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2 z ' sut to ' M ' M N uostrd N z ' sut to ' z ' sut to '

3 sl vrls vtor of : vrls surplus vtor of : uostrd s s s s s s

4 z sut to whr : ut ost of :out of : out of ( ' gr of h food ( utrt : rqurt for h utrt (t lst gr of h food ( gr of h utrt pr o ut of gr of wo/gr of h, h, h,...)...)...) h food utrt s rqurd pr d)

5 z ' sut to ssu s tr ( < ) d th r s. {,..., } [ ],...,

6 [ ] [ ] M M M M - vrl s :,...,, th opot of for......

9 3 / 3 2 / / 2

10 u

11 (,,3) 3 (,,3) (,,3) (2,,2) (,,) (2,,) (,2,) 2 (2,2,)

12 [ ] [ ] [ ] [ ] [ ] [ ] [ ] poltop P ),..., ( ),...,(,,...,,,...,,,...,, ) ( ) ( ) ( I I I

13 [ ] [ ] [ ] [ ] [ ] [ ] -,...,, otd for, ftr solvg ) ( I I

14 ,..., ,..., ( 2 3 )

15 s fs of Fdfd Th orrspodg s vrt of th ov poltop P Thr sts ost vtor suh tht s th uqu vtor stsfg ' '

16 d' d' ' ' d' ' d' ' ' ) ( ) ( ) (

17 Thr sts ost vtor suh tht s th uqu vtor stsfg ' ' P - Th (,..., - ) s th uqu pot R stsfg d' d' hprpl tht dfs trsto P trsto s uqu pot vrt 3 2

18 s ssu Th fs th, Howvr, 'sr ot lrl dpdt. ( ± d ) Df two pots d s - d vrt s vrt P, s.t. If 'sr lrl dpdt, th s fs. d for so d for sufft l sll suh tht > d > F d vrt vrt ot strt ov oto of > ± d 3 pots P. 2

19 ssu whr Lt Th ' N s, N N s, th optl optl ' vrts vrt wth lowst ost. ' N soluto. soluto. of P ' -λ λ -λ λ 2 3 λ (-λ ) 4 λ (-λ )(λ 2 2 (-λ 2 ) 3 ) λ (-λ )λ 2 2 (-λ ) (-λ 2 ) 3 α α 2 2 α 3 3 α α 2 α 3 λ (-λ )λ 2 (-λ ) (-λ 2 )

20 > othrws,, è, d, è w FS othrws,, FS ) th F s uoudd (è,, If è os ) è ( so utl è Irs è ) è ( (2) è () (2), () },..., : { ss FSfor Lt l l l

21 { } ss w s,...,,,,..., l l [ ] ss s lrl dpdt r of olus ll,...,, os (3), s th, st (3) w f r lrl dpdt,...,, (3) ) (,...,,,,..., Lt,,,, d d d d d d d d d d d l l l l l l l l l l l l l l l l > d d FS FS2

23 { } ,,, 3, 3 4,,, 3 FS [ ]' M s o - os 3 ) 2 2, 3 ( w ss, to To put 5 4, 3, > l

24 ( ) [ ] z z z Proof Thor, for... t optu r th w, for ll If : ost : : 2 z

25 [ ] [ ] [ ] [ ] [ ] [ ] optu glol s r ) s vrls (o -, For E z E E L L L L L L M O M L L L L M O M L M

26 produr g d opt : ' o';uoudd : ' o'; (wh thr os f ls g f d hoos ls spl for ll fdè > d pvot o ' s' whl opt ' o' d uoudd ' o' do for ll th opt : ' s' suh tht l th lgorth trts) < ; th uoudd : ' s' l l

27 4.. ot ss ( 3 2 ( ) ) 3 ( 2 ( ) 3 ) dtt tr

28 5. slt olu tht gvs ost gtv 6. oput > 7. pvot 8. rpt stps 3-7 utl for ll or ost gtv (or oputto)

29 ,,..., s vrls,othrws o s vrls orgl prol

30 d ll 's r drv out of th ss : o > : o fsl soluto to th orgl prol ut so 's r th ss : otu pvotg utl w gt ss wth th orgl vrls I th lst s, < pvot us (rll ( ) )

