Планирование эксперимента. Учебное пособие-1
.pdfɌɚɤɢɦ ɨɛɪɚɡɨɦ ȾɎɗ ɩɨɡɜɨɥɹɟɬ ɩɨɥɭɱɢɬɶ ɬɨɥɶɤɨ ɫɦɟɲɚɧɧɵɟ ɨɰɟɧɤɢ ɩɟɪɜɵɣ ɫɬɨɥɛɟɰ x ɬɚɤɠɟ ɫɨɜɩɚɞɚɟɬ ɫ ɜɨɫɶɦɵɦ ɫɬɨɥɛɰɨɦ x x x ɢ ɨɰɟɧɤɚb ɫɦɟɲɚɧɚ ɫb
ɇɨ ɟɫɥɢ ɞɨɩɭɫɬɢɬɶ ɱɬɨ ɞɨɫɬɚɬɨɱɧɨ ɨɩɪɟɞɟɥɢɬɶ ɬɨɥɶɤɨ ɱɟɬɵɪɟ ɤɨɷɮɮɢɰɢ
ɟɧɬɚ ɥɢɧɟɣɧɨɣ ɦɨɞɟɥɢ ɚ ɱɥɟɧɵ ɩɨɥɢɧɨɦɚ ɫ ɞɜɨɣɧɵɦɢ ɫɨɱɟɬɚɧɢɹɦɢ ɮɚɤɬɨɪɨɜ
ɩɪɟɧɟɛɪɟɠɢɦɨ ɦɚɥɵ ɬɨɝɞɚ ɫɨɱɟɬɚɧɢɟ ɜɫɟɯ ɬɪɟɯ ɮɚɤɬɨɪɨɜ ɬɟɦ ɛɨɥɟɟ ɦɚɥɨ ɬɨ ɩɥɚɧ ȾɎɗ ɛɭɞɟɬ ɩɨɥɟɡɟɧ Ʉɪɨɦɟ ɬɨɝɨ ɟɫɥɢ ɢɦɟɟɬɫɹ ɚɩɪɢɨɪɧɚɹ ɢɧɮɨɪɦɚ ɰɢɹ ɨ ɫɨɨɬɧɨɲɟɧɢɹɯ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɢɡ ɢɯ ɫɦɟɲɚɧɧɨɣ ɨɰɟɧɤɢ ɧɚɩɪɢɦɟɪ
b b | b b b | b b b | b
ɬɨ ɩɥɚɧ ɜɫɟɝɨ ɫ ɱɟɬɵɪɶɦɹ ɨɩɵɬɚɦɢ ɩɨɡɜɨɥɹɟɬ ɨɩɪɟɞɟɥɢɬɶ ɜɫɟ ɷɬɢ ɤɨɷɮɮɢ ɰɢɟɧɬɵ
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¦x u yu |
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ȿɫɥɢ ɨɤɚɡɵɜɚɟɬɫɹ ɱɬɨ ɩɨɫɬɪɨɟɧɧɵɣ ɧɚ ɨɫɧɨɜɟ ɩɥɚɧɚ ȾɎɗ ɩɨɥɢɧɨɦ ɧɟ ɭɞɨ
ɜɥɟɬɜɨɪɹɟɬ ɬɪɟɛɨɜɚɧɢɹɦ ɩɨ ɬɨɱɧɨɫɬɢ ɬɨ ɢɫɩɨɥɶɡɭɹ ɭɠɟ ɢɦɟɸɳɢɟɫɹ ɪɟɡɭɥɶɬɚɬɵ
ɨɩɵɬɨɜ ɩɥɚɧɚ ȾɎɗ ɩɥɚɧ ɥɟɝɤɨ ɞɨɫɬɪɚɢɜɚɟɬɫɹ ɞɨ ɩɥɚɧɚ ɉɎɗ Ⱦɪɨɛɧɵɟɮɚɤɬɨɪɧɵɟɷɤɫɩɟɪɢɦɟɧɬɵɫɨɯɪɚɧɹɸɳɢɟɫɜɨɣɫɬɜɚɩɨɥɧɵɯɮɚɤ
ɬɨɪɧɵɯ ɷɤɫɩɟɪɢɦɟɧɬɨɜ ɢ ɜ ɩɟɪɜɭɸ ɨɱɟɪɟɞɶ ɫɜɨɣɫɬɜɚ ɫɢɦɦɟɬɪɢɢ ɢ ɨɪɬɨɝɨɧɚɥɶ
ɧɨɫɬɢ ɩɨɥɭɱɢɥɢ ɧɚɡɜɚɧɢɟ ɪɟɝɭɥɹɪɧɵɟ ɞɪɨɛɧɵɟ ɪɟɩɥɢɤɢ
ɉɪɢk ɪɟɝɭɥɹɪɧɚɹ ɞɪɨɛɧɚɹ ɪɟɩɥɢɤɚ ɜɤɥɸɱɚɟɬ ɬɨɥɶɤɨ ɩɨɥɨɜɢɧɭ ɨɩɵɬɨɜ ɉɎɗ ɢ ɧɚɡɵɜɚɟɬɫɹ ɩɨɥɭɪɟɩɥɢɤɨɣ ɪɟɩɥɢɤɢ ɉɪɢk ɪɟɝɭɥɹɪɧɚɹ ɞɪɨɛɧɚɹ ɪɟɩɥɢɤɚ ɜɤɥɸɱɚɟɬ ɬɨɥɶɤɨ ɱɟɬɜɟɪɬɶ ɨɩɵɬɨɜ ɉɎɗ ɢ ɧɚɡɵɜɚɟɬɫɹ ɱɟɬɜɟɪɬɶɪɟɩɥɢɤɨɣ ɪɟɩɥɢɤɢ ɢ ɬ ɞ
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ k ɨɡɧɚɱɚɟɬ ɫɬɟɩɟɧɶ ɞɪɨɛɧɨɫɬɢ ɩɥɚɧɚ ȾɎɗ n k ɉɪɢ ɜɵ ɛɨɪɟ ɫɬɟɩɟɧɢ ɞɪɨɛɧɨɫɬɢ ɩɥɚɧɚ ɧɟɨɛɯɨɞɢɦɨ ɭɱɢɬɵɜɚɬɶ ɱɬɨ ɱɢɫɥɨ ɨɩɵɬɨɜ ɞɨɥɠ ɧɨ ɛɵɬɶ ɛɨɥɶɲɟ ɱɢɫɥɚ ɱɥɟɧɨɜ ɚɩɩɪɨɤɫɢɦɢɪɭɸɳɟɝɨ ɩɨɥɢɧɨɦɚ
Ɍɚɤ ɞɥɹ ɩɨɫɬɪɨɟɧɢɹ ɥɢɧɟɣɧɨɝɨ ɩɨɥɢɧɨɦɚ ɧɟɨɛɯɨɞɢɦɨ ɜɵɩɨɥɧɟɧɢɹ ɭɫɥɨɜɢɹ
n k t n
Ⱦɥɹ ɩɨɫɬɪɨɟɧɢɹ ɩɨɥɢɧɨɦɚ ɜɤɥɸɱɚɸɳɟɝɨ ɬɚɤɠɟ ɢ ɜɫɟ ɞɜɨɣɧɵɟ ɜɡɚɢɦɨɞɟɣ
