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¤å©±òâèå xi F
Ï®êà¦å¬ ï® èí¤óêöèèD ·ò® àêòèâí»© ý«å¬åíò ± í®¬å°®¬ i íå áó¤åò °åà«è§®E
â»âàòü ¤å©±òâèå ± í®¬å°®¬ k < iF „婱òâèòå«üí®D ﮱꮫüêó

@PFISA σ(x1 ) = c1 (x1 ),

ò® ïå°â®¬ó ý«å¬åíòó íåâ»ã®¤í® ®òê«®íÿòü±ÿ ± ¤å©±òâèÿ x1 íà x0 = 0F
Ïó±òü óòâå°¦¤åíèå ¤®êà§àí® ¤«ÿ ý«å¬åíòà ± í®¬å°®¬ i ’ 1F ’®ã¤à ¤«ÿ ý«å¬åíòà
± í®¬å°®¬ i
σ(xi ) ’ ci (xi ) = σ(xi’1 ) + ci (xi ) ’ ci (xi’1 ) ’ ci (xi )
= σ(xi’1 ) ’ ci (xi’1 ),
òFåF àêòèâí®¬ó ý«å¬åíòó ± í®¬å°®¬ i á姰৫è·í®D êàê®å ¤å©±òâèå °åà«è§®â»âàòü
" xi è«è xi’1 F Í® ﮱꮫüêó ï® ï°å¤ï®«®¦åíèþ èí¤óêöèè ý«å¬åíòó ± í®¬å°®¬ i’1
íåò â»ã®¤» °åà«è§®â»âàòü ¤å©±òâèÿ ¬åíüøåD ·å¬ xi’1 D ò®

σ(xi’1 ) ’ ci (xi’1 ) = σ(xi’1 ) ’ ci’1 (xi’1 ) +
+ci’1 (xi’1 ) ’ ci (xi’1 )
≥ σ(xi’k ) ’ ci’1 (xi’k ) +
+ci’1 (xi’k ) ’ ci (xi’k )
= σ(xi’k ) ’ ci (xi’k )

è àêòèâí®¬ó ý«å¬åíòó ± í®¬å°®¬ i íåò â»ã®¤» °åà«è§®â»âàòü ¤å©±òâèÿ ¬åíüøåD
·å¬ xi’1 D èD ±«å¤®âàòå«üí®D ¬åíüøå ·å¬ xi F
RP
Ï®êà¦å¬ òåïå°ü ï® èí¤óêöèèD ·ò® àêòèâí»© ý«å¬åíò ± í®¬å°®¬ i íå áó¤åò
°åà«è§®â»âàòü ¤å©±òâèå ± í®¬å°®¬ k > iF Ý«å¬åíòó ± í®¬å°®¬ n íåò â»ã®¤» °åàE
«è§®â»âàòü ¤å©±òâèåD ᮫üøå ·å¬ xn F
Ïó±òü óòâå°¦¤åíèå ¤®êà§àí® ¤«ÿ ý«å¬åíòà ± í®¬å°®¬ i + 1F ’®ã¤à ¤«ÿ ý«å¬åíòà
± í®¬å°®¬ i
@PFITA σ(xi ) ’ ci (xi ) = σ(xi+1 ) ’ ci+1 (xi+1 ) +
+ci+1 (xi ) ’ ci (xi ) =
= σ(xi+1 ) ’ ci (xi+1 ) + ci (xi+1 ) ’
’ci+1 (xi+1 ) + ci+1 (xi ) ’ ci (xi ) =
ri+1 xi+1

