JOURNAL OF IQRO – ЖУРНАЛ ИҚРО – IQRO JURNALI – volume 15, issue 02, 2025
ISSN: 2181-4341, IMPACT FACTOR ( RESEARCH BIB ) – 7,245, SJIF – 5,431
ILMIY METODIK JURNAL
Homidov Farhod Faxriddinovich
Osiyo Xalqaro Universiteti “Umumtexnik fanlar” kafedrasi o’qituvchisi
Farhod2708@mail.ru
IKKI NUQTALI BOG’LIQSIZ GRAFDA ANIQLANGAN MARKOV ZANJIRIGA MOS
KELUVCHI KVADRATIK STOXASTIK OPERATORLAR
Annotasiya:
Matematik biologiyaning masalalari kvadratik stoxostik operatorlarni o’rganishga
keladi. Bu operatorlar qurilishida o’lchovlar ishlatiladi. Ko’paytma o’lchovga ko’ra konstruksiya
bo’yicha qurilgan kvadratik stoxastik operatorlar dinamikasi o’rganilgan. Markov o’lchoviga
mos konstruksiya bo’yicha qurilgan kvadratik stoxastik operatorlar dinamikasi hali
o’rganilmagan. Shu sababli bu masalalarni o’rganish dolzarb hisoblanadi. Bu ishda biz Markov
zanjirlarini va Markov zanjirlariga mos keluvchi kvadratik stoxastik operatorlar ham alohida
o’rganilgan. Shu sababli biz novolterra tipidagi kvadratik stoxastik operatorlar uchun asosiy
masalani o’rganib chiqdik. Bunda Markov zanjiriga mos KSO dinamikasi o’rganish dolzarbdir.
Ikki nuqtali bog’liqsiz grafda mos keluvchi Markov zanjiri bo’yicha qurilgan kvadratik stoxastik
operatorlar o’rganilgan.
Kalit so'zlar:
Kvadratik stoxastik operatorning konsstruksiyasi E chekli va sanoqli to’plam
uchun kvadratik stoxastik operator konstruksiyasi,Potts modeli Gibbs o’lchovlari tadqiqot
obyektlari bo’lib ko’paytma o’lchovga mos keluvchi kvadratik stoxastik operatorlar,2 nuqtali
bog’liqsiz grafda aniqlangan Markov zanjiriga mos keluvchi kvadratik stoxastik
operatorlar,Markov zanjirlari,Markov zanjirlariga mos keluvchi kvadratik stoxastik operatorlar
hisoblanadi.
Kvadratik stoxastik operator
(
)
=
=
=
=
-
n
i
i
i
n
n
n
x
x
R
x
x
x
S
1
1
1
1
,
0
:
,...,
simpleksni o`ziga aks ettiruvchi akslantirish bo`lib quyidagi ko`rinishga egadir
(
)
n
k
x
x
p
x
V
j
n
j
i
i
k
ij
k
,.....
1
,
:
1
,
,
=
=
=
(1)
va bu yerda
-
k
ij
p
,
belgilarni avloddan-avlodga o`tish ehtimolligi bo`lib quyidagi shartlarni
qanoatlantiradi
,
0
,
k
ij
p
(
)
n
k
j
i
p
n
k
k
ij
,..
1
,
,
,1
1
,
=
=
=
(2)
JOURNAL OF IQRO – ЖУРНАЛ ИҚРО – IQRO JURNALI – volume 15, issue 02, 2025
ISSN: 2181-4341, IMPACT FACTOR ( RESEARCH BIB ) – 7,245, SJIF – 5,431
ILMIY METODIK JURNAL
Har bir
1
-
n
S
x
element
{
}
n
,..,
1
=
E
to`plamda aniqlangan ehtimollik o`lchovi bo`ladi.
Populyasiya ixtiyoriy
1
-
n
S
x
holatdan boshlanib, keyingi
Vx
holatga o`tadi va keyingi
x
V
2
holatga o`tadi va h.z….
Berilgan boshlang`ich
( )
1
0
-
n
S
x
nuqta uchun
( )
{ }
,..
2
,1
,
0
,
=
l
x
l
trayektoriya
( )
( )
( )
l
l
x
V
x
=
+
1
,
,..
2
,1
,
0
=
l
qonuniyat asosida, ya`ni (1) operatorning iteratsiyasi sifatida
aniqlanadi.
Matematik biologiyaning asosiy masalasi bo`lib berilgan kvadratik stoxastik operatorning
trayektoriyasining asimptorik holatini o`rganish bo`lib sanaladi. Bu masala Volterra kvadratik
stoxastik operatorlar sinfi uchun deyarli `toliq o`rganilgan.
Volterra kvadratik stoxastik operatorlar deb (1),(2) va qo`shimcha
,
0
,
=
k
ij
p
agar
{ }
n
k
j
i
j
i
k
,...,
2
,1
,
,
,
,
=
(3)
shart asosida aniqlanadi.
(3) shartning biologik ma`nosi juda sodda:
-
k
individ faqat
i
va
j
ota-onalarning birini
belgilarini takrorlashi mumkin.
R.N. Ganixo`jayevning ilmiy ishlaridan quyidagilar ma`lum
Volterra
(
)
( )
(
)
1
1
1
1
,..,
,..,
:
-
-
=
=
®
=
n
n
n
n
S
x
x
x
x
V
S
x
x
x
V
kvadratik stoxastik
operatorning umumiy ko`rinishi :
n
k
x
a
x
x
n
i
i
ki
k
k
,...,
1
,
1
1
=
+
=
=
(4)
va bu yerda
k
i
uchun
1
2
,
-
=
k
ik
ki
p
a
va
0
=
kk
a
munosabatlarni qanoatlantiradi.
Bundan tashqari
ik
ki
a
a
-
=
va
1
ki
a
munosabtlar ham o`rinli bo`ladi.
Volterra kvadratik stoxastik operatorlar nazariyasi Lyapunov funksiyalri va turnirlar nazariyalari
asosida rivojlantirildi. Ammo novolterra kvadratik stoxastik operatorlar sinfi haligacha to`liq
o`rganilmagan. Novolterra operatorlar sinfini o`rganish uchun qo`llash mumkin bo`lgan umumiy
nazariya yaratilmagan.
N.N. G`anixo`jayev va U.A. Roziqovlarning maqolalarida kvadratik stoxastik operatorlarni
konstruksiyasi quyidagicha keltirilgan.
(
)
L
,
L
– karrali qirralarga va halqalarga ega bo`lmagan graf berilgan bo`lsin.
L
– grafning
barcha uchlari va
L
– barcha qirralar to`plami bo`lsin.
JOURNAL OF IQRO – ЖУРНАЛ ИҚРО – IQRO JURNALI – volume 15, issue 02, 2025
ISSN: 2181-4341, IMPACT FACTOR ( RESEARCH BIB ) – 7,245, SJIF – 5,431
ILMIY METODIK JURNAL
Undan tashqari,
F
chekli to`plam berilgan bo`lsin.
F
to`plamni genotiplar to`plami (statistic
mexanika masalalarida
F
to`plamga spin qiymatlari to`plami ) deyiladi.
F
®
L
:
s
akslantirishga hujayra deyiladi. Barcha hujayralar to`plamini
W
bilan beilgilaymiz va
W
chekli
to`plamda aniqlangan barcha ehtimollik o`lchovlari to`plamini
(
)
F
L
,
S
orqali belgilab olamiz.
Berilgan
(
)
L
,
L
grafning maksimal bog`liq komponentalari
{ }
)
,...,
1
(
n
i
i
=
L
lar bo`lsin.
Ixtiyoriy ikkita
,
,
2
1
W
s
s
konfiguratsiya fiksirlaymiz va ular uchun quyidagi to`plamni
aniqlaymiz.
{
}
i
i
or
i
i
L
L
L
L
=
=
W
=
L
W
2
1
2
,
1
:
)
,
,
(
s
s
s
s
s
s
s
barcha
n
i
,...,
1
=
Biror
(
)
F
L
,
S
m
ehtimollik o`lchovi
W
da berilgan bo`lsin va bu o`lchov ixtiyoriy
W
s
hujayra uchun
( )
0
>
s
m
shartni qanoatlantirsin, ya`ni biror potensial bilan aniqlangan Gibbs
o`lchovi bo`lsin.
s
s
s
,
2
1
p
ehtimolliklarni quyidagi formula bialn aniqlaymiz.
L
W
L
W
=
hollarda
boshqa
,
0
)
,
,
(
agar
,
))
,
,
(
(
)
(
2
1
2
1
,
2
1
s
s
s
s
s
m
s
m
s
s
s
p
(5)
Ma`lumki, bu ehtimolliklar uchun quyidagi munosabatlar o`rinli bo`ladi.
