Планирование эксперимента. Учебное пособие-1
.pdfɑɟɦ ɨɩɪɟɞɟɥɹɟɬɫɹ ɜɵɛɨɪ ɫɬɪɭɤɬɭɪɵ ɦɨɞɟɥɢ"
Ɂɚɱɟɦ ɧɟɨɛɯɨɞɢɦ ɩɟɪɟɯɨɞ ɤ ɤɨɞɢɪɨɜɚɧɧɵɦ ɡɧɚɱɟɧɢɹɦ ɮɚɤɬɨɪɨɜ ɢ ɤɚɤ ɨɧ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ"Ʉɚɤɨɫɭɳɟɫɬɜɥɹɟɬɫɹɨɛɪɚɬɧɵɣɩɟɪɟɯɨɞɤɧɚɬɭɪɚɥɶɧɵɦ ɡɧɚɱɟɧɢɹɦ ɮɚɤɬɨɪɨɜ"
Ʉɚɤ ɨɩɪɟɞɟɥɹɟɬɫɹ ɡɧɚɱɢɦɨɫɬɶ ɮɚɤɬɨɪɨɜ"
ɑɬɨ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɪɚɫɲɢɪɟɧɧɚɹ ɦɚɬɪɢɰɚ ɩɥɚɧɚ" Ʉɚɤɚɹ ɱɚɫɬɶ ɪɚɫɲɢ ɪɟɧɧɨɣ ɦɚɬɪɢɰɵ ɩɥɚɧɚ ɹɜɥɹɟɬɫɹ ɫɨɛɫɬɜɟɧɧɨ ɦɚɬɪɢɰɟɣ ɩɥɚɧɚ"
Ɉɬ ɱɟɝɨ ɡɚɜɢɫɢɬ ɫɥɨɠɧɨɫɬɶ ɨɩɪɟɞɟɥɟɧɢɹ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɦɨɞɟɥɢ b b b "
ȼ ɱɟɦ ɡɚɤɥɸɱɚɟɬɫɹ ɨɫɧɨɜɧɚɹ ɢɞɟɹ ɨɪɬɨɝɨɧɚɥɶɧɨɝɨ ɩɥɚɧɢɪɨɜɚɧɢɹ ɷɤɫɩɟɪɢ ɦɟɧɬɚ"
ɋɮɨɪɦɭɥɢɪɭɣɬɟ ɭɫɥɨɜɢɟ ɨɪɬɨɝɨɧɚɥɶɧɨɫɬɢ ɦɚɬɪɢɰɵ
Ʉɚɤɚɹ ɦɚɬɪɢɰɚ ɧɚɡɵɜɚɟɬɫɹ ɞɢɚɝɨɧɚɥɶɧɨɣ"
ɉɪɢɜɟɫɬɢ ɜ ɨɛɳɟɦ ɜɢɞɟ ɦɚɬɪɢɱɧɭɸ ɮɨɪɦɭ ɡɚɩɢɫɢ ɪɟɡɭɥɶɬɚɬɨɜ ɪɚɡɥɢɱɧɵɯ ɨɩɵɬɨɜ
ɉɪɢɜɟɫɬɢ ɜ ɦɚɬɪɢɱɧɨɣ ɮɨɪɦɟ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɤɨɷɮɮɢɰɢɟɧ ɬɨɜ ɦɨɞɟɥɢ
Ɂɚɩɢɫɚɬɶ ɜ ɨɛɳɟɦ ɜɢɞɟ ɩɨɥɧɵɣ ɤɜɚɞɪɚɬɢɱɧɵɣ ɩɨɥɢɧɨɦ ɫ ɭɱɟɬɨɦ ɜɫɟɯ ɜɨɡɦɨɠɧɵɯ ɫɨɱɟɬɚɧɢɣ ɬɪɟɯ ɮɚɤɬɨɪɨɜ
Ʉɚɤ ɜɵɱɢɫɥɹɸɬɫɹ ɤɨɷɮɮɢɰɢɟɧɬɵ ɦɨɞɟɥɢ b b bm ɜ ɫɥɭɱɚɟ ɨɪɬɨɝɨ
ɧɚɥɶɧɨɝɨ ɩɥɚɧɢɪɨɜɚɧɢɹ ɷɤɫɩɟɪɢɦɟɧɬɚ"
ȼ ɱɟɦ ɨɫɧɨɜɧɚɹ ɢɞɟɹ ɦɟɬɨɞɚ ɧɚɢɦɟɧɶɲɢɯ ɤɜɚɞɪɚɬɨɜ ɞɥɹ ɨɰɟɧɤɢ ɤɨɷɮɮɢ ɰɢɟɧɬɨɜ ɪɟɝɪɟɫɫɢɨɧɧɨɣ ɦɨɞɟɥɢ"
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Ʉɚɤɨɣ ɫɦɵɫɥ ɢɦɟɟɬ ɮɭɧɤɰɢɨɧɚɥ |
a |
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S ¦ yu yu |
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ȽɅȺȼȺ
ɉɈɅɇɕȿ ɂ ȾɊɈȻɇɕȿ
ɎȺɄɌɈɊɇɕȿ ɗɄɋɉȿɊɂɆȿɇɌɕ
ɉɨɥɧɵɟ ɮɚɤɬɨɪɧɵɟ ɷɤɫɩɟɪɢɦɟɧɬɵ ɬɢɩɚQ
ɉɥɚɧɵ ɩɨɥɧɵɯ ɮɚɤɬɨɪɧɵɯ ɷɤɫɩɟɪɢɦɟɧɬɨɜ ɬɢɩɚ n ɉɎɗ n ɹɜɥɹɸɬɫɹ ɧɚɢɛɨɥɟɟ ɩɪɨɫɬɵɦɢ ɩɥɚɧɚɦɢ ɩɨɡɜɨɥɹɸɳɢɦɢ ɩɨɥɭɱɢɬɶ ɦɨɞɟɥɶ ɜ ɜɢɞɟ ɩɨɥɢɧɨɦɚ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ ɢ ɨɩɪɟɞɟɥɢɬɶ ɫɜɨɛɨɞɧɵɣ ɱɥɟɧ b n ɤɨɷɮɮɢɰɢɟɧɬɨɜ bi Cn ɤɨɷɮɮɢɰɢɟɧɬɨɜ bij ɩɪɢ ɪɚɡɥɢɱɧɵɯ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹɯ ɞɜɭɯ ɮɚɤɬɨɪɨɜ Cn ɤɨɷɮ ɮɢɰɢɟɧɬɨɜbijk ɩɪɢ ɪɚɡɥɢɱɧɵɯ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹɯ ɬɪɟɯ ɮɚɤɬɨɪɨɜ ɢ ɨɞɢɧ ɤɨ ɷɮɮɢɰɢɟɧɬb n Cnn ɩɪɢ ɜɡɚɢɦɨɞɟɣɫɬɜɢɢ ɜɫɟɯn ɮɚɤɬɨɪɨɜ
