chap2
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Ʉɨɷɮɮɢɰɢɟɧɬ, ɨɩɪɟɞɟɥɹɟɦɵɣ (II,6,6), ɨɛɨɡɧɚɱɚɸɬ ɢɧɨɝɞɚ IJb, ɚ (II,6,1) – IJɚ, ɟɫɥɢ ɧɟɬ ɨɛɴɟɞɢɧɟɧɢɣ ɪɚɧɝɨɜ IJɚ = IJb.
Ɉɛɪɚɬɢɦ ɜɧɢɦɚɧɢɟ ɧɚ ɬɨ, ɱɬɨ ɬɪɢ ɤɨɷɮɮɢɰɢɟɧɬɚ r, ȡ, IJ ɦɨɠɧɨ ɪɚɫɫɦɨɬɪɟɬɶ ɫ ɟɞɢɧɨɣ ɬɨɱɤɢ ɡɪɟɧɢɹ. Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ, ɩɭɫɬɶ, ɤɚɤ ɨɛɵɱɧɨ, ɢɦɟɟɬɫɹ ɫɨɜɨɤɭɩɧɨɫɬɶ ɢɡ N ɢɧɞɢɜɢɞɨɜ, ɤɚɠɞɵɣɢɡɤɨɬɨɪɵɯɦɨɠɟɬɛɵɬɶɨɯɚɪɚɤɬɟɪɢɡɨɜɚɧɫɩɨɦɨɳɶɸɡɧɚɱɟɧɢɣ ɞɜɭɯ ɩɪɢɡɧɚɤɨɜ X ɢ Y.
ȼɵɛɟɪɟɦ ɩɚɪɭ ɢɧɞɢɜɢɞɨɜ, ɧɚɩɪɢɦɟɪ, i ɢ j ɢ ɫɬɚɧɟɦ ɩɪɢɩɢɫɵɜɚɬɶ ɟɣ ɧɟɤɨɬɨɪɭɸ x – ɨɰɟɧɤɭ ɚij (ɤɨɧɤɪɟɬɢɡɚɰɢɹ ɨɰɟɧɨɤ ɛɭɞɟɬ ɞɚɧɚ ɧɢɠɟ), ɨɛɥɚɞɚɸɳɭɸ ɫɜɨɣɫɬɜɨɦ ɚɧɬɢɫɢɦɦɟɬɪɢɱɧɨɫɬɢ: ɚij= - ɚij Ⱥɧɚɥɨɝɢɱɧɨ ɜɜɟɞɟɦ ɭ – ɨɰɟɧɤɭ bij.
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Ɋɚɫɫɦɨɬɪɢɦ ɜɟɥɢɱɢɧɭ
¦¦aijbij
Ƚi j
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Ɇɵɭɠɟɜɢɞɟɥɢ (II,6,6), ɱɬɨɞɥɹɜɟɥɢɱɢɧɵ |
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ɟɫɥɢRi |
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aij ® |
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(ɝɞɟ R(x) |
– ɪɚɧɝɩɨ X i-ɝɨɷɥɟɦɟɧɬɚ) ɢɚɧɚɥɨɝɢɱɧɨɣ ɜɟɥɢɱɢɧɵ bij: Ƚ = IJ. |
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ɉɭɫɬɶ |
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aij = xj – xi, a, bij = yj – yi |
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ɬɨɝɞɚ |
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¦¦(xj xi )(y j yi ) 2N¦xi yi 2¦¦xi y j |
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ȿɫɥɢ ɩɨɥɨɠɢɬɶ |
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R(x) |
R(x) , ɚ |
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R( y) |
R( y) , ɬɨ ɦɨɠɧɨ ɚɧɚɥɨɝɢɱɧɨ ɩɪɟɞɵɞɭɳɟɦɭ |
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ɩɨɤɚɡɚɬɶ, ɱɬɨ Ƚ ɨɛɪɚɳɚɟɬɫɹ ɩɪɢ ɷɬɨɦ ɜ ȡ. ɗɬɨ ɪɚɫɫɦɨɬɪɟɧɢɟ ɫɨɫɬɚɜɢɬ ɞɥɹ ɱɢɬɚɬɟɥɹ ɫɚɦɨɫɬɨɹɬɟɥɶɧɨɟ ɭɩɪɚɠɧɟɧɢɟ 57.
Ɇɵ ɠɟ ɫɨɲɥɟɦɫɹ ɧɚ § 5 ɝɥɚɜɵ II, ɝɞɟ ɛɵɥɨ ɩɨɤɚɡɚɧɨ, ɱɬɨ ȡ ɹɜɥɹɟɬɫɹ r, ɩɪɢɦɟɧɟɧɧɵɦ ɤ ɪɚɧɝɚɦ, ɚ ɬɚɤ ɤɚɤ ɞɥɹ r ɪɚɫɫɦɨɬɪɟɧɢɟ ɩɪɨɜɟɞɟɧɨ, ɬɨ ɫ ɬɨɱɤɢ ɡɪɟɧɢɹ ɫɬɪɨɝɨɫɬɢ ɢɡɥɨɠɟɧɢɹ, ɜɵɤɥɚɞɤɢ ɞɚɧɧɨɝɨ ɭɩɪɚɠɧɟɧɢɹ ɜ ɬɟɤɫɬɟ ɤɧɢɝɢ ɧɟ ɹɜɥɹɸɬɫɹ ɧɟɨɛɯɨɞɢɦɵɦɢ. ȼ ɡɚɤɥɸɱɟɧɢɟ ɜɵɜɟɞɟɦɧɟɪɚɜɟɧɫɬɜɨ Ʉɨɲɢ.
Ɉɱɟɜɢɞɧɨɟ ɧɟɪɚɜɟɧɫɬɜɨ (A B )2 |
t 0 ɦɨɠɧɨɩɟɪɟɩɢɫɚɬɶ ɜɜɢɞɟ |
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ɉɨɥɚɝɚɹ |
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Aij |
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[122]
ɢɫɭɦɦɢɪɭɹɜɫɟɜɨɡɦɨɠɧɵɟ ɧɟɪɚɜɟɧɫɬɜɚ, ɩɨɥɭɱɢɦ:
84
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Ɍɚɤ ɤɚɤ ɥɟɜɚɹ ɱɚɫɬɶ ɪɚɜɧɚ 1, ɬɨ ɧɟɪɚɜɟɧɫɬɜɨ Ʉɨɲɢ ɞɨɤɚɡɚɧɨ. ɇɟɬɪɭɞɧɨ ɜɢɞɟɬɶ, ɱɬɨ ɨɧɨ ɩɪɟɜɪɚɳɚɟɬɫɹ ɜ ɪɚɜɟɧɫɬɜɨ, ɟɫɥɢ ɜɫɟ aij = Įbij (ɭɛɟɞɢɬɶɫɹ ɩɨɞɫɬɚɧɨɜɤɨɣ!), ɱɬɨ ɢ ɛɵɥɨ ɧɚɦɢ ɪɚɧɟɟ ɢɫɩɨɥɶɡɨɜɚɧɨ.
ɇɚɤɨɧɟɰ, ɪɚɫɫɦɨɬɪɢɦ ɫɥɭɱɚɣ, ɤɨɝɞɚ ɨɛɚ ɩɪɢɡɧɚɤɚ ɢɡɦɟɪɟɧɵ ɧɚ ɭɪɨɜɧɟ ɧɚɥɢɱɢɹ – ɨɬɫɭɬɫɬɜɢɹ.
ɉɭɫɬɶ ɢɧɞɟɤɫ 1 ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɧɚɥɢɱɢɸ, ɚ 2 ɨɬɫɭɬɫɬɜɢɸ ɩɪɢɡɧɚɤɚ, ɬɨɝɞɚ ɤɨɪɪɟɥɹɰɢɨɧɧɚɹ ɬɚɛɥɢɰɚɞɥɹɩɪɢɡɧɚɤɨɜ X ɢ Y ɩɪɢɧɢɦɚɟɬɜɢɞ:
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N11 |
N12 |
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N21 |
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Ʉɚɠɞɵɣ ɷɥɟɦɟɧɬ ɩɟɪɜɨɣ ɤɥɟɬɤɢ ɩɨɥɨɠɢɬɟɥɶɧɨɣ ɞɢɚɝɨɧɚɥɢ ɩɪɢ ɫɨɩɨɫɬɚɜɥɟɧɢɢ ɫ ɷɥɟɦɟɧɬɨɦ ɜɬɨɪɨɣɩɨɪɨɞɢɬ +1, ɜɫɟɝɨɬɚɤɢɯ +1 ɜ S ɜɨɣɞɟɬ N11·N22.
ɋɪɚɜɧɟɧɢɟ ɷɥɟɦɟɧɬɨɜɨɬɪɢɰɚɬɟɥɶɧɨɣɞɢɚɝɨɧɚɥɢɩɨɪɨɞɢɬ N12·N21, ɨɬɪɢɰɚɬɟɥɶɧɵɯ ɟɞɢɧɢɰ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ,
SN11N22 N12 N21;
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N(x2 )>N(x2 ) 1@; ɚ |
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Ⱥɧɚɥɨɝɢɱɧɨ: |
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[123]
ɬɟɩɟɪɶɤɨɷɮɮɢɰɢɟɧɬɄɟɧɞɷɥɚ, ɨɩɪɟɞɟɥɹɟɦɵɣ (II,6,5):
WN11N22 N12 N21
N(x1)N(x2 )N(y1)N(y2 )
ɬɚɤɢɦɨɛɪɚɡɨɦ, ɫɨɜɩɚɞɚɟɬɫɤɨɷɮɮɢɰɢɟɧɬɨɦ Ɏ (II,3,2).
ɗɬɨɬ ɪɟɡɭɥɶɬɚɬ ɩɪɨɹɫɧɹɟɬ ɫɦɵɫɥ ɮɨɪɦɚɥɶɧɨ ɜɜɟɞɟɧɧɨɝɨ ɪɚɧɟɟ ɤɨɷɮɮɢɰɢɟɧɬɚ ɤɨɧɬɢɧɝɟɧɰɢɢ.
7. ɗɧɬɪɨɩɢɣɧɵɟ ɦɟɪɵ ɜ ɫɨɰɢɨɥɨɝɢɱɟɫɤɨɦ ɚɧɚɥɢɡɟ
ɉɭɫɬɶ ɧɟɤɨɬɨɪɨɟ ɫɨɛɵɬɢɟ ɦɨɠɟɬ ɢɦɟɬɶ k ɪɚɡɥɢɱɧɵɯ ɢɫɯɨɞɨɜ Ai (i |
1,k) |
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ɤɨɬɨɪɵɯ ɨɛɨɡɧɚɱɢɦ ɱɟɪɟɡ P(Ai). əɫɧɨ, ɱɬɨ |
¦P(Ai ) 1. ɇɚɩɪɢɦɟɪ, |
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ɫɢɦɦɟɬɪɢɱɧɨɣɦɨɧɟɬɵ k = 2, Ⱥ1 — ɜɵɩɚɞɟɧɢɟ ɝɟɪɛɚ, A2 |
— ɪɟɲɤɢ, P(A ) |
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Ⱦɨɩɭɫɬɢɦ, ɱɬɨ ɦɵ ɯɨɬɢɦ ɩɪɟɞɫɤɚɡɚɬɶ |
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ɢɫɩɵɬɚɧɢɹ. ȿɫɥɢ |
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ɩɪɟɞɨɩɪɟɞɟɥɟɧ. ȿɫɥɢ k = 2, ɬɨ ɩɨɹɜɥɹɟɬɫɹ ɧɟɨɩɪɟɞɟɥɟɧɧɨɫɬɶ, ɤɨɬɨɪɚɹ ɦɚɤɫɢɦɚɥɶɧɚ ɩɪɢ Ɋ(A1) =
85
= Ɋ(A2). ȿɫɥɢ Ɋ(A1) > Ɋ(A2), ɬɨ ɱɟɦ ɛɨɥɶɲɟ Ɋ(A1), ɬɟɦ ɦɟɧɶɲɟ ɧɟɨɩɪɟɞɟɥɟɧɧɨɫɬɶ ɩɪɟɞɫɤɚɡɚɧɢɹ. ȼ ɩɪɟɞɟɥɟ, ɤɨɝɞɚ Ɋ(Ⱥ1) = 1 (Ɋ(A2) = 0), ɧɟɨɩɪɟɞɟɥɟɧɧɨɫɬɶ ɢɫɱɟɡɚɟɬ: ɜɨ ɜɫɟɯ ɢɫɩɵɬɚɧɢɹɯ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɢɫɯɨɞ A1.
