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
KVADRATIK STOXOSTIK OPERATORLAR
Annotasiya:
Kvadratik stoxastik operatorlar va novalterra kvadratik stoxastik operatorlar,
kvadratik stoxastik prosesslar, novalterra kvadratik stoxastik proseslarni o’rganish (E-chekli
bo’lganda). olingan natija matematik genetikaning masalalarini yechishda qo’llanilishi mukin.
Biologic masalalarnni yechishda ham novalterra kvadratik stoxastik prosesdan foydalaniladi.
Kalit so'zlar:
bu operatorlar nazariyasi , kvadratik stoxastik operatorlar, novalterra kvadratik
stoxastik operatorlar, tasodifiy proseslar, kvadratik stoxastik proseslar, novalterra kvadratik
stoxastik proseslardan iborat
Avvalombor shuni aytib o’tish kerakki ota-onadan tug’ilgan farzand o’g’il bo’ladigan bo’lsa, u
holda bu farzand kuchli (dominant) belgilarni to’liq o’z onasidan oladi, agar tug’iladigan farznd
qiz bo’ladigan bo’lsa, u holda u kuchli belgilarni ham otasidan ham onasidan aralash holda oladi.
Buni quyidagicha tushuntirish mumkin. Masalan, onaning sochlari qora bo’lib, otaning sochlari
sariq rangda bo’lsa, u holda ularndan tug’iladigan farzand albatta qora sochli bo’ladi, chunki
qora sochlar kuchli (dominant) belgi hisoblanadi.
Shuni ta’kidlash kerakki, biz qarayotgan sistema biologik sistema bo’ladi. Chunki faqatgina
biologik sistemadagina tug’iladigan “farzand” o’z “ota-ona”sining belgilarini takrorlaydi, ya’ni
o’z “ota-ona”siga o’xshamaydigan farzand tug’ilii mumkin mumkin bo’lmagan hodisadir.
E
vektor fazoni
F
vektor fazoga biror
F
E
U
®
:
aks ettirishi berilgan bo’lsin.
Ta’rif:Agar har qanday
E
y
x
,
va
K
b
a
,
uchun
)
(
)
(
)
(
y
U
x
U
y
x
U
b
a
b
a
+
=
+
munosabat o’rinli bo’lsa,u chiziqli aks ettirish yoki chiziqli operator deyiladi.Xususan,
F
fazo
sifatida
K
maydon olinsa, bunday aks ettirish chiziqli forma yoki chiziqli funksional deyiladi.
E
va
F
topolagik vektor fazolarning birini ikkinchisiga aks ettiruvchi
T
chiziqli operator
berilgan bo’lsin.
Agar
E
va
F
dagi topologiyalarga nisbatan
T
operator uzluksiz bo’lsa, u uzluksiz chiziqli
operator deyiladi.
1-teorema:
E
topologik vektor fazoni
F
topologik vektor fazoga aks ettiruvchi
T
chiziqli
operator uzluksiz bo’lishi uchun u nol nuqtada uzluksiz bo’lishi zarur va yetarli.
Isbot: (Zaruriyligi) O’z o’zidan ravshan.
(Yetarliligi)
q
q
=
T
(qulaylik uchun
E
va
F
fazolarning nol elementlarini bitta
q
harfi bilan
belgilaymiz) munosabatdan va
T
operatorning nol nuqtada uzluksizligidan
F
fazodagi
q
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
elementning ixtiyoriy
V
atrofi uchun
E
fazodagi
q
elementning shunday
U
atrofi mavjudligi
kelib chiqadiki , ular uchun ushbu
V
U
T
)
(
munosabat o’rinli bo’ladi. Agar
-
=
}
{
U
nolning atroflari bazisi va
E
x
ixtiyoriy element
bo’lsa, u holda
x
ning atroflari bazisi
}
{
U
x
+
ko’rinishga ega bo’ladi.Demak,
V
Tx
U
T
Tx
U
x
T
+
+
=
+
)
(
)
(
Bu munosabatlardan
T
operatorning ixtiyoriy
x
nuqtada uzluksiz ekanligi bevosita ko’rinib
turibdi.
1-natija:Agar
T
chiziqli operator biror
E
x
0
nuqtada uzluksiz bo’lsa, u holda
T
uzluksiz
chiziqli operatordir.
2-natija:Agar
E
va
F
metrikalangan topologik vector fazolar bo’lsa, u holda
T
chiziqli operator
uzluksiz bo’lishi uchun ushbu
q
q
®
®
n
n
Tx
x
munosabat bajarilishi zarur va yetarli.
Misol:
]
1,
0
[
C
F
E
=
=
topologik vektor fazoda T operatorni quyidagicha aniqlaymiz:
=
=
1
0
)
(
)
,
(
ds
s
x
s
t
K
Tx
y
Bu yerda
)
,
(
s
t
K
funksiyani
]
1,
0
[
]
1,
0
[
da uzluksiz deb faraz qilamiz. Bevosita ko’rinib
turibdiki , T operator
]
1,
0
[
C
fazoni
]
1,
0
[
C
fazoga aks ettiruvchi xhiziqli operatordir. Endi biz
]
1,
0
[
C
metrikalangan topologik vektor fazo bo’lgani uchun 2-natijaga ko’ra
q
q
®
®
n
n
Tx
x
Munosabat o’rinli ekanini ko’rsatish yetarli.
]
1,
0
[
]
1,
0
[
to’plam kompakt bo’lgani uchun
)
,
(
s
t
k
funksiya chegaralangan:
M
s
t
K
)
,
(
U holda
)
,
(
)
(
)
(
)
,
(
)
,
(
max
max
1
0
1
0
1
0
o
r
o
r
n
n
s
n
t
n
x
M
s
x
M
ds
s
x
s
t
K
Tx
=
=
Demak,
0
®
n
x
ekanligidan
0
®
n
Tx
kelib chiqadi.