,
0
,
2
1
s
s
s
p
s
s
s
s
s
s
,
,
1
2
2
1
p
p
=
va
W
=
s
v
s
s
1
,
2
1
p
barcha
W
2
1
,
s
s
.
(
)
F
L
,
S
da aniqlangan
( )
m
V
kvadratik stoxastik operator
(5) ehtimolliklar orqali
quyidagicha aniqlangan bo`lsin:
ixtiyoriy
)
,
(
F
L
S
l
o`lchov uchun
)
,
(
)
(
F
L
=
S
V
l
l
o`lchovni quyidagi formula
bilan aniqlaymz.
W
=
W
s
s
l
s
l
s
l
s
s
s
s
s
),
(
)
(
)
(
2
1
,
,
2
1
2
1
p
(6)
Shunday qilib, qachonki kvadratik stoxastik operatorlar qurilgan bo`lsa, Gibbs taqsimotlari hosil
bo`ladi.
(6) formula bilan kvadratik stoxastik operator Volterra kvadratik stoxastik operatori deyiladi,
agar
k
ij
p
,
koeffisiyentlar faqat
k
i
=
yoki
k
j
=
larda noldan farqli bo`lsa va boshqa barcha
hollarda nolga aylansa. Bunday modelning biologic interpritatsiyasi quyidagicha: yangi
tu`g`iladigan avlod faqat ota yoki onaning belgilarini takrorlashi mumkin.
2 Novolterra operatorni aniqlanishi
JOURNAL OF IQRO – ЖУРНАЛ ИҚРО – IQRO JURNALI – volume 15, issue 02, 2025
ISSN: 2181-4341, IMPACT FACTOR ( RESEARCH BIB ) – 7,245, SJIF – 5,431
ILMIY METODIK JURNAL
Bu paragrafda ushbu konktruksiya yordamida qurilgan novolterra operatorning dinamikasini
o`rganish bilan tanIshib chiqamiz.
)
,
(
L
G
L
=
– chekli graf va
{ }
m
i
i
,...,
2
,1
,
=
L
lar
G
grafning maksimal bo`g`liq
qism graflari bo`lsin.
i
i
L
F
=
W
bilan
{ }
m
i
i
,...,
2
,1
,
=
L
qism graf ustida aniqlangan barcha
konfiguratsiyalar to`plamini belgilaymiz.
i
W
to`plamda aniqlangan
i
m
ehtimollik o`lchovlari ixtiyoriy
i
W
s
konfiguratsiya uchun
( )
0
>
s
m
i
shartni qanoatlantirsin.
m
W
W
W
=
W
...
2
1
to`plamda
m
ehtimollik
o`lchovini quyidagi formula bilan aniqlaymiz
( )
( )
=
=
m
i
i
i
1
s
m
s
m
(7)
1
=
m
hol qurilgan kvadratik stoxastik operator Volterra kvadratik stoxastik operatori bo`ladi.
Teorema . (7) o`lchov yordamida qurilgan (6) kvadratik stoxastik operatorni
m
ta volterra
kvadratik stoxastik operatorlarga keltirish mumkin.
ISBOT.
(
)
W
=
m
j
j
j
j
,
,
,
2
1
K
va
(
)
W
=
m
y
y
y
y
,
,
,
2
1
K
konstruksiyalarni
fiksirlaymiz. Bu konstruksiyalarga mos
(
)
{
}
{
}
m
i
i
i
i
m
,...,
2
,
1
,
,
:
,
,
)
,
,
(
1
=
W
=
=
L
W
y
j
s
s
s
s
y
j
K
va
( )
L
W
+
=
=
hollarda
boshqa
,
0
)
,
,
(
agar
,
)
(
)
(
1
,
y
j
s
y
m
j
m
s
m
s
jy
m
i
i
i
i
i
i
i
p
(8)
tengliklarni hosil qilamiz. Biz bu yerda quyidagi tenglikdan foydalandik.
(
)
(
)
( )
{
}
( )
( )
(
)
=
=
=
+
=
=
W
m
i
i
i
i
i
m
i
m
i
i
i
i
i
i
m
G
1
,...,
1
,
,
:
,
,
1
1
,
,
y
m
j
m
s
m
y
j
m
y
j
s
s
s
K
.
Shunday qilib, (7) o`lchov yordamida (6) formula bilan aniqlangan kvadratik stoxastik operatorni
quyidagicha yozish mumkin
( )
(
)
( )
{
}
(
)
( )
(
)
(
)
( ) ( )
y
l
j
l
y
m
j
m
s
m
s
s
l
s
l
y
y
y
y
j
j
j
j
y
j
s
W
=
W
=
=
+
=
=
i
i
m
i
i
m
i
i
i
m
i
i
i
i
i
i
i
m
:
,...,
:
,...,
1
,
1
1
1
)
(
1
,...,
.
(9)
Quyidagicha belgilash kiritamiz
JOURNAL OF IQRO – ЖУРНАЛ ИҚРО – IQRO JURNALI – volume 15, issue 02, 2025
ISSN: 2181-4341, IMPACT FACTOR ( RESEARCH BIB ) – 7,245, SJIF – 5,431
ILMIY METODIK JURNAL
( )
(
)
W
+
-
=
W
+
-
=
=
i
k
m
i
i
i
k
k
m
i
i
i
X
,
,...,
,
,
,...,
1
1
1
:
,
1
1
1
,...,
,
,
,...,
s
s
s
w
s
s
w
s
s
w
s
s
w
s
s
l
s
l
(10)
(9) tenglikdan yuqoridagi belgilashga asosan quyidagi tenglikni hosil qilamiz
{
}
+
+
=
=
=
=
W
=
W
+
-
m
i
j
j
j
j
j
j
j
j
i
i
i
i
i
j
j
j
i
m
m
i
i
i
X
1
)
,
(
:
,...
,...
,
,..
:
,
)
(
)
(
1
)
(
)
(
)
(
)
(
)
(
1
1
1
1
y
m
j
m
s
m
y
m
w
m
w
m
s
l
y
j
s
w
s
s
y
y
j
j
j
j
w
s
s
w
+
-
+
+
+
-
m
i
i
m
i
i
i
i
m
m
i
i
y
y
y
y
j
j
j
m
w
m
w
m
y
y
l
j
j
j
j
l
,...,
,
,..
,...
1
1
1
1
1
1
1
1
)
(
)
(
)
(
)
,...
(
)
,...,
,
,...,
(
{
}
(
)
(
) (
)
=
+
+
-
=
m
i
i
m
m
i
j
j
j
j
j
j
j
j
j
j
j
y
y
y
y
l
j
j
l
y
m
j
m
s
m
y
j
s
,...,
,
,..
,...
)
(
)
(
1
)
(
1
1
1
1
1
,
{
}
+
+
=
+
-
+
-
=
m
i
i
j
j
j
m
m
i
i
m
i
j
j
j
j
j
j
j
j
i
i
i
i
s
s
s
s
y
j
s
y
y
j
j
j
j
y
m
j
m
s
m
y
m
w
m
w
m
,...,
,
,...,
1
)
,
(
,...
,...
,
,..
1
1
1
1
1
1
1
)
(
)
(
1
)
(
)
(
)
(
)
(
2
)
,...,
(
)
,...,
,
,
,...,
(
1
1
1
1
m
m
i
i
y
y
l
j
j
w
j
j
l
+
-
(11)
{
}
(
)
1
)
(
)
(
1
)
(
,...
,
,...,
1
,
1
1
1
=
+
+
-
=
m
i
i
j
j
j
m
i
j
j
j
j
j
j
j
j
s
s
s
s
y
j
s
y
m
j
m
s
m
tenglikni o`rinli ekanligini e`tiborga olsak, (11) formuladan quyidagiga ega bo`lamiz.
=
+
=
+
-
+
-
)
,...,
(
)
,....
,
,
,...
(
)
(
)
(
)
(
2
1
1
1
1
,...
,...
,
,..
,
1
1
1
1
m
m
i
i
i
i
i
i
i
m
m
i
i
X
y
y
l
j
j
w
j
j
l
y
m
w
m
w
m
y
y
j
j
j
j
w
+
-
+
-
+
=
+
-
+
-
m
i
i
m
i
i
m
i
i
m
i
i
y
y
y
y
j
j
j
j
y
y
w
y
y
l
j
j
w
j
j
l
,...
,
,...
,...
,
,...
1
1
1
1
1
1
1
1
1
1
1
1
)
,...
,
,
,...
(
)
,...
,
,
,...
(
=
+
+
+
-
+
-
+
-
W
)
,...,
(
)
,...
,
,
,...
(
)
(
)
(
)
(
2
1
,...
,
,...
,...
,
,...