ɑɟɦ ɛɨɥɶɲɟ k, ɬɟɦ ɦɟɧɟɟ ɨɩɪɟɞɟɥɟɧɧɵ ɩɪɟɞɫɤɚɡɚɧɢɹ, ɬɟɦ ɛɨɥɶɲɟ ɧɟɨɩɪɟɞɟɥɟɧɧɨɫɬɶ. ɉɨ
k
Ʉ. ɒɟɧɧɨɧɭ, ɦɟɪɨɣ ɧɟɨɩɪɟɞɟɥɟɧɧɨɫɬɢ ɹɜɥɹɟɬɫɹ ɜɟɥɢɱɢɧɚ E ¦P(Ai )log P(Ai ) , ɧɚɡɵɜɚɟɦɚɹ
i 1
ɷɧɬɪɨɩɢɟɣ. ȿɫɥɢ ɧɟɨɩɪɟɞɟɥɟɧɧɨɫɬɢ ɧɟɬ ɢ, ɫɤɚɠɟɦ, ɪɟɚɥɢɡɭɟɬɫɹ l-ɨɟ ɫɨɫɬɨɹɧɢɟ, ɬ.ɟ. Ɋ (Ⱥl) = 1, ɚ ɜɫɟ ɨɫɬɚɥɶɧɵɟ Ɋ (Ai) = 0, ɬɨ ȿ ɨɱɟɜɢɞɧɨ, ɨɛɪɚɳɚɟɬɫɹ ɜ ɧɭɥɶ. ɇɟɨɩɪɟɞɟɥɟɧɧɨɫɬɶ ɦɚɤɫɢɦɚɥɶɧɚ, ɟɫɥɢ ɜɫɟ ɢɫɯɨɞɵ ɪɚɜɧɨɜɨɡɦɨɠɧɵ, ɬ.ɟ. Ɋ (Ai) = 1/k. ɉɪɢ ɷɬɨɦ ȿmax = log k. ɑɟɦ ɛɨɥɶɲɟ k, ɬɟɦ ɛɨɥɶɲɟ ȿmax. ɂɬɚɤ, 0 ȿ log k.
ɉɭɫɬɶ N ɢɧɞɢɜɢɞɨɜ ɧɟɤɨɬɨɪɨɣ ɫɨɜɨɤɭɩɧɨɫɬɢ ɨɛɥɚɞɚɸɬ ɧɟɤɨɬɨɪɵɦ ɩɪɢɡɧɚɤɨɦ X, ɢ ɫɨɛɵɬɢɟ Ⱥi ɫɨɫɬɨɢɬ ɜ ɬɨɦ, ɱɬɨ ɡɧɚɱɟɧɢɟ ɩɪɢɡɧɚɤɚ ɪɚɜɧɨ xi. Ɉɛɨɡɧɚɱɢɦ ɱɟɪɟɡ Ni ɱɢɫɥɨ ɢɧɞɢɜɢɞɨɜ, ɭ ɤɨɬɨɪɵɯ X = xi. ȿɫɥɢ N ɞɨɫɬɚɬɨɱɧɨ ɜɟɥɢɤɨ, ɬɨ Ɋi = Ni/N, ɚ ȿ – ɦɟɪɚ «ɪɚɫɩɵɥɟɧɧɨɫɬɢ» ɪɚɫɩɪɟɞɟɥɟɧɢɹ. Ⱦɥɹ ɫɨɩɨɫɬɚɜɥɟɧɢɹ ɪɚɡɥɢɱɧɵɯ ɪɚɫɩɪɟɞɟɥɟɧɢɣ ɰɟɥɟɫɨɨɛɪɚɡɧɨ ɩɟɪɟɣɬɢ ɤ ɧɨɪɦɢɪɨɜɚɧɧɨɦɭ ɤɨɷɮɮɢɰɢɟɧɬɭ İ = ȿ/ȿmax. ȼɟɥɢɱɢɧɚ İ, ɩɪɢɧɢɦɚɸɳɚɹ ɡɧɚɱɟɧɢɹ ɦɟɠɞɭ 0 ɢ 1, ɹɜɥɹɟɬɫɹ ɚɧɚɥɨɝɨɦ ɞɢɫɩɟɪɫɢɢ.
[124]
ɉɟɪɟɣɞɟɦ ɤ ɞɜɭɯɦɟɪɧɵɦ ɪɚɫɩɪɟɞɟɥɟɧɢɹɦ ɞɥɹ ɩɪɢɡɧɚɤɨɜ X ɢ Y ɜ ɫɥɭɱɚɟ, ɤɨɝɞɚ ɷɦɩɢɪɢɱɟɫɤɢɣɦɚɬɟɪɢɚɥɫɜɟɞɟɧ ɜɤɨɪɪɟɥɹɰɢɨɧɧɭɸɬɚɛɥɢɰɭ {Nij}.
Ɍɟɩɟɪɶ
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E¦¦Pij log Pij , i 1 j 1
ɝɞɟ Ɋij = Nij/N ɢ ɫɭɦɦɢɪɨɜɚɧɢɟ ɜɟɞɟɬɫɹ ɩɨ ɜɫɟɦ ɤɥɟɬɤɚɦ ɤɨɪɪɟɥɹɰɢɨɧɧɨɣ ɬɚɛɥɢɰɵ. Ɂɞɟɫɶ ɢ ɞɚɥɟɟ ɦɵ ɧɟ ɭɤɚɡɵɜɚɟɦ ɨɫɧɨɜɚɧɢɟ ɥɨɝɚɪɢɮɦɚ, ɬɚɤ ɤɚɤ ɨɛɫɭɠɞɚɟɦɵɟ ɨɬɧɨɫɢɬɟɥɶɧɵɟ ɩɨɤɚɡɚɬɟɥɢ İɢȜ, ɨɬɧɟɝɨɧɟɡɚɜɢɫɹɬ.
ɍɩɪɚɠɧɟɧɢɟ 58. ɉɨɤɚɡɚɬɶ, ɱɬɨ ȿmax = log kl
ɍɩɪɚɠɧɟɧɢɟ 59. ɉɨɤɚɡɚɬɶ, ɱɬɨɬɟɩɟɪɶ
kl
N log N ¦¦Nij log Nij
Hi 1 j 1
N logkl
ɗɬɨ ɜɵɪɚɠɟɧɢɟ ɢɫɩɨɥɶɡɭɟɬɫɹɞɥɹɪɚɫɱɟɬɚ ɷɧɬɪɨɩɢɣɧɨɣɦɟɪɵɞɢɫɩɟɪɫɢɢ
Ɋɚɫɫɦɨɬɪɢɦ ɬɟɩɟɪɶ ɬɚɤ ɧɚɡɵɜɚɟɦɭɸ ɷɧɬɪɨɩɢɣɧɭɸ ɦɟɪɭ ɫɜɹɡɢ. ɇɟɨɩɪɟɞɟɥɟɧɧɨɫɬɶ Y- ɪɚɫɩɪɟɞɟɥɟɧɢɹ
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ɇɟɨɩɪɟɞɟɥɟɧɧɨɫɬɶ Y-ɪɚɫɩɪɟɞɟɥɟɧɢɹ ɭ ɢɧɞɢɜɢɞɨɜ ɫ X = ɯi, ɬɚɤ ɧɚɡɵɜɚɟɦɚɹ ɭɫɥɨɜɧɚɹ |
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ȼ ɢɬɨɝɨɜɭɸ ɭɫɥɨɜɧɭɸ ɧɟɨɩɪɟɞɟɥɟɧɧɨɫɬɶ ɤɚɠɞɚɹ ɫɬɪɨɱɤɚ ɬɚɛɥɢɰɵ ɞɚɟɬ ɜɤɥɚɞ c ɭɞɟɥɶɧɵɦ ɜɟɫɨɦ N(xi)/N, ɬ.ɟ. ɩɨɥɧɚɹɭɫɥɨɜɧɚɹɧɟɨɩɪɟɞɟɥɟɧɧɨɫɬɶ Y-ɪɚɫɩɪɟɞɟɥɟɧɢɹ:
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Ɇɟɪɨɣ ɫɜɹɡɢ ɦɟɠɞɭ ɩɪɢɡɧɚɤɚɦɢ X ɢ Y ɦɨɠɟɬ ɫɥɭɠɢɬɶ ɜɟɥɢɱɢɧɚ ɨɬɧɨɫɢɬɟɥɶɧɨɣ ɧɟɨɩɪɟɞɟɥɟɧɧɨɫɬɢ
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[125]
ɍɩɪɚɠɧɟɧɢɟ 60. Ɋɚɫɫɦɨɬɪɟɬɶ ɞɥɹ ɩɪɨɫɬɟɣɲɢɯ ɬɚɛɥɢɰ 2x2 ɫɥɭɱɚɣ ɨɬɫɭɬɫɬɜɢɹ ɫɜɹɡɢ ɢ ɩɨɤɚɡɚɬɶ, ɱɬɨȜ = 0. ɍɤɚɡɚɧɢɟ: ɢɫɩɨɥɶɡɨɜɚɬɶ, ɱɬɨ Nij = N(xi)N(yj)/N.
ɍɩɪɚɠɧɟɧɢɟ 61. Ɋɚɫɫɦɨɬɪɟɬɶ ɫɥɭɱɚɢ ɮɭɧɤɰɢɨɧɚɥɶɧɨɣ ɫɜɹɡɢ ɢ ɩɨɤɚɡɚɬɶ, ɱɬɨ Ȝ = 1. ɍɤɚɡɚɧɢɟ: ɭɱɟɫɬɶ, ɱɬɨɬɚɛɥɢɰɚ ɩɪɢɧɢɦɚɟɬɞɢɚɝɨɧɚɥɶɧɵɣ ɜɢɞ. ɂɬɚɤ, 0 Ȝ 1. ɑɟɦɛɨɥɶɲɟȜ, ɬɟɦ ɛɨɥɶɲɟɫɜɹɡɶɦɟɠɞɭ ɩɪɢɡɧɚɤɚɦɢ.
ɍɩɪɚɠɧɟɧɢɟ 62. ȼɵɱɢɫɥɢɬɶİɢȜy/x ɞɥɹɫɥɟɞɭɸɳɟɣ ɬɚɛɥɢɰɵ:
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Ɉɬɜɟɬ: İ = 0,872, ȿɭ/ɯ1 = 0,664, ȿɭ/ɯ2= 0,660, ȿɭ/ɯ3 = 0,582, Ȝy/x= 0,086.
