Дискретная математика. Учебное пособие-1

ȿɫɥɢ ɮɭɧɤɰɢɢ ɪɚɜɧɵ ɬɨ ɞɜɨɣɫɬɜɟɧɧɵɟ ɢɦ ɮɭɧɤɰɢɢ ɬɚɤɠɟ ɪɚɜɧɵ ɗɬɨ ɩɨɡ ɜɨɥɹɟɬ ɫ ɩɨɦɨɳɶɸ ɩɪɢɧɰɢɩɚ ɞɜɨɣɫɬɜɟɧɧɨɫɬɢ ɩɨɥɭɱɚɬɶ ɧɨɜɵɟ ɷɤɜɢɜɚɥɟɧɬɧɵɟ ɫɨɨɬɧɨɲɟɧɢɹ ɩɟɪɟɯɨɞɹ ɨɬ ɪɚɜɟɧɫɬɜɚ F1 F2 ɫ ɩɨɦɨɳɶɸ ɭɤɚɡɚɧɧɵɯ ɡɚɦɟɧ ɤ ɪɚɜɟɧ ɫɬɜɭ F 1 F 2 ɉɪɢɦɟɪɨɦ ɦɨɝɭɬ ɫɥɭɠɢɬɶ ɫɨɨɬɧɨɲɟɧɢɹ x x y x ɢ x x y x ɤɨɬɨɪɵɟ ɦɨɝɭɬ ɛɵɬɶ ɩɨɥɭɱɟɧɵ ɞɪɭɝ ɢɡ ɞɪɭɝɚ ɩɨ ɭɤɚɡɚɧɧɨɦɭ ɩɪɢɧɰɢɩɭ

ȻɍɅȿȼȺ ȺɅȽȿȻɊȺ ɂ ɌȿɈɊɂə ɆɇɈɀȿɋɌȼ

Ɋɚɧɟɟ ɛɵɥɢ ɨɩɢɫɚɧɵ ɛɭɥɟɜɵ ɚɥɝɟɛɪɵ ɦɧɨɠɟɫɬɜ ɬɨ ɟɫɬɶ ɚɥɝɟɛɪɵ ɜɢɞɚ B % U ɝɞɟ % U – ɛɭɥɟɚɧ ɦɧɨɠɟɫɬɜɚ U ɬɨ ɟɫɬɶ ɦɧɨɠɟɫɬɜɨ ɜɫɟɯ ɟɝɨ ɩɨɞɦɧɨɠɟɫɬɜ Ɉɛɳɢɣ ɬɟɪɦɢɧ «ɛɭɥɟɜɚ ɚɥɝɟɛɪɚª ɞɥɹ ɚɥɝɟɛɪ ɦɧɨɠɟɫɬɜ ɢ ɥɨɝɢɱɟɫɤɢɯ ɮɭɧɤɰɢɣ ɧɟ ɹɜɥɹɟɬɫɹ ɫɥɭɱɚɣɧɵɦ

Ɉɩɪɟɞɟɥɟɧɢɟ ȼɫɹɤɚɹ ɚɥɝɟɛɪɚ ɬɢɩɚ 2 2 1 ɧɚɡɵɜɚɟɬɫɹ ɛɭɥɟɜɨɣ ɚɥɝɟɛɪɨɣ ɟɫɥɢ ɟɟ ɨɩɟɪɚɰɢɢ ɭɞɨɜɥɟɬɜɨɪɹɸɬ ɫɨɨɬɧɨɲɟɧɢɹɦ – ɫɦ ɩɪɟɞɵɞɭɳɭɸ ɥɟɤɰɢɸ ȼ ɚɥɝɟɛɪɟ ɦɧɨɠɟɫɬɜ ɷɥɟɦɟɧɬɚɦɢ ɹɜɥɹɸɬɫɹ ɩɨɞɦɧɨɠɟɫɬɜɚ ɮɢɤɫɢɪɨɜɚɧɧɨɝɨ ɭɧɢɜɟɪɫɚɥɶɧɨɝɨ ɦɧɨɠɟɫɬɜɚ U ȼ ɷɬɨɣ ɚɥɝɟɛɪɟ ɨɩɟɪɚɰɢɹ ɩɟɪɟɫɟɱɟɧɢɹ ɫɨɨɬɜɟɬ ɫɬɜɭɟɬ ɤɨɧɴɸɧɤɰɢɢ ɨɩɟɪɚɰɢɹ ɨɛɴɟɞɢɧɟɧɢɹ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɞɢɡɴɸɧɤɰɢɢ ɚ ɨɩɟ ɪɚɰɢɹ ɞɨɩɨɥɧɟɧɢɹ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɨɬɪɢɰɚɧɢɸ ɋɚɦɨ ɦɧɨɠɟɫɬɜɨ U ɹɜɥɹɟɬɫɹ ɟɞɢ ɧɢɰɟɣ ɚ ɩɭɫɬɨɟ ɦɧɨɠɟɫɬɜɨ – ɧɭɥɟɦ ɋɩɪɚɜɟɞɥɢɜɨɫɬɶ ɫɨɨɬɧɨɲɟɧɢɣ – ɞɥɹ ɷɬɨɣ ɚɥɝɟɛɪɵ ɦɨɠɧɨ ɞɨɤɚɡɚɬɶ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ ɪɚɫɫɦɚɬɪɢɜɚɹ ɜ ɧɢɯ ɩɟɪɟɦɟɧɧɵɟ ɤɚɤ ɦɧɨɠɟɫɬɜɚ ɚ ɡɧɚɤɢ ɥɨɝɢɱɟɫɤɢɯ ɮɭɧɤɰɢɣ – ɤɚɤ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɨɩɟ-

ɪɚɰɢɢ ɧɚɞ ɦɧɨɠɟɫɬɜɚɦɢ ȼ ɨɞɧɨɣ ɢɡ ɩɪɟɞɵɞɭɳɢɯ ɥɟɤɰɢɣ ɨɬɦɟɱɚɥɨɫɶ ɜɡɚɢɦɧɨ ɨɞɧɨɡɧɚɱɧɨɟ ɫɨɨɬ

ɜɟɬɫɬɜɢɟ ɦɟɠɞɭ ɦɧɨɠɟɫɬɜɨɦ % U ɝɞɟ U ^a1 an ` ɢ ɦɧɨɠɟɫɬɜɨɦ Bn ɞɜɨɢɱ-

ɧɵɯ ɜɟɤɬɨɪɨɜ ɞɥɢɧɵ n Ʉɚɠɞɨɦɭ ɩɨɞɦɧɨɠɟɫɬɜɭ M U ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɞɜɨɢɱ

ɉɨɫɤɨɥɶɤɭ ɤɨɦɩɨɧɟɧɬɵ ɤɨɨɪɞɢɧɚɬɵ ɜɟɤɬɨɪɨɜ ɩɪɢɧɢɦɚɸɬ ɡɧɚɱɟɧɢɹ ɢɥɢ ɬɨ ɭɤɚɡɚɧɧɵɟ ɨɩɟɪɚɰɢɢ – ɷɬɨ ɩɪɨɫɬɨ ɥɨɝɢɱɟɫɤɢɟ ɨɩɟɪɚɰɢɢ ɧɚɞ ɞɜɨɢɱɧɵ ɦɢ ɩɟɪɟɦɟɧɧɵɦɢ ɩɨɷɬɨɦɭ ɨɩɟɪɚɰɢɢ ɧɚɞ ɜɟɤɬɨɪɚɦɢ ɟɫɬɟɫɬɜɟɧɧɨ ɧɚɡɜɚɬɶ ɩɨɤɨɦ ɩɨɧɟɧɬɧɵɦɢ ɥɨɝɢɱɟɫɤɢɦɢ ɨɩɟɪɚɰɢɹɦɢ ɧɚɞ ɞɜɨɢɱɧɵɦɢ ɜɟɤɬɨɪɚɦɢ

51

ɉɪɢɦɟɪ 8 Ⱦɚɧɵ ɜɟɤɬɨɪɵ a 01011 ɢ b 11010 ɇɚɣɬɢ a b a b a . Ɋɟɲɟɧɢɟ

a b 11011 a b 01010 a 00100 .

Ɂɚɦɟɬɢɦ ɱɬɨ ɩɨɞɨɛɧɵɟ ɨɩɟɪɚɰɢɢ ɧɚɪɹɞɭ ɫ ɥɨɝɢɱɟɫɤɢɦɢ ɨɩɟɪɚɰɢɹɦɢ ɧɚɞ ɩɟɪɟɦɟɧɧɵɦɢ ɜɯɨɞɹɬ ɜ ɫɢɫɬɟɦɭ ɤɨɦɚɧɞ ɥɸɛɨɣ ɫɨɜɪɟɦɟɧɧɨɣ ɗȼɆ

ȿɫɥɢ ɦɨɳɧɨɫɬɶ ɦɧɨɠɟɫɬɜɚ U ɪɚɜɧɚ n U n ɬɨ ɛɭɥɟɜɚ ɢɡɨɦɨɪɮɧɚ ɛɭɥɟɜɨɣ ɚɥɝɟɛɪɟ Bn ; .

