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Abstract A. V. Mikhalev, A. P. Mishina, Innite Abelian groups: methods and results, Fundamentalnaya i prikladnaya matematika 1(1995), 319{375.
The review paper is devoted to methods and results from the theory of in$nite Abelian groups. The content of the review: x 1 Some main de$nitions& x 2 Primary groups& x 3 Torsion free groups& x 4 Mixed groups& x 5 Classi$cation theorems& x 6 Quasi-isomorphisms& x 7 Endomorphism rings and groups& x 8 Groups of homomorphisms. Extension groups& x 9 Tensor products. Torsion products& x 10 Valuated groups& x 11 Varia.
x 0 : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : x 1 : : : : : : : : : : : : : : : x 2 : : : : : : : : : : : : : : : : : : : : : : : x 3 : : : : : : : : : : : : : : : : : : : : : : x 4 ! : : : : : : : : : : : : : : : : : : : : : : : x 5 #$ % : : : : : : : : : : : : : : : : : x 6 # $ : : : : : : : : : : : : : : : : : : : : : : : x 7 # (% ) $ . $ x 8 $ . !, : : : : : : : : x 9 - . : : : x 10 (valuated groups) : : : : : : : : : : x 11 0 : : : : : : : : : : : : : : : : : : : : : : : : :
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a (. . 5, ka = lb 5 %5 k, l), 3 5 ) , (x 1, . 5) hai G, 3 , ) a (x 0). J A = hai , r(A) = 1, (A) " a G. , 1, 2 1 > 2 ( 1 2), 74 5 1 = (k1 : : : ki : : :) 2 1 2 = = (l1 : : : li : : :) 2 2 , 5 ki > li , i = 1 2 : : :. @ 7, (!, ( 5 5 1 G \ $ (War~eld) :p, 282] IT(G) G, 7 (! 5 $ 1 | ! OT (G). J IT (G) = OT (G) G , G | 1 OT (G). B G OT(G) = 1 (, , 5 , (1 1 : : :)), , IT(G) 3 ( . 2. J G | 1, . 14 3 , 4 1 $ 3 , (x 1, . 2). #3 3 , 3 3 (:6], 86.7). $ 35 ( ( | , L D = Q, 4 L $ $ , , , F = Z, . x 1, . 3, 4). R. R. # :r, 55] ! 5 L , 3 ( 3 G = R, r(R ) = 1, 4 3 , ( 4 R). > ! (. :q, 123, 50, 35]).
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35 4 1 . > : , , | ) $, 3 , (. R ( (Arnold) :r, 132] R (, 5, (Arnold, Vinsonhaler) :r, 137]). ) ( (Butler) :p, 56]. B 5 3 ( 5, ) 5 ! ,, . :r, x 2] ( , :r, 164]), :s, x 6], :MV]. ( , . :b], :s, x 6], 3 :AV1], :AV2], :FM1], :FM2], :H[o], :Mu]. B 5 , 47 )5 (4 $ :AV1] (4 $ :AV2]. O 3 5 ) ( ) . O% ) (Mutzbauer) :s, 334] 5 , 5 , 3 ) , , , . Y, 5 4 5 3 R. R. # :s, 39]. :s, 163] 3, L ) ,: 1) 4 ( ), 3 ) , 3 L, 3 (. 4)K 2) ) L (! 5 475. G ( ( ) B1 - ( -3 1 ), Bext(G T) = 0 (x 8) 4 , , T . B1 - , ( , | (%, (Bican, Salce) :r, 164]). , B1 - . 2 (. . , ) 4 , ) B1 - , ( ,
3 :s, 162]. @ ( , 5 B1 -, R ( :a]. . 3 :s, x 6K 162, 163, 188, 196, 205] :r, 298], :AD], :BF2], :BSS], :DT], :FM1], :FM2], :FMR]. G B2 -, 7 47 ( ( 5 S S 0 = G0 G1 : : : G : : : G = G = G , G = < G , | ( , r(G+1=G) = 1 (8 < ) G+1 = G +B , B :s, 135]. B2 - B1 - , :r, 164]. B 7 6 @1 :s, 196]. @ 3 B1 - B2 - . 3 :R]. :FMR]
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, , ,. @ 5 5 75 ( . :s, x 5], :EMS2]. 4. , 3 34 (. . 2). @ 5 5 . :r, 45, 49, 78, 137], :s, 29, 65, 169, 337], :L], :MaV], :Pr]. 5. Y, 5 4 1. @ 4 3 :6, xx 96, 97], :o, x 7], :p, x 4], :q, x 4], :r, x 4], :s, x 4]. R( 5 :s, 128] A- , , A = fGg2! | 5 (. 3) (. . , 3 3 , () QHom(G G) 6= 0). Y , R( 5 A- V = A2A VA , VA | , AI (A) , A 2 A, I(A) | ( . B , AQ- , 7 (:6, x 94]) Q V = A2A VA W = A2A WA $ ( , VA = WA 4 A 2 A. ( P0 | 5 % Q h 5 : P0 = 1 e i , o(e ) = 1 . G i i i=1 :6, x 94], 4 $ : P0 ! G i (. . 5, , ( 3, ) (ei ) = 0. , ( , 3 , $5 Q, P0 Jp , p | ( Q(:6], 95.3, 37 ). J P = i2I Ai , Ai | ( , G |
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7 (:6], 27.3). 4 ( m 7 2m 35 7 m (:q, 346], . 3 :f, x 8]). 35 | 4 1, Jp (x 0) :9, x 32]. G 3, | 35 . 3 G 4 35 ( $ (x 1, . 2). , 4 5 %5 n > k > 1 3 , 4 A n, n k 5 n = r1 + + rk , ri > 1, i = 1 : : : k, 7 3 A = A1 Ak , Ai | 3 ri (:6], 90.2). B \ 55 3," 4 35 . :6, xx 90, 91] :9, x B.31, . 3]. 74 , 47 5 35 3 5. ) . 67 < :6, . XIII] ! . Y. # ! :s, 25] ( , $ ) J. R. 7 , :Bl]. R. . X J. R. 7 :s, 10] 3 ! 68 < :6, . XIII]: 5 (5 r1 : : : rk r10 : : : rl0, r1 + +rk = r10 +: : :+rl0 , , , 5 7 , 47 3 3 r1 : : : rk r10 : : : rl0
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A = Hom(A = Hom(BLA)), Q Q 5 , A = A , B = B 5 A = Ai , i j L B = Bj Ai , Bj 1 ! R. O. ( :q, 70, 72]. . 3 :AS2]. (Grosse) :p, 114] 5 A = Hom(A W), A | , W | . ,, C (Beyl, Hanna) :q, 121] , Hom(A Q=Z) = A ( 5 A ( 5 : A = Hom(A Q=Z) 4 , , , A). 4. Hom(A X) Hom(B X) 4 , X 3 A 6= B, =L , A = @0 Z Q, B = A Q (R. O. ( :q, 72]). B , C (Hill) :q, 223]. > | %( ! 34 < :6, . VIII]. 1. Y. :r, 14] , A, B | 3 , A Hom(A C) = Hom(B C) 4 , C 1, A B. # , 4 , 3 , = A 7 3 C1, Hom(A C1) = Hom(B C1 ) Hom(A A) = Hom(B B) , A B , 3 , B. = :AS1] , ( ( GCH, . x 2, . 3) 3 A ( . J. . 5 :P] , A, B | 3 % X | 3 5 5 5 1 A B, rp (Hom(X A)) = rp (Hom(X B)) ( rp (Hom(A X)) = rp (Hom(B X))) 4 p (x 0) 4 , X 2 X A = B. . R. # :Kr1] , % 5 A, B 74 C1 ,: : :, Cn , $ (x 6) Hom(A Ci) Hom(B Ci), i = 1 : : : n $ A B (. 34 < :6, . VIII]). 5. (Grosse) :p, 114] , . . ( H0 = A, H1 = = Hom(H0 W) = H1 (A W ),: : :, Hi+1 = Hom(Hi W) = Hi+1(A W ),: : :, A W | . A W - T , 74 % m > 0, s > 1, Hm = Hm+s '2H1 ker ' = 0. J ) W % , A | , , Hom(A W ) = Hi(A W) 4 i > 1. =(% (Schultz) :p, 260] G , H0 = G, Hi+1 = Hom(Hi G), i = 0 1 : : :, i: Hi ! Hi+2 ( i (hi)(hi+1 ) = hi+1(hi ) 4 5 hi 2 Hi , hi+1 2 Hi+1 ) $ i > 0. , G, , H1 = H3 .
:
353
( G | - , G = Hom(G Z), G = (G ) . > $, O , =5 (Eklof, Mekler, Shelah) :s, 217] G , 3 G ! G | $ . >, @5 (Eda, Ohta) :s, 213] %( ! , $ G ( , G, 7 ( , (:6], x 94). J 3 G ! Hom(Hom(G A) A) 5 G A $ , R( 5, (Albrecht, Goeters) :s, 133] 4 G A- . \ $ (War~eld) :p, 282] ! LF( ) 5 G , R-$ , R 1 (x 3, . 1), . 3 . R. # :r, 61]. , ( , LF( ) $ Hom(; R) , (4. > , ( ( $ ) | 7 , 5 . R ( (Arnold) :q, 90] , ( QD $ 5 (x 5, . 2) $ (x 5, . 3) $ ( 5 ). )
( ( (4 :o, 52], 47 QD 7 , 475 4 p- 5 , ( 5 , ) 4 , ( , , . B , ( \ $ , ( R ( (!4 ( . R. R. < :s, 94] ( 3 ) , ( D 5 A , 3 , 5 3 F , A=F | , 4 p- , | , % ( $ | $ ). # D 3 QD LF( ). B , ( R. R. < , LF( ), , (4 \ $, 3 (1 1 : : :) , (4 R ( QD. 5,, \ (Vinsonhaler, Wickless) :s, 384] , ( 5 , 3 7474 , ( \ $ , ( R (K %( , ( R. R. < . 6. B, C (Dugas, Herden) :r, 193] 4 , , ( !, (. 3), $5 5 , ,
( !,, 5 ,. (T F) , T | , , F | (!, , Hom(T F) = 0 4 5 T 2 T, F 2 F.
354
. . , . .
@ 5 5, Hom(A B) = 0, . :s, xx 4, 10]. J Hom(G Z) = 0, ( :s, 387] G 0-. ( X | ,- . B :s, 187] X? = fA j Hom(X A) = 0 8X 2 Xg
, X, ?X = fA j Hom(A X) = 0 8X 2 Xg | , X. B 5 . (?(X? ) X? ) , X, (?X (?X)? ) | , X. @ 5 X? ?X . :s, x 10], 3 :SVW]. (Goeters) :s, 252] !( ) , ?H, H? , (?H)? H. \ (Wickless) :Wi] TF ( ) ! ) G H () (?G)? = (?H)? . (x 1, . 5) G0 G 2 TF , ( ( , G0 G, (core). % G (4 $ . II. J A B, 74 G, G A G=A = B (, G = A B). G ! A 7 B. B ! G1 G2 A 7 B 4 " , 3 7 $ , 47, A 3 $ 347, 3 A,
47 3 ) B. 3 ) 5 3 , !, A 7 B % 3 , (. :6, x 49, 50], :5, x 61]). > ! Ext(B A) A 7 B. , 3 !,, ) 5 !4 A B. O (Mader) :p, 193] , ! , T 7 K $ ( ( ) ), 4 ) Ext(K T), 47
, . B 5 3 T K \ (C. Walker) :p, 277] , ! G T 7 K ) Ext(K T ), 47, , , ( , G $ (x 6) T K. @ 3 , ! G , T 7 Z(p1 ) ) Ext(Z(p1 ) T ) . :Mo2]. 1. > , Ext(B A) ( 7 , A B) :6, x 52]. , Ext(B A) = 0 $ ,
:
355
B 4 , A (
$ , A 4 , B) ( , B | (x 1, . 3) (
A | (x 1, . 4)). : Ext
M i2I
Y Y Y Ai G = Ext(Ai G)K Ext A Gi = Ext(A Gi) i2I
i2I
i2I
(:6], 52.2). B 4 , G , ( B! 0!A! C!0 (x 1, . 6) ( 0 ! Hom(G A) ! Hom(G B) ! Hom(G C) E! % % ! Ext(G A) ! Ext(G B) ! Ext(G C) ! 0
Hom(A G) E! 0 ! Hom(C G) ! Hom(B G) ! % % Ext(A G) ! 0 ! Ext(C G) ! Ext(B G) ! (:6], 51.3), , , ^ , ^ , , ^ , ^ | $ , %
$ , E E | (. :6], x 51). > ( 4 74 ( (5 5 3, 5 . 2. B 4 5 A, B Ext(B A) | (x 1, . 5K . :6], 54.6), ( , % , H Ext(Q=Z H) 3 (:6], 58.1, 0 (Rangaswamy) :p, 244]), . . ( , 37, H , ,. = (Schoeman) :q, 341] , Ext(B A) (x 1, . 5) 4 , A ( , ( B (x 1, . 3) | % 5 . @ 3 5 , Ext(B A) 4 , B ( $ , A). , A | ( , Ext(B A) | 4 , B (:6], x 52, @), )). @, Ext(B A) | 4 , A ( 4 , B) ( , B | ( A | ). Ext(B A) | % 4 , A
356
. . , . .
( 4 , B) ( , B | , , , ( A | ) | . 0 (Rangaswamy) :p, 244]. J G=tG | , Ext(G=tG tG) | (:6, . I, ) (Baer) 4], 3.3). Ext(G=tG tG) 3 ( , , ( (Nunke) :o, 162]). < , (Faticoni, Goeters) :s, 223] ! 5 ( (4, x 1, . 6), A, B Ext(B A) 3 (. 3 :s, 253] :Kr3], 2.5). R ( (Arnold) :A] , Ext(A A) | , A . B A, B \ $ :q, 375] Ext(B A), ( 47 Hom(B A). 1. X. # Ext(T A), T | , A | :q, 47], Ext(T G), T | % , G | ( (. :o, 17]). 0 (Rangaswamy) :p, 244] Ext(B Z) ( , B () | ! , 54 :5, . X]). B ( 5 !, 3 , :o, x 3], :p, x 8], :q, x 8], :r, x 8], :s, x 10], :6, x 52]. 3. , 4 !,, 1. X. # :o, 16]. O,, 3 (Meijer, Viljoen) :q, 300] , (x 1, . 4), p- , , 5 p 4, $ , Ext(B A). C :q, 243] ( ZFC + V=L, . x 2, . 3, 4), , G Ext(A G) 3 ( , , , , , A. , 3 C, =5 (Hiller, Shelah) :q, 236] , 7 G, , Ext(G Z) | . , 3 (Gr[abe, Viljoen) :p, 103, 104] 4 H Ext- ( ), 7 H, G = Ext(G H) (
G = Ext(H G)) 4 , G 2 H. B , 5 % 5 5 (x 1, . 5) | ) , (, Ext- , :p, 103]K Ext(A Z) = A ( , A | (,, C (Beyl, Hanna) :q, 121]). 4. 5 :p, 234] , L1 L2 | $ (x 6) , Ext(L1 C) = Ext(L2 C) 4 , C. . R. # :Kr3] (4 ! , 3 , A B , 47 5 % 5 5 5, Ext(A C) = Ext(B C) 4 , , C (:6], . IX, 43, . 1). R , 5 ) | $ A B. . R. # :Kr2] ! , 43
:
357
(. 3 :Go1]). R. Y. O :Mo1] , Ext(X C) = Ext(Y C) $ 5 X, Y 4 , C, Ext(tX C) = Ext(tY C) Ext(Xp C) = Ext(Yp C) ( | x 0) 3 p. J X, Y | p- X | ( (x 1, . 4), GCH (x 2, . 3) , Ext(X C) = Ext(Y C) (8C) X = Y. 5. (Gr[abe) :p, 102] ( ( E0(G H) = G, Ei+1(G H) = Ext(Ei(G H) H) (i = 0 1 : : :) Ext- q G, H. G H - ( H G- ) ( q, 74 % i, j (0 6 i < j), 5 Ei(G H) = Ej (G H). 04 , 7 H- 5 ( G- 5 ) $ H ($ G), 37 F, , Ei (F H) 6= 0 (
Ei (G F) 6= 0) 5 i. 6. 0 (Rangaswamy) :p, 244] A L- B ( B
R- A), Ext(B A) = 0. B 5 CB ( CA ) 5 A ( B), L- (5 (R- (5) 3 , CB ( CA). [ (G[obel) (Prelle) :r, 232] , X, Y X ? Y, Ext(X Y ) = 0 4 5 X 2 X, Y 2 Y. ( X? = fY j X ? Y g, ?X = fY j Y ? Xg. (X Y) :r, 232], Y = X? , X = ?Y. J (F G) | G = X? , F = ?(X? ) X, X (F G). @ 5 . 3 (Salce) :r, 343]. J A | p- (. . A = pA), B | p-, Ext(B A) = 0 :6, x 52, 1)]. 1. X. # :q, 46] 5 , 5 Ext(B A) = 0, , B | A | 4 , 3 , A B=tB | . G 1" , Ext(G T) 4 , , T . $$ :p, 111] , ) (:6], 101.1). W, , Ext(W Z) = 0, $ (Whitehead). B , \,5 , , , (. :6, . XIII, 79]). > $ (Eklof) :q, 153] , ) ,( ZFC + V=L (x 2, . 3, 4). ZFC + MA + :CH , 4 ( { > @1 7 \,5 7 {. - =5 (Shelah) :q, 346]: 3 \ 3 \,5 7 @1 "
358
. . , . .
ZFC ,. ( ) , O :r, 302]. @ 5 \,5 . 3 :q, 315], :EMS1]. 7. 74 ( 4 Ext(B A), , Pext(B A) Bext(B A). Pext(B A) 5 ) Ext(B A), 5 ! G AT 7 B, 5 A | . Pext(B A) = 1 n n=1 Ext(B A) (:6], 53.3). Pext(B A) = 0 4 , B ( 4 , A) ( , A ( B | % 5 ) | . :6], 3 53.4. 7 p- B, 475 % 5 , 5 Pext(B G) = 0 4 , , G, 7 , , GCH (O (Mekler) :r, 302]). = (Schoeman) :q, 341] G PT - ,, 4 , , T Pext(T G) = 0. PT - ,. :q, 341] 5 5 , 5 ! 4 PT -. Pext(B A) | 4 , B ( , A | PT -. ( , | PT -. ( F | , T | , G | ! T 7 F . T = tG G , 3, 3, G T 3 , ) g, G (g) = G=T (g + T), G (g) | ) g G, . . ( ( p- (x 1, . 4) p. > Ext(F T), ! G, 5 tG = T | , 4 Ext(F T) Bext(F T) (O3 :r, 298]). Bext(F T) t Ext(F T ). F , 5 Bext(F T) = 0 4 , , T , 4 B1 - 1 (. x 3, . 2). #$ (Keef) :Kee4] Bext(B A) Ext(B A) , A B | p-. ) A ! G A 7 B G, , ( 0 ! A ! G ! B ! 0 % 4 ( ( 0 ! A=p A ! G=p G ! B=p B ! 0 (x 1, . 4) 4 . , p Ext(B A) Bext(B A) ( | ), Bext(B A) p Ext(B A) Bext(B A) \ \ p Ext(B A) = 0. @ 5 5, 475 Ext(B A) 5 , . :o]{:s] :6, . IX].
:
x
9 0 .
359
I. ' A B 5 A B , , 47 3 3 a b (a 2 A, b 2 B), 47
! (x 1, . 3) | 3 (a1 + a2) b = a1 b + a2 b a (b1 + b2 ) = a b1 + a b2 a a1 a2 2 A, b b1 b2 2 B. @ \ (Whitney) (:9, x B.33, . 6]). L1. : L A B L = B AK (A B) C = A (B C)K i2I Ai j 2J Bj = ij (Ai Bj ) (:6], x 59). @, Z B = B 4 , B, Z(m) B = B=mB, Z(pr ) Z(ps ) = Z(pt ), t = min(r s). 2. J A B | (x 1, . 4), A B | . - 5 5 (x 1, . 5) | :r, 115]. B 4 , H G = Qp H p- ( , (x 4, . 2) H p. J A B | p-, A B | p-. J A | p-, pB = B, A B = 0 ( 4 A B = 0, A | p-, B | q- p 6= q). - 5 A, B , (:9, x B.33, . 6]). B a 2 A, b 2 B ) (x 3, . 1) (a b) = (a) + (b) (. :6, x 60, . 9]). 5 p- (. . 375 5 A, A = pA) 3 ( p-% , , :q, 80]. J U V | (x 1, . 5)
A B, ) u v (u 2 U, v 2 V ) 34 A B , $4 U V . > 3 4 5 A B, A B | (:9, x B.33, . 6]), 7 ) 3 . , Z(p) = U = A, Z(p) =V B = Z(p1 ), U V = Z(p), A B = 0. J A, B | p- A0, B 0 | 5 (x 2, . 1), A B = A0 B 0 (. :6], 61.1). @4 , 5 , , 5 % 5 (:6], 61.3). - p- A B $ , ( 5 ) A, p- (x 0) B :o, 90].
360
. . , . .
@ , : t(A B) = (tA tB)
(tA B=tB) (A=tA tB) (:6], 61.5). 3. R. R. < :q, 82] 5 5 , ( 47 # !-B) (Derry)-O(% (x 5, . 2) 3, (. :5, . XI, 57] :6, . X, 47]), 3 ( 7 ) 5 (. :r, 115]). O3 , - (Megibben, Toubassi) :q, 298] 5 G, K % , , (4 $ , ,, 47, 1, G K (:6], 104.3) G K. R. R. < :r, 115] , 3 A, 5 rp (A) 6 1 (x 0) 4 p, (4 , 3 (% (5 (x 5, . 3). 4. 1! (Lausch) :r, 283] 5 , mn $ 4 5 m n. :s, 310] , A r(A) = 8, A = B1 B2 = C1 C2 C3 , r(B1 ) = 4, r(Ci ) = 2 (i = 1 2 3) B1 | 5 2. \ $ (War~eld) :p, 282] , M | , A | 1, M = A G , G ( , 4 x 2 M 7 $ g 2 Hom(A M) ,, x 2 f(A), ) M = AHom(A M). (Nongxa) :r, 316] , (x 3, . 2) 1 , (. . (Z)). B ! C ! 0 | ( (, 5. J 0 ! A ! (A) | B ( ( ( ), 4 , G ( ( 0 ! A G ! B G ! C G ! 0 K 3 (:6], 60.4K C (Head) :p, 126]). J7 ,5 5 , . :6, . X] :o]{ :s], , , 7 (x 4, . 1) 5 , 5 . II. # Tor(A B) 5 A, B , , 47 3 3 , (m a b), m | % , a 2 A, b 2 B ma = 0 = mb, 47
! | 3 (m a1 b) + (m a2 b) = (m a1 + a2 b) (m a b1) + (m a b2) = (m a b1 + b2 )
:
361
(m na b) = (m a nb) = (mn a b) (:9, x B.33, .8], :6, x 62]). 1. Tor(A B) (m 0 b) = 0 = (m a 0). Tor(A B) = 0, 5 A, B | . B 4 5 Tor(A B) = Tor(tA tB) = Tor(tB tA) = Tor(B A)
M M M Ai Bj = Tor(Ai Bj ):
Tor
i2I
j 2J
ij
2. B 4 5 A, B Tor(A B) ,. J A B | p-, Tor(A B) | p-. J A | p-, B | q-, p q | , Tor(A B) = 0. @, : Tor(Z(m) G) = G:m] 4 , G 4 % m > 0K Tor(Q=Z G) = tGK Tor(Z(p1 ) G) = Gp (p- G, x 1, . 3). J A B | % p-, Tor(A B) | % ( (Nunke) :p, 217]). @ 5 5, Tor(A B) | 5 , . :p, 219], :r, 267, 253], :s, 268, 293, 295, 296, 298]. C (Hill) :r, 255] , 5 p- A, B Tor(A B) , , % 5 . 3. #$ (Keef) :s, 297] 5 , 5 p- G $ Tor(A B), A B | ( (x 2, . 2), :s, 298] , p- B TB 5 p- A, 5 A = Tor(A B). :s, 298] p- A Tor- , Tor(A A) = AK 74 % (x 1, . 4) Tor- 4 , (x 1, . 4). 4. < (Fuchs) (:6], . X, 50): 3 , A B, Tor(A C) = Tor(B C) 4 , % , C? C (Hill) :q, 223] , Tor(A X) = Tor(B X) 5 5 A, B 4 , % , , X, A B 4 4 7 ( m, ( ( (x 1, . 4), L (x 2,= .L2),B. $ mA # (Cutler) :r, 183] $5 m p- A, B, 5 Tor(A G) = Tor(B G) 4 , % , p- G. . 3 :s, 173]. B (5 ( 5 A, B, R, S #$ (Keef) :s, 297] , Tor(A B) = Tor(R S) ( , ( )5 5 4. \( Tor(A B) :p, 217]. B ! 5. J 0 ! A ! C ! 0 | ( (, 4 , G % ( ( 0 0 0 ! Tor(A G) ! Tor(B G) ! Tor(C G) ! A G ! B G ! C G ! 0
362
. . , . .
(:6], 63.1). J (A) B, 0 (Tor(A G)) | Tor(B G), 0 | ) $ (. :o, 93] :6], 63.2 60.4). B ( 5 Tor(A B) 3 , 5 :o]{:s], 3 :6, . X]. x
10 (valuated groups)
p- G , , v, . . $ % ,: 1) vx | 1 (x 2 G)K 2) vpx > vx ( , 1 < 1 < 1 4 )K 3) v(x + y) > min(vx vy)K 4) vnx = vx (n p) = 1 (. :t] :q, 331]). p- G: $ % h , 7
) x 2 G 74 p- h (x) (x 1, . 4). p- vp ( , , G $ %,
47 , , , p-. 8 , , p- vp , , 3 p (. :v]). 14 4 4 A 3 3( 4 B , p- 4 ) A ( 7 , p- , B 3 p (. :t] :v], 23). 2 A 4 A0 (. :v]) $ f, vp0 f(x) > vp x 4 5 x 2 A p. J f | $ vp0 f(x) = vp x 5 x 2 A p, f . A ( ,) B, A | B 3 A B | 3 5 . A , p- ( , (x 4, . 2) B !, 3 3 A 3 , ) b, vp b > vp (a + b) 4 a 2 A :q, 252]. # 5 - , 3 ) 5 5. F 5 % 5 5 , vp px = vp x + 1 4 5 x 2 F p, . 4 ( $ 5 5 5 % 5 R (, C, >. \ (Arnold, Hunter, E. Walker) :r, 135]. ) (Stanton) :r, 355] , , 3 , , 5 5 % 5 . 74 5 ! 5 5 % 5 p, 47 5 % 5 (. :v]).
