Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://www.philol.msu.ru/~humlang/articles/dopolnen.pdf
Äàòà èçìåíåíèÿ: Fri Apr 11 00:00:00 2003
Äàòà èíäåêñèðîâàíèÿ: Tue Oct 2 11:19:29 2012
Êîäèðîâêà:
>hihegbl_evgu_ ihykg_gby d jZ[hl_ << Djhf_jZ ©:jdlZg]_gk beb eh]bklZ D \hijhkm ^bZojhgbq_kdh]h kdZqdZ ª

"

< h[km`^Z_fhc jZ[hl_ HJ ijb\_^_gZ nhjfmeZ ^ey iehlghklb jZkij_ ^_e_gby p( x) = 1A ebg]\bklbq_kdhc bg_jlghklb x K mq_lhf h]jZgbq_g +
x

-

ghklb j_kmjkh\ bklhqgbdZ ebg]\bklbq_kdh]h \ha^_ckl\by BE< oZjZdl_jbam_fh]h \gmlj_ggbf khijhlb\e_gb_f R nhjfmeZ ijb jZkijhkljZg_gbb __ gZ h[eZklv \k_o ^_ckl\bl_evguo x (-, ) ijbgbfZ_l \b^
p( x) = A R+ x
1+ +


]^_

A

±

ghjfbjmxsbc fgh`bl_ev hij_^_ey_fuc bkoh^y < HJ jZkkfhlj_g qZklguc kemqZc k gZb[he__ lb
-

ba mkeh\by

ibqguf agZq_gb_f = 1 khhl\_lkl\mxsbf aZdhgm PbinZ >Ze__ \ HJ ihdZaZgh qlh j_r_gb_ ^bnn_j_gpbZevgh]h mjZ\g_gby >M dF (t ) = rF (t )(1 - F (t ) ) ijb = 2 [ebadh d nmgdpbb jZkij_^_e_gby aZ^Zdt

-



p ( x)dx = 1



\Z_fhc iehlghklvx

p( x) =

^_ qZklghc i_j_ibkdb ih HJ h^gbf ba dhjj_kihg^_glh\ [ue aZ^Zg \hijhk ihq_fm \u[jZgh bf_ggh agZq_gb_ = 2 >ey hl\_lZ ijboh^blky jZkkfhlj_lv gZb[he__ h[sbc kemqZc p( x) = A1+ ijb jZaebqguo agZq_gbyo >ey ¨o\hklh\´ jZkij_^_e_gby ijb x >> 1 ij_g_[j_]Z_f \gmlj_ggbf khijhlb\e_gb_f BE< BE< jZ[hlZ_l k fZehc gZ]jmadhc \ A j_`bf_ [ebadhf d j_`bfm ¨ohehklh]h oh^Z´ hldm^Z p( x) 1+ <
(0, )


A 1+ x

2

ijb khhl\_lkl\mxs_c ghjfbjh\d_ < oh

-

1+ x

h[eZklb
F (t ) =

fZeuo
p ( x )dx

agZq_gbc
A
1+

F(t dF (t ) = dt

-



x

-



x

dx =

A x




x

hldm^Z

1 1+ 1

]^_
F
1 1+

x x << -1 ,

BlZd \
1 1+ 1


A


h[eZklb fZeuo

x



\ gZqZe_ ijhp_kkZ >K

dF (t ) = rF (t ) dt



]^_

r=



Z

A


2 =1+ 1


Bkoh^y ba khh[jZ`_gbc kbff_ljbb jZkijhkljZgy_f nhjfmem gZ

h[eZklv x (-, ) : dF (t ) = rF (t )(1 - F (t ) ) Ijb = 1 ihemqZ_f = 2 . dt Bf_ggh wlhl kemqZc b jZkkfZljb\Z_lky \ HJ >K hkms_kl\ey_lky kh]eZkgh nmgdpbb jZkij_^_e_gby Dhrb Ijb > 2 jZkij_^_e_gb_ F(t) ]Zmkkh\h [1, k @ q_fm khhl\_lkl\m_l < 1,5 Ih^ wlhl kemqZc ih^iZ^Z_l eh]bklbq_kdh_ jZkij_^_e_gb_ k = 1 ( hq_gv \_ebdh G_]Zmkkh\uf jZkij_^_e_gbyf khhl\_lkl\m_l < 2 >lZf `_@ b > 1,5 Ih^ wlhl kemqZc ih^iZ^Z_l jZkkfhlj_gguc \ HJ kemqZc = 2 k oh^hf ijhp_kkZ >K [ebadbf d nmgdpbb jZkij_^_e_gby Dhrb kf jbk \HJ JZkkfhljbf mklhcqb\hklv jZkij_^_e_gby k iehlghklvx
p( x) = A 1+ x
1+

