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2015
ɇȺɍɑɇɕɃ ȼȿɋɌɇɂɄ ɆȽɌɍ ȽȺ
ʋ 220
ɍȾɄ 336
ɇȺɄɈɉɂɌȿɅɖɇɕȿ ɉȿɇɋɂɈɇɇɕȿ ɋɏȿɆɕ
ɋ ɊȺɁɅɂɑɇɕɆɂ ȾȿɄɊȿɆȿɇɌɇɕɆɂ ɎȺɄɌɈɊȺɆɂ
Ɇ.ɋ. ȺɅɖ-ɇȺɌɈɊ, ɋ.ȼ. ȺɅɖ-ɇȺɌɈɊ, ɇ.Ⱥ. ɈɅȿɇɑȿɇɄɈ
Ɋɚɫɫɦɚɬɪɢɜɚɸɬɫɹ ɧɚɤɨɩɢɬɟɥɶɧɵɟ ɩɟɧɫɢɨɧɧɵɟ ɫɯɟɦɵ ɫ ɭɫɬɚɧɨɜɥɟɧɧɵɦɢ ɪɚɡɦɟɪɚɦɢ ɜɡɧɨɫɨɜ ɩɪɟɧɭɦɟɪɚɧɞɨ ɢ
ɟɞɢɧɨɜɪɟɦɟɧɧɨɣ ɜɵɩɥɚɬɨɣ ɜ ɫɥɭɱɚɟ ɞɨɠɢɬɢɹ ɞɨ ɩɟɧɫɢɨɧɧɨɝɨ ɜɨɡɪɚɫɬɚ, ɨɬɥɢɱɚɸɳɢɟɫɹ ɨɬɧɨɲɟɧɢɟɦ ɤ ɧɚɫɥɟɞɨɜɚɧɢɸ ɢ
ɩɪɟɞɭɫɦɚɬɪɢɜɚɸɳɢɟ ɪɚɡɥɢɱɧɵɟ ɮɚɤɬɨɪɵ ɜɵɛɵɬɢɹ. Ɋɚɡɪɚɛɨɬɚɧɚ ɢɦɢɬɚɰɢɨɧɧɚɹ ɦɨɞɟɥɶ ɞɥɹ ɚɧɚɥɢɡɚ ɩɨɫɬɪɨɟɧɧɵɯ ɫɯɟɦ.
Ʉɥɸɱɟɜɵɟ ɫɥɨɜɚ: ɩɟɧɫɢɨɧɧɨɟ ɫɬɪɚɯɨɜɚɧɢɟ, ɭɪɚɜɧɟɧɢɟ ɛɚɥɚɧɫɚ, ɛɪɭɬɬɨ - ɩɪɟɦɢɹ, ɜɡɧɨɫɵ ɩɪɟɧɭɦɟɪɚɧɞɨ,
ɟɞɢɧɨɜɪɟɦɟɧɧɚɹ ɜɵɩɥɚɬɚ, ɩɟɧɫɢɨɧɧɵɟ ɫɯɟɦɵ ɫ ɭɫɬɚɧɨɜɥɟɧɧɵɦɢ ɜɡɧɨɫɚɦɢ.
ȼȼȿȾȿɇɂȿ
Ɋɚɫɫɦɨɬɪɟɧɧɵɟ ɜ ɞɚɧɧɨɣ ɪɚɛɨɬɟ ɧɚɤɨɩɢɬɟɥɶɧɵɟ ɫɯɟɦɵ ɫ ɟɞɢɧɨɜɪɟɦɟɧɧɨɣ ɜɵɩɥɚɬɨɣ ɜ ɫɥɭɱɚɟ
ɞɨɫɬɢɠɟɧɢɹ ɩɟɧɫɢɨɧɧɨɝɨ ɜɨɡɪɚɫɬɚ ɜ ɧɚɲɟɣ ɫɬɪɚɧɟ ɧɟ ɩɪɟɞɭɫɦɨɬɪɟɧɵ, ɯɨɬɹ ɜ ɦɢɪɨɜɨɣ ɩɪɚɤɬɢɤɟ ɞɟɹɬɟɥɶɧɨɫɬɢ ɇɟɝɨɫɭɞɚɪɫɬɜɟɧɧɵɯ ɉɟɧɫɢɨɧɧɵɯ Ɏɨɧɞɨɜ ɲɢɪɨɤɨ ɢɫɩɨɥɶɡɭɸɬɫɹ. ɉɨ ɩɨɩɭɥɹɪɧɨɫɬɢ ɷɬɢ
ɫɯɟɦɵ ɩɪɚɤɬɢɱɟɫɤɢ ɧɟ ɭɫɬɭɩɚɸɬ ɩɨɠɢɡɧɟɧɧɨɣ ɩɟɧɫɢɢ ɢ ɫɪɨɱɧɵɦ ɜɵɩɥɚɬɚɦ. ȼɟɪɨɹɬɧɨ, ɜ ɛɭɞɭɳɟɦ
ɬɚɤɢɟ ɫɯɟɦɵ ɩɨɹɜɹɬɫɹ, ɜɟɞɶ ɞɥɹ ɦɧɨɝɢɯ ɜɵɯɨɞ ɧɚ ɩɟɧɫɢɸ ɦɨɠɟɬ ɛɵɬɶ ɫɜɹɡɚɧ ɫ ɧɟɨɛɯɨɞɢɦɨɫɬɶɸ ɫɨɜɟɪɲɢɬɶ ɤɪɭɩɧɵɟ ɩɪɢɨɛɪɟɬɟɧɢɹ, ɧɚɩɪɢɦɟɪ, ɩɨɤɭɩɤɭ ɠɢɥɶɹ, ɛɵɬɨɜɨɣ ɬɟɯɧɢɤɢ ɢɥɢ ɠɟ ɞɨɪɨɝɨɫɬɨɹɳɟɟ
ɥɟɱɟɧɢɟ, ɢɥɢ ɠɟɥɚɧɢɟ ɩɨɠɢɥɵɯ ɪɨɞɢɬɟɥɟɣ ɨɛɟɫɩɟɱɢɬɶ ɛɭɞɭɳɟɟ ɫɜɨɢɯ ɧɟɫɨɜɟɪɲɟɧɧɨɥɟɬɧɢɯ ɞɟɬɟɣ.
Ɋɚɛɨɬɚ ɩɨɫɜɹɳɟɧɚ ɩɨɫɬɪɨɟɧɢɸ ɧɚɤɨɩɢɬɟɥɶɧɵɯ ɫɯɟɦ ɫ ɭɫɬɚɧɨɜɥɟɧɧɵɦɢ (ɮɢɤɫɢɪɨɜɚɧɧɵɦɢ) ɪɚɡɦɟɪɚɦɢ ɜɡɧɨɫɨɜ ɩɪɟɧɭɦɟɪɚɧɞɨ ɢ ɟɞɢɧɨɜɪɟɦɟɧɧɨɣ ɜɵɩɥɚɬɨɣ ɜ ɫɥɭɱɚɟ ɞɨɠɢɬɢɹ ɞɨ ɩɟɧɫɢɨɧɧɨɝɨ ɜɨɡɪɚɫɬɚ, ɤɨɬɨɪɵɟ ɨɬɥɢɱɚɸɬɫɹ ɨɬɧɨɲɟɧɢɟɦ ɤ ɧɚɫɥɟɞɨɜɚɧɢɸ. ȼ ɷɬɢɯ ɫɯɟɦɚɯ ɩɪɟɞɭɫɦɨɬɪɟɧ ɜɵɯɨɞ ɢɡ ɱɢɫɥɚ ɭɱɚɫɬɧɢɤɨɜ ɮɨɧɞɚ ɩɨ ɩɪɢɱɢɧɟ ɫɦɟɪɬɢ ɢɥɢ ɩɨ ɩɪɢɱɢɧɚɦ ɫɦɟɪɬɢ ɥɢɛɨ ɜɵɯɨɞ
ɧɚ ɩɟɧɫɢɸ ɩɨ ɢɧɜɚɥɢɞɧɨɫɬɢ.
Ⱦɥɹ ɤɚɠɞɨɣ ɫɯɟɦɵ ɩɨɫɬɪɨɟɧɨ ɭɪɚɜɧɟɧɢɟ ɛɚɥɚɧɫɚ, ɧɚɣɞɟɧɨ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɪɚɫɱɟɬɚ ɛɪɭɬɬɨ-ɩɪɟɦɢɢ ɢ ɫ ɩɨɦɨɳɶɸ ɢɦɢɬɚɰɢɨɧɧɨɣ ɦɨɞɟɥɢ ɜɵɱɢɫɥɟɧɨ ɟɟ ɡɧɚɱɟɧɢɟ ɩɪɢ ɪɚɡɥɢɱɧɵɯ ɡɧɚɱɟɧɢɹɯ ɩɚɪɚɦɟɬɪɨɜ, ɜɯɨɞɹɳɢɯ ɜ ɭɪɚɜɧɟɧɢɟ. ɂɫɯɨɞɹ ɢɡ ɩɨɥɭɱɟɧɧɵɯ ɱɢɫɥɟɧɧɵɯ ɪɟɡɭɥɶɬɚɬɨɜ ɫɞɟɥɚɧɵ
ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɜɵɜɨɞɵ ɢ ɪɟɤɨɦɟɧɞɚɰɢɢ.