:
363
C, 0 , >. \ (Hunter, Richman, E. Walker) :s, 282] , B | p-, p2 B = 0 B(m) = = fb 2 B j vb > mg = 0 % m > 0, B | % 5 5 . p2B = 0 !( B(!) = 0, ! | , 3 B 3 ( , , % 5 5 . B , 3 , , B, , p3 B = 0 B(7) = 0. :s, 283] 3 ( , 5 p- 5 ( 5 . (x 1, . 3) p- \( 5 p-5 (:6], 83.3). @ 5 5 \ $ (x 4, . 3) . :q, 252, 336], :r, 264, 309, 334] :k], :t]. @ 5 5, 5 , . , :q, x 10], :r, x 10], :s, x 11], , [ :g, . 7], 0 :t], :u], C, 0 , >. \ (Hunter, Richman, E. Walker) :k], 3 :Kee2]. R (, 0 (Arnold, Richman) :s, 141] 7 , (D- , , , D | 3 ). x
11 3
I. 9 . (x 1, . 5) (5 5 4 47 ,: 0) 4 | ,K 1) A | B, B | C, A | CK 2) A B C A | C, A | BK 3) A B, 4 , K A A=K B=KK 4) K A B, K | B A=K | B=K, A | B. , 5 5 ! ( !- ), 3 , , !- ( : A ! B), , 5 , 0){4) (. :10, x 1]). 1. !- 5 5. a) 8 ( !- 5 4 ( ). b) "- . B 3 p $ , ( ( 3, ,) (5 Mp = fk1p k2p : : :g , A ! B ( A " B), a = pkip b, a 2 A, b 2 B, a = pkip a1
364
. . , . .
a1 2 A ( 4 5 p kip 2 Mp ). , "- . . 0 5 :p, 29, 30, 31, 32]. "- : ( ( 3 Mp | 3 5 (5 )K (neatness) ( 3 Mp !( 1)K p- (Mp 5 (5 p 5 (5 5 )K ( 3 Mp , ,
4 ). c) - ) ( P - , . :p, 126], :q, 58]): A B ( , a = pk b (a 2 A, b 2 B, p 3 $ 3 P 5 , k | ( ( ) 7 % l > 0, pl a = pl+k a1 a1 2 A. N, B | A B, A B. ! , L 47, ( !- 5 4 , !- , ,. R. R. O % :q, 58] , ! 5 5 , - C , "- (. . A ! B ( , A B
A " B, " $ ). 5 5 4 !- 5 . Y. # ( :q, 45]. ( . Y. # ( J. . :r, 102]. d) ( M (
P) | (, 5 . A B !M - (!P - ) B, M ( P) L ( ) ( , ( 0 ! A ! B ! B=A ! 0 (x 1, . 6). 5 5 5 5 !- (#. \ (C. Walker) :p, 278]K :10], (1.16), (1.20)). J M | 5 5 , !M - G TEP-, , TEP (torsion extension property) G. @ TEP- 5 . :s, 196, 203], :FM2]. :Go2] N 5 , $5 $ , A 6= Q, 4 !N- !N - . J ! | - , , L ( ) B ! C ! 0, ( 5 5 ( , 0 ! A ! (A) ! B, 4 !- % (!- ). ( M! (
P! ) | 5 !-L 5 (!- 5) . @ 5 3 M! P! !- . :p, x 5], :q, x 5], :r, x 5], :s, x 7]. J ! | (, M! 5 5 (x 1, . 5), P! | 5 5 % 5 (:10, . 28{29]). !M !P (. !), M = M! ( P = P! ),
:
365
! | , 4 % ( ) !. !, 47 L ( ) , % ( ) . @ ( | L . 14 !M (
!P ) L ( ) . J ! L , 3 ( 3 4 , , (. 1, b)): 5 ) | M! 4 % 3 ( , (Y. . :q, 68]). G -!- % ( - !- ),
L ( ) ( 4 , , ( 0 ! H ! G ! G=H ! 0, H ! G. B 5 5 , L ( , ) ) :q, 22, 23, 105, 106, 119, 329], :r, 24, 25, 134], :s, 46, 104, 105, 106], :BL], :Ch2], :Do], ! 17 < (:6], . V) )5 . @ L 5 5 . 3 :Ch2], :c]. . . B :s, 19] 5 -p- L . ) Y. . :s, 76] 5 -!-L 5 4 , , (. c)) ! = p \ "p , p | $ , "p | ) "- (. b)), , Mp | ( , Mq 4 q 6= p | 3 . @ -p- 5 5 . :r, 149]. 2. 0! G A 7 B (x 8, II), 5 A !- , 4 ) Ext(B A), 47 ! Ext(B A). J ! | (, ! Ext(B A) = Pext(B A) (x 8, II, . 7). B 4 , !- !-L ( (!- () A ( , ! Ext(B A) = 0 (
! Ext(A B) = 0) 4 , B (:10, (1.6)]). J ! Ext(B A) = Ext(B A) (
, ! Ext(A B) = Ext(A B)) , A 4 , B, A !- (!- ). B , !-5 5 , !- 5 | 5 (:10, . 30]). B ( , !- !-5 (!- 5) 4 !- , (!- ,) , (:q, 28], ( R. . Y ). B A, B R. . Y :r, 31] ! 5 , 5 ) 4 5 !-5 5 !- 5 , !- , ! (!4 , 474 A B !-5 !- 5 () A-B- , . # :Ke], R. . Y :r, 31], . Y. # :r, 50], :s, 34], :Kom])K ( , 4 (% .
366
. . , . .
L
Q
J, $ ! Ext i2I Ci A = i2I ! Ext(Ci A) ( , ! | (R. R. L LO % :r, 71]K . x 8, II, . 1), . . A ! B ( 2 ) 2! A ! 2! B . , 47, ,: A-B- , A | 5 5 5 , B = f0g (. R. . Y :q, 28]). G ! B ! 0 | ( (, J 0 ! A ! , (A) ! G, 4 , , H ( 0 ! Hom(H A) ! Hom(H G) ! Hom(H B) ! ! ! Ext(H A) ! ! Ext(H G) ! ! Ext(H B) 0 ! Hom(B H) ! Hom(G H) ! Hom(A H) ! ! ! Ext(B H) ! ! Ext(G H) ! ! Ext(A H) (. x 8, II, . 1) :10, (1.3)]. @ 5 5 5 . 3 4 (4 J. . :w], 47 , 0. . R :r, 1], :s, 2] . . < :r, 109] . 1. J! :r, 29, 30], . R. 4 , :Se]. II. 0 . < % R, 7
3 , A R(A) , f: A ! B | $ , f(R(A)) R(B), ( (Nunke) :p, 217]). $ % 3 (C, O3 (Hill, Megibben) :p, 152]) (=( (Charles) :p, 59]). O (Marty) :p, 196] R(G) ( S(G)) G ( ), R (
S) | , , R(G=R(G)) = 0 (
S(S(G)) = S(G)). 0 , 47, % , (Gardner) :q, 183] . @, :10, x 2] , . J R(A) = A R(A) = 0, A R- R-
. Y , R(G) G | ) (! R- (5 G | 5 H, 5 R(G=H) = 0 (:10, (2.8)]). R. O. Y :q, 30] , R | A=R(A) | , A = R(A) A0 , A0 A. 0 R, A = R(A) A0 4 , A, . (, R (. . ,, R(A) = A 8A R(A) = 0 8A) 7 ( , 5 R- (5 L 5 5 (x 1, . 4) 5 p2 Gp , Gp | p-, | 3 5 ( (Gardner) :q, 182]).
:
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( A | (, 5 RA(G) = \ fker f j f 2 Hom(G X) X 2 Ag 4 , G. - RA | ( O, . !), fG j RA(G) = 0g | (!, , 37, A , ( 5 , ( , ). J R | (, 5 5 , R = RA A = fG j R(G) = 0g. >
3 (B, <,, =5 (Dugas, Fay, Shelah) :s, 189]). Q J A | , A 2 A ( A 2A X , X = X (8) | $ , , A , X, ! RA = RX . J R = pn: G ! pnG ( p ( n $ ), R = RZ (pn ) . J R = p! : G ! p! G (x 1, . 4), R = RQn Z (pn ) = RLn Z (pn ) . > ! + 1 4 , X p 6= RX :s, 189]. J A | 5 @1 - 5 (x 1, . 3), RA 9 :s, 208]. 0 , G 5 , 475 5 5 5. 5 5 5 3 , :10, x 2] 5 :p]{:s]. @ 3 . :10, (2.29)] :r, 31]. III. # 5 !, 5 ( . , 7 4 5 5 . G ( ), 4 5 ) a b 2 G 5 5 (a), (b) (x 3, . 1) (a) = (b) (
(a) 6 (b)) 7 $ () $ ) G, (a) = b :r, 27]. @ , , 3 ( (5 (:s, 16, 156], :Gr1], :GM]), ( ( , % ) (. :6, . II]). @ ,5 5 5 . :p, x 1], :q, x 1], :r, x 7], :s, x 8], :Kr1], :GM], :c], :Gr2], :Mi2]. # (Corner) :q, 134] ,, , ,, , % , p-. . 3 :p, 205], :Ch2]. 60-5 5 (. :o, x 5], :p, x 9]) ( % , 5 ( N- ( N | A, K A N - A, K | ( , , N \ K = 0). 0 , 7 N- , ( 3. , R. R. # :K] ! , 5 4 , N N- $ 3 ,
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(. :6], . II, 3). 3 :s, 151] N - , . . ( H, 5 N \ H = 0. @ ) . 3 :q, 113]. %( ( , 47 5 % ((4 :o, 52], :p, 62, 94], <,(! (Feigelstock) :q, 158{165], :r, 211, 212, 214{216], :s, 226] :4], J. Y. # % , :r, 51{54], :s, 36], R. Y. O :r, 75{77], B3 (Jackett) :r, 269{273], R ( (Arnold) :r, 132] .). C , O (Hirano, Mogami) :s, 279] % , , 3 3( , , (% ( <,(! :4]). @ )5 5 5 . :o]{:s], ( , $ 0 O 1962{1992 5, 3 ( :B], :Ch1], :Ch3], :Kee3], :MaV], :Mi1], :Re], :RF], :Sh], :SVW], :Wi].
4
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Abstract
N. M. Adrianov, Yu. Yu. Kochetkov, A. D. Suvorov, G. B. Shabat, Mathieu groups and plane trees, Fundamentalnaya i prikladnaya matematika 1(1995), 377{384.
The paper suggests a construction, associating a /nite group with a /nite plane tree. The complete list of plane trees, for which these groups are Mathieu ones, is presented. The arithmetico-geometric motivations of this construction are brie0y mentioned.
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# ATLAS] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R. A. Wilson. An ATLAS of nite groups. | Oxford Univ. Press, London, 1985. Con] M. Conder. The symmetric genus of the Mathieu groups // Bull. Lond. Math. Soc. | 23(1991). | N 5. | P. 445{453. Gor] D. Gorenstein. Finite groups. 2nd ed. | Chelsea Publising Company, N. Y., 1980. Gro] A. Grothendieck. Esquise d'un programme. | Preprint, 1984. Mat1] E. Mathieu. Memoire sur le nombre de valeurs que peut acquerir une fonction quand on y permet ses variables de toutes les manieres possibles. // Crelle J. | 1860. | 5. | 9{42. Mat2] E. Mathieu. Memoire sur l'etude des fonctions de plusiers quantites, sur la maniere de les formes et sur les substitions qui les laissent invariables. // Crelle J. | 1861. | 6. | 241{323. Mat3] E. Mathieu. Sur la fonction cinq fois transitive des 24 quantites // Crelle J. | 1873. | 18. | 25{46. Sh] G. Shabat. On the cartographical normalisation of Grothendieck dessins. | Preprint, LaBRI-universite 1, N93-8, Bordeaux. ShVo] G. Shabat, V. Voevodsky. Drawing curves over number elds. In: Grothendieck Festschrift. | Birkhauser, 1990, b. 3, s. 199{227. ShZv] G. Shabat, A. Zvonkin. Plain trees and algebraic numbers. | Contemp. Math. | 1994. | V. 178. | 233{275. ] . ! " # $ # %&$$#: '' ( ) ' *+, )-)*,.)/. | 0.: 0), 1985. 1,] 2. 1,,), 3. 4,+5 . 4 ") + ) %,-'. | 0.: 0), 1977. ) ! ! $* # 1995 .
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Abstract T. V. Golovatcheva, On the bandwidth dimension of nite-dimensional associative algebras over a eld, Fundamentalnaya i prikladnaya matematika 1(1995), 385{391.
In this paper the bandwidth dimension function on countable-dimensional algebras over a +eld is considered. Appropriate in+nite matrix representations of some rings which are algebras (including skew polynomial extensions of rings) are constructed. Therefore these rings have got zero bandwidth dimension.
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. / ( A#y D] yx = xy+1, x y F#x] f1 x x2 : : :g> . , x, ( x, y(f(x)) = D(f(x)) = @(f(x)) | f(x). L ( : A#y D] ! G(0) : 00 00 1 1 1 0 0 B CC BB 0 2 CC 1 0 B B C B CC : 1 0 0 3 (x) = B CC (y) = BB B 1 0 A 0 4 C @ @ A 0
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#1] K. R. Goodearl, P. Menal, J. Moncasi. Free and residually artinian regular rings // J. Algebra. To appear. #2] [. '( . N . | ".: C- , 1961. #3] D. V. Tjukavkin. Rings all of whose one-sided ideals are generated by idempotents // Communications in Algebra. | 1989. | V. 17. | N 5. | P. 1193{1198. #4] J. Hannah, K. C. O'Meara. A new measure of growth for countable-dimension algebra I. To appear. ( ) 1995 .
. .
. . . e-mail: [email protected]
, ! "#$ .
Abstract V. F. Edneral, Complex periodic solutions of autonomous ODE systems with analytical right sides near an equilibrium point, Fundamentalnaya i prikladnaya matematika 1(1995), 393{398.
The paper contains the proof of a theorem on the relation of frequencies of the periodic complex solutions of a nonlinear ordinary di-erential equation system resolved with respect to derivatives and having analytical right parts with the frequencies of periodic solutions of the corresponding linearized system in the neighborhood of an equilibrium point.
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1] A. D. Bruno (Brjuno). Analytical form of dierential equations. I. // Trans. Mosc. Math. Soc. | 1971. | V. 25. | P. 131{288. A. D. Bruno (Brjuno). Analytical form of dierential equations. II. // Trans. Mosc. Math. Soc. | 1972. | V. 26. | P. 199{239. 2] A. D. Bruno. Local Method in Nonlinear Dierential Equations. Part I | The Local Method of Nonlinear Analyses of Dierential Equations, Part II - The Sets of Analyticity of a Normalizing Transformation. | Springer Series in Soviet Mathematics. ISBN 3-540-18926-2, 1988. | 370 pages. ( " ) 1995 .
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Abstract M. I. Zelikin, Irregularity of optimal control in regular extremal problems, Fundamentalnaya i prikladnaya matematika 1(1995), 399{408.
Questions of typical discontinuity of optimal control in regular extremal problems of general position are discussed. Examples are given when there exists a countable set of accumulation points of switches.
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1] . . // . . -. . . | 1959. | N 2. | . 25{32. 2] Roxin E. The existence of optimal controls // Michigan Math. J. | 1962. | V. 9. | N 2. | P. 109{119. 3] (. ). * + ,-. /-+ , 0 0 + // 1, . 2 3. . . | 1984. | 6. 48. | N 4. | . 754{864. 4] Fuller A. T. Relay Control Systems Optimized for Various Performance Criteria // Automatic and Remote Control: Proc. First World Congress IFAC, Moscow, 1960. | Butterworths, London, 1961. | V. 1. | P. 510{519. 5] ) + . :. . - . | .: 2, 1969.
408
. .
6] Brunovsky P. Every normal linear system has a regular time-optimal synthesis // Math. Slovaca. | 1978. | V. 28. | N 1. | P. 81{100. 7] Brunovsky P. Existence of regular synthesis for general control probleme // J. Di;. Eq. | 1980. | V. 38. | P. 317{344. 8] < *. . = , // ( 2 3. | 1982. | 6. 267. | N 3. | . 532{535. 9] Sussman H. Synthesis, Presynthesis, Su>cient conditions for optimality abs. subanalytic sets // Non-linear Controllability and Optimal Control. Monograph Textbook Pure Appl. Math., 133 (ed. by H. Sussman). | N. Y.: Dekker, 1990. | P. 1{19. 10] < . 1., ) . . 30 .GKG K. + ,-. // 6- 1 2 . . . . | 1991. | 6. 197. | . 85{167. 11] Kupka I. The ubiquity of Fuller's phenomenon // Non-linear Controllability and Optimal Control. Monograph Textbook Pure Appl. Math., 133 (ed. by H. Sussman). | N. Y.: Dekker, 1990. | P. 313{350. 12] Zelikin M. I., Borisov V. F. Theory of Chattering Control with applications to Astronautics, Robotics, Economics and Engineering. | Birkhauser, Boston, 1994. 13] Kelley H. J., Kopp R. E., Moyer H. G. Singular Extremals // Topics in Optimization (ed. by G. Leitmann). | N. Y.: Acad. Press, 1967. | P. 63{103. 14] [ *. ., ) + . :., :-, 3. ., G \. . . ] . | .: 2, 1976. 15] (/ ]+ . ^., K . . 2/- / _ ,-. `
. // j . | 1968. | 6. 8. | N 4. | . 725{779. 16] ( . ., K . . =.
.
- ] . | .: 2, 1989. 17] K . ., + 2. [. + . | .: 2, 1993. ' ( 1995 .
, . .
. . .
, L , n + 1, I , ! n " L=I . $, % ( I ) ( .
Abstract
K. A. Zubrilin, On class of nilpotency of obstruction to embeddability of algebras satisfying Capelli identities, Fundamentalnaya i prikladnaya matematika 1(1995), 409{430.
In a /nitely generated algebra L satisfying Capelli identities of order n + 1 over an arbitrary /eld there exists a nilpotent ideal I such that the class of nilpotency of the ideal I is not greater than n and the quotient algebra L=I is embeddable. It is shown that this bound of class of nilpotency of obstruction (ideal I ) in the class of algebras of /nite signature cannot be improved.
,
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. 1. C - L, C | , n + 1. I L, L=I I ! n. 1995, 1, N 2, 409{430. c 1995 , !" \$ "
410
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$ !%
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415
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416
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16i1 <
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=
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F (b) | + C - F , $ + , $ ! p (b), 1 6 p 6 n% 2) / '(b) $ Wq C/ p(b)j 0 6 p 6 n] $ : f d = '(b)(f d), f 2 C/ p(b)j 0 6 p 6 n], d 2 Wq , ($ , Wq ,! F C Wq d ! 1 d)% 3) 2 f1 2 F(b), f2 2 C/ p(b)j 0 6 p 6 n], '(b)(f1 f2 d) = f1 f2 d,
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419
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X
16j1 < <jp 6q
dq jxj1 =b xj1 :::xjp =b xjp
1 6 p 6 q, p (b) dq (x1 : : : xq ) = 0, q < p 6 n, 0(b)jWq = id. $ Wq p (b), 0 6 p 6 n, , , ( x x. $ 5. = b x1 : : : xq , p (b) dq (x1 : : : xq ) x1 : : : xq , , Wq . - , , ( x x , b 2 D(L) p (b) dq (x1 : : : xq ) Wq . + $ , p (b), 0 6 p 6 n, Wq C/ p(b)j 0 6 p 6 n]- . D $,
p (b) ( r (b) dq (x1 : : : xq )) = r (b) ( p (b) dq (x1 : : : xq )): , 1 6 p r 6 q,
p (b) ( r (b) dq (x1 : : : xq )) = p(b) =
X
X
16j1 < <jr 6q
dq jxj1 =b xj1 :::xjr =b xjr =
X
16i1 <
dq jx1=bn1 x1 :::xq =bnq xq
ni = 0, i 2= fi1 : : : ip g fj1 : : : jr g, ni = 1, i 2 fi1 : : : ip g 4 4 fj1 : : : jr g, ni = 2, i 2 fi1 : : : ip g \ fj1 : : : jr g. + , C/ p(b)j 0 6 p 6 n]- Wq . 4 p (b), 0 6 p 6 n, Wq , p (b), 0 6 p 6 n, D(L=Bq+1 ). $ C/ p(b)j 0 6 p 6 n]- Wq p (b), 0 6 p 6 n, D(L=Bq+1 ), ' J '(b) D(L=Bq+1 )- Wq , ' 1) { 3). # , J '(b) 4). 4 -
420
. .
p (b), 0 6 p 6 n, , Wq n X
'(b)
= =
j =0
(;1)j j (b) g(bn j v x1 : : : xr ) = ;
n X
(;1)j j (b) g(bn j v x1 : : : xr ) = ;
j =0
q X
X
j =0
16i1 <
(;1)j
g(x0 x1 : : : xr )jx0 =bn;j vxi1 =b xi1 :::xij =b xij = 0:
# $ { q ( 3). D $ .
7. d | $ + J \ L. g(x0 x1 : : : xs) | 6,
x0 x1 : : : xq , q 6 s,
x1 : : : xq $ , , !
! x. gjx0 =dx1 =v1 :::xq =vq 2 Bq+1
v1 : : : vq | $ + L.
. 4 Bq+1 , L=Bq+1
q + 1. + J , d J \ L, , d=
r X i=1
hi
n X m=0
(;1)m m (ai) ani m vi ;
hi 2 F C D(L), ai 2 D(L), vi 2 L, 1 6 i 6 r. 8 J D(L=Bq+1 )- Wq ='(a1 ) '(ar ). + , , g(d x1 : : : xq : : : xs) 2 Wq (1 g(d x1 : : : xq : : : xs)) = = 1 g(d x1 : : : xq : : : xs). - , 4) 6 , r X
g(
i=1
hi
n X
m=0
(;1)m m (ai ) ani m v i x1 : : : xq : : : xs) = 0: ;
I hi, a hi , a J D(L) D(L=Bq+1 ). ; . > , J '(ai ) '(aj ), i 6= j, ' , . $ 7 J '(a1 ) '(ar ) $ J '(a(1) ) '(a(r) ), 2 Sr .
421
8. g | 6,
x1 : : : xq $ , , !
! x. ! v1 : : : vq 2 J \ L gjx1 =v1 :::xq =vq 2 Bq :
. @ , q. E , q = 1. @ $ , g . # , q ; 1. # , gjx1 =v1 :::xq;1 =vq;1 $ C- , hjt1=w1 :::tq;1=wq;1 , h(t1 : : : tq 1 ) | , t1 : : : tq 1 xq , t1 : : : tq 1, t1 : : : tq 1 | x, w1 : : : wq 1 | L. 4 7 , hjxq =vq 2 Bq . ; . ;
;
;
;
;
' 1. J \ L L #
n + 1. #$ 5 | C- C/ q (a)j a 2 D(L), 1 6 q 6 n], q (a), a 2 D(L), 1 6 q 6 n, | ' . E $,
0(a) = 1C . #$ J | 5- 5 C L, n X (;1)q q (a) an q v ;
q=0
a 2 D(L), v 2 L. Q, J \ L J \ L. ' 2. J \ L L ! n.
x4
% -
L0
#$ L, 6, 5, J , '( J (. J , $, $, 6 . 9 J , , D(L=L \ J) PI- . + , L L=L \ J, 5- $ 5 L . 4 C- L=L \ J C- 5 C L=J, C- D(L=L \ J) C- D(5 C L=J), 5- 5 C L=J, D(5 C L=J) 5- L=L \ J, D(5 C L=J) . 0
0
422
. .
9. ) 4" C - D(L=L \ J) P I - n(n + 3)=2 +,, " 1. ) 5 5- 5 H 5- 5 D(L=L \ J), H | D(L=L \ J),
$ !. . ) 4 J 5 C L , D(L=L \ J) n + 1 5 5- 5 D(L=L \ J). + , , D(L=L \ J) n + 1 X sgn( )x(n) y1 x(n 1)y2 yn x(0) = 0 ;
2Sn+1
Sn+1 f0 1 : : : ng. #
, $ n(n + 3)=2 JJ, 1.
) 5- 5 H P I- , H 5 n + 1. 4 B" , 5- $ 5 H . ; . #$ X | ' L , L | $ C- L=L \ J 6. ? L C- $ 6 X. 9 6 X $ , X, $ " 6. E $ lt (G) " G, ('( t . 9 lt (G), G | , ' g, $ lt (g) , ,. E $, g $" h (g < h), t lt (g) > lt (h) lt+j (g) = lt+j (h) ( j 2 N. $, g1 < g2 , g2 < g3, g1 < g3. = g1 < g2 , , g2 < g1 . 10. g | 6 + X .
,
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G, : n + 1
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, $
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. L Z- " 6 + X !, + # g X , g. . #$ g | 10. + 15 x 5 , g $ 0
0
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0
423
h(t1 : : : tn+1)jt1 =y1 :::tn+1 =yn+1 h | 6 X t1 : : : tn+1 , t1 : : : tn+1, yi , 1 6 i 6 n, | 6 X, , ' yi , ri , h , ti , si , " 0 6 r1 < < rn+1 r1 + s1 > > rn+1 + sn+1 : # $ L
n + 1, 0
X
2Sn+1
sgn( )h(y(1) : : : y(n+1) ) = 0:
> g h(y(1) : : : y(n+1)). D $, g < g ( ( 2 Sn+1 . > , g g $ X . # t( ) = max1 6 i 6 n + 1 (i) 6= i si + r(i) . + lt() (g) < lt() (g ) lt()+j (g) = lt()+j (g ) ( j 2 N. D $ .
11. 2 L | L=L \ J , 5- 5 L . . #$ X | ' L . > Y D(L=L \ J) 0
0
0
!(g1 : : : bgq : : : gr ), r = r(!) | $ , ! 6, g1 : : : bgq : : : gr | 6 X, $, ' , $" n. # $ X 6 , Y . 4 9 , 5- $ 5 H , H | D(L=L \ J), Y . ? L C- $ 6 X. # 10 6 X, $ $ 14 x 5, C- $ L . #$ g | 6 X, m 14. + g $ a1 am x, x 2 X, a1 : : : am | D(L=L \ J), !(g1 : : : bgq : : : gr ), ! 2 6, g1 : : : bgq : : : gr | 6 X. + J , g 14, , a1 : : : am m ; n Y . + , m > n, ' $ j, aj aj +1 : : : aj + m=n+1] 1 2 Y . # $ 5- $ , $" m g 0
0
;
424
. .
$ 5 L 5- , 6 X, ( $ X ( g, " $(, '( , $" " g. ; . 0
x5
. (
9 J , $ $ 3. + $(, x 4. #$ S2n | , ' f1 : : : 2ng. > X f1 : : : ng, Y | fn + 1 : : : 2ng. D Z X $, ZS2n T (Z) ' : X T (Z) = sgn( ) 2S2n (Z )Y
( Z, T (?) =
X
2S2n
sgn( )
sgn( ) | 2 S2n . * 1. 5 " ZS2n X X (;1) Z T(Z) = sgn( ) (9) j
j
Z X
2S2n (X )=X
Z X , ? X . . D 2 S2n Z( ) X ' : Z( ) = 1(Y ) \ X. + X T(Z) = sgn( ) (10) ;
2S2n Z Z ()
# ZS2n X X X sgn( ) = (;1) Z T(Z) = (;1) Z j
Z X
j
j
Z X
j
2S2n Z Z ()
=
X X
(;1) Z sgn( ) :
2S2n Z Z ()
j
j
425
9 P . - Z Z ()(;1) Z , Z Z( ), ? Z( ), 1, Z( ) = ?, 0, Z( ) | . @ Z( ) $ , (X) = X. # . 12. ( " k, 0 6 k 6 n, + hk " ZS2n , j
j
X
j
Z j=k
T(Z) = hk
X
sgn( )
2S2n (X )=X
Z k X . . 4 , X X X T (Z) = sgn( ) = Z =k Z =k S2n Z Z () X n (11) X X = Cmk sgn( ) : j
j
j
j
2
m=k
2S2n jZ ()j=m
# (10), J , ' k ' m Cmk . + $ $, m 2 f0 1 : : : ng X
2S2n jZ ()j=m
sgn( ) = tm
X
2S2n (X )=X
sgn( ) = = tm
X
2S2n jZ ()j=0
8 Rm , ' (i1 j1 ) (im jm ) 1 6 i1 < < im 6 n < j1 < < jm 6 2n. $, X
2S2n jZ ()j=m
sgn( ) =
+ tm = ; .
X
2Rm
X
2Rm
sgn( )
sgn( ) hk =
X
2S2n jZ ()j=0 n X m=k
Cmk tm :
sgn( ) :
sgn( ) :
426
. .
> , (11) 1. 4 , n X X X (;1) Z T (Z) = (;1)k T(Z) = j
Z X
=
n X
j
(;1)k
n X
k=0 Cmk
Z j=k
j
X
sgn( ) =
k=0 m=k 2S2n jZ ()j=m n X m X X = (;1)k Cmk sgn( ) = m=0 X k=0 2S2n jZ ()j=m
=
2S2n jZ ()j=0
sgn( ) :
13. - ZS2n Z- ! x1 : : : x2n, x1 x2n = x(1) x(2n) ! 2 S2n . T(Z) x1 x2n Z- "
" 2n , ! x1 : : : x2n, !
!, $ ! jZ j, | 2n ; jZ j. . 4 T (Z) , T(Z) =
X
2S2n (Z )Y
sgn( ) =
X
X
U Y 2S2n (Z )=U
sgn( )
U Y , jU j = jZ j. 8 X
X
U Y 2S2n (Z )=U
sgn( ) x1 x2n
Z X, U Y , jU j = jZ j. $, U X Y nU. ; . 8 ' , (xij ), xij , 1 6 i j 6 2n, | ' . # , e0 = x11 x2n2n e0 = = x1(1) x2n(2n). E JRS , S R f1 : : : 2ng jS j = jRj, $ , (xij ),
, S R. # , xij , 1 6 i j 6 2n, " JRS = 0, jS j = jRj > n+1. I , , (xij ) ( n. + 13 , T (Z) e0 = 0, jZ j 6= n.
427
4 12 1 , X
Z X
(;1) Z T(Z) e0 = j
j
n X q=0
=
(;1)q hq
X
sgn( ) e0 =
2S2n (X )=X (;1)q hq JXX JYY = (;1)n hn JXX JYY
n X q=0
= JXY JYX
$ hq JXX JYY = 0 q < n. + , JXX JYY = JXY JYX : (12) + $ $ $(. L $ $ $ $. 9 " ' . #$ G | h, x | $ " G. + , " x, ( " y G, " x
" y h. $ " x. D , (x y) + (x h) = (y h), " G . max (x h), x " G h | $ G, G. ; G " G, 0.
14. G |
, n | . ! n+1
#
G, $ !
# n,
# ,
, ,
.
G
G, ,
G,
n ,
#
G G ! n. 0
0
. = G ( n, G $ G. #$ G $" n. + X " G $ ( " G, n. 9 , $ G X. = " x X, " y G, ' x G, X. + , X " G , $ G. G G n. # G n $ . 0
0
0
0
0
428
. .
D $, (x G ) 6 n ( " x G. # $ G | $ G, ' " y, (x G ) = (x y) " y x G. + (x G ) = (x y), " x y X. I , (x y) 6 n. ; . 0
0
0
0
15.
#
G n + 1 $ !
# , $ !
n.
#
G n + 1 $ !
# , i-
#
G si ,
, i-
# , ri,
: 0 6 r1 < < rn+1 r1 + s1 > > rn+1 + sn+1
,
# $
,
.
