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r2 r 3
’àêè¬ ®á°à§®¬D ¬» íàø«è 󱫮âèå íà r1 è r2 D ï°è ê®ò®°®¬ @êàê ﮫó·àåò±ÿ è§
í੤åíí®ã® â»°à¦åíèÿ ï°è «þᮬ xA x1 = x2 F
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Çà¬åòè¬D ·ò® ï°è r1 = 1 󱫮âèå â»ã«ÿ¤å«® á» ±«å¤óþùè¬ ®á°à§®¬X
3 4 1
’+ ≥ 0.•
r 1 r2 r 3
ȱ±«å¤óå¬ â®ï°®± ® ò®¬D â êàêèµ ±«ó·àÿµ ¬®¦í® ãà°àíòè°®âàòü °åà«è§àöèþ
àêòèâí»¬è ý«å¬åíòà¬è °à§«è·í»µ ¤å©±òâè©F „«ÿ ýò®ã® ââå¤å¬ ôóíêöèè xk (˜n )
˜x
êàê °åøåíèÿ ó°àâíåíèÿ
@PFRQA (n ’ k + 1)ck (˜k ) ’ (n ’ k)ck+1 (˜k ) = cn (˜n ).
x x x
‚±å âà¦í»å ¤«ÿ íà± ±â®©±òâà ôóíêöè© xk (˜n ) ®ïè±»âàþò±ÿ ±«å¤óþùå© «å¬¬®©X
˜x
Ë嬬à 2.3.6. ”óíêöèÿ §àâè±è¬®±òè ¤å©±òâèÿ ï°®è§â®«üí®ã® ÀÝ ®ò ¤å©±òâèÿ
íàè«ó·øåã® ÀÝ ®ï°å¤å«åíà å¤èí±òâåíí»¬ ®á°à§®¬ ï°è ¤àíí®¬ ±®±òàâå À‘ è ±ò°®E
㮠⮧°à±òàåòF
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ÀÝ â®§°à±òàþòF Ê°®¬å ò®ã®D ï°è «þᮬ ¤å©±òâèè íàè«ó·øåã® ÀÝ °åøåíèå §à¤à·è
® ¤å©±òâèÿµ ¤°óãèµ ÀÝ å¤èí±òâåíí®F
‘«å¤óþùàÿ «å¬¬à ®ï°å¤å«ÿåòD â êàêèµ ±«ó·àÿµ ¬®¦í® ã®â®°èòü ® ò®¬D ·ò®
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¤åíí»µ ôóíêöè© xi (xn )F
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Ë嬬à 2.3.7. ‚ ±«ó·àå ±ò°®ã®© óï®°ÿ¤®·åíí®±òè °åà«è§ó嬻µ ¤å©±òâè© @â»E
·è±«åíí»µ ï® ó°àâíåíèþ @PFRQAA ¤âà °à§«è·í»µ ÀÝ íå áó¤óò °åà«è§®â»âàòü ®¤í®
è ò® ¦å ¤å©±òâèåF
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±òàò®·í»µ 󱫮âè© â»ï®«íåíèÿ ï°å¤ï®«®¦åíè© «å¬¬» «å¬¬ PFQFWF
Ë嬬à 2.3.8. „«ÿ â»ï®«íåíèÿ 󱫮âèÿ
ci (x) ’ ci+1 (x) > ci+1 (x) ’ ci+2 (x) ∀x ∈ X
RW
@èD êàê ±«å¤±òâèåD Ë嬬» PFQFWA ¤®±òàò®·í®D ·ò®á» ±óùå±òâ®âà«® òàê®å ∆ > 0D ·ò®

è
ri+1 ’ ri = ∆ ∀i = 1 · · · n ’ 1
∀r ∈ „¦, x ∈ X.
cxrr (r, x) < 0

’åïå°ü íà©¤å¬ íå®áµ®¤è¬»å 󱫮âèÿ ¤«ÿ ò®ã®D ·ò®á» °åà«è§óå¬»å ¤å©±òâèÿ
á»«è ±ò°®ã® óï®°ÿ¤®·åí»F
Ë嬬à 2.3.9. „®±òàò®·í»¬ 󱫮âèå¬ ¤«ÿ ò®ã®D ·ò®á» °à§í»å ÀÝ °åà«è§®â»E
âà«è °à§í»å ¤å©±òâèÿD ÿâ«ÿåò±ÿ â»ï®«íåíèå 󱫮âèÿ

@PFRRA ci (x) ’ ci+1 (x) > ci+1 (x) ’ ci+2 (x).