Ɉɛɨɡɧɚɱɟɧɢɟ n ɨɡɧɚɱɚɟɬ ɱɬɨ n ɮɚɤɬɨɪɨɜ ɜɚɪɶɢɪɭɸɬɫɹ ɧɚ ɞɜɭɯ ɭɪɨɜɧɹɯ ɗɬɢ ɭɪɨɜɧɢ ɮɚɤɬɨɪɨɜ ɛɟɡɪɚɡɦɟɪɧɵɟ ɢ ɫɨɨɬɜɟɬɫɬɜɭɸɬ ɦɢɧɢɦɚɥɶɧɨɦɭ ɢ ɦɚɤɫɢ ɦɚɥɶɧɨɦɭ ɡɧɚɱɟɧɢɹɦ ɢ ɑɢɫɥɨ ɜɫɟɜɨɡɦɨɠɧɵɯ ɫɨɱɟɬɚɧɢɣ ɡɧɚɱɟɧɢɣn ɮɚɤ ɬɨɪɨɜ ɚ ɡɧɚɱɢɬ ɢ ɱɢɫɥɨ ɨɩɵɬɨɜ ɢ ɱɢɫɥɨ ɫɬɪɨɤ ɩɥɚɧɚN n
ɉɭɫɬɶ ɩɨɥɢɧɨɦ ɩɪɟɞɫɬɚɜɥɹɟɬɫɹ ɜ ɜɢɞɟ
y b b x b x b x x b x b x
ɢ ɫɨɞɟɪɠɢɬm ɱɥɟɧɨɜ ɋɨɫɬɚɜɢɦ ɬɚɛɥɢɰɭ ɫɬɨɥɛɰɵ ɤɨɬɨɪɨɣ ɫɨɨɬɜɟɬɫɬɜɭ ɸɬ ɡɧɚɱɟɧɢɹɦ ɮɚɤɬɨɪɨɜ ɢɯ ɫɨɱɟɬɚɧɢɹɦ ɢ ɤɜɚɞɪɚɬɚɦ ɚ ɫɬɪɨɤɢ ɧɨɦɟɪɚɦ ɨɩɵ
ɬɨɜu |
Ⱦɥɹ ɞɜɭɯ ɮɚɤɬɨɪɨɜ ɱɢɫɥɨ ɨɩɵɬɨɜ ɉɨɫɥɟɞɧɢɣ ɫɬɨɥɛɟɰ ± |
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ɪɟɡɭɥɶɬɚɬɵ ɢɡɦɟɪɟɧɢɣ y |
y y y ɬɚɛɥɢɰɚ |
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Ɍ ɚ ɛ ɥ ɢ ɰ ɚ |
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ɉɥɚɧ ɉɎɗ |
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Ɉɤɨɧɱɚɧɢɟ ɬɚɛɥɢɰɵ
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¦xiu |
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ȼ ɩɟɪɜɵɣ ɫɬɨɥɛɟɰ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɣ x ɡɚɧɨɫɹɬɫɹ ɟɞɢɧɢɰɵ ɬɚɤ ɤɚɤ x ɮɢɤɬɢɜɧɵɣ ɮɚɤɬɨɪ ± ɜɨ ɜɫɟɯ ɢɡɦɟɪɟɧɢɹɯ ɩɪɢɧɢɦɚɟɬ ɡɧɚɱɟɧɢɟ ȼɨ ɜɬɨɪɨɣ ɫɬɨɥɛɟɰ ɡɚɧɨɫɹɬɫɹ ɟɞɢɧɢɰɵ ɫ ɱɟɪɟɞɭɸɳɢɦɢɫɹ ɡɧɚɤɚɦɢ ɧɚɱɢɧɚɹ ɫ Ɍɪɟɬɢɣ ɫɬɨɥɛɟɰ ɡɚɩɨɥɧɹɟɬɫɹ ɟɞɢɧɢɰɚɦɢ ɫ ɱɟɪɟɞɭɸɳɢɦɢɫɹ ɱɟɪɟɡ ɞɜɚ ɡɧɚɤɚɦɢ ɬɚɤɠɟ ɧɚɱɢɧɚɹ ɫ ɗɥɟɦɟɧɬɵ ɫɬɨɥɛɰɨɜ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɩɪɨɢɡɜɟɞɟɧɢɹɦ ɮɚɤɬɨɪɨɜ ɡɚɩɨɥɧɹɸɬɫɹɟɞɢɧɢɰɚɦɢɫɨɡɧɚɤɚɦɢɤɨɬɨɪɵɟɩɨɥɭɱɚɸɬɫɹɩɪɢɩɟɪɟɦɧɨɠɟɧɢɢ ɷɥɟɦɟɧɬɨɜ ɩɪɟɞɵɞɭɳɢɯ ɫɬɨɥɛɰɨɜ ɉɨɫɥɟɞɧɢɟ ɞɜɚ ɫɬɨɥɛɰɚ ɫɨɫɬɨɹɬ ɢɡ ɟɞɢɧɢɰ ɫɨ ɡɧɚɤɨɦ ɩɥɸɫ ɬɚɤ ɤɚɤ ɫɨɨɬɜɟɬɫɬɜɭɸɬ ɤɜɚɞɪɚɬɚɦ ɮɚɤɬɨɪɨɜ
ɋɨɛɫɬɜɟɧɧɨ ɩɥɚɧ ɷɤɫɩɟɪɢɦɟɧɬɚ ɨɛɪɚɡɭɸɬ ɞɜɚ ɫɬɨɥɛɰɚ ɜɵɞɟɥɟɧɧɵɟ ɭɬɨɥ
ɳɟɧɧɵɦɢ ɥɢɧɢɹɦɢ Ɉɫɬɚɥɶɧɵɟ ɫɬɨɥɛɰɵ ɧɨɫɹɬ ɜɫɩɨɦɨɝɚɬɟɥɶɧɵɣ ɯɚɪɚɤɬɟɪ
ɉɨɫɥɟɞɧɢɟ ɞɜɟ ɫɬɪɨɤɢ ɫɨɨɬɜɟɬɫɬɜɭɸɬ ɫɭɦɦɚɦ ɷɥɟɦɟɧɬɨɜ ɫɬɨɥɛɰɨɜ ɢ ɫɭɦ
ɦɚɦ ɤɜɚɞɪɚɬɨɜ ɷɥɟɦɟɧɬɨɜ ɫɬɨɥɛɰɨɜ ɢ ɩɪɢɜɟɞɟɧɵ ɜ ɞɨɩɨɥɧɟɧɢɟ ɤ ɬɚɛɥɢɰɟ Ɋɚɫ ɫɦɨɬɪɢɦ ɨɫɨɛɟɧɧɨɫɬɢ ɩɥɚɧɚ ɉɎɗ ɤɨɬɨɪɵɟ ɦɨɠɧɨ ɨɛɨɛɳɢɬɶ ɧɚ ɩɥɚɧɵ n ɫ ɥɸɛɵɦ ɱɢɫɥɨɦ ɮɚɤɬɨɪɨɜn
Ɋɚɡɥɢɱɧɵɯ ɫɬɨɥɛɰɨɜ ɜ ɬɚɛɥɢɰɟ ɥɢɲɶ ɱɟɬɵɪɟ ɋɬɨɥɛɰɵ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɤɜɚɞɪɚɬɚɦ ɮɚɤɬɨɪɨɜ ɫɨɜɩɚɞɚɸɬ ɫɨ ɫɬɨɥɛɰɨɦx ɗɬɨ ɨɛɳɢɣ ɪɟɡɭɥɶɬɚɬ ɞɥɹ ɜɫɟɯ
