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ó¤®â«åòâ®°ÿåò 󱫮âèÿ¬ rn < r— è lim rn = r— F ’®ã¤à â ±è«ó ®ï°å¤å«åíèÿ ôóíêöèè
n’∞
f (r)
f2 (r— ’) = lim f2 (rn ) = lim f (rn ) ¤ f (r— ) = f1 (r— ).
n’∞ n’∞

Àíà«®ãè·í® ï®ê৻âàåò±ÿD ·ò® f2 (r— +) ≥ f1 (r— ). ’àêè¬ ®á°à§®¬ ¬» ﮫó·àå¬ ¤â®©E
í®å íå°àâåí±òâ®

f2 (r— +) ≥ f1 (r— ) ≥ f2 (r— ’).

’àê®å ¦å íå°àâåí±òâ® âå°í® è ï°è ïå°å¬åíå f1 (r) íà f2 (r)X

f1 (r— +) ≥ f2 (r— ) ≥ f1 (r— ’).

‘«å¤®âàòå«üí® ôóíêöèè f1 (r) è f2 (r) ±®âïà¤àþò â® â±åµ ±â®èµ ò®·êൠíåï°åE
°»âí®±òè è ®ò«è·àþò±ÿ ò®«üê® â ò®·êൠ°à§°»âàF
Çà¬åòè¬D ·ò® ﮱꮫüêó ôóíêöèè fi (r) íå óá»âàþòD ò® ®íè °à§°»âí» íå ᮫ååD
·å¬ â ±·åòí®¬ ·è±«å ò®·åêD ±«å¤®âàòå«üí® ®áå ôóíêöèè fi (r)D i = 0, 1D ®¤í®§íà·í®
®ï°å¤å«åí» ï®·òè â±þ¤ó íà „¦F
„®êà§àòå«ü±òâ® óòâå°¦¤åíèÿ 2.2.5. Ï°å¤ï®«®¦è¬D ·ò® x ó¤®â«åòâ®°ÿåò
󱫮âèþ «å¬¬»X x ∈ [f (ˆ’), f (ˆ+)]F ‚®§ü¬å¬ «þá®å r < rF Ï®±ê®«üêó ï® Ë嬬å
r r ˆ
PFPFI ôóíêöèÿ f (r) ÿâ«ÿåò±ÿ íåóá»âàþùå©D ò® f (r) ¤ f (ˆ’) ¤ xD è òFêF cx (r, x)
r
VU
óá»âàåò ï® rD â»ï®«íåí®
x

c(ˆ, x) ’ c(ˆ, f (r)) =
r r cx (ˆ, y) dy
r
f (r)
x

cx (r, y) dy = c(r, x) ’ c(r, f (r)).
<
f (r)

‘«å¤®âàòå«üí®D âå°íà öåï®·êà íå°àâåí±òâ
@PQA σ(f (ˆ)) ’ c(ˆ, f (ˆ)) + c(ˆ, x)
r rr r
¤ σ(f (r)) ’ c(ˆ, f (r)) + c(ˆ, x)
r r
< σ(f (r)) ’ c(r, f (r)) + c(r, x).
Àíà«®ãè·í® ¬®¦í® ï®êà§àòüD ·ò® ∀ r > r òàê¦å â»ï®«íÿåò±ÿ öåï®·êà íå°àE
ˆ
âåí±òâ @PQAD òFåF @PQA âå°í® ¤«ÿ «þá®ã® rF
‘«å¤®âàòå«üí®D ±ï°àâ夫èâ®
σ (x) = inf (σ(f (r))) ’ c(r, f (r)) + c(r, x)
˜
r∈„¦

σ(f (ˆ)) ’ c(ˆ, f (ˆ)) + c(ˆ, x),
r rr r
=
·ò® è ò°åá®âà«®±ü ¤®êà§àòüF
„®êà§àòå«ü±òâ® óòâå°¦¤åíèÿ 2.2.6. Ï®±ê®«üêó ¦(·) " íåóá»âàþùè© ®ïåE
°àò®°D ò®
σ (x) = inf (σ(f (r)) ’ c(r, f (r)) + c(r, x))
˜
r∈„¦

= inf (ˆ (f (r)) ’ c(r, f (r)) + c(r, x))
σ
r∈„¦

= ¦(ˆ )(x) ≥ σ (x),
σ ˆ
·ò® è ò°åá®âà«®±ü ¤®êà§àòüF
„®êà§àòå«ü±òâ® «å¬¬» 2.2.7. ‚®§ü¬å¬ «þáóþ ï®±«å¤®âàòå«üí®±òü rn òàE
êóþD ·ò®
@PRA lim (σ(f (rn )) ’ c(rn , f (rn )) + c(rn , x)) = σ (x)
˜
n’∞