¤«ÿ «þỵ x ∈ X è i = 1, n ’ 2.
‚±ï®¬íè¬ òåïå°üD ¤«ÿ «þᮩ À‘ ¬» ®ï°å¤å«ÿ«è ±ó¬¬à°í»å §àò°àò» S(·) öåíE
ò°à íà °åà«è§àöèþ íåê®ò®°®ã® àã°åãè°®âàíí®ã® ¤å©±òâèÿF ηåâè¤í®D ·ò® â ±«ó·àå
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ãèèD íå ï°èáåãàÿ ê ±¬åøàíí»¬ @±¬åøàííàÿ ±ò°àòåãèÿ " ê®ã¤à öåíò° íà§íà·àåò
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íèå¬AF ‘«å¤óþùàÿ «å¬¬à ï°è⮤èò ¤®±òàò®·í»å 󱫮âèÿ ¤«ÿ ò®ã®D ·ò®á» §àò°àò»
ừè â»ïóê«®© ôóíêöèå©F
Ë嬬à 2.3.10. ‚»ï®«íåíèÿ 󱫮âèÿ

@PFRSA ci (x) ’ ci+1 (x) > ci+1 (x) ’ ci+2 (x).

¤«ÿ «þỵ x ∈ X è i = 1, n ’ 2 ¤®±òàò®·í® ¤«ÿ ò®ã®D ·ò®á» ±ó¬¬à°í»å §àò°àò»
öåíò°à ừè â»ïóê«®© ôóíêöèå©D è«èD ·ò® ò® ¦å ±à¬®åD öåíò°ó ừ® íåâ»ã®¤í®
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2.4. “íèôèöè°®âàíí»å ±è±ò嬻 ±òè¬ó«è°®âàíèÿ â àêòèâí»µ ±è±òå¬àµ
± áå±ê®íå·í»¬ ·è±«®¬ àêòèâí»µ ý«å¬åíò®â
‚ ¤àíí®¬ °à§¤å«å °à±±¬àò°èâàåò±ÿ §à¤à·à í൮¦¤åíèÿ ®ïòè¬à«üí®© ôóíêöèè
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À‘ ± ê®íå·í»¬ ·è±«®¬ ÀÝF Ðåøåíèå ¤àíí®© §à¤à·è íå®áµ®¤è¬® ¤«ÿ °åøåíèÿ §àE
¤à·è óï°àâ«åíèÿ ±®±òà⮬ À‘F
SH
Ïó±òü òèï àêòèâí®ã® ý«å¬åíòà è¬ååò íåê®ò®°®å íåï°å°»âí®å °à±ï°å¤å«åíèåD
±®±°å¤®ò®·åíí®å íà ®ò°å§êå ¤å©±òâèòå«üí®© ®±è „¦ = [r0 , r1 ] ± ôóíêöèå© °à±ï°åE
¤å«åíèÿ F (r)D ±ò°®ã® ⮧°à±òàþùå© è ¤èôôå°åíöè°ó嬮© íà „¦F ’®ã¤à r— = F (r)
áó¤åò è¬åòü °àâí®¬å°í®å °à±ï°å¤å«åíèå íà ®ò°å§êå [0, 1] òFêF