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-teorema:
F
E
,
haqiqiy topologik vektor fazolar bo’lib
T
additiv (ya’ni
Ty
Tx
y
x
T
+
=
+
)
(
xossaga ega bo’lgan) va uzluksiz aks ettirish bo’lsin. U holda
T
chiziqli operatordir.
Kvadratik stoxostik operatorlar: Populyatsiya-bir xil turdagi biologik elementlardan tashkil
topgan jamiyat. Bu jamiyat boshqa jamiyatlardan alohida yashash maydoniga ega va o’zaro erkin
chatishishlari mumkin va uzoq vaqt mavjud bo’ladi. Populyatsiyalar jinsiy yo’l bilan avlodlar
orqali bog’langan bo’lishi mumkin. Lekin jinsiy yo’l bilan ko’paymatdigan populyatsiyalar
mavjud.
Kvadratik stoxostik operatorlar tushunchasi birinchi marta 1924-yilda S.N.Bernshteyn
tomonidan kiritilgan.Bunday operatorlar matematik genetikaning ko’pgina modellarida tez-tez
uchraydi va bunday modellar o’rganilgan ko’pgina ilmiy maqolalar mavjud.Populyatsiya
genetikasiga yondashuv mustaqil populyatsiyaning evolyutsion operatorlarini aniq tavsiflash
masalasi loyihasidan kelib chiqadi.
Xardi-Vaynberg qonuni:
1.Avloddan-avlodga o’tganda genlar o’zgarmaydi;
2.Birinchi avloddan boshlab genotip chastotalari quyidagi formula bilan aniqlanadi:
=
-
=
=
-
=
=
2
2
2
)
1
(
2
)
1
(
2
q
p
z
pq
p
p
y
p
x
(chiziqli emas)
Jinsiy bog’lanmagan genlar
A
genning
n
- avlodda onadan paydo bo’lishlik ehtimoli(chastotasi)
)
(
n
P
a
deb belgilaymiz;
B
genning
n
- avlodda onadan paydo bo’lishlik ehtimoli(chastotasi)
)
(
1
n
P
a
-
deb belgilaymiz;
A
genning
n
- avlodda otadan paydo bo’lishlik ehtimoli(chastotasi)
)
(
n
P
e
deb belgilaymiz;
B
genning
n
- avlodda otadan paydo bo’lishlik ehtimoli(chastotasi)
)
(
1
n
P
e
-
deb belgilaymiz;
Ayollar
Erkaklar
XX
XY
Bolasi
XX
XY
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
1
0
2
1
2
2
1
1
1
1
)
2
1
(
)
(
))
0
(
)
0
(
(
3
2
))
0
(
2
)
0
(
(
3
1
)
0
(
)
1
(
),
1
(
)
0
(
2
1
,1
,
2
1
2
1
)
(
))
1
(
)
(
(
2
1
))
(
)
(
(
2
1
)
1
(
2
)
1
(
)
1
(
)
1
(
))
(
1
))(
(
1
(
)
1
(
)
(
))
(
1
(
))
(
1
)(
(
)
1
(
)
(
)
(
)
1
(
)
(
)
1
(
C
C
n
P
P
P
C
P
P
C
P
P
P
P
U
C
C
U
U
U
U
n
P
U
n
P
n
P
n
P
n
P
n
P
n
y
n
x
n
P
n
P
n
P
n
z
n
P
n
P
n
P
n
P
n
y
n
P
n
P
n
x
n
P
n
P
n
a
a
e
a
e
a
e
e
a
n
n
n
n
n
n
a
n
a
a
e
a
a
a
e
a
e
a
e
a
e
a
a
e
-
+
=
-
=
+
=
=
=
=
=
=
+
=
+
=
=
-
+
=
+
=
+
+
+
+
=
+
-
-
=
+
-
+
-
=
+
=
+
=
+
-
+
l
l
l
l
Xardi-Vaynberg qonuni asimptotik o’zgaradi.
Ta’rif:
1
)
0
(
-
n
S
x
uchun
),...
(
),
(
,
)
1
(
)
2
(
)
0
(
)
1
(
)
0
(
x
V
x
x
V
x
x
=
=
ketma-ketlik
)
0
(
x
ning
V
operator
ta’siridagi traektoriyasi deyiladi.
,
:
1
1
-
-
®
n
n
S
S
V
n
k
x
x
P
x
n
j
i
j
i
k
ij
k
,...
2
,1
,
1
,
.
'
=
=
=
(1.1)
=
"
=
n
k
k
ij
k
ij
j
i
P
P
1
,
,
,
,1
,
0
(1.2)
Ta’rif:(2.1) operator Volterra kvadratik stoxostik operator deyiladi agar
}
,
{
,
0
,
j
i
k
P
k
ij
=
(1.3)
shart bajarilsa .
Teorema:Har qanday Volterra kvadratik stoxastik operator (ya’ni (2.1),(2.2),(2.3)) quyidagi
ko’rinishga ega:
n
k
x
a
x
x
n
i
i
ki
k
k
,...,
2
,1
),
1
(
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
bunda
0
,
,1
,
=
-
=
ii
ki
ik
ki
a
i
k
a
a
a
Isbot:
1
1
1
1
2
1
2
2
0
1
0
2
1
)
(
2
1
2
)
1
(
)
)
1
2
(
1
(
)
2
1
(
)
2
(
,1
,
0
2
,
,
,
,
,
,
,
,
,
,
1
1
1
,
,
1
1
,
,
,
,
1
,
2
,
1
,
,
'
-
-
-
-
=
-
=
-
=
-
-
=
-
=
+
=
-
+
=
+
-
=
=
+
=
=
=
=
+
=
=
=
=
=
=
=
=
=
ki
k
ik
k
ik
k
ik
ik
i
ki
i
ki
k
ik
i
ki
k
ki
k
ik
k
ik
ki
n
i
i
ki
k
n
k
i
i
n
k
i
i
i
k
ik
k
i
k
ik
n
k
i
i
i
k
n
k
i
i
i
k
ik
k
k
kk
k
k
kk
i
kk
n
i
i
k
k
ik
k
k
kk
n
j
i
j
i
k
ij
k
a
P
P
P
a
P
P
P
P
P
P
P
a
x
a
x
x
P
x
x
P
x
x
x
P
x
P
x
k
i
P
P
x
x
P
x
P
x
x
P
x
Bizga
n
n
R
R
B
®
·
·
:
)
,
(
simmetrik chiziqli operator beilgan bo’lsin.
n
n
R
R
Q
®
:
akslantiruvchi
kvadratik operator quyidagi tenglik orqali aniqlanadi:
)
,
(
)
(
x
x
B
x
Q
=
.