= σ(xi+1 ) ’ ci (xi+1 ) + crx (r, x) dx dr ≥
ri xi

≥ σ(xi+1 ) ’ ci (xi+1 ),
òFåF àêòèâí®¬ó ý«å¬åíòó ± í®¬å°®¬ i íåò â»ã®¤» °åà«è§®â»âàòü ¤å©±òâèå xi+1
â¬å±ò® xi F Í® ﮱꮫüêó ï® ï°å¤ï®«®¦åíèþ èí¤óêöèè ý«å¬åíòó ± í®¬å°®¬ i + 1
íåò â»ã®¤» °åà«è§®â»âàòü ¤å©±òâèÿ ᮫üøåD ·å¬ xi+1 D ò® ﮱꮫüêó
σ(xi+1 ) ’ ci (xi+1 ) ≥ σ(xi+k ) ’ ci+1 (xi+k ) +
+ci+1 (xi+1 ) ’ ci (xi+1 )
≥ σ(xi+k ) ’ ci+1 (xi+k ) +
+ci+1 (xi+k ) ’ ci (xi+k )
= σ(xi+k ) ’ ci (xi+k ),
àêòèâí®¬ó ý«å¬åíòó ± í®¬å°®¬ i íåò â»ã®¤» °åà«è§®â»âàòü ¤å©±òâèå ᮫üøåD ·å¬
xi F
’àêè¬ ®á°à§®¬ ¬» ï®êà§à«èD ·ò® ï°è ®ï°å¤å«åíí®© ó°àâíåíèå¬ @PFIQA ±è±òå¬å
±òè¬ó«è°®âàíèÿ iE© àêòèâí»© ý«å¬åíò °åà«è§óåò ¤å©±òâèå xi F Ï®êà¦å¬ òåïå°üD
·ò® ¤àííàÿ ±è±òå¬à ±òè¬ó«è°®âàíèÿ ®ïòè¬à«üíàF
Ïó±òü ýò® íå òàêD ò®ã¤à ï® ô®°¬ó«å @PFIRA ±óùå±òâóþò òàêèå ê®í±òàíò» ±i ≥ 0D
·ò®
@PFIUA σ(xi ) = σ(xi’1 ) + ci (xi ) ’ ci (xi’1 ) + ±i ∀i = 1 . . . n.
Í® ò®ã¤à
n n i
@PFIVA (ck (xk ) ’ ck (xk’1 ) + ±k ) =
σ(xi ) =
i=1 i=1 k=1
n i n i
(ck (xk ) ’ ck (xk’1 )) + ±k =
=
i=1 k=1 i=1 k=1

n i n
(ck (xk ) ’ ck (xk’1 )) + (n ’ i + 1)±i ≥
=
i=1 k=1 i=1
n i n
(ck (xk ) ’ ck (xk’1 )) = σ (xi ),
= ˜
i=1 k=1 i=1
RQ
ï°è·å¬ °àâåí±ò⮠⮧¬®¦í® ò®«üê® å±«è â±å ±i = 0F Í® ï®±«å¤íåå â»°à¦åíèå å±òü
§àò°àò» –åíò°à íà ±òè¬ó«è°®âàíèå ï°è ±è±òå¬å ±òè¬ó«è°®âàíèÿ σ (·)D ó¤®â«åòâ®E
˜
°ÿþùå© @PFIQAD òFåF ýòà ±è±òå¬à ®ïòè¬à«üíàF
Êàê ¬®¦í® «åãê® ï®íÿòüD íåï°å°»âí»© àíà«®ã ôóíêöèè σ (·) è§ ô®°¬ó«» @PFIQA
˜
â»ã«ÿ¤èò ±«å¤óþùè¬ ®á°à§®¬ @x ∈ [xi’1 , xi ]AX
i’1
@PFIWA σ(x) = ci (x) ’ ci (xi’1 ) + (ck (xk ) ’ ck (xk’1 )) =
k=1
i’1
(ck (xk ) ’ ck+1 (xk )) .
ci (x) +
k=1

