CC BY
0TAQQOSLAMALAR .EYLER FUNKSIYASI.
Sharipova Madina Po'latovna
Osiyo Xalqaro Universiteti "Umumtexnik fanlar" kafedrasi o'qituvchisi saripovamadina807.m@gmail.com Latipova Shahnoza Salim qizi
Osiyo Xalqaro Universiteti "Umumtexnik fanlar" kafedrasi o'qituvchisi slatipova543@gmail.com
ARTICLE INFO
Qabul qilindi: 10-February 2024 yil Ma'qullandi: 15- February 2024 yil Nashr qilindi: 22- February 2024 yil
KEY WORDS
Butun sonlar halqasi,chegirmalar sinfi,modul,Eyler funksiyasi,Ferma teoremasi..
ABSTRACT
Maqolada Taqqoslamalar ularning xosslari o'rganilgan.TaqqosIamalarni Eyler va Ferma teoremalari orqali o'rganish.
Z-butun sonlar halqasi bo'lib, m>1 natural son bo'lsin. Ta'rif. Agar Z halqaga tegishli a va b
sonlarni m natural songa bo'lganda hosil bolgan qoldiqlar bir xil bo'lsa, yoki a-b ayirma m ga
bo'linsa, yoki a=b+mq tenglik o'rinli bo'lsa, u holda a va b sonlar m modul bo'yicha
taqqoslanadi deyiladi va uni a=b(mod m) ko'rinishda belgilanadi.
Taqqoslamalar quyidagi xossalarga ega:
10. Taqqoslama ekvivalent binar munosabat.
20.Bir xil modulli taqqoslamalarni hadma-had qo'shish (ayirish) mumkin. Bu ish n ta ai=bi(mod m), a2=b2(mod m),...,an=bn (mod m) taqqoslamalar uchun ham bajariladi, ya'ni ai±a2±...± an=(bi±b2±...±bn) (mod m) taqqoslamani hosil qilamiz.
Natija. Taqqoslamaning bir qismidagi sonni uning ikkinchi qismiga qarama-qarshi ishora bilan o'tkazish mumkin.
Natija. Taqqoslamaning ixtiyoriy qismiga modulga karrali sonni qo'shish mumkin. 30. Bir xil modulli taqqoslamalarni hadma-had ko'paytirish mumkin.
Natija. Taqqoslamaning ikki qismini (modulni o'zgartirmay) bir xil natural darajaga ko'tarish mumkin.
40. Modulni o'zgartirmagan holda taqqoslamaning ikki qismini bir xil butun songa ko'paytirish mumkin.
50.Agar x=y(mod m) bolsa, u holda ixtiyoriy butun koeffitsientli f(x)=aoxn+aixn-1+... +an-ix+an, f(y)=aoyn+aiyn-1+...+an-iy+an ko'phadlar uchun f(x)=f(y) (mod m) taqqoslama o>inli bo'ladi.
60.Agar bir vaqtda apbi (mod m)(i= 1, n ) va x= y (mod m) taqqoslamalar o>inli bolsa, u holda
ao xn+ai xn-1l+...+an-ix +an = bo yn + bi yn-1 +...+bn-i y+bn(mod m) taqqoslama o'rinli bo'ladi.
Natija. Taqqoslamada qatnashuvchi qo'shiluvchini o'zi bilan teng qoldiqli bo'lgan ikkinchi songa almashtirish mumkin.
70. Taqqoslamaning ikki qismini modul bilan o'zaro tub bo'lgan ko'paytuvchiga qisqartirish mumkin.
80. Taqqoslamaning ikki qismini va modulini bir xil musbat songa ko'paytirish, taqqoslamaning ikki qismi va moduli umumiy ko'paytuvchiga ega bo'lsa, u xolda bu taqqoslamaning ikki qismi va modulini bu umumiy ko'paytuvchiga boyish mumkin.
90. Agar taqqoslama bir necha Modul bo'yicha o'rinli bo'lsa, u holda bu taqqoslama shu modullarning eng kichik umumiy bo'linuvchisi bo'yicha ham o'rinli bo'ladi.
100. Agar taqqoslama biror m Modul bo'yicha o'rinli bo'lsa, u holda bu takdoslama modulning ixtiyoriy buluvchisi buyicha ham o'rinli bo'ladi.
ii0. Taqqoslamaning bir qismi va modulining EKUB bilan uning ikkinchi qismi va modulining EKUB o'zaro teng bo'ladi.
Barcha butun sonlarni m>l natural songa bo'lganda 0, 1, 2, ..., m-1 qoldiqlar hosil bo'ladi. Bunday har bir qoldiqqa butun sonlarning biror sinfi mos keladi. Ta'rif. m ga bo'linganda r ga teng bir xil qoldiq beradigan butun sonlar to'plami m modul
bo'yicha chegirmalar sinflari deyiladi va uni r kabi belgilanadi.
Ta'rif. Chegirmalar sinfining ixtiyoriy elementi shu sinfning chegirmasi deyiladi.