Berilgan
Q
kvadratik operator bo’yicha
B
-simmetrik chiziqli operator poyarizatsiya
munosabati yordamida quyidagicha bir qiymatli aniqlanadi:
[
] [
]
)
(
)
(
)
(
2
1
)
(
)
(
4
1
)
,
(
y
Q
x
Q
y
x
Q
y
x
Q
y
x
Q
y
x
B
-
-
+
=
-
-
+
=
.
n
R
fazoda esa har qanday kvadratik operator koordinata ko’rinishida quyidagicha bo’ladi:
=
=
=
n
j
i
j
i
k
ij
n
j
i
j
i
ij
x
x
p
x
x
p
x
Q
1
,
,
1
,
1
,
,
,..
)
(
,
bu yerda
n
n
R
x
x
x
x
=
)
,...
,
(
2
1
,
k
ij
p
,
lar esa strukturaviy koeffisientlar bo’lib,
k
ji
k
ij
p
p
,
,
=
o’rinli
deb hisoblash mumkin.
Ta’rif.
Q
kvadratik operator:
1. Elliptik tipda deyiladi, agar qandaydir chiziqli
f
funksional uchun
))
(
(
x
Q
f
kvadratik
forma musbat aniqlangan bo’lsa;
2. Parabolik tipda deyiladi, agar elliptik bo’lmasdan va mavjud
f
funksional uchun
))
(
(
x
Q
f
kvadratik forma manfiymas bo’lsa;
3. Giperbolik tipda deyiladi, agar yuqoridagi ikki hol bo’lmasa, ya’ni qolgan barcha hollarda,
xuddi shuningdek
0
f
uchun
))
(
(
x
Q
f
kvadratik forma o’z ishorasini almashtirib tursa.
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
Ta’rif. Quyidagi
=
=
=
=
-
0
,1
;
0
:
)
,...,
,
(
1
2
1
1
i
n
i
i
i
n
m
n
x
x
x
R
x
x
x
x
S
)
1
(
-
n
o’lchovli simpleks berilgan bo’lsin. Kvadratik operator
1
1
:
-
-
®
n
n
S
S
V
quyidagi tenglik
bilan aniqlanadi
( )
n
k
x
x
p
x
Vx
n
j
i
j
i
k
ij
k
k
,1
1
,
,
=
=
=
=
bunda
,1
,
,
0
1
,
,
,
,
,
=
=
=
n
j
i
k
ij
k
ji
k
ij
k
ij
p
p
p
p
1
2
1
)
,...,
,
(
-
=
n
n
S
x
x
x
x
Kvadratik stoxastik operatorlar chekli to’plamda berilgan barcha ehtimollik o’lchovlari to’plami
(simpleks)ni o’zini-o’ziga akslantiruvchi akslantirishdir.
Matematik genetikada
V
populyatsiyada evolyutsion operator deyiladi.
Populyatsiya
n
F
F
F
,...,
,
2
1
avlodlarning ketma-ketligi bilan farqlanadi.
Har xil turdagi genotiplarda populyatsiya amalgam oshmaydi. Populyatsiyaga kirishuvchi
har bir tur faqat qandaydir
n
tipga (ko’rinishga) kiradi, ya’ni
,...
2
,1
=
n
. Populyatsiyaning holati
bu turlarning
1
2
1
)
,...,
,
(
-
=
n
m
S
x
x
x
x
ehtimoli bo’ladi.
k
ij
p
,
-irsiyat koeffisienti, turning
tug’ilishiga bog’liq,
k
-tur
i
va
j
turlar bilan qo’shilishidagi koeffisieti. Tasodifiy
populyatsiyalar
x
ehtimollik holati
j
i
x
x
orqali ifodalanadi. Natijada
n
k
x
x
p
x
n
j
i
j
i
k
ij
k
,1
,
1
,
,
=
=
=
to’g’ridan-to’g’ri avlodlarning to’la ko’rinishdagi ehtimoli bo’ladi.
Agar biror avlodda populyatsiya
x
holatda bo’lsa, keyingi avlodda
Vx
x
=
bo’ladi. Ushbu
k
ji
k
ij
p
p
,
,
=
tengli tur jinsiga bog’liq emasligini bildiradi (germofroditizm).
Matematik genetikaning asosiy vazifalaridan biri
,..,
,
,
0
2
0
0
x
V
Vx
x
trayektoriyaning limitik holatiga bog’liq, ya’ni
V
ning aksiga bog’liq.
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
{
n
E
=
bo’lsin.
=
-
=
=
=
n
i
i
i
n
n
n
x
x
R
x
x
x
x
S
1
2
1
1
}
1
,
0
:
)
,...,
,
(
{
To’plam
)
1
(
-
n
o’lchovli simpleks deyiladi. Har bir
1
-
n
S
x
element
E
dagi ehtimollik
o’lchovi hisoblanadi va uni
n
elementdan tashkil topgan biologik (fizik) holat sifatida
interpritatsiya qilish (izohlash) mumkin.
Kvadratik stoxostik operator
1
1
:
-
-
®
n
n
S
S
V
quyidagi ko’rinishga ega:
=
=
n
j
i
j
i
k
j
i
k
x
x
P
x
V
1
,
,
,
'
:
(1.4)
Bunda
=
=
=
n
k
k
ij
k
ji
k
ij
k
ij
P
P
P
P
1
,
,
,
,
1
,
,
0
(1.5)
Hozirgi vaqtda volterra kvadratik operatorlar nazariyasi quyidagi shart asosida rivojlangan:
n
k
j
i
j
i
k
P
k
ij
,...,
2
,1
,
,
},
,
{
,
0
,
=
=
(1.6)
Novalterra tipidagi kvadratik stoxostik operatorlar.
