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719Furye qatori. Funksiyalarni Furye qatoriga yoyish
Maxsud Tulqin o'g'li Usmonov maqsudu32@gmail.com Mirzo Ulug'bek nomidagi O'zbekiston Milliy universiteti
Annotatsiya: Ushbu maqolada matematikaning eng muhim mavzularidan biri bo'lgan Furye qatori. Funksiyani Furye qatoriga yoyish tog'risida malumot keltirildi va mavjud muanmolar xal etildi. Agar f (x) funksiya [a;b] kesmada monoton bo'lsa yoki [a;b] kesmani chekli sondagi qismiy kesmalarga bo'lish mumkin bo'lsa va bu kesmalarning har birida f (x) funksiya monoton (faqat o'ssa yoki faqat kamaysa) yoki o'zgarmas bo'lsa, f (x) funksiyaga [a;b] kesmada bo'laklimonoton funksiya deyiladi. Agar f (x) funksiya [a;b] kesmada chekli sondagi birinchi tur uzilish nuqtalariga ega bo'lsa, f (x) funksiyaga [a;b] kesmada bo'lakli-uzluksiz funksiya deyiladi. Agar f (x) funksiya [a;b] kesmada uzluksiz yoki bo'lakli-uzluksiz bo'lib, bo'lakli-monoton bo'lsa f (x) funksiya [a;b] kesmada Dirixle shartlarini qanoatlantiradi deyiladi. Bu hollarda qo'yilgan masalalarni yechishda quyida biz o'rganadigan qatorlar nazariyasi katta ahamiyatga ega.
Kalit so'zlar: Furye qatori, Furye koeffitsiyentlari. Funksiyalarni Furye qatoriga yoyish.
Fourier series. Fourier series expansion of functions
Maxsud Tulqin oglu Usmonov maqsudu32@gmail.com National University of Uzbekistan named after Mirzo Ulugbek
Abstract: In this article, the Fourier series is one of the most important topics in mathematics. Information on the expansion of the function into the Fourier series was given and the existing problems were solved. If the function f (x) is monotone in the section [a;b] or if the section [a;b] can be divided into a finite number of partial sections, and in each of these sections the function f (x) is monotone (only if or only decreases) or is constant, the function f (x) is called a piecewise monotone function on the cross section [a;b]. If the function f (x) has a finite number of discontinuities of the first type on the section [a;b], then the function f (x) is called a piecewise-continuous function on the section [a;b]. If the function f (x) is continuous or piecewise-continuous in the cross section [a;b], and is piecewise monotone, then the function f (x) is said to satisfy the Dirichlet conditions in the cross section [a;b]. The
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theory of series, which we will study below, is of great importance in solving the problems posed in these cases.
Keywords: Fourier series, Fourier coefficients. Fourier series expansion of functions.
1. Furye qatori.
Faraz qilaylik, f (x) funksiya R-( ^, + da berilgan bo'lsin. Ma'lumki,
shunday T G R ^ ^ son topilsaki, v x e R da
f (x+T)-f (x)
tenglik bajarilsa, f (x) davriy funksiya, T * 0 son esa uning davri deyiladi.
Agar T * 0 son f (x) funksiyaning davri bo'lsa, u holda
kT (k = ±l+2,...) sonlar ham shu funksiyaning davri bo'ladi.
Agar f (x) va g (x) davriy funksiyalar bo'lib, T * 0 ularning davri bo'lsa,
f ( x )± g ( x), f ( x )• g (x), fM (g ( X )* 0)
funksiyalar ham davriy bo'lib, ularning davri T ga teng bo'ladi.
y — sin x, y — cos x funksiyalar T - davrli funksiya bo'lgan holda ushbu
(p(x)-acosax + bsinax saba- <■ rv^ttw
^ ' (ab,a o zgarmas, a*0)
T = 2n
funksiya ham davriy funksiya bo'lib, uning davri a bo'ladi. Haqiqatan
ham,
f 2n
p\ x +--
V a ,
— a cos (ax + + b sin (ax + 2n) — a cos ax + b sin ax — p (x)
bo'ladi.
t-, p(x~) — acosax + bsinax ,, , r , • , , ,
Bu ' sodda davriy funksiya bo lib, u garmonika deb
ataladi.
