Занг В.Б. Синергетическая экономика. Время и перемены в нелинейной экономической теории

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F(x, r, s) — k-ɩɚɪɚɦɟɬɪɢɱɟɫɤɚɹ ɛɢɮɭɪɤɚɰɢɨɧɧɚɹ ɡɚɞɚɱɚ. ɇɚɡɨɜɟɦ ɮɭɧɤɰɢɸ F

ɜɨɡɦɭɳɟɧɢɟɦ f, ɟɫɥɢ

F(x, r, 0, ..., 0) = f(x, r). (3.8.7)

Ⱦɥɹ ɪɟɲɟɧɢɹ ɩɪɨɛɥɟɦɵ ɤɥɚɫɫɢɮɢɤɚɰɢɢ ɦɵ ɢɳɟɦ k-ɩɚɪɚɦɟɬɪɢɱɟɫɤɨɟ ɫɟɦɟɣɫɬɜɨ F

ɜɨɡɦɭɳɟɧɢɣ ɮɭɧɤɰɢɢ f, ɨɛɥɚɞɚɸɳɟɟ ɬɟɦ ɫɜɨɣɫɬɜɨɦ, ɱɬɨ ɤɚɤɨɣ ɛɵ ɧɢ ɛɵɥɚ ɮɭɧɤɰɢɹ

f, ɥɸɛɨɟ ɟɟ ɜɨɡɦɭɳɟɧɢɟ ɷɤɜɢɜɚɥɟɧɬɧɨ F ɞɥɹ ɧɟɤɨɬɨɪɨɝɨ s ɜ ɨɤɪɟɫɬɧɨɫɬɢ ɧɭɥɹ. Ɍɨ

ɟɫɬɶ, ɞɥɹ ɩɪɨɢɡɜɨɥɶɧɨɝɨ ɜɨɡɦɭɳɚɸɳɟɝɨ ɱɥɟɧɚ hp(x, r, h) ɫɭɳɟɫɬɜɭɸɬ ɬɚɤɢɟ ɡɧɚɱɟɧɢɹ

ɩɚɪɚɦɟɬɪɨɜ s

(h),.. ., s

(h), ɱɬɨ ɞɥɹ ɦɚɥɨɝɨ s ɮɭɧɤɰɢɹ f + hp ɷɤɜɢɜɚɥɟɧɬɧɚ F. ɇɚɡɨɜɟɦ

ɬɚɤɭɸ F ɭɧɢɜɟɪɫɚɥɶɧɨɣ ɞɟɮɨɪɦɚɰɢɟɣ ɮɭɧɤɰɢɢ f. ɋɥɟɞɭɟɬ ɡɚɦɟɬɢɬɶ, ɱɬɨ ɱɢɫɥɨ k

ɩɚɪɚɦɟɬɪɨɜ, ɧɟɨɛɯɨɞɢɦɵɯ ɞɥɹ ɫɭɳɟɫɬɜɨɜɚɧɢɹ ɭɧɢɜɟɪɫɚɥɶɧɨɣ ɞɟɮɨɪɦɚɰɢɢ, ɡɚɜɢɫɢɬ

ɨɬ ɫɜɨɣɫɬɜ ɢɫɫɥɟɞɭɟɦɨɣ ɮɭɧɤɰɢɢ f. Ɍɚɤ, ɧɚɩɪɢɦɟɪ, ɦɨɠɧɨ ɩɨɤɚɡɚɬɶ, ɱɬɨ

ɹɜɥɹɟɬɫɹ ɭɧɢɜɟɪɫɚɥɶɧɨɣ ɞɟɮɨɪɦɚɰɢɟɣ ɜɢɥɤɢ.

Ɉɩɪɟɞɟɥɢɜ ɭɧɢɜɟɪɫɚɥɶɧɭɸ ɞɟɮɨɪɦɚɰɢɸ f, ɢɫɫɥɟɞɭɟɦ ɩɪɨɫɬɪɚɧɫɬɜɨ ɩɚɪɚɦɟɬɪɨɜ

ɞɟɮɨɪɦɚɰɢɢ R

, ɱɬɨɛɵ ɨɩɪɟɞɟɥɢɬɶ ɱɢɫɥɨ ɪɚɡɥɢɱɧɵɯ ɛɢɮɭɪɤɚɰɢɨɧɧɵɯ ɞɢɚɝɪɚɦɦ

{(

ɯ, r) : F(x, r, s) = 0}. Ⱦɥɹ ɭɧɢɜɟɪɫɚɥɶɧɨɣ ɪɚɡɜɟɪɬɤɢ ɜɢɥɤɢ (3.8.8) ɢɦɟɟɬɫɹ ɱɟɬɵɪɟ

ɨɫɧɨɜɧɵɯ ɛɢɮɭɪɤɚɰɢɨɧɧɵɯ ɞɢɚɝɪɚɦɦɵ, ɢɡɨɛɪɚɠɟɧɧɵɯ ɧɚ ɪɢɫ. 3.22, ɤɨɬɨɪɵɟ

ɜɨɡɧɢɤɚɸɬ ɩɪɢ ɢɡɦɟɧɟɧɢɢ s (Ƚɨɥɭɛɢɰɤɢɣ ɢ ɒɟɮɮɟɪ, 1984).

ɋɥɟɞɭɟɬ ɨɬɦɟɬɢɬɶ, ɱɬɨ ɟɫɥɢ ɜ ɦɨɞɟɥɶ ɜɜɟɫɬɢ ɬɪɢ ɢ ɛɨɥɟɟ ɩɚɪɚɦɟɬɪɨɜ, ɬɨ ɧɢɤɚɤɨɝɨ

ɧɨɜɨɝɨ ɩɨɜɟɞɟɧɢɹ ɦɵ ɧɟ ɨɛɧɚɪɭɠɢɦ. ɗɬɨ ɜɵɬɟɤɚɟɬ ɢɡ ɬɨɝɨ ɮɚɤɬɚ, ɱɬɨ (3.8.8)

ɹɜɥɹɟɬɫɹ ɭɧɢɜɟɪɫɚɥɶɧɨɣ ɪɚɡɜɟɪɬɤɨɣ ɜɢɥɤɢ.

Ɇɟɬɨɞɵ ɬɟɨɪɢɢ ɨɫɨɛɟɧɧɨɫɬɟɣ ɩɨɡɜɨɥɹɸɬ ɨɩɪɟɞɟɥɢɬɶ ɬɨɱɧɨɟ ɱɢɫɥɨ ɩɚɪɚɦɟɬɪɨɜ,

ɧɟɨɛɯɨɞɢɦɵɯ ɞɥɹ ɨɩɢɫɚɧɢɹ ɧɚɢɛɨɥɟɟ ɨɛɳɢɯ ɜɨɡɦɭɳɟɧɢɣ ɛɢɮɭɪɤɚɰɢɨɧɧɨɣ ɡɚɞɚɱɢ.

ɉɨɞɯɨɞ, ɨɫɧɨɜɚɧɧɵɣ ɧɚ ɬɟɨɪɢɢ ɨɫɨɛɟɧɧɨɫɬɟɣ, ɩɪɢɦɟɧɢɦ ɬɚɤɠɟ ɤ ɡɚɞɚɱɚɦ

ɭɫɬɨɣɱɢɜɨɫɬɢ. Ɂɚ ɞɚɥɶɧɟɣɲɢɦɢ ɫɜɟɞɟɧɢɹɦɢ ɨɬɫɵɥɚɟɦ ɱɢɬɚɬɟɥɹ ɤ ɤɧɢɝɚɦ

Ƚɨɥɭɛɢɰɤɨɝɨ ɢ ɒɟɮɮɟɪɚ (1984, 1988).

3.9 Ɍɟɨɪɢɹ ɤɚɬɚɫɬɪɨɮ

Ɋɚɫɫɦɨɬɪɢɦ ɞɢɧɚɦɢɱɟɫɤɭɸ ɫɢɫɬɟɦɭ

ɝɞɟ x

ɩɪɟɞɫɬɚɜɥɹɸɬ ɫɨɛɨɣ ɧɟɡɚɜɢɫɢɦɵɟ ɩɟɪɟɦɟɧɧɵɟ, ɚ r — ɩɚɪɚɦɟɬɪɵ (ɧɚɡɵɜɚɟɦɵɟ

ɜ ɬɟɨɪɢɢ ɤɚɬɚɫɬɪɨɮ ɨɛɵɱɧɨ ɭɩɪɚɜɥɹɸɳɢɦɢ).

ɋɢɥɶɧɨɟ ɩɪɟɞɩɨɥɨɠɟɧɢɟ, ɤɨɬɨɪɨɟ ɢɝɪɚɟɬ ɜɚɠɧɭɸ ɪɨɥɶ ɜ ɬɟɨɪɢɢ ɤɚɬɚɫɬɪɨɮ —

ɯɨɬɹ ɩɪɟɞɩɪɢɧɢɦɚɟɬɫɹ ɨɝɪɨɦɧɨɟ ɱɢɫɥɨ ɩɨɩɵɬɨɤ ɟɝɨ ɨɫɥɚɛɢɬɶ — ɫɨɫɬɨɢɬ ɜ ɬɨɦ, ɱɬɨ

ɫɢɫɬɟɦɚ (3.9.1) ɦɨɠɟɬ ɛɵɬɶ ɩɨɥɭɱɟɧɚ ɫ ɩɨɦɨɳɶɸ ɫɥɟɞɭɸɳɢɯ ɫɨɨɬɧɨɲɟɧɢɣ:

ɝɞɟ V — «ɩɨɬɟɧɰɢɚɥɶɧɚɹ» ɮɭɧɤɰɢɹ. ɗɬɨ ɩɪɟɞɩɨɥɨɠɟɧɢɟ ɨɡɧɚɱɚɟɬ, ɱɬɨ (3.9.1)

ɹɜɥɹɟɬɫɹ ɝɪɚɞɢɟɧɬɧɨɣ ɫɢɫɬɟɦɨɣ.

