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10

^_ F F1, c2, ..., cn), x = (x1, x2, ..., xn F[ ± kdZeyjgbc ^h[mlhd p1, ..., pn lZ S0 P\bf•jg• \_dlhjb klh\ip• kdeZ^_g• a dh_n•p•}gl•\ iSb g_\•^hfbo • \•evgbo qe_gZo kbkl_fb j•\gygv aZ^Zq•

éb1

ù

 

éa11

ù

 

éa12

ù

 

éa1n

ù

ê

ú

 

ê

ú

 

ê

ú

 

ê

ú

p0 = êb2

ú

,

p1 = êa 21

ú

,

p2 = êa 22

ú

, ..., pn

= êa 2n

ú .

êê... úú

 

êê...

úú

 

êê...

úú

 

êê...

úú

ëbn û

 

ëa m1 û

 

ëa m2 û

 

ëa mn û

IeZg [ [1, x2, ..., xn gZab\Z}lvky hihjgbf ieZghf hkgh\gh€

aZ^Zq• e•g•cgh]h ijh]jZfm\Zggy ydsh kbkl_fZ \_dlhj•\ Sj sh

\oh^ylv m jhadeZ^Zggy a ihablb\gbfb dh_n•p•}glZfb [j e•g•cgh g_aZe_`gZ

LZd yd \_dlhjb Sj } P\bf•jgbfb lh a \bagZq_ggy hihjgh]h ieZgm \bieb\Z} sh qbkeh ch]h ihablb\gbo dhfihg_gl•\ g_ fh`_ [mlb [•evr_ g•` P

Hihjgbc ieZg gZab\Z}lvky g_\bjh^`_gbf ydsh \•g kdeZ^Z}lvky j•\gh a P ihablb\gbo dhfihg_gl•\ m ijhlbe_`ghfm \biZ^dm \•g gZab\Z}lvky \bjh^`_gbf.

<eZklb\hkl• hkgh\gh€ aZ^Zq• e•g•cgh]h ijh]jZfm\Zggy

l•kgh ih\ yaZg• a \eZklb\hklyfb himdebo fgh`bg

G_oZc [1, x2, ..., xn ^h\•evg• lhqdb _\de•^h\h]h ijhklhjm (n.

Himdehx e•g•cghx dhf[•gZp•}x pbo lhqhd gZab\Z}lvky kmfZ

1x1+ 2x2+...+ nxn..

Fgh`bgZ gZab\Z}lvky himdehx ydsh jZahf a [m^v ydbfb ^\hfZ lhqdZfb \hgZ f•klblv i €ogx ^h\•evgm himdem e•g•cgm dhf[•gZp•x

LhqdZ [ himdeh€ e•g•cgh€ fgh`bgb gZab\Z}lvky dmlh\hx ydsh \hgZ g_ fh`_ [mlb iS_^klZ\e_gZ m \b]ey^• himdeh€ e•g•cgh€ dhf[•gZp•€ ydbo g_[m^v ^\ho •grbo ^h\•evgbo lhqhd ^Zgh€ fgh`bgb Fgh`bgZ ieZg•\ hkgh\gh€ aZ^Zq• e•g•cgh]h ijh]jZfm\Zggy }

himdehx ydsh \hgZ g_ ihjh`gy

G_ihjh`gy fgh`bgZ ieZg•\ hkgh\gh€ aZ^Zq• e•g•cgh]h ijh]jZfm\Zggy gZab\Z}lvky [Z]Zlh]jZggbdhf j•r_gv Z \kydZ dmlh\Z lhqdZ [Z]Zlh]jZggbdZ j•r_gv \_jrbghx.

Ydsh hkgh\gZ aZ^ZqZ e•g•cgh]h ijh]jZfm\Zggy fZ} hilbfZevgbc ieZg lh fZdkbfZevg_ agZq_ggy p•evh\Z nmgdp•y aZ^Zq• ijbcfZ} \

11

h^g•c a \_jrbg [Z]Zlh]jZggbdZ j•r_gv Ydsh fZdkbfZevg_ agZq_ggy p•evh\Z nmgdp•y ijbcfZ} [•evr g•` \ h^g•c \_Srbg• lh \hgZ ijbcfZ} ch]h m \kyd•c lhqp• sh } himdehx e•g•cghx dhf[•gZp•}x

pbo \_Srbg

Ydsh kbkl_fZ \_dlhS•\ S1, p2, ..., pk (k £ Q m jhadeZ^Zgg•

e•g•cgh g_aZe_`gZ • lZdZ sh [1p1+x2p2+...+xkpk = p0 ^_ \k• [j ³ lh lhqdZ [ [1, x2, ..., xk } \_Srbghx [Z]Zlh]jZggbdZ j•r_gv

