Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://new.math.msu.su/Sites/demosite/Uploads/cheb190.B9CE0A8D97DD4F9195F10B5F748FC85F.pdf Äàòà èçìåíåíèÿ: Wed Dec 4 16:24:37 2013 Äàòà èíäåêñèðîâàíèÿ: Sun Apr 10 01:52:43 2016 Êîäèðîâêà: IBM-866

..

-


VI I . . 1 190- ..

2011


519.2, 517, 531 22 C 56

. VI I. . . 1. 190- .. / .. , .. , .. , .. . .: - . -, 2011. 168 . ISBN 978 5 211 05652 7

190-

ISBN 978 5 211 05652 7

c - , 2011 .







16 2011 190 , (1821í1894). .. , , ; 16 . 1841 . . 1846 . 1847 , . 1849 , ; 1850 . 1856 . - . 1882 , , , . , - (1895), . . , .., .., .., .., .., .., ... , . 4




, , , , . , , . . . , , : . , , . .. .., . , , : . , . , . , : (1871), (1873), (1874); - (1860) (1877), (1893) , . 1944 , . : ( ..), , , 2010 . - .. , , . , , - .

5


1.

1



6


..

..
190- (1821í1894) . .. , .. - 1845 1846 . 1860 . , ( , 1845; , 1846; , 1867; , 1887). . , , . .., , , : , , , ... , , , , . . . ; . . ... n-1/2 , n . ; , , , .

7


1.

1
..
2

í . . -, , , , .

1



. . . , . . , . , . í , , . , í . , .

2


),

-, . {T , Z, Y , U, , L}. T í (0 < T Z = {Z (t), t [0, T ]} Y = {Y (t), t [0, T ]} í , U = {U (t), t [0, T ]} í .
1 2

10-01-00266. , ebulinsk@yandex.ru, , - . .. .

8


..,

X = {X (t), t [0, T ]}, , .. X = (T , Z, Y , U ). Z , Y , U , X , , . L í , . LT (U ) L(T , Z, Y , U, X ). . , , , , . (., , [7]). Z, Y , X ( , [5]).
1. UT = {U (t), t [0, T ]} , LT (UT ) = inf LT (UT ) ( LT (UT ) = sup LT (UT )), UT UT UT UT

(1 )

UT í . U = {UT , T 0} .

. , )

(1) , ( ) (1), ( .

~ ~ 2. U = {UT , T 0} , T , S 0 ~ ~ UT (t) = US (t), t min(T , S ).

3. U = (UT , T 0) ,
T

lim T

-1

LT (UT ) = lim T
T

-1

LT (UT ).

4. U = {UT , T 0} , . , , . ( ), [4] [6].

3



, (, , ..) : I. , II. III. , , . , c1 . c2 , 9


1.

p q = 1 - p. ( p = 0.) h r . k í k - . , k , k 1, F (§), (s) > 0 s > 0 . fn (x) í n , , x í . L(v ) = E[h(v - 1 )+ + r (1 - v )+ ], Gn (v , u) = (c1 - c2 )v + q L(v ) + c2 u + pL(u) + Efn-1 (u - 1 ), í . (., , [1]) n 1 fn (x) = -c1 x + min Gn (v , u) f0 (x) 0.
x v u

a+ = max(a, 0).

(2 )

v u, (2), ( n- ): v - x u - v .

4



, (2), . . = { ( c 1 , c 2 ) : 0 < c1 r ( 1 - ) - 1 , 0 < c2 r ( 1 - ) - 1 ( p + q ) } , I = { ( c 1 , c 2 ) : c 2 ( p + q ) c 1 } , I I = { ( c 1 , c 2 ) : c 2 < c1 - q r } , I I I = { ( c 1 , c 2 ) : (c 1 - q r ) + c 2 < ( p + q ) c 1 } , I I I = { ( c 1 , c 2 ) I I I : c 2 < p c 1 } , I I I = { ( c 1 , c 2 ) I I I : c 2 p c 1 } . - ‘ , 0 = {0 c1 r }, 0 = {0 c2 pr } k 1 k = {(c1 , c2 ) : r = { (c 1 , c 2 ) : r (p +
k k -1 i=1 k -1 i=0 k

< c1 r

i

i=0 k

i },

Ak = ik i , Ak = ik i ,

i ) < c1 r ( p +

i=1

i )} ,

. 10


..,

Gn (v , u) v u K (v ) = c1 - c2 + q L (v ), Tn (v ) = K (v ) + Sn (v ), Q(u) = c2 - c1 + pL (u) +
0 u-v ï 0

Sn (u) = c2 + pL (u) +
0



fn-1 (u - s)(s) ds

K (u - s ) (s ) d s,

R(u) = c2 (1 - ) + pL (u) + q

L (u - s)(s) ds,

V (v ) = c1 (1 - ) + L (v ).

, fn (x), n 1, . v, un , vn , u, u v, , ï ï^ ^

K (v ) = 0 , ï

Sn ( u n ) = 0 ,

Tn (vn ) = 0,

Q(u) = 0, ï

R (u ) = 0 , ^

V (v ) = 0 . ^

, F (v ) = (c2 + q r - c1 )/q (r + h), c2 c1 - q r , ï v = -. , K (v ) > 0 v , (c1 , c2 ) I I . ï . , F (v1 ) = (r - c1 )/(r + h) 0 v1 = - A1 , F (u1 ) = (pr - c2 )/p(r + h) 0 u1 = - A1 . - = {(c1 , c2 ) I I I : Sn (v ) < 0}, + = {(c1 , c2 ) I I I : Sn (v ) > 0} ï ï n n 0 = { ( c 1 , c 2 ) I I I : Sn ( v ) = 0 } . ï n K (v ) < 0 v < v K (v ) > 0 v > v , , Tn (v ) < Sn (v ) v < v ï ï ï Tn (v ) > Sn (v ) v > v. , . ï 1. (c1 , c2 ) - I I , v < vn < un , v > vn > un ï ï n + I ( c 1 , c 2 ) n . , .. F (§) .

5



vn (x) un (x) v u, (2). . 1. (c1 , c2 ) I un (x) = vn (x) = max(x, vn ). ^ vn , n 1, limn vn = v , F (v) = ^ (r -c1 (1 -))/(r + h). (c1 , c2 ) k , k 0, vn = - n k , vk+1
k

i F
i=0

(i+1)

(vk+1 ) =

r

k i=0

i - c1 , r+h

(3 )

F

i

í i- F . 11


1.

k n, 0 n = 1. 4, u1 < v1 < v v I 0 . ^ï , u1 (x) = v1 (x) = max(x, v1 )
f1 (x) = -c1 +

0, T1 (x),

x < v1 , x v1 .

(4 )

, fm (x), m n-1, (4), 1 m,

Tn (v ) = V (v ) +
0 Hm (v ) = (fm - fm-1 ) F (v ) fm (x) - fm-

v-vn-

1

Tn-1 (v - s) ds = Tn-1 (v ) + H

n-1

(v ),

(5 )

1

, Tn (vn-1 ) = H (vn-1) < 0, , (5) 1, vn un < vn < v . , ^ |V (vn )| =
vn -vn- 0
1

0, (x) = -Tm-1 (x), Tm (x) - Tm-1 (x),

x < vm-1 , vm-1 x < vm , x vm .
-1

< vn

Tn-1 (vn - s)(s) ds Tn-1 (v )F (vn - vn-1 ). ^ |V (vn )| 0, n , k 1. (4) 0 , vn-1 = -. limn vn = v k ^

, Tn-1 (v ) n-1(c1 + h), ^ = 1. , v = limn vn . ^ , vk = - Ak , v -v f1 (x) = L (x) (5) 0 n-1 , vk+1 (3) k . , 0 .

2. (c1 , c2 ) I I vn (x) = x un (x) = max(x, un ). un , n 1, limn un = u, ^ pF (u) + q F 2 (u) = [r (p + q ) - c2 (1 - )]/(r + h). ^ ^ (c1 , c2 ) k , k 0, un = - n k , uk+1
k

pF (u

k +1

)+
i=1

i F

(i+1)

(u

k +1

)=

r (p +

i ) - c2 . r+h

k i=1

3. (c1 , c2 ) I I I , n0 (c1 , c2 ), vn (x) = max(x, v ) ï ‘ un (x) = max(x, un ) n n0 , n < n0 un (x) = vn (x) = max(x, vn ). (c1 , c2 ) I I I , n0 = 1. , un , n n0 , - limn un = u. ï 2 3 - . [6] [3] [8]. , (c1 , c2 ) , / I . (c1 , c2 ) , , (c1 , c2 ) I I , x, 12


..,

n r . I I I : , ( I I I n 1, - n k ). - k , , , , .

6



= 1, {c1 0, c2 0}. , F . 4. vn (x) = un (x) = max(x, v ), (c1 , c2 ) I . (c1 , c2 ) I I ^ vn (x) = x un (x) = u, (c1 , c2 ) I I I vn (x) = max(x, v ) ï ï un (x) = max(x, u), n. ^ . . fn (x) n- . 2 3 , x lim n-1 fn (x) = lim n-1 fn (x).
n n l , fn (x) í n , , v I ( u (v , u) I I I I I ), ^ ^ ïï l , , . .

2. x n-1 (fn (x) - fn (x)) 0, n . (6 )

n 0 2. , fn (x) = fn (x) fn (x) = fn (x). , n

fn (x) - fn (x) =

l =1

l l (fn (x) - fn-1 (x))

l l 1 max |fn (x) - fn-1 (x)| max |fn- x x

l +1

0 (x) - fn-

l +1

(x)|.

, vn (. un ) v (. u u), ^ ï ^ > 0 n(), , , |vn - v | < , n > n. ^ ^ ^ ,
n-n ^ l =1 n l l |fn (x) - fn-1 (x)| (n - n)d ^ l l |fn (x) - fn-1 (x)| nb(x), ^

^ l =n -n +1

d < b(x) < . , (6). 13


1.

3. x
n

lim n-1 fn (x) = g0 ,




g0 = c1 ² + L(v ) (c1 , c2 ) I ; g0 = c2 ² + pL(u) + q 0 L(u - s)(s) ds, (c1 , c2 ) I I , ^ ^ ^ (c1 , c2 ) I I I g0 pL(u) + ï
0 u-v ïï

[c2 s + q L(u - s)](s) ds + ï

u-v ïï

[c1 s + (c2 - c1 )(u - v ) + q L(v )](s) ds. ïï ï

3 . .

7



, , F (x). (F1 , F2 ) = sup |F1 (x) - F2 (x)|.
x

(c1 , c2 ) I I I . v k uk Kk (v ) = 0 Qk (u) = 0, F Fk K (v ) Q(v ), , Qk v k v . ï 4. {Fk , k 1} í , Fk (u) = 0 < 1 u 0. (F, Fk ) 0, v k v uk u k . ï ï ^ ^ vk uk v k uk , Fk (s) = Fk (s). Fk (s) ï ï , [2]. 4, . 5. ^ P((F, Fk ) 0, k ) = 1, P(vk v, uk u, k ) = 1. ï ïï ï

U : . , xk-1 k - , k 2, (vk-1 - xk-1 )+ (uk-1 - max(xk-1 , vk-1 ))+ . ï ï ï xn U . 6. > 0 n() , P( ) > 1 - |u - un | < , |v - vn | < , xn < u + n n() ïï ïï ï . 14


..,

Gn (x) U . + n , n- , 1 = h(x - 1 ) + r (1 - x)+ . yn n- . n = h(yn - n )+ + r (n - yn )+ + c1 (vn-1 - ï n + + xn-1 ) + c2 (un-1 - max(xn-1 , vn-1 )) Gn (x) = k=1 gk , gk = Ek . ï ï 7. gn n g0 , 3. . 5. U , ..
n

lim n-1 Gn (x) = g

0



x.

3 7.


[1] ., . : , 1960. [2] ., . : , 1977. [3] Borgonovo E., Apostolakis G.E., A new importance measure for risk-informed decisionmaking // Reliability Engineering and System Safety, 2001, v. 72, p. 193í212. [4] Bulinskaya E., Some aspects of decision making under uncertainty // Journal of Statistical Planning and Inference, 2007, v. 137, p. 2613í2632. [5] Bulinskaya E., Sensitivity analysis of some applied probability mo dels // Pliska. Studia Mathematica Bulgarica, 2007, v. 18, p. 57í90. [6] Bulinskaya E., Sto chastic Insurance Mo dels, Their Optimality and Stability / Advances in Data Analysis, ed. Christos H. Skiadas, Birkhauser, Boston, Basel, Berlin, 2010, p. è 129í140. [7] ., : . .: , 1969. [8] .., // , 1990, . 2, 1, . 112í118.

15


1.


..
1

, . , .

1



, . , . , . . [1]í[8].

2



= (1 , 2 , . . . , k , . . . ). , . 1. A B , P(AB ) = P(A)P(B ) 1. A B , P(AB )P(A + B ) = P(AB )P(AB ) (2 ) (1 )

, AB , A + B , AB , AB , .. ( ) .
, ovinogradov@mail.ru, , - . .. .
1

16


..,

1. , A B , .. (1) . : 2. , . 3. , . , . . 2. (2) (1). . (1 , 2 , . . . , k , . . . ). k xk , s = k1 xk . s = 1, P(§) . s = 1, , ² (§, s). ² (A, s) A ² (A, s) = xk ,
k :{k A}

P (§) = ² (§, s) /s. 1 2. ² (§, s). P (§) = ²(§, s)/s , = 1 + 2 + 3 + 4 , i j = (i = j ) ² (1 , s) ² (2 , s) = ² (3 , s) ² (4 , s). , A = 1 + 4 , B = 1 + 3 .

3



1. a, b, c, d , ab = cd. S = a + b + c + d . . , aS = (a + c)(a + d). , (a + c)(a + d) S . 19 [9]: p 17


1.

, , , p. , , S , (a + c) (a + d) S . , .. S . . , pk = P(k ) = ak /bk , ak , bk ak /bk (1 k N ). ak /bk nk /n, n , .. , nk . 3. , n . . A pA = P(A) = nA /n. A B , (2) nAB nA+B = nAB nAB (3 ) .. nAB + nA 3 .
+B

+ nAB + nAB = n, 1 , n

.

3. 3 . , n - , , . , 6, , , n , . , p1 = p2 = p3 = 1/6, p4 = 1/2, . 4. n . .

. , , A B . (3). 2 , n , . 4 . 4. 3, 4 . , .. pk = 1/n (1 k N ). N = n.

5. . , n .

. ( .) n , .. n = pq (p > 1, q > 1). , A B , P(A) = p/n, P(B ) = q /n, P(AB ) = 1/n. (1) , A B , . . 4 5 6. , n . [1]. 18


..,

4



, N 1 , 2 , 3, . . . , N , pk = P(k ) > 0 (1 k N ). , N = 3 p1 , p2 , p3 . , p1 = (p1 + p2 )(p1 + p3 ) , .. p1 = (p1 + p2 )(1 - p2 ), , p1 + p2 = 1, , . , N 4. 2 . . 1. 1 , 2 , . . . , N -1 - . s . pk = P(k ) = k /s (1 k N - 1), pN = /s . . () . , 2 , . . 5. , . 2. 1 , 2 , . . . , N
-1

, s=


N -1 m=1

m + .

k 2 p k = P ( k ) = , 1 k N - 1; s . 3. 1 = 2 = § § § = N 1 pk = P(k ) = , s , N s = N - 2 + 2 3 2.
-2

33 pN = s 3

= 1, N

-1

= N =

2. 3 2 s ,

1 k N - 2;

p

N -1

= pN =

, N

. = 1 + 2 + 3 + 4 . . A B . N 1 3 : 1 N -1 , 3 N . 2 19


1.

1 = 2 = § § § = N -2 , 4 N 2

/2

= 2 = § § § = N

-2

. (4 )

² (1 , s) ² (2 , s) = ² (3 , s) ² (4 , s)

2 . N . , 1 , 2 , 3 , 4 (3) . , N -1 N 1 + 2 3 + 4 , (4) , .. . , N -1 1 , N 3 . (4) 3 3 (5 ) ( 2 + a)b = ( 2 + c)d, a, b, c, d , a 0, c 0, b > 0, d > 0, a + b + c + d = - 2. (5) (b - d) 3 2 = cd - ab. b = d, , N 3 2 . . b = d, (5) , a = c. N - 2 = a + b + c + d = 2(a + b), N . . (1 , 2 , . . . , N ) - pk = P(k ) > 0 (1 k N ). : P(1) = p1 (x) = x , x+q P(k ) = pk (x) = pk , x+q ( 2 k N ),

x q = p2 + § § § + pN . x . , p1 = . : x x ? 2, x 1 x = 1 + 2 + 3 + 4 , , : 1 = x (1 ), 2 = 2 (2 ), x x x x x x x 3 x = 3 (3 ), 4 = 4 (4 , . . . , N ). x , x x x
2 P(1 )P(x ) = P(3 )P(4 ) x x x

(6 )

(6) xp2 = p3 r , r = p4 + ... + pN . , x = p3 r /p2 , 2, x , P ( 1 ) = p3 r , p2 q + p3 r P ( k ) = p2 pk , p2 q + p3 r 2 k N.

, .. , . (6). a(x + b) = c, a, b, c . x. , x > 0, 1 x . X . 20


..,

, X . x , x X , x X . / , x , x . .. .


[1] Shiflett R. C. and Shultz H. S., An approach to independent sets // Mathematical Spectrum, 1979/80, v. 12, p. 11í16. [2] Eisenberg B. and Ghosh B. K., Independent events in a discrete uniform probability space // Amer. Statistician, 1987, v. 41, 1, p. 52í56. [3] Baryshnikov Y. M. and Eisenberg B., Independent events and independent experiments // Pro c. Amer. Math. So c., 1993, v. 118, 2, p. 615í617. [4] Chen Z., Rubin H. and Vitale R. A., Independence and determination of probabilities // Pro c. Amer. Math. So c., 1997, v. 125, 12, p. 3721í3723. [5] Chen Z., Rubin H. and Vitale R. A., Addendum to Independence and determination of probabilities // Pro c. Amer. Math. So c. 2001, v. 129, 9, p. 2817. [6] Skekely G. J. and Mori T. F., Independence and atoms // Pro c. Amer. Math. So c., 2001, v. 130, 1, p. 213í216. [7] Edwards W., Shiflett R. and Shultz H., Dependent probability spaces // Collega Mathematics Journal, 2008, v. 39, 3, p. 221í226. [8] Ionascu E. J. and Stancu A. A., On independent sets in purely atomic probability spaces with geometric distribution // Acta Mathematica Universitatis Comenianae, 2010, v. 79, 1, p. 31í38. [9] .., . : - , 1956.

21


1.

1
..
2

, ( ) , . .

1

B

(An , Bn ), n 1, , . , (1) , : , Mn = max{Y1 , . . . , Yn } n , [n] . (1) , [1]. [2], , , . , . . , [3] (. 2). , , , . Yn . ,
11-01-00050. , gold_ann@list.ru, , - . .. .
2 1

Yn , n 1, Yn = An Yn-1 + Bn , n 1, Y0 0, (1 )

22


.., ...

, (1), , . , (1) . , [4, ç8.4.1], ( Bn ), ( An ), , . . A. [2]. Yn , n 1, (1), 1) > 0, E A1 = 1, E A ln+ A1 < , 0 < E B1 < ; 1 2) B1 /(1 -A1 ) , ln A1 , A1 = 0, . : d 1) Y = A1 Y + B1 , Y (A1 , B1 ) , d -1 Y = j =1 Bj j=1 Ai ; i 2 3 4 5 ) ) ) ) (1) Y0 = Y , {Yn } ; d {Yn } Yn Y , n ; c > 0, , P(Y > x) cx- x ; Yn , =
1 d

P(



j

j =1 i=1

Ai y -1)y

- -1

dy ,

an = n-

1/

,
n

lim P(an Mn x) = exp(-c x- )

x > 0. , (1) An . , (1) ~ (An , Bn ) (An , Bn ), n 1, , 1, 2 , . , , (1) , , An .

2


t0

(Xt )

, X0 = x, (2 )

dXt = (c - d § Xt ) dt + Xt dWt ,

W , c R, d > 0, > 0. , (1), . 23


1.

[5, 5.2.1], (2) t- . 2 [6] c > 0, d > - 2 : Xt = e- a = -(d + :
(d+
2 2

t )t+W
t

x+c
0

e(d+

2 2

)s-W

s

ds ,

t 0.

2 2

), (2) t W
t

Xt = eat+

x+c
0

e-

as-W

s

ds ,

t 0.

(3 )

[6] : h(x) = 2 2c
-
2d 2

-1



2d +1 2

-1

x-

2d/2 -2

exp -

2c - x 2

1

,

x > 0,

2 = d2 + 1 = - a . 2 , X0 W , (3) .

(s) (s ) =

1. , W d a+W , W , A = e , a < 0, > 0. s -u = E A (u ) = E e . 2 2 -as - 2 s .

[7]. 1. A A = ea
d +

,

(4 )

, , , a < 0, > 0. (s) = 2 ( - s), = - a . 2 2. (s) = E As , A 2 2 2 2 a a . (s(u)), s(u) = - 2 + a -22 u s(u) = - 2 - a -22 u , a < 0, > 0, , A (4). 1. A = ea+ , , 2a , E - 2 +1 < , a < 0, > 0. X , (2), 2 c > 0 , d = -(a + 2 ). d 2 An = A, n 1, = - a . 2 d d

n = , W ; Tn = n=1 i , Yn = XTn , n 1. Yn , i [7]. 24


.., ...