ɫɬɜɢɹ ɮɚɤɬɨɪɨɜ ɧɟɨɛɯɨɞɢɦɨ ɱɬɨɛɵ
n k t n ɋn
ɝɞɟɋn ɱɢɫɥɨ ɞɜɨɣɧɵɯ ɫɨɱɟɬɚɧɢɣ ɢɡn
Ƚɪɚɮɢɱɟɫɤɨɟ ɢɡɨɛɪɚɠɟɧɢɟ ɩɥɚɧɨɜ ɉɎɗ ɢ ȾɎɗ ɜ ɬɪɟɯɦɟɪɧɨɦ ɮɚɤ ɬɨɪɧɨɦ ɩɪɨɫɬɪɚɧɫɬɜɟ ɩɪɟɞɫɬɚɜɥɟɧɨ ɧɚ ɪɢɫ
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ɊɢɫȽɪɚɮɢɱɟɫɤɨɟ ɢɡɨɛɪɚɠɟɧɢɟ ɩɥɚɧɨɜ ɉɎɗ ɢ ȾɎɗ ɜ ɮɚɤɬɨɪɧɨɦ ɩɪɨɫɬɪɚɧɫɬɜɟ
ɉɥɚɧ ɉɎɗ ɩɪɟɞɫɬɚɜɥɟɧ ɤɭɛɨɦ ɫ ɜɨɫɟɦɶɸ ɬɨɱɤɚɦɢ ɩɥɚɧɚ ɚ ɜɨɡɦɨɠɧɵɟ ɩɥɚɧɵ ȾɎɗ ± ɩɪɨɟɤɰɢɹɦɢ ɷɬɨɝɨ ɤɭɛɚ ɧɚ ɬɪɢ ɩɥɨɫɤɨɫɬɢ ɂɡ ɜɨɫɶɦɢ ɬɨɱɟɤ ɉɎɗ ɜɵɛɢɪɚɸɬɫɹ ɱɟɬɵɪɟ ɥɟɠɚɳɢɟ ɜ ɨɞɧɨɣ ɩɥɨɫɤɨɫɬɢ ɪɢɫ ɚ ɂɡ ɤɭɛɚ ɦɨɠɧɨ ɬɚɤɠɟ ɜɵɛɪɚɬɶ ɱɟɬɵɪɟ ɬɨɱɤɢ ɧɟ ɥɟɠɚɳɢɟ ɜ ɨɞɧɨɣ ɩɥɨɫɤɨɫɬɢ ɢ ɫɮɨɪɦɢ ɪɨɜɚɬɶ ɩɥɚɧ ȾɎɗ ɪɢɫ ɛ
Ɋɚɫɫɦɨɬɪɢɦ ɩɪɢɦɟɪ ɉɨɫɬɪɨɢɬɶ ɩɥɚɧ ȾɎɗ ɢ ɧɚɣɬɢ ɧɟɫɦɟɲɚɧɧɵɟ ɨɰɟɧɤɢ ɤɨɷɮɮɢɰɢɟɧɬɨɜb b b b b ɩɪɢ ɫɚɦɢɯ ɮɚɤɬɨɪɚɯ x x x x x ɢ ɫɦɟɲɚɧɧɵɟ ɨɰɟɧɤɢ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɩɪɢ ɩɚɪɧɵɯ ɫɨɱɟɬɚɧɢɹɯ ɮɚɤɬɨɪɨɜ Ⱦɨɩɭɫɤɚɹ ɱɬɨ ɜɫɟ ɬɪɨɣɧɵɟ ɢ ɨɞɧɨ ɱɟɬɜɟɪɬɧɨɟ ɫɨɱɟɬɚɧɢɹ ɮɚɤɬɨɪɨɜ ɩɪɟɧɟɛɪɟɠɢ ɦɨ ɦɚɥɵ ɫɨɯɪɚɧɢɬɶ ɨɪɬɨɝɨɧɚɥɶɧɨɫɬɶ ɩɥɚɧɚ
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɧɟɨɛɯɨɞɢɦɨ ɨɩɪɟɞɟɥɢɬɶ ɩɨɥɢɧɨɦ
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y b b x b x b x b x b x x b x x |
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b x x b x x b x x b x x |
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ɋɮɨɪɦɢɪɭɟɦ ɩɥɚɧ ɉɎɗ ɢɡ ɜɨɫɶɦɢ ɫɬɪɨɤ ɚ ɱɟɬɜɟɪɬɵɣ ɮɚɤɬɨɪ ɨɩɪɟɞɟ |
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ɥɢɦ ɫ ɩɨɦɨɳɶɸ ɝɟɧɟɪɢɪɭɸɳɟɝɨ ɫɨɨɬɧɨɲɟɧɢɹx |
x x x ɬɚɛɥɢɰɚ |
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ɋɬɨɥɛɟɰ y ɫɨɞɟɪɠɢɬ ɪɟɡɭɥɶɬɚɬɵ ɢɡɦɟɪɟɧɢɣ ȼ ɫɬɨɥɛɰɟ y ɩɪɢɜɟɞɟɧɵ ɡɧɚ |
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ɱɟɧɢɹ ɜ ɬɨɱɤɚɯ ɩɥɚɧɚ ɩɨɥɭɱɟɧɧɵɟ ɫ ɩɨɦɨɳɶɸ ɧɚɣɞɟɧɧɨɝɨ ɩɨɥɢɧɨɦɚ |
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Ɍ ɚ ɛ ɥ ɢ ɰ ɚ |