ȱﮫü§óÿ ï°å¤±òàâ«åíèå @PFIQA ¤«ÿ ôóíêöèè ±òè¬ó«è°®âàíèÿD ¬®¦í® íà©òè
¤èôôå°åíöèà«üí®å ó°àâíåíèåD ±âÿ§»âàþùåå ¤å©±òâèÿ xk è xk+1 ï°è ®ïòè¬à«üE
í®© ôóíêöèè ±òè¬ó«è°®âàíèÿ @屫è xk’1 < xk < xk+1 < xk+2 AD ® ·å¬ è ã®â®°èò
±«å¤óþùåå
“òâå°¦¤åíèå 2.3.1. Ïó±òü ï°è íåê®ò®°®© ®ïòè¬à«üí®© ôóíêöèè ±òè¬ó«è°®E
âàíèÿ kE© ÀÝ °åà«è§óåò ¤å©±òâèå xk D è ¤«ÿ â±åµ i = 1, n ’ 1 â»ï®«íÿåò±ÿ xi < xi+1 F
’®ã¤à âå°í® ±«å¤óþùåå ¤èôôå°åíöèà«üí®å ó°àâíåíèåX

@PFPHA (n ’ i ’ 1)(ci+2 (xi+1 ) ’ ci+1 (xi+1 )) ’ ci+1 (xi+1 ) =
(n ’ i)(ci+1 (xi ) ’ ci (xi )) ’ ci (xi ),

ã¤å ci (xi ) å±òü ôóíêöèÿ §àò°àò â ò®·êå xi ÀÝ òèïà ri F
„àíí®å óòâå°¦¤åíèå ¤àåò ⮧¬®¦í®±òü ¤«ÿ èòå°àòèâí®ã® ®ï°å¤å«åíèÿ ®ïòèE
¬à«üí® °åà«è§ó嬻µ ¤å©±òâè© ï°è ®ïòè¬à«üí®© ôóíêöèè ±òè¬ó«è°®âàíèÿX §íàÿ
®¤í® è§ íèµD ¬®¦í® íà©òè ¤å©±òâèÿ ±®±å¤íèµ àêòèâí»µ ý«å¬åíò®â è òF¤F ‚ ·à±òí®E
±òèD §íàÿ ¤å©±òâèåD °åà«è§ó嬮å nE¬ àêòèâí»¬ ý«å¬åíò®¬ ï°è ®ïòè¬à«üí®© ôóíêE
öèè ±òè¬ó«è°®âàíèÿD ¬®¦í® ®ï°å¤å«èòü ¤å©±òâèåD °åà«è§ó嬮å n ’1E¬ ý«å¬åíò®¬D
¤à«åå " n ’ 2E¬ ý«å¬åíò®¬D è òàê ¤à«üøå ¤® ïå°â®ã® àêòèâí®ã® ý«å¬åíòàF Τíàê®
µ®òå«®±ü á» è§áàâèòü±ÿ ®ò èòå°àòèâí®ã® ¬åµàí觬àD è òàêóþ ⮧¬®¦í®±òü ï°å¤®E
±òàâ«ÿåò ±«å¤óþùåå óòâå°¦¤åíèåF
“òâå°¦¤åíèå 2.3.2. Ïó±òü ï°è íåê®ò®°®© ®ïòè¬à«üí®© ôóíêöèè ±òè¬ó«è°®E
âàíèÿ kE© ÀÝ °åà«è§óåò ¤å©±òâèå xk D è ¤«ÿ â±åµ i = 1, n ’ 1 â»ï®«íÿåò±ÿ xi < xi+1 F
’®ã¤à ¤«ÿ «þỵ k è i âå°í® ±«å¤óþùåå ¤èôôå°åíöèà«üí®å ó°àâíåíèåX

@PFPIA (n ’ i)(ci+1 (xi ) ’ ci (xi )) ’ ci (xi ) =
(n ’ k)(ck+1 (xk ) ’ ck (xk )) ’ ck (xk ) = ’cn (xn ).