ɗɬɚ ɩɪɨɫɬɚɹ ɩɨ ɫɨɞɟɪɠɚɧɢɸ ɬɟɨɪɟɦɚ ɢɦɟɟɬ ɨɝɪɨɦɧɨɟ ɡɧɚɱɟɧɢɟ ɜ ɦɚɬɟɦɚ ɬɢɤɟ Ɉɧɚ ɩɨɡɜɨɥɹɟɬ ɡɚɦɟɧɢɬɶ ɬɟɨɪɟɬɢɤɨ-ɦɧɨɠɟɫɬɜɟɧɧɵɟ ɨɩɟɪɚɰɢɢ ɧɚɞ ɫɢɫɬɟ ɦɨɣ ɩɨɞɦɧɨɠɟɫɬɜ ɞɚɧɧɨɝɨ ɦɧɨɠɟɫɬɜɚ ɩɨɪɚɡɪɹɞɧɵɦɢ ɥɨɝɢɱɟɫɤɢɦɢ ɨɩɟɪɚɰɢɹɦɢ ɧɚɞ ɞɜɨɢɱɧɵɦɢ ɜɟɤɬɨɪɚɦɢ

ɉɨɯɨɠɚɹ ɩɨ ɮɨɪɦɭɥɢɪɨɜɤɟ ɧɨ ɡɧɚɱɢɬɟɥɶɧɨ ɨɬɥɢɱɚɸɳɚɹɫɹ ɩɨ ɫɦɵɫɥɭ ɬɟɨɪɟɦɚ ɫɭɳɟɫɬɜɭɟɬ ɞɥɹ ɦɧɨɠɟɫɬɜɚ ɜɫɟɯ ɥɨɝɢɱɟɫɤɢɯ ɮɭɧɤɰɢɣ n ɩɟɪɟɦɟɧɧɵɯ f x1 xn Ɉɛɨɡɧɚɱɢɦ ɷɬɨ ɦɧɨɠɟɫɬɜɨ P2 n Ɉɧɨ ɡɚɦɤɧɭɬɨ ɨɬɧɨɫɢɬɟɥɶɧɨ ɨɩɟɪɚ

ɰɢɣ ɢ ɫɥɟɞɨɜɚɬɟɥɶɧɨ ɨɛɪɚɡɭɟɬ ɤɨɧɟɱɧɭɸ ɛɭɥɟɜɭ ɚɥɝɟɛɪɭ P2 n ɤɨ ɬɨɪɚɹ ɹɜɥɹɟɬɫɹ ɩɨɞɚɥɝɟɛɪɨɣ ɛɭɥɟɜɨɣ ɚɥɝɟɛɪɵ ɥɨɝɢɱɟɫɤɢɯ ɮɭɧɤɰɢɣ

Ɍɟɨɪɟɦɚ ȿɫɥɢ ɦɨɳɧɨɫɬɶ ɦɧɨɠɟɫɬɜɚ U ɪɚɜɧɚ 2n U 2n ɬɨ ɛɭɥɟɜɚ ɚɥɝɟɛɪɚ % U ɢɡɨɦɨɪɮɧɚ ɛɭɥɟɜɨɣ ɚɥɝɟɛɪɟ ɮɭɧɤɰɢɣ P2 n .

Ɍɟɨɪɟɦɚ ɭɤɚɡɵɜɚɟɬ ɧɚ ɬɟɫɧɭɸ ɫɜɹɡɶ ɦɟɠɞɭ ɦɧɨɠɟɫɬɜɚɦɢ ɢ ɥɨɝɢɱɟ ɫɤɢɦɢ ɮɭɧɤɰɢɹɦɢ ɢ ɩɨɡɜɨɥɹɟɬ ɩɟɪɟɯɨɞɢɬɶ ɨɬ ɨɩɟɪɚɰɢɣ ɧɚɞ ɦɧɨɠɟɫɬɜɚɦɢ ɤ ɨɩɟ ɪɚɰɢɹɦ ɧɚɞ ɮɭɧɤɰɢɹɦɢ ɢ ɨɛɪɚɬɧɨ ȼ ɱɚɫɬɧɨɫɬɢ ɨɧɢ ɩɨɡɜɨɥɹɸɬ ɧɟɩɨɫɪɟɞɫɬɜɟɧ ɧɨ ɩɪɨɢɡɜɨɞɢɬɶ ɨɩɟɪɚɰɢɢ ɧɚɞ ɮɭɧɤɰɢɹɦɢ ɡɚɞɚɧɧɵɦɢ ɧɟ ɮɨɪɦɭɥɚɦɢ ɚ ɬɚɛɥɢ ɰɚɦɢ ɉɪɢɦɟɪ ɩɪɢɜɟɞɟɧ ɜ ɫɥɟɞɭɸɳɟɣ ɬɚɛɥɢɰɟ ɫɨɞɟɪɠɚɳɟɣ ɞɜɟ ɮɭɧɤɰɢɢ ɬɪɟɯ

ȾɇɎ ɂɇɌȿɊȼȺɅɕ ɂ ɉɈɄɊɕɌɂə

Ɍɟɨɪɟɬɢɤɨ-ɦɧɨɠɟɫɬɜɟɧɧɚɹ ɢɧɬɟɪɩɪɟɬɚɰɢɹ ɛɭɥɟɜɨɣ ɚɥɝɟɛɪɵ ɩɪɟɞɥɚɝɚɟɬ ɨɱɟɧɶ ɭɞɨɛɧɵɣ ɹɡɵɤ ɞɥɹ ɢɡɭɱɟɧɢɹ ɞɢɡɴɸɧɤɬɢɜɧɵɯ ɧɨɪɦɚɥɶɧɵɯ ɮɨɪɦ ȾɇɎ ɢ ɩɨɫɬɪɨɟɧɢɹ ɦɟɬɨɞɨɜ ɢɯ ɭɩɪɨɳɟɧɢɹ Ɋɚɫɫɦɨɬɪɢɦ ɤɪɚɬɤɨ ɨɫɧɨɜɧɵɟ ɩɨɧɹɬɢɹ ɫɜɹ ɡɚɧɧɵɟ ɫ ȾɇɎ

ȼɜɟɞɟɦ ɫɥɟɞɭɸɳɟɟ ɨɛɨɡɧɚɱɟɧɢɟ ɨɛɨɡɧɚɱɢɦ ɱɟɪɟɡ M f ɦɧɨɠɟɫɬɜɨ ɜɫɟɯ ɟɞɢ-

ɧɢɱɧɵɯ ɧɚɛɨɪɨɜ ɮɭɧɤɰɢɢ f Ɍɨɝɞɚ ɧɚɛɨɪ ɜɟɤɬɨɪ D ɢɡ ɦɧɨɠɟɫɬɜɚ Bn ɩɪɢɧɚɞ ɥɟɠɢɬ M f ɬɨɝɞɚ ɢ ɬɨɥɶɤɨ ɬɨɝɞɚ ɤɨɝɞɚ f D 1 Ɇɧɨɠɟɫɬɜɨ M f ɧɚɡɵɜɚɟɬɫɹ ɟɞɢ ɧɢɱɧɵɦ ɦɧɨɠɟɫɬɜɨɦ ɮɭɧɤɰɢɢ f ɚ ɮɭɧɤɰɢɹ f ɧɚɡɵɜɚɟɬɫɹ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟ ɫɤɨɣ ɮɭɧɤɰɢɟɣ ɦɧɨɠɟɫɬɜɚ M f Ʌɟɝɤɨ ɩɨɤɚɡɚɬɶ ɱɬɨ ɫɨɨɬɜɟɬɫɬɜɢɟ ɦɟɠɞɭ ɮɭɧɤ

ɰɢɹɦɢ ɢ ɢɯ ɟɞɢɧɢɱɧɵɦɢ ɦɧɨɠɟɫɬɜɚɦɢ ɹɜɥɹɟɬɫɹ ɢɡɨɦɨɪɮɢɡɦɨɦ

ȿɫɥɢ ɮɭɧɤɰɢɹ f x1 xn ɩɪɟɞɫɬɚɜɥɹɟɬɫɹ ɷɥɟɦɟɧɬɚɪɧɨɣ ɤɨɧɴɸɧɤɰɢɟɣ ɜɫɟɯ n ɩɟɪɟɦɟɧɧɵɯ ɬɨ ɦɧɨɠɟɫɬɜɨ M f ɫɨɞɟɪɠɢɬ ɪɨɜɧɨ ɨɞɢɧ ɷɥɟɦɟɧɬ ɦɧɨɠɟɫɬɜɚ Bn .

ȿɫɥɢ ɠɟ ɮɭɧɤɰɢɹ – ɷɥɟɦɟɧɬɚɪɧɚɹ ɤɨɧɴɸɧɤɰɢɹ k ɩɟɪɟɦɟɧɧɵɯ k n ɬɨ M f ɫɨ

ɞɟɪɠɢɬ 2n k ɞɜɨɢɱɧɵɯ ɧɚɛɨɪɨɜ ɗɬɨ ɨɛɴɹɫɧɹɟɬɫɹ ɬɟɦ ɱɬɨ ɜ ɬɚɤɨɦ ɫɥɭɱɚɟ n k ɩɟɪɟɦɟɧɧɵɯ ɧɟ ɜɯɨɞɹɳɢɯ ɜ ɷɬɭ ɤɨɧɴɸɧɤɰɢɸ ɧɟɫɭɳɟɫɬɜɟɧɧɵ ɞɥɹ ɮɭɧɤɰɢɢ f .

