Asymptotics of the Perron Eigenvalue and Eigenvector Using Max-Algebra

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(1)Asymptotics of the Perron Eigenvalue and Eigenvector Using Max-Algebra Marianne Akian, Ravindra Bapat, Stéphane Gaubert. To cite this version: Marianne Akian, Ravindra Bapat, Stéphane Gaubert. Asymptotics of the Perron Eigenvalue and Eigenvector Using Max-Algebra. [Research Report] RR-3450, INRIA. 1998. �inria-00073240�. HAL Id: inria-00073240 https://hal.inria.fr/inria-00073240 Submitted on 24 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. Asymptotics of the Perron eigenvalue and eigenvector using Max-algebra Marianne Akian , Ravindra Bapat , Stéphane Gaubert. No 3450 Juillet 1998 ` THEME 4. ISSN 0249-6399. apport de recherche.

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(4) Asymptotics of the Perron eigenvalue and eigenvector using Max-algebra . . Marianne Akian , Ravindra Bapat , Stéphane Gaubert Thème 4 — Simulation et optimisation de systèmes complexes Projets Meta2 Rapport de recherche n˚3450 — Juillet 1998 — 12 pages. Abstract: We consider the asymptotics of the Perron eigenvalue and eigenvector of irreducible nonnegative matrices whose entries have a geometric dependance in a large parameter. The first term of the asymptotic expansion of these spectral elements is solution of a spectral problem in a semifield of jets, which generalizes the max-algebra. We state a “Perron-Frobenius theorem” in this semifield, which allows us to characterize the first term of this expansion in some non-singular cases. The general case involves an aggregation procedure à la Wentzell–Freidlin. Key-words: Perron-Frobenius Theorem, Max-algebra, Perturbation of eigenvalues, Perturbation of linear operators, Asymptotics, Freidlin-Wentzell theory, Large deviations. (Résumé : tsvp) . . . Email: [email protected] Indian Statistical Institute, New Delhi, 110016, India. E-mail: [email protected] Email: [email protected]. Unité de recherche INRIA Rocquencourt Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France) Téléphone : 01 39 63 55 11 - International : +33 1 39 63 55 11 Télécopie : (33) 01 39 63 53 30 - International : +33 1 39 63 53 30.

(5) Asymptotiques de la valeur propre et du vecteur propre de Perron via l’algèbre max-plus Résumé : On s’intéresse à l’asymptotique de la valeur propre et du vecteur propre de Perron de matrices à coefficients positifs ou nuls, dépendant géométriquement d’un grand paramètre. Le premier terme du développement asymptotique de ces élements spectraux est solution d’un problème spectral sur un semi-corps de jets, qui généralise le semi-corps max-plus. Nous établissons un “théorème de PerronFrobenius” pour les jets, qui nous permet de caractériser le premier terme de ce développement dans des cas non-singuliers. Le cas général requiert une procédure d’agrégation à la Wentzell–Freidlin. Mots-clé : Théorème de Perron-Frobenius, Algèbre max-plus, Perturbation de valeurs propres, Perturbation d’opérateurs linéaires, Asymptotiques, Théorie de Freidlin-Wentzell, Grandes déviations.

(6) Asymptotics of the Perron eigenvalue and eigenvector using Max-algebra. 1. 3. Introduction. Let denote a nonnegative matrix, depending on a large real parameter . We consider the nonnegative spectral problem:.

(7) . "!# $%& (1) where denotes the set of nonnegative real numbers. When is irreducible, is unique, and it is called the Perron eigenvalue of (see e.g. [4, Ch. 2]). We call normalized Perron eigenvector the unique ' that satisfies (*)+ , )

(8) .- . In this. note, we adress the following problem: can we determine the asymptotic behavior of and / from that of ? We begin with an elementary large deviation type result, which extends the re sult given in [10] for

(9) 01 )32 . T HEOREM 1 (L ARGE. DEVIATION OF. ).. If the limits. < = )1 32 def. (2)

(10) 8 4+7:5+6 9 ; )32 exist for >?@A

(11) B-;C;C;CD8 , and if 1E

(12) 01 )32 is irreducible, then < T< C = ) W ) V W ) V Y ) X ) ) T < < 48+7:5+6 9 F,

