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Generic Asymptotics of Eigenvalues and Min-Plus Algebra

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(1)Generic Asymptotics of Eigenvalues and Min-Plus Algebra Marianne Akian, Ravindra Bapat, Stéphane Gaubert. To cite this version: Marianne Akian, Ravindra Bapat, Stéphane Gaubert. Generic Asymptotics of Eigenvalues and MinPlus Algebra. [Research Report] RR-5104, INRIA. 2004. �inria-00071479�. HAL Id: inria-00071479 https://hal.inria.fr/inria-00071479 Submitted on 23 May 2006. HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Generic Asymptotics of Eigenvalues and Min-Plus Algebra Marianne Akian — Ravindra Bapat — Stéphane Gaubert. N° 5104 February 2004. ISSN 0249-6399. ISRN INRIA/RR--5104--FR+ENG. THÈME 4. apport de recherche.

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(7)  .  . . Unité de recherche INRIA Rocquencourt Domaine de Voluceau, Rocquencourt, BP 105, 78153 Le Chesnay Cedex (France) Téléphone : +33 1 39 63 55 11 — Télécopie : +33 1 39 63 53 30.

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(37) ˆH‰.   B      

(38) @$ 9 

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(40) . 3. 7. ⊕ 5Y2 ⊕ 6Y ⊕ 13. hro w c j thrtltymsz"cihkjl] w.

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(42) ˆPˆ. 