Aytaylik, f (x) funksiya ^ da uzluksiz bo'lsin. Unda
/(.x)cosra", f(x)smnx (n = 1,2,3,...)
funksiyalar ham i da uzluksiz bo'lib, ular i da integrallanuvchi
bo'ladi. Bu integrallarni quyidagicha belgilaymiz:
" ( 2jc\ + b sin " ( 2xX
a cos a x +-- a x +--
_ I a ) _ _ I a ) _
a0 =~ K
1 n
1 j f ( x ) ^
^ -K
1 71
— f f(x)cosnxdx, (« = 1,2,...)
-K
1 71
bn = — |/(x)sin«xt/x. (« = 1,2,...)
an =-
K
(1)
Bu sonlardan foydalanib, ushbu
a.
— + cosnx + b sinnx)
2 n ' 2 n=1 (2)
qatorni ( uni trigonometrik qator deyiladi) hosil qilamiz.
(2) qator funksional qator bo'lib, uning har bir hadi garmonikadan iborat.
Ta'rif. (2) funksional qator f(x) funksiyaning Furye qatori deyiladi. (1) munosabatlar bilan aniqlangan
0 > 1' 1' 2 7 2 7 7 n 7 n 7
sonlar Furye koeffitsiyentlari deyiladi. 7.2. Funksiyalarni Furye qatoriga yoyish.
Demak, berilgan f (x) funksiyaning Furye koeffitsiyentlari shu funksiyaga
bog'liq bo'lib, (2) formulalar yordamida aniqlanadi, qator esa quyidagicha:
a œ
f ( x ) —0 + ^(a„ cos nx + bn sin nx )
2 n=1 .
belgilanadi.
1-misol. Ushbu f (x) = e ( K~ x ~K ,a* 0) funksiyaning Furye qatori topilsin. (1) formulalardan foydalanib, berilgan funksiyaning Furye koeffitsiyentlarini hisoblaymiz:
1 K 1 1
a0 = - j eaxdx = — (eaK - e-aK ) = — shan, k J„ anx ' an
aK
1 K
an = — I eax cos nxdx = n J
1 acos nx + n sin nx
K
2 . 2 a + n
= (-1 yL^—shax („ = 1,2,...), n a'+rr
1 K 1j
1T •>
bn = — | ea sin nxdx =
K
1 asin nx - n cos nx
K
2 . 2 a + n
= (-1 y-'L-^L-shan („ = 1,2,...).
7t er +tr
Demak,
funksiyaning Furye qatori
f ( x ) = ea
K
K
K
K
K
f (x) — eax ~ — + ^ (a cos nx + bn sin nx) —
2 n—1
2 shan
n
1 v (-1)\ • ^
+ —^-(acos nx - n sin nx )
bo'ladi.
Aytaylik, U holda
2a n—1 a2 + n2
f (x) funksiya [ n,n] da berilgan juft funksiya bo'lsin: f ( x) — f (x)
f (x>cosnxjuft, f (x>sinn toq (n —1,2,3,.. ) funksiya bo'ladi.
(1) formulalardan foydalanib, topamiz:
f (x) funksiyaning Furye koeffitsiyentlarini
Y n 1 0 n
an — — | f (x) cos nxdx — — j f (x) cos nxdx + j f (x) cos nxdx
n
2 } n
| f (x) cos nxdx + | f (x) cos nxdx
0 0
— [ /(x) cos nxdx (n = 0,1,2,...). ^ o
Y n 1 0 n
bn — — | f ( x) sin nxdx — — | f ( x) sin nxdx + j f ( x) sin nxdx
- j f ( x) sin nxdx + j f ( x) sin nxdx
= 0 («=1,2,...).
f ( x )
Demak, juft J ( ) funksiyaning Furye koeffitsiyentlari
an —
2
— ^ f i^x) cos nxdx (« = 0,1,2,...)
bn= 0 (« = 1,2,...)
bo'lib, Furye qatori
f (x) ~ 00+Z
a cosnx
bo'ladi.
f ( x )
Aytaylik, j ( ) funksiya
[-n,n]
da berilgan toq funksiya bo'lsin:
f (- x ) — --f ( x )
. U holda
f (x )• cos nx f (x )• sin nx ^^ (n —1,2,3,...)
juft
funksiya bo'ladi.