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ɉɥɚɧ ȾɎɗ |
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Ɂɧɚɱɟɧɢɹ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɩɨɥɢɧɨɦɚ ɫɨɫɬɚɜɥɹɸɬ |
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ȿɫɥɢ ɩɪɢɧɹɬɶ ɱɬɨ |
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b |b |
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b |b |
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b |b |
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ɬɨ ɩɨɥɢɧɨɦ ɢɦɟɟɬ ɜɢɞ |
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y x x x x x x x xx x x x x x x x
Ɂɧɚɱɟɧɢɹ ɩɨɥɢɧɨɦɚ ɜ ɬɨɱɤɚɯ ɩɥɚɧɚ ɩɪɢɜɟɞɟɧɵ ɜ ɩɨɫɥɟɞɧɟɦ ɫɬɨɥɛɰɟ ɩɥɚɧɚ ȾɎɗ ȼ ɧɚɲɟɦ ɫɥɭɱɚɟ ɬɨɱɧɨɫɬɶ ɟɝɨ ɞɨɫɬɚɬɨɱɧɨ ɜɵɫɨɤɚɹ
ȼ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɦ ɩɪɢɦɟɪɟ ɢɫɩɨɥɶɡɨɜɚɥɨɫɶ ɝɟɧɟɪɢɪɭɸɳɟɟ ɫɨɨɬɧɨɲɟɧɢɟ
x x x x
Ⱦɥɹ ɩɪɟɞɫɬɚɜɥɟɧɢɹ ɩɟɪɟɦɟɧɧɨɣ x ɜ ɨɛɳɟɣ ɫɥɨɠɧɨɫɬɢ ɦɨɠɧɨ ɢɫɩɨɥɶɡɨ ɜɚɬɶɜɨɫɟɦɶɝɟɧɟɪɢɪɭɸɳɢɯɫɨɨɬɧɨɲɟɧɢɣ ɱɟɬɵɪɟɜɡɹɬɵɯɫɨɡɧɚɤɚɦɢɩɥɸɫ ɢ ɱɟɬɵɪɟ ɫɨ ɡɧɚɤɨɦ ɦɢɧɭɫ
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x x x |
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x x x |
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ȼ ɨɛɳɟɦ ɫɥɭɱɚɟ ɞɥɹ ɩɪɨɢɡɜɨɥɶɧɨɣ ɩɨɥɭɪɟɩɥɢɤɢ n ɦɨɠɧɨ ɫɨɫɬɚ |
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ɜɢɬɶ ɋn Cn Cnn- ɝɟɧɟɪɢɪɭɸɳɢɯ |
ɫɨɨɬɧɨɲɟɧɢɣ ɝɞɟ |
ɋni |
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i n ɤɨɥɢɱɟɫɬɜɨ ɞɜɨɣɧɵɯ ɬɪɨɣɧɵɯ |
n ɫɨɱɟɬɚɧɢɣ ɢɡ |
n |
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ɮɚɤɬɨɪɨɜ Ɇɚɬɪɢɰɵ ɩɥɚɧɨɜ ɞɜɭɯ ɩɪɨɢɡɜɨɥɶɧɵɯ ɩɨɥɭɪɟɩɥɢɤ ɝɟɧɟɪɢɪɭɸɳɢɟ ɫɨ
ɨɬɧɨɲɟɧɢɹ ɤɨɬɨɪɵɯ ɨɬɥɢɱɚɸɬɫɹ ɬɨɥɶɤɨ ɡɧɚɤɚɦɢ ɧɟ ɫɨɞɟɪɠɚɬ ɨɞɢɧɚɤɨɜɵɯ ɫɬɪɨɤ ɢ ɨɛɴɟɞɢɧɟɧɢɟ ɬɚɤɢɯ ɩɨɥɭɪɟɩɥɢɤ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɩɥɚɧ ɉɎɗ
