x~ 2! x~0 = D(~x). 82 2 , & F G 0 /, . . F 0(D(x)) = F (x),
G0(D(x)) = G(x). 6 2
2( 2
, x~ (0 0), 1
0
(! /(
.
2 &&
& C 4 U x~,
!(/ x~ 0, K. D 2 K D = @x
@x | &&
D. .
x0 = D(x) | &. 5 1
(A B F G) 2
&
0
@x0 B (x)
0 0
A0(x0 (x)) = @x
A
(
x
)
B
(
x
(
x
))
=
@x
@x
F 0(x0(x)) = F (x)
G0 (x0(x)) = G(x):
2 L ! /
/ @i @j . <
2 L : U ! R2 C 3 , !(/ x~: (~x) = 0, @x
@xi
j
& A B F G & A0 B 0 F 0 G0 !( &:
A0 = A B 0 = B F 0 = F + h Ai G0 = G + h B i:
0
, (1), (2) ;
2 L.
*
;/ 2 !(
. C &&! ! Z
k
k
X
X
x_ = f0 (x) + fi (x)ui I (x(t)) = 0(x) + i(x)ui dt ! inf 0
i=1
x 2 Rn
i=1
f0 fi 2 Rn
ui 2 R 0 i 2 R:
: 13], & I 0 & (x), fi , i = 0 : : : k, . ; / fi i .
Z
I (x(t)) = (fi ) = i i = 0 : : : k:
, & 0 Z
Z
k
X
= (f0 ) + (fi )ui dt:
i=1
205
G & , . . = d!, & . 6
, (
!, 1
& 0
.
; 2
&&
0
! &
. G ;&&
&&
& = h dxi, Z
Z
(F 0 + G0u) dt = ((F + Gu) + h A + Bui) dt =
Z
Z
Z
Z
= (F + Gu) dt + h dxi = (F + Gu) dt + :
D / &, & F 0 G0 & F G.
1. 6
B (
, h B i = ;G. 2 G0 = 0, F 0 = F + h Ai, 0
(1) u. D 2(
0 C 2 0 L.
2. * 2 K L ! 2 & T X .
(1
2 0
S , (/ ;
. H
2
2
;
, 8]. < 2 0
dt 0
! &!, (! x.
2 N ! 0
/ /
& (x) C 3 U , R | !
0
/ (
/ R+. ;
2 N r 2 R 1
(A B F G) !( 2:
A0 = A B 0 = B F 0 = F G0 = G@ A0 = A B 0 = B F 0 = rF G0 = rG:
B
;
(x) 2 N ! 2 , 0
! &&
dt 0
(1x) . 5 &&
dx = (A + uB ) dt 0
(F + uG) dt ! , 2 N 2 ;
8]. D
, 0
& F G 0
! r 2 R
.
H, 2 K, L, N, R 2 ;
. 2 G , 01
! K, L, N, R. )0
G 0 2. H
G. 2 K, L,
206
. N, R G1, G2, G3, G4. B
, 2 0
G , ! 20
. <
G !, , 2 0
S , , 20
U R2 R .
. 8 gi 2 Gi gj 2 Gj i j = 1 : : : 4 9 gi0 2 Gi gj0 2 Gj :
gi gj = gj0 gi0
. B !2/ D 2 K, r 2 R, 2 N, 2 L D r = r D, r = r, = D = 0 D 0 = D@
0 T
D = 0 D 0 = @x
@x ( D)@
r = 00 r 00 = r :
.
.
@x
@x | &&
&&
& D. ! ,
L N G. , !2/ D 2 K,
r 2 R, 2 N, 2 L 0 00 2 L, 0 2 N, D = D 0 ,
D = D 0, r = r 00 . 2
.
g2G
g = g1 g2 g3 g4 gi 2 Gi i = 1 : : : 4
D
! 0
S 2(
0
, 0
/ 5
& A B F G x~ ! . 6 ; !(
1
.
. B
1
(A B F G) (A0 B 0 F 0 G0) 2 S , @A (~x) = @A0 (~x) B (~x) = B 0 (~x) @ 2 F (~x) = @ 2 F 0 (~x) @G (~x) = @G0 (~x):
@x
@x
@x2
@x2
@x
@x
! p S :
2F
@A
@
@G
p : (A B F G) 7! @x (~x) B (~x) @x2 (~x) @x (~x) 2 R11:
2 2 0
S 11-
P . 40
P R11 ii), iii). B
2
@2F
@A
@x (~x) a, B (~x) b, @x2 (~x)
f , - @G
@x (~x) g.
.
G S
P
. (A B F G) 2 S , g 2 G g(A B F G) =
= (A0 B 0 F 0 G0). D0 , p(A0 B 0 F 0 G0) = (a0 b0 f 0 g0 ) 0
-
,
,
.
.
207
p(A B F G) = (a b f g). 6 ! (a0 b0 f 0 g0) , g ;
2!(/ K, L,
N, R.
g = D 2 K. 2 M = @x
@x (~x) 2 GL(2 R) &&
D x~. D
, (a b f g) 2! !( 2:
a0 = MaM ;1 b0 = Mb f 0 = (M ;1)T fM ;1 g0 = gM ;1 :
g = 2 L. ) A(~x) = 0, (~x) = 0 @ , @x
T
@
@
@ (~x):
0
0
0
a = a b = b f = f + @x (~x) a + @x (~x) a g0 = g + bT @x
g = (x) 2 N. 5
a0 = (~x)a b0 = (~x)b f 0 = (~x)f g0 = (~x)g:
, , g = r 2 R. 5
a0 = a b0 = b f 0 = rf g0 = rg:
5 G 0
;
/ , 2(
(a0 b0 f 0 g0) 0
(a b f g). R
. 2
.
, (a0 b0 f 0 g0) (a b f g). ;
g 2 G D r ;
K, L, N, R.
5 2
, ;
g ,
@
2 @x
@x (~x), @x (~x), (~x), r. 6
@
@x / . 5 2,
G P 1
G GL(11).
2 ; GE . D 0
P ;&&
& G=GE . 9-
R.
GE 0 . G 2 ; 2 , 2 ! /
. : 0
!, ; . 6 11] 2/
, x_ = a0 (x) + a1 (x)u 2 ;
x .
& a1 adh (a0 )a1 ] = 0, h > 0, ad0 (a0 )a1 = a1 ,
adn+1(a0 )a1 = a0 adn(a0 )a1 ] n > 0. T ; / a0 a1, B A B ]] = 0 2/ .
/
2 R &&
&/. 6 2(
; .
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208
. ;
, C 1. D , / &&
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.
H
G=GE P 2
/ 2 .
1.
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g 2 G=GE
g(q) = q^ = (^a ^b f^ g^) 2 P
1
1
^
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a
B
. 4 .
2 0
/ ;
q^ 2 P , !(/ (3), Q. 40 ;
g 2 G=GE , q = (a b f g) q^ = (^a ^b f^ g^), ptr a
1
1
^
det
a
a^ = ;1 0 b = 0 f^12 = 0 g^ = 0
f^11 = bT fb ; 2gab
(4)
^f22 = det a(bT (a;1 )T fa;1 b ; 2ga;1 b):
(5)
6 f^11 = f^22 = 0 q^ 0
0 0
Q. 6 f^112 + f^222 6= 0
f^112 + f^222 /(
;
R. 2
0
1 0
, 0 q 2 P (
q^ 2 Q, 0( 2
q G=GE . D
2
, q^ . ; (
2
0 Q 0
2 P .
G f^11 = f^22 = 0, /
0
(F + uG) dt
&
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1
& (A B F G) / 2(
0
. 6
1
!(
2
:
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. Q
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.
209
! S 1 , ii)
a^)2 < 4 () ^a 2 (;2 2) a^211 = (tr
= R: 2
11
det a^
D
, ptrdeta a G. < / / A(x) & x~. 2
:
= ptr a :
det a
1
0 S 2, sin = f^11 ; cos = f^22 :
6
!( . 6 1 , 2
/(
& & 0
2 &! G , & F 0
C 2. ff^^1122 = ; tg 0
(4) (5) 0
q x~ & F 0 B @A ;1
det @A
@x ( @x ) B . & A1 B2 ; A2B1 , . . , A B .
) ! 2 G.
6
2! ! 0
P G=GE . 5
(1), (2) 2(
0
; .
3. 6 (1), (2) (
& u(t),
!( & (1). D , 2
, & J 2 2 . 6 ; (
. 6 ; , !( 2 2
inf J . 4 0
, 2
inf J ;
(
! H
l, (F + uG) dt .
l
1. ! T > 0, " x 2 U
%" u : 0 T ] ! 0 1], " " x_ (t) = A(x(t)) + B (x(t))u (t) x(0) = x
(6)
210
. & "
U
x(T ) = x(0) = x
TR
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.
0
.
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!.
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,
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x0 2 U
!
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nlim
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0
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6 2
6] B 1
&! &/ . ; & x~ & -/(
/ A B . D
, U !(
0.
1.
TV
x1 2 U
0
x 2U
T < TV 2
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l
fun(t)g !( 2. 0
! 1 (
, (! x0 x . ;
. .
n 21
l, u . D
, x x~, . . 2
. 6 0
1 ; 0 0
. 6 Tn 0
. .
Tn 2! !(! &
H
! !. H
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l
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lim
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& 1
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+R1
1
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1
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6
2
0
.
2.
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.
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0
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l
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l
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l
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l
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. 62
n 2 N, ZTn
V
(F + un G) dt < ; sup jF (x)j + sup jG(x)j T:
x2U
0
x2U
.
TV | 0
1. C !(
.
D x0 un Tn / x~.
.
T , 0
1 TV, (
x~
x0 . H
TZn +T
ZTn
(F + uG) dt 6 (F + uG) dt + T sup jF (x)j + sup jG(x)j < 0:
0
0
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, /(
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l
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1
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l
R
2 . 2
0
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S / 2(
0
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& (A B F G), p(A B F G) / 0 0
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, &
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0
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R
1 , / (
.
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. 1. &
, (1), (2), $"
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). inf J = ;1, Q .
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2
2
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3.
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A B
AB
C3
AB
x~
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;
+
;
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+
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A B ,
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rdAB (~x) = ;(rA2 )B1 (~x) = (1 0):
: AB & dAB . H & , 0 C 3 x~ Ox2 .
1
0
! @A
@x2 (~x) > 0. 2
1. C ! AB !, ! & F . AB x2 0 & F ! AB 5
dF . 6
@ x~. dx
@x2
2
AB , ; d2F (0) = @ 2 F (~x) = ; cos < 0:
dx22
@x22
, AB F = ; cos2 x22 + O(x32 ). C
;
2! x~ xV AB
hABi
u = ; hBBi 2 0 1]. 5 x_ = A + uB 2 . 6 & F + uG & J xV. )
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xV. 1 . 2
1 2 0
, / inf J = ;1. B ; 0
2
2 Q , !(
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1
; !( :
213
q
q
p
~)2p+2 ctg ~
2
2
~
~
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2 ctg ~
4
4
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6= 0@
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2 ctg ~
.
arcctg (0 ). 40 , ;
! ~ ! ! &! .
2.
(1), (2)
(A B F G) 2 S
p(A B F G)
Q
2 (~() 23 ]
inf J = ;1
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3 ! / 2/
0, 4 0/ 20
F/ 0 .
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h Q3 = ( ) 2 0 2 (~() 2) :
S
2W
Qi Q 2(
0
Q.
i
:0 q 2 Q 2 G=GE 0
P . 2 Pi P 2W
2 q 2 Qi ,
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Si
, / 2W
S . H 1{4 !(
.
5.
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U
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< !(! . 1
0
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01
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.
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, f w, 2
/
f = (w), 1
c > 0, / s 2 D jf (s)j < cjw(s)j. F
0
, f w, 2
/ f = X(w), a > b > 0,
/ s 2 D aw(s) 6 f (s) 6 bw(s) w 6 0@
aw(s) > f (s) > bw(s) w > 0:
p
:
& x1 x2 & r = x21 + x22
0 0
, ! ! / &.
> (1), (2) && !, . .
H = H0 + uH1. 5] (
u(x ) = argmax H (u x ) = argmax(H0(x ) + uH1(x )) = 0 H1 < 0 1 H1 > 0
u
u2#01]
H0 = ;F + h Ai H1 = ;G + h B i:
(7)
6
1
! . H
2
1 2.
X
_ = ; @H
=
r(F + uG) ; ( rA + rB u):
(8)
@x
4 , 0
R0 & !. B
& F
,
(
0
.
)
& H = 0. ; 0
;uH1 H0.
D
& H1 /! !
T X . 2 M.
217
2. M | % . M+ M;, "
T X
fx j dAB (x) > 0g fx j dAB (x) < 0g . f(x ) j x = 0 1 = 0 2 < 0g f(x ) j x = 0 1 = 0 2 > 0g. 7 " | " (x = 0 = 0).
. 6 T X H0 H1. )
(7) , dAB ;&&
!.
6 ; B2 F ; A2 G + B2 H0 ; A2 H1 0+F = 1
= d 1 ;BB21 ;AA12 H
H1 + G dAB ;B1 F + A1 G ; B1 H0 + A1 H1
AB
(9)
fH0 = 0 H1 = 0g, !( M, ;
= d 1 ;BB2 FF;+AA2 GG :
(10)
, $
+
AB
1
1
)
(10) 1
M+ M; . / T X 2! fx = 0 1 = 0 2 6 0g fx = 0 1 = 0 2 > 0g
.
6 dAB = 0 2 ;&&
(7) , . . / :
A1(G + H1) ; B1 (F + H0) = A1 (G + H1) ; B1 (F ; uH1) = 0:
(11)
A
G
H1 = 0, F = B11 (7).
6 x~ . 40
(7) x~ fx = 0 1 = 0g. !
2
.
0
, / / AB . 6 3 ! AB 0 x2, / & F AB x~ dxdF2 = 0,
d2 F2 = @ 2F2 = ; cos . ! F = ; 1 cos x2 + (r3 ).
2
2
dx2
@x2
3
A
G
3
1
, B1 = (r ). 5 2 ( 2 2 ), cos < 0. ,
F ; AB11G = X(x22 ) 6= 0. )
0
2 . 2
6 2
, 0(
M. (
(. 10]) 2/ 0. &&
0
@H
@u 0 1
. G & H = H0 + uH1, ux) = 0 adh H , ad0 H = H , adn+1 H = fH adn H g.