(1) formulalardan foydalanib, topamiz:
f ( x )
funksiyaning Furye koeffitsiyentlarini
to
to
n—1
^ 7 1 0 n
an = — I f (x) cos nxdx = — I f (x) cos nxdx + I f (x) cos nxdx
— 7 _— 7 0 _
1 "
= —[- f {x} cos nxdx + f {x} cos nxdx \ = 0 (« = 0,1,2,...) ^ o
Y 7 10 7
bn = — | f (x) sin nxdx = — | f (x) sin nxdx +1 f (x) sin nxdx
| f (x) sin nxdx
(/7=1,2,...).
Demak, toq f (x) funksiyaning Furye koeffitsiyentlari
a„= 0, (« = 0,1,2,...),
2 *
bn=—\f (x) sin/mix, (ri = 1,2, ...)
n I
bo'lib, Furye qatori
f (x) ~ Z bn si
sin nx
n=1
bo'ladi.
2-misol. Ushbu
f (x) = x2 (—7 < x <7)
' juft funksiyaning Furye qatori topilsin. Avvalo berilgan funksiyaning Furye koeffitsiyentlarini topamiz:
2_ 7'
a0 =-
7
| x2 dx
'2—2 = —7
3
a = — f x2 cos nxdx = —x
7Z{ 7
2 9 sin nx
_4_ 7n
x cos nx
n
7n
7 1 7 \
7 4 7
n7
| x sin nxdx
— fcosnxdx = (-1)" • . (n = 1,2, ...) n{ tr
Demak,
f ( x ) = x
2
funksiyaning Furye qatori
Л\ ? 7 cos nx
x)=x ~ 7"+4Z(—ST
bo'ladi.
3-misol. Ushbu f (x) = x ( 77 < x < 7 toq funksiyaning Furye qatori topilsin. Berilgan funksiyaning Furye koeffitsiyentlarini hisoblaymiz:
bn = — I x sin nxdx -
7
x cos nx
1 K
— I cos nxdx
V) J
_ 2 (—l)n
Demak,
bo'ladi.
f (x ) = x
funksiyaning Furye qatori
f (x) ~ Z(——)n—1_sin nx
>i=i n
Faraz qilaylik, f (x) funksiya [ p,p] (p > 0) segmentda uzluksiz bo'lsin. Ma'lumki, ushbu
n
t ——x p
almashtirish [ p,p] oraliqni [ n,n] ga o'tkazadi, ya'ni x o'zgaruvchi [ p,p] da o'zgarganda t o'zgaruvchi [ n,n] da o'zgaradi. Endi
f ( x ) — f | pt ] — p(t).
deymiz. Unda p(t) funksiya [ n,n] oraliqda berilgan uzluksiz funksiya bo'ladi. Bu funksiyaning Furye koeffitsiyentlari
an — -
n
1
- j p(t) cos ntdt, (« = 0,1,2,...)
^ -n
1 71
bn = — | p(t^smntdt (« = 1,2,...)
ni topib, Furye qatorini yozamiz:
p (t) ~ + 2 (a cos nt + bn sin nt)
n—1
Modomiki,
ekan, unda
n
t ——x p
P
fn ^ — x
V p J
bo'lib, uning koeffitsiyentlari
a0 ! 2 „-1
nn a cos n — x + b sin n — x
p p J
1 r 1 j
p n ^
an I PI -x
^ J
TT
cosn — xdx, (« = 0,1,2...) P
bn=—^p\—x sinn — xdx. (« = 1,2...)
.p J
bo'ladi. Natijada [ p,p] da berilgan f (x) funksiyaning Furye qatorini quyidagicha
f (x) ~ + 2
nnx 7 . nnx an cos--+ bn sin-
p P J
bo'lishini topamiz, bunda
-n
TO
n—1
a„ = —
1 if (x)
nnx 1 , „ , „ \ cos-ax yn = 0,1,2...)
1 P T17T
bn = — |/(x)sin— xdx (« = 1,2...)
4-misol. Ushbu
f (x ) = ex (-1 < x < 1)
funksiyaning Furye qatori topilsin.