Ɉɛɳɚɹ ɤɥɚɫɫɢɮɢɤɚɰɢɹ ɪɟɲɟɧɢɣ ɫɢɫɬɟɦɵ (3.9.2) ɦɨɠɟɬ ɛɵɬɶ ɩɪɨɜɟɞɟɧɚ ɧɚ ɨɫɧɨɜɟ

ɫɜɨɣɫɬɜɚ ɧɚɪɭɲɟɧɢɹ ɢɯ ɫɬɪɭɤɬɭɪɧɨɣ ɭɫɬɨɣɱɢɜɨɫɬɢ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ. Ɇɨɠɧɨ

ɨɩɪɟɞɟɥɢɬɶ ɬɨɱɤɢ, ɜ ɤɨɬɨɪɵɯ ɧɚɪɭɲɚɟɬɫɹ ɭɫɬɨɣɱɢɜɨɫɬɶ ɫɬɚɰɢɨɧɚɪɧɵɯ ɫɨɫɬɨɹɧɢɣ.

ɗɬɢ ɬɨɱɤɢ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ ɩɚɪɚɦɟɬɪɨɜ ɨɛɪɚɡɭɸɬ ɝɢɩɟɪɩɨɜɟɪɯɧɨɫɬɢ, ɜɞɨɥɶ ɤɨɬɨɪɵɯ

ɢɦɟɟɬ ɦɟɫɬɨ ɥɢɛɨ ɜɟɬɜɥɟɧɢɟ ɪɟɲɟɧɢɣ ɭɪɚɜɧɟɧɢɹ, ɥɢɛɨ ɮɭɧɤɰɢɹ V ɞɨɫɬɢɝɚɟɬ

ɚɛɫɨɥɸɬɧɨɝɨ ɦɢɧɢɦɭɦɚ ɧɟ ɦɟɧɟɟ, ɱɟɦ ɜ ɞɜɭɯ ɪɚɡɥɢɱɧɵɯ ɬɨɱɤɚɯ. Ⱦɪɭɝɢɦɢ ɫɥɨɜɚɦɢ,

ɩɪɢ ɩɟɪɟɫɟɱɟɧɢɢ ɷɬɢɯ ɝɢɩɟɪɩɨɜɟɪɯɧɨɫɬɟɣ ɩɪɨɢɫɯɨɞɢɬ ɩɟɪɟɯɨɞ ɢɡ ɨɛɥɚɫɬɢ ɫ ɨɞɧɢɦ

ɬɢɩɨɦ ɞɢɧɚɦɢɤɢ ɜ ɨɛɥɚɫɬɶ ɞɢɧɚɦɢɤɢ ɤɚɱɟɫɬɜɟɧɧɨ ɢɧɨɣ.

Ɍɟɨɪɢɹ ɤɚɬɚɫɬɪɨɮ ɧɚɯɨɞɢɬ ɦɧɨɠɟɫɬɜɨ ɩɪɢɥɨɠɟɧɢɣ ɜ ɪɚɡɥɢɱɧɵɯ ɨɛɥɚɫɬɹɯ ɧɚɭɤɢ.

ɉɪɢɦɟɪɵ ɩɪɢɥɨɠɟɧɢɹ ɷɬɨɣ ɬɟɨɪɢɢ ɤ ɫɨɰɢɚɥɶɧɵɦ ɫɢɫɬɟɦɚɦ ɦɨɠɧɨ ɧɚɣɬɢ, ɧɚɩɪɢɦɟɪ,

ɜ ɤɧɢɝɟ ȼɢɥɶɫɨɧɚ (1981). Ɇɵ ɞɚɞɢɦ ɧɟɫɤɨɥɶɤɨ ɬɚɤɢɯ ɩɪɢɦɟɪɨɜ ɜ ɝɥ. 4, ɬ. ɟ. ɩɪɨɜɟɞɟɦ

ɢɫɫɥɟɞɨɜɚɧɢɟ ɫɬɪɭɤɬɭɪɧɵɯ ɢɡɦɟɧɟɧɢɣ ɞɢɧɚɦɢɱɟɫɤɢɯ ɫɢɫɬɟɦ, ɤɨɬɨɪɵɟ ɨɩɪɟɞɟɥɹɸɬɫɹ

ɤɨɧɤɪɟɬɧɵɦɢ ɮɨɪɦɚɦɢ ɮɭɧɤɰɢɢ V(ɯ, r), ɝɞɟ ɯ ∈ R

ɢ r ∈ R

(ɫɦ. Ƚɢɥɦɨɪ, 1981).

ɉɪɟɠɞɟ ɜɫɟɝɨ, ɤɨɫɧɟɦɫɹ ɥɨɤɚɥɶɧɵɯ ɫɜɨɣɫɬɜ ɮɭɧɤɰɢɢ

ɋɜɨɣɫɬɜɚ ɷɬɨɣ ɮɭɧɤɰɢɢ ɨɩɪɟɞɟɥɹɸɬɫɹ ɪɹɞɨɦ ɬɟɨɪɟɦ ɮɭɧɤɰɢɨɧɚɥɶɧɨɝɨ ɚɧɚɥɢɡɚ —

ɬɟɨɪɟɦɨɣ ɨ ɧɟɹɜɧɨɣ ɮɭɧɤɰɢɢ, ɥɟɦɦɨɣ Ɇɨɪɫɚ ɢ ɬɟɨɪɟɦɨɣ Ɍɨɦɚ.

Ɍɟɨɪɟɦɚ ɨ ɧɟɹɜɧɨɣ ɮɭɧɤɰɢɢ ɭɬɜɟɪɠɞɚɟɬ, ɱɬɨ ɟɫɥɢ ɝɪɚɞɢɟɧɬ ∇

V ɧɟ ɪɚɜɟɧ ɧɭɥɸ ɜ

ɧɟɤɨɬɨɪɨɣ ɬɨɱɤɟ, ɬɨ ɦɨɠɧɨ ɩɨɞɨɛɪɚɬɶ ɬɚɤɨɟ ɝɥɚɞɤɨɟ (ɢɦɟɸɳɟɟ ɩɪɨɢɡɜɨɞɧɵɟ

ɩɪɨɢɡɜɨɥɶɧɨ ɜɵɫɨɤɨɝɨ ɩɨɪɹɞɤɚ) ɩɪɟɨɛɪɚɡɨɜɚɧɢɟ ɩɟɪɟɦɟɧɧɵɯ

ɱɬɨ V ɦɨɠɟɬ ɛɵɬɶ ɩɪɟɞɫɬɚɜɥɟɧɚ ɜ ɜɢɞɟ

ɝɞɟ ɫ — ɤɨɧɫɬɚɧɬɚ.

Ɉɩɪɟɞɟɥɟɧɢɟ 3.9.1. (Ʉɪɢɬɢɱɟɫɤɢɟ ɬɨɱɤɢ Ɇɨɪɫɚ.) ɋɬɚɰɢɨɧɚɪɧɵɟ ɬɨɱɤɢ, ɢɥɢ

ɤɪɢɬɢɱɟɫɤɢɟ ɬɪɚɟɤɬɨɪɢɢ, ɝɥɚɞɤɨɣ ɮɭɧɤɰɢɢ V(x) — ɷɬɨ ɬɨɱɤɢ, ɜ ɤɨɬɨɪɵɯ ∇

V = 0.

Ʉɪɢɬɢɱɟɫɤɢɟ ɬɨɱɤɢ, ɜ ɤɨɬɨɪɵɯ det V

≠ 0, ɝɞɟ V

= ∂

V|∂x

∂x

, ɧɚɡɵɜɚɸɬɫɹ

ɢɡɨɥɢɪɨɜɚɧɧɵɦɢ ɧɟɜɵɪɨɠɞɟɧɧɵɦɢ ɢɥɢ ɤɪɢɬɢɱɟɫɤɢɦɢ ɬɨɱɤɚɦɢ Ɇɨɪɫɚ.