@ýò® ¬®¦í® ±¤å«àòü â ±è«ó ®ï°å¤å«åíèÿ σ (x)AF ‚»áå°å¬ è§ ýò®© ï®±«å¤®âàòå«üí®E
˜
±òè ±µ®¤ÿùóþ±ÿ ﮤﮱ«å¤®âàòå«üí®±òü @ýò® ¬®¦í® ±¤å«àòü òFêF „¦ ±®¤å°¦èò±ÿ â
ꮬïêàêòå " ®ò°å§êå ¤å©±òâèòå«üí®© ï°ÿ¬®©A è â ¤à«üíå©øå¬ áó¤åò °à±±¬àò°èE
âàòü ýòó í®âóþ ï®±«å¤®âàòå«üí®±òüF ’àêè¬ ®á°à§®¬ ¬®¦í® ±·èòàòüD ·ò® ±óùå±òâóE
åò òàê®å r— D ·ò®
lim rn = r— .
n’∞
‘«å¤®âàòå«üí®D â ±è«ó íåï°å°»âí®±òè c(r, x) ï® ïå°â®¬ó à°ãó¬åíòóD è§ @PRA
è¬åå¬X
@PSA lim (σ(f (rn )) ’ c(rn , f (rn ))) = σ (x) ’ c(r— , x).
˜
n’∞

Í® â ±è«ó ®ï°å¤å«åíèÿ ôóíêöèè f (r) ¬» è¬åå¬
@PTA σ (f (r— )) ’ c(r— , f (r— )) ≥ σ (x) ’ c(r— , x).
˜ ˜
VV
Ê°®¬å ò®ã®D
@PUA lim (σ(f (rn )) ’ c(rn , f (rn )))
n’∞
≥ lim (σ(f (r— )) ’ c(rn , f (r— )))
n’∞
= σ(f (r— )) ’ c(r— , f (r— ))
= σ (f (r— )) ’ c(r— , f (r— )).
˜
Îáúå¤èíÿÿ òåïå°ü @PSAE@PUAD ﮫó·àå¬X
@PVA σ (x) ’ c(r— , x) = σ(f (r— )) ’ c(r— , f (r— )),
˜
òFåF â»á°àíí»© íà¬è r— ó¤®â«åòâ®°ÿåò óòâå°¦¤åíèþ «å¬¬»D ·ò® è ò°åá®âà«®±ü
¤®êà§àòüF
„®êà§àòå«ü±òâ® óòâå°¦¤åíèÿ 2.2.8. Ðà±±¬®ò°è¬ ¤âå °à§«è·í»µ ôóíêöèè
fi (r)D i = 1, 2D ê®ò®°»å ï°è⮤ÿò ê °à§«è·í»¬ ôóíêöèÿ¬
@PWA σi (x) ≡ min(σ(fi (r)) ’ c(r, fi (r)) + c(r, x))
˜
r∈„¦

è ïó±òü σ1 (x) è σ2 (x) °à§«è·àþò±ÿD òFåF ±óùå±òâóåò x— òàê®åD ·ò®
˜ ˜
σ1 (x— ) > σ2 (x— ).
˜ ˜
Ïó±òü ±®®òâåò±òâóþùèå ¬èíè¬ó¬» ¤®±òèãàþò±ÿ â ò®·êൠr1 è r2 ±®®òâåò±òâåíE
— —

í®D òFåF
@QHA σ(f1 (r1 )) ’ c(r1 , f1 (r1 )) + c(r1 , x— ) = σ1 (x— )
— — — —
˜
> σ2 (x— ) = σ(f2 (r2 )) ’ c(r2 , f2 (r2 )) + c(r2 , x— ).
— — — —
˜
‚ ±è«ó @PWA è ®ï°å¤å«åíèÿ r1 ¤®«¦í® â»ï®«íÿòü±ÿ



@QIA σ(f1 (r2 )) ’ c(r2 , f1 (r2 )) + c(r2 , x— ) ≥ σ1 (x— )
— — — —
˜
= σ(f1 (r1 )) ’ c(r1 , f1 (r1 )) + c(r1 , x— ).
— — — —