=
=
n
j
i
j
i
k
j
i
k
x
x
P
x
V
1
,
,
,
'
:
(1.7)
=
=
=
n
k
k
ij
k
ji
k
ij
k
ij
P
P
P
P
1
,
,
,
,
1
,
,
0
(1.8)
Ta’rif:Kvadratik operatorni (
1.7
),(
1.8
) novolterra kvadratik operator deymiz,agarda
n
k
j
i
j
i
k
P
k
ij
,...,
2
,1
,
,
},
,
{
,
0
,
=
=
(1.9)
bo’lsa
Bizga ma’lumki, novolterra kvadratik operatorlar faqat
3
n
bo’lganda mavjud bo’lib
2
=
n
bo’lganda (
1.7
) va (
1.8
) shartlar bir vaqtning o’zida bajarilmaydi.
Agar
3
=
n
bo’lsa, novalterra kvadratik operator quyidagi ko’rinishda bo’ladi:
+
+
=
+
+
=
+
+
=
2
1
2
2
2
1
'
3
3
1
2
3
2
1
'
2
3
2
2
3
2
2
'
1
2
2
2
:
x
x
x
bx
x
x
x
dx
ax
x
x
x
cx
x
x
V
b
a
(1.10)
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
Bunda
1
,
0
,
,
,
,
,
=
+
=
+
=
+
d
c
b
a
d
c
b
a
b
a
b
a
(1.11)
Ta’rif:
x
x
V
=
)
(
tenglamaning yechimi
V
ning qo’zg’almas nuqtasi deyiladi
Teorema:
x
x
m
m
=
®
)
(
lim
mavjud bo’lsa,
x
nuqta
V
uchun qo’zg’almas nuqta bo’ladi.
Isbot:faraz qilamiz quyidagi limit mavjud
C
x
Ax
x
x
x
x
x
x
x
x
x
x
x
x
x
V
I
x
x
P
x
x
V
x
V
x
x
x
V
x
n
n
n
n
n
n
j
i
j
i
k
ij
k
m
m
m
m
m
m
=
¶
¶
¶
¶
¶
¶
¶
¶
¶
¶
¶
¶
=
=
=
=
=
=
=
®
+
®
+
l
l
,
...
.....
..........
..........
...
)
)(
(
)
(
)
(
)
(
'
2
'
1
'
'
1
2
'
1
1
'
1
1
,
,
'
)
(
)
1
(
)
(
)
1
(
lim
lim
}.
0
:
{
}
,...,
0
:
{
,...,
2
,1
),
1
(
1
1
1
1
1
'
=
=
=
=
=
=
+
=
-
-
-
=
x
S
x
A
o
x
x
S
x
Г
n
k
x
a
x
x
n
m
n
n
m
n
i
i
ki
k
k
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
xos qiymat
Ta’rif:
)
)(
(
x
V
I
ning
Barcha xos qiymatlari moduli birlik aylanadan tashqarida bo’lsa,
x
qo’zg’almas nuqta giperbolik
nuqta:
-tortuvchi deyiladi agar barcha xos qiymatlarning moduli
1
dan kichik bo’lsa;
-itaruvchi deyiladi agar barcha xos qiymatlarning moduli
1
dan katta bo’lsa;
-sedlo deyiladi agar ba’zilarining moduli
1
dan kichik, ba’zilari katta.
Ta’rif:Agar
1
-
n
S
A
uchun
A
A
V
)
(
shart bajarilsa,
A
ga
V
ga nisbatan invariant to’plam
deyiladi.
}
:
)
(
{
)
(
A
x
x
V
A
V
=
Teorema:1)
1
-
n
S
ning barcha nuqtalari volterra kvadratik stoxostik operator uchun qo’zg’almas
bo’ladi;
2)
1
-
n
S
ning har bir
m
o’lchovli yog’i volterra kvadratik stoxostik operator uchun invariant
bo’ladi;
3)
1
-
n
S
ning har bir
-
m
o’lchovli yoqining ichi volterra kvadratik stoxostik operator uchun
invariant bo’ladi:
Volterra kvadratik stoxostik operator uchun Lyapunov funksiyasi.
)
(
:
1
)
(
1
)
0
(
1
1
+
-
-
-
=
®
m
m
n
n
n
x
V
x
S
x
S
S
V
Ta’rif: Uzluksiz funksional
R
S
n
®
-
1
:
j
Lyapunov funksiyasi deyiladi agar
)
0
(
x
"
uchun
)
(
)
(
lim
m
m
x
j
®
-mavjud bo’lsa.
Teorema:Volterra kvadratik stoxostik operator uchun
n
P
n
P
x
x
x
...
)
(
1
1
=
j
funksional Lyapunov
funksiyasi bo’ladi bunda:
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
k
P
a
P
P
n
i
i
ki
n
i
i
i
,...,
2
,1
,
0
,
0
1
1
=
=
=
=
Isbot:
)
(
)
(
)
(
)
)
(
1
)(
(
)
1
)(
(
)
1
(
)
(
1
,
0
,
0
,
)
1
(
)
(
))
1
(
...(
))
1
(
(
)
...(
)
(
))
(
(
)
(
)
(
)
1
(
1
1
1
1
1
1
1
1
1
1
1
1
1
1
'
'
1
'1
1
1
n
n
n
i
n
j
i
j
ij
n
j
n
i
i
j
ji
n
i
n
i
i
ji
j
i
i
k
n
i
i
i
n
i
P
i
P
n
j
n
i
i
ji
P
n
i
i
ni
n
P
n
i
i
i
P
n
P
x
x
x
x
P
a
x
x
P
a
x
x
a
P
x
P
P
a
a
p
a
x
a
x
x
a
x
x
a
x
x
x
x
V
x
i
j
n
n
j
j
j
j
j
j
j
j
j
-
=
+
=
+
=
+
=
=
+
+
=
=
=
+
=
=
=
=
=
=
=
=
=
=
=
=
Teotema:Volterra kvadratik stoxostik operatorlar uchun har qanday teskari traektoriya
yaqinlashuvchi bo’ladi.