Ta'rif. m Modul bo'yicha tuzilgan har bir chegirmalar sinfidan erkinlik bilan bittadan element
olib tuzilgan to'plamga m Modul bo'yicha chegirmalarning to'la sistemasi deyiladi.
Sinfning bitta chegirmasi m Modul bilan o'zaro tub bo'lsa, u holda bu sinfning barcha
elementlari ham m Modul bilan o'zaro tub bo'ladi.
• ^ I W l^H
Ta'rif. m Modul bilan o'zaro tub bo'lgan barcha chegirmalar sinfidan erkinlik bilan bittadan
chegirma olib tuzilgan to'plam chegirmalarning m Modul bo'yicha keltirilgan sistemasi deyiladi.
m modul bo'yicha chegirmalarning keltirilgan sistemasidagi elementlar sonini aniqlash uchun Eyler funktsiyasi deb ataluvchi ^(m) funksiyadan foydalanamiz.
Ta'rif. Agar quyidagi ikkita shart bajarilsa, u holda ^(m) sonli funktsiya Eyler funktsiyasi deyiladi: 1. 9(1) = 1.
2. ^(m) funktsiya m dan kichik va m bilan o'zaro tub bo'lgan natural sonlar soni. Ta'rif. Natural sonlar to'plamida aniqlangan f funktsiya uchun (m; n)=1 bo'lganda f(m-n)=f(m)-f(n) tenglik bajarilsa, u holda f funktsiyaga mul'tiplikativ funktsiya deyiladi.
Teorema. Eyler funktsiyasi mul'tiplikativ funktsiya bo'ladi.^(m) Eyler funksiyasini hisoblash formulalari quyidagilardan iborat: m=p tub son bo'lsa, u holda ^(p)=r-1 bo'ladi. m= ra (r-tub son, a-natural son) bo'lsa, u holda 9(pa)=pa-1-(p-1) bo'ladi.
m= Pia1 P 2 a2-PK '
bo'lsa, u holda
9(m) = 9(PiaiP2 a2-Pk ak )= m
1 -
P
i y
1-
P
2 y
1-
Pk
K
1
1
1
o'ladi.
Eyler teoremasi. Agar (a; m)=1 bo'lsa, u holda a9(m)=1(mod m) taqqoslama o'rinli bo'ladi.
Ferma teoremasi. Agar a son r tub songa bo'linmasa, u holda ap-1=1 (mod m) taqqoslama o'rinli bo'ladi.
Koeffitsientlari butun sonlardan iborat f(x)= ao xn+ +ai- •x"-1 ...an-ix+an ko'phad berilgan bo'lsin.
Ta'rif. Ushbu
f(x)=0(mod m) (ao son m ga bo'linmaydi, a*eZ, m>1) (1)
ko'rinishdagi taqqoslamani bir noma'lumli n- darajali taqqoslama deyiladi.
Ta'rif. Agar x=s bo'lganda
f(c)=0(mod m) (2)
taqqoslama to'g'ri bo'lsa, u holda s son (1) taqqoslamani qanoatlantiradi deyiladi.
Teorema. Agar s son (1) taqqoslamani qanoatlantirsa, u holda C chegirmalar sinfiga tegishli ixtiyoriy son ham (1) taqqoslamani qanoatlantiradi.
Ta'rif. Agar s son (1) taqqoslamani qanoatlantirsa, u holda C chegirmalar sinfi (1) taqqoslamaning echimi deyiladi.
m modul bo'yicha barcha chegirmalar sinfi 0,1,2,...,m-1 bo'ladi. Demak, m modulli taqqoslamani qanoatlantiruvchi sonlarni 0,1,2,..., m-1 sonlar ichidan qidirish lozim.
Ta'rif. Yechimlari to'plami ustma-ust tushgan taqqoslamalarni teng kuchli taqqoslamalar deyiladi.
Agar (1) taqqoslamaning ikki qismiga ixtiyoriy ko'phad qo'shilsa yoki har ikki qismini m Modul bilan o'zaro tub bo'lgan k songa ko'paytirilsa, yoki ikki qismi va modulini k natural songa ko'paytirilsa, u holda hosil bo'lgan taqqoslama berilgan taqqoslamaga teng kuchli bo'ladi.
Ta'rif. Ushbu ax=b(mod m) (a,beZ,VmeN) (3)
ko'rinishdagi taqqoslamaga bir noma'lumli birinchi darajali taqqoslama deyiladi.
Teorema. Agar (a;m)=1 bo'lsa, u holda (3) taqqoslama yagona echimga ega bo'ladi.
Teorema. Agar (a; m)=d bo'lib, b son d ga bo'linmasa, u holda (3) taqqoslama echimga ega emas.
Teorema. Agar (3) taqqoslamada (a; m)=d bo'lib, b son d ga bo'linsa, u holda (3) taqqoslama soni d ga teng bo'lgan ushbu
— m (d - 1)m
a, an--.....an--
dd
(5)
a b, . m.
echimlarga ega bo'lib, bundagi a echim — X = — (mod —) taqqoslamaning yagona
dad
echimi bo'ladi.