ȿɫɥɢ ɫɬɚɰɢɨɧɚɪɧɚɹ ɬɨɱɤɚ ɹɜɥɹɟɬɫɹ ɤɪɢɬɢɱɟɫɤɨɣ ɬɨɱɤɨɣ Ɇɨɪɫɚ, ɬɨ ɥɟɦɦɚ Ɇɨɪɫɚ

ɝɚɪɚɧɬɢɪɭɟɬ ɫɭɳɟɫɬɜɨɜɚɧɢɟ ɬɚɤɨɝɨ ɝɥɚɞɤɨɝɨ ɩɪɟɨɛɪɚɡɨɜɚɧɢɹ ɩɟɪɟɦɟɧɧɵɯ, ɱɬɨ

ɩɨɬɟɧɰɢɚɥ ɦɨɠɟɬ ɛɵɬɶ ɥɨɤɚɥɶɧɨ ɩɪɟɞɫɬɚɜɥɟɧ ɜ ɜɢɞɟ ɤɜɚɞɪɚɬɢɱɧɨɣ ɮɨɪɦɵ

ɝɞɟ Θ

— ɫɨɛɫɬɜɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɦɚɬɪɢɰɵ ɭɫɬɨɣɱɢɜɨɫɬɢ V

, ɜɵɱɢɫɥɟɧɧɵɟ ɜ ɬɨɱɤɟ

ɪɚɜɧɨɜɟɫɢɹ. ɉɟɪɟɯɨɞ ɤ ɧɨɜɨɦɭ ɦɚɫɲɬɚɛɭ ɞɥɢɧ ɜ ɧɨɜɵɯ ɤɨɨɪɞɢɧɚɬɚɯ z

= ɭ

°Θ

1/2

ɩɟɪɟɜɨɞɢɬ ɤɜɚɞɪɚɬɢɱɧɭɸ ɮɨɪɦɭ (3.9.5) ɜ ɤɚɧɨɧɢɱɟɫɤɭɸ ɮɨɪɦɭ Ɇɨɪɫɚ

Ɏɭɧɤɰɢɹ )(zM

ɧɚɡɵɜɚɟɬɫɹ i-ɫɟɞɥɨɦ Ɇɨɪɫɚ. ȼ ɫɨɫɬɨɹɧɢɢ ɪɚɜɧɨɜɟɫɢɹ ɥɨɤɚɥɶɧɵɦ

ɦɢɧɢɦɭɦɨɦ ɨɛɥɚɞɚɸɬ ɬɨɥɶɤɨ 0-ɫɟɞɥɚ Ɇɨɪɫɚ, ɬɚɤ ɱɬɨ ɬɨɥɶɤɨ ɬɚɤɢɟ ɫɟɞɥɚ ɹɜɥɹɸɬɫɹ

ɥɨɤɚɥɶɧɨ ɭɫɬɨɣɱɢɜɵɦɢ.

Ɉɩɪɟɞɟɥɟɧɢɟ 3.9.2. (ɇɟɦɨɪɫɨɜɵ ɤɪɢɬɢɱɟɫɤɢɟ ɬɨɱɤɢ.) Ʉɪɢɬɢɱɟɫɤɢɟ ɬɨɱɤɢ

ɮɭɧɤɰɢɢ V(x), ɜ ɤɨɬɨɪɵɯ det V

= 0, ɧɚɡɵɜɚɸɬɫɹ ɧɟɢɡɨɥɢɪɨɜɚɧɧɵɦɢ,

ɜɵɪɨɠɞɟɧɧɵɦɢ, ɢɥɢ ɧɟɦɨɪɫɨɜɵɦɢ ɤɪɢɬɢɱɟɫɤɢɦɢ ɬɨɱɤɚɦɢ.

ȿɫɥɢ ɩɨɬɟɧɰɢɚɥ ɡɚɜɢɫɢɬ ɨɬ ɨɞɧɨɝɨ ɢɥɢ ɛɨɥɟɟ ɭɩɪɚɜɥɹɸɳɢɯ ɩɚɪɚɦɟɬɪɨɜ ɝ, ɨɬ

ɷɬɢɯ ɩɚɪɚɦɟɬɪɨɜ ɡɚɜɢɫɹɬ ɢ ɦɚɬɪɢɰɚ ɭɫɬɨɣɱɢɜɨɫɬɢ V

, ɢ ɟɟ ɫɨɛɫɬɜɟɧɧɵɟ ɡɧɚɱɟɧɢɹ

. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɜɩɨɥɧɟ ɜɨɡɦɨɠɧɨ, ɱɬɨ ɞɥɹ ɨɩɪɟɞɟɥɟɧɧɵɯ ɡɧɚɱɟɧɢɣ

ɭɩɪɚɜɥɹɸɳɢɯ ɩɚɪɚɦɟɬɪɨɜ ɨɞɧɨ ɢɥɢ ɛɨɥɟɟ ɫɨɛɫɬɜɟɧɧɵɯ ɡɧɚɱɟɧɢɣ ɨɛɪɚɳɚɸɬɫɹ ɜ

ɧɭɥɶ. ȼ ɷɬɨɦ ɫɥɭɱɚɟ det V

= 0, ɬ.ɟ. ɭɫɥɨɜɢɹ, ɩɪɢ ɤɨɬɨɪɵɯ ɫɩɪɚɜɟɞɥɢɜɚ ɥɟɦɦɚ

Ɇɨɪɫɚ, ɛɨɥɟɟ ɧɟ ɜɵɩɨɥɧɹɸɬɫɹ. ȿɫɥɢ m ɫɨɛɫɬɜɟɧɧɵɯ ɡɧɚɱɟɧɢɣ Θ

(r),..., (Θ

(r) ɩɪɢ

r = r

ɨɛɪɚɳɚɸɬɫɹ ɜ ɧɭɥɶ, ɬɨ ɦɨɠɧɨ ɜɨɫɩɨɥɶɡɨɜɚɬɶɫɹ ɥɟɦɦɨɣ Ɍɨɦɚ ɨ ɪɚɫɳɟɩɥɟɧɢɢ

ɩɨɬɟɧɰɢɚɥɚ ɧɚ ɦɨɪɫɨɜɭ ɢ ɧɟɦɨɪɫɨɜɭ ɱɚɫɬɢ:

ɝɞɟ ɬ «ɩɥɨɯɢɯ» ɤɨɨɪɞɢɧɚɬ ɭ

(ɯ, r) (i = 1,..., m), ɫɜɹɡɚɧɧɵɯ ɫ ɨɛɪɚɳɟɧɢɟɦ ɜ ɧɭɥɶ m

ɫɨɛɫɬɜɟɧɧɵɯ ɡɧɚɱɟɧɢɣ Θ

(r), ɡɚɜɢɫɹɬ ɨɬ x ɢ ɨɬ r. «ɏɨɪɨɲɢɟ» ɤɨɨɪɞɢɧɚɬɵ y

m+j

(ɯ)

(j = 1, …,

ɩ–ɬ), ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɧɟɧɭɥɟɜɵɦ ɫɨɛɫɬɜɟɧɧɵɦ ɡɧɚɱɟɧɢɹɦ Θ

m+j

(r),

ɹɜɥɹɸɬɫɹ ɝɥɚɞɤɢɦɢ ɮɭɧɤɰɢɹɦɢ ɬɨɥɶɤɨ ɢɫɯɨɞɧɵɯ ɩɟɪɟɦɟɧɧɵɯ x. ȼ ɬɨɱɤɟ (ɯ

, r

)

ɦɚɬɪɢɰɚ ɭɫɬɨɣɱɢɜɨɫɬɢ ∂

|∂y

∂y

(1 ≤ i, j ≤ m) ɩɪɢɧɢɦɚɟɬ ɧɭɥɟɜɨɟ ɡɧɚɱɟɧɢɟ (ɜɫɟ

ɦɚɬɪɢɱɧɵɟ ɷɥɟɦɟɧɬɵ ɨɛɪɚɬɢɥɢɫɶ ɜ ɧɭɥɶ), ɬɨɝɞɚ ɤɚɤ (ɩ — m)×(ɩ — ɬ)-ɦɟɪɧɚɹ

ɦɚɬɪɢɰɚ ɭɫɬɨɣɱɢɜɨɫɬɢ ɮɭɧɤɰɢɢ Ɇɨɪɫɚ ɧɟɜɵɪɨɠɞɟɧɚ. ɉɪɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ

ɭɫɥɨɜɢɹɯ (k ≤ 5, ɧɟɫɢɦɦɟɬɪɢɱɧɨɫɬɢ ɢ ɨɬɫɭɬɫɬɜɢɢ ɞɪɭɝɢɯ ɞɨɩɨɥɧɢɬɟɥɶɧɵɯ ɭɫɥɨɜɢɣ ɭ

ɫɟɦɟɣɫɬɜɚ ɩɨɬɟɧɰɢɚɥɶɧɵɯ ɮɭɧɤɰɢɣ) ɬɟɨɪɟɦɚ Ɍɨɦɚ ɝɚɪɚɧɬɢɪɭɟɬ ɫɭɳɟɫɬɜɨɜɚɧɢɟ

ɝɥɚɞɤɨɣ ɡɚɦɟɧɵ ɩɟɪɟɦɟɧɧɵɯ, ɩɪɢɜɨɞɹɳɟɣ ɩɨɬɟɧɰɢɚɥ ɤ ɤɚɧɨɧɢɱɟɫɤɨɦɭ ɜɢɞɭ

ɝɞɟ ɮɭɧɤɰɢɹ CG(m) ɧɨɫɢɬ ɧɚɡɜɚɧɢɟ ɪɨɫɬɤɚ ɤɚɬɚɫɬɪɨɮɵ. ȼ ɬɚɛɥɢɰɟ 3.1 ɦɵ ɩɪɢɜɨɞɢɦ

ɜɫɟ ɤɚɧɨɧɢɱɟɫɤɢɟ ɪɨɫɬɤɢ ɤɚɬɚɫɬɪɨɮ ɞɥɹ ɫɥɭɱɚɟɜ k ≤ 5, ɤɨɬɨɪɵɟ ɫɨɨɬɜɟɬɫɬɜɭɸɬ

ɨɞɧɨɦɭ (m = 1) ɢɥɢ ɞɜɭɦ (m = 2) ɧɭɥɟɜɵɦ ɫɨɛɫɬɜɟɧɧɵɦ ɡɧɚɱɟɧɢɹɦ.