ɩɥɚɧɨɜ n Ɉɞɢɧɚɤɨɜɵɟ ɫɬɨɥɛɰɵ ɝɨɜɨɪɹɬ ɨ ɬɨɦ ɱɬɨ ɧɟɥɶɡɹ ɩɨɥɭɱɢɬɶ ɫ ɩɨɦɨ ɳɶɸ ɩɥɚɧɚ ɉɎɗ n ɪɚɡɞɟɥɶɧɵɟ ɨɰɟɧɤɢ ɤɨɷɮɮɢɰɢɟɧɬɚ ɦɨɞɟɥɢb ɢ ɤɨɷɮɮɢɰɢ ɟɧɬɨɜ ɩɪɢ ɤɜɚɞɪɚɬɚɯ ɮɚɤɬɨɪɨɜb b b ɢ ɬ ɞ Ⱦɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɤɨɷɮɮɢɰɢ ɟɧɬɨɜ ɩɪɢ ɤɜɚɞɪɚɬɚɯ ɮɚɤɬɨɪɨɜ ɢɫɩɨɥɶɡɭɸɬɫɹ ɩɥɚɧɵ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ȼ ɞɚɥɶ ɧɟɣɲɟɦɜɩɥɚɧɚɯ n ɫɬɨɥɛɰɵɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟɤɜɚɞɪɚɬɚɦɮɚɤɬɨɪɨɜɩɪɢɜɨ ɞɢɬɶɫɹ ɧɟ ɛɭɞɭɬ
ɑɢɫɥɨ ɪɚɡɥɢɱɧɵɯ ɫɬɨɥɛɰɨɜ ɪɚɜɧɹɟɬɫɹ ɱɢɫɥɭ ɮɚɤɬɨɪɨɜ ɢ ɢɯ ɪɚɡɥɢɱɧɵɯ ɫɨɱɟɬɚɧɢɣ ɬɨ ɟɫɬɶ ɱɢɫɥɭ ɫɬɪɨɤ ɩɥɚɧɚ ɱɢɫɥɭ ɨɩɵɬɨɜ N ɗɬɨ ɬɚɤɠɟ ɨɛɳɢɣ ɪɟ ɡɭɥɶɬɚɬ ɩɥɚɧɨɜ n ɋɥɟɞɨɜɚɬɟɥɶɧɨ ɫ ɩɨɦɨɳɶɸ ɉɎɗ n ɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ ɜɫɟ
ɤɨɷɮɮɢɰɢɟɧɬɵ ɥɢɧɟɣɧɨɝɨ ɩɨɥɢɧɨɦɚb b bn ɢ ɤɨɷɮɮɢɰɢɟɧɬɵ ɫɨ ɜɫɟɜɨɡ ɦɨɠɧɵɦɢ ɫɨɱɟɬɚɧɢɹɦɢ ɮɚɤɬɨɪɨɜ ɜɤɥɸɱɚɹ ɤɨɷɮɮɢɰɢɟɧɬ ɫn ɫɨɱɟɬɚɧɢɟɦb n
ɂɡ ɞɨɩɨɥɧɢɬɟɥɶɧɵɯ ɞɜɭɯ ɩɨɫɥɟɞɧɢɯ ɫɬɪɨɤ ɬɚɛɥɢɰɵ ɜɢɞɧɨ ɱɬɨ ɫɭɦɦɵ
ɷɥɟɦɟɧɬɨɜ ɫɬɨɥɛɰɨɜ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɮɚɤɬɨɪɚɦ ɢ ɫɨɱɟɬɚɧɢɸ ɮɚɤɬɨɪɨɜ ɪɚɜɧɵ ɧɭɥɸ ɋɭɦɦɵ ɷɥɟɦɟɧɬɨɜ ɨɫɬɚɜɲɢɯɫɹ ɫɬɨɥɛɰɨɜ ɪɚɜɧɵ ɱɢɫɥɭ ɨɩɵɬɨɜ N ȼ ɞɚɧ ɧɨɦ ɫɥɭɱɚɟ N ɋɭɦɦɵ ɤɜɚɞɪɚɬɨɜ ɷɥɟɦɟɧɬɨɜ ɜɫɟɯ ɫɬɨɥɛɰɨɜ ɪɚɜɧɵ N ɗɬɨ ɬɚɤɠɟ ɨɛɳɢɣ ɜɵɜɨɞ ɞɥɹ ɜɫɟɯ ɩɥɚɧɨɜ n Ʉɚɠɞɵɣ ɮɚɤɬɨɪ ɮɢɤɫɢɪɭɟɬɫɹ ɧɚ ɞɜɭɯ ɭɪɨɜɧɹɯ ɫɢɦɦɟɬɪɢɱɧɨ ɪɚɫɩɨɥɨɠɟɧɧɵɯ ɨɬɧɨɫɢɬɟɥɶɧɨ ɰɟɧɬɪɚ ɩɥɚɧɚ ɋɥɟɞɨɜɚ ɬɟɥɶɧɨ ɩɥɚɧɵ ɩɨɥɧɨɝɨ ɮɚɤɬɨɪɧɨɝɨ ɷɤɫɩɟɪɢɦɟɧɬɚ n ɨɬɜɟɱɚɸɬ ɬɪɟɛɨɜɚɧɢɹɦ ɨɪ ɬɨɝɨɧɚɥɶɧɨɝɨ ɩɥɚɧɢɪɨɜɚɧɢɹ ɢ ɭɫɥɨɜɢɟ ɪɚɜɟɧɫɬɜɚ ɧɭɥɸ ɫɤɚɥɹɪɧɨɝɨ ɩɪɨɢɡɜɟɞɟ ɧɢɹ ɞɜɭɯ ɜɟɤɬɨɪɨɜ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɞɜɭɦ ɥɸɛɵɦ ɫɬɨɥɛɰɚɦ ɦɚɬɪɢɰɵ ;ɤɪɨ ɦɟ ɫɬɨɥɛɰɨɜ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɤɜɚɞɪɚɬɚɦ ɮɚɤɬɨɪɨɜ ɤɨɬɨɪɵɟ ɜ ɩɥɚɧɚɯ n ɧɟ ɪɚɫɫɦɚɬɪɢɜɚɸɬɫɹ ɜɵɩɨɥɧɹɟɬɫɹ
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ɉɨɷɬɨɦɭ ɞɥɹ ɩɥɚɧɨɜ ɉɎɗ n ɤɨɷɮɮɢɰɢɟɧɬɵ ɧɚɯɨɞɹɬɫɹ ɩɨ ɩɪɨɫɬɵɦ ɮɨɪ ɦɭɥɚɦ
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ɢ ɫ ɭɱɟɬɨɦ ɬɨɝɨ ɱɬɨ ɜ ɫɥɭɱɚɟ ɩɨɥɧɨɝɨ ɮɚɤɬɨɪɧɨɝɨ ɷɤɫɩɟɪɢɦɟɧɬɚn
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Ɇɨɞɟɥɢ ɫ ɧɚɬɭɪɚɥɶɧɵɦɢ ɡɧɚɱɟɧɢɹɦɢ ɮɚɤɬɨɪɨɜ
ȼ ɦɨɞɟɥɹɯ ɩɨɥɭɱɚɟɦɵɯ ɫ ɩɨɦɨɳɶɸ ɦɟɬɨɞɨɜ ɬɟɨɪɢɢ ɩɥɚɧɢɪɨɜɚɧɢɹ ɷɤɫɩɟ ɪɢɦɟɧɬɚɤɚɤɩɪɚɜɢɥɨɢɫɩɨɥɶɡɭɸɬɫɹɤɨɞɢɪɨɜɚɧɧɵɟɡɧɚɱɟɧɢɹɮɚɤɬɨɪɨɜɢɜɵ ɱɢɫɥɹɸɬɫɹ ɤɨɷɮɮɢɰɢɟɧɬɵ ɩɚɪɚɦɟɬɪɵ b b b bm ɦɨɞɟɥɢ ɉɪɢ ɪɚɛɨɬɟ ɫ ɩɨɥɭɱɟɧɧɨɣ ɦɨɞɟɥɶɸ ɧɚɬɭɪɚɥɶɧɵɟ ɡɧɚɱɟɧɢɹ ɮɚɤɬɨɪɨɜ ɧɟɨɛɯɨɞɢɦɨ ɩɟɪɟɫɱɢɬɵ ɜɚɬɶ ɜ ɛɟɡɪɚɡɦɟɪɧɵɟ
ȿɫɥɢ ɩɪɢ ɪɚɛɨɬɟ ɫ ɦɨɞɟɥɶɸ ɜ ɩɨɥɭɱɟɧɧɨɟ ɜɵɪɚɠɟɧɢɟ ɫ ɛɟɡɪɚɡɦɟɪɧɵɦɢ ɡɧɚ