1
1
1
\
1
1
1
1
1
1
m
m
i
i
i
i
i
i
m
i
i
m
i
i
i
i
y
y
l
j
j
w
j
j
l
y
m
w
m
w
m
y
y
y
y
j
j
j
j
w
y
JOURNAL OF IQRO – ЖУРНАЛ ИҚРО – IQRO JURNALI – volume 15, issue 02, 2025
ISSN: 2181-4341, IMPACT FACTOR ( RESEARCH BIB ) – 7,245, SJIF – 5,431
ILMIY METODIK JURNAL
W
+
+
=
w
y
y
w
y
m
w
m
w
m
w
\
,
,
2
.
)
(
)
(
)
(
2
,
i
i
i
i
i
i
i
X
X
X
Demak (9) operatorni quyidagicha qayta yoziv olishimiz mumkin ekan
( )
( )
( )
+
+
=
W
w
y
y
w
w
w
y
m
w
m
w
m
\
,
,
,
,
2
i
i
i
i
i
i
i
i
X
X
X
X
(12)
va bu yerda
w
,
i
X
(10) formula bilan aniqlangan va
m
i
i
,...,
1
,
=
W
w
.
Ixtiyoriy
m
i
,...,
1
=
qiymat uchun
1
,
=
W
i
i
X
w
w
munosabatning o`rinli ekanligidan (12) tenglikdan
(9) kvadratik stoxastik operatorning quyidagi ko`rinishiga ega bo`lamiz
( )
( )
( )
( )
+
-
+
=
W
i
i
i
i
i
i
i
i
X
X
X
y
y
w
w
y
m
w
m
y
m
w
m
,
,
,
1
(13)
(4) operatorning ko`rinishi bilan (13) operator ko`rinishini solishtirsak har bir fiksirlangan
(
)
m
i
i
,...,
1
=
da (13) operatorning
( )
1
1
:
-
W
-
W
®
i
i
S
S
V
i
Volterra operatori bo`lishi
kelib chiqadi.
Teorema isbotlandi.
Biz foydalangan G grafda aniqlangan
F
spin qiymatlar to`plami
m
i
i
,...,
2
,1
,
=
W
uchun
ham aynan o`sha qiymatli konfiguratsiya bo`ladi. Qachonki har bir
i
L
qism graf o`zining
i
F
konfiguratsiyalar to`plamiga ega bo`lganda Teorema 2 ni yana umumiyroq hol uchun ham
isbotlash mumkin.
Novolterra operatorning taryektoriyasi
Ma`lumki, (4) ko`rinishdagi volterra operatorlarning trayektoriyasi to`la o`rganilgan. Bu ma`lum
nazariyadan foydalanib (9) ko`rinishda aniqlangan novolterra operator trayektoriyasini o`rganish
masalasi bilan shug`ullanamiz.
( )
( )
( )
( )
( )
y
m
w
m
y
m
w
m
y
j
i
i
i
i
i
a
+
-
=
,
belgilash olamiz. Har bir fiksirlangan
{
}
m
i
,..,
1
larda
( )
i
a
y
j
,
quyidagi xossalarni qanoatlantirishini son tekshirish mumkin
( )
( )
( )
.
1
,
,
,
,
-
=
i
i
i
a
a
a
y
j
y
j
y
j
(14)
JOURNAL OF IQRO – ЖУРНАЛ ИҚРО – IQRO JURNALI – volume 15, issue 02, 2025
ISSN: 2181-4341, IMPACT FACTOR ( RESEARCH BIB ) – 7,245, SJIF – 5,431
ILMIY METODIK JURNAL
Agar ixtiyoriy
{
}
m
i
,...,
1
uchun
( )
i
V
kvadratik stoxastik operatorning trayektoriyasini
asimptotik holatini bilgan holda
( )
®
®
l
X
X
i
i
l
,
*
,
,
w
w
ayta olsak, u holda (13) operatorni
ham trayektoriyasini
( )
( )
( )
®
®
*
l
l
,
s
l
s
l
aytish mumkin. Bu quyidagi chiziqli
tenglamalar sistemasini
( )
m
i
X
i
i
i
,...
1
,
,
*
,
:
=
W
=
=
W
*
w
s
l
w
w
s
s
(15)
yechish orqali topiladi.
Dastlab Volterra kvadratik stoxastik operatorlarni trayektoriyasi uchun ma`lum bo`lgan
natijalarni isbotsiz keltirib o`tamiz. Bunda biz quyidagicha belgilashlardan foydalanamiz.
Simpleks chegarasini
(
)
=
=
=
¶
=
-
-
0
:
,...,
1
1
1
1
n
j
j
n
n
n
x
S
x
x
x
S
, va simpleksning ichini
1
1
1
\
int
-
-
-
¶
=
n
n
n
S
S
S
kabi belgilaymiz.
Ixtiyoriy boshlang`ich
( )
1
0
-
n
S
x
nuqtani
olamiz va
( )
0
x
n
bilan
( )
{ }
l
x
trayektoriyaning limitik nuqtalar to`plamini belgilab olamiz.
Teorema 3.
1) Agar
( )
1
0
int
-
n
S
x
boshlag`ich nuqta qo`zg`olmas nuqta bo`lmasa ( ya`ni
( )
( )
0
0
x
Vx
),
u holda
( )
1
0
-
¶
n
S
x
n
.
2)
( )
0
x
n
limitik nuqtalar to`plami bitta nuqtadan iborat to`plam yoki cheksiz to`plam bo`lishi
mumkin.
3) Agar
1
int
*
-
n
S
x
volterra operatori uchun yakkalangan qo`zg`olmas nuqta bo`lsa, u holda
ixtiyoriy
( )
*
0
x
x
, boshlang`ich nuqta uchun
( )
{ }
l
x
trayektoriya yaqinlashunchi bo`ladi.
Natija 4. (13) novolterra operatori uchun
( )
0
x
n
limitik nuqtalar to`plami Teorema 3 ning 1)-3)
xossalari o`rinli.
ISBOT. 1) Teorema 3 ga ko`ra
0
*
~
,
=
w
i
X
shartni qanoatlantiruvchi kamida bitta
i
W
w
~
mavjud bo`ladi.
( )
0
>
s
l
o`rinli bo`lgani uchun (15) dan barcha
s
lar uchun shunday
w
s
~
=
i
topiladiki, u uchun
( )
0
*
=
s
l
tenglik o`rinli bo`ladi. Bu esa natijaning 1) tasdig`ini o`rinli
bo`lishini ko`rsatadi. 2) va 3) xossalar ham xuddi shunda (15) dan kelib chiqadi.
Endi volterra kvadratik stoxastik operatorlar uchun kiritilgan turnirlar nazariyasidan ayrim
tushunchalarni keltirib o`tamiz. Faraz qilaylik
k
i
larda
0
ki
a
bo`lsin. (4) formila
JOURNAL OF IQRO – ЖУРНАЛ ИҚРО – IQRO JURNALI – volume 15, issue 02, 2025
ISSN: 2181-4341, IMPACT FACTOR ( RESEARCH BIB ) – 7,245, SJIF – 5,431
ILMIY METODIK JURNAL
yordamida
n
uchli
n
G
to`la grafni aniqlaymiz.
n
G
grafning uchlari orasidagi yo`nalishni
quyidagicha aniqlaymiz. Agar
0
<
ki
a
bo`lsa, u holda graf
-
k
uchdan
-
i
uchga qarb
yo`nalgan deb va agar
0
>
ki
a
bo`lsa, u holda graf
-
i
uchdan
-
k
uchga qarab yo`nalgan deb
qaraladi. Ma`lumki to`la yo`naltirilgan graf turnir deb ataladi va
n
T
kabi belgilanadi. Agar
turnirning ixtiyoriy uchidan boshqa uchiga turnir yo`nalishi bo`yicha boorish mumkin bo`lsa
turnir kuchli turnir deyiladi.
Turnir uchlarini qayta nomerlash asosida
r
T
qism turnirni
n
T
turnirning dastlabki
r
ta
nuqtasidan tashkil topgan deb olish mumkin. Demak,
n
r
va
n
r
=
bo`ladi faqat va faqat agar
n
T
kuchli turnir bo`lsa.
Teorema 5. Faraz qilaylik
n
T
turnir kuchli bo`lmasin va
( )
1
0
int
-
n
S
x
bo`lsin. Agar
r
j
>
bo`lsa, u holda
®
l
da
( )
0
®
l
j
x
bo`ladi va bu yaqinlashish tezligi cheksiz kamayuvchi
geometric progressiya tezligida bo`ladi.
Natija 6.
( )
0
i
V
operatorga mos
( )
0
i
T
kuchli bo`lmagan turnir bo`luvchi
{
}
m
i
,...,
2
,1
0
son va
1
0
int
-
n
S
l
berilgan bo`lsin. U holda
0
0
~
i
i
W
s
,
W
s
uchun
( )
( )
0
®
s
l
l
munosabatni
qanoatlantiruvchi
0
0
~
~
i
i
W
W
qism to`plam mavjud va
®
l
da yaqinlashish tezligi cheksiz
kamayuvchi geometric progressiya tezligida bo`ladi.