Ɋɟɡɭɥɶɬɚɬɵ ɩɨ ɫɯɟɦɚɦ, ɭɱɢɬɵɜɚɸɳɢɦ ɨɞɢɧ ɮɚɤɬɨɪ ɜɵɛɵɬɢɹ, ɛɵɥɢ ɚɧɨɧɫɢɪɨɜɚɧɵ ɜ [1].
1. ɇȺɄɈɉɂɌȿɅɖɇɕȿ ɆɈȾȿɅɂ ȼ ɉȿɇɋɂɈɇɇɈɆ ɋɌɊȺɏɈȼȺɇɂɂ ɋ ɍɑȿɌɈɆ ɈȾɇɈȽɈ
ɎȺɄɌɈɊȺ ɋɆȿɊɌɂ
ɇɢɠɟ ɫɬɪɨɹɬɫɹ ɬɪɢ ɧɚɤɨɩɢɬɟɥɶɧɵɟ ɫɯɟɦɵ ɫ ɭɫɬɚɧɨɜɥɟɧɧɵɦɢ (ɮɢɤɫɢɪɨɜɚɧɧɵɦɢ) ɪɚɡɦɟɪɚɦɢ
ɜɡɧɨɫɨɜ ɩɪɟɧɭɦɟɪɚɧɞɨ ɢ ɟɞɢɧɨɜɪɟɦɟɧɧɨɣ ɜɵɩɥɚɬɨɣ ɜ ɫɥɭɱɚɟ ɞɨɠɢɬɢɹ ɞɨ ɩɟɧɫɢɨɧɧɨɝɨ ɜɨɡɪɚɫɬɚ,
ɤɨɬɨɪɵɟ ɨɬɥɢɱɚɸɬɫɹ ɨɬɧɨɲɟɧɢɟɦ ɤ ɧɚɫɥɟɞɨɜɚɧɢɸ. Ⱦɥɹ ɤɚɠɞɨɣ ɦɨɞɟɥɢ ɩɨɫɬɪɨɟɧɨ ɭɪɚɜɧɟɧɢɟ ɛɚɥɚɧɫɚ, ɧɚɣɞɟɧɨ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɪɚɫɱɟɬɚ ɛɪɭɬɬɨ-ɩɪɟɦɢɢ. ȼ ɨɫɧɨɜɭ ɫɨɫɬɚɜɥɟɧɢɹ ɭɪɚɜɧɟɧɢɹ ɛɚɥɚɧɫɚ
ɥɟɝ ɩɪɢɧɰɢɩ ɷɤɜɢɜɚɥɟɧɬɧɨɫɬɢ ɮɢɧɚɧɫɨɜɵɯ ɨɛɹɡɚɬɟɥɶɫɬɜ ɮɨɧɞɚ ɢ ɡɚɫɬɪɚɯɨɜɚɧɧɨɝɨ ɥɢɰɚ. ȼ ɨɛɳɟɦ
ɜɢɞɟ ɷɬɨɬ ɩɪɢɧɰɢɩ ɪɟɚɥɢɡɭɟɬɫɹ ɩɪɢɪɚɜɧɢɜɚɧɢɟɦ ɧɟɬɬɨ-ɩɪɟɦɢɣ, ɤɨɬɨɪɵɟ ɩɨɥɭɱɚɟɬ ɫɬɪɚɯɨɜɳɢɤ
(ɨɛɹɡɚɬɟɥɶɫɬɜɚ ɫɬɪɚɯɨɜɚɬɟɥɹ) ɤ ɨɛɳɟɣ ɫɭɦɦɟ ɜɵɩɥɚɱɢɜɚɟɦɵɯ ɩɟɧɫɢɣ (ɨɛɹɡɚɬɟɥɶɫɬɜɚ ɫɬɪɚɯɨɜɳɢɤɚ).
ȼɨ ɜɫɟɯ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɯ ɫɯɟɦɚɯ ɜɵɩɥɚɬɵ ɜ ɫɥɭɱɚɟ ɫɦɟɪɬɢ ɫɬɪɚɯɨɜɚɬɟɥɹ ɞɨ ɞɨɫɬɢɠɟɧɢɹ
ɢɦ ɩɟɧɫɢɨɧɧɨɝɨ ɜɨɡɪɚɫɬɚ ɛɭɞɭɬ ɩɪɨɢɡɜɨɞɢɬɶɫɹ ɧɟ ɜ ɦɨɦɟɧɬ ɫɦɟɪɬɢ, ɚ ɜ ɤɨɧɰɟ ɩɟɪɢɨɞɚ ɩɨɫɥɟɞɧɟɝɨ
ɜɡɧɨɫɚ, ɜ ɤɨɬɨɪɨɦ ɩɪɨɢɫɯɨɞɢɬ ɫɦɟɪɬɶ (ɢɧɚɱɟ ɝɨɜɨɪɹ, ɦɵ ɪɚɫɫɦɚɬɪɢɜɚɟɦ ɬɚɤ ɧɚɡɵɜɚɟɦɵɟ ɞɢɫɤɪɟɬɧɵɟ ɦɨɞɟɥɢ ɩɟɧɫɢɨɧɧɨɝɨ ɫɬɪɚɯɨɜɚɧɢɹ). Ⱦɨɥɝɨɫɪɨɱɧɨɟ ɫɬɪɚɯɨɜɚɧɢɟ ɭɱɢɬɵɜɚɟɬ ɞɨɯɨɞ ɨɬ ɢɧɜɟɫɬɢɪɨɜɚɧɢɹ ɫɨɛɪɚɧɧɵɯ ɩɪɟɦɢɣ ɢ ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ ɬɟɦ, ɱɬɨ ɩɪɢ ɪɚɫɱɟɬɚɯ ɩɪɢɧɢɦɚɟɬɫɹ ɜɨ ɜɧɢɦɚɧɢɟ
ɢɡɦɟɧɟɧɢɟ ɰɟɧɧɨɫɬɢ ɞɟɧɟɝ ɫ ɬɟɱɟɧɢɟɦ ɜɪɟɦɟɧɢ. ȼ ɱɚɫɬɧɨɫɬɢ, ɫɨɩɨɫɬɚɜɥɹɹ ɨɛɹɡɚɬɟɥɶɫɬɜɚ ɫɬɪɚɯɨɜɚɬɟɥɹ ɢ ɫɬɪɚɯɨɜɳɢɤɚ, ɬɪɟɛɭɟɬɫɹ ɩɪɢɜɨɞɢɬɶ ɢɯ ɤ ɨɞɧɨɦɭ ɦɨɦɟɧɬɭ ɜɪɟɦɟɧɢ [2-5]. Ⱦɥɹ ɧɚɲɢɯ ɫɯɟɦ ɜ
ɤɚɱɟɫɬɜɟ ɬɚɤɨɝɨ ɦɨɦɟɧɬɚ ɦɨɠɧɨ ɜɵɛɪɚɬɶ ɦɨɦɟɧɬ ɡɚɤɥɸɱɟɧɢɹ ɞɨɝɨɜɨɪɚ ɩɟɧɫɢɨɧɧɨɝɨ ɫɬɪɚɯɨɜɚɧɢɹ.
56
Ɇ.ɋ. Ⱥɥɶ-ɇɚɬɨɪ, ɋ.ȼ. Ⱥɥɶ-ɇɚɬɨɪ, ɇ.Ⱥ. Ɉɥɟɧɱɟɧɤɨ
ȼ ɷɬɨɦ ɩɭɧɤɬɟ ɢɫɩɨɥɶɡɭɸɬɫɹ ɫɥɟɞɭɸɳɢɟ ɨɛɨɡɧɚɱɟɧɢɹ:
x – ɜɨɡɪɚɫɬ ɡɚɫɬɪɚɯɨɜɚɧɧɨɝɨ;
y – ɜɨɡɪɚɫɬ ɜɵɯɨɞɚ ɧɚ ɩɟɧɫɢɸ (ɧɚɩɪɢɦɟɪ, ɦɭɠɱɢɧɵ-60, ɠɟɧɳɢɧɵ-55);
B – ɟɠɟɝɨɞɧɚɹ ɛɪɭɬɬɨ-ɩɪɟɦɢɹ, ɜɵɩɥɚɱɢɜɚɟɦɚɹ ɫɬɪɚɯɨɜɚɬɟɥɟɦ ɜ ɧɚɱɚɥɟ ɝɨɞɚ ɜɩɥɨɬɶ ɞɨ
ɜɵɯɨɞɚ ɧɚ ɩɟɧɫɢɸ;
P – ɟɞɢɧɨɜɪɟɦɟɧɧɚɹ ɧɟɬɬɨ-ɩɪɟɦɢɹ ɩɟɧɫɢɨɧɧɨɣ ɜɵɩɥɚɬɵ, ɩɪɢɜɟɞɟɧɧɚɹ ɧɚ ɞɟɧɶ ɡɚɤɥɸɱɟɧɢɹ ɞɨɝɨɜɨɪɚ;
α1 ,! , α y − x – ɤɨɷɮɮɢɰɢɟɧɬɵ ɧɚɝɪɭɡɤɢ ɩɨ ɝɨɞɚɦ ɞɟɣɫɬɜɢɹ ɞɨɝɨɜɨɪɚ;
i – ɫɥɨɠɧɚɹ ɝɨɞɨɜɚɹ ɩɪɨɰɟɧɬɧɚɹ ɫɬɚɜɤɚ;
υ = 1 (1 + i ) – ɞɢɫɤɨɧɬɢɪɭɸɳɢɣ ɦɧɨɠɢɬɟɥɶ;
PV – ɜɡɧɨɫɵ, ɜɨɡɜɪɚɳɚɟɦɵɟ ɜ ɫɥɭɱɚɟ ɫɦɟɪɬɢ ɢ ɩɪɢɜɟɞɟɧɧɵɟ ɧɚ ɞɟɧɶ ɡɚɤɥɸɱɟɧɢɹ ɞɨɝɨɜɨɪɚ ɫ
ɭɱɟɬɨɦ ɧɚɤɨɩɥɟɧɧɵɯ ɩɪɨɰɟɧɬɨɜ ɢ ɡɚ ɜɵɱɟɬɨɦ ɧɚɝɪɭɡɤɢ (ɜɵɩɥɚɱɢɜɚɟɬɫɹ ɜ ɤɨɧɰɟ ɩɟɪɢɨɞɚ ɫɦɟɪɬɢ).