(43)  !"#$ $#$. .ëo£¯hrz"j cdo jlZ/]>ty]HžXe/] g ]tuZ‚hvgdg\^mPtyjugdx e‚ty]p\^cdos³´q‚gne‚tŸasz ZEe/}Ÿz mr\^q/gd] \^] oXjŸz"mr}l}l]8tuq@mPo w cdo/ jlm λ = ρ ¥ (A) ¨ YZ‚]ŸrmPhr¥ g<¡ mv£WjlZ/]Ÿcdo‚ty]8}yjlcnmPomr£jlZ/]Ÿo/mP}u\Ÿhvgdcd² cdo/“£¯hPz±jlmr} t&cdo ®ˆH‰P°&cdt„jumpr] j„juZ/] £¤mPgngdm ¡ cdo/œZ‚mr\^mrP] o/]8c jxŸq/}lmrq6] }ujx , ®ˆPˆH° Schur(C, µλ, µA) = µ Schur(C, λ, A) , £¤mP}&thvgdyg cdλ,o/µjlZ/∈]&Rtlhvty\^e‚]¬z ZtuxEju\fZ@hkŒ6j mrλg ≤ ρ (AŒ6mvjuZ) £¤hvmro }w z µλmroE¢r≤]8oXρjucdmro‚hr(Ag‚hvo w )\$¨ cdos³9q/gde‚tasz ZEe/}ªz"mP\$q‚gn]8\$]8oXjlt cito/mvjhv\fŒ‚cnPe/mre‚t ,<z"mro@tyc w ] }l¥cdo/ „Schur \^cnos³9 q/¥ gde‚tasz ZEe/}z mr\^q/gd] \^w ] oXj tWmr£¶z"mr\^q/gd]"{Ÿ\Ÿhkjl}uciz"]Ht ¥ mr}ªz"mPoX¢P] os³ jlcnmPo‚Ì hvmvg¶jla/Zpz ZX\$e‚cd}osz"³9q/mrgd\^e‚tq/gdhv] o\^w ]8oPz j mrtoE¢rmv]8£ oP\$jlcncdmPoso‚³9q/hvg­gde‚astz ZE\Ÿe/hk}jl}uz cimrz"\^]Ht q/¥ gd¡ ] \^mre/] goXjltŒ@tl]„hk\^jlcdty]8£¤hrx o/cdo/rgd]8tlt ¨ ®ˆOr° Schur(C ∪ C , a) = Schur(C, Schur(C , a)) £¤mP}œhrgng L × L \^hvju}lcdz ]8t a ¥ hro w £¤mrw}$hvgdg w w cdt¯~mrcdoXj$tue/Œ‚tu]"jltœmv£cdo w ciz"]Ht w C, C w ⊂ L ¥ q‚}umk¢Ec w ] w jlZ‚hkw j jlZ/w ]Ÿasz ZEe/}„z"mr\^q/gd] \^]8oPj tw hv}l] ¡ ] gdg ] '‚o/] ®¯c £ Schur(C cit ]8gng ] '‚o/] juZ‚] oTjlZ/]^gn] £Ñj„Z‚hvo tuc ]fmr£$®ˆOr°¬]"{scitj t¬cn£ ¥ hvw o mro/gdx>c £ ¥ cnjltµ}lcnPZXj¬Z‚hvo w tuc w ]$,]"a){scitj tl° ¡ ¨ 4 £Wz"mre‚}ltu] ¥ ¥ ®ˆOr°¬cdtµh“z w gdhPtutuciz hvg \^Fhrcdose‚³9tlq/tygncie@hvt ohr]8gngnPcd\^] Œ/cn} o@hhkju®¤cdjumrZ/o]ªcgn] £Ñ] jCoXZ‚jucnhvjo x w¥ ¡ tyc Z/w ci]ªz Z“hvo citw ¡ juZ/]8]ªgngÅ}u©Ecdro/ZXm j ¡ Z‚o hv¥ o Œ6w mvjltyZ“c w cd]ªomvz £mr®oEˆH¢rP]8°¶oPhrjl}ucn]mPo‚e/hvo‚g6hrhv\fgdrŒ/]8cdŒ/r}le/hfmPe‚hvto } hkcnjlocnmPjuo‚Z/hv] g ] {sq/}u]Htutucdmro‚t ¥ ¡ cnjuZSw ] gd] \^]8oPj hv}lxpcnoXju]8}uq‚}u] jlhkjlcnmPo‚t&cnoju]8}u\Ÿtµmv£Wq@hkjuZ@t ¥ tu] ]œ£¤mr}µcdo‚tj hvo‚z <] ( ½¶hrg 6v2 )<£¤mr} \^mP}u]Œ‚hPz ©XP}umPe/o ° ¨ w Š cdo‚hvgdgnx ¥ cn£ K ⊂w L hvo c £ b citjuZ/] K ×K tue/Œ/\Ÿhkjl}ucn{„mv£ a ¥ ¡ ]ªtuZ‚hrgngstumr\^]"jlcn\^]8t ¡ }lc jl]ªhrŒ/e‚tucn¢P] gdx Schur(b, š ]o/a)m ¡ ¥ Pcdo‚cn¢Ptjl]ª]8tuh mr\^mr]£ rSchur(K, } hvq/ZœcnoXjl] a)}lq/¨}u] jlhvjucdmro‚tmv£@w juZ/] ¡ ] cdrZXjlt<hro w ]8cnP] oE¢khvgde/]8t mr£@w \^cno/³´q/gde‚t«asz ZEe/} z mr\^q/gd] \^] oXj t ¨ ½]"j G Œ6]ªh¬r} hvq/Z ¡ cnjuZ$ty] jw mr£‚o/m ]8t L ¥ gd]"j C Œ@]h¬tue/Œ‚tu]"j<mv£ L hvo ty] j N = L\C ¨ mr£ G ¥ ¡ ] ]8o/mvjl]œŒEx |p| jlZ/Z/]8]^}u] oEe/\fw Œ6] ] o/}mrmvju£ª]8t¬hv} juz Z‚t&]$mvz8£ hvp} w ¡cno@c hvjlZgdc jcdxo/cnmvju£Wcihvh g . ,i ) o‚Š‚mmr}w ]fhvgdcdg<o q‚Chkjl¥ Z‚c t ¨ ] p¨ ¥ =|p|(i =, . .#{0 tŒEu]"x j ¨ '‚® o‚ hrgdg g ju° Z/]$š q‚]ŸhvjuhvZ¦gitymcdoXjuw ]8]8}uo/q/mv}ljl]"]œj hkŒE≤jlx cnmPmo‚t≤Œ6] kgdm −jl¡ Z/]Ÿ1Z‚ty|hOe/¢PiŒ@] tyw]H∈žPe‚e‚hrC}] g o‚¢Pz ¥] ]œ} ¡ tymrcdmr£ o@t ¥ mr#mrŒsŒsj hvj hvcdo/cdo/] w] w ŒXŒXx xpw }l] ] gd]"q/jlgicnhro/z cno‚ju Z‚]$cdo/o‚cnmjuciw hv]8g t o‚mvj cno C ¨ ® p ∩ C o/] ] w o/mrw jµŒ6]Ÿh“pq@hk∩juZSC mv£ G° ¨ YZ/]$£¤mPgngdm ¡ cno‚p z"gihrtltucdz8hvg cnoXju]8}uq‚}u] jlhkjlcnmPoTmv£asz ZEe/} z mr\^q/gd] \^] oXj tcdthrocd\^\^] cihkjl]z mro‚tu]8žXe/]8o‚z"]mr£¶juZ‚]„r} hvq/ZcnoXjl] }lq/}u] jlhvjucdmro“mr£jlZ/]tyjlhv} ¨ ! (  G<  #    VU   !   — 4,687  6&  min. min. CC. min. 0. CC. 0. 0. 0. 0. k. C. C. . m. .  =    . C ⊂ C A.   Lλ ∈ R \ { } !   :  N   = ρL \(A  L  × L     R ) ≥ λ min  # 9 H min CC   G(Schur(C, λ, A))       U !     pmin   G(A)  5   p U    0        =  p0 ∩ N = p  !        7 p   G(Schur(C, λ, A))   p . . . . . ). |p|Schur(C,λ,A) = min |p0 |A − λ|p0 |C , $  :        G   2:         0   :    U    G(A)   2  :   ! 0           p              # 9  p      U>  p ∩  N  = cp0   G(A) !  5 c c0 ∩ N = c  #G(Schur(C,          λ,  A))  c    &  2 G(Schur(C, λ, A)). . $    9   . ÏÏ ØI‚âHá !J . .  ". . . !. |c|Schur(C,λ,A) = min |c0 |A − λ|c0 |C ,  

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