( dtd )h @H (@u
1
1
H 1
H 1
H 1
H
8
f g 2
2 .
B H_ 1 1
(7)
218
. @B
@A
@G
@F
A @x ; B @x ; A @x + B @x :
(12)
H
! fx = 0 1 = 0g 2 M+ , M; .
x = 0 1 = 0. A B G F / @A i = 2 = 0.
/ x~ (3) (12), H_ 1 = h ; @x
1
@F =
1 1 = dtd @x
B&&
(12)
t
=
0,
H
1
2
= u @@xF21 = u sin . H
sin 6= 0, ; 2 2
0 x~ 1. .
u H1 1 = 0. ! u 0, , x x~, 0.
2 2 M+ , M; . 6 0
2 dAB 6= 0. (9) (12). &&
;
A
;
uB
2
2
_
dAB H1 = A1 + uB1 A B ] H1 +
B
F
;
A
G
2
2
+ ;B F + A G A B ] ; dAB DA G + dAB DB F
1
1
H1 . 2 C , A + uB |
Q. D
, ;Q2 A B ] = d_ ; d tr @Q :
AB
AB @x
Q1
6 @Q
_
_
dAB H1 = dAB ; dAB tr @x H1 + C:
(13)
D M+ M; H1 = 0. , ;/ 2/
C , 2 0 C = 0.
H_ 1 = dAB
3.
C
C2
;
C = 21 sin x21 + cos x22 + (x)
(14)
@ 2
2 = (r)
= (r3 ) r = (r2 ) @x
2 ( 32 )
(14)
C < 0 U n fx~g
2 ( 2 )
(14)
C
x~
C1
x~
x~
x~
C2
X
H_ 1 = fH H1g = fH0 H1g =
8"
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0 + "!+ "
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219
B
. 2 & C ,
0(! i- , Ci , i = 1 : : : 4.
0
! 3 2/ M+ , M; 2
0 (
2 ( 2 ). C ; . 6 0
2 Ci ! ^Ci M, i = 1 : : : 4.
6 ! ! H1 1 . (13) dAB &&
C , t. ), H1 = 0 H_ 1 = dAB
!
_AB
_
d
@Q
d
_ 1 + C_ ; C d2_AB =
1
H1 = dt d ; tr @x H1 + ddAB H_ 1 ; tr @Q
H
@x
d
d
AB
AB
AB
AB
C_ : (15)
+
= ; dC tr @Q
AB @x dAB
D / ^Ci C = 0, (15) (14) / H1 1 = Dd A C + u Dd B C = d 1 A1 sin x1 + A2 cos x2 + (r3 )+ u(sin x1 + (r2 ))]:
AB
AB
AB
:;&&
u , 2 0 2 1.
)
H1 1 = 0 u = ; DDBA CC = ;x1 ; x2 +
+ ctg x2 + (r2 ). H
, 0 /
2
0 1]. 60
(14) !, ; ctg = ; xx221 + (r) u = ; x12 (x21 + x1x2 + x22) + (r2 ) < 1. 6 2 (;2 2) , u > 0
x2 < 0. 5 2, 2 0 ! ^C 3 ^C 4 . D C 3 & 20
x~, C 4 |
. 20
/ 2
.
6. ' )
(11) 1
&! H1 AB . C
, ; AG e1 i = hBF ; AG V i :
H1 = ;A1 GQ+ B1 F = hBFhQ
(16)
e1 i
hA + uB V i
1
6
BF ; AG Q = A + uB e1 = (1 0) 0 !2 V , Q. 5 2, (13) / AB !
2
.
C
; 2
u = 0. u = 0 2 (16) V = x? = (;x2 x1), H1 i
H1(t) = ;G(t) + F (t) hhBx
Ax i (t) + dAB (t)f (t). ; 0
(13),
?
?
220
. 0 / !(
! &!
f (t):
rF i
f_ + hr Aif + hx?F Ai2 hx? A ; Dx Ai + hhx
x? Ai = 0:
2 s(t), Zt ; Rt hrAi d ;
x
B
+
x
B
2
1
1
2
H1(t) = ;G + F ;x A + x A + dAB ; e s( ) d ;
2 1
1 2
0
1
;G + F ;x2B1 + x1B2 (0) :
dAB (0)
;x2 A1 + x1A2
6
. H
dtd (;x2 A1 + x1 A2) = hr Ai(;x2A1 + x1A2 ) +
Rt
; hrAi d
;e
0
tR2
hrAi d
t
e1
= ((;;xx22 AA11 ++xx11 AA22 )()(tt21)) + (r).
hx rF i = 2F + (r3 ). 5 2, s = 2?F
:
, A ; Dx A = (r2 ),
hx Ai + (r). , + (r3 )
Zt
1
H1(t) = ;x A + x A ;Fx2 + dAB ; 2F d ; ;dFx2 (0) + (r2 ): (17)
2 1
1 2
AB
0
C T X H0:
X
x_ = A _ = rF ; rA :
(18)
(2), (8), u = 0. C !
^ (t) (18), 0(! & H0. 1
+ ; = X ^ AB
AB
t+ t; . 6 H1(t+ ), H1 (t; ). H0 = 0 (11), H1 = ;G + BA11F = ; cos2 x2 + (x22 ).
. H1(t+ ) > 0 H1(t; ) < 0 2
6
1
U (dAB ), = ; xx12 . !, Ox2 . .&
/ . C 0
K = f(x1 x2) 2 U j ; xx12 2 ]g fx~g.
G U , 0
K AB x~. x~ 0
K 0
K+ = f(x1 x2) 2 K j dAB (x1 x2) > 0g,
K; = f(x1 x2) 2 K j dAB (x1 x2) < 0g. D 0
/ K+ K; (x1 x2) $ (dAB ) . x~ (dAB ) f0g ]. D
,
K x1 x2 & dAB 0 C 3.
,
.
221
0
! 2 0
K+ K; 2! 2 M+ M; T X . D ;/ / (10) ! & x , /, & dAB .
4.
(dAB )
f0g ]
C2
. C 0
2
(10). :
& / dAB 0 C 3 0
K
! (r2 ) = (d2AB ). )
0
Y. 2
C 0
KV 2 M, (
0
K+ K;
M (x = 0 = 0). 0
! 4 1
&
(dAB ) C 2. C X
x_ = A + B _ = r(F + G) ; ( r(A + B ))
(19)
8"
"
%" "
" & $
.
H0 + H1. u = 1 (2), (8).
H
(19), /(
0
KV t = 0. < 0 &
H0 + H1 t. 6 &&
&&
/ / x ;/ ! & C 2 / dAB t. , H1 & dAB t ;/ 0
0
C 2. &! H~ 1(dAB t) !( 2: H~ 1 = Ht1
t 6= 0 H~ 1 = H_ 1 t = 0. 6 , H1 = 0 t = 0, Y & H~ 1 0 C 1 .
2 ] . 6 & H~ 1 (dAB t) = (0 0). < / x . 6 ; H~ 1 = H_ 1 = 0. H
@ H~ 1 = lim H_ 1 (d 0) = lim C (d 0) = 1 (sin + cos 2)
dAB !0 d2AB AB
@dAB dAB !0 dAB AB
2
(20)
~
~
@ H1 = 0 @ H1 = 1 H1 = 1 sin 6= 0:
(21)
@
@t 2 1 2
& (dAB ) = (0 )
(
& t (dAB ), t (0 ) = 0 H~ 1(dAB t (dAB )) 0. < & 0 C 1. G1
! @t
2 @t
(22)
@dAB (0 ) = ;1 ; ctg @ (0 ) = 0:
222
. 6
& t 0
K t (dAB ) = dAB (;1 ; ctg 2 ) + (dAB )
(23)
1
2 C , dAB ! dAB ! 0.
7. )
3
2 ( 2 ). 0
T X , !(
2
(2), (8). 6 2 2 (
2/ 0 ^C 3 ^C 4 0
M. C 3
/ & 20
x~, C 4 | . 2
, / 2 0 ^C 3 . 0
, 2/ (2), (8), /(/ 2
0 ^C 3 , 2
.
x2 6 +
10
C 1
AB
z
z
u=1
;
3
-
z
- >o I
x-1
o
C 3
$. 4
u=0
;
AB
C , /(
^C 3 u = 1. H/ &! ! 2 0 C 3 ; , /(
x~ (. . 4). 0
, , /(/ ^C 3 u = 0, ! !
U . )
(18) ! 0
u = 0. <! & H1 (18) & H0 (17).
223
p | ^C 3 , ^ (t) | (18), /(
p t = 0. = X ^ ; 2
x~, x~ | & . ;
t < 0. 2 AB
t; , +
AB | t+ .
H
H1 &! ^ . C = 0
C_ . 6 , C_ > 0, dAB < 0 (15), H1 1(0) = dAB
p, H1 1(0) < 0, (
" > 0, H1(t) < 0 / t 2 (;" 0). tV = supft < 0 j H1 (t) > 0g. 6 6
tV > t+ .
5.
t 2 t; 0)
H1(t) < 0
. 0, (
t0 2 (t; 0), H1(t0 ) > 0. F
2( 0 , H1(t0 ) = 0 H1 (t) < 0 / t 2 (t0 0). D C (t0) < 0, dAB (t0) < 0,
C > 0. . 6 ; H_ 1(t0 ) = dAB
. 2
5 2, tV < t; . H1( tV) = 0 (17)
F = 12 (sin x21 ; cos x22) + (r3 ) = X(r2 ), dxAB2 (0) = X(1) dAB = x1 + (r2 ), 0 = ;X(r;2 )f;x2( tV)X(r2 )+ x1 ( tV)X(r2 )g + (r2 ) =) x2 ( Vt ) = X(1)x1 ( Vt ): (24)
5 2, ( tV) 0 .
(t) ^ C 1 C 4 t1 , t4 .
6.
t 2 t1 t4]
H1(t) = ;X(r)
. D ^C 4 x2 = ;X(r), x1 = X(r). (17), H1(t4 ) = ;&(1r2 ) fX(r3 ) + X(r)X(r2 ) + X(r2 )]g + (r2 ) = ;X(r).
t 2 t1 t4). C (13). 2 ;&&
{ =
H1 { . Q = A, t t4] dAB
2
_
1 +(r ) ; tr @A = x2 + (r) = (1). D ; (13) = ;A2 BdAB
@x x1
2
, 0
H1(t) {
R t { d Zt4 ; R dAB
d C
H1(t) = e t4 dAB ; e t4
d
+
H
(
t
)
1 4 :
dAB
+
+
.
.
t
D t t4] C > 0, ; / 2/ 0
(r). 5 H1(t4 ) = ;X(r), 2,
H1(t), ! ;X(r). 2
7. C ( Vt ) = ;X(r2 )
.
224
. . 6 (24) ( Vt ) 0 Vt < t4 .
Rt1
5 0
6 , tV < t1 , H_ 1(t) dt = H1(t1) = ; X(r).
t'
D H_ 1 = (r), ; t1 ; Vt = X(1). , C (t1 ) = 0, , (24) C_ = X(r2 ) tV t1]. ! Rt1
C ( tV) = ; C_ (t) dt = ; X(r2 ). 2
't
8. 4 " p &
^C 3 , V) " " 10 " C 2, ! " ( t
& " "! " x
~
C 3
)
0
8 2
/ &&
/ . & 0
7. 5
(24).
dx2
9.
10
x ! x~
dx1
. D
, dx2 10
, dx
1
dx2 = x2 + (r) = X(1):
(25)
dx1 x1
, !
10 0 x1 . Hd xx12
x2 1
x2
2
(25), dx1 = ( dx
dx1 ; x1 ) x1 = (1). < , x1 & x1 , , x1 ! 0. D
dx2 0
. 2
(25) dx
1
:
C 3 , 10 x~ U 2 (. . 4).
4 , 2 u = 0 0
/ 10 2 . 0
, 2 ! u = 1. C 2 / ; .
p | 2 0
^C 3 , ^ (t) | (19), /( p t = 0. 2 !
X ^ ^ &! . H
H1 &!
t ^ .
10.
H1(t) > 0
t<0
( )
U
2 t 0]
. 0, (
t0 < 0, H1(t0 ) 6 0 ( ) 0 U / 2 t0 0]. D ^C 3 C_ > 0, C_ < 0, dAB < 0. ; 2
H1 = H_ 1 = 0, H1 1 = dAB
2( 0 , H1(t0 ) = 0 H1(t) > 0 /
"
.
"
/
, "
& "
.
+ "+
+
.
225
C < 0.
t 2 (t0 0). D C ((t0)) > 0, dAB ((t0 )) < 0, ; H_ 1(t0 ) = dAB
. 2
6 0
2 !
10 ^10 M+ .
B, p | ^10 (x = 0 = 0), ^ (t) | (19), ^ (0) = p. 2 !
X ^ . H
H1 &! t ^ .
11.
H1(t) > 0
t<0
( )
U
2 t 0]
. 6
1
dAB (
. :! 10 0 dAB , ddxdAB
1 =
= 1 + (x1 ). 5 2, 10 1
&
= (dAB ) C 2. D
2
, d ddAB ,
. . (1).
= dinf (dAB ), = sup (dAB ). 6 0
8
AB
dAB
. !, 1
! 6.
: 10 0 0
K+ . & t 10 : t (dAB ) = t (dAB (dAB )). C ! ddtAB =
= @d@tAB + @t@ d ddAB . H
d ddAB = (1), ; (22) d lim!0 ddtAB =
AB
= ; 1 ; ctg (d lim!0 (dAB ))2 . (
0
AB
9.
6 0
3 7 & C ! 10 :
C2 = 1 ;sin + cos lim x222 = 1 ;sin + cos lim 2 (dAB ) < 0. B
lim
2
2
x1 !0 x1
x1 !0 x1
dAB !0
dt
1
; 2 sin < 0, d lim!0 dAB > 0. , t (dAB ) > 0 AB
dAB > 0.
D, / (19), /(/ 0
KV t = 0, & H1(t) / 0 t . D H_ 1 < 0 ^10, ; (
" > 0, H1(t) > 0 t 2 (;" 0). 2 U .
)
0
11 . 2
4 U &! (x), !(! ! :; 10] 2 0
0
10 11 | .
12.
(x)
U
C 3 ; 10
x~
C 3 ; 10
. C ! & , (! C 3 , 10 x~. 6 0
3 9 0 PC 1 / x~. 6 (dAB ) 1
&
(dAB ). D
, /
, "
+ "+
& "
+
.