Ydsh [ [1, x2, ..., xn \_SrbgZ [Z]Zlh]jZggbdZ j•r_gv lh

\_dlhSb Sj sh \•^ih\•^Zxlv ihablb\gbf [j m jhadeZ^Zgg•

e•g•cgh g_aZe_`g•

Kbfie_dkgbc f_lh^ \bj•r_ggy aZ^Zq• e•g•cgh]h ijh]jZfm\Zggy

<bj•r_ggy aZ^Zq• e•g•cgh]h ijh]jZfm\Zggy fh`gZ agZclb kbfie_dkgbf f_lh^hf I_jr g•` aZklhkh\m\Zlb p_c f_lh^ \Zjlh aZibkZlb \bo•^gm aZ^Zqm m nhjf• hkgh\gh€ aZ^Zq• EI ydsh \hgZ g_ fZ} lZdh€ nhjfb aZibkm

Kbfie_dkgbc f_lh^ \bj•r_ggy aZ^Zq• EI aZkgh\Zgbc gZ i_S_oh^• \•^ h^gh]h hihSgh]h ieZgm ^h •grh]h iSb ydhfm agZq_ggy p•evh\h€ nmgdp•€ a[•evrm}lvky iSb mfh\• sh ^ZgZ aZ^ZqZ fZ} hilbfZevgbc ieZg • dh`_g €€ hihSgbc ieZg } g_\bjh^`_gbf

AZagZq_gbc i_S_o•^ fh`eb\bc ydsh \•^hfbc ydbc g_[m^v \bo•^gbc hihSgbc ieZg Jha]eyg_fh aZ^Zqm ^ey ydh€ p_c ieZg fh`gZ [_aihk_j_^gvh aZibkZlb

G_oZc ihlj•[gh agZclb fZdkbfZevg_ agZq_ggy nmgdp•€

F = c1x1 + c2x2 + ... + cnxn

 

ì x1 + a

1m +1 x m +1 + ... + a1n x n = b1 ,

 

ï

+ a

2m +1 x m +1 + + a 2n x n = b 2 ,

 

ï x 2

iSb mfh\Zo

í

 

 

ï .......... .......... .......... .......... ...

 

 

ï

+ a mm +1 x m +1 + + a mn x n = b m ,

 

îx m

 

 

xj

³ 0, j =1, ...., n.

12

Lml Dij, bi lZ Fj L P M Q aZ^Zg• ihkl•cg• qbkeZ

P Q lZ Ei > 0).

<_dlhjgZ nhjfZ ^Zgh€ aZ^Zq• fZ} gZklmigbc \b]ey^ AgZclb fZdkbfmf nmgdp•€

 

 

 

 

 

 

n

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

F = åc jx j

 

 

 

 

 

 

 

(1.9)

 

 

 

 

 

 

j=1

 

 

 

 

 

 

 

 

 

 

iSb mfh\Zo

 

[1p1 + x2p2 + ... + xmpm + ... + xnpn = p0,

 

 

(1.10)

 

 

 

 

 

xj ³ 0, j = 1, ..., n,

 

 

 

 

 

 

(1.11)

 

é1ù

 

 

é0ù

 

é ù

 

 

éa1m+1 ù

 

 

é 1n ù

 

é 1 ù

 

 

 

 

 

 

0

 

 

 

 

 

 

a

 

 

b

 

ê ú

 

 

ê ú

 

ê ú

 

 

ê

ú

 

 

ê

ú

 

ê ú

^_ p

= ê0ú

;

p

= ê1ú

; ...; p

=ê0ú;

p

m+1

= êa2m+1

ú

; ...; p

=

êa2n

ú

; p

=êb2 ú.

1

ê...ú

 

2

ê...ú

m

ê...ú

 

ê ...

ú

n

 

ê ...