MISOL.
{ }
2
,1
=
L
va
=
L
bo`lgan
(
)
L
G
,
L
=
grafni qaraymiz. Spin qiymatlar to`plami
{ }
A
a
,
=
F
bo`lsin. Bu holda
(
)
( )
( )
( )
{
}
a
a
A
a
a
A
A
A
,
,
,
,
,
,
,
4
3
2
1
=
=
=
=
=
W
s
s
s
s
kabi bo`ladi. Quyidagi
to`plamlarni topamiz.
(
)
(
)
{
}
A
A
,
,
,
1
1
1
=
=
L
W
s
s
s
(
)
(
)
( )
{
}
a
A
A
A
,
,
,
,
,
2
1
2
1
=
=
=
L
W
s
s
s
s
(
)
(
)
(
)
{
}
A
a
A
A
,
,
,
,
,
3
1
3
1
=
=
=
L
W
s
s
s
s
(
)
(
)
( )
( )
( )
{
}
a
a
A
a
a
A
A
A
,
,
,
,
,
,
,
,
,
4
3
2
1
4
1
=
=
=
=
=
L
W
s
s
s
s
s
s
(
)
(
)
(
)
{
}
a
A
A
A
,
,
,
,
,
2
1
1
2
=
=
=
L
W
s
s
s
s
(
)
( )
{
}
a
A
,
,
,
2
2
2
=
=
L
W
s
s
s
(
)
(
)
( )
( )
( )
{
}
a
a
A
a
a
A
A
A
,
,
,
,
,
,
,
,
,
4
3
2
1
3
2
=
=
=
=
=
L
W
s
s
s
s
s
s
JOURNAL OF IQRO – ЖУРНАЛ ИҚРО – IQRO JURNALI – volume 15, issue 02, 2025
ISSN: 2181-4341, IMPACT FACTOR ( RESEARCH BIB ) – 7,245, SJIF – 5,431
ILMIY METODIK JURNAL
(
)
( )
( )
{
}
a
a
a
A
,
,
,
,
,
4
2
4
2
=
=
=
L
W
s
s
s
s
(
)
(
)
( )
{
}
A
a
A
A
,
,
,
,
,
3
1
1
3
=
=
=
L
W
s
s
s
s
(
)
(
)
( )
( )
( )
{
}
a
a
A
a
a
A
A
A
,
,
,
,
,
,
,
,
,
4
3
2
1
2
3
=
=
=
=
=
L
W
s
s
s
s
s
s
(
)
( )
{
}
A
a
,
,
,
3
3
3
=
=
L
W
s
s
s
(
)
( )
( )
{
}
a
a
A
a
,
,
,
,
,
4
3
4
3
=
=
=
L
W
s
s
s
s
(
)
(
)
( )
{
}
a
a
A
A
,
,
,
,
,
4
1
1
4
=
=
=
L
W
s
s
s
s
(
)
(
)
( )
{
}
a
a
a
A
,
,
,
,
,
4
2
2
4
=
=
=
L
W
s
s
s
s
(
)
(
)
( )
{
}
a
a
A
a
,
,
,
,
,
4
3
3
4
=
=
=
L
W
s
s
s
s
(
)
( )
{
}
a
a
,
,
,
4
4
4
=
=
L
W
s
s
s
Endi quyidagi ehtimolliklar berilgan bo`lsin.
( )
1
1
a
m
=
A
,
( )
2
1
a
m
=
a
,
( )
1
2
b
m
=
A
,
( )
2
2
b
m
=
a
1
,
0
,
,
,
2
1
2
1
2
1
2
1
=
+
=
+
b
b
a
a
b
b
a
a
.
(4.8) formulaga asosan quyidagilarga ega bo`lamiz.
( )
( ) ( )
1
1
2
1
1
b
a
m
m
s
m
=
=
A
A
( )
( ) ( )
2
1
2
1
2
b
a
m
m
s
m
=
=
a
A
( )
( ) ( )
1
2
2
1
3
b
a
m
m
s
m
=
=
A
a
( )
( ) ( )
2
2
2
1
4
b
a
m
m
s
m
=
=
a
a
(
)
(
)
( )
1
1
1
1
1
,
,
b
a
s
m
s
s
m
=
=
L
W
(
)
(
)
( )
( )
1
2
1
1
1
2
1
2
1
,
,
a
b
a
b
a
s
m
s
m
s
s
m
=
+
=
+
=
L
W
(
)
(
)
( )
( )
1
1
2
1
1
3
1
3
1
,
,
b
b
a
b
a
s
m
s
m
s
s
m
=
+
=
+
=
L
W
(
)
(
)
( )
( )
( )
( )
1
,
,
2
2
1
2
2
1
1
1
4
3
2
1
4
1
=
+
+
+
=
+
+
+
=
L
W
b
a
b
a
b
a
b
a
s
m
s
m
s
m
s
m
s
s
m
(
)
(
)
( )
( )
1
2
1
1
1
1
2
1
2
,
,
a
b
a
b
a
s
m
s
m
s
s
m
=
+
=
+
=
L
W
JOURNAL OF IQRO – ЖУРНАЛ ИҚРО – IQRO JURNALI – volume 15, issue 02, 2025
ISSN: 2181-4341, IMPACT FACTOR ( RESEARCH BIB ) – 7,245, SJIF – 5,431
ILMIY METODIK JURNAL
(
)
(
)
( )
2
1
2
2
2
,
,
b
a
s
m
s
s
m
=
=
L
W
(
)
(
)
( )
( )
( )
( )
1
,
,
2
2
1
2
2
1
1
1
4
3
2
1
3
2
=
+
+
+
=
+
+
+
=
L
W
b
a
b
a
b
a
b
a
s
m
s
m
s
m
s
m
s
s
m
(
)
(
)
( )
( )
2
2
2
2
1
4
2
4
2
,
,
b
b
a
b
a
s
m
s
m
s
s
m
=
+
=
+
=
L
W
(
)
(
)
( )
( )
1
1
2
1
1
3
1
1
3
,
,
b
b
a
b
a
s
m
s
m
s
s
m
=
+
=
+
=
L
W
(
)
(
)
( )
( )
( )
( )
1
,
,
2
2
1
2
2
1
1
1
4
3
2
1
2
3
=
+
+
+
=
+
+
+
=
L
W
b
a
b
a
b
a
b
a
s
m
s
m
s
m
s
m
s
s
m
(
)
(
)
( )
1
1
3
3
3
,
,
b
a
s
m
s
s
m
=
=
L
W
(
)
(
)
( )
( )
2
2
2
1
2
4
3
4
3
,
,
a
b
a
b
a
s
m
s
m
s
s
m
=
+
=
+
=
L
W
(
)
(
)
( )
( )
( )
( )
1
,
,
2
2
1
2
2
1
1
1
4
3
2
1
1
4
=
+
+
+
=
+
+
+
=
L
W
b
a
b
a
b
a
b
a
s
m
s
m
s
m
s
m
s
s
m
(
)
(
)
( )
( )
2
2
2
2
1
2
4
2
4
,
,
b
b
a
b
a
s
m
s
m
s
s
m
=
+
=
+
=
L
W
(
)
(
)
( )
( )
2
2
2
1
2
3
4
3
4
,
,
a
b
a
b
a
s
m
s
m
s
s
m
=
+
=
+
=
L
W
(
)
(
)
( )
2
2
4
4
4
,
,
b
a
s
m
s
s
m
=
=
L
W
Endi (4.5) formulaga asosan avloddan – avlodga o`tish koeffisiyentlarini yozamiz.
( )
(
)
(
)
( )
( )
1
,
,
1
1
1
1
1
,
1
1
1
=
=
L
W
=
s
m
s
m
s
s
m
s
m
s
s
s
p
.
0
2
1
1
,
=
s
s
s
p
0
3
1
1
,
=
s
s
s
p
0
4
1
1
,
=
s
s
s
p
( )
(
)
(
)
( )
( )
( )
1
1
1
1
2
1
1
2
1
1
,
,
,
1
2
1
b
a
b
a
s
m
s
m
s
m
s
s
m
s
m
s
s
s
=
=
+
=
L
W
=
p
( )
(
)
(
)
( )
( )
( )
2
1
2
1
2
1
2
2
1
2
,
,
,
2
2
1
b
a
b
a
s
m
s
m
s
m
s
s
m
s
m
s
s
s
=
=
+
=
L
W
=
p
0
3
2
1
,
=
s
s
s
p
JOURNAL OF IQRO – ЖУРНАЛ ИҚРО – IQRO JURNALI – volume 15, issue 02, 2025
ISSN: 2181-4341, IMPACT FACTOR ( RESEARCH BIB ) – 7,245, SJIF – 5,431
ILMIY METODIK JURNAL
0
4
2
1
,
=
s
s
s
p
( )
(
)
(
)
( )
( )
( )
1
1
1
1
3
1
1
3
1
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.