8"
"
%%
, "
"+
/ "+
,
-
"
+ .
.
226
. & (dAB ) 0 C 2 dAB 6= 0,
dAB = 0 .
C ^ M, (
^C 3 , ^10
(x = 0 = 0). : ^ 1
(10) 0
4 . C & :
d
(dAB ) = (dAB (dAB )). H
d ddAB = @d@AB + @
lim!0 @
@ ddAB . 6 dAB
@ = 0,
d = (1) lim d = lim @ . < / d dAB
dAB !0 d dAB dAB !0 @dAB
(
!, (
! d lim
(dAB ), AB !0
!.
4 , ^ | PC 1 / (x = = 0). H
& (x) 1
(18), (19), /( ^ . B
&&
&&
/ /. 2
6 0
12 & (x) N T X .
13.
(x)
U
!(x)
C1
. 0
0
N , . . 1
& dx. C , N &&
. 6 &&
& dx / , / & @t@ . @v@ |
.
X
@
@
@
@
d(dx) @t @v = (d ^ dx ) @t @v =
X 1 @ @x @ @x = 2 @t @v ; @v @t =
X @H @x @ @H 1 @H
1
= ;2
+ @v @ = ; 2 @v = 0
@x @v
N 0 H . D & d(dx) !
!2 , N . , (
& ! 2 C 2, d! = dx.
6 0
12 & (x) !i , i = 1 2 3, 1
/
2/, 0
/ 0 C 3 , ; , 10 . / ,
xlim
! = 0. 6 (x) & !i 2! !x
~ i
&! ! 2 C 1(U ). 2
> !(x) U ! F
. ; . > / 2
0 x~ 2
, 1
x_ 2 = (x2 ).
5
.
8"
"
.
8. )
2 4
227
B & / 2 & ~. ~ : (;2 2) ! ( 32 ) !( :
q
q
p
~)2p+2 ctg ~
(1 ; ctg ~)2 + 2 ctg ~ + ctg ~ p1 2 +arcctg (1pctg
2 ctg ~
4
4
q
;e
= 0
q
2
2
~
~
~
(1 ; ctg ) + ctg ; ctg 6= 0@
q
~
ctg ~
2 ~ ; ; arcctg ctgq ; 1 = 0 = 0:
ctg ; 1
2 ctg ~
;
;
;
.
arcctg (0 ). )
! ~ ! ! &! .
4 0
, 2 1,
4 0
T X &! F
.
2 ( 23 ). 5 0
3 C = ;X(r2 ). | 0
. !, ! 6. 2 0
K; , (! , K0 .
0 . 6 0
2 0
K0 K^ 0 2
M; . K0 K^ 0 dAB0 = dAB , 0 = . H
0 , ; ! 0
/ K0 K^ 0. C (19),
/(
t = 0 0
K^ 0. 0 & H0 + H1 . ; t,
/ | dAB0 0 .
! & t H1(dAB0 0 t (dAB0 0 )) 0. ;
0
f(dAB0 0 t (dAB0 0 ))g 2
/ K^ 1 2 M. H
; /. p0 | 0
K0 ,
!( dAB0 0 . p^0 2 K^ 0 | p0 M; , ^ (t) | (19), /( t = 0 p^0. 6 & dAB p^1 = ^ (t ) dAB0 0 . 2 / dAB1 1. 1
(23) Rt
Rt
dAB1 = dAB0 + d_AB dt = dAB0 + (1 + (r)) dt = ; dAB0 (ctg 02) + (dAB0 0),
0
0
! dAB ! 0. B
, 2 C 1 , dAB
0
0
Zt
x2(t ) = x2(0) + x_ 2 dt = x2 (0) + (r2 ):
0
(26)
228
. ,
(0) + (r2 )
tg + o(1):
1 = ; xx2((tt )) = ; ;d x2(ctg
=
;
(27)
2
0 ) + o(r)
0
1
AB0
6 , tg > 0, t = X(r), dAB1 = X(r), 1 = ;X(1).
, K1 / K^ 1 &! 0 C > 0, ; H1(t) > 0
. D / K^ 0 H_ 1 = dAB
0
/ t 2 (0 t ).
^ (t) | (18), /( p^1 2 K^ 1 t = 0. Vt =
C < 0, ; tV > 0.
= inf ft > 0 j H1(t) > 0g. 6 p^1 H_ 1 = dAB
1
2 ! X ^ ^ &! .
5
2
x~ . ;
;
t; > 0 1
! AB
, +
t+ > t; | ! AB . 6 6 tV < t+ .
14. Vt > t;
B
0
5.
H 0
14 , p^2 = ^ ( tV) 0 2 M; . 2 1
! ( tV) p2 . 4
tV 1
H1( tV) = 0. > H1(t) (17). 0
2
! Fx2( tV) + (r4 ) = dAB ( tV)(r2 ).
F = x1 (r) ; 21 cos x22 + (r3 ), dAB = x1 + (r2 ), x32( tV) + (r4 ) = x1( tV)(r2 ). ! ; xx21 (( 't't )) = (1).
dAB2 = dAB (x1( tV) x2( tV)), 2 = ; xx21 (( tt'')) .
4 20
P : (dAB0 0) 7! (dAB2 2 ) K^ 0 M; M;. D
, dAB2 = X(dAB0 ). ,
lim d = 0.
dAB !0 AB2
.
0
. 2& P & f0g ].
> dAB1 dAB2 0 1
. >! 1(dAB0 0 ) 0 0 1 (0 0) = ; tg0 (27). , & 2 (dAB1 1) 0
0
. !
0
.
Z, 20
P &&
dAB0 < 0.
AB2 2 )
, @@ ((ddAB
0 0 )
0
0! f0g ]. ;
@dAB2 = dAB2 + (r) = X(1), @2 = C (p0 ) + (r) = X(1).
@dAB0 dAB0
@0 C (p2 )
5 2, & 2 , f0g ],
&&
!(
&
0. 6 (27) lim
1(0 0 ) = ;1,
!0
0
229
lim (0 0) = 0. H (17) (dAB1 ! 0) () (dAB2 ! 0). !, 0 !1 1
d2 > 0, lim (d ) = ;1, lim (d ) = +1.
d
1
1 !;1 2 AB1 1
1 !+1 2 AB1 1
5 2, lim (0 0) = 2 max lim
2 (0 0) = ;1:
0 !+1 2
0 !0
.
2 max | (. . 5).
2 6
2 6
2 max
2 max
2
q
q
q
-
0 1 0 0 2
0
-
0
$. 5
4 0
, (
0 20
0 7! 2 (0 0). 6 ; (0 ) | 0 20
P. 2
,
1
T X .
G & & 2 (0 0) 0 & 2 = 0 (.
. 5 ), 2 = (0 0) . G 0 , 2 = 2(0 0 ) > 0 , (
! 2 = (0 0) (. . 5 ). 6, /
/ 0 (
.
C &! A2(x). G1
x~ (3) ;e1 = (;1 0). ; & A2 2
! C 3, /(! x~ !(! Ox2 .
s | 0
, 0( ; . C ! x_ = A, /(! s t = 0. 6 (3) 2
x~ . H
! x2 &! t. H
x_ 2(0) = A2 (s) = 0, x2 (0) = x2min . t+ = minft > 0 j A2 ((t)) = 0g, t; = maxft < 0 j A2((t)) = 0g,
x2max = min(x2(t+ ) x2(t; )). , x2max > 0 > x2min , 0 x^2 2 (x2min x2max) (
! + 2 (0 t+ ), ; 2 (t; 0),
x2( + ) = x2 ( ; ) = x^2. 6
1
2
x+1 = x1(( + )),
x;1 = x1(( ; )).
230
. R +
C I (^x2 ) = F ((t)) dt &!
x^2. H
I (x2min ) = 0. x^2 ;
dI = d + F ( + ) ; d ; F ( ;) = F ( + ) ; F ( ; ) =
dx^2 dx^2
dx^2
A2
A2
+
; 2
2
2
3
2
3 = 12 sin (x1 ) ;+ cos x2^2 + (r ) ; sin (x1 ) ;; cos x2^2 + (r ) :
;x1 + (r )
;x1 + (r )
dI =
D & A2 F =X(r2), , x^ !lim
2 x2min dx^2
dI = +1. x^2 = 0 dI = 1 sin (x; ; x+ + (r2 )) =
= x^ !lim
1
1
dx^2
2
2 x2max dx^2
dI
= ; X(r) < 0. dx^2 & I (^x2 ) (x2min 0) , (0 x2max) | . 6 ; ddIx^2 = AF2 ( + ) ; AF2 ( ; ) = 0. 0/ 2 x^
(28)
; tg = x^;2 +2 + (r):
x1 x1
2 I (^x2 ) Imin . 0
, Imin , (
.
.
Imin = ;X(r2 )
1
. I (^x2 ) = Imin = ;X(r2 ). W l, H 0 2 , 2 , (F + uG) dt < 0. C ! l
x_ = A + B , /(! ( + ). ; u = 1 X(r) ; , 1
! (Vx1 xV2). 6
x_ 2 = (r)
xV2 = x^2 + (r2 ), , xV1 = x;1 + (r2 ). 6 (Vx1 xV2) ! u = 0 1
( + ). H
; 4
I
(F + uG) dt =
('xZ1 x'2 )
(xZ1 x^2)
;
(F + G) dt +
.
(x+1 x^2 )
F dx + Z F dt =
A2 2
('x1 x'2 )
(x1 x^2)
2
2
= X(r)(r ) + (r )(r) + Imin = ;X(r2 ) < 0: 2
2
4 Imin = +X(r ), !-
(x+1 x^2)
15.
, ;
4
. I (^x2 ) = Imin = +X(r2 ). 0
, (
0 , 2 = 2 (0 0 ) > 0 .
.
231
.
, ( + ) ( ; ) 0 x_ = A. ; ;/ x~ (x;1 )2 + x^22 = X((x+1 )2 + x^22 ). :2 ; (28), ; xx^+12 = X(1). ; ( + ) 0 ! 0
K0 !, 1
! ; . 0 p0 = ( + ) , 20
P ( + ).
H, x1 (p0) = x+1 , x2(p0 ) = x^2. H (26) x2(p1 ) = x^2 + (r2 ), (27) (28) | x1(p1 ) = x;1 + o(r). 5 2,
0 p1, ( ; ) : d(p1 ( ; )) = o(r). o() / s ! x~, s & A2 x~.
C (t) (t) x_ = A, /(
t = ; p1 ( ; ) . :;&&
/ ; , ; (t) (t) t 2 ( ; + ) 1
(1
! , !(! o(r).
5 2, F , / :
R+
R+
F ((t)) dt = F ((t)) dt + o(r2 ) = Imin + o(r2 ) = X(r2 ). :
, 0 = AF2 (( + )) ; AF2 (( ; )) = AF2 (( + )) ; AF2 (p1) + o(r). D A2 = ;dAB + (r2 ),
; dFxAB2 (( + )) ; dFxAB2 (p1 ) = o(r2 ). ; (17), !(
&! H1(t) , H1( + ) =
Z +
1
Fx
2
+
+
= ;A x + A x ;F x^2( ) + dAB ( ) ; 2F dt + d (p1 ) + (r2 ) =
1 2
2 1
AB
= ; X(1r2 ) f;2dAB (( + ))(X(r2 ) + o(r2 ))g + (r2 ) = ;X(r):
! ( + ) . .
2 = ( tV) H1(2 ) = 0. H
H1( + ) =
R2 H_ 1
R2 (r)
+
=;
d
=
;
_
&(1) d = ; X(r). ! 2 = ( ) + X(1).
( + ) ( + )
C
0 p0 = ( + ) ( + ) o(r), ;
( + ) = 0 + o(1). ! 2 = 0 + X(1). D
, 0 2 s ! x~ 0 , 2 . ,
2 = 2(0 0 ) > 0 .
5 2, (
! 0 1 , 0 2 2(0 0 ) = 0 , 1
0 < 0 < 0 2 (. . 5). 5 (0 01 ) (0 02 ) | 0
20
P.
.&
0 = 01 . C dAB2 (dAB0 01 ) &! dAB0 . H/ 01 < 0 , , ;
;
;
232
. (
c < 1, ;c dAB0 > ;dAB2 . 0 = 02 ;dAB0 < ; dAB2 , c < 1.
d2AB0
@dAB2 =
@2 =
6 (0 0 1) @
> c12 > 1, @d
lim
d
0
AB0
dAB0 !0 2AB2
@d
d
AB2 6 c < 1, AB2 = 0. 6 (0 2 ) | @2 < 1,
= d lim!0 dAB
0
@0
@0
0
AB0
@dAB2 > 1. !, , , (0 ) = 2
0
0
@dAB0
1
2
, 0 0 .
5 2, 0 (0 0 1) 2
.
2 / 2/ 2
0 , ., , 12]. ) f0g ]. D
0 &
= (dAB )
C 1 .
5 2, 2 &&
!
!, ! 20
P !(! !
0 1. 2 ; ! 01, 1
M; | ^01. G p^0 2
! ^01 ,
p^1 2
! ! ^10 , p^2 | ^01. 10 ^10 !
1 0. 5
^ (18), (19), !(
^01 ^10 , 2! T X
2
(2), (8). 0 PC 1 / / ^01 ^10 . D
, , ; 2! (x = 0 = 0), N
T X . 5 x~ !-
;
.
B
0
N & F
! 0
13. 5 2, , N , . , 4,
!. )
0
15 . 2
B
2 4 !(
0
.
2 ( ~())
.
2 (~() 23 ] Imin = ;X(r2 )
2
Imin = +X(r )
B
Imin . 6 0
. 5 2,
2 (0 0) = 0 0
, !(
1.
5
.
4. .
02 , x~ -
.
4
.
, . 4
,
233
Y 2
&
4. H. .
(, ; 2.
+
1] . . . | .: ,
1975.
2] #$
%. . 1 -'( )( ) *( '+ '( ,- ,**' // /- 0 $ ,1. |
1977. | 2. 11, 3 1. | 4. 57{58.
3] 8 . ., 8 9. /., :
%. *),'' ,')$
*'0 $ ** ,')( 0
+ // 2
. | 2000.
4] ='$ 9. 4. . | .: >*. 0
.
0.-)'. '., 1961.
5] ='$ 9. 4., #'*- . >., >)
0 %. ., @ A. /. ')'+* ' ,')( ,**. | .: , 1969.
