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Ⱦɢɫɤɪɟɬɧɵɟ ɢɫɬɨɱɧɢɤɢ Ɇɚɪɤɨɜɚ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ
Ɋɢɫ Ƚɪɚɮɫɨɫɬɨɹɧɢɣɞɢɫɤɪɟɬɧɨɝɨ
ɦɚɪɤɨɜɫɤɨɝɨɢɫɬɨɱɧɢɤɚɩɪɢm
pp
p
pp
pp
pp
Ɇɚɬɪɢɰɚɩɟɪɟɯɨɞɧɵɯɜɟɪɨɹɬɧɨɫɬɟɣɹɜɥɹɟɬɫɹɫɬɨɯɚɫɬɢɱɟɫɤɨɣ±
ɤɚɠɞɚɹ ɫɬɪɨɤɚ ɦɚɬɪɢɰɵ ɉ ɯɚɪɚɤɬɟɪɢɡɭɟɬ ɩɨɥɧɭɸ ɝɪɭɩɩɭ ɫɥɭ
ɱɚɣɧɵɯɫɨɛɵɬɢɣɢɞɥɹɥɸɛɨɣɫɬɪɨɤɢɫɩɪɚɜɟɞɥɢɜɨɫɨɨɬɧɨɲɟɧɢɟ
m
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ji
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ɉɪɨɰɟɫɫ ɩɟɪɟɯɨɞɨɜ ɫɢɫɬɟɦɵ A ɢɡ ɫɨɫɬɨɹɧɢɹ ɜ ɫɨɫɬɨɹɧɢɟ
ɭɞɨɛɧɨɢɡɨɛɪɚɠɚɬɶɜɜɢɞɟɪɚɡɦɟɱɟɧɧɨɝɨɨɪɢɟɧɬɢɪɨɜɚɧɧɨɝɨɝɪɚɮɚ
ɫɨɫɬɨɹɧɢɣ ɜɟɪɲɢɧɵ ɤɨɬɨɪɨɝɨ ɩɪɟɞɫɬɚɜɥɹɸɬ ɫɨɛɨɣ ɤɨɧɤɪɟɬɧɵɟ
ɫɨɫɬɨɹɧɢɹɧɚɩɪɚɜɥɟɧɧɵɟɞɭɝɢɨɬɪɚɠɚɸɬɜɨɡɦɨɠɧɵɟɩɟɪɟɯɨɞɵɢɡ
ɫɨɫɬɨɹɧɢɹ ɜ ɫɨɫɬɨɹɧɢɟ ɚ ɜɨɡɥɟ ɞɭɝ ɭɤɚɡɚɧɵ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ
ɡɧɚɱɟɧɢɹɩɟɪɟɯɨɞɧɵɯɜɟɪɨɹɬɧɨɫɬɟɣɪɢɫ
a
a
ɇɚɝɪɚɮɚɯɬɚɤɨɝɨɪɨɞɚ ɜɟɪɨɹɬɧɨɫɬɢ ɡɚɞɟɪɠɤɢ
ɫɨɫɬɨɹɧɢɹɯ
ɞɢɚɝɨɧɚɥɶɧɵɟ ɷɥɟɦɟɧɬɵ ɦɚɬɪɢɰɵ Ȇ ɢɧɨɝɞɚ ɞɥɹ
a
i
a
ɫɢɫɬɟɦɵɜ
p
ii
ɩɪɨɫɬɨɬɵɧɟɩɪɨɫɬɚɜɥɹɸɬɩɨɫɤɨɥɶɤɭɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫɮɨɪɦɭɥɨɣ
ɨɧɢɹɜɥɹɸɬɫɹɞɨɩɨɥɧɟɧɢɟɦɞɨɟɞɢɧɢɰɵɜɫɟɯɨɫɬɚɥɶɧɵɯɷɥɟ
ɦɟɧɬɨɜiɣɫɬɪɨɤɢ
m
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ii ji
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j
ji
z
ɋɥɭɱɚɣɧɵɣ ɩɪɨɰɟɫɫ ɩɪɨɬɟɤɚɸɳɢɣɜɫɢɫɬɟɦɟ A ɧɚɡɵɜɚɟɬɫɹ
ɦɚɪɤɨɜɫɤɢɦ ɩɪɨɰɟɫɫɨɦ ɫ ɞɢɫɤɪɟɬɧɵɦ ɜɪɟɦɟɧɟɦ ɢ ɞɢɫɤɪɟɬɧɵɦɢ

Ʌɚɛɨɪɚɬɨɪɧɚɹ ɪɚɛɨɬɚ ʋ 4
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m
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m
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¹
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ɫɨɫɬɨɹɧɢɹɦɢ ɢɥɢ ɦɚɪɤɨɜɫɤɨɣ ɰɟɩɶɸ ɟɫɥɢ ɞɥɹ ɥɸɛɨɝɨ ɲɚɝɚ
ɭɫɥɨɜɧɵɟɜɟɪɨɹɬɧɨɫɬɢɫɨɫɬɨɹɧɢɣɫɢɫɬɟɦɵɜɛɭɞɭɳɟɦɩɪɢ
ɡɚɜɢɫɹɬɬɨɥɶɤɨɨɬɟɟ ɫɨɫɬɨɹɧɢɹ ɧɚɞɚɧɧɨɦɲɚɝɟɩɪɢ
ɡɚɜɢɫɹɬ ɨɬ ɬɨɝɨ ɧɚ ɤɚɤɨɦ ɲɚɝɟ ɩɪɢ
ɢ ɨɬɤɭɞɚ ɫɢɫɬɟɦɚ
kk
kk
kk!