ɋɥɟɞɭɟɬ ɡɚɦɟɬɢɬɶ, ɱɬɨ ɪɚɡɥɨɠɟɧɢɟ (3.9.7) ɜ ɨɤɪɟɫɬɧɨɫɬɢ (ɯ

, r

) ɜ R

× R

ɯɨɬɹ ɢ

ɫɩɪɚɜɟɞɥɢɜɨ, ɧɨ ɧɟ ɞɚɟɬ ɤɨɧɤɪɟɬɧɨɣ ɮɨɪɦɵ f

; ɪɚɡɥɨɠɟɧɢɟ (3.9.8) ɫɩɪɚɜɟɞɥɢɜɨ

ɬɨɥɶɤɨ ɜ ɨɤɪɟɫɬɧɨɫɬɢ x

ɜ R

, ɧɨ ɤɨɧɤɪɟɬɢɡɢɪɭɟɬ ɜɢɞ f

, ɧɚɡɵɜɚɟɦɵɣ ɪɨɫɬɤɨɦ

ɤɚɬɚɫɬɪɨɮɵ. ɇɚ ɫɚɦɨɦ ɞɟɥɟ Ɍɨɦ ɧɚɲɟɥ ɛɨɥɟɟ ɩɨɥɟɡɧɨɟ ɪɚɡɥɨɠɟɧɢɟ, ɢɦɟɧɧɨ: ɟɫɥɢ

— ɧɟɦɨɪɫɨɜɚ ɤɪɢɬɢɱɟɫɤɚɹ ɬɨɱɤɚ ɮɭɧɤɰɢɢ V (ɯ, r) ɩɪɢ r = r

, ɬɨ ɜ ɨɬɤɪɵɬɨɣ

ɨɤɪɟɫɬɧɨɫɬɢ (ɯ

, r

) ɜ R

× R

ɢɦɟɟɬ ɦɟɫɬɨ

Ɍɚɛɥɢɰɚ 3.1. ɗɥɟɦɟɧɬɚɪɧɵɟ ɤɚɬɚɫɬɪɨɮɵ Ɍɨɦɚ.

ɇɚɢɦɟɧɨɜɚɧɢɟ k Ɋɨɫɬɨɤ ȼɨɡɦɭɳɟɧɢɟ

1 x

±3

2 ±ɯ

ɯ + ɚ

3 x

ɯ + ɚ

+ ɚ

±5

4 ±ɯ

ɯ + ɚ

+ ɚ

5 x

ɯ + ɚ

+ ɚ

-4

3 x

y - ɭ

ɯ+ ɚ

y + ɚ

3 ɯ

ɭ + ɭ

ɯ+ ɚ

y + ɚ

4 ɯ

ɭ + ɭ

ɯ+ ɚ

y + ɚ

+ ɚ

-6

5 ɯ

ɭ - ɭ

ɯ+ ɚ

y + ɚ

+ ɚ

5 ɯ

ɭ + ɭ

ɯ+ ɚ

y + ɚ

+ ɚ

±6

5 x

y±y

ɯ+ ɚ

y + ɚ

xy + ɚ

+ ɚ

Ɏɭɧɤɰɢɹ Cat(m, k) ɧɚɡɵɜɚɟɬɫɹ ɮɭɧɤɰɢɟɣ ɤɚɬɚɫɬɪɨɮɵ ɢɥɢ ɉɪɨɫɬɨ ɤɚɬɚɫɬɪɨɮɨɣ.

Ʉɚɬɚɫɬɪɨɮɚ ɫɨɫɬɨɢɬ ɢɡ ɞɜɭɯ ɱɚɫɬɟɣ: ɪɨɫɬɤɚ ɤɚɬɚɫɬɪɨɮɵ CG(m) ɢ ɜɨɡɦɭɳɟɧɢɹ

Pert(m, k),

ɬ.ɟ., Cat(m, k) = CG(m) + Pert(m, k). ȼ ɬɚɛɥ. 3.1 ɩɟɪɟɱɢɫɥɟɧɵ ɤɚɧɨɧɢɱɟɫɤɢɟ

ɮɨɪɦɵ ɤɚɬɚɫɬɪɨɮ ɨɬ ɨɞɧɨɣ ɢ ɞɜɭɯ ɩɟɪɟɦɟɧɧɵɯ. Ɏɭɧɤɰɢɹ ɤɚɬɚɫɬɪɨɮɵ ɫɜɨɞɢɬɫɹ ɤ

ɪɨɫɬɤɭ ɤɚɬɚɫɬɪɨɮɵ ɩɪɢ ɪɚɜɟɧɫɬɜɟ ɮɢɡɢɱɟɫɤɨɝɨ ɭɩɪɚɜɥɹɸɳɟɝɨ ɩɚɪɚɦɟɬɪɚ r

ɜɟɥɢɱɢɧɟ

ɢɥɢ ɩɪɢ ɧɭɥɟɜɵɯ ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ ɭɩɪɚɜɥɹɸɳɢɯ ɩɚɪɚɦɟɬɪɚɯ a

(j = 1,... ,v).

ɂɬɚɤ, ɫ ɩɨɦɨɳɶɸ ɪɹɞɚ ɬɟɨɪɟɦ ɦɵ ɨɩɢɫɚɥɢ ɥɨɤɚɥɶɧɵɟ ɯɚɪɚɤɬɟɪɢɫɬɢɤɢ

ɩɨɬɟɧɰɢɚɥɚ. ȼ ɝɥ. 4 ɢ 8 ɦɵ ɜɨɫɩɨɥɶɡɭɟɦɫɹ ɧɟɤɨɬɨɪɵɦɢ (ɷɥɟɦɟɧɬɚɪɧɵɦɢ)

ɪɟɡɭɥɶɬɚɬɚɦɢ ɬɟɨɪɢɢ ɤɚɬɚɫɬɪɨɮ.

ɉɪɢɥɨɠɟɧɢɟ: ɇɟɤɨɬɨɪɵɟ ɡɚɦɟɱɚɧɢɹ ɨ ɬɟɨɪɢɢ ɛɢɮɭɪɤɚɰɢɣ

ɉɨɫɤɨɥɶɤɭ ɬɟɨɪɢɹ ɛɢɮɭɪɤɚɰɢɣ ɢɝɪɚɟɬ ɨɱɟɧɶ ɜɚɠɧɭɸ ɪɨɥɶ ɜ ɫɢɧɟɪɝɟɬɢɱɟɫɤɨɣ

ɷɤɨɧɨɦɢɤɟ, ɯɨɬɟɥɨɫɶ ɛɵ ɞɚɬɶ ɧɟɤɨɬɨɪɵɟ ɩɨɹɫɧɟɧɢɹ ɟɟ ɨɬɞɟɥɶɧɵɯ ɪɟɡɭɥɶɬɚɬɨɜ.

ɉɪɟɠɞɟ ɜɫɟɝɨ, ɨɛɫɭɞɢɦ ɧɟɤɨɬɨɪɵɟ ɨɛɳɢɟ ɬɟɨɪɟɦɵ, ɡɚɞɚɸɳɢɟ ɭɫɥɨɜɢɹ

ɜɨɡɧɢɤɧɨɜɟɧɢɹ ɛɢɮɭɪɤɚɰɢɣ. Ⱦɥɹ ɷɬɨɝɨ ɡɚɩɢɲɟɦ (3.7.1) ɜ ɜɢɞɟ

ɝɞɟ L — ɥɢɧɟɚɪɢɡɨɜɚɧɧɵɣ ɨɩɟɪɚɬɨɪ, ɚ N ɜɤɥɸɱɚɟɬ ɜ ɫɟɛɹ ɜɫɟ ɞɨɛɚɜɤɢ, ɧɟɥɢɧɟɣɧɵɟ

ɩɨ ɯ. ɗɤɜɢɜɚɥɟɧɬɧɵɣ ɫɩɨɫɨɛ ɫɮɨɪɦɭɥɢɪɨɜɚɬɶ ɷɬɭ ɩɪɨɛɥɟɦɭ ɫɨɫɬɨɢɬ ɜ ɬɨɦ, ɱɬɨɛɵ

ɨɩɪɟɞɟɥɢɬɶ r ɬɚɤɢɦ ɨɛɪɚɡɨɦ, ɱɬɨɛɵ ɦɨɠɧɨ ɛɵɥɨ ɜɵɞɟɥɢɬɶ ɨɬɞɟɥɶɧɨ ɱɚɫɬɶ J

, ɧɟ

ɡɚɜɢɫɢɦɭɸ ɨɬ r, ɢ ɧɟɩɪɟɪɵɜɧɭɸ ɱɚɫɬɶ, ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɭɸ r, ɬɚɤ ɱɬɨ

x -rx+ N(x, r) = 0, (3.Ⱥ.2)

ɝɞɟ

Ɍɟɨɪɟɦɚ 3.Ⱥ.1. Ɂɧɚɱɟɧɢɟ r

ɦɨɠɟɬ ɛɵɬɶ ɬɨɱɤɨɣ ɛɢɮɭɪɤɚɰɢɢ (3.Ⱥ.2) ɬɨɥɶɤɨ ɜ ɬɨɦ

ɫɥɭɱɚɟ, ɟɫɥɢ ɨɧɨ ɹɜɥɹɟɬɫɹ ɫɨɛɫɬɜɟɧɧɵɦ ɡɧɚɱɟɧɢɟɦ ɨɩɟɪɚɬɨɪɚ J

Ɉɛɪɚɬɧɨɟ ɭɬɜɟɪɠɞɟɧɢɟ ɧɟ ɜɫɟɝɞɚ ɫɩɪɚɜɟɞɥɢɜɨ. Ɇɵ ɝɨɜɨɪɢɦ, ɱɬɨ z

ɹɜɥɹɟɬɫɹ

ɫɨɛɫɬɜɟɧɧɵɦ ɡɧɚɱɟɧɢɟɦ ɤɜɚɞɪɚɬɧɨɣ ɦɚɬɪɢɰɵ L(r) ɚɥɝɟɛɪɚɢɱɟɫɤɨɣ ɤɪɚɬɧɨɫɬɢ k, ɟɫɥɢ

ɝɞɟ h(z

) ɧɟ ɪɚɜɧɨ ɧɭɥɸ.