ɇɚɩɨɦɧɢɦ ɫɥɟɞɭɸɳɢɟ ɚɤɬɭɚɪɧɵɟ ɨɛɨɡɧɚɱɟɧɢɹ (ɩɨɞɪɨɛɧɟɟ ɫɦ. [3], [5]):
lz – ɱɢɫɥɨ ɚɤɬɢɜɧɵɯ ɭɱɚɫɬɧɢɤɨɜ ɜ ɜɨɡɪɚɫɬɟ z ;
t px = l x + t l x – ɜɟɪɨɹɬɧɨɫɬɶ ɬɨɝɨ, ɱɬɨ ɥɢɰɨ ɜɨɡɪɚɫɬɚ x ɩɪɨɠɢɜɟɬ, ɩɨ ɤɪɚɣɧɟɣ ɦɟɪɟ t ɥɟɬ;
t qx = 1 − t px - ɜɟɪɨɹɬɧɨɫɬɶ ɬɨɝɨ, ɱɬɨ ɥɢɰɨ ɜɨɡɪɚɫɬɚ x ɭɦɪɟɬ ɜ ɬɟɱɟɧɢɟ ɛɥɢɠɚɣɲɢɯ t ɥɟɬ,
n −1
ax:n| = ¦ υ k k px – ɫɬɪɚɯɨɜɨɣ ɚɧɧɭɢɬɟɬ ɫɨ ɫɪɨɤɨɦ ɜɵɩɥɚɬ n = y − x ɥɟɬ ɩɪɟɧɭɦɟɪɚɧɞɨ.
k =0
1.1. ɇɚɤɨɩɢɬɟɥɶɧɚɹ ɫɯɟɦɚ ɫ ɜɡɧɨɫɚɦɢ m ɪɚɡ ɜ ɝɨɞ
ɇɚɦ ɭɞɨɛɧɟɟ ɧɚɱɚɬɶ ɫ ɷɬɨɣ ɨɛɳɟɣ ɫɯɟɦɵ. Ɉɫɬɚɥɶɧɵɟ ɩɨɥɭɱɚɬɫɹ ɤɚɤ ɱɚɫɬɧɵɟ ɫɥɭɱɚɢ.
Ɋɚɫɫɦɨɬɪɢɦ ɫɨɜɨɤɭɩɧɨɫɬɶ ɥɸɞɟɣ, ɤɨɬɨɪɵɟ ɫ ɜɨɡɪɚɫɬɚ x ɧɚɱɢɧɚɸɬ ɜɵɩɥɚɱɢɜɚɬɶ ɫɬɪɚɯɨɜɵɟ ɩɪɟɦɢɢ ɪɚɡɦɟɪɚ B ɜ ɧɚɱɚɥɟ ɤɚɠɞɨɝɨ ɩɟɪɢɨɞɚ ɜɩɥɨɬɶ ɞɨ ɜɵɯɨɞɚ ɧɚ ɩɟɧɫɢɸ, ɬ.ɟ. ɞɨ ɜɨɡɪɚɫɬɚ
y . ɉɟɪɢɨɞɢɱɧɨɫɬɶ ɜɧɟɫɟɧɢɹ ɩɪɟɦɢɣ ɨɩɪɟɞɟɥɹɟɬɫɹ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ ɫɬɪɚɯɨɜɚɬɟɥɟɦ ɢ, ɤɚɤ ɩɪɚɜɢɥɨ, ɷɬɨ ɤɚɠɞɵɣ ɦɟɫɹɰ, ɤɚɠɞɵɣ ɤɜɚɪɬɚɥ ɢɥɢ ɤɚɠɞɵɟ ɩɨɥɝɨɞɚ. ȼ ɫɥɭɱɚɟ ɞɨɫɬɢɠɟɧɢɹ ɜɨɡɪɚɫɬɚ y
ɡɚɫɬɪɚɯɨɜɚɧɧɵɟ ɥɢɰɚ ɩɨɥɭɱɚɸɬ ɟɞɢɧɨɜɪɟɦɟɧɧɭɸ ɜɵɩɥɚɬɭ ɪɚɡɦɟɪɚ P . ȼ ɫɥɭɱɚɟ ɫɦɟɪɬɢ ɜɧɭɬɪɢ
ɢɧɬɟɪɜɚɥɚ ɧɚɤɨɩɥɟɧɢɹ ɧɚɫɥɟɞɧɢɤɚɦ ɜɨɡɜɪɚɳɚɟɬɫɹ ɫɭɦɦɚ ɪɚɡɦɟɪɚ PV ɜ ɤɨɧɰɟ ɩɟɪɢɨɞɚ, ɜ ɤɨɬɨɪɨɦ
ɩɪɨɢɡɨɲɥɚ ɫɦɟɪɬɶ.
ɋɯɟɦɚ ɧɚɤɨɩɢɬɟɥɶɧɨɣ ɦɨɞɟɥɢ ɫ ɜɵɩɥɚɬɚɦɢ m ɪɚɡ ɜ ɝɨɞ ɩɪɟɞɫɬɚɜɥɟɧɚ ɧɚ ɪɢɫ. 1.1:
Ɋɢɫ. 1.1. ɋɯɟɦɚ ɦɨɞɟɥɢ ɫ ɜɵɩɥɚɬɚɦɢ m ɪɚɡ ɜ ɝɨɞ
ɉɨɫɤɨɥɶɤɭ (ɩɨ ɞɚɧɧɵɦ ɬɚɛɥɢɰɵ ɫɦɟɪɬɧɨɫɬɢ) ɮɭɧɤɰɢɹ lz ɢɡɜɟɫɬɧɚ ɬɨɥɶɤɨ ɞɥɹ ɰɟɥɵɯ ɡɧɚɱɟɧɢɣ z ,
ɚ ɜ ɧɚɲɟɦ ɫɥɭɱɚɟ ɢɦɟɸɬɫɹ ɢ ɞɪɨɛɧɵɟ, ɬɨ ɜɨɡɧɢɤɚɟɬ ɡɚɞɚɱɚ ɢɧɬɟɪɩɨɥɹɰɢɢ. Ⱦɥɹ ɧɚɲɟɣ ɡɚɞɚɱɢ ɛɵɥɚ ɢɫɩɨɥɶɡɨɜɚɧɚ ɥɢɧɟɣɧɚɹ ɢɧɬɟɪɩɨɥɹɰɢɹ [3], [5]: lx +t = (1 − t )lx + tlx +1 , x – ɰɟɥɨɟ, t ∈[0,1] .
ɇɚɤɨɩɢɬɟɥɶɧɵɟ ɩɟɧɫɢɨɧɧɵɟ ɫɯɟɦɵ…
57
Ȼɥɢɠɚɣɲɚɹ ɰɟɥɶ – ɩɨɤɚɡɚɬɶ, ɱɬɨ ɭɪɚɜɧɟɧɢɟ ɛɚɥɚɧɫɚ ɢɦɟɟɬ ɜɢɞ:
y − x −1
m −1
k =0
j =0
B ¦ (1 − α k +1 )¦υ
k+
j
m
j
B
PV + P y − x pxυ ( y − x ) ,
k qx ) =
m
m
( k px −
(1.1)
ɝɞɟ ɜɡɧɨɫɵ, ɜɨɡɜɪɚɳɚɟɦɵɟ ɜ ɫɥɭɱɚɟ ɫɦɟɪɬɢ, ɟɫɬɶ
m ( y − x ) −1
¦
PV =
k
« ( k −1) »
« m »
¬
¼
k =1
qx ¦ (1 − α « ( n −1) » )υ
« ( n −1) » ( n −1) mod m
« m »+
m
¬
¼
« m » +1
¬
¼
n =1
,
(1.2)
ɚ ¬« z ¼» – ɰɟɥɚɹ ɱɚɫɬɶ ɱɢɫɥɚ z .
