Планирование эксперимента. Учебное пособие-1
.pdfɑɢɫɥɟɧɧɨɟ ɡɧɚɱɟɧɢɟ ɤɨɷɮɮɢɰɢɟɧɬɚb ɨɩɪɟɞɟɥɹɟɬ ɦɨɳɧɨɫɬɶ ɩɪɢ ɧɭɥɟɜɵɯ ɩɪɢɜɟɞɟɧɧɵɯ ɡɧɚɱɟɧɢɹɯ ɮɚɤɬɨɪɨɜ ɤɨɬɨɪɵɟ ɫɨɨɬɜɟɬɫɬɜɭɸɬ ɧɚɬɭɪɚɥɶɧɵɦ ɡɧɚɱɟ ɧɢɹɦU ȼ ɢI Ⱥ ȾɟɣɫɬɜɢɬɟɥɶɧɨP U I ȼɬ
ɑɢɫɥɟɧɧɨɟ ɡɧɚɱɟɧɢɟ ɤɨɷɮɮɢɰɢɟɧɬɚ b ɨɩɪɟɞɟɥɹɟɬ ɜɟɥɢɱɢɧɭ ɭɜɟɥɢɱɟɧɢɹ ɦɨɳɧɨɫɬɢ ɩɪɢ ɢɡɦɟɧɟɧɢɢ ɧɚɩɪɹɠɟɧɢɹ ɫ ȼ ɞɨ ȼ ɉɪɢU ȼ ɦɨɳ ɧɨɫɬɶP U I ȼɬ ɬɨ ɟɫɬɶ ɧɚ ȼɬ ɛɨɥɶɲɟ ɱɟɦ ɩɪɢ ɧɚɩɪɹɠɟ ɧɢɢ ȼ
ɑɢɫɥɟɧɧɨɟ ɡɧɚɱɟɧɢɟ ɤɨɷɮɮɢɰɢɟɧɬɚ b ɨɩɪɟɞɟɥɹɟɬ ɜɟɥɢɱɢɧɭ ɭɜɟɥɢɱɟɧɢɹ ɦɨɳɧɨɫɬɢ ɩɪɢ ɢɡɦɟɧɟɧɢɢ ɬɨɤɚ ɫ Ⱥ ɞɨ Ⱥ ɉɪɢ I Ⱥ ɦɨɳɧɨɫɬɶ P U I ȼɬ ɬɨ ɟɫɬɶ ɧɚ ȼɬ ɛɨɥɶɲɟ ɱɟɦ ɩɪɢ ɬɨɤɟ Ⱥ
ɑɢɫɥɟɧɧɨɟ ɡɧɚɱɟɧɢɟ ɤɨɷɮɮɢɰɢɟɧɬɚ b ɨɩɪɟɞɟɥɹɟɬ ɜɟɥɢɱɢɧɭ ɭɜɟɥɢɱɟɧɢɹ ɦɨɳɧɨɫɬɢ ɩɪɢ ɨɞɧɨɜɪɟɦɟɧɧɨɦ ɢɡɦɟɧɟɧɢɢ ɧɚɩɪɹɠɟɧɢɹ ɢ ɬɨɤɚ ɫ ɧɭɥɟɜɵɯ ɩɪɢɜɟ ɞɟɧɧɵɯ ɡɧɚɱɟɧɢɣ ɞɨ ɦɚɤɫɢɦɚɥɶɧɵɯ ɡɧɚɱɟɧɢɣ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ ɭɜɟɥɢɱɟɧɢɟɦ ɧɚɩɪɹɠɟɧɢɹ ɢ ɬɨɤɚ ɜ ɨɬɞɟɥɶɧɨɫɬɢ ɉɪɢ ɦɚɤɫɢɦɚɥɶɧɵɯ ɡɧɚɱɟɧɢɹɯ ɧɚɩɪɹɠɟɧɢɹ U ȼ ɢ ɬɨɤɚ I Ⱥ ɦɨɳɧɨɫɬɶ P U I ȼɬ ɬɨ ɟɫɬɶ ɧɚ ȼɬ ɛɨɥɶɲɟ ɱɟɦ ɩɪɢ ɭɜɟɥɢɱɟɧɢɢ ɧɚɩɪɹɠɟɧɢɹ ɢ ɬɨɤɚ ɜ ɨɬɞɟɥɶɧɨɫɬɢ ɬɨ ɟɫɬɶ Ⱥ
ȼɢɞɧɨ ɱɬɨ ɦɨɞɟɥɶ ɫ ɭɱɟɬɨɦ ɞɜɨɣɧɵɯ ɫɨɱɟɬɚɧɢɣ ɮɚɤɬɨɪɨɜ ɜ ɬɨɱɤɚɯ ɜ ɤɨɬɨ ɪɵɯɩɪɨɢɡɜɨɞɢɥɢɫɶɢɡɦɟɪɟɧɢɹɞɚɟɬɚɛɫɨɥɸɬɧɨɬɨɱɧɵɣɪɟɡɭɥɶɬɚɬɉɪɨɜɟɪɢɦ ɧɚɫɤɨɥɶɤɨ ɬɨɱɧɵɣ ɪɟɡɭɥɶɬɚɬ ɛɭɞɟɬ ɩɨɥɭɱɚɬɶɫɹ ɩɪɢ ɬɚɤɢɯ ɡɧɚɱɟɧɢɹɯ ɮɚɤɬɨɪɨɜ ɩɪɢ ɤɨɬɨɪɵɯ ɧɟ ɩɪɨɢɡɜɨɞɢɥɢɫɶ ɢɡɦɟɪɟɧɢɹ ɇɚɩɪɢɦɟɪ ɩɪɢ ɧɚɩɪɹɠɟɧɢɢ ȼ ɢ ɬɨɤɟ Ⱥ ɱɬɨ ɜ ɩɪɢɜɟɞɟɧɧɵɯ ɛɟɡɪɚɡɦɟɪɧɵɯ ɡɧɚɱɟɧɢɹɯ ɫɨɫɬɚɜɥɹɟɬ ɫɨɨɬɜɟɬ ɫɬɜɟɧɧɨ
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ɉɪɨɜɟɪɹɟɦ ɡɧɚɱɟɧɢɟ ɦɨɳɧɨɫɬɢ ɜ ɷɬɨɦ ɫɥɭɱɚɟ |
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ɍɛɟɠɞɚɟɦɫɹ ɱɬɨ ɢ ɩɪɢ ɞɚɧɧɵɯ ɡɧɚɱɟɧɢɹɯ ɧɚɩɪɹɠɟɧɢɹ ɢ ɬɨɤɚ ɩɨɥɭɱɟɧɧɚɹ
ɦɨɞɟɥɶ ɞɚɟɬ ɩɪɚɜɢɥɶɧɵɣ ɪɟɡɭɥɶɬɚɬ Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɜ ɫɥɭɱɚɟ ɥɢɧɟɣɧɨɣ ɡɚɜɢɫɢ
ɦɨɫɬɢ ɨɬɤɥɢɤɚ ɨɬ ɮɚɤɬɨɪɨɜ ɩɨɥɭɱɟɧɧɚɹ ɦɨɞɟɥɶ ɡɚɞɚɜɚɟɦɚɹ ɩɨɥɢɧɨɦɨɦ ɩɟɪɜɨɝɨ
ɩɨɪɹɞɤɚ ɩɨ ɮɚɤɬɨɪɚɦ
y b b x b x b x x
ɞɚɟɬ ɬɨɱɧɵɣ ɪɟɡɭɥɶɬɚɬ
Ɋɚɫɫɦɨɬɪɢɦ ɩɪɢɦɟɪ ɫ ɬɪɟɦɹ ɮɚɤɬɨɪɚɦɢ