ɢɧɟ
k
ɩɪɢɲɥɚɜɷɬɨɫɨɫɬɨɹɧɢɟ
Ⱦɚɧɧɨɟɨɩɪɟɞɟɥɟɧɢɟɨɡɧɚɱɚɟɬɱɬɨɞɥɹɤɚɠɞɨɝɨɲɚɝɚkɜɟɪɨ
ɹɬɧɨɫɬɶ ɩɟɪɟɯɨɞɚ ɢɡɥɸɛɨɝɨɫɨɫɬɨɹɧɢɹ
ɜ ɥɸɛɨɟ ɫɨɫɬɨɹɧɢɟ
a
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ij «mɧɟɡɚɜɢɫɢɬ ɨɬɬɨɝɨɤɨɝɞɚɢɤɚɤɫɢɫɬɟɦɚ Aɨɤɚɡɚ
ɥɚɫɶ ɜ ɫɨɫɬɨɹɧɢɢ
k
ɜɟɪɨɹɬɧɨɫɬɶ ɩɟɪɟɯɨɞɚ ɫɢɫɬɟɦɵɢɡɫɨɫɬɨɹɧɢɹ
p
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ɡɚkɲɚɝɨɜɩɪɢɷɬɨɦ
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ɋɥɟɞɨɜɚɬɟɥɶɧɨ ɟɫɥɢ ɨɛɨɡɧɚɱɢɬɶ ɫɢɦɜɨɥɨɦ
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ɜ ɫɨɫɬɨɹɧɢɟ
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pp
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±ɢɫɯɨɞɧɵɟɜɟɪɨɹɬɧɨɫɬɢɩɟɪɟ
ɯɨɞɨɜ ɧɚ ɨɞɧɨɦ ɲɚɝɟ ɬɨ ɡɚɜɢɫɢɦɨɫɬɶ ɨɬ ɩɪɟɞɲɟɫɬɜɭɸɳɟɝɨ
k
ɝɨɦɨɦɟɧɬɚɦɨɠɧɨɜɵɪɚɡɢɬɜɜɢɞɟɪɟɤɭɪɪɟɧɬɧɨɣɮɨɪɦɭɥɵ
m
kk
ppp
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ji qi jq
q
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ɍɪɚɜɧɟɧɢɟɧɚɡɵɜɚɟɬɫɹɭɪɚɜɧɟɧɢɟɦɆɚɪɤɨɜɚɢɥɢɱɚɫɬ
ɧɵɦɫɥɭɱɚɟɦɭɪɚɜɧɟɧɢɹɄɨɥɦɨɝɨɪɨɜɚ±ɑɟɩɦɟɧɚɢɯɚɪɚɤɬɟɪɢɡɭɟɬ
ɨɞɧɨɪɨɞɧɭɸɢɥɢ ɝɨɦɨɝɟɧɧɭɸ ɦɚɪɤɨɜɫɤɭɸɰɟɩɶ ± ɰɟɩɶɭɤɨɬɨ
ɪɨɣ ɩɟɪɟɯɨɞɧɵɟ ɜɟɪɨɹɬɧɨɫɬɢ
ɩɨɫɬɨɹɧɧɵ ɢ ɧɟ ɡɚɜɢɫɹɬ ɨɬ ɧɨ
p
ji
ɦɟɪɚɲɚɝɚk
Ɏɨɪɦɭɥɭɨɛɵɱɧɨɩɪɟɞɫɬɚɜɥɹɸɬɜɦɚɬɪɢɱɧɨɦɜɢɞɟ
kk k
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ɝɞɟ
kk
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± ɟɫɬɶ kɲɚɝɨɜɚɹ ɦɚɬɪɢɰɚ ɩɟɪɟɯɨɞɨɜ ɪɚɜɧɚɹ kɣ
pp p
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ɫɬɟɩɟɧɢk «ɢɫɯɨɞɧɨɣɦɚɬɪɢɰɵ
ȿɫɥɢ ɨɛɨɡɧɚɱɢɬɶ
± ɜɟɪɨɹɬɧɨɫɬɶ ɧɚɱɚɥɶɧɨɝɨ ɫɨɫɬɨɹɧɢɹ
p
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ɫɢɫɬɟɦɵɧɚɧɭɥɟɜɨɦɲɚɝɟɞɥɹɜɫɟɯi «mɬɨɜɟɪɨɹɬɧɨɫɬɢ
j

Ⱦɢɫɤɪɟɬɧɵɟ ɢɫɬɨɱɧɢɤɢ Ɇɚɪɤɨɜɚ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ
k
ɫɨɫɬɨɹɧɢɣɫɢɫɬɟɦɵ ɧɚ kɦ ɲɚɝɟɫɜɹɡɚɧɵɫɩɟɪɟɯɨɞɧɵɦɢɜɟ
p
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ɪɨɹɬɧɨɫɬɹɦɢ
ȼɜɟɞɹ ɜɟɤɬɨɪ ɧɚɱɚɥɶɧɵɯ ɫɨɫɬɨɹɧɢɣ
k
ɪɟɤɭɪɪɟɧɬɧɵɦɫɨɨɬɧɨɲɟɧɢɟɦ
p
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kk
ppp
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iqiq
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pp p S
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ɜɵɪɚɠɟɧɢɟɦɨɠɧɨɩɪɟɞɫɬɚɜɢɬɶɜɦɚɬɪɢɱɧɨɣɮɨɪɦɟ
SSȆSȆ
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kk
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ɋɨɨɬɧɨɲɟɧɢɟɦɨɠɧɨɡɚɩɢɫɚɬɶɜɷɤɜɢɜɚɥɟɧɬɧɨɦɜɢɞɟ
m
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q
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ɢɥɢɜɦɚɬɪɢɱɧɨɣɮɨɪɦɟ
SSȆ
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ɍɪɚɜɧɟɧɢɹ ± ɬɚɤɠɟ ɧɚɡɵɜɚɸɬ ɭɪɚɜɧɟɧɢɹɦɢ Ɇɚɪ
ɤɨɜɚ
Ⱦɥɹ ɬɨɝɨ ɱɬɨɛɵ ɫɬɚɰɢɨɧɚɪɧɵɣ ɢɫɬɨɱɧɢɤ Ɇɚɪɤɨɜɚ ɝɨ ɩɨ
ɪɹɞɤɚ ɛɵɥ ɷɪɝɨɞɢɱɟɫɤɢɦ ɧɟɨɛɯɨɞɢɦɨ ɱɬɨɛɵ ɜ ɷɤɜɢɜɚɥɟɧɬɧɨɣ
ɟɦɭɨɞɧɨɪɨɞɧɨɣɦɚɪɤɨɜɫɤɨɣ ɰɟɩɢ ɫ ɜɨɡɪɚɫɬɚɧɢɟɦ ɧɨɦɟɪɚɲɚɝɚk
ɭɫɬɚɧɚɜɥɢɜɚɥɫɹ ɫɬɚɰɢɨɧɚɪɧɵɣɪɟɠɢɦ ɜɤɨɬɨɪɨɦ ɫɢɫɬɟɦɚ A ɩɪɨ
ɞɨɥɠɚɟɬɛɥɭɠɞɚɬɶɩɨ ɫɨɫɬɨɹɧɢɹɦɧɨɜɟɪɨɹɬɧɨɫɬɢɫɨɫɬɨɹɧɢɣɭɠɟ
ɨɬ ɧɨɦɟɪɚ ɲɚɝɚ ɧɟ ɡɚɜɢɫɹɬ Ɍɚɤɢɟ ɜɟɪɨɹɬɧɨɫɬɢ ɧɚɡɵɜɚɸɬɫɹ ɩɪɟ
ɞɟɥɶɧɵɦɢɮɢɧɚɥɶɧɵɦɢɧɟɡɚɜɢɫɹɬɨɬɧɚɱɚɥɶɧɨɝɨɫɨɫɬɨɹɧɢɹɫɢ
ɫɬɟɦɵɢɧɚɯɨɞɹɬɫɹɫɩɨɦɨɳɶɸɩɪɟɞɟɥɶɧɨɝɨɩɟɪɟɯɨɞɚ
k
of
k
OLP
ppa p
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Ɉɞɧɨɪɨɞɧɚɹɦɚɪɤɨɜɫɤɚɹɰɟɩɶɜɤɨɬɨɪɨɣ ɞɨɫɬɢɝɚɟɬɫɹ ɫɬɚɰɢ
ɨɧɚɪɧɵɣɪɟɠɢɦɧɚɡɵɜɚɟɬɫɹɪɟɝɭɥɹɪɧɨɣɢɥɢɷɪɝɨɞɢɱɟɫɤɨɣ
Ɋɟɝɭɥɹɪɧɚɹɰɟɩɶɭɞɨɜɥɟɬɜɨɪɹɟɬɫɥɟɞɭɸɳɢɦɭɫɥɨɜɢɹɦ
ȼɫɟ ɫɨɫɬɨɹɧɢɹ ɫɢɫɬɟɦɵ A ɹɜɥɹɸɬɫɹ ɬɪɚɧɡɢɬɢɜɧɵɦɢ ± ɢɡ