ȿɫɥɢ ɢɫɩɨɥɶɡɨɜɚɬɶ ɜ ɤɚɱɟɫɬɜɟ ɝɟɧɟɪɢɪɭɸɳɟɝɨ ɫɨɨɬɧɨɲɟɧɢɹ ɩɪɨɢɡɜɟɞɟɧɢɹ ɮɚɤɬɨɪɨɜ ɫɨ ɡɧɚɤɨɦ ɦɢɧɭɫ ɧɚɩɪɢɦɟɪ ɜ ɧɚɲɟɦ ɫɥɭɱɚɟ ɜɡɹɬɶ x x x x ɬɨ ɡɧɚɱɟɧɢɹ ɷɥɟɦɟɧɬɨɜ ɫɬɨɥɛɰɨɜ ɫɨɞɟɪɠɚɳɢɯ x x x x x x x x ɩɨɦɟɧɹ ɸɬ ɡɧɚɤ ɧɚ ɩɪɨɬɢɜɨɩɨɥɨɠɧɵɣ ɉɨɥɭɱɢɦ ɜɦɟɫɬɨ ɬɪɟɯ ɩɚɪ ɨɞɢɧɚɤɨɜɵɯ ɫɬɨɥɛɰɨɜ ɬɟ ɠɟ ɬɪɢ ɩɚɪɵ ɫɬɨɥɛɰɨɜ ɧɨ ɨɬɥɢɱɚɸɳɢɟɫɹ ɡɧɚɤɚɦɢ Ɍɨɝɞɚ ɫɨɜɦɟɫɬɧɵɦɢ ɨɰɟɧ ɤɚɦɢ ɛɭɞɭɬ ɹɜɥɹɬɶɫɹ
b b
b b
b b
ɉɪɢ ɜɵɛɨɪɟ ɝɟɧɟɪɢɪɭɸɳɢɯ ɫɨɨɬɧɨɲɟɧɢɣ ɜ ɩɥɚɧɚɯ ȾɎɗ n k ɠɟɥɚɬɟɥɶɧɨ ɞɥɹɞɨɩɨɥɧɢɬɟɥɶɧɵɯɮɚɤɬɨɪɨɜɢɫɩɨɥɶɡɨɜɚɬɶɝɟɧɟɪɢɪɭɸɳɢɟɫɨɨɬɧɨɲɟɧɢɹɫɨ ɞɟɪɠɚɳɢɟ ɩɪɨɢɡɜɟɞɟɧɢɹ ɦɚɤɫɢɦɚɥɶɧɨ ɜɨɡɦɨɠɧɨɝɨ ɤɨɥɢɱɟɫɬɜɚ ɨɫɧɨɜɧɵɯ ɮɚɤɬɨ ɪɨɜ ɢɡ ɤɨɬɨɪɵɯ ɫɬɪɨɢɬɫɹ ɩɥɚɧ ɉɎɗ q ɝɞɟq n k ɗɬɨ ɨɛɴɹɫɧɹɟɬɫɹ ɬɟɦ ɱɬɨ
ɱɟɦ ɜɵɲɟ ɩɨɪɹɞɨɤ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɮɚɤɬɨɪɨɜ ɬɟɦ ɱɚɳɟ ɨɧɢ ɛɵɜɚɸɬ ɛɥɢɡɤɢ ɤ
ɧɭɥɸ ɢ ɢɦɢ ɦɨɠɧɨ ɩɪɟɧɟɛɪɟɱɶ
ȼ ɫɜɹɡɢ ɫ ɬɟɦ ɱɬɨ ɜ ȾɎɗ n k ɡɧɚɱɟɧɢɹk ɮɚɤɬɨɪɨɜ ɨɩɪɟɞɟɥɹɸɬɫɹ ɩɨ ɝɟ ɧɟɪɢɪɭɸɳɢɦ ɫɨɨɬɧɨɲɟɧɢɹɦ ɩɪɟɞɫɬɚɜɥɹɸɳɢɦ ɩɪɨɢɡɜɟɞɟɧɢɹ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɞɪɭɝɢɯn k ɮɚɤɬɨɪɨɜ ɧɚɣɞɟɧɧɵɟ ɤɨɷɮɮɢɰɢɟɧɬɵ ɭɪɚɜɧɟɧɢɹ ɪɟɝɪɟɫɫɢɢ ɨɤɚɡɵ ɜɚɸɬɫɹ ɫɦɟɲɚɧɧɵɦɢ ɢ ɜɨɡɧɢɤɚɟɬ ɡɚɞɚɱɚ ɨɩɪɟɞɟɥɟɧɢɹ ɫɜɹɡɢ ɦɟɠɞɭ ɥɢɧɟɣɧɵɦɢ ɷɮɮɟɤɬɚɦɢ ɢ ɷɮɮɟɤɬɚɦɢ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɉɪɨɳɟ ɝɨɜɨɪɹ ɧɟɨɛɯɨɞɢɦɨ ɨɩɪɟɞɟ ɥɢɬɶ ɤɚɤɢɟ ɨɰɟɧɤɢ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɦɨɞɟɥɢ ɨɤɚɡɵɜɚɸɬɫɹ ɫɦɟɲɚɧɧɵɦɢ
Ɉɩɪɟɞɟɥɹɸɳɢɣ ɤɨɧɬɪɚɫɬ
ɉɪɢ ɢɫɩɨɥɶɡɨɜɚɧɢɢ ɩɨɥɭɪɟɩɥɢɤ ɬɨ ɟɫɬɶ ȾɎɗ n ɫɜɹɡɶ ɦɟɠɞɭ ɤɨɷɮɮɢ ɰɢɟɧɬɚɦɢ ɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ ɫ ɩɨɦɨɳɶɸ ɨɩɪɟɞɟɥɹɸɳɟɝɨ ɤɨɧɬɪɚɫɬɚ
Ɉɩɪɟɞɟɥɹɸɳɢɣ ɤɨɧɬɪɚɫɬ ɩɨɥɭɪɟɩɥɢɤɢ ɩɨɥɭɱɚɟɬɫɹ ɩɭɬɟɦ ɭɦɧɨɠɟɧɢɹ ɨɛɟɢɯ
ɱɚɫɬɟɣ ɝɟɧɟɪɢɪɭɸɳɟɝɨ ɫɨɨɬɧɨɲɟɧɢɹ ɧɚ ɟɝɨ ɠɟ ɥɟɜɭɸ ɱɚɫɬɶ Ⱥ ɬɚɤ ɤɚɤ ɞɥɹ ɥɸ ɛɨɝɨ ɮɚɤɬɨɪɚ xi ɢɫɩɨɥɶɡɭɸɬɫɹ ɬɨɥɶɤɨ ɞɜɚ ɤɨɞɢɪɨɜɚɧɧɵɯ ɡɧɚɱɟɧɢɹ ɥɢɛɨ
ɥɢɛɨ ɬɨ xi ɢ ɥɟɜɚɹ ɱɚɫɬɶ ɨɩɪɟɞɟɥɹɸɳɟɝɨ ɤɨɧɬɪɚɫɬɚ ɜɫɟɝɞɚ ɛɭɞɟɬ ɪɚɜɧɚ ɟɞɢɧɢɰɟ