ú

0

ê...ú

 

ê ú

 

 

ê ú

 

ê ú

 

 

ê

ú

 

 

ê

ú

 

ê ú

 

ë0û

 

 

ë0û

 

ë1û

 

 

ëamm+1

û

 

 

ëamnû

 

ëbmû

Lhfm sh E1p1 + b2p2 + ... + bmpm = p0 lh aZ \bagZq_ggyf hihSgh]h ieZgm [ E1; b2; ...; bm } hihSgbf ieZghf ^Zgh€ aZ^Zq•hklZgg• Q P dhfihg_gl•\ \_dlhSZ [ ^hj•\gxxlv gmex P_c ieZg \bagZqZ}lvky kbkl_fhx h^bgbqgbo \_dlhS•\ S1, p2, ..., pm yd• ml\hjxxlv [Zabk P\bf•Sgh]h ijhklhjm Lhfm dh`gbc a \_dlhS•\ S1, p2, ..., pm, ..., pn Z lZdh` \_dlhS S0 fh`mlv [mlb iS_^klZ\e_g• m \b]ey^• e•g•cgh€ dhf[•gZp•€ \_dlhS•\ ^Zgh]h [Zabkm

m

G_oZc p j = åxijpi , j = 0, 1, ..., n. IhdeZ^_fh

i=1

m

z j = åci xij , j = 1, 2, ..., n,

i=1

Dj = zj - cj , j = 1, 2, ..., n.

13

Hkd•evdb \_dlhSb S1, p2, ..., pm h^bgbqg• lh [ij = aij lZ

 

 

m

 

z j

= åci aij ,

 

 

i=1

 

 

m

 

ûj = åci a ij - c j .

 

 

i=1

HihSgbc ieZg [

[1', x2', ..., xm aZ^Zq• }

hilbfZevgbf ydsh

j ³ ^ey [m^v ydh]h M M Q

Ydsh k ^ey ^_ydh]h M

N lZ k_S_^ qbk_e Dik, i = 1, 2, ..., m,

g_fZ} ihablb\gbo Dik £ lh p•evh\Z nmgdp•y aZ^Zq•

g_ h[f_`_gZ gZ fgh`bg• €€ ieZg•\

Ydsh hihSgbc ieZg [ aZ^Zq• g_\bjh^`_gbc lZ k< 0, Ze_ k_S_^ qbk_e Dik } ihablb\g• g_ \k• Dik £ lh •kgm} hihSgbc ieZg

[ lZdbc sh ) [ ! ) [

I_j_jZoh\Zg• \eZklb\hkl• ^ha\heyxlv i_j_\•jblb qb } agZc^_gbc hihjgbc ieZg hilbfZevgbf • \by\blb ^hp•evg•klv i_j_oh^m ^h gh\h]h hihjgh]h ieZgm

>hke•^`_ggy hihSgh]h ieZgm gZ hilbfZevg•klv Z lZdh` ih^Zevrbc h[qbkex\Zevgbc iShp_k ajmqg•r_ \_klb ydsh mfh\b aZ^Zq• • ihqZldh\• ^Zg• hljbfZg• i•key \bagZq_ggy \bo•^gh]h hihSgh]h ieZgm aZibkZlb yd ihdZaZgh \ lZ[e

M klh\ip• F[ p•}€ lZ[ebp• aZibkmxlv dh_n•p•}glb iSb g_\•^hfbo p•evh\h€ nmgdp•€ sh fZxlv l• ` •g^_dkb sh • \_dlhSb ^Zgh]h [Zabkm

M klh\ip• S0 aZibkmxlv ihablb\g• dhfihg_glb \bo•^gh]h hihjgh]h ieZgm m gvhfm ` j_amevlZlb h[qbke_gv h^_j`mxlv

ihablb\g• dhfihg_glb hihjgh]h

ieZgm Klh\ip•

\_dlhS•\

Sj

iS_^klZ\eyxlv kh[hx dh_n•p•}glb

jhadeZ^Zggy pbo

\_dlhS•\

ih

\_dlhSZf ^Zgh]h [Zabkm

 

 

 

M lZ[e i_jr• P jy^d•\ \bagZqZxlvky \bo•^gbfb ^Zgbfb aZ^Zq• Z ihdZagbdb P]h jy^dZ h[qbkexxlv

M pvhfm jy^dm \ klh\ip• \_dlhSZ S0 aZibkmxlv agZq_ggy p•evh\h€

nmgdp•€ yd_ \hgZ ijbcfZ} iSb ^Zghfm hihSghfm ieZg• Z \ klh\ip•

\_dlhSZ Sj agZq_ggy

j = zj - cj.

14

LZ[ebpy <bo•^gbc hihjgbc ieZg

I

;Zabk

 

c[

p0

c1

c2

...

 

cr

 

...

cm

cm+1

...

ck

...

cn

 

 

p1

p2

...

 

pr

 

...

pm

pm+1

...

pk

...

pn

 

 

 

 

 

 

 

 

 

1

j1

 

c1

b1

1

0

...

 

0

 

...

0

a1(m+1)

...

a1k

...

a1n

 

2

j2

 

c2

b2

0

1

...

