1 Lubrication model for the flow driven by high surface tension 1 Emilia Rodica Borşa, Diana-Luiza Borşa

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1 Contents 1 Lubrication model for the flow driven by high surface tension 1 Emilia Rodica Borşa, Diana-Luiza Borşa 2 Structures bicatégorielles complémentaires 5 Dumitru Botnaru 3 Some accelerated flows for an Oldroyd-B fluid 29 Ilie Burdujan 4 A fuzzy algorithm for reliability simulation of an electric station 49 Simona Dzitac, Tiberiu Vesselenyi, Ioan Dzitac, Maria Parv 5 A seven equation model for relativistic two fluid flows-i 59 Sebastiano Giambò, Serena Giambò 6 A general mountain-pass theorem for local Lipschitz functions 71 Georgiana Goga 7 Some results on simultaneous algebraic techniques in image reconstruction from projections Lăcrămioara Grecu, Aurelian Nicola 8 Branching equation in the root subspace for equations nonresolved with respect to derivative and stability of bifurcating solutions Irina V. Konopleva, Boris V. Loginov, Yuri B. Rousak 9 Solvability of Hammerstein equations with angle-bounded kernel 109 Irina A. Leca v

2 vi 10 Numerical k - busy periods algorithms for Polling systems with semi - Markov switching Gheorghe Mishkoy, Diana Bejenari 11 Fragile Bits vs. Multi-Enrollment - a case study of iris recognition on Bath University Iris Database Nicolaie Popescu-Bodorin 12 Bifurcation in a nonlinear business cycle model 145 Carmen Rocşoreanu, Mihaela Sterpu 13 Existence and uniqueness of fuzzy solution for linear Volterra fuzzy integral equations, proved by Adomian decomposition method Hamid Rouhparvar, Tofigh Allahviranloo, Saeid Abbasbandy 14 On a certain differential inequality 163 Roxana Şendruţiu 15 Note on a paper on nonlinear inverse time heat equation in the unbounded region Nguyen Huy Tuan, Dang Duc Trong, Pham Hoang Quan 16 Stability of elastic elements of wing profile with time delay of bases reactions Petr A. Velmisov, Andrey V. Ankilov

3 ROMAI J., 5, 2(2009), 1 4 LUBRICATION MODEL FOR THE FLOW DRIVEN BY HIGH SURFACE TENSION Emilia Rodica Borşa, Diana-Luiza Borşa University of Oradea, Romania Student of Jacobs University, Bremen, Germany eborsa@uoradea.ro, dianalborsa@gmail.com Abstract In this paper the flow of a thin fluid film, driven by a surface tension, down an inclined plane, is considered. By using the Navier-Stokes equations for thin film flow, the continuity equation, the no-slip condition and the boundary conditions, we obtain the horizontal velocity, the vertical velocity and the governing equation of the film height. In general, the introduction of surface tension into standard lubrication theory leads to a nonlinear parabolic equation. Keywords: fluid mechanics, lubrication theory MSC: 76M INTRODUCTION The lubrication approximation of the Navier-Stokes equations (NSE) has been used to describe a multitude of situations. Our attention has been focussed on the situations where surface tension plays an important role, such as: rain running down a window, the evolution of drying paint layers or the spreading of a fluid drop on a surface. Research on lubrication equations with non-negligible surface tension appears into two distinct situations. Firstly, there are the physical studies where the work is directly motivated by a specific problem. In this case, after deriving the equation for film thickness, the mathematical treatment is generally limited to asymptotic or numerical methods. Secondly, there are the mathematical studies which are not directly motivated by a specific problem but delve into the lubrication equation in greater detail. In this paper we study the flow of a thin fluid layer lies on a plane which is at an angle α to the horizontal. The flow is driven by a surface tension. In this case the flow is governed by the nonlinear degenerate parabolic equation. Implicit in the derivation of this equation is the assumption that surface tension and gravity effects are of the same order. 1

4 2 Emilia Rodica Borşa, Diana-Luiza Borşa 2. MAIN RESULTS Consider the flow of a thin-layer, of an incompressible Newtonian fluid with constant density ρ and dynamic viscosity µ, down a plane inclined at an angle α to the horizontal (fig.1). The flow is driven simultaneously by gravity and a surface tension gradient Σ = γ x. The velocity is u = u(x, z, t) i +w(x, z, t) k. Fig. 1. In the thin-film approximation the NSE reduce to [1], [4]: 0 = 1 ρ p x + νu zz + g sin α, (1) 0 = 1 ρ p z g cos α, (2) where p is the pressure in the fluid, g is the gravitational acceleration. Here z = h(x, t) is the unknown equation of the free surface and ν = µ ρ is the coefficient of kinematic viscosity. Here the subscripts denote differentiation with respect to the corresponding variable. The motion of the fluid is governed by equations (1)-(2) and some initial and boundary conditions. The non-slip conditions must be satisfied: u = 0, at z = 0. (3) On the free surface z = h(x, t) the condition that the normal stress be equal to the atmospheric pressure p 0 reduces to p = p 0, at z = h(x, t). (4) The condition for the tangential stress (Marangoni effect) at the free surface [2] is µu z = Σ, at z = h(x, t). (5)

5 Lubrication model for the flow driven by high surface tension 3 and the kinematic boundary condition [3] The continuity equation is w = h t + uh x, at z = h(x, t). (6) u x + w z = 0. (7) From equation (2) and condition (5), we obtain for fluid pressure For z = h we have and thus By integrating (1) we get for u p = ρ g cosα z + C 1. C 1 = p 0 + ρ g cosα h, p = ρ g cosα (z h) + p 0. (8) u z = 1 ρ ν p x z 1 ν g sinα z + C 2. The boundary condition (5) implies for z = h C 2 = 1 µ Σ + 1 ν (g sinα 1 ρ p x)h and u z = 1 ν (g sinα 1 ρ p x)(h z) + 1 µ Σ. By integrating the last relation we obtain u = 1 ν (g sinα 1 ρ p x)(hz z2 2 ) + 1 µ Σz + C 3 and by using the condition (3) the horizontal velocity is u = ρg ( Σ 2µ ( sin α + h x cos α) z 2 + µ + ρg ) ρg h sin α µ µ h x h cos α z. (9) This may be used in the continuity equation (7) to determine w w z = u x, w(h) = z 0 u x dz and by using for z = 0 the relation (3) we have for the vertical velocity

6 4 Emilia Rodica Borşa, Diana-Luiza Borşa w = ρg 6µ h xx z 3 cos α+ ρg 2µ cos α [ h h xx + (h x ) 2] z 2 ρg 2µ h x z 2 sin α. (10) This expression together with the kinematic condition (6) leads to the governing equation for film height h(x, t) ρg 3µ cos α ( h 3 ) h x x = h t + Σ µ h h x + ρg µ sin α h2 h x. (11) If the thin fluid layer lies on a plane which is at an angle α to the horizontal then a surface tension driven flow is governed by the nonlinear degenerate parabolic equation (11). Appropriate form of equation (11) have been used to model fluid flows in a number of physical situations such as coating, draining of foams and the movement of contact lenses. 3. CONCLUSIONS In the lubrication approximation we have considered the flow of a thin layer on an inclined plane. The flow is driven simultaneously by gravity and a gradient of surface tension. This gradient implies a non-zero tangential stress boundary condition (Marangoni effect). We have estimated the response of the fluid to such a stress. We have determined the horizontal velocity, the vertical velocity and the governing equation for the film height, which is a nonlinear degenerate parabolic equation. This equation can be linearized [5] and in some special cases we can obtain implicit solutions of the linearized system. References [1] D. J. Acheson, Elementary Fluid Dynamics, Oxford University Press, United Kingdom, [2] E. Chifu, C.I. Gheorghiu, I. Stan, Surface Mobility of Surfactant Solutions. Numerical analysis for the Marangoni and the Gravity Flow in a Thin Liquid Layer of Triangular Section, Revue Roumaine de Chimie, 29(1984), [3] P. Grindrod, The Theory and Applications of Reaction-Diffusion Equations, Patterns and Waves, Clarendon Press, Oxford, [4] H. Ochendon, J. R. Ochendon, Viscous Flow, Cambridge University Press, [5] P. V. O Neil, Advanced Engineering Mathematics, Wadsworth Publishing Company Belmont, California, 1991.

7 ROMAI J., 5, 2(2009), 5 27 STRUCTURES BICATÉGORIELLES COMPLÉMENTAIRES Dumitru Botnaru Université Technique de la République de Moldavie, Chişinău, République de Moldavie dumitru.botnaru@gmail.com Abstract We examine the complete lattice of the bicategory structure in the category of the vector locally convex Hausdorff spaces and the category of the locally convex groups. We construct the complementary elements for some elements of this lattice. Keywords: bicategory structure, reflective subcategory, classes of hereditary end cohereditary morphisms, spaces with weak topology, complete spaces, Schwartz spaces, nuclear spaces, strict nuclear spaces MSC: 18 B30; 46 M 15. Résumé On examine la latice complète des structures bicatégorielles dans la catégorie des espaces vectoriels local-convexes Hausdorff et la catégorie des groupes local-convexes. On construit des éléments complémentaires pour une classe propre des éléments de cette latice. Mots clés: structure bicatégorielle, sous-catégorie reflective, classe de morphismes héréditaire et cohéréditaire, espace à topologie faible, espace complèt, espace Schwartz, espace nucléaire, espace strictement nucléaire. 1. INTRODUCTION 1.1 Dans une catégorie local et colocal-petite à limites projectives ou inductives, la latice des structures bicatégorielle B est complète avec le prime élément (E f, Mono) = (la classe de tous épis stricts, la classe de tous monos) et l ultime élément (Epi, M f ) = (la classe de tous épis, la classe de tous monos stricts). On considère (P 1, I 1 ) (P 2, I 2 ) si (P 1 P 2 ). Toute structure bicatégorielle (P, I) est déterminée uniquement par chacune de ses classes: par la classe de projections P, de même que par la classe d injections I. Ainsi, dans une pareille catégorie, pour deux éléments (P 1, I 1 ) et (P 2, I 2 ) de la latice B, notons par P 1 P 2 la classe de projections du minimum de ces deux éléments, et par P 1 P 2 - la classe des projections du maximum. 5

8 6 Dumitru Botnaru Il est clair que P 1 P 2 = P 1 P 2, et la structure bicatégorielle correspondante est (P 1 P 2, (P 1 P 2 ) ). D autre part, à la classe P 1 P 2 correspond la structure bicatégorielle ((I 1 I 2 ), I 1 I 2 ). Dans ces deux structures bicatégorielles, une des classes représente l intersection des classes correspondantes: P 1 P 2 et I 1 I 2. Mais l autre classe, en général, ne peut pas être construite d une manière si simple. On introduit dans l ouvrage la notion de classe héréditaire par rapport à une classe de morphismes de même par rapport à une structure bicatégorielle (Définitions 2.1 et 2.14). Ces notions permettent de décrire aussi la classe (I 1 I 2 ) comme composition des classes P 1 et P 2 (Théorème 3.2). Dans la catégorie C 2 V des espaces locallement convexes (topologiques vectoriels), de même dans la catégorie C 2 Ab des groupes local-convexes [15] sont connues les sous-catégories refléctives formées par rapport à chaque sousespace dont le foncteur reflécteur est exactement à gauche. Une telle souscatégorie R détermine uniquement une sous-catégorie coreflective K de telle manière que εr = µk où εr = {f C 2 V r(f) Iso}, µk = {f C 2 V k(f) Iso}. Une telle classe de morphismes sert de classe de projections d une structure bicatégorielle de droite (εr, (εr) ) et comme classe d injections d une structure bicatégorielle à gauche ((εr), εr). Ces classes sont nommées bicomplètes (Définition 4.1) et elles permettent de construire les structures bicatégorielles complémentaires (Théorème 5.4). 1.2 Catégories. Soient: C une catégorie abstraite, C 2 V la catégorie des espaces local-convexes (topologiques vectoriels) Hausdorff [15], C 2 Ab la catégorie des groupes localconvexes Hausdorff [16], D une des catégories C 2 V ou C 2 Ab, Th la catégorie des espaces Tikhonov. 1.3 Sous-catégories dans la catégorie D. Soient: Π la sous-catégorie des espaces complets à topologie faible [16], [12], Γ 0 la sous-catégorie des espaces complets, S la sous-catégorie des espaces à topologie faible, N la sous-catégorie des espaces nucléaires, sn la sous-catégorie des espaces strictement nucléaires [11], Sc la sous-catégorie des espaces Schwartz (voir [11]), M sous-catégorie des espaces à topologie Mackey [16]. 1.4 Morphismes orthogonaux. Le morphisme f est nommé orthogonal de haut au morphisme g, et g est nommé orthogonal de bas à f, et est noté

9 Structures bicatégorielles complémentaires 7 par f g si pour tout carré commutatif gu = vf, il existe un unique morphisme (diagonal) h de sorte que v = gh u = hf Figure 1.1 Mentionnons que l unicité du morphisme h avec les propriétés indiquées a lieu si f est un épi ou g est un mono. Pour deux classes de morphismes K et L de la catégorie C, nous écrirons K L, si tout morphisme de la classe K est orthogonal de haut à tout morphisme de la classe L. Notons K = {f C K f}, K = K MonoC, K = {f C f K}, K = K EpiC. 1.5 Latices. Soient: R la latice de toutes les sous-catégories reflectives non-nulles dans la catégorie D, R i la latice de toutes les sous-catégories reflectives dans la catégorie D qui contient la sous-catégorie Γ 0, R b la latice de toutes les souscatégories reflectives dans la catégorie D qui contient la sous-catégorie S, R c la latice de toutes les sous-catégories c-reflectives, Bic la latice des classes de morphismes bicomplets dans la catégorie D. 1.6 Structures bicatégorielle dans la catégorie D. Soient: (Epi, M f ) = (la classe de tous les épis, la classe de tous les monos stricts)=(opérateurs continus à image dense, inclusions topologiques à image fermée), (E f, Mono) = (la classe de tous les épis stricts, la classe de tous les monos)=(la classe des applications factorielles, des applications injectives), (E u, M p ) = (la classe de tous les épis universels; la classe de tous les monos précis)=(la classe des applications surjectives, la classe des inclusions topologiques),

10 8 Dumitru Botnaru (E p, M u ) = (la classe de tous les épis précis, la classe de tous le monos universels), (E p, M u) = ((M u), la classe de tous les monos universels à image fermée). Décrivons plus en détails les deux dernières structures bicatégorielles (voir [7], [9], étant donné qu elles jouent un rôle important dans l ouvrage. 1.7 THÉORÈME [8]. Soit f : (E, u) (F, v) un mono de la catégorie D. Les affirmations suivantes sont équivalentes: 1. f M u. 2. Toute fonctionel continu défini sur l espace (E, u) s extend par f. 3. Les topologies u et v (celle indue de l espace (F, v) sur l espace vectorial E) sont compatibles dans une et la même dualité. 1.8 THÉORÈME [8], [10]. Soit f : (E, u) (F, v) D, τ(v) - la topologie Mackey compatible avec la topologie v. Alors les affirmations suivantes sont équivalentes. 1. f E p (respectivement f E p ). 2. f est un épi (respectivement f E u ) et v =min {u; τ(v)}. 2. CLASSES DE MORPHISMES EXTRÉMAUX 2.1 Définition [1]. Soient A et E deux classes de morphismes de la catégorie C. La classe A s appelle E-héréditaire, si du fait que fg A et f E, il résulte que g A. Notion duale: La classe A s appelle E-cohéréditaire, si du fait que fg A et g E, il résulte que f A. Examinons des situations quand la classe des injections d une structure bicatégorielle est héréditaire, et la classe de projections est cohéréditaire par rapport à certaines classes de morphismes. 2.2 Définition. Une classe A s appelle stable de droite si du fait que a b = b a est un carré cocartésien et a A, il résulte que a A. Notion duale: la classe stable à gauche. 2.3 Exemples. 1. Soit (P, I) une structure bicatégorielle de droite (de gauche), A P, B I. Alors la classe I est A-cohéréditaire (la classe P est B-héréditaire). 2. Toute classe stable à droite est Epi-cohéréditaire. Dans toute catégorie C, la classe M u est Epi-cohéréditaire. Dans la catégorie D, la classe M p comme classe stable à droite est aussi Epi-cohéreditaire. 3. Dans la catégorie C 2 V est bien connue l affirmation suivante ([16], cap. VI, prop. 5): Soit (E, t) un espace complet. Alors l espace E reste complet dans toute topologie compatible à la topologie t et plus forte qu elle. Cette affirmation peut être généralisée de la manière suivante:

11 Structures bicatégorielles complémentaires 9 Soient R une sous-catégorie reflective dans la catégorie C 2 V et Γ 0 R. Si f : X Y E u M u et Y R, alors et X R. La dernière proposition est une simple conséquence de l affirmation suivante: la classe E u M u est Epi-cohéréditaire. Mentionnons que cette classe est aussi Mono-héréditaire. 2.4 LEMME. Soit (P, I) une structure bicatégorielle de droite. Alors la classe P est Epi-cohéréditaire. Démonstration. Soit p = fe (1) avec p P et e Epi. Considérons la (P, I) - factorisation de morphisme f Comme p i 1, il existe un morphisme h ainsi que f = i 1 p 1 (2) p 1 e = hp (3) i 1 h = 1 (4) De l hypothèse et de l égalité (3), il résulte que h Epi. Alors de l égalité (1), il résulte que i 1 Iso. Figure REMARQUE. La classe de projection P d une structure bicatégorielle de droite (P, I) de la catégorie C n est pas toujours C-cohéréditaire. Par exemple, dans la catégorie C 2 V, la classe εr pour R LEMME. Soit C une catégorie à carrés cocartésiens. Alors la classe Epi est M u -héréditaire. Démonstration. Soit uv Epi, u M u et fv = gv (5)

12 10 Dumitru Botnaru Figure 2.2 Construisons les carrés cocartésiens suivants: sur les morphismes u et f; u 1f = f u (6) u 2g = g u (7) sur les morphismes u et g; u 1u 2 = u 2u 1 (8) sur les morphismes u 1 et u 2. Alors les morphismes u 1, u 2, u 1, u 2, de même que u 1 u 2 (= u 2 u 1 ) sont monos. Nous avons u 1g uv (7) = u 1u 2gv (5) = u 1u 2fv (8) = u 2u 1fv (6) = u 2f uv et comme uv est un épi, déduisons que u 1g = u 2f (9) Ensuite u 1u 2f (8) = u 2u 1f (6) = u 2f u (9) = u 1g u (7) = u 1u 2g c est-à-dire u 1u 2f = u 1u 2g et comme u 1 u 2 est mono, déduisons que f = g. 2.7 Dans la catégorie Th des espaces Tikhonov existe la structure bicatégorielle (E u, M p ) =(applications surjectives, inclusions topologiques). LEMME. Soit M p I Mono. Alors la classe I n est pas Epi-cohéréditaire. Démonstration. Pour l espace X = (0; 2π) existent plusieurs extensions compactes: X peut être réalisé comme un sous-espace dense du cercle unitaire aussi du segment [0; 2π]. Soit Y un espace qui possède plusieurs extensions compactes, et b : Y Z - une d entre elles. Dans ce cas b = fβ Y

13 Structures bicatégorielles complémentaires 11 pour un certain morphisme f, où β Y est la compactification Stone-Čech. Figure 2.3 Dans cette égalité b, β Y M p Epi. Si f est un mono, alors il est un iso. 2.8 LEMME. Dans la catégorie Th la classe Epi n est pas (E u Mono)- héréditaire. Démonstration. Soit X un espace Tikhonov et Y un sous-espace dense à application canonique i : Y X. Soit dy et dx les mêmes ensembles dotés de la topologie discrète, et d Y : dy Y et d X : dx X les applications identiques qui sont continues. Alors l application d(i) : dy dx, générée de l application i, est aussi continue Figure 2.4 Dans l égalité id Y = d X d(i), l application id Y est épi, l application d X (E u Mono) et d(i) n est pas épi si Y X. 2.9 LEMME. Dans la catégorie C 2 V, la classe Mono n est pas (M u Epi) - cohéréditaire. Démonstration. Soit sur l espace vectorial E deux topologies local-convexes u et v comparables u > v, et qui ne sont pas compatibles avec l une et la même dualité: (E, v) (E, u) Soit, p 1 : (E, u) p(e, u) et p 2 : (E, v) p(e, v) Π-répliques des objets correspondants, où Π est la sous-catégorie des espaces local-convexes complets aussi avec la topologie faible.

14 12 Dumitru Botnaru Pour l application canonique f : (E, u) p(e, v), nous avons la diagramme commutative Il y a Figure 2.5 p(f)p 1 = p 2 f (10) où p 2 f est mono, et p 1 (M u Epi). Supposons que p(f) est mono. Dans ce cas p(f) est une inclusion topologique ([12], pr. 11, p.151). Mais de l égalité (10), il résulte que p(f) est épi. Donc p(f) est iso. Ce qui signifie que (E, u) = (E, v) la contradiction obtenue montre que p(f) n est pas mono LEMME. Dans la catégorie C 2 V la classe Epi n est pas (E u Mono)- héreditaire. Démonstration. Soit (E, u) un espace local-convexe, et (F, u ) un sousespace dense propre. Examinons sur les espaces vectoriels E et F les plus fines topologies local-convexes σ et σ. Nous avons la suivante diagramme commutative et les applications canoniques. De l égalité Figure 2.6 iσ F = σ E σ(i) comme iσ F est épi et σ E (E u Mono), il ne résulte pas que σ(i) est un épi. Ainsi σ(i), et avec lui i, et un iso ([12], pr. 10, p ) Soit K une sous-catégorie coreflective de la catégorie C avec le foncteur correspondant k : C K. Notons µk = {m MonoC k(m) IsoC}. Dual. Soit R est une sous-catégorie reflective avec le foncteur r : C R. Notons εr = {e EpiC r(e) IsoC}.

15 Structures bicatégorielles complémentaires 13 Dans une catégorie local-petite à limites projectives ((µk), µk) est une structure bicatégorielle de gauche. Dual: (εr, (εr) ) est une structure bicatégorielle de droite. Les morphismes de la classe (εr) sont nommés R-parfaits, et les morphismes de la classe εr sont nommés R-estensions [17]. LEMME [1]. La classe εr est Epi-cohéréditaire, et la classe µk est Monohéréditaire Soit C une catégorie à carrés cartésiens et cocartésiens, la classe M u stable à gauche, et R - une sous-catégorie monoreflective. Pour le morphisme arbitraire f : X Y C soit r X et r Y R-réplique des objets correspondants. Alors nous avons l égalité r(f)r X = r Y f (11) Sur le morphisme r(f) et r Y construisons le carré cartésien Alors r(f)v = r Y u (12) r X = vt (13) f = ut (14) pour certain morphisme t. Comme R (εr) il résulte que r(f) (εr). Ainsi, u (εr). Ainsi r Y est mono universel. Conformément à l hypothése concernant la classe M u, déduisons que v M u l est aussi. Alors de l égalité (13), conformément au lemme 2.6, t est un épi. On vérifie facilement que v est R-réplique de l objet P. Alors t εr. Figure 2.7 Ainsi, nous l avons démontré. THÉORÈME. Soit C une catégorie à carrés cartésiens et cocartésiens, la classe M u stable à gauche et R une sous-catégorie monoreflective.

16 14 Dumitru Botnaru Alors: 1. (εr, (εr) ) est une structure bicatégorielle de droite. 2. Pour tout morphisme f C l égalité f = ut est sa (εr, (εr) ) - factorisation. 3. f (εr ), alors et seulement alors quand le carré r(f)r X = r Y f est cartésien Nous examinerons encore une construction qui permet d obtenir des exemples de classes héréditaires (voir [10]). Soit (P, I) une structure bicatégorielle dans la catégorie C, et A - une classe de morphismes. Notons I (A) = {m MonoC a A ainsi que ma et ma I} P (A) = (I (A)) THÉORÈME. Soient C une catégorie local-petite à limites projectives, (P, I) - une structure bicatégorielle. 1. Si (A, A) est une structure bicatégorielle de gauche, alors (P (A), I (A)) est une structure bicatégorielle dans la catégorie C. 2. De plus, si A EpiC, alors a) I (A) est - cohérèditaire. b) I (A) est la plus petite classe A-cohéréditaire qui contient la classe I Définition. Soient A une classe de morphismes, et (E, M) - une structure bicatégorielle de droite dans la catégorie C. La classe A est nommée (E, M)-cohéréditaire si du fait que a = me est (E, M)-factorisation du morphisme a A, il résulte que m A. Notion duale. La classe A-héréditaire par rapport à la structure bicatégorielle de gauche (E, M) LEMME. Soient (P, I) et (E, M) deux structures bicatégorielles de droite et I M. La classe A est (E, M)-cohéréditaire, alors et seulement alors quand la classe A P est (E, M)-cohéréditaire. 3. COMPOSITION DES CLASSES DE MORPHISMES 3.1 Définition. Soient A et B deux classes de morphismes de la catégorie C. La composition des classes A et B s appelle la classe A B de tous les morphismes de la catégorie C de forme ab à éléments a A et b B pour laquelle existe la composition correspondante ab. 3.2 THÉORÈME. Soit (P, I) et (E, M) deux structures bicatégorielles de droite dans la catégorie C. Examinons les affirmations suivantes: 1. La classe I est E-cohéréditaire. 2. La classe I est (E, M)-cohéréditaire. 3. P E E P. 4. (E P, M I) est une structure bicatégorielle de droite dans la catégorie C.

17 Structures bicatégorielles complémentaires 15 Alors Démonstration Conformément aux définitions correspondantes Soit p P, e E et soit qu il existe la composition pe. Examinons (P, I)-factorisation du morphisme pe Soit pe = i 1 p 1 (15) i 1 = m 2 e 2 (16) (E, M)-factorisation du morphisme i 1. Il résulte des égalités ci-dessus que et comme e m 2, il existe un morphisme t ainsi que pe = m 2 (e 2 p 1 ) (17) e 2 p 1 = te (18) m 2 t = p (19) De l hypothèse et de l égalité (19) il résulte que m 2 I. De l égalité (18) il résulte que t Epi. Du lemme 2.4 et de l égalité (19), il résulte que m 2 P. Ainsi m 2 P I = Iso, et le morphisme pe peut être écrit avec m 2 e 2 E et p 1 P. pe = (m 2 e 2 )p 1 (20) Figure Les deux classes E P et M I sont fermées par rapport à la composition et (E P) (M I). Ainsi, il reste à démontrer que tout morphisme possède (E P, M I)-factorisation. Soit f C, et f = ip (21) est sa (P, I)-factorisation, i = me (22)

18 16 Dumitru Botnaru est (E, M) - factorisation du morphisme i, et m = i 1 p 1 (23) est (P, I)-factorisation du morphisme m. Conformément à l hypothèse le morphisme p 1 e peut être écrit p 1 e = e 2 p 2 (24) avec e 2 E et p 2 P. Nous avons i (22) = me (23) = i 1 p 1 e (24) = i 1 e 2 p 2 i = i1e2p2 (25) d où il résulte que p 2 I. Ainsi p 2 I P = Iso. Alors e 2 p 2 E, et de l égalité (24), déduisons que p 1 e E. Du lemme 2.4 il résulte, il que p 1 E. Comme m M de l égalité (23), il résulte que p 1 M. Donc p 1 E M = Iso, et le morphisme f peut être factorisé avec i 1 p 1 = m M I, et ep E P. f = (i 1 p 1 )(ep) (26) 4 2. Soit i I, et Figure 3.2 est (E, M)-factorisation du morphisme i. Soit i = me (27) i = t(e 1 p 1 ) (28) (E P, M I)-factorisation du même morphisme, où t M I, et e 1 E et p 1 P. De l égalité (28), comme i I, il résulte que p 1 Iso. Donc, les égalités (27) et (28) sont deux (E, M)-factorisations du morphisme i. Donc e = re 1 p 1 (29)

19 Structures bicatégorielles complémentaires 17 mr = t (30) pour un certain isomorphisme r. De la dernière égalité, comme r Iso, et t I, il résulte que m I. Figure COROLLAIRE. 1. Si (E P, M I) est une structure bicatégorielle de droite, f = ip (P, I)-factorisation du morphisme arbitraire f C, et i = me (E, M)-factorisation du morphisme i, alors f = m(ep) est (E P, M I)- factorisation du morphisme f. Figure Si (P, I) est une structure bicatégorielle, alors M I MonoC. 3.4 Exemples. 1. Soit R une sous-catégorie reflective non-nulle dans la catégorie D. Puisque la classe M u est Epi-cohéréditaire (exemple 2.3), elle est aussi (εr)-cohéréditaire. Ainsi les structures bicatégorielles (P, I) = (E p, M u ) et (E, M) = (εr, (εr) ) vérifient la condition 1 du Théorème 3.2. Donc ((εr) E p, (εr) M u ) est une structure bicatégorielle dans la catégorie D. 2. De même, ((εr) E u, (εr) M p ) est une structure bicatégorielle dans la catégorie D.