Yuqoridagi formulalardan foydalanib, f (x ) =e funksiyaning Furye koeffitsiyentilarini topamiz:
1 1 nn sin nnx - cos nnx
\exdx = e - e 1, a = \ ex cos nnxdx = „ „
J j 1 + n n
H-i)ni 1
1 + n n
^ (ecosnx-e 1 cosw;r) = (-1)" ——(n = l,2,...)
1
bn = J ex cos nnxdx
-1
= 1 + n2n2 (
sin nnx - nn cos nnx
1 + «v
nn(-1)n
= 1 + n 2n2
enn cos nn + nne cos nn\ =
(e-'-e) = (-lfl-^-J („ = 1,2,...) v 7 1 + n TT
Demak,
f (x) = ex (-1 < x < 1)
funksiyaning Furye qatori
e - e
TO
- + (e - e-1)!
(- 1)n (- 1)n+1 ——-—-cos nn + ——-—- nnsin nnx
1 + n n
1 + n n
bo'ladi.
Aytaylik, f (x)funksiya ^a'b da berilgan bo'lsin. ia'b segment a nuqtalar yordamida bo'laklarga ajratilgan. (a° = a a = .
Agar har bir (c'k (£-0,1,2,...,« l) /(x) fLinksiya differensial 1 anuvchi
bo'lib, x = ak nuqtalarda chekli o'ng
f'(ak+ 0) (k = 0,1,2,...,« —l) ^
va chap
f'(ak-0) (k = 0,1,2,...,«)
hosilalarga ega bo'lsa, f (x) funksiya b da bo'lakli-differensiallanuvchi deyiladi.
Endi Furye qatorining yaqinlashuvchi bo'lishi haqidagi teoremani isbotsiz keltiramiz.
a0 =
e
-1
e ~
2
n=1
Teorema. 2n davrli f (x) funksiya [ n,n] oraliqda bo'lakli-differensiallanuvchi bo'lsa, u holda bu funksiyaning Furye qatori
f (x) ~ — + 2 (a cos kx + b sin kx )
k—1
[ n,n] da yaqinlashuvchi bo'lib, uning yig'indisi
f ( x + 0) + f ( x-0) 2
ga teng bo'ladi.
• 7 tt uu J(x) — cosax (-n<x <n , a *n e Z) r , • • ^ . •
5-misol. Ushbu J v ' v ' funksiyaning Furye qatori
topilsin va u yaqinlashishga tekshirilsin.
Bu funksiyaning Furye koeffitsiyentlarini topamiz. Qaralayotgan funksiya juft
bo'lgani uchun
bn= 0 (n = 1,2,3,...)
bo'lib,
n \ sin an,
a — — j cos ax cos nxdx — j[cos (a - n ) x + cos (a + n ) x] dx -;(-1)n
1 1
-+-
a + n a - n
bo'ladi. Demak,
f (x)
sin an
n
1+K-1)"
1 1
a + n a - n
cos nx
Agar f (x) cosax funksiya teoremaning shartlarini bajarishini e'tiborga olsak,
unda
cos ax — -
sin an
n
1+j2(-1)"\—+— a n—1 V a + n a - n
cos nx
bo'lishini topamiz.
TO
Foydalanilgan adabiyotlar
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42. Усмонов,МТ. (2021). Абсолютная и условная сходимость несобственного интеграла. Признак Дирихле. Признак Абеля. «Science and Education» Scientific Journal, Tom-2, 66-76.
43. Усмонов,МТ. (2021). Решение систем линейных уравнений. «Science and Education» Scientific Journal, Tom-2, 131-138.
44. Usmonov, M.T. (2021). Matritsalar va ular ustida amallar. «Science and Education» Scientific Journal, Tom-2, 226-238.
45. Usmonov, M.T. (2021). Teskari matritsa. Teskari matritsani hisoblash usullari. «Science and Education» Scientific Journal, Tom-2, 292-302.
46. Usmonov, M.T. (2021). Bir jinsli chiziqli algebraik tenglamalar sistemasi. «Science and Education» Scientific Journal, Tom-2, 323-331.
47. Usmonov, M.T. (2021). Chiziqli fazo. Yevklid fazosi. «Science and Education» Scientific Journal, Tom-2, 121-132.
48. Usmonov, M.T. (2021). Vektorlarning skalyar ko 'paytmasi. «Science and Education» Scientific Journal, Tom-2, 183-191.