ɱɟɧɢɹɦɢ ɮɚɤɬɨɪɨɜ ɩɨɞɫɬɚɜɥɹɬɶ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ ɢɯ ɧɚɬɭɪɚɥɶɧɵɟ ɡɧɚɱɟɧɢɹ ɬɨ ɧɚɣɞɟɧɧɵɟ ɤɨɷɮɮɢɰɢɟɧɬɵ b b b bm ɧɟɨɛɯɨɞɢɦɨ ɩɟɪɟɫɱɢɬɵɜɚɬɶ ɉɪɢ ɷɬɨɦ ɢɡɦɟɪɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɮɭɧɤɰɢɢ ɨɬɤɥɢɤɚ ɨɞɢɧɚɤɨɜɵɟ ɤɚɤ ɩɪɢ ɢɫɩɨɥɶɡɨɜɚ ɧɢɢ ɤɨɞɢɪɨɜɚɧɧɵɯ ɡɧɚɱɟɧɢɣ ɮɚɤɬɨɪɨɜ ɫɨ ɫɜɨɢɦɢ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ ɬɚɤ ɢ ɩɪɢ ɢɫɩɨɥɶɡɨɜɚɧɢɢ ɧɚɬɭɪɚɥɶɧɵɯ ɡɧɚɱɟɧɢɣ ɮɚɤɬɨɪɨɜ ɫ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɦɢ ɢɦ ɤɨɷɮ ɮɢɰɢɟɧɬɚɦɢ Ɏɨɪɦɭɥɵ ɞɥɹ ɩɟɪɟɫɱɟɬɚ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɨɩɪɟɞɟɥɹɸɬɫɹ ɢɡ ɫɥɟɞɭ ɸɳɢɯ ɪɚɫɫɭɠɞɟɧɢɣ
ȼ ɩɨɥɭɱɟɧɧɨɦ ɜɵɪɚɠɟɧɢɢ ɞɥɹ ɩɨɥɢɧɨɦɚ ɫ ɛɟɡɪɚɡɦɟɪɧɵɦɢ ɮɚɤɬɨɪɚɦɢ y b b x bm xm ɡɚɦɟɧɢɦ ɛɟɡɪɚɡɦɟɪɧɵɟ ɮɚɤɬɨɪɵ x x xm ɢɯ ɜɵɪɚ
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ɋɪɚɜɧɢɜɚɹ ɩɨɫɥɟɞɧɟɟ ɜɵɪɚɠɟɧɢɟ ɫ ɜɵɪɚɠɟɧɢɟɦ ɜ ɤɨɬɨɪɨɦ ɢɫɩɨɥɶɡɭɸɬɫɹ ɧɚɬɭɪɚɥɶɧɵɟ ɡɧɚɱɟɧɢɹ ɮɚɤɬɨɪɨɜ y B B X Bm X m ɧɚɣɞɟɦ ɤɨɷɮɮɢɰɢ
ɟɧɬɵB B B Bm
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Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɩɪɢ ɢɫɩɨɥɶɡɨɜɚɧɢɢ ɜ ɩɥɚɧɟ ɉɎɗ n ɧɚɬɭɪɚɥɶɧɵɯ ɡɧɚɱɟ ɧɢɣ ɮɚɤɬɨɪɨɜ Xi PD[ɢ Xi PLQ ɜɦɟɫɬɨ ɛɟɡɪɚɡɦɟɪɧɵɯ ɢ ɧɟɨɛɯɨɞɢɦɨ ɩɟɪɟ
ɫɱɢɬɵɜɚɬɶ ɜɫɟ ɤɨɷɮɮɢɰɢɟɧɬɵ ɜɤɥɸɱɚɹ ɤɨɷɮɮɢɰɢɟɧɬB
ɗɬɨ ɨɛɴɹɫɧɹɟɬɫɹ ɬɟɦ ɱɬɨ ɩɪɢ ɢɫɩɨɥɶɡɨɜɚɧɢɢ ɜ ɩɥɚɧɟ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ ɧɚɬɭɪɚɥɶɧɵɯ ɡɧɚɱɟɧɢɣ ɮɚɤɬɨɪɨɜ Xi PD[ɢ Xi PLQ ɩɥɚɧ ɬɟɪɹɟɬ ɫɜɨɣɫɬɜɨ ɨɪɬɨɝɨ
ɧɚɥɶɧɨɫɬɢ ɉɨɷɬɨɦɭ ɧɟɫɦɨɬɪɹ ɧɚ ɬɨ ɱɬɨ ɢɡɦɟɪɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɜ ɨɩɵɬɚɯ ɜ ɷɬɨɦ
ɫɥɭɱɚɟ ɛɭɞɭɬ ɬɟ ɠɟ ɱɬɨ ɢ ɩɪɢ ɢɫɩɨɥɶɡɨɜɚɧɢɢ ɤɨɞɢɪɨɜɚɧɧɵɯ ɮɚɤɬɨɪɨɜ ɤɨɷɮɮɢ
ɰɢɟɧɬɵ ɞɨɥɠɧɵ ɪɚɫɫɱɢɬɵɜɚɬɶɫɹ ɩɨ ɞɪɭɝɢɦ ɮɨɪɦɭɥɚɦ
ɋɜɨɣɫɬɜɚ ɩɥɚɧɨɜ ɩɨɥɧɨɝɨ ɮɚɤɬɨɪɧɨɝɨ ɷɤɫɩɟɪɢɦɟɧɬɚ
Ⱥɧɚɥɢɡ ɩɥɚɧɨɜ ɩɨɥɧɨɝɨ ɮɚɤɬɨɪɧɨɝɨ ɷɤɫɩɟɪɢɦɟɧɬɚ n ɢ ɜ ɱɚɫɬɧɨɫɬɢ ɦɚɬ ɪɢɰɵ;ɩɨɡɜɨɥɹɟɬ ɫɞɟɥɚɬɶ ɜɵɜɨɞ ɱɬɨ ɷɬɢ ɩɥɚɧɵ ɨɛɥɚɞɚɸɬ ɫɥɟɞɭɸɳɢɦɢ ɫɜɨɣ ɫɬɜɚɦɢ
Ɉɪɬɨɝɨɧɚɥɶɧɨɫɬɢ ± ɫɭɦɦɚ ɩɚɪɧɵɯ ɩɪɨɢɡɜɟɞɟɧɢɣ ɷɥɟɦɟɧɬɨɜ ɥɸɛɵɯ ɞɜɭɯ ɪɚɡɥɢɱɧɵɯ ɫɬɨɥɛɰɨɜ ɪɚɜɧɚ ɧɭɥɸ
ɋɢɦɦɟɬɪɢɱɧɨɫɬɢ ± ɫɭɦɦɚ ɜɫɟɯ ɷɥɟɦɟɧɬɨɜ ɥɸɛɨɝɨ ɫɬɨɥɛɰɚ ɡɚ ɢɫɤɥɸɱɟ ɧɢɟɦ ɩɟɪɜɨɝɨ ɪɚɜɧɚ ɧɭɥɸ
ɇɨɪɦɢɪɨɜɚɧɧɨɫɬɢ ± ɫɭɦɦɚ ɤɜɚɞɪɚɬɨɜ ɷɥɟɦɟɧɬɨɜ ɥɸɛɨɝɨ ɫɬɨɥɛɰɚ ɪɚɜɧɚ ɱɢɫɥɭ ɨɩɵɬɨɜ