Ɍɟɨɪɟɦɚ 3.Ⱥ.2. ȿɫɥɢ z

(ɧɟ ɪɚɜɧɨɟ ɧɭɥɸ) ɹɜɥɹɟɬɫɹ ɫɨɛɫɬɜɟɧɧɵɦ ɡɧɚɱɟɧɢɟɦ J

(3.

Ⱥ.2) ɧɟɱɟɬɧɨɣ ɤɪɚɬɧɨɫɬɢ, ɬɨ z

— ɬɨɱɤɚ ɛɢɮɭɪɤɚɰɢɢ ɷɬɨɝɨ ɭɪɚɜɧɟɧɢɹ.

ȼ ɤɚɱɟɫɬɜɟ ɨɛɨɛɳɟɧɢɹ ɬɟɨɪɟɦɵ 3.Ⱥ.2 ɪɚɫɫɦɨɬɪɢɦ ɫɥɭɱɚɣ ɜɟɤɬɨɪɧɨɣ ɩɟɪɟɦɟɧɧɨɣ

ɯ ∈ ɏ, ɜɟɤɬɨɪɧɨɝɨ ɩɚɪɚɦɟɬɪɚ r ∈ Ɇ ∈ R

, r = (r

, r

,..., r

), F : ɏ × Ɇ ĺZ

ɝɞɟ ȼ ɢ A

(j = 1,..., n) — ɨɝɪɚɧɢɱɟɧɧɵɟ ɥɢɧɟɣɧɵɟ ɨɩɟɪɚɬɨɪɵ, N(0, r) = 0,

N(0, r) = 0. Ɍɨɱɤɭ r ɧɚɡɨɜɟɦ ɫɨɛɫɬɜɟɧɧɵɦ ɡɧɚɱɟɧɢɟɦ (ȼ, A

,..., A

), ɟɫɥɢ ɧɭɥɶ —

ɫɨɛɫɬɜɟɧɧɨɟ ɡɧɚɱɟɧɢɟ ɨɩɟɪɚɬɨɪɚ L(r).

Ɍɟɨɪɟɦɚ 3.Ⱥ.3. ɉɭɫɬɶ ɏ ɢ Z — ɛɚɧɚɯɨɜɵ ɩɪɨɫɬɪɚɧɫɬɜɚ, Ɇ — ɨɬɤɪɵɬɵɣ ɢɧɬɟɪɜɚɥ

ɢ F ∈ C

(M × X, Z) (ɬ ≥ 2). ȿɫɥɢ r

— ɩɪɨɫɬɨɟ ɫɨɛɫɬɜɟɧɧɨɟ ɡɧɚɱɟɧɢɟ (ȼ, A

,..., Am),

ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɟ ɧɟɧɭɥɟɜɨɦɭ ɫɨɛɫɬɜɟɧɧɨɦɭ ɜɟɤɬɨɪɭ ɭ

, ɬɨ (r, x) = (r

, 0) — ɬɨɱɤɚ

ɛɢɮɭɪɤɚɰɢɢ F(r, x) = 0. Ȼɨɥɟɟ ɬɨɝɨ, ɫɭɳɟɫɬɜɭɸɬ ɋ

n-1

ɮɭɧɤɰɢɢ

ɬɚɤɢɟ, ɱɬɨ ɞɥɹ ɞɟɣɫɬɜɢɬɟɥɶɧɵɯ v ɜɛɥɢɡɢ ɧɭɥɹ

ȼɫɟ ɧɭɥɢ F ɜɛɥɢɡɢ ɬɨɱɤɢ (r, 0) ɹɜɥɹɸɬɫɹ ɥɢɛɨ ɬɪɢɜɢɚɥɶɧɵɦɢ ɪɟɲɟɧɢɹɦɢ ɯ = 0,

ɥɢɛɨ ɡɚɞɚɸɬɫɹ ɜɵɪɚɠɟɧɢɟɦ (3.Ⱥ.4). ȿɫɥɢ F — ɚɧɚɥɢɬɢɱɟɫɤɚɹ ɮɭɧɤɰɢɹ ɜ ɨɤɪɟɫɬɧɨɫɬɢ

ɷɬɨɣ ɬɨɱɤɢ ɢɥɢ ɜ ɬɨɱɤɟ (r

, 0), ɬɨ ɬɚɤɨɜɵ ɠɟ r* (v), x*(v) ɜɛɥɢɡɢ v = 0.

ɗɬɚ ɬɟɨɪɟɦɚ ɩɪɢɧɚɞɥɟɠɢɬ ɑɭ ɢ ɏɟɣɥɭ (1982).

ɋɈȾȿɊɀȺɇɂȿ

4. Ɇɧɨɠɟɫɬɜɚ ɪɚɜɧɨɜɟɫɢɣ ɢ ɫɬɪɭɤɬɭɪɧɵɟ ɢɡɦɟɧɟɧɢɹ ɜ ɷɤɨɧɨɦɢɱɟɫɤɢɯ ɫɢɫɬɟɦɚɯ.......................... 78

4.1 Ɍɟɨɪɢɹ ɤɚɬɚɫɬɪɨɮ ɢ ɫɪɚɜɧɢɬɟɥɶɧɵɣ ɫɬɚɬɢɱɟɫɤɢɣ ɚɧɚɥɢɡ ............................................78

4.2 Ɇɨɞɟɥɢɪɨɜɚɧɢɟ ɪɟɝɢɨɧɚɥɶɧɨɣ ɞɢɧɚɦɢɤɢ......................................................................84

4.3 ɇɟɤɨɬɨɪɵɟ ɩɪɢɦɟɪɵ ɫɬɪɭɤɬɭɪɧɵɯ ɢɡɦɟɧɟɧɢɣ..............................................................87

4.3.1 Ⱦɟɥɨɜɵɟ ɰɢɤɥɵ ɜ ɦɨɞɟɥɢ Ʉɚɥɞɨɪɚ..........................................................87

4.3.2 ɍɩɪɚɜɥɟɧɢɟ ɪɟɫɭɪɫɚɦɢ.............................................................................89

4.3.3 Ⱦɢɧɚɦɢɱɟɫɤɢɣ ɜɵɛɨɪ ɜɢɞɚ ɬɪɚɧɫɩɨɪɬɚ ɢ ɛɢɮɭɪɤɚɰɢɢ .........................91

4.3.4 Ɇɧɨɠɟɫɬɜɚ ɪɚɜɧɨɜɟɫɢɣ ɜ ɦɨɞɟɥɢ ɪɨɡɧɢɱɧɨɣ ɬɨɪɝɨɜɥɢ ȼɢɥɶɫɨɧɚ .......92

4.4 Ȼɢɮɭɪɤɚɰɢɨɧɧɵɣ ɚɧɚɥɢɡ ɦɨɞɟɥɢ ɷɤɨɧɨɦɢɱɟɫɤɨɝɨ ɪɨɫɬɚ.............................................94

4.5 Ɍɟɨɪɢɹ ɨɫɨɛɟɧɧɨɫɬɟɣ ɜ ɷɤɨɧɨɦɢɱɟɫɤɨɦ ɚɧɚɥɢɡɟ ........................................................ 101

4.6 Ɂɚɦɟɱɚɧɢɹ.......................................................................................................................103

5. ɗɤɨɧɨɦɢɱɟɫɤɢɟ ɰɢɤɥɵ ................................................................................................................... 104

5.1 Ɍɟɨɪɢɢ ɷɤɨɧɨɦɢɱɟɫɤɢɯ ɰɢɤɥɨɜ ....................................................................................104

5.2 ɇɟɤɨɬɨɪɵɟ ɦɚɬɟɦɚɬɢɱɟɫɤɢɟ ɪɟɡɭɥɶɬɚɬɵ ɬɟɨɪɢɢ ɩɪɟɞɟɥɶɧɵɯ ɰɢɤɥɨɜ .......................110

5.2.1 Ɍɟɨɪɟɦɚ ɉɭɚɧɤɚɪɟ-Ȼɟɧɞɢɤɫɨɧɚ ɢ ɟɟ ɩɪɢɥɨɠɟɧɢɹ ɤ ɷɤɨɧɨɦɢɤɟ..........110

5.2.2 Ɍɟɨɪɟɦɚ ɏɨɩɮɚ ɨ ɛɢɮɭɪɤɚɰɢɹɯ.............................................................114

5.3 ɍɩɪɨɳɟɧɧɚɹ ɦɨɞɟɥɶ ɞɟɥɨɜɨɝɨ ɰɢɤɥɚ Ʉɟɣɧɫɚ..............................................................117

5.4 ɏɚɪɚɤɬɟɪ ɧɟɪɚɜɧɨɜɟɫɧɨɫɬɢ ɜ ɦɨɞɟɥɢ ɛɟɡ ɪɚɜɧɨɜɟɫɢɣ ................................................122

5.5 Ɇɨɧɟɬɚɪɧɵɟ ɰɢɤɥɵ ɜ ɨɛɨɛɳɟɧɧɨɣ ɦɨɞɟɥɢ Ɍɨɛɢɧɚ....................................................126

5.6 Ɉɫɰɢɥɥɹɰɢɢ ɜ ɝɢɛɪɢɞɧɨɣ ɦɨɞɟɥɢ ɪɨɫɬɚ ȼɚɧ ɞɟɪ ɉɥɸɝɚ............................................133