ɍɩɪɚɠɧɟɧɢɟ 63. Ɉɛɪɚɬɢɦɫɹ ɤ ɪɚɫɫɦɨɬɪɟɧɢɸ ɫɜɹɡɢ ɦɟɠɞɭ ɭɞɨɜɥɟɬɜɨɪɟɧɧɨɫɬɶɸ ɪɚɛɨɬɨɣ ɢ ɭɞɨɜɥɟɬɜɨɪɟɧɧɨɫɬɶɸ ɡɚɪɚɛɨɬɧɨɣ ɩɥɚɬɨɣ. Ⱦɥɹ ɬɚɛɥɢɰɵ 18 (ɪɚɛɨɬɧɢɤɢ ɜ ɜɨɡɪɚɫɬɟ ɞɨ 30 ɥɟɬ) ɧɚɣɬɢȜ.
Ɉɬɜɟɬ: ȿɭ/ɯ1 = 0,289, ȿɭ/ɯ2 = 0,396, ȿɭ/ɯ3 = 0,383, ȿɭ/ɯ = 0,348, Ȝ = 0,030.
ɍɩɪɚɠɧɟɧɢɟ 64. Ⱦɥɹɬɚɛɥɢɰɵ 19 (ɪɚɛɨɬɧɢɤɢɫɬɚɪɲɟ 30 ɥɟɬ) ɧɚɣɬɢȜ. Ɉɬɜɟɬ: Ȝ = 0,014. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɫɜɹɡɶ ɦɟɠɞɭ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɦɢ ɩɨɤɚɡɚɬɟɥɹɦɢ ɛɨɥɟɟ ɬɟɫɧɚɹ ɞɥɹ ɦɨɥɨɞɵɯ
ɪɚɛɨɬɧɢɤɨɜ. ȼ ɞɚɥɶɧɟɣɲɟɦ ɦɵ ɜɟɪɧɟɦɫɹ ɤ ɷɬɨɦɭ ɜɨɩɪɨɫɭ ɟɳɟ ɪɚɡ, ɢɫɩɨɥɶɡɭɹ ɞɪɭɝɢɟ ɦɟɬɨɞɵ ɫɬɚɬɢɫɬɢɱɟɫɤɨɝɨɢɡɭɱɟɧɢɹɫɜɹɡɟɣ (§ 8 ɝɥɚɜɵ II).
ɉɪɢɦɟɪ 24. ɉɪɟɞɫɬɚɜɥɹɟɬ ɧɟɫɨɦɧɟɧɧɵɣ ɢɧɬɟɪɟɫ ɡɚɞɚɱɚ ɨ ɫɜɹɡɢ ɢɧɬɟɝɪɚɥɶɧɨɣ ɭɞɨɜɥɟɬɜɨɪɟɧɧɨɫɬɢ ɫ ɱɚɫɬɧɵɦɢ ɭɞɨɜɥɟɬɜɨɪɟɧɧɨɫɬɹɦɢ (ɨɬɞɟɥɶɧɵɦɢ ɷɥɟɦɟɧɬɚɦɢ ɪɚɛɨɱɟɣ ɫɢɬɭɚɰɢɢ).
ȼ ɤɚɱɟɫɬɜɟ ɷɥɟɦɟɧɬɨɜ ɨɛɵɱɧɨ ɜɵɞɟɥɹɸɬ: 1) ɫɨɞɟɪɠɚɧɢɟ ɬɪɭɞɚ (ɫɨɜɨɤɭɩɧɨɫɬɶ ɬɪɭɞɨɜɵɯ ɮɭɧɤɰɢɣ, ɜɵɩɨɥɧɹɟɦɵɯ ɜ ɩɪɨɰɟɫɫɟ ɫɨɡɞɚɧɢɹ ɩɨɬɪɟɛɢɬɟɥɶɧɵɯ ɫɬɨɢɦɨɫɬɟɣ ɜ ɩɪɨɰɟɫɫɟ ɬɪɭɞɚ), 2) ɭɫɥɨɜɢɹ (ɮɚɤɬɨɪɵ, ɩɨɞ ɜɨɡɞɟɣɫɬɜɢɟɦ ɤɨɬɨɪɵɯ ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɬɪɭɞɨɜɚɹ ɞɟɹɬɟɥɶɧɨɫɬɶ: ɫɦɟɧɧɨɫɬɶ, ɮɢɡɢɱɟɫɤɚɹ ɧɚɝɪɭɡɤɚ, ɫɨɫɬɨɹɧɢɟ ɨɤɪɭɠɚɸɳɟɣɫɪɟɞɵɢɬ.ɞ.);
[126]
3) ɨɪɝɚɧɢɡɚɰɢɹ (ɫɨɜɨɤɭɩɧɨɫɬɶ ɦɟɪɨɩɪɢɹɬɢɣ, ɨɛɟɫɩɟɱɢɜɚɸɳɢɯ ɪɚɰɢɨɧɚɥɶɧɨɟ ɢɫɩɨɥɶɡɨɜɚɧɢɟ ɪɚɛɨɱɟɣ ɫɢɥɵ); 4) ɨɩɥɚɬɚ; 5) ɦɟɠɥɢɱɧɨɫɬɧɵɟ ɨɬɧɨɲɟɧɢɹ ɢ ɬ.ɞ.
Ɉɫɨɡɧɚɜɚɹ, ɱɬɨ ɱɟɥɨɜɟɤ ɧɟ ɦɨɠɟɬ ɬɨɱɧɨ ɨɩɪɟɞɟɥɢɬɶ ɜɤɥɚɞ, ɤɨɬɨɪɵɣ ɜɧɨɫɢɬ ɜ ɨɛɳɟɟ ɫɨɫɬɨɹɧɢɟ ɭɞɨɜɥɟɬɜɨɪɟɧɧɨɫɬɢ ɭɞɨɜɥɟɬɜɨɪɟɧɢɟ ɨɬɞɟɥɶɧɵɯ ɩɨɬɪɟɛɧɨɫɬɟɣ, ɦɵ ɨɬɤɚɡɚɥɢɫɶ ɨɬ ɦɟɬɨɞɚ ɪɚɧɠɢɪɨɜɚɧɢɹ ɪɚɡɥɢɱɧɵɯ ɮɚɤɬɨɪɨɜ. Ⱦɥɹ ɢɡɭɱɟɧɢɹ ɨɛɫɭɠɞɚɟɦɨɣ ɫɜɹɡɢ ɢɫɩɨɥɶɡɨɜɚɥɢɫɶ ɪɚɡɥɢɱɧɵɟ ɫɬɚɬɢɫɬɢɱɟɫɤɢɟ ɩɨɤɚɡɚɬɟɥɢ, ɤɨɬɨɪɵɟ ɜɵɱɢɫɥɹɥɢɫɶ ɞɥɹ ɪɚɫɩɪɟɞɟɥɟɧɢɣ ɫɨɜɨɤɭɩɧɨɫɬɢ ɜ ɫɥɭɱɚɟ, ɤɨɝɞɚ ɨɞɧɢɦ ɢɡ ɩɪɢɡɧɚɤɨɜ ɹɜɥɹɟɬɫɹ ɢɧɬɟɝɪɚɥɶɧɚɹ ɭɞɨɜɥɟɬɜɨɪɟɧɧɨɫɬɶ ɢ ɞɪɭɝɢɦ – ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨ-ɱɚɫɬɧɵɟ.
Ⱦɥɹ T ɢ Ȝ ɷɥɟɦɟɧɬɵ ɪɚɫɩɨɥɨɠɢɥɢɫɶ ɬɚɤ: ɫɨɞɟɪɠɚɧɢɟ ɬɪɭɞɚ, ɨɪɝɚɧɢɡɚɰɢɹ, ɨɩɥɚɬɚ, ɨɬɧɨɲɟɧɢɹ ɫ ɚɞɦɢɧɢɫɬɪɚɰɢɟɣ ɢ ɬ.ɞ. (ɫɦ. ɬɚɤɠɟ § 8 ɝɥ. II). Ɂɚɦɟɬɢɦ, ɱɬɨ ɩɪɢ ɢɧɬɟɪɩɪɟɬɚɰɢɢ ɫɥɟɞɭɟɬ ɭɱɢɬɵɜɚɬɶ, ɱɬɨ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɟ ɷɥɟɦɟɧɬɵ ɧɟ ɹɜɥɹɸɬɫɹ ɧɟɡɚɜɢɫɢɦɵɦɢ: ɫɨɞɟɪɠɚɧɢɟ ɬɪɭɞɚ, ɧɚɩɪɢɦɟɪ, ɧɟɥɶɡɹ ɫɱɢɬɚɬɶ «ɨɱɢɳɟɧɧɵɦ» ɨɬ ɜɥɢɹɧɢɹ ɡɚɪɩɥɚɬɵ, ɢɛɨ ɜ ɫɪɟɞɧɟɦ ɛɨɥɟɟ ɫɨɞɟɪɠɚɬɟɥɶɧɚɹ ɪɚɛɨɬɚ ɜɵɲɟ ɨɩɥɚɱɢɜɚɟɬɫɹ ɢ ɬ.ɞ. ɋɥɟɞɭɟɬ ɬɚɤɠɟ ɭɱɢɬɵɜɚɬɶ, ɱɬɨ ɪɟɱɶ ɢɞɟɬ ɨɛ ɨɰɟɧɤɚɯ ɷɥɟɦɟɧɬɨɜ, ɚ ɫɜɹɡɶ ɦɟɠɞɭ ɷɥɟɦɟɧɬɨɦ ɢ ɨɰɟɧɤɨɣ ɧɨɫɢɬ ɫɥɨɠɧɵɣ, ɨɩɨɫɪɟɞɫɬɜɨɜɚɧɧɵɣ
87
ɯɚɪɚɤɬɟɪ. ɇɚɩɪɢɦɟɪ, ɧɟɬ ɩɪɹɦɨɣ ɡɚɜɢɫɢɦɨɫɬɢ ɦɟɠɞɭ ɭɞɨɜɥɟɬɜɨɪɟɧɧɨɫɬɶɸ ɡɚɪɩɥɚɬɨɣ ɢ ɟɟ ɜɟɥɢɱɢɧɨɣ (ɜ ɧɚɲɢɯ ɢɫɫɥɟɞɨɜɚɧɢɹɯ ɛɵɥɨ ɭɫɬɚɧɨɜɥɟɧɨ ɧɚɥɢɱɢɟ U-ɨɛɪɚɡɧɨɣ ɡɚɜɢɫɢɦɨɫɬɢ25). Ɂɚɜɢɫɢɦɨɫɬɢ ɨɩɨɫɪɟɞɫɬɜɭɸɬɫɹ ɩɨɬɪɟɛɧɨɫɬɹɦɢ, ɩɪɢɬɹɡɚɧɢɹɦɢ. Ɍɚɤ, ɭɞɨɜɥɟɬɜɨɪɟɧɧɨɫɬɶ ɡɚɪɩɥɚɬɨɣ ɡɚɜɢɫɢɬ ɧɟ ɫɬɨɥɶɤɨ ɨɬ ɟɟ «ɚɛɫɨɥɸɬɧɨɣ» ɜɟɥɢɱɢɧɵ, ɫɤɨɥɶɤɨ ɨɬ ɞɨɫɬɢɠɟɧɢɹ «ɧɨɪɦɵ», ɜ ɤɚɱɟɫɬɜɟ ɤɨɬɨɪɨɣ, ɤɚɤ ɭɞɚɥɨɫɶ ɭɫɬɚɧɨɜɢɬɶ, ɜɵɫɬɭɩɚɟɬ ɫɪɟɞɧɹɹ ɩɪɨɝɪɟɫɫɢɜɧɚɹ ɪɟɮɟɪɟɧɬɧɨɣ ɝɪɭɩɩɵ (ɞɥɹ ɪɚɛɨɬɧɢɤɨɜ ɩɪɨɦɵɲɥɟɧɧɵɯ ɩɪɟɞɩɪɢɹɬɢɣ ɟɸ ɨɤɚɡɚɥɚɫɶ ɢɯ ɫɨɰɢɚɥɶɧɨɩɪɨɮɟɫɫɢɨɧɚɥɶɧɚɹ ɝɪɭɩɩɚ). ȼɨ ɜɫɹɤɨɦ ɫɥɭɱɚɟ ɧɚɦɢ ɭɫɬɚɧɨɜɥɟɧɚ ɬɟɫɧɚɹ ɤɨɪɪɟɥɹɰɢɹ ɦɟɠɞɭ ɭɞɨɜɥɟɬɜɨɪɟɧɧɨɫɬɶɸ ɡɚɪɩɥɚɬɨɣ ɪɚɛɨɱɢɯ ɞɚɧɧɨɣ ɝɪɭɩɩɵ ɢ ɱɢɫɥɨɦ ɪɚɛɨɬɧɢɤɨɜ, ɩɨɥɭɱɚɸɳɢɯ ɡɚɪɩɥɚɬɭɧɟɧɢɠɟɫɪɟɞɧɟɩɪɨɝɪɟɫɫɢɜɧɨɣ26.
