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fw(n) n n w w fw(n) n + 1 w = 0110100110010110... w(0) = 0 w(2n) = w(n) w(2n + 1) = 1 - w(n) n 0 : {0, 1} {0, 1} (0) = 01 (1) = 10 s = s(1)... s(n) {0, 1}n (s) = (s(1))... (s(n)) w0 = 0 w1 = 01 w2 = 0110 w3 = 01101001 wn+1 = (wn) n wn wn+1 w = w(0)w(1)w(2)... = 0110100110010110... wn = w(0)... w(2n - 1) n w = 010001010100... w(2n - 1) = 0 w(2n) = 1 - w(n) n : 0 0100, 1 01 (0, 1) R w(n) = (n + 1) + - n + . w = w(1)w(2)... q0 = 0 q1 = 01 qn = qn-1qn-2 q = 010010100100101001010... qn q : 0 01, 1 0 = (3 - 5)/ (m0, n0) w fw(m0, n0) m0n0 w X {1,..., n} N = {1, 2, 3, 4,...} N {0} X = (X,, ) X x y x, y X (x) (y) X (x) x X p: {1,..., n} {1,..., n} p {1,..., n} x p y p(x) p(y) {1,..., n} p: {1,..., n} {1,..., n} p(1) {1,..., n} p(2) p(n) p p: {1,..., n} {1,..., n} 1 2... n =, p(1) p(2)... p(n) = p(1)p(2)... p(n) n n Sn X X X N N (x) > (y) x y (1) < (3) < (5) <... (2) > (4) > (6) >... p: N N p(x) p(y) (x) (y) p(1) < p(3) < p(5) <...

p(2) x, y X x = y (x, y) {0, 1} 0, (x) < (y) (x, y) = 1, (x) > (y) t (x, y) = (x + t, y + t) x y t N0 (x, y) = (x + t, y + t) x N0 y N0 X X = (X,, ) y1,..., yn X y1 <... < yn {y1,..., yn} {1,..., n} {y,...,yn}(i, j) = (yi, yj) 1 i n 1 j n i = j y yn X = N X = {1,..., N} m1 m2 m1 < m2 {m1, m1 +1,..., m2} [m1, m2] [m1, m2] m2 - m1 + 1 i m2 - m1 + 1 1 j m2 - m1 + 1 [m,m2](i, j) = (m1 +i-1, m1 +j -1) Sn = [m1, m2] m1 m2 u u[m1, m2] u(m1)... u(m2) F F n F(n) = F Sn f(n) = |F(n)| f(n) n (f(n))n 2 F (f(n))n (fw(n))n n {0, 1} 2n n n! w fw(n) 2n f(n) n! w fw(n) n + 1 N Z N N Z Z f(n) n - C C n w (Gw(n)) w Gw(n) fw(n) w n fw(n + 1) w n + 1 u n + 1 w u u[1, n] u[2, n + 1] n 2 G(n) f(n) n f(n + 1) n + 1 F(n + 1) [1, n] [2, n + 1] G(n) n 312 123 4123 3124 v1 v2 v1 v2 d = (d1,..., dn-1) n d1 dn-1 u n w d = (d1,..., dn-1) m u = w(m + D0)... w(m + Dn-1) u(i) = w(m + Di-1) k i {1,..., n} D0 = 0 Dk = dj 1 k n - 1 j= w d fw(d) d d1 =... = dn-1 = 1 fw(d) = fw(n) kw(n) = supd fw(d) d n (kw(n)) w n d = (d1,..., dn-1) m = {m + D0,..., m + Dn-1} (i, j) = (m + Di-1, m + Dj-1) k i, j {1,..., n} i = j D0 = 0 Dk = dj 1 k n-1 j= d f(d) d d1 =... = dn-1 = 1 f(d) = f(n) k(n) = supd f(d) d n (k(n))n 2 kw(n) fw(n) k(n) f(n) kw(n) 2n k(n) n u n w m d u = w(m + d)w(m + 2d)... w(m + nd) u(i) = w(m + id) i n aw(n) n w (aw(n)) w n m d = {m + d, m + 2d,..., m + nd} a(n) n (a(n))n 2 aw(n) fw(n) a(n) f(n) w {0, 1} (x, y) = 0 w(x) = w(y) = 1 w(x) = w(y) = 0 x < y w(x) = w(y) = 1 x > y (x, y) = 1 w(x) = 1 w(y) = 0 w(x) = w(y) = x < y w(x) = w(y) = 0 x > y x < y (x, y) = w(x) [m1 +1, m1 +n+1] = [m2 + 1, m2 + n + 1] w[m1 + n, m1 + n] = w[m2 + 1, m2 + n] w n w n+1 w fw(n) = f(n + 1) kw(n) = k(n + 1) aw(n) = a(n + 1) {0, 1} 0 1 w = w(1)w(2)... w(n)... {0, 1} n N Rw(n) = 0, w(n)w(n + 1)w(n + 2)... = w(n + k)2-(k+1).

k w w i, j {1,..., n} w(i) < w(j) Rw(i) < Rw(j) w w w = (N,, w) w w(i)w(i + 1)...

w = 01101001... Rw(1) = 0, 01101001... Rw(2) = 0, 1101001... Rw(3) = 0, 101001... Rw(4) = 0, 01001...