ȼ ɱɚɫɬɧɨɫɬɢ, ɛɪɭɬɬɨ-ɩɪɟɦɢɹ ɜɵɱɢɫɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ:
B=
P y − x pxυ ( y − x )
y − x −1
m −1
¦ (1 − α )¦υ
k +1
k =0
k+
j
m
j =0
.
j
1
( k px − k qx ) − PV
m
m
(1.3)
ɋɨɝɥɚɫɧɨ ɪɢɫ. 1.1, ɩɪɢɜɟɞɟɧɧɚɹ ɫɬɨɢɦɨɫɬɶ ɜɡɧɨɫɨɜ ɤ ɦɨɦɟɧɬɭ ɡɚɤɥɸɱɟɧɢɹ ɞɨɝɨɜɨɪɚ (ɢɥɢ
ɬɟɤɭɳɚɹ ɫɬɨɢɦɨɫɬɶ ɩɨɬɨɤɚ ɜɡɧɨɫɨɜ) ɪɚɜɧɚ:
lx B(1 − α1 )υ 0 + l
+l
x +1+
1
m
1
x+
B(1 − α 2 )υ
m
+l
1 B (1 − α1 )υ
m
1+
1
m
+" + l
y −1+
1
m
2
x+
m
+ " + lx +1 B(1 − α 2 )υ 1 +
2 B (1 − α1 )υ
m
B(1 − α y − x )υ
y − x −1+
1
m
+" + l
y −1+
m −1
m
B(1 − α y − x )υ
y − x −1+
m −1
m
(1.4)
.
ȼ ɫɥɭɱɚɟ ɫɦɟɪɬɢ ɜɧɭɬɪɢ ɢɧɬɟɪɜɚɥɚ ( x, y) ɩɪɨɢɫɯɨɞɢɬ ɜɵɩɥɚɬɚ ɜɧɟɫɟɧɧɵɯ ɜɡɧɨɫɨɜ ɫ ɭɱɟɬɨɦ ɧɚɤɨɩɥɟɧɧɵɯ ɩɪɨɰɟɧɬɨɜ ɢ ɡɚ ɜɵɱɟɬɨɦ ɧɚɝɪɭɡɤɢ. ɉɪɢɜɟɞɟɧɧɚɹ ɫɬɨɢɦɨɫɬɶ ɷɬɢɯ ɜɵɩɥɚɬ ɤ ɦɨɦɟɧɬɭ ɡɚɤɥɸɱɟɧɢɹ ɞɨɝɨɜɨɪɚ ɪɚɜɧɚ:
(lx − l
) B(1 − α1 )υ + (l
0
x+
1
m
x+
1
m
−l
x+
2
m
) B[(1 − α1 )υ + (1 − α1 )υ ] + " +
1
+(l
x+
1
m
0
2
0
m
m
m −1 − l x +1 ) B[(1 − α1 )υ + (1 − α1 )υ + (1 − α1 )υ + (1 − α1 )υ
m −1
m
]+
m
+ " + (l y −1 − l
y −1+
1
m
) B[(1 − α y − x )υ
+(1 − α1 )υ 0 ] + " + (l
+(1 − α y − x )υ
y − x −1+
1
m
y −1+
m−2
m
y −1+
+ (1 − α y − x −1 )υ
m −1 ) B[(1 − α y − x )υ
y − x −1−
y − x −1+
1
m
2
m
+" +
(1.5)
+
m
+ (1 − α y − x )υ y − x −1 + " + (1 − α1 )υ 0 ],
ɝɞɟ l x + k + j − l x + k + j +1 , k = 0, y − x − 1 , j = 0,
ɦɟɠɞɭ ( x + k + j ) ɢ ( x + k + j + 1) .
−l
y − x −1
m −1
– ɱɢɫɥɨ ɜɵɛɵɜɲɢɯ ɩɨ ɩɪɢɱɢɧɟ ɫɦɟɪɬɢ ɜ ɩɟɪɢɨɞ
m
58
Ɇ.ɋ. Ⱥɥɶ-ɇɚɬɨɪ, ɋ.ȼ. Ⱥɥɶ-ɇɚɬɨɪ, ɇ.Ⱥ. Ɉɥɟɧɱɟɧɤɨ
ɉɪɢɜɟɞɟɧɧɚɹ ɫɬɨɢɦɨɫɬɶ ɜɵɩɥɚɬ ɩɨ ɨɤɨɧɱɚɧɢɢ ɩɟɪɢɨɞɚ ɫɬɪɚɯɨɜɚɧɢɹ ɫɨɫɬɚɜɢɬ ɜɟɥɢɱɢɧɭ:
Pl y υ ( y − x ) .
(1.6)
ɂɡ ɜɵɪɚɠɟɧɢɣ (1.4), (1.5) ɢ (1.6) ɫɨɫɬɚɜɢɦ ɭɪɚɜɧɟɧɢɟ ɛɚɥɚɧɫɚ:
lx B(1 − α1 )υ 0 + l
+l
+l
x +1+
y −1+
1
m
x+
B(1 − α 2 )υ
m
1
1+
m
+" + l
m −1 B (1 − α y − x )υ
y − x −1+
y −1+
m −1
m
1
m
2
x+
m
+ " + lx +1 B(1 − α 2 )υ 1 +
2 B (1 − α1 )υ
m
B(1 − α y − x )υ
y − x −1+
1
m
+" +
=
m
= (lx − l
+ (l
1
m
+l
1 B (1 − α1 )υ
x+
0
1 ) B (1 − α1 )υ + (l
m
1
x+
1
m
−l
x+
0
m
2 ) B[(1 − α1 )υ + (1 − α1 )υ ] + " +
m
1
m
x+
m −1
m
2
m
− lx +1 ) B[(1 − α1 )υ + (1 − α1 )υ + (1 − α1 )υ + (1 − α1 )υ
0
+ " + (l y −1 − l
y −1+
y − x −1
+ (1 − α y − x −1 )υ
1 ) B[(1 − α y − x )υ
1
m
]+
+" +
(1.7)
m
+ (1 − α1 )υ ] + " + (l
0
+ (1 − α y − x )υ
y − x −1−
m −1
m
y − x −1+
1
m
y −1+
m−2
m
−l
y −1+
m −1
m
) B[(1 − α y − x )υ
y − x −1+
2
m
+
+ (1 − α y − x )υ y − x −1 + " + (1 − α1 )υ 0 ] + Pl y υ m ( y − x ) .
ɉɪɢɦɟɧɹɹ ɥɢɧɟɣɧɭɸ ɢɧɬɟɪɩɨɥɹɰɢɸ ɤ ɩɪɚɜɨɣ ɱɚɫɬɢ, ɩɨɥɭɱɢɦ:
1
1
1
(lx − lx +1 ) B(1 − α1 )υ 0 + (lx − lx +1 ) B[(1 − α1 )υ 0 + (1 − α1 )υ m ] + " +
m
m
1
2
1
+ (lx − lx +1 ) B[(1 − α1 )υ 0 + (1 − α1 )υ m + (1 − α1 )υ m +
m
m −1
1
+(1 − α1 )υ m ] + " + (l y −1 − l y ) B[(1 − α y − x )υ y − x −1 +
m
+(1 − α y − x −1 )υ
y − x −1−
1
m
+ " + (1 − α1 )υ 0 ] +
2
1
y − x −1+
y − x −1+
1
m
m
+ (1 − α y − x )υ
+
(l y −1 − l y ) B[(1 − α y − x )υ
m
+(1 − α y − x )υ y − x −1 + " + (1 − α1 )υ 0 ] + Pl yυ ( y − x ) .
+
Ⱦɨɦɧɨɠɢɦ ɨɛɟ ɱɚɫɬɢ ɭɪɚɜɧɟɧɢɹ ɧɚ 1 lx . Ʌɟɜɚɹ ɱɚɫɬɶ ɩɪɟɨɛɪɚɡɭɟɬɫɹ ɜ
B
y − x −1
¦
k =0
m −1
(1 − α k +1 )¦ υ
j =0
k+
j
m
( k px −
j
k q x ),
m
(1.8)
ɇɚɤɨɩɢɬɟɥɶɧɵɟ ɩɟɧɫɢɨɧɧɵɟ ɫɯɟɦɵ…
59
ɚ ɩɪɚɜɚɹ – ɜ
« ( n −1) » ( n −1) mod m
+
m »¼
m
k
«
B m ( y − x ) −1
¬
q
(1
−
α
)
υ
¦
¦
x
(
k
1)
(
n
1)
−
−
«
»
«
»
m k =1 «¬ m »¼ n =1
« m » +1
¬
¼
+ P y − x pxυ ( y − x ) .
(1.9)
ɋɨɟɞɢɧɹɟɦ ɨɛɟ ɱɚɫɬɢ ɭɪɚɜɧɟɧɢɹ:
y − x −1
m −1
k =0
j =0
B ¦ (1 − α k +1 )¦υ
k+
j
m
( k px −
j
B
PV + P y − x pxυ ( y − x ) ,
k qx ) =
m
m
(1.10)
ɝɞɟ
PV =
m ( y − x ) −1
¦
k =1
k
« ( k −1) »
« m »
¬
¼
qx ¦ (1 − α « ( n −1) » )υ
« ( n −1) » ( n −1) mod m
« m »+
m
¬
¼
« m » +1
¬
¼
n =1
.
(1.11)
ɗɬɨ ɞɨɤɚɡɵɜɚɟɬ (1.1) - (1.3).