ɥɸɛɨɝɨ ɫɨɫɬɨɹɧɢɹ ɦɨɠɧɨ ɩɟɪɟɣɬɢ ɜ ɥɸɛɨɟ ɞɪɭɝɨɟ ɡɚ ɤɨɧɟɱɧɨɟ
ɱɢɫɥɨɲɚɝɨɜɆɚɬɟɦɚɬɢɱɟɫɤɢɷɬɨ ɨɡɧɚɱɚɟɬ ɱɬɨ ɫɭɳɟɫɬɜɭɟɬ ɧɟɤɨ

Ʌɚɛɨɪɚɬɨɪɧɚɹ ɪɚɛɨɬɚ ʋ 4
¸
¸
¸
¸¸
¸
ɬɨɪɨɟ ɡɧɚɱɟɧɢɟ ɲɚɝɚk ɞɥɹɤɨɬɨɪɨɝɨ ɜɫɟ ɷɥɟɦɟɧɬɵɦɚɬɪɢɰɵ
Ȇ
k
ɨɬɥɢɱɧɵɨɬɧɭɥɹ
ɐɟɩɶɧɟɹɜɥɹɟɬɫɹɧɢɰɢɤɥɢɱɟɫɤɨɣɧɢɩɨɝɥɨɳɚɸɳɟɣɐɢɤ
ɥɢɱɟɫɤɚɹ ɰɟɩɶ ± ɷɬɨ ɰɟɩɶ ɜ ɤɨɬɨɪɨɣɜ ɤɚɠɞɨɟɫɨɫɬɨɹɧɢɟ ɦɨɠɧɨ
ɩɨɩɚɞɚɬɶ ɱɟɪɟɡ ɨɩɪɟɞɟɥɟɧɧɨɟ ɧɟ ɫɥɭɱɚɣɧɨɟɱɢɫɥɨ ɲɚɝɨɜɩɟɪɢɨ
ɞɨɜ ɜɪɟɦɟɧɢ ɉɨɝɥɨɳɚɸɳɟɣ ɰɟɩɶɸ ɧɚɡɵɜɚɟɬɫɹ ɰɟɩɶ ɢɦɟɸɳɚɹ
ɩɨɤɪɚɣɧɟɣɦɟɪɟ ɨɞɧɨɩɨɝɥɨɳɚɸɳɟɟɫɨɫɬɨɹɧɢɟɩɨɩɚɜɜ ɤɨɬɨɪɨɟ
ɫɢɫɬɟɦɚɫɨɯɪɚɧɹɟɬɧɚɜɫɟɝɞɚɩɨɜɬɨɪɹɟɬɟɝɨɧɚɤɚɠɞɨɦɲɚɝɟɫɜɟ
ɪɨɹɬɧɨɫɬɶɸɪɚɜɧɨɣɟɞɢɧɢɰɟ ɩɪɢɱɟɦ ɢɡɥɸɛɨɝɨɫɨɫɬɨɹɧɢɹɦɨɠ
ɧɨɩɨɩɚɫɬɶɜɩɨɝɥɨɳɚɸɳɟɟɫɨɫɬɨɹɧɢɟɡɚɩɪɨɢɡɜɨɥɶɧɨɟɱɢɫɥɨɲɚ
ɝɨɜ
ɐɢɤɥɢɱɟɫɤɢɟɢ ɩɨɝɥɨɳɚɸɳɢɟ ɦɚɪɤɨɜɫɤɢɟɰɟɩɢɧɟ ɩɨɡɜɨɥɹ
ɸɬ ɚɞɟɤɜɚɬɧɨ ɨɩɢɫɵɜɚɬɶ ɪɟɚɥɶɧɵɟ ɞɢɫɤɪɟɬɧɵɟ ɢɫɬɨɱɧɢɤɢ ɢ ɩɨ
ɷɬɨɦɭɜɬɟɨɪɢɢɢɧɮɨɪɦɚɰɢɢɪɚɫɫɦɚɬɪɢɜɚɸɬɫɹɬɨɥɶɤɨɪɟɝɭɥɹɪɧɵɟ
ɰɟɩɢ
ɆɚɬɪɢɰɚȆ ɪɟɝɭɥɹɪɧɨɣ ɰɟɩɢ ɧɚɡɵɜɚɟɬɫɹ ɪɟɝɭɥɹɪɧɨɣɦɚɬɪɢ
ɰɟɣɩɟɪɟɯɨɞɧɵɯɜɟɪɨɹɬɧɨɫɬɟɣ
Ⱦɥɹɪɟɝɭɥɹɪɧɵɯɰɟɩɟɣɞɨɤɚɡɚɧɵɫɥɟɞɭɸɳɢɟɬɟɨɪɟɦɵ
ɌɟɨɪɟɦɚȿɫɥɢȆɟɫɬɶɪɟɝɭɥɹɪɧɚɹɦɚɬɪɢɰɚɩɟɪɟɯɨɞɧɵɯɜɟ
ɪɨɹɬɧɨɫɬɟɣ ɬɨ ɫɬɟɩɟɧɢ
Ȇ
k
ɩɪɢ
k of
ɫɬɪɟɦɹɬɫɹ ɤ ɫɬɨɯɚɫɬɢɱɟ
ɫɤɨɣɦɚɬɪɢɰɟ
S
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pp p
§·
¨¸
pp p
¨¸
¨¸
¨¸
pp p
©¹
m
m
m
ɝɞɟ
pp p S
± ɜɟɤɬɨɪɫɬɪɨɤɚ ɮɢɧɚɥɶɧɵɯ ɜɟɪɨɹɬɧɨ
m
ɫɬɟɣ
m
ɉɨɫɤɨɥɶɤɭ
¦
i
ɬɨɜɟɥɢɱɢɧɭɮɢɧɚɥɶɧɨɣɜɟɪɨɹɬɧɨɫɬɢ
p
i
p
i
ɦɨɠɧɨ ɢɧɬɟɪɩɪɟɬɢɪɨɜɚɬɶ ɤɚɤ ɨɬɧɨɫɢɬɟɥɶɧɨɟ ɜɪɟɦɹ ɩɪɟɛɵɜɚɧɢɹ
ɫɢɫɬɟɦɵɜɫɨɫɬɨɹɧɢɢ
ɩɪɢɨɱɟɧɶɛɨɥɶɲɨɦɱɢɫɥɟɲɚɝɨɜ
a
i
ɗɬɚɬɟɨɪɟɦɚɧɚɡɵɜɚɟɬɫɹɮɭɧɞɚɦɟɧɬɚɥɶɧɨɣɬɟɨɪɟɦɨɣɞɥɹɪɟ
ɝɭɥɹɪɧɵɯ ɰɟɩɟɣ Ɇɚɪɤɨɜɚ ɋɥɟɞɫɬɜɢɟɦ ɷɬɨɣ ɬɟɨɪɟɦɵ ɹɜɥɹɟɬɫɹ
ɩɪɟɞɟɥɶɧɨɟɫɨɨɬɧɨɲɟɧɢɟ

Ⱦɢɫɤɪɟɬɧɵɟ ɢɫɬɨɱɧɢɤɢ Ɇɚɪɤɨɜɚ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ
Ɍɟɨɪɟɦɚ ȿɫɥɢȆ ɟɫɬɶɪɟɝɭɥɹɪɧɚɹɦɚɬɪɢɰɚɩɟɪɟɯɨɞɧɵɯɜɟ
ɪɨɹɬɧɨɫɬɟɣ ɚ
S
± ɮɢɧɚɥɶɧɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɬɨSɟɫɬɶ ɟɞɢɧ
ɫɬɜɟɧɧɵɣɜɟɪɨɹɬɧɨɫɬɧɵɣɜɟɤɬɨɪɬɚɤɨɣɱɬɨ
SSȆ
ɍɪɚɜɧɟɧɢɟ ɩɨɡɜɨɥɹɟɬ ɨɩɪɟɞɟɥɹɬɶ ɜɟɥɢɱɢɧɵ ɩɪɟɞɟɥɶ
ɧɵɯ ɜɟɪɨɹɬɧɨɫɬɟɣ ɪɟɝɭɥɹɪɧɨɣ ɰɟɩɢ Ⱦɥɹ ɷɬɨɝɨ ɟɝɨ ɭɞɨɛɧɨ ɩɪɟɞ
ɫɬɚɜɢɬɶɜɪɚɡɜɟɪɧɭɬɨɦɜɢɞɟ
m
ppp
¦
iqiq
q
i «m
ɢɥɢ
pppp pp
°
pp p p p p
°
®
mm
mm
°
°
pp pp p p
mm mmm
¯
ɋɢɫɬɟɦɚ ɢɦɟɟɬ ɧɭɥɟɜɨɣ ɜɟɤɬɨɪ ɫɜɨɛɨɞɧɵɯ ɱɥɟɧɨɜ ɢ
ɫɥɟɞɨɜɚɬɟɥɶɧɨ ɛɟɫɤɨɧɟɱɧɨɟ ɱɢɫɥɨ ɪɟɲɟɧɢɣ ɫ ɬɨɱɧɨɫɬɶɸ ɞɨ ɩɨ
ɫɬɨɹɧɧɨɝɨ ɦɧɨɠɢɬɟɥɹ Ⱦɥɹ ɩɨɥɭɱɟɧɢɹ ɟɞɢɧɫɬɜɟɧɧɨɝɨ ɪɟɲɟɧɢɹ
ɧɟɨɛɯɨɞɢɦɨɨɬɛɪɨɫɢɬɶ ɨɞɧɨ ɧɚɩɪɢɦɟɪ ɩɨɫɥɟɞɧɟɟɭɪɚɜɧɟɧɢɟ ɫɢ
ɫɬɟɦɵɡɚɦɟɧɢɜɟɝɨɧɨɪɦɢɪɨɜɨɱɧɵɦɭɫɥɨɜɢɟɦ
m
p
¦
i
i
ȼ ɪɟɡɭɥɶɬɚɬɟ ɫɢɫɬɟɦɚ ɫ ɭɫɥɨɜɢɟɦ ɩɪɢɨɛɪɟɬɚɟɬ
ɨɤɨɧɱɚɬɟɥɶɧɵɣɜɢɞ
pppp pp
°
°
°
pp p p p p
mm
mm
®
°
pppp p p
mm mmm
°
°
¯
pp p
m

Ʌɚɛɨɪɚɬɨɪɧɚɹ ɪɚɛɨɬɚ ʋ 4
ɊɢɫȽɪɚɮɫɨɫɬɨɹɧɢɣɢɫɬɨɱɧɢɤɚ
ɋɢɫɬɟɦɚ ɢɦɟɟɬ ɟɞɢɧɫɬɜɟɧɧɨɟ ɪɟɲɟɧɢɟ ɢ ɨɞɧɨɡɧɚɱɧɨ
ɨɩɪɟɞɟɥɹɟɬɮɢɧɚɥɶɧɵɟɜɟɪɨɹɬɧɨɫɬɢ
pp p
ɞɚɸɳɢɟɜɫɭɦɦɟ
m
ɟɞɢɧɢɰɭ
Ɋɚɫɩɪɟɞɟɥɟɧɢɟ
pp p S
ɮɢɧɚɥɶɧɵɯ ɜɟɪɨɹɬɧɨ
m
ɫɬɟɣ ɧɚɡɵɜɚɟɬɫɹ ɷɪɝɨɞɢɱɟɫɤɢɦ ɪɚɫɩɪɟɞɟɥɟɧɢɟɦ ɋɨɨɬɜɟɬɫɬɜɟɧɧɨ
ɷɧɬɪɨɩɢɹ ɷɪɝɨɞɢɱɟɫɤɨɝɨ ɦɚɪɤɨɜɫɤɨɝɨ ɢɫɬɨɱɧɢɤɚ ɝɨ ɩɨɪɹɞɤɚ ɫ
ɭɱɟɬɨɦɜɜɟɞɟɧɧɵɯɨɛɨɡɧɚɱɟɧɢɣɞɥɹɩɟɪɟɯɨɞɧɵɯɢɮɢɧɚɥɶɧɵɯɜɟ
ɪɨɹɬɧɨɫɬɟɣɫɨɝɥɚɫɧɨɨɩɪɟɞɟɥɹɟɬɫɹɜɵɪɚɠɟɧɢɟɦ
mm
ORJ
HA p p p
¦¦
iji ji
ij
Ɋɚɫɱɟɬɧɵɟɡɚɞɚɧɢɹ
ɁɚɞɚɧɢɟɆɚɪɤɨɜɫɤɢɣ ɢɫɬɨɱɧɢɤ ɝɨ ɩɨɪɹɞɤɚ ɡɚɞɚɧ ɝɪɚ
ɮɨɦɫɨɫɬɨɹɧɢɣɩɪɟɞɫɬɚɜɥɟɧɧɵɦɧɚɪɢɫ
Ɍɪɟɛɭɟɬɫɹ ɩɨɫɬɪɨɢɬɶ ɦɚɬɪɢɰɭ ɩɟɪɟɯɨɞɧɵɯ ɜɟɪɨɹɬɧɨɫɬɟɣ
ɩɨɥɚɝɚɹɱɬɨɜɧɚɱɚɥɶɧɨɦɫɨɫɬɨɹɧɢɢɢɫɬɨɱɧɢɤɜɵɪɚɛɚɬɵɜɚɟɬɫɢɦ
ɜɨɥ
ɧɚɣɬɢ ɡɧɚɱɟɧɢɹ ɜɟɪɨɹɬɧɨɫɬɟɣ
a
kkk
ppp
ɡɚ ɩɟɪɜɵɟ
ɩɹɬɧɚɞɰɚɬɶɲɚɝɨɜɭɛɟɞɢɬɶɫɹɱɬɨɦɚɪɤɨɜɫɤɚɹɰɟɩɶɹɜɥɹɟɬɫɹɪɟɝɭ
ɥɹɪɧɨɣɧɚɣɬɢɷɪɝɨɞɢɱɟɫɤɨɟɪɚɫɩɪɟɞɟɥɟɧɢɟɜɟɪɨɹɬɧɨɫɬɟɣɪɚɫɫɱɢ