Ta'rif. Agar f(x) = aoxp+aixn-1 +...+an-i x+an ,a*eZ, r-tub son, aocon r ga bo'linmasa, u holda ushbu
f(x) = 0(mod p) (6)
taqqoslamaga tub modulli p-darajali bir nomat'lumli taqqoslama deyiladi. Teorema. Agar (6) taqqoslamada ao bosh koeffitsient r ga bo'linmasa, u holda (6) taqqoslama bosh koeffitsienti 1 ga teng bo'lgan boshqa bir taqqoslamaga teng kuchli bo'ladi.
Teorema. Agar f(x) va g(x) koeffitsientlari butun sonlardan iborat ko'phadlar bo'lsa, u
holda
f(x) = 0(mod p), (7)
f(x)-(xp-x)g(x) = 0(modp) (8)
taqqoslamalar teng kuchli bo'ladi.
Teorema. Darajasi n (n>r) bo'lgan r tub modulli taqqoslama darajasi r-1 dan katta bo'lmagan taqqoslamaga teng kuchli bo'ladi.
Teorema. Tub modulli n-darajali taqqoslama echimlari soni n tadan ortiq emas.
Foydalanilgan adabiyotlar ro'yxati:
1. Sharipova, M., & Latipova, S. (2024). TAKRORIY GRUPPALASHLAR. Development of pedagogical technologies in modern sciences, 3(3), 134-142.
2. Sharipova, M. (2024). TAQQOSLAMA TUSHUNCHASI VA UNING XOSSALARI. Current approaches and new research in modern sciences, 3(2), 68-78.
3. Sharipova, M. (2024). IKKI O'ZGARUVCHILI TENGSIZLIKLAR SISTEMASINI TAQQOSLAMALAR USULI BILAN YECHISH. Development and innovations in science, 3(2), 97105.
4. Sharipova, M. (2024). BIRINCHI DARAJALI TAQQOSLAMALARNI YECHISH USULLARI. Solution of social problems in management and economy, 3(2), 60-69.
5. Latipova, S., & Sharipova, M. (2024). KESIK PIRAMIDA MAVZUSIDA FOYDALANILADIGAN YANGI PEDAGOGIK TEXNOLOGIYALAR. 6X6X6 METODI, BBB (BILARDIM, BILMOQCHIMAN, BILIB OLDIM) METODLARI HAQIDA. Current approaches and new research in modern sciences, 3(2), 40-48.
6. Sharipova, M. (2024). IN THE FORM OF AN UNBOUNDED PARALLELEPIPED IN THE FIELD NONLOCAL BORDERLINE CONDITIONAL LINEAR THE REVERSE IS THE CASE. Science and innovation in the education system, 3(1), 105-116.
7. Sharipova, M. (2024). FUNCTIONAL SPACES. IN SHORT REFLECTION PRINCIPLE. Current approaches and new research in modern sciences, 3(1), 131-142.
8. Sharipova, M. (2024). A IS CORRECT OF THE INTEGRAL TO THE ECONOMY APPLICATIONS. Solution of social problems in management and economy, 3(1), 116-125.
9. Sharipova, M. (2024). ASYMMETRY AND KURTOSIS COEFFICIENTS. Theoretical aspects in the formation of pedagogical sciences, 3(1), 216-225.
10. Sharipova, M. (2024). TWO MULTIPLE OF THE INTEGRAL APPLICATIONS. Инновационные исследования в науке, 3(1), 135-140.
11. Sharipova, M. P. L. (2023). CAPUTA MA'NOSIDA KASR TARTIBLI HOSILALAR VA UNI HISOBLASH USULLARI. Educational Research in Universal Sciences, 2(9), 360-365.
12. Sharipova, M. P. (2023). MAXSUS SOHALARDA KARLEMAN MATRITSASI. Educational Research in Universal Sciences, 2(10), 137-141.
13. Madina Polatovna Sharipova. (2023). APPROXIMATION OF FUNCTIONS WITH COEFFICIENTS. American Journal of Public Diplomacy and International Studies (2993-2157), 1(9), 135-138.
14. Madina Polatovna Sharipova. (2023). Applications of the double integral to mechanical problems. International journal of sciearchers,2(2), 101-103.
15. Sharipova, M. P. L. (2023). FINDING THE MAXIMUM AND MINIMUM VALUE OF A FUNCTION ON A SEGMENT. American Journal of Public Diplomacy and International Studies (2993-2157), 1(9), 245-248.
16. Sharipova, M. P. (2023). FUNKSIYALARNI KOEFFITSIENTLAR ORQALI FUNKSIYALARNI YAKINLASHTIRISH HAQIDA MA'LUMOTLAR. GOLDEN BRAIN, 1(34), 102110.