. #$ x1 : : : xn+1 | " G, ( n. # ", , , (x1 h) > > (xn+1 h), h | $ G. 9 , " xi, " yi , yi i ; 1 (xi yi ) = n ; i + 1. I $
i, 1 6 i 6 n+1, $ " xi n. + " yi , 1 6 i 6 n + 1, . ; .
x6
* + 1 { 3
9 J " $ 2, 1 3. + 2. #$ 5, J | , x 3. 4 1 x 3 , $ J \ L ( n. 4 11 x 4 , 5- $ 5 L , L | L=L \ J. #$ X | ' L . # Y , Y D(L), , $ 11 x 4. > 5 5,
q (a1 am ), 0 6 q 6 n, 1 6 m 6 n(n + 3)=2. + B" , 9 x 4 J , D(L=L \ J) n 5 5 D(L=L \ J), , 5 - $ 5 L . + . + 2 , $, C , . 0
0
0
0
0
0
0
429
+ 1. # 1, C | . + 2 , C- L=I 5- L~ , 5 | , , 5- $ L~ I L $ " n. - $ /2] (. /3]), 1. + 3. #$ , | , L. #$ I | $ L, $ $" n, L=I . + D(L=I) . #$ f(t1 t2) | $ ( t1 , t2 JJ, 1, D(L=I). + f( ) x 2 I, x | ' L. # I n = 0. # (f( ) x)n = 0. # $ L | ,
n + 1, L $" n + 1. - , (f( ) x)n = 0 $ n. # 3.
(
1] . . // . | 1989. | . 28. | N 2. | !. 127{143. 2] & ' (. ). ( . ). *+ ,- . - // /. - . 0. | 1986. | . 41. | N 5. | !. 161{162. 3] &0 3. (0 . | *.: * , 1971. 4] '+ *. 6. 7 . 7 // * . . | 1989. | . 180. | N 6. | !. 798{808. 5] 0 (. . , 0 :; < (. // * . . | 1995. | . 186. | N 3. | !. 53{64. 6] *+ . ). = . - ,- // * . . | 1943. | . 13. | N 2{3. | !. 263{286. 7] * 6. . = ,- - +- // 6 . *>/. * ., -. | 1975. | N 6. | !. 118{119. 8] * 6. . = . + , . < - P I - // 6 . *>/. * ., -. | 1989. | N 2. | !. 17{20. 9] , + !. 6. 7 . - // !. . <0 . | 1989. | . 30. | N 1. | !. 134{144. 10] ? @. . = A< P I - - // . | 1974. | . 13. | N 3. | !. 337{360. 11] ? @. . , 0 :; < B . (. // ). 3 !!!?. ! . . | 1981. | . 45. | N 1. | !. 143{166.
430
. .
12] Amitsur S. A. A non-commutative Hilbert basis theorem and subrings of matrices // Trans. Amer. Math. Soc. | 149(1970), p. 133{142. 13] Irving R. S. AFne PI-algebras not embeddable in matrix rings // J. Algebra. | 82(1983), p. 94{101. 14] Procesi C. A formal inverse of the Caylay { Hamilton theorem // J. Algebra. | 107(1987), p. 63{74. 15] Procesi C. Rings with polynomial identities. | Dekker, N.Y., 1973. 16] Razmyslov Yu. P., Zubrilin K. A. Capelli identities and representations of Onite type // Commun. Algebra. | 22(14). | 1994. | P. 5733{5744. 17] Small L. W. An example in P I -rings // J. Algebra. | 17(1971), p. 434{436. & ' 1995 .
. .
. . .
, .
. ! . ! # #.
Abstract
E. A. Lapshin, The denition of acoustic signal propagation domain, Fundamentalnaya i prikladnaya matematika 1(1995), 431{454.
The propagation domain of solution of the Burgers equation with initial function which is equal to zero outside an arbitrary segment is considered. The precise and asymptotical estimations of solution carrier in arbitrary cutting are derived. An algorithm for the carrier determination for arbitrary initial function is presented. The results have numerical applications.
1 " = 0
(" = 0) @u ; u @u = 0 @z @ u (z = 0 ) = () u = ( + zu). " z, u(z ) $ % & . ' $ , . . % , ( %). () , z = 0 * ) (z = 0), ;1 + ; ; + = ;1 ; (u ) ; u z = + (u ) ; u z (1) d (z) = ; 1 (u; + u+ ) dz 2 ! ,,- ( N 95-02-03827a).
1995, 1, N 2, 431{454. c 1995 !", #$ \& "
432
. .
u; u+ (z) | $0 u+ u; | $ , . . u; = !lim u(z ) (z);0 u+ = !lim u(z )0 (z)+0 ; () | % & , * % ' % ;+ ) | % &, ' , + () 0 ;1 ;+ (u ;+ (). 1 % u; = ; ( ), u+ = + ( ). 2. 2$ 3 3. (4 $ $ 3 3.) 3 u(z = 0 ) = (),
8> 0 6 b > a < () = > k2 + v2 2 5b 0] : k1 + v1 2 50 a]:
()
4 $ k2 6 0 k1 6 0 v2 6 0 v1 > 0: k2 = 0 v2 = 0 $ $ $ | 3 3. 7 (1) d (z) = ; u; + u+ dz 2 +;v u (z) = k 1 ; u+ z 1 ; (z) = u k; v2 ; u;z
2
d (z) = ; u; + u+ dz 2 k 1 (z) + v1 + (10 ) u = 1;k z 1 u; = k21 ;(z)k +z v2 : 2 ) u+ u; , $ d (z) = ; 1 k1 (z) + v1 + k2 (z) + v2 dz 2 1 ; k1 z 1 ; k2 z k1 + k2 1 v d + 1 v 1 2 dz 2 1 ; k1z 1 ; k2z = ; 2 1 ; k1 z + 1 ; k2z (0) = 0 | z = 0:
433
8 $ %%&3 . p (z) = k ;1 k (v1 (1 ; z2 ) ; v2 (1 ; zk1 ) + (v2 ; v1 ) (1 ; zk1)(1 ; zk2 )) = 2 1 p p p p = k ;1 k (v1 1 ; zk2 + v2 1 ; zk1 )( 1 ; zk2 ; 1 ; zk1): 2
1
$ % (z) z k1 6= k2 . 4 $ k1 = k2 = k (z) d (z) + k (z) = ; 1 v1 + v2 dz 1 ; kz 2 1 ; kz (0) = 0 = C(z) (1 ; kz) (z) = ; 12 (v1 + v2 )z: k1 6= k2 % (z) ' ' : p p p p (z) = k ;1 k (v1 1 ; zk2 + v2 1 ; zk1 )( 1 ; zk2 ; 1 ; zk1) = 2 1 p p = k ;1 k v1p1 ; zk2 + vp2 1 ; zk1 ((1 ; zk2 ) ; (1 ; zk1 )) = (2) 1 ; zk2 + 1 ; zk1 2 p1 p = ; v1 p1 ; zk2 + vp2 1 ; zk1 z: 1 ; zk2 + 1 ; zk1 7 *3 $ (z) z, 3 3 3 . :' % (). 3 v1 v2 k1 k2 v1 = ;v2 > 0 k1 = k2 > 0: S1 S2 | * ) 3 z = 0. 2 p ; p1 ; k2z) p 1 ; k2 z p (z) = v2 z( 1 ; k2z + 1 ; k2 z 2 S1 = 12 v1a = ; 21 vk1 1
434
. .
2 S2 = 12 v2a = ; 21 vk2
S1 = v1 2 k2 = 2 : S2 v2 k1 2
8 (z0 ) = b, . .
p
p
v2z0p( 1 ; k2 z0 p; 1 ; k2 z0 ) = ; v2 : k2 1 ; k2z0 + 1 ; k2z0 ; < )3 z0 , z = z0 ' 3 b, . . 3 3 . p p p p z0 k2 ( 1 ; k2z0 ; 1 ; k2z0 ) = 1 ; k2z0 + 1 ; k2z0 p p 1 ; k2z0 (1 + k2z0 ) = 1 ; k2z0 (z0 k2 ; 1) p ;(1 + k2z0 ) = (1 ; k2z0 )(1 ; k2z0 ) (1 + k2 z0)2 = (1 ; k2 z0 )(1 ; k2 z0 ): $ , ' 3, $ ;(1 + k2 z0 ) > 0: (3) 8 1 + 2k2z0 + 2 k22z02 = 1 ; z0 k2(1 + ) + k22 z02 z0 k2(2 + 1 + ) = z02 k22( ; 2 ): 3 z0 k2 6= 0, +1+ (4) z0 = 2 k ( ; 2) : 2
; > 0 k2 < 0, $ z0 , $ % (4), )3 3 ; 2 < 0, . . S1 > S2 . , $ ; 2 < 0 (3). ( 3, 2 2 + + 2 + + + > 0: ; 1 + 2+;+2 1 = ; ; +2 = ; 2 ; ; 2 = , $, $ (z) = b )3 , % (4) 3 S1 > S2 . >$ (z) = a, )3 * 3 S1 < S2 , % + + ) : z0 = (2 (5) k2 (2 ; )
435
@ ) 3 z? 3 u(z = 0 ) = () (% ()). ; S1 > S2 , ) 50 z0), z0 = k22(+1+ ;2 ) , 3 z % & u( z)
8 0 6 b > a > > < k2 + v2 u( z) = > 1 ; k2z 2 5b (z)] > : k11; +k vz1 2 5 (z) a]: 1
C3 (z) % (2)
(6)
p
p
(z) = ; v1 p1 ; zk2 + vp2 1 ; zk1 z: 1 ; zk2 + 1 ; zk1 z = z0 * 3 :
8< 0 < b > a u( z0 ) = : k1 + v1 1 ; k1z0 2 5b a]:
( z~ = z ; z0 ~ = ; b % (2) $ v~2 = k~2 = 0, k~1 = 1;kk11z0 , v~1 = v1;1 +kk11z0b , $ , $ z > z0 , % &, * % ,
8> 0 6 (z) > a < u( z) = > k~1~ + v~1 : 1 ; k~ z~ 2 5 (z) a] 1
(z) = ~ (~z ) + b,
q
~ (~z ) = v~~1 ( 1 ; z~k~1 ; 1): k1 ; ) S1 < S2 , z 2 50 z0), + + ) z0 = (2 k (2 ; ) 2
3 % (6) . + + ) z = z0 = (2 k (2 ; ) 2
436
. .
3 3
8 0 6 b > a < u( z0 ) = : k2 + v2 1 ; k2z0 2 5b a]:
z > z0 u( z)
8> 0 6 b > (z) < u( z0 ) = > k~2~ + v~2 : 1 ; k~ z~ 2 5b (z)] 2 0 k~2 = 1 ;kk2 z v~2 = v12;+kk2za 2 0
2 0
(z) = ~ (~z ) + a z~ = z ; z0 ~ = ; a q ~ (~z ) = v~2 1 ; z~k~2 ; 1 : k~2 3. E z, 3 $ . 3 u(z = 0 ) = (. 1). k1 k2 | 3, b < 0, a > 0, v1 > v2 . v4
() k1
v1 b
B
O v2 k2
A
a
v3 . 1
1 , z ' B, ' A & $ . < ) , $ $ A B = a ; v4z = b ; v3 z
437
, . . ) & $ $ . = $ z (z) = b ; v3 z (7) (z) = a ; v4 z (8) . . < z ) 3 &. 3 v1 = ;v2 k1 = k2 . (7) ) (z) . 2, $
p
p
v2 z( p 1 ; k2z p; 1 ; k2 z = b ; v3z = b ; (v2 + k2b)z = b(1 ; k2z) ; v2 z: 1 ; k2z + 1 ; k2z
p
p
v z( 1 ; k z ; 1 ; k2z) = p p p 2 p 2 = b(1 ; k2z)( 1 ; k2z + 1 ; k2 z) ; v2 z( 1 ; k2z + 1 ; k2z) p p p 1 ; k2zv2 z( + 1) = b(1 ; k2 z)( 1 ; k2z + 1 ; k2 z) p v2 z( + 1) = b(1 ; k2z) + b (1 ; k2z)(1 ; k2 z) p (v2 ( + 1) + bk2)z ; b = b (1 ; k2z)(1 ; k2 z) v2( + 1) p + k (1 ; k2z)(1 ; k2z): 2 z ;1 = b :' $ $ C := v2(b+1) + k2.
p
Cz ; 1 = (1 ; k2z)(1 ; k2z) C 2z 2 + 1 ; 2Cz = k22 z 2 + 1 ; k2z( + 1) C 2 z ; 2C = k22 z ; k2( + 1) (C 2 ; k22 )z = 2C ; k2 ( + 1) 2 v2(b+1) + k2(1 ; ) z = 2C C;2k;2 (1k+2 ) = (9) v2 (+1) 2 + 2k v2 (+1) + k2(1 ; ) 2 2 2 b b ( $ z ) 3 Cz ; 1 > 0, $ z . I 3 &3 (7), (8), < $3 , * )3 (7). Cz ; 1 > 0 z > 0
438
. .
C = v2 (b+ 1) + k2 + ) : z = 2C c;2 k;2 (1 k22 $ C z, $
8 v (+1) 2 v (+1) >> 2 2 + b k2 (1 ; ) b 2 >> Cz ; 1 = c ;Ck22(1+)+k = > 0 2 2 < c ; k2 v (+1) +2k v (+1) + k2(1 ; ) 2 2 b b v (+1) >> (1 ; ) > 0: >> z = v (+1)22 b +v k(2+1) 2 : + 2k2 b + k2 (1 ; ) b 2
2
2
2
2
2
2
" 3 'J : 8> v ( + 1) 2 v ( + 1) + 2 b k2(1 ; ) > 0 >> 2 b < 2v2( + 1) >> b k22(1 ; ) > 0 >: v2( + 1) + 2v2( + 1) k + k2(1 ; ) > 0 2 2 b b 8> v ( + 1) 2 v ( + 1) + 2 b k2(1 ; ) 6 0 > 2 b >< 2v ( + 1) 2 >> b k22(1 ; ) 6 0 >: v2( + 1) + 2v2( + 1) k2 + k2(1 ; ) 6 0: 2 b b
C , $ v2 (b+1) > 0, . . v2 (b+1) = v2 ;b v1 > 0. C$, 3 * 8 v2( + 1) >< b +k22(1 ; ) > 0 (10) >: v2( + 1) + 2v2( + 1) k2 + k22(1 ; ) > 0 b b
8 2v2( + 1) >< + k2(1 ; ) 6 0 b >: v2( + 1) 2 + 2v2( + 1) k2 + k2(1 ; ) 6 0: 2 b b
8 3 $ k2 > 0 k2 < 0. 3 k2 > 0.
(11)
439
2 (10), (11) $ (10) , v2 (b+ 1) + k2 (1 ; ) > 0 (100) v2( + 1) 2 2v2( + 1) (11) , + k2 + k22 (1 ; ) 6 0: (110) b b 7$ k2 < 0. 8 2v2( + 1) >< + k2(1 ; ) > 0 (10) , v (b + 1) 2 2v ( + 1) >: 2 + 2 b k2 + k22 (1 ; ) > 0 b . . (11) < $ ). = , $ * ). 1. ; k2 > 0, * )3 (z) = ;v3 z + b $3 ) 3 ' (100), ' (110). ; ) k2 < 0, $3 ) 3 (1000). 1. k2 > 0 z % (9) z < k12 . 23 (z) = a ; v4z0 $ ) 1 $ * ). 2. ; k1 = k2 > 0, * )3 (z) = ;v4 z+a $3 ) 3 ' ; va2 (1 + ) + k2 ( ; 1) > 0 (12) v2(1 + ) 2 v2(1 + ) + k2 ( ; 1) 6 0: ; 2k (13) 2 2 a a ; ) k1 < 0, $3 ) 3 ; va2 (1 + ) + k2 ( ; 1) > 0 v2(1 + ) 2 ; 2k2 v2(1a+ ) + k22 ( ; 1) > 0: (14) a $3 % ;2 va2 (1 + ) + k2( ; 1) z= : (15) v2 (1+) 2 ; 2k v2 (1+) + k2 (2 ; ) 2 2 a a
2. k1 > k2 > 0 z % (15) z < k1 . 2
440
. .
2
r
. 2
> & & 3 $- 3. 8 , z = 0 % & ():
8 >< 0 < 0 > n () = > vj ; vj ;1 : j ; j;1 ( ; j;1) + vj;1 2 5j;1 = j ] j = 1Kn:
& & < ' z > 0. = . 8 ' )3 3 z ; 50 1) ) 50 z1] (z1 z2 ] : : : (zl 1). ) zi i = 1 l , $' ' z > zi & ' 3 & , | '3 &. < $ zi+1 & z = zi . ( ) ) (zi zi+1] i = 0 l, z0 := 0 zl+1 := 1, $ % , $ & z = zi . - $ ') . 441. 3, 4, 6 ' * ' . > & z = zi . 1) 3 &, ' , $' ' ( $ , ), $ ( . ))). & ) zi+1 ( . ' ' N 4). > & ' , ' ) < 3 $ : ' , & , (. 2)0 ) ' | & )
441
8 #: = 0 n: n 0 1 0 1 j | , j |
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1.
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v v :::v
v
2.
i
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? 7# ! 4.
6
Z > zi
=
zi
3.
+1
zi
?
5.
z
-
+1
?
6.
5 ! ( i i + 1] = . 8 =
z z
z
Z
60 .
5 ! ( i i +1] = i+1 z z
z
z
i
?
= +1 i
7.
> . | , # .
Z
(. 3). ; ) , & < $ . 2$ * : r
2
. 3
442
. .
z = zi $ , $ ) $ ) $. @ ) * : ' z = zi $ $ 3 . 7 $ ' ) m *3 S(m) < ($ , $ $3 , * &3 , &3 *3, $3 )3 )3 *3). L m , S(m) 3 , 3 . 1) zi . $ & z = zi;1 )* 3. 3 $ zi1 zi2 : : : 2 , z = zir , z > zir ' % $ ) , $ % ) 5zi;1 zir ]. 2 ) % , 3, ) & z > zi , zi = minfzi1 zi2 ::g (zi = 1, fzir gjr=12::: | )). C$ zir $ ) 5j j +1 ], * : 1) j j +1 vj +1 > vj . 2 j z zirj = k1 + zi;1 = vj +1 ; i;1 j j +1 ; vj kj | ) 5j j +1), 2) j , j +1 ) 2. 2, % (15), $ zirj = A2 ;;2k2AA++kkj ;(kkj;;1k ) + zi;1 j j j j ;1 ; + v v A = j ;j j +1 j 3) j , j +1 | ) 1. 2, % (9), $ 2B + kj ; kj +1 zirj = b2 + 2k + zi;1 j B + kj (kj ; kj +1) v; ; v+ B = j+1; j +1 : j j +1 2$ % ) (zi zi+1]. C k + 1 0 : : : k vj ( $ : vj+ vj; ) z = zi . 2' % &, * %
443
) (zi zi+1 ]. ( $ 3 ) $ . ; j , < * : j = j ; vj (z ; zi )
vj = vj :
; ) j | , v j = ; j
+
p1 ; z~k + v;p1 ; z~k p1 ; z~kj;1 + pj 1 ; z~k j j j ;1
j
k ( ; ) + v; + j (vj; ) = j ;1 j1 ; kj z~ j ;1
k ( ; ) + v+ (vj+ ) = j j1 ; kj z~ j z~ = z ; zi : j
2 " = 0 6
8 $ @ 2 " @V" " @@V2" + V" @V @ ; @z = 0 ;1 < < +1 0 < z 6 Z0 < +1 V" (z = 0 ) = f(): $ ) < $, , ) 51] (4. M. 7 , ;. >. O \> $ ) $, "). 7 ' ' 52]. R $ (1) 3 " > 0, ) (" = 0) $, . < $ ) $ , ) $ 0 < z 6 Z0 = (z) $ < . ) , $ ) V" (z ) =
N X n=0
vn (z )"n + o("N ):
(2)
444
. .
( ) * V" (z ) =
N
X
n=0
vn(z ) + wn(z ; " (z) ) "n + o("N )
(3)
% & vn(z ) ) , $ ) (2). 2 ) (3) ) ) (2), % & wn (z ) j j ! 1 ) 3 ( j ; (z)j > 0 > 0, 0 ", j ;" j " ! 0 ) 3 3 '3 $ ). 4 % & V" (z ) z > 0 ) 3 ')
N X V~ (N ) (z 0 ") = V (N ) (z ") + W (N ) (z ") = vn(z ) + wn(z ; " (z) ) "n n=0
) . 7$ , * <%%& ) (2), ) 3 $ (z). (2) (1) $ ". $
@v0 @v0 @v1 @v1 @v0 @2v0 @z ; v"0 @ + @z ; v0 @ ; @ v1 ; @2 " + # n;1 n ; v @vn ; v @v0 ; @ 2 vn;1 ; X v @vn;k "n + + o("N )=0 + @v k @ @z 0 @ n @ @ 5v0 (0 ) ; f()] +
N X
n=1
k=1
vn (0 )"n + o("N ) = 0:
(4) 8 (4) ' $3 $ o("N ) $ " $, <%%& ". 73, ) '3 <%%& < $, < @v0 ; v @v0 = 0 v (0 ) = f() 0 @z 0 @ @v1 ; v @v1 = @v0 v ; @ 2 v0 v (0 ) = 0 1 @z 0 @ @ 1 @2 .. . ;1 @vn ; v @vn ; v @v0 = @ 2 vn;1 ; nX @vn;k v (0 ) = 0: v 0 n k n @z @ @ @ @ k=1
445
C$ v0 (z ) ) $ (" = 0). @ f() $- , $ , $ v1 = v2 = : : : = vn = 0. 3 , * <%%& ) (3). 2 % & wn z ;" (z) $ 3 (z), < % & ' 3 3 $ , j ; (z)j $ . 1 < $ $ % & vn (z ) $ $ 3 $ = (z), < % & vn (z ) ) (3) ) 3 ) 3 $ (z): vn (z ) =
N 1 @ m v X n ( ; (z))m + O(( ; (z))N ) = m m! @ m=0 N 1 @ m v X n
=
m=0 m!
@m
m "m + O("N ):
(5)
C3 = ;" (z) , $ (R(z (z))) ' $ 3 $ % & R(z ) :
8< lim R(z ) = R (z (z)) 6 (z) (z);0 (R(z (z))) = : !lim R(z ) = R(z (z)) > (z): ! (z)+0
8 z = z < : = ; (z) :
2
"
@V (z ) = 1 @V (z ) @ " @ 2 @ V (z ) = 1 @ 2 V (z ) @2 "2 @ 2 @V (z ) = @V (z ) ; 1 d (z) @V (z ) @z @z " dz @ (1) z 2 @V ) ; @V = 0: ";1( @@ V2 + ddz(z) @V + V @ @ @z
(10 )
446
. .
=3 (5), (3) (10 ). M $ ", $
2 0
+ w ) @w0 + ";1 @@ w20 + ddz(z) @w + (v 0 0 @ @
@2w1 d (z) @w1 @v0 @w1 @v0
+ @ 2 + dz +(v0 + w0 ) @ + @ + @ + @ 0 @w0 @(v0 ) @w0 + w1 + @v @ ; @z ; @z + + m "m;1 + +o("N ;1 ) @ @w @v @v 2 2 @w0 1 0 + w w1 + @0 + 2 = @@ w22 + ddz(z) @w 2+ @ @ @ + @
@w1 2
) ; @ @v0 + @w (w ; 0 + (v0 ) @ @z @ @z m > 3 2 m @w0 w + m = @@ w2m + ddz(z) @w m @ + @ mX ;2 @wl w + @wm;1 w + @v0 + @vm (v + w ) ; @wm;1 : + 1 0 @ m;l @ @ @ 0 @z l=2
<%%& " , $ % & wn n = 0 1 : : : @ 2 w0 + d (z) @w0 + (v + w ) @w0 = 0 0 0 2 @ dz @ @ 2 @ w1 + d (z) @w1 + @ ((v + w )w ) = @ 2 dz @ @ 0 0 1 d (z) @v0 ; (v +w ) @v0 ; @v0 @w0 0 @(v0 ) + ; = @w 0 0 @z @z dz @ @ @ @ 2 @ w1 + d (z) @w2 + @ ((v + w )w ) = @ 2 dz @ @ 0 0 2
@v 2 ! @v @w @ @v @w 0 1 0 1 @w1 w1 + @ + @z @ ; @0 = @z ; @ w1 ; @ m > 3 @ 2 wm + d (z) @wm + @ ((v + w )w ) = 0 m @ 2 dz @ @ 0 @(v0) mX ;1 @w @w @w l m ;1 m ;1 wm;l + @ : = @z ; @ wm;l ; @ l=1
447
1 % & w0(z ) w1 (z ) $ , v" (z = 0 ) = f(), f() 3 * % & 8 V < f() = : 0 6 0 > a a = ; k (6) k + V 0 6 6 a V > 0 k < 0: 4 < $ d (z) = ; 1 p V v = 0 v = p V : 0 dz 2 1 ; zk 0 1 ; zk 1) w0 . ( % & w0 (z ) @ 2 w0 + d (z) @w0 + (v + w ) @w0 = 0 < 0 (7) 0 0 @ 2 dz @ @ @ 2 w0 + d (z) @w0 + (v + w ) @w0 = 0 > 0: (8) 0 0 @ 2 dz @ @ < $ ( * % & z). ( '$3 % & w0(z ) j j ! 0. : ' , $' 3 ') ~V (z ") = v0(z ) + w0 z ; "
= (z). L ') * : w0 (z ;0) + v0 = w0(z +0) + v0 : (9) (7), (8). 7
y(;) = v0 + w0, y(+) = v0 + w0. 2 ' ) . @ , (9) <%%& % & y(;) (z ) y(;) (z ) $ = 0, . . y(;) (z ;0) = y(;) (z +0). < % & w0(z ) % & v0 (z ) $ $3 % & y(z ), * @ 2 y + d (z) @y + @ 1 y2 = 0 ;1 < < 1: (10) @ 2 dz @ @ 2 " ) 3 lim y(z ) = v0 !+1 lim y(z ) = v0 : !;1
448
. .
' $ (10) , $ @y = ; 1 y2 + d (z) y + C(z) @ 2 dz C(z) | , 4' ' , $' y v0 % & @y @ 3
1 d (z) 1
2
)2 ; v0 + v0 v = ; 1 v v : C(z) = ylim y + y = (v !v0 2 dz 2 0 2 0 2 0 0 ; y ! v0 , C(z) $ ) ). 2 ' , % & y(z ) @y = ; 1 y2 + d (z) y + 1 v v : (11) @ 2 dz 2 0 0 ( $3 (6) @y = ; 1 y2 + 1 p V y: (12) @ 2 2 1 ; zk 8 (12) % & pV 1;zk ; 12 p V (+C0 (z))
: (A) 1 + e 1;zk @ C0(z) ' , $' * ') $ 3 ". z = 0 ' , $' 3 y=
Z1
;1
wn(0 )d = 0 n = 0 1 2 : : ::
= (13) n = 0 , $ C0 (z = 0) = 0. 1) w1. ( % & w1(z ) @ 2 w1 ; 1 p V @w1 + @ (yw ) = @w0 < 0 @ 2 2 1 ; zk @ @ 1 @z
(13)
(14)
@ 2 w1 ; 1 p V @w1 + @ (yw ) = @w0 ; @ 2 2 1 ; zk @ @ 1 @z (15) @v0 @v0 @v0 @ @v d (z) 0 ; @ @ ( w0 ) + @z ; dz @ ; v0 @ > 0:
449
$3 (6)
0 p V @v = 1 ;k kz 0 @ 1 ; zk dv0 = 1 V k d (z) = ; 1 p V : dz 2 (1 ; zk)3=2 dz 2 1 ; zk < (15) $
v =
(B)
@ 2 w1 ; 1 p V @w1 + @ (yw ) = @w0 ; k @ ( w ): (150) @ 2 2 1 ; zk @ @ 1 @z 1 ; kz @ 0 4 % (14), (15), (150) 3 % & % (A), w0(z ) = y(z ) < 0 w0(z ) = y ; v0 , > 0. = ' $ (14) (150 ) ' 1 ' , $' @w @ 3 w0 ! 0 w1 ! 0, $
Z @y @w1 + y ; 1 p V w 1= @ 2 1 ; zk @z d = (16) ;1 k 1 0 = y 2 1 ; zk ( + C0(z)) + C0(z) < 0 @w1 + y ; 1 p V w = Z @w0 ; k @ ( w ) d = 0 1 @ 21 1k; zk 1 @z 1 ; kz @ (17) p ( + C ( z )) 0 k 1 2 1;zk 0 = ye 2 1 ; zk ( ; C0(z) ; C0 (z)) > 0:
' 3, $' 3 $ % & V~ (1) (z ") ! (z) $3 ' $ $ O("). " &3 ' , @w1 (z +0) ; @w1(z +0) = @v0 (z (z) + 0) ; @v0 (z (z) ; 0) @ @ @ @ ' z 2 50 z0]. 1 (16) (17) $ $ % & w1(z ) , ) ' $ C00 (z) + 12 1 ;k zk C0 (z) = p k : V 1 ; zk :
p
p C0(z) = 1 ; zk(C0(0) ; V1 ln(1 ; kz)) = ; 1 V; kz ln(1 ; kz):
(18)
450
. .