P {r— ¤ x} = P {F (r) ¤ x} = P {r ¤ F ’1 (x)} =
= F (F ’1 (x)) = x, x ∈ [0, 1].
Îï°å¤å«è¬ ôóíêöèþ §àò°àò c(r— , x) = c(F ’1 (r— ), x) @¬» òàêè¬ ®á°à§®¬
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4ïå°å®ï°å¤å«è«è4 òèï àêòèâí®ã® ý«å¬åíòàAF ’®ã¤à ï® Ë嬬å PFIFP è ôóíêöèÿ
c(r— , x) ®á«à¤àåò â±å¬è ±â®©±òâà¬èD ê®ò®°»¬ ¤®«¦íà ó¤®â«åòâ®°ÿòü ôóíêöèÿ §àE
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Ê°®¬å ò®ã® ®·åâè¤í®D ·ò® ï°®è§âå¤ÿ ®á°àòíóþ §à¬åíó òèï®â â ®ïòè¬à«üí®© ±èE
±òå¬å ±òè¬ó«è°®âàíèÿ ¤«ÿ À‘ ± ôóíêöèå© §àò°àò c(r— , x)D ¬» ﮫó·è¬ ®ïòè¬à«üE
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±®µ°àíÿò±ÿF
‘«å¤®âàòå«üí® ¬» ï®êà§à«è âå°í®±òü ±«å¤óþùåã® óòâå°¦¤åíèÿX
“òâå°¦¤åíèå 2.4.1. Çà¤à·à í൮¦¤åíèÿ ®ïòè¬à«üí®© ôóíêöèè ±òè¬ó«è°®E
âàíèÿ ¤«ÿ ±«ó·àÿ ï°®è§â®«üí®ã® °à±ï°å¤å«åíèÿ òèï®â ÀÝ ±â®¤èò±ÿ ê §à¤à·å íàE
µ®¦¤åíèÿ ®ïòè¬à«üí®© ôóíêöèè ±òè¬ó«è°®âàíèÿ ¤«ÿ ±«ó·àÿ °àâí®¬å°í®ã® °à±E
ï°å¤å«åíèÿ òèï®â ÀÝF À è¬åíí®D ±óùå±òâóåò òàêàÿ ¬®í®ò®ííàÿ ôóíêöèÿ ï°å®á°àE
§®âàíèÿ òèï®â r = u(r)D ·ò® òèï» ÀÝ r è¬åþò °àâí®¬å°í®å °à±ï°å¤å«åíèåD è
®ïòè¬à«üíàÿ ôóíêöèÿ ±òè¬ó«è°®âàíèÿ ï°è òèïൠr ÿâ«ÿåò±ÿ ®ïòè¬à«üí®© ôóíêE
öèå© ±òè¬ó«è°®âàíèÿ ï°è òèïൠrF
ÈòàêD íà ®±í®âàíèè ï°å¤»¤óùåã® óòâå°¦¤åíèÿ ¬» áó¤å¬ â ¤à«üíå©øå¬ ï®«àE
ãàòüD·ò® òèï» àêòèâí»µ ý«å¬åíò®â è¬åþò ®¤èíàê®â®å ¤«ÿ â±åµ ý«å¬åíò®â °àâí®E
¬å°í®å °à±ï°å¤å«åíèå íà ¬í®¦å±òâå „¦ = [r0 , r1 ]X
r — ’r0
ï°è r ∈ „¦;
— r1 ’r0
P {r ∈ [r0 , r ]} =
èíà·å.
0
Êত»© è§ àêòèâí»µ ý«å¬åíò®â §íàåò ±â®© òèï â ò® â°å¬ÿD êàê öåíò°ó òèï»
è¬åþùèµ±ÿ ý«å¬åíò®â íåè§âå±òí»D è ®í â»íó¦¤åí °óê®â®¤±òâ®âàòü±ÿ âå°®ÿòí®±òE
í»¬ °à±ï°å¤å«åíèå¬F
‚ ±®®òâåò±òâèè ± @IFRRAE@IFRTA ¬àòå¬àòè·å±êè §à¤à·à â»ïè±»âàåò±ÿ â âè¤å ¤âóµ
§à¤à·X
@PFRTA ! §à¤à·à öåíò°à
E σ(f (r)) ’ min
f (r),σ(x)∈˜