JOURNAL OF IQRO – ЖУРНАЛ ИҚРО – IQRO JURNALI – volume 15, issue 02, 2025
ISSN: 2181-4341, IMPACT FACTOR ( RESEARCH BIB ) – 7,245, SJIF – 5,431
ILMIY METODIK JURNAL
0
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JOURNAL OF IQRO – ЖУРНАЛ ИҚРО – IQRO JURNALI – volume 15, issue 02, 2025
ISSN: 2181-4341, IMPACT FACTOR ( RESEARCH BIB ) – 7,245, SJIF – 5,431
ILMIY METODIK JURNAL
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JOURNAL OF IQRO – ЖУРНАЛ ИҚРО – IQRO JURNALI – volume 15, issue 02, 2025
ISSN: 2181-4341, IMPACT FACTOR ( RESEARCH BIB ) – 7,245, SJIF – 5,431
ILMIY METODIK JURNAL
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JOURNAL OF IQRO – ЖУРНАЛ ИҚРО – IQRO JURNALI – volume 15, issue 02, 2025
ISSN: 2181-4341, IMPACT FACTOR ( RESEARCH BIB ) – 7,245, SJIF – 5,431
ILMIY METODIK JURNAL
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1
=
b
bo`lsa, u holda
{
}
0
:
4
3
1
=
=
=
x
x
x
S
va
{
}
0
:
2
1
2
=
=
=
x
x
x
S
to`plamlar
(16) operatorning qo`zg`olmas nuqtalar to`plami bo`ladi va ixtiyoriy
2
1
)
0
(
S
S
x
boshlang`ich nuqtaning trayektoriyasi quyidagi munosabat qanoatlantiradi
<
>
®
1
2
1
2
lim
1
2
1
1
)
(
a
a
agar
S
agar
S
x
l
l
3. Agar
1
2
1
=
a
bo`lsa, u holda
{
}
0
:
4
2
3
=
=
=
x
x
x
S
va
{
}
0
:
3
1
4
=
=
=
x
x
x
S
to`plamlar
(16) operatorning qo`zg`olmas nuqtalar to`plami bo`ladi va ixtiyoriy
( )
4
3
0
S
S
x
boshlang`ich nuqtaning trayektoriyasi quyidagi munosabat qanoatlantiradi
JOURNAL OF IQRO – ЖУРНАЛ ИҚРО – IQRO JURNALI – volume 15, issue 02, 2025
ISSN: 2181-4341, IMPACT FACTOR ( RESEARCH BIB ) – 7,245, SJIF – 5,431
ILMIY METODIK JURNAL
( )
<
>
®
.
1
2
,
,1
2
,
lim
1
4
1
3
b
b
agar
S
agar
S
x
l
l
4.
Agar
1
2
2
1
1
=
=
b
a
bo`lsa, u holda
{
}
3
1
4
2
5
,
:
x
x
x
x
x
S
=
=
=
va
{
}
4
3
2
1
6
,
:
x
x
x
x
x
S
=
=
=
to`plamlar
(16) operatorning qo`zg`olmas nuqtalar to`plami
bo`ladi.
Markov zanjirlari
Bizga
,...
,
2
1
E
E
hodisalar berilgan bo`lsin va ularning ro`y berish ehtimolliklari
k
p
bo`lsin.
n
ta hodisaning birgalikda ro`y berish ehtimolligi shartli ehtimollik asosida
(
)
{
}
n
n
j
j
j
j
j
j
p
p
p
E
E
E
P
=
...
,...,
,
1
0
1
0
topiladi. Markov zanjirlarining sodda holini
qaraymiz, ya`ni ro`y berishi mumkin bo`lgan hodisa faqat o`zidan oldingi hodisaga bog`liq
bo`lgan holni qaraymiz.Bunda boshlang`ich
k
E
hodisa ehtimolligi
k
p
bo`g`liqsiz bo`lib, ammo
har bir
(
)
k
j
E
E
,
juftga fiksirlangan
jk
p
shartli ehtimollik mos kelsin, ya`ni biror tajribada
j
E
ro`y bergan bo`lsin, u holda keyingi tajribada
k
E
hodisani ro`y berish ehtimoli
jk
p
ga teng
bo`ladi. Dastlabki hodisa
k
E
ning ehtimoli
k
a
berilgan deb olsak,
(
)
{
}
jk
j
k
j
p
a
E
E
P
=
,
,
(
)
{
}
kr
jk
j
r
k
j
p
p
a
E
E
E
P
=
,
,
,
(
)
{
}
rs
kr
jk
j
s
r
k
j
p
p
p
a
E
E
E
E
P
=
,
,
,
, va
hokazo umuman olganda
(
)
{
}
n
n
n
n
n
j
j
j
j
j
j
j
j
j
j
j
j
p
p
p
p
a
E
E
E
P
1
1
2
2
1
1
0
0
1
0
...
,...,
,
-
-
-
=
.
(1)
jk
p
shartli ehtimollikni
j
E
holatdan
k
E
holatga o`tish ehtimolligi deb ataymiz va bu shartli
ehtimolliklardan tuzilgan
=
nn
n
n
n
n
p
p
p
p
p
p
p
p
p
P
K
M
O
M
M
K
K
2
1
2
22
21
1
12
11
(17)
matrisaga o`tish matrisasi deyiladi. Aniqlanishiga ko`ra
P
matrisa kvadrat matrisa bo`lib, har bir
satr elementlari yig`indisi birga teng. Bunday matrisalarga stoxastik matrisalar deb ataladi.
Ixtiyoriy stoxastik matrisa o`tish matrisasi bo`lib hisoblanadi va oldindan berilgan
{ }
k
a
taqsimot
bilan Markov zanjirini aniqlaydi.
(
)
L
,
L
– karrali qirralarga va halqalarga ega bo`lmagan graf berilgan bo`lsin.
L
– grafning
barcha uchlari va
L
– barcha qirralar to`plami bo`lsin.
JOURNAL OF IQRO – ЖУРНАЛ ИҚРО – IQRO JURNALI – volume 15, issue 02, 2025
ISSN: 2181-4341, IMPACT FACTOR ( RESEARCH BIB ) – 7,245, SJIF – 5,431
ILMIY METODIK JURNAL
Undan tashqari,
F
chekli to`plam berilgan bo`lsin.
F
to`plamni genotiplar to`plami (statistic
mexanika masalalarida
F
to`plamga spin qiymatlari to`plami ) deyiladi.
F
®
L
:
s
akslantirishga hujayra deyiladi. Barcha hujayralar to`plamini
W
bilan beilgilaymiz va
W
chekli
to`plamda aniqlangan barcha ehtimollik o`lchovlari to`plamini
(
)
F
L
,
S
orqali belgilab olamiz.
Berilgan
(
)
L
,
L
grafning maksimal bog`liq komponentalari
{ }
)
,...,
1
(
n
i
i
=
L
lar bo`lsin.
Ixtiyoriy ikkita
,
,
2
1
W
s
s
konfiguratsiya fiksirlaymiz va ular uchun quyidagi to`plamni
aniqlaymiz.
{
}
i
i
or
i
i
L
L
L
L
=
=
W
=
L
W
2
1
2
,
1
:
)
,
,
(
s
s
s
s
s
s
s
barcha
n
i
,...,
1
=
Biror
(
)
F
L
,
S
m
ehtimollik o`lchovi
W
da berilgan bo`lsin va bu o`lchov ixtiyoriy
W
s
hujayra uchun
( )
0
>
s
m
shartni qanoatlantirsin, ya`ni biror potensial bilan aniqlangan Gibbs
o`lchovi bo`lsin.
s
s
s
,
2
1
p
ehtimolliklarni quyidagi formula bialn aniqlaymiz.
L
W
L
W
=
hollarda
boshqa
,
0
)
,
,
(
agar
,
))
,
,
(
(
)
(
2
1
2
1
,
2
1
s
s
s
s
s
m
s
m
s
s
s
p
Ma`lumki, bu ehtimolliklar uchun quyidagi munosabatlar o`rinli bo`ladi.
,
0
,
2
1
s
s
s
p
s
s
s
s
s
s
,
,
1
2
2
1
p
p
=
va
W
=
s
v
s
s
1
,
2
1
p
barcha
W
2
1
,
s
s
.
(
)
F
L
,
S
da aniqlangan
( )
m
V
kvadratik stoxastik operator
s
s
s
,
2
1
p
ehtimolliklar orqali
quyidagicha aniqlangan bo`lsin:
ixtiyoriy
)
,
(
F
L
S
l
o`lchov uchun
)
,
(
)
(
F
L
=
S
V
l
l
o`lchovni quyidagi formula
bilan aniqlaymz.