Ɍɨɝɞɚ ɨɧɢ ɩɪɢɧɢɦɚɸɬ 2n k ɡɧɚɱɟɧɢɣ ɧɟ ɢɡɦɟɧɹɹ ɡɧɚɱɟɧɢɹ f Ɇɧɨɠɟɫɬɜɨ M f

ɬɚɤɨɣ ɮɭɧɤɰɢɢ ɧɚɡɵɜɚɟɬɫɹ ɢɧɬɟɪɜɚɥɨɦ

ɉɪɢɦɟɪ 9 Ɋɚɫɫɦɨɬɪɢɦ ɮɭɧɤɰɢɸ f x1 x2 x3 x4 x2 x4 ɢ ɧɚɣɞɟɦ ɟɟ ɢɧɬɟɪɜɚɥ ɉɪɟɠɞɟ ɜɫɟɝɨ ɡɚɦɟɬɢɦ ɱɬɨ ɞɜɟ ɩɟɪɟɦɟɧɧɵɯ x1 x3 ɹɜɥɹɸɬɫɹ ɧɟɫɭɳɟ ɫɬɜɟɧɧɵɦɢ ɗɬɨ ɩɨɡɜɨɥɹɟɬ ɫɪɚɡɭ ɨɩɪɟɞɟɥɢɬɶ ɤɨɥɢɱɟɫɬɜɨ ɟɞɢɧɢɱɧɵɯ ɧɚɛɨɪɨɜ

ɢɧɬɟɪɜɚɥ M x2 x4 ɩɨɤɪɵɜɚɟɬ ɦɧɨɠɟɫɬɜɨ M f ɢ ɜɫɟ ɟɝɨ ɩɨɞɦɧɨɠɟɫɬɜɚ

ɉɪɟɞɫɬɚɜɥɟɧɢɟ ɧɟɤɨɬɨɪɨɣ ɮɭɧɤɰɢɢ ɜ ɜɢɞɟ ȾɇɎ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɩɪɟɞɫɬɚɜ ɥɟɧɢɸ ɟɟ ɟɞɢɧɢɱɧɨɝɨ ɦɧɨɠɟɫɬɜɚ ɜ ɜɢɞɟ ɨɛɴɟɞɢɧɟɧɢɹ ɢɧɬɟɪɜɚɥɨɜ ɜ ɫɨɜɨɤɭɩɧɨɫɬɢ ɜɫɟ ɤɨɧɴɸɧɤɰɢɢ ȾɇɎ ɩɨɤɪɵɜɚɸɬ ɜɫɟ ɟɞɢɧɢɱɧɨɟ ɦɧɨɠɟɫɬɜɨ ɮɭɧɤɰɢɢ f . Ɉɛɪɚɬɧɨɟ ɬɚɤɠɟ ɜɟɪɧɨ ɟɫɥɢ ɜɫɟ ɷɥɟɦɟɧɬɵ ɧɟɤɨɬɨɪɨɝɨ ɢɧɬɟɪɜɚɥɚ Mk ɩɪɢɧɚɞɥɟ ɠɚɬ M f ɬɨ ɫɭɳɟɫɬɜɭɟɬ ȾɇɎ ɞɚɧɧɨɣ ɮɭɧɤɰɢɢ ɫɨɞɟɪɠɚɳɚɹ ɤɨɧɴɸɧɤɰɢɸ k .

53

ʚˈˍ˙ˋˢ ͒ Ǥ ʞˑˎːˑ˕˃ ˋ ˊ˃ˏˍː˖˕ˑ˔˕˟

Ɋɚɧɟɟ ɧɚɦɢ ɪɚɫɫɦɚɬɪɢɜɚɥɢɫɶ ɞɜɚ ɫɩɨɫɨɛɚ ɡɚɞɚɧɢɹ ɥɨɝɢɱɟɫɤɢɯ ɮɭɧɤɰɢɣ – ɬɚɛɥɢɱɧɵɣ ɢ ɫ ɩɨɦɨɳɶɸ ɮɨɪɦɭɥ Ɍɚɛɥɢɰɚ ɡɚɞɚɟɬ ɮɭɧɤɰɢɸ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ ɤɚɤ ɫɨɨɬɜɟɬɫɬɜɢɟ ɦɟɠɞɭ ɞɜɨɢɱɧɵɦɢ ɧɚɛɨɪɚɦɢ ɢ ɡɧɚɱɟɧɢɹɦɢ ɮɭɧɤɰɢɢ ɧɚ ɷɬɢɯ ɧɚɛɨɪɚɯ ɗɬɨɬ ɫɩɨɫɨɛ ɭɧɢɜɟɪɫɚɥɟɧ ɬɨ ɟɫɬɶ ɩɪɢɝɨɞɟɧ ɞɥɹ ɥɸɛɵɯ ɮɭɧɤɰɢɣ ɨɞ ɧɚɤɨ ɫɥɢɲɤɨɦ ɝɪɨɦɨɡɞɨɤ Ɏɨɪɦɭɥɚ – ɝɨɪɚɡɞɨ ɛɨɥɟɟ ɤɨɦɩɚɤɬɧɵɣ ɫɩɨɫɨɛ ɡɚɞɚɧɢɹ ɮɭɧɤɰɢɢ ɧɨ ɨɧɚ ɡɚɞɚɟɬ ɮɭɧɤɰɢɸ ɱɟɪɟɡ ɞɪɭɝɢɟ ɮɭɧɤɰɢɢ ɉɨɷɬɨɦɭ ɞɥɹ ɥɸ ɛɨɣ ɫɢɫɬɟɦɵ ɮɭɧɤɰɢɣ 6 ɜɨɡɧɢɤɚɟɬ ɟɫɬɟɫɬɜɟɧɧɵɣ ɜɨɩɪɨɫ ɜɫɹɤɚɹ ɥɢ ɥɨɝɢɱɟɫɤɚɹ ɮɭɧɤɰɢɹ ɩɪɟɞɫɬɚɜɢɦɚ ɮɨɪɦɭɥɨɣ ɜ ɷɬɨɣ ɫɢɫɬɟɦɟ ȼ ɩɨɡɚɩɪɨɲɥɨɣ ɥɟɤɰɢɢ ɛɵɥ ɩɨɥɭɱɟɧ ɩɨɥɨɠɢɬɟɥɶɧɵɣ ɨɬɜɟɬ ɞɥɹ ɫɢɫɬɟɦɵ 60 ^ ` ɬɟɨɪɟɦɚ ȼ ɞɚɧɧɨɣ ɥɟɤɰɢɢ ɛɭɞɟɬ ɩɨɤɚɡɚɧɨ ɤɚɤ ɪɟɲɚɬɶ ɷɬɨɬ ɜɨɩɪɨɫ ɞɥɹ ɩɪɨɢɡɜɨɥɶɧɨɣ ɫɢɫɬɟɦɵ 6.

ɎɍɇɄɐɂɈɇȺɅɖɇɈ ɉɈɅɇɕȿ ɋɂɋɌȿɆɕ

Ɉɩɪɟɞɟɥɟɧɢɟ ɋɢɫɬɟɦɚ ɮɭɧɤɰɢɣ 6 ɧɚɡɵɜɚɟɬɫɹ ɮɭɧɤɰɢɨɧɚɥɶɧɨ ɩɨɥɧɨɣ ɫɢ ɫɬɟɦɨɣ ɟɫɥɢ ɥɸɛɚɹ ɥɨɝɢɱɟɫɤɚɹ ɮɭɧɤɰɢɹ ɦɨɠɟɬ ɛɵɬɶ ɩɪɟɞɫɬɚɜɥɟɧɚ ɮɨɪɦɭɥɨɣ ɧɚɞ ɫɢɫɬɟɦɨɣ 6 ɹɜɥɹɟɬɫɹ ɫɭɩɟɪɩɨɡɢɰɢɟɣ ɮɭɧɤɰɢɣ ɷɬɨɣ ɫɢɫɬɟɦɵ

ɂɡ ɬɟɨɪɟɦɵ ɫɥɟɞɭɟɬ ɱɬɨ ɫɢɫɬɟɦɚ 60 ^ ` ɹɜɥɹɟɬɫɹ ɮɭɧɤɰɢɨɧɚɥɶɧɨ ɩɨɥɧɨɣ Ɋɚɜɧɵɦ ɨɛɪɚɡɨɦ ɮɭɧɤɰɢɨɧɚɥɶɧɨ ɩɨɥɧɚ ɥɸɛɚɹ ɫɢɫɬɟɦɚ 6 ɱɟɪɟɡ ɮɭɧɤ ɰɢɢ ɤɨɬɨɪɨɣ ɦɨɠɧɨ ɜɵɪɚɡɢɬɶ ɤɨɧɴɸɧɤɰɢɸ ɞɢɡɴɸɧɤɰɢɸ ɢ ɨɬɪɢɰɚɧɢɟ Ⱦɟɣ ɫɬɜɢɬɟɥɶɧɨ ɞɥɹ ɥɸɛɨɣ ɥɨɝɢɱɟɫɤɨɣ ɮɭɧɤɰɢɢ f ɢɡ ɬɚɤɨɣ ɫɢɫɬɟɦɵ ɫɥɟɞɭɟɬ ɫɨɫɬɚ ɜɢɬɶ ɛɭɥɟɜɭ ɮɨɪɦɭɥɭ ɚ ɨɧɚ ɨɛɹɡɚɬɟɥɶɧɨ ɫɭɳɟɫɬɜɭɟɬ ɫɨɝɥɚɫɧɨ ɬɟɨɪɟɦɟ ɢ ɩɨɬɨɦ ɜɵɪɚɡɢɬɶ ɜ ɧɟɣ ɤɨɧɴɸɧɤɰɢɸ ɞɢɡɴɸɧɤɰɢɸ ɢ ɨɬɪɢɰɚɧɢɟ ɱɟɪɟɡ ɮɭɧɤɰɢɢ ɫɢɫɬɟɦɵ 6. Ⱥɧɚɥɨɝɢɱɧɨ ɨɛɨɫɧɨɜɵɜɚɟɬɫɹ ɛɨɥɟɟ ɨɛɳɟɟ ɭɬɜɟɪɠɞɟɧɢɟ