ɉɪɢɦɟɪ 25. Ʉɨɷɮɮɢɰɢɟɧɬ Ȝ, ɨɩɪɟɞɟɥɟɧɧɵɣ ɜɵɲɟ, ɨɩɢɫɵɜɚɟɬ ɜɥɢɹɧɢɟ X ɧɚ Y. Ɇɵ ɨɛɨɡɧɚɱɢɦ ɟɝɨ Ȝy/x. Ⱥɧɚɥɨɝɢɱɧɨ
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ɦɨɠɧɨ ɜɜɟɫɬɢ ɤɨɷɮɮɢɰɢɟɧɬ O |
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Ex Ex / y |
, ɤɨɬɨɪɵɣ ɨɩɢɫɵɜɚɟɬ ɜɥɢɹɧɢɟ Y ɧɚ X. Ȝ |
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ɧɟɫɢɦɦɟɬɪɢɱɟɧ: ɜɨɨɛɳɟ ɝɨɜɨɪɹ Ȝy/x Ȝx/y ȿɫɥɢ ɢɡ ɫɨɞɟɪɠɚɬɟɥɶɧɨɝɨ ɚɧɚɥɢɡɚ ɹɫɧɨ, ɱɬɨ X ɦɨɠɟɬ ɜɥɢɹɬɶ ɧɚ Y ɢ Y ɧɚ X, ɬɨ ɰɟɥɟɫɨɨɛɪɚɡɧɨ ɜɵɱɢɫɥɢɬɶ ɨɛɚ ɤɨɷɮɮɢɰɢɟɧɬɚ. ɇɚɩɪɢɦɟɪ, ɭɞɨɜɥɟɬɜɨɪɟɧɧɨɫɬɶ ɪɚɛɨɬɨɣ (Y), ɜɥɢɹɟɬ ɧɚ ɭɞɨɜɥɟɬɜɨɪɟɧɧɨɫɬɶ ɫɩɟɰɢɚɥɶɧɨɫɬɶɸ (X) ɢ ɧɚɨɛɨɪɨɬ. ɉɨɷɬɨɦɭɦɵɜɵɱɢɫɥɹɟɦɨɛɚɤɨɷɮɮɢɰɢɟɧɬɚ, ɢɫɩɨɥɶɡɭɹ ɢɯ ɞɥɹ ɫɪɚɜɧɟɧɢɹ ɭɤɚɡɚɧɧɵɯ ɜɥɢɹɧɢɣ. Ɍɚɤ, ɜ ɤɨɧɤɪɟɬɧɨɦ ɢɫɫɥɟɞɨɜɚɧɢɢ ɪɚɛɨɱɢɯ ɂɥɶɢɱɟɜɫɤɨɝɨ ɫɭɞɨɪɟɦɨɧɬɧɨɝɨ ɡɚɜɨɞɚ (1974ɝ.) ɧɚɦɢ ɛɵɥɨ ɩɨɥɭɱɟɧɨɬɚɤɨɟɞɜɭɦɟɪɧɨɟɪɚɫɩɪɟɞɟɥɟɧɢɟɨɛɫɭɠɞɚɟɦɵɯɩɪɢɡɧɚɤɨɜ:
Ɍɚɛɥɢɰɚ31
ɋɜɹɡɶɦɟɠɞɭɭɞɨɜɥɟɬɜɨɪɟɧɧɨɫɬɶɸɪɚɛɨɬɨɣɢɭɞɨɜɥɟɬɜɨɪɟɧɧɨɫɬɶɸɫɩɟɰɢɚɥɶɧɨɫɬɶɸ
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X |
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y1 |
y2 |
y3 |
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x1 |
1105 |
30 |
110 |
1245 |
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x2 |
313 |
55 |
62 |
430 |
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x3 |
35 |
4 |
36 |
75 |
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N(yj) |
1453 |
89 |
208 |
1750 |
Ⱦɥɹ ɷɬɨɣ ɤɨɪɪɟɥɹɰɢɨɧɧɨɣ ɬɚɛɥɢɰɵ, ɨɤɚɡɵɜɚɟɬɫɹ, Ȝɭ/ɯ = 0,073, ɚ Ȝx/y = 0,057. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɦɨɠɧɨ ɩɪɟɞɩɨɥɨɠɢɬɶ, ɱɬɨ ɭɞɨɜɥɟɬɜɨɪɟɧɧɨɫɬɶ ɫɩɟɰɢɚɥɶɧɨɫɬɶɸ ɜ ɛɨɥɶɲɟɣ ɦɟɪɟ ɜɥɢɹɟɬ ɧɚ ɭɞɨɜɥɟɬɜɨɪɟɧɧɨɫɬɶ ɪɚɛɨɬɨɣ (ɩɪɟɞɩɪɢɹɬɢɟɦ), ɱɟɦ ɧɚɨɛɨɪɨɬ. ɉɨɞɱɟɪɤɧɟɦ, ɱɬɨ ɷɬɨ ɭɬɜɟɪɠɞɟɧɢɟ ɨɬɧɨɫɢɬɫɹ ɤ ɥɨɤɚɥɶɧɵɦ ɭɫɥɨɜɢɹɦ ɨɩɪɟɞɟɥɟɧɧɨɝɨ, ɜɟɫɶɦɚ ɫɩɟɰɢɮɢɱɟɫɤɨɝɨ ɩɪɟɞɩɪɢɹɬɢɹ. Ⱦɥɹ ɢɡɭɱɟɧɢɹ ɩɨɫɬɚɜɥɟɧɧɨɝɨ ɜɨɩɪɨɫɚ ɜ ɰɟɥɨɦ ɧɟɨɛɯɨɞɢɦɨ ɩɪɨɜɟɫɬɢ ɞɚɥɶɧɟɣɲɢɟ ɢɫɫɥɟɞɨɜɚɧɢɹ. ȼ ɧɚɲɭ ɡɚɞɚɱɭ ɡɞɟɫɶ ɜɯɨɞɢɥɨ ɨɡɧɚɤɨɦɥɟɧɢɟ ɫ ɢɞɟɟɣ ɦɟɬɨɞɚ ɢ ɬɟɯɧɢɤɨɣ ɜɵɱɢɫɥɟɧɢɹ.
ɍɩɪɚɠɧɟɧɢɟ 65. ȼɵɱɢɫɥɢɬɶ Ȝɭ/ɯ ɢ Ȝx/y ɞɥɹ ɬɚɛɥɢɰɵ ɢɡ ɭɩɪɚɠɧɟɧɢɹ 62 ɫɚɦɨɫɬɨɹɬɟɥɶɧɨ.
ɉɪɢɦɟɪ 26. ɗɧɬɪɨɩɢɣɧɵɣ ɚɧɚɥɢɡ ɫɨɰɢɚɥɶɧɵɯ ɫɬɪɭɤɬɭɪ.
ȼ ɲɟɫɬɢɞɟɫɹɬɵɟ ɝɨɞɵ Ɉ.ɂ. ɒɤɚɪɚɬɚɧ ɫ ɝɪɭɩɩɨɣ ɫɨɬɪɭɞɧɢɤɨɜ ɢɡɭɱɚɥ ɫɨɰɢɚɥɶɧɭɸ ɫɬɪɭɤɬɭɪɭ ɫɨɜɪɟɦɟɧɧɨɝɨ ɩɪɨɦɵɲɥɟɧɧɨɝɨ ɩɪɟɞɩɪɢɹɬɢɹ. Ɋɟɡɭɥɶɬɚɬɵ ɬɟɨɪɟɬɢɱɟɫɤɨɝɨ ɚɧɚɥɢɡɚ, ɛɚɡɢɪɭɸɳɟɝɨɫɹ ɧɚ ɡɧɚɱɢɬɟɥɶɧɨɦ ɷɦɩɢɪɢɱɟɫɤɨɦ ɦɚɬɟɪɢɚɥɟ, ɢɡɥɨɠɟɧɵ ɜ ɤɧɢɝɟ «ɉɪɨɛɥɟɦɵ ɫɨɰɢɚɥɶɧɨɣ ɫɬɪɭɤɬɭɪɵ ɪɚɛɨ-
25Ⱥɧɚɥɨɝɢɱɧɵɣ ɯɚɪɚɤɬɟɪ ɢɦɟɟɬ ɡɚɜɢɫɢɦɨɫɬɶ ɦɟɠɞɭ ɭɞɨɜɥɟɬɜɨɪɟɧɧɨɫɬɶɸ ɨɛɪɚɡɨɜɚɧɢɟɦ ɢ ɮɚɤɬɢɱɟɫɤɢɦ ɨɛɪɚɡɨɜɚɧɢɟɦ (ɨɛɫɥɟɞɨɜɚɥɢɫɶɪɚɛɨɬɧɢɤɢɩɪɨɦɵɲɥɟɧɧɵɯɩɪɟɞɩɪɢɹɬɢɣɝ. Ɉɞɟɫɫɵ).