ɬɚɬɶɷɧɬɪɨɩɢɸɢɢɡɛɵɬɨɱɧɨɫɬɶɢɫɬɨɱɧɢɤɚ
Ɋɟɲɟɧɢɟ ɋ ɭɱɟɬɨɦ ɨɛɨɡɧɚɱɟɧɢɣ ɜɟɥɢɱɢɧ ɜɟɪɨɹɬɧɨɫɬɟɣ ɧɚ
ɝɪɚɮɟɦɚɬɪɢɰɚɩɟɪɟɯɨɞɧɵɯɜɟɪɨɹɬɧɨɫɬɟɣɢɦɟɟɬɜɢɞ
a
a
a

Ⱦɢɫɤɪɟɬɧɵɟ ɢɫɬɨɱɧɢɤɢ Ɇɚɪɤɨɜɚ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ
ppp p p
§·§·
¨¸¨¸
ppp pp
Ȇ
¨¸¨¸
¨¸¨¸
ppp pp
©¹©¹
ɇɟɞɨɫɬɚɸɳɢɟɜɟɪɨɹɬɧɨɫɬɢɧɚɯɨɞɢɦɢɡɫɥɟɞɭɸɳɢɯɪɚɫɫɭɠɞɟɧɢɣ
± ɢɡɫɨɫɬɨɹɧɢɹ
ɫɜɟɪɨɹɬɧɨɫɬɶɸ
ɫɥɟɞɨɜɚɬɟɥɶɧɨ
p
p
± ɢɡɫɨɫɬɨɹɧɢɹ
aao
ɜɚɬɟɥɶɧɨ
ɫ ɜɟɪɨɹɬɧɨɫɬɹɦɢ
ɢ
p
± ɜɨɡɦɨɠɧɵɟ ɩɟɪɟɯɨɞɵ ɢɡ ɫɨɫɬɨɹɧɢɹ
aao
p
ɚ ɢɯ ɜɟɪɨɹɬɧɨɫɬɢ ɫɨɫɬɚɜɥɹɸɬ
ɢ
pp
ɜɨɡɦɨɠɧɵɞɜɚɩɟɪɟɯɨɞɚ± ɩɟɪɟɯɨɞ
a
ɢɩɟɪɟɯɨɞ
ɢ
ɬɚɤɠɟɜɨɡɦɨɠɧɵɞɜɚɩɟɪɟɯɨɞɚ±
a
pp
aao
pp
ɢ
p
p
ɫɜɟɪɨɹɬɧɨɫɬɶɸ
ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɫɥɟɞɨ
p
ɢ
ɟɫɬɶ
a
ɨɬɤɭɞɚ
p
aao
p
aao
aao
Ɍɚɤɢɦɨɛɪɚɡɨɦɦɚɬɪɢɰɚɩɟɪɟɯɨɞɧɵɯɜɟɪɨɹɬɧɨɫɬɟɣ
§·
¨¸
Ȇ
¨¸
¨¸
©¹
ȼ ɫɢɥɭ ɬɨɝɨ ɱɬɨ ɜɧɚɱɚɥɶɧɨɦ ɫɨɫɬɨɹɧɢɢ ɢɫɬɨɱɧɢɤ ɜɵɪɚɛɚ
ɬɵɜɚɟɬɫɢɦɜɨɥ
ɜɟɤɬɨɪɧɚɱɚɥɶɧɵɯɜɟɪɨɹɬɧɨɫɬɟɣɫɨɫɬɨɹɧɢɣ
a
ppp S
ɢ
ɢ
%Script_4_1
%Ɋɚɫɱɟɬ ɯɚɪɚɤɬɟɪɢɫɬɢɤ ɦɚɪɤɨɜɫɤɨɝɨ ɢɫɬɨɱɧɢɤɚ 1-ɝɨ ɩɨɪɹɞɤɚ
clear all; close all; clc
%Ɂɚɞɚɧɢɟ ɚɧɨɧɢɦɧɨɣ ɮɭɧɤɰɢɢ
%--------------------------fun=@(x) -x.*log2(x+(x==0));
%--------------------------%Ɂɚɞɚɧɢɟ ɦɚɬɪɢɰɵ ɨɞɧɨɲɚɝɨɜɵɯ ɩɟɪɟɯɨɞɧɵɯ ɜɟɪɨɹɬɧɨɫɬɟɣ
ɉ
P=[.5 .5 0; 0 .5 .5; .5 0 .5];

Ʌɚɛɨɪɚɬɨɪɧɚɹ ɪɚɛɨɬɚ ʋ 4
disp('Ɇɚɬɪɢɰɚ ɨɞɧɨɲɚɝɨɜɵɯ ɩɟɪɟɯɨɞɧɵɯ ɜɟɪɨɹɬɧɨɫɬɟɣ
ɉ:')
disp(P)
%Ɂɚɞɚɧɢɟ ɜɟɤɬɨɪɚ ɧɚɱɚɥɶɧɵɯ ɜɟɪɨɹɬɧɨɫɬɟɣ ɫɨɫɬɨɹɧɢɣ p0
p=[1 0 0];
disp('ȼɟɤɬɨɪ ɧɚɱɚɥɶɧɵɯ ɜɟɪɨɹɬɧɨɫɬɟɣ ɫɨɫɬɨɹɧɢɣ p0:')
disp(p)
%Ɂɚɞɚɧɢɟ ɱɢɫɥɚ ɲɚɝɨɜ Ʉ
K=15;
disp('ɑɢɫɥɨ ɲɚɝɨɜ K:')
disp(K)
%ȼɵɱɢɫɥɟɧɢɹ ɩɨ ɲɚɝɚɦ
for k=1:K
formatSpec = '%i-ɣ ɲɚɝ:\n';
fprintf(formatSpec,k);
disp('---------')
pk=p*P^k;
disp('ȼɟɤɬɨɪ ɜɟɪɨɹɬɧɨɫɬɟɣ ɫɨɫɬɨɹɧɢɣ:')
disp(pk)