ɇɚɩɪɢɦɟɪ ɞɥɹ ɩɨɥɭɪɟɩɥɢɤɢ ɢ ɝɟɧɟɪɢɪɭɸɳɟɝɨ ɫɨɨɬɧɨɲɟɧɢɹx x x ɩɨɥɭɱɢɦ ɨɩɪɟɞɟɥɹɸɳɢɣ ɤɨɧɬɪɚɫɬ
x x x
ɑɬɨɛɵ ɨɩɪɟɞɟɥɢɬɶ ɫ ɤɚɤɢɦɢ ɤɨɷɮɮɢɰɢɟɧɬɚɦɢ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɫɦɟɲɚɧɚ ɨɰɟɧɤɚ ɤɨɷɮɮɢɰɢɟɧɬɚ ɞɚɧɧɨɝɨ ɮɚɤɬɨɪɚ ɧɟɨɛɯɨɞɢɦɨ ɭɦɧɨɠɢɬɶ ɨɛɟ ɱɚɫɬɢ ɨɩɪɟ ɞɟɥɹɸɳɟɝɨɤɨɧɬɪɚɫɬɚɧɚɷɬɨɬɮɚɤɬɨɪɉɨɥɭɱɢɦɫɯɟɦɭɫɦɟɲɢɜɚɧɢɹɥɢɧɟɣɧɵɯ ɷɮɮɟɤɬɨɜ ɢ ɷɮɮɟɤɬɨɜ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɩɪɢ ɢɫɩɨɥɶɡɨɜɚɧɢɢ ɤɨɧɤɪɟɬɧɨɝɨ ɩɥɚɧɚ ȼ ɞɚɧɧɨɦ ɫɥɭɱɚɟ ɞɥɹ ɩɥɚɧɚ ɢ ɝɟɧɟɪɢɪɭɸɳɟɝɨ ɫɨɨɬɧɨɲɟɧɢɹ x x x ɩɨ ɥɭɱɢɦ ɢɡ ɨɩɪɟɞɟɥɹɸɳɟɝɨ ɤɨɧɬɪɚɫɬɚ ɫɥɟɞɭɸɳɢɣ ɩɨɪɹɞɨɤ ɫɦɟɲɢɜɚɧɢɹ ɮɚɤɬɨɪɨɜ
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x x |
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x |
x x |
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x |
x x |
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ɉɨ ɷɬɢɦ ɫɨɨɬɧɨɲɟɧɢɹɦ ɨɩɪɟɞɟɥɹɬɫɹ ɫɦɟɲɚɧɧɵɟ ɨɰɟɧɤɢ ɤɨɷɮɮɢɰɢɟɧɬɨɜ
b b
b b
b b
ȼ ɫɥɭɱɚɟ ɢɫɩɨɥɶɡɨɜɚɧɢɹ ɜ ɩɥɚɧɟ ȾɎɗ ɞɪɭɝɨɣ ɩɨɥɭɪɟɩɥɢɤɢ ɫ ɞɪɭɝɢɦ ɜɨɡɦɨɠɧɵɦ ɝɟɧɟɪɢɪɭɸɳɢɦ ɫɨɨɬɧɨɲɟɧɢɟɦ x x x ɩɨɥɭɱɢɦ ɨɩɪɟɞɟɥɹɸɳɢɣ ɤɨɧɬɪɚɫɬ
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x x x |
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ɉɨɪɹɞɨɤ ɫɦɟɲɢɜɚɧɢɹ ɮɚɤɬɨɪɨɜ ɢ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɫɦɟɲɚɧɧɵɟ ɨɰɟɧɤɢ ɨɤɚ |
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ɠɭɬɫɹ ɫɥɟɞɭɸɳɢɦɢ |
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x |
x x b b |
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x |
x x b b |
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x |
x x b b |
ȿɫɥɢ ɪɟɚɥɢɡɨɜɚɬɶ ɨɛɟ ɷɬɢ ɩɨɥɭɪɟɩɥɢɤɢ ɬɨ ɩɨɥɭɱɢɦ ɩɥɚɧ ɉɎɗ ɢ ɪɚɡ ɞɟɥɶɧɵɟ ɨɰɟɧɤɢ ɞɥɹ ɥɢɧɟɣɧɵɯ ɷɮɮɟɤɬɨɜ ɢ ɷɮɮɟɤɬɨɜ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ
Ɋɚɫɫɦɨɬɪɢɦ ɩɨɥɭɪɟɩɥɢɤɭ ɡɚɞɚɧɧɭɸ ɝɟɧɟɪɢɪɭɸɳɢɦ ɫɨɨɬɧɨɲɟɧɢɟɦ x x x x ɇɚɣɞɟɦ ɨɩɪɟɞɟɥɹɸɳɢɣ ɤɨɧɬɪɚɫɬ ɞɥɹ ɞɚɧɧɨɣ ɩɨɥɭɪɟɩɥɢɤɢ