d

baf_g_gbyf iZjZf_ljZ AgZq_gb_ R ^Ze__ ijbgbfZ_lky jZ\guf ihkdhevdm baf_g_gb_ R \ghkbl g_ijbgpbibZevgu_ ^ey gZr_]h jZkkfhlj_gby baf_g_gby fZkrlZ[Z >Ze__ lZd`_ g_ ^_eZ_lky jZagbpZ f_`^m x b t ihkdhevdm kh]eZkgh HJ x = t - t 0 Z t0 ijbgbfZ_f jZ\guf AgZq_gby A hij_^_e_ggu_
A= 1
+

kh]eZkgh
= 1
1+

mkeh\bx
1

+

-



p ( x)dt = 1

ih

nhjfme_
.

0,5
+

-



1+ x


0

ijb\_^_gu \ lZ[ebp_ 1 \ aZ\bkbfhklb hl
1+

1+ x

LZ[ ebp Z
0,5 1 1 = 0,3183 2 33 = 0,4135 4 1,5 4 33 = 0,7698 3



A

0 0

33 = 0,2067 8 3 8 33 = 1,5396

2 = 0,4502 4 3 1 = 0,7071 2

= 1+

1



2 1

A1 A


3

Ijhbgl_]jbjh\Z\
= 0,5

F (t ) =

\bkbfhklb ^ey oh^Z ^bZojhgbq_kdh]h kdZqdZ

-



x

p ( x )dx =

-



x

A 1+ x
1+




ihemqZ_f ke_^mxsb_ aZ

-

2 1 + x 2 x - 1 3 3 ln + F (t ) = 0,5 + sgn ( x ) 0,125 - arctg , (1) 8 1 - x + x 4 3

(

)

]^_

sgn ( z )

±

kb]gmfnmgdpby

1, _keb z > 0, sgn ( z ) = 0, _keb z = 0, - 1, _keb z < 0. 1 arctg x .

>

k

@

.

=1
F (t ) = 0,5 + (2)

=2
2 x - (1 + x )2 3 3 ln + F (t ) = 0,5 + sgn ( x ) 0,125 + arctg 8 1 - x + x 2 4 3 1 . (3)

=3
1 F (t ) = 0,5 [ ( x + 1) + ( x - 1)] + 4 x 2 + 2 x + 1 2 x ln , + 2arctg 2 x2 - 2 x + 1 1 - x
>

(4)

]^_

( z)

±

nmgdpby O_\bkZc^Z

1, _keb z 0, ( z) = 0, _keb z < 0.

k

@

.

GZg_k_gb_ aZ\bkbfhkl_c ih hl Zj]mf_glZ x gZ h^bg ]jZnbd g_p_e_khh[jZagh ihkdhevdm \k_ aZ\bkbfhklb oZjZdl_jbamxlky hlebqZxsbfbky agZq_gbyfb ihembgl_jd\Zjlbevghc rbjhlu Ijhba\h^gZy hl F(t) agZq_gb_ p(x)) ijb x = 0 jZ\gZ A GZ jbk ij_^klZ\e_gu aZ\bkbfhklb kh]eZkgh A1 hl agZq_gby Zj]mf_glZ x = x ]^_ A ± agZq_gb_ A ijb jZkkfZljb\Z_fhf Z A1 ± agZq_gb_ \u[jZgh ijhba\hevgh \ dZq_kl\_ wlZehgZ ^ey kjZ\g_gby Ih^h[gh_ ij_^klZ\e_gb_ iha\hey_l kjZ\gb\Zlv jZkij_^_e_gby ih ih\_^_gbx bo ©o\hklh\ª ijb x - eb[h x + .
A A ijb = 1