Ɍɟɨɪɟɦɚ ȿɫɥɢ ɜɫɟ ɮɭɧɤɰɢɢ ɮɭɧɤɰɢɨɧɚɥɶɧɨ ɩɨɥɧɨɣ ɫɢɫɬɟɦɵ 6 ɩɪɟɞɫɬɚɜɢɦɵ ɮɨɪɦɭɥɚɦɢ ɧɚɞ ɫɢɫɬɟɦɨɣ 6 ɬɨ ɫɢɫɬɟɦɚ 6 ɬɚɤɠɟ ɮɭɧɤɰɢɨɧɚɥɶɧɨ ɩɨɥɧɚ

ɉɪɢɦɟɪ

ɚ ɫɢɫɬɟɦɵ 61 ^ ` ɢ 62 ^ ` ɮɭɧɤɰɢɨɧɚɥɶɧɨ ɩɨɥɧɵ Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ ɫ ɩɨɦɨɳɶɸ ɡɚɤɨɧɨɜ Ⱦɟ Ɇɨɪɝɚɧɚ ɢ ɞɜɨɣɧɨɝɨ ɨɬɪɢɰɚɧɢɹ ɦɨɠɧɨ ɜɵɪɚɡɢɬɶ ɜ ɤɚɠ ɞɨɣ ɢɡ ɷɬɢɯ ɫɢɫɬɟɦ ɮɭɧɤɰɢɸ ɧɟɞɨɫɬɚɸɳɭɸ ɞɨ 60 ɱɟɪɟɡ ɨɫɬɚɥɶɧɵɟ ɞɜɟ

x1 x2 x1 x2 x1 x2 x1 x2 .

ɋ ɬɨɱɤɢ ɡɪɟɧɢɹ ɮɭɧɤɰɢɨɧɚɥɶɧɨɣ ɩɨɥɧɨɬɵ ɫɢɫɬɟɦɭ 60 ɫɥɟɞɭɟɬ ɫɱɢɬɚɬɶ ɢɡɛɵɬɨɱɧɨɣ ɨɧɚ ɫɨɯɪɚɧɹɟɬ ɫɜɨɣɫɬɜɨ ɩɨɥɧɨɬɵ ɢ ɩɪɢ ɭɞɚɥɟɧɢɢ ɢɡ ɧɟɟ ɤɨɧɴɸɧɤ ɰɢɢ ɢɥɢ ɞɢɡɴɸɧɤɰɢɢ Ɉɞɧɚɤɨ ɥɟɝɤɨ ɜɢɞɟɬɶ ɢɡ ɩɪɢɜɟɞɟɧɧɨɝɨ ɩɪɢɦɟɪɚ ɱɬɨ ɯɨɬɹ

54

ɫɢɫɬɟɦɵ 61 ɢ 62 ɧɟ ɹɜɥɹɸɬɫɹ ɢɡɛɵɬɨɱɧɵɦɢ ɡɚɬɨ ɮɨɪɦɭɥɵ ɜ ɧɢɯ ɩɨɥɭɱɚɸɬɫɹ ɝɨɪɚɡɞɨ ɞɥɢɧɧɟɟ ɡɚɦɟɧɚ ɨɞɧɨɣ ɨɩɟɪɚɰɢɢ ɧɚ ɞɪɭɝɭɸ ɜɧɨɫɢɬ ɜ ɮɨɪɦɭɥɭ ɫɪɚɡɭ ɬɪɢ ɥɢɲɧɢɯ ɨɬɪɢɰɚɧɢɹ;

ɛ ɫɢɫɬɟɦɵ 63 ^ ` ɲɬɪɢɯ ɒɟɮɮɟɪɚ ɢ 64 ^p` ɫɬɪɟɥɤɚ ɉɢɪɫɚ ɹɜɥɹɸɬ ɫɹ ɮɭɧɤɰɢɨɧɚɥɶɧɨ ɩɨɥɧɵɦɢ

x x x x p x; x1 x2 x1 p x2 p x1 p x2 x1 x2 x1 x2 x1 x2 .

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɫɢɫɬɟɦɚ 63 ɫɜɨɞɢɬɫɹ ɤ ɫɢɫɬɟɦɟ 61 ɚ ɫɢɫɬɟɦɚ 64 – ɤ ɫɢ

ɫɬɟɦɟ 62 ; ɜ ɫɢɫɬɟɦɚ 65 ^ 1` ɭɦɧɨɠɟɧɢɟ ɩɨ ɦɨɞɭɥɸ – ɫɥɨɠɟɧɢɟ ɩɨ ɦɨ

ɞɭɥɸ – ɫɦ ɩɭɧɤɬ ɥɟɤɰɢɢ ʋ ɹɜɥɹɟɬɫɹ ɮɭɧɤɰɢɨɧɚɥɶɧɨ ɩɨɥɧɨɣ ɉɨɫɤɨɥɶɤɭ x x 1 ɞɚɧɧɚɹ ɫɢɫɬɟɦɚ ɫɜɨɞɢɬɫɹ ɤ 60 .

ɇɚ ɫɜɨɣɫɬɜɚɯ ɷɬɨɣ ɫɢɫɬɟɦɵ ɨɫɬɚɧɨɜɢɦɫɹ ɩɨɞɪɨɛɧɟɟ

ȺɅȽȿȻɊȺ ɀȿȽȺɅɄɂɇȺ ɂ ɅɂɇȿɃɇɕȿ ɎɍɇɄɐɂɂ

Ɉɩɪɟɞɟɥɟɧɢɟ Ⱥɥɝɟɛɪɚ ɧɚɞ ɦɧɨɠɟɫɬɜɨɦ ɥɨɝɢɱɟɫɤɢɯ ɮɭɧɤɰɢɣ ɫ ɞɜɭɦɹ ɛɢ ɧɚɪɧɵɦɢ ɨɩɟɪɚɰɢɹɦɢ ɧɚɡɵɜɚɟɬɫɹ ɚɥɝɟɛɪɨɣ ɀɟɝɚɥɤɢɧɚ.

Ɂɚɦɟɱɚɧɢɟ Ɉɩɟɪɚɰɢɹ ɜɩɨɥɧɟ ɚɧɚɥɨɝɢɱɧɚ ɨɩɟɪɚɰɢɢ ɤɨɧɴɸɧɤɰɢɢ ɥɨ ɝɢɱɟɫɤɨɝɨ ɭɦɧɨɠɟɧɢɹ Ɉɞɧɚɤɨ ɨɩɟɪɚɰɢɹ ɢɦɟɟɬ ɫɨɜɟɪɲɟɧɧɨ ɞɪɭɝɨɣ ɦɚɬɟɦɚ ɬɢɱɟɫɤɢɣ ɫɦɵɫɥ ɱɟɦ ɞɢɡɴɸɧɤɰɢɹ ɫɨɨɬɜɟɬɫɬɜɭɸɳɚɹ ɮɭɧɤɰɢɹ ɪɚɧɟɟ ɛɵɥɚ ɧɚɡɜɚɧɚ ɧɟɪɚɜɧɨɡɧɚɱɧɨɫɬɶɸ ɉɨɷɬɨɦɭ ɧɢɤɚɤ ɧɟɥɶɡɹ ɫɱɢɬɚɬɶ ɚɥɝɟɛɪɭ ɀɟɝɚɥɤɢɧɚ ɢɧɨɣ ɮɨɪɦɨɣ ɡɚɩɢɫɢ ɛɭɥɟɜɨɣ ɚɥɝɟɛɪɵ

ȼ ɚɥɝɟɛɪɟ ɀɟɝɚɥɤɢɧɚ ɜɵɩɨɥɧɹɸɬɫɹ ɫɥɟɞɭɸɳɢɟ ɫɨɨɬɧɨɲɟɧɢɹ ɡɧɚɤ ɭɦɧɨ ɠɟɧɢɹ ɨɩɭɳɟɧ

Ʉɪɨɦɟ ɬɨɝɨ ɜɵɩɨɥɧɹɸɬɫɹ ɫɨɨɬɧɨɲɟɧɢɹ ɪɚɧɟɟ ɫɮɨɪɦɭɥɢɪɨɜɚɧɧɵɟ ɛɭɥɟ ɜɨɣ ɚɥɝɟɛɪɵ ɨɬɧɨɫɹɳɢɟɫɹ ɤ ɤɨɧɴɸɧɤɰɢɢ ɢ ɤɨɧɫɬɚɧɬɚɦ Ɉɬɪɢɰɚɧɢɟ ɢ ɞɢɡɴɸɧɤ ɰɢɹ ɜɵɪɚɠɚɸɬɫɹ ɬɚɤ

2.5.x x 1

2.6.x y xy x y .