ȿɫɥɢ ɞɥɹ ɷɬɨɣ ɫɯɟɦɵ ɩɨɥɨɠɢɬɶ PV = 0 , ɬɨ ɩɨɥɭɱɢɦ ɫɯɟɦɭ ɛɟɡ ɜɨɡɦɨɠɧɨɫɬɢ ɧɚɫɥɟɞɨɜɚɧɢɹ.
ȼ ɱɚɫɬɧɨɫɬɢ, ɢɡ (1.1) ɢ (1.3) ɩɨɥɭɱɢɦ, ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ, ɭɪɚɜɧɟɧɢɟ ɛɚɥɚɧɫɚ ɢ ɮɨɪɦɭɥɭ ɞɥɹ ɛɪɭɬɬɨɩɪɟɦɢɢ.
1.2. ɇɚɤɨɩɢɬɟɥɶɧɚɹ ɫɯɟɦɚ ɫ ɝɨɞɨɜɵɦɢ ɜɡɧɨɫɚɦɢ
Ʉɚɤ ɛɵɥɨ ɨɬɦɟɱɟɧɨ ɜɵɲɟ, ɟɫɥɢ ɜ ɩɪɟɞɵɞɭɳɟɣ ɫɯɟɦɟ ɩɨɥɨɠɢɬɶ PV = 0 , ɬɨ ɩɨɥɭɱɢɦ ɫɯɟɦɭ
ɛɟɡ ɜɨɡɦɨɠɧɨɫɬɢ ɧɚɫɥɟɞɨɜɚɧɢɹ. ȼ ɱɚɫɬɧɨɫɬɢ, ɞɥɹ ɝɨɞɨɜɵɯ ɜɡɧɨɫɨɜ ( m = 1) ɭɪɚɜɧɟɧɢɟ ɛɚɥɚɧɫɚ
ɢɦɟɟɬ ɜɢɞ:
y − x −1
Bax: y − x| = B ¦ α k +1υ k k px +P y − x pxυ y − x ,
(1.12)
k =0
ɚ ɟɠɟɝɨɞɧɚɹ ɛɪɭɬɬɨ-ɩɪɟɦɢɹ ɜɵɱɢɫɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ:
Pυ y − x y − x px
B=
ax: y − x| −
y − x −1
¦α
k =0
υ
k +1
.
k
k
(1.13)
px
Ⱥɧɚɥɨɝɢɱɧɨ, ɩɨɥɨɠɢɜ ɜ ɫɯɟɦɟ ɩ. 1.1 PV ≠ 0 , ɩɨɥɭɱɢɦ ɫɯɟɦɭ ɫ ɜɨɡɦɨɠɧɨɫɬɶɸ ɧɚɫɥɟɞɨɜɚɧɢɹ. ȼ ɱɚɫɬɧɨɫɬɢ, ɞɥɹ ɝɨɞɨɜɵɯ ɜɡɧɨɫɨɜ ( m = 1) ɭɪɚɜɧɟɧɢɟ ɛɚɥɚɧɫɚ ɢɦɟɟɬ ɜɢɞ:
Bax: y − x| = B
y − x −1
¦α
k =0
υ k k px + P y − x pxυ y − x + PV ,
k +1
(1.14)
ɝɞɟ ɜɡɧɨɫɵ, ɜɨɡɜɪɚɳɚɟɦɵɟ ɜ ɫɥɭɱɚɟ ɫɦɟɪɬɢ, ɟɫɬɶ
y − x −1
PV = B ¦
s =1
s
s −1
qx ¦ (1 − α k )υ k −1 ,
k =1
(1.15)
60
Ɇ.ɋ. Ⱥɥɶ-ɇɚɬɨɪ, ɋ.ȼ. Ⱥɥɶ-ɇɚɬɨɪ, ɇ.Ⱥ. Ɉɥɟɧɱɟɧɤɨ
ɚ ɟɠɟɝɨɞɧɚɹ ɛɪɭɬɬɨ-ɩɪɟɦɢɹ ɜɵɱɢɫɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ:
P y − x pxυ y − x
B=
ax: y − x| −
y − x −1
¦α
k =0
υ
k +1
k
k
px −
y − x −1
¦
s =1
s
s −1
qx ¦ (1 − α k )υ
.
(1.16)
k −1
k =1
Ɉɬɦɟɬɢɦ, ɱɬɨ ɩɪɢ m = 1 ɢ ɰɟɥɨɦ x ɥɢɧɟɣɧɚɹ ɢɧɬɟɪɩɨɥɹɰɢɹ ɧɟ ɩɨɧɚɞɨɛɢɬɫɹ.
2. ɇȺɄɈɉɂɌȿɅɖɇȺə ɉȿɇɋɂɈɇɇȺə ɋɏȿɆȺ ɋ ȼɈɁɆɈɀɇɈɋɌɖɘ ɇȺɋɅȿȾɈȼȺɇɂə ɂ
ȾȼɍɆə ɎȺɄɌɈɊȺɆɂ ȼɕȻɕɌɂə
Ɋɚɫɫɦɨɬɪɢɦ ɫɨɜɨɤɭɩɧɨɫɬɶ ɥɸɞɟɣ, ɤɨɬɨɪɵɟ ɜ ɜɨɡɪɚɫɬɟ ɯ ɧɚɱɢɧɚɸɬ ɜɵɩɥɚɱɢɜɚɬɶ ɜ ɧɚɱɚɥɟ ɝɨɞɚ ɫɬɪɚɯɨɜɵɟ ɩɪɟɦɢɢ ɪɚɡɦɟɪɚ B ɜɩɥɨɬɶ ɞɨ ɜɵɯɨɞɚ ɧɚ ɩɟɧɫɢɸ, ɬ.ɟ. ɞɨ ɜɨɡɪɚɫɬɚ y . ȼ ɫɥɭɱɚɟ ɞɨɫɬɢɠɟɧɢɹ ɷɬɨɝɨ
ɜɨɡɪɚɫɬɚ – ɩɨɥɭɱɚɸɬ ɟɞɢɧɨɜɪɟɦɟɧɧɭɸ ɜɵɩɥɚɬɭ ɪɚɡɦɟɪɚ P . ȼ ɫɥɭɱɚɟ ɫɦɟɪɬɢ/ɜɵɯɨɞɚ ɧɚ ɩɟɧɫɢɸ ɩɨ ɢɧɜɚɥɢɞɧɨɫɬɢ ɜɧɭɬɪɢ ɢɧɬɟɪɜɚɥɚ ɧɚɤɨɩɥɟɧɢɹ ɧɚɫɥɟɞɧɢɤɚɦ/ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ ɫɬɚɜɲɢɦ ɢɧɜɚɥɢɞɚɦɢ ɜɨɡɜɪɚɳɚɟɬɫɹ ɫɭɦɦɚ ɪɚɡɦɟɪɚ PV . Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, PV – ɜɡɧɨɫɵ, ɜɨɡɜɪɚɳɚɟɦɵɟ ɜ ɫɥɭɱɚɟ ɫɦɟɪɬɢ/ɜɵɯɨɞɚ ɧɚ ɩɟɧɫɢɸ ɩɨ
ɢɧɜɚɥɢɞɧɨɫɬɢ ɢ ɩɪɢɜɟɞɟɧɧɵɟ ɧɚ ɞɟɧɶ ɡɚɤɥɸɱɟɧɢɹ ɞɨɝɨɜɨɪɚ ɫ ɭɱɟɬɨɦ ɧɚɤɨɩɥɟɧɧɵɯ ɩɪɨɰɟɧɬɨɜ ɢ ɡɚ ɜɵɱɟɬɨɦ ɧɚɝɪɭɡɤɢ (ɜɵɩɥɚɱɢɜɚɟɬɫɹ ɜ ɤɨɧɰɟ ɝɨɞɚ ɫɦɟɪɬɢ/ɜɵɯɨɞɚ ɧɚ ɩɟɧɫɢɸ ɩɨ ɢɧɜɚɥɢɞɧɨɫɬɢ). ɋɯɟɦɚ ɧɚɤɨɩɢɬɟɥɶɧɨɣ ɦɨɞɟɥɢ ɫ ɜɨɡɦɨɠɧɨɫɬɶɸ ɧɚɫɥɟɞɨɜɚɧɢɹ ɢ ɞɜɭɦɹ ɮɚɤɬɨɪɚɦɢ ɜɵɛɵɬɢɹ ɩɪɟɞɫɬɚɜɥɟɧɚ ɧɚ ɪɢɫ. 2.1.
B
B
B
B
lx
l x +1
lx+2
l y −1
ly
x
x +1
x+2
y −1
y
α1 B
α2B
α3B
α y− x B
Pv
P
Ɋɢɫ. 2.1. ɋɯɟɦɚ ɦɨɞɟɥɢ ɫ ɜɨɡɦɨɠɧɨɫɬɶɸ ɧɚɫɥɟɞɨɜɚɧɢɹ ɢ ɞɜɭɦɹ ɮɚɤɬɨɪɚɦɢ ɜɵɛɵɬɢɹ
Ɍɟɤɭɳɚɹ ɫɬɨɢɦɨɫɬɶ ɜɡɧɨɫɨɜ ɫɨɫɬɚɜɢɬ ɜɟɥɢɱɢɧɭ:
lx B(1 − α1 )υ 0 + lx +1 B(1 − α 2 )υ1 + lx + 2 B(1 − α 3 )υ 2 + " + l y −1 B(1 − α y − x )υ y − x−1 .