Ⱦɚɧɚ ɷɥɟɤɬɪɢɱɟɫɤɚɹ ɰɟɩɶ ɫ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨ ɫɨɟɞɢɧɟɧɧɵɦɢ ɫɨɩɪɨɬɢɜɥɟɧɢ ɹɦɢ ɚɤɬɢɜɧɵɦ ɪɟɚɤɬɢɜɧɵɦ ɢɧɞɭɤɬɢɜɧɵɦ ɢ ɪɟɚɤɬɢɜɧɵɦ ɟɦɤɨɫɬɧɵɦ ɉɨ ɪɟ ɡɭɥɶɬɚɬɚɦ ɢɡɦɟɪɟɧɢɣ ɩɨɫɬɪɨɢɬɶ ɦɨɞɟɥɶ ɜ ɜɢɞɟ ɩɨɥɢɧɨɦɚ ɨɩɢɫɵɜɚɸɳɟɝɨ ɡɚɜɢ ɫɢɦɨɫɬɶ ɬɨɤɚ ɞɟɣɫɬɜɭɸɳɟɝɨ ɡɧɚɱɟɧɢɹ ɨɬ ɚɤɬɢɜɧɨɝɨ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɢɧɞɭɤɬɢɜ ɧɨɫɬɢ ɢ ɟɦɤɨɫɬɢI f R L C ɢɡɦɟɧɹɸɳɢɯɫɹ ɜ ɫɥɟɞɭɸɳɢɯ ɩɪɟɞɟɥɚɯ
ɚɤɬɢɜɧɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟR ± Ɉɦ ɢɧɞɭɤɬɢɜɧɨɫɬɶL ± Ƚɧ
ɟɦɤɨɫɬɶC ± ɦɤɎ Ɏ ɉɟɪɟɦɟɧɧɨɟ ɧɚɩɪɹɠɟɧɢɟ ɩɪɢɥɨɠɟɧɧɨɟ ɤ ɰɟɩɢU ȼ ɞɟɣɫɬɜɭɸɳɟɟ
ɡɧɚɱɟɧɢɟ ɧɟ ɢɡɦɟɧɹɟɬɫɹ
ɉɨɧɹɬɧɨ ɱɬɨ ɯɚɪɚɤɬɟɪ ɡɚɜɢɫɢɦɨɫɬɢ ɧɟɥɢɧɟɣɧɵɣ ɡɚɜɢɫɢɦɨɫɬɶ ɦɨɠɟɬ ɫɨ
ɞɟɪɠɚɬɶ ɹɜɧɨ ɜɵɪɚɠɟɧɧɵɣ ɦɚɤɫɢɦɭɦ ɬɨɤɚ ɫɥɭɱɚɣ ɪɟɡɨɧɚɧɫɚ ɧɚɩɪɹɠɟɧɢɹ ɞɥɹ
ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨ ɫɨɟɞɢɧɟɧɧɵɯ ɢɧɞɭɤɬɢɜɧɨɫɬɢ ɢ ɟɦɤɨɫɬɢ ɉɨɫɬɪɨɢɦ ɩɥɚɧ ɩɨɥɧɨɝɨ ɮɚɤɬɨɪɧɨɝɨ ɷɤɫɩɟɪɢɦɟɧɬɚ Ȼɭɞɟɦ ɫɱɢɬɚɬɶ ɱɬɨ ɢɡɦɟɪɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɬɨɤɚ ɫɨɜɩɚɞɚɸɬ ɫɨ ɡɧɚɱɟɧɢɹɦɢ
ɧɚɣɞɟɧɧɵɦɢ ɩɨ ɬɟɨɪɟɬɢɱɟɫɤɨɣ ɦɨɞɟɥɢI |
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xL ZL± ɢɧɞɭɤɬɢɜɧɨɟ ɪɟɚɤɬɢɜɧɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ
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± ɪɟɚɤɬɢɜɧɨɟ ɟɦɤɨɫɬɧɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ |
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f Ƚɰ ± ɱɚɫɬɨɬɚ ɩɟɪɟɦɟɧɧɨɝɨ ɧɚɩɪɹɠɟɧɢɹ
ɉɨ ɢɡɦɟɪɟɧɧɵɦ ɡɧɚɱɟɧɢɹɦ ɬɨɤɚ ɪɚɫɫɱɢɬɚɟɦ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɦɨɞɟɥɢ ȼɫɟ ɤɨɷɮɮɢɰɢɟɧɬɵ ɪɚɫɫɱɢɬɵɜɚɸɬɫɹ ɩɨ ɟɞɢɧɨɣ ɮɨɪɦɭɥɟ ɭɱɢɬɵɜɚɹ ɱɬɨ b b
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Ɋɚɫɲɢɪɟɧɧɚɹ ɦɚɬɪɢɰɚ ɩɥɚɧɚ ɉɎɗ ɫ ɪɟɡɭɥɶɬɚɬɚɦɢ ɢɡɦɟɪɟɧɢɣ ɩɪɢɜɟɞɟɧɚ |
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ɜ ɬɚɛɥɢɰɟ |
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ɉɥɚɧ ɉɎɗ ɫ ɪɟɡɭɥɶɬɚɬɚɦɢ ɢɡɦɟɪɟɧɢɣ |
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Ɂɧɚɱɟɧɢɟ b |
ɞɨɥɠɧɨɫɨɨɬɜɟɬɫɬɜɨɜɚɬɶɧɚɬɭɪɚɥɶɧɨɦɭɡɧɚɱɟɧɢɸɨɬ |
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ɤɥɢɤɚ ɬɨɤɚ ɜ ɧɭɥɟɜɨɣ ɬɨɱɤɟ ɬɨɱɤɟ ɜ ɤɨɬɨɪɨɣ ɩɪɢɜɟɞɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɮɚɤɬɨɪɨɜ
ɪɚɜɧɵ ɧɭɥɸ ɋɪɚɡɭ ɩɪɨɜɟɪɢɦ ɡɧɚɱɟɧɢɟ ɬɨɤɚ ɜ ɧɭɥɟɜɨɣ ɬɨɱɤɟ