23 (16), (17). 7 = ; p V ( + C ; 0(z)): 2 1 ; zk 2 (16)
(19)
@w1 ; p V 1 p V 1 ; e w = ; 1 p V 1 2k ; 2C 0 (z) : p + 1 0 @ 2 1 ; zk 2 1 ; zk 1+e 2 1 ; zk 1+e V 1 ; zk V 7 * ; 2p1; zk
dw1 + 1 ; e w = 1 2k ; 2C 0 (z) : p 1 0 @ 1 + e 1 + e V 1 ; zk 8 $ w1 = A( )(1 + e );2 e . 1 A( ) A0 ( ) = p 2k ; 2C00 (z) 1 +ee V 1 ; zk 1 2 ; e; (1 + ) ; 2C 0 (z)( ; e; ) + CK (z): A( ) = p 2k 1 0 V 1 ; zk 2 = , < 0 1 e 2 ; 0 ; K w1 = p 2k ; e (1 + ) ; 2C (z)( ; e ) + C (z) 1 0 (1 + e )2 : V 1 ; zk 2 (20) 23 (17). : 3 (19), $
@w1 ; 1 ; e w = e 2C 0 (z) + 2k C (z) + p2k : 0 @ 1 + e 1 1 + e 1 ; kz 0 V 1 ; zk : w1 = B( )(1 + e );2 e B 0 ( ) = (1 + e )(2C00 (z) + 1 ;2kkz C0(z)) + p2k (1 + e ) V 1 ; zk
2 + e ; e + C1(z): B 0 ( ) = ( + e )(2C00 (z) + 1 ;2kkz C0 (z)) + p 2k V 1 ; zk 2 C$, > 0
2 e 2k 2k w1 = 2C + 1 ; kz C0(z) ( +e )+ p + e ; e +C1 (z) (1+e )2 : V 1 ; zk 2 (21) 0 0
451
$ % & C0(z) % (18) (20) (21), $ 2k 1 w1 = p 2 ; e; (1+ ) ; V 1 ; zk 2 (200) ; K e 1 ; 1+ 2 ln(1 ; kz) ( ; e ) + C1 ( ) (1+e )2 < 0 2k 1 1 2 w1 = p ; e (1+ ) ; 1+ 2 ln(1 ; kz) ( ; e ) + V 1 ; zk 2 e (210) + C1 ( ) (1+e )2 > 0: L ') V~ (1)(z ") = (z) ) '3 v1 + w1 (z ;0) = v1 + w1(z +0): : 3 w1 (z ;0) = w1(z +0) v1 = v1 = 0: 4' CK1 (z) ' , $' 3 : 2k 1 1 e0 2 ;0 ;0 K p ; e (1+ ) ; 1+ ln(1 ; kz) ( ; e ) + C (z) 0 0 1 2 (1+e0 )2 = V 1 ; zk 2 0 1 e0 2 ; e0 (1+ ) ; 1+ 1 ln(1 ; kz) ( ; e0 ) +C (z) = p2k 0 0 1 2 (1+e0 )2 : V 1 ; zk 2 0 C3 0 = 21 ln(1 ; kz). : e;0 (1 + 0 ) ; e0 (1 ; 0) + 2 0 + 1 ; 12 ln(1 ; kz) e0 ; CK1(z) = p 2k V 1 ; kz p ; kz) + C1(z): ; e;0 1 + 21 ln(1 ; kz) + C1(z) = 2k ln(1 V 1 ; kz = (13) n = 1 C1(0):
Z +1 ;1
w1(z )d = 0:
= ) (200 ) (210) w1 , $ z = 0 % & w1(z ) $. <
Z +1 Z0 w1(z )d = 2 w1 (z )d = ;1Z 1 ;1 e Z0 0 4k e d =0: 2k 2 ; ; e ; + C d = C (0) =2 1(0) 1 2 (1 + e )2 V +1 (1 + e )2 +1 V
452
. .
: $ , $ C1(0) = 0. 23 % & C1(z). C1(z) ' ' , $' % & V~ (2)(z ") $ $3 ', $ $ O("2 ). " , @w2 (z ;0) ; @w2(z +0) = @v1 (z (z) + 0) ; @v1 (z (z) ; 0) (22) @ @ @ @ ' ' z 2 50 Z0]. ( % & w2 3
@w2 + d (z) + v + w w = Z @w1 ; @ ( 1 w2) d < 0 0 2 0 @ dz @ 2 1 ;1 @z @w2 + d (z) + v + w w = 0 2 0 @ dz Z @w1 @ 1 @v0 @ 2 = ; w ; ( w ) d > 0: 1 @z @ 2 1 @ @ ;1
2 : $ @w @ (22), $ % & C1(z)
Z 0 @w1 Z 0 @w1 @z d = @z d : ;1
+1
= z :
Z @w1 @w1 k k(1 + 21 ln(1 ; kz)) d = + ; V @ 2(1 ; zk) 2(1 ; zk) ; 2p1; +1 @z zk Z @w1 @w1 k 1 ln(1 ; kz)) k(1 + d : 2 = p @z + @ 2(1 ; zk) ; 2(1 ; zk) ; V 0
0
2 1;zk
;1
V + 1 ln(1 ; kz), 0 = 1 ln(1 ; kz). C3 = ; 2p1; 2 2 rz
Z @w1 0
+1
k
@z + 2(1 ; zk)
@(w1 )
@ ; w1
d =
@(w1 ) Z @w1 k = + 2(1 ; zk) @ ; w1 d : ;1 @z 0
3 Z 0 @w1 Z 0 @w1 k k ; 2(1 ; zk) w1 d = ; 2(1 ; zk) w1 d : ;1 @z +1 @z
453
w1 $ % (200) (210). 0 e 2k 1 k ( ; e; ; 2) ; k C (z) d = 0 p C (z) + 2 1 2(1 ; zk) 1 +1(1 + e ) Z 0 e V 1;zk2k2 1 ; kz k k 1 0 ; = 2 C1(z) + V p1 ; zk 2 1 ; kz ( ; e ) ; 2(1 ; zk) C1(z) d : ;1 (1 + e ) Z 0 e d = 10 (z) ; 2(1 ;k zk) C1 (z) 2 ;1 (1 + e )
Z 0 e Z 0 e 2 k ; = V (1 ; kz)3=2 2 ( ; e ; 2)d ; ;1 (1 + e )2 ( + e )d : +1 (1 + e ) : 1 2 k k 0 C1 (z) ; 2(1 ; zk) C1(z) = V (1 ; kz)3=2 1 ; 2 ln(1 ; kz) : 83 , $ C1(0) = 0. < 2 (1 ; 12 ln(1 p ; kz)) ; 1 : C1(z) = Vk 1 ; kz 0 0 23 % (20 ) (21 ) $ CK1(z) C1(z) $ 1 2k e 2 ; w1 = p ( ; 0) ; ( ; 0 )(1 + e ) > 0 V 1 ; kz (1 + e )2 2 w1 = p2k (1 +e e )2 12 ( ; 0)2 ; ( ; 0 )(1 + e ) 6 0: V 1 ; kz $ 3 & w1 % z. > 0 ; ; jw1j = p2k (1 +e e; )2 12 ( ; 0)2 ; (1 +e e; ) ( ; 0) 6 V 12k; zk 1 ; 2 ; p 6 (23) 2 e ( ; 0) + e ( ; 0) 6 V 1 ; zk 2k 2 2 + 1 : 6 p e2+0 + e1+1 0 6 V (12k ; kz) e2 e V zk 6 0 ; jw1j = p 2k (1 +e e )2 12 ( ; 0)2 ; (1 +e e; ) ( ; 0) 6 V 12k; zk 1 6 V p1 ; zk 2 e ( ; 0 )2 + e ( ; 0) 6 2k 2 2k 2 1 2k 2 1 1 6 p e2;0 + e1;0 6 V (1 ; kz) e0 e2 + e = V e2 + e : V zk
Z
454
. .
= , (B), . . $3 % (6), w1 & ; (z) k 2 1 w1 z 6 " V (1 ; zk) e2 + e > 0 ( 6 0 6 (z)): 4 '* $ $- & % & w1 ' 6 (z), (z) | & , * '
; (z) ( @v ) R z ( @v ) dz 2 1 w1 z 6 @ e z @ 2 + : " v0 e e 0
1 2
0
0
C3 z > z0 z0 { , & ' @v0 (z ): v0 = !lim v (z ), ( @v@0 ) = !lim l(z )+0 @ +0 0 3 W | $3. ; " 1 " 1 6 @v0 1 R z (W@v0 ) dz ; 2 1 ( @v0) e 2 z0 @ + e2 e ' " 2 50 " 1 ] ' 6 (z) j"w1j 6 W u"(z ) = v0 + w0 + w1" + O("2 ) w0 + O("2 ): 1 % & w0 * 3 O(") . C$, " 2 50 " 1 ] $3 O(") $ " = 0 & & . ( v = ;v , = ; , ) $ 3 "1 , $ ' " 2 50 "1 ] $3 O(") $ " = 0 & & . = ) $ , $ & , $ " = 0, " 2 50 "1], "1 = minf" 1 "1 g, $3 O(").
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517.5
, ! . ! " # $ $ $ % " ! .
Abstract M. K. Potapov, B. V. Simonov, On estimates for the modules of the smoothness of the functions with transformed Fourier series, Fundamentalnaya i prikladnaya matematika 1(1995), 455{469.
In this article the functions are considered which have generalized derivative in Weyl's sense. The lower and upper estimates for the modules of smoothness of these derivativesare expressed in terms of the modules of smoothnessof the function itself.
1
Lp (1 < p < 1) | 2- f(x),
0Z2 11=p kf kp = @ jf(x)jp dxA < 1" 0
w (f t)p | # ( Lp ) ( > 0) f(x) 2 Lp : 1 X w (f t)p = sup k (;1) f(x + ( ; )h)kp jhj6t
=0( ; 1) : : :( ; + 1) : =
!
- . /. " 0 % " 1 (" N NCJ000) - ! " 1 3 $ ! (" N 93{01{00240). 1995, 1, N 2, 455{469. c 1995 !, "# \% "
456
. . , . .
+ , f(x) - , f(x) a20 +
1 X
(an cos nx + bn sin nx)
n=1
# .
Sm (f x) = / 0 - 10 = A0 (x) 1m =
mX ;1 k=0
X
2m ;1
n=2m;1
1 X
m=0
Am (x)
(1)
Ak (x):
An (x)
m = 1 2 : : ::
M - , .2 , 1 X
n=0
an cos nx
# ann # 0 (n " 1) # # " 3 | , .2 , 1 X
n=1
an cos 2n;1x:
4 (t) - , 50 2], 5 2] .- # 2 (0 2) - ., 2 0 ,
Z2 0
(t)t dt < 1:
= k , # 0 < k < 1. 8 # :
= n =
Z2 1=n
(t) dt + nk
Z1=n
1=
(t)tk dt
n = 1 2 : : : |
0
0 , (t)" ( f) | - 2. 9 (1), 1 X ( f) n An(x)" n=1
f(n)g | .2 (0) =
457
0 :
Z2
(t) dt (n) =
1
Z
1=2n;1
(t) dt n = 1 2 : : ::
1=2n
; F () > 0 G() > 0, F () G() , 2
0 c, 2 , F() 6 cG(). ; F () G() G() F(), - F () G().
2
1. max(2 p) 6 < 1, k (t) - = k . f(x) 2 Lp ( f) '(x) 2 Lp ,
8 Z2 91= Z < = ; w (f t) dt + (f t)p dt
(t)t
(t)w p k + k + :
w (' )p: 0
2. 0 < 6 min(p 2), k (t) - = k ,
Z2
(t) dt
;k
Z 0
f(x) 2 Lp
(t)tk dt
Z
(t) dt
8 2 (0 ]:
(2)
2
Z2
(t)wk (f t)p dt < 1
(3)
0
'(x) 2 Lp ( f)
91= 8Z Z2 = < w (' )p : (t)wk + (f t)p dt + (t)t; wk + (f t)p dt : (4) 0
3. f(x) 2 Mk \ Lp , ) " 1 max(2 p) 6 6 < 1 1 < p 6 < 1$ %) " 2 0 < 6 6 min(2 p) 0 < 6 p < 1.
458
. . , . .
4. f(x) 2 3 \ Lp , ) " 1 max(2 p) 6 6 < 1 2 6 < 1$ %) " 2 0 < 6 6 min(2 p) 0 < 6 2. . (t) k - = k , (2). ). = 2 f(x) 2 L2 , '(x) 2 L2 ( f) %' , %"
Z2
8 > 0
(t)wk2 (f t)2 dt < 1
0
8Z 91=2 Z2 < = w (' )2 : (t)wk2+ (f t)2 dt + 2 (t)t;2 wk2+ (f t)2 dt : 0
%). = p f(x) 2 Mk \ Lp , '(x) 2 Lp ( f) %' , %"
Z2
8 > 0
0
(t)wkp (f t)p dt < 1
8Z 91=p 2 Z < = w (' )p : (t)wkp+ (f t)p dt + p (t)t;p wkp+ (f t)p dt : 0
). = 2 f(x) 2 3 \ Lp , '(x) 2 Lp ( f) %' , %"
Z2
8 > 0
0
(t)wk2 (f t)p dt < 1
8Z 91=2 Z2 < = w (' )p : (t)wk2+ (f t)2 dt + 2 (t)t;2 wk2+ (f t)2 dt : 0
459
3 !
1 (. 51]). 0 < 6 min(p 2) f(x) 2 Lp
Z2
(t)wk (f t)p dt < 1
0
'(x) 2 Lp ( f). 2 (. 52]). f(x) 2 Lp , 1 < p < 1 P1 A (x) . ( n n=0
0Z2 1 p=2 11=p kf kp @ X 1m2 dxA : m=0
0
3 (. 53]). f(x) 2 Lp , 1 < p < 1 P1 A (x), f g , n=1 n
n
jnj 6 M
X
n 2 N
2+1 ;1
m=2
jm ; m+1j 6 M
= 0 1 2 : : ::
( P1 n=1 n An (x) '(x) 2 Lp , k'kp 6 ckf kp , c f(x). 4 (. 51, 4]). f(x) 2 Lp , 1 < p < 1 P1 n=0 An (x). ( )
1 m X X An(x)kp
w f m1 m; k n An (x)kp + k p n=m+1 =1
m; kSm() (f x)kp + kf(x) ; Sm (f x)kp"
%) f(x) 2 M ,
X 1=p X 1=p m 1 w f m1 m; apnnp+p;2 + apn np;2 " p n=1 n=m+1
) f(x) 2 3,
w (f 2 )p 2 ;m
;m
X m =1
2 2
a 2
1=2 X 1 +
=m+1
a 2
1=2
:
460
. . , . .
5 (. 54]).
8 2 91=n > > Z1=n (t) dt + nk (t)tk dt> n = 1 2 : : : :1=n
0
: % n = 1 2 : : : ) n 6 n+1 $ %) n nk > n+1 =(n + 1)k $ ) 2n n .
4 # 1 + I=
Z
(t)wk + (f t)p dt +
0
Z2
(t)t; wk + (f t)p dt = I1 + I2 :
n | , 2;n+1 6 < 2;n. ? I1 . 4 . ), - I1
1 X
=n
2
;(k+)
( + 1)kS
(k+) 2
+ X ( + 1) p =n
(f)k
1
kf ; S2 (f)kp = I3 + I4:
? I4. @ 2 I4
0Z2 1 1 X @ X
=n
0
=
p=2 1=p 12m+1 2= ( + 1) dxA :
A , > p > 2 - -2 B # ,
0Z2" 1 1 =2#p= 1=p X X 12m+1 2= ( + 1) dxA I4 @ = n m = 0Z02" 1 m # 1=p 2= p=2 X X @ 12m+1 2= ( + 1) =2 dxA m = n = n 0Z02" 1 #p=2 1=p X @ 12m+1 22m dxA : 0
m=n
461
C n 5, , Bn = 2m =n , 2m 6 n < 2m+1 3. 8 # , 3, 2, 4 . ), I4 k
1 X
m=2n
m Am (x)kp w (' 2;n )p :
+ 0 , # , I4 , . I3 : I3
Z
1=2n
(t)t(k+) dtkS2kn+ (f)kp +
0Z2 p=2 1=p 1 X X @ + 12m 22m(k+) 2= ()2;2 (k+) dxA =n+1 0 m=n+1 0Z2 n p=2 1=p X 2;n @ 12m 22m(k+) 22n 2;2nk dxA + 0Z2* 10 m=1 =2+p= 1=p X X +@ 12m 22m(k+)2= ()2;2 (k+) dxA =n+1 m=n+1 0 0Z2 n p=2 1=p X 2;n @ 12m 22m 22m dxA + 0Z2* 01 m=11 2=+p=2 1=p X X = 2 12m 22m(+k) 2= ()2;2 (k+) dxA +@ 0
m=n+1 =m w (' 2;n)p + 0
0 2* 1=p + 1=Z2m; p= 2 Z 1 2 = X 2 2m(+k)
(t)t(k+) dt dxC +B 1m 2 @ A m = n +1 0 0 0 2* 1=p + 1=Z2m; p= 2 Z 1 2 = w (' 2;n)p + B@ X 12m22mk
(t)tk dt dxC A 1
1
w (' 2;n)p :
0
m=n+1
8 - , ,
0
I3 + I4 w (' 2;n)p :
462
. . , . .
8 I2. 0 , # ,
I1 . I2 2;n
nX +1 =1
()2;kkS2(k +) (f)kp +2;n
nX +1 =1
2 ()kf ; S2 (f)kp =I5 +I6 :
? I5 .
0Z2 p=2 1=p X I5 2;n @ 12m 22m(k+) 2= ()2;2k dxA 0=1Z2n0+1 m=1 1=p D E X X = 2 p= 2;n @ 12m 22m(k+) 2= ()2;2k dxA =1 m =1 0Z02" n+1 n+1 # 1=p 2= p=2 X X 2;n @ 12m 22m(k+) 2= ()2;2k =2 dxA = m m =1 0 02" # 1=p 1=Z2m; Z nX +1 2= p=2 12m 22m(+k)
(t)tk dt dxC 2;n B@ A nX +1
1
0 m=1 ;n w (' 2 )p :
1=2n+1
? I6. I6 2;n + 2;n
Z2
1=2n+1
nX +1 =1 n
X
=1
+
()2 kS2n+1 (f) ; S2 (f)kp
Z2 X 1
(t) dt
n X ;n
+2
2 ()kf ; S2n+1 (f)kp +
=1
0
1m m=n+2 Z2 X n
2 ()
Z2 X 1
12m 22m
2
m=
p=2 !=p dx
1m+1 2
0p=2 !=p dx
+
p=2 !=p dx
+
0 m=n+2 Z2X =2p= !=p n X n 2 2= 2 ;n 1m+1 2 () 2 dx =1 m = 0
463
k' ; S2n
+1
+2
;n
(')kp +
Z2 X n X mn m=1 =1
0
w (' 2;n)p:
w (' 2;n)p + 2;n
o=22= p=2 !=p
2 2=
1m+1 2 () 2
Z2 X n 0
m=1
dx
2m 2 m+1 2m;1
1
2
2
p=2 !=p dx
8 - , , I5 + I6 w (' 2;n)p : ?-D I3 , I4 , I5 , I6 # , : I 1= w (' )p : 8 1 ..
5 # 2
f(x) 2 Lp (3). 8 # 1 , 2 '(x) 2 Lp , ( f). 0 , (4). n | , 2;(n+1) 6 < 2;n. E # 4 . ) w (' )p w (' 2;n)p k
1 X
=2n
A (x)kp + ;n
+2
k X A (x)kp = I7 + 2;n I8: 2n ;1
=1
C 5 , Bn = n =2m+1 , 2m 6 n < 2m+1 , 3. 8 # , 3 2 (2), ,
0Z2 1 p=2 11=p X I7 k 1m 2m kp @ 12m 22m dxA 0 m=n+1 0m2=n+1 11=p m 1 = 2 X Z Z 1 2= p=2 B@ 12m
(t) dt dxC A 1 X
0
m=n+1
1=2m+1
464
. . , . .
0Z2 1 2=p=2 11=p 1 X X @ 12m 2m(k+) ( + 1)2; (k+) dxA : 0
=m
m=n+1
A , 6 2 6 p, 0 , B # , # :
0Z2" 1 =2#p= 11=p X X 2 = I7 @ j1mj2m(k+) ( + 1)2;(k+) dxA m=n+1 00 1="n+1 2 p=2 #=p11=p Z X X 2 = @ j1mj2m(k+) ( + 1)2;(k+) dx A =n+1 0 m=n+1 11= X !1= 0 1Z=2n 1 ( + 1)2; (k+) kS2(k +) (f)kp B @ (t)wk+ (f t)p dtCA : =n+1
0
? I8 . 0 0 , I7 , :
0Z2 n p=2 11=p X I8 @ 12m 22m; 22m dxA 0 02" m=1 2 #p=2 11=p 1=Z2m; Z Z ;1 2= n 2= X B@ 121 (t) dt + 12m+1
(t) dt 22m dxC A m=1 0 1 1=2m 0Z2" n;1 X 2= #p=2 11=p m X @ 12m+1 ()2 dxA m =0 =0 0Z02"n;1 n;1 =2#p= 11=p X X 2 = j1m+1j()2 @ dxA m= 0n0;1"=0 2 n;1 p=2 #=p11= Z X X 2 = dx A @ j1m+1j()2 =0 0 m= 11= nX !1= 0 Z2 ;1 ()2kf ; S2 (f)kp B@ (t)t; wk+ (f t)p dtCA : 1
1
=0
1=2n
?-D I7 I8 , (4). 8 2 ..
465
6 # 3 . ) +
I9 =
Z
(t)wk + (f t)p dt +
0
Z2
(t)t; wk + (f t)p dt:
m , m1+1 6 < m1 . 8 # 1 m X X I9 1 ()wk + (f 1 )p + m; 1()wk + (f 1 )p =m+1 =1
R
R
1=( ;1)
2
# 1(1) = (t) dt, 1 () =
(t) dt, = 2 3 : : :. 8 f 1 1= 0 4 . -): I9 + +
(
2 Mk ,
1=p X 1=p) 1 p (k++1)p;2 p p ;2 1 () an n + an n + =m+1 n=1 n = +1 ( 1=p m X X ; ;( k + ) p ( k + +1) p ;2 m 1 () an n + =1 n =1 ) X 1=p 1 p p ;2 an n = I10 + m; I11: n= +1 1 X
F ,
I10
+ +
;(k+)
1 X
=m+1 1 X
=m+1 1 X
=m+1
X
1 ()
;(k+)
1 () ;(k+) 1 ()
X 1
n=
X m
apnn(k++1)p;2
n=1X
n=m+1
=p
apnn(k++1)p;2
apn+1(n + 1)p;2
=p
+
=p
+
= I12 + I13 + I14:
? I12, I13 I14 . E I12. A fn g, I12 ;
X m
n=1
apn n(+1)p;2 pn
=p
:
> p, , - -2 B # I13 I14, : I13
X 1
n=m+1
apn n(k++1)p;2
X 1
=n
1()
;(k+)
p= !=p X 1
n=m+1
apnpn np;2
!=p
"
466 I14
. . , . .
X 1 n=m+1
apn+1 (n + 1)p;2
8 - ,
X n
I10
=m+1
X 1 n=m+1
8 I11 . F , I11 + +
m X
m =1 X m =1 X =1
1()
;k
1() 1()
1()
X
X mn=1
p= !=p X 1
apnpn np;2
apn n(k++1)p;2
apn+1 (n + 1)p;2
n=1
X
n=m+1
!=p
=p
=p
apn+1 (n + 1)p;2
n=m+1
an pn np;2
!=p
:
:
+ +
=p
= I15 + I16 + I17:
? I15 , I16 I17. 8 p 6 , , B # ,
I15 I16
X m
n=1
X m n=1
apn n(k++1)p;2
X m
apn+1(n + 1)p;2
=n
1()
X n =1
;k
1 ()
p= !=p X m
n=1
p= !=p mX+1
n=2
apnn(+1)p;2 pn apnpn n(+1)p;2
A fng, - I17 8 - , I11
m
X 1 n=m+2
m
X 1
n=m+2
apnpn np;2
apnnp;2 pn
=p mX +1 +
n=1
=p
:
apn pn n(+1)p;2
=p
?-D I10 I11 , : I
1= 9
4 . )
m
mX +1 n=1
apnpn n(+1)p;2
1=p X 1 +
I91= w (' )p
n=m+2
apnpn np;2
1=p
:
:
!=p
!=p
" :
467
(4). ) 3 .. G -) 4 # . 8 4 0, 3, 4
. -) 4 . ).
7 )
1. ; (t) = t;r;1 k > r > 0, # w (' )p , 1, 2, 3 ), -), ), # , - 55]. 2. H # - 55] .2 . ( ) | 50 2] , - .2 ( 1 ) 6 C1 ( 2 ) 0 6 1 6 2 6 2 (2 ) 6 C2( ) 0 6 6 # C1 C2 1 , 2 , . ?- k ) | H | f(x) 2 Lp P W(pA (x), , 1 ( f) , m=0 m '(x) 2 Lp " Hp | @ # | f(x) 2 Lp , w (f )p ( )" WH( p k) | H -@ # , . . f(x) 2 W(p k ) , w (' )p ( )" BH( p k) | f(x) 2 Lp ,
8 Z2 91= Z < = ; w (f t) dt + (f t)p dt
(t)t
(t)w p k + k + :
( ): 0
8 # , 2, 0 .2 0 W H( p k) BH( p k). I : 1). ; max(2 p) 6 < 1, WH( p k) BH( p k): 2). ; 1 < p 6 < 1, Mk \ WH( p k) Mk \ BH( p k): 3). ; 2 6 < 1, 3 \ WH( p k) 3 \ BH( p k): (t) . (2). 8 # 0 :
468
. . , . .
4). ; 0 < 6 min(p 2),
BH( p k) W H( p k):
5). ; 0 < 6 p < 1,
Mk \ BH( p k) Mk \ WH( p k):
6). ; 0 < 6 2,
3 \ BH( p k) 3 \ W H( p k):
7). J WH( 2 2 k) BH( 2 2 k)
.. 8). J Mk \ WH( p p k) Mk \ BH( p p k)
.. 9). J 3 \ WH( p 2 k) 3 \ BH( p 2 k)
.. 3. 0 < 6 min(p 2) 0 k (t) - . = (k+) . 1 = f1n g, # (n = 1 2 3 : : :)
;
n = n 1
Z2 1=n
;
(t)t
(k+)
dt + n
Z1=n
(k+)
(t)t
dt
1=
:
0
; f(x) 2 Lp ,
Z2 0
(t)wk + (f t)p dt < 1
2 '1 (x) 2 Lp , (1 f)
8Z 91= Z2 < = w ('1 )p : (t)wk + (f t)p dt + (t)t; wk + (f t)p dt : 0
4. ; f(x) 2 Mk+ \ Lp , , 0 2 0 < 6 p < 1:
5. ; f(x) 2 3 \ Lp , , 0 2 0 < 6 2:
*
469
1] ., . . ! // . ( $%). | 1979. | 3 (16) (31). | 295{312. 2] Littlewood J. E., Paley R. E. Theorems on Fourier series and power series // J. London Math. Soc. | 1931. | V. 6. | P. 230{233. 3] Marcinkiewcz J. Sur une nouvelle condition pour la convergence presque partout des series de Fourier // Ann. Scuola norme. Pisa. | 1939. | V. 8. | 239{240. 4] 2 . 3. 4 ! 5$ % 67. | 8. 39:9;9 22.06.1981, N 3031{81. 5] . ., . 4 < % ! -: 7$ 3! -: 7$ // 3. . -. 2. 1. . . | 1992. | N 4. | 44{53. ' ( 1995 .
- - . .
e-mail: [email protected]
- , ! " # . $
-
. %& & # ' ( - ) (
, & ' ! - ) ! . * , "##
( + , " ' "## + .
Abstract G. E. Puninski, Serial Krull-Schmidt rings and pure-injective modules, Fundamentalnaya i prikladnaya matematika 1(1995), 471{489.
A ring is called Krull-Schmidt if every 5nitely presented module over it can be decomposed into direct sum of modules with local endomorphism rings. The serial Krull-Schmidt rings are described as rings with the weak invariance condition. The classi5cation of indecomposable pure-injective modules over uniserial ring is simpli5ed and criteria for the existence of superdecomposable pure-injective module is given for semi-invariant case. Let T be the theory of all modules over e6ectively given invariant uniserial ring R with in5nite residue skew 5eld. It is shown that T is decidable if the question of invertibility of element from R can be solved e6ectively.
1
- 9]. , , ! " # " " " $%#. & "" (". (), *" + ",( $ # , ( . - "" # "" !" ", " 9], " , 7 # ' 8 ' ! ! # 7 ! (! JBX100). 1995, 1, N 2, 471{489. c 1995 , ! \# "
472
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-
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2.1
10, p. 177, Prop. 1.13]. M
, M . I# J(R) (J(M)) # ! 5(" R ( M). < " " J() R Tn n , 6 , : ( J(0) = J(R), J( + 1) = J() T J() = < J() , 6 ,. " U(R) # ! , $ R.
2.2
12, Prop. 5.5]. ! " , # J() . 8 ! " " " " , " $%#. E # ! , ! " # !, " \$". E , # " \$" " # ! \ " " $%#".
R # " -$ 9], " ! " R # " "
474
. .