20 18 Dumitru Botnaru 3. Soit S R. Ainsi la classe Mono est εr-cohéréditaire (lemme 2.6), et ((εr) E f, (εr) Mono) est une structure bicatégorielle dans la catégorie D. 4. ((εs) E p, (εr) M u ) = (E u, M p ). 5. ((εγ 0 ) E p, (εγ 0 ) M u ) = (E p, M u). 6. ((επ) E p, (επ) M u ) = (Epi, M f ). 3.5 Examinons, comme exemple au théorème précédent, la construction suivante. Soit (P, I) une structure bicatégorielle dans la catégorie D ayant les propriétés: 1. La classe I est Epi-cohéréditaire. 2. (I Epi, (I Epi) ) est une structure bicatégorielle de droite dans la catégorie D. La structure bicatégorielle de droite (I Epi, (I Epi) ) détermine une souscatégorie L de la catégorie D qui est (I Epi)-reflective L = (I Epi) (Π). Pour tout objet X de la catégorie D soit π X sa Π-réplique, et π X = m X l X. (31) (I Epi, (I Epi) )-factorisation de ce morphisme. Alors l X est L-réplique de l objet X. Notons par R I la latice de toutes les sous-catégories I-reflectives dans la catégorie D. Il est evident que R I = {R R L R}. Notons par B I la latice de toutes les structures bicatégorielles (E, M) pour lesquelles P E et la classe E est (I Epi)-héréditaire, ou, ce qui est la même chose, la classe E est I-hérèditaire. THÉORÈME. Les latices R I et B I sont antiisomorphes. Démonstration. Soit (E, M) un élément de la latice B I. Pour tout objet X de la catégorie D considérons L-réplique l X : X lx et (E, M)-factorisation du morphisme correspondant l X = s X r X. (32) La correspondance X r X définit la sous-catégorie R = M(L) connue une sous-catégorie E-reflective. De plus, comme l X I, il résulte que r X I aussi. Donc R R I. Ainsi nous avons établi la correspondance ϕ : B I R I, pour laquelle nous construirons celle inverse ψ : R I B I.

21 Structures bicatégorielles complémentaires 19 Soit R R I. Comme εr Epi, conformément à la première hypothèse concernant la classe I, elle est (εr)-cohéréditaire. Conformément au Théorème 3.2, le couple ((εr P, I (εr) ) est une structure bicatégorielle dans la catégorie D. Pour montrer qu elle appartient à la latice B I, il reste à démontrer que la classe (εr) P est (I Epi)-héréditaire. Soit ep = if, (33) où ep ((εr) P), c est-à-dire e εr, p P, et i I Epi. Comme p i du (33), il résulte l existence d un morphisme t avec les propriétés e = it, (34) f = tp. (35) La classe I Epi est stable à droite (hypothèse 2). Donc I Epi M u. Ainsi, dans l égalité (34) t Epi (lemme 2.6). Donc e εr et t Epi. Alors de l égalité (34), il résulte que t εr, et de l égalité (35) - que f (εr) P. De telle manière, nous avons montré que la classe (εr P) est (I Epi)- héréditaire. Donc ((εr) P, I (εr) ) est un élément de la latice B I. Figure 3.5 ϕψ = 1. Pour tout objet X de la catégorie D, L-replique l X appartient à la classe I. Ainsi ((εr P, I (εr) )-factorisation du morphisme l X coïncide avec (εr, (εr) )-factorisation (corollaire 3.3). Donc ϕψ est l application identique. ψϕ = 1. Soit maintenant (E, M) B I, R = M(L) et nous allons démontrer que E = (εr) P. (εr) P E. Comme P E, il reste de demontrer que εr E. Soit f : X Y εr. Alors gf = r X (36) pour un certain morphisme g, où r X est R-réplique de l objet X. Comme f Epi, déduisons que g εr εl I. Ainsi, r X E (l X = s X r X est

22 20 Dumitru Botnaru (E, M)-factorisation du morphisme l X ), g (I Epi) et comme la classe E est (I Epi)-héréditaire, il résulte que f E. Figure 3.6 E (εr) P. Soit f : X Y E, et f = ip (37) est (P, I)-factorisation du morphisme correspondant, où i : Z Y. Démontrons que i εr. Comme i (I Epi) M u Epi = επ, il résulte que π Z = gi, (38) pour un certain morphisme g. L égalité (38) peut être écrite aussi de la manière suivante (voir l égalité(31)): où i (I Epi), et m Z (I Epi). Donc gi = m Z l Z, (39) l Z = ti, (40) g = m Z t, (41) pour un certain morphisme t. L égalité (40), conformément à l égalité (32), peut être écrite ti = s Z r Z, (42) où i E, et s Z M. Donc r Z = hi, (43) t = s Z h, (44)

23 Structures bicatégorielles complémentaires 21 Figure 3.7 pour un certain morphisme h. L égalité (43) montre que i εr. Le théorème est démontré. 3.6 Examinons à présent certaines structures bicatégorielles qui vérifient les deux hypothèses du p (P, I) = (E p, M u ). Alors R I = R, L = Π. 2. (P, I) = (E p, M u), où M u est la classe des monomorphismes universels à l image fermée (voir [8], [9]). Il est évident que M u Epi = M u E u, et il est clair que cette structure bicatégorielle vérifie les hypothèses p Nous avons R I = R b, L = S. 3. (P, I) = (E u, M p ). Alors R I = R i, L = Γ 0. Figure REMARQUE. En ce qui concerne la latice R et ses sous-latices mentionnées dans le diagramme précédent (voir [5] et [9]).

24 22 Dumitru Botnaru 4. CLASSES DE MORPHISMES BICOMPLÈTES 4.1 Définition. La classe de morphismes B de la catégorie C sera nommée bicomplète si (B, B ) est une structure bicatégorielle de droite, et (B, B) est une structure bicatégorielle de gauche. 4.2 Remarque. Soit B une classe bicomplète de morphismes de la catégorie C, conformément à la définition, B E u M u. Ainsi, dans plusieurs catégories (par exemple, dans les catégories abéliennes) Iso est l unique classe bicomplète. 4.3 Dans la catégorie C 2 V de même que dans la catégorie C 2 Ab [15] existe une classe propre de sous-catégories réflectives qui contient la sous-catégorie S des espaces à topologie faible dont le foncteur reflecteur est exactement à gauche (voir [2-4], [10], [11], [14]). Une telle sous-catégorie R détermine de manière unique une sous-catégorie coreflective K avec les propriétés suivantes εr = µk. Ainsi εr E u M u et εr est la classe de projections de la structure bicatégorielle de droite (εr, (εr) ) et la classe d injections de la structure bicatégorielle de gauche ((µk), µk) = ((εr), εr). 4.4 THÉORÈME ([3], voir [7] Théorème 2.7). Soit R une sous-catégorie reflective dans la catégorie D avec le foncteur réflectif r : D R. Les affirmations suivantes sont équivalentes: 1. R est une sous-catégorie E u -reflective et r(m f ) M f. 2. R est une sous-catégorie E u -reflective et r(m p ) M p. 3. εr est une classe bicomplète. 4. Il existe une sous-catégorie coréflective K dans la catégorie D avec le foncteur coreflectif k : D K, ainsi que a) rk r; b) kr k. 5. Il existe une sous-catégorie coreflective K dans la catégorie D ainsi que εr = µk 4.5 Définition [3]. Une sous-catégorie reflective de la catégorie D qui vérifie les conditions équivalentes du théorème précédent est nommée c-reflective, et le couple de sous-catégories (K, R) est nommé couple conjugué de souscatégories. 4.6 Exemples. (M, S) est un couple conjugué de sous-catégories. Les souscatégories sn, Sh sont c-reflectives. La sous-catégorie N n est pas c-reflective. La sous-catégorie E u -reflective générée d un espace non-nul M p -injectif est c- reflective (voir [11], [14], [4]). 4.7 THÉORÈME [4]. 1. La correspondance R εr établit un isomorphisme de la latice complète R c des sous-catégories c-reflectives et de la latice Bic des classes bicomplètes de la catégorie D.

25 Structures bicatégorielles complémentaires Les éléments de ces latices R c et Bic forment des classe propres (ils ne sont pas d ensembles). 4.8 THÉORÈME. Soit C une catégorie à carrés cartésiens et cocartésiens, dans laquelle (E f, Mono) et (Epi, M f ) sont des structures bicatégorielles. 1. Pour toute classe bicomplète B, les couples (B E f, B ) et (B, M f B) sont des structures bicatégorielles dans la catégorie C. 2. Soit B 1 et B 2 deux classes bicomplètes distinctes dans la catégorie C. Alors les structures bicatégorielles (B 1 E f, B 1 ) et (B 2 E f, B 2 ) et les structures bicatégorielles (B 1, M f B 1 ) et (B 2, M f B 2 ) sont distinctes. Démonstration. 1. Conformément au lemme 2.6, la classe Mono est E u - cohéréditaire. Comme B E u, il résulte que la classe Mono est aussi B- cohéréditaire. Conformément au théorème 3.2, déduisons que (E f B, B ) est une structure bicatégorielle. Pour le couple (B, B M f ) la démonstration est duale. 2. Soit f B 1 et f / B 2. Alors f B 1 E f. Supposons que f B 2 E f. Alors ce morphisme se présente f = b 2 e avec b 2 B 2 et e E f. Comme f Mono, il résulte que e Mono. Alors e E f Mono = Iso, et f B Mentionnons encore d autres paires de structures bicatégorielles qui vérifient la condition 2 du Théorème 3.2. LEMME ([10], lemme 3.2). Soit R une sous-catégorie E u -reflective de la catégorie D. Alors: 1. r(m u ) M u. 2. r(mono) Mono THÉORÈME ([10], théorème 3.2). Soit (K, R) une paire conjuguée de sous-catégorie de la catégorie D, et (P, I) - une structure bicatégorielle. Alors les affirmations suivantes sont équivalentes: a) r(i) I; b) k(p) P Exemples. Soit (K, R) une paire conjuguée de sous-catégorie de la catégorie D. 1. Conformément au théorème 4.4, nous avons r(m f ) M f et k(e f ) E f. Ainsi, conformément au théorème 4.10, il résulte que r(mono) = Mono et k(epi) Epi i.e. r est un monofoncteur, et k est un épifoncteur. Le première affirmation résulte de le lemme 4.9 et la deuxième est une propriété topologique (voir [16]). 2. Conformément au théorème 4.4 pour la paire (K, R), nous avons r(m p ) M p. Donc r(e u ) E u, ce qui est évident. 3. Conformément à le lemme 4.9, nous avons r(m u ) M u. Donc k(e p ) E p. Cela résulte directement de la description de la classe E p (voir [8], [10]).

26 24 Dumitru Botnaru 4.12 THÉORÈME. Soit C une catégorie à carrés cortésiens et cocartésiens, dans laquelle la classe M u est stable à gauche, R une sous-catégorie monoréflective, et (P, I) - une structure bicatégorielle à droite dans la catégorie C ainsi que r(i) I. Alors 1. La classe I et (εr, (εr) ) - cohéréditaire. 2. ((εr) P, I (εr) ) est une structure bicatégorielle à droite dans la catégorie C. Démonstration. 1. Soit f I et examinons le diagramme du p Conformément à l hypothése r(f) I, et le carré r(f)v = r Y u est cartésien. Donc u I. Comme f = ut est (εr, (εr) ) - factorisation du morphisme f (théorème 2.12), il résulte que la classe I a la propriété respective. 2. Cette affirmation résulte des démonstrations précédentes et du théorème STRUCTURES BICATÉGORIELLES COMPLÉMENTAIRE 5.1 Dans une catégorie C local et colocal-petite à limites projectives ou inductives, la classe B des structures bicatégorielles est une latice complète avec un élément minimal et un élément maximal. Soit (P 1, I 1 ), (P 2 I 2 ) B. Considérons (P 1, I 1 ) (P 2 I 2 ) P 1 P 2. Comme une structure bicatégorielle de droite est uniquement déterminée par sa classe de projections, la relation (P 1, I 1 ) (P 2, I 2 ) sera notée plus simplement P 1 P 2. Ainsi E f est l élément minimal, et Epi est l élément maximal de la latice B. Mentionnons les suivantes règles simples: 1. P 1 P 2 = P 1 P P 1 P 2 P 1 P (P 1 P 2 ) = I 1 I (P 1 P 2 ) = I 1 I P 1 P 2 = (I 1 I 2 ). 5.2 En appelant à la terminologie de la théorie des latices, introduisons la notion (voir [13], cap. I 6). Définition. Les éléments (P 1, I 1 ), (P 2, I 2 ) de la latice B s appellent réciproquement complémentaires si P 1 P 2 = E f et P 1 P 2 = Epi. Mentionnons que, dans toute catégorie, les structures bicatégorielles (Epi, M f ) et (E f, Mono) sont réciproquement complémentaires. Conformément aux relations et 5.1.5, nous pouvons dire que les éléments (P 1, I 1 ) et (P 2, I 2 ) de la latice B sont réciproquement complémentaires si P 1 P 2 = E f et I 1 I 2 = M f.

27 Structures bicatégorielles complémentaires Soit B une classe bicomplète dans la catégorie C, (P 1, I 1 ) - une structure bicatégorielle avec la classe d injections I 1 (B, B ) - cohéréditaire; (P 2, I 2 ) - une structure bicatégorielle avec la classe de projections P 2 (B, B) - héréditaire. Alors, conformément au théorème 3.2 (P 1, I 1 ) = (B P 1, B I 1 ) est une structure bicatégorielle; et conformément au théorème dual 3.2, le couple (P 2, I 2 ) = (P 2 B, I 2 B) est aussi une structure bicatégorielle. THÉORÈME. 1. La classe I 2 est P 2 - cohéréditaire. 2. La classe P 1 est I 1 - héréditaire. 3. P 1 P 2 P 1 P P 1 P 2 P 1 P 2. Particulièrement, si les structures bicatégorielles (P 1, I 1 ) et (P 2, I 2 ) sont réciproquement complémentaires, alors le sont aussi les structures bicatégorielles (P 1, I 1 ) et (P 2, I 2 ). Démonstration. 1. Soit i 2 b I 2 avec i 2 I 2 et b B. Soit i 2 b = fp 2 (45) avec p 2 P 2. Comme p 2 i 2 il y a un morphisme g de telle manière que Dans l égalité (46), p 2 est un épi, donc le carré b = gp 2, (46) f = i 2 g. (47) g p 2 = 1 b (48) est cocartésien. Ainsi g B, et dans l égalité (47) i 2 I 2 et g B. Ainsi nous avons démontré que f I 2. Figure Dual. 3. Nous avons P 1 P 2 = (B P 1) (P 2 B ).

28 26 Dumitru Botnaru Soit que f P 1 P 2. Alors f = bp 1 avec b B et p 1 P 1. De même bp 1 P 2 et bp 1 B. De la relation bp 1 B, il résulte que b B. Donc b B B = Iso. Alors bp 1 P 2 et bp 1 P 1, i.e. bp 1 P 1 P Comme P 1 P 1, il reste à montrer que P 2 P 1 P 2. Soit p 2 P 2, et (P 1, I 1 ) - factorisation du morphisme p 2, p 2 = i 1 p 1 (49) i 1 = bb (50) (B, B) - factorisation du morphisme i 1. De l égalité (49), il résulte que i 1 P 2, et de l égalité (50), il résulte que b P 2, comme la classe P 2 est (B, B)- héréditaire. Ainsi dans l égalité p 2 = bb p 1 nous avons p 1 P 1, b B, et b P 2 B. Ainsi b, b, p 1 P 1 P 2 et bb p 1 P 1 P 2 Figure THÉORÈME. Soit C une catégorie à carrés cartésiens et cocartésiens, et B - une classe bicomplète de morphismes. Soit que (E f, Mono) et (Epi, M f ) sont des structures bicatégorielles dans la catégorie C. Alors 1. Les structures bicatégorielles (B E f, B ), (B, M f B) sont réciproquement complémentaires. 2. La classe M f B est Epi-cohérèditaire. 3. La classe B E f est Mono-héréditaire. Démonstration. 1. Les couples correspondants forment des structures bicatégorielles conformément au théorème 4.8. Ils sont complémentaires conformément au théorème précédent. 2 et 3 résultent du théorème COROLLAIRE. Dans la catégorie D, il existe une classe propre de structures bicatégorielles qui possède des compléments. Démonstration. Conformément aux théorèmes 5.4 et 4.7.

29 Structures bicatégorielles complémentaires 27 References [1] D. Botnaru, Reflective subcategories and right bicategory structures, Soviet. Math. Dokl., 15, 6 (1974), [2] D. Botnaru, Paires conjuguées de sous-catégories, Uspehy Mat. Nauk, 31, 3 (1976), (en russe). [3] D. Botnaru, Paires conjuguées de sous-catégories dans la catégorie des groupes local - convexes, Analyse fonctionnelle, Ulianovsck, 7 (1976) (en russe). [4] D. Botnaru, Sous-catégories réflectives et structures bicatégorielles de droite. Résumé de la thése du docteurès science, Moscou (1979) (en russe). [5] D. Botnaru, Composition et commutativité des foncteurs reflectifs, Analyse fonctionnelle, Ulianovsk, 21 (1983), (en russe). [6] D. Botnaru, Aspects catégoriaux des espaces vectorials local - convexes, Anales Scientifiques de l Université d Etat de Moldova, Série Sciences physico-mathématiques (2000), [7] D. Botnaru.,Cerbu O., Semireflexive product of two subcategories (to edit). [8] D. Botnaru., V. Gysin, Monomorphismes stables dans la catégorie des espaces local - convexes, Izv., Acad., Nauk, M.S.S.R., 1 (1973), 3-7 (en russe). [9] D. Botnaru, A. Turcanu, Les produit de gauche et de droite de deux sous-catégories. Acta et commentationes, V.III, Chisinau (2003), [10] D. Botnaru, A. Turcanu, On Giraux subcategories in locally convex spaces, ROMAI Journal, 1, 1(2005), [11] B. S. Brudovski, La topologie nucléaire associée, applications du type S et les espaces strict nucléaires. Soviet Math. Dokl., 178, 2 (1968), (en ruse). [12] A. Grothendieck, Topological Vector Spaces. Gordon and Breach, New York, [13] G. Grätzer, General Lattice Theory. Akademie-Verlag, Berlin, [14] V. A. Geiler, V. R. Gisin, Dualité généralisée pour les espaces local-convexes, Analyse fonctionnelle, Ulianovsk, 11 (1978), (en ruse). [15] P. Kenderov, Groupes Vectoriaux Topologiques, Mat. Sbornic, 81, (123), 3 (1970), (en russe). [16] A. P. Robertson, W. Robertson, Topological Vector Spaces, Cambridge University Press, [17] G. E. Strecker, On caracterization of perfect morphisms and epireflective hulls, Lecture Notes in Math., 378 (1974),

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31 ROMAI J., 5, 2(2009), SOME ACCELERATED FLOWS FOR AN OLDROYD-B FLUID Ilie Burdujan Department of Mathematics, University of Agricultural and Veterinary Medicine Ion Ionescu de la Brad, Iaşi, Romania burdujan Abstract This paper deals with an important problem in physics and engineering, namely with Taylor-Couette flow formation in an Oldroyd-B fluid filling the annular region between two infinitely long coaxial circular cylinders, due to a constant or time-dependent axial shear applied on the outer surface of the inner cylinder. The obtained solution is presented as the sum of the corresponding Newtonian solution and the non-newtonian contribution. Afterwards, it was specialized to give the solution for second grade fluids and Maxwell fluids, as well. Very simple forms of some exact solutions, that either have already been known or are obtained for the first time, are presented as limit cases of our solution. These results were established as limit cases of the solution of an initial-boundary problem in fractional derivatives which was obtained, in its turn, by using the Laplace and Hankel transforms. Keywords: Taylor-Couette flow, Oldroyd-B fluid, Maxwell fluid, second grade fluid MSC: 76A INTRODUCTION The wide variety of non-newtonian fluids arising in a large variety of industrial applications - such as chemical process industry (e.g. synthetic fibres, polymer solutions, petroleum production, etc.), food industries, construction engineering and so on - motivates the great interest in their study. It is well known that the analysis of the behavior of the fluid motion for the non- Newtonian fluids is more complicated and subtle in comparison with that of Newtonian fluids. For a wide class of flows of Newtonian fluids it is possible to give a closed form for analytical solution, while for non-newtonian fluids such solutions are seldom found. On the other hand, some of the mathematical models do not fit well with experimental data. That is why, some mathematical objects obtained by placing some fractional derivatives instead of some time derivatives into the rheological constitutive equations, that describe the rheological properties of some classes of materials, were tested. On this line we can quote the papers of Bagley [1], Friedrich [6], Makris and Constantinou [15], Glökle and Nonnenmacher [8], Mainardi [13], Mainardi and Gorenflo [14], Rossikhin and Y. A., Shitikova [17], [18] and so on. They had obtained results 29

32 30 Ilie Burdujan which are in a good agreement with experimental data. Unfortunately, an initial-boundary problem for an equation with fractional derivatives (shortly, IBPEFD) cannot be considered as being a mathematical model for a real dynamical system, because the fractional derivatives have no tensorial character. Nevertheless, some of its limit cases may be the mathematical models for real phenomena. Therefore, it becomes important to solve such an IBPEFD because its solution gives the possibility to find the solutions for all its limit cases, among them being the solutions of problems modelling real dynamical systems. Two important situations may arise in the limiting processes. A result of such a limit can be the disappearance of all fractional derivatives. As example, in the problem under consideration in the present paper (i.e., the IBPEFD [(7), (10), (12)]), the Newtonian solution is obtained by making the relaxation time λ and the retardation time λ r tend to zero. The second kind of results corresponds to the case when the orders of fractional derivatives, here α or/and β, tend to 1. This time the limiting process is considered in the sense of Schwartz s distribution theory with respect to some appropriately classes of testing functions. Indeed, the solution for ordinary Maxwell fluids is obtained from the before mentioned IBPEFD when λ r 0 and α 1. These remarks will be used in what follows in order to find the exact solution for Taylor-Couette flow of an incompressible Oldroyd-B fluid in a circular pipe. More exactly, the main purpose of this paper is to provide exact solutions for the velocity field and the shear stress corresponding to the large class of unsteady flows of incompressible Oldroyd-B fluids between two infinite coaxial circular cylinders, one of them being subject to a constant or time-dependent rotational shear stress. To this end, into the governing equations, corresponding to an Oldroyd-B fluid in the absence of body forces and a pressure gradient in the flow direction, some time derivatives are replaced by fractional derivatives. The obtained mathematical object was named by Tong and Liu [20] the governing equations of an incompressible generalized Oldroyd-B fluid. After making the similar replacement in the initial-boundary conditions, an IBPEFD is obtained. The governing equations for an incompressible generalized Maxwell fluid or a generalized second grade fluid are similarly obtained. The attribute generalized will be used here for designing the hypothetical fluids that would be characterized by such IBPEFDs. The solution of IBPEFD [(7), (10), (12)] is presented as a sum of the Newtonian solution and the corresponding non-newtonian contribution. It can be easily specialized to give the similar solutions for the second grade and Maxwell fluids. As it was already remarked, the Newtonian solutions can be also obtained as limit cases of general solutions. Furthermore, the non-newtonian contributions to the general solutions have been expressed in terms of the time derivatives of a Newtonian solution.

33 Some accelerated flows for an Oldroyd-B fluid MODEL AND BASIC EQUATIONS Let us consider an incompressible Oldroyd-B fluid at rest in the annular region between two infinitely long coaxial circular cylinders of radii R 1, R 2 (R 1 < R 2 ). The outer cylinder is always at rest, while at time t = 0 + the inner cylinder is suddenly set in rotation around its axis by a constant or a timedependent shear stress. The Cauchy stress tensor T for an incompressible Oldroyd-B fluid, is given by T = pi + S, S + λ DS ( ) Dt = µ DA A + λ r, (1) Dt where pi is the indeterminate spherical stress, S is the extra-stress tensor, A is the first Rivlin-Ericksen tensor, µ is the dynamic viscosity of the fluid, λ and λ r (< λ) are material constants, and DS Dt = ds dt + (V )S LS SLT, DA Dt = da dt + (V )A LA ALT. (2) Into above equation (2), V denotes the velocity, is the gradient operator, L is the velocity gradient and the superscript T indicates the transpose operation. For the problem under consideration, we assume a velocity field V and an extra-stress tensor S of the form V = V(r, t) = ω(r, t)e θ, S = S(r, t) (3) where e θ is the unit vector along the θ-direction of the cylindrical coordinate system r, θ and z. For such flows the constraint of incompressibility is automatically satisfied. Furthermore, if the fluid is at rest up to the moment t = 0, and therefore V(r, 0) = 0, S(r, 0) = 0, (4) then the governing equations for an Oldroyd-B fluid, in the absence of body forces and a pressure gradient in the flow direction, are given, for all r (R 1, R 2 ) and t > 0, by ( ) ( λ 2 ω(r, t) ω(r, t) 2 t 2 + = ν 1 + λ r t t r r r 1 ) r 2 ω(r, t), (5) ( 1 + λ ) ( ) ( τ(r, t) = µ 1 + λ r t t r 1 ) ω(r, t), (6) r where τ(r, t) = S rθ (r, t) is the nonzero shear stress, µ is the dynamic viscosity of the fluid, ρ is its constant density, ν(= µ/ρ) is the kinematic viscosity, and λ and λ r are the relaxation and retardation times.