6] Davydov A. A. Qualitative theory of Control Systems. | Providence, RI, 1991. |
Translations of Mathematical Monographs, vol. 141.
7] Fuller A. T. Constant-ratio trajectories in optimal control systems // Internat. J.
Control. | 1993. | Vol. 58, no. 6. | P. 1409{1435.
8] Jakubczyk B., Respondek W. Feedback classiGcation of analytic control systems in
the plane // Analysis of controlled dynamical systems (Lyon, 1990). | P. 263{273N
Progr. Systems Control Theory, vol. 8. | Boston: BirkhOauser Boston, 1991.
9] Kelley H. J. A second variation test for singular extremals // AIAA J. 2. | 1964. |
No. 8. | P. 1380{1382.
10] Kelley H. J., Kopp R. E., Moyer M. G. Singular extremals // Topics in optimization. | N.Y.: Acad. Press, 1967. | P. 63{101.
11] Krener A. J. Approximate linearization by state feedback and coordinate change //
Systems Control Lett. | 1984. | Vol. 5, no. 3. | P. 181{185.
12] Nitecki Z. Di^erentiable dynamics. | Cambridge: M.I.T. Press, 1971.
13] Zelikin M. I. On the singular arcs // Problems of Control and Information Theory. |
1985. | Vol. 14, no. 2.
14] Zelikin M. I., Borisov V. F. Theory of Chattering Control with Applications to Astronautics, Robotics, Economics and Engineering. | Boston: BirkhOauser, 1994.
' ( 2000 .
. . . . . 512.5+511
: , , , .
! "# $ " & " '#
# " $
$
(
# ). *"+ &# $
, ,- ", ".
Abstract
A. A. Chilikov, Taylor power series of algebraic functions over elds of positive
characteristics, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 1,
pp. 235{256.
In this work we show algorithmical solvability for the problem of calculation
of the Taylor power series for an algebraic function over a 5eld of positive characteristics. An e6cient algorithm for construction of a 5nite automaton solving this
problem is given.
1. , , , . ! !
, "
, | !
$ . % &
!
', "
$ .
!
, , ! $ . ( !
) )
! ' ( )
!
$
! $ !). %
& !
"
, , .
, 2001, 7, 7 1, . 235{256.
c 2001 !,
"#
$% &
236
. . 0 , $ $ $ ) !$ (" , $), "$
$ $ !
$ !
. 1 1$.
1$ !$ ,
" , ! ! !
C .
, & ! !)$ p. 2 & 3
!
& 1$, , !
, " "$ & ! ! . 4 !
!
: !
,
"$ x = !
(1)
x | !$ , = (1 : : : n) | , x | &
! #11 #nn , ! | !$ & !. 8
, &
!
& , ! .
9 "$ 1$ $ ! !)$ . 3 ) !
, " (1) ! !)$ .
( !
" ! ! p. !
)$ $ (1), !
! $ .
:" !
" " !3 $ )$ $ ! ! p.
;, !
$ ! ! !
$.
2. ! "
(
F | !, char F = p > 0. R F=#1 : : : #n], R | ! F==#1 : : : #n]].
?
0
i6s
X
i=0
ai ti = 0
(2)
ai R. @ | t(#1 : : : #n). A$ !$ x R 1$ t, (2). B !
, ! 2
2
0
237
1
1 : : : n, #nn . ? x !P # ! #1
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! $ h = f g.
(
Gh . %
X
h = f g = (" f) gp g :
0
0
0
0
256
. . ?
& , !
X
X
h = (" f) g(#p ) (" g )(#p )# 0
0
0
h=
X
#
X
(" f) g(#p )(" g )(#p ):
0
0
0
X
" h = (" f) g(#p )(" g )(#p ):
0
0
(25)
0
A (25) ! !
Gh Gf Gg Gg2 : : : Ggp 1 . G ) , , !
!$ $.
;
,
1] A. J. van der Poorten. Some facts should be better known, especially about rational
functions.
2] A. J. van der Poorten. Rational functions, diagonals, automata and arithmetic.
3] . . . | .: , 1986.
4] . &'. '(
. | .: , 1968.
5] *. +(,. p-
./ /
, p-
./0 .
-,. | .: ,
1982.
6] 1. 2. /. 3/
4(
5(
. | .: *
, 1993.
7] A. J. Belov, V. V. Borisenko, V. N. Latyshev. Monomial algebras. | NY: Plenum.
' ( ) 2000 .
T-
. . . . . 519.48
: , T-, , -
! ".
# $ ! % &' T-".
( " ! " T- 2(3), "!
)$ $ ! "". *+ % )! "+ $% +" "
1, 1
2 ] 2 , % 1 , 2 | "!, 1,
2 | ", " /$ T-, !$ " ! 0 "/ 0 ".
T
w
x x
w
x
x
w
w
Abstract
V. V. Shchigolev, On leading monomials of some T-ideals, Fundamentalnaya i
prikladnaya matematika, vol. 7 (2001), no. 1, pp. 257{266.
In this paper some analogs of the Gr'obner base for T-ideals are considered. A sequence of normal monomials of the T-ideal 2(3) is built so that the monomials are
independent w.r.t. the operation of monotonous substitution and the insertion operation. Also a theorem is proved stating that for algebras without 1 a multilinear
identity of the form 1 , 1 2 ] 2 , where 1 , 2 are variables and 1 , 2 are monomials, belongs to every T-ideal that is 8nitely based w.r.t. the inclusion relation of
the leading monomials.
T
w
x x
w
x
x
w
w
x 1. 1] T-
. ". #. $ 2] & '
.
( )
* & , *, .
( - *
* * * , & ,* . * T-.
/
* (. 0. 1
.
2
& K | , x1 : : : xn : : : | . ( '
), *: xi < xj
, 2001, 7, 9 1, . 257{266.
c 2001 !,
"#
$% &
258
. . i < j. 7 '
* - , & * ,
)
* -.
2
& S | x1 : : : xn : : :, , S (n) | , *, n, S0(n) | x(1) : : :x(n) , 2 Sn , Sn | * n. (2
9 '
S.)
1 K, *
'
S0(n) , P (n).
() jwj w ,
& w , **
* w. 7 u ,
u = u1xi u2xj u3 , i < j. <
S0n x1 : : :xn . 2 x 2 v, x | v.
=* ) c() ),
, & '
, x1 : : : xn : : :, * ) i degxi = degxi c().
( * S ), : xi1 : : :xin
xj1 : : :xjm & , &
& i1 : : : in **
* &
&) &
j1 : : : jm . > v u, , v * u (u v).
2 , * S, , v u,
v | u.
0 S '
), :
xi1 : : :xin xj1 : : :xjn , t fi1 : : : ing fj1 : : : jng, t(ik ) = jk , k = 1 : : : n, . ?
t
- F1 F2 F, fxi1 : : : xin g fxj1 : : : xjn g . > f1 2 F1 , f2 2 F2 f1
f2 '
-, , f1 f2 .
2
& F = K hx1 : : : xn : : :i | * * 1. =*
f 2 F f@ f, * ) I
F I@ = ff@: f 2 I g. 7, , I@,
& I. =, S(I) = S \ I@.
2* & S= ), : * '
B1 B2 B1 B2 (
B1 B2 ), ,
)
u1 2 B1 u2 2 B2 , u1 u2
(
u1 u2).
2
& (A 6) | * . /
a 2 A (A 6)- & fa1 : : : ang A, ,
ai, ai 6 a. <
(A 6) ,
,
)
a1 : : : an 2 A, ) '
a 2 A (A 6)- & fa1 : : : ang.
259
T-
( )
* * (S(I)= ) (S(I)= ) , I **
*
T-.
7 u 2 S (T- ) & ff1 : : : fk g F , u **
* (S )- ((S )- ) & ff@1 : : : f@k g.
B Tk(3) T- F, x1 x2 x3] : : :x3k;2 x3k;1 x3k ]. 2
& T (3) = T1(3) . ? ; S, )
xk xj xi xk xi xj (1)
(3)
i < j < k. 1 &, S n S(T ) ;.
( ), &
:
,
&
& w1 : : :wn : : : S n S(T2(3) ), *
i 6= j wi wj - ) D
& I | T- F ( 1), * (S(I)= ) . E I w1 x1 x2]w2, w1 w2 | .
x 2. 1
&, ) u 2 ; *
* r
Q
u = ui , ) ui vi xki , xki & ) i=1
vi vi , i < j )* ui & ) uj ,
ui , juij = 1.
2
&
& (u1 : : : ur ) u.
1. ; T (3).
.
2 . E w =
r
Q
= ui 2 ; **
* f 2 T (3) ,
i=1
(u1 : : : ur ) | . 2
& ui = vi xli . > ui , vi , , . > ui
, ui . 2 ) * i = 1 : : : r. E * w0 f 0 ,
),* f .
=
&, & w0 < w w0 | f, w00 | , w0 . 2
& w0 = xl 0 , w = xm ,
tQ
;1
l < m. ?
) , ,
t, =
ui u0t ,
i=1
260
. . Q
r
= u00t
ui , u0txm u00t = ut . H * *
i=t+1
, xl ut+1 : : : ur . 7 , xl . 7
&, xl u00t . 2'
),
*: ju00t j > 2 ju00t j = 1.
2
& * .E u00t ,
& xm , & ** u00t . ?
) u00t = vt00 xsxl ,
s > m. 2 xm u00t xsxl , xl 0 | xl 00 . ?
) w00 < w0 .
2
& * . E u00t = xl , xm u00t *, xl 0 xl 00 . ?
) w00 < w0 .
E T (3) , & & , f 0 2 T (3). 7 w0 s
Q
w0 = (xli xmi ), xli > xmi (xl1 xm1 : : : xls xms ) | i=1
.
r
Q
> & v 2 P (n) | & , v = vi , i=1
(v1 : : :vr ) | v. > &, ,
)
i1 < : : : < is , jvik j > 2 k = 1 : : : s, w0 v v
& & . 2'
&, ,
)
1 6 i1 < : : : < is;1 6 r, i 2= fi1 : : : is;1g jvi j = 1. =,
, P (n) * jv1j : : : jvr j. H&* '
-
, & & P (n). ?, r 6 n. =* i1 < : : : < is;1
* Cns;1 . ?
* & & *
jvi1 j : : : jvis;1 j. E jvi j 6 n * i = 1 : : : r, '
ns;1. ( Cns;1ns;1 & P (n).
/
, dn = dim(P (n) j P (n) \ T (3) ) T (3) , , , , 3]. 1 .
E , 1. ; = S n S(T (3) ).
( * T (3) (. 0. 1
4].
2
& I1 , I2 | F. E F I2 +
+I1 F | F F. 7 F F * u v, u v | F . 7
,
u2 v2 > u1 v1, u2 > u1 u2 = u1 v2 > v1. =* f 2 F F
f@ f. 0
u v & & L F F , * ) f 2 L u v 6= f.@
7 ),* 2. u v 2 F F , u v | I1 I2 . u v | F I2 + I1 F .
T-
261
. 2 . E ,
f 2
2 F I2 + I1 F, f@ = u v. 2
f ), :
f=
n
X
i=1
u i fi +
m
X
j =1
gj vj (2)
* ) i j 0 6= fi 2 I2 , 0 6= gj 2 I1 , ui vj | , ui > uj vi > vj i > j. < &, '
vj & & I2 . =
&, & vj ,
I2 l | & j, * vj | I2 . E vl = f@0 * f0 2 I2 . H
n0
m0
X
X
0
0
f = ui fi + gl f0 + gj vj + gl (vl ; f0 ) = ui fi + gj0 vj0 i=1
j =1
i=1
j =1
j=
6 l
& vj0 , & I2 , &, vl .
n
X
m
X
2
** '
, &* *. 2, ,
i, ui > u. 7 ui f@i & (2). E f@ > ui f@i > u v, . 2'
* i = 1 : : : n ui 6 u. 2
& ui0 = u, ui0 f@i0 &
(2). 2'
f@i0 6 v. E v &, f@i0 < v fi0 & v. 7
&, u v (2).
2
& & vj0 = v. > g@j0 > u, , , ,
i, ui = g@j0 . ( '
g@j0 v & (2). E f@ > g@j0 v > u v, .
2'
g@j0 6 u, u & & I1 , g@j0 < u gj0 & u. ?
) , u v (2) , (2) | . 1 .
. H &, u1 : : : un 2 F : : : F & & I1 F : : :F +F I2 F : : :F +: : : +
+ F : : : F In & , u1 : : : un & & I1 : : : In , w1 : : : wn > v1 : : : vn
* , ,
i = 1 : : : n, wl = vl l < i wi > vi .
2 '
E-.
2
& w = uv 2 P (n). ( & Iuv , * ), ), :
1) yx, y 2 v x 2 u,
2) u1u2 u3 u4]u5v1, c(u1u2 u3u4u5 ) = c(u), c(v1 ) = c(v),
3) u1v1 v2 v3 v4 ]v5, c(v1 v2 v3 v4v5 ) = c(v), c(u1 ) = c(u).
3. T2(3) \ P (n) Iuv \ P (n).
262
. . . 2
&
f = tt1 t2 t3]r1 r2 r3]r
tt1t2 t3 r1r2r3 r 2 P (n).
2
& w1 = t, w2 = t1t2 t3 , w3 = r1r2r3 , w4 = r. > ,
)
i j = 1 : : : 4, i < j wi v, wj u, f 2 Iuv 1). 7
&, &, ,
l = 1 : : : 4, i < l c(wi ) | c(u) i > l
c(wi ) | c(v). ( &, l > 3. Em
P
f = i tt1 t2 t3]i , i | r1 r2 r3 r i=1
*. > i '
1), i = 0i 00i , c(0i ) c(u)
(00i ) = c(v). ?
) t11 t2 t3]i 2 Iuv . 1 .
4. u v | T (3) w = uv 2
(
n
2 P ). w | Iuv , T2(3) , 3.
. 2 , & w = f,@ f 2 Iuv \ P (n). 2
& V0 | K, *
P (n), , 1), V1 | ,
*
, , 1). E P (n) = V0 V1 .
L, Iuv \ P (n) & '
. E w 2 V1 , w = f@0 , f0 f V0 .
H f0 2 Iuv . 2
f0 f0 =
n
X
i=1
uifi +
m
X
j =1
gj vj * ) i j c(ui) = c(u), c(vj ) = c(v), ui fi | 3), gj vj |
2) * ) 1 2 fi gj c(1 ) = c(v) c(2 ) = c(u). M
, fi 2 T (3) gj 2 T (3) . E
u v | n
m
i=1
j =1
X
X
f 0 = ui fi + gj vj
F T (3) + T (3) F . 2 2 * u v | T (3), ). 1 .