ɇɟɡɚɜɢɫɢɦɨɫɬɢ ± ɜɫɟ ɫɬɨɥɛɰɵ ɦɚɬɪɢɰɵ ;ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ n ɮɚɤ ɬɨɪɭ ɚ ɬɚɤɠɟ ɞɜɨɣɧɵɦ ɬɪɨɣɧɵɦ ɢ ɬ ɞ ɫɨɱɟɬɚɧɢɹɦ ɮɚɤɬɨɪɨɜ ɪɚɡɥɢɱɧɵ
ɇɚɫɵɳɟɧɧɨɫɬɢ ± ɨɛɳɟɟ ɱɢɫɥɨ ɨɩɵɬɨɜ N ɪɚɜɧɨ ɱɢɫɥɭ ɧɟɢɡɜɟɫɬɧɵɯ ɤɨ ɷɮɮɢɰɢɟɧɬɨɜ ɩɨɥɢɧɨɦɚm
Ɋɨɬɚɬɚɛɟɥɶɧɨɫɬɢ ± ɜɫɟ ɬɨɱɤɢ ɩɥɚɧɚ ɥɟɠɚɬ ɧɚ ɨɤɪɭɠɧɨɫɬɢ ɫɮɟɪɟ ɝɢɩɟɪ ɫɮɟɪɟ ɢ ɬɚɤɢɦ ɨɛɪɚɡɨɦ ɞɢɫɩɟɪɫɢɹ ɨɬɤɥɢɤɚ ɨɞɢɧɚɤɨɜɚ ɧɚ ɨɞɧɨɦ ɪɚɫɫɬɨɹ ɧɢɢ ɨɬ ɰɟɧɬɪɚ ɩɥɚɧɚ ɜ ɥɸɛɨɦ ɧɚɩɪɚɜɥɟɧɢɢ ɮɚɤɬɨɪɧɨɝɨ ɩɪɨɫɬɪɚɧɫɬɜɚ
ɗɬɢ ɫɜɨɣɫɬɜɚ ɩɨɡɜɨɥɹɸɬ ɨɩɪɟɞɟɥɹɬɶ ɤɨɷɮɮɢɰɢɟɧɬɵ ɩɨɥɢɧɨɦɚ ɩɨ ɩɪɨɫɬɵɦ
ɫɨɨɬɧɨɲɟɧɢɹɦ ɢ ɨɛɟɫɩɟɱɢɜɚɸɬ ɧɟɡɚɜɢɫɢɦɨɫɬɶ ɨɰɟɧɨɤ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɦɨɞɟɥɢ
ɂɧɬɟɪɩɪɟɬɚɰɢɹ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɦɨɞɟɥɢ
ȼ ɩɨɥɧɵɯ ɮɚɤɬɨɪɧɵɯ ɷɤɫɩɟɪɢɦɟɧɬɚɯ ɢɫɩɨɥɶɡɭɸɬɫɹ ɮɚɤɬɨɪɵ ɜ ɛɟɡɪɚɡɦɟɪ
ɧɨɦ ɤɨɞɢɪɨɜɚɧɧɨɦ ɜɢɞɟ ɤɚɠɞɵɣ ɢɡ ɧɢɯ ɜɚɪɶɢɪɭɟɬɫɹ ɜ ɨɞɢɧɚɤɨɜɵɯ ɩɪɟɞɟɥɚɯ
ɨɬ ± ɞɨ ɉɨɷɬɨɦɭ ɧɟɫɦɨɬɪɹ ɧɚ ɬɨ ɱɬɨ ɦɨɞɟɥɶ ɜ ɜɢɞɟ ɩɨɥɢɧɨɦɚ ɧɟ ɢɦɟɟɬ ɧɢ ɤɚɤɨɝɨ ɮɢɡɢɱɟɫɤɨɝɨ ɫɦɵɫɥɚ ɩɨɹɜɥɹɟɬɫɹ ɜɨɡɦɨɠɧɨɫɬɶ ɫɨɩɨɫɬɚɜɥɹɬɶ ɡɧɚɱɟɧɢɟ
ɨɞɧɨɝɨ ɤɨɷɮɮɢɰɢɟɧɬɚ ɫɨ ɡɧɚɱɟɧɢɟɦ ɞɪɭɝɨɝɨ ɢ ɨɩɪɟɞɟɥɹɬɶ ɩɨ ɜɟɥɢɱɢɧɟ ɤɨɷɮɮɢ
ɰɢɟɧɬɨɜ ɜɥɢɹɧɢɹ ɮɚɤɬɨɪɨɜ ɢ ɢɯ ɫɨɱɟɬɚɧɢɣ ɧɚ ɮɭɧɤɰɢɸ ɨɬɤɥɢɤɚ ȼɨɡɦɨɠɧɚ ɫɥɟɞɭɸɳɚɹ ɢɧɬɟɪɩɪɟɬɚɰɢɹ ɧɚɣɞɟɧɧɵɯ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɩɨɥɢ
ɧɨɦɚ
b ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɡɧɚɱɟɧɢɸ ɮɭɧɤɰɢɢ ɨɬɤɥɢɤɚ ɜ ɰɟɧɬɪɟ ɩɥɚɧɚ
bi ɪɚɜɟɧ ɩɪɢɪɚɳɟɧɢɸ ɮɭɧɤɰɢɢ ɩɪɢ ɩɟɪɟɯɨɞɟ ɡɧɚɱɟɧɢɹi ɝɨ ɮɚɤɬɨɪɚ ɫ ɧɭ ɥɟɜɨɝɨ ɭɪɨɜɧɹ ɧɚ ɜɟɪɯɧɢɣ ɷɬɨ ɜɤɥɚɞ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɝɨ ɮɚɤɬɨɪɚ ɜ ɡɧɚɱɟɧɢɟ ɮɭɧɤɰɢɢ
bij ɪɚɜɟɧ ɧɟɥɢɧɟɣɧɨɣ ɱɚɫɬɢ ɩɪɢɪɚɳɟɧɢɹ ɮɭɧɤɰɢɢ ɩɪɢ ɨɞɧɨɜɪɟɦɟɧɧɨɦ ɩɟ ɪɟɯɨɞɟ ɮɚɤɬɨɪɨɜi ɢ j ɫ ɧɭɥɟɜɨɝɨ ɭɪɨɜɧɹ ɧɚ ɜɟɪɯɧɢɣ ɷɬɨ ɜɤɥɚɞ ɩɚɪɧɨɝɨ
ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɮɚɤɬɨɪɨɜ ɢɥɢ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ ɜ ɡɧɚɱɟɧɢɟ
ɮɭɧɤɰɢɢ
bijk ɪɚɜɟɧ ɧɟɥɢɧɟɣɧɨɣ ɱɚɫɬɢ ɩɪɢɪɚɳɟɧɢɹ ɮɭɧɤɰɢɢ ɩɪɢ ɨɞɧɨɜɪɟɦɟɧɧɨɦ ɩɟ ɪɟɯɨɞɟ ɮɚɤɬɨɪɨɜi j ɢk ɫ ɧɭɥɟɜɨɝɨ ɭɪɨɜɧɹ ɧɚ ɜɟɪɯɧɢɣ ɷɬɨ ɜɤɥɚɞ ɬɪɨɣ
ɧɨɝɨ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɮɚɤɬɨɪɨɜ ɢɥɢ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ ɜ ɡɧɚ
ɱɟɧɢɟ ɮɭɧɤɰɢɢ ɢ ɬ ɞ
Ʉɨɷɮɮɢɰɢɟɧɬɵ ɦɨɞɟɥɢ ɱɚɫɬɨ ɧɚɡɵɜɚɸɬ ɷɮɮɟɤɬɚɦɢbi ± ɥɢɧɟɣɧɵɟ ɷɮɮɟɤ ɬɵ bij ± ɷɮɮɟɤɬɵ ɞɜɨɣɧɵɯ ɜɡɚɢɦɨɞɟɣɫɬɜɢɣ bijk ± ɷɮɮɟɤɬɵ ɬɪɨɣɧɵɯ ɜɡɚɢɦɨ