4

AgZq_gby

fu_ aZ\bkbfhklb ghjfbjh\Zebkv ih kh\iZ^_gbx b d\Zjlbeb qlh ijb[eb`_ggh wd\b\Ze_glgh bkihevah\Zgghc a^_kv ghjfbjh\d_ ih ijhba\h^ghc F(t) ijb x = 0 \\b^m g_agZqbl_evgh]h hldehg_gby aZ\bkbfhkl_c F(t hl ijyfhc \ kj_^g_c qZklb GZ jbk gZg_k_gu lZd`_ eh]bklbq_kdZy aZ\bkbfhklv b nmgdpby ghjfZevgh]h jZkij_^_e_gby lZd`_ k khhl\_lkl\mxs_c ghjfbjh\dhc ih ijhba\h^ghc F(0) qlh lj_[m_l mfgh`_gby Zj]mf_glZ eh]bklbq_kdhc nmgdpbb gZ 4 Z Zj]mf_glZ nmgdpbb ghj

A1 A

ijb\_^_gu \ lZ[ebp_ \ aZ\bkbfhklb hl





< HJ kjZ\gb\Z_

-

fZevgh]h jZkij_^_e_gby gZ

2 .


)W

D



D D


D



eh]bkl jZkij ghjf jZkij



[

Jbk





5





)W )W



[





D



D



eh]bkl jZkij



Jbk

ghjf jZkij





< HJ aZ\bkbfhklv ihemq_ggZy bgl_]jbjh\Zgb_f pbinh\hc iehlghklb A ; ( = 1) kjZ\gb\ZeZkv k aZ\bkbfhklvx ihemq_gghc iml_f p( x) = 2


1+ x





j_r_gby ^bnn_j_gpbZevgh]h mjZ\g_gby dF (t ) = rF 2 (t )(1 - F (t ) )2 ; ( = 2) b dt mklZgZ\eb\Zehkv qlh aZ\bkbfhklb [ebadb IhdZ`_f qlh ih^h[gu_ iZjZee_evgu_ aZ\bkbfhklb kms_kl\mxl b ^ey ^jm]bo khq_lZgbc b k\yaZgguo khhlghr_gb_f = 1 + 1 GZc^_f j_r_gby ^bnn_j_gpbZevgh]h mjZ\g_gby






dF (t ) = rF (t )(1 - F (t ) dt = 1,5 x= =2

) ijb gZqZevghf mkeh\bb
1 2 4 F (t ) - r 1 - F (t ) F(

F (- ) = 0 :

. t)

(5)

1 1 1 1 - F (t ) - - 2 ln . x= r 1 - F (t ) F (t ) F (t )

(6)


6 =3 1 1 x= r 2(1 - F (t )

)

2

-

3 3 1 - F (t - - 6 ln F (t ) 2 F 2 (t ) 1 - F (t ) F (t ) 1 +

) .

(7)

dF (t ) = rF (0)(1 - F (0) ) = 0,5 2 r < dt lZ[ebp_ ijb\_^_gu agZq_gby A ± ijhba\h^ghc dF (t ) aZ\bkbfhkl_c dt 1 b agZq_gby r h[_ki_qb\Zxsb_ agZq_gb_ A = agZq_gb_ A1 kh]eZkgh