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Ɋɚɡɧɨɫɬɶ ɫɨɫɬɚɜɥɹɟɬ ± Ⱥ ɉɨɝɪɟɲɧɨɫɬɶ ɨɬɥɢɱɢɟ ɨɬ ɬɨɤɚ ɜ ɧɭɥɟɜɨɣ ɬɨɱɤɟ ɜɫɟɝɨ
b |
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ɋ ɪɨɫɬɨɦ ɚɤɬɢɜɧɨɝɨ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɬɨɤ ɭɦɟɧɶɲɚɟɬɫɹ Ɍɨɤ ɨɛɪɚɬɧɨ ɩɪɨ ɩɨɪɰɢɨɧɚɥɟɧɚɤɬɢɜɧɨɦɭɫɨɩɪɨɬɢɜɥɟɧɢɸɊɟɚɤɬɢɜɧɵɟɫɨɩɪɨɬɢɜɥɟɧɢɹɧɟɢɡɦɟ ɧɹɸɬɫɹ
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b |
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ɋ ɪɨɫɬɨɦ ɢɧɞɭɤɬɢɜɧɨɫɬɢ ɬɨɤ ɪɚɫɬɟɬ ɚ ɧɟ ɭɦɟɧɶɲɚɟɬɫɹ ɤɚɤ ɤɚɡɚɥɨɫɶ ɛɵ
ɞɨɥɠɧɨ ɛɵɬɶ ɜɟɞɶ ɬɨɤ ɩɪɹɦɨ ɩɪɨɩɨɪɰɢɨɧɚɥɟɧ ɢɧɞɭɤɬɢɜɧɨɦɭ ɫɨɩɪɨɬɢɜɥɟɧɢɸ
Ɉɛɴɹɫɧɹɟɬɫɹ ɷɬɨ ɬɟɦ ɱɬɨ ɜ ɰɟɩɢ ɫ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɵɦ ɫɨɟɞɢɧɟɧɢɟɦ ɢɧɞɭɤɬɢɜɧɨ
ɝɨ ɢ ɟɦɤɨɫɬɧɨɝɨ ɫɨɩɪɨɬɢɜɥɟɧɢɣ ɜɨɡɧɢɤɚɸɬ ɪɟɡɨɧɚɧɫɧɵɟ ɹɜɥɟɧɢɹ ɋ ɪɨɫɬɨɦ ɢɧ
ɞɭɤɬɢɜɧɨɫɬɢ ɦɨɞɭɥɶ ɪɚɡɧɨɫɬɢ ɦɟɠɞɭ ɢɧɞɭɤɬɢɜɧɵɦ ɢ ɟɦɤɨɫɬɧɵɦ ɫɨɩɪɨɬɢɜɥɟɧɢ
ɹɦɢ ɭɦɟɧɶɲɚɟɬɫɹ ɪɟɚɤɬɢɜɧɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɭɦɟɧɶɲɚɟɬɫɹ ɫɥɟɞɨɜɚɬɟɥɶɧɨ ɬɨɤ
ɪɚɫɬɟɬ
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ɋ ɪɨɫɬɨɦ ɟɦɤɨɫɬɢ ɬɨɤ ɪɚɫɬɟɬ ɬɚɤ ɤɚɤ ɫ ɪɨɫɬɨɦ ɟɦɤɨɫɬɢ ɭɦɟɧɶɲɚɟɬɫɹ ɟɦ ɤɨɫɬɧɨɟɫɨɩɪɨɬɢɜɥɟɧɢɟɢɭɦɟɧɶɲɚɟɬɫɹɦɨɞɭɥɶɪɚɡɧɨɫɬɢɦɟɠɞɭɢɧɞɭɤɬɢɜɧɵɦ ɫɨɩɪɨɬɢɜɥɟɧɢɟɦ ɢ ɟɦɤɨɫɬɧɵɦ ɪɟɚɤɬɢɜɧɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɭɦɟɧɶɲɚɟɬɫɹ ɫɥɟɞɨ ɜɚɬɟɥɶɧɨ ɬɨɤ ɪɚɫɬɟɬ
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b |
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ɋ ɨɞɧɨɜɪɟɦɟɧɧɵɦ ɪɨɫɬɨɦ ɚɤɬɢɜɧɨɝɨ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɢ ɢɧɞɭɤɬɢɜɧɨɫɬɢ ɬɨɤ
ɭɦɟɧɶɲɚɟɬɫɹ ɧɨ ɷɬɨ ɭɦɟɧɶɲɟɧɢɟ ɧɟɡɧɚɱɢɬɟɥɶɧɨ ɬɚɤ ɤɚɤ ɫ ɪɨɫɬɨɦ ɢɧɞɭɤɬɢɜɧɨ
ɫɬɢ ɜɨɡɧɢɤɚɟɬ ɪɨɫɬ ɬɨɤɚ ɜ ɪɟɡɭɥɶɬɚɬɟ ɭɦɟɧɶɲɟɧɢɹ ɪɟɚɤɬɢɜɧɨɝɨ ɫɨɩɪɨɬɢɜɥɟɧɢɹ
ɱɬɨ ɨɫɥɚɛɥɹɟɬ ɭɦɟɧɶɲɟɧɢɟ ɬɨɤɚ ɡɚ ɫɱɟɬ ɪɨɫɬɚ ɚɤɬɢɜɧɨɝɨ ɫɨɩɪɨɬɢɜɥɟɧɢɹ
b |
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ɋ ɨɞɧɨɜɪɟɦɟɧɧɵɦ ɪɨɫɬɨɦ ɚɤɬɢɜɧɨɝɨ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɢ ɟɦɤɨɫɬɢ ɬɨɤ ɭɦɟɧɶ
ɲɚɟɬɫɹ ɧɨ ɷɬɨ ɭɦɟɧɶɲɟɧɢɟ ɦɟɧɶɲɟ ɱɟɦ ɩɪɢ ɪɨɫɬɟ ɚɤɬɢɜɧɨɝɨ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɜ
ɨɬɞɟɥɶɧɨɫɬɢ Ɋɨɫɬ ɟɦɤɨɫɬɢ ɩɪɢɜɨɞɢɬ ɤ ɭɦɟɧɶɲɟɧɢɸ ɟɦɤɨɫɬɧɨɝɨ ɫɨɩɪɨɬɢɜɥɟɧɢɹ
ɢ ɤ ɭɦɟɧɶɲɟɧɢɸ ɦɨɞɭɥɹ ɪɚɡɧɨɫɬɢ ɦɟɠɞɭ ɢɧɞɭɤɬɢɜɧɵɦ ɢ ɟɦɤɨɫɬɧɵɦ ɫɨɩɪɨɬɢɜ
ɥɟɧɢɹɦɢ ɭɦɟɧɶɲɟɧɢɸ ɪɟɚɤɬɢɜɧɨɝɨ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɢ ɡɧɚɱɢɬ ɤ ɧɟɤɨɬɨɪɨɦɭ ɪɨ
ɫɬɭ ɬɨɤɚ ȿɦɤɨɫɬɶ ɨɫɥɚɛɥɹɟɬ ɜɥɢɹɧɢɟ ɚɤɬɢɜɧɨɝɨ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɧɚ ɬɨɤ
b |
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Ɍɨɤ ɭɜɟɥɢɱɢɜɚɟɬɫɹ ɧɟɡɧɚɱɢɬɟɥɶɧɨ ɉɪɢ ɨɞɧɨɜɪɟɦɟɧɧɨɦ ɭɜɟɥɢɱɟɧɢɢ ɢɧɞɭɤ
ɬɢɜɧɨɫɬɢ ɢ ɟɦɤɨɫɬɢ ɫ ɨɫɧɨɜɧɨɝɨ ɭɪɨɜɧɹ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɝɨ ɧɭɥɸ ɜ ɛɟɡɪɚɡɦɟɪ ɧɨɣɲɤɚɥɟɧɚɦɚɤɫɢɦɚɥɶɧɵɣɪɚɡɧɨɫɬɶɩɨɦɨɞɭɥɸ xL xC ɩɪɢɜɵɛɪɚɧɧɵɯ
ɢɧɬɟɪɜɚɥɚɯɜɚɪɶɢɪɨɜɚɧɢɹɮɚɤɬɨɪɨɜɬɚɤɠɟɭɦɟɧɶɲɚɟɬɫɹɧɨɭɦɟɧɶɲɚɟɬɫɹɧɟ ɡɧɚɱɢɬɟɥɶɧɨ ɱɬɨ ɩɪɢɜɨɞɢɬ ɤ ɧɟɡɧɚɱɢɬɟɥɶɧɨɦɭ ɪɨɫɬɭ ɬɨɤɚ
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b |
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ɉɪɢ ɨɞɧɨɜɪɟɦɟɧɧɨɦ ɭɜɟɥɢɱɟɧɢɢ ɚɤɬɢɜɧɨɝɨ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɢɧɞɭɤɬɢɜɧɨɫɬɢ
ɢ ɟɦɤɨɫɬɢ ɫ ɨɫɧɨɜɧɨɝɨ ɭɪɨɜɧɹ ɧɚ ɦɚɤɫɢɦɚɥɶɧɵɣ ɩɨɥɧɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɭɜɟɥɢ
ɱɢɜɚɟɬɫɹ ɱɬɨ ɩɪɢɜɨɞɢɬ ɤ ɭɦɟɧɶɲɟɧɢɸ ɬɨɤɚ
ɇɚɣɞɟɦ ɪɚɫɱɺɬɧɵɟ ɩɨ ɦɨɞɟɥɢ ɡɧɚɱɟɧɢɹ ɬɨɤɚ ɩɪɢ ɪɚɡɥɢɱɧɨɦ ɭɱɟɬɟ ɜɡɚɢɦɨ
ɞɟɣɫɬɜɢɣ ɮɚɤɬɨɪɨɜ
Ȼɟɡ ɭɱɟɬɚ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɮɚɤɬɨɪɨɜ
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yc |
b b x b x b x |
x x x |
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ɋ ɭɱɟɬɨɦ ɞɜɨɣɧɵɯ ɜɡɚɢɦɨɞɟɣɫɬɜɢɣ ɮɚɤɬɨɪɨɜ |
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ycc b |
b x b x b x b x x b x x b x x |
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yc ɯ ɯ ɯ ɯ ɯ ɯ |
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ɋ ɭɱɟɬɨɦ ɞɜɨɣɧɵɯ ɢ ɬɪɨɣɧɨɝɨ ɜɡɚɢɦɨɞɟɣɫɬɜɢɣ ɮɚɤɬɨɪɨɜ |
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yccc |
b b x |
b x b x b x x b x x b x x b x x x |
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ycc |
ɯ ɯ ɯ |
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Ɋɚɫɫɱɢɬɚɧɧɵɟ ɩɨ ɦɨɞɟɥɢ ɡɧɚɱɟɧɢɹ ɬɨɤɚ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɭɫɥɨɜɢɹɦ ɩɪɨ ɜɟɞɟɧɢɹ ɤɚɠɞɨɝɨ ɨɩɵɬɚ ɞɥɹ ɬɪɺɯ ɪɚɡɥɢɱɧɵɯ ɜɚɪɢɚɧɬɨɜɭɱɟɬɚ ɜɡɚɢɦɨɞɟɣɫɬɜɢɣ ɮɚɤɬɨɪɨɜ ɩɪɢɜɟɞɟɧɵ ɜ ɬɚɛɥɢɰɟ