?
) 4. u v 2 ; uv 2 S . uv | (3)
T- T2 .
1. w1 : : :wn : : : S n S(T2(3) ), i 6= j wi wj - .
263
T-
. B ;0 w 2 S, ),
w = uv, u v 2 ;. 2
& D | w = uv 2 ;0, u v 2 ;. B wD , uD vD
, w, u v xi ! 1,
xi 2 D. 1 &, w0 = u0v0 , u0 v0 2 ;, w0 2 ;0 w0 w
(mod ), u0 u (mod ) v0 v (mod ). =
&, w0 w
(mod ), * D w0 wD . 2 , ,
)
u0 v0 , u0 uD , v0 vD u0 v0 = w0 . E u0 v0 2 ;,
u0 = u0 v0 = v0 . ?
) u0 uD , v0 vD u0 u (mod ), v0 v
(mod ). H* '
-
* 4, *
&, &
&
wn =
nY
;1
k=0
x4k+6x4k+3 x4n+5x4n+6x4n+3x2x4 x1
**
* . E .
nY
;1
k=0
x4k+8x4k+5 n 2 N
x 3. #
& * T- I , * * (S(I)= ) . ( NH & 0
{ (. 5]).
2. I | ! T- ! F 1,
! " (S(I)= ) . I
" !! w1x1 x2]w2, w1 w2 | .
. ( ), , & * '
&
: x, y, z, , & .
2
& 0 6= f 2 I \ P (d) . 2 ) ,
)
f1 : : : fN 2 F, * ) g 2 I * i g0, g0 fi g@0 g@.
(3)
0
2
& m = maxfdeg fi : i = 1 : : : N g. 7
f , ),* f xk ! wk = xk x(k;1)m+d+1 : : :xkm+d , k = 1 : : : d,
**
* w1 : : :wd .
H (3) , ,
S(I) m :
1) u,
2) uyv, uv , y & uv.
( xm 2 I F=I &
NH.
264
. . #
. H
h(x y 1 : : : n1 1 : : : n2 ) = uyv ;
n1
X
i=0
uiyhi 2 I \ P (m) (4)
1 : : :n1 x = u, 1 : : :n2 = v, juij = i 6 n1 = juj ; 1 ui | u.
2 un1+1 = u uyv h(x y 1 : : : n1 1 : : : n2 ).
? r & i = 0 : : : n1, '--
hi 0, ,
)
, ;1
. #
r = ;1. M x1 : : : xm x1. E (4) * '--
hi
xm1 2 I, NH *
& &
& F=I.
#
& , r . H jur+1 j 6 juj. 2
ur+1 ur+1 = ur xt. M& xt 6= y. B h0 h0i &
xt ! xm+1 xt h hi , i = 0 : : : r. >
r = 0, h00 = xm+1 h h00i = xm+1 hi , r > 1, h00 h00i
&
r ! r xm+1 h hi ,
i = 1 : : : r ; 1.
rP
;1
H h00 ; h0 = ur (xm+1 yhr ; yh0r ) + uiy(h00i ; h0i ). 2* i=0
x1 : : : xm+1 , y, x, y, ),
I:
xr+1 yxs ; xr yxs+1 s = n1 + n2 ; r + 1 6= 0, '--
hr . 2 i < r h0i h00i &, y. > 6= , F=I &
NH. > = , (5)
xr y x]xs 2 I
jr + sj 6 m ; 1.
H (5) , xp y x]xq 2 I, p = m + 2r, q = m + r + 2s. E char K = 0, (x + z)p y x + z](x + z)q
I. ( ,
pX
;1
qX
;1
xp;i;1zxi y x]xq + xpy x]xj zxq;j ;1 + xpy z]xq 2 I:
i=0
j =0
(6)
2, (6) I.
2 i > r xp;i;1zxiy x]xq 2 I - (5). 2
& &
i < r. . g1 = h(zxi y x]xs x : : : x) g10 =
= xn1 zxi y x]xs+n2 . E i < r, p ; n1 ; i ; 1 > r, &,
xp;n1;i;1 (g1 ; g10 )xq;s;n2 2 I. ?
) xp;n1;i;1 g10 xq;s;n2 =xp;i;1zxi y x]xq 2
2 I.
T-
265
", j > s, xp y x]xj zxq;j ;1 2 I - (5). 2
&
& j < s. . g2 = h(y x] xj zxr x : : : x) g20 = xn1 y x]xj zxr+n2 . E j < s, q ; j ; 1 ; r ; n2 > s xp;n1 (g2 ; g20 )xq;j ;1;r;n2 2 I. ?
) xp;n1g20 xq;j ;1;r;n2 =xpy x]xj zxq;j ;1 2
2 I. E , (6) I, &
xp z y]xq 2 I:
(7)
s
0
n
n
+
1
2
. g3 =h(z y]x x : : : x) g3 =x z y]x s ,
, &, p ; n1 > 2r > r, xp;n1 (g3 ; g30 )xq;n2 ;s 2 I:
(8)
p
;
n
0
q
;
n
;
s
p
q
2'
x 1 g3x 2 = x z y]x 2 I. H * xp1 z y]2 xq 2 I:
(9)
B F0 F 1, ) -
. 2
& & f 2 F, , (7),
(8), (9), f0p+q+1 2 I, f0 | &
'
F0 f.
H NH , ,
t, F0t I , ,
x1 x2]x3x4 x5]x6 : : :x3(t;1)x3(t;1)+1 x3(t;1)+2] 2 I.
2
& wk = wk0 x(k+1)(m+1) xm+k(m+1) * k = 0 : : : t ; 1, wk0 =
= x1+k(m+1) : : : xm;1+k(m+1) . E w0 : : :wt;1 **
* tQ
;1
wk0 x(k+1)(m+1) xm+k(m+1) ] I.
k=0
2 (3) ,
S(I) m :
1) ,
2) uxyv, uxv uyv y & x.
( F=I &
NH. ( uxyv =
t0
P
= iuyvi , c(uxyv) = c(uyvi ) * i. 2
**
x 0
i=1
t
P
v, uxyxk = uyxk+1, = i. > 6= 1, *
&
i=1
F=I &
NH. > = 1, ux y]xk 2 I:
(10)
( ux + z y](x + z)3k I.
( ,
3X
k ;1
ux y]xizx3k;i;1 + uy z]x3k:
(11)
i=0
> i > k, ux y]xizx3k;i;1 2 I (10). > i < k, I 3 ux y]xiz xk]x2k;i;1 = ux y]xizx3k;i;1 ; ux y]xi+kzx2k;i;1, (10)
ux y]xizx3k;i;1 2 I, 2k ; i ; 1 > k. H (11) uy z]x3k 2 I. 0 M*, uy z]x3k 2 I, ** NH, u1y z]u2 2 I \ P (s ) * & s0 . E .
266
. . M
, (S(I)= ) , F=I , (4).
=* ,*, jw1j jw2j > 1.
=
&, & I | T- F , x1 : : :xk xk+1 xk+2]xk+3 : : :xk+3+l x1 x2]x3 x4], k l > 1. 2,
(S(I)= ) m | &* & '
. 7 x2x1x3 : : :xm xm+2 xm+1 **
* x2 x1]x3 : : :xm xm+2 xm+1] I. ( * ,
**
* I, x2x1x3 : : :xm x3 : : :xm xm+2 xm+1 | I. E n xn 2 I, *
* x y]xm;2 = 0
xm;2 x y] = 0. H * . ( W K, *
fxiyxm;i;1 : i = 0 : : : m ; 1g. 2
W0 W , *
fxiyxm;i;1 : i = 1 : : : m ; 1g.
1 &, u1u2 u3]u4 2 W0 u1 u2]u3 u4] 2 W0, &
u1u2 u3u4 2 W , ju1j > k, ju4j > l u1 u2u3 u4 2 W . 7 , &, x y]xm;2 2= W0 . /
, x y]xm;2 2= I,
) (S(I)= ).
"
(. H. 1
*.
! 1] W. Specht. Gesetze in Ringen // Math. Z. | 1950. | Vol. 52, no. 5. | P. 557{589.
2] . . . !"# $#%& '( // '(
'(). | 1987. | * 5. | +. 597{641.
3] . . . /% (, ( 0% ) 0!"% PI-'(%. | 2.. . . )". 3.-. ). | /#), 1981.
4] 4. /. 5%6#. 7 #% % # " T-"' // +89. | 1963. | :. 4,
* 5. | +. 1122{1127.
5] G. Higman. On a conjecture of Nagata // Proc. Cambrige Philos. Soc. | 1956. |
Vol. 52, no. 1. | P. 1{4.
' ( ) 1998 .
; y v
; z w
+ z + x = 4t
; x u
y
. . . . . 511.3
: , .
, $ $
.
x y z t u v w
; x u
y
+
; y v
z
+
; z w
x
= 4 !
"t
Abstract
M. Z. Garaev, On the diophantine equation ; xy u + ; zy v + ; xz w = 4 , Funda-
mentalnaya i prikladnaya matematika, vol. 7 (2001), no. 1, pp. 267{270.
It is proved that the equation
positive integers
.
x y z t u v w
x
; x u
y
+
; y v
z
+
; z w
x
t
= 4 has no solutions in
t
1. 1] , n n2 +3n+9 n ; 3 3k + 2, x+ y + z =n
(1)
y z x
$% & & x y z. ( )% (1) $ n = ;6 (. (. +,, n = 1 | (.. . +. / & n 2 f;1 5g | 2. (. 3.
2] , $ (1) & n, 4& 3.
3] (1) ,& & x y z.
5 , n 2 f4k 8k ; 1 22m+1(2k ; 1) + 3g, k m , , (1) $% ,&
& x y z.
8 , 4% 4 ..
, 2001, 7, , 1, . 267{270.
c 2001 ,
!"
#$ %
268
. . 1. xu + yv + z w = 4t xyz
(2)
x y z t u v w (x y z) = 1.
. n1 n2 n3 | , n = n1 + n2 + n3, xn +yn +z n = 4t xn1 yn2 z n3 x y z t.
2. x u + y v + z w = 4t
(3)
y
z
x
x y z t u v w.
x
2. 1
(, 9 : ;. < , , x y z t u v w , (2) (x y z) = 1.
. , a) uvw
9 b) 9.
8 % a) uvw | 9 . = 4
. ,, u | 9 .
> xu + yv 0 (mod z(4t xy ; z w;1 ))
, v | 9 . @, w | . 9 .
/ , y, z | 9 . < ,, , z | 9
.
A y = 2k y1 , k | , , y1 | 9
, . <, u 9, vw 9, xu = ;yv (mod (4t xy ; z w;1))
% B , v
k t xy1 ; z w;1 ;
y
4
2
1 ;1
2
1 = 4t xy ; z w;1 = ;(;1)
=
y1
;1 = ;1:
1 ;1
2
= ;(;1)
y
y
y
1
< a).
8 , % b) uvw | 9 . + x y z
9& 9 . = ,, z 9, xy 9. A xy = 2k y1 , k | ,, y1 |
9 . 5 xu ;yv (mod z(4t xy ; z w;1))
; x u
y
+
; y v
z
+
; z w
x
= 4t
269
a) , w;1 k
;1
= ;1:
1 = 4t xy;;xyz w;1 = ; 4 2k ty2 y;1 z w;1 = ;(;1) 12 ;zy
1
1
< b).
5 1 .
y
x
3. 2
(, )% . 9 : ;. < ,
, x y z t u v w , (3). A,
. , (x y z) = 1.
A
(y z) = d1 (z x) = d2 (x y) = d3:
5 (x y z) = 1 , 4 , x1 y1 z1, x = d2d3x1 y = d1d3y1 z = d1 d2z1 :
(4)
< )
(d2x1 d1y1 ) = (d3x1 d1z1 ) = (d3 y1 d2z1 ) = 1:
(5)
< (5) (3) 9
d2x1 u + d3y1 v + d1z1 w = 4t:
(6)
d1y1
d2z1
d3 x1
B, (d2 z1 )v (d3x1)w 0 (mod du1 )
(d1 y1 )u (d3x1)w 0 (mod z1v ):
+ 9 (5) z1v 0 (mod du1 ) du1 0 (mod z1v )
du1 = z1v :
@
dw3 = y1u dv2 = xw1 :
270
. . < $ , 4 , X Y Z, x1 = X ( ) d2 = X ( ) y1 = Y ( ) d3 = Y ( ) z1 = Z ( ) d1 = Z ( ) :
< (6), ,
+ +
+ +
+ +
X ( ) + Y ( ) + Z ( ) = 4t X ( ) Y ( ) Z ( ) :
/ , (5) , (X Y Z) = 1. 3 $ % 1, 2.
. > , 2 , x u + y v + z w = 2t
y
z
x
uv
vw wu
vw
uv
vw wu
wu
v
vw
w
vw
w
wu
u
wu
u
uv
v
uv
uv
vw wu
uv
vw
vw
wu
wu
uv
uv
x y z t u v w | , , 9 u 6 v 6 w, u = v = 1.
D ) w = 1, ) , $%
,& & x y z t (. 3]). < & 49 & w $? A ) .
. , F . G. H 4, .
1] Erik Dofs. On some classes of homogeneous ternary cubic Diophantine equations //
Ark. Mat. | 1975. | V. 13. | P.
29{72.
P; 2] Maurice Craig. Integer values of xyz2 // J. Number Theory. | 1978. | V. 10. |
P. 62{63.
3] . . . ! "# // $% &'(. |
1997. | $. 218. | ). 99{108.
& ' ( 1998 .
. . 517.5
: , , , , .
! "
# "! #
" $%
, &# .
Abstract
V. V. Dubrovskii, On the asymptotics of spectral function for ordinary dierential selfadjoint operator, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001),
no. 1, pp. 271{274.
In this paper the author investigated the asymptotics of spectral function for
ordinary di-erential selfadjoint operator de.ned by regular boundary conditions.
+ , !0 1].
& i | , !0 1], i ( ) i( ) | 2 !0 1] , ,- i , . .
| -
,, i = i i , ( i j ) = ij )
-
, !0 1] , ( )) i | + , ( ) i( ) | 2!0 1] + , ,- i , . . ( + ) i = i i ,
R1
( i j ) = i /j = ij .