5.7 Ɉɩɬɢɦɚɥɶɧɚɹ ɩɟɪɢɨɞɢɱɟɫɤɚɹ ɩɨɥɢɬɢɤɚ ɡɚɧɹɬɨɫɬɢ......................................................138

5.8 Ɉɩɬɢɦɚɥɶɧɵɣ ɷɤɨɧɨɦɢɱɟɫɤɢɣ ɪɨɫɬ, ɫɜɹɡɚɧɧɵɣ ɫ ɷɧɞɨɝɟɧɧɵɦɢ ɮɥɭɤɬɭɚɰɢɹɦɢ......142

5.9 Ɂɚɦɟɱɚɧɢɹ ɨ ɜɨɡɦɨɠɧɵɯ ɩɨɫɥɟɞɭɸɳɢɯ ɛɢɮɭɪɤɚɰɢɹɯ ɩɪɟɞɟɥɶɧɵɯ ɰɢɤɥɨɜ .............145

5.10 Ʉɨɧɤɭɪɟɧɬɧɵɟ ɞɟɥɨɜɵɟ ɰɢɤɥɵ ɜ ɷɤɨɧɨɦɢɤɟ ɫ ɩɟɪɟɤɪɵɜɚɸɳɢɦɢɫɹ ɩɨɤɨɥɟɧɢɹɦɢ

— ɞɢɫɤɪɟɬɧɚɹ ɦɨɞɟɥɶ ...............................................................................................................149

4 Ɇɧɨɠɟɫɬɜɚ ɪɚɜɧɨɜɟɫɢɣ

ɢ ɫɬɪɭɤɬɭɪɧɵɟ ɢɡɦɟɧɟɧɢɹ ɜ

ɷɤɨɧɨɦɢɱɟɫɤɢɯ ɫɢɫɬɟɦɚɯ

Ɋɚɡɜɢɬɢɟ ɚɧɚɥɢɬɢɱɟɫɤɨɣ ɷɤɨɧɨɦɢɤɢ ɜ

ɧɚɩɪɚɜɥɟɧɢɢ ɫɪɚɜɧɢɬɟɥɶɧɨɣ ɞɢɧɚɦɢɤɢ ɞɨɥɠɧɨ

ɫɨɯɪɚɧɢɬɶɫɹ ɢ ɜ ɛɭɞɭɳɟɦ. ȿɫɬɶ ɧɚɞɟɠɞɚ, ɱɬɨ ɷɬɨɬ

ɩɭɬɶ ɩɪɢɜɟɞɟɬ ɤ ɪɟɲɟɧɢɸ ɦɧɨɝɢɯ ɩɪɨɛɥɟɦ ...,

ɞɚɠɟ ...ɝɥɨɛɚɥɶɧɵɯ ɩɪɨɛɥɟɦ ɷɤɨɧɨɦɢɱɟɫɤɨɝɨ

ɪɚɡɜɢɬɢɹ.

ɉ. Ⱥ. ɋɚɦɭɷɥɶɫɨɧ (1947)

Ɉɞɧɢɦ ɢɡ ɧɚɢɛɨɥɟɟ ɜɚɠɧɵɯ ɩɪɟɞɦɟɬɨɜ ɷɤɨɧɨɦɢɱɟɫɤɨɝɨ ɚɧɚɥɢɡɚ ɹɜɥɹɟɬɫɹ ɢɫɫɥɟɞɨɜɚɧɢɟ

ɜɥɢɹɧɢɹ ɢɡɦɟɧɟɧɢɣ ɜɧɟɲɧɢɯ ɩɚɪɚɦɟɬɪɨɜ ɧɚ ɩɨɜɟɞɟɧɢɟ ɷɤɨɧɨɦɢɱɟɫɤɢɯ ɩɟɪɟɦɟɧɧɵɯ. Ⱥɧɚɥɢɡ

ɩɨɞɨɛɧɵɯ ɷɮɮɟɤɬɨɜ

ɧɚɡɵɜɚɟɬɫɹ ɫɪɚɜɧɢɬɟɥɶɧɵɦ ɚɧɚɥɢɡɨɦ. ȼ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɬɨɝɨ,

ɨɫɭɳɟɫɬɜɥɹɟɬɫɹ ɚɧɚɥɢɡ ɫɬɚɬɢɱɟɫɤɨɣ ɢɥɢ ɞɢɧɚɦɢɱɟɫɤɨɣ ɦɨɞɟɥɢ, ɪɚɡɥɢɱɚɸɬ ɫɪɚɜɧɢɬɟɥɶɧɵɣ

ɫɬɚɬɢɱɟɫɤɢɣ ɢ ɫɪɚɜɧɢɬɟɥɶɧɵɣ ɞɢɧɚɦɢɱɟɫɤɢɣ ɚɧɚɥɢɡ. Ʉɨɝɞɚ ɫɢɫɬɟɦɚ ɭɫɬɨɣɱɢɜɚ,

ɫɪɚɜɧɢɬɟɥɶɧɵɣ ɞɢɧɚɦɢɱɟɫɤɢɣ ɚɧɚɥɢɡ ɧɨɫɢɬ ɧɚɡɜɚɧɢɟ ɩɪɢɧɰɢɩɚ ɫɨɨɬɜɟɬɫɬɜɢɹ ɋɚɦɭɷɥɶɫɨɧɚ.

ɋɪɚɜɧɢɬɟɥɶɧɵɣ ɚɧɚɥɢɡ ɜ ɬɨɦ ɜɢɞɟ, ɤɚɤ ɨɧ ɢɡɥɨɠɟɧ ɜ «Ɉɫɧɨɜɚɯ» ɋɚɦɭɷɥɶɫɨɧɚ, ɦɵ ɧɚɡɵɜɚɟɦ

ɬɪɚɞɢɰɢɨɧɧɵɦ ɫɪɚɜɧɢɬɟɥɶɧɵɦ ɚɧɚɥɢɡɨɦ. ȼ ɷɬɨɣ ɤɧɢɝɟ

ɦɵ ɧɚɦɟɪɟɧɵ ɢɡɭɱɢɬɶ ɬɟ ɩɪɨɛɥɟɦɵ

ɫɪɚɜɧɢɬɟɥɶɧɨɝɨ ɚɧɚɥɢɡɚ, ɤɨɬɨɪɵɟ ɬɪɚɞɢɰɢɨɧɧɵɣ ɫɪɚɜɧɢɬɟɥɶɧɵɣ ɚɧɚɥɢɡ ɨɛɯɨɞɢɬ.

4.1 Ɍɟɨɪɢɹ ɤɚɬɚɫɬɪɨɮ ɢ ɫɪɚɜɧɢɬɟɥɶɧɵɣ

ɫɬɚɬɢɱɟɫɤɢɣ ɚɧɚɥɢɡ

Ʉɚɤ ɭɠɟ ɛɵɥɨ ɫɤɚɡɚɧɨ ɜ ɝɥ. 2, ɢɡɥɨɠɟɧɢɟ ɋɚɦɭɷɥɶɫɨɧɨɦ «Ɉɫɧɨɜ ɷɤɨɧɨɦɢɱɟɫɤɨɝɨ ɚɧɚɥɢɡɚ»

ɛɚɡɢɪɭɟɬɫɹ ɜ ɰɟɥɨɦ ɧɚ ɞɜɭɯ ɜɟɫɶɦɚ ɨɛɳɢɯ ɝɢɩɨɬɟɡɚɯ. ɉɟɪɜɚɹ ɫɨɫɬɨɢɬ ɜ ɬɨɦ, ɱɬɨ ɭɫɥɨɜɢɹ

ɪɚɜɧɨɜɟɫɢɹ ɷɤɜɢɜɚɥɟɧɬɧɵ ɭɫɥɨɜɢɹɦ ɦɚɤɫɢɦɢɡɚɰɢɢ (ɦɢɧɢɦɢɡɚɰɢɢ) ɧɟɤɨɬɨɪɨɣ ɜɟɥɢɱɢɧɵ. ɗɬɚ

ɝɢɩɨɬɟɡɚ ɜ ɛɨɥɶɲɢɧɫɬɜɟ ɫɥɭɱɚɟɜ ɨɡɧɚɱɚɟɬ ɫɩɪɚɜɟɞɥɢɜɨɫɬɶ (ɬɪɚɞɢɰɢɨɧɧɨɝɨ) ɫɪɚɜɧɢɬɟɥɶɧɨɝɨ

ɫɬɚɬɢɱɟɫɤɨɝɨ ɚɧɚɥɢɡɚ, ɢɡ ɤɨɬɨɪɨɝɨ ɦɨɠɧɨ ɜɵɜɟɫɬɢ ɦɧɨɝɨ ɜɚɠɧɵɯ ɬɟɨɪɟɦ ɷɤɨɧɨɦɢɤɢ

. ȼɬɨɪɚɹ

ɝɢɩɨɬɟɡɚ ɫɨɫɬɨɢɬ ɜ ɬɨɦ, ɱɬɨ ɫɢɫɬɟɦɚ ɧɚɯɨɞɢɬɫɹ ɜ «ɭɫɬɨɣɱɢɜɨɦ» ɪɚɜɧɨɜɟɫɢɢ

ɥɢɛɨ ɜ ɞɜɢɠɟɧɢɢ. Ʉɚɤ ɩɨɤɚɡɚɧɨ ɜ ɝɥ. 2, ɢɡ ɜɬɨɪɨɣ ɝɢɩɨɬɟɡɵ ɫɥɟɞɭɟɬ ɫɩɪɚɜɟɞɥɢɜɨɫɬɶ

ɩɪɢɧɰɢɩɚ ɫɨɨɬɜɟɬɫɬɜɢɹ ɦɟɠɞɭ ɫɪɚɜɧɢɬɟɥɶɧɨɣ ɫɬɚɬɢɤɨɣ ɢ ɞɢɧɚɦɢɤɨɣ. ɏɨɬɹ, ɩɨ ɛɨɥɶɲɟɣ

ɱɚɫɬɢ, ɜ ɷɬɨɣ ɤɧɢɝɟ ɢɫɫɥɟɞɭɟɬɫɹ ɩɨɜɟɞɟɧɢɟ ɞɢɧɚɦɢɱɟɫɤɨɣ ɫɢɫɬɟɦɵ ɜ ɬɟɯ ɫɥɭɱɚɹɯ, ɤɨɝɞɚ ɧɟ