(13) GLN6IMPOQHKM J 6I K H J 1 1 Z C ; C ? C 1 (3) ) <RSRSR )UT < = ) 2 ) Indeed, !\[]^ $Z [_(`2;^ /,

(14) G- . Hence, + /, , which is bounded, has a < = limit point !a[cb ) [B- , and 6IH,J 2 b 2

(15) B- . It follows from (1) that d' also has a limit point e , which satisfies 6I2 H,J 1 )32 b 2

(16) fegb ) for >h

(17) E-i;C;CZCD8jC (4) Now, it is convenient to introduce the max-times semifield 1 kmlNn

(18) ?6IH,J#"8!#;-, . We recognize in (4) a spectral problem for the matrix 1 in A semiring o^pqrsqtsqu,q?v w is a set p equipped with two laws o+xiqyNwz{BxKr$y , o+xqyNw|z{&x,t$y , called addition and multiplication, respectively, such that o^pqrsqu}w is a commutative monoid, o+pqtsq?v w 1. is a monoid, the multiplication distributes over the addition, and the zero element 0 is absorbing for multiplication. A semifield is a semiring whose non zero elements have an inverse. In any semiring, we can define the matrix multiplication as usual (e.g. in , ).. 8 . RR n˚3450. ~|d oU$ w Ih t .

(19) 4. Marianne Akian , Ravindra Bapat , Stéphane Gaubert. the semifield kDlNn . The kDln analogue of the Perron-Frobenius theorem states that an irreducible matrix 1 has a unique eigenvalue, given by the maximal circuit mean kDlNn,1 , which, by definition, is the right hand side of (3) (see e.g. [2, Th. 3.100],[6, < kDlNn;1 holds for all limit points e of Fs, = . This §VI],[11, §3.7]). Thus, ej

(20) proves Theorem 1. < The above argument does not guarantee the convergence of + = , except when all the eigenvectors of 1 are proportional: this simple case is dealt with in §2. In §3, we show that if the non-zero entries of have asymptotic expansions of the form. , )32 ) 2 1 )32 . (5). then has an asymptotic expansion of the same form. This expansion is the unique eigenvalue of P)32 1 )32 , seen as a matrix with entries in a semifield of jets. When all the eigenvectors of the later matrix are proportional, the entries of : also have asymptotic expansions of the form (5). However, in general, (5) need not imply < the existence of the limits example:. b ) def

(21) 4+5^67:9+ , ) = , as shown by the following counter.

(22) 4+5+6`7:95 ++ ,, L. . =<. -

(23) }^ -

(24) . L. 4+5^687:

(25) "9 !$# ++ ,, L%. =<.

(26) C. In [1], we prove via an extension of the Puiseux expansion theorem that when the entries of have Dirichlet series expansions (see [13],[14, Ch. VI]), " and the entries of / also have Dirichlet series expansions. Then, a fortiori, the limit b

(27) b ) exists. It can be computed using an aggregation procedure. In §4, we only present the first step of this procedure, which is enough to determine b in some non-singular cases. The problem of computing the limits e and b arises in particular in Statistical Mechanics, when using the transfer operator methods at small temperatures &

(28) -'8 (see e.g. [3],[5]). Some of the results given below can be seen as partial extensions to the case of nonnegative matrices of the classical Freidlin-Wentzell singular perturbation results [9, Ch. 6] which deal with the special case of Markov. INRIA.

(29) Asymptotics of the Perron eigenvalue and eigenvector using Max-algebra. 5. matrices . Other max-algebra related (W.K.B. type) asymptotic results have been obtained in [7]. The proofs of the results presented here will be detailed in [1].. 2. When max-times spectral theory determines the asymptotics. Let A8!P;-K denote an arbitrary semiring. With a matrix 1 with entries in , we associate (as in conventional Perron-Frobenius theory) a digraph '1 with !

(30) . We say that 1 is irreducible if nodes -;C;C;CD8 , and set of arcs P>?@ 1 )32

(31) B '1 is strongly connected. K

(32) When the reflexive and transitive relation , defined by , is an order relation, and in particular, when \

(33) *AkDln , we defineO the Kleene star as the least upper bound of the monotone sequence LNMPOQM L , when it exists.