26Ɇɚɤɫɢɦɟɧɤɨ ȼ. ɋ., ɉɨɩɨɜɚ ɂ. Ɇ. Ɂɚɪɚɛɨɬɧɚɹ ɩɥɚɬɚ ɤɚɤ ɮɚɤɬɨɪ ɫɬɢɦɭɥɢɪɨɜɚɧɢɹ ɬɪɭɞɨɜɨɣɞɟɹɬɟɥɶɧɨɫɬɢ.— ȼɤɧ.: ɉɪɨɛɥɟɦɵɷɤɨɧɨɦɢɤɢɦɨɪɹɢɦɢɪɨɜɨɝɨɨɤɟɚɧɚ. Ɉɞɟɫɫɚ, 1973, ʋ 2
88
[128]
ɱɟɝɨ ɤɥɚɫɫɚ ɋɋɋɊ» (Ɇ., 1970). ɋɨɜɦɟɫɬɧɨ ɫ ɂ.ɇ. Ɍɚɝɚɧɨɜɵɦ Ɉ.ɂ. ɒɤɚɪɚɬɚɧ ɩɪɟɞɩɪɢɧɢɦɚɥ ɩɨɩɵɬɤɢ ɢɫɩɨɥɶɡɨɜɚɧɢɹ ɤɨɥɢɱɟɫɬɜɟɧɧɨɝɨ ɦɟɬɨɞɚ ɞɥɹ ɢɡɭɱɟɧɢɹ ɭɤɚɡɚɧɧɨɣ ɫɬɪɭɤɬɭɪɵ. Ɉɞɧɚ ɢɡ ɧɢɯ, ɫɜɹɡɚɧɧɚɹ ɫ ɩɪɢɦɟɧɟɧɢɟɦ ɷɧɬɪɨɩɢɣɧɨɝɨ ɚɧɚɥɢɡɚ, ɛɵɥɚ ɢɡɥɨɠɟɧɚ ɜ ɠɭɪɧɚɥɟ «ȼɨɩɪɨɫɵ ɮɢɥɨɫɨɮɢɢ» (1969, ʋ5) ɢ ɩɪɢɜɥɟɤɥɚ ɜɧɢɦɚɧɢɟ ɫɨɰɢɨɥɨɝɨɜ, ɢɧɬɟɪɟɫɭɸɳɢɯɫɹ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɤɨɥɢɱɟɫɬɜɟɧɧɵɯ ɦɟɬɨɞɨɜ ɜ ɫɨɰɢɚɥɶɧɵɯ ɢɫɫɥɟɞɨɜɚɧɢɹɯ. Ɋɚɫɫɦɨɬɪɢɦ ɟɟ ɫɭɬɶ ɩɪɢɦɟɧɢɬɟɥɶɧɨ ɤ ɮɚɤɬɢɱɟɫɤɢ ɪɟɚɥɢɡɨɜɚɧɧɨɣ ɢɫɫɥɟɞɨɜɚɬɟɥɹɦɢ ɩɪɨɝɪɚɦɦɟ, ɧɨ ɫ ɢɫɩɨɥɶɡɨɜɚɧɢɟɦ ɨɛɨɡɧɚɱɟɧɢɣ ɩɪɟɞɵɞɭɳɢɯ ɩɚɪɚɝɪɚɮɨɜ.
Ɉɫɧɨɜɧɚɹ ɡɚɞɚɱɚ, ɤɨɬɨɪɚɹ ɪɟɲɚɥɚɫɶ ɚɜɬɨɪɚɦɢ ɫ ɩɨɦɨɳɶɸ ɷɧɬɪɨɩɢɣɧɨɝɨ ɚɧɚɥɢɡɚ, ɫɨɫɬɨɹɥɚ ɜ ɜɵɞɟɥɟɧɢɢ ɫɜɨɣɫɬɜ (ɩɪɢɡɧɚɤɨɜ), ɨɩɪɟɞɟɥɹɸɳɢɯ ɧɟɨɞɧɨɪɨɞɧɨɫɬɶ ɢɡɭɱɚɟɦɨɣ ɫɨɰɢɚɥɶɧɨɣ ɫɬɪɭɤɬɭɪɵ. Ɂɚɞɚɱɚ ɪɚɫɫɦɚɬɪɢɜɚɥɚɫɶ ɜ ɬɪɟɯɦɟɪɧɨɦ ɩɪɨɫɬɪɚɧɫɬɜɟ, ɬ.ɟ. ɢɡ ɝɢɩɨɬɟɬɢɱɟɫɤɨɝɨ ɧɚɛɨɪɚ ɡɧɚɱɢɦɵɯ ɩɪɢɡɧɚɤɨɜ (ɨɧ ɛɵɥ ɫɨɫɬɚɜɥɟɧ ɧɚ ɨɫɧɨɜɟ ɩɪɟɞɜɚɪɢɬɟɥɶɧɨɝɨ ɚɧɚɥɢɡɚ, ɫɸɞɚ ɜɨɲɥɢ ɬɚɤɢɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ, ɤɚɤ ɨɛɪɚɡɨɜɚɧɢɟ, ɤɜɚɥɢɮɢɤɚɰɢɹ, ɩɨɥ, ɩɪɨɮɟɫɫɢɹ ɢ ɬ.ɞ. – ɜɫɟɝɨ 27 ɩɪɢɡɧɚɤɨɜ) ɚɜɬɨɪɵ ɜɵɞɟɥɹɥɢ ɤɚɠɞɵɣ ɪɚɡ ɬɪɨɣɤɭ ɩɪɢɡɧɚɤɨɜ, ɧɚɛɨɪ ɤɨɬɨɪɵɯ ɞɚɜɚɥ ɪɚɡɥɢɱɧɵɟ ɩɪɨɫɬɪɚɧɫɬɜɚ. ȼɫɟɝɨ ɬɚɤɢɯ ɩɪɨɫɬɪɚɧɫɬɜ ɦɨɠɧɨ ɜɵɞɟɥɢɬɶ C273 = 2925.
Ʌɨɝɢɤɚ ɢɫɫɥɟɞɨɜɚɧɢɹ ɬɚɤɨɜɚ. Ʉɚɠɞɵɣ ɢɧɞɢɜɢɞ ɞɚɧɧɨɣ ɫɨɜɨɤɭɩɧɨɫɬɢ ɹɜɥɹɟɬɫɹ ɧɨɫɢɬɟɥɟɦ ɪɚɡɥɢɱɧɵɯ ɩɪɢɡɧɚɤɨɜ. ɉɭɫɬɶ ɨɧ ɨɛɥɚɞɚɟɬ i-ɦ ɡɧɚɱɟɧɢɟɦ ɩɪɢɡɧɚɤɚ X, j-ɦ – ɍ, r-ɦ – Z (ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɨɝɪɚɧɢɱɟɧɢɟɦ, ɩɪɢɧɹɬɵɦ ɚɜɬɨɪɚɦɢ, ɦɵ ɪɚɫɫɦɚɬɪɢɜɚɟɦ ɩɪɨɫɬɪɚɧɫɬɜɨ, ɨɩɪɟɞɟɥɹɟɦɨɟ ɩɪɢɡɧɚɤɚɦɢ X, Y, Z). ɂɧɮɨɪɦɚɰɢɸ ɨɛ ɨɞɧɨɦ ɢɧɞɢɜɢɞɟ ɦɨɠɧɨ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɤɚɤ ɜɟɤɬɨɪ ɜ ɞɚɧɧɨɦ ɩɪɨɫɬɪɚɧɫɬɜɟ. ɋɨɜɨɤɭɩɧɨɫɬɢ ɢɡ N ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɯ ɢɧɞɢɜɢɞɨɜ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɫɨɜɨɤɭɩɧɨɫɬɶ N ɜɟɤɬɨɪɨɜ. ɂɡ ɜɫɟɯ ɜɨɡɦɨɠɧɵɯ ɩɪɨɫɬɪɚɧɫɬɜ (ɧɚɛɨɪɨɜ ɩɪɢɡɧɚɤɨɜ) ɧɭɠɧɨ ɜɵɞɟɥɢɬɶ ɬɚɤɨɟ, ɜ ɤɨɬɨɪɨɦ ɜɟɤɬɨɪɵ ɥɟɠɚɬ ɧɚɢɛɨɥɟɟ ɩɥɨɬɧɵɦɢ ɝɪɭɩɩɚɦɢ (ɧɚɛɨɪ ɩɪɢɡɧɚɤɨɜ ɧɚɢɛɨɥɟɟ ɪɟɡɤɨ ɞɢɮɮɟɪɟɧɰɢɪɭɟɬ ɫɨɜɨɤɭɩɧɨɫɬɶ ɢɧɞɢɜɢɞɨɜ). Ⱦɥɹ ɨɬɵɫɤɚɧɢɹ ɬɚɤɢɯ ɩɪɨɫɬɪɚɧɫɬɜ ɢ ɛɵɥ ɩɪɢɦɟɧɟɧ ɷɧɬɪɨɩɢɣɧɵɣ ɚɧɚɥɢɡ.
ɇɟɨɩɪɟɞɟɥɟɧɧɨɫɬɶ ɡɚɩɨɥɧɟɧɢɹ ɩɪɨɫɬɪɚɧɫɬɜɚ ɜɟɤɬɨɪɚɦɢ ɨɩɪɟɞɟɥɹɟɬɫɹ ɜɟɥɢɱɢɧɨɣ
k l m
E¦¦¦Pijr log Pijr , i 1 j 1 r 1
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Nijr |
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ɝɞɟ |
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(ɡɞɟɫɶ Nijr – ɱɢɫɥɨ ɢɧɞɢɜɢɞɨɜ, ɭ ɤɨɬɨɪɵɯ X = xi, Y = yi, Z = zi); |
i |
1,k; |
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ijr |
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j 1,l ; r |
1,m |
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[129] |
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ȿɫɥɢ ɜɟɤɬɨɪɵ ɪɚɜɧɨɦɟɪɧɨ ɡɚɩɨɥɧɹɸɬ ɩɪɨɫɬɪɚɧɫɬɜɨ, ɬɨ |
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logklm |
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ijr |
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klm |
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Emax E |
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Ɋɚɫɫɦɨɬɪɢɦ |
ɜɟɥɢɱɢɧɭ |
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. Ɍɚɤ ɤɚɤ 0 E Emax, |
ɬɨ 0 Į 1, ɩɪɢɱɟɦ |
Į |
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Emax |
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ɫɨɨɬɜɟɬɫɬɜɭɟɬ |
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= ȿmɚɯ, |
ɬ.ɟ. |
ɨɬɫɭɬɫɬɜɢɸ ɧɟɨɞɧɨɪɨɞɧɨɫɬɢ |
ɜ ɪɚɫɩɪɟɞɟɥɟɧɢɢ ɜɟɤɬɨɪɨɜ |
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(ɨɬɫɭɬɫɬɜɢɸ ɞɢɮɮɟɪɟɧɰɢɚɰɢɢ ɨɛɳɧɨɫɬɢ), ɚ Į = 1 ɫɨɨɬɜɟɬɫɬɜɭɟɬ E = 0, ɬ.ɟ. ɦɚɤɫɢɦɚɥɶɧɨɣ |
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ɧɟɨɞɧɨɪɨɞɧɨɫɬɢ (ɦɚɤɫɢɦɚɥɶɧɨɣ ɞɢɮɮɟɪɟɧɰɢɚɰɢɢ). |
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Ɉɱɟɜɢɞɧɨ, ɪɚɡɧɵɦ ɩɪɨɫɬɪɚɧɫɬɜɚɦ ɫɨɨɬɜɟɬɫɬɜɭɸɬ ɪɚɡɥɢɱɧɵɟ Į ɢ ɮɨɪɦɚɥɶɧɨ ɡɚɞɚɱɚ |
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ɫɜɨɞɢɬɫɹ ɤ ɨɬɵɫɤɚɧɢɸ ɩɪɨɫɬɪɚɧɫɬɜɚ ɫ ɦɚɤɫɢɦɚɥɶɧɵɦ Į. |
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ɍɩɪɚɠɧɟɧɢɟ 66. ɉɨɤɚɡɚɬɶ, ɱɬɨ |
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N log |
klm |
¦¦¦Nijr log Nijr |
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N logklm
89
ɇɚ ɷɦɩɢɪɢɱɟɫɤɨɦ ɦɚɬɟɪɢɚɥɟ ɥɟɧɢɧɝɪɚɞɫɤɢɯ ɫɨɰɢɨɥɨɝɨɜ ɜɟɥɢɱɢɧɚ Į ɨɤɚɡɚɥɚɫɶ ɦɚɤɫɢɦɚɥɶɧɨɣ ɞɥɹ ɧɚɛɨɪɚ ɩɪɢɡɧɚɤɨɜ «ɩɪɨɮɟɫɫɢɹ – ɤɜɚɥɢɮɢɤɚɰɢɹ – ɨɛɪɚɡɨɜɚɧɢɟ». ɂɦɟɧɧɨ ɜ ɷɬɨɦ ɩɪɨɫɬɪɚɧɫɬɜɟ ɜɟɤɬɨɪɵ ɥɟɠɚɬ ɧɚɢɛɨɥɟɟ ɩɥɨɬɧɵɦɢ ɝɪɭɩɩɚɦɢ, ɞɚɧɧɵɣ ɧɚɛɨɪ ɩɪɢɡɧɚɤɨɜ ɧɚɢɛɨɥɟɟ ɪɟɡɤɨ ɞɢɮɮɟɪɟɧɰɢɪɭɟɬ ɢɡɭɱɚɟɦɭɸ ɫɨɰɢɚɥɶɧɭɸ ɨɛɳɧɨɫɬɶ.
8.ɇɟɤɨɬɨɪɵɟ ɞɪɭɝɢɟ ɤɨɷɮɮɢɰɢɟɧɬɵ
ȼɞɚɧɧɨɦ ɩɚɪɚɝɪɚɮɟ ɦɵ ɪɚɫɫɦɨɬɪɢɦ ɧɟɫɤɨɥɶɤɨ ɫɬɚɬɢɫɬɢɱɟɫɤɢɯ ɤɨɷɮɮɢɰɢɟɧɬɨɜ, ɤɨɬɨɪɵɟ ɧɟ ɩɨɥɭɱɢɥɢ ɜ ɫɨɰɢɚɥɶɧɵɯ ɢɫɫɥɟɞɨɜɚɧɢɹɯ ɬɚɤɨɝɨ ɲɢɪɨɤɨɝɨ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ, ɤɚɤ, ɫɤɚɠɟɦ, r, ȡ, Ɍ ɢ ɞɚɠɟ Ș). Ɉɞɧɚɤɨ ɜ ɫɨɰɢɨɥɨɝɢɱɟɫɤɨɣ ɥɢɬɟɪɚɬɭɪɟ ɭɠɟ ɜɫɬɪɟɱɚɸɬɫɹ ɭɩɨɦɢɧɚɧɢɹ ɨɛ ɢɯ ɢɫɩɨɥɶɡɨɜɚɧɢɢ ɨɬɞɟɥɶɧɵɦɢ ɚɜɬɨɪɚɦɢ.
Ɇɵ ɫɱɢɬɚɟɦ ɰɟɥɟɫɨɨɛɪɚɡɧɵɦ ɪɚɫɫɦɨɬɪɟɬɶ ɨɩɪɟɞɟɥɟɧɢɹ, ɩɪɨɚɧɚɥɢɡɢɪɨɜɚɬɶ ɢɯ ɢ ɩɪɢɜɟɫɬɢ ɩɪɢɦɟɪɵ ɜɵɱɢɫɥɟɧɢɹ ɧɟɤɨɬɨɪɵɯ ɬɚɤɢɯ ɤɨɷɮɮɢɰɢɟɧɬɨɜ27. ɋ ɨɞɧɨɣ ɫɬɨɪɨɧɵ, ɷɬɨ ɩɨɤɚɠɟɬ ɱɢɬɚɬɟɥɸ, ɱɬɨ ɞɢɚɩɚɡɨɧ ɢɫɩɨɥɶɡɭɟɦɵɯ ɦɟɬɨɞɨɜ ɡɧɚɱɢɬɟɥɶɧɨ ɲɢɪɟ, ɱɟɦ ɦɨɠɟɬ ɩɪɟɞɫɬɚɜɢɬɶɫɹ ɩɨ ɨɫɧɨɜɧɨɣ ɦɚɫɫɟ ɩɭɛɥɢɤɚɰɢɣ, ɫ ɞɪɭɝɨɣ, ɩɨɡɜɨɥɢɬ ɛɨɥɟɟ ɫɜɨɛɨɞɧɨ ɨɪɢɟɧɬɢɪɨɜɚɬɶɫɹ ɜɧɚɭɱɧɵɯ ɫɬɚɬɶɹɯ.
[130]
g – ɤɨɷɮɮɢɰɢɟɧɬ Ƚɭɞɦɚɧɚ (ɞɥɹɧɨɦɢɧɚɥɶɧɵɯɲɤɚɥ)
Ʉɨɷɮɮɢɰɢɟɧɬ Ƚɭɞɦɚɧɚ ɧɟ ɹɜɥɹɟɬɫɹ ɫɢɦɦɟɬɪɢɱɧɵɦ: gyx gxy ȿɫɥɢ ɦɵ ɪɚɫɫɦɚɬɪɢɜɚɟɦ X ɤɚɤ ɧɟɡɚɜɢɫɢɦɵɣ (ɮɚɤɬɨɪɧɵɣ) ɩɪɢɡɧɚɤ, ɬɨ ɟɝɨ ɜɥɢɹɧɢɟ ɧɚ Y ɨɩɢɫɵɜɚɟɬɫɹ ɫ ɩɨɦɨɳɶɸ ɤɨɷɮɮɢɰɢɟɧɬɚ
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¦max Nij |
max N(yi ) |
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gyx |
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N max N(y j ) |
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ɝɞɟ max N(yj) – ɦɚɤɫɢɦɚɥɶɧɵɣ ɦɚɪɝɢɧɚɥ ɡɚɜɢɫɢɦɨɝɨ ɩɪɢɡɧɚɤɚ, ɚ max Nij – ɦɚɤɫɢɦɚɥɶɧɚɹ ɱɚɫɬɨɬɚ ɜ i-ɨɣ ɫɬɪɨɤɟ ɤɨɪɪɟɥɹɰɢɨɧɧɨɣ ɬɚɛɥɢɰɵ.
ȿɫɥɢ ɞɚɧɧɨɦɭ X ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɨɩɪɟɞɟɥɟɧɧɵɣ Y, ɬɨ ɜ ɫɬɪɨɤɟ ɥɢɲɶ ɨɞɧɚ ɱɚɫɬɨɬɚ ɫ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɦ ɦɚɪɝɢɧɚɥɨɦ, ɢɫɤɨɦɚɹ ɫɭɦɦɚ ɦɚɤɫɢɦɭɦɨɜ ɨɛɪɚɳɚɟɬɫɹ ɜ N, ɫɥɟɞɨɜɚɬɟɥɶɧɨ, gyx = 1
ȿɫɥɢ ɩɪɢɡɧɚɤɢ ɧɟɡɚɜɢɫɢɦɵ, ɬɨ |
Nij |
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N(xi )N(y j ), ɤɚɤ ɦɵ ɜɢɞɟɥɢ, ɢ ɦɚɤɫɢɦɚɥɶɧɚɹ |
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ɱɚɫɬɨɬɚ ɜ i-ɨɣ ɫɬɪɨɤɟ ɬɚɦ, ɝɞɟ ɦɚɤɫɢɦɚɥɟɧ Y-ɦɚɪɝɢɧɚɥ, ɬ.ɟ. |
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¦max Nij |
max N(y j |
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max N(y j ) |
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Ɍɟɩɟɪɶ gyx = 0. ɂɬɚɤ, 0 gyx 1. Ⱥɧɚɥɨɝɢɱɧɨ ɨɩɪɟɞɟɥɹɟɬɫɹ gxy, ɨɩɢɫɵɜɚɸɳɢɣ ɜɥɢɹɧɢɟ Y
ɧɚ X.
Ʉɨɷɮɮɢɰɢɟɧɬɵ Ƚɭɞɦɚɧɚ ɰɟɥɟɫɨɨɛɪɚɡɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ, ɟɫɥɢ ɢɡ ɫɨɞɟɪɠɚɬɟɥɶɧɵɯ ɫɨɨɛɪɚɠɟɧɢɣ ɹɫɧɨ, ɱɬɨ X ɦɨɠɟɬ ɜɥɢɹɬɶ ɧɚ Y (ɢ ɧɚɨɛɨɪɨɬ) ɢ ɷɬɨ ɜɥɢɹɧɢɟ, ɜɨɨɛɳɟ ɝɨɜɨɪɹ, ɧɟ ɫɢɦɦɟɬɪɢɱɧɨ.
ȼ ɬɟɯ ɫɥɭɱɚɹɯ, ɤɨɝɞɚ X ɧɟ ɦɨɠɟɬ ɜɥɢɹɬɶ ɧɚ Y (ɧɚɩɪɢɦɟɪ, X – ɤɜɚɥɢɮɢɤɚɰɢɹ, Y – ɜɨɡɪɚɫɬ), ɫɥɟɞɭɟɬ ɜɵɱɢɫɥɹɬɶ ɬɨɥɶɤɨ gxy (ɜɨɡɪɚɫɬɜɥɢɹɟɬ ɧɚ ɤɜɚɥɢɮɢɤɚɰɢɸ).
ɍɩɪɚɠɧɟɧɢɟ 67. Ɋɚɫɫɱɢɬɚɬɶ gyx ɢ gxy ɞɥɹ ɫɥɟɞɭɸɳɟɣ ɤɨɪɪɟɥɹɰɢɨɧɧɨɣ ɬɚɛɥɢɰɵ:
X |
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y1 |
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x1 |
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27 Ɉɛɡɨɪ ɪɹɞɚ ɞɪɭɝɢɯ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɦɨɠɧɨ ɧɚɣɬɢ ɜ ɤɧ.: ȿɥɢɫɟɟɜɚ ɂ.ɂ., Ɋɭɤɚɜɢɲɧɢɤɨɜ ȼ.Ɉ. Ƚɪɭɩɩɢɪɨɜɤɚ, ɤɨɪɪɟɥɹɰɢɹ, ɪɚɫɩɨɡɧɚɜɚɧɢɟ ɨɛɪɚɡɨɜ. Ɇ., 1977, ɝɥ. III, IV.
90
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45 |
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20 |
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30 |
65 |
Ɉɬɜɟɬ: gyx = 0,57; gxy = 1.
[131]
ɂɧɬɟɪɩɪɟɬɢɪɭɟɦ ɪɟɡɭɥɶɬɚɬ:
Ɂɚɞɚɧɢɟ Y ɨɞɧɨɡɧɚɱɧɨ ɨɩɪɟɞɟɥɹɟɬ X (ɫɦ. ɬɚɛɥɢɰɭ). ɋɨɨɬɜɟɬɫɬɜɟɧɧɨ gyx = 1; ɧɨ ɡɚɞɚɧɢɟ X ɧɟ ɨɩɪɟɞɟɥɹɟɬ ɟɳɟ Y (ɧɚɩɪɢɦɟɪ, ɟɫɥɢ X = x2, ɬɨ Y ɦɨɠɟɬ ɛɵɬɶ ɢ ɭ2, ɢ y3), ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ
gyx < 1
ɍɩɪɚɠɧɟɧɢɟ 68. 1. Ɂɚɩɢɫɚɬɶ ɥɸɛɭɸɞɢɚɝɨɧɚɥɶɧɭɸ ɬɚɛɥɢɰɭ ɢɭɛɟɞɢɬɶɫɹ, ɱɬɨ gyx = gxy = 1. 2. ɋɤɨɧɫɬɪɭɢɪɨɜɚɬɶ ɬɚɛɥɢɰɭ, ɞɥɹ ɤɨɬɨɪɨɣ gxy < 1, ɚ gyx = 1, ɂɧɬɟɪɩɪɟɬɢɪɨɜɚɬɶ ɪɟɡɭɥɶɬɚɬɵ
ɪɚɫɱɟɬɚ ɩɨ ɚɧɚɥɨɝɢɢ ɫ ɩɪɟɞɵɞɭɳɢɦ.