Îáúå¤èíÿÿ @QHA è @QIAD ﮫó·àå¬X
@QPA — — — — — —
σ(f1 (r2 )) ’ c(r2 , f1 (r2 )) > σ(f2 (r2 )) ’ c(r2 , f2 (r2 )).
Τíàê® â ±è«ó ®ï°å¤å«åíèÿ
f2 (r2 ) ∈ Argmax(σ(x) ’ c(r2 , x)),
x∈X
èD ±«å¤®âàòå«üí®D ¤®«¦í® â»ï®«íÿòü±ÿ
— — — —
σ(f (r2 )) ’ c(r2 , f (r2 )) ≥ σ(f (r1 )) ’ c(r2 , f (r1 )),
·ò®D ®¤íàê®D ï°®òèâ®°å·èò @QPAD ±«å¤®âàòå«üí® íàøå ï°å¤ï®«®¦åíèå ® ò®¬D ·ò®
σ1 (x) è σ2 (x) °à§«è·àþò±ÿD íåâå°í®D ·ò® è ¤®ê৻âàåò óòâå°¦¤åíèå óòâå°¦¤åíèÿF
˜ ˜

„®êà§àòå«ü±òâ® «å¬¬» 2.2.9. ‚ ±è«ó ®ï°å¤å«åíèÿ ôóíêöèè f (r) è ò®ã®D ·ò®
σ(f (r)) = σ (f (r))D è¬åå¬X
˜
@QQA σ (f (r1 )) ’ c(r1 , f (r1 )) ≥ σ (x2 ) ’ c(r1 , x2 );
˜ ˜

@QRA σ (f (r2 )) ’ c(r2 , f (r2 )) ≥ σ (x1 ) ’ c(r2 , x1 ).
˜ ˜
VW
‘ê«à¤»âàÿ @QQA è @QRA ± @PFUA ï°è i = 1, 2 è óï°®ùàÿ ﮫó·åíí®å íå°àâåí±òâ®D
ﮫó·àå¬X


@QSA c(r1 , x2 ) + c(r2 , x1 ) ≥ c(r1 , x1 ) + c(r2 , x2 );


x2 r 2

@QTA crx (r, x) dr dx ¤ 0.
x1 r 1


Í®D ﮱꮫüêó crx (r, x) < 0 è x1 < x2 D ï®±«å¤íåå íå°àâåí±òâ® ¬®¦åò â»ï®«E
íÿòü±ÿ ò®«üê® ï°è r2 ≥ r1 @§à¬åòè¬D ·ò® ®íè ¬®ãóò ±®âïà¤àòüAD ·ò® è ò°åá®âà«®±ü
¤®êà§àòüF
„®êà§àòå«ü±òâ® óòâå°¦¤åíèÿ 2.3.1. Ïó±òü σ(x) " ®ïòè¬à«üíàÿ ôóíêöèÿ
±òè¬ó«è°®âàíèÿD ï°è·å¬ xk å±òü ¤å©±òâèåD °åà«è§ó嬮å kE¬ àêòèâí»¬ ý«å¬åíò®¬D
à σ (x) " íàè«ó·øàÿ â ±¬»±«å §àò°àò ôóíêöèÿ ±òè¬ó«è°®âàíèÿD ï°è ê®ò®°®© àêE
˜
òèâí»© ý«å¬åíò òèïà rk °åà«è§óåò ¤å©±òâèå xk D ã¤å
˜
±
xk , k = i, i + 1;


@QUA x ’ ∆, k = i;
xk =
˜
k
 x + ∆, k = i + 1
k



ï°è ¬à«»µ §íà·åíèÿµ ∆ @§à¬åòè¬D ·ò® ï°è ¬à«»µ ∆ ï® ó±«®âèþ «å¬¬» xk è xk
˜
óï®°ÿ¤®·åí» ®¤èíàê®â»¬ ®á°à§®¬AF
‚ ±è«ó ò®ã®D ·ò® σ(·) ®ïòè¬à«üíàD ¤®«¦í® â»ï®«íÿòü±ÿ °àâåí±òâ®

n n
1
@QVA σ(xk ) ’ σ (˜k )
˜x
lim =0
∆’0 ∆
k=1 k=1


@屫è á» íå ừ® °àâåí±òâà íó«þD ò® ừ® Ỡ⮧¬®¦í® ó«ó·øåíèå σ(x)D ê®ò®°àÿD
®¤íàê®D ừà â»á°àíà ®ïòè¬à«üí®©AF
Í®D è±ï®«ü§óÿ @PFIQA è @QUAD è¬åå¬X

σ (˜i ) = σ(xi ) ’ (ci (xi ) ’ ci (xi ’ ∆));
˜x
σ (˜i+1 ) = σ(xi ) ’ (ci (xi ) ’ ci (xi ’ ∆))+
˜x
+ci+1 (xi+1 + ∆) ’ ci+1 (xi ’ ∆);
σ (˜k ) = σ(xk ) ï°è k ¤ i ’ 1;
˜x
σ (˜k ) = σ(xk ) ’ δ ï°è k > i + 1,
˜x