17. Sharipova, M. (2023, December). RELATIONSHIPS BETWEEN STRAIGHT LINES AND PLANES IN SPACE. In Международная конференция академических наук (Vol. 2, No. 12, pp. 60-66).
18. Sharipova, M. (2023). FRACTIONAL DERIVATIVES. Академические исследования в современной науке, 2(27), 106-113.
19. Sharipova, M. (2023). CORRECT PLACED AND CORRECT NOT PLACED ISSUES. Models and methods in modern science, 2(13), 115-121.
20. Sharipova, M. (2023). HEAT SPREAD EQUATION. Инновационные исследования в науке, 2(12), 50-56.
21. Madina Polatovna Sharipova. (2023). HIGH MATH SCORE AND INTERVAL ASSESSMENT. American Journal of Public Diplomacy and International Studies (2993-2157), 1(10), 420-424.
22. Madina Polatovna Sharipova. (2023). IN HIGHER MATHEMATICS, THE EXTREMUM OF A MULTIVARIABLE FUNCTION. American Journal of Public Diplomacy and International Studies (2993-2157), 1(10), 425-429.
23. Sharipova, M. P. (2024). ISSIQLIK TARQALISH TENGLAMASI UCHUN KOSHI MASALASI. GOLDEN BRAIN, 2(1), 525-532.
24. Latipova, S. (2024). YUQORI SINF GEOMETRIYA MAVZUSINI O'QITISHDA YANGI PEDAGOGIK TEXNOLOGIYALAR VA METODLAR. SINKVEYN METODI, VENN DIAGRAMMASI METODLARI HAQIDA. Theoretical aspects in the formation of pedagogical sciences, 3(3), 165173.
25. Latipova, S. (2024, February). SAVOL-JAVOB METODI, BURCHAKLAR METODI, DEBAT (BAHS) METODLARI YORDAMIDA GEOMETRIYANI O'RGANISH. In Международная конференция академических наук (Vol. 3, No. 2, pp. 25-33).
26. Latipova, S., & Sharipova, M. (2024). KESIK PIRAMIDA MAVZUSIDA FOYDALANILADIGAN YANGI PEDAGOGIK TEXNOLOGIYALAR. 6X6X6 METODI, BBB (BILARDIM, BILMOQCHIMAN, BILIB OLDIM) METODLARI HAQIDA. Current approaches and new research in modern sciences, 3(2), 40-48.
27. Latipova, S. (2024). 10-11 SINFLARDA STEREOMETRIYA OQITISHNING ILMIY VA NAZARIY ASOSLARI. Академические исследования в современной науке, 3(6), 27-35.
28. Latipova, S. (2024). HILFER HOSILASI VA UNI HISOBLASH USULLARI. Центральноазиатский журнал образования и инноваций, 3(2), 122-130.
29. Latipova, S. (2024). HILFER MA'NOSIDA KASR TARTIBLI TENGLAMALAR UCHUN KOSHI MASALASI. Development and innovations in science, 3(2), 58-70.
30. Latipova, S. (2024). KESIK PIRAMIDA TUSHUNCHASI. KESIK PIRAMIDANING YON SIRTINI TOPISH FORMULALARI. Models and methods in modern science, 3(2), 58-71.
31. Shahnoza, L. (2023, March). KASR TARTIBLI TENGLAMALARDA MANBA VA BOSHLANG'ICH FUNKSIYANI ANIQLASH BO'YICHA TESKARI MASALALAR. In " Conference on Universal Science Research 2023" (Vol. 1, No. 3, pp. 8-10).
32. qizi Latipova, S. S. (2024). CAPUTO MA'NOSIDAGI KASR TARTIBLI TENGLAMALARDA MANBA FUNKSIYANI ANIQLASH BO 'YICHA TO 'G 'RI MASALALAR. GOLDEN BRAIN, 2(1), 375-382.
33. Latipova, S. S. (2023). SOLVING THE INVERSE PROBLEM OF FINDING THE SOURCE FUNCTION IN FRACTIONAL ORDER EQUATIONS. Modern Scientific Research International Scientific Journal, 1(10), 13-23.
34. Latipova, S. (2024). GEOMETRIYADA EKSTREMAL MASALALAR. В DEVELOPMENT OF PEDAGOGICAL TECHNOLOGIES IN MODERN SCIENCES (Т. 3, Выпуск 3, сс. 163-172).
35. Latipova, S. (2024). EKSTREMUMNING ZARURIY SHARTI. В SOLUTION OF SOCIAL PROBLEMS IN MANAGEMENT AND ECONOMY (Т. 3, Выпуск 2, сс. 79-90).
36. Latipova, S. (2024). FUNKSIYANING KESMADAGI ENG KATTA VA ENG KICHIK QIYMATI. В CURRENT APPROACHES AND NEW RESEARCH IN MODERN SCIENCES (Т. 3, Выпуск 2, сс. 120-129).