ɪɚɛɨɬɚɟɬ ɜɬɨɪɚɹ ɝɢɩɨɬɟɡɚ, ɧɟ ɦɟɧɟɟ ɜɚɠɧɨ ɩɨɧɢɦɚɬɶ, ɱɬɨ ɩɪɨɢɡɨɣɞɟɬ, ɟɫɥɢ ɨɫɥɚɛɢɬɶ ɩɟɪɜɭɸ

ɝɢɩɨɬɟɡɭ. ɉɨɷɬɨɦɭ ɞɚɧɧɵɣ ɪɚɡɞɟɥ

ɩɨɫɜɹɳɟɧ ɩɟɪɜɨɣ ɝɢɩɨɬɟɡɟ.

Ƚɢɩɨɬɟɡɚ ɩɪɢɦɟɧɹɟɬɫɹ ɤ ɫɪɚɜɧɢɬɟɥɶɧɨɣ ɫɬɚɬɢɤɟ. ɂɫɩɨɥɶɡɨɜɚɧɢɟ ɷɬɨɝɨ ɨɛɳɟɝɨ ɦɟɬɨɞɚ

(ɧɚɩɪɢɦɟɪ, ɜ ɦɢɤɪɨɷɤɨɧɨɦɢɤɟ ɢ ɷɤɨɧɨɦɢɤɟ ɛɥɚɝɨɫɨɫɬɨɹɧɢɹ) ɩɨɡɜɨɥɢɥɨ ɩɨɥɭɱɢɬɶ

ɡɧɚɱɢɬɟɥɶɧɵɟ ɪɟɡɭɥɶɬɚɬɵ. ȼ ɫɥɭɱɚɟ ɤɨɝɞɚ ɪɚɜɧɨɜɟɫɧɵɟ ɡɧɚɱɟɧɢɹ ɩɟɪɟɦɟɧɧɵɯ ɦɨɠɧɨ

ɬɪɚɤɬɨɜɚɬɶ ɤɚɤ ɪɟɲɟɧɢɹ ɡɚɞɚɱɢ ɨɩɬɢɦɢɡɚɰɢɢ, ɫɬɚɧɨɜɢɬɫɹ ɜɨɡɦɨɠɧɵɦ ɨɞɧɨɡɧɚɱɧɨ ɨɩɪɟɞɟɥɢɬɶ

ɧɚɩɪɚɜɥɟɧɢɟ ɢɡɦɟɧɟɧɢɹ ɪɟɲɟɧɢɹ ɜ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬ ɫɞɜɢɝɚ ɩɚɪɚɦɟɬɪɨɜ. Ⱦɥɹ ɢɥɥɸɫɬɪɚɰɢɢ

ɩɪɢɜɟɞɟɦ ɞɜɚ

ɩɪɨɫɬɵɯ ɩɪɢɦɟɪɚ (ɋɚɦɭɷɥɶɫɨɧ, 1947).

Ɋɚɫɫɦɨɬɪɢɦ ɮɢɪɦɭ, ɞɥɹ ɤɨɬɨɪɨɣ ɡɚɞɚɧɵ ɮɭɧɤɰɢɢ ɫɩɪɨɫɚ ɢ ɩɪɨɢɡɜɨɞɫɬɜɟɧɧɵɯ ɢɡɞɟɪɠɟɤ.

ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɮɢɪɦɚ ɨɛɥɚɝɚɟɬɫɹ ɧɚɥɨɝɨɦ ɜɟɥɢɱɢɧɵ r ɧɚ ɟɞɢɧɢɰɭ ɩɪɨɞɭɤɰɢɢ. Ɍɨɝɞɚ

ɞɨɯɨɞ ɮɢɪɦɵ ɨɩɪɟɞɟɥɹɟɬɫɹ ɬɚɤ:

D = ɯɪ(ɯ) - ɋ(ɯ) - ɬɯ. (4.1.1)

Ɂɞɟɫɶ x, ɪ ɢ ɋ ɩɪɟɞɫɬɚɜɥɹɸɬ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɨɛɴɟɦ ɩɪɨɢɡɜɨɞɫɬɜɚ, ɰɟɧɭ ɩɪɨɞɭɤɰɢɢ ɢ

ɦɢɧɢɦɚɥɶɧɵɟ ɫɭɦɦɚɪɧɵɟ ɩɪɨɢɡɜɨɞɫɬɜɟɧɧɵɟ ɢɡɞɟɪɠɤɢ. Ɏɢɪɦɚ

ɨɩɪɟɞɟɥɹɟɬ ɭɪɨɜɟɧɶ

ɩɪɨɢɡɜɨɞɫɬɜɚ ɞɥɹ ɤɚɠɞɨɣ ɡɚɞɚɧɧɨɣ ɜɟɥɢɱɢɧɵ ɧɚɥɨɝɚ. ɉɪɢ ɤɚɠɞɨɣ ɡɚɞɚɧɧɨɣ ɧɚɥɨɝɨɜɨɣ ɫɬɚɜɤɟ

ɫɭɳɟɫɬɜɭɟɬ ɪɚɜɧɨɜɟɫɧɵɣ ɨɛɴɟɦ ɜɵɩɭɫɤɚ. Ɋɚɫɫɦɨɬɪɢɦ, ɤɚɤ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɢɡɦɟɧɟɧɢɟɦ

ɜɟɥɢɱɢɧɵ ɧɚɥɨɝɚ ɦɟɧɹɟɬɫɹ ɨɛɴɟɦ ɩɪɨɢɡɜɨɞɫɬɜɚ, ɨɩɪɟɞɟɥɟɧɧɵɣ ɮɢɪɦɨɣ.

ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɮɢɪɦɚ ɜɵɛɢɪɚɟɬ ɬɚɤɨɣ ɨɛɴɟɦ ɩɪɨɢɡɜɨɞɫɬɜɚ, ɤɨɬɨɪɵɣ ɦɚɤɫɢɦɢɡɢɪɭɟɬ

ɟɟ ɞɨɯɨɞ. Ɋɟɲɟɧɢɟ, ɦɚɤɫɢɦɢɡɢɪɭɸɳɟɟ D ɜ (4.1.1), ɛɭɞɟɬ ɪɚɜɧɨɜɟɫɧɨɣ ɜɟɥɢɱɢɧɨɣ.

ɇɟɨɛɯɨɞɢɦɵɦɢ ɢ

ɞɨɫɬɚɬɨɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ ɥɨɤɚɥɶɧɨɝɨ ɦɚɤɫɢɦɭɦɚ ɹɜɥɹɸɬɫɹ D

= 0, D

< 0.

ɂɡ D

= 0 ɢɦɟɟɦ

ɨɬɤɭɞɚ ɦɵ ɦɨɠɟɦ ɨɩɪɟɞɟɥɢɬɶ ɬɨɱɤɭ ɪɚɜɧɨɜɟɫɢɹ ɯ = J(r). Ⱦɢɮɮɟɪɟɧɰɢɪɨɜɚɧɢɟ (4.1.2) ɩɨ r

ɞɚɟɬ

Ɉɞɧɚɤɨ ɩɨɫɤɨɥɶɤɭ D

= [ɯɪ(ɯ) - ɋ(ɯ)]

ɞɨɥɠɧɨ ɛɵɬɶ ɨɬɪɢɰɚɬɟɥɶɧɵɦ, ɢɡ (4.1.3) ɢɦɟɟɦ

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɡɚɤɥɸɱɚɟɦ, ɱɬɨ ɟɫɥɢ ɮɢɪɦɚ ɧɚɯɨɞɢɬɫɹ ɜ ɪɚɜɧɨɜɟɫɢɢ ɞɨ ɢ ɩɨɫɥɟ

ɧɚɥɨɝɨɨɛɥɨɠɟɧɢɹ, ɭɜɟɥɢɱɟɧɢɟ ɧɚɥɨɝɚ ɜɫɟɝɞɚ ɜɵɡɨɜɟɬ

ɩɚɞɟɧɢɟ ɩɪɨɢɡɜɨɞɫɬɜɚ. ȼɵɲɟ ɦɵ ɧɟ ɨɩɪɟɞɟɥɹɥɢ ɜɢɞ ɮɭɧɤɰɢɣ ɪ(ɯ) ɢ ɋ(ɯ). ɇɚɲɟ

ɟɞɢɧɫɬɜɟɧɧɨɟ ɬɪɟɛɨɜɚɧɢɟ ɫɨɫɬɨɹɥɨ ɜ ɬɨɦ, ɱɬɨɛɵ ɡɚɞɚɱɚ ɢɦɟɥɚ ɪɟɝɭɥɹɪɧɨɟ ɦɚɤɫɢɦɚɥɶɧɨɟ