34 32 Ilie Burdujan By replacing some inner time derivatives by the fractional differential operators D α t and D β t (β α), the governing equations (5) and (6) of an incompressible Oldroyd-B fluid become (see [20]), for all r (R 1, R 2 ) and t > 0, ( (1 + λdt α ω(r, t) ) = ν(1 + λ r D β 2 t t ) r r (1 + λd α t )τ(r, t) = µ(1 + λ r D β t ) ( r 1 r where the fractional derivatives are defined by [16] D p t [f(t)] = 1 d Γ(1 p) dt t 0 r 1 r 2 ) ω(r, t), (7) ) ω(r, t), (8) f(τ) p dτ, 0 < p < 1, (9) (t τ) (here Γ( ) is Euler s Gamma function). The momentum equation (5) must be solved subject to the initial and boundary conditions respectively, ( 1 + λ t ω(r, 0) = ) τ(r 1, t) = µ ω(r, 0) t = 0, τ(r, 0) = 0, (10) ( ) ( ω(r1, t) 1 + λ r 1 ) ω(r 1, t) = ft a, (11) t r R 1 for r (R 1, R 2 ), t > 0, f R (a fixed real number), and a 0. As before, by replacing in (11) the inner derivatives with respect to t by the fractional differential operators Dt α and D β t (β α), we get (1 + λd α t )τ(r 1, t) = µ(1 + λ r D β t ) ( ω(r1, t) r ) 1 ω(r 1, t) R 1 = ft a, (12) for t > 0, a 0. Consequently, an IBPEFD, consisting of equations [(7), (10), (12)], is associated with the model of the Taylor-Couette flow of an Oldroyd-B fluid in an annulus due to a constant or time depending couple and characterized by simultaneously equations [(5), (10), (11)]. It will be solved by using the integral transform techniques. More exactly, the Laplace and finite Hankel transform are used to transform the IBPEFD [(7), (10), (12)] into an algebraic system. Moreover, the equations (7) and (8) contain as limit cases the governing equations of the so called (see [20]) generalized second grade and Maxwell models (i.e. the models obtained by replacing some inner time derivatives by some fractional differential operators in the governing equations of a second grade

35 Some accelerated flows for an Oldroyd-B fluid 33 or a Maxwell fluid), as well as the ordinary Oldroyd-B, Maxwell and second grade models. In this paper, we are especially interested in the cases a = 0 and a = 1. COMMENT. Of course, in order to enssure the dimensional consistency of equations (7) and (8), the material constants λ and λ r must have necessarily the dimensions of t α and t β, respectively. In some papers, e.g. [11], the authors (correctly) used λ α and λ β r instead of λ and λ r. However, for simplicity, we shall keep the notations λ and λ r, having in mind their correct significations. 3. EXACT SOLUTIONS FOR THE VELOCITY FIELD In what follows, we shall use the modified Hankel transform, with respect to r, defined by means of the Bessel functions of index 1 B(r, r n ) = J 1 (rr n )Y 2 (R 1 r n ) J 2 (R 1 r n )Y 1 (rr n ), where J 1 ( ), J 2 ( ), Y 1 ( ) and Y 2 ( ) are Bessel functions (of index 1 and 2), (r n ) n N is the increasing sequence of the positive roots of the transcedental equation J 1 (R 2 x)y 2 (R 1 x) J 2 (R 1 x)y 1 (R 2 x) = 0 (i.e. B(R 2, r n ) = 0 for all n N ). We shall denote by ω H (r n, t) the image of ω(r, t) by the modified Hankel transform defined by ω H (r n, t) = R 2 R 1 r ω(r, t) B(r, r n ) dr. (13) Standard computations, similar to those used for proving the identity ( ) in [5], give the identity R 2 [ 2 r B(r, r n ) R 1 r r r 1 ] r 2 ω(r, t) dr = [ ω(r1, t) = R 1 B(R 1, r n ) 1 ] ω(r r R 1, t) r 2 1 nω H (r n, t). (14) Multiplying Eq. (7) by rb(r, r n ), then integrating it with respect to r from R 1 to R 2 and taking into account the identity (14) it follows (1 + λd α t ) ω H(r n, t) t = f ρ ta R 1 B(R 1, r n ) νr 2 n(1 + λ r D β t )ω H(r n, t), t > 0, where ω H (r n, t), due to Eqs. (10), have to satisfy the initial conditions (15) ω H (r n, 0) = 0, n = 1, 2, (16)

36 34 Ilie Burdujan Now, applying the Laplace transform to Eq. (15), using the Laplace transform formula for fractional derivatives [16] and taking into account the initial conditions (16), as well as Eq. (A1) from Appendix, we find the Laplace image ω H (r n, q) of ω H (r n, t) as being ω H (r n, q) = 2f Γ(1 + a) 1 πρr n q 1+a q + λq 1+α + νrn(1 2 + λ r q β ). (17) We shall apply the inverse Laplace transform to (17) in order to obtain ω H (r n, t). However, for a suitable presentation of the final results, we firstly rewrite Eq. (17) in the form ω H (r n, q) = [ 2f 1 Γ(1 + a) πρr n q + νrn 2 q 1+a 1 λq 1+α + λ r νr 2 n q β q + λq 1+α + νr 2 n(1 + λ r q β ) ]. (18) In its turn, the last term in Eq. (18), can be written in an equivalent form as the series 1 λq 1+α + λ r νr 2 q 1+a n q β q(1 + λq α ) + νrn(1 2 + λ r q β ) = λq1+α + λ r νrn 2 q β q 1+a ( ) = k k νr2 n k!λ m r k=0m=0 λ m!(k m)! ( ) k k+1 λ r νr2 n k!λ m r λ m!(k m)! ( νrn) 2 k (1 + λ r q β ) k (λq) k+1 (q α + 1/λ) k+1 = k=0 q αm a (q α + 1/λ) k+1 q β m a k=0m=0 (q α + 1/λ) k+1, (19) where α m = mβ + α k 1 and β m = mβ + β k 2. Introducing (19) into (18), inverting the result by means of the inverse Laplace transform and using Eq. (A2), we find that ω H(r n, t) = 2f πρr n where F k (t) = m=0 and (see [12]) G a,b,c (d, t) = t 0 k k!λ m r m!(k m)! j=0 s a e νr2 n (t s) ds Re (ac b) > 0, d q a < 1. 2f πρr n Γ(1 + a) k=0 ( ) k t νr2 n F k (s)e νr2 n (t s) ds, λ 0 (20) [ G α,αm a,k+1 ( 1λ ), t + νrn 2 λ r λ G α,β m a,k+1 ( 1λ )], t (21) { } Γ(j + c)t a(j+c) b 1 Γ(j + 1)Γ(c)Γ[a(j + c) b] dj = L 1 q b (q a d) c, (22)

37 Some accelerated flows for an Oldroyd-B fluid 35 Recall that the inverse of the modified Hankel transform (13) is defined by f(r) = π2 2 rnj (R 2 r n )B(r, r n ) J2 2 (R 1 r n ) J1 2 (R 2 r n ) f H(r n ) = π2 r 2 2 nc F n f H (r n ), (23) n=1 where f H (r n ) denotes the image of f(r) by Hankel transform (13) and n=1 C F n = J 2 1 (R 2 r n )B(r, r n ) J 2 2 (R 1 r n ) J 2 1 (R 2 r n ) = D F nb(r, r n ). (24) By applying the inverse Hankel transform (23) to Eq. (20) we get, for the velocity field ω(r, t), the expression ω(r, t) = ω N,a (r, t) πf ρ Γ(1+a) where ω N,a (r, t) = πf ρ n=1k=0 ( ) k t νr2 n r n C F n F k (s)e νr2 n(t s) ds, λ 0 (25) t r n C F n s a e νr2 n (t s) ds, (26) n=1 represents the velocity field corresponding to a Newtonian fluid performing the same motion. In particular, by using (A3) one gets where ω N (r, t) = ω N,0 (r, t) = ϕ 0 (r) πf µ ϕ 0 (r) = f 2µ ( R1 R C F n e νr2 nt, (27) r n n=1 ) 2 ( R 2 ) 2 r r. Indeed, making λ = λ r = 0 into Eq. (17) and following the same way as before, we obtain (26). In particular, for a = 0 we attain to ω N,0 (r, t) given by (27). On the another hand, Eq. (27) gives ω N (r, t) t and consequently (26) becomes Further, for a = 1, one obtains ω N,1 (r, t) = = πf ρ r n C F n e νr2 nt n=1 ω N,a (r, t) = t a t ω N (r, t). (28) t 0 ω N (r, s) ds = 1 ω N (r, t)

38 36 Ilie Burdujan and the corresponding Newtonian solution takes the following form ω N,1 (r, t) = ϕ 0 (r)t + ϕ 1 (r) + πf 1 C F n e νr2 n t, (29) µν r 3 n=1 n where ϕ 1 (r) = A r + Br + Cr3 + E r ln r, with A = fr4 1 8R2µν 2 (2R2 2 R2 1 ), B = fr2 [ 1 4R 8R2µν ln R 2 (R2 2 R2 1 )2], C = f ( ) 2 R1, E = fr2 1 8µν R 2 2µν. It is important to notice that the Newtonian solution ω N (r, t) given by Eq. (27) is coincident with that obtained in [2] (p. 829) by a different technique. Finally, having in mind the expression (27) of the Newtonian solution ω N (r, t), it is easy to show that the general solution ω(r, t) can be written in a suitable form in terms of its time derivatives, namely Γ(1 + a) k=0m=0 + λ r λ Γ(1 + a) ω(r, t) = t a t ω N (r, t) k k!λ m r m!(k m)!λ k k+1 t ω N (r, t) G α,αm a,k+1 k k=0m=0 k!λ m r m!(k m)!λ k k+2 t ω N (r, t) G α,βm a,k+1 ( 1 λ, t ) + ( 1 λ, t ). 4. CALCULATIONS OF THE SHEAR STRESS (30) Applying the Laplace transform to Eq. (8) and using the initial conditions (10), we find that τ(r, q) = µ 1 + λ rq β 1 + λq α [ ω(r, q) r 1 r ω(r, q) ], (31) where τ(r, q) denotes the Laplace transform of τ(r, t). The last factor, containing the velocity field, may be obtained using Eq. (30) and (25) as well.

39 Some accelerated flows for an Oldroyd-B fluid 37 Indeed, from (30) we have where Γ(1 + a) k k=0m=0 + λ r λ Γ(1 + a) k=0m=0 ω(r, t) r 1 r ω(r, t) = ta t Ω N (r, t) k!λ m r m!(k m)!λ k k+1 t Ω N (r, t) G α,αm a,k+1 k k!λ m r m!(k m)!λ k k+2 t Ω N (r, t) G α,βm a,k+1 Ω N (r, t) = ω N(r, t) r ( 1 λ, t ) + ( 1 λ, t ), (32) 1 r ω N(r, t). (33) Applying the Laplace transform to Eq. (32) and having in mind Eq. (A4), it results that ω(r, q) 1 Γ(1 + a) ω(r, q) = r r q 1+a L{ t Ω N (r, t)} Γ(1 + a) k k!λ m r q αm a m!(k m)!λ k (q α k+1 L{ k+1 t Ω N (r, t)}+ + 1/λ) k=0m=0 + λ r λ Γ(1 + a) k k=0m=0 k!λ m r m!(k m)!λ k q βm a (q α + 1/λ) k+1 L{ k+2 t Ω N (r, t)}. Now, introducing (33) into (31), using the simple decomposition k=0 1 + λ r q β 1 + λq α = 1 + λ r λ q β q α + 1/λ q α q α + 1/λ (34) and applying the inverse Laplace transform we get, taking into account of (A5), the next form for the shear stress τ(r, t) = τ N,a (r, t) + µγ(1 + a)a ( 1 ) λ, t t Ω N (r, t) µγ(1 + a) [B k ( 1 ) λ, t t k+1 Ω N (r, t) λ r λ C k ( 1 ) ] λ, t t k+2 Ω N (r, t) where τ N,a (r, t) = µt a t Ω N (r, t) = = πfν rn[j 2 2 (rr n )Y 2 (R 1 r n ) J 2 (R 1 r n )Y 2 (rr n )] t J2 2 (R 1 r n ) J1 2 s a e νr2 n(t s) ds (R 2 r n ) k=0 0 (35) (36)

40 38 Ilie Burdujan represents the shear stress corresponding to a Newtonian fluid and A ( 1 ) λ, t = λr λ R α,β a 1 ( 1 ) λ, t R α,α a 1 ( 1 ) λ, t, R α,β (a, t) = a k t (k+1)α β 1 Γ((k + 1)α β), Re (α β) > 0, atα < 1, k=0 B k ( 1 λ, t ) = 1 λ k k m=0 C k ( 1 λ, t ) = 1 λ k k [ k!λ m r m!(k m)! G α,αm+α a,k+2 m=0 [ k!λ m r m!(k m)! G α,βm +α a,k+2 G α,αm a,k+1 ( 1 λ, t )], G α,βm a,k+1 ( 1 λ, t )]. ( 1 ) λ, t + λ r λ G α,α m +β a,k+2 ( 1 ) λ, t ( 1 ) λ, t + λr λ G α,β m +β a,k+2 ( 1 ) λ, t The equivalent form of the shear stress, resulting from (25) and (31) is +πfν Γ(1 + a) where τ(r, t) = τ N,a (r, t) + µγ(1 + a)a ( 1 ) λ, t t Ω N (r, t)+ n=1k=0 [ ( r n C F n ( νr2 n) k B k 1 ) λ, t + νrn 2 C F n = J 2(rr n )Y 2 (R 1 r n ) J 2 (R 1 r n )Y 2 (rr n ) J2 2 (R 1 r n ) J1 2 J1 2 (R 2 r n ), (R 2 r n ) λ r λ C ( 1 λ, t )] e νr2 nt, (37) Making a = 0 and 1 into (35) and (37), the shear stresses corresponding to f and ft into (10) are obtained. For instance, the Newtonian solutions are τ N (r, t) = τ N,0 (r, t) = τ 0 (r) + πf C F n e νr2 n t, (38) n=1 where τ N,1 (r, t) = tτ 0 (r) + τ 1 (r) πf ν C F n e νr2 n t, (39) n=1 τ 0 (r) = fr2 1 r 2, τ 1(r) = fr2 1 [ (R 2 2 r 2 ) 2 (R 2 2 R 2 1) 2] 8νR 2 2r 2.

41 5. LIMIT CASES Some accelerated flows for an Oldroyd-B fluid By taking the limit of Eqs. (25), (30), (35) and (37) as λ r 0, we get the similar solutions corresponding to the so-called generalized Maxwell fluids, namely ω(r, t) = = ω N,a (r, t) πf ρ Γ(1 + a) r n C F n k=0 n=1 k=0 = ω N,a (r, t) Γ(1 + a) 1 λ k G α,γ k a,k+1 ( 1 ) λ, t ( ) k νr2 t n G λ α,γk a,k+1 ( 1 ) λ, t e νr2 n (t s) ds = 0 0 k+1 t ω N (r, t), ( t τ(r, t) = τ N,a(r, t) µγ(1 + a) sω N (r, s)r α,α a 1 1 ) λ, t s µγ(1 + a) +πfνγ(1 + a) n=1k=0 ( ) r nc F νr 2 k [ n n λ G α,γk +α a,k+2 G α,γk a,k+1 ( 1 λ, t )] e νr2 n t = ( 1 λ, t ) = τ N,a (r, t) µγ(1 + a) t Ω N (r, t) R α,α a 1 ( 1 λ, t ) 1 k=0 λ k k+1 t [ Ω N (r, t) G α,γk a,k+1 ds+ ( 1 ) λ, t G α,γk +α a,k+2 ( 1 )] λ, t, where γ k = α k 1. Furthermore, by making λ 0 in (40) and (41) and taking into account of Eqs. (A6), the Newtonian solutions (40) (41) ω N,a (r, t) = t a t ω N (r, t), τ N,a (r, t) = µt a t Ω N (r, t) (42) are recovered. 2. By taking now α 1 into (40) and (41), the solutions for ordinary Maxwell fluids are obtained, namely πf ρ Γ(1 + a) n=1k=0 ( νr2 n λ = ω N,a(r, t) Γ(1 + a) ω(r, t) = ω N,a (r, t) ) k t r nc F n 1 0 k=0 λ k k+1 G 1, k a,k+1 ( 1 λ, t ) e νr2 n (t s) ds = ( t ω N (r, t) G 1, k a,k+1 1 ) λ, t, (43)

42 40 Ilie Burdujan 0 ( t τ(r, t) = τ N,a (r, t) µγ(1 + a) s Ω N (r, s)r 1, a 1 ) λ, t s +πfνγ(1 + a) 0 n=1k=0 ( ) r n C F νr 2 k n n λ [ ( t e νr2 n t G 1, k a,k+1 1 ) ( λ, t s G 1, k a+1,k+2 1 )] λ, t s µγ(1 + a) = τ N,a (r, t) µγ(1 + a) t Ω N (r, t) R 1, a ( 1 λ, t ) 1 k=0 λ k k+1 t ds+ ds = Ω N (r, t) [G 1, k a,k+1 ( 1 ) λ, t G 1, k a+1,k+2 ( 1 )] λ, t (44) Direct computations implying suitable grouping of terms shows that Eq. (43), corresponding to the special cases a = 0 and 1, can be respectively written in the forms (see also Eq. (A7), ω 0 (r, t) = ω N,0 (r, t) 2fλ ρ ω 1 (r, t) = ω N,1 (r, t) 2fλ ρ n=1 n=1 ( ) k ( νr2 n r n C F ne νr2 n t L 1 λ ( ) k ( νr2 n r n C λ F n e νr2 n t e νr2 n t L 1 q λq 2 + q + νr 2 n 1 λq 2 + q + νr 2 n. ), (45) ), (46) where ω 0 (r, t) and ω 1 (r, t) denote the velocity field respectively corresponding to a = 0 and a = 1. By using formulae (A8), one obtains ω 0 (r, t) = ω N (r, t) + πfλ µ Γ(1 + a) 1 C F n [e νr2n t + qn2eq n1t q n1e q ] n2t, (47) r n q n1 q n2 n=1 ω 1 (r, t) = ω N,1 (r, t) πfλ µν Γ(1 + a) 1 r 3 n=1 n [ C F n e νr2 n t q2 n2e qn1t qn1e 2 q ] n2t, (48) q n1 q n2 where q n1, q n2 are the solutions of equation λq 2 + q + νr 2 n = 0. Taking into account Eq. (27), the solutions (47) and (48) can be written in the following simpler forms: ω 0 (r, t) = f 2µ ( R1 R 2 ) 2 ( ) r R2 2 r n=1 rn 3 + πf µ 1 q n2e qn1t q n1e qn2t C F n, (49) r n q n1 q n2 n=1 ω 1 (r, t) = ft ( ) 2 ( ) R1 r R2 2 πf 1 C 2µ R 2 r µν F n + λ πf 1 µν n=1 rn 5 C F n q 2 n2e q n1t q 2 n1e q n2t q n1 q n2. (50)

43 Some accelerated flows for an Oldroyd-B fluid 41 A similar procedure, applied to Eq. (44), yields ( τ 0(r, t) = τ N (r, t) µr 1,0 1 ) λ, t Ω N (r, t)+ +πfνλ r nc F nl 1 q (51) ( n=1 λ q + 1 ), (q + νrn)(λq q + νrn) 2 λ ( τ 1 (r, t) = τ N,1 (r, t) µr 1, 1 1 ) λ, t Ω N (r, t)+ +πfνλ r n C F nl 1 1 (52) ( n=1 λ q + 1 ). (q + νrn)(λq q + νrn) 2 λ After a straightforward computation we get τ 0(r, t) = fr2 1 r 2 1 e t λ + πf λ n=1 C F n e q n1t e q n2t q n1 q n2, (53) τ 1 (r, t) = fr2 1 r 2 t t 1 e λ + πf 1 ν r 2 C F n n=1 n [1 qn2eq n1t q n1e q n2t q n1 q n2 ]. (54) 3. In the special case when λ 0 into (25), (30), (32) and (34), the solutions (see also Eq. (A6)) + πf ρ λ rγ(1 + a) n=1k=0m=0 = ω N,a (r, t) + λ r Γ(1 + a) ω(r, t) = ω N,a (r, t)+ k k!λ C m r ( νr 2 F n n) k+1 t s m!(k m)!γ(a β m ) a β m 1 e νr2 n(t s) ds = k k=0m=0 k!λ m t r m!(k m)! 0 0 s k+2 ω N (r, s) (t s)a β m 1 Γ(a β m ) ds (55)

44 42 Ilie Burdujan and t (t s) a β τ(r, t) = τ N,a(r, t) + µγ(1 + a)λ r sωn (r, t) ds Γ(a β + 1) πfνλ rγ(1 + a) r nc F n n=1 t = τ N,a(r, t) + µλ rγ(1 + a) +µγ(1 + a)λ r k=0m=0 k 0 k=0m=0 0 k!λ m r ( νr 2 n) k+1 m!(k m)! (t s) a β sωn (r, s) ds+ Γ(a β + 1) [ k k!λ k t r (t s) a β m 1 m!(k m)! Γ(a β m ) 0 [ t s a β m 1 Γ(a β m ) + s a βm β 1 λr Γ(a β m β) 0 ] (t s) a β m β 1 + λ r Γ(a β m β) s k+2 Ω N (r, s) ds (56) corresponding to a generalized second grade fluid are obtained. Of course, by taking λ r 0 into Eqs. (53) and (54), we again obtain the Newtonian solutions given by Eq. (42). Moreover, in the special case when β 1, Eqs. (55) and (56) reduce to the solutions for an ordinary second grade fluid, namely + πfλr ρ and Γ(1 + a)λr n=1k=0m=0 = ω N,a(r, t) + λ rγ(1 + a) +µγ(1 + a)λ r ω(r, t) = ω N,a (r, t)+ k k!λ m r ( νr 2 C F n n) k+1 t s m!(k m)!γ(k + a m + 1) k+a m e νr2 n (t s) ds = k=0m=0 k k!λ m t r m!(k m)! 0 k+2 s ω N (r, s) t τ(r, t) = τ N,a (r, t) + µaλ r (t s) a 1 s Ω N (r, t) ds πfνλ r Γ(1 + a) r n C F n n=1 [ t s k+a m 0 Γ(k + a m + 1) + λ r k=0m=0 0 k=0m=0 k k!λ m r m!(k m)! ( νr2 2) k+1 s k+a m 1 Γ(k + a m) = τ N,a(r, t) + µλ rat a 1 tω N (r, t)+ 0 (t s) k+a m Γ(k + a m + 1) ds (57) ] e νr2 n (t s) ds = [ k k!λ k r t k+a m m!(k m)! Γ(k + a m + 1) + λ t k+a m 1 r Γ(k + a m) for a 0, while for a = 0 we have πfνλ r r n C F n n=1 k=0m=0 τ(r, t) = τ N (r, t) + µλ rω N (r, t) k k!λ m r m!(k m)! ( νr2 2) k+1 t 0 [ s k m (k m)! + λr s k m 1 (k m 1)! ] k+2 t Ω N (r, t) ] e νr2 2 (t s) ds = (58) ] e νr2 n (t s) ds = (59)

45 Some accelerated flows for an Oldroyd-B fluid 43 +µλ r k=0m=0 = τ N (r, t) + µλ rω N (r, t))+ { } k k!λ k r t k m 1 m!(k m)! (k m 1)! [t + λ r(k m)] k+2 t Ω N (r, t). For a = 0 and 1 these solutions take the simple forms. Case a = 0. ω(r, t) = ω N (r, t) πf µ 1 n=1 r n C F n e νrnt νrnλ 2 r e νrn 2 t = = f ( ) νr R 1 R 2 2 r πf nt 2 1 C 2µ R 2 r µ n=1 r F n e 1 + νrnλ 2 r, n νr τ(r, t) = fr2 1 r 2 + πf nt 2 C F 1 n n=1 1 + νrnλ 2 e 1 + νrnλ 2 r = r νr 2 = τ N (r, t) + πf nt νrnλ 2 e 1 + νrnλ 2 r e νrn 2 t. r C F n n=1 (60) Case a = 1. ω(r, t) = ω N,1(r, t)+ + fλr 2µ ( ) 2 ( ) R1 R 2 2 r + πf R 2 r µν n=1 C F n r 3 n = f ( ) 2 ( ) R1 R 2 2 r (t λ 2µ R 2 r r ) πf µν τ(r, t) = fr2 1 r 2 t + πf ν 1 n=1 rn 2 C F n πf ν = τ N,1(r, t) + fr2 1[(R 2 2 r 2 ) 2 (R 2 2 R 1 ) 2 ] 8νR 2 2r 2 e νr2 n t + πf µν n=1 1 + νrnλ 2 r rn 3 C F ne 1 n=1 rn 3 C F n + πf µν n=1 νr nt 2 1 n=1 rn 2 C F 1 + νr ne nλ 2 r = πf 1 C F ν n e n=1 rn νrnλ 2 r rn 3 C F n e νrnt νrnλ 2 r = νrnt νrnλ 2 r, νrnt νrnλ 2 r e νrn 2 t. 4. In the special case when α 1 and β 1 into (25), (30), (35) and (37), the solutions (see also Eq. (A6)) for an Oldroyd-B fluid are obtained, (61)

46 44 Ilie Burdujan namely: ω(r, t) = ω N,a (r, t) Γ(1 + a) k k=0m=0 ( k!λ m r m!(k m)!λ k k+1 t ω N (r, t) G 1,m k a,k+1 1 ) λ, t + + λ r λ Γ(1 + a) k=0m=0 = ω N,a(r, t) πf ρ k k!λ m r m!(k m)!λ k k+2 t ω N (r, t) G 1,m k a 1,k+1 = Γ(1 + a) r n C F n n=1 k=0m=0 k k!λ m r m!(k m)! ( ) k νr2 n λ [G 1,m k a,k+1 ( 1 ) λ, t + νrn 2 λ r λ G 1,m k a 1,k+1 ( 1 )] λ, t, (62) τ(r, t) = τ N,a (r, t) + µγ(1 + a) λ ( r λ R λ 1, a 1 ) λ, t t Ω N (r, t) µγ(1 + a) k k!λ m r k=0m=0 m!(k m)!λ k [ G 1,m k a,k+1 ( 1/λ, t) + λ ( r λ G 1,m k a+1,k+2 ( 1/λ, t) G 1,m k a+1,k+2 1 )] λ, t k+1 t Ω N (r, t)+ +µγ(1 + a) λ r k k!λ m r λ k=0m=0 m!(k m)!λ k [ G 1,m k a 1,k+1 ( 1/λ, t) + λ ] r λ G 1,m k a,k+2 ( 1/λ, t) G 1,m k a,k+2 ( 1/λ, t) k+2 t Ω N (r, t) = = τ N,a(r, t) + µγ(1 + a) λr λ ( R λ 1, a 1 ) λ, t tω N (r, t)+ ( ) +πfνγ(1 + a) k r nc F k!λ m k r n νr2 n m!(k m)! λ + νrn 2 λ r λ n=1 k=0m=0 {[G 1,m k a,k+1 ( 1 ) λ [G, t 1,m k a 1,k+1 ( 1 ) λ, t + λ r λ λ G 1,m k a+1,k+2 ( 1 λ, t )] + + λ ( r λ G λ 1,m k a+1,k+2 1 )]} λ, t e νr2 n t (63) Using (A8) we respectively get, for a = 0 and a = 1, the following solutions. Case a = 0 ω(r, t) = ω N (r, t) πf µ ( 1 C r F n e νr2 nt q n2e qn1t q n1 e q ) n2t = n q n1 q n2 n=1 = f ( ) 2 ( ) R1 R 2 2 2µ R 2 r r + πf µ 1 q C n2 e qn1t q n1 e q n2t r F n, n q n1 q n2 n=1 (64)

47 Some accelerated flows for an Oldroyd-B fluid 45 τ(r, t) = τ N (r, t)+ J +fr 2 (rr n ) 2 r n J 1 (R 2 r n ) = 2f n=1 R 2 n=1 Case a = 1 J 2 (rr n ) r n J 1 (R 2 r n ) ω(r, t) = ω N,1 (r, t) fλ r 2µ [ (λ r νrn 2 1)e νr2 nt + (1 + λq n1)e qn1t (1 + λq n2 )e q ] n2t = λ(q n1 q n2 ) e t λ λ ( ) 2 ( ) R1 R 2 2 R 2 r r πf µν 1 n=1rn 3 = f ( ) 2 ( ) R1 R 2 2 2µ R 2 r r (t λ r ) πf µν e qn1t e q n2t. q n1 q n2 (65) ( C F n e νr2 nt λ q2 n2e qn1t qn1e 2 q ) n2t = q n1 q n2 1 n=1rn 3 [ C F n 1 λ q2 n2e qn1t qn1e 2 q ] n2t, q n1 q n2 (66) τ(r, t) = τ N,1 (r, t) + λ fr2 1 r 2 1 e t λ πf ν 1 n=1rn 2 C F n ( ) 1 e νr2 nt πf ν = fr2 1 r 2 1 n=1rn 2 C F n + πf ν t λ + λe t λ 1 n=1rn 2 C F n + πf ν 6. CONCLUSIONS 1 n=1rn 2 C F n ( 1 q n2e qn1t q n1 e q ) n2t = q n1 q n2 ( 1 q n2e qn1t q n1 e q ) n2t. q n1 q n2 (67) The main purpose of this work is to provide exact solution for the unsteady flow of an incompressible Oldroyd-B fluid filling the annular region between two infinitely long co-axial cylinders subject either to a constant or to a timedependent shear stress. Such solutions, obtained by using both the Hankel and Laplace transforms, are presented as a sum between the Newtonian solutions and the corresponding non-newtonian contributions. Furthermore, the non-newtonian contributions of the general solutions are also presented in equivalent forms, under series form in terms of the time derivative of the simplest Newtonian solution ω N. For λ r 0 and λ 0 these contributions

48 46 Ilie Burdujan tend to zero, such that the general solutions become Newtonian solutions corresponding to the given initial-boundary conditions. It is remarkable that the general solutions can be easily specialized to give both the similar solutions for generalized second grade and Maxwell fluids and the solutions for all ordinary fluids (Oldroyd-B, Maxwell and second grade) performing the same motions. Direct computations shows that the solutions which have been obtained certainly satisfy both the governing equations and all imposed initial and boundary conditions. Furthermore, the solutions corresponding to ordinary Maxwell and second grade fluids can be also obtained as limit cases of those for ordinary Oldroyd-B fluids. As regard the Newtonian solutions, given under simple forms (27), (29), (38) and (39), they can be obtained as limit cases of the previous solutions. From our general solutions, corresponding to non-newtonian fluids, it clearly results that the non-newtonian contributions of these solutions exponentially decrease in time, the motion of the non-newtonian fluids being well approximated, for large values of t, by the motion of the corresponding Newtonian fluid. Appendix A L{D p t f(t)} = qp L{f(t)} q 1 p f(0 + ); 0 < p < 1, (A1) G a,b,c (d, t) = L 1 { q b (q a d) c } ; Re(ac b) > 0, Re(q) > 0, d < 1, (A2) q a R2 R 1 (r 2 R 2 2) B(rr n ) dr = ( R2 R 1 ) 2 4 πrn 3, (A3) L{(f g)(t)} = L{f(t)} L{g(t)}, (A4) R a,b (c, t) = L 1 { q b q a c } ( s ν ), Re(a b) > 0, Re(q) > 0, E t (ν, a) = L. s a (A5) 1 lim λ 0λ k G a,b,k( 1/λ, t) = t b 1 Γ( b), lim 1 λ 0λ R a,b( 1/λ, t) = t b 1 Γ( b), (A6) 1 lim λ 0λ F a( 1/λ, t) = δ(t).

49 Some accelerated flows for an Oldroyd-B fluid 47 k=0 ( ) k νr2 n G 1, k a,k+1 ( 1λ ) ( ) 1 λ, t = λl 1 1 q a 1 λq 2 + q + νrn 2 (A7) ( ) e νr2 nt L 1 q λq 2 + q + νrn 2 = 1 ( νrn 2 e νr2 nt + q n2e qn1t q n1 e q ) n2t, ( ) q n1 q n2 e νr2 nt L 1 1 λq 2 + q + νrn 2 = 1 [ (νrn) 2 2 e νr2 nt + λ q2 n2e qn1t qn1e 2 q ] n2t, q n1 q n2 e at e bt = eat e bt a b. (A8) References [1] Bagley, R. L., A theoretical basis for the application of fractional calculus to viscoelasticity, J.Rheology, 27(1983), [2] Bandelli R., Rajagopal, K. R., Start-up flows of second grade fluids in domains with one finite dimension, Int. J. Non-Linear Mech., 30(1995), [3] Bandelli, R., Rajagopal, K. R., Galdi, G. P., On some unsteady motions of fluids of second grade, Arch. Mech., 47(1995), [4] Batchelor, G. K., An Introduction to Fluid Dynamics, Cambridge University Press, Cambridge, [5] Debnath, L., Bhatta. D., Integral Transforms and their Applications(second ed.), Chapman and Hall/CRC Press, New York, [6] Friedrich, Ch., Relaxation and retardation functions of the Maxwell model with fractional derivatives, Rheol. Acta, 30(1991), [7] Georgescu, A., Palese, L., Redaelli, A., A direct method and ite application to a linear hydromagnetic stability problem, ROMAI J., 1, 1(2005), [8] Glökle, W.G., Nonnenmacher, T. F., Fractional relaxation and the time-temperature superposition principle, Rheol. Acta, 33(1994), [9] Hayat, T., Khan, M., Wang, T., Non-Newtonian flow between concentric cylinders, Comm. Non-Linear Sci. Numer. Simm., 11(2006) [10] Heibig, A., Palade, L. I., On the rest state stability of an objective fractional derivative viscoelastic fluid model, J. Math. Phys. 49(2008), [11] Khan, M., Hyder Ali, S., Qi, H., Some accelerated flows for a generalized Oldroyd-B fluid, Nonlinear Analysis: Real world Applications (2007), doi: /j.nonrwa [12] Lorenzo, C. F., Hartley T. T., Generalized Functions for the Fractional Calculus, NASA/TP , [13] Mainardi, F., Fractional relaxation-oscillation and fractional diffusion-wave phenomena, Chaos, Solitons&Fractals 7, 9(1996), [14] Mainardi, F., Gorenflo, R., On Mittag-Leffler-type functions in fractional evolution processes, J. Comput. Appl. Math. 116, 2(2000),

50 48 Ilie Burdujan [15] Makris, N., Constantinou, M. C., Fractional derivative Maxwell model for viscous dampers, J. Struct. ASCE, 117, 9(1991), [16] Podlubny, I., Fractional Differential Equations, Academic Press, San Diego, [17] Rossikhin, Y. A., Shitikova, M. V., A new method for solving dynamic problems of fractional derivative viscoelasticity, Int. J. Engng Sci. 39(2000), [18] Rossikhin, Y. A., Shitikova, M. V., Analysis of dynamic behavior of viscoelastic rods whose rheological models contain fractional derivatives of two different orders, ZAMP 81, 6(2001), [19] Srivastava, P. N., Non-steady helical flow of a viscoelastic liquid, Arch. Mech. Stos., 18(1966), [20] Tong, D., Liu, Y., Exact solutions for the unsteady rotational flow of non-newtonian fluid in an annular pipe, Int. J. Eng. Sci., 43(2005), [21] Tong, D., Ruihe, Y., Heshan, W., Exact solutions for the flow of non-newtonian fluid with fractional derivative in an annular pipe, Science in China, Ser. G Physics, Mechanics & Astronomy, 48(2005), [22] Wood, W. P., Transient viscoelastic helical flows in pipes of circular and annular crosssection, J. Non-Newtonian Fluid Mech., 100(2001),

51 ROMAI J., 5, 2(2009), A FUZZY ALGORITHM FOR RELIABILITY SIMULATION OF AN ELECTRIC STATION Simona Dzitac, Tiberiu Vesselenyi, Ioan Dzitac, Maria Parv Energy Engineering Faculty, University of Oradea, Romania, Management and Technological Eng. Faculty, University of Oradea, Romania, Exact Sciences Faculty, Aurel Vlaicu University of Arad, and Cercetare Dezvoltare Agora/Agora R & D, Oradea, Romania, University of Agricultural Sciences and Veterinary Medicine, Cluj-Napoca, Romania simona.dzitac@gmail.com Abstract In this paper we present an applied study of reliability simulation for an electric station, based on a soft computing simulation method, namely using a fuzzy algorithm in MATLAB environment. This study revealed that the values of reliability obtained through this method is accurate compared to the values obtained by Monte Carlo method or by direct computation. Keywords: failure tree, reliability, fuzzy simulation, electric station (ES) MSC: 03E75, 93C INTRODUCTION The soft computing paradigm is based on fuzzy logic and is tolerant to imprecision, uncertainty, partial truth, and approximation. Fuzzy logic represents an extremely useful tool in modeling the behavior of electrical equipment. Fuzzy set theory is using multi state systems and multi criteria decisions, forming a mathematical instrument which is flexible and easily adaptable to reality. This theory is useful for modeling electromagnetic systems and also for energy equipment reliability evaluation [1, 10]. In reliability studies a bivalent operational evolution mode is generally accepted: normal operation state and failure state. In reality transitions between states are not swift, which implies a nuanced expression of system s performance (very good, good,..., medium, poor). This paper presents the development of reliability simulation software for electric stations. The software is based on failure trees method and is using fuzzy logic in the MATLAB environment. The MATLAB programming environment has predefined functions for development of fuzzy computing steps (fuzzyfication, inference, defuzzyfication) [6, 8]. These functions are linked to 2 external C++ modules, the inference system and the fuzzy engine. Typical structures of fuzzy inference systems can be represented by a model which reveals a correspondence between: crisp input value - membership functions 49

52 50 Simona Dzitac, Tiberiu Vesselenyi, Ioan Dzitac, Maria Parv - inference rules output characteristics - output membership functions - crisp output values. Similar studies and program variants are presented in [3] and [4]. In Section 2 we describe the development of a computer simulation program for the study of complex electric system s reliability, fuzzy algorithms, definition of asymmetric Gauss input and output membership functions, rule sets and result display methods. The third Section is focused on a case study for the electric station in Voivozi, (Bihor county) using the developed simulation program under MATLAB environment. Section 4 presents the conclusions, showing the importance and efficiency of fuzzy modeling in reliability analysis by comparing fuzzy and Monte Carlo methods also shown in equivalent reliability diagrams and highlighting the contribution of the authors. 2. DEVELOPMENT OF THE SIMULATION SOFTWARE USING FUZZY LOGIC A frequently used analysis method in a system s reliability study is based on failure probability evaluation. In this method the crisp values of failure probabilities for electrical components are generally used in order to compute the system s reliability, based on equivalent reliability diagrams [2,3,4,6,8]. 2.1 Definition of the input membership functions No. Linguistic variable Acronym Value 1. Not acceptable NA 0 2. Almost acceptable AA Close to acceptable CA Acceptable A Good G Almost very good AVG Very good VG 1 Table 1. The developed software is using Gaussian membership functions. For this kind of function the mean and the standard deviation (σ) must be specified. So for every component of the system seven degrees were defined (see Table 1), on a linear interval of failure and repair intensity values, and then the function values were established (λ and µ).

53 A fuzzy algorithm for reliability simulation of an electric station 51 Fig. 1. Schematic function blocks for fuzzy analysis set up. Fig. 2. Flowchart of the fuzzy program. The fuzzy method is presented schematically in Figure 1. The fuzzy analysis program generates input membership functions (on basis of specified failure intensity λ and repair intensity µ) and then generates the output membership functions and the rule set. The flowchart of the algorithm is presented in Figure 2. All program functions are launched from the d.fuzzy.m module. The Graphical User Interface (GUI) window is presented in Figure 3. The d.date.m module is launched on action of Input data button. The simulation data input includes λ and µ specification, data saving and data reload. The input system equation is introduced from the d.param module and the input λ and µ are instantiated from a separate window which allows as many parameters as many components were specified. The program also allows data saving (the data are saved in a.mat type file) in files with op-

54 52 Simona Dzitac, Tiberiu Vesselenyi, Ioan Dzitac, Maria Parv Fig. 3. Main program GUI (Translation: Fuzzy simulation...; Data; Inputs; Fuzzy simulation; Decision surfaces; Exit). tional names. The saved data can be reloaded in a separate interface from their files which contain all system parameters and also the system equation. Once established the input system parameter values for specific runs can be instantiated from the d.param.intr.m module. Mean values for membership functions are computed on basis of the relation F i = λ i λ i + µ i, (1) where i =1...7, is the number of the membership function according to the earlier defined grade. The standard deviation (σ) is computed with an asymmetric Gauss function based on the relations: σ i,1 = F i 1 F i, (2) 3 σ i,2 = F i F i+1. (3) 3 After the introduction or reload of data from saved files, we can reenter the main module which gives us two options: fuzzy simulation or decision surfaces display. After computing the input membership functions parameters for each system component the program generates these functions. The decision surface display facilitates the evaluation of the fuzzy outputs. 2.2 Definition of the output membership function In order to compute the output membership functions we start with the reduced system schematics from which the failure tree is generated. From

55 A fuzzy algorithm for reliability simulation of an electric station 53 the flowchart we can derive the system s characteristic equation and then the program generates the output membership function. 2.3 Definition of the rule set The rule set of the fuzzy inference system defines the way in which the inputs and the outputs are linked. The rules are described in form of logical relations having as variables linguistic degrees of the inputs and as operators the and and or logical operators. An example of fuzzy rule is: If elem1 is VW and elem2 is W and elem3 is A then the system is W. After the establishment of the rule set, the program can generate inference surfaces in the input-output space which are in fact the values of the outputs for the whole range of given inputs. Due to the limitations of 3D representation, these surfaces can be represented only as 2 inputs simultaneously, the remaining inputs being considered static for that case. The 2 inputs which we want to represent can be selected in the program interface. 2.4 Simulation results After generating the membership functions and the rule sets the program also generates the so called fuzzy inference system information structure. If this structure is used for a single run, then the crisp values of the inputs are specified and the evalfis function is used for the computation of the crisp output values. The program displays this value in a separate window. 3. CASE STUDY VOIVOZI ELECTRIC STATION Case study is performed for the normal form of Electric Stations (ES) Voivozi, Bihor County scheme. The evaluation of reliability is realized considering the Padurea Neagra user, positioned on BC1-20kV collector bar and the study criteria is considered in the absence of the consumer. Analyzing the operative mono cable 110 kv scheme of the Bihor County energy system, it can be concluded that Suplac is considered output and Oradea Vest and Marghita are considered inputs. Using statistical data representing median values of reliability indicators for the equipments in ES and also using the equivalent reliability diagram we have reduced the ES Voivozi scheme to an Equivalent Reliability Diagram (ERD) presented in Figure 7. This diagram was used to formulate the system of equa-

56 54 Simona Dzitac, Tiberiu Vesselenyi, Ioan Dzitac, Maria Parv Fig. 4. λ and µ parameter editing window for ES Voivozi. tion for the fuzzy simulation. The reduction of normal scheme has been made by transposing it in a scheme in which the elements are connected in series or parallel considering the dimensioning and the connection of elements. All feeds for Padurea Neagra consumer, on all path, from the source have been considered. In Figures 4-5 input data of analyzed electric station are presented. In Figure 6 the obtained membership function diagram are presented for ES Voivozi. Similarly we have representations of last nine elements of ERD. For computing output membership functions we start from the system scheme from which the failure tree is generated - Figures 7, 8. The characteristic equation of the system is deduced from the schemes presented in Figures 7, 8 and is given by relation F V OI = 1 (1 F 1 F 2 )(1 F 3 )(1 F 4 F 5 )(1 F 6 F 7 )(1 F 8 )(1 F 9 )(1 F 10 ) (4) Relation (4) is used by the program to generate the output membership function presented in Figure 9. In Figure 10 is presented, for example, a decision surface. The program displays the obtained output values in a separate window presented in Figure 11.

57 A fuzzy algorithm for reliability simulation of an electric station 55 Fig. 5. Simulation data editing window for ES Voivozi. Fig. 6. Membership functions for element 1. Fig. 7. The reduced scheme for ES Voivozi.

58 56 Simona Dzitac, Tiberiu Vesselenyi, Ioan Dzitac, Maria Parv Fig. 8. The failure tree for ES Voivozi. Fig. 9. Output membership functions generated for the analyzed system. Fig. 10. Output values for inputs 1 and 2.

59 A fuzzy algorithm for reliability simulation of an electric station 57 Fig. 11. Reliability output window for the reliability of ES Voivozi. 4. CONCLUSIONS The use of fuzzy sets theory in the study of the reliability of the electric energy systems and equipments is justified by the possibilities offered by the quantification and the modeling of the qualitative enounces - incomplete and altered information, subjective appreciations - in flexible forms, more close to the way of thinking that the engineers operates with. The program developed under MATLAB environment for the fuzzy simulation of reliability of electrical equipments permits the step by step definition of the fuzzy model and it is realized in a versatile manner, object oriented and modular. The program can make diverse simulations, in small times, for a given scheme, in the analyzed fuzzy intervals making possible the visualization of values range in which the non-reliability and the reliability of the system can evolve. In Table 2 we can see that the realized evaluations, obtained with the ES reliability fuzzy simulation program, are accurate, in comparison with the values obtained by Monte Carlo method and the direct ERD computation [2, 4]. ES/R FUZZY MONTE CARLO ( sim.) ERD VOIVOZI 0, , ,99947 Table 2. The development of ES fuzzy reliability simulation program by using the MATLAB programming environment, based on the failure tree method, application of this program for ES Voivozi, Bihor County, and the comparative evaluation with the Monte Carlo simulation method results and with the ERD analytical method results, are the contributions of the authors in this article.

60 58 Simona Dzitac, Tiberiu Vesselenyi, Ioan Dzitac, Maria Parv References [1] R. Billinton, R. Allan, Reliability Evaluation of Power Systems, Plenum Press, New York and London, [2] S. Dzitac, Contributions to modeling and simulation of reliability and disposability performance simulation for electric energy distribution systems, PhD Thesis, University of Oradea, [3] S. Dzitac, C. Hora, Fuzzy simulation in reliability analysis, Ann. of the Faculty of Eng. Hunedoara - Journal of Engineering, VII, 3(2009), [4] S. Dzitac, I. Felea, Application of fuzzy modeling in reliability analysis of the electric stations, International World Energy System Conference, Iasi, June 30 -July 2, [5] S. Dzitac, T. Vesselenyi, I. Dzitac, E. Valeanu, Electrical Power Station Reliability Modeling Procedure using The Monte Carlo Method, IFAC MCPL 2007, 4th IFAC Conference on Mangement and Control of Production and Logistic, Sibiu September, Romania, September 27-30, 2007, Preprints, ISBN , vol. III, [6] I. Felea, N. Coroiu,Reliability and maintenance of electrical equipment, Ed. Tehnica, Bucuresti, [7] E. Sofron, N. Bizon, S. Ionica, R. Raducu, Fuzzy control systems. Computer aided modeling and design, All, Bucuresti, [8] H. Tanaka, L. Fan, F. Lai, K. Toguchi, Fault-Tree Analysis by fuzzy probability, IEEE Transactions on Reliability, 32(5), , [9] T. Vesselenyi, S. Dzitac, I. Dzitac, M.-J. Manolescu, Fuzzy and Neural Controllers for a Pneumatic Actuator, Int. J. of Computers, Communications & Control (IJCCC), 2, 4(2007), , [10] L.A. Zadeh, D. Tufis, F.G. Filip, I. Dzitac (eds.), From Natural Language to Soft Computing: New Paradigms in Artificial Intelligence, Editura Academiei Roamane, 2008.

61 ROMAI J., 5, 2(2009), A SEVEN EQUATION MODEL FOR RELATIVISTIC TWO FLUID FLOWS-I Sebastiano Giambò, Serena Giambò Department of Mathematics, University of Messina, Italy giambo@dipmat.unime.it, sgiambo@unime.it Abstract An interface-capturing method is used to describe relativistic two-fluid flows. The conservation equations for the particle number of each fluid and for the total momentum-energy tensor of the mixture are the starting point of this approach. A model for relativistic two-fluid flow without friction and heat conduction and differential equations, plus additional algebraic relations, consistent with this model, are derived. The weak discontinuities propagating in this relativistic two-fluid system are examined and the expressions for the speeds of propagation are obtained. Keywords: general relativity, relativistic fluid dynamics, two-fluid mixtures, nonlinear waves MSC: 83C99, 80A10, 80A17, 76T99, 74J INTRODUCTION There are many topics in General Relativity where matter is represented as a mixture of two fluids. In fact, some astrophysical and cosmological situations need to be described by an energy tensor consisting of the sum of two or more perfect fluids. For most of the history of the universe, the dominant matter content is a mixture of matter and radiation [1]-[12]; other examples are a null fluid with string fluid [13], or a radiation fluid in addiction to a string fluid [14]-[17]. It was also shown [18]-[22] that an anisotropic relativistic fluid can be consistently described by two-perfect-fluid components and inflationary models have been deduced as mixtures of two relativistic fluids [23], [24]. Moreover, the acoustic modes [25] and the wave fronts [26], [27] have been studied in some of the cases quoted above. The purpose of this paper is to build up a relativistic formulation of some recent results on the classical dynamics of a mixture of two perfect fluids based on the papers of H. Guillard and A. Murrone [28]-[30]. The model presented here is a two-phase flow model, in which the entire flow domain is filled with a mixture of the two fluids. However, in this underlying two-phase model, the fluids are not mixed on the molecular level: the mixture consists of very small elements of the two pure fluids, arranged in an irregular pattern. So the fluid is a mixture in the macroscopic sense. 59

62 60 Sebastiano Giambò, Serena Giambò Moreover, both fluids are assumed to be present everywhere in the flow domain and the interfaces between the two fluids are considered as gradual transitions from fluid 1 to fluid 2. In this way, the concept of interface between the two fluids disappears from the model [28]-[32]. Each fluid still has its own particle number density, r k, its specific internal energy, ɛ k, and its energy density ρ k, [33], [34]: ρ k = r k (1 + ɛ k ), k = 1, 2. (1) In what follows, the units are such that the velocity of light is unitary: c = 1. Conversely, a single pressure, p, and a single four-velocity, u α, are assumed for the two fluids. Here, u α is the fluid unit four-vector defined to be futurepointing g αβ u α u β = 1, (2) where g αβ are the covariant components of Lorentz metric tensor with signature +,,,. In this paper we derive the complete system of governing differential equations and we determine the propagation speed of weak discontinuity wavefronts in this relativistic two-fluid model. The paper is organized as follows. Section 2 starts with a description of the relativistic mixture and the derivation of the flow equations. Section 3 analyzes the source term appearing in the flow equations. In Section 4, the evolution equation for the pressure is derived in order to perform the closure of the system of differential equations obtained in Section 2. In Section 5, the propagation of weak discontinuities admitted by the model under consideration are examined and the expressions for their speeds of propagation are obtained. Section 6 concerns a special case, that may be physically relevant, in which the expression of the velocity is the relativistic version of the Wallis formula [39]. 2. RELATIVISTIC FLOW MODEL The standard equations for simple relativistic fluid flow hold for the twofluid model. The total energy-momentum conservation is where the stress energy tensor is given by α T αβ = 0, (3) T αβ = rfu α u β pg αβ, (4) being r the total particle number density, f the relativistic total specific enthalpy f = 1 + h = 1 + ɛ + p r, (5)

63 A seven equation model for relativistic two fluid flows-i 61 with h = ɛ+ p r the classical specific enthalpy, of the mixture, ɛ and p denoting the total energy density and pressure, respectively. Moreover, the balance law for the total particle number is The projection of equation (3) along u α is α (ru α ) = 0. (6) u β α T αβ u α α ρ + (ρ + p) α u α = 0, (7) being ρ = r (1 + ɛ) the total energy density, whereas the spatial projection of equation (3) is γ λ β αt αβ rfu α α u λ γ αλ α p = 0, (8) where γ αβ = g αβ u α u β is the projection tensor onto the three-space orthogonal to u α (the rest space of an observer moving with four-velocity u α ). However, we have to determine suitable expressions for the bulk quantities r, ɛ, ρ and f. First, the volume fraction X and the mass fraction Y, Y = Xr 1 r of fluid 1 are chosen as field variables. The variables X and Y allow to define any bulk quantity; the particle number density r, the specific internal energy ɛ, the energy density ρ and the relativistic specific enthalpy f are defined as: r = X 1 r 1 + X 2 r 2, (9) ɛ = Y 1 ɛ 1 + Y 2 ɛ 2, f = Y 1 f 1 + Y 2 f 2, ρ = X 1 ρ 1 + X 2 ρ 2, rf = X 1 r 1 f 1 + X 2 r 2 f 2, (10) with X 1 = X, X 2 = 1 X, Y 1 = Y, Y 2 = 1 Y. (11) Thus, for regular solutions, the mathematical study of the model can be performed using the following set of seven independent field variables u α, r, p, X, Y. The governing system (6)-(8) is a set of five equations for seven variables. Thus, two more equations are to be determined in order to close the system.

64 62 Sebastiano Giambò, Serena Giambò Since all the bulk equations have already been used, the only option is to take into account quantities characterizing one of the two fluids. The first equation to be considered is, of course, the balance law for the particle number density for fluid 1. Using the partial density X 1 r 1, the corresponding equation is α (Xr 1 u α ) = 0. (12) From the conservation equation (12) (written using the relation Xr 1 = Y r) α (Y ru α ) = 0, taking into account equation (6), we obtain the following evolution law for the variable Y u α α Y = 0. (13) Observe that equation (12), together with equation (6), implies the following balance law for the particle number density of fluid 2 α (X 2 r 2 u α ) = 0, (14) where, as already said, X 2 = 1 X. At this point, it is clear that the only option in order to get one more equation, and then the closure of the governing system, is to determine the balance equation for the energy-momentum tensor of fluid 1 T αβ 1 = (ρ 1 + p) u α u β pg αβ. (15) Since exchanges of energy and momentum between the fluid components are allowed, there will be no local energy-momentum conservation for each fluid component separately. Then, the equation for the energy-momentum tensors of each of the two fluids, T αβ k, k = 1, 2, has the following form ( ) α X k T αβ k = F β k, k = 1, 2, (16) where F β k represents the loss and source term in the separate balance. Now, since the total energy-momentum tensor is conserved, according to (3), and taking into account the expression (15) of T αβ 1 and the relation T αβ = X 1 T αβ 1 + X 2 T αβ 2 (17) it is easily shown that F β 1 = F β 2 = F β.

65 A seven equation model for relativistic two fluid flows-i DERIVATION OF THE SOURCE TERM This section is devoted to handling the source term F β in equations (16). The projection along u α and the spatial projection of equation (16) for fluid 1 are, respectively, Xu α α ρ 1 + ρ 1 u α α X + X (ρ 1 + p) α u α = u α F α (18) and X { } (ρ 1 + p) u α α u β γ αβ α p pγ αβ α X = γ β αf α. (19) Equation (18), taking into account equations (1) and (12), yields the following equation ( ) Xr 1 u α 1 α ɛ 1 + p α = p u α α X + u α F α. (20) r 1 We assume the following axiom: the entropy S k of each fluid component is a function of the energy ɛ k and the specific volume 1/r k S k = S k (ɛ k, r k ), k = 1, 2. (21) By thermodynamic arguments, the derivatives of entropy can be related to some observable variables. Thus, we can write ( ) Sk = 1, ɛ k r k T k ( ) (22) Sk = p r k ɛ k rk 2T, k where T k is the temperature of fluid component k. From equation (??), it follows that T k ds k = dɛ k + pd 1 r k (23) and then ( ) T k u α α S k = u α 1 α ɛ k + p α r k. (24) We now also suppose that the entropy S k is conserved along the flow lines u α α S k = 0, (k = 1, 2). Thus, from equation (24) we can deduce that ( ) u α 1 α ɛ k + p α = 0 (25) r k

66 64 Sebastiano Giambò, Serena Giambò and equation (20) allows to write the following relation involving F α Next, using equations (19) and (8), we obtain u α F α = p u α α X. (26) X (r 1 f 1 rf) γ αβ α p prfγ αβ α X = rfγ β αf α. (27) Now, introducing the relativistic enthalpy concentration χ = f 1 f Y, we have r 1 f 1 = χ rf, (28) X thus, from (27) we deduce γ αβ F α = (χ X) γ αβ α p pγ αβ α X. (29) Therefore, using equations (26) and (29), the source term F β can now be computed as F β = (χ X) γ α β αp p β X, (30) which represents the relativistic formulation of the classical expression of the source terms obtained by Wackers and Koren [31], [32]. 4. PRESSURE EQUATION The derivation of a pressure equation is rather involved, as it requires the two energy equations (7) and (18). From equation (7), because ρ = r(1 + ɛ), we deduce that r 2 u α α ɛ + ρu α α r + r 2 f α u α = 0. (31) The total specific internal energy can be expressed in terms of variables r, p, X and Y by an equation of state. For this analysis, we use equations of state of the most general form, writing it as with ɛ 1 = ɛ 1 (r 1, p), ɛ 2 = ɛ 2 (r 2, p), (32) r 1 = Y X r, r 2 = 1 Y 1 X r. (33) Substituting (32) in equation (10) 2, the bulk specific internal energy ɛ can be written as ɛ (r, p, X, Y ) = Y ɛ 1 (r 1, p) + (1 Y ) ɛ 2 (r 2, p). (34)

67 A seven equation model for relativistic two fluid flows-i 65 Now, thanks to this last expression (34) of ɛ, and using equations (13) and (6), the following form of the bulk energy equation (31) is deduced r ɛ p uα α p + r ɛ ( X uα α X + p r 2 ɛ ) α u α = 0. (35) r Conservation of energy for fluid 1 (equation (20)), with equation (26) and (12), becomes r 1 u α α ɛ 1 + p X uα α X + p α u α = 0, (36) and, using (33) 1 and taking into account the equation of state (32) 1, equation (36) writes as ( ) ( ) ɛ 1 p r 1 p uα α p + X + r ɛ 1 1 u α ɛ 1 α X + p rr 1 α u α = 0. (37) X r From equations (35) and (37), we are able to deduce the following evolution equations for the pressure p and the volume fraction α, respectively u α α p + ω α u α = 0, (38) u α α X + ξ α u α = 0, where ω and ξ are defined by ω = ξ = ( ) ( p r 2 ɛ ɛ r r 1 1 ( r ɛ p r ɛ p r ɛ p X + p X ( r 1 ɛ 1 X + p X ) ɛ p rr 1 ɛ 1 r r 1 1 p ( ɛ r 1 1 X + p X ) ( r ɛ X ) ɛ ɛ rr 1 1 X p ( ) p r 2 ɛ r ) ɛ ɛ rr 1 1 X p p r 1 r ɛ 1 r. ), (39) To end this section, we note that the complete system of governing differential equations may be written in term of variables (u α, r, p, X, Y ) as u α α r = r α u α, rfu α α u β = γ αβ α p, u α α p = ω α u α, u α α X = ξ α u α, u α α Y = 0. (40)

68 66 Sebastiano Giambò, Serena Giambò 5. DISCONTINUITIES In a domain Ω of space-time V 4, let Σ be a regular hypersurface, not generated by the flow lines, being ϕ (x α ) = 0 its local equation. We set L α = α ϕ. As it will be clear below, the hypersurface Σ is a space-like one, i.e. L α L α < 0. In the following, N α will denote the normalized vector N α = L α L β L β, N α N α = 1. We consider a particular class of solutions of system (40) namely, weak discontinuity waves Σ, on which the field variables u α, r, p, X, Y are continuous, but, conversely, jump discontinuities may occur in their normal derivatives. In this case, if Q denotes any of these fields, then there exists [33],[38] the distribution δq, with support Σ, such that δ [ α Q] = N α δq, where δ is the measure of Dirac defined by ϕ with Σ as support, square brackets denote the discontinuity, δ being an operator of infinitesimal discontinuity; δ behaves like a derivative insofar as algebraic manipulations are concerned. Then, from the system (40), we obtain the following linear homogeneous system in the distributions N α δu α, δr, δp, δx and δy Lδr + rn α δu α = 0, rflδu α γ αβ N β δp = 0, Lδp + ωn α δu α = 0, LδX + ξn α δu α = 0, LδY = 0, (41) where L = u α N α. Moreover, from the unitary character of u α we have u α δu α = 0. (42) Now, we want to investigate the normal speeds of propagation of the various waves with respect to an observer moving with the mixture velocity u α. The normal speed λ Σ of propagation of the wave front Σ, described by a timelike word line having tangent vector field u α, that is with respect to the time direction u α, is given by [33]-[38] λ 2 Σ = L2 1 + L 2. (43)

69 A seven equation model for relativistic two fluid flows-i 67 The local causality condition, i.e. the requirement that the characteristic hypersurface Σ has to be timelike or null (or equivalently that the normal N α be spacelike or null, that is g αβ N α N β 0), is equivalent to the condition 0 λ 2 Σ 1. From the above equations (41), we first obtain the solution L = 0, which represents a wave moving with the mixture. For the corresponding discontinuities we find N α δu α = 0, δp = 0. (44) Since the coefficients characterizing the discontinuities exhibit five degrees of freedom, then system (41) admits five independent eigenvectors corresponding to L = 0 in the space of the field variables. From now on we suppose L 0. Equation (41) 2, multiplied by N β, give us rfln α δu α l 2 δp = 0, (45) where l 2 = 1 + L 2. As a consequence, (41) 3 and (45) represent a linear homogeneous system in the two scalar distributions N α δu α and δp, which may have non trivial solutions only if the determinant of the coefficients vanishes. Therefore, we find the equation H rfl 2 ωl 2 = 0, (46) which corresponds to the hydrodynamical waves propagating in such a twofluid system. Their speeds of propagation are given by λ 2 Σ = ω rf (47) and the condition 0 < ω rf 1 ensures their spatial orientation. The associated discontinuities can be written in terms of ψ = n α δu α as follows δu α = ψn α, δr = r l L ψ, δp = ω l L ψ, (48) δx = ξ l L ψ, δy = 0, where n α is the unitary space-like four-vector defined by n α = 1 l (N α Lu α ). (49)

70 68 Sebastiano Giambò, Serena Giambò Observe that if the above condition characterizing the space-like orientations of the surface is verified, then the governing equations represent a (not strictly) hyperbolic system. In fact, all velocities (eigenvalues) are real, and there is a complete set of eigenvectors in the space of field variables, i.e. seven independent eigenvectors (5 from L = 0 and 2 from H = 0), for the seven independent field variables u α, r, p, X and Y. 6. APPLICATION Now, we want to examine the application of the preceding method in order to determine weakly discontinuous solutions in the case of a mixture of two fluids of cosmological interest. To this end, we assume that each fluid (k = 1, 2) satisfy the equation of state of perfect gases: ɛ k = 1 X k p γ k 1 Y k r, k = 1, 2, (50) where γ k is the ratio of the specific heat capacities at constant pressure and volume of the k-th fluid. Using (50), (10) 2 writes as ( X ɛ = γ X ) p γ 2 1 r. (51) Then, (39) can be written, respectively, in the following form γ ω = 1 γ 2 p, Xγ 2 + (1 X) γ 1 ξ = X (1 X) γ 1 γ 2 Xγ 2 + (1 X) γ 1. Replacing this last expression (46) 1 of ω into the equation (47), we get (52) λ 2 Σ = 1 γ 1 γ 2 p. (53) rf X 1 γ 2 + X 2 γ 1 Recalling that the normal speeds of propagation of hydrodynamical waves, λ k, of each fluid k is given by λ 2 k = γ p k, k = 1, 2, (54) r k f k equation (53) can be rewritten under the form 1 rfλ 2 = X 1 Σ r 1 f 1 λ 2 + X 2 1 r 2 f 2 λ 2. (55) 2 Equation (55) represents the relativistic generalization of the formula due to Wallis [39], allowing to express the speed of acoustic modes for a two-fluid system as combination of the individual speeds of acoustic modes in each species.

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72 70 Sebastiano Giambò, Serena Giambò [23] W. Zimdahl, Reacting fluids in the expanding universe: a new mechanism for entropy production, M on. Not. R. Astron. Soc., 288 (1997), [24] W. Zimdahl, D. Pavon, R. Maartens, Reheating and causal thermodynamics, Phys. Rev., D 55 (1997), [25] J. P. Krisch, L. L. Smalley, Two fluid acoustic modes and inhomogeneous cosmologies, Class. Quantum Grav., 10 (1993), [26] M. Cissoko, Wavefronts in a relativistic cosmic two-component fluid, Gen. Rel. Grav., 30 (1998), [27] M. Cissoko, Wave fronts in a mixture of two relativistic perfect fluids flowing with two distinct four-velocities, Phys. Rev., D 63 (2001), [28] A. Murrone, Modèles bi-fluides à six et sept équations pour les écoulement diphasiques à faible nombre de Mach, PhD Thesis, Université de Provence, Aix-Marseille I, [29] A. Murrone, H. Guillard, A five equation reduced model for compressible two-phase flow computations, J. Comput. Phys., 202 (2005), [30] A. Murrone, H. Guillard, A five equation reduced model for compressible two phase flow problem, INRIA, Rapport de recherche 4778, [31] E. H. van Brummelen, B. Koren, A pressure-invariant conservative Godunov-type method for barotropic two-fluid flows, J. Comput. Phys., 185 (2003), [32] J. Wackers, B. Koren, Five-equation model for compressible two-fluid flow, Report MAS-E0414, (2004). [33] A. H. Taub, Relativistic Rankine-Hugoniot equations, Phys. Rev., 74 (1948), [34] A. Lichnerowicz, Relativistic fluid Dynamics, Cremonese, Roma, [35] G. Boillat, La propagation des ondes, Gauthier-Villas, Paris, [36] G. A. Maugin, Conditions de compatibilité pour une hypersurface singulière en mécanique relativiste des milieux continus, Ann. Inst. Henri Poincar, 24 (1976), [37] A. M. Anile, Relativistic fluids and magneto-fluids, Cambridge University Press, Cambridge, [38] Y. Choquet-Bruhat, Fluides relativistes de conductibilité infinie, Astronautica Acta, VI (1960), [39] G. B. Wallis, One-dimensional Two-Phase Flow, McGraw-Hill, New York, 1969.

73 ROMAI J., 5, 2(2009), A GENERAL MOUNTAIN-PASS THEOREM FOR LOCAL LIPSCHITZ FUNCTIONS Georgiana Goga N. Rotaru College, Constanţa, Romania georgia goga@yahoo.com Abstract A variant for locally Lipschitz functions of a general mountain pass principle, due to Ghoussoub and Preiss, which carries some information on the location of critical points, is proved by using the Borwein-Preiss variational principle as the main variational tool. Keywords: critical points, locally Lipschitz functionals, Borwein-Preiss variational principle, Clarke subdifferential MSC: 54C60, 58E30, 49J35, 49J INTRODUCTION In this paper we present a new proof of a general mountain-pass theorem for locally Lipschitz functions established in [5], giving information about the location of the critical points for these kind of functions. Unlike the method of localization used in [5], which replaces the Ghoussoub-Preiss techniques [6] by Ekeland s variational principle, our proof is based on the Borwein-Preiss variational principle [1] and a lemma of Choulli, Deville and Rhandi [3]. Let (E,. ) be a Banach space, S be a compact metric space, S 0 be a closed subset of S, C (S, E) be the Banach space of all E- valued bounded continuous mappings on S with the norm γ := sup γ (x). Let γ 0 C (S, E) be a fixed element and put x S Γ = {γ C (S, E) γ (s) = γ 0 (s), s S 0 }, c = inf sup γ Γs S f (γ (s)), where f is a real-valued function defined on E. We give now our main result. Theorem Let f : E R be a locally Lipschitz function and F be a closed nonempty subset of E. Assume that a) γ (S) F F c, γ Γ, where F c = {x E f (x) c}, 71

74 72 Georgiana Goga b) dist (γ 0 (S 0 ), F ) > 0, where dist (., F ) is the distance function to F in E. Then for every ε > 0 there exists x ε E such that i) dist (x ε, F ) < 3ε 2, ii) c f (x ε ) < c+ 5ε2 4, iii) dist (0, f (x ε )) 2ε, where f (x)is the Clarke subdifferential of f at x. Before the start of the proof of theorem 1, we recall some known results used in the sequel (the proofs can be found in standard convex analysis literature). 2. PRELIMINARIES Let X be a Banach space and Φ : X R a locally Lipschitz functional. For each x, v X, the generalized directional derivative of Φ at x in the direction v is Φ Φ (y + tv) Φ (y) (x; v) = lim sup. y x, t t 0 It follows, by the definition of locally Lipschitz functionals, that Φ (x; v) is finite and Φ (x; v) k x v, where k x is the Lipschitz constant of Φ on a neighborhood of x. Moreover, v Φ (x; v) is positively homogenous and subadditive and (x; v) Φ (x; v) is upper semi-continuous. The generalized gradient (Clarke subdifferential) of Φ at x is the multifunction Φ : X P (X ) = 2 X defined by Φ (x) = {x X Φ (x; v) x, v, for all v X}, where X is the dual space of X. It enjoys the following properties: 1) For each x X, Φ (x) is a non-empty, convex weak * compact subset of X. 2) For each x, v X, we have Φ (x; v) = max { x, v x Φ (x)}. 3) (Φ + Ψ) (x) Φ + Ψ, where Φ and Ψ are locally Lipschitz at x. Theorem 2.1. (Lebourg mean value theorem). If x and y are two distinct points in X, then there exists z = (1 t) x + ty, 0 < t < 1, such that Φ (y) Φ (x) Φ (z), y x. The notion of critical point of a locally Lipschitz functional is the following.

75 A general mountain-pass theorem for local Lipschitz functions 73 Definition 2.1. Let Φ : X R be locally Lipschitz. A point x X is a critical point of Φ if 0 Φ (x). A real number c is called a critical value of Φ if Φ 1 (c) contains a critical point x. We recall now some results on multivalued mappings. Let M and N be two topological spaces. Definition 2.2. A multivalued mapping T : M 2 N assigns to each point m M a subset T (m) of N. Definition 2.3. Let T : M 2 N be a multivalued mapping. is a map which T is upper semi-continuous (u.s.c.) if T 1 (A) is closed for all closed subsets A of N, where the preimage T 1 (A) is defined by T 1 (A) = {m M T (m) A }. T is lower semi-continuous (l.s.c.) if subsets A of N. T 1 (A) is open for all open T is continuous if is both lower and upper semi-continuous. Lemma 2.1. (Choulli, Deville, Rhandi). Let E be a Banach space, S be a compact metric space F : S 2 E be a weak * - upper semicontinuous multivalued mapping with weak *- compact and convex values and ε > 0. Denote: µ = inf { x x F (s), s S}. Then there exists a continuous function h : S E satisfying 1) h (s) 1, s S, 2) x, h (s) µ ε, s S, x F (s). Now recall the Borwein-Preiss variational principle, which is the variational tool of this article. Theorem 2.2. (Borwein-Preiss variational principle). Let E be a Banach space with the norm,and f : E R {+ } be a l.s.c. function bounded from below, let λ > 0 and let p 1. Assume that ε > 0 and z E satisfy f (z) < inf E f + ε. Then there exist y and a sequence (x i ) in E with x 1 = z and a function ϕ p : E R of the form ϕ p (x) := µ i x x i p, i=1

76 74 Georgiana Goga where µ i > 0 for all i = 1, 2,... and µ i = 1 such that a) x i y λ, i = 1, 2,..., b) f (y) + ( ε λ p ) ϕp (y) f (z), and c) f (x) + ( ε λ p ) ϕp (x) > f (y) + ( ε λ p ) ϕp (y), for all x E \ {y}. i=1 3. PROOF OF THE MAIN RESULT that Proof of Theorem 1.1. Let dist (γ 0 (S 0 ), F ) > ε (b) > 0. Take γ ε Γ such and put supf ( γ ε (s) ) < c + ε 2 (1) s S S 1 = { s S; dist ( γ ε (s), F ) ε }, Γ r = { y C (S, E) ; γ (s) γ ε (s) } r, s S0, c r = inf γ Γ r sup s S f (γ (s)), where S = S \ S 1. By (a), we obtain that there exists x ε γ (S) F such that f (x ε ) c. Since x ε γ (S), there exists s S such that x ε = γ (s), therefore f (x ε ) = f (γ (s)) c. (2) Now we prove the following. Claim. : c + ε 2 > c r c, r (0, ε). Proof. We have so c r supf ( γ ε (s) ) supf ( γ ε (s) ) (1) < c + ε 2, s S s S c r < c + ε 2. We assume now that c r < c. Then, for ε = 1 2 (c c r) > 0, there exists γ 1 Γ r such that supf (γ 1 (s)) < c r + ε = c r + 1 s S 2 (c c r) = 1 2 (c + c r) < c, therefore f (γ 1 (s)) sup s S f (γ 1 (s)) < c,

77 A general mountain-pass theorem for local Lipschitz functions 75 which is in contradiction with (2). Then, we have c r c and the Claim is proved. Let r = ε 8. By the Claim, since S S, we have supf ( γ ε (s) ) supf ( γ ε (s) ) < c + ε 2 c r + ε 2. (3) s S s S By compactness of S, it is easy to prove that the function Φ : Γ r R, Φ (x) = sup s S f (γ (s)) is locally Lipschitz in C (S, E) and, by (3), we have Φ ( γ ε) inf Γ r Φ + ε 2. Now, applying the Borwein-Preiss variational principle for the function Φ on Γ r with p = 2 and λ = ε 2 > 0, we obtain: there exists γ ε and a sequence (γ i ) in Γ r with γ 1 = γ ε, and a function ϕ 2 : Γ r R, defined by where µ i > 0, µ i = 1, such that i=1 ( ε 2 Φ (γ) + λ 2 ϕ 2 (γ) := µ i γ γ i 2, i=1 γ i γ ε λ, i = 1, 2,..., (4) ( ) ε 2 Φ (γ ε ) + ϕ 2 (γ ε ) Φ ( γ ε), (5) λ 2 ) ϕ 2 (γ) > Φ (γ ε ) + ( ε 2 λ 2 ) ϕ 2 (γ ε ), γ Γ r \ {γ ε }. (6) Now, (4) shows that γ i γ ε in Γ r and, by (6), γ ε is a local minimum of the function ( ) ε 2 Φ (γ) := Φ (γ) + ϕ 2 (γ) on C (S, E). By the necessary conditions for a local minimum and the formula for the Clarke directional derivative of the sum of two function, we obtain λ 2 0 Φ (γ ε ; v) Φ (γ ε ; v) + 8 lim γ γ ε i=1 µ i (γ γ i ) v, (7) v C (S, E). We use now Lemma 6, with F ( ) = f (γ ε ( )) and we obtain that there exists a continuous function h : S E such that f (γ ε (s) ; h (s)) d + ε, (8)

78 76 Georgiana Goga h (s) 1, s S, where d = inf s S dist (0, f (γ ε)). Since γ i γ ε, for t i 0 we have δ i := Φ (γ i + t i h) Φ (γ i ) t i Φ (γ ε ; h) (9) and Then, (7) and (10) imply µ i (γ ε γ i ) 0. (10) i=1 0 Φ (γ ε ; h) Φ (γ ε ; h). (11) Let s i be a maximum point of the function f 1 (γ i + t i h, ) over S, i.e. f 1 (γ i + t i h, s i ) = max s S f 1 (γ i + t i h, s). Then by the theorem 2, we have δ i f 1 (γ i + t i h, s i ) f 1 (γ i, s i ) t i = γ i, h max γ i, h = f 1 (γ i, s i ; h), where γ i γf 1 (γ i, s i ), γ i [γ i, γ i + s i h]. By (9) and (11) we obtain 0 Φ (γ ε ; h) Φ (γ ε ; h) f 1 (γ i, s i ; h). It is easy to see that f (γ (t) ; v (t)) = f1 (γ, t; v), γ, v C (S, E), t S. By compactness of S, we assume that s i s ε S and since γ i γ ε, by upper semicontinuity of f (, ), we obtain By Lemma 6, we have which means that Then, (12) and (13) give us Φ (γ ε ; h) f (γ ε (s ε ) ; h (s ε )) (8) d + ε. (12) γ i (s ε ), h (s ε ) µ ε, f (γ ε (s ε ) ; h (s ε )) µ ε. (13) µ ε d + ε µ < 2ε, and we have dist (0, f (γ ε )) < 2ε, which is (iii).

79 A general mountain-pass theorem for local Lipschitz functions 77 Since s ε S = S \ S 1, we have s ε / ints 1, therefore dist (x ε, F ) dist ( γ ε (s ε ), F ) + γ ε (s ε ) γ ε (s ε ) ε + λ = 3ε 2, which is (i). Now by (5), we have f ( γ ε) sup f (γ ε (s ε )) + 4 µ i γ ε (s ε ) γ i (s ε ) 2 s ε S i=1 sup f (γ ε (s ε )) + 4 ( µ γε i (s ε ) γ ε (s ε ) + γ ε (s ε ) γ i (s ε ) ) 2 s ε S i=1 sup s ε S f (γ ε (s ε )) + 16r 2 (1) c + ε ε2 64 = c + 5ε2 4, and (ii) is proved. References [1] J. Borwein, Q. Zhu, Techniques of Variational Analysis, Springer, New York 2005 [2] H. Brezis, N. Nirenberg, Remarks of finding critical points, Comm. Pure Appl. Math., Vol. XLIV (1991), [3] M. Choulli, R. Deville, A. Rhandi, A general mountain pass principle for nondifferentiable functions and applications, Rev. Math. Apl., 13 (1992), [4] F. H. Clarke, Non-smooth analysis and optimization, Wiley - Interscience [5] P. Georgiev, A short proof of a general mountain-pass theorem for locally Lipschity functions, Preprint, IC/95/369, International Centre for Theoretical Physics, Miramare- Trieste, 5 pp. (1995) [6] N. Ghousoub, D. Preiss, A general mountain pass principle for locating and classifying critical points, Ann. Inst. H. Poincare, 6 (1989), [7] N. Ghousoub, Multiplicity and Morse Indices for min-max critical points, J. Reine Angew. Math., 417 (1991), [8] N.K. Ribarska, T.Y. Tsachev, M.I. Krastanov, On the general mountain pass principle of Ghoussoub-Preiss, Mathematica Balkanica, 5 (1991).

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81 ROMAI J., 5, 2(2009), SOME RESULTS ON SIMULTANEOUS ALGEBRAIC TECHNIQUES IN IMAGE RECONSTRUCTION FROM PROJECTIONS Lăcrămioara Grecu, Aurelian Nicola Faculty of Mathematics and Computer Science, Ovidius University of Constanţa, Romania Abstract In this paper we make a comparative analysis of two projection based iterative algorithms for systems of linear equations arising from image reconstruction in computerized tomography: Kaczmarz s successive projection iteration (1937) and Simultaneous Algebraic Reconstruction Technique (SART; 1984). We start a theoretical analysis of the SART algorithm, which gives us the possibility to consider its extended and constrained versions. Systematic numerical experiments and comparisons are made on two phantoms widely used in image reconstruction literature, with the classical, extended and constrained versions of both Kaczmarz and SART methods. Keywords: image reconstruction MSC: 94A08, 68U INTRODUCTION Computerized tomography (CT) is a medical method that uses digital geometry processing to generate a three-dimensional image of the internals of an object from a large series of two-dimensional X-ray images taken around a single rotation axis. For the reconstruction of a two-dimensional image, the section coresponding to this image is scaned with X-rays sent forth from sources that move on an arc of a circle around that section (Fig. 1). The region ABCD coresponding to the two-dimensional image is divided in image elements (pixels). Each X-ray corresponds to a line in the scanning matrix, it s elements representing the length of the segments determined by the X-ray on each pixel (see Figure 2). 79

82 80 Lăcrămioara Grecu, Aurelian Nicola Figure 1. Fan beam scanning Figure 2. Construction of A and b So, if we use a number of M X-rays and a discretization of the image with N pixels, the scanning matrix A will be M N. The matrix A is sparse, rank-deficient, with a nonzero null space and is ill-conditioned. Measuring the intensities of the source emissions (S i ) and reception (in the detector D ji ) of the i-th X-ray, we obtain the component b i of the right hand side b of the future discreet model for the reconstruction problem. This technique that reduces the reconstruction to solving a least squares problem Ax b = min! (1) is called Algebraic Reconstruction. For solving the discret model (1) there have been developed a class of iterative methods based on projections, called Algebraic Reconstruction Techniques (ART). In this paper we shall make a comparative study between two methods: the Kaczmarz method (which uses orthogonal succesive projections on the hiperplanes of the equations of problem (1)) and the Simultaneous Algebraic Reconstruction Technique - SART (which uses simultaneous non-orthogonal projections on the hiperplanes mentioned before). This paper is organized as follows: in Section 2 we describe the Kaczmarz and SART algorithm, and the convergence results. In Section 3 we present some special theoretical results for the SART algorithm. A development of these results allows us to approach in Section 4 (at least from a formal point of view) the possibility of extending the SART method to an inconsistent problem of type (1) and/or combining it with constraining techniques. In Section 5 we present some reconstruction experiments of Kaczmarz algorithm and SART algorithm with and without constraints, with two phantoms frequently used in other articles. We shall use the following notations: LSS(A; b) = { x R N, Ax b = min! }, S (A; b) = { x R N, Ax = b }, x LS the minimum norm solution of (1), A i row i from matrix A, A j column j from matrix A, N(A) = { x R N, Ax = 0 }, R(A) = { Ax, x R N}, A T the transpose of A. P S will denote the orthogonal (Euclidean) projection onto the non-empty closed convex subset S R N.

83 Some results on simultaneous algebraic techniques in image reconstruction CONVERGENCE RESULTS FOR KACZMARZ AND SART ALGORITHM 2.1. KACZMARZ ALGORITHM The Kaczmarz method, based on orthogonal succesive projections on hiperplanes of problem (1) was proposed by it s author in [5]. Algorithm Kaczmarz (K). Initialization: x 0 R N Iterative step: x k+1 = (f 1 f M )(b; x k ), (2) where f i (b; x) = x x, A i b i A i 2 A i, i = 1,..., M. (3) It s convergence was analyzed in [5] and [11]. Theorem 2.1. For any x 0 R N, the sequence (x k ) k 0 generated by the algorithm (K) converges to x(x 0 ) R N. In the consistent case for (1), x(x 0 ) S(A; b) (with x(0) = x LS ) SART ALGORITHM SART was introduced, by Andersen and Kak in 1984 (see [1]), to reduce the salt and pepper noise commonly associated with ART-type reconstructions. This method consists of simultaneous application of the error correction terms as computed by ART for all rays in a given projection. In [1] the authors argue that a simultaneous application of the correction term for the rays in a particular view is preferable to the usual sequential one. Componentwise the SART algorithm can be written as follows x k+1 j = x k j + λ k M A ij i=1 M i=1 A ij N A ij j=1 ( b i A i x k), j = 1,..., N. (4) If we define the matrices V = diag (V 11,..., V NN ) and W = diag (W 11,..., W MM ) (5) with V jj = 1 W ii = M A ij, j = 1,... N, (6) i=1 N A ij, i = 1,..., M, (7) j=1

84 82 Lăcrămioara Grecu, Aurelian Nicola the algorithm (4) can be written in the following matricial form Algorithm SART. Initialization: x 0 R N Iterative step: ( x k+1 = x k + λ k V 1 A T W b Ax k). (8) In [4] Jiang and Wang proved the following convergence result. Theorem 2.2. For 0 λ k 2 and min (λ k, 2 λ k ) = +, the sequence n=0 (x k ) k 0 generated by the algorithm (SART) converges to x V,W (x 0 ) = x V,W LS + P V N(A) ( x 0 ), where PN(A) V is the projection onto N(A) w.r.t. the energy scalar product, V and x V,W LS is the solution with minimal V -norm of the problem Ax b W = min! (9) In the consistent case for (1), x V,W (x 0 ) S(A; b) (but, in general x V,W (0) x LS ). Remark 1. For the Kaczmarz algorithm we have lim k xk = x LS for x 0 = 0 in the consistent case. For SART algorithm we have lim k xk = x V,W LS (0) = xv LS in the consistent case and x 0 = 0, where x V LS is the minimal V -norm solution of (1). Example 1. For A = and b = 0 1 we obtain x LS = ( 1, 5 3, 1 3, 4 3), and x V LS = ( 1, 17 8, 1 8, 7 8) (for our problem we have V = diag(2, 1, 6, 3)). 3. SOME TEORETICAL RESULTS FOR SART In [8] the authors consider a general iterative method of the form x k+1 = T x k + Rb, x 0 R N, (10) where T : N N and R : N M. The following basic asumptions are introduced I T = RA, (11) if x N (A) then T x = x N (A), (12) if x R ( A T ) then T x R ( A T ), (13)

85 Some results on simultaneous algebraic techniques in image reconstruction y R m, Ry R ( A T ), (14) if T = T P R(A T ) then T < 1. (15) In this paper it is shown that if the method (10) satisfies these properties it is possible to extended it to an inconsistent problem of type (1) as well as combining it with special constraining techniques (see [6]). Now we shall prove some of these properties for the SART algorithm described in Section 2. But, because in the SART algorithm (8), appear the pozitive definite matrices V and W, from (6) and (7), the matrix A will be considered (as a linear operator) as follows A : ( R M,, V ) ( R M,, W ). (16) We will denote with A τ : ( R M,, W ) ( R N,, V ) the adjoint of A. Lemma 1. For A from (16) we have A τ = V 1 A T W, (17) R N = N(A) V R(A τ ), (18) where V represents the orthogonal direct sum with respect to, V. Proof. According to (16), A τ is characterized by From (19) we get and Ax, y W = x, A τ y V, x R N, y R M. (19) Ax, y W = W Ax, y = x, A T W y x, A τ y V = x, V A τ y, (20) x R n, y R M, A τ = V 1 A T W. (21) Because N(A) R N, R(A τ ) R N, for (18) we will prove only. Let x R N and x = PR V (Aτ ) (x). We define x = x x. We want to prove that x N(A) i.e. Ax = 0. If x = PR V (Aτ ) (x) we know that x x V R(A τ ), i.e. x x, y V = 0, y R(Aτ ) x x, V y = 0, y R(A τ ) x x, V V 1 A T W z = 0, z R M A ( x x ), W z = 0, z R M Ax, W z = 0, z R M Ax, t t R M Ax R M Ax = 0 Hence R N = N(A) + R(A τ ). (22)

86 84 Lăcrămioara Grecu, Aurelian Nicola Now we prove that N(A) V R(A τ ). Consider z R(A τ ) and y N(A), i.e. z = A τ x and Ay = 0. We obtain Hence z, y V = A τ x, y V = V A τ x, y = V V 1 A T W x, y = A T W x, y = W x, Ay = W x, 0 = 0. N(A) V R(A τ ). (23) From (22) and (23) we obtain (18). For λ k = λ, k 0 (24) in (8) we define the matrices T : N N and R : N M by T = I λv 1 A T W A, (25) and R = λv 1 A T W. (26) Lemma 2. The matrices T and R from (25) and (26) satisfy (i) I T = RA, (ii) if x N (A) then T x = x N (A), (iii) if x R (A τ ) then T x R (A τ ), (iv) y R M, Ry R (A τ ). Proof. (i) From (8) we have ( x k+1 = x k + λv 1 A T W b Ax k) = x k ( I λv 1 A T W A ) + λv 1 A T W b, thus according to (10) i.e. the equality (i). (ii) For x N(A), we obtain (iii) For x R (A τ ) we get from (25) (27) R = λv 1 A T W, (28) T = I λv 1 A T W A = I RA, (29) T x = (I RA) x = x RAx = x. (30) T x = ( I λv 1 A T W A ) x = (I λa τ A) x = x λa τ Ax R (A τ ). (31)

87 Some results on simultaneous algebraic techniques in image reconstruction (iv) From (26), we get for any y R M and the proof is complete. Ry = λv 1 A T W Ay = λa τ y R (A τ ), (32) Remark 2. From Lemma 2 we obtain that the matrices T and R from (25) and (26) satisfy the properties (11)-(14). Unfortunately we don t have yet a proof for property (15). 4. EXTENDED AND CONSTRAINED SART-LIKE ALGORITHMS Although (see Remark 2) we don t have yet a proof for a property analogous with (15) related to matrices T and R from (25) and (26), the fact that the properties (11)-(14) have been proved (see Lemma 2), as well as the experiments that we performed (see also Section 5) allowed us to consider the following extended and constrained versions of SART algorithm CONSTRAINED SART A box-constraint is a function C : R N R N, which is a metric projection operator onto the box [a, b] = [a 1, b 1 ]... [a N, b N ] R N, defined by x i, if x i [a i, b i ] (Cx) i = a i, if x i < a i. (33) b i, if x i > b i It satisfies Cx Cy x y, (34) if Cx Cy = x y then Cx Cy = x y, (35) if y Im (C) then Cy = y. (36) Algorithm Constrained SART (CSART). Initialization: x 0 Im(C) Iterative step: [ ( x k+1 = C x k + λ k V 1 A T W b Ax k)]. (37) 4.2. EXTENDED SART For Extended SART we shall use the model of Extended Kaczmarz algorithm from [9].

88 86 Lăcrămioara Grecu, Aurelian Nicola Algorithm Extended SART. Initialization: x 0 R N, y 0 = b Iterative step: y k+1 = y k µ k Ṽ 1 A W A T y k, (38) where Ṽ ii = j=1 b k+1 = b y k+1, (39) ( x k+1 = x k + λ k V 1 A T W b k+1 Ax k) (40) N 1 A ij, i = 1,... M, = W jj M A ij, j = 1,..., N. (41) 4.3. EXTENDED SART WITH CONSTRAINTS For the Extended SART with constraints we follow the model from the Constrained Kaczmarz Extended algorithm in [10]. Algorithm Constrained Extended SART. Initialization: x 0 Im(C), y 0 = b Iterative step: y k+1 = y k µ k Ṽ 1 A W A T y k, (42) x k+1 = C i=1 b k+1 = b y k+1, (43) [ ( x k + λ k V 1 A T W b k+1 Ax k)]. (44) 5. NUMERICAL EXPERIMENTS We have used in our experiments the Head and Mitochondrian phantoms described in the paper [3]. For each phantom we have the exact picture (N = 3969) x ex R 3969, i.e. with a pixel resolution, and a scanning matrix A : for Head phantom, respectively, A : for Mitochondrian phantom and a right hand side b R 1376, respectively, b R We also used the following wellknown error measures used in image reconstruction (see e.g. [3]): standard deviation, distance, relative error and normal equation residual, defined below. x ex = (x ex 1,..., xex N )T - the mitochondrian phantom x k = (x k 1,..., xk N )T - the current approximation x ex = 1 N N i=1 x ex i ; x k = 1 N N x k i i=1 current approximation, respectively - mean values of the exact image and

89 Some results on simultaneous algebraic techniques in image reconstruction Standard deviation = 1 N Distance = Relative error = N (x ex i x k i )2 i=1 N (x ex i x ex ) 2 i=1 N i=1 N x ex i x k i i=1 x ex i N (x k i xk ) 2 i=1 Residual = 1 N Ax k b (consistent) and 1 N A T (Ax k b) (inconsistent). In our experiments, beside SART, Constrained SART, Extended SART, and Constrained Extended SART algorithms we used the Kaczmarz original method (see [5], [11] with), Constrained Kaczmarz (CK, from [6]), Extended Kaczmarz (from [9]) and Contrained Extended Kaczmarz (from [10]). In Fig. 1 we present reconstructions for consistent case using Kaczmarz and SART for Head phantom. Similarly in Fig. 3 we show reconstructions for Mitochondrian phantom. Errors described above for these cases are shown in Figures 2 and 4. In Fig. 5 we present reconstructions for consistent case using Constrained Kaczmarz and Constrained SART for Head phantom. Similarly in Fig. 7 we show reconstructions for Mitochondrian phantom using constrains. Errors described above for these cases are shown in Figures 6 and 8. In Fig. 9 we present reconstructions for inconsistent case using Extended Kaczmarz and Extended SART for Head phantom. Similarly in Fig. 11 we show inconsistent reconstructions for Mitochondrian phantom. Errors described above for these cases are shown in Figures 10 and 12. In Fig. 13 we present reconstructions for inconsistent case using Constrained Extended Kaczmarz and Constrained Extended SART for Head phantom. Similarly in Fig. 15 we show inconsistent reconstructions for Mitochondrian phantom using constrains. Errors described above for these cases are shown in Figures 14 and 16.

90 88 Lăcrămioara Grecu, Aurelian Nicola Fig. 1. Consistent case reconstructions: exact solution, Kaczmarz, SART; no constraints Fig. 2. Errors for Head phantom, consistent case, no constraints Fig. 3. Consistent case reconstructions: exact solution, Kaczmarz, SART; no constraints

91 Some results on simultaneous algebraic techniques in image reconstruction Fig. 4. Errors for Mit phantom, consistent case, no constraints Fig. 5. Consistent case reconstructions: exact solution, Kaczmarz, SART; with constraints

92 90 Lăcrămioara Grecu, Aurelian Nicola Fig. 6. Errors for Head phantom, consistent case, with constraints Fig. 7. Consistent case reconstructions: exact solution, Kaczmarz, SART; with constraints

93 Some results on simultaneous algebraic techniques in image reconstruction Fig. 8. Errors for Mit phantom, consistent case, with constraints Fig. 9. Inconsistent case reconstructions: exact solution, Extended Kaczmarz, Extended SART; no constraints

94 92 Lăcrămioara Grecu, Aurelian Nicola Fig. 10. Errors for Head phantom, inconsistent case, no constraints Fig. 11. Inconsistent case reconstructions: exact solution, Extended Kaczmarz, Extended SART; no constraints

95 Some results on simultaneous algebraic techniques in image reconstruction Fig. 12. Errors for Mit phantom, inconsistent case, no constraints Fig. 13. Inconsistent case reconstructions: exact solution, Extended Kaczmarz, Extended SART; with constraints

96 94 Lăcrămioara Grecu, Aurelian Nicola Fig. 14. Errors for Head phantom, inconsistent case, with constraints Fig. 15. Inconsistent case reconstructions: exact solution, Extended Kaczmarz, Extended SART; with constraints

97 Some results on simultaneous algebraic techniques in image reconstruction Fig. 16. Errors for Mit phantom, inconsistent case, with constraints

98 96 Lăcrămioara Grecu, Aurelian Nicola References [1] Andersen A.H., Kak A.C., Simultaneous Algebraic Reconstruction Techniques (SART): a superior implementation of the ART algorithm, Ultrasonic imaging, 6(1984), [2] Censor Y., Stavros A. Z., Parallel optimization: theory, algorithms and applications, Numer. Math. and Sci. Comp. Series, Oxford Univ. Press, New York, [3] Herman, G. T., Image reconstruction from projections. The fundamentals of computerized tomography, Academic Press, New York, [4] Jiang M., Wang G., Convergence studies on iterative algorithms for image reconstruction, IEEE Trans. Medical Imaging, 22(2003), [5] Kaczmarz S., Angenaherte Auflosung von Systemen linearer Gleichungen, Bull. Acad. Polonaise Sci. et Lettres A (1937), [6] Koltracht I. and Lancaster P., Constraining strategies for linear iterative processes, IMA Journal of Numerical Analysis, 10(1990), [7] Natterer F., The Mathematics of Computerized Tomography, John Wiley and Sons, New York, [8] Nicola A., Petra S., Popa C., Schnörr C., On a general extending and constraining procedure for linear iterative methods, Preprint 9761(2009), IWR Heidelberg, Germany ( [9] Popa C., Extensions of block-projections methods with relaxation parameters to inconsistent and rank-defficient least-squares problems; B I T, 38(1)(1998), [10] Popa C., Constrained Kaczmarz extended algorithm for image reconstruction, Linear Algebra and its Applications, 429(2008), [11] Tanabe K., Projection Method for Solving a Singular System of Linear Equations and its Applications, Numer. Math., 17(1971),

99 ROMAI J., 5, 2(2009), BRANCHING EQUATION IN THE ROOT SUBSPACE FOR EQUATIONS NONRESOLVED WITH RESPECT TO DERIVATIVE AND STABILITY OF BIFURCATING SOLUTIONS Irina V. Konopleva, Boris V. Loginov, Yuri B. Rousak Ulyanovsk State Technical University, Russia; Ulyanovsk State Technical University, Russia; Canberra University, Australia Abstract For stationary and dynamic bifurcation problems some theorems on the inheritance of group symmetry of nonlinear equations are proved by A. M. Lyapunov and E. Schmidt branching equations in the root-subspaces (BEqRs), moving along the orbit of bifurcation point x 0. Also theorems about BEqRs of potential type reduction at the action of continuous group symmetry are proved. The case of isolated branching point is considered separately. Acknowledgement The obtained results are supported by grant RFBR Acad. Sci. of Romania No a and by the program Development of the Scientific Potential of Higher School of the Ministry of Education of Russian Federation (project No /6194). Keywords: Symmetry, potentiality, branching equations in the root subspace, stationary and dynamic bifurcation problems, stability MSC: 35B32, 35B INTRODUCTION In real Banach spaces E 1 and E 2 the nonlinear equation, i.e. bifurcation problem with small (numerical or functional) parameter ε Λ F (x, ε) = 0, F (x 0, ε) = 0, F x(x 0, ε) = B x0 + B x0 (ε), B 0 = B x0 = F x(x 0, 0) (1) is considered, where B x0 is a densely defined in E 1 Fredholm operator, D B D B(ε), B(ε) = F x(x 0, ε) + B x0, N(B x0 ) = span{ϕ i } n 1, ϕ i = ϕ i (x 0 ) is the zerosubspace (kernel) of the operator B 0, N (B x0 ) = span{ψ i } n 1, ψ i = ψ i (x 0 ) is the subspace of defect functionals, and {γ i } n 1, γ i = γ i (x 0 ) E1, {z i} n 1, z i = z i (x 0 ) the corresponding biorthogonal systems ϕ i, γ j = δ ij, ψ i, z j = δ ij. 97

100 98 Irina V. Konopleva, Boris V. Loginov, Yuri B. Rousak Bifurcating solutions of the equation (1) are considered as stationary solutions of the autonomous differential equation in Banach spaces E 1 and E 2 A dy dt = B x 0 y B x0 (ε)y R(x 0, y, ε), y = x x 0 R(x 0, 0, ε) 0, R(x 0, x x 0, ε) = 0( x x 0 ) (2) with closed densely defined Fredholm operators A and B = B x0. Definition 1.1. The solution y 0 (t) = x(t) x 0, t 0 of the equation (1) is Lyapunov stable if for any ε > 0 there exists δ > 0, such that for any solution y(t), y(0) y 0 (0) < δ for t > 0 the inequality y(t) y 0 (t) < ε is fulfilled and asymptotically stable if y(t) y 0 (t) 0 at t. The technique of branching equations in the root-subspaces (BEqR) was introduced in our articles [1] [5] with the aim of stability investigation of bifurcating solutions (see also the review articles [6, 7]). Everywhere below the terminology and notations of [6]-[8] are used. We will not stipulate in every case used here subordination conditions of densely defined Fredholm operators [6, 7], allowing to reduce the discussion to bounded operators: for a pair of densely defined Fredholm operators A : D A E 2, B : D B E 2, if D B D A, then A is subordinated to B, i.e. Ax Bx + x on D B ; or if D A D B then B is subordinated to A, i.e. Bx Ax + x on D A. It is supposed also that A and B haven t common zeros with the aim to avoid the complicated technique of nonfinished generalized Jordan chains (GJCh). Definition 1.2. ([8] [10]) Elements ϕ (s) k, s = 1, p k, k = 1, n form complete canonical GJSet (B(ε) GS), if Bϕ (s) k (x 0) = s 1 D p = det j=1 ϕ (s) B j ϕ (s j) k (x 0 ), B(ε) = B 1 ε + B 2 ε , [ k (x 0), γ l (x 0 ) = 0, s = 2, p k, pk B j ϕ (p k+1 j) k (x 0 ), ψ (1) l (x 0 ) j=1 ] 0, k, l = 1, n, ϕ k (x 0 ) = ϕ (1) k (x 0), ψ l (x 0 ) = ψ (1) l (x 0 ). This GJS is bicanonical, if the GJS of conjugate operator-function B B (ε) corresponding to elements {ψ l } n 1 (x 0) is also canonical and three-canonical, if in addition ϕ (j) i (x 0 ), γ (l) k (x 0) = δ ik δ jl, γ (l) p k (x k +1 l 0) = Bs ψ (p k+2 l s) k (x 0 ), z (j) i (x 0 ), ψ (l) k (x 0) = δ ik δ jl, z (j) i (x 0 ) = Φ = Φ(x 0 ) = (ϕ (1) 1 (x 0),..., ϕ (p 1) s=1 p i +1 j s=1 1 (x 0 ),..., ϕ (1) n B s ϕ (p i+2 j s) i (x 0 ), (x 0 ),..., ϕ (pn) n (x 0 )), (3) (4)

101 Branching equation in the root-subspace vectors γ = γ(x 0 ), Ψ = Ψ(x 0 ) and Z = Z(x 0 ) are defined analogously. Lemma 1.1. ([1, 2], [8] [10]) The elements of the basis of the kernel N(A) = span{φ i } m 1 (N(B) = span{ϕ i } n 1 ) can be chosen so that for the N(A ) = span{ ψ} m 1 (N(B ) = span{ψ} n 1 ) corresponding elements of B- and B -Jordan sets (A- and A -JS) of the operator-functions A εb and A εb (B µa and B µa ) would be three-canonical, i.e. φ (j) i, ϑ (l) k = δ ik δ jl, ζ (j) (l) i, ψ k = δ ik δ jl, j(l) = 1,..., q i (q k ), (5) ϑ (l) k = k B ψ(q +1 l) k, ζ (j) i = Bφ (q i+1 j) i, i, k = 1,..., m; ϕ (j) i, γ (l) k = δ ik δ jl, z (j) i, ψ (l) k = δ ik δ jl, j(l) = 1,..., p i (p k ), (6) γ (l) k = A ψ (p k+1 l) k, z (j) i = Aϕ (p i+1 j) i, i, k = 1,..., n. The conditions Aφ (s) i = Bφ (s 1) i, φ (s) i, ϑ (1) j = 0, s = 2,..., q i, i, j = 1,..., m; (Bϕ (s) i = Aϕ (s 1) i, ϕ (s) i, γ (1) j = 0, s = 2,..., p i, i, j = 1,..., n) determine B- JS (A-JS) uniquely. Its elements are linearly independent and form the basis of the root-subspaces K(A; B) (K(B; A)); k A = dim K(A; B) = m q i (k B = dim K(B; A) = n i=1 p i ) is called the root-number of the Fredholm point ε = 0 (µ = 0) of σ B (A) (σ A (B)). The relations (5),(6) allow to introduce the projectors [1, 2] p = m q i, ϑ (j) i φ (j) i =, ϑ φ : E 1 E k A 1 = K(A, B), i=1 j=1 q = m q i (j), ψ i ζ (j) i =, ψ (7) ζ : E 2 E 2,kA = span{ζ (j) i }, i=1 j=1 P = n Q = n p i i=1 j=1 p i i=1 j=1, γ (j) i, ψ (j) i ϕ (j) i z (j) i =, γ ϕ : E 1 E k B 1 = K(B; A), i=1 =, ψ z : E 2 E 2,kB = span{z (j) i } (where φ = (φ (1) 1,, φ(q 1) 1,, φ (1) m,, φ (q m) m ), and the vectors ϑ, ψ, ζ, ϕ, γ, ψ, z are defined analogously) generating the following direct sums expansions E 1 = E k A 1 +E k A 1, E 2 = E 2,kA +E ka, E 1 = E k B 1 +E k B 1, E 2 = E 2,kB +E 2, kb. The intertwining relations are realized Ap = qa on D A, Bp = qb on D B, BP = QB on D B, AP = QA on D A, (8) (9) (10)

102 100 Irina V. Konopleva, Boris V. Loginov, Yuri B. Rousak Aφ = A A ζ, Bφ = A B ζ, B ψ = AB ϑ, Bϕ = A B z, Aϕ = A A z, A ψ = A A γ) (11) with cell-diagonal matrices A A = (A 1,..., A m ), A B = (B 1,..., B m ) (A B = (B 1,..., B n ), A A = (A 1,..., A n )) where the q i q i -cells (p i p i - cells) have the forms A i = , B i = ( A i (B i ) have the same form as A i (B i )). The following relations for the operators A and B are valid N(A) E k A 1, AE k A 1 E 2,kA, A(E k A 1 D A ) E 2, ka, N(B) E k A 1, BE k A 1 E 2,kA, B(E k A 1 D B ) E 2, ka. (12) where the mappings B : E k A 1 E 2,kA, A= A : E k A 1 D A E 2, ka are one-to-one. Analogously, N(B) E k B 1, BE k B 1 E 2,kB, B(E k B 1 D B ) E 2, kb, N(A) E k B 1, AE k B 1 E 2,kB, A(E k B 1 D A ) E 2, kb. (13) and the mappings A : E k B 1 E 2,kB, B= B : E k B 1 D B E 2, kb are oneto-one. Thus the operators A and B (B and A) act as invariant pairs of subspaces E k A 1, E 2,kA and E k A 1, E 2, ka (E k B 1, E 2,kB and E k B 1, E 2, kb ). The first object of this article is to investigate the stability of stationary bifurcating solutions of (2) under intertwining condition of the nonlinear operator F in the equation by representations of some group G or individual operators. On this base the stability investigation of periodical bifurcating solutions at Poincaré-Andronov-Hopf bifurcation for differential equation non-resolved with respect to derivative can be realized F (p, x, ε) = 0, p = dx dt, F (0, x 0, ε) 0, F p(0, x 0, 0) = A x0 = A 0, F x(0, x 0, 0) = B x0 = B 0, F p(0, x 0, ε) = A 0 + A x0 (ε) = A(ε), F x(0, x 0, ε) = B 0 + B x0 (ε), D B0 = E 1, D B0 D(A(ε)) (14) under the same intertwining conditions. Stationary solutions to (2) are also stationary solutions to (14). The indicated problems similarly to [1]-[3] will be solved at the usage of BEqR techniques [1]-[5] presented in detail in [11].

103 Branching equation in the root-subspace STABILITY STUDY OF BIFURCATING SOLUTIONS UNDER GROUP SYMMETRY CONDITIONS OF NONLINEAR OPERATORS It is assumed that for continuously differentiable x and sufficiently smooth b (project no a), (y; ε) in some neighborhood of bifurcation point (x 0 ; 0); operator F allows the group G, i.e. there exist its representations L g in E 1 and K g in E 2 intertwining the operator F K g F (x, ε) = F (L g x, ε) (K g F (p, x, ε) = F (L g p, L g x, ε)). (15) Differentiation of this equalities by x and p at the (x 0 ; 0) gives the relations K g F x(x 0, ε) = F x(l g x 0, ε)l g, (K g A x0 = A Lgx 0 L g, K g B x0 = B Lgx 0 L g ) showing that operators B x0 and B x0 (ε) (A x0 and A x0 (ε)) possess the symmetry with respect only stationary subgroup of the point x 0. When x 0 is the stationary point of representation L g, the operators A 0, B x0, B x0 (ε) and R(x 0, y, ε) are intertwined by the pair L g, K g : K g A 0 = A 0 L g, K g B x0 = B x0 L g, K g B x0 (ε) = B x0 (ε)l g and K g R(x 0, y, ε) = R(x 0, K g y, ε) and the following partially generalized results of the article [2] are valid. Theorem 2.1. Assume that, in the general case of symmetry absence, the operator F is continuously differentiable with respect to x up to the order q, where q is the maximal length of B 0 -Jordan chains of the basic elements {φ i } m 1 N(A 0). If the Fredholm operator A 0 has a complete B 0 -Jordan set and the A 0 -spectrum σ A0 (B 0 ) lies in the left halfplane (even one point of σ A0 (B 0 ) gets into the right halfplane), then the trivial solution of the equation (1) is asymptotically stable (unstable). In these conditions the principle of linearized stability is fulfilled: the bifurcating stationary solution y(ε) from (x 0 ; 0) of the equation (1) is asymptotically stable if the A 0 -spectrum σ A0 (B x0 B x0 (ε) R y (x 0, y(ε), ε)) of the Frechét derivative B x0 B x0 (ε) R y (x 0, y(ε), ε) on the solution y(ε) lies in the left halfplane, and it is unstable if there exists even one point µ(ε) σ A0 (B x0 B x0 (ε) R y (x 0, y(ε), ε)) in the right halfplane. The last question is solved by the Newton diagram method applied to the BEqR of the eigenvalue problem (B x0 B x0 (ε) R y (x 0, y(ε), ε) µa 0 )ϕ = 0. (16) Here the relevant BEqR linear resolving system (LRS) [6] is constructed, the determinant of which determines the branching equation for eigenvalue bifurcation of the problem (16). If the bifurcating from x 0 solution y(ε) is represented by the series on a fractional degree ε 1/k this degree is denoted by ɛ, i.e. the problem about eigenvalue branching of Fredholm operators [8] is considered

104 102 Irina V. Konopleva, Boris V. Loginov, Yuri B. Rousak (B 0 B x0 (ɛ k ) R y (x 0, y(ɛ), ɛ k ) µa 0 )Φ = (B 0 R(ɛ) µa 0 )Φ = (B 0 ɛ k R k µa 0 )Φ = 0. k=1 (17) According to Lemma 1.1, we rewrite the equation (17) in the form of a system by using the Schmidt regularizator B0= B 0 + n, γ (1) j (x 0 ) z (1) j (x 0 ), Γ x0 = Γ 0 = B 1 0 B0 Φ ɛ k R k Φ = µa 0 Φ + k=1 s=1 the solution of which is sought in the form Φ = w+ n = w + v(x 0, ξ). Then B0 w + n p r p r r=1 ρ=2 ξ rρ B 0 ϕ (ρ) r = µa 0 w + µ n i=1 ξ s1 z s (1), ξ s1 = Φ, γ (σ) s (18) p r r=1 ρ=1 p r r=1 ρ=1 ξ rρ ϕ (ρ) r ) and using the relations B 0 ϕ (ρ) r = z (ρ r r +2 ρ), Γ 0 z (p r) r=1 ρ=1 ϕ (3) r,..., Γ 0z r (2) = ϕ (p r) r, Γ 0 z (1) r = ϕ (1) r, ϕ(p r+1) r = ϕ (1) r, z(pr+1) ξ rρ ϕ (ρ) r = w+ξ ϕ = ξ rρ A 0 ϕ (ρ) r + R(ɛ)(w + r = z r (1) one has r = ϕ (2), Γ 0z (p r 1) r r = w = (I Γ 0(µA 0 + R(ɛ))) 1 { p r r=1 ρ=2 ξ rρ ϕ (ρ) r + +µ n (ξ r1 ϕ (2) r + ξ r2 ϕ (3) r ξ rpr 1 ϕ(p r) r + ξ rpr ϕ (1) r ) + r=1 p r r=1 ρ=1 ξ rρ Γ 0R(ɛ)ϕ (ρ) r } = = p r r=1 ρ=2 ξ rρ ϕ (ρ) r + µ (ξ r1 ϕ (2) r + ξ r2 ϕ (3) r r= ξ rpr 1 ϕ(pr) r + ξ rpr ϕ (1) r )+ + p r r=1 ρ=1 ξ rρ Γ 0 R(ɛ)ϕ (ρ) r + Γ 0 (µa 0 + R(ɛ)))(I Γ 0 (µa 0 + R(ɛ))) 1 {...}. Substitution of the last expression in the second set of equalities (18) allows the system to determine ξ rρ, r = 1,..., n, ρ = 1,..., p n, w, γ (σ) s = 0, s = 1,..., n, σ = 1,..., p s (19)

105 or, with Γ 0 γ(σ) s form µξ sps + n p r r=1 ρ=1 ξ s2 + µξ s1 + n = Γ 0 A 0 ψ(p s+1 σ) s Branching equation in the root-subspace = ψ (p s+2 σ) s, ψ (p s+1) s = ψ (1) s, in coordinate ξ rρ R(ɛ)ϕ (ρ) r + (µa 0 + R(ɛ))(I Γ 0(µA 0 + R(ɛ))) 1 {...}, ψ (1) s = 0, p r r=1 ρ=1 ξ rρ R(ɛ)ϕ (ρ) r + (µa 0 + R(ɛ))(I Γ 0 (µa 0 + R(ɛ))) 1 {...}, ψ (p s) s = 0, ξ sps + µξ sps 1 + n p r ξ rρ R(ɛ)ϕ (ρ) r + (µa 0 + R(ɛ))(I Γ 0(µA 0 + R(ɛ))) 1 {...}, ψ (2) s = 0. r=1 ρ=1 Technical computations by using the expressions for (A 0 Γ 0 )κ γ (σ) s (20) through γ (τ) s allow us to calculate the determinant of the linear resolving system (19) ((20)) for ε = 0 : = ( µ) p p n ( 1) n. Hence by the method of Newton s diagram the µ-beqr has k(b, A) = (µ, ε) det n s=1 [ ] w, γ (σ) s = 0, s = 1,..., n; σ = 1,..., p s (21) p s roots µ j = µ j (ε), the signs of real parts of which determine the asymptotic stability of the solution x 0 (ε). Remark 2.1. The LRS (19) or (20) is E. Schmidt BEqR. According to [1, 2, 6, 11] the relevant A.M. Lyapunov s LRS can be constructed, in [6] their equivalence for nonlinearity case is proved. Assume now that the bifurcation point x 0 has nontrivial stationary subgroup. When G is Lie group G l = G l (a), a = (a 1,..., a l ) it is supposed to be l-dimensional differentiable manifold, satisfying the following conditions [15, 16],[11]-[14] c 1 ) The mapping a L g(a) x 0 acting from a neighborhood of the unique element in Banach space E 1 belongs to the class C 1, therefore Xx 0 E 1 for all infinitesimal operators Xx = lim t 1 [ L g(a(t)) x x ] in the tangent to L g(a) t 0 manifold Tg(a) l. c 2 ) The stationary subgroup of the element x 0 determines the representation L(G s ) of the local Lie group G s G l, s < l, with s-dimensional subalgebra Tg(a) s of infinitesimal operators. Thus the elements of the form in N(B x 0 ) form some m = (l s)-dimensional subspace, i.e. the bases in N(B x0 ) and in algebra Tg(a) s can be ordered so that ϕ k = ϕ k (x 0 ) = X k x 0, 1 k m, and X k x 0 = 0 for k m + 1. c 3 ) For all X Tg(a) l the mapping X : E 1 H is bounded in L(E 1, H)- topology. The dense embeddings E 1 E 2 H in a Hilbert space H are assumed with estimates u H α 2 u E2 α 1 u E1. To shorten the article, in the further presentation we will use and cite the works [12]-[14],[15, 16] without going into details.

106 104 Irina V. Konopleva, Boris V. Loginov, Yuri B. Rousak According to Theorem 1 (see also [1, 2, 11]) the stability of bifurcating from x = x 0 solution y(ɛ) is determined by the signs of the eigenvalues contained in σ A0 (B x0 F y (x 0, y(ɛ), ɛ k )), i.e. by the problem about eigenvalues branching of Fredholm operators B x0 y F y (x 0, y(ɛ), ɛ k )y µa 0 y = 0. (22) Lemma 2.1. LRS for the linear bifurcation problem (21) inherits the group symmetry of the equation (1) Proof Transformation of the equation (22) with the aid of E. Schmidt regularizer [8] gives Bx 0 Φ F y (x 0, y(ɛ), ɛ k )Φ µa 0 Φ = j=1 ξ j1 z (1) j (x 0 ). The substitution Φ = w + v(x 0, ξ) = w + ξ ϕ leads to the following system w = [ I Γ x0 (µa 0 + F y(x 0, y(ɛ), ɛ k )) ] 1 ( ξ ϕ (x0 )+ µ n p r ξ rρ ϕ (ρ+1) r (x 0 ) + F y(x 0, y(ɛ), ɛ k )v(x 0, ξ)), r=1 ρ=1 w, γ (σ) s = 0, s = 1,..., n; σ = 1,..., p s, where ϕ (pr+1) r (x 0 ) = ϕ (p 1) r (x 0 ) and the symbol means the omission of summands ξ r1, ϕ (1) r (x 0 ). The first equation of this system uniquely determines w = w(x 0, v(x 0, ξ), ɛ), while the second one leads to the BEqR LRS of the eigenvalue problem (22) t(x 0, v(x 0, ξ), ɛ) t j (x 0, v(x 0, ξ), ɛ)ϕ j (x 0 ) = P x0 w(x 0, v(x 0, ξ), ɛ) = 0. j=1 (23) The equation (22) for the bifurcation point (L g x 0, 0) is reduced to the system BL g x 0 L g (x x 0 ) = F L g y(l g x 0, L g y(ɛ), ɛ k )L g (x x 0 ) + µa 0 L g (x x 0 ) + n ξ s,σ = L g (x x 0 ), γ (σ) s (L g x 0 ). j=1 ξ j1 z (1) j (L g x 0 ), Analogously, the substitution L g (x x 0 ) = w + L g v(x 0, ξ) = w + v(l g x 0, ξ) gives the system K g F y(x 0, y(ɛ), ɛ k )(L 1 g K g Bx0 (L 1 g w + v(x 0, ξ)) = w + v(x 0, ξ)) + K g µa 0 (L 1 g w + v(x 0, ξ)) + n j=1 ξ j1 z (1) j (L g x 0 ), ξ sσ = w, γ (σ) s (L g x 0 ),

107 Branching equation in the root-subspace whence taking into account the relation Bx 0 v(x 0, ξ) = v (x 0, ξ), according to the unique solvability of the first equation, it follows L 1 g w = w(x 0, v(x 0, ξ), ɛ) = w = L g w(x 0, v(x 0, ξ), ɛ) together with the group symmetry inheritance theorem for BEqR- linear resolving system for (22) t(l g x 0, L g v(x 0, ξ), ɛ) P Lg x 0 w = P Lg x 0 L g w = L g P x0 w = L g t(x 0, L g v(x 0, ξ), ɛ). (24) Definition 2.1. [11]-[14] BEqR( LRS (23)) is an equation of potential type if in a neighborhood of the bifurcation point (x 0 ; 0) for the vector t(y, v(y, ξ), ɛ) = (t 11,..., t 1p1,..., t n1,..., t npn ) the equality t(y, v(y, ξ), ɛ) = d grad y U(y, ξ, ɛ) (25) holds with invertible operator d. Then the functional U(y, ξ, ɛ) is called the potential of BEqR (23) and the linear by ξ and nonlinear by y operator t as the pseudogradient of the functional U. In the article [11] (analogously [12]) the result about the reduction of LRS (23) for the eigenvalue problem (22) is proved. Theorem 2.2. Let in conditions c 1 ) c 3 ) the LRS (23) be of potential type, assume that its potential belongs to the class C 2 in some neighborhood of the bifurcation point (x 0 ; 0) and is invariant of the representation L g of the group G l (a), and let s be the dimension of stationary subgroup of the element x 0 and κ = l s > 0. Then 1 if κ = n then for all (ξ(ɛ), ɛ) (v(x 0, ξ(ɛ), ɛ)) from some neighborhood of zero in R n+1, LRS (23) is satisfied identically; 2 if κ < n and n 2, then the partial reduction of LRS takes place i.e. under accepted stipulation in c 2 ) condition about the enumeration of basic elements in E k B 1 the first k κ = p p κ equations are linear combinations of the other p κ p n. This theorem allows to reduce the order in LRS (23) on the equations quantity and by this the order of the determinant (21) as the bifurcation equation of the eigenvalue problem (20). Remark 2.2. Since the LRS (23) is equivariant only relative to stationary subgroup G s, the passage to the basis of irreducible invariant with respect to L g, g G s subspaces, analogously to [17], leads to the LRS (23) decomposition on independent systems. Thus the BEq for the finding µ(ɛ) is decomposed on separate factors that simplifies the Newton s diagram application. According to the results of [6] here we have the BEq reduction at once by Jordan chains.

108 106 Irina V. Konopleva, Boris V. Loginov, Yuri B. Rousak Remark A separate article will be devoted to the application of Theorem 2.1 to the investigation of bifurcating solutions stability at Poincaré- Andronov-Hopf bifurcation. 2. By using our previous results [18] the results obtained in this work can be generalized on the case of Noetherian operators. 3. By using the results [19, 20, 6], the stability results for stationary bifurcating solutions may be established in the presence of intertwining conditions of the equation (1). References [1] B.V. Loginov, On the stability of the solutions of differential equations with degenerate operator at the derivative, Izv. Akad. Nauk Uzbek SSR, fiz-mat., No.1, (1988) 29-32; Letter to the Editor, Izv. Akad. Nauk Uzbek SSR, fiz-mat., 2, (1988) 78. (in Russian) [2] B.V. Loginov, Yu.B. Rousak, Generalized Jordan structure in the problem of stability of bifurcation equations, Nonlinear Analysis. TMA, 17, 3 (1991) [3] L.R. Kim-Tyan, B.V. Loginov, Yu.B. Rousak, On solutions stability in differential equations with degenerate operator at the highest derivative in Banach spaces, Uzbek Math. J., 4(1999) (in Russian) [4] B.V. Loginov, Branching equation in the root subspace, Nonlinear Analysis, TMA, 32, 3(1998) [5] B.V. Loginov, I. V. Konopleva, Bifurcation systems in the root-subspace, their relation, symmetry and reduction possibilities, Proceedings of Int. Conf. Functional Spaces VI, Wroclaw, 2001, Poland, M. Dekker, 2003, [6] B. Karasözen, I. Konopleva, B. Loginov, Hereditary symmetry of resolving systems in nonlinear equations with Fredholm operators, Nonl. Anal. and Appl.: To V.Lakshnikantham on his 80th Birthday (Ravi P. Agarwal, Donal O Regan-eds.) Kluwer Acad. Publ. Dordrecht, 2(2003) [7] B.V. Loginov, Branching of the solutions of nonlinear equations and their group symmetry, Vestnik of Samara State Univ., 4(10)(1998), (in Russian) [8] M. M. Vainberg, V.A. Trenogin, Branching theory of solutions of nonlinear equations, Moscow, Nauka, 1969; Wolter Noordorf, Leyden, [9] B.V. Loginov, Yu.B. Rousak, Generalized Jordan structure in branching theory, In: Direct and Inverse Problems for PDEqs and Their Applications (M. Salakhitdinov -ed.) Fan, AN Uzbek SSR, Tashkent, 1978, (in Russian) [10] Yu.B. Rousak, Generalized Jordan structure in branching theory, PhD dissertation V.I. Romanovsky Mathematical University, Acad. Sci. Uzbek SSR, Tashkent, (in Russian) [11] I.V. Konopleva, B.V. Loginov, Yu.B. Rousak, Symmetry and Potentiality of branching equations in the root-subspaces for implicitly given stationary and dynamic bifurcation problems, Izv. Severo-Kaukaz. Nauchn. Center Visch. Shkoly, (2009)(in print), (in Russian). [12] I.V. Konopleva, B.V. Loginov, Yu.B. Rousak, Symmetry and Potentiality in general problem of branching theory, Izvestiya VUZ, Mathematics, 4(527) (2006), [13] B.V. Loginov, I.V. Konopleva, Yu.B. Rousak, Bifurcation and symmetry in differential equations nonresolved with respect to derivative, ROMAI Journal 3, 1(2007),

109 Branching equation in the root-subspace [14] I.V. Konopleva, B.V. Loginov, Bifurcation, Symmetry and Cosymmetry in Differential Equations Unresolved with Respect to the Derivative With Variational Branching Equations, Doklady Mathematics, 80, 1(2009) [15] N.I. Makarenko, On solutions branching for invariant variational equations, Russian Acad. Sci., Doklady Mathematics, 348, 3(1996), [16] N.I. Makarenko,Symmetry and cosymmetry of variational problems in waves theory, Proc. Int. School-Seminar Applications of symmetry and cosymmetry in theories of bifurcations and phase transitions, Sochi (August 2001), Rostov- on Don Univ., (2001) (in Russian) [17] B.V. Loginov, Branching Theory of Solutions of Nonlinear Equations Under Group Invariance Conditions, Tashkent, Fan, (in Russian) [18] B.V. Loginov,L.R. Kim-Tyan, Yu.B. Rousak, Modification of the Lyapounov-Schmidt method and the stability of solutions of differential equations with a singular operator of finite index multiplying the derivative, Russian Acad. Sci., Doklady Mathematics, 47, 3, (1993) (in Russian) [19] B.V. Loginov, I.V. Konopleva, Symmetry of resolving systems in degenerated functional equations, Proc. Int. Conf. Symmetry and Differential Equations (V.K. Andreev, V.V. Vasiliev - eds.), Krasnoyarsk, Inst. Math. Model. Siberian Branch. of RAS, (2000), [20] V.R. Abdullin, N.A. Sidorov, Interlaced equations in branching theory, Russian Acad. Sci., Doklady Mathematics, 377, 3, (2001)

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111 ROMAI J., 5, 2(2009), SOLVABILITY OF HAMMERSTEIN EQUATIONS WITH ANGLE-BOUNDED KERNEL Irina A. Leca Naval Academy Mircea cel Bătrân Constanţa, Romania Abstract The Browder-Gupta splitting of angle-bounded operators is interplayed with the elliptic super-regularization imbedding to study the solvability of Hammerstein operator equations. Keywords: monotone operators, Hammerstein equations MSC: 47H05, 47H30, 46B INTRODUCTION A nonlinear integral equation of Hammerstein type is one of the form: (1) u (x) + Ω K (x, y) f (y, u (y)) dy = h (x) where Ω is a measure space with a σ-finite measure dy, the given function h (x) and the unknown function u (x) are defined on Ω. In the operator-theoretic terms, the problem of determining a solutions of equation (1), with u and h lying in a given Banach space E of functions over Ω can be rewritten in the form of nonlinear operator equation (2) u + KNu = h with a linear integral operator (generalizing the Green function) (3) Kv (x) = Ω K (x, y) v (y) dy and a nonlinear superposition (or Nemîtskĭi) mapping (4) Nu (x) = (N f u) (x) = f (x, u (x)). In the sequel, we consider the homogeneous equation attached to (2) and E = X, the dual space of the real Banach separable space X. It is well-known that a Banach space is reflexive iff every bounded sequence has a weakly convergent subsequence. The weak convergence can be replaced by the weak convergence of subsequences of bounded sequences on X when X is a separable Banach space. Another reason for the choice of X as separable is to have conditions for the use of the Browder-Ton elliptic super-regularization argument. Let X be a real Banach space with its dual space X. Consider the operator Hammerstein equation: (5) u + KNu = 0, u X, 109

112 110 Irina A. Leca where the operators K : X X and N : X X are of monotone type. We start with the basic definitions: Let X be a real Banach space, X its dual space and denote by (x, x) the duality pairing between x X and x X. A mapping T : D (T ) X X is said to be monotone if (T x T y, x y) 0, x, y D (T ). More restrictive, T : D (T ) X X is strongly monotone if there is m > 0 such that: (T x T y, x y) m x y 2 X, x, y D (T ). 2. ANGLE-BOUNDED OPERATORS A significant concept introduced in the study of Hammerstein equations is the angle-bounded operators as a subclass of monotone operators. Definition 2.1. A linear monotone operator K : D (K) X X is said to be angle-bounded with the constant a 0, if (6) (Kx, y) (Ky, x) 2a (Kx, x) (Ky, y) 2, for all x, y D (K). For the sake of simplicity, we consider further that D (K) = X. Angle-bounded mappings generalize symmetric mappings. Indeed, the angleboundedness of K with a = 0 corresponds to the symmetry of K, i.e. (Kx, y) = (Ky, x) for all x, y X. As above, a linear mapping K : X X is said to be strongly monotone operator if there is a constant m > 0 such that (Kx, x) m x 2 X for all x X. It follows that every strongly monotone mapping is angle-bounded with a = K m since (Kx, y) (Ky, x) 2 K x y 2a (Kx, x) (Ky, y) 2, x, y X. We now relate the angle-boundedness to the cyclic monotonicity. An operator T : D (T ) X X is cyclic monotone if (T x 1, x 1 x 2 ) + (T x 2, x 2 x 3 ) (T x n, x n x n+1 ) 0, for all x i D (T ), i = 1, 2,..., n and all n N, where we set x n+1 = x 1. For n = 2 it reduces to usual monotonicity. A typical example of a cyclic monotone multivalued mapping is the subdifferential of a proper convex semicontinuous function (see [9], pp. 124). We are interested, in particular, when T : D (T ) X X is 3-monotone, that is, (T y T x, y z) (T x T z, z x), x, y, z D (T ). We proved in [7] that the Nemîtskĭi operators are 3-monotone. In a more general way, we can consider

113 Solvability of Hammerstein equations with angle-bounded kernel 111 Definition 2.2. An operator T : D (T ) X X is called 3-C-monotone if there is a constant C > 0 such that (T y T x, y z) C (T x T z, z x), x, y, z D (T ). We have the following equivalences: Theorem 2.1. Let K : X X be a linear monotone operator on the real Banach space X. The following three statements are equivalent: (a) K is 3-C-monotone, i.e. there is a constant C > 0 such that (7) (Kx Ky, z x) C (Ky Kz, y z), for all x, y, z X; (b) K satisfies the discriminant inequality (8) (Kv, w) 2 4C (Kv, v) (Kw, w), for all v, w X; (c) K is angle-bounded, i.e. there is a constant a > 0 such that (9) K (x, y) K (y, x) 2 4a 2 K (x, x) K (y, y), for all x, y X. Proof. Substituting in (7) v = y z and w = x z, by the linearity of K, we obtain (Kv, w) (Kw, w) C (Kv, v), for all v, w X. Replacing now v by tv, we obtain the inequality C (Kv, v) t 2 (Kv, w) t + (Kw, w) 0, for all v, w X, which is equivalent with the non-negativity of its discriminant, that is (a) (b). To establish the other equivalences, we now introduce [x, y] ± = 1 ((Kx, y) ± (Ky, x)), for all x, y X. 2 Since K is monotone, we have [x, x] + 0 for all x X and the generalized Schwarz inequality ensures that (10) [x, y] 2 + [x, x] + [y, y] +, for all x, y X. On the other hand, we have (11) (Kx, y) = [x, y] + + [x, y], for all x, y X. Moreover, the inequality (9), namely the angle-boundedness of K, can be written in the form: (12) [x, y] 2 a2 [x, x] + [y, y] +, for all x, y X. From (11) and the inequality (A + B) 2 2A 2 +2B 2 for real A, B, we derive (Kx, y) 2 2 ( 1 + a 2) [x, x] + [y, y] +, for all x, y X, that is (8) which is equivalent with the 3-C-monotonicity of K, as we proved early. Therefore, (c) (a). Finally, if K is 3-C-monotone and [x, y] = (Kx, y) [x, y] + then we obtain that [x, y] 2 2 (4C + 1) [x, x]2 + [y, y]2 +, for all x, y X, which is (9). Thus (a) (c), and the equivalences stated above are proved. The equivalence (a) (c) suggests us an extension of angle-boundedness.

114 112 Irina A. Leca Definition 2.3. A nonlinear mapping T : X X is said to be angle-bounded with a constant C > 0, if (T x T z, z y) C (T x T y, x y), for all x, y, z X. If y = z then an angle-bounded operator T is monotone. The angle-bounded operators were introduced by H. Amann [1] and applied to the investigation of Hammerstein equations especially by F.E. Browder - C.P. Gupta [4] and F.E. Browder [3]. We mention the detailed approaches of Hammerstein equations in the monographs [9],[11] and [12]. 3. THE SPLITTING OF ANGLE-BOUNDED OPERATORS We will develop the technique used in the proof of Theorem 2.1 to establish a splitting of linear angle-bounded operators, due to Browder-Gupta [4]. Our approach combines this splitting with the elliptic super-regularization method [5], recently improved in a general and simpler form by Berkovits [2]. Imbedding theorem of Browder and Ton. Let E be a real separable Banach space. Then there exists a separable Hilbert space H and a compact linear injection Ψ : H E such that Ψ (H) is dense in E. We define further the adjoint operator Ψ : E H by setting Ψ w, v = w, Ψv, v H, w E, where, stands for the inner product in H. Since Ψ (H) is dense in E, Ψ is also a linear compact injection. We describe the Browder-Gupta theorem for splitting the angle-bounded operators. Theorem 3.1. Let X be a real separable Banach space and K : X X a linear monotone operator which is angle-bounded with constant a 0. Then (A) there exist a separable Hilbert space H, a linear injection S : X H with its adjoint map S : H X and a skew-adjoint bounded operator B : H H such that K = S (I + B) S. (B) S : X H is compact, its range S (X) is dense in H, S : H X is also a compact injection and S 2 K ; (C) B : H H is a linear skew-adjoint operator, i.e., B = B, B a and I + B is bijective. Proof. It is sufficient to take E = X and change the names of the compact injections: S = Ψ : X H and S = Ψ : H X where the existence of the Hilbert space (H, (, )) is ensured by the Browder- Ton theorem. We carry out the proof of Theorem 2.1 and consider the null-

115 Solvability of Hammerstein equations with angle-bounded kernel 113 space Z = { x X [x, x] + = 0 }. Taking into account (11) and (12), we derive that [u, v] ± = 0 for all u Z and v Z and thereby [x + u, y + v] ± = 0 for all x, y X and all u, v Z. Consequently, the equivalence relation x y iff x y Z holds. Denote by H 0 the quotient (pre-hilbert) space X/Z with the equivalence classes defined above. We denote by U, V, W... the classes of H 0 and consider the Hilbert space H as the completion of H 0 with respect to the norm U 2 = [u, u] + = (Ku, u) for all U H 0 and u U. Moreover, on H the scalar product U, V = [u, v] + is defined, whatever are the representatives u U and v V. Indeed, from U, U = 0 it follows that [u, u] + = 0 for every u U, hence u Z, therefore U = 0. The injection S : X H defined by Su = U, u U, is the natural projection that assigns to each u X the corresponding equivalence class U in H 0. Therefore Su, Sv = [u, v] + = 1 ((Ku, v) + (Kv, u)). 2 The map S : X H is linear and continuous and by Su 2 H = Su, Su = (Ku, u) Ku u K u 2, it follows that S 2 K. In addition, since the range of S coincides with H 0, the range of S is dense in H and the adjoint map S : H X is injective. For U, V H 0, let us consider now the skew-symmetric part: b (U, V ) = [u, v] = 1 ((Ku, v) (Kv, u)) where u U and v V. 2 Taking into account of the angle-boundedness (12) of K, we have (13) b (U, V ) a U H V H, for all U, V H 0. As H 0 is dense in H, the bilinear form b : H H R is well-defined and induce a linear operator B : H H such that b (U, V ) = BU, V for all U, V H. Likewise, by (13) we have B a. Besides, because b (U, V ) = b (U, V ) it follows that B = B, i.e., B is a skew-adjoint operator. The splitting K = S (I + B) S follows from (Ku, v) = [u, v] + + [u, v] = Su, Sv + BSu, Sv = = (I + B) Su, Sv = (S (I + B) Su, v), u, v X. For simplcity, we denote further A = I + B : H H and prove: Propoziţia 3.1. The operators A and A 1 are strongly monotone. Moreover, A : H H is a bijection. Proof. Since B is skew-adjoint, we have BU, U = b (U, U) = 0 and (14) AU, U = (I + B) U, U = U 2 H, for all U H, which means the strong monotonicity of A. By B a, it follows that

116 114 Irina A. Leca (15) AU 2 H = (I + B) U, (I + B) U = U 2 H + BU 2 H ( 1 + a 2) U 2 H, for all U H. Combine (14) and (15) to derive (16) U, A 1 U = A ( A 1 U ), A 1 U = A 1 U 2 H ( 1 + a 2) 1 U 2 H, for all U H, that is, the strong monotonicity of A 1. Finally, from AU H U H it follows that A is injective and A (H) is closed. We prove also that A is surjective. Indeed, assuming A (H) H, there exists 0 V H with AU, V = 0 for U H. In particular, AV, V = 0 and (14) implies V = 0, hence A (H) = H. At the end of this section, we list some additional properties and comments related to the previous maps, required further down. Since we assume that the real separable Banach space is not generally reflexive, for the monotonicity of nonlinearity N : X X we will consider X as a subset of the bidual space X and identify the Hilbert space H with its dual H. 4. SOLVABILITY OF HAMMERSTEIN EQUATION We deal with the existence and uniqueness of solutions of Hammerstein equations. Recall that an operator T : D (T ) X X is hemicontinuous if it is continuous from each line segment of D (T ) to the weak topology of X. Further, T is called coercive if (T x, x) lim = +. x x The following basic criterion is used. Surjectivity theorem of Minty-Browder. An operator T : D (T ) X X hemicontinuous, monotone and coercive on a real reflexive Banach space X is surjective. Using above notations and the splitting of the angle-bounded operator K, obtained in the factorization Theorem 3.1, the nonlinear Hammerstein equation (5) becomes (17) u + S ASNu = 0, u X. Since S : H X is an injection, by u = S W, eq. (17) is equivalent to S W + S ASNS W = 0, W H, that is, S (I + ASNS ) W = 0. Taking again into account that S is injective, the equation (5) is equivalent to (18) W + ASNS W = 0, W H.

117 Solvability of Hammerstein equations with angle-bounded kernel 115 Since A : H H is a bijection, we consider the equivalent form of (17) (19) A 1 W + SNS W = 0, W H. for which we establish the following existence and uniqueness result. Theorem 4.1. Let X be a real separable Banach space, K : X X a linear, monotone, angle-bounded operator and N : X X a nonlinear hemicontinuous monotone mapping. Then the Hammerstein equation u + KNu = 0 has exactly one solution u X. Proof. As it has been explained above, taking account of the splitting of the angle-bounded operator K, given by Theorem 3.1, the equation (19) on the associated Hilbert space H to the operator K is equivalent with the initial equation (5) for u = S W. We define F = A 1 +SNS and show that F : H H is a hemicontinuous, strongly monotone operator. Indeed, for U, V H, F U F V, U V = A 1 (U V ), U V + (S U S V, NS U NS V ) holds and, due to the monotonicity of N and (16), we obtain (20) F U F V, U V = A 1 (U V ), U V ( 1 + a 2) 1 U V 2 H, i.e., F is strongly monotone. Moreover, F is hemicontinuous like N and F is coercive since it is strongly monotone. By the Minty-Browder surjectivity theorem of monotone mappings, the operator F : H H is bijective. In particular, there exists a unique solution U H of the equation F U = 0. To specify for later use some determinations of mappings above introduced, we mention that A : H H and S w = Sw for all w X. Furthermore, for the adjoint operator K = S A S we have the representation (21) K x = S A Sx for all x X. Later on, an approximation-solvability in Petryshyn s sense [6], for the Hammerstein equations will be outlined. We confine ourselves to a simple projection-solvability [8]. Let {X n } be a monotone increasing sequence of finite-dimensional subspaces of X such that X n is dense in X. Such a sequence {X n } is called a projectional system in X. Defining now the Hilbert subspaces {H n } by (22) H n = (S ) 1 K (X n ), n = 1, 2,.... we will prove that {H n } is also a projectional system in H. Indeed, from X n X n+1 it follows that H n H n+1 H and we have to show that H n is dense in H, that is, U, V = 0 for all V H n implies U = 0. In fact, for x X n, we have U, (S ) 1 K x AU, Sx = U, A SX = = 0 and, by (21), = 0. U, (S ) 1 K x Because we assumed that X n is dense in X and S : X H is continuous, we obtain AU, Sx = 0 for all x X. By assertion (B) in Theorem 3.1, the

118 116 Irina A. Leca range S (X) is dense in H and hence AU = 0. As A : H H is bijective, we conclude that U = 0. Concerning the operator equation (23) T u = f, u X, where T : X X is a (generally, nonlinear) hemicontinuous and monotone mapping, X a real separable Banach space and f X, we introduce a Galerkin method. Let {X n } be a projectional system in X. With the equation (23), we associate the sequence of finite-dimensional equations or Galerkin equations (24) (T u n, x) = (f, x), x X n. Definition 4.1. For a given f X, equation (23) is projectionally-solvable if the following conditions hold: (i) The original equation (23) has a unique solution; (ii) There is a number N such that for n N, the Galerkin equation (24) has a unique solution u n X n ; (iii) The Galerkin solutions converges, i.e., u n u in X as n. In a general framework, the approximation-solvability of (23) is related to the A-properness of mapping T, (see e.g. [10]). However, one of the simplest example of the projectionally-solvability of equation (23) holds in the case when T : X X is a strongly monotone operator. Returning to the original Hammerstein nonlinear equation (5) u + KNu = 0, u X, with the operators K : X X and N : X X of monotone type, we consider a Galerkin method, i.e. we study the approximate equations (25) (u n + KNu n, x) = 0, x X n, where {X n } is a projection system in X. Theorem 4.2. The Galerkin equation (25) has exactly one solution u n X n for every n N and the sequence {u n } converges in the norm topology of X to the solution of equation (5). Proof. The conclusion follows from the equivalence between the Galerkin solution of (25) and the construction of approximate solution U n H n of the equation: (26) F U n, V = 0, V H n where {H n } is the associate projection system given by (22). By Theorem 4.1, the Galerkin equation (26) has exactly one solution U n H n and U n U in H as n, where U H be the unique solution of the equation F U = 0. To complete the proof, we show that u n = S U n is a solution of the original Galerkin equation (25). More precisely, we look for u n K (X n ) and prove the equivalence between the equations (25) and (26).

119 Solvability of Hammerstein equations with angle-bounded kernel 117 As above in (26), we have F U n, W = A 1 U n, W + (NS U n, S W ), W H n. By (21), for every x X n we set K x = S A Sx in X. Moreover, to every x X n corresponds W = (S ) 1 K x in H, that is, W = A Sx. We compute A 1 U n, W = A 1 (S ) 1 u n, W = A 1 (S ) 1 u n, A Sx = = (S ) 1 u n, Sx = (u n, x) and (NS U n, S W ) = (Nu n, S A Sx) = SNu n, A Sx = = ASNu n, Sx = (S ASNu n, x) = (KNu n, x). Therefore F U n, W = (u n + KNu n, x). From H n = (S ) 1 K (X n ) and u n = S U n with U n H n it follows that u n K (X n ) and the equivalence between (25) and (26) is established. In the end, we prove the strong convergence of the sequence of Galerkin solutions. Let P n : H H n be the projection. Since {H n } is a projectional system in H, we have: (27) P n V V H 0 as n, for every V H. The restriction F n = P n F Hn has the same properties as F, from which the unique solvability of the equation F n U = 0 holds, as well. As above, this equation can be rewritten F n U, V = 0 for all V H n. We denote later by U n H n the solutions of this Galerkin method. Moreover, if U H is a solution of the initial equation F U = 0, then, by (27), we have F ( Pn U ) H 0 as n. Taking into account the strong monotonicity (20) and continuity of operator F : H H, we derive for approximate solutions the following estimate: U n P n U H ( 1 + a 2) Fn ( U n ) Fn ( Pn U ) H ( 1 + a 2) F ( P n U ) H 0 as n and thereby, U n U H S ( U Pn U H + Pn U U ) H 0 as n (the norm S is taken in the space L (H, X )), meaning that U n U in H. We set, as above, u n = S U n and u = S U. Then u X is a unique solution of Hammerstein equation u + KNu = 0 and the continuity of S implies that u n u in X as n. Acknowledgements.The author wants to express the gratitude to her Ph.D. advisor Dan Pascali for the valuable support in writting this paper. References [1] H. Amann, Zum Galerkin-Verfahren für die Hammerstein Gleichungen, Arch. Rational Mech. Anal. 35 (1969), ; MR 42#8345.

120 118 Irina A. Leca [2] J. Berkovits, A note on the imbedding theorem of Browder and Ton, Proc. Amer. Math. Soc. 131 (2008), ; MR 2004b: [3] F.E. Browder, Nonlinear functional analysis and nonlinear integral equa-tions of Hammerstein and Urysohn type, Contributions to nonlinear functional analysis (E.H. Zarantonello), , Academic Press, New York, 1971; MR 52# [4] F.E. Browder, C.P. Gupta, Monotone operators and nonlinear integral equations of Hammerstein type, Bull. Amer. Math. Soc. 75 (1969), ; MR 40#3381. [5] F.E. Browder, B.A. Ton, Nonlinear functional equations in Banach spaces and elliptic super-regularization, Math. Z. 105 (1968) ; MR 38#582. [6] P. M. Fitzpatrick, W. V. Petryshyn, Galerkin methods in the constructive solvability of nonlinear Hammerstein equations with applications in differential equations, Trans. AMS 238 (1978), ; MR 58#47H15. [7] I. A. Leca, Monotonicity of the Nemitsky operator, A XX-a Sesiune de Comunicări Ştiinţifice cu Participare Internaţională NAV-MAR-EDU 2007, Constanţa, 2007, [8] D. Pascali, An introduction to numerical functional analysis, Petryshyn s A-proper mapping theory, Lect. Notes Math. Ovidius Univ., Constanţa, [9] D. Pascali, S. Sburlan, Nonlinear mappings of monotone type, Edit. Academiei, Bucureşti and Sijthoff & Noordhoff Intern. Publ., Alphen aan den Rijn, 1978; MR 80g: [10] W.V. Petryshyn, Approximation-solvability of nonlinear functional and differential equations, Marcel Dekker, New York, 1991; MR 94f: [11] S.F. Sburlan, Gradul topologic, Edit. Academiei, Bucureşti, 1983; MR 86b: [12] E. Zeidler, Nonlinear functional analysis and its applications, Part II/A: Linear monotone operators; II/B: Nonlinear monotone operators, Spriger-Verlag, New York, 1990; MR 91b:47001,

121 ROMAI J., 5, 2(2009), NUMERICAL K - BUSY PERIODS ALGORITHMS FOR POLLING SYSTEMS WITH SEMI - MARKOV SWITCHING Gheorghe Mishkoy, Diana Bejenari Free International University of Moldova, Chişinău, Republic of Moldova gmiscoi@ulim.md, artemis85@mail.ru Abstract The queueing systems of Polling type are widely used in wireless networks with broadband centralized management (see, e.g., [1]). One of the important characteristics of these systems is the k - busy period [2]. In [3] it is shown that analytical results for k - busy periods can be viewed as generalizations of classical Kendall functional equation [4]. Unfortunately, analytical solution for such type of generalized equations does not exist. However, using the methodology of generalized priority systems and generalized algorithms elaborated in [3], numerical solutions with necessary required accuracy can be obtained. Some examples and numerical results are presented here. Acknowledgement. This work is supported partially by Russian Foundation for Basic Research (RFFI) grant RF and grant GF of the Federal Ministry of Education and Research (BMBF) of Germany. Keywords: k - busy period, numerical algorithms. system, LT-algebra MSC: 60K25, 68M20, 90B INTRODUCTION It is known that wireless networks have developed rapidly last years. For planning regional wireless networks, models and research methods of Polling systems are used. In this systems there is a common server for all users which serves proposed messages by users according to given rules. The main purpose of research of Polling systems is to determine the characteristics of systems development. But not always analytical formulas can be used directly, so great care is offered to numerical algorithms. In this paper, some examples of determining the k - busy period, that are obtained from numerical algorithms are presented. 2. THE K - BUSY PERIOD Definition 2.1. The period beginning with the exchange of user and ending when the system becomes free from the requirements of class k (messages from user k) is named a k - busy period [3]. 119

122 120 Gheorghe Mishkoy, Diana Bejenari By Π δ k is denoted the length of the k period, and by Π δ k (x) = P {Πδ k < x}, its distribution function. Further we will consider that π δ k (s) = 0 e sx dπ δ k (x) is the Laplace - Stieltjes transform of distribution function of k-period. The following result is known [3]. Theorem 2.1. The function π δ k (s) is determined from equation where π δ k (s) = c k(s + λ k λ k π k (s))π k (s), (1) π k (s) = β k (s + λ k λ k π k (s)), (2) and by c k (s) and β k (s) are denoted the Laplace - Stieltjes transforms of distribution functions C k (x) and B k (x) c k (s) = β k (s) = 0 0 e sx dc k (x), e sx db k (x). Functional equation (2) has no analytical solution, but it can be solved numerically with some required accuracy. Several numerical algorithms in C++ for solving functional equation (2) and (1), were created, depending on the type of distribution function taken by B k (x) and C k (x). 3. EXAMPLES In our examples we use the well known [5] types of distribution function: 1 Uniform distribution on [a, b], U(a, b). 0, x < a, F (x) = (x a)/(b a), a x b, 1, x > b,

123 Numerical k - busy periods algorithms for Polling systems with semi - Markov switching 121 E(x) = a + b 2, f(s) = 1 1 b a s (e sa e sb ). 2 Exponential distribution, Exp(λ). F (x) = 1 e λx, x > 0, E(x) = 1 λ, f(s) = λ s + λ. 3 Erlang distribution, Erl(λ, k). 0, x < 0, F (x) = x λ (λu)k 1 (k 1)! e λu du, x 0, 0 f(s) = E(x) = k λ, ( ) λ k. λ + s 4 Gamma distribution, Ga(λ,a). F (x) = Γ(α) = 0 λ α Γ(α) 0 α > 0, λ > 0, 0, x < 0, x α 1 e λx dx, x 0, x α 1 e λx dx, Γ(α) = (k 1)! for α = k, E(x) = α x,

124 122 Gheorghe Mishkoy, Diana Bejenari f(s) = ( ) λ α. λ + s Remark 1. For α = 1, Ga(λ, 1) = Exp(λ). Remark 2. For α = k, Ga(λ, 1) = Erl(λ, k). For this reason we chose only Erlang distribution. 5 Normal distribution, N(a, σ 2 ). F (x) = 1 σ 2π x e (u a)2 2σ 2 du, < x <, E(x) = a, V (x) = σ 2, f(s) = e ( σ2 s 2 sa). Remark 3. Normal standard distribution N(0, 1). Φ(x) = 1 2π x e t2 2 dt, < x <, E(x) = a = 0, V (x) = σ 2 = 1, f(s) = e s 2. Example 1. The type of distribution function taken by B k (x) and C k (x) is the Exponential distribution, so and B k (x) = 1 e b kx, x > 0, C k (x) = 1 e c kx, x > 0, with following parameters: λ k ={2, 3, 5, 6, 3, 8, 5, 4, 7, 3, 4, 2, 9, 7, 6, 4, 6, 3, 5, 2}, b k ={7, 5, 9, 4, 6, 2, 5, 4, 8, 6, 5, 3, 4, 5, 7, 6, 2, 4, 8, 6},

125 Numerical k - busy periods algorithms for Polling systems with semi - Markov switching 123 c k ={7, 5, 9, 4, 6, 2, 5, 4, 8, 6, 5, 3, 4, 5, 7, 6, 2, 4, 8, 6}. The results of the program are presented in Table 1. k π k (s) π δ k (s) k π k(s) π δ k (s) Table 1. Example 2. The type of distribution function taken by B k (x) is the Erlang distribution and by C k (x) is the Exponential distribution, so and B k (x) = x 0 0, x < 0, λ k (λ k u) k 1 (k 1)! e λ ku du, x 0, C k (x) = 1 e c kx, x > 0, with following parameters: λ k ={3, 5, 7, 6, 8, 4, 5, 7, 3, 6, 8, 5, 9, 7, 6, 2, 4, 1, 1, 1}, b k ={4, 4, 6, 7, 4, 8, 8, 3, 2, 5, 5, 7, 6, 4, 8, 6, 4, 2, 4, 6}, c k ={6, 7, 4, 3, 7, 6, 9, 7, 4, 5, 3, 2, 5, 6, 7, 5, 4, 3, 2, 8}, p k ={7, 5, 4, 3, 6, 5, 2, 5, 5, 7, 6, 5, 9, 8, 6, 5, 4, 6, 7, 6}. The results of the program are presented in Table 2.

126 124 Gheorghe Mishkoy, Diana Bejenari k π k (s) π δ k (s) k π k(s) π δ k (s) e e e e e e e e e e e e e e Table 2. Example 3. The type of distribution function taken by B k (x) is theerlang distribution and by C k (x) is Normal distribution, so B k (x) = x 0 0, x < 0, λ k (λ k u) k 1 (k 1)! e λ ku du, x 0, and C k (x) = 1 x 2π σ k e (u a) 2 2σ 2 k du, < x <, with following parameters: λ k ={0.6, 0.4, 0.6, 0.7, 0.3, 0.4, 0.6, 0.2, 0.6, 0.4, 0.7, 0.1, 0.3, 0.5, 0.7, 0.5}, b k ={0.2, 0.4, 0.7, 0.4, 0.6, 0.8, 0.6, 0.3, 0.4, 0.5, 0.3, 0.2, 0.5, 0.4, 0.6, 0.7}, c k ={0.5, 0.3, 0.5, 0.7, 0.5, 0.3, 0.4, 0.8, 0.5, 0.4, 0.2, 0.3, 0.4, 0.5, 0.3, 0.5}, p k ={2, 4, 3, 5, 4, 6, 5, 4, 3, 2, 4, 6, 5, 4, 7, 6}, σ k ={2, 4, 3, 5, 4, 6, 5, 4, 3, 2, 4, 6, 5, 4, 7, 6}. The results are in the Table 3.

127 Numerical k - busy periods algorithms for Polling systems with semi - Markov switching 125 k π k (s) π δ k (s) k π k(s) π δ k (s) Table 3. Example 4. The types of distribution function for B k (x) and C k (x) are given in the Table 6. We used the following notations: If the distribution function is Uniform we denote by the letter U, If the distribution function is Erlang I, If the distribution function is Exponential E, If the distribution function is Normal N. λ k ={0.2, 0.4, 0.7, 0.5, 0.6, 0.3, 0.4, 0.5, 0.6, 0.9, 0.2, 0.3} Required parameters for each distribution function are in the Table 4. and Table 5. U N U I N U I N I a = 4 σ = 0.6 a = 5 k = 2 σ = 0.9 a = 10 k = 3 σ = 0.4 k = 4 b = 1 a = 0.2 b = 2 a = 0.2 b = 2 a = 0.1 Table 4. U I U I U I N U I a = 14 k = 1 a = 9 k = 2 a = 3 k = 2 σ = 0.5 a = 4 k = 2 b = 7 b = 2 b = 1 a = 0.3 b = 1 Table 5.

128 126 Gheorghe Mishkoy, Diana Bejenari The results of the program are given in Table 6. k B k (x) C k (x) π k (s) π δ k (s) k B k(x) C k (x) π k (s) π δ k (s) 1 U N E I E U E U I N I E E U U I I N N U I U I E References Table 6. [1] V. M. Vishnevsky, O. V. Semenova, Polling Systems: The theory and applications in the broadband wireless networks, Moscow, Texnocfera, 2007 ( in Russian). [2] V. V. Rycov, Gh. K. Mishkoy, A new approach for analysis of polling systems. //Proceedings of the International Conference on Control Problems, Moscow, (2009), [3] Gh. K. Mishkoy, Generalized Priority Systems, Chisinau: Acad. of Sc. of Moldova, Stiinta, 2009, (in Russian). [4] H. Takagi, Queueing Analysis: Vacation and Priority Systems, North-Hodland, Elsevier Science Publ, vol. 1, [5] N. A. J. Hastings, J. B. Peacock, Statistical Distribution, Moskow, Statistics, 1980, (in Russian).

129 ROMAI J., 5, 2(2009), FRAGILE BITS VS. MULTI-ENROLLMENT - A CASE STUDY OF IRIS RECOGNITION ON BATH UNIVERSITY IRIS DATABASE Nicolaie Popescu-Bodorin Spiru Haret University, Bucharest, Romania Abstract This paper explores the use of fragile bits in the context of a recently proposed iris recognition methodology based on Circular Fuzzy Iris Segmentation and Gabor Analytic Iris Texture Binary Encoder. Iris images from Bath University Iris Database are encoded as iris codes at three different lengths (192, 512, 768 Bytes) and used to test the concept of fragile bits. Six iris recognition tests are presented in order to illustrate the efficiency of the recently proposed iris recognition methodology in both single-enrollment and multi-enrollment iris recognition scenarios. Three additional tests show that at least in the recognition scenarios presented here, using a certain type of fragile bits will determine a split of the set of genuine scores into two distributions having two completely different statistics, case in which apparent improvements of the recognition performance measured through decidability index and through Fisher s ratio are irrelevant. Also, some important differences between different types of fragile bits are stated here for the first time. Keywords: iris recognition, iris segmentation, circular fuzzy iris segmentation, Gabor analytic iris texture binary encoder, fragile bits; MSC: 68T10, 68U10, 68N99, 44A INTRODUCTION From the early stages of our PhD study we found that an open problem in iris recognition is the fact that we can t say for sure if a given iris database is or isn t a representative sample. To be more precise, if the length of the iris binary code is assumed to be 1024, then the numerical space representing the iris population counts more than 1.7E elements. Now, let s imagine a huge iris database containing, let s say, 1E + 12 images and assume that extraordinary iris recognition results have been proved using this database. We should use some sampling techniques enabling us to extrapolate these results to entire iris code population (and to other similar databases, in particular), despite the fact that representativity rate of our hypothetical database is nearly null (1E 296). Unfortunately, such techniques don t exist and consequently, in these circumstances, explaining the differences between theoretical and experimental results could prove to be difficult and misleading. 127

130 128 Nicolaie Popescu-Bodorin A possible example is the concept of fragile bits, initially introduced to explain the difference between experimentally determined False Reject Rates and the theoretically predicted values. As an alternative, we consider that the set of all available iris-codes is so sparse and scattered within the iris-code population that the matching between the bits of any two different irides happens only by chance. This is the first major hypothesis in our approach to iris recognition. It was initially formulated by Daugman in the early 90s but never fully exploited ever since. Present paper shows that accepting and following this hypothesis leads to theoretical and experimental iris recognition results agreeing each other. In this scenario we will give here some examples showing that masking the so called fragile bits don t necessary improve the iris recognition performance. The concept of fragile bits was proposed by Bolle and all [1] but the first work truly investigating this subject is very recent indeed (Hollingsworth and all [4]). The cited paper introduces fragile bits in two ways: first, as a mask of unstable bits that are flipping from 0 to 1 or vice versa in a chosen number of observations, and second, as a mask of the bits that are more susceptible to flip because of a phase instability phenomenon: in one particular observation, a bit is considered to be fragile if the corresponding complex value encoding the corresponding iris pixel is located close to the real or imaginary axis. But, in fact, the two cases describe two different concepts: - In the first case, fragile bits are defined as an a priori knowledge inferred from a given number of previous observation as a mask containing the pixels that are proven to be unstable by their values, but the causality of this instability is not evident. In this context, it is very important to answer the following questions: is this mask a feature of the iris or it is just a feature of the given set of observations? Does it depend on the iris texture or on the variability of the acquisition conditions? - As defined in the second case, the concept of fragile bits represents, in fact, an a posteriori knowledge about the current observation(s), a mask defined by a fuzzy membership assignment conditioned by the degree of closeness to the real/imaginary axis or, as mentioned in [4], by an adaptive thresholding of the lowest quartile calculated for the histogram of the set containing the absolute values corresponding to the complex representation of the unwrapped iris. - To test the concept of fragile bits in the first scenario, the current iris codes will be compared to the stored template gallery using the mask of fragile bits computed for the enrolled identities. The results of the tests could give the answers to the questions formulated above. But, the most important outcome of such a test will be to find out if the use of fragile bits is or isn t compatible with the hypothesis that the matching between the bits of any two different irides happens only by chance. - In the second scenario, testing the concept of fragile bits means to assume

131 Fragile Bits vs. Multi-Enrollment - a case study of iris recognition that any two iris codes will be compared using a mask computed at run-time. The best possible outcome of such a test would be an improvement in the system performance quantified as a narrowing of the genuine or /and imposter score distributions, an increase of the distance between the two classes of scores, or an improvement of other values calculated as evaluation criteria (decidability index, Fisher s ratio, [17]) We must clarify that we call a priori knowledge any piece of knowledge/data used to bring the biometric system in a fully functional ready state able to treat the client recognition request. One notable difference between the above two cases is that in the former, a priori knowledge includes the (stored) masks of fragile bits for all enrolled identities, while in the latter, a priori knowledge includes a method to produce (at run-time) a mask for any given pair of iris codes. From the beginning, it can be remarked that if the mask of fragile bits mainly depends on other things than the iris texture, at least one from the above tests will fail to improve recognition performance. 2. CIRCULAR FUZZY IRIS SEGMENTATION The segmentation algorithm presented here was proposed in [12] as an alternative to the currently available segmentation procedures that use integrodifferential Daugman operator [2], or Hough Transform [16], or active contours [3]. The most important difference between previous segmentation methods and CFIS is the dimension of the parameter space needed to be searched in order to find pupillary and limbic boundary: in CFIS procedure iris boundaries are found by solving one dimensional optimization problems. CFIS procedure consists in the following operations: pupil finding and limbic boundary circular approximation. Pupillary and limbic boundaries are assumed to be concentric circles. An anatomic argument for using this hypothesis is that since the pupil is nearly circular, there must be a circular concentric iris ring controlling the pupil movements. Such a circular iris ring is expected to play the most important role in iris recognition, despite the fact that it appears to be a rough approximation of the actual iris. A system requirement sustaining the above formulated hypothesis is that the segmentation routine must be fast and energy-efficient. Nearly lossless unwrapping of the iris can be computed using a polar / bipolar coordinate transform depending on the type assumed for the iris: concentric / eccentric circular ring. The latter is computationally more expensive than the former because the eccentricity varies from a sample to another and consequently, one bipolar mapping must be (re)computed for each sample (eye image). When the iris ring is assumed to be concentric, the polar mapping is computed once for all samples, during program initialization.

132 130 Nicolaie Popescu-Bodorin 2.1. FAST PUPIL FINDER ALGORITHM The proposed pupil finder algorithm can be stated as follows: Fast Pupil Finder Algorithm (N. Popescu-Bodorin): INPUT: the eye image IM; 1.Extract the pupil cluster (Fig.1.a): PC = fkmq(im,16); PC = (PC == min(pc(:))); 2.Compute horizontal and vertical Run Length quantization of PC (Fig.2.a-b): RLV(:,,j) = vrleq(pc); RLH(j,:) = hrleq(pc); 3.Compute the pupil indicator PI (Fig.2.c): [k, PI] = getpi(rlh, RLV); PI = find(pi == 1); PI = PI(1); 4.Extract available pupil segment through a flood-fill operation (Fig.3): P = imfill(pc, PI); 5.Fill the specular lights: P = rlefillsl(p); 6.Approximate the pupil by an ellipse; OUTPUT: The ellipse approximating the pupil; END. An example of finding the pupil indicator is stated as follows: function [k,pi] = getpi(rlh, RLV); k=16; PI = 0*RLH; While PI is the null matrix do: Compute the k-means quantization of RLV and RLH: RLHQ = fkmq(rlh, k); RLVQ = fkmq(rlv, k); Select the logical index of the highest cluster within RLHQ and RLVQ, respectively: LIH = ( RLHQ == max(rlhq(:)) ); LIV = ( RLVQ == max(rlvq(:)) ); Compute the binary matrix PI as logical conjunction of LIH and LIV: PI = LIH & LIV; k = k -1; EndWhile; END. We must clarify that the pupil indicator is found as a preimage corresponding to the maximum value of a fuzzy membership assignment describing the actual pupil as a subset of the pupil cluster: for each pixel within the pupil cluster, directional run-length coefficients encode the degree of membership of that pixel to the actual pupil. The argument is the fact that being (or containing) the most circular solid object within the pupil cluster, the actual

133 Fragile Bits vs. Multi-Enrollment - a case study of iris recognition pupil is the most resilient set to erosion [9] to be found in the pupil cluster. More details on this topic can be found in [12]. Fig. 1. Original eye image (a) and its 8-means quantization (b); The pupil cluster PC (c); Fig. 2. Vertical run-length uint8 quantization of the pupil cluster (a); Horizontal runlength quantization of the pupil cluster (b); The pupil indicator PI (c). Images are presented in binary or 8-bit complement. Fig. 3. Extracting available pupil segment through a flood-fill operation started from any pixel within the pupil indicator (a); Approximating the pupil by an ellipse (b) Also, the computation of the pupil indicator depends on a single parameter: a threshold for the (uint8) requantized horizontal and vertical run-length coefficients computed for the pupil cluster, threshold above which the membership of a pixel to the actual pupil is guaranteed. For these two reasons explained above, the proposed pupil finder procedure is a fuzzy approach that solves a one-dimensional optimization problem LIMBIC BOUNDARY APPROXIMATION The Fast Pupil Finder procedure presented above guarantees accurate pupil localization and enables us to unwrap the eye image (Fig.4.a - image from [15]) in polar coordinates (Fig.4.b) and also to practice the localization of the limbic boundary in the rectangular unwrapped eye image (Fig.4.c), obtaining an iris segment as in Fig.4.e.

134 132 Nicolaie Popescu-Bodorin Fig. 4. Iris segmentation stages Circular Fuzzy Iris Segmentation Procedure (N. Popescu-Bodorin): INPUT: the eye image IM; 1.Apply the Fast Pupil Finder procedure to find pupil radius and pupil center; 2.Unwrap the eye image in polar coordinates (UI - Fig.4.b) through a lossless pixel-to-pixel transcoding from the circles within the original image to the lines within the unwrapped iris area UI; 3.Strech the unwrapped eye image UI to a rectangle (RUI - Fig.4.c); 4.Compute three column vectors: A, B, C, where: A contains the means of the lines within UI matrix; B contains the means of the lines within RUI matrix; C contains the means of the lines within [A B] matrix; 5.Compute P, Q, R as being 3-means quantization of A, B, C, respectively (Fig.5); 6.For each line of the unwrapped eye image, count the votes given by P,Q and R. All the lines receiving at least two positive votes is assumed to belong to the actual iris segment; 7.Find limbic boundary and extract the iris segment (Fig.5, Fig.4.d, Fig.4.e); OUTPUT: pupil center, pupil radius, index of the line representing limbic boundary and the final iris segment; END.

135 Fragile Bits vs. Multi-Enrollment - a case study of iris recognition Membership assignment (to actual pupil / iris / non iris segment) LINES OUTSIDE THE IRIS IRIS SEGMENT PUPIL P Q 20 R Line (in unwrapped eye image) Fig. 5. Iris segmentation procedure: Line assignment (step 5 of the CFIS procedure) 3 fuzzy assignment to the pupil fuzzy assignment to the iris fuzzy indicator of the lines outside the iris combined fuzzy indicator for the above three zones combined crisp indicator for the above three zones fuzzy indicator for the iris boundaries LINES OUTSIDE THE IRIS IRIS PUPIL Lines within unwrapped iris area Fuzzy Boundaries LINES OUTSIDE THE IRIS IRIS PUPIL Lines within unwrapped iris area (transposed) Fig. 6. Fuzzy iris segment and fuzzy iris boundaries

136 134 Nicolaie Popescu-Bodorin Fig. 7. Circular Fuzzy Iris Segmentation Demo Program 2.3. EXAMPLES OF K-MEANS BASED FUZZY CLUSTERS AND FUZZY BOUNDARIES: FUZZY IRIS BAND, FUZZY IRIS BOUNDARIES The following two questions are among the most frequent questions asked by the readers of the previously published papers treating the above described segmentation procedure: why should we think that CFIS is a fuzzy procedure? what is fuzzy in CFIS procedure? This section is meant to answer these questions. Let us comment Fig.6 which shows what is happening with the vector B at the steps 4-5 of the CFIS procedure: hidden behind the combined crisp indicator function (crisp membership assignment) of three clusters like those marked in Fig.5 (pupil, iris and the area outside the iris) there are fuzzy membership assignment functions defined from the set of lines within rectangular unwrapped iris area (RUI) to each of the above mentioned regions and even to the iris boundaries. Hence, there is no doubt that area delimited between the fuzzy iris boundaries is the fuzzy iris band (within the rectangular unwrapped iris area RUI). Its preimage through the polar mapping is a circular fuzzy iris ring. Three fuzzy iris bands are determined using the vectors A, B, C. The final result is computed evaluating the chances that the lines within the unwrapped iris area to belong to the actual iris segment. This is done in the step 6 of the CFIS procedure by counting the votes received for each line within the unwrapped iris area as a member of a fuzzy iris band. More details regarding the fuzzification of the iris segment and boundaries can be found in [11].