Ijhba\h^gZy

dF (t ) dt

ijb

x=0

jZ\gZ

lZ[ebp_
A r



LZ[ ebp Z
1,5 1 r 8 8 2 1 r 16 16 3 1 r 64 64





)W


= 1,5


=2 =2



=1 =3 = 0,5





[



Jbk




7

GZ jbk gZg_k_gu ghjfZebah\Zggu_ ih dF (t ) aZ\bkbfhklb b khdt hl\_lkl\_ggh
j_gpbZevgh]h mjZ\g_gby dF (t ) = rF (t )(1 - F (t ) ) fh`_l [ulv bgl_jij_lbdt jh\ZgZ \ l_jfbgZo w\hexpbb fZl_jbb hl ]Zmkkh\uo ijbjh^guo kbkl_f d g_]Zmkkh\uf khpbZevguf q_j_a ijhf_`mlhqgu_ [bheh]bq_kdb_ kbkl_fu b ^Zevg_cr__ jZa\blb_ \k_ [he__ l\hjq_kdbo \b^h\ q_eh\_q_kdhc ^_yl_evghklb > k @ K jhklhf l\hjq_kdh]h gZiheg_gby \b^Z q_eh\_q_kdhc ^_yl_evghklb mf_gvrZ_lky iZjZf_lj jZkij_^_e_gby PbinZIZj_lh >lZf `_@, qlh \ khhl\_lkl\bb k nhjfmehc = 1 + 1 \_^_l d m\_ebq_gbx AgZq_gb_ = 0 k\hckl\_ggh jZa\blbx g_`b\hc fZl_jbb Kbkl_fZ \_^_l k_[y dZd _^bguc we_f_gl k\yav \ kbkl_f_ _^bgkl\_ggZy kbkl_fZ aZfdgmlZ gZ kZfh_ k_[y Ijb = 1 bf__f ]Zmkkh\m kbkl_fm hq_gv \_ebdh \ ij_^_e_ = qlh khhl\_lkl\m_l ]Zmkkh\m g_pbinh\hfm jZkij_^_e_gbx > k @ gZijbf_j eh]bkl_ Dhebq_kl\h k\ya_c \ kbkl_f_ kh\iZ^Z_l k dhebq_kl\hf we_f_glh\ kbkl_fu N l_ dZ`^uc we_f_gl aZfdgml gZ kZfh_ k_[y K\yab f_`^m we_f_glZfb hlkmlkl\mxl Ijb = 2 dhebq_kl\h k\ya_c \ kbkl_f_ jZ\gh N 2 l_ ^\mklhjhgg_c k\yavx k\yaZgu \k_ \hafh`gu_ \ kbkl_f_ iZju JZa\blb_ \ kbkl_f_ hkms_kl\ey_lky kh]eZkgh nmgdpbb jZkij_^_e_gby Dhrb beb [ebadhfm d g_fm Ijb = 3 kbkl_fZ hljZ`Z_lky \ dZ`^hf ba k\hbo we_f_glh\ l_ dZ`^uc we_f_gl ih^h[_g \k_c kbkl_f_ ©< dZ`^hc dZie_ \h^u hljZ`Z_lky hd_Zgª ©O\hkluª jZkij_^_e_gby _s_ [he__ ^ebggu_ q_f ©o\hkluª jZkij_^_e_gby Dhrb K ^Zevg_crbf jhklhf dhebq_kl\h k\ya_c khklZ\ey_l N kbkl_fZ \k_ [he__ njZdlZebabjm_lky


8

>jh[guf agZq_gbyf khhl\_lkl\mxl ijhf_`mlhqgu_ \ZjbZglu JZkkfhljbf ijbf_j k 1 < < 2 NjZdlZevghklv hlkmlkl\m_l dZ`^uc we_f_gl bf__l N -1 < N k\ya_c l_ we_f_gl k\yaZg g_ k dZ`^uf ba N we_f_glh\ kbkl_fu ±

n =1



N

qn = N

-1



AZf_gb\ kmffm jy^Z bgl_]jZehf ih
qn = -1 n
2 -


-

emqZ_f h^gh ba \hafh`guo ijb[eb`_gguo j_r_gbc dhlhjh]h \hajZklZ_l k jhklhf
n.

lhqghklv

AZdexqZy ^_eZ_f \u\h^ qlh oZjZdl_j ^bgZfbq_kdbo ijhp_kkh\ \ kbkl_f_ hij_^_ey_lky jZaf_jghklvx k\ya_c jZkij_^_e_gb_f \_jhylghkl_c mklZgh\e_gby k\ya_c f_`^m we_f_glZfb kl_i_gvx kZfhih^h[by \ kbkl_f_ KemqZc k = 2 gZb[he__ lbibq_g ihkdhevdm hl\_qZ_l jZkij_^_e_gbx PbinZIZj_lh k gZb[he__ lbibqguf agZq_gb_f = 1.

Ebl_jZlmjZ
OZclmg K> Ijh[e_fu dhebq_kl\_ggh]h ZgZebaZ gZmdb F GZmdZ FZl_fZlbq_kdZy wgpbdehi_^by / =e j_^ BF 1. 2.