37. Latipova, S. (2024). EKSTREMUMLARNING YUQORI TARTIBLI HOSILA YORDAMIDA TEKSHIRILISHI. IKKINCHI TARTIBLI HOSILA YORDAMIDA EKSTREMUMGA TEKSHIRISH. В SCIENCE AND INNOVATION IN THE EDUCATION SYSTEM (Т. 3, Выпуск 3, сс. 122-133).
38. Latipova, S. (2024). BIR NECHA O'ZGARUVCHILI FUNKSIYANING EKSTREMUMLARI. В THEORETICAL ASPECTS IN THE FORMATION OF PEDAGOGICAL SCIENCES (Т. 3, Выпуск 4, сс. 14-24).
39. Latipova, S. (2024). SHARTLI EKSTREMUM. В МЕЖДУРОДНАЯ КОНФЕРЕНЦИЯ АКАДЕМИЧЕСКИХ НАУК (Т. 3, Выпуск 2, сс. 61-70).
40. Latipova, S. (2024). KASR TARTIBLI HOSILALARGA BO'LGAN ILK QARASHLAR. В CENTRAL ASIAN JOURNAL OF EDUCATION AND INNOVATION (Т. 3, Выпуск 2, сс. 46-51).
41. Latipova, S. (2024). TURLI EKSTREMAL MASALALAR. BAZI QADIMIY EKSTREMAL MASALALAR. В CENTRAL ASIAN JOURNAL OF EDUCATION AND INNOVATION (Т. 3, Выпуск 2, сс. 52-57).
42. Latipova, S. (2024). FUNKSIYA GRAFIGINI YASASHDA EKSTREMUMNING QO'LLANILISHI. В CENTRAL ASIAN JOURNAL OF EDUCATION AND INNOVATION (Т. 3, Выпуск 2, сс. 58-65).
43. Latipova, S. (2024). BIRINCHI TARTIBLI HOSILA YORDAMIDA FUNKSIYANING EKSTREMUMGA TEKSHIRISH, FUNKSIYANING EKSTREMUMLARI. В CENTRAL ASIAN JOURNAL OF EDUCATION AND INNOVATION (Т. 3, Выпуск 2, сс. 66-72).
44. Boboqulova, M. (2024). FIZIKA O'QITISHNING INTERFAOL METODLARI. В CENTRAL ASIAN JOURNAL OF EDUCATION AND INNOVATION (Т. 3, Выпуск 2, сс. 73-82).
45. Murodov, O. (2024). INNOVATIVE INFORMATION TECHNOLOGIES AND NEW METHODS AND TOOLS FOR THEIR APPLICATION IN TODAY'S EDUCATION. В CENTRAL ASIAN JOURNAL OF EDUCATION AND INNOVATION (Т. 3, Выпуск 2, сс. 83-92).
46. qizi Sharopova, M. M. (2023). RSA VA EL-GAMAL OCHIQ KALITLI SHIFRLASH ALGORITMI ASOSIDA ELEKTRON RAQMLI IMZOLARI. RSA OCHIQ KALITLI SHIFRLASH ALGORITMI ASOSIDAGI ELEKTRON RAQAMLI IMZO. Educational Research in Universal Sciences, 2(10), 316-319.
47. Sharopova, M. M. qizi . (2023). JAVA TILI YORDAMIDA OB'EKTGA YUNALTIRILGAN DASTURLASH ASOSLARI BILAN TANISHISH. GOLDEN BRAIN, 1(34), 111-119.
48. Sharopova, M. (2023). CHOOSE: COMPOSITION OR INHERITANCE. Science and innovation in the education system, 2(13), 96-102.
49. Sharopova, M. (2023). JAVA PROGRAMMING IN THE LANGUAGE HERITAGE TO DO SYNTAX. Current approaches and new research in modern sciences, 2(12), 82-87.
50. Sharopova, M. (2023). ARRAY AND ARRAYS INSTALLATION. Development of pedagogical technologies in modern sciences, 2(12), 102-107.
51. Sharopova, M. (2023). CLASSES AGAIN APPLY. Solution of social problems in management and economy, 2(13), 106-111.
52. qizi Sharopova, M. M. (2023). INTRODUCING" PROGRAM CONTROL OPERATORS" IN THE JAVA PROGRAMMING LANGUAGE. Multidisciplinary Journal of Science and Technology, 3(5), 222-231.
53. qizi Sharopova, M. M. (2023). Working with folders in the JAVA programming language. Multidisciplinary Journal of Science and Technology, 3(5), 232-236.
54. Sharopova, M. (2024). CREATION OF A DATABASE FOR THE SYSTEM PLATFORM OF NON-GOVERNMENT EDUCATIONAL CENTERS. Current approaches and new research in modern sciences, 3(1), 185-191.
55. Sharopova, M. (2024). DSA ERI STANDARD. ELECTRONIC DIGITAL SIGNATURE OF GOST R 34.10-94. Theoretical aspects in the formation of pedagogical sciences, 3(1), 169-178.
56. Sharopova, M. (2024). COLLECTORS.(OBJECT CONTAINERS). Development of pedagogical technologies in modern sciences, 3(1), 93-101.