W
=
W
s
s
l
s
l
s
l
s
s
s
s
s
),
(
)
(
)
(
2
1
,
,
2
1
2
1
p
Shunday qilib, qachonki kvadratik stoxastik operatorlar qurilgan bo`lsa, Gibbs taqsimotlari hosil
bo`ladi.
Bu paragrafda ushbu konktruksiya yordamida qurilgan markov zanjiriga mos novolterra
operatorning dinamikasini o`rganish bilan tanishib chiqamiz.
)
,
(
L
G
L
=
– chekli graf va
{ }
m
i
i
,...,
2
,1
,
=
L
lar
G
grafning maksimal bo`g`liq
qism graflari bo`lsin.
i
i
L
F
=
W
bilan
{ }
m
i
i
,...,
2
,1
,
=
L
qism graf ustida aniqlangan
barcha konfiguratsiyalar to`plamini belgilaymiz.
JOURNAL OF IQRO – ЖУРНАЛ ИҚРО – IQRO JURNALI – volume 15, issue 02, 2025
ISSN: 2181-4341, IMPACT FACTOR ( RESEARCH BIB ) – 7,245, SJIF – 5,431
ILMIY METODIK JURNAL
i
W
to`plamda aniqlangan
i
m
ehtimollik o`lchovlari ixtiyoriy
i
W
s
konfiguratsiya uchun
( )
0
>
s
m
i
shartni qanoatlantirsin.
m
W
W
W
=
W
...
2
1
to`plamda
m
ehtimollik
o`lchovini quyidagi formula bilan aniqlaymiz
( )
(
)
( )
m
m
p
p
p
m
s
s
s
s
s
s
s
a
s
s
s
m
s
m
1
3
2
2
1
1
2
1
,...,
,
-
=
=
K
(18)
1
=
m
hol qurilgan kvadratik stoxastik operator Volterra kvadratik stoxastik operatori
bo`ladi.
(
)
W
=
m
j
j
j
j
,
,
,
2
1
K
va
(
)
W
=
m
y
y
y
y
,
,
,
2
1
K
konstruksiyalarni fiksirlaymiz. Bu
konstruksiyalarga mos
(
)
{
}
{
}
m
i
i
i
i
m
,...,
2
,
1
,
,
:
,
,
)
,
,
(
1
=
W
=
=
L
W
y
j
s
s
s
s
y
j
K
va
( )
( )
{
}
L
W
=
=
-
-
hollarda
boshqa
,
0
)
,
,
(
agar
,
,...,
1
,
,
:
,
,
1
1
,
1
1
3
2
2
1
1
3
2
2
1
y
j
s
s
a
s
a
y
j
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
jy
m
i
i
i
i
m
m
m
m
m
p
p
p
p
p
p
p
K
K
K
(19)
tengliklarni hosil qilamiz. Biz bu yerda quyidagi tenglikdan foydalandik.
(
)
(
)
( )
{
}
=
-
=
W
m
i
i
i
i
m
m
m
p
p
p
G
,...,
1
,
,
:
,
,
1
1
1
3
2
2
1
,
,
y
j
s
s
s
s
s
s
s
s
s
s
a
y
j
m
K
K
.
Shunday qilib, (3) o`lchov yordamida (6) formula bilan aniqlangan kvadratik stoxastik operatorni
quyidagicha yozish mumkin
( )
(
)
( )
( )
{
}
(
)
(
)
( ) ( )
y
l
j
l
s
a
s
a
s
s
l
s
l
y
y
y
y
j
j
j
j
y
j
s
s
s
s
s
s
s
s
s
s
s
s
s
s
s
W
=
W
=
=
-
-
=
=
i
i
m
i
i
m
i
i
i
m
m
m
m
m
m
i
m
p
p
p
p
p
p
:
,...,
:
,...,
,...,
1
,
,
:
,
,
1
1
1
1
1
1
1
3
2
2
1
1
3
2
2
1
,...,
K
K
K
.
(20)
Bu operatorning dinamikasini o`rganish murakkab bo`lgani uchun ba`zi hollarinigina qarab
o`tamiz.
1– hol: Agar
P
matrisaning hamma satrlari o`zaro teng bo`lsa, ya`ni
m
k
p
p
p
mk
k
k
=
=
=
1
,
...
2
1
JOURNAL OF IQRO – ЖУРНАЛ ИҚРО – IQRO JURNALI – volume 15, issue 02, 2025
ISSN: 2181-4341, IMPACT FACTOR ( RESEARCH BIB ) – 7,245, SJIF – 5,431
ILMIY METODIK JURNAL
bajarilsa, u holda markov zanjiriga mos keluvchi kvadratik stoxastik operator oldingi
paragrafda ko`rib o`tilgan ko`paytma o`lchovga mos keluvchi novolterra kvadratik stoxastik
operator bilan bir xil bo`ladi. Bu operatorning dinamikasi U.A. Rozikov va B.N.
Shamsiddinovlarning ilmiy ishlarida o`rganilgan.
2– hol
Bizga ikkita nuqtadan iborat va ularni tutashtiruvchi qirra mavjud bo`lmagan graf berilgan
bo`lsin, ya`ni
{ }
2
,1
=
L
va
=
L
bo`lgan
(
)
L
G
,
L
=
grafni qaraymiz. Spin qiymatlar
to`plami
{ }
A
a
,
=
F
bo`lsin. Bu holda
(
)
( )
( )
( )
{
}
a
a
A
a
a
A
A
A
,
,
,
,
,
,
,
4
3
2
1
=
=
=
=
=
W
s
s
s
s
kabi bo`ladi. Quyidagi
to`plamlarni topamiz.
(
)
(
)
{
}
A
A
,
,
,
1
1
1
=
=
L
W
s
s
s
(
)
(
)
( )
{
}
a
A
A
A
,
,
,
,
,
2
1
2
1
=
=
=
L
W
s
s
s
s
(
)
(
)
(
)
{
}
A
a
A
A
,
,
,
,
,
3
1
3
1
=
=
=
L
W
s
s
s
s
(
)
(
)
(
)
(
)
( )
{
}
a
a
A
a
a
A
A
A
,
,
,
,
,
,
,
,
,
4
3
2
1
4
1
=
=
=
=
=
L
W
s
s
s
s
s
s
(
)
(
)
(
)
{
}
a
A
A
A
,
,
,
,
,
2
1
1
2
=
=
=
L
W
s
s
s
s
(
)
(
)
{
}
a
A
,
,
,
2
2
2
=
=
L
W
s
s
s
(
)
(
)
(
)
(
)
( )
{
}
a
a
A
a
a
A
A
A
,
,
,
,
,
,
,
,
,
4
3
2
1
3
2
=
=
=
=
=
L
W
s
s
s
s
s
s
(
)
(
)
( )
{
}
a
a
a
A
,
,
,
,
,
4
2
4
2
=
=
=
L
W
s
s
s
s
(
)
(
)
( )
{
}
A
a
A
A
,
,
,
,
,
3
1
1
3
=
=
=
L
W
s
s
s
s
(
)
(
)
(
)
(
)
( )
{
}
a
a
A
a
a
A
A
A
,
,
,
,
,
,
,
,
,
4
3
2
1
2
3
=
=
=
=
=
L
W
s
s
s
s
s
s
(
)
(
)
{
}
A
a
,
,
,
3
3
3
=
=
L
W
s
s
s
(
)
(
)
( )
{
}
a
a
A
a
,
,
,
,
,
4
3
4
3
=
=
=
L
W
s
s
s
s
(
)
(
)
( )
{
}
a
a
A
A
,
,
,
,
,
4
1
1
4
=
=
=
L
W
s
s
s
s
(
)
(
)
( )
{
}
a
a
a
A
,
,
,
,
,
4
2
2
4
=
=
=
L
W
s
s
s
s
(
)
(
)
( )
{
}
a
a
A
a
,
,
,
,
,
4
3
3
4
=
=
=
L
W
s
s
s
s
JOURNAL OF IQRO – ЖУРНАЛ ИҚРО – IQRO JURNALI – volume 15, issue 02, 2025
ISSN: 2181-4341, IMPACT FACTOR ( RESEARCH BIB ) – 7,245, SJIF – 5,431
ILMIY METODIK JURNAL
(
)
( )
{
}
a
a
,
,
,
4
4
4
=
=
L
W
s
s
s
Endi quyidagi dastlabki ehtimollik
( )
1
a
a
=
A
,
( )
2
a
a
=
a
,
1
,
0
,
2
1
2
1
=
+
a
a
a
a
.
va
=
22
21
12
11
p
p
p
p
P
elementlari manfiymas stoxastik o`tish matrisasi berilgan bo`lsin.
(3) formulaga asosan quyidagilarga ega bo`lamiz.