49. Usmonov, M.T. (2021). Xos vektorlari bazis tashkil qiluvchi chiziqli operatorlar. «Science and Education» Scientific Journal, Tom-2, 146-152.
50. Usmonov, M.T. (2021). Chiziqli algebraik tenglamalar sistemasi va ularni еchish usullari. «Science and Education» Scientific Journal, Tom-2, 303-311.
51. Usmonov, M.T. (2021). Vektorlar. «Science and Education» Scientific Journal, Tom-2, 173-182.
52. Usmonov, M.T. (2021). Kvadratik forma va uni kanonik korinishga keltirish. «Science and Education» Scientific Journal, Tom-2, 153-172.
53. Usmonov, M.T. (2021). Arifmetik vektor fazo va unga misollar. «Science and Education» Scientific Journal, Tom-2, 109-120.
54. Usmonov, M.T. (2021). Chiziqli operatorlar va ularning xossalari. «Science and Education» Scientific Journal, Tom-2, 133-145.
55. Usmonov, M.T. (2021). Determinantlar nazariyasi. «Science and Education» Scientific Journal, Tom-2, 256-270.
56. Usmonov, M.T. (2021). Matritsa rangi. Matritsa rangini hisoblash usullari. «Science and Education» Scientific Journal, Tom-2, 280-291.
57. Usmonov, M.T. (2021). Autentification, authorization and administration. «Science and Education» Scientific Journal, Tom-2, 233-242.
58. Usmonov, M.T. (2021). Vektorlar nazariyasi elementlari. «Science and Education» Scientific Journal, Tom-2, 332-339.
59. Usmonov, M.T. (2021). EHTIMOLLAR NAZARIYASI. «Science and Education» Scientific Journal, Tom-1, 10-15.
60. Usmonov, M.T. (2021). Chiziqli algebraik tenglamalar sistemasi va ularni еchish usullari. «Science and Education» Scientific Journal, Tom-2, 333-311.
61. Usmonov, M.T. (2021). Bir jinsli chiziqli algebraik tenglamalar sistemasi. «Science and Education» Scientific Journal, Tom-21, 323-331.
62. Usmonov, M.T. (2021). Vektorlar nazariyasi elementlari. «Science and Education» Scientific Journal, Tom-2, 332-339.
63. Usmonov, M.T. (2021). Chiziqli fazo. Yevklid fazosi. «Science and Education» Scientific Journal, Tom-2, 121-132.
64. Usmonov M. T. & Qodirov F. E, BIR JINSLI VA BIR JINSLIGA OLIB KELINADIGAN DIFFERENSIAL TENGLAMALAR. AMALIY MASALALARGA TADBIQI (KO'ZGU MASALASI) , BARQARORLIK VA YETAKCHI TADQIQOTLAR ONLAYN ILMIY JURNALI: Vol. 2 No. 1 (2022): БАРКДРОРЛИК ВА ЕТАКЧИ ТАД^ЩОТЛАР ОНЛАЙН ИЛМИЙ ЖУРНАЛИ
65. Usmonov Maxsud Tulqin o'g'li, Sayifov Botirali Zokir o'g'li, Negmatova Nilufar Ergash qizi, Qodirov Farrux Ergash o'g'li, BIRINCHI VA IKKINCHI TARTIBLI HUSUSIY HOSILALAR. TO'LA DIFFERENSIAL. TAQRIBIY HISOBLASH , BARQARORLIK VA YETAKCHI TADQIQOTLAR ONLAYN ILMIY JURNALI: 2022: SPECIAL ISSUE: ZAMONAVIY UZLUKSIZ TA'LIM SIFATINI OSHIRISH ISTIQBOLLARI
66. Usmonov Maxsud Tulqin o'g'li, Sayifov Botirali Zokir o'g'li, Negmatova Nilufar Ergash qizi, Qodirov Farrux Ergash o'g'li, IKKI ARGUMENTLI FUNKSIYANING ANIQLANISH SOHASI, GRAFIGI, LIMITI VA UZLUKSIZLIGI , BARQARORLIK VA YETAKCHI TADQIQOTLAR ONLAYN
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ILMIY JURNALI: 2022: SPECIAL ISSUE: ZAMONAVIY UZLUKSIZ TA'LIM SIFATINI OSHIRISH ISTIQBOLLARI