(34) B> L ;C;C;CD> O is critical if When is the kDln semiring, we say that a circuit < T its mean geometric weight 1 ) < )WV 1 )WV)WX C;CZC81 ) T ) < attains the maximum in the right hand side of (3). The critical graph !"s1 is the subgraph of '1 , composed uniquely of the nodes and arcs in critical circuits. The strongly connected components L of the critical graph are called critical classes. We set 1_ #

(35) kDl nZ1 1 . Then, $1# exists. A column of %1# , whose index lies in a critical class, is called critical. The max-times spectral theorem (see [2, Th. 3.100], [6, Th. VI.10], [11, §3.7] and the references therein) states that if we select (arbitrarily) one critical column per critical class, we obtain a minimal generating set of the eigenspace of an irreducible matrix 1 . As an application of this result, we obtain: T HEOREM 2 (L ARGE DEVIATION OF ). If satisfies the assumptions of Theorem 1, and if 1 has a unique critical class, then. < = 4^7:5+6 9 ^ /, )

(36). . 1 # )3 2 O $1# O 2. for >h

(37) B-;C;C;CDj. where @ is an arbitrary node of this critical class.. RR n˚3450.

(38) 6. Marianne Akian , Ravindra Bapat , Stéphane Gaubert. O. . L $1# , and that it can be computed in Recall that %1# is equal to MPOQM , time using semiring versions of Gauss algorithm (see [2, Th. 3.20] and [12, Ch. 3,Algo. 3], respectively).. . 3. When the spectral theory of max-jets determines the asymptotics. . We denote by kDln the semifield with set of elements !

(39) P!P8! , equipped with the two laws.

(40) . ,A . , $

(41). ,A , . , A. if if if. P ,A .

(42) . !P . . . . (6). ,A , $

(43) 0 K C The zero element !#8! and the unit -i;-, will be denoted by !#Z- , respectively. This semifield was introduced in [8]. It is isomorphic to the semifield of asymptotic expansions of the form around

(44) , equiped with the usual addition. . and multiplication. We will say that a nonnegative real valued function of a large parameter has a first max-jet K A , and we will write h^ K A , if + :

(45) around

(46) (when K A

(47) `! , this means that h+|s

(48) *! for parge enough). The above definition and notation will be extended to matrices and vectors (entrywise). If and / have first max-jets f mkDln8 K and ` kmlNn8 respectively, it follows from (1) that ' has also a first max-jet B k l n , which satisfies

(49) . Thus, the max-jet of $ , if it exists, will be characterized in the particular cases when all the eigenvectors of are proportional. We next state a kDln analogue of the Perron-Frobenius theorem. For any subgraph ! of the digraph associated with a matrix 1 with entries in any semiring, we denote by 1 the matrix with entries 1 )32

(50) 1 )32 if >?@ is an arc of ! , and 1 )32

(51) ! otherwise. Given an eigenvector b BkmlNn? of a matrix 1j` kDlNn? , , the saturation graph m1}b is the subgraph of '1 with set of arcs P>?@ 1 )32 b 2

(52) kDlNn,1 b ) .. . . &.

(53) #. . . . $#. '. !. . . . . ". . %#. INRIA.

(54) Asymptotics of the Perron eigenvalue and eigenvector using Max-algebra. 7.

(55) G 81 mk lNn8 ! K . Clearly, has an eigenvector

(56) .D}bI kDln8 , with eigenvalue *

(57) c ?e , iff

(58) I

(59) (7) 1b0

(60) fegbs (the first identity is a spectral problem in kDln , the second identity is an ordiLet. nary nonnegative spectral problem). The saturation graph in general depends on the particular choice of b , but when 1 is irreducible, for all eigenvectors b of 1 , ! '1 m1 b , and any circuit of m1}b is contained in !"'1 . The matrix

(61) is block diagonal, the blocks being exactly the critical classes. We call basic classes of j

(62) B 81 the basic classes of

(63) in the usual sense, i.e. the classes with maximal Perron eigenvalue. We denote by ; the usual Perron eigenvalue of a matrix .. '. #. '. . !. #. T HEOREM 3 (“P ERRON -F ROBENIUS T HEOREM ” FOR MAX - JETS ). An cible matrix

(64) E 81 mkDln8 K admits the unique eigenvalue . a def

(65) . #.