Teorema:
)
,
,
(
,
,
2
1
2
1
,
'
m
s
s
s
s
s
s
s
Ф
G
V
V
x
x
P
x
=
W
=
operator volterra kvadratik stoxostik
operator bo’lishi uchun
G
graf bog’langan bo’lishi zarur va yetarli.
}
,
{
,
0
,
2
1
,
1
2
1
s
s
s
s
s
s
=
=
P
G
Л
F-kvadratik stoxostik operator.
E
F
P
n
E
k
kk
=
=
1
},
,...,
2
,1
{
,
F
-femele (ayollar to’plami)
M
-male (erkaklar to’plami)
(1.12)
"
=
=
=
=
=
=
k
M
j
F
i
M
j
i
F
j
i
k
M
j
i
F
j
i
k
P
P
n
E
F
E
M
k
ij
n
k
k
ij
;
;
,
0
,
;
,
;
0
,
0
,
;
,
;
0
,1
1
}
,...,
2
,1
{
\
,
0
,
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
Ta’rif:(3.6) shart bajarilsa, bunday operatorga
F
-kvadratik stoxostik operator deymiz.
Misol:
}
2
{
},
1
{
},
2
;1
;
0
{
=
=
=
M
F
E
1
,
0
,
,
0
0
0
0
0
1
0
0
0
0
1
1
1
1
1
2
,
22
2
,
11
2
,
12
2
,
02
2
,
01
2
,
00
1
,
22
1
,
11
1
,
12
1
,
02
0
,
01
1
,
00
0
,
22
0
,
11
0
,
12
0
,
20
0
,
01
0
,
00
=
+
+
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
c
b
a
c
b
a
P
cP
P
P
P
P
P
bP
P
P
P
P
P
aP
P
P
P
P
=
=
-
-
=
+
-
=
2
1
'
2
2
1
'
1
2
1
2
1
'
0
2
2
)
1
(
2
1
)
(
2
1
x
cx
x
x
bx
x
x
x
a
x
x
c
b
x
(1.13)
=
=
-
-
=
2
1
2
2
1
1
2
1
0
2
2
)
1
(
2
1
x
cx
x
x
bx
x
x
x
a
x
b
x
c
x
bc
bc
a
bc
bc
a
x
2
1
2
1
0
,
2
1
2
2
1
1
2
1
0
=
=
+
-
=
-
-
=
(
)
0
;
0
;1
,
2
1
;
2
1
;
2
1
2
+
-
b
c
bc
a
bc
0
,
2
1
,
2
1
1
2
1
2
1
=
=
=
=
+
+
a
c
b
c
b
a
b
c
2
1
)
(
x
x
x
=
j
=
0
;
2
)
4
(
0
,
0
2
2
1
bc
bc
x
bcx
bc
m
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
4
1
4
)
1
(
)
1
(
4
)
(
4
)
(
4
4
0
)
(
2
0
2
2
2
1
2
2
1
)
(
lim
-
-
=
+
+
=
®
x
a
x
x
c
b
x
bcx
x
m
m
j
Teorema:(
1.12
F
-kvadratik stoxostik operator yagona qo’zg’almas nuqta
(
)
0
;
0
;1
ga ega va
)
0
(
x
"
uchun
0
)
(
lim
=
®
m
m
x
Isbot:
0
)
(
2
2
)
(
)
(
2
)
(
1
)
1
(
1
®
=
=
+
m
m
m
m
x
b
x
bx
x
j
(
)
}
,...,
2
,1
{
}
2
;1
;
0
{
,
0
;
0
;1
)
(
n
E
E
x
m
=
=
®
}
,...,
1
{
}
,...,
1
{
},
0
{
\
1
1
m
m
M
m
F
E
F
+
=
=
=
=
-
-
=
=
+
=
=
+
=
n
k
x
x
P
x
x
x
P
x
m
i
m
m
j
j
i
k
ij
k
m
i
m
m
j
j
i
ij
,1
,
2
)
1
(
2
1
1
1
1
1
1
1
,
'
1
1
0
,
'
0
Teorema:
F
"
-kvadratik stoxostik operator yagona
3
2
1
n
0
,...,
0
,1
qo’zg’almas nuqtaga ega va
)
0
(
x
"
uchun
)
0
,...,
0
,1
(
)
(
lim
=
®
m
m
x
Isbot:
=
+
=
=
1
1
1
1
)
(
m
i
m
m
j
j
i
x
x
x
j
n
k
x
x
x
x
x
n
m
k
n
n
n
n
,1
),
(
2
0
)
4
1
(
)
(
0
))
(
(
)
(
)
(
)
(
2
)
(
2
)
(
)
1
(
=
+
j
j
j
j
1
1
:
-
-
®
n
n
S
S
V
=
=
n
j
i
j
i
k
ij
k
x
x
P
x
1
,
,
'
(1.14)
=
=
n
k
k
ij
k
ij
P
P
1
,
,
1
,
0
,
,
,
k
ji
k
ij
P
P
=
(1.15)
Ta’rif:Agar (
1.14
) , (
1.15
) kvadratik stoxostik operator quyidagi shartni qanoatlantirsa
}
,
{
,
0
,
j
i
k
P
k
ij
=
(1.16)
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
bunday operatorga qat’iy novolterra kvadratik stoxostik operator deyiladi.