ɞɟɣɫɬɜɢɣ ɢ ɬ ɞ ȼɟɥɢɱɢɧɚ ɤɨɷɮɮɢɰɢɟɧɬɚ ɩɨɡɜɨɥɹɟɬ ɤɨɥɢɱɟɫɬɜɟɧɧɨ ɨɰɟɧɢɬɶ ɷɮ
ɮɟɤɬ ɜɥɢɹɧɢɹ ɤɚɠɞɨɝɨ ɮɚɤɬɨɪɚ ɧɚ ɜɟɥɢɱɢɧɭ ɨɬɤɥɢɤɚ ɑɟɦ ɛɨɥɶɲɟ ɤɨɷɮɮɢɰɢɟɧɬ
ɬɟɦ ɫɢɥɶɧɟɟ ɜɥɢɹɟɬ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɣ ɮɚɤɬɨɪ ɢɥɢ ɜɡɚɢɦɨɞɟɣɫɬɜɢɟ ɮɚɤɬɨɪɨɜ ɧɚ
ɜɟɥɢɱɢɧɭ ɨɬɤɥɢɤɚ Ɂɧɚɤɢɤɨɷɮɮɢɰɢɟɧɬɨɜɨɩɪɟɞɟɥɹɸɬɯɚɪɚɤɬɟɪɷɬɨɝɨɜɥɢɹɧɢɹɉɨɥɨɠɢɬɟɥɶ
ɧɵɣ ɡɧɚɤ ɤɨɷɮɮɢɰɢɟɧɬɚbi ɝɨɜɨɪɢɬ ɨ ɬɨɦ ɱɬɨ ɫ ɭɜɟɥɢɱɟɧɢɟɦ ɡɧɚɱɟɧɢɹ ɮɚɤɬɨɪɚ ɪɚɫɬɟɬ ɡɧɚɱɟɧɢɟ ɨɬɤɥɢɤɚ ɨɬɪɢɰɚɬɟɥɶɧɵɣ ɡɧɚɤ ɧɚɨɛɨɪɨɬ ± ɭɛɵɜɚɟɬ Ɍɚɤɢɦ ɨɛɪɚ ɡɨɦ ɩɨɹɜɥɹɟɬɫɹ ɜɨɡɦɨɠɧɨɫɬɶ ɪɚɫɩɨɥɨɠɢɬɶ ɮɚɤɬɨɪɵ ɜ ɪɹɞ ɩɨ ɫɢɥɟ ɢɯ ɜɥɢɹɧɢɹ ɧɚ ɨɬɤɥɢɤ
ȿɫɥɢ ɭ ɷɤɫɩɟɪɢɦɟɧɬɚɬɨɪɚ ɢɦɟɸɬɫɹ ɚɩɪɢɨɪɧɵɟ ɫɜɟɞɟɧɢɹ ɨ ɯɚɪɚɤɬɟɪɟ ɞɟɣ
ɫɬɜɢɹ ɮɚɤɬɨɪɨɜ ɬɨ ɩɨɥɟɡɧɨ ɩɪɨɜɟɫɬɢ ɚɧɚɥɢɡ ɢ ɩɪɨɜɟɪɢɬɶ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɥɢ ɩɨ
ɥɭɱɟɧɧɚɹ ɦɨɞɟɥɶ ɷɬɢɦ ɫɜɟɞɟɧɢɹɦ
ɇɚɩɪɢɦɟɪ ɟɫɥɢ ɢɡɜɟɫɬɧɨ ɱɬɨ ɫ ɪɨɫɬɨɦ ɬɟɦɩɟɪɚɬɭɪɵ ɞɨɥɠɧɨ ɩɪɨɢɫɯɨɞɢɬɶ ɭɜɟɥɢɱɟɧɢɟ ɨɬɤɥɢɤɚ y ɚ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɣ ɤɨɷɮɮɢɰɢɟɧɬ ɪɟɝɪɟɫɫɢɢ ɢɦɟɟɬ ɡɧɚɤ
ɦɢɧɭɫ ɬɨ ɩɨɹɜɥɹɟɬɫɹ ɩɪɨɬɢɜɨɪɟɱɢɟ ɗɬɨ ɡɧɚɱɢɬ ɱɬɨ ɥɢɛɨ ɜ ɷɤɫɩɟɪɢɦɟɧɬɟ
ɞɨɩɭɳɟɧɚ ɨɲɢɛɤɚ ɢ ɨɧ ɞɨɥɠɟɧ ɛɵɬɶ ɩɪɨɜɟɪɟɧ ɥɢɛɨ ɧɟɜɟɪɧɵ ɚɩɪɢɨɪɧɵɟ ɩɪɟɞ
ɫɬɚɜɥɟɧɢɹ ɉɪɢ ɷɬɨɦ ɫɥɟɞɭɟɬ ɢɦɟɬɶ ɜ ɜɢɞɭ ɱɬɨ ɬɟɨɪɟɬɢɱɟɫɤɢɟ ɩɪɟɞɫɬɚɜɥɟɧɢɹ ɢɦɟɸɬ ɤɚɤ ɩɪɚɜɢɥɨ ɨɛɳɢɣ ɯɚɪɚɤɬɟɪ ɚ ɤɨɷɮɮɢɰɢɟɧɬbi ɨɬɪɚɠɚɟɬ ɜɥɢɹɧɢɟ ɮɚɤ ɬɨɪɚ ɬɨɥɶɤɨ ɜ ɜɵɛɪɚɧɧɨɣ ɨɛɥɚɫɬɢ ɮɚɤɬɨɪɧɨɝɨ ɩɪɨɫɬɪɚɧɫɬɜɚ ɂ ɧɟɢɡɜɟɫɬɧɨ ɦɨɠ ɧɨ ɥɢ ɪɚɫɩɪɨɫɬɪɚɧɢɬɶ ɪɟɡɭɥɶɬɚɬ ɧɚ ɞɪɭɝɢɟ ɨɛɥɚɫɬɢ Ʉɪɨɦɟ ɬɨɝɨ ɚɩɪɢɨɪɧɚɹ ɢɧ ɮɨɪɦɚɰɢɹ ɱɚɫɬɨ ɨɫɧɨɜɵɜɚɟɬɫɹ ɧɚ ɨɞɧɨɮɚɤɬɨɪɧɵɯ ɡɚɜɢɫɢɦɨɫɬɹɯ ɚ ɩɪɢ ɩɟɪɟɯɨɞɟ ɤ ɦɧɨɝɨɮɚɤɬɨɪɧɨɦɭ ɩɪɨɫɬɪɚɧɫɬɜɭ ɫɢɬɭɚɰɢɹ ɦɨɠɟɬ ɢɡɦɟɧɢɬɶɫɹ
ȿɫɥɢ ɟɫɬɶ ɭɜɟɪɟɧɧɨɫɬɶ ɜ ɬɨɦ ɷɤɫɩɟɪɢɦɟɧɬ ɩɪɨɜɟɞɟɧ ɤɨɪɪɟɤɬɧɨ ɬɨɝɞɚ ɞɥɹ
ɩɪɟɨɞɨɥɟɧɢɹ ɩɪɨɬɢɜɨɪɟɱɢɹ ɦɨɠɧɨ ɜɵɞɜɢɝɚɬɶ ɪɚɡɥɢɱɧɵɟ ɩɪɟɞɩɨɥɨɠɟɧɢɹ ɢ ɩɪɨ
ɜɟɪɹɬɶ ɢɯ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɨ ɉɨɥɭɱɟɧɧɚɹɦɚɬɟɦɚɬɢɱɟɫɤɚɹɦɨɞɟɥɶɜɜɢɞɟɩɨɥɢɧɨɦɚɩɨɡɜɨɥɹɟɬɜɵɹɜɢɬɶ
ɦɟɯɚɧɢɡɦɞɟɣɫɬɜɢɹɮɚɤɬɨɪɨɜɇɚɩɪɢɦɟɪɩɨɪɟɡɭɥɶɬɚɬɚɦɉɎɗɩɨɫɬɪɨɟɧɚɦɨ ɞɟɥɶ
y b b x b x b x x
ȿɫɥɢ ɡɧɚɤɢ ɤɨɷɮɮɢɰɢɟɧɬɨɜ b b b ɨɞɢɧɚɤɨɜɵɟ ɬɨ ɦɨɠɧɨ ɫɱɢɬɚɬɶ ɱɬɨ ɨɛɚ ɮɚɤɬɨɪɚ ɩɪɢ ɢɯ ɭɜɟɥɢɱɟɧɢɢ ɩɪɢɜɨɞɹɬ ɤ ɭɜɟɥɢɱɟɧɢɸ ɨɬɤɥɢɤɚ ɚ ɩɪɢ ɭɦɟɧɶ ɲɟɧɢɢ ɤ ɭɦɟɧɶɲɟɧɢɸ ɉɪɢɱɟɦ ɨɞɢɧ ɮɚɤɬɨɪ ɜɥɢɹɟɬ ɬɟɦ ɫɢɥɶɧɟɟ ɱɟɦ ɛɨɥɶɲɟ ɡɧɚɱɟɧɢɟ ɞɪɭɝɨɝɨ ɮɚɤɬɨɪɚ Ɏɚɤɬɨɪɵ ɩɪɢ ɫɨɜɦɟɫɬɧɨɦ ɜɨɡɞɟɣɫɬɜɢɢ ɜɥɢɹɸɬ ɧɚ ɨɬ ɤɥɢɤ ɫɢɥɶɧɟɟ ɱɟɦ ɩɪɢ ɪɚɡɞɟɥɶɧɨɦ ɜɨɡɞɟɣɫɬɜɢɢ