ȿɫɥɢ ɜ ɩɪɨɢɡɜɨɥɶɧɨɣ ɮɨɪɦɭɥɟ ɚɥɝɟɛɪɵ ɀɟɝɚɥɤɢɧɚ ɪɚɫɤɪɵɬɶ ɫɤɨɛɤɢ ɢ ɩɪɨɢɡɜɟɫɬɢ ɜɫɟ ɭɩɪɨɳɟɧɢɹ ɩɨ ɜɵɲɟɭɤɚɡɚɧɧɵɦ ɫɨɨɬɧɨɲɟɧɢɹɦ ɬɨ ɩɨɥɭɱɢɬɫɹ ɮɨɪɦɭɥɚ ɢɦɟɸɳɚɹ ɜɢɞ ɋɭɦɦɵ ɩɪɨɢɡɜɟɞɟɧɢɣ ɬɨ ɟɫɬɶ ɩɨɥɢɧɨɦ ɦɧɨɝɨɱɥɟɧ

55

ɩɨ ɦɨɞɭɥɸ Ɍɚɤɚɹ ɮɨɪɦɭɥɚ ɧɚɡɵɜɚɟɬɫɹ ɩɨɥɢɧɨɦɨɦ ɀɟɝɚɥɤɢɧɚ ɞɥɹ ɞɚɧɧɨɣ ɮɭɧɤɰɢɢ

Ɉɬ ɛɭɥɟɜɨɣ ɮɨɪɦɭɥɵ ɜɫɟɝɞɚ ɦɨɠɧɨ ɩɟɪɟɣɬɢ ɤ ɮɨɪɦɭɥɟ ɚɥɝɟɛɪɵ ɀɟɝɚɥɤɢ ɧɚ ɢɫɩɨɥɶɡɭɹ ɪɚɜɟɧɫɬɜɚ ɢ ɚ ɬɚɤɠɟ ɩɪɹɦɨɟ ɫɥɟɞɫɬɜɢɟ ɢɡ ɪɚɜɟɧɫɬɜɚ ɟɫɥɢ x1 x2 0 ɬɨ x1 x2 x1 x2 Ɉɧɨ ɜ ɱɚɫɬɧɨɫɬɢ ɩɨɡɜɨɥɹɟɬ ɡɚɦɟɧɹɬɶ ɡɧɚɤ ɞɢɡɴɸɧɤɰɢɢ ɡɧɚɤɨɦ ɜ ɫɥɭɱɚɹɯ ɤɨɝɞɚ ɢɫɯɨɞɧɚɹ ɮɨɪɦɭɥɚ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɋȾɇɎ

ɉɪɢɦɟɪ ɋɨɫɬɚɜɢɬɶ ɩɨɥɢɧɨɦɵ ɀɟɝɚɥɤɢɧɚ ɞɥɹ ɞɚɧɧɵɯ ɮɭɧɤɰɢɣ

Ɂɚɦɟɬɢɦ ɱɬɨ ɟɫɥɢ ɜ ɩɨɥɭɱɟɧɧɵɯ ɩɨɥɢɧɨɦɚɯ ɀɟɝɚɥɤɢɧɚ ɩɪɨɢɡɜɟɫɬɢ ɨɛ ɪɚɬɧɭɸ ɡɚɦɟɧɭ ɮɭɧɤɰɢɣ ɬɨ ɩɨɥɭɱɢɦ ɭɩɪɨɳɟɧɧɵɟ ɮɨɪɦɭɥɵ ɛɭɥɟɜɨɣ ɚɥɝɟɛɪɵ

Ɍɟɨɪɟɦɚ Ⱦɥɹ ɜɫɹɤɨɣ ɥɨɝɢɱɟɫɤɨɣ ɮɭɧɤɰɢɢ ɫɭɳɟɫɬɜɭɟɬ ɩɨɥɢɧɨɦ ɀɟ ɝɚɥɤɢɧɚ ɢ ɩɪɢɬɨɦ ɟɞɢɧɫɬɜɟɧɧɵɣ

ɋɭɳɟɫɬɜɨɜɚɧɢɟ ɬɚɤɨɝɨ ɩɨɥɢɧɨɦɚ ɩɨ ɫɭɬɢ ɭɠɟ ɞɨɤɚɡɚɧɨ ɚ ɞɥɹ ɞɨɤɚɡɚ ɬɟɥɶɫɬɜɚ ɟɝɨ ɟɞɢɧɫɬɜɟɧɧɨɫɬɢ ɞɨɫɬɚɬɨɱɧɨ ɩɨɤɚɡɚɬɶ ɫɭɳɟɫɬɜɨɜɚɧɢɟ ɜɡɚɢɦɧɨ ɨɞ ɧɨɡɧɚɱɧɨɝɨ ɫɨɨɬɜɟɬɫɬɜɢɹ ɦɟɠɞɭ ɦɧɨɠɟɫɬɜɨɦ ɜɫɟɯ ɮɭɧɤɰɢɣ ɩ ɩɟɪɟɦɟɧɧɵɯ ɢ ɦɧɨɠɟɫɬɜɨɦ ɜɫɟɯ ɩɨɥɢɧɨɦɨɜ ɀɟɝɚɥɤɢɧɚ

Ɉɩɪɟɞɟɥɟɧɢɟ Ɏɭɧɤɰɢɹ ɭ ɤɨɬɨɪɨɣ ɩɨɥɢɧɨɦ ɀɟɝɚɥɤɢɧɚ ɢɦɟɟɬ ɜɢɞ 6Di xi E ɝɞɟ ɩɚɪɚɦɟɬɪɵ Di E ɪɚɜɧɵ ɧɭɥɸ ɢɥɢ ɟɞɢɧɢɰɟ ɧɚɡɵɜɚɟɬɫɹ ɥɢɧɟɣɧɨɣ.

ȼɫɟ ɮɭɧɤɰɢɢ ɨɬ ɨɞɧɨɣ ɩɟɪɟɦɟɧɧɨɣ ɥɢɧɟɣɧɵ Ɍɚɤɠɟ ɥɢɧɟɣɧɵɦɢ ɹɜɥɹɸɬɫɹ ɮɭɧɤɰɢɢ ɷɤɜɢɜɚɥɟɧɬɧɨɫɬɶ ɢ ɫɭɦɦɚ ɩɨ ɦɨɞɭɥɸ

ɁȺɆɄɇɍɌɕȿ ɄɅȺɋɋɕ ɆɈɇɈɌɈɇɇɕȿ ɎɍɇɄɐɂɂ

Ɉɩɪɟɞɟɥɟɧɢɟ Ɇɧɨɠɟɫɬɜɨ Ɇ ɥɨɝɢɱɟɫɤɢɯ ɮɭɧɤɰɢɣ ɧɚɡɵɜɚɟɬɫɹ ɡɚɦɤɧɭ ɬɵɦ ɤɥɚɫɫɨɦ ɟɫɥɢ ɥɸɛɚɹ ɫɭɩɟɪɩɨɡɢɰɢɹ ɮɭɧɤɰɢɣ ɢɡ ɦɧɨɠɟɫɬɜɚ Ɇ ɫɧɨɜɚ ɩɪɢ ɧɚɞɥɟɠɢɬ Ɇ .

ȼɫɹɤɚɹ ɫɢɫɬɟɦɚ 6 ɥɨɝɢɱɟɫɤɢɯ ɮɭɧɤɰɢɣ ɩɨɪɨɠɞɚɟɬ ɧɟɤɨɬɨɪɵɣ ɡɚɦɤɧɭɬɵɣ ɤɥɚɫɫ ɚ ɢɦɟɧɧɨ ɤɥɚɫɫ ɜɫɟɯ ɮɭɧɤɰɢɣ ɤɨɬɨɪɵɟ ɦɨɠɧɨ ɩɨɥɭɱɢɬɶ ɫɭɩɟɪɩɨɡɢɰɢɹɦɢ ɮɭɧɤɰɢɣ 6 Ɍɚɤɨɣ ɤɥɚɫɫ ɧɚɡɵɜɚɟɬɫɹ ɡɚɦɵɤɚɧɢɟɦ 6 ɢ ɨɛɨɡɧɚɱɚɟɬɫɹ >6@ ȿɫɥɢ ɦɧɨɠɟɫɬɜɨ Ɇ – ɮɭɧɤɰɢɨɧɚɥɶɧɨ ɩɨɥɧɚɹ ɫɢɫɬɟɦɚ ɬɨ >M @ P2 .

ɉɪɢɦɟɪ

ɚ ɦɧɨɠɟɫɬɜɨ ɜɫɟɯ ɞɢɡɴɸɧɤɰɢɣ ɬɨ ɟɫɬɶ ɮɭɧɤɰɢɣ ɜɢɞɚ ɹɜɥɹ ɟɬɫɹ ɡɚɦɤɧɭɬɵɦ ɤɥɚɫɫɨɦ;

ɛ ɦɧɨɠɟɫɬɜɨ ɜɫɟɯ ɥɢɧɟɣɧɵɯ ɮɭɧɤɰɢɣ ɹɜɥɹɟɬɫɹ ɡɚɦɤɧɭɬɵɦ ɤɥɚɫɫɨɦ ɬɚɤ ɤɚɤ ɩɨɞɫɬɚɧɨɜɤɚ ɮɨɪɦɭɥ ɜɢɞɚ 6Di xi E ɩɨɫɥɟ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ ɞɚɟɬ ɮɨɪɦɭɥɭ ɬɚɤɨɝɨ ɠɟ ɜɢɞɚ

56

ȼɚɠɧɟɣɲɢɦ ɩɪɢɦɟɪɨɦ ɡɚɦɤɧɭɬɨɝɨ ɤɥɚɫɫɚ ɹɜɥɹɟɬɫɹ ɤɥɚɫɫ ɦɨɧɨɬɨɧɧɵɯ

b b1 bn ɜɵɩɨɥɧɹɟɬɫɹ a d b ɟɫɥɢ ɞɥɹ ɥɸɛɨɝɨ i 1 n ɜɵɩɨɥɧɹɟɬɫɹ ai d bi .

Ɂɞɟɫɶ ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɷɬɢɦ ɨɬɧɨɲɟɧɢɟɦ ɞɥɹ ɞɜɨɢɱɧɵɯ ɜɟɤɬɨɪɨɜ

Ɉɩɪɟɞɟɥɟɧɢɟ Ɏɭɧɤɰɢɹ f x1 xn ɧɚɡɵɜɚɟɬɫɹ ɦɨɧɨɬɨɧɧɨɣ ɟɫɥɢ ɞɥɹ ɥɸ ɛɵɯ ɞɜɭɯ ɞɜɨɢɱɧɵɯ ɧɚɛɨɪɨɜ ɞɥɢɧɵ n ɢɡ ɬɨɝɨ ɱɬɨ a d b ɫɥɟɞɭɟɬ f a d f b .