“±«®âèå â ï°å¤»¤óùå¬ óòâå°¦¤åíèè ÿâ«ÿåò±ÿ ±óùå±òâåíí»¬ ¤«ÿ í൮¦¤åíèÿ
°åøåíèÿD ﮱꮫüêó °åøåíèÿ @PFPIA íå ®áÿ§àòå«üí® ¬®í®ò®íí® óï®°ÿ¤®·åíí® â®§E
°à±òàþò ï°è ó«ó·øåíèè òèïà ÀÝ @·ò® ÿâ«ÿåò±ÿ íå®áµ®¤è¬»¬ ¤«ÿ °åøåíèÿAD è
è¬åíí® ï®ýò®¬ó ⮧¬®¦íà °åà«è§àöèÿ °à§«è·í»¬è àêòèâí»¬è ý«å¬åíòà¬è ®¤èE
íàê®â»µ ¤å©±òâè©D ® ·å¬ ±âè¤åòå«ü±òâóåò ±«å¤óþùè© ï°è¬å°F
Ï°è¬å° 4. Ðà±±¬®ò°è¬ êâरàòè·í»å ôóíêöèè §àò°àò c(ri , x) = ri x2 è ò°è
2


àêòèâí»µ ý«å¬åíòà ± òèïà¬è r1 = 1D r2 = 8/7 è r3 = 2F Ï®±ê®«üêó r1 < r2 <
r3 D ò® ôóíêöèè §àò°àò ó¦å óï®°ÿ¤®·åí» íó¦í»¬ ®á°à§®¬F Íà©¤å¬ ®ïòè¬à«üíóþ
RR
ôóíêöèþ ±òè¬ó«è°®âàíèÿ ¤«ÿ °åà«è§àöèè ±ó¬¬à°í®ã® ¤å©±òâèÿ xF Ïó±òü x— "
¯ i
¤å©±òâèåD ê®ò®°®å ï°è ýò®¬ °åà«è§óåò iE© àêòèâí»© ý«å¬åíòF
’àêè¬ ®á°à§®¬D íå®áµ®¤è¬® °åøèòü §à¤à·óX
3
@PFPPA σ(xi ) ’ min
x1 ,x2 ,x3
i=1
ï°è â»ï®«íåíèè 󱫮âè©X
@PFPQA x1 ¤ x2 ¤ x3 ;
3
@PFPRA xi = x;
¯
i=1

i
@PFPSA (ck (xk ) ’ ck (xk’1 )) .
σ(xi ) =
k=1

Ï°å¤ï®«àãàÿD ·ò® ®ã°àíè·åíèÿ @PFPQA ÿâ«ÿþò±ÿ ±ò°®ãè¬èD òFåF 屫è x1 , x2 è x3
ÿâ«ÿþò±ÿ °åøåíèå¬D ò® x1 < x2 < x3 D è§ @PFPPAE@PFPSA ﮫó·àå¬X
3
σ(xi ) = 3c1 (x1 ) + 2(c2 (x2 ) ’ c2 (x1 )) + (c3 (x3 ) ’ c3 (x2 ))
i=1
x2 x2 2 x2 1
3 2 1
=1 2
+3
’ ’
+
r1 r2 2 r 2 r3 2 r3
2
x2
37 71
+3
2 2
’ ’
= x1 + x2
28 84 4
5 2 5 2 (¯ ’ x1 ’ x2 )2
x
’ min .
x1 + x2 +
=
8 8 4 x1 ,x2