disp('Ɇɧɨɝɨɲɚɝɨɜɚɹ ɦɚɬɪɢɰɚ ɩɟɪɟɯɨɞɧɵɯ ɜɟɪɨɹɬɧɨɫɬɟɣ:')
disp(P^k)
end
%ȼɵɱɢɫɥɟɧɢɟ ɮɢɧɚɥɶɧɵɯ ɜɟɪɨɹɬɧɨɫɬɟɣ,
%ɷɧɬɪɨɩɢɢ ɢ ɢɡɛɵɬɨɱɧɨɫɬɢ ɢɫɬɨɱɧɢɤɚ
disp('ȼɵɱɢɫɥɟɧɢɟ ɮɢɧɚɥɶɧɵɯ ɜɟɪɨɹɬɧɨɫɬɟɣ:')
disp('---------------------------------')
%Ɉɩɪɟɞɟɥɟɧɢɹ ɩɨɪɹɞɤɚ ɦɚɬɪɢɰɵ Ɋ
[~,m]=size(P);
%Ɂɚɞɚɧɢɟ ɦɚɬɪɢɰɵ ɥɢɧɟɣɧɨɣ ɫɢɫɬɟɦɵ
A=P'-eye(m);
A(m,:)=ones(1,m);
disp('Ɇɚɬɪɢɰɚ ɥɢɧɟɣɧɨɣ ɫɢɫɬɟɦɵ:')
disp(A)
%Ɂɚɞɚɧɢɟ ɜɟɤɬɨɪɚ ɫɜɨɛɨɞɧɵɯ ɱɥɟɧɨɜ
b=[zeros(1,m-1) 1]';
disp('ȼɟɤɬɨɪ ɫɜɨɛɨɞɧɵɯ ɱɥɟɧɨɜ:')
disp(b)
%Ɋɟɲɟɧɢɟ ɫɢɫɬɟɦɵ ɥɢɧɟɣɧɵɯ ɭɪɚɜɧɟɧɢɣ
pf=linsolve(A,b);
disp('ȼɟɤɬɨɪ ɮɢɧɚɥɶɧɵɯ ɜɟɪɨɹɬɧɨɫɬɟɣ:')
disp(pf)
disp('ȼɵɱɢɫɥɟɧɢɟ ɷɧɬɪɨɩɢɢ ɢɫɬɨɱɧɢɤɚ:')
disp('-------------------------------')
disp('ɑɚɫɬɧɵɟ ɭɫɥɨɜɧɵɟ ɷɧɬɪɨɩɢɢ, ɛɢɬ/ɫɢɦɜɨɥ::')

Ⱦɢɫɤɪɟɬɧɵɟ ɢɫɬɨɱɧɢɤɢ Ɇɚɪɤɨɜɚ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ
Ɋɢɫ
Ƚɪɚɮɵɫɨɫɬɨɹɧɢɣɢɫɬɨɱɧɢɤɨɜ
ɚ
H_A_a=sum(fun(P),2);
disp(H_A_a)
disp('ɗɧɬɪɨɩɢɹ ɢɫɬɨɱɧɢɤɚ, ɛɢɬ/ɫɢɦɜɨɥ::')
H_A=sum(pf.*H_A_a);
disp(H_A)
disp('ɂɡɛɵɬɨɱɧɨɫɬɶ ɢɫɬɨɱɧɢɤɚ:')
x=1-H_A/log2(m);
disp(x)
Ɂɚɞɚɧɢɟ Ƚɪɚɮ ɫɨɫɬɨɹɧɢɣ ɦɚɪɤɨɜɫɤɨɝɨ ɢɫɬɨɱɧɢɤɚ ɝɨ
ɩɨɪɹɞɤɚɩɨɤɚɡɚɧɧɚɪɢɫɚ
Ɍɪɟɛɭɟɬɫɹ ɩɨɫɬɪɨɢɬɶ ɦɚɬɪɢɰɭ ɩɟɪɟɯɨɞɧɵɯ ɜɟɪɨɹɬɧɨɫɬɟɣ
ɩɨɥɚɝɚɹɱɬɨɜɧɚɱɚɥɶɧɨɦɫɨɫɬɨɹɧɢɢɢɫɬɨɱɧɢɤɜɵɪɚɛɚɬɵɜɚɟɬɫɢɦ
ɜɨɥ
ɧɚɣɬɢɡɧɚɱɟɧɢɹɜɟɪɨɹɬɧɨɫɬɟɣ
a
kkk
ppp
ɡɚɩɟɪɜɵɟɩɹɬɶ
ɲɚɝɨɜ ɭɛɟɞɢɬɶɫɹ ɱɬɨ ɦɚɪɤɨɜɫɤɚɹ ɰɟɩɶ ɹɜɥɹɟɬɫɹ ɪɟɝɭɥɹɪɧɨɣ
ɧɚɣɬɢ ɷɪɝɨɞɢɱɟɫɤɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɜɟɪɨɹɬɧɨɫɬɟɣ ɪɚɫɫɱɢɬɚɬɶ ɷɧ
ɬɪɨɩɢɸɢɢɡɛɵɬɨɱɧɨɫɬɶɢɫɬɨɱɧɢɤɚ
ɍɤɚɡɚɧɢɟɢɫɩɨɥɶɡɨɜɚɬɶɦɟɬɨɞɢɤɭɢɫɩɨɥɶɡɭɟɦɭɸɩɪɢɪɟɲɟ
ɧɢɢɡɚɞɚɧɢɹɚɬɚɤɠɟScript_4_1
a
a
a
a
a
a
ɛ
Ɂɚɞɚɧɢɟ Ƚɪɚɮ ɫɨɫɬɨɹɧɢɣ ɦɚɪɤɨɜɫɤɨɝɨ ɢɫɬɨɱɧɢɤɚ ɝɨ
ɩɨɪɹɞɤɚɩɨɤɚɡɚɧɧɚɪɢɫɛ
Ɍɪɟɛɭɟɬɫɹ ɩɨɫɬɪɨɢɬɶ ɦɚɬɪɢɰɭ ɩɟɪɟɯɨɞɧɵɯ ɜɟɪɨɹɬɧɨɫɬɟɣ