ã¤å


@QWA δ = σ(xi+1 ) + ci+2 (xi+1 + ∆) ’ ci+2 (xi+1 ) ’
σ(xi ) ’ (ci (xi ) ’ ci (xi ’ ∆)) + ci+1 (xi+1 + ∆) ’ ci+1 (xi ’ ∆) .
WH
’àêè¬ ®á°à§®¬D
n n
1
σ(xk ) ’ σ (˜k )
˜x
lim =
∆’0 ∆
k=1 k=1
n i’1
1
σ(xk ) ’ σ(xk ) + {σ(xi ) ’ (ci (xi ) ’ ci (xi ’ ∆))} +
lim
∆’0 ∆
k=1 k=1
{σ(xi ) ’ (ci (xi ) ’ ci (xi ’ ∆)) + ci+1 (xi+1 + ∆) ’ ci+1 (xi ’ ∆)} +
n
(σ(xk ) ’ δ) =
k=i+2
1
ci+1 (xi+1 ) ’ ci+1 (xi ) + (ci (xi ) ’ ci (xi ’ ∆)) +
lim
∆’0 ∆
((ci (xi ) ’ ci (xi ’ ∆)) ’ ci+1 (xi+1 + ∆) + ci+1 (xi ’ ∆)) +
(n ’ i ’ 1)(ci+1 (xi+1 ) ’ ci+1 (xi ) + ci+2 (xi+1 + ∆) ’ ci+2 (xi+1 ) ’
(’(ci (xi ) ’ ci (xi ’ ∆)) + ci+1 (xi+1 + ∆) ’ ci+1 (xi ’ ∆)) =
(n ’ i ’ 1)(ci+2 (xi+1 ) ’ ci+1 (xi+1 )) ’ ci+1 (xi+1 ) ’
(n ’ i ’ 1)(ci+1 (xi ) ’ ci (xi )) ’ ci+1 (xi ) + 2ci (xi )
èD ó·èò»âàÿ @QVAD ﮫó·àå¬X
(n ’ i ’ 1)(ci+2 (xi+1 ) ’ ci+1 (xi+1 )) ’ ci+1 (xi+1 ) =
(n ’ i)(ci+1 (xi ) ’ ci (xi )) ’ ci (xi ),
·ò® è ò°åá®âà«®±ü ¤®êà§àòüF
„®êà§àòå«ü±òâ® óòâå°¦¤åíèÿ 2.3.2. „àííàÿ «å¬¬à ÿâ«ÿåò±ÿ ±«å¤±òâèå¬
¬í®ã®ê°àòí®ã® ï°è¬åíåíèÿ Ë嬬» PFQFIF
„®êà§àòå«ü±òâ® óòâå°¦¤åíèÿ 2.3.3. Ï°å¤ï®«®¦è¬ ï°®òèâí®åD òFåF ·ò® ±óE
ùå±òâóåò òàê®å l ≥ 1D ·ò® x—
n’l’1 < xn’l = xn’l+1 = . . . = xn @êàê â±åã¤àD ï°å¤ï®«àE
— — —

ãàå¬ x— = 0AF ‚®§ü¬å¬ òàê®å ¬à«®å ∆ > 0D ·ò® x— ’ ∆ > x— n’l’1 è ®ï°å¤å«è¬
0 n’l
±
x— ï°è i < n ’ l;
i


@RHA x ’ ∆ ï°è n ’ l ¤ i < n;

xi =
 —i
 x + l∆ ï°è i = n.
i

‚ ±è«ó ï®±ò°®åíèÿ â»ï®«íÿåò±ÿ °àâåí±òâ®
n n
@RIA x— xi ,
=
i
i=1 i=1

òFåF ®áå ±è±ò嬻 ±òè¬ó«è°®âàíèÿ °åà«è§óþò ®¤í® è ò® ¦å ±ó¬¬à°í®å @è ±°å¤íååA
¤å©±òâèåF
Ïó±òü σ (x) " ôóíêöèÿ ±òè¬ó«è°®âàíèÿD ï®±ò°®åííàÿ ï® ô®°¬ó«å @PFIWA ¤«ÿ
˜
±è±ò嬻 {xi }n F ’®ã¤à â ±è«ó ï®±ò°®åíèÿ
i=1
±
σ(x— ) ï°è i < n ’ l;
i


σ(xn ) ’ δ ï°è n ’ l ¤ i < n;