Í൮¤è¬ 󱫮âèÿ ïå°â®ã® ï®°ÿ¤êàX
5
ï® x1 : x1 ’ x + x2 + x1 = 0;
¯
4
5
ï® x2 : x2 ’ x + x1 + x2 = 0.
¯
4
‚»°à¦àÿ è§ ýòèµ °àâåí±òâ x1 è x2 D ﮫó·àå¬X
4
x1 = x2 = x; ¯
13
5
x 3 = x ’ x 1 ’ x 2 = x.
¯ ¯
13
’àêè¬ ®á°à§®¬D ¬» ﮫó·è«èD ·ò® ïå°â»© è âò®°®© àêòèâí»å ý«å¬åíò» ï°è
«þᮩ ®ïòè¬à«üí®© ±è±òå¬å ±òè¬ó«è°®âàíèÿ è «þᮬ ±ó¬¬à°í®¬ ¤å©±òâèè °åàE
«è§óþò ®¤í® è ò® ¦å ¤å©±òâèåF
Çà¬åòè¬D ·ò® ï°è ¤°óãèµ òèïൠàêòèâí»µ ý«å¬åíò®â â êà·å±òâå °åøåíèÿ ®ïòèE
¬è§àöè®íí®© §à¤à·è ¬®¦í® ừ® Ỡﮫó·èòüD ·ò® ïå°â»© àêòèâí»© ý«å¬åíò ¤®«E
¦åí °åà«è§®â»âàòü á¡«üøåå ¤å©±òâèåD ·å¬ âò®°®©D ·åã® íå ¬®¦åò á»òü â ±è«ó Ëå¬E
®
¬» PFPFID èD òàêè¬ ®á°à§®¬D ï°è °åøåíèè ﮤ®áí»µ §à¤à· íå®áµ®¤è¬® ó·èò»âàòü
®ã°àíè·åíèÿ @PFPQAF•
Τíàê®D í屬®ò°ÿ íà ò®D ·ò® °à§«è·í»å àêòèâí»å ý«å¬åíò» ¬®ãóò °åà«è§®â»E
âàòü ®¤èíàê®â»å ¤å©±òâèÿD âå°í» ¤âå ±«å¤óþùèå «å¬¬»D ã®â®°ÿùèå ® ò®¬D ·ò®
RS
¤å©±òâèå ±à¬®ã® «ó·øåã® è§ àêòèâí»µ ý«å¬åíò®â ®ò«è·í® ®ò ¤°óãèµ è ¤å©±òâèå
±à¬®ã® µó¤øåã® è§ àêòèâí»µ ý«å¬åíò®â ±ò°®ã® ﮫ®¦èòå«üí®X
“òâå°¦¤åíèå 2.3.3. ‚ «þᮩ À‘ ®ïòè¬à«üí®å ¤å©±òâèå «ó·øåã® ÀÝ áó¤åò
®ò«è·àòü±ÿ ®ò ¤å©±òâè© â±åµ ¤°óãèµ ÀÝF
’àêè¬ ¦å ±ï®±®á®¬D êàêè¬ ¤®ê৻âàåò±ÿ ®ò«è·èå ¤å©±òâèÿD °åà«è§ó嬮㮠ï°è
®ïòè¬à«üí®© ±è±ò嬻 ±òè¬ó«è°®âàíèÿ «ó·øè¬ àêòèâí»¬ ý«å¬åíò®¬D ®ò ¤å©±òâè©
¤°óãèµ ý«å¬åíò®â @ï°å¤»¤óùåå óòâå°¦¤åíèåA ¤®ê৻âàåò±ÿ ò®ò ôàêòD ·ò® íàèµó¤E
øè© àêòèâí»© ý«å¬åíò °åà«è§óåò íåíó«åâ®å ¤å©±òâèåF
“òâå°¦¤åíèå 2.3.4. •ó¤øè© ÀÝ ï°è ®ïòè¬à«üí®© ±è±òå¬å ±òè¬ó«è°®âàíèÿ
áó¤åò â»áè°àòü íåíó«åâ®å ¤å©±òâèåF
Èíòå°ï°åòàöèÿ ¤àíí»µ óòâå°¦¤åíè© ±«å¤óþùàÿX ï°è íà«è·èè «þá®ã® íàá®°à
àêòèâí»µ ý«å¬åíò®â ï°è ®ïòè¬à«üí®© ôóíêöèè ±òè¬ó«è°®âàíèÿ â±å è§ íèµ áó¤óò
°åà«è§®â»âàòü íåê®ò®°»å ¤å©±òâèÿD ò® å±òü íè ®¤èí íèµ ï°è °àöè®íà«üí®¬ öåíò°å
íå ¤®«¦åí á»òü è±ê«þ·åí è§ ±è±ò嬻F „«ÿ ò®ã®D ·ò®á» ï®«í®±òüþ è±ï®«ü§®âàòü
⮧¬®¦í®±òè íàè«ó·øåã® àêòèâí®ã® ý«å¬åíòàD íå®áµ®¤è¬® ±¤å«àòü òàêD ·ò®á» åã®
¤å©±òâèå ®ò«è·à«®±ü ®ò ¤å©±òâè© ¤°óãèµ àêòèâí»µ ý«å¬åíò®âF
“·èò»âàÿ óòâå°¦¤åíèå Ë嬬» PFPFI è ô®°¬ó«ó PFIQD â ®áùå¬ ±«ó·àå §à¤à·à
ï®è±êà ®ïòè¬à«üí®© ±è±ò嬻 ±òè¬ó«è°®âàíèÿ ¤«ÿ ¤è±ê°åòí®ã® ±«ó·àÿ â»ïè±»âàE
åò±ÿ ±«å¤óþùè¬ ®á°à§®¬X