ɪɟɲɟɧɢɟ. ɗɬɨɝɨ ɞɨɫɬɚɬɨɱɧɨ ɞɥɹ ɬɨɝɨ, ɱɬɨɛɵ ɨɩɪɟɞɟɥɢɬɶ ɧɚɩɪɚɜɥɟɧɢɟ ɢɡɦɟɧɟɧɢɹ ɯ ɩɨɫɥɟ ɧɚɥɨ-

ɝɨɨɛɥɨɠɟɧɢɹ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɩɨ ɢɡɜɟɫɬɧɨɣ ɢɧɮɨɪɦɚɰɢɢ ɨ ɬɨɦ, ɱɬɨ ɮɢɪɦɚ ɦɚɤɫɢɦɢɡɢɪɭɟɬ

ɩɪɢɛɵɥɶ, ɦɵ ɦɨɠɟɦ

ɩɪɟɞɫɤɚɡɚɬɶ ɩɨɜɟɞɟɧɢɟ ɮɢɪɦɵ ɩɪɢ ɢɡɦɟɧɟɧɢɢ ɧɚɥɨɝɨɜɨɣ ɩɨɥɢɬɢɤɢ. ɗɬɨɬ

ɩɪɢɦɟɪ ɫɥɭɠɢɬ ɬɢɩɢɱɧɨɣ ɢɥɥɸɫɬɪɚɰɢɟɣ ɬɨɝɨ, ɱɬɨ ɦɵ ɩɨɧɢɦɚɟɦ ɩɨɞ ɫɪɚɜɧɢɬɟɥɶɧɵɦ

ɫɬɚɬɢɱɟɫɤɢɦ ɚɧɚɥɢɡɨɦ.

Ɍɟɩɟɪɶ ɪɚɫɫɦɨɬɪɢɦ, ɱɬɨ ɩɪɨɢɡɨɣɞɟɬ, ɟɫɥɢ ɝɢɩɨɬɟɡɵ, ɩɪɢɧɹɬɵɟ ɜ ɫɪɚɜɧɢɬɟɥɶɧɨɦ

ɫɬɚɬɢɱɟɫɤɨɦ ɚɧɚɥɢɡɟ, ɛɭɞɭɬ ɨɫɥɚɛɥɟɧɵ.

Ɋɚɫɫɦɨɬɪɢɦ ɡɚɞɚɱɭ ɨɩɬɢɦɢɡɚɰɢɢ

minf(x, r),

ɝɞɟ ɯ ɩɪɟɞɫɬɚɜɥɹɟɬ ɩɟɪɟɦɟɧɧɵɟ, ɚ r - ɩɚɪɚɦɟɬɪɵ. Ɇɢɧɢɦɭɦ f ɞɨɫɬɢɝɚɟɬɫɹ, ɤɨɝɞɚ

grad f = 0. (4.1.5)

Ɋɟɲɟɧɢɟ (4.1.5) ɞɚɟɬ ɬɨɱɤɭ ɪɚɜɧɨɜɟɫɢɹ, ɤɨɬɨɪɚɹ ɦɢɧɢɦɢɡɢɪɭɟɬ ɮɭɧɤɰɢɸ ɩɨɬɟɧɰɢɚɥɚ

f(x, r). ɉɪɢ ɢɡɦɟɧɟɧɢɢ r ɨɩɬɢɦɚɥɶɧɨɟ ɪɟɲɟɧɢɟ ɨɩɪɟɞɟɥɹɟɬ ɩɨɜɟɪɯɧɨɫɬɶ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ (x, r),

ɧɚ ɤɨɬɨɪɨɣ ɪɚɫɩɨɥɨɠɟɧɵ ɜɨɡɦɨɠɧɵɟ ɫɨɫɬɨɹɧɢɹ ɪɚɜɧɨɜɟɫɢɹ ɫɢɫɬɟɦɵ. ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ

ɬɪɚɞɢɰɢɨɧɧɵɦ ɫɪɚɜɧɢɬɟɥɶɧɵɦ ɫɬɚɬɢɱɟɫɤɢɦ ɚɧɚɥɢɡɨɦ ɩɪɢ

ɝɥɚɞɤɢɯ, ɦɟɞɥɟɧɧɵɯ ɢ ɦɚɥɵɯ

ɢɡɦɟɧɟɧɢɹɯ r ɦɵ ɦɨɠɟɦ ɨɠɢɞɚɬɶ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɝɥɚɞɤɢɯ ɦɚɥɵɯ ɢɡɦɟɧɟɧɢɣ ɯ. ȼ ɪɟɡɭɥɶɬɚɬɟ

ɬɪɚɟɤɬɨɪɢɹ ɪɚɜɧɨɜɟɫɢɹ ɜ ɩɪɨɫɬɪɚɧɫɬɜɟ (x, r) ɛɭɞɟɬ ɝɥɚɞɤɨɣ ɢ ɧɟ ɦɨɠɟɬ ɛɵɬɶ ɧɢɤɨɢɦ ɨɛɪɚɡɨɦ

ɫɤɥɚɞɱɚɬɨɣ.

ɉɭɫɬɶ ɜ ɩɨɥɨɠɟɧɢɢ ɪɚɜɧɨɜɟɫɢɹ ɜɬɨɪɚɹ ɩɪɨɢɡɜɨɞɧɚɹ ɮɭɧɤɰɢɢ f(x, r) ɪɚɜɧɚ ɧɭɥɸ ɢɥɢ

ɝɟɫɫɢɚɧ ɫɢɧɝɭɥɹɪɟɧ. ȼ ɷɬɢɯ ɫɥɭɱɚɹɯ ɩɨɥɨɠɟɧɢɹ, ɨɩɪɟɞɟɥɹɟɦɵɟ

ɭɫɥɨɜɢɟɦ (4.1.5), ɦɨɝɭɬ ɧɟ

ɛɵɬɶ ɨɩɬɢɦɚɥɶɧɵɦɢ. Ɍɚɤɢɟ ɬɨɱɤɢ ɪɚɜɧɨɜɟɫɢɹ ɢɡɜɟɫɬɧɵ ɤɚɤ ɨɫɨɛɵɟ, ɢ ɢɦɟɧɧɨ ɜ ɬɚɤɢɯ ɬɨɱɤɚɯ

ɢ ɜɛɥɢɡɢ ɧɢɯ ɦɨɠɧɨ ɧɚɛɥɸɞɚɬɶ ɧɟɨɛɵɱɧɨɟ ɩɨɜɟɞɟɧɢɟ ɫɢɫɬɟɦɵ. Ʉɚɤ ɩɨɤɚɡɚɧɨ ɜ ɝɥ. 3,

ɷɥɟɦɟɧɬɚɪɧɨɣ ɡɚɞɚɱɟɣ ɬɟɨɪɢɢ ɤɚɬɚɫɬɪɨɮ ɹɜɥɹɟɬɫɹ ɤɥɚɫɫɢɮɢɤɚɰɢɹ ɜɨɡɦɨɠɧɵɯ ɬɢɩɨɜ

ɨɫɨɛɟɧɧɨɫɬɟɣ. Ɍɟɨɪɢɹ ɤɚɬɚɫɬɪɨɮ ɢɦɟɟɬ ɞɟɥɨ ɫ ɜɧɟɡɚɩɧɵɦɢ ɢ ɞɢɫɤɪɟɬɧɵɦɢ ɢɡɦɟɧɟɧɢɹɦɢ

ɫɨɫɬɨɹɧɢɹ ɫɢɫɬɟɦɵ, ɤɨɬɨɪɵɟ ɹɜɥɹɸɬɫɹ

ɪɟɡɭɥɶɬɚɬɨɦ ɦɟɞɥɟɧɧɵɯ, ɝɥɚɞɤɢɯ ɢ ɦɚɥɵɯ ɢɡɦɟɧɟɧɢɣ

ɨɞɧɨɝɨ ɢ ɛɨɥɟɟ ɩɚɪɚɦɟɬɪɨɜ. ȼ ɫɥɭɱɚɟ ɱɢɫɥɚ ɭɩɪɚɜɥɹɸɳɢɯ ɩɚɪɚɦɟɬɪɨɜ ɜɟɤɬɨɪɚ r ɦɟɧɶɲɟɝɨ ɢɥɢ

ɪɚɜɧɨɝɨ 4, ɱɢɫɥɨ ɜɨɡɦɨɠɧɵɯ ɨɫɨɛɟɧɧɨɫɬɟɣ, ɜ ɬɨɩɨɥɨɝɢɱɟɫɤɨɦ ɫɦɵɫɥɟ, ɨɬɧɨɫɢɬɟɥɶɧɨ ɦɚɥɨ

(ɫɦ. ɝɥ. 3).

Ɋɚɫɫɦɨɬɪɢɦ ɩɪɨɫɬɨɣ ɩɪɢɦɟɪ — ɫɛɨɪɤɭ — ɨɞɧɭ ɢɡ ɷɥɟɦɟɧɬɚɪɧɵɯ ɤɚɬɚɫɬɪɨɮ Ɍɨɦɚ. Ɉɧɚ

ɧɚɯɨɞɢɬ ɧɚɢɛɨɥɟɟ ɲɢɪɨɤɨɟ; ɩɪɢɦɟɧɟɧɢɟ ɜ ɧɚɭɤɟ ɛɥɚɝɨɞɚɪɹ ɫɜɨɟɣ ɩɪɨɫɬɨɬɟ ɢ

ɬɢɩɢɱɧɨɫɬɢ.

Ɋɚɫɫɦɨɬɪɢɦ ɩɨɬɟɧɰɢɚɥɶɧɭɸ ɮɭɧɤɰɢɸ