67. Usmonov Maxsud Tulqin o'g'li. (2022). FURYE QATORI. FUNKSIYALARNI FURYE QATORIGA YOYISH. https://doi.org/10.5281/zenodo.6055125
68. Usmonov. M. T. ., & Qodirov. F. E. . (2022). DARAJALI QATORLAR. DARAJALI QATORLARNING YAQINLASHISH RADIUSI VA SOHASI. TEYLOR FORMULASI VA QATORI. IJTIMOIY FANLARDA INNOVASIYA ONLAYN ILMIY JURNALI, 8-20. Retrieved from http://www.sciencebox.uz/index.php/jis/article/view/1151
69. Usmonov. M. T. ., & Qodirov. F. E.. (2022). FURE QATORI VA UNING TADBIQLARI. IJTIMOIY FANLARDA INNOVASIYA ONLAYN ILMIY JURNALI, 21-33. Retrieved from http://www.sciencebox.uz/index.php/jis/article/view/1152
70. M.T Usmonov, M.A Turdiyeva, Y.Q Shoniyozova, (2021). SAMPLE POWER. SELECTION METHODS (SAMPLE ORGANIZATION METHODS). ООО НАУЧНАЯ ЭЛЕКТРОННАЯ БИБЛИОТЕКА , 59-60.
71. Усмонов,МТ, М.А.Турдиева (2021). ГЛАВА 9. МЕТОДЫ И СРЕДСТВА СОВРЕМЕННОЙ ЗАЩИТЫ КОМПЬЮТЕРНЫХ СЕТЕЙ. РИСКИ И ПРИНЦИПЫ ЗАЩИТЫ ИНФОРМАЦИИ В ЭЛЕКТРОННОЙ ПОЧТЕ. ББК 60 С69, Ст-99.
72. Усмонов,М.Т, J.M.Saipnazarov, K.B. Ablaqulov (2021 SOLUTION OF MATHEMATICAL PROBLEMS IN LOWER CLASSES. Книга: АКТУАЛЬНЫЕ ВОПРОСЫ СОВРЕМЕННОЙ НАУКИ И ОБРАЗОВАНИЯ, 167-177.
73. Усмонов М.Т. (2022). E-LEARNING И ЕГО РОЛЬ В СОВРЕМЕННОЙ СИСТЕМЕ ОБРАЗОВАНИЯ. : Special Issue_Ta'limni modernizatsiyalash jarayonlari muammolar va еchimlar». 168-171.
74. Usmonov. M. T. ., & Qodirov. F. E.. (2022). STOKS FORMULASI. SIRT INTEGRALLARI TADBIQLARI. IJTIMOIY FANLARDA INNOVASIYA ONLAYN ILMIY JURNALI, 34-45. Retrieved from https://sciencebox.uz/index.php/jis/article/view/1153
75. Usmonov M. T. The Concept of Compatibility, Actions on Compatibility. International Journal of Academic Multidisciplinary Research (IJAMR), Vol. 5 Issue 1, January - 2021, Pages: 10-13.
76. Usmonov M. T. The Concept of Number. The Establishment of the Concept of Natural Number and Zero. International Journal of Academic Information Systems Research (IJAISR), Vol. 4 Issue 12, December - 2020, Pages: 7-9.
77. Usmonov M. T. The Concept of Compatibility, Actions on Compatibility. International Journal of Engineering and Information Systems (IJEAIS), Vol. 4 Issue 12, December - 2020, Pages: 66-68.
78. Usmonov M. T. General Concept of Mathematics and Its History. International Journal of Academic Multidisciplinary Research (IJAMR). Vol. 4 Issue 12, December - 2020, Pages: 38-42
79. Usmonov M. T. Asymmetric Cryptosystems. International Journal of Academic Engineering Research (IJAER) ISSN: 2643-9085 Vol. 5 Issue 1, January -2021, Pages: 6-9.
80. Usmonov M. T. Basic Concepts of Information Security. International Journal of Academic and Applied Research (IJAAR) ISSN: 2643-9603 Vol. 5 Issue 1, January - 2021, Pages: 5-8.
81. Usmonov M. T. Communication Control Systems, Methodology. International Journal of Academic Engineering Research (IJAER) ISSN: 2643-9085 Vol. 5 Issue 1, January - 2021, Pages: 47-50.