ɉɪɢɦɟɪ

ɚ ɮɭɧɤɰɢɹ f x x ɦɨɧɨɬɨɧɧɚ; ɛ ɞɢɡɴɸɧɤɰɢɹ ɢ ɤɨɧɴɸɧɤɰɢɹ ɥɸɛɨɝɨ ɱɢɫɥɚ ɩɟɪɟɦɟɧɧɵɯ ɹɜɥɹɸɬɫɹ ɦɨɧɨ

ɬɨɧɧɵɦɢ ɮɭɧɤɰɢɹɦɢ; ɜ ɪɚɫɫɦɨɬɪɢɦ ɞɜɟ ɮɭɧɤɰɢɢ ɨɬ ɬɪɟɯ ɩɟɪɟɦɟɧɧɵɯ ɡɚɞɚɧɧɵɯ ɫɥɟɞɭɸɳɟɣ

ɬɚɛɥɢɰɟɣ

ɫɬɜɟɧɧɨɣ ɩɪɨɜɟɪɤɨɣ ɉɪɨɜɟɪɤɚ ɦɨɧɨɬɨɧɧɨɫɬɢ ɮɭɧɤɰɢɢ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ ɩɨ ɨɩɪɟɞɟɥɟɧɢɸ ɬɪɟ

ɛɭɟɬ ɚɧɚɥɢɡɚ ɬɚɛɥɢɰɵ ɮɭɧɤɰɢɢ ɢ ɦɨɠɟɬ ɨɤɚɡɚɬɶɫɹ ɞɨɫɬɚɬɨɱɧɨ ɬɪɭɞɨɟɦɤɨɣ ɉɨɷɬɨɦɭ ɜɟɫɶɦɚ ɩɨɥɟɡɧɨɣ ɞɥɹ ɭɫɬɚɧɨɜɥɟɧɢɹ ɦɨɧɨɬɨɧɧɨɫɬɢ ɹɜɥɹɟɬɫɹ ɫɥɟɞɭɸɳɚɹ ɬɟɨɪɟɦɚ ȼɫɹɤɚɹ ɛɭɥɟɜɚ ɮɨɪɦɭɥɚ ɧɟ ɫɨɞɟɪɠɚɳɚɹ ɨɬɪɢɰɚɧɢɣ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɦɨɧɨɬɨɧɧɭɸ ɮɭɧɤɰɢɸ ɨɬɥɢɱɧɭɸ ɨɬ ɤɨɧɫɬɚɧɬɵ ɧɚɨɛɨɪɨɬ ɞɥɹ ɥɸɛɨɣ ɦɨɧɨɬɨɧɧɨɣ ɮɭɧɤɰɢɢ ɨɬɥɢɱɧɨɣ ɨɬ ɢ ɧɚɣɞɟɬɫɹ ɩɪɟɞɫɬɚɜɥɹɸɳɚɹ ɟɟ ɛɭɥɟɜɚ ɮɨɪɦɭɥɚ ɛɟɡ ɨɬɪɢɰɚɧɢɣ

57

Ɍɟɨɪɟɦɚ ȼɫɹɤɚɹ ɛɭɥɟɜɚ ɮɨɪɦɭɥɚ ɧɟ ɫɨɞɟɪɠɚɳɚɹ ɨɬɪɢɰɚɧɢɣ ɩɪɟɞ ɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɦɨɧɨɬɨɧɧɭɸ ɮɭɧɤɰɢɸ ɨɬɥɢɱɧɭɸ ɨɬ ɤɨɧɫɬɚɧɬɵ ɧɚɨɛɨɪɨɬ ɞɥɹ ɥɸɛɨɣ ɦɨɧɨɬɨɧɧɨɣ ɮɭɧɤɰɢɢ ɨɬɥɢɱɧɨɣ ɨɬ ɢ ɧɚɣɞɟɬɫɹ ɩɪɟɞɫɬɚɜɥɹɸɳɚɹ ɟɟ ɛɭɥɟɜɚ ɮɨɪɦɭɥɚ ɛɟɡ ɨɬɪɢɰɚɧɢɣ

ɂɡ ɞɚɧɧɨɣ ɬɟɨɪɟɦɵ ɢ ɬɨɝɨ ɨɱɟɜɢɞɧɨɝɨ ɮɚɤɬɚ ɱɬɨ ɩɨɞɫɬɚɧɨɜɤɚ ɧɟɫɤɨɥɶɤɢɯ ɮɨɪɦɭɥ ɛɟɡ ɨɬɪɢɰɚɧɢɣ ɜ ɮɨɪɦɭɥɭ ɛɟɡ ɨɬɪɢɰɚɧɢɣ ɫɧɨɜɚ ɞɚɟɬ ɮɨɪɦɭɥɭ ɛɟɡ ɨɬɪɢ ɰɚɧɢɣ ɜɵɬɟɤɚɟɬ ɫɥɟɞɭɸɳɚɹ ɬɟɨɪɟɦɚ

Ɍɟɨɪɟɦɚ Ɇɧɨɠɟɫɬɜɨ ɜɫɟɯ ɦɨɧɨɬɨɧɧɵɯ ɮɭɧɤɰɢɣ ɹɜɥɹɟɬɫɹ ɡɚɦɤɧɭɬɵɦ ɤɥɚɫɫɨɦ

ɇɨ ɩɨɫɤɨɥɶɤɭ ɜɫɹɤɚɹ ɛɭɥɟɜɚ ɮɨɪɦɭɥɚ ɛɟɡ ɨɬɪɢɰɚɧɢɣ ɹɜɥɹɟɬɫɹ ɫɭɩɟɪɩɨɡɢ ɰɢɟɣ ɞɢɡɴɸɧɤɰɢɣ ɢ ɤɨɧɴɸɧɤɰɢɣ ɢɡ ɞɚɧɧɨɣ ɬɟɨɪɟɦɵ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ ɩɨɥɭɱɚ ɟɦ ɫɥɟɞɫɬɜɢɟ

ɋɥɟɞɫɬɜɢɟ Ʉɥɚɫɫ ɦɨɧɨɬɨɧɧɵɯ ɮɭɧɤɰɢɣ ɹɜɥɹɟɬɫɹ ɡɚɦɵɤɚɧɢɟɦ ɫɢɫɬɟɦɵ ɮɭɧɤɰɢɣ ^ 0 1`.

ɌȿɈɊȿɆɕ Ɉ ɎɍɇɄɐɂɈɇȺɅɖɇɈɃ ɉɈɅɇɈɌȿ

Ɍɟɩɟɪɶ ɩɟɪɟɣɞɟɦ ɤ ɪɚɫɫɦɨɬɪɟɧɢɸ ɨɫɧɨɜɧɨɝɨ ɜɨɩɪɨɫɚ ɩɨɫɬɚɜɥɟɧɧɨɝɨ ɜ ɪɚɦɤɚɯ ɞɚɧɧɨɣ ɥɟɤɰɢɢ ɤɚɤɨɜɵ ɧɟɨɛɯɨɞɢɦɵɟ ɢ ɞɨɫɬɚɬɨɱɧɵɟ ɭɫɥɨɜɢɹ ɮɭɧɤɰɢɨɧɚɥɶɧɨɣ ɩɨɥɧɨɬɵ ɞɥɹ ɩɪɨɢɡɜɨɥɶɧɨɣ ɫɢɫɬɟɦɵ ɮɭɧɤɰɢɣ 6" ȼɧɚɱɚɥɟ ɛɵɥɨ ɫɤɚɡɚ ɧɨ ɱɬɨ ɫɢɫɬɟɦɚ 6 ɩɨɥɧɚ ɟɫɥɢ ɤɨɧɴɸɧɤɰɢɹ ɞɢɡɴɸɧɤɰɢɹ ɢ ɨɬɪɢɰɚɧɢɟ ɹɜɥɹɸɬɫɹ ɫɭɩɟɪɩɨɡɢɰɢɹɦɢ ɮɭɧɤɰɢɣ ɢɡ ɫɢɫɬɟɦɵ 6 ɉɨɷɬɨɦɭ ɛɭɞɟɦ ɢɫɤɚɬɶ ɫɜɨɣɫɬɜɚ ɮɭɧɤ ɰɢɣ ɩɨɡɜɨɥɹɸɳɢɟ ɜɵɪɚɡɢɬɶ ɱɟɪɟɡ ɧɢɯ ɛɭɥɟɜɵ ɨɩɟɪɚɰɢɢ ɋɧɚɱɚɥɚ ɫɮɨɪɦɭɥɢɪɭɟɦ ɞɜɟ ɥɟɦɦɵ ɩɨɡɜɨɥɹɸɳɢɟ ɜɵɜɟɫɬɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɬɟɨɪɟɦɵ

Ʌɟɦɦɚ ɨ ɧɟɦɨɧɨɬɨɧɧɵɯ ɮɭɧɤɰɢɹɯ ȿɫɥɢ ɮɭɧɤɰɢɹ f x1 xn ɧɟɦɨɧɨ-

ɬɨɧɧɚ ɬɨ ɩɨɞɫɬɚɧɨɜɤɨɣ ɤɨɧɫɬɚɧɬ ɢɡ ɧɟɟ ɦɨɠɧɨ ɩɨɥɭɱɢɬɶ ɨɬɪɢɰɚɧɢɟ ɉɪɚɤɬɢɱɟɫɤɢ ɞɚɧɧɚɹ ɥɟɦɦɚ ɹɜɥɹɟɬɫɹ ɭɬɜɟɪɠɞɟɧɢɟɦ ɩɪɨɬɢɜɨɩɨɥɨɠɧɵɦ

