численные методыА.Б. САМОХИН, В.В. ЧЕРДЫНЦЕВ, А.А. ВОРОНЦОВ

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Ⱦɪɭɝɢɟ ɦɟɬɨɞɵ ɩɪɨɫɬɨɣ ɢɬɟɪɚɰɢɢ ɛɭɞɭɬ ɪɚɫɫɦɨɬɪɟɧɵ ɜ ɪɚɡɞɟɥɟ

5.3.2.

ɉɪɨɰɟɫɫ ɩɪɨɫɬɨɣ ɢɬɟɪɚɰɢɢ ɦɨɠɟɬ ɛɵɬɶ ɷɤɜɢɜɚɥɟɧɬɧɨ ɡɚɩɢɫɚɧ ɬɚɤɠɟ ɜ ɜɢɞɟ ɪɹɞɚ ɩɨ ɫɬɟɩɟɧɹɦ ɨɩɟɪɚɬɨɪɚ B ɬ ɟ ɜ ɜɢɞɟ ɬɚɤ ɧɚɡɵ ɜɚɟɦɨɝɨ ɪɹɞɚ ɇɟɣɦɚɧɚ

G	G	G	B	2 G	...	f	i G	(5.3.1.4)
x	b	Bb	B	b	...	¦	B b .	(5.3.1.4)
						i 0

ȿɫɥɢ ɦɚɬɪɢɰɚ B ɩɨɫɬɨɹɧɧɚ ɧɟ ɡɚɜɢɫɢɬ ɨɬ ɧɨɦɟɪɚ ɢɬɟɪɚɰɢɢ k ), ɬɨ ɬɚɤɨɣ ɢɬɟɪɚɰɢɨɧɧɵɣ ɩɪɨɰɟɫɫ ɧɚɡɵɜɚɟɬɫɹ ɫɬɚɰɢɨɧɚɪɧɵɦ

ɉɭɫɬɶ xG – ©ɝɢɩɨɬɟɬɢɱɟɫɤɨɟª ɬɨɱɧɨɟ ɪɟɲɟɧɢɟ ɫɬɪɨɝɨ ɭɞɨɜɥɟ

G	G	G	G(k )	x	(k )	x		– ɨɲɢɛɤɚ ɧɚ k -ɦ ɲɚɝɟ
ɬɜɨɪɹɸɳɟɟ x	Bx	b	ɚ 'x	x		x		– ɨɲɢɛɤɚ ɧɚ k -ɦ ɲɚɝɟ

ɉɨɞɫɬɚɜɥɹɹ ɜ ɮɨɪɦɭɥɭ ɩɪɨɫɬɨɣ ɢɬɟɪɚɰɢɢ ɩɨɥɭɱɚɟɦ ɞɥɹ ɫɨɨɬɧɨɲɟ

ɧɢɹ ɨɲɢɛɨɤ ɧɚ k 1	G
ɧɢɹ ɨɲɢɛɨɤ ɧɚ k 1	ɢ k -ɦ ɲɚɝɟ 'x
ɨɲɢɛɤɢ

(k 1) B'xG(k ) . Ⱦɥɹ ɧɨɪɦɵ

G(k 1)

G(k )

G(1)

Ɉɬɫɸɞɚ ɫɥɟɞɭɟɬ ɞɨɫɬɚɬɨɱɧɨɟ ɭɫɥɨɜɢɟ ɫɯɨɞɢɦɨɫɬɢ ɩɪɨɰɟɫɫɚ

ɩɪɨɫɬɨɣ ɢɬɟɪɚɰɢɢ

G 1.

Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ ɬɨɝɞɚ

'xG(k 1)

'xG(1)

Ɉɩɟɪɚɬɨɪ ɫ

G 1 ɧɚɡɵɜɚɟɬɫɹ ɫɠɢɦɚɸɳɢɦ ɚ ɩɪɨɰɟɫɫ

ɞɥɹ ɧɟɝɨ ɫɯɨɞɹɳɢɦɫɹ ɬ ɤ ɨɲɢɛɤɚ ɭɛɵɜɚɟɬ ɫ ɤɚ ɠɞɵɦ ɲɚɝɨɦ ɧɟɡɚɜɢɫɢɦɨ ɨɬ ɟɺ ɧɚɱɚɥɶɧɨɣ ɜɟɥɢɱɢɧɵ

ɋɩɟɤɬɪɚɥɶɧɵɦ ɪɚɞɢɭɫɨɦ ɦɚɬɪɢɰɵ ɤɨɧɟɱɧɨɦɟɪɧɨɝɨ ɨɩɟɪɚɬɨ

ɪɚ B ɧɚɡɵɜɚɟɬɫɹ U(B) max Ei ɝɞɟ Ei – ɫɨɛɫɬɜɟɧɧɵɟ ɱɢɫɥɚ

ɨɩɟɪɚɬɨɪɚ B ɫɦ Ⱦɥɹ ɥɸɛɨɣ ɧɨɪɦɵ ɫɩɪɚɜɟɞɥɢɜɨ ɫɨɨɬɧɨɲɟɧɢɟ U(B) d B .

Ⱦɨɤɚɡɵɜɚɟɬɫɹ ɱɬɨ ɧɟɨɛɯɨɞɢɦɵɦ ɢ ɞɨɫɬɚɬɨɱɧɵɦ ɭɫɥɨɜɢɟɦ ɫɯɨɞɢɦɨɫɬɢ ɩɪɨɰɟɫɫɚ ɩɪɨɫɬɨɣ ɢɬɟɪɚɰɢɢ ɹɜɥɹɟɬɫɹ

U(B) < 1,

(5.3.1.5)

ɩɪɢ ɷɬɨɦ ɢɬɟɪɚɰɢɢ ɫɯɨɞɹɬɫɹ ɧɟ ɯɭɠɟ ɝɟɨɦɟɬɪɢɱɟɫɤɨɣ ɩɪɨɝɪɟɫɫɢɢ ɫɨ ɡɧɚɦɟɧɚɬɟɥɟɦ q U(B) .

ɍɫɥɨɜɢɟ ɹɜɥɹɟɬɫɹ ɤɚɤ ɩɪɚɜɢɥɨ ɫɢɥɶɧɵɦ ɨɝɪɚɧɢɱɟɧɢ ɟɦ ɩɪɢ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨɦ ɩɪɢɦɟɧɟɧɢɢ ɦɟɬɨɞɚ ɤ

ɡɚɞɚɧɧɨɣ ɋɅȺɍ ȼɵɛɨɪ ɧɨɜɨɝɨ ɨɩɟɪɚɬɨɪɚ ɫ ɞɪɭɝɢɦ ɫɩɟɤɬɪɨɦ

ɩɪɢ ɷɤɜɢɜɚɥɟɧɬɧɨɫɬɢ ɢɫɯɨɞɧɨɣ ɫɢɫɬɟɦɟ ɦɨɠɟɬ ɡɧɚɱɢɬɟɥɶɧɨ ɪɚɫɲɢɪɢɬɶ ɨɛɥɚɫɬɶ ɫɯɨɞɢɦɨɫɬɢ ɩɪɨɰɟɫɫɚ ɩɪɨɫɬɨɣ ɢɬɟɪɚɰɢɢ ɫ ɟɝɨ ɭɱɚɫɬɢɟɦ

G x

(k 1) G(k )		k 0, 1, 2, ... .	(5.3.1.6)
Bx	b ,

ȼ ɤɚɱɟɫɬɜɟ ɭɫɥɨɜɢɹ ɜɵɯɨɞɚ ɢɡ ɜɵɱɢɫɥɢɬɟɥɶɧɨɝɨ ɩɪɨɰɟɫɫɚ ɩɨ ɞɨɫɬɢɠɟɧɢɢ ɡɚɞɚɧɧɨɣ ɬɨɱɧɨɫɬɢ ɪɟɲɟɧɢɹ H ɚɧɚɥɨɝɢɱɧɨ

ɞɢɭɫ B ɢɥɢ ɤɚɤɚɹ-ɥɢɛɨ ɨɰɟɧɤɚ ɞɪɭɝɨɣ ɧɨɪɦɵ B .