n
@PFPTA σ(xi ) ’ max ;
n {xi }i=1
i=1


i
@PFPUA (ck (xk ) ’ ck (xk’1 ) ∀i = 1 . . . n;
σ(xi ) =
k=1


@PFPVA xi+1 ≥ xi ∀i = 1 . . . n ’ 2;

n
@PFPWA xi = x.
¯
i=1

Ïå°åïè±»âàÿ §à¤à·ó ¬àê±è¬è§àöèè è è±ï®«ü§óÿ â»°à¦åíèå ¤«ÿ ôóíêöèè ±òèE
¬ó«è°®âàíèÿD ﮫó·àå¬
n n i
((ck (xk ) ’ ck (xk’1 ))
σ(xi ) =
i=1 i=1 k=1
n
(n ’ i + 1)((ci (xi ) ’ ci (xi’1 )) ’ max .
= n {xi }i=1
i=1

Ëàã°àí¦èàí
n
(n ’ i + 1)(ci (xi ) ’ ci (xi’1 ))+
L=
i=1
n n’2
» x’ µi (xi ’ xi+1 ).
xi +
¯
i=1 i=1
RT
Íå®áµ®¤è¬»¬ 󱫮âèå¬ ¤«ÿ ò®ã®D ·ò®á» ¬í®¦å±òâ® ¤å©±òâè© {xi }n ừ® °åE
i=1
øåíèå¬ §à¤à·è @PFPTAE@PFPWAD ÿâ«ÿåò±ÿ ±óùå±òâ®âàíèå òàêèµ ê®í±òàíò » ≥ 0 è
µi ≥ 0D ·ò®
‚L
= 0 ∀i = 1 . . . n,
‚xi
n
ï°è·å¬ xi = xD xi+1 ≥ xi ∀i = 1 . . . n ’ 2D à µi > 0 ò®«üê® ò®ã¤àD ê®ã¤à xi+1 = xi F
¯
i=1
ȱµ®¤ÿ è§ ýòèµ ó±«®âè©D ¤®«¦í® â»ï®«íÿòü±ÿ
‚L
@PFQHA = cn (xn ) ’ » = 0;
‚xn

‚L
@PFQIA = 2cn’1 (xn’1 ) ’ cn (xn’1 ) ’ µn’2 ’ » = 0;
‚xn’1

‚L
@PFQPA = nc1 (x1 ) ’ (n ’ 1)c2 (x1 ) ’ » + µ1 = 0;
‚x1

‚L
= (n ’ i + 1)ci (xi ) ’ (n ’ i)ci+1 (xi )’
@PFQQA ‚xi
’» + µi ’ µi’1 = 0
ï°è i = 2 . . . n ’ 2F
’àêè¬ ®á°à§®¬D 屫è xi òàê®â®D ·ò® xi’1 < xi < xi+1 D ò® ¬í®¦èòå«è Ëàã°àí¦à
µi’1 = µi = 0 è

@PFQRA (n ’ i + 1)ci (xi ) ’ (n ’ i)ci+1 (xi ) = cn (xn ).