82. Usmonov M. T. Compatibility between the Two Package Elements. Binar Relations and Their Properties. International Journal of Academic Multidisciplinary Research (IJAMR) ISSN: 2643-9670 Vol. 5 Issue 1, January - 2021, Pages: 52-54.
83. Usmonov M. T. Cryptographic Protection of Information. International Journal of Academic and Applied Research (IJAAR) ISSN: 2643-9603 Vol. 5 Issue 1, January - 2021, Pages: 24-26.
84. Usmonov M. T. Electronic Digital Signature. International Journal of Academic Pedagogical Research (IJAPR) ISSN: 2643-9123 Vol. 5 Issue 1, January -2021, Pages: 30-34.
85. Usmonov M. T. "Equal" And "Small" Relations. Add. Laws Of Addition. International Journal of Academic Information Systems Research (IJAISR) ISSN: 2643-9026 Vol. 5 Issue 1, January - 2021, Pages: 27-29.
86. Usmonov M. T. Establish Network Protection. International Journal of Academic Engineering Research (IJAER) ISSN: 2643-9085 Vol. 5 Issue 1, January -2021, Pages: 14-21.
87. Usmonov M. T. Fundamentals of Symmetric Cryptosystem. International Journal of Academic and Applied Research (IJAAR) ISSN: 2643-9603 Vol. 5 Issue 1, January - 2021, Pages: 36-40.
88. Usmonov M. T. General Concepts of Mathematics. International Journal of Academic Information Systems Research (IJAISR) ISSN: 2643-9026 Vol. 5 Issue 1, January - 2021, Pages: 14-16.
89. Usmonov M. T. Identification and Authentication. International Journal of Academic Pedagogical Research (IJAPR) ISSN: 2643-9123 Vol. 5 Issue 1, January -2021, Pages: 39-47.
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90. Usmonov M. T. Information Protection and Its Types. International Journal of Academic and Applied Research (IJAAR) ISSN: 2643-9603 Vol. 5 Issue 1, January - 2021, Pages: 1-4.
91. Usmonov M. T. Information Protection in Wireless Communication Systems. International Journal of Academic Pedagogical Research (IJAPR) ISSN: 2643-9123 Vol. 5 Issue 1, January - 2021, Pages: 61-64.
92. Usmonov M. T. Information protection supply. International Journal of Academic and Applied Research (IJAAR) ISSN: 2643-9603 Vol. 5 Issue 1, January -2021, Pages: 12-15.
93. Usmonov M. T. Information Security Policy. International Journal of Academic Pedagogical Research (IJAPR) ISSN: 2643-9123 Vol. 5 Issue 1, January -2021, Pages: 70-73.
94. Usmonov M. T. Information War. International Journal of Academic and Applied Research (IJAAR) ISSN: 2643-9603 Vol. 5 Issue 1, January - 2021, Pages: 79-82.
95. Usmonov M. T. International and National Legal Base in the Field Of Information Security. International Journal of Academic Pedagogical Research (IJAPR) ISSN: 2643-9123 Vol. 5 Issue 1, January - 2021, Pages: 7-14.
96. Usmonov M. T. Legal Legislative Basis for Detection of Information Crime. International Journal of Academic Engineering Research (IJAER) ISSN: 2643-9085 Vol. 5 Issue 1, January - 2021, Pages: 80-87.
97. Usmonov M. T. Mathematical Proofs. Incomplete Induction, Deduction, Analogy. The Concept Of Algorithm And Its Properties. International Journal of Academic Multidisciplinary Research (IJAMR) ISSN: 2643-9670 Vol. 5 Issue 1, January - 2021, Pages: 26-29.
98. Usmonov M. T. Means of Information Protection. International Journal of Academic and Applied Research (IJAAR) ISSN: 2643-9603 Vol. 5 Issue 1, January -2021, Pages: 27-30.
99. Usmonov M. T. Organization of E-Mail Protection. International Journal of Academic Engineering Research (IJAER) ISSN: 2643-9085 Vol. 5 Issue 1, January -2021, Pages: 36-40.
100. Usmonov M. T. Organizing Internet Protection. International Journal of Academic Engineering Research (IJAER) ISSN: 2643-9085 Vol. 5 Issue 1, January -2021, Pages: 24-28.