(66). kDlNn;1? C. . irredu-. (8). The characterization of the eigenspace is more subttle in kDln than in kDln . We will only need the following simple result.. . !. T HEOREM 4 (G EOMETRIC SIMPLICITY OF THE EIGENVALUE ). An irreducible matrix

(67) 81g* kmlNn K has a unique eigenvector (up to a proportionality factor) iff it has a unique basic class. An eigenvector ]

(68) D}b is obtained as follows: b is a column of 1# , whose index belongs to the basic class; is a positive eigenvector of

(69) . As a consequence of Theorems 3 and 4, we obtain:. . !. T HEOREM 5 (F IRST kDln8 K , then. ORDER ASYMPTOTICS ).. a%C Moreover, if has a unique basic class, then ' unique eigenvector of with sum - .. RR n˚3450. . If. . . has a first max-jet. (9). has a first max-jet, which is the.

(70) 8. 4. Marianne Akian , Ravindra Bapat , Stéphane Gaubert. Aggregated matrix. When the matrix has several basic classes, the determination of the limit eigenvector relies on an aggregation procedure, the first step of which we next describe. We denote by L ;C;CZCD the basic classes of . We set

(71) LNM ) M ) L and

(72) P-;C;C;CD8 I . Let ;C;C;CD be eigenvectors of the submatrices. < < ;CZC;CD8

(73)

(74) , respectively (for all subsets P-;C;C;CD , denotes the submatrix of ). The following key lemma is a consequence of the fact that I[ implies

(75) , for all irreducible nonnegative matrices and nonnegative vectors (see e.g. [4, Ch. 1,Th. 3.35]).. L EMMA 6. Any eigenvector of an irreducible matrix form

(76) , for some kmlNn8 , where. . . k l n ! K. is of the. . L L L (10) a a diag ;C;C;C C Here, is the identity matrix, and for all (possibly rectangular) matrices L the (possibly rectangular) block diagonal ZC;C;Cm O , diag L ;C;C;CD O denotes O L matrix with diagonal blocks ZC;C;Cm . In the sequel, we will identify the max-jet (resp. the matrix of max-jets) , A ) 2 )3 2 ). This allows us to write \

(77) to the function ! (resp. !. #"

(78) %$ , where 'a denotes the subgraph of ! 'a with set of nodes . In general, the remainder matrix $ has negative entries. Let & L ;C;C;CD&' denote the left eigenvectors of the submatrices. < < ;CZC;CD8

(79)

(80) , respectively, normalized by the condition & ) )

(81) - , for >

(82) -i;CZC;CD ( . We set &

(83) diag & L ZC;C;CD & * ) , ! + (! is the zero matrix). Left multiplying /

(84) ` by & , we obtain &-$ /

(85) cF/ . a 0 & C .

(86). def. Using Lemma 6, we obtain the following result. T HEOREM 7 (S ECOND ORDER and $ , respectively, then. $

(87). . ASYMPTOTICS ).. a . . If. . and $a have first max-jets. ? . (11). INRIA.

(88) Asymptotics of the Perron eigenvalue and eigenvector using Max-algebra. 9. . where

(89) '&-$ j kmlNn8 . Moreover, if has a unique basic class, then / has a first max-jet, which is the unique vector with sum - of the form ,< being an eigenvector of , and being defined in (10). def. Example 8. Consider the transfer matrix of the simplest one dimensional Ising model [3, Ch. 2]. . J#m & ' & J# . ' & & J #D . ' J#D . ' with `!P % C Setting

(90) can write the first J#D ]- ,

(91) J#D _ ! , c

(92) -' & , we < h L L < h L L. max-jet of as

(93) 1L with

(94)

(95) and 1

(96) ) L < + . We have a

(97) ?-?6IHKJ|0 ' ? . Thus, F6IHKJ|0 ' L ? . V When B ! , < L V V <

(98) P- , andV 1 # <

(99) ) + , $1#

(100) there is a unique critical class, ! L V < = V < L + . By Theorem ) 5, /

(101) L

(102) . By symmetry, if !, V < = L : the limit eigenvector at zero temperature is selected then

(103)

(104) by the sign of the external magnetic field . When

(105) ! , has two distinct critical classes ! L

(106) P- P !