(2.1)
ȼ ɫɥɭɱɚɟ ɫɦɟɪɬɢ/ɜɵɯɨɞɚ ɧɚ ɩɟɧɫɢɸ ɩɨ ɢɧɜɚɥɢɞɧɨɫɬɢ ɜɧɭɬɪɢ ɢɧɬɟɪɜɚɥɚ ( x, y ) ɩɪɨɢɫɯɨɞɢɬ
ɜɵɩɥɚɬɚ ɜɧɟɫɟɧɧɵɯ ɜɡɧɨɫɨɜ ɫ ɭɱɟɬɨɦ ɧɚɤɨɩɥɟɧɧɵɯ ɩɪɨɰɟɧɬɨɜ ɢ ɡɚ ɜɵɱɟɬɨɦ ɧɚɝɪɭɡɤɢ. Ⱥ ɢɦɟɧɧɨ,
ɜɵɩɥɚɬɚ ɫɭɦɦɵ
(d x + ix ) B(1 − α1 )υ 0 + (d x +1 + ix +1 ) B ª¬(1 − α1 )υ 0 + (1 − α 2 )υ 1 º¼ + " +
+(d y − 2 + iy −2 ) B[(1 − α1 )υ 0 + (1 − α 2 )υ 1 + " + (1 − α y − x −1 )υ y − x − 2 ],
(2.2)
ɝɞɟ d x + k , k = 0, y − x − 2 – ɱɢɫɥɨ ɜɵɛɵɜɲɢɯ ɩɨ ɩɪɢɱɢɧɟ ɫɦɟɪɬɢ ɜ ɩɟɪɢɨɞ ɦɟɠɞɭ x ɢ x + k , ix+k ,
k = 0, y − x − 2 – ɱɢɫɥɨ ɜɵɛɵɜɲɢɯ ɩɨ ɩɪɢɱɢɧɟ ɜɵɯɨɞɚ ɧɚ ɩɟɧɫɢɸ ɩɨ ɢɧɜɚɥɢɞɧɨɫɬɢ ɜ ɩɟɪɢɨɞ
ɦɟɠɞɭ x ɢ x + k .
ɇɚɤɨɩɢɬɟɥɶɧɵɟ ɩɟɧɫɢɨɧɧɵɟ ɫɯɟɦɵ…
61
ȼɵɩɥɚɬɵ ɩɨ ɨɤɨɧɱɚɧɢɢ ɩɟɪɢɨɞɚ ɫɬɪɚɯɨɜɚɧɢɹ ɫɨɫɬɚɜɹɬ ɜɟɥɢɱɢɧɭ:
Pl y υ y − x .
(2.3)
ɂɡ (2.1), (2.2) ɢ (2.3) ɫɨɫɬɚɜɢɦ ɭɪɚɜɧɟɧɢɟ ɛɚɥɚɧɫɚ:
lx B(1 − α1 )υ 0 + lx +1 B(1 − α 2 )υ 1 + lx + 2 B(1 − α 3 )υ 2 + " + l y −1 B(1 − α y − x )υ y − x −1 =
= (d x + ix ) B(1 − α1 )υ 0 + (d x +1 + ix +1 ) B ª¬ (1 − α1 )υ 0 + (1 − α 2 )υ 1 º¼ + " +
(2.4)
+( d y − 2 + i y − 2 ) B[(1 − α1 )υ 0 + (1 − α 2 )υ 1 + " + (1 − α y − x −1 )υ y − x − 2 ] + Pl y υ y − x .
ȼɵɩɢɲɟɦ ɹɜɧɨɟ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɟɠɟɝɨɞɧɨɣ ɛɪɭɬɬɨ-ɩɪɟɦɢɢ B ɜ ɬɟɪɦɢɧɚɯ ɮɭɧɤɰɢɢ ɞɟɤɪɟɦɟɧɬɧɨɣ ɬɚɛɥɢɰɵ ɫ ɞɜɭɦɹ ɮɚɤɬɨɪɚɦɢ (ɨɧɨ ɩɨɥɭɱɚɟɬɫɹ ɩɨɫɥɟ ɧɟɫɥɨɠɧɵɯ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ ɮɨɪɦɭɥɵ (2.4):
ɉɨɞɪɨɛɧɟɟ ɨ ɮɭɧɤɰɢɹɯ ɞɟɤɪɟɦɟɧɬɧɵɯ ɬɚɛɥɢɰ ɦɨɠɧɨ ɧɚɣɬɢ ɜ [3], [5], [6].
P y − x pxυ y − x
B=
ax: y − x| −
y − x −1
¦α
k =0
υ
k
k +1
k
px −
y − x −1
¦
s
s −1
s =1
qx ¦ (1 − α k )υ
,
(2.5)
k −1
k =1
ɝɞɟ
y − x −1
PV = B ¦
s =1
s
k −1
,
s −1 qx ¦ (1 − α k )υ
k =1
n −1
n
px = 1 −
¦ (d
j =0
x+ j
+ ix + j )
lx
– ɜɟɪɨɹɬɧɨɫɬɶ ɬɨɝɨ, ɱɬɨ ɡɚɫɬɪɚɯɨɜɚɧɧɵɣ ɜɨɡɪɚɫɬɚ x ɨɫɬɚɧɟɬɫɹ ɚɤɬɢɜɧɵɦ ɭɱɚɫɬɧɢɤɨɦ ɩɟɧɫɢɨɧɧɨɝɨ ɮɨɧɞɚ ɩɨ ɤɪɚɣɧɟɣ ɦɟɪɟ ɞɨ ɜɨɡɪɚɫɬɚ x + n ; n qx = 1 − n px – ɜɟɪɨɹɬɧɨɫɬɶ ɬɨɝɨ, ɱɬɨ ɡɚɫɬɪɚɯɨɜɚɧɧɵɣ ɜɨɡɪɚɫɬɚ x ɩɟɪɟɫɬɚɟɬ ɛɵɬɶ ɚɤɬɢɜɧɵɦ ɭɱɚɫɬɧɢɤɨɦ ɩɟɧɫɢɨɧɧɨɝɨ ɮɨɧɞɚ (ɩɨ ɩɪɢɱɢɧɟ ɫɦɟɪɬɢ ɢɥɢ ɢɧɜɚɥɢɞɧɨɫɬɢ) ɜ ɜɨɡɪɚɫɬɧɨɦ ɢɧɬɟɪɜɚɥɟ ɦɟɠɞɭ x ɢ x + n
n −1
ax: y − x| = ¦υ k k px , n = y − x
k =0
– ɫɬɪɚɯɨɜɨɣ ɚɧɧɭɢɬɟɬ ɩɪɟɧɭɦɟɪɚɧɞɨ ɫɨ ɫɪɨɤɨɦ ɜɵɩɥɚɬ n ɥɟɬ.
ɏɨɬɹ ɮɨɪɦɭɥɵ (2.5) ɢ (1.16) ɫ ɜɢɞɭ ɫɨɜɩɚɞɚɸɬ, ɨɧɢ ɜɫɟ-ɬɚɤɢ ɢɦɟɸɬ ɪɚɡɧɵɣ ɫɦɵɫɥ, ɩɨɫɤɨɥɶɤɭ ɜ (2.5) ɢɫɩɨɥɶɡɭɸɬɫɹ ɞɟɤɪɟɦɟɧɬɧɵɟ ɮɭɧɤɰɢɢ ɢ ɜɟɪɨɹɬɧɨɫɬɢ. ȼ ɱɚɫɬɧɨɫɬɢ, ax:n| ɩɨɧɢɦɚɟɬɫɹ ɤɚɤ ɫɬɪɚɯɨɜɨɣ ɞɟɤɪɟɦɟɧɬɧɵɣ ɚɧɧɭɢɬɟɬ ɩɪɟɧɭɦɟɪɚɧɞɨ ɫɨ ɫɪɨɤɨɦ ɜɵɩɥɚɬ n ɥɟɬ.