n=2 da qat’iy novolterra kvadratik stoxostik operator mavjud emas chunki:
0
,1
,
0
,
0
,1
2
,
11
2
,
11
1
,
12
1
,
11
0
,
00
=
=
=
=
=
P
P
P
P
P
n=3 da
b
a
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
3
,
22
3
,
23
3
,
13
3
,
12
3
,
11
2
,
22
2
,
23
2
,
13
2
,
12
2
,
11
1
,
22
1
,
23
1
,
13
1
,
12
1
,
11
,
0
,
0
,1
,
0
,
0
,1
,
0
,
,1
,
0
,
0
,
0
P
P
P
P
b
P
P
P
P
P
a
P
P
P
P
P
P
+
+
=
+
+
=
+
+
=
2
1
2
2
2
1
'
3
3
1
2
3
2
1
'
2
3
2
2
3
2
2
'
1
2
2
2
x
x
x
bx
x
x
x
dx
ax
x
x
x
cx
x
x
b
a
(1.17)
1
;1
,
,
,
,
,
0
=
+
=
+
=
+
b
a
b
a
d
c
b
a
d
c
b
a
(1.18)
+
+
=
+
+
=
+
+
=
2
1
2
2
2
1
3
3
1
2
3
2
1
2
3
2
2
3
2
2
1
2
2
2
x
x
x
bx
x
x
x
dx
ax
x
x
x
cx
x
x
b
a
)
(
2
1
)
(
2
)
(
)
(
)
(
1
3
2
3
1
2
1
2
3
2
1
2
2
3
2
3
1
2
1
2
3
2
1
2
2
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
d
c
x
b
a
x
x
+
+
+
+
+
=
+
+
+
+
+
+
+
+
=
b
a
Teorema: (3.11) operator ixtiyoriy (
1.18
) ni qanoatlantiruvchi parametrlar uchun yagona
qo’zg’almas nuqtaga ega.
Isbot: 1-hol:
1
,
1
0
,
0
3
1
3
2
1
2
=
+
+
-
-
=
x
x
x
x
x
x
a
a
0
1
)
1
2
(
3
2
3
2
3
2
2
=
-
+
+
+
+
x
cx
x
x
x
a
)
1
(
4
)
1
2
(
4
3
2
3
2
3
2
-
+
-
+
=
-
=
x
cx
x
ac
b
D
a
)
(
2
1
2
3
3
2
x
D
x
x
j
a
=
-
-
-
=
)
(
2
1
2
3
3
1
x
a
D
x
x
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
-
+
+
d
d
c
c
x
2
1
4
1
;
0
2
4
1
;
0
3
c yoki d=0
[ ]
1
;
0
3
x
)
(
,1
)
(
)
(
3
3
3
x
f
x
x
x
x
=
=
+
+
y
j
x
a
a
a
a
x
a
x
x
c
a
a
a
a
x
a
x
ad
x
f
=
-
+
+
+
-
-
+
+
+
-
+
-
+
-
+
+
+
-
+
-
=
)
(
2
2
)
(
2
1
4
)
1
(
4
)
1
(
4
)
(
2
1
4
)
1
(
4
)
1
(
4
)
(
2
2
a
a
a
a
a
a
a
a
a
a
a
a
[
]
0
...
1
4
)
1
(
4
)
1
(
4
)
(
1
)
1
(
2
)
(
1
0
2
'
+
+
+
-
+
-
-
+
-
+
-
=
a
x
a
x
ad
a
a
a
x
ad
x
f
x
a
a
a
funksiya har xilbo’lishi mumkin.Shuning uchun funksiyaning 2-tartibli hosilasini olib
tekshiramiz.
0
)
(
''
x
f
ekanligi kelib chiqadi;
2-hol:
0
,
0
=
a
a
sodda bo’ladi;
3-hol:
0
,
0
=
a
a
sodda bo’ladi;
4-hol:
0
,
0
=
=
a
a
+
=
+
=
+
=
2
2
1
3
3
1
2
3
2
3
2
2
3
1
)
(
2
2
x
x
x
x
x
dx
x
x
x
cx
x
)
5
4
(
2
8
5
4
)
5
3
7
(
,
)
5
4
(
2
1
5
)
7
5
3
(
,
2
5
3
*
1
*
2
*
3
-
-
+
-
=
-
-
+
-
=
-
=
c
x
c
x
x
Teorema isbotlandi.
Ta’rif: (3.13) , (3.14) , (3.15) tengliklar bilan berilgan operatorlarga ajraluvchi operatorlar
deyiladi.
=
-
-
=
®
n
j
i
j
i
k
ij
k
n
n
x
x
P
x
S
S
V
1
,
,
'
1
1
,
:
(1.19)
j
i
P
P
n
k
k
ij
k
ij
,
,1
,
0
1
,
,
"
=
=
(1.20)
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
k
j
i
b
a
P
jk
ik
k
ij
,
,
,
,
"
=
(1.21)
=
=
=
=
=
=
=
n
j
j
jk
n
i
i
ik
n
j
i
j
i
jk
ik
n
j
i
j
i
k
ij
k
x
b
x
a
x
x
b
a
x
x
P
x
1
1
1
,
1
,
,
'
)
)(
(
n
n
R
R
A
®
:
-chiziqli akslantirish
n
n
n
n
n
j
i
ij
R
x
x
x
x
R
x
x
x
x
A
a
A
=
®
=
=
=
)
,...,
,
(
)
,...,
,
(
:
,
)
(
'
'
2
'
1
'
2
1
1
,
n
k
x
a
x
A
n
i
i
ki
k
,1
,
:
1
'
=
=
=
(3.2)
0
jk
ki
b
a
j
i
b
a
n
k
jk
ki
,
,1
1
"
=
=
)
1
,...,
1,
1
(
,
1
...
1
......
1
...
1
=
=
T
j
T
Ab
AB
n=3 da
-
-
-
=
3
3
2
2
1
1
1
1
1
y
y
b
y
y
b
y
y
b
A
,
=
2
1
2
1
2
1
1
1
0
1
1
0
b
B
,
]
1
;
0
[
y
1-hol:
0
)
det(
)
det(
=
=
B
A
va barcha satrlari bir xil
n
k
b
a
x
k
k
k
,1
,
1
1
'
=
=
=
n
n
a
a
a
a
a
a
A
1
12
11
1
12
11
...
...
...
...
...
...
,
=
n
n
b
b
b
b
b
b
B
1
12
11
1
12
11
...