Ɂɚɦɟɬɢɦ, ɱɬɨ ɜɵɩɨɥɧɟɧɢɟ ɷɬɢɯ ɧɟɫɥɨɠɧɵɯ ɭɩɪɚɠɧɟɧɢɣ ɩɨɦɨɝɚɟɬ ɭɹɫɧɢɬɶ ɫɦɵɫɥ ɢ ɪɚɡɥɢɱɢɟ ɤɨɷɮɮɢɰɢɟɧɬɨɜ gyx ɢ gxy. Ɉɬɦɟɬɢɦ ɬɚɤɠɟ ɩɪɟɞɥɚɝɚɟɦɵɣ Ȼ. Ɇɢɪɤɢɧɵɦ ɩɨɞɯɨɞ ɤ ɨɛɪɚɛɨɬɤɟ ɫɨɰɢɨɥɨɝɢɱɟɫɤɨɣ ɢɧɮɨɪɦɚɰɢɢ28, ɤɨɬɨɪɵɣ ɦɨɠɟɬ ɛɵɬɶ ɢɫɩɨɥɶɡɨɜɚɧ ɞɚɠɟ ɞɥɹ ɫɥɭɱɚɹ ɧɨɦɢɧɚɥɶɧɵɯ ɲɤɚɥ. ȼ ɤɚɱɟɫɬɜɟ ɦɟɪɵ ɛɥɢɡɨɫɬɢ ɩɪɢɡɧɚɤɨɜ ɪɚɫɫɦɚɬɪɢɜɚɟɬɫɹ ɦɟɪɚ ɛɥɢɡɨɫɬɢ ɪɚɡɛɢɟɧɢɣ ɨɛɳɧɨɫɬɢ, ɨɫɭɳɟɫɬɜɥɹɟɦɵɯ ɷɬɢɦɢ ɩɪɢɡɧɚɤɚɦɢ.
Ʉɨɷɮɮɢɰɢɟɧɬ ɛɥɢɡɨɫɬɢ ɪɚɡɛɢɟɧɢɣ
Ɉɛɨɛɳɢɦ ɮɨɪɦɭɥɭ ɞɥɹ ɦɟɪɵ ɛɥɢɡɨɫɬɢ ɦɟɠɞɭ ɞɜɭɦɹ ɪɚɡɛɢɟɧɢɹɦɢ ɧɚ ɫɥɭɱɚɣ ɤɨɪɪɟɥɹɰɢɨɧɧɨɣ ɬɚɛɥɢɰɵ k × l. ȼ ɤɚɱɟɫɬɜɟ ɢɫɯɨɞɧɨɣ ɜɨɡɶɦɟɦ ɮɨɪɦɭɥɭ, ɩɪɢɜɨɞɢɦɭɸ Ȼ.Ƚ. Ɇɢɪɤɢɧɵɦ ɢ Ʌ.Ȼ. ɑɟɪɧɵɦ ɜ ɫɬɚɬɶɟ «Ɉɛ ɢɡɦɟɪɟɧɢɢ ɦɟɪɵ ɛɥɢɡɨɫɬɢ ɦɟɠɞɭ ɪɚɡɥɢɱɧɵɦɢ ɪɚɡɛɢɟɧɢɹɦɢ ɤɨɧɟɱɧɨɝɨ ɦɧɨɠɟɫɬɜɚ ɨɛɴɟɤɬɨɜ»29.
ȿɫɥɢ R ɢ S ɞɜɚ ɪɚɡɛɢɟɧɢɹ ɦɧɨɠɟɫɬɜɚ ɢɡ N ɷɥɟɦɟɧɬɨɜ ɢ R ɪɚɡɛɢɜɚɟɬ ɟɝɨ ɧɚ m, ɚ S ɧɚ n ɤɥɚɫɫɨɜ, ɩɪɢɱɟɦ ɜ i-ɨɦ ɤɥɚɫɫɟ |Ri| ɷɥɟɦɟɧɬɨɜ, ɚ ɜ j-ɨɦ |Si| ɷɥɟɦɟɧɬɨɜ, ɬɨ ɦɟɪɚ ɛɥɢɡɨɫɬɢ ɪɚɡɛɢɟɧɢɣ
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(Ɂɞɟɫɶ R S – ɩɟɪɟɫɟɱɟɧɢɟ ɤɥɚɫɫɨɜ R ɢ S). |
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Ɍɚɤ ɤɚɤ d |
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N(N 1) , ɧɨɪɦɢɪɨɜɚɧɧɚɹ ɦɟɪɚ G |
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, ɩɪɢɱɟɦ 0 į 1, ɝɞɟ į = 0 |
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ɫɨɨɬɜɟɬɫɬɜɭɟɬ
[132]
ɦɚɤɫɢɦɚɥɶɧɨɣ ɫɜɹɡɢ, į = 1 ɦɢɧɢɦɚɥɶɧɨɣ (ɨɬɫɭɬɫɬɜɢɟ ɫɜɹɡɢ).
Ⱦɥɹ ɤɨɪɪɟɥɹɰɢɨɧɧɨɣ ɬɚɛɥɢɰɵ {Nij}: ɩɪɢɡɧɚɤ X ɨɫɭɳɟɫɬɜɥɹɟɬ ɪɚɡɛɢɟɧɢɟ ɨɛɳɧɨɫɬɢ N ɧɚ k ɤɥɚɫɫɨɜ ɯi, ɜ ɤɚɠɞɨɦ ɢɡ ɤɨɬɨɪɵɯ N(xi) ɷɥɟɦɟɧɬɨɜ: ɩɪɢɡɧɚɤ Y ɧɚ l ɤɥɚɫɫɨɜ yi, ɜ ɤɚɠɞɨɦ ɢɡ
ɤɨɬɨɪɵɯ N(yj) ɷɥɟɦɟɧɬɨɜ. |
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Ɍɚɤ ɤɚɤ Nij |
N(xi ) N(y j ) , ɬɨ ɦɟɪɚ ɛɥɢɡɨɫɬɢ ɞɜɭɯ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɯ ɪɚɡɛɢɟɧɢɣ |
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ª k |
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«¦N 2 (xi ) ¦N 2 (y j ) 2¦¦Nij2 |
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N(N 1) |
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i 1 j 1 |
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Ⱦɨɩɭɫɬɢɦ, ɱɬɨ ɦɵ ɢɫɫɥɟɞɭɟɦ ɧɟɤɨɬɨɪɨɟ ɪɚɡɛɢɟɧɢɟ, ɨɫɭɳɟɫɬɜɥɹɟɦɨɟ Y, ɢ ɯɨɬɢɦ ɜɵɹɫɧɢɬɶ ɡɧɚɱɢɦɨɫɬɶ ɪɹɞɚ ɩɪɢɡɧɚɤɨɜ X(ɪ) (ɪ = 1, 2, ...) ɞɥɹ ɜɵɹɜɥɟɧɢɹ ɞɚɧɧɨɝɨ ɪɚɡɛɢɟɧɢɹ.
28Ɇɢɪɤɢɧ Ȼ.Ƚ. ɇɨɜɵɣ ɩɨɞɯɨɞ ɤ ɨɛɪɚɛɨɬɤɟ ɫɨɰɢɨɥɨɝɢɱɟɫɤɨɣ ɢɧɮɨɪɦɚɰɢɢ. – ȼ ɤɧ.: ɂɡɦɟɪɟɧɢɟ ɢ ɦɨɞɟɥɢɪɨɜɚɧɢɟ ɜ ɫɨɰɢɨɥɨɝɢɢ. ɇɨɜɨɫɢɛɢɪɫɤ, 1969.
29Ⱥɜɬɨɦɚɬɢɤɚ ɢɬɟɥɟɦɟɯɚɧɢɤɚ, 1970, ʋ5.
91
Ɂɧɚɱɢɦɨɫɬɶ ɏ(ɪ) ɛɭɞɟɬ ɬɟɦ ɛɨɥɶɲɟɣ, ɱɟɦ ɛɥɢɠɟ ɪɚɡɛɢɟɧɢɹ, ɬ.ɟ. ɱɟɦ ɦɟɧɶɲɟ į (ɏ(ɪ), Y) Łįp. Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɡɧɚɱɢɦɨɫɬɶ ɩɪɢɡɧɚɤɚ ɏ(ɪ) ɩɨ ɨɬɧɨɲɟɧɢɸ ɤ ɪɚɡɛɢɟɧɢɸ Y ɦɨɠɧɨ ɩɪɢɧɹɬɶ ɨɛɪɚɬɧɨ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨɣ ɪɚɫɫɬɨɹɧɢɸ įp. ɗɬɭ ɡɧɚɱɢɦɨɫɬɶ («ɫɢɥɭ ɜɥɢɹɧɢɹ») ɦɨɠɧɨ ɢɧɬɟɪɩɪɟɬɢɪɨɜɚɬɶ, ɫɥɟɞɭɹ Ȼ.Ƚ. Ɇɢɪɤɢɧɭ, ɤɚɤ ɦɟɪɭ ɫɜɹɡɢ ɦɟɠɞɭ ɩɪɢɡɧɚɤɚɦɢ. ɉɭɫɬɶ, ɧɚɩɪɢɦɟɪ, ɪɚɡɛɢɟɧɢɟ ɍ – ɷɬɨ ɫɨɰɢɚɥɶɧɨ-ɩɪɨɮɟɫɫɢɨɧɚɥɶɧɵɟ ɝɪɭɩɩɵ, ɚ ɏ(ɪ) – ɪɚɡɥɢɱɧɵɟ ɫɨɰɢɚɥɶɧɨɞɟɦɨɝɪɚɮɢɱɟɫɤɢɟ ɩɪɢɡɧɚɤɢ (ɩɪɨɮɟɫɫɢɹ, ɤɜɚɥɢɮɢɤɚɰɢɹ, ɨɛɪɚɡɨɜɚɧɢɟ, ɞɨɯɨɞ, ɦɟɫɬɨ ɠɢɬɟɥɶɫɬɜɚ ɢ ɬ.ɞ.), ɜɵɱɢɫɥɹɹ įp, ɦɵ ɦɨɠɟɦ ɨɩɪɟɞɟɥɢɬɶ ɡɧɚɱɢɦɨɫɬɶ (ɜɥɢɹɧɢɟ) ɪɚɡɥɢɱɧɵɯ ɏ(ɪ) ɞɥɹ ɜɵɹɜɥɟɧɢɹ Y-ɪɚɡɛɢɟɧɢɹ, ɜɵɞɟɥɢɬɶ ɧɚɢɛɨɥɟɟ ɢɧɮɨɪɦɚɬɢɜɧɵɟ ɩɪɢɡɧɚɤɢ.
Ɋɚɫɫɦɚɬɪɢɜɚɥɚɫɶ ɢ ɬɚɤɚɹ ɡɚɞɚɱɚ: ɩɭɫɬɶ Y – ɷɬɨ ɪɚɫɫɟɥɟɧɢɟ ɪɚɛɨɬɧɢɤɨɜ ɩɨ «ɡɨɧɚɦ ɞɨɫɬɭɩɧɨɫɬɢ ɩɪɟɞɩɪɢɹɬɢɹ»30, ɚ ɏ(ɪ) – ɧɟɤɨɬɨɪɵɟ ɫɨɰɢɚɥɶɧɨ-ɞɟɦɨɝɪɚɮɢɱɟɫɤɢɟ ɩɪɢɡɧɚɤɢ, ɡɧɚɱɢɦɵɟ ɞɥɹ ɪɚɫɫɟɥɟɧɢɹ. ɇɚɢɛɨɥɟɟ ɡɧɚɱɢɦɵɦ ɩɪɢɡɧɚɤɨɦ ɨɤɚɡɚɥɚɫɶ ɩɪɢɧɚɞɥɟɠɧɨɫɬɶ ɤ ɫɨɰɢɚɥɶɧɨ-ɩɪɨɮɟɫɫɢɨɧɚɥɶɧɨɣ ɝɪɭɩɩɟ.