 

0

 

...

0

a2(m+1)

...

a2k

...

a2n

 

.

.

 

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.

.

.

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.

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.

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.

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R

pr

 

cr

br

0

1

...

 

0

 

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ar(m+1)

...

ark

...

arn

 

.

.

 

.

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.

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.

.

.

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M

pm

 

cm

bm

0

0

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1

 

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am(m+1)

...

amn

...

amn

 

M+1

 

 

 

F0

0

0

...

 

0

 

...

0

m+1

...

k

 

...

n

 

AgZq_ggy zj

agZoh^blvky

yd

 

kdZeyjgbc ^h[mlhd \_dlhjZ Sj,

MQ lZ \_dlhjZ F[ = (c1, c2, ..., cm):

m

z j = åci aij , j = 1, 2, ..., n.

i=1

AgZq_ggy F0 ^hj•\gx} kdZeyjghfm ^h[mldm \_dlhjZ S0 lZ \_dlhjZ F[:

m

F0 = åci bi .

i=1

I•key aZih\g_ggy lZ[e \bo•^gbc hihSgbc ieZg i_j_\•jyxlv gZ hilbfZevg•klv >ey pvh]h ijh^b\eyxlvky _e_f_glb P]h jy^dZ lZ[ebp• < S_amevlZl• fh`_ fZlb f•kp_ h^bg a gZklmigbo lSvho \biZ^d•\

1)

j ³

 

^ey M P P Q iSb M

 

P ]j = cj lZ

\•^ih\•^gh

 

M Lhfm \ ^Zghfm \biZ^dm qbkeZ

j ³ ^ey \k•o M

\•^ ^h Q

 

 

 

 

2)

M ^ey ^_ydh]h M lZ mk• \•^ih\•^g• pvhfm •g^_dkm \_ebqbgb

aij £ 0 (i = 1, 2, ..., m);

 

 

3)

M

 

^ey ^_ydbo •g^_dk•\ M lZ ^ey

dh`gh]h lZdh]h M m

djZcgvhfm jZa• h^g_ a qbk_e Dij ihablb\g_

15

M i_Srhfm \biZ^dm gZ i•^klZ\• hagZdb hilbfZevghkl• \bo•^gbc hihSgbc ieZg } hilbfZevgbf < ^jm]hfm \biZ^dm p•evh\Z nmgdp•y g_ h[f_`_gZ a\_jom gZ fgh`bg• ieZg•\ Z \ lS_lvhfm \biZ^dm fh`gZ i_S_clb \•^ \bo•^gh]h ieZgm ^h gh\h]h hihSgh]h ieZgm iSb ydhfm agZq_ggy p•evh\h€ nmgdp•€ a[•evrblvky P_c i_S_o•^ \•^ h^gh]h hihSgh]h ieZgm ^h •grh]h a^•ckgx}lvky \bdexq_ggyf a \bo•^gh]h [Zabkm ydh]h g_[m^v a \_dlhS•\ • \\_^_ggyf m gvh]h gh\h]h \_dlhSZ

M ydhkl• \_dlhSZ sh \\h^blvky \ [Zabk

fh`gZ \aylb dh`_g a

\_dlhS•\ Sj sh fZ} •g^_dk M ^ey ydh]h

M G_oZc gZijbdeZ^

N • \bj•r_gh \\_klb \ [Zabk \_dlhS Sk >ey \bagZq_ggy \_dlhSZ sh i•^ey]Z} \bdexq_ggx a [Zabkm agZoh^ylv PLQ Ei/aik ^ey \k•o aik! G_oZc p_c f•g•fmf ^hky]Z}lvky iSb L U Lh^• a [Zabkm

\bdexqZxlv \_dlhS Sr Z qbkeh Drk gZab\Zxlv jhaj•rZxqbf _e_f_glhf Klh\i_pv • jy^hd gZ i_j_lbg• ydbo agZoh^blvky jhaj•rZxqbc _e_f_gl gZab\Zxlv gZijZ\eyxqbf.