ȿɫɥɢ ɡɧɚɤɢ ɤɨɷɮɮɢɰɢɟɧɬɨɜb ɢb ɨɞɢɧɚɤɨɜɵɟ ɚ ɡɧɚɤb ɩɪɨɬɢɜɨɩɨɥɨɠ ɧɵɣ ɬɨ ɬɚɤɠɟ ɨɛɚ ɮɚɤɬɨɪɚ ɩɪɢ ɢɯ ɭɜɟɥɢɱɟɧɢɢ ɩɪɢɜɨɞɹɬ ɤ ɭɜɟɥɢɱɟɧɢɸ ɨɬɤɥɢɤɚ ɚ ɩɪɢ ɭɦɟɧɶɲɟɧɢɢ ɤ ɭɦɟɧɶɲɟɧɢɸ ɧɨ ɜɥɢɹɧɢɟ ɨɞɧɨɝɨ ɮɚɤɬɨɪɚ ɨɫɥɚɛɟɜɚɟɬ ɫ ɪɨɫɬɨɦ ɞɪɭɝɨɝɨ Ɏɚɤɬɨɪɵ ɜ ɨɬɞɟɥɶɧɨɫɬɢ ɜɥɢɹɸɬ ɫɢɥɶɧɟɟ ɱɟɦ ɩɪɢ ɨɞɧɨɜɪɟɦɟɧ ɧɨɦ ɜɨɡɞɟɣɫɬɜɢɢ
Ʌɢɧɟɣɧɵɟ ɷɮɮɟɤɬɵ ɬɚɤɠɟ ɦɨɝɭɬ ɢɦɟɬɶ ɪɚɡɥɢɱɧɵɟ ɡɧɚɤɢ ɇɚɩɪɢɦɟɪ ɜɨɡ
ɦɨɠɧɵ ɫɢɬɭɚɰɢɢ
y b b x b x b x x
y b b x b x b x x
ȼ ɷɬɢɯɫɥɭɱɚɹɯ ɧɟɥɶɡɹ ɨɞɧɨɡɧɚɱɧɨ ɫɤɚɡɚɬɶ ɨɛɭɫɢɥɟɧɢɢ ɢɥɢ ɨɫɥɚɛɥɟɧɢɢ ɜɨɡɞɟɣɫɬɜɢɹ ɮɚɤɬɨɪɨɜ Ɇɨɠɧɨ ɥɢɲɶ ɩɪɟɞɩɨɥɨɠɢɬɶ ɱɬɨ ɜ ɩɟɪɜɨɦ ɫɥɭɱɚɟɩɪɢ ɜɡɚɢɦɧɨɦ ɜɥɢɹɧɢɢ ɧɚ ɨɬɤɥɢɤ ɮɚɤɬɨɪ x ɨɤɚɡɵɜɚɟɬ ɛɨɥɶɲɟɟ ɜɥɢɹɧɢɟ ɱɟɦ x ɬɚɤ ɤɚɤ ɡɧɚɤɢ b ɢb ɫɨɜɩɚɞɚɸɬ ȼɨ ɜɬɨɪɨɦ ɫɥɭɱɚɟ ɧɚɨɛɨɪɨɬ ɩɪɢ ɜɡɚɢɦɧɨɦ ɜɥɢɹɧɢɢ ɛɨɥɶɲɟɟ ɜɨɡɞɟɣɫɬɜɢɟ ɧɚ ɩɪɨɰɟɫɫ ɨɤɚɡɵɜɚɟɬ x ɬɚɤ ɤɚɤ ɡɧɚɤɢb ɢb ɫɨɜɩɚɞɚɸɬ
ɉɪɨɫɬɟɣɲɢɟ ɩɪɢɦɟɪɵ ± ɜɥɢɹɧɢɟ ɥɟɤɚɪɫɬɜɟɧɧɵɯ ɩɪɟɩɚɪɚɬɨɜ ɧɚ ɫɨɫɬɨɹɧɢɟ
ɛɨɥɶɧɨɝɨ ɜɥɢɹɧɢɟ ɭɞɨɛɪɟɧɢɣ ɧɚ ɪɨɫɬ ɪɚɫɬɟɧɢɣ ɩɪɢ ɢɯ ɪɚɡɞɟɥɶɧɨɦ ɢ ɫɨɜɦɟɫɬ
ɧɨɦ ɩɪɢɦɟɧɟɧɢɢ
ɉɨ ɪɟɡɭɥɶɬɚɬɚɦ ɧɚɛɥɸɞɟɧɢɣ ɡɚ ɫɨɫɬɨɹɧɢɟɦ ɛɨɥɶɧɨɝɨ ɦɨɠɧɨ ɩɨɫɬɪɨɢɬɶ ɦɨ
ɞɟɥɶ ɢ ɩɨ ɟɟ ɜɢɞɭ ɨɩɪɟɞɟɥɢɬɶ ɤɚɤɨɣ ɢɡ ɩɪɟɩɚɪɚɬɨɜ ɨɤɚɡɵɜɚɟɬ ɛɨɥɶɲɢɣ ɷɮɮɟɤɬ ɢ
ɛɵɫɬɪɟɟ ɩɪɢɜɨɞɢɬ ɤ ɜɵɡɞɨɪɨɜɥɟɧɢɸ Ʉɪɨɦɟ ɬɨɝɨ ɦɨɞɟɥɶ ɩɨɤɚɠɟɬ ɤɚɤ ɞɟɣɫɬɜɭ
ɸɬ ɩɪɟɩɚɪɚɬɵ ɩɪɢ ɢɯ ɫɨɜɦɟɫɬɧɨɦ ɩɪɢɦɟɧɟɧɢɢ ɬɨ ɟɫɬɶ ɰɟɥɟɫɨɨɛɪɚɡɧɨ ɥɢ ɫɨɜ
ɦɟɫɬɧɨɟ ɩɪɢɦɟɧɟɧɢɟ ɥɟɤɚɪɫɬɜ ɢɥɢ ɥɟɤɚɪɫɬɜɚ ɩɥɨɯɨ ɫɨɜɦɟɫɬɢɦɵ
ɉɪɢ ɩɪɢɧɹɬɢɢ ɪɟɲɟɧɢɹ ɩɨ ɪɟɡɭɥɶɬɚɬɚɦ ɦɨɞɟɥɢɪɨɜɚɧɢɹ ɧɟɨɛɯɨɞɢɦɨ ɭɱɢɬɵ
ɜɚɬɶ ɤɨɧɟɱɧɨ ɢ ɱɢɫɥɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɤɨɷɮɮɢɰɢɟɧɬɨɜ
Ɋɚɫɫɦɨɬɪɢɦ ɩɪɢɦɟɪ ɉɨ ɪɟɡɭɥɶɬɚɬɚɦ ɢɡɦɟɪɟɧɢɣ ɩɨɫɬɪɨɢɬɶ ɦɨɞɟɥɶ ɜ ɜɢɞɟ
ɡɚɜɢɫɢɦɨɫɬɢ ɚɤɬɢɜɧɨɣ ɦɨɳɧɨɫɬɢ ɨɬ ɧɚɩɪɹɠɟɧɢɹ ɢ ɬɨɤɚ ɜ ɷɥɟɤɬɪɢɱɟɫɤɨɣ ɰɟɩɢ
Pf U I ȼɵɛɟɪɟɦ ɞɢɚɩɚɡɨɧɵ ɜɚɪɶɢɪɨɜɚɧɢɹ ɮɚɤɬɨɪɨɜ
ɇɚɩɪɹɠɟɧɢɟU ȼ ɞɟɣɫɬɜɭɸɳɟɟ ɡɧɚɱɟɧɢɟ ɧɚɩɪɹɠɟɧɢɹ ɌɨɤI Ⱥ ɞɟɣɫɬɜɭɸɳɟɟ ɡɧɚɱɟɧɢɟ ɬɨɤɚ
ɉɨɫɬɪɨɢɦ ɩɥɚɧ ɩɨɥɧɨɝɨ ɮɚɤɬɨɪɧɨɝɨ ɷɤɫɩɟɪɢɦɟɧɬɚ ɞɥɹ ɞɜɭɯ ɮɚɤɬɨɪɨɜ Ȼɟɡ ɪɚɡɦɟɪɧɵɟ ɩɪɢɜɟɞɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɧɚɩɪɹɠɟɧɢɹ ɢ ɬɨɤɚ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ x ɢx
ɉɨɧɹɬɧɨ ɱɬɨ ɢɡɦɟɪɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɨɬɤɥɢɤɚ y ɦɨɳɧɨɫɬɢ ɞɨɥɠɧɵ ɩɨɥɧɨɫɬɶɸ