57. Sharopova, M. (2024). JAVA PROGRAMMING IN THE LANGUAGE FLOWING INPUT AND RELEASE. Solution of social problems in management and economy, 3(1), 84-93.
58. Tursunov, B. J., & Allanazarov, G. O. (2019). Perspektivnye tehnologii proizvodstva po uluchsheniyu kachestva benzina. Theory and practice of contemporary science, 3(45), 305308.
59. Турсунов, Б. Ж., & Алланазаров, Г. О. (2019). Перспективные технологии производства по улучшению качества бензина. Теория и практика современной науки, (3 (45)), 305-308.
60. Tursunov, B. Z. (2023). Analysis of Concepts About the Effect of an Explosion in Solid Wednesday. American Journal of Public Diplomacy and International Studies (2993-2157), 1(10), 296-304.
61. Tursunov, B. Z. (2023). Methods of Control of Explosion Energy Distribution in Rocks. Intersections of Faith and Culture: American Journal of Religious and Cultural Studies (29932599), 1(10), 108-117.
62. Tursunov, B. Z. (2023). WASTE-FREE TECHNOLOGY FOR ENRICHMENT OF PURIFIC COPPER-ZINC ORE. American Journal of Public Diplomacy and International Studies (29932157), 1(9), 288-293.
63. Tursunov, B. Z. (2023). ANALYSIS OF MODERN METHODS FOR OIL SLUDGE PROCESSING. American Journal of Public Diplomacy and International Studies (2993-2157), 1(9), 280-287.
64. Jumaev, K., & Tursunov, B. (2022, December). Environmentally friendly technology for obtaining fuel briquettes from oil waste. In IOP Conference Series: Earth and Environmental Science (Vol. 1112, No. 1, p. 012005). IOP Publishing.
65. Axmedova, Z. (2024). KOMPYUTER TESTLARINING MAQSADLARI, MAZMUNI VA TUZILISHI. Theoretical aspects in the formation of pedagogical sciences, 3(3), 211-222.
66. Axmedova, Z. (2024). NODAVLAT O'QUV MARKAZLARI TIZIMI PLATFORMASI UCHUN MOBIL ILOVA YARATISH. Академические исследования в современной науке, 3(6), 162179.
67. Axmedova, Z. (2024). NODAVLAT O'QUV MARKAZLARI TIZIMI PLATFORMASI UCHUN MA'LUMOTLAR BAZASINI YARATISH. Science and innovation in the education system, 3(3), 83-93.
68. Akhmedova, Z. (2024). STRUCTURES OF SMALL DATABASE MANAGEMENT SYSTEMS. Solution of social problems in management and economy, 3(1), 97-107.
69. Akhmedova, Z. (2024). DATA BY COMBINING MAIL THROUGH TO SEND METHODS. Theoretical aspects in the formation of pedagogical sciences, 3(1), 198-207.
70. Akhmedova, Z., & Rahmatova, N. (2024). LMS (LEARNING MANAGEMENT SYSTEM) LEARNING MANAGEMENT SYSTEM FEATURES. Science and innovation in the education system, 3(1), 85-94.
71. Akhmedova, Z. (2024). CREATION OF A DATABASE FOR THE SYSTEM PLATFORM OF NON-GOVERNMENT EDUCATIONAL CENTERS. Development of pedagogical technologies in modern sciences, 3(1), 106-116.
72. Akhmedova, Z. (2024). IPHONE OPERATIONAL IN THE SYSTEM APPLICATIONS TO CREATE INTENDED PROGRAMMING ENVIRONMENTS. approaches and new research in modern sciences, 3(1), 111-121.
73. Axmedova, Z. I. (2024). LEARNING MANAGEMENT SYSTEM IMKONIYATLARI. GOLDEN BRAIN, 2(1), 509-516.
MOBILE Current
У4. Axmedova, Z. I. (2023). MA'LUMOTLAR BAZASI
BOSHQARISH TIZIMLARI. GOLDEN
ENT OF INTERACTIVE ELEMENTS.
BRAIN, 1(34), 40-49.
75. Akhmedova, Z. (2023). CREATION AND PLACEM Solution of social problems in management and economy, 2(13), 120-128.
76. Ikromovna, A. Z. (2023). Programming Environments for Creating Mobile Applications on the Android Operating System. American Journal of Public Diplomacy and International Studies (2993-2157), 1(10), 305-309.
77. Akhmedova, Z. (2023). EDUCATIONAL MANAGEMENT SYSTEMS, ELECTRONIC EDUCATION: TASKS AND OPPORTUNITIES. Theoretical aspects in the formation of pedagogical sciences, 2(21), 171-177.
78. Ikromovna, A. Z. (2023). SQL (STRUCTURED QUERY LANGUAGE) CAPABILITIES OF THE STATISTICAL DATABASE LANGUAGE. Multidisciplinary Journal of Science and Technology, 3(5), 274-280.