ɩɨɥɚɝɚɹɱɬɨɜɧɚɱɚɥɶɧɨɦɫɨɫɬɨɹɧɢɢɢɫɬɨɱɧɢɤɜɵɪɚɛɚɬɵɜɚɟɬɫɢɦ
ɜɨɥ
ɧɚɣɬɢɡɧɚɱɟɧɢɹɜɟɪɨɹɬɧɨɫɬɟɣ
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ɡɚɩɟɪɜɵɟɩɹɬɶ

Ʌɚɛɨɪɚɬɨɪɧɚɹ ɪɚɛɨɬɚ ʋ 4
ɊɢɫȽɪɚɮɵɫɨɫɬɨɹɧɢɣɢɫɬɨɱɧɢɤɨɜ
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ɲɚɝɨɜ ɭɛɟɞɢɬɶɫɹ ɱɬɨ ɦɚɪɤɨɜɫɤɚɹ ɰɟɩɶ ɹɜɥɹɟɬɫɹ ɪɟɝɭɥɹɪɧɨɣ
ɧɚɣɬɢ ɷɪɝɨɞɢɱɟɫɤɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɜɟɪɨɹɬɧɨɫɬɟɣ ɪɚɫɫɱɢɬɚɬɶ ɷɧ
ɬɪɨɩɢɸɢɢɡɛɵɬɨɱɧɨɫɬɶɢɫɬɨɱɧɢɤɚ
ɍɤɚɡɚɧɢɟɢɫɩɨɥɶɡɨɜɚɬɶɦɟɬɨɞɢɤɭɢɫɩɨɥɶɡɭɟɦɭɸɩɪɢɪɟɲɟ
ɧɢɢɡɚɞɚɧɢɹɚɬɚɤɠɟScript_4_1
ɁɚɞɚɧɢɟȽɪɚɮ ɫɨɫɬɨɹɧɢɣ ɦɚɪɤɨɜɫɤɨɝɨ ɢɫɬɨɱɧɢɤɚ ɝɨ
ɩɨɪɹɞɤɚɩɨɤɚɡɚɧɧɚɪɢɫɚ
Ɍɪɟɛɭɟɬɫɹ ɩɨɫɬɪɨɢɬɶ ɦɚɬɪɢɰɭ ɩɟɪɟɯɨɞɧɵɯ ɜɟɪɨɹɬɧɨɫɬɟɣ
ɩɨɥɚɝɚɹɱɬɨɜɧɚɱɚɥɶɧɨɦɫɨɫɬɨɹɧɢɢɢɫɬɨɱɧɢɤɜɵɪɚɛɚɬɵɜɚɟɬɫɢɦ
ɜɨɥ
ɧɚɣɬɢ ɡɧɚɱɟɧɢɹ ɜɟɪɨɹɬɧɨɫɬɟɣ
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ɲɚɝɨɜ ɭɛɟɞɢɬɶɫɹ ɱɬɨ ɦɚɪɤɨɜɫɤɚɹ ɰɟɩɶ ɹɜɥɹɟɬɫɹ ɪɟɝɭɥɹɪɧɨɣ
ɧɚɣɬɢ ɷɪɝɨɞɢɱɟɫɤɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɜɟɪɨɹɬɧɨɫɬɟɣ ɪɚɫɫɱɢɬɚɬɶ ɷɧ
ɬɪɨɩɢɸɢɢɡɛɵɬɨɱɧɨɫɬɶɢɫɬɨɱɧɢɤɚ
ɍɤɚɡɚɧɢɟɢɫɩɨɥɶɡɨɜɚɬɶɦɟɬɨɞɢɤɭɢɫɩɨɥɶɡɭɟɦɭɸɩɪɢɪɟɲɟ
ɧɢɢɡɚɞɚɧɢɹɚɬɚɤɠɟScript_4.1
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ɁɚɞɚɧɢɟɆɚɪɤɨɜɫɤɢɣ ɢɫɬɨɱɧɢɤ ɝɨ ɩɨɪɹɞɤɚ ɡɚɞɚɧ ɝɪɚ
ɮɨɦɫɨɫɬɨɹɧɢɣɩɨɤɚɡɚɧɧɨɦɧɚɪɢɫɛ
Ɍɪɟɛɭɟɬɫɹ ɩɨɫɬɪɨɢɬɶ ɦɚɬɪɢɰɭ ɩɟɪɟɯɨɞɧɵɯ ɜɟɪɨɹɬɧɨɫɬɟɣ
ɩɨɥɚɝɚɹɱɬɨɜɧɚɱɚɥɶɧɨɦɫɨɫɬɨɹɧɢɢɢɫɬɨɱɧɢɤɜɵɪɚɛɚɬɵɜɚɟɬɫɢɦ
ɜɨɥ
ɧɚɣɬɢ ɡɧɚɱɟɧɢɹ ɜɟɪɨɹɬɧɨɫɬɟɣ
a
ɲɚɝɨɜ ɭɛɟɞɢɬɶɫɹ ɱɬɨ ɦɚɪɤɨɜɫɤɚɹ ɰɟɩɶ ɹɜɥɹɟɬɫɹ ɪɟɝɭɥɹɪɧɨɣ
ɧɚɣɬɢ ɷɪɝɨɞɢɱɟɫɤɨɟ ɪɚɫɩɪɟɞɟɥɟɧɢɟ ɜɟɪɨɹɬɧɨɫɬɟɣ ɪɚɫɫɱɢɬɚɬɶ ɷɧ
ɬɪɨɩɢɸɢɢɡɛɵɬɨɱɧɨɫɬɶɢɫɬɨɱɧɢɤɚ
ɍɤɚɡɚɧɢɟɢɫɩɨɥɶɡɨɜɚɬɶɦɟɬɨɞɢɤɭɢɫɩɨɥɶɡɭɟɦɭɸɩɪɢɪɟɲɟ
ɧɢɢɡɚɞɚɧɢɹɚɬɚɤɠɟScript_4_1
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