x x x x
Ɍɨɝɞɚ ɫɦɟɲɚɧɧɵɟ ɨɰɟɧɤɢ ɨɩɪɟɞɟɥɹɬɫɹ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ
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ȼɛɨɥɶɲɢɧɫɬɜɟɩɪɚɤɬɢɱɟɫɤɢɯɫɥɭɱɚɟɜɜɡɚɢɦɨɞɟɣɫɬɜɢɹɜɵɫɨɤɨɝɨɩɨɪɹɞɤɚ |
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ɦɚɥɨ ɜɥɢɹɸɬ ɧɚ ɮɭɧɤɰɢɸ ɨɬɤɥɢɤɚ ɢ ɢɦɢ ɨɛɵɱɧɨ ɩɪɟɧɟɛɪɟɝɚɸɬ ɉɨɷɬɨɦɭ ɠɟɥɚ
ɬɟɥɶɧɨ ɫɜɹɡɵɜɚɬɶ ɥɢɧɟɣɧɵɟ ɷɮɮɟɤɬɵ ɫ ɷɮɮɟɤɬɚɦɢ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɦɚɤɫɢɦɚɥɶɧɨ ɜɨɡɦɨɠɧɨɝɨ ɩɨɪɹɞɤɚ ɤɨɬɨɪɵɦɢ ɦɨɠɧɨ ɩɪɟɧɟɛɪɟɱɶ ɢ ɩɨɥɭɱɢɬɶ ɪɚɡɞɟɥɶɧɵɟ ɨɰɟɧɤɢ ɤɨɷɮɮɢɰɢɟɧɬɨɜb b bn
ȼ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɩɨɥɭɪɟɩɥɢɤɟ ɟɫɥɢ ɬɪɨɣɧɵɦɢ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹɦɢ ɦɨɠɧɨ ɩɪɟɧɟɛɪɟɱɶ ɬɨ ɥɢɧɟɣɧɵɟ ɷɮɮɟɤɬɵ ɛɭɞɭɬ ɪɚɡɞɟɥɟɧɵ ɉɚɪɧɵɟ ɜɡɚɢɦɨɞɟɣ ɫɬɜɢɹ ɨɤɚɡɚɥɢɫɶ ɫɦɟɲɚɧɧɵɦɢ
ɉɪɢ ɱɢɫɥɟ ɮɚɤɬɨɪɨɜ n t ɢɫɩɨɥɶɡɭɹ ɩɨɥɭɪɟɩɥɢɤɭ ɦɨɠɧɨ ɪɚɡɞɟɥɢɬɶ ɢ ɨɰɟɧɤɢ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɩɪɢ ɞɜɨɣɧɵɯ ɫɨɱɟɬɚɧɢɹɯ ɮɚɤɬɨɪɨɜ
Ɉɩɪɟɞɟɥɢɦ ɫɯɟɦɭ ɫɦɟɲɢɜɚɧɢɹ ɨɰɟɧɨɤ ɞɥɹ ɩɨɥɭɪɟɩɥɢɤɢ ɫ ɝɟ ɧɟɪɢɪɭɸɳɢɦ ɫɨɨɬɧɨɲɟɧɢɟɦx x x x x ɇɚɣɞɟɦ ɨɩɪɟɞɟɥɹɸɳɢɣ ɤɨɧɬɪɚɫɬ
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Ⱦɥɹ ɥɢɧɟɣɧɵɯ ɷɮɮɟɤɬɨɜ ɩɨɥɭɱɢɦ ɫɥɟɞɭɸɳɭɸ ɫɯɟɦɭ ɫɦɟɲɢɜɚɧɢɹ ɨɰɟɧɨɤ |
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Ⱦɥɹ ɷɮɮɟɤɬɨɜ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ
x x |
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ȼɫɟ ɩɚɪɧɵɟ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ ɨɤɚɡɵɜɚɸɬɫɹ ɪɚɡɞɟɥɟɧɵ ȿɫɥɢ ɬɪɨɣɧɵɦɢ ɫɨɱɟ
ɬɚɧɢɹɦɢ ɮɚɤɬɨɪɨɜ ɦɨɠɧɨ ɩɪɟɧɟɛɪɟɱɶ ɬɨ ɬɚɤɚɹ ɩɨɥɭɪɟɩɥɢɤɚ ɨɤɚɡɵɜɚɟɬɫɹ ɩɨɥɟɡ
ɧɨɣ
Ɉɛɨɛɳɚɸɳɢɟ ɨɩɪɟɞɟɥɹɸɳɢɟ ɤɨɧɬɪɚɫɬɵ
ɉɪɢ ɨɰɟɧɢɜɚɧɢɢ ɩɚɪɚɦɟɬɪɨɜ ɮɭɧɤɰɢɢ ɨɬɤɥɢɤɚ ɢɫɩɨɥɶɡɭɸɬɫɹ ɬɚɤɠɟ ɢ ɪɟ