Ɋɚɫɫɦɨɬɪɢɦ ɟɳɟ ɪɚɡ ɜɨɩɪɨɫ ɨ ɫɜɹɡɢ ɦɟɠɞɭ ɭɞɨɜɥɟɬɜɨɪɟɧɧɨɫɬɹɦɢ ɡɚɪɚɛɨɬɧɨɣ ɩɥɚɬɨɣ ɢ ɪɚɛɨɬɨɣ, ɢɫɩɨɥɶɡɭɹ ɞɥɹ ɟɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ ɨɛɫɭɠɞɚɟɦɭɸ ɦɟɪɭ (ɫɦ. ɩɪɢɦɟɪ ʋ 14 § 1 ɷɬɨɣ ɝɥɚɜɵ).
[133] |
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D = N(N–1)=1015056, į – 0,490.
ɍɩɪɚɠɧɟɧɢɟ 69. ɉɨɤɚɡɚɬɶ, ɱɬɨ ɞɥɹ ɬɚɛɥɢɰɵ 19 § 1 ɷɬɨɣ ɝɥɚɜɵ į = 0,528.
ȼ ɩɟɪɜɨɦ ɫɥɭɱɚɟ į ɦɟɧɶɲɟ, ɧɨ ɬɚɤ ɤɚɤ ɫɜɹɡɶ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɚ 1/ į ɬɨ ɨɧɚ ɛɨɥɶɲɟ, ɱɟɦ ɜɨ ɜɬɨɪɨɦ; ɬɚɤɢɦ ɨɛɪɚɡɨɦ, ɫɞɟɥɚɧɧɵɣ ɪɚɧɟɟ ɜɵɜɨɞ (§ 1) ɩɨɞɬɜɟɪɠɞɚɟɬɫɹ.
ǻ-ɤɨɷɮɮɢɰɢɟɧɬ (ɧɨɦɢɧɚɥɶɧɵɟ ɲɤɚɥɵ)
ȼ ɪɚɛɨɬɟ ɂ.Ⱥ. ɒɤɪɚɛɤɢɧɨɣ ɢ Ƚ.ɂ. ɋɦɢɪɧɨɜɨɣ «ɉɪɨɝɪɚɦɦɚ ɢɡɦɟɪɟɧɢɹ ɬɟɫɧɨɬɵ ɫɜɹɡɢ ɦɟɠɞɭ ɞɜɭɦɹ ɩɪɢɡɧɚɤɚɦɢ»31 ɩɪɟɞɥɚɝɚɟɬɫɹ ɞɥɹ ɢɡɦɟɪɟɧɢɹ ɫɜɹɡɢ ɢɫɩɨɥɶɡɨɜɚɬɶ ɦɨɞɭɥɶɧɵɣ ɤɨɷɮɮɢɰɢɟɧɬ ǻ.
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ɜɜɟɞɟɦ ɜɟɥɢɱɢɧɭ |
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Ɇɟɪɨɣ ɫɜɹɡɢ, ɬɨɱɧɟɟ, ɜɥɢɹɧɢɹ X ɧɚ Y, ɦɨɠɟɬ ɫɥɭɠɢɬɶ |
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ɉɨɤɚɠɟɦ ɷɬɨ. ȿɫɥɢ ɩɪɢɡɧɚɤɢ ɧɟɡɚɜɢɫɢɦɵ, ɬɨ nij |
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ɫɭɦɦɚ ɨɛɪɚɳɚɟɬɫɹ ɜ ɧɭɥɶ.
Ʌɨɝɢɤɚ ɢɡɦɟɪɟɧɢɹ ɫɜɹɡɢ ɬɚɤɨɜɚ: ɟɫɥɢ ɩɪɢɡɧɚɤɢɧɟɡɚɜɢ-
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30«Ɂɨɧɚ ɞɨɫɬɭɩɧɨɫɬɢ ɩɪɟɞɩɪɢɹɬɢɹ» ɨɩɪɟɞɟɥɹɟɬɫɹ ɜɪɟɦɟɧɟɦ, ɡɚɬɪɚɱɢɜɚɟɦɵɦ ɪɚɛɨɬɧɢɤɨɦ ɧɚ ɩɟɪɟɞɜɢɠɟɧɢɟ ɨɬ ɦɟɫɬɚ ɠɢɬɟɥɶɫɬɜɚ ɞɨ ɦɟɫɬɚ ɪɚɛɨɬɵ. ɉɨ ɧɨɪɦɚɦ ɝɪɚɞɨɫɬɪɨɢɬɟɥɶɫɬɜɚ ɜɵɞɟɥɹɸɬɫɹ ɱɟɬɵɪɟ ɡɨɧɵ: Ⱥ (ɞɨ 30 ɦɢɧ.), Ȼ (ɨɬ 30 ɞɨ 45 ɦɢɧ.), ȼ (ɨɬ 45 ɦɢɧ. ɞɨ ɱɚɫɚ), Ƚ (ɫɜɵɲɟ ɱɚɫɚ).
31Ⱥɧɚɥɢɡ ɫɨɰɢɨɥɨɝɢɱɟɫɤɨɣ ɢɧɮɨɪɦɚɰɢɢ ɫ ɩɪɢɦɟɧɟɧɢɟɦ ɗȼɆ, ɱ.1. Ɇ., 1973, ɫ.143-157
92
[134] |
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ɫɢɦɵ, ɬɨ ɩɪɢ ɢɡɦɟɧɟɧɢɢ X ɡɧɚɱɟɧɢɟ Y ɧɟ ɞɨɥɠɧɨ ɦɟɧɹɬɶɫɹ, ɬ.ɟ. ɱɢɫɥɚ ɢɧɞɢɜɢɞɨɜ |
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ɪɚɡɧɵɦɢ X ɩɪɢ ɮɢɤɫɢɪɨɜɚɧɧɨɦ Y ɞɨɥɠɧɵ ɛɵɬɶ ɩɪɢɦɟɪɧɨ ɨɞɢɧɚɤɨɜɵ, ɬ.ɟ. ɪɚɜɧɵɦɢ |
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ɠɟ X ɜɥɢɹɟɬ ɧɚ Y, ɬɨ nij ɞɨɥɠɧɵ ɨɬɥɢɱɚɬɶɫɹ ɨɬɫɪɟɞɧɟɝɨ nj . |
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Ɉɛɫɭɠɞɚɟɦɚɹ ɫɭɦɦɚ ɧɟ ɹɜɥɹɟɬɫɹ ɧɨɪɦɢɪɨɜɚɧɧɨɣ. ȼ ɪɚɛɨɬɟ ɒɤɪɚɛɤɢɧɨɣ ɢ ɋɦɢɪɧɨɜɨɣ ɜ
ɤɚɱɟɫɬɜɟ ɤɨɷɮɮɢɰɢɟɧɬɚ ɩɪɢ ɫɭɦɦɟ ɩɪɟɞɥɚɝɚɟɬɫɹ ɢɫɩɨɥɶɡɨɜɚɬɶ ɜɟɥɢɱɢɧɭ 1 . Ɉɞɧɚɤɨ ɤɚɤ ɥɟɝɤɨ k
ɜɢɞɟɬɶ, 'c S ɧɟ ɹɜɥɹɟɬɫɹ ɧɨɪɦɢɪɨɜɚɧɧɨɣ ɜɟɥɢɱɢɧɨɣ. k
Ⱦɥɹ ɧɨɪɦɢɪɨɜɤɢ ɧɟɨɛɯɨɞɢɦɨ ɧɚɣɬɢ ɦɚɤɫɢɦɚɥɶɧɨɟ ɡɧɚɱɟɧɢɟ ɫɭɦɦɵ S. Ɉɤɚɡɵɜɚɟɬɫɹ, ɱɬɨ ɨɧɨ ɞɨɫɬɢɝɚɟɬɫɹ ɜ ɫɥɭɱɚɟ ɩɨɥɧɨɣ ɫɜɹɡɢ (ɫɜɹɡɶ ɦɵ ɧɚɡɵɜɚɟɦ ɩɨɥɧɨɣ, ɟɫɥɢ ɤɚɠɞɨɦɭ X
ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɨɞɧɨ ɡɧɚɱɟɧɢɟ Y) ɢ ɪɚɜɧɨ 2 (m 1)(2k m) , ɝɞɟ m = min(k,l) – ɦɟɧɶɲɟɟ ɢɡ ɱɢɫɟɥ k
k ɢ l.
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɧɨɪɦɢɪɨɜɚɧɧɵɣ ɤɨɷɮɮɢɰɢɟɧɬ, ɨɩɢɫɵɜɚɸɳɢɣ ɜɥɢɹɧɢɟ X ɧɚ Y:
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ȼɫɟ ɪɚɧɟɟ ɪɚɫɫɦɨɬɪɟɧɧɵɟ ɡɞɟɫɶ ɤɨɷɮɮɢɰɢɟɧɬɵ ɩɪɢɦɟɧɢɦɵ ɞɚɠɟ ɞɥɹ ɧɨɦɢɧɚɥɶɧɵɯ ɲɤɚɥ. ɉɟɪɟɣɞɟɦ ɤ ɤɨɷɮɮɢɰɢɟɧɬɚɦ, ɤɨɬɨɪɵɟ ɢɫɩɨɥɶɡɭɸɬɫɹ ɩɪɢ ɧɚɥɢɱɢɢ ɭɩɨɪɹɞɨɱɟɧɢɹ ɡɧɚɱɟɧɢɣ ɩɪɢɡɧɚɤɨɜ.
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Ȗ-ɤɨɷɮɮɢɰɢɟɧɬ Ƚɭɞɦɚɧɚ |
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[135] |
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ɝɞɟ Ɋ – ɱɢɫɥɨ ɩɚɪ ɨɛɴɟɤɬɨɜ, ɭ ɤɨɬɨɪɵɯ ɨɛɚ ɩɪɢɡɧɚɤɚ ɭɩɨɪɹɞɨɱɟɧɵ ɜ ɨɞɢɧɚɤɨɜɨɣ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ, ɚ Q – ɬɨ ɠɟ, ɧɨ ɜɨɛɪɚɬɧɨɣ.
ɉɭɫɬɶ ɡɧɚɱɟɧɢɹ X ɢ Y ɜ ɤɨɪɪɟɥɹɰɢɨɧɧɨɣ ɬɚɛɥɢɰɟ ɜɵɩɢɫɚɧɵ ɜ ɨɞɢɧɚɤɨɜɨɣ ɩɨɫɥɟɞɨɜɚɬɟɥɶɧɨɫɬɢ. ȼɟɥɢɱɢɧɭ Ɋ ɦɨɠɧɨ ɜɵɱɢɫɥɢɬɶ ɤɚɤ ɫɭɦɦɭ ɪɟɡɭɥɶɬɚɬɨɜ ɭɦɧɨɠɟɧɢɹ ɱɚɫɬɨɬ ɤɚɠɞɨɣ
Ɍɚɛɥɢɰɚ 32
ɉɪɢɦɟɪ ɪɚɫɱɟɬɚ Ȗ-ɤɨɷɮɮɢɰɢɟɧɬɚ Ƚɭɞɦɚɧɚ
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93