" $%#. 0 " 9, Thm. 2.3] $ " - "! $ " + : ! ( % # , R , ! $ %. 0 , # , ( 13]. " R # . % - % x (1- -% ) '(x) # " % # R 9yK (Ky A = xKb), A | k l , Kb " l R, y = hy1 : : : yk i " #",. 5 M R $ m 2 M 6 M j= '(m) (M ! '(m)), " " $ mK = hm1 : : : mk i # M, ! mA K = mKb. 9 , ! " % ' $ # ! #6" "" , (" A) $ " $ $ " " x. 7 , -% ( + " " !" , ,. $ ! " # " A j xKb (A xKb). 5 1- -% ' 6 ' ! , " $ m # M # M j= '(m) M j= (m). 5 1- -% ' # " & , # !, ' , " ' ! ! '. 8(" ", 1- -% R, % # 6 $ " , # 6 L(R) " ! !, ! "! $ 6 | " * "" +, % . L" M R ' " 1- -% , '(M) = fm 2 M j M j= '(m)g # " - ( -% ') . 9 , ! - M " $ M (" ", , , ! !, # 6 L(M), + " % # 6 L(R). 5 ( 1- -% ' = \A j xKb" " 1- -% D' = \9zKt AKz t = 0 ^ x = KbzKt". , -% xa = 0 $ -% a j x, '(x) = \b j x" $ " D' bx = 0. 0 13, Ch. 8] ( ' 7! D' # #%# 6 , , 1- -% R, # " \ "" " ". hM mi, M ! " , m 2 M # " 1- -% ', " M j= '(m) 1- -% # M j= (m) ' ! . N 1- -% " # . , " # 1- -% a j x ^ xb = 0 hR=abR ai. 8(" 1- -% # " "", " #" , " " M $ m 2 M , ! M j= '(m) -% ' # $ (" . 8("
-
475
", 1- -% p(x), +," $ m M, # " 1- - # ! " ppM (m). 9 , ! 1- - $ !" " "" (" 1- -% . 0 ( , f : M ! N # " # , " f('(M)) = '(N) \ f(M) 1- -% '. 1 " ! " ! , $ " 15]. 8 M # " !"-*, " * " ", !", ( . 1 $ " " " " N, M "(" !" . 5 M "+" " (" !" #%# M) !"-* N "( + M !" !" " " $ " " ( " N 0, M "(" !" , "+" !" ( # N N 0 , " + " M). " # " # -' # M # ! " PE(M). 2.3 9]. #- " # -' " # , & ( .
5 1- - p ( "+" !"-* M, # + p $ m " " " ": " N j= p(n) $ n !"-* N, "+" !" ( M N, + m n. E " " !" #%# m, # " # -' # 1- - p # ! " PE(p). N # ( !"-* #% !"-* ! 1- - " $ . 5 !" M, ( $ m, PE(M) = PE(p), p = ppM (m).
3
-
Q" " $ " "" , e1 : : : en R , ! ei R Rei . R + ", $%# %" # ( ! " .
3.1.
% R r 2 ei Rej , ei ej 2 R . ) & "* : 1) ei R=rR & (2) %%- & ei ei R=rR 3) ei R=rR & -
476
. .
4) ei R=rR & (5) J(ei Rei )r rej Rej 6) ei Rei r rej Rej rej Rej ei Rei r.
. 8( "! eiR=rR ! eiR , . . 0 6= r 2 J(R). " M = ei R=rR p = ppM (ei ). " M ! " , 2.3 M $%# , PE(M) # (. E ei ( M, " " $ # (" - p. 7 1) , 2). 1) ) 3). " 7, Thm. 1.9] Mei $", ( " # " + # R" 16, p. 300, Cor. 1]: " . 9 , ! $%# M # " ( $ s 2 ei Rei , ! sr 2 rR (" $%# M ( ei . L" s 2 U(ei Rei ) sr = rt t 2 U(ej Rej ), ", ! ( s # %# M. 3) ) 4). " S = End(MR ) f g 2 S " ( $ s t 2 ei Rei . ( " ( "! , ! h(s) = t h 2 S. E !, ! hf = g S " " . 5 (, ! S | " . " f g 2 S 6 " " R ( "! , ! su = t u 2 ei Rei . ( " $ h 2 S, ! h(ei ) = u h(u) = ei . L" h(ei ) = u, fh = g, ! " . S", ! u 2 J(R) " ! h(u) = ei #(. $ # 6 # " ( , ! u 2 U(ei Rei ) h(u) = ei . " v | u $ ei Rei . E h(ei ) = h(uv) = h(u)v = ei v = v, " h " M ( $ v. , s = tv, $ f = gh. S", ! 4) ) 1). 3) ) 5). " s 2 J(ei Rei ) "" ei s $ M. ( " f g 2 S, ! f(s) = ei , g(ei ) = s. " s 2 J(R), ei 2= J(R), " ! #(. 0 " ! 0 = g(0) = g(r) = sr M, sr 2 rR. 5) ) 6). (". 5, Lem. 7.2]) L" # , ! , r = gru r = vrh g v 2 ei Rei , u h 2 ej Rej , ! g h 2 J(R). ( gvr = rt t 2 ej Rej . E r = gvrhu = rthu thu 2 J(ej Rej ), r = 0 | !. 6) ) 1). L" ei Rei r rej Rej , $ ei Rei " $%# M. E S " % # ei Rei , " , ( ( ) . 7 ! rej Rej ei Rei r. E ! 5, Lem. 7.12] ( # , ! J(ei Rei )r rJ(ej Rej ).
-
477
L" S = End(MR ) , " $ u 2 ei Rei , ! u ei ; u " ( %# M. L" u 2 J(ei Rei ), , ( , ur = rt t 2 J(ej Rej ). E (ei ; u)r = r(ej ; t) ej ; t 2 U(ej Rej ), " ( (ei ; u) " %# M | !. V ! #( " ! ei ; u 2 J(ei Rei ). 7 , u (ei ; u) 2 U(ei Rei ). R ! ur 2 rU(ej Rej ) (ei ; u)r 2 rU(ej Rej ) #( ( !. R , rR = urR + (ei ; u)rR rJ(R) rR, !. 2 # R , " ei ej # 6 " " $ r 2 ei Rej ei Rei r rej Rej , rej Rej ei Rei r.
3.2. 3 "* R: 1) R -$ 2) R . . 7# 5# -:% 1], 17] " , ! # ( ! " #% ei R=rR, r 2 ei Rej ei ej # 6 " " . <" " 3.1. 2 9 , ! " " " " - , " $%#. R + # , ! $ + " ! . 3.3. % V = Zi](2;i) 4 4 # , & 2 ; i R 4 R = r0 +r1 x+r2x2 +: : :, r0 2 V , ri 2 Qi] i > 1, rx = xr (\ " | ). ) R .
. F " R " "". 5 , " (2 + i);1x 2= xR x(2 + i);1 2= Rx, $ . 2 9 , ! R | " #" 2 ( " , " # % 2.1).
4 # $ -
, " ""% # (, !"-*, R. 5 # ( -
478
. .
ei 2 R hI J i, I ei R , J Rei R, # " ei - . ( I = ei R n I, J = Rei n J. 5 ( ei - (" 1- -% J =I = fxr = 0 ^ s j x : r 2 I s 2 J g f: sr j xr : r 2 I s 2 J g " ! " """ # ! # ( 1- - p = p(I J), ! # ! " " # ( 1- - R 7, Thm. 2.7]. # ei - 1- - R, , ! ei j x 2 p. 7 , ( "" ei - p = p(I J) ( " # ( !"-* N = N(p), " # ( !"-* R ! " " ". & 7, Thm. 2.10], "", ei - p = p(I J) ej - p0 = p(I 0 J 0) "" + N = N(p) N 0 = N(p0) #% , " $ r 2 ei Rej , r0 2 ej Rei , ! ) r 6= 0, I = rI 0 Jr = J 0 , ) r0 6= 0, r0I = I 0 J = J 0 r0 . 0# + $ 6 $ " (" e- # ! !# . :" """ ei - p(I J) # " " + " 4, . 2]: s r 6= s r + sr ", r 2 I r 2 I s 2 J s 2 J
()
" s = r = 0 " " , ! " ": s r 6= 0 ", r 2 I s 2 J
()
5 , 7, Thm. 3.2], 3, R . . 3] : # " " () """
ei - p(I J). 0 " + " " " .
4.1. 3 "* -
R:
1) ) () () " ei - 2) R -$ .
hI J i-
. 0 3.2 R " - , R , , ei - p = ppM (ei ) M = ei R=rR, r 2 ei Rej # (. " " 7, Cor. 2.6] $ : ab j xb 2 p " " a j x 2 p xb = 0 2 p a 2 Rei , b 2 ei R. 9 , ! a j x xb = 0 ! ab j xb " . 1) ) 2). 5" ! # (" ei - p = ppM (ei ) M = ei R=rR, r 2 ei Rej . 7 ! (". 6) abjeib 2 p
-
479
ajx xb = 0 2= p , a 2 Rei , b 2 ei R. 7# abjei b gab = b + rh , g h 2 R. " xb = 0 2= p, " , b 2= rR, "
r = bt t 2 J(R), b = ;bth + gab. " I = rR, J = J(Rei ), " " () ei hI J i. 5 " , " s r = 0 r 2 I , s 2 J , ds = ei d 2 R, " , r = 0 2 I, !. " hI J i " " (). ( r = ;bth, r = b, s = ga s = ei . S", ! r 2 I, r 2 I , s 2 J s r = s r + sr . E ( ga 2 J , a 2 J . Ra = Rei ajei 2 p, !. 2) ) 1). " ei - hI J i " (), " (), " s r = s r + sr , $ r 2 I, r 2 I , s 2 J, s 2 J ( "! , ! r r 2 ei Rej , s s 2 ek Rei . E r = r h s = gs , g 2 J(ek Rek ), h 2 J(ej Rej ), " (ek ; g)s r = s r h. " ( "! , ! s r ej Rej ek Rek s r . E # " R, ! 5, Lem. 7.12] J(ek Rek )s r s r J(ej Rej ), (ek ; g)s r = s r u u 2 U(ej Rej ) s r = 0. !. 2 R + ( ! ""% # (, !"-*, , 4] " ! .
4.2.
8 * # ( 4 # -' 4 R ( - ei - hI J i c (), ( " & . 9 # -' M ei 2 R , # Mei 6= 0, I = fr 2 ei R j mr = 0g J = fs 2 Rei j m 2 = Msg, m " & Mei .
9 , ! " " () !" ".
5 1--
0" $ % R . 5 a b 2 R ( '(a b) = \ab j xb". 9 , ! a 2 U(R) $ % (" " , a = 0 + " % xb = 0. R + # 2] # " ".
480
. .
5.1. : '(a b) & a b
-
R # 4 1--( R. . 0 " 3, R . . 1] 1- -% R $ ! * 1- -% '(ai bi) , ai bi 2 R. 7 , " % ' = '(a b) # ( Vn "! #, ( "! , ! '(a b) = i=1 '(ai bi) " " . ( 'i = '(ai bi). "" " + ", ! D' ! D'1 + : : : + D'n . L" D' ! D'i i, , " "" , 'i ! ', ! #(. 7 , ( "! , ! D' 6! D'i ", i. 9 , ! % D' " # hM bi, M = R=Rab " ( 3.2) $%#. % 2.3 1- - p = ppM (b) # (. E D'i 2= p, " 13, Thm. 4.29] " 1- -% 2 p, ! (D'1 ^ ) + : : : + (D'n ^ ) 2= p. 3, E. 1] 6 1- -% " , $ ^ (D'1 + : : : + D'n ) 2= p, D'1 + : : : + D'n 2= p. !. 2 5.2. % I , J # ai bi, i 2 I , cj dj, j 2 J & R. ; ^ '(ai bi) ! ^ '(cj dj )
i2I
j 2J
, * f : J 7! I , # '(af (j ) bf (j )) ! '(cj dj ) 4 j 2 J .
5.3.
< & a b c d R, c 2 J(R), '(a b) ! '(c d)
, d 2 bR a 2 Rc, ad = 0.
. R " " " D'(c d) ! D'(a b), ( # " cx = 0 ^ d j x ! ax = = 0 ^ b j x. 0 " 5.2 $ " , , # !, #(" : 1) cx = 0 ! ax = 0 cx = 0 ! b j xW 2) d j x ! ax = 0 d j x ! b j xW 3) cx = 0 ! ax = 0 d j x ! b j xW 4) d j x ! ax = 0 cx = 0 ! b j x. 0 " ! 1), "" $ 1 R=Rc ! a 2 Rc gc + bh = 1 , g h 2 R. " c 2 J(R), b 2 U(R), ! "", d 2 bR. V ! " ! 2), "" $ d R, ! ad = 0 d 2 bR. E ( ""( # , ! d 2 bR a 2 Rc " ! 3)W ad = 0 b 2 U(R) " ! 4). 2 5 , $ a b R ( a 6r b, " b 2 aR a r b, " aR = bR. 8(" R, % # 6 r ,
-
481
# Lr , #% ! ! (" , R. V ! " Ll . 5 # L !# Lb # ! L "!, " $ Lb | $ # L " +," (" L = A B , ! A " ", a 2 A, b 2 B a < b, ! # : hA B i 6 hA0 B 0 i, " A A0 ( , $ , B 0 B). I# 1 # ! "!, A = L. 5 "! bl = hA B i ( A(bl ) = A B(bl ) = B. S", ! Lb " 6 $ 1, , (" $ l 2 L " "! hA(l) B(l)i, A(l) = fa 2 L j a 6 lg, B(l) = L n A(l), ( "! , ! L Lb . L" L "( 6 $ l0 , l0 = 1 "" $ (" . # "" R hI J i, I "" , J "" R. 5.4. % f : Lr 7! Lbl "* ( , f(0r ) = 1 f(1r ) 6= 1. % I(f) = fb 2 R j A(br ) = A(f(br )) = 1g, I (f) = R n I(f), J(f) = fa 2 R j al 2 B(br ) 4 b 2 I (f)g, J (f) = R n J(f). ) hI(f) J(f)i | R. . " b 2 I(f) c 2 R. E # b 6r bc % f A(br ) A(bcr ). " f(br ) = 1, f(bcr ) = 1 bc 2 I(f). " R 0 2 I(f), I(f) R. E f(1r ) 6= 1, 1 2= I(f). E B(br ) " ", $ b 2 I (f) 6= ?, 0 2 J(f). L" I(f) , dal 2 A(br ) b 2 I (f). E al 6l dal , al 2 A(br ) a 2 J (f), !. E (" A(1r ) ", 1l 2 A(1r ), " 1 2 I (f), 1 2= J(f), " J(f) "" . 2 " p 1- - R. R( " p 6 '(p) (" R, ( b'(p)a a b 2 R, " '(a b) 2 p. " ( a " b " $ % '(a b) , $ -% , '(p) 6 # Lr Ll , " # ( # !. < p % f = f(p) : Lr 7! Lbl , ( br 2 Lr A(br ) = A(f(br )) = fal 2 Ll j b'(p)ag: R",
B(br ) = B(f(br )) = Ll n A(br ) (#( "). 1 , " # " " "(, "" , br al . " cl 6l al al 2 A(br )
482
. .
, $ a b c 2 R. E '(a b) 2 p, 5.3 '(a b) ! '(c b), '(c b) 2 p, " cl 2 A(br ). , b 2 R -% '(1 b) (" " , $ 1l 2 A(br ) 6= ? " ! % Lbl . 5.5. < " 1-- p R ( f = f(p) f(0r ) = 1. = - p , f(1r ) 6= 1. . " b d 2 R , ! br 6 dr , ! f(br ) 6 f(dr ), " A(br ) A(dr ). " al 2 A(br ), " '(a b) 2 p. 5.3 ! '(a b) ! '(a d), al 2 A(dr ) % f . E ", a 2 R % '(a 0) = \0 j x 0" (" " , A(0r ) = 1. L" - p , '(0 1) = \x = 0" 2= p, $ 0l 2= A(1r ) A(1r ) 6= 1. 2 " f = f(p) 1- - p. ( I(p) = I(f), I (p) = I (f) J(p) = J(f), J (p) = J (f). 9 , ! I(f) = fb 2 R j xb = = 0 2 pg. 5.6. % p 1-- R. ) hI(p) J(p)i R, # s r 6= 0 4 r 2 I (p), s 2 J (p). . R"" hI(p) J(p)i " # 5.4 5.5. (, ! ab = 0 , b 2 I (p), a 2 J (p). E al 2 A(dr ) d 2 I (p). L" $ b 6r d, # ab = 0 ! ad = 0. E al 2 A(dr ), '(a d) = \xd = 0" 2 p, d 2 I(p) | !. 7 ! d
5.7.
8 * # 1-- R ( f : Lr 7! Ll , "* "* : 1) f - 2) f(0r ) = 1 f(1r ) 6= 1- 3) s r 6= 0 4 r 2 I (f), s 2 J (f). 3 : p 7! f(p) : f 7! p(f).
b
. 0 " 5.5, 5.6 % f(p) 1- - p " 1){3). , # " # ", ! (
-
483
# . <" " , ! % f, + " 1){3), -% p f: '(c d) j '(c d) 2= pg, p = p(f) "". " $ . " ! (" $ (" "". 0 # " ", " # , # ( "! , ! "( . 5.1, ! '(a b) ! '(c d), '(a b) 2 p, '(c d) 2= p. 0 " " 5.3 b 6r d , # " : c 6l a ad = 0. p = p(f) ! al 2 A(br ) cl 2= A(dr ). 7# f f(br ) 6 f(dr ). L" c 6l a, cl 2 A(dr ), !. 7 ! a
6 ' $ -
, ! # " " # (, " "( , # (, , " , ( ! "", " # () 1- - p , " N(p). R (, ! 1- -% 6 p, " 2= p , 1- -% '1 '2 2= p ,, ! ! '1 '2 " 1- -% ' 2 p, ! ('1 ^ ')+('2 ^ ') 2= p. 0 " 13, Thm. 9.16] 1- - # " # ( , 6, % . R + , " # "" 6 1- -% 3, E. 1] 3, R . . 1].
6.1.
% p 1-- 2 = p 1--( . ) p '1 + '2 2 = p " 4 1--( '1 = '(a b) '2 = '(c d) 2= p 4, # ! '1 '2.
R + ( # # ! % " # (, 1- - . 6.2. 3 "* 1-- p R:
484
. .
1) p 2) ( f = f(p) 1){3) 5.7 , , 4) a b 2 R al 2 = A(br ), c 2 R , # f(br ) < f(cr ) al 2= A(cr ).
. 1) ) 2). ( , ! al 2= A(br ) a b 2 R "+" c 2 R , ! f(br ) < f(cr ) al 2= A(cr ). E al 2= A(br ), '(a b) 2= p b 2 I (p). (, ! 1- -% '(a b) 6 p, ! " ( !. 0 " 6.1 " ! " , ! '(c d) + '(e f) 2= p ", 1- -% '(c d) '(e f) 2= p ,, ! '(a b) ! '(c d) '(e f). 7 cl 2= A(dr ), el 2= A(fr ), ! "", d f 2 I (p). , " " 5.3, ! b 6r d f, A(br ) A(dr ) A(fr ) f. g = minl (c e), h = maxr (d f), 3, N. 1] '(g h) = = '(c d) + '(e f). S", ! h 2 I (p). L" a 2 J(p), a 2 B(sr ) ", s 2 I (p), " , ( A(br ) = A(dr ) = A(fr ) = A(hr ). " g l c g l e, gl 2= A(hr ) '(g h) 2= p, ! " . 7 ! a 2 J (p). E 3) 5.7 ! ad af ah 6= 0 5.3 c e 6l a. E g 6l a h >r b. L" al 2 A(dr ), , " c 6l a, ! cl 2 A(dr ), ! #(. 7 , al 2= A(dr ) ! al 2= A(fr ) A(hr ). ( % f A(br ) = A(dr ) = = A(fr ) = A(hr ). E cl el 2= A(dr ) = A(fr ), gl 2= A(hr ), '(g h) 2= p. 2) ) 1). 5" ! "" 6 1- -% p. ( . 3, R . . 1] ( "! , ! = '(a b). 0 ! "", al 2= A(br ). 4) d 2 R , ! f(br ) < f(dr ) al 2= A(dr ). 0 cl 2 A(dr ) n A(br ). S", ! c
% R . ) 4 & I R , # R=I . , I , .
.
S", ! 0 2 I. L" a 2 I, b 2 R an = 0, # " Ra aR aR Ra ! (ab)n = (ba)n = 0. R , I | . " cd 2 I c d 2 R, " (cd)m = 0. L" d 2 R , d2 2 Rcd, $ d 2 I # 6. V ! c 2 I, " c 2 dR. 0 " ! c = gd d = ch g h 2 J(R), c = gch, c = 0 5) 3.1.
-
485
(, ! I ( T J = n I n . E J 6= I. 0 " 5, Thm. 1.15] R=J " . E r 2 I rn = 0 n, r 2 J, I = J, !. 2
6.4.
8 "* & : 1) R 2) R # -' .
R
. 1) ) 2). (, ! Lr Ll "( $# ! , !" hQ 6i Lr Ll , ! q2 6= 0 q 2 Q. 5 ( $ Lr . L" I, 6.3, , R=I " # #"
. 5 " , , ! R | " , " # 6.3. L" R=I #" , % T 2.2 ! J() I , J( + 1) n I n = 0, R #"
( , ! #(. 0 " % 2.1 Lr "( , #% ! , !" . S", ! q2 6= 0 q 2 Q $ " ! . 7 ! 6.3 I 2 = I 6= 0. " r 2 I c r2 6= 0. E " R ! r = r0 r1 , r0 r1 2 I. 7 r0R rR. S", ! r02 r12 6= 0. 5 " , " r02 = 0, r2 = r0 r1r0r1 = 0 " R. 9 6 r0 = r00r01, r1 = r10r11 rij 2 I. E ! r00R r0R r0r10R rR. ( ( #, ! ( (. 7 , "! , ! hQ 6i Lr Ll . 5 br 2 Lr q 2 Q ( br r q, " ( br q $ # Q, ! 6 l . E # # $, 6 , !, " # ! " ( . < % f : Lr 7! Lbl . L" br r q q 2 Q, A(br ) " * A(abl ) ,, ! al r q. 0 " ! " A(br ) " * A(abl ) ,, ! al # Q "!, 6 "" + "!, # Q $ br . S", ! f(q) < f(q0 ) ", q q0 2 Q ,, ! q < q0 . <" " , ! " % " 1){4) # 5.7 ( 6.2. :" 1){2) "" # . 3). S", ! I(f) = fb 2 R j b >r q ", q 2 Qg J(f) = fa 2 R j a >l q ", q 2 Qg. L" b 2 I (f), a 2 J (f), br l q0 br
486
. .
2) ) 1). L" R " # ( !"-* M, 0 6= m 2 M, - p = ppM (m) " # (. 1. 5 % '(a b) 2= p , ! b 2 I (p), a 2 J (p), " $ d 2 I (p), c 2 J (p) , ! b n. L" | " " , ! # " 01 " 1 m- ", b
-
7 ) * +
487
A "" " ! + " # ! #6" ", . " R0 = Zhx0 x1 : : :i " , !" Z "! !" +, ,. R (, ! R &(( , " ( " % R0 , ! " , $ R ( $%% 6 . L" R , !" R0 ! R0 = Zx0 x1 : : :] !" "! (" +, ,. 9 , ! " ! " , #6" ", T(R) R, " "!" R # ! "!" # , " r = s R $ R0 " , T (R) j= \8x xr = xs". !, " "+" $%% " ( " " " ". 8 # #" $ " " "" #6" \$%% # "". R " ! " # ! .
! " 7.1.
) 4 &(( # .
. , ! $ " ! " "" " 8] ( $%% " . 2 5 , 1- -% ' ,, ! ! ' M Inv(' M) !" $ '(M)= (M), " ! " 1 " ! . S", ! " Inv(' ) > n # " " 89- (, ( # " . R + % ( ! $ ", -% . 7.2 (
$ ) 13, Cor. 2.13]. !"
& 4 .
! " 7.3.
< &(( ( 7.2 & 4 &(( .
. 1 " # # ! " "" " T (R) (# ! 7.1). 2 F R # " , " a 2 R Ra = aR, " R " .
488
. .
7.4.
% R # # R=J(R). ) " 4 1--( ' 4, # ! ' " M , Inv(' M) > 1, #.
. " R "!, " ! # , ! '(M) M 1- -% '. 3, R . . 1] ( "! , ! ' = '(a b) , a b 2 R. " m 2 '(M), " m0 ab = mb m0 2 M r 2 R. " R ! rb = bs abs = tab , s t 2 R. E m0 tab = m0 abs = mbs = mrb mr 2 '(M). 2
7.5. % R &(( # # & R &(( . ) T (R) 4 R .
. ( # R ( $%% " $ , ( ( 7.3). 7.4 " "!, $ ( $ ( Inv(' ) > 1. 7 , ( "! , ! " ( Inv(' ) > 1 1- -% ' . 7# # " Vi '(ai3,bi), . |1] , ! ( $%% "
' Vj '(cj dj). 0 " 5.1 " Inv(' ) > 1 # 6" ( '(a b) ! '(c d). 5.3 " ! $%% 6 " d 2 bR ! " " ". 0 " $ 6 , ! " " : br0, br1, : : : dr0, dr1, : : :. L" d = bri i, ( !. L" b = drj j, 6 " " rj . L" \ ", d 2 bR, d 2= bR, " \". E R , , # $, " ! # "". 2 7.6. % R &(( # # " & R , . ) 4 R . . 0 " 7.5 " ! $%% , $ a 2 R . " " " ar0 ar1 : : : (ri 2 R ) "" 0, 1. 0 " ! $ a , | . 2
,
1] . . . // . . | 1975. | ". 18, %&. 5. | '. 707{710.
-
489
2] +, - . /. 0. ',&1 2 -- 4 % , , % % . // ' . . 1. | 1992. | ". 31, N 6. | '. 655{671. 3] +, - . /. 0. 71 2 -- 4 % , & . // ", -. . -%. | 1994. | ". 56. | '. 1{13. 4] +, - . /. 0. 2 2 -- 4 % , &,& . // 9-& . ,. | 1994. | ". 49, %& 5. | '. 171{172. 5] Bessenrodt K., Brungs H. H., T:orner G. Right serial rings. Part 1. | Preprint. | 1990. 6] Camps R., Dicks W. On semilocal rings. // Israel J. Math. | 1993. | V. 81. | P. 203{211. 7] Eklof P., Herzog I. A Some model theory over a serial ring. | Preprint. | 1993. 8] Eklof P., Sabbagh G. Model completions and modules. // Ann. Math. Logic. | 1971. | V. 2, N 3. | P. 251{295. 9] Herzog I. A test for ;nite representation type. | Preprint. | 1993. 10] McConnell J. C., Robson J. C. Noncommutative noetherian rings. | New York, 1988. 11] Mohamed S. H., M:uller B. J. Continuous and discrete modules. | Cambridge, 1990. 12] M:uller B. J., Singh S. Uniform modules over serial rings. // J. Algebra. | 1991. | V. 144. | P. 94{109. 13] Prest M. Model theory and modules. | Cambridge, 1988. 14] Puninski G. E. Pure-injective modules over right noetherian serial rings. // Comm. Algebra, to appear. 15] Rothmaler Ph. A trivial remark on purity. // Seminarber. Fachber. Humbold Univ. Berlin. | 1991. | V. 112. | P. 112{127. 16] Stephenson W. Modules whose lattice of submodules is distributive. // Proc. Lond. Math. Soc., Ser. 3. | 1974. | V. 28, N 2. | P. 291{310. 17] War;eld R. B. Serial rings and ;nitely presented modules. // J. Algebra. | 1975. | V. 37, N 3. | P. 187{222. % & 1995 .
.
e-mail: [email protected]
P | , N , ( !). P # # $% &#, , %
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! $% & .
Abstract Bl. Sendov, Compulsory congurations of points in the plane, Fundamentalnaya i prikladnaya matematika 1(1995), 491{516.
Let P be a set of N points in a general position (no three points are collinear) on the plane. A subset of P may form a speci0c con0guration, say obtuse triangle or convex pentagon. There exist con0gurations of points, that compulsory emerge in every point set of great enough cardinality. In this paper, such compulsory con0gurations of points on the plane are considered.
1
P = fp1 p2 p3 : : : pN g , ( ). . P , - !" P, jP j = N | " P. $!" q 2= P, !" P . % P = P q , , !% , P " , P " . $ " % " . &" C - , ! C % . $ C - '%!(. Ck | k- . ) Ck !* , k > 3 , * , !% k-!%" . 1995, 1, N 2, 491{516. c 1995 !", #$ \& "
492
.
Ck (I) |
k- , 0! ! , , !% k-!%" , , % ! 1% k-!%" ! ! * I. & , C5 (0) | 1 ! !%" , ! " % % , . . . C3 | > . 3* , * " !% ", , !* '%!(* C3 . 4 > 2 , C3 . & '%!(* C , !" n, " ", n ! C - '%!(*. & ", n, * 1 , N(C ). & , C3=3 | " '%!(, N(C3=3 ) = 2. 3 % ", C4 " '%!( N(C4) = 4. 4 C " '%!( , ! " N(C ) = 1. , ! k-!%" ( % ) ! " '%!( . & P '%!( C , P ! !* ! : a) P '%!(* C , b) * q 2= P P = P q '%!(* C . & ",!* '%!( C N(C ). &", '%!( C , , N(C ). ) " , N(C ) 6 N(C ) '%!( C . " '%!( C , N(C )=N(C ). 3 % ! ", N(C3) = N(C3) = 2 N(C4) = N(C4) = 4 . . ! k-!%" k = 3 4 * %! '%!(. 6 '%!( C35=9 %! , N(C35=9 ) = 4 N(C35=9 ) = 3. 7 , % !%" !* C35=9 .