σ (xi ) =
˜
 σ(x— ) ’ δ + (cn (x— + l∆)’
n n

cn (x— ’ ∆)) ï°è i = n,


n

WI
ã¤å δ = (cn’l (x— ) ’ cn’l (x— ’ ∆)). Ì» §¤å±ü ÿâí® è±ï®«ü§®âà«è ò®ò ôàêòD ·ò®
n n
xn’l = xn’l+1 = . . . = xn F
— — —

Ðà±ïè±»âàÿ °à§í®±òü ¬å¦¤ó ±ó¬¬à°í»¬è §àò°àòà¬è íà â»øå ®ï°å¤å«åíí»µ
±è±òå¬àµ ±òè¬ó«è°®âàíèÿD ﮫó·àå¬
n n
σ(x— ) σ (xn ) = (l + 1)δ ’ (cn (x— + l∆) ’ cn (x— ’ ∆)) =
’ ˜
i n i
i=1 i=1
(l + 1)(cn’l (x— ) ’ cn’l (x— ’ ∆)) ’ (cn (x— + l∆) ’ cn (x— ’ ∆)) =
n n n n

(l + 1)cn’l (x— )∆ + o(∆) ’ ((l + 1)cn (x— )∆ + o(∆)) =
n n

(l + 1)(cn’l (x— ) ’ cn (x— ))∆ + o(∆).
n n


Í®D ﮱꮫüêó ï® ó±«®âèþ íà ôóíêöèè §àò°àò cn’l (x— ) > cn (x— )D ï°è ¤®±òàò®·í®
n n
¬à«»µ §íà·åíèÿµ ∆ áó¤åò â»ï®«íÿòü±ÿ
n n
@RPA σ(x— ) > σ (xi ),
˜
i
i=1 i=1

ò® å±òü íଠó¤à«®±ü ó¬åíüøèòü ±ó¬¬à°í®å ±òè¬ó«è°®âàíèÿ ï°è íå觬åíí®¬ ±ó¬E
¬à°í®¬ ¤å©±òâèèD ·åã® íå ¬®¦åò á»òü â ±âÿ§è ± ®ïòè¬à«üí®±òüþ ôóíêöèè σ(x)F
„®êà§àòå«ü±òâ® óòâå°¦¤åíèÿ 2.3.4. Ï°å¤ï®«®¦è¬ ï°®òèâí®åD òFåF ·ò® ±óE
ùå±òâóåò òàê®å l ≥ 1D ·ò® 0 = x— = x— = . . . = x— < x— F ‚®§ü¬å¬ ¬à«®å ∆ > 0 è
1 2 l l+1
®ï°å¤å«è¬ ±è±òå¬ó ¤å©±òâè© {xi }i=1 nX
±
∆ ï°è i ¤ l;


@RQA x— ’ l∆ ï°è i = l + 1;
xi =
 l+1
x— ï°è i > l + 1.

i


‚ ±è«ó ï®±ò°®åíèÿ â»ï®«íÿåò±ÿ °àâåí±òâ®
n n
@RRA x— = xi ,
i
i=1 i=1


òFåF ®áå ±è±ò嬻 ±òè¬ó«è°®âàíèÿ °åà«è§óþò ®¤í® è ò® ¦å ±ó¬¬à°í®å @è ±°å¤íååA
¤å©±òâèåF
Ïó±òü σ (x) " ôóíêöèÿ ±òè¬ó«è°®âàíèÿD ï®±ò°®åííàÿ ï® ô®°¬ó«å @PFIWA ¤«ÿ
˜
±è±ò嬻 {xi }n F ’®ã¤à â ±è«ó ï®±ò°®åíèÿ
i=1
±
σ(x— ) + c1 (∆) ï°è i ¤ l;
i


 σ(x— ) + c (x— ’ l∆)’

l+1 l+1
i
σ (xi ) =
˜
cl+1 (∆) + c1 (∆) ’ cl+1 (x— ) ï°è i = l + 1;
l+1


σ(xi ) + δ ï°è i > l + 1,




ã¤å

@RSA δ = cl+2 (x— ) ’ cl+2 (x— ’ l∆) + cl+1 (x— ’ l∆) ’ cl+1 (∆) +
l+1 l+1 l+1

c1 (∆) ’ cl+1 (x— ) = l∆(cl+l (x— ) ’ cl+2 (x— )) + o(∆)
l+1 l+1 l+1
WP
Ðà±ïè±»âàÿ °à§í®±òü ¬å¦¤ó ±ó¬¬à°í»¬è §àò°àòà¬è íà â»øå ®ï°å¤å«åíí»µ
±è±òå¬àµ ±òè¬ó«è°®âàíèÿD ﮫó·àå¬
n n
@RTA σ(x— ) ’ σ (xn ) =
˜
i

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