…±«è ¦å xk’1 < xk = xk+1 = . . . = xl < xl+1 D 1 ¤ k < l ¤ n ’ 1D ò® ê®í±òàíò»
µk’1 = µl = 0 èD êàê âè¤í® è§ @PFQQA @®á®§íà·àÿ x— = xk = . . . = xl AD è¬åå¬X
l
‚L
@PFQSA 0= =
‚xi
i=k
l
((n ’ i + 1)ci (xi ) ’ (n ’ i)ci+1 (xi ) ’
=
i=k
’» + µi ’ µi’1 ) =
l
((n ’ i + 1)ci (x— ) ’ (n ’ i)ci+1 (x— ) ’ ») =
=
i=k
= (n ’ k + 1)ck (x— ) ’ (n ’ l)cl+1 (x— ) +
l
((n ’ i + 1) ’ (n ’ (i ’ 1))ci (x— ) ’
+
i=k+1

’(l ’ k + 1)» =
= (n ’ k + 1)ck (x— ) ’ (n ’ l)cl+1 (x— ) ’
’(l ’ k + 1)» =
= (n ’ k + 1)ck (x— ) ’ (n ’ l)cl+1 (x— ) ’
’(l ’ k + 1)cn (x— ),
n
RU
±«å¤®âàòå«üí®
@PFQTA (n ’ k + 1)ck (x— ) ’ (n ’ l)cl+1 (x— ) = (l ’ k + 1)cn (x— ).
n

Ê°®¬å ò®ã®D íå®áµ®¤è¬®D ·ò®á» â»ï®«íÿ«®±ü µp ≥ 0 ï°è p = k . . . l ’ 1D è«èD
â»°à¦àÿ µp @àíà«®ãè·í® ò®¬óD êàê ýò® ±¤å«àí® â @PFQSAAD
µp = (n ’ k + 1)ck (x— )’
@PFQUA ’(n ’ p)cp+1 (x— ) ’ (p ’ k + 1)cn (x— ) ¤
n
¤ 0,
è«èD §à¬åíÿÿ cn (x— ) ï® ô®°¬ó«å @PFQTAD
n

@PFQVA (n ’ k + 1)ck (x— ) ’ (n ’ p)cp+1 (x— ) ¤
p’k+1
((n ’ k + 1)ck (x— ) ’ (n ’ l)cl+1 (x— )).
l’k+1
’àêè¬ ®á°à§®¬D ¤®êà§àíà
’å®°å¬à 2.3.5. Ïó±òü ï°è ®ïòè¬à«üí®© ôóíêöèè ±òè¬ó«è°®âàíèÿ â À‘ è§ n
ÀÝ iE© ÀÝ °åà«è§óåò ¤å©±òâèå x— F Ï°è x— < x— < x— â»ï®«íÿåò±ÿ ±«å¤óþùåå
i i’1 i i+1
°àâåí±òâ®X
@PFQWA (n ’ i + 1)ci (xi ) ’ (n ’ i)ci+1 (xi ) = cn (xn ).
Ï°è xk’1 < x— = xk = xk+1 = . . . = xl < xl+1 D 1 ¤ k < l ¤ n ’ 1 â»ï®«íÿåò±ÿ
@PFRHA (n ’ k + 1)ck (x— ) ’ (n ’ l)cl+1 (x— ) = (l ’ k + 1)cn (x— );
n