Ⱦɥɹ ɫɯɟɦɵ ɛɟɡ ɜɨɡɦɨɠɧɨɫɬɢ ɧɚɫɥɟɞɨɜɚɧɢɹ ( PV = 0 ) ɛɪɭɬɬɨ-ɩɪɟɦɢɹ, ɫɨɝɥɚɫɧɨ (2.5) ɪɚɜɧɚ:
B=
P y − x pxυ y − x
ax: y − x| −
y − x −1
¦α
k =0
υ
k +1
.
k
k
px
(2.5)
62
Ɇ.ɋ. Ⱥɥɶ-ɇɚɬɨɪ, ɋ.ȼ. Ⱥɥɶ-ɇɚɬɨɪ, ɇ.Ⱥ. Ɉɥɟɧɱɟɧɤɨ
3. ɂɆɂɌȺɐɂɈɇɇȺə ɆɈȾȿɅɖ ȾɅə ɇȺɏɈɀȾȿɇɂə ȻɊɍɌɌɈ-ɉɊȿɆɂɂ
Ɋɚɡɪɚɛɨɬɚɧɚ ɢɦɢɬɚɰɢɨɧɧɚɹ ɦɨɞɟɥɶ ɧɚ ɹɡɵɤɟ ɦɚɤɪɨɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ VBA ɩɪɨɝɪɚɦɦɵ
Excel, ɫ ɩɨɦɨɳɶɸ ɤɨɬɨɪɨɣ ɧɚ ɨɫɧɨɜɟ ɬɚɛɥɢɰɵ ɫɦɟɪɬɧɨɫɬɢ ɊɎ 2011 ɞɥɹ ɩɟɪɜɵɯ ɬɪɟɯ ɫɯɟɦ
ɢ ɬɚɛɥɢɰɵ ɜɵɛɵɬɢɹ ɞɥɹ ɫɯɟɦɵ ɫ ɞɜɭɦɹ ɮɚɤɬɨɪɚɦɢ ɛɵɥ ɩɪɨɜɟɞɟɧ ɪɚɫɱɟɬ ɟɠɟɝɨɞɧɨɣ ɛɪɭɬɬɨɩɪɟɦɢɢ ɩɪɢ ɪɚɡɥɢɱɧɵɯ ɡɧɚɱɟɧɢɹɯ ɩɚɪɚɦɟɬɪɨɜ, ɜɯɨɞɹɳɢɯ ɜ ɩɨɥɭɱɟɧɧɵɟ ɚɧɚɥɢɬɢɱɟɫɤɢɟ
ɮɨɪɦɭɥɵ:
– ɜɨɡɪɚɫɬ ɜɵɯɨɞɚ ɧɚ ɩɟɧɫɢɸ ɞɥɹ ɠɟɧɳɢɧ y = 55; ɞɥɹ ɦɭɠɱɢɧ y = 60;
– ɞɥɹ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɧɚɝɪɭɡɤɢ α1...α y-x ɪɚɫɫɦɨɬɪɟɧɵ ɞɜɚ ɜɚɪɢɚɧɬɚ ɢɯ ɜɵɛɨɪɚ:
ɚ) ɤɨɷɮɮɢɰɢɟɧɬ ɧɚɝɪɭɡɤɢ ɤɚɠɞɵɣ ɝɨɞ ɭɜɟɥɢɱɢɜɚɟɬɫɹ ɩɨ ɞɟɬɟɪɦɢɧɢɪɨɜɚɧɧɨɦɭ ɩɪɚɜɢɥɭ
(ɧɚɩɪɢɦɟɪ, ɤɨɷɮɮɢɰɢɟɧɬɵ ɧɚɝɪɭɡɤɢ ɨɛɪɚɡɭɸɬ ɚɪɢɮɦɟɬɢɱɟɫɤɭɸ ɩɪɨɝɪɟɫɫɢɸ ɫ ɪɚɡɭɦɧɵɦ
ɧɚɱɚɥɶɧɵɦ ɤɨɷɮɮɢɰɢɟɧɬɨɦ ɢ ɪɚɡɭɦɧɵɦ ɲɚɝɨɦ);
ɛ) ɤɨɷɮɮɢɰɢɟɧɬ ɧɚɝɪɭɡɤɢ ɦɟɧɹɟɬɫɹ ɫɥɭɱɚɣɧɨ ɫɨɝɥɚɫɧɨ ɪɚɜɧɨɦɟɪɧɨɦɭ ɪɚɫɩɪɟɞɟɥɟɧɢɸ ɧɚ
ɪɚɡɭɦɧɨɦ ɢɧɬɟɪɜɚɥɟ;
– ɡɧɚɱɟɧɢɹ lx + k , k = 0, y − x ɧɚɯɨɞɹɬɫɹ ɩɨ ɬɚɛɥɢɰɚɦ ɫɦɟɪɬɧɨɫɬɢ ɢ ɜɵɛɵɬɢɹ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ
ɞɥɹ ɦɭɠɱɢɧ ɢ ɞɥɹ ɠɟɧɳɢɧ;
– ɩɪɨɰɟɧɬɧɚɹ ɫɬɚɜɤɚ ɪɚɜɧɚ 3, 4, 5 ɢ 10%;
– ɪɚɡɦɟɪ ɟɞɢɧɨɜɪɟɦɟɧɧɨɣ ɜɵɩɥɚɬɵ ɦɨɠɟɬ ɩɪɢɧɢɦɚɬɶ ɥɸɛɵɟ ɧɟɨɬɪɢɰɚɬɟɥɶɧɵɟ
ɡɧɚɱɟɧɢɹ.
Ɋɟɡɭɥɶɬɚɬɵ ɪɚɫɱɟɬɚ ɩɨɤɚɡɚɥɢ, ɱɬɨ ɜɫɟ ɩɨɫɬɪɨɟɧɧɵɟ ɦɨɞɟɥɢ ɹɜɥɹɸɬɫɹ ɤɚɱɟɫɬɜɟɧɧɨ ɚɞɟɤɜɚɬɧɵɦɢ, ɚ ɢɦɟɧɧɨ:
1) ɱɟɦ ɦɟɧɶɲɟ ɩɟɪɢɨɞ ɧɚɤɨɩɥɟɧɢɹ, ɬɟɦ ɪɚɡɦɟɪ ɜɡɧɨɫɨɜ ɛɨɥɶɲɟ ɩɪɢ ɩɪɨɱɢɯ ɪɚɜɧɵɯ ɭɫɥɨɜɢɹɯ. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɪɚɡɦɟɪ ɜɡɧɨɫɨɜ ɞɥɹ ɠɟɧɳɢɧ ɜɫɟɝɞɚ ɛɨɥɶɲɟ, ɱɟɦ ɞɥɹ ɦɭɠɱɢɧ. ɉɨɫɤɨɥɶɤɭ
ɠɟɧɳɢɧɵ ɞɨɫɬɢɝɚɸɬ ɩɟɧɫɢɨɧɧɨɝɨ ɜɨɡɪɚɫɬɚ ɪɚɧɶɲɟ, ɱɟɦ ɦɭɠɱɢɧɵ, ɬɨ ɢ ɜɨɡɦɨɠɧɵɣ ɩɟɪɢɨɞ
ɧɚɤɨɩɥɟɧɢɹ ɭ ɠɟɧɳɢɧ ɜɫɟɝɞɚ ɦɟɧɶɲɟ;
2) ɱɟɦ ɦɟɧɶɲɟ ɠɟɥɚɟɦɚɹ ɫɭɦɦɚ ɧɚɤɨɩɥɟɧɢɹ, ɬɟɦ ɦɟɧɶɲɟ ɪɚɡɦɟɪ ɜɡɧɨɫɨɜ ɩɪɢ ɩɪɨɱɢɯ ɪɚɜɧɵɯ ɭɫɥɨɜɢɹɯ;
3) ɜ ɦɨɞɟɥɢ ɫ ɜɵɩɥɚɬɚɦɢ m ɪɚɡ ɜ ɝɨɞ: ɱɟɦ ɦɟɧɶɲɟ ɤɨɥɢɱɟɫɬɜɨ ɜɡɧɨɫɨɜ ɜ ɝɨɞ, ɬɟɦ ɜɵɲɟ ɢɯ
ɪɚɡɦɟɪ.
ɋɪɚɜɧɢɜɚɹ ɩɨɫɬɪɨɟɧɧɵɟ ɫɯɟɦɵ ɦɟɠɞɭ ɫɨɛɨɣ, ɦɨɠɟɦ ɨɬɦɟɬɢɬɶ:
ɚ) ɞɥɹ ɨɛɨɢɯ ɩɨɥɨɜ ɩɪɢ ɭɩɥɚɬɟ ɜɡɧɨɫɨɜ 1 ɪɚɡ ɜ ɝɨɞ ɢ ɭɱɟɬɟ ɬɨɥɶɤɨ ɫɦɟɪɬɧɨɫɬɢ «ɞɟɲɟɜɥɟ»
ɨɤɚɡɵɜɚɟɬɫɹ ɫɯɟɦɚ ɛɟɡ ɧɚɫɥɟɞɨɜɚɧɢɹ;
ɛ) ɞɥɹ ɫɯɟɦ ɫ ɧɚɫɥɟɞɨɜɚɧɢɟɦ ɢ ɭɱɟɬɨɦ ɬɨɥɶɤɨ ɫɦɟɪɬɧɨɫɬɢ ɞɥɹ ɨɛɨɢɯ ɩɨɥɨɜ
ɫɯɟɦɚ ɫ ɭɩɥɚɬɨɣ ɜɡɧɨɫɨɜ 1 ɪɚɡ ɜ ɝɨɞ «ɞɨɪɨɠɟ», ɱɟɦ ɫɯɟɦɚ ɯɨɬɹ ɛɵ ɫ ɞɜɭɦɹ
ɜɡɧɨɫɚɦɢ;
ɜ) ɞɥɹ ɫɯɟɦ ɫ ɧɚɫɥɟɞɨɜɚɧɢɟɦ ɢ ɭɩɥɚɬɨɣ ɜɡɧɨɫɨɜ 1 ɪɚɡ ɜ ɝɨɞ «ɞɟɲɟɜɥɟ» ɨɤɚɡɵɜɚɟɬɫɹ ɫɯɟɦɚ,
ɭɱɢɬɵɜɚɸɳɚɹ ɞɜɚ ɮɚɤɬɨɪɚ ɜɵɛɵɬɢɹ.
ɅɂɌȿɊȺɌɍɊȺ
1.
2.
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4.
5.
6.