@PFRUA f (r) ∈ Argmax(σ(x) ’ c(r, x)) ∀r ∈ „¦ ! §F ÀÝ
x


@PFRVA E f (r) = x @ã°àíè·í®å 󱫮âèåA,
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ã¤å E ®á®§íà·àåò ¬àòå¬àòè·å±ê®å ®¦è¤àíèå ï® òèïó àêòèâí®ã® ý«å¬åíòàF
Ì» áó¤å¬ °åøàòü §à¤à·ó @PFRUAE@PFRVAD ±·èòàÿD ·ò® §à¤à·à @PFRTA ¬àê±è¬è§àöèè
ï°èừè öåíò°®¬ ï°è °åøåíí®© §à¤à·å àãåíòà ÿâ«ÿåò±ÿ ò°èâèà«üí®© §à¤à·å© íà
¬àê±è¬è§àöèþ ôóíêöèè ®¤í®ã® ïå°å¬åíí®ã® @±°å¤íåã® ¤å©±òâèÿAF
SI
Ì®¦í® ừ® á» ï°å¤ï®«®¦èòüD ·ò® ï® ôóíêöèè f (x) ¬®¦í® â®±±òàí®âèòü íàèE
«ó·øóþ σ(x)D ï°è ê®ò®°®© ﮫó·àåò±ÿ ¤àííàÿ ôóíêöèÿ ¤å©±òâèÿF „®êà¦å¬ ýò®ò
ôàêòF „«ÿ ýò®ã® â±ï®¬íè¬D ·ò® ¤®«¦í® â»ï®«íÿòü±ÿ @PFRUAD è«è
σ(f (r— )) ’ c(r— , f (r— )) ≥ σ(f (r)) ’ c(r— , f (r))
@PFRWA
∀ r, r— ∈ „¦;

@PFSHA σ(f (r— )) ’ σ(f (r)) ≥ c(r— , f (r— )) ’ c(r— , f (r)).
Ïó±òü r— > rF Ð৤å«è¬ ®áå ·à±òè íå°àâåí±òâà @PFSHA íà ﮫ®¦èòå«üí®å r— ’ r è
⮧ü¬å¬ ï°å¤å« ï°è r ’ r— F Ï®«ó·è¬X
σ(f (r— )) ’ σ(f (r))
— —

σ (f (r ))f (r ) = lim—
r— ’ r
r’r

c(r— , f (r— )) ’ c(r— , f (r))
= cx (r— , f (r— ))f (r— ),
≥ lim—
r— ’ r
r’r

è«è ó·èò»âàÿD ·ò® f (r) > 0D ﮫó·àå¬
@PFSIA f (r— )σ (f (r— )) ≥ f (r— )cx (r— , f (r— )).
Ï°å¤ï®«®¦èâ r— < r è ï°®¤å«àâ òå ¦å â»ê«à¤êèD ¬» ﮫó·è¬ ®á°àòí®å íå°àE
âåí±òâ®D òFåF
@PFSPA f (r— )σ (f (r— )) ¤ f (r— )cx (r— , f (r— )),
·ò® â¬å±òå ± @PFSIA ï°è⮤èò ê °àâåí±òâó
@PFSQA f (r)σ (f (r)) = f (r)cx (r, f (r))
@â ±«ó·àå ±óùå±òâ®âàíèÿ ±®®òâåò±òâóþùèµ ï°®è§â®¤í»µAF
Ï°è â»ï®«íåíèè ¤®ï®«íèòå«üí®ã® 󱫮âèÿ
@PFSRA min(σ(f (r)) ’ c(r, f (r))) = 0
r

@ê®ò®°®å íå®áµ®¤è¬® ¤«ÿ °åøåíèÿ §à¤à· PFRTEPFRVD ·ò®á» ±®®òâåò±òâóþùàÿ ôóíêE
öèÿ σ(x) á»«à ®ïòè¬à«üí®©A ¬» ﮫó·àå¬ ®¤í®§íà·í®å ®ï°å¤å«åíèå σ(x) ï® f (r)
@®·åâè¤í®D ·ò® ï® σ(x) ¬» ¬®¦å¬ è±ï®«ü§óÿ ô®°¬ó«ó @PFRUA ®ï°å¤å«èòü f (r)AF ‚
±è«ó ò®ã®D ·ò® cx (r, x) > 0D ôóíêöèÿ σ(x) ò®¦å ¤®«¦íà ¬®í®ò®íí® â®§°à±òàòü @â
ò®·êൠ⮧°à±òàíèÿ §íà·åíè© ôóíêöèè f (r)AF
Çà¬åòè¬X ﮱꮫüêó íå®áµ®¤è¬»¬ 󱫮âèå¬ ¤«ÿ ò®ã®D ·ò®á» ý«å¬åíò òèïà
r â»áè°à« ¤å©±òâèå f (r) ÿâ«ÿåò±ÿ σ (f (r)) = cx (r, f (r))D ò®D °à±±¬®ò°åâ °à§íèE
öó σ(f (r)) ’ c(r, f (r)) è ï°®¤èôôå°åíöè°®âàâ åå ï® r ﮫó·è¬D ·ò® dr (σ(f (r)) ’ d