(107) , that are both basic. Theorem 7 allows us to determine the equivalent of . Indeed,

(108) &

(109) (the identity matrix), L

(110) L <. #"

(111) L . We obtain $

(112) $]

(113) D

(114) L <

(115)

(116) . Thus, and

(117)

(118) L L L

(119) B -i ,

(120) 0 , and ) + . L

(121). . . ———. Version française abrégée Soit une matrice ' à coefficients réels positifs ou nuls, définie au voisinage de

(122) . On considère le problème spectral (1) dans le cas où A est irréductible : est unique et il existe un unique ' vérifiant ( ) + , )

(123) - (voir par exemple [4, Ch. 2]). On cherche à déterminer les asymptotiques de " et à partir de celles de . En utilisant les résultats analogues au théorème de Perron-Frobenius dans le semi-corps k lNn

(124) . ?6IH,J#"8!#;-, , isomorphe au semi-corps max-plus (voir par exemple [2, Th. 3.100],[6, §VI],[11, §3.7]), on obtient les asymptotiques de type grandes déviations de s , et dans certains cas celles de .. $. RR n˚3450.

(125) 10. Marianne Akian , Ravindra Bapat , Stéphane Gaubert. T HÉORÈME 1. < Si les limites (2) existent et si 1

(126) 1 )32 est irréductible, alors = 4+5^687:9AF, existe. Elle est égale à la valeur propre de 1 dans AkmlNn , notée kDlNn,1 , donnée par le second membre de (3).. T<. Un circuit

(127) > L ;C;C;CD> O est dit critique si 1 ) < )WV 1 )WV)WX C;CZC81 ) T ) < réalise le maximum dans (3). On appelle graphe critique le graphe orienté formé des nœuds et arcs des circuits critiques. On appelle classe critique une L composante D k N l n fortement connexeO du graphe critique. On pose 1 #

(128) 1 1 , et on note 9 $1#

(129) O 1# l’étoile de Kleene de 1 # (la somme et les puissances sont dans kDln ).. 1 n’a qu’une = classe critique, alors 4^5+6 7:9 ^ )

(130) b ) où b

(131) Nb ) est l’unique vecteur propre de 1 dans kDln vérifiant 6 HKJ ) b )

(132) - . Celui-ci est proportionnel à n’importe quelle colonne de 1# d’indice critique. T HÉORÈME 2. Si. . . satisfait aux< hypothèses du théorème 1, et si. . . Afin d’obtenir des asymptotiques plus précises, on utilise le semi-corps de jets kDln (introduit dans [8]) composé de l’ensemble des couples K A , avec , c! ou

(133)

(134) ! , muni des lois (6). On dit que la fonction de admet un premier max-jet si +

(135) autour de

(136) . On note alors h^ K A . On étend cette notation aux vecteurs et matrices (coordonnée par coordonnée).. . . . . . $. . . !. T HÉORÈME 3. Si

(137) 1 kmlNn K , avec irréductible, alors a où a%

(138)

(139) kDlNn 1? est la valeur propre de dans kDln ,

(140) est la matrice obtenue à partir de en annulant les coefficients )32 tels que l’arc >?@ n’est pas dans le graphe critique, et où désigne la valeur propre de Perron. Si

(141) n’a qu’une classe basique, alors , l’unique vecteur propre de dans kDln de somme - . Celui-ci est de la forme D}b , où b est une colonne de 1# d’indice basique et est un vecteur propre positif de la matrice

(142) , P 3 ) 2 3 ) 2 2 obtenue en annulant les tels que 1 b kDlNnZ1 b ) .. #. #. #. . #. On appellera classes basiques de les classes basiques de

(143) . Si admet les classes basiques L ;C;C;C , alors tout vecteur propre de dans DkDln s’écrit

(144) , où est donné par (10). Dans (10), L ZC;C;CD sont des vecteurs propres de < < ZC;C;CD8

(145)

(146) , respectivement, désigne la f sous-matrice de ,. . INRIA.