( )
11
1
1
p
a
s
m
=
,
( )
12
1
2
p
a
s
m
=
,
( )
21
2
3
p
a
s
m
=
,
( )
22
2
4
p
a
s
m
=
(
)
(
)
( )
11
1
1
1
1
,
,
p
a
s
m
s
s
m
=
=
L
W
(
)
(
)
( ) ( )
1
12
1
11
1
2
1
2
1
,
,
a
a
a
s
m
s
m
s
s
m
=
+
=
+
=
L
W
p
p
(
)
(
)
( ) ( )
21
2
11
1
3
1
3
1
,
,
p
p
a
a
s
m
s
m
s
s
m
+
=
+
=
L
W
(
)
(
)
( ) ( ) ( ) ( )
1
,
,
22
2
21
2
12
1
11
1
4
3
2
1
4
1
=
+
+
+
=
+
+
+
=
L
W
p
p
p
p
a
a
a
a
s
m
s
m
s
m
s
m
s
s
m
(
)
(
)
( ) ( )
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12
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11
1
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2
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a
a
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21
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p
p
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(
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2
4
,
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p
p
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a
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L
W
JOURNAL OF IQRO – ЖУРНАЛ ИҚРО – IQRO JURNALI – volume 15, issue 02, 2025
ISSN: 2181-4341, IMPACT FACTOR ( RESEARCH BIB ) – 7,245, SJIF – 5,431
ILMIY METODIK JURNAL
(
)
(
)
( ) ( )
2
22
2
21
2
3
4
3
4
,
,
a
a
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p
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22
2
4
4
4
,
,
p
a
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=
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L
W
Endi (4) formulaga asosan avloddan – avlodga o`tish koeffisiyentlarini yozamiz.
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JOURNAL OF IQRO – ЖУРНАЛ ИҚРО – IQRO JURNALI – volume 15, issue 02, 2025
ISSN: 2181-4341, IMPACT FACTOR ( RESEARCH BIB ) – 7,245, SJIF – 5,431
ILMIY METODIK JURNAL
( )
(
)
(
)
( )
( ) ( ) ( ) ( )
21
2
4
3
2
1
3
4
1
3
,
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p
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p
JOURNAL OF IQRO – ЖУРНАЛ ИҚРО – IQRO JURNALI – volume 15, issue 02, 2025
ISSN: 2181-4341, IMPACT FACTOR ( RESEARCH BIB ) – 7,245, SJIF – 5,431
ILMIY METODIK JURNAL
( )
(
)
(
)
( )
( ) ( )
22
2
12
1
12
1
4
2
2
4
2
2
,
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2
p
p
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JOURNAL OF IQRO – ЖУРНАЛ ИҚРО – IQRO JURNALI – volume 15, issue 02, 2025
ISSN: 2181-4341, IMPACT FACTOR ( RESEARCH BIB ) – 7,245, SJIF – 5,431
ILMIY METODIK JURNAL
0
2
4
3
,
=
s
s
s
p
( )
(
)
(
)
( )
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21
22
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p
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W
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a
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s
JOURNAL OF IQRO – ЖУРНАЛ ИҚРО – IQRO JURNALI – volume 15, issue 02, 2025
ISSN: 2181-4341, IMPACT FACTOR ( RESEARCH BIB ) – 7,245, SJIF – 5,431
ILMIY METODIK JURNAL
0
1
4
4
,
=
s
s
s
p
0
2
4
4
,
=
s
s
s
p
0
3
4
4
,
=
s
s
s
p
( )
(
)
(
)
( )
( )
1
,
,
4
4
4
4
4
,
4
4
4
=
=
L
W
=
s
m
s
m
s
s
m
s
m
s
s
s
p
Bu hisoblangan ehtimolliklardan kvadratik stoxastik operatorning ko`rinishini topamiz. Agar
(
)
W
=
4
3
2
1
,
,
,
x
x
x
x
x
taqsimot berilgan bo`lsa, u holda
( )
x
V
x
=
kvadratik stoxastik
operatorning ko`rinishi quyidagicha bo`ladi.
+
+
+
+
+
=
+
+
+
+
+
=
+
+
+
+
+
=
+
+
+
+
+
=
4
3
22
4
2
22
2
12
1
22
2
3
2
22
2
4
1
22
2
2
4
4
4
3
21
3
2
21
2
4
1
21
2
3
1
21
2
11
1
21
2
2
3
3
4
2
22
2
12
1
12
1
3
2
12
1
4
1
12
1
2
1
12
2
2
2
3
2
11
1
4
1
11
1
3
1
21
2
11
1
11
1
2
1
11
2
1
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
:
x
x
p
x
x
p
p
p
x
x
p
x
x
p
x
x
x
x
p
x
x
p
x
x
p
x
x
p
p
p
x
x
x
x
p
p
p
x
x
p
x
x
p
x
x
p
x
x
x
x
p
x
x
p
x
x
p
p
p
x
x
p
x
x
V
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
(21)
va bu yerda
(
)
;1
,
2
,1
,
0
,
,
2
1
2
1
1
=
+
=
=
a
a
a
a
a
m
j
j
(21) operatorni dastlab simpleks chegaralarida qaraymiz. Operatorni ko`rinishidan simpleks
uchlarining qo`zg`olmas nuqta bo`lishi kelib chiqadi.
{
}
0
:
4
3
3
12
=
=
=
G
x
x
S
x
simpleks
chegarasida (21) operator quyidagi ko`rinishga o`tadi
(
)
(
)
(
)
(
)
=
=
-
+
=
-
+
=
=
=
=
+
=
+
=
0
0
1
2
1
1
2
1
0
0
2
2
:
4
3
1
12
2
2
2
11
1
1
4
3
2
1
12
2
2
2
2
1
11
2
1
1
x
x
x
p
x
x
x
p
x
x
x
x
x
x
p
x
x
x
x
p
x
x
V
Demak,
{
}
0
:
4
3
3
12
=
=
=
G
x
x
S
x
to`plam (21) operator uchun invariant to`plam bo`lib
hisoblanadi va bu simpleks chegarasida operator Volterra tipidagi operatori bo`ladi.
Xuddi shunday
{
}
{
}
0
:
,
0
:
4
2
3
13
4
3
3
12
=
=
=
G
=
=
=
G
x
x
S
x
x
x
S
x
JOURNAL OF IQRO – ЖУРНАЛ ИҚРО – IQRO JURNALI – volume 15, issue 02, 2025
ISSN: 2181-4341, IMPACT FACTOR ( RESEARCH BIB ) – 7,245, SJIF – 5,431
ILMIY METODIK JURNAL
{
}
0
:
3
2
3
14
=
=
=
G
x
x
S
x
,
{
}
0
:
4
1
3
23
=
=
=
G
x
x
S
x
,
{
}
0
:
3
1
3
24
=
=
=
G
x
x
S
x
,
{
}
0
:
2
1
3
34
=
=
=
G
x
x
S
x
simpleks chegaralarini
tekshirish mumkinki, bu chegaralar ham invariant to`plamlar bo`ladi va bu invariant to`plamlar
ustida (21) operator Volterra operatori bo`ladi.
Berilgan
=
22
21
12
11
p
p
p
p
P
o`tish matrisasining satrlari bir –biriga teng bo`lsa, ko`paytma
yordamida hosil qilinadigan novolterra operatoriga keladi. Matrisa stoxastik matrisa bo`lgani
uchun va aniqlik uchun
21
11
1
,
2
1
p
p
<
a
deb olamiz . Matrisaning stoxastikligidan
12
22
p
p
<
kelib chiqadi.
( )
2
1
x
x
x
+
=
j
funksiyani qaraymiz.
( )
(
)
+
+
+
+
+
+
=
+
=
3
2
11
1
4
1
11
1
3
1
21
2
11
1
11
1
2
1
11
2
1
2
1
2
2
2
2
x
x
p
x
x
p
x
x
p
p
p
x
x
p
x
x
x
x
V
a
a
a
a
a
j
(
)
+
+
=
+
+
+
+
+
+
2
2
1
4
2
22
2
12
1
12
1
3
2
12
1
4
1
12
1
2
1
12
2
2
2
2
2
2
x
x
x
x
p
p
p
x
x
p
x
x
p
x
x
p
x
a
a
a
a
a
(
)
+
+
+
+
+
+
+
+
2
2
1
3
2
1
4
1
1
4
2
22
2
12
1
12
1
3
1
21
2
11
1
11
1
2
2
2
2
x
x
x
x
x
x
x
x
p
p
p
x
x
p
p
p
a
a
a
a
a
a
a
a
(
) (
)(
) (
) ( )
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
j
=
+
=
+
+
+
+
=
+
+
+
+
2
1
4
3
2
1
2
2
1
3
2
4
1
4
2
3
1
.
Natijada
(
)
( ) ( )
,...