3



(Xt )t0 , (2). Yn = XTn , Tn = n=1 i , n 1, i 1 , 2 , . . . , , 2a , W , E - 2 +1 < . Y MtX = max[0,t] Xs , Mn = maxk=1,n Yk . , , [6]. (2) s(x) m. :
x y

s(x) =
1

exp -2

1

c-d§t dt 2 t2

dy ,

x ( 0 , ), § x 2 ,
2d

,
x

s (x) = exp -2



1

c-d§t dt 2 t2

= exp -

2c 2c 1 + § 2 2 x

x ( 0 , ).

m , (2), 2 x ( 0 , ). m (x) = 2 2 , x s (x) Xt , H h(x) = |m| = m((0, )). m (x) , |m| x ( 0 , ),

B. [6]. X0 = y (0, ) u(t)
t

lim |P(MtX u(t)) - F t (u(t))| = 0,

F (x) = exp -

1 |m|s(x)

, x ( 1 , + ).
2a2 2

2. : ² §

, ² = E .

. Tn = n=1 i , n 1, 1 , 2 , . . . i , , , Tn . (t) [0, t] . (t)/t 1/², t , .. Y , M (t) MtX u(t) Y P(M (t) u(t)) P(MtX u(t)). un n (un ) = e-1 ,
n
2 ï ï F H : F (x)/H (x) 2a2 , ï , n § F (un ) 1, n . n .

x . limn F

lim E H (un )

(n)

= lim E H (un )
n



(n) n

n

= lim H (un )
n

1 ²n

= e-

1 § ² §

2 2a2

,

. 25


1.

[8]. [3] . ~ ~d n . An = As , n 1, s > 0, = /s, = . ~ , An a/ . a, ². 3. = ²0 , E 0 = 1, ² > 0, a ². [8].
d

4



1. n = h, n 1, h > 0. An , . . Zn = ln An (ah, 2 h). h = 1 a [3], = - 22 , a/ (-, 0). ( í). , . [9]. [10] . [11] , , . ( ) , . , . 2. n , n 1, . , An . , P(Zn B ) = P(an + W (n ) B ) = = E I{an + W (n ) B } = E(E(I{an + W (n ) B }|n )) =

1 (u - an )2 exp - du = 2 2 n 2 n B 2 + - = B 0 21 x exp - (u2ax) exp(-x) dx du, 2x + (u - ax)2 p Z n (u ) = - x d x = exp - 2 2 x 2x 0 2 2 a = 22+a2 exp - 2+a |u| + 2 u . 2 =E

p(x) = ex + x < 0 p(x) = e- x x 0, , Zn + 26


.., ...

= 2 2 + a2 + a , 2 = 2 2 + a2 - a , 2 (5 )

2 = - a = (a2 +4a 2 -a)2 . , 2 2 .

An . [3], = - = (1 - )2 . (5),
2

2a2 2

, .. 2

3. ARC H - Xn =
n

+ X

2 n-1

,

n 1,

, > 0,

2 n , n 1, . Yn = Xn 2 2 (1) c An = n , Bn = n , n 1. n , An - 1 ( 2 , 21 ). [2], : ( + 1/2) = 1/2 (2)- . , An (4).

(s ) = E

As n

=

s

-

1 x2 x2s exp - 2 2

(2 ) dx =

s 0



ts

-

1 2

( 2 )s 1 e-t dt = s + 2

.

(4) (s) = ( - s) : ( - s + 1/2) = (2)2s- (s + 1/2). > 0. .. , .


[1] Vervaat W., On a sto chastic difference equation and a representation of non-negative infinitely divisible random variables // Adv. Appl. Probab., 1979, v. 11, p. 750í783. [2] de Haan L., Resnick S., Rootz‡n H., de Vries G., Extremal behaviour of solutions e to a sto chastic difference equation with applications to ARCH pro cesses // Sto chastic Pro cesses and their Applications, 1989, v. 32, p. 213í224. [3] .., .., // . . -, . ., 2008, 5, c. 6í10. [4] Embrechts P., Kluppelberg C., Mikosh T., Mo delling Extremal Events for Insurance and è Finance. Berlin: Springer, 2003. [5] ., . . : , 2003. [6] Kluppelberg C., Risk Management and Extreme Value Theory / Extreme Values in è Finance, Telecommunication and the Environment. Bo ca Raton:Clapman and Hall/CRC, 2002, p.101í168. 27


1.

[7] .., // . . -. . ., 2010, 6, c. 13í18. [8] .., // . . -. . ., 2012, 2 ( ) [9] Kozubowski T. J., Podgorski K., Log-Laplace distributions // Int. Math. J., 2003, v. 3, 4, p. 467í495. [10] Fr‡shet M., Sur les formules de r‡partition des revenus // Rev. Inst. Int. Statist, 1939, v. e e 7, p. 32-38. [11] Inoue T., On income distribution: the welfare implication of the general equilibrium mo del and the sto chastic pro cesses of income distribution formation: Ph.D.Thesis. University of Minnesota, 1978.

28


.., ...

1
..
2

, . .

1



() . [1] - ( ). . [2] Z+ R+ . [3, 4, 5]. [6]. d [7] d 2 R+ . d = 2. {Z (n)}, Z (n) = (Z1 (n), Z2 (n)), n 0,
x x P(Z1 (n) y1 , Z2 (n) y2 |Z1 (n - 1) = x1 , Z2 (n - 1) = x2 ) = F1 1 (y1 , y2 )F2 2 (y1 , y2 ), 2 F1 F2 T1 , T2 R+ . F1 F2 . , , , . . IX (, 17í20 2011) [8].

11-01-00050. , alebedev@mech.math.msu.su, , - . .. .
2

1

29


1.

2


() ()

{Z () (n)} c Fk Tk [1, 1 ] ½ [1, 2 ], > 0, k = pk , pk > 0, k 1, k = 1, 2. 1 [7] {Z () (n)} () [1, 1 ] ½ [1, 2 ]. ~ ~ () ~ () Z () = (Z1 , Z2 ). () . [3]. 1. q = (q1 , q2 ) (0, 1]2 , q = (1, 1), lim sup Fk (q1 1 , q2 2 ) < 1,
()

k = 1, 2

(1 )

u() = (u1 (), u2 ()) c


lim (1 - Fk (u())) = k [0, +],


()

k = 1, 2 , .
()



lim

()

(u()) = e-

(p1 1 +p2 2 )

. r = (r1 , r2 ), rk = qk k . , .

(r ) 0 ,



~ () () ~ ( ) () ~ () ~ () () (r ) = M(F1 (r )Z1 F2 (r )Z2 I{Z1 [1, r1 ], Z2 [1, r2 ]})+ ( ) ( ) ~ ~ () () ~ () ~ () +M(F1 (r )Z1 F2 (r )Z2 I{Z1 > r1 , Z2 [1, r2 ]})+ ~ () () ~ ( ) () ~ () ~ () +M(F1 (r )Z1 F2 (r )Z2 I{Z1 [1, r1 ], Z2 > r2 })+ ~ () () ~ ( ) () ~ () ~ () +M(F1 (r )Z1 F2 (r )Z2 I{Z1 > r1 , Z2 > r2 }) () () () () () () () () F1 (r )F2 (r )() (r ) + F1 (r )r1 F2 (r ) + F1 (r )F2 (r )r2 + F1 (r )r1 F2 (r )r2 ,



()

(r )

F1 (r )r1 F2 (r ) + F1 (r )F2 (r )r2 + F1 (r )r1 F2 (r ) 1 - F1 (r )F2 (r )
() ~
( )

()

()

()

()

()

()

r

2

()

()

0,

.

q (0, 1)2 . () (u()) = M(F1 (u())Z1 F2 (u( ~ () () ~ ( ) () ~ () +M(F1 (u())Z1 F2 (u())Z2 I{Z1 ~ () () ~ ( ) () ~ () +M(F1 (u())Z1 F2 (u())Z2 I{Z1 ~ () () ~ ( ) () ~ () +M(F1 (u())Z1 F2 (u())Z2 I{Z1
() () () () () ~ ( ) ~ () ~ () ))Z2 I{Z1 [1, r1 ], Z2 [1, r2 ]})+ ~ () > r1 , Z2 [1, r2 ]})+ ~ () [1, r1 ], Z2 > r2 })+ ~ () > r1 , Z 2 > r2 } ) ,



(u()) F1 (u())1 F2 (u())2 (1 - () (r )), () () (u()) () (r ) + () (r1 , 2 ) + () (1 , r2 ) + F1 (u())r1 F2 (u())r2 .
() (p1 1 +p2 2 )

(2 )

(2) Fk (u()) e-k , k = 1, 2, : e- lim inf
()

(u()) lim sup


()

(u()) e-

(q1 p1 1 +q2 p1 2 )

,

q . 30


.., ...
0 0 F1 , F2 [0, 1]2 , ( ) - H1 H2 [9, 10] , .. a1 (s) > 0, a2 (s), b1 (s) > 0, b2 (s), s > 0, v (s, x) = (a1 (s)x1 + b1 (s), a2 (s)x2 + b2 (s)), x = (x1 , x2 ), 0 lim F1 (v (s, x))s = H1 (x), 0 lim F2 (v (s, x))s = H2 (x),

s

s

k = 1, 2 .

(3 )

1. (1) f (s, q ) = (f1 (s, q ), f2 (s, q )) , 1 - Fk (f (, v (, x)) 1, 0 1 - Fk (v (, x))
s ()

.

(4 )

lim

()

p p (f (, v (, x)) = H1 1 (x)H2 2 (x). ()

. (3) (4) (1 - Fk (f (, v (, x))) - ln Hk (x). k = - ln Hk (x), u() = f (, v (, x)) 1, . , 1, x1 - 1 x2 - 1 0 , Fk (x1 , x2 ) = Fk , k = 1, 2 , 1 - 1 2 - 1 f (s, q ) = ((p1 s - 1)q1 + 1, (p2 s - 1)q2 + 1). , p1 = p2 = 1, .. .
0 0 1. F1 (x1 , x2 ) = x2 x2 , F2 (x1 , x2 ) = x1 x2 ( 1 2 ). 0 lim F1 (y1/s + 1, y2 /s + 1)s = e2y
1

+y

2

s

,

s

0 lim F2 (y1 /s + 1, y2 /s + 1)s = ey

1

+2y

2

,




lim

()

( + y1 , + y2 ) = e3(y

1

+y2 )

,

y 1 , y 2 0.

^ ~ () ~ () , Z () = ( - Z1 , - Z2 ) . . 1 = 10 500 (10, 10). . Z1 (n) = 9U1,
1/(2Z1 (n-1)+Z2 (n-1)) n

+ 1,

Z2 (n) = 9U2,

1/(Z1 (n-1)+2Z2 (n-1)) n

+ 1,

Ui,n , i = 1, 2, n 1, [0, 1]. 31


1.

. 1:
0 0 2. F1 (x1 , x2 ) = min{x2 , x2 }, F2 (x1 , x2 ) = min{x1 , x2 } ( 2 1 , 1, ). 0 lim F1 (y1 /s + 1, y2 /s + 1)s = em in{2y1 ,y2 }

s

,

s

0 lim F2 (y1 /s + 1, y2 /s + 1)s = em

in{y1 ,2y2 }

,




lim

()

( + y1 , + y2 ) = em

in{2y1 ,y2 }+min{y1 ,2y2 }

,

y 1 , y 2 0.

^ , Z () , . , 2 {(y1 , y2 ) R+ : 1/2 y2 /y1 2}. . 2 . Z2 = 1 + 3 Z1 - 1 Z2 = 1 + (Z1 - 1)2 /9, . Z1 (n) = 9 max U1,
1/(2Z1 (n-1)) n

, U2,

1/Z2 (n-1) n

+ 1, Z2 (n) = 9 max U1,

1/Z1 (n-1) n

, U2,

1/(2Z2 (n-1)) n

+ 1,

Ui,n , i = 1, 2, n 1, [0, 1]. ~ , Z () H (x) = exp{-(-x) }, x 0, > 0, H (x) = exp{-e-x } [9, 10]. G(x) = 1 - exp{-x/(1 - x)}, x [0, 1). 32


.., ...

. 2:
0 0 3. F1 (x1 , x2 ) = G2 (x1 )G(x2 ), F2 (x1 , x2 ) = G(x1 )G2 (x2 ), 0 lim F1 (z1 , z2 )s = exp{-2e-y1 + e-y2 }, 0 lim F2 (z1 , z2 )s = exp{-e-y1 + 2e-y2 },

s

s

zi = 1 -


1 yi + , 1 + ln s (1 + ln s)2

i = 1, 2 ,

lim

()

(w1 , w2 ) = exp{-3(e-y1 + e-y2 )},

-1 ( - 1)yi + , i = 1, 2 . 1 + ln (1 + ln )2 . wi = - . 3 .
0 0 4. F1 (x1 , x2 ) = min{G2 (x1 ), G(x2 )}, F2 (x1 , x2 ) = min{G(x1 ), G2 (x2 )}, ( 3) 0 lim F1 (z1 , z2 )s = exp -e- min{y1 -ln 2,y2 }

s

,

s

0 lim F2 (z1 , z2 )s = exp -e-

min{y1 ,y2 -ln 2}

,




lim

()

(w1 , w2 ) = exp - e-

min{y1 -ln 2,y2 }

+ e-

min{y1 ,y2 -ln 2}

.

, {(y1 , y2) : |y2 - y1 | ln 2}. 33


1.

. 3: . 4 . G((Z2 - 1)/9) = G2 ((Z1 - 1)/9) G2 ((Z2 - 1)/9) = G((Z1 - 1)/9), . , 1 ( ). . 2. F1 F2 (N ) 2 0 {1, . . . , N } , N 1, Fk (m) = Fk (m/N ), m = (m1 , m2 ) {1, . . . , N }2 , 0 0 F1 F2 (3) ak (s) = 1/s, bk (s) = 1.
N (N ) (N )

lim

(N )

(N - m1 , N - m2 ) = H1 (-m)H2 (-m).
()

. (3) (1-Fk (N -m1 , N -m2 )) - ln Hk (-m). k = - ln Hk (-m), u() = f (, v (, x)) 1, .
0 0 5. F1 (x1 , x2 ) = x2 x2 , F2 (x1 , x2 ) = x1 x2 , 1 2

N

lim

(N )

(N - m1 , N - m2 ) = e-

3(m1 +m2 )

.

^ ~ (N ) ~ (N ) , Z (N ) = (N - Z1 , N - Z2 ) . 2 4, . 34


.., ...

. 4:

3


()

0 {Z () (n)} c Fk (x1 , x2 ) = Fk (x1 , x2 ) Tk [1, +)2, 0 0 k = 1, 2, 1. , F1 , F2 (3), H1 H2 (.. ), b1 (s), b2 (s) , s . [9, 10]. , , . 1 [7] {Z () (n)} () [1, +]2. - ~ Z () . () . [4]. Fkl l- Fk , k = 1, 2, l = 1, 2. 0 0 (1 - Fkl (x)s ) dx,

k l (s ) =

s 1,

k , l = 1, 2 .

[4, 1í3], s 1, , , kl (s) ukl (s), s , 0 ukl (s) = (Fkl )-1 (1 - 1/s). (3) ukl (s) bl (s), kl (s) bl (s), s . k (s 1 , s 2 ) =
0 0 0 (1 - F1k (x)s1 F2k (x)s2 ) dx,

s1 , s2 1 ,

k = 1, 2 .

35


1.

, , : k (s1 , s2 ) 1k (s1 ) + 2k (s2 ), k = 1, 2 . (5 )

, s = (s1 , s2 ) ( ). () () () ~ () ²k = MZk , k = 1, 2. , ²1 , ²2 1. 1. > 0 ² .
() k

k (

1+

,

1+

), k = 1, 2,

. k ²
() k

~ () ~ () = Mk (Z1 , Z2 ) k (²

() 1

, ²2 ) 1k (²

()

() 1

) + 2k (²

() 2

).

(6 )

kl , > 0 Ckl , kl (s) Ckl s , s 1. : ² ² ²
() 0 () 1 () 2

C11 (² C12 (²

() 1) () 1)

+ C21 (²2 ) () + C22 (²2 )

()

(7 )

= max{²1 , ²

()

() 2

}, C0 = max{C11 + C21 , C12 + C22 }. (7) ²
() 0

C 0 ( ²

() 0)

,
() ()

²0 (C0 )1/(1-) . , > 0 ²1 , ²2 . (6), . 2. > 1 > 0 , s j (s
1+

()

,s

1+

ï )Fj0k (bk (s)) 0,
P ~ () Zk /bk () 1,

s , k = 1, 2 .

j, k = 1, 2,

(8 )

. > 1, > 0 (8).
~ ( ) 0 ~ ( ) 0 ~ () P(Zk > bk ()) = 1 - M F1k (bk ())Z1 F2k (bk ())Z2 () ï 0 () ï () () ï () ²1 F1k (bk ()) + ²2 F2k (bk ()) ²1 F1k (bk ()) + ² ï0 ï0 1 (1+ , 1+ )F1k (bk ()) + 2 (1+ , 1+ )F2k (bk ()),

() 2

ï0 F2k (bk ())

~ () P(Zk > bk ()) 0 (8). 0 0 < < 1. Fkl , Fj0k (bk (s))s 0, s . , ~ () ~ () Z1 , Z2 1 ..
0 0 ~ () P(Zk bk ()) (F1k (bk ())F2k (bk ()) 0,

.

, . 36


.., ...

, (8) s(b1 (s
1+

) + b2 (s

1+

ï ))Fj0k (bk (s)) 0,

s ,

j, k = 1, 2,

(9 )

(8) k kl . 3. b(s) , b1 (s)/b(s) p1 , b2 (s)/b(s) p2 , s , p1 , p2 0 , p p lim (v (b(), x)) = H1 1 (x)H2 2 (x).


.
0 lim (v (b(), x)) = lim M F1 (v (b(), x)) b1 () 0 F2 (v (b(), x)) ~ Z1
( )

0 F2 (v (b(), x))

~ ( Z2

)

=

0 = lim F1 (v (b(), x))

b2 ()

p p = H1 1 (x)H2 2 (x).

6.
0 F1 (x1 , x2 ) = (1 - e-x1 )2 (1 - e- x2 /3

),

0 F2 (x1 , x2 ) = (1 - e-x1 )(1 - e-

x2 /3 2

),

(3) a1 (s) = a2 (s) = 1, b1 (s) = ln s, b2 (s) = 3 ln s.
s s 0 lim F1 (x1 + ln s, x2 + 3 ln s)s = exp{-(2e- x
1

+ e-

x2 /3

lim

0 F2

(x2 + ln s, x2 + 3 ln s) = exp{-(e

s

-x

1

+ 2e

-x2 /3

)} , )} .

, (9) . b(s) = ln s, p1 = 1, p2 = 3. ,
3 lim (x1 + ln( ln ), x2 + 3 ln( ln )) = H1 (x)H2 (x) = exp{-(5e- x
1

+ 7e-

x2 /3

)} .

, .


[1] Lamperti J., Maximal branching pro cesses and long-range percolation // J. Appl. Probab., 1970, v. 7, 1, . 89í96. [2] .., // , 2005, . 50, 3, . 564í570. [3] .., // . .1. . , 2002, 6, . 55í57. [4] .., // , 2002, . 14, 3, . 143í148. 37


1.

[5] .., // . .1. . . 2005, 2, .47í49. [6] .., / , .: , 2009, . 4, 1, . 93í106. http://mech.math.msu.su/probab/svo dny2.pdf [7] .., // . .1. . . ( ). [8] .., / IX . : , 2011, . 46í50. [9] .., . .: , 1984. [10] ., ., ., . .: , 1989.

38


.., ...

1
..
2

, , , . , . .

1



() . [1] - ( ). . [2] Z+ R+ . [3, 4]. [6]. [7] d 2 Rd . + d = 2. d {Z (n)}, Z (n) = (Z1 (n), . . . , Zd (n)), n 0, P(Z1(n) y1 , . . . , Zd (n) yd |Z1 (n - 1) = x1 , . . . , Zd (n - 1) = xd ) = x x = F1 1 (y1 , . . . , yd ) . . . Fd d (y1 , . . . , yd ), xk , yk R+ . (1 )

Fk Tk Rd , 1 k d. + Fk , 1 k d, k - . , , , . (1) . , .
11-01-00050. , alebedev@mech.math.msu.su, , - . .. .
2 1

39


1.

( [7]), , , - [8, 9, 10]. . [11] . , G G(x1 , . . . , xd ) = C (G1 (x1 ), . . . , Gd (xd )), (2 )

Gi (x), 1 i d, , C (u1 , . . . ud ) d [0, 1] , [12]. , (2) . , , - ( ) [12, ç3.3.4], [13, ç7.5], : C s (u 1 , . . . , u d ) = C (u s , . . . u s ), 1 d s > 0. (3 )

, G C , Gs (x1 , . . . , xd ) = C (Gs (x1 ), . . . , Gs (xd )), s > 0, (4 ) 1 d Gs , .. - . C (u1 , . . . , ud ) = exp{-((- ln u1) + § § § + (- ln ud ) )
1/

},

1.

1. (3) - G. , ( -) C (u1 , . . . ud ) = (u
- 1

+§§§+u

- d

- d + 1)-

1/

,

> 0,

Gs , s. (4). , Fk Ck , 1 k d. (1). , (1) (4) :
d

Zk (n) =
j =1

Fj-1 Uj k

1/Zj (n-1) ,k ,n

,

(Uj,1,n , . . . , Uj,d,n ), n 1, Cj , 1 j d. (1) . 40


.., ...

1. Z (n) = (Z1 (n), . . . , Zd(n)) , 1 , (1 Z1 (n), . . . , d Zd (n)) Fk (y1 , . . . , yd ) = F (y1 /1 , . . . , yd/d )1/k ,

. . . , d > 0 Z (n) = 1 k d,

, Fk , 1 k d. , () [14]. 2. Zk (0) , Zk (n), 1 k d, n 1, . . [15], - . , ( ) . (1) Zk (n), 1 k d, n 1, , 1.8 [14] .