ɫɨɜɩɚɞɚɬɶ ɫ ɪɚɫɫɱɢɬɚɧɧɵɦɢ ɡɧɚɱɟɧɢɹɦɢ ɩɨ ɢɡɜɟɫɬɧɨɣ ɮɨɪɦɭɥɟ
P U I
Ɂɚɩɢɲɟɦ ɜ ɬɚɛɥɢɰɭ ɭɫɥɨɜɢɹ ɩɪɨɜɟɞɟɧɢɹ ɷɤɫɩɟɪɢɦɟɧɬɚ ɢ ɢɡɦɟɪɟɧɧɵɟ
ɡɧɚɱɟɧɢɹ ɦɨɳɧɨɫɬɢ ɚ ɬɚɤɠɟ ɡɧɚɱɟɧɢɹ ɦɨɳɧɨɫɬɢ ɪɚɫɫɱɢɬɚɧɧɵɟ ɩɨ ɩɨɥɭɱɟɧɧɨɣ
ɪɟɝɪɟɫɫɢɨɧɧɨɣ ɦɨɞɟɥɢ ɛɟɡ ɭɱɟɬɚ ɫɨɱɟɬɚɧɢɣ ɮɚɤɬɨɪɨɜ
yc b b x b x
ɢ ɫ ɭɱɟɬɨɦ ɞɜɨɣɧɵɯ ɫɨɱɟɬɚɧɢɣ ɮɚɤɬɨɪɨɜ
ycc b b x b x b x x
Ɍ ɚ ɛ ɥ ɢ ɰ ɚ
Ɋɚɫɲɢɪɟɧɧɚɹ ɦɚɬɪɢɰɚ ɩɥɚɧɚ ɉɎɗ ɫ ɪɟɡɭɥɶɬɚɬɚɦɢ ɢɡɦɟɪɟɧɢɣ ɢ ɪɚɫɱɟɬɨɜ
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Ɋɚɫɫɱɢɬɚɧɧɵɟ ɡɧɚɱɟɧɢɹ ɦɨɳɧɨɫɬɢ |
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ɂɡɦɟɪɟɧɧɵɟ |
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ɩɨ ɦɨɞɟɥɢ |
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ɛɟɡ ɭɱɟɬɚ |
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ɫ ɭɱɟɬɨɦ ɞɜɨɣɧɵɯ |
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ɇɨɦɟɪ |
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ɨɩɵɬɚ |
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ɫɨɱɟɬɚɧɢɣ |
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y >ȼɬ@ |
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ɮɚɤɬɨɪɨɜ |
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ɮɚɤɬɨɪɨɜ |
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y |
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>ȼɬ@ |
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yc >ȼɬ@ |
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Ʉɨɷɮɮɢɰɢɟɧɬɵ ɦɨɞɟɥɢb b b b ɩɨ ɢɡɦɟɪɟɧɧɵɦ ɡɧɚɱɟɧɢɹɦ ɦɨɳɧɨɫɬɢ |
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ɨɩɪɟɞɟɥɹɸɬɫɹ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ |
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Ɋɚɫɫɱɢɬɚɧɧɵɟ ɡɧɚɱɟɧɢɹ ɦɨɳɧɨɫɬɢ ɩɨ ɦɨɞɟɥɢ ɛɟɡ ɭɱɟɬɚ ɞɜɨɣɧɵɯ ɫɨɱɟɬɚɧɢɣ |
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yc |
x x |
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ɞɚɸɬ ɡɧɚɱɟɧɢɹ ɛɥɢɡɤɢɟ ɤ ɢɡɦɟɪɟɧɧɵɦ ɡɧɚɱɟɧɢɹɦ
ɉɪɢ ɭɱɟɬɟ ɞɜɨɣɧɵɯ ɫɨɱɟɬɚɧɢɣ ɮɚɤɬɨɪɨɜ
ycc x x ɯ ɯ
ɩɨɥɭɱɚɟɦ ɩɨɥɧɨɟ ɫɨɜɩɚɞɟɧɢɟ ɫ ɢɡɦɟɪɟɧɧɵɦɢ ɡɧɚɱɟɧɢɹɦɢ Ɂɧɚɤɢ ɜɫɟɯ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɩɨɥɨɠɢɬɟɥɶɧɵɟ ɗɬɨ ɨɡɧɚɱɚɟɬ ɱɬɨ ɫ ɪɨɫɬɨɦ
ɮɚɤɬɨɪɨɜ ɨɬɤɥɢɤ ɭɜɟɥɢɱɢɜɚɟɬɫɹ ɢ ɩɪɢ ɨɞɧɨɜɪɟɦɟɧɧɨɦ ɭɜɟɥɢɱɟɧɢɢ ɮɚɤɬɨɪɨɜ ɨɬ
ɤɥɢɤ ɭɜɟɥɢɱɢɜɚɟɬɫɹ