1,
0
,
1
=
+
n
x
x
n
n
j
j
(22)
tengsizlikni hosil qilamiz.
( )
0
2
1
+
=
x
x
x
j
bo`lgani uchun quyidan chegaralangan va kamayuvchi ketma ketlikning
limiti mavjud bo`ladi. Demak,
( )
x
j
funksiya (21) dinamik sistema uchun Lyapunov funksiyasi
bo`ladi.
a) Agar
( )
3
0
int
S
x
bo`lsa, u holda
( )
( )
1
0
<
x
j
bo`ladi.
( )
( )
0
lim
=
®
n
n
x
j
bo`ladi.
Haqiqatan ham, teskarisini faraz qilamiz.
( )
( )
0
lim
>
=
®
h
j
n
n
x
bo`lsin. U holda
(
)
( )
( )
( )
(
)
(
)
( )
( )
=
+
+
=
=
+
+
®
+
®
n
n
n
n
n
n
n
n
x
x
x
x
x
x
2
1
1
2
1
1
1
lim
lim
1
j
j
JOURNAL OF IQRO – ЖУРНАЛ ИҚРО – IQRO JURNALI – volume 15, issue 02, 2025
ISSN: 2181-4341, IMPACT FACTOR ( RESEARCH BIB ) – 7,245, SJIF – 5,431
ILMIY METODIK JURNAL
( )
( )
(
)
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( )
( )
n
n
n
n
n
n
n
n
n
n
n
n
n
x
x
x
x
x
x
x
x
p
p
p
x
x
p
p
p
x
x
2
1
3
2
1
4
1
1
4
2
22
2
12
1
12
1
3
1
21
2
11
1
11
1
2
2
1
2
2
2
2
lim
+
+
+
+
+
+
+
+
=
®
a
a
a
a
a
a
a
a
( )
( )
( ) ( )
( ) ( )
( ) ( )
( ) ( )
( )
( )
+
+
+
+
+
+
+
+
=
®
n
n
n
n
n
n
n
n
n
n
n
n
n
x
x
x
x
x
x
x
x
p
p
p
x
x
p
p
p
x
x
2
1
3
2
1
4
1
1
4
2
22
2
12
1
12
1
3
1
21
2
11
1
11
1
2
1
2
2
2
2
lim
1
a
a
a
a
a
a
a
a
( )
( )
(
)
( )
( )
(
)
( ) ( )
( ) ( )
( ) ( )
( ) ( )
n
n
n
n
n
n
n
n
n
n
n
n
x
x
x
x
x
x
p
p
p
x
x
p
p
p
x
x
x
x
3
2
1
4
1
1
4
2
22
2
12
1
12
1
3
1
21
2
11
1
11
1
4
3
2
1
2
2
2
2
a
a
a
a
a
a
a
a
+
+
+
+
+
=
+
+
B
u tenglikdan quyidagi tenglamalr sistemasiga kelamiz.
=
=
+
=
+
1
2
1
2
1
2
1
22
2
12
1
12
1
21
2
11
1
11
1
a
a
a
a
a
a
a
p
p
p
p
p
p
Bu tenglamalar sistemasidan quyidagi tengliklarni hosil qilamiz.
5
.
0
1
=
a
bo`ladi
1
2
1
=
+
a
a
tenglikdan
5
.
0
2
=
a
hosil bo`ladi.
Sistemaning birinchi tenglamasidan
21
11
21
11
11
5
.
0
5
.
0
p
p
p
p
p
=
+
=
va xuddi shunday
ikkinchi tenglamasidan
22
12
p
p
=
hosil bo`ladi bu yechimlar
21
11
1
,
2
1
p
p
<
a
,
12
22
p
p
<
deb olngan shartlarga zid. Demak, faraz noto`g`ri.
Bundan
( )
3
0
S
x
¶
w
munosabat kelib chiqadi.
b) Endi
( )
12
0
G
x
bo`lsin u holda yuqorida ko`rib o`tganimizdek, bu holda operator Volterra
operator bo`ladi. Demak, Volterra operatorlari uchun limitik nuqtalar to`plami simpleks
chegarasida bo`lishini hisobga olsak,
( )
3
0
S
x
¶
w
tasdiqqa ega bo`lamiz.
c) Endi boshlangi`ch nuqta
( )
{
}
4
,
3
,
2
,1
0
:
3
0
=
=
=
G
i
x
S
x
x
i
i
simpleks chegarasida
bo`lgan holni qaraymiz. (6) operatorning ko`rinishiga asosan
( )
3
int
S
V
i
G
ga ega bo`lamiz.
Yuqorida ko`rib o`tilgan a) punkt tasdig`iga ko`ra
( )
3
0
S
x
¶
w
.
21
11
2
,
2
1
p
p
>
a
holda
( )
4
3
x
x
x
+
=
y
funksiya uchun yuqorida mulohazalarni
takrorlasak, natijada
( )
3
0
S
x
¶
w
munosabatga ega bo`lamiz.
Yuqoridagi mulohazalarni umumlashtirib biz quyidagi teorema isbotladik.
JOURNAL OF IQRO – ЖУРНАЛ ИҚРО – IQRO JURNALI – volume 15, issue 02, 2025
ISSN: 2181-4341, IMPACT FACTOR ( RESEARCH BIB ) – 7,245, SJIF – 5,431
ILMIY METODIK JURNAL
Teorema 1.
1) Agar
21
11
1
,
2
1
p
p
<
a
bo`lsa,
( )
2
1
x
x
x
+
=
j
funksiya (21) operator uchun Lyapunov
funksiyasi bo`ladi.
2)
11
21
2
,
2
1
p
p
<
a
bo`lsa,
( )
4
3
x
x
x
+
=
y
funksiya (21) operator uchun Lyapunov
funksiyasi bo`ladi.
3) Ixtiyoriy
( )
3
0
S
x
uchun
( )
3
0
S
x
¶
w
.
Natija.
{ }
2
,1
=
L
va
=
L
bo`lgan
(
)
L
G
,
L
=
graf va spin qiymatlar to`plami
{ }
A
a
,
=
F
bo`lgan holga mos Markov zanjiriga mos kvadratik stoxastik operatorlar asimptotik Volterra
operatorlariga aylanadi, ya`ni bu operatorlarning o`zi Volterra operatori emas, ammo bu
operatorlar iteratsiyasi Volterra operatorlariga aylanadi.
ADABIYOTLAR
1.
Koshi Masalasi Yechimini Regulyarlashtirish
FF Homidov Educational Research in
Universal Sciences 2 (15 SPECIAL), 205-207
2.
Tekislikda momentli elastiklik nazariyasi sistemasi yechimi uchun somilian-betti formulasi
F.F Homidov Educational Research In Universal Sciences 2 (11), 132-136
3.
Elastiklik Nazariyasi Sistemasining Fundamental Yechimlari Matritsasini Qurish
F.F.Homidov Educational Research In Universal Sciences 2 (16), 300-302
4.
Koshi Masalasini Statika Tenglamalari Sistemasi Uchun Yechish
F. F Homidov GOLDEN
BRAIN 2 (6), 80-83
5.
Tekislikda Somilian–Betti Formulasi
F. F Homidov Educational Research in Universal
Sciences 3 (1), 587-589
6.
GARMONIK FUNKSIYALAR VA ULARNING XOSSALARI
H. F Faxriddinovich
PEDAGOG 7 (5), 511-521
7.
H.F Faxriddinovich PEDAGOG 7 (4), 281-290
8.
Boboqulova, M. X. (2025). O ‘TA O ‘TKAZUVCHANLIK.
Introduction of new innovative
technologies in education of pedagogy and psychology
,
2
(5), 60-67.
9.
Boboqulova,
M.
X.
(2025).
VODOROD
ATOMINING
KVANT
NAZARIYASI.
Introduction of new innovative technologies in education of pedagogy and
psychology
,
2
(5), 113-121.
10.
Boboqulova, M. X. (2025). IDEAL VA YOPISHQOQ SUYUQLIK. BERNULLI
TENGLAMASI.
Introduction of new innovative technologies in education of pedagogy and
psychology
,
2
(5), 122-129.
11.
The Cauchy problem for a system of moment e-elasticity theory existence sign of solution y
HF Faxriddinovich Multidisciplinary Journal of Science and Technology 4 (3), 433-440
12.
KOSHI MASALASINI STATIKA TENGLAMALARI SISTEMASI UCHUN YECHISH
FF Homidov GOLDEN BRAIN 2 (6), 80-83
13.
TEKISLIKDA SOMILIAN–BETTI FORMULASI
FF Homidov Educational Research in
Universal Sciences 3 (1), 587-589
14. У.У.Жамилов У.А.Розиков “О динамике строго неволътерростих квадратичных
стохастические операторов на двумерноле симплексе” . 2009 “Математический сборник”
Том 200 N:9 81-94 б