0
0
, (. !1]) 02 ; 4 1 ;1 6= 0
j i ; i;1 j > const n;1 0)
j i ( )j 6 const 8 2 !0 ] = 1 1
T
P
n
v
L
Tv
x
T
v
v v
P
T
u
T
u u
p x
x
L
P
u u dx
P
T
v
x
i
x
i
>
:
, 2001, 7, / 1, . 271{274.
c 2001 ,
!"
#$
%
P u
u
272
. . 0 ( N
X
i (x)ui(y) ;
N
X
u
i=1
i (x)vi (y):
v
i=1
0, - X
i ( ) i( )
u
i 6
x u
x
(1), (1) , T (
) T +P (
) + . 0
, o
K
T
T
K
O
x y K
x y P
T +P (x y )
=
T (x y ) ;
Z1
K
K
T (x z )p(z )KT +P (z y ) dz
(1)
0
K
T (x y )
T +P (
K
=
1 v (x)v (y)
X
i
i
)=
x y i;
i=1
1 u (x)u (y)
X
i
i
i=1
i;
, ( 1 (0 1). & (1) ;1
N = f j j j = ( N + N +1 )2 g
( (1) , ;N = f j Re = ( N + N +1 )2;1g
( , (1) j j;2) , L
K
N
X
i(x)ui (y)
u
i=1
=
N
X
1 Z
i( ) i( ) + p
2 ;1
v
i=1
x v
y
Z1
K
T (x z )p(z )KT +P (z y ) dz d:
;N 0
5, 6
(. !2]), , 1
1 Z Z
p
KT (x z )p(z )KT +P (z y ) d dz =
2 ;1
p
0 ;N
Z1 Z
p 1=p
1
= 2 KT (x z )p(z )KT +P (z y ) d dz 6
0 ;N
273
6 21
Z Z1
j T(
K
;N
T +P (
) jp
1=p
z y dz
0
Z
d
1=p
k(T ; E );1 kp k(T + P ; E );1 kp d
6 const
1
;N
Z1
6 const
1
)()
x z p z K
;1
(j j +
d
1=p
(1)
;1)2"p
o
n;1)(1;")2p;p
N(
Ci
6
6
= ( ;n+2 )
o N
= Im
0, .
1. 02 ; 4 1 ;1 6= 0 > 2 > " >
p.
1
1)
2)
N
X
u
i=1
n
i (x)ui (y)
=
i (x)ui (y)
=
u
i=1
N
X
N
X
v
i (x)vi (y) + o(N
v
i (x)vi (y) +
i=1
N
X
i=1
;n+2))
+ 1 ( ) + 2 ( ) + + k ( ) + ( ;(n;1)(k+1)+1)
1( ) 2( )
k( ) | , (1) .
8 !6] 2. N N
N : : : N
X
i=1
N
:::
N
o N
N
i=
N
X
i+
N
X
i=1
(
i
i) + o(N
pv v
i=1
;n+2 ):
3. & -
!6].
4. : !3{5] 6
!2] .
1] . . . | .: , 1969.
2] ! ". "., # $. %., & ". #'#. | .: "((, 1948.
3] $,#' -. -. ' ' ,#. .
// $ #1. | 1990. | 3. 26,
4 3. | 5. 533{534.
274
. . lp M
4] $,#' -. -. 8 ' ' 9 # #
5. 69{75.
(
) // $ #1. | 1992. | 3. 28, 4 1. |
5] $,#' -. -., &=# . 5. 8 ' ' ''1>9 '#9 # // $ #1. | 1993. | 3. 29, 4 5. | 5. 852{858.
6] "'# . ". ? ' @' ','#9 @= #9 ''1>9
# // $ 555A. | 1963. | 3. 150, 4 6. | 5. 1202{1205.
& ' 1997 .
0 2 ]
L2
. . . . . 517.51
: , .
, !" $!" 2 %0 2 ].
L
Abstract
M. G. Esmaganbetov, Minimization of exact constants in Jackson type inequalities and diameters of functions belonging to L2 %0 2], Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 1, pp. 275{280.
We obtain a series of results related to minimization of exact constans in Jackson
type inequalities as well as the diameters of functions belonging to 2 %0 2 ].
L
L2 | 2- 12
Z2
1
2
kf k2 = jf(x)j dx
0
" #
1
X
f k cos(kx + ') (0 0):
k=1
& L2 ' , ) "
*+"
f () 2 L2 (L02 L2 f 0 f):
,+ Sn;1 (f () - x) | ( > 0)-
" "
n ; 1 "
#
f () (x), , . , +. +' f () 2 L2 +
Tn . n ; 1
kf () (x) ; Tn (x)k =
En;1(f () ) = E(f () - K2Tn;1) = inf
Tn
21
1
X
(
)
(
)
2
2
= kf (x) ; Sn;1 (f - x)k =
k k (1)
k=n
, 2001, 7, - 1, . 275{280.
c 2001 !"#,
$%
&' (
276
. . K2Tn;1 | (2n ; 1)- +
L2 .
N- 2+, + +
H L2 3 +3) +:
dN dN (H L2 ) = inf sup inf kf ; uk
KN f 2H u2KN
N N (H L2) = inf
inf
sup kf ; Af k
N N (H L2) = inf
inf
sup kf ; Af k
KN A2L(L2 KN ) f 2H
KN A2L1 (L2 KN ) f 2H
L(L2 KN ) | ' + L2 N- KN , L1 (L2 KN ) | ' L(L2 KN ), . .
+
A + " KN , Af = f
+" f 2 KN .
X
1
r
r
k
!r (f- ) = sup k5h f(x)k = sup (;1) k f(x + (r ; k)h) |
06h6
06h6 k=0
+ +
"
r > 0 f(x).
2
2
(
)
r
& Wr (f - ) !rr (f () - t) 60 ] cos 2t , +"" 0 R
1r
(
)
r
B 0 !r (f - t) cos 2 t dt CC
Wr (f () - ) = B
B@ R
CA :
cos t dt
2
2
0
2
Hr(!(
)) | ' f 2 L(2) , Wr (f () - ) 6 !(
)
8. 9. & 61] ++ 3 ' En;1(f) !1(f () - t) 60 n ] sin nt. :) .+ ; ' . *. <
62,3], 8. >++
64], *. *. @
+
65], B. C. ,
66] .
c(r) | +'+
" , ")
" r. <
c(r)!r (f () - ) 6 Wr (f () - ) 6 !r (f () - )
+
Hr() +
" +
HD r(!) = f!r (f () - ) 6 !(
)g
+ " + . +
Hr(!), E Wr (f () - ) , -, + +" +.
+' .
277
...
F
+ +
HD r(!) +
G. H. G '
+' 1975 . . 2
+. + E 62{6] ++ +" '
!(
) + + +
+" . 8
+ " +
Hr (!(
)) ++ + '
!(
).
2 , +" 8. . 2 67], .+ +
KNT ) XrN (L2 L2 KN ) = sup WE(f
(2)
f 2L2 f 6=const r (f () - )
N, . . ++ " + (2) KN L2 N (8r > 1,
> 0, N = 1 2 3 : : :):
KNT ) XrN (L2 L2) = inf sup WE(f
(3)
(
KN f 2L2 f 6=const r (f ) - )
, +
(3) 3 +
Hr (!(
)) 0 < !(
) < 1. :, r = 1, = 0 1 2 : ::, 0 < 6 n , N = 2n ; 1,
n = 1 2 : : : (2) .
64].
1. r > 1, > 0, 0 < 6 n , N = 2n ; 1 N = 2n
( sin n )2 ; 2(
n)2 r2
1
!(
):
(4)
dN = N = N = n 4 2 2; 4(
n)2
Sn;1 (f- x).
. <
r > 1, , "" Zb X
n
a
0 < p 6 1, Z X
1
0
>
k=n
1
X
k=n
k=m
p p X
n Zb
juk (x)j dx
1
>
k=m a
juk jp dx
2r
r
2 sin kt2 k22k cos t
dt
2
>
k22k
p
1
1
Z
0
2 r X
Z
r
1
2 sin kt2 cos 2
t dt = k22k 2(1 ; cos kt) cos 2
t dt :
k=n
K
+, 64], 0 < 6 n "
Z
t dt
'(y) = cos yt cos 2
0
0
278
. . y. E + 2r
kh
2 sin 2 k22k
1
X
k5rh f () k2 =
k=1
, Z
(5)
r X
Z
r
1
2
2
! (f - t) cos 2
t dt > k k 2(1 cos nt) cos 2
t dt :
k=n
0
0
:3
(1) +
2
rr
()
3r
2 R
(
)
r
6 !r (f - t) cos 2 t dt 77
kf ; Sn;1(f)k 6 n1 664 0R ;
75 2
nt
cos t dt
2 sin
2
2
0
2
; +"" +"
f(x) = cos nx:
<
Sn;1 (Tn;1(x)) = Tn;1(x) +" Tn;1 (x) 2 KnT;1 Sn;1 (f- x) 2 K2Tn;1 +"
+3 f 2 L2 , +" +
Hr(!(
)) 2 R
3r
66 0 cos 2 t dt 77
!(
)
2n 6 2n;1 6 sup
kf ; Sn;1(f)k 6 n 64 R ;
75 : (6)
f 2Lr
2 sin nt2 2 cos 2 t dt
2
2
0
* + dN 6 N 6 N (6) + +" :
2
R cos t dt 3 r
77
66
2
0
dN 6 N 6 N 6 !(
)
75 :
6
n 4 R ; nt 2
2 sin 2 cos 2 t dt
2
(7)
0
& , .
2 R
3r
66 0 cos 2 t dt 77 T
R = !(
)
75 U2n+1 =
n 64 R ; nt 2
2 sin cos 2 t dt
0
8
2 R
3r 9
>
>
>
>
t
dt
cos
n
<
=
6
7
2
X
!(
)
6
7
0
= >Tn(x) = ck cos(kx + 'k ): kTn k 6 n 64 R ;
7
5>
k=1
>
2 sin nt 2 cos 2 t dt >
:
2
2
0
279
...
(2n + 1)- K2Tn+1 - +.
', Tn 2 Hr (!(
)). * + (7) 0 6 kt2 6 nt2 6 2
2 R
3r
()
r
66 0 !r (Tn - t) cos 2 t dt 77
64
R cos t dt 75 =
2
3r
2 0
Pn ;
r
R
2
r
2 sin kh2 k2c2k cos 2 t dt 77
66 06sup
h
6
t
k
=1
0
77 6
= 66
R cos d dt
4
5
2
0
2 R ;
3r
2
nt
r
6 n 2 sin 2 cos 2 t dt 77
6 664 0 R
75 kTnk 6 !(
):
cos d dt
2
2
1
2
2
2
0
2
E R Hr(!(
)). <
.
(. 67, . 347])
+
3 r2
2 R
66 0 2 cos 2 t dt 77
!(
)
d2n;1 > d2n > n 64 R ;
75 :
2 sin nt cos 2 d dt
0
:3
+ (7) + 1.
2. > 0, r > 0, N = 1 2 3 : ::, 0 < ! 6 1. XrN (L2 L2) = dN (Hr(!(
))- L2 ):
(8)
(
)
.
L2 Wr (f - ) = u > 0. ,+ f1 (x) =
()
;
1
= u !(
)f(x), Wr (f1 - ) = !(
), . . f1 2 Hr (!(
)). M
"
+'+3 + E(f- KN ) Wr (f () - ) KN ) 6 sup E(f- K ):
sup WE(fN
(
(f
f 2L2 r ) - ) f 2Hr
" ' " KN L2 N, + Xrn (L2 L2 ) 6 dN (Hr (!(
))- L2 ):
(9)
280
. . G , +" +3 f 2 Hr(!(
)) + +" +
Hr(!(
))
KN ) E(f- KN ) 6 WE(f(
(f ) - )
r
E +" ' KN , 3
+
, (9). < (8) . 9 1 2 +
. r > 1, > 0, 0 < 6 n , 0 6 !(
) 6 1, N = 2n ; 1
N = 2n ; n 2 ; 2(
n)2 ; r2
XrN (L2 L2) = n1 4 sin2 2; 4(
n)2
!(
):
1] . . " L2 0 2] // &. '. | 1967. | ,. 2, . 5. |
0. 513{522.
2] ," 3. 4. " " L2 // &. '. | 1977. | ,. 22, . 4. | 0. 535{542.
3] ," 3. 4. 0 " ' L2 //
&. '. | 1979. | ,. 25, . 2. | 0. 217{223.
4] 6" . " " L2
// 7 8 . 0 ". | 9
'
" -
" ",
1986. | 0. 3{10.
5] ;" 4. 4. " L2 " , < < " " <" // =. . . |
1991. | ,. 43, . 1. | 0. 125{129.
6] > " &. ?. " L2 " //
&< < @A8 ", < <, 'B, "<C< 90-D 0. &. 8. ,' ". | &., 1995. | 0. 124{125.
7] 9 . 7. , " <. | &.: , 1987.
) * + 1997 .
t- . . ,
. 517.986
: p- , p-, f -, f -.
! t- , "# $ f -, #%# $& '( ) $ 1. *+% & $$ $$& (f -) ")% $ & $.$ + ! t- .
Abstract
L. B. Luchishina, The properties of the t-adic integers, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 1, pp. 281{284.
In the article we introduce the t-adic algebra depending on certain f -type, generalizing usual Baer type for groups of rank 1. We consider the characteristics of
invariants of the constructed algebra (f -characteristics) and study some characteristics of the divisibility relations of the t-adic integers.
^ p, Q^ p Z, Zpm , Q, Z
, pm , , p- , p- , p | , m |
^ px], Q^ px] . Zx], Qx], Zpm x], Z
! " " , , pm , p- , p- .
f 2 Qx] . # Q^ px] f = r1 r2 : : : rs ,
% rj (j = 1 s) , . )
rj Ip (f ), ! , SIp (f ) =
= fr 2 Q^ px] j rjf r g 1]. I (f ) = Ip (f ).
p
# 1] ** f - f -
, ++ ,- * ! ! 1.
1. " fkr j r 2 I (f )g 1 * f -", kr = 1 *
^ px]. f - * (kr ) 1].
*! r 2= Z
, 2001, + 7, 3 1, . 281{284.
c 2001 !,
"#
$% &
282
. . 2. 1 (kr ) (lr ) * -, * kr = 1
! !, ! lr = 1 1].