Ɇɟɬɨɞ əɤɨɛɢ ɢ ɦɟɬɨɞ Ɂɟɣɞɟɥɹ

ɂɫɬɨɪɢɱɟɫɤɢ ɨɞɧɢɦɢ ɢɡ ɫɚɦɵɯ ɪɚɧɧɢɯ ɢɬɟɪɚɰɢɨɧɧɵɯ ɦɟɬɨɞɨɜ ɹɜɥɹɸɬɫɹ ɦɟɬɨɞ əɤɨɛɢ ɢ ɦɟɬɨɞ Ɂɟɣɞɟɥɹ ɤɨɬɨɪɵɟ ɦɨɝɭɬ ɛɵɬɶ ɩɪɟɞ ɫɬɚɜɥɟɧɵ ɜ ɜɢɞɟ ɦɨɞɢɮɢɤɚɰɢɢ ɦɟɬɨɞɚ ɩɪɨɫɬɨɣ ɢɬɟɪɚɰɢɢ ɉɟɪɟɩɢ ɲɟɦ ɜ ɫɥɟɞɭɸɳɟɦ ɜɢɞɟ:

(b a x

a x ... a x ),

a11

1n n

° 1

°....................................................

aii 1xi 1 aii 1xi 1 ain xn ), (5.3.2.1)

®xi

(bi ai1x1

aii

°....................................................

°x

a x

... a

° n

n1 1

n2 2

nn 1

n 1

ann

ɂɫɩɨɥɶɡɭɟɦ ɞɥɹ ɩɨɫɬɪɨɟɧɢɹ ɩɪɨɰɟɫɫɚ ɢɬɟɪɚɰɢɣ ɧɚ

G(0)

b ɩɪɢ k

0 ,

0,1, 2,...:

ɱɢɧɚɹ ɫ x

x(k 1)	1	(b a	x(k ) ... a			x(k )	a	1	x(k )	... a	x(nk )),

i	ai,i	i i,1	1	i,i 1		i 1	i,i		i 1	i,n

i 1, 2,..., n.										(5.3.2.2)
ȼ ɦɚɬɪɢɱɧɵɯ ɨɛɨɡɧɚɱɟɧɢɹɯ ɦɟɬɨɞ əɤɨɛɢ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ ɫɥɟ
ɞɭɸɳɢɦ ɨɛɪɚɡɨɦ ɉɪɟɞɫɬɚɜɢɦ C					D A ɝɞɟ			D — ɞɢɚɝɨɧɚɥɶɧɚɹ
ɦɚɬɪɢɰɚ ^D`i,i		^A`i,i ,i		1, 2,..., n ,^D`i, j			0,i z j . C — ɦɚɬɪɢɰɚ ɫ

ɧɭɥɟɜɨɣ ɝɥɚɜɧɨɣ ɞɢɚɝɨɧɚɥɶɸ Ɍɨɝɞɚ ɫɩɪɚɜɟɞɥɢɜɚ ɡɚɩɢɫɶ ɭɪɚɜɧɟ ɧɢɹ ɚɧɚɥɨɝɢɱɧɨ 5.3. ɝɞɟ

	D	1		D	1 G	(5.3.2.3)
B	D	C ,	b	D	b .	(5.3.2.3)
Ɇɚɬɪɢɰɚ D 1 — ɞɢɚɝɨɧɚɥɶɧɚɹ ɢ ^D 1`						1 ^A`i,i , i 1, 2,..., n.
					i,i

ɇɟɨɛɯɨɞɢɦɵɟ ɢ ɞɨɫɬɚɬɨɱɧɵɟ ɭɫɥɨɜɢɹ ɫɯɨɞɢɦɨɫɬɢ ɦɟɬɨɞɚ əɤɨɛɢ:

U(D 1(D A)) 1.

Ⱦɪɭɝɨɣ ɢɡɜɟɫɬɧɵɣ ɦɟɬɨɞ ɩɪɨɫɬɨɣ ɢɬɟɪɚɰɢɢ ɞɥɹ ɫɥɭɱɚɹ ɤɨɝɞɚ B ɫɬɪɨɢɬɫɹ ɧɚ ɨɫɧɨɜɟ ɦɚɬɪɢɰɵ ɫ ɧɭɥɟɜɨɣ ɝɥɚɜɧɨɣ ɞɢɚɝɨɧɚɥɶɸ — ɷɬɨ ɦɟɬɨɞ Ɂɟɣɞɟɥɹ Ɉɧ ɨɬɥɢɱɚɟɬɫɹ ɨɬ ɦɟɬɨɞɚ əɤɨɛɢ ɬɟɦ ɱɬɨ ɩɪɢ ɪɚɫɱɟɬɟ ɤɨɨɪɞɢɧɚɬ ɜɟɤɬɨɪɚ xG(k 1) ɧɚ ɬɟɤɭɳɟɣ ( k 1)-ɣ ɢɬɟɪɚɰɢɢ ɢɫɩɨɥɶɡɭɸɬɫɹ ɧɟ ɬɨɥɶɤɨ ɤɨɨɪɞɢɧɚɬɵ ɜɟɤɬɨɪɚ ɧɚ ɩɪɟɞɵɞɭɳɟɣ k -ɣ

ɢɬɟɪɚɰɢɢ xG(k ) ɧɨ ɢ ɭɠɟ ɪɚɧɟɟ ɧɚɣɞɟɧɧɵɟ ɧɚ ɬɟɤɭɳɟɣ ɢɬɟɪɚɰɢɢ ɤɨɨɪɞɢɧɚɬɵ ɜɟɤɬɨɪɚ xG(k 1) :

(k 1)

x(k 1)

... a

x(k 1)

x(k )

... a

x(nk )),

ai,i

i,1

i,i

i 1

i,i

i 1

i,n

k 0, 1, 2, ... .

1, 2, ..., n ;

(5.3.2.4)

ȼ ɦɚɬɪɢɱɧɵɯ ɨɛɨɡɧɚɱɟɧɢɹɯ ɷɬɨ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɩɪɟɞɫɬɚɜɥɟɧɢɸ

ɢɫɯɨɞɧɨɣ ɦɚɬɪɢɰɵ A ɤɚɤ

L D U ɝɞɟ L – ɧɢɠɧɹɹ ɬɪɟ

ɭɝɨɥɶɧɚɹ ɦɚɬɪɢɰɚ

D –

ɞɢɚɝɨɧɚɥɶɧɚɹ ɦɚɬɪɢɰɚ ^D`i,i

^A`i,i ,

1, 2, ..., n , ɢ U – ɜɟɪɯɧɹɹ ɬɪɟɭɝɨɥɶɧɚɹ ɦɚɬɪɢɰɚ

ȼ ɨɬɥɢɱɢɟ ɨɬ ɦɟɬɨɞɚ əɤɨɛɢ ɞɟɣɫɬɜɢɟ ɨɩɟɪɚɬɨɪɚ B ɧɚ ɜɟɤɬɨɪ ɩɪɟɞɵɞɭɳɟɣ ɢɬɟɪɚɰɢɢ ɪɚɡɞɟɥɹɟɬɫɹ ɡɞɟɫɶ ɧɚ ɞɜɟ ɱɚɫɬɢ

G(k )	D	1	G(k 1)	D	1 G(k )	,	(5.3.2.5)
Bx			Lx		Ux

ɢ ɩɪɨɰɟɫɫ ɟɝɨ ɜɨɡɞɟɣɫɬɜɢɹ ɧɨ ɧɟ ɪɟɡɭɥɶɬɚɬ ɧɟɥɶɡɹ ɫɜɟɫɬɢ ɤ ɜɨɡ ɞɟɣɫɬɜɢɸ ɤɚɤɨɣ-ɥɢɛɨ ɦɚɬɪɢɰɵ ɧɚ ɜɟɤɬɨɪ ɩɪɟɞɵɞɭɳɟɣ ɢɬɟɪɚɰɢɢ

Ɇɟɬɨɞ Ɂɟɣɞɟɥɹ ɯɨɪɨɲɨ ɚɥɝɨɪɢɬɦɢɡɢɪɭɟɬɫɹ ȿɫɥɢ ɢɡɜɟɫɬɧɚ ɫɤɨɪɨɫɬɶ ɫɯɨɞɢɦɨɫɬɢ ɦɟɬɨɞɚ ɧɟɬ ɧɟɨɛɯɨɞɢɦɨɫɬɢ ɯɪɚɧɢɬɶ ɨɛɚ ɜɟɤ ɬɨɪɚ xG(k 1) ɢ xG(k) .