...
...
...
...
...
1
1
:
-
-
®
n
n
S
S
V
)
,...,
,
(
1
1
12
12
11
11
n
n
b
a
b
a
b
a
2-hol:
0
)
det(
,
0
)
det(
=
B
A
B
-bir xil satrlardan tuzulgan
)
,...,
,
(
;
,...,
2
,1
,
)
0
(
)
0
(
2
)
0
(
1
)
0
(
1
1
'
n
n
i
i
ki
k
k
x
x
x
x
n
k
x
a
b
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
(
2
)
0
(
)
0
(
)
0
(
)
0
(
)
1
(
)
1
(
2
)
1
(
1
)
1
(
,
)
(
)
,...,
,
(
x
M
MMx
x
Mx
x
V
x
x
x
x
n
=
=
=
=
=
)
0
(
)
(
)
(
1
,
,
)
(
x
M
x
b
a
M
m
m
n
j
i
jk
ki
=
=
=
3-hol:
0
)
det(
)
det(
=
=
B
A
lekin barcha satrlari bir xil emas
)
)(
(
1
1
'
=
=
=
n
j
j
jk
n
i
i
ik
k
x
b
x
a
x
(1.22)
)
1
,...,
1,
1
(
,
,
=
=
T
j
jk
ik
k
ij
Ab
b
a
P
=
=
+
=
+
=
n
i
n
i
i
ki
k
i
ki
k
k
x
a
x
x
a
x
x
1
1
'
)
)
1
(
(
)
1
(
Teorema: (
1.22
) operator uchun
R
S
n
c
®
-
1
:
y
quyidagi
=
=
n
k
k
k
c
c
x
c
x
1
)
min(
)
(
y
funksional
Lyapunov funksiyasi bo’ladi agar
=
n
n
c
c
I
c
c
A
AC
M
M
1
1
yoki
n
n
c
c
I
c
c
B
M
M
1
1
bunda
1
,
0
ik
ik
b
a
=
=
1
0
0
0
1
0
0
0
1
K
K
K
K
K
K
K
I
E
=
®
)
(
)
(
lim
m
c
m
x
y
mavjud
Isbot:
Ic
Ac
=
=
=
=
=
=
=
=
=
=
=
=
n
k
n
i
n
i
n
i
i
i
i
n
j
ik
k
i
ik
n
k
k
n
j
j
jk
n
i
i
ik
n
k
k
k
k
c
x
c
x
a
c
x
a
c
x
b
x
a
c
x
c
x
1
1
1
1
1
1
1
1
1
'
'
)
(
)
(
y
Lemma:Agar
=
"
=
n
k
ik
i
i
a
a
1
,1
uchun bo’lsa, u holda
IC
AC
nolmas yechimga ega.
Isbot:
=
=
=
=
=
=
n
k
i
i
n
k
ik
n
k
k
ik
k
ik
c
a
a
a
a
c
a
1
1
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
2
1
2
1
2
1
,
,
'
s
s
s
s
s
s
s
s
x
x
P
x
=
(1.23)
m
s
s
s
s
m
=
®
W
...
],
1,
0
[
:
2
1
]
1,
0
[
:
,
1
®
W
W
=
W
=
i
i
m
i
i
m
U
=
=
n
i
i
i
1
)
(
)
(
s
m
s
m
(1.24)
Teorema: Agar (
1.23
) operator
(1.24)
o’lchovga mos qurilgan bo’lsa, bu operatorni
m
ta volterra
operatorlarga keltirib o’rganish mumkin.
Misol:
..
=
G
q
=
=
L
X
};
2
,1
{
}
:
{
};
,
{
2
Ф
X
b
a
Ф
®
=
W
=
s
)}
,
(
),
,
(
),
,
(
),
,
(
{
4
3
2
1
b
b
a
b
b
a
a
a
=
=
=
=
=
W
s
s
s
s
2
1
1
1
1
1
1
)
(
;
)
(
)};
(
),
{(
a
a
m
a
m
=
-
=
=
=
W
b
a
b
a
2
1
2
2
2
1
)
(
;
)
(
)};
(
),
{(
b
b
m
b
m
=
-
=
=
=
W
b
a
b
a
1
1
2
1
)
(
)
(
)
,
(
b
a
m
m
m
=
=
a
a
a
a
2
1
2
1
)
(
)
(
)
,
(
b
a
m
m
m
=
=
b
a
b
a
1
2
)
,
(
b
a
m
=
a
b
2
2
)
,
(
b
a
m
=
b
b
)
,
(
);
,
(
);
,
(
);
,
(
4
3
2
1
b
b
a
b
b
a
a
a
=
=
=
=
s
s
s
s
)}
,
(
),
,
(
{
)
,
(
2
1
2
1
b
a
a
a
=
=
=
W
s
s
s
s
}
,
{
)
,
(
3
1
3
1
s
s
s
s
=
W
}
,
,
,
{
)
,
(
4
3
2
1
3
2
s
s
s
s
s
s
=
W
=
=
=
W
=
W
=
W
W
W
=
0
,
0
,
)
,
(
(
)
(
,
))
,
(
(
)
(
)
,
(
,
0
)
,
(
,
))
,
(
(
)
(
2
1
1
1
1
1
2
2
1
2
1
2
1
1
2
1
2
1
2
1
,
2
1
b
b
a
b
a
s
s
s
s
s
s
m
s
m
s
s
s
s
m
s
m
s
s
s
s
s
s
s
s
m
s
m
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
3
3
:
S
S
V
®
+
+
+
+
=
+
+
+
+
=
+
+
+
+
=
+
+
+
+
=
3
2
2
2
4
3
2
4
2
2
4
1
2
2
2
4
'
4
4
1
1
2
4
3
1
3
2
1
2
3
1
2
2
3
'
3
4
1
2
1
4
2
1
3
2
2
1
2
1
2
2
2
'
2
3
1
1
1
4
1
1
1
3
1
1
2
1
1
2
1
'
1
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
b
a
b
a
b
a
b
a
b
b
a
a
b
a
a
b
a
b
b
a
b
a
a
b
(1.25)
+
=
+
=
4
3
2
2
1
1
x
x
X
x
x
X
+
=
+
=
4
2
2
3
1
1
x
x
Y
x
x
Y
)
(
2
)
(
2
)
(
2
2
2
2
)
(
2
1
4
1
2
1
3
1
2
2
1
4
2
1
3
1
1
4
1
1
3
2
1
2
2
1
'
2
'
1
'
1
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
x
X
+
+
+
+
+
+
=
+
+
+
+
+
=
+
=
a
a
a
a
a
a
+
=
+
=
)
2
(
)
2
(
1
2
2
2
'
2
2
1
1
1
'
1
X
X
X
X
X
X
X
X
a
a
+
=
+
=
)
2
(
)
2
(
1
2
2
2
'
2
2
1
1
1
'
1
Y
Y
Y
Y
Y
Y
Y
Y
b
b
=
=
)
(
)
,...,
,
(
)
(
2
1
i
i
m
s
m
s
s
s
m
s
m
)
,
(
)
,
(
2
1
)
(
2
)
(
1
X
X
X
X
m
m
®
)
(
)
(
2
)
(
1
)
(
1
1
lim
lim
m
m
m
m
m
X
X
X
X
+
=
=
®
®
Teorema:
(1.25)
operatorning traektoriyasi quyidagicha limitga ega:
1)
<
<
>
<
<
>
>
>
=
®
1
2
;1
2
),
1,
0
,
0
,
0
(
1
2
;1
2
),
0
,1
,
0
,
0
(
1
2
;1
2
),
0
,
0
,1
,
0
(
1
2
;1
2
),
0
,
0
,
0
,1
(
1
1
1
1
1
1
1
1
)
(
lim
b
a
b
a
b
a
b
x
x
m
m
2)
)
0
:
(
,1
2
4
3
1
=
=
=
=
x
x
x
S
b
}
0
:
{
2
1
2
=
=
=
x
x
x
S
invariant
<
>
®
1
2
,
1
2
,
)
(
1
2
1
1
)
(
lim
a
a
S
S
x
m
m
3)
}
0
:
{
,
0
2
4
2
3
1
=
=
=
=
x
x
x
S
a
}
0
:
{
3
1
4
=
=
=
x
x
x
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
<
>
®
1
2
,
1
2
,
1
4
1
3
)
(
lim
b
b
S
S
x
m
m
4)
1
2
;1
2
1
1
=
=
b
a
}
;
:
{
3
1
4
1
5
x
x
x
x
x
S
=
=
=
}
;
:
{
4
3
2
1
6
x
x
x
x
x
S
=
=
=
qo’zg’almas nuqtalar to’plami
i
k
a
ki
,
0
W
W
+
=
=
)
,
(
,
0
)
,
(
,
)