(147) Asymptotics of the Perron eigenvalue and eigenvector using Max-algebra. 11. LNM ) M ) ,

(148) P-i;CZC;CD8 , et l’étoile dans kmlNn est définie par la même formule que dans kmlNn . Soit &

(149) diag & L ;CZC;CD& ) ! + , où & L ;C;C;CD& désignent les vecteurs propres à gauche de < < ZC;C;CD8

(150)

(151) , respectivement, vérifiant & ) )

(152) B- . T HÉORÈME 4. Si j kDln8 ! K et $a

(153) .\ $ mk lNnQ &, , alors admet le développement asymptotique (11), où

(154) &-$ mk lNn? . Si n’a qu’une classe basique, alors , l’unique élément de kmlNn? de somme - , de la forme , où est un vecteur propre de , et est donné

(155). par (10).. References [1] M. Akian, R.B. Bapat, and S. Gaubert. Asymptotics of the Perron eigenvalue and eigenvector. In preparation, 1998. [2] F. Baccelli, G. Cohen, G.J. Olsder, and J.P. Quadrat. Synchronization and Linearity. Wiley, 1992. [3] R.J. Baxter. Exactly solved models in Statistical Mechanics. Academic Press, New York, 1982. [4] A. Berman and R.J. Plemmons. Nonnegative matrices in the mathematical sciences. Academic Press, 1979. [5] W. Chou and R.B. Griffiths. Ground states of one dimensional systems using effective potentials. Phys. Rev. B, 34:6219–34, 1986. [6] R.A Cuninghame-Green. Minimax algebra and applications. Advances in Imaging and Electron Physics, 90, 1995. [7] S.Yu. Dobrokhotov, V.N. Kolokoltsov, and V.P. Maslov. Quantization of the Bellman equation, exponential asymptotics and tunneling. In V.P. Maslov and S.N. Samborski˘ı, editors, Idempotent analysis, volume 13 of Adv. in Sov. Math. AMS, RI, 1992. [8] A.V. Finkelstein and M.A. Roytberg. Computation of biopolymers: a general approach to different problems. BioSystems, 30:1–20, 1993.. RR n˚3450.

(156) 12. Marianne Akian , Ravindra Bapat , Stéphane Gaubert. [9] M.I. Freidlin and A.D. Wentzell. Random Perturbations of Dynamical Systems. Springer, 1984. [10] S. Friedland. Limit eigenvalues of nonnegatives matrices. Linear Alg. and Appl., 74:173–178, 1986. [11] S. Gaubert and M. Plus. Methods and applications of (max,+) linear algebra. In R. Reischuk and M. Morvan, editors, STACS’97, number 1200 in LNCS, Lübeck, March 1997. Springer. [12] M. Gondran and M. Minoux. Graphes et algorithmes. Eyrolles, Paris, 1979. Engl. transl. Graphs and Algorithms, Wiley, 1984. [13] G.H. Hardy and M. Riesz. The general theory of Dirichlet’s series. Cambridge University Press, 1915. [14] J.P. Serre. Cours d’arithmétique. PUF, 1977.. INRIA.

(157) Unité de recherche INRIA Lorraine, Technopôle de Nancy-Brabois, Campus scientifique, ` NANCY 615 rue du Jardin Botanique, BP 101, 54600 VILLERS L ES Unité de recherche INRIA Rennes, Irisa, Campus universitaire de Beaulieu, 35042 RENNES Cedex Unité de recherche INRIA Rhône-Alpes, 655, avenue de l’Europe, 38330 MONTBONNOT ST MARTIN Unité de recherche INRIA Rocquencourt, Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex Unité de recherche INRIA Sophia-Antipolis, 2004 route des Lucioles, BP 93, 06902 SOPHIA-ANTIPOLIS Cedex. ´ Editeur INRIA, Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France) http://www.inria.fr ISSN 0249-6399.

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