@PFRIA (n ’ k + 1)ck (x— ) ’ (n ’ p)cp+1 (x— ) ¤
p’k+1
((n ’ k + 1)ck (x— ) ’ (n ’ l)cl+1 (x— ))
l’k+1
¤«ÿ â±åµ p = k, . . . , l ’ 1F
Íà ®±í®âàíèè ¤àíí®© ò宰嬻 ¬®¦í® ï®±ò°®èòü à«ã®°èò¬ í൮¦¤åíèÿ ®ïòèE
¬à«üí®© ôóíêöèè ±òè¬ó«è°®âàíèÿX
@IA Í൮¤è¬ §àâè±è¬®±òè ¤å©±òâè© â±åµ ÀÝ ®ò ¤å©±òâèÿ íàè«ó·øåã® ÀÝF
@PA Í൮¤è¬ §àâè±è¬®±òü §àò°àò íà ±òè¬ó«è°®âàíèå ®ò ±ó¬¬à°í®ã® °åà«è§óåE
¬®ã® ¤å©±òâèÿF
@QA Ðåøàÿ ¤èôôå°åíöèà«üíóþ §à¤à·ó @ï°®è§â®¤íàÿ ®ò °à§í®±òè ¤®µ®¤à ®ò àãE
°åãè°®âàíí®ã® ¤å©±òâèÿ è §àò°àò íà åã® °åà«è§àöèþAD í൮¤è¬ ®ïòè¬à«üE
í®å ¤å©±òâèå íàè«ó·øåã® ÀÝ è ®ïòè¬à«üíóþ ±è±òå¬ó ±òè¬ó«è°®âàíèÿF
x2
Ï°è¬å° 5. Ðà±±¬®ò°è¬ êâरàòè·í»å 觤尦êè c(ri , xi ) = D
n = 3D r1 = 1D
i
2ri
r2 è r3 ï°®è§â®«üí»D r1 < r2 < r3 F Îï°å¤å«è¬D â êàêèµ ±«ó·àÿµ ¬®¦åò á»òü x1 =
x2 < x3 @¤°óãè¬ â®§¬®¦í»¬ ±«ó·àå¬ ÿâ«ÿåò±ÿ x1 < x2 < x3 D è ï® Ë嬬ଠPFPFI
è PFQFQ èí»µ âà°èàíò®â á»òü íå ¬®¦åòAF Ï®«àãàå¬ k = 1D l = 2F
ȧ 󱫮âèÿ @PFQTA @ó·èò»âàÿD ·ò® x— = x1 = x2 A
(l ’ k + 1)cn (xn ) = (n ’ k + 1)ck (x— ) ’ (n ’ l)cl+1 (x— );

2c3 (x3 ) = 3c1 (x1 ) ’ c3 (x1 );
x3 x1
2 = 3x1 ’ ;
r3 r3
RV
r3 1
@PFRPA x1 3 ’
x3 = .
r3
2
Ê°®¬å ò®ã®D â ±è«ó @PFQUA ï°è p = 1
µp = ((n ’ k + 1)ck (x— ) ’ (n ’ p)cp+1 (x— ) ’ (p ’ k + 1)cn (xn )) ≥ 0;
3c1 (x1 ) ’ 2c2 (x1 ) ’ c3 (x3 ) ≥ 0;
x1 x3
3x1 ’ 2 ’ ≥ 0;
r2 r3
2 1
x1 3 ’ ’ x3 ≥ 0.
r2 r3
‚»°à¦àÿ x3 ·å°å§ x1 @è±ï®«ü§óÿ ô®°¬ó«ó @PFRPAAD
r3
2 1 1
x1 3 ’ ’ x1 3 ’ ≥ 0;
r2 r3 r3
2
2 1 1
3’’ 3’ ≥ 0;
r2 2 r3
4 1
3’ + ≥ 0.

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