I.
2 Ck
7 \ " '%!( Ck * !" k?" 0 6 ;5]. $ " ", % f(k), %, ,
493
", f(k) , ," k , !* ! %!%" . , f(3) = 2. 0 6 ! , f(4) = 4, 4. > , f(5) = 8. f(k) k > 6 . $. 0 , A. ) , ;4], ;5] , * ;4 2k;2 6 f(k) 6 2k (2.1) k;2 : A $. 0 , A. ) , , f(k) = 2k;2, " k = 3 4 5. $ (2.1) , '%!( Ck " k > 3. $ " 1% ' ;4]. C *% " % , % m P1 P2 : : : Pm , , " , ! , - % . 7 " ( . & '%!(* Pi1 Pi2 Pi3 : : : Pi !, !% Pi1 Pi2 Pi2 Pi3 : : : Pi ;1 Pi ! *, % !, *. $!" f1 (u v) | ", , " u , !* !!* '%!(*, v , !* % !!* '%!(*. % f1 (u v) = f1 (u ; 1 v) + f1 (u v ; 1) ; 1: (2.2) f1 (u ; 1 v) . 4 " v , !* % !!* '%!(*, (2.2) . 4 , ! '%!(, u ; 1 . E ** ! 1 ! '%!( !%!* ! % . % % !!* '%!(* v , ! , % !!* '%!(* v ; 1 . $ " ** !, !! " , % . ! Q = fQigri=1 , r = f1 (u v ; 1) , ! '%!(, u ; 1 . 7 Q u , !* !!* '%!(*, % , ! , v ; 1 , !* % !!* '%!(*. Q1 ! '%!(, u ; 1 . $!" Q0 | Q1 1 '%!(. 4 !% Q0Q1 ", , ! Q1Q2 , Q2 u ; 1 !* !!* '%!(* u F !% ", , Q0 Q1 Q2 Q3 : : : !* % !!* '%!(* v . (2.2). s
s
s
494
.
G % f1 (3 n) = n (2.2) !( !, 2n ; 4 f1 (n n) = n ; 2 + 1: (2.3) G f(n) = f1 (n n) ; 1 (2.3) ! (2.1). 3 (2.1) 2n;2 , ! n-!%" ;5]. 2.1
7 1 %' , ! '%!( Ck Ck (I)
, ! '-
%!( C3 . $!" | l | , ! ! . $ l " !. 4 l, !" h( l), !* , !"*, !%!* !" h( l) !"*. $!" % P ^ l) " ", u, , ! h( l) u = C(PF ! p q ! u P, !* ! (u + 2)-!%" , pq | , ! !% , p q , + < . , ^ l ) = C(PF ^ l). l " l P h( l ), C(PF ^ ^ $ P C(P F l) = C(PF ) " . P ^ F ) ! !" (u v) = C(PF ), % u = C(P ^ v = C(P F ;). P ", k = C(P), , k P, !* ! k-!%" . 3 % ! ", (u v) = C(PF ), maxfu vg 6 C(P) 6 u + v ; 2: I p = (p F u v), % p | , | u v | !" , , P " !"* C(PF ) = (u v). p , | , (u v) | ! p = (p F u v). A' ! ! " p, % " ! ( p u v !* .
495
, p = (p F u v) = (p ;F v u). & . 1 !: A, B, C : D, E. > %!%" ABC | ! !%" , 5 , A, 4 , B ! , ! C. C , " , , %!%" ABC \ " ! ! C. > %!%" AECD . 1 ! 11-!%" , AED | ! 7-!%" . 4
t
C
3
t
D 3@@5 A t
t
E
4;;2 B t
2. 1
( ) ! p = (p F u v) ", ( ", ) n = jpjmax (jpjmin ), , P jP j = n, * " !* !" (u v). , ! ! P " !"* , ! ! !% " . 3 % ! ", ! p = (p F u v) jpjmin = u + v ; 2: 7 ! '! " ! ! 0 , ) , ;4] !* . C ! p = (p F u v) u + v ; 2 jpjmax = (2.4) u;1 :
& . 2a ! Pa , '%!( C2n+1, . 2b | ! Pb , '%!( C2n. 3 % ! ", ! %!%" , Pa ! m-!%" , % m 6 2n. > " Pa
496
.
2n 2
2 2n
s
s
2n-2 4 s
4 2n-2 s
2n-4 6
6 2n-4
s
s
s
s
s
s
2. 2a
% (2.4)
jPaj = 2n + 2n + 2n + : : : +
1
3
5
2n = 22n;1: 2n ; 1
2n-1 2 s
(2.5)
s
2n-3 4 s
3 2n-2 s
2n-5 6 s
5 2n-4 s
s
s
s
s
2. 2b
C Pb . 2b jPbj = 2n ; 1 + 2n ; 1 + 2n ; 1 + : : : + 1 = 22n;2: 1 3 5
(2.6)
G (2.5) (2.6) ! " (2.1).
3 Ck (I ) $. 0 , '%!( Ck (I) ! I: k- , . . ! !% !% k-!%" . $!" ! I = q , k- q. 7 ;6] , N(C3 (0)) = 2 N(C4 (0)) = 4 N(C5 (0)) = 9:
497
C k > 7 C . C. K ;7] , N(Ck (0)) = 1, . . Ck (0) " '%!( k > 7: , Ck (q) | " '%!(, Ck (q+1), '%!( Ck (q) " , '%!( Ck (q ;1) " . C !" C . C. K ;7]:
3.1. k + 2 = 4m + r, r = 0 1 2 3 k Ck ((r + 4)2m;1 ; 4m ; r ; 1) .
> 7 -
# . $!" k + 2 = 4m + r, r = 0 1 2 3. ) ! C . C. K ! ;7], " % n 2n P = fp0 p1 p2 : : : p2 ;1 g * k-!%" , P ! (k + 4)2m;1 ; 4m ; r P. $!" an;1(i)an;2 (i) : : : a1(i)a0 (i) | " ( % i, 0 6 i < 2n, . . n
i = an;1(i)2n;1 + an;2(i)2n;2 + : : : + a1(i)21 + a0 (i)20 :
(3.7)
$!" c = 2n + 1, d(i) = an;1(i) + an;2(i)c + : : : + a1 (i)cn;2 + a0 (i)cn;1
(3.8)
P = fpi = (i d(i)) : i = 0 1 2 : :: 2n ; 1g: (3.9) $ * ;7]: a) L = fpi : i = 0 1 2 : :: 2n;1 ; 1g | P. b) R = fpi : i = 2n;1 2n;1+1 : : : 2n;1g | P | ! L. c) B = fpi : i = 0 2 4 : :: 2n ; 2g | P . d) T = fpi : i = 1 3 5 : :: 2n ; 1g | P | ! B. e) 7 L R B T * , !% !%. & , ! c B L. f) $ !% (c=2 ; 1 (cn ; 1)=2(c ; 1)) ! T B. g) 7 T , * , * B. L % B * , * T. h) 4 i j (mod 2s ), s | !" h 6 i (mod 2s), s+1 (' h *, , , * pi pj , ph .
498
.
" ! k-!%" Q P . > ", Q "* T , B. G , Q B, , * B L. $ 1 Q ! k-!%" L. L % , Q T, , * T L. M! " 1! ( !!, Q T, B. $!" pi1 pi2 : : : pi pi +1 pi +2 : : : pi + | Q B, % i1 < i2 < : : : < ix < ix+1 < : : : < ix+y d(i1 ) > d(i2 ) > : : : > d(ix ) < d(ix+1 ) < : : : < d(ix+y;1 ): $!" pi pi +1 (mod 2s ;1 )F l = 1 2 : : : x ; 1. %, ! h) !" Q, s1 < s2 < : : : < sx;1 an;s (il ) = 1 an;s (il+1 ) = 0: 0 , ", l ; 1 (' az (il ) s1 6 z < sl !*. ) " , ", 2l;1 ; 1 B n Q , pl pl+1 . % % e) B ! Q ", , 20 ; 1+21 ; 1+: : :+2x;2 ; 1+20 ; 1+21 ; 1+: : :+2y;1 ; 1 = 2x;1 +2y ; x ; y ; 1: 7 0! g)0 T ! Q 2x ;1 + 2y ; x0 ; y0 ; 1, % x + y + x0 + y0 = k. & (, P ! Q ", , 2x;1 + 2y + 2x0 ;1 + 2k;x;y;x0 ; k ; 2 > (r + 4)2m;1 ; 4m ; r % k + 2 = 4m + r r = 0 1 2 3, 1 ( . . 0 C . C. K ;7] , '%!( C7(0) " . &, % * , '%!( Ck ((r+4)2m;1;4m;r) " k > 6. 0 % * C . C. K ;7] , C6(0) | " '%!(. 7 " C6(0) . ) "* % "* % % " % !% !% " C. ;8] ! 20 , ! ! 6-!%" , , '%!( C6(0) " , N(C6(0)) > 20: (3.10) x
l
x
x
x
y
l
l
l
l
499
$ (. 3) \ " 20 , ! ! 6-!%" , (3.10) , N(C6(0)) 6 20: s s s
s
s
s
s s s s
s
s s s
s s s
s
s
s
2. 3. 4 20 ,
! !! 6-%
.
$!" ! I 0(modq) : k- q. L. M, $. C M. 7 ;1] ! % ! , '%!( Ck (0 mod q)) " * !" q k > 3, k > q + 3 k 2 (mod q):
4 C3
N '%!( C3 " 2(0 )? G ! 1% 3. >. M* ;2] $. 0 , A. ) , ;5]. C , '%!( C3 " * < . 3 % ", '%!( C3 " < , f() = N(C3 ) lim!;0 f() = 1. $. 0 , A. ) , ;4] , '%!( C3 " * < . 0 N(C3 ) N(C3 ), " % !" $. 0 , A. ) , ;4], ;5]. G 3. >. M* *" ", % (N), %, * '%!( N
500
.
, * !% > (N)F 0 6 < : 4 (N), % N(C3 ). & ! " (3) = 31 (4) = 21 (5) = 53 (6) = 23 " , N(C3=3 ) = 2 N(C3=2 ) = 3 N(C33=5 ) = 4 N(C32=3 ) = 2: & " , (7) = (8) = 23 :
0 $. 0 ,. G , N = 3 4 5 6 ! ! PN ( , " % !% N-!%" ), PN '%!(* C3 = (N), '%!( C3 > (N). C N = 7 8 1 . 7 1 ! % > 0 PN , PN '%!(* C3 , % = (N) + , '%!( C3 , % > (N) + . > ", N = 7 8 ! ! 1 " . 7 A. ) , ;14], $. 0 , A. ) , ;5] , (2n ) = (1 ; 1=n)
(4.11)
(2n + 1) > (1 ; 1=n) : $. 0 , A. ) , ;5] %*, 1 (N) = 1 ; n 2n;1 < N 6 2n: 7 ;9], ;10] , (9) = (10) = 75 (11) = (12) = : : : = (16) = 34 : 0 % % ! (4.12) N = 11 12 : :: 16. & ;12], ;13] !*!* !.
(4.12)
4.1.
1 (N) = 1 ; n
2n;1 + 2n;3 < N 6 2n
(4.13)
2 (N) = 1 ; 2n + 1
2n < N 6 2n + 2n;2
501 (4.14)
2 1 1 ; 2n + 1 < 6 1 ; n
1 2 n n ;2 N(C3 ) = 2 + 2 ; 1 1 ; n < 6 1 ; 2n + 1 : I " (4.13) (4.14),
. 0 '%!( * 1 " 1 , % (N). 0 '%!( 6 .
4.1
N(C3 ) = 2n ; 1
M! " ( ). I !" 1, " !
. % 4.1. ! C = fc1(o1 r1) c2(o2 r2) : : : cM (oM rM )g | # oi ri. $ # C % , # , # % C , , % % . & C 1, % . & C k, , # % , k ; 1. & , # , . ! # , , .
& . 4 !! !% . 6!% ( o1 o2 o3 * % 1. 6!% ( o4 o5 o6 o7 o8 * % 2, !% ( o3 o7 o8 o9 : : : o15 . % 4.2. ( V = fP C g = fp1 p2 : : : pN F c1 c2 : : : cM g,
N = jP j
M = jC j > N , , , , : a) &# P
C . + k,
k. b) , % 1. c) &# C # % .
502
.
r
O3 =P1
r
O14 =P9 r
O6
r
O13 =P8 r
r
O15 =P10 rO 4
r
O9 =P4
r
O1 O12 =P7
O8 =P3 r O2
r
O r 5 r
O10 =P5
r
O7 =P2 r
O11 =P6
2. 4
, *% % !% " '%!( ( P !% C, ! % !%) ! !* '%!(*. & . 4 '%!( 10 P1 : : : P10 15 !%. ) !% ( o1 ! !* '%!(* 7 10 !%. 3*!* !* '%!(* " !* '%!(* !% , !% . % 4.3. ! p q |
V = fP C g, o(pF q) | , #
p # q, o(qF p) | , # q # p. GD(pq) p q o(pF q)o(qF p),
,
GD(pq) = G(pq) = D(o(pF q)o(qF p)): & . 4 o(p4 F p2) = o1 o(p2 F p4) = o2 GD(p2 p4) = D(o2 o1 ).
503
! V = fP C g V 0 = fP 0 C 0g |
. - , V V 0 ' (V V 0 ),
pi p0i # P P 0, % i j G(pi pj ) G(p0i p0j ) .
% 4.4.
C ! 1 '%!( V V 0 ! " " pi p0iF i = 1 2 : : : N. % 4.5. / p q r, 0 , A(p q r) < . 1 GA(p q r) # 0
2 ;0 ) # G(qp) G(qr). / l l
0 < A(l l ) < , #
l l .
& . 4 GA(p2 p4 p8) = A(o1 o2 o4o6 ): % 4.6. 1 A(P )
P . 1 GA(V ) = GA(fP C)g
V = fP C g:
) !* . * 4.1.
V = fP C g V 0 = fP 0 C 0g p q r 2 C p0 q0 r0 2 C 0 | ,
,
GA(p q r) = GA(p0 q0 r0)
GA(fP C g) = GA(fP 0 C 0g):
% 4.7. ! dk (fP C g) | # % k
V = fP C g. 1 V h- , 0 < h < 21 , d1(fP C g) = 1 k C r(k) = hdk (fP C g). * 4.2. ! V = # 0 < h < 21 0 V .
fP C g
| . 2 V (h),
h-
# . , * t > 0 a b = tx + a, = ty + b !* !*
504
.
'%!(* 1 !* !* '%!(*. $!" fP (1) C (1)g = fP C g d1 (fP (1) C (1)g) = 1:
& ,% " !% ci (oi ri) % 1 C (1) ! % = h(x ; xi)=ri + xi = h(y ; yi )=ri + yi % oi = (xi yi). 0 , P (1). 4 !% % 1 C (1), !* !* '%!(* fP (2) C (2)g = fP (1) C (1)g = fP C g
k = 1. & ,% k ! !% cj (oj rj ) % k C (k) = hk (x ; xj )=rj + xj = hk (y ; yj )=rj + yj % oj = (xj yj ) hk = hdk (fP (k) C (k)g): 4 m | " % !% P, '%!( V (h) = = fP (m) C (m) g ! h- " ! ! * . * 4.3. P | V = fP C g | h- # # P , % % p q r P
jA(p q r) ; GA(p q r)j < 2 h
jA(P) ; GA(fP C g)j < 2 h:
# . 4 p q * % 1, % * % GD(pq) D(pq). 4 p q * % k ! % % !% % k ; 1, GD(pq) = D(pq). 4 p q * % * % k, ! !% % k ; 1, % !% ! D(pq) GD(pq) , h. 0 , " . % 4.8.
(N) = inf fA(P) : P = fp1 p2 : : : pN gg G(N) = inf fGA(fP C g) : P = fp1 p2 : : : pN gg: 3 #
P
C.
505
4.2.
! #
N
G(N) = (N):
(4.15)
# . 7 '%!( !% " % 1 !% !* !%. G 1% ! , G(N) 6 (N): (4.16) C , " (4.15) ",
G(N) > (N):
(4.17)
C! %, % % !" % N " > 0, G(N) < (N) ; :
(4.18)
G (4.18) ! , ! ! '%!(
fP C g,
GA(fP C g) < (N) ; , '%!( P = fp1 p2 : : : pN g GA(fP C g) < A(P ) ; : (4.19) )% 4.2 4.3 ! ! h- " '%!( fP C g, 1 fP C g, GA(fP C g) = GA(fP C g) > A(P) ; 2 h:
(4.20)
& (4.20) (4.19) h, (4.17) . )% 4.2 (N) ", " '%!( , ", '%!( . 7% 1% * , '%!( ! 1 " 1 V = fP C g jP j = N, GA(V ) = G(N) = (N): )! 1 " 1 ;13]. 4.2
& ! $. 0 , A. ) , ;5] % %'. $!" K (N ) | %' N (%' N , , * , ). I ( ) ( %' G ! !", ( ) .
506
.
) ! ;5], %' G * G = G1 + + G2 + : : : + Gn %' Gi !* : Gi , %' G, %' G " Gi (Gi " ). , Gi ( . 7 ;5] ;14] !* . * 4.4. K (N ) = G1 + G2 + : : :+ Gn |
n ,
N 6 2n : # . C " 1% !". $"! G1 ( , " , %' K (N ) A B, N1 N2 , , , %' G1 ! A B. & % G1 + G2 + : : : + Gn !(! G02 + G03 + : : : + G0n %' K 0 = K (N ) jA, K 0 | %' N1 , * !( N1 6 2n;1. L % N2 6 2n;1, ! N = N1 + N2 6 2n. G % ! " " E. & F 0 6 < 2 E | 1 o(0 0) ei ! . % 4.9. ! | 0 6 < 2 . $ #
T( ) = fz : z = aei' a real 6 ' < + gno(0 0) % . &#
% ( % ), % . (#, # Q %. T , Q 0 .
) !* . * 4.5. # Q # T % p q r Q D(pq) D(qr) #
# T ,
A(p q r) > ; : V = fP C g
* 4.6. ! , Q P
# , Q, # % % T1 (1 ), T2 (2 ),: : : , Tk (k ). GA(Q) 6 ; jQj 6 2k : (4.21)
507
$!" K (k) | %' , Q Gi, i = 1 2 : : : k | %' , Q, , * , ! % T (i ). > ! , K (k) = G1 +G2 +: : : Gk . C " , %' Gi (, 1 ( ! " % T(i ). & % 4.5 1 ! ! (4.21). " (4.21) ! 4.4. # .
% 4.10. 4 k
0 , # k+1 # 0 % 0
. % 4.11. 1 , .
0 -
$!" V = fP C g | '%!(. M! " V "* ( o1 o2 : : : os !% % 1 "* li 1 li 2 : : : li k(i), ( oi !% ci , i = 1 2 : : : s, " , P, ! ci . I , P , ! ci, k(i). & . 5 , '%!( 4 ( !* . * 4.7.
V = fP C g = fo1 o2 : : : osF li 1 li 2 : : : li k(i)F i = 1 2 : : : sg , 0
N = jP j
N = 2k(1) + 2k(2) + : : : + 2k(s): # . 4 !%! % 1 ! , ! 1% !% V . $!" k ; 1 !" ! % !% % 1 !% % 2, ! 1 !% ! k ; 1 2k;1 . O !% % 2 * ! !* !% % 1 k ; 1 , " !% % 2. ) " , ! !%! % 1 ! k 2k V . $ !( .
508
.
l4 1 O4 r r
O1
r
r
O3 l3 1
O2
l3 2
l2 1
l1 1 l l1 3 1 2 2. 5
* 4.8.
!
V = fP C g = fo1 o2 : : : osF li 1 li 2 : : : li k(i)F i = 1 2 : : : sg , Q = fo1 o2 : : : os g | #
1 V , = A(Q), Pi P | # ci , i = GA(Pi ) = maxf1 2 : : : sg 'i | # fli j F j = 1 2 : : : k(i)g foioj F i 6= j = 1 2 : : : sg, ' = minf'1 '2 : : : 'sg: +
GA(V ) = maxf ; 'g: # . $!" p q r | P, GA(p q r) = = GA(V ): 4 1 ! !% % 1, GA(p q r) = : 4 1 !% % 1, " | !% !% % 1, GA(p q r) = ; ': 4 ! % % !%, GA(p q r) = : 0 , " . * 4.9. l1 l2 : : : lk | , o c ,
A(li li+1 ) > 5 i = 1 2 : : : kF lk+1 = l1 , 2k Q c V = fQ C g,
GA(V ) 6 ; : # . $ , ! !% c 1. & ,% o1 1 o1 2 l1 = l1 1 , jo ; o1 1j = jo ; o1 2j = 2;1 c1 i = c(o1 i 3;1)F i = 1 2:
509
& ,% l2 i F i = 1 2 | , ( o1 i F i = 1 2 " l2 . o2 1 o2 2 l2 1, jo1 1 ; o2 1j = jo1 1 ; o2 2j = 2;2 , o2 3 o2 4 l2 2 , jo1 2 ; o2 3j = jo1 2 ; o2 4j = 2;2 c2 i = c(o2 i 3;2)F i = 1 2 3 4 = 22: & ,% k 2k pi = ok iF i = 1 2 : : : 2k !! !* ! !% , * !* !* '%!(* V , GA(V ) 6 ; . * 4.10. ! V = fP C g = fo1 o2 : : : osF li 1 li 2 : : : li k(i)F i = 1 2 : : : sg |
, Ni , i = 1 2 : : : s | % ci = c(oi ri) 1, Li = fti 1 ti 2 : : : ti s;1g | , oi oj 5 j 6= i, 'i j = A(ti j ti j +1)5 j = 1 2 : : : s ; 1, ti s = ti 1 , 'i 1 + 'i 2 + : : : + 'i s;1 = 1 GA(V ) = (1 ; 2=u) : +
0
Ni 6 2k (i)
k0(i)=(;u'i 1=2] ; 1)+ +(;u'i 2=2] ; 1)+ +: : :+(;u'i s;1=2] ; 1)+ : (4.22) # . $!" fi 1 i 2 : : : i q g f'i 1 'i 2 : : : 'i s;1g | !% f'i l g, !* ! (4.22) !* i j = 'i l > 4=u: $ ;ui j =2] ; 1 T( (Pi j + 2k=u)F 2 =u)F k = 1 2 : : : ;ui j =2] ; 1 (4.23) !% 2 =u, % Pi j 2 ;0 ) | !% ti m , 'i m !% i j . )% 4.8 * ! p p0 P, ! !% ci, GD(pp0 ) (4.23). $1! 4.6 ! (4.22). , ( ( ! !% ci " (Pi j + 2k=u)F j = 1 2 : : : q k = 1 2 : : : ;ui j =2] ; 1. li j , 1 6 j 6 k0 (i), % j " % . % % 4.9 , '%!( V = fP C g = fo1 o2 : : : osF li 1 li 2 : : : li k0(i) F i = 1 2 : : : sg '%!( ",, ! V , GA(V ) = GA(V ): , !* . * 4.11. 2
V = fP C g
V = fP C g = fo1 o2 : : : os F li 1 li 2 : : : li k0(i)F i = 1 2 : : : sg jP j > jP j GA(V ) = GA(V ):
510
.
" " k(i) 4.7 k0 (i) 4.10. & (, 4.11 ! 4.3. 2 # N G(N) = (N) = inf fGA(fP C g) : jP j = N fP C g
g:
)% 4.3 G(N) = (N) " " , '%!(. 6 , '%!( V = fP C g = fo1 o2 : : : osF li 1 li 2 : : : li k(i)F i = 1 2 : : : sg "* ( !% % 1 "* !* . 4.3 * 4.12. !
V = fP C g = fo1 o2 : : : osF li 1 li 2 : : : li k(i)F i = 1 2 : : : sg | , GA(V ) = (1 ; 2=u) ;u=2] = n = u=2 ; n. + jP j 6 2n 0 6 < 1=2 2n 6 u < 2n + 1
jP j 6 2n + 2n;2
1=2 6 < 1
2n + 1 6 u < 2n + 2:
# . )% 4.7
N = jP j =
s X i=1
2k(i)
4.10 k(i) = (;u'i 1=2] ; 1)+ + (;u'i 2=2] ; 1)+ + : : : + (;u'i s;1=2] ; 1)+ : M! %", ( , k(1) > k(2) > : : : > k(s) !(* s. a) 4 s = 2 '1 1 = '2 1 = 1, k(1) = k(2) = n ; 1
jP j = 2n:
b) 4 s = 3, !% * '1 1 '1 2 = 1 ; '1 1 '2 1 '2 2 = 1 ; '2 1 '3 1 = 1 ; '1 1 ; '2 1 '3 2 = '1 1 + '2 1 k(1) = (;u'1 1=2] ; 1)+ + (;u(1 ; '1 1 )=2] ; 1)+ k(2) = (;u'2 1=2] ; 1)+ + (;u(1 ; '2 1 )=2] ; 1)+ k(3) = (;u('1 1 + '2 1)=2] ; 1)+ + (;u(1 ; '1 1 ; '2 1 )=2] ; 1)+
4
511
2=u 6 '1 1 + '2 1 6 1 ; 2=u:
(4.24)
n ; 2 > k(1) > k(2) > k(3) jP j 6 3:2n;2 < 2n:
$!" k(1) = n ; 1. 0 " %, % '1 1 6 n + : (4.25) !. b.1) 4 0 6 < 1=2, (4.25) ! , '1 1 = 2n1+1 . G (4.24) ! '2 1 > n +1 ; 2n 1+ 1 > 2n 1+ 1 ! n ; 2 > k(2) > k(3) jP j 6 2n;1 + 2:2n;2 = 2n: 7 1 ! ! % , k(1) = n ; 1 k(2) = k(3) = n ; 2: b.2) $ 1=2 6 < 1, k(2) = n ; 2 , jP j 6 2n;1 + 2:2n;2 = 2n: $!" k(1) = k(2) = n ; 1. % % (4.25) '1 1 < n +1 1 '2 1 < n +1 1
512
.
(4.24) ! ) " ,
2=u 6 '1 1 + '2 1 < n +2 1 :
k(3) = (;u=(n + 1)] ; 1)+ + (;u(1 ; 2=u)=2] ; 1)+ = ;u=2] ; 2 = n ; 2
jP j 6 2:2n;1 + 2n;2 = 2n + 2n;2:
c) 4 s = 4, !. c.1) 4 0 6 < 1=2, ! jP j % k(1) = n ; 1 k(2) = n ; 2 k(3) = k(4) = n ; 3 %
jP j 6 2n: c.2) 4 1=2 6 < 1, ! jP j %
k(1) = n ; 1 k(2) = k(3) = n ; 2 k(4) = n ; 3 %
jP j = 2n + 2n;3 < 2n + 2n;2: $!" s ; 1, % s !. s.1) 4 0 6 < 1=2, ! jP j %
k(1) = n ; 1 k(2) = n ; 2 : : : k(s ; 1) = k(s) = n ; s + 1 %
jP j 6 2n;1 + 2n;2 + : : : + 2n;s+1 + 2n;s+1 = 2n: s.2) 4 1=2 6 < 1, ! jP j %
k(1) = n ; 1 k(2) = n ; 2 : : : k(s ; 2) = k(s ; 1) = n ; s + 2 %
k(s) = n ; s + 1
jP j = 2n;1 + 2n;2 + : : : + 2n;s+2 + 2n;s+2 + 2n;s+1 = 2n + 2n;s+1 6 2n + 2n;2:
0 , " . % 4.12. 1 N() , , V = fP C g, jP j = N() GA(V ) 6 :
513
, '! ( (N) N() ! * . G 4.12 ! * 4.13. = (1 ; 2=u) ,
N() 6 2n 2n 6 u < 2n + 1
N() 6 2n + 2n;2 2n + 1 6 u < 2n + 2: C ! !. * 4.14. = (1 ; 2=u) ,
N() > 2n 2n 6 u < 2n + 1
(4.26)
N() > 2n + 2n;2 2n + 1 6 u < 2n + 2: (4.27) # . $!" V | '%!( N ! !% % 1. G 4.7 4.10 s = 2 (. . 6) N = 2(u=2];1 + 2(u=2];1 = 2n ! ! (4.26).
O1
r
r
O2
2. 6
$!" V | '%!( N !% % 1. $!" o1 = (;1 0), o2 = (1 0) o3 = (0 tg 2n+1 ) | ( 1 ! (. . 7). % '1 1 = '2 1 = 2n 1+ 1 '1 2 = '2 2 = 1 ; 2n 1+ 1
514
.
r
O1
O3
r
r
O2
2. 7
'3 1 = 2n 2+ 1 '3 2 = 1 ; 2n 2+ 1 2n + 1 6 u < 2n + 2: (;u'i 1=2] ; 1)+ = 0 i = 1 2 3 (;u'i 2=2] ; 1)+ = n ; 1 i = 1 2 (;u'3 2=2] ; 1)+ = n ; 2: )% 4.7 4.10 , ! , N = 2n;1 + 2n;1 + 2n;2 = 2n + 2n;2: , (4.27) . & (, 4.13 4.14, , '! ( (N) N() , (N()) = , ! !*!* !. 4.4. (N) | , , N # , > (N)F 0 6 6 ,
2 (N) = 1 ; 2n + 1
(N) = 1 ; n +1 1
2n < N 6 2n + 2n;2
2n + 2n;2 < N 6 2n+1:
1 . $!" m (N) | ", , , * -
515
'%!( N m- , * !% > m (N)F 0 6 6 : $ m (N) " . " , m (m + 1) = =3: ) ! % $. 0 , A. ) , ;5], 3. C ( M. A* ! ;3] , m (2m ) = =2: & m (N) m > 2 ! N. & , =3 < 3(5) = arccos 71 < =2 " 3(6) = 3(7) = =2 .