(
)
(
)
(
1
,
y
j
s
y
j
s
y
m
j
m
s
m
s
jj
m
i
i
i
i
i
i
i
P
W
=
y
j
y
j
s
jy
s
,
,
'
x
x
P
x
=
=
+
=
=
W
}
,
{
,...,
1
1
1
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Misollar:
1) Limitik nuqtalar to’plamini toping.
=
)
(
0
x
w
?
>
+
=
-
=
0
),
1
(
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)
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0
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x
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) Quyidagi operatorni invariant to’plamlari topilsin?
-
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=
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)
1
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:
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x
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V
ta’rifga ko’ra
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m
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M
invariant deyiladi agar
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M
M
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)
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3) Operatorni qo’zg’almas nuqtalarini toping?
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)
1
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)
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Ta’rifga ko’ra
x
qo’zg’almas nuqta deyiladi
x
x
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x
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),
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x
V
1
,
0
,
,
3
2
1
3
2
1
=
+
+
x
x
x
x
x
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
-
-
=
+
-
=
+
+
=
2
1
3
1
3
2
1
1
1
1
1
1
x
x
x
x
x
x
=
+
=
=
+
0
0
3
1
3
1
3
2
x
x
x
x
x
x
2
3
2
1
0
;
0
S
x
x
x
=
=
=
demak qo’zg’almas nuqtalari 3 ta
3
2
1
,
,
e
e
e
4) Quyidagi operatorni
)
0
,1
,
0
(
*
=
x
dagi xos qiymatlarini toping?
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=
+
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=
+
+
=
)
1
(
)
1
(
)
1
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:
2
1
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x
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x
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x
x
V
-
-
-
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+
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-
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¶
¶
¶
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=
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x
x
I
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=
0
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0
1
1
1
0
0
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)
(
*
x
I
0
)
det(
=
-
E
I
l
0
0
0
1
1
1
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=
-
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l
l
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2
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3
2
1
=
=
=
=
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-
l
l
l
l
l
l
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
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
7.
H.F Faxriddinovich PEDAGOG 7 (4), 281-290
8.
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
9.
KOSHI MASALASINI STATIKA TENGLAMALARI SISTEMASI UCHUN YECHISH
FF Homidov GOLDEN BRAIN 2 (6), 80-83
10.
TEKISLIKDA SOMILIAN–BETTI FORMULASI
FF Homidov Educational Research in
Universal Sciences 3 (1), 587-589
11. 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
12. С.Х.Сирожиддинов М.Маматов Эхтимоллар назарияси ва математик статистика
Тошкент 1980
13. Boboqulova, M. X. (2025). QATTIQ JISMLARNING ERISH ISSIQLIGI. Introduction of
new innovative technologies in education of pedagogy and psychology, 2(4), 26-32.
14. Boboqulova, M. X. (2025). SUYUQ KRISTALLAR VA ULARNING XOSSALARI.
Problems and solutions at the stage of innovative development of science, education and
technology, 2(4), 42-49.
15. Boboqulova, M. X. (2025). VAVILOV-CHERENKOV EFFEKTINING FIZIK ASOSLARI
VA AMALIY QO ‘LLANILISHI. ИКРО журнал, 15(01), 282-284.