79. Ikromovna, A. Z. (2023). SQL (STRUCTURED QUERY LANGUAGE) STATISTICAL PACKAGES OF CAPABILITIES. Best Journal of Innovation in Science, Research and Development, 2(12), 781-787.
80. Zulxumor, A. (2022). IMPLEMENTATION OF INTERACTIVE COURSES IN THE EDUCATIONAL PROCESS. ILMIY TADQIQOT VA INNOVATSIYA, 1(6), 128-132.
81. Axmedova, Z. (2023). MOODLE TIZIMI VA UNING IMKONIYATLARI. Development and innovations in science, 2(11), 29-35.
82. Ikromovna, A. Z. (2023). USING THE USEFUL ASPECTS OF THE MOODLE SYSTEM AND ITS POSSIBILITIES. American Journal of Public Diplomacy and International Studies (29932157), 1(9), 201-205.
83. Ikromovna, A. Z. (2023). USING THE USEFUL ASPECTS OF THE MOODLE SYSTEM AND ITS POSSIBILITIES. American Journal of Public Diplomacy and International Studies (29932157), 1(9), 201-205.
84. Axmedova, Z. I. (2023). LMS TIZIMIDA INTERAKTIV ELEMENTLARNI YARATISH TEXNOLOGIYASI. Educational Research in Universal Sciences, 2(11), 368-372.
85. Murodov, O. (2024). DEVELOPMENT OF AN AUTOMATED PARAMETER CONTROL SYSTEM ROOMS AND WORKSHOPS BASED ON CLOUD TECHNOLOGIES. Академические исследования в современной науке, 3(2), 16-27.Murodov, O. T. R. (2023). Zamonaviy ta'limda axborot texnologiyalari va ularni qo 'llash usul va vositalari. Educational Research in Universal Sciences, 2(11), 481-486.
86. Муродов, О. Т. (2023). РАЗРАБОТКА АВТОМАТИЗИРОВАННОЙ СИСТЕМЫ УПРАВЛЕНИЯ ТЕМПЕРАТУРЫ И ВЛАЖНОСТИ В ПРОИЗВОДСТВЕННЫХ КОМНАТ. GOLDEN BRAIN, 1(26), 91-95.
87. Murodov, O. T. R. (2023). INFORMATIKA DARSLARINI TASHKIL ETISHDA INNOVATSION USULLARDAN FOYDALANISH. GOLDEN BRAIN, 1(32), 194-201.
88. Murodov, O. T. R. (2023). INFORMATIKA FANINI O 'QITISHDA YANGI INNOVATSION USULLARDAN FOYDALANISH METODIKASI. GOLDEN BRAIN, 1(34), 130-139.
89. Turakulovich, M. O. (2023). DEVELOPMENT AND INSTALLATION OF AN AUTOMATIC TEMPERATURE CONTROL SYSTEM IN ROOMS. International Multidisciplinary Journal for Research & Development, 10(12).
90. MURODOV, O. T. (2023). INNOVATIVE INFORMATION TECHNOLOGIES AND NEW METHODS AND TOOLS FOR THEIR APPLICATION IN TODAY'S EDUCATION. International Multidisciplinary Journal for Research & Development, 10(12).
91. Muradov, O. (2024, January). APPLICATION OF BASIC PRINCIPLES AND RULES OF INNOVATIVE PEDAGOGICAL TECHNOLOGIES TO EDUCATIONAL PROCESSES. In Международная конференция академических наук (Vol. 3, No. 1, pp. 46-55).
92. Muradov, O. (2024). BASIC PRINCIPLES AND RULES OF INNOVATIVE PEDAGOGICAL TECHNOLOGIES IN THE EDUCATIONAL PROCESS. Models and methods in modern science, 3(1), 84-93.
93. Muradov, O. (2024). APPLIED TO THE CURRENT TRAINING PROCESS REQUIREMENTS. Инновационные исследования в науке, 3(1), 54-63.
94. Murodov, O. (2024). DEVELOPMENT OF AN AUTOMATED PARAMETER CONTROL SYSTEM ROOMS AND WORKSHOPS BASED ON CLOUD TECHNOLOGIES. Академические исследования в современной науке, 3(2), 16-27.
95. Bobokulova, M. (2024). IN MEDICINE FROM ECHOPHRAPHY USE. Development and innovations in science, 3(1), 94-103.
96. Bobokulova, M. (2024). INTERPRETATION OF QUANTUM THEORY AND ITS ROLE IN NATURE. Models and methods in modern science, 3(1), 94-109.
97. Bobokulova, M. (2024, January). RADIO WAVE SURGERY. In Международная конференция академических наук (Vol. 3, No. 1, pp. 56-66).
98. Bobokulova, M. (2024). UNCERTAINTY IN THE HEISENBERG UNCERTAINTY PRINCIPLE. Академические исследования в современной науке, 3(2), 80-96.
99. Bobokulova, M. (2024). BLOOD ROTATION OF THE SYSTEM PHYSICIST BASICS. Инновационные исследования в науке, 3(1), 64-74.