1] A. Bialostocki, P. Dierker and B. Voxman. Some notes on the Erdos { Szekeres theorem // Discrete Mathematics. |1991. | V. 91. | P. 231{238. 2] Blumenthal L. M. Metric methods in determinant theory // Amer. Journal of Math. | 1939. | V. 61. | P. 912{922. 3] Danzer L., B. Grunbaum U ber zwei Probleme bezuglich konvexer Korper von P. Erdos and V. L. Klee // Math. Zeitschr. | 1962. | V. 79. | P. 95{99. 4] Erdos P., G. Szekeres A Combinatorial Problem in Geometry // Compositio Math. | 1935. | V. 2. | P. 463{470. 5] Erdos P., G. Szekeres On Some Extremum Problems in Elementary Geometry // Ann. Univ. Sci. Budapest. | 1960. | V. 3{4. | P. 53{62. 6] Harborth H. Konvex Funfecke in ebenen Punctmengen // Elem. Math. | 1978. | V. 33. | P. 116{118. 7] Horton J. D. Sets with no Empty 7-gons // C. Math. Bull. | 1983. | V. 26. | P. 482{484. 8] Rappaport D. Computing the Largest Empty Convex Subset of a Set of Points. | 1985. | ACM 0-89791-163-6/85/006/0161. | P. 161{167. 9] Sendov Bl. On a Conjecture of P. Erdos and D. Szekeres // Comptes Randus de l'Acad. Bulgare de Sci. | 1992. | V. 45. | N 12. | P. 17{20. 10] Sendov Bl. Optimal disposition of points in the plane with respect to the angles, determined by them // Discret Mathematics and Applications, Ed. K. Chimev & Sl. Shtrakov. | Blagoevgrad, 1993. | P. 10{24.
516
.
11] Sendov Bl. Compulsory Congurations of Points in Euclidean Plane // Advances in Parallel Algorithms, Ed. I. Dimov and O. Tonev. | Amsterdam: IOS Press, 1994. | P. 194{201. 12] Sendov Bl. Angles in a Plane Conguration of Points // Comptes Randus de l'Acad. Bulgare de Sci. (To appear) 13] Sendov Bl. Minimax of the Angles in a plane conguration of points. (To appear.) 14] Szekeres G. On an extremum problem in the plane // Amer. Journal of Math. | 1941. | V. 63. | P. 208{210. ( 1995 .
. . , R8 , - .
Abstract V. E. Balabaev, On one system of equations in octaves in eight dimensional Euclidean space, Fundamentalnaya i prikladnaya matematika 1(1995), 517{521.
This paper deals with the boundary properties of functions with values in algebra of octaves and satisfying an equation system in R8 similar to the Cauchy-Riemann system.
, , { , ! "1, 2]. ' ! , ( ! R8, { . * ( ( ! - , ! . ' R8 ( . + ( )= 1 ( ) 2( ) 3( ) 4 ( ) | ( (1 0)( ), ( ! C 4 . 0( 14 2 3 1 11 ; 2 ; 3 ; 4 = 0 1 + 2 + 13 ; 4 = 0 2 1 4 3 (1) 1 1 2 3 4 13 ; 4 + 1 + 2 = 0 11 + 2 ; 3 + 4 = 0 1 1 1 g z
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Abstract
A. Ya. Belov, The Nagata-Higman theorem for semirings, Fundamentalnaya i prikladnaya matematika 1(1995), 523{527.
This paper contains the proof of the Nagata-Higman theorem for semirings (with non-commutative addition in general). The main results are the following: Theorem. Let A be an l-generated semiring with commutative addition in which the identity xm = 0 is satis'ed. Then the nilpotency index of A is not greater than 2lm+1 m3 . Nagata-Higman theorem for general semirings. If an l-generated semiring satis'es the identity xm = 0 than every word in it of length greater than mm 2lm+1 m3 + m is zero.
, .
. . | -
!. "# $ N. . % X x : : :x 2 ! x1 : : :xm = (1) (1) (m) 6=id
. $ 1. * , #+ $ . , #+ ( 1995, 1, N 2, 523{527. c 1995 ) * & +,, -. & \0 "
524
. .
1. A | , M = Var(A), a1 : : : as | A, R | , ai . v R !, w
" w. . * $ # 0 ,
1 2 #. $ , 1 $ $ R . 2 % + 2 2 0, # # # , 1 , M 0 1 A . 3 , #+ 2. A # PIdeg(M) > 1. (PIdeg(M) , . . # , ). R
. & !
, R # # . 2 4 S f = 0. 4 1 \ " f . (8 , , x3 = 0, x2y+ + xyx + yx2 = 0.) f $ 2. , #+
3. S # f(t x)=0. (' t ! # , ! # x.) f | m x. S # f~ | f x. . ; :
X f~ = f(t x1 + + xm ) ; f(t x1 + + xbi + + xm ) + +
X f(t x i<j
i
1
+ + xbi + + xbj + + xm ) ; + (;1)m;1
X f(t x ): k
k
(= b, , #+$ .) 2 , 1 0 - : X f~ + f(t x1 + + xbi + + xm ) + i
-
+
X f(t x
1
i<j
525
+ + xbi + + xbj + + xm ) + =
= f(t x1 + + xm ) +
X f(t x i<j
1
+ + xbi + + xbj + + xm ) + :
, f | , 1 , , , f,~ #. 8 f~ #. 2
4. ( xm = 0 P x : : :x m = 0: (1)
( )
2
. ) , . ) 4 , 2 # $ $ . > x = y # : # x + z = y + t, # , z = t = 0 ( # + !). 3 8 -@ . 5. A | l- (
# ), # xm = 0. A #) 2lm+1 m3 . . ; , $ . 4 v 2lm+1 m3 . 4 B1] m- , m-# . 4 1 v m- v = v0 v1v2 : : :vm , v1 v2 vm . 3 # 2 Sm = id v = v0 v(1) : : :v(m) 0 v, , v, #. , : v = v0 v1v2 : : :vm = v0 (v1 v2 : : :vm +
Xv
6=id
(1) : : :v(m) ) =
= v0 (
Xv
(1) : : :v(m) ) = v0 0 = 0:
4 v. 3 8 -@ . 2
. , + 1 2 + 2 , $ 2 2.
526
. .
.
% R | , (x + y ; x ; y)z = = (x + y)(z ; z) = 0. 4 1 (x + y)z = (y + x)z,
. * 2 . E ,
. #+ + 2 $ . 6. x + y = y + x0 = 0: a b ax + bx = bx + ax xa + xb = xb + xa. . 0 = (a + b)(x + y) = ax + bx + ay + by. $ , bx + ax + ay + by = bx + a(x + y) + by = b(x + y) = 0. 4 1 bx + ax + (a + b)y = ax + bx + (a + b)y. 4 (a + b)x0 ,
. > xa + xb = xb + xa . 2 , 8 -@
7.0 * W | # 2lm+1 m3 , x + y = y + x0 = 0, 0 W x = xW = W x = x W = 0. 2 4 xm = 0. > (x1 + + xm )m (xm + + x1)m . > 2 ( P m , P m | . ., | m- ). 4
mi=1 vi mi=1 vi0 , # + mm | 2 $ xi . 4 1 vi0 = vmm ;i . (% $
P ,
). 4
P ;1 v , s0 = Pmm m mm v , r0 = Pmm ;1 v0 . ,
0 sk = ki=1 v , r = i k i i=m ;(k;1) i k i=k i k i=1 sk + rk = rk0 + s0k = 0, sk + vk = sk+1 , vk;1 + rk = rk;1, vk + s0k = s0k+1 , rk0 + vk;1 = rk0 ;1. 8. W ) k 2lm+1 m3. Wsk = = Wvk = W rk = Ws0k = Wrk0 = 0. W sk , rk , s0k , rk0 vk . . *
# Wsk , Wrk Wvk . * $ k. F k = 1 7. * sk + rk = 0 # Wrk . 4 W = uW 0, juj = 2lm+1 m3 , jW 0j > (k ; 1) 2lm+1 m3 . 4 # W 0sk;1 = W 0 vk;1 = W 0 rk;1 = W 0 s0k;1 = W 0 rk0 ;1 = 0. 3
W 0sk = W 0 s0k = W 0 vk W 0vk + W 0 rk = W 0sk + W 0 rk = W 0(sk + rk ) = 0: H W 0rk0 + W 0 s0k = W 0rk0 + W 0 vk = W 0 rk0 + W 0 sk = 0. x = W 0 rk , y = W 0 sk , x0 = W 0rk0 + . 2
-
527
9. (3 8 -@ + .) * l- # xm = 0, # ) mm 2lm+1 m3 + m . . , k 1 Pmmmm . 4 mm 2lm+1 m3 # $ # $
i=1 vi #. 2
. ) , 2
, 0 m! lm+1 m3 + m. , 1 # ( m) . ) . , , 0 0 . + x2 = x3 ( + # 2 2 ). ) $ #. F
$, 0 2 . * , !
(. . $ #) 8 -@ . ) J , 6, # J1, 1 # 1 . J2 J1 - A=J1 . Jk + # $ m. , 5 $ 6, 7 , Jk A2kl +1 m3 #. H H. *. 2K .
B1] A. J. Belov. Some estimations for nilpotence of nill-algebras over a Neld of an arbitrary characteristic and height theorem // Comm. in Algebra. | 1992. | 20(10). | P. 2919{2922. 2 $ * 1995 .
. .
512.553
S UR MR T S HomR (M U ), ! .
"# $ , % . T
M
Abstract G. M. Brodskii, Annihilators and weak topologies on modules, Fundamentalnaya i prikladnaya matematika 1(1995), 529{532.
For a bimodule S R we characterize modules R and submodules of S HomR ( ) for which the double annihilator conditions hold. We study the weak topologies induced on by . U
M
T
M U
M
T
, . 1] " . # Y X $ , X=Y $ . %
" " $ : L(X ), Lf (X ) Lf (X ) | $$ $
$ *, * $
* Q * $ * X + X | 2 X , X = X . - MR , " S UR , H S - hU (M ) = = HomR (M U ) A M J H IS (U ) = = TIm(S ! EndR (U ))+ AnnH (A) = fg 2 H j Ker g Ag AnnM (J ) = = g2J Ker g. # . , $ UR H - , A 2 L(M ) U : M=A ! U 2 H ( 2 /), : M ! M=A |
. . 0 UR H -
, Im g $ g 2 H . - 0 ! M=A !i U n
(1) # U n | $ , A 2 L(M ) $ , H - , 2 H k i 2 H (1 6 k 6 n), : M ! M=A |
. , k | , 2 U n. 3 S UR H -
, H - (1)
1995, 1, N 2, 529{532. c 1995 , ! \# "
530
. .
: U n ! U , ik 2 IS (U ), 1 6 k 6 n, ik | , 2 U n. 3
.
:
MR , S UR
H
hU (M ) -
UR H - ! 2) AnnM (Ann H (A)) = A # A 2 L(M ). 1)
6 R S " S UR . %
MR ,
T hU (M ), , $, T -# 2] 8 M , T . - $ 2 $ M $ " " fAnnM (J ) j J 2 Lf (T )g. 0 * * hU (M ) " $ $ CT HomR (M U ). 9 , $ T CT HomR (M U ). , $ T - M * , T hU (M ) 2, . . AnnM (T ) = 0. #
1.
:
MR , S UR H hU (M ) -
H -
! AnnH (AnnM (J )) = J # J 2 Lf (H )! 3) H \ CT HomR (M U ) = T # T 2 L(H ). 1)
S UR
2)
P
. 1) =) 2). - J = nk=1 Sgk 2 Lf (H ) n
i: M= AnnM (J ) ! U k i = gk , k = 1 2 : : : n+ : M ! M= AnnM (J ) |
. + k | , 2 U n. # J AnnH (AnnM (J )), " $ . - . ' 2 AnnH (AnnM (J )) ' = , : M= AnnM (J ) ! U . : H -
; S UR : U n ! U , i = ik 2 IS (U ), 1 6 kP6 n, ik | , 2 U n. < , $ ' = nk=1 ik gk 2 J . 2) =) 1). - H - (1) * P Sk i , $ A = AnnM (J ), J = nk=1 P 2 L # . f (H ). 2 AnnH (AnnM (J )) = J , $, = nk=1 k k i * 2* k 2 IS (U ). , $ = i, : U n ! U | , ik = k . 2) =) 3). # , $
CT HomR (M U ) =
J 2Lf (T )
Annh
U
(M ) (AnnM (J ))
" T
531
hU (M ) , ,
H \ CT HomR (M U ) =
J 2Lf (T )
AnnH (AnnM (J )):
(2)
= T H , 2 (2) 2) $, $ H \CT HomR (M U ) = T . 3) =) 2). #
3) T 2 Lf (H ) $ (2), $, $ AnnH (AnnM (T )) T . . S UR : 1) UR # ' S ! End R (U ) !
CT HomR (M U ) = T ( MR T hU (M )! CT HomR (M U ) = T ( MR # T hU (M ).
. 1) =) 2). : 3, 2.1] 1. 2) =) 3). $ . 3) =) 1). > i: MR ! UR 2 T = Si hU (M ). # T - M , hU (M ) = CT HomR (M U ) = T . > , UR ; . ? , $ $, MR = UR , i |
. UR , * EndR (U ) = IS (U ). 2. MR, S UR H hU (M ) 2)
3)
:
H -
, UR H - H - MR AB 5 ! 2) AnnM (AnnH (A)) = A AnnH (AnnM (J )) = J ( A M J H.
. 1) =) 2). @ , 1 H - $
UR , " , $ " AnnH : L(M ) ! L(H ) AnnM : L(H ) ! L(M ) " $$ $
* Lf (M ) Lf (H ). $ AB 5 M , 2:19 3]. 2) =) 1). $ , $ M AB 5 . > 1 $ H - $
UR . - , g 2 H Ker g = AnnM (Sg) M $ , Sg H $ . # (E )2 | ; * " $ *L * * * R- U = 2 E. # S EndR (U ) | , "
s 2 EndR (U ), 2 : $ P2 2E/.
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s
532
. .
MR D(M ) P S - hU (M ), 2 * * g: U ! M , $ Im g 2 E * 2 $ Dg /. : $ 2 ,
4] g
3.
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' .
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" .
1] Wisbauer R. Foundations of Module and Ring Theory. | Philadelphia, 1991. 2] Chase S. U. Function topologies on abelian groups // Ill. J. Math. | 1963. | V. 7. | N 4. | P. 583{608. 3] . . Hom !" #$ // %. . "& . ')$ *&. | 1983. | %. 46. | +. 164{186. 4] . . , &#-"&. * " #/. #* AB 5 // 0!$. "& . & . | 1983. | %. 38. | N 2. | +. 201{202. 5] Wu L. E. T. A characterization of self-injective rings // Ill. J. Math. | 1966. | V. 10. | N 1. | P. 61{65. 6] Chandler R. E., Koh K. Applications of a function topology on rings with unit // Ill. J. Math. | 1967. | V. 11. | N 4. | P. 580{585. 7] . . , #&' . !#1/. * " #/. // %!#12$&/ $'&: %$-. #$4. & 2 . '). 5$! '#. 6# . | 76 $*, 1988. | +. 16. % & 1995 .
. .
,
.
Abstract
V. V. Dubrovskii, Non-classical applications of Laplace operator, Fundamentalnaya i prikladnaya matematika 1(1995), 533{534.
Di%erentiable function is expanded on over&owing system in such a way that the expansion may be di%erentiated termwise in'nitely.
(
= ; + ( ) = f0 ) 2 1 ( ). : ( 0) = ( ++ : ( ) = (0 ) ( ) = (0 ) ( 0) = (0 ) = ; ( ) ( )=; ( ;; : tU
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1 (tN U )U + X 1 (tN U )U + ++ n++ n++ +; n+; n+; N N n++ n+; X 1 X 1 N + (tN;+ Un;+ )Un;+ + N N (t;; Un;;)Un;; n;+ n;; =
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" > 2 " $ $ '+ ' ' ( ), max j nj 6 n " $ ( 6 . 2 " ) $ n %$ *' &, n, ' &$ " $$ ##$% " . . 1$, ) ' $ 7. 8. 9 " \; ' " <" ", " ) ' $ $ ' ' $ $$ ( . 7 & ( $ >. 7. ? $ ) " + ( . n
U
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1] . . { . | : , 1992.
& ' 1995 .
. .
.
Abstract
V. P. Elizarov, Systems of linear equations over quasi-Frobenius rings, Fundamentalnaya i prikladnaya matematika 1(1995), 535{539.
A criterion of compatibility for a system of linear equations over quasi Frobenius ring is obtained.
R |
Am n X # = B # | (1) R. (1) ,
~ 2 Rm : (C~ A = ~0) =) (C~ B # = 0): 8C (n1) !
(n1) " (1), # R | " . $#
x1 x = x2
.
R = Z=2&x1 x2] (x21 x1x2 x22):
() )": M , " *+ (1) ) M
(n1) " . ! M ) &1]: R # M, + : 1) *+ a b 2 R ajb, bja2) a b 2 R , " ) ca = 0 cb = 0 c 2 R,
ax = b - # # M. .) /0 ) # , " 0
# M. 1 R ) ) 2+ (QF - ), *+0 0 0 I *+0 0 J Annr Annl I = I Annl Annr J = J . 3 ) 2+ )# &2, 0. 24] &3, 0. 13]. 1995, 1, N 2, 535{539. c 1995 , ! " \$ "" "
536
. .
1. 1. $ " ) , " QF - R # M (2 " jRj < 1 &4], ) 2 " 8 &5]). 1 0 QF - &2, . 337]. ( : ) * (1)
Y~ Am n = D~ (10 ) R ) , " (n1)
~ # = 0) 8C # 2 R(n) : (AC # = 0# ) =) (DC (n10 ) * " ) (1) (10) 0 0, 0 R | QF - . ; ) A#1 : : : A#n 2 R(m) A~ 1 : : : A~ m 2 Rn " ) ~ 1 : : : A~ m i +) " ) R(Rm) R Rn , # hA#1 : : : A#ni hA *: . . R | QF - , (1) (n1) m = 1 n = 1. . m = 1. ; c 2P Rn cA~ 1 = ~0 T n 8 c 2 i=1 Annl a1i R = Annl i=1 a1i R. 30 (n1)
Annl Pni=1 ai1 R Annl b1R. < R | QF - , b1 2 b1R Pni=1 ai1R
, ,
A~ 1 X # = b1 . n = 1. m = 1
a11x = b1 . ( " , b1 = a111 0 1 2 R. = ,
(n1) (1), *+
. = " , m = 2 (1) # ) a x=a 11 11 1 a21 x = a212:
u = c1 a11 = c2 a21, . . u 2 Ra11 \ Ra21, (c1 ;c2 ) 2 Annl aa11 . 21 <
(n1), c1a12 1 +(;c2 )a212 = 0, c1a12 (1 ; 2) = 0
1 ; 2 2 Annr (Ra11 \ Ra21) = Annr Ra11 + Annr Ra21 (
0 &3, 13.2.1 12.4.2]). / ui 2 Annr Rai1, i = 1 2, / 1 + u1 = 2 + u2 | 8
(1).
, ) m 1 + u1 = : : : = = m;1 + um;1 | 8
, : ) m ; 1 (1), 1 + u01 = m + um | 8
, : ) 1-0
537
QF-
m-0 (1), ui 2 Annr Rai1, i = 1 m, u01 2 Annr Ra11. ; (1) " ), "
1 + u01 + (Annr Ra11 \ Annr Ra1m )
" u1;u01 2 Annr
mX ;1 i=1
\
1 + u1 +
Rai1 +Annr (Ra11+Ram1 ) = Annr
m\ ;1 i=1
mX ;1 i=1
Annr Rai1 = 6 ?
Rai1\(Ra11+Ram1 ) :
g1 a11 + gm am1 = h1 a11 + + hm;1 am;1 1 . .) (h1 ; g1 )a11 + h2a21 + + hm;1 am;1 1 + (;gm )am1 = 0 0 (n1) " (h1 ; g1)a11 1 + h2a212 + + hm;1 am;1 1m;1 + (;gm )am1 m = 0:
# i " ) 1,
((h1 ; g1 )a11 + h2a21 + + hm;1 am;1 1 + (;gm )am1 ) 1 + + h2 a21u1 + + hm;1 am;1 1 u1 + (;gm )am1 u01 = 0
, +, (h2 a21 + + hm;1 am;1 1)u1 = gm am1 u01: !* (g1a11 + gm am1 )(u1 ; u01) = gm am1 u1 ; (h2 a21 + + hm;1 am;1 1)u1 = = (g1 + h1 )a11u1 = 0 " + . . R : 1) R | QF - 2) 8m n 2 N, 8A#1 : : : A#n 2 R(m) 8A~ 1 : : : A~ m 2 Rn Annr Annl hA#1 : : : A#ni = hA#1 : : : A#ni Annl Annr hA~ 1 : : : A~ m i = hA~ 1 : : : A~ m i~ 2 Rn (1) 3) 8m n 2 N, 8A 2 Rm n , B # 2 R(m) , D !" !", !" (n1), (10 ) !" !", !" (n10 ). . 2) =) 1). # m = 1 ) 2)
538
. .
n = 1 | , " Annr Annl I = I
Annl Annr J = J *+0 0 I 0 J R. 1 R ) &3, 13.2.1 13.2.3]. 2) =) 3). @
(n1) 8 Annl hA#1 : : : A#n i Annl hB # i. <0 " B # 2 hB # i Annr Annl hB # i Annr Annl hA#1 : : : A#ni = hA#1 : : : A#n i " ) " (1). 1 0 " ) (n10 ) (10). 3) =) 2). Annl hA#1 : : : A#niF # = 0, # (A1 : : : A#n)X # = F #
(n1). = " , F # = A#1 d1 + + A#n dn # # # di 2 R. 3 , F 2 hA1 : : : An i
Annr Annl hA#1 : : : A#n i hA#1 : : : A#ni: < + *"
" , " ) 2). ( ) / 0 " . 1) =) 2). n = 1 F # 2 Annr Annl hA#1 i )
3) =) 2) " , " A#1 X = F #
(n1). F # 2 hA#1 i , , Annr Annl hA#1 i = hA#1 i. n = 2
F # 2 Annr Annl (hA#1 i + hA#2 i) = Annr (Annl hA#1 i \ Annl hA#2 i) : ~ C~ 2 Annl hA#1 i ) B~ A#2 = C~ A#2 B~ ; C~ 2 Annl hA#1 i \ ; B ~ # = C~ F #. B ) ) \ Annl hA#2 i. + F # 0 BF # # # ~ #2 ) = BF ~ # . < +#
': Annl hA1 iA2 ! Annl hA1 iF # , # '(BA # # Annl hA1 iA2 | R ' | 02 ) R- , ~ #2 ) = BA ~ #2d, B~ 2 Annl hA#2 i &3, 13.2.1 : / d 2 R, " '(BA # # ~
12.4.2]. < +), B (F ; A2d) = 0, Annl hA#1 i(F # ; A#2 d) = 0, F # ; A#2 d 2 Annr Annl hA#1 i = hA#1 i F # 2 hA#1 i + hA#2i. ., Annr Annl hA#1 A#2 i = hA#1 A#2 i: ;
" ) 2). ( ) / " 0 " )
0 0 .
1] Camion P., Levy L. S., Mann H. B. Linear equations over a commutative ring // J. Algebra. | 1971. | V. 18. | N 3. | P. 432{446. 2] . : , !"# # $ ##. %. II. | &.: &#, 1979. 3] ' . & !"# # . | &.: &#, 1981.
QF-
539
4] ()* . . +#,,- ",$,- . ! *$ , $# ,! */#0 ,#" *- # # // 2.3# $ . ," . | 1993. | %. 48. | N 3. | 4. 197{198. 5] ()* . . ,),- */#0 ,#" *- !"#, .# 6,#7 ! # #,,- ",$ // ",! ,$ ,7 # .# !,7 $ $# . | 1995. | %. 1. | N 1. | 4. 229{254. & " ' 1995 .
. .
. . .
,
.
Abstract
N. K. Ioudu, Algorithmic solvability of zero divisor recognition in a certain class of algebras, Fundamentalnaya i prikladnaya matematika 1(1995), 541{544.
It is proved that the property of an element to be a one-side zero divisor is recognizable in algebras with one-side bounded conversion.
f A . 1], , , - . # A- , $% f , &$% . ' A - k: A = khx1 : : : xd i=I . * & X , +$% - , $
hX i. - , . I , N (h), h 2 khX i khX i = N k I , k, N | khX i. N (h) , . ( . 2]). # 1. A , $ u v 2 N \ hX i, N (uv) = N (uy)v , v = yv , y 2 X .
A = khX i=I , I = fxixj = ij xk xl ij 6= 0g $ $% . # L1 & , L2 | + , R1 | & , R2 | + . 4 (1). L1 , R1 , R2, L2 + | & & & $% $%. ', $% $, 0
1995, 1, N 2, 541{544. c 1995 , !" \$ "
0
542
. .
+ . 7 , 8 : A = khX i=I , X = Y Z , Y \ Z = ?, I = fyi zj = yk zl g, Y = L1 = R1, Z = L2 = R2. 4 (2). 9 I $ fxixj = ij xk xl ij 6= 0g
R2V \ F = ?, F & + ( . ), V | & , & : V = fxi j 9j xj xi 62 F g.
1. , , f . . , ! (2), f . . 4& $
+ (2) : + R2 \ L1 = ?, F L1 L2 , R2 V \ F = ?. & $ : N (uv) = N (N (uy)v ), u v v 2 hX i \ N , y 2 X : N (uy) = uy, N (uv) = uv, + $ , %$ . 7 N (uy) = uy , y 2 R2 , u 2 N , uy 62 N . v = tv (v 62 ?, & ), v = ytv t 2 V . ; y t 2 N $ R2V \ F = ?, N (uy)v = uy tv . . A = = Ln=0 An & A0 = k: fg = 0 =) f (0) = g(0) = 0, fP (0) = '(f ), ' : A ! A=A+ = k. g & di=1 xi gi . < , $% . 0
0
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1. " piP = N (fxi ). # ! gi 2 khX i, P d i=1 pi gi = 0 N di=1 xi gi 6= 0, ! -
h 2 khX i n I , N (fh) = 0.
. & , $% .
(=)) h N
Pd
x g , h 2 khX i n I i i i=1
543
d d d X X X N (fh) = N f xigi = N N (fxi )gi = N pigi = N (0) = 0: i=1
i=1
i=1
((=) & , h N (h) = = Pdi=1 xi hi, N Pdi=1 xihi 6= 0, h 62 I . ; gi & hi . < , d d d X X X 0 = N (fh) = N f xihi = N N (fxi )hi = pi hi i=1
i=1
i=1
. % pi khX i = T $%. - fpi gi=1d ( &, ) | fqigi=1d , 3. < , ( 2), , , & 1. 2. $ qi , i = 1 d =) qi-, , T . hX i: u < v =) uw < vw $ u v w 2 hX i, 1 $ , & , $% $ 2. ; . , & fpi gi=1d .
3. " fqig fpig T , fqigi=1d % () fpigi=1d T . T -P di=1 pigi = 0 ST MT = fp1 : : : pd gT . + + B . 4. " ST MT % B bj#, qi = 0.
544
. .
< & \ ". T = khX i | 78@ +, T - ST , T d MT rkST = d ; s, s | fqigi=1d , P ST = ker, : T d ! MT : (b1 : : : bd ) 7! di=1 pi bi. # 2. " N (fx1 ),: : : , N (fxd ) & fqigi=1d 1, . # P f | (Ann f )A % A di=1 xibij , bij 2 B , j 2 1 d , qj = 0. ) g (Ann f )A mdm+1 +1, m = deg f .
1] . . , . . , . . . . 2] .
.
! " , 1995, !%. .
'
(.
)
*.
|
.:
-*)- ./, 1988.
& ' 1995 .
. .
, A | , A = C T1 : :: Tm ] , C | 6 1, Sp2r (AX1 : : : Xn ]) (r > 2) ! " .
Abstract
V. I. Kopeiko, On the structure of the symplectic group over polynomial rings with regular coecients, Fundamentalnaya i prikladnaya matematika 1(1995), 545{548.
In this note we prove the following result. Let A be a ring of the geometric type or A = C T1 : : : Tm ] , where C is a regular ring and dim C 6 1. Then the group Sp2r (AX1 : : : Xn ]) (r > 2) is generated by elementary symplectic matrices.
1]. , ! 1. 1. , (X ) 2 Sp2r (AX1 : : : Xn]), r > 2, : 1. A | k, . . A = S ;1 C , C = kT1 : : : Tm ]=I | k- S | C 2. A = C T1 : : : Tm ] , C | " 6 1, K1 Sp(C ) = 0. # , (0) = 12r , $ " % $&
& .