ɬɟɨɪɟɦɟ ɨɛɪɚɬɧɨɣ ɤ ɬɟɨɪɟɦɟ ɋɦɵɫɥ ɟɟ ɡɚɤɥɸɱɚɟɬɫɹ ɜ ɬɨɦ ɱɬɨ ɞɥɹ ɮɭɧɤ ɰɢɢ f x1 xn ɫɭɳɟɫɬɜɭɟɬ ɬɚɤɚɹ ɩɨɞɫɬɚɧɨɜɤɚ n 1 ɤɨɧɫɬɚɧɬɵ ɱɬɨ ɮɭɧɤɰɢɹ ɨɫɬɚɜɲɟɣɫɹ ɨɞɧɨɣ ɩɟɪɟɦɟɧɧɨɣ ɹɜɥɹɟɬɫɹ ɨɬɪɢɰɚɧɢɟɦ

Ʌɟɦɦɚ ɨ ɧɟɥɢɧɟɣɧɵɯ ɮɭɧɤɰɢɹɯ ȿɫɥɢ ɮɭɧɤɰɢɹ f x1 xn ɧɟɥɢɧɟɣ ɧɚ ɬɨ ɫ ɩɨɦɨɳɶɸ ɩɨɞɫɬɚɧɨɜɤɢ ɤɨɧɫɬɚɧɬ ɢ ɢɫɩɨɥɶɡɨɜɚɧɢɹ ɨɬɪɢɰɚɧɢɣ ɢɡ ɧɟɟ ɦɨɠɧɨ ɩɨɥɭɱɢɬɶ ɞɢɡɴɸɧɤɰɢɸ ɢɥɢ ɤɨɧɴɸɧɤɰɢɸ

ɂɧɚɱɟ ɝɨɜɨɪɹ ɫɭɳɟɫɬɜɭɟɬ ɩɪɟɞɫɬɚɜɥɟɧɢɟ ɞɢɡɴɸɧɤɰɢɢ ɢ ɤɨɧɴɸɧɤɰɢɢ ɜ ɜɢɞɟ ɫɭɩɟɪɩɨɡɢɰɢɢ ɤɨɧɫɬɚɧɬ ɨɬɪɢɰɚɧɢɣ ɢ ɧɟɥɢɧɟɣɧɨɣ ɮɭɧɤɰɢɢ f .

58

Ɂɚɦɟɱɚɧɢɟ ɉɪɢ ɬɪɚɞɢɰɢɨɧɧɵɯ ɨɛɨɡɧɚɱɟɧɢɹɯ ɩɟɪɟɦɟɧɧɵɯ ɜ ɜɵɪɚɠɟɧɢɹɯ ɜɢɞɚ f x1 xn ɝɞɟ ɩɟɪɟɦɟɧɧɵɟ ɪɚɫɩɨɥɨɠɟɧɵ ɜ ɟɫɬɟɫɬɜɟɧɧɨɦ ɩɨɪɹɞɤɟ ɢɧɞɟɤ ɫɨɜ ɷɬɢ ɢɧɞɟɤɫɵ ɢɝɪɚɸɬ ɞɜɨɹɤɭɸ ɪɨɥɶ ɨɧɢ ɢɦɟɧɭɸɬ ɩɟɪɟɦɟɧɧɵɟ ɢ ɧɭɦɟɪɭɸɬ ɢɯ ɦɟɫɬɚ ɜ ɮɭɧɤɰɢɢ ɗɬɢ ɪɨɥɢ ɫɥɟɞɭɟɬ ɪɚɡɥɢɱɚɬɶ

Ⱦɜɟ ɭɤɚɡɚɧɧɵɟ ɥɟɦɦɵ ɩɨɡɜɨɥɹɸɬ ɩɨɥɭɱɢɬɶ ɜɫɟ ɛɭɥɟɜɵ ɨɩɟɪɚɰɢɢ ɫ ɩɨɦɨ ɳɶɸ ɧɟɦɨɧɨɬɨɧɧɵɯ ɮɭɧɤɰɢɣ ɧɟɥɢɧɟɣɧɵɯ ɮɭɧɤɰɢɣ ɢ ɤɨɧɫɬɚɧɬ ɗɬɨ ɟɳɟ ɧɟ ɩɨɥɧɨɬɚ ɜ ɬɨɱɧɨɦ ɫɦɵɫɥɟ ɫɥɨɜɚ ɬɚɤ ɤɚɤ ɤɨɧɫɬɚɧɬɵ ɫ ɫɚɦɨɝɨ ɧɚɱɚɥɚ ɩɪɟɞɩɨɥɚɝɚɥɢɫɶ ɞɚɧɧɵɦɢ Ɉɞɧɚɤɨ ɬɚɤɨɟ ɩɪɟɞɩɨɥɨɠɟɧɢɟ ɱɚɫɬɨ ɛɵɜɚɟɬ ɨɩɪɚɜɞɚɧɧɵɦ ɜ ɪɚɡɥɢɱɧɵɯ ɩɪɢɥɨɠɟɧɢɹɯ ɩɪɟɠɞɟ ɜɫɟɝɨ ɜ ɫɢɧɬɟɡɟ ɥɨɝɢɱɟɫɤɢɯ ɫɯɟɦ ɉɨɷɬɨɦɭ ɟɫɬɶ ɫɦɵɫɥ ɜɜɟɫɬɢ ɨɫɥɚɛɥɟɧɧɨɟ ɨɩɪɟɞɟɥɟɧɢɟ ɩɨɥɧɨɬɵ

Ɉɩɪɟɞɟɥɟɧɢɟ ɋɢɫɬɟɦɚ ɮɭɧɤɰɢɣ 6 ɧɚɡɵɜɚɟɬɫɹ ɮɭɧɤɰɢɨɧɚɥɶɧɨ ɩɨɥɧɨɣ ɜ ɫɥɚɛɨɦ ɫɦɵɫɥɟ ɟɫɥɢ ɥɸɛɚɹ ɥɨɝɢɱɟɫɤɚɹ ɮɭɧɤɰɢɹ ɦɨɠɟɬ ɛɵɬɶ ɩɪɟɞɫɬɚɜɥɟɧɚ ɮɨɪɦɭɥɨɣ ɧɚɞ ɫɢɫɬɟɦɨɣ 6 ^0 1` ɬɨ ɟɫɬɶ ɹɜɥɹɟɬɫɹ ɫɭɩɟɪɩɨɡɢɰɢɟɣ ɤɨɧɫɬɚɧɬ ɢ ɮɭɧɤɰɢɣ ɢɡ ɫɢɫɬɟɦɵ 6.

Ɉɱɟɜɢɞɧɨ ɱɬɨ ɢɡ ɨɛɵɱɧɨɣ ɩɨɥɧɨɬɵ ɫɢɫɬɟɦɵ ɫɥɟɞɭɟɬ ɟɟ ɫɥɚɛɚɹ ɩɨɥɧɨɬɚ

Ɍɟɨɪɟɦɚ ɩɟɪɜɚɹ ɬɟɨɪɟɦɚ ɨ ɮɭɧɤɰɢɨɧɚɥɶɧɨɣ ɩɨɥɧɨɬɟ Ⱦɥɹ ɬɨɝɨ ɱɬɨɛɵ ɫɢɫɬɟɦɚ ɮɭɧɤɰɢɣ 6 ɛɵɥɚ ɮɭɧɤɰɢɨɧɚɥɶɧɨ ɩɨɥɧɨɣ ɜ ɫɥɚɛɨɦ ɫɦɵɫɥɟ ɧɟɨɛ ɯɨɞɢɦɨ ɢ ɞɨɫɬɚɬɨɱɧɨ ɱɬɨɛɵ ɨɧɚ ɫɨɞɟɪɠɚɥɚ ɯɨɬ ɛɵ ɨɞɧɭ ɧɟɦɨɧɨɬɨɧɧɭɸ ɮɭɧɤ ɰɢɸ ɢ ɯɨɬɹ ɛɵ ɨɞɧɭ ɧɟɥɢɧɟɣɧɭɸ ɮɭɧɤɰɢɸ

Ⱦɨɤɚɡɚɬɟɥɶɫɬɜɨ

1.ɇɟɨɛɯɨɞɢɦɨɫɬɶ Ʉɥɚɫɫɵ ɦɨɧɨɬɨɧɧɵɯ ɢ ɥɢɧɟɣɧɵɯ ɮɭɧɤɰɢɣ ɡɚɦɤɧɭɬɵ ɢ ɫɨɞɟɪɠɚɬ ɢ ɉɨɷɬɨɦɭ ɟɫɥɢ 6 ɧɟ ɫɨɞɟɪɠɢɬ ɧɟɦɨɧɨɬɨɧɧɵɯ ɢɥɢ ɧɟɥɢɧɟɣɧɵɯ ɮɭɧɤɰɢɣ ɬɨ ɢɯ ɧɟɥɶɡɹ ɩɨɥɭɱɢɬɶ ɫ ɩɨɦɨɳɶɸ ɫɭɩɟɪɩɨɡɢɰɢɣ ɮɭɧɤɰɢɣ ɢɡ ɫɢɫɬɟɦɵ 6 ɢ ɤɨɧɫɬɚɧɬ

