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CL(n) L n L f : N0 N0 L L w w L L Rec Rec Gr(L) = lim (CL(n))1/n n O(f) f (f) (f) = O(f) (f) L r N j = 0,..., r-1 pj(n) j j j = j = 0 0 j < j n CL(n) = pn (n)n + O(n ), n = n mod r. n W Rec CS CF Reg AAP FAD APt AP MP AFact Fact Pref Rec AFact CS AP CF APt Reg AAP Pref W Fact MP FAD L Gr(L) j w n > 1 w w = w.. w w n, w - w.

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[+ ] [ > ] L(k, ) [L(k, +)] [ + ] k RT(k) 7 7 k RT(k) k=3 k=4 4 5 k-1 L(3, 2) L(2, 2+) L(2, 7/3) + x x < + L(k, ) L(2, 2+) L(k, ) k r pj(n)n j j nm n j Gr(L) j mj = Pd(L) A L (1) A L (2) A L (3) A L Pd(L) = m-1 m A k N O(Nk) G Ind(G) L L (N4) (N3) L k A N A 0 < < 1 Gr(L) (log(1/)Nk) (log(1/)N) G n m 0 < < 1 Ind(G) (log(1/)m) (log(1/)n) L A r A L r L A m Gr(L) A Pd(L) = m-1 Pd(L) A B N A B Ind(A) Ind(B) O(N4 + log(1/(N))N2) (N) N r j mj A f limn(f(n+1)/f(n)) limn(f(n+1)/f(n)) = limn(f(n+1)/f(n)) = 0 f = Gr(L) m = Pd(L) =1, m=0 =1, m>0 >1, m=0 >1, m>0 L (nPd(L)Gr(L)n) L M = M(L) {Mi} M M1 M2 ... Mi ... M, Mi = M i=1 Mi = M i Li Mi Li L L ... Li ... L1, Li = L.

i=1 Gr(Li) - Gr(L) -i L L L L TM (a) = ab (b) = ba TM M = {aaa, bbb} {ci(aba)a, di(bab)b, ci(bab)a, di(aba)b | i 0}, c, d - i(a) i(b).

M 32i+2 i i 0 Mi = M {a, b} 32 +2 M-1 = {aaa, bbb} i+1 Li Mi Gr(Li) = 1/2 m 0 (nm) s m 1 s m (ns) (nm) (1) (1) w L L u v uwv L w e(L) re(L) le(L) Gr(L) = Gr(re(L)) = Gr(e(L)) L CL(n)/Cre(L)(n) CL(n)/Ce(L)(n) r r Reg (1) (nn) > 1 (2) k k 3 (nk-2n) > 1 (1) (2) k k k k e(L(k, +)) = e(L(k, )) Gr(k, +) = Gr(k, ) < 2 2 = 2 7/3 [2+, 7/3] L(2, ) L(2, (7/3)+) 7/3 f : {1, 2, 3} {a, b} (7/3)+ u L (w1, w2) w1uw2 L u, v L n w : {1,..., n} w u v L(2, 2+) L(2, 2+) L(2, 2+) L(2, 2+) [2+, 7/3] L k! k! M L(k, ) Mm m N = (mCL(k,)(m))/k! Lm(k, ) O(N) (m, , k) O(N log N) O(N) 1 < < 2 xy L(k, ) (xy) (xy) (zt) |(zt)| < |xy|; |zt| |y| (4/3)+ |zt| < |y| (5/4)+ 2 2 Lm L(k, ) Lm 1 + Gr(Lm) m-1(-1) < Gr(L(k, )) L(3, 2) L(2, 3) m 2 k k 2 2+ 3 3+ (7/3)+ (5/2)+ 3 3+ 4 4+ 2 2+ 3 3+ 4 4+ m u |u| - |u| = m L(m)(k) L(k, ) r r m L(2)(k) m 3 O(m) O(m) k > 2m-3 L(m)(k) m = 3, 4, 5, 6 L(m)(k) k k 1.242 k L(k, ) (k, ) (k, ) > 2 [n+, n+1] n 2 1 1 1 1 k - + - + O, [n+, n+1], kn-1 kn k2n-2 k2n-1 2 (k, ) = 1 1 1 1 k - + + O, [(n+ )+, n+1].

kn-1 kn k2n-1 2 2+ (k-(k, )) k k 2 k [n+, n+1] 1 1 (k, n+) - (k, n) = + O, n-1 kn-2 k 1 1 (k, n+1) - (k, n+) = + O.

k2n-2 k2n-1 (k, ) = 2 1 1 1 (k+1, 2) = k - - + O ;

k k3 k5 1 1 1 1 (k, 2+) = k - - - + O.

k k3 k4 k5 k (k, ) 1 1 = 2 (k, 2+) - (k, 2) = 1 + + O( ) k2 k3 (k, 2) k + 1 1 (k+1, 2) - (k, 2+) = + O.

k4 k5 k < 2 n 3 k > n n + n-1 1 (k, ) = k+2-n- +O n-1 k k2, n n-1 1 (k, ) = k+1-n- +O.

n-1 k k2 =2 n = 2 < n < k n-1 n, k 2 < n < k + n n 1 (k, ) - (k, ) = 1 + O n-1 n-1 k2 + n n+1 1 1 (k, ) - (k, ) = + O n-1 n k k2 + n n 1 (k+1, ) - (k, ) = O.

n-1 n-1 k2 (k, ) k n 0 + k-n k-n n = lim (k, ) = lim (k, ).

n k-n-1 k-n-1 k k = n n+1 - n > 1 n+1 0 1.242 AP APt AAP AP w w Rk,(n) u u u L(k, ) u L(k, ) Rk,(n) k R3,2(n) u2 u (1) n (1 ) n {5, 7, 9, 10, 14, 17};

/ (1 ) n {1, 2, 3, 4, 6, 8, 11, 12, 13, 15, 16, 21} (2) n n R3,2(n) 0, n = 5, 7, 9, 10, 14, 17, R3,2(n) = 3, n = 1, 6, n = 2, 3, 4, 6, 8, 11, 12, 13, 15, 16, 21, R3,2(n) 12 K3,3 Rk,(n) u v u = yz v = zy y z p (1) p (5/2)+;

(2) p 5, 9, 11, 17 18 [(7/3)+, 5/2];

(3) m(a) m(b) m(aba) m(bab) m 0 [2+, (7/3)]; p = 2m p = 3 2m [(7/3)+, 5/2] n n 5, 9, 11, 18 k (1) k 4 = 2; (2) k 3 2+; (3) > RT( k/2 ) RT(k) k p k-1 + k (1) , p k-1 k-2 k+1 ( ) k < p < (k-1) p mod k = 0 2 k+3 ( ) p [(m-2)(k+1)+1, m(k-1)-1] m 2 p mod k = 0 ( ) p = 3k p = 4k;

k-1 + k-2 (2) , p [(m-1)(k+1)+1, m(k-2)-1] k-2 k-3 m [2, k-2];

2k-5 + k-3 (3) k 9 , p = 2k-7 2k-7 k-4 k-1 + k-2 + k k-1 , , k-1 k-2 k-2 k-3 k 7 p k (k2) p (k, , p) k p k 3 > RT(k) (1) (k, ) k-1 ( ) ;

k-2 + + 4 k-1 k-2 ( ) k = 6 [9, ] k 7 , ;

k-2 k-3 7 3 + 2k-5 k-3 ( ) k 9 , 2k-7 k-4 (2) (k, ) (3) k 9 (k, ) k(k-1)-1 k-1 k(k-1)-1 k-2 (5/2)+ k n n 18 n 27

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