Ⱦɨɫɬɚɬɨɱɧɵɦɢ ɭɫɥɨɜɢɹɦɢ ɫɯɨɞɢɦɨɫɬɢ ɦɟɬɨɞɨɜ əɤɨɛɢ ɢ Ɂɟɣɞɟ ɥɹ ɹɜɥɹɟɬɫɹ ɞɢɚɝɨɧɚɥɶɧɨɟ ɩɪɟɨɛɥɚɞɚɧɢɟ ɜ ɦɚɬɪɢɱɧɵɯ ɷɥɟɦɟɧɬɚɯ

q ai,i t ¦ ai, j , q 1, ɞɥɹ ɜɫɟɯ i 1, 2, ..., n ,

jzi

ɨɞɧɚɤɨ ɧɚ ɩɪɚɤɬɢɤɟ ɨɛɥɚɫɬɶ ɫɯɨɞɢɦɨɫɬɢ ɡɧɚɱɢɬɟɥɶɧɨ ɲɢɪɟ ɢ ɨɩ ɪɟɞɟɥɹɟɬɫɹ ɭɫɥɨɜɢɟɦ ɧɚ ɫɩɟɤɬɪɚɥɶɧɵɣ ɪɚɞɢɭɫ ɦɚɬɪɢɰɵɞɥɹ ɦɟɬɨɞɚ əɤɨɛɢ ɢ ɨɩɟɪɚɬɨɪɚ ɞɥɹ ɦɟɬɨɞɚ Ɂɟɣ ɞɟɥɹ Ⱦɥɹ ɪɟɲɟɧɢɹ ɋɅȺɍ ɫ ɥɟɧɬɨɱɧɵɦɢ ɦɚɬɪɢɰɚɦɢ ɦɟɬɨɞ Ɂɟɣ ɞɟɥɹ ɹɜɥɹɟɬɫɹ ɩɪɟɜɨɫɯɨɞɧɵɦ ɢɧɫɬɪɭɦɟɧɬɨɦ Ɍɚɤ ɞɥɹ ɫɢɦɦɟɬɪɢɱ ɧɵɯ ɩɨɥɨɠɢɬɟɥɶɧɨ ɨɩɪɟɞɟɥɟɧɧɵɯ ɦɚɬɪɢɰ ɨɧ ɛɭɞɟɬ ɜɫɟɝɞɚ ɫɯɨɞɹ ɳɢɦɫɹ Ɉɞɧɚɤɨ ɜɨɡɦɨɠɧɨ ɭɥɭɱɲɟɧɢɟ ɫɯɨɞɢɦɨɫɬɢ ɤɚɤ ɦɟɬɨɞɚ Ɂɟɣɞɟɥɹ ɬɚɤ ɢ ɥɸɛɨɝɨ ɞɪɭɝɨɝɨ ɦɟɬɨɞɚ ɩɪɨɫɬɨɣ ɢɬɟɪɚɰɢɢ ɫ ɩɨɦɨ ɳɶɸ ɢɡɥɨɠɟɧɧɨɝɨ ɧɢɠɟ ɦɟɬɨɞɚ ɨɩɬɢɦɚɥɶɧɨɝɨ ɫɩɟɤɬɪɚɥɶɧɨɝɨ ɩɚ ɪɚɦɟɬɪɚ

Ɇɟɬɨɞ ɨɩɬɢɦɚɥɶɧɨɝɨ ɫɩɟɤɬɪɚɥɶɧɨɝɨ ɩɚɪɚɦɟɬɪɚ Ɉɋɉ ɞɥɹ ɩɪɨɫɬɨɣ ɢɬɟɪɚɰɢɢ

Ɋɚɫɫɦɨɬɪɢɦ ɫɥɭɱɚɣ ɤɨɝɞɚ ɫɩɟɤɬɪ ɨɩɟɪɚɬɨɪɚ B ɜɵɯɨɞɢɬ ɡɚ ɝɪɚ ɧɢɰɵ ɟɞɢɧɢɱɧɨɝɨ ɤɪɭɝɚ ɧɚ ɤɨɦɩɥɟɤɫɧɨɣ E-ɩɥɨɫɤɨɫɬɢ ɫɨɛɫɬɜɟɧɧɵɯ ɱɢɫɟɥ ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɪɹɞ ɩɪɨɫɬɨɣ ɢɬɟɪɚɰɢɢ ɪɚɫɯɨɞɢɬɫɹ

Ɉɩɪɟɞɟɥɢɦ ɜɵɩɭɤɥɭɸ ɨɛɨɥɨɱɤɭ ɫɩɟɤɬɪɚ ɨɩɟɪɚɬɨɪɚ B ɤɚɤ ɜɵ ɩɭɤɥɭɸ ɡɚɦɤɧɭɬɭɸ ɤɪɢɜɭɸ ɧɚɢɦɟɧɶɲɟɣ ɦɟɪɵ ɩɨɥɧɨɫɬɶɸ ɨɯɜɚɬɵ ɜɚɸɳɭɸ ɫɩɟɤɬɪ ɨɩɟɪɚɬɨɪɚ ɧɚ E-ɩɥɨɫɤɨɫɬɢ Ⱦɨɤɚɡɵɜɚɟɬɫɹ ɱɬɨ ɟɫ ɥɢ ɬɨɱɤɚ 1 ɧɚɯɨɞɢɬɫɹ ɜɧɟ ɜɵɩɭɤɥɨɣ ɨɛɨɥɨɱɤɢ ɫɩɟɤɬɪɚ ɬɨ ɦɨɠɧɨ ɩɨɫɬɪɨɢɬɶ ɫɯɨɞɹɳɢɣɫɹ ɪɹɞ ɩɪɨɫɬɨɣ ɢɬɟɪɚɰɢɢ ɫ ɧɨɜɵɦ

			O k0
		O	1 k0
,PȜ			1 k0
,PȜ			~
	ȍB		Im O
	ȍB
	k0	ȕmax
ȕmin		ȕmax
ȕmin		r
		r
	Į
1		5HȜ	~	1	~
1		5HȜ	~	1	Re O
			r

		Ɋɢɫ 5.1
	Ⱦɚɞɢɦ ɤɨɧɫɬɪɭɤɬɢɜɧɵɣ ɫɩɨɫɨɛ ɩɨɫɬɪɨɟɧɢɹ
ɨɩɟɪɚɬɨɪɨɦ B f (B)	Ⱦɚɞɢɦ ɤɨɧɫɬɪɭɤɬɢɜɧɵɣ ɫɩɨɫɨɛ ɩɨɫɬɪɨɟɧɢɹ
ɬɚɤɨɝɨ ɫɯɨɞɹɳɟɝɨɫɹ ɪɹɞɚ ɉɪɢɦɟɦ
		B kI		G	b
		B kI			b
B			,	b		,	(5.3.3.1)
B		1 k	,	b	1 k	,	(5.3.3.1)