3. 2 - (kr )] f - (kr ) * f - 1]. f - t u : : :.
# 2] * ! - * *! ,- . ! ! Qft t- * *! f - t. 3
" f - t = (mr )], * ! ! Zft =
Q
^ px] + (r))=(r), mr = 1, Kr = Z
^ px]=(pmr r),
=
Kr , ! Kr = (Z
r2I (f )
mr < 1.
5
* * r = x + (r), , ! , r(r ) = 0,
Kr ^ pr ] mr = 1 r 2 Z
^ px].
Z
^ px].
Q^ pr ] mr = 1 r 2= Z
Zpmr r ] mr < 1:
7* 8, * p ! r 2 I (f )
p- -88, B = fr 2 I (f ) j r 2
^ px] n Z
^ px]g.
2Q
g = (gr )r2I (f ) 2 Zft = Q Kr . 9! r- gr r2I (f )
gr = a0 + a1 r + : : : + an;1nr ;1
(1)
! n = deg r(x), -88 ai (i = 1 n) | p- mr = 1 - Zpmr , mr < 1.
4. gr | r- - g = (gr )r2I(f ) p |
, + r. :;" kr , " pkr
-88 a0 a1 : : : an;1 (1) % r-"
- g.
5. : (kr ), +" r- ! - g 2 Zft * ! r 2 I (f ), " - g 2 Zft H (g) = (kr ).
1. < H (g) f -.
^ px], Z
^ pr ] = Q^ pr ] *** ,
1", r 2 Q^ px] n Z
- *
kr = 1.
B = fr1 r2 : : : rsg, ! ! Kr1 Kr2 : : : Krs Zft , " Btf = L Kr .
2. g
, H (g) = (mr ).
2 Zft ,
r2B
t = (mr )]. g 2 Btf t- 283
. g 2 Btf g = (gr ). 9! r 2 B ^ pr ] *** kr (gr ) = 1. r 2= B gr = 0 Z
kr = mr . 9 , H (g) - g
f -" (mr ).
), H (g) = (mr ) g = (gr )r2I (f ) , kr (gr ) = mr * !
r 2 I (f ). 1 , gr = 0 * ! r 2= B . 1",
r 2= B , - , -88 ! gr **
pmr , , mr < 1?, ! p,
, mr = 1. 9 , -88 gr = 0.
3. g h 2 Zft, t = (mr )]. H (g h) =
= (H (g) + H (h)) ^ (mr ).
. g = (gr )r2I(f ) , h = (hr )r2I(f ) . )
H (g) =
= (kr ), H (h) = (lr ) H (gh) = (sr ). ?8 ! r = r(x) 2 I (f ).
, mr = 1, ! kr lr ! ( 1). @ kr = lr = 1, gr hr 2 Q^ pr ] gr = hr = 0. 9!
A sr = kr + lr . @ kr < 1, lr = 1, sr = 1, ^ pr ] sr = kr + lr . @ kr < 1, lr < 1, gr hr 2 Z
A sr = kr + lr . , , mr < 1. # - kr < mr , lr < mr . - gr = pkr gr0 hr = plr h0r , ! gr0 h0r |
!. 9!
( kr +lr 0 0
gr hr kr + lr < mr gr hr = p
0
kr + lr > mr :
* , H (g h) = (H (g) + H (h)) ^ (mr ).
4. g 2 Zft. g = (gr )r2I(f )nB Zft
, H (g) = 0. g = (gr )r2B g 6= 0, g Zft .
. ), H (g) = 0 ! !, !
r- - g = (gr )r2I (f )nB " !. @
gr = a0 + a1 r + : : :+ an;1 nr ;1 | " ! Kr , +
ur = b0 + b1r + : : : + bn;1nr ;1, " ur gr = 1 Kr . 1", ! g r Q^ px] ! !, ! +
^ px], u1 g + v1 r = 1. 1 u1 v1 2 Q
^ px]
*+ ; p, Z
u g + v r = p :
(2)
p, uB gB + vB rB = 0 Zpx], % rB * r. 9! rBjuBgB , , rBjuB rBjgB. # ^ px]. 5
" ! g r rBjuB, u = rs + pl, ! r s l 2 Z
* " * (2), (rs + pl) g + v r = p ,
284
. . r(s g + v) + p l g = p . r , r s0 + l g = p ;1, ! ! .
^ px] , * r-, ur gr = 1 Kr ,
9! u g + v r = 1 Z
g = (gr )r2I (f )nB . @ g = (gr )r2B g =
6 0, g Zft
.
5. g h 2 Zft, t = (mr )]. g h , H (g) 6 H (h).
. @ g h, h = g u, ! H (h) = H (g u) =
= (H (g) + H (u)) ^ (mr ), , , H (g) 6 H (h).
), H (g) = H (h), H (g) = (lr ) = (kr ) = H (h), kr = lr
* r 2 I (f ).
r- ! * ! p ! !: gr = pkr gr0 ,
hr = plr h0r . 9! hr = plr h0r = pkr h0r = gr (gr0 );1 h0r . ? pkr = gr (gr0 );1
! gr0 .
@ H (g) < H (h), kr < lr * r 2 I (f ). 9! hr = plr h0r =
= pkr +sr h0r = pkr h0r psr = gr (gr0 );1 h0r psr .
# * ! h * ! g.
5
* 5 ** + *.
1. t- g1 g2 : : : gs
1
! d, "# H (d) = H (g ) ^ H (g2) ^ : : : ^
^ H (gs ).
2. t- g1 g2 : : : gs
1
! v, "# H (v) = H (g ) _ H (g2 ) _ : : : _
_ H (gs ).
1] A. Fomin, O. Mutzbauer. Torsion-free abelian -irredusible groups of nite rank //
Comm. Alg. | 1994. | Vol. 22. | P. 3741{3754.
2] . . . ! -"!# $ %& // " $". | 1989. | (. 28, ) 1. |*. 83{104.
' ( 1999 .
. . 514.762
: , , !" .
# $ %& & '&( %) ) $! , !" , '&(
( )& ( !" & * $! ! " 4.
Abstract
Yu. F. Pastukhov, Necessary conditions in the inverse variational problem, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 1, pp. 285{288.
In the paper the inverse variational problem in strati0ed speed spaces of arbitrary order is invariantly formulated, Lagrange cut is de0ned, and an analytic
reformulation of the problem is given. We give a necessary condition of a system of
ODE of even order not less than 4 to be a Lagrangian system.
Tk Xm | k Xm , kl : Tl Xm Tk Xm , l > k > 0, | (
k = 0 | | Xm ). , Xm | . U (v0 2xn;1) | v0 2xn;1 T2n;1Xm .
1. #
$ f : TkXm Tl Xm (0 6 k < l)
!
!
% , &' :
f
- Tl Xm
Tk Xm
Qid Qs
+ l
k
Tk Xm
2. f T2nXm ,
f = f (U ) = fvx2n 2 T2nXm j vx2n = f (vx2n;1 ) vx2n;1 2 U (v0 2xn;1)g
% () , $% f : T2n;1Xm U (v0 x2n;1) ! T2nXm :
, 2001, & 7, 1 1, . 285{288.
c 2001 ,
! "#
!!
$
286
. . 3. "L = "L (U ) = fvx2n 2 T2nXm j "(x)L(vx2n ) = 0 2 Rmg T2nXm L : n2n;1U ! R | %
$ +, L(x x_ : : : x(n)) | (x), % $% U (v0 2xn;1) T2n;1Xm , , -
&, % % $% "L T2n Xm .
. "(x)L : 22nn;1U ! R | +, (x) Xm T2nXm n
(n) X
"(x) L (x x_ : : : x(2n)) = (;1)k Dkt @L(x@x: :(k: ) x ) i = 1 m:
k=0
i
i
0
11], %
%3 (x) Xm T2nXm .
4 &' 11,2]: f : T2n;1Xm U (v0 2xn;1) ! T2nXm |
. 4'& U~ (v0 2xn;1) U (v02xn;1) %
$
+ L : n2n;1U~ ! R, f (U~ ) = "L (U~ ).
4.n #
+ L : Tn Xm ! R % %
n
$
vx 2 T Xm , (x) Xm 2L
det @x(n@) @x
(x x_ : : : x(n)) 6= 0
(n)
L(x x_ : : : x(n)) | + L : Tn Xm ! R (x).
0
11], %
%3 Xm .
( ). f : T2n;1Xm U (v0 2xn;1) ! T2nXm |
, U~ (v0 2xn;1) U (v0 2xn;1) | v0 2xn;1 2
2 T2n;1Xm , L : n2n;1U~ ! R | . f (U~ ) = "L (U~ )
, "(x)L f jU~ (v0 2 ;1 ) : T2n;1Xm U (v0 2xn;1) ! Rm
0 2 Rm U~ (v0 2xn;1).
k
n
x
i
287
5. f : T2n;1Xm U (v02xn;12)n;!1 T22nn;2n1;X1m | -
, v0x 2 T Xm . 4 f
% $% v0 2xn;1, ' U~ (v0 2xn;1) U (v0 2xn;1) %
$ + L : n2n;1U~ ! R, f (U~ ) = "L (U~ ).
7, +
%8 (
%3 3 ) %3 3 '
%
$
% 0{:$, 8
& 83 %3, $
;<=
&.
>
% : '& $% %
%
%3 ++%3 ? ;%, n > 1 ' % 3
% .
. x(2n) = fi(xk0 x_ k1 : : : x(2n;1)k2 ;1 ), i kl = 1 m, |
(x) Xm f : T2n;1Xm U (v0 2xn;1) ! T2nXm n > 1
! U~ (v0 2xn;1) U (v02xn;1) (U~ (v02xn;1) '), ' : U~ ! R2mn | '(U~ (v02xn;1)) = U~(x) (x0 : : : x(20 n;1)) R2mn
'(vx2n;1 ) = (xi0 x_ i1 : : : x(2n;1)i2 ;1 ) il = 1 m l = 0 2n ; 1:
n
X
X
:::j (x : : : x(n))x(k1) j1 : : : x(k ) j fi (x x_ : : : x2n;1) =
Ckij11:::k
n
i
n
r=1 n+16k1 :::k
r 6rn;1
n
r(n+1)6 P ki 6(r+1)n
r
r
r
r
i=1
:::jr (x x
Ckij11:::k
_ : : : x(n)) | n2n;1(U~(x) ),
r
2
n
;
1
2
mn
n : R
Rm(n+1) | :
n2n;1(xk0 x_ k1 : : : x(2n;1)k2n;1 ) = (xk0 x_ k1 : : : x(n)kn ):
!
. . fi (x : : : x2n;1) = ('0 ij (x : : : x(n)) + '1 ijp(x : : : x(n))x(n+1) p ) x(2n;1)j +
+ gi(x : : : x(2n;2))
'0 ij (x : : : x(n)) '1 ijp(x : : : x(n)), i j p = 1 m, | n2n;1U~ (x), n2n;1 : R2mn ! Rm(n+1) | ,
gi(x : : : x(2n;2)) | 22nn;;21U~(x) ,
288
22nn;;21 :
. . R2mn ! Rm(2n;1) | :
22nn;;21(xk0 : : : x(2n;1)k2 ;1 ) = (xk0 : : : x(2n;2)k2 ;2 ):
n
n
1] . . . . | !. "#$, & 1328-"-96.
2] . -. . / 0012 2. | 3.: 3,
1989.
% ! & 1998 .
-
. . 517.929
: - , , !
", # , $% , % &'.
-" '"# # ' %(
# ' !
".
Abstract
L. E. Rossovskii, Strongly elliptic dierence-dierential operators in semibounded cylinder, Fundamentalnaya i prikladnaya matematika, vol. 7 (2001), no. 1,
pp. 289{293.
In the paper we consider di/erence-di/erential operator in semibounded cylinder
and obtain necessary and su0cient conditions for a G1arding-type inequality using
a symbol of the operator.
- X
AR =
DR D
jjjj6m
Q = f(x1 x0) 2 Rn : x1 > 0 x0 2 Gg
G ; Rn;1
| P
( @G 2 C 1 n > 3),
1 ; 1 @ n
1
@
D = i @x1 : : : i @xn , R u(x) =
aj u(x1 + j x0), aj 2 C . %0
j
j
j
6
J
u(x1 + j x ) = 0 x1 + j 6 0.
&' AR () Q* , + u 2 C_ 1 (Q)
(1)
Re(AR u u)L2(Q) > c1kuk2W m (Q) ; c2kuk2L2 (Q) 2% 34 24" '
#( ,
" 5 95{01{00247.
, 2001, # 7, 5 1, . 289{293.
c 2001 ,
!"
#$ %
290
. . c1 > 0, c2 u. 0 W n (Q) 1 ), 23+ L2 (Q) P 3'
) m )4, kuk2W m (Q) =
kD uk2L2(Q) .
jj6m
5
- + + 6. 7. 1) ) 81]. : )) +, )
(, 3
+ . ; + ) , . ; , (0 d) G + , d 2= Z.
% + ) + ( )
- (. )2 82], 3 + < )( )-
+ 2 2 '
).
1. % R : L2 (Rn) ! L2 (Rn) :
X
Ru(x) =
aj u(x1 + j x0) aj 2 C :
jj j6J
(2)
= >k = f(x1 x0) 2 Rn : k ; 1 < x1 < kg. ? ) S
N
u 2 L2
>k (N = 1 2 : : :) 2 ' Rn, k=1
(2) )) ' S
S
N
N
R : L2
>k ! L2
>k .
k=1
k=1
;' UN : L2
S
N
k=1
>k ! LN2 (>1 ), (UN u)k (x) = u(x1 + k ; 1 x ) (x 2 >1 @ k = 1 : : : N ):
= RN ) N N (
(
(RN )km = am;k jm ; kj 6 J
0 jm ; kj > J:
0
S
N
S
N
(3)
A R : L2
>k ! L2
>k () k=1
k=1
2 RN )-
) LN2 (>1 ):
R = UN;1RN UN :
(4)
-
5, (2) (3)
(UN Ru)k =
=
X
jj j6J
N
X
aj u(x1 + j + k ; 1 x0) =
X
jm;kj6J
m=1:::N
291
am;k u(x1 + m ; 1 x0) =
(RN )km (UN u)m (x) = (RN UN u)k (x) (x 2 >1@ k = 1 : : : N ):
m=1
1. RN + RN (RN ) N = 1 2 : : :.
(R + R ): L2 (Rn) ! L2 (Rn) .