31 produr two - phs g Phs I : Phs II : d fsl : ' o';rdudt : ' o'; (Phs I st ths trodu rtfl ll spl w th ost f > ls g d opt to ' s' ) ss, Phs I th fsl : ' s' f rtfl vrl s th ss d ot drv out ( th rdudt : ' s' d ot th orrspodg row; ll spl wth orgl ; ; ost )

32 ξ ξ

33 ξ ξ

34 ξ ξ Ths pl gvs optu pot wh Phs I s fshd ( )

39 f. g - Th g. Lt gtv..so Tht s. soluto to Lt ( ) [ ] [ ] [ ] [ ] [ ] [ ] f g g g

40 Colu slto : Choos th lowst urd Th th { : z < } Row slto : : rg l l t s of t > I s of t, hoos th lowst urd olu to lv th ss lgorth trts ftr fvorl olu ft ur of lrgst d of vrl trg th ss durg th l out to tr th ss pvots. rtrr oss olu s.t. out to tr th ss > lp out to lv th ss ê p ê lp ĉ

41 [ ] [ ] [ ] Th othrws ot th l s,, f ) s th l (,, f f Lt p p p p p p p p p p

42 [ ] [ ] [ ] [ ] [ ] [ ] p p p p p p f g f g p Thrfor, s th l. or wh Not tht whr l) (fro th prvous othrws s ot th l,, f ) s th l (,, f f Lt l th ot

43 Howvr, st tr : 3rd tr : f f d d ust lss th for olu p th l q s th l d us sltd olu q s th rghtost olu th l), p p,,, s th l,,, s th l 2d tr : q p, f <, q p p lp < s s q q > ot t th l ot t th l q q q ( ot th l < ol for to tr th > ss otrdto q q, ( )

44 [ ] N N M M Dfto E z ð uostrd ð ð uostrd ð ð Dul Prl R,ð fsl soluto toð s ð th tr for), ( wth ss soluto optl If L

45 Thor If prl hs optl soluto, th dul hs optl soluto d thr osts r qul. Proof Lt Th. ( ( ) () d (2), s th l possl ost th dul. s optl th dul. C tdd to grl for of LP. ) () optl ost th prl. (2)

46 uostrd uostrd ) ( ) ( ) ( ) ( uostrd Proof th prl. s th dul of dul Th Thor dul (-) (-) N N M M uostrd uostrd Dul Prl

47 optl,, optl, uostrd uostrd Proof ) ( ) ( th dul :optl prl :optl Thor (opl tr slss) v u v u v v u u v u v u

48 Solv th dul for optu ð t optu, f ( Th solv ð If ( -) or or ( Othrws, ( ( ð ð ) for othr ð ) >, th. 's. ) >, th ( -) or or. ) for or th 's. rdud Thor (opl tr slss) :optl prl u ð :optl th dul v ð ( ( ð ) )

49 Shortst pth prol d ts dul w t s - s -w - s -w 2 - -w 3 t- -w 4 t- -w 5 s uostrd [ w w w w w ] 2 [ - ] 3 : flow fro vrt : flow dg, or : ost of dg Solv logst pth prol wth gtv wght - Dul [ ] [ - ] [ - ] s 4 t 5 s, t,, s t s w w w w w t

50 4 3 s t

51 Th t optl dul z, R soluto [ L ] Dul Spl lgorth : tg towrds prl fslt ( ) uostrd s E wth ss ( tr for), fsl soluto to Spl lgorth : tg prl towrds dul fslt ( ) fsl soluto ( ) worg dul fsl soluto ( ) worg

52 ( ) ( ) ss. tr th olu to th fd th d frst row Choos rg (2), d () Fro (2), fslt dul t To (), rs, to ost For th : pvot Sltg prl th of opot fsl to g orrspod row Slt soluto fsl) prl (ot fsl dul s th tlu w hv w ssu lgorth Spl Dul : l l l l l l z z z l l l l l < < < <

53 rdudt 4 3 s t

Extension Formulas of Lauricella s Functions by Applications of Dixon s Summation Theorem

Extension Formulas of Lauricella s Functions by Applications of Dixon s Summation Theorem Avll t http:pvu.u Appl. Appl. Mth. ISSN: 9-9466 Vol. 0 Issu Dr 05 pp. 007-08 Appltos Appl Mthts: A Itrtol Jourl AAM Etso oruls of Lurll s utos Appltos of Do s Suto Thor Ah Al Atsh Dprtt of Mthts A Uvrst