ɝɞɟ k – ɤɨɦɩɥɟɤɫɧɵɣ ɩɚɪɚɦɟɬɪ ɉɪɢ k z 1 ɢɫɯɨɞɧɵɟ ɭɪɚɜɧɟɧɢɹ


ɫ ɨɩɟɪɚɬɨɪɚɦɢ B ɢ B ɷɤɜɢɜɚɥɟɧɬɧɵ ȼɵɛɨɪɨɦ k ɩɨɩɪɨ
ɛɭɟɦ ɞɨɛɢɬɶɫɹ ɫɯɨɞɢɦɨɫɬɢ ɪɹɞɚ
ɉɭɫɬɶ :B – ɨɞɢɧ ɢɡ ɦɧɨɠɟɫɬɜɚ ɤɪɭɝɨɜ ɪɚɞɢɭɫɚ r ɩɨɥɧɨɫɬɶɸ
ɨɯɜɚɬɵɜɚɸɳɢɯ ɫɩɟɤɬɪ ɨɩɟɪɚɬɨɪɚ	B ɢ ɩɭɫɬɶ ɩɪɢ ɷɬɨɦ ɬɨɱɤɚ
1 :B Ɋɢɫ Ɉɱɟɜɢɞɧɨ ɱɬɨ :B ɜɤɥɸɱɚɟɬ ɜ ɫɟɛɹ ɜɵɩɭɤɥɭɸ ɨɛɨ
ɥɨɱɤɭ ɫɩɟɤɬɪɚ ȼɟɤɬɨɪ ɢɡ ɧɚɱɚɥɚ E	0 ɜ ɰɟɧɬɪ ɷɬɨɝɨ ɤɪɭɝɚ ɨɛɨɡɧɚ

ɱɢɦ k0 ɉɪɢ ɞɪɨɛɧɨ-ɥɢɧɟɣɧɨɦ ɩɪɟɨɛɪɚɡɨɜɚɧɢɢ ɫ k k0

ɤɪɭɝ :B		ɩɟɪɟɯɨɞɢɬ ɜ ɤɪɭɝ : ɫ ɰɟɧɬɪɨɦ ɜ ɬɨɱɤɟ E 0 ɢ ɪɚɞɢɭ
						B
ɫɨɦ	r		r	ȿɫɥɢ	r	1 ɬɨ ɪɹɞ ɫɯɨɞɢɬɫɹ
			r
		1 k0
		1 k0

ɤɚɫɚɧɢɹ				ɥɭɱɚ ɢɡ ɬ ɢ ɤɪɭɝɚ		ɂɡ		ɪɢɫ ɨɱɟɜɢɞɧɨ ɱɬɨ
	r		rGG		sin(D) ɢ ɫɥɟɞɨɜɚɬɟɥɶɧɨ		r	1.


		1	k0
		1	k0

Ɍɚɤɢɦ ɨɛɪɚɡɨɦ ɟɫɥɢ :B ɬɚɤɨɣ ɤɪɭɝ ɱɬɨ ɬɨɱɤɚ 1 :B ɢ ©ɜɢɞɢ ɦɵɣª ɢɡ ɬɨɱɤɢ 1 ɩɨɞ ɧɚɢɦɟɧɶɲɢɦ ɭɝɥɨɦ 2D ɬɨ ɤɨɦɩɥɟɤɫɧɨɟ ɪɚɫ ɫɬɨɹɧɢɟ ɞɨ ɰɟɧɬɪɚ ɷɬɨɝɨ ɤɪɭɝɚ ɟɫɬɶ ɨɩɬɢɦɚɥɶɧɵɣ ɩɚɪɚɦɟɬɪ ɞɥɹ ɫɯɨɞɢɦɨɫɬɢ ɚ ɫɤɨɪɨɫɬɶ ɫɯɨɞɢɦɨɫɬɢ ɪɹɞɚ ɧɟ ɯɭɠɟ ɱɟɦ ɭ ɝɟɨɦɟɬɪɢɱɟɫɤɨɣ ɩɪɨɝɪɟɫɫɢɢ ɫɨ ɡɧɚɦɟɧɚɬɟɥɟɦ sin(D) .

ɉɭɫɬɶ ɞɥɹ ɫɩɟɤɬɪɚ ^EQ` ɢɡɜɟɫɬɧɵ ɨɰɟɧɤɢ ɞɥɹ Emin , Emax - ɦɢɧɢɦɚɥɶɧɨɝɨ ɢ ɦɚɤɫɢɦɚɥɶɧɨɝɨ ɩɨ ɦɨɞɭɥɸ ɫɨɛɫɬɜɟɧɧɨɝɨ ɱɢɫɥɚɢɥɢ ɧɢɠɧɟɣ ɢ ɜɟɪɯɧɟɣ ɝɪɚɧɢɰɵ ɪɚɫɫɬɨɹɧɢɹ ɨɬ ɬ 0 ɞɨ ɨɛɥɚɫɬɢ ɪɚɫɩɨɥɨɠɟɧɢɹ ɫɩɟɤɬɪɚ ɜ ɫɥɭɱɚɟ ɧɟɩɪɟɪɵɜɧɨɝɨ ɫɩɟɤɬɪɚɥɶɧɨɝɨ ɦɧɨɠɟɫɬɜɚ . Ɍɨɝɞɚ ɟɫɥɢ ɜɟɫɶ ɫɩɟɤɬɪ ɨɩɟɪɚɬɨɪɚ ɪɚɡɦɟɳɚɟɬɫɹ ɜ ɤɪɭ ɝɟ :B ɧɚɬɹɧɭɬɨɦ ɧɚ ɬɨɱɤɢ Emin , Emax ɤɚɤ ɧɚ ɤɨɧɰɟɜɵɟ ɬɨɱɤɢ ɞɢɚ ɦɟɬɪɚ ɢ ɬɨɱɤɚ 1 :B ɞɥɹ ɨɩɬɢɦɚɥɶɧɨɝɨ ɩɚɪɚɦɟɬɪɚ ɜɟɪɧɚ ɩɪɨɫɬɚɹ ɩɪɢɛɥɢɠɟɧɧɚɹ ɮɨɪɦɭɥɚ

k0	Emin Emax .	(5.3.3.2)
	2

ȿɫɥɢ ɝɪɚɧɢɰɚ ɤɪɭɝɚ ɩɪɢɧɚɞɥɟɠɢɬ ɫɩɟɤɬɪɭ ɬɨ ɮɨɪɦɭɥɚ ɬɨɱɧɚɹ Ɍɨɱɧɚɹ ɨɧɚ ɬɚɤɠɟ ɢ ɜ ɫɥɭɱɚɟ ɜɟɳɟɫɬɜɟɧɧɨɝɨ ɫɩɟɤɬɪɚ Ɏɨɪ ɦɭɥɭ 5.3.3.2 ɦɨɠɧɨ ɭɥɭɱɲɢɬɶ ɭɱɢɬɵɜɚɹ ɛɨɥɟɟ ɬɨɱɧɭɸ ɤɨɧɮɢɝɭ ɪɚɰɢɸ ɫɩɟɤɬɪɚɥɶɧɨɣ ɨɛɥɚɫɬɢ ɧɚɩɪɢɦɟɪ ɟɫɥɢ ɨɛɥɚɫɬɶ ɪɚɫɩɨɥɨɠɟ ɧɢɹ ɫɩɟɤɬɪɚ – ɩɪɹɦɚɹ ɥɢɧɢɹ ɋ ɩɨɦɨɳɶɸ ɮɨɪɦɭɥɵ ɜɨ ɦɧɨɝɢɯ ɫɥɭɱɚɹɯ ɦɨɠɧɨ ɧɚɣɬɢ ɡɧɚɱɟɧɢɟ ɛɥɢɡɤɨɟ ɤ ɨɩɬɢɦɚɥɶɧɨɦɭ ɩɚɪɚɦɟɬɪɭ ɜ ɭɫɥɨɜɢɹɯ ɧɟɩɨɥɧɨɝɨ ɡɧɚɧɢɹ ɫɜɨɣɫɬɜ ɫɩɟɤɬɪɚ ɧɨ ɩɪɢ ɢɡɜɟɫɬɧɵɯ ɦɢɧɢɦɚɥɶɧɵɯ ɢ ɦɚɤɫɢɦɚɥɶɧɵɯ ɩɨ ɦɨɞɭɥɸ ɫɨɛɫɬɜɟɧɧɵɯ ɱɢɫɥɚɯ