c(r, f (r))) = σ (f (r))f (r) ’ cx (r, f (r))f (r) ’ cr (r, f (r)) = ’cr (r, f (r)) > 0D ò® å±òü
«ó·øè© ý«å¬åíò ﮫó·àåò ᮫üøóþ ï°èừü @°à§í®±òü ¬å¦¤ó ±òè¬ó«è°®âàíèå¬
è §àò°àòà¬èAF ȱµ®¤ÿ è§ ýò®ã® ¬®¦í® ±êà§àòüD ·ò® ¤«ÿ â»ï®«íåíèÿ 󱫮âèÿ @PFSRA
íå®áµ®¤è¬® è ¤®±òàò®·í®D ·ò®á» σ(f (r0 )) ’ c(r0 , f (r0 )) ≥ 0 @ﮱꮫüêó r1 > r0 AF ‚
·à±òí®±òèD ¬®¦í® ﮫ®¦èòü
@PFSSA σ(f (r0 )) = c(r0 , f (r0 )).
’àêè¬ ®á°à§®¬D σ(x) ®ï°å¤å«ÿåò±ÿ ï® f (r) â ±®®òâåò±òâèè ± ¤èôôå°åíöèà«üí»¬
ó°àâíåíèå¬ @PFSQA è ã°àíè·í»¬ 󱫮âèå¬ @PFSSAF
SP
Ï°è¬å° 6. Ïó±òü f (r) = rD c(r, x) = x2
F
’®ã¤à cx (r, x) = 2 x F Ï® ô®°¬ó«å @PFSIA
r r
f (r)
σ (f (r)) = σ (r) = cx (r, f (r)) = 2 r = 2 èD ±«å¤®âàòå«üí®D σ(x) = 2x + CF Ê®íE
±òàíòà C ﮤáè°àåò±ÿ è§ ó±«®âèÿ @PFSRAD ¤«ÿ â»ï®«íåíèÿ ê®ò®°®ã® íå®áµ®¤è¬® è
¤®±òàò®·í®D ·ò®á» ừ® â»ï®«íåí® @PFSSAD òFåF σ(f (r0 )) = c(r0 , f (r0 ))F ‚ ¤àíí®¬
±«ó·àå 2r0 + C = r0 D ò® ê®í±òàíòà C = ’r0 F•
Ðà±±¬®ò°è¬ §à¤à·óD ôèê±è°óþùóþ ±°å¤íè© â»ïó±ê íà ó°®âíå xF ’®ã¤à §à¤àE
¯
·à ® í൮¦¤åíèè ®ïòè¬à«üí®© ôóíêöèè ±òè¬ó«è°®âàíèÿ @ê®ò®°àÿ ®ï°å¤å«ÿåò±ÿ
®¤í®â°å¬åíí® ± ôóíêöèå© f (x)A â»ã«ÿ¤èò ±«å¤óþùè¬ ®á°à§®¬X


@PFSTA E σ(f (r)) ’ min
σ(x),f (r)


ï°è â»ï®«íåíèè 󱫮âè©X


@PFSUA E f (r) = x,
¯



@PFSVA σ (f (r)) = cx (r, f (r)),



@PFSWA σ(f (r0 )) = c(r0 , f (r0 )),

ã¤å E ®á®§íà·àåò ¬àòå¬àòè·å±ê®å ®¦è¤àíèåF
Çà¬åòè¬X

r1 r1
r1
@PFTHA ’
σ(f (r)) dr = rσ(f (r)) rσ (f (r))f (r) dr =
r0
r0 r0
r1
r1

= rσ(f (r)) rcx (r, f (r))f (r) dr;
r0
r0




r1

@PFTIA rcx (r, f (r))f (r) dr =
r0
r1
dc(r, f (r))
’ cr (r, f (r)) d r =
r
=
dr
r0
r1 r1
r1
’ rcr (r, f (r))d r ’
= rc(r, f (r)) c(r, f (r))d r.
r0
r0 r0


Ê°®¬å ò®ã®D
SQ
r1 r1
@PFTPA + (r1 ’ r0 )σ(f (r0 )) =
rσ(f (r)) = r1 σ(f (r))
r0 r0
r1