101. Usmonov M. T. Origin and Equal Strength Relationships between Sentences. Necessary and Sufficient Conditions. Structure of Theorem and Their Types. International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2643-640X Vol. 5 Issue 1, January - 2021, Pages: 45-47.
102. Usmonov M. T. PhysicalSecurity. International Journal of Academic and Applied Research (IJAAR) ISSN: 2643-9603 Vol. 5 Issue 1, January - 2021, Pages: 58-61.
103. Usmonov M. T. Practical Security Management. International Journal of Academic Engineering Research (IJAER) ISSN: 2643-9085 Vol. 5 Issue 1, January -2021, Pages: 71-74.
104. Usmonov M. T. Problem Solving In Primary Schools. International Journal of Academic Information Systems Research (IJAISR) ISSN: 2643-9026 Vol. 5 Issue 1, January - 2021, Pages: 72-83.
105. Usmonov M. T. Reproduction. The Laws of Reproduction. International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2643-640X Vol. 5 Issue 1, January - 2021, Pages: 36-40.
106. Usmonov M. T. Security Models. International Journal of Academic Pedagogical Research (IJAPR) ISSN: 2643-9123 Vol. 5 Issue 1, January - 2021, Pages: 18-23.
107. Usmonov M. T. Solving Problems In Arithmetic Methods. International Journal of Academic Information Systems Research (IJAISR) ISSN: 2643-9026 Vol. 5 Issue 1, January - 2021, Pages: 58-61.
108. Usmonov M. T. Stenographic Protection of Information. International Journal of Academic and Applied Research (IJAAR) ISSN: 2643-9603 Vol. 5 Issue 1, January - 2021, Pages: 31-35.
109. Usmonov M. T. Telecommunications and Network Security. International Journal of Academic Engineering Research (IJAER) ISSN: 2643-9085 Vol. 5 Issue 1, January - 2021, Pages: 57-61.
110. Usmonov M. T. The Concept of Compatibility, Actions on Compatibility. International Journal of Academic Multidisciplinary Research (IJAMR) ISSN: 26439670 Vol. 5 Issue 1, January - 2021, Pages: 10-13.
111. Usmonov M. T. The Concept Of National Security. International Journal of Academic and Applied Research (IJAAR) ISSN: 2643-9603 Vol. 5 Issue 1, January -2021, Pages: 73-75.
112. Usmonov M. T. The Concept of Number. The Establishment of the Concept of Natural Number and Zero. International Journal of Academic Multidisciplinary Research (IJAMR) ISSN: 2643-9670 Vol. 5 Issue 1, January -2021, Pages: 18-21.
113. Usmonov M. T. The Concept of Relationship. Characteristics of Relationships. International Journal of Academic Multidisciplinary Research (IJAMR) ISSN: 2643-9670 Vol. 5 Issue 1, January - 2021, Pages: 38-40.
114. Usmonov M. T. The Concept of Size and Measurement. International Journal of Academic Information Systems Research (IJAISR) ISSN: 2643-9026 Vol. 5 Issue 1, January - 2021, Pages: 36-40.
115. Usmonov M. T. The Emergence and Development of Methods of Writing All Negative Numbers. International Journal of Academic Information Systems Research (IJAISR) ISSN: 2643-9026 Vol. 5 Issue 1, January - 2021, Pages: 48-50.
116. Usmonov M. T. The Purpose, Function and History Of The Development Of Mathematical Science. International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2643-640X Vol. 5 Issue 1, January - 2021, Pages: 8-17.
117. Usmonov M. T. True and False Thoughts, Quantities. International Journal of Academic Information Systems Research (IJAISR) ISSN: 2643-9026 Vol. 5 Issue 1, January - 2021, Pages: 1-5.
118. Usmonov M. T. Virtual Protected Networks. International Journal of Academic Pedagogical Research (IJAPR) ISSN: 2643-9123 Vol. 5 Issue 1, January -2021, Pages: 55-57.
119. Usmonov M. T. What Is Solving The Problem? Methods of Solving Text Problems. International Journal of Engineering and Information Systems (IJEAIS) ISSN: 2643-640X Vol. 5 Issue 1, January - 2021, Pages: 56-58.