(0) (0 : : : 0) (Xi ! 0, i = 1 n). ' . ( 1 2 A | . *! B | A, s 2 B | 0 A, , + B=sB ! A=sA, ! , + . , ! ! , ! ! : B ! A # # Bs ! As . , B=sB = A=sA, Bb = Ab, b (s)-
. . , s | 0 A. 1995, 1, N 2, 545{548. c 1995 , !" \$ "
546
. .
1 ( ). , ' (B A s) (
, 2 Sp2r (A), r > 2. ) s 2 Ep2r (As ), " * : = , 2 Sp2r (B ), 2 Ep2r (A). 2 ! 2.4 5]. . , 1 ! ! ! 5, 3.3] ! 4 ( ! !
1]. 2 ( 5 ). A |
B - , B A, 2 Sp2r (AX ]), r > 2, (0) = 12r . ) M 2 Ep2r (AM X ]) ( M 2 Max(B ), 2 Ep2r (AX ]). 6! A = B 1], 4 ! ! . 2 2 ! ! m. 7 m = 0, ! 2] ( 1.2 3]). * , m > 0. * 5 , C | . * T = Tm , A0 = AX ] ( 4 AX1 : : : Xn ] AX ]), B = C T1 : : : Tm;1 ] T X1 : : :Xn ]. . , (B A0 T ) !! !. 8 C T ] | ! 6 2 T | , 5.2 4, . IV] C T ] T | ( . ; , K1 ;Sp C T ] T = 0, , ! T 2 2 Ep2r C T ] T T1 : : : Tm;1 ] X ] = Ep2r (A0T ). 6 , 9 = , 2 Ep2r (A0 ), 2 Sp2r (B ). * ! , (0) = 12r , (0) = (0);1 2 Ep2r (A). :: = = = (0)] (0);1 ] 2 Ep2r (A0) = Ep2r (AX ]), (0);1 2 Ep2r (B ) Ep2r (A0 ) ! . . + & $ 1, K1 Sp (C ) = 0 2, : Sp2r (AX1 : : : Xn ]) = Sp2r (A) Ep2r (AX1 : : : Xn]) : , , " Sp2r (A) = Ep2r (A) ! Sp2r (AX ]) = Ep2r (AX ]),
$' , " " . + , & , : K1 Sp (A) = K1 Sp (AX1 : : : Xn ]).
2 ! 2 , , ! (0);1, , (0) = 12r , , 5 2 Ep2r (AX ]) , C | . ' ; ! . !, ! 5, 6], !
547
2. C | " 6 1,
SK1 (C ) = 0, A = C T1 : : : Tm ] , 2 Shr (AX1 : : : Xn ]), r > 3. ) (0) = 1r , $ " % $& . . 8 2 5, 3.1] !, C = k | = ;, , ! (0) = 12r 9. > 4 , ! (0) = 12r 9, C | . 4 ! , ;++ , 4 , !. 3. ) ;C | , A = C T1 : : : Tm ], r > 2 Sp2r AX11 : : : Xs1 Y1 : : : Yn] % $ . 8 C= Nil(C )
; ! , ! , C = k | . ; ! ! ! s. 7 s = 0, 9. * , s > 0, ! 2 Sp2r (As n]) ( 4 As n] ! AX11 : : : Xs1 Y1 : : : Yn]). * X = Xs , B = As ; 1 n] S+ ( S; ) ! ! ! AX X ;1 ], 4! ( ! ( AX ] ( AX ;1]). ? U+ ( U; ) ! ! ! B X X ;1 ], 4! ( ! ( B X ] ( B X ;1 ]). . , S+ U+ , S; U; . @ ! :
B X X ;1 ] = As n] S+;1 AX X ;1 ]s ; 1 n] U+;1B X X ;1 ]: k(X ) T1 : : : Tm ] , k(X ) | ( 8 S+;1 AX X ;1 ] = +! X , ! , ; ; Sp2r S+;1 AX X ;1 ]s ; 1 n] = Ep2r S+;1 AX X ;1 ]s ; 1 n] :
*;! 2 Ep2r S+;1 AX X ;1]s ; 1 n] ;, , 2Ep2r U+;1 B X X ;1 ] . ? ,; 2 Ep U;;1B; X X ;1 ] , , 2r 4.1 1], 2 Ep2r B X X ;1 ] = Ep2r AX11 : : : Xs1 Y1 : : : Yn] . ;
;
1] . . ! // #. . | 1978. | (. 106. | N 1. | . 94{107.
548
. .
2] . . . ! 6 1// /#0. | 1992. | (. 47. | N 4. |
. 193{194. 3] Grunewald F., Mennicke J., Vaserstein L. On symplectic groups over polynomial rings // Math. Z. | 1991. | V. 206. | P. 35{56. 4] Lam T.-Y. Serre's conjecture // Lect. Notes Math. V. 635. | 1978. | P. 3{210. 5] Vorst T. The general linear groups of polynomial rings over regular rings // Commun. Algebra. | 1981. | V. 9. | N 5. | P. 499{509. 6] 5. 5. . ! // !. 50
6. . . | 1977. | (. 41. | N 2. | . 235{252. & ' 1995 .
. .
, 1], 2n , pn. " 1] \ " ,
%&& . ' pn .
Z
Z
Z
Abstract
A. S. Kuzmin, Polynomials of maximal period over primary residue rings, Fundamentalnaya i prikladnaya matematika 1(1995), 549{551.
The maximalitycriterion for the period of a polynomialover primary residue ring is proved. This criterion generalize the results of the paper 1], where the case of polynomials over 2n was considered, to the case of arbitrary primaryring pn. The criterion is based on the concept of \marked polynomial introduced in 1] and allows to verify the maximality of the period of a polynomial using only its coe.cients. Some su.cient conditions of maximality of the period of a polynomial over pn are given. R = pn |
pn, p | -
Z
Z
Z Z
. G(x) R ! " , #$ %&& , $ R. T(G) " G(x) ! R'x] ! # t $ : G(x) xt ; e (e | R). *! '2], G(x) R T (G) = T (G mod p) p " , 0 6 < n, T (G mod p) | G(x) mod p GF (p). - T (G) = T (G mod p) (. . = 0), G(x) ! '1]. *! (1) , ! G(x) m R pn;1 (pm ; 1). - T(G) = pn;1 (pm ; 1), G(x) ! ( ) '1]. / ! 0
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550
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p2 , p > 3, 8, p = 2. 1 '1] 0 %&& $ Z2n, !5$ " . - p > 3 F(x) | $ R, , F (x) p, " & $ F (x) + p D(x) deg D(x) < deg F(x) = m D(x) 6 0 (mod p) & ! " " p2. 1. F (x) | m R = Zpn , p > 3, F (x) mod p GF (p). F(x) | R , pY ;1 F(xp ) = F(x k ) k=0
| S R xp ; e = (x ; e) (x ; ) (x ; 2) : : : (x ; p;1 ): ! & " " "" #" S R, F(x) " $ 0 5 % 0 $ . 75 #" S ! 8 {:!#$ . ; " ! , ! F (x) mod p , F(x) # S m ! $ 1 : : : m
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j =1 P 2. G(x) = mk=0 gk xk | m R = Zpn , p > 3, G(x) | R
, : ) G(x) mod p | pm ; 1 GF(p), m X Y p! p ) G(x ) 6 P (gk xk )jk (mod p2): j ! : : : j ! P m k=0 m j =p m kj 0 (mod p) 0 k=0 k
k=0
k
551
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. G(x) =
Pm
xk | m R = Zpn, p > 3, T (G mod p) = pm ; 1" G(x) , # $p : ) g0 6 g0 (mod p2 ), ) m > p, G(x) | % . . ?! k=0 gk
! 5 " $ 8 , % '1] " p = 2.
1] . . // . . | 1993. | #. 184. | N 3. | &. 21{56. 2] Ward M. The arithmetical theory of linear recurring series // Trans. Amer. Math. Soc. | 1933. | V. 35. & " ' 1995 .
. . , R S R. ! S , " .
Abstract
V. L. Kurakin, Binomial presentation of linear recurring sequences, Fundamentalnaya i prikladnaya matematika 1(1995), 553{556.
It is proved that any linear recurring sequence over commutative local Artinian ring R can be presented as a linear combination of binomial sequences over some Galois extension S of R. If the roots of the binomial sequences belong to the +xed coordinate set of S , then this presentation is unique.
u = (u(0) u(1) : : :) | S e. G(x) = cm xm + c1 x+c0 S!x] u # v = G(x)u, v(i) = cm u(i + m) + + c1 u(i + 1) + c0 u(i) i > 0: % Ann(u) = G(x) S!x] G(x)u = 0 # u. & Ann(u) ' ( ( )** e) G(x), u # (+, ), G(x) | . - LS (G) . ' / +, S / G(x). ak] (a S, k > 0), #0
(# ak] (i) = 0 0 6 i < k ak] (i) = ki ai;k i > k ( a0 = e), a. 1. a S , m > 1. LS ((x a)m ) | S - a0] : : : am;1] . . %
( (x a)a0] = 0, (x a)ak] = ak;1] , k > 1, , a0] : : : am;1] LS ((x a)m ). # +, 2
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2
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2
2
;
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u LS ((x a)m ) ## u(0 m 1) = (u(0) : : : u(m 1)). 3# , ak] (0 m 1) = = (0 : : : 0 e : : : ), e / # k , k 0 m 1. 2. u = c0 a0] + + cm;1 am;1] , a ck S . c = (c0 : : : cm;1 0 0 : : :). Ann(u) = = H(x a) H(x) Ann(c) . . H(x) = h0 + h1x + S!x]. 4 mX ;1 X H(x a)u = ht (x a)t ck ak] = 2
;
;
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t>0
i=0
v = H(x)c. 5 , H(x a)u = 0 , v(0) = : : : = v(m 1) = 0 , ) , v = H(x)c = 0. u LS (F ), F (x) = (x a1)m1 : : :(x ar )m (x ai )m , i 1 r, . 6 !1] LS (F ) = LS ((x a1)m1 ) LS ((x ar )m ) u = u1 + + ur , ui LS ((x ai)m ), Ann(u) = Ann(u1) : : : Ann(ur ) = Ann(u1) : : : Ann(ur ): (1) 4 1, 2 0, +, u ## . / aik] , i = 1 r, 0 6 k < mi , ) 0 ' # +, u. 8 +, . / / ' # #, . , 9 / F(x) . R | e, J = RadR, R = R=J. : G(x) R!x] # ;, G(x) R. 8 ) S = R!x]=(G(x)) . : ' , R S. < S ( ; R !2]. ' R
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1] . . // . . | 1991. |
. 3. | ". 4. | $. 105{127.
2] Ganske G., McDonald B. R. Finite local rings // Rocky Mountain J. of Math. | 1973. | V. 3. | N 4. | P. 521{540. 3] . . + ,- / // 01 . . | 1993. |
. 48.
| ". 3. |
$. 197{198.
& " ' 1995 .
PI- , . .
. . .
: P I - R M , N . , , R- M N !, R , M .
Abstract V. T. Markov, On P I -rings having a faithful module with Krull dimension, Fundamentalnaya i prikladnaya matematika 1(1995), 557{559.
The main result: if a P I -ring R has a faithful left module M with Krull dimension, then its prime radical N is nilpotent. Moreover if the left modules M and N are )nitely generated then R has left Krull dimension which is equal to Krull dimension of the module M .
1] , PI- R M, rad(R)
R= rad(R)
. 2] P I-, , ! Kdim(RR) = Kdim(M) ( Kdim(M) | $% % M, &' ( 3]). * %+ %% %:
1. R | PI - , M | R-
. : (a) rad(R) | ! (b) " R- M (rad(R)) #, R
Kdim(RR) = Kdim(M). - % %
(
1. $
, " . *+ , ! * , , ( 093-1544), 0! , * ( JBX100), ! INTAS. 1995, 1, N 2, 557{559. c 1995 !, "# \% "
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- PI- % , %0+
( !11
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. | 1, | , . 5 N & % ( ' ( ' . 5 (a) 1 0+ 1 & &+ ( %. 2. N | - R, M | R, %
, % f(x1 : : : xd ) &'' , " 1, f(a1 : : : ad)m = 0 )" a1 ::: ad 2 R m 2 M . % n 2 N, N nM = 0. 7 ( % 2 % d f. - 1
d % % % $% Ndim(M), % % %
( , % M ('4 , Ndim(M) + , + Kdim(M)). 8 % ( 0+% 1. 5]. $ M | , %
, 0 = B1 B2 : : : M | %, M S=1 M1 M2 : : : | )% M , M = i=1 Bi , % i j 2 N, Mi Mi+1 + Bj . 7% (&) 1 % 0+% % 1]: N = rad(R), R0 = R=N. : N n = 0 % n 2 N. -( M 0 = (M=NM) (NM=N 2M) N n;1M. * , M 0 ( R0-, ( $%, M. ; % %, M 0 | R0 -, M N | (
, M 0 ( . -( 1, 1, ( 1 0, R | , 2. R | * M | # R- . R ' '" M . ; 2 ( % 3 2] 0+ &0 %:
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559
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' . 1. $ R & %, 0- + $% , + ( . . 0+
0 $%) R-. - F | , V | & fe1 e2 : : :g, T = F x]] | 1 '
' %. - V T -, % xe1 = 0, xei = ei+1 i > 1. ?+ & ' !
y1 = x y2 : : : T , (
% & R = Fy1 y2 : : :] $%, V | R-. 2. P I- R, 0+ , R= rad(R) $%. - fy0 y1 : : :g | ( !
T ,
1, V 0 = V Ff. @ 0 ! 1 f'i i = 1 2 : : :g V 0 ,
' 0+ & : 'i (f) = ei , 'i (ej ) = yi ej . : & EndF (V 0), (
% f'i g, &
. 3. $ R, 0+
, rad(R)
. 7 ! 1 f'i g + 0+: 'i (f) = ei , 'i (ej ) = ej ;1 1 < j < i + 1, 'i (ej ) = 0 j = 1 j > i. A & $. 8. B &( % .
1] . . P I - // . . 8. | !, 1989. | C. 99{103. 2] Amitsur S. A., Small L. W. Finite-dimensional representations of P I -algebras // J. Algebra. | 1990. | V. 133. | N 2. | P. 244{248. 3] Gordon R., Robson J. C. Krull dimension. | Mem. Amer. Math. Soc., 133. | 1973. 4] Procesi C. Rings with polynomial identities. | N. Y. et al.: Marcel Dekker, 1973. 5] Lenagan T. H. Reduced rank in rings with Krull dimension // Ring Theory (Antwerp 1978). | N.Y.: Marcel Dekker, 1979. | Lecture Notes in Pure and Appl. Math. V. 51. | P. 123{131. 6] /0 1. 2., . . 3 P I -, 45 0 ! 6!4 17, 7 7!7 8 // 9! : . . | 1993. | . 48. | N 6. | ;. 141{142. ' ( 1995 .
. , .
345 . . , -
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Abstract D. Mikhalin, I. Nikonov, The false coin identi cation problem, Fundamentalnaya i prikladnaya matematika 1(1995), 561{563.
The problem of how to identify a false coin using minimal number of weighings on a balance (with scales and without weights) is under consideration. It is unknown whether the false coin is heavier or lighter. Determining the di*erence in weight is not required.
, , . " # , , , | ,
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n 3 2; 1 .) : , . & , , n 3 2; 1 . " , k n . ; n . 2 j (1 6 j 6 n) | . 2 (a1 a2 : : : a ) # k . - a 0, i- = 1, = ;1, . : + , , . 5 # n , . . n 6 3 . > , , , 3 2; 3 . 2 , , , ,
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+ +. 5 n = 3 2+ 1 . 5 . 5 : , , , . 5 m1 m2 |
;1 . 5 m1 > 3 2 + 1 . : , , # m1 , k ; 1 ;1 . , m1 6 3 2 + 1 . : m2 > 3 ;1. : m2 ,
m2 > 3 ;1. % , . : m2 . 2 3 ;1 , # . :
, , , , m2 . : m2 , , 1 ;1, 2m2 6 2 3 ;1. , m2 6 3 ;1. 2 m2 > 3 ;1. 5 . , k n , n 6 3 2; 1 . " > : ). (. / . k
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. .
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512.55
A , " A((x')) %& % % , % A | & . ( ) * &, A((x')) | & . + &,- & % %& & % '.
Abstract
K. I. Sonin, Von Neumann regular skew-Laurent series rings, Fundamentalnaya i prikladnaya matematika 1(1995), 565{568.
Let ' be an automorphism of 2nite order. Then the skew-Laurent series ring A((x')) is von Neumann regular i3 A is semisimple Artinian. The third equivalent condition is that A((x')) is semisimple Artinian. The same result for strong regularity is proved in the case of an arbitrary automorphism '.
. A((x ')) A c ' P!1, . 101] , i 1 i=m ai x , x | , m 2 Z , ai 2 A, c ' ( (, :
1 X i=t
ai xi
1 X j =r
bj xj =
1 X X
k=t+r i+j =k
ai 'i (bj ) xk :
!3] ' 1. ,
1.
'n
2.
& :
1 : (1) A((x ')) | !" (2) A((x ')) | " (3) A | . (1) (2) (3)
n > 1,
A((x ')) | !" A((x ')) | ' ( A | ' ( .
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1995, 1, N 2, 565{568. c 1995 !, "# \% "
-
566
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1 !2, . 2.13]. A0 A1 A2 : : : | ' ) ! A, fei j 0 6 iL <m1g A, ' * m i=0 ei A = Am .
2. A | !, ' | +( A, I | * , , , A '(I ) I , A * ' P fei j 0 6 i < 1g, ' 'i (ei ) 2= ik;=01 'k (ek )A i.
. 2 ( 3 ( 1 L '1 , fas j s 2 S g, ' J = s2S as A IAP,1 , J | A,
' k=0 'k (J ) I 6= A. b0 fas j s 2 S g, bi fas j s 2 S g n fb0 ::: bi;1g , ' bi 2= Ji , P P
Ji = ik=0 ij;=01 'k (bj )A 6= A. 2 m- (m > 1) 3 bm (, as fas j s 2 S g n fb0 ::: bm;1g 'm (as)2 Jm , J ( ' ( ';m (Jm ), ' ' A. 2 ei , 0 6 i < 1, ' Ld 1 Ld b A d. 2 i >1, ' e A = i i i=0 P i=0 P i ; 1 i k ' (ei ) 2 k=0 ' (ek )A, 'i (ei ) 2 Ji , 'j (ej ) 2 'j jk=0 bk A Ji , P 0 6 j 6 i ; 1, , 'i (bi ) 2 Ji ( 'i (eq ) 2 'i qk=0 bk A Ji
q 6 i ; 1), ' ' bi .
3. R = A((x ')) | !, A * ' , P fei j 0 6 i < 1g, , ' 'i (ei ) 2= ik;=01 'k (ek )A.
.P/ ,
, 1i=0 'i (ei )xi
( z = 2 R . 4 P j (t 6 0), ' gR = zR, gz = z , g = 1 g x j j =t 5 xi : gt'i (ei;t ) + ::: + g0 'i (ei ) + ::: + gi'i (e0 ) = 'i (ei ), 0 6 i < 1. 7 , ' g0 'i (ei ) = 'i (ei ). 8 ( 5 h 2 R, ' Pg r = zh , , h0 ::: hPr r 2 A, ' g0 = k=0 'kP (ek )hk . 2': 'r+1 (er+1 ) = g0'r+1 (er+1 ) = k=0 'r (ek )hk 'r+1 (er+1 ) 2 rk=0 'k (ek )A, '
' .
4. A | !, R = A((x ')) | .
567
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, ( 5 a b 2 Rop : a b = ba, ba | 1 .
6. - ! Aop ((x ';1 )) (+ ! A((x '))op . < ( . 2. (1) ) (3): A((x ')) , ( A((x '))op , , ', 6, Aop ((x ';1 )). 2 A L fas j s 2 S g, ' J = s2S asA | P i A. 2 (, ' 1 i=0 ' (J ) 6= A. 2 (, ' 5 Pt i , b0 : : : bt 2 A a0 : : : at fas g, ' i=0 ' (ai )bi = 1 ( ( ' ' ', A | , P bi = 0, i ). 2( z = ti=0 ai xi. > 5 AopP ((x ';1)), , 5 1 g = k=0 gk xk 2 Aop ((x ';1)), g0 6= 0, ' z g = 0, | ( 5 . 8 , , 5 1 x, ' : ';i (g0)ai = 0, 0 6 i 6 t, g01 = 0. ; ', A , '. / ' (1) ) (3) 1. , , ' ' . (3) ) (2): 1 , ' 1 ? . (2) ) (1): . @ @. . A1 @. @. 8 .
568
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1] L. H. Rowen. Ring Theory. | London: Academic Press, 1988. 2] K. R. Goodearl. Von Neumann Regular Rings. | London: Pitman, 1979. 3] . . . ! // # . $. % &. | 1995. | (. 1. | N 1. | . 315{318. ' ( 1995 .
=6 . .
517.95
= f( ) j 0 6 '(*$ + *$ , ( *$ , $ - $ . -$ / 0 + = 0 , = 0 . !" | " (' 2",+% 2 1 ( ). 3 "-* , ,* 2 "* "'. + , 3 2, ' .-*/ , "& -*/ 2",+$ .
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Abstract I. V. Tomina, The rst regularized trace for a power of the Laplace operator on the rectangular triangle with the angle =6 in case of Dirichlet problem, Fundamentalnaya i prikladnaya matematika 1(1995), 569{572.
p
Consider the Hilbert space = 2 ( ), where = f( ) j 0 6 3 6 6 6 (2 ; p 3) 3g. Let be the self-adjointnon-negative operatorfrom to which is generated by the spectral Dirichlet problem 0 + = 0 on = 0 on . For 2 1 ( ) let the operator : ! take each 2 to the product . In this paper concrete formulas for the 7rst regularized trace of the operator + , 3 2, are given for di8erent classes of essentially bounded functions . H
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;u = u D u = 0 @D (1) p p D = f(x y) j 0 6 y 3 6 x 6 (2 ; y 3)=3g | . "# $ M(x y) 2 R2 p p 6 Mj (xj yj )p2 R2, j = p0 5, x0 = x, y0 = y, x1 = (x+y 3)=2, y1 = (x 3; y)=2,px2 = (x ; y 3)=2, y2 = (x 3 + y)=2 j 2 f0 1 2g xj +3 =p ; xj , yj +3 = = 3 ; yj . ) * +, g 2 L2 (K) K = -0 ] -0 = 3] | , H0 , j = 0 5 g(Mj ) = g(M) .. M 2 D. f 2 L2(D) fe(Mj ) = f(M) M 2 D j = 0 5/ , fe 2 H0. 0 h | * + K, ,1 2 : M 2 D n @D h(Mj ) = 1 j = 0 2 3 5 h(Mj ) = ;1 j = 1, 4/ M 2 @D j = 0 5 h(Mj ) = 0. 1995, 1, N 2, 569{572.
c 1995 !"#, $ \& "
570
. .
6 2 , f 2 L2 (D) (m n) 2 Z2
(hfe 'mn )1 = -1 + (;1)m+n ](f vmn)0
(2)
2 2 ( )0 ( )1 | 7 p L (D) L (K), 'mn (x y) = sin mx sin ny 3, vmn (x y) = 'mn (x y) ; 'mn (x1 y1 ) + 'mn (x2 y2 ) , # ,
vmn (x y) = 'mn (x y) ; 'm1 n1 (x y) + 'm2 n2 (x y)
(3)
m1 = (3n ; m)=2, n1 = (n + m)=2, m2 = (3n + m)=2, n2 = (n ; m)=2. 0 J = f(m n) 2 Z2 j n > m > 0 (;1)m+n = 1g, V = fvmn j (m n) 2 J g. 0 7 , (m n) 2 J vmn 2 * + $ (1), ,1 2 mn = m2 + 3n2 . 9 m +n (2), (3) hevmn vmn ( K (;1) p = 1), : vmn = ;vm1 n1 = vm2 n2 , ;b f(2 4 3=)'mn j m n > 0g | ()<0=) L2 (K), ,1 , .
p 1. Vb f(2 12=)vmn j (m n) 2 J g 4
L2(D).
0 c 7 V L2 (D) -1], 7 V | -2]. > 7 1 $ 2 # -2]. 0 T | + 7$ # 7$ L2 (D), # 7$ $ $ (1)/ > 0 (m n) 2 J vmn 2 * + T , c ,1 2 mn . 0 P | # L2 (D) * +, p 2 L1 (D)/ (m n) 2 J ) (p) | 2 7 T + P , 7
(mn () 2 $ 2 , j mn (p) ; mn j 6 const. P bmn 2 ? i 2 f0 1 2g F Z $
9P i
2F
(mn)
(i)
2F bmn S 2 C (i- ), 2 7 i- 1-
() (mn)
. @ S (0) , S (1) , S (2) , * ( 7), 7 A
7 , S (i) = klim !1
X
bmn i 2 f0 2g/ S (1) = lim
M !1 N !1
(mn)2F m2 +(i+1)n2 6k
X
bmn :
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7 * 7 , :
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p2. > 3=2 p 2 L1 (D)
571
(4), mn (p) = 8 3(Pvmn vmn )0 =2, (p) = 0. p p 7 -3] 1. 0 Eb = fp (2 4 3 m n =)emnjm n > 0g | )<0= L2 (K), emn (x y) = = cos mx cos ny 3, n = 1 n 6= 0 0 = 1=2.P -g] | $ $ C * + g 2 L2(K) )<0= Eb : g(x y) mn>0 m n amn (g)emn (x y), p amn (g) = 4 3(g emn )1 =2 / i 2 f0 1 2g Sg(i) (x y) i- -g], # i- Dgi R2. f 2 L2(D) (m n) 2 Z2 * (fe emn )1 = = -1+(;1)m+n ](f umn )0 , umn (x y) = emn (x y)+emn (x1 y1 )+emn (x2 y2) = = emn (x y) + em1 n1 (x y) + em2 n2 (x y). 9 umn = um1 n1 = um2 n2 , (;1)m+n = 1 umn 2 H0, g 2 H0 (;1)m+n = ;1 amn (g) = 0. 2 D vmn = (wmn + 2umn )=4, wmn = 3 ; u2m0 ; u3n;m0 ; ;pu3n+m0 ; u02n ; u0n+pm ; u0n;m + u2m2n + 2u3nm, (m n) 2 J 8 3(Pvmn vmn)0 =2P= (2 3=2 )(p wmn)0 + amnP (~p)=2. 0 G(g) (2) a (g), B (g) mn k n>m>0 n>0 n aknn(g) g 2 H0 , k 2 f0 1g. g 2 H0, X R2, i 2 f0 1p2g : g 2 Qi -X], -g] i- X/ Xk f(t kt= 3) j 0 6 t 6 =(k+1)g, X 0 = X0 X1 . 3. 1) g 2 H0, (0 0) 2 Dg2 k = 0 1 9Bk (g) 2 C , G(g) = -2Sg(2) (0 0) ; 3B0 (g) ; 3B1 (g) + 2 5a00(g)]=6 2) g 2 Qi-X 0 ] p R (i) i 2 f0 1 2g, k = 0 1 Bk (g) = 2 0 Sg (t kt= 3) dt. . > 3=2, p 2 L1 (D). 1) 9G(~ p) 2 C , (4) p ZZ 2 mn (p) = 23 p(x y)wmn (x y) dx dy (p) = G(~p)=2: (5) D 2) (0 0) 2 Dp2~ p~ 2 Qi -X 0 ] - i 2 f0 1 2g, 9G(~ p) 2 C (p) (5) ! " p 1 Z S (i) (M)(M) ds + 5 3 ZZ p(x y) dx dy (6)
(p) = 61 Sp(2) (0 0) ; ~ 2 X p~ 2 D p (M) = 3 X1 , (M) = 1 X0 @D n X1 . , , p ! @D, . . " M 2 @D lim p(M 0) = D 3M !M = p(M), (6) Sp(~i) (M) # p(M) X 0 | @D. . >P3=2, p ! D ! % &: a) p~ 2 Q2-D], b) jamn(~p)j < 1, c) p # ( mn>0 C (D) -4] > 1 @p=@ = 0 @D ( | @D). * 0
0
572
. .
Sp(~i) p, X 0 | @D.
(4) mn (p) (5) (p) (6)
&
Sp(2) ~
. F 1 7 , p
p
Ub f(2 4 12=) m n =mn umn j n > m > 0 (;1)m+n = 1g
)<0= L2 (D), 1 2 7 * +$ $ < $ ;u = u D @u=@ = 0 @D (7) ,1 2 7 mn = m2 + 3n2/ A mn $ + 7 0, n ; m m2 + n2 . ),, 7: , 1, -3] *
< $ (7). 0 A $ # , :7 7 D -1]. F 2 G. G. 2 2 .
-1] H F. C. I 6 . | ".: G71 : , 1991. -2] Makai E. // Studia Scientiarum Mathematicarum Hungarica. | 1970. | N 5. | P. 51{62. -3] 2 $ G. G. // R><.| 1991. | D. 46. | G7 . 3. | =. 187{188. -4] 9 G. F., 0 T. I. ) 7 . U.2. | >.: < , 1973. ( ) 1995 .