I•key \b^•e_ggy gZijZ\eyxqh]h jy^dZ • gZijZ\eyxqh]h klh\ipy agZoh^ylv gh\bc hihSgbc ieZg [ • dh_n•p•}glb jhadeZ^Zggy \_dlhS•\ pj q_S_a \_dlhSb gh\h]h [Zabkm sh \•^ih\•^Zxlv gh\hfm hihSghfm ieZgm Ihablb\g• dhfihg_glb gh\h]h hihSgh]h ieZgm h[qbkexxlv aZ nhjfmehx

 

ì

 

 

 

 

b r

 

, ijb i ¹ r,

 

ïb i

-

 

 

a ik

 

 

a rk

b i

ï

 

 

 

 

 

(1.12)

'= í

b r

 

 

 

 

 

 

ï

 

 

, ijb i = r,

 

 

 

 

 

îï a rk

 

 

 

 

Z dh_n•p•}glb jhadeZ^Zggy \_dlhS•\ Sj q_S_a \_dlhSb gh\h]h [Zabkm sh \•^ih\•^Zxlv gh\hfm hihSghfm ieZgm aZ nhjfmehx

 

ì

 

 

 

a rj

 

, ijb i ¹ r,

 

ïa ij -

 

 

a ik

 

 

a rk

a ij

ï

 

 

 

 

(1.13)

'= í

 

 

 

 

 

 

 

ï a rj

, ijb i

= r.

 

ï

 

 

 

 

 

 

 

 

 

 

î a rk

 

 

 

 

h^gh]h hihSgh]h ieZgm ^h •grh]h a\h^blvky ^h kbfie_dk lZ[ebp• ^h •grh€ ?e_f_glb gh\h€

16

I•key h[qbke_ggy Ei lZ Dij a]•^gh lZ €og• agZq_ggy aZghkylv m lZ[e

?e_f_glb P]h jy^dZ p•}€ lZ[ebp• fh`mlv [mlb h[qbke_g• Z[h aZ nhSfmeZfb

 

 

 

 

F0' = F0 - k(br/ark),

 

 

 

 

 

 

(1.14)

 

 

 

 

 

j' =

j -

k(arj/ark),

 

 

 

 

 

(1.15)

Z[h gZ i•^klZ\• €o \bagZq_ggy

 

 

 

 

 

 

 

 

 

 

LZ[ebpy ± Hihjgbc ieZg

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

i

;Zabk

c[

p0

c1

c2

...

cr

...

cm

cm+1

...

ck

...

cn

p1

p2

...

pr

...

pm

pm+1

...

pk

...

pn

 

 

 

 

1

j1

c1

b'1

1

0

 

...

a'1r

...

0

a'1(m+1)

...

0

 

...

a'1n

2

j2

c2

b'2

0

1

 

...

a'2r

...

0

a'2(m+1)

...

0

 

...

a'2n

.

.

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.

.

.

 

.

.

.

.

.

.

.

 

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r

pk

ck

b'r

0

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...

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.

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1

a'm(m+1)

...

0

 

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a'mn

m+1

 

 

F'0

0

0

 

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z'r-cr

...

0

z'm+1-cm

...

0

 

...

z'n-cn

GZy\g•klv ^\ho kihkh[•\ agZoh^`_ggy _e_f_gl•\ P]h jy^dZ ^ha\hey} a^•ckgx\Zlb dhglShev ijZ\bevghkl• h[qbke_gv sh ijh\h^ylvky

A \bieb\Z} sh iSb i_S_oh^• \•^ h^gh]h hihSgh]h ieZgm ^h

•grh]h gZc[•evr ^hp•evgh \\_klb \ [Zabk \_dlhS Sj sh fZ} •g^_dk M iSb ydhfm fZdkbfZevgbf aZ Z[khexlghx \_ebqbghx } qbkeh j(br/arj), j>0, arj! H^gZd a f_lhx kijhs_ggy h[qbkex\Zevgh]h

iShp_km gZ^Ze• [m^_fh \_dlhS sh \\h^blvky \ [Zabk \bagZqZlb

\boh^yqb a fZdkbfZevgh€ Z[khexlgh€ \_ebqbgb \•^}fgbo qbk_e

j.

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lZdbc `_ •g^_dk yd • fZdkbfZevg_ a qbk_e Fj sh \bagZqZxlvky

^Zgbfb qbkeZfb

j ( j<0).

Hl`_ i_S_o•^ \•^ i_S_oh^m \•^ h^g•}€

17

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ïìk1 (w1 ,

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í ............................

 

 

NhjfZevgh

 

ï

 

, w N ) = 0;

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îcm (w1 ,

 

ìkm+1 (w1

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nmgdp•x

f(w) = f(w1,w2,...,wN ^_ Z

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S

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g_e•g•cgbo

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lZ m \b]ey^• g_j•\ghkl_c

ï

 

 

í ............................

 

ï

(w1 , , w N )

³ 0.

 

îcp

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f(w) → min;

cj(w) = 0; j = 1, ..., m;

ck(w) ³ 0; k = m+1, ..., p.

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w = u v,

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