Ɍ ɚ ɛ ɥ ɢ ɰ ɚ
Ɋɚɫɫɱɢɬɚɧɧɵɟ ɩɨ ɦɨɞɟɥɢ ɡɧɚɱɟɧɢɹ ɬɨɤɚ ɜ ɤɚɠɞɨɦ ɨɩɵɬɟ
ɩɪɢ ɪɚɡɥɢɱɧɵɯ ɜɚɪɢɚɧɬɚɯ ɭɱɟɬɚ ɜɡɚɢɦɨɞɟɣɫɬɜɢɣ ɮɚɤɬɨɪɨɜ
ɢ ɢɯ ɫɪɚɜɧɟɧɢɟ ɫ ɢɡɦɟɪɟɧɧɵɦɢ ɡɧɚɱɟɧɢɹɦɢ
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Ɋɚɫɫɱɢɬɚɧɧɵɟ ɩɨ ɦɨɞɟɥɢ ɡɧɚɱɟɧɢɹ ɬɨɤɚ |
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ɂɡɦɟɪɟɧɧɵɟ |
Ȼɟɡ ɭɱɟɬɚ |
ɋ ɭɱɟɬɨɦ |
ɋ ɭɱɟɬɨɦ |
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ɇɨɦɟɪ |
ɞɜɨɣɧɵɯ |
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ɡɧɚɱɟɧɢɹ ɬɨɤɚ |
ɞɜɨɣɧɵɯ |
ɞɜɨɣɧɵɯ |
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ɨɩɵɬɚ |
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ɢ ɬɪɨɣɧɵɯ |
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y>$@ |
ɜɡɚɢɦɨɞɟɣɫɬɜɢɣ |
ɜɡɚɢɦɨɞɟɣɫɬɜɢɣ |
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ɜɡɚɢɦɨɞɟɣɫɬɜɢɣ |
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cc |
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y >$@ |
y >$@ |
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ɂɡ ɬɚɛɥɢɰɵ ɜɢɞɧɨ ɱɬɨ ɱɢɫɬɨ ɥɢɧɟɣɧɚɹ ɦɨɞɟɥɶ ɞɚɟɬ ɡɚɦɟɬɧɵɟ ɪɚɫɯɨɠ
ɞɟɧɢɹ ɫ ɢɡɦɟɪɟɧɧɵɦɢ ɡɧɚɱɟɧɢɹɦɢ Ɇɚɤɫɢɦɚɥɶɧɨɟ ɪɚɫɯɨɠɞɟɧɢɟ ɡɞɟɫɶ ɱɬɨ
ɫɨɫɬɚɜɥɹɟɬ ɨɬ ɢɡɦɟɪɟɧɧɨɝɨ ɨɩɵɬ ɦɢɧɢɦɚɥɶɧɨɟ ɪɚɫɯɨɠɞɟɧɢɟ
ɫɨɫɬɚɜɥɹɟɬ ɨɬ ɢɡɦɟɪɟɧɧɨɝɨ ɡɧɚɱɟɧɢɹ ɨɩɵɬ ɋɪɟɞɧɟɟ ɪɚɫɯɨɠɞɟɧɢɟ ɜɨ
ɜɫɟɯ ɜɨɫɶɦɢ ɨɩɵɬɚɯ ɍɱɟɬ ɞɜɨɣɧɵɯ ɜɡɚɢɦɨɞɟɣɫɬɜɢɣ ɡɚɦɟɬɧɨ ɭɦɟɧɶɲɚɟɬ ɪɚɫɯɨɠɞɟɧɢɹ ɫ ɢɡɦɟ
ɪɟɧɧɵɦɢ ɡɧɚɱɟɧɢɹɦɢ ɤɨɬɨɪɵɟ ɫɨɫɬɚɜɥɹɸɬ ɨɬ ɢɥɢ ɨɩɵɬ ɞɨ ɢɥɢ ɨɩɵɬ Ɂɧɚɱɟɧɢɹ ɪɚɫɫɱɢɬɚɧɧɵɟ ɢ ɫ ɭɱɟɬɨɦ ɬɪɨɣɧɨɝɨ ɜɡɚɢ ɦɨɞɟɣɫɬɜɢɹ ɞɚɸɬɩɪɚɤɬɢɱɟɫɤɢ ɩɨɥɧɨɟɫɨɜɩɚɞɟɧɢɟ ɫ ɪɚɫɱɟɬɧɵɦɢɡɧɚɱɟɧɢɹɦɢ Ɇɚɤɫɢɦɚɥɶɧɨɟɪɚɫɯɨɠɞɟɧɢɟɜɷɬɨɦɫɥɭɱɚɟ ɱɬɨɫɨɫɬɚɜɥɹɟɬɧɚɢɛɨɥɶɲɢɣ ɩɪɨɰɟɧɬ ɜɨ ɜɬɨɪɨɦ ɨɩɵɬɟ
ɉɪɢɦɟɪ ɩɨɤɚɡɵɜɚɟɬ ɱɬɨ ɢ ɜ ɫɥɭɱɚɟ ɧɟɥɢɧɟɣɧɨɝɨ ɩɪɨɰɟɫɫɚ ɧɟ ɜɫɟɝɞɚ ɟɫɬɶ
ɧɟɨɛɯɨɞɢɦɨɫɬɶ ɩɟɪɟɯɨɞɢɬɶ ɤ ɩɥɚɧɚɦ ɝɨ ɩɨɪɹɞɤɚ
Ⱦɪɨɛɧɵɣ ɮɚɤɬɨɪɧɵɣ ɷɤɫɩɟɪɢɦɟɧɬQ N
ɋ ɪɨɫɬɨɦ ɤɨɥɢɱɟɫɬɜɚ ɮɚɤɬɨɪɨɜ n ɪɟɡɤɨ ɪɚɫɬɟɬ ɱɢɫɥɨ ɨɩɵɬɨɜ N ɜ ɩɥɚɧɚɯ ɩɨɥɧɨɝɨ ɮɚɤɬɨɪɧɨɝɨ ɷɤɫɩɟɪɢɦɟɧɬɚ Ɋɟɚɥɢɡɚɰɢɹ ɩɥɚɧɨɜ ɉɎɗ ɫɬɚɧɨɜɢɬɫɹ ɫɥɨɠ ɧɨɣ ɢɥɢ ɩɪɚɤɬɢɱɟɫɤɢ ɧɟɜɨɡɦɨɠɧɨɣ