σ (f (r))f (r) dr + (r1 ’ r0 )c(r0 , f (r0 )) =
= r1
r0
r1
dc(r, f (r))
’ cr (r, f (r))
= r1 dr +
dr
r0

+(r1 ’ r0 )c(r0 , f (r0 )) =
r1
r1
’ r1
= r1 c(r, f (r)) cr (r, f (r)) dr +
r0
r0

+(r1 ’ r0 )c(r0 , f (r0 )) =
r1
r1
’ r1
= rc(r, f (r)) cr (r, f (r)) dr.
r0
r0

Îáúå¤èíÿÿ @PFTHAE@PFTPAD ﮫó·àå¬X
r1

@PFTQA (r1 ’ r0 ) E σ(f (r)) = rcr (r, f (r)) ’ r1 cr (r, f (r)) + c(r, f (r)) dr =
r0
r1

((r ’ r1 )cr (r, f (r)) + c(r, f (r))) dr =
=
r0
r1 r1

(r ’ r1 )cr (r, f (r)) dr.
c(r, f (r)) dr +
=
r0 r0

Ì®¦í® §à¬åòèòüD ·ò® ïå°â»© èíòåã°à« â ï®±«å¤íå¬ â»°à¦åíèè ®òâå·àåò §à
±òè¬ó«è°®âàíèå ï°è ïå°±®íèôèöè°®âàíí®© ±è±òå¬å ±òè¬ó«è°®âàíèÿ ± ôóíêöèå©
¤å©±òâèÿ f (r)D à â® âò®°®¬ èíòåã°à«å ﮤ»íòåã°à«üí®å â»°à¦åíèå â±åã¤à ᮫üøå
íó«ÿ òFêF r ¤ r1 D à cr (r, x) < 0 è °àâí® âå«è·èíå ïå°åï«àò àêòèâí»¬ ý«å¬åíòଠ§à
è±ï®«ü§®âàíèå íåóíèôèöè°®âàíí®© ±è±ò嬻 ±òè¬ó«è°®âàíèÿ ï°è ôóíêöèè ¤å©E
±òâèÿ f (r)F
Êàê ¬®¦í® §à¬åòèòüD â ¬èíè¬è§è°ó嬮¬ â»°à¦åíèè ¬» è§áàâè«è±ü ®ò ôóíêE
öèè σ(x)D è òåïå°ü ¬èíè¬è§àöèþ ¬®¦í® ﰮ⮤èòü ò®«üê® ï® ®¤í®© ôóíêöèè "
ï® f (x)F ÈòàêD íଠíम ¬èíè¬è§è°®âàòü èíòåã°à« @PFTQA ï°è ®ã°àíè·åíèè @PFSUAF
‘¤å«àå¬ ýò®D ⮱ﮫü§®âàâøè±ü ¬åò®¤®¬ ¬í®¦èòå«å© Ëàã°àí¦àX

@PFTRA L = E σ(f (r)) ’ »(E f (r) ’ x) =
¯

r1
1
((r ’ r1 )cr (r, f (r))
=
r 1 ’ r0
r0

+c(r, f (r)) ’ »f (r)) dr + »¯.
x
SR
Íå®áµ®¤è¬»¬ 󱫮âèå¬ ¤«ÿ ¬èíè¬è§àöèè @PFTRA ÿâ«ÿåò±ÿ â»ï®«íåíèå 󱫮âèÿ
Ý©«å°à @±¬F ‘72“AX


@PFTSA (rcr (r, f (r)) ’ r1 cr (r, f (r)) + c(r, f (r)) ’ »f (r))) = 0,
‚f
è«è

@PFTTA (r ’ r1 )cxr (r, f (r)) + cx (r, f (r)) = ».

Ê®í±òàíòà » í൮¤èò±ÿ è§ ó±«®âèÿ E f (r) = xF
¯
c(x)
Çà¬åòè¬D ·ò® ¤«ÿ ±«ó·àÿ §àò°àò âè¤à c(r, x) = + C ó°àâíåíèå @PFTTA ï°èE
r
®á°åòàåò âè¤

@PFTUA ’r1 cxr (r, f (r)) = ».