ɋɯɨɞɢɦɨɫɬɶ ɤɚɠɞɨɝɨ ɢɡ ɪɚɫɫɦɨɬɪɟɧɧɵɯ ɦɟɬɨɞɨɜ ɩɪɨɫɬɨɣ ɢɬɟ ɪɚɰɢɢ ɡɚɜɢɫɢɬ ɨɬ ɤɨɧɤɪɟɬɧɨɝɨ ɜɢɞɚ ɢɫɯɨɞɧɨɣ ɦɚɬɪɢɰɵ ɚ ɬɨɱɧɟɟ

ɨɬ ɫɜɨɣɫɬɜ ɟɺ ɫɩɟɤɬɪɚ Ɇɨɠɧɨ ɩɪɢɜɟɫɬɢ ɩɪɢɦɟɪɵ ɦɚɬɪɢɰ ɞɥɹ ɤɨ ɬɨɪɵɯ ɫɯɨɞɢɬɫɹ ɬɨɥɶɤɨ ɨɞɢɧ ɢɡ ɪɚɫɫɦɨɬɪɟɧɧɵɯ ɦɟɬɨɞɨɜ ɨɞɧɚɤɨ ɤɨɦɛɢɧɚɰɢɹ ɦɟɬɨɞɚ ɩɪɨɫɬɨɣ ɢɬɟɪɚɰɢɢ Ɂɟɣɞɟɥɹ ɢɥɢ əɤɨɛɢ ɫ ɦɟɬɨ ɞɨɦ ɨɩɬɢɦɚɥɶɧɨɝɨ ɫɩɟɤɬɪɚɥɶɧɨɝɨ ɩɚɪɚɦɟɬɪɚ Ɉɋɉ ɩɨɡɜɨɥɹɸɬ ɞɨ ɛɢɬɶɫɹ ɫɯɨɞɢɦɨɫɬɢ ɜ ɫɥɭɱɚɹɯ ɤɨɝɞɚ ɤɚɠɞɵɣ ɢɡ ɷɬɢɯ ɦɟɬɨɞɨɜ ɩɨ ɨɬɞɟɥɶɧɨɫɬɢ ɪɚɫɯɨɞɢɬɫɹ

Ɋɚɫɫɦɨɬɪɢɦ ɩɪɢɦɟɧɟɧɢɟ ɦɟɬɨɞɚ Ɉɋɉ ɧɚ ɩɪɢɦɟɪɚɯ ɤɨɧɤɪɟɬɧɵɯ ɦɚɬɪɢɱɧɵɯ ɡɚɞɚɱ

ɉɭɫɬɶ ɷɥɟɦɟɧɬɵ ɦɚɬɪɢɰɵ A ɩɪɢ n 2 ɫɥɟɞɭɸɳɢɟ a11 2 ,

a22 2 ,	a12 3,	a21 7 . Cɨɛɫɬɜɟɧɧɵɟ ɱɢɫɥɚ ɦɚɬɪɢɰɵ B (5.3.1.2)
ɪɚɜɧɵ E	4 , E	2	6 ɢ ɪɚɫɩɨɥɚɝɚɸɬɫɹ ɩɨ ɪɚɡɧɵɟ ɫɬɨɪɨɧɵ ɨɬ ɬɨɱ
1		2

ɤɢ 1 ɧɚ ɩɪɹɦɨɣ ɩɪɨɯɨɞɹɳɟɣ ɱɟɪɟɡ ɧɟɺ ȼ ɷɬɨɦ ɫɥɭɱɚɟ ɬɨɱɤɚ 1 ɩɪɢɧɚɞɥɟɠɢɬ ɜɵɩɭɤɥɨɣ ɨɛɨɥɨɱɤɟ ɫɩɟɤɬɪɚ ɢ ɞɪɨɛɧɨ-ɥɢɧɟɣɧɵɦ ɩɪɟɨɛɪɚɡɨɜɚɧɢɟɦ ɧɟɥɶɡɹ ɞɨɛɢɬɶɫɹ ɫɯɨɞɢɦɨɫɬɢ ɢɬɟɪɚɰɢ ɨɧɧɨɝɨ ɩɪɨɰɟɫɫɚ ɋɨɛɫɬɜɟɧɧɵɟ ɠɟ ɱɢɫɥɚ ɦɚɬɪɢɰɵ əɤɨɛɢ ɪɚɜɧɵ Y1 2.3i , Y2 2.3i ɡɞɟɫɶ i – ɦɧɢɦɚɹ ɟɞɢɧɢɰɚ ɢ ɬɨɱɤɚ 1 ɧɚɯɨɞɢɬɫɹ ɜɧɟ ɜɵɩɭɤɥɨɣ ɨɛɨɥɨɱɤɢ ɫɩɟɤɬɪɚ Ɍɨ ɠɟ ɫɚɦɨɟ ɦɨɠɧɨ ɭɬɜɟɪɠɞɚɬɶ ɢ ɨ ɫɩɟɤɬɪɟ ɨɩɟɪɚɬɨɪɚ Ɂɟɣɞɟɥɹ Ɉɞɧɚɤɨ ɧɟɩɨɫɪɟɞɫɬ ɜɟɧɧɨɟ ɩɪɢɦɟɧɟɧɢɟ ɦɟɬɨɞɚ əɤɨɛɢ ɢɥɢ Ɂɟɣɞɟɥɹ ɧɟ ɩɪɢɜɟɞɺɬ ɤ ɫɯɨ ɞɹɳɟɦɭɫɹ ɪɹɞɭ ɬ ɤ YQ !1 ɢ ɧɟ ɜɵɩɨɥɧɹɟɬɫɹ Ɂɚɤɥɸɱɚɹ

ɫɩɟɤɬɪ YQ ɜ ɤɪɭɝ :Y ɫ ɰɟɧɬɪɨɦ ɜ ɬ 8	ɩɪɢɯɨɞɢɦ ɤ ɫɯɨɞɹɳɟɦɭɫɹ
ɦɟɬɨɞɭ əɤɨɛɢ – Ɉɋɉ ɫ ɩɚɪɚɦɟɬɪɨɦ k	8 Ⱦɥɹ ɦɟɬɨɞɚ Ɂɟɣɞɟɥɹ -

Ɉɋɉ ɨɩɬɢɦɚɥɶɧɵɣ ɩɚɪɚɦɟɬɪ k 1 ɩɪɢɜɨɞɢɬ ɤ ɛɵɫɬɪɨ ɫɯɨɞɹɳɟɦɭ ɫɹ ɩɪɨɰɟɫɫɭ Ɋɟɲɟɧɢɟ ɋɅȺɍ ɫ ɩɪɚɜɨɣ ɱɚɫɬɶɸ bQ 1,i 1, 2 ɢ

ɬɨɱɧɨɫɬɶɸ H 10 5 ɞɨɫɬɢɝɚɟɬɫɹ ɡɚ m 20 ɢɬɟɪɚɰɢɣ ɪɹɞɚ ɇɚɨɛɨɪɨɬ ɟɫɥɢ ɦɚɬɪɢɰɚ əɤɨɛɢ ɨɩɟɪɚɬɨɪ Ɂɟɣɞɟɥɹ ɢɦɟɸɬ

ɫɩɟɤɬɪ ɜɵɩɭɤɥɚɹ ɨɛɨɥɨɱɤɚ ɤɨɬɨɪɨɝɨ ɫɨɞɟɪɠɢɬ ɬ 1 ɬɨ ɧɢɤɚɤɢɟ ɦɨɞɢɮɢɤɚɰɢɢ ɷɬɢɯ ɦɟɬɨɞɨɜ ɧɟ ɩɪɢɜɟɞɭɬ ɤ ɫɯɨɞɹɳɟɦɭɫɹ ɩɪɨɰɟɫɫɭ ɉɪɢɦɟɧɟɧɢɟ ɦɟɬɨɞɚ Ɉɋɉ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ ɤ ɢɫɯɨɞɧɨɣ ɦɚɬɪɢɰɟ ɜ ɜɢɞɟ ɦɨɠɟɬ ɩɪɢɜɟɫɬɢ ɜ ɷɬɨɦ ɫɥɭɱɚɟ ɤ ɫɯɨɞɢɦɨɫɬɢ Ɍɚɤɨɜɚ