ɋɪɨɫɬɨɦ n ɪɚɫɬɟɬɢɱɢɫɥɨɜɡɚɢɦɨɞɟɣɫɬɜɢɣɮɚɤɬɨɪɨɜ±ɱɢɫɥɨɞɜɨɣɧɵɯ ɬɪɨɣɧɵɯ ɢ ɬ ɞ ɜɡɚɢɦɨɞɟɣɫɬɜɢɣ ȼɡɚɢɦɨɞɟɣɫɬɜɢɹ ɜɵɫɨɤɢɯ ɩɨɪɹɞɤɨɜ ɤɚɤ ɩɪɚɜɢ ɥɨ ɦɚɥɨ ɜɥɢɹɸɬ ɧɚ ɮɭɧɤɰɢɸ ɨɬɤɥɢɤɚ ɢ ɢɦɢ ɩɪɟɧɟɛɪɟɝɚɸɬ Ɉɛɵɱɧɨ ɩɪɢ n ! ɜ ɦɨɞɟɥɶ ɜɤɥɸɱɚɸɬ ɧɟ ɜɫɟ ɚ ɬɨɥɶɤɨ ɧɟɤɨɬɨɪɵɟ ɞɜɨɣɧɵɟ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɜɡɚɢ ɦɨɞɟɣɫɬɜɢɹ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ ɜɵɛɢɪɚɟɦɵɟ ɩɨ ɤɚɤɢɦ ɬɨ ɫɨɨɛɪɚɠɟɧɢɹɦ ɢɫɯɨɞɹ ɢɡɡɧɚɧɢɣɩɪɨɛɥɟɦɧɨɣɨɛɥɚɫɬɢȼɫɟɦɢɬɪɨɣɧɵɦɢɜɡɚɢɦɨɞɟɣɫɬɜɢɹɦɢɜɡɚɢɦɨ ɞɟɣɫɬɜɢɹɦɢ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɢ ɬɟɦ ɛɨɥɟɟ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹɦɢ ɛɨɥɟɟ ɜɵɫɨɤɢɯ ɩɨ ɪɹɞɤɨɜ ɩɪɚɤɬɢɱɟɫɤɢ ɜɫɟɝɞɚ ɩɪɟɧɟɛɪɟɝɚɸɬ
ɉɨɷɬɨɦɭ ɤɨɥɢɱɟɫɬɜɨ ɥɢɲɧɢɯ ɨɩɵɬɨɜ ɫɬɚɧɨɜɢɬɫɹ ɱɪɟɡɦɟɪɧɵɦ Ʉɨɧɟɱɧɨ ɥɢɲɧɢɟ ɨɩɵɬɵ ɩɨɥɟɡɧɵ ɞɥɹ ɩɪɨɜɟɪɤɢ ɚɞɟɤɜɚɬɧɨɫɬɢ ɦɨɞɟɥɢ
ɇɨ ɫɥɢɲɤɨɦ ɛɨɥɶɲɨɟ ɢɯ ɤɨɥɢɱɟɫɬɜɨ ɧɟ ɨɩɪɚɜɞɵɜɚɟɬ ɬɪɭɞɧɨɫɬɢ ɪɟɚɥɢɡɚɰɢɢ ɂɡ ɬɚɛɥɢɰɵ ɜɢɞɧɨ ɱɬɨ ɫ ɪɨɫɬɨɦ ɱɢɫɥɚ ɮɚɤɬɨɪɨɜn ɤɨɥɢɱɟɫɬɜɨ ɨɩɵɬɨɜ
ɞɥɹ ɩɨɫɬɪɨɟɧɢɹ ɥɢɧɟɣɧɨɣ ɦɨɞɟɥɢ ɚ ɬɚɤɠɟ ɦɨɞɟɥɢ ɭɱɢɬɵɜɚɸɳɟɣ ɜɫɟ ɞɜɨɣɧɵɟ
ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɮɚɤɬɨɪɨɜ ɢ ɞɚɠɟ ɭɱɢɬɵɜɚɸɳɟɣ ɜɫɟ ɬɪɨɣɧɵɟ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ
ɫɬɚɧɨɜɢɬɫɹ ɱɪɟɡɦɟɪɧɵɦ
ɇɚɩɪɢɦɟɪ ɞɥɹ ɉɎɗ ɫ n ɧɟɨɛɯɨɞɢɦɨ ɩɪɨɜɟɫɬɢ ɢɡɦɟɪɟɧɢɹ Ⱦɥɹ ɥɢ ɧɟɣɧɨɣ ɦɨɞɟɥɢ ɧɟɨɛɯɨɞɢɦɨ ɜɫɟɝɨ ɢɡɦɟɪɟɧɢɣ Ⱦɥɹ ɩɨɫɬɪɨɟɧɢɹ ɦɨɞɟɥɢ ɜɤɥɸ ɱɚɸɳɟɣ ɜɫɟ ɞɜɨɣɧɵɟ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɞɨɫɬɚɬɨɱɧɨ ɢɡɦɟɪɟɧɢɣ ɂ ɞɚɠɟ ɟɫɥɢ ɜɤɥɸɱɢɬɶ ɜ ɦɨɞɟɥɶ ɢ ɜɫɟ ɬɪɨɣɧɵɟ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɬɨ ɧɟɨɛɯɨɞɢɦɨ ɟɳɟ ɢɡɦɟ ɪɟɧɢɣ ɢɬɨɝɨ ɜɫɟɝɨ ɢɡɦɟɪɟɧɢɹ ɱɬɨ ɬɚɤɠɟ ɡɧɚɱɢɬɟɥɶɧɨ ɦɟɧɶɲɟ ɢɡɦɟɪɟɧɢɣ ɜ ɉɎɗ
Ɍ ɚ ɛ ɥ ɢ ɰ ɚ
Ɋɨɫɬ ɱɢɫɥɚ ɨɩɵɬɨɜ ɉɎɗ ɫ ɭɜɟɥɢɱɟɧɢɟɦ ɤɨɥɢɱɟɫɬɜɚ ɮɚɤɬɨɪɨɜ
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ɉɨɷɬɨɦɭ ɞɥɹ ɫɨɤɪɚɳɟɧɢɹ ɱɢɫɥɚ ɨɩɵɬɨɜ ɩɟɪɟɯɨɞɹɬ ɤ ɞɪɨɛɧɨɦɭ ɮɚɤɬɨɪɧɨɦɭ ɷɤɫɩɟɪɢɦɟɧɬɭ ȾɎɗ ɨɛɨɡɧɚɱɚɟɦɨɦɭ n k ɜɤɥɸɱɚɸɳɟɦɭ ɬɨɥɶɤɨ ɱɚɫɬɶ ɩɨɥɧɨ ɝɨ ɮɚɤɬɨɪɧɨɝɨ ɷɤɫɩɟɪɢɦɟɧɬɚ q ɞɥɹ q ɨɫɧɨɜɧɵɯ ɮɚɤɬɨɪɨɜ ɝɞɟ q n k
ɢ k ɞɨɩɨɥɧɢɬɟɥɶɧɵɯ ɫɬɨɥɛɰɨɜ ɪɚɡɦɟɪɧɨɫɬɢ q ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɨɫɬɚɥɶɧɵɦ k ɮɚɤɬɨɪɚɦ