’åïå°ü à«ã®°èò¬ í൮¦¤åíèÿ ®ïòè¬à«üí»µ ôóíêöè© ±òè¬ó«è°®âàíèÿ è ¤å©E
±òâèÿ ¬®¦í® §àïè±àòü â ±«å¤óþùå¬ âè¤åX
@IA ȱﮫü§óÿ ¤èôôå°åíöèà«üí®å ó°àâíåíèå @PFTTA @è«è â ·à±òí®¬ ±«ó·àå
PFTUAD í൮¤è¬ ôóíêöèþ f (r) â §àâè±è¬®±òè ®ò ê®í±òàíò» » > 0 @êàê ¬í®E
¦èòå«ü Ëàã°àí¦à â §à¤à·å ¬èíè¬è§àöèè ï°è ®ã°àíè·åíèè E f (r) ’ x ≥ 0AY
¯
@PA Í൮¤è¬ §íà·åíèå » è§ ó±«®âèÿ E f (r) = xD ã¤å E f (r) ⻷豫ÿåò±ÿ ï®
¯
ô®°¬ó«å @PFTQAY
@QA Çíàÿ f (r)D í൮¤è¬ ôóíêöèþ ±òè¬ó«è°®âàíèÿ σ(x)D è±ï®«ü§óÿ ¤èôôåE
°åíöèà«üí®å ó°àâíåíèå @PFTTA è ã°àíè·í®å 󱫮âèå @PFSSA σ(f (r0 )) =
c(r0 , f (r0 )).
Çà¬åòè¬D ·ò® ¤«ÿ ôóíêöè© §àò°àò âè¤à c(r, x) = g(x) + C ±ó¬¬à ¤âóµ ±«àãàåE
r
¬»µ â @PFTTA rcxr (r, f (r)) + cx (r, f (r)) = 0 è íå®áµ®¤è¬®å 󱫮âèå ¬èíè¬ó¬à @PFSTA
ïå°åïè±»âàåò±ÿ â ᮫åå ï°®±ò®¬ âè¤åX

@PFTVA ’r1 cxr (r, f (r)) ’ » = 0.

‚ ýò®¬ ±«ó·àå

@PFTWA » = ’r1 cxr (r, f (r))

è ¬èíè¬à«üí»å ±°å¤íèå §àò°àò» íà ±òè¬ó«è°®âàíèå °àâí»
r1
1
@PFUHA E σ(f (r)) = ’ r1 cr (r, f (r)) dr.
r 1 ’ r0
r0


’àêè¬ ®á°à§®¬D ¬» ¤®êà§à«è ±«å¤óþùåå óòâå°¦¤åíèå @â ï°å¤ï®«®¦åíèè ôóíêE
öèè §àò°àò âè¤à c(r, x) = c(x) + CAX
r
“òâå°¦¤åíèå 2.4.2. …±«è ôóíêöèÿ f (r) ÿâ«ÿåò±ÿ ôóíêöèå© ¤å©±òâèÿ ¤«ÿ
®ïòè¬à«üí®© ôóíêöèè ±òè¬ó«è°®âàíèÿD ò® íà â±åµ ó·à±òêൠ⮧°à±òàíèÿ ôóíêöèè
f (r) â»ï®«íÿåò±ÿ ±«å¤óþùåå ¤èôôå°åíöèà«üí®å ±®®òí®øåíèåX

@PFUIA » = ’cxr (r, f (r)),
SS
ã¤å » " íåê®ò®°àÿ íå®ò°èöàòå«üíàÿ ê®í±òàíòàF Ï°è ýò®¬ ±°å¤íèå §àò°àò» íà ±òèE
¬ó«è°®âàíèå °àâí»
r1
1
@PFUPA σ(f (r)) = ’ rmax cr (r, f (r)) dr,
E
rmax ’ rmin
r0
ã¤å rmax è rmin E ±®®òâåò±òâåíí® íè¦íÿÿ è âå°µíÿÿ ã°àíèö» èíòå°âà«àD íà ê®ò®°®¬
±®±°å¤®ò®·åí® °àâí®¬å°í®å °à±ï°å¤å«åíèå òèï®â ÀÝF
Ï°è¬å° 7. Ðà±±¬®ò°è¬ ôóíêöèþ §àò°àò c(r, x) = xr F ȱµ®¤ÿ è§ @PFTVAD è¬åå¬X

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