2

C

[2] ( ). ï ï G1 G2 , G1 (x) G2 (x) -1 -1 x. , G1 (u) G2 (u), u (0, 1). : G1 G2 , 1 G1 2 G2 , 1 2 (..). , , [16]. , (). , . Z (n) n , ( ) . 3. 0 0 , Fkl Fkl 1 k , l d, n n n 1, , . . Z (0) Z (0) , Z (0) Z (0) .. :
d

Z (n) =

k

(
j =1

Fjk )-1

1/Zj (n-1) Uj,k,n

d

,

Z (n) =

k

(Fjk )
j =1

-1

Uj

1/Zj (n-1) ,k ,n

,

(5 )

(Uj,1,n , . . . , Uj,d,n ), Cj , 1 j d. (Fjk )-1 (u) (Fjk )-1 (u), u Z (n) Z (n) ..

n 1, (0, 1) 1 j, k d, (5) n n n 1. n . 41


1.

3



[7]
min{inf T1 , inf T2 } x0 > 0,

(6 ) (7 )


x x F1 0 (y1 , y2), F2 0 (y1 , y2 )

.

(7) , (6) :
d inf Tk x0 > 0.

(8 ) ..) -

k =1

( ), ( , . (s) [7] Tk = d=1 Tj,k Tk , j
(s) Tk

s Fk . , = Tk s Tk {x0 }. , - s > 0. , .

. , F s 0 1 , : = (a1 , b1 ] ½ § § § ½ (ad , bd ] . . p s (y 1 , . . . , y d ) = d (F s (y 1 , . . . , y d ) ) . y1 . . . yd

1. F Rd , d 2, F s , s > d - 1, .

, ps (y ) , us , F (y ), F , 1- d- ( , ). s > d - 1, F s . , y , s > d - 1. , F s . , ~ F , F ~ . F s F s . ~ s s > d-1, F , , F s . , F s s > d - 1. 42


.., ...

1. F - Rd , d 2, F s , s > 0, . . 0 < s1 < s2 . - G = F s1 /d . F s1 = Gd , F s2 = Gds2 /s1 , .. G , d - 1. 1 . Z (n)
d

S=
k =1

x(k) : x(k) Tk , 1 k d ,

Z (0) Rd \{0} Z (n) S n 2 .. S + Z (n) Z (n - 1) = x x ( 0 , + )d . Sk S O xk , 1 k d. , S ( ). x0 = (x0 , . . . , x0 ). , a = (a1 , . . . , ad ) , A = [a1 , +) ½ § § §½ [ad , +) A = (0, a1 ] ½ § § § ½ (0, ad ] P(Z (n) A|Z (n - 1) = x0 ) > 0, P(Z (n) A |Z (n - 1) = x0 ) > 0. (9 ) Rd : (x1 , . . . , xd ) = max{|x1 |, . . . , |xd |}. (k) Fk k = lim sup uP(
u (k )

> u ),
d k =1

1 k d. k < e- , Z (n) -

1. (8), (9) .

.
d d

G (u ) =
k =1

P(

(k )

u) =

Fk (u, . . . , u),
k =1

ï = lim supu uG(u) d=1 k < e- . , 0 < k - < e M > a , G(u) e- /u u M . (1) :
d

P( Z (n) u|Z (n - 1) = x) =

k =1

x Fk k (u, . . . , u) G

x

(u ).

g (x) = max{ln( x /M ), 0} ²(x) = M(g (Z (n))|Z (n - 1) = x) - g (x), ²(x)
0 + 0 +

(1 - G

Me

g (x )

(M ev )) dv - g (x)

= + ln - Ei(- eg

(1 - exp{- eg

(x) -v

(x)

e }) dv - g (x) =

),

43


1.

Ei Ei(y ) = -
+ -y

e-t dt, t

y < 0,

Ei(y ) 0, y -. + ln < 0, , > 0, N M , ²(x) < - x > N . x N M(g (Z (n))|Z (n - 1) = x) . , [17, ç4.2] . [17, ç2]. = d=1 Fk (a1 , . . . , ad ) > 0. k x = (x1 , . . . , xd ) V = [x0 , N ]d S . , Z (n) Z (n - 1) = x0 , , Z (n) Z (n - 1) = x - x0 ( , x = x0 ), . Z (n) Z (n - 1) = x. B S P(Z (n) B |Z (n - 1) = x) = P( B ) P( B A) P( B A, A ) = P( B A)P( A ) P( B A) N . x = x0 . S : (B ) = P( B | A) 0 < p < P( A) N , P(Z (n) B |Z (n - 1) = x) > p(B ). , Z (n) ( V ). Z (n) [17]. 2. (8), (9) M Z (n) . P(
(k ) (k )

< 1 k d,

> u ) = o ( 1 / u ), u , k = 1 , 2 .

2. , - .

4



[7] , Fk , 1 k d, ( ) [13, ç7.5], [18, ç5.2]. Rd , + ( ). Fkl Fkl (y ) = exp{-ckl (y - bkl ) 0,
-k

}, y > bkl , y bkl 44

k , ckl > 0,

bkl 0,


.., ...

Ck . :
x Fk (y1, . . . , yd ) = Fk (x- 1/k

(y1 - bk1 ) + bk1 , . . . , x-

1/k

(yd - bkd ) + bkd ), x > 0.

, :
d

Zk (n) =
j =1

Z
(j )

1/j j

(n)(k (n) - bj k ) + bj
(j )

(j )

k

,

(1 0 )

(j )(n) = (1 (n), . . . , d (n)), n 1, Fj , 1 j d. , 1 , . , (8) , bkl > 0. , (3): P( (k) > u) = 1 = 1 - Ck (exp{-ck = 1 - exp{u-k ln - ln Ck (e-ck1 , . . , + , - ln Ck (e- k = 0,
c

- Fk (u, . . . u) = -k }, . . . , exp{-ckd (u - bkd )-k }) = 1 (u - bk 1 ) Ck (exp{-ck1 (1 - bk1 /u)-k }, . . . , exp{-ckd (1 - bkd /u) . , e-ckd )u-k , u .

-k

} )}

k1

, . . . , e-

c

kd

0 < k < 1, ), k = 1, k > 1.

(1 1 )

, (11) 1. , k > 1, . k = 1. 1. Ck (u1 , . . . , ud ) = u1 . . . ud ( ). k = d=1 ckl . l 2. Ck (u1 , . . . , ud ) = min{u1 , . . . , ud } ( ). k = d=1 ckl . l 3. Ck (u1 , . . . , ud) = exp{-((- ln u1 ) + § § § + (- ln ud ) ) ( ).
d 1/ 1/

},

1

k =
l =1

c

kl

.

- . 45


1.

,b,c (y ) =

exp{-c(y - b) 0,

-

}, y > b; y b, r, s > 0. (1 2 )

, c > 0, b 0. r ,b,c (sy ) ,
b/s,cr /s


(y ),

(k ),(bkl ),(ckl ) ( ) . - , , : (),(bkl ),(ckl ) , k = . 1. a1 , . . . , ad > 0
(k ),(bkl /al ),(ckl ak /al k )


(x1 , . . . xd )

(k ),(bkl ),(ckl )

(a1 x1 , . . . ad xd ),

d x R+ .

. Z kl , bkl , ckl , 1 k , l d. Z (n) = (Z1 (n)/a1 , . . . Zd (n)/ad ). (k ),(bkl ),(ckl ) (a1 x1 , . . . ad xd ). , 1, Z a Fk (x1 , . . . xd ) = Fk k (a1 x1 , . . . ad xd ) . (12)
a Fk (x1 , . . . xd ) = Fk k (a1 x1 , . . . ad xd ) = a = Ck k (k ,bk1 ,ck1 (a1 x1 ), . . . , k ,bkd ,ckd (ad xd )) = = Ck (ak ,bk1 ,ck1 (a1 x1 ), . . . , ak ,bkd ,ckd (ad xd )) = k k = Ck (k ,bk1 /a1 ,ck1ak /ak (x1 ), . . . , k ,bk1 /ad ,ck1ak /ak (xd )), 1
d

Z 3. a > 0
(k ),(bkl /a),(ckl /ak
-1

(k ),(bkl /al ),(ckl ak /al k )



.

)

(x)

(k ),(bkl ),(ckl )

(ax).

, bkl = 0. (8) , .. . , S W (n) = (W1 (n), . . . , Wd (n)), Wk (n) = ln Zk (n). 2. bkl = 0 k > 1, W (n) . . = min k > 1. (10) :
d

Wk (n) =
j =1

Wj (n - 1) (j ) + k (n) , j

k (n) = ln k (n).
d

(j )

(j )

|Wk (n)|

j =1

|Wj (n - 1)| + j 46

d

j =1

|k (n)|,

(j )


.., ...

W (n - 1 ) + (n), W (n)
d

(n) =
j =1

(j ) (n) ,

(1 3 )

(n) , M (n) < . g (x) = x ²(x) = M(g (W (n))|W (n - 1) = x) - g (x), (13) , > 0, N > 0, ²(x) < - x > N . x N M(g (W (n))|W (n - 1) = x) . , [17, ç4.2] . [17, ç2] , 1, Z (n), V = [e-N , eN ] S . W (n) Z (n) S \{0}. . 1 Z (n) 500 Z (0) = (1, 1) , 1 = 2 = 3, c11 = c22 = 2, c12 = c21 = 1, b11 = b12 = b21 = b22 = 0.

. 1:

5



{Z () (n)} c () Fkl (y ) = Fkl (y ), > 0, Fkl . ( ) () . 47


1.

() . [5]. 2. Z () (n) , ï > 1, v > 0, ckl > 0, Fkl ,0,v Fkl (y ) ckl y -, y , lim () (x1/(-1) ) = (),(0),(ckl ) (x), x Rd . (1 4 ) +


^ . ² = 1/(-1) , ² = ² . Z () (n) = () () () () ^ ^ Z (n)/², (x) = (²x) , (),(0),(ckl ) , . ^ () Fkl (y ) = Fkl (²y )² ,0,ckl (y ), . (1 5 ) ^ () , Fkl ,0,v (12) Fkl ,0,v . ï , Fkl (y ) ckl y -, y , b > 0, w > max ckl , ^ () ^ () Fkl ,b,w , Fkl ,b/²,w , Fkl ,1,w > b-1 . , 3, ^ (),(0),(v) () (),(1),(w) . ^ {() } S \{0} , , S \{0}. 10.1 [17] , , (15) , , .
0,c
kl

3. Fkl = , ( 3).

, (14) -

, , Fkl (y ) = 1 - (y /bkl ) 0,
-

, y > bkl ; y bkl

> 1, bkl > 0, Fkl (y ) ,0,bkl (y ) y > 0. ( ) .


[1] Lamperti J., Maximal branching pro cesses and long-range percolation // J. Appl. Probab., 1970, v. 7, 1, p. 89í96. [2] .., // , 2005, . 50, 3, . 564í570. [3] .., // . .1. . , 2002, 6, . 55í57. [4] .., // , 2002, . 14, 3, . 143í148. 48


.., ...

[5] .., // . .1. . , 2005, 2, .47í49. [6] .., / , .: , 2009, . 4, 1, . 93í106. http://mech.math.msu.su/probab/svo dny2.pdf [7] .., // . .1. . ( ). [8] Balkema A.A., Resnick S.I., Max-infinite divisibility // J. Appl. Probab., 1977, v. 14, 2, p. 309í311. [9] Resnick S.I., Extreme values, regular variation and point pro cesses. Springer, 1987. [10] ., max- // , 1992, . 37, 1, . 173í175. [11] Jirina M., Sto chastic branching pro cesses with continuous state space // Czech. Math. J., 1958, v. 8, 2, p. 292í313. [12] Nelsen R., An intro duction to copulas / Lecture Notes in Statistics. Springer, 1999, v. 139. [13] McNeil A.J, Frey R., Embrechts P., Quantitative risk management. Princeton University Press, 2005. [14] .., .., . .: , 2008. [15] Marshal l A., Olkin I., Domains of attraction of multivariate extreme value distributions // Ann. Probab., 1983, v. 11, 1, p. 168í177. [16] ., ., : . : , 1983. [17] .., . .: , 1999. 440 . [18] .., . .: , 1984.

49


1.

1
..
2

x(t), N , . t = t(N ), N , ‡ .

1



, , . , . () , . , . (x1 , . . . , xN ) N , (x1 , . . . , xN ) (x1 , . . . , xN ), {x1 , . . . , xN } {x1 , . . . , xN } . , , . [2]. , , [1], . .
, 09-01-00761 11-01-90421. , manita@mech.math.msu.su, , - . .. .
2 1

50


..,

, . (. [3, 4, 5, 6, 7, 8, 9]). (. [10, 11, 12]) (.., ) , , . [13], . , , . , , , x(t) = (x1 (t), . . . , xN (t)) N t = t(N ). ‡ : , , . , t = t(N ), t . x(t) = (x1 (t), . . . , xN (t)) y (t) = (y1 (t), . . . , yN (t)) . y (t) , yi (. (3) ). , , . , [3, 4, 5, 6, 7, 8, 9] . t = t(N ) N , " " . , , .. , , . , . : x(t) y (t) , ‡ . ( 3.1) , " " . , , , , ( ), . 5 , k - . y (t). . [3] 51


1.

. [4, 5] . [6] . [7] k - . [8] . [9] .

2


x(t) = (x1 (t), . . . , xN (t)) RN , t R+ ,



. x(t) y (t) .. . . , N , xi (t) R1 i- t. NN = {1, . . . , N }. (x(t), t 0), ~ ~~ , F , P . () {y a,s }
aRN , s0

([14, 15]) t s, y a,s (s) = a,

a a y a,s (t) = (y1 ,s (t), . . . , yN,s(t)) RN ,

y (t). . a ~ ~~ , yk ,s (t) , F , P : a a yk ,s (t) = yk ,s (t, ) . ~ () {n }
n =1

(m = m ( )) ~

0 = 0 < 1 < 2 < § § § ,

() x(0) = (x1 (0), x2 (0), . . . , xN (0)), . , {y a,s }, {n } 1 x(0) . n= ( , F , P ), (i1 , j1 ), (i2 , j2 ), . . . , (in , jn ), . . . (1 )

(i, j ) , i, j NN , i = j . = ((i1 , j1 ), (i2 , j2 ), . . . , (in , jn ), . . .) 52


..,

in ( ) = in jn ( ) = jn . ~ ~ ~ (, F , P) = ½ , F ½ F , P ½ P , (x(t, ), t 0), = ( , ). ~ (x(t), t 0) : (s, ), ~ s [n ( ), n+1 ( )), k NN , ~ ~ xjn ( ) (n ( ), ) = xin ( ) (n ( ) - 0, ), ~ ~ xm (n ( ), ) = xm (n ( ) - 0, ) ~ ~ m NN \ {jn ( )} . , x(t) : . , 0 < 1 < 2 < § § § : n ( ) (i, j ) j i: (xi , xj ) (xi , xi ) . (2 ) xk (s, ) = y
x(n ( )),n ( ) ~ ~ k

(k , k+1 ) x(t) y (t). , x(t) y (t). , . , : () N , t ; () N , t = t(N ) t(N ) ( ). ‡ , , (), () 3.2. M. {n } 1 n= , , {n - n-1 } 1 n= , : P (n - n-1 > s) = exp(- s), s > 0.

M , (x(t), t 0) RN Ly + Ls , 0 > 0,

y L0 y (t), Ls . y (t) = (y1 (t), . . . , yN (t)):

r dy i = N

N

j =1

(yi - yj ) dt + dBi , 53

i = 1, . . . , N .

(3 )


1.

r R, > 0, B (t) = (B1 (t), . . . , BN (t)) N - . , y (t) , . , , r < 0, , r > 0. r = 0 y (t) B (t) , [8]. , x(t), , N , r , . MN (r, , ). MN (rN , N , N ) ( N ) x(t) = (x1 (t), x2 (t), . . . , xN (t)) RN , , , N .

3



. , "" .

3. 1



, "" "" (x1 , . . . , xN ) V : R R+ , M (x) :=
1 N N N

1 V (x) := N -1

N

m=1

(xm - M (x))2 ,

xm x = (x1 , . . . , xN ). m=1

V . x RN : R+ R+ , t 0
x RN (t) := E V (x(t)).

(4 )

x , RN (t(N )) t = t(N ). , , x RN (t).

3.1. x(t) MN (rN , N , N ). > 0, rN , N , N , , 1) aN = 0, N aN = - 2 rN , (5 ) N (N - 1 )
2 x x RN (t) = N a-1 (1 - exp (-aN t)) + exp (-aN t) RN (0), N

54


..,

2) aN = 0

x 2 x RN (t) = N t + RN (0).

x , RN (t) : dx 2 x R (t) = N - aN RN (t) . dt N , MN (rN , N , N ), , . , MN (rN , N , N ) 3.1.

1. (1) (2), , = 2. , k - [7]. > 0 (. [8]). 2. 3.1 , N rN 0. , " , " (N = 0 & rN = 0) , x(t) N - N B (t). , 2 (6 ) RNN B (t) = RNN B (0) + N t . . " , " (N > 0 & rN = 0) , [8]. " , " (N = 0 & rN = 0) , 4 . ( 3.1) .

3. 2



y (t) x(t) , N , t : t . , : (x1 , . . . , xN ) (x1 + a, . . . , xN + a), , t . , , y (t) = (yi (t), i NN ) x (t) = (x (t), i NN ), i , :
yi (t) = yi (t) - M (y (t)),

x (t) = xi (t) - M (x(t)), i

(. [8]). , , , [3], , . - (. [16]). 55


1.

3.2 ( y (t) ). y (t) . 1) : 1 d M (y (t)) = d B (t) , N B (t) . 2) rN > 0 ( ), y (t) t . 3) rN < 0 ( ), y (t) , , t , . 4) rN = 0 y (t) N - , y (t) t . 1 4 3.2 , 2 3 . y (t) (3) () N [17]. 2, (N = 0) y y 3.1. , RN (t) = RN (t) = EV (y (t)) dy y 2 R (t) = N + 2rN RN (t) , dt N , W (y ) = EV (y ) y (t). , 2 3 (. [17]). rN < 0 ²N y (t) " ". , y , 3.1: RN (t) 2 N /(2rN ), (t ). , , , x (t) 3.2. , M (x(t) M (x(t) , M (y (t)). M (x(t) , . x (t) , . , 3.3 , . 3.3 ( y (t) 1) 2rN - N (NN 1) 0, - t . 2) 2rN - N (NN 1) < 0, - t ). x (t) x (t) , , , . 56


..,

3.2 3.3, , , , rN < 0 rN < 2NNN 1) . (- , rN > 0 ( rN < 0).

3. 3



, , , . ‡ , N , t = t(N ) . 3.4 ( ). b R > 0 , 2b - = 0, > 0. MN (b/N 2 , , ), , rN = bN
-2

N = 1, 2 , . . . ,
N

. 1: 2b - < 0 ( ) t(N ) x 0, RN (t(N )) 2 t(N ). I. 2 N II.

x , N = , N = . , sup RN (0) < .

t(N ) x c1 > 0 , RN (t(N )) b, (c1 ) 2 t(N ), N2 b, (c) > 0 , d b, (0) = 1, dc b, (c) < 0, (c > 0), b, (+) = 0. N 2.

t(N ) 2 x , RN (t(N )) N2 - 2b 2: 2b - > 0 ( ) t(N ) x I. 0, RN (t(N )) 2 t(N ). N2 III-S. II.

t(N ) x c1 > 0 , RN (t(N )) b, (c1 ) 2 t(N ), N2 d b, (0) = 1, dc b, (c) > 0, (c > 0), b, (+) = +. t(N ) , N2 2 N 2 t(N ) x RN (t(N )) exp (2b - ) . 2b - N2

III-R.

. ‡ I y (t). y (t) (. (3)) / rN = b/N 2 , I, x (3). RN (t(N )) , 57


1.

, , (. (6)). , [5], I . , / ‡ x . I I RN (t(N )) b, (c1 ), 1 2. 1 I I . 1 I I I-S , t(N ), . , c1 , I I I I-S. 2 , I I I-R x(t). 2 I I I I-R. , () : , . , b 1 2, . 2b - = 0. 3.5. b > 0 > 0 , 2b - = 0, > 0. MN (
N b N (N -1)

, , ),

N = 1, 2 , . . . ,

(7 )

x , sup RN (0) < . t(N ) x RN (t(N )) 2 t(N ),

N .

3.5 "" , . x RN (t(N )), , I. , , . . (. [18, 19, 20, 21, 5, 22]. , , , .

3. 4



. 3.6 ( ). MN (rN , N , N ), N = 1, 2, . . . , , , 58


..,

1) lim inf N > 0,
2) aN = N (NN 1) - 2rN 0. - x , , sup RN (0) < . N N x 2 RN (t(N )) N t(N ). x RN (t(N )) x RN (t(N )) N

. I. aN t(N ) 0, II. III-S. III-R. aN t(N ) c = 0, aN t(N ) +, aN t(N ) -,

1 - exp(-c) 2 N t(N ). c

2 N . aN 2 x RN (t(N )) N exp (|aN | t(N )). |aN |

3. c, II 3.6, f (c ) = f (c ) < 0 , (c = 0), f (-) = +, f (0 ) = 1, f (+ ) = 0 . 1 - e-c , c (8 )

3. 5



, , , "" : rN = r
N

+ rN ,

N = N + N ,

N = N + N .

, aN = aN + aN . qN = aN /aN .

, 3.6. . N N aN = - 2 rN , aN = - 2 rN . N (N - 1 ) N (N - 1 )

3.7 ( ). M N (r
N

, N , N ),

N = 1, 2 , . . . ,

, rN , N , N 1 2 3.6, aN = 0 N . - ) |qN | + qN 1 = O (1), N , , MN (rN , N , N ) MN (rN , N , N ) x t = t(N ) I RN, x RN : x x 2 RN, (t(N )) (N )2 t(N ), RN (t(N )) N t(N ) ; 59


1.