100. Bobokulova, M. (2024). THE ROLE OF NANOTECHNOLOGY IN MODERN PHYSICS. Development and innovations in science, 3(1), 145-153.
101. Boboqulova, M. X. (2023). STOMATOLOGIK MATERIALLARNING FIZIK-MEXANIK XOSSALARI. Educational Research in Universal Sciences, 2(9), 223-228.
102. Xamroyevna, B. M. (2023). ORGANIZM TO 'QIMALARINING ZICHLIGINI ANIQLASH. GOLDEN BRAIN, 1(34), 50-58.
103. Bobokulova, M. K. (2023). IMPORTANCE OF FIBER OPTIC DEVICES IN MEDICINE. Multidisciplinary Journal of Science and Technology, 3(5), 212-216.
104. Khamroyevna, M. B. (2023). PHYSICO-CHEMICAL PROPERTIES OF BIOLOGICAL MEMBRANES, BIOPHYSICAL MECHANISMS OF MOVEMENT OF SUBSTANCES IN THE MEMBRANE. Multidisciplinary Journal of Science and Technology, 3(5), 217-221.
105. Bobokulova, M. K. (2024). TOLALI OPTIKA ASBOBLARINING TIBBIYOTDAGI AHAMIYATI. GOLDEN BRAIN, 2(1), 517-524.
106. Behruz Ulug'bek o'g Q. li.(2023). Mobil ilovalar yaratish va ularni bajarish jarayoni. International journal of scientific researchers, 2(2).
107. Karimov, F. (2022). ANIQ INTEGRALNI TAQRIBIY HISOBLASH. ЦЕНТР НАУЧНЫХ ПУБЛИКАЦИЙ (buxdu. uz), 14(14).
108. Quvvatov, B. (2024). GLOBAL IN VIRTUAL LEARNING MOBILE APP CREATION INFORMATION SYSTEMS AND TECHNOLOGIES. Science and innovation in the education system, 3(1), 95-104.
109. Quvvatov, B. (2024). SQL DATABASES AND BIG DATA ANALYTICS: NAVIGATING THE DATA MANAGEMENT LANDSCAPE. Development of pedagogical technologies in modern sciences, 3(1), 117-124.
110. Quvvatov, B. (2024). CONSTRUCTION OF SPECIAL MODELS THROUGH DIFFERENTIAL EQUATIONS AND PRACTICAL SOLUTIONS. Solution of social problems in management and economy, 3(1), 108-115.
111. Quvvatov, B. (2024). FINDING SOLUTIONS OF SPECIAL MODELS BY INTEGRATING INTEGRAL EQUATIONS AND MODELS. Current approaches and new research in modern sciences, 3(1), 122-130.
112. Quvvatov, B. (2024). WEB FRONT-END AND BACK-END TECHNOLOGIES IN PROGRAMMING. Theoretical aspects in the formation of pedagogical sciences, 3(1), 208-215.
113. Behruz Ulug'bek o'g, Q. (2023). USE OF ARTIFICIAL NERVOUS SYSTEMS IN MODELING. Multidisciplinary Journal of Science and Technology, 3(5), 269-273.
114. Behruz Ulugbek og Q. (2023). TECHNOLOGY AND MEDICINE: A DYNAMIC PARTNERSHIP. International Multidisciplinary Journal for Research & Development, 10(11).
115. Quvvatov, B. (2024). DIFFERENTSIAL TENGLAMALAR VA AMALIY ECHIMLAR ORQALI MAXSUS MODELLARNI QURISH. Menejment va iqtisodiyotda ijtimoiy muammolarni hal qilish , 3 (1), 108-115.
116. Behruz Ulug'bek o'g', Q. (2023). SUN'IY NERV TIZIMLARIDAN MODELLASHDA FOYDALANISH. Fan va texnologiyaning ko'p tarmoqli jurnali , 3 (5), 269-273.
117. Behruz Ulug'bek og', Q. (2023). TEXNOLOGIYA VA TIBBIYOT: DlNAMIK HAMKORLIK. Tadqiqot va ishlanmalar bo'yicha xalqaro multidisipliner jurnali , 10 (11).
118. Quvvatov, B. (2024). ALGEBRAIK ANIQLIGI YUQORI BOLGAN KVADRATUR FORMULALAR. GAUSS KVADRATUR FORMULALARI. Models and methods in modern science, 3(2), 114-125.
119. Quvvatov, B. (2024). ALGEBRAIK ANIQLIGI YUQORI BOLGAN KVADRATUR FORMULALAR. ORTOGONAL KOPHADLAR. Инновационные исследования в науке, 3(2), 47-59.
120. Quvvatov, B. (2024, February). ALGEBRAIK ANIQLIGI YUQORI BOLGAN KVADRATUR FORMULALAR. REKURSIV TRAPETSIYALAR QOIDASI. In Международная конференция академических наук (Vol. 3, No. 2, pp. 41-51).