ɉɥɚɧ ȾɎɗ ɫɨɞɟɪɠɢɬ ɦɟɧɶɲɟɟ ɱɢɫɥɨ ɨɩɵɬɨɜ ɢ ɫɬɪɨɢɬɫɹ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚ ɡɨɦ ɂɡ ɜɫɟɯ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɯn ɮɚɤɬɨɪɨɜ ɜɵɛɢɪɚɸɬq n k ɨɫɧɨɜɧɵɯ ɢ ɞɥɹ
ɧɢɯ ɫɬɪɨɢɬɫɹ ɩɥɚɧ ɉɎɗ q ɗɬɨɬ ɩɥɚɧ ɞɨɩɨɥɧɹɟɬɫɹk ɫɬɨɥɛɰɚɦɢ ɫɨɨɬɜɟɬɫɬɜɭ ɸɳɢɦɢɨɫɬɚɜɲɢɦɫɹ k ɮɚɤɬɨɪɚɦɑɢɫɥɨɡɧɚɱɟɧɢɣɜɷɬɢɯɫɬɨɥɛɰɚɯɪɚɜɧɨq Ɂɧɚɱɟɧɢɹ ɷɬɢɯ ɫɬɨɥɛɰɨɜ ɜɵɛɢɪɚɸɬɫɹ ɧɟ ɩɪɨɢɡɜɨɥɶɧɨ ɚ ɬɚɤɢɦ ɨɛɪɚɡɨɦ ɱɬɨɛɵ ɫɨɯɪɚɧɹɥɚɫɶ ɨɪɬɨɝɨɧɚɥɶɧɨɫɬɶ ɩɥɚɧɚ ɗɬɨ ɞɨɫɬɢɝɚɟɬɫɹ ɫ ɩɨɦɨɳɶɸ ɬɚɤ ɧɚɡɵɜɚɟ ɦɨɝɨ ɝɟɧɟɪɢɪɭɸɳɟɝɨ ɫɨɨɬɧɨɲɟɧɢɹ ɤɨɬɨɪɨɟ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɩɪɨɢɡɜɟɞɟɧɢɟ ɧɟ ɦɟɧɟɟ ɞɜɭɯ ɢ ɧɟ ɛɨɥɟɟn k q ɨɫɧɨɜɧɵɯ ɮɚɤɬɨɪɨɜ Ɂɧɚɱɟɧɢɹ ɞɨɩɨɥɧɢɬɟɥɶ
ɧɵɯɫɬɨɥɛɰɨɜɩɨɥɭɱɚɸɬɫɹɤɚɤɪɟɡɭɥɶɬɚɬɩɨɷɥɟɦɟɧɬɧɨɝɨɭɦɧɨɠɟɧɢɹɡɧɚɱɟɧɢɣ ɨɫɧɨɜɧɵɯ ɮɚɤɬɨɪɨɜ
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɨɛɨɡɧɚɱɟɧɢɟ ȾɎɗ n k ɨɡɧɚɱɚɟɬ ɱɬɨ ɪɚɫɫɦɚɬɪɢɜɚɟɬɫɹ n ɮɚɤɬɨɪɨɜ ɞɥɹ ɜɵɛɪɚɧɧɵɯn k q ɨɫɧɨɜɧɵɯ ɮɚɤɬɨɪɨɜ ɫɬɪɨɢɬɫɹ ɩɥɚɧ ɉɎɗ q ɮɚɤɬɨɪɵ ɜɚɪɶɢɪɭɸɬɫɹ ɧɚ ɞɜɭɯ ɭɪɨɜɧɹɯ ɚk ɨɡɧɚɱɚɟɬ ɤɨɥɢɱɟɫɬɜɨ ɞɨɩɨɥɧɢɬɟɥɶ ɧɵɯ ɫɬɨɥɛɰɨɜ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɨɫɬɚɜɲɢɦɫɹ ɮɚɤɬɨɪɚɦ ɑɢɫɥɨ ɷɥɟɦɟɧɬɨɜ ɷɬɢɯ ɞɨɩɨɥɧɢɬɟɥɶɧɵɯ ɫɬɨɥɛɰɨɜ ɪɚɜɧɨ n k q
ɉɪɢ k ɩɥɚɧ ȾɎɗ n ɫɨɤɪɚɳɚɟɬɫɹ ɜɞɜɨɟ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ ɩɥɚɧɨɦ ɉɎɗn ɢ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɦɚɬɪɢɰɭ
D ȾɎɗ n DɉɎɗ n B n
ɝɞɟDɉɎɗ n ɦɚɬɪɢɰɚ ɩɥɚɧɚ ɉɎɗn ɞɥɹn ɨɫɧɨɜɧɵɯ ɮɚɤɬɨɪɨɜ
B n ɜɟɤɬɨɪ ɫɬɨɥɛɟɰ ɪɚɡɦɟɪɧɨɫɬɢ n ɩɨɫɬɪɨɟɧɧɵɣ ɩɨ ɜɵɛɪɚɧɧɨɦɭ ɝɟ
ɧɟɪɢɪɭɸɳɟɦɭ ɫɨɨɬɧɨɲɟɧɢɸ ɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɣ ɞɨɩɨɥɧɢɬɟɥɶɧɨɦɭ ɮɚɤɬɨɪɭ ɧɟ
ɜɨɲɟɞɲɟɦɭ ɜ ɱɢɫɥɨ ɨɫɧɨɜɧɵɯ
ɉɪɢk ɩɥɚɧ ȾɎɗ n ɫɨɤɪɚɳɚɟɬɫɹ ɜ ɱɟɬɵɪɟ ɪɚɡɚ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ ɩɥɚ
ɧɨɦ ɉɎɗn ɢ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɦɚɬɪɢɰɭ
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ɝɞɟDɉɎɗ n ɦɚɬɪɢɰɚ ɩɥɚɧɚ ɉɎɗ n ɞɥɹn ɨɫɧɨɜɧɵɯ ɮɚɤɬɨɪɨɜ
B n ɢ B n ɞɜɚ ɜɟɤɬɨɪ ɫɬɨɥɛɰɚ ɪɚɡɦɟɪɧɨɫɬɢ n ɩɨɫɬɪɨɟɧɧɵɟ ɩɨ ɜɵ
ɛɪɚɧɧɵɦ ɝɟɧɟɪɢɪɭɸɳɢɦ ɫɨɨɬɧɨɲɟɧɢɹɦ ɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɞɜɭɦ ɞɨɩɨɥɧɢɬɟɥɶ