) qN MN (rN , N , N ) c =


1, N , , MN (rN , N , N ) t = t(N ) II x x RN, RN 0:
x 2 RN (t(N )) f (c) N t(N ) ,

x RN, (t(N )) f (c) (N )2 t(N ),

MN (rN , N , N ) MN (rN , N , N ) t = t(N ) III-S III-R x x RN, RN , . aN = 0, 3.7, . , MN (rN , N , N ) (7) 2b - = 0, MN (rN , N , N ) = M
N b N2

f (c) (8). ) 0 < lim inf qN < lim sup qN < + ,
N N

, , .

aN = 0, aN = N 2 (2b-1) , b . 3.5 N MN (rN , N , N ) I:
x RN, (t(N )) 2 t(N ),

t(N ) .

, 3.6, , MN (rN , N , N ) I, I I I I I-S: t1 (N ) = ‡ o(N 3 ), t2 (N ) uN 3 t3 (N )/N 3 ,
x RN (t1 (N )) 2 t1 (N ), x RN (t2 (N )) f (2b u) 2 t2 (N ) (1 - e- x RN (t3 (N )) 2b u

)

2 3 N, 2b

2 N 2b

3

. , u + I I I I I-S.

4



3.1 3.6, ( 3.4, 3.5 3.7) . 3.1 x RN (t).

4. 1

3.1

2.3 [8], . t = 60


..,

max {m : m t}. , t = max {i : i t}. x RN (t) = E V (x(t)) E (§) = E E E § | {j } 4.1. E V (x(t) | { j } 1 j=
2 N = A, 2 rN t -2 t i=0
t

j =1

| t

.

ht N

-i

A
-1

t

-1

§ § § Ai+1 Bi +

+ hN A,t B + ht A,t A N hN := 1 - Am A,t Bm
t

+ B,t +

-1

x § § § A1 A0 RN (0) ,

, N (N - 1 ) := exp ( 2rN (m+1 - m ) ) , 0 0, := exp ( 2rN (t - t ) ) , := Am - 1, B,t := A,t - 1 .

(9 ) (1 0 ) (1 1 )

ç 4.2, . (11) Bm B,t 4.1, E V (x(t) | {j }
j =1

=

2 N ht A,t A 2 rN N

t

-1

§ § § A1 A0 +
t -2 t i=1
t

+ (1 - hN ) A,

hNt

-i

A

t

-1

§ § § Ai+1 Ai +

+ (1 - hN ) hN A,t A + ht A,t A N
j =1
t

-1

+ (1 - hN ) A,t - 1 +

-1

x § § § A1 A0 RN (0) .

(9) (10), E V (x(t) | {j } =
2 N ht exp(2rN t) + 2 rN N

+ (1 - hN )

t -2 i=1

ht N

-i

exp 2rN (t - i ) +
t

(1 2 ) (1 3 ) (1 4 )

+ (1 - hN ) hN exp 2rN (t -

-1

)+

+ (1 - hN ) exp 2rN (t - t ) - 1 +
x + ht exp(2rN t) RN (0) . N

E (§ | t ) . , (. ç 4.2). 61


1.

4.2. (t , t 0) N . t - m , t -: P {t - m (x, x + dx) | t = n} = n C
m-1 n-1

xn

-m

(t - x) tn

m-1

dx,

x (0, t), m = 1, n.

E (§ | t = n) (12)í(14), (1 - hN ), ,
n t

hn-i N
i=1

e
0

2rN x

nC

i-1 n-1

xn-i (t - x) tn

i-1

t

=t

-1 0

e

2rN x

n

hN x + (t - x) t

n-1

dx .

, E (V (x(t) | t = n) =
2 N -1 + hn exp(2rN t) + N 2 rN t

+ (1 - hN ) t-

1 0

e2r

N

x

n

1-

(1 - hN ) x t

n-1

dx +

x + hn exp(2rN t) RN (0) . N

(s) := Es

t

= exp ( N t (s - 1) )

t . t ,
2 N E (V (x(t)) = -1 + (hN ) exp(2rN t) + 2 rN t

+ (1 - hN ) t

-1 0 x RN (

e2r 0) .

N

x





1-

(1 - hN ) x t

dx +

+ (hN ) exp(2rN t)

, (s) = N t exp ( N t (s - 1) ),
t

t-

1 0

e2r

N

x





1-

(1 - hN ) x t

t

dx = t-

1 0 t

e2r

N

x

N t exp -N t §

(1 - hN ) x t

dx =

, (5), aN =

N N (N -1)

- 2rN . ,

= N exp (2rN x - N (1 - hN ) x ) dx = 0 N 1 - e-aN t , aN = 0, a N = N t, aN = 0,
aN t

(hN ) exp(2rN t) = exp( 2rN t - N t (1 - hN ) ) = e- 62

.


..,

E (V (x(t)). aN = 0 E (V (x(t)) = e-
aN t 2 N -1 2 rN (1 - hN ) N + 1 aN 2 2 rN x RN (0) + N 2rN aN 2 x RN (0) + N 1 - aN x RN (0) +

+ e-

aN t

+ =
aN t

- e-

aN t

= e- = e-

aN t

1 - e- e-
aN t

=

aN t

.

aN = 0 2rN = N (1 - hN ), , E (V (x(t)) =
x RN (0) + 2 N -1 + 1 + 2 rN

+ (1 - hN ) N t = = 3.1 .
x 2 RN (0) + N t .

4. 2



-, {i } 1 x(t): i= Fm = ((x(s), s m ) , {i } 1 ) , m = 0, 1, . . . i= Fm- = ((x(s), s m - 0) , {i } 1 ) , m = 1, 2, . . . . i= F
0- 0-

= ( {i }

i=1

). ,
m-

F

F0 § § § F

Fm F

(m+1)-

F

m+1

§§§

4.1 . 4.3. > 0 , m N E (V (x(m )) | F hN = 1- N (N - 1 ) (0, 1).
m-

) = hN V (x(m - 0)) ,

. [8]. , , [7]. [7], , 4.3 . (. 1), , , : = 2. 4.4. (y (t), t 0) V (y (t)) = V (y (0)) exp (2rN t) + 63
2 N (exp (2rN t) - 1) . 2 rN


1.

4.4 . 4.1. x(t) , {i } 1 y (t), i= m N E (V (x(m+1 - 0)) | Fm) = V (x(m )) exp (2rN (m+1 - m )) + 2 + N (exp (2rN (m+1 - m )) - 1) = 2 rN 2 = V (x(m )) Am + N Bm 2 rN (9) (11). 4.3, ,

2 N E (V (x(m+1 )) | Fm) = hN Am V (x(m )) + hN Bm . (1 5 ) 2 rN (15) , ,

E (V (x(m+1 )) | F

m-1

) = E (E (V (x(m+1 )) | Fm) | F = hN = hN = h2 N

E V (x(n ) | {j }
j =1

)= 2 Am E (V (x(m )) | Fm-1) + N hN Bm = 2 rN 2 2 Am hN Am-1 V (x(m-1 )) + N hN Bm-1 + N hN Bm = 2 rN 2 rN 2 Am Am-1 V (x(m-1 )) + N h2 Am Bm-1 + hN Bm . N 2 rN ,
m-1 n = hN An -1 x § § § A1 A0 RN (0) +

2 +N 2 rN

n-2 i=0

hn-i An N

-1

§ § § Ai+1 Bi + hN Bn

-1

.

t > 0, , (t , t] . , t . 4.4, 4.1. , , , "=" , , " ". 4.2 , t , . 4.2 . 4.4. , (y (t), t 0), k = m d (y k - y m ) = dY
km

rN N

N

j =1 km

(yk - ym ) dt + N (dBk - dBm ) , dt + N 2 dBkm , 64

= rN Y


..,

Y

km

m2 (t) = E (Ykm (t)) , m2 (t): d 2 m2 = 2rN m2 + 2N , dt , m2 (t) = m2 (0) e2r
N

(t) = yk (t) - ym (t), Bkm (t), t 0
2

. -

t

+

2 N rN

e2r

N

t

-1 .

(1 6 )

, V (x) := , V (y (t)) := 1 N -1
N

m=1

(xm - M (x))2 = 1 (N - 1)N

1 (N - 1)N (Y
k
k
(xk - xm )2 ,

(t))2 .

(16), 4.4.

4. 3

3.4, 3.5, 3.6 3.7

3.6 . 1 3.1 . 3.4, , 3.6. b, (c1 ) = f ( c1( - 2b) ), f (c) , (8). 3.5 . 2 3.1. 3.7 3.6 .

5



-, 1, k - (k -), k > 2. . , (2-) 2. - k - , , . k - , [7, 8]. 3 , , y (t) , : dyi = rN yi dt + N dBi , 65 i = 1, . . . , N , (1 7 )


1.

, (3), rN R, N > 0, B (t) = (B1 (t), . . . , BN (t)) N - . , y (t), (17), . , rN < 0 N -. (17) , (3). (17) (3) , y (t) , , . , 3.2 (. 2í4) 3.3 y y x x . 3.3í3.5 .


[1] Mitra D., Mitrani I., Analysis and optimum performance of two message-passing parallel pro cessors synchronized by rollback // Performance Evaluation, 1987, v. 7, p. 111í124. [2] Bertsekas D.P., Tsitsiklis J.N., Parallel and Distributed Computation: Numerical Metho ds. Belmont: Athena Scientific, 1997. [3] Manita A., Shcherbakov V., Asymptotic analysis of a particle system with mean-field interaction // Markov Pro cesses Relat. Fields, 2005, v. 11, N. 3, p. 489í518. [4] Malyshev V., Manita A., Asymptotic Behaviour in the Time Synchronization Mo del / Representation Theory, Dynamical Systems, and Asymptotic Combinatorics AMS, American Mathematical So ciety Translations Series 2: Advances in the Mathematical Sciences, 2006, v. 217, p. 101í115. [5] ., ., // , 2005, . 50, 1, . 150í158. [6] .., // , 2006, . 42, 3, . 78í96. [7] .., // , 2008, . 53, 1, . 162í168. [8] Manita A., Brownian Particles Interacting via Synchronizations // Communications in Statistics Theory and Metho ds, 2011, v. 40, N. 19-20, p. 3440í3451. [9] Manita A., On Markovian and non-Markovian Mo dels of Sto chastic Synchronization / Pro ceedings of The 14th Conference "Applied Sto chastic Mo dels and Data Analysis" (ASMDA), 2011, Rome, Italy, p. 886í893. [10] Manita A., Simonot F., Clustering in Sto chastic Asynchronous Algorithms for Distributed Simulations / Lecture Notes in Computer Science, 2005, v. 3777, p. 26-37. [11] Manita A., Simonot F., On the cascade rollback synchronization // arXiv.org:math/0508533, 2005. [12] .., // , 2006, . 61, 5, . 187-188. 66


..,

[13] .., // , 2007, . 14, 6, . 1001í1021. [14] ., . 2. .: , 1963. [15] .., , .: , 1975. [16] .., . : - , 2007. [17] .., , .: , 1969. [18] .., . . .: , 1980. [19] Grimmett G., Percolation. Berlin: Springer-Verlag, 1999. [20] .., // . , 2003, . 73, 4, . 637í640. [21] .., " " // . , 2003, . 74, 4, . 637í640. [22] .., .., // , 2008, . 53, 4, c. 798-í809.

67


1.

1
..
2

, . .

1



. . H = (V , E ), V = V (H ) , , E = E (H ) V , . n-, n . V H = (V , E ) , E (H ) . H , . H (H ). (H ) r , , H r -. n > l 2 . H (n, l)-, n-, l H ( ) H . , H (n, 2)-, , n- . (n, l)- , , . S (l, n, m) n- m , l ( ) . , S (l, n, m) (n, l)- , l = 2, . 1853 . [1]. , , .
( 09-01-00294), ( -8784.2010.1) -3429.2010.1. 2 , gold-amber@yandex.ru, , - . .. .
1

68


..,

, , [2] [3]. (n, l)- . 2001 . . , . , . . [4] m (n, r, l), (n, l)- , r . , m (n, r, l) : m (n, r, l) = min {|E (H )| : H (n, l)-, (H ) > r } .

m (n, r, l), m (n, r, l), . 2001 . . (. [4]) m (n, r, l). 1. (.. , . , . , . , [4]) n 3, l 2, r 2 2n3l m (n, r, l) 2 (n)
l/(l-1) l

r

n-1

ln(er )

l/(l-1)

.

(1 )

(1) (n)l : (n)l = n(n - 1) . . . (n - l + 1). 2009 , . [6] . 2. (.. , . , [6]) r 2 l 2 n0 = n0 (r, l), n > n0 m (n, r, l) 10r
2

2r n n2

l/(l-1)

.

(2 )

, 2010 . (. [7]). 3. (.. , . , [7]) r 2 l 2 n1 = n1 (r, l), n > n1 m (n, r, l) (4 er )l (n r n ln r )l
/(l-1)

.

(3 )

2010 . , . . [8]. 4. (. , . , . , [8]) r0 = r0 (n, l), n > l 2, r > r0 m (n, r, l) 2


100(n) l!

2 l

1/(l-1)

10(n - 1) l-1

l/(l-1)

r

n-1

ln r

l/(l-1)

.

(4 )

, m (n, r, l) (. (2), (3), (4)), l ( ). . (1) l, , , l (, l n). 69


1.

2



m (n, r, l), l 3. 5. C , n > l 3 r 2 m (n, r, l) C n2l , , r n, m (n, r, l) C
/(l-2)

r

n-1

ln r

l/(l-2)

.

(5 )

n2l l!

1/(l-2)

r

n-1

ln r

l/(l-2)

.

(6 )

5 . m (n, r, l). (3) (1), (2) (5) l r n. , (3) Or
nl/(l-1) l/(l-1)

n

,

(1) r (2) r
nl/(l-1) 2l/(l-1) nl/(l-1) (2l2 +l)/(l-1)

n

,

n

,

(5) r
nl/(l-2) 2l/(l-2)

n

.

, , (4) . (1), l/(l-1) 1/(l-1) l/(l-1) 2n3l 100(n)2 10(n - 1) l < . l! l-1 (n)l , l 3, n 3: 100(n) l!
2 l 1/(l-1)

10(n - 1) l-1

l/(l-1)

<

100n2l 6
l/(l-1) l/(l-1) l

1/(l-1)

1 0n 2

l/(l-1)



50 3

1/2

5

3/2 3

n < 50n3 < n = n3l
l/(l-1)

21 2

n7

n2l .

+1 l/(l-1)

=

nl

2n3l < (n)

, (4) (1), 4. , r n l (4) 70


..,

(6): (4) r (n-1)l/(l-1) , (6) r (n-1)l/(l-2) . , , (1) (5), n > l 3 r 2. (1) (5) n >n
3l2 l-1 3l2 l-1

-

2l l-2

((n)l )-1 (r
(n-1)l (l-1)(l-2)

n-1

ln r )
+

-l/[(l-1)(l-2)] -
4 l-2

> .

-

2l l-2

-l -

r

= n2l

3 l-1

-3 -

r

(n-1)l (l-1)(l-2)

, , , n lln r = o(l ln n) (5) (1) . , r n, (6). n lln r l ln n , , . , n0 (r, l), n1 (r, l) r0 (n, l) 2, 3 4 . , m (n, r, l), . 1. l 3, r 2 n (3). 2. n > l 3 r , (4). 3. , n l n lln r l ln n, , , (1). 4. , n l , r < n (5). 5. , n l , r n (6).
n ln r l

= o(l ln n), = o(l ln n),

n ln r l

, m (n, r, l), n, r l.

3



5. l/(l-2) m (n, r, l) C1 n2l/(l-2) r n-1 ln r , (7 ) C > 0 , n > l 3, r 2. (7) n- H , : 1) (H ) > r , 2) H (n, l)-, 71


1.

2) H (7). (n) H (N , n, p). KN n N , H (N , n, p) (n) (n) KN , KN : H (N , n, p) p 1 - p. G(n, p), 60- . . (. [9]). . [10], [11], [12]. H (N , n, p) : N= n2l r
2(n-1)

(ln r )2 (4.1)2

1 l-2

,

p = (4.1)

r

n-1 N n

ln r

N.

(8 )

. . 1. 1 P ( (H (N , n, p)) r ) < . 4

. , P ( (H (N , n, p)) r ) = P { r - H (N , n, p)} ,



r - . , P


{

r - H (N , n, p)}


r i=1



P (


r - H (N , n, p)) =


(1 - p)

(

Ni ( ) n

),

Ni ( ) i .
r

i=1

Ni ( ) n

r Ni ( ), i=1 Ni ( ) = N , r r N/n-1 . , , [13]. ,

P ( (H (N , n, p)) r ) N



( 1 - p )r ( =r
N

N / r -1 n

) r N ( 1 - p )r (
n

N / r -1 n

)<

N /r - 1 exp -p r n

exp - 4.1 (ln r )N r

N/r -1 n N n

.

(9 )

72


..,

(9). 1 + x > ex/(1+x) x > -1, r
n N/r -1 n N n

1-

rn N

n

exp -

r n2 N - rn

e-

4/3

.

(1 0 )

. , N (. (8)) r n2 N . r n N/n N/4 (.. 5 n 4). (10). , P ( (H (N , n, p)) r ) < r r
N N

exp - 4.1 (ln r )N r
N

n

N/r -1 n N n
N 16

r
-2

exp - 4.1 (ln r )N e-

4/3


exp -

17 (ln r )N 16

=r

-

1 . 4

N r n2 32. 1 . 2. 1 P (H (N , n, p) (n, l)-) < . 4 . (n, l)- , , l . , (8) , P (H (N , n, p) (n, l)-) = (4.1) (r
2 n-1

N l
2 2

N -l n-l n2 N

2

p2 =
l

((n)l )2 ln r ) N (4.1)2 (r l!(N )l
2 2

n-1

1 ln r ) N l!

=

= (4.1)2 2 . 3.

n2l r 2n-2 (ln r ) l ! N l -2

2



(4.1)2 1 1 1 = <. 2 l! (4.1) l! 6 4

P |E (H (N , n, p))| 2 . , : X

N p n

1 <. 4

H (N , n, p) N , p . n , P (X 2 EX ) e- 73
3 EX/8

.


1.

, , [11]. N 3N p= P |E (H (N , n, p))| 2 p exp - 8n n = exp - 3 . . 1, 2, 3 , H (N , n, p) 1/4 : -, r -, -, (n, l), , -, 2 N p = 8.2 N r n
n-1

3 (4.1)r 8

n-1

ln r N

r

-1.5 r

n-1

2

-12

1 <. 4

ln r = O n2l

/(l-2)

r

n-1

ln r

l/(l-2)

.

(7) . m (n, r, l): m (n, r, l) C
2

n2l l!

1/(l-2)

r

n-1

ln r

l/(l-2)

,

(1 1 )

C2 , r n > l 3. H (N , n, p), N p -: N= n2l r l!
2(n-1)

(ln r )2 (8.2)

2

1 l-2

,

p = (4.1)

r

n-1 N n

ln r

N.

(1 2 )

, 1, 2, 3. (9) P ( (H (N , n, p)) r ) r
N

exp - 4.1 (ln r )N r

n

N/r -1 n N n

.

(10) , r n2 N . N (. (12)) N r 2 n, r n. , N n2 r (10) . , 1 : P ( (H (N , n, p)) r ) < r r
N N

exp - 4.1 (ln r )N r
N

n

N/r -1 n N n
N 16

r
-2

exp - 4.1 (ln r )N e-

4/3


exp -

17 (ln r )N 16

=r

-

1 . 4

1 . 74


..,

(l - 1)- H (N , n, p), , : 2 N N -l P (H (N , n, p) (n, l)-) p2 l n-l (4.1)2 n2l r 2n-2 (ln r ) l ! N l -2
2



(4.1)2 1 =. 2 (8.2) 4

2 . 3 , . . 1, 2, 3, , , H (N , n, p) 1/4 : -, r -, -, (n, l)-, , -, N p = 8.2 N r 2 n
n-1

ln r = O

n2l l!

1/(l-2)

r

n-1

ln r

l/(l-2)

.

(11) . 5 .


[1] Steiner J., Combinatorische Aufgabe // Journal fur die Reine und Angewandte è Mathematik, 1853, v. 45, p. 181í182. [2] Assmus E.F. Jr., Key J.D., Steiner Systems // Designs and Their Co des, Cambridge University Press, 1994, p. 295í316. [3] Beth T., Jungnickel D., Lenz H. Design Theory, 2nd ed., Cambridge University Press, Cambridge, 1986. [4] Kostochka A.V., Mubayi D., Rèd l V., Tetali P., On the chromatic number of set systems o // Random Structures and Algorithms, 2001, v. 19, 2, p. 87í98. [5] Grable D., Phelps K., Rèd l V., The minimum independence number for designs // o Combinatorica, 1995, v. 15, p. 175í185. [6] Kostochka A.V., Kubmhat M., Coloring uniform hypergraphs with few edges // Random Structures and Algorithms, 2009, v. 35, 3, p. 348í368. [7] Kostochka A.V., Rèd l V., Constructions of sparse uniform hypergraphs with high o chromatic number // Random Structures and Algorithms, 2010, v. 36, 1, p. 46í56. [8] Bohman T., Frieze A., Mubayi D., Coloring H-free hypergraphs // Random Structures and Algorithms, 2010, v. 36, 1, p. 11í25. [9] Erds P., R‡nyi A., On the evolution of random graphs // Publications of the o e Mathematical Institute of of the Hungarian Academy of Sciences, 1960, v. 5, 1í2, p. 17-61. 75


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[10] Bol lob‡s B., Random graphs, Cambridge University Press, Cambridge, 2001. a [11] Jansen S., Luczak T., Rucinski A., Random graphs, Wiley-Interscience, New York, 2000. [12] Karonski M., Luczak T., Random hypergraphs / Combinatorics, Paul Erdos is eighty, ‡ Bolyai So ciety Mathematical Studies, v. 2, eds. D. Miklos, V.T. Sos, T. Szonyi, Springer, ‡ ‡ 1996, p. 283í293. [13] .., .., , .: , 2005.

76


..,

1
..
2

Zd . , e . , , (d = 1, 2). Zd , , .

1



. . . , , , () [4, 4], . Zd . , e , , . , . . ( ) . [1], . , , [2, 3, 4]. H = A+ 0 , A = (a(z )), z Zd , T l2 (Zd ), 0 0 = 0 0 , 0 = 0 (§) d - Z , , . (., , [2, 3, 4]).
10-01-00266. , yarovaya@mech.math.msu.su, , - . .. .
2 1

77


1.

, a(§) z Zd |z |2 a(z ) < . Z Z2 , Zd d 3. [2, 3, 4] , d 3 c > 0 ( > c ) , . c . c . , d = 1, 2, , d = 1, 2 c = 0, > 0. [4] c = G-1 , G 0 0. [2, 3, 4] d 3 G0 < (, , c > 0) z Zd |z |2 a(z ) < . , G0 , , , (. ) . , , .., a(z ) z -(d+) , (0, 2). , , Z Z2 . , . > 0, Z Z2 ( ), > c = G-1 > 0. 0 . 2, A, Zd . . 3 [2, 3, 4], , A , . 4 . 4.2 : , , Z, (0, 1), Z2 , (0, 2). 5 , , .

2

A

a(§) l1 (Zd ).
z

|a(z - z )| =

z



|a(z - z )| = c,

78


..,

c = a

l

1

:=

z Z

d

|a(z )|, (., , [6]) (Au)(z ) :=
z Z
d

a(z - z )u(z )

(1 )

p [1, ] A : lp (Zd ) lp (Zd ), A A1: a(z ) 0, z = 0 a(0) = -
z =0 l
p

c.

a(z ) 0. a(z - z )u(z ), |a(z - z )| = |a(0)|, (2 )

A (Au)(z ) - a(0)u(z ) = , [6]
z =z


z Zd ,z =z

|a(z - z )| =
lp

z =z

A - a(0)I

|a(0)|,

p [1, ].

p (A) A : lp (Zd ) lp (Zd ) p (A) {z C : |z - a(0)| |a(0)|}, , , A2: a(z ) = a(-z ) z Zd , A . A2 A l2 (Zd ), . (3) , 2 (A) [2a(0), 0]. , 2.1. a(§) l1 (Zd ) A1. (1) p [1, ] A : lp (Zd ) lp (Zd ), (2) (3). , A2 , A l2 (Zd ), 2 (A) (4). A a(§). () ( ) =
z Z
d

p [1, ].

(3 )

(4 )

a(z )ei

z ,

,

[- , ]d ,

(5 )

79


1.

§, § Zd . A2 (§) ( ) =
z Z
d

a(z ) cos z , ,

[- , ]d ,

, , . A1 , ( ) = a(z ) (cos z , - 1) 0, [- , ]d . (6 )
z Zd ,z =0

, (0) = 0, z = 0 (§ ). 1. a(§) , z Zd z1 , z2 , . . . , zk Zd , z = k=1 zi a(zi ) = 0 i = 1, 2, . . . , k . i a(§). A3: a(§) . Xt , t 0, Xt t. A Xt A1íA3. p(h, x, y ) , x = 0 h y p(h, x, y ) = a(x, y )h + o(h) = a(y - x)h + o(h) for y = x, p(h, x, x) = 1 + a(x, x)h + o(h) = 1 + a(0)h + o(h). , , [4], p(t, x, y ) t p(t, x, y ) = Ap(t, x, y ), G (x, y ) :=
0

Zd (1) ,

p(t, x, y ) = y (x).

(7 )

e-t p(t; x, y )dt,

> 0.

(8 )

A1 xZd p(t, x, y ) t y (., , [4]). , (7) p(t, , y ) = ~ p(t, x, y )ei(x,).
xZ
d

(7) t p(t, , y ) = ()p(t, , y ), ~ ~ p(0, , y ) = ei(,y) . ~

, p(t, , y ) = e()t ei(,y) , ~ t 0 p(t, x, y ) = 1 (2 )
d [ - , ]
d

e

( )t+i( ,y -x)

d ,

x, y Zd .

80


..,

(8): G (x, y ) = G0 = G0 (0, 0). 2. Xt , t 0, (), G0 < , , G0 = . 1 (2 )
d

[ - , ]

d

ei(, y-x) d , - ( )

x, y Zd ,

0.

(9 )

3


z Z
d

a(§) A4: |z |2 a(z ) < . , , z Zd |a(z )| < . A4 , . , () A1íA4 = 0 . 3.1. a(§) A1íA4, (§) , () - ||2 , > 0. 3.1 [4, 2.1.2]. , , = 0 () ( ) = (0) , .. | (0), | ||2 , > 0. d 1 x, y Zd d p(t; x, y ) d as t , (1 0 ) t2 d = ((2 )d Dd )- 2 Dd = |det (0)| (., ., [5] [4]). [5] (8) (10). 3.2. X = (Xt )t0 Zd A, A1íA4. p(t; x, y ) (7), X d 3 d = 1, 2. 81
1

[- , ]d ,

(0), + o (| | 2 ), 2


1.

4



H : [- , ]d R, H (x) = H (-x), x [- , ]d , (1 1 ) 0 < H0 H (x) H 0 , x [- , ]d , (1 2 )

H0 H 0. , a(§) : A5: a(z ) =
H

( |z | ) z
d+

|z |

, (0, 2), z Zd z = 0. 1 < > d. |z | a(z ) < > 0.

(., , [7, 4.642]),

z Zd ,z =0

(12)
z Z ,z =0
d

, a(0) a(0) = - a(z ) 0,

z Z ,z =0

d

a(§) A1. (11) a(§) A2. A4 a(§) 0 < < 2. , : 4.1. a(z ) A5, A1 A2, A4 : |z |2 a(z ) = .

z Zd ,z =0

(5) a(§). A1, A2 A5 a(§) Zd . ei z , z , () a(§) d- [- , ]d . , 4.2. a(z ) A5, () , d- [- , ]d . , () (6). , A5 () a(§) z H ( ) = |z |-(d+) (cos z , - 1) , [- , ]d . (1 3 ) |z | d
z Z ,z =0

82


..,

4.1. (0, 2) a(§) A5, () a(§) () -C || , C . 4.1 (13) , [9, I, 5, 2.6]. 4.2. X = (Xt )t0 a(§) A5, p(t; x, y ) (7). , d = 1 0 < < 1 d 2 0 Zd A. X < < 2. [- , ]d ,

. p(t) = p(t; 0, 0). (9) 4.1 G0 =
0

p(t) dt =

1 (2 )

d [ - , ]
d

d 1 (-()) (2 )

d [ - , ]
d

d , C | |

C > 0.

< d, 4.2.

5



A , . f (u) := 0 bn un , bn 0 n= n = 1, b1 < 0 n bn = 0, , r = f (r) (1) < , r N, = 1 = f (1), f (r) r - f . ²t (y ) t y , , t = 0 , x, ²0 (y ) = y (x). ²t = yZd ²t (y ). , t x0 = 0 ²t (0) > 0 , , x0 = 0, h p(h; 0, y ) = a(y )h + o(h) y = 0; n = 1 (, ); (, n = 0) p (h; n) = bn h + o(h); ( ) 1 - y=0 a(y )h - n=1 bn h + o(h). - [8], , , , -(a(0) + b1 ). [2, 3, 4], , , > c = G-1 . 0 83 (1 4 )


1.

c . , . H H := A + 0 . (1 5 ) [10], H , A, lp (Zd ), 1 p . p = 2. 5.1. H (15), A A1íA3 A5. (i) H : l2 (Zd ) l2 (Zd ) ( ) ; (ii) H , (14); (iii) (14), H . .. .


[1] Borovkov A., Borovkov K., Asymptotic Analysis of Random Walks. Heavy-Tailed Distributions, Cambridge University Press, Cambridge, 2008. [2] Albeverio S., Bogachev L.V., Yarovaya E.B. Asymptotics of branching symmetric random walk on the lattice with a single source // C. R. Acad. Sci., 1998, Paris. S‡r. I., v. 326, e 9, . 975í980. [3] . ., . ., // , 1998, . 363, 4, c. 439í442. [4] . ., . M.: . -, 2007. [5] ., . M: , 1969. [6] . ., // , . ., 1985, . 49, 3, c. 652í671. [7] Gradshteyn I., Ryzhik I., Table of Integrals, Series, and Pro ducts, 5th ed., Academic Press, Inc., 1994. [8] ., . . 2. .: , 1984, 752 c. [9] Zygmund A., Trygonometric Series. Vol. I, 2nd ed., Cambridge University Press, Cambridge, 1959. [10] . . // . ., 2010, . 55, 4, c. 705í731. 84


2.

2



85


2.

..
.. . 1856 , , , , , . 1853 , .. , . , . , , . . . .. .., .., .., .., , . , . .. . 1855 , , , . .. , , , . .., .., .. . .. , . .. , , .. . .. . , - , , .

86


.., ?

?1
.K.
2

. C2 ? , . (r ) = {v = ur (|z |2 )}. (5,5), (4,2), (3,3) (2,2). , , Qn (t) = tg(n arctg(t)), .

1



C2 (z , w = u + iv ) , aut , , . aut {v = 0}, , , - , .. S = {|z |2 + |w 2| = 1}. . , - , , , . , , aut -, , . - 1907- ([1], [2]). , , . aut - aut , , . , , . , , , .. , aut , , -. .
11-01-00495- 11-01-12033---2011. , vkb@strogino.ru, , - . .. .
2 1

87


2.

aut ? , , , .. . , -, 6, 7 8 . ([2]) - v = |z |2 + 2 Re(hz 4 z 2 ) + . . . , ï ï (z , z , u). h = 0, . , , ( ). , - , ( ). , , , .. 2 Re(hz 4 z 2 ) , ï ( ) , ([4]). , . , , . , , , , , , , . ([3]). , . , ( ), . . . ( ) 6, 7 8. , . , 5- ? , , . : , 8-, ? , , . , , . = {v = m (z , z ) + . . . }, ï m (z , z m 2 ï 2 Re(pz m ) ( ), , m. z z 1, u ï m. , m = 2, , m > 2. , , , ï Q = {v = m (z , z )}. , aut 88


.., ?

aut0 Q0 . aut0 Q0 . 1996- . ([5]). , - . , .. , Sm = { v = | z |
2m

}.

. 2005- ([6]). , , , , - . , Sm {z = 0, v = 0}, . , Sm m S1 (z z , w w ). {z = 0} .

2

v = ur(|z|2).

v = uF (z , z , u). ï , (r ) = { v = u r ( | z | 2 ) } X1 = 2 Re iz z X2 = 2 R e w w , (2 ) (1 )

: (z eit1 z , w w ) (z z , w t2 w ).

aut0 (r )0 . , r (t) (r ) , . , r = const. . w = 0 . Im ln w = arctg v = arctg(r (|z |2 )), u

(z z , w ln w ) v = arctg(r (|z |2 ) = (|z |2 ). , , ... , , |z |2 = t, t (t) + (t). 89


2.

, , , r (t) = tg((t)). , r (t) . , r (0) = 0. , v = u(r0 + r (|z |2 )) Im((1 - ir0 )w ) = ur (|z |2 ). w (1 - ir0 )w v = u r (| z | 2 ) = u ~ r (| z | 2 ) . 1 + | r 0 | 2 + r 0 r (| z | 2 )

r (t) = rs ts + . . . , r ~ r (t) = ~ rs ts + . . . 2 1 + | r0 |

, r (t) t = 0 , z |r (0)|z r (t) = t + . . . , = ‘1. , (r ) = { v = u r ( | z | 2 ) } , r (t) r (t) = t + .. r (0) = 1. , 2 R e f (z , w ) + g (z , w ) z w r2 2 r3 3 t + t + ..., 2! 3! (3 )

aut 0 , .. : L = (i + r ( |z |2 )g (z , w ) + (-i + r ( |z |2 )g (z , w)+ ïï ï ïï ï 2 r ( | z | 2 )u (z f (z , w ) + z f (z , w ) ) = 0 ï w = u(1 + ir ( |z |2 )). (4 )

, g (0, 0) = 0. (4) z = 0 ï f (0, u) = a(u), g (0, u) = ub(u), g (z , w ) = w b(w ) + 2i w a(w )z , ï
Im b(u) = 0. fz (0, u) = c(u), (4) z ï z = 0 ï f (z , w ) = a(w ) + c(w )z + (2iw a (w ) - r2 a(w ))z 2 . ï ï

(4) |z |2 |z |4 , c (w ) = w b (w ) r2 b(w ) +i + iC. 2 4

C , .. , a b, X1 . . . (4) f g , , (4) z m z n , ï 90


.., ?

.. (m, m), (m, m + 1) (m + 1, m). b, a. , a b . . w tw , c t, , a(w ) = aj w j b(w ) = bj w j (4), . a- (a(w ) = Aw m , b = 0) b- (a = 0, b(w ) = B w n ). b-. B , B = 1. Xb (r2 , n) = 2 Re R = 1 + ir (|z |2 ), = arg R = arctg r,
n +1

2n + ir2 n zw +w 4 z

n +1

w

.

|R|2 = 1 + r 2 ,

r = tg .

(4) u ï Re iRR
n +1


2

+ 2 r ( | z | 2 ) | z |

2n + ir2 R 4

n

=0

|z |2 t (1 + r (t)2 ) sin(n(t)) = r (t)t(n cos(n(t)) - r2 sin(n(t))). 2 (5 )

n = 0 (5) , (2), , n > 0. (5) r (t)t = (1 + r (t)2 ) n-
r2 2

1 ctg(n arctg t)

(6 )

, tg(n arctg t) n, . (6) t + r2 t2 +
2 9r2 + 2r3 - 4 + 4n2 3 t + ... 12

(6) n , t n , r (t) (6) n. (6), (t), r2 dt n ctg(n) - d = 2 t . , (t) = t + o(t), , (t): r2 sin(n(t)) = nt exp (t) . (7 ) 2
3

91


2.

, n = 1, 2, 3, . . . (7), r = rb (t; r2 , n). r2 r (t) = t + 2 rb (t; r2 , n) = rb (t; r2 , n), , , r2 = ~~ ~

r2 R r (t) = tg((t)) t2 + . . . r2 , , n = n. ~

a-, .. a(w ) = Aw m , A C, b(w ) = 0. ï ï 2 Re (Aw m + Aw m (2im - r2 )z 2 ) + 2iAz w z
m+1

w

.

L, (z , z ), , ï (m, m + 1) (m + 1, m). , 2Aum+1 z , |z |2 t, ï
m -r (t)R(t)m + (r2 + 2im)r (t)tRït) + |R(t)|2 Rït) = 0, ( (

1 + r (t)2 = cos(2m(t)) - r2 t r sin(2m(t)) = 2mt. m = 0, (9) , (8) , r2 = 0 (t) = - r2 = 0 (t) = t. m = 1, 2, 3, . . . , (9) (t) = (8) 1 1 - (2mt)
2

(8 ) (9 )

1 ln(1 - r2 t), r2

(1 0 )

(1 1 )

1 arcsin(2mt), 2m

(1 2 )

=

1 1 - (2mt)2 - r2 t

,

(1 3 )

, r2 = 0. , m > 0 (8), (9) r2 = 0, r = ra (t, m) = tg 1 arcsin(2mt) . 2m (1 4 )

m, , . a- b-. r2 = 0, (7) (t) (t) = 1 arcsin(nt), n 92


.., ?

n = 2m a-. , ra (t, m) = rb (t, 2m, 0) = tg 1 arcsin(2mt) . 2m (1 5 )

2 t3 , r2 = 2 13 2 ( 12 r2 + n ), r2 = 0. a- 3 m = 0 r2 = 0, .. (t) = t. (16). v = ur ( |z |2 ), r (0) = 1, , , , , (, r2 , r3 ) . , , , . .. , .

, , r2 = 0 a- (t) = - r12 ln(1 - r2 t) b-, .. 1 . (1 6 ) (t)t = n ctg(n(t)) - r22

, t = 0 s, , . v = ur (|z |2 ) = u( ts + r
s +1

(ts ) + . . . ).

(4) z = 0 , g (0, 0) = 0, ï g (z , w ) = b(w )w , Im b(u) = 0. f ~ ~ f (z , w ) = c(w )z + f (z , w ), f (z , w ) = fj (w )w j j = 0, 2, 3, . . . (1 8 ) (1 7 )

L = 0, , (b, c) ~ . (.. (m, m)), f L = 0 2r (|z |2 )|z |4 Re ~ f (z , w ) z = 0 .

, , , , ~ ~ f , , f = 0. (b, c). b(w ) = bj w j , c (w ) = cj w j , b(w ) = bn+1 w n+1 , c(w ) = cn w n . 2 R e cn w n z + bn+1 w z 93
n +1

w

.

(1 9 )


2.

b(w ) = b0 c(w ) = 0, 2 Re b0 w . w (bn+1 , cn ) bn+1 = 0, L = 0 , cn = 0, , , bn+1 = 0. , , bn+1 = 1, cn = K + iM . , |z |2 = t (1 + r (t)2 sin(n(t)) = 2r (t)t(K cos(n(t)) - M sin(n(t)). n = 0, K = 0 2 Re iM z +w z w , (2 0 )

(2). n > 0. (20), 2 sin(n(t)) = (C t)
n 2K

K ln(sin(n(t))) - 2M (t) = ln(C t), C = const, n Mn (t) . K

exp

, nts = C , K =
n 2s
1 n 2K

t

n 2K

,

, C = n s . sin(n(t)) = nts exp(2M s(t)). (2 1 )

r (t), (t) = arctg(r (t)) s, (t) = (ts ), ( ) = + 2 2 3 3 + + ... 2! 3! (2 2 )

, r , sin(n ( )) = n exp(2M s ( )). (2 3 )

(s 2, n 1, M R) = ( , s, n, M ): 2 = 4M s, 3 = 36(M s)2 + n2 , . . .
2 2

(2 4 )

2M s

, ( ) , 2 (2 5 )

sin(n ( )) = n exp

(7) s = 1 (t) = (t, s, n, 2 ). 94


.., ?

( , s1 , n1 , M1 ) = ( , s2 , n2 , M2 ), , s2 = s1 , .. , M2 = M1 , .. 2 = 4M s; n2 = n1 , .. 3 = 36(M s)2 + n2 . , (, s, 2 , 3 ) , . , (, s 2, n 1, M R) v = ur (|z |2), . v = u tg( (|z |2s )), = (t, s, n, 2 ). X3 = 2 R e 2n + i2 n zw +w 4s z
n +1

(2 6 )

w

.

v = ur (|z |2 ), r t = 0 , , . v = ur ( |z |2 ) (, s, n, 2 ). s > 1, s = 1, v = ur ( |z |2s ), r ( ) = t + r22 2 + . . . , r ( ) = tg( ( )), 2 = r2 (B) s = 1. .. , s, s = 1, s = 2, 3, . . . .

3



= {v = ur (|z |2 )} , r . aut 0 - . , ‘1. . (AB) = {v = ur ( |z |2 )} r (t) = tg 1 arcsin(2mt) 2m m = 1, 2, 3 . . .

X3 = 2 Re mz w
2m

+w z

2m+1

X4 = 2 Re w m (1 + 2imz 2 )

+ 2iz w z X5 = 2 Re w m (i + 2mz 2 ) + 2z w z

m+1

m+1

w w w

, , .

, w = 0. , (X2 , X3 , X4 , X5 ) . 95


2.

(, m). (A) = {v = ur ( |z |2 )} r (t) = tg - 1 ln(1 - r2 t) , r2 = 0 r (t) = tg(t). r2

X3 = 2 R e ( 1 - r 2 z 2 ) + 2iz z X4 = 2 Re i(1 + r2 z 2 ) + 2z z w w w w , .

, (X1 , X2 ). : w = 0. (, r2 ), r2 - . (B) = {v = ur ( |z |2 )}, , r (t) = tg((ts )), sin(n(t)) = nt exp , s > 1, n X3 = 2 R e r2 (t) 2

, r2 = 0. 2n + ir2 n zw +w 4s z w

n +1

, . (, s, n) r2 . (O) , . , dim aut 0 5 , r (0) = 0 , dim aut 0 3. dim aut0 0 5 , r (0) = 0 , dim aut0 0 3. (AB), (A), (B) . (AB) , , . : , ? , . 96


.., ?

aut , , C2 . . , X4 X5 (AB ) , . , .. aut{v = u(|z |2 )} = aut{v = u(-|z |2 )} . (AB) (A) w = 0. (B) , , .. (AB) -, (A) (B) , (O) . . : (O), (A), (B), (AB). , , (AB), (AB) . , , - Rh, ‘, .

4

C

3

C3 , aut ? -, [2] 15. , - , 16 , . .. , 15-. , , . 5 , , , , . .. 5 , ( ) . (.. ), , [7], 8. , [8], 10. ( , ), . , : , 15-. 15 , .. Im w = |z1 |2 + |z2 |
2



Im w = |z1 |2 - |z2 |

2

10. 97


2.

, 8- , v = (z1 z2 + z2 z1 ) + |z1 | ï ï
4

. 3. : (Im w )2 = (Im z1 )2 + (Im z2 ) , 10- . 5- . . § 11 15? § , 10-, ? § , 15- ? , = {Im w = Pm (z1 , z1 , z2 , z2 ) + O (m + 1)}, ï ï (2 7 )

P m 4 , O (m + 1) m + 1 . Q = {Im w = Pm (z1 , z1 , z2 , z2 )} ï ï (27). aut0 0 aut0 Q0 . aut0 Q0 Pm , . , , .. . ., . , , . C2 . , .


[1] Poincare H, Les fonctions analytiques de deux variables et la representation conforme // Rend. Circ. Mat. Palermo, 1907, p. 185í220. [2] Chern S.S., Moser J.K., Real hypersurfaces in complex manifold // Acta Math., 1974, v.133, 3í4, p.219í271. [3] Cartan E., Sur la geometrie pseudo-conforme des hypersurfaces de l'espace de deux variables complexes // Ann. Math. Pura Appl., 1932, v. (4) 11, p.17í90 (Oeuvres II, 2 , 1 2 3 1 í 1 3 0 4 ). 98


.., ?

[4] .., // . , 1991, . 182, 2, . 203í219. [5] .., C2 . - . .., , 1996. [6] Kol lar M., Normal forms for hypersurfaces of finite type in C2 // Mathematical Research Letters, 2005, v. 12, p. 897í910. [7] . ., , C3 // . ., 1999, . 33, 1, c. 68í71. [8] Fels G., Kaup W., Classification of Levi degenerate homogeneous CR-manifolds in dimension 5 // Acta Math., 2008, v. 201, 1, p. 1í82.

99


2.

Lp, 1 p 1
..2, ..
3

Lp , 1 p . .

[1] Lp , 1 < p < . Lp , 1 p . , .

1


§ Lp , 1 p , - 2 - x,
2

:

1 p < ||f ||p =

0

|f (x)| dx

p

1 p

< ,

p = ||f ||p = max |f (x)|;
0x2

§ L0 - f Lp , , p

2

f (x)dx = 0;
0

§ sn (f )- f Lp , 1 s n (f ) =
1

2

f (x + t)
0

sin(n + 1 )t 2 dt; n = 0, 1, 2, ...; t 2 sin 2

( 09-01-00175) ( -32-52-2010.1). 2 , mkpotapov@mail.ru, , - . .. . 3 , simonov-b2002@yandex.ru, , .

100


.., .., ...

§ Vl (f )- - f (x), V 0 (f ) = s 0 (f ), V l (f ) = §f
()

1 sl (f ) + ... + s l

2l-1

(f ) , l = 1, 2, ...;

- f (x) ( > 0);

§ En (f )p - f Lp Tn (x) , n (n = 0, 1, 2, ...)), En (f )p = inf ||f -
Tn

Tn ||p ;
()

§ Wp - f L0 , p f () L0 . p f Lp : (f ) = h
=0

(-1)

f (x + ( - )h),

. = 1 = 0, = = 1, = (-1)..(- +1) 2. ! (f , )p - ,

(f , )p = sup || (f )||p. h
| h |

- f L0 p K (f , , , p) = inf
g W
() p

||f - g ||p + ||g

()

||p .

G (f , ), F (f , ) ,

F (f , ) G(f , ) , F (f , ) C, f , , C G(f , ). F (f , ) G(f , ) G(f , ) F (f , ), F (f , ) G(f , ).

1.1. [2] f L0 , 1 p , n = 0, 1, 2, .... p | | V n (f )| | p | | f | | p . 1.2. [3] > 0.
=0

(-1)



= 0.

1.3. [3] f L0 , g L0 , 1 p , > 0, > 0. p p (a) (f + g ) = (f ) + (g ); h h h (b) (f )) = + (f ); hh h (c) || (f )||p ||f ||p. h 101


2.

1.4. [3] 1 p , > 0, Tn - , n (n N ). (a) h : 0 < |h| n ,
( || Tn ||p n- ||Tn) ||p ; h

(b)
( ||Tn) ||p n || Tn ||p .
n

1.5. [2] f L0 , 1 p , k N , n = 0, 1, .... p ||f - Vn (f )||p En (f )p
k

f,

1 n+1

.
p

1.6. [4] f L0 , 1 p , > 0, m = 0, 1, .... p ||V
2m+
1

(f ) - V2m (f )||p 2

-m

||V

() 2m+

1

(f ) - V

() 2m

(f )| | p .

1.7. [3] Tn (x)- n (n = 1, 2, ...), 1 p .
( ||Tn) ||p (n + 1) ||Tn ||p .

2


2n - 1

2.1. f L0 , 1 p , > 0, n N . p


f,

p

n- ||V

() n

(f )| | p + | | f - V n (f )| | p .

(2.1)

. , h n N || (f )||p || (Vn (f ))||p + || (f - Vn (f ))||p = J1 + J2 . h h h 1.3 (), , J 2 | | f - V n (f )| | p . 1.4 (), h , 0 < |h| J1 n- ||V , h (0 < |h| || (f )||p n- ||V h
() n 2n-1

, ,

(f )| | p .

2n-1

),

() n

(f )| | p + | | f - V n (f )| | p .

(2.1). (2.1). 102


.., .., ...

1.5, A1 = ||f - Vn (f )||p
[]+1

f,

1 n+1

.
p

, , A1
[]+1

f,

2n - 1

.
p

1.3 (b) (), , A1 sup ||h
2n-1

[]+1-

|h|

(f ) ||p h

|h|

sup || (f )||p = h
2n-1



f,

2n - 1

.
p

1.4 (b), A2 = ||V
() n

(f )||p n ||

2n-1

( V n ( f ) )| | p . 2n - 1 2n - 1

1.1, A2 n ||(Vn , ||f - Vn (f )||p + n- ||V
() n
2n-1

(f ))||p n ||

2n-1

(f )||p n



f,

.
p

(f )||p = A1 + n- A2



f,

.
p

(2.1). 2.1 . , 2.1 [4]-[5].

3









-

3.1. f L0 , 1 p , > 0, 0 < . p (f , )p K (f , , , p). . (0, ] n ,
2n-1

(3.1)
2n+1

<

. f L0 , V p

n +1

(f )

() Wp

.
n +1

K (f , , , p) ||f - V ||f - V
n +1

(f )| | p + | | V

() n +1

(f )| | p

2.1, , , K (f , , , p)


(f )||p + (n + 1)- ||V 2n + 1

() n +1

(f )| | p .

f,

p

(f , )p ,

103


2.

(3.1). (3.1). () g Wp . , 1.3 (), (f , )p (f - g , )p + (g , )p = I1 + I2 . 1.3 (), , I1 ||f - g ||p . I2 . (0, ] m , 2m+2 -1 < 2m+1 -1 . I3 =


g,

2m+1 -1

.
p

1.3 () (), I3


g - V2m (g ),
2m (g ) ||p

2
m+1

-1

+
p



V2m (g ),

2
m+1

||g - V

+

sup
|h|

2m+1 -1

|| (V2m (g ))||p = i21 + i22 . h
2m+1 -1

-1

p



1.4 (), 1.1, h , 0 < |h| , i22 2
=m -m

,

||V

() 2m

(g )| | p = 2

-m

||V2m (g

()

)| | p 2

-m

||g

()

||p .
()

1.6 1.1, , i21 ||V2 (g ) - V
=m 2
+1

(g )| | p
()

=m

1 || V2 (g ) - V 2
()

2

+1

(g )

||p



1 || V2 (g 2

)-V

2

+1

(g

) ||p 2

-m

||g

()

||p .

i21 i22 , I3 2
-m

||g

()

||p . ,
p

(g , )p , I1 I2 , I2 2
-m

g,

2

m+1

-1
()

||g

()

||p ||g

||p . ||p .

(f , )p | | f - g | | p + | | g

()

g Wp , (f , )p K (f , , , p), (3.1). 3.1 .

()

, 3.1 [6], [8]. 104


.., .., ...

4


=1

4.1. f L0 , g L0 , 1 p , f (x) p p b sin (1 (2 (3 (4 (5 ) ) ) ) ) x, > 0 , k N , n = 0, 0)p = 0, + g , )p (f , )p + )p (f , t)p , (f ,t)p (f,)p , 0 < t (f , )p ( + 1) (f , > (f , (f (f ,
1 n +1 p

a cos x +

1, .... (g , )p , 0 < < t, < t , )p , > 0,
n +1

(6) En (f )p f , (7 ) (8 ) (9 ) (1 0 )



1 (n+1)


=1

-1

E -1 (f )p ,

(4.1)

(f , )p (f , )p , > 0, (f ,) (f ,)p p , 0 < , 2
-n

||V

() 2n

(f )| | p f ,

1 2n p





2

-

1 |ak | + |bk | f , k )p k

-

=n k

||V

() 2

(f )| | p ,
=k +1

=1

|a | + |b | +

|a | + |b |.

. (1), (2) (3) 1.2 1.3 () . (4). 3.1, K (f , t, , p) K (f , , , p) (f , )p (f , t)p , t t (4). (5) (3) (4). (6). , n 1, n = 2m, n = 2m - 1, m N . En (f )p , J = En (f )p E2m-1 (f )p ||f - Vm (f )||p . 2.1, J


f,

2m - 1

.
p

(4) , , J


f,

1 n+1

,
p

(4.1) . n = 0. En (f )p , A = E0 (f )p ||f - V0 (f )||p ||f - V1 (f )||p + ||V1 (f ) - V0 (f )||p = A1 + A2 . V0 (f ) = S0 (f ) = 0, V1(f ) = S1 (f ) V 2k > (k N ). 105
(2k ) 1

(f ) = (-1)k V1 (f ), A2 = ||V

(2k ) 1

(f )| | p ,


2.

A | | f - V 1 (f )| | p + | | V 2.1,

(2k ) 1

(f )| | p .

A 2k (f , )p . 1.3 () (), (4) , , 2k (f , )p = sup ||2k h
| h | -

( (f ))||p sup || (f )||p = (f , )p (f , 1)p . h h
| h |

(4.1) . (4.1). n = 0, 1, ... m , 2m+1 - 1 n + 1 < 2m+2 - 1. (3) , 2.1, J =


f,

1 n+1

p





f,

2
m+1

-1

p

||f - V

2m (f )||

p

+2

-m

||V

() 2m

(f )||p = j1 + j2 .

1.5, , j1 E2m (f )p . V0 (f ) = 0,
m

A2 = ||V

() 2m

(f )| | p

²=0

||V

() 2²

(f ) - V

() [2²-1 ]

(f )| | p .

1.7,
m

A2

²=0

2² ||V2² (f ) - V

[2²-1 ]

(f )| | p .

1.5, ,
m

A2 , j2 2
m+1

2² E[
²=0

2²-1 ]

(f )p .

m -m ²=0

2² E[

2²-1 ]

(f )p .

J2

-m ²=0

2 E[

²

2²-1 ]

1 (f )p (n + 1)

n +1


=1

-1

E -1 (f )p .

(4.1) . 106


.., .., ...

(7). 1.3 () (), (f , )p = sup || h
| h | -

( (f ))||p sup || (f )||p = (f , )p , h h
| h |

. (8). (0, ] n , n+1 < . (4) , n A= (f , )p n


f,

1 n+1

.
p

(4.1), ,
n +1

A


=1

-1

E -1 (f )p .

, (4.1),
n +1

A


=1

-1





1 f,

n +1

=
p =1



- -1

f , (

1 p 1 )

.

, (4) (3) , , A f , (
1 n +1 n +1 p 1 ) n +1 =1



- -1



f , (

n +1 p ) n +1



(f , )p ,

. (9). (9) (2.1). . (4) , 2.1, J=


f,

1 2n

p





f,

2
n +1

-1

p

2

-n

||V

() 2n

(f )||p + ||f - V2n (f )||p
2
+1

2

-n

||V

() 2n

(f )| | p +

=n

||V2 (f ) - V

(f )| | p .

1.6, , J 2
-n

||V

() 2n

(f )| | p +

=n

2

-

||V

() 2

(f ) - V

() 2 +1

(f )| | p

=n

2

-

||V

() 2

(f )| | p ,

. (10). 1 ak =
2

1 f (x) cos k xdx =

2

f (x) - Tk-1 (x) cos k xdx
0

0

Tk-1 , k - 1, f (x) Lp , , Ek-1 (f )p = ||f - - Tk-1 ||p , |ak | Ek-1 (f )p . 107


2.

(6) (4) , , |ak | , |bk |


f,

1 k

.
p

f,

1 k

.
p

(10). . 4 2.1, J =


f,

1 k

p





f,

2k - 1

p

k

-

||V

() k

(f )| | p + | | f - V k (f )| | p = J 1 + J 2 .

1.5, , J2 Ek (f )p ||f - Sk (f )||p , J1 k
- 2k -1 =1 =k +1

|a | + |b |.

|a | + |b | .

(10). 4.1 . , , 4.1, [3] - [8]. . 3 4 , 2.1 1 f , 2n 1 p f , n p . -


[1] .., .., .., Lp , 1 < p < / . 105- .., .: , 2011, . 6, 1, . 90í110. [2] ., , .: , 1965. [3] Taberski R., Differences, mo duli and derivatives of fractional orders // Comment. Math. Prace Mat., 1976/77, v. 19, 2, p. 389í400. [4] .., .., // , 2008, . 199, 9, .107í146. [5] .., . , , 1985. 108


.., .., ...

[6] Butzer P.L., Duckhoff H., Gorlich E., Steus R.L., Best trigonometric approximation, fractional order derivatives and Lipschitz classes // Can. J. Math., 1977, v. 29, p. 781í 793. [7] .., ., // Publications de L'institut Math. (), 1979, . 26(40), . 215í228. [8] Simonov B., Tichonov S., Sharp Ul'yanov-type inequalities using fractional smo othness // Journal of Approximation Theory, 2010, v. 162, p. 1654í1684.

109


2.

1
..
2

, , . . , . , . .


[1], , . , . , . . , . . , . , . , . , . , . 1 , , , , ,
09-01-00175. , vferdorov@rambler.ru, , - . .. .
2 1

110


..,

. . , , . , . 6 , , , . , . , . , , . , , . , . 1 ( .). B[0, 1] f : [0, 1] R - , , .. . : f (x) g (x) x [0, 1] . [0, 1] . , . U0 0 U1 1 B[0, 1] {rn } = Q[0, 1] [0, 1]. fn,m (x) = 1, x (rn - 1/m, rn + 1/m), fn,m (x) = 0, x (rn - 1/m, rn + 1/m). n N / {fn,m } fn,m 0 (m ) . m1 N, f1,m1 U0 . f1,m1 f2,m f1,m1 (m ) , n m2 N, f1,m1 f2,m U0 , .. gn k =1 fk ,mk U0 n N. {gn } U0 gn 1 (n ) . n N gn U1 . , U0 U1 = , , B[0, 1] . 2 ( ..). K(R) R. : f g , f (x) g (x) x R. . , . 111


2.

, K (R ). 0 (x) = 1, x [-1, 1], 0 (x) = 0, x / 1 [-2, 2], [-2, -1] [1, 2]. n,k (x) n 0 (x) sin k x, n,k (x) 1 (x - n), n,k (x) n,k (x) + n,k (x) n, k N. S = {n,k } K(R), T = k0 {(n,k , n,k )} K(R) ½ K(R) F (, ) = + K(R). F (T ) = S , S K(R), T K(R) ½ K(R) T = T (0, 0). F , F (T ) F (T ) = S = S . , F (0, 0) = 0 F (T ), 0 S . , F , , / K(R).

1



E R, x y , , . x y , x y y x, Z E , z E , z 0. x 0 E K E , .. , Z . K E , . . Z = 0, E , , , : x y y x, x = y . E (. [2, . 26] [3, . 258]). E [x, y ] {z E | x z y }. x y , [x, y ] = . [x, y ] , [-x, x] . A E E , A [x, y ]. e K , [-e, e] E , . 1. E K . e K , K . , [-e, e] , x E > 0, , x [-e, e] || < . , -e x e || < . , e + x K | | < . , x E > 0, , e + x K || < , e + x 0 e - x 0 || < . x [-e, e] || < , , [-e, e] . 112


..,

A E , x, y A [x, y ] A. or(A) x,y A [x, y ], , A, A. A B B E , z = (1 - t)x + ty A, x, y B , x, y A. . , K K , x, y K x + y , x, y . , L E , = L K K . , Z K K . , Z . 1. () M E y E , x y (x y ) x M . () M ( ) M E. sup(M ) ( inf (M )) M E () () M , .. () y E , y z (y z ) () z E M. , M E E () , .. (). , (), () () . 2. , A E () B E E , x B y A, , x y (y x). E0 E1 E1 , x E1 y E0 , , x y (. [4, . 344] [5, . 301]). E0 E1 , E1 . E0 E1 E1 E0 + K1 = E0 - K1 . K1 E1 , E0 + K1 = E0 - K1 E0 . 3. E ( y n N ), x, y E nx x 0. .. E , K E , . E K , ( ) E , K , 113


2.

( ) (. [6, . 38], [3, . 280]). , K - E . E , y 0 x ty t 0 x t1 y , t1 = inf {t 0 | x ty }. , . , nx y n N. z = supn 1 {nx}. z + x = supn 1 {(n + 1)x} z , , x 0. 2. (K ) * K E * K . , ‘x K , (‘x) (x) = 0 K . Z (K ) x K -x K , / / K , (x) = 1 (-x) x (K ) -x (K ) , . . (K ) Z . / / 0 . , = 1.

3. * L K * K . * wrn (x) = , K K , ‘ K , . . ‘(x) 0 x K . , 0 x K , , L . , L , ‘ K , , K .

4. E , x, y E sup(x, y ) inf (x, y ). E , () A E () sup(A) (inf (A)). . E K E1 K1 , f1 : E E1 , : a) f1 , , .. ker(f1 ) = 0 , f1 (K ) K1 f1 (Z ) = Z1 ; b) f1 E E1 , .. () x = sup(A) (y = inf (A)) A E , () f1 (A) E1 f1 (x) = sup f1 (A) (f1 (y ) = inf f1 (A)); c) E0 = f1 (E ) E1 , .. z E1 B , C E , z = inf f1 (B ) = sup f1 (C ). 114


..,

[7, . 130] , .. , E E1 , , .

2



E . x 0 x + x0 x0 x E R x0 E 0 R, E [8, . 446]. , , (, x) x (0, 0), . E , , . 4. E , B0 , . , E . x x + x0 , , x 0 x 0 = 0 , . x0 , x0 E U > 0, x0 U || < , .. U . , , U . , (, x) x > 0 V , x U x V || . V = {x | x V , || }, V V . , V , V U c . , , x x + x0 . U B0 , x0 . U B0 , (, x) x . , V = ²U , ² , 0 < ²|0| < 1, U , 0 x U x V , .. x 0 x . 1. U0 E : 1) U, V U0 U V U0 ; 2) U U0 U V V U0 ; 3) U U0 = 0 U U0 ; 4) U U0 ; 5) U U0 V U0 , , V U , 115


2.

() E , U0 ( U0 ). , , U0 : 6) U U0 V U0 , V U U U0 + x xV, U0 . . E x x + x0 , x E , Ux {V E | V = U + x, U U0 }. max E , A E , : x A U Ux , U A. . (1) . , max . U max , U0 , (2) U U0 . , max E , U0 . , (6). (6) , U U0 V U0 , V U , .. y U , U Uy . U U0 U0 {y U | U Uy } , U0 max . 0 U , U0 = . , y U0 , U - y U0 . (f ) V U0 , , U - y Uz z V . Vy = V + y , Vy Uy U Uz z Vy . U0 Vy U0 . , U0 max , , U max . T E , U U0 . 0 , U (1), (2), (3), (4), (5), (6). U 0 0 U0 , U0 (f ). a , A E , A F F E F . , a . , (, x) x (0, 0). U Ua V {v E | 0 [-1, 1]v U }. V = . , V Ua . 0 F E v V F . , v V F . [-1, 1]v U F F , N F , [-1, 1]v + N U F , F N0 F , [-1, 1]N0 N , .. [-1, 1](v + N0 ) U F . V v + N0 V F . 116


..,

, a Ua 0 E . T max c . min T . min T . , min (1), (2), (3), (4), (5), (6) . U0 (6), min . 5. E , . A E , U U0 > 0, A U || < . 1 2 E , .. 1 2 , 2 1 . , (E , 1 ) , (E , 2 ). 5. A E E , {xn } A {n }, lim n = 0, lim n xn = 0 . , A E U U0 , > 0, , A E || < . N N, |n | < n N . , {xn } A, n xn U n N. , , A E . 1 U U0 , n N n A U . 1 {xn } A, n xn U . , / 1 { n xn } E . 1. A E E , {xn } A . A E E , {xn } A {n }, lim n = 0, E lim n xn = 0 . E max min , (. ). max , min . st(A) y A [0, y ] = K (A - K ) A E E . A E , st(A) A. , st(A B ) = st(A) st(B ). A E sth(A) (A + Z ) st(A). 117


2.

6. E K E , Z K . E , B = {Ui }iI E E /Z , i I j I , st(Uj ) Ui . , Z = 0, .. K E , E . E , () E () [9, . 28]. 6. E : (a) C = {Vj }j J E {sth(Vj ) | j J } E ; (b) D = {Wj }j J E , st(Wj ) Wj j J . (c) S = {si }iI T = {ti }iI , , 0 si ti i I , T lim T = 0 S lim S = 0.

B = {Ui }iI E , , C = {Vj }j J E . i, j I k J , , Vk + Z Ui Uj . st(Vk ) st(Uj ) Ui , , sth(Vk ) Ui . (a). Wj sth(Vj ), (a) (b). , (b), S = {si }iI T = {ti }iI 0 si ti i I T lim T = 0. j J k I , , ti Wj i k . si st(Wj ) Wj i k . S lim S = 0. , E . B = {Ui }iI E i I , j I st(Uj ) Ui . tj Uj sj st(Uj ) \ Ui , 0 sj tj j I . , I B. T = {tj }j I lim T = 0, S = {sj }j I . 2. E A E E , sth(A) E . , [x, y ] E . , U E V U U E . > 0, A V || < . sth(A) = sgn()sth(||A) sgn()sth(V ) = 118


..,

sgn()V = V U , , sth(A) E . , E , sth(S ) S E E . 3. E K E . , , E , K E . D = {Vj }j J E , . K (V j - K ) st(Vj ) V j . , D = {V j }j J K E Z . , . 7. () E , K E , () , . , E . E (E ) ( ) , .. () E . E (E ) : , (x) (x) x K , K = -K (K = -K ) () , K E (E ) E , E ( E , E ). , E (E ) () . E o = K - K E E. E , , , E o E , .. E o E . , E E o E , E o . E , E o E . 7. E : a) K E ( ) E o E ; b) E (E , E o ) . 119


2.

(a) , K M M , M [3, . 34]. , dimR M = dimR (M K ) = n. e1 , . . . , en M K . y n 0, e1 , . . . , en M K . x M K , z = (1 - 1/n)x + y /n M K . -nx (y - x) , , x 0. (a), . E o E , E = E o 6 [9, . 33] E . , E = E o , , E o E . 4. E . K E , E . 2 spR (K ) = K - K E o = K - K spR (K ) = Z . spR (K ) * E = Z . , spR (K ) , , E o , E . 8. D E ( ) E , x, y D z D , x, y z ( x, y z ). D E E x0 E , U E y D , Dy U + x0 , Dy {x D | x y } D , . E lim D D E , sup M inf M M E , , Z , (mo d Z ). 8. D E E , K E , x0 = lim D , sup D = x0 (mo d Z ). , x1 E D , y x x1 x Dy y D . Dy x0 D y [16, . 104] , y x0 x1 . x0 = sup D (mo d Z ). E [10, . 166], C (X ) X , , . 120


..,

, C (X ). . 2. , E . , D E (E , E ), E . . 8, , D lim D = 0, D K . , U E , D . [9, . 29], U . y D U , Dy {x D | 0 x y } U . D U = . , (D + K ) U = . , y (x + K ) U x D , [0, y ] U U . 0 x y , x U . , D + K . x, y D , z D , z x z y . (x + K ) (y + K ) z + K , . . D + K , , , , . , D + K U , D . 5. , E D , sup D = x0 . , sup f (D ) = f (x0 ) f K , lim f (D ) = f (x0 ) A E . , 6 [9, . 33] E = K - K . D x0 sup f (D ) = f (x0 ) f K . 2 D x0 E . A E , > 0 U E , f (U ) < f A. D x0 Dy , f (x0 ) - f (Dy ) < f A. , lim f (D ) = f (x0 ) A E . , 2. 8. 9. . ) D E x0 = sup D x0 = lim D E ( E .

sup D = sup Dx x D , , Dx . z E D , 121


2.

Dx [x, z ], , , 1 [9, . 29] Dx E . f (Dx ) R f K . 6 [9, . 33] E E = K - K , Dx E , E . E [3, . 183], x0 = lim Dx . 8, x0 = sup Dx , , 2 , x0 = lim Dx E .

3



9. S E E , Sk = {si }i k E . S (), S : I E , I , ( ), .. si sj (si sj ) i j . , A A S ( A S ) S , a A ia I , si a (si a) i ia . S ( o-) E lim S = x, A , S , B , S , , inf A = sup B = x (mo d Z ) (. [11, . 55] [4, . 42], [5, . 41] [6, . 39]). S ( r -) E lim S = x, a K R+ , inf = 0, , a S - x, -a S - x. a K S , r - (. [1, . 473], [4, . 167], [5, . 82]). , E . S E o- x E , D K , inf (D ) = 0, , D S - x -D S - x. , D = A - B , A B o-, inf (D ) = inf (A - x) + inf (x - B ) = 0. A = x + D B = x - D , S . , E , o- S x E , D , inf (D ) = 0, S - x. 122

S = {si }iI E S : I E , I S (i) = si E i I (. [11, . 40] [1, . 279]). S S E .


..,

E , lim S lim S S = {si }iI , lim S
k I

inf {sup Sk } ,

lim S

sup{inf Sk } .
k I

Sk = {si }i k S . , S E o- , E limS = limS (mo d Z ). 10. () S E x E , , () sup S = x (inf S = x) (mo d Z ). S = {si | i I } s0 , A , S , B , S , , inf A = sup B = x (mo d Z ). a A ia I , , si a i ia . i I j i, ia , si sj a i I a A. , si s0 . , x si i I , , , x b b B , x s0 . , , x = sup S (mo d Z ). . 11. S = {si }iI R = {rj }j J x y , S R ( R S ), x y . , (mo d Z ). A R B S , inf A = y sup B = x (mo d Z ). a A ja J , rj a j ja , b B ib I , si b i ib . , jb J , sib rj j jb . , a A b B ib I jb J , b sib rjb a. , b a a A b B . x y . 12. E S1 , S2 E R, lim S1 + lim S2 = lim (S1 + S2 ) , lim S1 = lim(S1 ) (mo d Z ) .

A1 A2 S1 = {s11 | i1 I1 } i S2 = {s22 | i2 I2 } , B1 B2 i S1 S2 , inf A1 = sup B1 inf A2 = sup B2 . A1 + A2 = {a1 + a2 | (a1 , a2 ) A1 ½ A2 } S1 + S2 = {s11 + s22 | (i1 , i2 ) I1 ½ I2 }, i i B1 + B2 = {b1 + b1 | (b1 , b2 ) B1 ½ B2 } S1 + S2 , (mo d Z ) inf (A1 + A2 ) =
(a1 ,a2 )(A1 ½A2 )

inf

(a1 + a2 ) = inf 123

a1 A1 a2 A

inf (a1 + a2 ) =
2


2.

sup sup (b1 + b2 ) =
b1 B1 b2 B2

sup
(b1 ,b2 )B1 ½B2

(b1 + b2 ) = sup(B1 + B2 ) .

I1 ½ I2 , .. (i1 , i2 ) (j1 , j2 ), i1 j1 i2 j2 . > 0 A1 S1 , B1 S1 , inf A1 = sup B1 (mo d Z ). A1 S1 , B1 S1 , inf (A1 ) = sup(B1 ) (mo d Z ). < 0. , . 13. E () sup A (inf A) A R, sup(Ax) = (sup A)x (inf (Ax) = (inf A)x) (mo d Z ) x K . , A R a0 = sup A. ax a0 x a A x K . , ax y a A, n N a A, a0 - 1/n a, , , (a0 - 1/n)x y . n(a0 x-y ) x n N. , a0 x y , . . sup(Ax) = a0 x. , . 6. E r - o- . 3. C2 : (z1 , z2 ) (w1 , w2 ), Re z1 < Re w1 , Re z1 = Re w1 Re z2 2 C , . (1/n, 0) (0, 1) n N, inf (1/n, 0) > 0 , 13 . Re w2 . ,

14. E , K E . R S E , (lim )(lim S ) = lim(S ) (mo d Z ). K , .. E = K - K , , S c > 0 r K . , lim = 0 , lim S = s0 , lim S = lim(c - )(r - S ) - cr + 0 r + cs0 , , R+ S K . , 12 13. , U = {j | j J }, V , inf U = sup V = 0 . A S = {si | i I }, B S , inf A = sup B = s0 (mo d Z ). , V B , . , , , U A 124


..,

S = {j si | (j, i) J ½ I }, V B S . , 13 : inf (U A) =
(u,a)U ½A

inf

(ua) = inf inf (ua) = inf U inf A = 0 s0 =
uU aA

= sup V sup B = sup sup(v b) =
vV bB

sup
(v,b)V ½B

(v b) = sup(V B )

(mo d Z ) .

, lim S = 0 s0 (mo d Z ). 10. , E , S E lim S E , : a) lim S S E (mo d Z ) Z E ; b) S = {si | i I }, si = s0 E i I , lim S = s0 (mo d Z ); c) T S S E lim T = lim S (mo d Z ). E . , , E , .. d) lim(S ) = (lim )(lim S ) (mo d Z ) S E R; e) lim(S + T ) = lim S + lim T (mo d Z ) S, T E , E . , , (d) (e) . 3. E , , o- (r -) , .. E o- (r -) . . 12 14 o- (r -) E , .. E o- (r -) . 11 K E o- (r -).

11. A E o- (r -) E , S E , o- (r -) x A, , x A [1, . 289]. A E o- (r -) A E , o- (r -) , A. B = E \ A o- (r -) A E o- (r -) E . o E , o- , ( o-). r , r - , ( r ). 125


2.

o E c , [5, . 46] [6, . 40]. [12] [13], r . [1, .475] [14, . 419] [15], . . 4. o (r ) E E , o- (r ) E . . o ( r ). A = Ai o- Ai o , o- S A lim S Ai , Ai , , , lim S A, .. o- E . A = A1 A2 A1 , A2 E . S A S1 A1 , lim S = lim S1 A1 A A1 . S , A2 , lim S A2 A A2 . A. , o E , 11 K E [x, y ] E o . , x0 E Z K , [x, x] = Z . , S E s0 = lim S o . U o s0 , S1 S S1 E \ U . S1 S , s0 , , E \ U . , . , o E . , S E , .. , . [16, . 97] , , o . r . 12. T = {tj }j J S = {si }iI T S , : J I , tj = s(j ) j J , i I k J , (j ) i j k. S E E ( ) x E , T S R T , x = lim T . , o, o-, r -, r . 126


..,

3 . X , Y , Z f : X ½ Y Z , [1, . 288]. , S = {si }iI T = {tj }j J x X y Y . G = {gl }lL F = {f (si , tj )}(i,j )I ½J , G(l) = gl = f (si(l) , tj (l) ) l L. S1 = {si(l) }lL S , T1 = {tj (l) }lL T , S2 = {si(ln ) }nN T2 = {tj (lm ) }mM , x y . H = {hn,m }(n,m)N ½M , H (n, m) = hn,m = f (si(ln ) , tj (lm ) ) (n, m) N ½ M , G f f (x, y ).

7. , o- E o-

E , (r -) , .. (r -) .

15. E S E o (r -) x E o (r ) x. , 4 , S , x o (r ), S . , .. S o (r ). Ox o (r ) x R S , , Ox . , R , x o (r ), . , S E , o , , , * . , * E , , A E o . , A A E S A. A . A, , A S < A . A < A , , , E . o , , . , (. [17, . 280], [18, . 802], [19, . 360]).

4



K - E (.. ) . . , (. [4, . 163]), o- , .. 127


2.

{xnm }(n,m)N½N , o- limn limm xnm = x, {xn,mn }nN , rnr o- limn xn mn = x. o- . , o- . . 13. E . C E ( ), , .. . , E , {Si }iI Si = {si,j }j Ji , limi lim Si = x, S : I ½ iI Ji E x, S (i, ) si,(i) iI Ji . E , . o- r - E o- r -, o- r - E o- r -. . , , S x. S x, U x T , , U . , T , x. , [11, . 43]. , , , , . . , . 5. E o- (o) , {Ai }iI Ai K , inf Ai = 0, a(i) A : I ½ iI Ai E o- (o-) , A(i, a) . a iI Ai . . o- (o-) . {Ai }iI Ai K , inf Ai = 0. 10 Ai o- , , o- . o- (o-) limi lim Ai = 0 . , A : I ½ iI Ai E , A(i, a) a(i), i I a iI Ai , o- (o-) . . {Si }iI o- Si = {sij }j Ji , o- lim Si = xi lim xi = x. Ai 128


..,

, inf Ai = 0, Ai Si - xi -Ai Si - xi . A(i, a) = a(i), i I a iI Ai , o- . A S (i, ) - x -A S (i, ) - x, i I iI Ji , A S - x -A S - x. , S o- E o- lim S = x. o-. T = {tl }lL S (i, ). : L I ½ iI Ji , tl = s(l) . (l) = (il , l ), l = {l (i)}iI . I = {i I | i = il , (l) = (il , l ) , l L} , Ji = {j Ji | j = l (i) , (l) = (il , l ) , l L} . i I k L, il i l k , j Ji ki Ji , l (i) j l ki . , I I I Ji Ji Ji i I . , Si = {si,l (i) }lL Si . o- Si Ri = {rni }ni Ni Si , o- lim Ri = xi i I , rni = si,ln ni Ni . {yn }nN i {xil }lL , o- lim yn = x, yn = xiln n N . R {Riln }nN o- qundhrq x T . o- (o-) 5 r - (r -).

8. E r - (r ) , {ai }iI K {i }iI i R+ , inf i = 0, A : I ½ iI i E r (r -) , A(i, ) (i)ai iI i . 16 ( ). o- (o-) E o lim S = 0, = T (n, ) = n s(n) , n N ) lim T = 0. o- (o-) . S = {si }iI - (o-) {n } R T : N ½ I N E , I N , o- (o-

17 ( ). o (o-) E , S = {si }iI E {n }nN R, lim n = 0, T : I ½ NI E , T (i, ) = (i) si , i I NI , o- (o) lim T = 0. 129

, n N o- (o-) lim n S = 0 , o- (o-) T : N ½ I N E , T (n, ) = n s(n) , n N a AN , o- (o-) lim T = 0.


2.

18 ( ). S E o- (o-) E o- (o-) lim S = x0 , r - (r -) lim S = x0 . A E S - x0 , -A S - x0 , inf A = 0. 16 B : N ½ AN E , B (n, a) = n an , a AN , o- lim B = 0. , B E , .. B(m,b) , m N b AN , , B(m,b) [0, c] 1 c n ma b. c K . an n a = {an } A, S - x0 , 1 1 -a S - x0 . {- n c} S - x0 , {- n c} S - x0 . , o- x0 r - x0 . . o- o-. 9. o- (o-) E r - (r -).

o- (o-) lim n si = 0 i I , o- (o-) T : N ½ NI E , T (i, ) = (i) si , i I NI , o- (o-) lim T = 0.

18 y , o- (o-) E o- (o-) S = {si }iI lim S = x0 xn = sin , r - (r -) lim xin = x0 . o- (o-) o (o-) E . , 18 o- (o-), 17. r (r -) , , o- (o) r - (r -) , . 19 ( ). o- (o-) E {Si }iI o- (o-) Si = {sij }j Ji E , r - (r -) . , 18, i I o- 1 Si = {sij }iJi xi E ci K , , { n ci } 1 Si - xi , {- n ci } Si - xi . 17 , T : N ½ NI E , T (i, ) = 1i) ci , (i, ) I ½ NI , o- lim T = 0. ( , T E , .. T(i,) , , T(i,) [0, c] c K , (i, ) I ½ NI . > 0 li = li (, ) Ji , sij - xi (i) [-ci , ci ] j li . sij - xi [-c, c] j li . 130


..,

, 19 17 18 , o- (o-). , .. o- (o-) E {Si }iI o- (o-) Si , E o- (o-). X , U x X V x, U . . . [11, . 44] , , {Si }iI Si = {sij }j Ji xi = lim Si , lim T = x, T (i, ) = si,(i) , (i, ) I ½ iI Ji , lim xi = x. . . [17, . 280] : E , , A E , S , x A, , x A. . 6. E , K E . E o- (r -) , o- (r ) o- o (r - r ). . , E o- (r -), E , . o- (r ) . , o- (r -) o- o (r - r ). , o- (r ) o- o (r - r ), o- o (r - r ) , , E o- (r - ). 10. o- (r -) E , K E , . , E o (r -), 6 o- (r -) o- (r -). , 7 E o- (r -) . , o- (r -). 131


2.

5



E . b E , . , , [15]. b . Bb b 0 U E , [a, b] E , .. [a, b] E > 0, [a, b] U || < . E b , .. b . b E , E b E . b , . , U E , b , , , . , b . 7. E b r , .. b = r . . S = {si }iI r - . a K = {i}iI R+ , , si i [-a, a]. U E b , > 0, [-a, a] U || < . k I , 0 < i < i k . si U i k . , S b . r , b r . , , [a, b] E r . {xn } [a, b]. , , a b K - K . a = a1 - a2 , b = b1 - b2 , a1 , a2 , b1 , b2 K . c = a1 + a2 + b1 + b2 , {xn } [-c, c] , , n R+ , lim n = 0, n xn n [-c, c] r - lim n xn = 0 . lim n xn = 0 r , , [a, b] r . b , r . 20. f : E R b , . , f E b , > 0 U Ub , |f (U )| < . [a, b] E 0 > 0, [a, b] U || . , f ([a, b]) < / . , e f , f -1 (-1, 1) , . , f -1 (-1, 1) b . 132


..,

21. E E = n=1 Ei Ei , i E Ei. , P1 : E E1 . , P1 P1 ([a, b]) = [P1 (a), P1 (b)] [a, b]. , U E - , P1 1(U ). , P1 . , [a1 , b1 ] E1 , [a1 , b1 ] ½ {0} E . U E , P1 (U ) [a1 , b1 ]. , E . . 14. {xn } E E , o- sn n k =1 xk , n N. , {xn } E 1 , e K n R+ , n =1 n < xn n [-e, e]. 8. E e K . (E , b ) , E . (E , b ) , {xn } K 1 . . pe [-e, e]. e , pe , b , [-e, e] b . , e b , [-e, e] [x, y ] , , b . , E e = b . pe , E , , . , , [9, . 29] b . K , , e [0, 2e] , K . x K , y = (1 - 1/n)x + e/n K [3, . 53]. -nx (e - x) , x 0. {xn } K 1 , .. xn n e, n n n > 0 n=1 n < . pe (sn - sm ) n=m+1 k , sn k =1 xk . , E , o- s = lim sn (mo d Z ). , {xn }, 133


2.

, pe (xn+1 - xn ) n . xn+1 - xn n [-e, e] zn n e - (xn+1 - xn ) K . , {xn }, 1 zk . zn 2n e, k= o- u = lim un 8 u = sup{un } K , un = n=1 zk . k
m+n

0 lim pe (u - um ) b . 2

u - um = sup
n

zk
k =m+1

2

k =m+1

k e ,

k =m+1

k , ,

11. E , e K , E b = E o . , E 4. 8 (E , pe ) , [9, . 30], E b = K - K = E o . 12. E e K . E b , E . , K E , E . E b , 8 E . , E , 8 E b . , E . , [-e, e] E , , , E c b . C (X ), L (X ) , , E , K . 1 e K E , , e K b . , , , , , , p < [20, . 771]. Lp (X ) 1 b 8 . E , K E , e K E
e

{x E | n N | -ne

x

ne} =

n =1

n[-e, e]

, b . Ee [-e, e]. {Ee | e K } . , e1 , e2 K c > 0, , e1 ce2 , Ee1 Ee2 . 134


..,

E , , E , , Ee E , e K , . 9. E K E , C K K . (E , b ), b , , {Ee | e C }. . , b , Ee E . U b , [-e, e], e C . U Ee Ee , , . , b . , C K [x, y ] E n[-e, e], e C n N. [x, y ] . , , , b . 13. E , K . (E , b ), b , . 9 . b U E , U = eC e [-e, e], e R+ . E 6 (b) , st(U ) U . x st(U ) = K (U - K ), y U K e C , 0 x y e e. x U . , b E , E o E . , E . , , , . . Ee1 Ee2 Ee1 , , Ee2 . , , E = Lp (X ), p > 0, , .. f g , f (x) g (x) . . Y X . 13 , Lp (X ) , , b . , 0 f g, f , g Y . Ef Eg . 22. E . : a) b ; 135


2.

b) E , ; (b) , 13 b . , 2 E , , E . b , , b E .


[1] ., . : , 1984. [2] ., . .: , 1959. [3] ., . .: , 1971. [4] . ., . ., . ., . .-.: , 1950. [5] . ., . .: , 1961. [6] ., . , . .: , 1967. [7] . ., , // . 105- . . . . , 2011, . 6, 1, . 126í142. [8] Klee V.L., Convex sets in linear spaces // Duke Math. J., 1951, v. 18, p. 443í466. [9] . ., // . . 1, . , 2008, 4, . 26í36 [10] ., . .: , 1966. [11] Birkhoff G., Mo ore-Smith convergence in general topology // Ann. of Math., 1937, v. 38, 1, p. 39í56. [12] Mo ore E.H., On the faundations of the theory of linear integral equations // Bull. Am. Math. So c., 1912, v. 18, p. 334í362. [13] K . ., // . ., 1937, v. 2, 44, . 121í168. [14] Gordon H., Relative uniform convergence // Math. Ann., 1964, v. 153, p. 418í427. [15] Namioka I., Partially ordered linear topological spaces // Memors Am. Math. So c., 1957, v. 24, Am. Math. So c. Providence. [16] ., . .: , 1981. 136


..,

[17] Kel ley J., Convergence in topology // Duke Math. J., 1950, v. 17, p. 277í283. [18] .., . : , 1951. II. [19] . ., . : , 1974. [20] Klee V.L., The support property of a convex set in linear normed space // Duke Math. J., 1948, v. 15, p. 767í772.

137


3.

3



138


..

..
.. . . , ( ) . . , . . , . , . .. : , , . , , - . - . , - , .. , . ( ), , , , , . , , . . .. . , , , , . , .. , ; , , , . ... .. , .

139


3.

, , 1
..2, ..
3

, , . . , , . , , . .


.. , [1, 2]. , , , , .. . , , .. . , , , . , .. , . , ( ), . . , .
, 11-01-00322. , m1shok7@mail.ru, , - . .. . 3 , kuleshov@mech.math.msu.su, akule@pisem.net, , - . .. .
2 1

140


.., .., , ,

. , , . .

1



R. , , (. 1). . 2.

. 1: , . [3]-[6] í , 2 = R. , . . , , . , .

2


. .

. í , , , : d = k s , dt d = -k s + ös , dt 141 d = -ös . dt

(1 )


3.

s í , k = k (s) í ( ), ö = ö(s) í . , e1 , e2 , e3 dei = [ ½ ei ] , dt i = 1, 2 , 3 . (2 )

í . (1) (2); ( ) : = k s + ös . , í , ö = 0 = k s . (3 )

O xy , . - : r = r (s) = x(s)ex + y (s)ey , (s) = dx dy dr = ex + ey ds ds ds

ex O x. (s) í , O x O y cos sin . (s) = cos ex + sin ey = dx dy ex + ey , ds ds (4 )

dx dy = cos , = sin . ds ds , d d = (- sin ex + cos ey ) = k (s) , ds ds

d = k (s ). (5 ) ds , k = k (s) x = x(s), y = y (s). . . : , , . , 142




.., .., , ,

í . . . [7] :


=



+



. , s

. s K (s), , k (s). , : s = s(t).

1. : = (K - k ) sez , ez í , . . . O xy z , í A z P ez P (. 2). O xy z P ez P , , = K (s)sez (. (3)). A z P ,, , = k sez . í , , , . ,


(6 )

=



-



= K (s)sez - k (s)sez = (K - k )sez

. . , . , A B ( , ). A k (s) A : s = s(t). 2. K (s) A K = k cos , í . 143 (7 )


3.

. 2: . . : O ex ey ez í ez , , ; A í A A1 1 ez í (. 3). , A B , , - AB . : = 1 1 + 1 1 . , =


(8 )

+



,

í O ex ey ez , í A , í A O ex ey ez . A , (. (3), (6)):


= -k se .




=

1

+

1

,

1 í A A1 1 ez , 1 í A 1 ez O ex ey ez . 1 , 1 = 1 , í ez ; 1 í A1 1 ez , 1 = K sez , K í . :

= K sez + 1 - k s . 144


.., .., , ,

. 3: 2. , = -1 sin + 1 cos : = 1 + k s sin 1 - k s cos ez + K sez . (8) , , ez K - k cos = 0 .

3

,

. , , . Gx1 x2 x3 , . Gx1 x2 x3 G , Gx2 , . Gx3 , Gx1 . e1 , e2 , e3 (. 4). A B í . A Gx2 C1 A. , s í : s = R. B Gx2 C2 B (. 4). - A : - GA = r1 = R sin e1 - ( + R cos ) e2 , - B : - - GB = r2 = ( + R cos ) e2 - R sin e3 . 145


3.

. 4: .
r2 - r1 , (r1 ) (r2 ) . :

-R sin 2 + R cos + R cos -R sin R cos R sin 0 -R sin -R cos 0

= 0.

cos = - R cos . R + 2 cos (9 )

p = /R, (9) : cos cos = - . (1 0 ) 1 + 2p cos , p p (0, +). (10), - B : - - GB = R p + (2p2 - 1) cos 1 + 2p cos e2 - R (1 + (2p - 1) cos ) (1 + (2p + 1) cos ) e3 . 1 + 2p cos (1 1 )

, (11) , cos 1 cos - ,1 , ( 2p + 1 ) : - arccos - 1 (2 p + 1 ) arccos - 146 1 (2 p + 1 ) ,


.., .., , ,

- arccos -

1 (2 p + 1 )

arccos -

1 ( 2p + 1 )

.

, . . , , Gx1 x2 x3 . X , Y , Z í Gx1 x2 x3 . : X - R sin -R sin R cos Y + R (cos + p) R 2pR (1 + 2p cos + cos2 ) - 1 + 2p cos R sin Z (1 + (2p - 1) cos ) (1 + (2p + 1) cos ) =0 1 + 2p cos 0

, , - sin X + cos Y + n=- sin sin + (1 + 2p cos )
2 2

(1 + (2p - 1) cos ) (1 + (2p + 1) cos )Z + R (1 + p cos ) = 0. cos sin + (1 + 2p cos ) e3 .
2 2

e1 +

e2 +

+

(1 + (2p - 1) cos ) (1 + (2p + 1) cos ) sin + (1 + 2p cos )
2 2

. , : cos = (n § e3 ) = (1 + (2p - 1) cos ) (1 + (2p + 1) cos ) sin + (1 + 2p cos )
2 2

.

k = 1/R. , (7), A : K= (1 + (2p - 1) cos ) (1 + (2p + 1) cos ) R sin + (1 + 2p cos )
2 2

.

, K A . O xy z . O = 0. O x = 0, O z . O y O x O z . í A O x. (5) d d = = K ( ) , ds R d d = d 147 (1 + (2p - 1) cos ) (1 + (2p + 1) cos ) sin + (1 + 2p cos )
2 2

.


3.

= (). (4), xA = xA () yA = yA (). , A . , = () , (1 + (2p - 1) cos ) (1 + (2p + 1) cos ) d sin2 + (1 + 2p cos ) p A B p. . 5-6 p = 1. í
2

. A B p = 1/2 A, B.

. 5: p = 1/2.

. 6: p = 1. , p (12) . , p = 1/2 (.. ) (12) (. [5, 6]) = 2 arcsin 2 sin 2 3 - arcsin sin 2 3 cos
2

,

148


.., .., , ,

2R 3 xA () = 9

sin 2 arcsin sin + arcsin 2 + 2 sin 1 + 2 cos , 2 2 3 3 cos 2 2R 3 2 2 8R 3 2 y A ( ) = sin - ln cos ,- << . 9 2 9 2 3 3

B : 2R 3 xB () = 9 arcsin 2 sin + arcsin 2 3 sin 2 3 cos - 2 sin 2 1 + 2 cos , (1 + cos )

2

7R 3 2R 3 2R 3 2 2 y B ( ) = - + ln cos ,- << . 2 9 9 2 3 3 9 cos 2 p = 1/ 2 (12) . . . - - GA A n , V , . , - V = M g zG = -M g GA § n = M g R (1 + p cos ) sin2 + (1 + 2p cos )
2

.

. V : V = M gR (2p2 - 1) sin cos .

sin2 + (1 + 2p cos )

2 3/2

, : = 0 = /2. V , , . = 0 : V
=0

=

M g R (2 p2 - 1 ) . (1 + 2p)3

, = 0 , 2p2 - 1 > 0 2p2 - 1 < 0. , = /2 : V , 2 p2 - 1 > 0 . 2p2 - 1 = 0, .. p = 1 .
= /2

M gR = - 2 p2 - 1 22

/ 2 .

= /2 , 2p2 - 1 < 0 ,

149


3.

4



, , . .


[1] .., 5 . .4. . .-.: , 1948. 255 . [2] .., . .: , 1955. 929 . [3] Schatz P., Rhythmusforschung und Technik. Verlag Freies Geistesleben. Stuttgart, 1998. 196 p. [4] Dirnbèk H., Stachel H. The Development of the Oloid // Journal of Geometry and Graphics, o 1997, v. 1, p. 105í118. [5] Kuleshov A.S., Hubbard M., Peterson D.L., Gede G., On the motion of the Oloid toy // Pro ceedings of XXXIX International Summer Scho ol-Conference APM 2011. SaintPetersburg: IPME. P. 275í282. [6] .., // . .. , 2011, 4, 2, c. 191í192. [7] .., .., .., .., . .: , 2010. 430 .

150


.., .., ..,


..1, ..2, ..
3

, . , , , . , . , , , .


. , . [1]-[4]. .. , . , , . , , , , , . , , , , - . , .
, samson@imec.msu.ru, , . . . . . 2 , seliutski@imec.msu.ru, , . . . . . 3 , alexey_mih@inbox.ru, ..-..
1

151


3.

: ; . , .

1

.

, . .

1. 1

.

, n- . Oi (i = 0, ..., n), O0 On . , - , O0 O1 On-1 On O0 On . O0 xy , O0 On (. 1). , F, Ok .

. 1: : s O0 On , r i- , i i- Ox. , , : n di cos i = s
i=1 n i

- O0 Oi , d

i



(1 )

di sin i = 0

i=1

s . (1) (1 , ..., n ) , 152


.., .., ..,

(n - 2). P:(1 , ..., n ) . : G = 0, 1 d1 sin 1 ... dn sin n G= , = ... d1 cos 1 ... dn cos n n rank (G) P rank (G) = 1, O0 O1 O2 ...On = 2, T Rn-2 . rank (G) = 1 . , , i = m i , m i Z, i 1, ..., n, ( O0 On ).

1. 2


, (2 )

, rk = 0. 1. P:(1 , ..., n ) M rk = 0

i = 1 + m i , m i Z, i = 1, ..., k j = k+1 + m j , m j Z, j = k + 1, ..., n

(2) , , (. 2). Ok F .

. 2: . :
k k

xk , yk rk : xk = (2) , rk O Ok :
k k i=1

di cos i , yk =
i=1

di sin i .

xk = - yk =

i=1 k

di sin i i = -
k

i=1

(-1)mi di
i

i

sin cos

1

(3 ) (4 )

di cos i i =
i=1 i=1

(-1)mi di

1

153


3.

(2) , rk Ok On :
n n

xk =
i=k +1 n

di sin i i =
i=k +1 n

(-1)mi di

i

sin

k +1

(5 ) (6 )

yk = -

i=k +1

di cos i i = -

i=k +1

(-1)mi di

i

cos

k +1

, , rk = 0. k + 1, n: sin(i - j ) = 0. di

P , 1 = k+1 + m , m Z . : i 1, k , j , , (2): sin i i + dj sin j j = - dv sin v
v=i,j v

, sin(1 - k+1 ) = 1, k + 1). : rk = 0
k i=2 n

di cos i i + dj cos j j = -

(7 ) dv cos v
v

v=i,j

i , j . 0. v (v =

d1 sin 1 1 + ... + dk sin k k = 0 d1 cos 1 1 + ... + dk cos k k = 0 dk+1 sin k+1 k+1 + ... + dn sin n n = 0 dk+1 cos k+1 k+1 + ... + dn cos n n = 0



(8 )

di sin(i - 1 ) i = 0 di sin(i -
k +1

) i = 0



i = 1 + m i , m i Z, i = 1, ..., k (9 ) j = k+1 + m j , m j Z, j = k + 1, ..., n

i=k +2

, . 1. (2) k = 1 (k = n - 1), , F O1 (On-1), O O1 (On-1On ).

1. 3

..

.. [4, 5]. - , O P , P A AB . , B , PA A (. 3). . , V O B . , 154


.., .., ..,

. 3: .. Fn , A ( ). : s O B , d1 , d2 , d3 , r - OA, 1 , 2 , 3 O x. : Fn r = 0 , : Fn = {-Fn sin ( + 3 ) , Fn cos ( + 3 )} = 2 - 3 - + r = {d3 sin 3 3 , -d3 cos 3 3 } (1 1 ) , (10) : Fn d3 cos 3 = 0 (1 0 )

, (11) . ) cos = 0, .. AB , , Fn . ) Fn = 0. , , , , . 2 = . ) 3 = 0. , . sin (1 - 2 ) = 0, 1 = 2 + k , k Z . , O P P A . : 1) 1 = 2 + 2 k , k Z ( OP PA , . 4) 2) 1 = 2 + + 2 k , k Z ( OP PA , . 4) 155


3.

. 4: , , (), , (.. s < d2 + d3 - d1 ). , , .

2



, P1 , . . . , PN fi (x1 , y1 , z1 , . . . , xN , yN , zN , l) = 0, i = 1, . . . , k , k < 3N , (1 2 )

l (xj , yj , zj - rj Pj ). l (12) R3N Q(l) n = 3N - k . N Q(l) N fi fi fi xj + yj + zj = 0, i = 1, . . . , k . (1 3 ) xj yj zj j =1 (13) k n. q1 , . . . , qn . , l = l M Q(l ) , (13) k - 1, .. 1 , . . . , k-1 , : fk = xj
k -1 i=1

fi fk i , = xj yj

k -1 i=1

fi fk i , = yj zj

k -1 i=1



i

fi , j = 1, . . . , N zj

(1 4 )

, , , k + 3N 156


.., .., ..,

(12,14) k + 3N (x1 , y1 , z1 , . . . , xN , yN , zN , l, 1 , . . . , k-1) . (12) , q1 , . . . , qn+1 . , l = 0, M q1 = 0, . . . , qn+1 = 0. qk : (q1 , . . . , qn+1 , l) = 0, (13) :
n +1

(1 5 )

i=1

qi = 0. qi

(1 6 )

, M , .. q1 = 0, . . . , qn 1 = 0, , (16) qi . +

, = 0 i j = 0, [4], q1 , . . . , qn+1 l qq , M (15) :
2 2 2 2 q1 + . . . + qm - qm+1 - . . . - qn +1

2

- l = 0.

(1 7 )

Q(l) M q1 , . . . , qn+1 l = 0 [6]. n = 1 [1,2]. , l = 0, , - . m l Q(l), (17), . m = n + 1 (m = 0) , n- l > 0 (l < 0), l < 0 (l > 0). 0 < m < n + 1 n- ( n = 1 ), - n- m. Q(l) n = 1 ( M ) l > 0, l < 0, n = 2 l > 0, l < 0, n > 2 1 < m < n l > 0, l < 0. , 0 < m < n + 1 . M , , , .

3



, . , , , . 157


3.

. (q) = n+ (q1 , . . . , qn+1 ) = i=11 ai qi + O ( q 2 ) q 0, : A = a2 + . . . + 1 2 a2 - am+1 - . . . - a2 +1 = 0, : m n ) Al > 0 q = ‘q , :
qi = ai

l , i = 1, . . . , m A + o(1 ) l , j = m + 1, . . . , n + 1 A + o(1 )

l 0; ) Al < 0 . (18) , l > 0, l < 0 , l = 0. q = ‘q , , ( ) m A . , Q(l) , , Q(l) . A = 0 , (., , [3]). (n = 1). , , (. 5). , (M ) = 0 , 2 (M ) = 0 ; (M ) > 0 2

qj = -aj

(1 8 )

() (.1). l = 0 , .. ( (M ) < 0 , (M ) > 0 - ). . l < 0, l > 0 Q(l) . .., ( ) . , , [3]. 158


.., .., ..,

. 5:

4



Pi Ri Ri = Ri + Ri , Ri . Ri , , , .
-1 fk = fk - k=1 i fi , (t) Q(l) , i (t). : Ri = Ci1 Kn 2 + Ci2 + O ( |l|) l 0, i = 1, . . . , N , f Ci2 = Ci2 (q) C2 , rk = 0, 0 < C3 Ci1 (q), C1 ,C i l. f


Kn Ci1 = Ci1 (q) C1 , , 2 ,C3

, rk = 0, Ri i Q(l) , . = const Kn = 0, 1 Kn l 0. ,
|l |

Kn = 0, 1 + . . . + m - m+1 - . . . - n+1 = 0. 2 2 2 2 (n - 1). = const Kn = 0, l 0 Pi . , . , , , . , . , . , , . 159


3.


[1] .. // , 1999, . 63, 5, . 770í774. [2] .., .. // , 2005, 4, . 13í17. [3] .. // , 2008, 6, . 10í13. [4] .., .. // - , 2009, 27, c. 168í173. [5] .. // - , , 2007. 128 . [6] Milnor J. Morse Theory. Princeton University Press, Princeton, New Jersey, 1963.

160


Abstracts

Abstracts
Empirical asymptotically optimal p olicies
Bulinskaya E.B., Professor, Faculty of Mathematics and Mechanics, MSU, e-mail: ebulinsk@yandex.ru
The aim of the pap er is to establish the optimal control of applied sto chastic mo dels under incomplete information. To this end we prop ose a three-step algorithm for construction of so-called empirical asymptotically optimal p olicies. For illustration we treat an inputoutput mo del arising in various domains of applied probability such as insurance, inventory and queueing theory.

Prime numb ers and indep endence
Vinogradov O.P., Professor, Faculty of Mathematics and Mechanics, MSU, e-mail: ovinogradov@mail.ru
For discrete probability spaces the pap er p oses the question of existence or nonexistence of (nontrivial) indep endent events. For finite probability spaces in a numb er of cases the answer is p ositive if certain integer parameter is a prime numb er or comp osite numb er.

Analysis of recursive random sequence indexes through continuous time pro cesses
Goldaeva A.A., Assistant, Faculty of Mathematics and Mechanics, MSU, e-mail: gold_ann@list.ru
The pap er prop oses a new approach in which we consider the sto chastic difference sequences like the sequences of observations (in deterministic or random moments of time) of continuous time pro cess which satisfies the sto chastic differential equation. The main results are explicit formula for the index of the tail and the upp er b ound of the extremal index.

Limit theorems for stationary distributons of two-typ e maximal branching pro cesses
Lebedev A.V., Asso ciate Professor, Faculty of Mathematics and Mechanics, MSU, e-mail: alebedev@mech.math.msu.su 161


Abstracts

We consider the maximal branching pro cesses with two typ es of particles, previously intro duced by the author. Limit theorems for stationary distributions of such pro cesses are proved.

Multityp e maximal branching pro cesses with extreme value copulas
Lebedev A.V., Asso ciate Professor, Faculty of Mathematics and Mechanics, MSU, e-mail: alebedev@mech.math.msu.su
We consider the maximal branching pro cesses with several typ es of particles in the case when the offspring numb ers multivariate distributions have extreme value copulas. The ergo dic theorem, monotonicity prop erty and limit theorem for stationary distributions in the p ower tails case are proved. Particular attention is paid to multivariate Fr‡chet e distributions.

Phases in time evolution of multi-dimensional interacting diffusions with synchronization
Manita A.D., Asso ciate Professor, Faculty of Mathematics and Mechanics, MSU, e-mail: manita@mech.math.msu.su
We consider multi-dimensional Markov pro cesses x(t), t 0, describing dynamics of N sto chastic particles with combined interaction of the mean field+synchronization typ e. We find time scales t = t(N ), N , on which the particle system x(t) exhibits different b ehavior.

Partial Steiner systems and random hyp ergraphs
Shabanov D.A., Assistant, Faculty of Mathematics and Mechanics, MSU, e-mail: gold-amber@yandex.ru
The work deals with a well-known problem in extremal combinatorics concerning partial Steiner systems. By using random hyp ergraphs we obtain new upp er b ounds for the minimum p ossible numb er of edges of a partial Steiner system with high chromatic numb er.

Symmetric branching walks with heavy tails
Yarovaya E.B., Asso ciate Professor, Faculty of Mathematics and Mechanics, MSU, e-mail: yarovaya@mech.math.msu.su 162


Abstracts

We consider a continuous-time symmetric branching random walk on Zd , where the particles are b orn and die at a single lattice p oint. The underlying random walk is assumed to b e symmetric. Moreover, corresp onding transition rates of the random walk have heavy tails. As a result, the condition of the finite variance of jumps is not valid, and random walk may b e transient even on low-dimensional lattices, that is, for d = 1, 2. Conditions of transience for a random walk on Zd and limit theorems for the numb ers of particles b oth at an arbitrary p oint of the lattice and on the entire lattice are obtained.

Can the stabilizer b e eight-dimensional?
Beloshapka V.K., Professor, Faculty of Mathematics and Mechanics, MSU, e-mail: vkb@srogino.ru
The main result is full description of the automorphisms Lie algebra of the real hyp ersurfaces (r ) = {v = ur (|z |2 )} in C2 . The dimension of such Lie algebras can b e equal 2, 3, 4 and 5. The dimension of the stabilizer can b e equal 2, 3 and 5. Some general conjecture was discussed.

Mo dules of smo othness of p ositive orders of functions from space Lp , 1 p
Potapov M.K., Professor, Faculty of Mathematics and Mechanics, MSU, e-mail: mkpotapov@mail.ru Simonov B.V., Asso ciate Professor, Volgograd State Technical University, e-mail: simonov-b2002@yandex.ru

This pap er systematizes basic prop erties of mo dules of smo othness of p ositive orders of functions from space Lp , 1 p . The pro of of all statements is based on equivalence of the mo dules of smo othness to its constructive characteristic.

On the maximal top ology of convergence in partially ordered space
Fedorov V.M., Asso ciate Professor, Faculty of Mathematics and Mechanics, MSU, e-mail: vferdorov@rambler.ru
In this pap er we study the maximum top ology in a partially ordered spaces defined the ordered, relatively uniform and order-b ounded convergence. These top ologies are affine. It is pro of that the condition of stellar regularity the corresp onding top ology convergence in the maximum top ology is equivalent stellar convergence. It is provided sufficient conditions under which these top ologies are linear. We also give description of top ologies with via inductive limit.

163


Abstracts

Motion of the rigid b o dy consisting of two disks
Itskovich M.O., Student, Faculty of Mathematics and Mechanics, MSU, e-mail: m1shok7@mail.ru Kuleshov A.S., Asso ciate Professor, Faculty of Mathematics and Mechanics, MSU, e-mail: kuleshov@mech.math.msu.su, akule@pisem.net
We present a kinematic analysis of motion of the rigid b o dy consisting of two identical discs whose symmetry planes are at right angle. The no-slip constraints of this b o dy are integrable, hence the system is essentially holonomic. In this pap er we present analytic expressions for the tra jectories of the ground contact p oints of the system. Equilibria of the b o dy on the plane are found and their stability conditions are obtained.

On features of equilibriums of critical systems
Samsonov V.A., Professor, Institute of Mechanics, MSU, e-mail: samson@imec.msu.ru Selutski Yu.D., Asso ciate Professor, Institute of Mechanics, MSU, e-mail: seliutski@imec.msu.ru Mikhalev A.A., PhD, e-mail: alexey_mih@inbox.ru
Kinematical chains are considered, for which the set of virtual displacements of one or several p oints is empty. It is shown that such mechanisms have dead equilibrium p ositions not dep ending up on the direction of force applied to one of these p oints. Mechanisms with singularity of different typ es are analyzed, where the surgery of the configurational manifold o ccurs at a certain (critical) value of parameter of constraints. It is shown that this surgery is top ologically describ ed with Morse surgery.

164






. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 .., . . . . . . . . . . . . . . . 8 .., . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 .., . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 .., . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 .., . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 .., . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 .., . . . . . . . . . . . . . 68 .., . . . . . . . . . . 77

2. . . . . . . . . . . . . . . . . . . . . 85 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 .., ? . . . . . . . . . . . . . . . . . . . . . . 87 .., .., Lp , 1 p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 165




.., . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

3. . . . . . . . . . . . . . . . . . . . . . . . . 138 .. . . . . . . . . . . . . . . . . . . . 139 .., .., , , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 .., .., .., . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

Abstracts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

166




I. 1. .. , .. . 2. .. , .. . I I. 1. .. , .. . 2. .. , .. . I I I. 1. .. , .. . 2. . .. . 3. . .. -. IV. 1. . .. . 2. . .. , .. . 3. . .. , .. , .. . V. 1. . .. , .. . 2. . .. , .. . 3. . .. . VI. , .. . . 1. 105- .. . .. , .. . 2. 100- .. . .. .. 3. 100- .. . .. .. -




VII. . . 1. 190- .. .. , .. , .. , .. .

-: ..

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