. ;' 4 4 )4 u 2 L2(Rn0).
5 )+ h 2 R N ) uh , uh (x) = u(x1 + h x ),
S
N
>k . % (4), 2 L2
k=1
Re(Ru u)L2(
Rn = Re(Ruh uh)L
)
2
;S
N
k=1
k
=
= Re(UN Ruh UN uh )LN2 (1 ) = 21 ((RN + RN )UN uh UN uh )LN2 (1 ) >
> ckUN uh k2LN2 (1) = ckuk2 ; SN = ckuk2L2( n) L2
k=1
k
R
c > 0 N = 1 2 : : :.
; L2 (Rn) 2 + ) 4 u 2 L2(Rn). 7 ).
%+ ) 4 C, Re(Ru u)L2( n) = (Re r( )~u u~)L2 ( n) P
r( ) = r(1 ) =
aj eij1 | . =,
R
R
jj j6J
2 ' (R + R ): L2 (Rn) ! L2 (Rn) () 2 . =4 1 2. + RN P RNij
aj e 1 > 0 (1 2 R).
N = 1 2 : : :. Re
jj j6J
2. !
. AR ! Q* ,
292
. . Re
X
X
jjjj=m jj j6J
aj eij + > 0 ( 2 R 0 6= 2 Rn):
. & +. = Qk = Q \ >k )
N
S
N 4 )4 u 2 C_ 1
Qk . A
k=1
V = Un u | )-
) C_ 1N (Q1 ). E (4), X
Re(AR u u)L2(Q) = Re
(R D u Du) SN
=
= Re
= Re
k )
(
jjjj6m
X
k=1
jjjj6m
(UN R D u UN D u)LN2 (1 ) =
X
jjjj6m
(RN D V DV )LN2 (1 ) = Re
X
(RN D V DV )LN2 (Q1 ) :
jjjj6m
6, kukL2(Q) = kV kLN2 (Q1 ) , kukW m (Q) = kV k2W mN (Q1 ) . A) ,
) (1) X
Re
(RN D V D V )LN2 (Q1 ) > c1 kV k2W mN (Q1 ) ; c2 kV k2LN2 (Q1 ) 2
2
2
jjjj6m
) c1 > 0, c2 N V 2 C_ 1N (Q1).
% P
4 ( Q* 1 + D RN D , ) (.
jjjj6m
P
81{3]), (RN + RN ) + 0 6= 2 Rn 2
jjjj=m
), 3 N . ; 2
X
Re r () + > 0 ( 2 R 0 6= 2 Rn):
jjjj=m
5. % u 2 C_ 1 (Q). E %<, Re
X
jjjj=m
DR D u u
% 4 X
k1 > 0.
jjjj=m
L2 (Q)
X
=
Rn :
(Re r(1 ) u~ u~)L2 (
jjjj=m
Re r () + > k1jxij2m ( 2 R 2 Rn)
; 2 = 1 . A
X
jjjj=m
Re r(1 ) + u~ u~
R > k kjj u~kL Rn :
L2 ( n)
1
m
2
2(
)
)
-
293
1 %<, )2 ()+ m
+ W (Q), Re
X
jjjj=m
D R D u u
L2 (Q)
> k1kuk2W m (Q) :
=) <+ '
Re(AR u u)L2(Q) > k1 kuk2W m (Q) ; k2kukW m (Q) kukW m;1 (Q) :
=4 + tkukW m;1 (Q) 6 c(kukW m (Q) + tm kukL2(Q) )
p1p2 6 t;m p21 + tm p22 (p1 p2 2 R@ t > 0)
, Re(AR u u)L2(Q) > (k1 ; k3t;1 )kuk2W m (Q) ; k4 t2m;1kuk2L2(Q) :
; t > 0 ), (k1 ; k3t;1 ) > 0, (1).
A ).
; - | ) , | +
+: 2 Rn, 43 , 2 R, 43 (. 82]). E ), 2 ) , ) 4
, 2') ( ) = 1 ), ) + (.
"
1] Skubachevskii A. L. The rst boundary value problem for strongly elliptic dierential-dierence equations // J. Dierential Equations. | 1986. | Vol. 63, no. 3. |
P. 332{361.
2] . . !"#$ -%!!&&#$ '( "#& // )#. *#+&. | 1996. | ,. 59, . 1. | /. 103{113.
3] 12 ). 3. 4 $ $$56&( &+#( %!!&&#$ '( "#& //
)#&+. 7. | 1951. | ,. 29, . 3. | /. 615{676.
& ' 1998 .
,
. . 517.9
: , , , ,
.
!!"
$!!", $ $!!"
%
&'.
Abstract
Y. T. Silchenko, Linear dierential equation with non-densely dened operator
coecient, generating a non-analytical semigroup, Fundamentalnaya i prikladnaya
matematika, vol. 7 (2001), no. 1, pp. 295{300.
The solvability of the Cauchy problem for the linear di.erential equation with
operator coe/cient is established when this coe/cient is not densely de0ned and
generates a semigroup with a singularity.
1. (1)
v0 + Av = f(t) (0 < t 6 1) v(0) = v0
"# E. % A | ' ' "( D(A) = D, (* ' "' A;1 , f(t) | '
t > 0 E
v0 | ' + E. (1) '
'
,0 1] , v0 (t) Av(t) *( ''
(0 1] '' (1). , ,1], "' ' ' ' "
.
% (1) '# '# . . , t > 0 * -
U(t) = exp(;tA), "(*
1
&
1223 (
4 01{01{00408).
, 2001, 7, 4 1, . 295{300.
c 2001 ! "#$,
%&
'( )
296
. . 1) U(t) | ' ' E D (t > 0)2
2) U(t + ) = U(t)U() (t > 0)2
3) t!+0
lim U(t)v = v v 2 D2
4) U(t) U 0(t) = ;AU(t)2
5) U(t) A D2
6) '' kU(t)k 6 Mt; exp(;!t) kU 0(t)k 6 Mt; exp(;!t)
(2)
'# ! > 0, > 1, > 1.
6 7
-
U(t) '
,2] A( ). 8, + A "
.
8 1. 1) A;1
2) U(t) A( ), A,
(2) ! > 0, 0 6 < 1, + 1 6 3) kf(t + 9t) ; f(t)k 6 ct; j9tj" " 2 ( ;;1 1], 2 ,0 1)
4) v0 2 D(A ) 2 ( 1], D: = E , v0 2 D(A ) 2 (minf2 ;;1 g 1], D: 6= E ( A ,2]).
! (1) ", ! #
Zt
v(t) = U(t)v0 + U(t ; s)f(s) ds:
(3)
0
% (1) ,3].
8 , A (D: = E), = 1
= 0. ; " A ,4]. < + " , j arg j < ,
jj > R > 0 A * (A + I);1 k(A + I);1 k 6 cjj;r
(4)
r = 1. = ( = 0, = 1. < " ,5] ,
A * ( > ) (4)
' r 2 (0 1]. < + = 1 ; r, 2 ,0 1),
= 1 + , " 2 ( 1], 2 ,0 1 ; ), = 1. < ,6] ' (4)
"
f: Re > ;c(1 + j Imj)r1 g c > 0 r 6 r1 :
@ r = r1 = 1r ; 1, = 2 + 1 (1) , r 2 ( 23 1], 2 ,0 21 ), " 2 (2 1], = 1, = 0.
...
297
8, ' #'(
1. 8 >, '# "# ' r, ' ' . 8 ,2] ' ' , '# + ' ' >7 " ( ,2] ' ' +, < 2).
< 1 '
+ | > "' ("' , 1 +. < 1 (
' A, v0 2 D(A ) 6 1. =
1 ,3{6].
2. @7 ' 1. 8 (*#
.
1. U(t) A( ), A. " ! , #
(3).
. @ v(t) | (1). 6 ' > v0 (s) + Av(s) = f(s). @ -(
U(t ; s), s # x t ; y
(0 < x < t ; y < t). <'
' + ( 4) '), U(y)v(t ; y) ; U(t ; x)v(x) =
Zt;y
x
U(t ; s)f(s) ds:
(5)
@ Av(t ; y), U(t ; x) ' t > 0 '#
x y, , 3) ', > (, (5) x y ! +0. @+ * ' (3). C (3) .
Rt
(3) g(t) = U(t ; s)f(s) ds.
2.
g(t) 2 D 0
# f(t) $ 3) 1. kAg(t)k 6 ct;; kf k" kf k" kf kC0" = kf kC0 +
sup
06t<t+t61
kf(t + 9t) ; f(t)k t; kf kC0 = sup kt f(t)k:
06t61
(6)
9t"
;
,3], ( g(t) ; g(t) = ,dD(s)]'(s), D(s) = A;1 U(t(1 ; s)), '(s) = f(ts). 6
.
R1
0
298
. . g(t) > '# Sn =
2n X
k=1
k
k
;
1
D 2n ; D 2n
' 2kn :
;' j
kA,D(s + 9s) ; D(s)]k 6 c t(;1)+(1;j9s
)(1 ; s)+(1;)
(" 2 ,0 1] k'(s + 9s) ; '(s)k 6 ct";s; j9sj":
<'" 1 ; " < < 1;; . 6 kA(Sn+1 ; Sn )k 6 ct; 2;(n+1)(+";1) n = 0 1 : : ::
F, kAS0 k 6 ct;;kf kC0 , Ag(t) = limASn (6).
3.
t > 0 % 2 # g(t) ##
g0 (t) = f(t) ; Ag(t):
(7)
. @ f(t) ' t > 0. 6 ( g(t) > Zt
g(t) = A f(t) ; U(t)A f(0) ; U(t ; s)A;1 f 0 (s) ds:
;1
;1
0
C + g(t) t > 0 (7).
@ f(t) ( G7. <7 ' t+1=n
(
Z
Z
f (s) ds = f t + ns ds
f (t) = f(t) 0 6 t 6 1 fn (t) = n
f(1) t > 1
t
0
1
Zt
gn(t) = U(t ; s)fn (s) ds:
C0"
,
0
@ + f (t) 2
kfnk" 6 kf k" , fn (t) ! f(t), gn (t) ! g(t) E t > > 0. H fn (t) ' ' t > 0, + + > "( gn(t) gn0 (t) = fn (t) ; Agn (t). @>, '# gn0 (t) > #
( t > > 0). J
+ 299
...
G7
" ="
kf k" 6 ckf k"" =" kf k1;
C0 0 < "0 < " 6 1. @ + fn (t) ; f(t), ,
> > C0"
, ;;1 < "0 < " (
(6)):
0
0
0
0
Zt
A U(t ; s),fn (s) ; f(s)] ds 6 ct;; kfn ; f k"0
0
6
" ="
6 ct;;kfn ; f k"" =" kfn ; f k1;
C0 :
0
0
@ kfn ; f k" 6 2kf k" , # , Agn(t) ! Ag(t) E t > > 0. @+
gn0 (t) ! f(t) ; Ag(t) ' t > 0. 8(
> '.
v1 (t) = U(t)v0 (3). H
v1 (t)
(* (1). < ,2] , + ' t > 0, v0 2 D(A ), 4) ' 1. @+ v(t) (3) (1). = ' 1.
3. 6 1 -' '
' . @v + (;1)m @ 2m v = f(t x) t 2 (0 1] x 2 ,0 1]
(8)
@t
@x2m
v(n ) (0) + v(n ) (1) + T v = 0 = 1 2 : : : 2m ; r
(9)
Z1
0
'k (x)v(x) dx = 0 k = 1 2 : : : r
(10)
v(0 x) = v0 (x) x 2 ,0 1]:
(11)
% | ' , j j + j j 6= 0, 0 6 n1 6 : : : 6 n 6
6 n +1 6 : : : 6 2m ; 1, T | ' ' '
Wpk ;1(0 1) p 2 ,1 1), = 1 2 : : : 2m ; r, 0 6 r 6 2m,
f'k (x)g | ''# '# ,0 1]. 8, (9){(10) (
' ,5]. K( " 2m Lp (0 1). <7 + A = (;1)m dxd 2m "( D, * v(x) Wp2m (0 1), (*# (9){(10) (+ > Lp ). 6 (8){(11) 7
" (1).
,7] , A " j arg j < ; ,
jj > R > 0 > 0 * , '1
(4) r = 1 + 2mp
; 2nm ( n 300
. . 'k (x)). @+ ,5, . 3, x 1] A > A( )
= 1 ; r, = 2 ; r. L n 2 , p1 1p + 2m), 2 ,0 1), "
'' ' 1. @+ 2. $ :
1) n, (. ,7]), p1 6 n < p1 + 2m
2) # f(t x) $ '
t Lp ( x)
kf(t + 9t x) ; f(t x)k 6 cj9tj"t;
1 1] 2 ,0 1)
" 2 ( 2nm ; 2mp
3) v0 (x) 2 D.
!2m(8){(11) " v = v(t x), @ v
# @v
@t @x2m t > 0 Lp .
1] . . . | !.: #$, 1967.
2] *$ +. ,., $ -. .. /01 0* 1 234 0 $2, 536 4 3 // $ .
5. | 1986. | ,. 27, 9 4. | . 93{104.
3] $ -. .. > 4* , 536 *$ 4 // ?@# /. | 1968. |
,. 183, 9 2. | . 292{295.
4] Da Prato G., Sinestrati E. DiBerential operators with non dense domain // Annali
della scuola normale superiore. Di Pisa. | 1987. | Vol. 14. | P. 285{344.
5] F$ . F. { 5. | H$: I, 1985.
6] Favini A., Yagi A. Abstract second order diBerential equations with applications //
Funkcialaj Ekvacioj. | 1995. | Vol. 38. | P. 81{99.
7] *$ +. ,. > 4 4*
// K0 0. !$. | 1997. | 9 6 (421). | . 32{36.
* !+ 1998 .
. . . ,
. . . ... !. . "
1 (5 1999 .).
. ., . ., . . : "#, $ %, %.
, , , . ,
, .
! "# " . $
() . , , # ( ' ,
# ), ( )
{+
"# .
2 (12 1999 .).
'. . (( . )*). + " $$*,.
3 (19 1999 .).
* . '. . $.
( ,. -. .
/0 /0 "# (
, "# "# "
# 1 . 1 , 2 (" #(
3
), , #
, (
,
, (
,
'
. 4 ( , .
302
4 (2 $ 1999 .).
0%1 2. 3. 4# , *# %
56 #.
8 (
, # . ) # . 9 !
#
4 ( (
). $ 3 (
,
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