2.Ⱦɨɫɬɚɬɨɱɧɨɫɬɶ ɉɭɫɬɶ 6 ɫɨɞɟɪɠɢɬ ɧɟɦɨɧɨɬɨɧɧɭɸ ɢ ɧɟɥɢɧɟɣɧɭɸ ɮɭɧɤɰɢɸ Ɍɨɝɞɚ ɩɨ ɥɟɦɦɟ ɩɨɞɫɬɚɧɨɜɤɨɣ ɤɨɧɫɬɚɧɬ ɢɡ ɦɨɧɨɬɨɧɧɨɣ ɮɭɧɤɰɢɢ ɩɨɥɭɱɚɟɦ ɨɬɪɢɰɚɧɢɟ ɚ ɡɚɬɟɦ ɩɨ ɥɟɦɦɟ ɢɡ ɧɟɥɢɧɟɣɧɨɣ ɮɭɧɤɰɢɢ ɫ ɩɨɦɨɳɶɸ ɨɬɪɢ ɰɚɧɢɣ ɢ ɤɨɧɫɬɚɧɬ ɩɨɥɭɱɚɟɦ ɞɢɡɴɸɧɤɰɢɸ ɢ ɤɨɧɴɸɧɤɰɢɸ

ɉɪɢɦɟɪ

ɚ ɫɢɫɬɟɦɚ 66 ^ ` ɮɭɧɤɰɢɨɧɚɥɶɧɨ ɩɨɥɧɚ ɜ ɫɥɚɛɨɦ ɫɦɵɫɥɟ ɬɚɤ ɤɚɤ ɨɩɟ ɪɚɰɢɹ ɧɟɥɢɧɟɣɧɚ ɤɚɤ ɢ ɤɨɧɴɸɧɤɰɢɹ ɚ ɨɩɟɪɚɰɢɹ ɫɥɨɠɟɧɢɟ ɩɨ PRG2 ɧɟ ɦɨɧɨɬɨɧɧɚ;

ɛ ɜ ɮɭɧɤɰɢɨɧɚɥɶɧɨ ɩɨɥɧɨɣ ɫɢɫɬɟɦɟ 63 ^ ` ɟɞɢɧɫɬɜɟɧɧɚɹ ɮɭɧɤɰɢɹ – ɲɬɪɢɯ ɒɟɮɮɟɪɚ – ɧɟɥɢɧɟɣɧɚ ɢ ɧɟɦɨɧɨɬɨɧɧɚ

Ⱦɥɹ ɮɨɪɦɭɥɢɪɨɜɤɢ ɧɟɨɛɯɨɞɢɦɵɯ ɢ ɞɨɫɬɚɬɨɱɧɵɯ ɭɫɥɨɜɢɣ «cɢɥɶɧɨɣª ɩɨɥ ɧɨɬɵ ɜ ɨɬɥɢɱɢɟ ɨɬ ɫɥɚɛɨɣ ɧɭɠɧɨ ɜɜɟɫɬɢ ɪɹɞ ɨɩɪɟɞɟɥɟɧɢɣ ɨɩɢɫɵɜɚɸɳɢɯ ɟɳɟ ɬɪɢ ɡɚɦɤɧɭɬɵɯ ɤɥɚɫɫɚ ɮɭɧɤɰɢɣ

59

Ɉɩɪɟɞɟɥɟɧɢɟ Ɏɭɧɤɰɢɹ f x1 xn ɧɚɡɵɜɚɟɬɫɹ ɫɨɯɪɚɧɹɸɳɟɣ ɧɨɥɶ ɟɫɥɢ ɜɵɩɨɥɧɹɟɬɫɹ f 0 0 0 ɢ ɫɨɯɪɚɧɹɸɳɟɣ ɟɞɢɧɢɰɭ ɟɫɥɢ ɜɵɩɨɥɧɹɟɬɫɹ f 1 1 1.

Ɉɛɚ ɞɚɧɧɵɯ ɤɥɚɫɫɚ ɮɭɧɤɰɢɣ ɹɜɥɹɸɬɫɹ ɡɚɦɤɧɭɬɵɦɢ ɱɬɨ ɩɪɨɜɟɪɹɟɬɫɹ ɩɨɞ ɫɬɚɧɨɜɤɨɣ ɤɨɧɫɬɚɧɬ ɜ ɫɭɩɟɪɩɨɡɢɰɢɢ Ɋɚɜɧɵɦ ɨɛɪɚɡɨɦ ɡɚɦɤɧɭɬɵɣ ɤɥɚɫɫ ɨɛɪɚɡɭ ɸɬ ɫɚɦɨɞɜɨɣɫɬɜɟɧɧɵɟ ɮɭɧɤɰɢɢ ɬɚɤɢɟ ɱɬɨ f x1 xn f x1 xn

Ɍɟɨɪɟɦɚ ɜɬɨɪɚɹ – ɨɫɧɨɜɧɚɹ – ɬɟɨɪɟɦɚ ɨ ɮɭɧɤɰɢɨɧɚɥɶɧɨɣ ɩɨɥɧɨ ɬɟ Ⱦɥɹ ɬɨɝɨ ɱɬɨɛɵ ɫɢɫɬɟɦɚ ɮɭɧɤɰɢɣ 6 ɛɵɥɚ ɮɭɧɤɰɢɨɧɚɥɶɧɨ ɩɨɥɧɨɣ ɜ ɫɢɥɶ ɧɨɦ ɫɦɵɫɥɟ ɧɟɨɛɯɨɞɢɦɨ ɢ ɞɨɫɬɚɬɨɱɧɨ ɱɬɨɛɵ ɨɧɚ ɫɨɞɟɪɠɚɥɚ ɧɟɥɢɧɟɣɧɭɸ ɮɭɧɤɰɢɸ ɧɟɦɨɧɨɬɨɧɧɭɸ ɮɭɧɤɰɢɸ ɮɭɧɤɰɢɸ ɧɟ ɹɜɥɹɸɳɭɸɫɹ ɫɚɦɨɞɜɨɣ ɫɬɜɟɧɧɨɣ ɮɭɧɤɰɢɸ ɧɟ ɫɨɯɪɚɧɹɸɳɭɸ ɧɨɥɶ ɮɭɧɤɰɢɸ ɧɟ ɫɨɯɪɚɧɹɸɳɭɸ ɟɞɢɧɢɰɭ

ʚˈˍ˙ˋˢ ͒ ʹǤ ʮˊ˞ˍ ˎˑˆˋˍˋ ˒˓ˈˇˋˍ˃˕ˑ˅

1. ɉɊȿȾɂɄȺɌɕ

Ɉɩɪɟɞɟɥɟɧɢɟ ɉɪɟɞɢɤɚɬɨɦ P x1 xn ɧɚɡɵɜɚɟɬɫɹ ɮɭɧɤɰɢɹ P M n o B ɝɞɟ M – ɩɪɨɢɡɜɨɥɶɧɨɟ ɦɧɨɠɟɫɬɜɨ ɚ B – ɨɩɪɟɞɟɥɟɧɧɨɟ ɪɚɧɟɟ ɞɜɨɢɱɧɨɟ ɦɧɨɠɟ ɫɬɜɨ ^0;1`.

ɂɧɚɱɟ ɝɨɜɨɪɹ n ɦɟɫɬɧɵɦ ɩɪɟɞɢɤɚɬɨɦ ɨɩɪɟɞɟɥɟɧɧɵɦ ɧɚ ɦɧɨɠɟɫɬɜɟ Ɇ ɧɚɡɵɜɚɟɬɫɹ ɞɜɭɡɧɚɱɧɚɹ ɮɭɧɤɰɢɹ ɨɬ n ɚɪɝɭɦɟɧɬɨɜ ɢɡ ɩɪɨɢɡɜɨɥɶɧɨɝɨ ɦɧɨɠɟɫɬɜɚ Ɇ. Ɇɧɨɠɟɫɬɜɨ Ɇ ɧɚɡɵɜɚɟɬɫɹ ɩɪɟɞɦɟɬɧɨɣ ɨɛɥɚɫɬɶɸ ɩɪɟɞɢɤɚɬɚ ɩɟɪɟɦɟɧ ɧɵɟ x1 xn – ɩɪɟɞɦɟɬɧɵɦɢ ɩɟɪɟɦɟɧɧɵɦɢ ȼ ɩɪɢɧɰɢɩɟ ɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ ɩɪɟ ɞɢɤɚɬ ɤɚɤ ɮɭɧɤɰɢɸ P M1 u M n o B ɬɨ ɟɫɬɶ ɞɨɩɭɫɬɢɬɶ ɱɬɨ ɩɟɪɟɦɟɧɧɵɟ ɩɪɢ ɧɢɦɚɸɬ ɡɧɚɱɟɧɢɹ ɢɡ ɪɚɡɥɢɱɧɵɯ ɦɧɨɠɟɫɬɜ – ɜ ɧɟɤɨɬɨɪɵɯ ɫɥɭɱɚɹɯ ɷɬɨ ɨɤɚɡɵɜɚɟɬɫɹ ɭɞɨɛɧɵɦ

Ⱦɥɹ ɥɸɛɵɯ M ɢ n ɫɭɳɟɫɬɜɭɟɬ ɜɡɚɢɦɧɨ ɨɞɧɨɡɧɚɱɧɨɟ ɫɨɨɬɜɟɬɫɬɜɢɟ ɦɟɠɞɭ n ɦɟɫɬɧɵɦɢ ɨɬɧɨɲɟɧɢɹɦɢ ɢ n-ɦɟɫɬɧɵɦɢ ɩɪɟɞɢɤɚɬɚɦɢ ɧɚ ɦɧɨɠɟɫɬɜɟ M ɨɩɪɟɞɟɥɹɟɦɨɟ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ Ʉɚɠɞɨɦɭ n-ɦɟɫɬɧɨɦɭ ɨɬɧɨɲɟɧɢɸ R ɫɨɨɬ