ɦɚɬɪɢɰɚ ɫ ɷɥɟɦɟɧɬɚɦɢ a11 5, a22 0.7 , a12	4 , a21			2 , ɞɥɹ ɤɨ
ɬɨɪɨɣ ɫɨɛɫɬɜɟɧɧɵɟ ɱɢɫɥɚ ɦɚɬɪɢɰɵ E	1.5	, E	2	0.8 ɚ ɫɨɛ
1			2

ɫɬɜɟɧɧɵɟ ɱɢɫɥɚ ɦɚɬɪɢɰɵ -Y1 1.5, Y2 1.5 ɉɪɢɦɟɧɟɧɢɟ ɦɟ ɬɨɞɨɜ əɤɨɛɢ ɢ Ɂɟɣɞɟɥɹ ɢ ɢɯ ɦɨɞɢɮɢɤɚɰɢɣ ɞɚɸɬ ɪɚɫɯɨɞɹɳɢɣɫɹ ɩɪɨɰɟɫɫ ɬ ɤ ɬɨɱɤɚ 1 ɩɪɢɧɚɞɥɟɠɢɬ ɜɵɩɭɤɥɨɣ ɨɛɨɥɨɱɤɟ ɫɩɟɤɬɪɚ ɉɪɢɦɟɧɟɧɢɟ ɠɟ ɦɟɬɨɞɚ Ɉɋɉ ɤ ɩɪɨɫɬɨɣ ɢɬɟɪɚɰɢɢ ɫ ɦɚɬɪɢɰɟɣɞɚɟɬ ɛɵɫɬɪɨ ɫɯɨɞɹɳɢɣɫɹ ɪɹɞ Ɋɟɲɟɧɢɟ ɋɅȺɍ ɫ ɬɨɱ

ɧɨɫɬɶɸ H 10 5 ɞɨɫɬɢɝɚɟɬɫɹ ɡɚ m 9 ɢɬɟɪɚɰɢɣ ɪɹɞɚ ɉɪɢɦɟɧɟɧɢɟ ɦɟɬɨɞɚ Ɉɋɉ ɧɚɢɛɨɥɟɟ ɭɫɩɟɲɧɨ ɜ ɬɨɦ ɫɥɭɱɚɟ ɤɨ

ɝɞɚ ɫɩɟɤɬɪ ɨɩɟɪɚɬɨɪɚ B ɜ ɥɨɤɚɥɢɡɨɜɚɧ ɜ ɧɟɛɨɥɶɲɨɣ ɨɤɪɟɫɬ ɧɨɫɬɢ ɫ ɰɟɧɬɪɨɦ ɜ ɬ k0 ɜɞɚɥɢ ɨɬ ɬɨɱɤɢ 1 Ɍɨɝɞɚ ɩɪɢɦɟɧɟɧɢɟ ɷɬɨɝɨ ɦɟɬɨɞɚ ɫ ɨɩɬɢɦɚɥɶɧɵɦ ɩɚɪɚɦɟɬɪɨɦ k k0 ɹɜɥɹɟɬɫɹ ɫɚɦɵɦ ɭɞɚɱ ɧɵɦ ɫɪɟɞɢ ɨɞɧɨɲɚɝɨɜɵɯ ɫɬɚɰɢɨɧɚɪɧɵɯ ɦɟɬɨɞɨɜ ɢ ɩɪɢɜɨɞɢɬ ɤ ɛɵ ɫɬɪɨ ɫɯɨɞɹɳɟɦɭɫɹ ɪɹɞɭ ɩɪɨɫɬɨɣ ɢɬɟɪɚɰɢɢ ȼ ɤɚɱɟɫɬɜɟ ɩɪɢɦɟɪɚ ɪɚɫɫɦɨɬɪɢɦ ɋɅȺɍ ɫ ɦɚɬɪɢɰɟɣ a11 7 , a22 3, a12 4 , a21 1. ȼ ɷɬɨɦ ɩɪɢɦɟɪɟ ɞɥɹ ɦɚɬɪɢɰ ɢ ɢɦɟɟɦ ɫɥɟɞɭɸɳɢɟ ɫɨɛɫɬɜɟɧ

ɧɵɟ ɱɢɫɥɚ E	4 , E	2	4 ɢ Y		0.436i ,		Y	0.436i Ɂɧɚɱɟɧɢɟ
1		2		1			2
ɨɩɬɢɦɚɥɶɧɨɝɨ ɩɚɪɚɦɟɬɪɚ k0					E1 E2	4	ɩɟɪɟɜɨɞɢɬ ɜ ɞɚɧɧɨɦ
ɨɩɬɢɦɚɥɶɧɨɝɨ ɩɚɪɚɦɟɬɪɚ k0					2	4	ɩɟɪɟɜɨɞɢɬ ɜ ɞɚɧɧɨɦ
					2
ɫɥɭɱɚɟ ɬɨɱɤɭ E	4 ɜ ɤɨɬɨɪɨɣ ɧɚɯɨɞɢɬɫɹ ɜɟɫɶ ɫɩɟɤɬɪ ɦɚɬɪɢɰɵ B ɜ

ɬɨɱɤɭ E 0 ɜ ɤɨɬɨɪɨɣ ɧɚɯɨɞɢɬɫɹ ɫɩɟɤɬɪ ɦɚɬɪɢɰɵ B Ɍɚɤɢɦ ɨɛɪɚ

ɡɨɦ ɫɤɨɪɨɫɬɶ ɫɯɨɞɢɦɨɫɬɢ ɪɹɞɚ ɫ ɦɚɬɪɢɰɟɣ

	Ɋɟɲɟɧɢɟ
ɜ ɞɚɧɧɨɦ ɫɥɭɱɚɟ ɨɱɟɧɶ ɜɵɫɨɤɚɹ ɬ ɤ U(B) 0

ɋɅȺɍ ɫ ɬɨɱɧɨɫɬɶɸ ɞɨ ɦɚɲɢɧɧɨɣ ɤɨɧɫɬɚɧɬɵ ɞɨɫɬɢɝɚɟɬɫɹ ɡɚ m 2 ɢɬɟɪɚɰɢɢ Ɋɟɲɟɧɢɟ ɬɨɣ ɠɟ ɡɚɞɚɱɢ ɦɟɬɨɞɚɦɢ əɤɨɛɢ ɢ Ɂɟɣɞɟ ɥɹ ɬɪɟɛɭɟɬ ɝɨɪɚɡɞɨ ɛɨɥɶɲɟɝɨ ɤɨɥɢɱɟɫɬɜɚ ɢɬɟɪɚɰɢɣ - m 48 ɢ m 23 ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ Ⱦɥɹ ɦɟɬɨɞɚ əɤɨɛɢ ɩɪɢɦɟɧɟɧɢɟ Ɉɋɉ ɧɟ ɞɚɫɬ ɭɥɭɱɲɟɧɢɹ ɫɯɨɞɢɦɨɫɬɢ ɬ ɤ ɰɟɧɬɪ ɫɩɟɤɬɪɚ ɢ ɬɚɤ ɧɚɯɨɞɢɬɫɹ ɜ

ɬɨɱɤɟ 0 ɢ ɨɩɬɢɦɚɥɶɧɵɣ ɩɚɪɚɦɟɬɪ k0		Y1 Y2	0 Ⱦɥɹ ɦɟɬɨɞɚ ɠɟ

	2

Ɂɟɣɞɟɥɹ ɫɩɟɤɬɪ ɨɩɟɪɚɬɨɪɚ ɨɬɥɢɱɚɟɬɫɹ ɨɬ ɫɩɟɤɬɪɚ ɦɚɬɪɢɰɵ (5 ɢ ɢɫɩɨɥɶɡɨɜɚɧɢɟ ɦɟɬɨɞɚ Ɂɟɣɞɟɥɹ-Ɉɋɉ ɫ ɨɩɬɢɦɚɥɶɧɵɦ ɩɚ ɪɚɦɟɬɪɨɦ k0 0.076 ɬ ɟ ɪɹɞɚ ɫ ɨɩɟɪɚɬɨɪɨɦ

ɩɪɢɜɨɞɢɬ ɤ ɭɦɟɧɶɲɟɧɢɸ ɬɪɟɛɭɟɦɨɝɨ ɤɨɥɢɱɟɫɬɜɚ ɢɬɟɪɚ ɰɢɣ – m 15.