Rw(4) < Rw(1) < Rw(3) < Rw(2) w w w[1, 4] = 2431 {0, 1} 2134 w w[1, 4] = 21 w w w w w 2431 2134 w w 3 u[1, n] = v[1, n] u[1, n - 1] = v[1, n - 1] n + 1 n [2, n + 1] = n + 1 n [1, n] = n 2 n + 1 (n) = (n/d)2d d|n n + 1 2 n P (n + 1) = (t) 2n-t t= 3 n + 1 fu(n) f (n + 1) P (n + 1) = 2n(n - + O(n2-n/2)), u (0, 1) R w = w w (sn) sn = n + - n + (sn) (i) < (j) si < sj = [m1 + 1, m1 + n + 1] = [m2 + 1, m2 + n + 1] w[m1 + 1, m1 + n] = w[m2 + 1, m2 + n] f(n) = fw(n - 1) = n G(n) Gw(n - 1) k(n) = n n a(n + 1) = (n + 1) (p), p= (p) {1,..., p - 1} p k(n) = n 2n a(n) = w w = 010001010100...

n n Bn Sn Bn (m1, m2) = w[m1 + 1, m1 + n] = w[m2 + 1, m2 + n] w[m1, m1 + n] = w[m2, m2 + n] w(m1) = w(m2) Sn (m1, m2) = w[m1 + 1, m1 + n] = w[m2 + 1, m2 + n] w[m1, m1 + n] = w[m2, m2 + n] w(m1) = w(m2) f(n + 1) - f(n) = |Bn| + |Sn|.

f(n) |Bn| |Sn| Bn Sn n bn = |Bn| sn = |Sn| bn = |Bn| b2 = 1 b3 = 2 b4 = 2 b5 = 2 bn = 1 5 2t + 1 n 6 2t bn = 2 6 2t + 1 n 5 2t+ sn = |Sn| s3 = 2 sn = 2t n = 32t t 1 sn = n = 3 2t f(2) = 2 f(3) = 3 f(4) = 7 f(5) = 9 f(6) = n 7 n + 6 2t - 1, 5 2t < n 6 2t t 1;

f(n) = 2n + 2 2t - 2, 6 2t < n 10 2t t 0.

w w = 0110100110010110... n N{0} Rw(n) Rw(n) = 0, w(n)w(n + 1)w(n + 2)... = w(n + k)2-(k+1).

k = (N{0},, ) x y Rw(x) Rw(y) 0 01, 1 10 x = (x1,..., xn) Rn y = (y1,..., yk) Rk x y = (x1,..., xn, y1,..., yk) Rn+k : R R2 x x, + 1, x 2 (x) = x x, - 1, x > 2 ((x1,..., xn)) = (x1)... (xn) x Rn y Rk (x y) = (x) (y) T (0) = 0 t0 = T (0) = t1 = (t0) t2 = (t1) t3 = (t2) (T (n))n 0 n tn = (T (0),..., T (2n - 1)) 1 1 1 3 T (0) = 0 T (1) = 1 T (2) = T (3) = - T (4) = T (5) = - T (6) = - 2 2 4 4 3 1 7 3 5 T (7) = T (8) = T (9) = - T (10) = - T (11) = T (12) = - 4 8 8 8 8 7 3 5 T (13) = T (14) = T (15) = - T (16) = 8 8 8 = (N {0},, ) x y T (x) T (y) T (i) < T (j) Rw(i) < Rw(j) (N {0}, , ) (x, y) = (x, y) x x y y x (x) {0, 1} x x = (x)2(k) 2k (x)2(k) {0, 1} k= (x)2(k) = 0 k > log2 x t: {0, 1} {0, 1} t N {0} x y x = y x = y p (x)2[0, p - 1] = (y)2[0, p - 1] (x)2(p) = (y)2(p) (x, y) = (x)2(p) t((x)2[0, p - 1]), t 2 p = t (x, y) t t t = t x y z t(x) < t(y) < t(z) t(x) > t(y) > t(z) t t f (n) = 2log n.

t G (n) t t n G(n) 2log n n = 2h G(n) 2h t t k (n) = 2n-1.

t n m d = {m+d, m+2d,..., m+nd} a (n) n (a (n))n 2 t t 22h, n = 2h + a (n) = t 22h+1, 2h + 2 n 2h+ a (n) a (n) 2n-1, t t a (n) t t t0 a (n) = a (n) t0 t n 2 t1 a (n) = 2n-1 n t 2n

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