ɩɥɢɤɢ ɛɨɥɟɟ ɜɵɫɨɤɨɣ ɞɪɨɛɧɨɫɬɢ ɋɢɫɬɟɦɚ ɫɦɟɲɢɜɚɧɢɹ ɥɢɧɟɣɧɵɯ ɷɮɮɟɤɬɨɜ ɢ ɷɮɮɟɤɬɨɜ ɜɡɚɢɦɨɞɟɣɫɬɜɢɣ ɦɟɠɞɭ ɫɨɛɨɣ ɩɨɥɭɱɚɟɬɫɹ ɛɨɥɟɟ ɫɥɨɠɧɨɣ ɱɟɦ ɩɪɢ ɩɪɢɦɟɧɟɧɢɢ ɩɨɥɭɪɟɩɥɢɤ Ⱦɥɹ ɧɚɯɨɠɞɟɧɢɹ ɫɢɫɬɟɦɵ ɫɦɟɲɢɜɚɧɢɹ ɫɬɪɨɹɬɫɹ ɨɛɨɛ ɳɚɸɳɢɟ ɨɩɪɟɞɟɥɹɸɳɢɟ ɤɨɧɬɪɚɫɬɵ
ɉɪɢ ɩɨɫɬɪɨɟɧɢɢ ɱɟɬɜɟɪɬɶ ɪɟɩɥɢɤɢ ɢɫɩɨɥɶɡɭɸɬɫɹ ɞɜɚ ɝɟɧɟɪɢɪɭɸɳɢɯ ɫɨɨɬ
ɧɨɲɟɧɢɹ ɢɡ ɤɨɬɨɪɵɯ ɮɨɪɦɢɪɭɸɬɫɹ ɞɜɚ ɨɩɪɟɞɟɥɹɸɳɢɯ ɤɨɧɬɪɚɫɬɚ ɗɬɢ ɤɨɧɬɪɚ
ɫɬɵ ɩɟɪɟɦɧɨɠɚɸɬɫɹ ɩɨɩɚɪɧɨ ɢ ɩɨɥɭɱɚɸɬɫɹ ɨɩɪɟɞɟɥɹɸɳɢɟ ɤɨɧɬɪɚɫɬɵ ɜɬɨɪɨɝɨ ɭɪɨɜɧɹ Ɉɛɨɛɳɚɸɳɢɣ ɨɩɪɟɞɟɥɹɸɳɢɣ ɤɨɧɬɪɚɫɬ ɛɭɞɟɬ ɫɨɫɬɨɹɬɶ ɢɡ ɢɫɯɨɞɧɵɯ ɤɨɧɬɪɚɫɬɨɜ ɢ ɪɟɡɭɥɶɬɚɬɚ ɢɯ ɩɨɩɚɪɧɨɝɨ ɩɟɪɟɦɧɨɠɟɧɢɹ ɬɨ ɟɫɬɶ ɟɞɢɧɢɰɚ ɜ ɥɟɜɨɣ ɱɚɫɬɢ ɛɭɞɟɬ ɪɚɜɧɚ ɭɠɟ ɬɪɟɦ ɩɪɨɢɡɜɟɞɟɧɢɹɦ ɮɚɤɬɨɪɨɜ
ȼ ɪɟɩɥɢɤɟ ɜɢɞɚ n ɪɟɩɥɢɤɚ ɞɥɹ ɬɪɟɯ ɞɨɩɨɥɧɢɬɟɥɶɧɵɯ ɮɚɤɬɨɪɨɜ ɢɫ ɩɨɥɶɡɭɸɬɫɹ ɬɪɢ ɝɟɧɟɪɢɪɭɸɳɢɯ ɫɨɨɬɧɨɲɟɧɢɹ ɢɡ ɤɨɬɨɪɵɯ ɩɨɥɭɱɚɸɬɫɹ ɬɪɢ ɢɫ ɯɨɞɧɵɯ ɨɩɪɟɞɟɥɹɸɳɢɯ ɤɨɧɬɪɚɫɬɚ ɗɬɢ ɤɨɧɬɪɚɫɬɵ ɫɧɚɱɚɥɚ ɩɟɪɟɦɧɨɠɚɸɬɫɹ ɩɨ ɩɚɪɧɨ ɩɨɥɭɱɚɸɬɫɹ ɨɩɪɟɞɟɥɹɸɳɢɟ ɤɨɧɬɪɚɫɬɵ ɜɬɨɪɨɝɨ ɭɪɨɜɧɹ ɡɚɬɟɦ ɢɫɯɨɞɧɵɟ
ɤɨɧɬɪɚɫɬɵ ɫɧɨɜɚ ɩɟɪɟɦɧɨɠɚɸɬɫɹ ɩɨ ɬɪɢ ɢ ɩɨɥɭɱɚɟɬɫɹ ɤɨɧɬɪɚɫɬ ɬɪɟɬɶɟɝɨ ɭɪɨɜɧɹ ɧɚɢɜɵɫɲɟɝɨ ɞɥɹ ɪɟɩɥɢɤɢ n Ɉɛɨɛɳɚɸɳɢɣ ɨɩɪɟɞɟɥɹɸɳɢɣ ɤɨɧɬɪɚɫɬ ɫɨ ɫɬɚɜɥɹɟɬɫɹ ɩɭɬɟɦ ɩɟɪɟɱɢɫɥɟɧɢɹ ɜɵɪɚɠɟɧɢɣ ɞɥɹ ɤɨɧɬɪɚɫɬɨɜ ɜɫɟɯ ɭɪɨɜɧɟɣ ɜɤɥɸ ɱɚɹ ɢɫɯɨɞɧɵɟ ɨɩɪɟɞɟɥɹɸɳɢɟ ɤɨɧɬɪɚɫɬɵ
ȼ ɨɛɳɟɦ ɫɥɭɱɚɟ ɜ ɪɟɩɥɢɤɟ n k ɮɨɪɦɢɪɭɟɬɫɹk ɢɫɯɨɞɧɵɯ ɨɩɪɟɞɟɥɹɸɳɢɯ ɤɨɧɬɪɚɫɬɨɜ ɨɩɪɟɞɟɥɹɸɳɢɯ ɤɨɧɬɪɚɫɬɨɜ ɩɟɪɜɨɝɨ ɭɪɨɜɧɹ ɂɡ ɢɫɯɨɞɧɵɯ ɤɨɧɬɪɚ ɫɬɨɜ ɫɧɚɱɚɥɚ ɩɨɩɚɪɧɵɦ ɩɟɪɟɦɧɨɠɟɧɢɟɦ ɩɨɥɭɱɚɸɬ ɤɨɧɬɪɚɫɬɵ ɜɬɨɪɨɝɨ ɭɪɨɜɧɹ ɡɚɬɟɦ ɩɟɪɟɦɧɨɠɟɧɢɟɦ ɩɨ ɬɪɢ ɬɪɟɬɶɟɝɨ ɭɪɨɜɧɹ ɢ ɬ ɞ ɞɨ ɤɨɧɬɪɚɫɬɚ ɧɚɢɜɵɫɲɟɝɨ k ɝɨ ɭɪɨɜɧɹ ɤɨɬɨɪɵɣ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɩɪɨɢɡɜɟɞɟɧɢɟ ɜɫɟɯk ɢɫɯɨɞɧɵɯ ɨɩɪɟ ɞɟɥɹɸɳɢɯ ɤɨɧɬɪɚɫɬɨɜ Ɉɛɨɛɳɚɸɳɢɣ ɨɩɪɟɞɟɥɹɸɳɢɣ ɤɨɧɬɪɚɫɬ ɫɨɫɬɚɜɥɹɟɬɫɹ ɩɭ ɬɟɦ ɩɟɪɟɱɢɫɥɟɧɢɹ ɜɵɪɚɠɟɧɢɣ ɞɥɹ ɜɫɟɯ ɫɮɨɪɦɢɪɨɜɚɧɧɵɯ ɤɨɧɬɪɚɫɬɨɜ
Ɋɚɫɫɦɨɬɪɢɦ ɩɪɢɦɟɪ ɪɟɩɥɢɤɚ ɦɨɠɟɬ ɛɵɬɶ ɡɚɞɚɧɚ ɝɟɧɟɪɢɪɭɸɳɢɦɢ ɫɨɨɬɧɨɲɟɧɢɹɦɢ
x x x x x x x
Ɉɩɪɟɞɟɥɹɸɳɢɦɢ ɤɨɧɬɪɚɫɬɚɦɢ ɪɟɩɥɢɤɢ ɹɜɥɹɸɬɫɹ
x x x x x x x
ɉɟɪɟɦɧɨɠɢɜ ɨɩɪɟɞɟɥɹɸɳɢɟ ɤɨɧɬɪɚɫɬɵ ɩɨɥɭɱɢɦ ɬɪɟɬɶɟ ɫɨɨɬɧɨɲɟɧɢɟ
x x x
Ɉɛɨɛɳɚɸɳɢɣ ɨɩɪɟɞɟɥɹɸɳɢɣ ɤɨɧɬɪɚɫɬ ɛɭɞɟɬ ɜɤɥɸɱɚɬɶ ɬɪɢ ɩɪɨɢɡɜɟɞɟɧɢɹ
x x x x x x x x x x
ɋɯɟɦɭ ɫɦɟɲɢɜɚɧɢɹ ɨɰɟɧɨɤ ɧɚɯɨɞɢɦ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɵɦ ɭɦɧɨɠɟɧɢɟɦ ɨɛɨɛɳɚɸɳɟɝɨ ɨɩɪɟɞɟɥɹɸɳɟɝɨ ɤɨɧɬɪɚɫɬɚ ɧɚx x x ɢ ɬ ɞ
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Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɩɨɥɭɱɟɧɚ ɜɟɫɶɦɚ ɫɥɨɠɧɚɹ ɫɢɫɬɟɦɚ ɫɦɟɲɢɜɚɧɢɹ ɨɰɟɧɨɤ ȼɫɟ
ɥɢɧɟɣɧɵɟ ɷɮɮɟɤɬɵ ɨɤɚɡɚɥɢɫɶ ɫɦɟɲɚɧɧɵɦɢ ɫ ɧɟɫɤɨɥɶɤɢɦɢ ɩɚɪɧɵɦɢ ɬɪɨɣɧɵɦɢ
ɜɡɚɢɦɨɞɟɣɫɬɜɢɹɦɢ
ɋɭɳɟɫɬɜɭɟɬ ɬɚɤɨɟ ɩɨɧɹɬɢɟ ± ɪɚɡɪɟɲɚɸɳɚɹ ɫɩɨɫɨɛɧɨɫɬɶ ɞɪɨɛɧɨɣ ɪɟɩɥɢɤɢ
ɤɨɬɨɪɚɹ ɨɩɪɟɞɟɥɹɟɬɫɹ ɦɢɧɢɦɚɥɶɧɵɦ ɱɢɫɥɨɦ ɷɥɟɦɟɧɬɨɜ ɜɯɨɞɹɳɢɯ ɜ ɨɩɪɟɞɟɥɹ
ɸɳɢɣ ɤɨɧɬɪɚɫɬ ɑɟɦ ɜɵɲɟ ɪɚɡɪɟɲɚɸɳɚɹ ɫɩɨɫɨɛɧɨɫɬɶ ɬɟɦ ɛɨɥɶɲɟ ɜɨɡɦɨɠɧɨ
ɫɬɟɣ ɭ ɞɪɨɛɧɨɣ ɪɟɩɥɢɤɢ ɞɥɹ ɩɨɫɬɪɨɟɧɢɹ ɪɟɝɪɟɫɫɢɨɧɧɨɣ ɦɨɞɟɥɢ Ɋɚɡɪɟɲɚɸɳɚɹ ɫɩɨɫɨɛɧɨɫɬɶ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ ɞɪɨɛɧɨɣ ɪɟɩɥɢɤɢ ɨɤɚɡɚɥɚɫɶ ɧɢɡɤɚɹ ɉɨɥɶɡɨ ɜɚɬɶɫɹ ɬɚɤɨɣ ɪɟɩɥɢɤɨɣ ɦɨɠɧɨ ɥɢɲɶ ɜ ɬɨɦ ɫɥɭɱɚɟ ɟɫɥɢ ɜɫɟ ɩɚɪɧɵɟ ɜɡɚɢɦɨɞɟɣ ɫɬɜɢɹ ɛɥɢɡɤɢ ɤ ɧɭɥɸ ɢ ɦɨɠɧɨ ɨɝɪɚɧɢɱɢɬɶɫɹ ɥɢɧɟɣɧɨɣ ɦɨɞɟɥɶɸ
y b ¦bi xi
i
ȿɫɥɢɧɟɥɶɡɹɩɪɟɧɟɛɪɟɱɶɩɚɪɧɵɦɢɜɡɚɢɦɨɞɟɣɫɬɜɢɹɦɢɬɨɫɥɟɞɭɟɬɩɪɢɦɟ ɧɢɬɶ ɪɚɫɫɦɨɬɪɟɧɧɭɸ ɜɵɲɟ ɩɨɥɭɪɟɩɥɢɤɭ ɫ ɝɟɧɟɪɢɪɭɸɳɢɦ ɫɨɨɬɧɨɲɟɧɢɟɦ x x x x x ɗɬɨ ɩɨɬɪɟɛɭɟɬ ɜ ɞɜɚ ɪɚɡɚ ɛɨɥɶɲɟɝɨ ɤɨɥɢɱɟɫɬɜɚ ɬɨɱɟɤ ɩɥɚɧɚ ɷɤɫ ɩɟɪɢɦɟɧɬɚɩɨɫɪɚɜɧɟɧɢɸɫɱɟɬɜɟɪɬɶɪɟɩɥɢɤɨɣɧɨɩɨɡɜɨɥɢɬɪɚɡɞɟɥɢɬɶɜɫɟɥɢ ɧɟɣɧɵɟ ɷɮɮɟɤɬɵ ɢ ɩɚɪɧɵɟ ɜɡɚɢɦɨɞɟɣɫɬɜɢɹ
Ɋɚɫɫɦɨɬɪɢɦ ɜ ɤɚɱɟɫɬɜɟ ɩɪɢɦɟɪɚ ɬɚɤɠɟ ɩɥɚɧɢɪɨɜɚɧɢɟ ɪɟɩɥɢɤɢ ɫ ɛɨɥɟɟ ɜɵ
ɫɨɤɨɣ ɫɬɟɩɟɧɶɸ ɞɪɨɛɧɨɫɬɢ ɚ ɢɦɟɧɧɨ ɪɟɩɥɢɤɭ ɬɢɩɚ
ȼɵɛɟɪɟɦ ɝɟɧɟɪɢɪɭɸɳɢɟ ɫɨɨɬɧɨɲɟɧɢɹ
x x x x x x x x x x x x x
Ɉɩɪɟɞɟɥɹɸɳɢɦɢ ɤɨɧɬɪɚɫɬɚɦɢ ɷɬɨɣ ɪɟɩɥɢɤɢ ɛɭɞɭɬ ɫɨɨɬɧɨɲɟɧɢɹ
x x x x x x x x x x x x x
ɍɦɧɨɠɚɹ ɢɯ ɩɨɩɚɪɧɨ ɩɨɥɭɱɢɦ ɲɟɫɬɶ ɨɛɨɛɳɚɸɳɢɯ ɤɨɧɬɪɚɫɬɨɜ ɜɬɨɪɨɝɨ ɭɪɨɜɧɹ
x x x x x x x x x x x x x x x x x x x x x