ɉɭɫɬɶ ɪɚɫɫɦɨɬɪɟɧɧɚɹ ɦɚɬɪɢɰɚ ɩɪɨɞɨɥɠɟɧɚ ɧɚ ɛɨɥɶɲɭɸ ɬɪɟɯ ɞɢɚɝɨɧɚɥɶɧɭɸ ɦɚɬɪɢɰɭ ɫ n 100 ɢ ɬɚɤɢɦɢ ɠɟ ɷɥɟɦɟɧɬɚɦɢ ɬ ɟ ɧɚ ɝɥɚɜɧɨɣ ɞɢɚɝɨɧɚɥɢ ɱɟɪɟɞɭɸɬɫɹ ɡɧɚɱɟɧɢɹ 7 ɢ 3 ɚ ɧɚ ɞɜɭɯ ɫɨɫɟɞɧɢɯ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ 4 ɢ 1 ɋɩɟɤɬɪ ɢɫɯɨɞɧɨɣ ɦɚɬɪɢɰɵ ɫɭɳɟɫɬɜɟɧɧɨ ɬɪɚɧɫɮɨɪɦɢɪɭɟɬɫɹ ɢɡ ɬɨɱɤɢ ɜ ɩɪɨɬɹɠɟɧɧɭɸ ɨɛɥɚɫɬɶ ɧɚ ɤɨɦɩɥɟɤɫ ɧɨɣ ɩɥɨɫɤɨɫɬɢ ɧɨ ɩɪɢ ɷɬɨɦ ɡɧɚɱɟɧɢɟ ɨɩɬɢɦɚɥɶɧɨɝɨ ɩɚɪɚɦɟɬɪɚ ɩɨɥɭɱɟɧɧɨɝɨ ɩɨ ɮɨɪɦɭɥɟ ɫ ɭɱɚɫɬɢɟɦ ɦɢɧɢɦɚɥɶɧɨɝɨ ɢ ɦɚɤɫɢɦɚɥɶɧɨɝɨ ɩɨ ɦɨɞɭɥɸ ɫɨɛɫɬɜɟɧɧɨɝɨ ɱɢɫɥɚ ɦɚɬɪɢɰɵ Bɨɫɬɚɟɬɫɹ ɧɟɢɡɦɟɧɧɵɦ k0 4 ɗɬɨ ɫɩɪɚɜɟɞɥɢɜɨ ɞɥɹ ɥɸ ɛɨɣ ɬɪɟɯɞɢɚɝɨɧɚɥɶɧɨɣ ɦɚɬɪɢɰɵ ɩɨɥɭɱɟɧɧɨɣ ɬɚɤɢɦ ɩɟɪɢɨɞɢɱɟ ɫɤɢɦ ɩɪɨɞɨɥɠɟɧɢɟɦ ɢɡ ɦɚɥɨɣ ɦɚɬɪɢɰɵ Ɉɞɧɚɤɨ ɷɬɨ ɡɧɚɱɟɧɢɟ k0 ɜɫɟ ɠɟ ɩɪɢɛɥɢɠɟɧɧɨɟ ɜ ɫɢɥɭ ɬɨɝɨ ɱɬɨ ɦɚɬɪɢɰɚ ɧɟ ɹɜɥɹɟɬɫɹ ɩɨɥɨ ɠɢɬɟɥɶɧɨ ɨɩɪɟɞɟɥɟɧɧɨɣ ɢ ɞɪɭɝɢɟ ɤɨɦɩɥɟɤɫɧɵɟ ɫɨɛɫɬɜɟɧɧɵɟ ɱɢɫɥɚ ɜɵɯɨɞɹɬ ɡɚ ɩɪɟɞɟɥɵ ɤɪɭɝɚ ɧɚɬɹɧɭɬɨɝɨ ɧɚ [Emin ,Emax ] ɤɚɤ ɧɚ ɞɢɚ ɦɟɬɪ Ɉɩɵɬɧɵɦ ɩɭɬɟɦ ɞɥɹ ɫɪɚɜɧɢɬɟɥɶɧɨ ɦɚɥɵɯ ɦɚɬɪɢɰ ɫ n 10 ɡɧɚɱɟɧɢɟ ɨɩɬɢɦɚɥɶɧɨɝɨ ɩɚɪɚɦɟɬɪɚ ɦɨɠɧɨ ɭɬɨɱɧɢɬɶ ɞɨ k0 5.9 ɢ ɷɬɨ ɡɧɚɱɟɧɢɟ ɨɫɬɚɟɬɫɹ ɩɪɚɤɬɢɱɟɫɤɢ ɧɟɢɡɦɟɧɧɵɦ ɞɥɹ ɜɫɟɯ ɛɨɥɶɲɢɯ ɦɚɬɪɢɰ ɬɚɤɨɝɨ ɜɢɞɚ Ⱦɥɹ ɩɚɪɚɦɟɬɪɨɜ k0 4 ɢ k0 5.9 ɢ ɬɨɱɧɨɫɬɢ

ɪɟɲɟɧɢɹ H	10 5 ɩɨɥɭɱɚɟɦ ɫɨɨɬɜɟɬɫɬɜɟɧɧɨ ɱɢɫɥɨ ɬɪɟɛɭɟɦɵɯ ɢɬɟ
ɪɚɰɢɣ m	129 ɢ m 46 . ȼɩɟɱɚɬɥɹɸɳɢɣ ɪɟɡɭɥɶɬɚɬ ɞɥɹ ɞɚɧɧɨɣ ɡɚ

ɞɚɱɢ ɩɪɢɧɨɫɢɬ ɦɟɬɨɞ Ɂɟɣɞɟɥɹ-Ɉɋɉ ȿɫɥɢ ɞɥɹ ɨɛɵɱɧɨɝɨ ɦɟɬɨɞɚ Ɂɟɣɞɟɥɹ ɱɢɫɥɨ ɢɬɟɪɚɰɢɣ m 190 ɬɨ ɫ ɩɪɢɦɟɧɟɧɢɟɦ Ɉɋɉ ɩɪɢ k0 0.17 ɱɢɫɥɨ ɬɪɟɛɭɟɦɵɯ ɢɬɟɪɚɰɢɣ ɫɧɢɠɚɟɬɫɹ ɞɨ m 6 !

Ʉɨɧɟɱɧɨ ɡɚɞɚɱɚ ɨɩɪɟɞɟɥɟɧɢɹ ɫɩɟɤɬɪɚ ɦɚɬɪɢɰɵ ɜ ɨɛɳɟɦ ɫɥɭɱɚɟ ɧɢɱɟɦ ɧɟ ɩɪɨɳɟ ɡɚɞɚɱɢ ɪɟɲɟɧɢɹ ɋɅȺɍ ɩɪɹɦɵɦɢ ɦɟɬɨɞɚɦɢ Ɉɞ ɧɚɤɨ ɞɥɹ ɪɹɞɚ ɦɚɬɪɢɰ ɩɪɢɛɥɢɠɟɧɧɨɟ ɡɧɚɱɟɧɢɟ ɨɩɬɢɦɚɥɶɧɨɝɨ ɩɚ ɪɚɦɟɬɪɚ k0 ɞɥɹ ɦɟɬɨɞɚ Ɉɋɉ ɜ ɩɪɢɦɟɧɟɧɢɢ ɤ ɩɪɨɫɬɨɣ ɢɬɟɪɚɰɢɢɧɚɯɨɞɢɬɫɹ ɜɟɫɶɦɚ ɩɪɨɫɬɨ ɱɟɪɟɡ ɟɺ ɤɨɷɮɮɢɰɢɟɧ ɬɵ ɇɚɩɪɢɦɟɪ ɞɥɹ ɛɨɥɶɲɨɣ ɬɪɟɯɞɢɚɝɨɧɚɥɶɧɨɣ ɦɚɬɪɢɰɵ ɫ ɞɜɭɦɹ ɩɨɫɬɨɹɧɧɵɦɢ ɞɢɚɝɨɧɚɥɹɦɢ ɜɨɡɥɟ ɝɥɚɜɧɨɣ ɢ ɫ ɱɟɪɟɞɭɸɳɢɦɢɫɹ ɡɧɚ

ɱɟɧɢɹɦɢ a ɢ b ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɧɚ ɝɥɚɜɧɨɣ ɞɢɚɝɨɧɚɥɢ Ⱦɥɹ ɬɚɤɨɣ ɦɚɬɪɢɰɵ A ɜ ɡɧɚɱɟɧɢɟ ɨɩɬɢɦɚɥɶɧɨɝɨ ɩɚɪɚɦɟɬɪɚ ɜ ɫɪɚɜɧɨ k0 (a b)2 1 ɢ ɟɫɥɢ A – ɩɨɥɨɠɢɬɟɥɶɧɨ ɨɩɪɟɞɟɥɟɧɧɚɹ ɦɚɬɪɢɰɚ ɬɨ ɷɬɨ ɡɧɚɱɟɧɢɟ ɬɨɱɧɨɟ ɗɬɨ ɧɟ ɡɧɚɱɢɬ ɱɬɨ ɞɥɹ ɥɸɛɨɣ ɦɚɬɪɢɰɵ ɬɚɤɨɝɨ ɬɢɩɚ ɦɨɠɧɨ ɩɨɫɬɪɨɢɬɶ ɫɯɨɞɹɳɢɣɫɹ ɢɬɟɪɚɰɢɨɧɧɵɣ ɩɪɨɰɟɫɫ ɧɨ ɟɫɥɢ ɦɨɠɧɨ ɞɨɛɢɬɶɫɹ ɫɯɨɞɢɦɨɫɬɢ ɬɨ ɩɪɢ ɬɚɤɨɦ k k0 ɦɟɬɨɞ ɫɯɨɞɢɬɫɹ

Ʉɪɨɦɟ ɬɨɝɨ ɞɥɹ ɮɢɡɢɱɟɫɤɢɯ ɢ ɬɟɯɧɢɱɟɫɤɢɯ ɡɚɞɚɱ ɨɛɥɚɫɬɶ ɥɨ ɤɚɥɢɡɚɰɢɢ ɫɩɟɤɬɪɚ ɨɩɟɪɚɬɨɪɚ ɱɚɫɬɨ ɢɡɜɟɫɬɧɚ ɬ ɤ ɨɧɚ ɫɨɨɬɜɟɬɫɬɜɭ ɟɬ ɮɢɡɢɱɟɫɤɢ ɧɟɪɟɝɭɥɹɪɧɵɦ ɢ ɪɟɡɨɧɚɧɫɧɵɦ ɪɟɲɟɧɢɹɦ

ɉɪɟɨɛɪɚɡɨɜɚɧɢɟ ɨɩɟɪɚɬɨɪɚ ɦɨɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɜ ɭɫ ɥɨɜɢɹɯ ɧɟɩɨɥɧɨɣ ɢɧɮɨɪɦɚɰɢɢ ɨɛ ɟɝɨ ɫɩɟɤɬɪɟ Ɍɚɤ ɧɚɩɪɢɦɟɪ ɟɫɥɢ ɢɡɜɟɫɬɧɚ ɜ ɬɨɱɧɨɫɬɢ ɬɨɥɶɤɨ ɨɞɧɚ ɝɪɚɧɢɰɚ ɜɟɳɟɫɬɜɟɧɧɨɝɨ ɫɩɟɤɬɪɚ Ȼɨɥɟɟ ɨɩɪɟɞɟɥɟɧɧɨ ɩɭɫɬɶ ɢɡɜɟɫɬɧɨ ɱɬɨ ɫɨɛɫɬɜɟɧɧɵɟ ɱɢɫɥɚ EQ ɧɚ

ɯɨɞɹɬɫɹ ɧɚ ɢɧɬɟɪɜɚɥɟ > M ,m@ ɢ ɡɧɚɱɟɧɢɟ M t1 ɢɡɜɟɫɬɧɨ ɬɨɱɧɨ ɚ ɞɥɹ m ɢɡɜɟɫɬɧɨ ɥɢɲɶ ɱɬɨ m 0,1 Ɍ ɤ ɞɥɹ ɞɚɧɧɨɝɨ ɫɥɭɱɚɹ U(EQ ) t1 ɬɨ ɪɹɞ ɩɪɨɫɬɨɣ ɢɬɟɪɚɰɢɢ ɪɚɫɯɨɞɢɬɫɹ ɧɨ ɜ ɫɢɥɭ ɬɨɝɨ ɱɬɨ 1 :B ɦɨɠɧɨ ɩɨɫɬɪɨɢɬɶ ɫɯɨɞɹɳɢɣɫɹ ɪɹɞ Ⱦɟɣɫɬɜɢɬɟɥɶɧɨ ɩɪɢɧɢ

ɦɚɹ k		M ɩɨɥɭɱɚɟɦ ɫɯɨɞɹɳɢɣɫɹ ɪɹɞ ɩɪɨɫɬɨɣ ɢɬɟɪɚɰɢɢ ɞɥɹ ɨɩɟ
			ª	m M º
ɪɚɬɨɪɚ B ɫɩɟɤɬɪ ɤɨɬɨɪɨɝɨ ɥɟɠɢɬ ɧɚ ɢɧɬɟɪɜɚɥɟ «0,					» ɩɪɢɱɟɦ
			¬	1 M ¼
	m M
		1 ɬ ɟ U(EQ ) 1 Ɇɨɠɧɨ ɩɨɤɚɡɚɬɶ ɬɚɤɠɟ ɱɬɨ ɜ ɭɫɥɨɜɢɹɯ
	1 M	1 ɬ ɟ U(EQ ) 1 Ɇɨɠɧɨ ɩɨɤɚɡɚɬɶ ɬɚɤɠɟ ɱɬɨ ɜ ɭɫɥɨɜɢɹɯ
	1 M
ɧɟɨɩɪɟɞɟɥɟɧɧɨɫɬɢ ɞɚɧɧɨɣ ɡɚɞɚɱɢ 0 m 1			ɥɭɱɲɢɣ ɪɟɡɭɥɶɬɚɬ ɞɚɫɬ

k M / 2.

ȿɫɥɢ ɞɚɠɟ ɩɪɢɯɨɞɢɬɫɹ ɞɟɬɚɥɶɧɨ ɢɫɫɥɟɞɨɜɚɬɶ ɫɩɟɤɬɪ ɡɚɞɚɱɢ ɞɥɹ ɩɨɫɬɪɨɟɧɢɹ ɛɵɫɬɪɨ ɫɯɨɞɹɳɟɝɨɫɹ ɢɬɟɪɚɰɢɨɧɧɨɝɨ ɩɪɨɰɟɫɫɚ ɬɨ ɨɞ ɧɚɠɞɵ ɟɝɨ ɩɨɫɬɪɨɢɜ ɦɨɠɧɨ ɡɚɬɟɦ ɦɧɨɝɨɤɪɚɬɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɞɥɹ ɪɚɫɱɟɬɨɜ ɫ ɪɚɡɥɢɱɧɵɦɢ ɢɫɬɨɱɧɢɤɚɦɢ - ɩɪɚɜɵɦɢ ɱɚɫɬɹɦɢ b .

ɉɪɟɢɦɭɳɟɫɬɜɚ ɠɟ ɛɵɫɬɪɨ ɫɯɨɞɹɳɢɯɫɹ ɢɬɟɪɚɰɢɨɧɧɵɯ ɩɪɨɰɟɫ ɫɨɜ ɩɟɪɟɞ ɩɪɹɦɵɦɢ ɦɟɬɨɞɚɦɢ ɢɡɜɟɫɬɧɵ ɗɬɨ

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