Ⱥɥɶ-ɇɚɬɨɪ Ɇ.ɋ., Ⱥɥɶ-ɇɚɬɨɪ ɋ.ȼ., Ɉɥɟɧɱɟɧɤɨ ɇ.Ⱥ. ɇɚɤɨɩɢɬɟɥɶɧɵɟ ɦɨɞɟɥɢ ɜ ɩɟɧɫɢɨɧɧɨɦ ɫɬɪɚɯɨɜɚɧɢɢ //
ɋɛ. ɬɟɡɢɫɨɜ ȼɫɟɪɨɫɫɢɣɫɤɨɣ ɤɨɧɮɟɪɟɧɰɢɢ "ɉɪɢɤɥɚɞɧɚɹ ɬɟɨɪɢɹ ɜɟɪɨɹɬɧɨɫɬɢ ɢ ɬɟɨɪɟɬɢɱɟɫɤɚɹ ɢɧɮɨɪɦɚɬɢɤɚ". ɂɉɂ ɊȺɇ
2012. ɋ. 5-7
Ⱥɥɶ-ɇɚɬɨɪ Ɇ.ɋ., Ʉɚɫɢɦɨɜ ɘ.Ɏ., Ʉɨɥɟɫɧɢɤɨɜ Ⱥ.ɇ. Ɉɫɧɨɜɵ ɮɢɧɚɧɫɨɜɵɯ ɜɵɱɢɫɥɟɧɢɣ: ɮɚɤɬɵ, ɮɨɪɦɭɥɵ, ɩɪɢɦɟɪɵ,
ɡɚɞɚɱɢ ɢ ɬɟɫɬɵ. ɑ. I. – Ɇ.: Ɏɢɧɚɧɫɨɜɵɣ ɭɧɢɜɟɪɫɢɬɟɬ, 2012.
Ȼɚɭɷɪɫ ɇ., Ƚɟɪɛɟɪ ɏ., Ⱦɠɨɧɫ Ⱦ., ɇɟɫɛɢɬɬ ɋ., ɏɢɤɦɚɧ Ⱦɠ. Ⱥɤɬɭɚɪɧɚɹ ɦɚɬɟɦɚɬɢɤɚ. – Ɇ.: əɧɭɫ-Ʉ, 2001.
Ʉɚɫɢɦɨɜ ɘ.Ɏ. Ɏɢɧɚɧɫɨɜɚɹ ɦɚɬɟɦɚɬɢɤɚ. – Ɇ.: ɘɪɚɣɬ, 2014.
Ʉɚɫɢɦɨɜ ɘ.Ɏ. ȼɜɟɞɟɧɢɟ ɜ ɚɤɬɭɚɪɧɭɸ ɦɚɬɟɦɚɬɢɤɭ (ɫɬɪɚɯɨɜɚɧɢɹ ɠɢɡɧɢ ɢ ɩɟɧɫɢɨɧɧɵɯ ɫɯɟɦ). – Ɇ.: Ⱥɧɤɢɥ,
2001.
ɑɟɬɵɪɤɢɧ ȿ.Ɇ. Ⱥɤɬɭɚɪɧɵɟ ɪɚɫɱɟɬɵ ɜ ɧɟɝɨɫɭɞɚɪɫɬɜɟɧɧɨɦ ɩɟɧɫɢɨɧɧɨɦ ɢ ɦɟɞɢɰɢɧɫɤɨɦ ɫɬɪɚɯɨɜɚɧɢɢ. – Ɇ.: Ⱦɟɥɨ,
2009.
ɇɚɤɨɩɢɬɟɥɶɧɵɟ ɩɟɧɫɢɨɧɧɵɟ ɫɯɟɦɵ…
63
ACCUMULATIVE PENSION SCHEMES WITH VARIOUS DECREMENTAL FACTORS
Al-Nator M.S., Al-Nator S.V., Olenchenko N.A.
We consider accumulative defined contribution pension schemes with a lump sum payment on retirement. These
schemes differ in relation to inheritance and provide various decremental factors. A simulation model was developed to
analyze the constructed schemes.
Keywords: pension insurance, balance equation, gross premium, premium prenumerando, lump sum. defined contribution pension schemes.
REFERENCES
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Al-Nator M.S., Al-Nator S.V., Olenchenko N.A. Nakopitelnyy modeli v pensionnom strakhovanii // Sbornik tezisov Vserosseiskoy konferentsii “Prikladnaya teoriya veroyatnostei i teoreticheskaya informatika”, IPI RAN,
2012, pp. 5-7. (In Russian).
Al-Nator M.S., Kasimov Yu.F., Kolesnkov A.N. Osnovy finansovykh vychisleniy: fakty, formuly, primery,
zadachi i testy. Chast I. Moscow, Finansovyy universitet, 2012. (In Russian).
Bowers N., Gerber H., Jones D., Nesbitt C., Hickman J. Aktyarnaya matematika. M. Yanus-K, 2001.
(In Russian).
Kasimov Yu.F. Finansovaya matematika. Ɇ. Urait, 2014. (In Russian).
Kasimov Yu.F. Vvedenye v aktyarnyu matematiku (strakhovanya jizni i pensionnikh skhem). M. Ankil, 2001.
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Chetyrkin E.M. Aktyarnyy rashuty v negosudarstvennom pensionnom i meditsiskom starakhovanii. M. Delo,
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ɋɜɟɞɟɧɢɹ ɨɛ ɚɜɬɨɪɚɯ
Ⱥɥɶ-ɇɚɬɨɪ Ɇɭɯɚɦɦɟɞ ɋɭɛɯɢ, 1968 ɝ.ɪ., ɨɤɨɧɱɢɥ ɮɢɡɢɤɨ-ɦɚɬɟɦɚɬɢɱɟɫɤɢɣ ɮɚɤɭɥɶɬɟɬ ɊɍȾɇ (1992), ɤɚɧɞɢɞɚɬ ɮɢɡɢɤɨ-ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ ɧɚɭɤ, ɞɨɰɟɧɬ ɤɚɮɟɞɪɵ ɩɪɢɤɥɚɞɧɨɣ ɦɚɬɟɦɚɬɢɤɢ Ɏɢɧɚɧɫɨɜɨɝɨ ɭɧɢɜɟɪɫɢɬɟɬɚ ɩɪɢ ɉɪɚɜɢɬɟɥɶɫɬɜɟ ɊɎ, ɚɜɬɨɪ ɛɨɥɟɟ 50 ɧɚɭɱɧɵɯ ɩɭɛɥɢɤɚɰɢɣ, ɨɛɥɚɫɬɶ ɧɚɭɱɧɵɯ ɢɧɬɟɪɟɫɨɜ – ɚɤɬɭɚɪɧɚɹ ɦɚɬɟɦɚɬɢɤɚ, ɬɟɨɪɢɹ
ɪɢɫɤɚ, ɫɬɨɯɚɫɬɢɱɟɫɤɢɟ ɦɨɞɟɥɢ ɜ ɮɢɧɚɧɫɚɯ.
Ⱥɥɶ-ɇɚɬɨɪ ɋɨɮɶɹ ȼɥɚɞɢɦɢɪɨɜɧɚ, ɨɤɨɧɱɢɥɚ ɮɢɡɢɤɨ-ɦɚɬɟɦɚɬɢɱɟɫɤɢɣ ɮɚɤɭɥɶɬɟɬ ɊɍȾɇ (1992), ɤɚɧɞɢɞɚɬ
ɮɢɡɢɤɨ-ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ ɧɚɭɤ, ɞɨɰɟɧɬ ɤɚɮɟɞɪɵ ɬɟɨɪɢɢ ɜɟɪɨɹɬɧɨɫɬɟɣ ɢ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɫɬɚɬɢɫɬɢɤɢ Ɏɢɧɚɧɫɨɜɨɝɨ
ɭɧɢɜɟɪɫɢɬɟɬɚ ɩɪɢ ɉɪɚɜɢɬɟɥɶɫɬɜɟ ɊɎ, ɚɜɬɨɪ ɛɨɥɟɟ 20 ɧɚɭɱɧɵɯ ɩɭɛɥɢɤɚɰɢɣ, ɨɛɥɚɫɬɶ ɧɚɭɱɧɵɯ ɢɧɬɟɪɟɫɨɜ – ɚɤɬɭɚɪɧɚɹ
ɦɚɬɟɦɚɬɢɤɚ, ɬɟɨɪɢɹ ɪɢɫɤɚ, ɫɬɨɯɚɫɬɢɱɟɫɤɢɟ ɦɨɞɟɥɢ ɜ ɮɢɧɚɧɫɚɯ, ɚɤɬɭɚɪɧɵɟ ɪɚɫɱɟɬɵ.
Ɉɥɟɧɱɟɧɤɨ ɇɚɬɚɥɶɹ Ⱥɥɟɤɫɚɧɞɪɨɜɧɚ, ɨɤɨɧɱɢɥɚ ɆȽɌɍ ȽȺ (2010), ɨɛɥɚɫɬɶ ɧɚɭɱɧɵɯ ɢɧɬɟɪɟɫɨɜ – ɚɤɬɭɚɪɧɚɹ
ɦɚɬɟɦɚɬɢɤɚ, ɬɟɨɪɢɹ ɪɢɫɤɚ, ɚɤɬɭɚɪɧɵɟ ɪɚɫɱɟɬɵ.
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