ɧɵɦ ɮɚɤɬɨɪɚɦ ɧɟ ɜɨɲɟɞɲɢɦ ɜ ɱɢɫɥɨ ɨɫɧɨɜɧɵɯ
ȼ ɨɛɳɟɦ ɫɥɭɱɚɟ ɩɥɚɧ ȾɎɗn k ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɦɚɬɪɢɰɭ
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ɝɞɟD |
ɉɎɗ |
n k ɦɚɬɪɢɰɚ ɩɥɚɧɚ ɉɎɗn k |
ɞɥɹn k ɨɫɧɨɜɧɵɯ ɮɚɤɬɨɪɨɜ |
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n k ɜɟɤɬɨɪ ɫɬɨɥɛɰɵ ɪɚɡɦɟɪɧɨɫɬɢ n k ɩɨɫɬɪɨɟɧɧɵɟ ɩɨ |
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ɜɵɛɪɚɧɧɵɦ k ɝɟɧɟɪɢɪɭɸɳɢɦ ɫɨɨɬɧɨɲɟɧɢɹɦ ɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ k ɞɨɩɨɥɧɢ ɬɟɥɶɧɵɦ ɮɚɤɬɨɪɚɦ ɧɟ ɜɨɲɟɞɲɢɦ ɜ ɱɢɫɥɨ ɨɫɧɨɜɧɵɯ
ɉɭɫɬɶ ɧɟɨɛɯɨɞɢɦɨ ɩɨɫɬɪɨɢɬɶ ɦɨɞɟɥɶ ɞɥɹ ɬɪɟɯ ɮɚɤɬɨɪɨɜ ɜ ɜɢɞɟ ɩɨɥɢɧɨɦɚ
y b b x b x b x b x x b x x b x x b x x x
ɉɨɫɬɪɨɢɦ ɩɥɚɧ ȾɎɗ ɬɚɛɥɢɰɚ
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Ⱦɥɹ ɮɚɤɬɨɪɨɜ x ɢ x ɡɧɚɱɟɧɢɹ ɡɚɞɚɸɬɫɹ ɤɚɤ ɢ ɞɥɹ ɉɎɗ ɚ ɡɧɚɱɟɧɢɹ ɞɥɹ ɮɚɤɬɨɪɚx ɨɩɪɟɞɟɥɹɸɬɫɹ ɩɨ ɝɟɧɟɪɢɪɭɸɳɟɦɭ ɫɨɨɬɧɨɲɟɧɢɸx x x
ȼ ɩɥɚɧɟ ɜɨɫɟɦɶ ɫɬɨɥɛɰɨɜ ɢ ɩɪɢ ɷɬɨɦ ɱɟɬɵɪɟ ɩɚɪɵ ɫɬɨɥɛɰɨɜ ɨɞɢɧɚɤɨɜɵɟ
ɉɨɫɥɟɞɧɢɣ ɫɬɨɥɛɟɰ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɣ ɬɪɨɣɧɨɦɭ ɫɨɱɟɬɚɧɢɸ ɮɚɤɬɨɪɨɜ ɧɚɪɭɲɚ
ɟɬ ɭɫɥɨɜɢɟ ɨɪɬɨɝɨɧɚɥɶɧɨɫɬɢ ɩɥɚɧɚ ɬɚɤ ɤɚɤ ɫɭɦɦɚ ɟɝɨ ɷɥɟɦɟɧɬɨɜ ɤɚɤ ɢ ɭ ɫɬɨɥɛ ɰɚx ɧɟ ɪɚɜɧɚ ɧɭɥɸ
ȼ ɨɪɬɨɝɨɧɚɥɶɧɨɦ ɠɟ ɩɥɚɧɟ ɬɨɥɶɤɨ ɨɞɢɧ ɫɬɨɥɛɟɰ ɦɨɠɟɬ ɢɦɟɬɶ ɫɭɦɦɭ ɷɥɟ
ɦɟɧɬɨɜ ɧɟ ɪɚɜɧɭɸ ɧɭɥɸ ɉɨɷɬɨɦɭ ɩɨɫɥɟɞɧɢɣ ɫɬɨɥɛɟɰ ɢɫɤɥɸɱɚɟɬɫɹ ɢɡ ɪɚɫɫɦɨɬ
ɪɟɧɢɹ ɢ ɤɨɷɮɮɢɰɢɟɧɬɵ ɦɨɞɟɥɢ ɧɚɯɨɞɹɬɫɹ ɩɨ ɩɪɨɫɬɵɦ ɮɨɪɦɭɥɚɦ ɤɚɤ ɢ ɞɥɹ ɩɥɚ
ɧɚ ɉɎɗ Ⱦɨɩɭɫɤɚɹ ɱɬɨ ɬɪɨɣɧɨɟ ɫɨɱɟɬɚɧɢɟ ɮɚɤɬɨɪɨɜ ɩɪɟɧɟɛɪɟɠɢɦɨ ɦɚɥɨ ɦɨɠɧɨ
ɨɩɪɟɞɟɥɢɬɶ ɤɨɷɮɮɢɰɢɟɧɬ
N
¦x u yu b u N
ɝɞɟu ɧɨɦɟɪ ɨɩɵɬɚN ɤɨɥɢɱɟɫɬɜɨ ɨɩɵɬɨɜ ɢɥɢ ɫɬɪɨɤ ɩɥɚɧɚ ȾɎɗ
Ɉɫɬɚɥɶɧɵɟ ɤɨɷɮɮɢɰɢɟɧɬɵ ɨɩɪɟɞɟɥɹɟɦɵɟ ɩɨ ɚɧɚɥɨɝɢɱɧɵɦ ɫɨɨɬɧɨɲɟɧɢɹɦ ɢɡ ɡɚ ɬɨɝɨ ɱɬɨ ɜ ɩɥɚɧɟ ɟɫɬɶ ɨɞɢɧɚɤɨɜɵɟ ɫɬɨɥɛɰɵ x ɢx x ɢx x ɢx ɛɭɞɭɬ ɹɜɥɹɬɶɫɹ ɫɨɜɦɟɫɬɧɵɦɢ ɨɰɟɧɤɚɦɢ ɥɢɧɟɣɧɵɯ ɷɮɮɟɤɬɨɜ ɢ ɷɮɮɟɤɬɨɜ ɜɡɚɢɦɨ ɞɟɣɫɬɜɢɹb b b b b b Ɉɰɟɧɤɢ ɤɨɬɨɪɵɟ ɧɟɥɶɡɹ ɪɚɡɞɟɥɢɬɶ ɧɚɡɵ ɜɚɸɬɫɹ ɫɦɟɲɚɧɧɵɦɢ ɗɬɢ ɫɦɟɲɚɧɧɵɟ ɨɰɟɧɤɢ ɟɫɬɶ ɪɟɡɭɥɶɬɚɬ ɬɨɝɨ ɱɬɨ ɜ ɩɥɚɧɟ ȾɎɗ ɦɚɥɨ ɨɩɵɬɨɜ ɬɨɥɶɤɨ ɱɟɬɵɪɟ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɜɫɟɯ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɩɨɥɢɧɨɦɚ