Проектирование составов бетонных смесей с помощью современных информационных технологий. Монография
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Ɍɚɤɨɟ ɜɵɞɟɥɟɧɢɟ ɝɪɭɩɩ ɭɫɥɨɜɧɨ ɩɨɫɤɨɥɶɤɭ ɩɨɫɬɪɨɟɧɧɵɟ ɫɬɚ ɬɢɫɬɢɱɟɫɤɢɦɢ ɦɟɬɨɞɚɦɢ ɦɚɬɟɦɚɬɢɱɟɫɤɢɟ ɦɨɞɟɥɢ ɦɨɝɭɬ ɢɡɭɱɚɬɶɫɹ ɜ ɞɚɥɶɧɟɣɲɟɦ ɫ ɩɨɦɨɳɶɸ ɧɚɩɪɢɦɟɪ ɥɢɧɟɣɧɨɝɨ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ
Ɇɚɬɟɦɚɬɢɱɟɫɤɨɟ ɦɨɞɟɥɢɪɨɜɚɧɢɟ ɞɨɥɠɧɨ ɩɪɨɜɨɞɢɬɶɫɹ ɬɨɥɶɤɨ ɧɚ ɨɫɧɨɜɟ ɢɧɮɨɪɦɚɰɢɢ ɨ ɤɨɧɤɪɟɬɧɨɣ ɬɟɯɧɨɥɨɝɢɱɟɫɤɨɣ ɫɢɬɭɚɰɢɢ Ⱦɥɹ ɞɟ ɬɟɪɦɢɧɢɪɨɜɚɧɧɵɯ ɦɨɞɟɥɟɣ ɧɟɨɛɯɨɞɢɦɨ ɤɚɤ ɩɪɚɜɢɥɨ ɩɪɟɞɫɬɚɜɥɟɧɢɟ ɨɛ ɢɯ ɮɢɡɢɤɨ ɯɢɦɢɱɟɫɤɨɣ ɩɪɢɪɨɞɟ ɢ ɡɚɜɢɫɢɦɨɫɬɹɯ ɭɩɪɚɜɥɹɸɳɢɯ ɧɚɛɥɸɞɚɟɦɵɦ ɩɪɨɰɟɫɫɨɦ ɉɪɢ ɩɨɫɬɪɨɟɧɢɢ ɫɬɚɬɢɫɬɢɱɟɫɤɢɯ ɦɨɞɟɥɟɣ ɦɨɠɧɨ ɨɝɪɚɧɢɱɢɬɶɫɹ ɫɜɟɞɟɧɢɹɦɢ ɨ ɬɨɦ ɤɚɤ ɢɡɦɟɧɟɧɢɟ ɬɟɯɧɨɥɨɝɢɱɟ ɫɤɢɯ ɮɚɤɬɨɪɨɜɏɜɥɢɹɟɬ ɧɚ ɤɨɧɟɱɧɨɟ ɫɜɨɣɫɬɜɨɍɗɬɢ ɫɜɟɞɟɧɢɹ ɦɨɠɧɨ ɩɨɥɭɱɢɬɶ ɬɨɥɶɤɨ ɧɚɛɥɸɞɟɧɢɟɦ ɡɚ ɜɵɛɪɚɧɧɨɣ ɫɢɫɬɟɦɨɣ ɩɪɢɱɟɦ ɩɨɧɹ ɬɢɟ ɧɚɛɥɸɞɟɧɢɟ ɡɞɟɫɶ ɫɥɟɞɭɟɬ ɬɪɚɤɬɨɜɚɬɶ ɲɢɪɨɤɨ ² ɤɚɤ ɫɨɛɫɬɜɟɧ ɧɨ ɧɚɛɥɸɞɟɧɢɟ ɢ ɤɚɤ ɷɤɫɩɟɪɢɦɟɧɬ
ɋɬɨɯɚɫɬɢɱɟɫɤɢɟ ɦɨɞɟɥɢ ɩɨɥɭɱɚɸɬ ɫ ɩɪɢɦɟɧɟɧɢɟɦ ɷɤɫɩɟɪɢɦɟɧ ɬɚɥɶɧɨ ɫɬɚɬɢɫɬɢɱɟɫɤɢɯ ɦɟɬɨɞɨɜ ɨɛɪɚɛɚɬɵɜɚɹ ɞɚɧɧɵɟ ɩɚɫɫɢɜɧɨɝɨ ɢɥɢ ɚɤɬɢɜɧɨɝɨ ɷɤɫɩɟɪɢɦɟɧɬɚ ɪɢɫ ɉɪɢ ɩɚɫɫɢɜɧɨɦ ɷɤɫɩɟɪɢ ɦɟɧɬɟ ɩɪɨɜɨɞɹɬ ɨɩɵɬɵ ɫ ɩɨɨɱɟɪɟɞɧɵɦ ɜɚɪɶɢɪɨɜɚɧɢɟɦ ɤɚɠɞɨɝɨ ɮɚɤɬɨ ɪɚ ɢɥɢ ɧɚɛɢɪɚɸɬ ɧɟɨɛɯɨɞɢɦɵɣ ɫɬɚɬɢɫɬɢɱɟɫɤɢɣ ɦɚɬɟɪɢɚɥ ɧɟ ɩɪɨɜɨɞɹ ɫɩɟɰɢɚɥɶɧɵɯ ɨɩɵɬɨɜ Ⱥɤɬɢɜɧɵɣ ɷɤɫɩɟɪɢɦɟɧɬ ɫɬɚɜɢɬɫɹ ɩɨ ɡɚɪɚɧɟɟ ɫɨɫɬɚɜɥɟɧɧɨɦɭ ɩɥɚɧɭ ɬ ɧ ɩɥɚɧɢɪɨɜɚɧɢɟ ɷɤɫɩɟɪɢɦɟɧɬɚ ɩɪɢ ɷɬɨɦ ɩɪɟɞɭɫɦɚɬɪɢɜɚɟɬɫɹ ɢɡɦɟɧɟɧɢɟ ɜɫɟɯ ɢɡɭɱɚɟɦɵɯ ɮɚɤɬɨɪɨɜ > @
Ɉɛɪɚɛɨɬɤɚ ɩɨɥɭɱɟɧɧɵɯ ɞɚɧɧɵɯ ɩɪɨɜɨɞɢɬɫɹ ɫ ɩɨɦɨɳɶɸ ɤɨɪɪɟɥɹ
ɰɢɨɧɧɨɝɨ ɢ ɪɟɝɪɟɫɫɢɨɧɧɨɝɨ ɚɧɚɥɢɡɨɜ
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ȼɵɞɜɢɠɟɧɢɟ |
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ǸȐș ǹȚȘȈȚȍȋȐȧ ȗȖșȚȘȖȍȕȐȧ ȔȈȚȍȔȈȚȐȟȍșȒȖȑ ȔȖȌȍȓȐ |
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Ȼɨɥɟɟɜɵɫɨɤɭɸɫɬɭɩɟɧɶɩɨɡɧɚɧɢɹɫɢɫɬɟɦɵɨɛɟɫɩɟɱɢɜɚɟɬɷɤɫɩɟɪɢ ɦɟɧɬ ɩɪɢ ɤɨɬɨɪɨɦ ɢɫɫɥɟɞɭɟɦɵɟ ɩɪɨɰɟɫɫɵ ɜɨɫɫɨɡɞɚɸɬɫɹ ɜ ɧɟɨɛɯɨɞɢ ɦɵɯ ɭɫɥɨɜɢɹɯ ɗɤɫɩɟɪɢɦɟɧɬ ɩɨɡɜɨɥɹɟɬ ɩɨɫɬɪɨɢɬɶ ɛɨɥɟɟ ɫɨɜɟɪɲɟɧɧɵɟ ɦɨɞɟɥɢ ɱɟɦ ɦɨɞɟɥɢ ɩɨɥɭɱɟɧɧɵɟ ɜ ɪɟɡɭɥɶɬɚɬɟ ɬɨɥɶɤɨ ɧɚɛɥɸɞɟɧɢɹ
ɋɩɨɫɨɛɵ ɨɩɪɟɞɟɥɟɧɢɹ ɫɨɫɬɚɜɚ ɛɟɬɨɧɚ ɪɚɫɫɦɨɬɪɟɧɧɵɟ ɜ ɪɹɞɟ ɪɚ ɛɨɬ > @ ɨɫɧɨɜɚɧɵ ɧɚ ɫɪɟɞɧɢɯ ɡɚɜɢɫɢɦɨɫɬɹɯ ɇɚ ɩɪɨɢɡɜɨɞ ɫɬɜɟ ɩɨɫɥɟ ɩɪɨɜɟɞɟɧɢɹ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɨɩɵɬɨɜ ɷɬɢ ɡɚɜɢɫɢɦɨɫɬɢ ɭɬɨɱɧɹɸɬ ɢ ɬɟɦ ɫɚɦɵɦ ɩɨɜɵɲɚɸɬ ɬɨɱɧɨɫɬɶ ɪɚɫɱɟɬɚ ɫɨɫɬɚɜɚ ɛɟɬɨɧɚ Ɉɞɧɚɤɨɜɪɚɫɫɦɨɬɪɟɧɧɵɯɫɩɨɫɨɛɚɯɜɨɢɡɛɟɠɚɧɢɟɫɥɨɠɧɵɯɦɚɬɟɦɚɬɢ ɱɟɫɤɢɯ ɪɚɫɱɟɬɨɜ ɢ ɜɫɥɟɞɫɬɜɢɟ ɧɟɞɨɫɬɚɬɨɱɧɨɣ ɢɧɮɨɪɦɚɰɢɢ ɨ ɜɡɚɢɦɧɨɦ ɜɥɢɹɧɢɢ ɦɧɨɝɢɯ ɮɚɤɬɨɪɨɜ ɧɚ ɫɜɨɣɫɬɜɚ ɛɟɬɨɧɚ ɱɢɫɥɨ ɭɱɢɬɵɜɚɟɦɵɯ ɮɚɤɬɨɪɨɜ ɨɝɪɚɧɢɱɟɧɨ ɱɬɨ ɜ ɢɡɜɟɫɬɧɨɣ ɦɟɪɟ ɩɪɟɩɹɬɫɬɜɭɟɬ ɞɚɥɶɧɟɣɲɟ ɦɭ ɩɨɜɵɲɟɧɢɸ ɬɨɱɧɨɫɬɢ ɪɚɫɱɟɬɚ
ɉɪɢɦɟɧɟɧɢɟ ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ ɦɨɞɟɥɟɣ ɤɨɬɨɪɵɟ ɩɨɥɭɱɚɸɬ ɜ ɪɟ ɡɭɥɶɬɚɬɟ ɩɪɟɞɜɚɪɢɬɟɥɶɧɵɯ ɨɩɵɬɨɜ ɢ ɜ ɤɨɬɨɪɵɯ ɦɨɠɧɨ ɭɱɢɬɵɜɚɬɶ ɛɨɥɶɲɨɟɤɨɥɢɱɟɫɬɜɨɮɚɤɬɨɪɨɜɞɟɣɫɬɜɭɸɳɢɯɜɤɨɧɤɪɟɬɧɵɯɭɫɥɨɜɢɹɯ ɩɪɨɢɡɜɨɞɫɬɜɚ ɩɨɡɜɨɥɹɟɬ ɧɟ ɬɨɥɶɤɨ ɭɬɨɱɧɢɬɶ ɬɟɯɧɨɥɨɝɢɱɟɫɤɢɟ ɪɚɫɱɟ ɬɵ ɧɨ ɢ ɭɫɩɟɲɧɨ ɭɩɪɚɜɥɹɬɶ ɤɚɱɟɫɬɜɨɦ ɛɟɬɨɧɚ ɢ ɩɪɨɦɵɲɥɟɧɧɵɦ ɩɪɨ ɢɡɜɨɞɫɬɜɨɦ ɠɟɥɟɡɨɛɟɬɨɧɧɵɯ ɢɡɞɟɥɢɣ ɜɧɨɫɹ ɧɟɨɛɯɨɞɢɦɵɟ ɤɨɪɪɟɤɬɢɜɵ ɜ ɩɪɨɰɟɫɫ ɩɪɢ ɥɸɛɨɦ ɢɡɦɟɧɟɧɢɢ ɜɯɨɞɧɵɯ ɩɚɪɚɦɟɬɪɨɜ
Ɇɚɬɟɦɚɬɢɱɟɫɤɢɟɦɨɞɟɥɢɩɨɥɭɱɚɸɬɜɪɟɡɭɥɶɬɚɬɟɩɪɚɜɢɥɶɧɨɫɩɥɚ ɧɢɪɨɜɚɧɧɨɝɨɷɤɫɩɟɪɢɦɟɧɬɚɢɩɪɢɦɟɧɹɸɬɜɬɟɯɤɨɧɤɪɟɬɧɵɯɭɫɥɨɜɢɹɯ ɞɥɹ ɤɨɬɨɪɵɯ ɨɧɢ ɩɨɥɭɱɟɧɵ Ⱦɥɹ ɞɪɭɝɢɯ ɭɫɥɨɜɢɣ ɬɪɟɛɭɟɬɫɹ ɩɪɨɜɟɪɹɬɶ ɩɪɢɦɟɧɢɦɨɫɬɶ ɬɨɣ ɢɥɢ ɢɧɨɣ ɦɨɞɟɥɢ ɢ ɜɧɨɫɢɬɶ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɤɨɪ ɪɟɤɬɢɜɵ
ȼ ɦɨɞɟɥɹɯ ɧɚɪɹɞɭ ɫ ɨɰɟɧɤɨɣ ɫɜɨɣɫɬɜ ɩɨ ɫɬɚɧɞɚɪɬɧɵɦ ɦɟɬɨɞɢɤɚɦ
ɦɨɝɭɬ ɢɫɩɨɥɶɡɨɜɚɬɶɫɹ ɢ ɞɪɭɝɢɟ ɤɪɢɬɟɪɢɢ ɜ ɧɚɢɛɨɥɶɲɟɣ ɦɟɪɟ ɫɨɨɬɜɟɬ ɫɬɜɭɸɳɢɟ ɮɢɡɢɤɟ ɹɜɥɟɧɢɣ ɢ ɫɩɨɫɨɛɫɬɜɭɸɳɢɟ ɩɨɜɵɲɟɧɢɸ ɬɨɱɧɨɫɬɢ ɪɚɫɱɟɬɨɜ
ɉɪɨɰɟɫɫ ɩɨɥɭɱɟɧɢɹ ɦɧɨɝɨɮɚɤɬɨɪɧɵɯ ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ ɦɨɞɟɥɟɣ ɜɤɥɸɱɚɟɬ ɧɟɫɤɨɥɶɤɨ ɷɬɚɩɨɜ ɩɨɤɚɡɚɧɧɵɯ ɧɚ ɪɢɫ
Ɉɫɧɨɜɧɨɣ ɢɫɯɨɞɧɵɣ ɫɨɫɬɚɜ ɛɟɬɨɧɚ ɧɚɡɧɚɱɚɟɬɫɹ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɭɤɚɡɚɧɢɹɦɢ > @ ȼ ɨɛɹɡɚɬɟɥɶɧɨɦ ɩɨɪɹɞɤɟ ɨɫɧɨɜɧɨɣ ɫɨ ɫɬɚɜ ɛɟɬɨɧɚ ɞɨɥɠɟɧ ɛɵɬɶ ɩɪɨɜɟɪɟɧ ɢ ɨɬɤɨɪɪɟɤɬɢɪɨɜɚɧ ɨɩɵɬɚɦɢ
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ȼɵɛɨɪ ɮɚɤɬɨɪɨɜ ɢ ɢɧɬɟɪɜɚɥɨɜ ɢɯ ɜɚɪɶɢɪɨɜɚɧɢɹ |
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ȼɵɛɨɪ ɩɥɚɧɚ ɢ ɭɫɥɨɜɢɣ ɩɪɨɜɟɞɟɧɢɹ ɷɤɫɩɟɪɢɦɟɧɬɚ |
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Ɋɚɫɱɟɬ ɜɫɟɯ ɫɨɫɬɚɜɨɜ ɛɟɬɨɧɧɨɣ ɫɦɟɫɢ ɢ ɩɪɨɜɟɞɟɧɢɟ ɷɤɫɩɟɪɢɦɟɧɬɚ |
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ɩɨ ɜɵɛɪɚɧɧɨɦɭ ɩɥɚɧɭ |
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Ɉɛɪɚɛɨɬɤɚ ɪɟɡɭɥɶɬɚɬɨɜ ɷɤɫɩɟɪɢɦɟɧɬɚ ɫ ɩɨɥɭɱɟɧɢɟɦ ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ |
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ɡɚɜɢɫɢɦɨɫɬɟɣ ɫɜɨɣɫɬɜ ɛɟɬɨɧɚ ɢ ɛɟɬɨɧɧɨɣ ɫɦɟɫɢ ɨɬ ɜɵɛɪɚɧɧɵɯ ɮɚɤɬɨɪɨɜ |
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Ɋɚɫɱɟɬ ɨɫɧɨɜɧɨɝɨ ɢɫɯɨɞɧɨɝɨ ɫɨɫɬɚɜɚ ɛɟɬɨɧɧɨɣ ɫɦɟɫɢ |
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ǸȐș ȅȚȈȗȣ ȗȖȓțȟȍȕȐȧ ȔȕȖȋȖȜȈȒȚȖȘȕȣȝ |
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ȔȈȚȍȔȈȚȐȟȍșȒȐȝ ȔȖȌȍȓȍȑ |
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ɉɨɫɥɟɩɪɨɜɟɞɟɧɢɹɨɩɵɬɨɜɜɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢɭɤɚɡɚɧɧɨɣɩɥɚ ɧɨɦ ɢɯ ɩɨɜɬɨɪɹɸɬ ɜ ɨɛɪɚɬɧɨɣ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ ɩɨɥɭɱɚɹ ɬɚɤɢɦ ɨɛɪɚɡɨɦ ɩɨ ɞɜɚ ɧɚɛɥɸɞɟɧɢɹ ɪɟɡɭɥɶɬɚɬɚ ɜ ɤɚɠɞɨɦ ɨɩɵɬɟ
Ɉɛɴɟɦ ɡɚɦɟɫɚ ɜ ɤɚɠɞɨɦ ɨɩɵɬɟ ɭɫɬɚɧɚɜɥɢɜɚɸɬ ɫ ɭɱɟɬɨɦ ɱɢɫɥɚ ɨɩɪɟɞɟɥɹɟɦɵɦɯɚɪɚɤɬɟɪɢɫɬɢɤɛɟɬɨɧɚɩɪɨɱɧɨɫɬɟɣɩɪɢɫɠɚɬɢɢɪɚɫɬɹ ɠɟɧɢɢ ɢɡɝɢɛɟ ɢ ɬ ɞ ɉɪɢɝɨɬɨɜɥɟɧɢɟ ɛɟɬɨɧɧɨɣ ɫɦɟɫɢ ɮɨɪɦɨɜɚɧɢɟ ɨɛɪɚɡɰɨɜ ɢɫɩɵɬɚɧɢɟ ɛɟɬɨɧɧɨɣ ɫɦɟɫɢ ɢ ɡɚɬɜɟɪɞɟɜɲɟɝɨ ɛɟɬɨɧɚ ɜɵɱɢɫ ɥɟɧɢɟ ɮɚɤɬɢɱɟɫɤɢɯ ɪɚɫɯɨɞɨɜ ɦɚɬɟɪɢɚɥɨɜ ɜɵɩɨɥɧɹɸɬ ɜ ɨɛɵɱɧɨɦ ɩɨ ɪɹɞɤɟ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɭɤɚɡɚɧɢɹɦɢ > @
ǪȣȉȖȘ ȔȍȚȖȌȐȒȐ ȗȓȈȕȐȘȖȊȈȕȐȧ ȥȒșȗȍȘȐȔȍȕȚȈ
Ɉɞɧɨɣ ɢɡ ɨɫɧɨɜɧɵɯ ɡɚɞɚɱ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɯ ɢɫɫɥɟɞɨɜɚɧɢɣ ɹɜ ɥɹɟɬɫɹ ɨɩɪɟɞɟɥɟɧɢɟ ɱɢɫɥɟɧɧɵɯ ɡɧɚɱɟɧɢɣ ɩɚɪɚɦɟɬɪɨɜ ɧɟɨɛɯɨɞɢɦɵɯ ɞɥɹ ɪɚɫɱɟɬɚ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ ɦɨɞɟɥɟɣ ɢ ɩɨɞɬɜɟɪɠɞɟ ɧɢɹ ɚɞɟɤɜɚɬɧɨɫɬɢ ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ ɦɨɞɟɥɟɣ
Ɍɪɚɞɢɰɢɨɧɧɵɟ ɦɟɬɨɞɵ ɢɫɫɥɟɞɨɜɚɧɢɣ ɫɜɹɡɚɧɵ ɫ ɷɤɫɩɟɪɢɦɟɧɬɨɦ ɤɨɬɨɪɵɣ ɬɪɟɛɭɟɬ ɛɨɥɶɲɢɯ ɡɚɬɪɚɬ ɫɢɥ ɢ ɫɪɟɞɫɬɜ ɬɚɤ ɤɚɤ ɨɫɧɨɜɚɧ ɧɚ ɩɨɨɱɟɪɟɞɧɨɦ ɜɚɪɶɢɪɨɜɚɧɢɢ ɨɬɞɟɥɶɧɵɯ ɧɟɡɚɜɢɫɢɦɵɯ ɩɟɪɟɦɟɧɧɵɯ
ɜ ɭɫɥɨɜɢɹɯ ɤɨɝɞɚ ɨɫɬɚɥɶɧɵɟ ɩɟɪɟɦɟɧɧɵɟ ɫɨɯɪɚɧɹɸɬɫɹ ɧɟɢɡɦɟɧɧɵɦɢ ɋɥɨɠɧɨɫɬɶɬɚɤɨɝɨɷɤɫɩɟɪɢɦɟɧɬɚɨɩɪɟɞɟɥɹɟɬɫɹɱɢɫɥɨɦɜɫɟɜɨɡɦɨɠɧɵɯ ɤɨɦɛɢɧɚɰɢɣ ɡɧɚɱɟɧɢɣ ɩɟɪɟɦɟɧɧɵɯ ɮɚɤɬɨɪɨɜ ȿɫɥɢ ɢɫɫɥɟɞɭɟɬɫɹ k ɮɚɤɬɨɪɨɜɞɥɹɤɚɠɞɨɝɨɢɡɤɨɬɨɪɵɯɫɭɳɟɫɬɜɭɟɬɪ ɭɪɨɜɧɟɣɬɨɱɢɫɥɨ ɤɨɦɛɢɧɚɰɢɣ ɛɭɞɟɬ ɪɚɜɧɨ ɪk ɧɚɩɪɢɦɟɪ ɞɥɹ ɫɢɫɬɟɦɵ ɫ ɮɚɤɬɨɪɚɦɢ
ɧɚ ɭɪɨɜɧɹɯ ɢɦɟɟɬɫɹ ɫɨɫɬɨɹɧɢɣ ɉɪɹɦɨɣ ɩɟɪɟɛɨɪ ɜɜɢɞɭ ɨɝɪɨɦ ɧɨɝɨ ɱɢɫɥɚ ɫɨɫɬɨɹɧɢɣ ɧɟɪɚɰɢɨɧɚɥɟɧ ɚ ɢɧɨɝɞɚ ɧɟɜɨɡɦɨɠɟɧ ɩɨɷɬɨɦɭ ɩɪɢɛɟɝɚɸɬ ɤ ɩɪɨɰɟɞɭɪɟ ɩɥɚɧɢɪɨɜɚɧɢɹ ɷɤɫɩɟɪɢɦɟɧɬɚ
ȼɧɚɫɬɨɹɳɟɟɜɪɟɦɹɜɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɯɢɫɫɥɟɞɨɜɚɧɢɹɯɲɢɪɨɤɨ ɩɪɢɦɟɧɹɸɬɫɹ ɦɟɬɨɞɵ ɩɥɚɧɢɪɨɜɚɧɢɹ ɷɤɫɩɟɪɢɦɟɧɬɚ Ȼɥɚɝɨɞɚɪɹ ɨɩɬɢ ɦɚɥɶɧɨɣ ɨɪɝɚɧɢɡɚɰɢɢ ɩɥɚɧɢɪɨɜɚɧɢɹ ɷɤɫɩɟɪɢɦɟɧɬɚ ɩɨɹɜɥɹɟɬɫɹ ɜɨɡ ɦɨɠɧɨɫɬɶ ɫ ɦɢɧɢɦɚɥɶɧɵɦɢ ɡɚɬɪɚɬɚɦɢ ɦɚɬɟɪɢɚɥɶɧɵɯ ɜɪɟɦɟɧɧɵɯ ɢ ɥɸɞɫɤɢɯ ɪɟɫɭɪɫɨɜ ɩɨɥɭɱɢɬɶ ɜɫɸ ɢɧɮɨɪɦɚɰɢɸ ɧɟɨɛɯɨɞɢɦɭɸ ɞɥɹ ɩɨɫɬɪɨɟɧɢɹ ɚɞɟɤɜɚɬɧɵɯ ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ ɦɨɞɟɥɟɣ ɢɫɫɥɟɞɭɟɦɵɯ ɨɛɴɟɤ ɬɨɜ > @
Ɇɟɬɨɞɵ ɩɥɚɧɢɪɨɜɚɧɢɹ ɷɤɫɩɟɪɢɦɟɧɬɨɜ ɩɨɡɜɨɥɹɸɬ ɨɫɭɳɟɫɬɜɢɬɶ ɩɟɪɟɜɨɞ ɪɟɡɭɥɶɬɚɬɨɜ ɫ ɹɡɵɤɚ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɦɨɞɟɥɢ ɧɚ ɹɡɵɤ ɤɨɬɨ ɪɵɦ ɜɥɚɞɟɟɬ ɷɤɫɩɟɪɢɦɟɧɬɚɬɨɪ ɉɪɢɧɹɬɢɟ ɪɟɲɟɧɢɣ ɧɚ ɤɚɠɞɨɦ ɷɬɚɩɟ ɷɤɫɩɟɪɢɦɟɧɬɢɪɨɜɚɧɢɹ ɩɪɢɜɨɞɢɬ ɤ ɰɢɤɥɢɱɟɫɤɨɦɭ ɩɨɜɬɨɪɟɧɢɸ ɷɬɚɩɨɜ ɱɬɨ ɩɪɢɞɚɟɬ ɝɢɛɤɨɫɬɶ ɜɫɟɦɭ ɩɪɨɰɟɫɫɭ ɷɤɫɩɟɪɢɦɟɧɬɢɪɨɜɚɧɢɹ ɢ ɨɛɟɫɩɟ ɱɢɜɚɟɬ ɨɛɪɚɬɧɭɸ ɫɜɹɡɶ
ɉɥɚɧɢɪɨɜɚɧɢɟ ɷɤɫɩɟɪɢɦɟɧɬɚ ɩɨɡɜɨɥɹɟɬ ɢɫɩɨɥɶɡɭɹ ɚɩɩɚɪɚɬ ɬɟɨ ɪɢɢ ɜɟɪɨɹɬɧɨɫɬɟɣ ɢ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɫɬɚɬɢɫɬɢɤɢ ɨɛɨɫɧɨɜɚɬɶ ɩɪɢɧɹɬɵɟ ɪɟɲɟɧɢɹ ɢ ɩɨɤɚɡɚɬɶ ɢɯ ɩɪɟɢɦɭɳɟɫɬɜɚ ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ ɞɪɭɝɢɦɢ ɜɨɡ ɦɨɠɧɵɦɢ ɪɟɲɟɧɢɹɦɢ > @
Ɍɟɯɧɢɤɚ ɩɥɚɧɢɪɨɜɚɧɢɹ ɷɤɫɩɟɪɢɦɟɧɬɚ ɬɚɤɨɜɚ ɧɚ ɤɚɠɞɨɦ ɲɚɝɟ ɫɬɚ
ɜɢɬɫɹ ɧɟɛɨɥɶɲɚɹ ɫɟɪɢɹ ɨɩɵɬɨɜ ɜ ɤɚɠɞɨɦ ɢɡ ɤɨɬɨɪɵɯ ɜɚɪɶɢɪɭɸɬɫɹ ɩɨ
ɨɩɪɟɞɟɥɟɧɧɵɦ ɩɪɚɜɢɥɚɦ ɜɫɟ ɮɚɤɬɨɪɵ Ɇɚɬɟɦɚɬɢɱɟɫɤɚɹ ɨɛɪɚɛɨɬɤɚ ɪɟ ɡɭɥɶɬɚɬɨɜ ɷɤɫɩɟɪɢɦɟɧɬɚ ɩɨɡɜɨɥɹɟɬ ɜɵɪɚɛɨɬɚɬɶ ɭɫɥɨɜɢɹ ɩɪɨɜɟɞɟɧɢɹ ɫɥɟɞɭɸɳɟɣ ɫɟɪɢɢ ɨɩɵɬɨɜ ɧɚɩɪɚɜɥɟɧɧɵɯ ɤ ɞɨɫɬɢɠɟɧɢɸ ɨɩɬɢɦɭɦɚ
Ɉɫɨɛɟɧɧɨɫɬɢ ɩɥɚɧɢɪɨɜɚɧɢɹ ɷɤɫɩɟɪɢɦɟɧɬɚ
x ɫɬɪɟɦɥɟɧɢɟ ɤ ɦɢɧɢɦɢɡɚɰɢɢ ɨɛɳɟɝɨ ɱɢɫɥɚ ɨɩɵɬɨɜ
xɨɞɧɨɜɪɟɦɟɧɧɨɟ ɜɚɪɶɢɪɨɜɚɧɢɟ ɜɫɟɦɢ ɩɟɪɟɦɟɧɧɵɦɢ ɨɩɪɟɞɟ ɥɹɸɳɢɦɢ ɩɪɨɰɟɫɫ ɩɨ ɫɩɟɰɢɚɥɶɧɵɦ ɩɪɚɜɢɥɚɦ ² ɚɥɝɨɪɢɬɦɚɦ
xɜɵɛɨɪ ɱɟɬɤɨɣ ɫɬɪɚɬɟɝɢɢ ɩɨɡɜɨɥɹɸɳɟɣ ɩɪɢɧɢɦɚɬɶ ɨɛɨɫɧɨɜɚɧ
ɧɵɟ ɪɟɲɟɧɢɹ ɩɨɫɥɟ ɤɚɠɞɨɣ ɫɟɪɢɢ ɷɤɫɩɟɪɢɦɟɧɬɨɜ
ɉɥɚɧɢɪɨɜɚɧɢɟ ɷɤɫɩɟɪɢɦɟɧɬɚ ɦɨɠɟɬ ɫɥɭɠɢɬɶ ɨɫɧɨɜɨɣ ɞɥɹ ɚɜɬɨɦɚ ɬɢɡɢɪɨɜɚɧɧɨɝɨ ɩɪɨɟɤɬɢɪɨɜɚɧɢɹ ɤɚɤɨɝɨ ɥɢɛɨ ɩɪɨɰɟɫɫɚ Ⱦɥɹ ɷɬɨɝɨ ɧɟɨɛɯɨɞɢɦɨ ɧɚɥɢɱɢɟ ɦɚɬɟɦɚɬɢɱɟɫɤɨɝɨ ɨɩɢɫɚɧɢɹ ɨɛɴɟɤɬɚ ɩɪɨɟɤɬɢɪɨ ɜɚɧɢɹ ɢɥɢ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɦɨɞɟɥɢ ɩɨɥɨɠɟɧɧɨɣ ɜ ɨɫɧɨɜɭ ɤɨɦɩɶɸɬɟɪ ɧɨɣ ɦɨɞɟɥɢ ɧɚ ɩɨɞɯɨɞɹɳɟɦ ɹɡɵɤɟ ɩɪɨɝɪɚɦɦɢɪɨɜɚɧɢɹ
Ɍɚɤɚɹɦɨɞɟɥɶɞɨɥɠɧɚɨɬɪɚɠɚɬɶɮɭɧɤɰɢɨɧɚɥɶɧɨɟɜɡɚɢɦɨɞɟɣɫɬɜɢɟ ɷɥɟɦɟɧɬɨɜ ɢ ɢɯ ɫɨɟɞɢɧɟɧɢɣ ɩɪɨɫɬɪɚɧɫɬɜɟɧɧɵɟ ɫɜɹɡɢ ɢ ɪɚɫɩɨɥɨɠɟɧɢɟ ȼ ɩɪɨɰɟɫɫɟ ɢɡɦɟɪɟɧɢɣ ɩɨɫɥɟɞɭɸɳɟɣ ɨɛɪɚɛɨɬɤɢ ɞɚɧɧɵɯ ɚ ɬɚɤɠɟ ɮɨɪɦɚɥɢɡɚɰɢɢ ɪɟɡɭɥɶɬɚɬɨɜ ɜ ɜɢɞɟ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɦɨɞɟɥɢ ɜɨɡɧɢɤɚɸɬ ɩɨɝɪɟɲɧɨɫɬɢ ɢ ɬɟɪɹɟɬɫɹ ɱɚɫɬɶ ɢɧɮɨɪɦɚɰɢɢ ɫɨɞɟɪɠɚɳɟɣɫɹ ɜ ɢɫɯɨɞ ɧɵɯ ɞɚɧɧɵɯ ɉɪɢɦɟɧɟɧɢɟ ɦɟɬɨɞɨɜ ɩɥɚɧɢɪɨɜɚɧɢɹ ɷɤɫɩɟɪɢɦɟɧɬɚ ɩɨɡ ɜɨɥɹɟɬ ɨɩɪɟɞɟɥɢɬɶ ɩɨɝɪɟɲɧɨɫɬɶ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɦɨɞɟɥɢ ɢ ɫɭɞɢɬɶ ɨ ɟɟ ɚɞɟɤɜɚɬɧɨɫɬɢ ȿɫɥɢ ɬɨɱɧɨɫɬɶ ɦɨɞɟɥɢ ɨɤɚɡɵɜɚɟɬɫɹ ɧɟɞɨɫɬɚɬɨɱɧɨɣ ɬɨ ɩɪɢɦɟɧɟɧɢɟ ɦɟɬɨɞɨɜ ɩɥɚɧɢɪɨɜɚɧɢɹ ɷɤɫɩɟɪɢɦɟɧɬɚ ɩɨɡɜɨɥɹɟɬ ɦɨ ɞɟɪɧɢɡɢɪɨɜɚɬɶ ɦɚɬɟɦɚɬɢɱɟɫɤɭɸ ɦɨɞɟɥɶ ɫ ɩɪɨɜɟɞɟɧɢɟɦ ɞɨɩɨɥɧɢɬɟɥɶ ɧɵɯ ɨɩɵɬɨɜ ɛɟɡ ɩɨɬɟɪɢ ɩɪɟɞɵɞɭɳɟɣ ɢɧɮɨɪɦɚɰɢɢ ɢ ɫ ɦɢɧɢɦɚɥɶɧɵɦɢ
ɡɚɬɪɚɬɚɦɢ ɐɟɥɶ ɩɥɚɧɢɪɨɜɚɧɢɹ ɷɤɫɩɟɪɢɦɟɧɬɚ ² ɧɚɯɨɠɞɟɧɢɟ ɬɚɤɢɯ ɭɫɥɨɜɢɣ
ɢ ɩɪɚɜɢɥ ɩɪɨɜɟɞɟɧɢɹ ɨɩɵɬɨɜ ɩɪɢ ɤɨɬɨɪɵɯ ɭɞɚɫɬɫɹ ɩɨɥɭɱɢɬɶ ɧɚɞɟɠ ɧɭɸ ɢ ɞɨɫɬɨɜɟɪɧɭɸ ɢɧɮɨɪɦɚɰɢɸ ɨɛ ɨɛɴɟɤɬɟ ɫ ɧɚɢɦɟɧɶɲɢɦɢ ɡɚɬɪɚɬɚ ɦɢ ɬɪɭɞɚ ɚ ɬɚɤɠɟ ɩɪɟɞɫɬɚɜɢɬɶ ɷɬɭ ɢɧɮɨɪɦɚɰɢɸ ɜ ɤɨɦɩɚɤɬɧɨɣ ɢ ɭɞɨɛ ɧɨɣ ɮɨɪɦɟ ɫ ɤɨɥɢɱɟɫɬɜɟɧɧɨɣ ɨɰɟɧɤɨɣ ɬɨɱɧɨɫɬɢ
ȼɫɜɨɟɣɪɚɛɨɬɟɦɵɛɭɞɟɦ ɪɚɫɫɦɚɬɪɢɜɚɬɶɦɟɬɨɞɢɤɭɩɨɥɧɨɝɨɮɚɤ ɬɨɪɧɨɝɨ ɷɤɫɩɟɪɢɦɟɧɬɚ > @ ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɞɚɧɧɨɣ ɦɟɬɨɞɢɤɨɣ ɩɪɢ ɩɪɨɜɟɞɟɧɢɢ ɷɤɫɩɟɪɢɦɟɧɬɨɜ ɜɫɟ ɮɚɤɬɨɪɵ ɜɚ ɪɶɢɪɭɸɬɫɹ ɧɚ ɬɪɟɯ ɭɪɨɜɧɹɯ ɫɪɟɞɧɟɦ ɨɫɧɨɜɧɨɦ ɧɭɥɟɜɨɦ ɜɟɪɯɧɟɦ ɢ ɧɢɠɧɟɦ ɩɪɢ ɷɬɨɦ ɭɪɨɜɧɢ ɨɬɫɬɨɹɬ ɨɬ ɨɫɧɨɜɧɨɝɨ ɧɚ ɨɞɢɧɚɤɨɜɭɸ ɜɟ ɥɢɱɢɧɭ ɧɚɡɵɜɚɟɦɭɸ ɢɧɬɟɪɜɚɥɨɦ ɜɚɪɶɢɪɨɜɚɧɢɹ'ɏ.
Ⱦɥɹ ɭɩɪɨɳɟɧɢɹ ɡɚɩɢɫɟɣ ɢ ɩɨɫɥɟɞɭɸɳɢɯ ɪɚɫɱɟɬɨɜ ɭɪɨɜɧɢ ɮɚɤɬɨ ɪɨɜ ɩɟɪɟɜɨɞɹɬ ɜ ɤɨɞɨɜɵɣ ɧɨɪɦɚɥɢɡɨɜɚɧɧɵɣ ɦɚɫɲɬɚɛ ɫ ɩɨɦɨɳɶɸ ɫɥɟɞɭɸɳɟɝɨ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ
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ɝɞɟxi² ɡɧɚɱɟɧɢɟiɝɨ ɮɚɤɬɨɪɚ ɜ ɧɨɜɨɦ ɤɨɞɨɜɨɦ ɦɚɫɲɬɚɛɟ 'ɏi² ɢɧɬɟɪɜɚɥ ɜɚɪɶɢɪɨɜɚɧɢɹiɝɨ ɮɚɤɬɨɪɚ
ɏi — ɡɧɚɱɟɧɢɟiɝɨ ɮɚɤɬɨɪɚ ɜ ɫɬɚɪɨɦ ɧɚɬɭɪɚɥɶɧɨɦ ɦɚɫɲɬɚɛɟ ɏi0 — ɨɫɧɨɜɧɨɣ ɭɪɨɜɟɧɶiɝɨ ɮɚɤɬɨɪɚ
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɜɧɨɜɨɦ ɦɚɫɲɬɚɛɟ ɜɟɪɯɧɢɣ ɭɪɨɜɟɧɶ ɮɚɤɬɨɪɚ ɛɭ ɞɟɬ ɪɚɜɟɧ ɧɢɠɧɢɣ ɭɪɨɜɟɧɶ ɪɚɜɟɧ í ɨɫɧɨɜɧɨɣ ɭɪɨɜɟɧɶ ɪɚɜɟɧ ɇɢɠɧɹɹ ɝɪɚɧɢɰɚ ɢɧɬɟɪɜɚɥɚ ɜɚɪɶɢɪɨɜɚɧɢɹ ɧɟ ɦɨɠɟɬ ɛɵɬɶ ɦɟɧɶɲɟ ɩɨɝɪɟɲɧɨɫɬɢ ɨɩɪɟɞɟɥɟɧɢɹ ɮɚɤɬɨɪɚ ɜɟɪɯɧɹɹ ɝɪɚɧɢɰɚ ɢɧɬɟɪɜɚɥɚ ɜɚɪɶɢ ɪɨɜɚɧɢɹ ɨɩɪɟɞɟɥɹɟɬɫɹ ɭɫɥɨɜɢɟɦ ɧɟɜɵɯɨɞɚ ɢɡ ɨɛɥɚɫɬɢ ɨɩɪɟɞɟɥɟɧɢɹ ɮɚɤɬɨɪɚ ɂɧɬɟɪɜɚɥ ɜɚɪɶɢɪɨɜɚɧɢɹ ɤɨɪɪɟɤɬɢɪɭɟɬɫɹ ɜ ɯɨɞɟ ɷɤɫɩɟɪɢɦɟɧ
ɬɚ ɟɫɥɢ ɧɟ ɭɞɨɜɥɟɬɜɨɪɹɟɬɫɹ ɭɫɥɨɜɢɟ ɚɞɟɤɜɚɬɧɨɫɬɢ ɦɨɞɟɥɢ ɗɤɫɩɟɪɢɦɟɧɬɵ ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɤɨɥɢɱɟɫɬɜɚ ɮɚɤɬɨɪɨɜ ɜɵɩɨɥɧɹ
ɸɬɫɹ ɩɨ ɫɩɟɰɢɚɥɶɧɨɦɭ ɩɥɚɧɭ ɩɪɢɥ ɉɪɢ ɩɪɨɜɟɞɟɧɢɢ ɷɤɫɩɟɪɢɦɟɧɬɨɜ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɜɵɛɪɚɧɧɵɦ
ɩɥɚɧɨɦ ɰɟɥɟɫɨɨɛɪɚɡɧɨ ɨɩɵɬɵ ɜ ɧɭɥɟɜɵɯ ɬɨɱɤɚɯ ɤɨɝɞɚ ɜɫɟ ɮɚɤɬɨɪɵ ɧɚɯɨɞɹɬɫɹ ɧɚ ɨɫɧɨɜɧɨɦ ɭɪɨɜɧɟ ɪɚɜɧɨɦɟɪɧɨ ɪɚɫɩɪɟɞɟɥɹɬɶ ɦɟɠɞɭ ɜɫɟ ɦɢ ɨɫɬɚɥɶɧɵɦɢ > @
ɉɨɫɥɟ ɩɪɨɜɟɞɟɧɢɹ ɷɤɫɩɟɪɢɦɟɧɬɨɜ ɩɨɥɭɱɚɸɬ ɦɚɬɟɦɚɬɢɱɟɫɤɢɟ ɡɚ ɜɢɫɢɦɨɫɬɢ ɜ ɜɢɞɟ ɩɨɥɢɧɨɦɚnɣ ɫɬɟɩɟɧɢ
Ⱦɥɹ ɩɥɚɧɨɜ ɩɟɪɜɨɝɨ ɩɨɪɹɞɤɚ
ɚ ɞɜɭɯɮɚɤɬɨɪɧɵɣ ɷɤɫɩɟɪɢɦɟɧɬ
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ɛ ɬɪɟɯɮɚɤɬɨɪɧɵɣ ɷɤɫɩɟɪɢɦɟɧɬ |
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ɜ ɜ ɨɛɳɟɦ ɜɢɞɟ |
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Ⱦɥɹ ɩɥɚɧɨɜ ɜɬɨɪɨɝɨ ɩɨɪɹɞɤɚ |
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ɚ ɞɜɭɯɮɚɤɬɨɪɧɵɣ ɷɤɫɩɟɪɢɦɟɧɬ |
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ɛ ɬɪɟɯɮɚɤɬɨɪɧɵɣ ɷɤɫɩɟɪɢɦɟɧɬ
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ɝɞɟi j k² ɩɨɪɹɞɤɨɜɵɟ ɧɨɦɟɪɚ ɮɚɤɬɨɪɨɜ ɭi — ɢɫɫɥɟɞɭɟɦɨɟ ɫɜɨɣɫɬɜɨ ɛɟɬɨɧɚ ɯi² ɢɫɯɨɞɧɵɟ ɮɚɤɬɨɪɵ
b0, bi, ɢ bij² ɤɨɷɮɮɢɰɢɟɧɬɵ ɭɪɚɜɧɟɧɢɣ ɤɨɬɨɪɵɟ ɜɵɱɢɫɥɹɸɬɫɹ
ɩɨ ɫɥɟɞɭɸɳɢɦ ɮɨɪɦɭɥɚɦ
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ɡɞɟɫɶɭu² ɡɧɚɱɟɧɢɟ ɢɫɫɥɟɞɭɟɦɨɝɨ ɫɜɨɣɫɬɜɚ ɛɟɬɨɧɚ ɜuɦ ɨɩɵɬɟ xiu² ɡɧɚɱɟɧɢɟiɝɨ ɮɚɤɬɨɪɚ ɜuɦ ɨɩɵɬɟ xju² ɡɧɚɱɟɧɢɟjɝɨ ɮɚɤɬɨɪɚ ɜuɦ ɨɩɵɬɟj i
N² ɱɢɫɥɨ ɨɩɵɬɨɜ ɜ ɩɥɚɧɟ ɡɚ ɢɫɤɥɸɱɟɧɢɟɦ ɨɩɵɬɨɜ ɜ ɧɭɥɟɜɵɯ ɬɨɱɤɚɯ ɞɥɹ ɬɪɟɯɮɚɤɬɨɪɧɨɝɨ ɷɤɫɩɟɪɢɦɟɧɬɚN = ɞɥɹ ɱɟɬɵ ɪɟɯɮɚɤɬɨɪɧɨɝɨ ɷɤɫɩɟɪɢɦɟɧɬɚN =
ȼɟɥɢɱɢɧɚ ɦɨɞɭɥɹ ɤɨɷɮɮɢɰɢɟɧɬɚ ɦɨɞɟɥɢ ɭɤɚɡɵɜɚɟɬ ɧɚ ɫɢɥɭ ɜɥɢɹ
ɧɢɹ ɮɚɤɬɨɪɚ ɚ ɡɧɚɤ ² ɧɚ ɧɚɩɪɚɜɥɟɧɢɟ ȼɟɥɢɱɢɧɚ ɤɨɷɮɮɢɰɢɟɧɬɚ ɪɚɜɧɚ
ɜɤɥɚɞɭ ɞɚɧɧɨɝɨ ɮɚɤɬɨɪɚ ɩɪɢ ɩɟɪɟɯɨɞɟ ɮɚɤɬɨɪɚ ɫ ɧɭɥɟɜɨɝɨ ɭɪɨɜɧɹ ɧɚ ɤɪɚɣɧɢɣ ɭɪɨɜɟɧɶ ² ɷɮɮɟɤɬ ɮɚɤɬɨɪɚ ɨɫɧɨɜɧɨɣ ɷɮɮɟɤɬ ɝɥɚɜɧɵɣ ɷɮ ɮɟɤɬ
ɉɨɫɥɟɩɨɥɭɱɟɧɢɹɭɪɚɜɧɟɧɢɣɩɪɨɢɡɜɨɞɹɬɩɪɨɜɟɪɤɭɨɬɥɢɱɢɹɤɨɷɮ ɮɢɰɢɟɧɬɨɜ bi ɨɬ ɧɭɥɹ ɢ ɩɪɢɝɨɞɧɨɫɬɢ ɭɪɚɜɧɟɧɢɣ ɞɥɹ ɨɩɢɫɚɧɢɹ ɢɫɫɥɟ
ɞɭɟɦɵɯ ɡɚɜɢɫɢɦɨɫɬɟɣ ȼɵɛɨɪ ɫɯɟɦɵ ɫɬɚɬɢɫɬɢɱɟɫɤɨɝɨ ɚɧɚɥɢɡɚ ɢ ɪɚɫ ɱɟɬɧɵɯɮɨɪɦɭɥɡɚɜɢɫɢɬɨɬɬɢɩɚɢɫɩɨɥɶɡɨɜɚɧɧɨɝɨɩɥɚɧɚɢɜɢɞɚɩɨɥɭ ɱɚɟɦɵɯ ɭɪɚɜɧɟɧɢɣ
ɉɨɥɭɱɟɧɧɵɟ ɜ ɪɟɡɭɥɶɬɚɬɟ ɪɚɫɱɟɬɨɜ ɡɚɜɢɫɢɦɨɫɬɢ ɭɪɚɜɧɟɧɢɹ ɭɬɨɱɧɹɸɬɫɹ ɩɪɨɢɡɜɨɞɫɬɜɟɧɧɵɦɢ ɷɤɫɩɟɪɢɦɟɧɬɚɦɢ ɢ ɜ ɞɚɥɶɧɟɣɲɟɦ ɦɨ ɝɭɬ ɢɫɩɨɥɶɡɨɜɚɬɶɫɹ ɞɥɹ ɪɟɲɟɧɢɹ ɪɚɡɥɢɱɧɵɯ ɩɪɨɢɡɜɨɞɫɬɜɟɧɧɵɯ ɡɚɞɚɱ ɫɜɹɡɚɧɧɵɯ ɫ ɤɨɪɪɟɤɬɢɪɨɜɤɨɣ ɢ ɨɩɬɢɦɢɡɚɰɢɟɣ ɫɨɫɬɚɜɚ ɛɟɬɨɧɚ
ǴȍȚȖȌȣ șȚȈȚȐșȚȐȟȍșȒȖȑ ȖȉȘȈȉȖȚȒȐ ȌȈȕȕȣȝ
ɉɪɢɩɥɚɧɢɪɨɜɚɧɢɢɷɤɫɩɟɪɢɦɟɧɬɚɜɰɟɥɹɯɭɦɟɧɶɲɟɧɢɹɱɢɫɥɚɜɵ ɯɨɞɧɵɯ ɩɚɪɚɦɟɬɪɨɜ ɦɚɬɟɦɚɬɢɱɟɫɤɨɣ ɦɨɞɟɥɢ ɦɨɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɤɨɪ ɪɟɥɹɰɢɨɧɧɵɣ ɚɧɚɥɢɡ > @ Ⱦɥɹ ɷɬɨɝɨ ɧɭɠɧɨ ɢɡɦɟɪɢɬɶ ɜɫɟ ɩɚɪɚ ɦɟɬɪɵɡɚɬɟɦɨɰɟɧɢɬɶɤɨɪɪɟɥɹɰɢɸɨɬɫɟɹɬɶ ɱɚɫɬɶɩɚɪɚɦɟɬɪɨɜɢɫɬɪɨ ɢɬɶ ɦɨɞɟɥɶ ɞɥɹ ɦɢɧɢɦɚɥɶɧɨ ɜɨɡɦɨɠɧɨɝɨ ɢɯ ɱɢɫɥɚ
Ʉɨɷɮɮɢɰɢɟɧɬ ɩɚɪɧɨɣ ɤɨɪɪɟɥɹɰɢɢ ɹɜɥɹɟɬɫɹ ɨɛɳɟɩɪɢɧɹɬɨɣ ɜ ɦɚ ɬɟɦɚɬɢɱɟɫɤɨɣ ɫɬɚɬɢɫɬɢɤɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɨɣ ɫɜɹɡɢ ɦɟɠɞɭ ɞɜɭɦɹ ɫɥɭ ɱɚɣɧɵɦɢ ɜɟɥɢɱɢɧɚɦɢ Ⱦɥɹ ɩɪɨɜɟɪɤɢ ɡɧɚɱɢɦɨɫɬɢ ɤɨɷɮɮɢɰɢɟɧɬɚ ɩɚɪ ɧɨɣ ɤɨɪɪɟɥɹɰɢɢ ɩɨɥɶɡɭɸɬɫɹ ɬɚɛɥɢɰɟɣ ɤɪɢɬɢɱɟɫɤɢɯ ɡɧɚɱɟɧɢɣ
ɉɨɫɥɟ ɪɟɚɥɢɡɚɰɢɢ ɷɤɫɩɟɪɢɦɟɧɬɚ ɩɨ ɜɵɛɪɚɧɧɨɦɭ ɩɥɚɧɭ ɩɪɨɜɨɞɹɬ ɨɛɪɚɛɨɬɤɭ ɪɟɡɭɥɶɬɚɬɨɜ ɷɤɫɩɟɪɢɦɟɧɬɚ ɫ ɩɨɫɬɪɨɟɧɢɟɦ ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ ɡɚɜɢɫɢɦɨɫɬɟɣ ɫɜɨɣɫɬɜ ɛɟɬɨɧɧɨɣ ɫɦɟɫɢ ɨɬ ɜɵɛɪɚɧɧɵɯ ɮɚɤɬɨɪɨɜ
Ɂɚɬɟɦ ɩɪɨɢɡɜɨɞɹɬ ɩɪɨɜɟɪɤɭ ɨɬɥɢɱɢɹ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɭɪɚɜɧɟɧɢɣbi
ɨɬ ɧɭɥɹ ɢ ɩɪɢɝɨɞɧɨɫɬɢ ɭɪɚɜɧɟɧɢɣ ɞɥɹ ɨɩɢɫɚɧɢɹ ɢɫɫɥɟɞɭɟɦɵɯ ɡɚ ɜɢɫɢɦɨɫɬɟɣ ȼɵɛɨɪ ɫɯɟɦɵ ɫɬɚɬɢɫɬɢɱɟɫɤɨɝɨ ɚɧɚɥɢɡɚ ɢ ɪɚɫɱɟɬɧɵɯ ɮɨɪ ɦɭɥ ɡɚɜɢɫɢɬ ɨɬ ɬɢɩɚ ɢɫɩɨɥɶɡɨɜɚɧɧɨɝɨ ɩɥɚɧɚ ɢ ɜɢɞɚ ɩɨɥɭɱɚɟɦɵɯ ɭɪɚɜ ɧɟɧɢɣ
ɋɬɚɬɢɫɬɢɱɟɫɤɢɣ ɚɧɚɥɢɡ ɥɢɧɟɣɧɵɯ ɡɚɜɢɫɢɦɨɫɬɟɣ ɩɪɨɢɡɜɨɞɢɬɫɹ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ
ɋɪɟɞɧɢɟ ɡɧɚɱɟɧɢɹ ɩɚɪɚɦɟɬɪɚ ɨɩɬɢɦɢɡɚɰɢɢ ɜ ɪɚɡɥɢɱɧɵɯ ɨɩɵ ɬɚɯ ɧɨ ɩɪɢ ɧɟɢɡɦɟɧɧɵɯ ɡɧɚɱɟɧɢɹɯ ɮɚɤɬɨɪɨɜ ɬɨ ɟɫɬɶ ɤɨɝɞɚ ɜɫɟ ɮɚɤ ɬɨɪɵ ɭɞɟɪɠɢɜɚɸɬɫɹ ɧɚ ɨɞɧɢɯ ɢ ɬɟɯ ɠɟ ɭɪɨɜɧɹɯ ɢɦɟɸɬ ɪɚɡɥɢɱɧɵɟ ɜɟɥɢɱɢɧɵ ɜ ɫɢɥɭ ɫɬɚɬɢɫɬɢɱɟɫɤɨɣ ɨɛɭɫɥɨɜɥɟɧɧɨɫɬɢ ɷɤɫɩɟɪɢɦɟɧɬɚ
ɋɭɦɦɚɪɧɚɹ ɩɨɝɪɟɲɧɨɫɬɶ ɩɚɪɚɦɟɬɪɚ ɨɩɬɢɦɢɡɚɰɢɢ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɫɨ ɫɬɚɜɥɹɟɬ ɨɲɢɛɤɭ ɜɨɫɩɪɨɢɡɜɨɞɢɦɨɫɬɢ ɨɞɧɨɝɨ ɢɡ ɨɩɵɬɨɜ ɩɥɚɧɚ ɉɭɫɬɶ ɩɨ ɤɚɤɨɣ ɬɨ ɫɬɪɨɤɟ ɦɚɬɪɢɰɵ ɩɥɚɧɢɪɨɜɚɧɢɹ ɩɪɨɜɨɞɢɬɫɹ n ɩɨɜɬɨɪɧɵɯ
ɨɩɵɬɨɜ ɬɨɝɞɚ ɩɨ ɪɟɡɭɥɶɬɚɬɚɦ ɬɚɤɢɯ ɨɩɵɬɨɜ ɜɵɱɢɫɥɹɟɬɫɹ ɨɲɢɛɤɚ ɜɨɫ
ɩɪɨɢɡɜɨɞɢɦɨɫɬɢ
ɉɨ ɪɟɡɭɥɶɬɚɬɚɦ ɨɩɵɬɨɜ ɜ ɧɭɥɟɜɵɯ ɨɫɧɨɜɧɵɯ ɬɨɱɤɚɯ ɨɩɪɟɞɟɥɹɸɬ
ɚ ɫɪɟɞɧɟɟ ɚɪɢɮɦɟɬɢɱɟɫɤɨɟ ɡɧɚɱɟɧɢɟ
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ɛ ɞɢɫɩɟɪɫɢɸ ɨɲɢɛɤɢ |
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ɜ ɫɪɟɞɧɟɟ ɤɜɚɞɪɚɬɢɱɟɫɤɨɟ ɨɬɤɥɨɧɟɧɢɟ ɯɚɪɚɤɬɟɪɢɡɭɸɳɟɟ ɨɲɢɛɤɭ
ɨɩɵɬɚ
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ɝ ɫɪɟɞɧɸɸ ɤɜɚɞɪɚɬɢɱɟɫɤɭɸ ɨɲɢɛɤɭ ɤɨɷɮɮɢɰɢɟɧɬɨɜ |
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ɝɞɟ y u² ɡɧɚɱɟɧɢɟ ɢɫɫɥɟɞɭɟɦɨɝɨ ɫɜɨɣɫɬɜɚ ɛɟɬɨɧɚ ɜ ɧɭɥɟɜɨɣ ɬɨɱɤɟ ɜuɦ ɨɩɵɬɟ
n ² ɱɢɫɥɨ ɨɩɵɬɨɜ ɜ ɧɭɥɟɜɨɣ ɬɨɱɤɟ ɱɢɫɥɨ ɩɨɜɬɨɪɧɵɯ ɨɩɵɬɨɜ
ɨɞɧɨɣ ɫɬɪɨɤɢ ɩɥɚɧɚ
N² ɱɢɫɥɨ ɨɩɵɬɨɜ ɜ ɩɥɚɧɟ ɡɚ ɢɫɤɥɸɱɟɧɢɟɦ ɨɩɵɬɨɜ ɜ ɧɭɥɟɜɵɯ
ɬɨɱɤɚɯ ɩɨɜɬɨɪɧɵɯ ɨɩɵɬɨɜ
Ⱦɚɥɟɟ ɨɩɪɟɞɟɥɹɸɬ ɪɚɫɱɟɬɧɨɟ ɡɧɚɱɟɧɢɟ ɤɪɢɬɟɪɢɹ ɋɬɶɸɞɟɧɬɚ tɪ
ɩɨ ɮɨɪɦɭɥɟ
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ɢ ɫɪɚɜɧɢɜɚɸɬ ɞɥɹ ɤɚɠɞɨɝɨ bi ɩɨɥɭɱɟɧɧɨɟ ɡɧɚɱɟɧɢɟ tɪ ɫ ɬɚɛɥɢɱɧɵɦ t ɩɪɢ ɱɢɫɥɟ ɫɬɟɩɟɧɟɣ ɫɜɨɛɨɞɵn í
ȿɫɥɢ tɪ t ɬɨ ɩɪɢ ɡɚɞɚɧɧɨɦ ɭɪɨɜɧɟ ɡɧɚɱɢɦɨɫɬɢ D ɤɨɷɮɮɢɰɢɟɧɬ
ɫɱɢɬɚɸɬ ɪɚɜɧɵɦ ɧɭɥɸ ɚ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɣ ɟɦɭ ɱɥɟɧ ɭɪɚɜɧɟɧɢɹ ɨɬ ɛɪɚɫɵɜɚɸɬ ɇɚɱɢɧɚɬɶ ɩɪɨɜɟɪɤɭ ɫɥɟɞɭɟɬ ɫ ɧɚɢɦɟɧɶɲɟɝɨ ɩɨ ɚɛɫɨɥɸɬ ɧɨɦɭ ɡɧɚɱɟɧɢɸ ɤɨɷɮɮɢɰɢɟɧɬɚ ɬɚɤ ɤɚɤ ɜ ɫɥɭɱɚɟ ɟɝɨ ɡɧɚɱɢɦɨɫɬɢ ɧɚɞɨɛɧɨɫɬɶ ɜ ɩɪɨɜɟɪɤɟ ɨɫɬɚɥɶɧɵɯ ɜɟɥɢɱɢɧ ɨɬɩɚɞɟɬ ɉɨɫɥɟ ɨɬɛɪɚɫɵɜɚ ɧɢɹ ɧɟɡɧɚɱɢɦɵɯ ɱɥɟɧɨɜ ɩɨɥɭɱɚɸɬ ɭɬɨɱɧɟɧɧɨɟ ɭɪɚɜɧɟɧɢɟ ɜɵɪɚɠɚɸ ɳɟɟ ɡɚɜɢɫɢɦɨɫɬɶ ɢɫɤɨɦɨɝɨ ɩɚɪɚɦɟɬɪɚ ɨɬ ɮɚɤɬɨɪɨɜ ɯɚɪɚɤɬɟɪɢɡɭɸɳɢɯ ɫɨɫɬɚɜ ɢ ɫɜɨɣɫɬɜɚ ɛɟɬɨɧɧɨɣ ɫɦɟɫɢ
ɉɪɨɜɟɪɤɭ ɡɧɚɱɢɦɨɫɬɢ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɪɟɝɪɟɫɫɢɢ ɦɨɠɧɨ ɨɫɭɳɟ ɫɬɜɢɬɶ ɢ ɫ ɩɨɦɨɳɶɸ ɞɨɜɟɪɢɬɟɥɶɧɨɝɨ ɢɧɬɟɪɜɚɥɚ ɤɨɬɨɪɵɣ ɜɵɱɢɫɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ
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ɝɞɟt² ɬɚɛɥɢɱɧɨɟ ɡɧɚɱɟɧɢɟ ɤɪɢɬɟɪɢɹ ɋɬɶɸɞɟɧɬɚ
Ʉɨɷɮɮɢɰɢɟɧɬ ɪɟɝɪɟɫɫɢɢ ɡɧɚɱɢɦ ɟɫɥɢ ɟɝɨ ɚɛɫɨɥɸɬɧɚɹ ɜɟɥɢɱɢɧɚ ɛɨɥɶɲɟ ɞɨɜɟɪɢɬɟɥɶɧɨɝɨ ɢɧɬɟɪɜɚɥɚ
Ɂɚɬɟɦ ɩɨɥɭɱɟɧɧɨɟ ɭɪɚɜɧɟɧɢɟ ɩɨɞɜɟɪɝɚɟɬɫɹ ɩɪɨɜɟɪɤɟ ɧɚ ɩɪɢɝɨɞ ɧɨɫɬɶ ɚɞɟɤɜɚɬɧɨɫɬɶ Ⱦɥɹ ɷɬɨɝɨ ɜɵɱɢɫɥɹɸɬ ɞɢɫɩɟɪɫɢɸ ɚɞɟɤɜɚɬɧɨɫɬɢ ɢɥɢ ɨɫɬɚɬɨɱɧɭɸ ɞɢɫɩɟɪɫɢɸ ɩɨ ɮɨɪɦɭɥɟ
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ɝɞɟyu² ɡɧɚɱɟɧɢɟ ɢɫɫɥɟɞɭɟɦɨɝɨ ɫɜɨɣɫɬɜɚ ɛɟɬɨɧɚ ɜuɦ ɨɩɵɬɟ
yu ² ɡɧɚɱɟɧɢɟ ɢɫɫɥɟɞɭɟɦɨɝɨ ɫɜɨɣɫɬɜɚ ɛɟɬɨɧɚ ɜ uɦ ɨɩɵɬɟ ɜɵ
ɱɢɫɥɟɧɧɨɟ ɩɨ ɭɬɨɱɧɟɧɧɨɦɭ ɭɪɚɜɧɟɧɢɸ
m — ɱɢɫɥɨ ɡɧɚɱɢɦɵɯ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɜɤɥɸɱɚɹb Ɉɩɪɟɞɟɥɹɸɬ ɪɚɫɱɟɬɧɨɟ ɡɧɚɱɟɧɢɟ ɤɪɢɬɟɪɢɹ Ɏɢɲɟɪɚ Fɪ
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